Optical Phenomena Associated with the Orientation of Elongated Particles in a Fluid Flow

A. Gubankov, Leningrad

Introduction

Let particles of an elongated form in a colloidal solution be suspended in a liquid at rest or moving with a velocity that is the same at all points. Then, as a consequence of Brownian motion, the orientation of such particles will be completely random; in this case the liquid is fully isotropic and has no preferential directions in the optical sense.

If, however, there is some velocity gradient in the liquid, then different forces will act on different points of a long particle. Because of this, the orientation of the particle relative to the velocity field of the liquid will change. In particular, if the particles may be regarded as very thin rods, whose thickness can be neglected in comparison with their length, then the behavior of the particles in a fluid flow can be determined from purely kinematic considerations, without the aid of the equations of hydrodynamics. It will be shown below that, in this case, in the stationary state and in the absence of Brownian motion, all particles will be oriented in a definite way along the direction of the liquid velocity. If one wishes to take the thickness of the particles into account, a hydrodynamic calculation must be carried out; it will be shown below that in this case the particles do not remain in a state of equilibrium, but rotate continuously. This, incidentally, is obvious, since even on a particle oriented along the direction of the velocity there acts a rotating moment, which forces it to leave the state of apparent equilibrium. This is the fundamental difference between “hydrodynamic” orientation and the orientation of particles in some potential field. Therefore the motion of particles in a fluid flow cannot be described by means of a potential function, even a fictitious one.

Thus, on the one hand, Brownian forces act on the particles, tending to impart to them a chaotic orientation; on the other hand, hydrodynamic forces act, under whose influence the particles will move in a definite manner. As a result of the interaction of both factors, a certain statistical orientation of the particles will be established, with each interval of directions

for the particle axes there will correspond a definite relative number of particles; thus a distribution function of the particles over angles is introduced.

If there is a preferred orientation of the particles, then the solution will behave optically like a crystal; certain directions will play the role of optical axes. In this case the solution will exhibit double refraction; light that is linearly polarized, after passing through such a solution, will become elliptically polarized.

The practical interest of studying these phenomena is beyond doubt for the following reasons.

1. A solidified flow of a colloidal solution constitutes a polaroid—an artificial film that polarizes light—used in the construction of automobile headlights and windows, three-dimensional cinema, etc.

2. From the double refraction of a colloidal solution flowing around a hydrodynamic model one can study the distribution of liquid velocities and the character of the flow around the body.

3. Optical measurements make it possible to determine the sizes and shapes of particles and to study various properties and processes in colloidal solutions and in solutions of substances with complex molecules, for example, the aging of a solution\(^6\), etc.

I. ORIENTATION OF PARTICLES

In the first part of the present work the distribution function of particles over angles is calculated.

Such a calculation can be carried out only for certain special cases. The differential equation of the distribution function is obtained from the condition that the number of particles whose axes fall into a given elementary solid angle is constant (stationary state). To formulate this condition it is necessary, in turn, to consider two factors determining the motion of the particles: the hydrodynamic forces and the forces of Brownian motion.

In § 1 we consider the motion in a liquid flow of a very thin particle. This restriction is removed in § 2. The following paragraph is auxiliary in character: in it the rotational resistance entering the diffusion coefficient is calculated. The differential equation for the distribution function is derived in § 4. In §§ 5 and 6 the approximate integration of this equation is considered.

§ 1. Motion of a thin particle in a liquid flow

Let us consider the question of the orientation of elongated particles in a liquid flow, namely in a flow specified by the equations

\[ V = W = 0;\quad U = Hy;\quad H = \mathrm{const}, \tag{1.1} \]

i.e., with a constant velocity gradient directed along the \(y\)-axis, and with a velocity directed along the \(x\)-axis (\(U, V, W\) are the components of the velocity along the \(x, y, z\) axes). The origin of coordinates, as is evident from the equations, is placed at the center of gravity of the particle (at the origin the velocity of the liquid is zero). The orientation of the particle may be specified by two angles \(\Phi\) and \(\theta\) (Fig. 1); \(\theta\) is the angle between the axis of the particle and the axis \(OZ\), \(\Phi\) is the angle between the plane passing through these two axes and the plane \(ZOY\). It is required to find the distribution function \(N(\Phi, \theta)\,d\Omega\). Let us first suppose that the particles of our colloidal solution are so elongated that their transverse dimensions may be neglected and the particle may be regarded as a thin rod. Such a rod will follow the motion of the liquid, since its inertia may be neglected, and no forces other than hydrodynamic ones act on the rod. The influence of Brownian motion we shall for the present not consider.

The velocity of the liquid at a certain point \(M\) on the axis of the rod is

\[ U = Hy = H\xi \sin\theta \cos\Phi, \]

where \(\xi\) is the distance of \(M\) from the center of the rod. Resolve this velocity into three components. The component perpendicular to the plane \(ZOM\) is

\[ U_{\Phi}=H\xi \sin\theta \cos^{2}\Phi, \]

since the normal to the plane \(OM\) makes an angle \(\Phi\) with the direction \(V\). The component directed along the particle is

\[ U_{r}=H\xi \sin^{2}\theta \sin\Phi \cos\Phi. \]

Similarly, the component of the velocity in the plane \(ZOM \perp OM\) is

\[ U_{\theta}=H\xi \sin\theta \cos\theta \sin\Phi \cos\Phi. \]

The components \(U_{\Phi}\) and \(U_{\theta}\) give rise to rotation of the particle about its center, namely \(U_{\Phi}\) to an increase of the angle \(\Phi\), or precession, and \(U_{\theta}\) to an increase of the angle \(\theta\), or nutation. Indeed, these velocities are at all times perpendicular to the rod, and for any point of it are proportional to the distance from the center \(\xi\).

The angular velocities of the rod are obtained if the rotational velocities of the point \(M\) are divided by the corresponding radii of rotation. For precession, i.e. rotation about the axis \(OZ\), the radius of rotation of the point \(M\) will be \(\xi\sin\theta\); consequently,

\[ \dot{\Phi}=\frac{U_{\Phi}}{\xi\sin\theta}=H\cos^{2}\Phi. \tag{1.2} \]

For nutation, i.e. rotation about the axis \(OX\), the radius of rotation is equal to \(\xi\), and therefore

\[ \dot{\theta}=H\sin\theta\cos\theta\sin\Phi\cos\Phi. \tag{1.3} \]

The disturbance of the motion of the liquid will occur only owing to the velocity component \(U_{r}\), which therefore plays the role

when determining the viscosity of a colloidal solution; for us, however, it is of no interest.

Integrating equation (1.2) from \(t_o\) to \(t\) gives

\[ \tg \Phi = H(t-t_o), \tag{1.4} \]

Dividing (1.3) by (1.2), we obtain

\[ -\frac{d\theta}{\sin\theta\cos\theta}=\frac{\sin\Phi}{\cos\Phi}\,d\Phi, \]

whence

\[ \tg\theta\cos\Phi=\mathrm{const}. \tag{1.5} \]

For a more visual illustration of the results obtained, let us introduce new angular coordinates (Fig. 1), namely \(\lambda\)—the angle between the planes \(XOY\) and \(XOM\), and \(\beta\)—the angle between \(OM\) and \(OX\). From the spherical triangle \(ZMM'\) we have

\[ \ctg\lambda=\tg\theta\cos\Phi;\quad \ctg\beta=\tg\Phi\cos\lambda. \]

Then the equations of motion in the new coordinates take the following form:

from (1.5), \(\ctg\lambda=\mathrm{const}\), i.e. \(\lambda=\mathrm{const}\),

from (1.4), \(\ctg\beta=H(t-t_o)\cos\lambda\).

