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Grain boundary curvature and graingrowth kinetics with particle pinningSina Shahandeh a & Matthias Militzer aa The Centre for Metallurgical Process Engineering , The Universityof British Columbia , Vancouver , BC , V6T 1Z4 , CanadaPublished online: 14 Jun 2013.

To cite this article: Sina Shahandeh & Matthias Militzer (2013) Grain boundary curvature andgrain growth kinetics with particle pinning, Philosophical Magazine, 93:24, 3231-3247, DOI:10.1080/14786435.2013.805277

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Grain boundary curvature and grain growth kinetics with particlepinning

Sina Shahandeh* and Matthias Militzer

The Centre for Metallurgical Process Engineering, The University of British Columbia,Vancouver, BC V6T 1Z4, Canada

(Received 25 February 2013; final version received 7 May 2013)

Second-phase particles are used extensively in design of polycrystalline mate-rials to control the grain size. According to Zener’s theory, a distribution ofparticles creates a pinning pressure on a moving grain boundary. As a result,a limiting grain size is observed, but the effect of pinning on the detail ofgrain growth kinetics is less known. The influence of the particles on themicrostructure occurs in multiple length scales, established by particle radiusand the grain size. In this article, we use a meso-scale phase-field model thatsimulates grain growth in the presence of a uniform pinning pressure. Thecurvature of the grain boundary network is measured to determine the drivingpressure of grain growth in 2D and 3D systems. It was observed that the graingrowth continues, even under conditions where the average driving pressure issmaller than the pinning pressure. The limiting grain size is reached when themaximum of driving pressure distribution in the structure is equal to thepinning pressure. This results in a limiting grain size, larger than the onepredicted by conventional models, and further analysis shows consistency withexperimental observations. A physical model is proposed for the kinetics ofgrain growth using parameters based on the curvature analysis of the grainboundaries. This model can describe the simulated grain growth kinetics.

Keywords: grain growth; microstructure change; computer simulation

1. Introduction

In polycrystalline materials, second-phase particles impede motion of grain boundaries.This phenomenon has been utilized in important technological applications in whichgrain size determines physical and mechanical properties of the materials. Particlepinning is a multi-scale problem, since the interaction of grain boundaries with theparticles occurs on the scale of the particle diameter (1–100 nm). On the other hand,evolution of the grain structure has the length scale of the grain size (1–100 μm). Theseminal model by Zener and Smith [1] takes this fact into account by introducing theparticle pinning pressure opposing the driving pressure of the grain boundary curvature.The model can be used to predict the limiting grain size for a given particle size andfraction.

*Corresponding author. Email: sina@shahandeh.com

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The pinning pressure can be obtained using a variety of geometrical assumptions[2]. In conventional models, a uniform distribution of uni-size particles is assumed,resulting in a constant pinning pressure [3–5]. The limiting grain size in these theoriesis then obtained by equating the constant pinning pressure with the driving pressure ofgrain growth using mean-field assumptions [5,6].

Although there is an extensive body of research on ideal grain growth, the kineticsand topological aspects of grain growth under the influence of drag and pinning pres-sures has received less attention. Initially, Hillert [7] introduced the pinning pressure ofparticles into a kinetic relationship for growth of a grain surrounded by grains of a rep-resentative mean radius and obtained a kinetic relationship assuming a self-similar grainsize distribution. Later, Hunderi and Ryum [8] used a numerical technique to model theevolution of the grain size distribution as it approaches the limiting grain size. Alterna-tively, Andersen et al. [9] proposed a phenomenological relationship between averagegrain size and the kinetics of grain growth. This kinetic model is frequently used to fitexperimental results [10,11]. The aforementioned models, however, have always somedegrees of freedom to be adjusted by the experimental data and do not capture com-plexities of grain growth under influence of pinning. In addition, it is difficult to findthe pinning pressure for a given distribution of particles and this yields potentially inac-curate adjustment of pinning pressure to fit the kinetics data.

