Influence of diffusional stress relaxation on growth of stoichiometric precipitates in binary systems

Acta Materialia 54 (2006) 4575–4581 www.actamat-journals.com

Influence of diffusional stress relaxation on growth of stoichiometric precipitates in binary systems J. Svoboda a

a,*

, E. Gamsja¨ger b, F.D. Fischer

b

Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zˇizˇkova 22, CZ-616 62, Brno, Czech Republic b Institute of Mechanics, Montanuniversita¨t Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria Received 6 February 2006; accepted 25 April 2006 Available online 28 August 2006

Abstract Precipitation is usually connected with a significant misfit strain which may drastically influence the precipitation kinetics. Very recently it was demonstrated by Svoboda et al. [Svoboda J, Gamsja¨ger E, Fischer FD. Philos Mag Lett 2005;85:473] that the stress fields due to misfit strains can be effectively relaxed by the diffusive transport of vacancies in the matrix and their annihilation or generation at the precipitate/matrix interface. This idea has provided motivation to develop a new model for the simultaneous precipitate growth and misfit stress relaxation in binary systems. The actual state of the precipitate is described by its effective radius and by the thickness of the layer of the matrix atoms deposited at the precipitate/matrix interface. For these parameters the evolution equations are derived by application of the thermodynamic extremal principle. The model is applied to the Fe–C system by considering the growth of a cementite precipitate in the ferritic matrix. The influence of the stress relaxation on the precipitate growth kinetics is demonstrated.  2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Diffusion; Vacancies; Phase transformation kinetics; Precipitation; Stress relaxation

1. Introduction During precipitation significant elastic stress fields can be formed in and around the precipitates due to misfit strains. The formation of the stress fields is coupled with negative mechanical driving forces, which can prohibit the growth of precipitates. Generally it is assumed that the stress fields can be relaxed by matrix plasticity and so the negative driving force is significantly reduced; see, for example, the well-known experiments by Eikum and Thomas [2] and some further references in Ref. [1]. With a decreasing length scale the role of distinct dislocations increasingly comes into the play. Different concepts have been applied to investigate this effect. Liu and Jonas [3] introduced a coherency loss parameter C, which ranges from C = 0 for full coherency to C = 1, when a sufficient

*

Corresponding author. E-mail address: [email protected] (J. Svoboda).

number of interface dislocations is assumed so that a semi-coherent interface at equilibrium is obtained. The critical value C* is found from a minimum of the free energy including a surface energy term meeting the energy stored in the interface dislocations. This concept was applied by Liu and Jonas [4] and later by Popov et al. [5] for Ti carbonitrides. A more sophisticated and somewhat fashionable concept is to meet the length scale by a distinct parameter within the framework of gradient plasticity; see the application for a spherical precipitate by Gao [6]. From the point of view of physical reality the concept of crystal plasticity can be considered as a proper concept dealing with the activation of slip systems on individual slip planes. Ohashi’s pioneering work [7] is mentioned, recently published for a matrix with an inclusion subjected to global tension; however, the case of a misfitting precipitate is only being elaborated. All these concepts, however, fail for nano-sized precipitates, because the matrix plasticity caused by the motion of dislocations is not a sufficiently fine tool for the relaxation of the highly localized stress fields.