Fig. 1

Thus, in its motion the particle asymptotically approaches the direction \(OX\), remaining all the time in the plane \(ZOM\); as \(t=\infty\), \(\ctg\beta=\infty\), i.e. \(\beta=0\). Consequently, in the stationary state, which practically sets in not at infinity but rather soon, all particles will be parallel to the axis \(OX\).

§ 2. Allowance for the Thickness of Particles

For particles that are not very long, it is necessary to take their thickness into account. In this case it is impossible to obtain solutions from purely kinematic considerations. We shall therefore consider the hydrodynamic equations of motion of an ellipsoidal particle derived by Jeffery\(^3\). In view of the cumbersomeness of this derivation, we shall indicate only its principles.

Jeffery uses not the integral equations obtained with the aid of Oseen’s formula\(^2\), but the ordinary differential equation of hydrodynamics

\[ \eta\nabla^2 u-\frac{\partial p}{\partial x} = \rho\left(\frac{\partial u}{\partial t}-\omega_3 v+\omega_2 w\right), \]

here \(\omega_1,\ \omega_2\) and \(\omega_3\) are the angular velocities of the particle, \(p\) is the pressure, \(\eta\) the viscosity. Analogous equations may be written for \(v\) and \(w\) by cyclic permutation. If one neglects powers of the particle velocities relative to the liquid higher than the first, then we obtain

\[ \eta \nabla^2 u=\frac{\partial p}{\partial x};\quad \eta \nabla^2 v=\frac{\partial p}{\partial y};\quad \eta \nabla^2 w=\frac{\partial p}{\partial z}. \]

The quantities \(u,\ v,\ w\) are connected by the equation of continuity

\[ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0. \]

The boundary conditions in solving these equations follow from the condition of flow past the particle: at any point of the particle surface the velocity of the liquid is equal to the velocity of this point.

Since Jeffery considers the particles as triaxial ellipsoids with semiaxes \(a,\ b\), and \(c\), the boundary conditions on the surface

\[ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1, \]

have the form:

\[ u=\omega_2 z-\omega_3 y;\quad v=\omega_3 x-\omega_1 z;\quad w=\omega_1 y-\omega_2 z. \]

Far from the particle, however, there must be the prescribed undisturbed motion of the liquid.

Solving the equations, we obtain \(u,\ v\) and \(w\) for any point of the liquid in the most general case of the presence of ellipsoidal particles, and find expressions for the moments of the forces acting on the particle from the side of the liquid. If it is assumed that no other forces act on the particles, then in the stationary state these hydrodynamic moments must also be equal to zero. From this condition three equations are obtained for determining the components of the angular velocity of an ellipsoid moving under the action of hydrodynamic forces. For the particular case when: 1) the undisturbed motion is given by equations (1.1), and 2) the ellipsoids are ellipsoids of revolution, from Jeffery’s calculation in Euler angles we obtain the equations of motion of the particles:

\[ \dot{\Phi}\cos\theta+\dot{\psi}=-\frac{1}{2}\cos\theta, \]

\[ (a^2+b^2)\dot{\theta}=H(a^2-b^2)\sin\theta\cos\theta\sin\Phi\cos\Phi, \tag{1.6} \]

\[ (a^2+b^2)\dot{\Phi}=H(a^2\cos^2\Phi+b^2\sin^2\Phi). \tag{1.7} \]

Pure rotation of the particle does not affect its orientation; therefore only equations (1.6) and (1.7) need be considered. For \(b=0\) these equations pass into (1.2) and (1.3).

The integrals of equations (1.6) and (1.7) are:

\[ \tg\Phi=\frac{a}{b}\tg\frac{Habt}{a^2+b^2},\qquad \tg\theta=\frac{ca}{\sqrt{a^2\cos^2\Phi+b^2\sin^2\Phi}}. \]

These equations indicate a periodic motion of the particle axis with period

\[ T=\frac{2\pi(a^2+b^2)}{Hab}. \]

In the stationary state the particles will no longer be oriented in a definite way, but will rotate with a variable angular velocity depending on the position of the particle. The particle distribution function is, obviously, inversely proportional to the angular velocity for given values of the angles \(\Phi\) and \(\theta\). The maximum \(N(\Phi,\theta)\) coincides with the direction for which the angular velocities are minimal: each particle passes through this position most slowly and, consequently, remains in it longer.

§ 3. Calculation of the resistance of a cylinder rotating in a liquid about a transverse axis

If at some point of the liquid a force \(F\) is applied, then the resulting velocity field at small distances from the point of application of the force can be calculated with the aid of Oseen’s equations

\[ u=\frac{F_x}{8\pi\eta}\left(\frac{1}{r}+\frac{x^2}{r^3}\right);\quad v=\frac{F_x}{8\pi\eta};\quad w=\frac{F_xxz}{8\pi\eta r^3}; \tag{1.8} \]

the force is directed along the \(x\)-axis.

Oseen’s equations can be used to calculate the resistance of bodies moving in a liquid. For this purpose, considering the motion in a coordinate system connected with the body, one must mentally replace the body by such a system of forces applied to the liquid that the disturbance of the motion of the liquid created by these forces would be equivalent to the disturbance created by the body. Obviously, the resultant of this system of forces will be equal to the resistance experienced by the body from the liquid, with the opposite sign. This resultant may be a force, a moment, or a system of forces, depending on the shape and character of the body’s motion.

To determine the equivalent system of forces one uses the well-known law of the flow of bodies by a liquid—on the surface of the body the velocity of the liquid must be equal to zero. This velocity is composed of the velocity of the undisturbed motion and the velocity created by the equivalent system of forces. The latter velocity, regarding the forces for the time being as known, can be determined by Oseen’s formula, and in the case of a continuous system of forces one has to integrate over the entire region of their application. Equating to zero the total velocity of the liquid at the surface of the body, we obtain an equation for the equivalent forces. If separate forces are applied to the body, then the equation will be ordinary; if, however, a system of continuous forces acts, then the equation will be integral. Solving the equation, we find the equivalent forces and, integrating over the entire

areas of their application, obtain the resistance experienced by the body. In solving, it must be kept in mind that the disturbance produced by the body must tend to zero at a sufficient distance from the body.

In solving concrete problems one has only to satisfy approximately the above-stated conditions for an equivalent system of forces. In particular, one has to require that the resultant velocity of the fluid be equal to zero not at every point of the surface of the body, but only on the average over a certain section, or over several sections. The integral equations are solved approximately, by representing the unknown force in the form of a polynomial with undetermined coefficients.

By the method described one may obtain the well-known Stokes formula for the motion of a sphere in a viscous medium, \(F=6\pi r v\eta\), and for its derivation it suffices to apply only one force at the center of the sphere. Burgers1 also derives formulas for the resistance of an elongated ellipsoid of revolution moving along its axis of revolution, and for the transverse motion of a cylinder.

Let us now pass to the problem of interest to us, that of determining the resistance of a cylinder rotating in a fluid about a transverse axis.

Let a cylinder of length \(2a\) and radius \(b\) rotate with angular velocity \(\omega\) about an axis passing through its center of gravity and perpendicular to the axis of the cylinder. Let us take a coordinate system attached to the moving body, with origin at the center of the cylinder; direct the \(OX\) axis along the axis of the cylinder, and \(OZ\) along the axis of rotation (Fig. 2). Then the undisturbed velocity of the fluid at a point \(M\) on the axis of the cylinder, at a distance \(x\) from its center, will be \(v=\omega x\) and is directed parallel to \(OY\). We apply an equivalent system of forces, likewise directed parallel to the \(OY\) axis and continuously distributed along the axis of the cylinder. Let a force \(g(\xi)d\xi\) act on an element of length of the axis \(d\xi\).

Fig. 2

Then, on the basis of (1.8), the velocity of the fluid produced by this force at some point \(N\) of the surface of the cylinder is

\[ v_\xi=\frac{g(\xi)d\xi}{8\pi\eta}\left\{\frac{1}{[(x-\xi)^2+b^2]^{\frac{1}{2}}}+\frac{z^2}{[(x-\xi)^2+b^2]^{\frac{3}{2}}}\right\}. \tag{1.9} \]

Here \(x\) and \(z\) are the coordinates of the point \(N\).

To obtain the velocity produced by the whole equivalent system of forces, expression (1.9) must be integrated with respect to \(\xi\) from \(-a\) to \(+a\).

Here it is not possible to require that the resultant velocity of the liquid be equal to zero at every point of the surface of the cylinder; we shall confine ourselves only to the requirement that the mean value of the velocity along the perimeter of any transverse section be equal to zero. We shall obtain this mean value along the perimeter if in (1.9) we put \(z=b\cos\vartheta\), integrate with respect to \(\vartheta\) from \(0\) to \(2\pi\), and divide by \(2\pi\). Since

\[ \int_0^{2\pi}\cos^2\vartheta\,d\vartheta=\pi, \]

the mean velocity of the liquid along the perimeter, produced by the force \(g(\xi)d\xi\), is

\[ \bar v_z=\frac{g(\xi)d\xi}{8\pi\eta} \left\{ \frac{1}{[(x-\xi)^2+b^2]^{\frac12}} + \frac{b^2}{2[(x-\xi)^2+b^2]^{\frac32}} \right\}. \]

And, by virtue of the condition imposed,

\[ \frac{1}{8\pi\eta}\int_{-a}^{a} g(\xi)d\xi \left\{ \frac{1}{[(x-\xi)^2+b^2]^{\frac12}} + \frac{b^2}{2[(x-\xi)^2+b^2]^{\frac32}} \right\} =wx, \tag{1.10} \]

For the solution of this integral equation we put

\[ g(\xi)=8\pi\eta\omega a \left[ B_1\left(\frac{\xi}{a}\right)+B_2\left(\frac{\xi}{a}\right)^3 \right]. \tag{1.11} \]

We take for \(g(\xi)\) terms with odd powers of \(\xi\), because, in view of the central symmetry of the rotational velocities, it must be

\[ g(-\xi)=-g(\xi). \]

If (1.11) is substituted in (1.10), then the integral (1.10) is easily evaluated, and we obtain

\[ \left[ B_1\left(\frac{x}{a}\right)+B_2\left(\frac{x}{a}\right)^3 \right]\ln\frac{4(a^2-x^2)}{b^2} -(B_1-B_2)\frac{x}{a} -\frac{8}{3}B_2\left(\frac{x}{a}\right)^3 =wx. \tag{1.12} \]

In order to determine the undetermined coefficients \(B_1\) and \(B_2\) by equating the multipliers of equal powers of \(x\), we replace

\[ \ln\frac{4(a^2-x^2)}{b^2} \]

by

\[ \lg\frac{4a^2}{b^2}-\frac{x^2}{a^2} = 2\sigma-\frac{x^2}{a^2} \]

and substitute this expression in (1.12). Neglecting powers of \(\frac{x}{a}\) higher than the 4th, we obtain

\[ 2\sigma \left[ B_1\frac{x}{a}+B_2\left(\frac{x}{a}\right)^3 \right] -(B_1-B_2)\frac{x}{a} -\left(B_1+\frac{8}{3}B_2\right) \left(\frac{x}{a}\right)^3 =\frac{x}{a}. \]

Equating the coefficients at like powers of \(\dfrac{x}{a}\), we obtain two equations for \(B_1\) and \(B_2\), from which we determine them. Substituting the values of \(B_1\) and \(B_2\) into (1.11), one can find \(g(\xi)\). The moment of the equivalent system of forces is determined from the equation

\[ M=\int_{-a}^{+a} g(\xi)\,\xi\,d\xi . \]

Omitting the calculations, we give the final result

\[ M=16\pi\eta\omega a^3\left(\frac{1}{3}B_1+\frac{1}{5}B_2\right) \simeq \frac{8\pi\eta\omega a^3}{3\left(\ln \frac{2a}{b}-0.8\right)}; \]

or, dividing by \(\omega\), we obtain the coefficient of resistance to rotation

\[ R_{\omega}= \frac{8\pi\eta a^3}{3\left(\ln \frac{2a}{b}-0.8\right)} . \tag{1.13} \]

We shall need this expression later, since it enters into the diffusion coefficient.

§ 4. Allowance for Brownian Motion

The equations of motion of a particle derived in § 2 are valid only as a first approximation for the case when the principal influence on the motion of the liquid is exerted by hydrodynamic forces.

Fig. 3

For a more exact consideration of the question, however, one must take into account the action of Brownian motion, which tends to orient the particles chaotically. When the particles rotate, their ends move over a sphere of radius \(a\). If a particle is oriented inside an elementary solid angle bounded by the values of the angles \(\theta\) and \(\theta+d\theta\), \(\Phi\) and \(\Phi+d\Phi\), then (Fig. 3) its end lies on the area \(ABCD\). The action of Brownian motion on the orientation of particles may be regarded as diffusion of the ends of the particles (points) over the surface of a sphere. Let us apply to this case the ordinary diffusion equation. The diffusion flux through \(AB\) will be

\[ D\frac{\partial N}{\partial \theta}\,d\Phi . \]

Here the diffusion coefficient \(D\) depends on the intensity of Brownian motion, i.e. on the temperature of the solution and on the resistance coefficient

rotation \(R_{\omega}\) [formula (1.13)]. From the theory of Brownian motion,

\[ D=\frac{kT}{R_{\omega}}. \]

Thus \(D\) depends on the viscosity and on the dimensions of the particle. On the other hand, transport will also take place owing to the motion of the particles under the action of the fluid flow. The total transport of the ends of the particles through \(AB\) is

\[ t_{\theta}\,d\Phi=\left(N\dot{\theta}\sin\theta-D\frac{\partial N}{\partial\theta}\right)\cdot d\Phi . \tag{1.14} \]

By entirely analogous reasoning we obtain, for the transport through \(AD\),

\[ t_{\Phi}\,d\theta=\left(N\dot{\Phi}\sin\theta-\frac{D}{\sin\theta}\cdot\frac{\partial N}{\partial\Phi}\right)\cdot d\theta . \tag{1.15} \]

For the stationary state the number of particles entering any solid angle and leaving it will be the same, and therefore for the flux \(t\) one may write the equation of continuity

\[ \frac{\partial t_{\theta}}{\partial\theta}+\frac{\partial t_{\Phi}}{\partial\Phi}=0, \]

or, substituting \(t_{\theta}\) and \(t_{\Phi}\) from (1.14) and (1.15),

\[ \frac{\partial}{\partial\theta}\left(N\dot{\theta}\sin\theta-D\frac{\partial N}{\partial\theta}\right) +\frac{\partial}{\partial\Phi}\left(N\dot{\Phi}\sin\theta-\frac{D}{\sin\theta}\cdot\frac{\partial N}{\partial\Phi}\right)=0. \tag{1.16} \]