Particle pinning has been simulated with a variety of computational tools such asthe finite element method [12], the vertex method [13] in 2D, Monte Carlo models in2D [14,15] and 3D [16]. Grain growth with particle pinning has also been simulatedwith phase-field models in 2D [17,18] and 3D [19]. In all of these simulations, particlesand grains are discretized on the same numerical grid. Therefore, these approaches havelimitations to study the role of very fine particles or small particle volume fractions. Tosolve this problem, one can model particle pinning using a dual-scale methodology. Inthis approach, the pinning pressure is obtained for a given particle size distributioneither from an analytical [2] or microstructural evolution model [20,21] at the smallscale. Then, the pinning pressure is implemented in the meso-scale model to modify thevelocity of grain boundaries without resolving the particles. Apel et al. [22] introducedthe pinning pressure using an effective mobility that depends on the driving pressuresuch that it becomes zero when the driving pressure is equal or lower than the pinningpressure. Grain growth simulations in 2D result in a limiting grain size and abnormalgrain growth depending on the pinning pressure [23,24]. In a previous work [24], wehave developed a phase-field model that implements pinning pressure as a back stresson the grain boundaries. The local energy density of grains at the grain boundary isshifted in order to create an opposite pressure on the moving interface. This methodol-ogy takes into account the pinning pressure directly and can also be extended to avelocity dependent drag pressure.

Although curvature of grain boundaries is the driving force for grain growth, mostanalytical and experimental studies focus only on the grain size. One reason is the factthat measuring grain boundary curvature is challenging both experimentally and inmathematical models. For instance, indirect measurements of curvature have been pro-posed using stereological relations [25–27]. Interface curvature has also been calculatedfor discrete data-sets using tessellation [28] and for structures with diffuse interfacesusing differential calculus [29]. Measuring curvature of the grain boundary network inour 2D phase-field simulations showed that at the pinned grain structure, the average

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driving pressure is smaller than the pinning pressure [24]. Therefore, a larger limitinggrain size was predicted compared to conventional theories.

Evolution of grain boundary curvature and its application to describe the kinetics ofgrain growth with pinning is presented in this work. We use a phase-field method tosimulate the evolution of grain boundary networks during grain growth with and with-out pinning pressure. The pinning pressure is introduced as a phenomenological frictionpressure that opposes the motion of grain boundaries uniformly across the structure.Curvature of the grain boundary network is analysed directly from the diffuse grainboundary profile to obtain the driving pressure distribution of the system. We first pres-ent the results of both 2D and 3D ideal grain-growth simulations to obtain a relation-ship between the average driving pressure and grain growth rate. Then, 3D simulationsin the presence of particle pinning pressure are performed and grain boundary curvaturein 2D and 3D systems is analysed. Based on this analysis, a physical model is proposedfor the kinetics of grain growth with pinning. We conclude by comparing computersimulation results with the available experimental data.

2. Simulation methodology

2.1. Phase field modelling

We use a multi-phase-field model that was originally proposed by Chen and Yang [30].This model has been extensively used for modelling of ideal grain growth [30–34] andgrain growth in the presence of particle pinning [17,19,35]. In a multi-phase-fieldmodel, each grain is assigned a field variable, called order parameter, gi that has a valueof 1 inside the grain i and 0 outside. The order parameters transition smoothly from 1to 0 at the grain boundaries with a known profile [36]. In this model, grain boundarymovement is related to the rate of change of order parameters at each point (@gi=@t).Grain boundary velocity v matches the phenomenological relation v ¼ MrK, where Mand r are the grain boundary mobility and energy, respectively, and K is the grainboundary curvature.

The pinning pressure can be included in the phase-field model as a driving pressureopposing the motion of the interface. In this approach, the pinning pressure Pz isapplied depending on the direction of the grain boundary motion. If the driving pressureis smaller than the pinning pressure, the applied “friction” pressure is equal to the driv-ing pressure, causing the grain boundary to stop. The detailed derivation of the phase-field equation with a friction pressure has been presented in our previous work [24].The evolution of the grain structure is obtained by solving a system of phase-field dif-ferential equations, given by:

@gi@t

¼ �L f 0i � P0z

� �; ð1Þ

where L is a kinetic constant f 0i and P0z are terms that distribute the driving and pinning

pressure across the diffuse grain boundary:

f 0i ¼ m g3i � gi þ 3giXpj 6¼i

g2i

!� jr2gi ð2Þ

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P0z ¼

3gið1� giÞsgnðf 0i ÞPz j f 0i j[j3gið1� giÞPzjf 0i j f 0i j � j3gið1� giÞPzj

�ð3Þ

Here, m and j are constants that together with L determine the grain boundary energy,thickness and mobility such that one can quantitatively relate physical properties of agiven material to the phase-field parameters [36]. In this model, all grain boundarieshave the same grain boundary energy and mobility, i.e. an isotropic system is consid-ered. Equation (3) determines whether the local driving pressure f 0 is larger or smallerthan the local pinning pressure and adjusts the opposing pressure accordingly.

