IV Transport Phenomena. Lecture 21: Solids and Concentrated Solutions

Transcription

1 IV Transport Phenomena Leture 21: Solids and Conentrated Solutions MIT Student (and MZB) Marh 28, Transport in Solids 1.1 Diffusion The general model of hemial reations an also be used for thermally ativated diffusion. Figure 1: Partile diffusion by thermally ativated transitions Here the exess hemial potential ats like the potential energy of partile state. Thermally ativated transition without drift or bias implies a random walk phenomena where the diffusivity is a funtion of mean-average time between steps and is given by: 1

2 Leture 21: Transport in solids and onentrated solutions Diffusivity x 2 D = (1) 2τ τ =mean time between transitions The mean average transition time is a funtion of the potential energy gap between the transition state and stable original state. ex ex k B T τ = τ 0 e ( µ TS µ ) γ TS = τ0 (2) γ 1 T = attempt frequeny for transitions, and reall, µ ex = k B T ln γ. τ 0 Finally, we an now write diffusivity of solids in terms of ativity oeffiients. x 2 γ D = 2τ0 γ TS (3) We go ahead and onsider a few speifi ases to further simplify the diffusivity expression Dilute limit Commonly aepted hypothesis here is that γ and γ TS do not depend on onentration,. ETS E min γts = e k B T and γ = e k B T x 2 = = E A D e k B 2 T (4) τ 0 where E A = E TS E min is the ativation energy barrier Ideal solid solution (Lattie gas) Model: Consider a lattie gas model where the transition state requires two vaan ies. Then we have, 2

3 Leture 21: Transport in solids and onentrated solutions γ E min e kb T γ = (1 / max) E TS e kb T TS = (1 / max) 2 Figure 2: Lattie gas transition state with two required vaanies D = D 0 (1 max ) (5) The fator (1 ) an be understood as the ondition probability max that the target site ( state 2 ) after the step is vaant, given that partile starts at a ertain position ( state 1 ) General ase γ D = f(, T,σ,..) =D 0 (6) γ TS Example: σ =stress tensor. Then E A = EA 0 + σ : ɛ A, where ɛ A is the ativation strain tensor, whih desribes the shape of the transition state. = stress-assisted diffusion in solids 3

4 Leture 21: Transport in solids and onentrated solutions 1.2 Drift We now look at diffusion when there is a gradient in hemial potential as a funtion of x. Figure 3: Partile drift due to gradient in hemial potential Figure 4: Partile rossing the transition state 4

5 Leture 21: Transport in solids and onentrated solutions 1 V = : ell volume max (7) A x : ell area (8) V x = A x (9) Flux, F = R, where R =reation rate for net drift in the x diretion A x = ( (µex (x) µ(x x ))/k T (µex (x) µ(x+ R R e TS B x 2 e TS ))/k 2 B T ) (10) 0 R 0 = 1 (sine the probability of making the transtion from barrier is 1 ) 2τ 0 2 Assume µ(x) is slowly varying at the moleular sale. x x µ(x) = µ(x ± ) µ(x) ± (11) 2 2 x x µ µ(x) =k B T ln a(x) =k B T ln(γ/ max ) and k B T x 1 x 1 [ 2 2 F (x) = e k B T k T e B 2τ 0 A x γ TS (x) µ µ( x ) µ(x+ x ) sinh( x µ(x) 2k = B T x ) e k B τ T 0 A x γ TS (x)v γ(x) x µ = sinh( ) τ 0 A x γ TS (x) 2k B T x x 2 γ(x) (x) µ(x) = ( )( ) 2τ 0 γ TS (x) k B T x D µ = ( ) (12) k B T x ] Here, µ x =generalized/thermodynami fore. From a fundamental postulate of nonequilibrium thermodynamis, we know: F = M µ (13) x where M is the mobility (veloity/fore=1/drag) 5

6 Leture 21: Transport in solids and onentrated solutions This implies the Einstein relation 1 Mk B T = D (14) The mobility of a traer is thus generally related to its diffusivity, even in a onentrated solution (or solid). Example: Solid solution/lattie gas µ = k B T ln( ) max µ [ 1 1 ] = = k B T + x max x =( k B T ) 1 x max D F (x) = (1 ) x max (15) We see that the thermodynami driving fore blows up as max due to strong effet of exluded volume. However, the traer diffusivity goes to zero in the same limit due to the lak of available vaanies for partile steps, D = D 0 (1 ). This leads to a remarkable anellation max of nonlinear effets, suh that the hemial diffusivity is preisely onstant for all onentrations and equal to the traer diffusivity of partiles or holes in isolation: F = D 0 (16) x Fik s law 1 Einstein originally derived this relation for the speial ase of Brownian motion of a dust partile in air by equating the mobility (drift veloity per gravitational fore) with the inverse of the Stokes drag oeffiient for rigid sphere in visous flow, M 1 =6πηR. This allowed him to predit the diffusivity D = k B T/6πηR in terms of the fluid visosity η and the partile size R in good agreement with experiments later done by Perrin (who earned the Nobel Prize as a result, due to the importane of this theoretial verifiation in establishing the moleular nature of matter). 6

7 Leture 21: Transport in solids and onentrated solutions More generally in a rystal, if we assume D = D 0 (1 max ) for a lattie gas, then for any model of µ(, x,...), µ F = M x D 0 max µ = ( )( )(1 ) (17) k B T max max x A onservation law based on this flux F + =0 t x yields the suitable form of the Cahn-Hilliard equation for a solid solution or lattie gas. (See leture 38) 2 Conentrated Solution Theory Reall, hemial potential in a onentrated solution is given by: µ = kt ln(γ) = kt ln γ +kt ln {{} } exess (18) Therefore, the flux F in a onentrated solution is: µ F = M x D µ = kb T x D [ kb T ln γ ] = kb T + k B T x (19) This is rewritten as, Here, D hem has ontributions from two effets. F = D hem () (20) x 7

8 Leture 21: Transport in solids and onentrated solutions [ ln γ ] Dhem = D }{{} 1 + (21) ln Fik s law }{{} onentrated solution effets where D = D 0 γ γts 8

9 MIT OpenCourseWare Eletrohemial Energy Systems Spring 2014 For information about iting these materials or our Terms of Use, visit: