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Ionic Conductivity in Solid Salts¹
W. Seith
The electrical conductivity of solid salt-like compounds has for a number of years attracted the attention of numerous investigators. A large number of interesting experimental works and frequently sharply contradictory theories constitute an almost immense literature in this field of physics². From time to time review articles on this question have appeared. They noted the steady progress of our knowledge in the indicated direction and especially clearly revealed the author’s own point of view in interpreting various questions¹. In the present review we shall consider the development of our knowledge of the electrical conductivity of solid nonmetallic substances up to the beginning of 1936, which has led to the unification of present-day ideas about the mechanism of current passage in these compounds. In view of the fact that these ideas at the same time serve for the creation of new views on the structure of solids, their significance to a considerable extent goes beyond the limits of the present topic.
Since it became known that salt crystals are built of ions forming an ionic lattice, attempts began to be made, on the basis of this conception, to draw conclusions also about the mechanism of current transport in these systems. But here investigators encountered a whole series of difficulties. The lattice energy holding the ions in their positions is considerably greater than those values of the work of detachment which are obtained on the basis of studies of electrical conductivity. Thus, for example, the energy required for separating the ions of the NaCl lattice and carrying them to an infinite distance from one another is 182,000 cal/mol, whereas, as will be shown later, the work of detachment of the Na ion in the lattice, calculated from electrical-conductivity data, amounts to only ~45,000 cal, and for the Cl ion only slightly more than 50,000 cal per ion². Further, in a crystalline lattice all the points of which are occupied, one can perhaps explain self-diffusion, but not the transport of current by the exchange of places between equally charged combined ions. There are only two possibilities for avoiding these difficulties, namely, either by assuming that
¹ Z. Elektrochem., 42, 635, 1936. Translated by V. P. Zhuse.
² The bibliography includes only works used in the present review.
in an ideal crystal lattice an exchange of places between neighboring ions cannot occur, and conductivity can arise only in those places where this ideal lattice structure is disturbed (defective sites of the crystal)3, or else by imagining that ions, even in an ideally regular crystal lattice, when they possess considerable energy, can leave their places in the lattice and, leaving behind an empty site (“hole”), temporarily occupy positions in the intercrystalline space (“Zwischengitterplatz”)4. Before proceeding to a discussion of these questions, let us first consider the experimental data; later we shall give them a theoretical interpretation.
Faraday’s Law and the Transference Number
The first question, the solution of which is of decisive importance, is the question of the nature of the carriers of electric charges in crystalline substances. Gabe and Tolochko5, and later Bruni and Scarpa6, were able to prove the applicability of Faraday’s laws in the electrolysis of solid BaCl₂ or AgJ, and, consequently, the electrolytic nature of the electrical conductivity in these substances. Only the experimental skill of Tubandt and his collaborators7 made it possible to generalize these facts. It was shown that a large number of the substances investigated are indeed ionic conductors. However, solid salts differ from melts and aqueous solutions in that in them, instead of an approximately equal participation of both sorts of ions in the transport of current, practically only ions of one sign move, while the oppositely charged ions remain immobile.
Consequently, for ions of one sign the transference number is equal to 1, and for the other it is equal to zero; i.e., what occurs here is so-called unipolar conductivity. Tubandt’s experiments showed that cylinders pressed from the salt under investigation and brought into contact with one another can be quantitatively separated from each other after heating and passage of current; at the same time, the replacement of one kind of ion by another during electrolysis can readily be determined by simple weighing of the cylinders before and after the experiments. On the other hand, most experiments on the electrolysis of solid salts showed that the metal is deposited in the form of thin dendrite-like threads, rapidly growing through the crystal from cathode to anode, which in turn leads to a short circuit of the system. It turned out, however, that this phenomenon is not observed with AgJ. By using protective cylinders of α-AgJ between the salt under investigation and the cathode, it is possible in many cases to eliminate the formation of metallic threads8. Although these works are widely known, it will nevertheless be appropriate here to cite a few examples both to illustrate the accuracy of the method and to show that even this accuracy of the results in the case of Ag₂S led to an erroneous conception of the mechanism of conductivity.
For the determination of transference numbers a whole series of cylindrical-
of the salt under investigation, prepared by pressing and clamped between electrodes. In the investigation of, for example, \(Ag_2S\), the following system is used:
\[ \mathrm{Pt\text{-}cathode}/\alpha\text{-}\mathrm{AgJ}/\mathrm{Ag}_2S/\mathrm{Ag}_2S/\mathrm{Ag}_2S/\mathrm{Ag\text{-}anode}. \]
The constancy of the weight of the middle cylinder makes it possible to judge the correct course of the experiment.
As an example, Table 1 gives experimental data for two substances investigated.
TABLE 1
| AgJ (160°) | AgS (180°) | |
|---|---|---|
| 1. Weight of silver deposited in the Ag-voltameter | 0.6477 | 0.2896 |
| 2. Weight of silver deposited from AgJ on the Pt-cathode | 0.6476 | 0.2899 |
| 3. Weight of the salt cylinder at the cathode before the experiment | 4.3918 | 2.3713 |
| 4. Weight of the salt cylinder at the cathode after the experiment | 4.3919 | 2.3714 |
| 5. Weight of the middle cylinder of salt before the experiment | 2.5334 | 1.3032 |
| 6. Weight of the middle cylinder of salt after the experiment | 2.5332 | 1.3034 |
| 7. Weight of the salt cylinder at the anode before the experiment | 3.9973 | 1.1476 |
| 8. Weight of the salt cylinder at the anode after the experiment | 3.9972 | 1.1476 |
| 9. Loss in weight of the Ag-anode | 0.6479 | 0.2894 |
The agreement of rows 1, 2, 9, as well as the equality of rows 3, 4, 7, and 8 of the table, indicates that, in the substances considered, the transport of current is due exclusively to silver ions and, consequently, pure cationic conductivity takes place. If iodine ions were also moving, then \(3\) would be \(>4\) and \(7<8\); if there were some fraction of electronic conductivity, \(1\) would be \(>2\) and \(9\).
By experiments similar to those described, it was established that the purely cationic conductors include: \(AgCl\), \(AgBr\), \(\alpha\)-\(AgJ\), \(AgNO_3\), etc., while the anionic conductors include \(PbF_2\), \(PbCl_2\), \(PbBr_2\), \(BaF_2\), \(BaCl_2\), \(BaBr_2\); \(\beta\)-\(Ag_2S\) is a mixed conductor in which the electronic part of the electrical conductivity gradually increases with increasing temperature up to the transformation point at \(179^\circ\), when transition into the ion-conducting \(\alpha\)-form occurs.
In \(\gamma\)-\(CuJ_2\) a gradual transition is observed from pure electronic conductivity to pure ionic conductivity (Fig. 1). Simultaneous mobility of anions and cations was found in \(PbJ_2\), which was later confirmed by another method. Since \(PbJ_2\) forms, with \(AgJ\), a mixed crystal that melts easily at low temperature, \(AgJ\) cannot be used as the protective electrode. Instead, in this case \(PbCl_2\) can be successfully used for this purpose.
The results of experiments carried out with PbJ₂ at 270° are summarized in Table 2.
TABLE 2
| 1. Pt cathode PbCl₂ I |
−0.035 g | In the Ag voltameter, 0.0106 g Ag was deposited, which corresponds to 0.01017 g Pb or: 0.01247 g J or: 0.00350 g Cl J − Cl = 0.00897 g Pb + Cl = 0.01367 g |
| 2. PbCl₂ II PbCl₂ III |
0.0000 " | |
| 3. PbJ₂ I | 0.0012 " | |
| 4. PbJ₂ IV | 0.0010 " | |
| 5. PbJ₂ III PbCl₂ IV |
0.0012 " | |
| 6. PbCl₂ V Ag anode |
+0.0035 " |
Since PbCl₂ is an anionic conductor, then in the case of purely anionic electrical conductivity of PbJ₂, cylinders 3 and 5 should respectively have decreased and increased their weight by 0.00897 g; with cationic conductivity, however, they should respectively have increased or decreased their weight by 0.01367 g. The actual change in the weight of cylinders 3 and 5 by ±0.0012 g makes it possible to calculate the transport number for lead
\[ \frac{0.0012+0.00897}{0.01367+0.00897}=0.450. \]
In the temperature range studied, both kinds of ions take part in the transport of current to an equal extent. Other examples of bipolarly conducting salts include NaCl, NaF, and KCl. However, determination of transport numbers in these cases is considerably more difficult, since the precipitated alkali metal readily evaporates at the temperature of the experiments. It is therefore necessary to resort to various experimental techniques to reduce errors. Nevertheless, the data obtained often differ from one another to a rather considerable degree⁹. Thus, for example, Phipps and Leslie determined the transport number for Cl′ ions in NaCl at 600° and found it to be 0.056–0.095, whereas Jost gives 0.55. In NaF the fraction of anions participating in the electrical conductivity, within the temperature range from 550–625°, increases from 0.004 to 0.139, and in KCl, between 435–600°, from 0.044 to 0.116. The exact value of the transport numbers is of great importance for elucidating the mechanism of conductivity. In mixed crystals and double compounds several anions or cations are often mobile. Several such compounds
Fig. 1. Temperature dependence of the ionic and electronic parts of the electrical conductivity of γ-CuJ₂ (according to Tubandt, Rindtorff, and Jost).
also investigated by Tubandt and his collaborators. Thus, for example, in a mixed crystal of PbCl$_2$—PbBr$_2$ at a temperature of 250°, the participation of the Br ion in the transport of current increases with increasing concentration of PbBr$_2$ as follows: at 25%—0.206; 50%—0.49; 75%—0.886. The participation in current transport, it is understandable, may strongly differ from the relative content of the given ion in the mixed crystal. For the mixed crystal AgCl—NaCl at 280° it was found that the transport number of Na at 25% NaCl content is 0.015; at 50%—0.35, and at 75% NaCl—0.055. On the other hand, in mixed crystals there is often an almost identical mobility of both corresponding ions, as, for example, in the case of $\alpha$-AgJ—CuJ.
The value of the transport numbers for components of a mixed crystal makes it possible, as we shall see later, on the basis of the diffusion coefficient of one of the components present in a slight concentration, to draw a conclusion about the self-diffusion of the principal component.
Temperature Dependence of Electrical Conductivity
The ionic conductivity of solid salts increases strongly with increasing temperature. In the region of high temperatures, the experimental data are especially well described by the equation of Rashev and Hinrichsen$^{10}$, first applied to this phenomenon, the van ’t Hoff equation:
$$ \sigma = Ae^{-\frac{B}{T}} \tag{1} $$
or
$$ \ln \sigma = \ln A - \frac{B}{T}. \tag{2} $$
Equation (1) may be rewritten in the following form:
$$ \sigma = Ae^{-\frac{Q}{RT}}, \qquad B=\frac{Q}{R}, \tag{3} $$
where a parallel is drawn between $\sigma$ and the rate of a chemical reaction, and where $Q$ may be regarded as the value of the energy corresponding to the heat of activation and which is called the energy of loosening (Ablösungsenergie) or the “heat of loosening” (Auflockerungswärme).
