On the Relationship Between the Electrical Strength of Crystals and Their Surface Energy
Abstract Generated abstract
This paper examines the proposed connection between the electrical breakdown strength of crystals and their surface energy, treating breakdown as a form of crystal destruction involving the formation of new surfaces. Surface energies of alkali halide single crystals and solid solutions are determined by mutual grinding and drilling methods, and the results indicate that impurity addition lowers surface energy, with minima near equimolar compositions in the systems studied. The author argues that both surface energy and electrical strength correlate with the number of gram-molecules per unit volume, expressed as density divided by molecular weight, and uses this relation to estimate breakdown strengths for several alkali halide crystals lacking experimental data. Comparisons with lattice energy and with liquid dielectrics suggest that the dependence is characteristic of crystal lattice properties rather than universal across dielectric materials.
Full Text
On the Relationship Between the Electrical Strength of Crystals and Their Surface Energy
P. A. Savintsev
Electrical breakdown is, in the final analysis, the destruction of a crystal associated with the formation of new surfaces. Surface energy characterizes the amount of work that must be performed in forming a unit of new surface. It therefore makes sense to seek a relationship between the electrical strength of a crystal and its surface energy.
It is of interest to find a physical quantity that is connected both with the electrical strength of a crystal and with its surface energy. In this connection it is necessary to consider the question of determining the surface energy of crystals.
The surface energy of crystals $\sigma$ has been calculated theoretically by a number of investigators [1, 2]. The results obtained by different authors are presented in Table 1, from which it is evident that, for the same salts, different values of $\sigma$ have been found, sometimes differing greatly from one another. Therefore the experimental determination of surface energy is of great interest.
Table 1
Values of surface energies in $\mathrm{erg}/\mathrm{cm}^2$ according to calculations by different authors for the face (100).
| No. | Authors | Crystals | Cl | Br | J |
|---|---|---|---|---|---|
| 1 | Born and Stern | Na | 150 | 119 | 96 |
| 1 | Born and Stern | K | 107 | 92 | 75 |
| 2 | Frenkel | Na | 164 | 140 | 107 |
| 2 | Frenkel | K | 121 | 118 | 83 |
| 3 | Zhdanov | Na | 210 | 174 | — |
| 3 | Zhdanov | K | — | 115 | 113 |
| 4 | Bimoller | Na | 87 | 70 | 51 |
| 4 | Bimoller | K | 91 | 77 | 59 |
| 5 | Lennard-Jones and Dent | Na | 93 | 90 | 88 |
| 5 | Lennard-Jones and Dent | K | 75 | 73 | 72 |
| Molten salts at the melting temperature | Na | 114 | 106 | 87 | |
| Molten salts at the melting temperature | K | 97 | 89 | 85 |
There are several methods for determining the surface energy of crystals. However, each of them can be applied only to a certain group of bodies, and as yet no method suitable for all bodies is known.
We measured the surface energy of crystals by two methods: drilling and mutual grinding, the theory of which was developed by V. D. Kuznetsov [2].
Let, in the mutual grinding of two crystals, identical powder particles of size \(b^3\) be formed. The number of particles in the volume \(v\) of the ground crystal will be \(N=\dfrac{v}{b^3}\). The surface of each powder particle will be
\(S=6b^2\). For the formation of one particle the energy expended will be \(S\sigma=6b^2\sigma\), and for the dispersion of the crystal
\[ NS\sigma=\frac{6v\sigma}{b}. \]
If \(v_1\) and \(v_2\) are the volumes of the ground layers of the first and second crystals, and \(\sigma_1\) and \(\sigma_2\) are their surface energies, then, assuming that in the mutual grinding of two crystals the energy spent on dispersion is distributed equally between the rubbing crystals and that the sizes of the powder particles of the first and second crystals are identical, one may write
\[ v_1\sigma_1=v_2\sigma_2. \]
V. D. Kuznetsov tested this dependence on polycrystals of alkali-halide salts and then calculated the surface energy of single crystals.
We determined the surface energies of single crystals KCl, NaCl, KBr, and NaBr. As a crystal with known surface energy we chose a KCl single crystal, for which \(\sigma_1=108\) erg/cm\(^2\). This value was obtained theoretically by Born and Stern and confirmed in a number of earlier works by V. D. Kuznetsov [2]. Our results are presented in Table 2.
Table 2
Surface energies of alkali-halide single crystals, determined by the method of mutual grinding
| Measured quantity | KCl | NaCl | KBr | NaBr |
|---|---|---|---|---|
| Ratio of surface energies \(\dfrac{\sigma_2}{\sigma_1}\) | 1 | 1.45 | 0.91 | 1.20 |
| Surface energy | 108 | 156 | 98 | 129 |
In subsequent experiments we determined the surface energy of KCl—NaCl single crystals by the methods of mutual grinding and drilling [2, 3]. When drilling a crystal under a constant load, in our case \(P=2.5\) kg, a conical pit of radius \(r\) is formed. After \(n\) revolutions of the drill the radius of the pit increases by \(\Delta r\); the drill also descends by the same amount. The work of the load will be equal to \(\Delta U_1=\alpha P\Delta r\), where \(\alpha\) is a coefficient depending on the type of drill.
When the radius of the conical pit is increased by \(\Delta r\), its volume increases by \(\Delta v=\pi r^2\Delta r\). If it is assumed that during drilling particles are formed in the form of cubes with edge \(x\), then in the volume \(\Delta v\) the number of particles will be equal to
\[ N=\frac{\pi r^2\Delta r}{x^3}. \]
The lateral surface of these particles will be
\[ S=6x^2N=\frac{6\pi r^2\Delta r}{x}. \]
The energy expended on the formation of the indicated particles is
\[ \Delta U_2=S\sigma=\frac{6\pi r^2\Delta r}{x}\sigma. \]
If it is assumed that all the energy \(\Delta U_1\) goes into the comminution of the crystal, then one may write
\[ \Delta U_1=\Delta U_2;\qquad \alpha P\Delta r=\frac{6\pi r^2\Delta r\sigma}{x}. \]
Hence
\[ P=\frac{6\pi\sigma r^2}{\alpha x}. \]
As a result of experiments, V. D. Kuznetsov established that the dimensions of the powder particles formed during drilling are proportional to the radius of the pit, i.e.,
\[ x=\beta r. \]
Then
\[ P=\frac{6\pi\sigma}{\alpha\beta}r. \]
Drilling experiments showed that there is a linear relation between \(P\) and \(r\), i.e. \(r=r_0+aP\), where \(r_0\) is a quantity depending on the degree of blunting of the drill, and \(a\) is the angular coefficient on the drilling diagram. For an ideally sharp drill, disregarding the initial chipping, one may write \(r=aP\). Then
\[ P=\frac{6\pi\sigma}{\alpha\beta}aP. \]
Hence
\[ a\sigma=\frac{\alpha\beta}{6\pi}=\mathrm{const}, \]
\[ a\sigma=\mathrm{const}. \]
It follows that the ratio of the surface energies of crystals is inversely proportional to the ratio of the angular coefficients of the straight lines on the drilling diagrams, i.e.,
\[ \frac{\sigma_1}{\sigma_2}=\frac{a_2}{a_1}. \tag{1} \]
If the drilling experiments are carried out under a constant load, then between the diameter of the pit and the number of revolutions of the drill \(n\) there is the relation
\[ \begin{aligned} d_1&=d_{01}+a_1'n &&\text{for the first crystal and}\\ d_2&=d_{02}+a_2'n &&\text{for the second crystal.} \end{aligned} \]
In this case expression (1) is transformed into the form
\[ \frac{\sigma_1}{\sigma_2}=\frac{a_2'}{a_1'}. \]
The angular coefficients on the drilling diagrams, \(a_1\) and \(a_2\), can be determined if the difference between the diameters of the pits is found, for example, at \(n_1 = 50\) and \(n_2 = 100\) revolutions of the drill.
If for the first crystal the resulting difference of diameters is equal to \(\Delta d_1\), and for the second crystal to \(\Delta d_2\), then
\[ \frac{\sigma_1}{\sigma_2}=\frac{\Delta d_2}{\Delta d_1}. \]
This dependence was confirmed by the experiments of V. D. Kuznetsov with a number of crystals.
We used the last formula to determine the surface energies of mixed crystals.
In view of the fact that KCl—NaCl solid solutions decompose at temperatures below \(495^\circ\) [4], we were able to grow and investigate transparent single crystals only with concentrations up to 10% NaCl in KCl, and also up to 10% KCl in NaCl. The results of experiments carried out jointly with V. V. Kutsapalenko [3] are given in Table 3, from which it is clear that the surface energy of mixed crystals decreases with increasing impurity percentage. Experiments carried out with cloudy crystals of high impurity concentrations in this system showed that the surface energy has its lowest value at 50% KCl in NaCl.
Table 3
Measurement of the surface energy of KCl—NaCl single crystals by the method of mutual grinding and drilling
\((\text{in erg}/\text{cm}^2)\)
| Method of measurement | % KCl in NaCl | % KCl in NaCl | % KCl in NaCl | % KCl in NaCl | % KCl in NaCl | % NaCl in KCl | % NaCl in KCl | % NaCl in KCl | % NaCl in KCl | % NaCl in KCl |
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 2.5 | 5 | 7.5 | 10 | 0 | 2.5 | 5 | 7.5 | 10 | |
| Drilling | 143 | 80 | 65 | 45 | 40 | 113 | 81 | 67 | 62 | 45 |
| Mutual grinding | 156 | 64 | 59 | 49 | 41 | 108 | 69 | 66 | 54 | 40 |
We then measured the surface energy of single crystals of the KCl—KBr [3] and KBr—NaBr systems.
Table 4
Surface energy \(\sigma\) of KCl—KBr and KBr—NaBr crystals, determined by the method of mutual grinding
\((\text{in erg}/\text{cm}^2)\)
| % KBr in KCl | \(\sigma\) | % KBr in NaBr | \(\sigma\) |
|---|---|---|---|
| 2.5 | 78 | 10 | 85 |
| 5 | 82 | 20 | 70 |
| 10 | 66 | 30 | 66 |
| 15 | 65 | 40 | 62 |
| 20 | 58 | 50 | 60 |
| 30 | 45 | 60 | 56 |
| 50 | 50 | 70 | 58 |
| 70 | 52 | 80 | 61 |
| 90 | 88 |
The results of these measurements are given in Table 4. These data show that the surface energy of single crystals decreases when impurities are introduced into them and reaches a minimum value at 50% of each component. It is interesting to note that the electrical strength of single crystals of solid solutions changes according to approximately the same law [5].
The surface energy \(\sigma\) of the single crystals we studied can be calculated from the formula \(\sigma = k \dfrac{D}{M}\), where \(D\) is the density of the single crystal, \(M\) is the molecular weight of the crystal, and \(k\) is a constant. The ratio \(\dfrac{D}{M}\) is the number of gram-molecules of the substance per unit volume. The surface energy of a single crystal of a solid solution at 50% of each component can be found from the formula \(\sigma = k \dfrac{D}{M_1 + M_2}\), where \(D\) is the density of the alloy, and \(M_1\) and \(M_2\) are the molecular weights of the substances forming the solid solution.
The values of the surface energy \(\sigma\), calculated from these formulas, are given in Table 5.
Table 5
Surface energy of crystals, calculated from formulas
| Crystals | \(D\) | \(M\) or \(M_1 + M_2\) | \(\sigma\), erg/cm\(^2\), experiment | \(\sigma\), erg/cm\(^2\), calculation |
|---|---|---|---|---|
| KCl | 1.99 | 74.5 | 108 | 104 |
| NaCl | 2.17 | 58.5 | 156 | 148 |
| KBr | 2.76 | 119 | 98 | 88 |
| NaBr | 3.20 | 103 | 129 | 124 |
| 0.5 KCl + 0.5 KBr | 2.37 | 195 | 50 | 52 |
| 0.5 KBr + 0.5 NaBr | 3.0 | 223 | 60 | 56 |
It is assumed that electrical strength is related to surface energy. It is therefore of interest to determine whether there is a relation between the electrical strength \(E_{\mathrm{pr}}\) and the number of gram-molecules per unit volume, \(\dfrac{D}{M}\).
A comparison of electrical strength with the number \(\dfrac{D}{M}\) is made in Table 6 and in Fig. 1. The values of \(E_{\mathrm{pr}}\) were taken by us from work [6]. From Fig. 1 and Table 6 it can be seen that the electrical strength \(E_{\mathrm{pr}}\) is related to the number of gram-molecules per unit volume \(\dfrac{D}{M}\) by the relation
\[ E_{\mathrm{pr}} = k \frac{D}{M}. \]
Using this relation, we determined the electrical strength of the crystals LiCl, LiBr, CsCl, CsBr, and CsJ, for which experimental values of \(E_{\mathrm{pr}}\) are not yet known. These data are given in Table 6.
Table 6
Comparison of the electrical strength of alkali-halide crystals with the number of gram-molecules per unit volume
| Crystal | $E_{\mathrm{br}}$, MV/cm, experiment | $D$ | $M$ | $\dfrac{D}{M}\cdot 10^3$ | $E_{\mathrm{br}}$, MV/cm, theory |
|---|---|---|---|---|---|
| LiF | 3.1 | 2.00 | 25.94 | 77 | |
| NaF | 2.4 | 2.73 | 42.0 | 65 | |
| KF | 1.8 | 2.37 | 38.10 | 41 | |
| NaCl | 1.5 | 2.17 | 58.46 | 37 | |
| KCl | 1.0 | 1.99 | 74.56 | 26 | |
| RbCl | 0.8 | 2.76 | 120.9 | 23 | |
| NaBr | 1.0 | 3.20 | 103 | 31 | |
| KBr | 0.7 | 2.76 | 119 | 22 | |
| RbBr | 0.6 | 3.36 | 165 | 20 | |
| NaJ | 0.8 | 3.66 | 150 | 24 | |
| KJ | 0.6 | 3.11 | 166 | 19 | |
| RbJ | 0.5 | 3.55 | 212 | 15 | |
| LiCl | — | 2.07 | 42.4 | 49 | 1.9 |
| LiBr | — | 3.46 | 86.8 | 40 | 1.4 |
| CsCl | — | 3.99 | 168 | 24 | 0.8 |
| CsBr | — | 4.44 | 213 | 21 | 0.8 |
| CsJ | — | 4.51 | 260 | 17 | 0.5—0.6 |
Fig. 1. Dependence of the electrical strength $E_{\mathrm{br}}$ on the number of gram-molecules per unit volume $\dfrac{D}{M}$
($D$—density, $M$—molecular weight of the substance).
It should be noted that the electrical strength of the LiCl crystal was calculated earlier by E. K. Zavadovskaya [6], proceeding from the relation between \(E_{\mathrm{pr}}\) and the energy of the crystal lattice. The value of \(E_{\mathrm{pr}}\) obtained by us for LiCl coincided with that found by E. K. Zavadovskaya. A comparison of the electrical strength, the crystal-lattice energy [6], and the number of gram-molecules per unit volume is made in Table 7 and in Fig. 2. These results make it possible to assert the existence of a direct relationship between the number of gram-molecules per unit volume and the lattice energy.
Fig. 2. Relationship between the crystal-lattice energy \(U\) and the number of gram-molecules per unit volume \(\dfrac{D}{M}\).
