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Investigation of bulk diffusion and surface kinetics by the electrical resistance method
Scripta METALLURGICA
Vol. 19, pp. 1385-1390, 1985 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved
INVESTIGATION OF BULK DIFFUSION AND SURFACE KINETICS BY THE ELECTRICAL RESISTANCE METHOD E.M.Tseitlin, AoF.Vjatkin Institute of Problems of Microelectronics Technology and Superpure Materials, Academy of Sciences of the USSR, Chernogolovka, Moscow district 142432, USSR (Received June 14, 1984) (Revised September 3, 1985) Introduction The technique based on the linear dependence between the electrical resistivity of metal J~ and concentration of diffusing impurity C
jo = ~
+ ~.£
(1)
where: ~ is the resistivity of metal free of impurities under question, and is the contribution into resistivity from I% of the impurity, has been used for a long time to study the kinetics of metal-impurity interaction EI-6]. Relationship (I) usually holds for dilute solid solutions and is restricted by the necessity of using metallic specimens with a small free path of electrons e~R , where R is the distance in which ~ C is significant, i.e. R V g ~ C . Relationship (I) is local and, apparently, it is valid for the whole specimen volume only with a uniform distribution of impurities. This occurs when interaction kinetics are controlled by the surface reaction on the external surface. This situation is usually realized in thin foils, where bulk diffusion is not a limiting process. Generally, when kinetics are determined not omly by the surface process, but also by the bulk diffusion, concentration gradients are high and the electmical resistivity of a specimem can be f o u n d b y summation of those of a set of parallel conductors with a cross-section of d d = d x d ~ over the whole area of the cross-section ~" .
::t::E
-'
This expressio_n makes it possible to obtain an exact formula for the dependence ~(t) or jO(C# , where ~I~/ is the meau impurity concentration known from the solution of diffusion equation, only for the case of diffusion in a thin plate for long times ~ [6]. There are papers [4-61 where analysis was made by a numerical method for bodies with the simplest geometry. This analysis was carried out, however, firstly under the assumption of diffusion control of the whole process aud, s~condly, ~ t h o u t specifying the limits of use of the linear approximation j o I ~ ) = ~ + ~ C , which may differ for absorption and desorpticn of impurities. The ~resent study attempts, to analytically obtain the dependence between jo and C for the ccmbinatlon of variation intervals of ~ and ~ interesting from the viewpoint of kinetics. The problem of ~as saturation: C(:Z".O):O and C(,Z', °°) =Co Let us write down expression (2) in a dimensionless form:
~= 1385 0036-9748/85 $3.00 + .00 Copyright (c) 1985 Pergamon Press Ltd.
(3)
1386
BULK DIFFUSION
Vol. 19, No. 12
w h e r e : T(X, F.) = c/:~, ~-//Co i s t h e d ~ n e n s i o n l e s s c o n c e n t r a t i o n , So =/~Co/~o is the i m p u r i t y c o n t r i b u t i o n i n t o e l e c t r i c a l r e s i s t i v i t y , Co i s the e q u i l i b ~ i ~ concentration of the absorbed impu~ity with respect ;o. surro~u~J~s,~ ~XJ~/K i s t h e d i m e n s i o n l e s s d i f f u s i o n t i m e (FO~Lrier c r i t e r i o n ) , X = - ~ - , ~ ) i s t h e impurity diffusion coefficient, R is the half-width of the specimen in the direction X When S o , T < i ~ 4 ) , t h e n one c a n u s e t h e e x p a n s i o n o f t h e f r a c t i o n t o be a v e r aged as follows:
/ Hence
= J-5oT.SZ.T~-S~T~...
= _~÷'~ ( - i ) " ( 5 o T ) " .
"'"
t ÷S ° ' T
(5)
~-
trtJ = "where:p/t/=~, ~:J~/t)/Joo =I÷So'LP i s t h e l i n e a r a p p r o x i m a t i o n f o r t h e e l e c t rical resistivity of the specimen. Condition (4) for the convergency of series (6) physically means, that t h e imp~Lrity c o n t r i b u t i o n t o t h e e l e c t r i c a l resistivity s h o u l d be l e s s t h a n that of the metal itself. For the case of gas saturation, when T / X , ~ / ~ ~ i t i s e q u i v a l e n t t o t h e a s s u m p t i o n t h a t S o < t 0 E x p a n s i o n ( 6 ) means t h a t w i t h i n a p o s sible error of ~/~/ one_can substitute the electrical resistivity ~ for its linear approximation ~ , this erro.r bei~ the