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Acta Materialia 61 (2013) 3370–3377

Modeling and predicting microstructure evolution in lead/tin alloyvia correlation functions and stochastic material reconstruction

Yang Jiao a,b,⇑, Eric Padilla b, Nikhilesh Chawla b,⇑

a Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, NJ 08544, USAb Materials Science and Engineering, Arizona State University, Tempe, AZ 85287-6206, USA

Received 26 December 2012; received in revised form 17 February 2013; accepted 17 February 2013Available online 7 March 2013

Abstract

The binary lead/tin (Pb/Sn) alloy is widely used as an interconnect in microelectronics. The physical properties of this heterogeneousmaterial critically depend on its complex bulk microstructure containing Pb-rich and Sn-rich phases, which can be both laminar andglobular. In this paper, we devise a procedure to model and predict the microstructure evolution (i.e. coarsening) in a Pb–Sn alloy agedat elevated temperatures below its melting point using statistical morphological descriptors, i.e. the two-point correlation functions S2

associated with the phases. We verify via phase-field simulations that the growing length scale characterizing microstructure coarseningcan be well captured by the corresponding correlation functions, which enables us to predict the S2 of intermediate microstructures giventhe initial and final microstructures. Stochastic material reconstruction techniques are employed to generate virtual three-dimensionalmicrostructures that are consistent with the predicted correlation functions, which are quantitatively compared with the actual alloymicrostructures when available.� 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Microstructure evolution; Heterogeneous materials; Correlation functions; Stochastic microstructure reconstructions

1. Introduction

The binary lead/tin alloy, a two-phase heterogeneousmaterial, has been studied for decades due to its impor-tance in electronic packaging applications. The physicalproperties of such a heterogeneous material are determinedby the associated properties of individual constituentphases as well as the complex morphologies of these phases– the material microstructure – which can be quantitativelycharacterized by certain statistical morphological descrip-tors, i.e. the spatial correlation functions of the alloyphases [1–3].

1359-6454/$36.00 � 2013 Acta Materialia Inc. Published by Elsevier Ltd. All

http://dx.doi.org/10.1016/j.actamat.2013.02.026

⇑ Corresponding authors at: Princeton Institute for the Science andTechnology of Materials, Princeton University, Princeton, NJ 08544, USA(Y. Jiao). Tel.: +1 480 965 2402; fax: +1 480 727 9321 (N. Chawla).

E-mail addresses: Yang.Jiao.2@asu.edu (Y. Jiao), Nikhilesh.Chawla@asu.edu (N. Chawla).

Binary lead/tin alloys are widely used as solders [4]. Inparticular, a eutectic alloy of 63% tin (Sn), 37% lead (Pb)has been used as an interconnect due to its unique lowmelting point (183 �C), good wettability and excellentmechanical properties. The eutectic microstructure con-tains a Pb-rich phase and a Sn-rich phase, which can pos-sess both laminar and globular morphologies. The salientmicrostructural features such as the width and extent ofthe laminar phases as well as the size and spatial distribu-tion of the globular phases can significantly affect the over-all mechanical properties of the alloy [5–7].

At temperatures below the eutectic melting point, theenhanced diffusion of Pb and Sn atoms can lead to signifi-cant coarsening in the alloy, which lowers the total interfa-cial energy [8]. This coarsening process can be accuratelymodeled by the Cahn–Hilliard equation [9,10]. A heat treat-ment (e.g. annealing) can then be employed to “tune” theeutectic microstructure to achieve desirable material perfor-mance. On the other hand, coarsening could also produce an

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Y. Jiao et al. / Acta Materialia 61 (2013) 3370–3377 3371

undesirable degradation of material properties such as lossof strength or the disappearance of grain-boundary pinningeffects. Moreover, the rate of coarsening increases with tem-perature and it is of particular importance to quantitativelymodel and predict the degree of coarsening in the design ofmaterials for high temperature applications.

