LATTICE DEFECT ENTROPIES IN IONIC CRYSTALS

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LATTICE DEFECT ENTROPIES IN IONIC

CRYSTALS

F. Bénière

To cite this version:

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LATTICE DEFECT

ENTROPIES IN IONIC CRYSTALS

Physique des Materiaux, Institut Universitaire de Technologic, (UniversitC de Rennes), 22302 Lannion, France

Rbum6. - Les entropies des defauts de reseau dans les cristaux ioniques sont calculCes selon une methode theorique simplifiee a partir des constantes thermodynamiques et mkcaniques du solide. La methode est appliquee a l'evaluation des entropies de formation des : defauts de Schottky, paires de lacunes et associations impurete-lacune, ainsi qu'au calcul de l'entropie de solubilite des impuretbs dans les cristaux ioniques.

Abstract. - The entropies of lattice defects in ionic crystals are computed by a simplified theoretical method from the thermodynamical and mechanical constants of the solid. The method is applied to the evaluation of the entropies of formation of : Schottky defects, vacancy pairs and impurity-vacancy pairs, as well as to the calculation of the entropy of solubility of impurities in ionic crystals.

1. Introduction. - The different entropies of lat-

tice defects in ionic crystals are : ₄

lEll

- Entropy of formation of point defects (vacancies and interstitials) ;

- Entropy of association of the defects (vacancy pairs, impurity-vacancy pairs) ;

- Entropy of solubility of impurities (hornovalent and aliovalent ions) ;

-

Entropy of migration of the ions (cation and anion).

A A A A A Q A

A A A A A A A

A A m A A A A

Some years ago, all those parameters were only A A A A A considered as a part of the so-called preexponential

factor which appears in the expression of any trans-

port process. Indeed, it was difficult to go further

d

m

&I

because of the lack of accuracy in the determinations FIG. 1. - The lattice defect entropies can be measured at of these preexponential factors. Now on the contrary, thermodynamical equilibrium : Number of entering the due to the considerable improvement of the experi- lattice = number of escaping from the l a z e .

El

can be

mental techniques, the reported values are in close any impurity or any point defect. agreement between the different laboratories. Because

of the reliability of these new results and the amount of information which is contained in the entropies, it is worthwhile to analyse these parameters in deeper detail.

Our results for the entropies o f : formation of Schottky defects and solubility of impurities have been reported recently in refs [I] and

121 respectively.

Oiily the main outlines will be recalled in this paper which will focus on the further significance of entropies. Before, it seems useful to define the exact meaning of the entropies as derived from experiment.

2. Experimental definition of the entrepies. - Let us consider a perfect crystal composed of atoms A and surrounded by atoms B (Fig. la). When the crystal

is heated at a temperature T, B diffuses into A (Fig. lb). Then, B is called a point defect of the crystal A. An equilibrium is reached when the mole fraction x of B becomes equal to a value called solid solubility. This is true whatever the nature s f B, which can be as well an impurity, or a vacancy, or an interstitial atom, or a F-center, etc

...

The condition of equilibrium may be expressed in terms of chemical potentials as :

PB(B) = PB(A) (1) that is, the chemical potential of B is the same in both phases. Neglecting the solubility of A in B and for small solubilities of B in A, one gets

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LATTICE DEFECT ENTROPIES IN IONIC CRYSTALS C7-26 1 which can be rewritten in the form : is the free enthalpy (or Gibbs free energy) of one mole

where R is the gas constant (which is to be replaced by Boltzmann's constant if AGO is expressed in eV), T the absolute temperature and AGO the standard

0 0

Gibbs free energy (= /+(A)

-

pB(B)).

