Elastic Properties of Polycrystalline Specimens of Solid Solutions of the KCl–KBr and KBr–KJ Systems

V. N. Belomestnykh, Yu. N. Sukushin

(Presented by the scientific seminar of the Department of Radiation Chemistry)

Recently, a number of authors have begun to develop methods for determining important physicochemical characteristics of solid substances (surface energy, crystal-lattice energy, bond energy, activation energy of the monomolecular stage of decomposition, etc.) on the basis of measuring their elastic properties [1–5]. Agreement has been shown between experimental and calculated data for one group of substances, and a discrepancy has been indicated between calculated and observed values for another (\(Zn\), \(Cd\), and others, whose hexagonal lattice is characterized by an anomalously large ratio of the axes \(c/a\)). In order to apply these methods to as broad a class of substances as possible, for most of which it is difficult to obtain single crystals acceptable for dimensional measurements, it became necessary to investigate the elastic properties of polycrystalline specimens. As the objects of investigation, solid solutions of alkali-halide compounds \(KCl\), \(KBr\), and \(KJ\) (the systems \(KCl - KBr\) and \(KBr - KJ\)) were chosen. These compounds are convenient models for investigation, since they have a simple structure and large single crystals, which makes it possible to test certain existing methods for calculating the elastic moduli of polycrystalline specimens from the elastic constants of single crystals.

To study the elastic properties of solid polycrystalline solutions as a function of the porosity and composition of the components, the velocities of longitudinal and transverse elastic ultrasonic waves were measured. The work was carried out by the pulse method. The principle of measuring the ultrasonic velocity is based on comparing the transit time of a pulse in the specimen under study and in a liquid reference standard. The velocity is determined with an accuracy of \((0.5 \div 5)\%\) from the expression:

\[ V_{e,t}=\frac{\ell}{h}V^{(\mathrm{ref})}, \tag{1} \]

where \(V_{e,t}\) are, respectively, the longitudinal and transverse ultrasonic velocities in the specimen; \(\ell\) is the length of the specimen; \(h\) is the length of the column of reference liquid; \(V^{(\mathrm{ref})}\) is the ultrasonic velocity in the reference liquid. From the measured \(V_e\) and \(V_t\), the dynamic Young’s modulus \(E\) and shear modulus \(G\) were calculated [6]:

\[ E=\frac{V_e^2\rho(1+\sigma)(1-2\sigma)}{1-\sigma}; \qquad G=V_t^2\rho, \tag{2} \]

where

\[ \sigma=\frac{V_e^2-2V_t^2}{2\left(V_e^2-V_t^2\right)} \]

is Poisson’s ratio, and \(\rho\) is the density of the specimen.

Formulas (2) for determining \(\sigma\) and \(E\) require that the geometric dimensions of the specimen satisfy the conditions for propagation of a wave of length \(\lambda\) in the specimen as in an inorganic medium. For this purpose, in order to select the optimal specimen dimensions when determining the elastic moduli at an ultrasonic frequency \(f=1.67\) MHz, measurements were carried out analogously to those performed in [7]. Taking into account the micrometer error \((0.01\ \mathrm{mm})\) in determining \(\ell\) and \(h\), and the error in the sound velocity in the liquid reference standard \((\pm 2\ \mathrm{m/sec})\)

and, proceeding from the requirement that the samples be small in size, the relative errors in determining \(G\) (\(<5\%\)) and \(E\) (\(<5\%\)) can be achieved on samples of length \(l=15\) mm and diameter \(d=15\) mm. The samples of all the systems we studied had lengths from 11 to 15 mm and diameter \(d=14,36\) mm, which provided for KCl (as the substance with the highest longitudinal velocity of ultrasound among the systems we studied) the following ratio between the diameter of the samples and the wavelength:

\[ d = 6\lambda . \]

Samples of polycrystalline solid solutions were prepared as follows. A single crystal of the solid solution, grown by the Kyropoulos method, was crushed, finely ground, sieved, and dried in small quantities in vacuum for 48 hours. The powder prepared in this way was pressed in a press mold of special design. The pressure was varied from 200 kg/cm\(^2\) to 1200 ÷ 1500 kg/cm\(^2\), and the holding time under the press was 10 ÷ 30 minutes. As a result, a homogeneous sample was obtained and, in the case of pressing in vacuum with heating, with a density approaching that of a single crystal. For each system, from 6 to 10 samples of different density were prepared. The density of the pressed samples was determined in two ways:

1) directly from measurement of the geometrical dimensions and the weight of the sample;

2) from the volume of bidistilled water displaced by samples coated with paraffin of known specific gravity. For most samples the densities differed by (2 ÷ 3)%, and the maximum deviation did not exceed 5%.