higher, the higher are fC mad ~o • S i n c e c o n c e n t r a t i o n ~ s d i e n ~ s a r e m , ~ - ~ a n f o r t h e c a s e o f d i f f u s i o n control of kinetics, then without losing generality, a n ~ , a l y s i s o~ c o n v e r g e n c y o f s e r i e s ( 6 ) c a n be c a r r i e d o u t u n d e r t h e a s m m p t i o n o f d i f f u s i o n c o n t r o l of the gas saturation kinetics. This series proves to converge well only for s a m l l t i m e s ~ - ~ O ; t h i s was n o t t a k e n i n t o a c c o u n t i n C6], w h e r e a s i m i l a r e x p a n s i o n was s u p p o s e d t o be u s e d i n t h e g a s s a t u r a t i o n p r o b l e m f o r a ~ t i m e s . In the 4~4tial stage (~o~O.J) the diffusion process covers only a narrow ~pace r e g i o n X ~ V ~ and s m a l l p a r m a e t e r s a r e q u a n t i t i e s ~ and ~ - ~ rather than ~FIx,Fe)~ [ 0 , 1 ] . This means t h a t a l l members of s e r i e s (6), rare o f t h e same order of mmllness with respect to ~ , an i s t h e s e c o n d o n e , i . e . ~ ( ~ ) ~ ~ and the relative error of~dete~ination of ~ ~rom t h e l i n e a r a p p r o x £ a m t i o n i s
e
ual t o
at expression be r e p r e s e n t e d i n ~-'0 o" ~ . t h e f o r m o f a c o ny ~ r g e n ~ s e r ~ e s : _ a ~z ~ s ~ --'- u7 ~- ~•÷ , .~/* 1c" ~ 0 - - ~r J Oe~, _T_.t.c'÷ ,,~,~/S;,-2~".~,÷ ) , ' P 'IS, + . . . ) + 0 / ~ 1 (,7) --'0 . , . 1 ¢ ~ "'" Pot large t~esF~ >tO.t , corresponding to a regular r e . e , when the d i f fusion process ocou~e over the whole g e o m e t r i c a l body ~8], f o r gem s a t u r a t i o n "T'-p't sad {t-T)'-(/-co/ are shall p a r a m e t e r s . T h e r e f o r e , it is r e a s o n a b l e t o use the expansion as follows: : ~
* Z'Z.,¢-~),
where:linearSapp:o~.ation,:(.,°.-J°o)/J°. = ~/r'('° : ~ ¢ 1/ f o r w~ So. ";:/.= ~ :~¢-.,c.{/-t ) i s the Since f o r a r e g u l a r regime /t-T/e'"-H-P_) n the members of s e r i e s ( 8 ) a r e i n f~J~'Lt]~ small q u a n t i t i e s of the o r d e r o f / / - ~ ) " , thus JJa&LoatJJ~ both r a p i d convergency and the p o s s i b i l i t y of u s i n g the Ztaaear appro=zLamtion ~.=~.~ f o r the determination of concentration- ~ _~ /~ ~ , a T ~ t - - , " - - ~ - ~ _ ~/.~ -~'J--"-', . . . . . The ez~or of detezadJa~tion of impu~.ity c o n c e n t r a t i o n a c c o r ~ N ~ ~o ~ne linear a p p r o ~ 4 - - , t i o n o f series (7) and (8) c a n be e x p r e s s e d a s f o l l o w s . T h i n u l a t e : f o r ,~o':~J
It
is seen that
L/=
) So . r
use of the linear
[o>0.t (~'zo. 1 ) . =aX~Llele~i~ed: f o r Fo ' O•I I'.OX~K u , . c,U., er: f o r >o.i :
_
- Is s . ] . . , . o
appro~4~a~tion i s more ~ u s t i f i e d ,n~ ~,'* ~'~+ ~,- =S(~'~' - t)" .~. 0['/-21 zl _.~
(9)
in the range
(11)
Vol. 19, No, 12
BULK DIFFUSION
1387
where/~i -2.4048 i s t h e 1 - s t r o o t o f t h e B e s s e l f u n c t i o n
~
t
For are: (I-~)~(F-v~or Fo ~ o As a consequence o f tain a series similar to o o n v e r g i ~ f o r az~ So ,
Qz/ !
0.6
]
I
~o/x),~"-t= C~qq6
C(~ O)=Co and C ( I , = o ) =0
o.7 I
and~T f o r o f gas s a t u r a t i o n , t h e s m a l l p a t t e r e r s ~ Fo > 0 ~ t h i s , f o r small t i m e s (Fo<0.1) from (8) one may ob( 7 ) , however, wi~h r e s p e c t t o p a r m m t e r ( ~ - ~ ) and since always 5 ~ .
o.8 I
~
o.9
o..g5
I
YlG°I 4E Dependence between ~ and ~o , which was p l o t t e d a c c o r d i n g t o t h e e x p r e s s i o n s (9) and (10), which o b t a i n e d u ~ e r the conditions of d i f f u s i o n c o n t r o l l e d a b s o r p t i o n o f impuity into a thin plate
~So=0.5).
5
I
Qt
I
I
I
I
O..d
I
Fo
tO
For a r e g u l a r regime o f g a s e v o l u t i o n (Fo> 0.1 ), i n s t e a d o f e x p r e s s i o n (8) s e r i e s (6) i s u s e d , which g i v e s f o r t h i n p l a t e , p a r a l l e l e p i p e d and c y l i n d e r t h e r e s u l t s , s i ~ L l a r t o (10, 11, 12)(which a r e o b t a i n e d from t h e l a t t e r one by t h e s u b l t i t u t i o n f-~-~and3-'-5e). This a p p r o a c h t o t h e problem o f g a s e v o l u t i o n ( a t Pc> 0 . 1 ) base d on e x p a n s i o n (5) i s r e s t r i c t e d by t h e n e c e s s i t y o f f u l f i l l i n g c o n d i t i o n ( 4 ) , t h i s be~-~ e q u i v a l e n t f o r t h e b o d i e s u n d e r question to the followiz~ requirement: ~' (13) ~-o.6)~ ~-
So'~ ~=(o.
