The equilibrium shape of a misfitting precipitate

Thompson, M.E.; Su, Chun-Ming; Voorhees, Peter W.

doi:10.1016/0956-7151(94)90036-1

Cited by 263 publications

(118 citation statements)

References 35 publications

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“…We have seen that below the bifurcation point, the shape of the precipitate is elongated diamond like, while with increase in the precipitate size the morphology turns towards a twisted diamond-like shape which is also referred to as S-shape precipitates by Voorhees et al [11] In their study, they mention that these S-shapes of precipitates are qualitatively consistent with shape of precipitates observed experimentally in Mg-stabilized ZrO 2 . [51] Lanteri et al [52] have stated that the ZrO 2 precipitates acquire tetragonal morphologies and associate it with the accommodation of the coherent strains at the interface due to a lattice misfit between the precipitate (t-ZrO 2 ) and the matrix (c-ZrO 2 ).…”

Section: F Comparison With Experimental Resultssupporting

confidence: 74%

“…Voorhees et al [10] and Thompson et al [11] give numerical predictions for the equilibrium morphologies of a precipitate in a system with dilatational and tetragonal misfit in an elastically anisotropic medium (cubic anisotropy). In this work, the authors discretize the interface coordinates in terms of the arc-length and use this to write the total energy of the system, that is a sum of the elastic and the interfacial energies.…”

Section: Introductionmentioning

confidence: 99%

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Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses

Bhadak

Sankarasubramanian

Choudhury

2018

Metall Mater Trans A

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Coherency misfit stresses and their related anisotropies are known to influence the equilibrium shapes of precipitates. Additionally, mechanical properties of the alloys are also dependent on the shapes of the precipitates. Therefore, in order to investigate the mechanical response of a material which undergoes precipitation during heat treatment, it is important to derive the range of precipitate shapes that evolve. In this regard, several studies have been conducted in the past using sharp-interface approaches where the influence of elastic energy anisotropy on the precipitate shapes has been investigated. In this paper, we propose a diffuse interface approach which allows us to minimize grid-anisotropy-related issues applicable to sharp-interface methods. In this context, we introduce a novel phase-field method where we minimize the functional consisting of the elastic and surface energy contributions while preserving the precipitate volume. Using this technique, we reproduce the shape bifurcation diagrams for the cases of pure dilatational misfit that have been studied previously using sharp-interface methods and then extend them to include interfacial energy anisotropy with different anisotropy strengths which has not been a part of previous sharp-interface models. While we restrict ourselves to cubic anisotropies in both elastic and interfacial energies in this study, the model is generic enough to handle any combination of anisotropies in both the bulk and interfacial terms. Further, we have examined the influence of asymmetry in dilatational misfit strains along with interfacial energy anisotropy on precipitate morphologies.

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Section: F Comparison With Experimental Resultssupporting

confidence: 74%

Section: Introductionmentioning

confidence: 99%

Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses

Bhadak

Sankarasubramanian

Choudhury

2018

Metall Mater Trans A

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“…We have verified the static numerical solutions from our boundary integral formulation of the elasticity problem against the analytic solutions in [4]. We have also compared our time-dependent and multi-precipitate computations for the case of homogeneous, cubic elastic media against the solutions found using the elastic solver introduced by Voorhees et al [43,41,42]. The different solution techniques agree up to numerical resolution.…”

Section: Convergence Testsmentioning

confidence: 84%

“…Most previous work on simulating microstructural evolution in elastic media has focused either on the case of homogeneous elasticity with cubic anisotropy, e.g., [43,41,42,27,45,44] or inhomogeneous, isotropic elasticity, e.g., [15,20]. In the former, the elastic constants of the two phases are anisotropic, with cubic symmetry, but are identical (elastically homogeneous) in the two phases.…”

Section: Introductionmentioning

confidence: 99%

“…In the former, the elastic constants of the two phases are anisotropic, with cubic symmetry, but are identical (elastically homogeneous) in the two phases. This case is much easier to treat than the inhomogeneous case because the elastic fields can be calculated by direct calculation of an integral and so it is not necessary to solve any equations to obtain the elastic fields [43,41,42]. In the latter case, the elastic constants of the two phases are different but the elasticity is assumed to be isotropic.…”