1359-6454/$30.00  2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.04.039

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Experimental and theoretical studies have been published by Dahmen and co-workers [7–9], demonstrating that the stress fields formed in and around the precipitates can be effectively relaxed by generation or annihilation of vacancies at the precipitate/matrix interfaces accompanied by the diffusion of vacancies and their annihilation or generation in the matrix. It was clearly shown that no dislocation plasticity contributes to the stress relaxation. Inspired by these experimental findings Svoboda et al. [1] developed a non-equilibrium model with one relaxation parameter d, the thickness of a layer of matrix atoms deposited at the interface, and used the thermodynamic extremal principle (see, e.g. Svoboda et al. [10]) to find an evolution equation for the parameter d. The quantitative agreement between the developed theoretical model and experiments is very good. It should be mentioned that this concept differs from other relaxation concepts such as that proposed by Roitburd and co-workers (see, e.g. Ref. [11]) by addressing a direct physical meaning to the relaxation parameter d, and an explicit calculation of the dissipation due to the development of this parameter. The aim of the paper is to present a model for the growth of the stoichiometric precipitate in binary systems driven by both chemical and (negative) mechanical driving forces. The mechanical driving force is allowed to be changed by the stress field relaxation due to the transport of vacancies. The model is used for simulations of the growth of the cementite precipitate in the Fe–C system. 2. Model assumptions and definitions Let us assume a binary system with one substitutional component M and one interstitial component X forming a solid solution for low mole fractions of X as well as a stoichiometric compound M1bXb, with b being a fixed rational number b 2 (0, 1). For a stoichiometric compound denoted as MAXB with A and B being positive integers b = B/(A + B). The unit cell is represented by one M1bXb precipitate of radius q surrounded by the matrix of radius R and of the mean chemical composition characterized by x, the mole fraction of the component X (see Fig. 1). Due to volumetric misfit a stress field is built up in the unit cell. Let the precipitate grow by diffusion of the component X towards the precipitate. The interface between the precipitate and the matrix as well as the matrix itself are supposed to act as sources and sinks for vacancies. The vacancies can be generated or annihilated in the matrix, diffuse in the matrix from or to the precipitate, and due to their annihilation or generation at the interface the misfit stress can be effectively relaxed. The generation of vacancies at the precipitate/matrix interface accompanied by vacancy diffusion from the precipitate causes the deposition of a layer of atoms of thickness d at the interface. The thickness d is negative in the case of vacancy annihilation at the interface accompanied by their diffusion to the precipitate. In the framework of this approach the actual state of the system is described uniquely by the values of q and d. The

Fig. 1. Schematic of the diffusional zone with one inclusion and the deposited layer of matrix atoms of thickness d at the interface.

respective evolution equations for these state parameters can be derived by application of the thermodynamic extremal principle. The system is considered to be closed, and the conservation law for the X component can be expressed as bq3 ðR3  q3 Þx þ ¼ const: Xm Xp

ð1Þ

The value of the constant is given by the initial configuration of the system. The fixed molar volume (volume corresponding to one mole of atoms) of the precipitate is Xp. The molar volume of the matrix, Xm, is given by Xm ¼ ð1  xÞXM þ xXX

ð2Þ

with XM and XX being the fixed partial molar volumes of components M and X in the matrix. The misfit (transformation) strain eT in the radial direction is given by the relation 3

ð1 þ eT Þ ¼

Xp =ð1  bÞ Xm =ð1  xÞ

ð3Þ

The terms Xp/(1  b) and Xm/(1  x) represent the volumes containing one mole of atoms of the component M in the precipitating phase and in the matrix, respectively. The misfit strain eT then corresponds to the situation that the precipitate grows exclusively by diffusion of component X in the matrix, and the component M is immobile. The misfit strain depends on the chemical composition of the matrix, since the lattice parameter increases due to the deposition of the interstitial atoms of component X. Putting Eqs. (1) and (2) together one can determine x = x(q). Using Eq. (3) one can determine eT = eT(x), and consequently eT = eT(q) can also be obtained. 3. Total Gibbs energy of the system The total Gibbs energy G of the system consists of the chemical part Gchem and the mechanical part Gmech. The chemical part is given by

