Computational Materials Science 67 (2013) 27–34

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Computational Materials Science

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Effect of second phase mobile particles on polycrystalline grain growth:A phase-field approach

Ashis Mallick ⇑Department of Mechanical Engineering, Tezpur University, Tezpur 784 028, India

a r t i c l e i n f o

Article history:Received 6 January 2012Received in revised form 11 July 2012Accepted 8 August 2012Available online 29 September 2012

Keywords:Phase-fieldPolycrystalline grain growthMobile particlesActive parameter tracking

0927-0256/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.commatsci.2012.08.022

⇑ Fax: +91 3712 267005.E-mail address: mal123_us@yahoo.com

a b s t r a c t

The control of grain size is of great interest to the material science community since the properties andperformance of materials are strongly affected by the size of grains. Grain boundary mobility plays a keyrole in the grain growth process. In this context, grain boundary mobility is strongly influenced by thepresence of second phase particles. In this work, we examine the effect of mobile monodisperse secondphase particles on the kinetics of polycrystalline grain growth using a phase-field theory. The governingequations for the evolution of the order parameters are obtained from a thermodynamically consistenttheory of phase transformations. For each grain and particle, separate phase-field variables are consid-ered. The active parameter tracking (APT) algorithm [S. Vedantam, B.S.V. Patnaik, Phys. Rev. E 73(2006) 016703] is used for considering the large number of phase-field variables. A bicrystal model withdense particles is examined first, and then extended to polycrystalline simulations. The accuracy of themodel was examined by comparing with the Zener relation for limiting mean radius Rlim in 2D graingrowth with immobile particles. The results obtained from the model are in very close agreement withthe results from literature.

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1. Introduction

It is well known that the microstructure of materials plays animportant role on the mechanical and physical behaviour of mate-rials. Thus, the characterisation of microstructure has been wellstudied over the past several decades. Particularly, the mechanicalproperties of metals and alloy system are adversely affected by thesize of grains [1–3]. For this reason, controlling the grain size is animportant goal for materials scientists and materials designers [4].Second phase particles strongly affect the grain boundary mobility,and thus control grain growth in polycrystalline materials [5]. Tillrecently, the major attention has been focused on the effect ofimmobile second phase particles for controlling the grain growth.However, it has been known that particles may also be mobile[6]. There have been very few studies on the influence of mobilesecond phase particles on the grain growth. While the theory forthe influence of solute atoms on grain boundary motion is almosthalf a century old [7], more recently Gottstein and Shvindlerman[8] and Verhasselt et al. [9] have studied the effect of mobile sec-ond phase particles on a single grain boundary. However, sincepolycrystalline materials are far more complex, there has been in-creased interest in computational approaches to this problem. A

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few studies of this problem have employed the Monte Carlo meth-od [10] and cellular automata method [11].

Recently, the use of phase-field theories for the study of graingrowth has been gaining popularity due to several advantages overthe other numerical approaches [12–14]. In this approach, grainorientations are assumed to take a finite set of discrete valuesand each orientation is represented by a unique order parameter.The total energy of the system is then written as a sum of the freeenergy of individual grains, and the grain boundary energy repre-sented by gradients of the order parameters. Ginzburg–Landautype kinetic equations are then employed to obtain a set of partialdifferential equations governing the evolution of the order param-eters. To study the effect of second phase immobile particles ongrain growth, Moelans et al. [15,16] modified the free energy termby considering an additional phase-field variable for describing allthe particles. The same form of free energy has been used by Suwaet al. [17] for three dimensional study of polycrystalline graingrowth in presence of immobile particles. In connection with thestudy of grain growth using phase-field method, till date, therehas been no work dealing with the polycrystalline grain growthin presence of second phase mobile particles.

The main objective of this paper is to employ a phase-field the-ory for studying the influence of monodisperse mobile secondphase particles on polycrystalline grain growth in a two-dimen-sional setting. In that context, the interaction between grain bound-ary and a dense distribution of mobile particles in a bicrystal system

Fig. 1. Representation of free energy density of one order parameter (a) outside ofthe particle and (b) inside of the particle.

