Computational Materials Science 126 (2017) 228–237

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Computational Materials Science

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Compression twinning and structural phase transformation of singlecrystal titanium under uniaxial compressive strain conditions:Comparison of inter-atomic potentials

http://dx.doi.org/10.1016/j.commatsci.2016.09.0340927-0256/� 2016 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail addresses: alig.sunil@gmail.com (S. Rawat), nilanjan@civil.iitkgp.ernet.in

(N. Mitra).

Sunil Rawat ⇑, Nilanjan MitraIndian Institute of Technology Kharagpur, Kharagpur 721302, India

a r t i c l e i n f o

Article history:Received 21 July 2016Received in revised form 22 September2016Accepted 23 September 2016

Keywords:Compression twinningPhase transformationHCP metalsTitaniumMolecular dynamics

a b s t r a c t

We perform molecular dynamics simulations to simulate the c-axis compression of single crystal Ti athigh strain rates under uniaxial strain conditions. Since it is well known that molecular dynamics simu-lations heavily rely upon the type of potential used, a comprehensive study is presented in which fourdifferent commonly utilized potentials for Ti (Ackland, Mishin, Kim and Hennig) are evaluated againsttheir abilities to demonstrate different variants of compression twins, dislocation structures and struc-tural phase transformation. We find that f10 �11g and f11 �22g twins activate for Ackland and Mishinpotentials, while only f10 �11g twins activate for Kim potential. No compression twin systems activatefor Hennig potential. The c-vector analysis of unknown structure generated with Ackland, Mishin andKim potentials shows that the unknown structure has f10 �11g twin-like orientations and the structurefactor analysis gives a signature of pressure-induced x phase for the twin-like oriented unknown struc-ture. No signature of twin-like oriented unknown structure andx phase is observed for Hennig potential.The large amount of dislocation density is observed for Ackland potential followed by Mishin, Kim andHennig potentials. The presence of compression twins and high dislocation density for Ackland, Mishinand Kim potentials suggest that the c-axis deformation is accommodated by twins and slip together,while only slip accommodates the c-axis deformation for Hennig potential. Based on these observationsand as well as on the formulation of the above mentioned potentials, Kim potential is being recom-mended for use under c-axis uniaxial compressive strain loading situations.

� 2016 Elsevier B.V. All rights reserved.

� � � � �

1. Introduction

Light weight, high specific strength and stiffness makes Ti(hexagonal-close-pack metal at ambient temperature and pressureconditions) a promising material. Titanium (in metallic form aswell as in alloy form) is used primarily in the space industry whereweight savings are crucial. It is also a major component of variousdefense applications (such as Ti-6Al-4V alloy being used for man-ufacture of various fighter jets as well as external ballistic cover forvarious vehicle armaments) primarily because of its high strengthand stiffness. Apart from enhanced mechanical properties, Ti is alsowell known for its wear resistance, corrosion resistance as well asbio-compatibility. However, the HCP structure of the materialresults in strong anisotropy in mechanical properties [1]. The mostcommonly observed slip systems in a single crystal Ti are basalðf0001gh11 �20iÞ, prismatic ðf10 �10gh11 �20iÞ, pyramidal hai

ðf1011gh1120iÞ and pyramidal hcþ ai ðf1011gh1123iÞ [2,3].The basal slip systems have a Burgers vector in the basal planeand therefore, cannot accommodate the c-axis deformation. Thenon-basal slip system (pyramidal hcþ aiÞ is hard to activate (2–3times harder than hai slip) due to its high value of critical resolvedshear stress (CRSS) [2,3]. Therefore, an additional mechanism isrequired to accommodate the deformation of c-axis. This addi-tional deformation mechanism has been proposed as twinning[4]. It should be mentioned that deformation twinning plays animportant role on ductility, fracture strength, texture evolution[5] and strain hardening [6,7].

The compression twins activated under c-axis compression aref11 �22g, {11 �24} and {10 �11}. The {11 �22} compression twins aremost frequently observed at ambient temperature and pressureconditions [8,9]. The {10 �11} twins activate at elevated tempera-tures [8] and {11 �24} activate at low temperatures [10]. The activa-tion of {10 �11} twins at room temperature has also been reported inmicro-pillar compression experiments on single crystal titaniumalloy [11] and dynamic compression experiments on commercially

