Scripta Materialia 55 (2006) 461–464

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Effective growth law from three-dimensional grain growthsimulations and new analytical grain size distribution

Peter Streitenberger* and Dana Zollner

Otto-von-Guericke-Universitat Magdeburg, Institut fur Experimentelle Physik, Abteilung Materialphysik,

Universitatsplatz 2, 39106 Magdeburg, Germany

Received 16 March 2006; revised 4 May 2006; accepted 9 May 2006Available online 5 June 2006

A new analytical grain size distribution function is derived, which is based on a quadratic approximation of the average self-sim-ilar volumetric rate of change as a function of the relative grain size as it has been determined from Monte Carlo Potts model sim-ulations of three-dimensional normal grain growth. The analytical grain size distribution function yields an excellent representationof the simulated grain size distribution.� 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Grain growth; Kinetics; Microstructure; Monte Carlo techniques; Theory

The calculation and availability of analytical grainsize distributions that are derived from or, at least, areconsistent with the basic principles of curvature-drivengrain growth kinetics are still a fundamental issueof grain growth theory. According to Hillert [1] normalgrain growth can be characterized by a self-similarscaled grain size distribution function (GSDF), whichsatisfies a continuity equation in size space that is associ-ated with an average or effective growth law _R :¼ dR=dtfor all individual grains of size R [2,3]. In particular,Hillert assumed for the effective growth law the form

_R ¼ ~ak1

Rc

� 1

R

� �; ð1Þ

where Rc = Rc(t) is a time-dependent critical grain sizedefined by _RðR ¼ RcÞ ¼ 0, k is the kinetic constant ofcurvature-driven grain boundary motion and ~a a con-stant. By application of the coarsening theory of Lifshitzand Slyozov [4] and Wagner [5] (LSW) to grain growthas characterized by Eq. (1), Hillert obtained his well-known scaled GSDF [1], which, however, has never beenobserved, either experimentally or by computer simula-tions [6–10].

For two-dimensional grain growth Brandt et al. [8](cf. also Ref. [11]) and others [12–14] attempted to mod-

1359-6462/$ - see front matter � 2006 Acta Materialia Inc. Published by Eldoi:10.1016/j.scriptamat.2006.05.009

* Corresponding author. E-mail: peter.streitenberger@physik.uni-magdeburg.de

ify Eq. (1) while retaining the LSW method andobtained analytical GSDFs, which were in good agree-ment with the GSDFs of the simulations. This approachhas been generalized by the present authors to three-dimensional grain growth in Refs. [9,10].

Alternatively, it was recently also attempted [15] tomodify the LSW procedure while retaining the growthlaw Eq. (1). This formulation, however, suffers fromsome inconsistencies leading to a diverging thirdmoment of the resulting GSDF associated with an infi-nite average grain volume.

In the present paper it is shown that an adequatemodification of the effective growth law allows also aconsistent modification of the LSW procedure. In par-ticular, based on the effective growth law derived fromthe Monte Carlo (MC) simulation data of three-dimen-sional grain growth in Refs. [9,10], a new analyticalGSDF is derived, which is fully consistent with therequirement of total-volume conservation and the exis-tence of a finite average grain volume. The obtainedanalytical GSDF represents the simulation data ofthree-dimensional grain growth very well.

Hillert’s growth law, Eq. (1), is a special form of thegeneral property that the effective volumetric rate ofchange V �1=3 _V depends in the self-similar state of coars-ening only on the relative grain size x = R/hRi [16–18].That is, V �1=3 _V ¼ fR _R ¼ fkHðxÞ, where R is the radiusof the grain-volume equivalent sphere, hRi = hRi(t) isthe average grain size, H(x) a dimensionless time-invari-ant function and f = (48p2)1/3. Eq. (1) corresponds to

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Figure 2. Self-similar volumetric rate of change R _R versus number ofneighbouring grains s = s(x). The curve represents the least-squares fitof Eq. (3) to the MC simulation data (crosses).

Figure 3. Number of neighbouring grains s = s(x) versus relative grainsize x divided into size classes. The curve represents the least-squares fitof the quadratic polynomial Eq. (4) to the MC simulation data (crosseswith error bars).