The simulations were preformed in 2D and 3D domains using periodic boundarycondition. Equation (1) was solved using a forward Euler finite-difference method. Eachgrain has a unique order parameter. Since each simulation starts with thousands ofgrains, a sparse data structure was used to only store order parameters with a non-zerovalue at each grid point. Calculations were performed only for those active orderparameters. In 2D, 5 active order parameters and in 3D, 8 order parameters were used.This provides sufficient order parameters to accurately calculate topological changes inthe grain boundary network even up to a rare merger of two quadruple points. Higherorder finite-differencing stencils [37] were used to minimize anisotropic effects causedby the numerical grid. The simulation parameters are grid spacing of Dx ¼ 1, time stepof Dt ¼ 0:1, L ¼ 1, m ¼ 2 and j ¼ 3 for 2D and Dx ¼ 1, Dt ¼ 0:1, L ¼ 1, m ¼ 1 andj ¼ 2 for 3D. This results in physical parameters of M ¼ 2:598; r ¼ 1:155 for 2D andM ¼ 3; r ¼ 0:667 for 3D systems [36].

Initial grain structures were created by populating 1% of the grid points in the struc-ture with nuclei seeds of gi ¼ 1 and solving Equation (1) with Pz ¼ 0 for up to 4000time steps. This microstructural evolution results in a self-similar ideal grain size distri-bution. The resulting grain structure is then used as the initial condition for simulationswith different values of Pz ranging from 0:010 to 0:030.

2.2. Grain boundary curvature

A new technique is used to measure the curvature of grain boundaries in the phase-fieldsimulations. This method creates less noisy results compared to other methods given in[29]. The limitation of this method, however, is that it can only be applied when ananalytical relationship for the diffuse interface profile is known in the curvilinear coor-dinate system.

Consider parallel iso-gi surfaces for a boundary between grain i and j that has asmall thickness compared to the radius of curvature. The normal direction ni is definedas a unit vector normal to an iso-gi surface pointing from inside the grain i to the out-side. At any point on the grain boundary [38]:

r2gi ¼@2gi@n2i

þ @gi@ni

Kgi : ð4Þ

Here ni is the distance along the normal direction and Kgi is a scalar field equal tothe sum of the principal curvatures (K1 þ K2) on an iso-gi surface. Kgi ¼ �Kgj holds

for any point on the grain boundary. For calculating the curvature at each grid point,

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the value of r2gi is determined numerically. The partial derivative terms in Equation(4) are known analytically from the gi profile given by [36]:

gi ¼1

21þ tanh

ffiffiffiffiffiffiffiffiffiffi2jm

ni

r !" #: ð5Þ

For statistical sampling, Kgi is determined for all order parameters at all grid pointsfalling in the range of 0:2\gi\0:8. Then, at each grid point on the grain boundary themagnitude of curvature, K is stored in a list provided that the difference between thecurvature of two order parameters is less than 10%, i.e. jKgi j � jKgj j\0:1jKgi j. This cur-vature list is then used to determine the curvature distribution of the grain boundarynetwork. Since the interface thickness is the same for all grain boundaries, thefrequency of a value in the curvature list is proportional to the length of the grainboundary network having that value. As a result, the arithmetic average of thediscretized data in the curvature list equals to the average curvature (�K) of the grainboundary network. We define the average driving pressure for grain growth in thestructure, �Pd , as the sum of the driving pressures along the length of the entire networkdivided by the total surface of the grain boundaries:

�Pd ¼ r�K ¼ r

RKdSRdS

: ð6Þ

Equation (4), gives inaccurate curvature values close to the junctions where morethan two order parameters have non-zero values. The error is due to the fact that theanalytical profile obtained for the equilibrium boundary profile (Equation (5)) is notvalid at the junctions. Therefore, the value of curvature at junctions is neglected here,as is the effect of triple line tensions on the grain growth kinetics. This assumption isjustified as long as the grain size is much larger than the interface thickness, which isalso a necessary condition for the validity of the phase-field simulations in general.

3. Results and discussion

3.1. Ideal grain growth

We first present the results of ideal grain-growth simulations and measure the associatedcurvature distribution of the grain boundaries. This provides a benchmark for the modeland determines the relationship between grain growth kinetics and the curvaturedistribution. The grain growth process is driven by the curvature of grain boundaries.Figure 1 shows snapshots of grain boundary curvature values for 2D structures. Thelocal curvature of the grain boundary network is not necessarily constant across a givengrain boundary segment. It is influenced not only by the grain size and the number offaces, but also by the configuration of the neighbours of a grain.