28 A. Mallick / Computational Materials Science 67 (2013) 27–34

was studied first. In contrast to Moelans et al. [15,16], a unique or-der parameter was introduced in the evolution equation for eachparticle. This proves to be more realistic approach when particlesare mobile in order to avoid interpenetration of two or more parti-cles. A smart algorithm called active parameter tracking algorithmproposed by Vedantam and Patnaik [18] was employed to over-come the computational burden for the consideration of large num-ber of grains and particles. The velocity of the mobile particles istaken to be proportional to the driving force on the particle whicharises from the curvature of the grain boundary (GB). When thedriving force of GB is small then particles move with the GB andslowing it down. On the other hand, when the driving force is largethen the GBs have a tendency to detach from the particles and movewith a higher velocity. The details of the interaction between the GBand the particles have been discussed in our earlier publication[19]. The effects of particle mobility, the area fraction of particles,and the size of particles on the grain growth process are analysedsystematically. The results of grain growth in presence of secondphase mobile particles are compared with the grain growth of sin-gle-phase polycrystalline materials.

2. Model description

We begin with continuous field variables g1(x, t), g2(x, t), . . . ,gQ(-x, t) (referred to as order parameters) which are the functions ofmaterial points x and time t. The order parameters represent thevolume fraction of grains and distinguish the different orientationof the grain matrix. In the phase-field method, grain growth in apolycrystalline material is described by the continuous evolutionof order parameters. The evolution of the order parameters is spec-ified by the time dependent Ginzburg–Landau equations for eachof the Q order parameters

@gi

@t¼ �Li

dFdgi

; i ¼ 1; 2; . . . ; Q ; ð1Þ

where Li are mobility coefficients and F is the total free energy func-tional whose reduction causes the possibility of microstructuralevolution. The total free energy is chosen in the same form of Fanand Chen [20] with additional phase-field variables gk describingeach particle [19],

F ¼Z

V

XQ

i¼1

�12ag2

i þ14

bg4i

� �þ cXQ

i¼1

XQ

j¼1

g2i g

2j þ e

XQ

i¼1

XQþp

k¼Qþ1

g2i g

2k þ

XQ

i¼1

ji

2jrgij

2

( )dV :

ð2Þ

The first three terms are local free energy density inside of grainmatrix, while the fourth term is a free energy density for particleswith a constant size and shape. The gradient term of the Eq. (2) isan interfacial energy of phase-field variables which penalises sharpinterfaces between neighbouring grains. The gradient energy coef-ficients, ji are positive constants and the values are same in an iso-tropic setting. The coefficients a, b, c, and e are phenomenologicalparameters and are chosen to be a positive constants. It is to benoted that Moelans et al. [15,16] proposed a similar form of freeenergy for immobile particles. However, they include only oneadditional phase-field variable to represent all the particles, astheir particles were immobile and do not evolve with time. In con-trast to their model, the present model is based on the presence ofmobile particles. Hence, it is convenient to use separate phase-fieldvariable for each particle, as the particles are continuously chang-ing their position with time. The concept of choosing separatephase-field variables will help to track each particle and avoidinterpenetration between two neighbouring particles. Thus, gi,i = 1, . . . ,Q represent the grains, and gk, k = Q + 1, . . . ,Q + p representp number of particles. Inside the grain matrix, the local free energydensity has 2Q minima with an equal potential well depth at

(g1,g2, . . . ,gQ) = (±1,0, . . . ,0), (0,±1, . . . ,0), . . .. , (0,0, . . .±1) represent-ing 2Q possible orientations of the grains. However, inside the par-ticles, the local free energy has one minimum equal to zero for allgi. The gradient term is non-zero only at grain boundaries and atthe interface of the particle matrix. The value of order parametersgi, inside a particle is equal to zero. Fig. 1 represents the shape offree energy density outside and inside of particles for one phasefield variable.

