An invariant formulation of multicomponent diffusion in crystals

Scripta METALLURGICA

Vol. 17, pp. 927-932, 1983 Printed in the U.S.A.

Pergamon Press Ltd. All rights reserved

AN INVARIANT FORMULATIONOF MULTICOMPONENT DIFFUSION IN CRYSTALS J. W. Cahn Center for Materials Science National Bureau of Standards Washington, DC 20234 F. C. Larch~ Universite de Montpellier 2 34060 Montpellier Cedex, France

(Received April 25, 1983)

Formulations of multicomponent diffusion equations relate diffusional fluxes to gradients in either con~)ositions or chemical potentials [ l - l O ] . All these quantities depend in some way on conventional choices. The magnitude of the fluxes depends on the frame of reference. Many coordinate systems have been proposed. For fluids they are a l l arbitrary. For solids there is a natural one based on the crystal l a t t i c e [1,2,5-7,10]. Nonetheless many others have been proposed as useful for special applications. For example, some frames are defined so that the fluxes of a l l the species sum to zero, or have such a condition as a consequence. Since fluxes can be defined in terms of number of molecules, volumes [4] or mass, zero summed fluxes would lead to several different coordinate systems depending on which is chosen. A constraint on the fluxes has the effect of making them interdependent. Such an i n t e r dependence is a feature of the gradients as well. Mole fractions sum to unity. Composition gradients, expressed in mole fractions per unit length sum to zero. Whenthe individual chemical potentials are defined t h e i r gradients are linked by the Gibbs-Duhem equation [3,10]. In an N con~oonent system, a linear relation between N fluxes and N gradients leads to N2 phenomenological coefficients that are not independent because the gradients are not independent and sometimes the fluxes are not either. A reduction of the system of equations into (N-I) equations linking N-l independent fluxes (or combinations of fluxes) to N-l independent gradients (or combinations of gradients) can be done in many ways, with varying results. We present here one more such formulation which grew out of a thermodynamic study. It incorporates some in~oortant thermodynamic properties that result from constraints that c r y s t a l l i n i t y imposes. Consistent with this c r y s t a l l i n i t y i t uses the l a t t i c e as a coordinate system. The resulting formulation showed some unexpected invariances to what appeared to be an awkward conventional choice. In the thermodynamic treatment for substitutional solid solutions chemical potentials of individual species are not definable I l l , 1 2 ] . T h i s reflects our i n a b i l i t y to insert a species substitutionally into the structure without taking something else out. This something else may be vacancies, which we have chosen to he a species. A quantity MjK akin to a chemical potential difference is definable. It corresponds to the free energy change for the insertion of a species J and removal of a species K from a site. At equilibrium the MjK are constant throughout the system. They obey the equation M +M +M =0 IJ JK KI

I,J,K = 1,2 . . . . N

(1)

Hence

M = 0 II

(2)

and

M = -M IJ JI

(3)

927 0036-9748/83 $3.00 + .00 Copyright (c) 1983 Pergamon Press Ltd.

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With these conditions only N-] of the MjK are independent. Each flux is defined relative to the l a t t i c e in terms of molecular number per unit time and per unit area in terms of l a t t i c e parameters. I f every diffusive iump is an exchange of atoms, rings of atoms, or atoms with vacancies these fluxes must sum to zero [lO],

o

I

: 0

(4)

Thus there are N-l independent fluxes. We now can define a set of phenomenological quantities BIJ(K) by the N equations - J I = J ~K BIJ(K)VMJK Because o f e q u a t i o n (2) we can remove the r e s t r i c t i o n

I = I ..... N

(5)

on the summation o v e r J

N

-Jl = a~l BId(K)vMJK

I = I ..... N

(6)

with BIK(K) as yet undefined. There are (N-l) gradients, a l l independent. The choice of the species to serve as K is arbitrary. It can be vacancies, or any one of the species. Had we chosen a different species, say L, the equation would have been -J

I

:

N Z B VM J=l IJ(L) JL

(7a)

What then is the relationship between the BIJ(K) and BId(L)? Substituting equation ( I ) into (6) we obtain (7b)

"JI = ZBIJ(K)VMjL - VMKLZBIJ(K) J J Since the gradients are independent we equate equations 7a and b term by term. BIJ(K ) = BIJ(L ) for all I,J,K,L

When JCK

(a)

Since K and L could be any species we may define a two index symbol Bid = BIJ(I ) = BId(2 ) . . . . = BIJ(N)

(9)

When J = K we obtain BIK(K) - ~BIJ(K) = BIK(L) which when combined with (9) becomes

Equation (4) implies

ZBIj =0 J

(lO)

ZBIj = 0

(ll)

I

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Instead of N3 BIJ(K~, there are only N2 BIj that do not depend on which species has been chosen as K. Furthermor~ ~quations (I0) and (11) impose 2N-I restrictions on the BIj. ~Because Z Z BIj = 0 there are not 2N independent equations). As a result there are (N-l) ~ independent IJ coefficients to be determined. Any N-I equations "JI =

