CLEAVAGE WORK OF MICA CRYSTALS

M. S. METSIK

The splitting of mica crystals into thin sheets is the most labor-intensive operation in the mica-processing industry.

Despite the existence of a number of studies on the measurement of the forces and energies of cleavage of mica crystals, it must be acknowledged that, both theoretically and experimentally, this process has been studied quite insufficiently.

Mantrov [1, 2] and Loginov [3] determined the forces splitting mica crystals by measuring the loads necessary to move a cylindrical glass rod inserted into the split. Samples of muscovite and phlogopite, both air-dry and wetted with various liquids, were subjected to splitting.

They showed that the splitting force increases in proportion to the width and thickness of the crystal being split, decreases when the crystal is wetted with water, and increases when xylene and benzene are introduced into the split. Lazarev [5] used a dynamic method to measure the work of splitting. The latter was evaluated as the loss of energy of a physical pendulum that carried out the splitting, and on average for muscovite and phlogopite crystals was equal to 2140 erg/cm². He also established the fact that the work decreased by approximately a factor of 2 after the preliminary introduction of a small amount of water into a notch in the crystal, but did not consider its dependence on the rate of splitting, the dimensions, and the type of mica crystals.

Obreimov [4] measured the surface energy of mica by splitting muscovite in vacuum and in air with the aid of a glass wedge. He determined the free energy as one half of the elastic energy of a bent plate. The elastic bending was evaluated by observing interference bands in the region of splitting. According to Obreimov’s measurements, the free energy of mica crystals in air is 1500 erg/cm², and in vacuum—20000 erg/cm². Kuznetsov and Bolshanina [11] found an error in Obreimov’s calculations and, after correcting it, obtained for the free energy in air and in vacuum, respectively, 375 and 5000 erg/cm².

Repeating Obreimov’s experiments in our laboratory showed that the determination of the free energy by this method requires further corrections.

Kuznetsov [11] measured the surface energy of mica crystals by grinding them with rock salt and for the plane (001) obtained 31 erg/cm². This quantity is at least an order of magnitude smaller than the values of the surface energy obtained by direct cleavage of crystals.

Such is the list of the principal studies devoted to the question under consideration. They establish certain experimental facts, which in a number of cases require verification, and leave entirely unclear the question of the physical mechanism of the cleavage process.

In the present work, the range of experimental facts concerning the process of cleavage of mica crystals is expanded, indicating the dependence of the cleavage work—

dependence on the geometric dimensions of the specimens, the rate of cleavage, humidity, pressure, and air temperature. These facts make it possible to express definite judgments about the physical mechanism of the processes occurring on the surfaces of mica crystals during their cleavage.

Method

The cleavage work of mica crystals was studied by us on apparatuses in the form of an adhesiometer, and physical and “mathematical” pendulums.

Experiments with pendulums showed that a considerable and difficult-to-control part of the energy is spent on elastic bending of the plate. With pendulums it is impossible to measure accurately the rate of separation and to carry out cleavage at low rates.

Therefore, the greater part of the measurements of cleavage work was performed on an adhesiometer according to B. V. Deryagin [6]. The main part of the adhesiometer used by us was a metal flat plate—the specimen holder—rotating about an axis \(O\) (Fig. 1), parallel to one of its edges.

Fig. 1. Adhesiometer. Diagram of cleavage of a mica crystal on the adhesiometer.

A specimen in the form of a rectangular plate 20–30 mm wide with a tab at the end was glued to a wooden or metal strip and fastened to the holder by means of a special device. From the side of the tab the specimen was split approximately to the middle, and a load of weight \(P\) was suspended from the separated plate.

The cleavage region of the specimen was placed on the axis of the instrument. Then the specimen was rotated about the axis until motion of the cleavage zone was detected. By recording the angle \(\varphi\) between the plane of the crystal and the vertical, the specific work of cleavage can be calculated as follows: suppose that separation began at \(O\) (Fig. 1) and the separation zone moved by an infinitely small segment \(\Delta l\) at \(O\). In this case the load \(P\) descends by an amount \(\Delta h\). The specific work of separating a plate of width “\(b\)” is expressed as:

\[ A_0=\frac{P\Delta h}{b\Delta l}=\frac{P}{b}(1-\cos\varphi). \tag{1} \]

The magnitude of the errors of measurements by this method was 8–10%. In studying the dependence of the cleavage work on the velocity, the specimen was placed horizontally, and a given velocity was set by the magnitude of the load \(P\).