Substitution of the total free energy expression (2) into the evo-lution Eq. (1) yields the following set of Q coupled non-linear par-tial differential equation for the evolution of gi.

@gi

@t¼ �Li �agi þ bg3

i þ 2cgi

XQ

j–1

g2j þ 2egi

XQþp

k>Qþ1

g2k � jir2gi:

!;

i ¼ 1;2; . . . ;Q : ð3Þ

For the evaluation of grain growth kinetics, the Eq. (3) is solvednumerically by discretizing in space and time. The left term of Eq.(3) is discretized explicitly by Euler’s first forward differencescheme while the gradient term on the right hand side is discret-ized using a central difference scheme. For the simulations, an ac-tive parameter tracking (APT) algorithm has been employedwithout restricting the number of grains. The advantage of thisalgorithm is the reduction of simulation time without affecting

A. Mallick / Computational Materials Science 67 (2013) 27–34 29

the simulation results. That is made possible, because in this algo-rithm, at every time step only nonzero (active) order parametersare evolved in each grid point. While, in the classical algorithm,all order parameters are involved at every grid points in the evolu-tion process. It is to be noted that a small number of non-zero orderparameters are present at any given grid point. Hence, instead ofsolving a large number of equations at any given grid point, a lim-ited number of equations are to be solved in APT algorithm. The de-tails of the simulation procedure and the algorithm have beendiscussed in our earlier publications [14,19].

Ashby and Gentamore [6] experimentally observed that parti-cles were dragged by the grain boundaries (GBs). Hence, it can con-stitutively be assumed that the driving force for the motion of theparticles arises from the grain boundary which can be proportionalto the curvature of the grain boundary

vp ¼ mpcbK; ð4Þ

where mp is the mobility of the particle, cb is the grain boundary en-ergy per unit area, and K is curvature of the grain boundary. It is tobe noted that particle mobility strongly depends on the size andshape of the particle, and the transportation mechanism [9]. Whenparticles are mobile, the following expression is given for the solidinclusion with bulk diffusion in the matrix as the dominant mech-anism of atomic transport:

mp ¼DvolX

kbTr3 ; ð5Þ

with Dvol is the coefficient of bulk diffusion in the matrix, X is theatomic volume, kb is the Boltzmann constant, T is the absolute tem-perature, and r is the radius of particle. The above expression sug-gests that the mobility of particles is adversely affected with theincrease of particle size. In contrast to the motion of a particle,the velocity of the curvature driven grain boundary is given by:

vgb ¼ mbcbK; ð6Þ

where, mb is the mobility of grain boundary. The interaction be-tween particle and the GB depends on the relative magnitudes ofthe particle mobility and the grain boundary mobility. Particlescan move only when they lie on the grain boundary. If particlemobility (mp) is higher than grain boundary mobility (mb) thenthe grain boundary will not detach from the particle. In that case,the grain boundary and particle will move together. On the otherside, if the particle mobility (mp) is smaller than grain boundarymobility (mb) then first the GB will be pinned by the particle, andat later stages may detach from the particle.

Considering normal displacement associated with each seg-ment of the grain boundary, the motion of the k-th particle maybe expressed explicitly in terms of order parameters by the follow-ing constitutive equation [19]:

drk

dt¼ M

r3p

giyyg2

ix� 2gixgiygixy

þ gixxg2

ix

g2ixþ g2

iy

� �3=2

8><>:

9>=>;; ð7Þ

where rk is the position vector of the centre of the k-th particle andM ¼ DvolXcb=kbT is the mobility coefficient.

The values of gi and the new location of particles obtained fromEqs. (3) and (7) respectively which are updated in each time stepexplicitly. Particles interpenetrations are prevented by employinga repulsive force between particle i and j as,

f ij ¼C

jri � rjj2nij; with i–j: ð8Þ

where C is a constant and nij ¼ ðri � rjÞ=jri � rjj is a unit vector in thedirection of line joining the centres of two particles. This force is

insignificant when the particles are apart but repels overlappingparticles.

