Phase-field modeling on the diffusion-driven processes in metallic conductors and lithium-ion batteries
PPhase-field modeling on the diffusion-driven processes inmetallic conductors and lithium-ion batteries
Zur Erlangung des akademischen Grades
Doktor der Ingenieurwissenschaften von der KIT-Fakultät für Maschinenbau desKarlsruher Instituts für Technologie (KIT)genehmigte
Dissertation von
M. Tech. Jay Santoki geboren am 20.06.1992in Jam-Jodhpur (Indien)Tag der mündlichen Prüfung: 10.09.2020Hauptreferent: Prof. Dr. rer. nat. Britta NestlerKorreferent: Prof. Dr.-Ing. habil. Marc Kamlah a r X i v : . [ phy s i c s . a pp - ph ] F e b bstract Diffusion-driven processes are important phenomena of materials science in the field of en-ergy conversion and transmission. During the conversion from chemical energy to electricalenergy, the species diffusion is generally linked to the rate of exchange, and hence to theperformance of the conversion device. Alternatively, the transmission of the electric fielddiffuses the species when it passes through any medium. The consequences of this effectcan be regulated to attune surface nano-patterns. Otherwise, uncontrolled morphologiesmay lead to permanent degradation of the metallic conductors. Therefore, the understand-ing of the material behavior, in the presence of the driving forces of the diffusional species,is of scientific interest. The presented dissertation proposes to investigate one exampleof species diffusion in each case, during energy conversion and transmission. Specifically,the objective of the study is to explore the lithium insertion into the cathode electrodeof lithium-ion batteries and the morphological evolution of inclusions, while propagatingunder the electromigration in the metallic conductors.During insertion, lithium manganese oxide spinel, a cathode electrode material oflithium-ion batteries, shows a coexistence of Li-rich and Li-poor phases. For an enhancedunderstanding of the mechanism of a two-phase coexistence, a mathematical model ofphase separation is derived, which is based on the Cahn-Hilliard equation. To begin with,the geometrical shape polydispersity of an isolated particle is considered to investigate themesoscopic effect of the surface curvature. The simulation results show that the onset ofthe phase separation preferentially occurs in high-curvature regions of the particle. Further-more, the elliptical particle with a higher aspect ratio is subjected to the onset of the phaseseparation, prior to the particles with a lower aspect ratio. Finally, the effect of the variationof the parameters on the charge dynamics is discussed. The study is further extended tomultiple particle systems, so as to understand the influence of various microstructural de-scriptors, such as particle size, porosity, and tortuosity, on the transport mechanism. A lineardependence of the transportation rate is observed with tortuosity. The slope of this linearrelation is independent of the particle size, but shows some interdependency with porosity.Furthermore, the presented results suggest that systems consisting of smaller particles areclosely follow the surface reaction limited theory while larger particles tend towards theibstract iibulk-transport limited theory derived for planar electrodes. In order to identify the promis-ing hierarchically structured electrodes, the presented simulation results could be utilizedto optimize the experimental efforts.The electromigration-induced morphological evolution of inclusions (voids, precipi-tates, and islands) has recently been scrutinized in terms of the efficient design of theinterconnects and surface nanopatterns. To understand the morphological evolutions, aphase-field model is derived to account for the inclusions migrating under the externalelectric field. Initially, the insights gained from the numerical results of the isotropic inclu-sions corroborate the findings from the linear stability analysis. Additionally, the numericalresults can elegantly elucidate the transition of a circular inclusion to a finger-like slit. Thesubsequent drift of the slit is characterized by shape invariance, along with a steady-stateslit width and velocity, which scale with the applied electric field as E − / ∞ and E / ∞ , respec-tively. The results obtained from the phase-field simulations are critically compared withthe sharp-interface solution. The repercussions of the study, regarding the prediction of voidmigration in flip-chip Sn-Ag-Cu solder bumps and the fabrication of channels with desiredmicro / nano dimensions, are discussed. The study is further extended to anisotropic inclu-sions migrating in { } , { } , and { } crystallographic planes of face-centered-cubiccrystals. Based on the numerical results, morphological maps are constructed in the planeof the misorientation angle and, the conductivity contrast between the inclusion and thematrix. The simulations predict a rich variety of morphologies, which includes steady-stateand time-periodic morphologies, as well as zigzag oscillations and an inclusion breakup.Furthermore, the influence of the variation in conductivity contrast and the misorientationis observed to be influential on the morphological evolution of the time-periodic oscillations,steady-state shapes, and the way inclusions break apart. Finally, the numerical results ofthe steady-state dynamics, obtained for anisotropic inclusions, are critically compared withisotropic analytical and numerical results.The presented dissertation demonstrates that the phase-field methods are able to ele-gantly capture the essential physics of the diffusion-driven phenomena discussed above. urzfassung Diffusionsgetriebene Prozesse sind wichtige Phänomene der Materialwissenschaft im Bere-ich der Energieumwandlung und -übertragung. Während der Umwandlung von chemis-cher Energie in elektrische Energie ist die Speziesdiffusion im Allgemeinen mit der Aus-tauschrate und folglich mit der Leistung der Umwandlungsvorrichtung verbunden. Al-ternativ diffundiert die Übertragung des elektrischen Feldes durch die Spezies, wenn siedurch irgendein Medium verläuft. Die Konsequenzen dieses Effekts können reguliertwerden, um Oberflächen-Nanomuster abzustimmen. Andernfalls können die unkontrol-lierten Morphologien zu einer dauerhaften Verschlechterung der metallischen Leiter führen.Daher ist das Verständnis des materiellen Verhaltens bei Vorhandensein der treibendenKräfte von Diffusionsspezies von wissenschaftlichem Interesse. Die vorgestellte Disserta-tion schlägt eine Untersuchung von jeweils einem Beispiel der Speziesdiffusion währendder Energieumwandlung und -übertragung vor. Ziel der Studie ist es insbesondere, sowohldie Lithiumeinfügung von Lithium-Ionen-Batterien in die Kathodenelektrode als auch diemorphologische Entwicklung von Einschlüssen zu untersuchen, während sie sich unter derElektromigration in den metallischen Leitern ausbreiten.Lithium-Manganoxid-Spinell, ein Kathodenelektrodenmaterial von Lithium-Ionen-Batterien, zeigt während des Einfügens eine Koexistenz von Li-reichen und Li-armenPhasen. Für ein besseres Verständnis des Mechanismus einer zweiphasigen Koexistenzwird ein mathematisches Modell der Phasentrennung abgeleitet, das auf der Cahn-Hilliard-Gleichung basiert. Zunächst wird die geometrische Formpolydispersität eines isoliertenPartikels betrachtet, um den mesoskopischen Effekt der Oberflächenkrümmung zu unter-suchen. Die Simulationsergebnisse zeigen, dass der Beginn der Phasentrennung bevorzugtin Bereichen auftritt, in denen das Partikel eine starke Krümmung aufweist. Weiterhinwird das elliptische Teilchen mit einem höheren Querschnittsverhältnis dem Einsetzen derPhasentrennung vor den Teilchen mit einem niedrigeren Querschnittsverhältnis ausgesetzt.Abschließend wird der Einfluss der Variation der Parameter auf die Ladungsdynamik disku-tiert. Die Studie wird weiter auf mehrere Partikelsysteme ausgedehnt, um den Einfluss ver-schiedener mikrostruktureller Deskriptoren wie Partikelgröße, Porosität und Tortuosität aufden Transportmechanismus zu verstehen. Bei Tortuosität wird eine lineare Abhängigkeitiiicknowledgements ivder Transportrate beobachtet. Die Steigung dieser linearen Beziehung ist unabhängig vonder Partikelgröße, zeigt jedoch eine gewisse Abhängigkeit von der Porosität. Darüber hinauslegen die vorgestellten Ergebnisse nahe, dass Systeme, die aus kleineren Partikeln bestehen,der durch Oberflächenreaktionen begrenzten Theorie genau folgen, während größere Par-tikel zu der durch Massentransporte begrenzten Theorie tendieren, die für planare Elektro-den abgeleitet wurde. Um die hierarchisch strukturierten Elektroden zu identifizieren, kön-nten die vorgestellten Simulationsergebnisse verwendet werden, um den experimentellenAufwand zu optimieren.Die durch Elektromigration induzierte morphologische Entwicklung von Einschlüssen(Hohlräume, Ausfällungen und Inseln) wurde kürzlich im Hinblick auf die effiziente Ausle-gung der Verbindungen und Oberflächen-Nanomuster untersucht. Um die morphologis-chen Entwicklungen zu verstehen, wird ein Phasenfeldmodell abgeleitet, um Einschlüssezu berücksichtigen, die unter dem externen elektrischen Feld wandern. Die Erkenntnisseaus den numerischen Ergebnissen zu isotropen Einschlüssen bestätigen zunächst die Ergeb-nisse der linearen Stabilitätsanalyse. Zusätzlich können die numerischen Ergebnisse denÜbergang eines kreisförmigen Einschlusses zu einem fingerartigen Schlitz elegant erläutern.Die nachfolgende Drift des Schlitzes ist durch eine Forminvarianz zusammen mit einer sta-tionären Schlitzbreite und -geschwindigkeit gekennzeichnet, die mit dem angelegten elek-trischen Feld jeweils als E − / ∞ und E / ∞ skaliert werden. Die Ergebnisse aus Phasenfeld-simulationen werden kritisch mit der Lösung mit scharfen Grenzflächen verglichen. DieAuswirkungen der Studie auf die Vorhersage einer Hohlraumwanderung in Flip-Chip-Sn-Ag-Cu-Lötperlen und die Herstellung von Kanälen mit gewünschten Mikro- / Nanodimen-sionen werden diskutiert. Die Studie wird weiter auf anisotrope Einschlüsse ausgedehnt,die in { } , { } und { } kristallografischen Ebenen von flächenzentrierten kubis-chen Kristallen wandern. Basierend auf numerischen Ergebnissen werden morphologis-che Karten in der Ebene des Fehlorientierungswinkels und des Leitfähigkeitskontrasts zwis-chen dem Einschluss und der Matrix erstellt. Die Simulationen sagen eine Vielzahl vonMorphologien voraus, darunter stationäre und zeitperiodische Morphologien sowie Zick-Zack-Oszillationen und eine Einschlussauflösung. Darüber hinaus wird beobachtet, dassder Einfluss der Variation des Leitfähigkeitskontrasts und der Fehlorientierung Einfluss aufdie morphologische Entwicklung der zeitperiodischen Schwingungen, der stationären For-men und der Art und Weise hat, wie Einschlüsse auseinander brechen. Schließlich werdendie numerischen Ergebnisse der stationären Dynamik, die für anisotrope Einschlüsse erzieltwurden, kritisch mit isotropen analytischen und numerischen Ergebnissen verglichen.Die vorgestellte Dissertation zeigt, dass die Phasenfeldmethoden die wesentliche Physikder oben diskutierten diffusionsgetriebenen Phänomene elegant erfassen können. cknowledgements This thesis is possible due to the encouragement and support of countless people. Here, Itake the opportunity to thank a few that have made this thesis possible with a plethora ofmemories to remember for life.I begin my deepest gratitude to Prof. Dr. Britta Nestler for providing an opportunity to bea part of her research group. During technical discussions, her helpful insights are a sourceof inspiration, which keeps me learning and growing during this journey. The freedomprovided by her, during the research work has garnered interest in me to investigate trulyexceptional phenomena in materials science and ultimately in search of a better world. Iam extremely grateful for her constructive comments, quality discussions, and support. Iwould like to extend my gratitude to Prof. Dr. Marc Kamlah to be the second referee of mythesis committee. His intriguing comments on my work have not only improved the qualityof the research, but also helped me to develop scientific reasoning and consequently, tobecome an independent researcher.I have had the good fortune to find some truly exceptional collaborators. Special thanksto Dr. Daniel Schneider for his continues feedback and numerous discussions. The interac-tions with Dr. Arnab Mukherjee has allowed me to observe the multi-facets of the phase-field modeling from close corners. I also acknowledge very thought-provoking discussionswith Dr. Fei Wang, Dr. Michael Selzer, Dr. Oleg Tschukin, Prof. Leslie Mushongera, and Dr.Sebastian Schulz during the initial stages of PhD. I am extremely indebted to my colleaguesand collaborators, Walter Werner, Andreas Reiter, Sumanth Nani, Dr. Prince Gideon, PaulHoffrogge, Yinghan Zhao, Nishant Prajapati, Simon Daubner, Dr. Tao Zhang, ChristophHerrmann, Dr. Ephraim Schoof, Dr. Felix Schwab, Dr. Ramanathan Perumal, Dr. AmolSubhedar, Nikhil Kulkarni, Mehwish Huma Nasir, and Fridolin Haugg for many a technicaldiscussions from which I have gained immensely. Furthermore, I am greatly obliged to allthe group members for providing a thoughtful ambiance, which has inspired me to strivefor continuous improvement.I am grateful to the technical staff in the group, Christof Ratz and Halil Bayram fortheir support in both hardware and software related issues which helped me go about myvontents viwork smoothly. A special thanks to Leon Geisen for his editorial assistance and the Ger-man translation of the abstract. I also express gratitude to the secretariat at IDM of theHochschule Karlsruhe and IAM-CMS of the Karlsruhe Institute Technology, especially, Ms.Claudia Hertweck-Maurer, Ms. Stephanie Mueller and Ms. Inken Heise for all their helpwith the administrative work.It would not have been possible to endure the rough patches alone without the support,love, and empathy of my family and friends. Thank you for being there for me every time Ineeded someone to count on. In addition, I request a sincere apology for not being able tospend as much time as you would have wanted in the last few years. I dedicate my thesisto my family and friends, who are the saviors of my life.Finally, I acknowledge the financial supports during different stages of my PhD. Thefirst phase of the research was funded by the cooperative graduate school "Gefuegestruk-turanalyse und Prozessbewertung" of the Ministry of the State of Baden-Wuerttemberg andpartially through the initiative "Mittelbau". The second phase of my research was fundedby the German Research Foundation (DFG) under Project ID 390874152 (POLiS Cluster ofExcellence). ontents
Abstract iKurzfassung iiiAcknowledgments vI Introduction and Literature review 11 Introduction and literature review 3
II Methods : Phase-field formulation 192 Phase-field model 21
III Results and Discussion:Phase separation in lithium-ion batteries 65 ontents ix α (cid:48)(cid:48) . . . . . . . . . . . 825.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 IV Results and Discussion:Electromigration in metallic conductors 997 Motion of isotropic inclusions 101 χ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.3.2 Significance of β and the current crowding . . . . . . . . . . . . . . . . . 1137.4 Finger-like slit propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.4.1 Specific features of finger-like slits . . . . . . . . . . . . . . . . . . . . . . 1167.4.2 Sharp-interface analysis for slit profile . . . . . . . . . . . . . . . . . . . . 1197.4.3 Comparison with the sharp-interface description . . . . . . . . . . . . . 1217.4.4 Effect of conductor line width, w . . . . . . . . . . . . . . . . . . . . . . . 1237.4.5 Selection of slit width and velocity . . . . . . . . . . . . . . . . . . . . . . 1237.5 Discussion and experimental correlation . . . . . . . . . . . . . . . . . . . . . . . 1267.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1338.2.1 Steady-State Inclusion Dynamics . . . . . . . . . . . . . . . . . . . . . . . 1348.2.2 Effect of misorientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.2.3 Morphological Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1428.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 V Conclusions and Future Directions 16510 Conclusions and future directions 167
VI Appendices 175Appendix A Interface profiles 177Appendix B Preparation of electrode microstructure with a single particle 179Appendix C Preparation of electrode microstructure with several particles 181Appendix D Phase separation in Electrode along with homogeneous mixture inElectrolyte 185Appendix E Calculation of tortuosity 187List of Symbols and Abbreviations 189List of Figures 199 ontents xii
List of Tables 205Bibliography 207 art IIntroduction and literature reviewhapter 1Introduction and literature review
During their operations, many material systems encounter a movement of atoms, moleculesor charge carriers. In response to the established gradient in the concentration of thesespecies or the external driving force for diffusion, the movement in different materials isspecific. Hence, the diffusion coefficient or the diffusivity is defined to reflect this materialproperty. The diffusivity is generally measured for a given pair of species and pairwise fora multi-species system. For instance, carbon dioxide in the air has a diffusion coefficientof 16 mm / s, and in water, its diffusion coefficient is 0.0016 mm / s [ ] . The higher thediffusivity of one species, with respect to another, the faster they diffuse into each other.These are the examples of diffusion, in order to minimize the presence of concentrationgradients, as shown in Figure 1.1(a). However, it is also possible that the diffusion insome cases can take place against the concentration gradients, where the region of highconcentration accumulates further species from the lower one to grow even higher [ ] , asshown in Figure 1.1(b). As an example, consider a hot mixture of water and phenol. At hightemperatures, the phenol and the water may mix to form a single thermodynamic phase.However, when the mixture is cooled, the clusters of the phenol-rich phase and the water-rich phase are expected. Hence, this necessitates rigorous investigations to understanddiffusion in materials.Diffusion plays an important role in many processes of materials science, including itsbenefits as well as adverse nature. For instance, it can be applied as a process of diffusionbonding, in order to weld or join metals [ ] , and sintering to shape materials in the pow-der metallurgy [ ] . On the other hand, the diffusion can be critical in processes such asdimensional changes, due to a pore formation in the grain boundaries [ ] . In addition, thediffusion in adjoining layers is prevented by a chemical isolation in integrated circuits, soas to avoid a transgranular movement of the species, while maintaining electrical connec-tions with the help of barrier metal films [ ] . Therefore, it is important to understand thematerial behavior under the external driving forces for the species diffusion.3hapter 1. 4 B AAAAAB B BB BAAAB BBBAB B AAAAA AAAB B BBB B BAAAB BBBABB B A A AABB AB AAAAABAAAB BBABBB A A AA BB A A AA AAABAAABBBB AAAABBAAABBBB AABAABBBBB A A AA (a)(b) BABA AAA BB BB BB B BBBABAAA BB B
Figure 1.1: Schematics of diffusion processes: (a) in the direction of the concentration gradient and (b)against the concentration gradient.
There are several external driving forces for the species diffusion. For instance, theappearance of a species flux, as a consequence of a thermal gradient, the so-called Soreteffect, is important, where the large temperature gradients are present such as in nuclearreactors [ ] . In addition, the diffusion can be induced by centrifugal forces [ ] , gravity [ ] ,hydrostatic pressure and mechanical stresses [ ] . However, diffusion associated with theelectric field is the most provocative field of research in many communities [ ] .During the energy conversion from the chemical energy to the electrical energy, variousphenomena can be observed, due to the species diffusion. Gas diffusion layers, for instance,are the key components to the various types of fuel cells, which essentially act as an elec-trode that facilitates the diffusion of reactants across the active area of the membrane [ ] .In addition, diffusion-controlled electrodeposition [ ] leads to a dendrite formation at theelectrode surfaces. Furthermore, the diffusion is reported to be prominent in the batterysystems [
14, 15 ] .In addition to energy conversion, the electric field diffuses the species when it passesthrough any medium. In several applications, this effect leads to significant consequences.For instance, the passage of an intense electric current in thin-film metallic conductors,known as electromigration, may cause open circuits [ ] , due to the voids and short circuits [
17, 18 ] , caused by the formation of hillocks. On the contrary, the electric field can behelpful to determine the chloride diffusivity in concrete and the depth of the penetrationin cement pastes [ ] . Furthermore, the microstructures in polymers can be monitored byhapter 1. 5the electric field to obtain a lamellar pattern [ ] .The presented dissertation proposes to investigate one example of each aspect of thespecies diffusion, namely during the energy conversion and during the transmission of theelectric field in a medium. Specifically, the objective of the study is to explore the lithium in-sertion into the cathode particles of lithium-ion batteries and the morphological evolution ofinclusions under the electromigration in metallic conductors. In the following paragraphs,a literature review on both topics is performed separately, which is necessary to establish asound platform for the methods and results presented in the later chapters. The miniaturization of portable electronic devices, the storage systems for renewable energysources, and achieving light weight of the electrical vehicles require high specific energy,high specific power, and a low cost of the energy storage systems. One of the probable so-lutions for such future needs, in the form of electrochemical energy storage devices, is thebattery system [ ] . Batteries are classified into two categories, based on its reversibility,the primary batteries and the secondary batteries. Primary batteries are non-rechargeable,while secondary batteries are rechargeable. A rechargeable battery can be used over manycycles throughout its lifespan. The increment in the number of possible charge / dischargecycles before performance degradation is a major concern, with reference to the improve-ment of battery life. Figure 1.2 shows a comparison of different rechargeable battery sys-tems, with their typical operational range of specific power and specific energy.Due to their versatile applications with higher specific power and specific energy,lithium-ion batteries are one of the promising candidates of rechargeable batteries [ ] .The lithium-ion battery consists of intercalated compounds, compared to the metalliclithium used in a non-rechargeable lithium battery [ ] . The three primary functional com-ponents of a lithium-ion battery are the positive and negative electrodes and electrolytes.Generally, the negative electrode (anode) of a conventional lithium-ion battery is madeof graphite, silicon or carbon-based compounds [ ] . The positive electrode (cathode) isof a metal oxide, and the electrolyte is of a lithium salt in an organic solvent [ ] . Thelithium ions are stored as guest species in the crystal lattice of the electrode materials. Theelectrodes in lithium-ion batteries are a mixture of lithium-ion-conducting and electron-conducting particles which form a porous composition to sustain the transport of electronsand lithium ions to and from the active sites.During the discharging (or charging) process, the battery delivers (or consumes) a cur-rent spontaneously (or deliberately) from the external power supply, due to a electrochem-hapter 1. 6 Figure 1.2: Comparison of the specific energy and specific power of common rechargeable batteries,along with Li-ion batteries. Reprinted with permission from Ref. [ ] . ical voltage difference between the anode and the cathode, E = E A − E C , (1.1)where E is the electrochemical voltage of the open circuit and E A and E C are the respectiveanode and cathode voltages. This delivered current is the transport of electrons from theanode to the cathode, through an external circuit, while lithium ions transport from theanode to the cathode inside the battery (Figure 1.3). During discharge, lithium ions aretransferred from the anode to the cathode, through the electrolyte. At the anode side:Li Discharge −−−−− (cid:42)(cid:41) −−−−−
Charge Li + + e – .The negatively charged electrons minimize their energy, when they are transferred fromthe anode to the cathode. At the cathode side:Li + + e – Discharge −−−−− (cid:42)(cid:41) −−−−− Charge
Li.The lithium-ion inflow continues until the potential difference between the anode and thecathode vanishes. Due to the reversibility, the battery can be charged up again through theexternal power supply.hapter 1. 7
Figure 1.3: Schematic diagram of a lithium-ion battery unit, while discharging, attached to an externaldevice, such as a mobile phone. In the fully charged state, the battery reaches its electrochemical voltagethreshold (E max ) and the battery is ready for the application. During the discharging process, lithium-ions deport from the lithiated anode to the delithiated cathode, through the electrolyte. When thepotential difference between the anode and cathode vanishes, the ionic transfer stagnates. Then, thebattery is discharged and requires an external supply to be charged again. However, the battery capacity decays upon electrochemical cycling, as a result of severalnon-reversible losses [ ] . In addition, other aspects to improve the battery performancewith an improved battery life [ ] , such as the energy density, the stability of the operatingtemperature, the low self-discharge, the safety, the charging time, the output power, andthe cost of the technology [
27, 28 ] , are of scientific interest to the increasing demand forhigh-performance lithium-ion batteries. The provision of high performance requires an un-derstanding of the fundamental physical and chemical processes at various levels of batteryoperational and non-operational conditions. One such process is phase separation, whichis the focus of the presented work. Most of the popular anode (graphite) and cathode materials (one of three [ ] : a layeredoxide such as lithium cobalt oxide, a polyanion such as lithium iron phosphate or a spinelsuch as lithium manganese oxide) show phase-separated states during lithium intercala-tion. The selection of the particular battery components, including the cathode material,the anode material, and the electrolyte, depends on various characteristics associated witha specific range, including the discharge rate, the battery cycle time, the battery weight, thebattery life, the battery cost, and the operating temperature [
31, 32 ] . The typical charac-hapter 1. 8 (a) (b) c Figure 1.4: Multiple phase coexistence in graphite materials. (a) Open-circuit voltage for a lithiuminsertion into graphite. The experimental data from Ref. [ ] confirm multiple voltage plateaus, alongwith the results from the simulations (solid curve), and the voltage corresponds to the homogeneousfree energy density (dashed curve). (b) Experimental image (above), along with simulation prediction(below) [ ] . In the experiment and the simulation, three distinct co-existent phases are observed atthe same temporal position. Reprinted with permission from Ref. [ ] . teristics of few of the electrode materials are highlighted in the following paragraphs.Graphite exhibits multiple voltage plateaus during (de-)intercalation [ ] , as shown inFigure 1.4(a). It has complex thermodynamics [ ] and at least three clear stages with ex-perimentally observed fronts of color-changing phase transformations, which can be seenin the above inset of Figure 1.4(b). In addition, graphite can form multilayered structureswith different energies. In fact, each layer of the graphite may show a distinct phase sepa-ration behavior [ ] . This complicates the mechanism of the phase separation even further,when considered that a multilayered material and some complex morphologies, such as acheckerboard pattern [ ] , can be observed. Therefore, this multiple phase coexistence isa unique characteristic of anode materials, in particular graphite.Cathode material such as lithium-iron-phosphate (LFP) shows a two-phase coexistence [
38, 39 ] . In a particle, photon energies of two phases, LiFePO and FePO , have distinctbehaviors under the X-ray absorption spectrum, as shown in Figure 1.5(a). This can beutilized for locating different regions of phase-separated states in (b), which shows the ex-perimental evidence of the phase separation behavior in the porous electrode consists ofseveral LFP particles. In the porous electrodes, the electrolytic solution permeates throughthe void spaces of porous matrices. Depending upon a particle situation, it can be regardedas completely lithiated, completely delithiated, or as an active particle under phase sep-aration transition. In the LFP electrode, the population of active particles varies widely,ranging from nearly particle-by-particle to concurrent (all particles simultaneously) inter-hapter 1. 9 (b) (c)(a) Figure 1.5: Two-phase coexistence in lithium-iron-phosphate particles of the porous electrode. X-rayabsorption spectra of fully lithiated and delithiated particles of LFP are shown in (a), which is consideredfor a reference, so as to identify these regions. The state-of-charge map is shown in (b), which is producedby fitting a linear combination of the spectrum to every single pixel. The red color represents the lithiatedfraction, while the green color indicates the delithiated region. Transmission electron microscopy (TEM)image of the same electrode region, which can be utilized to identify the boundary of each particle.Reprinted with permission from Ref. [ ] . calation. The active population depends on various factors influencing phase separationsuch as intercalation rates [ ] . Therefore, understanding the influence of these factors isof technological relevance.The mechanism of phase separation is also reported in other electrode materials, such aslithium cobalt oxide [ ] , silicon [ ] , and lithium titanate [ ] . However, the presentedwork explores lithium manganese oxide spinel (LMO) cathode material. For numerousreasons, LMO is a promising candidate for a cathodic material, including the fact that it isa cheaper, highly abundant material, which is environmentally favorable [ ] . However, astruggle with a capacity fade is a shortcoming [ ] . An important source of the capacityfade is undergoing phase separation during battery operation, which leads to a prematurebattery failure [
41, 45–47 ] . The finding of a two-phase coexistence is a breakthrough inscientific knowledge about LMO particles, obtained from ample experimental evidence [ ] .Figure 1.6 shows complex thermodynamics of phase separation in LMO materials. Sub-stantial attempts are directed to understand the phase separation mechanisms of Li x Mn O .There are several ranges of x, where two-phase coexistence is reported [
40, 46 ] . For in-stance, Phase B and Phase C are depicted from x ≈ ≈ (a) (b)(d)(c) Figure 1.6: Two-phase coexistence in lithium-manganese oxide spinel. (a) Changes in the lattice pa-rameters, with lithium concentration x for Li x Mn O ( ≤ x ≤ ) . (b) Changes in the volumefraction, with x for ( ≤ x ≤ ) . (c) Dependencies of the lattice parameters on x for ( ≤ x ≤ ) .(d) The calculated phase diagram, as a function of temperature, and the lithium concentration x for ( ≤ x ≤ ) . Reprinted with permission from Refs. [
40, 41 ] . x ≈ [
40, 41, 46 ] and strongdependencies on various parameters (Figure 1.6(b) and (c)), the two-phase coexistence isclearly eminent in LMO materials. Therefore, the study of two-phase coexistence in LMOis considered in the present work.hapter 1. 11 The coexistence of two phases in LMO, as a cathode electrode, is extensively explored inexperiments [
41, 45–47 ] . Systematic numerical modeling of phase separation may comple-ment to the scientific understanding of the mechanism and helps to optimize the processparameters efficiently and economically [ ] .The modeling of two-phase coexistence in electrodes has a long-standing literature [
49, 50 ] . The intercalation in LMO follows shrinking-core kinetics, where the interfaceboundaries travel in the direction of lithium flux from the surface [ ] . Specific schemes,such as a flexible sigmoidal function [ ] , a trapping concept [ ] , are employed for theinterface tracking. However, the classical Cahn-Hilliard equation-based phase-field modelsare mostly employed to avoid cumbersome interface tracking [ ] and embrace the exis-tence of a high concentration gradient between the Li-rich and the Li-poor phases elegantly [ ] .The phase-field models [
51, 55–60 ] considered for the phase separation during inter-calation are substantial for a single particle. These models focus on the isolated particles,which can be applicable to planar electrodes [ ? ] . In addition, the electrodes in manybattery systems consist of a porous structure, composed of nanoparticles [ ] . There aresome theoretical and numerical studies performed to account essential features of phase-separating porous electrodes [
37, 63–65 ] . Even though extensive research is being carriedout [
66, 67 ] to understand the phase separation mechanism, the relative importance of thematerial properties, the operational conditions, and the microstructural properties on thephenomenon are still unclear. Therefore, this work is intended to postulate such techno-logical insights into the phase-separating LMO materials through phase-field models. When a metallic conductor is subjected to an external electric field, two distinct forces canbe apparent in the medium.1. Direct Force ( F d ): Direct action of the external field on the charge of the migratingspecies.2. Wind Force ( F w ): Electrons migrating in the opposite direction to electric field collidewith the diffusing species and transfer the momentum to the species. This stimulatesthe motion of the species, in the direction of the electron propagation [ ] .hapter 1. 12 Void movement Void movementElectron movementElectron movement(a) (b)
Figure 1.7: Schematic of tiny void movement in the opposite direction of the electric wind, due to adisplacement of the species, (a) before the species move and (b) after the species move.
Both forces are proportional to the applied electric field and can be expressed as F total = F d + F w = eZ s E ∞ , (1.2)where e denotes the electron charge, Z s = Z d + Z w is the effective valence combining thevalences of the electrostatic ( Z d ) and the electron wind ( Z w ), and E ∞ denotes the appliedelectric field. Therefore, the resultant force dictates the movement of the species. Gener-ally, the later force is prominent in the field of “electromigration (EM),” [ ] , as shown inFigure 1.7. For simplicity, the term electron wind force therefore often refers to the neteffect of these two electrical forces. In modern semi-conducting chips, a dense array of narrow, thin-film metallic conductors,called interconnects, are the crucial constituents of the integrated circuits (ICs), which con-nect active elements of microelectronics [ ] . Due to the very large scale integrated (VLSI)circuits [ ] , the interconnects are subject to intense electric current densities. This raisesserious reliability concerns in the microelectronics industry, noticeable in the form of anelectrical failure of the interconnects, due to electromigration. Furthermore, this is onlyexpected to worsen with the current trend of continuing miniaturization [ ] . Therefore,understanding the difficulties associated with electromigration is important for the efficientdesign of the batteries and the prolongation of the lifetime of the interconnects.Postmortem analyses of the damaged lines reveal the presence of multiple voids alongthe line, often of various shapes and sizes [ ] , some of which have grown critical, leadingto failure [ ] . Furthermore, in situ studies endorse the failure to be a dynamic process [ ] . Typically, voids nucleate at current crowding zones [ ] or at structural heterogeneoushapter 1. 13 (a)(b) Figure 1.8: (a) SEM images of a cross section of a Cu interconnect, at various time intervals, during thecharacterization of in situ electromigration. (b) Resistance of a structure shown in (a). The sequence ofimages in (a) indicates an electromigration-induced void movement with morphological shape changes,while (b) shows their relevance to the resistance. Reprinted with permission from Ref. [ ] . sites, such as grain boundaries (GBs), or triple junctions [
75, 78, 79 ] . Following nucleation,the voids may escape the grain boundary, migrate along or across the line in a self-similarmanner or evolve into various time-dependent configurations [ ] . Although manyevents precede the final failure, the lifetime of the interconnect is governed by the rate-limiting step [ ] . In addition, void shape changes, such as bifurcation [ ] and coalescence [ ] , are a major lifetime-inhibiting factor. The physical dimensions of the voids are directlyrelated to the electrical resistance, as shown in Figure 1.8, and in turn to the reliability ofthe interconnects. A large enough void may increase the resistance dramatically, and mayeven lead to the open circuits and the eventual failure of the interconnects [ ] .The electromigration-induced degradation behavior can be observed by various in-situ experimental methods, such as transmission electron microscopy [ ] , scanning electronmicroscopy [ ] , and X-ray microscopy [ ] . However, accelerated lifetime tests [ ] arecommonly employed to study reliability and performance predictions of the interconnects.Under these tests, the interconnect samples are stressed considerably more severe than theirhapter 1. 14operating conditions, in order to speed up the process. After the Black’s [ ] derivation, anempirical relation to predict the time to failure ( T T F ) can be expressed as
T T F = A it j n e ( E a / k B T ) , (1.3)where A it represents a constant which encompasses the material and geometrical propertiesof the interconnect, j is the current density, k B represents the Boltzmann constant, T is thetemperature, E a denotes the activation energy, and n represents an exponent that describesthe dependencies (1 ≤ n ≤
2) of the failure mechanism on the current density.
Contrary to the disturbance of voids in interconnects, the second application of electro-migration, an absorption of metallic substances, in the form of precipitates, provides aconstructive advantage to the performance of the interconnects. A strategical introductionof the desired metal precipitates into the interconnects significantly enhances the materialproperties related to electromigration [ ] . For instance, a small number of precipitates ofAl Cu are known to improve the reliability of Al-Cu interconnects [ ] . These precipitatesserve as reservoirs of Cu, for a depleted interconnect solution [ ] . Thus, it is expected thatthe precipitates increase the reliability of the interconnects by redistribution [ ] .Figure 1.9 shows the growth of a precipitate at a cathode end. Furthermore, it is possiblethat all precipitates are accumulated at one end of the electrode, during operation [ ] .Then, a current reversal can be considered for the redistribution of the precipitates. Thiscurrent reversal process further enhances the effectiveness of the precipitates [ ] . Thepresence of the precipitates, in the interconnect lines, modifies the actual time to failure [ ] , which can be expressed as T T F wp = T T F + d p A p , (1.4)where T T F is the time to failure, without precipitates, including the incubation periodof precipitate drifts, d p denotes the factor associated with the geometrical properties ofthe precipitates, and A p reflects the operating conditions such as temperature and currentdensity. The pattern formation, directed by an electric field, has experienced great success in ap-plications like block copolymers [
97, 98 ] , EM-induced liquid metal flow [ ] , growth ofhapter 1. 15 (a)(b)(c)(d) Figure 1.9: TEM images of the growth of an Al Cu precipitate, at the cathode end, during electromigra-tion testing. The sequence (a) to (d) indicates the progress of time. Reprinted with permission from Ref. [ ] intermetallic compound [ ] , and grain-boundary grooving [ ] . In addition,the surface nanopatterning is the third intriguing application of electromigration, in therapidly advancing field of surface engineering [ ] , which has similar characteristicsto void-interconnect mechanisms. This means that the atomic clusters (islands) are ad-sorbed on a crystalline substrate, under the action of an external electric field. The islandsevolve due to the momentum transfer by the collision of the electrons [ ] , as shown inFigure 1.10. Adatom migration along the surface of the island is reported to be a dominantmass transport mechanism [ ] .Understanding the influence of the materialistic properties and the characteristics of theexternal fields is of scientific importance for the desired dynamics of the islands, to tailorhapter 1. 16 (a)(e)(c) (d)(b)(f) Δy = -44.9 nmΔt = 1080 sΔy = 51.8 nmΔt = 1080 sΔy
Figure 1.10: Experimental evidence of the movement of monolayer islands, due to electromigration. (a)and (b) show the upward displacement of the island and the downward movement of the current, while(c) and (d) show the downward displacement of the island and the upward movement of the current.the schematics of the island in (e) and (f) illustrate the displacement, due to electromigration. The bluearrows indicate a direct current direction, while the red arrows indicate the direction of the electronwind. Reprinted with permission from Ref. [ ] the surface resistivity. Surprisingly, the islands are reported to glide in or against the currentdirection, at a constant velocity, depending on the reconstruction of the island [ ] , andwhether the electron flow interacts with the substrate or the island. To be consistent withhapter 1. 17the direction of the void migration, defined in section 1.2.1, the islands in the followingwork are presumed to migrate in the opposite direction of the electron flow. Hereafter,‘inclusion’ is used as a general terminology to refer to the voids (insulator), precipitates(conductivity different from the matrix), and islands (conductivity identical to the matrix),otherwise stated explicitly. The analytical [ ] and numerical models [ ] have been employed exten-sively for the electromigration of inclusions. Sharp interface analyses are presented to an-alyze the characteristics of a presumed shape in Refs. [ ] . Linear stability analysesare exerted for the prediction of the inclusion morphology [ ] . Pertaining to thecomplexity, theoretical analyses have focused on the stability of a presumed shape [ ] ,which lacks to provide any information regarding the shape after an instability, or the pre-diction accuracy decreases as a geometry deviates from a presumed shape. Alternately,the sharp-interface numerical models, including Monte-Carlo [ ] , continuum [ ] ,and direct dynamical models [ ] , provide important technological insights into thephenomena such as many-inclusion patterns [ ] , a phase diagram of anisotropy strength,the inclusion size [ ] , nanowire patterns [ ] , and the analysis of the oscillatory dynam-ics of inclusions [ ] .Due to its capability to implicitly track the interface, the phase-field methods are ex-tensively employed to study the motion of electromigration-induced isotropic inclusion insingle crystals [ ] . In addition, relatively few studies are performedto understand the motion of anisotropic inclusions [ ] . However, the literatureis devoid of exquisite comparisons of analytical findings with the phase-field results. Inaddition, the shape-changing mechanism, observed during the migration of inclusions, iscomplex and obscure. Therefore, the presented work is intended to understand this mech-anism through the electromigration-based phase-field model. In recent decades, phase-field models have become an ideal simulation tool for numerousapplications. In the phase-field approach, a continuous variable is defined to represent theentire microstructure, which is also known as the order parameter or phase-field param-eter. The order parameter itself holds a nearly constant value in the bulk domains andvaries continuously across a finite but narrow interface. Due to the presence of a mostelegant feature, the diffuse interface, these models provide an efficient framework to ad-hapter 1. 18dress the time-dependent free boundary problems. In addition, the boundary conditionsare incorporated implicitly to avoid cumbersome interface tracking. Therefore, the move-ment of complex boundaries and stringent boundary conditions can be managed readily.The transformation of the order parameter works on optimization principles, such as theminimization of the free energy or the maximization of the system entropy, which leads tomorphological changes in the microstructure. The Cahn-Hilliard model [ ] is importantin the field of conserved order parameters, while the Allen-Cahn model [ ] is relevant inthe field of non-conserved order parameters.The thesis is organized as follows. Chapter 2 presents a review of the basic frameworkof the phase-field models, by considering theoretical advancements from the microscopicproperties to the mesoscopic description. Thereafter, Chapters 3 and 4 respectively derivethe models of two previously described phenomena, namely, two-phase coexistence in LMOmaterials and the motion of inclusions, due to electromigration. The numerical results ofphase separating LMO electrodes are presented in Chapters 5 and 6. Subsequently, thestudy of various aspects of the electromigration-induced motion of inclusions is presentedin Chapters 7, 8, and 9. In Chapter 10, the thesis concludes with a brief summary and afew possible future directions. art IIMethods : Phase-field formulationhapter 2Phase-field model In the past decades, the understanding of many natural phenomena has been obtainedthrough experimental investigations. Afterwards, these experimental findings have beenable to motivate the development of theories one way or the other. Thereafter, the ex-ponential increase in computational resources has recently encouraged the advancementof numerical techniques based on a unique theoretical framework, which may untanglethe intricacies associated with the physical process under investigation. The resulting en-hanced understanding may assure an optimal design, a better performance prediction, andan efficient employment of the resources.The applicability of the numerical modeling ranges from several pico-seconds to thou-sands of years in time scale and on the atomistic level (nanometers) to the level of theuniverse (kilometers) in length scale. Figure 2.1 illustrates some of the computationalmethods with their typical range of length and time scale used in the literature. In ad-dition, communication across the models is often considered to improve the accuracy ofthe employed method. For instance, simulations of molecular dynamics are considered inthe fiber-reinforced thermosets to estimate material properties. Then, the curing processdetermines the eigenstresses. On the basis of these local eigenstresses, the phase-field frac-ture simulations are considered to predict the occurrence of microcracks in Ref. [ ] . Theselection of a specific computational method for a concerned phenomenon is a matter ofchoice, based on its feasibility and effectiveness.Phase-field modeling is one of such computational technique recognized for the pastcouple of decades. This mesoscopic approach is employed in diverse avenues of research,including multicellular systems in biology [ ] , multicomponent fluid flow in fluid dynam-ics [ ] , alloy solidification in materials science [ ] , among others. As depicted fromFigure 2.1, the typical dimensions of the utilized phase-field methods are on the mesoscalelevel. Therefore, before introducing the phase-field model, a preface is presented to relate21hapter 2. 22 Figure 2.1: Estimated physical length scale and time scale of the various computational methods, utilizedfor numerical modeling. Reprinted from the work of Stan [ ] with permission from Elsevier. the mesoscale description with the microscopic properties. The origins of the phase-field methodology have been considerably influenced by the mean-field theory of phase transformation. Thus, some of the foundations can be interpreted bythe mean-field theory, which is based on the framework of statistical thermodynamics. Thistheory provides the necessary groundwork for the concept of an order parameter and its re-lation to the system free energy, commonly adopted in the phase-field theories. The presentsection is motivated by Refs. [ ] and some parts of their works are represented here,according to the present context.For simplicity, consider a binary mixture of two species, A and B. This mixture is scat-tered on a region which consists of N m small volume elements, known as lattice sites, whichare identified by an index α . These lattice sites carry either the A or B species. The statevariable c α is defined to identify the state of the cell, i.e. c α = N B = (cid:80) N m α = c α . In a simple lattice model, the total energyhapter 2. 23(Hamiltonian) H of such a system with A and B states is expressed as H { c α } = B m N m (cid:88) α = c α + B m N m (cid:88) α = M m (cid:88) α (cid:48) (cid:54) = α c α c α (cid:48) , (2.1)where α (cid:48) denotes the neighboring interstitial site, c α (cid:48) indicates the state of the neighboringinterstitial lattice site α (cid:48) , and M m represents the coordination number, which accounts forthe number of neighboring lattice sites. In addition, B m represents the internal systemenergy coefficient, which is independent from neighboring interactions amongst interstitiallattice sites. The interaction between the neighboring lattices α and α (cid:48) is quantified by thetwo-body interaction field B m . The first term in Eq. (2.1) comprises the contribution of theindividual lattices and the second term represents the interaction between the neighboringlattices. The estimation of the system free energy, by means of the Hamiltonian Eq. (2.1),requires the calculation of the grand canonical partition function Z gc ( T , µ ) = (cid:88) c α ∈{ } exp (cid:130) − H { c α } − µ (cid:80) α c α k B T (cid:140) , (2.2)where µ is the chemical potential, T denotes the absolute temperature, and k B is the Boltz-mann constant. In addition, Z gc is related to the grand canonical potential function g ( T , µ ) and the free energy can be calculated from the following expressions: g ( T , µ ) = − k B T ln Z gc ( T , µ ) , (2.3) N B = − ∂ g ( T , µ ) ∂ µ , (2.4) F ( T , N B ) = g ( T , µ ) + N B µ ( T , N B ) . (2.5)Note that the primary objective of the presented analysis is to estimate the free energy F ( T , N B ) , which can also be accomplished from the canonical partition function. However,the estimation of the canonical partition function imposes mathematical intricacies. Toovercome these difficulties, instead of using the canonical partition function, which is afunction of the concentration, the free energy is calculated from the grand canonical parti-tion function, a function of the chemical potential.Landau [ ] postulated a phenomenological approach for phase transformation to an-alyze the transition from one state to the other state, which can be viewed as a growth of astate variable. After that, an important scalar state variable, referred to as order parameter , c is introduced to describe the transition and state of collective sites. This parameter istreated as an average thermodynamic property of the state, such as symmetry, concentra-tion, and density, which differentiates different states, c = N B N m . (2.6)hapter 2. 24Due to the two-body interaction term in the Hamiltonian Eq. (2.1), the calculation of thegrand canonical partition function is tedious. To overcome the difficulties, the mean-fieldapproximation allows to replace the two-body term in the Hamiltonian (2.1) by a one-bodyterm, considering a small fluctuation in c α around the average concentration c , by utilizingthe relation c α = c + δ c α , where δ c α / c (cid:28) δ c α are the fluctuations of c α , comparedto the average. After substituting this relation into the two-body term of Eq. (2.1), theapproximation consisting of the one-body term can be expressed as N m (cid:88) α = M m (cid:88) α (cid:48) (cid:54) = α ( c + δ c α )( c + δ c α (cid:48) ) = − N m M m c + M m c N m (cid:88) α = c α . (2.7)As δ c α / c (cid:28)
1, the second-order fluctuations δ c α δ c α (cid:48) are neglected. In addition, the re-lations Σ N m α = Σ M m α (cid:48) (cid:54) = α = N m M m / Σ N m α = Σ M m α (cid:48) (cid:54) = α δ c α = − N m M m c + M m Σ N m α = c α are utilized.Substituting Eq. (2.7) into the Hamiltonian Eq. (2.1), the equivalent Hamiltonian, consist-ing of the one-body term, can be written as H { c α } = ( B m + M m cB m ) N m (cid:88) α = c α − N m M m c B m . (2.8)The two-body term is eliminated from the Hamiltonian (2.1), by introducing a field vari-able c . Therefore, the effects of the neighboring sites are incorporated through the aver-aged field variable. By substituting Eqs. (2.8) and (2.6) into Eq. (2.2), the resultant grandcanonical partition function can be obtained as Z gc ( T , µ ) = exp (cid:18) N m M m c k B T B m (cid:19) N m (cid:89) α = (cid:88) c α ∈{ } exp (cid:18) − ( B m + B m M m c − µ ) k B T c α (cid:19) = exp (cid:18) N m M m c k B T B m (cid:19) (cid:130) + exp (cid:18) − B m + B m M m c − µ k B T (cid:19)(cid:140) N m (2.9)Inserting this relation into Eq. (2.3), the grand canonical potential can be expressed as g ( T , µ ) = − N m B m M m c − N m k B T ln (cid:130) + exp (cid:18) − B m + B m M m c − µ k B T (cid:19)(cid:140) . (2.10)Substituting this relation and Eq. (2.4) into Eq. (2.6), the expression for the chemicalpotential can be obtained as µ = B m + B m M m c + k B T ln (cid:129) c − c (cid:139) . (2.11)Finally, from Eqs. (2.5), (2.10), and (2.11), the free energy of the system can be expressedas F ( T , N B ) / N m = (cid:129) B m + B m M m (cid:139) c − (cid:129) B m M m (cid:139) c ( − c )+ k B T (cid:0) c ln ( c ) + ( − c ) ln ( − c ) (cid:1) . (2.12)hapter 2. 25 B AAA AA AAAB B BBB B BAAAB BBBAB
Figure 2.2: Schematic of a lattice consisting of a mixture of A and B species, which forms bonds betweenA–A, B–B, and A–B.