J. Svoboda et al. / Acta Materialia 54 (2006) 4575–4581

Gchem

4pq3 gp 4pðR3  q3 Þ½ð1  xÞlM ðxÞ þ xlX ðxÞ ¼ þ 3Xm 3Xp

ð4Þ

The Gibbs energy of the precipitate phase per mole of atoms is gp, and lM and lX are the chemical potentials of components M and X in the matrix. These quantities can be calculated by standard procedures. Using the relation x = x(q), Gchem can be expressed as a function of q: Gchem = Gchem(q). Denoting the surface energy density by c, the mechanical part of the Gibbs energy is given by [1]  2 4p 3 d q E  f ðm; H Þ eT þ Gmech ¼ þ 4pq2 c; 3 q 3H ð5Þ f ðm; H Þ ¼ ðH þ 2Þ þ ðH  4Þm The quantities E and m are the Young’s modulus and Poisson’s ratio of the matrix. The quantity H allows for an elastic contrast of the precipitate and the matrix with H Æ E being the Young’s modulus of the precipitate. The Poisson’s ratio of the precipitate is assumed to be the same as that of the matrix; for details see Fischer et al. [12]. The first term in Eq. (5) represents the elastic strain energy. It is important to keep in mind that eT = eT(q). Finally the total Gibbs energy of the system is given as Gðq; dÞ ¼ Gchem ðqÞ þ Gmech ðq; dÞ

ð6Þ

4. Gibbs energy dissipation Four dissipative processes are considered in the present model:    

diffusion of component X, migration of the precipitate/matrix interface, diffusion of vacancies in the diffusive zone q 6 r 6 R, generation or annihilation of vacancies at the precipitate/matrix interface.

Moreover, it is assumed that the vacancies can be generated or annihilated within the matrix zone without any dissipation. The diffusion of the component X coupled with the migration of the precipitate/matrix interface, enables the growth of the precipitate. We can assume that the atoms of the component X are collected uniformly in the whole matrix. Then the radial flux of component X in the matrix jX(r) at a distance r from the system center is given by R3  r3 q2 jX ðrÞ ¼ ðx=Xm  b=Xp Þ  3  q_ ð7Þ R  q3 r 2 where the overdot denotes the total time derivative. Furthermore, we assume that insignificant gradients of x are formed in the matrix. The derivation of Eq. (7) can be taken from Appendix A. The dissipation due to diffusion of the component X in the matrix is given as (for details see, e.g., Svoboda et al. [13])

4pRg T Xm Q1 ¼ xDX

Z

R

4pRg T Xm q3 j2X r2 dr  xDX q

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x b  Xm Xp

2

q_ 2

ð8Þ

where Rg is the gas constant, T is the absolute temperature, and DX is the tracer diffusion coefficient of component X in the matrix. The dissipation term Q1, expressed by Eq. (8), involves tacitly q  R. The dissipation due to migration of the precipitate/ matrix interface can be expressed with the interface mobility MI as Q2 ¼

4pq2 2 q_ MI

ð9Þ

If a uniform generation or annihilation of vacancies is assumed in the whole diffusive zone, then the radial diffusive flux of vacancies jV(r) in the matrix is given by (see also Appendix A) jV ðrÞ ¼ jM ðrÞ ¼

V_ d 1 R3  r3 q2    4pq2 Xm R3  q3 r2

ð10Þ

The volume Vd of the matrix material, transported to the precipitate by the flux of atoms jM(r) (being the counterflux to the flux of vacancies jV(r)) and deposited at the precipitate/matrix interface due to vacancy generation at the interface follows as i 4p h 3 ðq þ dÞ  q3 ¼ 4pðq2 d þ qd 2 þ d 3 =3Þ Vd ¼ ð11Þ 3 Then the Gibbs energy dissipation Q3, connected with the diffusion of vacancies in the matrix, can be determined again for q  R as Z R 4pRg T Rg T V_ 2d Q3 ¼ ð12Þ j2V r2 dr  cV D V q 4pDM Xm q The concentration of vacancies in the matrix is denoted as cV, the diffusion coefficient of vacancies in the matrix as DV, and cVDV is estimated by DM/Xm with DM being the tracer diffusion coefficient of M in the matrix. A detailed explanation of this estimation can be found in Ref. [1] and Eqs. (6), (7) and (16) thereof. Differentiation of Eq. (11) and its insertion into Eq. (12) give Q3 ¼