3. Simulation parameters and procedure

For isotropic grain growth, we use the value of gradient energycoefficient ji = j = 2 and kinetic energy coefficient (related withmobility) Li = L = 1 for all i. The lattice step size and time step arechosen to be, respectively, Dx = 2.0, Dt = 0.25. Periodic boundaryconditions are applied in both the x- and y- directions.

Based on the Zener relationship [5], for a given area fraction thefinal average grain size, Rlim is proportional to the particle size. Re-cently, Chang and his co-researchers [21] have reported a system-atic analysis of the effect of second phase particle size on the graingrowth. In their analysis, it has been shown that small secondphase particles are more efficient than large particles to controlthe grain growth for a given volume fraction and morphology ofparticles. However, this relationship is imparted only for theimmobile particles. For mobile second phase particles, final aver-age grain size after growth process will not only depend on the sizeof particles, but also depend on the mobility of the particles for agiven area fraction of particles. To avoid an unrealistic interactionbetween grain boundary and particles, a certain area of minimumparticle size has been chosen. In the present simulation, the parti-cle sizes of radius 2.5, 3.0, 3.5 and 4.0 grid points are taken. Fig. 2represents the discretized shape of particles for radius, rp = 2.5, 3,and 4 respectively. The special phase-field variables, gk = 1 insideof the discretized grid points (gp) of particles, and gk = 0 outsideof the particle’s gp.

An active parameter threshold s = 10�6 has been chosen andperiodic boundary conditions are applied in the entire simulationprocess. For the visualisation of microstructure, we define afunction

wðx; tÞ ¼XjRji¼1

g2niðx; tÞ; ðni;gni

Þ 2 R; ð9Þ

whose values are 1.0 inside of the grain matrix representing whiteand significantly smaller positive values across the grain boundariesdisplayed as dark gray-level. The value of w(x, t) = 0, indicating par-ticles appeared as dark black spot. The sum is taken over the neigh-bouring active parameters. The area fractions of particles weretaken from 1% to 15% of the total area of grains present. In the pres-ent simulation with the limitation of memory requirements of com-puter, we consider domain sizes of 256 � 256 (for bicrystal)512 � 512 and 1012 � 1012 grid points.

4. Results and discussion

In this section, we present our simulation result for two-dimen-sional microstructural evolution of grain growth in presence of sec-ond phase mobile particles. Also, we present few simulation resultof single-phase grain growth to examine the effect of mobile andimmobile particles in the microstructural evolution process. Forease of simulation, the parameters a = b = c/2 = 1 have been chosenfollowing Vedantam and Patnaik [18]. While, following Moelanset al. [16] the parameter e = 1 has been taken for extra fieldvariables.

Fig. 3 represents one quarter of a circular grain embedded byanother grain at different time step. Initially mobile particles (6%area fraction) are randomly distributed inside and outside the cir-cular grain. The simulation domain is 256 � 256 grid points. Theradius of particles, rp = 1.5 (grid units), the mobility coefficient ofparticles, M = 100, and the initial radius of the circular grain,R = 180 (grid units) are taken in the simulation. The movement ofthe grain boundary is due to the force arising from its curvature to-

Fig. 2. Discretisation of particles. The radius of particles are rp = 2.5, 3.0 and 4.0 from left to right.

Fig. 3. Grain shrinkage of bicrystal evolution in presence of 6% vf mobile particles with M = 100 and rp = 1.5. (a) Initial microstructure, (b) after 4000 ts and (c) after 8000 ts.

30 A. Mallick / Computational Materials Science 67 (2013) 27–34

wards the centre of curvature. Hence, a circular grain is expected toshrink and the other will grow. It has been observed that as simu-lation time increases, the grain boundary drags particles towardsthe centre of circular grain. Due to the dragging of particles bythe grain boundary, a clear particle free band behind the grainboundary was observed. This dragging of particles by grain bound-ary is quite natural if the particles are mobile, and is comparablewith the experimental evidence of Ashby and Gentamore [6]. Itcan also be noted that particles will move along with the grainboundaries, and can never be unpinned as long as the mobility ofparticles is greater than the migration rate of the grain boundaries[22].

