Ltur 4: Diffusion: Fik s sond law Today s topis Larn how to ddu th Fik s sond law, and undrstand th basi maning, in omparison to th first law. Larn how to apply th sond law in svral pratial ass, inluding homognization, intrdiffusion in arburization of stl, whr diffusion plays dominant rol. In last ltur, w larnd Fik s first Law: d ( ) J = -D (Unit: D: m /s; J: numbr/m /s) d Whr D = a ν 6 -ΔG /RT = 6 Γ a Fik s first law applis to stady stat systms, whr onntration kps onstant. But in many ass of diffusion, th onntration howvr hangs with tim, how to dsrib th diffusion kintis in ths ass --- dmanding Fik s sond Law. ontinud from last ltur, w will larn how to ddu th Fik s sond law, and undrstand th manings whn applid to som pratial ass. Lt s onsidr a as lik this d j() j(+d) +d d W an dfin th loal onntration and diffusion flu (through a unit ara, ) at position as: (,t), J() w hav d() = [ J ( ) J ( + d )]d t, J(+d) = J() + dj d
d(, t) thn, w hav = - dj dt d or, rwrit it in this format with rplaing d t (, ) = - J d ( ) ( ) from th first law: J = -D = -D d thn, w hav (, t) J = - = D This is th Fik s sond law. () In thr-dimnsional spa, it an b writtn as: = D. t stady (quilibrium) stat, w hav t (, )/dt = (maning no onntration hang) Thn, solving Eq. () givs d -D = J = onstant --- bak to th Fik s first law. d So, Fik s first law an b onsidrd as a spifi (simplifid) format of th sond law whn applid to a stady stat. Now, lt s onsidr two ral pratial ass, and s how to solv th Fik s sond law in ths spifi ass. as. Homognization: (non-uniform à uniform) onsidr a omposition profil as suprimposd sinusoidal variation as shown blow, whr th solid lin rprsnts th initial onntration profil (at t=), and th dashd lin rprsnts th profil aftr tim τ.
t= β _ l t = τ t t=, sin = + β l with th Fik s sond law, = D, whr D is th diffusion offiint, a onstant. t tim t, (,t) = + β sin l t τ Whr τ = l / D, τ is dfind as th rlaation tim. l Th largst Dviation of onntration at =, whr sin l =, th maimum. It is an ponntial day, th longr th wavlngth (l), th longr th rlaation tim (τ), thn th slowr day. Short wavlngth dis fast. That s why shaking always hlps spd up th disprsion, baus it nabls wid sprading (smallr l) of th stuff (lik partils) you try to disprs. as. Intrdiffusion (th arburization of stl):doping of stl with arbon Situation a): Doping with fid amount of dopant onsidr a thin layr of B dpositd onto, through annaling at high tmpratur, w will b abl to gt th onntration profil at diffrnt tims, from thr thn w an dtrmin th diffusion offiint, D 3
B Infinit intgration of th Fik s sond law, W hav (,t) = α t / 4 () = D s th B diffuss into, th total amount of B is fid (, t) d = N = onstant Thn, α t / 4 d = α D ( ) d( ) = N To solv th abov quation, lt s dfin y = thn w hav, α D y dy = N sin y dy = thn, α = N D so Eq. () an now b writtn as 4
(,t) = N / 4 (3) as dtrmind by this diffusion kintis quation, th onntration profil of arbon at various tims will b lik this t t 3 > t > t t t 3 Th abov diffusion is on-dirtion ( à + ). But if w tnds it to two-way, from - to + (lik a droplt dissolvd into a solution) with dopant at =, thn w hav t t t > t α = (,t) = N D N / 4, (-, ) Situation b): Doping with a fid surfa onntration (.g. arburization of stl) H 4 /O = F arbon onntration profil shown at diffrnt tims, arbonization thiknss is dfind as th diffusion dpth at ½( s + ), whih is = onsidr a ral ampl: arbon diffusion in austnit (γ phas of stl) at, D=4 - m s -, arbonization of. mm thik layr rquirs a tim of a. sonds, or 7 min. 5
Th solution of th Fik s sond law an b obtaind as follows, th surfa is in ontat with an infinit long rsrvoir of fid onntration of s. For <, hoos a oordinat systm u. diffus s u du = X Th fid amount of dopant pr ara is s du=n, whih diffus toward right. Thn using Eq. (3) abov, th slab du ontributs to th onntration at is d(, t) = du s u /4 So, all th slabs from u= - to totally ontribut (, t) = d (, t) = s u /4 du Dfining y = u, thn, (, t) = s / y dy, = s [ y dy - / y dy ] = s [ - / y dy ] = s [ - rf ( )] Whr rror funtion rf (z) = z y dy 6
onsidring boundary onditions: ( = ) = s, onstant, fid. ( = ) =, orrsponding to th original onntration of arbon isting in th phas, rmains onstant in th far bulk phas at =. (, t) = s ( s )rf ( ) th onntration profil shown abov follows this diffusion quation. Now lt s onsidr Intrdiffusion as shown blow, whih rprsnts mor gnral ass. > + t = t = Solving th Fik s sond law givs (,t) = ( + + ) ( ) rf ( ) Intrdiffusion is popular btwn two smi-infinit spimns of diffrnt ompositions,, whn thy ar joind togthr and annald, or mid in as of two solutions (liquids). Many ampls in prati fall into th as of intrdiffusion, inluding two smiondutor intrfa, mtal-smiondutor intrfa, t. 7