ATOMIC TRANSPORTON THE ANALYSIS OF IONIC CONDUCTIVITY

(1)

HAL Id: jpa-00216933

https://hal.archives-ouvertes.fr/jpa-00216933

Submitted on 1 Jan 1976

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ATOMIC TRANSPORTON THE ANALYSIS OF

IONIC CONDUCTIVITY

C. Murthy, P. Pratt

To cite this version:

(2)

A

TOM/C TRANSPORT

ON THE ANALYSIS

OF IONIC CONDUCTIVITY

C. S. N. MURTHY (*) and P. L. PRATT

Department of Metallurgy and Materials Science Imperial College London SW7 2BP, England

RBsumB.

-

Dkfauts de rkseau et paramktres de migration obtenus par l'accommodation des donnies de conductivites ioniques totales ont kt6 testes par deux diffirentes mkthodes de moindres carr6s. Les sommes minimales des carrks des diffkrences entres les conductivitks calcul6es et mesu- rkes semblent dependre fortement de la prkcision des computations. Les courbes sont t r b plates et beaucoup d'accords sont ainsi possibles. Les valeurs calculQs de cette facon n'ont qu'une signi- fication qualitative, ou du mieux semi-quantitative, en particulier en ce qui concerne les entropies de formation de pairs Schottky et de migration de lacunes cationiques.

Abstract.

-

Defect formation and migration parameters obtained from the fitting of total ionic conductivity data have been tested adopting two distinct least-squares routines. The minimised sum of the squares of the difference between calculated and measured conductivity seems to depend very much on the accuracy of the computing. The minima themselves are extremely flat, and thereby give rise to many possible fits. The values that emerge from such calculations have only a qualitative or at best semi-quantitative significance, particularly for the entropies involved in the formation of Schottky pairs and in the cation vacancy migration.

1. Introduction. - Starting from the simple model due t o Koch and Wagner [I], it has been customary to determine the defect energy parameters by fitting straight lines to the ionic conductivity data. However, the linear regions to which the conductivity equations were fitted turned out to be rather short owing to a slight upward curvature at high temperatures and to the concavity of the conductivity plot at low temperature. Refinements of analysis involving defect interactions were first made by Teltow [2], by Etzel and Maurer [3] who allowed for nearest-neighbour defect interactions between divalent cation impurities and cation vacancies, and by Lidiard 14, 51, who pointed out that long range interactions could be included by using the Debye-Hiickel theory. Recently the defect interaction problem has been looked into in more detail [6] and the small differences in the vacancy concentrations in pure NaCl [7], obtained by using Debye-Hiickel and physical cluster theory activity coefficients, seem to be unimportant in the analysis of ionic conductivity data. Further improvements of interpretation were made by considering the association and precipitation of divalent cation impurities [8] and the contribution from anion vacancies [9, 101 at high temperatures. Such complexities have led to attempts to analyse the whole conductivity curve as a sum of exponentials by least-squares methods, by Beaumont and Jacobs [ l l ] on KCI, by Dawson and Barr [12] on KBr and by Fuller and Reilly [I31 on

(*) Now at the University of Genoa, Genoa, Italy.

RbC1, in order to compute precise enthalpies and entropies for comparison with theoretical calculations 1141. This has been adopted to a great extent [15 to 221 in spite of the fact that a fairly large number of parameters are involved and the normal equations for least-squares fitting are highly ill-conditioned when dealing with sums of exponentials. In view of the many sophisticated least-squares routines and also of the improved precision of recent experimental data, there is a danger of placing too much confidence in the best-fit values obtained with the aid of a computer from conductivity data alone. Remaining discre- pancies such as an anomalously high conductivity near the melting temperature [16, 17, 221, and the difficulty of determining parameters for minority carriers as in KBr [12], AgCl [19] and NaCl [16, 231, have been attributed to the inadequacy of the model leading to the suggestion of additional mechanisms including Frenkel defects, and the trivacancy and excess conductance models [I 71.

