Lecture 3: Diffusion: Fick s first law


 Jonathan Goodwin
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1 Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th Fick s first law, and undrstand th maning of diffusion cofficint and diffusion lngth. In last two lcturs, w larnd th thrmodynamics, concrnd mainly with stabl or quilibrium systms. Th study of matrials Kintics, lik phas transformation, concrns thos mchanism by which a systm attmpts to rach th quilibrium stat and how long it taks. On of th most fundamntal procsss that control th rat at which many transformation occur is th diffusion, as w will larn in th following svral lcturs and us throughout th whol smstr. Gnral rviw of diffusion: Diffusion dscribs th sprad of particls (which can also b atoms, molculs) through random motion usually (but not always) from rgions of highr concntration to rgions of lowr concntration. Th tim dpndnc of th statistical distribution in spac is givn by th diffusion quation. Th concpt of diffusion is tid to that of mass transfr drivn by a concntration gradint, but diffusion can still occur whn thr is no concntration gradint (but thr will b no nt flux). History: Th concpt of diffusion mrgd from physical scincs. Th paradigmatic xampls wr hat diffusion, molcular diffusion and rownian motion. Thir mathmatical dscription was laboratd by Josph Fourir in 8, dolf Fick in 855 and by lbrt Einstin in 905. Spcifically, atomic diffusion is a diffusion procss whrby th random thrmallyactivatd movmnt of atoms in a solid rsults in th nt transport of atoms. For xampl, hlium atoms insid a balloon can diffus through th wall of th balloon and scap, rsulting in th balloon slowly dflating. Othr air molculs (.g. oxygn, nitrogn) hav lowr mobilitis and thus diffus mor slowly through th balloon wall. Thr is a concntration gradint in th balloon wall, bcaus th balloon was initially filld with hlium, and thus thr is plnty of hlium on th insid, but thr is rlativly littl hlium on th outsid (hlium is not a major componnt of air). Th rat of transport is govrnd by th diffusivity and th concntration gradint. Diffusion in solid: In th crystal solid stat, diffusion within th crystal lattic occurs by ithr intrstitial or substitutional mchanisms and is rfrrd to as lattic diffusion. In intrstitial lattic diffusion, a diffusant (such as C in an iron alloy), will diffus in btwn th lattic structur of anothr crystallin lmnt. In substitutional lattic diffusion (slfdiffusion for xampl), th
2 atom can only mov by substituting plac with anothr atom. Substitutional lattic diffusion is oftn contingnt upon th availability of point vacancis throughout th crystal lattic. Diffusing particls migrat from point vacancy to point vacancy by th rapid, ssntially random jumping about (jump diffusion). S blow in this Lctur for dtails about intrstitial or substitutional. Diffusion is drivn by dcras in Gibbs fr nrgy or chmical potntial diffrnc. s a simpl illustration of this, considr th figur blow, whr two blocks of th sam  solid solution, but with diffrnt compositions (concntrations), ar wldd togthr and hld at a tmpratur high nough for long rang diffusion to occur. Cas : along concntration gradint  rich  rich μ G G a G μ G m For cas : μ > μ > => gos from to => gos from to Th molar fr nrgy diagram of th  alloy is shown at right: th molar fr nrgy of ach part is givn by G and G, and th initial total fr nrgy of th wldd block is G a, upon th diffusion as illustratd on th lft (so as to liminat th concntration diffrnc), th total fr nrgy dcrass towards G m, th fr nrgy charactristic of a homognous systm. In this cas, th dcras in fr nrgy is producd by and diffusion along th concntration gradint. Howvr, diffusion nds not always b along th concntration gradint. For xampl, in th alloy systms that contain a miscibility gap, th fr nrgy curv can hav a ngativ curvatur at low tmpratur, as shown blow in Cas. In such a cas, and atoms would diffus against th concntration gradint (i.., toward th rgions of high concntration as shown at lft blow), though this diffusion is still a natural (spontanous) procss as it rducs th total fr nrgy from G a to G m as shown. For cas : μ > μ > => gos from to => gos from to
3 Cas : against concntration gradint  rich  rich μ G G a G μ G m Diffusion is always along chmical potntial gradint! Diffusion stops whn chmical potntial of all spcis vrywhr ar sam. In most cass, chmical potntial incrass with incrasing concntration, so it is convnint to xprss diffusion in trm of concntration. Now lts considr Intrstitial diffusion vs. Substitutional diffusion: Intrstitial Substitutional (by vacancy) Considrd at low concntration Now Lt s dduc th Diffusion Equation for intrstitial diffusion basd on th following illustration, moving from position to. 3
4 x x+a c (x) ~ concntration at x, dc c (x+a) = c(x) + a ~ concntration at x+a μ(x) ~ chmical potntial at x, ΔG ~ activation barrir d µ( x) μ(x+a) =μ(x) + a ~ chmical potntial at x+a Th forward rat  G G ; RT  s last Lctur for how to dduc this quation with an assumption that ΔG << RT, and this is usually tru for diffusion, whr th fr nrgy chang within short distanc is small. ΔG = G (x+a)  G (x) = μ(x+a)  μ(x) (hr ΔG is only contributd by th chang in chmical potntial ) Now G RT = µ ( x + a) µ ( x) a dµ = RT RT RT d µ( x) (as dfind abov, μ(x+a) =μ(x) + a ) Lt ν b vibrational frquncy, thn an atom will hav nough nrgy in unit tim (pr scond) to ovrcom th barrir ν can dfin as Γ.  G tims, i.., th numbr of jump of stp a pr scond, which w Γ = ν  G In a thrdimnsional spac (lattic), an atom can jump in six quivalnt dirctions (±x, ±y, ±z), ν so th numbr of jumps toward right (along x ) in on scond is. So, th nt forward rat for an atom to jump from x to x+a is 4
5 Rat = ν a dµ dµ ( ) (whr <0) RT Lt n(x) b th numbr of atoms (solut) pr unit ara at x, thn th nt flux along x. (i.., numbr of atom passing through unit ara in unit tim along x ) is ν J = n(x) a dµ ( ) () RT Sinc n(x) = a c(x), Eq. () can b writtn as J =  c ( x) dµ RT () Now lt s considr th xprssion of chmical potntial as w larnd in th cours of Thrmodynamics: μ(x) = μ 0 + RTlna(x) = μ 0 + RT ln γ cx ( ) whr a(x) is th activity of solut (atom) at x, γ is th activity cofficint, a constant. Thn, w hav dµ RT dc ( x) = c(x) (3) Substituting Eq. (3) into Eq. (), w hav J =  dc ( x) (4) Lt s dfin D = Thn w hav dc ( x) J = D as th intrinsic diffusion cofficint (Unit: D: cm /sc; J: numbr/cm /sc) This is th Fick s first law of diffusion. 5
6 Now, lt s considr diffusion in a thrdimnsional spac, i.., random walk of an atom: s dfind on pag 4, Γ = ν  G Combining with D = W hav D = Γ a = Thn w hav  G Γ= ν = D / a (Γ: numbr of jump an atom mak pr scond) For a random walk in thrdimnsions, an atom travls an avrag distanc of stps (i.., n tims of jump) with a stp lngth of a. n a aftr n Thus, aftr tim t, th atom jumps Γ t tims (stps). So it movs a distanc of r = a Dt Γ t = a a = D t.4 D t This is usually dfind as th diffusion lngth.
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