Simulations of the Kirkendall-Effect-Induced Deformation of Thermodynamically Ideal Binary Diffusion Couples with Genera
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AL diffusion was once considered a process of direct atomic exchange until E.O. Kirkendall discovered the motion of inert markers during diffusion.[1–3] The marker motion provided solid evidence that substitutional diffusion occurs instead via vacancymediated mechanisms. In the presence of concentration gradients among the components of an alloy, vacancies preferentially exchange with the fast diffuser that has a higher successful hop frequency. As a result, a net vacancy flux from regions rich in slow diffusers to those rich in fast diffusers arises to compensate for the imbalance of atomic fluxes. During this process, lattice sites are reconstructed to supply or accommodate excess or depleted vacancies. The generation and annihilation of lattice sites lead to a lattice plane shift and consequently motion of inert markers, which is observed as the Kirkendall effect. In addition to the marker motion, the Kirkendall effect also results in various types of deformation. Note the terminology Kirkendall effect in this article is used broadly to encompass phenomena induced by unbalanced interdiffusion, not strictly limited to the marker motion as it was originally defined. Such a broadened usage is prevalent in the broader HUI-CHIA YU, Assistant Research Scientist, and A. VAN DER VEN and K. THORNTON, Associate Professors, are with the Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI 48109. Contact e-mail: [email protected]. Manuscript submitted January 16, 2011. Article published online July 21, 2012 METALLURGICAL AND MATERIALS TRANSACTIONS A
materials science community today; for example, the void formation resulting from unbalanced interdiffusion has been referred to as the ‘‘Kirkendall void’’ or ‘‘Kirkendall porosity.’’ Shortly after Kirkendall’s discovery, Darken’s analysis,[4] which linked the marker motion to interdiffusion, became widely accepted. This model implicitly assumes that a crystalline solid contains a sufficient amount of vacancy sources and sinks such that vacancies can be generated or eliminated everywhere in a crystal to maintain vacancy concentration at the equilibrium value. It was generally agreed that the marker motion observed experimentally was consistent with a one-dimensional incompressible plastic flow stemming from unequal atomic diffusivities.[5–7] However, because diffusion processes are often not fully one dimensional, the plastic deformation is not necessarily confined to one dimension. Instead, deformations including bulging and grooving near the diffusion interface,[8–12] bending of a plate diffusion couple[13–15] and void formation and growth [16–22] are commonly observed in experiments. To model the structural deformation during interdiffusion, models that account for vacancy generation and elimination and that also incorporate plastic deformation due to local volumetric changes[5–7] were proposed. In these models, plastic deformation is described by a Newtonian flow. Moreover, general models have been developed to include diffusion and drift (convection) by Danielewski et
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