This constant, just like the constant $A$, at first could not be determined from other known quantities. Equation (2), represented graphically in the coordinates $\ln \sigma$ and $\frac{1}{T}$, gives a straight line (Fig. 2). This rectilinear portion of the curve may cover a considerable temperature interval, as for example in the case of TlBr and TlCl. With sufficient expansion of the temperature region, deviations from the rectilinear course are nevertheless observed.
Near the melting point, in most cases, a bend is observed
toward larger values of electrical conductivity. This phenomenon is caused by impurities and can be greatly reduced by using purer specimens for the investigation. This section of the straight line, therefore, has nothing in common with the intrinsic conductivity of the salt under study, but is due to the presence of eutectics with a low melting temperature.
At low temperatures a break in the straight line is also usually observed; after the break, the course of the electrical conductivity as a function of temperature in the coordinates \(\ln \sigma\) and \(1/T\) is again well represented by a straight line (Fig. 3).
These facts led Smekal\(^{11}\) to interpret the observed dependence of \(\sigma\) on \(T\) as the result of the superposition of two partial conductivities, each of which may be represented by an equation of the form (1) (Fig. 3). The course of the complete curve is represented in this case by an equation of the form:
Fig. 2. Electrical conductivity of TlCl and TlBr (according to Philips and Partridge).
Fig. 3. Electrical conductivity of NaCl crystals (according to Lehfeldt).
\[ \sigma = A_1 e^{-\frac{B_1}{T}} + A_2 e^{-\frac{B_2}{T}} . \tag{4} \]
When plotted on a logarithmic scale, as is usually done, the curve representing the total conductivity, at some distance from the bend, approaches the straight lines corresponding to the course of the partial conductivities. Smekal considers that the steeper section of the conductivity curve, corresponding to the larger separation work, is due to the conductivity of the lattice itself, i.e., to the motion of dissociated ions within the lattice itself, whereas the more gently sloping section of the curve corresponds to conductivity caused by ions produced at low temperatures at mechanical inhomogeneities of the lattice. These ions may mov—
to take place at sites where the regularity of the crystal lattice is disturbed, chiefly on internal surfaces arising as a result of structural irregularities that exist in every real crystal.
The possibility of this point of view is indicated by the fact that the steep part of the curve is excellently reproducible and its slope[^12] may be regarded as a material constant, whereas the magnitude and the slope of the flat part of the curve vary from specimen to specimen.
Depending on the preliminary treatment, the electrical conductivity of two specimens at one and the same temperature may differ by several orders of magnitude (Fig. 4). For the most part, however, the slope of these lower, flat portions of the curve is the same, which indicates that in equation (4) only the magnitude \(A\) changes, while \(B\) remains almost constant.
Since curves \((\ln \sigma,\; 1/T)\) with two rectilinear portions of different slope are sometimes also observed in those cases where the unipolar character of the conductivity has been unquestionably proved, one must, together with Smeckal, accept that one and the same kind of ions may be bound in the lattice by two different kinds of bonds and may behave in the crystal in two ways: as an ion situated in a regular position of rest in the crystal lattice, and as an ion situated in an irregular position (Lockerion), for example on the surface of some mechanical inhomogeneity, and thereby give rise both to structurally insensitive conductivity and to structurally sensitive defect conductivity (Störleitung).
Fig. 4. Electrical conductivity of NaCl. 1 — Seelen, 2, 3 — Fippes, Cook.
It should be noted that the predominant significance which Smeckal ascribes to “distorted sites” in the electrical conductivity of real crystals, on the basis of a whole series of recent data, appears to be considerably exaggerated. Inhomogeneities and “distorted sites” undoubtedly play an essential role, but it seems to us unlikely that they are the principal factor determining the course of the process.
Fippes’s assumption[^12] that the two portions of the curve are due to the different participation of anions and cations in the conductivity has been proved only for \(\mathrm{PbJ_2}\). In the case of bipolar conductivi-
the curve of the temperature dependence of the electrical conductivity can no longer be described by a two-term equation. In this case it is necessary to introduce into consideration from three to four pairs of constants \(A\) and \(B\), especially when it is necessary to take proper account of all possibilities. However, it is hardly possible to calculate accurately the values of the constants of a three- or four-term equation from experimentally obtained curves of the temperature variation of electrical conductivity. The conductivity of mixed crystals containing two unipolarly conducting conductors may also be represented by two terms. For a mixed crystal containing 80 mol.% AgBr and 20 mol.% CuBr, the temperature variation of the electrical conductivity is represented, according to Reinhold and Schulz \(^{13}\), by the equation:
\[ \sigma=\sigma_{\mathrm{Ag}\cdot}+\sigma_{\mathrm{Cu}\cdot}=850e^{-\frac{42300}{T}}+7.3e^{-\frac{1860}{T}}. \]
It may happen, however, that the mobilities of both identical ions are equal, as, for example, in the case of the mixed crystal AgJ—CuJ, and then a one-term equation is sufficient.
Defect Conductivity
When measuring electrical conductivity at low temperatures, and consequently in the region of defect conductivity, it should be borne in mind that the “true conductivity” exists only at the first instant after the current is switched on, since the rapidly arising polarization soon lowers the value of the conductivity to a certain definite constant value, called the residual conductivity. This polarization arises not at the electrodes, but is due to the formation, inside the crystal, of a volume charge, which can also be measured after the external field has been switched off \(^{14}\). Measurement of the initial current, or extrapolation of the temporal decrease of the current to zero, is in most cases impossible.
Beran and Kvithner \(^{15}\) proposed a method for measuring the polarization voltage caused by the formation of a space charge, in which the magnitude of the electromotive force of polarization is determined directly. The disappearance of the polarization that has formed can be greatly slowed down if the passage of reverse current is not allowed. In this case the disappearance of the volume electric charge occurs only owing to diffusion, i.e. a process proceeding much more slowly than the transfer of charges by an electric current. A certain potential difference \(V_0\) is applied to the specimen under investigation, and the establishment of a stationary residual current is awaited; after this, the electrode of the specimen is very quickly connected to another, smaller source of voltage \(V\). The voltage is selected in such a way that the electrometer connected to the other electrode gives no deflection, indicating that the magnitude of the polarization \(P\) is exactly equal to \(V\). From this it is easy to calculate also the true conductivity of the specimen.
Both the true and the residual conductivity can be well represented by means of an exponential formula of the form (1) (Fig. 5). The shape of the curve representing the dependence of the electrical conductivity of a salt crystal on temperature may be strongly affected by impurities of foreign substances. In this direction, as early as 1897, C. Fritsch^16 published experiments indicating a considerable increase in the conductivity of PbCl$_2$ upon the addition of small amounts of NaCl. It must be noted, however, that both the method of investigation and the explanation left much to be desired.
Le Blanc^17 and, later, Ketscher^18 point to an already previously found dependence of the electrical conductivity on the preliminary treatment of the specimen by drying, annealing, and compression, used in the preparation of pellets.^19 Ketscher established that the electrical conductivity of PbCl$_2$ upon addition of 0.001% NaCl increases by approximately a factor of 50. Gyulai^20 also investigated PbCl$_2$ both in pure form and with the addition of a small amount of KCl; according to this investigation, not only the constant $A$ changes, but also the work of extraction—from 10,860 to 8,720 cal. As the experiments of Tubandt and Reinhold^21 on the determination of transport numbers showed, impurities in such cases do not take part in the transfer of current. Thus the increase in conductivity occurs at the expense of an increase in the number of defective sites in the crystal, possessing a smaller value of the work of extraction.
Fig. 5. True ($K_w$) and residual ($K_d$) electrical conductivities (after Beran and Quittner).
Figure 6 gives the results of measurements by Lefeld of KCl with slight additions of foreign substances. This effect was also observed in the experiments of Geweshe,^22 who studied the electrical conductivity of NaNO$_3$ single crystals. The electrical conductivity of the very same substance, solidified after melting already in the form of a crystal, proved to be considerably higher. Pressed plates likewise have much better conductivity than single crystals of the same substance.^23 Gotals^24 indicated that the addition of 6% by volume of pure quartz sand increases the internal surface of a NaNO$_3$ specimen so strongly that the electrical conductivity increases almost twofold. Smekal and Quittner^25 compared the conductivity of NaCl single crystals obtained by crystallization from solution and from the melt. At 90° the crystals obtained from solution had a conductivity of the order of $10^{-16}\ \Omega^{-1}\,\mathrm{cm}^{-1}$, whereas those obtained from the melt had a conductivity of the order of $10^{-14}\ \Omega^{-1}\,\mathrm{cm}^{-1}$. Crystals obtained from solution evidently possess a much more perfect structure than crystals formed from the melt.
Lefeld\(^{26}\) showed that crystals, when an electric current is passed through them for a long time, are freed from impurities that disturb their regular structure and thereby are improved.
Deformation can also be the cause of the formation in a crystal of places with defective structure (“damaged places”). Gyulai and Hartly\(^{27}\) determined the conductivity of NaCl crystals during their simultaneous deformation. Under stepwise loading of a rock-salt crystal within the range from 20 to 700 kg/cm\(^2\), after each increase in the load a jump in conductivity is observed, which, however, disappears after several minutes. On unloading, no effect is observed, and under a new load the effect appears only if its magnitude exceeds that of the preceding one. A. Ioffe\(^{28}\) thinks that the jump in electrical conductivity occurs not because of an increase in the number of mobile ions, but is due to a shift of the space charge. Gyulai, however, in one of his subsequent papers argues against this point of view.
Fig. 6. Defect conductivity of KCl crystals (according to Lebedev): 1—initial material, 2—C \(\sim 0.3\) mol. % Cu, 3—also \(\sim 0.3\) mol. % Pb.
Stepanov\(^{30}\) was able to prove that the explanation of the jump in electrical conductivity should indeed be sought in the instantaneous increase of the true conductivity of the crystal, and not in a decrease of the electromotive force of polarization. Somewhat later, Boros and Gyulai\(^{31}\) used the magnitude of the jump in electrical conductivity as a measure of the decrease in strength during annealing of a crystal strengthened by plastic deformation. If a crystal that has already once been subjected to the action of a load is annealed before the next loading, then a jump in conductivity is again observed, even under conditions in which the magnitude of the first load is not exceeded. These secondary jumps in conductivity are small at first; however, with increasing duration and temperature of annealing, their magnitude approaches that which was observed under the first loading.
On the basis of the foregoing one may conclude that defect conductivity depends strongly on all changes in the structure of the specimen under investigation, and consequently is also closely connected with the more gross disturbances of the regularity of the crystal structure and is caused by a comparatively small number of ions with a low work of separation. The ratio of the number of these ions
with respect to the number of ions of the crystal lattice itself is, according to Smekal,^32 about 0.0001; the work of detachment is approximately 0.4 of that for lattice ions.
Intrinsic conductivity, or lattice conductivity
The upper part of the curve depicting the temperature dependence of electrical conductivity (in the coordinates $\ln \delta$, $\frac{1}{T}$) is distinguished by the fact that it does not depend on the structure, i.e. the conductivity in this region depends little, or even not at all, on the prior history of the specimen. Whereas the mechanism of defect conductivity is very complex and only with difficulty permits theoretical calculations, attempts at a theoretical treatment of the intrinsic conductivity of a perfect lattice are considerably more successful. A whole series of regularities has been established here. If one tries to form an idea of the general mechanism of conductivity in the extremely diverse behavior, in this respect, of various salts, then it is very useful to consider the phenomenon of electrical conductivity near the melting point. If it is assumed that the electrical conductivity of the melt of salts of strong electrolytes is, approximately, the same, then the jump in conductivity observed upon the transition of the crystal into the melt, and considered by Gewecke^33 as a measure of the loosening of the crystal upon melting, makes it possible to judge the changes in those obstacles that the conducting ions encounter in their motion in different crystal lattices.
