HAL Id: jpa-00216930
https://hal.archives-ouvertes.fr/jpa-00216930
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.
ON THE INITIAL STAGE OF IMPURITY-VACANCY COMPLEX AGGREGATION IN NaCl : CaCl2
A. Kessler
To cite this version:
A. Kessler. ON THE INITIAL STAGE OF IMPURITY-VACANCY COMPLEX AGGREGA- TION IN NaCl : CaCl2. Journal de Physique Colloques, 1976, 37 (C7), pp.C7-291-C7-296.
�10.1051/jphyscol:1976770�. �jpa-00216930�
ON THE INITIAL STAGE OF IMPURITY-VACANCY COMPLEX AGGREGATION IN NaCl : CaCI,
A. KESSLER
2, Institute of Physics, University of Stuttgart, FRG
R6sum6. - Une analyse est presentee de mesures publiees antkrieurement ainsi que de nouvelles mesures de l'affaiblissement du maximum de perte didectrique provoque par les complexes lacune- impuretk dans le NaCl et dans le CaClz. Elle montre (1) qu'il existe, a c8tB de I'aggregation fond&
sur une nucleation homoghe, un processus d'aggregation dfi aux dislocations et (2) qu'un ajustement des courbes d'affaiblissement mesurees sur une equation donnee de vitesse de reaction n'a de signifi- cation physique que si les valeurs des paramMres de I'kquation sont compatibles avec la realit6 physique. Compte tenu de ceci, il apparait que des dimkres sont form& au premier pas de l'agglom6ration.
Abstract. - On analysis of some earlier published [21] and of some new measurements of the decay of the dielectric loss maximum caused by the impurity-vacancy complexes in NaC1: CaClz is presented. It shows, that (1) there exists besides the aggregation based on a homogeneous nuclea- tion, an aggregation process due to dislocations and that (2) a fit of the measured decay curves by a given reaction rate equation is physically meaningfull only, if the values of the parameters of this equation are compatible with the physical reality. Taking this into account it appears that dimers are formed as the first steps in the agglomeration.
Introduction. - The existence in polar crystals of aliovalent impurity-vacancy complexes and their role in the conductivity, in the dielectric loss, in the impurity diffusion etc. is generally accepted [I]. There is in consequence no doubt that the initial stage of the precipitation of aliovalent impurities can be but an agglomeration of complexes. It is the merit of Cook and Dryden [2, 31 to have undertaken the first attempts to study this agglomeration in detail. Based on an analysis of the decay with annealing time of the dielectric loss maximum caused by complexes, they arrived at the conclusion that the first step in this process is the formation of so called trimers.
This notion was since confirmed by various investi- gators for a variety of substances and admixtures (see survey articles [4,5]).
This result is surprising. The simultaneous encounter of three complexes is surely far less probable than that of two only and no plausible reason has yet been given why dimers should not exist. To the contrary, Capelletti and De Benedetti [6], who used the more accurate and sensitive method of ionic thermocurrents for their investigations found that the beginning of the decay in KCl: SrCI, can be fitted by a second order kinetics equation. It was then realized, that eventually a second order process could be masked due to a low binding energy of the dimers 171 and that by taking into account the migra- tion of the complexes which is to precede the asso- ciation [8] or the back reaction (dissociation of dimers) which necessarilly is due to take place 191,
experimental data can be fitted by appropriate second order kinetics equations.
It appears, that the interpretations of the experi- ments are contradictory. This is shown especially by some recently published comments on this issue [lo].
It is the discussion of this aspect of the problem we want t o put the emphasis on in this talk, rather than on the support given to the idea of dimer formation by the presented experimental data, because they, too, turned out to be insufficient to give an answer in all the details needed to form a definitive picture of the agglomeration.
