DEPENDENCE OF THE ELECTRICAL CONDUCTIVITY OF BOROSILICATES ON THE CONCENTRATION OF METAL IONS

R. L. MÜLLER

1. All refractory inorganic dielectrics are, to a certain extent, ionized bodies. This accounts for the ionic conductivity observed in such dielectrics. The latter, however, cannot be studied in any depth without taking into account those chemical factors which determine not only ionization, but also the degree of electrolytic dissociation characteristic of the given dielectrics. In this respect we shall not be mistaken if we cease to regard the electrical conductivity of dielectrics and the phenomena associated with it as a chapter of pure physics, and instead consider them as a component part of the electrochemistry of solids. In other words, any study of the ionogenic properties of dielectrics requires of the investigator a sufficient acquaintance with the accumulated experience in the field of electrochemistry. Unfortunately, we do not yet possess this at the present day. It is precisely this unfortunate circumstance that explains why, in both our domestic and foreign literature, the most primitive mechanical notions of the nature of the ionic conductivity of dielectrics are current. Here we have in mind also the mechanical theory of the conductivity of complex alkali glasses of Bell and von Lengyel [1] and, in connection with this, their erroneous determination of transport numbers in glasses [2]; we also have in mind the idea, widespread among us, of the “neutralization” of two alkali oxides in glasses [3]; to this should also be added Appen’s failure to understand the phenomenon of superposition of electrical conductivity in alkali glasses [4].

Instead of opening here a discussion on these questions, we shall use this forum in order to illuminate the most important problem for all of us—the nature of the concentration dependence of the ionic electrical conductivity of dielectrics—limiting our analysis to questions of the electrical conductivity of the simplest glasses containing only one oxide of an alkali metal. Such a limitation is justified by the fact that misunderstanding precisely the simplest cases of electrical conductivity leads the majority of investigators to erroneous judgments about the electrical conductivity of more complex systems.

1. There exists an opinion that the rapid increase in the electrical conductivity of alkali glasses with increasing content of alkali oxide in them is caused by loosening of the network structure as a result of bond ruptures. It has recently been shown that, in the case of boron glasses, the introduction of an alkali oxide is not only not accompanied by rupture of the valence bond between boron atoms, but the opposite phenomenon is observed—an increase in the number of valence bonds. As a result of the transition of boron atoms from a trivalent neutral state to a tetravalent negatively ionized state, the bond network is strengthened, the melting temperature rises, the critical temperatures increase, and boron glass becomes chemically more stable [5].

The foregoing can be clearly represented in the form of Fig. 1, which pertains to the system \( \mathrm{B_2O_3}—\mathrm{Na_2B_4O_7} \) [6]. The abscissa gives the molar percentage content of borax. On the left ordinate (lower curve) is plotted the rate of dissolution at \(100^\circ\mathrm{C}\), expressed in \(10^5\)-fold value of \(g/\mathrm{cm}^2\cdot\mathrm{sec}\). On the right ordinate (upper curve) is plotted the logarithm of the molar electrical conductivity at \(250^\circ\mathrm{C}\). It is evident from the figure that, whereas the electrical conductivity of borate glasses increases hundreds of thousands of times with increasing sodium oxide content, dissolution under the same conditions is slowed by a factor of one hundred. Such a sharp decrease in dissolution indicates strengthening of the glass-forming network. As for electrical conductivity, its exponential increase is not determined by the degree of strength of the network of valence bonds, which determine the viscosity of glass-forming bodies. Electrical conductivity, however, is determined mainly by the state of the ionogenic component. Of this we are convinced by a careful analysis of the experimental data on electrical conductivity.

Fig. 1

1. For a correct understanding of the experimental material on the electrical conductivities of glasses, it is necessary to dwell briefly on the features of glass-forming materials from the electrochemical point of view.

Owing to the low dielectric constants of \( \mathrm{B_2O_3} \) (3.2) and \( \mathrm{SiO_2} \) (3.8), sodium and potassium ions, even in the melt, are strongly bound to ionized oxygen atoms. This is evident from the fact that the magnitude of the energy of their

\[ W_{ac}=-\frac{e^2N}{r}\simeq150\div180\ \text{kcal/mole} \]

retains a still large value in a medium with dielectric permeability 3–4, incomparably larger than in an aqueous solution with dielectric permeability about 80. Therefore, even at a temperature of about \(2000^\circ\mathrm{K}\), the degree of dissociation of alkali cations still remains very small. Here, of course, a melt with a low concentration of alkali oxides is assumed, for only in this case does the melt retain a low dielectric permeability, close to that observed for \( \mathrm{B_2O_3} \) and \( \mathrm{SiO_2} \). Such melts with a small content of alkali oxides, in their electrochemical properties, are close to solutions of electrolytes with low dielectric permeability. Solvation of ions under these conditions is extremely weak, and the electrostatic interaction of cations with ionized oxygen atoms will be large; consequently, they will be paired. The exception will be an insignificant fraction of cations possessing a statistically excessive dissociation energy [7].