Let us substitute here the expressions for \(\dot{\theta}\) and \(\dot{\Phi}\), determined from the equations of motion of a particle under the action only of hydrodynamic forces (1.6) and (1.7), and, to abbreviate the notation, introduce the designation

\[ m=\frac{a^{2}-b^{2}}{a^{2}+b^{2}}. \]

Then

\[ \begin{aligned} \dot{\theta}&=\frac{1}{2}mH\sin\theta\cos\theta\sin2\Phi,\\ \dot{\Phi}&=\frac{1}{2}H(1+m\cos2\Phi). \end{aligned} \tag{1.17} \]

Substituting these expressions into (1.16) and regrouping the terms, we obtain the differential equation with respect to \(N\)

\[ D\left(\frac{\partial^{2}N}{\partial\theta^{2}} +\frac{\cos\theta}{\sin\theta}\cdot\frac{\partial N}{\partial\theta} +\frac{1}{\sin^{2}\theta}\cdot\frac{\partial^{2}N}{\partial\Phi^{2}}\right) = \]

\[ = mH\left[\sin\theta\cos\theta\sin2\Phi\,\frac{\partial N}{\partial\theta} +\left(\frac{1}{m}+\cos2\Phi\right)\cdot\frac{\partial N}{\partial\Phi} -3N\sin^{2}\theta\sin2\Phi\right]. \tag{1.18} \]

It is required to solve this equation under the additional obvious conditions that \(N > 0\) and, of course, on the entire surface of the sphere; then the problem of the orientation of the particles will be solved. Unfortunately, this equation is solved only in certain particular cases, which we shall now consider.

§ 5. Approximate integration of the orientation equation

The equation (1.18) can be integrated in two limiting cases: if \(D \gg \frac{mH}{2}\) and if \(\frac{mH}{2} \gg D\). If \(D \gg \frac{mH}{2}\), which corresponds to the case of strong Brownian motion and a small velocity gradient in the fluid flow, then one may entirely discard the 2nd term in equation (1.18)* and obtain

\[ \frac{\partial^2 N}{\partial \theta^2} + \frac{\cos \theta}{\sin \theta}\cdot\frac{\partial N}{\partial \theta} + \frac{1}{\sin^2 \theta}\cdot\frac{\partial^2 N}{\partial \varphi^2} =0. \]

The unique solution of this equation satisfying the supplementary conditions is \(N=\mathrm{const}\), i.e. the absence of any orientation of the particles. This solution, obvious without any calculations, is of no interest to us. A more exact solution can be obtained if \(N\) is sought in the form of the series

\[ N=1+\sigma N_1+\sigma^2N_2+\sigma^3N_3+\cdots, \tag{1.19} \]

where

\[ \sigma=\frac{mH}{2D}. \]

Here \(N\) is normalized so that the 1st term is equal to 1. Substituting (1.19) into (1.18), after rather long and painstaking calculations we obtain

\[ N_1=\frac{1}{2}\sin^2\theta \sin 2\Phi, \]

\[ N_2=\frac{1}{16}\sin^4\theta-\frac{8}{3m}\sin^2\theta \cos 2\Phi-\sin^4\theta \cos 4\Phi. \]

For small \(\sigma\) it is sufficient to restrict ourselves to two terms

\[ N=1+\frac{\sigma}{2}\sin^2\theta \sin 2\Phi. \tag{1.20} \]

It follows from this that, independently of the ratio \(\frac{b}{a}\), the preferred orientation of the particles will be in the direction \(\theta=90^\circ,\ \Phi=45^\circ\). As we shall see below, this is brilliantly confirmed by experiment.

If \(\frac{mH}{2} \gg D\), which corresponds to weak Brownian motion, then in the first approximation one may use the calculation carried out in § 2. The case of weak, but nevertheless non-negligible

Brownian motion cannot be rigorously analyzed. To obtain an approximate solution, let us suppose that the majority of particles are oriented near the directions: \(\theta=90^\circ\), \(\Phi=90^\circ\); in this case, for convenience of exposition, put

\[ \Phi=90^\circ-\varphi=90-\xi\cdot(2\sigma)^{-\frac13};\qquad \theta=90^\circ-\vartheta=90-\eta\cdot(2\sigma)^{-\frac13} \tag{1.21} \]

and

\[ \frac{1}{m}-1=\frac{2b^2}{a^2-b^2}=\mu=2\rho\,(2\sigma)^{-\frac23}. \]

Here \(\xi\), \(\eta\), and \(\rho\) are, evidently, sufficiently small quantities. Now substitute into equation (1.18)

\[ \sin\theta\cos\theta\sin2\Phi \simeq 2\xi\vartheta;\qquad \cos2\Phi \simeq -1+2\varphi^2;\qquad \sin^2\theta\sin2\Phi \simeq 2\varphi \]

and, passing, moreover, from the derivatives with respect to \(\theta\) and \(\Phi\) to derivatives with respect to \(\xi\) and \(\eta\) according to (1.21), we obtain

\[ \frac{\partial^2 N}{\partial \xi^2} + \frac{\partial^2 N}{\partial \eta^2} + (\xi^2+\rho)\frac{\partial N}{\partial \xi} + \xi\eta\frac{\partial N}{\partial \eta} + 3\xi N = 0. \]

A very crude approximation is given by the expression

\[ N=ce^{-\alpha(\xi-\gamma)^2-\beta\eta^2}. \tag{1.22} \]

Here \(c\), \(\alpha\), \(\beta\), and \(\gamma\) are constants of integration. The maximum orientation will be in the direction \(\theta=90^\circ\), \(\Phi=90-\gamma(2\sigma)^{-\frac13}\). Let us emphasize once more that (1.19) is valid for \(\sigma\leqslant1\), while (1.22) is valid for \(\sigma\gg1\), so that for mean values of \(\sigma\) the spatial problem remains unresolved.

These two cases have been considered by us mainly in order to justify the transition to the plane case.

§ 6. Plane case

From the preceding paragraph it is clear that the preferential orientation of the particles is always obtained for \(\theta=90^\circ\) and for \(\Phi\) within the limits from 45 to \(90^\circ\). Therefore, approximately, one may assume that all particles are oriented in the plane \(XOY\), i.e. \(\theta=90^\circ\). For this so-called plane case, equation (1.16) is simplified as follows:

\[ \frac{\partial}{\partial\Phi} \left( N\dot{\Phi} - D\frac{\partial N}{\partial\Phi} \right) = 0. \tag{1.23} \]

Integrating equation (1.23) once, we have

\[ N\dot{\Phi} - D\frac{\partial N}{\partial\Phi} = \mathrm{const}. \tag{1.24} \]

Substituting here \(\Phi = 90 - \varphi\) and, from (1.17),

\[ \Phi=\frac{1}{2}mH(1+\mu-\cos 2\varphi); \]

we obtain

\[ \frac{dN}{d\varphi}+\sigma N(1+\mu-\cos 2\varphi)=\mathrm{const}. \]

Integration gives the general solution

\[ N=\frac{c}{\psi(\varphi)}\int_{\varphi}^{\varphi+\pi}\psi(\varphi)\,d\varphi,\quad \text{where }\psi(\varphi)=e^{\sigma(1+\mu)\varphi+\frac{1}{2}\sigma\sin 2\varphi}. \tag{1.25} \]

According to Beder, for extreme values of \(\sigma\) equation (1.23) can be integrated approximately, and then its solution is obtained not in the form of an unevaluated integral, but in the more convenient form of a series.