S. Rawat, N. Mitra / Computational Materials Science 126 (2017) 228–237 229

pure titanium [12]. Within a fixed time interval, as the intensity ofcompressive load is increased in the crystal along the direction of‘c’-axis, compression twins are formed along with activation of cer-tain slip mechanisms. Phase transformation from a to x has alsobeen reported in experiments which occurs when slip along withdeformation twinning is unable to counter the applied load in thesingle crystal. It is well known that the a tox phase transformationoccurs at ambient temperature and high pressures [13], but the ori-entation relationship (OR) between a andx phases under differentloading conditions is still an active area of research. A number ofORs has been proposed for a tox transition under different loadingconditions [14–18]. These ORs are proposed based on eitherexperimental studies of poly-crystalline Zr and Ti [14–16] orcomputational studies on Ti [17,18] under different loadingconditions. The experimentally reported ORs [14–16] are(0001)ak(10 �11)x; [10 �10]a k [�1011]x known as variant I and(0001)a k (1 �210)x; [1 �210]a k [0001]x known as variant II.The OR (0001)ak(10 �11)x; [10 �10]a k [11 �23]x proposed by Songet al. [16] is a subset of variant I [19]. The computationallyreported ORs [17,18] are (0001)ak(0 �111)x; [11 �20]a k [01 �11]xknown as TAO-I which is variant I and (0001)ak(10 �10)x;[10 �10]a k [11 �23]x.

In this study, we perform c-axis deformation of perfect singlecrystal titanium under uniaxial strain conditions at room temper-ature using four commonly utilized inter-atomic potentials for tita-nium [20–23]. The uniaxial strain conditions are obtained in highvelocity impact experiments [24,25] in which compressive stresswaves generated at the point of impact propagate towards the freeends of the target material. The passage of these shock waves gen-erates an uniaxial compressive strain conditions [25] within thetarget material leading to creation of defects (dislocations, twins,etc.) [25,26] and phase transformation [26–28].

Very few research is present which addresses the atomic levelstudy of deformation of Ti under different loading conditions[18,29,30]. These studies have been performed with differentinter-atomic potentials. For example, Zong et al. [18] studied theshock compression of Ti single crystals using Hennig potential[23]; Ren et al. [29] studied the compression of nanopillars underuniaxial stress conditions using Ackland potential [20]; grainboundary motion under uniaxial tension and compression loadingconditions has been studied by Zong et al. [30] using Mishin poten-tial [21].

However, there is no general consensus as to which potentialsuits Ti single crystals under certain type of loading conditions.To our knowledge, there is no atomistic study on the deformationbehaviour (compression twins and phase transformation) of bulkTi single crystals at room temperature under uniaxial strain/stressconditions. The question is: which potential we should use to studythe deformation of bulk Ti single crystals at room temperatureunder uniaxial strain conditions? It is well known that results ofmolecular dynamics (MD) simulations depend upon type ofinteratomic potentials being used. Thereby this simulation studyof c-axis compression under uniaxial strain conditions has beencarried out with 4 well known inter-atomic potentials for titanium[20–23] with an objective of enumerating differences between thepotential with regards to demonstration of deformation behaviour(which includes compression twinning along with phase transfor-mations). It is expected that this comprehensive research will helpother researchers choose proper potentials of Ti for these type ofloading situations as well as other type of loading situations.

2. Computational method

We have simulated the c-axis compression of perfect singlecrystal titanium under uniaxial strain conditions (ezz – 0,

exx ¼ eyy ¼ 0Þ using LAMMPS [31]. The simulation domain contains4.6 � 105 atoms with a side length of 20 nm in each direction. TheX-, Y- and Z-directions of the simulation domain correspond toh2 �1 �10i, h01 �10i and h0001i, respectively. The periodic boundaryconditions have been used in each direction and velocity-Verlet algorithm has been used to integrate the equation of motion with atime-step of 2 fs. To control the temperature and pressure, Nose-Hoover thermostat and barostat have been used. The system is firstequilibrated at 300 K and 0 bar pressure using an NPT ensemble upto 40 ps and then a strain rate of 109 s�1 is applied to deform thesystem at a constant temperature of 300 K using an NVT ensemble.The simulations at 109 s�1 strain rate have been performed to savecomputational time. To study the strain-rate sensitivity on theactivation of compression twins and dislocation slip, the calcula-tions have also been performed at 108 s�1 strain rate. The atomicdata is then post-processed using crystal analysis tool (CAT) [32]and basal plane vector (BPV) [33] analyses. The results are visual-ized in Ovito [34].

Typically many body potentials prescribed for HCP elements areembedded-atom method (EAM) type (such as [21]), Finnis-Sinclair(F-S) type (such as [20]), RGL type ([35]) and Modified embedded-atom method (MEAM) type (advancement over the EAM type suchas [22,23]). The semi-empirical type EAM potential [21] has beendeveloped for Ti by fitting to a large experiment database as wellas ab initio data. A Finnis-Sinclair formalism of the second momentapproximation to the tight-binding theory has been utilized for theF-S type potential developed [20]. Mathematically this formulationturns out to be equivalent to the EAM potentials and energy isexpressed in terms of short range pair functionals with no explicitangular dependence. This lack of angular dependency poses severallimitations to these models (typically for modelling of hcp metals[36]) and has been corrected using modified version of the EAMpotential (MEAM potential). The MEAM potentials include theangular dependent electron density to model the effects of bondbending (which the EAM potentials typically lack). The new gener-ation of MEAM (referred to as 2NN-MEAM) considers the secondnearest neighbour interactions while the original MEAM considersonly the first nearest neighbour interaction which is prone tostructural stability issues [37].

