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Clarification and generalization of corrections to the Darken diffusion equations
Scripta
METALLURGICA
Vol. 21, pp. 33-38, 1987 P r i n t e d in the U.S.A.
P e r g a m o n J o u r n a l s , Ltd. All r i g h t s r e s e r v e d
CLARIFICATION AND GENERALIZATION OF CORRECTIONS TO THE DARKEN DIFFUSION EQUATIONS J.S. Kirkaldy Institute for Materials Research, McMaster University Hamilton, Canada, L8S 4MI (Received (Revised
S e p t e m b e r 12, 1986) O c t o b e r 20, 1986)
Since Bardeen and Herring (i) drew attention to the shortcomlngs of the Darken analysis (2) relating intrinsic to tracer coefficients in a countercurrent binary, local equilibrium process investigators have sought a quantificatlon of the corrections as they appear within expressions for the mutual diffusion coefficient D and the Kirkenda11 velocity v (3-11). Noteworthy are the contributions of Msnning who presented explicit expressions based upon correlation effects within an alloy model strongly normalized to tracer behavior (the random alloy model). All subsequent approaches are in some way related to his offerings (7-ii). Manning's initial viewpoint was strongly atomlstic, and so in the words of Lidiard "the arguments leading to these various relations are complex and rather special, with the result that the theoretical assessment of their nature and accuracy is hard" (11). Because of this lack of transparency, subsequent workers have sought greater rigor and a degree of clarity through extensive use of basic phenomenology. In our view, none of these attempts have been completely successful. Furthermore, an element of confusion has arisen because the investigators have not, in relation to various formulas, made unequivocal statements as to when and where the vacancies are conserved and equivalently, whether the mobility matrices are unique or nonunique. One of our main aims is to remove this confusion by pointing out that there are two distinct problems which only by careful argument can be related. These concern the Darken Equations under strict local equilibrium, binary (A-B) interdiffusionwith non-conserved vacancies and ternary A-B-V interdiffusion with strictly conserved vacancies. In this reassessmant, we obtain rigorous and more general formulations of Manning's equations for v and ~ and present a new derivation of the mobility matrix for A-B-V interdiffusion with vacancies conserved. The Rigorous Local E~uilibriumPhenomenolo~y In all cases, investigators start with the conventional definition of the intrinsic coefficients D a and ~ via the fluxes in a binary local equilibrium lattice-fixed or Kirkendall drift frame of reference, viz., Ja = -Da N VX a ; Jb = -Db N V ~
(i)
where N is the number of sltes per unit volume and the X's are mole fractions of atoms a and b. Since the binary Gibbs-Duhem equation in terms of chemical potentials, ~, with V~V = 0 is XaV~ a + ~ V ~ b
= 0
(2)
Jb = - ~ V ~ b
(3)
we can also define intrinsic mobilities via Ja = -LaVUa ; with kT La Da = -~ Xq # ;
kT Lb DD = -~Xbb ~
(4)
33
0036-9748/87 Copyright
(c)
1987
$3.00 + .00 Pergamon Journals
Ltd.
34
DARKEN DIFFUSION EQUATIONS
Vol.
21, No. 1
and d£nTa =
¢ =
1
+ d~tnXa
d~nTb
d£n7
1 + d£-----~b - 1 + d£nX
(5)
where the 7's are activity coefficients. The structural question as to when D a and ~ metallic solids is answered in our closlng section.
exist in
The conventional transformation to the laboratory frame of Eqs. 1 and 3 yields the mutual coefficient and the Kirkendell velocity
kT .Xb
X
5 - Da% + % x a - -~q- La + ~-~ ~) a
(65
b
and kT~La ~ ~Ca v = (Da . Db)$VX a = - N ~ - ~ ) ~ - - x
(7)
where Ca is in atoms/unit volume (2). The central theoretical problem is to express the intrinsic diffusion coefficients or mobilities in terms of the tracer coefficients DE and ~ . The Darken solution to this problem (25 is well-known so will not be repeated here. However, as Bardeen and Herrln E (1) and Le Claire (35 among others have pointed out, there are delicate correlation effects which have been ignored, and which can be expressed as non-zero off-dlagonal terms in an appropriate OnsaEer mobility matrix (12,13). Howard and Lidlard (45, expanding' on the work of Le Claire (35, have expressed the corrections to the Darken equatlon in terms of an L-matrlx corresponding to the quaternary local equilibrlma system A-A"-B-B ~ and make very clear throughout their analysis, with X a - X A + XA, and = ~ + XR,, that they reEard Lab as an independent, non-zero, and therefore unique c~efflclent. This p~oposltlonwas recently challenged by Kirkaldy (105 who insisted that the only unique representation is given by Eqs. 3 where
La-'La;%b'&;Lab'%a'0
(85
Howard and Lidiard obtained the corrected Darken equations in the limit of A* and B* as dilute tracers
~/~b -
,
XBD:
kT
L~,
+ XADB + ~ - {XB(X. ~
LAB
. .LeB,
LAB
'~"5 + ~ A t ' ~ " - - XA 5}
(95
and
,
,
kT.{(x.~** ~ +~- ~ .
g5 oN" %,
v/,~ A - D ~ - %
which clearly involve only two independent L-coefflclents.
