angular-dependent interatomic potential for the cu–ta...
TRANSCRIPT
Angular-dependent interatomic potential for the Cu–Ta system and itsapplication to structural stability of nano-crystalline alloys
G.P. Purja Pun a, K.A. Darling b, L.J. Kecskes b, Y. Mishin a,⇑aDepartment of Physics and Astronomy, George Mason University, MSN 3F3, Fairfax, VA 22030, USAbUS Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA
a r t i c l e i n f o
Article history:Received 5 June 2015Revised 17 August 2015Accepted 22 August 2015
Keywords:Interatomic potentialAtomistic simulationImmiscible alloysCu–Ta systemGrain boundaries
a b s t r a c t
Atomistic computer simulations are capable of providing insights into physical mechanisms responsiblefor the extraordinary structural stability and strength of immiscible Cu–Ta alloys. To enable reliable sim-ulations of these alloys, we have developed an angular-dependent potential (ADP) for the Cu–Ta systemby fitting to a large database of first-principles and experimental data. This, in turn, required the devel-opment of a new ADP potential for elemental Ta, which accurately reproduces a wide range of propertiesof Ta and is transferable to severely deformed states and diverse atomic environments. The new Cu–Tapotential is applied for studying the kinetics of grain growth in nano-crystalline Cu–Ta alloys with differ-ent chemical compositions. Ta atoms form nanometer-scale clusters preferentially located at grainboundaries (GBs) and triple junctions. These clusters pin some of the GBs in place and cause a drasticdecrease in grain growth by the Zener pinning mechanism. The results of the simulations are well con-sistent with experimental observations and suggest possible mechanisms of the stabilization effect of Ta.
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1. Introduction
Recent years have seen a rapidly growing interest in immisciblemetallic alloys due to their favorable combination of high mechan-ical strength and extraordinary stability at high temperatures[1–11]. The improved thermal stability is especially important fornano-crystalline alloys, which are characterized by a strongtendency for rapid grain growth at elevated temperatures. In par-ticular, Cu–Ta has emerged as one of the most promising immisci-ble alloy systems for high-temperature, high-strength applications[8,10,9,7]. Cu and Ta have different crystalline structures (FCC andBCC, respectively) and negligible mutual solubility in the solidstate [12]. High-energy mechanical alloying is capable of forcinga significant amount of Ta into a metastable FCC solid solution withCu [8,7]. During subsequent thermal processing, Ta precipitates ina variety of structural forms ranging from atomic-level coherentclusters to incoherent particles reaching a few nanometers and lar-ger in size.
It is the formation of the Ta clusters that is believed to beresponsible for the extraordinary properties of the mechanicallyalloyed Cu–Ta alloys. Such clusters tend to form at grain bound-aries (GBs), pinning them by the Zener mechanism [13] andpreventing grain growth even in nano-crystalline alloys with a
strong capillary driving force for coarsening. In addition, the clus-ters themselves contribute to the strengthening of the material,making it significantly stronger than predicted by the Hall–Petchmodel of GB hardening [14,15]. While the exact mechanisms ofthese effects remain the subject of active current research, it islikely that the nanometer and sub-nanometer size clusters restraindislocation glide in the Cu lattice similar to the Guinier–Prestonzones [16,17] in Al(Cu) alloys. Furthermore, the cluster located atGBs are likely to increase the resistance to GB sliding, which isone of the important deformation and accommodation mecha-nisms in nano-grained materials [18].
Atomistic computer simulations offer a powerful tool for gain-ing deeper insights into the physical mechanisms of the Ta effect.Such simulations typically employ the molecular dynamics (MD)and Monte Carlo (MC) methods and rely on semi-empirical inter-atomic potentials to describe atomic interactions [19]. Our previ-ous simulations of the Cu–Ta system [7] utilized an interatomicpotential [20] that had been developed with emphasis on surfacephenomena, such as wetting and dewetting of Cu thin films depos-ited on Ta substrates. While this potential does capture basictrends in this system, quantitative investigations of immisciblebulk alloys require a more accurate potential developed for a widerrange of applications. Indeed, since the development of the previ-ous potential [20], more first-principles data became availableand potential generation and testing methods have been signifi-cantly improved. The goal of this work is to develop a new and
http://dx.doi.org/10.1016/j.actamat.2015.08.0521359-6454/! 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
⇑ Corresponding author.E-mail address: [email protected] (Y. Mishin).
Acta Materialia 100 (2015) 377–391
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improved Cu–Ta potential taking advantage of the richer first-principles databases and more advanced fitting and testingmethodologies existing today.
The strategy for achieving this goal is as follows. We will adoptthe existing embedded-atom method (EAM) [21] potential for Cu[22], which has been successfully tested in manymaterials applica-tions. A new potential will be constructed for Ta. The previouslydeveloped angular-dependent potential (ADP) [23] had a flaw:due to an error in the fitting program, some of the elastic constantsof Ta are different from those reported in the paper. In addition, themelting temperature of Ta predicted by that potential is signifi-cantly higher than experimental. Several other potentials havebeen developed for Ta [24–28], the most recent ones being theEAM potentials by Li et al. [27] and Ravelo et al. [29]. The RaveloEAM potential [29] is especially accurate and transferable as willbe discussed below. However, we prefer the ADP potential format[30,23,20,31,32], which we believe to be more appropriate for BCCtransition metals than the regular EAM. We therefore set the goalof constructing a new ADP potential for Ta, and along the way toimprove on some of the properties of the Ravelo potential [29],such as the melting temperature, the surface energies and theinterstitial formation energies.
The development of the ADP Ta potential will be describedin Sections 2 and 3 and the testing results will be reported inSection 4. We compare the predictions of the new potential withfirst-principles calculations and the potentials by Li et al. [27]and Ravelo et al. [29]. Having a Ta potential, we then cross it withEAM Cu [22] by fitting the mixed-interaction functions in the ADPformat. The fitting and testing results for the binary Cu–Ta poten-tial are presented in Section 5. The ability of the new potential tomodel the Cu–Ta system is demonstrated in Section 6 by studyingstructural stability of nano-crystalline Cu–Ta alloys. Two types ofalloys are tested, one modeling the highly non-equilibrium statewith a random dispersion of Ta atoms created by mechanical alloy-ing, and the other mimicking the heat-treated alloys containing Taclusters. The results of this study are well consistent with experi-mental observations and reveal some interesting effects such asthe strong pinning of GBs by the Ta clusters and the formationand growth of Ta clusters in random alloys by the mechanism ofGB diffusion.
2. Potential format
In the ADP formalism [30,23,20,31,32], the total energy of a sys-tem is expressed by
Etot ¼12
X
i;jði–jÞUsisj ðrijÞ þ
X
i
Fsi ð!qiÞ þ12
X
i;aðla
i Þ2 þ 1
2
X
i;a;bðkabi Þ
2
% 16
X
i
m2i ; ð1Þ
where indices i and j enumerate atoms and superscriptsa; b ¼ 1; 2; 3 represent Cartesian components of vectors and ten-sors. In Eq. (1), Usisj ðrijÞ is the pair interaction energy between anatom i of chemical species si located at position ri and an atom jof chemical species sj located at position rj ¼ ri þ rij. The secondterm is the energy of embedding an atom of chemical species si inthe host electron density !qi induced at site i by all other atoms.The host electron density is given by
!qi ¼X
j–i
qsjðrijÞ; ð2Þ
whereqsjðrijÞ is the atomic electron density function of a neighboring
atom j of chemical species sj. The first two terms in Eq. (1) constitutethe standard embedded-atommethod (EAM) format [21].
