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Correlation factor in tracer diffusion for high tracer concentrations
J. P&s. Ch. Mi& Ffint&inGrrrtI$itnin.
vol. 46, No. 12, pp. 13614370,
0022~349’1/85 t3.M) + .cKJ Q I985 ~PlmLod
1985
CORRELATION FACTOR IN TRACER DIFFUSION FOR HIGH TRACER CONCE~~TIONS HIROSHISATO,TAKUMAISHIKAWA~and RYOICHIKIKUCHI$ School of MaterialsEngineering,PurdueUniversity,WeatLafayette,IN 47907, U.S.A. (Received 26 March 1985; accepted 30 May 1985)
Al&act-The meaning of cross terms in the Onsagerequation for diffusion in rn~ti~rn~nent systems is &rifled basedon the PathProbabiIitymethod of irreversiblestatistic&mechanics.The probIemsinvoIved in increasingthe concentrationof traceratomsin tracerdiKusionexperimentsarediscw&. SOmecomments on the comlation function approachwith respectto the presentproblemare also added.
1. ~ODU~ON
intem~o~ between a pair of atoms are exactly the same and do not depend on the species of atoms (til The crosstermsin the Onsagerequations for diffision = 8 = s$. Under such a condition, the distribution of in multicomponent systemsare often a source of difficulties. A cross term indicates an effect of a finite the two kinds of atoms is completely random at any amount of flow of one specieson the Bow of another. temperature and the jump probability of atoms does The effect is most commonly discussed in terms of not depend on their local environments 14, 51. Alirreversiblethe~~~i~ [ 1,2]. However,in such though a treatment of mom general solid solution does not necessa@ introduce any additional difhculty, the a case, the Onsager matrix elements are not derived “ideal solution” we adopt here has a conceptual adfrom a microscopic point of view and, hence, are not vantage in that physical correlation phenomena [4] due explicitly given. Moreover, the Onsager equations are to mutual interactions between two kinds of atoms defined under the steady state condition and the Onneed not be taken into account in the interpretation. sager matrix elements are evaluated with respect to the As such an ideal solution, we deal with a case. in which ~~b~urn lotion of constituents at the reference the two constituents are each isotopes. Radioactive plane. Etecause of this, all the matrix elements include isotopes are oRen utilized as tracers to investigate the the effect of finite flows of constituents in a self-conmechanism of diflusion. However, the dependence of sistent fashion 133. It is rather difficult therefore to the tracer dilfusion coefficient on boundary conditions evaluate the contribution of individual Onsager matrix of measurement and on the density of tracer atoms elements (including cross terms) simply based on a has o&n been a matter of ~mroversy [ I]. In order to qualitative ~e~~y~i~ formalism, which often answer this question, it is necesmry to know the comleads to confusion in dealing with dil%sion phenomposition dependence of each Onsager matrix element ena. The advantages of the Path Probability method and their relative contributions to the tracer diffusion (PPM) of irreversible statistical mechanics when apcoefficient under given boundary conditions. plied to difhrsion problems are that the driving forces In evaluating diffusion coefficients for a variety of can be clearly identified and the Onsager matrix can experiments, an approach based on correlation funcbe analytically derived so that the relations among tions is often utilized 171.Thii is based on the concept measurable quantities are un~biguously unde~~ that, in the linear range of ditKrsion, the 810~ is the [3]. The purpose of the present treatment is to clarify time correlation of the velocity of particles being such confusing situations by evaluating the Onsager watched, evaluated under the equilibrium distribution matrix elements quantitatively in the simplest possible of particles. In most cases, the method is utilized to examples. In order to fir&l1 the above purpose, we deal with evaluate the drift motion of a single particle. However, diffusion problems in a two component system by a if the method is utilized to evahmte the dril? motion vacancy mechanism utilizing a pair approximation of of an assembly of a finite number of particles, diffithe Path Probability method. Further, we assume a culties similar to the evaluation of the Onsager matrix case of ideal solution, and the number of vacancies in elements are encountered. We elucidate these relations and answer these questions in an analytical fashion, the system is assumed to be negligibly small. The ideal In order to deal with transport problems by the PPM solution here is meant to be a solid solution in which satisfactorily, two conversion rehnions with respect to the averaging process have been found nv, the instantaneous distribution conversion process (3, 41 t Presentadd= PhysicsDepartment,Tokyo Instituteof and the time conversion process [3,4,8]. However, in Technology, Oh-Okayama, Meguro-ku, Tokyo, Japan. $ Permanentaddress:HughesResearchLaboratories,Mal- the case of treating an ideal solution, only the time ibu, CA 90265. conversion process is requited, and the application of 1361
H. NATO
1362
et al.
Here, we take any appropriately defined driving force of diffusion as cc. The advantage of the PPM over irreversible thermodynamics is that the PPM derive not only Lbs but also the expressions of cr,‘s in terms of microscopic (state) variables. Another important consequence of the PPM is that iuls can be any driving 2. DEFINITION OF THE CORRELATION FACTOR force of diffusion other than the chemical potential The concept of the correlation factor has been ex- gradient and that L;s are independent of the types of the driving force [ 13, 161. We refer to such properly tensively discussed for selfditlusion [9-l 11. However, defined driving forces such as the electric field gradient the extension of the definitions of the correlation factor for ionic motion [ 161as general&d chemical potential for general diffusion problems has not been necessarily gradients. This relation among diGsrent types of driving unique. forces allows us to compare flows properly under difThe correlation factor f was originally defined for ferent types of driving forces such as the derivation of the self-diffusion as the ratio of the tracer diffusion the Haven ratio. &‘s are evaluated with respect to the coefficient & and the selfdiffision coefficient Ds as equilibrium distribution and, hence, in terms of the [121 equilibrium state variables. Equilibrium state variables 0,=&f (1) are calculated by the Cluster Variation method (CVM) of equilibrium statistical mechanics (31. Vacancies serve only as the medium for the motion of particles. In the limit of nearly perfect crystals (small number of The reference system used in describing diffusion is vacancies), however, Ds is represented by the di&sion that of a laboratory frame fixed at a crystal lattice plane. coefficient 41, or the diffusion coefficient of the asFor ideal solutions, Lis are given, for the body centered sembly of particles making the random walk (or the cubic structure (with the coordination number 20 = 8), diffusion coefficient of a vacancy which makes the ranas dom walk in such a system). Therefore, the definition off; eqn (1), has been replaced for selfaifision by L,, = W,xdrl{l - 2w,-G/S}, [9-l I] the time conversion process to the present problem does not change the qualitative feature of the result. Therefore, here, we limit ourselves to the treatment by the original formalism of the PPM.
DT = &f:
(2)
In other words, the correlation factorfrepresents the efficiency of the motion of a tracer atom (or an assembly of tracer atoms) for a long distance diffusion with respect to the random walk motion of the same atom. Ordinarily, however, the correlation factor is calculated in the limit of negligible tracer concentrations 19-l 11. This definition off can then be extended uniquely to general diffusion problems, and was utilized in our earlier calculations off in binary alloys [13-l 51 and superionic conductors [ 16, 171. 3. ONSAGER EQUATION FOR DIFFUSION FOR BINARY IDEAL SOLUTIONS The Onsager equations for diffusion for two component systems can be written as
*p2= -L*,li, L2
=
-
L22iY2,
(3)
=
w2-%x2{l
-
2w2-wq,
(3
L,2 = L2, = 2ww2w,xdS, s = 2(x, + xz)(w, + w2) + 5(w,x, + w2xz).
These expressions arc derived from more general expressions, eqn (4.12) of Ref. [ 181, for example, by making
where Q’S indicate the interaction potentials between nearest neighboring i-j pair of atoms (condition for the ideal solution) and by making the temperature gradient zero. The choice of 0 for tii in eqn (6) does not introduce any deviation from generality in the present case [4, 51. The quantities ~0, x, and x2 indicate the concentration of vacancies, atoms of species 1 and species 2, respectively, and wI and wr are jump frequencies of species 1 and species 2, respectively defined as 14, 51
wi = (jie-8”’
(7)
L2,.
Here, apI and @2represent the flows of the two species, 1 and 2, respectively, and dr, and dl2indicate the driving forces for the two species, where dots indicate the derivatives of (Y,and a2 with respect to the space. The quantities a’s are given in terms of the generalized chemical potential p’s as Cri=p~i
L22
1 (B=~T.~=10r2).
(4)
For reference, some notations used here and in related papers [3-5, 181in deriving the Onsager equations are listed in Table 1. Note the inclusion of the second term in L,, and & in eqn (5) (2w&S or 2w2xJ.S) which is equal to the amount of LIZ = L2,. This is the effect of the induced flow on the diagonal elements of the Onsager matrix and is introduced by the requirement that the distribution of atoms is kept unchanged irrespective of the existence of a finite amount of flow. In order to show the situation, the derivation of the On-
1363
Correlation factor in tracer dithuion Table 1. Glossarv probability of having constituent i on a lattice site. In disordered alloys, q indicates the density of ith species probability of having a constituent i on a lattice she and a constituent j on a nearest neighbor lattice site interaction energy between a nearest neighbor pair of constituents i and j tfi + EBB - 2tm l/4 (c~ - cm); we assume ciV= 0 (kT)-‘; k is the Roltzmann constant; 2’the absolute temperature the concentration gradient of constituent i due to the existence of the concentration gradient B+ = K,, = exp(+r,,)
particle to move in a sequence of time, its motion cannot be independent of the motion of others. Because L;s are self-consistently determined with respect to the equilibrium distribution in the presence of finite amounts of flows, 0, and 9z, in the derivation of the Onsager equation, not only the cross terms, but also the diagonal terms LII and Lz2, are affected by the presence of +r and 4. These effects under various boundary conditions are what we would like to discuss here.
