Position of the Optic Axes in a Liquid Showing Flow Double Refraction

HAROLD G. JERRARD Department of Physics, The University, Southampton, England

(Received April 2, 1957)

HEN a velocity gradient is established in a liquid contain­ing asymmetric molecules or particles, the shearing forces

which occur can produce a measurable amount of double refrac­tion. The most satisfactory apparatus for quantitative measure-ments is one in which the liquid is subjected to shear between two concentric cylinders, one of which rotates, while the other is stationary.1 By this means a uniform velocity gradient is produced across the gap. Under such conditions the flowing liquid is rendered optically biaxial. Thus three different principal axes Ox, Oy, Oz defining the extinction directions in the liquid exist, together with three principal refractive indexes nx, ny, and nz. One of these axes, say Oz, coincides with the direction of the common axis of the cylinders, and the other two, mutually perpendicular, lie in a plane perpendicular to this direction. Further, in general one axis, say Ox, is inclined at an angle x (the angle of extinction) to the direc­tion 01 of the streamlines. Because the direction of observation must be along Oz, the only observed birefringence is given by Δ n = n x – n y , and it is often stated that the flowing liquid behaves as a uniaxial crystal. Also in the literature there sometimes appears to be a confusion between extinction directions and optical axes. In this note the positions of the index ellipsoid2 and the optical axes are considered.

If a liquid contains particles, which are assumed to be uncharged rigid ellipsoids of revolution of which the lengths of the major and minor axis are 2a1 and 2α2, respectively, then it may be shown3

that for small concentrations

In this equation n0 is the refractive index of the liquid, NP is the number of particles per unit volume, Axy is an optical factor which

W

FIG. 1. (a) Trace of index ellipsoid in plane yOx and general position of optic axes Oa and Oa'. The directions OX and 02 lie along and perpendicular to the flow streamlines in the annulus. (b) Positions of axes for a liquid in flow behaving as a negative biaxial crystal when nx–nv is positive and negative.

depends on the optical polarizabilities of the particles, and B is an orientation factor which depends on the dimensions and diffusion constants of the particles. The corresponding equation involving nx and nz is

The index ellipsoid is given by the equation

with its axes coinciding with the directions Ox, Oy, and Oz. The ellipsoid has two circular cross sections inclined at angles ψ and π—ψ, respectively, to Ox; these are shown as SOS and SOS' in Fig. 1(a), which represents the elliptical section of the ellipsoid cut by the yOx plane. The value of ψ is determined by the points of intersection of the circle of radius nz with the ellipse, i.e., from the equations

Thus

which by Eqs. (1) and (2), when the factors Axy and Axz are evaluated becomes

In Eq. (3), σ is the ratio of the velocity gradient to the diffusion constant, α and β are the polarizabilities along the major and minor particle axes, and b= (a1

2–a22)/(a1

2+a22). Using the value

of B given by Scheraga et al.,4 then

The optic axes Oa and Oa' are inclined at angles (π/2±ψ) to Ox and the optic axial angle is 2ψ.

By Eq. (4), if σ is small then ψ≏45°. Also with σ small, x≏4 5 ° so that the optic axes lie almost along, and perpendicular to, the direction 01 of the streamlines. When b(a+6β)/(a-β) is positive, values of ψ<45° occur and the liquid behaves as a negative or positive biaxial crystal according to whether nx>nz>ny or nx<nz<nv, respectively. For negative values of b(a+6β)(a—β), then ψ>45° and the same equalities make the liquid behave as a positive or negative crystal, respectively. As σ increases, x de-

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JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 47, NUMBER 8 AUGUST, 1957

creases towards zero and ψ approaches 0° or 90°. The positions of of the optic axes with respect to the extinction directions Ox and Oy are shown in Fig. 1 (b) for a negative crystal. In experimental work on flow double-refraction only the sign corresponding to nx– nu is considered and it is customary to designate a liquid as positive or negative according to the value of this quantity, irrespective of the crystallographic definition.

1 H. G. Jerrard, Rev. Sci. Instr. 26, 1007 (1955). 2 A. Johannsen, Manual of Petrographic Methods (McGraw-Hill Book Company, Inc., New York, 1918). 3 A. Peterlin and H. A. Stuart, Z. Physik 112, 1 (1939). 4Scheraga, Edsall, and Gadd, J. Chem. Phys. 19, 1101 (1951).

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