462 P. Streitenberger, D. Zollner / Scripta Materialia 55 (2006) 461–464

the linear function H HillertðxÞ ¼ ~aðx=xc � 1Þ, wherexc = Rc/hRi is the scaled critical grain size.

Recently we have shown by MC Potts model simula-tions of three-dimensional normal grain growth that,contrary to Hillert’s assumption, H(x) is a non-linearfunction, which in the range of observed grain sizescan be represented by the quadratic polynomial [9,10](Fig. 1)

R _R ¼ kHðxÞ ¼ a2x2 þ a1xþ a0: ð2ÞA qualitative understanding of the non-linear behav-

iour of R _R ¼ kHðxÞ can be gained on the basis of recentresults of Hilgenfeldt et al. [19] and Glicksman [20], whohave shown by considering three-dimensional spacefilling polyhedral networks that the average volumetricrate of change of grains of relative size x is solely a func-tion of its average number of faces or neighbourss = s(x), which can be approximated by the expression

R _R ¼ C0 þ C1 �ffiffiffiffiffiffiffiffisðxÞ

p: ð3Þ

While the square-root dependence on s(x) is asymp-totically exact [19], it represents, by appropriate choiceof the coefficients C0 and C1, the simulation data verywell also in the numerically relevant range s(x) 6 45[10,20] (Fig. 2). Eq. (3) can be considered as the three-dimensional analogue to the von Neumann–Mullinstopological law in two dimensions [20,21].

If we accept Eq. (3), then the simulation results forR _R in Figure 1 and the failure of Hillert’s GSDF to agreewith simulations mean that s(x) must be a non-linearfunction of order three or higher or, at least, a non-bino-mial quadratic function. Figure 3 shows a plot of s(x)representing an average value over 10 runs after 500MC time steps of our three-dimensional MC Potts mod-el simulation [9,10]. In fact, the least-squares fit of aquadratic polynomial to the simulation data clearly yieldsa non-binomial parabolic function (cf. also [22–26])

sðxÞ ¼ s2x2 þ s1xþ s0 ¼ ðpxþ qÞ2 þ d: ð4ÞEq. (3) shows in conjunction with Eq. (4) a behaviour

that is consistent with the weak non-linearity of theeffective growth law derived from the time-developmentof the simulated microstructure [10] (cf. Fig. 1). This isdifferent from an assumption in Ref. [27] according towhich x � ðV =hV iÞ1=3 � eC þ eD ffiffiffiffiffiffiffiffi

sðxÞp

, which would

Figure 1. Self-similar volumetric rate of change R _R versus relativegrain size x. Least-squares fit of Eq. (2) (solid curve) to the MCsimulation results (stars with error bars) and plot of Eq. (3) with s(x)from Eq. (4) (dotted curve).

imply a quadratic binomial for s(x) and consequently,via Eq. (3), a linear volumetric rate of change givingagain Hillert’s GSDF [27].

Both expressions Eqs. (2) and (3) together with Eq.(4) approximate the numerical data of the effectivegrowth law very well (cf. Fig. 1) and yield, despite theirdifferent behaviour for very large x, very similar GSDFs,as we will see. While Eq. (3) together with Eq. (4) can beused for numerical integration, the quadratic approxi-mation Eq. (2) has the advantage of being feasible byan analytical treatment. In any case, only small devia-tions from linearity in Eq. (2) are required for agreementof the calculated GSDF with the simulation results.

For the analytical treatment Eq. (2) is transposed tothe new variable u = R/Rc = x/xc

R _R ¼ kHðuÞ ¼ a2x2cðu2 � 1Þ þ a1xcðu� 1Þ; ð5Þ

where xc = Rc/hRi and Rc = Rc(t) is the critical grainsize defined by _RðR ¼ RcÞ ¼ 0. Following the standardprocedure [1–4,6,12,16,18] the dimensionless scaledgrowth law UðuÞ ¼ du=ds ¼ _R= _Rc � u is introduced,where s = ln (Rc/Rc0) represents a dimensionless loga-rithmic time scale. U(u) is related to H(u) in Eq. (5) byHðuÞ ¼ eC�1u½UðuÞ þ u�, where eC ¼ k=ðRc