Table 7
Comparison of electrical strength, crystal-lattice energy [6], and the number of gram-molecules per unit volume
| Crystal | \(E_{\mathrm{pr}},\ \dfrac{\mathrm{Mv}}{\mathrm{cm}}\) | Lattice energy, kcal/mol | Number of gram-molecules per unit volume \(\cdot 10^{3}\) |
|---|---|---|---|
| LiF | 3.3 | 240 | 77 |
| NaF | 2.4 | 215 | 65 |
| KF | 1.8 | 190 | 41 |
| NaCl | 1.5 | 180 | 37 |
| KCl | 1.0 | 164 | 26 |
| RbCl | 0.8 | 160 | 23 |
| NaBr | 1.0 | 172 | 31 |
| KBr | 0.7 | 158 | 22 |
| RbBr | 0.6 | 152 | 20 |
| NaJ | 0.8 | 160 | 24 |
| KJ | 0.6 | 149 | 19 |
| RbJ | 0.5 | 144 | 15 |
It would also be interesting to estimate from this point of view the electrical strength of solid solutions. Analogously to the dependence of surface energy on the number
\[ \frac{D}{M_1 + M_2} \]
we found the dependence of the electrical strength \(E_{pr}\) on the number
\[ \frac{D}{M_1 + M_2}. \]
The results of our calculations, in which we took the values of \(E_{pr}\) from work [5], are given in Table 8.
Table 8
Determination of the electrical strength of solid solutions
| Composition | D | \(M_1 + M_2\) | \(E_{pr}\), MV/cm, experiment | \(E_{pr}\), MV/cm, theory | \(\dfrac{D}{M_1 + M_2}\) |
|---|---|---|---|---|---|
| 0.5 NaCl + 0.5 NaBr | 2.7 | 161 | 0.45 | 0.64 | 16 |
| 0.5 KJ + 0.5 KBr | 2.9 | 286 | 0.40 | 0.4 | 10 |
| 0.5 KCl + 0.5 KBr | 2.8 | 195 | 0.65 | 0.52 | 13 |
| 0.5 NaJ + 0.5 KJ | 3.4 | 316 | 0.42 | 0.44 | 11 |
| 0.5 NaBr + 0.5 KBr | 3.0 | 223 | 0.75 | 0.56 | 14 |
They allow one to conclude that the electrical strength is related to the ratio
\[ \frac{D}{M_1 + M_2}. \]
In addition, we found a dependence of the electrical strength of certain liquid dielectrics on the number of gram-molecules per unit volume (Fig. 3). The values of \(E_{pr}\), density, and molecular weight were taken from the works of A. A. Vorob’ev and A. F. Kalganov [7].
Fig. 3. Dependence of the electrical strength \(E_{pr}\) of liquid dielectrics on the number of gram-molecules per unit volume \(\dfrac{D}{M}\).
It is seen from Figs. 1 and 3 that the character of the dependence of the electrical strength on the number of gram-molecules per unit volume is different for alkali-halide crystals and liquid dielectrics.
CONCLUSIONS
-
The dependence of the electrical strength of alkali-halide crystals on the number of gram-molecules per unit volume has been found.
-
The electrical strength \(E_{pr}\) has been calculated for single crystals of LiCl, LiBr, CsBr, CsCl, and CsI, for which no experimental data on the value of \(E_{pr}\) are available.
-
It has been shown that the electrical strength of a solid solution at 50% of the first and second components depends on the ratio
\[ \frac{D}{M_1 + M_2}. \]
- The dependence of the electrical strength of alkali-halide crystals on the number of gram-molecules per unit volume confirms the views developed by the school of Prof. A. A. Vorob’ev that electrical strength is determined by quantities characterizing the crystal lattice [8].
LITERATURE
- M. Born and O. Stern. Berl. Ber., p. 901, 1919; Ya. I. Frenkel. Electrical Theory of Solids, 1924; V. Zdanon. A. Erschow. Z. f. Phys. 94, 241, 1935.
- Kuznetsov V. D. Surface Energy of Solids. 1954.
- Savintsev P. A. and Kutsepalenko V. V. Proceedings of SFTI, issue 34, 170, 1955.
- Kurnakov N. S. and Zhemchuzhnyi S. F. Collected Selected Works of N. S. Kurnakov. Vol. I, 186, 1937.
- Vorob’ev A. A. and Trubitsyn A. M. Present issue of Izvestiya TPI, p. 113.
Trubitsyn A. M. Present issue of Izvestiya TPI, p. 119.
Kislina A. N. Present issue of Izvestiya TPI, p. 125. - Zavadovskaya E. K. Reports of the Academy of Sciences of the USSR 82, No. 5, 1952.
- Vorob’ev A. A. and Kalganov A. F. ZhTF 29, issue 11, 1955.