TABLE 3
| NaNO₃ | LiNO₃ | KCl | PbCl₂ | SnCl₂ | NaCl | CdCl₂ |
|---|---|---|---|---|---|---|
| 20 000 | 10 000 | 9000 | 5000 | 4000 | 3000 | 200 |
| TlCl | ?J | AgCl | AgBr | AgJ | CuBr |
|---|---|---|---|---|---|
| 160 | 100 | 30 | 5 | 0.9 | 0.7 |
On the basis of consideration of Table 3 it should be noted that there are salts which, in the solid state, conduct better than in the liquid state.
At low temperatures the difference in the values of the electrical conductivity of different salts becomes still greater, since those salts which possess high electrical conductivity have a small work of detachment $(B)$, i.e. with lowering temperature and
electrical conductivity decreases more slowly than in salts which, at high temperatures, possess insignificant electrical conductivity.
TABLE 4
| Salt | Melting temperature | \(\sigma\) (at m.p.) | \(A\) | \(\varepsilon\) | \(B\) | Observer |
|---|---|---|---|---|---|---|
| LiF | 842 | \(0.6\cdot 10^{-2}\) | \(4\cdot 10^{7}\) | 2.20 | 25 500 | Lehfeldt |
| LiCl | 606 | \(1.5\cdot 10^{-2}\) | \(5\cdot 10^{7}\) | 1.65 | 19 000 | " |
| NaF | 992 | \(1.7\cdot 10^{-3}\) | \(1.5\cdot 10^{6}\) | 2.25 | 26 000 | " |
| NaCl | 800 | \(1.3\cdot 10^{-3}\) | \(1\cdot 10^{6}\) | 1.90 | 22 000 | " |
| NaBr | 735 | \(1.3\cdot 10^{-3}\) | \(1\cdot 10^{6}\) | 1.78 | 20 600 | " |
| NaJ | 661 | \(4.0\cdot 10^{-3}\) | \(1.5\cdot 10^{6}\) | 1.42 | 16 500 | " |
| KF | 846 | \(8.0\cdot 10^{-4}\) | \(3\cdot 10^{7}\) | 2.35 | 27 200 | " |
| KCl | 768 | \(2.0\cdot 10^{-4}\) | \(2\cdot 10^{6}\) | 2.06 | 23 900 | " |
| KBr | 728 | \(2.0\cdot 10^{-4}\) | \(1.5\cdot 10^{6}\) | 1.97 | 22 800 | " |
| KJ | 680 | \(1.5\cdot 10^{-4}\) | \(3\cdot 10^{5}\) | 1.77 | 20 500 | " |
| RbCl | 712 | \(5.0\cdot 10^{-5}\) | \(3\cdot 10^{6}\) | 2.12 | 24 600 | " |
| RbBr | 680 | \(3.5\cdot 10^{-5}\) | \(1.8\cdot 10^{6}\) | 2.03 | 23 500 | " |
| TlCl | 427 | \(5.0\cdot 10^{-3}\) | \(2.5\cdot 10^{3}\) | 0.79 | 9160 | " |
| TlBr | 457 | \(5.0\cdot 10^{-3}\) | \(1.7\cdot 10^{3}\) | 0.80 | 9280 | " |
| AgCl | 455 | \(1.0\cdot 10^{-1}\) | \(3\cdot 10^{6}\) | 0.96 | 11 100 | Tubandt |
| AgBr | 422 | \(6.0\cdot 10^{-1}\) | \(3\cdot 10^{6}\) | 0.89 | 10 300 | " |
| \(\alpha\)-AgJ | 522 | 25 | 5.5 | 0.05 | 593 | " |
| Ag\(_2\)HgJ\(_4\) | \(4\cdot 10^{2}\) | 0.37 | 4300 | Ketelaar | ||
| PbCl\(_2\) | 501 | \(5\cdot 10^{-3}\) | 6.6 | 0.47 | 5480 | Zeyt |
| PbJ\(_2\) | 402 | \(3\cdot 10^{-5}\) | \(1.2\cdot 10^{5}\) | 1.30 | 15 000 | " |
| \(9.8\cdot 10^{4}\) | 0.40 | 4680 | " |
Table 4 gives several values of the constants \(A\) and \(B\), which refer to the upper part of the electrical-conductivity curve, i.e., characterize the intrinsic conductivity of the crystal lattice. In Fig. 7, for greater clarity, some of the corresponding curves are given, showing the dependence of electrical conductivity on temperature, borrowed for the most part from the work of Lehfeldt \(^{34}\).
As can be seen from the table, a whole series of regularities is present here, especially clearly observed for the carefully investigated halide compounds of the alkali metals. In Fig. 8
graphically represents the dependence of the work of extraction (in eV) on the anion radius according to Lefeld. The polarizability of the anion, increasing with its size, according to Fajans ^35 and Rehns ^36, promotes an increase in the mobility of the cation, just as this is caused by the polarizing action of the cation itself, increasing in the sequence: Rb, K, Na, Li.
Fig. 7. Dependence of electrical conductivity on temperature.
Electrical Conductivity and Diffusion
By self-diffusion of a definite kind of ions in a crystal is meant the diffusion of these ions through some plane drawn in the crystal, it being assumed that the ions on one side of this plane are marked in some way.
For such mixing of identical particles, the characteristic circumstance is that the total concentration of ions in the given section of the crystal does not change. The number of ions participating in this phenomenon, and the resistance which they must overcome, are the same as in electrolytic conduction. There must be a close connection between the electrical conductivity of a unipolar conductor, or the shares of participation of ions of one kind in the electrical conductivity of a bipolar conductor, and the diffusion constant of this ion. Nernst ^37 and, somewhat later, Einstein ^38 derived for aqueous solutions the following relation:
\[ D=\frac{RT}{N}B, \tag{5} \]
where \(D\) is the diffusion coefficient, \(B\) is the mobility of the ion under the action of a force equal to unity, \(N\) is Avogadro’s number, \(R\) is the gas constant, and \(T\) is the absolute temperature. This equation was applied by Hevesy ^39, and also by Tubandt, Reinhold, and Jost ^40, to solid ionic conductors. K. Wagner ^41, in a thorough theoretical
experimental work once again showed that equation (5) corresponds to the relations in solid ionic conductors. Only in a small number of cases is it possible to determine directly the self-diffusion of ions in solid salts. In view of the fact that the majority of salts are unipolar conductors, one of the simple ways of determining diffusion is the determination of electrical conductivity. Thus, for example, in PbCl\(_2\) it was possible to determine the self-diffusion of lead ions with the aid of radioactive lead, and it was found that lead chloride is a practically pure anionic conductor. The directly measured fraction of participation of chlorine ions in the electrical conductivity is represented by equation \(4^2\). The temperature dependence of the self-diffusion of lead ions is represented by the expression:
Fig. 8. Dependence of the work of detachment in eV on the radius of anions in halide compounds of alkali metals (according to Leffeld).
\[ \sigma = 6{,}55 e^{-\frac{5480}{T}} . \tag{6} \]
The temperature dependence of the self-diffusion of lead ions is represented by the expression:
\[ D = 6{,}76 \cdot 10^{5} e^{-\frac{17900}{T}} \ \mathrm{cm}^{2}\cdot d^{-1}. \tag{7} \]
The difference in the exponents of \(e\) indicates a different magnitude of the detachment work. The transport numbers for lead ions, calculated on this basis, are, at 90, 273, and 484°—\(10^{-10}\), \(10^{-5}\), and \(10^{-3}\), respectively. The fraction of participation of lead ions in current transport lies near the melting point below the accuracy limit of the direct determination of transport numbers according to Tubandt \(^{43}\). It was possible to compare both methods for lead iodide. The temperature dependence of the electrical conductivity of lead iodide in the range from 155 to 376° follows the equation:
\[ \sigma = 9{,}78 \cdot 10^{-4} \cdot e^{-\frac{4680}{T}} + 1{,}15 \cdot 10^{5} \cdot e^{-\frac{15000}{T}}, \tag{8} \]
Self-diffusion of lead ions, measured in the range from 114 to 315°, is represented by the equation:
\[ D = 3.43 \cdot 10^5 \cdot e^{-\frac{15000}{T}} \tag{9} \]
From equations (8) and (9) we see that the second term in the expression for the temperature dependence of the electrical conductivity, which corresponds to the steep part of the curve (Fig. 9), is due to the mobility of lead ions. The part of the curve \(\ln \delta,\ \frac{1}{T}\) with the smaller slope characterizes the participation of iodine ions in the electrical conductivity.
Fig. 9. \(K_{\mathrm{PbI}_2}\) — conductivity of \(\mathrm{PbI}_2\)
\(K_{\mathrm{J}'}\) — contribution to the conductivity of \(\mathrm{J}'\)-ions
\(K_{\mathrm{Pb}''}\) — contribution to the conductivity of \(\mathrm{Pb}''\)-ions
\(D_{\mathrm{Pb}''}\) — self-diffusion of \(\mathrm{Pb}''\)-ions.
In this case, the ratio of the second term of equation (8) to the sum of both terms gives the value of the transport number for lead ions, which was also determined by Tubandt, Reinhold, and Liebold \(^{44}\). The values of the transport numbers are given in Table 5. Also in that
TABLE 5
| Temperature | 155 | 194 | 228 | 255 | 270 | 290 | 338 | 376 |
|---|---|---|---|---|---|---|---|---|
| Tubandt . . . . . | — | — | — | 0.38 | 0.45 | 0.67 | — | — |
| Seith . . . . . | 0.004 | 0.03 | 0.12 | 0.30—0.35 | 0.40—0.50 | 0.55—0.65 | 0.79—0.85 | 0.99—1.00 |
case where there is no possibility of using radioactive indicators, the given method may be applied, since self-diffusion is determined on the basis of investigating the diffusion of related elements.
Brown\(^{45}\) determined the mobility of ions in Ag\(_2\)S, making it possible for small amounts of Cu and Se to diffuse in this substance. The diffusion rates of Cu and Se in Ag\(_2\)S as functions of temperature are well determined by the equations:
\[ D_{\mathrm{Cu}}=40.3\cdot e^{-\frac{1590}{T}}, \tag{10} \]
\[ D_{\mathrm{Se}}=59.3 e^{-\frac{10120}{T}}. \tag{11} \]
The work of detaching the anion, as is evident from the value \(B=10\,120\), is considerably greater than that of the cation, for which \(B=1590\). The transport number for the anion may be calculated from the following equation:
\[ u=\frac{2D_{\mathrm{Se}}}{D_{\mathrm{Se}}+D_{\mathrm{Cu}}}. \tag{12} \]
The value of the transport number reaches, at
\[ \begin{array}{rcl} 177^\circ & \cdots\cdots\cdots\cdots\cdots & 0.9\cdot 10^{-8}\\ 571^\circ & \cdots\cdots\cdots\cdots\cdots & 0.6\cdot 10^{-4}\\ 694^\circ & \cdots\cdots\cdots\cdots\cdots & 2.2\cdot 10^{-4}\\ 836^\circ & \cdots\cdots\cdots\cdots\cdots & 0.7\cdot 10^{-3} \end{array} \]
The diffusion rate determined by adding small quantities of related elements, generally speaking, of course, cannot correspond exactly to the self-diffusion of the principal substance; however, the deviations can be determined if, according to Tubandt, Reinhold, and Jost\(^{46}\), the participation of the ions of the principal substance and the participation of the ions of the added related element in the electrical conductivity are determined from measurements of the transport numbers in the resulting mixed crystal. The coefficient of self-diffusion of the ions of the principal substance will stand in the same ratio to the diffusion coefficient of the impurity as the ratio of the shares of participation of both kinds of ions in the mechanism of electrical conductivity.
Diffusion in AgCl is determined with Na and Cu ions as indicators. At \(238^\circ\), the values of the diffusion coefficients of Na and Cu differ greatly from each other: \(D_{\mathrm{Na}}=3.5\cdot10^{-6}\ \mathrm{cm}^2/\mathrm{day}\) and \(D_{\mathrm{Cu}}=2.1\cdot10^{-2}\ \mathrm{cm}^2/\mathrm{day}\). If, as indicated above, one calculates the coefficient of self-diffusion of Ag ions, then one obtains well-agreeing values for \(D_{\mathrm{Ag}}\): \(0.9\cdot10^{-4}\) and \(2.3\cdot10^{-4}\ \mathrm{cm}^2/\mathrm{day}\).
In most cases the difference between the rate of self-diffusion of the ions of the principal substance and the diffusion of impurity ions is only very slight. If Cu ions are introduced into \(\alpha\)-AgJ, then the obtained
the value of the diffusion coefficient will be only 1.35 times greater than the self-diffusion coefficient of Ag ions. Tubandt, Reinhold, and Jost, for comparison, also calculated the self-diffusion coefficients from the Nernst equation using electrical conductivity.
These data are given in Table 6 \((D_{\text{calculated}})\).
TABLE 6
Self-diffusion coefficient of Ag ions in AgCl from measurements of electrical conductivity and diffusion
\((\text{in } \mathrm{cm}^2 \cdot d^{-1})\)
| Temperature | 454° | 500° | 551° | 594° | 651° | 701° | 744° |
|---|---|---|---|---|---|---|---|
| \(D_{\text{calculated}}\) | 1.48 | 1.68 | 1.89 | 2.04 | 2.22 | 2.35 | 2.46 |
| \(D_{\text{calculated}}\) | 2.14 | 2.68 | 3.32 | 3.86 | 4.60 | 5.25 | 5.85 |
| \(D_{\text{corrected}}\) | 1.53 | 2.00 | 2.48 | 2.88 | 3.46 | 3.98 | 4.19 |
| \(\alpha\) | 0.71 | 0.75 | 0.75 | 0.75 | 0.75 | 0.76 | 0.72 |
Under \(D_{\text{corrected}}\) appear the values of the self-diffusion coefficients calculated from the Cu diffusion data. The values of the self-diffusion coefficients found by the two methods agree to within the constant factor \(\alpha\). Equation (4) must therefore be rewritten as:
\[ D = \frac{RT}{N} B \cdot \alpha . \tag{5a} \]
The theoretical value of the constant coefficient \(\alpha\), which also appears in other cases (CuJ, PbJ, etc.), cannot be explained from the standpoint of K. Wagner’s theory.
The mechanism of chemical reactions in the solid phase can be successfully clarified by studying electrical conductivity. Reinhold and Möring determined the rate of formation of a film of \(\beta\)-Ag\(_2\)S on a silver wire placed in liquid sulfur between 130 and 170°. The dependence of the rate of formation of this film (which gives tarnish colors) on temperature is represented by the equation:
\[ k = 17 \cdot e^{-\frac{10500}{T}} . \]
The specific electrical conductivity of \(\beta\)-Ag\(_2\)S in this temperature interval depends very strongly on the deviation of the composition of the preparation from the stoichiometric one. In the case where the composition of the substance corresponds to the stoichiometric formula, the temperature dependence of the electrical conductivity is determined by the equation:
\[ \sigma = 6 \cdot 10^{6} e^{-\frac{6800}{T}}, \]
whereas with an excess of sulfur (when the wire is in contact with sulfur)
\[ \sigma = 8 \cdot 10^{8} e^{-\frac{10700}{T}} . \]
Pure \(\beta\)-Ag\(_2\)S at 99% is an electronic conductor, whereas [[unclear: continuation cut off on page]]
the mechanism of formation of a film of silver sulfide on Ag is identical with the mechanism of conductivity of a preparation containing sulfur in excess.
The case of formation of a CuJ film on copper placed in iodine vapors was investigated at \(195^\circ\) by Hagemann and K. Wagner\(^{48}\). The rational rate coefficient of this reaction\(^{49}\), determined directly from the increase in weight, proved to be equal to \(k = 3.4 \cdot 10^{-10}\) equiv. cm\(^{-1}\) sec\(^{-1}\).
From the fact that the coefficient characterizing the rate of formation of the “tarnish film,” calculated from data on the ionic part of the electrical conductivity, proved to be equal to \(3.8 \cdot 10^{-10}\) equiv. cm\(^{-1}\) sec\(^{-1}\), it may be concluded that the mechanism of film formation consists in the fact that Cu ions and electrons continuously diffuse through the already formed CuJ layer and react with iodine on its outer side.
Dependence on Direction
The electrical conductivity of regular crystals, as A. Joffe showed, does not depend on the orientation of the crystal with respect to the direction of the current\(^{50}\). However, in crystalline systems with unequal axes, as for example in quartz, anisotropy is well expressed.
Zeit\(^{51}\) determined the conductivity of single crystals of PbJ\(_2\) in various directions and found that the dependence of electrical conductivity on direction is well expressed. His data are given in Fig. 10. As is clear, these data differ greatly from the results of investigation of pressed specimens (Fig. 9).
Fig. 10. Dependence of the electrical conductivity of PbJ\(_2\) on direction.
In Fig. 10 curve 1 corresponds to conductivity along the \(C\) axis, 2a and 2b to conductivity in the direction perpendicular to the \(C\) axis, whereas 3a and 3b represent the results of investigating the participation of iodine and lead ions in the conductivity of a pressed PbJ\(_2\) specimen.
Self-diffusion of lead ions, determined by the indicator method with the aid of radioactive lead, is almost the same in both directions. These data are quite understandable if one bears in mind that PbJ\(_2\) has a layered lattice in which a layer of lead atoms alternates with two layers of iodine atoms. The large iodine ions, which require considerably less energy to leave their place than lead ions (9000 cal and 3000 cal respectively),
move predominantly along the layers, which is consistent with the great structural sensitivity of this part of the conductivity. The smaller lead ions, on the contrary, move mainly perpendicular to the layer, between the readily polarizable iodine ions forming strongly loosened layers, just as easily as along the layer.
It should be noted that not only the total electrical conductivity, but also the transport numbers depend on the direction in the crystal. Thus the difference in electrolytic conductivity is explained by the simultaneous presence of two kinds of ions in the lattice, and not by different mobility of one definite ion in different directions.
Electrical conductivity of $\alpha$-Ag$_2$S
The explanation of the mechanism of conductivity in $\alpha$-Ag$_2$S is one of the most interesting chapters in the study of the electrical properties of nonmetallic solid substances, since the experiments carried out in this field seem, at first glance, so mutually contradictory that it is very difficult to imagine the existence of theories unifying them.
Tubandt, Eggert, and Schibbe$^{52}$ established in 1921 that, in the electrolysis of solid $\alpha$-Ag$_2$S, silver is liberated at the anode in strict accordance with Faraday’s law, whereas no substance is liberated at the cathode.
On the basis of these experiments it was concluded that $\alpha$-Ag$_2$S is a purely cationic conductor. What was surprising, however, was that the electrical conductivity of $\alpha$-Ag$_2$S was approximately 200 times greater than that of the best solid electrolytes already known earlier$^{55}$. The study of diffusion, on the contrary, gave a perfectly normal value of the diffusion coefficient; moreover, the value of the electrical conductivity calculated on its basis, by the Nernst equation, was considerably smaller than that measured directly.
Tubandt and Reinhold$^{54}$ were obliged to conclude from this that, besides the mobile Ag ions, the presence of which is determined in diffusion measurements, there must exist a small number of extraordinarily mobile ions, which are responsible for the high electrical conductivity of $\alpha$-Ag$_2$S.
However, these ions should not take part in the process of homogenization during the diffusion of Cu ions from Cu$_2$S into Ag$_2$S, since they can move predominantly only along disturbed sites of the lattice. This assumption requires wholly inconceivable values of mobility for these fast ions$^{55}$. Jost, as well as Tubandt and Reinhold in their time, came to the conclusion that, in order to explain this difficulty, it would be necessary to assume the participation of electrons in the electrical conductivity of $\alpha$-Ag$_2$S, if the experimental verification of Faraday’s law (determination of transport numbers) did not point so definitely and unambiguously to
exclusively ionic mechanism of conductivity in this substance.
An elucidation of these questions was obtained by K. Wagner[^56]. He proceeded from the assumption that the ionic conductivity of $\alpha$-Ag$_2$S in fact has the magnitude obtained on the basis of calculating the diffusion coefficient of Ag ions. The observed dependence of the electrical conductivity of $\alpha$-Ag$_2$S on the elasticity of sulfur vapor in the surrounding space[^57] indicates that, simultaneously with ionic conductivity, electronic conductivity also exists, since from the theoretical point of view it is unlikely that so strong a dependence of ionic conductivity on the elasticity of sulfur vapor could occur.
On the contrary, such a dependence, as is known, is observed in electronically conducting Cu$_2$O, whose conductivity, as Duinwald and Wagner showed, depends to a great extent on the pressure of oxygen[^57].
On this basis, the experiments to determine transport numbers for $\alpha$-Ag$_2$S, carried out by Tubandt, Eggert, and Schibbe, despite the fact that possible errors seemed completely excluded, must be critically reconsidered once again. The scheme of their experiments was as follows: Ag—anode/$\alpha$-Ag$_2$S/AgJ/Pt—cathode. When current passed, a decrease in the weight of the anode and an increase in the weight of the cathode were observed, equal to the weight of the silver deposited in the coulometer; this was explained by the fact that during electrolysis, for 96,540 coulombs, 1 g-equiv of silver is transported from the anode through $\alpha$-Ag$_2$S and AgJ to the cathode. Wagner showed, however, that another possibility also exists, in which the transport number of silver ions in $\alpha$-Ag$_2$S will not be equal to unity, but may be considerably smaller. The mechanism of the passage of current through the system of pellets in determining transport numbers can be represented as follows: electrons move predominantly from the boundary between AgJ and Ag$_2$S to the anode, and only an insignificant number of Ag ions moves in the opposite direction, whereas the current from the interface between AgJ and $\alpha$-Ag$_2$S is carried to the cathode by Ag ions. Therefore, at the interface, ions and electrons combine, which ultimately leads to the appearance of a corresponding excess of sulfur.
