a new approach to modelling of non-steady grain growth

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Acta Materialia 55 (2007) 4467–4474

A new approach to modelling of non-steady grain growth

J. Svoboda a, F.D. Fischer b,c,*

a Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zizkova 22, CZ-616 62 Brno, Czech Republicb Institute of Mechanics, Montanuniversitat Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria

c Materials Center Leoben Forschung GmbH, Franz-Josef-Straße 13, A-8700 Leoben, Austria

Received 6 March 2007; received in revised form 4 April 2007; accepted 5 April 2007Available online 6 June 2007

Abstract

Grain coarsening is a phenomenon common to all polycrystalline materials at elevated temperatures. During the last 50 years a num-ber of sophisticated models have been developed for grain-coarsening kinetics. The authors have shown that Onsager’s extremal principlerepresents a systematic way of deriving evolution equations for parameters characterizing thermodynamic systems. If the grain radii arechosen as those parameters, the application of Onsager’s principle reproduces Hillert’s classical evolution equations for the radii of indi-vidual grains (multigrain concept). The observed or calculated ensemble of grains is usually classified by a grain radii distribution func-tion involving a certain number of parameters. The novelty of the paper is represented by the direct application of Onsager’s principle tothe radii distribution function by derivation of the evolution equations for its parameters (distribution concept). The kinetics of systemswith bimodal and different monomodal starting distribution functions are calculated by means of both multigrain and distribution con-cepts, and the results of simulations are compared and discussed. The dissipation of the grain coarsening process is evaluated, and it isshown that the width of the distribution function decisively influences the coarsening kinetics.� 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Grain growth; Modelling; Thermodynamics; Simulation; Extremal principle

1. Introduction

Grain coarsening is a dissipative process driven by thedecrease of the total Gibbs energy of grain boundariesfor the case of closed systems under constant temperatureand pressure. The problem of grain coarsening concernsa wide field of materials and has been a favourite topicof research during the last 50 years [1–5]. Although in theliterature the term ‘‘grain growth’’ has mainly been usedfor this phenomenon, we prefer the term ‘‘grain coarsen-ing’’, since it reflects much better the fact that during theprocess only the large grains grow and the small grainsshrink and disappear.

The second law of thermodynamics admits all processeswith positive entropy production (positive dissipation) butdoes not allow the determination of a distinct unambigu-

1359-6454/$30.00 � 2007 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2007.04.012

* Corresponding author. Tel.: +43 3842 402 4000; fax: +43 3842 46048.E-mail address: [email protected] (F.D. Fischer).

ous evolution path of the system [6]. In 1931 Onsager for-mulated an extremal principle [7], according to which theunambiguous path of the thermodynamic system corre-sponds to a constrained maximum of dissipation in anypart of the system at any time. From the thermodynamicpoint of view Onsager’s principle can be understood as astrong form of the second law of thermodynamics allowingthe determination of the unambiguous evolution path ofthe system.

In 1991 Onsager’s principle was first formulated in termsof discrete parameters characterizing the state of the system[8]. The time dependence of these parameters is aimed atdescribing the system evolution in a natural way. Theapplication of the principle provides a systematic conceptfor the derivation of the evolution equations for theseparameters for a wide class of problems in the context oflinear non-equilibrium thermodynamics [9].

The theoretical treatment of grain coarsening is possibleonly by accepting some model assumptions, which simplify

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4468 J. Svoboda, F.D. Fischer / Acta Materialia 55 (2007) 4467–4474

the reality, but still keep the important features of graincoarsening. One of these usual model assumptions is theapproximation of each grain by a sphere of the same vol-ume. Then the system of grains can be described by theradii Rk of individual grains, and the evolution of Rk canbe treated by application of Onsager’s principle [5]. Thistreatment is called, in this paper, ‘‘the multigrain concept’’.