The remaining terms in Eq. (1) represent the non-central char-acter of bondings and include the dipole vectors
lai ¼
X
j–i
usisj ðrijÞraij; ð3Þ
and quadrupole tensors
kabi ¼X
j–i
wsisj ðrijÞraijr
bij; ð4Þ
where the trace of kabi is
mi ¼X
akaai : ð5Þ
In these equations, usisj ðrÞ and wsisj ðrÞ are additional potentialfunctions of the radial distance between atoms. For the binaryCu–Ta system, the construction of an ADP potential requires 13potential functions: UCuCuðrÞ; UTaTaðrÞ, UCuTaðrÞ; qCuðrÞ; qTaðrÞ,FCuð!qÞ; FTað!qÞ; uCuCuðrÞ, uTaTaðrÞ; uCuTaðrÞ; wCuCuðrÞ, wTaTaðrÞ, andwCuTaðrÞ. In this work, the ADP potential functions for pure Ta willbe constructed separately and then crossed with an existing EAMpotential for pure Cu developed in [22] to obtain a binary potential.
3. Fitting of the ADP potential for Ta
The pair-wise interaction function UTaTaðrÞ, the host electrondensity qTaðrÞ, and the quadrupole interaction function wTaTaðrÞwere represented by cubic splines. For each spline, not only thevalues of the function but also positions of the nodes were adjus-table parameters. Each function was parameterized by five nodes,giving a total of 30 fitting parameters. To impose a smooth cutoffat a distance rc , the splines were multiplied by the truncation func-tion wððr % rcÞ=hÞ, were
wðxÞ ¼x4
1þx4 if x < 0;0 if x P 0:
(ð6Þ
The truncation introduces two more parameters rc and h, bring-ing the total number of fitting parameters to 32. In this work, wedid not use the dipole term. It was found that it did not improvethe potential quality enough to warrant its inclusion.
The embedding energy FTað!qÞ was obtained by inverting Eq. (1)to ensure that the energy E per atom of BCC Ta follows Rose’s equa-tion of state postulated in the form
EðaÞ ¼ E0 1þ axþ f 3a3 þ f 4a4! "e%ax; ð7Þ
where
x ¼ aa0
% 1 ð8Þ
and
a ¼ %9V0BE0
# $12
: ð9Þ
Here, a is the cubic lattice parameter and E0; a0 and V0 are the equi-librium values of the cohesive energy, lattice parameter and atomicvolume, respectively. In Eq. (7), the coefficients f 3 and f 4 could alsobe treated as adjustable parameters. However, in this work we usedthe values f 3 ¼ %0:035744 and f 4 ¼ %0:020879 optimized byRavelo and co-workers [29]. Note that this parameterization ofthe potential functions guarantees an exact fit to E0; B and a0.
The potential parameters were fitted to the elastic moduli,vacancy formation energy and energies of low index surfaces ofBCC Ta and energies of several alternate crystalline structures ofTa. The parameters were optimized by minimizing the sum ofweighted mean-squared deviations of computed properties from
378 G.P. Purja Pun et al. / Acta Materialia 100 (2015) 377–391
the respective experimental and/or first-principles values. A two-step optimization process was applied. First, a preliminary opti-mization was implemented utilizing a genetic algorithm developedin our previous work [31], followed by a more refined fit using asimulated annealing method.
4. Testing results for the Ta potential
The optimized potential functions are plotted in Fig. 1. The pairpotential and embedding energy function are plotted in the effec-tive pair format [33,34], although the host electron density !q hasnot been scaled and the minimum of the embedding energy occursat !q ¼ 1:074599.
Some of the properties of BCC Ta predicted by the potentialare summarized in Table 1 in comparison with experimental data,first-principles calculations and predictions of EAM potentials[29,27]. All properties, except for the melting point, were com-puted at absolute zero temperature. For defect energies, thestructures were relaxed and the results were tested for conver-gence with respect to the system size. The formation energiesof vacancies (Ef
v) and interstitials (Efi ) were computed for a set
of different system sizes, plotted against the inverse of the num-ber of atoms and extrapolated to the infinite size [35]. Thevacancy migration energy Em
v was calculated in a periodic super-cell containing 1024 atoms using the nudge elastic band (NEB)method [36,37]. The activation energy of self-diffusion wasobtained by Q ¼ Ef
v þ Emv . The generalized stacking fault energies
were computed in the gamma-surface mode [38]. Namely, twohalf-crystals were translated with respect to each other in a givendirection parallel to a chosen crystal plane. After each incrementof translation, the structure was relaxed with respect to atomicdisplacements in the x direction normal to the fault plane whilekeeping the y and z components of atomic coordinates fixed.Most of the computations reported here used the LAMMPS simu-lation program [39].
Table 1 shows that the ADP potential accurately reproduces theelastic constants c11 and c12 but slightly overestimates c44. Fig. 2presents phonon dispersion relations for BCC Ta computed at300 K by MD simulations [40] with the ADP potential. In thismethod, the dynamical matrix is reconstructed from correlationsbetween atomic displacements and is subject to a Fourier transfor-mation in high-symmetry directions in the reciprocal space. Thesimulations were performed in a periodic supercell containing16&16&16 primitive unit cells. The phonon dispersion curves arein qualitative agreement with experimental measurements byneutron scattering [41]. While low phonon frequencies are repro-duced by the potential accurately, the maximum phonon fre-quency at point H of the zone boundary is underestimated whilethe frequencies at points P and N are overestimated. Similar devi-ations were found with other potentials. For example, the maxi-mum frequency in the phonon density of states predicted by thepresent ADP potential and the two EAM potentials developed byRavelo et al. [29] are, respectively, 5.05, 4.56 and 4.25 THz. Theexperimental value is 5.03 THz. Note that phonon frequencies werenot included when fitting the potentials.
The vacancy migration energy predicted by the ADP potential(0.82 eV) overestimates the accepted experimental value of0.7 eV but falls within the range of first-principles calculations.The activation energy of self-diffusion is consistent with experi-mental measurements and first-principles calculations. For inter-stitials, the potential underestimates the first-principlesformation energy for the h100i dumbbell orientation, as do allother potentials [29,27]. However, the energies of the h110i andh111i interstitials are reproduced more accurately than with otherpotentials. The energy ctw of the 211f gtwin boundary is in
good agreement with the first-principles value (no experimentaldata is available). The energies cs of low-index surfaces wereincluded in the potential fit and are in good agreement withfirst-principles calculations. The ranking of the surfaceenergies is also consistent with first-principles calculations:
(a)
−0.5
0.0
0.5
1.0
1.5
2.0
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
Ene
rgy
(eV
)
Distance (Å)
(b)
0.00
0.02
0.04
0.06
0.08
0.10
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
Ele
ctro
n de
nsity
Distance (Å)
(c)
−5.0
0.0
5.0
10.0
15.0
20.0
25.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Ene
rgy
(eV
)
�ρ (arbitrary units)
(d)−0.05
0.000.050.100.150.200.250.300.350.40
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
Ene
rgy
(eV
/Å2 )
Distance (Å)
Fig. 1. Potential functions of the ADP Ta potential: (a) pair interaction, (b) electrondensity, (c) embedding energy, and (d) quadrupole interaction function.
G.P. Purja Pun et al. / Acta Materialia 100 (2015) 377–391 379
csh110i < csh100i < csh211i < csh111i. The EAM potentials[29,27] significantly underestimate the surface energies.