The diffusion of silver from the anode through $\alpha$-Ag$_2$S to the interface continuously makes up for this deficiency of silver atoms. However, this is possible only when small current-density values are used.
Jost and Ruter[^58], after Wagner had pointed out the possibility of the mechanism of the phenomenon considered above, showed experimentally that, with a systematic increase in current density, the decrease in the weight of the silver anode is indeed always less than would follow from the coulometer data, and that sulfur is liberated at the boundary between AgJ and Ag$_2$S. Further proof of the insignificant participation of Ag ions in current transport was given by K. Wagner[^59]. Wagner measured the electromotive force in the circuit: Pt/sulfur/
$\alpha$-Ag$_2$S/Ag, and it turned out to be equal to $E_0 n_{\mathrm{Ag}}$, where $E_0$ is the value of the e.m.f. calculated from the chemical affinity on the assumption of perfectly ionic conductivity, and $n_{\mathrm{Ag}}$ is the coefficient determining the share of participation of Ag ions in the conduction mechanism. The value of $E_0$ is approximately 0.2 V, whereas the measured value of the e.m.f. of the cell reaches only 0.002–0.005 V, which evidently corresponds to 1–3% ionic conductivity.
Almost simultaneously with the appearance of K. Wagner’s work, Tubandt and Reinhold $^{60}$ showed that $\alpha$-Ag$_2$S, $\alpha$-Ag$_2$Se, and $\alpha$-Ag$_2$Te are mixed conductors with a predominance of electronic conductivity. With the aid of these data, the initial discrepancy between the results of the investigation of diffusion and of conductivity is fully clarified.
$\alpha$-Ag$_2$S is predominantly an electronic conductor. The greater part of the current (99%) is carried by electrons, despite the fact that the number of free electrons possessing great mobility is small in comparison with the number of moving ions. The arrangement of the ions in the crystal lattice is similar to that described below for $\alpha$-AgJ $^{61}$. The presence of electronic conductivity is also confirmed by observations of the Hall effect $^{62}$.
Electrical Conductivity of $\alpha$-AgJ
One of the best ionic conductors is $\alpha$-AgJ. This substance has the peculiarity that its conductivity is considerably higher at temperatures below the melting point than in the molten state. The temperature coefficient of conductivity is very small, which leads one to suppose that the work of tearing Ag ions away is very small. On the other hand, the diffusion coefficient of silver ions is very large and agrees well with the value of the electrical conductivity. Consequently, we are dealing here with a conduction mechanism that presupposes a very high mobility of silver ions. In interpreting this mechanism, opinions long diverged $^{65}$. Smekal attempted to explain the high conductivity of $\alpha$-AgJ exclusively by the great mobility of ions situated in an “irregular” position; however, the investigations of Bloch and Jost $^{64}$, carried out on layers of AgJ $10^{-5}$ cm thick, which showed that their electrical conductivity is no greater than the electrical conductivity of compact material, as well as the observations of Hevesy and Rinecker $^{65}$ on the homogenization of mixed crystals of AgJ—CuJ by means of diffusion, indicate precisely the opposite. A whole series of theoretical considerations also speaks against Smekal’s view, although the arguments are for the most part not very convincing. The clarification of the conduction mechanism of $\alpha$-AgJ belongs to Strock $^{66}$, who used for solving the problem an X-ray investigation of the structure of the $\alpha$-AgJ lattice. The investigation showed that the $\alpha$-AgJ lattice is built of iodine ions arranged in a body-centered cubic lattice, while the small silver ions are distributed quite irregularly in
in the spaces between iodine atoms. Their great mobility is therefore due to the fact that the number of possible sites for them in the lattice is considerably greater than their number, and the high electrical conductivity is possible because all silver ions take part equally in the transfer of current, and not only those which are located at “special” sites of the lattice. The same mechanism of conductivity was established by Ketelaar⁶⁷ also for Ag₂HgJ₄. Here two silver ions and one mercury ion have at their disposal only four possible sites in the lattice for transitions, as follows from the X-ray determination of the structure. In Fig. 11 the change of electrical conductivity with temperature is shown for α-AgJ and Ag₂HgJ₄. The upper portion of the curve for Ag₂HgJ₄ corresponds to the formula:
$$ \sigma = 400 \cdot e^{-\frac{4300}{T}}. $$
Fig. 11. Electrical conductivity of AgJ and Ag₂HgJ₄.
Thermoelectromotive force and thermolysis
If two metallic conductors connected to one another are immersed in solutions, and they are connected to an instrument measuring the potential difference, then the latter will indicate the presence of an electromotive force, the magnitude of which depends on the concentration of ions in the solutions. The scheme of such a concentration cell may be represented in this form:
$$ \mathrm{Me}\mid(\mathrm{Me})\,c_1\mid(\mathrm{Me})\,c_2\mid\mathrm{Me}, $$
where \(c_1\) and \(c_2\) are the concentrations of ions in the solutions.
In a solid salt the number of ions is almost always the same, but the number of ions capable of moving in the crystal lattice depends strongly on temperature. Rasch and Hinrichsen⁷⁰ long ago pointed out that from the equation
$$ \sigma = Ae^{-\frac{Q}{RT}} \tag{3} $$
one can determine the degree of thermal dissociation
$$ \frac{\sigma_\vartheta}{\sigma_\infty}=e^{-\frac{Q}{RT}}, $$
since for \(T=\infty\), \(\sigma=A\). This corresponds to the ratio \(\Lambda/\Lambda_{\infty}\) for solutions, where \(\Lambda_{\infty}\) is the equivalent electrical conductivity at infinite dilution. Consequently, one can imagine a cell composed of solid salts in which, instead of different concentrations, different temperatures will be maintained. The scheme of such a cell may be represented as
\[ \mathrm{Me}\,|\,\mathrm{MeX}_{T_1}\,|\,\mathrm{MeX}_{T_2}\,|\,\mathrm{Me}. \]
Despite the complexity of the mechanism by which the emf arises in such a cell, it is nevertheless easy to detect its connection with ionic electrical conductivity. According to Reinhold\(^{69}\), who described the investigations referred to here in a series of papers, the thermoelectromotive force arising in such a cell, just as in metallic systems, is a complex quantity.
Here two components may be distinguished: the so-called heterogeneous part of the thermoelectromotive force of the cell, arising at the “junction,” corresponding to the Peltier effect in metals, and the homogeneous part of the thermoelectromotive force, whose origin is due to the temperature drop along the middle, homogeneous part of the cell. The magnitude of the heterogeneous component of the thermoelectromotive force can be determined from the equations of K. Wagner\(^{70}\) and Reinhold\(^{69}\). For example, in the case of a cell composed of silver halide compounds, this component amounts to only an insignificant part of the measured thermoelectromotive force, whereas the greater part of it is due to the homogeneous effect. In \(\mathrm{PbCl_2}\) and \(\mathrm{PbBr_2}\) the second component of the thermo-emf (homogeneous) is even larger than the measured thermoelectromotive force, since the heterogeneous effect has the opposite sign.
The magnitude of the effect, reaching \(0.30\ \mathrm{V/grad}\) for \(\alpha\)-\(\mathrm{AgJ}\), is well demonstrated by the following experiment of Reinhold. Between two silver electrodes several plates of \(\alpha\)-\(\mathrm{AgJ}\), placed one on top of another, are clamped, and the circuit is closed through a measuring instrument. If the ends of the circuit are maintained at temperatures of 220 and \(490^\circ\), then a current of \(2\ \mathrm{mA}\) flows in the circuit, carrying the corresponding amount of silver from the hot electrode to the cold one. The theoretical derivation of the magnitude of the homogeneous component of the thermoelectromotive force in a cell of solid ionic conductors is given by K. Wagner:
\[ \frac{d\varphi}{dT}=-\frac{Q^*}{zRT}. \tag{13} \]
Here: \(Q^*\) is the heat of transport of the corresponding ion (Überführungswärme), and \(z\) is its valence. A mutual connection between the homogeneous component of the thermoelectromotive force and the mobility of ions should be expected, if only on the basis of the existing parallelism between the magnitude of the thermoelectromotive force and the work of escape. For the silver halide compounds \(\mathrm{AgCl}\), \(\mathrm{AgBr}\), \(\mathrm{AgJ}\), ...
...the value of \(\frac{dx}{dT}\) is, respectively, 0.85, 0.62, and 0.30, while \(Q^*\) is 22000, 20600, and 1200. A more profound clarification of the question belongs to Reinhold, who investigated the Ludwig–Soret effect. If, along a mixed crystal—for example, a double cation conductor with a common anion—a certain temperature drop is created, then a stationary concentration distribution is established in it.
The magnitude of the concentration gradient is determined, according to K. Wagner, by the following expression:
\[ \frac{dx_2}{dT}=\frac{x_1x_2(E_1^*-E_2^*)}{RT^2}, \tag{14} \]
where \(x_1\) and \(x_2\) are the molar concentrations of the components, while \(E_1^*\) and \(E_2^*\) are the elementary transport energies.
Since the electrical conductivity is determined by an equation of the form (4), and the transport number, as was shown above, is the ratio of one term to the sum of two terms, the Ludwig–Soret effect arises only in the case when \(Q_1^*\) is not equal to \(Q_2^*\), i.e., in other words, when the transport numbers change with temperature.
The effect is the greater, the greater the difference \(E_1^* - E_2^*\) or \(Q_1^* - Q_2^*\). Reinhold additionally verified this on the mixed crystals \(\mathrm{Cu_2S—Ag_2S}\) and \(\mathrm{CuBr—AgBr}\). He was able to show that a stationary concentration distribution in the case of a temperature drop along the specimen is attained only when the concentrations and the corresponding temperatures, plotted along the coordinate axes, form a curve along which the transport numbers are everywhere the same (Fig. 12). In this case the comparatively less mobile component will be correspondingly enriched. In other words, this means that ions with the larger value of \(E^*\) move in the direction of the temperature drop, while ions with smaller values of \(E\) move in the opposite direction.
Fig. 12. Curves of equal transport numbers for the mixed crystal \(\mathrm{Cu_2S—Ag_2S}\) (according to Reinhold). \(I\)—curves of equal transport numbers, \(II\)—concentration distribution in the region of the temperature drop.
A particularly large effect is observed in those cases where the salt has a transformation point at a temperature lying within the limits of the temperature difference imposed on the specimen, and where a strong change in the transport numbers occurs before and after the transformation point. The assumption that the heats of transport and the separation works are identical is not confirmed, despite the fact that the differences \(E_1^* - E_2^*\) and \(Q_1 - Q_2\) are equal to each other. Especially inte-
similar relations were found by Reinhold and Schulze for the mixed crystal CuBr—AgBr. In this case the sign of the thermolytic effect and, correspondingly, of the temperature coefficients of the transport numbers changes within a certain concentration interval and in a certain temperature region in such a way that a stationary state is reached.
Theoretical Considerations
As was already mentioned earlier, there was no lack of attempts at a theoretical treatment of the data on the investigation of electrical conductivity.