Phase-field modeling techniques, which simulate themicrostructure evolution according to the Cahn–Hilliardequation (see Section 3.1), are widely used to study coars-ening and phase separation in binary systems [11–15].Specifically, an appropriate length scale k associated withthe evolving microstructure can be defined, which satisfiesthe following scaling relation for the coarsening processesthat are entirely diffusion controlled (i.e. diffusion is theslowest process) [8]:

kðtÞ3 � k30 � t ð1Þ

where k0 is the length scale associated with the initialmicrostructure and t is the time duration of the coarseningprocess. It was first established for the distribution of grainsize in a supercooled system that had undergone Ostwaldripening [16]. For systems that are not diffusion controlled(e.g. the association and disassociation of atoms are slowerthan their diffusion), the exponent associated with the char-acteristic length scale would be different. Using phase-fieldsimulations, it has been verified that for systems evolvingaccording to the Cahn–Hilliard equation, the associated ksatisfies Eq. (1). It is noteworthy that there is no uniquechoice of k. In other words, a variety of different character-istic length scales satisfying Eq. (1) can be chosen for thebinary system of interest, which are mutually consistentwith one another. For example, for alloys containing par-ticle-like phases, the effective particle radius can be usedas k; while for alloys with laminar phases, the volume-to-surface ratio of the phases can be a natural choice for k.

Although phase-field modeling has proven to be verypowerful and successful in many aspects, it is difficult todirectly use simulated microstructures to quantitativelymodel the ones obtained experimentally. This is becausethe spatial correlations of the initial density fluctuationsleading to phase separation and coarsening in the systemare generally not known a priori. Gaussian random fieldsare often employed to approximate the actual density pro-files [13], which might not be sufficient to capture certainsubtle initial correlations in the system that persist andare manifested in the coarsened microstructures.

In this paper, we devise a methodology to model andpredict microstructure evolution in Pb–Sn alloy aged atelevated temperatures below its melting point, using thetwo-point correlation function S2 (defined below) associ-ated with either Pb-rich or Sn-rich phases and stochasticmaterial reconstruction techniques [17]. In particular, usingphase-field simulations, we show that a growing length scalek(t) is well defined via the S2 associated with the microstruc-ture of interest. This in turn enables us to predict the two-point correlation function of intermediate microstructuresgiven the initial and final microstructures. We extract such

length scales by analyzing two-dimensional (2-D) imagesof Pb37Sn63 alloy samples that were isothermally aged at175 �C up to 216 h and construct a general functional formof S2 that, when appropriately parameterized with k(t), canaccurately approximate and predict the corresponding S2 ofmicrostructures aged for different time periods. Stochasticmaterial reconstruction techniques are then employed togenerate virtual three-dimensional (3-D) microstructuresconsistent with the predicted S2, which are quantitativelycompared with the experimentally obtained material mor-phology to verify the accuracy of our approach. Our proce-dure is complementary to the phase-field modelingapproaches. The rest of the paper is organized as follows.In Section 2, we present mathematical definitions of the sta-tistical microstructural descriptors employed in the paper,including the two-point correlation functions S2 and the lin-eal-path function L. In Section 3, we describe in detail ourprocedure for modeling and predicting microstructure evo-lution in Pb–Sn alloy using S2 and the stochastic reconstruc-tion techniques, which are used to verify the accuracy of ourmodels and predictions. In Section 4, we offer concludingremarks.

2. Statistical morphological descriptors

In general, the microstructure of a two-phase heteroge-neous material (e.g. a binary alloy) can be uniquely deter-mined by specifying the indicator functions associatedwith each individual phase [1], i.e.

I ðiÞðxÞ ¼1 x in phase i

0 otherwise

8><>: ð2Þ

where i = 1 and 2. The volume fraction of phase i is thengiven by

ui ¼ hI ðiÞðxÞi ð3Þ

where h�i denotes the ensemble average over many indepen-dent material samples or volume average over a single largesample if it is spatially “ergodic” [1,2]. The two-point cor-

relation function SðijÞ2 ðx1; x2Þ associated with phases i and j

is defined as

SðijÞ2 ðx1; x2Þ ¼ hI ðiÞðx1ÞI ðjÞðx2Þi ð4Þ

which is also the probability that two randomly selectedpoints x1 and x2 fall into phase i and j respectively. For abinary heterogeneous material, there are in total three dif-ferent S2, including two autocorrelation functions (i = j)and one cross-correlation function (i – j). However, ithas been shown that only one of them is independent[1,18] and the remaining two functions can be explicitlyexpressed in terms of the independent one. Thus, it issufficient for us to consider the S2 of either the Pb-richphase or the Sn-rich phase. In the subsequent discussions,we will only employ S2 associated with the Pb-rich phase

3372 Y. Jiao et al. / Acta Materialia 61 (2013) 3370–3377

of the alloy and will drop the superscripts (i.e. the phase

indicators) in SðijÞ2 ðx1; x2Þ for convenience.For statistically homogeneous and isotropic materials,

i.e. those possessing rotational invariance symmetry andhaving no preferred centers, which are the focus of thispaper, the associated S2 only depends on the relative scalardistance between the two points [1,2], i.e.