The simulations in this manuscript have been performed usingfour interatomic potentials: Finnis-Sinclair many-body potential[20], embedded atom method (EAM) potential [21], 2NN-MEAMpotential [22] and spline-based modified embedded atom method(MEAM) potential [23] to study the compression twins and phasetransformation in single crystal titanium subjected to c-axis uniax-ial compression. The EAM potentials [20,21] underestimate thestacking fault energy (SFE) obtained in experiments. The experi-mentally reported SFE for titanium is 300 mJ m�1 [38], while thefitted values are 64 mJ m�1 and 56 mJ m�1 for Ackland [20] andMishin [21] potentials, respectively. The MEAM potentials [22,23]do a better job by fitting the SFE with a high value in comparisonto EAM potentials, but both of them still underestimate the exper-imental value. The SFEs for Kim and Hennig potentials are213 mJ m�1 [22] and 170 mJ m�1 [23], respectively. Note thatKim potential [22] has high SFE compared to other potentials.

3. Results and discussion

3.1. Stress-time profile

Fig. 1 shows the time evolution of stress developed along axialand transverse directions due to c-axis compression of single crys-tal titanium with four inter-atomic potentials [20–23]. The axialstress (rzz) for all potentials increases with time, reaches a peakand then drops. The drop in stress indicates the onset of plastic

0

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70

Acklandσzzσxxσyy

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0 40 80 120 160 200

En

gin

eeri

ng

str

ess

(GP

a)

Mishinσzzσxxσyy

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

0 10 20 30 40 50 60 70

Kimσzzσxxσyy

0 10 20 30 40 50 60

0 10 20 30 40 50 60

0 10 20 30 40 50 60

0 40 80 120 160 200 240 280

Time (ps)

Hennigσzzσxxσyy

Fig. 1. Stress developed as a function of time for four different potentials.

230 S. Rawat, N. Mitra / Computational Materials Science 126 (2017) 228–237

deformation. Both, Ackland [20] and Mishin [21] potentials, havesame peak stresses (33 GPa), but Mishin potential yields earlierin comparison to Ackland potential. The Kim potential followsthe similar trend as obtained with Ackland and Mishin, but in thiscase yielding is delayed and corresponds to a highest peak stress(66 GPa). The stress-time profile obtained with Hennig’s potentialis completely different from those obtained with other potentials[20–22]. Initially, it follows the similar trend as obtained withother potentials. After some time point (80 ps), the nature ofstress-time profile changes and the stress increases slowly. Thestress drops at 212 ps with a peak value of 28 GPa which is lowerthan that obtained with Ackland and Mishin potentials. The calcu-lations for rate of stress drop (drzz=dt) shows that the drop in stressfor Mishin potential (4.5 GPa/ps) is fastest followed by Ackland(4.1 GPa/ps) and Kim (3.7 GPa/ps) potentials in comparison toslowest stress drop for Hennig potential (0.39 GPa/ps). A strongstrain hardening is seen for Kim potential followed by almost equalstrain hardening for Ackland and Mishin potentials. The strainhardening is small for Hennig potential in comparison to Ackland,Mishin and Kim potentials.

3.2. Structural analysis

To identify the structural changes due to deformation, we per-form CAT analysis [32] which gives information about the parentHCP structure, stacking faults and unknown structure. Fig. 2(a)shows the time evolution of number fraction corresponding toHCP structure, stacking faults and unknown structure. It is seenin Fig. 2(a) that the number fraction corresponding to HCP struc-ture decreases with increase in time. The decrease in HCP structurecorresponds to the creation of an unknown structure and stackingfaults (Ackland and Mishin) or creation of only unknown structure(Kim and Hennig). The presence of stacking faults for Ackland andMishin potentials is due to the low stacking fault energies (SFEs)predicted by Ackland and Mishin potentials since low SFE of thematerials leads to the formation of stacking faults [39]. For Acklandpotential, the number fraction belonging to stacking faults is largecompared to that for Mishin potential. The number fractionbelonging to unknown structure is largest for Mishin potentialand smallest for Hennig potential.