_RcÞ ¼ C=x2c ,

and C ¼ k=ðhRih _RiÞ ¼ const is the growth constant cor-responding to the parabolic growth law hRi2 � hRi20 ¼ð2k=CÞt [2,3]. Defining the dimensionless parametersc = Ca1/(kxc) and a = a2xc/a1, U(u) is represented bythe two-parameter function

Figure 4. The scaled growth law Eq. (6) for D P 0 and different valuesof the parameters a and c. 1: D = 0 for a = 0 (c = 4, u0Hillert = 2, LSW–Hillert state [1]), 2: D > 0, 3: D = 0 for a > 0 (c < 4, u0 > 2, generalizedLSW state [12]).

Figure 5. Plot of c versus a. The allowed values of the parameters c,a in Eq. (11) are represented by the shaded region (D > 0) and theboundary curve (D = 0). Indicated are the actual parameter valuesresulting from various fitting procedures.

P. Streitenberger, D. Zollner / Scripta Materialia 55 (2006) 461–464 463

UðuÞ ¼ cu½u� 1þ aðu2 � 1Þ� � u ¼ � 1

uðAu2 þ Buþ CÞ;

A ¼ 1� ca; B ¼ �c; C ¼ cð1þ aÞ: ð6Þ

For a = 0, Eq. (6) reduces to the linear one-parameterexpression of the LSW–Hillert theory [1–4].

In the scaling state of coarsening the grain size distri-bution can be described by the self-similar function[12,16,18]

F ðR; tÞ ¼ V 0

aDhuDiRDþ1c

/ðuÞ ¼ V 0

aDhxDihRiDþ1f ðxÞ; ð7Þ

where the differently scaled GSDFs /(u) and f(x) areinterrelated by f(x) = /(x/xc)/xc. Eq. (7) fulfils explicitlythe requirement of conservation of the total volume,V 0 ¼

R10

aDRDF ðR; tÞdR ¼ const. Here, D = 1, 2 or 3 isthe spatial dimension of the system, aD the volume ofthe D-dimensional unit sphere and huni ¼

R10

un/ðuÞdu,respectively hxni ¼

R10

xnf ðxÞdx are the n-th momentsof the corresponding normalized scaled GSDFs. In par-ticular, from the definition of x it follows hxi = 1, andxc = 1/hui.

In the statistical mean-field theory of grain growth[2,3] it is assumed that F(R,t) obeys the continuity equa-tion in size space,

oF ðR; tÞot

þ o

oR_RF ðR; tÞ� �

¼ 0; ð8Þ

where in the present paper the effective growth law _R isgiven by Eqs. (2) and (5), respectively. Inserting Eq. (7)into Eq. (8), the continuity equation transforms into anordinary differential equation, whose solution is[12,16,18]

/ðuÞ ¼ � DUðuÞ exp D

Z u

0

du0

Uðu0Þ

� �ð9Þ

with U(u) given by Eq. (6). Eq. (9) implies that /(u)vanishes for large u fast enough so that limu!1U(u)�/(u) = 0, and from the normalization condition it fol-lows limu!0U(u)/(u) = �D.

The integration over 1/U(u) in Eq. (9) is performedfor the case that

D ¼ 4AC � B2 ¼ 4cð1� caÞð1þ aÞ � c2 P 0; ð10Þ

which corresponds to a behaviour of U(u) as shown inFigure 4 yielding

/ðuÞ¼ DCD2Au

ðAu2þBuþCÞ1þD2A

� expDB

AffiffiffiffiDp arctan

Bþ2AuffiffiffiffiDp

� ��arctan

BffiffiffiffiDp� �� �

:

ð11Þ

For the limiting case D = 0, that is for c = 4(1 + a)/[4a(1 + a) + 1] (cf. the curves 1 and 3 in Figure 4),U(u) exhibits a double root at u0 = 2(1 + a) correspond-ing to the generalized LSW procedure already consid-ered in Refs. [9,10,12,13], where Eq. (11) reduces tothe one-parameter function

/ðuÞ ¼ aua0 expðaÞuðu0 � uÞ2þa exp � au0

u0 � u

� �; u 6 u0

¼ 0; u > u0;

u0 ¼ 2ð1þ aÞ; a ¼ Dð1þ 2aÞ2;

