Materials Science and Engineering A 385 (2004) 166–174

Modelling of kinetics in multi-component multi-phasesystems with spherical precipitates

I: Theory

J. Svobodaa, F.D. Fischerb,e,f , P. Fratzlc,e,f,1, E. Kozeschnikd,f,∗a Institute of Physics of Materials, Academy of Sciences of the Czech Republic,Zizkova 22, CZ-61662, Brno, Czech Republic

b Institute of Mechanics, Montanuniversit¨at Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austriac Institute of Metal Physics, Montanuniversit¨at Leoben, Jahnstraße 12, A-8700, Leoben, Austria

d Institute for Materials Science, Welding and Forming, Graz University of Technology, Kopernikusgasse 24, A-8010 Graz, Austriae Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, Jahnstraße 12, A-8700 Leoben, Austria

f Materials Center Leoben, Franz-Josef-Straße 13, A-8700 Leoben, Austria

Received 22 January 2004; received in revised form 3 June 2004

A

inciple isp phases int precipitates of differentc©

K

1

hootmtsise

P

ter-nenty beges

truc-e theim-

cip-n ofTheasesas adientn of achesod

0d

bstract

A new model for the evolution of the precipitate structure derived by means of application of the thermodynamic extremum prresented. The model describes the evolution of the radii and of the chemical composition of individual precipitates of different

he multi-component system. In connection with a proper theory of nucleation, the model is able to describe the evolution of thetructure in the classical stages of nucleation, growth and coarsening as well as interaction of precipitates of different phases,hemical composition and of different sizes via diffusion in the matrix.2004 Elsevier B.V. All rights reserved.

eywords:Modelling; Diffusion; Phase transformation; Thermodynamics; Precipitation

. Introduction

The evolution of the precipitate microstructure in alloysas been studied for many years and is generally described asccurring in three stages, which can, however, considerablyverlap[1,2]. First, precipitates nucleate and start to grow,he change in the chemical composition of the supersaturatedatrix being still negligible. Then, during the growth stage of

he precipitates, the supersaturation of the matrix decreasesubstantially. Finally, in the coarsening stage, large precip-tates grow at the expense of smaller ones and the matrixupersaturation is very low and gradually decreases towardsquilibrium. The picture is complicated by the fact that the

∗ Corresponding author. Tel.: +43 316 873 7181; fax: +43 316 873 7187E-mail address:ernst.kozeschnik@iws.tugraz.at (E. Kozeschnik).

1 Present address: Max-Planck-Institute of Colloids and Interfaces, 14424otsdam, Germany

evolution of the microstructure is influenced by misfit inactions between precipitates and that, for multi-compoalloys, the chemical composition of the precipitates machanging (and far from their equilibrium values) at all staof phase separation.

There are a number of approaches to model microstural evolution. The most detailed approach is to describalloy at the atomic level using for example Monte Carlo sulations of the Ising model (for reviews see[3,4]). While thisapproach gives important details of the diffusion and preitation processes, it is much too detailed for a descriptiomulti-component, multi-particle, multi-phase processes.usual method is to use a continuum description of the phwith a boundary in between, which is either describedsharp interface (sharp interface models) or by a steep grain a continuous phase variable, such as the concentratiogiven atomic species (phase-field models). Both approaare able to predict microstructural evolution with go

921-5093/$ – see front matter © 2004 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2004.06.018

J. Svoboda et al. / Materials Science and Engineering A 385 (2004) 166–174 167

accuracy (for a recent review, see[5]). Both the phase-fieldmethods[6] and the sharp interface models[7] have beendeveloped sufficiently far to be applied to many-particlesystems. The time-consuming simulations are usuallyperformed for two-phase systems in two dimensions. Inprincipal, there is no restriction to apply this concept to threedimensional multiphase configurations as real structuralmaterials.

There are attempts published that treat complex precipita-tion reactions in terms of an extended Johnson–Mehl–Avramitheory, e.g., refs.[8,9]. To the knowledge of the authors,however, a simpler tool for an approximate - but for thepractice still sufficiently accurate - prediction of the evolutionof a complex precipitate structure, which also rigorouslytakes into account multi-component thermodynamics, doesnot exist. Therefore, a new model is developed, based onapplication of the Onsager extremum principle[10] withina mean field approach, which simultaneously keeps thecomplexity and the most important features of the realmicrostructure. This paper is restricted to microstructureswith a low volume fraction of different kinds of nearlyspherical precipitates with stochiometric composition, asit is e.g., the case in low alloy steels. In the present form,the model delivers the evolution equations of all relevantparameters like size distribution or chemical composition ofa orieso s oft

2e

ions( latedb ves tooli ed in[ era-t era-t ass vityn nglec fu-so ewr alsor con-v inm

atep raturea ns ong ., thet nso e

of the total Gibbs energy dissipation,Q, can be expressed bymeans ofqi andqi (Q = Q(q1, q2, . . . , qK, q1, q2, . . . , qK)).In case thatQ is a positive definite quadratic form of the ratesqi (the kinetic parameters), the evolution of the system isgiven by the set of linear equations with respect to ˙qi [11]:

∂G

∂qi= −1

2

∂Q

∂qi, i = 1, . . . , K (1)

This treatment can be applied to the system considered in theintroduction.

