PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
NAMED AFTER S. M. KIROV

Vol. 257                1973

RELATIONSHIP BETWEEN THE SHEAR MODULUS AND THE SURFACE ENERGY OF ALKALI-HALIDE COMPOUNDS

V. N. Belomestnykh, Yu. N. Sudushkin

Abstract

Using experimental data on the temperature dependence of the elastic constants of single crystals of the alkali-halide series (excluding \(CsF\)), the applicability of Kachinskii’s formula [1] is considered for determining the temperature dependence of the surface energy of the \((100)\), \((110)\), and \((111)\) faces of these crystals. The results of the calculation are used to understand the anisotropy of physicomechanical properties.

Surface energy is one of the most important physicomechanical properties of a solid and the principal characteristic of its surface. The concept of surface energy and the use of its values are widely employed in a number of important physicochemical theories (topochemistry, adsorption theory, the energetic theory of heterogeneous catalysis, etc.). However, at present there are no reliable experimental methods for measuring it for solid crystalline bodies. Therefore, investigations of the relationship between surface energy and certain characteristics of solids are of undoubted interest for determining its values by indirect methods.

It has been shown [1, 2] that the shear modulus is determined by the volume density of the interaction energy of particles, and that it is related to the surface energy of metals by the relation

\[ \sigma = \beta G r, \tag{1} \]

where \(\beta\) is a constant coefficient, \(G\) is the shear modulus, and \(r\) is the interatomic distance.

Relations of this kind between the energy characteristics of solids and their elastic properties (for example, [2, 3])

were checked by us for alkali halide compounds and for certain solid solutions based on them; the calculations showed that the resulting values of the lattice energy and bond energy are in good agreement with tabulated values [4]. As a further extension of the indicated correlations, it is meaningful to extend (1) to other classes of compounds for various crystallographic directions as a function of temperature:

\[ \sigma_{(hkl)}(T)=\beta G_{[hkl]}(T) z_{[hkl]}(T), \tag{2} \]

where \(\sigma_{(hkl)}(T)\) is the temperature dependence of the surface energy for different faces of the crystal;

\[ G_{[hkl]}(T) \]

is the temperature dependence of the shear modulus in the crystallographic direction;

\[ z_{[hkl]}(T) \]

is the temperature dependence of the shortest interatomic distance in this direction.

Verification of (2) for crystals of the alkali halide series is facilitated by the fact that, on the one hand, experimental data are available for them on the elastic constants as functions of temperature, and, on the other, there is a rather substantial number of works devoted to the theoretical calculation of surface energy. The available experimental data on surface energy, as a rule, are in poor agreement with the theoretical values [5, 6].

The shear moduli at a given temperature were determined from the elastic constants using the known relations [7].

The dependence of the interionic distance on temperature was calculated using the coefficient of linear expansion \(L(T)\):

\[ z(T)=z_0\left(1+L(T)T\right), \tag{3} \]

where \(z_0\) is the interionic distance at \(0^\circ\mathrm{K}\).

The quantity \(\beta\) was taken to be 0.06, as in [8].

The relations between the shortest interionic distances in different crystallographic directions, as is known, are:

\[ z_{[110]}=\sqrt{2}\,z_{[100]}, \qquad z_{[111]}=\sqrt{3}\,z_{[100]} \]

and

\[ z_{[110]}=\sqrt{2}\,z_{[100]}, \qquad z_{[111]}=\frac{\sqrt{3}}{2}\,z_{[100]} \]

for compounds with structures of the NaCl and CsCl types, respectively.

Table 1 gives the results of calculations of the surface energy of the (100) and (110) faces of crystals of the alkali-halide series at \(0^\circ\mathrm{K}\), of the (100) face at the melting temperature,\(^{1)}\) and of the temperature coefficient of the surface energy of the (100) face according to the calculations of S. N. Zadumkin [9] and as obtained experimentally for the melt [10].

Calculations of these same quantities by (2) for the faces of the cube, rhombic dodecahedron, and octahedron at temperatures \(0^\circ\mathrm{K}\), \(293^\circ\mathrm{K}\), and \(T_{\mathrm{m}}\) are presented in Table 2. To determine the shear modulus at \(0^\circ\mathrm{K}\) and \(T_{\mathrm{m}}\), experimental data on the temperature dependences of the elastic constants (elastic coefficients) extrapolated to these temperatures were used, with the exception of \(\mathrm{LiJ}\), for which estimated values of the elastic constants at \(0^\circ\mathrm{K}\) and the temperature coefficient were taken from [11]. Data on \(\tau_0\), \(d(T)\), and \(T_{\mathrm{m}}\) were taken from [12–14].

Comparison of the results of the two tables shows, in general, that calculation by (2) gives the correct distribution in the magnitudes of the surface energies of compounds of the alkali-halide series, and that the values obtained for the surface energy and its temperature coefficients are acceptable.