Figure 2 shows the distribution of the grain boundary curvature at different times. Aself-similar distribution is observed when K is normalized by �K, i.e. all curves at differ-ent times collapse on a master curve. The average curvature �K is related to the averageequivalent grain radius, �R, using a proportionality constant, a1:

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�K ¼ a1�R: ð7Þ

Here, a1 ¼ 0:20� 0:02 is obtained for the self-similar 2D ideal grain growthstructure.

The kinetic data is analysed using the method of Burke and Turnbull [39]. Kineticsof grain growth is characterized using the average grain size that is proportional to theaverage driving pressure such that:

d�R

dt¼ a2M �Pd ¼ Ma2a1

1�R

ð8Þ

Figure 1. (colour online) Grain boundary network in 2D ideal grain growth simulations. Thevalues of the curvature field are mapped on the structure after (a) 20,000 time steps and (b)50,000 time steps. Please refer to the online version of the figure for the colourbar.

Figure 2. (colour online) The self-similar grain boundary curvature distribution for 2D idealgrain growth simulations.

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where a2 is a proportionality constant related to the nature of the averaging process.Integrating Equation (8) gives the parabolic grain growth relationship:

�R2 � �R20 ¼ 2a1a2Mrt ¼ kMrt: ð9Þ

Here, k ¼ 2a1a2 is introduced as the grain growth rate constant. Figure 3 shows thesquare of the average equivalent grain radius as a function of normalized times ¼ Mrt. The simulation results performed with different values for initial grain size,grain boundary mobility, energy and thickness confirm parabolic grain growth withk ¼ 0:29� 0:02. This result is in good agreement with recent 2D simulations where thefollowing values were obtained: 0:27 with phase field [34], 0:27 with vertex model[40], 0:29 with Monte Carlo simulation [41] and 0:26 with a front-tracking model [42].

The values of a1 and k are measured independently from the grain boundary curva-ture distribution and the grain growth kinetics, respectively. Therefore, according toEquation (9), a2 ¼ 0:69� 0:10 is obtained.

A similar approach is applied to 3D ideal grain growth to obtain the kinetic con-stants. The kinetic constant k ¼ 0:45� 0:04 is determined from the slope of �R2 vs. sfor 3D grain growth simulations. The value of k ¼ 0:45 in this work is comparable withthe results of other studies, e.g. 0:5 [43] and 0:49 [33] for phase-field simulations, 0:43for Monte Carlo simulations [44] and 0:42 for a modified level-set model [45]. Lazaret al. used a new model based on a direct incorporation of the MacPherson–Srolovitzrelation in 3D [46] and obtained k ¼ 0:39, which is smaller than the value obtained inthis work.

Theoretical models require substantial approximations for the geometry of grains inorder to calculate the grain growth constant. Rios and Glicksman suggest k ¼ 0:5 [47]in 2D as a first approximation. They use a topological proxy for grains called “averageN-hedra” [48] to relate topological and metric properties of a grain in the mean fieldtheory. The average N-hedra theory suggests k ¼ 0:23 [49] for 3D. Mullins [50] usedexperimentally obtained data for the size and topological distribution of grains andsuggested k ¼ 0:5 for 3D.

Figure 3. (colour online) Equivalent grain radius as a function of normalized time (s ¼ Mrt) fordifferent simulations in 2D.

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Figure 4(a) shows a grain boundary curvature map superimposed on the structure.The self-similar curvature distribution in 3D, shown in Figure 4(b), resembles the 2Ddistribution. The value of a1 ¼ 0:27� 0:03 is obtained from the 3D curvature distribu-tions using Equations (6) and (7). Consequently, comparing k ¼ 0:45 with the a1 resultsin a2 ¼ 0:81� 0:16 for 3D grain growth.

The value of a1 is related to both geometric factors of grain shape and the statisticalfactors of grain size distribution. To illustrate this fact, assume a large ensemble ofgrains, each with an equivalent radius Ri and a grain boundary area of Si. The averagecurvature of each grain, Ki, can be measured using Equation (6) by integrating overarea of the grain considering a grain as a polyhedra, Si ¼ siR2

i and Ki ¼ ki=Ri where siand ki are solely geometrical constants. For instance, compared to a sphere, si is alwaysbigger than 4p and ki smaller than 2, depending on the number of faces and shape of agrain. Average curvature of the entire network can be written as:

�K ¼P

SiKiPSi

¼P

sikiRiPsiR2

i

ð10Þ

One can use simplifying assumptions for the shape of grains, e.g. a regular polyhe-dra or an average N-hedra as proposed by Glicksman [48], to evaluate si and ki. How-ever, as it is evident on the right-hand side of Equation (10), a statistical factor notedby the ratio

PRi=P

R2i enters the relationship for the average curvature which is a

function of the grain size distribution. Abovementioned analysis allows one to derive a1analytically for the scaling distribution. However, here, we are content to use thenumerical evaluation.

