Real Polymer Chains with Large Frictional Interactions

Real Polymer Chains with Large Frictional Interactions. —Let us first consider the simpler problem of linear translation of the molecule through the solvent. We shall, however, view the motion of the solvent with reference to the molecule. Near its center the solvent is very nearly stationary in this sense, but as we move outward its velocity increases. The velocity will pass through a maximum before it finally, perhaps at some distance beyond the outer fringes of the molecule, reaches the unperturbed relative velocity of the surrounding medium. (The occurrence of this maximum is merely a consequence 

Of major importance is the fact that the specific character of polymer chains of a given type enters the relationship (17) only through the effective size of one of its beads as indicated by the ratio f/970. Even the effect of this factor vanishes when the total internal resistance to flow is sufficiently large. Hence, in this limit, which will include nearly all actual cases of interest (see Sec. 4), the molecular frictional coefficient should depend only on the size /s and not otherwise on the nature of the polymer. Accordingly, we choose to let 

The detailed hydrodynamic treatment of Kirkwood and Riseman yields the following relationship 

In the molecular weight range of interest x will exceed 10 and usually 

The estimation of f from Stokes law when the bead is similar in size to a solvent molecule represents a dubious application of a classical equation derived for a continuous medium to a molecular phenomenon. The value used for f above could be considerably in error. Hence the real test of whether or not it is justifiable to neglect the second term in Eq. (19) is to be sought in experiment. It should be remarked also that the Kirkwood-Riseman theory, including their theory of viscosity to be discussed below, has been developed on the assumption that the hydrodynamics of the molecule, like its thermodynamic interactions, are equivalent to those of a cloud distribution of independent beads. A better approximation to the actual molecule would consist of a cylinder of roughly uniform cross section bent irregularly into a random, tortuous configuration. The accuracy with which the cloud model represents the behavior of the real polymer chain can be decided at present only from analysis of experimental data. 

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