Solution in semi infinite diffusion couples (error function analysis)

Transcription

1 Soluon n sem nfne dffuson couples (error funcon analyss) Le us consder now he sem nfne dffuson couple of wo blocks wh concenraon of and I means ha, n a A- bnary sysem, s bondng beween wo blocks made of pure A and an alloy of A. Unlke he prevous case, here, because of he dfference n he composon, dffuson wll be drven by he chemcal poenal graden. However, we shall show laer ha he soluon we are gong o derve, can be used only n he case where he concenraon and he chemcal poenal dfference of he end members s no much. Tha means dffuson coeffcens do no vary sgnfcanly wh he composon.

2 Sem nfne means ha, we anneal for a ceran annealng me such ha he end of he nal maerals are no affeced by he dffuson of elemens. Alhough he meanng of he sem nfne ndcaes ha a good par of he end of he dffuson couple should no be affeced bu acually even wh one unaffeced aomc layer n he end s suffcen o consder he sysem as sem nfne dffuson couple! Ths s mporan snce oherwse we can apply he relaon derved here o deermne he dffuson parameer or o calculae he concenraon profle from he known dffuson parameer. For he sake of solvng he Fck s second law, we dvde he alloy wh he composon no n numbers of very hn slces. We consder he dsrbuon of elemen from one slce. Suppose from he slce,, whch s a he dsance of from he nal conac plane,. Then he dsrbuon of elemen followng he soluon for hn flm, we can wre ( ) ep 4 where M ( mole / m )

3 d 4 ep ) ( Smlarly, f we consder he dsrbuon of elemens from all he slces and supermpose, we shall ge he oal dsrbuon of elemen a a parcular poson as n 4 ep Furher consder Snce we have consdered very hn slces, we can wre d d ep ep d d Tha means y replacng When and when

4 Snce hs negraon s no very sraghforward and akes very long me o deermne one sngle value, s epressed n erms of error funcon. Advanage of usng hs error funcon s ha one can deermne he values from a able. Error funcon of Z s defned as z ( z) ep( ) d Noe here ha z s acually he upper lm of he negraon. Lower lm of he negraon should be zero. So he epresson n he prevous slde should be rewren such ha can be replaced by he error funcon ( ) ep( ) d ep( ) d Followng, we can wre n erms of error funcon as ( ) snce (-z) (z)

5 So from he prevous relaon, s apparen ha he sgn of wll depend on whch sde of he, we are neresed o calculae. Furher, f he composon profle s jus he oppose compared o he frs eample, can be shown ha he relaon becomes Noe: We always use a superscrp for he concenraon o denoe rgh hand sde of he couple and - for he lef hand sde of he couple.

6 Tabulaed error funcon values z ( z) ep( ) d z z z z... 3! 5! 73! ( z) ( z)

7 In he prevous case, we consdered he dffuson beween he dffuson couple of wo dfferen blocks wh composon and or oherwse - and. Tha means n boh he cases, was a couple beween one pure elemen and an alloy of A. Error funcon soluons gven prevously can be rewren nerms of normalzed concenraon profles as In some cases, s possble ha wo alloys of A are couples. Now f we consder he dffuson couple beween - and, where > -, he relaon can be wren as shown n Fg. a he relaon can be wren as shown n Fg. a or

8 Furher, he relaon n he case of dffuson couple, where - and and - > as shown n Fg. b (n he prevous slde), can be epressed as or In he case of carburzng, we can see ha he relaon wll be he same, rrespecve of he sde on whch carburzng s done, snce n he case of Fg. a, s posve bu n Fg. b, s negave o S o S S o S o S For o

9 Few mporan noes So from he prevous relaons, s apparen ha he sgn of (wheher posve or negave) wll depend on n whch sde of he, we are neresed o calculae. Noe agan ha wll be more or less he consan a any poson s calculaed. I was also one of he assumpons for hs dervaon. From he error funcon analyss above, we can see ha, a from he relaon, comes ( ) from he relaon, S ( S ) comes ( ) S ( ) ( ) I ndcaes ha n a sysem, where wo blocks wh dfferen concenraons are coupled, a, whch s bascally he nal conac plane, he concenraon wll be always average of he concenraon of he nal maerals. Noe ha we need o locae anyway he nal conac plane afer he dffuson process, snce n he equaon s acually measured from he locaon of he nal conac plane. Ths also ndcaes ha hs equaon can only be used when end pars of he couple s no affeced by he dffuson process, snce oherwse a, wll have anoher average value dependng on how much of he end members are affeced.

10 P P Q Q In he case of carburzng, he surface concenraon should always be he same. Noe agan ha wll be more or less he consan a any poson s calculaed. I was also one of he assumpons for hs dervaon. efore proceedng furher for calculaon usng error funcon, one needs o check f ndeed you can locae he poson of by akng average of he end member concenraons. Ths poson should be possble o locae even by equalzng he areas P and Q (snce he loss from one sde of he couple should be equal o he gan n anoher sde). If he composon profle s no symmerc, can be found mahemacally by d ( )d When for a parcular he negrals have he same values, should be fed as poson.

11 Alhough looks very rval, fndng hs plane may no be ha easy n mos of he cases. Snce nal conac plane s an unque plane (only one value s possble), should be possble o fnd from any of he concenraon profles of elemens A or, as shown n he prevous slde. However, from our eperence, we have seen ha s possble o locae or fnd a sngle value from any of he profles, only when molar volume changes deally n he sysem, as shown n he fgure above. Noe ha he molar volume daa are requred o calculae he concenraon. When devaes (posvely or negavely) from he dealy, one fnds wo dfferen values from wo dfferen composon profles. fference depends on he devaon from he dealy. I feels as f he sysem los s nal conac plane. I wll be dscussed furher n deals a laer sage, wh eamples. So he lmaon of he error funcon analyss s no only ha dffuson coeffcen s more or less he consan bu also he molar volume does no change much wh he composon or changes deally.