To simplify further discussions on the free energy, considering the relations B m + B m M m / = X k B T ref , B m M m / = X k B T ref , and F ( T , N B ) / ( N m k B T ref ) = f ( T , c ) , ,the system free energydensity is expressed as f ( T , c ) = X c + X c ( − c ) + TT ref (cid:0) c ln ( c ) + ( − c ) ln ( − c ) (cid:1) , (2.13)which is in normalized form. The term energy density, is used to refer to its normalizedform throughout the document.To understand the different characteristics of the free energy equation (2.13), weconsider X = T = T ref for simplicity, and additionally designate the two terms ∆ H = X c ( − c ) and − ∆ S = c ln ( c ) + ( − c ) ln ( − c ) in Eq. (2.13), which respectivelyare the change in the internal energy and the entropy of the system. In the forthcomingparagraphs, these terms are extensively utilized to distinguish typical features of the freeenergy. In order to describe these characteristics, a simple lattice system of a binary solutionis considered, as shown in Figure 2.2.The structure of a binary mixture contains three types of bonds, A–A, B–B, and A–B,with the bond energies (cid:34) AA , (cid:34) BB , and (cid:34) AB respectively, which are illustrated schematically.After mixing, the change in the internal energy of the system can be expressed as ∆ H = (cid:34) t P AB , (2.14)where P AB is the total number of A–B bonds and (cid:34) t is the difference between the A–B bondenergy and the average of the A-A and B-B bond energies: (cid:34) t = (cid:34) AB − ( (cid:34) AA + (cid:34) BB ) . (2.15)hapter 2. 26 (a) (b)(c) (d) Figure 2.3: Free energy curves for different values of (cid:34) t . The plots correspond to (a) X = , (b) − ,(c) , and (d) in Eq. (2.13) , while considering X = and T = T ref . As the spontaneous change in the entropy is always positive ∆ S >
0, the following casescan be obtained by considering the value of ∆ H :1. If (cid:34) t =
0, it implies that ∆ H =
0, which means that the solution is ideal. In this case,A–B bonds are equally favorable to A–A and B–B bonds, based on the internal energyargument, which indicates that the species are arranged randomly. As the change inthe internal energy is treated to be negligible for an ideal solution ∆ H =
0, the systemfree energy only contains the entropy term, which is shown schematically in Figure2.3(a). However, this type of behavior is seldom observed in practice and usually, theinternal energy changes with the mixing, i.e. ∆ H (cid:54) = (cid:34) t < ∆ H < f ( T , c ) , the species prefer to be surrounded by species ofanother type, which results in mixing, which increases the number of A–B bonds P AB ,compared to the ideal solution. The internal energy and the enthalpy are in consensushapter 2. 27to keep the free energy of the mixture at the lowest, as shown in Figure 2.3(b). Thissignifies that maintaining the isolated regions of the A or B species is energeticallyunfavorable, and that the system rather prefers a state of a homogeneous mixture.3. More complicated situations arise for (cid:34) t > ∆ H > P AB tends to decrease, compared to an ideal solution, for − ∆ S (cid:29) ∆ H ,the system still prefers mixing, as shown in Figure 2.3(c). This is due to the factthat the system free energy curve has a positive curvature at all points. However,on the other hand, when − ∆ S is comparable or lower than ∆ H , the competitionbetween these two establishes a double well in the resultant free energy curve, asshown in Figure 2.3(d). The existence of the concavity in the curve indicates thata homogeneous mixture is energetically unfavorable in a certain region. Instead,phase-separated states of two distinguished colonies are expected in the system. Thisbehavior of concavity is a focus of the presented work, which is extensively exploredin the forthcoming sections.Based on the mean-field approximation, the order parameter c is assumed to be ho-mogeneous over the entire system. However, the concavity in Figure 2.3(d) indicates thatthe homogeneous distribution does not ensure the system free energy to be minimal in acertain region. Therefore, the mean-field theory alone is inadequate to confiscate the non-homogeneous behavior. As an extension, the Ginzburg-Landau free-energy functional canbe considered to address the concavity in the system free energy. The mean-field-free energies discussed in the previous section 2.1 only consider an averageof the bulk properties in a material. This type of free energy can be employed for a sys-tem that is infinite in extent and has uniform thermodynamic properties throughout. Thisrestricts its applicability to the extent of considering the finiteness of the phase † . More im-portantly, the relevance of this approach is devoid of the multiple phases and interfaces sep-arating these phases, which are inexplicable with the mean-field approximation. However,the interfaces are the most important features, governing the formation of microstructuresin most materials. Their migration and interaction are central to the manifold applications [ ] . Therefore, this section aims to incorporate interfacial properties into the mean-fieldfree energy, resulting in the Ginzburg-Landau free energy functional. † A phase can be defined as a distinct and homogeneous form of matter, separated from other forms byits surface. hapter 2. 28To facilitate the incorporation of interfacial properties into the system free energy andenable spatially dependent phase changes, a spatial dependency is incorporated into the or-der parameter field, c ( x α ) , defined in section 2.1, which was only associated with bulk prop-erties. To formulate a free energy which encompasses the interface, the Hamiltonian intro-duced in Eq. (2.1) must be redefined extensively. Correspondingly, the spatial-dependentHamiltonian reads as H { c ( x α ) } = B m N m (cid:88) α = c ( x α ) + B m N m (cid:88) α = M m (cid:88) α (cid:48) (cid:54) = α c ( x α ) c ( x α (cid:48) ) . (2.16)By considering the two-body term, the interaction between the phases in Eq. (2.16) encom-passes both the bulk phases and the interface. Note that the contribution of the interface isnot separate and implicit in the expression. The two-body term, represented as an interac-tion between two different interstitial sites c ( x α ) and c ( x α (cid:48) ) , can be simplified to H { c ( x α ) } = B m N m (cid:88) α = c ( x α ) + B m N m (cid:88) α = M m (cid:88) α (cid:48) (cid:54) = α (cid:16) c ( x α ) + c ( x α (cid:48) ) − (cid:0) c ( x α ) − c ( x α (cid:48) ) (cid:1) (cid:17) = B m N m (cid:88) α = c ( x α ) + B m M m N m (cid:88) α = c ( x α ) − B m N m (cid:88) α = M m (cid:88) α (cid:48) (cid:54) = α (cid:0) c ( x α ) − c ( x α (cid:48) ) (cid:1) . (2.17)The first two terms on the right-hand side of Eq. (2.17) represent the contribution of theindividual phases, while the third term corresponds to the interaction across the phases,which can be attributed to the neighbor sites. In a cubic lattice setup, wherein the coor-dination number M m =
6, the interaction of any site c α is restricted to its neighbor sites,front c ( x α F ) , back c ( x α B ) , top c ( x α T ) , bottom c ( x α O ) , right c ( x α R ) , and left c ( x α L ) sites.Therefore, the expression in the third term of Eq. (2.17) can be expanded as M m (cid:88) α (cid:48) (cid:54) = α (cid:0) c ( x α ) − c ( x α (cid:48) ) (cid:1) = (cid:23) a (cid:168)(cid:130) (cid:0) c ( x α ) − c ( x α F ) (cid:1) (cid:23) a + (cid:0) c ( x α ) − c ( x α R ) (cid:1) (cid:23) a + (cid:0) c ( x α ) − c ( x α T ) (cid:1) (cid:23) a (cid:140) + (cid:130) (cid:0) c ( x α ) − c ( x α B ) (cid:1) (cid:23) a + (cid:0) c ( x α ) − c ( x α L ) (cid:1) (cid:23) a + (cid:0) c ( x α ) − c ( x α O ) (cid:1) (cid:23) a (cid:140)(cid:171) , (2.18) ≈ (cid:23) a (cid:8) | ∇ c ( x α ) | (cid:9) . (2.19)Here ∇ ( • ) indicates the gradient of the respective field and (cid:23) a is the lattice parameter, whichis the distance of the neighboring lattice sites. In the present approximation, it is assumedto be a constant. Such an approximation might lead to the omission of several spatialmicroscopic details and misestimating properties associated with it. However, for the ma-terials which consist of a directionally independent lattice parameter, which is of interesthapter 2. 29to the present work, the employed assumption is valid. Each term in the round brackets ofEq. (2.18) represents the magnitudes of the one-sided gradients of the order parameter indiscrete form, at the point α , which is replaced by a continuous description in Eq. (2.19).Based on the approach provided by Ginzburg and Landau [ ] , the order parameter,which is defined non-continuously, considering the microscopic properties in the Hamilto-nian Eq. (2.16), can be transformed to spatially continuous, in the mesoscopic length scale.In the mesoscopic description, the discrete microscopic summation over " α " can thereforebe replaced by the continuous operators of the form N m (cid:88) α = ⇒ (cid:90) V Ω d V Ω (cid:23) a , (2.20)where V Ω is the volume of the system and (cid:82) V Ω ( • ) d V Ω is the integral over the volume of therespective function. The devision by (cid:23) a is intended to encapsulate the microscopic prop-erties in the mesoscopic limit, by considering the volume of an element. This descriptionenables the transition of the piecewise description of the properties to the continuum limit.By substituting Eqs. (2.19) and (2.20) into the Hamiltonian Eq. (2.16), the resultant con-tinuously defined Hamiltonian, which is identified as an internal energy, can be expressedas E ( c ( x )) = (cid:90) V Ω (cid:110) B m c ( x ) + B m M m c ( x ) − B m (cid:23) a | ∇ c ( x ) | (cid:111) d V Ω (cid:23) a . (2.21)The first two terms in the equation are the bulk terms, which only consist of the currentposition dependency, while the third term is the gradient term, which encloses the interfa-cial properties across the neighbor phases. Note that the two-body interaction term in theHamiltonian Eq. (2.16) is replaced by the gradient term in Eq. (2.21). In addition, this termis nearly vanishing in the bulk phases and only varies significantly at the interfaces, wherethere is appreciable change of the order parameters. Therefore, this term can be linked withthe surface energy of the phases. In addition to this contribution, the role of the entropy ∆ S ( c ( x ) , T ( x )) should be included to translate the internal energy into system free energy.Conventionally, the total number of microscopic configurations correlates to the entropyterm, which can be formulated by employing Sterling’s approximation. If the free energycontribution of the bulk phase is represented by f ( c ( x ) , T ( x )) , after the inclusion of theentropy, the resulting free energy functional is expressed as F (cid:0) c ( x ) , T ( x ) , ∇ c ( x ) (cid:1) = (cid:90) V Ω (cid:16) X c ( x ) + X c ( x ) (cid:0) − c ( x ) (cid:1) + κ | ∇ c ( x ) | + T ( x ) T ref (cid:0) c ln ( c ) + ( − c ) ln ( − c ) (cid:1) (cid:17) d V Ω . (2.22) = (cid:90) V Ω (cid:129) f ( c ( x ) , T ( x )) + κ | ∇ c ( x ) | (cid:139) d V Ω , (2.23)hapter 2. 30where κ k B T ref = − B m (cid:23) a is the gradient energy coefficient. This formulation, which isbased on a mesoscopic description, consistently allows to represent the properties of thebulk phases ( f ( c ( x ) , T ( x )) ) and the interfaces separating them ( κ/ | ∇ c ( x ) | ). The derivedEq. (2.23) is commonly referred to as total free energy functional, where the order param-eter spatially varies continuously. In this sense, the discrete description of microscopicproperties, in the form of the Hamiltonian (2.16), are transformed to the mesoscopicallydefined, spatially continous free energy functional (2.23), in the present section. How-ever, this form of Hamiltonian is not followed further. Instead, the free energy functionalserves as a starting point for many phenomena which are modeled using the phase-fieldmethodology [ ] . In thermodynamics, it is of interest to know the change in free energy, δ F ( c , ∇ c ) , of agiven system with some activity , δ c . The activity can be regarded as either an addition (orremoval) of species or a growth (or shrinkage) in at least one of the phases. A chemicalpotential is introduced as a proportionality constant, so as to relate the activity with itseffect on the free energy [ ] . Based on a similar rationale, the chemical potential can beexpressed as µ = δ F ( c , ∇ c ) δ c = ∂ (cid:0) f ( c ) + κ | ∇ c | (cid:1) ∂ c − ∇ · ∂ (cid:0) f ( c ) + κ | ∇ c | (cid:1) ∂ ∇ c . (2.24)Here δ ( • ) /δ c is the variational derivative of the free energy functional F ( c , ∇ c ) = (cid:90) V Ω (cid:129) f ( c ) + κ | ∇ c | (cid:139) d V Ω , (2.25)with the order parameter c . Even though the functional still varies spatially and may containtemperature dependencies similar to Eq. (2.23), these variables are implicitly incorporatedand will be omitted from the expression for a convenience, without loss of generality.Since the activity is related to the chemical potential from Eq. (2.24), the activity isanother means of describing the state of a system. If the activity or chemical potential isvery low, the matter is reluctant to change its form, place or orientation, and the system is inequilibrium. In equilibrium, in which all competing influences are balanced, the system isstable. In the present sense, this is due to the negligible chemical potential. The variationalderivative of the free energy functional [ ] is expressed as0 = δ F ( c , ∇ c ) δ c = ∂ f ( c ) ∂ c − κ ∇ · ∇ c . (2.26)hapter 2. 31This equilibrium is sustained for a particular system configuration. However, any finite ac-tivity might interrupt the developed equilibrium and the chemical potential will be nonzero.This demands an investigation to minimize the free energy functional. Eventually, the sys-tem might attain a new equilibrium with an improvised configuration. To dissect thesecharacteristics, it is crucial to understand some specific characteristics of the order param-eter, which is described in the next section. It is important to note that the scalar state variable, which is identified as an order parameter c , in the Ginzburg-Landau theory [ ] , possesses a characterizing property which aidsin distinguishing different phases. This parameter represents the fraction of the system,exhibiting a particular property or a state, which changes spatially along with the system.For instance, in the solid to liquid transformation, an order parameter can be assigned todifferentiate the solid and the liquid phase. Eventually, considering the transformation,wherein the solid changes entirely to liquid, the order parameter, pertaining to the originalsystem, completely vanishes. In other words, the order parameter is considered as a non-conserved parameter. Consequently, it becomes unphysical to directly relate this orderparameter with the concentration, where the conservation of mass should be achieved.In order to formulate the free energy functional for a conserved parameter, Cahn andHilliard [ ] derived an expression, identical to Eq. (2.23), for concentration, , in their pi-oneering work, similar to the Ginzburg-Landau [ ] free energy functional, by replacingthe non-conserved order parameter. Despite the differences in the nature of the thermo-dynamical properties, the free energy functional alone does not discriminate against theconservancy of the order parameter. Therefore, the expressions derived in sections 2.1and 2.2 are equally applicable to both conserved and non-conserved order parameters. Inaddition, it is important to note that some salient features remain identical to these or-der parameters, such as the fact that it assumes a definite value in the bulk phases, whileit spatially varies in the narrow, but finitely defined interface regions, which separate thephases.As the free energy functionals could be considered identical to conserved and non-conserved order parameters, for non-equilibrium dynamics, there should be another meansto separate these properties. This can be achieved through the dynamic equations, whichare the time-dependent evolution of the order parameter in a non-equilibrium. The kinet-ics of these quantities are typically formulated as a Langevin-type equation [ ] , whichevolves to minimize the free energy functional. In the next sections, these kinetic equationsare consequently described for conserved and non-conserved order parameters.hapter 2. 32 Dynamic equations for the order parameter fields are called conserved if they take to be of aflux-conserving form. For instance, when the order parameter is required to be conserved,in terms of variables, like the concentration of the species during spinodal decomposition,as shown in Figure 2.4. This implies that a spatial integral of the field on the entire vol-ume should a constant for a closed system. Therefore, the concentration will be assigned inthe forthcoming descriptions to identify the conservative property of the order parameter.Under a non-equilibrium condition, in addition to spatial dependency, the concentrationexhibits a temporal evolution. The concentration evolution is modeled through the diffu-sion equation. In his pioneering work, Fick [ ] provided a time-dependent concentrationevolution in which the fluxes J should obey the conservation of mass in the form ∂ c ∂ t = − ∇ · J , (2.27)where t is the time and ∇ · ( • ) denotes the divergence of the vector. The diffusional fluxes arethe driving force that stimulates the matter movement. In Fick’s analogy, these forces aredirectly associated with a gradient of the chemical potential, which is linked to a gradientof the concentration. Hence, the fluxes can be expressed as J = − D ∇ c , (2.28)where D denotes the diffusion coefficient, which is a material property. This equation iscrucial in the dilute solution limit. Note that similar to these expressions, Fourier’s law ofheat conduction in pure materials can be derived by considering the temperature and thethermal conductivity. However, this lacks to separate the bulk phase and is inadequate toprovide any information about the interfaces, which is central to the phase-field commu-nity [ ] . For such a system, the chemical potential should be based on the free energyfunctional in Eq. (2.24).To account for such effects, Cahn and Hilliard [ ] proposed a model of phase separat-ing binary systems. This equation can be derived from the basic laws of classical thermody-namics, with the consideration of the diffusion between two types of A and B species, whichconsist of the concentrations c A and c B , respectively. According to the linear phenomeno-logical relations of the thermodynamics of fluxes, J A = − M AA ∇ µ A − M AB ∇ µ B , (2.29) J B = − M BA ∇ µ A − M BB ∇ µ B , (2.30)where M AA , M AB , M BA , and M BB are the mobilities, due to interaction among the A and Bspecies, µ A and µ B are the chemical potentials of A and B, respectively. The net flux of thehapter 2. 33 (d) c (b) (c) (d)(a) Figure 2.4: Schematic of an evolution, containing a conserved order parameter, concentration c, duringspinodal decomposition. The figures from left to right show the evolution of time, with the coarseningof smaller grains to larger ones. component B, opposed by component A, can be expressed as J = J B − J A . (2.31)A convenient notation for the binary system reads as c = c B = − c A , where c refers to theconcentration. Then the Gibbs-Duhem relationship reads as ( − c ) ∇ µ A + c ∇ µ B =
0. (2.32)Substituting Eqs. (2.29), (2.30), and (2.32) into Eq. (2.31), the net flux can be expressedas J = − M ∇ µ , (2.33)where the mobility M is given by M = ( − c )( M BB − M AB ) + c ( M AA − M BA ) and the chem-ical potential, µ = µ B − µ A can be expressed as the difference from B to A. The chemicalpotential can be obtained from the variational derivative of the system free energy func-tional, expressed in Eq. (2.24). Substituting the flux equation (2.33) into the conservationequation (2.27), the resultant expression can be written in the form of ∂ c ∂ t = ∇ · (cid:0) M ∇ µ (cid:1) , (2.34)which is designated as Cahn-Hilliard equation. To understand the analogy of this equation,two components are considered in the present section, for simplicity. However, this equa-tion can be extended to consider the multiple components along the lines of Ref. [ ] ,employed for a ternary system. Thereafter, the Cahn-Hilliard equation became prominent innumerous applications, such as spinodal decomposition [ ] , intercalation in lithium-ionbatteries [ ] , and electromigration in metallic conductors [ ] .hapter 2. 34 (a) (b) (c) (d) ϕ Figure 2.5: Schematic of an evolution containing a non-conserved domain parameter φ during crystalgrowth, as the solidification of a pure melt. The figures from left to right show the time evolution, withthe formation and growth of a dendritic pattern. The conservation of an order parameter is not always desired. In fact, some of the phasetransformations are driving to the energy minimum, through the loss (or gain) of phasefractions. For instance, crystal growth, due to the solidification of a pure melt, is one ofthe examples of a non-conserved order parameter, as shown in Figure 2.5. Therefore, thefraction of a crystal over the entire domain might not be constant during solidification. Toassociate the non-conservative properties, this parameter will be referred to as the domain(indicator) parameter φ . In accordance with the non-equilibrium thermodynamics [ ] ,the temporal evolution of the domain parameter is directly related to the variation deriva-tive of the functional expressed in Eq. (2.23). A relaxation coefficient τ p can be introducedas a propotionality constant. Therefore, the resultant time-dependent Ginzburg-Landauequation [ ] can be expressed as ∂ φ ∂ t = − τ p δ F ( φ , ∇ φ ) δφ . (2.35)Note that the free energy functional F ( φ , ∇ φ ) can be obtained similar to Eq. (2.23).Independantly, Allen and Cahn [ ] derived a similar expression to the time-dependentGinzburg-Landau equation (2.35). In the phase-field community, this equation is thus re-ferred to as the Allen-Cahn equation [ ] .As discussed in section 2.5, the Cahn-Hilliard equation (2.34) seems to be the obvi-ous choice for the order parameters that observe the mass conservation. However, theAllen-Cahn equation (2.35) is employed for a few of such cases, by incorporating spe-cific features like the redistribution-energy technique [ ] . This provision contributesvolume-preserving properties to the order parameter, by compensating a bulk-phase frac-tion by means of addition or subtraction [ ] . One of the primary hypotheses for suchhapter 2. 35an approach is that the former contains a fourth-order spatial derivative, while the latterhas second-order spatial derivative. In many numerical techniques like finite-differenceschemes, the computational time multiplies by manifolds, with an increase in the order ofthe spatial derivatives. Therefore, the Allen-Cahn equation might reduce the computationalefforts and improves the efficiency of the algorithms. However, to preserve the volumeanomalously, an additional term in the free energy allows counteraction to the source orthe sink of the phases. It is identified that this approach might constitute a physically im-probable and thermodynamically incoherent transformation mechanism [ ] . However,this establishes an appropriate and theoretically consistent final microstructure in equilib-rium. Therefore, the volume-preserving Allen-Cahn equation can only be employed wherethe final microstructure is of a prime objective, compared to the transformation mechanism.The elegance of the Cahn-Hilliard and the Allen-Cahn equations lies in their applicabil-ity. As they are directly related to the free energy, which can be readily extended to themultiphysical phenomenon, by considering a respective contribution in a free energy Li-aponuv functional [ ] . The challenging research goal is then to derive such functionals,which can sufficiently capture the physical phenomenon. In addition, it is also plausibleto compose an algorithm that embraces both conserved and non-conserved order parame-ters simultaneously [ ] for a phase transformation such as solidification of a binary Fe-Cmelt, which requires a non-conserved state variable and conserved concentrations of Feand C components or electromigration-induced grain boundary grooving [ ] . Therefore,these two equations, commonly known as phase-field equations, are employed extensivelyfor numerous applications [ ] . The specific feature of phase-field modeling is the introduction of the diffuse interface, re-placing the sharp one. One common practice in all phase-field models is that they considerone or several phase-fields, which describe the physical states such as the density, the form,the orientation or the order of any geometry under investigation. The physical state vari-able, the phase field, assumes a fixed and predetermined value in the bulk phases, whilevarying smoothly from one bulk value to the other, along with the interface of finitely de-fined width. This uniquely defined diffuse interface provides a remarkable advantage ofimplicit interface tracking, by averting the cumbersome techniques, as in the case of thesharp-interface models. In the next chapters, the phase-field models are derived on thebasis of the principles described in the present chapter for the concerned phenomena, i.e.,two-phase coexistence in cathode materials of lithium-ion batteries in Chapter 3 and inclu-sion motion under electromigration in Chapter 4. hapter 3Phase-field model for cathode particlesof Lithium-ion battery
During the charging and discharging of a battery, electrons transfer from the outer circuitand the positively charged lithium ions shuttle from one electrode to the other electrodethrough the electrolyte internally. The three-dimensional structural network of the elec-trode hosts the Li-ions, and the reversible insertion process of the Li-ions into the hostcompounds takes place at both ends of the electrodes [ ] . As discussed in section 1.1,most of the electrodes of lithium-ion batteries, such as lithium manganese oxide (LMO) [ ] observe the mechanism of phase separation during the insertion, which is of scientificinterest due to its direct implications as an important source of capacity fade. Also, thecoexistence of two phases in LMO is extensively explored in experiments [
40, 41, 45–47 ] .Systematic numerical modeling of phase separation may complement to the scientific un-derstanding of the mechanism and helps to optimize the process parameters efficiently andeconomically [ ] .To represent the two-phase coexistence in LMO cathode particles, a phase-field modelis derived based on Cahn-Hilliard equation in Section 3.1. Thereafter, the boundary con-ditions are explained in Sections 3.2 and 3.3. Followed by the implementation strategies,which are presented in Section 3.4. Finally, the chapter concludes with the outline andmodel applicability in Section 3.5. The lithium species migrate inside the particles after being deposited at the interface ofthe electrodes. In a closed system like lithium-ion batteries, these species obey mass con-servation laws. Therefore, a concentration parameter c ( x , t ) ∈ (
0, 1 ) can be introduced to37hapter 3. 38indicate the local state of the diffusion with time t and spatial coordinates x . Note thatthe concentration c in the present case is the mole fraction of occupied sites by the lithiumspecies to the total available sites in the electrode particles † . The two-phase coexistence inthe particles is due to the nature of phase-separating systems. Within the miscibility gap ofa phase-separating system, spinodal decomposition can take place at intermediate values of c , in which the system is unconditionally unstable with spontaneous phase decomposition,which initiates Li-rich and Li-poor phases. The free energy densities of such a system aredescribed in the following paragraphs. The occurrence of such phase decomposition can be understood by the diffusion of Li ina manner that allows it to instigate a Li-poor and a Li-rich region. These two regions areseparated by a steep concentration gradient called an interface. Furthermore, the motionof the interface is governed by the Li flux at the surface of the particle. Therefore, in theelectrochemical systems, the change in the concentration distribution leads to the alterationof the concentration gradient and, as a consequence, to the reshaping of the concentrationinto a changed minimum energy state. In turn, the concentration and the concentrationgradient are implicitly coupled. The combined change in the concentration gradient ∇ c and the concentration c are related by the system free energy functional by integratingover the volume V Ω and the surface S Ω of the form, F ( c , ∇ c ) = (cid:90) V Ω (cid:167) f ( c ) + κ | ∇ c | (cid:170) d V Ω + (cid:73) S Ω f S ( c ) d S Ω , (3.1)to model the process of phase separation. Here f ( c ) is the chemical free energy density,1 / κ | ∇ c | denotes the gradient energy density, and f S ( c ) represents the surface energy. Inaddition, κ is the gradient energy coefficient, which controls the interface thickness betweenthe adjacent Li-rich and Li-poor phases.The model focuses on a representative elementary volume (REV) of an electrode. Theset of indicator parameters ψ = [ ψ , ..., ψ N ] , ψ a ∈ {
0, 1 } , ∀ a ∈ {
1, . . . , N } is defined torecognize different phases. For instance, ψ = ψ a =
0, where a (cid:54) =
1) representselectrode particles, while ψ = ψ a =
0, where a (cid:54) =
2) indicates the electrolyte.Thus, the indicator parameter identifies multiple regions of the same physical properties as † Alternatively, the concentration can be defined as a molar concentration of the total lithium speciesexpressed in moles divided by the volume under consideration [ ] . In this case, the concentration c variesfrom 0 < c < c max , where c max denotes the maximum concentration in the cathode materials. hapter 3. 39a single phase. In general, the chemical free energy density can be expressed as f ( c ) = N (cid:88) a = h ( ψ a ) f chem a ( c ) (3.2)where h ( ψ a ) is the interpolation function and f chem a ( c ) denotes the chemical-free energydensity of the phase a . Here, as the phases are defined by a step function of indicatorparameters, meaning, the interface between the different phases is sharply defined, a first-order interpolation h ( ψ a ) = ψ a can be utilized [ ] .The chemical-free energy density f chem a ( c ) is a function of local field parameters, suchas the local lithium-ion concentration c , externally fixed parameters like the absolute tem-perature T , and the material properties like internal energy coefficients. The coexistenceof lithium-rich and poor phases in several Li intercalation compounds is of scientific impor-tance to investigate the phase separating behavior. To capture this effect, in the presentstudy, the regular solution model is considered for the expression of the chemical free en-ergy density equivalent to Eq. (2.13) of the form, f chem a ( c ) = α (cid:48) a c + α (cid:48)(cid:48) a c ( − c ) + TT ref (cid:8) c ln ( c ) + ( − c ) ln ( − c ) (cid:9) . (3.3)where α (cid:48) a and α (cid:48)(cid:48) a are the regular solution parameters associated with the internal energy, T ref denotes the reference temperature and ln ( • ) is the natural logarithm of the respectivesystem variable. The effect of variation in α (cid:48)(cid:48) a on the free energy density curves is discussedin section 2.1. However, to perceive the significance of the parameter α (cid:48) a , consider a Fig-ure 3.1. When α (cid:48) a =
0, the free energy curve is symmetric to the center c = ( c = ) is equals to the totally occupied sites ( c = ) by thelithium in the intercalating material. Otherwise, a nonzero α (cid:48) a should be considered, whichregulates the relative heights of the two wells of the free energy, as shown in Figure 3.1.This free energy density (3.3) is considered [ ] with nonzero α (cid:48) a and α (cid:48)(cid:48) a for the study oftwo-phase coexistence in the active particles of cathode material in lithium-ion batteries.Furthermore, the gradient energy coefficient can be expressed as an interpolation betweenthe phases of the form, κ = N (cid:88) a = h ( ψ a ) κ a (3.4)where κ a is the gradient energy coefficient of the phase a . In the phase-field formulation, the optimization of an objective functional (3.1), formulatedin terms of the energy densities is of primary concern. Towards such intent, performing thehapter 3. 40 -0.2-0.10.00.10.0 0.2 0.4 0.6 0.8 1.0 f r ee ene r g y den s i t y , f a c he m ( c ) c α′ a =-0.1, α′′ a = 2.6 α′ a = 0.0, α′′ a = 2.6 Figure 3.1: (a) Effect of the parameter α (cid:48) a on the free energy curves. The green curve represents thediminishing α (cid:48) a where two energy wells are at same energy height and the curve is symmetric withc = . The blue curve indicates a nonzero free energy coefficient α (cid:48) a considered for the two-phasecoexistence in intercalating materials, otherwise stated. variational derivative of the free energy functional and rearranging the terms, δ F ( c , ∇ c ) = (cid:90) V Ω (cid:150) δ f ( c ) δ c δ c + δ (cid:8) / κ | ∇ c | (cid:9) δ c δ c (cid:153) d V Ω + (cid:73) S Ω δ f S ( c ) δ c δ c d S Ω = (cid:90) V Ω (cid:150) ∂ f ( c ) ∂ c δ c + κ ∂ ( ∇ c ) ∂ ∇ c · ∇ δ c (cid:153) d V Ω + (cid:73) S Ω ∂ f S ( c ) ∂ c δ c d S Ω = (cid:90) V Ω (cid:150) ∂ f ( c ) ∂ c δ c − κ ∇ (cid:18) ∂ ( ∇ c ) ∂ ∇ c (cid:19) · δ c + κ ∇ · (cid:18) ∂ ( ∇ c ) ∂ ∇ c δ c (cid:19) (cid:153) d V Ω + (cid:73) S Ω ∂ f S ( c ) ∂ c δ c d S Ω (3.5) = (cid:90) V Ω (cid:150) ∂ f ( c ) ∂ c − κ ∇ · ∇ c (cid:153) δ c d V Ω + (cid:73) S Ω (cid:20) ∂ f S ( c ) ∂ c − κ ( n · ∇ c ) (cid:21) δ c d S Ω (3.6)where n denotes the inward pointing unit normal to the particle surface and divergencetheorem is operated to the third term of Eq. (3.5), which converts the volume integral toa surface integral in Eq. (3.6). It is important to differentiate the gradient energy and thesurface energy. The second term in the volume integral corresponds to interface widthbetween Li-rich and Li-poor phases, while the terms in the surface integral associated withthe surface energy.The understanding of the surface energy is significant to consider heterogeneous nucle-ation [ ] . As the lithiation takes place at the surface of the particles, the phase separationinitiates from the surface. However, it might be probable to observe heterogeneous nu-hapter 3. 41cleation of phase separation at the interface. This behavior can be incorporated from thesurface energy, f S ( c ) . The minimization of the system free energy is ensured by equatingthe functional derivative to zero, δ F ( c , ∇ c ) =
0. As the variations in the domain and onthe boundary are represented by the spatially smooth functions, both terms multiplying δ c should vanishes independently. Therefore, Eq. (3.6) yields the boundary condition, (cid:20) ∂ f S ( c ) ∂ c − κ ( n · ∇ c ) (cid:21) δ c = S Ω , (3.7)on the particle surface. Furthermore, the quantity in the volume integral of Eq. (3.6) isreferred as the chemical potential, µ = ∂ f ( c ) ∂ c − κ ∇ · ∇ c , (3.8)which is an objective functional to optimize [ ] . The first term refers to the chemicalpotential corresponding to the bulk free energy density µ chem . For a phase, the bulk chemicalpotential is expressed as, µ chem a ( c ) = ∂ f chem a ( c ) ∂ c = α (cid:48) a + α (cid:48)(cid:48) a ( − c ) + TT ref (cid:8) ln ( c ) − ln ( − c ) (cid:9) . (3.9)Furthermore, the second term denotes the chemical potential corresponding to the gradientfree energy density, µ grad . For a phase, the gradient chemical potential is expressed as, µ grad a ( ∇ c ) = − κ a ∇ · ∇ c . (3.10) The driving force for the diffusion of each species in the electrochemical systems is thegradient of the chemical potential of the species. The Onsager relation defines the fluxdistribution inside the particle of the form J = − M ∇ µ , (3.11)where µ denotes the chemical potential of the system expressed in Eq. (3.8) and M is theatomic mobility in the form of a diagonal matrix. If the mobility is direction-dependent, asfor instance in lithium-iron phosphate (LFP) [ ] , M is represented by unequal compo-nents of the diagonal matrix. Since the mobility in LMO is isotropic, the matrix reduces toa scalar prefactor of the form [ ] , M = c ( − c ) (cid:32) T ref T N (cid:88) a = h ( ψ a ) D a (cid:33) , (3.12)hapter 3. 42where D a is the diffusion coefficient of phase a .The time evolution of the lithium-ion diffusion can be obtained from the Onsager’s re-lation for non-equilibrium thermodynamics. As the diffusion follows mass conservationof species, the continuously-defined concentration c ( x , t ) takes spatially and temporallydependent form, ∂ c ∂ t = − ∇ · J , (3.13)Note that by substituting the flux Eq. (3.11) in the mass conservation Eq. (3.13) yields theclassical Cahn-Hilliard equation of the form ∂ c ∂ t = ∇ · ( M ∇ µ ) on V Ω . (3.14)This partial differential equation determines the evolution of concentration c ( x , t ) for giveninitial and boundary conditions. The solution of the partial differential equation (3.14) under the boundary conditions is ofa primary objective to represent the lithium diffusion. It can be feasible to utilize analyticalmethods to solve non-complex partial differential equations under trivial boundary condi-tions. However, with the increase in the complexity of the equations, it became arduousto employ analytical theories. In such cases, these equations need to solve with numericalalgorithms. Even for numerical schemes like Fourier-spectral and finite-difference meth-ods, it is challenging to employ a boundary condition on the surfaces which are inside thesimulation box. Furthermore, the complications escalate with considering particle of a cur-vaceous geometry. For such cases, the general boundary condition of arbitrary geometricalshapes need to implement through some special techniques such as the smoothed boundarymethod [ ] . In smoothed-boundary method, a domain parameter ψ , is registered to differentiate thecathode particle and the electrolyte, and interpolate between different phases. The domainparameter ψ indicates the particle by ψ = ψ = < ψ < ψ , ψ ∂ c ∂ t = ψ ∇ · ( M ∇ µ ) , (3.15)hapter 3. 43 Diffuse particle interface0 . < ψ < . n = ∇ ψ | ∇ ψ | Particle ψ = 1 . ψ = 0 . (cid:0)(cid:0)(cid:0)(cid:18)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:82)(cid:64)(cid:64)(cid:64)(cid:82) Figure 3.2: Schematic diagram of a cathode particle, surrounded by the electrolyte in a lithium-ionbattery. The domain parameter ψ indicates the particle by ψ = , the electrolyte by ψ = andthe diffuse particle surface region by < ψ < . The surface normals in the diffuse region are usedto implicitly incorporate the boundary condition into the evolution equation. to incorporate the smoothed boundary method. Using the relation − M ∇ µ = J , the evolu-tion equation can be rearranged as ψ ∂ c ∂ t = ∇ · ( ψ M ∇ µ ) + ∇ ψ · J ∂ c ∂ t = ψ ∇ · ( ψ M ∇ µ ) + | ∇ ψ | ψ n · J (3.16) = ψ ∇ · ( ψ M ∇ µ ) + | ∇ ψ | ψ J n ,where n = ∇ ψ / | ∇ ψ | is the inward normal to the particle surface, and J n is the boundaryflux in normal direction to the particle.For the smoothed boundary method, the chemical potential corresponding to the chem-ical free energy density remains unchanged from Eq. (3.9), while the chemical potentialcorresponding to the gradient energy density changes from Eq. (3.10) to µ grad ( ∇ c ) = − κ ∇ · ∇ c = − (cid:18) κψ ∇ · ψ ∇ c − κψ ∇ ψ · ∇ c (cid:19) . (3.17)hapter 3. 44The second term on the right side of Eq. (3.17) can be further simplified as ∇ ψ · ∇ c = | ∇ ψ | n · ∇ c . According to Eq. (3.7), n · ∇ c is linked to preferential surface wetting energy, f S ( c ) responsible for heterogeneous nucleation. However, in the case of isotropic surfaceenergy and homogeneous nucleation, this term vanishes identically, i.e., ∇ ψ · ∇ c =
0. (3.18)Finally, considering Eqs. (3.8), (3.16), and (3.17), the resultant evolution equation is ex-pressed of the form, ∂ c ∂ t = ψ ∇ · ψ M ∇ (cid:168) ∂ f ( c ) ∂ c − (cid:18) κψ ∇ · ψ ∇ c − κψ ∇ ψ · ∇ c (cid:19)(cid:171) + | ∇ ψ | ψ J n , (3.19)here the Neumann flux condition can be operated in the last term of the equation by con-sidering appropriate choice of function, J n . When a natural (no-flux) boundary conditionis feasible, i.e. J n =
0, the last term vanishes.
