The Unit Cell Reconstruction and Related Thermal Activation Process within Coherent Twin Boundary Migration in Magnesium
The
Unit
Cell
Reconstruction and
Related
Thermal
Activation
Process within
Coherent
Twin
Boundary
Migration in Magnesium
Xiao ‐ Zhi
Tang* ,1 , Qun Zu , and Ya ‐ Fang
Guo* ,1 Institute of Engineering
Mechanics,
Beijing
Jiaotong
University,
Beijing
China School of Mechanical
Engineering,
Hebei
University of Technology,
Tianjin
China
Abstract By analyzing the interface defect loop nucleation and the interface disconnection expansion in dynamic simulations, the elementary migration process of coherent twin boundary of magnesium is identified to be independent unit cell reconstruction. The atomistic pathways of the unit cell reconstruction prove their collective behavior as a stochastic response to thermal fluctuation at a stressed state, and also the onset mechanism of interface disconnection gliding: predominant pure ‐ shuffle basal ‐ prismatic transformation along with atomistic shear movements. The athermal shear strength, the migration barrier, the critical length of disconnection dipole and other parameters characterizing the thermal activation process are reported. Keywords:
Coherent
Twin
Boundary,
Interface
Disconnection
Gliding,
Basal ‐ prismatic Transformation,
Potential
Energy
Surface,
Migration
Barrier,
Athermal
Shear
Strength
Main manuscript Introduction
Based on profuse investigations on the (cid:4668)101(cid:3364)2(cid:4669) twin (known as tension twin) in hexagonal close ‐ packed metals [1], recently researchers pointed out with substantial evidence and rigorous analysis, that (cid:4668)101(cid:3364)2(cid:4669) twin is shuffling ‐ controlled at the nucleation stage [2, After the nucleation stage, the migration of CTB is a glide ‐ shuffle mechanism which involves both shear and shuffle [1]. The interface disconnection with the same topological character of twinning dislocation on (cid:4668)101(cid:3364)2(cid:4669) CTB is responsible for this glide ‐ shuffle mechanism [1]. The interface disconnection on (cid:4668)101(cid:3364)2(cid:4669) CTB has been widely confirmed in transmission electron microscopy (TEM) observations [4 ‐ and the mobility of such disconnection is supported by atomistic simulations [9 ‐ which directly contributes to twin boundary migration. Well, this is not the whole story of (cid:4668)101(cid:3364)2(cid:4669) CTB migration.
The widely accepted convention is, twinning dislocation strongly interacts with precipitates so that twins are pinned down, and in this way hardening is induced [15, But the interface disconnection on (cid:4668)101(cid:3364)2(cid:4669) CTB does not. In magnesium ‐ based alloys, such precipitation hardening is surprisingly weak: accompanied by interface disconnection gliding, tension twin engulves precipitates or even bypasses them [17 ‐ Also in pure hexagonal close ‐ packed (HCP) metals, (cid:4668)101(cid:3364)2(cid:4669) CTB deviates largely from the twinning plane [20 ‐ In these deviated boundaries, basal/prismatic (BP) interface coexists with CTB confirmed by many TEM observation [6, ‐ In atomistic simulations CTB and BP interface can transform into each other [11, Specifically, BP interface is supposed to be strongly associated with atomic shuffling mechanism [3, Therefore, researchers began to question the conventional glide ‐ shuffle mechanism of the migration of (cid:4668)101(cid:3364)2(cid:4669) CTB [32 ‐ Some believe that it is solely accomplished by atomic shuffling [35]. The atomic shuffling mechanism in the case of (cid:4668)101(cid:3364)2(cid:4669) twin particularly refers to basal ‐ prismatic transformation [1, of HCP lattice, which leads directly to unit cell reconstruction (UCR) [20, of tension twin. Until now, a complete description and a unified theory for (cid:4668)101(cid:3364)2(cid:4669) CTB migration at different spatio ‐ temporal scales are still not presented. By analyzing the migration process in dynamic simulations, this study examined the connection between UCR mechanism and interface disconnection gliding.