The evolution equations for Rk obtained from Onsager’sprinciple are equivalent to those derived first by Hillert [1]by means of heuristic considerations in 1965. Since thattime the models on grain coarsening have been refined,and the treatment has become more and more complicated.At this point it is necessary to mention the approachesbased on irreversible thermodynamics: for example, Rioset al. [10–12] have an ongoing discussion about grain sizedistribution with important results from three-dimensionalcomputer simulations and experimental grain size distribu-tions (see, e.g. Rios et al. [13]); and Glicksman [14] andRios and Glicksman [15,16] question the usual sphericalgrain approximation by the introduction of grain topology.

The grain structure is usually characterized by a grainradii distribution function f(R) = dN/dR, where N is thenumber of grains in the system with radii 6R. In the pres-ent paper, the non-steady evolution of the system is treatedby means of well-established grain radii distribution func-tions involving a set of parameters. Then the evolution ofthe system is given by the evolution of the parameters. Inthis paper, this treatment is called ‘‘the distributionconcept’’.

The goal of this paper is to derive the evolution equa-tions for the parameters of the distribution function byapplication of Onsager’s principle without the necessityto know the evolution equations for the radii Rk of individ-ual grains. The equations thus derived are employed to pre-dict the kinetics of systems with monomodal and bimodalgrain radii distributions. The results of simulations basedon both the multigrain concept and the distribution con-cept are compared and discussed in detail.

2. Evolution equations for radii of individual grains – the

multigrain concept

First let us repeat the derivation of evolution equationsfor radii of individual grains from Onsager’s principle (fordetails see Ref. [5]).

Let the system be composed of a set of grains approxi-mated by spheres of radii Rk, k = 1, . . .,N. Then the totalGibbs energy of the system is given by

G ¼ 1

24pc

XN

k¼1

R2k : ð1Þ

The quantity c is the Gibbs energy of the grain boundariesper unit area. If a grain boundary of a unit area and ofmobility M migrates by a velocity v under the action of athermodynamic force F, then the dissipation is given bythe product Fv. By assuming the proportionality relation

v = MF, the dissipation becomes a quadratic function ofv as v2/M. During the growth of a spherical grain k thegrain boundary velocity v coincides with _Rk, and thus thetotal dissipation in the system follows as [5,9]

Q ¼ 1

2

4pM

XN

k¼1

R2k

_R2k : ð2Þ

The dot symbol is used for the time derivative. The factor1/2 in Eqs. (1) and (2) takes into account that each grainboundary is common to two grains. The total volume ofthe grains in the system must be conserved, and thus

4p3

XN

k¼1

R3k ¼ const:) 4p

XN

k¼1

R2k

_Rk ¼ 0: ð3Þ

According to Onsager’s principle, the evolution path ofthe system corresponds to the maximum of the total dissi-pation in the system, Q, constrained by _Gþ Q ¼ 0 andother constraints, in this case only one constraint expressedby Eq. (3). The necessary condition for the constraint max-imum of Q is given by

o

o _Rk

Qþ aðQþ _GÞ þ k4pXN

k¼1

R2k

_Rk

!¼ 0; k ¼ 1; . . . ;N

ð4Þwith _G ¼ 4pc

PNk¼1Rk

_Rk and a and k being Lagrange multi-pliers. The partial derivatives in Eq. (4) can be performedas

4pM

R2k

_Rk þ a4pM

R2k

_Rk þ 4pcRk

� �þ k4pR2

k ¼ 0; k ¼ 1; . . . ;N :

ð5ÞEach term in Eq. (5) can be multiplied by _Rk and all equa-tions can be summarized with the result

4pM

XN

k¼1

R2k

_R2k þ a

4pM

XN

k¼1

R2k

_R2k þ 4pc

XN

k¼1

Rk_Rk

!