3. System description

The system consists of a matrix and precipitates. We as-sumes substitutional components andp interstitial compo-nents in the system,n= s+ p, and the numbers are fixed. LetNi (i = 1, . . ., n) be a fixed number of moles of componenti in the system,m the number of precipitates in the system,ρk (k = 1, . . ., m) the radius of a precipitatek, uki (k = 1, . . .,m, i = 1, . . ., n) the mean site fraction of componenti in theprecipitatek, andUk the fixed structure parameter of the pre-cipitatek (e.g., in the case of M23C6,Uk = 6/23). We assumethat the ratio of interstitial to substitutional components int d thef

∑

eso sub-sa aresv onem ciesi si-t owtho ionali st bep

m

c

Tg

N

ll precipitating phases in all stages. The classical thef nucleation, growth and coarsening are the limit case

he model.

. Thermodynamic extremum principle andvolution equations

The extremum principle substituting transport equatequations for the heat conduction) has been first formuy Onsager in 1931[10]. In 1991, Svoboda and Turek hahown[11], that the principle can be used as a handy

n the development of models. The treatment present11] is derived for a closed system under constant tempure and pressure, which is a good approximation of opion conditions for a majority of systems. This principle wuccessfully used by the authors e.g., in modelling of caucleation and growth, sintering, creep in superalloy sirystals, grain growth, Ostwald ripening, diffusion or difional phase transformation (see e.g., refs.[12–16]) and byther groups[17,18]. In all cases, the principle offered nesults being compatible with existing knowledge, buteproduced classical results, however, derived in a moreenient way (e.g., for grain growth or Ostwald ripeningulti-component system).Let qi (i = 1, . . ., K) be the suitable independent st

arameters of a closed system under constant tempend external pressure. Then under several assumptioeometry of the system and/or coupling of processes, etc

otal Gibbs energy of the systemGcan be expressed by meaf the state parametersqi (G=G(q1,q2, . . .,qK)), and the rat

he precipitates corresponds to a fixed stoichiometry, anollowing relations hold:

s

i=1

uki = 1,n∑

i=s+1

uki = Uk, k = 1, . . . , m (2)

For simplicity, we assume that the partial molar volumf the substitutional components are the same for alltitutional components in all phases (Ωi = Ω, i = 1, . . ., s)nd partial molar volumes of all interstitial componentsmall and set to zero (Ωi = 0, i = s + 1, . . ., n). Then, theolume corresponding to one mole of site positions (toole of substitutional atoms, if the site fraction of vacan

s negligible),Ω, is independent of the chemical compoion and of the phase. To capture the fact that during grf precipitates significant stresses can be built up, addit

nformation about the expansion of each precipitate murovided to the model.

The mean concentration in the precipitatek, cki, k= 1, . . .,, can be introduced as

ki = uki

Ω, k = 1, . . . , m, i = 1, . . . , n (3)

he number of moles of the componenti in the matrixN0i isiven by

0i = Ni −m∑

k=1

4πρ3kcki

3, i = 1, . . . , n (4)

168 J. Svoboda et al. / Materials Science and Engineering A 385 (2004) 166–174

and the total number of moles of substitutional componentsin the matrix is

N0 =s∑

i=1

N0i (5)

Then the mean concentrationc0i of componenti in the matrixis given by

c0i = N0i

ΩN0, i = 1, . . . , n (6)

The state of the system is uniquely described by indepen-dent state parametersm, ρk, k = 1, . . ., mandcki, k = 1, . . .,m, i = 2, . . ., s, s+ 2, . . ., n. The concentrationsc0i in the ma-trix can be calculated from independent state parameters byusingEqs. (2)–(6). The concentrationsck1 andcks+1 can becalculated fromEqs. (2) and (3). The independent parametersdevelop in time and describe the system evolution.

4. The total Gibbs energy of the system

Let µ0i (i = 1, . . ., n) be the chemical potential of com-ponenti in the matrix andµki (k = 1, . . ., m, i = 1, . . ., n)be the chemical potential of componenti in the precipitatek s ofta , e.g.,b lG

G

wc las-t andi ld pa-r ivenb

∂G

∂cki= 4πρ3

k

3(µki − µ0i − µk1 + µ01),

k = 1, . . . , m, i = 2, . . . , s (9)

∂G

∂cki= 4πρ3

k

3(µki − µ0i − µks+1 + µ0s+1),

k = 1, . . . , m, i = s + 2, . . . , n (10)

Eqs. (8)–(10)represent generalized driving forces for theindependent state parameters. Note that the rate of radiuschange (Eq. (8)) is driven by capillarity, by mechanicaldriving force and by chemical driving force. On the otherhand, the rates in chemical composition change (Eqs. (9) and(10)) are driven only by the corresponding chemical drivingforce.