The uncovered connection is successfully testified by stress ‐ dependent migration pathways of CTB on its potential energy surface (PES). Simulation
Methodology
The
CTB migration in magnesium was investigated at K by Molecular
Dynamics (MD) and at K by Molecular
Statics (MS), adopting the
EAM potential developed by Liu et al [38]. The simulation system contains (cid:2873) atoms, and the
CTB could be considered infinite large since Periodic
Boundary
Condition (PBC) was applied along two in ‐ plane directions. Therefore no surface effect exists. To find out the effects of simulation technique and eliminate them in the defect analysis, two ways of loading were adopted: displacing one of the outermost fixed atom layers to apply shear stress, and displacing all free atoms (sandwiched between the two fixed atom layers) to apply shear strain to the whole system. The strain rate in both loading ways is set to (cid:2876) s (cid:2879)(cid:2869) . We are not aiming to reproduce the CTB behavior at experimental strain rates and room temperature. Contrarily, extremely high strain rate in MD and low temperatures applied here are quite necessary to reveal the existence of different evoluanalycontrstress R3.1 T FTsimureferssimusimuthe nis a lono STmigrafor mstill eution pathwayze them to frolled, strains ‐ strain curv Results and
Dhe elementa
Fig. The sticThe plastic lations (poins to migratiolations we lations, thernext section.ow ‐ probabilT in MD procesmigration in exists becausays in the pfind the conn ‐ controlled)ves are plotte Discussion ary migratio ck ‐ slip behavdeformationnt A, C, D) aon in entiretycould see brmal fluctuat The ST neeity event. T. In MS simss [41, isentirety [43se after the sphase spacenections. By) and two sed in Fig. on process viors [39, was initiand by what y) in all MS sboth mechations benefiteds cooperatherefore it imulations, th purely stres3]. Thereforesystem is pla associated y the two loasimulation te0] of CTB reflated by Diswe named simulations anisms (point the DG metive thermals not dominhere is no ss/strain ‐ drive the ST hasastically defowith CTB stading ways mechniques (lected in sheconnection as Simultan(point E, G).nt B in MDechanism for activationsnant in MD thermal eneven. A defecs priority in ormed, the cructures, somentioned aMD, MS), reear stress fluGliding (DGeous
Transfo.
Only in streD F in Mr the reasonon the whosimulations ergy, so thect ‐ free CTB aMS simulatcooperative o that we caabove (stressepresentativ uctuation. G) in all Mormation (Sess ‐ controlleMS In M presented iole CTB whic(for example elementarat K is ideations. The
Dnucleation oan s ‐ ve D T, ed D in ch le ry al G of elementary processes couldn’t be always perfect, and the stress/strain distribution is more likely to be inhomogeneous. Stress ‐ controlled model actually applies larger local shear strain by introducing greater relative parallel displacement between (cid:4668)101(cid:3364)2(cid:4669) atom layers than strain ‐ controlled model does, therefore the yield stresses are lower in stress ‐ controlled models for both MD and MS simulations. Furthermore, the variety of migration mechanisms arises from the inherent randomness of the atom ‐ position iterations in stress ‐ controlled model. The strain ‐ controlled model updates atom positions artificially, so the randomness declines. Observing two mechanisms in one simulation (MD and MS promises inherent differences between them and thus promises comparison. The details of the two migration mechanisms are in the following. In elastic stage in MDs, by Common
Neighbor
Analysis (CNA) [44], a defect is observed to appear randomly on the CTB.
Fig. shows them by coloring atoms according to their y ‐ coordinates. The defect is named as an embryo (Fig. Generally the embryo lasts less than a femtosecond and disappears. If a certain area is full of embryos and the embryos inside evolve further, this part of CTB migrates and interface disconnections are left (Fig. to (c)). All the five disconnection loops in Fig. have mobility.
From
Fig. to (e), parts of the loops annihilate. When they are all annihilated by each other, the migration process finishes (after Fig. not shown). On the x ‐ y cross section of the CTB, the disconnection has the same topological character of 〈101(cid:3364)1(cid:3364)〉 twinning dislocation (TD) in TEM observations [1,
The migration process is also the same as described in previous atomistic simulations [9, Accompanied by increasing stress, new embryos nucleate on the migrated CTB (Fig.
