Shape factors in modeling of precipitation

Materials Science and Engineering A 441 (2006) 68–72

Shape factors in modeling of precipitation E. Kozeschnik a,b,∗ , J. Svoboda c , F.D. Fischer b,d,e a

Graz University of Technology, Institute for Materials Science, Welding and Forming, Kopernikusgasse 24, A-8010 Graz, Austria b Materials Center Leoben Forschungsgesellschaft mbH, Franz-Josef Straße 13, A-8700 Leoben, Austria c Academy of Science of the Czech Republic, Institute of Physics of Materials, Zizkova 22, Cz-61662 Brno, Czech Republic d Montanuniversit¨ at Leoben, Institute of Mechanics, Franz-Josef Straße 18, A-8700 Leoben, Austria e Austrian Academy of Sciences, Institute for Materials Science, Jahnstraße 12, A-8700 Leoben, Austria Received 20 January 2006; received in revised form 8 June 2006; accepted 25 August 2006

Abstract Recently, a model for the growth and dissolution kinetics of precipitates in multi-component multi-phase environments has been developed. In the original treatment, the evolution equations for the size and chemical composition of the precipitates have been derived for spherical precipitates. In this paper, shape factors are introduced, which provide extensions of the existing formalism to needle-shaped and disc-shaped precipitate geometries. The mathematical formalism can be seamlessly incorporated into the existing model. The growth kinetics of precipitates with different shapes are discussed and compared to the spherical case. © 2006 Elsevier B.V. All rights reserved. Keywords: Precipitates; Precipitation kinetics; Shape factor

1. Introduction In two recent papers [1,2], a new approach to the modeling of precipitation kinetics in multi-component systems has been developed. With the simplification that the precipitates have a spherical shape, the Gibbs free energy G of a given volume of matter with m precipitates and n components has been expressed as   n m n m     4πρk3 G= N0i μ0i + cki μki + 4πρk2 γk . λk + 3 i=1 i=1 k=1 k=1 (1) The subscripts ‘0’ denote quantities related to the matrix. The index ‘k’ denotes quantities related to the precipitate; ρk is the precipitate radius, γ k is the specific interfacial energy, and λk accounts for the contribution of mechanical energy due to the misfit volume between the precipitate and the matrix. N0i is the number of moles of component i in the matrix, μ0i and μki are ∗ Corresponding author at: Graz University of Technology, Institute for Materials Science, Welding and Forming, Kopernikusgasse 24, A-8010 Graz, Austria. Tel.: +43 316 873 4304; fax: +43 316 873 7187. E-mail address: [email protected] (E. Kozeschnik).

0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.08.088

the average values of chemical potentials in the matrix and in the precipitates and cki are the average values of concentrations in the precipitates. In the derivation of the model, it has been assumed that the radius and the mean chemical composition of each precipitate evolve according to the thermodynamic extremal principle of maximum entropy production [3]. In non-equilibrium systems under a constant temperature and pressure, the available Gibbs energy is dissipated by three mechanisms: (i) interface migration (Q1 ), (ii) diffusion in the matrix (Q2 ) and (iii) diffusion in the precipitate (Q3 ). These quantities are discussed in detail in Refs. [1,2]. Some experimental observations indicate, that the shape of precipitating phases are often far from the spherical one and they can be very well approximated by needle-shaped to discshaped geometries with a fixed aspect ratio known from the observations. The main aim of the paper is to develop shape factors, incorporate them into the fundamental expressions for G, Q1 , Q2 and Q3 and provide an approximate treatment of precipitation kinetics for needle-shaped to disc-shaped precipitates. Finally, the extended evolution equations are presented, and the influence of the precipitate shape on precipitation kinetics is discussed in detail on the base of comparison with the spherical case.

E. Kozeschnik et al. / Materials Science and Engineering A 441 (2006) 68–72

69

m  n  4πOk RTρ3 (ρ˙ k (cki − c0i ) + ρk c˙ ki /3)2 k

Q3 =

c0i D0i

k=1 i=1

,

(5)

where R is the gas constant, D0i and Dki are the diffusion coefficients in the matrix and in the precipitates, Mk the interface mobility and T is the absolute temperature. The dependence of λk on h is dealt with in Refs. [4,5] and it is not repeated here. From these expressions, the evolution equations are obtained according to the proceduredescribed in Ref. [1], resulting in k sets of linear equations n−1 j=1 Aij yj = Bi (i = 1, . . ., n − 1), which describe the evolution of the system. For a fixed k, y1 ≡ ρ˙ k and yi ≡ c˙ ki (i = 2, . . ., n − 1) and the coefficients of the linear set of equations are n

A11 =

 (cki − c0i )2 Kk + Ok RTρk , Mk c0i D0i

Fig. 1. Illustration of the shape parameter h = H/D.

Ok RTρk2 A1i = Ai1 = 3

2. Shape parameter h The precipitate shape is approximated by a family of cylinders with the height H and the diameter D. A single shape parameter h is used to describe the precipitate shape given by h = H/D (see Fig. 1). The quantity ρk has the meaning of the equivalent precipitate radius guaranteeing the same volume of the spherical and the cylindrical precipitate. With this definition, small values of h represent discs, whereas large values of h represent needles. In the following treatment it is assumed that the shape parameter h remains constant during evolution of each precipitate.

The shape factors are introduced into the original expressions for the Gibbs energy G and the dissipation rates Q1 , Q2 , Q3 in such a way that they relate all the quantities in terms of the ratio between the cylindrical and the spherical precipitates of the same volume. One individual shape factor is introduced into each of the original terms G, Q1 , Q2 and Q3 . The basic expressions with the new shape factors Sk , Kk , Ik and Ok are given as (compare Eqs. (7), (11), (13) and (22) in Ref. [1]):   n m n    4πρk3 G= N0i μ0i + cki μki λk + 3 +

m  k=1

Q1 =

4πSk ρk2 γk ,

A1i = Ai1 =

(2)

Mk

k k



(7)

Ok RTρk2 3



cks+1 − c0s+1 cki+1 − c0i+1 − c0i+1 D0i+1 c0s+1 D0s+1

RTρk3 Aij = 45

(8)

 5Ok Ik δij + cki Dki 3c0i D0i   Ik 5Ok + + ck1 Dk1 3c01 D01



(i = 2, . . . , s, j = 2, . . . , s), RTρk3 Aij = 45



(9)

 Ik 5Ok δij + cki+1 Dki+1 3c0i+1 D0i+1   Ik 5Ok + + cks+1 Dks+1 3c0s+1 D0s+1



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

(10)

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

(11)

with the coefficients of the right-hand side: n

 −2Sk γk − λk − cki (μki − μ0i ), ρk

(12)

i=1

,

m  n  4πIk RTρ5 c˙ 2

k ki

k=1 i=1

ck1 − c01 cki − c0i − c0i D0i c01 D01

(i = s + 1, . . . , n − 1),

B1 =

m  4πKk ρ2 ρ˙ 2 k=1

Q2 =

i=1

k=1



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

3. Shape factors in the evolution equations

i=1

(6)

i=1

45cki Dki

,

ρk (μki − μ0i − μk1 + μ01 ) 3

(3)

Bi = −

(4)

ρk (μki+1 − μ0i+1 − μks+1 + μ0s+1 ) 3 (i = s + 1, . . . , n − 1).

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

(13)

Bi = −

(14)

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The Kronecker delta δij has its usual meaning and s is the number of substitutional components in the system. 4. Evaluation of the shape factors

V

4.1. Shape factor for the surface of the precipitate Giving D the length “1”, the volume of the cylindrical precipitate is πh/4, and its surface is Sp = π(h + 1/2). A sphere of the same volume has a radius ρk = (3h/16)1/3 and a surface Ss = 4π(3h/16)2/3 . Thus, the shape factor Sk relating the surface of the cylindrical precipitate to the spherical precipitate is given by Sk =

Sp = 0.7631h1/3 + 0.3816h−(2/3) . Ss

(15)

