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Acta Materialia 56 (2008) 4896–4904

A model for evolution of shape changing precipitatesin multicomponent systems

J. Svoboda a, F.D. Fischer b,c,*, P.H. Mayrhofer d

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, Roseggerstraße 12, A-8700 Leoben, Austriad Department of Physical Metallurgy and Materials Testing, Montanuniversitat Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria

Received 8 April 2008; received in revised form 3 June 2008; accepted 5 June 2008Available online 18 July 2008

Abstract

Recently the authors introduced a concept of shape factors to extend an already established model for the growth and coarseningkinetics of spherical precipitates in multicomponent multiphase environments to needle- and disc-shaped geometries. The geometry ofthe precipitates is kept in the original version of the concept to be self-similar with a given fixed aspect ratio. In the present treatment,the aspect ratios of individual precipitates are treated as independent evolving parameters. The evolution equations of each precipitate,described by its effective radius, mean chemical composition and the aspect ratio, are derived by application of the thermodynamic extre-mal principle. The driving force for the evolution of the aspect ratio of the precipitate stems from the anisotropic misfit strain of theprecipitate and from the orientation dependence of the interface energy. The model is used for the simulation of the precipitation ofTi3AlN and Ti2AlN in Ti–Al–0.5 at.% N matrix.� 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Multicomponent diffusion; Phase transformation kinetics; Precipitation; Titanium aluminides; Aspect ratio

1. Introduction and motivation

In a series of papers a new approach to modelling ofprecipitation kinetics in multicomponent systems has beendeveloped [1] and applied with success to several practicalcases [2,3]. The model is based originally on spherical pre-cipitates. The concept of ‘‘shape factors” outlined in Ref.[4] has allowed investigating cylindrical precipitates withthe height Hk and the diameter Dk of the precipitate k

with the shape parameter (aspect ratio) hk = Hk/Dk. Thecorresponding equivalent radius of a sphere, qk = Dk/bk

with bk = (16/(3hk))1/3, follows from the same volume ofthe cylinder and of the sphere. The aspect ratio hk, how-

1359-6454/$34.00 � 2008 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2008.06.016

* Corresponding author. Address: Institute of Mechanics, Mont-anuniversitat Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria.Tel.: +43 3842 402 4001; Fax: +43 3842 46048.

E-mail address: mechanik@mu-leoben.at (F.D. Fischer).

ever, has been kept as a constant quantity during the evo-lution process. In reality the preferred shape and habit ofa precipitate are determined by the requirement that thesum of the elastic strain energy and the surface energyis minimized for a given volume of a precipitate; see,e.g., Khachaturyan et al. [5]. This means that the surfaceenergy dominates for precipitates in the nano-scale afternucleation, since the surface energy scales with the surfacearea compared to the volume relevant for the strainenergy. As a consequence spherical precipitates, minimiz-ing the surface energy, nucleate and grow. With theincreasing size the elastic strain energy, which is a func-tion of the misfit strain and the elastic behaviour,becomes relevant. If the misfit strain and/or the elasticbehavior are anisotropic, the shape of the precipitate devi-ates more and more from the original spherical shape.Specially, the case of elastic anisotropy was investigatedin detail by Vorhees et al., see, e.g., Ref. [6]. Of course,

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J. Svoboda et al. / Acta Materialia 56 (2008) 4896–4904 4897

also the dependence of the interface energy on the crystal-lographic orientation (or, in other words, the anisotropyof the interface energy) alone would control the develop-ment of the particle shape, see, e.g., the according refer-ences in Ref. [6]. The application of the phase-fieldapproach to find the shapes of precipitates is presentedby Hu et al. [7], where a set of order parameters is intro-duced in addition to the concentration field and impliedinto the Gibbs energy. The anisotropy of the chemicalenergy of interfaces was studied in detail by Muller [8]for Zn-precipitates in an Al-rich Al–Zn solid.

The main goal of this paper is the development of thekinetic model for the evolution of precipitates with vari-able aspect ratio. The aspect ratio evolution may stemfrom anisotropic misfit strain together with different elas-tic properties of the matrix and the precipitate and theanisotropy of the interface energy. The according ener-getic contribution to the total Gibbs energy is added tothe chemical part of the Gibbs energy. Dissipation inthe system is assumed due to interface migration and dif-fusion. The thermodynamic extremal principle [9] allowsfinding the evolution equations for the parameters ofthe precipitates including the shape parameter. In contrastto the phase-field approach [7] our concept is based onlyon real physical quantities and well-defined geometry ofobjects. The model is used for the simulation of the pre-cipitation kinetics of Ti3AlN and Ti2AlN in the Ti–Al–0.5 at.% N matrix, and the results of simulations are com-pared with experiments presented in the open literature.

2. Shapes of precipitates

Often spheroidal precipitates are observed with anequator radius a and the length aa of the half-axis ofrotation, a denominates the aspect ratio. For an arbitrarya and a it is possible to find an equivalent cylindrical pre-cipitate with the same volume and by inserting hk fora. Then the diameter of the cylinder is given byDk = (16/3)1/3a.