þ k4pXN

k¼1

R2k

_Rk ¼ 0: ð6Þ

Using the expression for _G and Eqs. (2) and (3), Eq. (6)can be rewritten into the form

2Qþ að2Qþ _GÞ ¼ 0) 2þ 2aa

Qþ _G ¼ 0: ð7Þ

Comparing Eq. (7) with _Gþ Q ¼ 0 provides immediatelythe equation for a:

2þ 2aa¼ 1) a ¼ �2:

Insertion of the value of a into Eq. (5) enables to simplifythese equations as

R2k

_Rk ¼ 2Mð�cRk þ kR2kÞ; k ¼ 1; . . . ;N : ð8Þ

Insertion of Eq. (8) into Eq. (3) allows the evaluation of k,and then the evolution equations for individual grains read

J. Svoboda, F.D. Fischer / Acta Materialia 55 (2007) 4467–4474 4469

_Rk ¼ 2cM1

RC

� 1

Rk

� �; k ¼ 1; . . . ;N ð9Þ

with RC ¼PN

i¼1R2i =PN

i¼1Ri being a critical radius. Grainswith the radius RC neither grow nor shrink. One can ob-serve that large (supercritical) grains grow while small (sub-critical) grains shrink and must be eliminated from thesystem if they reach zero radius. As the total number ofgrains N decreases and the total volume of grains remainsconstant during the process, the mean volume as well as themean radius of the grains R increase; a detailed treatment ispresented in Fischer et al. [5].

Let f(R, t)t = 0, be a starting grain radii distributionfunction. Then one can generate a sufficiently large set ofgrain radii Rk and integrate Eq. (9) in time. At any time t

the distribution function f(R, t) can be constructed fromthe values Rk(t).

3. Evolution equations for parameters of distribution function

– the distribution concept

A new idea can be followed by applying Onsager’s extre-mal principle directly to the distribution function itself. Letthe distribution function depend on n parametersa1(t), . . .,an(t), which cause the evolution of the distributionfunction

fsðR; tÞ ¼ f ðR; a1ðtÞ; . . . ; anðtÞÞ: ð10ÞTo avoid taking into account the condition of a fixed vol-ume of the system as a constraint, we assume that all con-sidered distribution functions f meet the conditionZ 1

0

R3f dR ¼ 1 ð11Þ

at any time, i.e. the system has a fixed volume 4p/3.The well-known continuity relation of the distribution

function f in the radii space reads as

ofotþ oðf _RÞ

oR¼ 0: ð12Þ

Eq. (12) can be integrated at a fixed time asZ 1

R

ofot

dR0 ¼ ðf _RÞjR � ðf _RÞj1 ¼ ðf _RÞjR ð13Þ

since ðf _RÞj1 ¼ 0 due to finite _Rj1 and fj1 = 0. Eq. (10)yields

ofot¼Xn

i¼1

ofoai

_ai ð14Þ

and insertion of Eq. (14) into Eq. (13) enables to calculate_R as

_RðRÞ ¼

Pni¼1

_ai

R1R

ofoai

dR0

f: ð15Þ

Eq. (15) represents a key equation of the treatment, as it al-lows expressing of the total dissipation in the system bymeans of _a1; . . . ; _an.

Similarly to Eqs. (1) and (2), the total Gibbs energy ofgrains is given by

G ¼ 2pcZ 1

0

R2f dR ð16Þ

and the total dissipation in the system due to grain bound-ary migration follows as

Q ¼ 2pM

Z 1

0

R2f _R2 dR

¼ 2pM

Z 1

0

R2

f

Xn

i¼1

_ai

Z 1

R

ofoai

dR0 !2

dR: ð17Þ

According to Onsager’s principle the evolution of thesystem corresponds to the maximum of the total dissipa-tion in the system Q constrained by _Gþ Q ¼ 0 (no otherconstraints exist for _a1; . . . ; _an). The treatment, analogousto that presented in the previous section, leads to evolutionequations of the system given by a set of linear equationswith respect to _ai; i ¼ 1; . . . ; n (for details see Ref. [9])

1

2

oQo _ai¼ � oG

oaiorXn

j¼1

1

2

o2Q

o _ai o _aj_aj ¼ �

oGoai

; i ¼ 1; . . . ; n

ð18Þwith

oGoai¼ 2pc

Z 1

0

R2 ofoai

dR; i ¼ 1; . . . ; n ð19Þ

1

2

oQo _ai¼ 2p

M

Z 1

0

R2

f

Z 1

R

ofoai

dR0Xn

j¼1

_aj

Z 1

R

ofoaj

dR0 !