(a) (b)

Figure 4. (colour online) (a) Curvature map of the grain boundary network for ideal graingrowth at tn = 20,000. (b) Distribution of grain boundary curvature at different time steps.

Table 1. Model parameters of ideal grain growth kinetics in Equation (9).

α1 α2 k= 2α1α2

2D 0.20 ± 0.02 0.69 ± 0.10 0.29 ± 0.023D 0.27 ± 0.03 0.81 ± 0.16 0.45 ± 0.04

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Consistency of the results from this study with other simulations is a benchmark forthe accuracy of the model. In addition, the curvature analysis here provides a precisedefinition for the average driving pressure and its relationship with the grain growthkinetics, which is not evident in other studies. The parameters a1 and a2 are constantsapplicable to ideal grain growth in any system, as summarized in Table 1.

3.2. Grain growth with particle pinning pressure

3.2.1. Limiting grain size

In this study, the pinning pressure is applied on initial structures that are obtained fromideal grain growth. With the pinning pressure, the structure evolves and grain growthcontinues until the limiting grain structure is reached. The average grain radius �R as afunction of time is shown in Figure 5(a) for 3D simulations with different pinning pres-sures. As shown in Figure 5(b), the limiting grain size, �Rlim is inversely proportional tothe pinning pressure with a slope of 0:60r.

An inverse relationship was also observed in 2D simulations [24]. Measuring thecurvature distribution of the limiting structure reveals that the average driving pressurein the system is not equal to the pinning pressure, but it is lower by a scaling factor, b:

�Pd;lim ¼ a1;limr�Rlim

¼ bPz: ð11Þ

For 2D grain growth, a1;lim ¼ 0:26 and b ¼ 0:34 [24]. To obtain a1 and b at thelimiting structure in 3D, the curvature distribution of the grain boundary network isanalysed. The average curvature is smaller than the pinning pressure andb ¼ �Pd=Pz ¼ 0:85 for 3D. At the limiting grain structure, a1;lim ¼ 0:50 is measured bycomparing the average curvature with 1=�R using Equation (7). The ratio a1;lim=b ¼ 0:59is close to the slope of the line in Figure 5(b) confirming that a1;lim and b obtained from

(a) (b)

Figure 5. (colour online) (a) Equivalent grain radius as a function of time for 3D grain growthsimulations with different pinning pressures. (b) Limiting grain size as a function of inversepinning pressure.

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the curvature measurements produce the same limiting grain size (from Equation (11))as obtained from the microstructural measurements.

3.2.2. Evolution of curvature

Pinning pressure modifies the dynamics of the grain boundary network evolution. Toprovide more insight into the kinetics of grain growth with pinning pressure, the evolu-tion of grain boundary curvature is investigated. The grain boundary curvature distribu-tion changes from the self-similar state of ideal grain growth. Figure 6 shows grainboundary network curvature and curvature distribution histograms for three time stepsduring 3D grain growth.

There is a stark transition from the initial curvature distribution in Figure 6(a) to thedistribution with pinning pressure in Figure 6(b) that occurs relatively fast compared tothe time required to completely stop grain growth. Points on the grain boundary net-work with higher curvature than Pz=r move; however, they also affect areas with lowercurvature. Figure 6(a) and (b) show two points on the grain boundary network witharrows (I) and (II) that have higher and lower driving pressure than Pz, respectively.Movement of grain boundary at point (I) continues due to the high curvature. In com-parison, the curvature at point (II) increases as the neighbouring grains disappear andthe adjacent triple line moves, demonstrating that topological transitions propagatethrough the grain boundary network.

The curvature evolution and stagnation mechanism is different in 2D. Figure 7shows the curvature distribution and the corresponding microstructures during 2D graingrowth under pinning pressure of Pz=r ¼ 0:0087. Initially, in this structure, only 32%of the length of the grain boundary network have a higher curvature than the Pz=rvalue and, thus, are able to move. Growth of the grain structure continues until 75% of

(a) (b) (c)

Figure 6. (colour online) Grain structure and associated curvature distributions for 3D systemwith pinning pressure of Pz ¼ 0:01. (a) initial structure, (b) partially pinned structure at time step2000 and (c) final frozen structure. Dashed green lines are placed at the �K and solid red lines aresuperimposed at the value of Pz=r on the curvature distributions.

