Rethinking battery degradation in presence of surface effects: mechanical versus electrochemical peformance mediated by charging condition
RRethinking battery degradation in presence of surface effects:mechanical versus electrochemical peformance mediated bycharging condition
Amrita Sengupta, Jeevanjyoti ChakrabortyFebruary 23, 2021
Abstract
Surface stresses, in nano-sized anode particles undergoing chemomechanical interactions,have a relaxing effect on the diffusion-induced stresses thus improving the mechanicalendurance of the particles, whereas, the compressive effect of surface stresses degradesthe electrochemical performance. However, this straightforward prediction of an improvedmechanical performance is challenged in this work. Silicon nanowires undergoing hugevolumetric changes during lithiation, may undergo significant axial length-increase, whichserves as an important criterion in determining the mechanical performance of SiNWs.Interestingly, surface stresses tend to reduce the length-increase under potentiostatic chargingcondition, but under galvanostatic charging, the length-increase gets enhanced, thus degradingthe mechanical performance of the SiNWs. To further make the study more inclusive, thenanowire is modelled with a constraining material at its core, and a competitive analysisis presented for the overall performance of the anode particles under the combined effectsof surface stresses and constraining material. The mathematical model is based on largedeformation theory, considering two-way coupling of diffusion-induced stresses and stress-enhanced diffusion. It is hoped that this study will provide a fresh perspective in designingnext-generation lithium-ion battery particles.
Keywords: lithium-ion battery; silicon; nanoparticles; nanowire; surface stress; length-increase;mechanical performance; electrochemical performance, finite deformation1 a r X i v : . [ c ond - m a t . s o f t ] F e b Introduction
The development of commercially-usable lithium-ion batteries (LIBs)[1–3] led to a sudden up-surge in their demand over the last few decades. Due to its large-scale applications ranging fromelectronic gadgets to aircrafts, the focus has now shifted towards improving the storage capacityand energy density of LIBs. In order to fulfill the growing demand for high storage capacity,high energy and power densities, silicon (Si) as an anode material, with a theoretical capacityof 4200 mAh g − , is the best replacement for traditionally-used graphite (having a theoreticalspecific capacity of 372 mAh g − ) [4–6]. The drawback associated with the usage of Si as anodeis the large expansion in volume (upto 310 %) of the anode particles upon insertion of lithium,and back to original volume upon dis-insertion [7–10]. This cyclic change in volume duringeach charging-discharging cycle leads to structural degradation[11–17] of the anode particles,eventually lowering their electrochemical performance[18, 19]. The structural degradation couldbe in the form of buckling instabilities [20–24], fracture[25–31], altered material properties of theanode particles [32], crack initiation [33], and a less-studied issue of length-increase in cylindricalparticles[34]. To mitigate these problems, researchers have come up with the solution of usingnano-sized silicon anode particles [35–42].With advancement in computational and manufacturing techniques, the research on differentkinds of nano-structured anode particles have gained momentum, for example solid and hollowspherical particles, nanowires, and nanotubes [43–51]. Nanostructured Si particles forming theanode in LIBs provide enough empty spaces around the particles to accommodate the largevolume expansion associated with lithiation. Further, the time required for Li atoms to reach thecore of the Si nanoparticles are less than that for larger particles. Thus, using nanoparticles, thetime taken to reach full lithiation decreases. It is important to note here that, since the Li atomtakes very less time to reach the Si core, the diffusion-induced stresses (DISs) generated within aSi particle (due to concentration-inhomogeneity of Li) is significantly low for nanoparticles. Thelowering of DISs eventually lowers the chances of failure through fracture, thus increasing thestructural integrity of the particles. Beside design perspectives, nano-wires and nano-tubes haveenhanced electrochemical performance due to their robust electrical contact with the metalliccurrent collector which enable each particle to contribute to the capacity [43, 44]. A numberof studies indicate the prominence of surface stress in nano-particles[52] as the prime reasonbehind their structural durability [53–60]. Furthermore, the effect of surface stress is dependenton the particle size (with decrease in particle size surface stress effect increases[61, 62]), as wellas the charging rate (with increase in charging rate, surface stress effect increases) [56]. Hence,in order to maximize the mechanical endurance, we might decrease the particle size and increasethe charging/discharging rate as much as possible but, it comes with a constraint in terms of theelectrochemical performance of the electrode, or the battery as a whole. It has been observedfor spherical Si anode particles undergoing small deformation, that the presence of surfacestress degrades the electrochemical performance of the battery, and with increase in surfacestress, the degradation increases [63]. The suppressing or compressive effect due to surfacestress at the surface of the particle, inhibits lithium uptake, thus lowering the charge capacity ofthe battery [64]. Therefore, the surface stresses associated with nanoparticles give rise to twocompetitive phenomena where, on one hand, the particles become mechanically robust withdecreasing tensile stresses, and on the other hand, the electrochemical performance is affectednegatively. Strikingly, a decreasing tensile stress alone does not ensure an upgraded mechanicalperformance. Although it does ascertain a reduction in fracture and crack initiation, but anothercrucial mechanical phenomenon in case of axially unconstrained Si nanowires/nanotubes is thelength-increase of the particles because of volumetric expansion during lithiation [34]. Thelength-increase, along with the DISs, together will be considered as the mechanical performanceof the electrode particles in our