ELECTRO-OPTICAL PROPERTIES OF COLLOIDS

E. V. Shpolsky

1. It is known that transparent isotropic bodies, under the action of an electric field, become birefringent, with the optical axis parallel to the direction of the field (the Kerr effect). If, therefore, the substance under study is placed in a plane capacitor and a beam of light is passed through it, linearly polarized by a Nicol prism with the plane of vibration making an angle of \(45^\circ\) with the direction of the field, then the emerging beam will prove to be elliptically polarized. Elliptical polarization is observed because components with vibrations parallel to the field and perpendicular to it propagate with different velocities; as a result, in the emergent beam there is found a phase difference between these mutually perpendicular components:

\[ \varphi = 2\pi \frac{(n_p - n_n)l}{\lambda}, \tag{1} \]

where \(n_p\) and \(n_n\) are the refractive indices of the ordinary and extraordinary components, \(l\) is the path length of the light, and \(\lambda\) is the wavelength. The double refraction may be positive or negative, depending on the sign of the difference of the refractive indices \(n_p - n_n\). Experiment has shown that the phase shift is, with great accuracy, proportional to the square of the electric-field strength. Characteristic of the effect is the Kerr constant \(B\), which is determined by the electro-optical properties of the substance itself,

\[ B = \frac{n_p - n_n}{\lambda E^2}. \tag{2} \]

The magnitude of this constant for pure substances is very small. It has its largest value in liquids, but even in this case its order of magnitude is \(10^{-8}\)—\(10^{-5}\), for example:

Recently it has become clear that most colloids and suspensions, as well as certain protein compounds, exhibit an enormous electro-optical (and also magneto-optical) effect. The Kerr constant in these cases exceeds that for nitrobenzene by \(10^3\)—\(10^7\) times. Whereas, in order to obtain a phase shift of \(2\pi\) in the case of nitrobenzene, fields approaching breakdown voltage and very large path lengths are required, in the anomalous cases mentioned, phase differences several times greater than \(2\pi\) are obtained in fields of the order of \(100\ V/\text{cm}\) (or \(1000\) gauss for the magneto-optical effect) and with a path length of only a few centimeters. Among such substances anomalous in the electro-optical respect, colloidal solutions of vanadium pentoxide, solutions of the proteins of the mosaic-disease virus, and especially colloidal solutions of bentonite\(^2\) are noteworthy (bentonite is a special kind of clay, occurring in large quantities also in the USSR). In view of the fact that the remarkable electro-optical properties of colloidal solutions and proteins are of outstanding interest from the most varied points of view, including practical ones, we shall set forth below the results of work in this direction carried out by the American physicist H. Mueller at the Massachusetts Institute of Technology. The results of these works have so far been published mainly in the form of short preliminary communications. However, the method of investigation and the application of bentonite for the study of electric fields have been described in full.

1. Many of the colloids exhibiting anomalously high values of the Kerr constant are unsuitable for practical applications for the following reasons\(^4\): a) they are insufficiently transparent; b) the Kerr effect in these colloids shows saturation, and despite the fact that the Kerr constant is large for small field values, the maximum attainable value of the phase difference proves to be smaller than that obtained even with nitrobenzene; c) owing to the existence of conductivity, the colloids heat up noticeably even in weak fields; this heating leads to the appearance of convective currents, which in turn give rise to flow birefringence\(^5\), which is superposed on the electro-optical effect and distorts it.

The indicated shortcomings are least manifested in specially prepared colloidal solutions of bentonite, for which reason precisely these solutions were chiefly used by Mueller. Nevertheless, even in the case of bentonite the use of constant electric fields is impossible, since the colloidal particles carry an electric charge, as a result of which they move in the field and coagulate at one of the electrodes. In view of this, in his work Mueller used alternating fields and, together with Zakhman, developed a special method for investigating the Kerr effect in alternating fields.

Mueller describes the preparation of bentonite solutions and their properties in the following way: “Aqueous solutions are obtained by dis-

of the introduction of yellow bentonite gels. These gels were prepared by Prof. D. A. Gauzer and Dr. D. S. Le-Bo. Only those sols in which the mean diameter of the clay particles is less than \(20\,m\mu\) prove suitable. Dilution of the gel is continued until the spontaneous double refraction disappears and the relaxation time of the streams becomes less than \(0.01\) sec. This occurs at a concentration of about \(2\%\) by weight.

