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Acta Materialia 60 (2012) 6961–6971

Multiscale simulations on the coarsening of Cu-rich precipitates in a-Feusing kinetic Monte Carlo, molecular dynamics

and phase-field simulations

David Molnar a,b,⇑, Rajdip Mukherjee c,d,⇑, Abhik Choudhury c, Alejandro Mora a,Peter Binkele a, Michael Selzer c,d, Britta Nestler c,d, Siegfried Schmauder a,b

a Institute for Materials Testing, Materials Science and Strength of Materials, University of Stuttgart, Pfaffenwaldring 32, 70569 Stuttgart, Germanyb Stuttgart Research Center of Simulation Technology (SRC SimTech), SimTech Cluster of Excellence, University of Stuttgart, 70569 Stuttgart, Germany

c Institute of Applied Materials, Karlsruhe Institute of Technology (KIT), Haid-und-Neu-Str. 7, 76131 Karlsruhe, Germanyd Institute of Materials and Processes, Kalsruhe University of Applied Sciences, Moltkestrasse 30, 76133 Karlsruhe, Germany

Received 2 July 2012; received in revised form 21 August 2012; accepted 23 August 2012Available online 13 October 2012

Abstract

The coarsening kinetics of Cu-rich precipitates in an a-Fe matrix for thermally aged Fe–Cu alloys at temperatures above 700 �C isstudied using a kinetic Monte Carlo (KMC) simulation and a phase-field method (PFM). In this work, the KMC approach adequatelycaptures the early stage of the system evolution which involves nucleation, growth and coarsening, while the PFM provides a suitableframework for studying late-stage coarsening at large precipitate volume fraction regimes. Hence, both models complement each otherby transferring the results of KMC along with precipitate–matrix interface energies from a broken-bond model to a quantitative PFMbased on a grand chemical potential formulation and the CALPHAD database. Furthermore, molecular dynamics simulations provideinformation on the structural coherency of the precipitates and hence justify the sequential parameter transfer. We show that our PFMcan be validated quantitatively for the Gibbs–Thomson effect and that it also predicts the coarsening kinetics correctly. It is found thatthe kinetics closely follow the LSW (Lifshitz–Slyozov–Wagner) law, whereas the coarsening rate constant increases with an increase involume fraction of precipitates.� 2012 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Multiscale; Precipitation; Kinetic Monte Carlo; Molecular dynamics; Phase-field methods

1. Introduction

a-Fe alloyed with Cu among other elements findsapplication in many areas, e.g. as pipe material in powerplants. The alloying with Cu yields an increased flowstress which can be attributed to solid-solution strength-

1359-6454/$36.00 � 2012 Acta Materialia Inc. Published by Elsevier Ltd. All

http://dx.doi.org/10.1016/j.actamat.2012.08.051

⇑ Corresponding authors. Addresses: Institute for Materials Testing,Materials Science and Strength of Materials, University of Stuttgart,Pfaffenwaldring 32, 70569 Stuttgart, Germany (D. Molnar), Institute ofApplied Materials, Karlsruhe Institute of Technology (KIT), Haid-und-Neu-Str. 7, 76131 Karlsruhe, Germany (R. Mukherjee).

E-mail addresses: david.molnar@imwf.uni-stuttgart.de (D. Molnar),rajdip@gmail.com (R. Mukherjee).

ening and particle strengthening due to the interactionof dislocations with Cu atoms and Cu precipitates withinthe material, respectively [1,2]. The latter strengtheningeffect depends on the thermal treatment of the materialas well as on the service conditions. At elevated temper-atures (above 300 �C), Cu precipitates form within theFe matrix on a relatively short timescale, yielding first astrengthening of the material. However, as the particlesundergo coarsening with time, the failure mechanismdue to tensile loading may change, which is undesirablefor safety reasons. The computational modelling of theprecipitate coarsening behaviour requires an understand-ing of the physical processes on the atomistic scale as well

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6962 D. Molnar et al. / Acta Materialia 60 (2012) 6961–6971

as on intermediate length scales in order to predict mate-rial properties on the macroscopic scale.

The classical theory of coarsening of precipitates due toOstwald ripening was proposed by Lifshitz and Slyozov [3]and Wagner [4], and is hence known as LSW theory.Although the theory is valid for an infinitesimally dilutesecond-phase particle, both theoretically [5–9] and experi-mentally [10–14] it is found that for higher volume frac-tions of precipitates, the temporal power law is governedby a rate constant larger than predicted by theory. Further-more, the size distribution is broader and has a smalleramplitude compared to the distribution predicted by thetheory. In this context, our multiscale approach is justifiedto predict quantitatively the coarsening kinetics for highvolume fractions of precipitates along with coalescenceevents in Fe–Cu systems.