Each cylindrical precipitate k can be described either bythe height Hk and the diameter Dk or by the aspect ratio hk

and the equivalent radius qk of the sphere of the same vol-ume as that of the cylinder. The corresponding relationsare

Dk ¼ bkqk ð1ÞH k ¼ bkhkqk ð2Þ

For the time derivatives, marked by a dot, one finds

_Dk ¼ bkð _qk � qk_hk=ð3hkÞÞ ð3Þ

_H k ¼ bkðhk _qk þ 2qk_hk=3Þ ð4Þ

The reader should note that, in addition to concepts out-lined in Refs. [1–4], we have now further time-dependentquantities hk, the evolution equations for which must beadded to the system of equations outlined in Ref. [4], Sec-tion 3.

3. Model

We assume that each precipitate k is described by its equiv-alent radius qk, its average values of concentrations cki ofcomponents i in the precipitate and by its aspect ratio hk.

3.1. Total Gibbs energy of the system

Similar to Refs. [1–4] the total Gibbs free energy G of afixed amount of matter with n components and m precipi-tates can be expressed as

G ¼Xn

i¼1

N 0il0i þXm

k¼1

4pq3k

3kkðhkÞ þ

Xn

i¼1

ckilki

!

þXm

k¼1

4pq2k �

b2k

8cH

k þ 2hkcDk

� �ð5Þ

The first term is the chemical part of the Gibbs energy ofthe matrix, the second term corresponds to the stored elas-tic energy and to the chemical part of the Gibbs energy ofthe precipitates and the third term represents the total pre-cipitate/matrix interface energy. The subscripts ‘‘0” denotequantities related to the matrix, e.g., N0i is the number ofmoles of component i in the matrix and l0i its chemical po-tential in the matrix. The quantity kk(hk) accounts for thecontribution of elastic strain energy due to the volume mis-fit between the precipitate and the matrix, and lki are thevalues of chemical potentials in the precipitates corre-sponding to cki. We distinguish between the interface en-ergy cD

k at the mantle of the cylinder and cHk at the

bottom and top of the cylinder.There exists a well-elaborated concept to calculate the

elastic strain energy due to a misfit eigenstrain in a spheroidalinclusion, going back to Eshelby’s seminal work, see, e.g.,Refs. [10,11]. The precipitate is geometrically defined by a

and a. The elastic properties of the matrix are E (Young’smodulus) and v (Poisson’s ratio). We assume an elastic con-trast C to the precipitates with the Young’s modulus CE,and, for sake of simplicity, the same Poisson’s ratio m. Themisfit eigenstrain is characterized by the isotropic strain dand an additional eigenstrain component Dd in the directionof the axis of rotation of the spheroidal precipitate. Finallyby inserting hk for a the factor kk(hk) reads as

kkðhkÞ ¼Ed2

1� mF 0ðhk; DCÞ þ F 1ðhk; DCÞðDd=dÞ½

þF 2ðhk; DCÞðDd=dÞ2i

ð6Þ

with DC = C � 1. The functions F0(hk; DC), F1(hk; DC) andF2(hk; DC) are found by evaluating the ‘‘Eshelby scheme”

outlined in Refs. [10,11]. A constant Poisson’s ratiom = 0.3 is used in Fig. 1a–c, where the reader can find thecurves of the three functions. Furthermore, the F0, F1, F2

can be approximated by logFi = Pi (x) � log(1 + DC) withPi = Ai + Bix + Cix

2 + Dix3 and x = loghk. The coefficients

Ai, Bi, Ci, Di are given in Table 1.

Fig. 1. Demonstration of the three functions F0, F1, F2 in relation (6) for the strain energy shape factor kk. Corresponding polynomial approximations aregiven in Table 1.

Table 1Coefficients for the polynominal approximations of the strain energyshape factor kk

DC A0 B0 C0 D0

F 0*

2 0.15213 0.02259 0.14431 �0.037041 0.10659 0.00684 0.08206 �0.024580 0 0 0 0�(1/2) �0.14764 0.00713 �0.04934 0.0203�(2/3) �0.25413 0.01088 �0.06344 0.02883

F 1*

2 �0.0235 0.61991 �0.07026 �0.124481 �0.07182 0.53386 �0.12737 �0.090960 �0.18097 0.39194 �0.18913 �0.02825�(1/2) �0.32795 0.26766 �0.20332 0.02259�(2/3) �0.43324 0.20606 �0.19257 0.04182

F 2*

2 �0.51336 0.90915 �0.15008 �0.202551 �0.5703 0.79368 �0.18961 �0.167760 �0.69643 0.59089 �0.22465 �0.09462�(1/2) �0.86066 0.40330 �0.22061 �0.02615�(2/3) �0.97484 0.30903 �0.20331 0.00383

* log[Fi � (1 + DC)] = Ai + Bix + Cix2 + Dix

3, x = loga, i = 0,1,2.

