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Free energy and point defect distribution for heavily doped AgCl by an integral equation method
A. Allnatt, E. Loftus Allnatt
To cite this version:
A. Allnatt, E. Loftus Allnatt. Free energy and point defect distribution for heavily doped AgCl by an integral equation method. Journal de Physique Colloques, 1980, 41 (C6), pp.C6-94-C6-96.
�10.1051/jphyscol:1980625�. �jpa-00220063�
JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 7 , Tome 41, Juillet 1980, page C6-94
Free energy and point defect distribution for heavily doped AgCl by an integral equation method (*)
A. R. Allnatt and E. Loftus Allnatt
Department of Chemistry, University of Western Ontario, London, Ontario N6A 5B7, Canada.
RCsumC. - Au moyen des Cquations hypernetted chain on a calculC 1'Cnergie libre et les fonctions de distribution radiale dans le cas de AgCl dopi: par CdCl, (1-5 %). On a comparC la fraction d'impuretCs qui existe en paires lacune-impureti: et l'knergie libre avec la thCorie basCe sur la loi d'action de masse.
Abstract. - The free energy and radial distribution functions for vacancies and impurities were calculated from the hypernetted chain integral equations for AgCl doped with CdC1, (1-5 %). The fraction of impurities in nearest- neighbour vacancy-impurity pairs (but not in larger clusters) and the free energy were compared with the mass action theory description.
1. Introduction. - Properties of AgCl doped with CdC12 and many similar systems have been studied extensively at low dopant concentrations i.e. less than 0.1 mole
%.
An important feature is that an appreciable fraction of impurities and vacancies are in nearest-neighbour pairs which do not contribute to the electrical conductivity. The fraction is calculated using the law of mass action as described by Lidiard [I].The analysis generally works quite well. There is a small number of experiments for high dopant concentrations of 1
%
or more for which theories which work at the low concentrations are less satis- factory. For example, for AgCl and AgBr the ionic conductivity including high concentration data can be interpreted [2] to indicate that the fraction of pairs is close to zero, which is not expected from an extra- polation of the successful analysis [3] at low concen- trations. A similar situation exists for thermopower, and there are also high concentration anomalies in Cd2+ and Mn2+ diffusion [4]. Possible sources of these difficulties include the presence of unexpected migration mechanisms, the failure of the mass action- based description of defect distribution, or the failure of some other aspect of the kinetic analysis. Here we examine the second possibility. We compare the mass action description with that afforded by radial dis- tribution functions and free energies calculated from an integral equation known to be rather accurate for ionic solutions [5].The model, identical to that in [6], assumes a Cou- lombic interaction between defects at all distances except for the use of an experimental binding energy for a nearest-neighbour (n.n.) vacancy-impurity pair.
The calculations are for AgCl doped with CdC12 at 250 OC.
(*) Research supported by N.S.E.R.C., Canada.
2. Mass action description. - Lidiard [I] gave an approximate expression for the defect contribution to the Helmholtz free energy, A, for a system of equal concentrations, p, of impurities and vacancies of which a fractionp are in n.n. pairs.p was determined by the law of mass action derived from A . However, at high concentrations there will be numbers of defect triplets in which each defect is n.n. to at least one other oppositely charged defect. Since the kinetic properties of such triplets will be different from those of pairs and also in an attempt to improve the accuracy of A it becomes necessary to treat triplets as a distinct species. The extended expression for the reduced excess free energy F = - 2 AIBk, T, using Debye- Huckel theory for interactions between species with a net charge, is
F = - 4 p l n ( l - p - q) - 2 p ( p + 4913)
+
-I- 4 p(1 - p - 2 q/3)3/2 X ~ T ( X ) , (1) where
q is the fraction of impurities in triplets, B is the number of unit cells, a the anion-cation distance, D the dielectric constant, and b = e2/Dk, T. Two inde- pendent mass action equations for formation of pairs and triplets determine p and q, as described earlier [6].
Figure 1 shows, as a function of p, values ofp assum- ing only pairs are present, and values of p and q assuming pairs and triplets are present. Clearly, at impurity concentrations greater than 112
%
the con- centration of triplets is not negligible. No doubt at 5%
concentration there are appreciable concentrations
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980625
FREE ENERGY AND POINT DEFECT DISTRIBUTION FOR HEAVILY DOPED AgCl
8
t
p (pairs only)and triplets)
Fig. 1. - Fraction of pairs, p, and triplets, q , calculated from law of mass action compared with the fraction of pairs p(HNC) from the HNC equations, as functions of concentration p at 250 OC.
of larger n.n. clusters in which each defect is n.n. to at least one oppositely charged defect. The difficulties with the mass action approach are therefore : (a) the number of equations rises rapidly past the triplet stage if we try to be systematic, (b) the use of Debye- Huckel theory and the treatment of configurational entropy become less plausible at high concentrations.
