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Clustering of divalent cation-vacancy pairs in alkali halide crystals
E. Lilley
To cite this version:
E. Lilley. Clustering of divalent cation-vacancy pairs in alkali halide crystals. Journal de Physique Colloques, 1980, 41 (C6), pp.C6-429-C6-431. �10.1051/jphyscol:19806109�. �jpa-00220015�
JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 7 , Tome 41, Juiffet 1980, page C6-429
Clustering of divalent cation-vacancy pairs in alkali halide crystals
E. Lilley
Materials Science Division, School of Engineering and Applied Sciences, University of Sussex, Falmer, Sussex, England
RBsumB. - L'analyse par Unger et Perlman des rksultats expkrimentaux concernant la formation d'agrkgats de paires bivalentes cation-lacune s'est revClke insuffisante pour distinguer entre la formation de dimbres et le modkle de Dryden pour la formation de trim&res, compte tenu des conditions experimentales. Dans l'ensemble, les rksul- tats correspondent A une cinktique du troisikme ordre dans la rkgion preckdant le plateau.
Abstract. - The analysis of experimental results of the clustering of divalent cation-vacancy pairs by Unger and Perlman is shown here to be unable to distinguish between their model of dimer formation and the earlier model of' Dryden for trimer formation under the conditions where it has been applied. In general, experimental results appear to correspond to overall third order kinetics in the pre-plateau region.
Although many experimental studies of clustering of divalent cation-vacancy pairs [M' +V-] have been carried out during the last 20 years, the types of clusters formed and the order of the kinetics is still in doubt.
In principle the experiments are quite simple. Typi- cally an alkali halide crystal containing about 100 ppm divalent cations is heated to 300 OC to distribute the M + + ions and the compensating cation vacancies ' throughout the crystal and then it is quickly cooled to room temperature. Virtually a11 of the vacancies associate with the M f + ions to form pairs. By aging at a temperature, usually between room temperature and 100 O C , the [M+ +V-] pairs diffuse and react with each other to form clusters such as dimers (2 pairs) and trimers (3 pairs). The [M++V-] concen- tration can be measured with ITC [l] or dielectric loss [2] as a function of time to determine the order of kinetics and thereby deduce the type of clusters formed.
The original work in this field was performed by Dryden and co-workers [2, 31 who made a n extensive study of clustering in several alkali halideldivalent cation systems. They invariably found that their results corresponded to third order kinetics i.e. log plots of half life versus composition measured iso- thermally gave linear plots with a slope of two.
This led them to propose the formation of trimers according to
where A , represents an [ M + + V L ] pair and A, represents a trimer. Some plots of concentration [ M + + V P ] versus log time showed a plateau which was seen to be a consequence of the back reaction.
However it should be noted that these workers
often chose for analysis kinetic data in which the plateaus were at very low [ M + + V P ] concentrations.
The direct formation of trimers by a third order reaction has not however received complete accep- tance. On mechanistic grounds the formation of dimers seems preferable and so Crawford [4] and Strutt and Lilley [5] advocated a dimer-trimer model.
The latter computor fitted several sets of data includ- ing some of Cook and Dryden's to a dimer-trimer model with apparent success.
More recently in a series of papers, Unger and Perlman have proposed that dimers (A,) form in the pre-plateau region of the aging plots [6-101,
They explicitly took account of the back reaction in their rate equation which they then solved to get,
where X is the relative concentration of [Mf 'V-]
pairs (X = 1 at t = 0), X , is the value of X a t the plateau, k is the forward rate constant, t is the time, No is the concentration of [Mf +V-] pairs in the crystal a t t = 0 assuming no clustering during quenching.
This equation was then used to plot experimental data for KCI/Eu++, KCIjSr", KCl/Ba++ and NaC1/Cat+. In these cases they obtained linear graphical plots from which they concluded that the
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19806109
C6-430 E. LILLEY
pre-plateau region corresponded to essentially second order kinetics due to dimer, not trimer formation.
A feature of this analysis is that it must include a plateau. This is not the case for the experimental data analysed by Dryden and co-workers to establish third order kinetics.
An exchange of views by Cook and Dryden [ll]
and Unger and Perlman [9] failed to lead to any resolution of this problem. Unger and Perlman maintained that the analysis of Dryden and co-workers failed to explicity take into account the back reaction in equation (1) so that the experimental data looked third order but was in fact second order. Since then the Unger and Perlman analysis has been adopted by other authors in studies of NaCl/Mn [12, 131 and now the pre-plateau formation of dimers appears to be becoming widely accepted.
The disagreement discussed above is primarily about data analysis so let us consider the two models further. The questions we seek to answer are (i) how significantly does the back reaction in equation (1) affect the analysis in the work of Dryden and co- workers and (ii) is the Unger and Perlman type of kinetic plot based on equation (3) able to distinguish between second and third order kinetics ? In the following we shall deal with numerical values obtained from equations, not experimental data.