The precipitation of Cu in a-Fe has been observedexperimentally [15–17] by means of small-angle neutronscattering as well as by high-resolution tomography, andhas also been modelled by applying a kinetic Monte Carlo(KMC) approach [18]. On the other hand, moleculardynamics (MD) simulations have been performed in orderto investigate the interactions of edge dislocations with Cuatoms in a solid solution [19] as well as with Cu precipitates[20,21], confirming the experimentally observed strengthen-ing in a Cu-containing alloy [22].

Both KMC and MD simulations are limited to smallsample sizes, of the order of tens of nanometers. In orderto reach higher length scales and to simulate late-stagecoarsening, the phase-field method (PFM) becomes a nec-essary tool. A wide range of phenomena described byphase-field methods can be found in Refs. [23–28]. Severalattempts have been made to simulate microstructure evolu-tion using the PFM along with atomistic simulations[29,30], but not for the Fe–Cu precipitation system.Recently, the CALPHAD (CALculation of PHAse Dia-grams) database has been used for the thermodynamicdescription for PFM in order to quantitatively predict themicrostructure evolution in precipitation systems. Mostof these studies of coarsening kinetics have been performedin 2-D [31–33] and only a few in 3-D [34–36]. For the Fe–Cu system, limited work has been done using quantitativephase-field modelling and these studies mostly involve thespinodal regime [37–39]. In our study, the integration ofKMC with the PFM makes it possible to study precipitatenucleation as well.

In the following section, a sequential multiscaleapproach is described in detail, followed by the modellingschemes of KMC (Section 3.1), MD (Section 3.2) andPFM (Section 3.3). In Section 4, the results of the simula-tions are discussed. Section 5 closes the paper with conclu-sions derived from the simulation results.

2. Multiscale approach

The multiscale approach applied within this study is of asequential type, i.e. simulation methods are connected via

appropriate parameter transfers. In this study, we have cho-sen to transfer the particle arrangement at late KMC precip-itation stages to the PFM. By coupling the two methodssequentially, their advantages can be exploited, circumvent-ing simultaneously their particular disadvantages. Interfaceenergies derived from a broken-bond model (BBM) are fur-ther input data for the PFM. In order to provide structuralinformation of the precipitates, MD relaxation simulationsare performed since the KMC approach cannot account forthis due to the rigid lattice (see Section 3.2).

3. Simulation methods and applied models

3.1. Cu precipitation: KMC simulations

The process of Cu precipitation in a-Fe is simulated by aKMC method which is based on a thermally activatedvacancy diffusion on a rigid body-centred cubic (bcc) crystallattice model (RLM) [18]. Although in nature Cu has theface-centred cubic structure, Cu clusters with sizes smallerthan 2 nm are coherently embedded on a-Fe lattice sites[40,41], justifying the RLM. The KMC simulation used inthis study was first proposed by Soisson et al. [18]. Adetailed description can be found in Refs. [18,42]. As start-ing configuration of L = 128 lattice constants yieldsN = 2L3 = 4,194,304 lattice sites and a cubic box with anedge length of 36.7 nm. The box surfaces have normals inthe {10 0} directions and periodic boundary conditionsare set in all directions. Fe atoms are replaced randomlyby Cu atoms to obtain Fe–Cu solid solutions with 1, 2, 5and 10 at.% Cu, respectively. An empty site represents a sin-gle vacancy and the annealing temperature is set to 700 �C.

The chemical binding between atoms is described byfirst- and second-nearest neighbour pair interactions

eðiÞFe-Fe; eðiÞCu-Cu and eðiÞFe-Cu with i 2 {1,2}, where i denotes the

ith-nearest neighbour (see Fig. 1). The energies eðiÞFe-Fe and

eðiÞCu-Cu; i 2 f1; 2g were estimated from the cohesive energies

of the pure metals assuming eð2ÞFe-Fe ¼ eð1ÞFe-Fe=2 and

eð2ÞCu-Cu ¼ eð1ÞCu-Cu=2 (see also Ref. [43]). The thermallyactivated position exchange between the vacancy V and aneighbouring atom A (with A = Fe or Cu) is given bythe jump frequencies:

CA;V ¼ mA exp �DEA;V

kBT

� �; ð1Þ

with T and kB being the temperature and the Boltzmannconstant, respectively. mA denotes the attempt frequency,which is estimated using the diffusion constants of the puremetals. The activation energies for migration, which de-pend on the local configuration, are given by:

DEA;V ¼ ESP;A-X

X

eA-X �X

Y

eV-Y; ð2Þ

where ESP,A is the energy at the saddle-point between Aand V, eA-X are the interaction energies of the first- and sec-ond-nearest neighbours of A (X atoms), and eV-Y are the