Section: Introductionmentioning

confidence: 99%

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Microstructural Evolution in Orthotropic Elastic Media

Leo

Lowengrub

Nie

2000

Journal of Computational Physics

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We consider the problem of microstructural evolution in binary alloys in two dimensions. The microstructure consists of arbitrarily shaped precipitates embedded in a matrix. Both the precipitates and the matrix are taken to be elastically anisotropic, with different elastic constants. The interfacial energy at the precipitatematrix interfaces is also taken to be anisotropic. This is an extension of the inhomogeneous isotrpic problem considered by H.-J. Jou et al. (1997, J. Comput. Phys. 131, 109). Evolution occurs via diffusion among the precipitates such that the total (elastic plus interfacial) energy decreases; this is accounted for by a modified Gibbs-Thomson boundary condition at the interfaces. The coupled diffusion and elasticity equations are reformulated using boundary integrals. An efficient preconditioner for the elasticity problem is developed based on a small scale analysis of the equations. The solution to the coupled elasticity-diffusion problem is implemented in parallel. Precipitate evolution is tracked by special non-stiff time stepping algorithms that guarantee agreement between physical and numerical equilibria. Results show that small elastic inhomogeneities in cubic systems can have a strong effect on precipitate evolution. For example, in systems where the elastic constants of the precipitates are smaller than those of the matrix, the particles move toward each other, where the rate of approach depends on the degree of inhomogeneity. Anisotropic surface energy can either enhance or reduce this effect, depending on the relative orientations of the anisotropies. Simulations of the evolution of multiple precipitates indicate that the elastic constants and surface energy control precipitate morphology and strongly influence nearest neighbor interactions. However, for the parameter ranges considered, the overall evolution of systems with large numbers of precipitates is primarily driven by the overall reduction in surface energy. Finally,

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Simulation of Microstructural Evolution Using the Field Method

Wang

Chen

2002

Characterization of Materials

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Morphological pattern formation and its spatio‐temporal evolution are among the most intriguing natural phenomena governed by nonlinear complex dynamics. These phenomena form the major study subjects of many fields including biology, hydrodynamics, chemistry, optics, materials science, ecology, geology, geomorphology, and cosmology, to name a few. In materials science, the morphological pattern is referred to as microstructure, which is characterized by the size, shape, and spatial arrangement of phases, grains, orientation variants, ferroelastic/ferroelectric/ferromagnetic domains, and/or other structural defects. These structural features usually have an intermediate mesoscopic length scale in the range of nanometers to microns. Microstructure plays a critical role in determining the physical properties and mechanical behavior of materials. Today the primary task of material design and manufacture is to optimize microstructures— by adjusting alloy composition and varying processing sequences—to obtain the desired properties. Microstructural evolution is a kinetic process that involves the appearance and disappearance of various transient and metastable phases and morphologies. The optimal properties of materials are almost always associated with one of these nonequilibrium states, which is “frozen” at the application temperatures. Theoretical characterization of the transient and metastable states represents a typical nonequilibrium nonlinear and nonlocal multiparticle problem. This problem generally resists any analytical solution except in a few extremely simple cases. With the advent of and easy access to high‐speed digital computers, computer modeling is playing an increasingly important role in the fundamental understanding of the mechanisms underlying microstructural evolution. In this article, methods recently developed for simulating the formation and dynamic evolution of complex microstructural patterns are reviewed. First, a brief account of the basic features of each method, including the conventional front‐tracking method (also referred to as the sharp‐interface method) and the new techniques without front‐tracking are given. Detailed description of the simulation technique and its practical application focuses on the field method since it is the most familiar and it is a more flexible method with broader applications. The need to link the mesoscopic microstructural simulation with atomistic calculation as well as with macroscopic process modeling and property calculation is also briefly discussed.

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The equilibrium shape of a misfitting precipitate

Cited by 263 publications

References 35 publications

Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses

Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses

Microstructural Evolution in Orthotropic Elastic Media

Simulation of Microstructural Evolution Using the Field Method

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