Similar to Neumann condition, the Dirichlet condition can be implemented on the particlesurface through smoothed-boundary method by manipulating the classical Cahn-Hilliardequation (3.14). To incorporate the Dirichlet condition, the chemical potential correspondsto the gradient energy Eq. (3.17) can be rewritten as, µ grad ( ∇ c ) = − (cid:20) κψ ∇ · ψ ∇ c − κψ ∇ ψ · ∇ c (cid:21) , = − (cid:150) κψ ∇ · ψ ∇ c − κψ (cid:110) ∇ ψ · ∇ ( c ψ ) − c ∇ ψ · ∇ ψ (cid:111)(cid:153) , = − (cid:150) κψ ∇ · ψ ∇ c − κψ (cid:110) ∇ ψ · ∇ ( c ψ ) − c n | ∇ ψ | (cid:111)(cid:153) . (3.20)where c n is the Dirichlet boundary condition imposed at the location where | ∇ ψ | containsfinite value, i.e. particle surface. By substituting Eq. (3.20) into Eq. (3.8), the final evolutionequation can be written as, ∂ c ∂ t = ψ ∇ · ψ M ∇ (cid:40) ∂ f ( c ) ∂ c − (cid:130) κψ ∇ · ψ ∇ c − κψ (cid:0) ∇ ψ · ∇ ( c ψ ) − c n | ∇ ψ | (cid:1)(cid:140)(cid:41) .(3.21)Note that the Dirichlet concentration boundary condition can be employed at the last termof the form, c n .hapter 3. 45It is worth noticing that these equations (3.19) and (3.21) can be combined to specify theNeumann flux and Dirichlet concentration boundary conditions, J n and c n simultaneously,which can be referred as Robin boundary condition. The final form of the evolution equationis expressed as, ∂ c ∂ t = ψ ∇ · (cid:150) ψ M ∇ (cid:168) ∂ f ( c ) ∂ c − (cid:129) κψ ∇ · ψ ∇ c − κ W c ψ (cid:0) ∇ ψ · ∇ ( c ψ ) − c n | ∇ ψ | (cid:1)(cid:139)(cid:171)(cid:153) + W J | ∇ ψ | ψ J n , (3.22)where W J and W c are the spatially dependent weights for Neumann flux and Dirichlet con-centration boundary conditions respectively. In addition, W c + W J =
1, these factors discrim-inate Neumann flux and Dirichlet concentration boundary conditions imposed on differentregions of particle surface.The evolution equations (3.19), (3.21), and (3.22) are solved everywhere in the do-main, i.e. in the region of the particle ( ψ =
1) as well as in the electrolyte ( ψ = ν = × − , is added to the denominators to prevent divi-sion by zero in the region ψ =
0. Yu et al. [ ] described the effect of different values of ν on the simulation results in detail. The boundary conditions at the particle surface can be encompassed into the main evolutionequation through the smoothed-boundary method, as described in section 3.2. However,the boundary conditions on the separator end can be realized by considering a REV element,as shown in Figure 3.3. As the conditions are provided at the boundary of the simulationdomain, the main evolution equation (3.14) need not to be modified. A concentrationequation in the form of Dirichlet condition is prescribed on one side of the boundary, whileon the opposite end a Neumann condition is considered. The remaining two boundariesfor 2D (four boundaries for 3D) simulations are considered periodically repetitive. Thecomplete expression can be written in the form, c ( t , i = N x , j , k ) = c ps , ∂ c /∂ x ( t , i = j , k ) =
0, on ∂ V Ω (3.23) c ( t , i , j + N y , k ) = c ( t , i , j , k ) c ( t , i , j , k + N z ) = c ( t , i , j , k ) .Where c ps is the prescribed concentration equation at the boundary and N x , N y , and N z arethe number of grid points in x-, y- and z-directions ( i , j , k ∈ x ) respectively. Similarly, ahapter 3. 46 Figure 3.3: Simulation setup of Dirichlet concentration or Neumann flux boundary conditions at theseparator on 2D microstructure in (a) and 3D microstructure in (b) for the diffusion of lithium speciesin the electrode. flux equation in the form of Neumann condition can be prescribed on one of the boundaryinstead of the first provision in the equation set (3.23) in the form, M ∂ µ∂ x ( t , i = N x , j , k ) = J ps , (3.24)where J ps denotes the prescribed flux at the boundary. Similar to Eq. (3.22), it is also possi-ble to combine the Dirichlet concentration (first equation of the set (3.23)) and Neumannflux (Eq. (3.24)) boundary conditions to obtain the Robin condition at the separator.It is worth noticing that the boundary conditions at the particle surface c n and J n and atthe separator c ps and J ps are not confined to be constant values. Alternatively, these mightbe variables, which encompass the function of local concentration, time, space, and otherparameters. For computational convenience, the variables in the evolution equations and the boundaryconditions are considered non-dimensional as,ˆ x = x L , ˆ t = D L t , ˆ J n = LD J n . (3.25)where L is the reference length for normalization, D is the diffusion coefficient of lithiumspecies in the electrode material, and the notation ˆ • defines the normalized quantity of thehapter 3. 47 z k − / z k z k +1 / x i x i +1 / x i − / y j y j − / y j +1 / ∆ y ∆ z ∆ xz xy scalar quantities c, ψ vector quantities ∇ c x , ∇ { ψ } x ∇ c y , ∇{ ψ } y ∇ c z , ∇ { ψ } z Figure 3.4: Arrangement of the scalar and vector quantities on a staggered grid. A single simulationcell is shown with employed notation. respective entities.Standard finite-difference numerical schemes are employed to solve the partial differen-tial equations on a rectangular grid. A single simulation cell is displayed in Figure 3.4 withutilized nomenclature in the present study at position ( i , j , k ) . The spatial derivatives arediscretized by the finite-difference scheme. The gradient of concentration in all directions x , y , and z at a discrete time n , for example, is computed as (cid:8) ∇ c x (cid:9) ni + / j , k = c ni + j , k − c ni , j , k ∆ x + (cid:79) ( ∆ x ) , (3.26) (cid:8) ∇ c y (cid:9) ni , j + / k = c ni , j + k − c ni , j , k ∆ y + (cid:79) ( ∆ y ) , (3.27) (cid:8) ∇ c z (cid:9) ni , j , k + / = c ni , j , k + − c ni , j , k ∆ z + (cid:79) ( ∆ z ) , (3.28)where, ∆ x , ∆ y , and ∆ z are the simulation cell widths in x -, y - and z -directions, respec-tively and (cid:79) ( • ) represents order of approximation. The chosen finite difference formulationapproximates the first derivatives to first order. Here, an increment (or decrement) in thesubscript indicates the next (or previous) cell in the respective direction. The superscriptdenotes the time step n . This scheme is implemented on a staggered grid, in which thescalar variables are stored in the cell centers, whereas the vectors are located at the cellfaces (see Figure 3.4). As a consequence, a scalar, at the cell centers to be multiplied withhapter 3. 48a vector, is translated to the cell faces. For example, a contribution of ∇ · ( ψ ∇ c ) in an x -direction at the position ( i , j , k ) , is calculated as (cid:148) ∇ · (cid:0) ψ ∇ c (cid:1)(cid:151) nx = (cid:168) (cid:0) { ψ } i + j , k + { ψ } i , j , k (cid:1) c ni + j , k ∆ x − (cid:0) { ψ } i + j , k + { ψ } i , j , k + { ψ } i − j , k (cid:1) c ni , j , k ∆ x + (cid:0) { ψ } i , j , k + { ψ } i − j , k (cid:1) c ni − j , k ∆ x (cid:171) . (3.29)The contributions in y - and z -direction are calculated analogously and the sum of all resul-tants provides the value of ∇ · ( ψ ∇ c ) of the form, (cid:148) ∇ · (cid:0) ψ ∇ c (cid:1)(cid:151) n = (cid:150)(cid:168) (cid:0) { ψ } i + j , k + { ψ } i , j , k (cid:1) c ni + j , k ∆ x + (cid:0) { ψ } i , j , k + { ψ } i − j , k (cid:1) c ni − j , k ∆ x − (cid:0) { ψ } i + j , k + { ψ } i , j , k + { ψ } i − j , k (cid:1) c ni , j , k ∆ x (cid:171) + (cid:168) (cid:0) { ψ } i , j + k + { ψ } i , j , k (cid:1) c ni , j + k ∆ y + (cid:0) { ψ } i , j , k + { ψ } i , j − k (cid:1) c ni , j − k ∆ y − (cid:0) { ψ } i , j + k + { ψ } i , j , k + { ψ } i , j − k (cid:1) c ni , j , k ∆ y (cid:171) + (cid:168) (cid:0) { ψ } i , j , k + + { ψ } i , j , k (cid:1) c ni , j , k + ∆ z + (cid:0) { ψ } i , j , k + { ψ } i , j , k − (cid:1) c ni , j , k − ∆ z − (cid:0) { ψ } i , j , k − + { ψ } i , j , k + { ψ } i , j , k − (cid:1) c ni , j , k ∆ z (cid:171)(cid:153) . (3.30)Similarly, the terms containing spatial derivative in the evolution equation (3.22) can becalculated.The temporal derivative in the evolution equation is discretized by an explicit Eulerscheme, ∂ c ∂ ˆ t = c n + i , j , k − c ni , j , k ∆ t + (cid:79) ( ∆ t ) , (3.31)which approximates the time derivative by a forward difference of first order. Here, super-scripts n + n represent the values at the next and the current time steps, the subscripts i , j , k indicate the spatial position and ∆ t is the difference between the current and the nexttime step. The numerical stability is ensured by a limiting time step [ ] , ∆ t < ∆ x d + ˆ M max ˆ κ , (3.32)hapter 3. 49where, ˆ κ is the dimensionless gradient energy coefficient, d denotes the dimensions of thesimulation study, ˆ M max represents the maximum of dimensionless mobility ˆ M defined inEq. (3.12). ∆ x is the width of the unit cubic simulation cell. In this chapter, the phase-field method for the phase separation in LMO cathode particlesis described. The regular solution free energy density is considered to represent the phaseseparation. Above a critical value of the regular solution parameter, the phase separation ismore favorable than the homogeneous mixture, which explains the two-phase coexistencein LMO cathode particles. The driving force for the movement of the interface separat-ing these two phases is considered through boundary conditions. These conditions on theparticle surfaces are employed through the smoothed-boundary method, which is expectedto capture the geometry of complex-shaped particles elegantly. The validation of the ap-proach and the numerical results obtained for an isolated single particle is explained inChapter 5. The model is further extended to consider boundary conditions at the separator,which is intended to capture multiple particle configurations irrespective of particle size,shape, and orientations in an electrode. The numerical results of this model are presentedin Chapter 6. hapter 4Phase-field model for inclusionmorphology under electromigration
Most of the present analytical and numerical theories which are based on sharp interfacedescription suffer from three significant limitations. Firstly, analytical theories only allowthe criteria of the onset of bifurcation or can deduce the characteristics of the assumedsteady-state shape [ ] . Moreover, even if a numerical technique is employed, itrequires an explicit tracking of the interface boundaries [
81, 163 ] . This is a cumbersometheoretical challenge, especially since the interface evolves continuously and with complexgeometrical shapes [ ] . Finally, the responses in a local electric field due to the dynamicchange in the inclusion shape are neglected [
82, 83 ] . In the present work, a phase-fieldmodel is derived to investigate the temporal evolution of the inclusions propagating in theconductor. The elegance of the phase-field method lies in its ability to simulate movingboundary problems without having to track the interfaces explicitly. In the following sec-tion, a free energy functional is described to track a morphologically evolving inclusion ina conductor. A phase-field model is employed to examine the electromigration-driven dynamics of trans-granular inclusion. The schematic of an inclusion subjected to an external electric field ispresented in Figure 4.1. A conserved order parameter c is introduced to demarcate thematrix denoted by c = c =
1. The tradi-tional sharp interface between the matrix and the inclusion is replaced by a narrow regionof finite thickness. In this region, the order parameter varies smoothly between 0 and 1. Inthis way, the variable c fulfills a dual role in tracking the species concentration and the in-51hapter 4. 52 Figure 4.1: A schematic diagram of the simulation setup describes the inclusion with the order parameterc = , in the interconnect material c = , which is subjected to the electric potential + φ ∞ at the anode,and − φ ∞ at the cathode. terface between the matrix and the inclusion. The smooth interface provides a remarkableadvantage in phase-field modeling to avoid the tedious task of explicit tracking as opposedto the sharp interface counterpart.The free energy of the system F ( c , ∇ c ) is expressed as a function of the order parameterand its gradient as F ( c , ∇ c ) = (cid:90) V Ω (cid:149) f ( c ) + κ | ∇ c | (cid:152) d V Ω , (4.1)where f ( c ) represents the bulk free energy, κ is the gradient energy coefficient, and | • | denotes the norm of the vector.It is worth noticing that the choice of the free energy function, f ( c ) , can produce asignificant effect on the physical behavior of the interface. Hence, it should be selectedappropriately. The regular solution model derived in section 2.1 is a well-known free energyfunction for a broad range of applications. In that, the position of the two wells is decisiveto obtain equilibrium values based on optimal energy state. For symmetric energy curve( X =
0) and T = T ref in Eq. (2.13), the minima of the free energy are the equilibriumvalues † . For instance, at X = c L = c H = † For asymmetric curves, a Maxwell construction can be employed to determine the equilibrium values, hapter 4. 53In addition, X = X increases above a critical limit,the minima starting to drift apart progressively. For a very high X , the two equilibriumvalues coincide c L = c H =
1. These values are convenient to indicate the bulk phases [ ] . However, the numerical calculations are unstable at these points dueto logarithmic terms in Eq. (2.13). An alternative form of the free energy can be expressedas Landau polynomials [ ] of the form, f L ( c ) = N l (cid:88) l = L l c l , (4.2)where l indicates the index, N l represents the last term, and L l is the coefficient of the l -thterm of the Landau polynomial. The simplest form of the polynomial is the double-wellfunction, f dw ( c ) = c ( − c ) .The double-well function represents an approximation of the Van der Waals [ ] near the critical point, and has been employed extensively in the phase-field models. How-ever, when the model is developed solely for interface tracking purposes, this has led to thefrequently observed spontaneous volume shrinkage phenomenon. Whereby, the high vol-ume phase allows significant infiltration into the low volume phases and can ultimatelycause the complete disappearance of the lower volume phases [ ] . In addition, themovement of the lower volume phase inside the higher one may further enhance this effect.Therefore, to minimize these losses over the duration of a simulation, it requires a reconsid-eration of the free energy function. An alternative energy function for interface tracking ap-plications in the form of the double-obstacle can be considered. The diffuse interface of theorder parameter follows cosine function in the double-obstacle, while hyperbolic-tangent isexpected in the former case, see Appendix A. This provides an extremely controlled diffuseinterface width and less dispersion of inclusion volume. Consequently, lower change in in-clusion volume leads to a stoppage of the spontaneous volume shrinkage phenomena. Inaddition, this function is adequate to track larger movements of the lower volume phase.Therefore, the obstacle-type function may prove useful for interface tracking applicationsof the phase-field model where the nature of the simulated phenomena introduces phasecontinuity concerns like the inclusion migration in the metal conductors.In addition, the obstacle-type free energy gains a computational advantage, by allowingthe solution needs to be determined merely in the interface region, over the double-welltype formulation [ ] . The obstacle-type free energy density f ob ( c ) , as shown in Fig-ure 4.2, is considered with equal minima at c = f ( c ) = f ob ( c ) = X A c ( − c ) + I ( c ) , (4.3) which is further discussed in details in section 5.4. Furthermore, these values are pertinent to flat interfaces.Alternatively, the curvature correction should be added according to Gibbs-Thomson law [ ] . hapter 4. 54 f ( c ) c Obstacle-type free energy
Figure 4.2: Double-obstacle type free energy f ob ( c ) as a function of c. The function is symmetric aboutcenter c = and has two free energy minima at 0 and 1. where X A sets the barrier height of the free energy density, and I ( c ) is the indicator functionexpressed as I [ ] = (cid:40)
0, for 0 ≤ c ≤ ∞ , for c < c >
1. (4.4)The morphological evolution of inclusion can be recognized by the competition betweenthe electromigration force and the capillary force. The capillarity alone prefers circularshape inclusion. While the external electric field instigates species transport, which leads toshape alterations. In addition, the capillary force seeks uniform curvature and consequentlydiffuses the species to reduce any curvature gradient along the inclusion surface. Therefore,the shape of the inclusion is governed by the relative strength of these two forces, whichcan be incorporated into the diffusional fluxes.
The evolution of the diffusing species follows a continuity equation ∂ c ∂ t = − ∇ · J i , (4.5)where t denotes time, ∇ · ( • ) represents the divergence of the vector and J i is the net fluxof the diffusing species i . The flux J i is expressed as a linear combination of the flux andhapter 4. 55driving force within the framework of irreversible thermodynamics as [
69, 171 ] J i = − M ii ∇ ( µ + eZ i φ ) − M ie eZ w ∇ φ (4.6)where M ii and M ie are the mobilities related to the diffusivity of the species i and interactionbetween species i and electron respectively. ∇ ( • ) represents the gradient of a scalar, µ is the chemical potential, Z i denotes the valence of the diffusing species, Z w denotes themomentum exchange effect between the electrons and the species, e denotes the electriccharge and φ is the electrical potential. The term ( µ + eZ i φ ) combined is referred as theelectrochemical potential [ ] . To perceive the driving forces arising due to the imposedelectric field, Eq. (4.6) can be rearranged to give J i = − M ii ∇ µ − M ii eZ i ∇ φ − M ie eZ w ∇ φ (4.7)The second term in the previous equation is the result of the direct electrostatic force, whilethe third term is the electron wind force, which reflects the cross-effect arising from the in-teraction between the diffusing species and the conducting electrons. In conductors, metalspecies are shielded by the negative electrons so that the direct electrostatic force is muchless than the wind force [
69, 173 ] . The dominance of the wind force is further corroboratedby experimental observation which suggests the movement of the diffusing species in thedirection of the electron flow [
68, 174 ] . Therefore, the second and third terms are com-bined through effective valence, Z s . Hence considering the effective valence in Eq. (4.7)and substituting in Eq. (4.5), the modified Cahn-Hilliard equation can be expressed as, ∂ c ∂ t = ∇ · (cid:128) M ii ∇ (cid:0) µ + eZ s φ (cid:1)(cid:138) (4.8)The chemical potential µ can be obtained from the variational derivative of the system freeenergy, µ = δ F δ c . (4.9)The surface diffusion is dominant compared to the other forms of mass transport mecha-nisms at the operating conditions. Therefore, the surface effective charge Z s and the atomicmobility M ii are restricted at the surface by selecting a bi-quadratic form in c , expressed as, M ii ( c ) = D s f θ ( θ ) c ( − c ) , (4.10)where D s denotes the surface diffusion coefficient. Here f θ ( θ ) is the anisotropy function,given by [ ] , f θ ( θ ) = (cid:128) + A cos (cid:0) m ( θ + (cid:36) ) (cid:1)(cid:138) ( + A ) . (4.11)hapter 4. 56In the above equation, A represents the strength of anisotropy, which describes the superi-ority of the maximum value of diffusion compared to the minimum along the surface. Inaddition, m is the parameter related to the grain symmetry. Specifically, 2 m denotes thenumber of crystallographic directions of fast diffusion sites in the plane of inclusion mi-gration. Therefore, based on a total number of locations for fast diffusivity, m =
1, 2, and3 are characterized by twofold, fourfold, and sixfold symmetry respectively, as shown inFigure 4.3, while m = θ = tan − (cid:0) ∂ c ∂ y / ∂ c ∂ x (cid:1) isthe angle formed by the local tangent at the inclusion surface and (cid:36) is the misorientationangle formed in the clockwise direction by the position of the maximum diffusion site withthe perpendicular to the external electric field. The flux of the charge carrier, i.e. electron, J e is given by J e = − M ei ∇ ( µ + eZ i φ ) − M ee eZ w ∇ φ (4.12)where M ei and M ee are the phenomenological coefficients and e is the electron charge. Thecurrent inside the conductor is entirely due to the imposed electron wind and the cross effectdue to mass flux i.e. the first term is negligible. In addition, the timescale of relaxation ofcharge is much faster compared to the diffusion process. Hence, the current continuityequation translates into Laplace equation as ∇ · J e = ∇ · [ M ee eZ w ∇ φ ] =
0. (4.13)Comparing the previous equation with Ohm’s Law, the conductivity function can be writtenas, σ ( c ) = M ee eZ w . Therefore, the potential field is calculated from the Laplace equation as ∇ · [ σ ( c ) ∇ φ ] =
0, (4.14)where σ ( c ) is the electrical conductivity dependent on order parameter c , interpolatedbetween the inclusion σ icl and the matrix σ mat as, σ ( c ) = σ icl h ( c ) + σ mat [ − h ( c )] . (4.15)Any smooth function that satisfies h ( c ) | c = = h ( c ) | c = = h ( c ) = c , h ( c ) = c ( − c ) , and h ( c ) = c ( − c + c ) . The linear interpolation function ( h ( c ) = c ) is considered in the present workfor computational convenience and the form of interpolation functions does not alter theresults as long as the interface width is small.hapter 4. 57 Figure 4.3: Schematic of anisotropy in surface diffusivity with (a) twofold, m =
1, (b) fourfold, m = = (cid:36) for each case is defined with respect to the direction of theperpendicular to the external electric field. hapter 4. 58 To facilitate comparison and physical significance, the relations of phase-field model pa-rameters should correspond to sharp-interface theories. In this section, a linkage betweenthe two methods is established.
In the phase-field model, the physical properties of the phenomenon such as interfacialenergy γ s and interfacial width δ s are related to the double-obstacle barrier height X A andthe gradient-energy coefficient κ .The spatial gradient of the order parameter along the interface can be equated to the freeenergy density function to derive interfacial width, rearranging Euler-Lagrange Eq. (A.3),of the form d x = (cid:118)(cid:116) κ f ( c ) d c , (4.16)Substituting f ( c ) from Eq. (4.2) into Eq. (4.16) and integrating along the interface from c = c = ⇒ x = x = δ s . The interface width can be obtained as, δ s = π (cid:118)(cid:116) κ X A . (4.17)It is important to note that this relation is applicable to obstacle type free energy density,where the interface width can be strictly related to κ and X A . However, for well-type freeenergy density, the diffuse interface reaches infinity. Therefore, to determine the interfacewidth, an approximation should be considered by limiting the change in order parameterupto a few orders.The system free energy expressed in Eq. (4.1) identically vanishes in the bulk phaseswhere c = γ s = (cid:90) V Ω (cid:110) f ob ( c ) + κ | ∇ c | (cid:111) d V Ω . (4.18)This expression can be calculated numerically to obtain the interfacial energy for a complexsystem of the free energy densities [ ] . Alternatively, the derivation of an analytical re-lation is also plausible for the simple systems considered for the presented case. To derivethe expression, assuming that the spatial dependency of the order parameter c is restrictedhapter 4. 59 -1.4-1.2-1.0-0.8-0.6-0.4-0.20.00.0 0.5 1.0 1.5 2.0 2.5 A m p li t ude , l n time, t × simulation plotline fit ( A t / A ) Figure 4.4: Evolution of the sinusoidal amplitude with time. A denotes the initial amplitude of a pertur-bation with frequency k. At time t, A t denotes the temporal-dependent amplitude of the perturbations.The points are obtained from phase-field simulations, which are approximated by a linear line. Theslope of the line indicates − D s δ s Ω γ s k / ( k B T ) . to X-direction, writing one-dimensional form of the previous equation, γ s = (cid:90) ∞−∞ (cid:168) f ob ( c ) + κ (cid:129) d c d x (cid:139) (cid:171) d x . (4.19)Substituting the Euler-Lagrange Eq. (4.16) into the above equation, γ s = (cid:112) κ (cid:90) (cid:198) f ob ( c ) d c . (4.20)Substituting f ob ( c ) from Eq. (4.2) into Eq. (4.20) and integrating, the final form of theinterfacial energy can be expressed as, γ s = π (cid:112) (cid:112) κ X A . (4.21)It is evident from these relations (4.17) and (4.21) that the physical properties of the sys-tem, the interface width and the interfacial energy can be recovered in a phase-field model.Furthermore, by manipulating the energy barrier height X A = γ s /δ s and the gradient co-efficient κ = γ s δ s /π desired values of interface width δ s and the interfacial energy γ s canbe achieved. Surface diffusivity related to the Mullins’ constant B (= D s δ s Ω γ s / k B T ) is obtained by consid-ering the dampening of a sinusoidal perturbation A sin( k x ) under surface diffusion [ ] ,hapter 4. 60 c Figure 4.5: Decay in the amplitude of the sinusoidal wave with time sequence from (a) to (c). as shown in Figure 4.5. Here, A represents the amplitude of the perturbation at time t =
0, and k denotes the frequency of the perturbation. According to the Mullins’ law forthe curvature-driven surface diffusion [ ] , the progressive dampening of the perturbationis expressed by ∂ y ∂ t = − (cid:18) D s δ s Ω γ s k B T (cid:19) ∂ y ∂ x . (4.22)The analytical solution of this equation, which is valid for A k <
1, provides the expressionfor the amplitude of the perturbation: A t ( t ) = A e x p (cid:150)(cid:18) − D s δ s Ω γ s k B T (cid:19) k ( t ) (cid:153) . (4.23)Thus, the value of the parameter D s / k B T is obtained by the plot of ln ( A t / A ) vs. t − t asshown in Figure 4.4, which provides the slope = (cid:128) − D s δ s Ω γ s k B T (cid:138) k . Eqs. (4.8) and (4.14) constitute the coupled partial differential equations to describe theinclusion dynamics under electromigration. It has been reported in Ref. [ ] via formalasymptotic analysis, that at the sharp-interface limit i.e. when the interface width tends tozero, these coupled equations recover the motion by surface laplacian of the curvature andthe electric potential. In addition, these equations are normalized for the computationalconvenience [ ] with λ (cid:48) =
210 nm being the length scale and τ (cid:48) = λ (cid:48) k B T /γ s Ω D s δ s beingthe time scale. The physical values of the material properties and simulation conditions arelisted in Table 4.1.hapter 4. 61 Table 4.1: The following SnAgCu material parameters [ ] and experimental conditions are adoptedin the model. Atomic volume Ω = × − m Surface diffusivity D s = × − m / sThickness of surface layer δ s = × − mSurface energy γ s = / m Temperature T =
419 KElectron charge e = × − cBoltzmann constant k B = × − J / KElectrical resistivity ρ mat = × − Ω mAn explicit finite-difference scheme is utilized for the implementation of the model. Thespatial derivatives of the coupled Eqs. (4.8) and (4.14) are discretized utilizing a combina-tion of forward, backward, and central differences on a staggered grid, leading to second-order accuracy. A first-order explicit Euler technique is employed to discretize the temporalderivative. Isolate boundary conditions are imposed on the order parameter c , for all direc-tions. The Dirichlet boundary conditions are set at the right and left ends, for the electricpotential φ in the form of prescribed constant value, and are isolate at the top and bottom.The Laplace equation is solved iteratively employing the conjugate gradient scheme. Thetermination criterion is selected such that the maximum permissible difference in the elec-tric potential is less than 1 × − at a given position between the current and the previousiteration.The model is implemented into the in-house software package PACE3D version 2.2.0 [ ] . In this work, the study is intended to track the complete temporal morphologicaltransition of a circular inclusion. This may require a extremely large simulation domain ofthe order of 5000 − c and φ takes place only in the vicinity of the inclusion, due to the surfacediffusion. The computational cost is restrained within reasonable limits by utilizing theso-called moving window approach [ ] . Hence, a simulation starts with a fixed domainsize. When any inclusion part reaches the center of the simulation domain, for each unitcell ( ∆ x ) migration of the inclusion, a grid point is removed from the rear end and iscorrespondingly added in front of the inclusion.hapter 4. 62 Some specific features of the developed model for a transgranular inclusion propagationare discussed as follows:1. The presence of an electric field in the conductors can induce motion of the diffusingmetal species in two ways: (1) the direct electrostatic force and (2) the electron-wind force. Firstly, the direct electrostatic force which drives the species towardsthe negative terminal (cathode). Second, the negatively charged electrons acceler-ated in the direction of the positive terminal (anode), which are colliding with thespecies, thus transferring momentum results in the advancement of the species inthe direction of electron flow. This contribution is termed as electron wind. At theoperating current densities of interconnects, that is of the order of 10 − A / m ,electron-wind force is the dominant of the two. Therefore, within the framework ofirreversible thermodynamics, the cross-effect arising due to the interaction betweenthe conducting electrons and the diffusing species is considered. This term has beenadded phenomenologically in the diffusion equation (4.8).2. Although the inclusion nucleation, which is due to electromigration, is directly relatedto the reliability of the interconnects [ ] , the process of inclusion nucleation isneglected. Instead, the studies are conducted to specific characteristics associatedwith the shape evolution of the inclusion.3. The metallic conductor is assumed to be a single crystal, such that the inclusion-grainboundary interactions are neglected.4. The passage of an electric current, through a conductor, produces local heating nearsharp corners and bends [ ] . Therefore, thermomigration may accompany elec-tromigration. In the presented work, such effects of Joule heating are neglected.The inclusion shape changes are explored as a result of the competition between therelative magnitude of the electromigration force and the surface capillary force.5. The surface diffusivity at the inclusion-matrix interface is very high in the operatingtemperature conditions ( T <
500 K ) [ ] . Hence, the surface diffusion is assumedto be the only atomic transport mechanism, and the bulk diffusion is neglected.6. In general, the conductor materials might contain more than a single inclusion [ ] .The local electric fields of different inclusions may interact with each other and mayalter the overall dynamics of the inclusion evolution. In the presented dissertation, astudy of morphological dynamics of an isolated inclusion is performed neglecting theeffect of neighboring inclusions.hapter 4. 63 In this chapter, the phase-field method for morphological evolution of inclusion under theexternal electric field is described. The double-obstacle type of free energy density is con-sidered to represent the temporal evolution of inclusions. The Laplace equation is employedto facilitate the distribution of the electric field in the simulation domain. Besides, numeri-cal strategies considered for model implementation are discussed. The special provision inthe mobility equation (4.10) of the model allows simulating isotropic as well as anisotropicmobilities of inclusions. The results obtained for isotropic inclusions are presented in Chap-ter 7, which also compares sharp-interface theories with numerical results. Thereafter, re-sults on anisotropic inclusions consisting of two-fold symmetry are described in Chapter 8,followed by fourfold and sixfold symmetries in Chapter 9. art IIIResults and Discussion:Phase separation in lithium-ion batterieshapter 5Surface irregularities of a cathodeparticle
Phase-field literature, which models the Li transport within a single particle [
56, 57, 184 ] , isample. Previous models simulate particle geometry with various simplifications to performthe study in 1D [
51, 60 ] or 2D [ ] . There are few exceptional works simulatedin 3D, which are limited to regularly shaped geometries, such as spherical particles [ ] . The experimental results [ ] reveal that the electrode consists of particles thatshow a shape polydispersity. The effect of particle geometry should be considered to obtainsimulated results that are closer to the real electrode particle [ ] . Stein and Xu [ ] described the influence of particle geometry on the Li concentration profile in ellipsoidalparticles. Guo et al. [ ] reported that the difference in maximum and minimum Liconcentration is larger in elliptical particles than in spherical particles. Chakraborty et al. [ ] found that there are important differences between the plastic stretches in cylindricalparticles and those in spherical particles, investigated earlier by Cui et al. [ ] .The presented work has two objectives. The first objective is to study the insertion dy-namics of an irregularly shaped particle, to imitate a real particle in physical existence. Thesecond objective is to consider the effect of various parameters influencing the charge dy-namics. The foundation of the model was developed in chapter 3. Based on that model, nextsection 5.2 describes employed boundary conditions to study the surface irregularities pre-sented in this chapter. In addition, the theory is validated with commercial multi-physicalsoftware COMSOL Multiphysics by considering a particular case from a benchmark study [ ] . Thereafter, the simulation results are presented in section 5.3 with plausible justifica-tion in section 5.4. Afterward, the numerical study is extended in section 5.5 to investigate67hapter 5. 68the effect of various material parameters on the transportation mechanism. In section 5.6,the chapter is concluded with some remarks on the results. Some parts of this chapter arepublished in the journal Modelling and Simulation in Materials Science and Engineering [ ] . For a simplicity, a temporally independent lithium flux J n is considered at the particle surfaceof the form, J n = J · n = (cid:40) − CR N ex d × for c s < c s =
1. (5.1)The parameter C refers to the C-rate, and x C-rate measures the Li insertion or extractionrate at which complete charging or discharging takes place within 1 / x hours [ ] . Further-more, R is the radius of the reference particle, c s is the concentration at the surface of theparticle, d defines the dimension of the simulation study, e.g., d = N ex = (cid:40) Particle circumference × Reference particle areaParticle area × Reference particle circumference for the 2D case,
Particle surface area × Reference particle volumeParticle volume × Reference particle surface area for the 3D case. (5.2)The constant Neumann flux boundary condition in Eq. (5.1) on the arbitrary geometricalshapes is implemented through the evolution equation (3.19) for the Li diffusion. In addi-tion, homogeneous nucleation from the surface is considered in the form of Eq. (3.18). Forthe 2D (or 3D) simulation study, a circular (or spherical) particle with radius R = µ mis considered as a reference particle along with other parameter set is provided in Table 5.1.A study is conducted to observe the Li diffusion from the surface of the particles, which arereported here. The general boundary condition of arbitrary geometrical shapes is implemented through thesmoothed boundary method [ ] , in which a domain parameter, ψ , is registered todifferentiate and interpolate between different phases. The simulation is initialized withthe domain parameter ψ , continuously defined in the entire domain. The field quantity ψ takes the value 1.0 for the bulk cathodic particle, 0.0 for the electrolyte and varies smoothlyin the boundary between the particle and the electrolyte. The specific features of a diffuseinterface between the particle and the electrolyte are responsible for the configuration ofhapter 5. 69 Table 5.1: Material properties of LiMn O cathodic intercalation materials [ ] and operational con-ditions. Parameter Symbol Value UnitDiffusion coefficient D × − m / sLength scale L µ mGradient energy coefficient κ × − m Regular solution parameter α (cid:48) -0.1 - α (cid:48)(cid:48) T ref
300 Kthe surface normals, as shown in Figure 3.2. As a consequence, the particle surface normalis exploited into the flux boundary condition given in Eq. (5.1).In the present chapter, particle morphology is considered to be constant during lithia-tion. Therefore, the simulations are performed in two independent steps. In the first step,the particle is formed from the Allen-Cahn equation presented in Appendix B. The resultantdomain encompasses a smooth interface between the cathode particle and the electrolyte.In the second step, the smooth surface is exploited to implicitly incorporate the flux bound-ary condition in the Cahn-Hilliard equation (3.19), in which lithiation inside the particleobtained from the first step is considered.The model simulation of the species flux infiltrates from the lithium abundant electrolyteto the cathode particle, through the exterior surface. A simulator is developed to obtain thenumerical solution of the compound partial differential equation of the Cahn-Hilliard equa-tion combined with a smoothed boundary method. By employing standard message passinginterface (MPI) concepts, a parallel three-dimensional solver for large-scale computationson a rectangular mesh is realized as described in section 3.4.It is important to note that owing to the smoothed boundary method, the electrolyte isconsidered as LMO material as well. Even though the evolution equation (3.19) is solvedeverywhere in the domain, i.e. in the region of the particle ( ψ =
1) as well as in the elec-trolyte ( ψ = ψ = · − x S O C ( i n % ) Present study ( (cid:15) p /R = 0 . .
03 0 .
05 0 .
07 0 .
09 0 . (cid:15) p /R S O C ( i n % ) Present study (∆ x = 1 . × − )finite element software Figure 5.1: Effect of the dimensionless cell width and the interface thickness coefficient on the SOCvalue: Comparison of the present approach with results obtained by the finite element software COMSOLMultiphysics ® . The simulation studies are performed for LMO cathodic particle at a lithiation of default 1C-rate discharging condition. Therefore, the inflow of the lithium flux fills up the particle.The natural logarithmic terms ln ( c ) (or ln ( − c ) ) in the chemical-free energy formulationare singular at a concentration of c = c i = c = [ ] presented a smoothed boundary method based on a finite differencescheme for various applications, such as the mechanical equilibrium equation, phase trans-formation with the presence of additional boundaries and the diffusion equation with Neu-mann and Dirichlet boundary conditions. In this section, the presented approach in sec-tion 3.2 is validated by considering a particular case in a phase-separating particle. Duringthe insertion of the lithium species, a concentration gradient develops inside the particlefrom the higher concentration at the surface to the lower concentration at the center [ ] .As a consequence, the surface reaches the threshold value of c = = / V Ω (cid:82) V Ω ψ c ) of 100% for 1 C-rate species inflow.Therefore, the latest SOC attained by different values of simulation cell width of sphericalparticles are compared, with the 1D computations performed in Ref. [ ] by using thesoftware COMSOL Multiphysics ® , which is based on the finite element method. The resultsobtained from the presented model are in agreement with the study of Huttin and Kamlah [ ] . The comparison suggests that at most a dimensionless cell width of ∆ x = × − is acceptable, to obtain a sufficient convergence of the simulation results, see Figure 5.1.Moreover, the calculated concentration value deviates further from the benchmark valueas the interface thickness increases. The results ensure convergence for a thin interfacethickness, which is in agreement with literature [ ] . Therefore, the simulation study iscarried out at ∆ x = × − for 3D and at 1 × − for 2D studies with the interfacethickness coefficient ε p / R = To begin with, a 2D simulation study with a varying aspect ratio of ellipses of equal area isconsidered. The ellipses with the aspect ratios γ =
1, 2, 3, 4, and 5 with major semiaxes are1.00, 1.42, 1.74, 2.00, and 2.25 µ m, respectively in Figure 5.2 with initial filling c i = c i = γ =
1, while at 18.56% SOCfor γ =
5. Furthermore, the circular particle (i.e., γ =
1) evolves with a continuous phaseseparation across the interface, while two Li-rich islands form at the maximum curvaturepoints in the ellipses (i.e., γ =
2, 3, 4, and 5). The Li-rich islands continue to grow furtherat the particle surface. In Figure 5.2, beyond 60% SOC, the Li-poor phase is surroundedby the Li-rich phase in all particles, meaning they eventually follow the “core-shell model",and the core shrinks as the shell enlarges with the evolution of time, which is called the“shrinking-core model". Contrary to c i = c i =
10% 60% 90%State of charge (SOC) γ = 1 γ = 2 γ = 3 γ = 4 γ = 5 20.38%19.57%19.12%18.81%18.56% c Figure 5.2: Temporal evolution corresponding to a 10%, 60%, and 90% state of charge, including anintermediate concentration profile at the initiation of a phase separation inside 2D elliptical particles,with aspect ratios γ =
1, 2, 3, 4, and 5. The gold and magenta colors respectively correlate to the Li-richand Li-poor phases. The particles are filled with an initial concentration of c i = lowed by an ellipse and eventually transforms into a circle as it shrinks, which can be seenin Figure 5.4(b), (c) and (d) respectively.A 2D simulation study of five elliptical particles simultaneously immersed in the elec-trolyte, see Figure 5.5. The particles are filled with an initial concentration of c i = γ =
1, 2, 3, 4 and 5 with 1.0, 1.54, 2.13, 2.72, and3.35 µ m major semiaxes and 1.00, 0.77, 0.71, 0.68, and 0.67 µ m minor semiaxes ellipses,respectively. The circumference to area ratio of the particles is fixed. Therefore, N ex =
30% 50% 70% 90%State of charge (SOC) γ = 1 γ = 2 γ = 3 γ = 4 γ = 5 c Figure 5.3: Concentration profiles corresponding to 30%, 50%, 70%, and 90% state of charge inside2D elliptical particles, with aspect ratios γ =
1, 2, 3, 4, and 5. The gold and magenta colors correlateto the Li-rich and Li-poor phases, respectively. The particles are filled with an initial concentration ofc i = Li-rich islands, which are marked in red color, develop at the convex surface, which cor-responds to the maximum curvature of the particle. The islands progress to form a singlecontinent at 40% SOC and eventually, the core shrinks with time.For various initial concentrations, a comprehensive study is performed as a function ofaspect ratio γ for a SOC at which phase separation is expected, SOC PS , see Figure 5.6(a).The curve fit suggests the empirical relation,SOC PS = ( − c i + ) γ ( c i − ) for 0.00 < c i < PS is the SOC at the onset of phase separation. It is evident that increasingthe initial concentration reduces the difference in the values of SOC PS between the higherand the lower aspect ratio particles continuously. As a consequence, at a certain initialhapter 5. 74 ψ . . . c . . . Figure 5.4: Concentration profiles of an irregularly shaped particle, when the SOC ranges from approx-imately 19% to 90%. The images correspond to 19%, 36%, 77% and 90%. The red color, ψ = ψ = concentration, namely c i = The local concentration characterizes a condition of the particles, which corresponds to thezones I, II and III in Figure 5.8, which determine a phase-segregated state or a homogeneousstate during intercalation. In the phase equilibrium diagram of the LMO particle, thesezones are related to spinodal and miscibility gaps. In Figure 5.8, the spinodal gap is thelocus of points where the curvature of the chemical free energy density changes its sign,i.e. from convex to concave, at point B and from concave to convex, at point C. Let c L and c H be the two concentration levels of a phase-segregated state for the Li-rich and the Li-poor phases, respectively. In electrode particle ( ψ = µ chem1 and the grand-chemical potential G chem1 = f chem1 − c µ chem1 for both concentrations, c L and c H , as µ chem1 ( c L ) = µ chem1 ( c H ) , (5.4) f chem1 ( c L ) − c L µ chem1 ( c L ) = f chem1 ( c H ) − c H µ chem1 ( c H ) . (5.5)The simulation study is carried out at an absolute temperature of T =
300 K. The corre-sponding chemical free energy density plot in Figure 5.8 represents the double-well struc-ture, which entails the presence of a concavity between the two wells, and hence the pres-ence of miscibility and spinodal gaps. The concentration of the miscibility gap fulfills thehapter 5. 75 ψ . . . c . . . . Figure 5.5: Concentration profiles of five particles, simultaneously immersed in an electrolyte, whenthe SOC ranges from approximately 30% to 90%, displayed with an increment of 20%, until it reaches96.37% in the last frame (f). In the entire domain, the domain parameter ψ is continuously defined.The yellow color, ψ = ψ = equilibrium conditions expressed in Eqs. (5.4) and (5.5), which are marked by the points Aand D. Between these points, the free energy can be decreased by decomposing the homo-geneous state of higher energy to a phase-segregated state of lower energy. The spinodalgap concentrations between the zero curvature points of the free energy curve, marked aspoints B and C, render the homogeneous distribution states unstable and force the particleinto phase segregation. In Figure 5.8, the red dotted line refers to the common tangentconstruction and represents the minimum energy line.With the application of flux, a concentration gradient is established from the surface tothe center of the particle. Therefore, the phase segregation initiates from the surface of theparticle. The generation and increase in the pre-existing phase boundary between the Li-rich and the Li-poor phases are energetically costly, and the system follows a path with theminimum phase boundary possible. The increase in the phase boundary portion leads toan energy penalty. Thus, the equilibrium states tend towards a minimum phase boundaryinside the particle, following the evolution in the direction of the minimum free energypath. As a consequence, the inner phase shrinks by assuming an irregular shape with acurved boundary, at the beginning, which then transforms into an ellipse, and eventuallyinto a circle, as shown in Figure 5.4.hapter 5. 76 γ S O C P S ( i n % ) c i = 0.03 c i = 0.05 c i = 0.11 c i = 0.13 1 2 3 4 5929394 Aspect ratio, γ S O C PE ( i n % ) c i = 0.03 c i = 0.05 c i = 0.11 c i = 0.131 2 3 4 50 . . . . γ t P S ( i n × s ec o nd s ) c i = 0.03 c i = 0.05 c i = 0.11 c i = 0.13 1 2 3 4 52 . . . γ t PE ( i n × s ec o nd s ) c i = 0.03 c i = 0.05 c i = 0.11 c i = 0.13 (a) (b)(c) (d)2 Figure 5.6: The SOC (or duration) at which phase separation initiates, SOC PS in (a) (or t PS in (c)) andcompletes SOC PE in (b) (or t PE in (d)) are plotted as a function of the aspect ratio of the 2D ellipticalparticles for the initial concentrations c i = i . In Figure 5.2, the uniform curvature along the surface of the circular particle ( γ = γ = (a) (b) (c)(d) (e) (f)20% 40%60% 80% 90% Figure 5.7: Concentration profiles of an irregularly shaped particle, when the SOC ranges from approx-imately 20% to 90%. The images correspond to 20%, 40%, 60%, 80% and 90%. The opaque silvercolored figure in (a) indicates the contour of the particle surface at the value ψ = = and it represents the lithium-rich phase. The bluecolor is a contour of c = and the blue color cavity inside the lithium-rich phase corresponds to thelithium-poor phase. the surroundings of the endpoints of the major semiaxis, in contrast to the minor semiaxis,which is in agreement with the literature [ ] . This comparison suggests thatmore sites are available for the applied flux, compared to the sites that host lithium speciesnear the major semiaxis. This triggers a phase separation process during the insertion asseen in Figure 5.2.The difference between the maximum and minimum concentration across the surfaceof the particle increases with the aspect ratio, as can be seen in Table 5.2. Therefore, themaximum concentration point reaches the spinodal gap with less SOC for particles witha higher aspect ratio. Hence, the SOC required to initiate a phase separation decreaseswith the aspect ratio. Similar to Figure 5.7, the higher curvature region accumulates morehapter 5. 78 F r eee n e r g y d e n s i t y , f c h e m T e m p e r a t u r e , T / T r e f concentration, c I II III II II II III II IMiscibility gapSpinodal gap
Figure 5.8: The phase diagram shows various zones, which are characterized by the miscibility gap andthe spinodal gap. Zones I, II and III respectively correspond to homogeneous states, nucleation states,and phase-separated states. The solid blue line in the bottom graph is a chemical-free energy density,f chem1 at absolute temperature, T = K. The projections of the miscibility gap, on the free energydensity curve, indicate the equilibrium conditions expressed in Eqs. (5.4) and (5.5). The projections ofthe spinodal gap, on the free energy density curve, indicate zero curvature points. The dotted red line inthe bottom graph corresponds to the minimum energy path, widely known as Maxwell construction. hapter 5. 79
Table 5.2: The maximum and minimum concentration, along the surface of the particles, for the initialconcentrations c i = and , at 18% SOC. The maximum concentration observed in a regionwith maximum curvature, while the minimum concentration is observed in a region with minimumcurvature in ellipses. Aspect ratio, γ Maximum concentration, c Minimum concentration, cc i = c i = c i = c i = Figure 5.6(a) shows the increment in SOC PS for higher aspect ratio particles with a rise ininitial concentration c i . An explanation is the decrease in the difference between a maxi-mum and a minimum concentration at higher c i , see Table 5.2. Furthermore, an increase in c i results in less concentration required locally to reach the spinodal zone, and drive phaseseparation. If the initial concentration c i , in all particles of Figure 5.3 falls in the spinodalzone, phase separation is initiated everywhere at the surface of the particles simultaneously,regardless of the curvature variation across the particle surface. Furthermore, the particlewith the higher c i requires less time to reach the same SOC compared to the lower. There-fore, the duration required to observe the onset of phase separation (i.e., t PS ) decreases forhigher initial concentration, as reported in Figure 5.6(c).Although the initiation of the phase separation process is affected by the initial con-hapter 5. 80centration of c i , the completion of phase separation is independent, as depicted in Fig-ure 5.6(b). During the spinodal decomposition, the particle attains two concentration levels(i.e., Li-rich and Li-poor phases), irrespective of the initial concentrations. In other words,the phase separation process hinders the effect of initial concentration. Therefore, the effectof different initial concentrations on the SOC required to complete the phase separation (i.e., SOC PE ) is not observed. In addition, the effect of variation of c i on the duration untilthe phase separation observed t PE is reported in Figure 5.6(d). Obviously, the increase in c i shows additional leverage to SOC, which does not require to accumulate during operation.As a consequence, the increase in c i leads to a decrease in the time to complete the phaseseparation t PE . γ S O C P S ( i n % ) C C C γ S O C PE ( i n % ) C C C γ S O C P S ( i n % ) M M M γ S O C PE ( i n % ) M M M (a) (b)(c) (d)2 Figure 5.9: Variation of C-rate as a function of the aspect ratio of the elliptical particles for SOC at whichthe onset of phase separation starts SOC PS in (a) and ends SOC PE in (b). The variation of mobility ˆ Mas a function of SOC PS in (c) and SOC PE in (d). hapter 5. 81 Ideally in the quasi-static condition (i.e., C-rate C (cid:28) [ ] , the phase separation startswhen the average concentration reaches the spinodal point B and ends outside the miscibil-ity gap (point D in Figure 5.8). In fact, the local concentrations at all positions are almostequal to the average concentration in quasi-static conditions. However, higher values ofC-rate stimulate a steeper concentration gradient from the surface to the center. Therefore,a local concentration at the surface reaches the spinodal point before the average concen-tration. The difference between these two increases with flux. As a result, SOC requiredto observe the onset of phase separation SOC PS decreases for higher flux rates, which isdepicted in Figure 5.9(a). Contrarily, the phase separation ends when the Li-poor phase iscompletely depleted by the Li-rich phase. The slope of the concentration gradient increaseswith the C-rate. As a consequence, the presence of gradient penalizes the SOC required toend the phase separation SOC PE on a higher side, which can be seen in Figure 5.9(b).On the one hand, mobility can be related to the inherent ability of the system to ho-mogenize the concentration distribution by transferring species in order to eliminate anygradients developed during operation. On the other hand, C-rate enhances the concentra-tion gradient at the surface by depositing the flux of lithium species. The interplay betweenthe deposition and transfer rates characterizes the equilibrium states. At the elevated mo-bility, the species transfer rate is more prominent compared to the species deposition. Asa consequence, the species disperse immediately after the deposition at the surface. Thus,a particle requires higher SOC to observe phase separation at higher mobility as depictedin Figure 5.9(c). Furthermore, the slope of concentration decreases at the agile mobility,which results in decreased SOC PE (see Figure 5.9(d)). Conversely, the lower mobility pro-vides comparatively enough time for the accumulation of the species, which are depositedat the particle surface. Hence, SOC PS decreases and SOC PE increases for sluggish mobilityas an effect from the increased slope of the concentration gradient.A very high mobility (or very low C-rate) corresponds to the steady-state, in whichphase separation starts at spinodal point B in Figure 5.8, for all particles irrespective ofcurvature. In other words, the curvature effects are suppressed in steady-state. As themobility decreases (or C-rate increases), the curvature effects become prominent as a resultof the development of concentration gradients. For instance, the increase in the differencebetween SOC PS for the 1.0 ˆ M and 1.2 ˆ M with aspect ratio γ is plausible due to the particlecurvature.hapter 5. 82 . . . . − . − .