Fig. 1. Schematic representation of the bcc lattice and the interactionenergies used in the model (Fe = black, Cu = blue, vacancy = yellow).(For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

D. Molnar et al. / Acta Materialia 60 (2012) 6961–6971 6963

binding energies between the vacancy and its first-nearestneighbours (Y atoms) (see Fig. 1). For each first-nearestneighbour of the vacancy V, the jump frequenciesC1, . . . ,C8 are calculated. Applying a rejection-free resi-dence time algorithm [1,18], one of the eight weighted jumppossibilities is selected. This procedure is repeated over 1011

times during the simulation of precipitation. The real tem-poral scale is obtained from:

treal ¼cV;sim

cV;real

� �tMC; with tMC ¼

X8

j¼1

Cj

!�1

; ð3Þ

where cV,sim and cV,real denote the vacancy concentrationsin the simulation and in the real material, respectively.tMC is the average residence time. The energetic parametersof the KMC simulations are listed in Ref. [44]; these, inturn, are based on Refs. [43,45–47].

3.2. Structural coherency: MD simulations

In order to provide information on the coherency of theembedded Cu precipitates, MD relaxation simulations arecarried out using the IMD (ITAP Molecular Dynamics)code [48] which allows for massively parallel computations,yielding elastic constants, stress and pressure tensors forrelaxed structures. For metals, embedded atom method(EAM) potentials describe the atomic interactions as theyinclude an additional embedding term, besides pair interac-tions /ij, accounting for the local electron charge density inthe lattice, i.e.

V ¼ 1

2

Xi–j

/ijðrijÞ þX

i

U iðniÞ with ni ¼Xi–j

qijðrijÞ; ð4Þ

where Ui describes the energy of embedding atom i in adensity ni, which is the sum of contributions qj from neigh-bours j at distances rij. To describe the Fe–Cu system,EAM potential the recently published by Bonny et al.[49] is applied. Starting from a lattice where all Fe andCu atoms have the same lattice constant, i.e. the latticeconstant of Fe, the structure is relaxed to T � 0 K(15,000 MD steps with 2 fs per step). During relaxation,the Cu atoms tend to move into an energetically preferredconfiguration. In order to give the atoms more time to doso, the structure can be heated up to 300 K (50,000 MDsteps) and kept at this temperature for another 50,000MD steps before relaxation by applying the NpT ensemble(constant number of particles, constant pressure and tem-perature). In any case, the surrounding a-Fe matrix will af-fect their relaxation depending on the size of the Cuprecipitate. The results will be discussed in Section 4.2.

3.3. Particle coarsening: PFM

3.3.1. Model description

The PFM applied for the investigation of precipitationin the Fe–Cu system is based on the grand-potential func-tional [50]:

XðT ; l;/Þ ¼Z

XWðT ; l;/Þ þ �~að/;r/Þ þ 1

�~wð/Þ

� �� �dX;

where X is the total grand potential, W is the grand chem-ical potential density, T is the temperature, l is the chemi-cal potential, / ¼ /a;/b . . . N is the phase-field vectorconsisting of the N phase-field variables and � is the inter-face width. The energy densities ~a and ~w together representthe interface energy of the system, where the former is thegradient energy and the latter the interface potential. Thephase evolution is determined by the phenomenologicalminimization of the grand potential functional.

The concentration fields are obtained by a mass conser-vation equation for each concentration field ci, from the setof K � 1 independent concentration variables, K being thenumber of components in the system. The evolution equa-tion for the N phase-field variables (/a, a = 1, . . . ,N) canbe written as:

s�@/a

@t¼ � r � @~að/;r/Þ

@r/a

� @~að/;r/Þ@/a

� �� 1

�

@~wð/Þ@/a

� @WðT ; l;/Þ@/a

� K; ð5Þ

where K is the Lagrange parameter maintaining the con-straint

PNa¼1/a ¼ 1. The gradient energy density ~að/;r/Þ

has the form:

~að/;r/Þ ¼XN ;Na;b¼1

ða<bÞ

c½acðqabÞ�2jqabj

2;

where qab = (/a$ /b � /b$/a) is a normal vector to the abinterface. ac(qab) describes the anisotropy of the evolving

6964 D. Molnar et al. / Acta Materialia 60 (2012) 6961–6971

phase boundary, which for the studies in the presentinvestigation have been assumed to be ac(qab) = 1 formodelling isotropic systems. The obstacle type potential~wð/Þ, which was also previously described in Ref. [51,52],can be written as:

~wð/Þ ¼ 16

p2

XN ;Na; b ¼ 1

ða < bÞ

c/a/b;

where c is the interface energy. The parameter s in Eq. (5) is

written as:

PN ;N

a<bsab/a/bPN ;N

a<b/a/b

, where sab is the relaxation constant

of the ab interface. It is chosen in a manner such that theinterface kinetics vanishes [50]. The evolution equationfor the concentration fields can be derived as:

@ci

@t¼ r �

XK�1

j¼1

Mijð/Þrlj

!: ð6Þ

Here, Mij(/) is the mobility of the interface formulated byan interpolation of the individual phase mobilities as:

Mijð/Þ ¼XN�1

a¼1

Maijgað/Þ:

Each of the Maij is defined using the expression:

Maij ¼ Da

ij

@cai ðl; T Þ@lj

:

The function ga(/) interpolates the mobilities. Daij are the

interdiffusivities in each phase a. Both the evolution equa-tions (Eqs. (5) and (6)) require information about thechemical potential l.

We write the grand potential density W, as an interpola-tion of the individual grand potential densities Wa, whereWa are functions of the chemical potential l and of the tem-perature T in the system:

WðT ; l;/Þ ¼XN

a¼1

WaðT ; lÞhað/Þ with ð7Þ

WaðT ; lÞ ¼ faðcaðlÞ; T Þ �XK�1

i¼1

licai ðl; T Þ;

where fa(ca(l), T) is the free energy density of the phase a.The concentration ca

i ðl; T Þ is an inverse of the functionla

i ðc; T Þ for every phase a and component i. From Eq. (7)the following relation can be derived:

ci ¼XN

a¼1

cai ðl; T Þhað/Þ; ð8Þ

using @WðT ;l;/Þ@li

¼ �ci.

3.3.2. Calibration of CALPHAD data

For the Fe–Cu system, we reduce the system to a two-phase binary system where the independent concentrationis identified by c = cFe. Similarly, we reduce the chemicalpotential vector l to l, which defines an independent chem-

ical potential with respect to Fe. The phase-field vectorcontains only two components: / = (/1, /2). We start theconstruction of free energies of the respective phases withthe following type of expressions for the free energies:

f aðT ; cÞ ¼ AaðT Þc2 þ BaðT Þcþ EaðT Þ;where the coefficients Aa(T), Ba(T) and Ea(T) are functionsof temperature T. By using a polynomial formulation, weaim to fit a simplified form for the free energies utilizingthe data obtained from the CALPHAD database. We

can determine the terms Aa(T) as @2f a

@c2 jceq� 1

V m

@2Ga

@c2 jceq, com-

puted at the equilibrium concentration of the phase at thetemperature T, where Ga(T,c) is the free energy functionobtained from CALPHAD. Next we derive the chemical

potential leq ¼ 1V m

@Ga

@c jceqfrom the database and compute

Ba(T) by equating the first derivative of the constructed freeenergies to the chemical potential, giving:

BaðT Þ ¼ leq � 2AaðT Þceq:

The only term left out is Ea(T), which is fitted by equatingit to the grand potential, which yields:

EaðT Þ ¼ Weq � AaðT Þc2eq;

with Weq ¼ 1V mðGaðT ; ceqÞ � leqcÞ. With these equations we

can adequately fit all the terms in the free energies at thegiven temperature T. For a non-isothermal description, itis essential to derive the equations in the neighbourhoodof the temperature one is simulating and perform a fittingin the temperature space. In most cases, a linear tempera-ture fit suffices. The chemical potential can be derived interms of the concentration c and volume fractions /1,/2

of the phases by using the relation (8). The phase concen-trations can be written as functions of the chemical poten-

tial as caðl; T Þ ¼ l�BaðT Þ2AaðT Þ

� �. Using the constraint in Eq. (8),

the following relation for l derives:

l ¼cþ

XN

a¼1

BaðT Þhað/Þ2AaðT ÞXN

a¼1

hað/Þ2AaðT Þ

: ð9Þ

This value of l is used both in the evolution equations forthe phase-field and in the concentration equations.

3.3.3. Mobilities for diffusion

To derive the mobilities, we require the second deriva-tives of the free energies with respect to the concentrations.This can be realized through the diffusion equation for abinary alloy, written as:

@ca

@t¼ r � ðMarlÞ;

where Ma is defined as D @ca

@l , in order to derive Fick’s law inthe bulk. Taking the partial derivative of ca(l,T) yields:

@ca

@l¼ 1

2AaðT Þ :

