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Unformatted text preview: VIII. Phase Transformations Lecture 39: Reaction-limited Phase Separation MIT Student Last time we presented the classical Cahn-Hilliard theory for phase trans formations in closed systems characterized by a conserved order parameter (concentration). In this lecture we adapt the model to electrochemical sys tems by including Faradaic surface reactions. The resulting model describes evolution of a conserved parameter in an open system that is in contact with an infinite reservoir at fixed chemical potential. This model is a gener alization of the Allen-Cahn equation which describes the evolution of non- conserved order parameters during phase transformation. 1 Phase transformation during intercalation and adsorption The Cahn-Hilliard model, which was derived in the previous lecture, is: c = ( t Mc ) (1) = ( c )- K c where is a diffusional chemical potential for an inhomogeneous system. Two boundary conditions are imposed. The first is a variational boundary condition (see 2009 notes for derivation): n K c + % s ( c ) = (2) s is surface tension and may v ( ary with concen ) tration and orientation. Phys ically, this boundary condition avoids discontinuity in bulk chemical poten tial at the boundary by prohibiting gradients (bulk phase interface). True surface chemical potential should come from a separate surface contribution to free energy, which we have neglected so far. 1 Lecture 39: Reaction-limited phase separation 10.626 (2011) Bazant The second boundary condition equates the ux across the boundary to a reaction rate: R = n F % =- n Mc (3) Butler-Volmer is a logical choice for reaction rate kinetics: R = R e (1- ) e/kT- e- e/kT (4a) Recall that the variables...
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