5. The rate of the total Gibbs energy dissipation inthe system

During the system evolution, the total Gibbs energy dissi-pates. Let us consider three dissipative processes:

1 Migration of interfaces characterized by their mobilities

2 ter-

3 by

n isg

Q

W icalc ed orc re-c

j

E intsa sto-c

tesi

Q

. All chemical potentials can be expressed as functionhe concentrationscki (k = 0, . . ., m, i = 1, . . ., n), and it isssumed, that they are known (they can be calculatedy means of the CALPHAD approach[19]). Then the totaibbs energy of the system,G, is given by

=n∑

i=1

N0iµ0i +m∑

k=1

4πρ3k

3

(λk +

n∑i=1

ckiµki

)

+m∑

k=1

4πρ2kγk (7)

hereγk is the interface Gibbs energy density andλk ac-ounts for the contribution of the elastic energy (and pic work) due to the volume expansion of precipitatess an input information, see theAppendix A. The partiaerivatives ofG with respect to the independent stateameters (m is assumed to be fixed for the moment) are gy:

∂G

∂ρk= 8πρkγk + 4πρ2

k

[λk + 1

Ωµk1 − µ01

+Uk(µks+1 − µ0s+1)+s∑

i=2

cki(µki − µ0i − µk1

+µ01) +n∑

i=s+2

cki(µki − µ0i − µks+1 + µ0s+1)

,

k = 1, . . . , m (8)

Mk, k = 1, . . ., m.Diffusion of all components in the precipitates characized by diffusion coefficientsDki, k = 1, . . ., m, i = 1, . . .,n.Diffusion of all components in the matrix characterizeddiffusion coefficientsD0i, i = 1, . . ., n.

The total rate of dissipation due to interface migratioiven by

1 =m∑

k=1

4πρ2k ρ

2k

Mk

(11)

e can assume that during the evolution of the chemomposition in precipitates all components are depositollected uniformly. Then the radial diffusive flux in the pipitate is given by

ki = − rcki

3, 0 ≤ r ≤ ρk (12)

qs. (2) and (3)automatically ensure appropriate constramong the diffusive fluxes in the precipitate to maintainhiometry.

The total dissipation rate due to diffusion in precipitas given in[11] as

2 =m∑

k=1

n∑i=1

∫ ρk

0

RT

ckiDki

4πr2j2ki dr

=m∑

k=1

n∑i=1

4πRTρ5k c

2ki

45ckiDki

(13)

J. Svoboda et al. / Materials Science and Engineering A 385 (2004) 166–174 169

whereR is the gas constant andT is the absolute temperature.In the integrand inEq. (13), the local value of the concentra-tion was replaced by its mean valuecki. As the rates ˙cki areconstrained by

s∑i=1

cki = 0 andn∑

i=s+1

cki = 0 (14)

(seeEqs. (2) and (3)), Q2 can be expressed by means ofindependent rates ˙cki as

Q2 =m∑

k=1

4πRTρ5k

45

s∑

i=2

c2ki

ckiDki

+n∑

i=s+2

c2ki

ckiDki

+(∑s

i=2cki)2

ck1Dk1+(∑n

i=s+2cki)2

cks+1Dks+1

](15)

The rate of change of the number of moles of componenti inthe precipitatek is given by

Nki = 4πρ2k(ρkc0i − J∗

ki) (16)

whereJ∗ki is the radial flux of the componenti in the matrix

at the interface. The number of molesNki in the precipitateis given by

N

D

N

Tj xi d( no

J

T sportt , seee

(

T toE andste la-t ctuals

witha n a

sphere of radiusZ. Then the diffusive fluxJki in the matrixsurrounding the precipitate is given by

Jki = J∗ki

ρ2k

r2

Z3 − r3

Z3 − ρ3k

, ρk < r ≤ Z (21)

Of course, the fluxJki is only valid in a spherical configurationwith no disturbance of this rather simplified geometry, finallyrequesting a dilute distribution of precipitates. The problemsolution is much more difficult, if the diffusion fields of theindividual particles interact, see e.g., Vorhees and Glicksman[22,23] for details. Finally, the contribution of the diffusivefluxes as outlined below should be seen in the sense of a firstapproximation.