4898 J. Svoboda et al. / Acta Materialia 56 (2008) 4896–4904

3.2. Total dissipation in the system

During the system evolution the total Gibbs energy dis-sipates by changing into the heat being drained off the sys-tem and/or causing the increase of the configurationalentropy of the system [9]. The total dissipation Q = Q1 +Q2 + Q3 is assumed to be due to the interface migration

(Q1) and diffusive fluxes inside the precipitates (Q2) andthe matrix (Q3).

3.2.1. Dissipation due to the interface migration

The dissipation due to the interface migration can becalculated as [9]

Q1 ¼Xm

k¼1

1

2pD2

k

ð _H k=2Þ2

MDk

þ pHkDkð _Dk=2Þ2

MHk

" #ð7Þ

with MDk being the interface mobility at the mantle of the cyl-

inder and MHk that at the bottom and top of the cylinder rep-

resenting the precipitate k. The dissipation can be directlyexpressed in the variables qk, cki, hk (k = 1, . . . ,m; i = 1, . . . ,n) and their rates by direct application of Eqs. (1)–(4) as

J. Svoboda et al. / Acta Materialia 56 (2008) 4896–4904 4899

Q1 ¼Xm

k¼1

pb4kq

2k

_q2khk

hk8MD

kþ 1

4MHk

� �þ _qk

_hkqkhk

6MDk� 1

6MHk

� �þ

_h2kq

2k

118MD

kþ 1

36hk MHk

� �264

375ð7aÞ

Fig. 2. Cylindrical model for a precipitate, notation of fluxes jDki; j

D0i and

jHki ; j

H0i .

3.2.2. Dissipation due to the diffusion in the precipitates

The dissipation due to the diffusion in the precipitates canbe evaluated by the general formula (see, e.g., Refs. [1,9])

Q2 ¼ RTXm

k¼1

Xn

i¼1

ZV k

j2ki=ðckiDkiÞdV ð8Þ

The symbol Vk stands for the region occupied by the pre-cipitate k, the quantity R is the gas constant, T the absolutetemperature, jki is the diffusive flux and Dki is the tracer dif-fusion coefficient of component i in the precipitate k. Forthe further treatment it is necessary to express the dissipa-tion Q2 in terms of variables qk, cki, hk (k = 1, . . . ,m;i = 1, . . . ,n) and their rates – see later.

3.2.3. Dissipation due to the diffusion in the matrix

The dissipation due to the diffusion in the matrix can beevaluated by the general formula (see, e.g., Refs. [1,9])

Q3 ¼ RTXn

i¼1

ZV 0

j20i=ðc0iD0iÞdV ð9Þ

The symbol V0 stands for the region occupied by the ma-trix, j0i is the diffusive flux and D0i is the tracer diffusioncoefficient of component i in the matrix. For the furthertreatment it is again necessary to express the dissipationQ3 in terms of variables qk, cki, hk (k = 1, . . . ,m;i = 1, . . . ,n) and their rates – see later.

4. Mathematical treatment of the model

4.1. Boundary conditions for diffusive fluxes

A diffusive process occurs in the precipitate as well as inthe matrix. There is a jump (c0i � cki) in the concentrationof component i at the interface, where c0i is its value on thematrix side and cki its value on the precipitate side. Wemust distinguish between two sets of flux components nor-mal to the interface, one at the mantle of the cylinder, jD

0i

and jDki, and a further one at the bottom and top of the cyl-

inder, jH0i and jH

ki , see Fig. 2. The following jump conditionsare valid

jD0i � jD

ki ¼ ð _Dk=2Þðc0i � ckiÞ ð10ÞjH

0i � jHki ¼ ð _Hk=2Þðc0i � ckiÞ ð11Þ

for details of the derivation see Section 3 of Ref. [12].For the fluxes on the precipitate side of the interface one

can set

jDki ¼ �ðDk=6Þ _cki ð12Þ

jHki ¼ �ðHk=6Þ _cki ð13Þ

which can be verified by the balance of the fluxes and themoles Nki in the precipitates with their rates _Nki. The fluxeson the matrix side of the interface follow by combination ofEqs. (10)–(13) as

jD0i ¼ �ðDk=6Þ _cki þ ð _Dk=2Þðc0i � ckiÞ ð14Þ

jH0i ¼ �ðH k=6Þ _cki þ ð _H k=2Þðc0i � ckiÞ ð15Þ

We can now insert the relations (1)–(4) and obtain theboundary fluxes jD

ki, jHki , jD

0i and jH0i in terms of qk, cki, hk

and their rates

jDki¼�bkqk _cki=6 ð16Þ

jHki ¼�bkqk _ckihk=6 ð17Þ

jD0i¼bk½ _qkðc0i�ckiÞ=2�qk _cki=6�qkðc0i�ckiÞ _hk=ð6hkÞ� ð18Þ

jH0i¼hkbk½ _qkðc0i�ckiÞ=2�qk _cki=6þqkðc0i�ckiÞ _hk=ð3hkÞ� ð19Þ

4.2. Dissipation in the precipitates

The flux distribution jki in the precipitate k is given inthe analogy with [1] by

divðjkiÞ ¼ � _cki ð20Þ

and by the boundary conditions (16) and (17).A general solution of the problem can be written in the

form

jki ¼ qk _cki~jðhk; x=ðbkqkÞÞ¼ qk _cki~jðhk; r=ðbkqkÞ; z=ðbkqkÞÞ ¼ qk _cki~jðhk; f; nÞ ð21Þ

where x = x(r,z) is the polar coordinate vector measuredfrom the center of the precipitate k. Insertion of Eq. (21)into Eq. (20) provides a differential equation for the dimen-sion-free flux ~j as