4pRg T ½ð2qd þ d 2 Þ2 q_ 2 þ 2ð2qd þ d 2 Þðq þ dÞ2 q_ d_ DM Xm q 4 ð13Þ þ ðq þ dÞ d_ 2 

In the framework of linear thermodynamics one can express the dissipation connected with the generation or annihilation of vacancies at the precipitate/matrix interface as V_ 2d 4pq2 U 4p 2 2 ¼ 2 ½ð2qd þ d 2 Þ q_ 2 þ 2ð2qd þ d 2 Þðq þ dÞ q_ d_ qU 4 þ ðq þ dÞ d_ 2 

Q4 ¼

ð14Þ

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J. Svoboda et al. / Acta Materialia 54 (2006) 4575–4581

The material parameter U characterizes the ideality of the vacancy sources and sinks at the precipitate/matrix interface. With U ! 1 ideal sources and sinks at the interface are simulated, because Q4 ” 0 for an arbitrary value V_ d . In contrast, by choosing U ! 0 the dissipation term Q4 ! 1 for any nonzero values of V_ d and the sources and sinks for vacancies at the interface get automatically inactive enforcing V_ d  0. The total dissipation Q in the system is then given as Q ¼ Q1 þ Q2 þ Q3 þ Q4 ð15Þ 5. Evolution equations for the system The evolution equations for the system can be derived by application of the thermodynamic extremal principle according to Refs. [12,13] leading to oG 1 oQ oG 1 oQ ¼ and ¼ ð16Þ oq 2 oq_ od 2 od_ The partial derivatives of Q, Eqs. (8), (9), (13) and (14), and of G, Eqs. (4)–(6), can be determined analytically as "  2 oQ Rg T Xm q x b 2 ¼ 8pq  oq_ xDX Xm Xp    1 Rg T 1 2 2 þ þ þ Þ ð2qd þ d q_ MI q3 DM Xm U q4 #   Rg T 1 2_ 2 þ ð17Þ ð2qd þ d Þðq þ dÞ d þ 3 q DM Xm U q4   oQ Rg T 1 2 2 þ ¼ 8pq ð2qd þ d 2 Þðq þ dÞ q_ q3 DM Xm U q4 od_    Rg T 1 4_ þ 3 þ ð18Þ ðq þ dÞ d q DM Xm U q4 To calculate oG/oq one requires some relations following from the previous equations dq R3  q3 XM 1 ¼   2 dx 3q Xm ðx  bXm =Xp Þ !1=3 deT XX Xp ¼ dx 3Xm ð1  bÞð1  xÞ2 Xm and by using the Gibbs–Duhem equation in the form ð1  xÞ dldxM þ x dldxX ¼ 0   d ð1  xÞlM þ xlX l X M  lM X X ¼ X dx Xm X2m Then oG/oq follows after some algebra as

2 gp ð1  xÞlM þ xlX ðlM XX  lX XM Þ oG ¼ 4pq2 4   oq Xm XM Xp Xm   2c d ðx  bXm =Xp Þ þ þ Ef ðm; H Þ eT þ q q 0 !1=3 13 d 2X ðx  bX =X Þ X X m p p A5 @ eT þ  3q 3XM ðR3 =q3  1Þ ð1  bÞð1  xÞ2 Xm ð19Þ

