lectures18-20 - Self-Diffusion Through Vacancy Mechanism...

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Unformatted text preview: Self-Diffusion Through Vacancy Mechanism Self-diffusion: diffusion of atoms in a pure material (e.g., Al atoms in pure Al) or binary and multicomponent compounds in which the diffusion of different atoms is limited to their own sublattices (e.g. Mg and O diffusion in MgO) Recall for a cubic lattice 2 a D s Γ = γ where γ is the geometrical factor ( γ =1 for FCC and BCC, and 1/8 for diamond structure), Γ s is the jump frequency of an atom to a particular nearest neighbor site, and a is lattice parameter If the vacancy concentration is x v w x v s = Γ w – the exchange rate or frequency between an atom and a vacancy ( ) ( ) ( ) kT H k S kT G x v v v v ∆ − ∆ = ∆ − = exp exp exp ( ) ( ) ( ) kT H k S kT G w m m m ∆ − ∆ = ∆ − = exp exp exp ν ν ν – vibration frequency (~ the Debye frequency) ∆ S m , ∆ H m – vacancy migration entropy and enthalpy Temperature dependence of self-diffusion coefficient (i) Self-diffusion in elemental solids ∆ S v , ∆ H v – vacancy formation entropy and enthalpy ∆ G m ∆ G coordinates Therefore, ( ) kT H D kT H H k S S a D o m v m v ∆ − = ∆ + ∆ − ∆ + ∆ = exp exp exp 2 ν γ ∆ + ∆ = k S S a D m v o exp 2 ν γ m v H H H ∆ + ∆ = ∆ Activation enthalpy Typical values for various parameters in metals: ( ) 10 ~ 4 v − m T x k S S 2 1 ~ m v − ∆ + ∆ V 3 1 ~ v e H − ∆ Typical values in semiconductors: k S 10 ~ v ∆ V 5 3 ~ v e H − ∆ V 1 ~ m e H ∆ 1 13 10 ~ − s v s cm T D m 2 8 10 ~ ) ( − 1 13 10 ~ − s v s cm T D m 2 12 10 ~ ) ( − (i) Self-diffusion in ionic compounds There are at least two self-diffusion coefficients in ionic compounds, one for each sublattice. Let’s use the NaCl as an example. The self-diffusion coefficients of Na + and Cl- are given by or ( ) ( ) ( ) ( ) kT H k S kT H k S a wx a D s s m m 2 exp 2 exp exp exp Na , Na , 2 V 2 Na Na ∆ − ∆ ∆ − ∆ = = + + + + ν 2 Na , Na s m H H Q ∆ + ∆ = + + 2 Cl , Cl s m H H Q ∆ + ∆ = − − ∆ + ∆ = + + k S S a D s 2 exp Na m, 2 Na o, ν ∆ + ∆ = − − k S S a D s 2 exp Cl m, 2 Cl o, ν ( ) kT Q D D + + + − = Na Na o, Na exp ( ) ( ) ( ) ( ) kT H k S kT H k S a wx a D s s m 2 exp 2 exp exp exp Cl , Cl , m 2...
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This note was uploaded on 01/19/2010 for the course MAT SCI 503 taught by Professor Chen during the Spring '02 term at Pennsylvania State University, University Park.

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lectures18-20 - Self-Diffusion Through Vacancy Mechanism...

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