Streaming Birefringence of Rigid Macromolecules in General Two-Dimensional Laminar Flow

Abstract

Quantitative expressions for the direction of the angle of isocline and for the amount of birefringence due to a dilute solution of rigid macromolecules in a general two‐dimensional laminar flow are derived. If E is the principal strain rate and Λ₀ is the angle between the streamline direction and the direction of the principal strain rate axis, the angle of isocline, measured from the first principal strain rate axis is $$\mathrm{\chi =-(E}\sin {\text{2Λ}}_0{\text{/6D){1−(E}}^2{\text{/27D}}^2\mathrm{)[}{\sin }^2{\text{2Λ}}_0{\text{+(24b}}^2\text{/35)]+···],}$$ where b=[(a₁²—a₂²) / (a₁²+a₂²)] is a shape factor for an ellipsoid of revolution of semimajor axis a₁ and semiminor axis a₂ and D is the rotary diffusion constant for this ellipsoid. The amount of birefringence is $${\text{Δn=(4π/15)(cGEb/nD) {1−(E}}^2{\text{/18D}}^2\mathrm{)[}{\sin }^2{\text{2Λ}}_0{\text{+(6b}}^2\text{/35)]+···},}$$ where n is the mean index of refraction of the solution, c the volume concentration of the macromolecules, and G=g₁—g₂ is the optical anisotropy of the ellipsoids. It is seen that if the principal strain rate is not at 45° to the streamline at the point of observation, this will make itself felt in the position of the angle of isocline before it influences the amount of birefringence. Detailed expressions for the effect of polydispersity show that there is a simple relationship between the birefringence and angle of isocline measured in Couette flow and these quantities measured in general two‐dimensional flow only if (a) Λ₀=45° in the general flow or (b) the birefringence and angle of isocline values are linear with strain rate.