ZBIjVMjK J

I = 1,2 . . . . . (N-l)

(12)

w i l l contain this number of parameters. I f they have been obtained either experimentally or theoretically the coefficients for any other choice of K are known. To i11ustrate this let us write two forms of the equations for a two-component alloy with vacancies (N=3). First choosing v for the K species we have -Jl = B11VMIv + B12VM2v -J2 = B21VM1v + B22VM2v -Jv = BvlVMIv + Bv2VM2v

(13)

Alternatively choosing 2 for the K species we have -Jl = Bli VM12 + BlvVMv2 -J2 = B21VM12 + B2vVMv2 -Jv = BvlVM12 + BvvVMv2

(14)

Note that because of equation ( I l l the fluxes sum to zero. To show that the flux expressions are the same we subtract the two expressions for J1 from each other 0 = B11V(MIv - M12) + BI2VM2v - BlvVMv2 Using equations (1), (3), and (I0) we see that this is an identity. The formulation is therefore invariant to the choice of the species K. It is also consistent with what we call the network constraint imposed by the l a t t i c e . The l a t t i c e is not only a natural coordinate system for the fluxes, but its thermodynamic consequences lead to the definitions of the MjK rather than individual chemical potentials. Two obvious candidates in solids for the species K are the major species and the vacancies. We prefer the major species which usually has the smallest uncertainty in its concentration. In fluids i t often has the smallest gradient in chemical potential. Vacancyconcentrations are usually not known at a l l . A common assumption that vacancy chemical potentials can be defined and be set to zero everywhere within a solid is based on assuming that unstressed vacancy sources and sinks are distributed within the solid on a scale small compared to diffusion distances. Under these conditions the Miv depend entirely on the concentrations of the chemical species. For many diffusion problems these assumption are not valid. Whenthey are valid choosing the vacancy as the K species has many advantages. Let us compare this formulation with one for fluids where the chemical potentials of each species is defined and measurable. We might write "JI = ZBIJV~J (15) J In equation (5) a l l gradients were independent. They could be varied individually and therefore each BIj could be uniquely determined, e.g., as the ratio of the flux of species I to the gradient of Mn~, when a l l the other qradients were held zero. The gradient of chemical potentials ~ n not be i n d i v i d u a l l y v a r i e d . Fromthe Gibbs-Duhem equation we obtain the relationship

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0 = ZCjvpj J

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(16)

We cannot single out one term in (15) by holding all other gradients zero. Experiments can only be performed on combinations of gradients that obey (16). EQuation (15) has meaning only for such combinations. There are many equations that reduce to equation (15) only when equation (16) holds. We can, for instance, a r b i t r a r i l y choose a set of constant ~I and rewrite equation

(15)[10]

-JI : Z(BIJ - xICj)vpJ J

(17)

Equation (17) is indistinguishable from equation (15) for the experimentally accessible gradients obeying (16). We can choose ~I to obtain particularly useful forms. I f we choose X T : ZBILIZCK L K

(18)

and define BIj = BIj- ~ICj = BIj - (Cj/ZCL)ZSIK L K

(19)

we obtain -Jl : ZBIjVUj J

(20)

and ZBIj : 0 J

(21)

As a result of equation (21) we can a r b i t r a r i l y choose one component K and rewrite equation (20) -JI = Z(BIjVUj) " VUKZBIj = ZBIjV(UJ" UK) (22) J J J Agren [ I 0 ] has chosen a particularly symmetrical form of (22)

-Ji = ZBIjV(p J- pl)

(23)

J but he unnecessarily set the diagonal terms in B to zero. Equations (21) and (22) are equivalent respectively to equations (lO) and (12), especially i f one notices that the (~j - ~ ) have the same properties as the MjK in equations ( I ) - ( 3 ) . Clearly the BIj in equati6n (2)) are the same as in equation (20) and cannot depend on our choice of K. The sum in equation (22) is over (N-I) independent gradients, the J = K term being zero. Equation (22) could be used as a starting point to derive equations (20) and (21) just as for solids. Equation (20) is of the same form as equation (15). We have replaced a restriction on the gradients (equation (16)) with a restriction on the coefficients (equation (21)). The effect of this restriction is that the flux really depends only on (N-I) gradients of the difference of chemical potential (equation (22)). There are alternative ways of dealing with the restriction of equation (16). We can solve for V~K and eliminate i t from equation (15) [5]. The same result is achieved by setting

~I = BIK/CK and obtain

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-JI = Z(BIJ - (Cj/CK)BIK)VUJ J

931

(24)

defining B'

IJ(K)

=

BIj

. (Cj/CK)

BIK

(25)

we obtain -JI = ZB'IJ(K)VUJ J with

B'

(26)

= 0

(27)

IK(K)

I t is remarkable that, while the B in a l l three equations (20), (22) and (23) are the same, the B' in equation (26) are different and depend on which species is taken for K. The d i f f e r ence is seen i f one compares equation (21) with (27). Because of (27), there is no term containing vuK in (26). Even though some terms in the series of equation (22) and (23) are i d e n t i c a l l y zero, i f we rearrange these equations and use (21) we obtain equation (20) which contains the gradients of every u. Because of equation (27) we have (N-l) independent gradients in equation (26). The gradient of PK given by V, K

: -(I/C ) Z C V, K JcK J J

does not contribute to the sum in (26). other terms.