In view of the fact that the mechanism of current transport by ions and the mechanism of self-diffusion are in principle one and the same, and that the magnitude of the electrical conductivity and the diffusion coefficient are related by the Nernst equation, it is first necessary briefly to mention investigations devoted to elucidating only the mechanism of diffusion. Deshman and Langmuir\(^{72}\) give a semi-empirical formula for the diffusion coefficient containing only one material constant—the heat of loosening (Auflockerungswärme):
\[ D = \frac{Q a^{2}}{N h}\cdot e^{-\frac{Q}{R T}} . \tag{15} \]
Here \(a\) is the lattice constant, \(N\) is Avogadro’s number, and \(h\) is Planck’s constant. The range of application of this equation is, however, limited to only a few examples. Brauer\(^{75}\) attempted to approach the problem from another side. He proceeded from Lindemann’s ideas on the melting of crystals, according to which breakdown of the lattice and, with it, melting occur when the amplitude of the thermal vibrations of the atoms begins to exceed a certain known value. In this case the ratio of the diffusion coefficient at a given temperature \((D_T)\) to the diffusion coefficient immediately below the melting temperature \((D_{T_s})\) will be equal to:
\[ \frac{D_T}{D_{T_s}} = e^{3b^2\left(1-\frac{T_s}{T}\right)}, \tag{16} \]
where \(b\) is a numerical coefficient equal to \(\sim 2\).
Similar to equation (16) is the equation proposed by Limpt\(^{74}\) in his theory of recrystallization; here the diffusion velocity is connected with the atomic frequency \(\nu\) and the lattice constant \(a\) in the following way:
\[ D = \frac{\pi a^{2}\nu}{6}\cdot e^{-\frac{3 b^{2} T_s}{T}} . \tag{17} \]
Recently this theory was developed by him in greater detail\(^{75}\). Underlying all these considerations is the idea that only lattice atoms located-
occupying a regular position of rest. Ioffe chose another path[^76][^78]. He considers that only a definite number of ions are mobile, which may be regarded as “dissociated.” These ions move in the space between the elements of the lattice and therefore do not occupy regular lattice sites. As an example of such motion one may cite the electrolytic displacement of Li through NaNO$_3$.
In this case the quantity $Q$ contained in equation (3),
\[ k = A e^{-\frac{Q}{RT}}, \]
must consist of two terms, the first of which depends on dissociation, and the second on the mobility of the ions. Frenkel[^4] adheres to this conception and ascribes to atoms or ions the following possible types of thermal motion in the crystalline lattice:
- Vibrations of atoms about a regular (regular) or irregular (irregular) position of equilibrium.
- Dissociation, i.e. the transition of an atom from a definite regular site in the lattice to an irregular position in the intercrystalline space.
- Displacement in the intercrystalline space.
- Association, i.e. the process inverse to dissociation.
- Displacement of free sites (“holes”) of the lattice by the transition into them of the nearest particles.
On the basis of this conception, dissociation takes place in such a way that an ion situated at a definite regular lattice site dissociates into an ion occupying an irregular position between the elements of the lattice, and into an empty site in the lattice, i.e. a “hole.” On the basis of a consideration of possibilities 1, 2, and 3, Frenkel arrives at formula (16):
\[ D = \frac{a^2}{6\sqrt{g \tau_0 \tau_0'}} e^{-\frac{u_0 + u_0'}{2kT_0}}, \tag{18} \]
where $a$ is the lattice constant, $g$ is a numerical coefficient equal to $\sim 1$, $\tau_0$ and $\tau_0'$ are the periods of free vibrations of atoms in the regular and irregular states, respectively, and $u_0$ and $u_0'$ are the transition energies for phenomena 2 and 3. In deriving the formula it is assumed that the probability of “exchange of places” per unit time is equal to $\alpha = e^{-\frac{u_0}{kT}}$ and $D' = \frac{1}{b}\frac{a^2}{\tau''}$. Formula (18) can be transformed, as Nernst and Einstein point out, into a formula for determining the electrical conductivity.
Braunbek[^77] showed that it is possible theoretically to obtain a quantitative dependence of the ionic conductivity of NaCl on temperature, without using the conceptions of intermediate “irregular” sites in the lattice introduced into Frenkel’s consideration.
of the “damaged” sites of the Smekal lattice. The spatial treatment of the phenomenon is simplified by introducing the assumption that only the Na ions execute linear oscillations, while the chlorine ions remain immobile at their sites in the crystal lattice. As is known, in the NaCl lattice the Na ion is located at the center of an octahedron whose vertices are occupied by six Cl ions.
The smallest value of the work required to remove a Na ion from the octahedron will occur in the 8 directions connecting the center of the octahedron with the centers of the faces of the octahedron; in doing so the Na ion will occupy a new site, freed in the same way. Braunbek determined the probability that such a process occurs during the period of oscillation of the Na ion, and, using the value of the self-diffusion coefficient, obtained the following expression for the electrical conductivity:
\[ \sigma=\frac{2e^{2}}{3\tau ac\varphi_{0}}\,e^{-\frac{\varphi_{0}}{kT}}, \tag{19} \]
where \(e=4.77\cdot10^{-10}\) CGSE; \(a=5.63\cdot10^{-8}\,\text{cm}\), \(k=1.37\cdot10^{-16}\,\text{erg}/2\,\text{rad}\). Strictly speaking, instead of \(\tau\) one should substitute the duration of the transition; however, instead of this one substitutes the period of oscillation of the Na ion in the NCl lattice, \(\tau_{0}=2.1\cdot10^{-13}\,\text{sec}\). \(C\) is the value of the curvature of the potential curve at the point where we assume the position of the Na ion. For NaCl \(C=8.3\) (the melting temperature of NaCl—1073°). Instead of the potential barrier overcome in the exchange of places, the value of the potential barrier for the phenomenon of melting has been substituted\(^{78}\).
Using the equation (19) given above, one can represent the value of \(\sigma\) in the following form:
\[ \ln\sigma=27.7-\frac{10400}{T}. \]
Seelen\(^{14}\) experimentally found that in the temperature range from 20 to 500°
\[ \ln\sigma=25.9-\frac{10700}{T}. \]
The agreement, as follows from comparison of these formulas, is quite good. The constants of the formula given above have the same values as those found also for the electrical conductivity in the region of low temperatures, although the latter should not be ascribed to the mechanism indicated above. In the region of the lattice’s intrinsic conductivity the constants \(A\) and \(B\) are respectively equal to: \(10^{6}\) and 22000. Owing to the very strong simplification, the substitution made above is not suitable for fully explaining these phenomena, although we have tacitly assumed the existence of vacant sites in the lattice, since the site to which the ion passes must already be free.
On the basis of the ideas of Frenkel and Jost[^9], using the methods of probability theory, he came to the conclusion that the number of ions in irregular positions in the intercrystalline space must be equal to:
\[ n_1 \approx Ne^{-\frac{E}{2kT}} \tag{20} \]
or
\[ \ln \frac{n_1}{N} \approx -\frac{1}{2}\frac{E}{2kT}, \tag{21} \]
where \(N\) is the total number of ions. This conclusion agrees fully with Frenkel’s data, obtained by another method. The particles situated in the special position \((n_1)\), however, still cannot diffuse, since in order to acquire this possibility they must overcome a potential barrier; therefore, in the exponent, in addition to \(\frac{E}{2}\), there also appears \(u\). To estimate the magnitude of the diffusion coefficient, Jost assumed that each empty site (“hole”) is surrounded by six ions. The total number of ions bordering “holes” is thus \(6n_1\). Jost showed that near a “hole” the coefficient is determined by the following expression:
\[ D_{\text{hole}}=\frac{1}{6}\left(a\cdot\frac{v}{2}\right)e^{-\frac{u}{kT}}, \tag{22} \]
where \(a\) is the lattice parameter, and \(v\) is the mean velocity of the ions. This number must further be multiplied by \(\frac{6n_1}{N}\), i.e., by the ratio of the number of ions surrounding “holes” to the total number of ions in the lattice. This ratio is equal to:
\[ 6e^{-\frac{e}{2kT}}. \]
Then
\[ D=\frac{1}{6}\left(d\frac{v}{2}\right)e^{-\frac{u}{kT}}\cdot 6e^{-\frac{E}{2kT}} =\frac{dv}{2}e^{-\frac{\left(\frac{E}{2}+u\right)}{kT}}. \tag{23} \]
If it is assumed that \(d=3\cdot10^{-4}\ \text{cm}\) and \(v=5\cdot10^{-4}\ \text{cm/sec}^{-1}\), then \(A=7.5\cdot10^{4}\ \text{cm}^2\ \text{sec}^{-1}\), or \(65\ \text{m}^2\cdot d^{-1}\). The obtained value of \(A\) determines only the rate of diffusion. However, using the Nernst equation (5) \(D=BkT\), where \(B\) is the mobility of a diffusing ion under the action of a force equal to unity, we can pass to the equation of electrical conductivity. The coefficient before the exponential term in the electrical-conductivity equation has almost the same value as \(\frac{dv}{2}\), if \(D\) is expressed in \(\text{cm}^2\cdot d^{-1}\).
The equation representing the temperature dependence of electrical conductivity, derived on the basis of Jost’s assumptions, has the following form:
\[ \sigma \simeq (10 \text{ to } 100)e^{-\frac{\frac{E}{2}+u}{kT}} . \tag{24} \]
In a whole series of cases the value of the constant \(A\) does indeed lie between 10 and 100. There are, of course, also significant deviations in one direction or another, which still await explanation. After determining \(A\), Jost attempted to estimate the values of \(E\) and \(u\) for the NaCl lattice model. To transfer an ion from the site \(111\) to the position \(\frac{1}{2}\ \frac{1}{2}\ \frac{1}{2}\), it is necessary to expend an energy equal to
\[ 2.27\,\frac{e^2}{a} \simeq 13 \text{ eV}, \]
or \(300\,000\ \text{cal/mol}\), whereas the experimentally determined value of the heat of separation reaches only \(40\,000\ \text{cal/mol}\).
Fajans and Reiss \(^{80,81}\) were the first to point out the importance of the mutual polarization of ions. Mutual polarization causes differences in the work of extracting ions, which in turn often gives rise to practically unipolar electrical conductivity. An ion will move the more easily, the smaller its charge, the slighter its deformability, and the greater its deforming action on neighboring ions \(^{82}\). This explains the fact that good conductors always possess unipolar conductivity. Asymmetric electric fields, in which the polarization forces may reach the same magnitude as the Coulomb forces, arise especially readily in the presence of “spoiled” places in the lattice (cracks, ruptures, atoms and ions in the intercrystalline space). If the crystal is considered as a dielectric continuum, then
\[ E_{\mathrm{pol}}=-\frac{e^2}{r_0}\cdot\frac{\varepsilon-1}{\varepsilon}, \tag{25} \]
where \(\varepsilon\) is the dielectric constant of the crystal. If one takes \(\varepsilon=6\), and \(r_0=\frac{a}{2}\), then \(E_{\mathrm{pol}}=-1.67\,\frac{e^2}{a}\).
Finally we obtain that:
\[ E=(2.27-1.67)\frac{e^2}{a}\simeq 2.5 \text{ eV}\simeq 60000\ \text{cal/mol}, \tag{26} \]
which approximately agrees with the experimental results.