In all of these cases, the nature of the data (the number of points or the temperature interval for the measurements, the purity, divalent cation doping and in some cases anion doping) and the details of the criteria adopted in getting the best-fit parameters, especially for different samples of the same material, vary considerably and no detailed comparisons between them are available [16]. Such a comparison would be useful because, in addition to their sophisti- cation, the available least-square routines differ appreciably in their efficiency. Furthermore, a critical

(3)

C7-308 C. S. N. MURTHY AND P. L. PRATT assessment of the estimated parameter Iimits and a

statistical measure of the goodness of fit is lacking except in the case of the silver halides, for which a constrained fitting has been carried out only in an intermediate temperature range [21] and the chi square test has been applied to check the goodness of fit [24]. In this paper we have analysed two sets of data obtained by Jacobs and Pantelis [17] on zone- refined KC1 (Run 102-2) and on 5N purity KC1 (Run B4-6) ; some results from a composite run constructed by combining the 92 points from B4-6 with data from other runs are included in the Tables. In figure 1,

RG. 1. - Conductivity plots for run 102-2 (A) and the compo- site 136 point ruu ( 0 ) .

log,, aT is plotted against temperature for the run 102-2 and for the composite run. Two different least- squares routines have been adopted for the fitting, and, to check the uniqueness of the defect parameters for different sets of data, all the parameters have been varied freely.

2. Model calculation, fitting procedure and tests. -

These calculations are based on the conventional Schottky defect model [5] which allows for the association of cation vacancies with divalent cation impurity ions on nearest-neighbour cation sites and takes into account the long range Coulomb interactions between defects by using Debye-Hiickel activity coefficients in the mass action equations for defect

concentrations and by using the Onsager-Pitts mobility correction. It is assumed that neither cation vacancy complexes nor stable cation-anion vacancy pairs contribute to the conductivity and that all the enthalpies and entropies are temperature independent.

On the Schottky model [5], the conductivity, a, is given by

a = a ( T ; h , , s,, Ah,, As,, Ah,, As,, C , x,y) (1)

where T is the temperature, h, and s, the enthalpy and entropy of formation of Schottky defects, Ah,,, and

AS^,^

the enthalpy and entropy of migration of cation and anion vacancies, C the mole fraction of divalent impurities, and

x

and y the enthalpy and entropy of association. Those values of the parameters are desired which will give an optimum fit of the data to the model. If the random errors of the measurements are normally distributed, the least-squares fit will also be the maximum likelihood fit, and as the parameters are mostly non-linear, the fitting is generally done by a trial-and-error process.

With a trial set of parameters the conductivity, (aT),, is predicted at each temperature corresponding to the observed values, (oT),, and the function,

is minimised with the sum over all the data points. This is carried out by a nonlinear least-squares routine by varying the parameters until no further reduction of the sum can be obtained.

(4)

ON THE ANALYSIS OF IONIC CONDUCTIVITY 0 - 3 0 9 ponding to a probability (1-a) is defined by all combi-

nations of the xi for which

h

where S = min S, F(,

-,,

(n, m-n) is the upper (1-a) point of variance distribution for n and (m-n) degrees of freedom, where m and n are the number of data points and parameters respectively. The procedure is to choose a confidence probability (1-a), and, using the above inequality to determine the critical value of

S corresponding to (1-a), then to vary the parameters one at a time by trial and error to determine the upper and lower limits of each parameter where S assumes the critical value. Alternatively, in the vicinity of the least-squares estimate, by virtue of the linearized Taylor series expansion, one-parameter-at-a-time confidence limits for xi are

where t(,-,) (m-n) is the two tailed (1-a) point of the Student's t distribution, S, is the standard error of estimate given by

Se = [~[(m-n)]'I2 ( 5 )

and Cjj is the diagonal element of the inverse matrix C of the least-square normal equations. Comparison of the confidence limits computed from equations (3)

and (4) affords a measure of the local nonlinearity of the model. As suggested by Stone [30], within the applicability of linear theory, the ttue maximum limits for each parameter individually are given by support- plane confidence intervals,

Marquardt's routine prints out all this information directly and there is an option of force-off to calculate and print the confidence limits for any set of parameters.