Thim =~=~ply r e d u c e s t h e r e g i o n o£ i n v e s t i g a t i o n o f t h e gas e v o l u t i o n k i n e t i c s a t ~))A r . T h e r e f o r e , i t i s o~ i n t e r e s t t o f i n d t h e c o n n e c t i o n between J~ and a t S~>)~ . I n t h i s case ~or i n t e r m e d i a t e t i m e s , when a r e g u l a r regime i s a l r e a d ~ o b s e r v e d and one more c o n d i t i o n 5o °~>)~ i s f u l f i l l e d f o r t h i n p l a t s and p a r a l l e l e p i p e d , one can o b t a i n t h e a n a l y t i c a l a p p r o x i m a t i o n f o r b o t h d i f f u s i o n c o n t r o l l e d k i n e t i c s and f o r t h e m ~ e d one. ThAn plate
t h e d i f f u m i o n c o n t r o l l e d gas e v o l u t i o n a t F o >0.1 [ 7 , 8 ] T(X, Fo) ~" ~ ' ~ ( ~,Fo) CO:S-~-x =A c o 3 ,z X , whereA When A > > t ~(~)=(( ox = /'C~w= t __~ A For
~,,o).
(14)
1388
BULK DIFFUSION
Vol.
19, No. 12
Lon~ ~arallele~ioed 31milarly
to the ~ . previous
case for the diffusion c o n t r o l l e d ~ s
evolution,
~ ( ~ ) = ( 1~, / t + A-d~ C Od ~u ' ~ x ' C ~ V ) , -~ ,,= _ _ ~o a ~ / _ a where ~ = ¢~)o'~(~). (15) Hence at/1>>.~ ~'(~.)=~.~"z'~'~ " ~ , whereK(J)z~-~ . (16) Under mixed c o n t r o l o f t h e gas e v o l u t i o n i n (17) at Fo>0.1
Y +
R . C¢:~.,u, • ,
where:/~//x,y/~x,~J are t h e f i r s t c ~ _ ~ a c t e r i s t i c r o o t s o f t h e d i f f u s i o n e q u a t i o n f o r r e l i a b l e s x , y and ~ " = , ~ R / ~ i s t h e c r i t e r i o n o f B l o t , ~ is the c o n s t a n t o f a s u r f a c e r s e c t i o n ; ~ / ~ ) may be w r i t t e n i n t h e s i m p l e s t form f o r t h e s p e c ~ e n w~th a s q u a r e c r o s s - s e c t i o n ( e x p . l ~ a ) , and
~(~/= # / ~ o/~,~,
a
•
/~--/./~
f
). ;o~/~).
(1~)
Using e x p r e s s i o n (17) f o r the specimens w i t h a s q . ~ e c r o s s - s e c t i o n (~x=)%~=)~) ~u ~ 1 0 ] the d a t a were a n a l y z e d on the k~.uet£cs o f d e c ~ r b t w i z a t i o n o f N i i n a hydrogen atmosphere. Use o f the e l e c t r i c a l resistivity t e c h n i q u e ~or the s t u d y o f k i n e t i c s o f gas s a t u r a t i o n i s i l l u s t r a t e d i n the f o l l o w £ n ~ e x p e r i m e n t s . A b s e n t i o n o f h~dro~en by c o ~ e r Up t o now t h e k i n e t i c s o f d i s s o l u t i o n o f gaseous t ~ d r o g e n i n copper have b e e n s t u d i e d f o r t e m p e r a t u r e s above 670 K and i s c h a r a c t e r i z e d by t h e mixed k ~ u e t i o a l r eg im e ( 0 . 1 ~ B i ~ I O 0 ) a t R~-I ~ [ 1 1 ] . ~n t h e p r e s e n t s t u d y , w~th t h e help of electrical resistivity technique, the k~uetics of hydrogen dissolution i n c o p p e r a r e s t u d i e d f o r c o p p e r i n c o n c e n t r a t e d HC1 (C~n~=27 wt~) al~ 373 K. S ~J ~le c r y s t a l c o p p e r s p e o ~ n e n s , 99.999 a t . ~ p u r e , were~l~eed w ~ t h ~ = ~ s ~ / ~ . ~ = -1 7 0 0 , a l e n g t h o f 30 ,m and a r e c t ~ a r c r o s s - s e c t i o n l x 2 , --, To v e r i f y t h e v a l i d i t y o f a l ~ l e e ~ d e p e n d e n c e b e t w e e n ~ and ~ i n d ~ f f u s i o n a n n e a l ~ . ~ f o r r e l a t i v e l y l a r g e t i m e s F o > 0 . 2 , we s i m u l t a n e o u s l y meas u r e d t h e r e s i d u ~ e l e c t r i c a l ~ e s i e t i v i t y a t 4 . 2 K and h y d r o g e n c o n t e n t E by the m a s s - s p e c t r o m e t r i c hydrogen degassin~ technique with cont~uuous e ~ a c u a t i o n [ 1 2 , 1 3 ] . FIG.2 shows t h e ~ r a p h o f t h i s dependence c o _ n ~ i ~ i ~ i n t h e e x p e r ~ e n t e l i n t e r v a l s t u d i e d t h e l i n e a r d e p e n d e n c e b e t w e e n ~ and C . The c o n t r i b u t i o n o f ~ y d r o g e n t o t h e e l e c t r i c a l r e ~ i s t i v i t y o f Cu K=(2.2 + 0 . 1 ) ~ v R c m / l a t . ~ ~ms d e t e r m i n e d , t h i s b e i n g somewhat h i g h e r t h a n K=(1.50 + 0.~5)~uR-om/lat.~ t a k e n from ~142, where t h e y u s e d t h e ~ n o r e a s e o f t o t a l p r e s s u r e on d e g a s s ~ o f s p e c ~ a e n i n a c l o s e d volwae t o m e a sure t h e h y d r o g e n c o n t e n t ; t h i s t e c h n£que amy eoaewhat e x a ~ a e r a t e the--h~d~ogen amount. The b ~d r o~ en c o n c e n t r a t i o n ~ r e a c h e s t h e e q u i l i b r i u m v a l u e Cs a c c o r d i n 8 t o t h e d ep en d e nc e ( F I G . 3 ) ae f o l l o w s : = t~ . e ~
Cs ,.. where [s]=
~ocor~
~noe 1 ~ . ~
~
~'~Y,: +
to the ex ~ _ r ~ n } ~ m
2
~i,,
a~i~O=4.4,~lO
/ - / C t .t ) ,
*'-" ,o,~
(18)
a 281~
a
"" ,
~¢,,'{~. 8,,, &v_/,
.,- (~xQ.~)_., B~.~.9~.6;.,~,=(6.7+9.6).l.o-.~t.~.
