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Inverse Ripening and Rearrangement of Precipitates under Chemomechanical Coupling
Computational Materials Science 130 (2017) 292–296
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Letter
Inverse Ripening and Rearrangement of Precipitates under Chemomechanical Coupling Reza Darvishi Kamachali ⇑, Christian Schwarze Interdisciplinary Centre for Advanced Materials Simulation (ICAMS), Ruhr-University Bochum, 44801 Bochum, Germany
a r t i c l e
i n f o
Article history: Received 21 November 2016 Received in revised form 10 January 2017 Accepted 16 January 2017
Keywords: Phase transformation Coherent precipitation Ripening Elasticity
a b s t r a c t A coupling between diffusional and mechanical relaxation raised from composition-dependent elastic constants, and its effects on the evolution of precipitates with finite misfit strain are investigated. Inverse ripening has been observed where smaller precipitate grows at the expense of a larger one. This occurs due to fluxes generated under elastically-strained solute gradients around precipitates that 6 scales with Rr where R and r are the precipitate radius and the radial coordinate, respectively. Both isotropic and anisotropic dependency of elastic constants on the composition were considered. The latter leads to the emergence of new patterns of elastic anisotropy and rearrangement of precipitates in the matrix. Ó 2017 Elsevier B.V. All rights reserved.
The effect of internal stress on the kinetics of precipitation and distribution of precipitates in solid state is a long-standing problem with many applications in metallurgy and materials science [1,2]. The internal stress is generated as a result of lattice mismatch between the precipitate and matrix. Although classical theory of ripening does not include the effect of stress [3], there have been extensive studies which take the elasticity into account (see the review paper by Fratzl et al. [4] and references therein). The elastic energy produced during precipitation influences different aspects of precipitation via diffusion: A spatial flux of atoms is expected el
under the gradients of elastic energy, r @f@c with f
el
¼ 12 ½ij ij
C ½kl kl , that adds to the fluxes under concentration gradients. During phase separation, elastic heterogeneity between two phases has been shown to have significant effects on the morphology of the evolving phases [5,6] and kinetics of their transformation [7]. The space dependency of elastic properties is also discussed in some literature. In particular, mechanically-driven fluxes originating from size mismatch of alloying elements (Vegard’s law) have been widely investigated [8–11]. In this study, we focus on the less investigated case of local coupling between stiffness of materials and local composition varying at a transformation front. This coupling exists due to the fact that the atomic bonds between solute and solvent atoms are often of different strength. Thus, the local elastic constants of the solution should ijkl
be a function of local composition, C ijkl ðcðx; tÞÞ. In this case, the ⇑ Corresponding author. E-mail address: [email protected] (R. Darvishi Kamachali). http://dx.doi.org/10.1016/j.commatsci.2017.01.024 0927-0256/Ó 2017 Elsevier B.V. All rights reserved.
mechanical relaxation can be achieved by means of diffusion, leading to local softening/stiffening of the materials. Although the possibility of this effect has been introduced long ago [10,12], it was largely neglected until recently, when we discussed its influence on the precipitation in NiTi shape memory alloys [13]. Furthermore, Steinbach and co-workers have shown that a ‘strained equilibrium’ may exist [14], i.e. stabilized concentration gradients will be produced in the neighbourhood of a single precipitate, if elastic constants of the matrix are composition dependent. In this work we investigate the effect of ‘chemomechanical’ coupling due to composition-dependent elastic constants on the ripening of precipitates. New mechanisms of inverse ripening and rearrangement of the precipitates are discovered in the presence of this coupling. We study precipitation and ripening of d0 (Al3Li) in aluminium8 at.% lithium system, at 473 K, but the effect is general and can be extended to a large class of materials. The precipitate holds a volumetric transformation strain about 1% and has a coherent interface energy of 0.014 J m2 [15]. Both precipitate and aluminium matrix are cubic crystals with nearly isotropic stiffness with Zener indexes of 0.77 and 1.29, respectively [16]. Ab-initio studies [17] and experimental measurements [18] reveal that the elastic constants of aluminium–lithium solution depend on the lithium solute content, anisotropically. In accordance with these observations, a linear relation
C ijkl ðcÞ ¼ C ijkl 1 þ jijkl c 0
ð1Þ
is considered for small composition variation in the matrix. Whilst C11 and C44 increase (positive coupling factor), C12 decreases (negative coupling factor) with increasing the solute content. Thus
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293
jijkl (at.%1) is the anisotropic coupling matrix with j11 ¼ j22 ¼ j33 ¼ 0:005, j12 ¼ j13 ¼ j23 ¼ 0:023 and j44 ¼ j55 ¼ j66 ¼ 0:052. For comparison, two ‘isotropic’ cases of coupling with jijkl ¼ j ¼ 0:01 and 0.05 at.%1 were also considered in this study. are elastic constants of pure aluminium and c ¼ cðx; tÞ is the C ijkl 0 local lithium concentration. Since the elastic constants depend on the composition, thus the elastic anisotropy of the matrix also becomes a function of composition that will be discussed in the following. We treat the precipitate as stoichiometric phase. The total flux of the lithium atoms in the matrix is given by
h i 1 kl kl J ¼ Drc M r ½ij ij jijkl C ijkl 0 ½ 2