4. CORRELATION FAnOR UNDER VARIOUS BOUNDARY mNDlTIONS
Yij = @G,-’
= q/t
c
4J’
m-t
= @pi;represent the chemical potential of the ith snecies. a, is given bv ecn (A. 16a) of Ref. 131 coc&ination number of the lattice, ‘forbee, i~-= 8 flow of the ith species across a reference plane in the k direction (in the direction of the chemical potential gradient) and in the -x direction, respectively flow of the ith species in one direction under the equilibrium condition = &e-” jump frequency of ith species
In evaluating the correlation factor, we regard the second species of the constituents of the binary ideal solution as the tracer, and we can assume that w1 = wr (the instantaneous distribution conversion process is, therefore+ not required in this case [4]). The correlation factorfcan then be derived, after reducing a2 in eqn (3) to make it solely a function of h2, by [3, 81 *2
=
+2h'2xdc2
=
'%/(DR~z)
=
-f&2
@a)
OI
&/DR
(Y+i - Y-,)/Y, 6
ln
Y/LvryY, the symbol 6 refers to the deviation from YuYjvYwi the equilibrium value (Ref. [ 141)
sager equations for diffusion for the binary ideal solution [eqn (S)] is shown in detail in the Appendix, because this point has been a source of controversy. In Fig. 1, LII, L22 and L,z = Lz, (actually LI&,, etc.) for w,/wr = 1 are plotted as a function of composition (x, + x2 = 1). (Unless wI = w2, the instantaneous distribution conversion process is required to obtain correct results even in an ideal solution. However, in the approximation presented in Ref. [4], the conversion process does not atEit the results of ideally disordered solid solutions even if wI # w2.) An important point to be noted here is the appearance of the cross terms L12 = &, even in the absence of interactions among atoms. The contribution of the cross terms amounts to approximately 10% of the diagonal terms at the intermediate compositions. The existence of the cross terms in an ideal solution is often not recognized. The appearance of the cross terms in the absence of mutual interactions among constituent particles is, however, essentially due to the fact that atoms of the species 2 cannot jump into lattice sites which have been occupied by atoms of the species 1. This represents an implicit interaction between atoms of species 1 and species 2, appearing in the form of the time correlation (the sequence of motion) of motion of particles. For a
=
h.
(W
The value& = 1, therefore, indicates the random walk of the species 2. Note that the normalized flow, a2, in eqn (8) is defined in terms of the generalized chemical potential gradient rather than the concentration gradient. The basis for eqn (8) in relating LIr and h2 follows from the relation [ 131
when both species 1 and 2 are isotopes and distributed at random among themselves.
‘2
Y IA
06
x_
E 9
0.2 Ll2’
=
L2I
11
o.o= 0.0
0.2
0.4
0.6
0.8
1.0
COMPOSITION Fig.
1. Relative contribution of the cross term LL2= &, in ideally disordered systems (w, = ~2).
1364
H.
SATO
(1) &r = c& This condition corresponds, for example, to a case in which both the species 1 and 2 have the same electric charge, and an external electric field is applied. By measuring the current due to the species 2, @r can be obtained. From eqn (3), *pz= -L*,a, = -(La
- Lucuz +
L22b2.
(9)
Using the relations eqn (5) and the condition w, = w2, ‘k2 = ~pz/w&x~ = -{ 1 - 2wrx,/S + 2w,x,/S}&r = -&
(w, = wr)
(10)
and thus fi=
1.
The result is independent of the value of x2. In other words, if the same driving force is applied both to matrix atoms and to tracer atoms, the motion of the tracer atoms is a random walk same as the total assembly of atoms. This is a natural result, but is against a belief that the motion of tracer atoms is correlated. It should be remembered that, in order for this result to hold, the existence of the cross term is necessary. (2) &I = 0 The case (1) indicates that, in order to measure f2, it is necea%uy to apply a driving force only to the second species. This type of case would be realized, for example, if only the species 2 has an electric charge and an external electric field is applied as a driving force, and the ionic conductivity of the second species is measured. This type of case can also be real&d if the species 2 diffuses under its own concentration gradient as long as its concentration is extremely small, just as in ordinary tracer diffusion measurement. Under the condition of &, = 0 in eqn (3), a2 = -L&.
et al (20) as the expected value of the correlation factor [8]. That f2reaches 1 as x2 - 1 is quite natural if one thinks of the fact that, as x2 - 1, the diEusion of the species 2 approaches the self-diffusion. The result is shown in Fig. 2 as the curve (a). It would be worthwhile to mention here that information with respect to the correlation factor f2 is included in the expression of L22. Thermodynamically, the term L12 = L2, is commonly taken for the discussion of the correlation factor. It is true that Lt2 is the source of (time) conelation of isotope atoms with atoms of the matrix as discussed earlier, but this effect is reflected to the term L22 in determining the efficiency of motion of tracer atoms with respect to the random walk motion. In the limit of x2 = 0, at which the correlation factor is generally evaluated, L12 - 0 and, therefore, the Liz term itself is not directly related to the correlation factor. It is to be pointed out here that the evaluation of the correlation factor by the random walk theory is to evaluate the efficiency of the drifl motion of a single particle under equilibrium [9-l 11. The curve (a) also represents the relative change of the charge diffusion coefhcient DC(= akT/(ne2), where c is the ionic conductivity, k Boltzmann constant, T temperature, n the density of conduction ions per unit volume and e the charge of a conduction ion) with composition when only the species 2 has the charge. If the measurement of the Haven ratio HR = &/DC is carried out in such a mixture with a small addition of tracer ions of the species 2, HR tends to 1 in the limit of x2 - 0 (Fig. 3). This result, however, holds for more general cases, as can easily be seen in the foregoing arguments. If one disregards the effect of induced flows in the L22 term, which is equivalent to make L12 = 0 in the Onsager equation, the result is the curve (c) in Fig. 2.
(11)
In other words, this case is equivalent to evaluating L22. By definition, G2 represents the drift motion of particle(s) of the species 2 under equilibrium. Based on eqn (8), therefore, *2 = cp,l(w2w2) f2 =
= -11
- 2w2x*IS]~2,
1 - 2wzx,/s = 1 - (2/9)x,.
(12a)
In this case,& starts as 7/9 or (2~ - 1)/(2w + 1) and tends to 1 linearly as x2 is increased from 0 to 1. The value (2~ - 1)/(2w + 1) corresponds to the value of the correlation factor at x2 = 0 expected from the pair approximation of the original PPM [8]. After converting to long time averaging (the time conversion process [S]), however, the value becomes (2~ - 2)/
TRACER CONCENTRATION
Fig. 2. Dependence of the correlation factorh on the concentration of tracer atoms (species 2) calculated under dilTenznt boundary conditions. (a) Cut= 0 [case 2)]. (b) xl& = -x& [Gibbs-Duhem relation (case 3)]. (c)i, = 0, ~5~2 = 0 (neglecting the contribution of the f&back effectto L& (Appendix ii).
Correlation factor in tracer diffusion
1365
corresponds to the return probability P,), while Z2 is the probability that any other atom on the surrounding sites which forms the pair with the tagged atom of the species 2 across the vacancy jumps into the vacancy, and represents the escape probability of the vacancy P2 = (2~ - 1)Z2 or the probability of the forward jump of the atom of the species 2. The eqn ( 12b) in the limit ofxz - 0 f
=
’
I
OL
0
I
0.2
0.4 Composition
I
I
0.6 (X,)
0.6
I
(2w - l)Pe (2w 1P2 2+(2w-3)Z2=2P,+(2w-l)P,
(13)
thus indicates the efficiency of the forward motion or the drift motion of a single tagged atom under the equilibrium distribution at the time instant. In this respect, the calculation of L22 or the linear expansion
1.0
Fig. 3. Haven ratio for ions of the species 2 as a function of x2.
A small addition of tracer ions of the species 2 is assumed.
/0
The meaning of this curve is discussed in the later seo
tion, because this feedback effect in the diagonal term is not well recognized. For a future reference, a more general expression forfi in eqn ( 12a) is given as follows: According to eqn (A20), the most general expression forf2 is f
=
(2~
-
lXZ2
2
+
x2A)
+
2tw,/(w1
+
2 + (20 - 3)(Z, + x2A)
--_-
- --- Q
A
w2W2
’
(12b) where
x,wi
z2=c
j
w2
+
(i=
L2)
(12d
WJ
and
A.=w2_w2
+
WI w2
w,
+
w2
*
(124
Because wl = w2 in the present case, we have A = 0. Also, Z2 is equal to l/2 and eqn ( 12b) becomes identical with eqn ( 12a). In interpreting diffusion phenomena in terms of PPM, the function Z2 defined in (12~) plays an important role. In Fig. 4a, a vacancy, its surrounding atoms and a tagged atom of the species 2 are shown. This should be interpreted to be a time instant at which a tagged atom of the species 2 has just jumped out from the central site, replacing with a vacancy at its neighboring site. Therefore, the probability of the jumping back of the tagged atom into the vacancy it has just replaced is calculated here in competition with other atoms on the surrounding sites. The quantity xj specifies the species of atoms (the probability of finding an atom of the* species) on the surrounding site. As is clear from thii 6gure, the quantity 1 - Z2 represents the jump back probability of the tagged atom (which
Fig. 4. Definition of 22 and Z2 + {z in terms of escape prob. abilityP.andmtumprobabilityofatag@atomofthespef5es 2 (with *). White and hawhcd circle indicate atoms of the species I and 2, mrpectively. Figure Ya) repmsents a case where the number of atoms of the species 2 is negligibly small while (b) represents a case where the number of atoms of species 2 is large and the tagead atom cannot be disti~ from the rest of atoms of species 2.
