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Acta Materialia 60 (2012) 984–993

Shear induced chemical mixing in heterogeneous systems

Yinon Ashkenazy a,b,⇑, Nhon Q. Vo a, Daniel Schwen a, Robert S. Averback a,Pascal Bellon a

a Department of Materials Science and Engineering, Seitz Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana,

IL 61801, USAb Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel

Received 17 October 2011; received in revised form 6 November 2011; accepted 7 November 2011Available online 15 December 2011

Abstract

Shear-induced mixing in heterogeneous Cu alloy systems was investigated by molecular dynamics simulation. Each system, whichcomprised a single spherical particle within a Cu matrix, was subjected to cyclical shearing events to high strains at 100 K. The particlesinvestigated were Cu, Ag, Ni, Fe, Nb, and V. fcc particles were observed to undergo “superdiffusive” mixing with dislocations crossingthe particle–matrix interface. The initial rate of mixing in these systems increased quadratically with particle radius. bcc particles showeddifferent behaviors. For Nb and V dislocations did not cross into the particles and the initial rates of mixing increased linearly with par-ticle radius. The Nb particles remained spherical but developed amorphous Cu–Nb shells at the particle–matrix interface. The V particleshowed some faceting, but it too formed amorphous layers on non-facetted interfaces. The Fe particle behaved similarly to Nb and Vparticles initially, but more like fcc particles at high strains. The molecular dynamics results are compared with experimentalobservations.� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Severe plastic deformation; Homogenization/solutionization; Molecular dynamics; Simulation; Copper alloys

1. Introduction

Nanostructured materials are gaining widespread inter-est for a number of applications where bulk quantities ofmaterial are required, including structural, hydrogen stor-age, shape memory and magnetic applications. In most ofthese situations top-down processing schemes, such assevere plastic deformation (SPD) (ball milling, accumula-tive roll bonding (ARB), equal channel angular extrusion(ECAE), wire drawing or high pressure torsion (HPT))are of practical interest [1]. These methods are all knownto refine grain sizes to the nanometer length scale [2]. Theresulting microstructures, however, can be quite distinct,as these methods impose different deformation modes,

1359-6454/$36.00 � 2011 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2011.11.014

⇑ Corresponding author at: Racah Institute of Physics, Hebrew Univer-sity of Jerusalem, Jerusalem 91904, Israel. Tel.: +972 524291968.

E-mail address: yinonashkenazy@gmail.com (Y. Ashkenazy).

ranging from simple shear for HPT and ECAE to planestrain for ARB, thus stabilizing different crystallographictextures. Also of interest is the phase stability of these mate-rials under SPD. Many two phase alloys, such as Cu–Ag [3]and Cu–Fe [4,5], are known to become homogeneous singlephase alloys under SPD at suitably low temperatures andseveral intermetallic compounds can become chemicallydisordered and even amorphous under these conditions[6]. These different behaviors are generally rationalized bythe concept that atoms relocate during SPD by ballisticmixing processes [7], although the details of these processesare not generally known. This suggests that most binary sys-tems will become homogeneous during SPD, unless the tem-perature is sufficiently high for thermally activated diffusionprocesses to contribute, usually at temperatures at whichvacancies become mobile. These models have been dis-cussed elsewhere, and interested readers are referred toMartin and Bellon [8]. One of the deficiencies of all of these

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Y. Ashkenazy et al. / Acta Materialia 60 (2012) 984–993 985

models is that SPD performed at very low temperaturesdoes not always lead to homogenization. Ni–Ag [9], Cu–Ta [10] and Cu–Mo [11] are examples of such behavior,for which it was shown that only small, but measurable,extensions of the terminal solubility were possible duringcryo-milling at temperatures approaching 100 K.

The failure of some binary systems to homogenize underSPD remains a significant unanswered question in thedevelopment of this processing method. Two suggestionsto resolve this question are that (i) ballistic mixing is nottruly “ballistic” and that the mechanical mixing of atomsis influenced by the heat of mixing of the constituent alloyelements and (ii) flow localization in the softer phase pre-vents atomic mixing across the interface. The first of thesewas indicated in atomic simulations, first using molecularstatics on two-dimensional amorphous alloys [12] and thenlater using fully three-dimensional (3-D) molecular dynam-ics (MD) simulations of model fcc metals [13]. The lattersuggestion of flow localization has also been observed in3-D MD simulations [14,15]. Other questions regardingshear mixing involve the roles of the crystal and interfacestructures, as these strongly influence crystalline plasticity[16]. The present study is directed at examining the mecha-nisms of shear mixing in these highly heterogeneous sys-tems by performing MD simulation on various two phasealloys. In this study we examine the mixing of initiallyspherical particles in Cu, including the fcc metals Ag andNi and the bcc metals Fe, Nb and V. These various systemsthus examine wide ranges of heat of mixing, differences inelastic properties, atomic size and crystalline structure.For comparison we also examine the mixing of a Cu parti-cle in Cu to provide a reference for these other systems.Our results illustrate, as perhaps should be expected, therelative importance of different mechanisms varies fromone system to another, however, we will show that a keyfactor in understanding shear mixing is whether dislocationcan transfer from one phase to another. While shear mixingtakes place in either case, the mechanisms and dynamics ofmixing for the two are dramatically different.