ð2Þ
where D ¼ 1:2 1018 m2 s1 is diffusion coefficient at 473 K [19], D M ¼ RT is atomic mobility with R the gas constant, ij are the total strains and
ij
are the eigenstrains. Eq. (1) is used in evaluation of el
¼ 0) at each time step and Eq. (2) mechanical equilibrium (r j @f @ ij gives temporal evolution of composition (c_ þ r J ¼ 0). These two equations are temporally coupled that results in the chemomechanical cross-coupling effect. For interface kinetics, we use the phasefield method as detailed in [20,21]. We apply a finite difference scheme to solve phase-field and diffusion equations. An iterative spectral solver with periodic boundary conditions is employed to obtain the mechanical equilibrium [22]. Cubic simulation boxes with 64 grid cells are considered with a grid spacing of 1 nm and time stepping of 0.25 s. A linearised phase diagram is assumed where the chemical driving force is proportional to the deviation from equilibrium matrix composition, DGch ¼ m DS ðc c1 Þ where m ¼ 60 K at.%1 is the slope of the curve in the T–c phase diagram, 5
1
3
DS ¼ 9:7315 10 J K m is the entropy of formation of the precipitate phase [23], and c1 is the equilibrium composition at a flat interface. The precipitates grow from nuclei with negligible size. Thus, any presumption of size and shape is avoided. Elasticity and coupling accompany the transformation from the beginning. Across the interface, Reuss homogenization scheme is applied. The elastic energy, diffusion potentials and driving forces have been derived and computed, accordingly. More details of the simulation set-up and derivations will be given elsewhere soon [24]. The interface kinetics reproduced by the phase-field equation follows the Gibbs-Thomson relation
V ¼ rK þ DGch þ DGel Mb
ð3Þ
where V is interface velocity, M b is interface mobility, r is interface energy, K is interface curvature and DGi s are driving forces. In general, the chemical driving force is in favour of precipitation from a super-saturated matrix, whilst the elasticity can suppress or accelerate the growth depending on the global mechanical equilibrium. In the current simulations, since the volume fraction of the precipitate is much less than 50%, elasticity (along surface tension) suppresses its growth and DGch is the only positive driving force. Fig. 1 summarizes the results of single precipitate simulations. The black dots in Fig. 1(a) show growth due to chemical driving force, neglecting elasticity, where the volume fraction of the precipitate is only influenced by its curvature. The elastic contribution, once included, suppresses growth and equilibrium volume fraction (red1 dots). Considering chemomechanical coupling, the effects of elasticity are enforced in both regards. Fig. 1(b) shows the concentration profile at equilibrium t = 10,000 s. In the presence of chemomechanical coupling, a concentration gradient is stabilized around the precipitate, whilst without the coupling those gradients in the 1 For interpretation of color in Fig. 1, the reader is referred to the web version of this article.
Fig. 1. (a) Evolution of a single precipitate and (b) concentration profiles from the centre of the precipitate (t = 10,000 s) are shown.
matrix vanish. This is a result of non-vanishing elastically-driven fluxes around the precipitate which balance with the Fick’s diffusion (Eq. (2)). The analytical solution of Eq. (2) will be discussed shortly. Note that elasticity without the coupling corrects equilibrium composition at the interface but does not influence solute distribution. The chemomechanical coupling also corrects the equilibrium composition of the matrix raising its concentration, as shown in Fig. 1(b), by pushing the solute atoms into the matrix. Fig. 2 shows the results of simulations for a pair of precipitates with different coupling situations. The precipitates nucleate at different time steps that gives the initial size difference in the growth stage. The conventional theory of ripening [3] predicts that precipitates compete with respect to their size. This is indeed observed in our simulations if there is no chemomechanical coupling. Fig. 2(a) compares evolution of precipitates with and without elasticity. Compared to normal ripening, elasticity results in deceleration of the growth and acceleration of the ripening in the later stages. The cross-coupling between diffusion and mechanical relaxation, however, changes the ripening behaviour, entirely: As soon as the matrix is depleted and the precipitates ‘feel’ each other via diffusion of the solute atoms, the larger precipitate shrinks until the two precipitates equilibrate with respect to their size (Figs. 2(b) and (c)). This is called ‘inverse ripening’ in the current letter in contrast to conventional ripening in which larger precipitates grow at the expense of smaller ones. Similar behaviour has been observed for j ¼ 0:05 at.%1. Two mechanical effects on the kinetics of transformation are (i) the direct offset of elastic energy between precipitate and matrix phases (third term in Eq. (3)) and (ii) the indirect modification of solute flux (entering second term in Eq. (3)). Whilst the first effect is always present upon elastic interaction, the latter is only a consequence of the chemomechanical coupling introduced in this study. The results show that the interfacial energy offset cannot lead to inverse ripening. In fact, elasticity even accelerates the ripening process when there is no chemomechanical interaction (Fig. 2(a)). These observations acknowledge the significance of
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Fig. 3. (a) Configuration of precipitates at t = 37,500 s and (b) concentration profiles (centre-to-centre) at different time steps are shown for j = 0.01 at.%1.