The snapshots for structural changes for these potentials areshown in Fig. 2(b). It is seen in Fig. 2(b) that for Ackland, Mishinand Kim potentials, the atomic arrangement of unknown structureis observed in a specific fashion indicating c-axis rotation whichcould be a result of phase transformation for which different ORs[14–18] have been proposed. This kind of atomic arrangement isnot observed for Hennig potential.

3.2.1. Misorientation angle distributionThe distribution of c-axis misorientation is a very useful way to

get a signature of any c-axis rotation. The c-axis misorientation iscalculated using the initial c-vector and instantaneous c-vectorobtained by BPV analysis [33]. Fig. 3 shows the distribution ofc-axis misorientation at different time points. The low anglemisorientations (<10�) are not shown in Fig. 3 for clear visualiza-tion of high angle misorientation peaks. It is seen in Fig. 3 thatexcept for Hennig potential, the peaks occur in the range of50–70� for Ackland, Mishin and Kim potentials. For Ackland poten-tial, peaks also occur in the range of 40–50� at 200 ps in addition tothe 50–70� peaks. While for Mishin and Kim potentials, additionalpeaks occur in the range of 80–90�. The analysis for 40–50� peaksfor Ackland potential and 80–90� peaks for Mishin and Kim poten-tials show that these peaks belong to an unknown structure. Peaksfor the Kim potential are observed to be much sharper in compar-ison to the Mishin potential. In fact the flat peak nature observed inMishin potential is similar to that in Ackland potential with the dif-ference being observance of splitting of the peaks. This phe-nomenon of flat peaks and/or peak splitting may be due toinability of these potentials to account for directionality of thebond which is typically accounted for in MEAM potentials. Notethat the number fraction corresponding to 50–70� peaks increaseswith increase in deformation. We know that the theoretical misori-entations for {10 �11} h�1012i and f11 �22g h�1 �123i twin systems[40] are 57.2� and 64.4�, respectively. Thereby we focus on thepeaks in the range of 50–70� which are possibly created due toactivation of compression twins or phase transformation [41]. Tounderstand whether these peaks (50–70�) belong to HCP structureor unknown structure, we further compute the number fractioncorresponding to HCP and unknown structures in the range of50–70� misorientation angle. Fig. 4 shows the evolution of number

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ion

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Mishin

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Hennig

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Kim

Hennig

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Stacking faults

Kim

Hennig

0 220 240 260 280

(a)

(b)

Fig. 2. (a) Time evolution of number fraction belonging to HCP structure, stacking faults and unknown structure. (b) Structural changes for Ackland (200 ps), Mishin (160 ps),Kim (240 ps) and Hennig (280 ps) potentials.

S. Rawat, N. Mitra / Computational Materials Science 126 (2017) 228–237 231

fraction corresponding to HCP and unknown structures for atomswhich have c-axis misorientation in the range of 50–70�. For easycomparison, the number fraction corresponding to total HCP andunknown structure is also shown. It is seen in Fig. 4 that in all threecases, the unknown structure has significant contribution to thepeaks in the range of 50–70�, while HCP structure gives a verysmall contribution to these peaks in the misorientation angle dis-tribution. Mishin potential shows maximum contribution ofunknown structure to these peaks followed by Ackland and Kimpotentials.

In the next sections, we analyze the 50–70� orientationsobtained for HCP and unknown structures by performing (a) twin

analysis for activation of possible twin systems which includesactivation of twin systems, twin variants and quantitative informa-tion of twinned volume (b) analysis for unknown structure whichincludes c-vector analysis and structure factor calculations.

3.2.2. Twin analysisIn the previous section (Section 3.2.1), we have seen that the

small fraction of HCP atoms has c-axis misorientation in the rangeof 50–70� indicating the possible activation of compression twins.Since the theoretical misorientation angle for {10 �11} h�1012i and{11 �22} h�1 �123i twin systems lies in the range of 50–70�, the

Ackland115 ps

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Kim220 ps

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Hennig 220 ps

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280 ps

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10 20 30 40 50 60 70 80 90

Fig. 3. Distribution of c-axis misorientation at different time points.

Ackland

Total HCPHCP (50-70o)

Total unknownUnknown (50-70o)

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Kim

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Fig. 4. Time evolution of number fraction corresponding to the peaks in the range of 50–70� in the misorientation angle distribution. For comparison, the number fractioncorresponding to total HCP and unknown structure is also shown.

232 S. Rawat, N. Mitra / Computational Materials Science 126 (2017) 228–237

activation of both these twin systems can be expected. Table 1shows the possible six twin variants of {10 �11} h�1012i and{11 �22} h�1 �123i twin systems. To distinguish the twin variantsfrom each other for two twin systems, each variant is representedby Vij, where ‘i’ refers to twin system type and ‘j’ refers to varianttype of that twin system. The Schmid factors for loading and con-straint directions are also calculated and tabulated in Table 1.