10 BBB Concentration, c F r eee n e r g y d e n s i t y , f c h e m T /T ref = 0 . T /T ref = 1 . T /T ref = 1 .
05A C DA C DA C D 0 0 . . . . − . − . − .
10 BBB Concentration, c F r eee n e r g y d e n s i t y , f c h e m . α . α . α A C DA C DA C D (a) (b)2
Figure 5.10: The effect of variation in (a) temperature T / T ref and (b) free energy parameter α (cid:48)(cid:48) on freeenergy density plots. The points A and D represent free energy local minima obtained from Eqs. (5.4)and (5.5), while B and C are spinodal points where the curvature of the plot changes its sign. α (cid:48)(cid:48) During operating conditions, the battery is seldom subjected to a constant temperature.Thus, understanding the effect of variation in operating temperature on the phase separa-tion dynamics has technological implications. The free energy density for three differentvalues of the temperature T / T ref is plotted in Figure 5.10(a). The particle, which is initiallyfilled with concentration value of c i = PS increases with temperature T / T ref , which can be seenin Figure 5.11(a). On the contrary, the rise in temperature shortens the miscibility gap, i.e.,a shift in point D towards the lower concentration values (see Figure 5.10(a)). As the pointD is closely related to SOC PE , the effect of a variation of the temperature T / T ref is justifiedin Figure 5.11(b).Figures 5.11(c) and (d) show the effect of variation of the free energy parameter α (cid:48)(cid:48) .In Figure 5.10(b), it can be seen that the miscibility and spinodal gaps enlarge with anincrease in α (cid:48)(cid:48) , the behavior is opposite to the temperature T / T ref . Therefore, SOC PS de-creases and SOC PE increases for higher α (cid:48)(cid:48) , which shows an opposite tendency compared tothe temperature T / T ref .hapter 5. 83 γ S O C P S ( i n % ) T /T ref = 0.95
T /T ref = 1.00
T /T ref = 1.05 1 2 3 4 591939597 Aspect ratio, γ S O C PE ( i n % ) T /T ref = 0.95
T /T ref = 1.00
T /T ref = 1.051 2 3 4 5171921 Aspect ratio, γ S O C P S ( i n % ) α α α γ S O C PE ( i n % ) α α α (a) (b)(c) (d)2 Figure 5.11: Variation of the temperature T / T ref as a function of the aspect ratio of the elliptical particlesfor SOC at which the onset of phase separation starts SOC PS in (a) and ends SOC PE in (b). The variationof free energy parameter α (cid:48)(cid:48) as a function of SOC PS in (c) and SOC PE in (d). Finally, Figure 5.5 and Figure 5.7 exhibits the capability of the method to simulate mul-tiple particles simultaneously and three-dimensional particles with an irregular shape im-mersed in an electrolyte. As the particle boundary condition is implicitly defined in thesmoothed boundary method, this method can be applied to particles with almost any geom-etry. Hence, this technique is very powerful and convenient to solve differential equationsin complex geometries with complex boundary conditions that are often difficult to mesh.hapter 5. 84
A phase-field study is performed to simulate phase separation during the insertion pro-cess. The presented model employs two coexisting phases, which is validated with a bench-mark. In addition, the increase in mesh resolution conveys the convergence of the presentedmethod. In the finite-difference framework, the smoothed boundary method enables themodeling of particles with almost any geometry. The effects of the particle geometry on theconcentration evolution are explored numerically. For a constant flux boundary conditionat the particle surface, the free energy density and the chemical potential are discussed indetail. Based on that, the spinodal and miscibility gaps are estimated.The simulation results show that the growth of the phase boundary between the Li-poorand Li-rich phases, inside the particle, leads to an energy penalty. As a consequence, themorphological evolution of the concentration profile suggested a minimum phase boundarypathway. With the application of flux, the particle surface with a higher curvature prefer-entially accumulates lithium species. Therefore, phase segregation starts in the vicinity ofthe regions with a higher curvature. The reason for the inhomogeneous phase separationacross the particle surface is explained in detail, by means of miscibility and spinodal gaps.Furthermore, the elliptical particle with a higher aspect ratio is subjected to the onset ofthe phase separation, prior to the lower ones.Finally, it is evident from Figure 5.5 that the smoothed boundary method can be ap-plicable to multiple particles simultaneously. However, the significance of the simulationinvolving multiple particles is restricted due to the boundary conditions are applied on theparticle surfaces. Therefore, the influence of neighboring particles is limited. In fact, thetransportation of species in any particle is independent of the other particles and thereforecould be simulated separately of the others, as shown in Figure 5.3. Hence, the concentra-tion profiles are identical for the densely packed particles and the sparsely packed, whichshould not be the case generally. This provokes an extension of the simulation study toidentify the significance of the microstructural properties of the electrodes consists of manyparticles. Therefore, these issues are addressed in the next chapter. hapter 6Morphological descriptors in multipleparticle porous electrodes
High-performance batteries necessitate electrodes of superior active material and moreimportantly, optimized porous microstructures [
62, 66 ] . Recent advancements in manu-facturing facilities provide great control over the particle size and the complex-tortuousmicrostructure of electrodes [ ] . Thus, electrode engineering allows producing high-density electrodes without compromising its rate capability. The physical microstructure ofthe electrode has a direct impact on battery performance measures such as cyclic stability,intercalation rates, and power density amongst others [ ] . Therefore, it is importantto quantify the parameters that control electrode microstructure, such as particle shape,size, porosity, and tortuosity. For instance, an experimental study [ ] reported that therate performance of the graphite electrode to be a function of particle size and porosity ofthe electrode. In addition, images from X-ray tomography suggest that the microstructuralinhomogeneities largely influence the direction of lithium transport [ ] .Systematic numerical studies of porous electrodes may complement the scientific under-standing of the hierarchical microstructures and help to optimize the process parametersefficiently and economically [ ? , 203 ] . Despite some theoretical and numerical studiesare performed to account essential features of phase-separating porous electrodes [
37, 63–65 ] , the literature providing exclusive information on the microstructural properties of theelectrodes is scarce. For instance, Ferguson and Bazant [ ] showed a change in effec-tive diffusivity relates to the system porosity and tortuosity. Besides, Orvananos et al. [ ] demonstrated the impact of electrode architecture in terms of ionic and electronic connec-tivities by considering interactions between active particles. Furthermore, Vasileiadis et al. [ ] investigated that the tortuosity, particle sizes, porosity, and electrode thickness influ-ence the capacity of Li Ti O (LTO) electrodes as a function of C-rate. However, most ofthe studies utilize an approximation by the Bruggeman relation [ ] for the calculation85hapter 6. 86of tortuosity, instead of estimating the tortuosity based on the electrode microstructure un-der investigation [ ] . In addition, these porous electrode phase-field models arefocused on LiFePO or Li Ti O electrode materials. Relatively few models have been de-veloped for LMO materials [
51, 55, 59, 60 ] , which are limited to single particles. Therefore,this chapter aims to provide crucial measures to define the electrode microstructures andthe relation of those attributes to the performance of LMO electrodes. Additionally, themicrostructural properties, specifically the tortuosity, are estimated numerically to providea better understanding of the transport mechanism [ ] .In the present chapter, the transportation rates of porous electrode structures with thetwo-phase coexistence in LMO particles is investigated. In this initial effort, neglectingthe reaction kinetics at the electrode-electrolyte interface, the focus is based on the diffu-sional properties of the electrode and the electrolyte. In addition, an explicit treatment ofmechanical effects due to misfit strains at the phase boundary is avoided. Instead, the em-phasis is placed on the influence of various microstructural properties such as particle sizes,porosity, and tortuosity of electrode consists of ellipsoid-like particles with a focus on thetransport mechanism. The chapter organized as follows: Section 6.1 illustrates consideredsimulation setup to investigate the electrode microstructures. Few typical cases of insertiondynamics are discussed in section 6.2 with highlighting the underlying mechanism. Also,the obtained results are analyzed and compared with known relations. Thereafter, a nu-merical study of various morphology descriptors is performed in section 6.3. Finally, thechapter is concluded by a brief discussion on the applicability of the presented results inSection 6.4. Some parts of this chapter are submitted for publication as a journal article. The generic model of phase separation for multiphase systems of LMO material is describedin chapter 3. In the present chapter, the evolution equation (3.14) is solved numerically withconsidering a constant concentration boundary condition (3.23), as shown in Figure 3.3.As a demonstration of the model, the results are obtained for considering only two phases inthe present chapter, the electrode and the electrolyte with highlighting the broad applicabil-ity. Particularly, the simulation results are presented for propylene carbonate + M LiClO (lithium perchlorate) as an electrolyte and Li x Mn O as an electrode. The utilized valuesof parameters for the simulation study are expressed in Table 6.1. The characterization ofthe electrode microstructure consists of several particles is described in Appendix C.The simulation setup consists of cathode particles and the electrolyte. The respectiveregions are filled with their equilibrium concentration values to avoid any self-generateddriving force to the diffusing species, which may hinder the effect of external lithium driv-hapter 6. 87 Table 6.1: Simulation conditions and material properties of propylene carbonate +
1M LiClO electrolyteand Li x Mn O electrode. Parameter Symbol Value UnitReference Temperature T ref [ ] KAbsolute Temperature T 300 [ ] KReference length scale L × − [ ] m s − Gradient energy coefficient κ × − [ ] m Regular solution parameters α (cid:48) [ ] - α (cid:48)(cid:48) -5.2 [ ] -Electrolyte Parameters:Diffusion coefficient D × − [ ] m s − Gradient energy coefficient κ Regular solution parameter α (cid:48)(cid:48) c i ( i , j , k , t = ) = c i ( i , j , k , t = ) = c ps ( i = N x , j , k , t ) ( = Figure 6.1 shows insertion in particles aligned perpendicular to the current collector in (a)and parallel to the current collector in (b) for equal porosity and particle size. Firstly, theonset of the development of the Li-rich phase is observed near the separator initially, wherethe constant concentration boundary condition is employed as evident in (a2) and (b2). Onthe one hand, the coexistence of two-phases can be observed for a greater depth of electrodehapter 6. 88 (a1) (a2) t=1.28 h, SOC=40% (a3) t=3.85 h, SOC=70% (a4) t=7.70 h, SOC=90% (a5) t=10.26 h, SOC=93% (b1) (b2) t=1.28 h, SOC=28% (b3) t=5.13 h, SOC=52% (b4) t=17.11 h, SOC=87% (b5) t=23.10 h, SOC=93%
Figure 6.1: Representative cases of insertion of species in a porous electrode with µ m depth alongwith current collector under a constant concentration boundary condition at the separator in two-dimensional space. (a) shows insertion in particles aligned to the species flow, i.e. perpendicular tothe current collector, while (b) shows particle parallel to the current collector. The leftmost columndemonstrates simulation geometry, while the other four columns depict phase-separated morphologiesfor various time-steps and SOCs. in the particles perpendicular to the current collector in (a3). On the other hand, evenafter a longer period, the coexistence of two-phases is observed for a shorter length spanof the electrode in (b3), along with completely intercalated and deintercalated particlesnear the separator and the current collector respectively. The former case corresponds tomore concurrent intercalation, while the latter case rather follows the particle-by-particleintercalation [ ] .Within a single particle, due to the diffusing species conveniently cover the depth ofalmost the complete electrode region in the former case, phase separation initiates nearlyfrom the whole surface of a particle, which is in contrast to the latter where phase separationobserved mainly at the regions oriented toward the separator as shown in (b4) comparedto (a4). Therefore, the core-shell type phase separation within a single particle is moreprominent in the former case as shown in (a4), while a phase-separated traveling frontfrom the separator to the current collector is apparent in the latter case in (b4) for theporous electrode.The porous structure of an electrode is of interest in battery applications due to highsurface area yet minimum energy loss [ ] . Another type of structure that is widely underscrutiny is the planar electrode, as utilized in solid-state thin-film batteries [ ] . Thethin film electrode layer can be viewed as a solid foil, which is a continuous film of activeelectrode material directly deposited on the current collector. Therefore, the electrolytecan not infiltrate the electrode, rather it is in contact with the planar surface. The lithiumhapter 6. 89 (a) (b) Figure 6.2: Average of concentration c as a function of the depth of the electrode from the separator forvarious SOC values. The concentration profiles of the porous electrode (Figure 6.1(a) and (b) correspondto (a) and (b) of the present Figure respectively) is compared with the planar single electrode particlemodel. species have to travel through this film via diffusion after being deposited at the interface.The numerical results of the planar electrode and the porous electrode are comparedin Figure 6.2. The concentration evolution of the planar electrode shows a clearly definedregion of interface with a lithium-rich phase near the separator and lithium-poor phasenear the current collector. However, even though more fraction of region near the separatorcorresponds to the Li-rich phase during the concentration evolution in the porous electrode,no clearly defined interface is detectable along the depth of the electrode. In addition,Figure 6.2(a) shows concentration profiles of the porous electrode with electrode particlesaligned perpendicular to the separator, while (b) shows concentration profiles of electrodeparticles aligned parallel to the separator. The concentration profiles in the latter caseoscillate more compared to the former case (red and green curves in Figure 6.2(a) and (b)).This can be justified from the fact that the particles parallel to the separator apparently arestacked layer-wise as shown in Figure 6.1(b1). As the phase separation in a particle startsfrom the section towards the separator, several particles at the same distance in an affectedlayer initiate the phase separation simultaneously. Due to the enhanced diffusivity of theelectrolyte, the lithium flux traverses through the surrounding of the particles. This initiatesthe phase separation in the next layer prior to the end of phase separation in the previouslayer as shown in Figure 6.1(b3). Therefore, the wave-like pattern can be observed inFigure 6.2(b).hapter 6. 90 (a)(b)
Figure 6.3: (a) Change in SOC of the electrode and (b) fraction of total particles transformed to Li-rich with time for various particle sizes obtained from the phase-field model (PFM). Here τ denotestortuosity, ρ indicates porosity of the system, and R denotes particle size whose area is equivalent tothe area of the circular particle of radius R. Along side, the evolution of SOC for the planar electrodefrom PFM and the analytical relation for bulk transport limited theory j Cot ∝ t − / (Cottrell [ ] line) are displayed for the comparison in (a). The solid dark-colored lines in (b) are guide to eye forthe results from PFM, while light-colored lines are the relations obtained from Johnson-Mehl-Avrami(JMA) equation (6.2) . hapter 6. 91 Li-poor Li-richInterface Surface Li-richSurfaceInterfaceLi-poor(a) (b)
Figure 6.4: Schematic of (a) bulk diffusion and (b) surface reaction limited transportation dynamics.The black arrows indicate dominant driving force, while red arrows illustrate sluggish response. Thethick blue line represents electrode surface, while dotted-black line indicates interface between the Li-richand Li-poor regions.
Simplified kinetics of the system is studied by measuring the net species insertion responseof the material under a prescribed concentration at the simulation boundary. The SOC ofthe porous electrode increases rapidly during the initial phase and shows a plateau subse-quently as shown in Figure 6.3(a), which indicates that the flux (rate of change of SOC) ofdiffusing species of the presented model decays with time. The obtained results are com-pared with the analytical relations from bulk-transport and surface-reaction limited diffu-sion [ ] , which are the standard techniques employed to measure the change in speciescurrent in a controlled potential environment in electrochemistry.The bulk-transport limited theory assumes the diffusion of species in the bulk electrodebeing slower than the deposition at the surface, as illustrated in Figure 6.4(a). Therefore,the transportation rate (or species flux) is controlled by bulk diffusion. A simplistic rela-tion of species flux with time in a bulk-transport limited electrode under a potentiostaticcondition is described by Cottrell equation [ ] , j Cot = k Cot t − / , (6.1)where j Cot (in h − units) is the flux of lithium species measured as the rate of change of SOCand k Cot (in h − / units) is the proportionality constant associated with operating conditionsand electrode properties. The Cottrell equation is commonly considered for planar elec-trodes in Potentiostatic Intermittent Titration Technique (PITT) to measure the diffusivity [ ] . Therefore, the Cottrell equation (6.1) for k Cot = − / is plotted with the resultshapter 6. 92obtained from planar electrode for comparison. After the deposition at the particle surface,the species transport through the bulk of the electrode. Due to the lower diffusivity of theelectrode, species transport through the bulk of the electrode is the transportation rate-limiting factor, which correlates to the Cottrell equation (6.1), as shown in Figure 6.3(a).In addition, the results obtained from the porous electrode of various particle sizes are pre-sented. Compared to electrodes of smaller particles, it is evident that the response fromthe electrode of larger particles ( R = µ m) tends towards the planar electrode and theCottrell line. Therefore, it can be inferred that the bulk diffusion-limited transportation ismore prominent in larger particles compared to the smaller ones.Another widely recognized theory of the species diffusion is the surface-reaction limitedtransportation, which considers the bulk diffusion to be more favorable compared to thespecies deposition at the surface, as shown in Figure 6.4(b). Therefore, the transportationrate is controlled by the surface reaction, in which the nucleation events are a primary fo-cus. Due to dominant bulk diffusion, ideally, the nucleated particles can be regarded asa completely transformed to the Li-rich phase, otherwise consisting of Li-poor phase. Ananalytical relation of the transformation rate with time can be obtained by invoking fewassumptions such as, the nucleation is assumed to occur randomly over the entire untrans-formed portion and the previously nucleated particle does not influence the likelihood ofnucleation around that particle. With these conditions, Johnson-Mehl-Avrami (JMA) equa-tion [ ] relates the transformed fraction in two-dimensional systems as, ζ = − e N c t , (6.2)where ζ denotes transformed fraction and N c is the constant associated with nucleation rate.The JMA equation defines the transformation of the particles to follow a sigmoidal shape.Figure 6.3(b) shows results obtained from PFM for three different particle sizes comparedwith the relations from the JMA equation. As the particle size increases, the transformationrate deviates from the sigmoidal shape considerably. Therefore, the electrode with smallerparticles prefers surface reaction limited transportation.Ultimately, these two theories, utilized in Ref. [ ] for intercalation compounds, canbe viewed as special cases of the presented numerical model. As an inference from the com-parison, the transition of SOC evolution between two different regimes, the bulk-transportand the surface-reaction limited theories can be induced by changing the electrode charac-teristics such as the size of the particles.hapter 6. 93
10 15 20 25 30 35 1 2 3 4 t i m e , t PE ( i n h ) tortuosity ( τ ) ρ = 0.55, R = 0.61 µ m ρ = 0.55, R = 0.72 µ m ρ = 0.55, R = 0.83 µ m ρ = 0.55, R = 1.00 µ m Figure 6.5: Effect of tortuosity τ for different particle sizes R as a function of the time required tocomplete the phase separation process t PE , where the SOC stagnates. The complete process of phase separation is displayed in Figure 6.1 for two representa-tive systems: (a) electrode particles aligned perpendicular and (b) parallel to the separa-tor. System (a) consumes t PE = t PE = t PE .One of the parameters that quantify the geometrical particle distribution is the tortuosity ofthe system τ . Based on studies of Refs. [ ] , the technique to measure the tortuosityis described in Appendix E. For the provided representative cases in Figure 6.1, tortuosity τ = t PE = τ = t PE = τ and maintaining the porosity and the mono-disperse particle sizes of thesystems consistent, t PE is linearly related to tortuosity τ , which is plotted as the red curvein Figure 6.5. Therefore, the increase in tortuosity of the system linearly increases the timerequired to end the phase separation, t PE .hapter 6. 94 The rate of change of SOC and fraction transformed relate to the particle sizes R , as dis-played in Figure 6.3(a) and (b) respectively, which shows the numerical result of systemscontaining several particles of a given size with maintaining equal tortuosity and porosity,where R denotes particle size whose area is equivalent to the area of the circular particle ofradius R . The system of smaller particles (blue curve) tends towards the surface-reactionlimited line in (b), while that of the larger particles (green curve) is situated close to thebulk-transport limited line (black curve) in (a). The discrepancy can be explained by con-sidering the distance diffusing species have to travel after being deposited at the surfaceof the particle. The diffusing species avail more time to migrate from the surface to reachthe center in the larger particles. Therefore, bulk transport is the rate-limiting factor inthe larger particles, which is analogous to the bulk-transport limited model. This effectdiminishes with the reduction in particle size. Therefore, smaller particles observe higherspecies flux (rate of change of SOC) in the initial stage and saturate quickly, similar to thesurface-reaction limited model.Figure 6.5 shows the effect of different mono-disperse particle sizes as a function oftortuosity τ and t PE with maintaining porosity. For different particle sizes, as shown inFigure 6.3, the system of smaller particles reaches saturation earlier compared to largerparticles. Apart from that, linearity in the τ and t PE relation is observed for all systems ofvarious particle sizes. Furthermore, τ vs t PE lines are parallel for different particle sizeswith the same porosity. However, the porosity certainly affects the time of t PE , which isdescribed in the next section. During the formation of electrode packing, the selection of an optimum mass fraction ofhost material has gathered much attention due to its implications on energy density andperformance of the battery. The mass fraction of the host material is directly linked to theporosity of the electrode. Therefore, understanding the effect of variation in porosity on thephase separation dynamics has technological implications. Figure 6.6 shows the relation of τ and t PE with the porosity ρ for a consistent particle size R . A linear relation of t PE with τ can be observed for respective values of porosity.The decrease in porosity results in the increased mass fraction of host material anddecreased ion carriers in the form of electrolytes. Hence, the system of lower porosityrequires more time t PE to obtain the SOC plateau. Also, the slope of the linear relationship( τ with t PE ) is not constant, instead increases with decreasing porosity. This implies thatthe system of higher τ exhibits appreciable increment in t PE compared to the lower oneshapter 6. 95
10 20 30 40 50 1 2 3 4 t i m e , t PE ( i n h ) tortuosity ( τ ) ρ = 0.35, R = 0.61 µ m ρ = 0.45, R = 0.61 µ m ρ = 0.55, R = 0.61 µ m ρ = 0.65, R = 0.61 µ m Figure 6.6: Effect of porosity ρ for different tortuosity τ as a function of the time required to completethe phase separation process t PE , where the SOC stagnates. for the same porosity difference.Finally, Figure 6.7 demonstrates the capability of the employed model that the methodcan be readily extended to simulate three-dimensional systems. The 3D electrode is con-sidered for the numerical experiment to study the transport mechanism. The constant con-centration at the separator initiates the formation of the Li-rich phase from the outer sur-face of the particles, otherwise consists of the Li-poor phase throughout, as shown in (b).The Li-rich phase continues to grow further at the surface of the particles in (c), (d), and(e). Eventually, the lithium-poor phase has been eliminated by the lithium-rich phase in(f). Note that the 3D case shows a significantly different pattern compared 2D case dur-ing transportation in the particles. Even though particles are aligned parallel, it shows atransportation mechanism more like the perpendicular 2D case than the parallel 2D case.This is due to the fact that the tortuosity here ( τ = (a) (b) (c) (d) (e) (f) Figure 6.7: Insertion of species in a cathode electrode with µ m depth along with current collectorunder a constant concentration boundary condition at the separator in three-dimensional space. (a)demonstrates simulation geometry, while others depict phase-separated morphologies for various time-steps. The images correspond to SOC (b) 58% (t = ), (c) 70% (t = ), (d) 85% (t = ),(e) 90% (t = ), and (f) 93% (t = ). The red color represents the lithium-rich phase. Theblue color cavity inside the lithium-rich phase corresponds to the lithium-poor phase. A numerical result of the porous electrode is presented to understand the effect of vari-ous microstructural properties such as particle size, porosity, and tortuosity on the lithiumtransport mechanism. In this chapter, the phase separation mechanism in LMO particles hasbeen successfully demonstrated in a multiple particle model system. Ellipsoid-like particlesare considered as an example, however, the model can be readily applicable to particlesof complicated geometries. According to the diffusional properties of electrode and elec-trolyte, a study is conducted on transportation rate dependence with various morphologicaldescriptors of electrode microstructures.The obtained results suggest that the transportation rate of the system is strongly relatedto the tortuous pathways formed by the particle orientation, which can be quantified by thetortuosity parameter τ . When controlling the lithium concentration at the separator, thelithium transportation rate is observed to be linearly related to the tortuosity. Furthermore,the slope of this linear relation is independent of the particle size, while the slope alters withhapter 6. 97a change in the porosity of the electrode. Therefore, the tortuosity, the porosity, and theparticle size can be suitable descriptors for the characterization of electrode morphologies.Furthermore, the evolution of the state of charge of the obtained results for the porousstructures of mono-disperse particles are compared with the bulk-transport and surface-reaction limited theories. The results suggest that systems consisting of smaller particlesare limited by surface reaction, while larger particles tend towards the bulk-transport lim-ited theory derived for planar electrodes. In order to identify the promising hierarchicallystructured electrodes, the presented simulation results could be utilized to optimize theexperimental efforts. art IVResults and Discussion:Electromigration in metallic conductorshapter 7Motion of isotropic inclusions Phase-field methods are extensively applied to study isotropic inclusion propagations underelectromigration. In the stimulating work by Mahadevan and Bradley [ ] , the develop-ment of a slit-like feature from an edge perturbation is studied by the phase-field model dueto electromigration. The work focused on the physical mechanism of slit growth at the lo-cation of a preexisting notch and subsequent propagation transverse to the line. As a result,the change in volume of the inclusion depends strongly on the applied current. Contrarily,experimental evidence also corroborates that the motion of volume-preserving inclusionsalong the conductor line [
16, 214 ] , which instigated interest in modeling community inrecent years. Bhate et al. [ ] presented the morphological evolution of a geometricalshape under the influence of surface energy, electric and stress fields on single isolated in-clusions. Thereafter, Barrett et al. [ ] studied the migration, splitting, and coalescenceof inclusions along the metallic conductors. Furthermore, Baˇnas et al. [ ] presented re-sults of geometrical shape evolution in three dimensions space. There are instances wherepropagation of a finger-like slit with shape and volume preserved is observed in SnAgCusolder bumps [ ] , which has not received attention from a modeling perspective untilnow. Furthermore, a critical comparison between the solutions obtained from phase-fieldand sharp-interface methods is lacking.Therefore, the emphasis of the presented chapter is to compare the results from thephase-field model derived in Chapter 4 with the sharp-interface analysis to investigate theselection of slit width and velocity of an initially circular inclusion at an applied electricfield. More importantly, the second objective of the study is to investigate the geometri-cal morphologies of inclusions from the phase-field model, which is otherwise unfeasiblewith sharp-interface theories. The sharp-interface analysis is considered in section 7.1 toderive a critical limit of circular inclusions stability with assuming equal conductivities tothe matrix. Afterward, the assumption of equal conductivities is relieved in section 7.2.Thereafter, results from the phase-field numerical study are compared with linear stability101hapter 7. 102 E ∞ E t V n IslandIsland surface J s Conductor θ V n V Figure 7.1: Schematic of a circular island, subjected to an external electric field E ∞ , in an infiniteconductor domain. analysis in section 7.3. Subsequently, the description of the finger-like slits obtained fromthe numerical study is presented in section 7.4. Also, the characteristics associated with theslit shapes are derived from sharp-interface model and a critical comparison with numericalresults are also presented. Finally, the chapter is concluded by its practical implications insection 7.5 with subsequent discussion of the important results in section 7.6. Some partsof this chapter are published in the Journal of Electronic Materials [ ] . The stabilty analysis of a circular dislocation loop propagation, under the assumption of asteady-state, is derived by Yang et. al [ ] . For brevity, the basic steps of the derivationsare casted here for islands migrating in a conductor. In the present context, the term, islanddescribes a cluster of material entrapped in the metallic conductor consists of equivalentconductivity to that of the conductor.Consider an island, translating along the length of the conductor under an externalelectric field, as shown in Figure 7.1. The migration of the island is due to the result ofa mass transport flux, which is induced by surface electromigration, and by the capillaritygiven by the Nernst-Einstein relation: J s = D s δ s Ω k B T (cid:16) − eZ s E t + Ω γ s d κ s d s (cid:17) , (7.1)where J s represents the number of species passing per length of island surface per unittime, D s is the surface diffusivity, δ s is the thickness of the surface layer, Ω denotes thehapter 7. 103atomic volume, e is the electron charge, Z s represents the effective valence at the surface, k B is the Boltzmann constant, T denotes the absolute temperature, γ s is the surface energy, κ s represents the local curvature of the surface, and s denotes the arc length along theisland surface. The capillary-mediated flux is driven by the gradients of curvature, alongthe island surface, while the electromigration flux is dictated by the tangential componentof the electric field along the island surface E t , given by E t = − E ∞ sin θ , (7.2)which is responsible for the island transmission [ ] . E ∞ is the applied electric field, θ isthe tangent angle of the island surface.Mass conservation relates the surface divergence of the flux to the normal velocity asfollows: d J s d s = V n Ω . (7.3)By assuming a steady state, the normal velocity can be related to the velocity along appliedelectric field as V n = V cos θ . (7.4)The curvature along the island surface is the fraction of change in tangent angle to thelength of line segment, which can be expressed as, κ s = d θ d s . (7.5)Substituting Eqs. (7.1), (7.2), (7.4), and (7.5) into the mass conservation equation (7.3),the resultant expression can be written as, V cos θ = D s δ s k B T (cid:130) eZ s E ∞ cos θ κ s + Ω γ s κ s dd θ (cid:18) κ s d κ s d θ (cid:19)(cid:140) . (7.6)This equation can be non-dimensionalized using the relation, (cid:2) /κ s (cid:3) = (cid:2) / ˆ κ s (cid:3) D s δ s k B T V eZ s E ∞ (7.7)Here ˆ κ s denote the non-dimensional curvature of the island. Substituting Eq. (7.7) intoEq. (7.6), the shape equation can be expressed in a non-dimensional form as,dd θ (cid:18) ˆ κ s dˆ κ s d θ (cid:19) + χ (cid:18) − κ s (cid:19) cos θ =
0. (7.8)where the dimensionless number χ is defined as, χ = (cid:18) D s δ s k B T V eZ s E ∞ (cid:19) eZ s E ∞ Ω γ s . (7.9)hapter 7. 104The non-linear ordinary differential equation (7.8) determines the steady-state shapes.Note that, heretofore, the expressions are equally applicable to non-circular islands. Movingon to circular shaped islands, their stability can be analyzed by introducing perturbationsto the surface in the form, ˆ κ s = + (cid:34) (cid:99) ( θ ) (7.10)where (cid:34) is the small perturbation parameter and (cid:99) ( θ ) denotes the perturbation function.Substituting Eq. (7.10) into Eq. (7.8), the growth of perturbations can be expressed as,d (cid:99) d θ + χ (cid:99) cos θ =
0. (7.11)This non-trivial expression is similar to Mathieu equation. A MATLAB based subroutine isdeveloped with bpv4c algorithm to seek eigenvalue of Eq. (7.11). The second order ordinarydifferential equation splits into two first order equations with initial guess function andboundary conditions are employed in the form of a cosine function. The lowest eigenvalueis estimated as χ = R i , the steady-statevelocity [ ] can be determined by using κ s = / R i and d κ s / d s =
0. Therefore, substitutingthese relations in Eq. (7.6), the velocity of a circular island can be expressed as, V = − D s δ s R i k B T eZ s E ∞ (7.12)Therefore, the dimensionless number in Eq. (7.9) can be expressed for steady-state circularislands of the form, χ = eZ s E ∞ R i Ω γ s . (7.13)This equation represent relative strength of the capillarity Ω γ s / R i and the electromigrationforce eZ s E ∞ . The capillarity attempts to reduce any curvature gradient in an ultimate aimto form a circular shape, while the atomic transfer due to electromigration in the directionof electron wind alters the preexisting circular shape of the island. Therefore, the islandshape is controlled by the parameter χ , which represents the competition between theelectromigration force and the capillarity. A critical limit of the electric field, at which acircular island maintains its shape is defined by χ = A direct projection of external electric field at the infinity is considered on the surface ofinclusion in previous section 7.1(see Eq. (7.2)). This can be justified for homogeneous in-clusions. However, for heterogeneous inclusions, a general expression of electrical potentialin a matrix containing a circular inclusion is determined. Then, this expression is extendedto a circular inclusion with small perturbations on the surface. Finally, a stability limit ofa circular inclusion is determined from the basic species transport relation. The presentedtheory is in coherence with the work of Ref. [ ] . For the brevity and completeness, theapproach with necessary assumptions is described. Consider a circular inclusion of radius R i in an infinite matrix subject to an external electricfield, E ∞ . In the present study, the plane polar coordinate system ( r , ϑ ) is considered withthe origin at the center of the inclusion. r represents a distance of a point in the system (i.e.,matrix + inclusion) and ϑ corresponds to an angle formed by the point under considerationat the origin with the direction of electric field. The electrical potential at any far point fromthe inclusion is stated as, φ ( r , ϑ ) | r →∞ = − E ∞ r cos ϑ . (7.14)It is easy to perceive that the inclusion of conductivity σ icl disrupts the electrical potentialdistribution in the matrix of different conductivity, σ mat , inside and nearby the inclusion.The spatial distribution of the electrical potential can be obtained by Laplace’s equation, incylindrical coordinates with azimuthal symmetry,1 r ∂∂ r (cid:18) r ∂ φ ( r , ϑ ) ∂ r (cid:19) + r ∂∂ ϑ (cid:18) ∂ φ ( r , ϑ ) ∂ ϑ (cid:19) =
0. (7.15)The general solution of the above equation can be derived by the variable separationmethod,hapter 7. 106 φ ( r , ϑ ) = ∞ (cid:88) l = (cid:129) A l r l + B l r l (cid:139) ( C l cos l ϑ + D l sin l ϑ ) . (7.16) A l , B l , C l , and D l are the real coefficients, which need to be determined by known condi-tions. At the far distance, above equation of the electrical potential should correspond toEq. (7.14). Thus the equation reduced to the form, φ ( r , ϑ ) = Ar cos ϑ + Br cos ϑ , (7.17)containing only cos ϑ term. Furthermore, as the origin is at the center of the inclusion, thesecond term diverges at the origin. Therefore, electrical potential outside and inside of thesphere is expressed separately as, φ in ( r , ϑ ) = A in r cos ϑ , (7.18) φ out ( r , ϑ ) = A out r cos ϑ + B out r cos ϑ . (7.19)Comparing above equation with the asymptotic limit at the far distances, the expression A out = − E ∞ can be obtained. It is easy to notice that, from the continuity equation, on theinclusion surface, ∂ φ in ∂ r (cid:12)(cid:12)(cid:12)(cid:12) r = R i = β ∂ φ out ∂ r (cid:12)(cid:12)(cid:12)(cid:12) r = R i and ∂ φ in ∂ ϑ (cid:12)(cid:12)(cid:12)(cid:12) r = R i = ∂ φ out ∂ ϑ (cid:12)(cid:12)(cid:12)(cid:12) r = R i . (7.20)Here β = σ mat /σ icl . The resultant electrical potential reads, φ ( r , ϑ ) = (cid:40) − β ( + β ) E ∞ r cos ϑ , for r ≤ R i , − E ∞ r cos ϑ + E ∞ R i r ( − β )( + β ) cos ϑ for r ≥ R i . (7.21)Note that both expressions are satisfied at the surface of the inclusion. Using the conformal mapping to express complex potential (Eq. (7.21)) for the circularinclusion at point z = x + i y in complex form, Φ in ( z ) = − β ( + β ) E ∞ z , (7.22) Φ out ( z ) = − E ∞ z + ( − β )( + β ) E ∞ R i z . (7.23)hapter 7. 107Here Φ ( z ) = φ + i ψ , where ψ is the orthogonal function to the electrical potential. Hereto-fore, the linear stability of a perfectly rounded inclusion is considered. Now, a small pertur-bation ( (cid:34) (cid:28) ) is introduced to the surface of inclusion. In the present part, an expressionof electric field is derived for the matrix containing inclusion of circular shape with smallperturbations around the surface. The mapping of inclusion surface, z ( R i , ϑ ) = x + i y = R i ( cos ϑ + i sin ϑ ) + (cid:34) ∞ (cid:88) n = a n ( cos n ϑ − i sin n ϑ ) , (7.24)here a n are real constants, which associated with the strength of surface perturbations. Theexpression for the electrical potential needs to be updated, as the small perturbations on theinclusion surface modify the electrical potential distribution. From the arguments based ona convergence of the electrical potential inside and outside of the inclusion, the expressionof electrical potential can be given by, Φ in ( z ) = − β ( + β ) E ∞ z + (cid:34) ∞ (cid:88) n = C n z n , (7.25) Φ out ( z ) = − E ∞ z + ( − β )( + β ) E ∞ R i z + (cid:34) ∞ (cid:88) n = D n z − n . (7.26)Here C n and D n are coefficients, which are determined by substituting z in the above equa-tion and applying continuity conditions, Eq. (7.20), on the surface of the inclusion, aftertransforming, from ∂ φ in /∂ r = β ( ∂ φ out /∂ r ) to ψ in = β ψ out by the Cauchy-Riemann con-dition (i.e., ∂ φ/∂ r = ∂ ψ/∂ ϑ ) and from ∂ φ in /∂ ϑ = ∂ φ out /∂ ϑ to φ in = φ out . Therefore,the resultant electrical potential on the surface of inclusion, Φ in = (cid:168) − β ( + β ) E ∞ R i e i ϑ − (cid:34) β ( + β ) E ∞ ∞ (cid:88) n = a n e − in ϑ (7.27) − (cid:34) β ( − β )( + β ) E ∞ ∞ (cid:88) n = a n e in ϑ (cid:171) , Φ out = − E ∞ R i e i ϑ + ( − β )( + β ) E ∞ R i e − i ϑ − (cid:34) β ( + β ) E ∞ ∞ (cid:88) n = a n e − in ϑ . (7.28)Here e i ϑ = cos ϑ + i sin ϑ . The mapping function z can be expressed for a transient inclusion, which drifts in the ma-trix. The mapping function of the inclusion, whose position changes with time can be givenby,hapter 7. 108 Figure 7.2: Schematic of a line segment of a surface of inclusion. z ( R i , ϑ , t ) = R i e i ϑ + S ( t ) + (cid:34) ∞ (cid:88) n = a n ( t ) e − in ϑ . (7.29)where S ( t ) is the spatial displacement function related to path traveled by the inclusion. Inthe present case, the coefficients a n vary with time as the inclusion shape transforms as itevolves.Consider a small segment on the inclusion surface as shown in Figure 7.2. Length ofthe segment can be expressed as,d s = | d z | = (cid:12)(cid:12)(cid:12)(cid:12) d z d ϑ d¯ zd ϑ (cid:12)(cid:12)(cid:12)(cid:12) / d ϑ = R i (cid:32) − (cid:34) R i ∞ (cid:88) n = na n cos ( n + ) ϑ (cid:33) d ϑ . (7.30)Here, ¯ • is a complex conjugate of the respective function. The curvature of the segmentcan be expressed as, κ s = d θ d s = dd s (cid:130) tan − (cid:18) − d x / d s d y / d s (cid:19)(cid:140) . (7.31)where θ is an angle formed by line segment with the perpendicular to external electricfield. Simplifying above equation by substituting the expressions for d y / d s , d x / d s andd s / d ϑ from the Eqs. (7.29) and (7.30), one obtains κ s = R i (cid:32) + (cid:34) R i ∞ (cid:88) n = n ( n + ) a n cos ( n + ) ϑ (cid:33) . (7.32)Now, the electric field can be evaluated from Eqs. (7.28) and (7.30) as, E t = − d φ out d s = (cid:168) − β ( + β ) E ∞ sin ϑ − (cid:34) β ( − β )( + β ) E ∞ R i ∞ (cid:88) n = na n sin n ϑ − (cid:34) β ( + β ) E ∞ R i ∞ (cid:88) n = a n − ( n − ) sin n ϑ (cid:171) , (7.33)It is easy to verify that by putting (cid:34) = [ ] .hapter 7. 109The normal velocity at any point on the line segment is given by, V n = Ω d J n d s = V cos ( ◦ + θ ) + V sin ( ◦ + θ ) = d y d t d x d s − d x d t d y d s (7.34)Substituting Eqs. (7.29) and (7.30) in the above equation and integrating, the resultantequation can be given by, Ω J s = − ˙ SR i sin ϑ − (cid:34) R i ∞ (cid:88) n = ˙ a n ( n + ) sin ( n + ) ϑ + (cid:34) ˙ S ∞ (cid:88) n = a n sin n ϑ . (7.35)Substituting Eqs. (7.32), (7.33), and (7.35) into the basic mass transport equation, Ω J s = D s δ s k B T (cid:18) − eZ s E t + Ω γ s d κ s d s (cid:19) . (7.36)In addition, the obtained equation is equivalently applicable to the perfectly circular in-clusion. Therefore, by substituting (cid:34) =
0, one finds a relation of a velocity of a perfectlycircular inclusion, ˙ S = V = − D s δ s k B T β ( + β ) eZ s E ∞ R i . (7.37)and the equation of evolution of inclusion surface reduced to, − ˙ a n − = D s δ s k B T β ( + β ) eZ s E ∞ R i n (cid:130) ( n − ) a n − − ( + β ) χ β n ( n − ) a n − + (cid:18) ( − β )( + β ) n + (cid:19) a n (cid:140) , (7.38)where, a dimensionless parameter is defined as χ = eZ s E ∞ R i / ( Ω γ s ) . At the equilibrium ofinclusion shape, i.e. ˙ a n − =
0, a set of an infinite number of homogeneous linear equationsare obtained of the form, ( n − ) a n − − ( + β ) β χ ct n ( n − ) a n − + (cid:18) n − β + β + (cid:19) a n = n ≥
2. (7.39)For all integer values of n ≥
2, a homogeneous linear equation can be obtained in theform of coefficients, a n − , a n − , a n , β , and χ ct . The purpose of these equations is to finda relation between β and χ ct for non-trivial coefficients, a n ’s. Only a subset of an infinitenumber of equations is considered due to restrictions on the computational and analyticalapproaches. In other words, finite set of equations are considered with assumption that co-efficients beyond a N (i.e., a N , a N + ,...) are zero, where N indicates the number of equationsconsidered for the study. The analytical solution can be obtained by substitution of vari-ables up to five equations. Beyond that, the difficulty to obtain analytical relation increasesexponentially with an increase in the number of equations. Therefore, a numerical tool isdeveloped on MATLAB platform by using vpasolve algorithm. The convergence study of theresultant relation is plotted in the Figure 7.3. The derived relation between β and χ ct inthe present section (Eq. (7.39)) is considered further for the comparison with the resultsobtained from the phase-field method in Figure 7.4.hapter 7. 110 -70-50-30-10 10 0 0.2 0.4 0.6 0.8 1 e rr o r ( i n % ) β N=2N=4N=10N=30 -0.04-0.02 0 0.02 0.04 0.6 0.7 0.8
Figure 7.3: Convergence of the numerical solution for linear stability analysis. N indicates the numberof equations considered for the study. The error is calculated on the basis of relation obtained from 36equations.