~ ~A
"
(io)
Here (45
(11)
LAB - Lab But in view of Eqs. 8, whence LAB
LAB. xA)
0, these become (105
k T {XB LAA*
LBB*}
(12)
and
.
,_
%B*
(13)
na -- n ~ + n~, ; ~b -- % B + %B*
(14)
v/~VXA DA
,+kT{L~* De ~- XA,
x~, }
where (i0)
Lidiard and ManninE have strenuously objected to this modification (i0). However, these objections will Be allayed in the followlnE section t~rrough the presentation of a transparent Eeneral-
Vol.
21, No.
1
DARKEN
DIFFUSION
EQUATIONS
55
ization of the Manning equations based upon the necessary condition that LAB = 0. Evaluation of LAA, and ~ B * The pertinent pheno~enological equations in the Kirkendall frame of reference pertaining to A - A * B interdlffuslon are (i0) JB = -LBBV~B
(15)
JA "
-LAAV~A - LAAeVUA*
(16)
JA* =
-LA*AV~ A - LA*A*V~A,
(17)
where Ja is disaggregated t o JA + JA*' and since B atoms do not distinguish between A and A*, L ~ n - L h by deflnition. There is a similar set for B-B*-A diffusion. Consider a countercurrent A-~ exp~rlment with the concentration and chemical potential of dilute tracer A* initially uniform with D~ > i~ as indicated in Fig. i. At the optima of the A* curves, V~A, = 0, so applying F..qs. 7, 15 and ,
JA* JA
-
LA*A
XA*Jv(I- fA)
XA* (Da-Db) (i- fA) dP
,-
LAA
JA
(18)
DA
where fA is the A* tracer correlation factor and VX A has been cancelled between numerator and denominator on the right. JA, has been evaluated as the fraction of the vacancy flux intercepted by uniform A* atoms per u~it time (-X**Jv) multiplied by the fraction of such contacts which are not random (1-fA), and therefore contributing to an A* flux in the uniform A* regions. This, in our view, is the--correct and very limited context of the so-called " ~ wind". Now, in view of dilute A*, LA, A << LAA , so from Eqs. 14, D a ffikTLAA~/NX A, and
LA,A . LAA, ffi k'TNXAXA,(Da_Db) (l_fA)
(19)
~*B
(20)
and similarly " ~B*
= ~N ~ , ( D b _ D a )
(l_fB)
Substituting these into Eq. 13 and comparin E with Eq. 7, we obtain v/~Vx A = v a
- ~
= (DA. - DB,)/?
(21)
where = XAf A + xBf s
(22)
Next, substituting Eq. 21 i n t o 19 and 20, and these into 12, we obtain
If we define the average tracer coefficlent as
= D,~XA + D~XB
(24)
then we can invoke detailed balance for a local equilibrium A*-B*-A-B solution in the form DAI (l-f A) = DB/(I-f B) = ~ / ( I - ~ )
(25)
where the right follows from the left equality. The latter, after cross-multiplylng and multip l y i n g by X A ~ , , s t a t e s that the probability of B *-vacancy exchanges associated with the vacancy wind produced in local A* motions (cf. Eq. 18) is equal to the probability of A*vacancy exch=,ges associated locally with B* motions since each trajectory set is the other~ inverse set. Vacancy binding effects are assumed as above to be included in fA and fB" Thus with simultaneous application of Eqs. 25. Eq. 23 can be written in general as
D/~ " X.BD~ + XAD ~ + XAXB(D~-DB*)2(1-'f)/fD with the intrinsic coefficients
(26)
56
DARKEN
DA= D~{[I +
DIFFUSION
EQUATIONS
Vol.
21, No.