In carrying out the experiments, attention was paid to the careful preparation of specimens with minimal edge delaminations.

In cleavage on the adhesiometer, only such detachments were taken into account for which the difference in thickness of the plates was less than 1 micron. The thickness variation was checked by means of a polarizing microscope for mica developed by us.

Each point on the graphs was obtained by averaging the results of 5–10 experiments on detaching plates of similar thicknesses (variations within 5–10 microns) under identical experimental conditions.

The thickness was measured with an accuracy of up to 1–2 microns.

Dependence of the detachment work on the thickness of the specimen

Figure 2 presents the dependence of the cleavage work \(A_0\) of muscovite crystals on the thickness of the plate being detached, \(h\). It is seen that at a detachment velocity of \(0.1\)—\(0.2\) mm/sec the work is practically independent of the thickness.

Under ordinary conditions of cleavage in air without special treatment of the crystal, \(A_0 \simeq 700\) erg/cm\(^2\).

On carefully trimmed mica crystals with minimal edge delaminations and with abundant wetting with water of the edge perpendicular to the cleavage front, the work at low velocities proves to be of the order of 300–350 erg/cm\(^2\). The same order of magnitude is found for the cleavage work of a crystal in air saturated with water vapor.

If, however, a leaf not completely detached from an untreated crystal is freed from the action of the cleaving force, then in the region close to the cleavage front it again adheres to the crystal. For its secondary separation, an expenditure of energy of the order of 250–300 erg/cm\(^2\) is required.

A similar picture is sometimes observed for several tens of successive cleavages of the crystal over the course of several hours.

Fig. 2. Dependence of the detachment work on thickness at cleavage velocities equal to:
\(\bullet \sim 0.1,\ \times \sim 0.4,\ \circ \sim 1.0\) mm/sec.

The circumstance indicated above makes it possible to regard the value 250–300 erg/cm\(^2\) as close to the equilibrium work of dividing the crystal into two parts. The increased value of the specific cleavage work of mica crystals whose edge was not subjected to special treatment may be explained by edge coupling of the individual elementary leaves composing the crystal.

Such interlacing of layers always occurs when the crystal is trimmed with scissors or with a guillotine and is revealed in the form of an edge “tree.”

Treatment with water of the crystal edge weakens the influence of the edge effect. This same fact explains the increase in the specific cleavage work of specimens with decreasing width. At detachment velocities,

exceeding 0.1–0.2 mm/sec, over a considerable range of thicknesses the work increases with thickness proportionally to the latter to a power close to unity,

\[ A_0 = ch, \tag{2} \]

where \(c\) is a constant.

Dependence of Cleavage Work on Velocity

Measurements were made of the cleavage work of muscovite crystals in air at a temperature of 20°C, a pressure of 725 mm Hg, a humidity of 40–50%, and at various velocities of advance of the cleavage front. The results obtained in these experiments are presented in Fig. 3.

Fig. 3. Dependence of cleavage work on velocity for a plate thickness of \(20 \pm 5\) microns.

It is seen that with increasing velocity from 0.15–0.20 mm/sec the cleavage work on the adhesiometer increases rather rapidly, and beginning at a velocity of 1–1.5 mm/sec it slows down, then asymptotically approaches a certain limiting value that depends on the thickness of the plate. In the indicated range of velocities one may assume that:

\[ A_0 = c_1 v, \tag{3} \]

where the coefficient \(c_1\) increases slightly with the thickness of the specimen.

The influence of velocity on the cleavage work of mica crystals is not due to any elastic irreversible phenomena (hysteresis, relaxation), but can be understood on the basis of ideas about the electrification of the crystal surfaces at the moment of separation [6].

The cleavage of crystals that have been kept for a long time in moist air, at velocities of the order of 0.1 mm/sec, does not depend on velocity.

Influence of Humidity

The magnitude of the cleavage work of mica crystals is strongly affected by the humidity of the air. For a quantitative estimate of this influence, the adhesiometer was placed under a glass bell jar. A definite humidity was set by sulfuric-acid solutions and by drying with calcium chloride. The experiments were carried out for 6 humidity levels, at a temperature of 20°, a cleavage velocity of the order of 0.2 mm/sec, and with the specimen kept under the bell jar for 20 hours.

From the graph in Fig. 4 it is evident that the cleavage work in “dry air” is greatest, decreases sharply at a relative humidity of 1.5% to 700–800 erg/cm², and then decreases slowly with humidity according to an almost linear law. At 100% relative humidity the specific cleavage work of muscovite crystals is 300–350 erg/cm², i.e., it is close to the equilibrium value.

The increase in work as the humidity decreases down to approximately 1% may be attributed to drying of the crystal edge and to enhancement of the edge effect. The further increase in work at still lower humidities indicates the significant role, in the cleavage process, of surface phenomena, in particular adsorption.

Most probably, adsorption of water molecules on the fresh surfaces of a mica crystal is accompanied by a rapid decrease in the electrification of these surfaces, and consequently in the work of their separation.

Effect of Air Pressure

In these experiments, air dried with calcium chloride was admitted under the vacuum bell jar. Cleavage was carried out at low speeds.

From the graph in Fig. 5 it is evident that, beginning with a pressure of 300 mm Hg, the cleavage work increases up to approximately a pressure of 500 mm Hg. With a further decrease in pressure down to 1 mm Hg, the cleavage work is practically independent of pressure and is equal to the cleavage work in air dried with sulfuric acid at a pressure of 725 mm Hg.

The effect of pressure on the cleavage work is explained by us by the fact that, with a change in air pressure, its humidity changes; i.e., pressure acts through humidity.

Fig. 4. Dependence of the cleavage work of muscovite crystals on air humidity at a temperature of 20°C.

Fig. 5. Cleavage work of muscovite crystals at various pressures of air dried with CaCl₂.

Therefore one should expect that in water vapor boiling at room temperature \((P_{\mathrm{H_2O}} \simeq 20\ \text{mm Hg})\) the cleavage work should be the same as at atmospheric pressure and 100% relative humidity.

This assumption is confirmed by the results of special experiments.

In water vapor boiling at room temperature in vacuum, the dependence of the cleavage work on velocity has a form analogous to the similar dependences obtained at atmospheric pressure.

On the basis of the views expressed, the coincidence of the cleavage work of mica crystals in atmospheric air dried with sulfuric acid and in air dried with calcium chloride at pressures of several mm Hg becomes understandable. In both these cases the elasticity of the water vapor proves to be approximately the same.

Influence of temperature

The experiments were carried out in a vacuum drying oven on phlogopite crystals. Cleavage occurred at velocities greater than 0.2 mm/sec. The humidity of the air was not controlled.

Figure 6 presents the dependences obtained for the cleavage work on temperature at pressures of 725 and 5 mm Hg.

The increase in cleavage work with temperature at atmospheric pressure can be explained by a decrease in the relative humidity of the air.

At low pressures and temperatures above \(100^\circ\mathrm{C}\), the cleavage work of phlogopite crystals decreases as a result of the appearance of irreversible thermal cleavage in vacuum, caused by their swelling.

Fig. 6. Influence of temperature on the cleavage work of phlogopite crystals at pressures:

1 — 725 mm Hg, 2 — 5 mm Hg.

Discussion of the experimental results and their theoretical interpretation

From the experimental material presented in this study it is seen that the cleavage work of mica crystals:

a) increases with increasing rate of separation;

b) at appreciable (moderate) velocities increases with increasing thickness of the plate being separated;

c) decreases with increasing relative humidity of the air;

d) does not depend on the air pressure within the limits 725–2 mm Hg, if the elasticity of the water vapor remains unchanged;

e) depends on the temperature at which separation is carried out.

The only satisfactory explanation of these facts can be an explanation based on the idea of electrification of the crystal surfaces at the moment of their separation.

There are a number of direct indications of electrical phenomena observed during the cleavage of mica crystals.

Let us name some of them.

1. Obreimov first observed [4] strong electric fields between mica sheets during the cleavage of crystals in vacuum. He also recorded the curious fact of the sticking together of parts of the crystal when they came into contact.

after cleavage, so that for their secondary separation a force close to the original one was required.

1. Krotova and Karasev [9] discovered, during the splitting of muscovite crystals along cleavage planes in vacuum, electron emission from the crystal surfaces, with electron energies up to 15 keV.

2. We also established that immediately after cleavage in moist air, mica crystals possess increased surface electrical conductivity, which gradually decreases with time, sometimes by tens of times.

3. Freshly cleaved mica plates have dielectric losses at high and low frequencies two or more times greater than the losses in “old” plates.

4. We also observed that, when mica crystals are rapidly split in the dark under ordinary conditions, a gas glow occurs in the region of splitting, differing for different micas.