The total rate of dissipation due to diffusive fluxes in thematrix assumingZ ρk is

Q3 =m∑

k=1

n∑i=1

∫ Z

ρk

RT

c0iD0i4πr2J2

ki dr

≈m∑

k=1

n∑i=1

4πRTρ3k(ρk(cki − c0i) + ρkcki/3)2

c0iD0i(22)

Again, in the integrand inEq. (22), the local value of theconcentration is replaced by its mean valuec0i. Furthermore,the higher powers ofρk/Z have been neglected. ExpressingQ

Q

T+ eters( fQ areg

ki = 4πρ3kcki

3(17)

ifferentiation ofEq. (17)gives

˙ki = 4πρ2

kckiρk + 4πρ3k cki

3(18)

he second term inEq. (18) represents−4πρ2kj

∗ki, j∗

ki =ki|r=ρk - seeEq. (12)), wherej∗

ki represents the diffusive flun the precipitate at its surface. Comparison ofEqs. (16) an18)makes it possible to expressJ∗

ki as a linear combinatiof ρk andcki as

∗ki = ρk(c0i − cki) − ρkcki

3(19)

he same result can be achieved by applying the tranheorem to the mass flux leading to the jump condition.g., Lehner[20] or Fried and Gurtin[21],

J∗ki − j∗

ki) = ρk(c0i − cki) (20)

he diffusive fluxes inEq. (20) are constrained dueqs. (2)–(6), which ensure mass balance in the systemtochiometry in the precipitates. As the concentrationscki inhe precipitates as well as the concentrationsc0i in the matrixvolve in time,Eq. (20)does not represent some fixed reions among the diffusive fluxes, but depends on the atate of the system.

The fluxes in the matrix are assumed to be radialuniform deposition or collection of components withi

3 by means of independent rates leads to

3 =m∑

k=1

4πRTρ3k

[s∑

i=2

(ρk(cki − c0i) + ρkcki/3)2

c0iD0i

+n∑

i=s+2

(ρk(cki − c0i) + ρkcki/3)2

c0iD0i

+(ρk(ck1 − c01) − ρk

∑si=2cki/3

)2c01D01

+(ρk(cks+1 − c0s+1) − ρk

∑ni=s+2cki/3

)2c0s+1D0s+1

](23)

he total rate of dissipation in the system,Q = Q1 + Q2Q3, can be expressed by independent kinetic param

usingEqs. (11), (15) and (23)). The partial derivatives owith respect to the independent kinetic parameters

iven by

∂Q

∂ρk= 8πρ2

k

[1

Mk

+ RTρk

n∑i=1

(cki − c0i)2

c0iD0i

]ρk

+ 8πRTρ4k

3

[s∑

i=2

(cki − c0i

c0iD0i− ck1 − c01

c01D01

)cki

+n∑

i=s+2

(cki − c0i

c0iD0i− cks+1 − c0s+1

c0s+1D0s+1

)cki

,

k = 1, . . . , m (24)

170 J. Svoboda et al. / Materials Science and Engineering A 385 (2004) 166–174

∂Q

∂cki= 8πRTρ4

k

3

(cki − c0i

c0iD0i− ck1 − c01

c01D01

)ρk

+ 8πRTρ5k

45

[(1

ckiDki

+ 5

c0iD0i

)cki

+s∑

j=2

(1

ck1Dk1+ 5

c01D01

)ckj

,

k = 1, . . . , m, i = 2, . . . , s (25)

∂Q

∂cki= 8πRTρ4

k

3

(cki − c0i

c0iD0i− cks+1 − c0s+1

c0s+1D0s+1

)ρk

+ 8πRTρ5k

45

[(1

ckiDki

+ 5

c0iD0i

)cki

+n∑

j=s+2

(1

cks+1Dks+1+ 5

c0s+1D0s+1

)ckj

,

k = 1, . . . , m, i = s + 2, . . . , n (26)

6

ni pen-d

a

T notd s ofkuy flf

A

A

A1i = Ai1 = RTρ2k

3

(cki+1 − c0i+1

c0i+1D0i+1− cks+1 − c0s+1

c0s+1D0s+1

),

i = s + 1, . . . , n − 1, (31)

Aij = RTρ3k

45

[(1

ckiDki

+ 5

c0iD0i

)δij

+(

1

ck1Dk1+ 5

3c01D01

)],

i = 2, . . . , s, j = 2, . . . , s (32)

Aij = RTρ3k

45

[(1

cki+1Dki+1+ 5

c0i+1D0i+1

)δij

+(

1

cks+1Dks+1+ 5

c0s+1D0s+1

)],

i = s + 1, . . . , n − 1, j = s + 1, . . . , n − 1

(33)

Aij = Aji = 0, i = 2, . . . , s, j = s + 1, , n − 1 (34)

SinceQ is a positive definite quadratic form in the indepen-dent kinetic parameters, seeEqs. (11), (15), (23), symmetryand positive definiteness of the matrixAij is guaranteed au-t

y

B

I

B

w

F

r (sumo thei

B

B

. Evolution equations

For a fixed number of precipitates,m, the system evolutios given by a set of linear equations with respect to indeent kinetic parameters according toEq. (1)as