(28)

The effect of this gradient is i m p l i c i t in all the

Unlike crystal|ine solids, there is no natural frame of reference for defining the fluxes in f l u i d s . I f one frame (denoted by primes) l o c a l l y moves relative to the other by a velocity V which may depend on position and time, the instantaneous local fluxes are related by JI , = JI _ VCI

(29)

I f we have determined the fluxes in some a r b i t r a r i l y chosen (primed) reference frame, we can always choose a local V given by j

i

v = -~ i/ZcI and find a moving reference frame in which equation (4) holds. equation (20) yields equation (11).

(30) In this new frame of reference

I f we expect Onsagers relationships to hold and i f we have chosen to formulate diffusion in a way that implies (21), i . e . , starting either with (22) or with (17) and (18), i t is necessary but not sufficient that we choose the frame given by (30). With any other frame (11) does not hold, implying that the Onsager relations can not hold. The Onsager relations can not hold for the formulation that leads to equations (24)-(27). Agren [ I 0 ] has postulated without proof that for an exchange mechanism (including exchange with vacancies) in the l a t t i c e frame the coefficients in equation (23) (and hence in (12), (20) and (22) as well) would be symmetrical. Proving that the Onsager r e l a t i o , s hold in multicomponent diffusion is d i f f i c u l t and s t i l l controversial [13-15]. The naive [16] method of choosing fluxes and forces from an expression for entropy production has long been known [15] not to guarantee a symmetric B matrix. The proof should be undertaken for the formulations in terms of the BIJ and a reference frame based on the crystal l a t t i c e i t s e l f for substitutional diffusion.

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The advantages of a formulation of diffusion having the demonstrated invariances is the f l e x i b i l i t y of alternate formulations. The species K can be a r b i t r a r i l y chosen without affecting the coefficients. This is in contrast to the commonly used formulation in equation (23) to (27). For fluids equation (22), or alternatively equation (20) with the restriction imposed by equation (21) unambiguously defines each BIj. On the other hand equation (15) with the restriction of equation (16) permits a wide range of sets of coefficients that are physically indistinguishable. This as we have seen can be used to advantage in derivations, but is awkward to use in solving diffusion problems. The equations (26) with the restriction in (27) is commonly used. I t suffers from the serious d i f f i c u l t y that the coefficients depend on which species is chosen for K. For substitutional diffusion in crystalline solids, the physical situation is much simpler. Our i n a b i l i t y to define meaningful individual chemical potentials leads us immediately to an invariant formulation. That the same formulation exists for f l u i d s , where there are alternate choices may give us more insight into the advantages.

Acknowledgment We are grateful to John R. Manning for continued encouragement and valuable c r i t i c i s m , and to Roland de Wit, Mats H i l l e r t and Didier de Fontaine for helpful discussions. References I. 2. 3. 4. 5. 6. 7. 8. 9. lO. 11. 12. 13. 14. 15. 16.

J. Bardeen and C. Herring, Atom Movements, ASM, p. 87, (1950). J. E. Lane and J. S. Kirkaldy, Can. J. Phys. 42, 1643 (1964). J. S. Kirkaldy and G. R. Purdy, Can. J. Phys. 40, 208 (1962). T. O. Ziebold and A. R. Cooper, Acta Met. 13, 465 (1965). Y. Adda and J. P h i l i b e r t , "La Diffusion dans les Solides," Presse Universitaires de France, Paris, p. 313ff and Apendix V, p. 351 (1966). J. R. Manning, Can. J. Phys. 46, 2633 (1968). J. R. Manning, "Diffusion Kinetics for Atoms in Crystals," D. van Nostrand, Princeton, p. 221 (1968). J. R. Manning, Met. Trans. l , 499 (1970). J. M. Sanchez and D. de Fontaine, "The Phenomenological Equations for Multicomponent Diffusion," Technical Report 7506, UCLA School of Engineering and Applied Science (1975). John Agren, J. Phys. Chem. Solids, 43, 421 (]982). F. Larch~ and J. W. Cahn, Acta Met. 21, I051 (1973), 26, 1579 (1978). F. Larch6 and J. W. Cahn, Acta Met. 30, 1835 (1982), B. D. Coleman and C. Truesdell, J. Chem. Phys. 33, 28 (1960). R. F. Sekerka and W. W. Mullins, J. Chem. Phys. 33, 28 (1980). W. W. Mullins and R. F. Sekerka, Scripta Met. 15, 29 (1981). S. R. de Groot, "Thermodynamics of Irreversible Processes," North Holland, Amsterdam p. 5-9 (1951).