ð12Þ

showing a finite cut-off at u0.From the asymptotic behaviour of Eq. (11), that

is limu!1/(u) � u�(1+D/A) and limu!1uD/(u) �u�D(1/A�1)�1, it follows that the moment huDi of theGSDF, which according to Eq. (7) is intimately relatedto the average grain volume and the requirement oftotal-volume conservation, is only finite if A < 1, thatis ca > 0. For that reason, the only permissible solutionfor the limiting case A = 1, that is for a = 0, is given byEq. (12) at c = 4, that is the LSW–Hillert solution with afinite cut-off at u0 = u0Hillert = 2. Therefore, Eq. (11) rep-resents a two-parameter family of GSDFs, which fulfilthe requirement of volume conservation in conjunctionwith the existence of the D-th moment huDi of theGSDF if the parameters a and c obey the conditions(cf. Fig. 5)

c 64ð1þ aÞ

4að1þ aÞ þ 1; a > 0; ð13aÞ

c ¼ 4; a ¼ 0: ð13bÞIn Figure 6 Eq. (11) is plotted for D = 3 and some

selected values of the parameters a and c, after rescalingto the measurable scaled size variable x = R/hRi =xcu = u/hui. For a plot of Eq. (12) see [10,12,13].

Figure 6. The analytical GSDF Eq. (11) for D = 3 and D > 0 forselected values of the parameters a and c.

Figure 7. Comparison of the analytical GSDF, Eq. (11), with theGSDF resulting from MC Potts model simulation of three-dimen-sional grain growth (diamonds) [10]. Solid black line: Eq. (11) forparameter values a,c resulting from Eq. (2) and the scaling require-ment. Dotted black line: Direct fit of Eq. (11) to the simulated GSDF(diamonds). Solid grey line: GSDF based on Eq. (3).

464 P. Streitenberger, D. Zollner / Scripta Materialia 55 (2006) 461–464

In a first comparison, the two parameters a and c inthe GSDF Eq. (11) are determined from the parametersa1, a2 and xc of the effective growth law, Eq. (2), fitted tothe simulated microstructure (see Fig. 1). While thisgives for a immediately a = a2xc/a1 = 0.4277, c is deter-mined self-consistently by the scaling requirement thatthe scaled critical grain size xc = 1.2171 of the simulatedmicrostructure [9,10] has to be the same asxc ¼ xcða; cÞ ¼ 1=

Ru/ðuÞdu following from the GSDF

Eq. (11), yielding c = 1.517 (cf. also Fig. 5). The result-ing analytical GSDF is in excellent agreement with theGSDF of the MC Potts model simulation (Fig. 7). Theresulting growth constant, C = kxcc/a1 = 10.418, agreesalso very well with the simulation result C = 10.475.

In a second comparison, the two parameters inEq. (11) are determined by a direct least-squares fit ofEq. (11) to the GSDF of the MC simulation givinga = 0.5204 and c = 1.4601. The resulting analyticalGSDF Eq. (11) represents the simulation data also verywell (Fig. 7). The value of the scaling parameterxc ¼ 1=

Ru/ðuÞdu ¼ 1:20998 is also in good agreement

with the above value from the simulation. The calcula-tion of the growth constant C, however, is not possiblewithout recourse to the effective growth law, Eq. (2).

The coefficients in Eq. (2) are not longer independentof each other leading to the modified expressionR _R ¼ ðkc=CÞ½xcðx� xcÞ þ aðx2 � x2

c�, which fits the sim-ulation data in Figure 1 for C = 11.043.

Finally, the GSDF resulting from Eq. (3) with s(x)given by Eq. (4), is calculated for D = 3 from Eq. (9)by numerical integration (Fig. 7).

All the fitted parameter pairs a and c are very close tothe boundary curve for D = 0 in Figure 5, which indi-cates that the one-parameter function Eq. (12) is virtu-ally a just as good representation of the GSDF as thetwo-parameter expression Eq. (11).

The authors would like to thank the Deutsche Fors-chungsgemeinschaft for financial support under GrantNumber GKMM 828.

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