Lecue 17: Kineics of Phase Gowh in a Two-componen Sysem: descipion of diffusion flux acoss he α/ ineface Today s opics Majo asks of oday s Lecue: how o deive he diffusion flux of aoms. Once an incipien nucleus has eached is ciical size (*), he suface enegy ha esics he developmen of he new phase become insignifican and he kineics fo gowh ae becoming dominaed by he limiing kineic mechanism, i.e., he migaion o jumping of aoms fom α maix o paicle. If he phase gowh equies no long-ange diffusion of aoms, hen he ae of gowh is conolled by he ae of aomic ansfe acoss he gowing paicle ineface. This is usually he case of single-componen phase ansfomaion as we discussed in lecue 15. Howeve, fo he wo-componen phase ansfomaion (paiculaly in he case of dilue soluion of one phase dispesed in anohe), gowh of he mino phase usually equies long-ange diffusion. 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. Ineface Limied Gowh: In his case, gowh is limied by how fas aoms can ansfe acoss he α/ ineface and no he ae a which aoms can be anspoed o he gowing ineface. This is equivalen o gowh whee no long-ange diffusion is equied (like ha descibed in Lecue 15 fo he single-componen sysem). Diffusion Limied Gowh: In his case, he gowh ae is limied by he diffusiviy, i.e., how fas he necessay aoms ae ansfe fom he α maix o he gowing -paicles. In geneal, he ae of diffusion anspo falls off vey quickly wih empeaue. S A B B S α α S aom Two-componen sysem Concenaion of B aoms a beginning, single-componen sysem (Lecue 15) C 0, X A >> X B 1
The following kineics eamen applies only o he dilue-soluion of α 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 α/ ineface: J = M ( C ) (1) Whee M = M ' RT defined as an ineface paamee, a measue of he anspo kineics of aoms acoss he α/ ineface, C has he uni of #/cm 3, M has he uni of cm/sec. Deiving he diffusion flux via Fick s law: Hee again he plo descibing he concenaion of B as a funcion of he adial coodinae fom he cene of he paicle of adius : C phase C C 0 α maix fla ineface adial coodinae Fick s fis Law: J = -D dc d The peinen diffusion equaion is Fick s second law c = D C (Assume D is consan) Assuming a quasi-seady sae in he α phase, c =0 C =0 In hee dimensions, in Caesian coodinaes, we have = + + x y z
In spheical pola coodinae: x = sinφcosθ z y = sinφsinθ z = cosφ φ Fo a spheically symmeic case θ y No dependence on φ and θ x = d d + d d Thus, d c d C =0 + dc d =0 Then we have: C( ) = a + b Whee a, b ae consans. Now conside wo limiing condiions: 1. As, C( ) = C 0, hen we have b = C 0 a. As, C( ) = C, hen we have C = +C 0, hen we have a = (C -C 0 ) ( C C0) So, C( ) = C 0 + = C 0 - ( C0 C ) ; This assumes ha concenaion in he maix fa away fom he gowing paicle is C 0. In geneal, howeve, hee will be ohe paicles, all compeing fo B aoms a same ime. The ne effec is ha he aveage concenaion in he bulk is lowe han C 0 as descibed in he diagam below. Also, his aveage concenaion in he bulk is ime dependen, now maked as C. paicle C 0 paicle C C C 3
So he above equaion can be e-wien as C( ) = C - ( C C) Then he concenaion gadien in α nex o paicle is dc C C ( ) = d = Now, wih he Fick s fis Law, we have dc J = J = D( ) d = = DC ( C) ( ) Hee we use J jus in ode o disinc he flux fom he ohe wo as deduced in Eq. (1) and (3) Since his flux also descibes he diffusion of B aoms acoss he α/ ineface, i mus be equal o he flux as descibed above in Eq. (1) J = M ( C ) (1) The hid way o deive he diffusion flux: As B aoms coss he α/ ineface, he adius of inceases. In ime ineval d, he adius inceases by d, he volume of inceases by d. The composiion in his egion changes fom C o, and he # of B aoms aived in ime d in he volume elemen is ( C ) d d, as >>C The aea hough which B aoms aived is uni ime, i.e., he flux, is J = C d d C d = (3) d, hus, # of B aom cossing he α/ ineface pe uni aea pe Hee we use J jus in ode o disinc he flux fom he ohe wo as deduced in Eq. (1) and () In a quasi-seady sae, all hee fluxes J, J, J as deduced above in Eqs. (1)()(3) ae equal, J = J = J o d DC ( C) C = = MC ( Cα) d 4
DC ( C) Fis, fom = MC ( Cα ), we have DC + MCα C = D + M Le s examine wo limiing cases: 1. when M >> D: Then C This is he diffusion limied case, whee he consumpion of B aoms aound he paicle is so apid ha he local concenaion of B eaches he equilibium concenaion of B in α phase,. C In his case, hee is vey small buildup of B aoms nea he paicles. C. when D >> M: Then C C This is he ineface limied case, whee he consumpion of B aoms aound he paicle is so slow compaed o he long-ange diffusion flux fom he bulk α phase ha he local concenaion of B emains appoximaely he same as he bulk concenaion of B in α phase, C. C C In his case, diffusion is fas and gowh is ineface-conolled. Thee is a lage buildup of B nex o. We & Zene examined diffusion limied gowh, Tubull examined ineface limied gowh (Lecue 15). Nex Lecue, we will addess he geneal case ha consides boh he wo kineics pocesses. 5