Modeling gas diffusion into metals with a moving-boundary phase transformation
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ATOMIC diffusion with a moving phase boundary is a classical problem, which has been treated analytically and numerically by many techniques.[1] In general, the problem involves the solution of Fick’s second law of diffusion, subject to a mass conservation condition at the moving phase boundary, which accounts for discontinuities in the diffusivity and diffusant concentration at the boundary (as shown schematically in Figure 1). Applications of the movingboundary diffusion problem to materials science are numerous: second-phase precipitation and dissolution,[2–7] diffusive alloying,[8–12] oxidation,[13] transient liquid-phase bonding,[14] and solidification problems[15] have all been described by considering appropriate moving-boundary diffusion equations. The mathematics of the moving-boundary diffusion problem are complicated, and analytical solutions exist only for simplified situations; these typically involve boundary motion in one dimension in an infinite medium.[1] Many numerical methods based on the finite-difference or finite-element techniques have thus been proposed to solve such problems. For analogous problems of heat conduction across a phase boundary, these methods have been extended to multiple dimensions and applied to practical engineering problems.[1,16] In solidstate diffusion problems, two main finite-difference techniques have been employed. The movable-grid method, as originally proposed by Murray and Landis,[17] fixes the boundary at a mesh point, which moves with the boundary. Away from the boundary, the mesh in each phase is contracted or expanded such that the grid is equally spaced on either side of the boundary. This method has been used extensively by Heckel and coworkers[5,12,18,19] to model metallurgical phenomena. In contrast to the movable-grid method, the more efficient fixed-grid method uses an unchanging, evenly spaced mesh, and allows the phase boundary to traverse between adjacent grid points.[20] Lagrange polynomials are used to approximate the concentration gradient in the vicinity of the boundary. In heat transfer, this method has also been extended
to two or three Cartesian coordinates.[16] Zhou and North[21] have used the fixed-grid approach to describe dissolution of plate-morphology phases; Lee and Oh[22] have considered issues of mass balance error and extended the model to multicomponent diffusion. The moving-boundary problem as regards gas diffusion into metals has, to date, primarily been applied to simple situations, such as growth of surface layers. When analytical solutions do not apply to these problems, simple one-dimensional numerical methods assuming an infinite medium have proven sufficient. However, there are several contemporary problems in which a specimen of finite size is fully transformed by introduction of a diffusing species at the surface. For example, hydrogen storage alloys are converted to a hydride phase by diffusion of hydrogen from the surface.[23,24] Efficient hydrogen storage requires a complete phase transformation on charging and complete
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