XA(D~-D;I(1-"f'I/f'D] andDB = D;O~ + XB(D;-D~I(1-~')/~]
I
(27)
W i t h ~ = f, the tracer value in a pure material, and one of the defining conditions of Manning's random alloy model, these reduce to his well-known expressions. We emphasize, however, that subject to the aforementioned conditions on fA' fB' Eqs. 26 and 27 are exact and independent of the thermodynamic and kinetic model. In view of Manning's strong objection to our insistence upon a diagonal L-matrix in the A-B system it came as a surprise that our structure generates his corrections. We can only conclude that his interpretation of the L-matrices is different than the conventional one used here (cf. co--~nt of Lidlard quoted above). Transformation to the A-B-V Conserved Vacancy Experiment From Anthony's general phenomanological scheme for conserved vacancies (14) and JV = -(JA + JB )
(28)
we can rigorously obtain the ternary flux set with Vuv as an independent force
XA JA = -(LAA - ~
LAB)VUA + [LAA + LAB(I +
LAA + LAB ~ B + LAB JV ffi [ XA ~ ]XA VUA - [LAA + 2LAB + ~ B
)]VUV
(29)
+ ~ ( L B B + LAB)]V~V
(30)
and symmetrically
%, JB = - ( ~ B " XAA LAB)PUB + [~B + LAB(I + -'~') XA ]VuV
(31)
Since the vacancies are conserved, the lattice and laboratory frames are coincident. Now it is the conventional intelligence that the binary A-B Darken-Kirkendall structure as developed in previous sections can be obtained from Eqs. 29, 30 and 31 by setting V~V - 0 on the right. We will now take this to be the case, delaying the Justification to the following section. The key question is: can this transformation be reversed in general? The answer is clearly, no, for a diagonal 2x2 matrix cannot be disaggregated into a 2x2 non-diagonal matrix in the absence of rules or constraints. Such information must be provided by one or more new postulates, pertaining in this case to the A-B-V process (cf. Refs. 5, 8, 9, Ii). The present structure provides a particularly transparent and plausible route to a solution of the ternary problem provided the vacancy solution is dilute and henrian (no association). Under this circumstance, Eqs. 5 remain valid and from Eqs. 27, the intrinsic coefficient in the form
(321
D A ffiD A* + XA(D A - D;)(1-fA)/~ and Eq. 29 with VUv = 0, we can write
XA.
LAB N ~ "~
[DA-++ XAD+~ (1-fA) ~
* (1-fAI XADB _f ]
(33)
Now the ternary A-A*-V d i f f u s i o n problem with conserved vacancies has been solved exactly yielding the L-matrix (4, 15, 161 NXA*
(lf ] ; LA,A, = ~ LAA " NxA kT IDa%+ XAD A f-----l) and
NXAXA,DA LAA, ffiLA, A = kT(XA+XA,)
[DA + XA,D A*
]
( 34 )
(35)
where f is the tracer correlation factor. ~rom the form of the vacancy wind term of Eq. 18 in relation to a comparison of the second term in Eq. 33 and the second term of LAA in Eqs. 34, one concludes that both of the latter are to be uniquely associated with the interception of the conserved vacancy flux in these alternate experimental situations. Thus the ratio of the two correlatlun terms can reasonably 5e inferred as ~l-fA)f/¢l-flf , whence we can conclude that in the alloy case
Vol.
21, No.
1
DARKEN DIFFUSION EQUATIONS
LAA - - ~ - [ i
+ XA--~----] and ~ B
=
37
kT$ [l +
_ f
]
(36)
Furthermore, the last term in Eq. 33 must now be associated with the cross coefficient and therefore . N XA--D~ . B .(l-fA) . . N * (I-EB) (37) LAB ~ f and h A kT~ X A ~ D A f Now, applying Eqs. 25 simultaneously, we obtain
_ (z-fA) (l-f~) LAB = LBA
=
kT¢ XAXBD
"{ (l-f)
~
D~ ~
= kT#" XAXB
:
(38)
~-
Onsager Reciprocity appears on account of detailed balance or microscopic reversibility (12) as implied by Eqs. 25, and to complete the matrix,
LAA =
kT~b
[i +
---Df
"] ; LBB =
kT~b
[i +
__ Df
.]
(39)
Again, these have the same f o r m a s Manning's relations, where for the random alloy model, ~ -~ f. Because of our assumption of the absence of vacancy association it could be concluded that our L-matrlx expressed in terms of • does not have the generality of the expressions for v and D. However, on comparison of our result with that of Lidlard (ii), which claims generality, it is fair to conjecture that Eqs. 3~ and 39 are also general. .Diagnostics on Mechanism We now e~plore the Justification for the analytic continuation from a ternary conserved vacancy model to the binary local equilibrium model used in the previous section. Since we are assuming that within certain limits the Kirkendall drift is structure insensitive, we can illustrate the argument without loss in generality by considering that the creep process is effected via an appropriately dense array of parallel grain boundaries lying perpendicular to the diffusion direction. Now if the ternary formalism of the previous section is to agree with the binary description, the tlme-dependent forms of the unmodified equations 29 and 31 (V~V # 0) as applied to the perfect crystal regions must be somehow equivalent to Eqs. 1 with strictly equilibrated vacancies. Using our previous results, Eqs. 29 and 31 with internally conserved A, B and V transform with X v ÷ 0 to the local, low gradient approximation
~x2 and similarly for ~ ,
~
~x2
with
~xv. (~A-D~)* . ~2XA ~ ~2Xv ~t
T
~x2
~ X v ~x2
(41)
Now the local equilibrium condition can be applied to these equations in the strong form ~ / ~ t = ~2Xv/%x2 - 0 which implles the impossible result that ~XA/%t = 0. If, on the other hand, we apply the weaker condition, ~Xv/~ t = 0, allowing curved quasl-stationary vacancy profiles between defects, then we obtain the equations
~t
--
A
~
~)x2
; -~t
=
~
9x 2
(42)
To understand this quasl-steady result, which implies ~ -~XA, it is necessary to recognize that for one-dimenslonal diffusion in the absence of sources an~ sinks (Fig. 2) the eigenvalues of the D-matrlx in Eqs. 40 and 41 are (15,16) X 1 ---
~/fX V and ~2 ffiD~D~/~
(43)
38
DARKEN
DIFFUSION
EQUATIONS
Vol.
