TEMPERATURE CHANGES IN YOUNG’S MODULUS OF ALKALI-HALIDE SOLID SOLUTIONS

A. A. Botaki

Huntington [1] pointed out that, according to the theory of Leibfried and Hahn [2], the temperature dependence of the elastic constants of ionic crystals is determined mainly by the thermal expansion of the crystal lattice. As the temperature rises, the interionic distance increases, weakening the forces of electrostatic interaction between ions, in consequence of which Young’s modulus decreases.

Thermal expansion obeys the law [3]

\[ \frac{\Delta l}{l}=\frac{\alpha}{\kappa}U(T), \tag{1} \]

where \(\alpha\) is the coefficient of linear expansion, \(U(T)\) is the mean energy of the oscillator, \(\kappa\) is Boltzmann’s constant, and \(\Delta l/l\) is the relative elongation. At sufficiently high temperatures \(U(T)\) is proportional to \(\kappa T\), and, consequently, \(\Delta l/l\) is proportional to \(T\), i.e., the length of a body is a linear function of the temperature of the body \(T\).

In solid solutions of the type NaCl—NaBr, KCl—KBr, the ions Br\(^{-}\) and Cl\(^{-}\) are randomly distributed among the anion vacancies [4, 5]. Such a system may be regarded as a substance with an impurity introduced by substitution, or as an alkali-halide crystal with two anionic “isotopes” having very different masses [6].

For sufficiently high temperatures, the energy of the oscillators of a solid solution must likewise be proportional to \(\kappa T\), and, consequently, according to formula (1), one should expect a linear dependence of the change in the body length for an alkali-halide solid solution on temperature.

A number of authors [7, 8] have shown that, for temperatures above the Debye temperature, pure alkali-halide crystals have a linear dependence of Young’s modulus on temperature. Since the temperature changes in Young’s modulus are determined by the thermal expansion of the crystal lattice, on the basis of the foregoing there is reason to suppose,

that for the solid solutions of the systems under consideration there should likewise be a linear dependence of Young’s modulus on temperature. However, upon a more detailed consideration of the question, attention should be paid to the fact that the formation of the crystal lattice of a solid solution is accompanied by a change in the lattice periods of the components and by a change in the interaction forces of the ions of the solid solution as compared with the interaction of these same ions in the pure components. This is confirmed, above all, by the fact that the energy of the crystal lattice changes.

In the temperature behavior of Young’s modulus, what is important is not the interaction force of the ions itself, but the rate of its change with increasing interionic distance. This is confirmed by the fact that, according to the general theory, the elastic coefficients are determined by the second derivatives of the strain energy with respect to distance.

In the work of M. S. Ivankina [9] it is shown that the coefficient of linear expansion of the solid solutions under consideration has a value greater than the additive one. An increase in the coefficient of linear expansion in solid solutions, as compared with pure salts, indicates a decrease in the rate of change of the attractive forces.

Indeed, at sufficiently high temperatures an ion is positioned in such a way that its kinetic energy of thermal motion is equal to its potential energy. Since the kinetic energy is proportional to \(kT\), then when the temperature is raised by \(\Delta T\), the ion must change its position so that its kinetic energy again becomes equal to its potential energy. In this case, the larger the coefficient of linear expansion, the greater the distance over which the ion will move away. The same change in the potential energy of an ion in a crystal with a large coefficient of linear expansion occurs over a greater distance than in a crystal with a small coefficient of thermal expansion. This means that in the former case the attractive forces decrease more slowly than in the latter. Therefore, although upon heating a solid solution the distance between ions will change more substantially than follows from the law of additivity, it may turn out that, owing to the change in the law governing the decrease of the interaction forces between ions, the rate of change of Young’s modulus with temperature will be less than additive.

Using a two-part piezoquartz resonator at a frequency of 125 kHz [7, 8], measurements of the sound velocity and Young’s modulus were carried out by the method described by us in [8].