study. The entire analysis is performed for two different charging2onditions: first, galvanostatic case where the charging/discharging rate is kept constant, andsecond, potentiostatic case where the interfacial voltage drop at the electrode-electrolyte interfaceis kept constant. The effect of surface stresses on the length-increase for different chargingconditions has been analyzed in detail over very interesting and competitive results. To make ourstudy comprehensive, we introduce a constraining material at the inner core of the Si nanotube,the properties of which modulates the length-increase of the particle [34] which is already underthe influence of surface stresses at the outer periphery.The primary contribution of this work is an analysis of how charging condition plays avital role in the surface-stress-induced competition between mechanical and electrochemicalperformances associated with reduced particle size in nano-structured cylindrical Si anode parti-cles in a coupled, electro-chemo-mechanical environment, undergoing large deformation duringeach charging/discharging cycle due to insertion/dis-insertion of Li atoms. While developingthe model for the Si nano-particle (owing to its huge volumetric expansion and contraction),we consider the modified surface stress formulation [56] used to calculate the surface stressesin cylindrical nanoparticles undergoing finite deformation. Additionally, we also include thepossibility of plastic deformation during lithiation/delithiation in our model, as considered insimilar studies done previously [20, 34, 41, 65–67]. The rest of the paper is organized as follows.In Section 2.1 we set up the model geometry to be used in the analysis. Section 2.2 has twosections: first, Section 2.2.1 which covers the chemical and electrochemical aspects, and definesthe modelling conditions for galvanostatic and potentiostatic charging conditions; second, Section2.2.2 which covers the mechanical aspects with special focus on the surface stress at the outerperiphery, in an abridged form. In Section 3, we discuss the results separately for galvanostaticcase (Section 3.1) and for potentiostatic case (Section 3.2). The qualitative resemblance ofthe results with those obtained by previous researchers [53–55, 66] validates our computationalmodel for further study. The crucial part of this work lies in the extensive analysis of howcharging condition is primarily responsible for the overall performance of the battery underdecreasing particle size. Finally in Section 4 we summarize the important results. We consider a single, cylindrical, hollow silicon anode particle in the cylindrical ( r, θ, z ) coordinatesystem, having inner radius R i , outer radius R , and height L , undergoing axisymmetricdeformation due to axisymmetric lithiation and delithiation. The inner core has a constrainingmaterial of radius R i , whose material properties are defined by the thickness of the core( r d = R i /R ) and its yield strength σ f ; rest of the properties are similar to that of the electrode.The Li influx occurs radially through the outer periphery. Deformation in the axial directionis allowed. We consider a coupled-phenomenon of diffusion-induced stress and stress-enhanceddiffusion. The particle is free to grow in the axial direction, keeping the radial symmetry intact.A schematic diagram depicting the model is shown in Fig. 1. The governing differential equations characterizing the chemo-mechanical phenomenon of lithia-tion and de-lithiation of silicon anode particles are stated in the following sub-sections, coveringthe chemical and electrochemical aspects first, and then the mechanical aspect. The evolution ofdiffusion-induced stresses have been studied extensively by previous researchers [34, 41, 66–69].Because of the availability of detailed literature, we introduce the important expressions and3 onstraining materialElectrode material x y (a) (b) (c)Exaggerated view of the surface film Li + ionsLi atoms after reductionNeglecting the surface film Figure 1: A schematic representation of the model set up; (b) and (c) shows the top views of aquadrant, where (b) is without and (c) is with surface-stress effects. These can be understoodextensively through the results and discussions.equations in their non-dimensionalized form, where the scheme of non-dimensionalization ispresented below.˜ r = rR , ˜ z = zL , ˜ D = DD , ˜ t = D R t, ˜ u = uR , ˜ w = wL , ˜ J r, = V Si m R D J r, , ˜ σ r,θ,z, eff = V Si m R g T σ r,θ,z, eff , ˜ σ r,θ,z, eff = V Si m R g T σ r,θ,z, eff , ˜ µ ,s = 1 R g T µ ,s . (1)Here r and z are the radial and axial coordinates of any point in the reference configuration;( u, v, w ) are the displacements in radial, circumferential, and axial directions, respectively (notethat axisymmetric diffusion and deformation leads to axisymmetric growth, and hence, v ≡ J r and J are the flux of lithium at any radial position r and at the surface r = R ; σ r,θ,z arethe Cauchy stresses in radial, cicumferential, and axial directions, respectively; σ r,θ,z are thePiola-Kirchhoff stresses (PK1) in radial, cicumferential, and axial directions, respectively; lastly, µ is the chemical potential. The constants D , V Si m , R g , T are part of the list of mechanicalproperties and parameters given in Table 1. The lithiation/delithiation into/ out of the anode particle is governed by the conservationequation: ∂c∂ ˜ t = − ∂ ˜ J r ∂ ˜ r − ˜ J r ˜ r . (2)Here the flux of Li ( ˜ J r ) is given as ˜ J r = − ˜ Dc ∂ ˜ µ∂ ˜ r , (3)4able 1: Mechanical properties and parameters used in the modelMaterial property or parameter Value A , parameter used in activity constant − − [66] B , parameter used in activity constant − − [66] D , diffusivity of Si 1 × − m s − [70]˙ d , characteristic strain rate for plastic flow in Si 1 × − s − [66] Y , modulus of elasticity of pure Si 90 .
13 GPa [71] m , stress exponent for plastic flow in Si 4 [68] R g , universal gas constant 8 .
314 JK − mol − R , initial radius of unlithiated Si electrode 50 nm, 200 nm T , temperature 300 K V Si m , molar volume of Si 1 . × − m mol − [66] x max , maximum concentration of of Li in Si 4.4 α , coefficient of diffusivity 0 .
18 [72] η , coefficient of compositional expansion 0 . η E , rate of change of modulus of elasticitywith concentration − . ν , Poisson’s ratio of Si 0 .
28 [66] σ f , initial yield stress of Si 0 . ± .