Contours of w(x, t) are plotted in Fig. 4 for particle free evolution(first row), evolution in presence of mobile particles (second row)and evolution in presence of immobile particles (third row) at sim-ulation time steps 500 (initial microstructure), 12,000 and 24,000.It is to be noted that simulation starts from initial sub-cooled liquidstate using random values for unique order parameters at everygrid point. Until the initial grain formation, the growth kinetics isgoverned by the basic algorithm. For sake of convenience, it is as-sumed that the formation of grains was initiated after 500 timesteps. Particles are represented by black spots, while grains areshown bright. The grain boundaries are indicated by gray lines. To-tal number of particles, p = 100, the mobility coefficient of parti-cles, M = 50, and the size of each particle, rp = 3.0 (grid units)were chosen in the simulation. It is apparent that almost all theparticles (mobile and immobile) are located on the grain bound-aries. This situation suggests that second phase particles having atendency either to move with the grain boundaries or to pin thegrain boundaries. Furthermore, it can be seen that there are a greatproportion of smaller and larger grains when the evolution is inpresence of particles. The crowd of small grains are located wheremost of the particles are present. This phenomenon is more pro-nounced when the particles are immobile. It implies that grainboundaries are arrested by the particles, while boundaries move

freely when away from particles. In contrast, when particles aremobile then particles move along with the grain boundaries duringthe evolution of grain growth. In this situation, the grain growth islikely to be uniform in comparison with that of immobile case. Mo-bile particles have a tendency of congregation. From the observa-tion of contours, it can be seen that the particle free graingrowth process (first row) exhibits a larger average grain size atevery time step as compare to the grain growth in presence of par-ticles. The presence of particles adversely affects the grain growthprocess and it is more pronounced when particles are immobile.

The time dependent average diameter of the grains have beencalculated here using the standard mathematical formula,D ¼

ffiffiffiffiffiffiffiffiffiffiffiffi4A=p

p, where A is the time dependent average area of grains

calculated from the total area of the domain divided by the numberof grains [23]. The area of the GB is negligible compared to the areaof grain matrix and is hence ignored in the calculation of averagearea A. In the simulations, it was assumed that initial microstruc-ture formed after 500 ts and particles are nucleated. The effectsof particle mobility on the kinetics of grain growth in presence ofmobile particles are depicted in Fig. 5. The simulation domain sizewas considered to be 512 � 512 grid points. In order to make samearea fraction of the mobile particles, the number of particles andtheir sizes were taken to be the same. It can be observed that thegrain growth is increased with the increase of particle mobilityfor an identical number and size of particles. Fig. 6 illustrates theeffect of area fraction of particles in the grain growth processes.The size and the mobility coefficient of particles are identical. Assecond phase particles restrict the grain boundary motion, thegrain growth is adversely affected by the increase of area fractionof particles. Thus, for the same simulation time, the average diam-eter of the grains decreases with increasing area fraction of secondphase particles. This simulation results are consistent with manytheoretical and experimental observations reported by many dis-tinguished researchers [16,24,25]. In the present simulation, wecould not find the limiting average grain diameter where the grain

MP

IM

NP

(a) (b) (c)Fig. 4. Simulated result for two-dimensional grain growth of a particle free (NP), containing mobile particles (MPs) and immobile particles (IMs) in a polycrystalline system of512 � 512 gp domain. (a) Initial microstructure, (b) after 10,000 ts and (c) 24,000 ts.

Fig. 5. The effect of particle mobility on the average diameter of grain as a functionof simulation time. The number of particles (p = 100) and their sizes (rp = 3.0 gp) aresame for all cases. The results compared with particle free case and in presence ofimmobile particles.