The migration mechanism described above is DG. In the other mechanism (ST in Fig. migration in entirety requires that the CTB is full of embryos and the subsequent transition is cooperative. Lattice transformations took place uniformly across the
CTB, without any disconnections throughout the process.
Migration in this way definitely involves no dislocation mechanism. Embryo, as small as a single lattice, is a precursor to the transformation of HCP lattice, literally the
UCR.
Therefore the elementary migration process in both mechanisms is confirmed to be UCR.
This is the first evidence used for our interpretation of CTB migration.
Fig. sttimalo2
Snapshots train of is ps fong 〈112(cid:3364)0〉 of DG mech7 (point A) afor DG and Thhanism (a ‐ e) and at strain0.45 ps for SThe number osequ and ST mec of (poT. X ‐ axis is aof color bar iuence of CTBhanism (g ‐ i)oint B) sepalong 〈101(cid:3364)1(cid:3364)〉 indicates theB. captured inrately. The t 〉 direction ae neighborinn MD taketotal elapsednd z ‐ axis is ng position n d Fglen basalIn
FigplottedeforCTB. and tequayellowdiscoplanetransplanerequiig. (a ‐ b) Thgliding. (c ‐ e)gthening (atthat the inWe dug dl ‐ prismatic tg. strain ‐ ced by the darmation, andIn Fig. the prismatil to (cid:2024)/4 (cid:3398) t w atoms aronnections ses separatesformation: te in the twinire shear buhe basal ‐ pris) Coupling
CTtomistic shuncluded angldeeper on thransformatiocontrolled mashed line. Td atom C is tbefore migic plane is in tan (cid:2879)(cid:2869) (cid:3435)(cid:1855) √3⁄ re originally wept these ly. This clethe atoms ofn, no more, ut only the asmatic transfTB migrationffling) and roe (cid:2016) is larger he propertieon, and its cmodel is illusTwo tagged the index ofration the bndexed by s (cid:1853)(cid:3439) , essentiabelong to four atomsearly and uf prismatic pno less. Thatomic shuff formation crn to shear deotating (accuthan the aces of UCR inconnections strated in (aatoms (A anf relative traasal plane oolid line. Thally caused btwo adjaces (CTB migrandoubtedly plane in the pe basal ‐ prisfling [30, by inteformation bumulated shtual value fon CTB migrawith interfaa), where ornd B) are shoanslation of of the twin ihey have a iby c/a ratio nnt prismaticates over), tdemonstraparent exactmatic transf1, So, wterface discoby two comphear movemor visual cleation, for theace disconneriginal positown as an inthe grains ps indexed byncluded angnot equal toc planes. Afthey constituates the batly are the atformation itwhere is theonnection ponents: ents). Note arance. e evidence oection glidingion of CTB ndex of sheaparallel to thy dashed lingle (cid:2016) which o √3 . Red anfter interfacute the basaasal ‐ prismattoms of basaself does noe shear whicof g. is ar he ne is nd ce al ic al ot ch indeemigrathe uinto planeand rbecauprismthe mseconsimuTparamzero ‐ bandthe smechmethdisco(c). Hdimepreseembred exists in tation coupleuntransformbasal plane e in the twinrotating. Butuse the lengmatic, and vimain effect nd evidence Fig. (a) Shultaneous trTo uncover meters and atemperature (CINEB) mesystem dimehanisms at dhod [48 ‐ onnection dipHere one elension of thented valueryo is a tranthe case of (cid:4668) ed with sheaed prismaticwith a lengn. The twinnt the shear mgthening is ce versa. Twof disconneused for ouhear stress ‐ dansformatiothe transitioathermal she potential eethod [47]. ension to odifferent strThe NEB papole and a Cment represe sample as are obtainnsition state (cid:4668)101(cid:3364)2(cid:4669) twinr deformatioc plane in tgth of (cid:1863) , it hning shear (cid:2016)′ movement oa pure ‐ shufwinning sheaection glidinr interpretatdependent mon (K, L, M) aimages ofon processeear strengthenergy