08 GPa [73] λ s , surface Lam´e constant 3.5 Nm − [61] µ s , surface Lam´e constant -6.23 Nm − [61] k , reaction rate constant 1 × − m / mol − / s − [63] F , Faraday’s constant 96500 Cmol − φ ref , reference potential 0.88 V [69] x Li + , Li ion concentration in electrolyte 1000 molm − [63]5here ˜ D = exp (cid:16) αV Si m σ θ R g T (cid:17) is the non-dimensionalized diffusivity and c is the concentration of Li.The chemical potential ˜ µ is decomposed into stress-independent and stress-dependent parts as:˜ µ = µ R g T + ln( γ c) (cid:124) (cid:123)(cid:122) (cid:125) stress-independent + ˜ µ S + ˜ µ S + ˜ µ S (cid:124) (cid:123)(cid:122) (cid:125) stress-dependent , (4)where γ = 1(1 − c ) , (5)and the expressions of the terms ˜ µ S , ˜ µ S , ˜ µ S have been properly derived in Eq. (12)-(25) of[66]. The expression for γ as described in Eq. (5) is chosen for a reason, as explained in theAppendix. Initial and boundary conditions involving the constraining material
Equation (2) issolved both for the constraining material as well as the electrode material (here, silicon). Inboth these domains, it needs an initial condition and two boundary conditions. We start with aLi-free particle, thus c (˜ r,
0) = 0 in both the domains. At the centre of the cylinder, the flux isalways zero. Therefore, ˜ J r (0 , ˜ t ) = 0 at the centre. At the interface of the constraining materialand the electrode, we have continuity of concentration and flux. Due to very low coefficientof volumetric expansion of the constraining material, the concentration of Li within it and itsexpansion in volume are low. The concentration levels also depend upon the diffusivity of thematerial, although in the present model, diffusivities in both the domains are considered tobe the same. The boundary condition at the outer periphery is determined by the method ofcharging the battery: first, galvanostatic charging condition where the C-rate is kept constant;second, potentiostatic charging condition where the interface voltage drop remains constant. Weelaborate on these charging conditions in the following sections. Interface voltage-drop
We know, while charging, the Li ions in the electrolyte reduces toform Li atoms at the surface of the anode, before diffusion. Thus, the electrochemical reactionat the surface of the anode can be written as [69]e − | electrode surface + Li + | electrolyte charging − (cid:0) ======== (cid:1) − discharging Li | electrode surface (6)Let ¯ µ Li , ¯ µ Li + , and ¯ µ e − denote the electrochemical potentials of Li atoms (on electrodesurface), Li + ions (in the electrolyte), and electrons (on electrode surface). We define the drivingforce , commonly known as the ‘overpotential’ for the chemical reaction (Eq. (6)), by takingthe difference between the electrochemical potentials of the reactants and products, divided byFaraday’s constant, given as η o = 1 F [¯ µ Li − (¯ µ Li + + ¯ µ e − )] , (7)for charging, and η o = 1 F [(¯ µ Li + + ¯ µ e − ) − ¯ µ Li ] , (8)for discharging. The nature of the overpotential drives the reaction towards a particular direction,as follows:- when η o >
0, Li oxidizes to form Li + ,- when η o <
0, Li + reduces to form Li, and- when η o = 0, the reaction is in equilibrium condition.6urther, we may write the electrochemical potentials ¯ µ Li , ¯ µ Li + , and ¯ µ e − , in terms of chemicalpotentials and electric potentials as ¯ µ i = µ i + z i F φ i , (9)where µ i is the chemical potential of the i th charged species, z i is its valence, and φ i is the electricpotential field where the charged species i exists. In our case, the electrochemical potential ofeach species is expressed as ¯ µ Li = µ Li , (10a)¯ µ Li + = µ Li + + F φ e , (10b)¯ µ e − = µ e − − F φ a . (10c)Here, Li atoms are neutral species, Li + ions have a valence of (+1), and e − have a valence of( − φ e and φ a are the electric potentials in the electrolyte and anode, respectively.At this point, we make two important assumptions:First, the electrolyte and the electrode are assumed to be infinite reservoirs of Li + ions andelectrons, respectively, and their activities are equal to 1. We know, the chemical potentialof any species is given by µ i = µ i + R g T ln( a i ), where a i is the activity of the species, and µ i is the reference chemical potential. Therefore in Eq. (10), µ Li + = µ + and µ e − = µ − , but µ Li = µ + ˆ µ Li , where ˆ µ Li is a function of concentration, stress, and temperature (refer Eq. (4)).Second, the electric potentials φ e and φ a are assumed to be constant within the electrolyteand the anode. The variations occur only in the electrode-electrolyte surface.Based on the above assumptions, we re-write Eq. (10) as¯ µ Li = µ + ˆ µ Li , (11a)¯ µ Li + = µ + + F φ e , (11b)¯ µ e − = µ − − F φ a . (11c)Now using Eq. (11) in (7) for charging, we have η o = ˆ µ Li F − µ + + µ − − µ F + ∆ φ, (12)where ∆ φ = φ a − φ e is the “voltage drop” across the electrode-electrolyte interface.At equilibrium, η o = 0, thus from Eq. (12) we have the equilibrium interfacial voltage drop∆ φ eq = φ ref − ˆ µ Li F , (13)where φ ref = µ + + µ − − µ F is the reference electric potential.From Eq. (12) and (13) we can write η o = ∆ φ − ∆ φ eq . (14)The overpotential η o is related to the net current density I n (refer Eq. (17)) in a non-equilibrium electrochemical reaction by the well-known Butler-Volmer equation, I n = I (cid:20) exp (cid:18) − β F η o R g T (cid:19) − exp (cid:18) (1 − β ) F η o R g T (cid:19)(cid:21) , (15)7here I is the concentration-dependent exchange current density, given as I = F k x m x (1 − β )Li + (1 − c s ) β c (1 − β ) s . (16)Here c s is the Li concentration at the anode surface, x Li + is the Li-ion concentration in theelectrolyte, k is the heterogeneous reaction rate constant, and β is taken to be 0.5. Galvanostatic condition
In case of galvanostatic charging/discharging of the battery, wedecide a constant C-rate N , and the anode particles are charged/discharged with a constantcurrent density I n , given as [63] I n = F R x m N , (17)where F is Faraday’s constant, and x m = x max /V Si m is the maximum moles of Li per molarvolume of Si. The flux density at the cylindrical surface ˜ J is related to I n as [63]˜ J = I n R F D x m . (18)It is important to note that it is this relation which is used as the second boundary conditionfor the governing differential equation for diffusion (Eq. (2)).Using Eq. (15) we find out the overpotential as η o = 2 R g TF sinh − (cid:18) − I n I (cid:19) (19)for charging, and η o = 2 R g TF sinh − (cid:18) I n I (cid:19) (20)for discharging.Therefore from Eq. (14), we have the interfacial voltage drop as∆ φ = φ ref − ˆ µ Li F + 2 R g TF sinh − (cid:18) ± I n I (cid:19) (21)The interfacial voltage drop, being dependent upon various factors such as stress, concentration,particle-size, temperature, C-rate, etc. is sensitively modulated when any of these factors vary.In the Results and Discussion section we analyze the effects of particle size and C-rate on ∆ φ . Potentiostatic condition
In case of potentiostatic charging/discharging of the battery,the charging/discharging occurs at a constant interface potential difference ∆ φ (Eq. (12)),whereas the charging/discharging rate of Li may vary. In the present case, we set the interfacepotential difference ∆ φ at 5.31 mV [63].From the Butler-Volmer equation (Eq. (15)) the constant current density I n at which theanode particles are charged/discharged is given as I n = − I sinh (cid:20) F R g T (∆ φ − ∆ φ eq ) (cid:21) (22)8 .2.2 Mechanical aspect Pertaining to the axisymmetric situation, the radial component of the mechanical equilibriumequation in the reference configuration is given as: ∂ ˜ σ r ∂ ˜ r + ˜ σ r − ˜ σ θ ˜ r = 0 , (23)Considering linear-elastic, isotropic behaviour, the strain-energy density function in the referenceconfiguration, assuming small deformation in the elastic domain, takes the form: W = J c Y (1 + ν ) (cid:20) ν − ν ( E ekk ) + E ejk E ekj (cid:21) , (24)where ˜ Y = ˜ Y (1 + η E x max c ) is the concentration-dependent modulus of elasticity; ν is thePoisson’s ratio; J c = 1 + 3 ηx max c gives the ratio of final to initial volumes related to thestress-free, unconstrained swelling/contraction behaviour of the Si particle on diffusion, in/outof the anode, respectively. Here x max is the maximum number of moles of Li per mole of Si( x max = 4 .