The quantitative information of twinned volume in the simula-tion domain can be obtained by computing the twin volume frac-tion (TVF) which, for an atomic system, is defined as the ratio oftotal number of twinned atoms (Ntw) to the total number of atoms(N) in the simulation domain. It is an experimentally measurablequantity and therefore, one can compare the evolution trend ofthe twin volume fraction with that observed in the experiments.

Fig. 5 shows the time evolution of the total twin volume fractionand twin variant volume fraction for {10 �11} h�1012i twin systemand total twin volume fraction for {11 �22} h�1 �123i twin system.The twin variant volume fraction for {11 �22} h�1 �123i twin systemis not shown in Fig. 5 since the volume fraction of the activatedvariants is very small.

It is seen in Fig. 5 that all six variants of {10 �11} h�1012i twinsystem are activated for Ackland, Mishin and Kim potentials, whileembryonic twins of all variants of {11 �22} h�1 �123i twin system areactivated for Ackland and Mishin potentials but no activation of{11 �22} h�1 �123i twin system for Kim potential. In this regard itshould be pointed out that the activation of {10 �11} h�1012i twinsystem has been reported in micro-pillar compression experiments

Table 1Schmid factor and misorientation angle for compression twin systems.

Twin type Twin variants Schmid factor C-axis misorientation (degree)

h0001i h2 �1 �10i h0 �110if10 �11g V11 f10 �11g h�1012i 0.42 0.31 0.10 57.22

V12 f�1011g h10 �12i 0.42 0.31 0.10

V13 f01 �11g h0 �112i 0.42 0 0.42

V14 f0 �111g h01 �12i 0.42 0 0.42

V15 f�1101g h1 �102i 0.42 0.31 0.10

V16 f1 �101g h�1102i 0.42 0.31 0.10

f11 �22g V21 f11 �22g h11 �2 �3i 0.45 0.11 0.34 64.40

V22 f�1 �122g h�1 �12 �3i 0.45 0.11 0.34

V23 f�12 �12g h�12 �1 �3i 0.45 0.45 0.34

V24 f1 �212g h1 �21 �3i 0.45 0.45 0.34

V25 f�2112g h�211 �3i 0.45 0.11 0

V26 f2 �1 �12g h2 �1 �1 �3i 0.45 0.11 0

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115 155 195

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115 155 195

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in v

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V11V12V13

V14V15V16

Fig. 5. Time evolution of total twin volume fraction and twin variant volume fraction for {10 �11} h�1012i and {11 �22} h�1 �123i twin systems.

S. Rawat, N. Mitra / Computational Materials Science 126 (2017) 228–237 233

at room temperature [11]. It can also be observed from Table 1 thatthe Schmid factor for {11 �22} h�1 �123i is high compared to that for{10 �11} h�1012i, but the twins belonging to {10 �11} h�1012i twinsystem dominate the twinned volume. In addition to this, only{10 �11} h�1012i twin system, which has low Schmid factor com-pared to {11 �22} h�1 �123i twin system, activates for Kim potential.This suggests that the activation of twin systems does not followthe Schmid factor criterion. Note that the activation of compres-sion twins is expected for Ackland and Mishin potentials due tothe low stacking fault energy. The Kim potential which has highstacking fault energy compared to rest three potentials also showsactivation of compression twins, but there are no traces of com-pression twins for Hennig potential even with low stacking faultenergy compared to Kim potential. This demonstrates that Hennigpotential may not be suitable for these type of loading conditions.

It is seen in Fig. 5 that the total twin volume fraction for {11 �22}h�1 �123i twin system is very small compared to that for {10 �11}h�1012i twin system for both (Ackland and Mishin) potentials. Allvariants of {10 �11} h�1012i twin system get activated for Ackland,Mishin and Kim potentials indicating that the activation of twinvariants of a given twin system follows the Schmid factor criterion.However, the activated twin variants with equal Schmid factor (c.f.

Table 1) do not contribute equally to the total twin volume fractionin either case suggesting that the evolution of twin variants doesnot depend on the Schmid factor. Recent molecular dynamics sim-ulations [42] on the evolution of {10 �12} tension twins in magne-sium also observed that the Schmid factor does not play any rolein the evolution of twin variants. For both Ackland and Mishinpotentials, the twin volume fraction saturates at �6%. However,the evolution characteristics of twin variant volume fraction andtotal twin volume fraction are different for both potentials. Thetotal twin volume fraction for Ackland potential increases rapidlyup to 4%, increases slowly and then saturates. Note that theincrease in twin volume fraction is due to the nucleation of newtwins and growth of the pre-existing twins. For Mishin potential,the total twin volume fraction increases rapidly up to 2%, saturatesfor a time period and then increases linearly. Finally, it saturates at6%. The evolution of total twin volume fraction for Kim potential isdifferent from that obtained for Ackland and Mishin potentials. Inthis case, it increases rapidly up to 0.42% and then decreases.