2. Simulation methods

The simulations were performed using dilute Cu alloysas a model system, represented by embedded atom methodpotentials [17–20] (Cu–V potential received from M.J.Demkowicz, private communication). The MD codeLAMMPS [21] and visualization using OVITO [22] wereemployed for these simulations. The samples in this studywere created by introducing a spherical particle inside theCu matrix in a random orientation. Different orientationsgave similar results. The system was then annealed at600 K at a hydrostatic pressure of 500 MPa for 100 psbefore bringing it to a temperature of 100 K at a pressureof 0 Pa. This procedure was repeated 10 times in order toensure stabilization of the particle–matrix interface. Thesesamples were then sheared cyclically under biaxial com-pression along the 10 0i directions of the Cu matrix (x

direction). In each cycle the sample was deformed to anengineering strain of 20% using compression first in the x

and y directions, followed by similar strains in the y–z

and z–x directions. Strains in each direction were modifiedby up to 2% in order to preserve the shape of the compu-tational cell at the end of each cycle. The strain rate was2 � 109 s�1. The results were insensitive to changes in thetotal strain accumulated per cycle between 15% and 25%and up to a factor of 5 slower strain rates. System size,i.e. the number of Cu matrix atoms, was chosen so thatthere were no observable effects on the mixing rates ofthe particle for strains up to 50. This was achieved bychoosing a system size that was at least twice the diameterof the particle for particles above 8 nm, up to 1.5 � 106 forthe largest systems (particles of 12 nm). For smaller parti-cles the matrix was more than three times larger than theparticle diameter with the number of atoms always above105.

3. Results

3.1. Atomic mixing in single phase alloys

3.1.1. MC simulation

We begin to describe atomic mixing during SPD by treat-ing the hypothetical case of a single component fcc metalunder the idealized conditions that dislocations simply passthrough the material, shifting slabs of atomic planes oneBurgers vector at a time. We at first neglect possible compli-cations of a dislocation interacting with other dislocations orgrain boundaries (GB). While this situation does not occurin any real material, it provides a reference for characterizingforced atomic mixing as complexity is added. Since in ourfirst approximation dislocations do not interact weemployed the approach described previously in [23], usingMonte Carlo (MC) simulations. A fcc structure with peri-odic boundary conditions is thus created using a rhombohe-dral computational cell, and pairs of dislocations withopposite Burgers vectors are passed through the structurealong one of the 12 {11 1}h110i slip systems of the fcc struc-ture. The particular slip system is selected randomly. Pairs ofdislocations are required by the periodic boundary condi-tions. The distance between the pairs is also chosen ran-domly, ranging from one Burgers vector to half the lengthof the computational cell. We characterize the forced mixinghere in three ways. The first is the mean relative shear-induced diffusion coefficient, which is defined as [14],

DðRÞ¼ 1

6

@r2ðR; t1; t2Þ@e

¼ 1

6

@

@e

XN

i¼1

XzðR;tÞj¼1

ðRijÞt2�ðRijÞt1

h i2 !,

2XN

i¼1

zðR; iÞ ð1Þ

where Rij is the vector distance between atom pairs i and jat times t1 and t2, and Z is the number of such pairs. D(R)represents the movement of atoms relative to other atomsseparated by a distance R. Previously we showed using

986 Y. Ashkenazy et al. / Acta Materialia 60 (2012) 984–993

MD that D(R) increases linearly with pair separation insingle crystalline Cu during uniaxial compression. The rea-son for the linear dependence is simply that a dislocationmust past between two atoms to effect their relative mo-tion, and the probability of such an event is proportionalto their distance apart when the deformation is homoge-neous. This mixing is referred to as superdiffusive mixing,as the root mean square separation distance between pairsof atoms grows superlinearly with time [24]. Furthermore,by measuring D(R) the distance over which dislocationstravel can be inferred from the pair separation distance atwhich D(R) deviates from this superdiffusive regime.Fig. 1 illustrates this behavior for our MC simulation ofshearing. The behavior of D(R) is observed to fit a curveof the form 6D(R) = kRb(1 � R/L) where b is the Burgersvector, k is a geometric factor �1, and L is the periodic sys-tem size. At small r the curve is linear, as expected. Thequadratic behavior at large r derives from the periodicboundaries.