surfaDefects alone lattice aresses were athways andCTB segmentsents a smallong x ‐ axis ned only froe (K07 in Fi ? In the conon, this queshe parent hhas to rotat ′ in Fig. (eof atoms in Hffle mechanar is mainly cng is a puretion of CTB mminimum enand disconnef transition ses of ST andh, we explorece (PES) usinong 〈101(cid:3364)1(cid:3364)〉 dlong 〈112(cid:3364)0〉 detected bycorrespondt which is allest size CTBis way largeom the struig. befventional distion can behas a lengthe by angle (cid:2016) ) is brought HCP lattices ism transforcontributed e ‐ shuffle tramigration. nergy paths (ection glidinstates. d DG, calcued the pathwng the climbdirection are 〉 direction. y autonomoding atomic iso two ‐ elemB in unit of laer than threuctures shofore the CTiagram of gr explained. h of (cid:1866) . After (cid:2016) to align wboth by theis actually nrming the b by atomic sansformation(MEP) separag (Q, R, S). (ulate both tways of migbing image ne prohibitedThe final stous basin climages of a ment long arattice spacinee elementsown in the TB is fully arain boundarIn Fig. (c ‐ er transformewith the basae lengtheninnot that largebasal into thshuffling, ann. This is thately of b)(c) Atomiche activatiogration on thnudged elastd by reducintates of botimbing (ABCtwo ‐ elemenre in Fig.
The actuas, but all thfigure. In Sctivated.
Thry e), ed al ng e, he nd he c on he ic ng th C) nt ) ‐ al he T, he atomembrstatediscoFig. expadefecour fCTB reconrelaticase threeslightand rshear T mistic saddleryos. The ac.
Across the onnections a4(c)).
NEB paryo.
This is thAs said, an ed be any sizeas the tranding on timct gliding onfirst and thirmigration nstructs HCPionship of (cid:4668)101(cid:3364)2(cid:4669) Ce evidences tly different responsible r. he thermal Fig. ( ‐ point statetivation potsaddle poinand the embaths prove the third evidmbryo is a pe and shapeansformationme scale. Thn the interfard evidencesreally meanP lattice, thnot
ThCTB migratioare indicatfrom the cofor the more activation p (a) Activation (K09) has ential energnt, the two ebryos inside that the intedence used fprecursor toe (in a certan between e nucleationce is constits mentionedn? Althoughere is still his is clearly n has an exting the glidonventional e effective p process n potential e a more symgy is embryos on taccomplisherface discofor our inter UCR.
Tracinin range), atwo nuclean of loops retuted by sucd above. Whh the basashear in threflected intended meane ‐ shuffle mone. Here tpart of an inenergy. (b) Pmmetric GB V. In DG, emthe edge of ah the transitonnection is pretation ofng back to Find the discoation eventsequires nothccessive
UCRhat does
UCal ‐ prismatic he UCR, to n Fig. (c ‐ e)ning than prmechanism athe atomic snterface discPotential enestructure dmbryo is theactivated CTtion (state
Rinherently ef CTB migratg. disconnonnection gs of differehing but emRs. This is inCR in the catransformamake the ). In this wareviously proapplied on (cid:4668) shuffling is pconnection, wergy reductioifferent from saddle ‐ poinTB evolve intR06 to R15 ievolved fromion. nection loopgliding can bnt ‐ size loopbryos, so linn accord witase of (cid:4668)101(cid:3364)2 ation alreadorientationay, UCR in thoposed [ ]. A CTBpredominanwhile not thon. m nt to in m ps be ps ne th dy al he All is nt, he The activation potential energy as a function of shear stress is plotted in Fig.
Liner fittings enable us to describe the stress dependency of the two migration mechanism. To be noted the calculated barrier is for a system with dimension of two CTB elements, so the intrinsic activation parameters can be gotten by the derived line for one CTB element:
The athermal shear strength (cid:2028) (cid:3028) is MPa.