Fig. 2. Evolution of pair precipitates (a) without coupling and with (b) isotropic ij coupling, j = 0.01 at.%1 and (c) kappa = a.i. values are shown.
the latter contribution for the inverse ripening that is discussed in the following. As a result of chemomechanical coupling, the composition around a precipitate of radius R is influenced by elasticallydriven fluxes as formulated in Eq. (2). For a quasi-steady state, c_ ¼ r J ¼ 0, we have obtained the concentration profile for r P R2
6 R R cðrÞ ¼ c0 g 0 j þ ðg 0 j g 1 Þ þ Oðj2 Þ r r
ð4Þ
2
Vmb where g 0 ¼ 6GmRT and g 1 ¼ c0 cR . Here Gm is the shear modulus of the matrix, V m is the molar volume of the matrix phase, c0 is the matrix composition far from the precipitate, cR is the equilibrium composition at the surface of the precipitate and, b is a materials constant
b¼ ¼
3 p Bp 3Bp þ 4Gm
p
k
ð1 þ mp Þ mpp
ð5Þ
k
ð1 þ mp Þ mpp þ 2ð1 2mm Þ mkmm
in which B is the bulk modulus, k is the Lame’s first coefficient, m is the Poisson’s ratio and the subscripts m and p relate to matrix and precipitate phase, respectively. p is the transformation strain of the To solve c_ ¼ r J ¼ 0 for a two- or many-particle system with elasticity is not straightforward. Similar to our previous work [14], however, one can find the solution using Eq. (2) for a single precipitate. In this solution, c0 presents supersaturation of the matrix which varies during the ripening. Here we assume elastically isotropic precipitate and matrix and obtain cðrÞ to the first order approximation in j. In [14], we only considered a single precipitate at the steady state and solved for strong condition J ¼ ~ 0, that was sufficient for demonstration of static concentration gradient. 2
precipitate. Eq. (4) expresses quasi-steady strained concentration gradients around a precipitate when the elastic constants of the matrix depend on the solute concentration, locally. The strained concentration gradients are demonstrated in our simulations of the single precipitate (Fig. 1(b)) and the pair of precipitates as 6 shown in Fig. 3(b). The new terms, g 0 j Rr þ g 0 j Rr, in Eq. (4) are the outcomes of the chemomechanical coupling effect (j – 0) in which g 0 is a positive coefficient and independent of the sign of the transformation strain, p . Using Eq. (4), the solute flux at the surface of each precipitate reads
@c 1 J ¼ D ¼ D ð5g 0 j þ g 1 Þ R @r R r¼R
ð6Þ
that shows the additional flux, 5DgR0 j, in the presence of the chemomechanical coupling. Thus, if 5g 0 j þ g 1 > 0, larger fluxes are conducted towards smaller precipitates and make them grow. Note that, unlike curvature or elasticity effect on the composition (Gibbs-Thompson effects), the chemomechanical coupling does not influence equilibrium composition at the interface (cðr ¼ RÞ ¼ cR ). Nevertheless, it results in solute fluxes at the surface of the precipitate. This is because of the higher order scaling of the 6 first coupling term, g 0 j Rr . From the energetic point of view, during the inverse ripening, the total interface energy maximizes. There is, therefore, a competition between increasing interfacial energy and decreasing energy due to the coupling. Neglecting the effect of elasticity on cR , one can estimate g 1 ¼ D aR with supersaturation D ¼ c0 c1 and cap2rV c
illarity length a ¼ RTp 1 , where V p is the molar volume of the precipitate [3]. Hence, the condition for the inverse ripening can be approximated as 5g 0 j þ D > aR. This is, indeed, true for our system of study as the interface energy, r, is small. In agreement with Eq. (6), we have found that the steeper concentration gradients form next to the smaller precipitate. This leads to formation of asymmetric concentration profiles between the two precipitates as
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295
Fig. 4. Initial (a) and equilibrium stage of twofold-precipitates interaction are shown for (b) no coupling, (c and d) isotropic coupling (j = 0.01 at.%1), and (e and f) anisotropic coupling (a.i. values). The isosurfaces show the inner surface of the hollow precipitate at equilibrium. The cross sections are normal to x axes. Yellow corresponds to the precipitate phase. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
shown in Fig. 3(b). Thus once the two precipitates ‘feel’ each other through the composition field, the larger precipitate, with a softer concentration gradient around, acts like a source for the smaller precipitate until the fluxes in the system are cancelled and the two precipitates equilibrate with respect to their size. For larger r, we have found that the inverse ripening does not occur (not shown here) because the interface energy dominates and g 1 becomes largely negative. For j < 0, the inverse ripening becomes j ¼ gR1 and normal ripening will be impossible and for j ¼ 0, @c @r r¼R recovered. Interestingly, it is found that the sum of total interfacial and elastic energy increases monotonically during the inverse ripening. This is in contrast to the previous studies of inverse ripening, where the elastic contribution (in certain conditions) compensates for the interface energy and leads to stabilization of the precipitates (see the review paper by Fratzl et al. [4] and references therein). In particular, Johnson and coworkers [25,26] and Mitazaki et al. [27] have shown that elasticity influences the growth and for some combination of thermal/physical parameters there might be an inverse ripening possible. They consider the correction into the boundary condition (interface composition) due to the elastic interaction.3 This is the last term in Eq. (3) that, as shown in Fig. 2 (a), has not led to the inverse ripening in the current set-up. In the presence of the chemomechanical coupling a new mechanism of inverse ripening becomes indeed possible: The additional ‘chemical’ fluxes in the (elastically) strained gradients of the solute concentrations (Eq. (4)) overcome the chemical fluxes due to the curvature effects, when the interface energy is small enough. In fact, the chemomechanical coupling raises the absolute amount of supersaturation felt by the precipitate and for certain combinations of the thermal/physical parameters satisfying 5g 0 j þ D > aR, the inverse
3 Note that previous studies on precipitation with transformation strains are often built upon global energy minimization and presumed set-up and geometry of precipitates. The temporal evolution here, however, follows a local minimization path where elasticity accompanies the entire precipitation process from early stages of the growth, where the energy landscape develops upon transformation and elastic interactions.