The second characteristic of shear mixing is the rate ofcreating new A–B bonds per unit strain, dNAB/de, whereA and B refer to the two elements of a binary system. Thisquantity depends on the original configuration of the sys-tem. In the present work component B comprises a singlesphere embedded in matrix A. For pure Cu we identifyatoms in the sphere as CuB and atoms in the matrix asCuA. The results from the MC simulation are shown inFig. 2, where the mixing rate dNAB/de is plotted as a func-tion of the number of atoms in the sphere using log–logscales. It is observed that the slope of the line is approxi-mately 2/3, illustrating that the mixing scales as R2. Thisscaling law is understood by noting that the probabilityfor a dislocation to intersect the sphere is proportional tor, and a dislocation cutting through the sphere creates a lineof A–B bonds at the circumference, which is also propor-tional to r, the result yielding the number of new A–B bondsscaling as R2 [24]. We emphasize that if the precipitates were

Fig. 1. Relative diffusion coefficient comparing MC (dashed line) and MD(solid line) simulations. The dotted line indicates a fit to 6D(R) = kRb

(1 � R/L), where L is the length of the computational cell (14.5 nm), b

equals 2.55 and k is a geometric factor used as a fitting parameter(k = 1.02 ± 0.01).

mixing with the matrix by ordinary Fickian diffusion therate would be linear in R, and not R2.

The last characteristic of interest concerns the evolutionof the interface morphology. Again, this characteristicdepends on the initial configuration. In a previous workwe showed that for a planar (100) interface the interfaceinitially acquires a long wavelength roughness but that itremains rather sharp. New A–B bonds are largely associ-ated at low strain with the creation of new interface area,rather than intermixing, i.e. long wavelength Fourier com-ponents dominate the interface power spectrum [25]. Theresults for the spherical precipitates are shown in Fig. 3for a strain of e = 6 and in Fig. 4 for a strain of e = 60.It is seen that the sphere first becomes distorted with littleintermixing, similar to planar interfaces, but eventually itbecomes homogeneous within the matrix.

3.1.2. MD simulations

We next report the characteristics of shear-induced mix-ing in pure Cu, but now computed using MD simulations.These results are shown in Figs. 1–4. The main finding isthat MD yields nearly identical results for pure Cu as thepreviously described MC simulations. First, Fig. 1 illus-trates that the superdiffusive properties of shear mixingare nearly the same for the two simulations. This result issignificant since it shows that dislocations glide over dis-tances at least as large as the computational cell. If thiswere not the case, but rather, for example, the dislocationsformed sessile dislocation tangles, then the superdiffusivebehavior would only be observed at shorter length scalesand D(R) would become independent of R at larger lengthscales [24]. The fact that no such dislocation tangles wereformed can be observed directly in Fig. 4. The values ofD(R) were calculated after strains of 0.6, 1.2 and 6 andno differences were found.

The magnitude of the shear mixing rate dNAB/de is alsoquite similar for the two types of simulations. This is inter-esting in view of the very different ways the shearing wasapplied. For the MC simulations planar slabs are simplyshifted in a random way, whereas in MD the cell is

Fig. 2. Mixing rate of a spherical Cu particle within a Cu matrix. NAB

refers to the number of bonds created between particle and matrix atomsand dNAB/de is the rate at which these are created per unit strain. Resultsof MD simulations are shown in black and MC in gray. The linear fit linesoverlap. The abscissa is the initial number of atoms in the particle.

Fig. 3. Morphology of spherical particles in Cu after a strain of 6.0. The left-most configuration is from a MC simulation of a Cu precipitate in a Cumatrix. The rest of the configurations are from MD simulations of precipitates in a Cu matrix. The particle material is indicated in each panel. Colorindicates the structural factors for the matrix atoms; blue are fcc atoms and gray are atoms positioned in a locally non-pure fcc environment (e.g. astacking fault or next to a point defect). For the particle atoms color indicates local mixing; green indicates that most of the neighbors are from the particleand red indicates that most atoms are from the matrix. (For interpretation of the references to colour in this figure legend, the reader is referred to the webversion of this article.)