4900 J. Svoboda et al. / Acta Materialia 56 (2008) 4896–4904

~divð~jÞ ¼ �bk ð22Þ

We work now in a polar dimension-free coordinate systemwith f being the radial coordinate and n the axial coordi-nate. The radius of the mantle of the precipitate isfD = 1/2. The axial coordinates of the top and bottomare nH = ± hk/2. The operator ~div is the divergence-opera-tor in the f,n coordinate system.

In analogy to the treatment in Section 4.1 one can for-mulate the boundary conditions for ~j by using Eqs. (16),(17) and (21) as

f ¼ 1=2; �hk=2 6 n 6 hk=2 : ~jD ¼ �bk=6 ð23Þ0 6 f 6 1=2; n ¼ �hk=2 : ~jH ¼ �bkhk=6 ð24Þ

The analytical solution of the differential Eq. (22) for theflux ~j in the precipitate, fulfilling the boundary conditions(23) and (24), is given by the components

~jf ¼ �bk=3 � f; ~jn ¼ �bk=3 � n ð25ÞThe dissipation in the precipitates is given by applying

Eqs. (8), (21), (23) and (24) as

Q2 ¼ RTXm

k¼1

Xn

i¼1

q5k _c2

kib3k

ckiDki

Z hk=2

�hk=2

Z 1=2

0

2pf~j2dfdn ð26Þ

For ~j given by Eq. (25) one can perform the integration andexpress Q2 in the form resembling Eq. (13) in [1] for aspherical precipitate

Q2 ¼ RTXm

k¼1

Xn

i¼1

4pIkq5k _c2

ki

45ckiDkið27Þ

with

Ik ¼ 45b3k

Z hk=2

0

Z 1=2

0

f~j2dfdn

¼ 0:424h4=3k þ 0:636h�2=3

k ð28Þ

The quantity Ik is the shape factor corresponding to Q2 andagrees with that reported in Ref. [4], Eq. (22).

4.3. Dissipation in the matrix

In the analogy with Ref. [1] we assume that the flux dis-tribution j0i in the matrix is a superposition of the fluxes j0ki

around all precipitates being

j0i ¼Xm

k¼1

j0ki ð29Þ

From the solution presented in Ref. [1] it is evident thatonly in the nearest vicinity of the precipitates the magni-tude of fluxes is dominant and the dissipation is decisive.Thus, one can calculate the dissipation in the matrix givenby Eq. (9) as

Q3 ¼ RTXm

k¼1

Xn

i¼1

ZV 0

j20ki=ðc0iD0iÞdV ð30Þ

For the determination of the flux j0ki around each precipitateone must solve in the analogy with Ref. [1] the equation

divðj0kiÞ ¼ 0 ð31Þ

with boundary conditions (18), (19) and j0ki ¼ 0 ininfinity.

According to the structure of the boundary conditions(18) and (19) one can split the flux j0ki into two contribu-tions j0ki = j0ki,1 + j0ki,2, namely

j0ki;1 ¼ 3½� _qkðc0i � ckiÞ þ qk _cki=3�~j1ðhk; f; nÞ ð32Þ

and

j0ki;2 ¼ qkðc0i � ckiÞ � ð _hk=hkÞ~j2ðhk; f; nÞ ð33Þ

To fulfil Eq. (31) one can assume

~div~j1 ¼ ~div~j2 ¼ 0 ð34Þ

Again the divergence operator ~div in the f,n coordinatesystem is used.

One can formulate the boundary conditions for ~j1 and ~j2

in analogy to the treatment in Section 4.2 by using Eqs.(18), (19), (32) and (33) as

f¼1=2; �hk=26n6hk=2 : ~j1D¼�bk=6;~j2D¼�bk=6 ð35Þ06 f61=2; n¼�hk=2 : ~j1H ¼�bkhk=6;~j2H ¼bkhk=3 ð36Þ

No analytical solution for the fluxes ~j1;~j2 in the matrix canbe given. We look for numerical solutions of~D/1 ¼ ~D/2 ¼ 0 with ~D being the Laplace operator in thef,n coordinate system, according to the boundaryconditions

f¼1=2; �hk=26n6hk=2 : o/1=of¼o/2=of¼�bk=6 ð37Þ06 f61=2; n¼�hk=2 : o/1=on¼�bkhk=6; o/2=on¼bkhk=3 ð38Þ

Finally, the fluxes ~j1;~j2 are given by ~j1 ¼ ~grad/1;~j2 ¼ ~grad/2.