ripening becomes possible. Onuki and Nishimori simulated spinodal decomposition in binary alloys when there is a modulus inhomogeneity introduced by composition-dependent elastic constants [28] and shear moduli [29] of the two phases. Whilst it is difficult to compare their 2D simulations with the current results, they have observed slow coarsening microstructures deviating from classical theory of ripening which they could not explain. It is also found that during the phase separation softer phase forms a percolated network and surrounds the harder one. In addition to the inverse ripening, rearrangement of the precipitates is observed. This is known as a general result of elastic interactions, but upon chemomechanical coupling this effect has been found to be much more pronounced showing new features of alignment of the precipitates. In order to study these features, a thought experiment has been conducted in which two precipitates interact in a spherical symmetry as one of them wraps the other, separated by a layer of matrix phase. Fig. 4(a) shows the cross section of the initial configuration. Without the coupling, normal ripening occurs as expected (Fig. 4(b)) during which the smaller precipitate in the centre disappears. There are again two cases of chemomechanical coupling with j ¼ 0:01 at.%1 and jijkl = a.i. values studied. The cross sections and isosurfaces in Figs. 4(c)–(f) show the results of the simulations. Activating the chemomechanical cross-coupling, the smaller precipitate in the centre becomes stable and the inner surface of the hollow precipitate evolves (grows or shrinks) in certain spatial orientations indicating possible spatial ordering of the precipitates arrays. There are certain crystallographic directions for which two interacting particles attract or repel each other. For the isotropic coupling shown in Figs. 4(c) and (d), repulsion along primary axis is observed. The anisotropy of the interaction in this case is a result of weak anisotropy of the matrix and d0 phase. For the anisotropic coupling the interaction becomes much more complex: It is found that new patterns of elastic anisotropy emerge in which the precipitates repel each other within the x-y, x-z and y-z planes whilst attracting each other along the diagonals. Figs. 4(e) and (f) demonstrate the results of the anisotropic coupling. These results propose formation of new arrangements of the precipitates in the
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presence of the chemomechanical interaction. Alignment of the precipitates along softer crystallographic orientations of the matrix has been reported by different authors [30–32]. The chemomechanical coupling not only influences the existing interaction but also can generate new patterns of elastic anisotropy as shown in Fig. 4. Current set of the coupling factors confirmed by ab initio calculations leads to a softening along h1 1 1i directions in the aluminium rich matrix. As a matter of fact, due to the mechanism of inverse ripening, precipitates which are sources of the internal stress are preserved to some extent and the elastic interaction between them is naturally prolonged. In addition, the matrix phase becomes nonuniform because of the spatial composition/stiffness variation within the stress fields. This heterogeneity is long-range following the distribution of elastic energy. Detailed analysis of large-scale simulations that demonstrate inverse ripening and new patterns of the precipitate alignment are under investigation and the results will be published soon [24]. Experimental study of these phenomena are also under development and will be presented in the future. In summary, we have investigated a chemomechanical coupling resulting from composition-dependent elastic constants and its effect on the kinetics of ripening and rearrangement of precipitates. Simultaneous coupled evolution of the concentration and the stress fields around the precipitates results in formation of the strained concentration gradients around the precipitates which follow the form of stored elastic energy in the matrix. The quasi-steady state solution of strained concentration profile around the precipitates has been obtained. Although equilibrium composition at the interface, cðr ¼ RÞ ¼ cR , is not affected by the coupling, an additional flux of the solute atoms is generated due to the coupling which leads to a new mechanism of inverse ripening. Furthermore, the local softening/stiffening around the precipitates leads to the formation of new patterns of elastic anisotropy in the matrix phase. This has been demonstrated by a thought simulation in our study. Both features of the inverse ripening and rearrangement of the precipitates are expected to be applicable to a wide range of alloys in which diffusion and composition dependency of the elastic constants are significant.