I t /S L l a t l ~ J t ~
:KZ,O+_.O.7)=IO L / S t
~le
ooaJaOa.QeB
with thf ex~t3Lpolation of the ~ temperature data of [11]~-10-'~ exp ( - 3 ) . 4 / R ~ ~ - ~ ) r = ~ j x = 2 . 1 l O ' ° m [ S . R s t £ m a t i o n o£ t h e r ~ t e c o n s t a n t , w h i c h i s dete~ed 5y the d i f f t t e i o n of~H ~ i n HC1 g i v e e ~ Hc~ ~ l O - ~ / s > ~ X and does n o t e f f e c t on the p e n e t r a t i o n o f H~ i n t o Cu.
Vol. 19, No. 12
BULK DIFFUSION
1389
/
,
! I ~M
5
,
0
S
~0
fo"(po ),
p
|
Is
9..c,',,
FIG.2 Dependence b e t w e e n • and C i n t h e r u n o f d i s s o l u t i o n o f h y d r o g e n i n Cu from HCI for the times Po~ 0.2 at 373 K.
I
I
I
,
o
I
5
i
L
L
I
,
.~
~o~t. 5
PIG. 3 Dependence on time o f t h e r e l a t i v e amount o f i m p u r i t y t - f l o s required to reach its equilibrium c o n c e n t r a t i o n Cs i n t h e p r o cess of dissolution of hydrogen in Cu from HCI at 373 K.
~onclusiona The a n a l y t i c a l d e p e n d e n c e b e t w e e n t h e e l e c t r i c a l r e s i s t i v i t y of a specimen p and mean c o n c e n t r a t i o n o f d i f f u s i o n i m p u ~ i t y ~ i s d e t e r m i n e d f o r a number o f o a s e s i n t e r e s t i n g from t h e v i e w p o i n t o f s t u d ~ i ~ a b s o r p t i o n and d e s o r p t i o n k i n e t i c s . The l i m i t s o f v a l i d i t y o f a l i n e a r d e p e n d e n o e ) ° = ~ = ~ + K £ f o r t h e s t u d y o f d i f f u s i o n were a n a l y z e d . The l i n e a r dependence b e t w e e n jo and ~ i n t h e h y d r o g e n a b s o r p t i o n by Ou from HC1 was e x p e r i m e n t a l l y v e r i f i e d . Acknowled~oments The a u t h o r s a r e g r a t e f u l t o A.Yu.Kasumov f o r t h e m easurem ent s o f r e s i d u a l electrical resistivity a t 4 . 2 K and h i s i n t e r e s t t o t h i s s t u d y . References 1. R . M . B a r r e r , D i f f u s i o n i n and t h r o u g h s o l i d s , Cambridge ( 1 9 4 1 ) . 2. S.D.Gem-zriken, I . Y a . D e k h t y a r ' , D i f f u s i o n i n m e t a l s and a l l o y s i n a s o l i d h a s e , S t a t e p u b l i s h i n g House o f p h y s i c o - m a t h e m a t i c a l l i t e r a t u r e , Moscow
~196o).
3. B . ~ , E . G e b h a ~ d t , Case and K o h l e n s t o f f i n M e t a l l e n , S p r i ~ e r - V e r l a g , B e r l i n H e i d e l b e r g New-YoTk ( 1 9 7 6 ) . 4. L.MoBorukhin, E o S . S h p i c h i n e t s k i y , Z a v . l a b . , ~.~, NolO, 1196 ( 1 9 7 1 ) . 5. K.Yamakawa, M.Tada, F.E.Fu~ita, Jap.J.Appl.P~s., 15, N~11, 769 (1976). 6. J.H.Nslter, J.M.Nitt, Let.Trans., 12A, Nat1, 1877 (Y981). 7. H.S.Ca~slaw, J . C . J a e g e r , C o n d u c t i o n o f Heat i n S o l i d s , C l a r e n d o n P r e s s , Oxford (1959). 8 . A.V.Lykov, Y u . A . i i k h a l l o v , T h e o r y o f Heat and Mass T r a n s f e r , G o s e n e r g o i ~ d a t ,
Moscow Lenim~-ad (1963).
9. A . P . P r u d n i k o v , Yu.A.~xTchkov, O . I . M a r i t o h e v , I n t e g r a l s E l e m e n t a r y f u n c t i o n s , Nauka, Moscow ( 1 9 8 1 ) .
and s e r i e s .