Discussion of some earlier published data on complex agglomeration in NaCl : CaCI,. - It is tacidly assumed by most authors dealing with our subject, that the agglomeration takes place under homogeneous and isotropic conditions. This sup- position does not seem to be always justified. As-grown alkali halide crystals, e. g., show a fairly high density of dislocations (- lo7 cm-2 1111) and these attract complexes 1121. In consequence the formation of zones of increased impurity concentration (Cotterel clouds [13]) with an enhanced local agglomeration is to be expected around the dislocations. This effect is e. g. the basis of the well-known decoration tech- nique [14, 151 used to make dislocations visible. One characteristic feature of this process is a lag of the set in of the agglomeration behind the beginning of the anneal. If however the agglomeration obeys a quasichemical kinetic equation (see e. g. [4]) as it is proposed by most of the investigators, the decay
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1976770
C7-292 A. KESSLER is due to exhibit the highest rate in the beginning ; the rate decreases then steadilly till equilibrium is reached. If we look under this point of view at the decay curves obtained for NaCI : CaCI, by Cook and Dryden [2] reproduced by figure la, the suspicion arrises that aggregation at dislocations plays indeed a role. At loy impurity concentrations there is a considerable delay of the beginning of the decay and only if the impurity concentration is high enough it is obviously overplayed by the homogeneous aggre- gation.
ro E 1 3 . 6 r l ~ - ' m ~ l fr ca2*
-
m. - - -
2. 2.2x10-'rno1 3 6 8 ~ 1 0 - ~ m o l fr. fr. ca2' co2'x 0.5 -
d
-
ma
FIG. 1. - Decay of tg 6mx caused by complexes in NaCl : CaCl2 at 20 C according Cook and Dryden [2] as a function of
annealing time t (a), as a function of t 2 / 3 (b).
The concentration N(t) of impurities precipitated at the dislocations after an annealing time t is accord- ing to Cottrell and Bilby [16]
No is the initial impurity concentration, K is for a given system a constant, p is the dislocation density and D is the impurity diffusion coefficient. Eq. (1) was, it is true, originally developed to explain pre- cipitation in metals. The situation in polar crystals is though, in principle, much alike, the difference being mainly in the diffusion mechanism. It could be shown in fact, that e. g. conductivity data obtained during the annealing of supersaturated NaC1: BaCl, crystal samples with different dislocation densities [ l l ]
can be satisfactorilly fitted by eq. (1) and that the rate of decay is proportional to the dislocation density p. Further an electron-microscope investiga- tion by means of a technique developed by Bethge 1171 revealed that precipitates occured usually in a dis- location core and that the pattern of the decoration of the thermally etched surfaces of the samples inter- preted according Bethge gives evidence for the existence of Cottrell clouds.
The attempt was thus made to represent the data of figure l a as a
tg ~,,(t)ltg amX(0) = [l - N(t)INoI
.
t2I3 plot (Fig. 1 b).
It is seen, that for the lowest impurity concentration of 6.8 x mol fr. of CaC1, one gets a straight line over practically the whole range of decay and for 2.2 x molfr. there is still a straight line over
3
of the range of decay. As the diffusion coeffi- cient is increasing with the impurity concentration [18]the higher rate of precipitation in the latter case is also in accord with eq. (1). For the highest CaCI, concentration eq. (1) is already not obeyed.
[I:vi
2. 3. 1. 3.6xf0-'mol 2 . ~ ~ 1 0 ~ 6.8~0.' mol mol fr. fr. fr. ca2' co2' ca2+5
0 )-
0.2300 LOO
T a d K 1
FIG. 2. - Relative equilibrium concentration of unassociated complexes as a function of the annealing temperature Tan*
according Cook and Dryden [2].
In the discussed paper also equilibrium values of the concentration of unassociated complexes are given (see Fig. 2). Let us assume that the effect of the Cottrell cloud formation is only an increase of the complex concentration. The equilibrium ratio of free and of agglomerated complexes should then allow in principle a check, whether dimers or trimers are formed. If n,(T), n,(T) and n,(T) denote the equilibrium concentrations of complexes, dimers and of trimers resp., c the concentration of all the complexes involved, whether associated or not, and K(T) = K. exp(U/kT) denotes the respective reaction constant with an activation energy U, the equations
FIG. 3. - Reaction constant calculated from the data of figure 2 under the assumption of dimer formation (a), trimer
formation (b).
should apply. In figures 3a and 3b the logarithm of K(T) vs.103/T as calculated from some random choosen points from the curves of figure 2 is given.
As U is assumed to be independent of c, parallel straight lines should fit the data calculated by means of the valid equation. For the second order equation there is a modest fit, alone it is not that better than the fit for the third order equation to justify a deci- sion between the two models. Only the activation energy which would follow from figure 3b seems far too high to be real. The binding energy of simple complexes reported by different authors is 0.29 eV [19], 0.31 eV [20] and 0.67 eV [21] resp. and the binding energy of agglomerates is hardly expected to exceed it. One would expect thus rather the second order equation to be right.