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the initial grains disappear and the movement of all segments comes to a halt. Duringthis process, two distinct classes appear in the curvature distribution. A large fraction ofthe grain boundary network length has zero curvature, as shown in Figure 7c. A smallerpeak appears at the limiting curvature of Pz=r. The average driving pressure is aboutthree times smaller than the pinning pressure at the frozen final structure (b ¼ 0:34 inEquation (11)).

The value of �K is smaller than Pz=r for the limiting structure regardless of thedimension (Figures 6(c) and 7(c)), resulting in b\1. This is due to a fraction of bound-ary segments that have lower Pd than Pz. These low curvature segments still evolvebecause if the neighbouring segments move, their curvature may change. In otherwords, junctions maintain topological connectivity of the network and allow areas ofthe grain boundary network that are immobile at earlier times to evolve later. This pro-cess is different in 2D and 3D. In 2D, fraction of completely flat boundaries increaseswith time whereas in 3D, the fraction of low curvature segments decreases with time.

Changes in the curvature distribution alter the total driving pressure of grain growthfor a given grain size. The curvature distribution is not self similar during grain growthwith pinning and regions with an initial driving pressure below Pz also eventuallyevolve. Therefore, the value of a1 ¼ �K�R (Equation (7)) changes during the growth pro-cess. The increase in a1 is accompanied by formation of the peak at Pz=r in the curva-ture distribution. In 2D, a1 increases from 0:20 for the ideal structure to a1;lim ¼ 0:26 atthe limiting grain structure. In 3D, a1 increases from 0:27 to a1;lim ¼ 0:50. The higherincrease in 3D is due to the sharper peak in the curvature distribution compared to 2D(Figures 6 and 7). The evolution of the grain boundary network and the change in theaverage driving pressure has a significant effect on the overall kinetics of grain growththat cannot be described with conventional models.

(a) (b) (c)

Figure 7. (colour online) Grain structure and associated curvature distributions for a 2D systemwith pinning pressure of Pz ¼ 0:010. (a) initial structure, (b) partially pinned structure at time step100; 000 and (c) final frozen structure. Green line shows the average curvature (Pd=r) and redline is at Pz=r.

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3.2.3. Kinetics of grain growth

Based on the above curvature analysis, a physical model can be presented to describethe overall kinetics of grain growth and the limiting grain size. Combining Equations(6) and (7) gives the average driving pressure for grain growth. In accordance withEquation (8), the rate of growth of �R is proportional to the available driving pressurewhen the effective pinning pressure is subtracted:

d�R

dt¼ Ma2 �Pd � bPzð Þ ¼ Ma2

a1r�R

� bPz

h ið12Þ

The scaling constant b is added since at the limiting size, d�R=dt ¼ 0 and Equation(11) has to hold. The kinetics of grain growth is obtained by integrating Equation (12):

��R

bPz� a1rlnða1rþ bPz

�RÞb2P2

z

" #�R

�R0

¼ a2Mt: ð13Þ

�R starts at an initial grain size �R0 and asymptotically reaches the limiting grain size:

�Rlim ¼ a1;limb

rPz

: ð14Þ

Figure 8 shows the average equivalent grain radius as a function of time for 2D and3D systems with Pz ¼ 0:010. Simulation results are plotted in comparison with differentmodels. The only model that can describe the kinetics and limiting grain size accuratelyis Equation (13) where the constants are obtained from the curvature distribution. For2D, a1 ¼ 0:26, b ¼ 0:34 and a2 ¼ 0:70 was used and for 3D, a1 ¼ 0:50, b ¼ 0:85 anda2 ¼ 0:80. Although the value of a1 changes from the initial structure to the limiting

(a) (b)

Figure 8. (colour online) Kinetics of grain growth with pinning compared to analytical models.Red lines are Equation (13) in which the constants are derived from curvature measurements. (a)2D system with Pz ¼ 0:010, r ¼ 1:155 and M ¼ 2:598. (b) 3D system with Pz ¼ 0:010,r ¼ 0:667 and M ¼ 3.

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structure, using the constant a1;lim for the entire curve produces satisfying results. Thereason is that the transition of a1 from the value for the ideal grain growth structure tothe value for the structure under pinning pressure happens quickly, compared to thetime required to reach the limiting grain size. Figure 6(a) and (b) shows the formationof a peak at Pz=r just after 2000 time steps, compared to 128,000 time steps for reach-ing the limiting grain size.