Generalized stacking faults energies for the 110f g and211f gfault planes are displayed in Fig. 3. The first-principlesgamma-surfaces [29,23] are shown for comparison. For the sakeof completeness, we also include the earlier first-principles calcu-lations by Frederiksen and Jacobsen [42] for the h111if110g fault,which are unrelaxed and thus predictably overestimate the recentrelaxed calculations [29,23]. The maxima along the gamma sur-faces represent unstable stacking fault energies cus, which are sum-marized in Table 1. The ADP predictions are in excellent agreement
with first-principles calculations, even though the latter were notincluded in the potential fit. Unstable stacking fault energies areimportant for simulations of core effects in dislocation glide.
Equilibrium energies DE of several alternate structures of Ta rel-ative to the BCC ground state are summarized in Table 2. Theagreement between the ADP energies and first-principles data isreasonable. To facilitate comparison with other potentials, we plotthe energies predicted by the ADP and EAM potentials versus thefirst-principles data (Fig. 4). The bisecting line in this plot is the lineof perfect agreement. Despite the scatter of the points, all poten-tials show a strong correlation with first-principles results. The
Table 1Properties of BCC Ta computed with the present ADP potential in comparison with experimental data, first-principles calculations and previouslydeveloped EAM potentials [29,27]. The properties marked by an asterisk were included in the potential fit. The values marked by a dagger werecomputed in this work using the published potentials.
Property Experiment Ab initio ADP EAM1 EAM2 EAM3
a0 (Å)⁄ 3.3026q; 3.3039l 3.26p; 3.3086i 3.304 3.304 3.304 3.30263.315e; 3.3k
3.27864f; 3.311m
E0 (eV)⁄ %8.1r %8.1 %8.1 %8.1 %8.089B (GPa) 194.2a; 198l 195.58k; 194f 194.7 194.7 195.9 179.2
190.95j; 196m
c11 (GPa)⁄ 266.3a 266.0v 265; 259.6c; 258.7k 268.9 263 267 247.5y
249.68j 257t
c12 (GPa)⁄ 158.2a 160.94v 159i; 165.2c; 168.8k 157.6 161 160 143.6y
161.59j 163t
c44 (GPa)⁄ 87.4a 82.47v 74t; 64.2c; 67.8k 90.4 82 86 86.570.26j 71t
E fv (eV)⁄ 2.2–3.1s 3.22b; 2.99h; 2.95g 3.06 2.43 2.00 2.76
2.91d 3.14u
Emv (eV) 0.7s 0.76b; 0.83h 0.82 0.99 1.08 1.240.86 1.48u
Q (eV) 3.8–4.39s 3.98b 3.88 3.42 3.08 4.00
E fi h100i (eV) 7.00u 5.86 5.34y 5.89y
E fi h110i (eV) 6.38u 5.65 4.55y 5.12y
E fi h111i (eV) 5.83u 5.62 4.44y 5.05y
Tm (K) 3293 3296 3033 3080ctw (J/m2) 0.304b 0.28 0.29y 0.30y 0.42y
cus (J/m2):f110gh111i 0.840b 0.782 0.864 0.803f110gh110i 1.947b 2.041 2.181y 2.182y
f110gh100i 1.947b 2.041 2.194 2.208f211gh110i 4.079b 3.927 3.422y 3.312y
f211gh111i 0.947b 0.918 1.017 0.975cs (J/m2):f100g' 3.142b; 3.097n; 2.97p 2.991 2.28 2.40 2.03f110g' 2.881b; 3.087n; 2.58p 2.755 1.97 2.04 1.77f111g' 3.312b; 3.455n 3.236 2.53 2.74 2.20y
f211g 3.270b; 3.256n 3.113 2.32y 2.48y 2.02y
1;2 Ref. [29].a Ref. [64]b Ref. [23]c Ref. [65]d Ref. [66]e Ref. [67]f Refs. [68,69]g Ref. [70]h Ref. [71]i Ref. [72]j Ref. [73]k Ref. [74]l Ref. [75]
m Ref. [76]n Ref. [77]p Ref. [78]q Ref. [79]r Ref. [80]s Ref. [81]t Ref. [82]u Ref. [83]v Ref. [84]3 Ref. [27]
380 G.P. Purja Pun et al. / Acta Materialia 100 (2015) 377–391
scatter is more significant for the potential by Li et al. [27]. Thealternate structures were included in this analysis as a test oftransferability of the potentials to various atomic environmentsdifferent from the BCC structure. It is important to note that boththe ADP and EAM potentials correctly predict the metastable b-Uand b-W structures to be closest in energy to the BCC ground state.The b-phase of Ta (b-U prototype) is often observed experimentally
in deposited thin films [26,43,44]. The proposed ADP potentialshould be suitable for thin film simulations involving the b-phase.
We next discuss the performance of the potential under largestructural distortions. The pressure–volume relation computedfor BCC Ta at 0 K is in good agreement with experiment andfirst-principles calculations up to 1 TPa (Fig. 5). As a further testbeyond the limits of linear elasticity, we performed a static uniax-ial compression of BCC Ta at 0 K in three different directions.Stress–strain relations under such conditions are relevant to shockwave propagation and were computed by Ravelo et al. [29]. Thematerial was compressed in a chosen x-direction up to e ¼ 40%while keeping fixed dimensions in the y and z directions. Theresults are shown in Fig. 6 in comparison with first-principles cal-culations [29]. While marked deviations are observed under com-pressions exceeding 20%, the overall agreement with first-principles calculations is good and on the same level of accuracyas with the EAM potentials [29].
Several other tests were conducted to evaluate the transferabil-ity of the potential to highly distorted configurations. One of theminvolved homogeneous twinning and anti-twinning deformationpaths, which are relevant to mechanical behavior of Ta. This defor-mation carries a BCC crystal back to itself but with a twinning oranti-twinning orientation relative to the initial state. A detailedcrystallographic description of this path can be found in Ref. [23].
0
1
2
3
4
5
6
Γ H P Γ N
Fre
quen
cy (
TH
z)
[ζ,0,0] [ζ,ζ,ζ] [ζ,ζ,ζ] [0,ζ,ζ]
Fig. 2. Phonon dispersion relations for BCC Ta computed with the ADP potential at300 K (solid lines). The circles (() and triangles (M) represent experimental datafrom Ref. [41].
0.0
0.5
1.0
1.5
2.0
2.5
0.0 0.2 0.4 0.6 0.8 1.0
Ene
rgy
(J/m
2 )
Displacement
(110)⟨100⟩
DFTDFTADP
(a)
0.0
0.5
1.0
1.5
2.0
2.5
0.0 0.2 0.4 0.6 0.8 1.0
Ene
rgy
(J/m
2 )
Displacement
(110)⟨110⟩
DFTADP
(b)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.2 0.4 0.6 0.8 1.0
Ene
rgy
(J/m
2 )
Displacement
(110)⟨111⟩
ADPDFTDFTDFT
(c)
0.0
1.0
2.0
3.0
4.0
5.0
0.0 0.2 0.4 0.6 0.8 1.0
Ene
rgy
(J/m
2 )
Displacement
(211)⟨110⟩
DFTADP
(d)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.2 0.4 0.6 0.8 1.0
Ene
rgy
(J/m
2 )
Displacement
(211)⟨111⟩
DFTDFTADP
(e)Fig. 3. Cross-sections of gamma surfaces for BCC Ta computed with the present ADP potential in comparison with first principles (DFT) calculations (( [23], ! [29] andM [42]): (a) h100if110g, (b) h110if110g, (c) h111if110g, (d) h110if211g, (e) h111if211g. The displacements are normalized by the period in the respective direction.