First let us consider equation (1) for trirner for- mation. The rate equation including the back reaction is
where 1 1 , and t13 are the concentrations of M + + ions existing as pairs and trimers and k , and k, are the forward and backward rate constants respec- tively. Equation (4) can be expressed, changing to relative concentrations, as the integral
With this equation we have computed sets of numerical data, i.e. X , and t values for various X , plateau levels. A family of calculated curves is shown in a plot of X , versus log t in figure 1. It is quite clear that when X , is small e.g. X , < 0.1, that the aging curve is very close to the ideal third order without back reaction down to X = 0.15. Recalling that Dryden and co-workers measured the half life for such reactions, figure 1 reveals the differences at X = 0.5 between the pure third order data and that for various back reactions. For the direct formation of trimers the half life analysis is satisfactory for plateaus at X , < 0.3 giving an error well within the expected experimental error. This indicates the analysis of Dryden where no correction is made for the back
I I I
10 lo1 1 oa 1 o9
LOG TlME (ARBITRARY UNITS)
Fig. 1. - Family of computed kinetic curves for a thud order reaction with a back reaction. For small values of X, the back reaction has a negligible effect over most of the reaction.
reaction, is therefore quite satisfactory at low values of X,. (Moreover, we believe that their corrections for the back reaction are also satisfactory.)
The second question of whether Unger and Perlman plots can differentiate between second and third order reactions each involving a back reaction, can be answered using the same data computed from equation (5) for figure 1. These values of X,, X , and t can now be plotted using equation (3) as an Unger and Perlman kinetic plot, i.e. the third order data is used in the second order plot. As an example, one set of values is shown in figure 2 for X , = 0.5 which shows a slight curvature. However, by choosing a slightly different X , plateau position value than the correct value, e.g. X , = 0.51 instead of X , = 0.5
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 TlME (ARBITRARY UNITS)
Fig. 2. - Unger and Perlman type of plot. 0 points represent computed third order data for X, = 0.50.
+
points represent modified data using an erroneous X, = 0.51 which leads to a linear plot.CLUSTERING OF DIVALENT CATION-VACANCY PAIRS IN ALKALI HALIDE CRYSTALS C6-431
but still using the X, t data generated for
X,
= 0.5, this small deviation can be almost completely removed (Fig. 2). It should be remembered that in analysing experimental data,X,
must be judiciously chosen and it is never likely to be more accurate than 1%.
Unger and Perlman even represent their experimental data for several chosen X , values, showing that one value gives the best straight line.We must conclude that the Unger and Perlman analysis of kinetic data exhibiting plateaus unfortu- nately is unable to distinguish between dimer and trimer formation. Similarly the computer fitting of curves to a dimer-trimer model (5) is not likely to be definitive. Nevertheless, with experimental data having plateaus at very low levels, i.e. X, + 0 then the distinc- tion between dimer and trimer formation can be made. For example, numerical data for equation (5) would thed give non linear plots for the following (a)
an Unger and Perlman type of plot (b) a second order 1/X - 1 versus t plot (c) a second order half life versus N i l plot. Considering the published experimental data in the light of our work, the overall reaction appears to be third order not second order.
It is however, quite possible that a small concentration of dimers form as a precursor to the trimer so that the overall kinetics are third order. In this case it is experimentally very difficult to prove the existence of initial dimer formation using a kinetic approach.
Another subject of controversy concerns the post- plateau region of the clustering curves. Cook and Dryden (21 proposed that in this region pentamers were formed. Unger and Perlman [lo] have suggested that trimers were formed. Recent experimental results show that this post-plateau region corresponds to the formation of precipitates of a second phase [14].
DISCUSSION
Comment. - J. S. DRYDEN.
A comparison of the activation energy for the aggregation of the bivalent ion-vacancy pairs and the enthalpy obtained by classical diffusion methods for the jump of the same bivalent ion into an adjacent vacancy provides information about the stability of dimers. In NaCl : Pb2+ this comparison yields an association energy of 0.13 eV. PbZi is much larger than a Nai ion. We have recently obtained a value of 0.87 eV for the activation energy of the aggregation which is close to the saturation diffusion value for Ca2+ in NaCl (0.85 eV) reported to the Berlin Confe- rence by Machida and Fredericks. Suggesting that if the bivalent ion is of the same size as the cation it substitutes, the dimer has a short lifetime.
Comment. - A. KESSLER.
The fact that the same set of data can be fitted as well by a second order as by a third order equation does not surprise, because the interval of the fit is arbitrary. This ambiguity can be removed by the requirement, that the activation energy of the reaction constant be compatible with the activation energy of I-V-complex migration, for instance. There are, however, still other reasons which make the evaluation of the data difficult : the crystals contain necessarily a considerable amount of dislocations where agglomeration takes place. In consequence the data can be fitted especially in the beginning and for lower concentrations with a quite different type of law.
References
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Proc. Phys. Soc. Lond. 80 (1962) 479.
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[5] STRUTT, J. E. and L n ~ m , E., Proc. Seventh Int. Symp. on the Reactivity of Solids (Chapman and Hall) 84, 1972.
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[I31 GUERRERO, A. L., JAIN, S. C. and PRATT, P. L., Phys. Starus Solidi (a) 49 (1978) 353.
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