H. SATOet al.
1366
coefficient of & is that of the drift motion of the assembly of isotope atoms. At a finite concentration of tracer atoms (xz # 0), fi becomes f
2
=
(2w
-
IV2
+
2IwMw2
+
( = -Lu(1
%)I%
2(1 - 2,) + (2w - l)Z,
*2 = -L2,
*
=
f2w
-
fW2
+
52)
2 + (2~ - 3)(Z2 + 5;) ’ 5;
=
x2w2 w2 + w2
(1%
&2.
(184
This equation can be rewritten into a well known relation as
This is further reduced, by introducing eqn (5) for Ltj's and using wI = ~2, to tir, = -{l
&
x2L21
X,
Xl L22
(14)
Although wl is numerically equal to w2, we distinguish w, and w2for the clarity of the ~ument. In analogous to eqn (13), eqn (14) represents the drift motion of a group of isotope atoms under the equilibrium distribution. If the ““feedback effect” is dismgarded, the expression for& becomes [eqn (A.23)]
- - & - L& XIJ
- 2w&s)&
(19)
or
q2+2Ap&2,
(20)
w
*
fi = (2w - 1)/(2w + I) This equation should be compared with eqn (13). In Fig. 4(b), a physical situation which represents 22 + S; is illustrated in comparison with that for Z2. It is clear that, analogous to that in eqn (13), this represents the drift motion of a group of isotope atoms at a time instant t from an initial equilibrium distribution, and the 5; term represents the indisting~shability among isotope atoms (see Appendix). The difference between eqns ( 14) and ( 15) represents that while eqn ( 15) rep resents the motion of atoms in which the equilibrium distribution is maintained before and after the jump, while in the latter the equilibrium distribution after the jump is not imposed.
(3) Gibbs-Duhem Relation The condition (2) cannot be utilized in a system in which the species 2 diffuses into an assembly of the species f under their con~n~tion gradient and their concentration is high, because the concentration gradient of the species 2 must create the concentration gradient of species 1 since condition xi + x2 = 1 and hence -i-, = -x2
(16)
must hold (under the assumption of negligible amount of vacancies). On the other hand, under the equilibrium condition, the Gibbs-Duhem relation XI& + x&2 = 0
(17)
should hold. Because cases close to the equilibrium condition are handled in diffusion, the Gibbs-&hem relation is often assumed to hold in the treatment of diffusion. Equation ( 15) relates ii to &. and 4pzin eqn (3) can then be solely expressed in terms of &, as
irrespective of the value of x2. Or, more generally, based on eqns (18a), (A.8) and (A.9) f
= 2
120
-
1X22
+
x24
2 + (20 - 3)(Z2 + x2A) *
(21)
Here, A is defined in eqn (12d). Because A = 0 in the present case, fi =
2
1”y2“;z2 W
(22)
follows. in other words, Gibbs-Duhem relation gives the identical result for fi as eqn (13) independent of the value of x2. The result is shown in Fig. 2 as the curve (b). It should be noted, however, that the concept of “local ~~lib~urn” should be limited within the plane perpendicular to the gradient and the Gibbs-Duhem relation, eqn (17), does not hold strictly in difhtsion phenomena. Rather, the Gibbs-Duhem relation should be treated as a given boundary condition that can ap proximately hold in such a case as interdif&ion. Under the Gib~~em relation, the diffision coefficient of a single isotope atom is measured while the evaluation of flow by means of b2 [eqn (1 l)] corresponds to the measurement of the diffusion coefhcient of the assembly of isotope atoms. The condition, eqn ( 17), requires that the motion of atoms of the species 2 should be compensated by the motion of species 1 atoms and that the ~~~bution of the species 1 and 2 as a whole should remain constant. In other words, the GibbsDuhem relation imposes a very strong requirement with respect to the driving forces Ly,and &s. Therefore, ifthere existsan independent driving force for diffusion of 1 and 2 species such as the vacancy concentration gradient as in a certain type of demixing problem, [ 19]
Correlation factor in tracer diffusion the imposition of the Gibbs-Duhem relation results in paradoxical results. On the other hand, the imposition of the Gibbs-Duhem relation to some type of treatment of difhtsion results in a great deal of simplification (41 and yet gives a correct value for the correlation factor (in the limit ofx, - 0). All our previous derivations [8- 171 of the correlation factor for various many body diffusion problems by the PPM were made under this boundary condition. 5. CORRELATION FUNCI’ION APPROACH In some experiments such as inelastic scattering of a neutron from liquids, the diffusion coefficient is evaluated in terms of the drift motion of particle(s) under equilibrium through the use of the correlation function based on the linear response theory. The flow (diffusion coefficient) is related to the velocity-velocity correlation of a tagged atom at time 0 and t evaluated under the equilibrium condition, (v(0) = v(t))=. For this purpose, we define the following hmctions which represent drift motions under equilibrium:
r
(‘*kfi
Vi(O) . Vj(t))Tdt,
in the previous sections in that both evaluate the flow in terms of the drift motion, as long as atoms of the species 2 correspond to tagged atoms in the correlation function. The correlation function approach, however, clearly shows that the flow is directly related to the time correlation of the motion of particles, and the time average has to be taken for the evaluation of the correlation factor, while in the original PPM, Lz2 is evaluated in terms of the ensemble average [8]. The functions 4k, c$’and &’are evaluated for lattices with two sublattices such as simple cubic or body centered cubic structure as [20]
Wd (27b)
(27~)
(24)
Here, vr indicates the velocity during flight for a jump diffusion process, and rf and r, indicate the time of flight and mean residence time in the jump dithtsion process. The correlation factors are then defined as
(25)
CW
(23)
4 = k
1367
i,j
4R =
(v(O)* W)ydt dm
and
f’=&_3”-l &2w+
1’
W-W
v,(O)& =+s+#d=&++dd,
(26a)
where
The value offcorresponds to that derived by the pair approximation of the original PPM andf’ to that after the time conversion [8]. The evaluation of 4 and hence the correlation factor for an assembly of tracer atoms
(26b) Here, we deal with a special case in which the number of vacancies is negligible and, hence, their motion is random. Suffixes used above are defined as follows: v = vacancy; T = tracer atom; p = particle (chemically similar tracer atom); NT = number of tracer atoms; N,, = number of particles; R = random; GCE = only consecutive exchanges of T and v, i.e., v on the next nearest neighboring site never returns to tracer atoms; s = self; and d = distinct. Then 4’ indicates the drift motion of a single tagged atom, & the same quantity if only consecutive exchanges of T and v are considered, 4 the same quantity for an assembly of NT tracer atoms, 4, = 4’ the self-correlation function and & indicates that if the correlation between only different atoms is considered. The evaluation of the functions 4, = #, & and r$ corresponds to that of & for corresponding cases
is what we are concerned here. Equations (28a), (b) and (c) are equivalent to eqn (2) in terms of Dr and &. The relation between g’s and the diffusion coefficient is given by (vfTf)Z
6(rr + rr&
z-z 1: 6 - tf
(29) Here Ir = r+rr corresponds to the jump distance and 1/tf (tf = Tf + Tm) corresponds to the jump frequency in the random walk theory. The relation between Dr, and +R in eqn (26) is based on the linear response theory that, in the linear range, the flow is determined by the drift motion evaluated under the equilibrium condition times the driving force.
H. SATOet al.
1368
In the derivation of 4 in eqn (27c), each jump is treated equally Born the equilibrium distribution to a nearest neighboring vacancy. Therefore, it is clear that the evaluation of the drift motion eqn (27~) is equivalent to that in eqn (13). On the other hand, the evaluation of Q in eqn (25) is complicated, and except for the fact that 4 + 4~ as Nr - Np, no reliable direct derivation of this quantity has been found. However, in view of the correspondence of these functions and h2 in the previous section,
3 = k
$
(“?
v,(O) * v,(t))T,-df,
(30)
,J
which corresponds to + in eqn (25), can be calculated by the original PPM. However, a problem arises in that whether the feedback effect should be included. Because the drift motion of a time instant from the equilibrium distribution is calculated in the evaluation of 4 in eqn (30), it is tempting to use eqn ( 15) without the feedback effect. However, because the time correlation effect is involved in deriving the correlation functions, and because each jump is treated as a jump from the equilibrium distribution in the derivation of & in eqn (27c), it should be understood that the feedback effect to restore the equilibrium distribution at each jump is imposed in the calculation. In other words eqn (12b) rather than eqn ( 15) should be used for the evaluation of $. In the limit x1 = 1 or x2 = 1, the feedback effect does not enter and their derivations are far simpler. 6. SUMMARY The meaning of cross terms in tbe Onsager equation for diffusion has been examined in detail based on the pair approximation of the PPM of irreversible stat&&l mechanics. Such effects had been examined based on the irreversible thermodynamics. Because the cross terms (or the elements of the Onsager matrix in general) cannot be derived horn the atomistic model by irreversible thermodynamics, the microscopic mechanism for the appearance of cross terms has never been clearly understood earlier. Examinations in the present paper are based on binary ideal solutions so that effects due to mutual interactions among particles can be eliminated to clarify the key concept of the correlation factor. The present paper clearly shows that nonnegligible cross terms exist even under the condition of ideal solution. The appearance of the cross terms under the condition of ideal solution is due to the fact that atoms of the species 2 cannot jump into lattice sites which have been occupied by atoms of the species 1. Depending on the boundary conditions imposed in actual measurements, relative contribution of cross terms and diagonal terms to the flow varies. Such a dependence of flow or the diffusion coefficient on boundary conditions is shown for several typical cases. In particular, the houndary condition (2) or the evaluation of J& corresponds to the evaluation of diffision coefficient from the drift motion of an assembly of atoms under
equilibrium. The boundary condition (3) or the imposition of the Gibbs-Duhem relation corresponds to the evaluation of the dill&ion coefficient under the condition that the relation of interditlWon between isotope atoms and matrix atoms holds strictly and corresponds to the evaluation of the motion of individual atoms. The two conditions, therefore, give the identical result in the limit x2 - 0. The existence of the feedback effect in the diagonal elements of the Onsager matrix is not well recognixed and is often disregarded. The effect is essentially due to finite values of the cross terms and represents the effect of restoring the equilibrium distribution after each jump of atoms. This contibution in ideal solution amounts to ~10% of the diagonal terms in the intermediate composition range. A derivation of the feedback effect is shown for the case of a binary ideal solution in Appendix. Acknowledgements-The content of the present paper is mainly a result of discussions of one of the authors (H.S.) with Professor H. .Sclnnalxried and Professor K. Fur&e of Institut Eir Physikalische Chemie und Elektrochemie, Universitit Hannover during his stay there in I98 1. He thanks them for their valuable mggestions and their ho&a&. He also thanks the Humboklt~Foundation in West Germany for the generous support which made his stay at the Institut possible. Discussions with Dr. M. Kabumgi and the help of Mr. S. A. Akbar in pmparmgthemanusciiptarehighlyappre&ed. The work was supported by the United States Department of Energy under grant number DE-EGO2-84ER45 133. REFERENCES
1. Albright J. G., J. Phys. Chem. 72, 11 (1968). 2. Home F. H., In Physics of Superionic Conductors and Electrode Materials (Edited by J. W. Perram), NATO AS1 Series, Series B, Vol. 92, p. 257. Plenum Press, New York (1983). 3. Sato H., In Nontraditional Methods in Dt@kon (Edited by G. E. Murch, H. K. Bimbaum, and J. R. Cost), pp.