21,
No.
I
with A I >> A2, and it is these which establish the profiles, and therefore the gradients, near to and-far from the origin, respectively (15-17). If we assume for this structure that the vacancies exactly equilibrate at the grain boundaries then according to Eq. 41 with dXv/dt = 0 the curvature of the vacancy profile within a grain is given by
(44) which in proportion to its own average concentration is of the same magnitude as in X A or X n. We note flnally that subject to Eq. 44, V~v/VB A can everywhere be made as smell as we w i s h e r the boundary or defect density is sufficiently"increased. Since this is also the condition for a local equilibrium blnary structure based on Eqs. 1 with D a and ~ physically defined, we conclude that the analytic continuation of a ternary conserved vacancy structure to the binary structure via the first order constraint V~ V = 0 is ~alid. Note that the latter argument generalizes to three dimensional arrays of de£ects where ~2/~x2 ÷ V 2. Acknowledgements The author is grateful to Drs. D.J. Young of the University of N.S.W., A.B. Lidiard of Harwell, J.E. Lane in Melbourne, G.V. Kidson of W.N.R.E. and J.R. Manning of N.B.S. for many critical and helpful comments. Thanks are also due to K. Balasubramanian of McMaster University for helpful discussions and for checking soma of the calculations. References 1. 2. 3. 4. 5. 6. 7. 8. 9. i0. 11. 12. 13. 14. 15. 16. 17.
J. Bardeen and C. Herring, Atom Movements, ASM, Cleveland, 1951, p. 87. L.S. Darken, Trans. AIME, 175, 184 (1948). A.D. LeClaire, Prog. Metal Phys., ~, 265 (1953). R.E. Howard and A.B. Lidiard, Pep. Prog. Phys., 27, 161 (1964). J.R. Manning, b~ta Met., 15, 847 (1967). J.R. Manning, Diffusion Kinetics for Atoms in Crystal, D. Van Nostrand Inc., Princeton, N.J., 1968. K.P. Gurov, A.V. Nazarov and K.K. Anandybov, Fiz. Metal. Metalloved, 32, 176 (1971). Th. Heumann, J. Phys. F. ~, 1997 (1979). M.A. Dayananda, A~taMet., 29, 1151 (1981). J.S. Kirkaldy in Solute-Defect Interactlon - Theory and Experiment, Eds. S. Salmoto, G.R. Purdy and G.V. Kidson, Pergamon Press, Toronto~ 1986. Discussions by A.B. Lidlard and J~R. Y~,n~ng. A.B. Lidlard, Acta Y~t., submltted 1986. L. Onsager, Phys. Rev., 37, 405; 38~ 2265 (1931). L. Onsager, Ann. N.Y. Acad. Sci., 46, 241 (1945-46). T.R. Anthony in Diffusion in Solids, Ed. A.S. Nowick and J.J. Burton, Academic Press, New York, 1975. J.S. g/rkaldy, D.J. Young and J.E. Lane, Acta Met., in press, 1986. J.S. Kirkaldy and D.J. Young, Diffusion in the Condensed State, Institute of Metals, London, 1986, in press. J.S. Kirkaldy, Can. J. Phys. 36, 899 (1958).
XA~0
I I " i
Jv
Fig. i: Redistribution of tracer A* due to the "vacancy wlnd" JV created by the countercurrent flow of A and B at local equilibrium. Note that tracer B would flow in the same direction as A so that LBB * has the opposite sign to LAA, .
0 Fig. 2: Redistribution of initially ~miform conserved vacancies due to countercurrent A and B fluxes. Note how the back force of vacancies in the alloy diffusion zone symmetrlzes the_~A and s profiles, u - ¢ ~ ( D ~ V ~ / B ) t and V - ~,DVt •
METALLURGICA
Vol. 21, pp. 33-38, 1987 P r i n t e d in the U.S.A.