4.2. Shape factor for interface migration of the precipitate Giving D the length “1”, we assume that the interface velocities vr , vz are vz = Ah/2 for z = h/2, vz = −Ah/2 for z = −h/2 and vr = A/2 for r = 1/2 (A being a proper constant) to ensure that the value of the parameter h remains constant during evolution of the precipitate. Then, the dissipation due to interface migration corresponding to the cylindrical precipitate is given by    2   π Ah 2 A2 π 2 1 A Q1p = = + πh (h + 2h). Mk 2 2 2 8Mk (16) The rate of the volume change of the precipitate for these interface velocities equals 3πAh/4. The interface velocity of the spherical precipitate of the same volume as the cylindrical precipitate corresponding to the same rate of volume change is ρ˙ k = A(3h/16)1/3 . The dissipation due to interface migration of the spherical precipitate is given by Q1s =

4πρk2 ρ˙ k2 Mk

=

4πA Mk

2

3h 16

4/3 .

(17)

Then the shape factor for interface migration of the cylindrical precipitate k follows as Kk =

Q1p = 0.2912h2/3 + 0.5824h−(1/3) . Q1s

(18)

4.3. Shape factor for diffusion inside the precipitate Let us assume a spherical precipitate with the radius ρk in which a component is diffusing with the flux vector j = Br (B being a proper constant) with r being the position vector with the length r and the origin in the centre of the precipitate. Then the divergence of the flux j yields: div j = 3B

in the whole precipitate. The dissipation due to diffusion in a homogeneous volume V follows as

2 (20) Q = Θ (j) dV

(19)

and in the case of the spherical precipitate as  5/3

ρk 4π 2 5 4π 2 3h 2  2 Q2s = 4πΘ B Θρk = B Θ r (j ) dr = , 5 5 16 0 (21) where Θ is a proper constant, such that the spherical precipitate has the same volume as the cylindrical one and D is chosen as a length unit. The total flux out of the sphere is 3πBh/4. We solve now the differential Eq. (19) for j = grad u with u being a differentiable function and a cylindrical geometry (0 ≤ r ≤ 1/2, −h/2 ≤ z ≤ h/2). As boundary conditions ˜ = B; r = 1/2 : jr = B/2, ˜ we chose with B z = h/2 : jz = ˜Bh/2, z = −h/2 : jz = −Bh/2, ˜ so that the total flux out of the cylinder is equal to that of the sphere. If we interpret u as the temperature, we can immediately engage a finite element program (here ABAQUS [6]) to solve the stationary heat conduction problem with given heat fluxes as above on the boundary yielding both u and the flux j = grad u. Axisymmetric bilinear rectangular elements are employed. The solution procedure is standard. Q2p is evaluated and we finally obtain Ik =

Q2p ≈ 0.4239h4/3 + 0.6453h−(2/3) . Q2s

(22)

4.4. Shape factor for diffusion outside the precipitate To calculate the dissipation due to diffusion in the matrix during the growth of the spherical precipitate we assume that any component diffuses in the matrix with the radial flux j = Cρk2 r /r 3 , C being a proper constant. Then, the divergence of the flux is div j = 0

(23)

in the whole matrix and j = 0 at infinity. The total dissipation in the case of a spherical precipitate according to Eq. (20) is

∞ 3 Q3s = 4πΨ r 2 j 2 dr = 4πC2 Ψρk3 = πC2 Ψh, (24) 4 ρk where ψ is a proper constant, such that the spherical precipitate has the same volume as the cylindrical one. We solve now the differential Eq. (23) for j = grad u with u being a differentiable function and as topology an infinite body with a cylindrical cavity (0 ≤ r ≤ 1/2, −h/2 ≤ z ≤ h/2). As ˜ = C(3h/16)1/3 the same boundary condition we choose with B one as in the problem of Section 4.3, so that an amount of ˜ 3πBh/4 flows out of the body into the cavity. The same procedure as outlined in Section 4.3 is followed. Q3p is evaluated and we finally obtain: Ok =

Q3p ≈ 1.0692h2/3 . Q3s

(25)

E. Kozeschnik et al. / Materials Science and Engineering A 441 (2006) 68–72

Fig. 2. Shape factors as a function of the shape parameter h = H/D.