Fig. 4. As Fig. 3 but after a strain of 60.

Y. Ashkenazy et al. / Acta Materialia 60 (2012) 984–993 987

compressed biaxially in a cyclical manner, as describedabove. We tested whether the strain increment employedin each cycle influenced the results; no significant changewas observed for strains per cycle of e = 0.45, 0.6 and 0.75as described in Section 2. This suggests that shear mixingin homogeneous systems is not strongly affected by the

details of the shearing process, as long as the shearingremains distributed over multiple slip systems. While thestrains per cycle used here are comparable with that usedduring ECAE, they can be larger during HPT or ARB,although the strain paths during the latter methods for het-erogeneous structures are not yet well characterized [25].

Fig. 5. Rate of A–B bond formation as a function of particle size. Thelines are logarithmic fits to the data. Dark gray, fcc–fcc systems; light gray,fcc–bcc systems.

Fig. 6. Number of A–B bonds as a function of strain for 6 nm particles inCu, normalized to the number of bonds for an isolated atom in the matrix.

988 Y. Ashkenazy et al. / Acta Materialia 60 (2012) 984–993

The similarity in the mixing rates between the MC and dif-ferent MD strain cycles, however, do suggest that the dislo-cation glide in MD is irreversible, i.e. dislocations are notstored and then glide back along the same slip plane laterin the shear cycle. Lastly, as observed with the MC simula-tions, the spherical Cu particle first roughens during shear-ing and eventually mixes in the MD simulations.

3.2. Atomic mixing in two phase alloys

3.2.1. Cu–Ag and Cu–Ni

We add heterogeneity into the system by first consider-ing the shear mixing of Cu with spherical fcc precipitatesof Ni and Ag. Ni is closely lattice matched with Cu (seeTable 1) and has a very small heat of mixing with Cu.The shear modulus of Ni, however, is considerably higherthan that of Cu. Ag, on the other hand, is significantlyoversized in Cu, has a moderate heat of mixing with Cu,but has a significantly smaller shear modulus than eitherCu or Ni (see Table 1). The characteristics of the shear mix-ing of these two systems are presented in Figs. 3–6. Asshown in the cross-sections of Figs. 3 and 4, the qualitativemixing behavior of these fcc systems is quite similar. This ismade more quantitative in Figs. 5 and 6. The initial mixingrates at the Cu–Ag and Cu–Ni interfaces are seen toincrease, like pure Cu, as the square of the particle radiusand they all have nearly the same magnitude. We alsoexamined the magnitude of mixing as a function of strainto large strains e = 60; these data appear in Fig. 6. Hereagain it is observed that the magnitudes of shear mixingin these fcc systems are very similar, varying by less thana factor of two. We had previously shown that the heatof mixing does influence shear mixing [13,14], presumablyaccounting for the somewhat smaller shear mixing in Cu–Ag than in the hypothetical Cu–Cu system.

Lastly we examined whether strain localization is impor-tant in these systems by calculating the initial relative diffu-sion coefficients between nearest neighbor atom pairs,D0

AA, D0BB and D0

AB (see Eq. (1)). D0AA thus considers the rela-

tive displacements of all nearest neighbor Cu–Cu atoms,while D0

AB considers relative displacements of nearest neigh-bor Cu–Ag(Ni) pairs. Initial values of D0

AB accordinglyexamine the relative atomic motion of pairs at the matrix–precipitate interface. These quantities characterize the extentof dislocation glide in the specific regions [26]. The results arereported in Fig. 7, where it is observed that D0

AA (in the

Table 1Properties of the simulated Cu alloys.

System Heat of mixing (eV atom

A B B in A A

Cu Ag 0.38 0.Cu Ni 0.27 0.Cu Fe 0.55 0.Cu V 0.75 0.Cu Nb 1.3 0.

matrix) is nearly the same for all systems, but that initiallyfor the activity within the particle D0

BB(Ag) > D0BB(Cu) >

D0BB(Ni), showing that dislocation activity depends initially

on the shear modulus, as was noted previously [14,15]. Note,however, that D0

BB(Ni) increases with strain, becomingnearly equal to D0

AA. This is mainly due to the reductionin size of the Ni precipitate; as it shrinks to a few monolay-ers dislocation activity within it is controlled by activitywithin the matrix. Indeed the rate D0

BB/e decreases withincreasing precipitate size. The large value of DAB for the

�1) Volume mismatch (DV V�1)

in B B in A A in B

33 0.53 �0.2819 �0.16 0.0645 0.1 0.035 0.7 0.16 1.1 �0.25

Y. Ashkenazy et al. / Acta Materialia 60 (2012) 984–993 989

Cu particle in a Cu matrix at small strains is a consequenceof the initial misorientation of the particle. After a strain ofabout unity the particle orients in the matrix and the threerelative diffusion coefficients become equal. The Ni and Agparticles similarly orient after small strain increments. Themorphology of the Cu, Ag and Ni particles in Cu as theyevolve during shearing are shown in Figs. 3 and 4.