D. Molnar et al. / Acta Materialia 60 (2012) 6961–6971 6965

4. Simulation results

4.1. KMC simulations and the BBM

The KMC simulations have been performed asexplained in Section 3.1 for 1, 2, 5 and 10 at.% Cu. At anannealing temperature of 700 �C, 6 � 1011 KMC steps yieldthe formation of Cu clusters within the a-Fe matrix. Fig. 2ashows that at the beginning of the ageing process a largenumber of particles forms in relatively short time due tothe high annealing temperature. Hence, these clusters con-tain only small amounts of Cu atoms, i.e. their mean radiusis small. After reaching a maximum, the number of parti-cles decreases again. Nevertheless, the mean radiusincreases (Fig. 2b), a fact that can be attributed to Ostwaldripening, i.e. to the coarsening of particles. Since at thehigh temperatures considered here, there are almost noclusters for the case of 1 at.% Cu, the corresponding dataare not considered in the following. With the KMCmethod, precipitates with a mean radius of 1.5 nm can beobtained in a reasonable amount of computing time(O (days)). However, reaching mean particle radii of acouple of nanometers would increase the computation timetremendously, justifying the application of PFM in thecoarsening regime. Fig. 3a shows the atomic configurationsat two different ageing times where “peak” denotes thestates with the maximum number of particles (peaks inFig. 2a) and “end” denotes the end states of the KMC sim-ulations for 2, 5 and 10 at.% Cu, respectively.

Since at the peak position only small clusters haveformed, they are hardly visible among the completelysolved Cu atoms. In contrast, at the end of the simulationthe precipitates are clearly visible. At this point, theremaining matrix has already reached thermal equilibrium,i.e. almost no further nucleation is expected and Ostwaldripening can be considered as the dominant precipitationprocess. Transferring information from KMC to PFMcan be done in two ways. One possibility is to transferthe whole atomic configuration, i.e. the positions of allCu atoms, at certain precipitation states to PFM simula-tions. By this means, no information is lost. The otherway is to transfer statistical information on the atomic con-

Fig. 2. (a) Number of particles and (b) their mean radius, as a function of thermstatistical analysis inaccurate. Therefore, the 1 at.% Cu KMC results are not f

figuration at certain precipitation states, i.e. fitted distancedistributions and radius distributions as shown in Fig. 3b,c.This study provides a means to validate transfer of an iden-tical particle arrangement, and hence is very useful for per-forming large-length-scale simulations with the PFM.

4.1.1. Interface energies: BBM

In order to obtain the interface energy needed in thePFM simulations, the BBM is applied [53,54]. Within thismodel, the first- and second-nearest neighbours are consid-ered and all atoms contribute to the interface energy whosecoordination number changes when inserting an interface.Deriving the interface energies for interfaces oriented inthe [10 0], [110] and [111] directions without relating themto a specific interface area reads:

Eint½100� ¼

1

2�4e1

AA � 2e2AA þ 8e1

AB þ 4e2AB � 4e1

BB � 2e2BB

� �ð10Þ

Eint½110� ¼

1

2�2e1

AA � 2e2AA þ 4e1

AB þ 4e2AB � 2e1

BB � 2e2BB

� �ð11Þ

Eint½111� ¼

1

2�5e1

AA � 6e2AA þ 10e1

AB þ 12e2AB � 5e1

BB � 6e2BB

� �:

ð12Þ

Eqs. (10) and (11) can also be found in Ref. [55]. Calculat-ing the interface energies using the activation energies fromthe KMC simulations (see Section 3.1) yields the interfaceenergies shown in Table 1. Thus the preferred interfaces ofCu clusters are {110} interfaces. This is in very good agree-ment with results that can be found in Refs. [18,42].

4.2. MD simulations

From experiments [40,41] it is known that small Cu clus-ters (<2 nm) are coherently embedded in the a-Fe matrix,while bigger clusters undergo a structural change, i.e.bcc! 9R! 3R! fcc [40,41]. In this study, precipitateswith radii from 2 to 14 lattice constants, i.e. with radiiranging from 0.57 to 4.00 nm are analysed in order to pro-vide qualitative information on the necessity of embeddingthe structural transition into the PFM simulations.Therefore, each precipitate is sufficiently relaxed accordingto the modelling scheme described in Section 3.2. After

al ageing time. In the case of 1 at.% Cu, almost no particles form, makingurther considered.

Fig. 3. (a) Arrangement of Cu atoms at two distinct times for 2, 5 and 10 at.% Cu. Up to the point at which the number of particles reaches the maximumvalues (see Fig. 2), i.e. at the “peak” positions, very small clusters have formed and the resulting images looks nearly like solid solutions. At the “end” ofthe simulation, several large particles (radii > 1 nm) have formed, while the remaining solid solution is in thermal equilibrium. Fitted Gaussians representthe distance distributions (b) in lattice constants (a = 0.2867 nm) and the “peak” positions of the radius distributions (c) for 2, 5 and 10 at.% Cu, whereasLSW distributions are assumed at the “end” positions.

Table 1Interface energies of Fe–Cu interfaces for different orientations obtainedby evaluating Eqs. (10)–(12) for A = Fe and B = Cu. By relating thecalculated interface energies to the orientation-dependent areas per atom,the interface energies per unit area are obtained.