Vol. 19, pp. 1385-1390, 1985 Printed in the U.S.A.
Pergamon Press Ltd. All rights reserved
INVESTIGATION OF BULK DIFFUSION AND SURFACE KINETICS BY THE ELECTRICAL RESISTANCE METHOD E.M.Tseitlin, AoF.Vjatkin Institute of Problems of Microelectronics Technology and Superpure Materials, Academy of Sciences of the USSR, Chernogolovka, Moscow district 142432, USSR (Received June 14, 1984) (Revised September 3, 1985) Introduction The technique based on the linear dependence between the electrical resistivity of metal J~ and concentration of diffusing impurity C
jo = ~
+ ~.£
(1)
where: ~ is the resistivity of metal free of impurities under question, and is the contribution into resistivity from I% of the impurity, has been used for a long time to study the kinetics of metal-impurity interaction EI-6]. Relationship (I) usually holds for dilute solid solutions and is restricted by the necessity of using metallic specimens with a small free path of electrons e~R , where R is the distance in which ~ C is significant, i.e. R V g ~ C . Relationship (I) is local and, apparently, it is valid for the whole specimen volume only with a uniform distribution of impurities. This occurs when interaction kinetics are controlled by the surface reaction on the external surface. This situation is usually realized in thin foils, where bulk diffusion is not a limiting process. Generally, when kinetics are determined not omly by the surface process, but also by the bulk diffusion, concentration gradients are high and the electmical resistivity of a specimem can be f o u n d b y summation of those of a set of parallel conductors with a cross-section of d d = d x d ~ over the whole area of the cross-section ~" .
::t::E
-'
This expressio_n makes it possible to obtain an exact formula for the dependence ~(t) or jO(C# , where ~I~/ is the meau impurity concentration known from the solution of diffusion equation, only for the case of diffusion in a thin plate for long times ~ [6]. There are papers [4-61 where analysis was made by a numerical method for bodies with the simplest geometry. This analysis was carried out, however, firstly under the assumption of diffusion control of the whole process aud, s~condly, ~ t h o u t specifying the limits of use of the linear approximation j o I ~ ) = ~ + ~ C , which may differ for absorption and desorpticn of impurities. The ~resent study attempts, to analytically obtain the dependence between jo and C for the ccmbinatlon of variation intervals of ~ and ~ interesting from the viewpoint of kinetics. The problem of ~as saturation: C(:Z".O):O and C(,Z', °°) =Co Let us write down expression (2) in a dimensionless form:
~= 1385 0036-9748/85 $3.00 + .00 Copyright (c) 1985 Pergamon Press Ltd.
(3)
1386
BULK DIFFUSION
Vol. 19, No. 12
w h e r e : T(X, F.) = c/:~, ~-//Co i s t h e d ~ n e n s i o n l e s s c o n c e n t r a t i o n , So =/~Co/~o is the i m p u r i t y c o n t r i b u t i o n i n t o e l e c t r i c a l r e s i s t i v i t y , Co i s the e q u i l i b ~ i ~ concentration of the absorbed impu~ity with respect ;o. surro~u~J~s,~ ~XJ~/K i s t h e d i m e n s i o n l e s s d i f f u s i o n t i m e (FO~Lrier c r i t e r i o n ) , X = - ~ - , ~ ) i s t h e impurity diffusion coefficient, R is the half-width of the specimen in the direction X When S o , T < i ~ 4 ) , t h e n one c a n u s e t h e e x p a n s i o n o f t h e f r a c t i o n t o be a v e r aged as follows:
/ Hence
= J-5oT.SZ.T~-S~T~...
= _~÷'~ ( - i ) " ( 5 o T ) " .
"'"
t ÷S ° ' T
(5)
~-
trtJ = "where:p/t/=~, ~:J~/t)/Joo =I÷So'LP i s t h e l i n e a r a p p r o x i m a t i o n f o r t h e e l e c t rical resistivity of the specimen. Condition (4) for the convergency of series (6) physically means, that t h e imp~Lrity c o n t r i b u t i o n t o t h e e l e c t r i c a l resistivity s h o u l d be l e s s t h a n that of the metal itself. For the case of gas saturation, when T / X , ~ / ~ ~ i t i s e q u i v a l e n t t o t h e a s s u m p t i o n t h a t S o < t 0 E x p a n s i o n ( 6 ) means t h a t w i t h i n a p o s sible error of ~/~/ one_can substitute the electrical resistivity ~ for its linear approximation ~ , this erro.r bei~ the higher, the higher are fC mad ~o • S i n c e c o n c e n t r a t i o n ~ s d i