S2ðx1; x2Þ ¼ S2ðjx2 � x1jÞ ¼ S2ðjrjÞ ¼ S2ðrÞ ð5Þwhere r = x2 � x1 and r = |r|. S2 (r) gives the probabilitythat two randomly chosen points, separated by distancer, both fall into the phase of interest. It is also the proba-bility that the end points of a line segment of length r, whenrandomly thrown down into the system, both fall into thephase of interest. At r = 0, S2 gives the probability of a ran-domly selected point lying in the phase of interest, i.e. thevolume fraction of the associated phase u. At large r val-ues, the probabilities of finding the two points in the phasesof interest are independent of one another, leading to theasymptotic value u2.

It is also useful to introduce the autocorrelation func-tion f(r) associated with a given S2(r), i.e.

f ðrÞ ¼ S2ðrÞ � u2

uð1� uÞ ð6Þ

We note that f(r) contains the same microstructuralinformation as the associated S2(r) once the volume frac-tion is given. At r = 0, f(r) = 1 and at large r values, f(r)approaches its asymptotic value 0. The reader is referredto Ref. [1] for detailed discussions of S2 and the more gen-eral n-point correlation functions Sn.

Another correlation function on the two-point levelfrequently employed to characterize microstructures ofstatistically homogeneous and isotropic materials is thelineal-path function L(r), which provides the probabilityof finding a randomly oriented line segment with length r

entirely lying in the phase of interest [1,19]. We note thatL(r) contains nontrivial linear topological connectedness(i.e. clustering) information of the system of interest andit is dimension independent. In other words, one can obtainL(r) of a 3-D material from the associated 2-D slices [1].This quantity will be employed in the sequent sections toquantitatively ascertain the accuracy of our 3-D reconstruc-tions of the material microstructure from predicted S2.

3. Modeling and predicting microstructure evolution in Pb–

Sn alloy

In this section, we present our procedure to model andpredict microstructure evolution in Pb–Sn alloy using thetwo-point correlation functions S2. We first show viaphase-field simulations that a growing length scale k canbe defined via S2. This length scale is extracted from the2-D images of Pb37Sn63 alloy samples that were isother-mally aged at 175 �C up to 216 h. Specifically, we will showthat the coarsening in our alloy samples is dominated byEq. (1). A general functional form of S2 parameterized with

k(t) is constructed that can accurately approximate andpredict the corresponding S2 of microstructures aged fordifferent time periods. Stochastic material reconstructiontechniques are then employed to generate virtual 3-Dmicrostructures consistent with predicted S2, which arequantitatively compared with the actual alloy microstruc-tures when available to verify the accuracy of ourapproach.

3.1. Growing characteristic length scale captured by S2

Phase-field modeling [13] is employed to simulate themicrostructure evolution (i.e. coarsening) in a 2-D hypo-thetical binary system undergoing spinodal decompositionstarting from Gaussian random concentration (density)fluctuations. Specifically, the Cahn–Hilliard equation

@w@t¼ D0r2lw ð7Þ

is numerically solved using the Euler forward method on aregular grid [20,21], where D0 denotes the compositionalmotility (with unit length2/time) and lw � dF/dw is thedimensionless chemical potential associated with a phe-nomenological free energy F given by

F ¼Z

dr1

2ðrwÞ2 � w2

2þ w4

4

� �ð8Þ

In Eq. (8), w is the dimensionless concentration fielddefined via

w � ½cAðr; tÞ � cBðr; tÞ�=2c ð9Þwhere ci (i = A,B) is the concentration of species i and c isthe critical composition (in moles per unit volume). Wenote that the Cahn–Hilliard equation (7) has the form ofa conservation law, and thus the domain separation pro-cess it describes converses the total concentration of theconstituents. The phenomenological free energy F is con-structed to explicitly take into account spatial inhomogene-ity in the system. Specifically, the term ($w)2 in Eq. (8)accounts for interface in the system undergoing domainseparation and the remaining two terms sensitively capturelocal density fluctuations. The numerical scheme we used tosolve the Cahn–Hilliard equation (7) closely follows thatdescribed in Ref. [20], to which the reader is referred formore details. For numerical convenience, we have setD0 = 1 in our simulations since the time and space unitsare not important to the determination of growing lengthscales.