Interface energies

orientation [100] [110] [111]Eint [J] 0.248 0.149 0.396area per atom a2

ffiffi2p

2 a2ffiffiffi3p

a2

c Jm2

0.483 0.410 0.447

6966 D. Molnar et al. / Acta Materialia 60 (2012) 6961–6971

relaxation, the neighbours of each atom within a certaindistance are counted. For the bcc structure, a cut-off dis-tance between the second- (0.2855 nm) and third-(0.4038 nm) nearest neighbour of the corresponding atomis an appropriate choice, as the first- (0.2473 nm) and sec-ond-nearest neighbours lie very close to each other. In aperfect bcc crystal, the counting would yield 14 nearestneighbours (NN = 14) in total for each atom, i.e. 8 first-and 6 second-nearest neighbours. Visualizing the relaxedstructures, atoms with NN = 14 are shown as red points(see Fig. 4). Deviations are shown as spheres. The colour-ing is explained in Table 2. The visualization shows thatsmall precipitates (r 6 4 lattice constants) are coherentlyembedded within the a-Fe matrix, whereas bigger precipi-tates show severe deviations in their interiors. However,the surfaces remain approximately in the bcc structure,i.e. the coordination number of surface atoms does not

change. Hence, the transition from bcc to fcc within theclusters does not have to be taken into account in this firstcoupling of KMC and PFM simulations and the precipi-tates in the PFM simulation can be assumed as perfectlycoherent.

4.3. PFM simulations

4.3.1. Data transfer to the PFM

We use a quantitative PFM based on the grand chemicalpotential formulation (described in Section 3.3). The modelparameters used for our simulations are listed in theirdimensional form in Table 3. As described in Section 4.1,the size and location of each precipitate cluster obtainedin the KMC simulations are transferred as input for thePFM simulations. In PFM simulations, we use a computa-tional box of 1283 grid points where the grid spacing Dx isequal to the lattice constant resulting in the same systemsize as used for the KMC simulation described in Section3.1. Assuming isothermal ageing conditions, the free energyfor the Fe–Cu system at 1100 K (827 �C) is obtained fromthe CALPHAD database. We fit this free energy data tothe simplified form of the free energy functions using theprocedure described in Section 3.3.2 and calculate thechemical potential l, which is the driving force for diffu-sion, using Eq. (9).

In our system, the compositions of equilibrium Cu-richprecipitates and Fe-rich matrix are very close to pure Cu

Fig. 4. Coherency of the precipitates after relaxation at regions close to the surface. The radii range from 0.57 nm (2 lattice constants) to 4.00 nm (14lattice constants) where the lattice constant of Fe is taken [49,56]. Small precipitates are perfectly coherent while deviations develop in bigger spheres.Nevertheless, at the precipitates’ surfaces the deviations remain small. This behaviour is also found if the structure is kept at 300 K for some time beforerelaxation (14, 300 K). Thus, the structural transition does not necessarily have to be incorporated into the PFM simulations in this study.

Table 2Atom colouring with respect to the number of atoms within a specifiedcut-off radius rc. In bcc crystals, NN = 14 neighbours have to be found.Deviations (NN – 14) indicate defects within the single crystal. Anadditional identification of atoms with NN = 12 neighbours is due to thefact that the corresponding fcc structure would have 12 neighbours withinrc.

Atom type Number of neighbours Colour

Cu NN = 14 RedCu NN < 14 & n – 12 OrangeCu NN > 14 PurpleCu NN = 12 Yellow

Table 3Parameters used in the PFM simulations. All the parameters are defined inSection 3.3.

Parameters Values Parameters Values

Aa 1.71 � 1011 J m�3 s 6.89 � 1018 J m�4

Ab 1.39 � 1011 J m�3 D 10�16 m2 s�1

Ba �3.33 � 1011 J m�3 � 7.0 � 10�10 mBb �6.43 � 109 J m�3 c 0.41 J m�2

Ea 1.13 � 1011 J m�3 Dx 2.8665 � 10�10 mEb �5.04 � 1010 J m�3 Dt 5 � 10�5 s

D. Molnar et al. / Acta Materialia 60 (2012) 6961–6971 6967

and pure Fe, respectively. From the BBM calculations (seeTable 1), it is found that the lowest interface energybetween a Cu cluster and the surrounding Fe matrix,among several possible interfaces, is 0.410 J m�2 ([1 10]interface). We use this interface energy value as a represen-tative and approximated input parameter for the PFM sim-ulations. Internal structural changes within the precipitates

as discussed in Section 4.2 may affect the coarseningkinetics only towards the end of the simulation (meanradius >3 nm) when distortions get more severe. Hence,in the present PFM simulations we assume the absence ofelastic strain fields, which will be incorporated in our futurestudy.