The competition between the EM wind force and the capillary force, defined by the non-dimensional parameter χ , determines the stability of the circular inclusion. Accordingly,there exists a critical value χ ct , above which the circular inclusion can no longer be sta-ble and loose its circular shape. The theoretical value of the dimensionless parameter, χ ct = β =
1) is obtained by computing the eigenvalue ofMathieu-type equation (7.11), obtained in section 7.1. The derived value is in agreementwith linear stability analysis presented in section 7.2. Furthermore, as shown in Figure 7.4, χ ct shows a strong dependency with conductivity ratio, β . Even though, this theory pro-vides information regarding the stability of circular inclusion. However, it serves no infor-mation about the shape of the inclusion after the collapse from the circular shape. Theseinferences can be understood by the analysis of the phase-field simulations.Inset of images from phase-field simulations in Figure 7.4 shows the influence of thenon-dimensional parameters χ and β on inclusion equilibrium shapes. On the one hand,the competition between electromigration wind force and the capillary force decides theconsequences on the inclusion shape. The non-dimensional parameter χ determines therelative strength of the electromigration force compared to the capillarity. On the otherhand, β identifies maximum and minimum values of E x and E y for a known external elec-hapter 7. 111 χ ( E M f o rc e / C a p ill a r i t y ) β χ Figure 7.4: Dependence of an inclusion equilibrium shape on the dimensionless parameter, χ and theconductivity ratio β . The black solid curve corresponds to χ ct , which is obtained by linear stabilityanalysis. This line identifies whether an inclusion to retain its circular shape or evolves to subsequentcollapse. The black dashed line indicates lines of action on which the phase-field simulations are ob-tained. The inset of images are the snapshots from the phase-field model.Table 7.1: Values of parameters considered for the simulation sets of an isotropic inclusions to studysignificance of χ and β . χ β Set: 1 5.0 to 60.0 3Set: 2 7.5 1 to 10000Set: 3 15.0 1 to 10000tric field E ∞ , which is responsible for the change of local electric field distribution. Thesignificance of these two parameters is discussed in the following paragraphs. The simula-tion parameter set considered for this study is summarized in Table 7.1. χ The shape of the inclusion is governed according to the competition between the electro-migration wind force and the capillary force, as it is evident from the non-dimensionalparameter χ . The capillary force seeks a uniform curvature and consequently diffusesspecies to reduce any curvature gradient along the inclusion surface. Electromigration, onthe other hand, instigates an atomic transport in the direction of the electron wind force,hapter 7. 112 L i Anode Cathodee − e − R i u wt/τ (cid:48) = 0 t/τ (cid:48) = 0 . × t/τ (cid:48) = 0 . × t/τ (cid:48) = 0 . × t/τ (cid:48) = 1 . × t/τ (cid:48) = 1 . × t/τ (cid:48) = 2 . × (a)(b) t/τ (cid:48) = 0 t/τ (cid:48) = 1 . × t/τ (cid:48) = 2 . × t/τ (cid:48) = 3 . × t/τ (cid:48) = 5 . × t/τ (cid:48) = 6 . × t/τ (cid:48) = 7 . × Figure 7.5: (a) A circular inclusion migrates by preserving its shape, and (b) evolves its shape to thefinger-like slit. The results correspond to the dimensionless parameter χ = for (a), and χ = for (b) keeping the conductivity contrast constant, β = . The arrows show the direction of the electronwind, from the cathode to the anode. Here u is the half-slit width and w denotes the line width of theconductor. leading to its shape alterations. In the present case, this corresponds to the diffusion of thespecies, from the right towards the left surface.The migration of the circular inclusion, at a low electromigration force, is shown inFigure 7.5(a). From the phase-field simulations, the inclusion surface is extracted as anisoline of value c = β and the current crowding V x , β = 1 t ( τ ' ) V ( λ ' / τ ' ) V x , β = 3 V x , β = 10000 Figure 7.6: Velocity of the centroid of the inclusions along the direction of external electric field, V forconductivity ratios, β =
1, 3, and 10000, and the dimensionless parameter χ = . The inset ofimages are the snapshots from the phase-field model in equilibrium. To understand the significance of β , simulations along two horizontal dotted lines shownin Figure 7.4 are considered. According to linear stability analysis, all inclusions on thelower line should migrate maintaining their circular shapes, while shape alterations areexpected on the right part of the upper line.From the phase-field simulations, the migration of the initially-circular inclusions forlower electromigration force ( χ = β . However, the velocities of the inclusions are modified by β , whichis in agreement with the literature [ ] and linear stability analysis (see Eq. (7.37)).In fact, the adjustment of the inclusion velocity approximately corresponds to the factor of2 β / ( + β ) . This infers that the β increases velocity of the circular inclusions, as inclu-sions of heterogeneous conductivities ( β >
1) travels faster compared to its homogeneous( β =
1) counterpart. Therefore, the increase in conductivity ratio improves propagationspeed, up to two times for a circular inclusion.During the inclusion migration at higher electromigration force ( χ = Figure 7.7: The change in local electric field distribution during the evolution of the inclusion. Thedistribution of electric field components E x in first column and E y in second column are shown for β =
1, 3, and from top to bottom respectively. The electric field components are normalized by theexternal electric field E ∞ . The electric field distribution remains unchanged for β = . On the contrarily,although it remains nearly constant at the distance from the inclusion, while inside and around theinclusion, electric field changes effectively for β = and . The contour of maximum value ofE x ( = β = . In addition, E x enhances localelectric field up to two times, while E y forms a quadrupole pattern around the inclusion of β = . gration force perceives the length of the inclusion along the traverse direction to the electricfield as a resistance in the path of electron wind. The higher electromigration force reducesthe resistance in the path by altering inclusion shape to the formation of a finger-like slit(see inset at the right side of Figure. 7.4). However, these arguments are only valid for thehomogeneous systems, meaning, of similar conductivities ( β = E x and E y , in the longitudinal and traverse directionsrespectively.The distribution of E x in Figure 7.7(c1) describes an increase in the local electric fieldnear the inclusion in the vertical direction, while a decrease in the horizontal direction. Theasymmetric allocation of the electric field promotes local species diffusion from the lowerelectric field to the higher one. Therefore, the local atomic flux for β = β = E x (i.e., E x / E ∞ = β = χ ct with β , obtained by linear stability analysis in Figure 7.4. Thestiff gradient in χ ct on the left side of Figure 7.4 can be explained by the relative strengthof the external electric field and the local change in the electric field. The higher electronwind force due to further increase in the dimensionless parameter χ for the constant β establishes a higher velocity of inclusions. This promotes protrusion to the surface of theinclusion with the same conductivity ratios for higher χ .As evident, the finger-like slits are the frequently observed morphology in isotropic in-clusions. Therefore, to analyze the characteristics of the finger-like slits, only the occurrenceof single slits is considered for a detailed analysis in the following section. The phase-fieldsimulations are performed for parameter set β = χ = Gan et al. [ ] observed that inclusions nucleate and grow at the intermetallic compound(IMC) and the grain boundary interface. Afterward, the inclusions migrate into the conduc-tor lines in a transgranular manner, which might be subjected to shape alterations [ ] .hapter 7. 116 p / p ( × ) t/ τ ' y x = 5 × = 6 × = 7 × ( λ ') ( λ ' ) t/ τ 't/ τ 't/ τ ' (a)(b) φ Figure 7.8: (a) Temporal evolution of the perimeter as a function of time, from the initial circularinclusion to the finger-like slit. (b) Contour plot of the slit, at three different times indicating thatthe slit propagates with an invariant shape. The results are shown for the dimensionless parameter χ = . Kraft et. al. [ ] reported that a slight change in the void shape can be amplified by theelectron wind leading to a slit. Perhaps, the most pervasive form of inclusion migrationis the transition to a slit-like morphology. The presence of finger-shaped (also known aspancake-type) inclusions have been reported in Refs. [
77, 216 ] . The shape changes during the transition of the circular inclusion to a stationary slit can becaptured effectively by tracking the temporal inclusion perimeter of the phase-field simu-lation, as shown in Figure 7.8(a). The perimeter p is calculated from the slit profile of theisoline with the value c = p . As the protrusion initiates, the inclusion perimeter increases with time, untilhapter 7. 117 ElectromigrationCapillary
Figure 7.9: A schematic diagram of constricted neck region at t /τ (cid:48) = × in Figure 7.8. The bluearrows show atomic mass transport due to the capillary force, while the red arrows are the flux due tothe electron wind. a maximum at t /τ (cid:48) = × . At this stage, the junction at the end of the parallel region ofthe slit and its circular end become constricted as shown in Figure 7.9. The surface is con-cave, with respect to the neighboring regions on either side. Thus, the flux from either side,which is induced by the curvature gradient, in addition to the electromigration-induced fluxfrom the cathode side, leads to the momentary decrease in the inclusion perimeter. At a laterstage ( t /τ (cid:48) > × ), the inclusion perimeter saturates to a constant value, which indicatesthe time-invariant migration of the inclusion. The time-invariant drift of the inclusion is themanifestation of the equilibrium of the capillary-induced and electromigration-induced fluxat every point on the surface. The invariance of the inclusion shape is also evident from thecomplete overlap of the slit profile, at the late stages, as shown in Figure 7.8(b). The slitprofiles have been translated by a factor V t , where V is the steady-state velocity.The interplay between the capillary force and the electromigration wind force is furtherreflected by changing the applied potential, after the equilibration of the slit profile. Duringoperational conditions, the interconnect lines are seldom subjected to a constant electricfield. Thus, understanding the effect on the inclusion shape, which is due to a change inthe electric field, has technological implications. Figure 7.10(a) depicts the change of theinclusion shape, when the applied potential is abruptly changed from φ ∞ = − χ = φ ∞ = − χ = V ( × . λ ' / τ ' ) ( × ) Tip VelocityBase Velocity t/ τ ' u ( λ ' ) ( × ) t/ τ ' (a)(b) φ φ φ φ Figure 7.10: The change in the electrical potential at time t /τ (cid:48) = × results in the alteration of(a) the base velocity, the tip velocity, and (b) the width of the slit propagation. The dotted-black linein the graph represents the change in the electrical potential, from φ ∞ = χ = ) to φ ∞ = χ = ). The slit adjusts the width and the velocity according to the externalelectric field, and evolves to a steady-state finger-like shape. librated electromigration and capillary forces. With the decrease in electromigration flux,the capillary-mediated flux from the rear end towards the cathode end becomes prominent,leading to a decrease in the inclusion perimeter. In the second cycle, the species transportfaster from the rear end, due to the momentarily steep capillary force, which leads to aninitially faster base velocity, in contrast to the former. The transport, which is mediatedby the capillary, leads to an increase in the slit width, in the attempt to minimize the cur-vature gradient. This process can be understood as follows: The curvature at the tip ofthe slit is approximately κ s ≈ / u , and the curvature gradient d κ s / ds ≈ / u . Since thedecrease in E ∞ leads to a decrease in the electromigration flux (first part in Eq. (7.1)), thecorresponding decrease in the capillary part is achieved by an increase in the width of theslit. The temporal evolution of the slit width u , with the change in the electric field cycleis illustrated in Figure 7.10(b). The slit width is calculated from the averaged values of thehapter 7. 119difference between the ordinates of the slit profile. The width increases until a new equi-librium is established between the two counteracting forces, after which the entire surfaceagain drifts with a uniform steady-state velocity.The above results imply a strong dependence of the selection of the slit width and ve-locity on the electric field, which is discussed in the last part of this section (in subsec-tion 7.4.5). E t θ x slit frontslit surface J s J s , s tipwakewake yE ∞ u Figure 7.11: Schematic diagram of the finger-like slit front, subjected to an external electric field, in aninfinite interconnect domain. The origin lies at the point of intersection between the tangent to the slittip and a straight line through the flat wake.
Transgranular slit propagation, under the assumption of a steady-state, is derived bySuo et. al. [ ] . For brevity, the basic steps of the derivations are described. A completedescription can be found in Refs. [ ] .Consider a slit front, translating along the length of the conductor, with a time-invariantshape. The coordinate system is assumed to move in the frame of reference, which is at-tached to the slit, with the origin coinciding with the flat region, as shown in Figure 7.11.The slit migrates as a result of a mass transport flux, which is induced by the surface elec-tromigration and the capillarity. Therefore, the Nernst-Einstein relation (7.1), mass con-servation equation (7.3), and the normal velocity (7.4) are equally applicable to finger-likeslit propagation. However, to generalize the tangential electric field component Eq. (7.2),which includes the effect due to distinct conductivities in the matrix and the inclusion, theelectric field along the slit surface can be expressed as, E t = − β + β E ∞ sin θ , (7.40)hapter 7. 120which is responsible for the slit transmission [ ] .Substituting Eq. (7.4) into Eq. (7.3), and utilizing the relation d y = d s cos θ , the resul-tant expression can be written as, d J s d y = V Ω . (7.41)Integrating once, J s = V Ω y + C st . (7.42)Here C st denotes the integration constant, which is determined from the fact that in theflat wake, i.e., d κ s / d s = y =
0, substituting this relation in the Nernst-Einstein rela-tion (7.1), the flux in the flat wake can be expressed as, J s | y = = D s δ s Ω k B T eZ s β + β E ∞ (7.43)Equating Eqs. (7.42) and (7.43), the resultant flux, J s = V Ω y + D s δ s Ω k B T eZ s β + β E ∞ . (7.44)For a circular inclusion of radius u , which is migrating maintaining its shape, the velocityrelation can be given by Eq. (7.12), V = − D s δ s uk B T eZ s β + β E ∞ (7.45)Henceforth, the derivation digresses from the work of Suo et al. [ ] . According to Yaoet. al. [ ] , the shape evolution changes the velocity simultaneously. The parameter ξ can then be introduced to reflect the discrepancy between the velocities of the circularinclusion with the radius u , and the finger-like slit with the same half slit width. Thus, theslit propagation velocity relative to the circular one is defined by V = ξ V . (7.46)The value ξ > < ξ < y = u . Consider a small segment d s , whose tangent makes an angle θ withthe y-axis. Let q = sin θ = ( − d x / d y ) / (cid:112) + ( d x / d y ) . Note that d y / d s = ( − q ) / and κ s = d θ / d s = d q / d y . The curvature gradient writes asd κ s d s = (cid:18) d κ s d y (cid:19) (cid:129) d y d s (cid:139) = ( − q ) / d q d y . (7.47)hapter 7. 121Substituting Eqs. (7.1),(7.40),(7.45),(7.46),and (7.47) into Eq. (7.44) results in1 η ( − q ) / d q d Y + q + ξ Y =
1, (7.48)where Y = y / u and a dimensionless number, η is expressed as η = eZ s β E ∞ u Ω γ s ( + β ) . (7.49)The dimensionless group η reflects the relative strength between the electromigration windforce and the capillary force. It is worth noticing that even though both dimensionless pa-rameters, χ and η determines relative strength between electromigration force and capil-larity, the former provides condition for the stability of the circular inclusion, while letterhelps to find physical characteristics associated to the slit, which is elaborated further inthe forthcoming paragraphs.For the flat wake, the vanishing slope and curvature provide the boundary conditions: q =
1, d q / d Y = Y =
0. (7.50)The sharp-interface equation (7.48) is solved under the boundary conditions in Eq. (7.50),by using a combined shooting and Runge-Kutta methods. The value of η is held fixedand shooting for the value of ξ , such that the angle at the slit tip is zero (i.e., q = Y = u = (cid:18) η Ω γ s ( + β ) Z s e β E ∞ (cid:19) / , (7.51) V = ξ D s δ s k B T ( Z s e β E ∞ ) / ( η Ω γ s ) / ( + β ) / . (7.52) The parameter χ (Eq. (7.13)) is introduced to provide a condition for the circular inclusionstability. If χ exceeds a critical value χ ct , an initially circular inclusion collapses into aslit. However, the parameter contains no information about the dimensions of the newlyformed slit, and about the propagation velocity after the breakdown [ ] . By incorporat-ing the value η into the sharp-interface model, all corresponding slit characteristics can bedetermined. η is an input parameter in sharp-interface analysis. While, in the phase-fieldmodel, the surface energy, the electric field and the ratio of the initial inclusion to the linewidth are given as the input, and the slit selects its width and velocity accordingly. Hence, η is a priori unknown quantity. To facilitate a comparison of the slit characteristics, namelyhapter 7. 122 η w/R i PFMLine fit
Figure 7.12: The values of η , obtained from the phase-field model, together with a line fit accordingto Eq. (7.53) are plotted as a function of w / R i . A smaller values of w / R i result in wider slit width forsame applied electric field. the slit profile, the width, and the velocity obtained from the sharp-interface and diffuse-interface formalism, the value of η needs to be extracted from the phase-field simulation.Furthermore, from Figure 7.10, it is evident that an increase (or decrease) in the electricfield leads to a finer (or wider) slit. The same is also apparent from Eq. (7.51). However,it is worth noting that E ∞ and u appear as a product in the numerator, in the expressionof η . Since the cause ( E ∞ ) and the effect ( u ) act in the opposite way, it is inexplicablebeforehand whether an increase (or decrease) in E ∞ entails a corresponding increase (ordecrease) in η .Additionally, it is to be noted that the sharp-interface analysis does not consider theeffect of the line width on the selection of the final width and the velocity of the slit. How-ever, the phase-field simulations reveal a strong dependency of the line width on the finalslit characteristics. In Figure 7.12, η is shown as a function of the interconnect width w to the inclusion radius R i . η exhibits a steep increase, with a corresponding decrease inthe values of w / R i . This implies that the slit originating from the higher values of w / R i re-sult in narrower slit width for the same value of the applied electric field. The relationshipbetween the two parameters η and w / R i is fitted by using an exponential function: η = e − w / R i + η in the present work are obtained from simulations where E ∞ and w / R i arehapter 7. 123simultaneously changed. Moreover, such a procedure further illuminates the effect of theline width. w The most interesting result of the present study is the dependence of line width on the finalslit characteristics. Further, the line width has a significant impact on the inclusion shapeand, as a consequence, the performance of the interconnect material. Xia et al. [ ] com-puted critical values of χ , at which a circular inclusion collapses, as a function of linewidth analytically. Kraft et al [ ] reported shape changes of inclusions with line widthexperimentally. In the present study, the dependence of line width on the slit characteris-tics can be justified by the analysis of the electric field. On one hand, at elevated electricfield collisions of electrons on the inclusion surface may overcome the surface energy andconsequently result in the occurrence of shape changes in Figure 7.5(b). On the otherhand, electrons flow at lower electric field colliding on the inclusion may not overcome thesurface energy and, therefore, circular inclusions maintain their shape as seen in Figure7.5(a). Furthermore, decreasing the line width of the interconnect motivates enhanced in-teraction between the electric field vectors and the inclusion surface locally, as depicted inFigure7.13(b) and (c). Therefore, the enhanced electric field in the vicinity of the inclusionsurface, in Figure 7.13(a), provokes electromigration-induced mass flux from the inclusiontip to the inclusion wake. As a consequence, the slit width, and accordingly η , increases,while the velocity decreases, for the shorter line width.A comparison of the slit profiles at the tip, obtained from the numerical solution ofEq. (7.48), and from the the phase-field simulations for different η are presented in Fig-ure 7.14. The results reveal a good agreement with the sharp-interface analysis across allvalues of η . Another point to be noted is that the asymmetry in the profile, relative to thecircular inclusion, decreases with an increasing η . Next, the dependence of the electric field strength on the selection of the slit width andthe velocity is addressed. The slit width, which corresponds to different values of η andthe dependency on the field strength, is plotted in Figure 7.15(a). An excellent agreementbetween the sharp interface relation Eq. (7.51) and the phase-field simulations is observed.The slit width scales as E − / ∞ , implying a narrower width at higher field strengths.The slit velocity is measured after it attains the equilibrium. Note that in the steady-state, every point on the slit moves at a constant velocity V . In Figure 7.15(b), the velocityhapter 7. 124 e − e − e − e − e − e − Electromigration-inducedmass fluxflat wakeflat wake (a)(b)(c)
E/E ∞ Figure 7.13: Schematic diagram of the interaction between the slit surface and the electron wind in (a),normalized electric field vectors are superimposed on the steady-state slit profile for w / R i = , η = in (b) and w / R i = , η = in (c) under the dimensionless parameter χ = . hapter 7. 125 -1.0-0.50.00.51.00.0 1.0 2.0 3.0 y x SI η = 2.0PFM η = 2.0SI η = 2.4PFM η = 2.4SI η = 2.8PFM η = 2.8SI η = 3.6PFM η = 3.6 -0.30.00.33.2 3.4 3.6 ( λ ') ( λ ' ) Figure 7.14: A comparison of the front part of the slit profile from the phase-field model with the sharp-interface analysis for different values of η . All dimensions obtained from the phase-field model arenormalized by the slit width u. is plotted as a function of the electric field strength, for different values of η . The phase-field results show good agreement with the sharp-interface relation in Eq. (7.52). The slitvelocity scales as E / ∞ , which indicates that the slit migrates faster at higher electric fieldstrengths.Yao et. al. [ ] reported that the inclusion propagation velocity changes simultane-ously if the inclusion collapses to a finger-like slit. They introduced a parameter ξ , as theratio of the velocities of a finger-like slit to the circular inclusion of radius equal to halfslit width. The value ξ > < ξ < η and ξ , which isnothing but the relationship between the slit width and the velocity. For each simulationresult, the parameter ξ can be associated with a unique value of η . The relation betweenthe two parameters η and ξ , as displayed in Figure 7.16 is fitted with ξ = e − η + e − η . (7.54)The slit velocity changes slowly for η >
2, while it increases rapidly for the lower values.The parameter ξ approaches the value of unity, at η = η = u ( λ ' ) E -1/2 PFM η = 2.4SI η = 2.4PFM η = 2.8SI η = 2.8PFM η = 3.1SI η = 3.1 V ( × . λ ' / τ ' ) E ( × -4 ) PFM η = 2.4SI η = 2.4PFM η = 2.8SI η = 2.8PFM η = 3.1SI η = 3.1 (a)(b) ∞∞ Figure 7.15: A comparison of the phase-field simulation results with the sharp-interface analysis for (a)the half slit width and (b) the velocity of the slit, as a function of the external electric field, for differentvalues of η . The plots suggest the power-law dependence u ∝ E − / ∞ and V ∝ E / ∞ . The half slit widthand the velocity are calculated in the steady-state part of the motion.Table 7.2: A comparison of the values of the void size and the velocity, for eutectic SnAgCu solder bumpsobtained from phase-field, and experiments. Only the experimental results from Zhang et al. [ ] areconsidered for the comparison with presented phase-field and sharp-interface methods. PFM Experiment [ ] void size (2 u ) 0.80 − µ m 2.44 µ mvelocity ( V ) 1.38 − µ m / h 4.40 µ m / h The movement of voids, as in cavities, is one of the applications of species diffusion due toelectromigration in the interconnects. The evidence is reported that the metallic lines oftenhapter 7. 127 ξ η SILine fitPFM
Figure 7.16: The dependence of ξ on η obtained from sharp interface, and the phase-field model. Thesolid line represents exponential fit to the sharp-interface data. observe a dramatic change in the resistance and even the performance of the conductors isbeing affected due to sudden alteration of rounded voids to complex shapes [ ] . Althoughthe results presented in this chapter examined the case of the transgranular voids migratingalong the metal line, which previously was considered uncritical in terms of failure [ ] , the present study is applicable in the flip-chip solder bumps. The nucleated voidsat the current crowding zone propagate in the direction of the electron wind, along theinterface between the solder and the intermetallic compound.In order to make a comparison between the magnitude of the void size and the velocityobserved in the solder bumps and in the phase-field model, the experimental data at thecurrent density 1.1 × A / m from [ ] as shown in Table 7.2 are utilized. Since thedescriptions of the phase-field, in Chapter 4 correspond to a single-component system, thematerial parameters for Sn are assumed to be the dominant diffusing species. Furthermore,it is important to emphasize that the effective valence Z s of Sn is reported to be 17. However,to compare this with the simulation, the results are obtained at Z s = E / ∞ (from Eq. (7.52)), the same law is validfor the latter case on the dimensional ground. As the velocity is inversely proportional tothe failure time of the interconnect material. Therefore, the exponent in the Black’s law isexpected to be n = / [ ] . Since the motion of the voids is governed by the fluxdivergence, the diffusivity along the void surface in addition to that of the passivated in-terface dictates the void velocity. However, in most cases, the void surface diffusivity ismuch greater than the interface diffusivity of the passivated layer due to which the resultpresented here is expected to agree considerably in the presence of the dielectric interface.In addition to the technological implications in the efficient design of interconnects, theresults of the present study can be exploited in the fabrication of wires with a high aspectratio, ranging from the nano- to microscale. Since the width of the slit scales as E − / ∞ , thefeatures of the desired dimension can be achieved by tuning the inclusion radius and E ∞ .The pattern formation, which is induced by the electric field, already enjoys a great dealof success in modulating the morphology in block copolymers [
97, 98 ] , metal conductors [ ] , fluids [ ] , and more recently in an electromigration-induced flow in liquidmetals [ ] . Sharp-interface models, both analytical as well as numerical approaches, have been em-ployed extensively in the past to simulate electromigration-induced inclusion migration [ ] . Analytical methods (section 7.1 and 7.2) provide infor-mation with regard to the onset of the shape bifurcation of circular inclusions. Subsequentchanges in the shape, however, are inexplicable. In the present context of slit propagation,the derivation in Section 7.4.2 assumes a slit shape and quantifies information regardingthe selection of the steady-state width and velocity for a given electric field. It does notreveal the dependence of the initial inclusion radius to the line width on the selection offinal slit width (and hence velocity). The presented diffuse-interface approach, on the otherhand, allows one to track the entire temporal events elegantly, leading from circular inclu-sion to slit transition without requiring to track the interface explicitly. Thus, in additionto the electric field, the effect of the initial inclusion to the line width ratio is elucidatedthrough the phase-field simulations, as shown in Figure 7.12, which otherwise is obscurehapter 7. 129in the sharp-interface analysis.Numerical sharp-interface methods have experienced a great deal of success in simu-lating inclusion migration [ ] , inclusion to slit transition, and concurrent faceting [ ] and void coalescence [ ] , amongst others. The studies hitherto have been limitedto two dimensions, owing to the onerous task of interface tracking. In addition, the elec-tric field on the inclusion / slit surface is approximated by the local tangential component − ( β / ( + β )) E ∞ sin θ . This limitation of sharp-interface theories is eluded in the pre-sented results by solving the Laplace equation to obtain the electric field distribution atevery spatial coordinate. Moreover, the effect of the current crowding at sharp corners andbends, if any, is inherently captured (section 7.3.2). Besides, the effect due to distinct con-ductivities in the matrix and the inclusion is incorporated (section 7.3.1). Furthermore,the model presented here can be readily extended to three dimensions, without any fur-ther complexity, albeit with an increased computational cost. The ability of the phase-fieldmethod to predict damage in polycrystalline interconnect in three dimensions has beenexemplified recently [ ] .Insights of phase-field simulations on the morphological evolution of initially circularinclusions are considered in the present chapter. A morphological map is constructed inthe plan of β and χ , as shown in Figure 7.4. Circular inclusions and finger-like slits areobserved for the isotropic inclusions. To this end, the effect of diffusional anisotropy, be-tween the inclusion and the matrix, can further be utilized towards efficient guidance ofthe morphological patterns [ ] . Therefore, anisotropic inclusions are considered in thenext chapters. Particularly, anisotropic inclusions of two-fold symmetry are investigated inChapter 8, followed by fourfold and sixfold symmetries in Chapter 9. hapter 8Motion of anisotropic inclusions consistof 2-fold symmetry The phase-field literature focused on the development of inclusion morphology is limited [ ] . The majority of the works are established based on finger-like slit formation [ ] , wedge-shaped inclusion [ ] , crack-type feature [ ] , and / or cardioid-type morphology [ ] . However, the literature is devoid of a variety of inclusionmorphology. This limitation is likely due to assumptions of isotropic diffusivity. More im-portantly, the morphological influence by differing the conductivities for inclusions andmatrices is absent. In addition, the inclusions have been known to undergo complex topo-logical transitions such as splitting [ ] , coalescence [ ] and coarsening [ ] . Facetingis another common feature that has been observed experimentally [
74, 228 ] , which can notbe explained by diffusional isotropy.These observations can be explained by the consideration of either surface energyanisotropy or adatom mobility anisotropy. Most interconnect materials are face-centered-cubic (FCC) metals with a strong crystallographic direction-dependent surface energy [ ] and adatom mobility [ ] . In the present work, diffusional anisotropy and isotropic sur-face energy are considered, due to the fact that the intensity of anisotropy in adatom mo-bility for the facet development during electromigration-induced surface diffusion is muchhigher than that of the surface energy [ ] . Moreover, the faceted inclusions areknown to revert back to circular shapes when the electric field is switched off, further high-lighting the dominance of anisotropy in species diffusion over surface energy [ ] .131hapter 8. 132 The common finding of the previous studies is that the morphological evolution of theinclusion can be characterized by six dimensionless parameters.1. χ = eZ s E ∞ R i / Ω γ s , denotes the ratio of surface electromigration force to capillaryforce. Lower values of χ promote rounded shape while higher values distort theshape into the slit. The effect of this parameter is discussed in chapter 7.2. Λ L = w / R i defines the ratio of line width w to the initial inclusion radius R i . Larger Λ L leads to finer and faster propagating slit and vice versa [ ] . The significance ofthis parameter is addressed in subsection 7.4.4.3. A characterizes the strength of anisotropy in species diffusion.4. m denotes the grain symmetry parameter.5. (cid:36) related to misorientation of the fast surface diffusion direction with respect to aperpendicular to the external electric field, E ∞ .6. Finally, β = σ mat /σ icl depicts the ratio of conductivity of the matrix ( σ mat ) to that ofthe inclusion ( σ icl ) .A complete understanding of the inclusion dynamics requires an exploration of the com-plete six-dimensional parameter space of χ , Λ L , A , m , (cid:36) , and β . Such an exercise is cer-tainly computationally expensive and does not help in unraveling the effect of individualparameters. A considerable simplification of the problem is then to explore the two or three-dimensional parametric space. Kuhn et al. [ ] studied the effect of χ and A on inclusiondynamics on the substrate using a continuum-based numerical approach. They proposed amorphological map bifurcating the regions of steady-state, oscillatory-state, zigzag motion,and breakup amongst others. Dasgupta et al. [ ] further extended the work to ex-plore χ , m , and (cid:36) space. Maroudas and coworkers [
81, 164, 232 ] have performed a seriesof systematic study using front tracking dynamical simulations to unravel the dynamics ofedge inclusions in χ , Λ L , A , m , and (cid:36) five-dimensional space. They highlighted the stabil-ity of wedge-shaped inclusions [ ] , facet-selection mechanisms [ ] and propagation ofsoliton-like feature [ ] on the inclusion surface. β While the above-mentioned parameters have received much attention until now, the studyon the role of β is relatively scarce. The conductivity contrast, β can significantly influencehapter 8. 133inclusion dynamics and morphology. For instance, careful consideration of the conductivitycontrast between the inclusion and matrix is important to ascertain the contribution of bulkand surface electromigration using marker motion technique [ ] . In interconnect lines,conductive species might be trapped inside the inclusions leading to a finite conductivity [ ] . Therefore, understanding the importance of β is of scientific interest. Haoand coworkers [ ] studied the stability of a circular inclusion analytically in χ - β space using linear stability analysis. In addition, Li and Chen [ ] studied the effect ofconductivity on the electromigration-driven motion of an elliptical inclusion analytically.Furthermore, Li et al. [ ] focused on the role of χ - β on elliptical and wedge-shapedinclusions using the phase-field method. However, these studies have one or more of thefollowing limitations: Although the linear stability analysis sheds light on the stability ofthe circular inclusion, it does not provide further information about the topological tran-sition. Moreover, sharp-interface based front tracking methods usually approximate theelectric field by a local surface projection instead of solving the Laplace equation therebyneglecting the effect of current crowding. Furthermore, previous phase-field studies haveonly considered the effect of inclusion size [ ] . The only study [ ] which consid-ered the effect of conductivity contrast is applicable to edge inclusions migrating on { } planes of FCC metals.The aim of the present chapter is thus to systematically investigate the unexplored topo-logical changes in β - (cid:36) space. In addition, the emphasis is laid on inclusion dynamics on { } textured single crystal. The phase-field model developed in chapter 4 illustrates theemployed numerical method in the form of the phase-field model. Few typical cases of mor-phological evolution of the inclusion highlighting the underlying mechanism are discussedin section 8.2. The presented results suggest that while the shape of the inclusion stronglydepends on the conductivity contrast, the stability is dictated by the misorientation. Thechapter is summarized by a brief discussion on the applicability of the presented results insection 8.3, followed by discussion in section 8.4. Some parts of this chapter are publishedin the Journal of Applied Physics [ ] . The twofold symmetry in the inclusion is a result of the two fast diffusion locations existingin the plane. For instance, in FCC crystals, the symmetry axes corresponding to 〈 〉 crystallographic directions have higher surface diffusivity compared to 〈 〉 and 〈 〉 directions [ ] . Therefore, the number of 〈 〉 axes in a plane decides the number offast diffusion axes in the inclusion. As shown in Figure 8.1, (110) plane contains twocrystallographic directions [ ¯110 ] and [ ] , which are responsible for the fast diffusion.hapter 8. 134 Figure 8.1: Schematic of an inclusion in (110) crystallographic plane. The directions of dominantatomic diffusivity are represented by black arrows. The red color plane is the surface of the film per-pendicular to the circular inclusion. The double-headed arrows define positions of maximum surfacediffusivity.
Similarly, each plane in { } family ( m =
1) contains two of 〈 〉 directions. Note thatthe fast diffusion sites are always opposite to each other in the same axis for all possibleplanes of twofold symmetry.A systematic study of the dynamics of a circular inclusion as a function of β and (cid:36) isperformed. The parameters χ , Λ L , A , and m are held constant, and the numerical valuesutilized in this chapter are summarized in Table 8.1. Due to the symmetry in the shape of thepreexisting inclusion, the parameter space for (cid:36) can be reduced to a range of 0 ◦ − ◦ (= ◦ / m ) , without compromising the features of the inclusion morphology. Few typicalmigration modes are discussed next. When the electric field aligns along the fast diffusion direction (i.e., (cid:36) = ◦ ), the speciesat the inclusion front are displaced faster leading to the formation of a protrusion as shownin Figure 8.2(a). With time, the protrusion progressively assumes a narrow slit-like shape,which eventually migrates with an invariant shape with a high aspect ratio elongated inthe direction of the electric field. The slit formation can be attributed to the high value ofthe parameter χ due to which the capillary force is unable to compete with the surfaceelectromigration force.hapter 8. 135 Table 8.1: Values of parameters considered for the simulation study for an inclusion in { } crystallo-graphic plane. Parameter Value χ Λ L A m (cid:36) ◦ to 90 ◦ β (a) (b) (c) Figure 8.2: Inclusion morphology for conductivity ratio, β = in (a), in (b) and in (c) at themisorientation angle, (cid:36) = ◦ . The inclusions with surface contours are presented with time, t( τ (cid:48) ).The increasing darkness of the contour indicates inclusion evolution. The inclusions are displaced inspace for better visual inspection. The inclusion for β = shows a nanowire morphology, while wedgeshape is observed for . Higher β leads to the slit with a lower aspect ratio in Figure 8.2(b). On increasing β ,the tangential component of the electric field along the inclusion surface becomes moreprominent due to current crowding, as shown in Figure 7.7. Hence, the species transportedrelatively faster from the front end accumulate at the diametrically opposite ends, which areperpendicular to the applied field corresponding to the slowest species mobility regions. Asa result, the rear end becomes progressively thicker and flatter. On further increasing of β = β is shown in Figure 8.3(red curve). For comparison, the velocity of a shape-preserving circular inclusion obtainedfrom Ho’s analytical expression similar to Eq. (7.45) (green curve) and that migrating un-hapter 8. 136 V ( λ ' / τ ' ) β anisotropic PFMisotropic circular Analytical isotropic PFM Figure 8.3: The velocities of the centroid of steady-state inclusions for two-fold anisotropy (m = ), (cid:36) = ◦ (in red graph) and for isotropic (m = ) surface diffusion (in blue graph) as a function of theconductivity ratio β . The green curve corresponds to a steady-state velocity of the circular inclusion ofisotropic surface diffusivity, derived from Eq. (7.45) considering the diffusion coefficient D s . The inset ofimages show inclusion morphology in steady state. β governs the inclusion shapes and the steady-statevelocities. der isotropic diffusion (blue curve) obtained from the phase-field simulations ( m = E t = E ∞ g θβ ( θ , β ) , (8.1)where g θβ ( θ , β ) denotes a function of shape ( θ ) and conductivity contrast ( β ). The sur-face mass flux expressed for isotropic inclusions in Eq. (7.1) is modified for the anisotropicinclusion of the form, J s = D s f θ ( θ ) δ s Ω k B T (cid:16) − eZ s E t + Ω γ s d κ s d s (cid:17) , (8.2)The electric field projection Eq. (8.1) is substituted in the flux equation (8.2) and subse-hapter 8. 137quently into the mass conservation equation (7.3), to obtain normal velocity of the form, V n = D s δ s k B T dd s (cid:150) f θ ( θ ) (cid:168) Ω γ s d κ s d s − q s E ∞ g θβ ( θ , β ) (cid:171)(cid:153) , (8.3)The normal velocity in the phase-field governing Eqs. (4.8) and (4.14) reverts to the aboveexpression as the interface thickness approaches zero, has already been proved in Refs. [ ] via formal asymptotic analysis. The first term in the curly parenthesesrepresents the contribution due to capillarity or curvature-gradient, while the second termdenotes the effect of electromigration. The electric field along the surface Eq. (8.1) is afunction of both, the shape (given by θ ) and the conductivity ratio ( β ). In general, thefunction g θβ ( θ , β ) cannot be written down as an analytically closed form solution for anyarbitrary inclusion geometry. For a shape-preserving circular inclusion (d κ s / d s = g θβ ( θ , β ) = − β cos θ / ( + β ) ) migrating due to isotropic diffusion ( m = f θ ( θ ) = [ ] and is plottedin Figure 8.3 (green color). The non-linear decrease in the velocity with 1 /β is evident fromthe above expression. For the same magnitude of applied electric field and initial radius ofthe inclusion, a shape-preserving circular inclusion due to isotropic diffusion (green curve)will migrate faster than a triangular or slit-shaped inclusion induced as a result of two-fold anisotropic diffusion (red curve). Although a similar non-linear decrease with 1 /β isobserved due to anisotropic diffusivity (red curve), the velocity is lowered roughly by afactor of three.In absence of diffusional anisotropy (blue curve), however, a slit has a faster propaga-tion velocity than a circular one (green curve) indicating the amplification of the factor g θβ ( θ , β ) . At the lowest 1 /β (i.e., β = /β ( ≈ − /β . In addition, on comparisonof the red and the blue curve, it appears that the effect of anisotropy f θ ( θ ) is to furtherslow down the velocity of inclusion migration. With the incorporation of misalignment of the fast diffusion sites to the electric field, (cid:36) = ◦ (keeping β = t = τ (cid:48) (Figure 8.4(a)). The upper edge develops a boomerang appearance at t = τ (cid:48) , with theupper vertex growing perpendicular to the field direction. The upper edge of the advancinginclusion reverts to the flat surface at t = τ (cid:48) and, consequently, the inclusion propagateshapter 8. 138preserving the shape.
50 100 150 200 t =10 τ 't =75 τ 't =150 τ 't =225 τ 't =300 τ ' (a)
100 150 200 250 300 t =580 τ 't =665 τ 't =750 τ 't =835 τ 't =920 τ ' (b) x ( λ ' )x ( λ ' ) Figure 8.4: The complex shape dynamics of inclusions at misorientation angle (cid:36) = ◦ during themorphological evolution for the conductivity ratio β = in (a) and in (b). The snapshots ofthe inclusions are shifted upwards in time. The solid red line represents a position of the valley duringevolution. The inclusion for β = shows time-periodic oscillations with rounded hills and valleys. Whilea steady-state inclusion morphology is observed for β = . Another interesting case arises for (cid:36) = ◦ and β = t = τ (cid:48) ). A newprotrusion emerges at the forepart at t = τ (cid:48) , the amplitude of which increases withtime, and shifts towards the left. The amplitude of the protrusion at the rear end decreasessimultaneously. At t = τ (cid:48) , the inclusion reverts back to the shape that was observed at t = τ (cid:48) , implying the completion of a period of wave-like motion on the surface of theinclusion. The cycle is repeated indefinitely thereafter. The time-periodic dynamics can beinferred by studying the temporal evolution of the normalized inclusion perimeter as shownhapter 8. 139in Figure 8.5 (green curve). After the initial transient regime, which is characterized by anincrease of the perimeter, the inclusion undergoes a time-periodic oscillatory state withconstant amplitude.