P e r g a m o n J o u r n a l s , Ltd. All r i g h t s r e s e r v e d
CLARIFICATION AND GENERALIZATION OF CORRECTIONS TO THE DARKEN DIFFUSION EQUATIONS J.S. Kirkaldy Institute for Materials Research, McMaster University Hamilton, Canada, L8S 4MI (Received (Revised
S e p t e m b e r 12, 1986) O c t o b e r 20, 1986)
Since Bardeen and Herring (i) drew attention to the shortcomlngs of the Darken analysis (2) relating intrinsic to tracer coefficients in a countercurrent binary, local equilibrium process investigators have sought a quantificatlon of the corrections as they appear within expressions for the mutual diffusion coefficient D and the Kirkenda11 velocity v (3-11). Noteworthy are the contributions of Msnning who presented explicit expressions based upon correlation effects within an alloy model strongly normalized to tracer behavior (the random alloy model). All subsequent approaches are in some way related to his offerings (7-ii). Manning's initial viewpoint was strongly atomlstic, and so in the words of Lidiard "the arguments leading to these various relations are complex and rather special, with the result that the theoretical assessment of their nature and accuracy is hard" (11). Because of this lack of transparency, subsequent workers have sought greater rigor and a degree of clarity through extensive use of basic phenomenology. In our view, none of these attempts have been completely successful. Furthermore, an element of confusion has arisen because the investigators have not, in relation to various formulas, made unequivocal statements as to when and where the vacancies are conserved and equivalently, whether the mobility matrices are unique or nonunique. One of our main aims is to remove this confusion by pointing out that there are two distinct problems which only by careful argument can be related. These concern the Darken Equations under strict local equilibrium, binary (A-B) interdiffusionwith non-conserved vacancies and ternary A-B-V interdiffusion with strictly conserved vacancies. In this reassessmant, we obtain rigorous and more general formulations of Manning's equations for v and ~ and present a new derivation of the mobility matrix for A-B-V interdiffusion with vacancies conserved. The Rigorous Local E~uilibriumPhenomenolo~y In all cases, investigators start with the conventional definition of the intrinsic coefficients D a and ~ via the fluxes in a binary local equilibrium lattice-fixed or Kirkendall drift frame of reference, viz., Ja = -Da N VX a ; Jb = -Db N V ~
(i)
where N is the number of sltes per unit volume and the X's are mole fractions of atoms a and b. Since the binary Gibbs-Duhem equation in terms of chemical potentials, ~, with V~V = 0 is XaV~ a + ~ V ~ b
= 0
(2)
Jb = - ~ V ~ b
(3)
we can also define intrinsic mobilities via Ja = -LaVUa ; with kT La Da = -~ Xq # ;
kT Lb DD = -~Xbb ~
(4)
33
0036-9748/87 Copyright
(c)
1987
$3.00 + .00 Pergamon Journals
Ltd.
34
DARKEN DIFFUSION EQUATIONS
Vol.
21, No. 1
and d£nTa =
¢ =
1
+ d~tnXa
d~nTb
d£n7
1 + d£-----~b - 1 + d£nX
(5)
where the 7's are activity coefficients. The structural question as to when D a and ~ metallic solids is answered in our closlng section.
exist in
The conventional transformation to the laboratory frame of Eqs. 1 and 3 yields the mutual coefficient and the Kirkendell velocity
kT .Xb
X
5 - Da% + % x a - -~q- La + ~-~ ~) a
(65
b
and kT~La ~ ~Ca v = (Da . Db)$VX a = - N ~ - ~ ) ~ - - x
(7)
where Ca is in atoms/unit volume (2). The central theoretical problem is to express the intrinsic diffusion coefficients or mobilities in terms of the tracer coefficients DE and ~ . The Darken solution to this problem (25 is well-known so will not be repeated here. However, as Bardeen and Herrln E (1) and Le Claire (35 among others have pointed out, there are delicate correlation effects which have been ignored, and which can be expressed as non-zero off-dlagonal terms in an appropriate OnsaEer mobility matrix (12,13). Howard and Lidlard (45, expanding' on the work of Le Claire (35, have expressed the corrections to the Darken equatlon in terms of an L-matrlx corresponding to the quaternary local equilibrlma system A-A"-B-B ~ and make very clear throughout their analysis, with X a - X A + XA, and = ~ + XR,, that they reEard Lab as an independent, non-zero, and therefore unique c~efflclent. This p~oposltlonwas recently challenged by Kirkaldy (105 who insisted that the only unique representation is given by Eqs. 3 where
La-'La;%b'&;Lab'%a'0
(85
Howard and Lidiard obtained the corrected Darken equations in the limit of A* and B* as dilute tracers
~/~b -
,
XBD:
kT
L~,
+ XADB + ~ - {XB(X. ~
LAB
. .LeB,
LAB
'~"5 + ~ A t ' ~ " - - XA 5}
(95
and
,
,
kT.{(x.~** ~ +~- ~ .
g5 oN" %,
v/,~ A - D ~ - %
which clearly involve only two independent L-coefflclents.