4, therefore, in Li x Si, 0 ≤ x ≤ .
4) and c = x/x max represents the spatio-temporallyvarying concentration of Li inside Si particle. The PK1 stresses (˜ σ r , ˜ σ θ , ˜ σ z ) can be derived from W [34] and are expressed in terms of the elastic part of the Lagrangian strains ( E er , E eθ , E ez ) as:˜ σ r = J c ˜ Y (1 + ν )(1 − ν ) [(1 − ν ) E er + ν ( E eθ + E ez )] 2 E er + 11 + ∂ ˜ u/∂ ˜ r , (25a)˜ σ θ = J c ˜ Y (1 + ν )(1 − ν ) [(1 − ν ) E eθ + ν ( E er + E ez )] 2 E eθ + 11 + ˜ u/ ˜ r , (25b)˜ σ z = J c ˜ Y (1 + ν )(1 − ν ) [(1 − ν ) E ez + ν ( E eθ + E er )] 2 E ez + 11 + ∂ ˜ w/∂ ˜ z . (25c)Instead, the description of E er , E eθ , E ez are connected to the kinematics of the particle [20]. Boundary conditions involving constraining material
There is no deformation at thecenter of the constraining material, i.e. ˜ u | ˜ r =0 = 0; at the interface ( x = R i ) there is continuityof radial displacements and radial stresses between the constraining material and silicon. In theabsence of surface stress at the outer periphery of the Si particle, the radial stress ˜ σ r | ˜ r =1 = 0,since we assume a negligible effect of the electrolyte and SEI on the anode particles. However,when we consider a particle of outer radius <
50 nm, the surface stresses can no longer beneglected, and in such cases, at the periphery,˜ σ r e r = − (cid:98) ˜ σ θ R w V Si m R g T e r , (26)where (cid:98) ˜ σ θ is the PK1 surface stress in the circumferential direction, the mathematical formulationof which is presented in details in [56]. There is no constraint in the axial direction, and hencethe axial force over the entire domain (from x = 0 to x = R ) is zero. Mathematically,2 π (cid:90) RiRo ˜ σ z ˜ rd ˜ r (cid:124) (cid:123)(cid:122) (cid:125) core constraining material + 2 π (cid:90) RiRo ˜ σ z ˜ rd ˜ r (cid:124) (cid:123)(cid:122) (cid:125) outer electrode material = 0 (27)9 Results and Discussions
The governing equations as obtained in Section 2 are highly coupled and therefore, are solvednumerically using the PDE interface of COMSOL Multiphysics 5.3a simulation software. Theresults have been discussed in two separate sections: first, for the galvanostatic charging con-dition; second, for the potentiostatic charging condition. For future reference, we define thestate-of-charge or SOC as the average concentration of lithium within the Si particle, withrespect to its reference configuration, and is mathematically expressed as: SOC= (cid:16)(cid:82) R c πr dr (cid:17)(cid:16)(cid:82) R πr dr (cid:17) .In the galvanostatic condition, we consider a complete charging-discharging cycle (chargingupto 80-90 % SOC) to undertsand the electrochemical performance better. However, in thepotentiostatic case, the anode particle attains an equilibrium SOC during lithiation, and nofurther charging increases the SOC beyond that. Detailed discussions are done in the respectivesections. Since the detailed analysis on the evolution of stresses can be found in earlier literature [66, 67],we start our discussion with the plots analyzing the electrochemical performance of the electrodeparticle. (a) (b)
Figure 2: Variation of interface voltage-drop with (a) C-rate (keeping particle size constant at 50nm) and (b) particle size (keeping C-rate constant at 20C), plotted against SOC for a completecharging-discharging cycle.The electrochemical performance in case of galvanostatic condition can be best judged bythe interface voltage-drop variation. As mentioned earlier, interface voltage-drop is dependenton various factors, most important among them being particle size and charging rate. Figures 2(a) and (b) shows how a variation in particle sizes and charging rates affect the interface-voltageprofiles.As charging/discharging rate determines the net current density and the influx rate (Eq.(17) and (18)), it directly affects the overpotential value, as well as the equilibrium potential.Thus to examine the effect of C-rate on the overall voltage-drop profile, we plot ∆ φ against SOCat various C-rates (2 (a)), for a particle of radius 50 nm, considering no surface stresses. As10 .2 0.4 0.6 0.80.70.80.911.11.2 (a) (b) (c) (d) Figure 3: Keeping C-rate constant at 20C and particle size 15 nm, we plot (a) interface voltage-drop against SOC, with and without surface stress, along with the contribution of surface stress(inset figure), (b) overpotential against SOC, with and without surface stress, (c) equilibriumpotential against SOC, with and without surface stress, and (d) chemical potential at the surfaceof Si particle against SOC, with and without surface stress. N (charging/discharging rate) increases, the net current density I n increases, which enhancesthe overpotential magnitude, thus decreasing ∆ φ during charging and increasing ∆ φ duringdischarging. The voltage drop profile becomes broader with increasing C-rate. Next, to examinethe effect of particle size on the voltage-drop profile, we plot ∆ φ against SOC for various R values (2 (b)), keeping C-rate fixed at 20 C. It is observed that, as the particle size decreases,the net current density for a given C-rate (refer Eq. (17)) decreases, which further decrease themagnitude of the overpotential. This increases the interfacial voltage drop during charging anddecreases it during discharging, hence narrowing the gap in voltage-drop cycle.The surface stress effects become more significant with decrease in the size of the anodeparticles [53, 54, 56]. Therefore, to analyze the effect of surface stress on ∆ φ , we plot ∆ φ againstSOC (Fig. 3 (a)), with and without considering surface stress, for a particle of radius 15 nm,charged/discharged at 20C rate. It is observed that the voltage drop profile shifts downwarddue to the effect of surface stress. In the inset, we see the contribution of surface stress in ∆ φ ;the shift, as we can see, is more or less constant throughout, except for a brief period whenLi concentration inside anode is very low. Now, we would try to correlate this shift with the11 eactants: Li + and e - Product: Li, w/o-SSProduct: Li, w/SS w/o-SS G w/SS G Surface energy, *S W G (Spontaneous reaction)Time G i bb s F r ee E n e r gy Figure 4: A schematic representation of the variation of Gibbs free energy with time, with andwithout considering surface stress, for the reaction: Li + + e − → Li at the surface of the anodeparticle. [ W ∗ s is the interface energy per unit area in the reference configuration [56].]overall electrochemical performance. To understand the shift, we plot overpotential against SOC(Fig. 3 (b)), with and without surface stress, equilibrium potential against SOC (Fig. 3 (c)),with and without surface stress, and chemical potential at the surface of Si particle against SOC(Fig. 3 (d)), with and without surface stress. Clearly, overpotential trends in both the casesof with and without surface stress, overlaps, except for very low SOCs, which means, surfacestress does not affect the overpotential directly. The effect of surface stress on overpotentialis indirect, through the surface concentration terms in I , and therefore, is negligible. Next,from the equilibrium potential trends, we witness a similar negative shift as observed in caseof interfacial voltage-drop, which implies the shift in ∆ φ is a direct consequence of the shift in∆ φ eq . From Eq. (13), we know ∆ φ eq = φ ref − ˆ µ Li F , where the value of φ ref is constant. Therefore,the shift in ∆ φ eq occurs due to a shift in the chemical potential at Si surface, given by ˆ µ Li (seFig. 