G.P. Purja Pun et al. / Acta Materialia 100 (2015) 377–391 381
The deformation is described by a shear parameter which is zero inthe initial state and attains the values of 1=
ffiffiffi2
pand %
ffiffiffi2
pfor the
twin and anti-twin orientations, respectively. The energy along thispaths computed with the ADP potential is shown in Fig. 7 in com-parison with first-principles calculations [23]. The potential accu-rately reproduces the maximum anti-twinning energy (1.04 eV/atom) and slightly underestimates the maximum twinning energy(0.15 eV/atom).
The potential was also tested for three volume-conservingdeformation paths: orthorhombic, tetragonal and trigonal.Orthorhombic deformation is implemented by elongation of theBCC cubic unit cell along the ½001* direction with simultaneouscompression along the ½110* direction to preserve the equilibriumBCC volume. The respective deformation parameter p was definedin [45]. For the tetragonal deformation path (Bain path), the unitcell is elongated in the ½001* direction with simultaneous compres-sion in the ½100* and ½010* directions by equal amounts. The defor-mation is measured by the parameter p ¼ c=a with p ¼ 1 for theBCC structure and p ¼
ffiffiffi2
pfor the FCC structure. Further elongation
produces a local energy minimum corresponding to a body-centered tetragonal (BCT) structure. Finally, for the trigonal defor-mation we compress the BCC structure along the ½111* axis with
simultaneous elongation in perpendicular directions. The deforma-tion parameter p is the Lagrangian strain described in [46]. Thedeformation starts with p ¼ 0 (BCC structure) and creates the sim-ple cubic (SC) and FCC structures at p ¼ 2 and p ¼ 4, respectively.Comparison of the ADP and EAM potentials with first-principlescalculations is shown in Fig. 8. Given that no information aboutthese deformation paths was included in the potential fit, theagreement with first-principles data is quite reasonable. The ADPpotential demonstrates about the same level of accuracy as thepotentials by Ravelo et al. [29]. The potential by Li and co-workers [27] is somewhat less accurate. In particular, for thetetragonal deformation path it predicts a local minimum of energyfor the FCC structure instead of a maximum.
To evaluate thermal properties predicted by the ADP potential,the linear thermal expansion factor was computed as a function oftemperature (Fig. 9). The simulations utilized a periodic cubicsupercell containing 16000 atoms. MD simulations were executedin the NPT (fixed temperature and pressure) ensemble in which allthree dimensions of the supercell were allowed to fluctuate inde-pendently to ensure zero components of pressure in all three direc-tions. The cubic lattice parameter a was averaged over a 1 ns longMD run and the linear expansion coefficient was defined asða% artÞ=art , where art is the lattice parameter at room temperature
Table 2Equilibrium energies DE per atom (relative to the cohesive energy of BCC Ta) and the respective lattice parameters a0 of alternate structures of Ta computed with the ADPpotential in comparison with EAM potentials [29,27] and first-principles calculations [23]. SH is the simple hexagonal phase and L12 is the FCC structure with one unrelaxedvacancy per cubic unit cell. The structures marked by an asterisk were included in the potential fit.
Structure Ab initio ADP EAMa EAMb EAMc
DE (eV) a0 (Å) DE (eV) a0 (Å) DE (eV) a0 (Å) DE (eV) a0 (Å) DE (eV) a0 (Å)
BCC (A2)* 0 3.2398 0 3.304 0 3.304 0 3.3026 0 3.304b-U (Ab) 0.041 9.9974 0.073 10.2441s 0.038y 10.1505y,s 0.076 10.1097y,s 0.026y 10.1084y,s
b-W (A15) 0.060 5.1805 0.075 5.3482 0.035y 5.2670y 0.086y 5.2727y 0.029y 5.2716y
x (C32) 0.249 4.7611 0.143 4.7197r 0.148y 4.7442y,! 0.215y 4.6898y,m 0.166y 4.7530y,a
FCC (A1)* 0.303 4.1256 0.124 4.2456 0.134y 4.1850y 0.137 4.2039y 0.167y 4.1556y
HCP (A3)* 0.385 2.9112 0.151 2.9841l 0.139y 2.9632y,l 0.137y 2.9726y,l 0.176y 2.9452y,lSH (Af)* 0.757 2.7715 0.811 2.7010 0.920y 2.7284y 1.127y 2.9032y 0.932y 2.7453y
SC (Ah) 1.137 2.6710 1.064 2.5920 1.134y 2.5958y 1.382y 2.6638y 1.153y 2.6374y
Cu3Au (L12) 1.116 3.9120 1.246 3.9801 1.113y 3.8903y 1.138y 4.1057y 1.016y 3.8370y
DC (A4)* 3.128 5.9096 2.931 6.2362 2.867y 5.4476y 2.274y 6.2665y 3.075y 5.5242y
a;c Ref. [29].bRef. [27].l ideal c/aa c/a = 0.5790.m c/a = 0.5792.! c/a = 0.5827.r c/a = 0.5876.y computed in this work.s c/a = 0.5290.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
EA
M a
nd A
DP
ene
rgy
(eV
)
DFT energy (eV)
EAM1EAM2EAM3ADP
Fig. 4. Energies of alternate structures of Ta predicted by ADP and EAM potentialsversus the results of first-principles (DFT) calculations [23]. Potentials: EAM1 [29],EAM2 [29], EAM3 [27] and ADP (present work). The bisecting dashed line is the lineof perfect correlation. Details of the structures are given in Table 2.
0
200
400
600
800
1000
0.40 0.50 0.60 0.70 0.80 0.90 1.00
Pre
ssur
e (G
Pa)
V/V0
ADPDFTDFTExpExp
Fig. 5. Pressure–volume relation for BCC Ta computed with the present ADPpotential in comparison with experiments (( [85], M [75,86]) and first-principles(DFT) calculations (þ [87], r [88]).
382 G.P. Purja Pun et al. / Acta Materialia 100 (2015) 377–391
(300 K). The ADP potential demonstrates a good agreement withexperimental data up to the temperature of 1200 K butunderestimates the experimental lattice constant at highertemperatures.
The melting temperature Tm of BCC Ta was computed by twomethods: the phase coexistence method and the interface velocitymethod. The phase coexistence method is described in detail else-
where [47,48]. In short, a simulation block containing a planesolid–liquid interface is equilibrated by a micro-canonical MD sim-ulation and the equilibrium temperature and pressure (definedhere as the trace of the stress tensor averaged over the system)are calculated. The simulation is repeated several times starting
(a)
0
100
200
300
400
500
600
700
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Str
ess
(GP
a)
ε⟨100⟩
Pxx (ADP)Pzz (ADP)Pxx (DFT)Pzz (DFT)
(b)
0 50
100 150 200 250 300 350 400 450
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Str
ess
(GP
a)
ε⟨110⟩
Pxx (ADP)Pyy (ADP)Pzz (ADP)Pxx (DFT)Pyy (DFT)Pzz (DFT)
(c)
0 50
100 150 200 250 300 350 400 450
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Str
ess
(GP
a)
ε⟨111⟩
Pxx (ADP)Pzz (ADP)Pxx (DFT)Pzz (DFT)
Fig. 6. Stress components as functions of strain during uniaxial compression of BCCTa calculated with the present ADP potential in comparison with first-principles(DFT) calculations [29]. Crystallographic directions: (a) x : ½100*; y : ½010*; z : ½001*.(b) x : ½110*; y : ½!110*; z : ½001*. (c) x : ½111*, y : ½!110*; z : ½!1 !12*.
