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Unformatted text preview: IV Transport Phenomena Lecture 21: Solids and Concentrated Solutions MIT Student (and MZB) March 28, 2011 1 Transport in Solids 1.1 Diffusion The general model of chemical reactions can also be used for thermally activated diffusion. Figure 1: Particle diffusion by thermally activated transitions Here the excess chemical potential acts like the potential energy of par ticle state. Thermally activated transition without drift or bias implies a random walk phenomena where the diffusivity is a function of mean-average time between steps and is given by: 1 Lecture 21: Transport in solids and concentrated solutions 10.626 (2011) Bazant- E ∴ TS E min γ e k T k T TS = B and γ = e- B Δ x 2 E = =- Δ A ⇒ D e k T 2 B (4) τ whereΔ E A = E TS- E min is the activation energy barrier. 1.1.2 Ideal solid solution (Lattice gas) Model: Consider a lattice gas model where the transition state requires two vacan cies. Then we have, Diffusivity Δ x 2 D = (1) 2 τ τ =mean time between transitions The mean average transition time is a function of the potential energy gap between the transition state and stable original state. ex ex k B T τ = τ e ( µ TS- µ ) = τ γ TS (2) γ 1 ∝ T = attempt frequency for transitions, and recall, µ ex = k B T ln γ . τ Finally, we can now write diffusivity of solids in terms of activity coeffi cients....
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