Dynamics of Dilute Polymer Solutions

This Section will concentrate on the long-range collective motions of chains, under both steady and oscillating shear, but restricted to linear viscoelastic properties. Two reviews by Ferry and by Bixon describe progress on experimental and theoretical fronts respectively up to 1976. 

Just as the Gaussian chain is the basic paradigm of the statistics of polymer solutions, so is its extension to the bead-spring model still basic to current work in the held of polymer dynamics. The two limiting cases of free draining (no hydrodynamic interaction between beads, characterized by the draining parameter A = 0) and non-free draining (dominant hydrodynamic interaction, A= CO, due to Rouse and Zimm, respectively, are sufficiently familiar that the approach is often known as the Rouse-Zimm model.  

Experimentally, most dilute polymer solutions in 0-solvents fit the Zimm theory best, whereas as the concentration and/or the solvent quality is increased the behaviour becomes more Rouse-like (A is decreased). Several theories exist to describe physical properties for intermediate values of A. 

In the Zimm theory, the flow perturbations and the co-operative hydrodynamic interactions between segments are treated using the Oseen tensor, pre-averaged for simplification. Pyun and Fixman (PF) avoided this approximation by a perturbation solution of the Kirkwood diffusion equation up to second order. One of the consequences was that [equation (3)] was re-evaluated (see Table 1). 

Now Yoshizaki and Yamakawa°° have extended the calculation to third-order terms, but with the Oseen tensor pre-averaged. In this way a precise lower bound for Og was obtained, close to that obtained by Auer and Gardner using Kirkwood-Riseman theory. The paper by Bixon and Zwanzig performs an infinite order calculation based upon the PF treatment. Using a numerical method, they obtain g 2.76 x 10 , between the Zimm and PF values. For flexible polymers (as for rigid rods) the pre-averaging of the hydrodynamic interaction tensor thus introduces only a small error the effect on the spectrum of relaxation times is more dramatic cf. columns 3 and 4 of Table 2), and the relaxation time of the slowest mode (proportional to 1/A/) is more than twice as slow. This difference should be detectable experimentally. 

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The expressions for the stress tensor together with the equations for the moments considered as additional variables, the continuity equation, and the equation of motion constitute the basis of the dynamics of dilute polymer solutions. This system of equations may be used to investigate the flow of dilute solutions in various experimental situations. Certain simple cases were examined in order to demonstrate applicability of the expressions obtained to dilute solutions, to indicate the range of their applicability, and to specify the expressions for quantities which were introduced previously as phenomenological constants. 

Pokrovskii VN, Tonkikh GG (1988) Dynamics of dilute polymer solutions. Fluid Dyn 2 3 (1) 115—121... 

Theories for polymer dynamics of dilute polymer solutions include the elastic (Hookean) spring model (Kuhn, 1934) which considers that the system is mechanically equivalent to a set of beads attached with a spring. The properties are then based on a spring constant between beads and the friction of beads through solvent. The viscosity of a Hookean system is then described by... 

Having seen the general background of Brownian motion, we shall now discuss the dynamics of a polymer in solution. As we have seen in Chapter 2 the static properties of a polymer can be represented by a set of beads connected along a chain. It is natural to model the dynamics of the polymer by the Brownian motion of such beads. Such a model was first proposed by Rouse and has been the basis of the dynamics of dilute polymer solutions. 

DYNAMICS OF DILUTE POLYMER SOLUTIONS moving plate... 

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