11 .11 .21 .31 .41 .51 .61 .7 500 1000 1500 2000 2500 p / p t/ τ ' β = 1.0 β = 1.0 ϖ = 75, β = 1.0 Figure 8.5: The evolution of normalized perimeter of inclusions with misorientation angles (cid:36) = ◦ , ◦ , and ◦ for β = . The inclusions undergo time-periodic oscillations. The period of oscillationsdecreases while the amplitude increases with a decrease in (cid:36) . The normalized inclusion perimeter isdefined as the instantaneous perimeter (p) over the initial perimeter (p ). The oscillatory dynamics was found to be dependent on both, (cid:36) and β . This can beunderstood considering the following two cases. Firstly, for the same β , a decrease in the (cid:36) decreases the period as shown in Figure 8.5, which implies accelerated wave propagationon the inclusion surface. In addition, the amplitude of the oscillations, which signifies thelimit of shape variation, increases with a decrease of (cid:36) . The state with higher amplitudeentails greater shape deviations than that of the lower. The highest perimeter corresponds toshape with the valley at the center and two peaks on either side, while the lowest perimeterrelates to a single peak at the rear end. Furthermore, it is to be noted that the time elapsedin reaching the periodic state decreases with a decrease in (cid:36) . Secondly, as presented inFigure 8.6, increasing the conductivity ratio β , for the same (cid:36) increases the amplitude, theperiod of oscillation and the time taken to reach the periodic state simultaneously.All inclusions breakup for misorientation (cid:36) = ◦ (Figure 8.7). Owing to the misalignedfast diffusivity sites, both species flux directions became prominent. This provokes specieshapter 8. 140
11 .11 .21 .31 .41 .51 .6 500 1000 1500 2000 2500 p / p t/ τ ' β = 1.0 β = 1.5 ϖ = 82, β = 3.0 ϖ = 82, Figure 8.6: The evolution of perimeter of inclusions with misorientation angle (cid:36) = ◦ for β = , 1.5,and 3. The inclusions observe time-periodic oscillations. The period of oscillations increases with anincrease in β , in addition, to increase in the amplitude. (a) (b) Figure 8.7: Inclusion morphology for conductivity ratio, β = in (a) and in (b) at the mis-orientation angle (cid:36) = ◦ . The inclusions with surface contours are presented with time, t( τ (cid:48) ). Theincreasing darkness of the contour indicates inclusion evolution. The inclusions are displaced in spacefor better visual inspection. The preferential elongation in traverse direction for β = ruptureslower edge, while the enlargement along the electric field for β = break at the upper. flow in two separate directions, which ultimately leads to a necking instability. However,the morphological evolution strongly depends upon the conductivity ratio, β . For instance,some typical cases for (cid:36) = ◦ are shown in Figure 8.7, where inclusions breach uniquelydepending on the conductivity ratio. The preferential elongation in the traverse direction tothe electric field for β = β =
100 200 300 t =153 τ ' x ( λ ' ) t =177 τ 't =199 τ 't =239 τ 't =273 τ '
100 200 300 t =311 τ ' x ( λ ' ) t =345 τ 't =384 τ 't =441 τ 't =510 τ '
100 200 300 t =15 τ ' x ( λ ' ) t =65 τ 't =89 τ 't =114 τ 't =132 τ ' Figure 8.8: The complex shape dynamics of inclusions at misorientation angle (cid:36) = ◦ during themorphological evolution for the conductivity ratio β = . The snapshots of the inclusions are shiftedupwards in time. For lower (cid:36) = ◦ (Figure 8.8), the high diffusivity regions at the two diametrically op-posite ends are perpendicular to the applied field direction, which migrates faster relativeto the rest of the inclusion. This results in a triangular shape with the apex at the rear end( t = τ (cid:48) ). The flat front undergoes a transition to a boomerang shape ( t = τ (cid:48) ), fol-lowed by a pinch-off at the two ends and ultimately breaks up into three smaller inclusions( t = τ (cid:48) ). All three inclusions experience a similar boomerang morphological transition.It has been shown by Ho [ ] (similar to the expression in Eq. (7.37)) that under isotropicdiffusion a smaller inclusion has a faster migration velocity than a larger one. Although theparent inclusion is larger than the two daughter inclusions, the faster-growing slit tips ofthe parent (due to anisotropic diffusion) relative to the rear end of the daughters, lead to acoalescence as evident in Figure 8.8 ( t = τ (cid:48) ). The present results are in agreement withhapter 8. 142the self-consistent numerical simulations of Cho et al. [ ] , who showed that the inversedependence of velocity on radius is violated due to diffusional anisotropy. A complex inter-action between the broken parts of the inclusions follows thereafter engendering successivebreakup and coalescence. At t = τ (cid:48) in Figure 8.8 the initially circular inclusion has splitinto six smaller ones. In addition, it is evident that the position of the inclusion parts aresymmetrical with respect to the longitudinal (migration) direction. Σ ϖ ( deg ) β Breakup Time Periodic Steady State
Figure 8.9: Morphological map of inclusion migration modes as a function of β and (cid:36) for anisotropictwofold symmetry. The triangular, square and circular points correspond to steady-state, time-periodic,and breakup morphology respectively. The solid-black colored line is a guide to the eye. The effect of (cid:36) and β on the inclusion evolution can be summarized in the form of amorphological map as presented in Figure 8.9. Higher (cid:36) favors a shape-preserving steady-state drift. For the chosen values of χ =
15 and A =
10, the dominant electromigrationforce promotes a slit or triangular-like shape depending on the conductivity contrast. Thesteady-state region diminishes as the conductivity of the inclusion approaches that of thematrix. For intermediate (cid:36) values and conductivity ratio (1 /β ) greater than 0.3, the inclu-hapter 8. 143sion undergoes a time-periodic dynamics. At (cid:36) less than about 70 ◦ , the inclusions becomeunstable and experience a complex cycle of break up and coalescence for all conductivityratios. For steady-state migration, the slit forming propensity of the inclusion changes from beingalong the line to perpendicular to the line as β increases (Figure 8.2). Such perpendiculargrowth of slits transverse to the line is expected to prove fatal to the lifetime of the line;especially for the case of smaller Λ L when the inclusion radius is comparable to the linewidth which enhances the current crowding effect. The results on the steady-state migra-tion (Figure 8.3) are an important extension to the analytical theory of Ho [ ] , whichwas developed for inclusion motion under isotropic diffusion. Compared to a circular in-clusion, the triangular and slit-shaped ones engendered due to anisotropic diffusion haveabout three times lower velocity. Moreover, the two-fold anisotropy in diffusivity lowersthe steady-state velocity approximately by a factor of seven in comparison to the slits orig-inating due to isotropic diffusion. E ∞ E ∞ aa bb cc J EM J EM J EM J EM (a) Stable (b) Unstable Figure 8.10: Schematic of electromigration-induced atomic flux (a) from the apex to the base, and (b)from the base to the apex on the surface of the wedge-type inclusions corresponding to Figure 8.2(c) andFigure 8.8 (t = τ (cid:48) ). The former migration mode leads to steady-state propagation, while the latterinstigates inclusion breakup. The red region highlights the locality of high diffusion. Two kinds of triangular-shaped inclusions were found during the course of evolutiondepending on the value of the misorientation angle as shown in Figure 8.10. First, as inFigure 8.2(c), where the apex is located at the migrating front while the rear end is per-pendicular to the direction of the electric field. Second, in which the shape of the trian-gular inclusion is exactly reversed with the apex forming at the rear end as in Figure 8.8( t = τ (cid:48) ). While the former shape is found to be stable, the latter subsequently disinte-hapter 8. 144grate into a number of daughter inclusions. The stability of the triangular-shaped inclusionscan be understood in terms of the mass flux along the surface (Figure 8.10). In the formercase, electromigration induces mass transport from a to b and a to c, while no mass trans-port takes place along the edge bc as it is perpendicular to the field direction, as shown inFigure 8.10(a). As a result, the inclusion migrates along the line preserving the shape. Incontrast, in the latter case (Figure 8.10(b)) mass transport from b to a and c to a leads toslit-like growth towards line edges. The slit subsequently pinch-off at the thinnest sectiondue to necking instability.The importance of the misorientation angle on the stability of the inclusion as evidentin Figure 8.9 needs to be emphasized. The assumed two-fold anisotropic atomic mobilityresults in two fast diffusing directions. However, the starting condition of a pre-existingcircular inclusion renders the high diffusivity rear end effectively insignificant due to theunidirectional motion of species from the cathode to the anode end. Beyond a criticalmisorientation angle, both the fast diffusivity sites become operative. As a consequence,these ends migrate faster relative to the rest of the surface leading to the onset of instability.The focus of the chapter has been understanding the dynamics of inclusion under elec-tromigration in thin-film metallic conductors. The case of β = [ ] . The case of β = [ ] investigatedsingle layer island dynamics on conducting { } , { } , and { } -oriented substrates,with a slightly different parameterization. While the present chapter is reported for a pa-rameter χ which denotes the ratio of wind force to capillary force and has been held fixed,Kumar et al. [ ] defined the length scale as the radius of the island for which the cap-illary force balances the electron wind force. The island radius measured in units of thedefined length scale was varied. In the present work, this implies varying χ by varying theisland radius while holding the applied electric field constant. The nanowire formation inthe present study obtained from parameter set (cid:36) = ◦ , β = m = χ =
15, thuscorresponds to φ = ◦ , m = R E = (cid:112) π l E of their article, where l E = (cid:112) γ s Ω / q s E ∞ and R E is the square root of island area.hapter 8. 145 The numerical results are presented on the migration of circular inclusion in a { } -oriented single crystal of face-centered-cubic metals under the action of an external electricfield. The simulations predict a rich variety of morphologies, which include steady-state,time-periodic, and inclusion breakup. The amplitude and the frequency of time-periodicoscillations are strongly dependent on the values of conductivity contrast β and misorien-tation angle (cid:36) . Furthermore, higher β promotes a transverse elongation, while similarconductivities lead to a slit-like feature along the direction of the electric field. Finally, amorphological map is constructed delineating the dependence of various migration modeson conductivity contrast and misorientation. Results presented here have important im-plications on void dynamics in interconnects and fabrication of nanostructures of desiredfeatures and dimensions.The focus of the present work has been on inclusion migration along { } planes. Astraightforward extension would be to address the dynamics and morphologies of inclu-sion, migrating along { } and { } planes i.e. four and six-fold diffusional anisotropyrespectively. This study is performed in the Chapter 9. hapter 9Motion of anisotropic inclusions consistof 4-fold and 6-fold symmetries Even though numerical approaches employed for the electromigration-driven inclusionmorphology are ample, there are only a few works focused on the conductivity contrast.For instance, Li et. al. [ ] studied the effect on the stability of the elliptical inclusion, andfurther highlighted the morphological evolution of crack and wedge-shaped inclusions in-fluenced by various conductivity contrasts. However, this literature is devoid of a systematicguideline of a conductivity contrast, in terms of a morphological map. In an attempt, resultspresented in chapter 8 obtain the morphological map for mobility anisotropy with two-foldsymmetrical inclusions, as shown in Figure 8.9. Therefore, in the present chapter, the mor-phological evolution of the higher-order (fourfold and sixfold) symmetrical inclusions areconsidered. It has two objectives: (i) to study the effect of change in the conductivity con-trasts between the inclusion and matrix and (ii) to critically compare the results obtained byfourfold and sixfold symmetrical inclusions with those of isotropic and twofold inclusions.This chapter demonstrates a detailed description and the analysis of the numerical re-sults obtained from phase-field simulations presented in chapter 4 on the electromigration-induced dynamics of fourfold and sixfold inclusions at a constant volume considering m = Table 9.1: Values of parameters considered for the simulation study inclusions in { } and { } crystallographic planes Name of the parameter symbol valueElectric field to capillary force ratio χ Λ L A m fourfold: 2sixfold: 3Misorientation angle (cid:36) fourfold: 0 ◦ to 45 ◦ sixfold: 0 ◦ to 30 ◦ Conductivity ratio β Figure 9.1: Schematic of an inclusion in (100) crystallographic plane in (a) and (111) plane in (b).The directions of dominant species diffusivity are represented by black arrows. The red color plane isthe surface of film perpendicular to the circular inclusions.
The diffusional anisotropy in an inclusion is a result of the distinct fast diffusion locationsin the plane. For instance, see Figure 9.1(a), (100) plane contains four crystallographicdirections [ ] , [ ] , [ ] and [ ] , which are responsible for fast diffusion. Similarly,(111) plane consists of six fast diffusion directions [ ] , [ ¯110 ] , [ ] , [ ] , [ ¯101 ] and [ ] , as shown in Figure 9.1(b). In fact, each planes in { } and { } families containhapter 9. 149four and six of 〈 〉 directions and referred as fourfold and sixfold respectively. Due tothe symmetry of initial inclusion shape, the parameter space of the misorientation (cid:36) canbe reduced to a range from 0 ◦ to 45 ◦ (= ◦ / m ) for fourfold symmetry and from 0 ◦ to30 ◦ for sixfold symmetry, without compromising the features of the inclusion morphology.In the systematic study of fourfold and sixfold symmetrical inclusions, the morphologicallycomplex dynamics of inclusion shape is obtained as a function of the misorientation angle (cid:36) and the conductivity ratio β = σ mat /σ icl , where σ mat denotes conductivity of matrix and σ icl represents the conductivity of the inclusion.Several inclusion morphologies can be observed during its propagation underelectromigration-induced surface diffusion. Figure 9.2(a) and (b) present the morphologi-cal map of different migration modes in β and (cid:36) plane obtained by the phase-field simula-tions for anisotropic fourfold and sixfold symmetries of species diffusion respectively. Theinclusions assume various morphologies such as a breakup, steady-state, time-periodic, andzigzag oscillations for sixfold symmetry, while only the former three migration modes can beidentified for the fourfold symmetrical inclusions. Dynamics of the different regions withina particular migration mode in the morphological map can be recognized by the distinctcharacteristics during propagation. These aspects of the morphological map are describedextensively in the forthcoming paragraphs. At low (cid:36) , inclusions breakup can be observed for all values of conductivity contrast infourfold symmetry, as compared to only for conductivity contrast for sixfold symmetry. Aninclusion breaks up due to two specific ways: retention and elongation. Both cases can beobserved for fourfold symmetry, while only the former is prominent for sixfold symmetry.As a representative case, the characteristics of an inclusion migrating for fourfold sym-metry at (cid:36) = ◦ is described. For all β , the inclusions of (cid:36) = ◦ and m = β <
6) separates atthe first valley formation (Figure 9.3(a)) due to insufficient species transport between twohills. Secondly, the wavy increase in perimeter (Figure 9.3(c)) demonstrates breakup dueto retention, where an inclusion of higher conductivity ratio ( β >
6) surpasses the firsthapter 9. 150 (a)(b)(a)
Figure 9.2: Morphological map of inclusion migration modes as a function of β and (cid:36) for anisotropicfourfold (m = ) symmetry in (a) and sixfold (m = ) symmetry in (b). The triangular, square, pen-tagonal, and circular points correspond to steady-state, time-periodic, Zigzag oscillations and breakupmorphology respectively. The solid-black colored lines are a guide to the eye to differentiate betweendifferent modes of migration, while the dotted lines separate distinct characteristics of a particular mi-gration mode. Semi-log plot is employed for the morphological map of the sixfold symmetry for thebetter visual representation. hapter 9. 151 (a) (b) p / p t/ τ′ Inclusion breakupInclusion breakup (c)
Elongation breakup (m = 2, = 1 and β ϖ = 0)Retention breakup (m = 2, β ϖ = 0)=10000 and
Figure 9.3: Representative dynamics of inclusion breakup due to elongation in (a) and retention in(b). The inclusion breakups are shown for fourfold inclusion (m = ) at misorientation angle (cid:36) = ◦ during the morphological evolution at conductivity ratio β = and 10000 respectively. The inclusionswith surface contours are presented with time, t ( τ (cid:48) ) . The inclusions are displaced in space for bettervisual inspection. (c) represents evolution of the inclusions perimeter (normalized by initial inclusionperimeter) until breakup. valley without breakage (Figure 9.3(b) at t = τ (cid:48) ). Consequently, the inclusion attendsseveral hills and valleys during its propagation before breakage. On increasing β , the tan-gential component of the electric field along the inclusion surface becomes more prominent,which encourages to maintain its traverse elongation. Due to that, the species diffusivityreduces at the last hill of the rear end of the inclusion. Also, the inclusion continues to formnew hills and valleys, with maintaining last hill intact. Which leads to break up induced bythe retention of species at the last hill (in Figure 9.3(b) at t = τ (cid:48) ).hapter 9. 152
200 300 x ( λ ′ ) t=634 τ 't=697 τ 't=760 τ 't=823 τ 't=886 τ 't=949 τ 't=1012 τ ' (a)
400 500 600 x ( λ ′ ) t=751 τ 't=828 τ 't=905 τ 't=982 τ 't=1059 τ 't=1136 τ 't=1213 τ ' (b) x ( λ ′ ) t=2489 τ 't=2585 τ 't=2681 τ 't=2777 τ 't=2873 τ 't=2969 τ 't=3065 τ ' (c) Figure 9.4: The time-periodic shape dynamics of inclusions from crawling in (a) to gliding in (b) and(c). The inclusions are shown for fourfold symmetry (m = ) at misorientation angle (cid:36) = ◦ duringthe morphological evolution of an inclusion of conductivity ratio β = in (a), 6 in (b), and 10000in (c). The snapshots of the inclusions are shifted upwards in time. The solid red line is an attempt torecognize inclusion gliding by locating the position of the valley during evolution. The inclusion withequal conductivity to the matrix shows no appreciable gliding. The inclusion of higher conductivitycontrast shows relatively more gliding compared to the lower one. It is important to note that, when the fast diffusion sites of the 2-fold ( (cid:36) = ◦ ) and6-fold ( (cid:36) = ◦ ) inclusions align to the electric field, then a steady-state morphology canbe observed. Contrarily, inclusions of the 4-fold undergo inclusion breakups when the fastdiffusion sites align the electric field ( (cid:36) = ◦ ). Reason for this can be explained by the posi-tion of the other fast diffusion sites which are not aligned to the electric field. There are twofast diffusion sites in the 4-fold inclusions exactly perpendicular to the electric field, whichare not the case for 2-fold and 6-fold inclusions. Small deviations in species flux might leadto broken symmetry with the axis of the external electric field spontaneously. Furthermore,this slight change in the alignment of the axis instigates one of the fast diffusion sites inthe perpendicular direction to the electric field slightly ahead compared to its diametricalcounterpart. As a consequence, this fast diffusion site became favored for species diffusion.Which in turn alters the direction of the inclusion propagation altogether and eventuallyleads to breakups. In fact, when the fast diffusion sites are perpendicular to the electricfield ( (cid:36) = ◦ for all cases), then most likely the inclusions exhibit breakup.hapter 9. 153 The intermediate values of (cid:36) promotes time-periodic oscillations for both the symmetries(blue regions in Figure 9.2). Only low conductivity contrast ( β <
Change of position of a particular valley (red lines in Figure 9.4) can be utilized as a mea-sure of gliding during evolution. The inclusion with no conductivity contrast ( β =
1) re-flects crawling motion due to a negligible change in position of a particular valley, while β = t = τ (cid:48) in Figure 9.4(a) and t = τ (cid:48) in Figure 9.4(c), different evolution-ary pathways are observed during the motion. For that purpose, the inclusion evolution forboth types of time-periodic oscillations at the end of their respective cycles are shown inFigure 9.5. The complete elimination of the last hills (for β = β = t = τ (cid:48) in Figure 9.5(a). This behavior is demonstrated for β = β = β irrespective of the type of motion.hapter 9. 154 (a) (b)
977 980984 987991 994998 10011005 10081012
Figure 9.5: (a) Crawl and (b) Glide time-periodic shape dynamics of fourfold inclusions at misorien-tation angle (cid:36) = ◦ during the morphological evolution at conductivity ratios β = and 10000respectively. The inclusions with surface contours are presented with time, t ( τ (cid:48) ) . The inclusions aredisplaced in space for better visual inspection. Furthermore, the evolution of perimeters of crawling and gliding inclusions are shownin Figure 9.6(c) and (d) respectively. On the one hand, the increase in the conductivitycontrast increases the amplitude and the period of crawl. On the other hand, the increasein the conductivity contrast increases the period, while decreases the amplitude of glide.In addition, time to reach the time-periodic cycle is approximately equal for various β inthe crawling dynamics, while it increases with β for the gliding motion. A similar trend isobserved for various β of the sixfold symmetrical inclusions for both types of motion.Next, the effect of misorientation in the fourfold and sixfold symmetrical inclusionson the different types of motion are highlighted in Figure 9.7. A completely analogousbehavior is observed between the fourfold and sixfold inclusions. For instance, amplitudeof crawl motion of fourfold inclusion depreciates and period enhances with (cid:36) , which issimilar to the sixfold inclusions. Similarly, the akin trend is observed between the fourfoldand sixfold inclusions during gliding dynamics. It is important to note that the inclusionsat highest (cid:36) in fourfold (blue curves in Figure 9.7(a) and (b)) and in sixfold symmetry(green curves in Figure 9.7(c) and (d)) show oscillations of highest periods compared totheir counterparts. This behavior can be rationalized from the location of the time-periodicregion in the morphological map. Meaning, steady-state morphology is situated above thetime-periodic region as shown in Figure 9.2. Steady-state morphology can be perceived astime-periodic oscillation with an infinite period.hapter 9. 155 p / p t/ τ′ p / p t/ τ′ (c) (d)(a) (b) Crawl time-periodic ( m = 2, ϖ = 30) = 1.0 and Crawl time-periodic ( m = 2, ϖ = 30) = 1.5 and β Crawl time-periodic ( m = 2, ϖ = 30) = 3.0 and β Crawl time-periodic ( m = 2, ϖ = 30) = 1.0 and Crawl time-periodic ( m = 2, ϖ = 30) = 1.5 and β Crawl time-periodic ( m = 2, ϖ = 30) = 3.0 and ββ Glide time-periodic ( m = 2, ϖ = 30) = 4.5 and Glide time-periodic ( m = 2, ϖ = 30) = 6.0 and β Glide time-periodic ( m = 2, ϖ = 30) =10000 and ββ Glide time-periodic ( m = 2, ϖ = 30) = 4.5 and Glide time-periodic ( m = 2, ϖ = 30) = 6.0 and β Glide time-periodic ( m = 2, ϖ = 30) =10000 and ββ V ( λ ′ / τ ′ ) t/ τ′ β V ( λ ′ / τ ′ ) t/ τ′ Figure 9.6: Effect of conductivity contrast during the morphological evolution of crawling and glidingtime-periodic oscillations. The inclusions are shown for misorientation angle (cid:36) = ◦ and variousvalues of conductivity contrast β . The top row corresponds to the centroid velocity of the inclusions,while the bottom row shows normalized perimeter as a function of the crawling motion in the leftcolumn and the gliding dynamics in the right column. The dotted lines in the upper panel of the graphare mean centroid velocity of the respective solid curves. Based on number of valleys during the evolution, the time-periodic oscillations can be dis-criminated between the 1-cycle or 2-cycle as shown in Figure 9.8. During evolution, a 1-cycle inclusion commutes between no valley point in one part of the cycle (near t = τ (cid:48) in Figure 9.8(a)) to one valley in the other part, while between one to two valleys (near t = τ (cid:48) in Figure 9.8(b)) in 2-cycle. Evolution of perimeter and number of valley pointsare displayed in Figure 9.8(c) and (d) respectively. From the comparison, it is evident thatnumber of valleys is directly associated with the value of the perimeter. Therefore, pres-ence of two valley points during oscillations certainly increases the perimeter as shown inFigure 9.8(c).Difference in dynamics of 1-cycle and 2-cycle can be further clarified from the influ-ence of β . The evolution of perimeter for 1-cycle and 2-cycle time-periodic oscillations arehapter 9. 156 p / p t/ τ′ p / p t/ τ′ (a) (b)(c) (d) Crawl time-periodic ( m = 3, ϖ = 5) = 1.0 and Crawl time-periodic ( m = 3, ϖ = 10) = 1.0 and β Crawl time-periodic ( m = 3, ϖ = 15) = 1.0 and ββ Glide time-periodic ( m = 3, ϖ = 5) = 6.0 and Glide time-periodic ( m = 3, ϖ = 10) = 6.0 and β Glide time-periodic ( m = 3, ϖ = 15) = 6.0 and ββ Glide time-periodic ( m = 2, ϖ = 26) = 6.0 and β Glide time-periodic ( m = 2, ϖ = 30) = 6.0 and β Crawl time-periodic ( m = 2, ϖ = 26) = 1.0 and β Crawl time-periodic ( m = 2, ϖ = 30) = 1.0 and β p / p t/ τ′ p / p t/ τ′ Figure 9.7: Evolution of the perimeter as a function of (cid:36) . The top row corresponds to fourfold, whilethe bottom row represents sixfold symmetrical inclusions. The left column shows results for crawling,while right column associates gliding during time-periodic oscillations. displayed in Figure 9.9(a) and (b) respectively. Increasing β enhances the amplitude ofperimeter alongside the increment in the period for 1-cycle oscillations, which is contraryto 2-cycle. Furthermore, a significant disparity exists at the mean perimeter. The meanperimeter is approximately equal for all cases of 1-cycle, while the value of mean perimeterincreases with β for 2-cycles. For all values of conductivity contrast, the inclusions of higher misorientation in fourfoldsymmetry and sixfold symmetry (red regions in Figure 9.2) obtain steady-state morpholo-gies after initial adjustments as shown in Figure 9.10. Faceted-wedge and seahorse mor-phologies are observed for sixfold symmetry, while only a former case is observed at four-fold symmetry. A representative morphological evolution of faceted-wedge and seahorsepattern is shown in Figure 9.10(a) and (b) respectively. The inclusion perimeter graph (inFigure 9.10(c)) shows a monotonously increasing perimeter for faceted-wedge shape, whilehapter 9. 157
200 300
200 300 x ( λ ') t= τ 't= τ 't= τ 't= τ 't= τ 't= τ 't= τ ' (a) t= τ 't= τ 't= τ 't= τ 't= τ 't= τ 't= τ ' x ( λ ') (b) p / p t/ τ ′ β = 3 and
012 500 1000 1500 2000 N o . o f V a ll e y s t/ τ ′ (c)(d) m = 2, ϖ m = 2, ϖ = 15)= 15) β = 1 and m = 2, ϖ m = 2, ϖ = 15)= 15) β = 1 and β = 3 and Figure 9.8: Inclusion dynamics of 1-cycle time-periodic in (a) and 2-cycle time-periodic in (b). Theinclusion morphologies are shown for fourfold inclusion at misorientation angle (cid:36) = ◦ at the con-ductivity ratios β = and 3 respectively. The snapshots of the inclusions are shifted upwards in time.(c) represents the evolution of the normalized inclusions perimeter and (d) shows the number of valleys(locations of negative curvature) during the inclusion propagation. a wavy behavior is observed for seahorse-shaped inclusions.Comparison of the steady-state morphologies across the isotropic and anisotropic(twofold, fourfold, and sixfold) cases are conducted on the inclusions propagating withmaintaining symmetricity along the electric field during migration. After the inclusions at-tain an invariant shape, the steady-state velocities of the centroid for fourfold and sixfoldsymmetries are obtained from phase-field simulations for different values of β as shown inFigure 9.11 (blue and green curves respectively). For comparison, the velocity of a shape-preserving circular inclusion (pink curve) is derived from Ho’s analytical formula similar toEq. (7.37). Along with, velocities of inclusions migrating under isotropic diffusion ( m = m =
1) obtained from phase-field simulations of chapters 7 and 8are presented.On the one hand, the morphologies at fourfold symmetry reveal progressively incre-hapter 9. 158 p / p t/ τ ′ m = 2, = 1.5 and = 15) ϖ β m = 2, = 3.0 and = 15) ϖ β m = 2, = 3.5 and = 15) ϖ β p / p t/ τ ′ m = 2, = 1.5 and = 30) ϖ β m = 2, = 3.0 and = 30) ϖ β m = 2, = 3.5 and = 30) ϖ β (a) (b) Figure 9.9: Evolution of perimeter for (a) 1-cycle and (b) 2-cycle time-periodic oscillations as functionof β . The inclusion morphologies are shown for fourfold inclusion at misorientation angle (cid:36) = ◦ and ◦ respectively. The dotted lines in the graph are mean perimeter of a complete period of a respectivesolid curves. ment in the width of the inclusion and decrement in the length with the increase in theconductivity contrast β , in addition to the increase in the centroid velocities, as shown inthe blue graph. On the other hand, a sudden shrinkage in the length (along with expansionin width) and decrement in the velocity of the inclusion from conductivity contrast β = β = β = β = β = Zig-zag oscillations are observed only for sixfold symmetry with lower misorientation andlower conductivity ratio (1 /β ≈ m = β =
6, and (cid:36) = ◦ . In contrary to the time-periodicoscillations, where a straight edge on one side remains permanently impervious, whilehills and valleys on the other side undergo shape changes during propagation. Zigzaghapter 9. 159 (a)(b) (c) p / p t/ τ ′ Faceted-wedge ( m = 3, β = 1 and ϖ = 20)Seahorse ( m = 3, β = 10000 and ϖ = 20) Figure 9.10: Steady-state morphological evolutions of faceted-wedge in (a) and seahorse pattern in(b). The inclusion dynamics are shown for sixfold symmetry at misorientation angle (cid:36) = ◦ andconductivity ratios β = and 10000 respectively. The inclusions with surface contours are presentedwith time, t ( τ (cid:48) ) . The inclusions are displaced in space for better visual inspection. (c) represents theevolution of the normalized inclusions perimeter until it attains equilibrium shape. oscillations are a result of the facet associated with the straight edge changes from theupper to the lower facet and reverses during evolution. The species diffusion from rear-endprioritize the formation of a rounded end (Figure 9.12(a2) from t = τ (cid:48) to 1500 τ (cid:48) )before further elongation of straight edge. The straight edge undergoes an alteration afteran undefined time period. Variation of the perimeter is demonstrated in Figure 9.12(b).The zig-zag pattern was reported previously in the homogeneous substrate in Ref. [ ] .Dotted lines in Figure 9.12(a) are the preferential facet directions obtained from thecontinuum theory [ ] . A facet y = mx , oriented at an angle θ with the per-pendicular direction to the electric field is considered for the stability analysis, as shown inhapter 9. 160 V ( λ ′ / τ ′ ) β anisotropic 2-fold PFManisotropic 4-fold PFManisotropic 6-fold PFMisotropic PFMisotropic analytical Figure 9.11: The velocities of the centroid of steady-state inclusions for twofold anisotropy (m = ), (cid:36) = ◦ (in red graph), fourfold anisotropy (m = ), (cid:36) = ◦ (in blue graph), sixfold anisotropy(m = ), (cid:36) = ◦ (in green graph), and for isotropic (m = ) surface diffusion (in the black graph) asa function of the conductivity ratio β . The inset of images correspond to steady-state morphologies ofrespective m and β . The magenta curve corresponds to a steady-state velocity of the circular inclusionof isotropic surface diffusivity, derived from [ ] considering the diffusion coefficient D s . Evidently, β governs the inclusion shapes and the steady-state velocities. Figure 9.13. Using the relation from Eq. (7.3) and d x = − d s sin θ , the mass conservationof surface flux can be expressed as, d y d t = − Ω d J s d x . (9.1)This governing equation relates the height of the facet during inclusion propagation. Sub-stituting the Nernst-Einstein relation (8.2) into the governing equation 9.1, the resultantexpression can be written as,d y d t = D s δ s E ∞ eZ s k B T (cid:168) β + β f θ ( θ ) cot θ + d f θ ( θ ) d θ ( + cot θ ) / d y d x − Ω γ s eZ s E ∞ ( + cot θ ) d y d x (cid:171) . (9.2)where cot θ = − d y / d x . A dispersion relation is derived from Eq. (9.2) to predict the growthhapter 9. 161 Zigzag oscillations ( m = 3, β = 6 and ϖ = 2) τ ' p / p (a1) (a2)(b)300 400 500 600 600 700 800 900x ( λ ')t=700 τ 't=800 τ 't=900 τ 't=1000 τ 't=1100 τ 't=1200 τ ' t=1300 τ 't=1400 τ 't=1500 τ 't=1600 τ 't=1700 τ 't=1800 τ 'x ( λ ') Figure 9.12: (a) Zigzag oscillations of the sixfold symmetrical inclusion at misorientation angle (cid:36) = ◦ and conductivity ratio β = . The snapshots of the inclusions are shifted upwards in time. Thegray dotted lines represent preferential facet orientations predicted by the linear stability theory. (b)represents the evolution of the normalized inclusions perimeter. or decay rate of the small perturbations on the facet expressed as, ω d ( k d ) = D s δ s E ∞ eZ s k B T (cid:168) − β + β f θ ( θ ) cot θ + d f ( θ ) d θ ( + cot θ ) / k d − Ω γ s eZ s E ∞ ( + cot θ ) k d (cid:171) , (9.3)For a facet to be unconditionally stable, the perturbation frequency ω d ( k d ) should be neg-hapter 9. 162 θ E t E ∞ Facet dsdx dy xy Figure 9.13: Schematic of a facet of an inclusion. Thick black line indicates the facet, which is subjectedto an external electric field E ∞ . The projection of electric field on the surface of facet is E t . ative. As the second term in Equation 9.3 is always negative, therefore the sufficient con-dition for the facet stability is f θ ( θ ) cot θ + d f θ ( θ ) d θ >
0. (9.4)Using this expression, the preferential facet orientations θ ∗ = θ is derived for the requiredanisotropy parameters. For instance, these orientations are displayed as the dotted lines inFigure 9.12(a). It is evident that the inclusion morphology precisely trails the preferentialfacet directions. The numerical results are presented on the migration of circular inclusion in { } and { } -oriented crystallographic planes under the action of the external electric field. Mor-phological maps are constructed in the plane of misorientation angle and conductivity con-trast based on numerical results. The steady-state, time-periodic oscillations, zig-zag, andinclusion breakups are observed. In addition, these various migration modes can be furtherclassified as stable faceted or seahorse morphology, crawling-based or gliding-based time-periodic oscillations, and elongation or retention breakups. Furthermore, the influence ofvariation in conductivity contrast and misorientation on the dynamics of the time-periodicoscillations are discussed. Finally, the steady-state dynamics obtained with the fourfold andsixfold symmetrical inclusions in the present study are critically compared with the twofoldsymmetry, isotropic analytical and numerical results.The simulations predict reach a variety of morphologies during propagation. Someof the morphologies are already reported in the literature, such as steady-state facetedhapter 9. 163inclusions [ ] , time-periodic crawls by formation and destruction of hills [ ] , zigzag-type motion [ ] , and inclusion breakups due to elongation [ ] . Other morphologiesare introduced to the scientific community for the first time in the present chapter, such astime-periodic glides without formation and destruction of hills, the breakup of inclusionsdue to species retention at the last valley, and seahorse-like inclusions. art VConclusions and Future Directionshapter 10Conclusions and future directions Diffusion-driven processes during energy conversion and transmission of electric field arehighlighted. Particularly, two-phase coexistence in cathode particles of lithium-ion batteriesand electric-field induced inclusion migration in metallic conductors are investigated in thepresented work. Phase-field models are formulated to obtain an enhanced understandingof the physical phenomena. Numerical results show that phase-field methods can efficientlyapprehend the essential thermodynamics in inclusion migration in metallic conductors andtwo-phase coexistence in cathode particles. In addition, the results enhance the currentunderstanding of the phenomena. In the following paragraphs, findings from numericalstudies and their applicability are highlighted along with possible future directions.
The presented method in Chapter 3 based on classical Cahn-Hilliard is capable to modelthe evolution of two-phase coexistence. In finite-difference framework, the derived modelis parallelized through message passing interface (MPI) algorithms. The increase in meshresolution conveys the convergence of the presented method, which is validated with abenchmark study. The reason for the two-phase coexistence, the presence of the miscibilitygap is discussed in detail with the help of free energy density and the chemical potential.
Smoothed boundary method enables the modeling of particles with almost any geometry.A simulation study is demonstrated on a cathode particle during discharge in Chapter 5,which is applicable to a constant flux boundary condition at the particle surface. Effectsof the particle geometry on concentration evolution are explored numerically. Numerical167hapter 10. 168results show that the phase segregation starts in the vicinity of the regions with highercurvature. This is due to the fact that particle surface with a higher curvature preferentiallyaccumulates lithium species and exhibit the spinodal decomposition earlier to other regionsof the surface, with the application of constant flux at the particle boundary. Therefore, theelliptical particle with a higher aspect ratio is subjected to the onset of the phase separation,prior to the lower particle.Microstructructure modeling on cathode particles of lithium-ion batteries can be helpfulfor better understanding and tailoring the material properties influencing the mechanismsunderneath the actual process. Specifically, numerical studies are performed on C-rate,mobility, temperature and the energy parameter. It has been depicted that the C-rate andthe species mobility related to deposition and transportation rates respectively, while theoperating temperature and free energy parameters are regulating the miscibility gap. Theresults from the performed study can be utilized to accelerate or decelerate by tuning theprocess of phase separation. For instance, higher values of energy parameter and C-rateaccelerate the phase separation, while higher values of temperature and mobility suppressthe process.While numerical study in Chapter 5 is focused on constant lithium flux at the particlesurface, the model is certainly not restricted to it. A straightforward extension is to considertime, concentration, or spatial-dependent flux at the particle surface in the form Neumannflux boundary condition expressed in Eq. (3.19). In addition, a study can be extendedto numerically investigate phase separation mechanism under the Dirichlet concentrationboundary condition (Eq. (3.20)) at the particle surface [ ] . Finally, the described methodcan be generalized to incorporate mechanical effects. The development of stresses, due todiffusion gradients in the particle pose a threat to the life expectancy of the battery andmay ultimately lead to crack formation [ ] and permanent degradation of the material. A numerical study performed in Chapter 6 can be employed for the performance predictionof tortuous structures based on diffusional properties. For instance, the SOC evolutionof the obtained results for the porous structures of mono-disperse particles are comparedwith the bulk-transport and surface-reaction limited theories. Results show that the smallermono-disperse particles tend to surface-reaction limited theory, while the larger ones tendtowards bulk-transport limited theory of the planar electrode. Thus, the lithium transportrate can be efficiently controlled through an appropriate selection of electrode particle sizes.Similarly, an extension of numerical findings presented here can be considered to pre-pare a guideline which tunes the transportation rates by selecting the appropriate values ofhapter 10. 169structural descriptors, such as mono-disperse particle size R , porosity ρ , and tortuosity τ . Inaddition, the porous structures are characterized numerically in the current investigationsas described in Appendix C, which allows to control the desired geometrical properties ofelectrode microstructure. This procedure in combination with the presented model can beemployed as a numeric prototype for preliminary investigations and testing, which mightreduce the experimental efforts.A number of interesting directions can be pursued hereafter. For instance, focus ofthe present study is on the effect of various geometrical parameters such as monodisperseparticle size, porosity, and tortuosity on the charge dynamics. A straightforward extensionis to investigate other crucial parameters such as electrode overpotential (value of c ps ),electrode thickness, size polydispersity, and shape of the particles. These parameters havephysical significance, which should be the subject of future investigations. In addition,a constant concentration condition is employed at the separator in the form of Eq. (3.23).This can be extended to be a time, local concentration, and / or spatially-dependent function.Furthermore, numerical investigations of Neumann flux boundary conditions in the formof Eq (3.24) are perhaps interesting future endeavors. Electromigration-induced morphological evolution of inclusions (voids, precipitates, andhomoepitaxial islands) is considered for a numerical study, which has acquired recentscrutiny for the efficient design of the interconnects and surface nanopatterns. In Chap-ter 4, a phase-field method is derived from the Cahn-Hilliard equation coupled with theLaplace equation of the electric field. This allows the account of different electrical conduc-tivities of inclusions and the matrix. Distinct conductivities alter the local distribution ofthe electric field in the vicinity of the inclusion. Longitudinal component of the electric fieldenhances up to two times, while the perpendicular component forms a quadrupole patternaround the inclusion. Furthermore, the developed model is employed to investigate diffu-sional isotropy and anisotropy of inclusions migrating in { } , { } and { } planesof face-centered-cubic crystal, which resembles twofold, fourfold, and sixfold symmetryrespectively. Phase-field study in Chapter 7 is intended to study isotropic inclusions, which successfullycorroborate the findings from the linear stability analysis. In addition, the numerical re-sults can estimate the shape of the inclusion after diversion from circular shapes, whichhapter 10. 170is arduous to obtain from analytical theories. The transition of a circular inclusion to afinger-like slit is elucidated. Following an initial transient regime, the inclusion attains anequilibrium shape with a narrow parallel slit-like body, which contains a circular rear end,and a parabolic tip. It is identified that the subsequent drift of the inclusion is characterizedby shape invariance. In addition, selection of steady-state shape and velocity are implicitlyrelated. However, the steady-state slit width and velocity are determined to scale with theapplied electric field as E − / ∞ and E / ∞ , respectively.Velocity of a slit with half-slit width u modifies by a factor of ξ compared to circularinclusion with a radius of u . In addition, the slit characteristics are significantly influencedby three non-dimensional parameters, namely, the ratio of conductor line width to inclusionradius, Λ L = w / R i , velocity discrepancy coefficient ξ , and competition between the electricfield and capillarity, η . Effect of these parameters on slit profiles are analyzed and theresults obtained from phase-field simulations are critically compared with sharp-interfacesolution. These results reveal a good agreement across all values. Repercussions of thestudy, in terms of prediction of inclusion migration in the flip-chip Sn-Ag-Cu solder bumpsand the fabrication of channels with desired micro / nanodimensions, are discussed.The presented results are mostly focused on the case where an initial circular void trans-forms into a single slit. It is also possible for an initially circular inclusion to break up intoa number of independent slits, as shown in Figure 7.4. The reason for this is due to higherelectromigration flux. For instance, consider the constricted concave neck region in Fig-ure 7.9. Capillary flux from both neighboring convex regions is able to flatten the neck.But, higher electromigration flux transports material from the right to the neck and drain-ing it from the left. Eventually, this mass transport causes a thinning of the neck, whichwould lead to a pinch-off event. The daughter slit subsequently propagates independentlyfrom the parent inclusion. The trailing parent inclusion may again undergo a shape bi-furcation, ejecting a relay of daughter slits each of which may or may not propagate withthe same width and velocity. Consequently, coalescence and coarsening become importantevents. Preliminary phase-field simulations indicate the possibility of such events, as shownin top insets of Figure 7.4. However, a detailed exposition requires further investigationsand should be reported in future publications. Several inclusion morphologies are observed during its electromigration-induced motionunder anisotropic surface diffusion. Morphological maps are constructed from microstruc-tural evolution obtained from phase-field study. These are categorized as: Firstly, steady-state inclusions are the stable faceted morphologies attained after initial alterations. Sec-hapter 10. 171ondly, time-periodic oscillations are the repetition of the sequence of morphology with afixed period. Thirdly, in contrast to time-periodic oscillations where one straight edge re-mains unaffected, zig-zag oscillations are a result of the facet associated with straight edgechanges from upper to lower facet and reverses repeatedly. Finally, inclusions breakup arethe result of the rupture or disintegration of the inclusion while propagating under theexternal electric field.