3. Results and discussion.

-

In all the analyses, following Jacobs and Pantelis (JP), the entropy of association, q, was neglected and an eight parameter

model was used. The converged set of parameters, appropriate to the Schottky model, together with, S, the sum of the squares of the deviations of loglo

oT

at the minimum, S,, the standard error of estimate, and N,,, the number of positive deviations of

are given in tables I and I1 for the runs 102-2 and B4-6 respectively. The starting set of parameters given in the upper rows in column 2 of table I was used for all sets of data and all the parameters were varied freely from these values to check the consis- TABLE I

Fitted and the fit parametersfor 77-points 102-2 run ; zone refined KC1

Parameter (*)

-

hs Ss Ahc Ase Aha Ass C X 104

s

103

s,

N P ~ Initial

values (a) Powell's VAOSA accuracy of S

10-3 10-4 10-5 10-6 - - - - 2.301 2.300 2.285 2.281 2.236 2.281 0.444 0.445 0.667 0.712 0.538 0.773 0.699 0.698 0.698 0.698 0.686 0.698 0.223 0.222 0.099 0.073 0.200 0.396 1.55 1.553 1.567 1.580 1.598 1.605 1.067 1.068 0.954 0.923 1.085 0.906 0.238 0.240 1.005 1.365 0.253 1.973 0.435 0.422 0.217 0.195 0.300 0.063 7.09 3.22 2.65 2.60 9.98 2.58 3.20 2.16 1.96 1.94 3.80 1.93 5 32 37 38 45 36

(*) Enthalpies and entropies are given in eV and meV/K respectively and C in ppm.

(a) The second row values for each parameter correspond to the second set of starting parameters.

( b ) From sample output run on IBM 7094.

(5)

C. S. N. MURTHY AND P. L. PRATT Parameter (*) - hs ss Ahe Asc Aha AS^ C X 104 S (* *) TABLE I1

Fitted and the fit parameters for 92-Point B 4-6 run : 5 N Purity KC1

Powells' VAO5A accuracy of S

10-3 10-4 10-5 10-6 -

-

2.042 1.956 1.955 1.927 -0.061 -0.197 -0.152 -0.185 0.670 0.666 0.663 0.660 0.270 0.277 0.249 0.245 1.266 1.286 1.286 1.292 0.956 1.010 0.986 0.996 0.541 0.474 0.622 0.633 0.590 0.598 0.590 0.592 8.47 3.48 3.25 3.23 E Test - 2.08 i 0.44 0.10 & 0.42 0.67 k 0.06 0.26 & 0.26 1.26

+

0.06 0.93 k 0.15 0.63 k 1.34 0.58 L- 0.04 3.46

Marquardt's NLIN 2 convergence criteria

E Test (a) vc Test ( b ) VE Test (3 JP (e)

- - -

-

%

limits (*)

Linear model Nonlinear model Parameter (equation (4)) (equation (3))

-

(1.92, 2.23) (2.079, 2.080) h, (1.87, 1.92) (1.894, 1.895)

,

(- 0.14, 0.16) (0.007, 0.008) ss (- 0.17, - 0.09) (- 0.131,

-

0.130) (0.65, 0.69) (0.669, 0.670) Ahc As, (0.17, 0.35) (0.258, 0.259) 79 (0.16, 0.18) (0.1 820, 0.1 825) Aha (1.24, 1.28) (1.262, 1.263) (0.88, 0.99) (0.935, 0.936) 51

Ass

C (0.15, 1 .lo) (0.624, 0.630)

X

(0.57, 0.60) (0.583, 0.585)