3.2.2. bcc Precipitates in Cu

bcc precipitates in Cu behave very differently from fccprecipitates under shear in a number of ways. This isclearly observed in Figs. 3 and 4, which show the structureof the systems after strains of 6 and 60. The first observa-tion is that V and Nb particles undergo very little deforma-tion during shear, retaining their equiaxed morphologyeven at strains up to 60. These particles, and Fe, however,do reorient during the early stages of strain at e < 10, sim-ilarly to the fcc particles, but now with a Kurdjimov–Sachs(KS) orientation. Deformation in the Fe is at first negligi-ble, like V and Nb, but by a strain of 60, the particle hasundergone considerable shear. These results are quantifiedin Fig. 7, where it is observed that D0

BB for Cu–V and Cu–Nb remains very small during shearing, indicating that fewdislocations pass through the precipitates. D0

BB for an Feprecipitate, on the other hand, is initially small, but aftera strain of �6 it begins to increase. The visualization of thisprocess presented in the Supplementary material showsthat initially dislocations accumulate at the Cu–Fe inter-face, but as the particle rotates into alignment with the

Fig. 7. Nearest neighbors D0 for AA, BB and AB for simulated systems asrespectively.

Cu matrix dislocations begin cutting through the Fe parti-cle. Then, as the particle becomes increasingly deformed,the rate of dislocations passing through the Fe increasesuntil it reaches the rate in bulk Cu at e � 60. We note thatfor total strains e > 10 bcc Fe forms a coherent fcc interfacewhen the strain increment exceeds 15% in the shear cycle(by transformation along the Bain path), allowing directtransfer of dislocations from the matrix to the particleand back to the matrix. This results in significant shapechanges of the particle, and it appears to explain theincreased plasticity of the particle. The particle regainsthe bcc structure when the stress is reduced later in thecycle.

A second interesting feature observed in Figs. 3 and 4 isthe formation of a highly disordered (amorphous) shellaround the Nb precipitate, which grows with strain. Anamorphous region is also observed around the V precipitate,but in V it is less uniform, and one region of the precipitatebecomes highly faceted at the {111}Cu||{110}V interface.These observations were reproduced in several simulationsusing somewhat different starting conditions. Examinationof our models for the Cu–Nb and Cu–V systems reveals thatthe amorphous phase is meta-stable during shearing at2 � 109 s�1 for uniform solute concentrations as low as4 at.% Nb and 8 at.% V. No amorphous shell is observedin Cu–Fe, as might be expected owing to the small size misfit.Average compositions at the precipitate interfaces are shownin Fig. 8. Here the concentration of the precipitate compo-nent is plotted as a function of distance from the center of

a function of strain, as solid dark gray, black, and dashed gray lines,

Fig. 9. Relocations of V atom during shearing. Lines show displacementvectors for atoms moving more than 0.7 nm. Coloring is according to thedisplacement with light gray (blue online) atoms moving less than 0.1 nmand dark gray (red online) ones moving more than 0.7 nm. (Forinterpretation of the references to colour in this figure legend, the readeris referred to the web version of this article.)

990 Y. Ashkenazy et al. / Acta Materialia 60 (2012) 984–993

mass of the precipitate. The higher amorphization thresholdfor Cu–V illustrates why the Cu–Nb interface develops acomplete amorphous shell, whereas Cu–V can only forman amorphous region where the local composition is higherthan average.

The shear mixing at the Cu–V interface was furtherexamined to determine why some regions were facetedand chemically sharp while others were amorphous andchemically mixed. This morphology is, in fact, reminiscentof that reported in Cu–Nb after wire drawing [27]. Fig. 9explains this behavior quite clearly. Dislocations glidingon Cu slip systems, at a tangent to the particle, efficientlytransport V away from this region. The displacement ofV is indicated in the figure by small arrows. As can be seen,V is “shaved” from the particle along these low index inter-faces by dislocation glide, and swept away. Fig. 9 showsfew V atoms at this interface. Since dislocations move pri-marily in the Cu, it also not surprising that the shear mix-ing profiles shown in Fig. 8 are highly asymmetrical, with Vand Nb mixing into the Cu, but not vice versa.

