Diffusion in a regular solid solution
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Diffusion in a regular solid solution was investigated. A thin plate of isotropic solid of constant surface molal fraction was considered. The regular solution is described by the parameter a. When a is positive, the attraction between unlike atoms is greater than the attraction between like atoms, and conversely. The depths of penetration and average molal fraction for a given period increase with increasing a. The separation of two curves for given a increase with increasing surface molal fraction. The diffusion coefficient is positive only if a is greater than —\/X(l — X). Diffusion is an important process in the manufacturing industry. It alters the structure of materials and their physical and mechanical properties. The diffusion coefficient is a major parameter that affects mass transfer. It is influenced by temperature, stress, concentration, etc. The diffusion coefficient increases with temperature according to the Arrhenius equation.1 Diffusion-induced stresses depend upon the geometry of the specimen and boundary conditions.2"10 The diffusion coefficient enhanced by chemical stresses was found to be proportional to concentration.'' Diffusion dependent on concentration for various cases was investigated by Crank.12 Solid solutions classified as ideal (Raoultian) are described according to an activity coefficient y with y = 1 or y ¥= 1, respectively.13 Margules14 proposed to express the activity coefficient as a power series with argument molal fraction. One special case of nonideal solution is the regular solution for which ln(y) is proportional to the quadratic term of molal fraction. Hildebrand15 found that the regular solution has a nonzero heat of formation and an ideal entropy of formation. Such a regular solution was observed in the system thallium and tin.16 We investigated the effect of chemical stresses on diffusion11 in which the diffusion coefficient is linearly proportional to the hydrostatic stress. It prompted us to examine the diffusion in a regular solid solution for which the diffusion coefficient is a function of activity coefficient. The problem is considered a thin plate of isotropic material in the region — a =£ x =£ a as shown in the inset of Fig. l(a). The molal fraction Xs (or concentration) at the boundary surfaces x = ±a is maintained constant. The driving force of mass transfer is the gradient of chemical potential. The chemical potential in a regular solution with the stress-free condition is written13
yu,(0' is constant, y is the activity coefficient, and X is the mobile molal fraction, a is a parameter to describe the solution, when a is zero, the result becomes the ideal solution. When a exceeds zero, the chemical potential is less than that of the ideal solution, and then the attraction between unlike atoms is greater than the attraction between like atoms. When a is less than zero, the chemical potential and attractions between atoms have the opposite trend to those for a > 0. According to Einstein17 and Darken,18 the flux at the point x is proportional to the gradient of
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