In most practical cases, in a first but good approximation, x depends exponentially on temperature. The temperature dependence of x may therefore be written in the form :

with the constants S and H not depending on temperature. Equations (3) and (4) lead to identify the experimental parameter S to be the entropy As0 and H to be the enthalpy AH0. However, let us notice that if the enthalpy AH0 slightly depends on temperature (as it does very probably), it may be developed to the first order as :

Then the experimental entropy S has the following meaning :

s

= AS'

+

~ ( A H O ) / ~ T (6)

namely, S would contain the first derivative of the enthalpy with respect to temperature, though x would not depart from a pure Arrhenius equation.

The most direct way to determine S and H is there- fore to measure x at several temperatures.

In the case of the impurity solubility, a precise method consists in surrounding A with B and to let the impurity diffuse until the solubility limit is reached (Fig. 1). By marking B with a radioactive isotope, counting of the radioactivity of B in A readily gives

x at the temperature of diffusion [3].

In a similar way, the mole fraction of Schottky defects may be derived from the knee temperature observed in the plots of the self-diffusion or of the ionic conductivity of doped crystals. If C is the mole fraction of divalent cations doped in an alkali halide, the simple equation :

x = C (7)

holds in a first approximation at the knee temperature. Results obtained by different workers for several kinds and concentrations of impurity are consistent [4] and allow a direct determination of S.

Finally, the point emphasized here is that the defect entropies can be determined straight from direct experiments. Now that the entropies are obtained, let us recall their physical meaning.

3. Physical definition of the entropies.

-

By definition, the standard chemical potential of the solute :

of B in solid solution in A, excluding the mixing

entropy and any interaction B-B. is therefore an energy composed of the two parts :

- p i s the energy per mole in the bonds of B with the surrounding atoms of the lattice in an hypothetical static state ;

-

~3

is the energy per mole in the vibrations of the bonds between B and the surrounding atoms A

of the lattice in the real dynamical state.

In all the calculations, we shall consider temperatures T higher than TE the Einstein temperature namely :

T > TE = hoE/k (9)

where m, is Einstein mean pulsation and k and h the Boltzmann and Planck constants. As a matter of fact, the experimental entropies are determined from experiments carried out at much higher temperature than TE. Then, the vibration entropy of a pure solid is given with a good first order approximation [5] by :

S = 3 R [Ln (kT/h~+) - 3 Ln (N)

+

I]

.

(10) The procedure of the calculation will always be the same. At equilibrium, the defect entropy AS0 defined in the equation (3) is the change of the entropy

-

S defined in the equation (S), namely :

or, more generally, the entropy change between two states.

Finally, the calculation of the defect entropies redu- ces to the calculation of some modes of pulsation. In the following applications, the latter will be estimated either from a thermodynamical approach, or from a mechanical one, or from both when possible.

4. Results.

-

4.1 ENTROPY OF SOLUBILITY OF HOMOVALENT IONS. - It was obtained [2] for example, the solubility entropy of KC1 in NaCl :

where oc,-(,,,, and ocl-(,,cl, are the Einstein pulsations of the C1- ion in the perfect crystals of KC1 and NaCI, respectively.

4 . 1 . 1 Thermodynamical approach. - The two Eins- tein pulsations in each compound were computed by fitting the experimental specific heat. The numerical result was :

ASso1

-

- 0.2 k

4.1 .2 Mechanical approach. - The second method consisted in calculating the modes of pulsation by deriving the force constants from the second derivative of the potential energy. We obtained :

[

'KC']

-

- 0.3

k

(13)

AS,,, = - k Ln

(4)

modulus, respectively.

The order of magnitude, namely this very small solubility entropy, is confirmed by the experimental value of about 0 k [6].

4 . 2 ENTROPY OF FORMATION OF SCHOTTKY DEFECTS.

-

4.2.1 Thermodynamical approach.

-

The modes of pulsation altered by the proximity of the vacancies were identified as the modes of pulsation of the ions in the liquid state [I]. This gave for the (molecular) entropy of formation of a Schottky defect in the ionic crystals of NaC1-type :

where L is the (molecular) heat of fusion, T, the temperature of fusion and Sf the (molecular) entropy of fusion. The numerical results reported in table I show first, that all the alkali halides have nearly the same value, and second, that this value is rather close to the experimental results.

