Module Fick s laws of diffusion
Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms ec. (m) posiion parameer dc/d concenraion gradien Fick s firs law saes ha he flu is proporional o he negaive concenraion gradien., diffusion coefficien is he proporionaliy consan. Minus sign comes from he fac ha maer flows down he concenraion gradien. I is no necessarily rue in all he cases. Maer may also diffuse up he concenraion gradien, which is called uphill diffusion. As can be seen in he ne module ha no he concenraion bu he chemical poenial gradien is imporan o know he direcion of flu.
Fick s firs law can direcly be applied in he case of seady sae, as shown in he eample below. Seady sae means ha here will no be any change in he composiion profile wih ime. If we pass carburizing gas hrough he pipe as shown in he figure and decarburizing gas ouside he pipe, a seady composiion profile may develop. oncenraion gradien can be calculaed following: dc d i d o o d i From his, one can calculae he flu from he known diffusion coefficiens of carbon or he diffusion coefficien from he flu deermined.
However, jus he Fick s firs law may no be useful o relae he flu and he concenraion profile, since, in mos of he cases, sysems develop non seady sae concenraion profile. I means ha concenraion a a paricular posiion changes wih ime. For eample, if we keep concenraion of carbon on one of he Fe surfaces anneal, composiion profile will change wih ime, as shown in he figure. We can apply Fick s firs law direcly o evaluae he concenraion profile ha develops during diffusion in his case, since, as shown in he figure, composiion changes wih annealing ime a a paricular posiion. So we need o have a relaion, which can relae ime along wih he concenraion and he posiion. For ha Fick s second law is derived. I is derived using conservaion of mass and Fick s firs law.
Le us consider a very hin slab in he maerial. J 1 is he incoming flu, J is he ougoing flu So he amoun of carbon coming in shor ime δ is J 1 δ (mole/m ) and going ou is J δ (here J 1 >J ) If he slab hickness is, hen δ ( J1 J ) δ Eq. 1 Furher, flu change in he hin slab can be considered linear and we can wrie J J J1 J1 J Eq. From Eq. 1 and Eq. J Using Fick s firs law ( J ):
Soluion o he Fick s second law Soluion of he Fick s second law depends on iniial and boundary condiions. Furher,, in some cases, may be considered consan. Tha means does no change wih concenraion or posiion. However, his condiion mees only in very limied cases. In mos of he pracical eamples, is a funcion of concenraion. In his case, soluion o he Fick s second law is complicaed. So in he beginning, we shall solve he Fick s second law for consan. Soluions are mainly for wo differen ypes of condiions, small and large ime values. When diffusion annealing ime is small, soluion is epressed wih inegrals or error funcions. For long annealing ime, i is epressed in erms of rigonomerical series. Noe ha he long or shor annealing ime is raher relaive. y saying long annealing ime, we mean ha he complee sample is affeced by he diffusion process and may lead o homogenizaion. y saying shor annealing ime, we mean ha eperimens are conduced such ha whole maerial is no affeced by he diffusion process.
Soluion for a hin film source Le us consider ha is consan. Tha means does no change wih he composiion. Since a a paricular locaion changes coninuously wih annealing ime, or a paricular changes is locaion coninuously. From he assumpion, we can sae ha is he same a any locaion. The meaning of he above saemen will be more clear, when we shall consider he change of wih he change in concenraion. Le us consider he siuaion, when a very hin film of elemen is sandwiched beween maerial A and hen annealed a elevaed emperaure.
Noe: One migh ask, how hin i is? y saying hin we mean ha he amoun of maerial is very low and even afer oal miing (full homogenizaion) elemen can be considered as impuriies in he maerial A. Tha means, afer deposiion of on A, he chemical poenial gradien is negligible. In oher sense, we can consider his as an eample of diffusion in he absence of any driving force (alhough small driving force will be here from enropy of miing) Acual meaning will be clear, when we discuss abou he aomic mechanism of diffusion! We can consider ha is consan, since he maerial in which i diffuses has fied composiion. For consan, he Fick s nd law can be wrien as As shown in he previous slide, i is seen ha he elemen disribuion afer he diffusion can be epressed by eponenial relaion. Following he Fick s second law he disribuion of elemen in A can be epressed as ( ) 0 ep 1/ where 4 o is a consan This relaion is developed based on he fac ha composiion profile shows eponenial decay from he hin film source.
The correcness of he soluion can be checked by differeniaion of he equaion wih respec o and and hen using hem in he Fick s second law o find he equal values on boh he sides. Furher, he boundary condiion ha 0, a a 0 and, a 0 a 0 also mee. Now one migh ge confused, when we say, since concenraion of elemen (X /V m 1/V m ) can be infinie anyime and will have some definie value even a compleely pure sae. Here is noional and means ha he elemen is no dilued by any oher elemen, which furher means ha i is in pure sae and for sysem i means ha i has infinie source of elemen before saring he eperimen. We shall show ha he absolue values of () or o are no imporan bu he raio ()/ o is imporan, where his soluion is acually used o calculae he racer or impuriy diffusion coefficien of he species.
Toal maerial (mole/m ) ha was sandwiched before annealing can be found following M + d So we can wrie M + 0 ep 1/ d 4 Furher, we consider λ means ( ) λ d d Since inegraion gives, we ge π M ( λ ) dλ 0 π + 0 ep M ep Replacing his for M, we find ( ) π 4
vs. describes he disribuion of elemens. d /d describes (following he Fick s law) he change in flu wih respec o disance a a paricular annealing ime. d d d d vs. I eplains he rae of change of elemen The negaive values indicae he region, where i loses he elemen and posiive value indicaes ha he region where i gains he elemen. Noe ha he region where i is losing or gaining he elemen depends on he ime of annealing.
The change in disribuion of elemens wih he increase in ime is shown in he figure. M ( ) ep π 4 Facor comes from he fac ha elemens diffuse boh he sides from he source
If he elemen diffuses o one direcion hen he facor should no be considered. M 4 ep ) ( π M 4 ln ) ( ln π From he calculaed slope, one can deermine he diffusion coefficien from he known annealing ime.