The solution procedure is straightforward and needs nofurther explanation. Only one comment seems to be usefulwith respect to the singular flux behavior near the corner-lines f = 1/2, n = ± hk/2. Here, the solution / is propor-tional to dc, 1/2 < c < 1, d being the distance to a pointinside the matrix near the corner (see, e.g., Ref. [13]). Theintegration of (o//od)2, in a small circular region with theradius �d yields a quantity �d2c�1 which obtains the value zeroin the corner, since 2c > 1. Therefore, in the case of a finemesh no numerical problem occurs due to this singularbehavior.

The dissipation in the matrix (Eq. (30)) is given byapplying Eqs. (32)–(38), already written with respect tothe dimension-free coordinate system, as

Q3 ¼ RTXn

k¼1

Xm

i¼1

4pq3kb

3k

c0iD0i� ð _qkðcki � c0iÞ þ qk _cki=3Þ2 � Ok1

h�ð _qkðcki � c0iÞ þ qk _cki=3Þ � qkðcki � c0iÞ � ð _hk=hkÞ � Ok2

þðqkðcki � c0iÞ � ð _hk=hkÞÞ2 � Ok3

ið39Þ

with

J. Svoboda et al. / Acta Materialia 56 (2008) 4896–4904 4901

Ok1 ¼ 9b3k

Z~V 0

~j21 d~V ð40Þ

Ok2 ¼ 6b3k

Z~V 0

~j1 � ~j2 d~V ð41Þ

Ok3 ¼ b3k

Z~V 0

~j22 d~V ð42Þ

The quantities Ok1, Ok2, Ok3 are three shape factors corre-sponding to Q3. The space occupied by the matrix in thef,n coordinate system is denoted by ~V 0.

Finally the shape factors Oki, i = 1,2,3, obtained by fit-ting of the numerically calculated curves, read

Ok1 � 0:881h�0:122k ð43Þ

Ok2 � 0:377h0:195k ð44Þ

Ok3 � 0:072h0:331k ð45Þ

Obviously, a mistake happened in the previous paper [4],Section 4.4, where Ok ¼ 1:0692h2=3

k is reported. Since Ok

should be identical to Ok1 given by Eq. (43), we ask thereader to replace Ok, Eq. (25) of [4] by Ok1 and are sorryabout this mistake.

5. Evolution equations

To obtain the evolution equations for the system one canfollow the procedure described in Ref. [1]. First of all weintroduce constraints following from the fixed stoichiome-try of the precipitates and the mass conservation law forall components in the system. By using these constraints(Eqs. (2)–(6) in Ref. [1]) the precipitate k can be describedby a set of n independent parameters, qk, cki (i = 2, . . ., s,s + 2, . . . ,n) and hk, k = 1, . . . ,m. The total Gibbs energyG (Eq. (5)) can also be expressed by this set. The total dissi-pation Q can be expressed by the same set of parameters andtheir rates. Then, according to the thermodynamic extremalprinciple [9], the evolution of the system is given by

1

2

oQo _qk¼ � oG

oqk;

1

2

oQo _cki¼ � oG

ockiði ¼ 2; . . . ; s; sþ 2; . . . ; nÞ;

1

2

oQ

o _hk

¼ � oGohk

; ðk ¼ 1; . . . ;mÞ ð46Þ

10-2

10-1

100

101

102

103

104

105

10-9

10-8

10-7

Hk

Dkρ

k

k/m, D

k/m, H

k/m

time/s

ρ

Fig. 3. Simulated evolution of precipitate parameters qk, Dk andHk inTi3AlN at 800 �C with starting value of hk = 1.

representing m sets of linear equations for _qk, _cki (i =2,. . . , s, s + 2, . . . ,n) and _hk.

For a fixed k one can replace y1 � _qk, yi � _cki (i = 2,. . . , s), yi � _ckiþ1 (i = s + 1, . . . ,n � 1) and yn ¼ _hk and writethe set of linear equations in the form

Pnj¼1Aijyj ¼ Bi,

(i = 1, . . . ,n), with the coefficients

A11¼b4khk

hk

32MDk

þ 1

16MHk

� �þOk1RT qk

Xn

i¼1

ðcki�c0iÞ2

c0iD0ið47Þ

A1i¼Ai1¼Ok1RT q2

k

3

cki�c0i

c0iD0i�ck1�c01

c01D01

� �ði¼2; . . . ;sÞ ð48Þ

A1i¼Ai1¼Ok1RT q2

k

3

ckiþ1�c0iþ1

c0iþ1D0iþ1

�cksþ1�c0sþ1

c0sþ1D0sþ1

� �ði¼ sþ1; . . . ;n�1Þ ð49Þ

Aij¼RT q3

k

45

Ik

ckiDkiþ 5Ok1

c0iD0i

� �dijþ

Ik

ck1Dk1

þ 5Ok1

c01D01

� �� ði¼2; . . . ;s;j¼2; . . . ;sÞ ð50Þ

Aij¼RT q3

k

45

Ik

ckiþ1Dkiþ1

þ 5Ok1

c0iþ1D0iþ1

� �dij

�

þ Ik

cksþ1Dksþ1

þ 5Ok1

c0sþ1D0sþ1

� �ð51Þ

ði¼ sþ1; . . . ;n�1;j¼ sþ1; . . . ;n�1ÞAij¼Aji¼0 ði¼2; . . . ;s;j¼ sþ1; . . . ;n�1Þ ð52Þ