For small \(\sigma\), putting \(\Phi=90-\varphi\), we seek \(N\) in the form of the series

\[ N=1+\sum_{\nu=1}^{\infty}\sigma^\nu g_\nu(\varphi). \]

Here \(N\) is normalized so that

\[ \int_{0}^{\pi}N(\varphi)\,d\varphi=\pi, \]

i.e. the mean value \(\overline{N(\varphi)}=1\). We obtain

\[ N=1+\frac{\sigma}{2}\sin 2\varphi+\cdots . \tag{1.26} \]

Fig. 4

Formula (1.20) gives the same result for the spatial problem if one sets \(\theta=90^\circ\) in it.

For large \(\sigma\) we seek \(N\) in the form of the series

\[ N=\sum_{1}^{\infty}\frac{1}{\sigma^\nu}f_\nu(\varphi), \]

and find

\[ N=\frac{2c}{2\sigma}\left( \frac{1}{\sin^2\varphi} +\frac{1}{2\sigma}\frac{\cos\varphi}{\sin^3\varphi} +\frac{1}{4\sigma^2}\frac{5-4\sin^2\varphi}{\sin^2\varphi} \right). \tag{1.27} \]

By numerical integration of (1.25), Beder\(^4\) constructed a series of curves for various \(\sigma\) (Fig. 4). These curves illustrate quite well the results of the theory of hydrodynamic orientation.

particles in a planar position, and can be used for comparing theory with experiment and for practical calculations.

II. Optical Phenomena

A liquid in which an orientation of some particles has been created possesses optical anisotropy.

If we restrict ourselves to consideration of the planar case (§ 6.1), then the medium may be characterized by two refractive indices. To determine these optical constants it is necessary to calculate the projections of the polarization vector \(P\) (the dipole moment per unit volume) on the principal optical axes. The value of the projections will, of course, depend on the distribution function of the particles, the form of which was clarified in the first part of the work.

The dipole moment of a unit volume of the medium under investigation must be calculated as the sum of the values of \(P\) for the solvent and for the colloidal particles. Knowing the polarization vector, it is easy to set up an equation for the refractive indices, which can be tested experimentally.

In §§ 1 and 2 the projections of the polarization vectors \(P_1\) and \(P_2\) of the system “oriented particles—liquid” on the principal optical axes are calculated; the case is considered in which the electric field is directed along an optical axis. With the aid of the result of the first two paragraphs, in § 3 the equation of double refraction is set up. These equations are applied in § 4 to the case, considered in Part 1, of the hydrodynamic orientation of particles in a liquid. In § 5 certain corrections are introduced and the direction of the optical axes of the system is determined.

§ 1. Dipole Moment of the Solvent

As a model of the molecule we shall take an ellipsoid of revolution. In this case the polarization properties of the molecule are described by two coefficients: the polarizability \(\alpha_1'\),1 for an electric field parallel to the axis of revolution, and \(\alpha_2'\), for a field perpendicular to this axis.

Let the electric field \(E'\) be directed along the first principal optical axis. In this case the formula for the field acting on the dipole is applicable: \(E' = E + \dfrac{4}{3}\pi P\), where \(P\) is the dipole moment per unit volume.

The dipole moment of a molecule situated at an angle \(\theta'\) to the direction of the field is equal to \(\alpha_1' E' \cos \theta' + \alpha_2' E' \sin \theta'\). The projection of the moment of the molecule on the direction of the field is equal to \(\alpha_1' E' \cos^2 \theta' + \alpha_2' E' \sin^2 \theta'\).

For the projections of the dipole moment per unit volume, \(\alpha_1'\) and \(\alpha_2'\) (the polarization vectors), on the directions of the optical axes, in that case—

case, when the electric field coincides with these directions, we obtain: for the first direction

\[ \alpha_1'=\left(E+\frac{4}{3}\pi P_1\right) I_1', \tag{2.1} \]

where

\[ I_1'=\int_0^\pi(\alpha_1'\cos^2\theta' + \alpha_2'\sin^2\theta')\,dN' \tag{2.2} \]

for the second direction

\[ \alpha_2'=\left(E+\frac{4}{3}\pi P_2\right) I_2' \tag{2.3} \]

\[ I_2'=\int_0^\pi(\alpha_1'\sin^2\theta' + \alpha_2'\cos^2\theta')\,dN' \tag{2.4} \]

§ 2. Dipole moment of colloidal particles

Formulas (2.1)—(2.4) can also be used for colloidal particles, if only the depolarizing action of each particle upon itself is taken into account. This leads, as calculations show, to a change in the formula for the field acting on the dipole. Namely, instead of \(E'=E+\frac{4}{3}\pi P\), for the 1st direction we obtain

\[ E'=E+\frac{4}{3}\pi(1+\varepsilon_1)P \]

and for the 2nd direction

\[ E'=E+\frac{4}{3}\pi(1+\varepsilon_2)P. \]

Here the quantities \(\varepsilon_1\) and \(\varepsilon_2\) depend on the ratio of the semiaxes of the ellipsoid\(^1\) or on its eccentricity \(l\).

Thus the projection of the dipole moment, induced by a field parallel to direction 1, onto this direction will be

\[ \left(E+\frac{4\pi}{3}P_1\right)(\alpha_1\cos^2\theta+\alpha_2\sin^2\theta) +\frac{4\pi}{3}P_1(\varepsilon_a\alpha_1\cos^2\theta+\varepsilon_b\alpha_2\sin^2\theta) \]

and

\[ \alpha_1=\left(E+\frac{4\pi}{3}P_1\right)I_1+\frac{4\pi}{3}P_1H_1, \]

\(^1\) Namely:

\[ (1+\varepsilon_1)=3\left(\frac{1}{l^2}-1\right) \left(\frac{1}{2l}\lg\frac{1+l}{1-l}-1\right); \quad (1+\varepsilon_2)= \]

\[ =\frac{3}{2}\left(\frac{1}{l^2}-\frac{1-l^2}{2l^2}\lg\frac{1+l}{1-l}\right). \]

Here

\[ l=1-\frac{b^2}{a^2}. \]

where

\[ I_1=\int_0^\pi(\alpha_1\cos^2\theta+\alpha_2\sin^2\theta)\,dN \tag{2.5} \]

and

\[ H_1=\int_0^\pi(\varepsilon_1\alpha_1\cos^2\theta+\varepsilon_2\alpha_2\sin^2\theta)\,dN . \tag{2.6} \]

For a field parallel to direction 2, we similarly have

\[ \alpha_2=\left(E+\frac{4\pi}{3}P_2\right)I_2+\frac{4\pi}{3}P_2H_2, \]

where

\[ I_2=\int_0^\pi(\alpha_1\sin^2\theta+\alpha_2\cos^2\theta)\,dN, \tag{2.7} \]

\[ H_2=\int_0^\pi(\varepsilon_1\alpha_1\sin^2\theta+\varepsilon_2\alpha_2\cos^2\theta)\,dN . \tag{2.8} \]

The action of the other colloidal particles and of the medium on the given colloidal particle is neglected.