The measurements were performed on specimens cleaved in the 001 direction from single crystals grown by the Kyropoulos method. The specimens had dimensions of the order of \(1.5 \times 1.75 \times 15\) mm. The two-part resonator was placed in a chamber with double walls, through which transformer oil was circulated; the temperature was maintained by a Vobser type \(И\)-8 thermostat.

The apparatus used made it possible to measure the frequency, mass, and length with an error of less than 0.1%. The error in setting the temperature was \(0.5^\circ\)C. The density of solid solutions of different compositions, required for calculating Young’s modulus, was taken from [10]. After measurement, the specimens were subjected to chemical analysis by potentiometric titration, and their composition was determined with an accuracy of up to 0.5%.

Figures 1 and 2 present the results of measurements of Young’s modulus as a function of temperature for the solid solutions NaCl—NaBr, KCl—KBr. The graphs show that in the temperature interval from \(0^\circ\) to \(110^\circ\)C a linear dependence of Young’s modulus \(E\) on temperature \(t\) is well maintained.

Fig. 1. Temperature dependence of Young’s modulus for different compositions of the solid solution NaCl—NaBr.

Axis labels and in-figure labels:

• \(E \times 10^{-10}\), dyn/cm\(^2\)
• \(t^\circ\mathrm{C}\)
• NaCl; NaBr (mol. %):
  \(100; 0\), \(71; 29\), \(46; 54\), \(22; 78\), \(0; 100\)

Fig. 2. Temperature dependence of Young’s modulus for different compositions of the solid solution KCl—KBr.

Axis labels and in-figure labels:

• \(E \times 10^{-10}\), dyn/cm\(^2\)
• \(t^\circ\mathrm{C}\)
• KCl; KBr (mol. %):
  \(100; 0\), \(92.5; 7.5\), \(71; 29\), \(45; 55\), \(22; 78\), \(0; 100\)

Fig. 3. Diagrams of \(-dE/dt\) versus composition for the NaCl—NaBr and KCl—KBr systems.

Fig. 4. Young’s modulus—composition diagrams for the NaCl—NaBr and KCl—KBr systems.

Fig. 5. Dependence of the speed of sound on temperature for different compositions of the NaCl—NaBr solid solution.

Fig. 6. Dependence of the speed of sound on temperature for different compositions of the KCl—KBr solid solution.

The slope of the curves (the quantity \(-\dfrac{dE}{dt}\)) changes continuously on going from one composition of the solid solution to another. As Fig. 3 shows, the \(-\dfrac{dE}{dt}\)—composition diagram has negative nonadditivity, which does not contradict the proposition put forward by Huntington. Figure 4 shows the dependence of Young’s modulus on composition for the solid solutions NaCl—NaBr and KCl—KBr at \(0^\circ\)C. As is seen from Figs. 1, 2, the “depth” of the nonadditivity in Fig. 4 depends on temperature and increases as the temperature is lowered. The maximum deviation from additivity in both systems occurs for equimolar compositions. The greatest deviation from additivity is found for the NaCl—NaBr alloy. This is consistent with other properties of the solid solutions under consideration. The NaCl—NaBr alloy has, in comparison with KCl—KBr, a larger value of the heat of formation and of the coefficient of linear expansion (11, 12).

Figures 5 and 6 show the dependence of the sound velocity \(C\) on temperature. The dependence is linear. However, as we pointed out in [8], simultaneous linear dependences of \(E\) and \(C\) on \(t\) are incompatible, since they are related by

\[ E = c^2 \rho \tag{2} \]

where \(\rho\) is the density of the specimen. Calculation showed that, in the temperature interval considered, for NaCl the deviation from linearity at the end of the interval amounts to \(0.2\%\) of the measured quantity. With a scatter of the experimental points of \(0.2\)—\(0.3\%\) of the measured quantity, this nonlinearity cannot be detected in the graphs.

REFERENCES

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