Mathematical Research in Materials Science Opportunities and Perspectives (1993) / Chapter Skim
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4 EVOLUTION OF MICROSTRUCTURES
4 EVOLUTION OF MICROSTRUCTURES
Pages 34-51
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From page 34... ...
Spinodal decomposition happens without an energy barrier for nucleation (second-order phase transition often modeled by the Cahn-Hilliard equation; Cahn and Hilliard, 1958~; otherwise, there is a barrier to nucleation of regions of the new phases determined by competition between an increase in surface energy and a reduction in bulk energy. Nucleation is often studied using statistical mechanics on lattice models.
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From page 35... ...
Examples include the growth of a new phase into an old one, domain growth in spinodal decomposition, or in an ordering system, grain growth, and solid-state or liquidphase sintering. Driving forces for the motion are bulk or surface energy reduction, and the response is governed by `diffusion or interface control.
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From page 36... ...
term to the evolution equation, viscosity solutions of a Hamilton-Jacobi partial differential equation, phase-field methods for an order parameter, and order-disorder transformation via Monte Carlo simulations with a Q-state Potts model on a lattice. There are both theoretical and computational versions of most of these methods, although many computational methods are fully developed only for one-dimensional interfaces in 2-space.
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From page 37... ...
This question has been studied in the materials science literature for some time, but the confusion between motion by mean curvature and the type of motion that soap films undergo (involving diffusion of gas and instantaneous rearrangement to a locally area-minimizing configuration) seems to have clouded the issue.
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From page 38... ...
A NIST metallurgist has said of it, "l found new materials science problems to solve after seeing the ideal too! to tackle them with." SHAPE EVOLUTION CONTROLLED BY SURFACE DIFFUSION Surface shape evolution due to surface diffusion leads to an equation in which the normal velocity is proportional to the surface Laplacian of the mean curvature (in the 38
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From page 39... ...
Examples of important practical situations modeled in part by the isotropic surface evolution equation described above are sintering, grain growth in thin films, the breakup of thin films into an island structure, and texture development in thin-film growth from vapor. These situations arise in such materials science processes as molecular beam or liquid-phase epitaxy, and deposition by ion beam or sputtering or chemical vapor.
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From page 40... ...
These have been studied using methods based on boundary integrals that range from large-scale numerical simulation to qualitative approaches based on renormalization group ideas developed in studies of critical phenomena (see Chapter 8~. The mathematical challenges are to overcome computational limitations, extract asymptotic behavior from the relevant integral equations, and take into account that one envisions a statistical process (such as an ensemble of initial conditions)
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From page 41... ...
is a laudable goal, practical difficulties make computational simulation of crystal furnaces unattainable with current resources. Convection occurs in the melt and the description of crystal growth requires the solution of the Navier-Stokes equations with both heat and solute transport.
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From page 42... ...
Citing this example does not imply advocating the theory (or the "asymptotics beyond all orders" concept, which does not contribute to the real case) ; it is offered only to point out that this issue is likely to be resolved by the combined efforts of materials scientists and mathematical scientists.
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From page 43... ...
Once the correct physics had been identified, it was necessary to solve the resulting mathematical models. This turned out to be a subtle issue in singular perturbation theory, which involved asymptotics beyond all ~ ~ ~ T ~ ~ "d ~ ~ _ ~ ~ · · .
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From page 44... ...
The stresses that are presumed to give rise to these phenomena are caused by compositional inhomogeneity, external loads, or particle misfit strains. The elastic stresses manifest themselves in the diffusional and mechanical field equations and in the boundary conditions for diffusion.
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From page 45... ...
The evolution situation is often formulated as a set of nonlinear partial differential equations with boundary conditions on moving boundaries; see Starchy and Cahn (1992) for a simple example.
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From page 46... ...
In the geometrically nonlinear theory, the free energy depends on the nonlinear Cauchy-Green elastic strain, since it is invariant under all rigid body rotations. Of the "crystallographic theory of martensite," which is nonlinear but strictly phenomenological, and the earlier work of Khachaturyan and Roitburd, which uses energy minimization but is geometrically linear.
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From page 47... ...
. Future challenges for the mathematical sciences are to create more efficient algorithms that will make possible the computation of more complex microstructures, and to extend error analysis to multidimensional contexts where there is not a unique solution.
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From page 48... ...
, but these methods fall short of comprehensively treating defective crystals. Magnetostriction is the phenomenon whereby magnetization produces deformation and, conversely, deformation produces magnetization.
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From page 49... ...
The macroscopic properties of the new superconducting materials are more complicated than those of metallic superconductors due to anisotropy, layered character, short coherence lengths, and high values of the upper critical magnetic field. As a consequence, the macroscopic equations for the superconducting order parameter, namely, the Ginzburg-Landau equations, are different from and more complex than those used to describe metallic materials.
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From page 50... ...
The study of the interaction of Abrikosov vortices with structural defects of various types such as point defects, dislocations, grain boundaries (see Chapter 5) , and twin planes, all of which can serve as pinning sites, may also be an opportunity for mathematical scientists.
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From page 51... ...
There has been a recent surge of activity in macroscopic modeling for superconductivity. The mathematical methods involved include the use of asymptotic analysis, free-boundary formulations, variational principles, and finite-element approximations.
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