203-235. TMSAIME Conference Pmceedings Volume (1984). 4. Sato H., Akbar S. A. and Murch G. E., Dtjusion in Solids: Recent Developments (Edited by G. E. Murch and M. A. Dayananda), pp. 67-95. TMS-AIME Conference Proceedings Volume (1985). 5. Wada K., Suzuki A., Sato H. and Kikuchi R., J. Phys. Chem. Solids, 46, 1 I95 (1985). 6. Sato H. and Kikuchi R.. Mass Transport Phenomena in Ceramics, Materials Scknce Research (Edited by A. R. g and A. Heuer), p. 149. Plenum Press, New York I. Egelstalf P. A., An Introduction to the Liquid State. Academic Press, New York/London (1967). 8. Sato H. and Kikuchi R., Phys. Rev. B 28,648 (1983). 9. Manning J. R., Di$usion Kinetics forAtoms in Crystal. Van Nostrand, New York (1968). 10. L&aim A. D., In Physical Cherrustry,Advanced Treatise, Vol. X, Solid State (Edited by H. E&g, D. Henderson, and W. Jest), D. 281. Academic Press New York (1970). 11. Peterson N. L., In Solid State Phystcs, Vol. 22 (Ed&I by F. Seitz, D. Turdbull, and H. Ehrenreich), p. 409. Academic Press, New York/London ( 1968). 12. Bardeen J. and Herring C., In Atomic Movements, p. 57. ASM, Cleveland ( 195 1). 13. Kikuchi R. and S&J H., J. Chem. Phys. 51, 161 (1969). 14. Kikuchi R and Sato H., J. Chem. Phys. 53,2702 ( 1970). 15. Kikuchi R. and Sato H., J. Chem. Phys. 57,4962 (1972).
1369
Correlation factor in tracer ditfusion 16. Sato H. and Kikuchi R., J. Chem. Phys. 55,677 (1971). 17. Kikuchi R. and Sato H., J. Chem. Phys. 55,702 (197 1). 18. Kikuchi R., Ishikawa T. and Sato H., Physica A 123,227 (1984). 19. Ishikawa T., Sato H., Kikuchi R. and Akbar S. A., J. Am. Ceramic Sot. 68, 1 (1985). 20. Funke K., private communication. The expressions for functions, eqns (23)-(26), are due to him.
Q,*,
statements made. in the text. In Refs. [31and [81,the derivation of the Onsager equation for ditfusion under the steady state. condition was made for three component systems consisting of A, B and B* (B* being the isotope of B) by the original PPM. The instantaneous distribution conversion process is not taken into account here. The present treatment corresponds simply to the case for x* (the composition of A) = 0, but with mutual interactions among atoms. Explanations of symbols used are given in Table 1. For ideal solutions, the relations G, + wi and Qi + Xi follow in the present approximation. (i) Derivation of the Onsager matrix elements L,. The equation of llow (Ref. [3], eqn (A. 15)) for a bii system derived by the pair approximation PPM in the linear range is given by ‘I’I = -h, +
Qzrt21,
92
Qd12,
=
42
#2, = -*12,
+
Al
=
=-
Q2*2
=
-QI&
{
2
I
+(I
-R)-
/
ix,
W’,l = ~21~221+
a2 + (1 - RP’21,
64.6)
Q2b2.
4Q2
---2 6,
+
$2
.
4Q, $1 + $2
dr,+(1
-R)acGz. I
2
64.7)
For &, an equivalent expression is obtained Because 0, = q,ti,QQ, and 4 = ~2G22QQ2, the Onsager matrix elements are obtained from eqn (A.7) and the equivalent expression for *r,
Il-%&m. I+(,_R)_)G,P,_ 6, + 4
(A.2)
-
1
1-a
-
Here, ei is the normalized flow delined in eqn (8) for the species i, ir, is the chemical potential gradient, Q, indicates the quantity mlated to the probability of finding the ith species on the neamst neighboring site of a specitled site, &, is called the deviation quantity and is antisymmetric with respect to the exchange of i and j. (Both i and j am used to specify the species) Without the existence of the second term on the right hand side, eqn (A. 1) indicates the random walk of particles of the ith species under the driving force &, and the second term indicates the effect of flows of other species which is determined by the nature of the diffusion path The quantity Jj, is proportional to the driving force &jin the linear approximation [3]. The steady state condition that the distribution of atoms does not change with time (eqn (A.20) of Ref. [3]) leads to a relation between #,, and tiU %NJ,z + & + (1 -
+
By substituting (A.6) into (A.5), we obtain
(A.1)
9422= 0.
-R)s.
From (A. I) and (A.2), on the other hand,
APPENDIX
The derivation of the Onsager equationfor difksion for two component systems The purpose of this section is to provide background to the
-R)s+(l
-(I
kc,+ $2
-L”= {, _d%!-+__+)}QoQ”l’ -(,
(A.8a)
(A.9a) -J522
=
(A.8b)
-L,,(@,
-L,s = -L,2(e)
= -L2,.
(A.9b)
The expression for Lzz is then from eqn (A.8b)
_
1
(A.3)
_a G, + 4
+(,
-@?&
$1 + 4
-&‘={, -+!%&-+~)}BQ2~2~
R - 2/(2w - 1).
-(,
Hem, G,‘s are jump frequencies including the bond breaking effect as defined in Table 1. From eqn (A.3), we get
(A. 10) Therefore, from eqn ( 12a)
$12 = &
K~22i22
-
%k)
+
(1
-
RX&z*2
-
%U,)l.
(A.4) Therefore, &, can be eliminated from eqn (A. l), and 8, = -&, - - GQ2 a2 + - GQ2 ci, 6, + G2 I& + 4 -(I
-R)m+(l
I
-R)$$, 2
I
2
(A.3
GzQ, -=-+6, + +
QQ, 9, + $2
62Q2 -a 6, + $2
(A.12)
1370
H. SATOet al. 1
and, by defining Z, as [eqn (12c)], z2
=
” Q,w -+$2 + $1
(2~ - l)(Z, + Q*A) + 2 a =
Q24
(2
(20
-
3)(z2
+
2
Q2k))
(A. 19b) ’
62 + 4 (A.‘3)
G2Q1 -+-= 6, + $2
+
+2Q2 4 + 4
1 -z2
(A.14)
(ii) Feedback efh in L2. The feedback effect in LX is evaluated as follows. In eqn (A.5) qr2= --h2 - - *IQ1 iu, + - *2Q, &2 G, f 62 4, + $2
we obtain 4Q, 6, + $2
-=
1-zz-52,
&here (A.16) Therefore *2Q, _+
%Qz _ l-z2-~2+*
G, + $2
56, + $2
_ (1 -R) E2
(AX)
(A.20)
+ (1 -R) E.
We first impose the condition iy, = 0. Under this condition, the term including 9, in the right hand side in eqn (A.20) then represents tbe induced current of the species 1 by the existence of a2 which would create the feedback effect in &. By eliminating this team, eqn (A.20) becomes
.
I
[ 1-(1-R)*
-
-a2
2
I
= 1 - Z2 - Q2A2,
2
G2Q1
lo, +
$2 I
&y1. (A.21)
(A.17) By using eqn (A. 154,eqn (A.2 1) then becomes
where (A. 18)
‘=-
-Ra+[l
(2w - lXZ2 + i-2) 2 + (2~ - 3)(& + f2;).
