Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation
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1 Lecue 8: Kineics of Phase Gowh in a Two-componen Sysem: geneal kineics analysis based on he dilue-soluion appoximaion Today s opics: In he las Lecues, we leaned hee diffeen ways o descibe he diffusion flux of B aoms acoss he / ineface aound he paicle, and hese hee fluxes should be equal each ohe. Fo he wo-componen phase ansfomaion (paiculaly in he case of dilue soluion of phase dispesed in phase, gowh of he phase (paicle usually equies long-ange diffusion of B aoms owads o he paicle. In his case, he gowh ae can be deemined by wo diffeen ae-limiing pocesses: Ineface Limied Gowh and Diffusion Limied Gowh. Boh of hese wo pocesses ae empeaue dependen --- ypically he gowh ae is Ahenius ype wih gowh becoming vey slow a low empeaues. When M >> D, hen --- The gowh falls ino he diffusion limied case, whee hee is vey small buildup of B aoms nea he paicles. When D >> M, hen --- The gowh falls ino he ineface limied case, whee hee is lage buildup of B aoms nea he paicles. Howeve, in a moe geneal case, M ~ D, he phase gowh is deemined by boh he long-ange diffusion of B aoms fom he maix owads o he paicle and he diffusion acoss he / ineface. Today s opic is o lean how o descibe he kineics of such a geneal phase gowh. The following kineics eamen applies only o he dilue-soluion of phase conaining small mola facion of phase, i.e., mola facion of B (X B << mola facion of A (X A. In las Lecue, we deived he diffusion flux of B aoms acoss he / ineface in equaions: J = M ( ( Whee M = M ' RT defined as an ineface paamee, a measue of he anspo kineics of aoms acoss he / ineface, has he uni of #/cm, M has he uni of cm/sec. dc J = D( ρ d ρ = = D ( ( J = d d d = ( d
2 In a quasi-seady sae, all hee fluxes J, J, J as deduced above in Eqs. ((( ae equal, J = J = J o d D ( = = M ( d D ( Fis, fom = M (, we have D + M = (i D + M Fom his equaion we have wo limiing cases: When M >> D, hen --- The gowh falls ino he diffusion limied case, whee hee is vey small buildup of B aoms nea he paicles. When D >> M, hen --- The gowh falls ino he ineface limied case, whee hee is lage buildup of B aoms nea he paicles. Now le s deal wih he geneal case, whee boh he long-ange diffusion of B aoms fom he maix owads o he paicle and he diffusion acoss he / ineface will be consideed. A =, befoe he phase ansfomaion begins, he maix concenaion of B aoms is ; When he ansfomaion is complee, he maix concenaion of B aoms will be. As assumed a he vey beginning, he oiginal soluion is dilue, o he volume facion of is much less han.. Now, we define he facion ansfomed, x(, as V ( x( = V ( =, V (= <<. whee V is he uni volume of phase. Now, V ( ( = ( - V (( - (ii --- inceased # of B aoms wihin he phase (paicles equals o he deceased # of B aoms wihin he phase (now wih a volume of - V ( Since >>, and V ( <<. (he dilue soluion assumpion We have V ( ( => V ( =
3 V (= = Thus, x( = (iii Now, assuming hee ae n paicles (of adius of pe uni volume, hen, V ( = 4 π n Then, Eq. (ii n ( = ( ( V ( Again, Since >>, and V ( <<. (he dilue soluion assumpion, we have n (iv Diffeeniaion of Eq. (iv wih espec o leads o d d n = (v d d Also, Diffeeniaion of Eq. (iii wih espec o leads o ( d = d d o d ( = ( (vi d d ombining Eq. (v and (vi gives, 4 d d π n ( = ( (vii d Also, we have J = J = J o d D ( = = M ( d
4 So, we can e-wie Eq. (vii as 4 π n M ( = ( d Submiing wih Eq. (i, we have n M D + M { } = D + M d O n M D ( = D + M d (viii Fom Eq. (iv, we have = ( n hen wih Eq. (iii, we have ( ( = = x ( n n o / / / = ( ( x (ix n Also wih Eq. (iii, we have, = = x (x Now, wih Eq. (x, we can e-wie Eq. (viii as d ( ( D + M = 4 π nmd ( ( D + M nmd x = { + } 4 π n M ( x D( x = Subsiuing wih Eq. (ix, we have 4
5 d = + 4 / / / π n / / / M ( ( x ( x D ( ( x ( x n n Now le s se new paamees K 6 π n ( / = [ ] and K 48 π n ( / = [ ] Then we have d = + MK x ( x DK x ( x / / (xi Fom his equaion, i is no possible o expess x as an explici funcion of. Rahe, we can show ha is an explici funcion of x. Tha is, we can deemine he ime equied fo he ansfomaion o pogess o a given exen, in em of facion ansfomed, x(, as defined a he vey beginning above. Se y = x, hen Eq. (xi can be e-wien as dy ydy d = + MK ( y DK ( y The can be expessed as y y y = ( + ln[ ] + ( an ( + A MK DK ( y MK DK Whee A is a consan Submiing back wih y= x /, we have + x + x x + = ( + ln[ ] + ( an [ ] + A MK DK ( x MK DK / / / / Now, consideing he fac: when =, x=, hen we can deduce he value of he consan A A = = ( an MK DK ( ( 6 MK DK Submiing back A ino he equaion, we have π + x + x x + π = ( + ln[ ] + ( [an ( ] MK DK ( x MK DK 6 / / / / (xii 5
6 Eq. (xii can be e-wien as / / + x + x ln[ ] / ( x = ( + + ( / / x + π MK DK x + π MK DK an ( an ( 6 6 Then, a plo of vs. / x + π an ( 6 / / + x + x ln[ ] / ( x gives a saigh line / x + π an ( 6 wih slope ( + and inecep MK DK ( MK DK Now conside siuaions: If MK << DK, ineface ansfe much slowe han diffusion: slop is MK, inecep MK If MK >> DK, diffusion much slowe han ineface ansfe: slope is DK, inecep DK So, fom eal expeimens: A negaive inecep ( DK A posiive inecep ( MK inecep indicaes diffusion limied gowh, and aio = = ; slope inecep indicaes ineface limied gowh, and aio = = slope 6
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