The deviation of the measuring points from the straight line at 103/T < 2.5 is due to the increasing inaccuracy of 1 - tg 6,,(Ta,,)/tg 6,,(403O) and the deviation for 1 0 3 / ~ > 3.0 probably due to a syste- matic error in the tg 6,,(Tan,) values in consequence of the overlap of the maximum by a new loss peak which is formed in due course of the agglomeration.
Cook and Dryden report, that after annealing at 130 OC and subsequently quenching the samples a relaxation maximum is found at -- 230 Hz for 20 OC, which is attributed to complexes. It is the maximum, the decay of which they investigated. After a storage of the samples for same time the tg 6,, of this peak decreased and a second peak occured at a higher frequency. In due course of the storage it increased further and shifted to still higher frequencies till it reached about 630 Hz. As it does not seem likely that trimers should cause a relaxation loss also this fact speaks rather against trimers.
Some new experimental data on the complex agglomeration in NaCl : CaCl,. - The analysis of the experimental data just given shows, that more favourite experimental conditions should be sought, mainly to avoide the influence of dislocation. NaCl crystals with a higher CaCl, content were thus inves- tigated. The high CaCI, concentration was expected to secure moreover a higher rate of decay and a decreased distance a complex is to travel to join another one (cf. [8]).
To avoid under these conditions the occurance of higher agglomerates in fresh samples they were annealed at temperatures over 3000C. After the anneal they were quenched to room temperature and put immediatelly into the already heated measuring cell. That way the first data could be obtained in fractions of a minute. The measurement of the tg 6 was performed by means of a General Electric 1 605 A impendence bridge. It was made sure by an investi- gation of the temperature dependence of the tg 6 of freshly quenched samples that there is only the peak due to complexes. With the proceeding of the agglomeration though the loss maximum that follows annealing (cf. preceeding section) overlaps somewhat the complex maximum and causes in the later stages of the decay an increasing error.
100 200 300 400 500 I
t tmin I
FIG. 4. - Decay of the tg 6mx due to complexes in NaCl : CaC12 as a function of annealing time at 60 C (A). Reaction constant calculated from the tg 6mx and its derivative under the supposition of dimer formation ($), trimer formation ($) and no
backreaction.
A typical decay curve is shown in figure 4. The maximum decay rate occures at the beginning and the rate decreases continuously. This shows that an attempt can be made to fit a kinetic equation to the data. As the tg 6,, is proportional to the number of complexes,
{
l/tg 6,,(t) - l/tg 6,,(0)) was plotted against t for a fixed impurity concentration and various fixed temperatures (Fig. 5a) and for samples with different impurity contents at a fixed tempe- rature (Fig. 5b). It is seen, that at the beginning the curves can be fitted by straight lines through the origin as is to be expected in the case of a second order kinetics process before the backreaction makes itself felt. But this fit by itself does not prove the second order of the decay. Also for third order kine- tics a fit is obtained and even the deviations of the measuring points from the straight lines do not provide a means for a decision ; the only difference is a longer time interval in which the third order kinetics fits the data.To eliminate the inaccuracy of the data and its even- tual influence on the fitting the smooth tg a,,-curve
C7-294 A. KESSLER
through the measuring points of figure 4 was esta- blished and the ratio of the derivative of tg dm, to
2 0 the square and to the third power resp. of l/tgdmx
determined. The results are found in figure 4. If the alternative is not strictly second or third order kinetics, these results could indicate that there is first a second order kinetics followed by a third order kinetics.
G 15 This would be in agreement with the measurements
- i!
of Capelletti and De Benedetti [6] on KC1: Sr2+a and the theory of Crawford [7].
a
C
1 The linear portions in figures 5a and 5b should
-
1
give the value of the reaction constant K(T). In
-
10 figure 6 the In K(T), as determined from the slopesx .c. in figure 5a is plotted against 103/T. The representation
w E
0,
C
\
-
5 c = const.