The activation volume of elementary migration process Ω (cid:3032) is ‐ ‐ eV/MPa. The activation energy of elementary migration process at stress free condition ∆(cid:1847) (cid:2868) is eV. The migration barrier at stress free condition (cid:2011) (cid:2998) is mJ/m . Clearly, the existence of disconnection dipole raises the barrier at certain stresses. The increment per unit length of disconnection ∆(cid:1847) (cid:3031)(cid:3031) reflects the slope difference of the two fitted lines and is stress ‐ dependent. Apparently, if the final state of transition is not in a lower energy state, satisfying the barrier does not realize the transition. That means the migrated
CTB is unstable unless local shear stress make it a lower energy state. The potential energy reductions as a function of shear stress are plotted in Fig.
For simultaneous
UCR, the reduction in unit of energy density is also linearly stress ‐ dependent. Therefore the effective ranges of four physical factors for thermal migration are confirmed: Stress:
MPa ( (cid:2028) (cid:3033) ) < (cid:2028) < MPa ( (cid:2028) (cid:3028) ). The corresponding energy reduction: mJ/m < (cid:2011) ∗ < mJ/m . The corresponding migration barrier: mJ/m > (cid:2011) > mJ/m . Considering the temperature and strain rate effect, all the calculated values basically ensure the self ‐ consistency within our model. Due to the unavoidable calculation inaccuracies, they are not preferred in precise comparisons, but valid for qualitative analysis. Apparently the existence of disconnection dipole makes the energy reduction less effective, the critical length of a dipole could be analyzed in thermodynamics (the pure mechanical factors are not considered). At stress of MPa ( (cid:2028) (cid:3033) for disconnection dipole of two two ‐ element CTB), energy reduction of simultaneous UCR is eV. Since the slopes of the two mechanisms (red and blue dash lines in Fig. are quite the same, the eV is supposed to be stress ‐ independent, as well as the (cid:2011) (cid:3031)(cid:3031)∗ =0.041 eV/ (cid:2868) =0.064 eV/nm. At a certain stress, the critical length of disconnection dipole (cid:1864) (cid:3030)(cid:3031)(cid:3031) in the unit of CTB element is (cid:3082) (cid:3279)(cid:3279)∗ ∗(cid:2870)(cid:3039) (cid:3116) (cid:3082) ∗ ∗(cid:3002) (cid:3116) (cid:3397) 1 (cid:3404) (cid:2868).(cid:2868)(cid:2872)(cid:2869)(cid:2869).(cid:2870)(cid:2869)(cid:3400)(cid:2869)(cid:2868) (cid:3127)(cid:3120) (cid:3099)(cid:2879)(cid:2868).(cid:2868)(cid:2874) (cid:3397) 1 (1) According to Eq. (1),
MPa is the athermal stress for disconnection dipole nucleation ( (cid:2028) (cid:3028)(cid:3031)(cid:3031) ), and the inhomogeneous distribution on CTB is still indispensable. At stress of MPa ( (cid:2028) (cid:3028) ), the (cid:1864) (cid:3030)(cid:3031)(cid:3031) is CTB elements. At stress of MPa ( (cid:2028) (cid:3033) ), the (cid:1864) (cid:3030)(cid:3031)(cid:3031) is CTB elements (12.90 nm), and the activation potential energy is calculated to be around eV (2.5 meV/atom). Definitely the thermal energy (cid:1863) (cid:3003) (cid:1846) at room temperature (26 meV) is enough large. So theoretically (cid:2028) (cid:3033) becomes the lower limit of the migration stress window for bulk magnesium, where the characteristic length of CTB is usually at micrometer scale. As we can see, potential energy barrier is generally small and wouldn’t be a problem. Stress level, critical length of disconnection dipole, and inhomogeneous distributions of physical quantities are the three requirements for shuffle ‐ induced migration. Once again, all the conclusions here do not involve the disconnections along 〈101(cid:3364)1(cid:3364)〉 direction. In practice, disconnection gliding has to be analyzed as a loop for a more comprehensive understanding. It is clear that a single embryo corresponds to a transition state in the migration process of a CTB element.
Energetically and mechanically it is hard to finish migration for just one CTB element.