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0
Ene
rgy
(eV
/ato
m)
Shear strain
anti-twinning
twinning
DFTADP
Fig. 7. Energy along the homogeneous twinning and anti-twinning deformation-paths in BCC Ta calculated with the present ADP potential in comparison with first-principles (DFT) calculations [23].
(a) 0
50
100
150
200
250
1.0 1.2 1.4 1.6 1.8 2.0
E [m
eV/a
tom
]
p
BCT
ADPEAM1EAM2EAM3DFT
(b) 0
50 100 150 200 250 300 350 400 450 500
0.8 1.0 1.2 1.4 1.6 1.8 2.0
E [m
eV/a
tom
]
p
BCC
FCC
BCT
ADPEAM1EAM2EAM3DFT
(c) 0
200 400 600 800
1000 1200 1400 1600 1800
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
E [m
eV/a
tom
]
p
BCC
SC
FCC
ADPEAM1EAM2EAM3DFT
Fig. 8. Energy along three homogeneous deformation paths of Ta computed withinteratomic potentials in comparison with first-principles (DFT) calculations [45].(a) Orthorhombic deformation path. (b) Tetragonal deformation path. (c) Trigonaldeformation path. Potentials: EAM1 [29], EAM2 [29], EAM3 [27] and ADP (presentwork). For each deformation path, the volume remains fixed at its equilibrium valuein BCC Ta.
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0 500 1000 1500 2000 2500 3000 3500
Line
ar th
erm
al e
xpan
sion
(%
)
Temperature (K)
Tm
ADPExperiment
Fig. 9. Linear thermal expansion factor (%) of BCC Ta relative to room temperaturecomputed with the present ADP potential in comparison with experimental data[89]. The experimental melting temperature is indicated.
G.P. Purja Pun et al. / Acta Materialia 100 (2015) 377–391 383
from slightly different initial states and the temperature–pressurerelation thus obtained is extrapolated to zero pressure to deter-mine Tm. In this work we used a simulation block with dimensions49:6& 99:9& 47:6 Å containing 12240 atoms. The solid–liquidinterface was parallel to the (111) plane and normal to the longestdimension of the simulation block. A linear fit to the pressure–tem-perature points shown in Fig. 10 gives the melting temperature of3301+ 0:5 K. As an independent check, Tm was evaluated by NPTsimulations using the same simulation block but annealed isother-mally at several temperatures above and below the expected melt-ing point. In such simulations, the solid–liquid interface migratesuntil one of the two phases disappears. The rate of migration wasevaluated by the average energy change per unit time and extrap-olated to zero rate (Fig. 11). This calculation gives Tm ¼ 3291+ 4 K.Both numbers are close to the experimental melting point of3293 K. Each method has its advantages and shortcomings. In thephase coexistence method, the state of stress is generally non-hydrostatic and the trace of the stress tensor is not an adequatemeasure of residual stresses in the system [49]. This is one of thesources of the statistical scatter of the points in Fig. 10. On theother hand, the velocity method creates non-equilibrium statesof the system. With the information available to us, we estimatethe melting point by averaging the two numbers to obtain3296 K. The melting temperatures predicted by the EAM potentials[29] (3033 and 3030 K) are in less accurate agreement withexperiment.
5. ADP potential for the Cu–Ta system
5.1. Potential format and fitting
For the binary Cu–Ta system, the mixed pair potential UCuTaðrÞwas represented by a Lenard–Jones potential with additional poly-nomial terms:
UCuTaðrÞ ¼ E0a12r
& '12% a6
r
& '6þ
X11
i¼1; i–6
air
& 'i( )
þ a0
" #
wr % rch
& ';
ð10Þ
where wðxÞ is the cutoff function defined by Eq. (6). The fittingparameters of UCuTaðrÞ are rc , h and the ai’s. The mixed quadrupoleinteraction function wCuTaðrÞ was postulated in the form
wCuTaðrÞ ¼ wr % rch
& 'q1e
%q2r þ q3ð Þ; ð11Þ
where q1; q2 and q3 are three additional fitting parameters. The cut-off parameters rc and h remain the same as in Eq. (10). Several alter-nate functional forms of UCuTaðrÞ andwCuTaðrÞwere also tested, but it
was found that Eqs. (10) and (11) offered the best accuracy of thefitting.
Since the pure Cu potential [22] is in the EAM format, therespective dipole and quadrupole interactions were set to zero.During the fitting of the mixed interactions, all potential functionswere subject to invariant transformations [34,50,47] that did notaffect properties of pure Cu and Ta but provided additional degreesof freedom for the optimization of mixed interactions. Such trans-formations involved three coefficients sTa; gCu and gTa, which wereused as additional fitting parameters.
The potential functions (10) and (11) were fitted to first-principles formation energies of several imaginary Cu–Ta com-pounds with cubic structures (Table 3). Although none of thesecompounds exists on the Cu–Ta phase diagram, their role is to rep-resent various structural and chemical environments that mayoccur during applications of the potential. As in the case of pureTa, the fitting parameters were optimized by a combination of agenetic algorithm and simulated annealing. The optimized
3280
3285
3290
3295
3300
3305
3310
3315
3320
-0.2 -0.1 0.0 0.1 0.2
Tem
pera
ture
(K
)
Pressure (GPa)
Fig. 10. Melting temperature as a function of pressure obtained by solid–liquidphase coexistence simulations for BCC Ta using the present ADP potential. The lineis a linear fit to find the melting point at zero pressure.
-8
-6
-4
-2
0
2
4
6
8
3000 3100 3200 3300 3400 3500 3600
Ene
rgy
(eV
/ato
m/n
s)
Temperature (K)
Fig. 11. Energy rate (measure of interface velocity) as a function of temperature inMD simulations of solidification and melting of BCC Ta with the present ADPpotential. The line is a linear fit to find the melting point.
Table 3Equilibrium formation energies (in eV per atom) of Cu–Ta compounds computed withthe present ADP potential in comparison with the first-principle calculations. Thecompounds marked by an asterisks were included in the potential fit.
Phase Structure Ab initio ADP
CuTa B2⁄ 0.1920a; 0.200b; 0.139c 0.07994L1'
0 0.1235a; 0.260b; 0.075c 0.23447L1'
1 0.2612a; 0.269b; 0.228c 0.30669B32 0.4841a 0.35980B1⁄ 0.7789a; 0.788b 0.69585B3⁄ 1.7421a; 1.755b 1.86985
Cu3Ta L1'2 0.3443a; 0.372b; 0.350c 0.23956
D019 0.1653a; 0.193b 0.18674D011 0.1971a 0.20504D022 0.1172a; 0.193b 0.28788D0'
3 0.1161a; 0.143b 0.30295A15 0.3739a 0.43389
Ta3Cu L1'2 0.1490a; 0.158b; 0.126c 0.23590
D019 0.1822a; 0.189b 0.25064D011 0.1393a 0.14615D022 0.1596a; 0.166b 0.27967D0'
3 0.1833a; 0.191b 0.19087A15 0.1168a 0.28516
Layer CuCuCuTaTaTa⁄ 0.387c 0.19685CuTaTa⁄ 0.352c 0.17293CuCuTa⁄ 0.219c 0.28657
Cu2Ta C11b 0.0959a 0.13506C11f 0.063c 0.18855
Ta2Cu C11b 0.1627a; 0.046c 0.09871C11f 0.141c 0.09216
a Ref. [51]b Ref. [52]c Ref. [20]
384 G.P. Purja Pun et al. / Acta Materialia 100 (2015) 377–391
mixed-interaction functions UCuTaðrÞ and wCuTaðrÞ are shown inFig. 12 together with the respective elemental functions. It shouldbe emphasized that, due to the invariant transformations men-tioned above, the Cu and Ta pair potentials look in this plot differ-ent from their original forms. In particular, the pair interactionUTaTaðrÞ looks different from the curve plotted in Fig. 1. Neverthe-less, when the binary potential developed here is applied to pureCu or pure Ta, all properties of these metals are exactly the sameas when using the original elemental potentials.