A1n¼An1¼b4kqk

hk

48MDk

� 1

48MHk

� �

þOk2RT q2k

2hk

Xn

i¼1

ðcki�c0iÞ2

c0iD0ið53Þ

Ain¼Ani¼Ok2RT q3

k

6hk

cki�c0i

c0iD0i�ck1�c01

c01D01

� �ði¼2; . . . ;sÞ ð54Þ

Ain¼Ani¼Ok2RT q3

k

6hk

ckiþ1�c0iþ1

c0iþ1D0iþ1

�cksþ1�c0sþ1

c0sþ1D0sþ1

� �ði¼ sþ1; . . . ;n�1Þ ð55Þ

Ann¼b4kq

2k

1

72MDk

þ 1

144hkMHk

� �þOk3RT q3

k

Xn

i¼1

ðcki�c0iÞ2

c0iD0ih2k

ð56Þ

B1¼�b2

k

4ðcH

k þ2hkcDk Þ=qk�kk�

Xn

i¼1

ckiðlki�l0iÞ ð57Þ

Bi¼�qk

3ðlki�l0i�lk1þl01Þ ði¼2; . . . ;sÞ ð58Þ

Bi¼�qk

3ðlkiþ1�l0iþ1�lksþ1þl0sþ1Þ

ði¼ sþ1; . . . ;n�1Þ ð59Þ

Bn¼�qk

3

dkk

dhkþ b2

k

12hkcH

k �hkcDk

� �ð60Þ

The Kronecker delta dij has its usual meaning.We would like to remark that coefficients correspond-

ing to Eqs. (50) and (51) in the previous paper (Eqs.(9), (10) in Ref. [4]) contain an incorrect factor 3 in thedenominators.

To determine dkk/dhk one needs to know the derivativesdFi/dhk with the notation from Section 3.1 as

4902 J. Svoboda et al. / Acta Materialia 56 (2008) 4896–4904

dF i

dhk¼ dP i

dx� F ihk ð61Þ

After the evaluation of the rates _qk, _cki (i = 2, . . . , s,s +2, . . . ,n) and _hk, (k = 1, . . . ,m), the time integration stepcan be proceeded and the evolution in time can be per-formed by repetition of the procedure.

6. Simulation of the precipitation kinetics of Ti3 AlN and Ti2AlN and discussion of results

Let us consider as a representative example the devel-opment of nitride precipitates in L10-ordered Ti–Al–0.5at.% N matrix. This kind of particles (Ti2AlN andTi3AlN) was experimentally investigated in detail by Tianand Nemoto [14]. The precipitates appear at 800 �C.Therefore, the following data are related to this tempera-ture. TiAl has a tetragonal structure. The matrix withoutnitrogen is supposed, as we assume that nearly all nitro-gen is deposited quite quickly in precipitates. The Ti3AlNprecipitates have a cubic lattice, the Ti2AlN ones have ahexagonal lattice. Therefore, the misfit strain of Ti2AlNin the basal plane must be averaged in the procedure athand.

The elastic properties of all phases can be taken fromSchafrik [15], and the misfit strain of precipitates can becalculated from Schuster and Bauer [16] and transferredto d and Dd. Tian and Nemoto [14] used data from [16]for calculation of misfit strain; however, they made a mis-take in evaluation of e111 influencing Dd for Ti2AlN. There-fore, we decided to calculate the structure parameters andelastic properties of all phases by means of ab initiocalculations.

Ab initio calculations are based on the density-func-tional theory, using the VASP code [17,18], in conjunctionwith the generalized-gradient approximations projectoraugmented wave potentials [19]. Relaxation convergenceof 1.0 meV for ions and 0.1 meV for electrons, recipro-cal-space integration with a Monkhorst–Pack scheme[20], energy cutoff of 500 eV, and tetrahedron methodwith Blochl corrections [21] for the energy were used in

10-2

10-1

100

101

102

103

104

105

1

5

4

3

2

hk

time/s

Fig. 4. Simulated evolution of aspect ratio hk in Ti3 AlN at 800 �C withstarting value of hk = 1.

the calculations. Relaxations for both the lattice parame-ter as well as the ion positions were performed. A k-points grid of 3 � 3 � 3 was used for all calculations ofthe supercells. The volumes, total energies and bulk mod-uli (B) at equilibrium were obtained by a least-square fitof the calculated total energy over volume curves employ-ing the Birch–Murnaghan’s equation of state [22]. Theelastic moduli E are estimated from the bulk moduli B

and Poisson ratio m, which is calculated after the approx-imation introduced by Korzhavyi et al. [25] using theDugdale–MacDonald–Gruneisen constant [26]. The posi-tions of atoms in the supercells can be found at the homepage http://institute.unileoben.ac.at/mechanik/research/Shapefactor2supercells.pdf.