-(Z,+QzA)]
{l -(l’-R)(l-(Zz+aA)]]
-(20-
l)-2&
(A.23)
(A. 19a)
Or
A=-
(A.22)
and, hence,
By introducing & and A2 into eqn (A. 11) -1
xP2[1 - (1 - R)(1 - Z, - 1;)] = -(Z, + si)dlz
+ (2~ - l)[ 1 - (z, + Q2)l
((2~ - 1) _’ (2~ L 3)[1 - (Z, + Q2A)]}
Here, Z2 + c2 represents the escape probability under the equilibrium condition of a taggaJ atom of the species 2 including the indistinguishability among atoms of the species 2. The di%rence between eqn (12b) and eqn (A.23) then rep resents the f&back effect.
vol. 46, No. 12, pp. 13614370,
0022~349’1/85 t3.M) + .cKJ Q I985 ~PlmLod
1985
CORRELATION FACTOR IN TRACER DIFFUSION FOR HIGH TRACER CONCE~~TIONS HIROSHISATO,TAKUMAISHIKAWA~and RYOICHIKIKUCHI$ School of MaterialsEngineering,PurdueUniversity,WeatLafayette,IN 47907, U.S.A. (Received 26 March 1985; accepted 30 May 1985)
Al&act-The meaning of cross terms in the Onsagerequation for diffusion in rn~ti~rn~nent systems is &rifled basedon the PathProbabiIitymethod of irreversiblestatistic&mechanics.The probIemsinvoIved in increasingthe concentrationof traceratomsin tracerdiKusionexperimentsarediscw&. SOmecomments on the comlation function approachwith respectto the presentproblemare also added.
1. ~ODU~ON
intem~o~ between a pair of atoms are exactly the same and do not depend on the species of atoms (til The crosstermsin the Onsagerequations for diffision = 8 = s$. Under such a condition, the distribution of in multicomponent systemsare often a source of difficulties. A cross term indicates an effect of a finite the two kinds of atoms is completely random at any amount of flow of one specieson the Bow of another. temperature and the jump probability of atoms does The effect is most commonly discussed in terms of not depend on their local environments 14, 51. Alirreversiblethe~~~i~ [ 1,2]. However,in such though a treatment of mom general solid solution does not necessa@ introduce any additional difhculty, the a case, the Onsager matrix elements are not derived “ideal solution” we adopt here has a conceptual adfrom a microscopic point of view and, hence, are not vantage in that physical correlation phenomena [4] due explicitly given. Moreover, the Onsager equations are to mutual interactions between two kinds of atoms defined under the steady state condition and the Onneed not be taken into account in the interpretation. sager matrix elements are evaluated with respect to the As such an ideal solution, we deal with a case. in which ~~b~urn lotion of constituents at the reference the two constituents are each isotopes. Radioactive plane. Etecause of this, all the matrix elements include isotopes are oRen utilized as tracers to investigate the the effect of finite flows of constituents in a self-conmechanism of diflusion. However, the dependence of sistent fashion 133. It is rather difficult therefore to the tracer dilfusion coefficient on boundary conditions evaluate the contribution of individual Onsager matrix of measurement and on the density of tracer atoms elements (including cross terms) simply based on a has o&n been a matter of ~mroversy [ I]. In order to qualitative ~e~~y~i~ formalism, which often answer this question, it is necesmry to know the comleads to confusion in dealing with dil%sion phenomposition dependence of each Onsager matrix element ena. The advantages of the Path Probability method and their relative contributions to the tracer diffusion (PPM) of irreversible statistical mechanics when apcoefficient under given boundary conditions. plied to difhrsion problems are that the driving forces In evaluating diffusion coefficients for a variety of can be clearly identified and the Onsager matrix can experiments, an approach based on correlation funcbe analytically derived so that the relations among tions is often utilized 171.Thii is based on the concept measurable quantities are un~biguously unde~~ that, in the linear range of ditKrsion, the 810~ is the [3]. The purpose of the present treatment is to clarify time correlation of the velocity of particles being such confusing situations by evaluating the Onsager watched, evaluated under the equilibrium distribution matrix elements quantitatively in the simplest possible of particles. In most cases, the method is utilized to examples. In order to fir&l1 the above purpose, we deal with evaluate the drift motion of a single particle. However, diffusion problems in a two component system by a if the method is utilized to evahmte the dril? motion vacancy mechanism utilizing a pair approximation of of an assembly of a finite number of particles, diffithe Path Probability method. Further, we assume a culties similar to the evaluation of the Onsager matrix case of ideal solution, and the number of vacancies in elements are encountered. We elucidate these relations and answer these questions in an analytical fashion, the system is assumed to be negligibly small. The ideal In order to deal with transport problems by the PPM solution here is meant to be a solid solution in which satisfactorily, two conversion rehnions with respect to the averaging process have been found nv, the instantaneous distribution conversion process (3, 41 t Presentadd= PhysicsDepartment,Tokyo Instituteof and the time conversion process [3,4,8]. However, in Technology, Oh-Okayama, Meguro-ku, Tokyo, Japan. $ Permanentaddress:HughesResearchLaboratories,Mal- the case of treating an ideal solution, only the time ibu, CA 90265. conversion process is requited, and the application of 1361
H. NATO
1362
et al.
Here, we take any appropriately defined driving force of diffusion as cc. The advantage of the PPM over irreversible thermodynamics is that the PPM derive not only Lbs but also the expressions of cr,‘s in terms of microscopic (state) variables. Another important consequence of the PPM is that iuls can be any driving 2. DEFINITION OF THE CORRELATION FACTOR force of diffusion other than the chemical potential The concept of the correlation factor has been ex- gradient and that L;s are independent of the types of the driving force [ 13, 161. We refer to such properly tensively discussed for selfditlusion [9-l 11. However, defined driving forces such as the electric field gradient the extension of the definitions of the correlation factor for ionic motion [ 161as general&d chemical potential for general diffusion problems has not been necessarily gradients. This relation among diGsrent types of driving unique. forces allows us to compare flows properly under difThe correlation factor f was originally defined for ferent types of driving forces such as the derivation of the self-diffusion as the ratio of the tracer diffusion the Haven ratio. &‘s are evaluated with respect to the coefficient & and the selfdiffision coefficient Ds as equilibrium distribution and, hence, in terms of the [121 equilibrium state variables. Equilibrium state variables 0,=&f (1) are calculated by the Cluster Variation method (CVM) of equilibrium statistical mechanics (31. Vacancies serve only as the medium for the motion of particles. In the limit of nearly perfect crystals (small number of The reference system used in describing diffusion is vacancies), however, Ds is represented by the di&sion that of a laboratory frame fixed at a crystal lattice plane. coefficient 41, or the diffusion coefficient of the asFor ideal solutions, Lis are given, for the body centered sembly of particles making the random walk (or the cubic structure (with the coordination number 20 = 8), diffusion coefficient of a vacancy which makes the ranas dom walk in such a system). Therefore, the definition off; eqn (1), has been replaced for selfaifision by L,, = W,xdrl{l - 2w,-G/S}, [9-l I] the time conversion process to the present problem does not change the qualitative feature of the result. Therefore, here, we limit ourselves to the treatment by the original formalism of the PPM.
DT = &f:
(2)
In other words, the correlation factorfrepresents the efficiency of the motion of a tracer atom (or an assembly of tracer atoms) for a long distance diffusion with respect to the random walk motion of the same atom. Ordinarily, however, the correlation factor is calculated in the limit of negligible tracer concentrations 19-l 11. This definition off can then be extended uniquely to general diffusion problems, and was utilized in our earlier calculations off in binary alloys [13-l 51 and superionic conductors [ 16, 171. 3. ONSAGER EQUATION FOR DIFFUSION FOR BINARY IDEAL SOLUTIONS The Onsager equations for diffusion for two component systems can be written as
*p2= -L*,li, L2
=
-
L22iY2,
(3)
=
w2-%x2{l
-
2w2-wq,
(3
L,2 = L2, = 2ww2w,xdS, s = 2(x, + xz)(w, + w2) + 5(w,x, + w2xz).
These expressions arc derived from more general expressions, eqn (4.12) of Ref. [ 181, for example, by making
where Q’S indicate the interaction potentials between nearest neighboring i-j pair of atoms (condition for the ideal solution) and by making the temperature gradient zero. The choice of 0 for tii in eqn (6) does not introduce any deviation from generality in the present case [4, 51. The quantities ~0, x, and x2 indicate the concentration of vacancies, atoms of species 1 and species 2, respectively, and wI and wr are jump frequencies of species 1 and species 2, respectively defined as 14, 51
wi = (jie-8”’
(7)
L2,.
Here, apI and @2represent the flows of the two species, 1 and 2, respectively, and dr, and dl2indicate the driving forces for the two species, where dots indicate the derivatives of (Y,and a2 with respect to the space. The quantities a’s are given in terms of the generalized chemical potential p’s as Cri=p~i
L22
1 (B=~T.~=10r2).
(4)
For reference, some notations used here and in related papers [3-5, 181in deriving the Onsager equations are listed in Table 1. Note the inclusion of the second term in L,, and & in eqn (5) (2w&S or 2w2xJ.S) which is equal to the amount of LIZ = L2,. This is the effect of the induced flow on the diagonal elements of the Onsager matrix and is introduced by the requirement that the distribution of atoms is kept unchanged irrespective of the existence of a finite amount of flow. In order to show the situation, the derivation of the On-
1363
Correlation factor in tracer dithuion Table 1. Glossarv probability of having constituent i on a lattice site. In disordered alloys, q indicates the density of ith species probability of having a constituent i on a lattice she and a constituent j on a nearest neighbor lattice site interaction energy between a nearest neighbor pair of constituents i and j tfi + EBB - 2tm l/4 (c~ - cm); we assume ciV= 0 (kT)-‘; k is the Roltzmann constant; 2’the absolute temperature the concentration gradient of constituent i due to the existence of the concentration gradient B+ = K,, = exp(+r,,)
particle to move in a sequence of time, its motion cannot be independent of the motion of others. Because L;s are self-consistently determined with respect to the equilibrium distribution in the presence of finite amounts of flows, 0, and 9z, in the derivation of the Onsager equation, not only the cross terms, but also the diagonal terms LII and Lz2, are affected by the presence of +r and 4. These effects under various boundary conditions are what we would like to discuss here.