Chapter 8 exhibits a study to delineate the effect of conductivity con-trast ( β ) and misorientation of surface diffusion anisotropy ( (cid:36) ) on inclusion propagatingin a { } -oriented single crystal of face-centered-cubic metals. When high diffusivity sitesalign perpendicular to the electric field ( (cid:36) ≈ ◦ ), then the inclusions are likely to breakupfor all conductivity contrast. Higher values of (cid:36) establish a steady-state morphology,while intermediate (cid:36) exhibits time-periodic oscillation. Formation of various migrationmodes is explained in detail with plausible reasons. For instance, inclusion morphologiesfor a steady-state are compared with time-periodic oscillations in Figure 8.4 and inclusionbreakup in Figure 8.10.The conductivity ratio of β is found to be influential in determining the migration mode,and more importantly, the shapes while propagation. For instance, the slit forming propen-sity of the inclusion changes from being along the line to perpendicular to the line as β increases for steady-state inclusions. In addition, the results on steady-state migration(Figure 8.3) are an important extension to the analytical theory of Ho [ ] , which wasdeveloped for inclusion motion under isotropic diffusion. In steady-state morphologies,compared to a circular inclusion due to isotropic diffusion, the triangular and slit-shapedinclusions instigated due to two-fold anisotropic diffusion have about three times lowervelocity. A phase-field numerical study is presented in Chapter 9to investigate inclusions migrating along { } and { } planes of face-centered-cubiccrystal. In particular, the emphasis is laid on understanding the effect of conductivity con-trast ( β ) between the inclusion and the matrix. These results demonstrate that the elevated β increases traversal lengths of steady-state inclusions. In addition to that, time-periodicoscillations of inclusions transform their crawling motion to gliding kinetics at elevated β . Furthermore, the breakup of inclusions due to elongation substitutes the retention ofspecies at a higher β . Finally, inclusions of only lower β undergo zigzag motion, while theircounterparts of higher values break apart.Significance of β can be further extended by adjusting the conductivity contrast duringinclusion migration. Electrical conductivity is a material property, which may vary withhapter 10. 172 -0.20.00.20.40.6 0 500 1000 1500 2000 t ( τ ' ) V ( λ ' / τ ' ) (a)(b) p / p β = 10000 β = 10000 to 1 t ( τ ' ) β = 1 Vy, β = 1 Vy, β = 10000 Vy, β = 10000 to 1 Vx, β = 10000 to 1 Vx, β = 10000 Vx, β = 1m = 3, ϖ = 10, andm = 3, ϖ = 10, andm = 3, ϖ = 10, and Figure 10.1: The complex shape dynamics of sixfold inclusion at misorientation angle (cid:36) = ◦ duringthe morphological transformation. (a) shows evolution pathway of inclusion on the perimeter curve asa function of time for conductivity ratio, β = , , and the alteration of the conductivity ratio from β = to after t = τ (cid:48) . The inset of images show island morphology during propagation. (b)shows evolution history of the velocity components V x aligned and V y transverse to the external electricfield. external parameters, such as temperature [ ] . Therefore, understanding the effectof conductivity alteration during the inclusion migration has scientific importance. Fig-ure 10.1 depicts morphologies for β = t = τ (cid:48) , the inclusion revamps its longitudinal velocity im-mediately to match with the newly developed situation. This is in accordance with thehapter 10. 173inspection during the change in the electric field, as shown in Figure 7.10. Eventually, theinclusion catches up, in perimeter and shape (green), with the propagation of its counter-part, which is fully developed in the homogeneous system (red). Note that both inclusions(red and green) are identical in shape during its steady-state propagation. However, this istrue only for a few cases. For instance, segregated inclusions may not reassemble further.In addition, the inclusion may be in a strange position and reestablishment may lead tofurther deviation and breakup. These investigations could be performed in future work.A number of interesting directions can be pursued hereafter. The nanostructures can bestabilized by reducing the temperature (so that the species diffusivity is inhibited) when thedesired morphology has been achieved. Other cases would correspond to heteroepitaxialislands i.e. island and the substrate of dissimilar metals. However, effect of misfit strain hasto be additionally accounted in such cases to faithfully capture the island dynamics similarto the front tracking simulations of Dasgupta et al. [ ] .Other driving forces such as thermal gradient can act in conjunction with or counteractthe effect of electromigration [ ] . The Soret effect can be modeled by appendinga temperature gradient-dependent species diffusion term in Eq. (4.8) and supplementingit by a temperature Laplace equation. Thermomigration has been a subject of a number ofprevious studies [ ] . Interestingly boomerang-like topological transition and sub-sequent splittings as in Figure 8.8 has also been reported during thermomigration [ ] .A combined understanding of thermomigration and electromigration, however, is still ininchoate stages.Single crystals have been considered in the present study. Most commercial intercon-nects are polycrystalline with grain boundary networks. Migrating inclusions can either bepinned to or penetrate into the grain boundary [
75, 90 ] . Modeling effort in this directionwould require a coupled solution to the Cahn-Hilliard, Allen-Cahn and Laplace equationsas employed in Refs. [ ] . In addition, isolated inclusions are investigated in thepresent work. Generally, the metallic conductors are consist of several inclusions simulta-neously. The subsequent events of coalescence and splitting between these might lead toshrinkage and / or growth of the inclusions. These issues should be a subject of future work.Inclusion morphologies and dynamics can be tailored in order to form various interest-ing nanopatterns by monitoring the conductivity contrast. Even though a fixed value of theelectric field is employed in the present work, numerous morphologies can be observed.Variety of morphologies can be further enhanced by regulating the strength of the electricfield or the size of the inclusion. In addition, the incorporation of other fields, such aselasticity flourishes the richness of patterns. Effect of these fields could be addressed inupcoming works. art VIAppendicesppendix AInterface profiles At equilibrium, the partial derivative of the free energy functional expressed in Eq. (4.1)should vanishes. Consequently, δ F ( c , ∇ c ) δ c = ∂ f ( c ) ∂ c − κ ∇ · ∇ c =
0. (A.1)Assuming that the spatial dependency of the order parameter c is restricted to X-direction,writing one-dimensional form of the above equation, ∂ f ( c ) ∂ c − κ d c d x =
0. (A.2)Multiplying both sides of Eq. (A.2) by d c / d x and further manipulations would realize, theequation of the form, d c d x = (cid:118)(cid:116) κ (cid:198) f ( c ) . (A.3)This expression is referred as Euler-Lagrange relation. To obtain interface profile of double-well free energy density, consider a simplified form, f dw ( c ) = c ( − c ) . Plugging this intoEq. (A.3) and taking integral, the interface profile is expressed as, c = + (cid:18) tanh x (cid:112) κ (cid:19) . (A.4)This implies that the well-type free energy assumes hyperbolic-tangent function at the in-terface. To determine interface profile of double-obstacle, considering the free energy ofthe form f ob ( c ) = X A c ( − c ) , similar manipulations to the well type, the interface profile isexpressed as, c = + (cid:32) cos (cid:118)(cid:116) X A κ x (cid:33) . (A.5)177 ppendix BPreparation of electrode microstructurewith a single particle The smooth profile of the domain parameter φ in the simulation domain is obtained bysolving the non-conserved Allen-Cahn equation [ ] , τ p ∂ φ ∂ t = − ∂ f dw ( φ ) ∂ φ + ε p ∇ · ∇ φ (B.1)for a few initial time steps, where ε p is related to the interface thickness, τ p (= L / D ) de-notes relaxation coefficient, and f dw ( φ ) is a double-well free energy function expressed as f dw ( φ ) = φ ( − φ ) . The irregularly shaped particle is obtained by a strategical distri-bution of rectangles and ellipses, as shown in Figure B.1. Furthermore, an originally sharpdomain boundary is smoothed by Eq. (B.1) to yield a diffuse interface with a finite thicknessgiven by 0 < φ < φ is abolished thereafter. To discriminate betweenthe stationary and evolving particles, the evolving parameter φ are referred to stationaryparameter ψ in the main body of the dissertation (specifically, in Chapters 3, 5, and 6).179hapter B. 180 (a) (b) (c) (d) (e) (f) Figure B.1: Morphological evolution of irregularly shaped particle. The time increment follows (a) to (f).The blue region indicates electrolyte ( φ = ) and yellow color represents electrode particle ( φ = ).Except (a), other insets from (b) to (f) consist of diffuse interface between the electrode particle and theelectrolyte. ppendix CPreparation of electrode microstructurewith several particles The electrode structure is characterized numerically by employing volume preserved tech-nique on multiphase Allen-Cahn formulation. Each particle is considered as a separatephase freely evolving in the liquid electrolyte phase. A brief procedure is outlined in thefollowing paragraphs.The inhomogeneous system consists of different phases whose identity is distinguishedby its physical state or orientation. In the present description, a multi-phase Allen Cahnmodel is considered to study the evolution of multiphase systems whose interfacial energydecreases with preserving their volume. The evolution of a general system containing N phases is governed by Ginzburg-Landau free energy, F = (cid:90) V Ω f ( φ , ∇ φ ) d V Ω = (cid:90) V Ω (cid:129) ε p a ac ( φ , ∇ φ ) + ε p w ac ( φ ) + g ac ( φ ) (cid:139) d V Ω , (C.1)where f ( φ , ∇ φ ) is the free energy density, V Ω is the domain under consideration, a vec-tor phase-field parameter φ ( x , t ) = [ φ ( x , t ) , . . . , φ N ( x , t )] , where φ a ∈ [
0, 1 ] , ∀ a ∈{
1, 2, . . . , N } , x represents spatial coordinates and ε p denotes the thickness of the diffuseinterface. The gradient energy density is expressed as [ ] , a ac ( φ , ∇ φ ) = N , N (cid:88) a , b = ( a < b ) γ ab a ab ( φ , ∇ φ ) | q ab | (C.2)where γ ab denotes the interfacial energy density between a and b phases, q ab = φ a ∇ φ b − φ b ∇ φ a , which defines the gradient vector in the normal direction to a − b interface. Thefunction a ab ( φ , ∇ φ ) represents the interface anisotropy. For the present study, a faceted181hapter C. 182anisotropy is considered of the form, a ab ( φ , ∇ φ ) = max ≤ i ≤ η ab (cid:168) q ab | q ab | · η i , ab (cid:171) , (C.3)where { η i , ab | i =
1, . . . , η ab } represents position vectors in the Wulff shape of phase a withrespect to phase b . The bulk energy density potential w ac ( φ ) is assumed to be a multiob-stacle type, w ac ( φ ) = π N , N (cid:88) a , b = ( a < b ) γ ab φ a φ b + N , N , N (cid:88) a , b , l = ( a < b < l ) γ abl φ a φ b φ l . (C.4)The higher order interfacial energy term γ abl penalizes the ghost phase occurrences at theinterfaces between two phases. The additional bulk free energy density g ac ( φ ) is responsi-ble for the volume preservation, can be expressed as, g ac ( φ ) = N (cid:88) a = Υ a h ( φ a ) , (C.5)where the function h ( φ a ) = φ a ( φ a − φ a + ) interpolates the free energy density terms Υ a between the bulk phases. The time-dependent antiforce terms Υ ( t ) ∈ { Υ , . . . , Υ N } arecalculated at each timestep to maintain the phase volume equal to initially prescribed. Fora detailed algorithm, the readers are suggested to Ref. [ ] . Finally, the evolution of thephases is governed by a general model for multi-phase Allen-Cahn type equation as, τ p ε p ∂ φ a ∂ t = − δ f ( φ , ∇ φ ) δφ a − N N (cid:88) a = δ f ( φ , ∇ φ ) δφ a . (C.6)where τ p denotes the relaxation coefficient, ∂ ( • ) /∂ t is partial derivative with time t and δ ( • ) /δφ a denotes functional derivative. The second term in the above equation ensuresthe constraint (cid:80) Na = φ a ( x , t ) = ∇ φ a · n =
0, where n is the normalvector at the wall of the simulation boundary. In other words, the phases form a 90 ◦ angleat the boundary. To specify other contact angles, an extension is made in the form of − ε p ∂ a ( φ , ∇ φ ) ∂ ∇ φ a · n − ∂ f b ( φ ) ∂ φ a − (cid:168) N N (cid:88) φ a = (cid:130) − ε p ∂ a ( φ , ∇ φ ) ∂ ∇ φ a · n − ∂ f b ( φ ) ∂ φ a (cid:140)(cid:171) =
0, (C.7)where f b ( φ ) = (cid:80) N φ a = ( γ ab h ( φ a )) denotes the boundary function and the terms γ ab representthe interfacial energy density between a phase and the boundary, the value of these parame-ters controls the contact angle between the boundary and different phases from 0 ◦ (completenon-wetting) to 180 ◦ (complete wetting). The term in the parentheses in equation (C.7)ensures the sum of all phases at any location in the simulation domain is unity.hapter C. 183 ±15° ±45° ±75° ϕ index Figure C.1: Represetative cases of morphological evolution of initially circular electrode particles toprescribed geometrical shape and orientations. The dominant values of φ a (i.e. where φ a > ) aredemontrated in one frame with a parameter φ index , where φ to φ are separate electrode particlesand φ corresponds to the electrolyte. The electrode mictrostructure is obtained from random distribution of N − φ a and the last phase φ N denotes the electrolyte. These phases are al-lowed to evolve under N equations (C.6) with volume preserved for non-wetting boundarycondition (C.7). The particles evolve to a geometry specified by the position vectors η i , ab to the vertices in equation (C.3). To obtain the microstructure for the presented work inChapter 6, four ( ± ± ) and four ( ± ± ) vertices for 2D simulations and eight ( ± ± ± ) , eight ( ± ± ± ) , and eight ( ± ± ± ) vertices for 3Dsimulations are considered. For each particles, these vertices are rotated at a specified ori-entation θ in xy plane for 2D ( θ in xy, θ in yz, and θ in xz planes for 3D) domain. Forinstance, Figure C.1 shows particles are rotated using a random number generator in therange of ± ◦ , ± ◦ , and ± ◦ . The obtained setups are then utilized for the study ofinsertion of lithium species in the particles completely submerged in the liquid electrolyte,which is described Chapter 6. ppendix DPhase separation in Electrode along withhomogeneous mixture in Electrolyte The behavior of two-phase coexistence in the electrode compounds are well known [
41, 45–47 ] . However, instead of phase separation, a homogeneous distribution is apparent in theelectrolyte. Both behaviors can be achieved by careful consideration of the free energyparameter α (cid:48)(cid:48) a in Eq. (3.3) and the gradient energy parameter κ a in Eq. (3.4). The Cahn-Hilliard equation is employed for the study of phase separating behavior in ample literature [
51, 55–60 ] . Therefore, the objective of the present analysis is only to demonstrate that fora specific choice of these parameters, the Cahn-Hilliard equation exactly corresponds to theFick’s dilute solution model [ ] .Considering a special case, where any phase a consists the regular solution parameter α (cid:48)(cid:48) a = κ a =
0. The chemical potential of such systemcan be expressed as, µ a = α (cid:48) a + TT ref ( ln c − ln ( − c )) . (D.1)Here µ a is the chemical potential in the phase a . The species flux can be written as, J a = − M a ∇ µ a = − D a TT ref ∇ c . (D.2)Here J a and M a represent species flux and mobility in the phase a respectively. Eq. (D.2)is equivalent to the equation responsible for the homogeneous mixture. Therefore, param-eters α (cid:48)(cid:48) a = κ a = ppendix ECalculation of tortuosity The electrode particles are surrounded by the liquid electrolyte, which conducts species fluxin the electrode. The species diffusivity of the electrode is much lower than the electrolyte [ ] . Also, the insertion and the extraction of species in the electrode particles takes placeat the interface between the electrode and the electrolyte. Therefore, understanding oftortuous pathways in the electrolyte is of scientific interest. In the present study, instead ofassuming a Bruggeman relation [ ] , a computational tool is employed to calculate thetortuosity of the electrolyte microstructure considering infinite resistive electrode particles [ ] .The tortuosity is calculated in the x-direction, where the species flux propagates, fromthe separator to the current collector. A potential difference φ ∞ is applied between twoends, which is predefined as 1.0V at separator and 0.0V at the current collector for thepresent investigation as shown in Figure E.1. While no-flux boundary conditions applied onthe remaining two (four for 3D) boundaries. The potential distribution inside the simulationdomain is calculated using the Laplace equation of the form, ∇ · ( − σ ety ∇ φ ) =
0. (E.1)Where σ ety is the electrical conductivity of the electrolyte. Afterwards, the current density i = σ ety ∇ φ can be estimated at the each pixels (voxels in 3D). The total current flow fromthe cross-section perpendicular to the x-direction can be obtained as a summation, I m = (cid:90) A ⊥ i ⊥ dA ⊥ . (E.2)Where A ⊥ is the area of the cross-section surface under consideration. Note that, as the no-flux conditions prescribed at the remaining boundaries, due to conservation, the measuredcurrent flow at any cross-section perpendicular to the x-direction should be nearly equal.If not then the meaningful conclusions can not be drawn from the presented method. For187ist of symbols and abbreviations 188 ϕ Figure E.1: Schematic of potential distribution in the electrolyte during the tortuosity calculation. instance, in the case where there is no path for species to flow across when electrode parti-cles completely block the path by forming an entire blockage in the perpendicular directionto the x-axis.The tortuosity is a measure of conductive pathways deviated from the ideal straightchannels of uniform cross-section. The tortuosity can be calculated based on the actualcurrent flow in the microstructure in contemplation to the current flow in the ideal pathwithout any discontinuities, τ = I i I m , (E.3)where I i is the ideal current flow, determined from the analytical expression I i = σ ety φ ∞ A ⊥ / l and l is the length of the domain in the direction of the x-axis. ist of Symbols and Abbreviations The list describes several symbols that are used within the document. ∆ x , ∆ y , and ∆ z simulation cell widths in x -, y - and z -directions respectively, . . . . . . . . 47 γ ab (or γ abl ) interfacial energy density between a and b (or a , b , and l ) phases, . . . . . 181 α index of a lattice site, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 α (cid:48) the neighboring interstitial lattice site, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 α (cid:48) a and α (cid:48)(cid:48) a regular solution parameters of phase a , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .39 β conductivity ratio, σ mat /σ icl , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106 φ phase-field parameter set for Allen-Cahn equation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 ψ indicator parameter set, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 J effective diffusional mass fluxes, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 J A mass flux of A species, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 J B mass flux of B species, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 J i flux of the diffusing species i , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 J e flux of the charge carrier, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 n inward pointing unit normal to the surface, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 q ab gradient vector in the normal direction to a − b interface, . . . . . . . . . . . . . . . . . . . . . . . 181 x α spatial position of the α site, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 x α B neighbor on the back side of the α site, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28189ist of symbols and abbreviations 190 x α F neighbor on the front side of the α site, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 x α L neighbor on the left side of the α site, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 x α O neighbor on the bottom side of the α site, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 x α R neighbor on the right side of the α site, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 x α T neighbor on the top side of the α site, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 x α (cid:48) spatial position of the α (cid:48) site, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 χ dimensionless number associated with the steady state shapes of an island, . . . . . . . 103 χ dimensionless number associated with the stability of circular island / inclusion, . . . 104 χ critical value of χ , above which circular island / inclusion is unstable, . . . . . . . . . . . 104 δ c α (or δ c α (cid:48) ) fluctuation of c α (or c α (cid:48) ) compared to the average, . . . . . . . . . . . . . . . . . . . . . . 24 ∆ H change in internal energy of the system, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 ∆ S change in entropy of the system, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 ∆ t time step increment, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 δ s interface width, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ε interface thickness coefficient, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 ε p parameter related to interface thickness in Allen-Cahn equation, . . . . . . . . . . . . . . . . . 179 η slit characteristic parameter, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 γ aspect ratio of ellipse, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 γ s interfacial energy, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58ˆ • normalized quantity of the respective entities, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46ˆ κ s non-dimensional curvature of the island surface, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103ˆ M max maximum of dimensionless mobility ˆ M , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 κ gradient energy coefficient, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 κ a gradient energy coefficient of phase α , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 κ s curvature along the island / inclusion surface, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103ist of symbols and abbreviations 191 λ (cid:48) reference length scale for normalization, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Λ L ratio of line width w to the initial inclusion radius R i , . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 (cid:79) order of approximation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 (cid:23) a lattice parameter, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 µ chemical potential, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 µ chem chemical potential corresponding to the bulk free energy density, . . . . . . . . . . . . . . . 41 µ grad chemical potential corresponding to the gradient free energy density, . . . . . . . . . . . 41 µ A chemical potentials of the A species, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 µ B chemical potentials of the B species, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 µ chem a chemical potential corresponding to the bulk free energy density of a phase, . . . . 41 µ grad a chemical potential corresponding to the gradient free energy density of a phase, . 41 ν denominator coefficient, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Ω atomic volume, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ∂ V Ω simulation domain boundary surfaces, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 φ ∞ electrical potential boundary condition, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 φ local electrical potential, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 φ a evolving indicator parameter for phase a in Allen-Cahn equation, . . . . . . . . . . . . . . . . 181 π Archimedes’ constant, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 ψ orthogonal function to the electrical potential, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 ψ a stationary indicator parameter for phase a , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ρ porosity, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 ρ mat electrical resistivity of the matrix, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 σ electrical conductivity dependent on order parameter c , . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 σ ety electrical conductivity of the electrolyte, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 σ icl conductivity of an inclusion, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56ist of symbols and abbreviations 192 σ mat conductivity of the matrix, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 τ tortuosity, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 τ (cid:48) reference time scale for normalization, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 τ p relaxation coefficient, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 θ angle formed by the local tangent at the inclusion surface, . . . . . . . . . . . . . . . . . . . . . . . . . 56 Υ a time-dependent antiforce term for phase a , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 (cid:34) perturbation parameter, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 (cid:34) t difference between the A–B bond energy and the average of A-A and B-B bond energies,25 (cid:34) AA bond energy between A–A species, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 (cid:34) AB bond energy between A–B species, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 (cid:34) BB bond energy between B–B species, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 (cid:99) perturbation function, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 (cid:36) misorientation angle, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ξ discrepancy coefficient, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 ζ transformed fraction, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 A strength of anisotropy, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 a index of the phase, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 A amplitude of the perturbation at time t =
0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 a n surface perturbation coefficient corresponds to n th -term, . . . . . . . . . . . . . . . . . . . . . . . . . 107 A t amplitude of the perturbation at time t , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 a ab a-b interface anisotropy function, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 B Mullins’ constant, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 B m the internal system energy parameter, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 B m two-body interaction field from neighboring lattices, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23ist of symbols and abbreviations 193 C C-rate, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 c concentration / order parameter, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 c H higher concentration level of a phase-separated state, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 c L lower concentration level of a phase-separated state, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 c α state of the lattice site α , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 c A concentration of A species, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 c B concentration of B species, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 c i initial concentration filling, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 c n Dirichlet concentration boundary condition at the particle surface, . . . . . . . . . . . . . . . . . 44 c s concentration at the surface of the particle, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 c α (cid:48) state of the neighboring interstitial lattice state α (cid:48) , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 d dimensions of the simulation study, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 D a diffusion coefficient of phase a , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 D s surface diffusion coefficient, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 e electron charge, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 E t tangential component of the electric field along the island / inclusion surface, . . . . . 103 E x , E y local electric field components in x and y-directions, . . . . . . . . . . . . . . . . . . . . . . . . . . 110 E ∞ applied electric field, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 F system free energy, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 f system free energy density, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 f dw double-well free energy function, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 f ob obstacle-type free energy density, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 f S surface energy density, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 f θ anisotropy function, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 g grand canonical potential function, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23ist of symbols and abbreviations 194 g ac additional bulk energy density, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 g θβ function of shape and conductivity contrast, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 G chem a grand-chemical potential of phase a , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 H Hamiltonian of the lattice model, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 h interpolation function, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 I indicator function for the obstacle-type free energy density, . . . . . . . . . . . . . . . . . . . . . . . . . 54 i complex number, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 i , j , k ∈ x spatial location of a simulation grid point in the x-, y-, and z-directions respec-tively, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 I m (or I i ) measured (or ideal) current flow, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 J n Neumann flux boundary condition at the particle surface, . . . . . . . . . . . . . . . . . . . . . . . . . . 43 J s surface atomic flux, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 j Cot
Cottrell flux, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 J ps Neumann flux boundary condition at the separator, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 k frequency of the sinusoidal perturbation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 k B Boltzmann constant, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 k Cot
Cottrell flux-time proportionality constant, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 L reference length-scale for normalization, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 M effective diffusional mobility, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 m grain symmetry parameter, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 M m co-ordination number, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 M AA mobility of the A species, due to the interaction with the A species, . . . . . . . . . . . . . . . 32 M AB mobility of the A species, due to the interaction with the B species, . . . . . . . . . . . . . . . 32 M BA mobility of the B species, due to the interaction with the B species, . . . . . . . . . . . . . . . 32 M BB mobility of the B species, due to the interaction with the B species, . . . . . . . . . . . . . . 32ist of symbols and abbreviations 195 M ee mobility of electrons due to interaction with the electrons, . . . . . . . . . . . . . . . . . . . . . . . 56 M ei mobility of electrons due to interaction with species i , . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 M ie mobility of species i due to the interaction with the electrons, . . . . . . . . . . . . . . . . . . . . 55 M ii mobility of species i due to the interaction with the species i , . . . . . . . . . . . . . . . . . . . . . 55 N total number of phases, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 n discrete time, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 N B the total number of specified species, occupied in the lattice sites, . . . . . . . . . . . . . . . . . 22 N c nucleation rate coefficient, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 N m total number of lattice sites, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 N x , N y , and N z number of grid points in x-, y- and z-directions, . . . . . . . . . . . . . . . . . . . . . . . 45 N ex characteristic extension, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 p transitory perimeter of the inclusion, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 p initial perimeter of the inclusion, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 P AB total number of A–B bonds, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 R particle whose area is equivalent to the area of the circular particle of radius R , . . . . . 90 r , ϑ polar coordinates of a point, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 R radius of the reference particle, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 R i radius of circular island / inclusion, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 S displacement function, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 s arc length along the island / inclusion surface, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 S Ω surface of the domain, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 T absolute temperature, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 t time, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 t PE time attained at the end of phase separation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 t PS time at the onset of phase separation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76ist of symbols and abbreviations 196 T ref reference temperature, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 u half-slit width, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 V velocity along the external electric field, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 V steady-state velocity of an island / inclusion, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 V Ω volume of the system or simulation domain, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 V n velocity along the surface normal, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 w line width of the conductor, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 w ac bulk energy density potential, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 W c spatially dependent weight for Dirichlet concentration boundary condition, . . . . . . . 45 W J spatially dependent weight for Neumann flux boundary condition, . . . . . . . . . . . . . . . . 45 X first free energy density parameter, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 X second free energy density parameter, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 X A barrier height of the obstacle-type free energy density, . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Z i valence of the diffusing metal species i , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Z s effective valence, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Z w momentum exchange effect between the electrons and the diffusing species, . . . . . . 55 Z gc grand canonical partition function, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 η i , ab position vectors in the Wulff shape, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 c ps Dirichlet concentration boundary condition at the separator, . . . . . . . . . . . . . . . . . . . . . . 45 f chem a regular solution free energy density of phase a , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392D two-dimensional, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .713D three-dimensional, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71EM electromigration, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12GB grain boundaries, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13IMC intermetallic compound, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115ist of symbols and abbreviations 197JMA Johnson-Mehl-Avrami, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90LFP lithium iron phosphate, LiFePO , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41LMO lithium manganese oxide, LiMn O , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70LTO lithium titanate, Li Ti O , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85MPI message passing interface, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69PACE3D Parallel Algorithms for Crystal Evolution in 3D, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61PFM phase-field model, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90PITT Potentiostatic Intermittent Titration Technique, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91REV representative elementary volume, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38SOC state of charge, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71SOC PE state of charge attained before the end of phase separation, . . . . . . . . . . . . . . . . . . . 76SOC PS state of charge at the onset of phase separation, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ist of Figures Cu precipitate, at the cathode end . . . . . 151.10 Experimental evidence of the movement of monolayer islands, due to electro-migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1 Estimated physical length scale and time scale of the various computationalmethods, utilized for the numerical modeling . . . . . . . . . . . . . . . . . . . . . 222.2 Schematic of a lattice consisting of a mixture of A and B species . . . . . . . . . 252.3 Free energy curves for different values of (cid:34) t . . . . . . . . . . . . . . . . . . . . . . 262.4 Schematic of an evolution, containing a conserved order parameter, concen-tration c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5 Schematic of an evolution containing a non-conserved domain parameter φ . 343.1 Effect of the parameter α (cid:48) a on the free energy curves. . . . . . . . . . . . . . . . . 40199ist of figures 2003.2 Schematic diagram of a cathode particle, surrounded by the electrolyte in alithium-ion battery. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3 Simulation setup of Dirichlet concentration or Neumann flux boundary con-ditions at the separator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.4 Arrangement of the scalar and vector quantities on a staggered grid. . . . . . . 474.1 Schematic diagram of the simulation setup describes the inclusion in the ma-trix subjected to external electric potential . . . . . . . . . . . . . . . . . . . . . . 524.2 Double-obstacle type free energy density f ob ( c ) as a function of c . . . . . . . . 544.3 Schematic of anisotropy in surface diffusivity of inclusions . . . . . . . . . . . . 574.4 Evolution of the sinusoidal amplitude with time. . . . . . . . . . . . . . . . . . . . 594.5 Decay in the amplitude of the sinusoidal wave with time . . . . . . . . . . . . . . 605.1 Effect of the dimensionless cell width and the interface thickness coefficienton the SOC value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.2 Temporal evolution of isolated 2D elliptical particles with an initial concentra-tion of c i = c i = c i = c i on the phase separation dynamics . . . . . . . . 765.7 Concentration profiles of 3D irregularly shaped particle . . . . . . . . . . . . . . 775.8 Phase diagram indicating various zones characterized by the miscibility andthe spinodal gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.9 Variation of C-rate and mobility as a function of the aspect ratio of the ellipticalparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.10 Effect of temperature T / T ref and free energy parameter α (cid:48)(cid:48) on free energydensity plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.11 Variation of the temperature and free energy parameter as a function of theaspect ratio of the elliptical particles . . . . . . . . . . . . . . . . . . . . . . . . . . 83ist of figures 2016.1 Representative cases of insertion of species in 2D porous electrode under aconstant concentration boundary condition at the separator . . . . . . . . . . . . 886.2 Average of concentration c in porous and planar electrodes as a function ofthe depth of the electrode from the separator . . . . . . . . . . . . . . . . . . . . . 896.3 Evolution of SOC and fraction transformed from PFM compared with analyti-cal relations for bulk-transport and surface-reaction limited theories . . . . . . 906.4 Schematic of bulk diffusion and surface reaction limited transportation dy-namics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.5 Effect of tortuosity τ on transportation dynamics for different particle sizes R . 936.6 Effect of porosity ρ on transportation dynamics for different tortuosity τ . . . 956.7 Insertion of species in a 3D cathode electrode under a constant concentrationboundary condition at the separator . . . . . . . . . . . . . . . . . . . . . . . . . . 967.1 Schematic of a circular island, subjected to an external electric field E ∞ , in aninfinite conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1027.2 Schematic of a line segment of a surface of inclusion. . . . . . . . . . . . . . . . . 1087.3 Convergence of the numerical solution for linear stability analysis . . . . . . . . 1107.4 Dependence of an inclusion equilibrium shape on the dimensionless parame-ter, χ and the conductivity ratio β . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.5 Representative case of steady-state circular inclusion and circular inclusion toslit transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.6 Velocity of the centroid of the inclusions along the direction of external electricfield, V for conductivity ratios, β =
1, 3, and 10000 . . . . . . . . . . . . . . . . 1137.7 Change in local electric field distribution due to current crowding near inclu-sion surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.8 Temporal evolution of the perimeter as a function of time from the initialcircular inclusion to the finger-like slit and contour plots of the steady-stateslit at three different times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.9 Schematic diagram of constricted neck region with EM force and capillarity . . 1177.10 Effect of change in the electrical potential on propagation velocity and widthof the slit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.11 Schematic diagram of the finger-like slit front . . . . . . . . . . . . . . . . . . . . 119ist of figures 2027.12 Non-dimensional parameter η as a function of conductor line width to inclu-sion radius ratio w / R i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1227.13 Schematic diagram and simulation results with electric field vectors . . . . . . . 1247.14 Comparison of the front part of the slit profile from the phase-field model withthe sharp-interface analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.15 Comparison of the phase-field results with the sharp-interface analysis for thehalf slit width and the velocity of the slit as a function of the external electricfield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267.16 Dependence of ξ on η obtained from sharp interface, and the phase-field model1278.1 Schematic of an inclusion in (110) crystallographic plane . . . . . . . . . . . . . 1348.2 Inclusion morphology for conductivity ratio, β =
1, 3, and 10000 at the mis-orientation angle, (cid:36) = ◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358.3 Velocities of the centroid of steady-state inclusions for two-fold anisotropy( m = (cid:36) = ◦ and for isotropic ( m =
0) surface diffusion (numericaland analytical) as a function of the conductivity ratio β . . . . . . . . . . . . . . . 1368.4 Complex shape dynamics of inclusions at misorientation angle (cid:36) = ◦ forthe conductivity ratio β = (cid:36) = ◦ , 82 ◦ , and 75 ◦ for β = (cid:36) = ◦ for β =
1, 1.5, and 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1408.7 Inclusion morphology for conductivity ratio, β = (cid:36) = ◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1408.8 Complex shape dynamics of inclusions at misorientation angle (cid:36) = ◦ for theconductivity ratio β = β and (cid:36) foranisotropic twofold symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1428.10 Schematic of electromigration-induced atomic flux from the apex to the base,and from the base to the apex on the surface of the wedge-type inclusions . . . 1439.1 Schematic of inclusions in (100) and (111) crystallographic planes . . . . . . . 1489.2 Morphological map of inclusion migration modes as a function of β and (cid:36) foranisotropic fourfold ( m =
2) and sixfold ( m =
3) symmetries . . . . . . . . . . . 150ist of figures 2039.3 Representative dynamics of inclusion breakup due to elongation and retention 1519.4 Time-periodic shape dynamics of inclusions from crawling to gliding motion . 1529.5 Crawl (for β =
1) and Glide (for β = (cid:36) = ◦ . . . . . . . . . . . . . . . 1549.6 Effect of conductivity contrast during the morphological evolution of crawlingand gliding time-periodic oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 1559.7 Evolution of the perimeter as a function of (cid:36) during time-periodic oscillations 1569.8 Inclusion dynamics of 1-cycle and 2-cycle time-periodic oscillations . . . . . . . 1579.9 Evolution of perimeter for 1-cycle and 2-cycle time-periodic oscillations asfunction of β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1589.10 Steady-state morphological evolutions of faceted-wedge and seahorse pattern 1599.11 Velocities of the centroid of steady-state inclusions for twofold, fourfold, andsixfold anisotropies, compared with isotropic surface diffusion (analytical andnumerical) as a function of the conductivity ratio β . . . . . . . . . . . . . . . . . 1609.12 Zigzag oscillations of the sixfold symmetrical inclusion at misorientation angle (cid:36) = ◦ and conductivity ratio β = β = ist of Tables O cathode and operational conditions . . . . . . 695.2 Maximum and minimum concentration, along the surface of the particles,for the initial concentrations c i = + M LiClO electrolyte and Li x Mn O electrode. . . . . . . . . . . . . . . . . . . . . . 877.1 Values of parameters considered for the simulation sets of an isotropic inclu-sions to study significance of χ and β . . . . . . . . . . . . . . . . . . . . . . . . . 1117.2 Comparison of the values of the void size and the velocity, for eutecticSnAgCu solder bumps obtained from phase-field, and experiments . . . . . . . 1268.1 Values of parameters considered for the simulation study for an inclusion in { } crystallographic plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359.1 Values of parameters considered for the simulation study inclusions in { } and { } crystallographic planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 148205 ibliography [ ] S. P. Cadogan, G. C. Maitland, and J. M. Trusler. Diffusion coefficients of CO andN in water at temperatures between 298.15 K and 423.15 K at pressures up to 45MPa. J. Chem. Eng. Data , 59(2):519–525, 2014. [ ] R. Krishna. Uphill diffusion in multicomponent mixtures.
Chem. Soc. Rev. ,44(10):2812–2836, 2015. [ ] B. Derby and E. Wallach. Theoretical model for diffusion bonding.
Met. Sci. ,16(1):49–56, 1982. [ ] R. L. Coble. Sintering crystalline solids. I. Intermediate and final state diffusionmodels.
J. Appl. Phys. , 32(5):787–792, 1961. [ ] R. Balluffi and L. Seigle. Effect of grain boundaries upon pore formation and dimen-sional changes during diffusion.
Acta Metall. Mater. , 3(2):170–177, 1955. [ ] K. Holloway and P. M. Fryer. Tantalum as a diffusion barrier between copper andsilicon.
Appl. Phys. Lett. , 57(17):1736–1738, 1990. [ ] M. Rahman and M. Saghir. Thermodiffusion or soret effect: Historical review.
Int.J. Heat Mass Tran. , 73:693–705, 2014. [ ] T. Mashimo. Self-consistent approach to the diffusion induced by a centrifugal fieldin condensed matter: Sedimentation.
Phys. Rev. A , 38(8):4149, 1988. [ ] T. Mashimo, M. Ono, X. Huang, Y. Iguchi, S. Okayasu, K. Kobayashi, and E. Naka-mura. Gravity-induced diffusion of isotope atoms in monoatomic solid Se.
EPL-Europhys. Lett. , 81(5):56002, 2008. [ ] M. J. Aziz. Pressure and stress effects on diffusion in Si. In
Defect and DiffusionForum , volume 153, pages 1–10. Trans. Tech. Publ., 1998. [ ] H. Mehrer.
Diffusion in solids: fundamentals, methods, materials, diffusion-controlledprocesses , volume 155. Springer-Verlag, Berlin Heidelberg, 2007.207ibliography 208 [ ] L. Cindrella, A. M. Kannan, J. Lin, K. Saminathan, Y. Ho, C. Lin, and J. Wertz. Gasdiffusion layer for proton exchange membrane fuel cells-A review.
J. Power Sources ,194(1):146–160, 2009. [ ] C. Lupo and D. Schlettwein. Modeling of dendrite formation as a consequenceof diffusion-limited electrodeposition.
J. Electrochem. Soc. , 166(1):D3182–D3189,2019. [ ] P. M. Panchmatia, A. R. Armstrong, P. G. Bruce, and M. S. Islam. Lithium-ion diffusionmechanisms in the battery anode material Li + x V − x O . Phys. Chem. Chem. Phys. ,16(39):21114–21118, 2014. [ ] B. Tian, J. `Swiatowska, V. Maurice, C. Pereira-Nabais, A. Seyeux, and P. Marcus.Insight into lithium diffusion in conversion-type iron oxide negative electrode.
J.Phys. Chem. C , 119(2):919–925, 2015. [ ] J. E. Sanchez Jr., L. T. McKnelly, and J. W. Morris Jr. Slit morphology of electromigra-tion induced open circuit failures in fine line conductors.
J. Appl. Phys. , 72(7):3201–3203, 1992. [ ] J. Lienig and G. Jerke. Current-driven wire planning for electromigration avoidancein analog circuits. In
Proceedings of the 2003 Asia and South Pacific Design AutomationConference , ASP-DAC ’03, pages 783–788. IEEE, Kitakyushu, 2003. [ ] F. Fantini, J. Lloyd, I. De Munari, and A. Scorzoni. Electromigration testing of inte-grated circuit interconnections.
Microelectron. Eng. , 40(3-4):207–221, 1998. [ ] T. Luping and L.-O. Nilsson. Rapid determination of the chloride diffusivity in con-crete by applying an electric field.
ACI Mater. J. , 89(1):49–53, 1993. [ ] K. Amundson, E. Helfand, X. Quan, S. D. Hudson, and S. D. Smith. Alignment oflamellar block copolymer microstructure in an electric field. 2. Mechanisms of align-ment.
Macromolecules , 27(22):6559–6570, 1994. [ ] K. Kordesch. Electrochemical energy storage. In
Comprehensive Treatise of Electro-chemistry , pages 123–190. Springer US, New York, 1981. [ ] C. A. DeForest and K. S. Anseth. Advances in bioactive hydrogels to probe and directcell fate.
Annu. Rev. Chem. Biomol. , 3:421–444, 2012. (Original source: JohnsonControl–SAFT 2005 and 2007). [ ] A. Dey. Lithium anode film and organic and inorganic electrolyte batteries.
ThinSolid Films , 43(1-2):131–171, 1977.ibliography 209 [ ] C. M. Hayner, X. Zhao, and H. H. Kung. Materials for rechargeable lithium-ion bat-teries.
Annu. Rev. Chem. Biomol. , 3:445–471, 2012. [ ] M. Silberberg.
Chemistry: The Molecular Nature of Matter and Change . McGraw-HillHigher Education, New York, 2014. [ ] R. Deshpande, M. Verbrugge, Y.-T. Cheng, J. Wang, and P. Liu. Battery cycle lifeprediction with coupled chemical degradation and fatigue mechanics.
J. Electrochem.Soc. , 159(10):A1730–A1738, 2012. [ ] J. M. Tarascon and M. Armand. Issues and challenges facing rechargeable lithiumbatteries.
Nature , 414(6861):359–367, 2001. [ ] D. K. Kim, P. Muralidharan, H.-W. Lee, R. Ruffo, Y. Yang, C. K. Chan, H. Peng, R. A.Huggins, and Y. Cui. Spinel LiMn O nanorods as lithium ion battery cathodes. NanoLett. , 8(11):3948–3952, 2008. [ ] J. Vetter, P. Novák, M. R. Wagner, C. Veit, K. Möller, J. O. Besenhard, M. Win-ter, M. Wohlfahrt-Mehrens, C. Vogler, and A. Hammouche. Ageing mechanisms inlithium-ion batteries.
J. Power Sources , 147(1-2):269–281, 2005. [ ] M. M. Thackeray, J. O. Thomas, and M. S. Whittingham. Science and applications ofmixed conductors for lithium batteries.
MRS Bull. , 25(3):39–46, 2000. [ ] M. S. Whittingham. Lithium batteries and cathode materials.
Chem. Rev. ,104(10):4271–4301, 2004. [ ] C. Liu, F. Li, L.-P. Ma, and H.-M. Cheng. Advanced materials for energy storage.
Adv.Mater. , 22(8), 2010. [ ] T. Ohzuku, Y. Iwakoshi, and K. Sawai. Formation of lithium − graphite intercalationcompounds in nonaqueous electrolytes and their application as a negative electrodefor a lithium ion (shuttlecock) cell. J. Electrochem. Soc. , 140(9):2490, 1993. [ ] T. R. Ferguson and M. Z. Bazant. Phase transformation dynamics in porous batteryelectrodes.
Electrochim. Acta , 146:89–97, 2014. [ ] T. Ohzuku, Y. Iwakoshi, and K. Sawai. Formation of lithium-graphite intercalationcompounds in nonaqueous electrolytes and their application as a negative electrodefor a lithium ion (shuttlecock) cell.
J. Electrochem. Soc. , 140(9):2490–2498, 1993. [ ] R. B. Smith, E. Khoo, and M. Z. Bazant. Intercalation kinetics in multiphase-layeredmaterials.
J. Phys. Chem. C , 121(23):12505–12523, 2017.ibliography 210 [ ] Y. Li, F. El Gabaly, T. R. Ferguson, R. B. Smith, N. C. Bartelt, J. D. Sugar, K. R. Fen-ton, D. A. Cogswell, A. D. Kilcoyne, T. Tyliszczak, M. Z. Bazant, and W. C. Chueh.Current-induced transition from particle-by-particle to concurrent intercalation inphase-separating battery electrodes.
Nat. Mater. , 13(12):1149, 2014. [ ] M. Tang, W. C. Carter, and Y.-M. Chiang. Electrochemically driven phase transitions ininsertion electrodes for lithium-ion batteries: Examples in lithium metal phosphateolivines.