~ ~A
"
(io)
Here (45
(11)
LAB - Lab But in view of Eqs. 8, whence LAB
LAB. xA)
0, these become (105
k T {XB LAA*
LBB*}
(12)
and
.
,_
%B*
(13)
na -- n ~ + n~, ; ~b -- % B + %B*
(14)
v/~VXA DA
,+kT{L~* De ~- XA,
x~, }
where (i0)
Lidiard and ManninE have strenuously objected to this modification (i0). However, these objections will Be allayed in the followlnE section t~rrough the presentation of a transparent Eeneral-
Vol.
21, No.
1
DARKEN
DIFFUSION
EQUATIONS
55
ization of the Manning equations based upon the necessary condition that LAB = 0. Evaluation of LAA, and ~ B * The pertinent pheno~enological equations in the Kirkendall frame of reference pertaining to A - A * B interdlffuslon are (i0) JB = -LBBV~B
(15)
JA "
-LAAV~A - LAAeVUA*
(16)
JA* =
-LA*AV~ A - LA*A*V~A,
(17)
where Ja is disaggregated t o JA + JA*' and since B atoms do not distinguish between A and A*, L ~ n - L h by deflnition. There is a similar set for B-B*-A diffusion. Consider a countercurrent A-~ exp~rlment with the concentration and chemical potential of dilute tracer A* initially uniform with D~ > i~ as indicated in Fig. i. At the optima of the A* curves, V~A, = 0, so applying F..qs. 7, 15 and ,
JA* JA
-
LA*A
XA*Jv(I- fA)
XA* (Da-Db) (i- fA) dP
,-
LAA
JA
(18)
DA
where fA is the A* tracer correlation factor and VX A has been cancelled between numerator and denominator on the right. JA, has been evaluated as the fraction of the vacancy flux intercepted by uniform A* atoms per u~it time (-X**Jv) multiplied by the fraction of such contacts which are not random (1-fA), and therefore contributing to an A* flux in the uniform A* regions. This, in our view, is the--correct and very limited context of the so-called " ~ wind". Now, in view of dilute A*, LA, A << LAA , so from Eqs. 14, D a ffikTLAA~/NX A, and
LA,A . LAA, ffi k'TNXAXA,(Da_Db) (l_fA)
(19)
~*B
(20)
and similarly " ~B*
= ~N ~ , ( D b _ D a )
(l_fB)
Substituting these into Eq. 13 and comparin E with Eq. 7, we obtain v/~Vx A = v a
- ~
= (DA. - DB,)/?
(21)
where = XAf A + xBf s
(22)
Next, substituting Eq. 21 i n t o 19 and 20, and these into 12, we obtain
If we define the average tracer coefficlent as
= D,~XA + D~XB
(24)
then we can invoke detailed balance for a local equilibrium A*-B*-A-B solution in the form DAI (l-f A) = DB/(I-f B) = ~ / ( I - ~ )
(25)
where the right follows from the left equality. The latter, after cross-multiplylng and multip l y i n g by X A ~ , , s t a t e s that the probability of B *-vacancy exchanges associated with the vacancy wind produced in local A* motions (cf. Eq. 18) is equal to the probability of A*vacancy exch=,ges associated locally with B* motions since each trajectory set is the other~ inverse set. Vacancy binding effects are assumed as above to be included in fA and fB" Thus with simultaneous application of Eqs. 25. Eq. 23 can be written in general as
D/~ " X.BD~ + XAD ~ + XAXB(D~-DB*)2(1-'f)/fD with the intrinsic coefficients
(26)
56
DARKEN
DA= D~{[I +
DIFFUSION
EQUATIONS
Vol.
21, No.