4. CORRELATION FAnOR UNDER VARIOUS BOUNDARY mNDlTIONS
Yij = @G,-’
= q/t
c
4J’
m-t
= @pi;represent the chemical potential of the ith snecies. a, is given bv ecn (A. 16a) of Ref. 131 coc&ination number of the lattice, ‘forbee, i~-= 8 flow of the ith species across a reference plane in the k direction (in the direction of the chemical potential gradient) and in the -x direction, respectively flow of the ith species in one direction under the equilibrium condition = &e-” jump frequency of ith species
In evaluating the correlation factor, we regard the second species of the constituents of the binary ideal solution as the tracer, and we can assume that w1 = wr (the instantaneous distribution conversion process is, therefore+ not required in this case [4]). The correlation factorfcan then be derived, after reducing a2 in eqn (3) to make it solely a function of h2, by [3, 81 *2
=
+2h'2xdc2
=
'%/(DR~z)
=
-f&2
@a)
OI
&/DR
(Y+i - Y-,)/Y, 6
ln
Y/LvryY, the symbol 6 refers to the deviation from YuYjvYwi the equilibrium value (Ref. [ 141)
sager equations for diffusion for the binary ideal solution [eqn (S)] is shown in detail in the Appendix, because this point has been a source of controversy. In Fig. 1, LII, L22 and L,z = Lz, (actually LI&,, etc.) for w,/wr = 1 are plotted as a function of composition (x, + x2 = 1). (Unless wI = w2, the instantaneous distribution conversion process is required to obtain correct results even in an ideal solution. However, in the approximation presented in Ref. [4], the conversion process does not atEit the results of ideally disordered solid solutions even if wI # w2.) An important point to be noted here is the appearance of the cross terms L12 = &, even in the absence of interactions among atoms. The contribution of the cross terms amounts to approximately 10% of the diagonal terms at the intermediate compositions. The existence of the cross terms in an ideal solution is often not recognized. The appearance of the cross terms in the absence of mutual interactions among constituent particles is, however, essentially due to the fact that atoms of the species 2 cannot jump into lattice sites which have been occupied by atoms of the species 1. This represents an implicit interaction between atoms of species 1 and species 2, appearing in the form of the time correlation (the sequence of motion) of motion of particles. For a
=
h.
(W
The value& = 1, therefore, indicates the random walk of the species 2. Note that the normalized flow, a2, in eqn (8) is defined in terms of the generalized chemical potential gradient rather than the concentration gradient. The basis for eqn (8) in relating LIr and h2 follows from the relation [ 131
when both species 1 and 2 are isotopes and distributed at random among themselves.
‘2
Y IA
06
x_
E 9
0.2 Ll2’
=
L2I
11
o.o= 0.0
0.2
0.4
0.6
0.8
1.0
COMPOSITION Fig.
1. Relative contribution of the cross term LL2= &, in ideally disordered systems (w, = ~2).
1364
H.
SATO
(1) &r = c& This condition corresponds, for example, to a case in which both the species 1 and 2 have the same electric charge, and an external electric field is applied. By measuring the current due to the species 2, @r can be obtained. From eqn (3), *pz= -L*,a, = -(La
- Lucuz +
L22b2.
(9)
Using the relations eqn (5) and the condition w, = w2, ‘k2 = ~pz/w&x~ = -{ 1 - 2wrx,/S + 2w,x,/S}&r = -&
(w, = wr)
(10)
and thus fi=
1.
The result is independent of the value of x2. In other words, if the same driving force is applied both to matrix atoms and to tracer atoms, the motion of the tracer atoms is a random walk same as the total assembly of atoms. This is a natural result, but is against a belief that the motion of tracer atoms is correlated. It should be remembered that, in order for this result to hold, the existence of the cross term is necessary. (2) &I = 0 The case (1) indicates that, in order to measure f2, it is necea%uy to apply a driving force only to the second species. This type of case would be realized, for example, if only the species 2 has an electric charge and an external electric field is applied as a driving force, and the ionic conductivity of the second species is measured. This type of case can also be real&d if the species 2 diffuses under its own concentration gradient as long as its concentration is extremely small, just as in ordinary tracer diffusion measurement. Under the condition of &, = 0 in eqn (3), a2 = -L&.
et al (20) as the expected value of the correlation factor [8]. That f2reaches 1 as x2 - 1 is quite natural if one thinks of the fact that, as x2 - 1, the diEusion of the species 2 approaches the self-diffusion. The result is shown in Fig. 2 as the curve (a). It would be worthwhile to mention here that information with respect to the correlation factor f2 is included in the expression of L22. Thermodynamically, the term L12 = L2, is commonly taken for the discussion of the correlation factor. It is true that Lt2 is the source of (time) conelation of isotope atoms with atoms of the matrix as discussed earlier, but this effect is reflected to the term L22 in determining the efficiency of motion of tracer atoms with respect to the random walk motion. In the limit of x2 = 0, at which the correlation factor is generally evaluated, L12 - 0 and, therefore, the Liz term itself is not directly related to the correlation factor. It is to be pointed out here that the evaluation of the correlation factor by the random walk theory is to evaluate the efficiency of the drifl motion of a single particle under equilibrium [9-l 11. The curve (a) also represents the relative change of the charge diffusion coefhcient DC(= akT/(ne2), where c is the ionic conductivity, k Boltzmann constant, T temperature, n the density of conduction ions per unit volume and e the charge of a conduction ion) with composition when only the species 2 has the charge. If the measurement of the Haven ratio HR = &/DC is carried out in such a mixture with a small addition of tracer ions of the species 2, HR tends to 1 in the limit of x2 - 0 (Fig. 3). This result, however, holds for more general cases, as can easily be seen in the foregoing arguments. If one disregards the effect of induced flows in the L22 term, which is equivalent to make L12 = 0 in the Onsager equation, the result is the curve (c) in Fig. 2.
(11)
In other words, this case is equivalent to evaluating L22. By definition, G2 represents the drift motion of particle(s) of the species 2 under equilibrium. Based on eqn (8), therefore, *2 = cp,l(w2w2) f2 =
= -11
- 2w2x*IS]~2,
1 - 2wzx,/s = 1 - (2/9)x,.
(12a)
In this case,& starts as 7/9 or (2~ - 1)/(2w + 1) and tends to 1 linearly as x2 is increased from 0 to 1. The value (2~ - 1)/(2w + 1) corresponds to the value of the correlation factor at x2 = 0 expected from the pair approximation of the original PPM [8]. After converting to long time averaging (the time conversion process [S]), however, the value becomes (2~ - 2)/
TRACER CONCENTRATION
Fig. 2. Dependence of the correlation factorh on the concentration of tracer atoms (species 2) calculated under dilTenznt boundary conditions. (a) Cut= 0 [case 2)]. (b) xl& = -x& [Gibbs-Duhem relation (case 3)]. (c)i, = 0, ~5~2 = 0 (neglecting the contribution of the f&back effectto L& (Appendix ii).
Correlation factor in tracer diffusion
1365
corresponds to the return probability P,), while Z2 is the probability that any other atom on the surrounding sites which forms the pair with the tagged atom of the species 2 across the vacancy jumps into the vacancy, and represents the escape probability of the vacancy P2 = (2~ - 1)Z2 or the probability of the forward jump of the atom of the species 2. The eqn ( 12b) in the limit ofxz - 0 f
=
’
I
OL
0
I
0.2
0.4 Composition
I
I
0.6 (X,)
0.6
I
(2w - l)Pe (2w 1P2 2+(2w-3)Z2=2P,+(2w-l)P,
(13)
thus indicates the efficiency of the forward motion or the drift motion of a single tagged atom under the equilibrium distribution at the time instant. In this respect, the calculation of L22 or the linear expansion
1.0
Fig. 3. Haven ratio for ions of the species 2 as a function of x2.
A small addition of tracer ions of the species 2 is assumed.
/0
The meaning of this curve is discussed in the later seo
tion, because this feedback effect in the diagonal term is not well recognized. For a future reference, a more general expression forfi in eqn ( 12a) is given as follows: According to eqn (A20), the most general expression forf2 is f
=
(2~
-
lXZ2
2
+
x2A)
+
2tw,/(w1
+
2 + (20 - 3)(Z, + x2A)
--_-
- --- Q
A
w2W2
’
(12b) where
x,wi
z2=c
j
w2
+
(i=
L2)
(12d
WJ
and
A.=w2_w2
+
WI w2
w,
+
w2
*
(124
Because wl = w2 in the present case, we have A = 0. Also, Z2 is equal to l/2 and eqn ( 12b) becomes identical with eqn ( 12a). In interpreting diffusion phenomena in terms of PPM, the function Z2 defined in (12~) plays an important role. In Fig. 4a, a vacancy, its surrounding atoms and a tagged atom of the species 2 are shown. This should be interpreted to be a time instant at which a tagged atom of the species 2 has just jumped out from the central site, replacing with a vacancy at its neighboring site. Therefore, the probability of the jumping back of the tagged atom into the vacancy it has just replaced is calculated here in competition with other atoms on the surrounding sites. The quantity xj specifies the species of atoms (the probability of finding an atom of the* species) on the surrounding site. As is clear from thii 6gure, the quantity 1 - Z2 represents the jump back probability of the tagged atom (which
Fig. 4. Definition of 22 and Z2 + {z in terms of escape prob. abilityP.andmtumprobabilityofatag@atomofthespef5es 2 (with *). White and hawhcd circle indicate atoms of the species I and 2, mrpectively. Figure Ya) repmsents a case where the number of atoms of the species 2 is negligibly small while (b) represents a case where the number of atoms of species 2 is large and the tagead atom cannot be disti~ from the rest of atoms of species 2.