I I I I I I I
-
20 40 60 80 5
(4
t [minl Yc 2.0-
-
FIG. 6. - Reaction constant calculated from the linear approxi- mations of the initial decay, c = 1 mol % i. m.
yields, as is to be expected for a thermal stimulated process, a straight line and gives an activation energy of about 0.37 eV. This value should be compatible with the activation energy of the diffusion coefficient D(T) of Cazf. The only known experimental value is 0.9 eV [22] for undoped crystals. But D(T) for aliovalent impurities depends on the impurity con- centration [1, 181 : It increases with the impurity concentration, but the activation energy should decrease at the same time. Thus experimental values for zink and magnesium in undoped samples are 1.02 eV [23] and 0.95 eV [24] whereas in doped crys- tals they are 0.51 eV [23] and 0.66 eV [25] resp. If we are not too particular the value of 0.37 eV might thus be acceptable, if the limited accuracy of our
(b) t [ r n i n ~ figure is taken into account. Also a comparison
of K ( T ) for different impurity concentrations suits FIG. 5. - Representation of the measured tg Grnx = f ( t ) the known dependence of D(T) on the concentration.
data under the supposition of dimer formation and no back- seen in figure 5b, K(T) is increasing with con-
reaction : (a) for a constant CaClz-content and different anneal-
ing temperatures, (b) for a constant annealing temperature and centration and the increase is higher at
different CaClz-content. low concentrations. as it should.
Analysis of the initial rate of decay.
(Tan, temperature of agglomeration,
A
rate of decay of the tg dm,)N cN(ca2+) Tan" "N
&XN
( ) (
tg) (5)
mol
%
OC min: CN+I \ tg Smxlv+ I n ~ + 1 ,(*) From tg 6 measurements of NaCl : CaClz stored for a long periode of time [26], it is known that CaC12 is built-in substitu- tionally only up to roughly 0.05 mol % of CaC12 in the crystal.
Finally the initial rate of decay
was determined for samples with different impurity concentrations and the data summarized in table I.
The figures show, that the rate of decay of the complex concentration is in good agreement with the second order kinetics of eq. (3).
Summary. - The evaluation of an observed complex concentration decay consists basically in the attempt to find a kinetic equation, which would fit the expe- rimental data. It is believed, that such an equation depicts in principle the physical reality of the observed process. We have seen though, that it was possible to fit the data with at least two different models.
The reason of this ambiguity is, in our opinion, the backreaction and possibly still other factors too, due to which the solution of the kinetics can in reality be approximated by a power series. Given a certain
inaccuracy of the data, one gets then in a certain restricted interval a second order kinetics fit, in a longer interval a third order kinetics fit or by a more accurate method different intervals in which the one and the other fit. The fit alone has in conse- quence hardly any physical meaning.
To reach a fit, appropriate values of some constants contained in the tested equations are to be found or emerge from the procedure. This makes the result of the fitting still more dubious except when the physical meaning of the constants is known. In that case namely their value, especially as a function of some proper parameters as e. g. temperature or impurity concentration can be checked against values of the same constant determined from other kind of measurements. Only a fit which is reached by means of realistic values of these constants can be of physical relevance.
If we consider under this point of view the known results, one would tend to accept a basically second order kinetics, modified though by the way the migra- tion of the complexes is realized. As the migration depends on the complex concentration and on the dislocation density of the crystals, the agglomeration should be to a high extent structure sensitive.
References LIDIARD, A. B., Encyclopedia of Physics, Vol. XXII
(Springer Berlin) 1957.
COOK, J. S., DRYDEN, J. S., Austr. J. Phys. 13 (1960) 260.
COOK, J. S., DRYDEN, J. S., Proc. Phys. Soc. 80 (1962) 479.
HARTMANOVA, M., Phys. status Solidi (a) 7 (1971) 303.
CAPELLETTI, R., Proc. Symposium on thermal and photo- stimulated currents, Dallas, 1975 (Elchem. Soc. Inc., Princeton) 1976.
CAPELLETTI, R., DE BENEDETTI, E., Phys. Rev. 165 (1968) 981.
CRAWFORD, J. H. Jr, J. Phys. & Chern. Solids 31 (1970) 399.
WINTLE, H. J., Phys. Rev. 179 (1969) 769.
UNGER, S., PERLMAN, M. M., Phys. Rev. B 10 (1974) 3692 ; ibid. B 12 (1975) 809.
COOK, J. S.. DRYDEN. J. S.. Phvs. Rev. B 12 (1975) 5995 and UNGER, S., PERLMAN, M: M., ibid. p. 5597.