The resulting yellowish liquid is transparent for thicknesses up to \(10\ \text{cm}\). The Kerr constant for it in a 60-cycle alternating field is about \(10\) CGSE. The saturation value of the double refraction is so large that already along a path of \(1\ \text{cm}\) the phase shift exceeds \(2\pi\). Saturation occurs at field strengths of about \(150\ \text{V}/\text{cm}\); for fields of \(100\ \text{V}/\text{cm}\) the heating effect becomes noticeable after about 1 min. Heating produces no interference, since for photography an exposure of \(1\) sec is sufficient. The value of the Kerr constant and the magnitude of the limiting phase shift vary depending on the size of the particles, the concentration, the temperature, and the frequency of the electric field. With some practice one can readily learn to set any suitable value of the Kerr constant by increasing or decreasing the concentration, but large values of \(\varphi\) can be obtained only with sufficiently small particles.”

1. The usual method employed for studying the Kerr effect consists in compensating the arising phase shift \(\varphi\) by one compensator or another, so that the light emerging from the compensator is again linearly polarized and can be extinguished by the analyzer. The phase shift is then read directly from the indication of the compensator. In the case of alternating fields, however, this method does not make it possible to find the complete characteristic of the phenomenon. If the field varies according to the harmonic law \(E=E_0\cos\omega t\), then the phase shift, proportional to the square of the field strength, will be

\[ \varphi=\varphi_0\cos^2\omega t=\frac{1}{2}\varphi_0(1+\cos 2\omega t). \tag{3} \]

Thus, the phase shift varies with a frequency twice that of the field. The applicability of relation (3), however, is limited to cases in which the field frequency is such that the period of oscillation \(1/\omega\) considerably exceeds the so-called “relaxation time” of the particles, i.e., the time required for a particle to turn over. This is explained by the fact that the cause of the Kerr effect lies\(^{1}\) in the orientation under the action of the field of anisotropic molecules or of suspended particles present in the colloidal solution. This time depends on the viscosity of the medium \(\eta\) and the radius of the particles \(r\); according to Debye’s theory, for spherical particles its order of magnitude is

\[ \tau=\frac{8\pi\eta r^3}{kT}. \]

For aqueous solutions and for room temperature this gives

\[ \tau \simeq 10^{3} r^{3}. \]

Since bentonite particles have dimensions of the order of \(10^{-5}\)—\(10^{-6}\) cm, for them \(\tau \sim 10^{-12}\)—\(10^{-15}\) sec. Thus, at frequencies of 100 cycles/sec, relation (3) should be applicable. If, however, the applied field has a high frequency, as is often the case in technical uses of the effect, then two cases are possible: a) the particles have a permanent dipole moment—in this case, at a sufficiently high field frequency the particles do not have time to follow the changes of the field, and no birefringence is observed; b) the particles acquire an induced dipole moment in the field. In precisely this case the moment of the orienting couple is proportional to \(E^{2}\); the mean value of this couple does not vanish at high frequency, and a certain birefringence arises, having, however, a constant magnitude. It can be shown that in both cases the phase shift is expressed by the following general formula:

\[ \varphi = A + B \cos(2\omega t - \psi), \tag{4} \]

i.e., it consists of a constant part, generally speaking dependent on \(E_{0}\) and \(\omega\), and a variable part, whose amplitude \(B\) also depends on \(E_{0}\) and \(\omega\). In addition, the change of phase \(\varphi\) lags behind the change of the field by an angle \(\psi\), also dependent on \(E_{0}\) and \(\omega\). This formula describes the effect in both of the cases indicated above. For low frequencies \(A = B\), \(\psi = 0\), and (4) passes into (3); in the case of particles with a permanent dipole moment at high frequencies \(A = B = 0\) and \(\psi = \pi\). For nonpolar particles at high frequencies \(B = 0\), \(A\) does not depend on \(\omega\), and \(\psi\) approaches \(\pi/2\).