(*) Correspond to the R 4-6 run for column 6 set of parame-

ters. The second row values given for hs, s,, s, are for a fit in

i

, , A A ; A , i * , . , 10

1

which C was kept fixed. See the text. o

L

d 12

0

5.5 $ A

.J

while mentioning the type and series of the computing 0.95 097 0.99 1.01 203 10s

machine used when presenting computer-fitted results. 1000/~( LI

3.2 RUN B4-6.

-

Table I1 shows the results on run B4-6. For this run also S is decreased as the accuracy is increased but the changes in C, s, and Asc are much smaller than in run 102-2. The mean set of parameters reported by Jacobs and Pantelis for

h,, s,, As, and C given in the last column of table I1

do not compare well with any of the other sets in the table. Column 7 corresponds to the check on the VAOSA solution through NLIN 2 and in column 8

we show the converged parameters obtained using the Jacobs and Pantelis mean values of column 10 as the starting set. It should be pointed out [27] that when the ye test is passed, presumably the value of S is

minimized within the limits of the rounding error. However, as the present calculations were performed on a CDC machine with fourteen digit accuracy, the values should still be quite reliable.

Instead of showing the detailed deviation plots, corresponding to all the sets of parameters in table 11, from which one can draw merely qualitative conclusions concerning the nonrandom behaviour of error distribution, the number of positive deviations are given in the last row ; at high temperatures, the deviations are large in magnitude as shown in figure 5. An impartial judgement about the goodness of a fit may be obtained from the standard chi square test [24] but we have not tried this because all the fits correspond to the complete conductivity plot. The fits are different over different ranges of temperature and thus it may be difficult to get a normal distribution. For three input values of C, namely 0.5, 0.89 and 1.1 1 ppm, allowing all parameters to vary, the converged values for

C

are 0.64, 0.68 and 0.70 ppm respecti-

FIG. 5. - High temperature conductivity (x, a, and correspond to 8 points from 102-2, original B4-6 and 136 point composite runs respectively) and deviations (@, @ and 0 are unconstrained fits from NLlN 2 for 10 2-2, original B4-6 and composite runs respectively ; L I and A correspond to JP mean set of results for B4-6 and 136 point composite run). Missing points

indicate their coincidence with other points.

vely whereas the other parameter values are similar to those given in column 6 and 8 of table 11. The corresponding values for

S^,

are 3.45, 3.44 and 3.44 units) respectively. This reproducibility of converged values for the concentration and the ability to predict both a higher concentration and a smaller h,

value compared to run 102-2, as expected from the conductivity plots (figures 1 and 5), suggest that the minima found are reliable.

(8)

ON THE ANALYSIS O F IONIC CONDUCTIVITY C7-313

the various estimated parameters for column 6 of table 11. All the parameters are highly correlated and since the correlation of C with other parameters in the case of the B4-6 data is somewhat less important compared to that of run 102-2 the convergence of C in this case is rather consistent.

4. Conclusions. - We may now draw some conclusions on a very general basis for the defect energy parameters we have obtained. Having checked the solutions obtained by the two routines by treating the solution from one as a starting set for the other (columns 6 and 9 of table I, columns 2 and 7 of table 11), and since it is hard to see from the deviation plots, except for a low accuracy of that this set of parameters could be discarded, consideration of all the analyses at their face value leads to the results given in column 2 of table V. The limits in column 2 include all the analysed values, and by analogy with precision limits, we may call them the support-plane confidence limits which are realistic only if the parameters are not

constrained. The nature of the parameters and their limits given in the third column may be taken in the same spirit ; these were obtained by omitting the sets of values given in columns 5, 6 and 9 of table I, and thus they correspond to a somewhat constrained procedure which is reasonable in view of the judgement of the relative amount of impurity content. An enforcement in different parameters separately or together results in different sets and an average of these would lead to supposedly precise values (column 4) for the enthalpy and entropy of formation of a Schottky pair and the entropy of cation vacancy migration.

Even though the experimental data may cover a range as wide as five orders of magnitude in alkali halides, and may be precise up to four significant digits corresponding to each exponential term in the model, the data may not appear to support the model for a number of different reasons. Marquardt (') has pointed out that the limited regions over which the various

( 2 ) Marquardt, D. W., private communication.