3 (d)). It is seen from Fig. 3 (d) that ˆ µ Li becomes less negative when surface stress isconsidered. We know, the chemical potential is given as: µ i = (cid:20) ∂G∂n i (cid:21) S,T,j (cid:54) = i , (28)where µ i is the chemical potential of species i , defined as the change in Gibbs free energy ( G )of the system due to an infinitesimal change in the number of species i in the system, keepingentropy ( S ), temperature ( T ), and other species constant. We also know, that a reaction isspontaneous when ∆ G is negative. The positive shift of ˆ µ Li in Fig. 3 (d) implies that the changein Gibbs free energy during the reduction reaction (Li + + e − → Li) at the surface of the anodeparticle is less negative in the presence of surface stress, than in its absence (Fig. 4). Thishappens because of the presence of surface energy at the anode surface, which increases the Gibbsfree energy at the product side. Therefore, when surface stress comes into play, the reduction ofLi + ions to form Li atoms become less spontaneous than the case without surface stress. Since thereduction reaction determines the level of charging, we can state that the charging and hence, theelectrochemical performance is affected negatively in the presence of surface stress. Further, westudy how this downward shift of ∆ φ can be modulated by varying the particle radius and C-rate.12 (a) (b) Figure 5: Effect of surface stress varies with (a) particle size (keeping C-rate constant at 20C),and (b) C-rate (keeping particle size constant at 50 nm); plotted against SOC for a completecharging-discharging cycle.It is well-known from previous works [56] how surface stresses and hence the DISs getmodulated by the variation in particle sizes and charging rates. As mentioned earlier, interfacevoltage-drop is dependent on various factors, most important among them being particle sizeand charging rate. In Fig. 5 (a), we observe the variation of (∆ φ w / o − SS − ∆ φ w / SS ) with SOC fordifferent particle radii, keeping charging rate fixed at 20C for each of the cases. As the particlesize decreases, the surface stress effect increases, i.e. the magnitude of negative shift as observedin Fig. 3 increases. This means, as we reduce the size of the anode particle, the surface energyincreases and the electrochemical performance of the particle decreases. Next, in Fig. 5 (b),we plot the variation of (∆ φ w / o − SS − ∆ φ w / SS ) with SOC for different charging rates, keepinginitial particle size same as R = 50 nm for each of the cases. We observe that at chargingrate 2C, there is no effect of surface stress, but as we increase the charging rate, the surfacestress effect becomes more and more prominent. Although the overall magnitude does not varymuch with increase in C-rate, the gap between charging and discharging magnitudes increases.Therefore, for very high C-rates (6C and 20C), the electrochemical performance reduces, butthe reduction is comparable, and is independent of the charging rate. This result is consistentwith the effects of increasing influx rate on surface stress as witnessed in an earlier study [56].Similar to Fig. 5 (b), the effects of charging/influx rate on the surface stress was only vis-ible during the initial stages of lithiation when the concentration inhomogeneity of Li is very high.In all the previous plots (Fig.3-Fig.5) the yield strengths of both the inner-core (constrainingmaterial) and the Si electrode were the same. Further, it is to be noted that the thickness ofthe constraining material is also kept constant at 0.3. To understand how the constrainingmaterial-properties affect the mechanical and electrochemical performances of the electrodeparticle, we vary the ratio of yield strengths of the constraining material to that of Si ( σ f σ f )from 0.01 to 10 (by 4 orders of magnitude). The electrochemical performance is gauged by theelectrode-potential trends, whereas the mechanical performance is reckoned by the amount oflength-increase of the axially-unconstrained particle.Figure 6 shows the (a) variation of electrode potential ∆ φ , and (b) variation of length-increase ∂w∂z over a complete charging-discharging cycle, for ratios of yield strengths σ f σ f varying from0.01 to 10. The charging (and discharging) occurs at a rate of 20C, and the particle size is13 (a) (b) Figure 6: (a) Variation of electrode potential, and (b) variation of length-increase over a completecharging-discharging cycle, for various yield-strengths of the constraining material. The chargingrate is kept at 20C and the particle size is taken to be 15 nm.taken as 15 nm in each of the cases. We have neglected the effect of surface stresses in thisfigure, so as to focus only on the effects of constraining material-properties. The increasing yieldstrength of the constraining material at the core shifts the electrode potential ∆ φ towards thedownward direction (Fig.6(a)), indicating an increase in the surface energy and thus, decrease inthe electrochemical performance of the electrode, albeit for ratios of 1 and below, this reductionis not significant. On the other hand, the length-increase of the electrode particle reduces due toan increase in the yield strength of the constraining material, from 0.01 to 10. The change inthe length-increase, however, remains within 1 - 5 %. It is important to note that the positionof the constraining material, as well as its thickness play very crucial roles in determining theamount of length-increase. An extensive analysis on this topic exists in Figures 3 and 4 of [34].Our motivation behind considering a constraining material at the inner-core is to understandthe effects of constraining material and surface stresses, individually and simultaneously, in thesame study. (a) (b) Figure 7: (a) Variation of electrode potential with SOC, and (b) variation of length-increase withSOC for two different values of σ f σ f , without (solid lines) and with (dashed lines) consideringsurface stresses. The charging rate is kept at 20C and the particle size is taken to be 5 nm.14ow that we have delineated the significance of yield strength of the constraining materialon the electrode potential and length-increase, we need to understand the effect surface stresseshave on both of them. Figure 7 shows the effect of surface stresses on the (a) electrode potentialand (b) length-increase for two different values of yield-strengths of the constraining material( σ f σ f =0.1, 10) and thickness r d = 0 .
