Abstract
As a preparative step to treat the intrinsic viscosity of a stiff chain macromolecule in a solution, we have investigated the hydrodynamic behavior of a slightly curved rigid rod in a shear field.
Regarding the longitudinal axis of the rod as a differentiable space curve, we can represent the curve as follows using the Bouquet's formula: r(s)=r(0)(s)+κ1r(1)(s)+κ12r(2)(s)+κ1'r(3)(s)+κ1κ2r(4)(s)+ …, where κ1 and κ2 are the curvature and the torsion of the curve at the middle point of it and the prime stands for the differentiation with respect to s, the contour length of the curve. The force distribution F on the longitudinal axis when the rod is in the shear field can be estimated by the Oseen's method: 1/8πμ∫a-aT(s, s0)·F(s)ds=-Vr(s0), where a is the half length of the rod, . μ the viscosity of the fluid, T the interaction tensor and Vr is the relative velocity to the rod. The Brownian motion is not taken into account here. Expanding T, F and Vr into power series of (κ1, κ1', …; κ2, κ2', …), we can obtain a set of equations instead of the above equation. The first equation of the set is identical with that for the straight rod derived by Burgers. The motion of the rod is to be determined upon the conditions of the balance of the force and of the moment of force.
Concrete calculation is made of a rod curved like a circular arc, and the intrinsic viscosity for the case is estimated as follows:
[μ]=2/3πρa/b1/σ-1.80√1+(a/b)2(κ1a)2σ-2.13/10(σ-1.80),
where ρ and b are the density and the radius of the cross-section, of the rod and σ=log2a/b.