Typical matrix of correlations among estimated parameters

Parameter - hs S s Ahc Asc Aha

Ass

c

(9

X

(*) The first row corresponds to the B 4-6 run pertaining to the parameters in column 6 of table I1 and the completematrix is given in this case. The second row is that of the composite run, and the last row is for 102-2 run corresponding to the parameters of column 7 of table I.

Defect energy parameters for KC1

maximum limits covering unconstrained (2nd column) and somewhat constricted (column 3) analyses.

(9)

C7-314 C. S. N. MURTHY AND P. L. PRATT

conduction processes occur may be an important fac- tor, while Barr and Lidiard [14] have drawn attention to the difficulties in computing arising from the compa- rable magnitudes of the anion and cation migration enthalpies in the intrinsic region, and of the cation migration enthalpy and the enthalpy of association in the extrinsic region. Finally, the eight parameter model itself may be inadequate, and, if independent physical evidence for other transport processes, is forthcoming, further terms may have to be added to the model. Under these circumstances, it may be hard to assess the accuracy of the values of the parameters determined by fitting the whole conductivity curve. As Chandra and Rolfe [18] have shown, there is a need to include both cation- and anion-doped conductivity results and preferably diffusion results too if meaning- ful values of the parameters are to be obtained. In the recent work of Beni&re et al. [31], data from transport processes, including both conductivity and diffusion, of pure, cation-doped and anion-doped KC1 have been

analysed in a simple way considering explicitly the vacancy pair contribution to the diffusion. The close agreement between the values for KC1 reported in the last three columns of table V, which were obtained from such combined studies, shows the importance of proceeding in this way. Computer fitting of ionic conductivity data alone is unlikely to lead to reliable values.

5. Acknowledgment.

-

We are extremely grateful to Professor P. W. M. Jacobs and to Dr. P. Pantelis for kindly sending us their data and for their help by correspondence and by discussions. Special thanks are due to Dr. I. M. Boswarva for helpful discussions and to Mr. Evans for the continued help in computing. One of us (C. S. N. M.) is indebted to the Department of Metallurgy and Materials Science for financial support and acknowledges gratefully stimulating discussions he had with Dr. A. B. Lidiard, Dr. J. A. D. Matthew and Professor F. G. Fumi.

References

[I] KOCH, E. and WAGNER, C., 2. Phys. Chem. B38 (1937) 295. [2] T E L T O ~ , J., Ann. Phys. 5 (1949) 63, 71.

[3] ETZEL, H. W. and MAURER, R. J., J. Chem. Phys. 18 (1950) 1003.

[4] LIDIARD, A. B., Phys. Rev. 94 (1954) 29.

[5] LIDIARD, A. B., Handbuch der Physik ed. by S. Flugge, vol. 20, Part 2, pp. 246 (Springer verlag, Berlin) 1957. [6] ALLNATT, A. R. and LOFTUS, E., J. Chem. Phys. 59 (1973)

2541.

[7] YUEN, P. S. and ALLNATT, A. R., J. Phys. C 8 (1975) 2213. [8] DREYFUS, R. W. and NOWICK, A. S., J. Appl. Phys. 33 (1962)

473 ; Phys. Rev. 126 (1962) 1367.

[9] ALLNATT, A. R. and JACOBS, P. W. M., Trans. Faraday Soc. 58 (1962) 116.

[lo] KIRK, D. L. and PRATT, P. L., Proc. BY. Ceram. Soc. 9

(1967) 215.

[ l l ] BEAUMONT, J. H. and JACOBS, P. W. M., J. Chem. Phys. 45 (1966) 1496.

1121 DAWSON, D. K. and BARR, L. W., Phys. Rev. Lett. 19 (1967) 844.

[13] FULLER, R. G. and REILLY, M. H., Phys. Rev. Lett. 19 (1967) 113.