3. As observed in Fig.3(a), the electrode potential shiftsdownward as a result of surface stresses, the trend is similar here for both σ f σ f = 0.1 and 10. Themagnitude of this shift is not significantly different for the two values of yield-strengths. Comingto Fig. 7(b), we observe an interesting trend. The length-increase is enhanced when surface stressis considered, and the magnitude of this increase in ∂w∂z is higher for higher yield strengths of theconstraining material. This means, as the ratio of yield-strengths σ f σ f increases, the effect ofsurface stress on ∂w∂z increases, and it becomes more and more detrimental. Given the compressiveeffect of surface stress at the outer-periphery, this result is quite counter-intuitive, however athorough invetigation reveals that when surface stress is considered, the radial expansion of theouter-periphery decreases as compared to without surface stress, as there exists a compressivestress at the outer-periphery in the radial direction. Since the volume-expansion due to lithiationis similar in both the cases (without and with surface stress), the length increase ∂w∂z is enhancedin the latter, to compensate for the reduced radial-expansion. Now, in addition to this, when theyield strength of the constraining material is increased, the radial compressive stress increases atthe inner-periphery of the electrode and hence, the radial expansion is further subdued from boththe inner and outer peripheries of the electrode particle. As a combined effect, the magnitude of ∂w∂z increases further. (a) (b) Figure 8: (a) Variation of ∆ (cid:0) ∂w∂z (cid:1) with σ f σ f for different particle sizes, keeping charging rateconstant at 20C. (b) Variation of ∆ (cid:0) ∂w∂z (cid:1) with σ f σ f for different charging rates, keeping particlesize constant at 15 nm.The increase in ∂w∂z as an effect of surface stress can be represented by ∆( ∂w∂z ) which isdefined as ∆( ∂w∂z ) = ( ∂w∂z ) w/SS − ( ∂w∂z ) w/o-SS . Figure 8 depicts the variation of ∆( ∂w∂z ) with σ f σ f fordifferent (a) particle sizes, and (a) charging rates. The length-increase is considered at 80% SOC.From Fig. 8(a), it is evident that ∆( ∂w∂z ) increases with increase in σ f σ f , and with decrease inparticle sizes. This result is consistent with a previous work [56] where surface stress increases indecrease in particle size. On the contrary, in Fig. 8(b) we observe a decrease in the magnitudeof ∆( ∂w∂z ) with increase in C-rate. Although this result might seem like contradicting previousresults [56], but if we look closely at the results obtained in [34], we would find that the over-allmagnitude of length-increase reduces with increase in charging rates. The detailed explanation15f this phenomenon exists in the cited document. Figure 9: Variation of SOC against time for particles of R = 50 and 15 nm, with and withoutconsidering surface stress effects.In potentiostatic condition, as ∆ φ eq nears ∆ φ , the overpotential value becomes less and lessnegative. We have set a minimum value for overpotential as − . η o <
0, the reaction diesdown at an overpotential of − . η o = 0. Here, for the sake of obtaining numericalresults we have considered a cut-off value for η o as − . (a) (b) (c) Figure 10: Variation of (a) overpotential (b) equilibrium potential, and (c) surface chemicalpotential, with and without surface stress, plotted against SOC for a potentiostatic chargingcase in a particle of size 15 nm.When surface stress is absent, the equilibrium SOC for both the particles (50 and 15 nm)are equal. Intuitively, time taken by Li to diffuse into a larger particle is greater than that for asmall particle. So, the reduction reaction at the anode surface and evolution of Li concentrationinside the anode particle is independent of particle size. On consideration of surface stress, the16quilibrium SOC for both the particles drop to a lower value, and this drop is higher for thesmaller particle as surface stress is higher in case of 15 nm particle, as compared to that in 50nm. Hence, as the charge content within the anode particle decreases when surface stress isconsidered, we can directly conclude a degradation in the electrochemical performance due tothe presence of surface stress. During the course of this study it has come to our notice that theequilibrium SOC is also determined by the form of µ (stress-independent chemical potential)considered, and a detailed discussion of the same is done in the Appendix section.To understand the effects of surface stress better, in Fig. 10, we plot against SOC (a) theoverpotential, (b) the equibrium potential, and (c) the chemical potential, with and withoutsurface stress for a particle of initial radius 15 nm. The interface voltage-drop being constant inpotentiostatic case, the change in overpotential occurs due to the change in equilibrium potential.Furthermore, the change in equilibrium potential is reflected as a change in the chemical potentialand hence, change in the Gibbs free energy at the anode surface where Li ions get reduced toform Li atoms. Similar to the obsevations in case of galvanostatic charging condition (Fig. 3),the chemical potential becomes less negative when surface stress is considered, and thus, thespontaneity of Li ions to form Li atoms decreases. This affects the charging and electrochemicalperformance negatively. (a) (b) Figure 11: (a) Variation of SOC with time for different values of σ f σ f ; (b) variation of ∂w∂z with σ f σ f at the equilibrium SOC, without (solid lines) and with (dashed lines) considering surfacestresses. The particle size is considered to be 15 nm.Until now, for the potentiostatic case, the yield strengths of both the constraining materialand Si were equal, and we were only studying the effect of surface stress on the electrochemicalperformance. Just as we proceeded in the galvanostatic case, we would like to see the effects ofmaterial properties of the constraining material on the electrochemical performance, as well asmechanical performance (which is demonstrated by the length-increase of the particle, similar toSection 3.1). Figure 11(a) shows the variation of SOC with time, for different values of σ f σ f (1, 2,5, 10), without (solid lines) and with (dashed lines) considering the effects of surface stress. Thegap between the solid and the dashed lines reduces as the ratio of yield strengths increases. Also,with increase in σ f σ f value, the equilibrium SOC decreases. For the mechanical performance,Fig.11(b) shows the variation of ∂w∂z at equilibrium SOC with σ f σ f , without (solid line) and with(dashed line) considering surface stress. Quite opposite to what we obtained in galvanostaticcharging case, the length-increase magnitudes decrease when surface stress in considered; thisoccurs because the equilibrium SOC reached with surface stress is lower than that without17urface stress (refer Fig.9) and hence, the overall volumetric expansion of the electrode particleis low due to low lithium concentration. Additionally, the gap between the solid and the dashedlines decreases with increase in σ f σ f . This is consistent with the result in Fig. 11(a), which alsosees a reduction in the gap between the solid and dashed lines. dd rr wz wz Decreasing mechanical performance
Decreasing electrochemical performance eq eq (nm) (V) (a) ffffff eq wz wz eq Decreasing mechanical performanceDecreasing electrochemical performance (nm) (V) (b) dd rr wz wz Eq. SOC Eq. SOC
Increasing mechanical performance Decreasing electrochemical performance (nm) (c) ffffff wz wz Increasing mechanical performance Decreasing electrochemical performance (nm)
Eq. SOC Eq. SOC (d)