Fig. 1 shows a snapshot of the simulated microstructure(left panel) and the autocorrelation function f(r) (middlepanel) for the associated binary microstructure (not shown)by trivially thresholding the original microstructure. Theautocorrelation function is related to the two-point correla-tion function S2 via Eq. (6) and is an oscillating functionwith exponentially decaying peak/valley values. In particu-lar, the distance k associated with the first minimum of S2,roughly corresponding to the mean separation distance

Fig. 1. Left panel: a snapshot of the microstructure of a hypothetical binary system evolving according to the Cahn–Hilliard equation. Middle panel: theautocorrelation function f(r) that is trivially related to S2 via Eq. (6) associated with the microstructure shown in the left panel. The distance k associatedwith the first minimum of f(r) can be used a length scale characterizing the coarsening process. Right panel: k as a function of time t, which satisfies thescaling relation (1).

Y. Jiao et al. / Acta Materialia 61 (2013) 3370–3377 3373

between the laminar structures in the system, can be usedas a length scale characterizing the coarsening process. Itis well known that for our model system evolving accordingto the Cahn–Hilliard equation, an appropriate characteris-tic length scale of the system should satisfy the scaling rela-tion (1). The right panel of Fig. 1 shows k as a function oftime t, from which one can clearly see that the scaling rela-tion applies, indicating that k is indeed an appropriatecharacteristic length scale of the system.

3.2. Predicting microstructure evolution via S2

We now apply the same analysis to the experimentallyobtained 2-D images of Pb37Sn63 alloy samplesisothermally aged at 175 �C for different time periods. ThePb–Sn samples were melted and quenched in water. Thesamples were then sectioned, mounted and polished. Thiswas followed by optical microscopy. Fig. 2 shows opticalmicroscope images representative of the microstructuresfor t = 0, 65 and 120 h.

The optical images are converted to 256-bit grayscaleimages using MatLab, which are then thresholded (withthreshold value 185) to produce the binary images shownin Fig. 2.

One can clearly see from Fig. 2 the development ofcoarsening in this system: the somewhat lamellar structures

Fig. 2. 2-D images of Pb37Sn63 alloy samples isothermally aged at 175 �C

grow in size and length and become more globular. The lin-ear dimension of our images is �250 lm. For each agingtime t (including t = 0), five independent microstructuresare analyzed to obtain the corresponding S2, from whichwe extract the growing length scale k, i.e. the distance asso-ciated with the first minimum of S2.

The left panel of Fig. 3 shows the autocorrelation func-tion associated with the microstructures of the Pb37Sn63alloy samples aged for different t, from which the growinglength scale k is extracted. The right panel of Fig. 3 shows kas a function of t, which satisfies the scaling relation (1) to ahigh accuracy. This suggests that the microstructure evolu-tion in the binary system is diffusion controlled and can bewell described by the Cahn–Hilliard equation (7).

Moreover, we find that if the autocorrelation functionsassociated with microstructure aged for different t arerescaled by k(t), i.e. f(x) = f[r/k(t)], these functions allapproximately collapse onto a universal curve, which hasthe following functional form:

f ðrÞ ¼ expð�ar=kÞ cosðpr=kþ bÞ= cosðbÞ ð10Þwhere a = 3.5 and b = 0.60 are fitting parameters for thebinary Pb–Sn systems we considered and

k ¼ k30 þ ðk

3f � k3

0Þttf

� �1=3

ð11Þ

for different time periods. The linear size of the system is �200 lm.

Fig. 3. Left panel: autocorrelation functions f(r) associated with microstructures aged for different t, from which the growing length scale k is extracted.Right panel: k as a function of t, which satisfies the scaling relation (1). The unit for length (i.e. r and k) is pixel, and 1 pixel � 0.67 lm. The unit for time(i.e. t) is hour.