In the PFM, the atomic mobility M can be derived fromthe interdiffusivity D (described in Section 3.3.3). D is cal-culated from the self-diffusivity values of each componentusing Darken’s equation:

D ¼ X FeD�Cu þ X CuD�Fe � 10�16 m2 s�1; ð13Þwhere D�Cu and D�Fe are the self-diffusivities, and XCu andXFe are the mole fractions for Cu and Fe, respectively.Self-diffusivities of Fe and Cu in a-Fe as a function of tem-perature are reported in the literature [57,58]. We calculatethe value of D at 1100 K from these available experimentaldata using Eq. (13).

4.4. Phase-field results

The equilibrium compositions of precipitate and matrixin the Fe–Cu system at a particular temperature across aflat interface can be calculated using the tie lines from thefree energy CALPHAD data. These compositions changefrom their equilibrium values across a curved interfacedue to the Gibbs–Thomson effect. This shift in the sur-rounding matrix composition provides the driving forcefor precipitate coarsening. Phase-field models can ade-quately capture the Gibbs–Thomson effect for the precipi-tate–matrix microstructure [59,60]. Therefore, it is anideal tool for studying coarsening phenomena. In our

6968 D. Molnar et al. / Acta Materialia 60 (2012) 6961–6971

three-dimensional domain, the precipitates are nearlyspherical because of the isotropic interface energy. Theshift in composition at the precipitate side Dcp is given by:

Dcp ¼ 2cj

np cpeq � cm

eq

� � ; ð14Þ

where

np ¼@2f a

@c2

� �ppt

¼ 2AaðT Þ;

and the shift in composition at the matrix side Dcm followsas:

Dcm ¼ ðnp=nmÞDcp:

In the above equations, j is the interface curvature (inverseof radius), cp

eq and cmeq are the equilibrium compositions

across a flat interface at the precipitate and the matrix side,respectively, and c is the interface energy.

Phase-field simulations are performed with a singlespherical precipitate in a three-dimensional domain forthree different precipitate sizes. For each simulation, weuse the equilibrium precipitate composition obtained fromthe free energy function using a common tangent construc-tion, with a very small supersaturation in the matrix. Dur-ing the simulation, the precipitate and matrix compositionswith time evolution reach the composition correspondingto the equilibrium values across a curved interface.

In Fig. 5, we have plotted the shift in composition on theprecipitate side from its equilibrium composition, Dcp, as afunction of interface curvature. The slope of the line is

found to be 2c= np cpeq � cm

eq

� �� �. This test confirms the abil-

ity of our model to predict the Gibbs–Thomson effectcorrectly.

Fig. 6 shows the precipitate–matrix microstructures attimes t = 0 s and t = 175 s for 2, 5 and 10 at.% Cu, respec-tively. Microstructures at t = 0 s correspond to the “end”

results of the KMC simulations (see Fig. 3). In all systems,the large particles grow at the expense of smaller particles.

Fig. 5. Change in precipitate composition from its equilibrium compo-sition across a flat interface (Dcp) is plotted against the inverse ofprecipitate radius. The straight line for Dcp is given by the Gibbs–Thomson equation (see Eq. (14)). The data points represent Dcp measuredat the centre of the spherical precipitates obtained from three differentPFM simulations.

The mean radius increases with time, while the number ofparticles decreases with time. During this time interval,after an initial transient, the total volume of the particlesremains constant for each system, which implies that thematrix supersaturation is zero and the system is in thecoarsening regime.

Fig. 7a–c shows the particle size distributions (PSDs) att = 0 s and t = 175 s for the three systems. In all the cases,the size distribution shifts to its right and the distributionalso broadens with time. An important aspect of LSW the-ory is dynamic scaling behaviour, which means the wholemicrostructure is self-similar during steady-state coarsen-ing when scaled by the mean precipitate radius. The PSDpredicted by LSW holds only for spherical particles in aninfinitesimally small volume fraction of precipitates. Asmentioned in Section 1, several modified theoretical andexperimental results showed that, with a higher volumefraction of precipitate, the distribution becomes broaderand has a smaller amplitude. Fig. 7d shows that the PSDsscaled by the mean radius obtained at different times, forthe Cu 5 at.% system, follow this trend.