dR;

i ¼ 1; . . . ; n ð20Þ

1

2

o2Qo _ai o _aj

¼ 2pM

Z 1

0

R2

f

Z 1

R

ofoai

dR0Z 1

R

ofoaj

dR0 dR;

i; j ¼ 1; . . . ; n: ð21Þ

4. Kinetics of system with initial Rayleigh’s distribution

The measured grain radii distribution functions canoften be well approximated by Rayleigh’s distribution [16]:

fRðR; cðtÞÞ ¼8

3ffiffiffipp c5=2R expð�cR2Þ ¼ Kc5=2R expð�cR2Þ:

ð22ÞThe distribution function is normalized to ensure Eq. (11)for arbitrary positive values of c(t) standing for a1(t),n = 1. It can easily be shown that grain coarsening corre-sponds to a decrease in c. The distribution functions resem-bling Rayleigh’s distribution have also been derivedtheoretically by taking the spatial grain size correlationand topology effects into account [17].

Eq. (20) yields

1

2

dQR

d _c¼ 2p _c

M

Z 1

0

R2

fR

Z 1

R

dfR

dcdR0

� �2

dR ¼ 5pK _c

16Mc5=2ð23Þ

3.5

4.0

4.5

tal G

ibbs

ene

rgy multigrain concept

distribution concept

a


and Eq. (19) provides

dGR

dc¼ 2pc

Z 1

0

R2 dfR

dcdR ¼ pcK

2c1=2: ð24Þ

The kinetics of the system follows from Eq. (18) as

_c ¼ � 8

5cMc2: ð25Þ

The mean grain radius RR is given by

RR ¼Z 1

0

RfR dR�Z 1

0

fR dR ¼ffiffiffipp

2c1=2: ð26Þ

Then the kinetics of the mean grain radius RR results fromcombining Eqs. (25) and (26)

RR_RR ¼ �

ffiffiffipp

2c1=2

ffiffiffipp

4c3=2_c ¼ � p

5

_cc2

¼ p5

cM or R2Rjt � R2

Rj0 ¼2p5

cMt ð27Þ

which is consistent with the well-known parabolic law forthe grain coarsening.

In this one-parametric case c can be replaced by RR

(using Eq. (26)) in Eq. (22). After a proper normalizationof the grain radii distribution function fR and of the grain

0 1 2 3 40.0

0.2

0.4

0.6

t = 0 0.2 0.4 0.6 0.8 1

multigrain concept distribution concept

Dis

trib

utio

n fu

nctio

n

Grain radius

0 1 2 3 4Grain radius

0.00

0.02

0.04

0.06

0.08

0.10

0.12

t = 1


Dis

trib

utio

n fu

nctio

n

a

b

Fig. 1. Comparison of the evolution of monomodal grain radii distribu-tions by application of multigrain and distribution concepts. (a) Individualcurves corresponding to the times denoted in the figure. (b) Detailed plotof both distribution functions for time t = 1.

radius R by RR, the general non-steady distribution conceptdegenerates to the well-known steady-state concept.

Using Eqs. (23)–(27), the total dissipation in the system,described by Rayleigh’s distribution and characterized bythe mean grain radius RR, is given by

QR ¼1

2

dQR

d _c_c ¼ � dGR

dc_c ¼ 4p2c2M

15R3R

: ð28Þ

To allow a simple demonstration of the simulationresults c = 1 and M = 1 are set in all simulations. Twostudies of the system evolution are presented with thesame starting Rayleigh’s distribution given by c(0) = 1in Fig. 1. The first simulation covers the distribution con-

0.0 0.2 0.4 0.6 0.8 1.0

3.0

To

Time

0.00 0.02 0.04 0.06 0.08

4.5

4.6

4.7

Tot

al G

ibbs

ene

rgy

Time


0.0 0.2 0.4 0.6 0.8 1.0

1

2

3

4 multigrain concept distribution concept

Tot

al d

issi

patio

n

Time

b

c

Fig. 2. The time dependence of the total Gibbs energy and the totaldissipation corresponding to simulations presented in Fig. 1. (a) TotalGibbs energy. (b) Detailed plot of the total Gibbs energy for small times.(c) Total dissipation.