H. SATOet al.
1366
coefficient of & is that of the drift motion of the assembly of isotope atoms. At a finite concentration of tracer atoms (xz # 0), fi becomes f
2
=
(2w
-
IV2
+
2IwMw2
+
( = -Lu(1
%)I%
2(1 - 2,) + (2w - l)Z,
*2 = -L2,
*
=
f2w
-
fW2
+
52)
2 + (2~ - 3)(Z2 + 5;) ’ 5;
=
x2w2 w2 + w2
(1%
&2.
(184
This equation can be rewritten into a well known relation as
This is further reduced, by introducing eqn (5) for Ltj's and using wI = ~2, to tir, = -{l
&
x2L21
X,
Xl L22
(14)
Although wl is numerically equal to w2, we distinguish w, and w2for the clarity of the ~ument. In analogous to eqn (13), eqn (14) represents the drift motion of a group of isotope atoms under the equilibrium distribution. If the ““feedback effect” is dismgarded, the expression for& becomes [eqn (A.23)]
- - & - L& XIJ
- 2w&s)&
(19)
or
q2+2Ap&2,
(20)
w
*
fi = (2w - 1)/(2w + I) This equation should be compared with eqn (13). In Fig. 4(b), a physical situation which represents 22 + S; is illustrated in comparison with that for Z2. It is clear that, analogous to that in eqn (13), this represents the drift motion of a group of isotope atoms at a time instant t from an initial equilibrium distribution, and the 5; term represents the indisting~shability among isotope atoms (see Appendix). The difference between eqns ( 14) and ( 15) represents that while eqn ( 15) rep resents the motion of atoms in which the equilibrium distribution is maintained before and after the jump, while in the latter the equilibrium distribution after the jump is not imposed.
(3) Gibbs-Duhem Relation The condition (2) cannot be utilized in a system in which the species 2 diffuses into an assembly of the species f under their con~n~tion gradient and their concentration is high, because the concentration gradient of the species 2 must create the concentration gradient of species 1 since condition xi + x2 = 1 and hence -i-, = -x2
(16)
must hold (under the assumption of negligible amount of vacancies). On the other hand, under the equilibrium condition, the Gibbs-Duhem relation XI& + x&2 = 0
(17)
should hold. Because cases close to the equilibrium condition are handled in diffusion, the Gibbs-&hem relation is often assumed to hold in the treatment of diffusion. Equation ( 15) relates ii to &. and 4pzin eqn (3) can then be solely expressed in terms of &, as
irrespective of the value of x2. Or, more generally, based on eqns (18a), (A.8) and (A.9) f
= 2
120
-
1X22
+
x24
2 + (20 - 3)(Z2 + x2A) *
(21)
Here, A is defined in eqn (12d). Because A = 0 in the present case, fi =
2
1”y2“;z2 W
(22)
follows. in other words, Gibbs-Duhem relation gives the identical result for fi as eqn (13) independent of the value of x2. The result is shown in Fig. 2 as the curve (b). It should be noted, however, that the concept of “local ~~lib~urn” should be limited within the plane perpendicular to the gradient and the Gibbs-Duhem relation, eqn (17), does not hold strictly in difhtsion phenomena. Rather, the Gibbs-Duhem relation should be treated as a given boundary condition that can ap proximately hold in such a case as interdif&ion. Under the Gib~~em relation, the diffision coefficient of a single isotope atom is measured while the evaluation of flow by means of b2 [eqn (1 l)] corresponds to the measurement of the diffusion coefhcient of the assembly of isotope atoms. The condition, eqn ( 17), requires that the motion of atoms of the species 2 should be compensated by the motion of species 1 atoms and that the ~~~bution of the species 1 and 2 as a whole should remain constant. In other words, the GibbsDuhem relation imposes a very strong requirement with respect to the driving forces Ly,and &s. Therefore, ifthere existsan independent driving force for diffusion of 1 and 2 species such as the vacancy concentration gradient as in a certain type of demixing problem, [ 19]
Correlation factor in tracer diffusion the imposition of the Gibbs-Duhem relation results in paradoxical results. On the other hand, the imposition of the Gibbs-Duhem relation to some type of treatment of difhtsion results in a great deal of simplification (41 and yet gives a correct value for the correlation factor (in the limit ofx, - 0). All our previous derivations [8- 171 of the correlation factor for various many body diffusion problems by the PPM were made under this boundary condition. 5. CORRELATION FUNCI’ION APPROACH In some experiments such as inelastic scattering of a neutron from liquids, the diffusion coefficient is evaluated in terms of the drift motion of particle(s) under equilibrium through the use of the correlation function based on the linear response theory. The flow (diffusion coefficient) is related to the velocity-velocity correlation of a tagged atom at time 0 and t evaluated under the equilibrium condition, (v(0) = v(t))=. For this purpose, we define the following hmctions which represent drift motions under equilibrium:
r
(‘*kfi
Vi(O) . Vj(t))Tdt,
in the previous sections in that both evaluate the flow in terms of the drift motion, as long as atoms of the species 2 correspond to tagged atoms in the correlation function. The correlation function approach, however, clearly shows that the flow is directly related to the time correlation of the motion of particles, and the time average has to be taken for the evaluation of the correlation factor, while in the original PPM, Lz2 is evaluated in terms of the ensemble average [8]. The functions 4k, c$’and &’are evaluated for lattices with two sublattices such as simple cubic or body centered cubic structure as [20]
Wd (27b)
(27~)
(24)
Here, vr indicates the velocity during flight for a jump diffusion process, and rf and r, indicate the time of flight and mean residence time in the jump dithtsion process. The correlation factors are then defined as
(25)
CW
(23)
4 = k
1367
i,j
4R =
(v(O)* W)ydt dm
and
f’=&_3”-l &2w+
1’
W-W
v,(O)& =+s+#d=&++dd,
(26a)
where
The value offcorresponds to that derived by the pair approximation of the original PPM andf’ to that after the time conversion [8]. The evaluation of 4 and hence the correlation factor for an assembly of tracer atoms
(26b) Here, we deal with a special case in which the number of vacancies is negligible and, hence, their motion is random. Suffixes used above are defined as follows: v = vacancy; T = tracer atom; p = particle (chemically similar tracer atom); NT = number of tracer atoms; N,, = number of particles; R = random; GCE = only consecutive exchanges of T and v, i.e., v on the next nearest neighboring site never returns to tracer atoms; s = self; and d = distinct. Then 4’ indicates the drift motion of a single tagged atom, & the same quantity if only consecutive exchanges of T and v are considered, 4 the same quantity for an assembly of NT tracer atoms, 4, = 4’ the self-correlation function and & indicates that if the correlation between only different atoms is considered. The evaluation of the functions 4, = #, & and r$ corresponds to that of & for corresponding cases
is what we are concerned here. Equations (28a), (b) and (c) are equivalent to eqn (2) in terms of Dr and &. The relation between g’s and the diffusion coefficient is given by (vfTf)Z
6(rr + rr&
z-z 1: 6 - tf
(29) Here Ir = r+rr corresponds to the jump distance and 1/tf (tf = Tf + Tm) corresponds to the jump frequency in the random walk theory. The relation between Dr, and +R in eqn (26) is based on the linear response theory that, in the linear range, the flow is determined by the drift motion evaluated under the equilibrium condition times the driving force.
H. SATOet al.
1368
In the derivation of 4 in eqn (27c), each jump is treated equally Born the equilibrium distribution to a nearest neighboring vacancy. Therefore, it is clear that the evaluation of the drift motion eqn (27~) is equivalent to that in eqn (13). On the other hand, the evaluation of Q in eqn (25) is complicated, and except for the fact that 4 + 4~ as Nr - Np, no reliable direct derivation of this quantity has been found. However, in view of the correspondence of these functions and h2 in the previous section,
3 = k
$
(“?
v,(O) * v,(t))T,-df,
(30)
,J
which corresponds to + in eqn (25), can be calculated by the original PPM. However, a problem arises in that whether the feedback effect should be included. Because the drift motion of a time instant from the equilibrium distribution is calculated in the evaluation of 4 in eqn (30), it is tempting to use eqn ( 15) without the feedback effect. However, because the time correlation effect is involved in deriving the correlation functions, and because each jump is treated as a jump from the equilibrium distribution in the derivation of & in eqn (27c), it should be understood that the feedback effect to restore the equilibrium distribution at each jump is imposed in the calculation. In other words eqn (12b) rather than eqn ( 15) should be used for the evaluation of $. In the limit x1 = 1 or x2 = 1, the feedback effect does not enter and their derivations are far simpler. 6. SUMMARY The meaning of cross terms in tbe Onsager equation for diffusion has been examined in detail based on the pair approximation of the PPM of irreversible stat&&l mechanics. Such effects had been examined based on the irreversible thermodynamics. Because the cross terms (or the elements of the Onsager matrix in general) cannot be derived horn the atomistic model by irreversible thermodynamics, the microscopic mechanism for the appearance of cross terms has never been clearly understood earlier. Examinations in the present paper are based on binary ideal solutions so that effects due to mutual interactions among particles can be eliminated to clarify the key concept of the correlation factor. The present paper clearly shows that nonnegligible cross terms exist even under the condition of ideal solution. The appearance of the cross terms under the condition of ideal solution is due to the fact that atoms of the species 2 cannot jump into lattice sites which have been occupied by atoms of the species 1. Depending on the boundary conditions imposed in actual measurements, relative contribution of cross terms and diagonal terms to the flow varies. Such a dependence of flow or the diffusion coefficient on boundary conditions is shown for several typical cases. In particular, the houndary condition (2) or the evaluation of J& corresponds to the evaluation of diffision coefficient from the drift motion of an assembly of atoms under
equilibrium. The boundary condition (3) or the imposition of the Gibbs-Duhem relation corresponds to the evaluation of the dill&ion coefficient under the condition that the relation of interditlWon between isotope atoms and matrix atoms holds strictly and corresponds to the evaluation of the motion of individual atoms. The two conditions, therefore, give the identical result in the limit x2 - 0. The existence of the feedback effect in the diagonal elements of the Onsager matrix is not well recognixed and is often disregarded. The effect is essentially due to finite values of the cross terms and represents the effect of restoring the equilibrium distribution after each jump of atoms. This contibution in ideal solution amounts to ~10% of the diagonal terms in the intermediate composition range. A derivation of the feedback effect is shown for the case of a binary ideal solution in Appendix. Acknowledgements-The content of the present paper is mainly a result of discussions of one of the authors (H.S.) with Professor H. .Sclnnalxried and Professor K. Fur&e of Institut Eir Physikalische Chemie und Elektrochemie, Universitit Hannover during his stay there in I98 1. He thanks them for their valuable mggestions and their ho&a&. He also thanks the Humboklt~Foundation in West Germany for the generous support which made his stay at the Institut possible. Discussions with Dr. M. Kabumgi and the help of Mr. S. A. Akbar in pmparmgthemanusciiptarehighlyappre&ed. The work was supported by the United States Department of Energy under grant number DE-EGO2-84ER45 133. REFERENCES
1. Albright J. G., J. Phys. Chem. 72, 11 (1968). 2. Home F. H., In Physics of Superionic Conductors and Electrode Materials (Edited by J. W. Perram), NATO AS1 Series, Series B, Vol. 92, p. 257. Plenum Press, New York (1983). 3. Sato H., In Nontraditional Methods in Dt@kon (Edited by G. E. Murch, H. K. Bimbaum, and J. R. Cost), pp.