Deryagin and Krotova [6] have convincingly shown that similar electrical phenomena are observed in many cases of detachment.

They were also the first to develop a theory of the electrical processes observed during the detachment, chiefly, of polymer films from solid bodies.

Deryagin and Krotova [6] indicate the following possible mechanisms for the dissipation of surface electric charges:

a) electrical discharge in the gas in the region of detachment;
b) autoelectron emission from the surfaces;
c) surface electrical conductivity.

In [6] the action of chiefly the first factor is analyzed in detail. Applying this theory to the process of splitting mica crystals, we are inclined to draw the following conclusions.

1. During the splitting of mica crystals, electrification of the newly formed surfaces occurs. The separation of parts of the crystal is accompanied not only by work against molecular forces, but also by work to overcome the interaction of electrostatically charged surfaces (Fig. 7). Individual elements of the crystal surface attract one another, like the plates of a charged capacitor.

Fig. 7. Pattern of electrification of the surfaces of a mica crystal during splitting.

1. With comparatively slow detachment (up to speeds on the order of 1 mm/sec) in moist air, the principal mechanism for the dissipation of electrostatic charges is the mechanism of surface electrical conductivity. In this case, as the detachment speed increases, the work will increase.

At sufficiently high detachment speeds, the electrification density does not have time to decrease appreciably during the time in which the parts of the crystal move apart to distances equal to the mean distances between the mosaic-distributed charges on the surface. From this moment the electric field in the region of splitting becomes strongly nonuniform and rapidly decr—

decreases with distance. Under these conditions the work no longer depends on the velocity. The considerations set forth above qualitatively explain the general picture of the change in cleavage work with velocity and its independence of air pressure.

1. The influence of the thickness of the cleaved-off plate on the magnitude of the cleavage work is explained by the fact that the thicker the plate, the smaller its curvature and the larger the dimensions of the cleavage region in which electrostatic forces act. Therefore, with increasing thickness or elasticity of the plate, the work of its separation will increase. On the basis of all the foregoing, let us develop a general theory explaining the dependence of the cleavage work of mica crystals on the conditions of cleavage.

Owing to electrical conductivity, both the mosaic-like, randomly distributed alternating-sign regions on each surface and the charges of the separated surfaces can be neutralized.

In both cases the process of decrease in the density of electrification can be described as follows. The current \(\dfrac{d\sigma}{dt}\) flowing along the surface will be proportional to the density of electrification,

\[ \frac{d\sigma}{dt}=-a\sigma \qquad \text{or} \qquad \frac{d\sigma}{\sigma}=-a\,dt . \]

Integrating, we obtain:

\[ \sigma=\sigma_0 \exp(-at) \tag{4} \]

Obviously, the surface electrical conductivity of fresh cleavages of mica crystals in an external field will vary according to the same law. This has been confirmed by our special experiments. With sufficient prevalence on the surface of charged regions of one sign of charge, the force \(F\) of their interaction can be expressed as the attractive force of the plates of a parallel-plate capacitor (Fig. 7).

For a strip of unit width we obtain:

\[ dF=+k_2\sigma^2\,dx=+k_2\sigma_0^2 \exp(-2at)\,dx . \tag{5} \]

Since \(x=vt\), where \(v\) is the velocity of advance of the cleavage front, then, integrating (5) from 0 to \(x_m\), for the force separating the strip we have:

\[ F=\frac{k_2\sigma_0^2 v}{2a} \left[ 1-\exp\left(-\frac{2ax_m}{v}\right) \right]. \tag{6} \]

At \(x=x_m\) the field between the separated surfaces can no longer be regarded as homogeneous, and the interaction force practically falls to zero. The value \(x_m\) depends on the thickness of the plate. From a theoretical analysis of the conditions of separation it follows that the shape of the plate in the separation region is described by an equation of the form [12]:

\[ y=k_3\frac{x^3}{h^3}. \tag{7} \]

Measurements of the change in the coordinate \(x\) as the wedge splitting the crystal in vacuum advances, performed by Obreimov [4], also confirm equation (7).

From (7)

\[ x_m=k_4 h y_m^{1/3}. \tag{8} \]

The end of the plate being detached from the crystal on the adhesiometer is arranged vertically.

Under these conditions, advancement of the cleavage front by an amount \(x\) will be accompanied by work equal to \(F \cdot x\). Therefore the specific work

\[ A_0=\frac{Fx}{x}=F \ \text{erg}/\text{cm}^2 \]

is determined by an expression similar to the expression for \(F\).