Fig. 6. Effect of area fraction of particle on the grain growth as a function of time.The size (rp = 3.0) and mobility coefficient (M = 50) of particles are same for allcases.

A. Mallick / Computational Materials Science 67 (2013) 27–34 31

growth processes are almost stopped. However, the situation ofarresting of grain growth is possible if the mobility of particles isvery low or particles are totally immobile [19]. Fig. 7 representsthe effect of particle size on the average grain diameter at differenttime steps. The area fraction of mobile particles, fa = 1%, and the

mobility coefficient of particles, M = 50 are considered to be samefor all the cases. The results are also compared with the particlefree grain growth process. It has been observed that the presenceof particles restricts the grain growth process. However, for thesame area fraction of particles, the changes in particle size cannot

Fig. 7. Effect of particle size on the grain growth as a function of time. The volumefraction (fa = 1.0%) and mobility coefficient (M = 50) of the particles are same foreach cases.

Fig. 8. Linearisation of grain growth as a function of simulation time for differentarea fraction of particles. The particle radius (rp = 3.0) and mobility coefficient(M = 50) are same.

32 A. Mallick / Computational Materials Science 67 (2013) 27–34

predict any clear conclusion on the effect of grain growth. More orless same average diameters of the grains were observed whensimulation time, N = 15,000 ts. At time step, N = 30,000 ts, the max-imum average grain diameter was observed when particle size,rp = 3.5, and followed by rp = 4.0, 2.5 and 3.0. However, when par-ticle size, rp = 4.0 (i.e. larger particle), the grain growth increasesrapidly after time step, N = 30,000 ts, and attains the maximumaverage grain diameter at N = 40,000 ts. The relationship betweenthe number of pinning particles per unit volume (p), volume frac-tion of the particles (fv), and the diameter of pinning particles (d)can be expressed as p / fv=dn [26]. Thus, the decrease in particlesize for a given volume fraction or area fraction means increasingthe number of pining particles. The particles are assumed to beon the grain boundaries and strongly pinning them. As a result,the grain growth processes are adversely affected with increasingthe number of particle in a given area fraction, as long as the par-ticles are not detached from the grain boundaries. However, thisagreement is applicable for immobile particles only. The mobileparticles move along with the grain boundaries instead of pinningthe grain boundaries. The mobile particles have a tendency to re-duce the grain boundary mobility. The effective grain boundarymobility depends on the size as well as the number of particles.The effective grain boundary mobility (Meff), grain boundary mobil-ity (MB), particle mobility (mp) and the number of mobile particles(np) can be expressed as, Meff ¼ MB=ð1þ npMB=mpÞ [27]. This rela-tionship suggests that the effective grain boundary mobility willdecrease if the number of mobile particles is increased. As a result,the driving force for the grain growth process will be reduced. Inthis situation, the overall grain growth should be affected ad-versely. This can be explained in a more general way that the in-crease of number of particles means increase the number ofparticle-boundary interactions which cause a decline in the graingrowth. For the same volume fraction, the number of particles isincreased at the expense of particle size. From Eqs. (5) and (7),one can predict that the particle mobility should enhance withthe reduction of particle size (mp / M=d3). The increase in particlemobility increases the effective grain boundary mobility. Hence,the reduction of particle size for a given area fraction can improvethe grain growth process. The above discussions on the kinetics ofgrain growth in presence of mobile particles can be summarisedas: (a) for a given volume fraction, the grain growth can be de-clined by the reduction of particle size due to increase in numberof particles, and (b) on the other hand, growth process may alsobe enhanced by the reduction of particle size due to increase in

particle mobility. Hence, it is hard to predict the effect of changesin particle size or the number of particles on the controlling ofgrain growth for a given area fraction (fa) and mobility coefficient(M) of the particles. The control of grain growth in presence of mo-bile particles is a competition between the size and the number ofparticles. It is interesting to note that this phenomenon is in vari-ance with the case of immobiles particles as observed by Changet al. [21]. In their observation, smaller immobile particles aremore efficient to control the grain growth as compare to largerimmobile particles for same area fraction and morphology. Thekinetics of grain growth of the average grain diameter is given bya simple power law, Dn � Dn

o . In the presence of mobile particles,Novikov [28] approximated the grain growth exponent, n = 3.0.Thus, the expression of grain growth in the presence of mobile par-ticles can be expressed as:

D3 � D3o ¼ kt; ð10Þ

where Do corresponds to average initial diameter of grains, k is theproportionality constant, and t is the time. Fig. 8 shows a plot ofhD3 � D3

oi versus simulation time associated with the plot inFig. 6. Almost linear curves are observed in all cases. This is anagreement with the proposed approximation of grain growth expo-nent, n = 3, when growth is affected by the mobile particles [28].However, a distinct deviation in the curve (designated by solidsquares) is associated when larger area fraction of particles(fa = 6%) is present. This discrepancy is most likely due to the con-tinuous shape distortion of grains during grain growth with higharea fraction of particles.

To show the accuracy of the present simulation with the refer-enced work [16], the grain growth results in presence of immobileparticles are represented in Fig. 9. The plot shows the ratio of aver-age arrested grain radius (Rlim) to particle radius (rp) as a functionof area fraction (fa) of monodisperse second phase immobile parti-cles. Keeping in mind the memory restriction and time consumedfor simulation, the system size is restricted to 1024 � 1024 gridpoints. For a large number of grains present in the system, the ef-fect of grain boundaries is assumed to be negligible. From the Ze-ner relationship of pinning pressure offered by the second phaseparticles, the mean final grain size Rlim can be expressed byRlim ¼ br=f b

v , where fv is volume fraction, r is the radius of particle,b and b are coefficients. The value of b = 4/3 and b = 1 gives for themaximum critical grain size in three-dimensional setting. How-ever, much controversy exists for the accurate description of thevalues of coefficients, and thus many attempts were made to im-

Fig. 9. Variation of normalised limiting radius as a function of area fraction ofparticles.

Fig. 10. Comparison of normalised grain size distribution in different system ofevolution processes.

A. Mallick / Computational Materials Science 67 (2013) 27–34 33

prove the Zener relationship [25,29,30]. In spite of this, the valuesof coefficient depend on the particle volume fraction as well as dis-tribution of particles. For high volume fraction of particles, most ofthe particles are located on the grain boundaries, while for low vol-ume fractions, particles are located randomly on grain boundary aswell as inside the grain matrix. The value of b = 1.8 and b = 0.33 forhigh volume fraction and b = 0.222 and b = 0.93 for low volumefraction was proposed in Ref. [31]. The Monte Carlo (MC) two-dimensional simulation results predicts the coefficients, b = 1.73and b = 0.5 [32] or b = 1.2 and b = 0.55 [26]. The front tracking ver-tex 2D model gives b = 0.5 [33] and b = 0.46 [34]. For low area frac-tion of particles, the value of b = 1.3 was estimated by Chen et al.[35] with an assumption of b = 0.5. Very recently, in two-dimen-sional phase-field simulation, when all particles are located onthe grain boundaries, the relationship between mean limiting ra-dius (Rlim), particle radius (r) and area fraction (fa) of particle hasbeen predicted by Moelans et al. [16] as Rlim ¼ 1:28r=f 0:5

a . Basedon the present simulation, the linear fit of log Rlim=r versus log(1/fa) plot (not shown here) predict the value of coefficients b = 1.18and b = 0.48 which are in very good agreement with results givenin Refs. [16,33–35].