'
KUPCA, S., HARTMANOVA, M., VLASAK, G., Czech. J.
Phys. B 19 (1969) 789.
FRANK. W.. Phvs. Status Solidi 29 (1968) 767.
COTTRELL, A. H., Rep. on strength of solids, Phys. Soc.
Lond. 1948. o. 30.
[14] HARVEY, K. B., Phil. Mag. 8 (1963) 435.
[15] AMERLINCKX, S., Sol. Stat. Phys. Suppl. 6 (1964) 55.
[16] COTTEREL, A. H., BILBY, B. A., Proc. Phys. Soc. A 62 (1949) 49.
[17] BETHGE, H., Phys. Status Solidi2 (1 962) 3.
[18] KENESHEA, F. J., FREDERICKS, W. J., J. Phys. & Chem.
Solids 26 (1965) 501.
[19] TRNOVCOVA, V., Czech. J. Phys. B 19 (1969) 663.
[20] KANZAKI, H., KIDO, K., TAMURA, S., OKI, S., J. Phys.
SOC. Japan 20 (1965) 2305.
[21] MURIN, A. N., BANASEWICH, S. N., GRUSHKO, Yu. S., Sov. Phys. Solid State 3 (1962) 1762.
[22] BANASEWICH, S. N., LURE, G. B., MURIN, A. N., Fiz.
Tverd. Tela 2 (1960) 80.
[231 ROTHMAN, S. J., BARR, L. W., ROWE, A. H., SELWOOD, P. G., Phil. Mag. 14 (1966) 501.
[24] STEWARD, W. H., REED, C. A., J. Chern. Phys. 43 (1965) 2890.
[25] LURE, B. G., MURIN, A. N., BRIGEWICH, R. F., Fiz. Tverd.
Tela 4 (1 962) 1957.
C261 KESSLER, A., MARIANI, E., Czech. J. Phvs. B22 (1972) 77.
C7-296 A. KESSLER
DISCUSSION F. AGULLO-LOPEZ. - I would like to ask two ques-
tions :
a) Our data on NaCl : Mn (90 ppm) indicate no difference in the kinetics of dipole aggregation after a few percent plastic deformation. Have you any comments on this ?
6 ) I feel that a definite prediction on the kinetics of dipole aggregation cannot be made unless the ini- tial state of the system is well known. Optical data on NaCl : Pb and KC1 : Pb show that even after a very severe quenching substantial amounts of simple dipole aggregates are observed except possibly for very low doped crystals. Any comments ?
A. KESSLER. - There are conceivable various reasons : e.g. different values of the parameters rele- vant for the agglomeration, which would shift the treshold for the predomination of the homogeneous nucleation. As to my knowledge evidence exists for the said process in NaCl : BaC1, [ll], in NaCl : SrC1, (see contribution abstract booklet pg. 175 by M. Hart- manova and G. Vlasak) and as shown here, in NaCl : CaCl,. These earth-alkali metals have ionic radii of 1.35, 1.13 and 0.99 A resp., wheras Mn has only 0.80 A.
As far as the second question is concerned I can only confirm, that by annealing and quenching one cannot remove higher aggregates completely. We have tried to prevent their formation, by quickly
cooling the crystals after their growth. Nevertheless the tg6,,, of the 1
%
doped crystal was (cf. Table I) not much higher than that of the 0.01%
doped crystal.But a repeated annealing and quenching after the measurement of the decay restored the original values of the tga,,,. This makes us believe, that if there were precipitates present, their number was too small for a measurable influence on the agglomeration.
L. SLIFKIN. - 1 will later show some results on AgCl doped with Mn2 + which, analyzed by 3 different criteria, indicate that whether the process proceeds by 2nd or 3rd order kinetics may depend strongly on the initial degree of supersaturation (of course AgCl : Mn is not the same as NaCl : Ca).
A. KESSLER. - I am far from implying, that under all circumstances either a 2nd order or a 3rd order equation fit must be possible.
Whether the results you mention can be compared with ours is, as you point out yourself, questionable : in the first place there is, if I am not mistaken a far higher solubility limit in AgC1. But if you go through some details which could not be mentioned in the oral presentation because of lack of time, you will find that there is room for an interpretation, that a third order process is overlapping after some time the initial second order process (nucleation). And that could easily be in line with your findings.