∂G

∂ρk= −1

2

∂Q

∂ρk, k = 1, . . . , m (27)

nd

∂G

∂cki= −1

2

∂Q

∂cki, k = 1, . . . , m, i = 2, . . . , s,

s + 2, . . . , n (28)

he matrix of the set of linear equations is, however,ense, and it can be decomposed for individual valueinto m sets of linear equations of dimensionn − 1. Lets denote for a fixedk y1 ≡ ρk, yi ≡ cki, i = 2, . . ., s andi−1 ≡ cki, i = s+ 2, . . ., n. Then the matrixAij of the set oinear equations

∑n−1j=1Aijyj = Bi, i = 1, . . ., n− 1, is given

or a fixedk as

11 = 1

Mk

+ RTρk

n∑i=1

(cki − c0i)2

c0iD0i(29)

1i = Ai1 = RTρ2k

3

(cki − c0i

c0iD0i− ck1 − c01

c01D01

),

i = 2, . . . , s (30)

omatically.The right hand side of the set of equations is given b

1 = −2γkρk

− λk − 1

Ωµk1 − µ01 + Uk(µks+1 − µ0s+1)

−s∑

i=2

cki(µki − µ0i − µk1 + µ01)

−n∑

i=s+2

cki(µki − µ0i − µks+1 + µ0s+1) (35)

n a much more convenient formB1 can be expressed as

1 = −2γkρk

+ F (36)

here

= −λk −n∑

i=1

cki(µki − µ0i) (37)

epresents the standard expression for the driving forcef the mechanical and chemical driving forces) acting on

nterface (see e.g.,[24,25]).Furthermore

i = −ρk

3(µki − µ0i − µk1 + µ01), i = 2, . . . , s (38)

i = −ρk

3(µki+1 − µ0i+1 − µks+1 + µ0s+1),

i = s + 1, . . . , n − 1 (39)

J. Svoboda et al. / Materials Science and Engineering A 385 (2004) 166–174 171

7. Evolution of the number of precipitates

The numbermof precipitates in the system (or the num-ber per unit volume) is an important characteristic quantityof the system influencing the transformation kinetics and themechanical properties of the system. The number of precip-itates increases due to the nucleation of precipitates duringthe nucleation stage and decreases due to the dissolution ofprecipitates during the coarsening stage. In the following, webriefly describe a theory of the nucleation process in a multi-component system used in our simulations. According to theknowledge of the authors, a general and approved nucleationtheory for multi-component systems is still not available (seee.g.,[26]).

In the case of a binary system, the steady state nucleationratemSS is given by the classical nucleation theory as

mSS = Zβ∗NC exp

(−∆G∗

kBT

)(40)

with Z being the Zeldovich factor (as defined in ref.[26]),β∗ the atomic attachment rate for the critical nucleus,NCthe number of available nucleation sites,kB the Boltzmannconstant and(G∗ is the nucleation barrier given by

∆G∗ = fγ3k (41)

f fhf nb ntBc

β

w

ρ

w efi blea thes y of Ba omics

n isc asesa ul-t ctori cle-a ualc canc es ort d of

the precipitate as well as for the different values of the matrixdiffusion coefficients of the components.

In the case of the above binary A-B system, the precipi-tates do not change the chemical composition (˙ckB ≡ 0). Ifwe apply the concept ofSection 6, the growth rate of theprecipitate is given by

ρk = F − (2γk/ρk)

RTρku0BD0BΩ. (44)

Considering a multi-component system without any changein chemical composition in the precipitates, the growth rateof the precipitate is given by

ρk = F − (2γk/ρk)

RTρk

[n∑

i=1

(cki − c0i)2

c0iD0i

]−1

(45)

In the binary system, the kinetics of nucleation as well asthe growth of precipitates are controlled by diffusion of Batoms in the matrix (note the productu0BD0B in Eqs. (42)and (44)). If one assumes that the nucleation of precipitatesin multi-component systems is also controlled by the samecombination of diffusive processes as the growth of the pre-cipitates, one can replace the productu0BD0B in Eq. (42)by

the expression[Ω∑n

i=1(cki − c0i)2/c0iD0i]−1

, andEq. (42)can be rewritten for the multi-component system as:

β

N )a

le-a ratem

m

w

τ

T p-id

m

8

and-i ticlesi oft on-c haves item is

F2

is the shape factor of the nucleus (f = 16π/3 in the case oomogeneous nucleation of spherical nuclei andf < 16π/3

or heterogeneous nucleation).F is the driving force givey Eq. (37). In a binary system, A-B with a dilute componeand precipitates consisting of only B atoms,β∗ can be

alculated as

∗ = 4πρ∗2

a2

D0Bu0B

a2(42)

ith

∗ = 2γkF

(43)

herea is the lattice spacing andρ∗ is the critical radius. Thrst factor inEq. (42)corresponds to the number of availatomic sites on the surface of the critical nucleus andecond factor represents the mean exchange frequenctoms between the matrix and the critical nucleus per atite at the interface.