§ 3. Fundamental equation of double refraction

We can now write the expression for the projections of the total polarization vector of the solution on the principal axes for the case when the electric field is directed along one of these axes:

\[ P_1=\alpha_1+\alpha_1'=\left(E+\frac{4\pi}{3}P_1\right)(I_1+I_1')+\frac{4\pi}{3}P_1H_1, \tag{2.9} \]

\[ P_2=\alpha_2+\alpha_2'=\left(E+\frac{4\pi}{3}P_2\right)(I_2+I_2')+\frac{4\pi}{3}P_2H_2. \tag{2.10} \]

Let us use the Maxwell–Hertz equation, valid for the case when the field \(E\) is parallel to one of the principal directions,

\[ (n^2-1)E=4\pi P . \tag{2.11} \]

Eliminating \(P\) from (2.9) and (2.11), and then from (2.10) and (2.11), we obtain

\[ \frac{n_1^2-1}{n_1^2+1}=\frac{4\pi}{3}\, \frac{I_1+I_1'}{1-\frac{4\pi}{3}H_1} \quad\text{and}\quad \frac{n_2^2-1}{n_2^2+2}=\frac{4\pi}{3}\, \frac{I_2+I_2'}{1-\frac{4\pi}{3}H_2}. \tag{2.12} \]

Let us suppose that the concentrations of the solution are so weak,

which makes it possible to neglect terms with \(N^2\). But \(I\) and \(H\) are proportional to \(N\), and therefore equations (2.12) may be rewritten in the form

\[ \frac{n_1^2-1}{n_1^2+2} = \frac{4\pi}{3} \left( I_1+I'_1+\frac{4\pi}{3} I'_1 H_1 \right), \]

\[ \frac{n_2^2-1}{n_2^2+2} = \frac{4\pi}{3} \left( I_2+I'_2+\frac{4\pi}{3} I'_2 H_2 \right). \tag{2.13} \]

Let us suppose that the solvent molecules are not oriented; then \(I'_1=I'_2=I'\),

\[ \frac{4\pi}{3} I'=\frac{n_0^2-1}{n_0^2+2}, \]

where \(n_0\) is the refractive index of the pure solvent. Denote

\[ \Delta=\frac{n^2-1}{n^2+2} \]

and, subtracting one equation (2.13) from the other, we obtain

\[ \Delta_1-\Delta_2 = \frac{4\pi}{3}(I_1-I_2) + \frac{4\pi}{3}\Delta'(H_1-H_2). \tag{2.14} \]

This general equation of the birefringence of a solution is applicable to all cases of particle orientation. Thus the birefringence depends on two factors: 1) the optical anisotropy of the individual colloidal particles, characterized by the terms with \(I_1-I_2\) (if the particles are isotropic, then \(I_1=I_2\) and this term vanishes); 2) the elongated form of the colloidal particles, characterized by the term with \((H_1-H_2)\Delta'\) (for spherical particles this term vanishes).

If \(n_1-n_2\) is small, then from (2.13) it follows that

\[ \Delta_2-\Delta_1 = (n_1-n_2) \frac{n_1+n_2}{(n_1^2+2)(n_2^2+2)} = (n_1-n_2)\frac{2n_0}{(n_0^2+2)^2}; \]

substituting into (2.14), we obtain the fundamental equation of birefringence in the form

\[ (n_1-n_2)\frac{n_0}{(n_0^2+2)^2} = \frac{2\pi}{3}(I_1-I_2) + \frac{2\pi}{3}\frac{n_0^2-1}{n_0^2+2}(H_1-H_2). \tag{2.15} \]

Knowing \(n_1-n_2\), we easily find from the well-known formula

\[ G=l(n_1-n_2) \]

the path difference between the ordinary and extraordinary rays. \(l\) is the thickness of the layer. The quantity \(G\) is measured directly in experiment.

Let us now apply the general formulae of birefringence to the case of hydrodynamic orientation.

§ 4. Calculation of birefringence under hydrodynamic orientation of particles for small \(\sigma\)

If second powers of \(\sigma\) may be neglected, then the distribution function takes the form (from 1.26)

\[ N=1+\frac{\sigma m}{2}\sin 2\varphi . \]

Let us assume that one of the principal optical axes coincides with the direction of preferential orientation of the particles. Then the angle \(\theta\) entering into the basic formula for birefringence is the angle between the axis of the particle and the direction of preferential orientation.

In Part I we saw that preferential orientation is observed at \(\varphi=45^\circ\); consequently,

\[ \theta=\varphi-45^\circ \quad \text{or} \quad \varphi=\theta+45^\circ . \tag{2.16} \]

Substituting \(dN\), determined from (1.26) with allowance for (2.16), into formulas (2.5)—(2.8) and carrying out the integration, we obtain the values \(I_1, I_2, H_1\), and \(H_2\). Substituting these quantities into (2.15), we establish the formula for birefringence for hydrodynamic orientation of particles at small \(\sigma\)

\[ \frac{1}{N}(n_1-n_2)\frac{D}{N}\frac{n_0}{(n_0^2+2)^2} = \frac{\pi}{3}\frac{a^2-b^2}{a^2+b^2} \times \]

\[ \times \left[ \alpha_1-\alpha_2+ \frac{n_0^2-1}{n_0^2+2} (\varepsilon_a\alpha_1-\varepsilon_b\alpha_1) \right]. \tag{2.17} \]

In an entirely analogous way one can carry out the calculation also for the case of large \(\sigma\), using the approximate formula (1.27) derived for this case, and also for any \(\sigma\), if formula (1.25) is used. The position of the principal axes may be determined from the preferential orientation of the particles, by finding \(\max N\) from formulas (1.26) and (1.27).

§ 5. A more exact theory of birefringence

In the theory set forth above for the optical properties of a flow of solution, the following approximations were made.

1. We assumed that one of the principal optical axes of the solution coincides with the direction of preferential orientation of the particles, which is not entirely obvious.

2. We completely ignored the third dimension, assuming that all particles are oriented in one plane. In order to judge the agreement of the two-dimensional problem with experiment, it is sufficient to consider formula (1.20) for the case of small \(\sigma\) analyzed by us in § 4, i.e. of strong Brownian motion. If \(\sigma\) is very small,

then \(N\) will differ little from unity for all values of \(\theta\); thus it may rather be assumed that the particles are uniformly distributed with respect to the angle \(\theta\), than that they are concentrated in the plane \(\theta = 90^\circ\).

A theory which does not make the indicated approximations was developed by Beder\(^4\). Beder’s calculations consist in deriving the value of the tensor of the dielectric constant from a knowledge of the polarizability tensor of the particle and the law of distribution of the particles. We shall give only the results of Beder’s theory.

1. The angle between the \(OX\) axis, i.e. the direction of the flow velocity, and the first axis,

\[ \chi = \frac{1}{2}\operatorname{arc\,tg}\frac{B}{A}, \tag{2.18} \]

where

\[ A=\int_0^{2\pi} N(\varphi)\cos 2\varphi\,d\varphi;\qquad B=\int_0^{2\pi} N(\varphi)\sin 2\varphi\,d\varphi. \]

1. The tensor of the dielectric constant

\[ \varepsilon = \begin{vmatrix} a+b\theta & 0 & 0\\ 0 & a-b\theta & 0\\ 0 & 0 & a \end{vmatrix} \]

where

\[ a=1-\frac{4\pi k_1}{\frac{4\pi}{3}k_1-1};\qquad b=\frac{k_2}{\left(\frac{4\pi}{3}k_1-1\right)^2};\qquad \theta=\sqrt{A^2+B^2}, \]

\[ k_1=\frac{4}{3}\pi(2\alpha_1+\alpha_2);\qquad k_2=\frac{2}{3}(\alpha_1-\alpha_2). \]

\(\alpha_1\) and \(\alpha_2\) are the polarizabilities of an individual particle.