Jost calculated that for the ionic part of the conductivity of \(\alpha\)-Ag\(_2\)S, \(E=0.24\). This value is in order of magnitude agreement with the experimental data (\(E=0.14\ \text{eV}\)).
Of particular interest is the result of calculating the potential barrier overcome by Ag ions in passing into a neighboring “hole.” The height of the barrier turned out to be of the usual order of magnitude and equal to
2 eV. Since, however, the experimental data for $\alpha$-Ag$_2$S showed that the true value of the height of the potential barrier is much smaller, Jost suggested that in $\alpha$-Ag$_2$S the “exchange of places” preferentially takes place in the intercrystalline space. When an ion passes from one place in the intercrystalline space to another, $u = 0.45$ eV, if one takes into account that the ions situated near the place through which the ion diffusing through this lattice plane passes are repelled by the latter from one another. Since the assumptions underlying the calculation are very schematic, agreement can be expected only as to the order of magnitude.
Almost at the same time as Jost carried out the above calculation of the energy quantities, Wagner$^{83}$ was engaged in constructing a picture of the mechanism of the “exchange of places” of ions in the crystal lattice, which he built on the basis of the ideas developed by Schottky and Wagner in their theory of ordered solutions$^{87}$.
When publishing data on electrical conductivity it is usually assumed that the substance investigated is sufficiently well defined if the degree of purity of the preparation and the method of its preparation are known, and discrepancies between the results of different investigators are always attributed to differences in these two factors. However, Bedeker$^{85}$ had already found that the electrical conductivity of CuJ, which is an electronic semiconductor, depends strongly on the partial pressure of iodine vapor in the atmosphere surrounding the sample, or, in other words, on the excess of iodine, relative to the stoichiometric composition, in the sample under investigation. Steinberg$^{86}$ established that the Hall effect also depends on the same factor.
Gudden$^{87}$ concluded, on the basis both of the above and of other numerous investigations, that the electrical conductivity of such electronic semiconductors as CuJ, Cu$_2$O, NiO, UO$_2$, is due to an excess of the negative component.
Wagner developed a picture of the mechanism of ionic and electronic conductivity in polar compounds, based on the theory of “disordering” of the crystal lattice, which had already been considered in connection with the general theory of ordered solutions. In doing so he proceeded from three possibilities for the formation of an excess of component (B) in the lattice of the compound (AB).
The excess atoms B may be located between the main atoms of the lattice (interstitial type, or intercrystalline type).
Cases are possible in which all atoms of component B form a regular lattice (Teilgitter), while a certain number of sites in the lattice formed by atoms of component A, corresponding to the number of excess atoms B, remains unoccupied. In other words, empty sites (Leerstellentypus) are formed in the places of the missing atoms A.
Finally, a third case is possible, when replacement occurs
of individual atoms A by excess atoms of component B (substitution type).
The third case, with respect to polar compounds, must be singled out. In an ionic lattice there cannot occur an excess of ions of one sign, since the electrical neutrality of the lattice must be observed. An excess of cations entails the simultaneous appearance of an excess of electrons, while an excess of anions must be connected with the appearance of empty places in the lattice (“holes”). These empty places may be formed by the transition of cations into a state of higher valence, or of anions into a state of lower valence. In addition, the formation of “defective places” in a crystal of perfectly stoichiometric composition can be explained by considering the following equilibria:
-
“ideally perfect” crystal $\rightleftarrows$ quasi-free electron $+$ “hole”.
-
“ideally perfect” crystal $\rightleftarrows$ cations between the normal sites of the lattice $+$ empty places in the cation lattice.
-
“ideally perfect” crystal $\rightleftarrows$ anions in the intercrystalline space $+$ empty places in the anion lattice.
Here it should be especially emphasized that the number of “defective” lattice sites (Fehlstellen) is determined, according to Wagner, by the thermodynamic equilibrium established between the two sides of the schemes given above.
The existence of an “ideally perfect” crystal is therefore not practically possible. The magnitude of the electrical conductivity is determined by the position of the equilibrium and is connected with the presence of “defective” places in the crystal lattice.
On the basis of the point of view set forth above, Schmekal regards “defective” places in the same way as places where conducting ions arise. Their number determines the degree of imperfection of the real crystal in comparison with the “ideally perfect” one.
K. Wagner^88 established in all 5 classes, combining 12 different possibilities for the formation of “disorder” in the crystal lattice, as well as 12 mechanisms of electrical conductivity, which follow from the 2 types of deviations from stoichiometric composition indicated above and the 3 possibilities for the formation of “defective” places in compounds of strictly stoichiometric composition. Since an excess or deficiency of one of the components of a compound determines the magnitude of the electrical conductivity, the latter depends on the partial pressure of the negative component in the external atmosphere surrounding the specimen; and the direction and magnitude of this dependence make it possible to create a classification of practical cases on this basis. Below we give Wagner’s classification of the various types of conductivity in the form in which it is given in his original work:
- Electron conductivity (Überwiegende Elektronenleitung), decreasing with increasing partial pressure of the negative component.
Excess of metal = cations + electrons.
Electronic “excess” conductivity.
- Electronic conductivity increasing with increasing partial pressure of the negative component.
Excess of the negative component = anions + vacant sites (“holes”) in a band of quantum states completely filled with electrons. Hole conductivity (Elektronen-Defektleitung).
- Electronic conductivity independent of the pressure of the negative component.
Equality of the equivalent concentrations of free electrons and “holes.”
- Excess cationic conductivity. Equality of the concentrations of anions in the intercrystalline space and of vacancies in the anion lattice.
In cases 1 and 2 there is an excess of ions of one component, which entails the presence of different numbers of quasi-free electrons or vacant sites in a band of quantum states completely filled with electrons, i.e. “holes.” In all these cases a predominant electronic conductivity is observed, owing to the greater mobility of the electrons; moreover, the magnitude of the conductivity decreases or increases with increasing partial pressure of the negative component, depending on whether cations or anions are present in excess.
As an example of the first case one may cite ZnO and CdO, whose electrical conductivity was recently thoroughly investigated by Baumbach and Wagner ^89. In these substances the electrical conductivity falls with increasing oxygen pressure (Fig. 13).
The question of whether “excess” electronic or hole (“defect”) conductivity occurs here was resolved by K. Wagner ^88 on the basis of a study of the thermoelectromotive force of a couple composed of two specimens of the corresponding semiconductor with different electrical conductivities (i.e. with different contents of excess oxygen). Wagner showed that ZnO possesses “excess” electronic conductivity.
The second case was practically realized by Dünwald and Wagner ^90 in the example of Cu₂O, whose conductivity increases very strongly with increasing partial pressure of oxygen (Fig. 14). At high temperatures, along with electronic conductivity, ionic conductivity also appears, and the copper ions are mobile. At 1000° the transport number for copper ions is equal to \(4 \cdot 10^{-4}\). Since, however, the mobility of Cu ions is very small in comparison with the mobility of electrons, their number must nevertheless be fairly considerable. The sign of the thermoelectromotive force indicates the defect conductivity of Cu₂O.
Whereas Baumbach and K. Wagner in their investigations sought to obtain the corresponding equilibrium state in the specimen by keeping it for a prolonged time at the measurement temperature in an oxygen atmosphere, other investigators ^94 “quenched” a specimen heated to a high temperature inst—
with slow cooling, and thus investigated the so-called “frozen” equilibrium. The results of both groups of investigators agree fairly well. According to Jouze and Kurchatov \(^{91}\), and also Weibel \(^{92}\), stoichiometric \(\mathrm{Cu_2O}\) is an insulator with \(\sigma = 10^{-10}\ \Omega^{-1}\,\mathrm{cm}^{-1}\), whereas upon
Fig. 13. Dependence of the electrical conductivity of CdO on the oxygen pressure (after Baumbach and Wagner).
saturation of the specimen with excess \(\mathrm{O_2}\), the electrical conductivity at room temperature increases to \(10^{-2}\ \Omega^{-1}\,\mathrm{cm}^{-1}\) \(^{93}\). Copper oxide \(\mathrm{CuO}\) is the most representative of the third group, with an established equilibrium between “holes” and quasi-free electrons. Baumbach, Dünwald, and Wagner \(^{97}\) showed that the electrical conductivity of \(\mathrm{CuO}\) does not depend on the oxygen pressure.
Fig. 14. Dependence of the electrical conductivity of \(\mathrm{Cu_2O}\) on pressure (after Dünwald and Wagner).
Scheme (I): “ideally perfect” crystal \(\rightleftarrows\) electrons \(=\) conductivities \(+\) “holes,” may for this case be written as follows:
\[ 2\mathrm{Cu}^{\bullet\bullet} \rightleftarrows \mathrm{Cu}^{\bullet} + \mathrm{Cu}^{\bullet\bullet\bullet}. \]
For the last two cases, 4 and 5, it is characteristic that the concentration of mobile electrons is exceptionally small in comparison with the number of mobile ions. These groups include all crystals with purely ionic or, at least, predominantly ionic conductivity, such as, for example, AgCl or BaCl₂.
The classification considered above, of course, covers only typical limiting cases and therefore by no means exhausts all possibilities. To explain the properties of Ag₂S, whose electrical conductivity, according to Tubandt and Reinhold⁹⁴, decreases with increasing sulfur vapor pressure, one may, for example, make the following assumption.
Alongside the equilibrium of cations in the intercrystalline space and of vacant sites (cationic), there exists in the crystal a certain number of quasi-free electrons and “holes” (the role of “holes” may, for example, be played by singly charged sulfur ions). An increase in the elasticity of sulfur vapor entails a decrease in the number of cations in the intercrystalline space, and also of quasi-free electrons, which in turn leads to a decrease in electrical conductivity. In salts possessing bipolar conductivity, when considering the mechanism of electrical conduction one must take into account the formation of “defective” sites in both ionic lattices (anion and cation).
According to the data of O. Kubaschewski⁹⁵, in an investigation undertaken at Seitz’s suggestion, lead iodide has such a strong dependence of electrical conductivity on iodine vapor pressure that one may suppose, alongside ionic conductivity, the presence of electronic conductivity. However, there is as yet no unambiguous solution to this problem.
K. Hauffe and K. Wagner⁹⁶ showed that the dependence of the electrical conductivity of copper iodide (CuJ) on iodine vapor pressure, measured in the range from 40 to 300°, as well as measurements of the thermoelectromotive force and the Hall effect, undoubtedly point to “hole” electronic conductivity of this substance. Alongside this, the same authors established the presence in this substance also of cationic conductivity, which at 200° and an iodine vapor pressure of 40 mm Hg reaches the value \(1.8 \cdot 10^{-4}\ \Omega^{-1}\cdot\mathrm{cm}^{-1}\).
Tubandt and Boduin⁹⁷ give, for the electrical conductivity of CuJ at 200° in high vacuum, after it has lost part of its stoichiometric iodine, the value \(5 \cdot 10^{-4}\ \Omega^{-1}\cdot\mathrm{cm}^{-1}\). Thus in this case as well the ionic conductivity is dependent on pressure, which indicates the presence of vacant sites in the lattice of Cu ions.