e n ~ s a r e m , ~ - ~ a n f o r t h e c a s e o f d i f f u s i o n control of kinetics, then without losing generality, a n ~ , a l y s i s o~ c o n v e r g e n c y o f s e r i e s ( 6 ) c a n be c a r r i e d o u t u n d e r t h e a s m m p t i o n o f d i f f u s i o n c o n t r o l of the gas saturation kinetics. This series proves to converge well only for s a m l l t i m e s ~ - ~ O ; t h i s was n o t t a k e n i n t o a c c o u n t i n C6], w h e r e a s i m i l a r e x p a n s i o n was s u p p o s e d t o be u s e d i n t h e g a s s a t u r a t i o n p r o b l e m f o r a ~ t i m e s . In the 4~4tial stage (~o~O.J) the diffusion process covers only a narrow ~pace r e g i o n X ~ V ~ and s m a l l p a r m a e t e r s a r e q u a n t i t i e s ~ and ~ - ~ rather than ~FIx,Fe)~ [ 0 , 1 ] . This means t h a t a l l members of s e r i e s (6), rare o f t h e same order of mmllness with respect to ~ , an i s t h e s e c o n d o n e , i . e . ~ ( ~ ) ~ ~ and the relative error of~dete~ination of ~ ~rom t h e l i n e a r a p p r o x £ a m t i o n i s
e
ual t o
at expression be r e p r e s e n t e d i n ~-'0 o" ~ . t h e f o r m o f a c o ny ~ r g e n ~ s e r ~ e s : _ a ~z ~ s ~ --'- u7 ~- ~•÷ , .~/* 1c" ~ 0 - - ~r J Oe~, _T_.t.c'÷ ,,~,~/S;,-2~".~,÷ ) , ' P 'IS, + . . . ) + 0 / ~ 1 (,7) --'0 . , . 1 ¢ ~ "'" Pot large t~esF~ >tO.t , corresponding to a regular r e . e , when the d i f fusion process ocou~e over the whole g e o m e t r i c a l body ~8], f o r gem s a t u r a t i o n "T'-p't sad {t-T)'-(/-co/ are shall p a r a m e t e r s . T h e r e f o r e , it is r e a s o n a b l e t o use the expansion as follows: : ~
* Z'Z.,¢-~),
where:linearSapp:o~.ation,:(.,°.-J°o)/J°. = ~/r'('° : ~ ¢ 1/ f o r w~ So. ";:/.= ~ :~¢-.,c.{/-t ) i s the Since f o r a r e g u l a r regime /t-T/e'"-H-P_) n the members of s e r i e s ( 8 ) a r e i n f~J~'Lt]~ small q u a n t i t i e s of the o r d e r o f / / - ~ ) " , thus JJa&LoatJJ~ both r a p i d convergency and the p o s s i b i l i t y of u s i n g the Ztaaear appro=zLamtion ~.=~.~ f o r the determination of concentration- ~ _~ /~ ~ , a T ~ t - - , " - - ~ - ~ _ ~/.~ -~'J--"-', . . . . . The ez~or of detezadJa~tion of impu~.ity c o n c e n t r a t i o n a c c o r ~ N ~ ~o ~ne linear a p p r o ~ 4 - - , t i o n o f series (7) and (8) c a n be e x p r e s s e d a s f o l l o w s . T h i n u l a t e : f o r ,~o':~J
It
is seen that
L/=
) So . r
use of the linear
[o>0.t (~'zo. 1 ) . =aX~Llele~i~ed: f o r Fo ' O•I I'.OX~K u , . c,U., er: f o r >o.i :
_
- Is s . ] . . , . o
appro~4~a~tion i s more ~ u s t i f i e d ,n~ ~,'* ~'~+ ~,- =S(~'~' - t)" .~. 0['/-21 zl _.~
(9)
in the range
(11)
Vol. 19, No, 12
BULK DIFFUSION
1387
where/~i -2.4048 i s t h e 1 - s t r o o t o f t h e B e s s e l f u n c t i o n
~
t
For are: (I-~)~(F-v~or Fo ~ o As a consequence o f tain a series similar to o o n v e r g i ~ f o r az~ So ,
Qz/ !
0.6
]
I
~o/x),~"-t= C~qq6
C(~ O)=Co and C ( I , = o ) =0
o.7 I
and~T f o r o f gas s a t u r a t i o n , t h e s m a l l p a t t e r e r s ~ Fo > 0 ~ t h i s , f o r small t i m e s (Fo<0.1) from (8) one may ob( 7 ) , however, wi~h r e s p e c t t o p a r m m t e r ( ~ - ~ ) and since always 5 ~ .
o.8 I
~
o.9
o..g5
I
YlG°I 4E Dependence between ~ and ~o , which was p l o t t e d a c c o r d i n g t o t h e e x p r e s s i o n s (9) and (10), which o b t a i n e d u ~ e r the conditions of d i f f u s i o n c o n t r o l l e d a b s o r p t i o n o f impuity into a thin plate
~So=0.5).
5
I
Qt
I
I
I
I
O..d
I
Fo
tO
For a r e g u l a r regime o f g a s e v o l u t i o n (Fo> 0.1 ), i n s t e a d o f e x p r e s s i o n (8) s e r i e s (6) i s u s e d , which g i v e s f o r t h i n p l a t e , p a r a l l e l e p i p e d and c y l i n d e r t h e r e s u l t s , s i ~ L l a r t o (10, 11, 12)(which a r e o b t a i n e d from t h e l a t t e r one by t h e s u b l t i t u t i o n f-~-~and3-'-5e). This a p p r o a c h t o t h e problem o f g a s e v o l u t i o n ( a t Pc> 0 . 1 ) base d on e x p a n s i o n (5) i s r e s t r i c t e d by t h e n e c e s s i t y o f f u l f i l l i n g c o n d i t i o n ( 4 ) , t h i s be~-~ e q u i v a l e n t f o r t h e b o d i e s u n d e r question to the followiz~ requirement: ~' (13) ~-o.6)~ ~-
So'~ ~=(o.