For a certain area, thermal fluctuations have to activate all the elementary migration processes inside (indexed by embryos) to make cooperative transition physically possible, and local shear stress has to lower the energy state of migrated CTB with the given ratio of (disconnection length)/(the area) to make the cooperative transition energetically possible. If all the physical conditions are satisfied, statistically we would always witness disconnection gliding when (cid:4668)101(cid:3364)2(cid:4669) CTB migrates.
The thermal energy at K (26 meV) is enough high for the elementary migration process (8 meV/atom). However the
CTB deviated from original position is not an equilibrium state without shear stress (at least ~510 MPa is needed according to Fig.
Therefore (cid:4668)101(cid:3364)2(cid:4669)
CTB is thermally stable at room temperature or even higher. The shear stress along 〈101(cid:3364)1(cid:3364)〉 produces asymmetry of the barriers rendering directional bias to the embryo nucleation, and also makes the MEP favorable by system evolution. At stress below athermal strength, the Brownian motion of CTB at high temperature [39, is expected. At stress above athermal strength, CTB migration turns into pure stress ‐ driven process. The nature of nucleation bias of embryos lies in the lattice transformation from parent to twin. Two misfit strain components are produced by the shear strain/stress loading [35]. Therefore, to accommodate the local misfit strains, the direction of CTB migration is certain and reflects on the bias of embryo nucleation. Generally the stick ‐ slip dynamics of CTB is characterized by a saw ‐ tooth time/strain/grain ‐ translation dependence of the stress and the motion in a stop ‐ and ‐ go manner: stress fluctuates periodically with equal peak values and CTB migrates to the nearest neighboring position in each slip event. This is not reflected well in Fig. High strain rate and low temperature were applied to realize simultaneous UCR mechanism, but the accumulated deformation is also released in an abrupt way that the peak values varies drastically and migration spans several neighboring positions. At a lower strain rate of (cid:2875) s (cid:2879)(cid:2869) , standard stop ‐ and ‐ go manner is recovered at K. The peak value is MPa, significantly smaller than the other temperatures (155
MPa at K and MPa at K).
The temperature sensitivity suggests that free energy calculations should be carried out for addressing accurate thermo ‐ mechanical characteristics. Existing theory of boundary migration [39] describes the rate dependency by treating the interface defect nucleation as governed by its activation energy. While it does not work well for (cid:4668)101(cid:3364)2(cid:4669) CTB migration at room temperature, for what NEB calculations above already proves:
Whether activation energy ∆(cid:1833) or migration barrier (cid:2011) is barely larger than (cid:1863) (cid:3003) (cid:1846) . What triggers the migration is not a reduced barrier, but a lower final ‐ state energy which is quite sensitive to stress/strain. Yet analytical approaches independent of activation energy needs to be applied here. Conclusions In conclusion, by investigations on the kinetics and MEPs of (cid:4668)101(cid:3364)2(cid:4669) CTB migration, we found the successive unit cell reconstruction mainly composed by atomic shuffling is the onset mechanism of the interface disconnection gliding, and the seemingly stochastic behavior of the elementary migration processes reflects the stress and temperature effects on activation parameters. Thus we describe the (cid:4668)101(cid:3364)2(cid:4669) CTB migration as following outlines: Pure ‐ shuffle basal ‐ prismatic transformation along with shear movements intrinsically induced by c/a ratio constitutes the interface disconnection gliding. Thermal activation of disconnection loop is the collective behavior of the elementary migration processes. Shear stress makes unit cell reconstruction physically realizable not by reducing the energy barrier, but lowering the final energy state, and meantime, shorten the critical length of a disconnection dipole along 〈101(cid:3364)1(cid:3364)〉 direction. Overall, the intrinsic shuffling mechanism and the efficacy of mobile interface disconnection are both included in one framework presented in this article. Atomic simulations at realistic laboratory strain rates and free energy calculations can be available soon, providing more conclusive evidence. Acknowledgement
This work was supported by National
Natural
Science
Foundation of China (No. and
No.
XZT sincerely appreciates the hospitality of the Nuclear
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THE
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