The essential structural difference between network refractory structures and low-melting molecular solutions introduces no changes into the processes of ion association. This is understandable, since the process of association of ions in molecular solutions proceeds statistically by means of molecular diffusion; while the process of association of alkali ions and ionized oxygen atoms in the network structure of a glass melt proceeds statistically with no less success by way of elementary diffusion.

alkali ions and, at high temperatures, also by the more complex diffusion of all the constituent parts of network-valence-bonded atoms. Elsewhere it has been shown that the mechanism of complex diffusion of atomic nodes in a network must proceed by means of statistical switching of bonds [8].

However, the interaction of the ionogenic component of borosilicates is not limited to the process of association of cations and ionized oxygen atoms. Owing to the same low dielectric constant, intense interaction occurs between the dipoles of ionogenic structural units

\[ \cdots \mathrm{B}^{-}\mathrm{O}_{4/2}\mathrm{M}^{+} \cdots \]

in boric anhydride, or—

\[ \cdots \mathrm{SiO}_{3/2}\mathrm{O}^{-}\mathrm{M}^{+}— \]

in silica.

According to the known relation [9], the interaction energy of two dipoles \(W_{pp}\) lies within the limits:

\[ -\frac{28.9\,p^{2}}{\varepsilon r_{\text{\AA}}^{3}} \leq W_{pp} \leq -\frac{14.45\,p^{2}}{\varepsilon r_{\text{\AA}}^{3}}\ \frac{\text{kcal}}{\text{mole}} . \]

Taking into account the limits of the dielectric constant \(3 \leq \varepsilon \leq 4\) and of the distances between the centers of the ionized atoms \(\mathrm{Li}^{+}\mathrm{B}^{--}\) and \(\mathrm{K}^{+}\mathrm{B}^{--}\), \(1.66 \leq d_{\text{\AA}}^{2} \leq 2.21\), we obtain the limits of the molar interaction energies of pairs of dipoles (for \(r_{\text{\AA}} \simeq 3\ \text{\AA}\)) equal to:

\[ 8.4 \leq W_{pp} \leq 39.8\ \text{kcal/mole}. \]

These values are sufficiently large and are appreciably greater than the value of \(RT\) even at two thousand degrees Kelvin (\(4\ \text{kcal/mole}\)).

Thus, already in the melt, association of ionogenic structural nodes must take place. According to the law of mass action, as the content of ionogenic polar nodes in the network increases (with increasing content of metallic ions), the number of associated groups will increase, just as the sizes of the associated groupings of the ionogenic nodes of the network will grow. It is not difficult to imagine that, as a result of sufficient increase in the content of associating polar nodes in the network, their interlacing must occur. Figure 2 gives a schematic representation of such a transition from individual dipoles and a few associated groups (Fig. 2a) to enlarged associated groupings interlaced with one another.

The fine atom-valence network permeates everything—both the dipolar ionogenic nodes shown in the drawing as small circles and the white field, representing the non-ionogenic medium of boric anhydride or silica. This fine atom-valence structure of the glass must be combined with coarser amicrondisperse branch-structural formations. In the case of alkali glasses, these formations must be amicrondisperse, since such glasses are optically homogeneous. In iz-

Fig. 2

with a known increase in the interaction energy of the dipolar nodes of the network, sufficiently large associated groupings can form to cause opalescence in the glass. This is observed in glassy borates of alkaline-earth metals [10].

In full agreement with the considerations set forth here, one may expect in alkali glasses the presence of three concentration regions characterized by specific features of the dissociation energies of polar groups:

a) The region of the lowest content of ionogens (Fig. 2a). There are few associated groupings. Dissociation of the cations occurs into a nonpolar anhydride medium. The dissociation energy \(E_1\) is high. Low electrical conductivity with a high temperature coefficient.

b) The region of higher concentrations. Some increase in the dissociation energy \(E_2\) owing to coarsening of the associated groupings (Madelung coefficient \(>1\) (Fig. 2b). Some decrease in electrical conductivity.

c) The region of formation and growth of interlacings from associated structural ionized nodes of the atom-valent network (Fig. 2c).

Through conductivity in a continuous ionogenic medium. A subsequent decrease in the dissociation energy with increasing concentration of polar groups. A sharp increase in electrical conductivity and a decrease in its temperature coefficient [12]. Dissociation of cations in a polar medium requires relatively less energy because of the interaction of the dissociated cations with the surrounding dipoles.