Acknowledgements RDK acknowledges financial support from German Research Foundation (DFG) for Grant DA 1655/1-1, within the priority program SPP1713 ‘‘Strong Coupling of Thermo-chemical and Thermo-mechanical States in Applied Materials”. We thank Ingo Steinbach for helpful discussions. References [1] A.J. Ardell, Metal. Trans. A 16 (1985) 2131. [2] J.W. Martin, Precipitation Hardening: Theory and Applications, ButterworthHeinemann, 2012. [3] I.M. Lifshitz, V.V. Slyozov, J. Phys. Chem. Solids 19 (1961) 35. [4] P. Fratzl, O. Penrose, J. Lebowitz, J. Stat. Phys. 95 (1999) 1429. [5] J.K. Lee, Mater. Sci. Eng. A 238 (1997) 1. [6] S.Y. Hu, L.Q. Chen, Acta Mater. 49 (2001) 1879. [7] M. Cottura, B. Appolaire, A. Finel, Y. Le Bouar, Scripta Mater. 108 (2015) 117. [8] J.D. Eshelby, Solid Stat. Phys. 3 (1956) 79. [9] A.G. Khachaturyan, Theory of Structural Transformations in Solids, Courier Corp., 2013. [10] J.W. Cahn, Acta Metal. 9 (1961) 795. [11] J.W. Cahn, Acta Metal. 10 (1962) 179. [12] F. Larché, J.W. Cahn, Acta metal. 33 (1985) 331. [13] R. Darvishi Kamachali, E. Borukhovich, N. Hatcher, I. Steinbach, Model. Sim. Mater. Sci. Eng. 22 (2014) 034003. [14] R. Darvishi Kamachali, E. Borukhovich, O. Shchyglo, I. Steinbach, Phil. Mag. Lett. 93 (2013) 680. [15] S. Baumann, D. Williams, Scripta Metal. 18 (1984) 611. [16] M. Mehl, Phys. Rev. B 47 (1993) 2493. [17] A. Taga, L. Vitos, B. Johansson, G. Grimvall, Phys. Rev. B 71 (2005) 014201. [18] W. Müller, E. Bubeck, V. Gerold, in: C. Baker (Ed.), Aluminium-Lithium Alloys, vol. III, London Inst. of Metals, 1986. [19] W.D. Callister, D.G. Rethwisch, Fundamentals of Materials Science and Engineering: An Integrated Approach, John Wiley & Sons, 2012. [20] I. Steinbach, M. Apel, Phys. D: Nonlin. 217 (2006) 153. [21] I. Steinbach, Model. Sim. Mater. Sci. Eng. 17 (2009) 073001. [22] www.OpenPhase.de. [23] S.W. Chen, C.H. Jan, J.C. Lin, Y.A. Chang, Metal Trans. A 20 (1989) 2247. [24] C. Schwarze, A. Gupta, T. Hickel, R. Darvishi Kamachali, 2017 (submitted for publication). [25] W.C. Johnson, Acta Metal. 32 (1984) 465. [26] W.C. Johnson, P.W. Voorhees, D.E. Zupon, Metal. Trans. A 20 (1989) 1175. [27] T. Miyazaki, K. Seki, M. Doi, T. Kozakai, Mat. Sci. Eng. 77 (1986) 125. [28] H. Nishimori, A. Onuki, Phys. Rev. B 42 (1990) 980. [29] A. Onuki, H. Nishimori, Phys. Rev. B 43 (1991) 13649. [30] Y. Wang, L.-Q. Chen, A. Khachaturyan, Acta Metal. Mater. 41 (1993) 279. [31] C.H. Su, P.W. Voorhees, Acta Mater. 44 (1996) 2001. [32] V. Vaithyanathan, L.Q. Chen, Acta Mater. 50 (2002) 4061.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Letter
Inverse Ripening and Rearrangement of Precipitates under Chemomechanical Coupling Reza Darvishi Kamachali ⇑, Christian Schwarze Interdisciplinary Centre for Advanced Materials Simulation (ICAMS), Ruhr-University Bochum, 44801 Bochum, Germany
a r t i c l e
i n f o
Article history: Received 21 November 2016 Received in revised form 10 January 2017 Accepted 16 January 2017
Keywords: Phase transformation Coherent precipitation Ripening Elasticity
a b s t r a c t A coupling between diffusional and mechanical relaxation raised from composition-dependent elastic constants, and its effects on the evolution of precipitates with finite misfit strain are investigated. Inverse ripening has been observed where smaller precipitate grows at the expense of a larger one. This occurs due to fluxes generated under elastically-strained solute gradients around precipitates that 6 scales with Rr where R and r are the precipitate radius and the radial coordinate, respectively. Both isotropic and anisotropic dependency of elastic constants on the composition were considered. The latter leads to the emergence of new patterns of elastic anisotropy and rearrangement of precipitates in the matrix. Ó 2017 Elsevier B.V. All rights reserved.
The effect of internal stress on the kinetics of precipitation and distribution of precipitates in solid state is a long-standing problem with many applications in metallurgy and materials science [1,2]. The internal stress is generated as a result of lattice mismatch between the precipitate and matrix. Although classical theory of ripening does not include the effect of stress [3], there have been extensive studies which take the elasticity into account (see the review paper by Fratzl et al. [4] and references therein). The elastic energy produced during precipitation influences different aspects of precipitation via diffusion: A spatial flux of atoms is expected el
under the gradients of elastic energy, r @f@c with f
el
¼ 12 ½ij ij
C ½kl kl , that adds to the fluxes under concentration gradients. During phase separation, elastic heterogeneity between two phases has been shown to have significant effects on the morphology of the evolving phases [5,6] and kinetics of their transformation [7]. The space dependency of elastic properties is also discussed in some literature. In particular, mechanically-driven fluxes originating from size mismatch of alloying elements (Vegard’s law) have been widely investigated [8–11]. In this study, we focus on the less investigated case of local coupling between stiffness of materials and local composition varying at a transformation front. This coupling exists due to the fact that the atomic bonds between solute and solvent atoms are often of different strength. Thus, the local elastic constants of the solution should ijkl
be a function of local composition, C ijkl ðcðx; tÞÞ. In this case, the ⇑ Corresponding author. E-mail address: [email protected] (R. Darvishi Kamachali). http://dx.doi.org/10.1016/j.commatsci.2017.01.024 0927-0256/Ó 2017 Elsevier B.V. All rights reserved.