3374 Y. Jiao et al. / Acta Materialia 61 (2013) 3370–3377

where k0 and kf are respectively the length scale in the as-processed and final aged microstructures, and tf is the asso-ciated time of aging. We note that it has been shown [22]that the phenomenological form of f(r) (Eq. (10)) can verywell capture the salient morphological features of lamellarstructures as those in the Pb–Sn alloys, which we elaboratebelow.

Fig. 4 shows the rescaled f(r) as well as the universalcurve given by Eq. (10). The exponential decay of thelong-range correlations in the systems is captured by thecorresponding monotonically decreasing exponential func-tion in Eq. (10); and the short-range correlations mani-fested as the oscillations are approximated by the cosinefunctions in Eq. (10) [22,23]. This parameterized generalfunctional form of rescaled f(r) enables one to fit the spe-cific spatial correlations in the systems [22], which origi-nated from the density (concentration) fluctuations in thebeginning of the phase separation and are manifested inthe coarsened microstructures. Importantly, it also enablesone to predict f(r) associated with microstructures aged forintermediate t for which no experimental data are avail-

Fig. 4. Rescaled autocorrelation functions associated with microstruc-tures aged for different t, which approximately collapse onto a universalcurve after rescaled by the appropriate k(t).

able. We note that in general, one can approximate anytwo-point correlation functions by a given set of basis func-tions as discussed in Ref. [22]. However, we find that Eq.(10) is a sufficiently good approximation for the Pb–Sn sys-tem (as well as the microstructures generated from phase-field simulations) and thus, we do not include additionalbasis functions. Stochastic material reconstruction tech-niques can then be employed to generate virtual micro-structures consistent with the predicted f(r) (i.e. S2),which we discuss in the subsequent section.

3.3. Stochastic material microstructure reconstruction

We use the general Yeong–Torquato reconstructionprocedure [17] to generate virtual 3-D microstructuresfrom the correlation functions at a specific time of agingt predicted by Eq. (10). In particular, the reconstructionproblem is formulated as an “energy” minimization prob-lem, with the energy functional E defined as follows:

E ¼X

r

½S2ðrÞ �fS2ðrÞ� ð12Þ

where fS2ðrÞ is the predicted (target) correlation functionand S2(r) is the corresponding function associated with atrial microstructure. The simulated annealing method [24]is usually employed to solve the aforementioned minimiza-tion problem. Starting from an initial trial microstructure(i.e. old microstructure), which contains a fixed numberof voxels for each phase consistent with the volume frac-tion of that phase, two randomly selected voxels associatedwith different phases are exchanged to generate a new trialmicrostructure. Relevant correlation functions are sampledfrom the new trial microstructure and the associated energyis evaluated, which determines whether the new trial micro-structure should be accepted or not via the probability:

paccðold! newÞ ¼ min 1; expEold

T

� �exp

Enew

T

� �� �ð13Þ

Fig. 6. The lineal-path functions associated with the virtual microstruc-tures (lower panels in Fig. 5) and the corresponding microstructures of theactual Pb–Sn alloy (upper panels in Fig. 5). The unit for length (i.e. r) ispixel, and 1 pixel � 0.67 lm.

Y. Jiao et al. / Acta Materialia 61 (2013) 3370–3377 3375

where T is a virtual temperature that is chosen to be highinitially and slowly decreases according to a coolingschedule [17,21]. The above process is repeated until E issmaller than a prescribed tolerance, which we choose tobe 10�10 here. Generally, several hundred thousand trialsneed to be made to achieve such a small tolerance.

A key step in the reconstruction process is to efficientlycompute the correlation functions from the trial micro-structure. Since an extremely large number of trial micro-structures are generated, it is highly undesirable tocompletely re-compute the functions for each trial. Instead,we use the correlation functions of the old microstructureto obtain those for the new microstructure. In addition,we consider the voxels as “particles” of a lattice–gas systemand employ the lattice-point method [25,26] to efficientlycompute S2 associated with each trial microstructure. Spe-cifically, the voxels of the phase of interest are grouped andthe separation distance between a pair of voxels within thegroup is computed. These distances are properly binnedand normalized to obtain the autocorrelation function.The reader is referred to Ref. [25] for details of this method.