The last partial derivative follows as   oG d 2 Ef ðm; H Þ ¼ 8pq eT þ od 3 q

ð20Þ

The set of two linear algebraic equations (Eqs. (16)) in q_ _ having a positive definite symmetric matrix, can be and d, _ and the rates can solved with respect to the rates q_ and d, be integrated numerically in time. 6. Results and discussion The model is applied to the Fe–C system to simulate the growth of spherical cementite Fe3C in a ferritic matrix. The molar volumes of ferrite and cementite are calculated from the temperature-dependent lattice parameters which can be found in Ref. [14]. At temperature T = 973 K, corresponding to 27 K below the eutectoid temperature, the molar volume of the cementite is XP = 5.992 · 106 m3 mol1. The partial molar volume of iron in ferrite is XFe = 7.306 · 106 m3 mol1. Due to the small solubility of carbon in ferrite the partial molar volume of carbon in ferrite is not available from experimental data, and therefore the value for austenite XC = 4.670 · 106 m3 mol1 is used in the calculations. For the initial composition of the matrix given by x0 = 2 · 103 or by x0 = 4 · 103, Eq. (3) with b = 0.25 provides in both cases practically the same value eT = 0.030. The Young’s modulus E of the matrix can be found in Ref. [15] as E = 208.06 Æ [1.06–4.98 · 104 K1 T] GPa, and m is approximately 0.3. The elastic properties of cementite can be estimated as Ep = 160 GPa and mP  0.3 [16]. The values of E Æ f(m, H) = 190 GPa and R = 106 m are used in the calculations. The initial precipitate radius q = 5 · 109 m corresponds to a slightly supercritical value. The interface energy c is chosen to be c = 0.2 J m2. We suppose that during the nucleation the growth of the precipitate by means of size fluctuations occurs much slower than during its deterministic growth and, thus, the stress is totally relaxed for the initial precipitate radius. This implies the initial value of d = 1.5 · 1010 m. The tracer diffusion coefficient of iron in the matrix is given by DFe = 1.6 · 104 Æ exp(2.4 · 105 J mol1/RgT) m2 s1 [17]. The tracer diffusion coefficient of carbon DC follows a more compli˚ gren [18] and cated relationship, which can be found in A is repeated here: DC ¼ 2 106  expð84100 J mol1 =Rg T Þ      2 15309K  exp 0:5898  1 þ arctan 1:4985  m2 s1 p T

We assume that MI ! 1 m2 s kg1, and the quantity U is considered as a parameter controlling the intensity of the stress relaxation during the precipitate growth. The values of chemical potentials lFe and lC and of the molar Gibbs energy of the cementite gFe3 C are calculated by using SGTE-data [19,20]. The results of simulations are summarized in Figs. 2–9. The first series of figures (Figs. 2–5) is plotted for x0 =

J. Svoboda et al. / Acta Materialia 54 (2006) 4575–4581

-3

x0 = 2x10

-7

precpitate radius ρ / m

1.5x10

2

-1

U = ∞ m s kg -20 2 -1 U = 10 m s kg 2 -1 U = 0 m s kg

-7

1.0x10

-3

2.0x10 mole fraction x

-3

x0 = 2x10

2

-1

U = ∞ m s kg -20 2 -1 U = 10 m s kg 2 -1 U = 0 m s kg

-3

1.5x10

-3

1.0x10

-5

10

-4

10

-3

10

-2

10

-1

0

10 10 time / s

1

2

10

3

10

10

-3

Fig. 4. Evolution of the mole fraction x in the matrix for x0 = 2 · 103 and different values of parameter U.

elastic energy per precipitate volume / Jm

2 · 103 and using the set of values of U 2 {0, 1020, 1} m2 s kg1. It is demonstrated in these figures that the activity of sources and sinks for vacancies at the precipitate/matrix interface drastically influences the precipitate growth kinetics. In Fig. 2 it can be seen that for U ! 1 m2 s kg1 the precipitate reaches its maximum size after approximately 100 s. The precipitate growth is slowed down by a factor of about 20 for U = 1020 m2 s kg1, and the precipitate is practically prohibited from growth for U = 0 m2 s kg1. The respective evolution of the deposited layer thickness d and that of the mole fraction x are depicted in Figs. 3 and 4. The equilibrium mole fraction in the matrix is reached after different times for U ! 1 and U ! 1020 m2 s kg1 or not at all for U = 0 m2 s kg1. The evolution of the total elastic energy per precipitate volume is shown in Fig. 5. Note, that the initial value of d is chosen in such a way that no stress occurs at the time t = 0 s. The negative mechanical driving force due to the stress state and the interface energy c gets equilibrated soon with the positive chemical driving force for U = 0 m2 s kg1. Since there is no stress relaxation, the precipitate growth is inhibited. The second series of figures (Figs. 6–9) is plotted for x0 = 4 · 103 and the same values of U 2 {0, 1020, 1} m2