mechanical relaxation can be achieved by means of diffusion, leading to local softening/stiffening of the materials. Although the possibility of this effect has been introduced long ago [10,12], it was largely neglected until recently, when we discussed its influence on the precipitation in NiTi shape memory alloys [13]. Furthermore, Steinbach and co-workers have shown that a ‘strained equilibrium’ may exist [14], i.e. stabilized concentration gradients will be produced in the neighbourhood of a single precipitate, if elastic constants of the matrix are composition dependent. In this work we investigate the effect of ‘chemomechanical’ coupling due to composition-dependent elastic constants on the ripening of precipitates. New mechanisms of inverse ripening and rearrangement of the precipitates are discovered in the presence of this coupling. We study precipitation and ripening of d0 (Al3Li) in aluminium8 at.% lithium system, at 473 K, but the effect is general and can be extended to a large class of materials. The precipitate holds a volumetric transformation strain about 1% and has a coherent interface energy of 0.014 J m2 [15]. Both precipitate and aluminium matrix are cubic crystals with nearly isotropic stiffness with Zener indexes of 0.77 and 1.29, respectively [16]. Ab-initio studies [17] and experimental measurements [18] reveal that the elastic constants of aluminium–lithium solution depend on the lithium solute content, anisotropically. In accordance with these observations, a linear relation
C ijkl ðcÞ ¼ C ijkl 1 þ jijkl c 0
ð1Þ
is considered for small composition variation in the matrix. Whilst C11 and C44 increase (positive coupling factor), C12 decreases (negative coupling factor) with increasing the solute content. Thus
R. Darvishi Kamachali, C. Schwarze / Computational Materials Science 130 (2017) 292–296
293
jijkl (at.%1) is the anisotropic coupling matrix with j11 ¼ j22 ¼ j33 ¼ 0:005, j12 ¼ j13 ¼ j23 ¼ 0:023 and j44 ¼ j55 ¼ j66 ¼ 0:052. For comparison, two ‘isotropic’ cases of coupling with jijkl ¼ j ¼ 0:01 and 0.05 at.%1 were also considered in this study. are elastic constants of pure aluminium and c ¼ cðx; tÞ is the C ijkl 0 local lithium concentration. Since the elastic constants depend on the composition, thus the elastic anisotropy of the matrix also becomes a function of composition that will be discussed in the following. We treat the precipitate as stoichiometric phase. The total flux of the lithium atoms in the matrix is given by
h i 1 kl kl J ¼ Drc M r ½ij ij jijkl C ijkl 0 ½ 2
ð2Þ
where D ¼ 1:2 1018 m2 s1 is diffusion coefficient at 473 K [19], D M ¼ RT is atomic mobility with R the gas constant, ij are the total strains and
ij
are the eigenstrains. Eq. (1) is used in evaluation of el
¼ 0) at each time step and Eq. (2) mechanical equilibrium (r j @f @ ij gives temporal evolution of composition (c_ þ r J ¼ 0). These two equations are temporally coupled that results in the chemomechanical cross-coupling effect. For interface kinetics, we use the phasefield method as detailed in [20,21]. We apply a finite difference scheme to solve phase-field and diffusion equations. An iterative spectral solver with periodic boundary conditions is employed to obtain the mechanical equilibrium [22]. Cubic simulation boxes with 64 grid cells are considered with a grid spacing of 1 nm and time stepping of 0.25 s. A linearised phase diagram is assumed where the chemical driving force is proportional to the deviation from equilibrium matrix composition, DGch ¼ m DS ðc c1 Þ where m ¼ 60 K at.%1 is the slope of the curve in the T–c phase diagram, 5
1
3
DS ¼ 9:7315 10 J K m is the entropy of formation of the precipitate phase [23], and c1 is the equilibrium composition at a flat interface. The precipitates grow from nuclei with negligible size. Thus, any presumption of size and shape is avoided. Elasticity and coupling accompany the transformation from the beginning. Across the interface, Reuss homogenization scheme is applied. The elastic energy, diffusion potentials and driving forces have been derived and computed, accordingly. More details of the simulation set-up and derivations will be given elsewhere soon [24]. The interface kinetics reproduced by the phase-field equation follows the Gibbs-Thomson relation
V ¼ rK þ DGch þ DGel Mb
ð3Þ
where V is interface velocity, M b is interface mobility, r is interface energy, K is interface curvature and DGi s are driving forces. In general, the chemical driving force is in favour of precipitation from a super-saturated matrix, whilst the elasticity can suppress or accelerate the growth depending on the global mechanical equilibrium. In the current simulations, since the volume fraction of the precipitate is much less than 50%, elasticity (along surface tension) suppresses its growth and DGch is the only positive driving force. Fig. 1 summarizes the results of single precipitate simulations. The black dots in Fig. 1(a) show growth due to chemical driving force, neglecting elasticity, where the volume fraction of the precipitate is only influenced by its curvature. The elastic contribution, once included, suppresses growth and equilibrium volume fraction (red1 dots). Considering chemomechanical coupling, the effects of elasticity are enforced in both regards. Fig. 1(b) shows the concentration profile at equilibrium t = 10,000 s. In the presence of chemomechanical coupling, a concentration gradient is stabilized around the precipitate, whilst without the coupling those gradients in the 1 For interpretation of color in Fig. 1, the reader is referred to the web version of this article.
Fig. 1. (a) Evolution of a single precipitate and (b) concentration profiles from the centre of the precipitate (t = 10,000 s) are shown.