Several 2-D slices of the virtual microstructures associ-ated with f(r) with different aging times generated usingthe reconstruction procedure are visually compared withthe corresponding 2-D images of the actual Pb–Sn alloymicrostructures in Fig. 5. It can be seen that the salient geo-metrical and topological features of the actual alloy micro-structures, e.g. the shape/size and connectedness of thephases, are very well reproduced in the reconstructions.The corresponding microstructures are also quantitativelycompared to one another by measuring the associated lin-eal-path functions L(r), which provide linear connectedness

Fig. 5. 2-D slices of virtual microstructures associated with f(r) with different tiactual Pb–Sn alloy microstructures (lower panel) for purposes of comparison.

information in the system. Fig. 6 shows L(r) for the actualalloy microstructures and the reconstructions. It can clearlybe seen that the reconstructions accurately produced thedegree of clustering in the system, which is determined bythe coarsening process. This verifies the accuracy of ourgeneral form of f(r) (cf. Eq. (10)) as well as the reconstruc-tion procedure. It is noteworthy that the correlation lengthin L(r), i.e. the distance r beyond which L is smaller than aprescribed tolerance, can also be used as a proper lengthscale characterizing the coarsening in the system.

With the accuracy of our procedure verified, we employit to predict microstructures associated with aging times forwhich no experimental data are available. Fig. 7 shows 3-D

me periods of aging (upper panel) and the corresponding 2-D images of theThe linear size of the system is �100 lm.

Fig. 7. Virtual 3-D microstructures associated with f(r) with different time periods of aging generated using the reconstruction procedure. The linear size ofthe system is �80 lm (�120 pixels).

3376 Y. Jiao et al. / Acta Materialia 61 (2013) 3370–3377

virtual microstructures aged for t = 8 h, 80 h and 180 hgenerating from the predicted correlation functions f(r)(cf. Eq. (10)). One can clearly see that coarsening developsin the 3-D microstructure, which is controlled by the char-acteristic length scale k(t) in our model.

We note that in general a two-point correlation functionf(r) alone is not sufficient to uniquely determine a micro-structure, and thus the reconstructions entirely based onf(r) usually lead to structures that are distinct from the tar-get system. However, our purpose here is not to uniquelyreconstruct a single microstructure that perfectly matchesthe target image. Instead, we are interested in producinga set of 3-D microstructures that statistically representthe key morphological features of the 2-D target image.In other words, it is sufficient for the optimization recon-struction procedure to find a local optimum instead ofthe true global optimum associated with the target image,which has been shown to be extremely difficult [27]. Inour case, the microstructures associated with these localoptima very well capture the salient features of the targetimage, as quantitatively ascertained by the associated lin-eal-path functions.

4. Conclusions

In this paper, we have devised a procedure to model andpredict microstructure evolution in Pb–Sn alloy aged at175 �C for different time periods using the associatedtwo-point correlation functions S2. Specifically, we haveverified via phase-field simulations that an appropriategrowing length scale k(t) characterizing microstructurecoarsening can be defined via the corresponding correla-tion functions, which enables us to predict S2 associatedwith intermediate microstructures given the initial and finalmicrostructures. Stochastic material reconstruction tech-niques are employed to generate virtual 3-D microstruc-tures consistent with the predicted correlation functions,which are quantitatively compared with the actual alloymicrostructures when available.

Although our approach is developed for a particularsystem (i.e. Pb37Sn63 alloy), we expect that it could beapplied to model a wide range of binary systems whichevolve according to the Cahn–Hilliard equation. However,one might need to construct different functional forms ofS2 similar to that given by Eq. (10) for different systems,in order to properly take into account the associated intrin-sic spatial correlations. Once such a functional form isobtained, one can employ the same stochastic reconstruc-tion procedure presented in the paper to generate realisticvirtual material microstructures for further analysis.

In future work, we would like to obtain 3-D microstruc-tural data of the Pb–Sn alloy via X-ray tomography [28] toquantitatively ascertain the accuracy of the predictedmicrostructures. More generally, we would like to developa general physics-based statistical microstructure modelingprocedure via correlation functions that enables one tocharacterize and predict microstructure evolution on multi-ple time and length scales, such as the nucleation, growthand propagation of damage in structural materials.

Acknowledgements

The authors would like to thank Tao Han (PrincetonUniversity) for valuable suggestions on phase-field model-ing. EP and NC are grateful for financial support fromthe Semiconductor Research Corporation (SRC).

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