In the case of a multi-component system, the situatioomplicated by the fact that precipitates of different phnd of different chemical compositions may nucleate sim

aneously, which do not allow to use the simple second fan Eq. (42). In a first step, one has to decide about the nution of the specific types of precipitates with their individhemical composition. With this information at hand, onealculate the driving forceF (Eq. (37)). In the next step, thecond factor used inEq. (42)must be modified to account fhe differences in chemical composition of the matrix an

∗ = 4πρ∗2

a4Ω

[n∑

i=1

(cki − c0i)2

c0iD0i

]−1

(46)

ote that for the binary dilute A-B system,Eqs. (42) and (46re identical.

Based onEqs. (40), (41) and (46), the steady state nuction ratemSScan be evaluated. The transient nucleation

˙ nuc is given by

˙ nuc = mSSexp(−τ

t

)(47)

ith τ being the incubation time calculated from

= 1

2β∗Z2(48)

he dissolution rate ˙mdis is given by the number of precitates whose radiusρk falls below a predefined valueρk,disuring a time unit. The total rate ˙m finally follows as

˙ = mnuc − mdis (49)

. Discussion

Obviously there is an ongoing interest in the understng of the growth and/or coarsening of second phase parn a matrix, see e.g.,[27]. These authors apply the solutionhe classical diffusion equation with given equilibrium centrations at a moving interface. As Svoboda et al.hown in[25] this is only acceptable in the case of an infinobility of the interface. The solution for finite mobility

172 J. Svoboda et al. / Materials Science and Engineering A 385 (2004) 166–174

described in[25] for a binary system. The general solutionfor the multi-component system with finite interface mobilityseems to be extremely complicated. In the present approach,the solution of the problem with finite mobility is not difficultat all, since it requires only taking into account the dissipationtermQ1. (seeEq. (11)).

When solving the problem of diffusional transformationsthree different approaches for the treatment of diffusion canbe applied:

1 The most comprehensive description based on the solu-tion of the Fick’s second law (offers the time evolution ofconcentration profiles)[27–29].

2 Stationary solutions assuming a constant deposi-tion/collection rate of components within the wholediffusional zone (offers the steady state concentrationprofiles)[30].

3 Global formulation of the problem by means of the ther-modynamic extremum principle which enables one to de-termine the kinetics of the system without the necessityof knowing the concentration profiles and the contact con-ditions for chemical potentials at the migrating interface.This approach is fully consistent with the mean field ap-proach.

Both approaches 1 and 2 are suitable for a detailed solutiono offersa hichi

hichl thepf

∑

T thes theirp s forv rtions )a

∑

i ionala x int e anys n theo omsi them iont

eroa e the

same partial molar volume and all interstitial elements tohave a zero partial molar volume, there is never any volumeflux in the system. This means that no deformation due todiffusion or phase transformation is reflected by the model.However, local straining near individual precipitate due totransformation strains can be taken into account by a factorλk, seeEq. (7)and theAppendix A.

In many cases, the precipitates take on shapes, such ascubes, discs or needles that deviate from the spherical geom-etry. This requires some non-trivial, however, possible mod-ifications of the model at four places, namely

1 In the calculation of the surface energy term and the elasticenergy and plastic work term due to the volume expansionof precipitates inEq. (7).

2 In the calculation of the dissipation rate in the precipitate(Eq. (15)).

3 In the calculation of the dissipation rate in the matrix(Eq. (22)).

4 In the calculation of the dissipation rate due to interfacemigration (Eq. (11)).

This can be managed by introduction of correction factors(shape factors), which could be calculated in preprocessingfor a given precipitate shape. Moreover, these shape factorscan also be dependent on the volume fraction of the precip-i elo

9

ump mi-c largen d ofd trix.T r ofp ion ofe po-s ablet ipi-t ng. Itm pre-c icalc tialo rt IIo

A

theA meo rialsC sup-p , by

f selected problems. On the other hand, the approach 3n easy approximate solution with a sufficient accuracy w

s suitable for application to complex systems.The present model involves a number of constraints w

ead to implicit bonds amongst the diffusional fluxes inrecipitates and in the matrix. FromEqs. (12) and (13)it

ollows that

s

i=1

jki = 0 andn∑

i=s+1

jki = 0 (50)

his means, that during diffusion in the precipitates bothubstitutional and interstitial atoms may only exchangeositions. This implies that there are no sources and sinkacancies in the interiors of precipitates. Such an asseeems to be reasonable. IfEqs. (2), (3), (5), (6), (14), (19nd (21)are used, the relation

s

i=1

Jki = 0 (51)

s obtained. This means that the sum of fluxes of substituttoms in the matrix is zero, which means no vacancy flu

he matrix. Thus, the present model does not assuminks or sources for vacancies in the whole system. Other hand, there is no constraint to fluxes of interstitial at

n the matrix. The interstitial atoms can be collected inatrix without any restrictions and transported by diffus

o the growing precipitates.Since the sum of all fluxes of substitutional atoms is z

nd all substitutional components are assumed to hav

tates, which would not limit the applicability of the modnly to small volume fractions of the precipitates.

