Kirkwood-Riseman approximation solutions

Dynamic light scattering from dilute solutions provides the value of the diffusion coefficient, which can be converted to hydrodynamic radius J h,star of the star polymer. The ratio Rh/(Rg) characterizes the compactness of the macromolecule for the uniform hard sphere impenettable for the flow, it is Rb/Rg= (5/3) 1.29, whereas for the Gaussian coil, Rh/(Rg) = 3a- / /8 0.66. For ideal stars (without excluded-volume interactions), the ratio Rh/(Rg) can be derived within the Kirkwood-Riseman approximation, which gives the value of Rh/(Rg) 0.93. Reported experimental values of the Rh/(Rg) ratio for star polymers and starlike block copolymer micelles are usually found close... 

The extremely simple expressions for intrinsic viscosity at zero gradient and sedimentation coefficient containing in the denominator the square root of molecular weight agree very well with experimental data of theta solutions in the whole range from fully drained to impermeable coil. They turned out to be very good approximations of the more complicated expressions derived by Kirkwood-Riseman and Debye-Bueche ° and hence were used for the calculation of intrinsic viscosity of polydisperse samples. 

Hie cylinder model gave no theoretical lalue of ]i/] for short flexible chains (AL<2.278 and Ad>0.1) owing to the nature of the kernel in the integral equation. To evaluate ]i/] for such chains, Yoshizaki et adopted the touched-bead model, that is, a discrete chain consisting of N beads of diameter db whose centers are located on the continuous KP or HW contour. These authors showed ](/] to be represented approximately by the sum of the solution of the Kirkwood-Riseman integral equation and the contribution of N beads as Einstein spheres. The result for the KP chain may be expressed in the form... 

The Zimm theory includes an alternative treatment in which frictional resistance to motion of the beads in the bead-spring chain is dominated by the viscous drag from other beads in the same chain (dominant hydrodynamic interaction. Fig. 9-5-11). The interaction is treated approximately as in the theory of Kirkwood and Riseman for the intrinsic viscosity of dilute polymer solutions, by use of the equilibrium-averaged Oseen tensor for the influence of the motion of one bead on another the average distances between pairs of beads are supposed to correspond to those in a 0-solvent. 

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