4579

8

1.0x10

7

7.5x10

-3

x0 = 2x10 7

5.0x10

2

7

2.5x10

0.0 -5 10

-1

U = ∞ m s kg -20 2 -1 U = 10 m s kg 2 -1 U = 0 m s kg

-4

10

-3

10

-2

10

-1

0

10 10 time / s

1

2

10

3

10

10

Fig. 5. Evolution of the elastic strain energy per precipitate volume for x0 = 2 · 103 and different values of parameter U.

-8

5.0x10

0.0 -5

10

-4

10

-3

10

-2

10

-1

0

10 10 time / s

1

10

2

10

-7

2.5x10

3

10

deposited layer thickness d / m

0.0 -9

-1.0x10

-3

x0 = 2x10

-9

-2.0x10

2

-9

2.0x10

-7

-1

-3

1.5x10

x0 = 4x10

2

-1

U = ∞ m s kg -20 2 -1 U = 10 m s kg 2 -1 U = 0 m s kg

-7

1.0x10

-8

5.0x10

0.0 -5 10

U = ∞ m s kg -20 2 -1 U = 10 m s kg 2 -1 U = 0 m s kg

-3.0x10

precpitate radius ρ / m

-7

Fig. 2. Evolution of the precipitate radius q for x0 = 2 · 103 and different values of the parameter U, describing the activity of sources and sinks for vacancies at the precipitate/matrix interface.

-4

10

-3

10

-2

10

-1

0

10 10 time / s

1

10

2

10

3

10

Fig. 6. Evolution of the precipitate radius q for x0 = 4 · 103 and different values of parameter U.

-9

-4.0x10

-9

-5.0x10

-5

10

-4

10

-3

10

-2

10

-1

0

10 10 time / s

1

10

2

10

3

10

Fig. 3. Evolution of the deposited layer thickness d of atoms at the precipitate/matrix interface for x0 = 2 · 103 and different values of parameter U.

s kg1 are used. From the figures one can conclude that the chemical driving force is always sufficient for a significant growth of the precipitate in the first 0.2 s. Then, the further growth of the precipitate is significantly influenced by stress relaxation. The final size of the precipitate is

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J. Svoboda et al. / Acta Materialia 54 (2006) 4575–4581

d and q define a certain volume Vd given by Eq. (11). As this volume remains fixed, the absolute value of d decreases with increasing precipitate radius q.

deposited layer thickness d / m

0.0

-9

-2.0x10

-3

x0 = 4x10

7. Conclusions 2

-1

U = ∞ m s kg -20 2 -1 U = 10 m s kg 2 -1 U = 0 m s kg

-9

-4.0x10

-9

-6.0x10

-5

10

-4

10

-3

10

-2

10

-1

0

10 10 time / s

1

10

2

10

3

10

Fig. 7. Evolution of the deposited layer thickness d of atoms at the precipitate/matrix interface for x0 = 4 · 103 and different values of parameter U.

-3

-3

x0 = 4x10

4.0x10 mole fraction x

2

-1

U = ∞ m s kg -20 2 -1 U = 10 m s kg 2 -1 U = 0 m s kg

-3

3.0x10

-3

2.0x10

-3

1.0x10

-5

10

-4

10

-3

10

-2

10

-1

0

10 10 time / s

1

10

2

10

3

10

Fig. 8. Evolution of the mole fraction in the matrix x for x0 = 4 · 103 and different values of parameter U.