The qualitative picture of shear mixing derived fromFig. 4 helps to explain the more quantitative results in Figs.5–7. First, it can be observed in Fig. 5 that the mixing ratefor the bcc precipitates does not scale as R2, as for the fccprecipitates, but rather as R. The mixing, therefore, is con-sistent with a diffusive rather than a superdiffusive process.This is not altogether surprising, since superdiffusive mix-ing results from the long-range correlation of atomic dis-placements associated with dislocation glide. We notedpreviously that the length scale of superdiffusive mixing isrestricted to the mean free path of the dislocations [24].In amorphous materials, moreover, superdiffusive behavioris only observed over a very small length scale, typicallyless than 1 nm [28]. Since mixing for bcc and fcc precipi-tates scales as R and R2, respectively, the difference in mix-ing between the fcc and bcc particles increases asprecipitate size increases. It is interesting, however, thatthe mixing for smaller particles, 63 nm, is nearly the samefor all precipitates.

Fig. 8. Concentrations of Nb and V as a function of distance from the center othe figure.

Finally, we note that Fig. 6 shows that the rates of mix-ing of fcc precipitates begin to decrease at large strains, asthe particle dissolves and the number of AB bonds satu-rates. For V and Nb the strains are not sufficient for thisto occur. The Cu–Fe system is interesting as it shows posi-tive curvature of the mixing curve. This can be understoodby our observation above that dislocations begin to betransferred through the Fe particle with increasing strain,due to the formation of coherent interfaces during theshear cycle. The mixing behavior, therefore, switches fromdiffusive to superdiffusive.

4. Discussion

It is now well recognized that during severe plasticdeformation of heterogeneous structures, such as twophase alloys, the length scales of the inhomogeneities aregreatly diminished by uniform distortion of the structure.For example, during ARB stacks of different layers are

f mass of the particle after strains of 1, 20 and 40. The strain is specified in

Y. Ashkenazy et al. / Acta Materialia 60 (2012) 984–993 991

thinned in plane strain, before significant mixing takesplace at the interfaces [29]. Similar behavior occurs duringwire drawing[27], and for ball milling the micron sizedpowder particles are continuously flattened, fractured andrebound [30], again reducing the length scale of the micro-structure. During these deformation processes the inter-faces remain relatively sharp at first, but eventually theinterface structure becomes unstable [29] and shearingacross interfaces becomes dominant. The current investiga-tion has considered these final stages of homogenizationwhen the microstructure has become refined, due to mixingacross the interfaces.

The primary finding of this work is that while shear mix-ing at low temperatures is a consequence of dislocationglide in all systems studied, the detailed mechanisms gov-erning local forced atomic rearrangements can be very dif-ferent. In pure Cu the mixing mechanism found by MDsimulations for random shearing is essentially what wasproposed in our previous MC simulation [23]. Atomic mix-ing is effected by the shifting of atomic planes on the vari-ous slip planes. This highly correlated motion, as wepointed out previously [14], results in superdiffusive mixing.The Cu–Ni and Cu–Ag systems are observed to behavesimilarly to pure Cu, although the mixing rates are some-what diminished for these two heterogeneous systems. Itis interesting to note that deformation, or the relative diffu-sion coefficient D0

BB, in the Ni particle is initially smallerthan in Cu. This appears to explain the smaller mixing rate,i.e. not all dislocations arriving at the Cu–Ni interfacetransfer through the Ni particle. For Ag D0

BB is significantlylarger than D0

AA. In this case, however, the dislocationsnucleate at the Cu–Ag interface, and only cross the Ag par-ticle. There is some experimental validation of these find-ings obtained from measurements of shear mixing in Cu–Ag during HPT deformation. Panah et al. reported thatshear mixing of a Cu–Ag eutectic alloy, which had a com-positional wavelength of �160 nm, showed homogeniza-tion after a strain of e � 700 [31]. This value is in goodagreement (a factor of two) with the predictions of theMC model, which, as we noted, gives similar results tothe MD simulations.