- Vorob’ev A. A. Present issue of Izvestiya TPI, p. 173.
Tomsk Polytechnic Institute.
ERRATA
| Page | Line | Printed | Should read |
|---|---|---|---|
| 5 | 15 from top | Ziderman | Zingerman |
| 9 | 2 from bottom | breakdown voltage | breakdown strength |
| 13 | 4 from top | $$\varepsilon_m \simeq 4\,\frac{h}{2\pi}$$ | $$\varepsilon \simeq 4\,\frac{h}{2\pi}\,\omega$$ |
| 24 | caption under figure | fig. | fig. 9. |
| 50 | 14 from top | $$-\frac{1}{M_L^{ak}}=\left\{\frac{1}{M}\right.$$ | $$-\frac{1}{M_L^{ak}}=\left\{\frac{1}{M_{-}}\right.$$ |
| 53 | 10 from top | $$-\frac{df^{1/2}}{d\varepsilon}$$ | $$-\frac{df_2^{1}}{d\varepsilon}$$ |
| 57 | 5 from top | $$(1-\sqrt{1-\frac{1}{1.3\,\varepsilon}})\,e^{\frac{q}{E}(\varepsilon-I)^2}$$ | $$(1-\sqrt{1-\frac{1}{1.3\,\varepsilon}})\,e^{\frac{q}{E}(\varepsilon-I)^2}$$ |
| 64 | Equation (87) | $$l_2(\varepsilon_2,\varepsilon_1)=l_1 e^{-\frac{1}{l_0}}$$ | $$l_2(\varepsilon_2,\varepsilon_1)=l_1 e^{-\frac{l_1}{l_0}}$$ |
| 65 | 18 from bottom | $$I \div 2\varepsilon_p$$ | $$I+2\varepsilon_p$$ |
| 85 | 7 from bottom | $$10^{14}\cdot\frac{\pi\,0.52\cdot10^{-2}}{4},\;10^{-}$$ | $$10^{14}\cdot\frac{\pi\,0.52\cdot10^{-2}}{4}\;10^{-2}$$ |
| 111 | 2 from top | producibility | conductivity |
| 111 | 8 and 10 from top | activation | activation |
| 120 | 5 from top | strength chemical | strength from chemical |
| 135 | ref. 6 | p. ... | p. 113 |
| 147 | 11 from bottom | $$\varepsilon=1000$$ | $$\varepsilon\simeq1000$$ |
| 148 | caption to Fig. 4 | $u_{\text{disch.}}, u_{\text{break.}}$ | $u_{\text{disch.}};\;u_{\text{break.}}$ |
| 204 | Fig. 1 | KBr | RbBr |
| 253 | caption to Fig. 6 | $2-f=4\cdot10^3$ Hz; $3-f=10^2$ Hz | $2-f=4\cdot10^2$ Hz; $3-f=10^3$ Hz |
| 273 | 7 from top | KCu | KCl |
| 280 | caption to Fig. 4 | $210^5$ | $2\cdot10^5$ |
| 294 | 1 from bottom | from 10 | from $10^{-4}$ at $f=2\cdot10^7$ to $6\cdot10$ |
| 282 | table | table 2 | |
| 324 | 3 from top | p. 269 | p. 251 |
| 336 | 16 from bottom | [3,4] | [3] |
| 337 | 10 from top | [5] | [4] |
| 337 | 4 from bottom | [6] | [5] |
| 337 | 1 from bottom | [7] | — |
| 338 | 2 from top | [8] | [6] |
| 338 | 4 and 5 from top | [9,10] | [7,8] |
| 339 | 8 from bottom | [11,12] | [9,10] |
| 340 | 14 from top | [13] | [11] |
| 347 | 23 from bottom | [14] | [12] |
| 347 | 3,4 from bottom | [15] | — |
| 348 | 17 from bottom | [16] | [13] |
| 359 | 4 from top | $$\psi_0[M]x=\mathrm{Const}$$ | $$\psi_0[M]_x=\mathrm{Const}$$ |
| 451 | 1 from bottom | mechanical | technical |