Möller and Sackmann\(^3\) demonstrated the applicability of formula (4) in the case of a colloidal solution and one of the viruses of mosaic disease by the following experiment. A Kerr cell containing the colloid was placed between crossed Polaroids and illuminated with monochromatic light from a mercury lamp burning at 60 periods. An alternating field was applied to the cell, with a frequency only slightly exceeding 60 cycles, and with an intensity sufficient to produce a phase displacement of \(\pi/2\). The light that had passed through the cell and analyzer fell on a photocell; the photocurrent was amplified and recorded by means of a cathode oscillograph. Samples of such a record are shown in Fig. 1. The main curve of the type \(\sin^{2} x\) corresponds to the 60-period variation of the intensity of the mercury lamp; the superposed “ripple” is due to the Kerr effect. The frequency of this ripple is twice the frequency of the field; its amplitude decreases with increasing field frequency and is always less than the amplitude of the main \(\sin^{2} x\) curve. The latter shows that \(B < A\).

At a certain critical frequency the ripple disappears and does not reappear until significantly higher frequencies. From this fol—

implies that, beginning with this critical frequency, the variable part of the phase shift turns to zero, i.e. in formula (4) \(B=0\), and only the constant part remains. This constant part can be measured accurately in the usual way with the aid of a compensator.

If, however, the compensator is used at frequencies below the critical one, its readings give only the average value of the phase shift. Therefore the usual method is unsuitable for investigating the Kerr effect in alternating fields, since for a complete characterization of the phenomenon it is necessary to find the three parameters \(A\), \(B\), and \(\psi\), which cannot be determined from a single compensator reading.

Fig. 1.

1. Analysis of visual measurements of the Kerr effect with the aid of a compensator leads to the following results. If, between the nicols crossed at an angle of \(45^\circ\) to the direction of the field, in addition to the Kerr cell one also introduces a compensator, then the mean intensity of the light transmitted by such a system is equal to

\[ \overline{I}=\frac{1}{2} I_0\{1-J_0(B)\cos(kx+A)\}, \tag{5} \]

where \(I_0\) is the intensity transmitted by parallel nicols in the absence of a field in the cell, \(A\) and \(B\) are constants entering formula (4) for the phase shift, \(J_0\) is the Bessel function of zero order, \(x\) is the displacement of the compensator wedges, and \(k\) is the calibration constant of the compensator. The Bessel function \(J_0(B)\), for small values of the argument, i.e. \(B\), has values close to unity, and formula (5) in this case takes the form

\[ \overline{I}\simeq \frac{1}{2} I_0[1-\cos(kx+A)]. \tag{5′} \]

Small values of \(B\) are always obtained at high field frequencies, and at low frequencies—in fields of weak intensity. When these conditions are observed, according to formula (5′), a system of minima should be obtained corresponding to the condition

\[ kx_n+A=2\pi n. \]

From this one can determine the magnitude and sign of the constant part of the phase shift \(A\). However, the amplitude of the variable part \(B\) and the phase difference \(\psi\) in formula (4) cannot be determined from visual observations. In addition, the determination of the constant part \(A\) becomes inaccurate when \(B\) is sufficiently large.

1. To determine all three parameters \(A\), \(B\), and \(\psi\), Möller and Sackmann developed a convenient photoelectric method. The essence of this method is as follows. The intensity of light that has passed through crossed nicols and a Kerr cell is equal to

\[ I=I_0 \sin^2 \varphi = \]

\[ = I_0 \sin^2 [A+B\cos(2\omega t-\psi)]. \tag{6} \]

If this light falls on a photocell with a linear characteristic, then the photocurrent strength is expressed by the analogous formula

\[ i=i_0 \sin^2 [A+B\cos(2\omega t-\psi)]. \tag{6'} \]

The problem is to determine \(A\), \(B\), and \(\psi\), using the shape of the curve of this photocurrent. In doing so, however, the following difficulty arises. In a colloidal solution of bentonite, the constant part of the phase displacement \(A\) can easily be made equal to from \(5\cdot 2\pi\) to \(7\cdot 2\pi\), but at the same time, at a field frequency of 60 cycles/sec., \(B\) also has a large value. If \(A\) and \(B\) are simultaneously large, then the current curve becomes very complicated, as is seen from Fig. 2, where several examples of oscillographic recording of the photocurrent are given.