In the case of bcc precipitates, V or Nb, dislocations donot transfer from the Cu matrix to the particles. As a con-sequence, these particles remain equiaxed and do notdevelop the long wavelength roughness observed in fcc sys-tems. Nevertheless, these particles do undergo shear mixingin Cu. The interfaces between these particles and the Cumatrix become highly disordered with shearing, and amor-phous shells are created around the particles. The exactmixing process is complex, but it is clear that dislocationspile up at the Cu–particle interface and create high levelsof stress. The interface relieves these high stresses by localatomic rearrangements and mixing. Similar “shuffling”

rearrangements at GB have been discussed with regard tothe onset of plasticity in nanocrystalline materials [32]. Atheterogeneous interfaces, however, the heat of mixingand relative atomic size also play a role, since they influ-

ence the local energy barriers that must be overcome dur-ing relaxation. Vo et al. showed that in fcc metals withvery high heats of mixing such local atomic rearrangementsin dislocation cores can lead to precipitation during SPD,even during shearing at 100 K [13]. The mixing mechanismin fact resembles that in amorphous materials, in whichshearing results in local relaxations without any long-rangecorrelations in the atomic motion. For this reason the shearmixing is diffusive, rather than superdiffusive, as revealedby the mixing rate dNAB/de plotted in Fig. 5.

There is also some experimental verification of thesesimulations. For example, Sauvage et al. examined colddrawing of both V [33] and Nb [27] filaments in Cu. Forthe Cu–V sample these authors reported that after at a truestrain of 9.2 an �2 nm wide mixed region was observed byatom probe tomography at the Cu–V interfaces. This levelof mixing agrees roughly with the simulations here (Fig. 8).These studies also showed faceting at some interfaces. Fac-eting is clearly illustrated in the simulations in Figs. 9 and10, even after a strain of 200. It is noteworthy that little V isfound in the Cu matrix near these faceted regions. Theexperiments in this work did not, however, observe anamorphous shell. We cannot attribute this difference tothe very low temperature of the simulations (100 K), sinceadditional simulations at 400 K also showed amorphousshell formation. A similar study on wire drawing of Cu–Nb, on the other hand, did show amorphization at theCu–Nb interface [27]. In that work it was reported thatthe amorphous regions were associated with the finest pre-cipitate microstructures and that it was due to intermixingof Cu and Nb. Our simulations agree with these findings.The experimental study also reported that some interfacesof Cu–Nb remained crystalline and sharp, suggesting thatthe semi-coherent interfaces were more stable. We showedelsewhere (Ashkenazy, Vo, Schwen, Averback and Bellon,unpublished results) that semi-coherent interfaces, such asKurdjumov–Sachs or Nishiyama–Wassermann interfaces,are more stable against shear mixing than high energyinterfaces, but these were not studied here. For the smallparticles employed here, with small radii of curvature, thesestable interfaces do not form.

As can be clearly seen in Fig. 10, after a strain of e = 200a highly mixed amorphous shell surrounds the Nb particle.It can also be observed that Nb diffuses over long distancesin Cu, away from the amorphous shell. Presumably thislonger range Nb transport is effected by its superdiffusivemotion in the Cu matrix. Similar behavior is seen forCu–V. Shear mixing of these systems can thus be considereda slow diffusive process at the interface coupled with a fastdiffusive process away from the particles. For an array ofnanoparticles we can, therefore, expect a rapid exchangeof solute atoms between particles, with the final solubilitydetermined by how efficiently solute atomscan be released from the precipitate, i.e. the value ofdNAB/de. This illustrates how solubilities can be increasedin two phase alloys, without complete homogenization,even at low temperatures when thermally activated diffusion

Fig. 10. Morphology of the Cu–V (left) and Cu–Nb (right) systems after a stain of 200. V and Nb atoms are color coded according to their CSP factor toillustrate whether they are bcc or highly distorted sites. Cu atoms are indicated by a small gray (blue online) sphere only if in a non-ideal fcc site (e.g. astacking fault or next to a point defect). Cu atoms in fcc sites are not plotted. (For interpretation of the references to colour in this figure legend, the readeris referred to the web version of this article.)

Fig. 11. Mixing of a 6 nm Fe precipitate in Cu as a function of strain.Mixing is calculated using the number of Cu–Fe bonds normalized to thenumber of bonds for a fully dissolved Fe atom (N(mix) = NAB(all)/NAB(1)) and divided by the initial number of atoms in the precipitate (N0).

992 Y. Ashkenazy et al. / Acta Materialia 60 (2012) 984–993

is suppressed. Further details regarding the stability of twophase alloys under shear will be reported elsewhere.