The calculated elastic moduli are E = 198.3 GPa andm = 0.21 for the matrix, E = 231.8 GPa and m = 0.26 forTi3AlN and E = 238.8 GPa and m = 0.27 for Ti2AlN. Weapply the temperature dependence for the elastic moduliby Schafrik [15] who reports �0.0342 T (in GPa), with T

in K, as a temperature correction term for TiAl. Finallywe use for 800 �C the following data with v = 0.25 for allphases:

TiAl: E = 162 GPa, X = 9.79 � 10�6 m3 mol�1.Ti3AlN: E = 195 GPa, C = 1.31, d = 0.042,

Dd = � 0.044, X = 8.49 � 10�6 m3 mol�1.Ti2AlN: E = 202 GPa, C = 1.35, d = 0.055,

Dd = � 0.082, X = 8.01 � 10�6 m3 mol�1.The molar volume is denoted by X.The authors are not aware of data for the surface energy

terms cHk , cH

k for the precipitate/parent phase interfaces.The value of the coherent interface energy terms is assumedto be very small and is set to zero in simulations.

The chemical driving forces �Pn

i¼1ckiðlki � l0iÞ from Eq.(57) are not at disposal in the thermodynamic databases.The values of the chemical driving forces were, therefore,chosen as free parameters so that they provide the growthkinetics being in agreement with experiments [14].

The diffusion coefficients DTi = 1.01 � 10�18 m2 s�1 andDAl = 6.26 � 10�20 m2 s�1 in the matrix are calculatedaccording to Ref. [23]. The diffusion coefficient of N inthe matrix DN = 4.55 � 10�15 m2 s�1 is calculated as that

10-2

10-1

100

101

102

103

104

105

10-9

10-8

10-7

Dk

k

Hk

k/m, D

k/m, H

k/m

time/s

ρ

ρ

Fig. 5. Simulated evolution of precipitate parameters qk, Dk and HkinTi3AlN at 800 �C with starting value of hk = 10.

10-2

10-1

100

101

102

103

104

105

4

5

6

7

8

9

10h k

time/s

Fig. 6. Simulated evolution of aspect ratio hk in Ti3 AlN at 800 �C withstarting value of hk = 10.

10-2

10-1

100

101

102

103

104

105

106

10-9

10-8

10-7

10-6

Dk

Hk

k

k/m, D

k/m, H

k/m

time/s

ρ

ρ

Fig. 7. Simulated evolution of precipitate parameters qk, Dk and Hk inTi2AlN at 800 �C with starting value of hk = 1.

10-2

10-1

100

101

102

103

104

105

106

0.1

1

0.3

0.4

0.6

0.8

0.2

h k

time/s

Fig. 8. Simulated evolution of aspect ratio hk in Ti2 AlN at 800 �C withstarting value of hk = 1.

J. Svoboda et al. / Acta Materialia 56 (2008) 4896–4904 4903

of B in TiAl [24] due to a lack of data. As DN is about fiveorders of magnitude higher than DAl, its value does notinfluence the kinetics of the system significantly. The inter-face mobility is set to be infinite.

Precipitates of Ti3AlN appear as ‘‘cigar-type” ones withan aspect ratio hk � 5 and Hk � 130 nm after 105 s anneal-ing at 800 �C [14]. The Ti2AlN precipitates show a thinplate-like shape geometry with Dk � 500 nm after3.6 � 106 s annealing at 800 �C [14].

The results of simulations are presented in Figs. 3–8. Weassumed fixed chemical composition in the precipitates aswell as in the matrix, so that the chemical driving force inEq. (57) can be kept constant. Then _cki � 0, and the systemof equations to be integrated in time is reduced only to thatfor variables qk and hk (only Eqs. (47), (53), (56), (57), (60)are used).

For Ti3AlN the starting values qk = 1 nm and hk = 1(results of simulations in Figs. 3 and 4) and qk = 1 nm andhk = 10 (results of simulations in Figs. 5 and 6) are chosen.From those figures it is evident that the aspect ratio hk

changes very quickly (during about 10 s) to its stationaryvalue hk = 4.5 (see Figs. 4 and 6). After that time periodthe kinetics of qk, Dk and Hk obeys the parabolic law (seeFigs. 3 and 5). The final values of all parameters agree wellwith observations in Ref. [14].