The average grain size is an important scaling factor for the sta-tistics of the evolution of grains at different simulation time steps.However, the distribution of grain sizes in a particular time stepcan give us the information about the nature of size distributionfor different mechanisms of grain growth. Thus, the temporal evo-lution of normalised grain size distribution of particle free grainsand in presence of second phase mobile and immobile particlesare presented in Fig. 10. The simulation domain size was512 � 512 grid points, and the time steps, N = 20,000 were consid-ered. It can be seen that the peak of normalised grain size distribu-tion shifts towards the smaller grain size when particles arepresent, and it is more pronounced for immobile particles. Thissuggests that there are a greater proportion of smaller and largergrains when particles are present. In case of particle free graingrowth process, smaller grains gradually disappear as the neigh-bouring grains can easily grow. However, second-phase particleshelp to prevent the disappearance of smaller grains and this affectsthe grain size distribution. The present result of size distribution isin good agreement with the results presented in Refs. [16,21,34–38]. However, it should be kept in mind that there are some impor-tant limitations of 2D simulation of grain growth over 3D simula-tion. In some respects, 2D simulations in polycrystalline systemsdeviate from experimental findings of bulk polycrystalline sam-

ples. In 2D simulation, nearly all particles are inhabited on thegrain boundaries. Thus, all particles effectively contribute to con-trol the migration of grain boundaries. However, in real cases manyparticles are located inside the grain matrix. Furthermore, in 2Dsystems, when particles interact with the grain boundaries, thegrain boundaries have a tendency to become straight (as grainboundary represented by lines). Thus, the particles have limitedtendency to detach from the grain boundaries. Hence, in general,2D simulations overestimate the rate of control of grain growth.The above limitations can be overcome by 3D simulations. How-ever, in 3D simulations, the required number of grid points shouldbe proportional to the third power of the 2D system size which ismuch more computationally expensive. Moelans et al. [16] re-ported that 105 times larger number of grid points is required toobtain a reliable value of the grain size, if the simulations are per-formed in 3D.

5. Summary and concluding remarks

A phase-field model for 2D polycrystalline grain growth in pres-ence of mobile particles has been developed successfully. The evo-lution equation for the grain growth has also been modified byconsidering additional phase-field variables for each mobile parti-cle. The ability of APT algorithm reduces the computer require-ment and successfully enables us to consider a large number ofphase-field variables in the simulation process. The distributionof particles in the simulation was described by spatially dependentparameters. The effects of particle size, area fraction of particlesand particle mobilities in the grain growth processes are analysed.The increase of the area fraction of particles retarded grain growthwhile increase of mobility accelerates growth process. The attrac-tion force between particles and grain boundaries is proportionalto the size of particles. The growth kinetics affected by mobile par-ticles are approximated by a power law: hRn � Rn

oi ¼ kt, where thevalue of growth exponent, n is greater than 2.0 (for ideal isotropicGB energy with particle free case, n = 2.0 [25]). The final meangrain size in presence of immobile particles (all particles are as-sumed to be on the grain boundary) is expressed by Zener typerelation Rlim ¼ br=f b

a with coefficients b = 1.18 and b = 0.48 whichgives a very good agreement with the literature results. The inter-action between particles and grain boundary affects the distribu-tion of normalised grain size, and the peak shifts toward thesmaller grain size. This is due to the effect of second phase particleslying on the grain boundaries due to which the smallest grains can-not disappear in the growth process. This effect is much more pro-

34 A. Mallick / Computational Materials Science 67 (2013) 27–34

nounced in case of immobile particles than that of mobile particles.For a given area fraction and morphology, the smaller inert parti-cles are more efficient than larger particles for controlling the graingrowth. However, this prediction is not always true when particlesare mobile. The model predicts the final grain size of a polycrystal-line system which depends on the nature and area fraction of par-ticles present. The use of the APT algorithm in the presentsimulation allows a large number of phase-field variables whichhelps us to consider a large number of mobile particles in the sys-tem. This algorithm also reduces the simulation time drastically ascompared to the classical algorithm.

Acknowledgement

The author is grateful to Dr. Srikanth Vedantam, Department ofEngineering Design, Indian Institute of Technology, Madras, Indiafor encouragement, support, stimulating discussion and interestin this work.

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