4.2.2 Mechanical approach. - The altered modes of pulsation were independently calculated from the change of the force constant of the ions close to the vacancy ; This led to the relation :

potential and 6V/V the relaxation volume. The first term accounts for the suppression of bonds when the vacancy is created. The second term accounts for the change of the equilibrium distance between the first and second neighbours of the vacancy. This relation also gives results in good agreement with the experimental data of table I when the theoretical relaxation volumes are used in eq. (15), but not when the experimental relaxation volumes are considered.

4.3 ENTROPY OF SOLUBILITY OF ALIOVALENT IONS.

- We started [2] from the basic equation (1).

Considering the case, for example, of the solid solubility of MCl, in KCl, phase B is then the pure MCl, whereas phase A is the KC1 lattice. When the solubility limit is reached, the total mole fraction of M + + is equal to C . This is composed of two parts : the non- associated ions in mole fraction = C(l

-

p) and the impurity vacancy complexes in mole fraction PC, where p is the association degree.

Development of the chemical potential leads to :

pB(*) = & + +

+

RTLn C

+

2

&,-

+

Actually, the dissolution of one MCl, molecule entails the formation of one cation vacancy (whose mole fraction was already called x). Finally, we

Theoretical and experimental values for the lattice defect entropies in ionic crystals in k (Boltzmann constant) unit. The data relative to formation of vacancies and solubility of impurities are taken from the original works quoted in refs. [I] and [2].

Entropy of

-

Formation of point defects e. g. : Schottky defects in NaC1,

KC1, KBr, KI

Solubility of an homovalent ion e. g. : KC1 in NaCl

Solubility of an aliovalent ion e. g. : K,SO, in KCl, e. g. : YCl, in NaCl

Formation of a vacancy pair from the isolated vacancies in NaC1, KC1, KBr, KI

Formation of an impurity-vacancy complex from isolated vacancy and impurity in KC1 Bif + Cd+ + S r + + Theoretical results, thermodynamical approach - Theoretical result mechanical

approach Experimental result

- -

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LATTICE DEFECT ENTROPIES IN IONIC CRYSTALS C7-263 concluded, as far as the modes of pulsation were

concerned, that the entropy of solubility of aliovalent ions is essentially due to the vacancy formation.

Subsequently, the entropy of solubility of any divalent ion in any alkali halide should be :

while for any trivalent ion, it should be twice as much :

The few available experimental data seem to support this very simple view (Table I).

4 . 4 ENTROPY OF FORMATION OF A VACANCY PAIR FROM THE ISOLATED VACANCIES.

-

Let US consider

the equilibrium of vacancy pair formation : cation vacancy

+

anion vacancy FI. vacancy pair which gives for the chemical potential :

0

ppr

+

RTLn

xi

=

P:+

+

RTLn x,

+

where the symbols have the following meaning ; p,, : chemical potential of the vacancy pairs having one given orientation among the z distinct orientations ;

j.~,,+ and p,- : respective chemical potentials of the

cation and anion vacancies ;

xl

: mole fraction of those vacancy pairs ; x+ and x- : respective mole fractions of cation and anion vacancies. In the NaCl lattice there are six equivalent orientations, so that the total number of vacancy pairs is

x' = 6 xf

.