Thim =~=~ply r e d u c e s t h e r e g i o n o£ i n v e s t i g a t i o n o f t h e gas e v o l u t i o n k i n e t i c s a t ~))A r . T h e r e f o r e , i t i s o~ i n t e r e s t t o f i n d t h e c o n n e c t i o n between J~ and a t S~>)~ . I n t h i s case ~or i n t e r m e d i a t e t i m e s , when a r e g u l a r regime i s a l r e a d ~ o b s e r v e d and one more c o n d i t i o n 5o °~>)~ i s f u l f i l l e d f o r t h i n p l a t s and p a r a l l e l e p i p e d , one can o b t a i n t h e a n a l y t i c a l a p p r o x i m a t i o n f o r b o t h d i f f u s i o n c o n t r o l l e d k i n e t i c s and f o r t h e m ~ e d one. ThAn plate
t h e d i f f u m i o n c o n t r o l l e d gas e v o l u t i o n a t F o >0.1 [ 7 , 8 ] T(X, Fo) ~" ~ ' ~ ( ~,Fo) CO:S-~-x =A c o 3 ,z X , whereA When A > > t ~(~)=(( ox = /'C~w= t __~ A For
~,,o).
(14)
1388
BULK DIFFUSION
Vol.
19, No. 12
Lon~ ~arallele~ioed 31milarly
to the ~ . previous
case for the diffusion c o n t r o l l e d ~ s
evolution,
~ ( ~ ) = ( 1~, / t + A-d~ C Od ~u ' ~ x ' C ~ V ) , -~ ,,= _ _ ~o a ~ / _ a where ~ = ¢~)o'~(~). (15) Hence at/1>>.~ ~'(~.)=~.~"z'~'~ " ~ , whereK(J)z~-~ . (16) Under mixed c o n t r o l o f t h e gas e v o l u t i o n i n (17) at Fo>0.1
Y +
R . C¢:~.,u, • ,
where:/~//x,y/~x,~J are t h e f i r s t c ~ _ ~ a c t e r i s t i c r o o t s o f t h e d i f f u s i o n e q u a t i o n f o r r e l i a b l e s x , y and ~ " = , ~ R / ~ i s t h e c r i t e r i o n o f B l o t , ~ is the c o n s t a n t o f a s u r f a c e r s e c t i o n ; ~ / ~ ) may be w r i t t e n i n t h e s i m p l e s t form f o r t h e s p e c ~ e n w~th a s q u a r e c r o s s - s e c t i o n ( e x p . l ~ a ) , and
~(~/= # / ~ o/~,~,
a
•
/~--/./~
f
). ;o~/~).
(1~)
Using e x p r e s s i o n (17) f o r the specimens w i t h a s q . ~ e c r o s s - s e c t i o n (~x=)%~=)~) ~u ~ 1 0 ] the d a t a were a n a l y z e d on the k~.uet£cs o f d e c ~ r b t w i z a t i o n o f N i i n a hydrogen atmosphere. Use o f the e l e c t r i c a l resistivity t e c h n i q u e ~or the s t u d y o f k i n e t i c s o f gas s a t u r a t i o n i s i l l u s t r a t e d i n the f o l l o w £ n ~ e x p e r i m e n t s . A b s e n t i o n o f h~dro~en by c o ~ e r Up t o now t h e k i n e t i c s o f d i s s o l u t i o n o f gaseous t ~ d r o g e n i n copper have b e e n s t u d i e d f o r t e m p e r a t u r e s above 670 K and i s c h a r a c t e r i z e d by t h e mixed k ~ u e t i o a l r eg im e ( 0 . 1 ~ B i ~ I O 0 ) a t R~-I ~ [ 1 1 ] . ~n t h e p r e s e n t s t u d y , w~th t h e help of electrical resistivity technique, the k~uetics of hydrogen dissolution i n c o p p e r a r e s t u d i e d f o r c o p p e r i n c o n c e n t r a t e d HC1 (C~n~=27 wt~) al~ 373 K. S ~J ~le c r y s t a l c o p p e r s p e o ~ n e n s , 99.999 a t . ~ p u r e , were~l~eed w ~ t h ~ = ~ s ~ / ~ . ~ = -1 7 0 0 , a l e n g t h o f 30 ,m and a r e c t ~ a r c r o s s - s e c t i o n l x 2 , --, To v e r i f y t h e v a l i d i t y o f a l ~ l e e ~ d e p e n d e n c e b e t w e e n ~ and ~ i n d ~ f f u s i o n a n n e a l ~ . ~ f o r r e l a t i v e l y l a r g e t i m e s F o > 0 . 2 , we s i m u l t a n e o u s l y meas u r e d t h e r e s i d u ~ e l e c t r i c a l ~ e s i e t i v i t y a t 4 . 2 K and h y d r o g e n c o n t e n t E by the m a s s - s p e c t r o m e t r i c hydrogen degassin~ technique with cont~uuous e ~ a c u a t i o n [ 1 2 , 1 3 ] . FIG.2 shows t h e ~ r a p h o f t h i s dependence c o _ n ~ i ~ i ~ i n t h e e x p e r ~ e n t e l i n t e r v a l s t u d i e d t h e l i n e a r d e p e n d e n c e b e t w e e n ~ and C . The c o n t r i b u t i o n o f ~ y d r o g e n t o t h e e l e c t r i c a l r e ~ i s t i v i t y o f Cu K=(2.2 + 0 . 1 ) ~ v R c m / l a t . ~ ~ms d e t e r m i n e d , t h i s b e i n g somewhat h i g h e r t h a n K=(1.50 + 0.~5)~uR-om/lat.~ t a k e n from ~142, where t h e y u s e d t h e ~ n o r e a s e o f t o t a l p r e s s u r e on d e g a s s ~ o f s p e c ~ a e n i n a c l o s e d volwae t o m e a sure t h e h y d r o g e n c o n t e n t ; t h i s t e c h n£que amy eoaewhat e x a ~ a e r a t e the--h~d~ogen amount. The b ~d r o~ en c o n c e n t r a t i o n ~ r e a c h e s t h e e q u i l i b r i u m v a l u e Cs a c c o r d i n 8 t o t h e d ep en d e nc e ( F I G . 3 ) ae f o l l o w s : = t~ . e ~
Cs ,.. where [s]=
~ocor~
~noe 1 ~ . ~
~
~'~Y,: +
to the ex ~ _ r ~ n } ~ m
2
~i,,
a~i~O=4.4,~lO
/ - / C t .t ) ,
*'-" ,o,~
(18)
a 281~
a
"" ,
~¢,,'{~. 8,,, &v_/,
.,- (~xQ.~)_., B~.~.9~.6;.,~,=(6.7+9.6).l.o-.~t.~.