Figure 12: Competitive plots for galvanostatic charging condition (a , b) and Potentiostaticcharging condition (c ,d). In (a) and (c), core radius of the constraining material r d is varied(0.1, 0.3) and ( σ f σ f ) is kept constant at 10. In (b) and (d), the ( σ f σ f ) is varied (0.1, 1, 10) and r d is kept constant at 0.3. The solid lines depict mechanical performance, and the dashed linesdepict the electrochemical performance.From previous literature [53–56, 74] it was established that to make the anode particles saferagainst mechanical failures in the form of instabilities, fracture, delamination of SEI, and losingcontact with the current collector, we should design anode particles with dimensions less than100 nm. This conclusion was based solely on the fact that surface stresses shift the DISs inthe negative direction, hence rendering the particles mechanically safer. In the present model,by considering the length-increase phenomenon, which is another mechanical development, wechallenge the previous conclusion, based on two negative outcomes: first, the electrochemicalperformance of the particles degrades as we reduce their sizes; second, in galvanostatic chargingcondition, the length-increase of the cylindrical particles increases with increase in surface stress18i.e. decrease in the size of the particles). However, in case of potentiostatic charging condition,owing to low lithium concentration within the particles, the length-increase magnitude decreasesas we reduce the particle-size. Now, when we add the influence of a constraining materialat the inner core, the over-all effects of the surface stress and constraining material on themechanical and electrochemical performance of the electrode particles needs to be investigated,in tandem. The effects of the constraining material could be analyzed by varying the physicalproperties (for eg. the radial thickness of the constraint), or the material properties (for eg.the yield strength of the material). In order to arrive at a competitive analysis, we plot aparameter each representing mechanical and electrochemical aspects, under the combined effectsof physical/material properties of the constraining material and surface stress (by varying theparticle size). Figures 12(a) and (b) are for the case of galvanostatic charging. The difference ∆(quantity) isdefined as (quantity) with surface stress − (quantity) without surface stress . Therefore, as the magnitudeof ∆ increases, it would imply that the effect of surface stress increases. For Fig.12(a) we varythe core radius r d , keeping the yield strength ratio ( σ f σ f ) constant at 10. It is important tonote that the magnitude of ∆( ∂w∂z ) is positive (discussed earlier for Fig. 7), which means themechanical performance degrades in the presence of surface stress. Similarly, the magnitude of∆(∆ φ eq ) is negative, which means the electrochemical performance degrades when surface stressis considered. The particle size is varied from 5 to 100 nm. As the particle size decreases, themagnitude of Delta increases, for both length-increase ( ∂w∂z ) and equilibrium potential (∆ φ eq ).Hence, with increase in surface stresses, both the mechanical and electrochemical performancesdegrade. This decrease in performances is further aided by an increase in the core radius of theconstraining material.Next, if we keep the core radius constant and change the yield strength of constrainingmaterial instead, we observe that with increase in the yield strength of the constraining materialas compared to that of Si, the effect of surface stresses increase, i.e. the magnitude of ∆ increases,thus decreasing both the mechanical and electrochemical performances. Figures 12(c) and (d) are for the case of potentiostatic charging condition. Contrary to ourobservations for the galvanostatic cases, here we observe a negative value for the magnitude of∆( ∂w∂z ) , which means the mechanical performance improves with the advent of surface stress.However, the electrochemical performance degrades here as well. The particle size is varied from15 to 100 nm. As the particle size decreases, the surface stresses become more prominent, whichleads to an increased mechanical performance and a reduced electrochemical performance. Theeffect of core radius on surface stress is opposite to what we had observed in the galvanostaticcase. Here, with increase in core radius of the constraining material, the effect of surface stressdecreases.When we keep the core radius constant, and vary the yield strength ratio ( σ f σ f ), again weobserve an opposite effect from that of galvanostatic case. As σ f σ f increases, the effect of surfacestress decreases. We developed a fully coupled electro-chemo-mechanical formulation for a nano-structured,cylindrical silicon anode particle undergoing finite deformation due to lithiation/de-lithiationto study the effects of surface stress on the electrochemical and mechanical performances of19he anode particles. We analyze the effects of increasing surface stress under two differentcharging/discharging conditions: first, galvanostatic case with constant charging/dischargingrate; second, potentiostatic case with constant interface voltage drop.As observed by previous researchers [53–56, 74], the compressive effect of surface stressrelaxes the diffusion-induced stresses within the anode particle, thus increasing their structuralintegrity. However, when we considered the phenomenon of length increase of the particles uponlithiation, and how surface stress affects it, we obtain an interesting result based on the kindof charging/discharging condition. Most importantly, it has been observed that the growingsurface stress magnitudes can have a detrimental effect on the electrochemical performance ofthe battery. It is important to highlight the four major observations: • As surface stress gets significant, the surface energy of the anode particles increases.During charging, as lithium ions reduce to form lithium atoms at the surface of the anodeparticles, the chemical potential at the Si surface becomes less negative. This implies,that the electrochemical reaction becomes less spontaneous in the presence of surfacestress. This manifests itself (a) by a negative shift of the interfacial voltage-drop curve incase of galvanostatic charging, and (b) by reaching a lower equilibrium SOC in case ofpotentiostatic charging, when compared to without surface stress case. • As we decrease the size of the particle, the magnitude of the surface stress increases.Therefore, with a decreasing particle size, the negative shift in interfacial voltage-dropincreases for galvanostatic case, and the reduction in equilibrium SOC increases forpotentiostatic case. • With increase in magnitude of surface stress for a decreasing particle size, the magnitudeof length-increase of the particles enhances for a galvanostatic case, and reduces for apotentiostatic case. However, as observed from previous literature, in both the cases,the compressive effect of surface stress relaxes the diffusion-induced stresses within theanode particle, apparently giving an impression of improvement in structural integritybeing directly proportional to decreasing particle-size. This straight-forward inference ischallenged through the results in the present work. In galvanostatic case, an unrestrictedreduction in particle-size may lead to an overwhelmed length-increase, which has negativeconsequences on the mechanical performance of the battery. • The introduction of a constraining material at the core of the Si nanotube makes theinvestigation of the length-increase phenomenon complete. As we increase the yield-strength or/and the core radius of the constraining material (neglecting surface stress),the length-increase magnitude decreases.Finally, the two competitive phenomena of increasing mechanical safety and degradingelectrochemical performance can be compared to choose the best possible particle sizes for apotentiostatic case. Whereas, for a galvanostatic case, the particle sizes should not be madesignificantly small.