A new model for the simultaneous precipitate growth and misfit stress relaxation is presented. The system consists of one spherical precipitate embedded in the matrix. The actual state of the system is described by two parameters: q, the precipitate radius; and d, the thickness of the layer of matrix atoms deposited at the precipitate/matrix interface. The total Gibbs energy of the system is calculated as the sum of the chemical Gibbs energy and of the mechanical Gibbs energy consisting of the elastic strain energy and the interface energy. The total Gibbs energy is dissipated by    

the diffusion of the interstitial component; the migration of the precipitate/matrix interface; the diffusion of vacancies in the matrix; the generation or annihilation of vacancies at the precipitate/matrix interface.

The vacancies are assumed to be generated or annihilated in the matrix without any dissipation. The evolution equations for the parameters q and d are derived by means of the thermodynamic extremal principle. Based on the model, the growth of a cementite precipitate embedded in the ferritic matrix is simulated. The influence of the stress relaxation kinetics on the precipitate growth is discussed.

elastic energy per precipitate volume / Jm

-3

Acknowledgements ¨ sterreichische ForschungsFinancial support by the O fo¨rderungsgesellschaft mbH, the Province of Styria, the Steirische Wirtschaftsfo¨rderungsgesellschaft mbH and the Municipality of Leoben under the frame of the Austrian Kplus Programme is gratefully acknowledged.

8

1.5x10

-3

x0 = 4x10

8

1.0x10

2

7

5.0x10

0.0 -5 10

-1

U = ∞ m s kg -20 2 -1 U = 10 m s kg 2 -1 U = 0 m s kg

-4

10

-3

10

-2

10

-1

0

10 10 time / s

Appendix A

1

10

2

10

3

10

Fig. 9. Evolution of the elastic strain energy per precipitate volume for x0 = 4 · 103 and different values of parameter U.

smaller in the case of no stress relaxation and the matrix remains supersaturated. It is necessary to note that under the absence of stress relaxation the absolute value of d decreases with increasing precipitate radius q (compare Figs. 6 and 7). This is due to the fact that the initial value of d is assumed to correspond to a totally relaxed stress state for the initial value of the precipitate radius q. The values of

The mass conservation in the bulk for a component i (i = X or M) in the bulk follows, e.g. from Ref. [21], Section 3, as   d xi v ¼ div~ ji ðA:1Þ þ xi div~ dt Xm The quantity ~ v is the material velocity, div means the divergence operator and ~ ji is the diffusive flux of the component i. If the material is rather incompressible or ~ v itself is very small as in the case at hand, then the term xi div~ v can be neglected. If the atoms of a component X are deposited uniformly, then one may assume that the concentration x/Xm changes everywhere in the same way. This means finally that div~ jX must be a constant quantity. For the

J. Svoboda et al. / Acta Materialia 54 (2006) 4575–4581

spherically symmetric case this is fulfilled by the following radial component of the flux of component X: 3

jX ðrÞ ¼ jX ðqþ Þ 

3

2

R r q  R 3  q3 r 2

ðA:2Þ

which meets also the boundary condition that jX(R) = 0. The mass conservation at the interface at r = q follows, e.g. from Ref. [21], Section 3, as jX ðqþ Þ  jX ðq Þ ¼ ðxðqþ Þ=Xm  xðq Þ=Xp Þq_

ðA:3Þ

The radius q+ is on the right side of the interface, q on the left side. Since there is no flux inside the precipitate, jX(q) ” 0 and x(q) = b. Furthermore we assume x(q+) = x and thus jX ðqþ Þ ¼ ðx=Xm  b=Xp Þq_

ðA:4Þ

If the volume rate V_ d of the matrix matter is transported to the precipitate, then jM ðqÞ ¼ V_ d =4pq2 is the radial flux of component M at the interface. Analogously to Eq. (A.2) the corresponding radial flux jM(r) is then jM ðrÞ ¼

V_ d 1 R3  r 3 q 2    2 4pq Xm R3  q3 r2

ðA:5Þ

which meets also the boundary condition that jM(R) = 0.

4581

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