203-235. TMSAIME Conference Pmceedings Volume (1984). 4. Sato H., Akbar S. A. and Murch G. E., Dtjusion in Solids: Recent Developments (Edited by G. E. Murch and M. A. Dayananda), pp. 67-95. TMS-AIME Conference Proceedings Volume (1985). 5. Wada K., Suzuki A., Sato H. and Kikuchi R., J. Phys. Chem. Solids, 46, 1 I95 (1985). 6. Sato H. and Kikuchi R.. Mass Transport Phenomena in Ceramics, Materials Scknce Research (Edited by A. R. g and A. Heuer), p. 149. Plenum Press, New York I. Egelstalf P. A., An Introduction to the Liquid State. Academic Press, New York/London (1967). 8. Sato H. and Kikuchi R., Phys. Rev. B 28,648 (1983). 9. Manning J. R., Di$usion Kinetics forAtoms in Crystal. Van Nostrand, New York (1968). 10. L&aim A. D., In Physical Cherrustry,Advanced Treatise, Vol. X, Solid State (Edited by H. E&g, D. Henderson, and W. Jest), D. 281. Academic Press New York (1970). 11. Peterson N. L., In Solid State Phystcs, Vol. 22 (Ed&I by F. Seitz, D. Turdbull, and H. Ehrenreich), p. 409. Academic Press, New York/London ( 1968). 12. Bardeen J. and Herring C., In Atomic Movements, p. 57. ASM, Cleveland ( 195 1). 13. Kikuchi R. and S&J H., J. Chem. Phys. 51, 161 (1969). 14. Kikuchi R and Sato H., J. Chem. Phys. 53,2702 ( 1970). 15. Kikuchi R. and Sato H., J. Chem. Phys. 57,4962 (1972).
1369
Correlation factor in tracer ditfusion 16. Sato H. and Kikuchi R., J. Chem. Phys. 55,677 (1971). 17. Kikuchi R. and Sato H., J. Chem. Phys. 55,702 (197 1). 18. Kikuchi R., Ishikawa T. and Sato H., Physica A 123,227 (1984). 19. Ishikawa T., Sato H., Kikuchi R. and Akbar S. A., J. Am. Ceramic Sot. 68, 1 (1985). 20. Funke K., private communication. The expressions for functions, eqns (23)-(26), are due to him.
Q,*,
statements made. in the text. In Refs. [31and [81,the derivation of the Onsager equation for ditfusion under the steady state. condition was made for three component systems consisting of A, B and B* (B* being the isotope of B) by the original PPM. The instantaneous distribution conversion process is not taken into account here. The present treatment corresponds simply to the case for x* (the composition of A) = 0, but with mutual interactions among atoms. Explanations of symbols used are given in Table 1. For ideal solutions, the relations G, + wi and Qi + Xi follow in the present approximation. (i) Derivation of the Onsager matrix elements L,. The equation of llow (Ref. [3], eqn (A. 15)) for a bii system derived by the pair approximation PPM in the linear range is given by ‘I’I = -h, +
Qzrt21,
92
Qd12,
=
42
#2, = -*12,
+
Al
=
=-
Q2*2
=
-QI&
{
2
I
+(I
-R)-
/
ix,
W’,l = ~21~221+
a2 + (1 - RP’21,
64.6)
Q2b2.
4Q2
---2 6,
+
$2
.
4Q, $1 + $2
dr,+(1
-R)acGz. I
2
64.7)
For &, an equivalent expression is obtained Because 0, = q,ti,QQ, and 4 = ~2G22QQ2, the Onsager matrix elements are obtained from eqn (A.7) and the equivalent expression for *r,
Il-%&m. I+(,_R)_)G,P,_ 6, + 4
(A.2)
-
1
1-a
-
Here, ei is the normalized flow delined in eqn (8) for the species i, ir, is the chemical potential gradient, Q, indicates the quantity mlated to the probability of finding the ith species on the neamst neighboring site of a specitled site, &, is called the deviation quantity and is antisymmetric with respect to the exchange of i and j. (Both i and j am used to specify the species) Without the existence of the second term on the right hand side, eqn (A. 1) indicates the random walk of particles of the ith species under the driving force &, and the second term indicates the effect of flows of other species which is determined by the nature of the diffusion path The quantity Jj, is proportional to the driving force &jin the linear approximation [3]. The steady state condition that the distribution of atoms does not change with time (eqn (A.20) of Ref. [3]) leads to a relation between #,, and tiU %NJ,z + & + (1 -
+
By substituting (A.6) into (A.5), we obtain
(A.1)
9422= 0.
-R)s.
From (A. I) and (A.2), on the other hand,
APPENDIX
The derivation of the Onsager equationfor difksion for two component systems The purpose of this section is to provide background to the
-R)s+(l
-(I
kc,+ $2
-L”= {, _d%!-+__+)}QoQ”l’ -(,
(A.8a)
(A.9a) -J522
=
(A.8b)
-L,,(@,
-L,s = -L,2(e)
= -L2,.
(A.9b)
The expression for Lzz is then from eqn (A.8b)
_
1
(A.3)
_a G, + 4
+(,
-@?&
$1 + 4
-&‘={, -+!%&-+~)}BQ2~2~
R - 2/(2w - 1).
-(,
Hem, G,‘s are jump frequencies including the bond breaking effect as defined in Table 1. From eqn (A.3), we get
(A. 10) Therefore, from eqn ( 12a)
$12 = &
K~22i22
-
%k)
+
(1
-
RX&z*2
-
%U,)l.
(A.4) Therefore, &, can be eliminated from eqn (A. l), and 8, = -&, - - GQ2 a2 + - GQ2 ci, 6, + G2 I& + 4 -(I
-R)m+(l
I
-R)$$, 2
I
2
(A.3
GzQ, -=-+6, + +
QQ, 9, + $2
62Q2 -a 6, + $2
(A.12)
1370
H. SATOet al. 1
and, by defining Z, as [eqn (12c)], z2
=
” Q,w -+$2 + $1
(2~ - l)(Z, + Q*A) + 2 a =
Q24
(2
(20
-
3)(z2
+
2
Q2k))
(A. 19b) ’
62 + 4 (A.‘3)
G2Q1 -+-= 6, + $2
+
+2Q2 4 + 4
1 -z2
(A.14)
(ii) Feedback efh in L2. The feedback effect in LX is evaluated as follows. In eqn (A.5) qr2= --h2 - - *IQ1 iu, + - *2Q, &2 G, f 62 4, + $2
we obtain 4Q, 6, + $2
-=
1-zz-52,
&here (A.16) Therefore *2Q, _+
%Qz _ l-z2-~2+*
G, + $2
56, + $2
_ (1 -R) E2
(AX)
(A.20)
+ (1 -R) E.
We first impose the condition iy, = 0. Under this condition, the term including 9, in the right hand side in eqn (A.20) then represents tbe induced current of the species 1 by the existence of a2 which would create the feedback effect in &. By eliminating this team, eqn (A.20) becomes
.
I
[ 1-(1-R)*
-
-a2
2
I
= 1 - Z2 - Q2A2,
2
G2Q1
lo, +
$2 I
&y1. (A.21)
(A.17) By using eqn (A. 154,eqn (A.2 1) then becomes
where (A. 18)
‘=-
-Ra+[l
(2w - lXZ2 + i-2) 2 + (2~ - 3)(& + f2;).
-(Z,+QzA)]
{l -(l’-R)(l-(Zz+aA)]]
-(20-
l)-2&
(A.23)
(A. 19a)
Or
A=-
(A.22)
and, hence,
By introducing & and A2 into eqn (A. 11) -1
xP2[1 - (1 - R)(1 - Z, - 1;)] = -(Z, + si)dlz
+ (2~ - l)[ 1 - (z, + Q2)l
((2~ - 1) _’ (2~ L 3)[1 - (Z, + Q2A)]}
Here, Z2 + c2 represents the escape probability under the equilibrium condition of a taggaJ atom of the species 2 including the indistinguishability among atoms of the species 2. The di%rence between eqn (12b) and eqn (A.23) then rep resents the f&back effect.