We can justify (b) at low concentrations by an approach based on Mayer cluster theory [6], but this approach is unuseable at high concentrations.
3. Radial distribution function description. - p2 g+-(r) is the probability of finding an impurity and a vacancy on particular sites a distance r apart, and p2 g + + (r) is the probability for a pair of vacancies or a pair of impurities at the same sites. We calculated these radial distribution functions, g, by a variational solution [7] of the hypernetted chain integral equa- tions for 0.01
<
p $ 0.05. Full details will be sub- mitted elsewhere. Typical results, denoted g(HNC), are shown in figure 2 together with g(DHX), the well- known extended Debye-Huckel approximation [5] :g+ +(r) = exp {
+
b expi- x(r - a~)llr[l+
xa111
,(3) where a, =
@.
Apart from the inadequacy of (3) the noteable feature is that g + + is less than 1 for the n.n. separation but is greater than 1 at the second n.n.where, for p 2 0.02, it exceeds g+
-.
This is qualita- tively consistent with the formation of triplets and larger n.n. clusters as discussed in paragraph 2.1.5 2.5 3:5
r l a
Fig. 2. - Radial distribution functions as functions of rju. The first six nearest-neighbour separations are indicated by vertical lines at the bottom. a) p = 0.03, b ) p = 0.01.
All experience with other systems [5] suggests that solutions of the H N C equations will be rather accu- rate. However, for the fraction p of impurities in n.n.
pairs (and not members of triplets, etc.) of interest in transport theory only an approximate relation is available :
The approximation arises because a condition like not member of a triplet introduces a triplet correlation function which is not known and therefore avoided
C6-96 A. R. ALLNATT AND E. LOFTUS ALLNATT
through a suitable approximation [8]. An identical situation exists in ionic solution theory [9]. The adequacy of the approximation is unknown in both fields.
The p values calculated from the HNC results, denoted p(HNC) in figure 1, do not support the sug- gestion that p approaches zero at high concentrations.
The rise in p(HNC) as p increases, in contrast to the mass action result, may be due to the formation of fewer larger aggregates than expected from the law of mass action or to the inadequacies of eq. (4).
Finally we compare the reduced free energy calcu- lated from the mass action and HNC approaches in figure 3. For the mass action theory we include calcu- lations for 7 = 1 as well as for .r given by eq. (2). It is well-known that the corrections introduced by taking 7
+
1 compared with z = 1 are not accurate at high concentrations [lo] and it is sometimes argued that it is better to take .r = 1. The discrepancy between the two results gives some idea of the uncertainties inherent in the simple theories. The close similarity between results for the results with 7 = 1 including pairs and triplets and those obtained from the HNC equation is probably fortuitous. Although the free energy is not of much direct interest in this particularsystem a similar calculation for defect interactions in pure CaF, might be helpful in clarifying ideas about the various contributions to the free energy near the
1
H N CFig. 3. - Reduced free energy, E , as a function of p at 250 "C for the HNC theory, and for the mass action theory with pairs and triplets using z = 1 (as marked) and using z of eq. (2) (marked r 9 1).
phase transition to the superionic state ; we are inves- tigating this point.
DISCUSSION
Question. - M. J. GILLAN. Reply. - A. R. ALLNATT.
The idea of using the HNC equation to calculate Yes, I agree that there is a real difficulty. It probably defect concentrations in fluorite compounds is very also arises for AgBr where the interstitial-vacancy attractive. But I fear there may be a serious difficulty pair may not be stable at the nearest-neighbour which stems form the fact that pairs of unlike defects separation. This problem will arise in any statistical are unstable against recombination. This means that mechanical treatment of these systems. The correct a proper interaction energy may be hard to define. way of dealing with these instabilities at small sepa-
rations is not yet clear.
References
[I] LIDIARD, A. B., Phys. Rev. 94 (1954) 29. [6] ALLNATT, A. R. and LOFTUS, E., J. Chem. Phys. 50 (1973) 2550.
[2] FRYER, G. M., Philos. Mag. 34 (1976) 217. [7] OLIVARES, W. and MCQUARRIE, D. A,, J. Chem. Phys. 65 (1976) [3] CORISH, J. and JACOBS, P. W. M., J. Phys. Chem. Solids 33 3604.
(1972) 1799. [8] ALLNATT, A. R., Adv. Chem. Phys. 11 (1966) 1.
[4] SUPITZ, P. and WEIDMANN, R., Phys StatusSolidi27(1968)631. [9] FRIEDMAN, H. L. and LARSEN, B., J. Chem. Phys 70 (1979) 92.
[5] OUTHWAITE, C. W., Statistical Mechanics, ed. by K. Singer, [lo] MCQUARRIE, D. A., Statistical Mechanics (Harper Row) 1976.
Vol. 2 (Chemical Society, London) 1975, p 188.