XA(D~-D;I(1-"f'I/f'D] andDB = D;O~ + XB(D;-D~I(1-~')/~]
I
(27)
W i t h ~ = f, the tracer value in a pure material, and one of the defining conditions of Manning's random alloy model, these reduce to his well-known expressions. We emphasize, however, that subject to the aforementioned conditions on fA' fB' Eqs. 26 and 27 are exact and independent of the thermodynamic and kinetic model. In view of Manning's strong objection to our insistence upon a diagonal L-matrix in the A-B system it came as a surprise that our structure generates his corrections. We can only conclude that his interpretation of the L-matrices is different than the conventional one used here (cf. co--~nt of Lidlard quoted above). Transformation to the A-B-V Conserved Vacancy Experiment From Anthony's general phenomanological scheme for conserved vacancies (14) and JV = -(JA + JB )
(28)
we can rigorously obtain the ternary flux set with Vuv as an independent force
XA JA = -(LAA - ~
LAB)VUA + [LAA + LAB(I +
LAA + LAB ~ B + LAB JV ffi [ XA ~ ]XA VUA - [LAA + 2LAB + ~ B
)]VUV
(29)
+ ~ ( L B B + LAB)]V~V
(30)
and symmetrically
%, JB = - ( ~ B " XAA LAB)PUB + [~B + LAB(I + -'~') XA ]VuV
(31)
Since the vacancies are conserved, the lattice and laboratory frames are coincident. Now it is the conventional intelligence that the binary A-B Darken-Kirkendall structure as developed in previous sections can be obtained from Eqs. 29, 30 and 31 by setting V~V - 0 on the right. We will now take this to be the case, delaying the Justification to the following section. The key question is: can this transformation be reversed in general? The answer is clearly, no, for a diagonal 2x2 matrix cannot be disaggregated into a 2x2 non-diagonal matrix in the absence of rules or constraints. Such information must be provided by one or more new postulates, pertaining in this case to the A-B-V process (cf. Refs. 5, 8, 9, Ii). The present structure provides a particularly transparent and plausible route to a solution of the ternary problem provided the vacancy solution is dilute and henrian (no association). Under this circumstance, Eqs. 5 remain valid and from Eqs. 27, the intrinsic coefficient in the form
(321
D A ffiD A* + XA(D A - D;)(1-fA)/~ and Eq. 29 with VUv = 0, we can write
XA.
LAB N ~ "~
[DA-++ XAD+~ (1-fA) ~
* (1-fAI XADB _f ]
(33)
Now the ternary A-A*-V d i f f u s i o n problem with conserved vacancies has been solved exactly yielding the L-matrix (4, 15, 161 NXA*
(lf ] ; LA,A, = ~ LAA " NxA kT IDa%+ XAD A f-----l) and
NXAXA,DA LAA, ffiLA, A = kT(XA+XA,)
[DA + XA,D A*
]
( 34 )
(35)
where f is the tracer correlation factor. ~rom the form of the vacancy wind term of Eq. 18 in relation to a comparison of the second term in Eq. 33 and the second term of LAA in Eqs. 34, one concludes that both of the latter are to be uniquely associated with the interception of the conserved vacancy flux in these alternate experimental situations. Thus the ratio of the two correlatlun terms can reasonably 5e inferred as ~l-fA)f/¢l-flf , whence we can conclude that in the alloy case
Vol.
21, No.
1
DARKEN DIFFUSION EQUATIONS
LAA - - ~ - [ i
+ XA--~----] and ~ B
=
37
kT$ [l +
_ f
]
(36)
Furthermore, the last term in Eq. 33 must now be associated with the cross coefficient and therefore . N XA--D~ . B .(l-fA) . . N * (I-EB) (37) LAB ~ f and h A kT~ X A ~ D A f Now, applying Eqs. 25 simultaneously, we obtain
_ (z-fA) (l-f~) LAB = LBA
=
kT¢ XAXBD
"{ (l-f)
~
D~ ~
= kT#" XAXB
:
(38)
~-
Onsager Reciprocity appears on account of detailed balance or microscopic reversibility (12) as implied by Eqs. 25, and to complete the matrix,
LAA =
kT~b
[i +
---Df
"] ; LBB =
kT~b
[i +
__ Df
.]
(39)
Again, these have the same f o r m a s Manning's relations, where for the random alloy model, ~ -~ f. Because of our assumption of the absence of vacancy association it could be concluded that our L-matrlx expressed in terms of • does not have the generality of the expressions for v and D. However, on comparison of our result with that of Lidlard (ii), which claims generality, it is fair to conjecture that Eqs. 3~ and 39 are also general. .Diagnostics on Mechanism We now e~plore the Justification for the analytic continuation from a ternary conserved vacancy model to the binary local equilibrium model used in the previous section. Since we are assuming that within certain limits the Kirkendall drift is structure insensitive, we can illustrate the argument without loss in generality by considering that the creep process is effected via an appropriately dense array of parallel grain boundaries lying perpendicular to the diffusion direction. Now if the ternary formalism of the previous section is to agree with the binary description, the tlme-dependent forms of the unmodified equations 29 and 31 (V~V # 0) as applied to the perfect crystal regions must be somehow equivalent to Eqs. 1 with strictly equilibrated vacancies. Using our previous results, Eqs. 29 and 31 with internally conserved A, B and V transform with X v ÷ 0 to the local, low gradient approximation
~x2 and similarly for ~ ,
~
~x2
with
~xv. (~A-D~)* . ~2XA ~ ~2Xv ~t
T
~x2
~ X v ~x2
(41)
Now the local equilibrium condition can be applied to these equations in the strong form ~ / ~ t = ~2Xv/%x2 - 0 which implles the impossible result that ~XA/%t = 0. If, on the other hand, we apply the weaker condition, ~Xv/~ t = 0, allowing curved quasl-stationary vacancy profiles between defects, then we obtain the equations
~t
--
A
~
~)x2
; -~t
=
~
9x 2
(42)
To understand this quasl-steady result, which implies ~ -~XA, it is necessary to recognize that for one-dimenslonal diffusion in the absence of sources an~ sinks (Fig. 2) the eigenvalues of the D-matrlx in Eqs. 40 and 41 are (15,16) X 1 ---
~/fX V and ~2 ffiD~D~/~
(43)
38
DARKEN
DIFFUSION
EQUATIONS
Vol.