1. The path difference:

\[ G=l\left(\sqrt{\varepsilon_1}-l-\sqrt{\varepsilon_2}\right)\simeq l\frac{b}{\sqrt{a}}\,\theta . \]

It is of interest to compare with this theory the results obtained from Sadron’s simplified reasoning. For example, for small \(\sigma\), substituting \(N\) from (1.26) into (2.18), we obtain

\[ \chi=\frac{1}{2}\operatorname{arc\,tg}\frac{2m}{\sigma}. \]

According to the approximate theory, the principal optical axis coincided with the direction of the preferred orientation of the particles, i.e. for small \(\sigma\) it was situated at an angle of \(45^\circ\) to the direction of flow of the liquid:

\[ \chi' = 45^\circ . \]

The discrepancy between the two values is:

\[ \chi' - \chi = 45^\circ - \frac{1}{2}\operatorname{arc tg}\frac{2m}{\sigma} = 45^\circ - \frac{1}{2}\left(90^\circ - \operatorname{arc tg}\frac{\sigma}{2m}\right), \]

or, expanding \(\operatorname{arc tg}\frac{\sigma}{2m}\) in a series and retaining only the first term, we obtain:

\[ \chi' - \chi = \frac{\sigma}{4m}, \]

i.e., the discrepancy is the smaller, the smaller \(\sigma\) is. For example, when \(\frac{\sigma}{m}=0.05\), \(\chi' - \chi \approx 0.7^\circ\). Comparison of the expressions obtained for the path difference is more difficult because of the difference in calculation methods. It should be noted that Beder’s calculation is inferior to Sadron’s calculation in that in the former the refractive index of the solvent and the thickness of the particles are taken into account.

The results of both Beder’s and Sadron’s calculations agree quite well with experiment, despite all the approximations made. At present there is no need for a more thorough calculation, since we do not know exactly the shape and size of the particles.

III. EXPERIMENTAL DATA

§ 1. Experiments of Freundlich et al.

The first quantitative investigations of optical phenomena were carried out by Freundlich et al.^6 with a colloidal solution of vanadium pentoxide \(V_2O_5\). This substance is convenient for investigation, since it gives very elongated particles of more or less the same shape and size, depending only on the age of the sol. Freundlich’s apparatus is shown in Fig. 5. In the glass trough \(1\) a stream of colloidal solution of vanadium pentoxide is produced through tube \(2\) from vessel \(3\). The pressure in vessel \(3\), and consequently the rate of flow, are regulated by means of vessel \(4\). Vessel \(3\) is placed in thermostat \(5\), in which one or another temperature is maintained. The thermostat stands on three legs with setting screws \(6\) and can rotate about one of them, so that trough \(1\) can be adjusted relative to the optical axis of the instrument. The optical arrangement of the apparatus is shown in Fig. 6. Here \(A\) is a slit giving a narrow beam of light, \(L\) is the objective, \(P\) is the polarizer (a Glan prism), \(C\) is the trough with the liquid stream, \(Z\) is a Babinet–Soleil compensator for the path difference, \(W\) is Leiser’s half-shadow apparatus for accurate

Fig. 5

the compensator, \(N\)—an analyzer for determining the angle of rotation of the plane of polarization by the flow, \(R\)—a Lippich half-prism for precise adjustment of the analyzer.

Fig. 6

This apparatus made it possible to measure the angle of rotation of the plane of polarization \(\Gamma\) and the optical path difference of the rays \(\Delta\). The absolute value of the flow velocity was not measured; the relative value of the velocity could be established from the amount of liquid that had flowed through, using measuring cylinder 7 (Fig. 5).

We present the results of Freundlich’s work.

1. The phenomenon of aging of a colloidal solution was discovered. As the age of the solution increases, \(\Gamma\) and \(\Delta\) increase. This can be explained by an increase in the size of the particles, since, according to formula (1.13), the particle size enters in the cube into the numerator of the rotational resistance factor \(R\), i.e., into the denominator of the diffusion coefficient \(D\), and particles of larger size are less subject to the disorienting action of Brownian motion. This aging depends strongly on the contamination of the solution and on temperature.

2. The limiting value of the anisotropy of the solution is proportional to the content of \(\mathrm{V_2O_5}\).

3. The anisotropy of the flow increases with increasing velocity: for young sols linearly, and for old ones—along a curve with saturation.

4. With increasing temperature the anisotropy decreases.

When these experiments were carried out, the theory of the phenomenon had not yet been developed; now, however, all these results can be readily explained. For example, it follows from the theory that the anisotropy is proportional to the concentration of the colloid [formula (2.17)]; with increasing temperature, as a result of Brownian motion, disorientation of the particles occurs, and the anisotropy decreases. With increasing particle size the diffusion coefficient \(D\) decreases, and the anisotropy increases. The anisotropy also increases with increasing flow velocity, i.e., with the velocity gradient \(H\), and for small \(\sigma\), which in these experiments corresponds to young sols with small particle size, this dependence is indeed linear [formula (2.17)].

No quantitative agreement between the data of these experiments and the theory should be expected, since in the experiments Poiseuille flow of the liquid occurred, whereas the theory applies to a flow with a constant velocity gradient.

§ 2. Observation of the Vortex Cross

The same investigators observed the so-called vortex cross\(^7\). The apparatus is shown schematically without construc-

typical details in Fig. 7. The \(V_2O_5\) solution was placed between two glass cylinders \(M\) and \(N\). The outer cylinder \(N\) rotates together with the metal cylinders \(E\) in ball bearings \(F\), driven by an electric motor \(D\) through a belt transmission. The speed of the electric motor was regulated, and the number of revolutions of cylinder \(N\) was measured by means of a counter \(Z\). Along the axis of the cylinders, from below, a broad beam of light was admitted. The cylinders were placed between two crossed Nicols \(R\) and \(R_1\), of which \(R_1\) was fixed, while \(R\) could rotate. Cylinder \(M\) was suspended immovably from a holder with a ring \(P\) by means of the wider part \(T\), and could be adjusted. Cylinder \(N\) was adjusted by means of setting screws \(K\). First the polarizer \(R\) was set to darkness with cylinder \(N\) at rest; then rotation was imparted to the cylinder. If

Fig. 7

Fig. 8

one looks at the cylinders from above, the layer of \(V_2O_5\) solution is seen in the form of a ring. With the cylinder at rest and the polarizers crossed, this ring will be dark; when the cylinder rotates, four dark bands are observed (Fig. 8) against the background of a brighter ring. If these bands are prolonged to the center of the ring, a rectangular cross is obtained; therefore this phenomenon is called the vortex cross. The rays of the cross make a certain angle with the direction of the plane of polarization of both Nicol prisms. This angle was read on the limb \(L\) (Fig. 7).

As a result of the measurements made by Freundlich and others, the following was established.

1. The cross angle \(\psi\)—the larger of the angles between the minimum of illumination and the plane of polarization of one of the Nicol prisms—does not depend on the thickness of the liquid layer or on the concentration of the sol; it increases with increasing velocity gradient and with the age of the sol, and decreases with increasing temperature.

1. With the passage of time the angle \(\psi\) increases; this value oscillates* between \(45\) and \(90^\circ\).

All these regularities are in full agreement with the data of the theory. Indeed, between the cylinders there is produced a flow of liquid with a constant velocity gradient equal to \(H=\pi\omega\dfrac{r_1+r_2}{r_1-r_2}\), where \(r_1\) and \(r_2\) are the radii of the cylinders.

The difference from the example considered earlier is only that the gradient is directed not along a fixed axis, but along the radius. Each element of the ring of solution \(\varepsilon\) (Fig. 8) represents, as it were, an anisotropic crystal. It resolves the ray polarized by the prism \(R\) into ordinary and extraordinary rays, the direction of the plane of polarization of each of which is no longer perpendicular to the plane of polarization of the prism \(R\); therefore there will be no complete extinction of light. It will occur only in the case when the principal optical axis of the solution coincides with the plane of polarization of one of the prisms. But the optical axis of the solution, coinciding with the direction of preferential orientation of the particles, according to the theory makes an angle \(\varphi\) from \(0\) to \(45^\circ\) with the direction of the flow velocity, i.e. in the present case with the tangent to the ring. Hence it is clear (Fig. 8) that

Fig. 9

for some value of \(\psi\), oscillating between \(45\) and \(90^\circ\), the optical axis of the given element of the solution coincides with one of the planes of polarization of the prisms, and a minimum of light is obtained. The dependences of \(\psi\) on the velocity gradient, the age of the sol, and the temperature also agree completely with the theory.