5.2. Properties of the Cu–Ta system
The imaginary compounds included in the potential fit had therelatively simple crystalline structures such as B1, B2, B3, L10, L11,L12 and D03. In addition, layered structures CuCuCuTaTaTa, CuCuTaand CuTaTa (see [20] for details) were also included. This set ofcompounds samples very diverse structures and different stoi-chiometries across the entire composition range between Cu andTa.
Three sources of first-principles data for the structural energieswere available for this work: the first-principles calculations byHashibon et al. [20] and the quantum–mechanical databases devel-oped by the groups of Curtarolo (AFLOW) [51] and Wolverton(OQMD: Open Quantum Materials Database) [52]. Certain discrep-ancies were found between the formation energies reported by thethree sources, apparently arising from different choices of calcula-tion parameters. To give an idea about the scatter of the data, weplot the predictions of Refs. [51,52] against each other for a set ofsimple structures selected for this work (Fig. 13(a)). The root-mean-square deviation between these datasets is 0.072 eV andPearson’s correlation factor is 0.997. In view of these discrepancies,the first-principles energies used as input for fitting and testing ofthe potential were obtained by averaging the numbers comingfrom all sources available for any particular compound.
We were unable to achieve a perfect fit to all compounds cho-sen as targets. It was especially difficult to reproduce the formation
energies of the layered structures without significant sacrifices inaccuracy with respect to the simpler structures. We thereforechose the strategy of putting less weight on the layered structuresand spreading the fitting errors more or less uniformly over theremaining compounds. Table 3 summarizes the results of fittingand testing. While for some compounds the agreement withfirst-principles data is good, for others the discrepancies can reachup to , 0:1 eV. Nevertheless, Fig. 13(b) demonstrates that theentire set of the ADP formation energies exhibits a significant cor-relation with first-principles data. The root-mean-square deviationbetween the ADP and first-principles energies is 0.089 eV and Pear-son’s correlation factor is 0.978.
The zero-temperature energy of dilute solutions predicted bythe ADP potential is 0.6771 eV for Cu in BCC Ta and 1.66471 eVfor Ta in FCC Cu. Both energies are large enough to ensure thatthe mutual solid-state solubility between Cu and Ta be very smallin agreement with the experimental phase diagram [12]. Calcula-tion of the Cu–Ta phase diagram predicted by this potential isbeyond the scope of this work. It is interesting to note, however,that the liquid Cu–Ta alloys exhibit a highly non-uniform structureconsisting of nanometer-scale Ta clusters embedded in the Cumatrix. Similar thermodynamically stable nano-colloidal struc-tures were found with the previous ADP potential [20] and wereexplained by a strong stabilizing effect of the Cu/Ta interface cur-vature [53]. The reproducibility of this unusual structure withtwo different potentials suggests that it is likely to be real andnot an artifact of an interatomic potential.
6. Structural stability of Cu–Ta alloys
6.1. Methodology of simulations
A 3D polycrystalline Cu sample was constructed by the Voronoitessellation method. Initially, 32 grain centers were selected as
(a)−0.5
−0.4
−0.3
−0.2
−0.1
0.0
0.1
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
Ene
rgy
(eV
)
Distance (Å)
Cu
CuTa
Ta
(b)−0.05
0.000.050.100.150.200.250.300.350.40
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
Ene
rgy
(eV
/Å )
Distance (Å)
Ta
CuTa
2
Fig. 12. Plots of: (a) mixed pair interaction function and (b) mixed quadrupoleinteraction function for the binary Cu–Ta system in comparison with elementalinteraction functions for Ta and Cu.
(a)0.00.20.40.60.81.01.21.41.61.82.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
E (
eV/a
tom
)
E (eV/atom)
(b)0.00.20.40.60.81.01.21.41.61.82.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Ab
initi
o en
ergy
(eV
/ato
m)
ADP energy (eV/atom)
Fig. 13. Formation energies (relative to equilibrium FCC Cu and BCC Ta) of selectedCu–Ta compounds listed in Table 2. (a) First-principles data from AFLOW [51](abscissa) and OQMD [52] (ordinate). (b) Predictions of the ADP potential versusaverage first-principles data. The bisecting dashed line is the line of perfectcorrelation.
G.P. Purja Pun et al. / Acta Materialia 100 (2015) 377–391 385
points of a regular FCC lattice but were subsequently randomizedby adding random displacements. This ensured random sizes ofthe grains in the polycrystal. The Voronoi tessellation was per-formed using the free open-source program voro++ [54,55]. Afterthe tessellation, the system was uniformly scaled in all three direc-tions to the dimensions of 40& 40& 40 nm with periodic bound-ary conditions in all three directions. Next, the polyhedra werefilled with FCC Cu with crystallographic axes randomly orientatedin space. Some overlap was initially allowed between the lattices ofneighboring grains. Subsequently, the GB structures were opti-mized by identifying pairs of atoms separated by less than 0.6 ofthe equilibrium first-neighbor distance in FCC Cu and replacingthem by a single atom. Further technical details of this procedurecan be found in [56,57]. To further relax the GBs, the polycrystalwas heated to the temperature of 300 K in 0.4 ns and annealedfor another 0.4 ns by NPT MD simulations. These steps resultedin a 32-grain periodic polycrystalline sample containing about5.4 million Cu atoms.
The initial grain sizewas estimated by d0 ¼ ðV=NgÞ1=3 - 12:5 nm,where Ng ¼ 32 is the number of grains and V is the system volume.During the subsequent grain growth simulations, the grain size dwas defined by d ¼ d0X0=X, where X is the number of GB atomsand X0 is the value of X in the initial state. To determine X for a givenMD snapshot, the latter was quenched by static energy minimiza-tion and analyzed for bond angle distortions using the OVITO visu-alization tool [58]. The bond angle analysis identifies atoms withFCC and HCP environments, which correspond to the perfect FCClattice and intrinsic stacking faults, respectively. The remainingatoms were attributed to GBs and their number was X. The implicitassumption of this calculation of d is that the GBwidth remains con-stant,whichmaybe a source of some error as theGBwidthmay varyduring the simulations.
Two types of Cu–Ta alloys were prepared. First, to model thenon-equilibrium solid solution created by mechanical alloying, aset percentage of Ta atoms was introduced by random substitutionof Cu atoms using a random numbers generator. As a result, the Taatoms were randomly scattered over the system with the samedensity regardless of the GBs or grain interiors. Second, more equi-librium states of the alloy were prepared by semi-grand canonicalMonte Carlo simulations [59] at zero pressure. In such simulations,the temperature T and chemical potential difference Dl betweenTa and Cu are fixed while the distribution of Ta atoms over the sys-tem can vary to reach local thermodynamic equilibrium. The trialmoves include displacements of randomly selected atoms by a ran-dom amount in a random direction with simultaneous random re-assignment of the chemical species of the chosen atom to either Taor Cu. Simultaneously, the dimensions of the simulation block in allthree directions are changed independently by random amountswith corresponding re-scaling of atomic coordinates. The trialmove is accepted or rejected by a Metropolis algorithm. Furthertechnical details of this method can be found in [60–62]. The sim-ulation temperature was chosen to be 800 K at which no significantgrain growth occurred during the Monte Carlo process, yet equilib-rium could be reached by an affordable number of Monte Carlosteps. The chemical potential difference Dl was adjusted to createseveral alloys with chemical compositions up to a few at.%Ta. Itshould be noted that these compositions are significantly higherthan the Ta solid solubility limit in Cu. Thus, the alloys createdby the Monte Carlo process are metastable and are intended tomodel the experimental alloys after a thermal treatment.