cept (solid lines) and the second one the multigrain con-cept for 25,000 starting grains (dotted lines). The individ-ual curves in Fig. 1a correspond to grain radiidistributions for times t = 0, 0.2, 0.4, 0.6, 0.8 and 1, whilethe curves in the detailed Fig. 1b correspond only to timet = 1. In Fig. 2, the time dependences of the total Gibbsenergy G (Fig. 2a and b) and the total dissipation

1

2

o2Q

o _Ao _c1

¼ pAKc1=21

4M

Z 1

0

R c3=21 expð�c1R2Þ � c3=2

2 expð�c2R2Þh i

ð3� 2c1R2Þ expð�c1R2Þ

Ac5=21 expð�c1R2Þ þ ð1� AÞc5=2

2 expð�c2R2ÞdR ð34Þ

1

2

o2Q

o _Ao _c2

¼ pð1� AÞKc1=22

4M

Z 1

0


2 expð�c2R2Þh i

ð3� 2c2R2Þ expð�c2R2Þ


2 expð�c2R2ÞdR ð35Þ

Q � � _G (Fig. 2c) are plotted for both simulations. It isevident from Fig. 2c that the initial dissipation accordingto the multigrain concept is higher than that according tothe distribution concept. This is in agreement with expec-tations, since a higher number of degrees of freedomallows a finer description of the system evolution and con-sequently a higher dissipation. However, in advancedstages, when the grain radii distributions for both con-cepts begin to deviate more and more, the multigrain con-cept provides a lower dissipation (see Fig. 2a). Asdiscussed later, this is due to a narrowing of the distribu-tion function treated by the multigrain concept (seeFig. 1b).

5. Kinetics of the system with initial bimodal distribution

For the treatment of systems with a bimodal grain radiidistribution [18], Fig. 10b of [19], one can use a linearcombination of two Rayleigh’s distributions

fB ¼ KR½Ac5=21 expð�c1R2Þ þ ð1� AÞc5=2

2 expð�c2R2Þ� ð29Þmeeting again the condition (11) of the constant volume ofthe system for arbitrary values of parameters from the region0 6 A 6 1, c1 > 0, c2 > 0. Using Eq. (19), one can write

oGoA¼ 2pcK

Z 1

0

R3 c5=21 expð�c1R2Þ � c5=2

2 expð�c2R2Þh i

dR

¼ pcK c1=21 � c1=2

2

h ið30Þ

oGoc1

¼ 2pcAKc5=21

Z 1

0

R3 5

2c1

� R2

� �expð�c1R2ÞdR ¼ pcAK

2c1=21

ð31ÞoGoc2

¼ 2pcð1� AÞKc5=22

Z 1

0

R3 5

2c2

� R2

� �expð�c2R2ÞdR

¼ pcð1� AÞK2c1=2

2

: ð32Þ

Eq. (21) yield

1

2

o2Q

o _A2¼ pK

2M

�Z 1

0


2 expð�c2R2Þh i2

Ac5=21 expð�c1R2Þ þ ð1�AÞc5=2

2 expð�c2R2ÞdR

ð33Þ

1

2

o2Qo _c2

1

¼ pA2Kc1

8M

�Z 1

0

Rðð3� 2c1R2Þexpð�c1R2ÞÞ2

Ac5=21 expð�c1R2Þþ ð1�AÞc5=2

2 expð�c2R2ÞdR

ð36Þ

1

2

o2Qo _c1o _c2

¼ pAð1� AÞKc1=21 c1=2

2

8M

�Z 1

0

Rð3� 2c1R2Þð3� 2c2R2Þ expð�ðc1 þ c2ÞR2ÞAc5=2

1 expð�c1R2Þ þ ð1� AÞc5=22 expð�c2R2Þ

dR

ð37Þ

1

2

o2Qo _c2

2

¼ pð1� AÞ2Kc2

8M

�Z 1

0

Rðð3� 2c2R2Þ expð�c2R2ÞÞ2


2 expð�c2R2ÞdR:

ð38Þ

The integrals in Eqs. (33)–(38) must be calculated numeri-cally. The kinetics of the system is then given by Eq. (18).