I t /S L l a t l ~ J t ~
:KZ,O+_.O.7)=IO L / S t
~le
ooaJaOa.QeB
with thf ex~t3Lpolation of the ~ temperature data of [11]~-10-'~ exp ( - 3 ) . 4 / R ~ ~ - ~ ) r = ~ j x = 2 . 1 l O ' ° m [ S . R s t £ m a t i o n o£ t h e r ~ t e c o n s t a n t , w h i c h i s dete~ed 5y the d i f f t t e i o n of~H ~ i n HC1 g i v e e ~ Hc~ ~ l O - ~ / s > ~ X and does n o t e f f e c t on the p e n e t r a t i o n o f H~ i n t o Cu.
Vol. 19, No. 12
BULK DIFFUSION
1389
/
,
! I ~M
5
,
0
S
~0
fo"(po ),
p
|
Is
9..c,',,
FIG.2 Dependence b e t w e e n • and C i n t h e r u n o f d i s s o l u t i o n o f h y d r o g e n i n Cu from HCI for the times Po~ 0.2 at 373 K.
I
I
I
,
o
I
5
i
L
L
I
,
.~
~o~t. 5
PIG. 3 Dependence on time o f t h e r e l a t i v e amount o f i m p u r i t y t - f l o s required to reach its equilibrium c o n c e n t r a t i o n Cs i n t h e p r o cess of dissolution of hydrogen in Cu from HCI at 373 K.
~onclusiona The a n a l y t i c a l d e p e n d e n c e b e t w e e n t h e e l e c t r i c a l r e s i s t i v i t y of a specimen p and mean c o n c e n t r a t i o n o f d i f f u s i o n i m p u ~ i t y ~ i s d e t e r m i n e d f o r a number o f o a s e s i n t e r e s t i n g from t h e v i e w p o i n t o f s t u d ~ i ~ a b s o r p t i o n and d e s o r p t i o n k i n e t i c s . The l i m i t s o f v a l i d i t y o f a l i n e a r d e p e n d e n o e ) ° = ~ = ~ + K £ f o r t h e s t u d y o f d i f f u s i o n were a n a l y z e d . The l i n e a r dependence b e t w e e n jo and ~ i n t h e h y d r o g e n a b s o r p t i o n by Ou from HC1 was e x p e r i m e n t a l l y v e r i f i e d . Acknowled~oments The a u t h o r s a r e g r a t e f u l t o A.Yu.Kasumov f o r t h e m easurem ent s o f r e s i d u a l electrical resistivity a t 4 . 2 K and h i s i n t e r e s t t o t h i s s t u d y . References 1. R . M . B a r r e r , D i f f u s i o n i n and t h r o u g h s o l i d s , Cambridge ( 1 9 4 1 ) . 2. S.D.Gem-zriken, I . Y a . D e k h t y a r ' , D i f f u s i o n i n m e t a l s and a l l o y s i n a s o l i d h a s e , S t a t e p u b l i s h i n g House o f p h y s i c o - m a t h e m a t i c a l l i t e r a t u r e , Moscow
~196o).
3. B . ~ , E . G e b h a ~ d t , Case and K o h l e n s t o f f i n M e t a l l e n , S p r i ~ e r - V e r l a g , B e r l i n H e i d e l b e r g New-YoTk ( 1 9 7 6 ) . 4. L.MoBorukhin, E o S . S h p i c h i n e t s k i y , Z a v . l a b . , ~.~, NolO, 1196 ( 1 9 7 1 ) . 5. K.Yamakawa, M.Tada, F.E.Fu~ita, Jap.J.Appl.P~s., 15, N~11, 769 (1976). 6. J.H.Nslter, J.M.Nitt, Let.Trans., 12A, Nat1, 1877 (Y981). 7. H.S.Ca~slaw, J . C . J a e g e r , C o n d u c t i o n o f Heat i n S o l i d s , C l a r e n d o n P r e s s , Oxford (1959). 8 . A.V.Lykov, Y u . A . i i k h a l l o v , T h e o r y o f Heat and Mass T r a n s f e r , G o s e n e r g o i ~ d a t ,
Moscow Lenim~-ad (1963).
9. A . P . P r u d n i k o v , Yu.A.~xTchkov, O . I . M a r i t o h e v , I n t e g r a l s E l e m e n t a r y f u n c t i o n s , Nauka, Moscow ( 1 9 8 1 ) .
and s e r i e s .