EMA5001 Lecture 3 Steady State & Nonsteady State Diffusion - Fick s 2 nd Law & Solutions
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1 EMA5 Lecue 3 Seady Sae & Noseady Sae ffuso - Fck s d Law & Soluos
2 EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law Seady-Sae ffuso Seady Sae Seady Sae = Equlbum? No! Smlay: Sae fuco (e.g.,, ) does NOT chage wh me ffeece: Ne flu (o e eaco ae) Zeo () e flu fo equlbum sae vs. o-zeo e flu fo seady sae Fck s s Law If seady sae dffuso does NOT chage wh me does NOT chage wh me Fo -, f cosa, wha s he (=) coceao pofle ude seady sae? Slope = d ( ) d d ( ) ( ) Lea coceao pofle
3 EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law 3 Noseady-Sae ffuso Noseady Sae oceao chages wh boh Locao (, y, z) Tme () Take a small slce a locao δ : Thckess of he slce : Flu o he slce : Flu ou fom he slce I small me peod δ, he chage of coceao ha slce We have A A A +δ +δ Aea A δ
4 EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law Fck s d Law oued fom p. 3 As δ ad δ, we have Theefoe Ivokg Fck s s Law We have.e., If cosa, smply o 4 Fck s d Law
5 EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law Implcaos of Fck s d Law Two coceao pofles oes he coceao he specfed ego cease o decease wh me? 5 ceases wh me deceases wh me
6 EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law 6 Specal ase Homogezao ffuso o elmae o local coceao vaao Smples case: =, vaes susodally s l Assumpo: cosa Soluo o Fck s d Law akes he fom l s ep, whch s elaao me l Low β Hgh l Low Hgh = = τ The amplude of he vaao ep
7 Specal ase Sp-o opa fo Slco Wafe EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law 7 ffuso Sem-Ife a w/ Fed Amou of Toal opa opg slco suface wh boo o phosphoous sp-o dopas ad dffuse a 8- o Fck s d Law Assumpo: cosa Ial codo ( =, = ) = ; ( >, = ) = ouday codo Zeo coceao fa away fom suface ( ) = If he oal amou of dopa s fed of N, he < < 3 3 N 4, ep
8 EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law 8 Specal ase Ife ffuso ouple ffuso Ife ffuso ouple Two dlue alloys of A welded ogehe Fck s d Law Assumpo: cosa Ial codo ( >, = ) = ; ( <, = ) = ouday codo: (, > ) = ; ( -, > ) = Soluo s, ef () () ef() < < whch eo fuco, ef s gve by z ef ( z) ep( y ) dy -
9 Specal ase abuzao & ecabuzao of Seel EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law 9 ffuso Sem-Ife a w/ osa Suface oceao Icease/decease cabo coceao suface abuzao: H 4 /O amosphee a elevaed empeaue (fo F γ-fe) ecabuzao: vacuum a elevaed S empeaue abuzao of seel ouday/ial codos: < < 3 abuzao: ( = ) = S ; ( ) = 3 ecabuzao: ( = ) = ; ( ) = Soluos ae abuzao, S ( S ) ef 3 < < 3 ecabuzao ecabuzao of seel, ef
10 EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law ffuso Legh Eample: abuzao of seel Fo eo fuco, f ef (z) =.5, z.5 Theefoe, fo coceao pofle, S ( S ) ef Whe ef.5 Idcag Theefoe, z S,.5 S S ffuso Legh - chaacesc legh a maeal wh whch epeeces sgfca chage due o dffuso abuzao of seel If 3 Wha s he elaoshp bewee,, ad 3 assumg cosa?
11 EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law Mcoscopc Vew of ffuso Legh () Fo a esal aom Each adom ump wh dsplaceme of Afe umps, oal dsplaceme veco s To oba absolue dsplaceme legh afe umps, we have Ogal poso Poso afe umps 3... ) ( ) (
12 EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law Mcoscopc Vew of ffuso Legh () oue fom p. Successful umpg occus oly o he eaes eghbo, he fo =,,, whch s he umpg dsace fo a (esal) aom o s eaes eghbo. osde whch s he agle bewee ad The dsplaceme afe umps wll be Now cosde adom umpg of a lage amou of aoms: Each aoms umps fo mes, ad he aveage dsplaceme amog all aoms, os os,,, os
13 EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law 3 Mcoscopc Vew of ffuso Legh (3) oued fom p. Fo aveage ove a lage amou of aoms, we have os, Theefoe, If s he successful ump fequecy, ad f he umps ake me, ( ) Fom eale devao abou dffuso of esal aoms, ( ) 6 os Theefoe, The aveage (oo mea squae) dsplaceme afe me fo adom walk s,
14 EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law 4 Homewok Poe 3 d Ed, Eecse.,.3,.6 ue Feb 3 class
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