The Cu–Fe system is interesting because its mixing char-acter changes with increasing strain. Initially Cu–Febehaves similarly to the Cu–V and Cu–Nb systems: D0

BB

is very small, showing little dislocation motion within theprecipitate, and the mixing is diffusive rather than superdif-fusive. As the strain increases above �6, however, disloca-tions begin crossing the Cu–Fe interface, which is no longersharp, and the shear mixing rate increases rapidly. Shearmixing in the Cu–Fe system has recently been studied usingHPT [5]. The samples used for HPT were prepared by firstusing wire drawing to reduce the length scale of the micro-structure. Many of the features reported in that study areconsistent with the current findings. First, the experimentsshowed that the mixing begins with Fe dissolving into theCu and that the Fe filaments did not deform as much asthe Cu matrix. This is also seen in the simulations fromthe relative values of D0

AA and D0BB and by inspection of

the composition profiles. A direct comparison of the exactmixing rate between the experiments and simulations is dif-ficult, owing to the different starting microstructures, butthe experiments show nearly complete homogenization ofthe Fe filaments after a shear strain of e � 300. These fila-ments had a diameter of �11 nm. The results of the simu-lations of mixing are shown in Fig. 11. Here is seen that a6 nm Fe particle is mixed after a strain of �100. We canexpect that a larger particle would require additional strainfor dissolution. For example, for CuA–CuB, which showssuperdiffusive mixing, the strain for dissolution increaseslinearly with radius. For Fe in Cu, as discussed above,the behavior is neither purely superdiffusive nor purely dif-fusive, however, by assuming it is either superdiffusive,which scales with R2, or diffusive, which scales with R,we can set lower and upper bounds of 200 and 400 forthe strain required for homogenization of 11 nm features.These bounds bracket the experimental strain of 300.

Finally, we note that the experimental incorporationrate of Fe in the Cu matrix was observed to be independentof strain rate and temperature. While the strain rates wereorders of magnitude smaller than those used in the simula-tions, the experimental findings offer some justification forapplying MD simulations to practical situations. The largedifference in strain rates, however, may explain some of thediscrepancies between experimental results and simula-tions. For example, the partial amorphous phase observedat Cu–V interfaces by MD but not experimentally may verywell be caused by the high strain rate.

5. Conclusion

The simulations presented here illustrate the wide rangeof behaviors that two phase alloys may undergo duringshear mixing at low temperatures. First we note that bothfcc and bcc spherical particles become oriented in Cu under

Y. Ashkenazy et al. / Acta Materialia 60 (2012) 984–993 993

prolonged shear deformation. For Cu, Ag, Ni and Fe thisalignment enables dislocations to transfer from the Cumatrix to the particle. Mixing in this case can be describedas superdiffusive, where the mixing rate dNAB/de dependson the square of the particle radius and not linearly, aswould be expected for a diffusive process. Most of thenew bonds created during the early stages of mixing aredue to distortion of the shape of the particle and theincrease in interface area, rather than mixing across theinterface. For the V and Nb particles a quite differentbehavior is observed. Dislocations do not transfer acrossthe interface and the particles maintain their initial shape.Pile-up of dislocations at the interfaces, however, resultsin local relaxations that induce mixing between the particleand matrix. This process results in diffusive mixing, withdNAB/de increasing linearly with particle radius. For theNb particle the introduction of Nb atoms into Cu resultsin an amorphous shell surrounding the particle. At theperiphery of this shell, which remains crystalline, Nb atomsare efficiently transported into the Cu matrix by a superdif-fusive mixing process, creating an asymmetrical mixingprofile. V particles behave similarly to Nb particles inCu, however, in this case a process of “shaving” wasobserved at low energy interfaces, which efficiently carriesV away from the Cu–V interface and prevents amorphiza-tion there. The difference between Nb and V is attributed tothe lower solubility of Nb in the crystalline Cu lattice andthe larger atomic radius of Nb compared with V. Compar-ison of the mixing rates and interface structures obtainedexperimentally with those predicted by the simulationsshows overall good agreement.

Acknowledgements

This work was supported under grant DOELANL76604-001-10 which is part of the Center forMaterials at Irradiation and Mechanical Extremes, anEnergy Frontier Research Center and funded by the USDepartment of Energy, Office of Science.

Appendix A. Supplementary material

The supp1ementary material comprises a movie showingmixing in of a 6 nm Fe particle embedded within a Cumatrix. Cu atoms are blue and gray if at a defect site. Featoms are colored according to the local environment, withgreen corresponding to mostly Fe neighbors and red

corresponding to mostly Cu neighbors. The movie showsthe evolution of the particle from a strain of 1–60 at incre-ments of 0.6, with each second corresponding to anincrease of 0.6 in the total accumulated strain. Supplemen-tary data associated with this article can be found, in theonline version, at doi:10.1016/j.actamat.2011.11.014.

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