5. Discussion In the present treatment, precipitates with non-spherical shapes are approximated by a family of cylinders with a single shape parameter h = H/D. The functions obtained from the analysis are exact solutions in the case of the shape factors Sk and Kk . The shape factor Sk represents the ratio between the surface area of a cylinder and a sphere of the same volume and Kk is related to the Gibbs energy dissipation due to interface migration. The shape factors Ik and Ok are related to the diffusive fluxes inside and outside the precipitate depending also on the precipitate geometry. These shape factors are approximate solutions obtained from a numerical finite element analysis. Fig. 2 shows the values of the shape factors as a function of the shape parameter h. Since the shape factors Sk and Kk are directly related to the volume-to-surface ratio of the cylinder and this ratio has a minimum at h = 1, Sk and Kk also show a minimum at this value. The shape factor Ik is related to the diffusion inside the precipitate and has its minimum at a value of h ≈ 0.8724. For this value of h, the chemical composition of the precipitate can most easily change. In contrast to the shape factors Sk , Kk and Ik , which show a pronounced minimum, the shape factor Ok continuously increases with increasing h (large values of h correspond to needle-shaped precipitates). This indicates that, for a given driving force, disc-shaped precipitates grow much faster than needle-shaped precipitates. Since it is difficult to study the effect of the shape factors on the growth kinetics in a complex system, it is advantageous to make some simplifications: (i) the interface mobility is assumed to be very high (Mk → ∞), (ii) the precipitate has a fixed stoichiometry (˙cki = 0) and (iii) there are no mechanical stresses acting on the precipitate (λk = 0). In this case, only the shape factors Sk and Ok are relevant to describe the shape-dependence of the growth kinetics of the precipitate. Accordingly, using Eqs. (6) and (12) the rate of change of the precipitate radius ρ˙ k is

71

Fig. 3. Scaled growth rate as a function of the shape parameter h = H/D and the equivalent precipitate radius ρk .

given by Ok RTρk

n  (cki − c0i )2 i=1

c0i D0i

n

· ρ˙ k = −

2Sk γk  − cki (μki − μ0i ), ρk i=1 (26)

 where − ni=1 cki (μki − μ0i ) represents the chemical driving force DF . Fig. 3 shows the calculated growth rates of precipitates according to Eq. (26) scaled to the growth rate of the cylindrical precipitate with h = 1. The values DF = 108 J/m3 and γ k = 0.15 J/m2 have been used in this analysis. According to Eq. (26), the growth rate of the precipitate gets the higher, the lower the shape factors Ok and Sk become. Fig. 2 shows that Sk has a minimum at h = 1, i.e. for the equiaxed cylinder. This result is expected since this shape provides an optimum surface-to-volume ratio and only minimum surface area is created during growth of the equiaxed cylinder. Fig. 3 reflects these findings: For small precipitates, the growth rate has a maximum close to h = 1. However, the influence of Sk becomes the weaker, the larger the precipitate grows. For large values of ρk , the chemical driving force DF dominates over the influence of interface curvature, and the right-hand side of Eq. (26) approaches a constant value. In that case the growth rate ρ˙ k is inversely proportional to the shape factor Ok . Since Ok steadily increases with increasing h, the growth rate steadily decreases inversely with increasing h. Although this result might seem counter-intuitive at first sight, from the viewpoint of a pure diffusional collection mechanism for precipitate growth, the disc represents the most favorable shape because the diffusion distances to grow the precipitate become a minimum. Finally, it is important to emphasize that the present treatment does not provide any conclusion on the optimum shape of a growing precipitate. Although the disc-shape appears to be a favorable shape in terms of diffusional growth kinetics, real precipitate geometries are usually determined by manifold conditions such as anisotropy of the interfacial energy, anisotropy of the elastic constants or strongly heterogeneous nucleation conditions. For instance, although many precipitates form as