3.2.3. Analysis for unknown structureIn Section 3.2.1, we have seen that the unknown structure has

significant contribution to 50–70� peaks in the misorientation

234 S. Rawat, N. Mitra / Computational Materials Science 126 (2017) 228–237

angle distribution (c.f. Fig. 4). To understand whether the c-vectorof these atoms follows the same orientation as the twins, we per-form the c-vector analysis. Fig. 6 shows the time evolution ofunknown structure which satisfies the c-vector conditions forf10 �11g twin variants. The number fraction belonging to totalunknown structure contributing to 50–70� peaks is also shownfor comparison.

It is seen in Fig. 6 that significant amount of unknown structuresatisfies the c-vector conditions for f10 �11g twin variants (twin-like oriented structure). This twin-like oriented unknown structuremight indicate the presence ofx phase. The a! x transformationat room temperature occurs over a wide range of pressure from2.9 GPa [43] to 11 GPa [44,45] depending on the purity of thematerial. Since in pristine Ti single crystal studied here, there areno pre-existing impurities and defects and therefore higher stressis required to nucleate the defects (c.f. Fig. 1). Once yielding occurs,the a ! x transformation occurs following compression twins. Forthe inter-atomic potentials, the pressure developed due to c-axisdeformation (after yielding) is more than 15 GPa (Fig. 1) which isgreater than the transition pressure for a! x phase. Therefore,the twin-like oriented unknown structure could be an x phase.

To confirm the presence ofx phase, we perform structure factorðSðkÞ ¼ P

n exp ik � rn, where k: reciprocal lattice vector, rn: atomicpositions) calculation [46] for the twin-like oriented unknownstructure (Fig. 6). Fig. 7 shows the structure factor calculated fortwin-like oriented unknown structure for Ackland, Mishin andKim potentials. The calculations have been performed for different(hkl) planes corresponding to a [47,48] and x [49] phases.

It is seen in Fig. 7 that the (111) peak for x phase dominatesover the other peaks for all three cases of Ackland, Mishin andKim potentials indicating presence of x phase. It should be notedthat for Ackland, Mishin and Kim potentials where the compres-sion twin systems activate, the twin-like oriented unknownstructure is generated. While for Hennig potential for which nocompression twin system activates, the unknown structure gener-ated does not have any orientation relationship. The shock com-pression experiments on polycrystalline Zr [Supplementalmaterial Fig. S1 [41]] shows that the higher volume fraction of xphase is correlated with higher reorientations and the reorienta-

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mb

er f

ract

ion

Tim

Mishin

Fig. 6. The evolution of unknown structure which satisfies c-vector conditions for varianunknown structure contributing to 50–70� peaks is also shown for comparison.

tion of the matrix resulting possibly due to twinning takes placebefore this phase transformation. This is in good agreement withour observations with Ackland, Mishin and Kim potentials wherethe activated compression twins assist the x phase transforma-tion. In f10 �11g twinning, the c-axis of the parent matrix is rotatedby 57.2� and therefore the basal plane of the parent becomes par-allel to the f10 �11g twin plane. This also results in the reorienta-tion of the prismatic plane by 57.2� leading to h10 �10i kh�1011i.This indicates the OR between a and x as f0001gakf10 �11gx;h10 �10iakh�1011ix which is variant I proposed by [15–17].

3.3. Dislocation analysis

To understand the evolution of dislocation density, we performdislocation analysis using Dislocation Extraction Algorithm (DXA)in Ovito [34]. Fig. 8 shows the time evolution of dislocation densityfor all four cases of inter-atomic potentials.

Based on the Burgers vector analysis, there are three types ofdislocations present in system: (a) 1

3 h1 �210i dislocations on the

prismatic plane (b) 13 h1 �100i partial dislocations on the basal plane

[50] and (c) 13 h11 �23i dislocations on the pyramidal plane. In addi-

tion to these dislocations, there are some ‘‘other” dislocationswhich are not identifiable by DXA. The Ackland potential showshighest total dislocation density among all the four cases, whileleast is observed for Hennig potential. For Ackland, Mishin andKim potentials, 1

3 h1 �100 i partial dislocations show significant con-tribution to the total dislocation density, while for Hennig poten-tial their contribution is negligible. In contrast, the pyramidal13 h11 �23i dislocations have significant contribution to the total dis-location density for Hennig potential, while their contribution tothe total dislocation density is negligible for Ackland, Mishin andKim potentials. It is interesting to note that the Ackland, Mishinand Kim potentials which show significant contribution of partialdislocations to the total dislocation density lead to the creationof twins (c.f. Fig. 5), while no twins are observed for Hennig poten-tial which shows negligible contribution of partial dislocations. Inaddition to this, the evolution trend of the total dislocation densityis quite similar to that for the evolution of total twin volume frac-