This interaction reduces to a polarization displacement in the surrounding polar groups \(\mathrm{B}^{-}\mathrm{O}_{1/2}^{-}\mathrm{M}^{+}\) or \(\mathrm{SiO}_{3/2}\mathrm{O}^{-}\mathrm{M}^{+}\) of the metal ions. Effectively, such displacements are equivalent to an orientation of dipoles [13], and as a whole the entire phenomenon has the character of solvation of the dissociated cations by the surrounding polar groupings.

1. At one time Frenkel proposed an approximate method for calculating the just-described solvation effect for molecular-dispersed media [14]. Carrying out an analogous calculation for an ion-valent-atomic medium gives the following values of the dissociation energies \(\Delta\Phi\) for cases of dissociation in a medium poor in polar groups (\(\varepsilon = 6\)) and in a polar medium (\(\varepsilon = 18\)) [15]:

These approximate calculations not only confirm the theoretical considerations set forth above, but at the same time show that the electrical conductivity of glasses will increase from potassium to lithium owing to the decrease in dipole moments and the consequent reduction of the interaction of the dipoles with one another [15].

5. The foregoing is in agreement with the experimentally observed regularities of the electrical conductivities of glasses.

The logarithmic curve of the molar electrical conductivity \(\Lambda\), plotted against the content in boron-sodium glass of polar groups

\[ [\mathrm{Na}] = [\ddot{\mathrm{B}}^{-}\mathrm{O}_{4/2}\,\mathrm{Na}^{+}] \]

has a minimum, confirming the phenomenon of association of polar groups (Fig. 3). As for the steep increase in electrical conductivity after a certain critical concentration has been reached, it is evidently due to the average statistical convergence of polar groups in the glass. This is indicated by the curve of the mean distance between polar lattice sites,

Fig. 3

Visible labels in the figure: \(-\lg \Lambda_{250^\circ \mathrm{C}}\); \(\lg \Lambda\); \(\vartheta\); conductivity of the nonpolar medium; conductivity of the polar medium; \(-\lg[\mathrm{Na}]\); \(\chi,\ \ell\).

\[ \vartheta = \left(6 \cdot 10^{23}[\mathrm{Na}]\right)^{-1/3}\cdot 10^{8}\ \text{\AA}. \]

The rapid increase in electrical conductivity begins near the value \(\vartheta = 6\ \text{\AA}\), very close to the distance between the centers of two polar groups separated by one nonpolar group:

\[ \ddot{\mathrm{B}}^{-}\mathrm{O}_{4/2}\mathrm{Na}^{+} \ \cdots\ \mathrm{BO}_{4/2} \ \cdots\ \ddot{\mathrm{B}}^{-}\mathrm{O}_{3/2}\mathrm{Na}^{+} \]

In full accordance with this, in the critical region the ratio \(\gamma\) of nonpolar structural sites \(\mathrm{BO}_{3/2}\) to polar \(\mathrm{B}^{-}\mathrm{O}_{4/2}\mathrm{M}^{+}\) is close to six [16]. This means that the increase in electrical conductivity begins at glass compositions corresponding statistically to rupture of the last layers of nonpolar \(\mathrm{BO}_{3/2}\) elements, which until then had octahedrally isolated the polar structural elements \(\mathrm{B}^{-}\mathrm{O}_{4/2}\mathrm{M}^{+}\) from one another.

6. Let us consider in more detail the logarithmic concentration dependence of the molar electrical conductivity on the content of polar \(\mathrm{B}^{-}\mathrm{O}_{4/2}\mathrm{M}^{+}\) at

200 and 300°C. In Fig. 4, in addition to this dependence, curves are also given for the dependence of the dissociation energy on the same content of polar groups in the glass. The initial considerable, 630-fold increase in the content of the ionogenic component is accompanied by an insignificant, only 6-fold change in the molar electrical conductivity at 200°C. A further, one-hundred-times smaller (only 6-fold) change in the sodium content is accompanied by an extremely steep, forty-million-fold increase in molar electrical conductivity. Correspondingly, the dissociation energy \(\psi_\varphi\), changing relatively little at first, in the region of the steep rise in electrical conductivity decreases sharply, by more than a factor of two.

Fig. 4

A detailed analysis of the numerical experimental material confirms that the jump in electrical conductivity is determined mainly by the decrease in the dissociation energy in the concentration region of the polar constituent parts of the glass sufficient to ensure direct interaction of the polar structural elements.

1. If the totality of data on the dependence of the dissociation energy \(\psi_\varphi\) on the concentration of metallic ions in borates of various ...

metals in logarithmic form (Fig. 5), then in the region of the steep rise of electrical conductivity one obtains the well-known linear logarithmic dependence of the dissociation energy on the volume concentration of alkali ions.

\[ \begin{aligned} \lg \psi_{\varphi} &= b - x \lg [M] \quad \text{or}\\ \psi_{\varphi}[M]^x &= \mathrm{const.} \end{aligned} \tag{1} \]

This dependence was established experimentally more than 20 years ago [6]. Here, for boron glasses, \(x = 0.5\), and \(\mathrm{const.} = 3.5\ \text{kcal/mol}\) (for sodium glasses [6] and 3.9 for potassium and lithium glasses [17]).