According to the theory of M. Born [15] and Ya. I. Frenkel [16], for the halide salts of alkali metals

\[ \frac{\sigma_0(110)}{\sigma_0(100)} = 2.5; \qquad \frac{\sigma_0(111)}{\sigma_0(100)} = 5.81, \]

whereas the experimental data of V. D. Kuznetsov [5, 6] give

\[ \frac{\sigma(110)}{\sigma(100)} = \sqrt{2}; \qquad \frac{\sigma(111)}{\sigma(100)} = \sqrt{3}. \]

Calculation by (2) shows that the ratios of the surface energies of crystal faces vary with changes in temperature and in the composition of the compound, while remaining approximately constant at a given temperature for a series of compounds with the same metallic ion.

Although the temperature dependence of the surface energy of crystals of the form (2) is not entirely correct, nevertheless the conclusions from the results of Table 2 for the compounds considered do not contradict known facts. For all compounds at \(0^\circ\mathrm{K}\), the surface energy of the (100) faces proved to be the smallest, which agrees with the presence of cleavage planes for them along the cube faces. However, with a change in temperature the minimum value of the surface energy of the (100) face is not retained for some compounds. Thus, for example, for

\(^{1)}\) In this work \(T_{\mathrm{m}}\) is understood as a temperature whose numerical value is as close as desired to the true one.

for lithium compounds, already at room temperature the surface energies of the (100) and (110) faces approximately coincide. From the simplest experiments it is known that, when the (100) face of a LiF single crystal is drilled, cracks arise at an angle of \(45^\circ\), i.e., along the directions of the (110) planes. The observation of secondary cleavage planes (110) under certain conditions is indicated, for example, in Gilman’s review [17]. For crystals of \(\mathrm{NaCl}\), \(\mathrm{NaBr}\), \(\mathrm{NaJ}\), \(\sigma(100)\) and \(\sigma(110)\) become equal near the melting temperature. In the experiments of M. A. Bolshanina [6] with impacts of balls, for rock-salt crystals an inversion of the direction of crack development was indeed observed, though at \(200^\circ\mathrm{C}\).

The temperature dependence of the surface energy of crystal faces, according to (2), may be regarded as linear. Indeed, the shear moduli have a linear temperature dependence (with the exception of a small temperature range near \(0^\circ\mathrm{K}\)) and decrease monotonically with its increase, while the nonlinearity \(L(T)\) and its contribution to the overall temperature dependence \(\sigma(T)\) are quite small. From Table 2 it is seen that

\[ \frac{d\sigma}{dT}=\mathrm{const} \]

for like faces of crystals with the same cation, which, incidentally, was to be expected in calculations by (2), since the change of elastic anisotropy with temperature in alkali-halide compounds is determined primarily by the type of metallic ion [18].

At the melting temperature the surface energy of the (100) face for most compounds proved to be in satisfactory agreement with that calculated by Bimüller [19] and experimentally determined by Eger [10].

In comparison with experimental values of the surface energy, one may note both satisfactory agreement (\(\sigma_{(100)} = 300\ \mathrm{erg}\cdot\mathrm{cm}^{-2}\) for \(\mathrm{NaCl}\) at room temperature [5]) and poor agreement (\(\sigma_{(100)} = 340\ \mathrm{erg}\cdot\mathrm{cm}^{-2}\) for \(\mathrm{LiF}\) at \(196^\circ\mathrm{C}\)) [17]. It is necessary to note that the calculation by (2) is performed with the greatest accuracy for \(\sigma_{(100)}\) at room temperature, since in this case \(G_{(100)}\) is given directly by experiment and extrapolation of the elastic constants to limiting temperatures is absent.

Thus, there is a quite real possibility of determining the surface energy from elastic constants by indirect methods, for example, ultrasonic ones. In this case no destruction is required of the surface of the specimen or of its internal parts, on which the majority of experimental methods for determining surface energy are based.

Table 1

Surface energy of the faces (110), (100) of alkali-metal halides according to the calculations of the authors for 0 K and \(T_{\mathrm{m}}\), in erg·cm\(^{-2}\), and its temperature coefficient in erg·cm\(^{-2}\)·deg\(^{-1}\)

Results of calculation by (2): $\sigma$ in erg·cm$^{-2}$ for three temperatures and $-d\sigma/dT$ in erg·cm$^{-2}$·deg$^{-1}$

This is especially important in the subsequent extension of the described method to compounds that undergo decomposition under mechanical fracture, for which, moreover, the method of heats of solution is also inapplicable. Determining the surface energy through the shear modulus makes it possible to examine, in a comparatively simple way, its dependence on various external conditions, for example, temperature. Taking into account the possibility of determining the lattice energy and the binding energy through the elastic characteristics of a substance, one may conclude that the energetics of certain solids can be determined within the framework of a single experimental method.

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