None of the conventional models is consistent with the simulation data. One reasonfor the discrepancy is the fact that to stop grain growth, the average driving pressure isnot equal to the pinning pressure. The limiting grain size is reached when the maximumof the driving pressure distribution reaches Pz. Failing to account for the distribution ofdriving pressure causes an overestimation of the effect of the pinning pressure on graingrowth. Other reasons for the discrepancies are simplified assumptions for the evolutionof the grain size distribution or reliance on ideal grain growth to explain evolution of asystem with pinning pressure.

In most experimental studies, grain size can be readily measured. To explain thekinetics of grain growth, Andersen and Grong [9] proposed the following relationshipfor the growth rate of �R:

d�R

dt¼ M

a1r�R

� Pz

� �ð15Þ

However, the pinning pressure is usually unknown due to the challenges measuringparticle size and fraction accurately. Therefore, a fitting procedure is frequently appliedto find Pz from the limiting grain size using Equation (15). This procedure can producea fitting curve on the experiments; however, the estimated value of Pz may not be accu-rate. For instance, we apply this procedure on our simulation data pretending not toknow the value of Pz. The rate Equation (15) is integrated from an initial grain size toobtain the average grain radius as a function of time. The result of the fitting is shownin Figure 9. Based on the Andersen-Grong model, the best fit on the data is obtainedwith Pz ¼ 0:0035 and a1 ¼ 0:21. With this procedure, a pinning pressure is obtainedthat is three times smaller than the actual value. For comparison, a curve with

Figure 9. (colour online) Fitting the Andersen-Grong model, Equation (15), on the simulationdata for a 3D system with Pz ¼ 0:010, r ¼ 0:667 and M ¼ 3. Dashed line is Equation (13).

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Pz ¼ 0:010 that gives the same limiting size is also plotted, but it does not describe thekinetics correctly since a1 is unrealistically high. Further, based on measurements byPatterson and Liu [25], Rios [6] suggested a1 ¼ 0:47. Using this value for a1 and theactual value of Pz ¼ 0:010 leads to a clear underestimation of the limiting grain size.

Hillert used a mean field assumption to derive his analytical model. The drag pres-sure is introduced into the kinetic relationship for growth of a grain with radius Rembedded in a matrix of grains with a critical radius Rcr. The drag pressure Pz opposesthe driving pressure for growth or shrinkage of the grain [7]:

dR

dt¼ M ar

1

Rcr� 1

R

� � Pz

�ð16Þ

In Equation (16), the � sign depends on the radius of each grain. It is negativewhen ar=R\ar=Rcr � Pz for large grains and positive when ar=R[ar=Rcr þ Pz forsmall grains. Grains with sizes between these two margins neither grow nor shrink, i.e.dR=dt ¼ 0. Hillert’s equation with a ¼ 1 and Rcr ¼ 9=8�R provides a good fit on indi-vidual grain growth rate for ideal grain growth in 3D, as shown in Figure 10(a).

Figure 10(b) shows the individual growth rate of grains as a function of normalizedradius for two systems with different pinning pressures. The rates are obtained for atime step in which �R=�Rlim ¼ 0:60. In this condition, simulation data collapse approxi-mately on the same scatter region. Analytically, dR=dt can be obtained from Equation(16) with a given Rcr. Unlike for ideal grain growth, Rcr is not constant during graingrowth with pinning. Rcr changes due to the lack of self similarity in the grain size dis-tribution. To obtain Rcr, we numerically fit a curve on the grain size distribution, f , andtogether with Equation (16), solve for Rcr such that the conservation of volume is satis-fied, i.e.

(a) (b)

Figure 10. (colour online) Normalized growth rate of individual grains as a function of size for3D simulations. a) Ideal grain growth; the solid lines is Hillert’s equation (Equation (16)) witha ¼ 1 and Rcr ¼ 9=8�R. b) Grain growth with pinning; two sets of data for Pz ¼ 0:010 andPz ¼ 0:012 are plotted at time steps when �R=�Rlim ¼ 0:6. Solid lines are Hillert’s equation witha ¼ 1 and Rcr ¼ 0:91�R.

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Z 1

0

R2f ar1

Rcr� 1

R

� � Pz

�dR ¼ 0: ð17Þ

Rcr is found to be lower than 9=8�R in the presence of pinning pressure. For thecases shown in Figure 10(b), Equation (16) is plotted for both systems using the valueof Rcr ¼ 0:91�R with solid lines. The two curves are also very close, suggesting thatRdR=dt scales with �R=�Rlim for different Pz. The simulation data follow the trend sug-gested by Hillert’s model; however, grains in the intermediate range have higher abso-lute growth rates compared to Equation (16) that suggests zero growth. As �Rapproaches �Rlim, individual grain growth rates decrease and the horizontal section of thecurve with dR=dt ¼ 0 extends to higher and lower R=�R.