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E. Kozeschnik et al. / Materials Science and Engineering A 441 (2006) 68–72

almost ideal spheres, such as Al3 Sc precipitates in the Al-Sc system [7] or gamma-prime precipitates (Ni3 Al, L12 -ordered fcc) in Ni-Al alloys [8], perovskite-type carbides and nitrides in TiAl have pronounced needle-shape, see e.g. Refs. [9,10]. And earlystage precipitates in Al–Cu alloys (GP-zones) have pronounced disc-shape due to volumetric misfit and preferred growth along the elastically soft 10 -directions of the Al matrix [11]. In the context of the present treatment, the precipitate shape must be considered as an input quantity from microscopic observations and determined by external constraints. Once the parameter h is determined, the shape factors developed in this work provide an approximate treatment of the growth kinetics of the precipitate taking into account deviations from ideal spherical shape. Of course a fair objection may be brought to the present model, why the parameter h is not considered as a free parameter which develops according to an evolution equation derived from the thermodynamic extremal principle. However, to carry out this task it is necessary to extend the existing dependences G = G(ρk , cki , hk ) and Q = Q(ρk , cki , hk , ρ˙ k , c˙ ki ) to G = G(ρk , cki , hk ) and Q = Q(ρk , cki , hk , ρ˙ k , c˙ ki , h˙ k ) (the index k is used also for parameter h to include different types of precipitates in the system with different h parameters—in the present treatment this is implicitly included in dependence of shape factors on k). In the present model an orientation independent interface energy density γ k and isotropic transformation strains (leading to λk = λk (hk )) are assumed, and thus the only dependence of G on hk is incorporated in the dependence of the shape factor Sk on hk . This provides a certain driving force −∂G/∂hk , which is, however, zero for hk = 1 representing the equilibrium shape of the precipitate. There are two major effects, however, which may counteract the tendency of hk approaching the value 1. The first effect is an orientation dependent interface energy density γ k (providing a different dependence of Sk on hk than given by Eq. (15)). The second one is an anisotropic transformation strain leading to λk = λk (hk ). The possibility of calculation of λk = λk (hk ) is outlined in Ref. [4]. The treatment of the evolution of the shape parameter hk is beyond the scope of the present paper. It is currently under investigation and it will be published in a consecutive paper.

6. Summary Based on a recently published model for the growth of spherical precipitates, shape factors are developed and introduced into the existing formalism that account for non-spherical precipitate shapes. The precipitates are approximated by a family of cylinders with a single shape parameter h. The results indicate that, for large precipitates, the growth rates are highest for disc-shaped precipitates with h < 1, whereas the growth rate steadily decreases for needle-shaped precipitates with h > 1. For small precipitates, the effect of interface energy becomes prominent and the maximum growth rates are obtained for equiaxed precipitates with h ≈ 1. It is emphasized that the present treatment does not provide information on the optimum shape of growing precipitates. The shape parameter h must be considered as an input quantity which is determined by external constraints. Acknowledgements ¨ Financial support by the Osterreichische Forschungsf¨orderungsgesellschaft mbH, the Province of Styria, the Steirische Wirtschaftsf¨orderungsgesellschaft mbH and the Municipality of Leoben under the frame of the Austrian Kplus Programme is gratefully acknowledged. References [1] J. Svoboda, F.D. Fischer, P. Fratzl, E. Kozeschnik, Mater. Sci. Eng. A 385 (1/2) (2004) 166–174. [2] E. Kozeschnik, J. Svoboda, F.D. Fischer, Calphad 28 (4) (2005) 379–382. [3] J. Svoboda, I. Turek, F.D. Fischer, Phil. Mag. 85 (2005) 3699–3707. [4] F.D. Fischer, H.J. B¨ohm, Acta Mater. 53 (2) (2005) 367–374. [5] F.D. Fischer, H.J. B¨ohm, E.R. Oberaigner, T. Waitz, Acta Mater. 54 (2006) 151–156. [6] Commercial Finite Element software ABAQUS, http://www.hks.com. [7] E.A. Marquis, D. Seidman, Acta Mater. 53 (2005) 4259–4268. [8] P. Haasen, R. Wagner, Metall. Trans. 23A (1992) 1901–1914. [9] W.H. Tian, T. Sano, M. Nemoto, Phil. Mag. A 68 (1993) 965–976. [10] W.H. Tian, M. Nemoto, Intermetallics 13 (2005) 1030–1037. [11] L. L¨ochte, A. Gitt, G. Gottstein, I. Hurtado, Acta Mater. 48 (2000) 2969–2984.