6.2. Simulation results
To model grain growth in pure Cu, the initial sample was heatedto the temperature of 1200 K in 1.2 ns and annealed at this temper-
ature for at least 10 ns in the NPT MD ensemble. Typical snapshotsbefore and after the annealing are shown in Fig. 14. The GBs arerevealed by the bond angle analysis using OVITO [58]. The imagesclearly demonstrate that significant grain growth has occurredduring the anneal (see online supplementary material [63] for ani-mations of the grain growth). The dynamics of grain growth arequantified in Fig. 15 by plotting the average grain size d as a func-tion of time. The points represent individual MD snapshots. Thescatter of the points tends to increase with the grain size, espe-cially when the sample is left with only 3–4 grains. The plot showsthat the grain size approximately doubles during the first 10 ns.
We now discuss the results for the Cu–Ta alloys. During theMonte Carlo simulations, the Ta atoms introduced into Cu tendedto form nanometer-size clusters. Some of them appeared insidethe grains and were coherent with the FCC lattice. However, theoverwhelming majority of the clusters formed at GBs and triple
Fig. 14. Nanocrystalline Cu (a) before and (b) after a 10 ns long isothermal MDsimulation at 1200 K. The GBs are revealed by bond angle analysis using OVITO [58].Dark blue atoms represent perfect FCC structure.
386 G.P. Purja Pun et al. / Acta Materialia 100 (2015) 377–391
junctions as illustrated in Fig. 16(a). This microstructure is wellconsistent with experimental TEM observation that also showthe preference of Ta clusters in nano-grained Cu samples [8].
To quantify the cluster size, the size distribution histogramswere computed. To this end, all Cu atoms were removed fromthe relaxed MD snapshot and the remaining Ta atoms were pro-cessed by the cluster analysis algorithm implemented in the OVITOtool [58]. This analysis enabled us to construct the distributionfunction of the number of atoms in the clusters. This function isre-plotted in Fig. 17 as the normalized probability versus the clus-ter size dc . The latter was defined by dc ¼ ðNcV=NÞ1=3 , where Nc isthe number of Ta atoms in the cluster and N is the total number ofatoms in the system. The plots are somewhat noisy because thestatistics were obtained from only one snapshot (correspondingto the given simulation time) and the number of Ta atoms is rela-tively small due to the low Ta concentration. The first peak atdc < 0:5 nm is caused by single Ta atoms and their small groupsof 2–3 atoms. The second peak is caused by the nano-clusters,which have a typical size of about 0.7 nm. Note that the numberof clusters (the peak hight) tends to increase with the alloy concen-tration with only a small effect on the cluster size. This trend, andin fact the very existence of the peak, suggest that the clusters con-stitute a well-defined and well-reproducible structural feature ofthe alloy and are not caused by mere statistical fluctuations inthe Ta distribution.
As demonstrated in Fig. 15(a), the addition of Ta in the form ofclusters can drastically reduce the grain growth. While the addi-tion of 0.1 at.%Ta has little effect, in the alloy with 0.3 at.%Ta thegrain size increases only slightly and soon reaches a stagnation[63]. The alloys with P 1 at.%Ta exhibit no grain growth withinthe uncertainty of the calculations. The final structure of the0.3 at.%Ta alloy after a 10 ns anneal is illustrated in Fig. 16(b). Mostof the clusters remain at GBs, suggesting strong pinning. It is inter-esting to note, however, that a number of clusters can now be
found inside larger grains that underwent some growth. Anima-tions of the grain growth [63] confirm that such cluster initiallyformed at GBs, which were then able to break away from the clus-ters and leave them behind inside the grains. The strong pinning ofGBs by Ta clusters is consistent with experimental studies ofmicrostructure and thermal stability of nano-crystalline Cu–Taalloys [8,9,7]. Such studies indicate that the precipitation of Taclusters and larger particles at GBs drastically reduces the graingrowth by the Zener pinning mechanism. The slight increase inthe cluster size in the 4 at.%Ta alloy suggested by Fig. 17(b) canbe explained by some coarsening due to Ta ‘‘short-circuit” diffusionalong GBs. While the time scale limitation of MD prevents us fromobserving further growth of these cluster, their trend for coarsen-ing is in agreement with the experimental observation of a largespectrum of cluster sizes, from a nano-meter size to a few tens ofnanometers, with larger clusters located at GBs [8].
(a) 10
15
20
25
30
0 2 4 6 8 10
Gra
in s
ize
[nm
]
Time [ns]
Cu0.1 at.% Ta0.3 at.% Ta0.5 at.% Ta1.0 at.% Ta2.0 at.% Ta4.0 at.% Ta
(b) 10
15
20
25
30
0 2 4 6 8 10
Gra
in s
ize
[nm
]
Time [ns]
Cu0.1 at.% Ta0.5 at.% Ta1.0 at.% Ta2.0 at.% Ta
Fig. 15. Average grain size versus time for several compositions of Ta in Cu–Taalloys during isothermal anneals at 1200 K by NPT MD simulations. (a) Clustered Ta.(b) Randomly dispersed Ta.
Fig. 16. Nanocrystalline Cu-0.3at.%Ta alloys (a) before and (b) after a 10nsisothermal MD simulation at 1200 K. The GBs are revealed by bond angle analysisusing OVITO [58]. The Ta atoms are shown in green and are larger than Cu atoms.
G.P. Purja Pun et al. / Acta Materialia 100 (2015) 377–391 387
We now turn to alloys with randomly dispersed Ta atoms. Inthis case, the addition of Ta also reduces the grain growth(Fig. 15(b)), but the effect is not as strong as in the previous case.For example, while the addition of 1 at.%Ta in the form of clusterspractically stops the grain growth (cf. Fig. 15(a)), the alloy with arandom distribution of the same amount of Ta still shows some
grain growth for the first 5 ns. In fact, a slight grain growth is evenseen in the random 2 at.%Ta alloy [63].
Another interesting feature of Fig. 15(b) is that, for all composi-tions larger than 0.1 at.%Ta, the grain growth stops after about 5 ns,as it typically does also in the alloys with clusters. A plausibleexplanation of this effect is the formation of Ta clusters at the mov-ing and stationary GBs during the anneal of random alloys. This isconsistent with the images and animations [63] of the alloy struc-ture. As an example, Fig. 18 shows a typical snapshot revealing ini-tial stages of cluster formation in the random Cu-1at.%Ta alloy. Thecluster formation is further confirmed by the cluster size distribu-tion plotted in Fig. 19. As before, the sharp peak at small clustersizes (< 0:3 nm) represents individual Ta atoms and their 2–3atom clusters created by pure chance during the sample prepara-tion. After the 10.8 ns anneal, a new peak begins to form reflectingthe formation of larger clusters with the size of about 0.45 nm. Iflonger anneals could be afforded by our computation resources,we expect that the peak would grow in height and shift towardslarger cluster sizes of about 0.8 nm. But already the observationof the early stages of peak formation combined with MD snapshotsprovide evidence for the clustering of the initially dispersed Taatoms. These newly formed clusters eventually pin the GBs andstop the grain growth. This observation is consistent with experi-ments [8] and a possible mechanism for the formation of Ta clus-ters in GB locations during the heat treatment of mechanicallyalloyed Cu–Ta alloys [8].