matrix vanish. This is a result of non-vanishing elastically-driven fluxes around the precipitate which balance with the Fick’s diffusion (Eq. (2)). The analytical solution of Eq. (2) will be discussed shortly. Note that elasticity without the coupling corrects equilibrium composition at the interface but does not influence solute distribution. The chemomechanical coupling also corrects the equilibrium composition of the matrix raising its concentration, as shown in Fig. 1(b), by pushing the solute atoms into the matrix. Fig. 2 shows the results of simulations for a pair of precipitates with different coupling situations. The precipitates nucleate at different time steps that gives the initial size difference in the growth stage. The conventional theory of ripening [3] predicts that precipitates compete with respect to their size. This is indeed observed in our simulations if there is no chemomechanical coupling. Fig. 2(a) compares evolution of precipitates with and without elasticity. Compared to normal ripening, elasticity results in deceleration of the growth and acceleration of the ripening in the later stages. The cross-coupling between diffusion and mechanical relaxation, however, changes the ripening behaviour, entirely: As soon as the matrix is depleted and the precipitates ‘feel’ each other via diffusion of the solute atoms, the larger precipitate shrinks until the two precipitates equilibrate with respect to their size (Figs. 2(b) and (c)). This is called ‘inverse ripening’ in the current letter in contrast to conventional ripening in which larger precipitates grow at the expense of smaller ones. Similar behaviour has been observed for j ¼ 0:05 at.%1. Two mechanical effects on the kinetics of transformation are (i) the direct offset of elastic energy between precipitate and matrix phases (third term in Eq. (3)) and (ii) the indirect modification of solute flux (entering second term in Eq. (3)). Whilst the first effect is always present upon elastic interaction, the latter is only a consequence of the chemomechanical coupling introduced in this study. The results show that the interfacial energy offset cannot lead to inverse ripening. In fact, elasticity even accelerates the ripening process when there is no chemomechanical interaction (Fig. 2(a)). These observations acknowledge the significance of
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Fig. 3. (a) Configuration of precipitates at t = 37,500 s and (b) concentration profiles (centre-to-centre) at different time steps are shown for j = 0.01 at.%1.
Fig. 2. Evolution of pair precipitates (a) without coupling and with (b) isotropic ij coupling, j = 0.01 at.%1 and (c) kappa = a.i. values are shown.
the latter contribution for the inverse ripening that is discussed in the following. As a result of chemomechanical coupling, the composition around a precipitate of radius R is influenced by elasticallydriven fluxes as formulated in Eq. (2). For a quasi-steady state, c_ ¼ r J ¼ 0, we have obtained the concentration profile for r P R2
6 R R cðrÞ ¼ c0 g 0 j þ ðg 0 j g 1 Þ þ Oðj2 Þ r r
ð4Þ
2
Vmb where g 0 ¼ 6GmRT and g 1 ¼ c0 cR . Here Gm is the shear modulus of the matrix, V m is the molar volume of the matrix phase, c0 is the matrix composition far from the precipitate, cR is the equilibrium composition at the surface of the precipitate and, b is a materials constant
b¼ ¼
3 p Bp 3Bp þ 4Gm
p
k
ð1 þ mp Þ mpp
ð5Þ
k
ð1 þ mp Þ mpp þ 2ð1 2mm Þ mkmm
in which B is the bulk modulus, k is the Lame’s first coefficient, m is the Poisson’s ratio and the subscripts m and p relate to matrix and precipitate phase, respectively. p is the transformation strain of the To solve c_ ¼ r J ¼ 0 for a two- or many-particle system with elasticity is not straightforward. Similar to our previous work [14], however, one can find the solution using Eq. (2) for a single precipitate. In this solution, c0 presents supersaturation of the matrix which varies during the ripening. Here we assume elastically isotropic precipitate and matrix and obtain cðrÞ to the first order approximation in j. In [14], we only considered a single precipitate at the steady state and solved for strong condition J ¼ ~ 0, that was sufficient for demonstration of static concentration gradient. 2
precipitate. Eq. (4) expresses quasi-steady strained concentration gradients around a precipitate when the elastic constants of the matrix depend on the solute concentration, locally. The strained concentration gradients are demonstrated in our simulations of the single precipitate (Fig. 1(b)) and the pair of precipitates as 6 shown in Fig. 3(b). The new terms, g 0 j Rr þ g 0 j Rr, in Eq. (4) are the outcomes of the chemomechanical coupling effect (j – 0) in which g 0 is a positive coefficient and independent of the sign of the transformation strain, p . Using Eq. (4), the solute flux at the surface of each precipitate reads
@c 1 J ¼ D ¼ D ð5g 0 j þ g 1 Þ R @r R r¼R
ð6Þ
that shows the additional flux, 5DgR0 j, in the presence of the chemomechanical coupling. Thus, if 5g 0 j þ g 1 > 0, larger fluxes are conducted towards smaller precipitates and make them grow. Note that, unlike curvature or elasticity effect on the composition (Gibbs-Thompson effects), the chemomechanical coupling does not influence equilibrium composition at the interface (cðr ¼ RÞ ¼ cR ). Nevertheless, it results in solute fluxes at the surface of the precipitate. This is because of the higher order scaling of the 6 first coupling term, g 0 j Rr . From the energetic point of view, during the inverse ripening, the total interface energy maximizes. There is, therefore, a competition between increasing interfacial energy and decreasing energy due to the coupling. Neglecting the effect of elasticity on cR , one can estimate g 1 ¼ D aR with supersaturation D ¼ c0 c1 and cap2rV c
illarity length a ¼ RTp 1 , where V p is the molar volume of the precipitate [3]. Hence, the condition for the inverse ripening can be approximated as 5g 0 j þ D > aR. This is, indeed, true for our system of study as the interface energy, r, is small. In agreement with Eq. (6), we have found that the steeper concentration gradients form next to the smaller precipitate. This leads to formation of asymmetric concentration profiles between the two precipitates as
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Fig. 4. Initial (a) and equilibrium stage of twofold-precipitates interaction are shown for (b) no coupling, (c and d) isotropic coupling (j = 0.01 at.%1), and (e and f) anisotropic coupling (a.i. values). The isosurfaces show the inner surface of the hollow precipitate at equilibrium. The cross sections are normal to x axes. Yellow corresponds to the precipitate phase. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