Fig. 2.

The matter is greatly simplified if one chooses conditions such that \(A\) and \(B\) are less than \(\frac{1}{2}\pi\). In this case the right-hand side of equality (6) can be expanded in a power series and only the first terms of the expansion retained. This gives

\[ i=i_0\left[\sin^2 \frac{1}{2}A+\frac{1}{2}B\sin A\cos(2\omega t-\psi)\right]. \tag{7} \]

Formula (7) shows that the photocurrent consists of a constant part

\[ i_1=i_0\sin^2 \frac{1}{2}A \]

and a variable part

\[ i_2=\frac{1}{2}B\sin A\cos(2\omega t-\psi) \]

with amplitude equal to \(\frac{1}{2}B\sin A\). The frequency of the variable part of the photocurrent is twice the frequency of the applied field, and its phase coincides with the phase of the variable part of the phase displacement in the Kerr effect. The constant and

the variable part of the photocurrent is easy to measure separately. Namely, if the photocurrent is measured with a galvanometer, it will give a deflection proportional to the constant component, whereas if an amplifier for alternating current is used, then the amplified part will be delayed by a capacitor, and the variable part will be amplified. In this way, knowing $i_1$ and $i_2$, one can compute $A$ and $B$.

To satisfy the condition necessary for the indicated separation ($A$ and $B<\pi/2$), one can simply reduce the length of the Kerr cell accordingly.

The method described also makes it possible to find the phase difference $\psi$. For this it is sufficient to connect the horizontal deflecting plates of a cathode oscilloscope with the electrodes of the Kerr cell, and the vertical plates with the photocurrent amplifier. As a result, a Lissajous figure will be obtained on the screen, corresponding to the superposition of mutually perpendicular oscillations $E\cos\omega t$ and $i_2\cos(2\omega t-\psi)$. This figure has a form resembling the letter $U$ when $\psi=0$, and the curve $\infty$ when $\psi=90^\circ$. In Fig. 3 a Lissajous figure is presented for an intermediate value of the phase difference; the magnitude $\psi$ is determined from the position of the point of intersection on the vertical axis.

Fig. 3.

1. Large values of the Kerr constant in the case of colloidal solutions and proteins can be conveniently applied to the study of electric fields by means of the electro-optical effect. In principle, the method is based on the following considerations. The phase shift in the Kerr effect, other conditions being the same, is uniquely determined by the field strength $E$

\[ \varphi=2\pi lBE^2, \tag{8} \]

where $l$ is the length of the path of the ray in the liquid. Therefore, by determining at each point $\varphi$ and the direction of the optical axis, one can find the magnitude and direction of the field at that point. In the case of colloidal solutions, as was indicated, a phase displacement several times exceeding $2\pi$ can readily be obtained, in view of which this method proves practically feasible.

Let the field make an angle $\theta$ with the plane of vibration of the ray passed through the polarizer; then the intensity of the light, after passing through the system polarizer—Kerr cell—analyzer, will be

\[ I=\sin^2 2\theta \sin^2 \frac{1}{2}\varphi; \]

where \(\varphi\) is the phase shift due to the electro-optical effect. From this formula it follows first of all that at all points of the field where \(\theta = 0^\circ\) or \(90^\circ\), \(I = 0\). Thus, in the case of a nonuniform field there must result a system of dark lines, representing the geometric loci of points at which the direction of the field coincides with the direction of the polarizer or analyzer. Möller calls these dark lines “isoclinic,” or “lines of equal inclination.” Obviously, they are not affected by the magnitude of the field strength. Therefore they are of secondary interest, and they are usually eliminated by the method indicated below.

Another system of dark lines corresponds to values of the phase shift \(\varphi\) equal to \(2\pi\) or to multiples of \(2\pi\). In accordance with the formula

\[ \varphi = 2\pi lBE^2 \]

such values of the phase shift, for a constant length of the condenser, are obtained for definite “critical” values of the field strength \(E_1, E_2,\ldots\). For colloidal solutions the relation (8) proved inapplicable; for them there holds a somewhat more complicated relation

\[ \varphi = 2\pi lBf(E^2), \tag{9} \]

where the form of the function \(f\) is different for different colloids. Nevertheless, in this case too there exist critical values of the field strength at which \(\varphi = k \cdot 2\pi\). These values, however, are not the same for different wavelengths, as a result of which the lines obtained are colored. Möller called these lines “isochromatic,” or “isodynamic.” With their aid one can determine the field strength at any point in the following way. If the potential of the electrodes is changed a certain number of times, then the field strength at any point changes by the same factor. The isochromatic lines will thereby be displaced. By following these changes under the influence of definite changes in the potential, one can assign to the various isochromatic lines the corresponding values \(E_i\).