In the experimental study [14], the formation of very thinplates of size Dk = 500 nm of Ti2AlN are observed for thetime t = 3.6 � 106 s. Using the values of the parameters dand Dd, provided by ab initio calculations, the simulationsdo not reproduce the experimental results, leading to thefinal value hk � 2. This is evidently due to the fact that Ddhas a large negative value, which favours the formation of‘‘cigar-type” precipitates similar to those of Ti3AlN. Theincrease of the specific interface energy cD to 1 J m�2 doesnot cause a significant change.

We have analyzed the possible reasons of the disagree-ment between simulations and experimental observationsin the case of Ti2AlN. One can easily imagine that the elasticaccommodation of misfit strain state leads to very high val-ues for the stress components in the precipitate and in thematrix. However, the c-TiAl matrix is in the position toaccommodate the misfit strain by plastification, especiallyat the rather high temperature of 800 �C. In fact one canobserve several dislocations as reported in Ref. [14] for thesystem with Ti2AlN precipitates. Specifically, the plate-likeTi2AlN precipitates favour plastic accommodation, sincetheir equator plane is situated on the (11 1) planes of the c-TiAl matrix, which are its crystallographic slip planes. A firstimpression on the stress state can be obtained by calculatingthe equivalent stress rv on the matrix side of the interface forthe elastic behaviour. Analytical expressions for the stressstate, taking into account both the elastic contrast C andthe misfit strain contributions d and Dd, are available inRef. [11]. One finds for Ti2AlN with a thin plate geometrya maximum value of rv = ETiAl (2.98d + 1.06Dd) and forTi3AlN under the assumption of a very high aspect ratio amaximum value of rv = ETiAl (1.44d + 0.28Dd), in both casesin the equator plane. Inserting the according misfit datayields rv = 0.077ETiAl for Ti2AlN and rv = 0.048ETiAl forTi3AlN. One can now conclude that specifically in the caseof Ti2AlN the misfit at the bottom and top of the cylindrical

4904 J. Svoboda et al. / Acta Materialia 56 (2008) 4896–4904

precipitate can be decreased by the matrix plastification by afactor of, say, five leading to d = 0.011 and Dd = � 0.038 sofinally the equivalent stress rv obtains the magnitude of theactual field strength of the TiAl matrix.

Of course a question arises why the misfit has not beenrelaxed also in the case of Ti3AlN. First of all no disloca-tion activity is visible in systems with Ti3AlN. Further rea-sons are a shorter time period in the case of the Ti3AlNexperiment and a much better reinforcement of the matrixby very fine ‘‘cigar-type” Ti3AlN precipitates than by quitecoarse Ti2AlN plates.

The simulations have been repeated for Ti2AlN systemby using the modified misfit data d = 0.011 andDd = � 0.038. The starting values qk = 1 nm and hk = 1are chosen, and the results of simulations are presentedin Figs. 7 and 8. They are in a good agreement with exper-imental observations [14].

An interesting question arises whether the shape ofnuclei can also be predicted. The authors believe that it ispossible in the frame of cluster dynamics; however, it willbe extremely cumbersome. In principle one can expressthe nucleation barrier as a function of the additional vari-able hk and one can also estimate the rate of fluctuations ofhk. Then one must solve the Fokker–Planck equation in atleast two dimensions for a given driving force and orienta-tion-dependent interface energy.

7. Summary

In contrast to the previous work [4], the shape evolutionof each precipitate k is given by the aspect ratio hk, which isconsidered as a free time-dependent parameter. The totalGibbs energy of the system has to be extended by termsexpressing the dependence of the elastic strain energy andof the precipitate surface energy on hk. Both provide thedriving force for the precipitate shape evolution. The totaldissipation Q in the system has to be extended by termsdepending on _hk causing the dissipation due to interfacemigration and bulk diffusion in the matrix. The final setof evolution equations resembles the original set of evolu-tion equations [1] modified by proper shape factors depen-dent on hk.

The model is applied to simulations of precipitation ofTi3AlN and Ti2AlN in Ti–Al–0.5 at.% N matrix. The sim-ulations indicate that the aspect ratio hk reaches veryquickly its stationary value, and after that period the kinet-ics of the system obeys the parabolic law. The results ofsimulations are in agreement with the experimental studyby Tian and Nemoto [14] for Ti3AlN. For Ti2AlN theagreement with experiments [14] is obtained under theassumption of a significant misfit stress relaxation.

Acknowledgements

The authors are grateful to T. Antretter, Leoben, forperforming the finite element calculations. Furthermore,the authors mention that the numerous calculations aswell as the numerical fitting of the curves in Fig. 1a–cwere performed by E. Gamsjager and E.R. Oberaigner,both Leoben. Finally, the authors express their thanksto T. Waitz, Vienna, who helped to interprete the experi-mental results for Ti2AlN, Ti3AlN and to clarify the misfitstrains.

Financial support by Materials Center Leoben Fors-chung GmbH (project SP19) and by Research Plan of Insti-tute of Physics of Materials (Project CEZ:AV0Z20410507)is gratefully acknowledged. The works have been also sup-ported by the Grant Agency of the Academy of Sciences ofthe Czech Republic – Grant No. IAA200410601.

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