An initial bimodal grain radii distribution can beobtained by choosing A(0) = 0.003, c1(0) = 25 andc2(0) = 1. The evolution of the parameters is depicted inFig. 3. One can observe that the contribution correspond-ing to c1 gradually decreases and disappears at timet = 0.027741 (A! 0 at that time; see Fig. 3a). The valueof the parameter c1 increases drastically (see Fig. 3b),which correlates with a significant refinement of grains cor-responding to the c1-contribution. The c2-contributionexhibits gradual coarsening (decreasing of the value of c2;see Fig. 3c) and survives as the only one. Starting witht = 0.027741, the evolution of the system can be describedby Eq. (25).

The time evolution of the distribution corresponding tothe evolution of the parameters A, c1 and c2 (calculatedalong the distribution concept) is depicted by the solid linesfor times t = 0, 0.005, 0.01, 0.015, 0.02, 0.025 and 0.027741

0.00 0.01 0.02 0.031E-11

1E-9

1E-7

1E-5

1E-3

Par

amet

er A

Time

0.00 0.01 0.02 0.0310

100

1,000

10,000

100,000

Par

amet

er c

1

Time

0.00 0.01 0.02 0.030.96

0.97

0.98

0.99

1.00

Par

amet

er c

2

Time

a

b

c

Fig. 3. Evolution of parameters of the bimodal distribution functioncalculated up to the time 0.027741. (a) Parameter A. (b) Parameter c1. (c)Parameter c2.

Fig. 4. Comparison of the evolution of bimodal grain radii distributionsby the application of multigrain and distribution concepts. Individualcurves correspond to the times denoted in the figure.


in Fig. 4. This evolution is compared with that based on themultigrain concept for the same starting bimodal distribu-tion and the same plot times (dotted lines).

6. Kinetics of system with Hillert’s distribution

The well-known Hillert’s theory [1] provides the distri-bution function fH being the steady-state solution of Eq.(9) as

fHðR;RCðtÞÞ ¼L

R4C

R=RC

2�R=RCð Þ5 exp �62�R=RC

� �; 0 6 R < 2RC;

otherwise fH ¼ 0:

(

ð39Þ

with L = 503.88289 to conform with Eq. (10). The kineticsof the steady-state grain coarsening is then given by theevolution equations for the critical radius RC(t) and/orthe mean radius RH ¼ 8RC=9 (see Hillert [1]) as

_RC ¼cM2RC

and _RH ¼32cM

81RH

: ð40Þ

It is well known that Eqs. (39) and (40) describe per-fectly the evolution of grain radii distribution obtainedby the present multigrain concept in the steady state. Ifany different reasonable grain radii distribution functionis used as the starting one, the distribution treated by themultigrain concept approaches the distribution given byEq. (39) (see the discussion by Fischer et al. [5]). In otherwords, the grain radii distribution function given by Eqs.(39) and (40) is a long-term solution of the grain coarseningtreated by the multigrain concept.

The distribution concept can also be applied to Hillert’sdistribution given by Eq. (39). The function involves onlyone parameter RC. Then Eqs. (19) and (20) provide

1

2

dQH

d _RC

¼ 2p _RC

M

Z 2RC

0

R2

fH

Z 2RC

R

dfH

dRC

dR0� �2

dR ¼ 4pL _RC

27e3MRC

ð41ÞdGH

dRC

¼ 2pcZ 2RC

0

R2 dfH

dRC

dR ¼ � 2pLc

27e3R2C

: ð42Þ

Using Eq. (18), Hillert’s equation (40) is reproduced.Applying Eqs. (40)–(42), the dissipation in the system

described by Hillert’s distribution and characterized bythe mean grain radius RH is given by

