7-Vacancy+Chemical+Potential

7-Vacancy+Chemical+Potential - Vacancy Chemical Potential...

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98 Vacancy Chemical Potential If a crystal is chemically inhomogeneous and/or internally strained, then one must use chemical potential gradients instead of vacancy concentration gradients to describe vacancy flow. Fick’s first law for one-dimensional diffusion would be written as J v = ± D v c v kT d μ v dx where v is the vacancy chemical potential, instead of J v = ± D v dc v dx . We need to define the vacancy chemical potential for a chemically inhomogeneous and internally strained system. We start with the basic definition ± v = ²± G ² n v which is the free energy change associated with inserting vacancies at one particular point in the solid.
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99 When n v = Nx v vacancies are introduced, the free energy changes by ± G = Nx v ± g v + NkT x v ln x v + 1 ² x v () ln 1 ² x v () {} so the chemical potential is ± μ v = ²± G ² n v = 1 N ²± G ² x v = ± g v + kT ln x v 1 ³ x v ´ µ · ¸ ¹ . But we know that at equilibrium ±² G ± x v = 0 so 0 = ± g v + kT ln x v eq 1 ² x v eq ³ ´ µ · ¸ and then x v eq 1 ± x v eq = exp ± ² g v kT ³ ´ µ · ¸ x v eq ¹ exp ± ² g v kT ³ ´
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This note was uploaded on 12/01/2011 for the course MS&E 206 taught by Professor Nix during the Spring '08 term at Stanford.

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7-Vacancy+Chemical+Potential - Vacancy Chemical Potential...

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