Assuming the grain size distribution is self-similar, which is obviously not true inpresence of pinning, Hillert obtained the following relationship for the growth of Rcr

[7]:

dR2cr

dt¼ M

2ar 1� PzRcr

ar

� 2

ð18Þ

Equation (18) is integrated from an initial grain size and �R ¼ 8=9Rcr is plotted inFigure 8(b). The curve coincides with the simulation data at earlier stages of graingrowth, but deviates before reaching the limiting grain size.

4. Conclusions

In this paper, we used a phase-field method to simulate grain growth in the presence ofa uniform pinning pressure. Our simulation technique allows the pinning pressureobtained from small-scale calculations to be incorporated into large-scale grain growthsimulations. Relationships for the kinetics of grain growth and the limiting grain sizewere obtained by analysing the curvature of the grain boundaries. Two main results canbe concluded from this study. The first result is related to the relationship between pin-ning pressure and limiting grain size for 3D systems. In our simulations, the pinningpressure was introduced as a phenomenological friction pressure, but in reality it isdetermined by the particle distribution. We can assume that on average, a spatially uni-form particle distribution exerts a constant pinning force of Pz on the entire grainboundary network. The pinning pressure depends on volume fraction, fp, and radius, rp,of the particles. For example, a pinning pressure of Pz ¼ 3:3rfp=rp was obtained fromfinite element simulations of a grain boundary interacting with an ensemble of particles[12]. This relationship for the pinning pressure can be combined with the relationshipfor the limiting grain size, �Rlim ¼ 0:60r=Pz, obtained from the present phase-field simu-lations (Figure 5), resulting in �Rlim ¼ 0:18rrp=fp. On the other hand, Manohar [2] com-piled experimental results of various systems and showed that for systems with lowvolume fraction of fine particles, the limiting grain size is �Rlim ¼ 0:17rrp=fp. Therefore,our simulation accurately replicates the experimental results.

The average driving pressure of grain growth was defined by measuring the curva-ture of the grain boundary network. �Pd is related to the grain size using a constant a1.We showed that at the limiting structure, the average driving pressure for grain growth

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is not equal but smaller than the pinning pressure (�Pd ¼ bPz). The scaling factor,b ¼ 0:34 for 2D and b ¼ 0:85 for 3D systems, is independent of the pinning pressurein the investigated range. While the average driving pressure is lower than Pz, the high-est curved point in the grain boundary network has a driving pressure equal to Pz andis the last point to stop moving. The general relationship for limiting grain size(Equation (14)) obtained from curvature analysis can be applied to any system withisotropic grain boundary properties. The ratio of a1;lim=b ¼ 0:59 obtained from curva-ture analysis in 3D agrees well with the results deduced from simulated microstructuresand is, thus, consistent with a wide range of experimental data listed in Manohar’sreview.

Another core result of this study is related to describing the kinetics of graingrowth. While conventional models of particle pinning fail to capture the kinetics ofgrain growth, the proposed curvature-based model that provides an accurate descriptionof the grain growth kinetics. We establish a clear relationship between the average grainsize and the average driving pressure by analysing curvature of the grain boundary net-work that is described by the constant a1. We found that the rate of grain growth,d�R=dt, is proportional to the average driving pressure with the proportionality constanta2 (Equation (8)). The value of a2 obtained from ideal grain growth can also be appliedto grain growth with pinning pressure. Consequently, the proposed model (Equation(12)) with the three parameters (a1, a2 and b) accurately describes the kinetics of graingrowth with pinning. These parameters can be used for isotropic systems with uniformpinning pressure in which the initial structure is close to the ideal grain structure. Sincethe parameters are function of the grain size distribution and the curvature distribution,they may not be generalized to systems with large variation in grain size, bimodal grainsize distribution or inhomogeneous particle dispersion. For such systems, one can usethe proposed phase-field model to simulate the structure starting from a desired grainsize structure with spatial and temporal variation in the pinning pressure.

Furthermore, this study demonstrates that the evolution of grain boundary networksunder a uniform pinning pressure is fundamentally different in 2D or 3D due to the dif-ferent curvature distributions. Therefore, one cannot obtain quantitative informationabout 3D systems from 2D simulations.

AcknowledgementsThe authors would like to acknowledge the funding from Natural Science and EngineeringResearch Council of Canada and Evraz Inc. NA. The simulations were performed on theCompute Canada’s WestGrid computer cluster.

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