[14] BARR, L. W. and LIDIARD, A. B., Physical Chemistry, an

Advanced Treatise ed. by H. Eyring, D. Henderson

and W. Jost, vol. 10, chap. 3, (Academic Press, New York) 1970.

[15] ALLNATT, A. R. and PANTELIS, P., Solid State Commun. 6

(1968) 309.

[16] ALLNATT, A. R., PANTELIS, P. and SIME, S. J., J. Phys. C 4 (1971) 1778.

[17] JACOBS, P. W. M. and PANTELIS, P., Phys. Rev. B 4 (1971) 3757.

[la] CHANDRA, S. and ROLFE, J., Can. J. Phys. 48 (1970) 397, 412 ; Zbid. 49 (1971) 2098 ; Zbid. 51 (1973) 236. [I91 CORISH, 3. and JACOBS, P. W. M., J. Phys. & Chem. Solids

33 (1972) 1799.

[20] BROWN, N. and JACOBS, P. W. M., J. Physique Colloq. 34 (1973) C 9-437.

1211 ABOAGYE, J. K. and FRIAUF, R. J., Phys. Rev. B 11 (1975) 1654.

[22] FULLER, R. G., MARQUARDT, C. L., REILLY, M. H. and WELLS, J. C., Phys. Rev. 176 (1968) 1036.

[23] NELSON, V. C. and FRIAUF, R. J., J. Phys. & Chem. Solids

31 (1970) 825.

1241 FRIAUF, R. J., J. Physique Colloq. 34 (1973) C 9-403. [25] VAO5A HarweIl Library Routine.

1261 POWELL, M. J. D., Harwell Report AERE-R5947 (1968). [27] IBM Share Library Distribution No. 309401 NLIN 2. [28] MARQUARDT, D. W., J. SOC. Znd. AppI. Math. 11 (1963)

431.

[29] WOLBERG, J. R., Prediction analysis (D. Van Nostrand Co. Inc.) 1967.

[30] STONE, H., J. R. Statist. Soc. B 22 (1960) 84.

[31] BENIBRE, M., CHEMLA, M. and BENJ~RE, F., J. Phys. & Chem.

Solids 37 (1976) 525.

[32] Box, M. J., DAVIES, D. and SWANN, W. H., Nonlinear

Optimisation Techniques. I. C. I. Monograph No. 5

(10)

ON THE ANALYSIS OF IONIC CONDUCTIVITY

DISCUSSION

A . R. ALLNATT.

-

I wish to mention that in my P. PRATT.

-

Correct. We are delighted to find our experience the performance of computer fitting selves in agreement with Prof. Friauf, Prof. BCnibre, analysis is not satisfactory if only pure crystals are and all the participants in the discussion.

used, but is satisfactory if analysis is performed on pure crystals and on doped crystals in which the impurity content has been accurately determined, for example by isotope dilution. In the case of NaCl this proadure has given values for the enthalpy and entropy of Schottky defect formation and the enthalpy and entropy of cation migration in good agreement with the recent independent determination of BCnihre, Chemla and BCni6re (J. Phys. & Chem. Solids 1976). Because of the relatively small anion transport number the anion migration parameters cannot be determined satisfactorily from the conductivity analysis for NaCl. Details can be found in a paper by Allnatt, Pantelis, and Sime (J. Phys. C : Solid St. Phys. 1972). F. BBNIERE. - Finally, the emerging conclusion outlined by Prof. Pratt, and on which all the participants in the discussion are agreeing, emphasizes the need of combining the tracer self-diffusion coefficient of both ions to the ionic conductivity, if possible in pure and in doped crystals, in order to obtain reliable

values for all the defect parameters.

R. J. FRIAUF. - Let me paraphrase your moral. Computer fitting is a dangerous business and must be carried out with great care. It is also vital to use all other available information, such as diffusion coeffi- cients. And whereever the plot of residual deviation us. parameter value does not show any minimum, then there is almost certainly some difficulty.