AS and JC acknowledge the financial support of Ministry of Human Resource Development,Government of India. JC thanks the DST-INSPIRE program of the Government of India(DST/INSPIRE Faculty Award/2016/DST/INSPIRE/04/2015/002825) as well as the ISIRDFunding (IIT/SRIC/ME/MFD/2017-18/70) from IIT Kharagpur for the computational resources.20 eferences [1] K Mizushima, PC Jones, PJ Wiseman, and John B Goodenough. Lixcoo2 (0¡ xˆa ©½ Solid State Ionics , 3:171–174,1981.[2] M Stanley Whittingham. Intercalation chemistry and energy storage.
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Chemical potential for non-ideal mixture
The non-dimensional flux of the diffusing species in the host material is given by:˜ J r = − ˜ Dc ∂ ˜ µ∂ ˜ r , (29)where ˜ µ is the non-dimensionalized form of the chemical potential of the system ( µ ). In acoupled chemo-mechanical system, the diffusion being affected by the mechanical stresses, µ becomes a function of both concentration and mechanical stresses ( µ = µ + µ s , where µ is thestress-indepdendent part of the chemical potential and µ s is the stress-dependent part). Keepingaside the stress-enhanced diffusion for the moment, we concentrate on the different forms of µ considered by different researchers, and their impact on the present problem.Chemical potential is the change in free energy of the system due to an infinitesimal changein the concentration of the diffusing species, mathematically described in Eq. (28).For an ideal mixture, ∆ G idealmix = R g T [ x A ln x A + x B ln x B ] , (30)where x A and x B are the mole fractions of the two components in the mixture.For a non-ideal/real mixture,∆ G realmix = R g T [ x A ln x A + x B ln x B + κx A x B ] . (31)The extra term in Eq. (31) is due to the deviation from ideality, and κ is the dimensionlessinteraction parameter.In the present case, ∆ G = R g T [ c ln c + (1 − c )ln(1 − c ) + κc (1 − c )] , (32)which when partially differentiated with respect to c gives the chemical potential as: µ = R g T [ κ (1 − c ) + ln c − c ] . (33)This form of µ has been used by Lu et. al. [63].Haftabaradaran et. al. [72] and Cui et, al. [66] studied solutions with high solute concen-trations. In the continuum modelling of diffusion in solids, the stress-independent part of thechemical potential is expressed as [72] µ = µ s − µ v , (34)where µ i = µ i + R g T ln a i ( i = s , v). Here µ is the free energy of the host solid having 1 mole ofvacant interstitial sites, and µ is the free energyof the same system when the interstitial sitesare occupied by the diffusing atoms; a i is the interaction energy, and is expressed as: a s = γ s c a v = γ v (1 − c ) , (35)where γ s and γ v are the activity coefficients of the solute atoms and vacancies, respectively.Therefore, chemical potential becomes µ = µ + R g T ln γ s cγ v (1 − c ) , (36)where µ = µ − µ is a constant. Equation (36) can be further simplifies to give µ = µ + R g T ln( γc ) , (37)26 (a) (b) Figure 13: (a) Variation of stress-independent part of chemical potential with SOC for differentforms of µ ( c ) as considered by Cui et. al. [66], Lu et. al. [63], and Bucci et. al. [73]; (b)SOC achieved by a particle of size 50 nm, in the potentiostatic charging condition, consideringdifferent forms of µ ( c ).where γ = 11 − c exp (cid:18) ∆ τ R g T (cid:19) such that, ∆ τ = 2( A − B ) c − A − B ) c [66] and µ = µ .Cui et. al. [66] used this form of stress-independent chemical potential and worked on thestress-dependent µ -form.Bucci et. al. [73] and Leo et. al. [69] on the other hand, followed Verbrugge and Koch’s model[75] and defined µ as R g T ln γ = N (cid:88) n =2 Ω n n c n − , (38)where Ω n are a set of coefficients, determined experimentally.In Fig. 13 (a) we plot the different forms of stress-independent part of chemical potentialagainst SOC. While the evolution of µ and µ are steep and comparable, the variationof µ . Lu with the growing concentration of Li is very slow. These differnet forms of µ providesatisfactory results in case of galvanostatic charging/discharging condition. But, in case ofpotentiostatic charging, it is observed that the equilibrium SOC reached is directly affected bythe form of µ considered. As illustrated in Fig. 13 (b), when we use µ and µ , weachieve a very low value of equilibrium SOC (below 6 %). On the contrary, using µ . Lu weachieve an equilibrium SOC upto 33 %. Hence, in the present study, for better representation ofresults, we consider the form of µ0