Taking (8) into account, formula (6) is transformed into (9)

\[ A_0=k_5\frac{v}{a}\left[1-\exp\left(-\frac{2ak_4 y_m^{1/3}h}{v}\right)\right]. \tag{9} \]

If to (9) one adds a term allowing for the work of molecular forces, i.e., the free energy of the newly formed crystal surfaces \(2\sigma_0\), then for the specific work of cleavage we obtain:

\[ A_0=2\sigma_0+k_5\frac{v}{a}\left[1-\exp\left(-\frac{2k_4ah y_m^{1/3}}{v}\right)\right]. \tag{10} \]

Depending on the conditions of detachment, expression (10) may be simplified. We shall indicate the following three limiting cases.

1. The detachment rate is high, and the air humidity is insignificant (\(a\)—small).

In this case

\[ \frac{2k_4ahy_m^{1/3}}{v}=\beta\ll 1. \]

Expanding \(e^{-\beta}\) in a series and restricting ourselves to two terms of the expansion, we obtain:

\[ A_0=2\sigma_0+k_5k_4y_m^{1/3}h=2\sigma_0+k_6h, \tag{11} \]

where \(k_6\) is a constant, since \(y_m\) is determined by the topography of the electrostatic charges on the surface and, at sufficiently high rates, does not depend on the rate.

It follows from expression (11) that at high detachment rates the work does not depend on the rate and increases in proportion to the thickness of the detached plate.

These conclusions are confirmed by the experiments described above.

1. At considerable air humidity (\(a\)—large) and low detachment rates, \(\exp(-\beta)\simeq 1\), increasing only slightly with the thickness of the plates—\(h\).

From (9)

\[ A_0=2\sigma_0+k_5\frac{v}{a}, \tag{12} \]

where the coefficient \(k_5\) depends only weakly on \(h\).

A regularity of the form (12) is indeed observed experimentally at an air humidity of 50–60% for detachment rates of \(0.2\)–\(1\ \text{mm/sec}\).

1. When the air is saturated with water vapor (\(a\)—very large), at low temperature and low detachment rate (\(v<0.15\ \text{mm/sec}\)),

\[ k_5\frac{v}{a}\simeq 0 \quad \text{and} \quad A_0=2\sigma_0 \tag{13} \]

In our experiments \(A_{o\,\min} \simeq 300\ \text{erg}/\text{cm}^2\); consequently, \(\sigma_0 \simeq 150\ \text{erg}/\text{cm}^2\)—a quite reasonable order of magnitude for the free energy of a crystal. The theory developed loses its validity when the distance between the surfaces being separated becomes comparable with the mean distances between charged regions on the surfaces and the field becomes nonuniform.

But the work in a nonuniform field is small in comparison with the work in a uniform field, and neglecting it does not introduce substantial errors into the results.

In conclusion I consider it my pleasant duty to express my gratitude to B. V. Deryagin for a number of suggestions by which I was guided in carrying out the work, and to V. D. Kuznetsov for discussion of the results of the study.

REFERENCES

1. M. I. Mantrov, ZhETF, 4, 10, 1057–62, 1934.
2. M. I. Mantrov, Vestnik elektropromyshlennosti No. 10, 35–40, 1934.
3. G. I. Loginov, ZhTF 8, 21, 1857–71, 1938.
4. I. Obermoff, Proc. Roy. Soc., Ser. A 127, N A 805, 290–7, 1930.
5. V. P. Lazarev, Zhurnal fizich. khimii, vol. 7, 3, 320–6, 1936.
6. B. V. Deryagin and N. A. Krotova, Adhesion, Academy of Sciences of the USSR, Institute of Physical Chemistry, Moscow–Leningrad, 1949.
7. E. K. Lashev, Mica. Promstroiizdat, 1948.
8. M. S. Mechik, Proceedings of Irkutsk University, vol. 8, issue 1, 45–53, 1953.
9. N. A. Krotova and B. V. Karasev, DAN SSSR, XCII, 3, 607–610, 1953.
10. N. A. Kaptsov, Electrical Phenomena in Gases and Vacuum. Gostekhizdat, 1947.
11. V. D. Kuznetsov, Surface Energy of Solids. GITTL, Moscow, 1954.
12. A. N. Dinnik, Handbook of Technical Mechanics. Gostekhizdat, p. 438, 1949.

Irkutsk, State University