From eq. (S), eq. (17) may be rewritten as :

XI

--

'ASP, AH,,

x+ x- -exp(,R)exp(-F) with :

AS,, = S,, - (Sv+

+

S v - ) = S,, - Ss ₍₁₉₎ since S,, the formation entropy of a Schottky defect is defined as the sum of the formation entropies of the two individual vacancies. Ss was calculated [l] from the basic equation :

where o, was the Einstein pulsation of the perfect lattice averaged on the cation and anion, while ok was the pulsation of the modes of the ions close to a vacancy in direction of the vacancy. Let us apply the same approximation for S,, : All modes of pulsation are taken to be equal to the Einstein pulsation, except the ten modes of pulsation of the first neighbours of the vacancy pair which point towards a vacancy (Fig. 2). This gives :

FIG. 2. - Formation of a vacancy pair from the isolated vacan- cies. There are twelve modes of pulsation altered around the free vacancies (among which eight are shown on the plane) for only ten around the vacancy pair (six on the plane of the

figure).

which leads from eq. (19) and (20) to our final result :

AS,, =

-

Ss/6

L A

Tt follows from the numerical values for S, of table I that in nearly all the alkali halides the formation entropy of a vacancy pair from the isolated vacancies should be close to - 2 k. Unfortunately, at the present time there exists no reliable experimental determination of this parameter to check our theory.

4 . 5 ENTROPY OF FORMATION OF AN IMPURITY- VACANCY COMPLEX FROM ISOLATED IMPURITY A N D VACANCY. -Applying the same treatment to the equilibrium of association :

divalent ion

+

vacancy f impurity-vacancy complex leads to equations very similar to eqs (17), (18) and (19).

We once again use the same approximation, namely that all pulsations are equal to the Einstein pulsations except the three pulsations of the foreign ion, the six pulsatiolls of the first neighbours of the impurity and the six pulsations of the first neighbours of the vacancy. Subsequently, there is no change in the modes of pulsation between the initial and final states as shown on figure 3. Finally, one should expect no change in the vibration entropy :

FIG. 3. - Formation of an impurity-vacancy complex. The numbers of altered modes of pulsation close to the foreign ion

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KC1 about which the most numerous data are available. The general trend is well that the experimental values are small, but still not equal to zero, which shows that a deeper analysis is needed.

5. Conclusion. - Lattice defect entropies in ionic crystals begin to be known from experiment with some confidence. One or two decades ago, the ionic crystals-

values for the enthalpies, which are now clearly understood from the theoretical point of view. The present very simplified approach shows that the entropies can also be interpreted, at least t o the first order.

It becomes then possible to evaluate the whole Gibbs free energy, which is the parameter of real interest in thermodynamics of lattice defects.

References

[I] BENI~RE, F., J. PhysiqueLett. 36 (1975) L 9. [71 REISFELD, R. and HONIGBAUM, A., J. Chem. Phys. 48

[2] BENIERE, F., J. Physique Lett. 37 (1976) L 177. (1968) 5565.

131 BfiNIkR~, M., cHEMLA, M. and BfiNIkRE, F., J. phys. & [81 KRAUSE, J. L. and FREDERICKS, W. J., J. Physique Colloq.

Chem. Solids 37 (1976) 525. 34 (1973) C 9-25.

[91 FULLER, R. G., MARQUARDT, C. L., REILLY, M. H. and [4] HUDDART, A., and WHITWORTH, R. W., Phil. Mag. 27

(1973) 107. _[lo] WELLS, J. C., Phys. Rev. 176 (1968) 1036.

JACOBS, P. W. M. and PANTELIS, P., Phys. Rev. 4 (1971)

[5] FRIEDEL, J., J. Physique Lett. 35 (1974) L 59. 3757.

[6] BARRETT, W. T. and WALLACE, W. E., J. Am. Chem. Soc. [ l l ] CHANDRA, S. and ROLFE, J., Can. J. Phys. 48 (1970) 412. 76 (1954) 366. [I21 TRNOVCOVA, V., Czech. J. Phys. B 19 (1969) 663.

DISCUSSION

P. W. M. JACOBS. - An entropy change for Schot- F. BBNIERE. - The change in vibrational frequency tky defect formation of 10 k seems to be rather large. in this case is (10 k = S,) would be by a factor 2 only On a simple Einstein model this would correspond to (= exp 10112). The Einstein model holds in a quite a change in vibrational frequency by a factor of 5. good first order approximation at temperatures higher