§ 3. Beder’s Experiments

Beder’s apparatus\(^8\) is basically analogous to that shown in Fig. 7; the difference is only in the optical scheme, which is shown

Fig. 10. \(1\) and \(4\)—\(D=0.1\ \mathrm{sec}^{-1}\);
\(2\) and \(3\)—\(D=0.2\ \mathrm{sec}^{-1}\)

Fig. 11.

in Fig. 9. Here: 1—a point osmium lamp, 2—a condenser, 3—a cuvette with water for absorption of heat rays, 4—a lens, 5—a Nicol (polarizer), 6—a lens, 7—the cylinders according to Fig. 7, 8—a lens, 9—a Babinet compensator, 10—lenses, 11—a Nicol (analyzer), 12—a screen.

Beder measured the path difference \(\theta\) and the angle between the directions of the principal optical axis and the flow velocity \(\chi\) for various values of \(a\) for a cellulose solution and plotted the experimental points obtained on a graph (Fig. 10). The theoretical curves are also plotted on the same graph. These curves were constructed to within a constant parameter and then superposed on the graph so as to obtain the best possible agreement of the curve with the experimental data. This made it possible to determine this constant parameter and to calculate the coefficient of rotational diffusion \(D\).

Beder treated in the same way Sinha’s experimental results for determining \(\chi\) for colloidal solutions of rubber and various polystyrenes (Fig. 11), and the experiments described by us of Freundlich for determining \(\theta\) for solutions of vanadium pentoxide of different ages (Fig. 12).

As can be seen from all three figures, the agreement of the theoretical curves with the experimental points, obtained even before the creation of the theory and used to test it, is sufficiently good, which serves as the best confirmation of the correctness of the theory.

§ 4. Sadrone’s Experiments

Sadrone’s apparatus\(^8\) is in principle the same as Beder’s, but the construction of the cylinders and the optical arrangement are considerably simplified, owing to which on this apparatus it is possible to determine only the direction of the optical axes of the solution, but not the path difference of the rays. Fig. 13 schematically shows Sadrone’s apparatus. Here \(A\) and \(P\) are Nicol prisms, \(S\) is the light source, \(L_1\) and \(L_2\) are lenses, \(A\) are cylinders, and the inner cylinder is rotated by means of pulley \(B\).

Fig. 12. \(1\)—\(V_2O_5\), 6.99 days; \(2\)—\(V_2O_5\), 4.03 days; \(3\)—\(V_2O_5\), 0.97 day

Fig. 13

The adjustment of the velocity gradient of the liquid here was carried out

as by changing the number of revolutions of the cylinder within the limits from 0.2 to 60 rev/sec, and also by using inner cylinders of different radii \(R_1=24.2;\ 25;\ 24.98\) and \(25.28\) mm with the radius of the outer cylinder \(R=25.5\) mm.

Fig. 14

The experiment was carried out in the following order. First, with the rotor of the instrument stationary, the prisms \(A\) and \(P\) were set to darkness; then the rotor was started and both prisms together were turned through a certain angle until complete darkness. As was already said above, in this case the direction of the principal optical axes of the solution must coincide with the planes of polarization of both prisms and can thus be determined. The results obtained fully confirm the data of the calculation made by Sadron.

To confirm his theory in the part concerning the determination of the difference in path, Sadron specially processes the results of the latest experiments of Signer, who investigated solutions of polystyrenes and nitrocellulose in various solvents. If one takes one and the same colloidal substance and dissolves it in different solvents under otherwise equal conditions, then the variable quantities in the formula for double refraction (2.17) will be only the diffusion coefficient and the refractive index of the solvent \(n_0\). Let us assume that \(D\) can be represented in the form \(D=\dfrac{D_0}{\mu}\), where \(D_0\) depends only on the suspended particles, and \(\mu\) only on the solvent. Then formula (2.17) can be rewritten in the form

\[ \frac{1}{NH}\frac{n_1-n_2}{\mu}\frac{n_0}{(n_1^2+2)^2} = A+B\frac{n_0^2-1}{n_0^2+2}, \tag{3.1} \]

where

\[ A=\frac{\pi}{3}\frac{a^2-b^2}{a^2+b^2}\frac{1}{D_0}(\alpha_1-\alpha_2) \tag{3.2} \]

and

\[ B=\frac{\pi}{3}\frac{a^2-b^2}{a^2+b^2}\frac{1}{D_0}(\varepsilon_a\alpha_1-\varepsilon_b\alpha_2) \tag{3.3} \]

are, under the given experimental conditions, constant coefficients. Therefore, plotting along the abscissa axis \(\dfrac{n_0^2-1}{n_0^2+2}\), and along the ordinate axis

\(\dfrac{1}{NH\varphi}(n_1-n_2)\dfrac{n_0}{(n_0^2+2)^2}\), obtained from experiment, we should obtain straight lines, provided Sadron’s theory is correct. In Fig. 14, for polystyrenes with molecular weights 193,000 and 175,000, and in Fig. 15, for nitrocellulose dissolved in various substances, graphs constructed in this way are presented. The experimental points corresponding to one dissolved substance but to different solvents do indeed fall well on straight lines, i.e., Sadron’s theory is correct to a sufficient degree.

With such a treatment of the experimental results, all parameters that are difficult to determine are excluded, as are all systematic errors of the instruments; moreover, the need to make any absolute measurements disappears (one may confine oneself to relative measurements), which greatly simplifies the readings.

Fig. 15

Let us note that \(B<0\) always; therefore the straight lines in the graphs are always inclined downward. If, at the same time, \(A>0\), then formula (3.1) may give a positive, zero, or negative Maxwell effect (double refraction), depending on \(n_0\). This is in full agreement with experiments on various solutions.

From graphs such as those shown in Figs. 14 and 15, \(A\) and \(B\) can easily be determined, since

\[ \frac{A}{B}=\frac{n^2-1}{n^2+2}=-\frac{\alpha_1-\alpha_2}{\varepsilon_a\alpha_1-\varepsilon_b\alpha_2} \]

from (3.2) and (3.3).

The relation obtained makes it possible to determine \(n_1, n_2\) from a known \(\dfrac{a}{b}\), or the shape of the particles \(\dfrac{a}{b}\) from known \(n_1\) and \(n_2\). Thus, from relative optical measurements one can obtain important characteristics of the particles of a solution.

REFERENCES

1. Burgers, On the motion of small particles of elongated form, suspended in a viscous liquid.
2. Oseen, Hydrodynamik, 1927.
3. Jeffry, Proc. Roy. Soc., A 102, 169, 1922–1923.
4. Boeder, Z. Physik, 75, 258, 1932.
5. Sadron, Journ. Physique, 8, 481, 1937.
6. Freundlich, Stapelfeld u. Zocher, Z. Physik, 114, 161, 1925.
7. Freundlich, Stapelfeld u. Zocher, Z. Physik, 114, 190, 1925.
8. Sadron, Journ. Physique, 7, 263, 1936.

1. By a single prime we shall denote quantities corresponding to the solvent; the same quantities corresponding to the colloidal particles are denoted by the same letter without a prime. ↩↩