Total unknown (50-70o)Twin-Like oriented

e (ps)

0

0.1

0.2

0.3

0.4

0.5

220 230 240 250 260 270 280

Kim

ts of f10 �11g twins as a function of time. The number fraction corresponding to total

0

0.02

0.04

0.06

(100)

(001)

(101)

(110)

(111)

(200)

(002)

(102)

(201)

(211)

(112)

(220)

(202)

(300)

(103)

(301)

(004)

α ω α/ω

α/ω

ω α α/ω

α α/ω

ω α/ω

ω α ω α ω α

Ackland (220 ps)

0

0.01

0.02

0.03

(100)

(001)

(101)

(110)

(111)

(200)

(002)

(102)

(201)

(211)

(112)

(220)

(202)

(300)

(103)

(301)

(004)

α ω α/ω α/ω ω α α/ω α α/ω ω α/ω ω α ω α ω α

Mishin (220 ps)

0

0.01

0.02

0.03

(100

)

(001

)

(101

)

(110

)

(111

)

(200

)

(002

)

(102

)

(201

)

(211

)

(112

)

(220

)

(202

)

(300

)

(103

)

(301

)

(004

)

α ω α/ω α/ω ω α α/ω α α/ω ω α/ω ω α ω α ω αStr

uct

ure

Fac

tor

(hkl) planes

Kim (278 ps)

Fig. 7. Per atom structure factor for Ackland, Mishin and Kim potentials.

0

2e+17

4e+17

6e+17

Ackland

0

2e+17

4e+17

6e+17

115 135 155 175 195 215

0

1e+17

2e+17

3e+17

115 135 155 175 195 215

Dis

loca

tio

n d

ensi

ty (

m-2

)

Time (ps)

MishinTotal1/3<1-210>1/3<1-100>1/3<1-213>

Others

0

1e+17

2e+17

3e+17

115 135 155 175 195 215

0

5e+15

1e+16

1.5e+16

2e+16

Kim

0

5e+15

1e+16

1.5e+16

2e+16

220 230 240 250 260 270 280

0

1e+15

2e+15

3e+15

4e+15

5e+15

220 230 240 250 260 270 280

Hennig

0

1e+15

2e+15

3e+15

4e+15

5e+15

220 230 240 250 260 270 280

Fig. 8. The evolution of dislocation density with four different inter-atomic potentials.

S. Rawat, N. Mitra / Computational Materials Science 126 (2017) 228–237 235

tion (c.f. Fig. 5) indicating a correlation between the creation oftwins and dislocation density. The Ackland and Mishin potentialswhich have large total twin volume fraction (c.f. Fig. 5) show hightotal dislocation density in comparison to the Kim and Hennigpotentials which have small and zero twin volume fraction, respec-tively. For Ackland potential, the other dislocations dominate overthe partial dislocations, while for Mishin potential they competewith each other. Similar competition between the other and partialdislocations also occurs for the Kim potential.

4. Discussion

We simulate the c-axis deformation of perfect single crystaltitanium under uniaxial strain condition at room temperature withfour inter-atomic potentials at two different strain rates. Theresults presented in the manuscript is for a strain rate of 109 s�1

whereas discussion of similar behaviour is demonstrated at108 s�1 in the supplemental material. The objective was to under-

stand the deformation behaviour of single crystal titanium sub-jected to c-axis uniaxial compressive strain loading situationwith respect to the potential used for the study. The deformationbehaviour is characterized in terms of the stress-time response,structural evolution, activation of compression twin systems andthe twin variants of the activated twin systems, twin volume frac-tion, phase transformation, activation of dislocation slip and dislo-cation density. The stress-time response for Hennig potential isfound to be entirely different from that obtained with Ackland,Mishin and Kim potentials. For Kim potential, yielding is signifi-cantly delayed with a highest peak stress in comparison to Acklandand Mishin potentials. The activation of twin systems does not fol-low Schmid factor criterion. All variants of f10 �11g compressiontwins activate for Ackland, Mishin and Kim potentials, while allvariants of f11 �22g twins activate only for Ackland and Mishinpotentials. This means that activation of twin variants for a twinsystem follows the Schmid factor criterion. However, the evolutionand dominance of an individual twin variant are not dictated by