QH ¼1

2

dQH

d _RC

_RC ¼ �dGR

dRC

_RC ¼512pLc2M

ð27eÞ3R3H

: ð43Þ

7. Comparison of kinetics of systems with Rayleigh’s and

Hillert’s distributions

The total dissipation expressing the rate of decrease ofthe total Gibbs energy in the system provides valuable

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

Hillert Rayleigh

Dis

trib

utio

n fu

nctio

n

Grain radius

Fig. 5. Comparison of Rayleigh’s and Hillert’s distribution functions forsystems of the same volume and the same grain boundary area.


information on the kinetics of the system. This acts as amotivation to compare the total dissipation in systems withRayleigh’s distribution, QR, and Hillert’s distribution, QH,by using Eqs. (28) and (43). First we assume identical meangrain radii in both systems, i.e. RR ¼ RH (see Eqs. (26) and(40)), and obtain QR/QH = 1.28379. The ratio differs from1 to a surprising degree. Therefore it is necessary to per-form a second evaluation for systems with the same volumeand the same grain boundary area by using the additionalconditionZ 1

0

R2f dR ¼ 1 ð44Þ

for f ” fH (Eq. (39)) and f ” fR (Eq. (22)). One finds the val-ues of RC = 0.92914266 and c = 9p/16, to which the meanradii RH ¼ 0:82590459 and RR ¼ 2=3 of Hillert’s and Ray-leigh’s distributions correspond. Both distributions arecompared in Fig. 5. After the insertion of the values ofRH and RR one obtains the ratio QR/QH = 2.44094, whichis drastically different from 1. Thus, one can conclude thatthe grain radii distribution has a strong impact on the graincoarsening kinetics.

8. Discussion

The reason why the kinetics of the system with Ray-leigh’s distribution is significantly faster than that withHillert’s distribution evidently originates from the fact thatRayleigh’s distribution is markedly wider than Hillert’s (seeFig. 5). The narrower a distribution function, the smallerthe deviation of each grain radius from the critical oneand the lower the magnitude of the rate of the radius (seeEq. (9)) and the total dissipation.

The rather complicated interaction amongst grains isreflected in the grain radii distribution function. An advan-tage of the distribution concept lies in the fact that itreflects the interaction amongst grains purely phenomeno-logically only by choosing the proper family of distributionfunctions. The higher dissipation in the case of Rayleigh’sdistribution compared with Hillert’s supports the experi-mental observations that this distribution often fits the real

grain structures much better than Hillert’s. On the otherhand, one cannot overestimate the role of the distributionconcept by presenting it as a tool for understanding themathematics and physics behind the complicated interac-tion amongst grains. The distribution concept is rather ahandy tool for phenomenological description of graingrowth being much more flexible than recent phenomeno-logical concepts.

To remain as realistic about the grain coarsening as pos-sible, the following three steps are recommended:

� the determination of a family of grain radii distributionfunctions involving a reasonable number of parametersfrom experimental observations;� the determination of the initial values of the parameters;� the determination of the time evolution of the parame-

ters by means of the distribution concept.

9. Summary

The achievements in the paper can be summarized asfollowing:

� Based on Onsager’s principle, multigrain and distribu-tion concepts are developed. The equations for the evo-lution of individual grains derived within the frameworkof the multigrain concept coincide with Hillert’s classicalequations. The new distribution concept allows thetreatment of the evolution of the grain radii distributionwithout the necessity of knowing the evolution equa-tions for individual grains.� The distribution concept is applied to the monomodal

and bimodal Rayleigh-type grain radii distribution func-tions. The results of simulations along the distributionconcept are compared with those calculated by meansof the multigrain concept. The distribution concept isalso applied to the treatment of the steady-state graincoarsening with Rayleigh’s and Hillert’s distributionfunctions. Their comparison clearly indicates that thekinetics of the system depends decisively on the widthof the distribution function.

Acknowledgements

Financial support by Materials Center Leoben Fors-chung GmbH (Project SP19) and by Research Plan ofInstitute of Physics of Materials (Project CE-Z:AV0Z20410507) is gratefully acknowledged.

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a new approach to modelling of non-steady grain growth

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