Ann. Rev. Mater. Res. , 40(1):501–529, 2010. [ ] A. K. Padhi, K. S. Nanjundaswamy, and J. B. Goodenough. Phospho-olivines aspositive-electrode materials for rechargeable lithium batteries.
J. Electrochem. Soc. ,144(4):1188–1194, 1997. [ ] G. Li, A. Yamada, Y. Fukushima, K. Yamaura, T. Saito, T. Endo, H. Azuma, K. Sekai,and Y. Nishi. Phase segregation of Li x Mn O ( < x < ) in non-equilibriumreduction processes. Solid State Ionics , 130(3):221–228, 2000. [ ] A. V. der Ven, C. Marianetti, D. Morgan, and G. Ceder. Phase transformations andvolume changes in spinel Li x Mn O . Solid State Ionics , 135(1):21–32, 2000. [ ] A. Van der Ven, M. K. Aydinol, and G. Ceder. First-principles evidence for stageordering in Li x CoO . J. Electrochem. Soc. , 145(6):2149–2155, 1998. [ ] X. H. Liu, J. W. Wang, S. Huang, F. Fan, X. Huang, Y. Liu, S. Krylyuk, J. Yoo, S. A.Dayeh, A. V. Davydov, S. X. Mao, S. T. Picraux, S. Zhang, J. Li, T. Zhu, and J. Y.Huang. In situ atomic-scale imaging of electrochemical lithiation in silicon.
Nat.Nanotechnol. , 7(11):749, 2012. [ ] A. Vasileiadis, N. J. J. de Klerk, R. B. Smith, S. Ganapathy, P. P. R. M. L. Harks, M. Z.Bazant, and M. Wagemaker. Toward optimal performance and in-depth understand-ing of spinel Li Ti O electrodes through phase field modeling. Adv. Funct. Mater. ,28(16):1705992, 2018. [ ] T. Ohzuku, M. Kitagawa, and T. Hirai. Electrochemistry of manganese dioxide inlithium nonaqueous cell: III. X-ray diffractional study on the reduction of spinel-related manganese dioxide.
J. Electrochem. Soc. , 137(3):769–775, 1990. [ ] W. Liu, K. Kowal, and G. C. Farrington. Mechanism of the electrochemical insertionof lithium into LiMn O spinels. J. Electrochem. Soc. , 145(2):459–465, 1998. [ ] X. Q. Yang, X. Sun, S. J. Lee, J. McBreen, S. Mukerjee, M. L. Daroux, and X. K. Xing.In situ synchrotron X-ray diffraction studies of the phase transitions in Li x Mn O cathode materials. Electrochem. Solid St. , 2(4):157–160, 1999.ibliography 211 [ ] R. Malik, A. Abdellahi, and G. Ceder. A critical review of the Li insertion mechanismsin LiFePO electrodes. J. Electrochem. Soc. , 160(5):A3179–A3197, 2013. [ ] M. Landstorfer and T. Jacob. Mathematical modeling of intercalation batteries at thecell level and beyond.
Chem. Soc. Rev. , 42(8):3234–3252, 2013. [ ] D. Grazioli, M. Magri, and A. Salvadori. Computational modeling of Li-ion batteries.
Comput. Mech. , 58(6):889–909, 2016. [ ] M. Huttin and M. Kamlah. Phase-field modeling of stress generation in electrodeparticles of lithium ion batteries.
Appl. Phys. Lett. , 101(13):133902, 2012. [ ] S. Huang, F. Fan, J. Li, S. Zhang, and T. Zhu. Stress generation during lithiation ofhigh-capacity electrode particles in lithium ion batteries.
Acta Mater. , 61(12):4354–4364, 2013. [ ] A. D. Drozdov. A model for the mechanical response of electrode particles inducedby lithium diffusion in Li-ion batteries.
Acta Mech. , 225(11):2987–3005, 2014. [ ] J. W. Cahn and J. E. Hilliard. Free energy of a nonuniform system. I. Interfacial freeenergy.
J. Chem. Phys. , 28(2):258–267, 1958. [ ] T. Zhang and M. Kamlah. A nonlocal species concentration theory for diffusionand phase changes in electrode particles of lithium ion batteries.
Continuum Mech.Therm. , 30(3):553–572, 2018. [ ] P. Stein, Y. Zhao, and B.-X. Xu. Effects of surface tension and electrochemical re-actions in Li-ion battery electrode nanoparticles.
J. Power Sources , 332:154–169,2016. [ ] C. V. D. Leo, E. Rejovitzky, and L. Anand. A Cahn-Hilliard-type phase-field theoryfor species diffusion coupled with large elastic deformations: Application to phase-separating Li-ion electrode materials.
J. Mech. Phys. Solids , 70:1–29, 2014. [ ] M. J. Welland, D. Karpeyev, D. T. O’Connor, and O. Heinonen. Miscibility gap closure,interface morphology, and phase microstructure of 3D Li x FePO nanoparticles fromsurface wetting and coherency strain. ACS nano , 9(10):9757–9771, 2015. [ ] J. Santoki, D. Schneider, M. Selzer, F. Wang, M. Kamlah, and B. Nestler. Phase-field study of surface irregularities of a cathode particle during intercalation.
Model.Simul. Mater. Sc. , 26(6):065013, 2018. [ ] A. C. Walk, M. Huttin, and M. Kamlah. Comparison of a phase-field model for inter-calation induced stresses in electrode particles of lithium ion batteries for small andfinite deformation theory.
Eur. J. Mech. A-Solid , 48(1):74–82, 2014.ibliography 212 [ ] J. Newman and W. Tiedemann. Porous-electrode theory with battery applications.
AIChE J. , 21(1):25–41, 1975. [ ] A. S. Arico, P. Bruce, B. Scrosati, J.-M. Tarascon, and W. Van Schalkwijk. Nanostruc-tured materials for advanced energy conversion and storage devices.
Nat. Mater. ,4:366–377, 2005. [ ] P. Bai and G. Tian. Statistical kinetics of phase-transforming nanoparticles in LiFePO porous electrodes. Electrochim. Acta , 89:644–651, 2013. [ ] R. B. Smith and M. Z. Bazant. Multiphase porous electrode theory.
J. Electrochem.Soc. , 164(11):E3291–E3310, 2017. [ ] W. C. Chueh, F. El Gabaly, J. D. Sugar, N. C. Bartelt, A. H. McDaniel, K. R. Fenton,K. R. Zavadil, T. Tyliszczak, W. Lai, and K. F. McCarty. Intercalation pathway in many-particle LiFePO electrode revealed by nanoscale state-of-charge mapping. NanoLett. , 13(3):866–872, 2013. [ ] W. Dreyer, C. Guhlke, and R. Huth. The behavior of a many-particle electrode in alithium-ion battery.
Physica D , 240(12):1008–1019, 2011. [ ] B. Orvananos, R. Malik, H.-C. Yu, A. Abdellahi, C. P. Grey, G. Ceder, and K. Thornton.Architecture dependence on the dynamics of nano-LiFePO electrodes. Electrochim.Acta , 137:245–257, 2014. [ ] I. A. Blech. Electromigration in thin aluminum films on titanium nitride.
J. Appl.Phys. , 47(4):1203–1208, 1976. [ ] P. S. Ho and T. Kwok. Electromigration in metals.
Rep. Prog. Phys. , 52(3):301–348,1989. [ ] K. N. Tu. Recent advances on electromigration in very-large-scale-integration of in-terconnects.
J. Appl. Phys. , 94(9):5451–5473, 2003. [ ] R. de Orio, H. Ceric, and S. Selberherr. Physically based models of electromigration:From Black’s equation to modern TCAD models.
Microelectron. Reliab. , 50(6):775–789, 2010. [ ] A. Vairagar, S. Mhaisalkar, A. Krishnamoorthy, K. Tu, A. Gusak, M. A. Meyer, andE. Zschech. In situ observation of electromigration-induced void migration in dual-damascene Cu interconnect structures.
Appl. Phys. Lett. , 85(13):2502–2504, 2004. [ ] S.-K. Lin, Y.-C. Liu, S.-J. Chiu, Y.-T. Liu, and H.-Y. Lee. The electromigration effectrevisited: Non-uniform local tensile stress-driven diffusion.
Sci. Rep.-UK , 7(1):3082,2017.ibliography 213 [ ] E. Arzt, O. Kraft, W. D. Nix, and J. E. Sanchez Jr. Electromigration failure by shapechange of voids in bamboo lines.
J. Appl. Phys. , 76(3):1563–1571, 1994. [ ] S. P. Riege, J. A. Prybyla, and A. W. Hunt. Influence of microstructure on electro-migration dynamics in submicron Al interconnects: Real-time imaging.
Appl. Phys.Lett. , 69(16):2367–2369, 1996. [ ] A. W. Hunt, S. P. Riege, and J. A. Prybyla. Healing processes in submicron Al inter-connects after electromigration failure.
Appl. Phys. Lett. , 70(19):2541–2543, 1997. [ ] T. Miyazaki and T. Omata. Electromigration degradation mechanism for Pb-free flip-chip micro solder bumps.
Microelectron. Reliab. , 46(9-11):1898–1903, 2006. [ ] T. Marieb, P. Flinn, J. C. Bravman, D. Gardner, and M. Madden. Observations ofelectromigration induced void nucleation and growth in polycrystalline and near-bamboo passivated Al lines.
J. Appl. Phys. , 78(2):1026–1032, 1995. [ ] A. V. Vairagar, S. G. Mhaisalkar, A. Krishnamoorthy, K. N. Tu, A. M. Gusak, M. A.Meyer, and E. Zschech. In situ observation of electromigration-induced void migra-tion in dual-damascene Cu interconnect structures.
Appl. Phys. Lett. , 85(13):2502–2504, 2004. [ ] J. Lloyd. Black’s law revisited-Nucleation and growth in electromigration failure.
Microelectron. Reliab. , 47(9-11):1468–1472, 2007. [ ] M. R. Gungor and D. Maroudas. Theoretical analysis of electromigration-inducedfailure of metallic thin films due to transgranular void propagation.
J. Appl. Phys. ,85(4):2233–2246, 1999. [ ] J. Cho, M. R. Gungor, and D. Maroudas. Theoretical analysis of current-driven inter-actions between voids in metallic thin films.
J. Appl. Phys. , 101(2):023518, 2007. [ ] J. Cho, M. R. Gungor, and D. Maroudas. Current-driven interactions between voidsin metallic interconnect lines and their effects on line electrical resistance.
Appl.Phys. Lett. , 88(22):221905, 2006. [ ] C. Liao, K. Chen, W. Wu, and L. Chen. In situ transmission electron microscopeobservations of electromigration in copper lines at room temperature.
Appl. Phys.Lett. , 87(14):141903, 2005. [ ] N. Claret, C. Guedj, L. Arnaud, and G. Reimbold. Study of void growth in 120 nmcopper lines by in situ SEM.
Microelectron. Eng. , 83(11-12):2175–2178, 2006.ibliography 214 [ ] G. Schneider, M. A. Meyer, E. Zschech, G. Denbeaux, U. Neuhäusler, andP. Guttmann. In situ X-ray microscopy studies of electromigration in copper inter-connects. In
AIP Conf. Proc. , volume 683, pages 480–484, 2003. [ ] T. Nitta, T. Ohmi, T. Hoshi, S. Sakai, K. Sakaibara, S. Imai, and T. Shibata. Evaluatingthe large electromigration resistance of copper interconnects employing a newly de-veloped accelerated life-test method.
J. Electrochem. Soc. , 140(4):1131–1137, 1993. [ ] J. R. Black. Electromigration–A brief survey and some recent results.
IEEE T. Electron.Dev. , 16(4):338–347, 1969. [ ] T. Shaw, C.-K. Hu, K. Lee, and R. Rosenberg. Copper migration and precipitate dis-solution in aluminum / copper lines during electromigration testing. Mater. Res. Soc.Symp. P. , 428:187–199, 1996. [ ] Q. Ma and Z. Suo. Precipitate drifting and coarsening caused by electromigration.
J. Appl. Phys. , 74(9):5457–5462, 1993. [ ] C. Witt, C. Volkert, and E. Arzt. Electromigration-induced Cu motion and precipita-tion in bamboo Al-Cu interconnects.
Acta Mater. , 51(1):49–60, 2003. [ ] S. K. Theiss and J. Prybyla. In situ study of Al Cu precipitate evolution during elec-tromigration in submicron Al interconnects.
Mater. Res. Soc. Symp. P. , 428:207–212,1996. [ ] R. Rosenberg. Inhibition of electromigration damage in thin films.
J. Vac. Sci. Tech-nol. , 9(1):263–270, 1972. [ ] C. Witt.
Electromigration in bamboo aluminum interconnects . PhD thesis, Universityof Stuttgart, 2001. [ ] R. Spolenak, O. Kraft, and E. Arzt. Effects of alloying elements on electromigration.
Microelectron. Reliab. , 38(6-8):1015–1020, 1998. [ ] C. Tao, W. Cullen, and E. Williams. Visualizing the electron scattering force in nanos-tructures.
Science , 328(5979):736–740, 2010. [ ] A. Mukherjee, R. Mukherjee, K. Ankit, A. Bhattacharya, and B. Nestler. Influenceof substrate interaction and confinement on electric-field-induced transition in sym-metric block-copolymer thin films.
Phys. Rev. E , 93:032504, 2016. [ ] A. Mukherjee, K. Ankit, A. Reiter, M. Selzer, and B. Nestler. Electric-field-inducedlamellar to hexagonally perforated lamellar transition in diblock copolymer thinfilms: Kinetic pathways.
Phys. Chem. Chem. Phys. , 18:25609–25620, 2016.ibliography 215 [ ] I. Dutta and P. Kumar. Electric current induced liquid metal flow: Application tocoating of micropatterned structures.
Appl. Phys. Lett. , 94(18):184104, 2009. [ ] M. Park, S. Gibbons, and R. Arróyave. Phase-field simulations of intermetalliccompound evolution in Cu / Sn solder joints under electromigration.
Acta Mater. ,61(19):7142–7154, 2013. [ ] P. Zhou and W. C. Johnson. A diffuse interface model of intermediate-phase growthunder the influence of electromigration.
J. Electron. Mater. , 40(9):1867, 2011. [ ] V. Attari, S. Ghosh, T. Duong, and R. Arroyave. On the interfacial phase growthand vacancy evolution during accelerated electromigration in Cu / Sn / Cu microjoints.
Acta Mater. , 160:185–198, 2018. [ ] A. Mukherjee, K. Ankit, R. Mukherjee, and B. Nestler. Phase-field modeling of grain-boundary grooving under electromigration.
J. Electron. Mater. , 45(12):6233–6246,2016. [ ] S. Chakraborty, P. Kumar, and A. Choudhury. Phase-field modeling of grain-boundarygrooving and migration under electric current and thermal gradient.
Acta Mater. ,153:377–390, 2018. [ ] A. Yamanaka, K. Yagi, and H. Yasunaga. Surface electromigration of metal atomson Si(111) surfaces studied by UHV reflection electron microscopy.
Ultramicroscopy ,29(1):161–167, 1989. [ ] R. Yongsunthon, C. Tao, P. Rous, and E. Williams. Surface electromigration andcurrent crowding. In
Nanophenomena at Surfaces , pages 113–143. Springer-Verlag,Berlin Heidelberg, 2011. [ ] J.-J. Métois and M. Audiffren. An experimental study of step dynamics under theinfluence of electromigration: Si(111).
Int. J. Mod. Phys. B , 11(31):3691–3702,1997. [ ] M. F. G. Hedouin and P. J. Rous. Relationship between adatom-induced surface re-sistivity and the wind force for adatom electromigration: A layer Korringa-Kohn-Rostoker study.
Phys. Rev. B , 62:8473–8477, 2000. [ ] J.-J. Métois, J.-C. Heyraud, and A. Pimpinelli. Steady-state motion of silicon islandsdriven by a DC current.
Surf. Sci. , 420(2-3):250–258, 1999. [ ] W. Wang, Z. Suo, and T.-H. Hao. A simulation of electromigration-induced trans-granular slits.
J. Appl. Phys. , 79(5):2394–2403, 1996.ibliography 216 [ ] M. Schimschak and J. Krug. Electromigration-induced breakup of two-dimensionalvoids.
Phys. Rev. Lett. , 80:1674–1677, 1998. [ ] T.-H. Hao and Q.-M. Li. Linear analysis of electromigration-induced void instabilityin Al-based interconnects.
J. Appl. Phys. , 83(2):754–759, 1998. [ ] H. Mehl, O. Biham, O. Millo, and M. Karimi. Electromigration-induced flow of islandsand voids on the Cu(001) surface.
Phys. Rev. B , 61(7):4975, 2000. [ ] O. Pierre-Louis and T. Einstein. Electromigration of single-layer clusters.
Phys. Rev.B , 62(20):13697, 2000. [ ] P. Kuhn, J. Krug, F. Hausser, and A. Voigt. Complex shape evolution ofelectromigration-driven single-layer islands.
Phys. Rev. Lett. , 94(16):166105, 2005. [ ] D. Dasgupta and D. Maroudas. Surface nanopatterning from current-driven assemblyof single-layer epitaxial islands.
Appl. Phys. Lett. , 103(18):181602, 2013. [ ] A. Kumar, D. Dasgupta, and D. Maroudas. Surface nanopattern formation dueto current-induced homoepitaxial nanowire edge instability.
Appl. Phys. Lett. ,109(11):113106, 2016. [ ] D. Dasgupta, A. Kumar, and D. Maroudas. Analysis of current-driven oscillatorydynamics of single-layer homoepitaxial islands on crystalline conducting substrates.
Surf. Sci. , 669:25–33, 2018. [ ] J. Santoki, A. Mukherjee, D. Schneider, M. Selzer, and B. Nestler. Phase-field studyof electromigration-induced shape evolution of a transgranular finger-like slit.
J.Electron. Mater. , 48(1):182–193, 2019. [ ] F. Haußer, S. Rasche, and A. Voigt. The influence of electric fields on nanostructures-simulation and control.
Math. Comput. Simulat. , 80(7):1449–1457, 2010. [ ] L. Baˇnas and R. Nürnberg. Phase field computations for surface diffusion and voidelectromigration in R . Comput. Vis. Sci. , 12(7):319–327, 2009. [ ] Y. Li, X. Wang, and Z. Li. The morphological evolution and migration of inclusionsin thin-film interconnects under electric loading.
Compos. Part B-Eng. , 43(3):1213–1217, 2012. [ ] D. N. Bhate, A. Kumar, and A. F. Bower. Diffuse interface model for electromigrationand stress voiding.
J. Appl. Phys. , 87(4):1712–1721, 2000.ibliography 217 [ ] M. Mahadevan and R. M. Bradley. Simulations and theory of electromigration-induced slit formation in unpassivated single-crystal metal lines.
Phys. Rev. B ,59:11037–11046, 1999. [ ] Z. Suo, W. Wang, and M. Yang. Electromigration instability: Transgranular slits ininterconnects.
Appl. Phys. Lett. , 64(15):1944–1946, 1994. [ ] Y. Yao, Y. Wang, L. M. Keer, and M. E. Fine. An analytical method to predictelectromigration-induced finger-shaped void growth in SnAgCu solder interconnect.
Scripta Mater. , 95:7–10, 2015. [ ] J. W. Barrett, H. Garcke, and R. Nürnberg. A phase field model for the electromigra-tion of intergranular voids.
Interface Free Bound. , 9(2):171–210, 2007. [ ] J. Santoki, A. Mukherjee, D. Schneider, and B. Nestler. Role of conductivity on theelectromigration-induced morphological evolution of inclusions in { } -orientedsingle crystal metallic thin films. J. Appl. Phys. , 126(16):165305, 2019. [ ] S. Allen and J. Cahn. A microscopic theory for antiphase boundary motion and itsapplication to antiphase domain coarsening.
Acta Metall. Mater. , 27(6):1085–1095,1979. [ ] M. Stan. Discovery and design of nuclear fuels.
Mater. Today , 12(11):20–28, 2009. [ ] F. K. Schwab.
Curing simulations of a fibre-reinforced thermoset on a micro- and nano-scale . PhD thesis, Karlsruher Institut für Technologie (KIT), 2019. [ ] M. Nonomura. Study on multicellular systems using a phase field model.
PloS One ,7(4):e33501, 2012. [ ] J. Kim. Phase-field models for multi-component fluid flows.
Commun. Comput. Phys. ,12(3):613–661, 2012. [ ] B. Nestler, H. Garcke, and B. Stinner. Multicomponent alloy solidification: Phase-field modeling and simulations.
Phys. Rev. E , 71:041609, 2005. [ ] M. Huttin.
Phase-field modeling of the influence of mechanical stresses on chargingand discharging processes in lithium ion batteries . PhD thesis, Karlsruher Institut fürTechnologie (KIT), 2014. [ ] D. A. Porter, K. E. Easterling, and M. Sherif.
Phase transformations in metals andalloys, (revised reprint) . CRC press, Boca Raton, Florida, 2009. [ ] N. Provatas and K. Elder.
Phase-field methods in materials science and engineering .Wiley-VCH, Berlin, 2010.ibliography 218 [ ] L. D. Landau. On the theory of phase transitions. In D. T. Haar, editor,
CollectedPapers of L.D. Landau , pages 193–216. Pergamon, Oxford, 1965. [ ] N. Moelans, B. Blanpain, and P. Wollants. An introduction to phase-field modelingof microstructure evolution.
Calphad , 32(2):268–294, 2008. [ ] V. Ginzburg and L. Landau. On the theory of superconductivity.
Zh. Eksp. Teor. Fiz. ,20:1064–1082, 1950. [ ] L.-Q. Chen. Phase-field models for microstructure evolution.
Ann. Rev. Mater. Res. ,32(1):113–140, 2002. [ ] B. Nestler and A. Choudhury. Phase-field modeling of multi-component systems.
Curr. Opin. Solid St. M. , 15(3):93–105, 2011. [ ] W. J. Boettinger, J. A. Warren, C. Beckermann, and A. Karma. Phase-field simulationof solidification.
Ann. Rev. Mater. Res. , 32(1):163–194, 2002. [ ] D. S. Lemons and A. Gythiel. Paul Langevin’s paper "On the Theory of BrownianMotion" [ "Sur la théorie du mouvement brownien," CR Acad. Sci.(Paris) 146, 530–533 (1908) ] . Am. J. Phys , 65(11):1079–1081, 1997. [ ] A. Fick. V. On liquid diffusion.
Philos. Mag. Series 4 , 10(63):30–39, 1855. [ ] E. B. Nauman and D. Q. He. Morphology predictions for ternary polymer blendsundergoing spinodal decomposition.
Polymer , 35(11):2243–2255, 1994. [ ] J. W. Cahn. On spinodal decomposition.
Acta Metall. Mater. , 9(9):795–801, 1961. [ ] L. Anand. A Cahn-Hilliard-type theory for species diffusion coupled with largeelastic-plastic deformations.
J. Mech. Phys. Solids , 60(12):1983–2002, 2012. [ ] I. Prigogine.
Introduction to thermodynamics of irreversible processes , volume 7. In-terscience Publishers, New York, 1961. [ ] A. Schmid. A time dependent Ginzburg-Landau equation and its application to theproblem of resistivity in the mixed state.
Phys. kondens. Materie , 5(4):302–317,1966. [ ] H. Schmidt. The onset of superconductivity in the time dependent Ginzburg-Landautheory.
Z. Physik , 216(4):336–345, 1968. [ ] P. K. Amos, E. Schoof, J. Santoki, D. Schneider, and B. Nestler. Limitations of pre-serving volume in Allen-Cahn framework for microstructural analysis.
Comp. Mater.Sci. , 173:109388, 2020.ibliography 219 [ ] B. Nestler, F. Wendler, M. Selzer, B. Stinner, and H. Garcke. Phase-field model formultiphase systems with preserved volume fractions.
Phys. Rev. E , 78(1):011604,2008. [ ] A. Novick-Cohen and L. A. Segel. Nonlinear aspects of the Cahn-Hilliard equation.
Physica D , 10(3):277–298, 1984. [ ] B. Halperin, P. Hohenberg, and S.-K. Ma. Renormalization-group methods for criticaldynamics: I. Recursion relations and effects of energy conservation.
Phys. Rev. B ,10(1):139, 1974. [ ] A. Mukherjee.
Electric field-induced directed assembly of diblock copolymers and grainboundary grooving in metal interconnects . PhD thesis, Karlsruher Institut für Tech-nologie (KIT), 2019. [ ] N. Moelans, B. Blanpain, and P. Wollants. Quantitative phase-field approach forsimulating grain growth in anisotropic systems with arbitrary inclination and mis-orientation dependence.
Phys. Rev. Lett. , 101:025502, 2008. [ ] G. K. Singh, G. Ceder, and M. Z. Bazant. Intercalation dynamics in rechargeablebattery materials: General theory and phase-transformation waves in LiFePO . Elec-trochim. Acta , 53(26):7599–7613, 2008. [ ] M. E. Gurtin. Generalized Ginzburg-Landau and Cahn-Hilliard equations based ona microforce balance.
Physica D , 92(3-4):178–192, 1996. [ ] L. Hong, L. Liang, S. Bhattacharyya, W. Xing, and L. Q. Chen. Anisotropic Li interca-lation in a Li x FePO nano-particle: a spectral smoothed boundary phase-field model. Phys. Chem. Chem. Phys. , 18:9537–9543, 2016. [ ] H.-C. Yu, H.-Y. Chen, and K. Thornton. Extended smoothed boundary method forsolving partial differential equations with general boundary conditions on complexboundaries.
Model. Simul. Mater. Sc. , 20(7), 2012. [ ] W. Yang, W. Wang, and Z. Suo. Cavity and dislocation instability due to electriccurrent.
J. Mech. Phys. Solids , 42(6):897–911, 1994. [ ] L. Xia, A. Bower, Z. Suo, and C. Shih. A finite element analysis of the motion andevolution of voids due to strain and electromigration induced surface diffusion.
J.Mech. Phys. Solids , 45(9):1473–1493, 1997. [ ] M. R. Gungor and D. Maroudas. Electromigration-induced failure of metallic thinfilms due to transgranular void propagation.
Appl. Phys. Lett. , 72(26):3452–3454,1998.ibliography 220 [ ] M. Plapp. Phase-field models. In
Multiphase Microfluidics: The Diffuse InterfaceModel , volume 538, pages 129–175. Springer, Vienna, 2012. [ ] I. Steinbach. Phase-field model for microstructure evolution at the mesoscopic scale.
Ann. Rev. Mater. Res. , 43(1):89–107, 2013. [ ] H. Emmerich, H. Löwen, R. Wittkowski, T. Gruhn, G. I. Tóth, G. Tegze, andL. Gránásy. Phase-field-crystal models for condensed matter dynamics on atomiclength and diffusive time scales: an overview.
Adv. Phys. , 61(6):665–743, 2012. [ ] J. Van der Waals. Thermodynamische Theorie der Kapillarität unter voraussetzungstetiger Dichteänderung.
Z. Phys. Chem. , 13(1):657–725, 1894. [ ] J. D. van der Waals. The thermodynamic theory of capillarity under the hypothesisof a continuous variation of density.
J. Stat. Phys. , 20(2):200–244, 1979. [ ] P. Yue, C. Zhou, and J. J. Feng. Spontaneous shrinkage of drops and mass conserva-tion in phase-field simulations.
J. Comput. Phys. , 223(1):1–9, 2007. [ ] K. Tu. Electromigration in stressed thin films.
Phys. Rev. B , 45(3):1409, 1992. [ ] J. E. Guyer, W. J. Boettinger, J. A. Warren, and G. B. McFadden. Phase field modelingof electrochemistry. I. Equilibrium.
Phys. Rev. E , 69(2):021603, 2004. [ ] H. Ceric and S. Selberherr. Electromigration in submicron interconnect features ofintegrated circuits.
Mat. Sci. Eng. R , 71(5):53–86, 2011. [ ] S. Lau, J.-Y. Feng, J. Olowolafe, and M.-A. Nicolet. Iron silicide thin film formationat low temperatures.
Thin Solid Films , 25(2):415–422, 1975. [ ] C. Joshi, T. Abinandanan, R. Mukherjee, and A. Choudhury. Destabilisation ofnanoporous membranes through GB grooving and grain growth.
Comp. Mater. Sci. ,139:75–83, 2017. [ ] W. W. Mullins. Flattening of a nearly plane solid surface due to capillarity.
J. Appl.Phys. , 30(1):77–83, 1959. [ ] L. Zhang, S. Ou, J. Huang, K. N. Tu, S. Gee, and L. Nguyen. Effect of current crowdingon void propagation at the interface between intermetallic compound and solder inflip chip solder joints.
Appl. Phys. Lett. , 88(1):012106, 2006. [ ] J. Hötzer, A. Reiter, H. Hierl, P. Steinmetz, M. Selzer, and B. Nestler. The parallelmulti-physics phase-field framework Pace3D.
J. Comput. Sci.-Neth. , 26:1–12, 2018.ibliography 221 [ ] A. Vondrous, M. Selzer, J. Hötzer, and B. Nestler. Parallel computing for phase-fieldmodels.
Int. J. High Perform. C. , 28(1):61–72, 2014. [ ] I. A. Blech and E. S. Meieran. Electromigration in thin Al films.
J. Appl. Phys. ,40(2):485–491, 1969. [ ] A. T. Huang, A. M. Gusak, K. N. Tu, and Y.-S. Lai. Thermomigration in SnPb compositeflip chip solder joints.
Appl. Phys. Lett. , 88(14):141911, 2006. [ ] P. S. Ho. Motion of inclusion induced by a direct current and a temperature gradient.
J. Appl. Phys. , 41(1):64–68, 1970. [ ] K. Zeng, R. Stierman, T.-C. Chiu, D. Edwards, K. Ano, and K. N. Tu. Kirkendall voidformation in eutectic SnPb solder joints on bare Cu and its effect on joint reliability.
J. Appl. Phys. , 97(2):024508, 2005. [ ] J. Christensen and J. Newman. Stress generation and fracture in lithium insertionmaterials.
J. Solid State Electr. , 10(5):293–319, 2006. [ ] Y. Xie, M. Qiu, X. Gao, D. Guan, and C. Yuan. Phase field modeling of silicon nanowirebased lithium ion battery composite electrode.
Electrochim. Acta , 186:542–551,2015. [ ] M. Klinsmann.
The effects of internal stress and lithium transport on fracture in storagematerials in lithium-ion batteries . PhD thesis, Karlsruher Institut für Technologie(KIT), 2016. [ ] S. J. Harris, E. K. Rahani, and V. B. Shenoy. Direct in situ observation and numericalsimulations of non-shrinking-core behavior in an MCMB graphite composite elec-trode.
J. Electrochem. Soc. , 159(9):A1501–A1507, 2012. [ ] Z. Guo, J. Zhu, J. Feng, and S. Du. Direct in situ observation and explanation oflithium dendrite of commercial graphite electrodes.
RSC Adv. , 5(85):69514–69521,2015. [ ] C. Lampe-Onnerud, J. Shi, P. Onnerud, R. Chamberlain, and B. Barnett. Benchmarkstudy on high performing carbon anode materials.
J. Power Sources , 97:133–136,2001. [ ] K. Takahashi and V. Srinivasan. Examination of graphite particle cracking as a failuremode in lithium-ion batteries: a model-experimental study.
J. Electrochem. Soc. ,162(4):A635–A645, 2015.ibliography 222 [ ] Z. Guo, L. Ji, and L. Chen. Analytical solutions and numerical simulations ofdiffusion-induced stresses and concentration distributions in porous electrodes withparticles of different size and shape.
J. Mater. Sci. , 52(23):13606–13625, 2017. [ ] P. Stein and B. Xu. 3D Isogeometric Analysis of intercalation-induced stresses inLi-ion battery electrode particles.
Methods, Comput and Mech, Appl. , 268:225–244,2014. [ ] J. Chakraborty, C. P. Please, A. Goriely, and S. J. Chapman. Combining mechanicaland chemical effects in the deformation and failure of a cylindrical electrode particlein a Li-ion battery.
Int. J. Solids Struct. , 54:66–81, 2015. [ ] Z. Cui, F. Gao, and J. Qu. A finite deformation stress-dependent chemical potentialand its applications to lithium ion batteries.
J. Mech. Phys. Solids , 60(7):1280–1295,2012. [ ] X. Zhang, W. Shyy, and A. Marie Sastry. Numerical simulation of intercalation-induced stress in Li-ion battery electrode particles.
J. Electrochem. Soc. ,154(10):A910–A916, 2007. [ ] J. Crank.
The mathematics of diffusion . Oxford university press, Oxford, 1979. [ ] M. Ebner, D.-W. Chung, R. E. García, and V. Wood. Tortuosity anisotropy in lithium-ion battery electrodes.
Adv. Energy Mater. , 4(5):1301278, 2014. [ ] T. Marks, S. Trussler, A. J. Smith, D. Xiong, and J. R. Dahn. A guide to Li-ion coin-cellelectrode making for academic researchers.
J. Electrochem. Soc. , 158(1):A51–A57,2011. [ ] H. Buqa, D. Goers, M. Holzapfel, M. E. Spahr, and P. Novák. High rate capabil-ity of graphite negative electrodes for lithium-ion batteries.
J. Electrochem. Soc. ,152(2):A474–A481, 2005. [ ] S. J. Harris, A. Timmons, D. R. Baker, and C. Monroe. Direct in situ measurements ofLi transport in Li-ion battery negative electrodes.
Chem. Phys. Lett. , 485(4):265–274,2010. [ ] S. J. Harris and P. Lu. Effects of inhomogeneities-nanoscale to mesoscale-on thedurability of Li-ion batteries.
J. Phys. Chem. C , 117(13):6481–6492, 2013. [ ] S. Santhanagopalan, Q. Guo, P. Ramadass, and R. E. White. Review of modelsfor predicting the cycling performance of lithium ion batteries.
J. Power Sources ,156(2):620–628, 2006.ibliography 223 [ ] W. Lai and F. Ciucci. Mathematical modeling of porous battery electrodes-Revisit ofNewman’s model.
Electrochim. Acta , 56(11):4369–4377, 2011. [ ] T. R. Ferguson and M. Z. Bazant. Nonequilibrium thermodynamics of porous elec-trodes.
J. Electrochem. Soc. , 159(12):A1967–A1985, 2012. [ ] V. D. Bruggeman. Berechnung verschiedener physikalischer Konstanten von hetero-genen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörperaus isotropen Substanzen.
Ann. Phys. , 416(7):636–664, 1935. [ ] J. Joos, T. Carraro, A. Weber, and E. Ivers-Tiffée. Reconstruction of porous electrodesby FIB / SEM for detailed microstructure modeling.
J. Power Sources , 196(17):7302–7307, 2011. [ ] M. Ender, J. Joos, T. Carraro, and E. Ivers-Tiffée. Quantitative characterizationof LiFePO cathodes reconstructed by FIB / SEM tomography.
J. Electrochem. Soc. ,159(7):A972–A980, 2012. [ ] M. Ender, J. Joos, T. Carraro, and E. Ivers-Tiffée. Three-dimensional reconstructionof a composite cathode for lithium-ion cells.
Electrochem. Commun. , 13(2):166–168,2011. [ ] T. F. Fuller, M. Doyle, and J. Newman. Simulation and optimization of the duallithium ion insertion cell.
J. Electrochem. Soc. , 141(1):1–10, 1994. [ ] M. Minakshi, D. Appadoo, and D. E. Martin. The anodic behavior of planar andporous zinc electrodes in alkaline electrolyte.
Electrochem. Solid St. , 13(7):A77–A80,2010. [ ] F. Cottrell. Residual current in galvanic polarization regarded as a diffusion problem.
Z. Phys. Chem. , 42:385–431, 1903. [ ] C. J. Wen, B. Boukamp, R. A. Huggins, and W. Weppner. Thermodynamic and masstransport properties of "LiAl".
J. Electrochem. Soc. , 126(12):2258–2266, 1979. [ ] R. W. Balluffi, S. Allen, and W. C. Carter.
Kinetics of materials . John Wiley & Sons,Hoboken, New Jersey, 2005. [ ] Y.-C. Joo and C. V. Thompson. Evolution of electromigration-induced voids in singlecrystalline aluminum lines with different crystallographic orientations.
Mater. Res.Soc. Symp. P. , 309:351–356, 1993. [ ] W. Wang, Z. Suo, and T. Hao. A simulation of electromigration-induced transgranularslits.
J. Appl. Phys. , 79(5):2394–2403, 1996.ibliography 224 [ ] H. Gan, W. Choi, G. Xu, and K. Tu. Electromigration in solder joints and solder lines.
JOM , 54(6):34–37, 2002. [ ] S. Shingubara, Y. Nakasaki, and H. Kaneko. Electromigration in a single crystallinesubmicron width aluminum interconnection.
Appl. Phys. Lett. , 58(1):42–44, 1991. [ ] O. Kraft, S. Bader, J. Sanchez Jr., and E. Arzt. Observation and modelling ofelectromigration-induced void growth in Al-based interconnects.
Mater. Res. Soc.Symp. P. , 308:267–372, 1993. [ ] O. Kraft, U. Möckl, and E. Arzt. Shape changes of voids in bamboo lines: a newelectromigration failure mechanism.
Qual. Reliab. Eng. Int. , 11(4):279–283, 1995. [ ] T. Zaporozhets, A. Gusak, K. Tu, and S. Mhaisalkar. Three-dimensional simulationof void migration at the interface between thin metallic film and dielectric underelectromigration.
J. Appl. Phys. , 98(10):103508, 2005. [ ] Z. Choi, R. Mönig, and C. Thompson. Effects of microstructure on the formation,shape, and motion of voids during electromigration in passivated copper intercon-nects.
J. Mater. Res. , 23(2):383–391, 2008. [ ] D. Du and D. Srolovitz. Electrostatic field-induced surface instability.
Appl. Phys.Lett. , 85(21):4917–4919, 2004. [ ] V. Gill, P. Guduru, and B. Sheldon. Electric field induced surface diffusion andmicro / nano-scale island growth. Int. J. Solids Struct. , 45(3):943–958, 2008. [ ] M. D. Morariu, N. E. Voicu, E. Schäffer, Z. Lin, T. P. Russell, and U. Steiner. Hierar-chical structure formation and pattern replication induced by an electric field.
Nat.Mater. , 2(1):48, 2003. [ ] Y. Wang, Y. Yao, and L. M. Keer. Surface diffusion induced shape evolution of multiplecircular voids under high current density.
J. Appl. Phys. , 121(20):205111, 2017. [ ] A. Mukherjee, K. Ankit, M. Selzer, and B. Nestler. Electromigration-induced surfacedrift and slit propagation in polycrystalline interconnects: Insights from phase-fieldsimulations.
Phys. Rev. Applied , 9:044004, 2018. [ ] A. Kumar, D. Dasgupta, C. Dimitrakopoulos, and D. Maroudas. Current-drivennanowire formation on surfaces of crystalline conducting substrates.
Appl. Phys.Lett. , 108(19):193109, 2016. [ ] E. Arzt, O. Kraft, and U. Möckl. Electromigration damage in conductor lines: Recentprogress in microscopic observation and mechanistic modelling.
Mater. Res. Soc.Symp. P. , 338, 1994.ibliography 225 [ ] B. Sun, Z. Suo, and W. Yang. A finite element method for simulating interfacemotion–I. Migration of phase and grain boundaries.
Acta Mater. , 45(5):1907–1915,1997. [ ] O. Kraft and E. Arzt. Electromigration mechanisms in conductor lines: void shapechanges and slit-like failure.
Acta Mater. , 45(4):1599–1611, 1997. [ ] M. Giesen and S. Dieluweit. Step dynamics and step-step interactions on the chiralCu(5 8 90) surface.
J. Mol. Catal. A-Chem. , 216(2):263–272, 2004. [ ] M. R. Gungor and D. Maroudas. Non-linear analysis of the morphological evolutionof void surfaces in metallic thin films under surface electromigration conditions.
Surf.Sci. , 415(3):L1055–L1060, 1998. [ ] D. Maroudas. Dynamics of transgranular voids in metallic thin films under electro-migration conditions.
Appl. Phys. Lett. , 67(6):798–800, 1995. [ ] M. R. Gungor and D. Maroudas. Current-induced non-linear dynamics of voids inmetallic thin films: morphological transition and surface wave propagation.
Surf.Sci. , 461(1-3):L550–L556, 2000. [ ] E. D. Koronaki, M. R. Gungor, C. I. Siettos, and D. Maroudas. Current-induced wavepropagation on surfaces of voids in metallic thin films with high symmetry of surfacediffusional anisotropy.
J. Appl. Phys. , 102(7):073506, 2007. [ ] Z. Li and N. Chen. Electromigration-driven motion of an elliptical inclusion.
Appl.Phys. Lett. , 93(5):051908, 2008. [ ] S.-B. Liang, C.-B. Ke, M.-B. Zhou, and X.-P. Zhang. Morphological evolution andmigration behavior of the microvoid in Sn / Cu interconnects under electrical fieldstudied by phase-field simulation. In , pages 260–265. IEEE, Changsha, 2015. [ ] S. Liang, C. Ke, W. Ma, M. Zhou, and X. Zhang. Numerical simulations of migrationand coalescence behavior of microvoids driven by diffusion and electric field in solderinterconnects.
Microelectron. Reliab. , 71:71–81, 2017. [ ] M. Mahadevan and R. M. Bradley. Phase field model of surface electromigration insingle crystal metal thin films.
Physica D , 126(3-4):201–213, 1999. [ ] J. Cho, M. R. Gungor, and D. Maroudas. Electromigration-driven motion of mor-phologically stable voids in metallic thin films: Universal scaling of migration speedwith void size.
Appl. Phys. Lett. , 85(12):2214–2216, 2004.ibliography 226 [ ] J. Krug and H. T. Dobbs. Current-induced faceting of crystal surfaces.
Phys. Rev. Lett. ,73:1947–1950, 1994. [ ] Y.-T. Cheng and M. W. Verbrugge. Evolution of stress within a spherical insertionelectrode particle under potentiostatic and galvanostatic operation.
J. Power Sources ,190(2):453–460, 2009. [ ] D. Chen.
Microscopic investigations of degradation in lithium-ion batteries . PhD thesis,Karlsruher Institut für Technologie (KIT), 2012. [ ] P. Goli, H. Ning, X. Li, C. Y. Lu, K. S. Novoselov, and A. A. Balandin. Thermal prop-erties of graphene–copper–graphene heterogeneous films.
Nano Lett. , 14(3):1497–1503, 2014. [ ] K. Mohsin, A. Srivastava, A. Sharma, C. Mayberry, and M. Fahad. Temperature sen-sitivity of electrical resistivity of graphene / copper hybrid nano ribbon interconnect:A first principle study. ECS J. Solid State Sc. , 6(4):P119–P124, 2017. [ ] N. Somaiah, D. Sharma, and P. Kumar. Electric current induced forward and anoma-lous backward mass transport.
J. Phys. D-Appl. Phys. , 49(20):20LT01, 2016. [ ] W. Chen, Y. Peng, X. Li, K. Chen, J. Ma, L. Wei, B. Wang, and Y. Zheng. Phase-field study on geometry-dependent migration behavior of voids under temperaturegradient in UO crystal matrix. J. Appl. Phys. , 122(15):154102, 2017. [ ] W. Chen, Y. Zhou, S. Wang, F. Sun, and Y. Zheng. Phase field study the effects ofinterfacial energy anisotropy on the thermal migration of voids.
Comp. Mater. Sci. ,159:177–186, 2019. [ ] L. Zhang, M. R. Tonks, P. C. Millett, Y. Zhang, K. Chockalingam, and B. Biner. Phase-field modeling of temperature gradient driven pore migration coupling with thermalconduction.
Comp. Mater. Sci. , 56:161–165, 2012. [ ] S. Y. Hu and C. Henager Jr. Phase-field simulation of void migration in a temperaturegradient.
Acta Mater. , 58(9):3230–3237, 2010. [ ] N. Prajapati, M. Selzer, B. Nestler, B. Busch, and C. Hilgers. Modeling fracture cemen-tation processes in calcite limestone: a phase-field study.