shown in Fig. 3(b). Thus once the two precipitates ‘feel’ each other through the composition field, the larger precipitate, with a softer concentration gradient around, acts like a source for the smaller precipitate until the fluxes in the system are cancelled and the two precipitates equilibrate with respect to their size. For larger r, we have found that the inverse ripening does not occur (not shown here) because the interface energy dominates and g 1 becomes largely negative. For j < 0, the inverse ripening becomes j ¼ gR1 and normal ripening will be impossible and for j ¼ 0, @c @r r¼R recovered. Interestingly, it is found that the sum of total interfacial and elastic energy increases monotonically during the inverse ripening. This is in contrast to the previous studies of inverse ripening, where the elastic contribution (in certain conditions) compensates for the interface energy and leads to stabilization of the precipitates (see the review paper by Fratzl et al. [4] and references therein). In particular, Johnson and coworkers [25,26] and Mitazaki et al. [27] have shown that elasticity influences the growth and for some combination of thermal/physical parameters there might be an inverse ripening possible. They consider the correction into the boundary condition (interface composition) due to the elastic interaction.3 This is the last term in Eq. (3) that, as shown in Fig. 2 (a), has not led to the inverse ripening in the current set-up. In the presence of the chemomechanical coupling a new mechanism of inverse ripening becomes indeed possible: The additional ‘chemical’ fluxes in the (elastically) strained gradients of the solute concentrations (Eq. (4)) overcome the chemical fluxes due to the curvature effects, when the interface energy is small enough. In fact, the chemomechanical coupling raises the absolute amount of supersaturation felt by the precipitate and for certain combinations of the thermal/physical parameters satisfying 5g 0 j þ D > aR, the inverse
3 Note that previous studies on precipitation with transformation strains are often built upon global energy minimization and presumed set-up and geometry of precipitates. The temporal evolution here, however, follows a local minimization path where elasticity accompanies the entire precipitation process from early stages of the growth, where the energy landscape develops upon transformation and elastic interactions.
ripening becomes possible. Onuki and Nishimori simulated spinodal decomposition in binary alloys when there is a modulus inhomogeneity introduced by composition-dependent elastic constants [28] and shear moduli [29] of the two phases. Whilst it is difficult to compare their 2D simulations with the current results, they have observed slow coarsening microstructures deviating from classical theory of ripening which they could not explain. It is also found that during the phase separation softer phase forms a percolated network and surrounds the harder one. In addition to the inverse ripening, rearrangement of the precipitates is observed. This is known as a general result of elastic interactions, but upon chemomechanical coupling this effect has been found to be much more pronounced showing new features of alignment of the precipitates. In order to study these features, a thought experiment has been conducted in which two precipitates interact in a spherical symmetry as one of them wraps the other, separated by a layer of matrix phase. Fig. 4(a) shows the cross section of the initial configuration. Without the coupling, normal ripening occurs as expected (Fig. 4(b)) during which the smaller precipitate in the centre disappears. There are again two cases of chemomechanical coupling with j ¼ 0:01 at.%1 and jijkl = a.i. values studied. The cross sections and isosurfaces in Figs. 4(c)–(f) show the results of the simulations. Activating the chemomechanical cross-coupling, the smaller precipitate in the centre becomes stable and the inner surface of the hollow precipitate evolves (grows or shrinks) in certain spatial orientations indicating possible spatial ordering of the precipitates arrays. There are certain crystallographic directions for which two interacting particles attract or repel each other. For the isotropic coupling shown in Figs. 4(c) and (d), repulsion along primary axis is observed. The anisotropy of the interaction in this case is a result of weak anisotropy of the matrix and d0 phase. For the anisotropic coupling the interaction becomes much more complex: It is found that new patterns of elastic anisotropy emerge in which the precipitates repel each other within the x-y, x-z and y-z planes whilst attracting each other along the diagonals. Figs. 4(e) and (f) demonstrate the results of the anisotropic coupling. These results propose formation of new arrangements of the precipitates in the
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presence of the chemomechanical interaction. Alignment of the precipitates along softer crystallographic orientations of the matrix has been reported by different authors [30–32]. The chemomechanical coupling not only influences the existing interaction but also can generate new patterns of elastic anisotropy as shown in Fig. 4. Current set of the coupling factors confirmed by ab initio calculations leads to a softening along h1 1 1i directions in the aluminium rich matrix. As a matter of fact, due to the mechanism of inverse ripening, precipitates which are sources of the internal stress are preserved to some extent and the elastic interaction between them is naturally prolonged. In addition, the matrix phase becomes nonuniform because of the spatial composition/stiffness variation within the stress fields. This heterogeneity is long-range following the distribution of elastic energy. Detailed analysis of large-scale simulations that demonstrate inverse ripening and new patterns of the precipitate alignment are under investigation and the results will be published soon [24]. Experimental study of these phenomena are also under development and will be presented in the future. In summary, we have investigated a chemomechanical coupling resulting from composition-dependent elastic constants and its effect on the kinetics of ripening and rearrangement of precipitates. Simultaneous coupled evolution of the concentration and the stress fields around the precipitates results in formation of the strained concentration gradients around the precipitates which follow the form of stored elastic energy in the matrix. The quasi-steady state solution of strained concentration profile around the precipitates has been obtained. Although equilibrium composition at the interface, cðr ¼ RÞ ¼ cR , is not affected by the coupling, an additional flux of the solute atoms is generated due to the coupling which leads to a new mechanism of inverse ripening. Furthermore, the local softening/stiffening around the precipitates leads to the formation of new patterns of elastic anisotropy in the matrix phase. This has been demonstrated by a thought simulation in our study. Both features of the inverse ripening and rearrangement of the precipitates are expected to be applicable to a wide range of alloys in which diffusion and composition dependency of the elastic constants are significant.
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