In order to remove the isoclinic lines and leave only the isochromatic ones, it is convenient to make use of the fact that the position of the isochromats does not depend on the position of the crossed Nicols. But since, when the crossed Nicols are rotated, the isoclinic lines are correspondingly displaced, by making the Nicols rotate sufficiently rapidly one can remove the isoclinic lines. This can be done still more simply with the aid of a “circular polarizer,” i.e., the system: polarizer—quarter-wave plate—Kerr cell—quarter-wave plate—analyzer. For this system the intensity of the transmitted light is proportional to \(\sin^2 \frac{1}{2}\varphi\), and only isochromatic lines are observed.

If the pattern of the distribution of the isochromatic lines, coinciding with the pattern of the distribution of lines of equal field strength, is photographed with a long exposure, then one can obtain a suffi-

exact sharpness, without resorting to stroboscopic illumination. According to formula (5), the average intensity transmitted by the system in the presence of a compensator is equal to

\[ \overline{I}=\frac{1}{2} I_0[1-J_0(B)\cos(kx+A)]. \]

If the compensator is absent, then this formula takes the form

\[ \overline{I}=\frac{1}{2} I_0[1-J_0(B)\cos A]. \]

At a sufficiently high frequency of the field, \(B\) is very small and \(J_0(B)=1\); therefore, for certain values of \(A\), sharp minima are obtained.

Fig. 4.

Fig. 4 shows photographs of the field between two parallel cylinders at a field frequency of \(10\,000\) cycles/sec\(^{-1}\). As can be seen, the field pattern has quite sufficient sharpness.

This method can be made quantitative. In the same example of the field between two parallel cylinders, Müller showed that, from the form of the isochromats, it is possible to determine the value of the field strength at the midpoint between the cylinders with an accuracy of up to \(1^\circ\). For details we refer to the original work \(^{4}\).

The exceptional simplicity of the experimental means with which such investigations of the field can be carried out deserves attention. Here is how Müller describes the arrangement of the experiment: “In our experiments the Kerr cell was a glass jar with parallel windows measuring \(6 \times 6\) cm. The electrodes were nickel rods about \(5\) mm in diameter and \(50\) mm long. They were held parallel to one another by means of two lucite plates. The alternating voltage was supplied through two wires with rubber insulation. Two polaroid plates served as polarizer and analyzer. A mercury lamp operating on alternating current was used as the light source. For observations in monochromatic light we used a green light filter. By means of a lens the light beam was made parallel; a second lens focused the transmitted light on the objective of a photographic camera. The exposure with the film used by us (Agfa-Superpan) was 1 sec.”

It should be thought that the remarkable electro-optical properties of colloids recently discovered will find varied practi-

technical applications, all the more so because the various kinds of shortcomings associated with their use (heating due to conductivity, etc.) can be considerably mitigated.

REFERENCES

1. For a review of earlier works and the theory, see
   R. Ladenburg, Die elektrische Doppelbrechung. Müller-Pouillets, Lehrbuch der Physik, II Aufl. B. II, 2, p. 2214, Braunschweig, 1929.
2. J. Errera, Overbeck and Sack, J. de Chimie Physique, 32, 681, 1935.
3. H. Mueller and B. Sackmann, Journ. Optical Soc. Am., 32, 309, 1942.
4. H. Mueller, Journ. Optical Soc. Am. 31, 286, 1941.
5. See, for example, Gubanov, Uspekhi fizich. nauk, XXI, 1939.
6. P. Debye, Polar Molecules. Transl. by N. K. Shchofro, Moscow, 1931, p. 119.
7. See, for example, G. Bruhat, Cours d’Optique, p. 429, Masson & Cie Éditeurs, Paris, 1931.