Fig. 5. Dependence of the dissociation energy on the concentration of metallic ions. Stable state, 200°C. Borates: 1—lithium, 2—sodium, 3—potassium, 4—rubidium, 5—cesium, 6—silver.

A general empirical formula was obtained for boron glasses rich in alkali oxides, giving the dependence of the molar electrical conductivity on temperature and concentration:

\[ \Lambda = C \exp\left(-\frac{a}{T\sqrt{[M]}}\right)\, \frac{\Omega^{-1}\,\text{cm}^{2}}{\text{mol}}, \tag{2} \]

where \(a = 930 \pm 50\), \(T\) is the absolute temperature, and \([M]\) is the molar volume concentration of alkali ions.

An analogous dependence was established for sodium silicate glasses. In contrast to boron glasses, for silicate glasses in expression (1) the exponent is \(x = 0.25\) [11], [18].

Expression (1) is inconvenient from the physical point of view, since in passing from borates to silicates the dimensionality of the constant changes. Therefore, a more rational linear dependence is proposed here for the dissociation energy \(\psi_{\varphi}\) on the ratio \(\gamma\) of the number of nonionized network nodes \(BO_{3/4}\) or \(SiO_{1/2}\) to the number of ionized \(B^{-}O_{1/4}M^{+}\) or \(SiO_{3/2}O^{-}M^{+}\):

\[ \psi_{\varphi} = \psi_{0} + \beta \gamma, \]

where

\[ \gamma=\frac{[\mathrm{BO}_{3/2}]}{[\mathrm{B}^{-}\mathrm{O}_{1/2}\mathrm{M}^{+}]} \quad \text{or} \quad \gamma=\frac{[\mathrm{SiO}_{4/2}]}{[\mathrm{SiO}_{3/2}\mathrm{O}^{-}\mathrm{M}^{+}]} \tag{3} \]

In this case, for \(\gamma \leq 6\) we have the following values of the constants \(\psi_0\) and \(\beta\):

Thus, both in borates and in silicates, when all structural nodes of the glass-forming network are ionized \((\gamma=0)\), the dissociation energy becomes close to the mean value \(\psi_\varphi=\psi_0 \simeq 28\) kcal/mol.

Fig. 6. Dependence of the dissociation energy on the magnitude of the atomic–ionic ratio.

Stable state at \(262^\circ\mathrm{C}\).

Borates: 1—lithium, 2—sodium, 3—potassium.

It should also be pointed out that the points approach a single straight line on the graph when considering the linear dependence of the dissociation energy \(\psi_\varphi\) on the magnitude of the atomic–ionic ratio \(\gamma\) (Fig. 6). In this case, for alkali-metal borates a single linear dependence is obtained:

\[ \psi_\varphi = 27.0 + 6.25\gamma \ \frac{\text{kcal}}{\text{mol}}, \tag{4} \]

and for the molar electrical conductivity the generalized expression is

\[ \Lambda = C \cdot \exp\left(-\frac{6800+1.588\gamma}{T}\right)\ \frac{\mathrm{ohm}^{-1}\mathrm{cm}^{2}}{\mathrm{mol}} . \tag{5} \]

Finally, an expression practically suitable for any borosilicates of alkali metals will now be

\[ \Lambda = C \cdot exp\left(-\frac{7000+0.25\,\beta\gamma}{T}\right)\frac{\mathrm{ohm}^{-1}\mathrm{cm}^{2}}{\mathrm{mole}}. \tag{6} \]

Conclusions

1. In addition to the temperature dependence, the electrical conductivity of borosilicates is determined by the concentration of ionized constituent parts of the ionic-atomic-valence structural network. At the same time, the decisive role is played by the concentration dependence of the magnitude of the dissociation energy of ionogenic polar nodes of the network. The latter are not only very weakly dissociated electrolytically, but are also in an associated state in accordance with the law of mass action.

2. A weak change in molar electrical conductivity is observed in borosilicates in the region of low concentrations of alkali ions, and a steep increase in electrical conductivity in the region of elevated concentrations. At low concentrations the conductivity is caused by the motion of dissociated cations outside the polar medium; at high concentrations through-conductivity is observed in the polar medium of ionized constituent parts of the valence-atomic network.

The transition concentration region corresponds to the appearance of through-conductivity owing to the formation of continuous bonding between the ionized elements of the atomic-valence network of borosilicates.

1. A new mathematical expression has been proposed for the concentration dependence of the dissociation energy of alkali cations in the polar medium of borosilicates, and for a generalized formula of the molar electrical conductivity of borosilicates.

References

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Kemerovo, Mining
Institute.

Errata