The molar composition of the components was determined by the method of potentiometric titration with an accuracy of 0.5 ÷ 1%.

The experimentally determined Young’s modulus and shear modulus as functions of the molar concentration of the components for samples of different porosity are presented in the table. In the region of small porosity values (up to 5%), the elastic moduli depend linearly on it; moreover, the relative decrease of the elastic moduli per each percent of porosity for

of the systems studied proved to be approximately the same. This circumstance makes it possible, with practically sufficient accuracy, to determine the elastic moduli at a given porosity \(L(\rho)\) by extrapolating the zero-porosity values of the moduli \(L(0)\) according to the formula:

\[ L(\rho)=L(0)(1-\alpha\rho), \tag{3} \]

where \(\rho\) is the porosity, \(\alpha = 0.033\).

The linear dependence of Young’s modulus and the shear modulus on porosity for some of the systems studied was preserved up to \(\rho = (12 \div 15)\%\); however, since in this case the determined transverse ultrasonic velocity approached the lower limit of the range of velocities measured by the instrument \((800\ \text{m/sec})\), the accuracy of the measurement decreased (because of the need to reduce the length of the specimen). If the obtained dependence is objective, then, using its linear character, one can obtain \(L(0)\) without having at one’s disposal the density value of the single crystal.

Indeed, carrying out measurements at two values of the density of a polycrystalline pressed specimen, \(\rho_2\) and \(\rho_1\), we shall have different values of the elastic moduli

\[ L(\rho_1)=L(0)(1-\alpha\rho_1), \tag{4} \]

and

\[ L(\rho_2)=L(0)(1-\alpha\rho_2) \]

taking into account that

\[ \rho_i=\frac{\rho-\rho_i}{\rho}\,100\%, \tag{5} \]

where \(\rho\) is the density of the single crystals,

one can obtain an expression for the density of the single crystal:

\[ \rho=\frac{100\alpha(\rho_1-A\rho_2)}{(1-100\alpha)(A-1)},\qquad A=\frac{L(\rho_1)}{L(\rho_2)}. \tag{6} \]

If in (8) one substitutes the values of the elastic moduli measured in pressed specimens with 1% and 5% porosity, then at \(\alpha = 0.033\) the density \((\text{g}/\text{cm}^3)\) of the single crystals of the components, from the measured Young’s modulus and shear modulus, respectively, ...

will be (in parentheses the density of the single crystal, measured by the method of hydrostatic weighing, is indicated): \(KCl\)—1.99, 1.96, (1.982); \(KBr\)—2.69, 2.64, (2.735); \(KJ\)—3.06, 3.12, (3.130).

Thus, knowing the elastic properties of polycrystalline pressed specimens, one can determine the density of single crystals of inhomogeneous solids, for which ordinary methods of measurement are, for one reason or another, impracticable.

The graph presents the values of Young’s modulus \(E(0)\) and the shear modulus \(G(0)\), extrapolated to zero porosity, as well as the values of these moduli calculated according to the Voigt–Reuss–Hill (VRH) scheme [8] from the elastic constants measured by us for single crystals of solid solutions of the same compounds.