The mean particle radii R are obtained at different stagesof coarsening for each system and are plotted as a functionof ageing time in Fig. 8a. In Fig. 8b, the number of precip-itates for each of these three systems is plotted as a functionof time. For the 2 at.% Cu system, the number of particlesin the computational domain reduces from about 55 att = 0 s to 27 at t = 175 s. The small number of particlesin a domain makes this system statistically unreliable forpredicting the overall trend in the temporal evolution ofthe microstructure. Each denucleation event during evolu-tion causes a gradual decrease in mean radius and suddenincrease in the effective supersaturation in the matrixbecause of the small total number of particles. These eventsappear as steps in the mean radius vs. time plot. For the10 at.% Cu system, the initial number of particles (att = 0 s) is 260. However, this number decreases rapidly to126 at t = 10 s due to a large number of coalescence events,and then decreases slowly to 30 at t = 175 s. Denucleationevents of a group of small particles appear as steps only ata later stage of evolution. For the 5 at.% Cu system, thenumber of particles decreases slowly from 166 at t = 0 sto 33 at t = 175 s. The larger number of particles than inthe 2 at.% Cu composition and the smaller number of coa-lescence events than in the 10 at.% Cu composition makethis system more favourable for studying coarsening kinet-ics following the LSW law.

According to the LSW theory, a cube of mean radiusRðtÞ should increase linearly with time t during coarseningregime:

RðtÞ ¼ ðRð0Þ3 þ KtÞ1=3; ð15Þ

where Rð0Þ is the mean radius at t = 0 s and K is the coars-ening rate constant. In Fig. 8c and d, we fit RðtÞ for the 5and 10 at.% Cu systems, respectively, using the aboveequation in the time interval between t � 0 s (after the ini-tial transient) and t = 120 s when the simulation domain

Fig. 6. Microstructures obtained at t = 0 s (a,b,c) and t = 175 s (d,e,f) for 2, 5 and 10 at.% Cu systems (first, second and third columns, respectively).

Fig. 7. Particle size distributions (PSDs) for (a) 2, (b) 5 and (c) 10 at.% Cu at the beginning of the PFM simulations (start = red) and after 175 s(end = blue). (d) PSDs scaled by mean radius, for the Cu 5 at.% system, at three different times (0, 100, 175 s). The line shows the distribution obtainedfrom the LSW theory. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

D. Molnar et al. / Acta Materialia 60 (2012) 6961–6971 6969

Fig. 8. (a) The precipitate mean radius ðRÞ, and (b) the number of particles (np) is plotted as a function of time for 2, 5 and 10 at.% Cu. The temporalevolution of the R for (c) 5 and (d) 10 at.% Cu, fitted with LSW kinetics at the time interval indicated by the green datapoints. Inside this time interval bothsystems contain sufficient precipitates for good statistics. (For interpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)

6970 D. Molnar et al. / Acta Materialia 60 (2012) 6961–6971

contains a reasonable number of particles to describe thecoarsening kinetics (see Fig. 8b). For the 5 at.% Cu system,the mean precipitate radius closely follows the LSWkinetics after an initial transient and the fit is good in thistime interval. We found Rð0Þ ¼ 1:72 nm and KLSW =0.1024 nm3 s�1 for this system. The value of Rð0Þ is highercompared to the end results of KMC, because very smallprecipitates undergo a denucleation process during the ini-tial transient.

For the 10 at.% Cu system, the number of coalescenceevents is very high, which leads to an increase in R. Fur-thermore, the particles do not remain spherical in shape(see Fig. 6f) during time evolution. The fit in this regimeis not as good as in the 5 at.% Cu system, because theinherent assumptions of the LSW theory do not hold goodfor the 10 at.% Cu system. We found Rð0Þ ¼ 1:86 nm andKLSW = 0.2389 nm3 s�1 for this system. For our systems,we observe the rate constant K to increase with increasingprecipitate volume fraction, which is also reported in theliterature.

5. Conclusions

In the present survey, we employ KMC and PFM simu-lations to provide a quantitative description of the differentstages of precipitation by suitably using the strengths ofeach method. While KMC is able to model the process ofearly nucleation and growth adequately, late stage Ostwaldripening is more appropriately treated with the PFM.

Using this approach we are able to capture not only thecoarsening regime with both models, but also the coales-cence events at high volume fractions of precipitates whichare retrieved in the PFM simulations. It is found that thekinetics closely follows the LSW temporal power law inthis regime, but the coarsening rate increases with anincrease in volume fraction of precipitates. It is noteworthythat while analytical theories such as LSW can adequatelydescribe ripening kinetics for specific particle morphologiesand configurations, deviations from the assumptionsrequire the employment of simulations. For this reason,computational modelling approaches such as the KMCand the PFM are useful.

Acknowledgements

D.M. and S.S. would like to thank the German Re-search Foundation (DFG) for financial support of the pro-ject within the Cluster of Excellence in SimulationTechnology (EXC 310/1) at the University of Stuttgart.A.M., P.B., S.S., R.M., A.C. and B.N. would like to thankthe German Research Foundation (DFG) for financialsupport of the projects SCHM746/101-1 and NE822/12-1.

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