21,
No.
I
with A I >> A2, and it is these which establish the profiles, and therefore the gradients, near to and-far from the origin, respectively (15-17). If we assume for this structure that the vacancies exactly equilibrate at the grain boundaries then according to Eq. 41 with dXv/dt = 0 the curvature of the vacancy profile within a grain is given by
(44) which in proportion to its own average concentration is of the same magnitude as in X A or X n. We note flnally that subject to Eq. 44, V~v/VB A can everywhere be made as smell as we w i s h e r the boundary or defect density is sufficiently"increased. Since this is also the condition for a local equilibrium blnary structure based on Eqs. 1 with D a and ~ physically defined, we conclude that the analytic continuation of a ternary conserved vacancy structure to the binary structure via the first order constraint V~ V = 0 is ~alid. Note that the latter argument generalizes to three dimensional arrays of de£ects where ~2/~x2 ÷ V 2. Acknowledgements The author is grateful to Drs. D.J. Young of the University of N.S.W., A.B. Lidiard of Harwell, J.E. Lane in Melbourne, G.V. Kidson of W.N.R.E. and J.R. Manning of N.B.S. for many critical and helpful comments. Thanks are also due to K. Balasubramanian of McMaster University for helpful discussions and for checking soma of the calculations. References 1. 2. 3. 4. 5. 6. 7. 8. 9. i0. 11. 12. 13. 14. 15. 16. 17.
J. Bardeen and C. Herring, Atom Movements, ASM, Cleveland, 1951, p. 87. L.S. Darken, Trans. AIME, 175, 184 (1948). A.D. LeClaire, Prog. Metal Phys., ~, 265 (1953). R.E. Howard and A.B. Lidiard, Pep. Prog. Phys., 27, 161 (1964). J.R. Manning, b~ta Met., 15, 847 (1967). J.R. Manning, Diffusion Kinetics for Atoms in Crystal, D. Van Nostrand Inc., Princeton, N.J., 1968. K.P. Gurov, A.V. Nazarov and K.K. Anandybov, Fiz. Metal. Metalloved, 32, 176 (1971). Th. Heumann, J. Phys. F. ~, 1997 (1979). M.A. Dayananda, A~taMet., 29, 1151 (1981). J.S. Kirkaldy in Solute-Defect Interactlon - Theory and Experiment, Eds. S. Salmoto, G.R. Purdy and G.V. Kidson, Pergamon Press, Toronto~ 1986. Discussions by A.B. Lidlard and J~R. Y~,n~ng. A.B. Lidlard, Acta Y~t., submltted 1986. L. Onsager, Phys. Rev., 37, 405; 38~ 2265 (1931). L. Onsager, Ann. N.Y. Acad. Sci., 46, 241 (1945-46). T.R. Anthony in Diffusion in Solids, Ed. A.S. Nowick and J.J. Burton, Academic Press, New York, 1975. J.S. g/rkaldy, D.J. Young and J.E. Lane, Acta Met., in press, 1986. J.S. Kirkaldy and D.J. Young, Diffusion in the Condensed State, Institute of Metals, London, 1986, in press. J.S. Kirkaldy, Can. J. Phys. 36, 899 (1958).
XA~0
I I " i
Jv
Fig. i: Redistribution of tracer A* due to the "vacancy wlnd" JV created by the countercurrent flow of A and B at local equilibrium. Note that tracer B would flow in the same direction as A so that LBB * has the opposite sign to LAA, .
0 Fig. 2: Redistribution of initially ~miform conserved vacancies due to countercurrent A and B fluxes. Note how the back force of vacancies in the alloy diffusion zone symmetrlzes the_~A and s profiles, u - ¢ ~ ( D ~ V ~ / B ) t and V - ~,DVt •