. Summary

Based on the application of the thermodynamic extremrinciple the evolution equations for the precipitaterostructure have been derived. The model admits aumber of interacting precipitates of different sizes anifferent phases embedded in the multi-component mahe model accounts for the evolution of the total numberecipitates and of the size and mean chemical compositach precipitate. The evolution of the mean chemical comition of the matrix is also calculated. The model is applico simulations of all classical stages of evolution of precate microstructure i.e., nucleation, growth and coarseniay, however, also account for misfit effect between the

ipitates and the matrix or for the influence of the chemomposition on the stability of the precipitate. The potenf the model is illustrated for examples presented in paf this paper.

cknowledgement

The authors are indebted to the Kplus program ofustrian government for financial support within the fraf the strategic projects SP9 and SP11 of the Mateenter Leoben. The work on the project was alsoorted by Grant agency AS CR (project No. K-1010104)

J. Svoboda et al. / Materials Science and Engineering A 385 (2004) 166–174 173

Kontakt—Austrian–Czech Scientific Cooperation (projectNo. A-15/03) and is performed in the framework of the Re-search Plan of AS CR (No. Z-2041904).

Appendix A

Since a precipitate usually shows a misfit with respect tothe matrix, the misfit accommodation enforces some additiveenergy storage. In the case of an elastic material the additionalenergy is stored in the system as elastic strain energy depend-ing linearly on the volume of the precipitate and non-linearlyon the shape factor of the precipitate. In the case of a spher-ical precipitate the shape factor achieves the value 1.0, andλk is for the same elastic properties in the precipitate and thematrix

λ0k = ε2

0E

(1 − ν)(A.1)

E is the Young’s modulus,ν the Poisson’s ratio and the misfitvolume strain is 3ε0. Since in the case of a dilute distributionof the precipitates the interaction energy terms between twoindividual precipitates are very small and can be neglected,the total stored strain energy can be calculated as the sumof the strain energy contributions of each individual precip-i k toEB tw ter-a nitee ate-m entd

da-t ap-p asticwd licitc Fi-s esa n-h rSz

S

T itse

λ

N .0f -t thatf

the interval 0.199≤ S≤ 1.0 and reaches again the value 1for S= 1. The consequence is that for rather stiff materialsplastification may lead to a higher total work compared withthe purely elastic case.

Of course, if plasticity occurs, the superposition of thestress fields due to the individual precipitates is in principlenot allowed. Therefore, a safe distance between the particlesis necessary to permit the simplified procedure withf(S) asoutlined above. A minimum distancedp for uniformly dis-tributed spherical particles with a similar radiusR,

dp = (1 + ∆−1/3)R/S1/3 (A.4)

assures the validity of this simplified procedure within thesame amount of accuracy as the average yield stressσf isestimated. This is expressed by a factor(, (1− ∆)σy ≤ σf ≤(1 + ∆)σy; σy is the actual yield stress.

It is important to note that the plastic contribution to thetotal work is here added to the elastic energy, since both en-ergy terms must be exerted by the growing precipitate: theelastic energy as a recoverable energy term and the plasticwork finally as heat.

Usually λk often plays a minor role as compared to thechemical contribution inEq. (7). However, this statementis not true in general. The situation may, e.g., changesignificantly, if one considers two precipitates in a nearn mayb thewo[

R

ns in

iley-

429.witz,

03)

173

997)

997)

[

[[ 90)

[[[ 002)

[ 75.

tate. Much literature does exist on this topic going bacshelby’s work, see e.g., Pineau[31] and Bohm et al.[32].oth papers deal with a mismatch strainε0 being constanithin the precipitate. It should be mentioned that the inction energy in the case of spherical particles in an infilastic medium is exactly zero. In the literature this stent is known as the “Bitter–Crum” theorem, for a reciscussion see Fratzl et al.[3].

The situation is much more difficult, if the accommoion enforces any plastification of the matrix. A practicalroach is the numerical calculation of the elastic and plork terms, see, e.g., the work by Leitch und Shi[33] forifferent shapes of precipitates. With respect to an expalculation of the work terms it is referred to a paper bycher and Oberaigner[34] dealing with spherical precipitatnd assuming a constant valueσf of the yield stress in a noardening matrix. According to[34] a plastic softening factois introduced, which achieves the valueS= 1, if the plasticone outside of the precipitate just starts to develop,

= σf (1 − ν)

Eε0,0 ≤ S ≤ 1 (A.2)

he specific total workλk, done by the precipitate due toxpansion, follows withλ0

k from Eq. (A.1)as

k = λ0kf (S), f (S) = 2S(1 − ln S) − S2 (A.3)

ote thatf(S) starts withf(0) = 0 and achieves the value 1or S= 0.199. UsuallyS lies in this interval for a precipitaion at rather high temperatures. It is interesting to note(S) obtains a maximum valuefmax = 1.456 forS= 0.567 in

eighbourhood. In this case, the shape of the precipitatee influenced significantly by the mechanical fields, seeorks by Vorhees and co-workers, e.g., Thornton et al.[35],r the studies by Gross and co-workers, e.g., Muller et al.