\[ E_{\varphi}= \frac{(c_{11}-c_{12}-3c_{44})(c_{11}+2c_{12})} {2c_{11}+3c_{12}+c_{44}}, \]

\[ E_{\rho}= \frac{5c_{44}(c_{11}-c_{12})(c_{11}+2c_{12})} {c_{44}(3c_{11}+c_{12})+(c_{11}-c_{12})(c_{11}+2c_{12})}, \]

\[ G_{\varphi}=\frac{c_{11}-c_{12}+3c_{44}}{5}; \qquad G_{\rho}=\frac{5c_{44}(c_{11}-c_{12})}{4c_{44}+3(c_{11}-c_{12})} \tag{7} \]

\[ E_{\mathrm{VRH}}=\frac{E_{\varphi}+E_{\rho}}{2}; \qquad G_{\mathrm{VRH}}=\frac{G_{\varphi}+G_{\rho}}{2}. \]

According to our results, both moduli exhibit a slight nonadditivity as a function of the composition of the solid solution. The deviation of the elastic modulus from the additive value toward a decrease for solid solutions or chemical compounds formed with absorption of heat was indicated, in a qualitative consideration of the question of the relation between the interaction energy of the components and the elastic constants of solid solutions and chemical compounds, in works [9, 10]. There is good agreement between the extrapolated values of the moduli and those calculated according to the VRH scheme. The moduli calculated for \(KCl\), \(KBr\), and \(KJ\), with an accuracy not exceeding 1.8%, coincide with the moduli calculated according to the same scheme using computer data from other sources, while the extrapolated values agree well (to within 3%) with the experimental data [3].

Fig. 1. Young’s modulus and shear modulus of polycrystalline samples of solid solutions of the systems KCl–KBr and KBr–KI.

Table No. 1. Dependence of the elastic properties of solid solutions KCl–KBr and KBr–KI on porosity.

A good correlation between the elastic moduli extrapolated to zero porosity and the calculated values indicates that the elastic moduli measured in this way may be taken as characteristics of isotropic solid solutions.

The obtaining of objective elastic moduli makes it possible to pose the inverse problem—the determination of the elastic constants \(C_{ij}\) from data on \(E\) and \(G\) obtained for polycrystals; the possibility of solving this problem using computers is currently being developed.

The values of the elastic constants of an anisotropic solid are necessary for calculating, by the method of [2], the binding energy and the frequencies of natural vibrations [11].

CONCLUSIONS

1. It has been shown that in pressed polycrystalline specimens of solid solutions of the systems \(KCl - KBr\), \(KBr - KJ\), the elastic properties are determined mainly by porosity and not by composition.

2. Comparison of the experimental results obtained with the result of calculation according to the theoretical VRH scheme gives grounds to believe that the values of the elastic moduli extrapolated to zero porosity are characteristics of solid solutions as an isotropic elastic medium.

3. The dependence of the elastic properties on porosity (in the region of its small values) is linear in character and is described, for all the compounds studied, independently of composition, by the expression

\[ L(\rho) = L(0)(1 - a,033\rho). \]

1. The dependence of Young’s modulus and the shear modulus on composition in the systems studied has a character close to additive.

2. A scheme has been proposed for the experiment and for the treatment of experimental data that makes it possible to determine the density of the single crystal and its anisotropic elastic properties from measurements only on polycrystalline specimens.

References

1. E. G. Kuczynski, J. Appl. Phys., 24, No. 9, 1250 (1953).

2. E. R. Fitzgerald, Phys. Letters, 10, No. 1, 42 (1964).

3. E. R. Fitzgerald, T. W. Wright, Phys. stat. solidi, 24, 37 (1967).

4. S. V. Nemilov, Doklady of the Academy of Sciences of the USSR, 181, No. 6, 1427 (1968).

5. S. V. Nemilov, ZhTKh, 42, No. 6, 1391 (1968).

6. L. Bergmann, Ultrasonics and Its Applications in Science and Technology, IL, 1957.

7. T. G. Gavrilov, E. S. Osternik, Factory Laboratory, XXXV, No. 3, 346, 1969.

8. W. Mason, Lattice Dynamics, Mir, Moscow, 1968.

9. V. D. Kuznetsov, Physics of the Solid State, Vol. I, Vol. II, 1937, 1941.

10. P. N. Sirota, Proceedings of the Sector of Physico-Chemical Analysis, Institute of General and Inorganic Chemistry, Academy of Sciences of the USSR, 21, 96, 1952.

11. M. Born and M. Göppert-Mayer, Theory of the Solid State, ONTI, 1938.