7. Discussion and conclusions
We have developed an ADP potential for the Cu–Ta system byfitting to a large database of first-principles and experimental dataand employing a combination of a genetic algorithm and simulatedannealing for parameter optimization. We utilize an available EAMCu potential [22] but construct a new ADP potential for Ta. Wechose the ADP potential format [30,23,20,31,32] over the regularcentral force EAM for two reasons. First, the ADP format offersadditional adjustable parameters that can be utilized for moreaccurate fitting to target properties. Second, and perhaps moreimportant, the ADP format captures the directional component ofchemical bonding and is more appropriate for BCC transition met-als. For this reason, even if the accuracy of fitting to target proper-ties is the same, an ADP potential can be expected to be moretransferable to a wider set of properties of Ta and Ta alloys in com-parison with regular EAM. Extensive tests show that the newpotential accurately reproduces many properties of Ta and is trans-ferable to severely deformed states and diverse atomic environ-ments different from the equilibrium BCC lattice. The potential
(a)0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
Pro
babi
lity
Cluster size [nm]
0 ns10 ns
(b)0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.4 0.6 0.8 1.0 1.2 1.4 1.6
Pro
babi
lity
Cluster size [nm]
0 ns10 ns
Fig. 17. Ta cluster size distribution in Cu–Ta alloys before and after a 10 ns annealat 1200 K. (a) Cu-0.1at.%Ta alloy. (b) Cu-4at.%Ta alloy. The initial clusters werecreated by a semi-grand canonical Monte Carlo simulation at 800 K.
Fig. 18. Snapshot of the Cu-2at.%Ta alloy after a 10.8 ns isothermal MD simulationat 1200K. The initial state was created by purely random dispersion of Ta atoms.The GBs are revealed by bond angle analysis using OVITO [58]. The Ta atoms areshown in green and are larger than Cu atoms.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.3 0.4 0.5 0.6 0.7 0.8
Pro
babi
lity
Cluster size [nm]
0 ns10.8 ns
Fig. 19. Ta cluster size distribution in Cu-1at.%Ta alloy before and after a 10.8 nsanneal at 1200 K. The initial state was created by purely random dispersion of Taatoms.
388 G.P. Purja Pun et al. / Acta Materialia 100 (2015) 377–391
should be widely applicable to simulations of thermal andmechanical properties of Ta in the future. It is more accurate thanmost of the previously proposed Ta potentials and is comparable inaccuracy with the recently developed EAM potential by Ravelo andco-workers [29]. In fact, the ADP potential shows some improve-ments over [29] with respect to surface energies, interstitial forma-tion energies and the melting point. For example, the meltingtemperature predicted by the ADP potential is within a few degreesfrom the experimental value. Given that the ADP format is physi-cally more appropriate for Ta, we believe that in the long run,the proposed potential will demonstrate a better reliability in awide range of applications than could not be achieved within thestandard EAM framework [21].
To obtain the binary Cu–Ta potential, the elemental Cu and Tapotentials have been crossed by fitting the mixed-interaction func-tions to first-principles energies for a set of imaginary intermetalliccompounds that do not exist on the phase diagram but sample var-ious atomic environments and chemical compositions.
The main application of the proposed Cu–Ta potential thatmotivated the present work is the problem of structural stabilityand strength of nano-crystalline Cu–Ta alloys. Detailed atomisticsimulations of these properties are the subject on ongoing workthat will be reported elsewhere. In the present paper, we confineourselves to some preliminary tests intended to demonstrate someof the capabilities of the potential. We focused on the grain growthretardation effect caused by Ta alloying. It was found that Ta pre-cipitates in the form of atomic-scale clusters preferentially locatedat GBs and triple junctions, forming a microstructure observed inrecent experiments [8] by transmission electron microscopy.Although the mutual solid solubility of Cu and Ta is extremelysmall, the high-energy cryogenic ball milling forced Ta atoms intothe Cu lattice forming a far-from equilibrium solution. Theoreti-cally, the Ta and Cu atoms are expected to completely separateduring the subsequent annealing of the alloy. In reality, due tothe slow diffusion of Ta in the Cu lattice [7], the separation can onlyoccur to a limited degree. It is reasonable to assume that the pro-cess starts with a spinodal decomposition of the solution resultingin the formation of a set of Ta clusters coherent with the Cu matrix.In the perfect lattice, such clusters are unlikely to grow much fur-ther. However, in the presence of GBs, dislocations and otherdefects providing high-diffusivity paths (‘‘short-circuit” diffusion),the clusters may grow in size and eventually lose coherency withthe matrix. In the present simulations, this was confirmed by theobservation of coarsening of the clusters located at GBs, a processwhich is similar to Ostwald ripening. The clusters never reachedthe large sizes often observed in experiments due to the limitedtime scale of MD simulations (tens of ns). A further confirmationcomes from the finding that, in alloys with a totally random distri-bution of Ta atoms (imitating the as-milled states of the material),Ta clusters spontaneously begin to nucleate and grow at GBs bydiffusion-controlled redistribution of Ta atoms. Although thisagreement between the simulations and experiments is reassuring,further work is needed for a quantitative comparison. This requirescalculations of Ta GB diffusivities and detailed investigations of thecluster nucleation and growth kinetics depending on the alloycomposition, temperature and GB type.
Besides this kinetic factor, there must be a thermodynamic pref-erence for the cluster nucleation at GBs (heterogeneous nucle-ation). Indeed, the preferential formation of clusters at GBs wasobserved in semi-grand canonical Monte Carlo simulations, inwhich the re-distribution of atoms is implemented by an artificialswitching process that does not involve diffusion. Validation of theheterogeneous nucleation hypothesis requires a special studyinvolving calculations of the GB and interface free energies andthe elastic strain energies associated with the cluster formation.
The present simulations also reproduce the drastic retardationeffect of Ta on the grain growth kinetics known from experimentalstudies [8] as well as previous simulations [7]. The fact that nearlyall Ta atoms located at GBs exist in the form of clusters is a strongindication that the retardation effect is due to the Zener pinningmechanism. It is most likely that in this class of alloys, other stabi-lization mechanisms, such as solute drag or reduction in the GBfree energy by segregation, are less important. The rare GBs thatwere able to break away from the Ta clusters remained pure Cuboundaries and only stopped when encountered other clusters.
Finally, previous simulations [7] have revealed a significantincrease in tensile strength caused by Ta alloying. While this con-clusion is in agreement with experiment [8], the exact strengthen-ing mechanism responsible for this effect remains elusive. As wasmentioned in Section 1, the preservation of the small grain sizealone cannot explain the strength in excess of the Hall–Petch rela-tion [8]. Atomistic simulations of the deformation process mayprovide insights into the effect of the GB clusters on deformationmechanisms leading to the high strength of Cu–Ta alloys.
Acknowledgments
We are grateful to Dr. Ramon Ravelo for providing first-principles data for the gamma surfaces and tensile compressionstresses from Ref. [29] in numerical format, which enabled directcomparison with the ADP predictions in Figs. 3 and 6. This workwas supported by the U.S. Army Research Office under a contractnumber W911NF-15-1-0077.
Appendix A. Supplementary data
Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.actamat.2015.08.052. These data include animations of Figures 14, 16 and 18 of thisarticle.
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