36] or Schmidt[37].

eferences

[1] R. Wagner, R. Kampmann, P.W. Voorhees, Phase TransformatioMaterials, Wiley-VCH, Weinheim, 2001.

[2] K. Binder, P. Fratzl, Phase Transformations in Materials, WVCH, Weinheim, 2001.

[3] P. Fratzl, O. Penrose, J.L. Lebowitz, J. Stat. Phys. 95 (1999) 1[4] R. Weinkamer, P. Fratzl, H.S. Gupta, O. Penrose, J.L. Lebo

Phase Transitions (2004) in press.[5] K. Thornton, J. Agren, P.W. Voorhees, Acta Mater. 51 (20

5675.[6] L.Q. Chen, Annu. Rev. Mater. Res. 32 (2002) 113.[7] N. Akaiwa, K. Thornton, P.W. Voorhees, J. Comput. Phys.

(2001) 61.[8] J.D. Robson, H.K.D.H. Bhadeshia, Mater. Sci. Technol. 13 (1

631.[9] J.D. Robson, H.K.D.H. Bhadeshia, Mater. Sci. Technol. 13 (1

640.10] L. Onsager, Phys. Rev. I 37 (1931) 405;

L. Onsager, Phys. Rev. II 38 (1931) 2265.11] J. Svoboda, I. Turek, Philos. Mag. B 64 (1991) 749.12] J. Svoboda, I. Turek, V. Sklenicka, Acta Metall. Mater. 38 (19

573.13] J. Svoboda, H. Riedel, Acta Metall. Mater. 40 (1992) 2829.14] J. Svoboda, P. Lukas, Acta Mater. 48 (2000) 2519.15] J. Svoboda, F.D. Fischer, P. Fratzl, A. Kroupa, Acta Mater. 50 (2

1369.16] F.D. Fischer, J. Svoboda, P. Fratzl, Philos. Mag. 83 (2003) 10

174 J. Svoboda et al. / Materials Science and Engineering A 385 (2004) 166–174

[17] S.P.A. Gill, M.G. Cornforth, A.C.F. Cocks, Int. J. Plast. 17 (2001)669.

[18] A.C.F. Cocks, S.P.A. Gill, J.-Z. Pan, Advances in Applied Mechan-ics, Academic Press, San Diego, 1999.

[19] J. Agren, F.H. Hayes, L. Hoglund, U.R. Kattner, B. Legendre, R.Schmid-Fetzer, Z. Metallkd. 93 (2002) 128.

[20] F.K. Lehner, Deformation Processes in Minerals, Ceramics andRocks, Unwin Hyman, London, 1990 (Chapter 11).

[21] E. Fried, M.E. Gurtin, J. Stat. Phys. 95 (1999) 1361.[22] P.W. Vorhees, M.E. Glicksman, Acta Metall. 32 (19842001).[23] P.W. Vorhees, M.E. Glicksman, Acta Metall. 32 (1984) 2013.[24] M. Hillert, Acta Mater. 47 (1999) 4481.[25] J. Svoboda, F.D. Fischer, P. Fratzl, E. Gamsjager, N.K. Simha, Acta

Mater. 49 (2001) 1249.[26] K. Russell, Adv. Colloid Interface Sci. 13 (1980) 205.[27] F.J. Vermolen, C. Vuik, S. van der Zwaag, Mater. Sci. Eng. (2002)

14, 238 A.

[28] J.O. Andersson, L. Hoglund, B. Jonsson, J.Agren, Fundamentalsand Applications of Ternary Diffusion, Pergamon Press, New York,1990.

[29] N. Fujita, H.K.D.H. Bhadeshia, M. Kikuchi, Metall. Mater. Trans.33A (2002) 3339.

[30] A. Prikhodovski, M. Maubach, I. Hurtado, D. Neuschutz, Steel Res.70 (1999) 496.

[31] A. Pineau, Acta Metall. 24 (1976) 559.[32] H.J. Bohm, F.D. Fischer, G. Reisner, Scripta Mater. 36 (1997) 1053.[33] B.W. Leitch, S.-Q. Shi, Modell. Simul. Mater. Sci. Eng. 4 (1996)

281.[34] F.D. Fischer, E.R. Oberaigner, ASME J. Appl. Mech. 67 (2000) 793.[35] K. Thornton, N. Akaiwa, P.W. Voorhees, in: P.E.A. Turchi, A Go-

nis (Eds.), Phase Transformations and Evolution in Materials, TMSWarrendale, 2000, 73.

[36] R. Muller, S. Eckert, D. Gross, Arch. Mech. 52 (2000) 663.[37] I. Schmidt, Comput. Mater. Sci. 22 (2001) 333.