The success achieved thanks to the works of K. Wagner consists in the development of the theory of the “mechanism” of site exchange “between atoms” (Platzwechselmechanismus), with the aid of which it has been possible to explain a whole series of experimental data. With the aid of the theory of “site exchange” it has been possible, on the one hand, to explain the mechanism of current transport through a perfect ionic lattice, and on the other—
bring into correspondence the mobility of ions due to the presence of “defective” sites in the lattice, whose origin is due to thermal motion and whose number is determined by a kind of thermodynamic equilibrium, with the mobility due to the presence of “defective” sites in the form of impurities, ruptures, and other mechanical damage to the crystal.
These considerations relate primarily to the mechanism of electrical conductivity in the region of high temperatures; however, they can also be extended to low temperatures.
On the basis of the experiments and calculations of Jost described above, Schottky \(^{98}\) developed his views on the mechanism of conductivity in solid salts, which constitute a valuable contribution to this field of knowledge. As the working model the NaCl lattice was chosen, since already after Jost’s work it had been shown that \(\mathrm{Ag_2S}\) represents a special, insufficiently characteristic case. First of all Schottky, using Jost’s method and taking into account the polarization energy of the ions by one another, calculated the activation work, which proved to be equal to \(1.6\ \mathrm{eV}\), i.e. in order of magnitude in good agreement with the experimental data. Subsequently Schottky went considerably further than Jost. Jost assumed that in a unipolarly conducting crystal “defective” sites are formed only by the mobile kind of ions, while the other ions firmly occupy their places in the crystal lattice. This leads first of all to the consequence that in such a crystal the number of “defective” sites is equal to the number of vacant sites.
If, on the contrary, one assumes, as Schottky does, that both kinds of ions may take part to the same extent in the formation of “defective” sites, then one can imagine a system in which the number of vacant sites will be many times greater or smaller than the number of sites occupied in the intercrystalline space, and this without abandoning the necessary electrical neutrality of the crystal as a whole. Only the number of vacant sites in the cation and anion lattices must be the same, as well as the number of cations and anions in the intracrystalline space. It should be noted that the concepts developed can be extended not only to a heterogeneous crystal of a bipolar salt, but also to a homogeneous crystal built from one kind of atoms, as for example a metallic one; in this case it is assumed that such a crystal contains in the intercrystalline space either an excess of vacant sites or, conversely, of atoms of the substance. What has been said above can be reduced to the following two equilibria:
-
An atom in the proper (“regular”) position of rest \(\rightleftarrows\) a vacant site in the lattice (Gitterlücke) + an atom in an “irregular” position (in the intercrystalline space).
-
An atom in a “regular” position of rest \(\rightleftarrows\) a vacant site in the lattice + an atom internally adsorbed.
Schottky speaks directly of the “solubility of intracrystalline particles” and of the “solubility of vacant sites,” and the “product of their solubilities” determines the work of “disordering—
“lattice positions.” The assumption that the number of vacant sites in the lattice and of particles in the “irregular” position are unequal goes so far beyond the bounds of general conceptions that the new treatment of the problem gives a picture much more consistent with the experimental data. Thus, for example, it is very unlikely that, in such a closely packed lattice as the NaCl lattice, containing no excess atoms of any kind, the electrical conductivity should be due to the motion of cations in the intercrystalline space.
As a result of a calculation, the details of which cannot be given here, Schottky found that:
\[ \begin{aligned} \mathrm{I}\quad & RT \ln x_{(\mathrm{Na})}\cdot x_{[\mathrm{Na}]} = \sim - E_{(\mathrm{Na})} - E_{[\mathrm{Na}]} = \Delta E_{\mathrm{Na}}, \tag{27}\\ \mathrm{II}\quad & RT \ln x_{(\mathrm{Cl})}\cdot x_{[\mathrm{Cl}]} = \sim - E_{(\mathrm{Cl})} - E_{[\mathrm{Cl}]} = \Delta E_{\mathrm{Cl}}. \tag{28} \end{aligned} \]
Here \(x_{(\mathrm{Na})}\) is the ratio of the number of sodium ions in the “irregular” position (i.e., in the intracrystalline space) \(n_{(\mathrm{Na})}\) to the number of all sodium ions \((n)\), i.e.,
\[ x_{(\mathrm{Na})}=\frac{n_{(\mathrm{Na})}}{n}, \]
\(x_{[\mathrm{Na}]}\) is the value of the corresponding ratio for vacant sites in the cation lattice.
\(E_{(\mathrm{Na})}\) is the change in energy when one Na ion, at absolute-zero temperature, passes from the regular into the irregular position. \(E_{[\mathrm{Na}]}\) is the corresponding change in energy when a vacant site in the Na ionic lattice is filled. The symbols \(x_{(\mathrm{Cl})}\), \(x_{[\mathrm{Cl}]}\), \(E_{(\mathrm{Cl})}\), and \(E_{[\mathrm{Cl}]}\) have the same meaning, respectively.
If in equation (27) the number of “irregular” sites (sites in the intercrystalline space) is equal to the number of vacant sites, then:
\[ \ln x_{(\mathrm{Na})}=-\frac{1}{2}\frac{\Delta E}{RT}. \tag{29} \]
Equation (29) is identical with the expression first given by Jost. To determine all four concentrations of interest to us, namely the concentrations of cations and anions in the intercrystalline space and the concentrations of vacant sites in the cation and anion lattices, the equations given are insufficient.
Additional equations are necessary. One of them can be obtained if the stoichiometric composition and electrical neutrality of the crystal are assumed. Then:
\[ \mathrm{III}\quad x_{(\mathrm{Na})}-x_{[\mathrm{Na}]}=x_{(\mathrm{Cl})}-x_{[\mathrm{Cl}]}. \tag{30} \]
Schottky obtained two other equations by introducing the Born lattice energy, \(E_g\), into the discussion:
\[ \mathrm{IV}\quad RT \ln x_{(\mathrm{Na})}x_{(\mathrm{Cl})}=-[E_{(\mathrm{Na})}+E_{(\mathrm{Cl})}]+E_g, \tag{31} \]
\[ \mathrm{V}\quad RT \ln x_{[\mathrm{Na}]}x_{[\mathrm{Cl}]}=-E_g+[E_{[\mathrm{Na}]}+E_{[\mathrm{Cl}]}]. \tag{32} \]
These five equations are quite sufficient for determining all four concentrations. In doing so, one must bear in mind that the calculation gives, instead of the values of the corresponding concentrations, only the values of the following expressions:
\[ (E_{(\mathrm{Na})}-E_{[\mathrm{Na}]}),\quad (E_{[\mathrm{Na}]}+E_{(\mathrm{Cl})}),\quad (E_{(\mathrm{Na})}+E_{(\mathrm{Cl})}),\quad \text{and}\quad (E_{[\mathrm{Na}]}+E_{[\mathrm{Cl}]}). \]
Especially transparent relations are given somewhat below, and the following quantities occur in them:
\[ K=e^{-\frac{E_g-(E_{[\mathrm{Na}]}+E_{[\mathrm{Cl}]})}{RT}}, \]
\[ k_1=e^{-\frac{(E_{(\mathrm{Na})}+E_{[\mathrm{Cl}]})-E_g}{RT}}, \]
\[ k_2=e^{-\frac{(E_{(\mathrm{Cl})}+E_{[\mathrm{Na}]})-E_g}{RT}}. \]
The quantities \(k_1\) and \(k_2\) deserve special attention, since a new combination of energies enters the exponent of \(e\), making it possible to clarify the mechanism of electrical conductivity. The expressions \(E_{\text{cation}}+E_{[\text{anion}]}\) and \(E_{[\text{cation}]}+E_{\text{anion}}\), denoted by Schottky as conjugate energies and representing, respectively, the sums of the energy of a cation in the intercrystalline space and of an empty anion site, or of an anion in the intercrystalline space and a cation “hole,” determine, depending on whether they are greater or less than the lattice energy \(E_g\), the values of \(k_1\) and \(k_2\) (which, respectively, may be considerably less or greater than 1).
On this basis one can distinguish four typical cases for the concentration of “disturbed” sites in the lattice (in other words, for the “degree of disorder” of the lattice (A—cation; B—anion)).
\[ \begin{array}{llll} \mathrm{I} & k_1 \gg 1; & k_2 \ll 1 & \text{then } x_{[A^+]} \text{ and } x_{(A^*)} \text{ predominate} \\ \mathrm{II} & k_1 \ll 1; & k_2 \gg 1 & \text{then } x_{[B^-]} \text{ and } x_{(B^-)} \text{ predominate} \\ \mathrm{III} & k_1 \gg 1; & k_2 \gg 1 & \text{then } x_{[A^+]} \text{ and } x_{(B^-)} \text{ predominate} \\ \mathrm{IV} & k_1 \ll 1; & k_2 \ll 1 & \text{then } x_{[A^+]} \text{ and } x_{[B^-]} \text{ predominate} \end{array} \]
In other words, four mechanisms of electrical conductivity are possible:
I. Empty sites in the cation lattice and an equal number of cations in the intercrystalline space in an “irregular” position.
II. Empty sites in the anion lattice and an equal number of anions in the intercrystalline space.
III. An equal number of anions and cations in the intercrystalline space.
IV. An equal number of empty sites in the anion and cation lattices.
The model considered by Jost belongs to case I. The circumstance that the anionic and cationic conductivities in crystals are of the same order of magnitude made it possible for Schottky to conclude that the conductivity in them is entirely due to the displacement of vacant anionic and cationic sites (case IV).
Since the numbers of anions and cations participating in the transport of current are equal, the deviation of the transport numbers from 0.5 is due to the different mobility of anions and cations. A rough estimate of the magnitudes of the energy barriers and of the electrical conductivity also indicates the predominant participation of vacant lattice sites in the mechanism of conductivity, although, on the other hand, it says nothing about the actual concentration of these vacant sites. However, an exact determination of the concentration of disordered atoms—that is, atoms not sitting in their own places—and consequently of vacant sites in the lattice, since a strictly stoichiometric composition of the crystal is assumed, would be entirely possible if it were possible to determine with great accuracy the exponent in the dissociation-energy equation (~9).
The results achieved in the field we have touched upon may be formulated as follows:
the mechanism of ionic conductivity has been studied to such an extent that it is now possible to indicate in advance the path along which the theoretical calculation of each separate case should be carried out; but at the same time, for the accurate performance of these calculations, more reliable experimental data are necessary.
At present it is urgently necessary to carry out experimental investigations to determine which of the theoretically possible mechanisms of lattice “disordering” considered above actually occur. One such investigation was recently carried out by K. Wagner and Beyer^99 with AgBr.
The purpose of the work was to determine which of the mechanisms of “disordering,” Frenkel’s or Schottky’s, occurs in AgBr. In other words, whether the number of vacant sites is equal to the number of ions in the intercrystalline space (i.e., in irregular positions), or whether the number of the former is considerably greater. Since an increase in temperature results in an increase in the degree of disorder of the lattice, in the second case one should expect a decrease in the average number of molecules in the elementary crystal cell. However, both at room temperature and at 400° this number, within the accuracy of the measurements (~1%), remained unchanged, whereas if the Schottky mechanism were taking place one would have expected a decrease of approximately 16%. Consequently, in AgBr the number of vacant sites in the lattice is approximately equal to the number of silver ions in the intercrystalline space, which corresponds to the picture developed by Frenkel.
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