236 S. Rawat, N. Mitra / Computational Materials Science 126 (2017) 228–237

the Schmid factor criterion. The variants activated do not con-tribute equally to the total twin volume fraction even with equalSchmid factor. No compression twin systems activate for Hennigpotential. The unknown structure created due to deformation isquantified by c-vector and structure factor analyses. The c-vectoranalysis shows that the unknown structure has f10 �11g twin-likeorientations and structure factor analysis gives a signature ofpressure-induced x phase for the twin-like oriented unknownstructure. This indicates the orientation relationship between par-ent andx phase as f0001gakf10 �11gx; h10 �10iakh�1011ix. No sig-nature of twin-like orientated unknown structure and x phase isobserved for Hennig potential. The large amount of dislocationdensity is observed for Ackland potential followed by Mishin,Kim and Hennig potentials. The presence of compression twinsand high dislocations density suggest that the c-axis deformationis accommodated by twins and slip together for Ackland, Mishinand Kim potentials, while only slip accommodates the c-axis defor-mation for Hennig potential.

Based on previous experimental tests carried out on Zr [Supple-mental material [41]], it has been demonstrated that the materialexhibits twinning as well as shows phase transformation at higherpressures under shock loading. There is no available experimentaldata for Ti but the plastic behaviour of Ti is very similar to that of Zrdue to their close electronic structures [51]. From the above simu-lations, it is clear that Hennig potential does not perform wellunder this type of loading situation as it fails to activate the com-pression twins and phase transformation. Ackland, Mishin and Kimpotentials do a better job in terms of the activation of compressiontwins and transformation of a to x phase in comparison to Hennigpotential. From a qualitative perspective the Ackland and Mishinpotentials give similar behaviours like the Kim potential in beingable to predict twinning and phase transition but if a quantitativeperformance evaluation is carried out, obviously Kim is bettersince it has a stacking fault energy closer to experimental observa-tions and also it accounts for angular dependent electron density toaccount for bond bending. It has been discussed in Section 2 thatMishin and Ackland potentials underestimate the stacking faultenergy which is important for splitting of dislocations into Shock-ley partials and can affect the activation of preferred slip/twin sys-tems and dislocation mobility [52]. Lower the stacking faultenergy, higher will be the splitting distance between the partials(i.e. wide dislocation core) leading to activation of slip at lowerstrain value. This is why Mishin and Ackland potentials yield atlower strain and stress values compared to the Kim potentialwhere yielding occurs at higher strain and stress values due tothe high stacking fault energy. As stacking fault energy for Kimpotential is closer to the experimental value, it should give realisticdislocation core structure in contrast to the Mishin and Acklandpotentials. The activation of a particular twin system during thephase transformation also depends on the crystallography of thephase transformation [52]. This suggests that the activation off10 �11g twin system should be preferred for a tox phase transfor-mation. Both Ackland and Mishin potentials show the activation off10 �11g and f11 �22g twin systems during the a to x phase trans-formation, but the OR for a tox phase corresponds to f10 �11g twinsystem. While for Kim potential, only f10 �11g twin system acti-vates during the a to x phase transformation. Therefore, basedon circumstantial evidence, Kim potential should be consideredas the best choice to simulate the deformation of Ti under uniaxialc-axis compressive strain loading situations.

5. Conclusions

In summary, we test the four interatomic potentials under uni-axial strain conditions to characterize the response of single crystal

titanium in terms of deformation twinning, dislocation activity andphase transformation. The main conclusions of this study are asfollows:

� The activation of compression twin systems does not follow theSchmid factor criterion. All variants of f10 �11g twins activatefor Ackland, Mishin and Kim potentials, while all variants off11 �22g twins activate for Ackland and Mishin potentials. Thevariants activated do not contribute equally to the total twinvolume fraction even with equal Schmid factor.

� Although the activation of twin systems does not follow Schmidfactor criterion, but the activation of twin variants of a twin sys-tem follows the Schmid factor criterion. The evolution and dom-inance of an individual twin variant is not dictated by theSchmid factor criterion.

� The orientation relationship for the a andx phase is found to bef0001gakf10 �11gx; h10 �10iakh�1011ix which in good agree-ment with the experiments.

� The Kim potential with stacking fault energy closer to experi-mental observations should be considered as the best choiceto simulate the deformation of Ti under c-axis uniaxial com-pressive strain loading conditions.

Based on the simulations performed here at two strain rates andat a given temperature under one type of loading conditions, aboveconclusions have been drawn. To check the suitability of thesepotentials further, the simulations should be performed under dif-ferent loading conditions and along different loading directionswith a focus on deformation twinning and structural phase trans-formation. It would also be interesting to test these potentialsunder some complex loading situations such as shock compressionalong different loading directions.

Acknowledgments

We thank the referee for his comments which have definitelyimproved the paper. We acknowledge the Centre for TheoreticalStudies for the use of computational facility where simulationshave been carried out.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.commatsci.2016.09.034.These data include MOL files and InChiKeys of the mostimportant compounds described in this article.

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