Entanglement Friction

If a steady systematic motion is imposed, those topological arrangements which present barriers to configuration relaxation will come into play. As in dilute solutions the chains are continually drawn into a disturbed distribution of configurations by the external motions and respond by diffusing collectively toward an equilibrium distribution. In concentrated systems each chain must 

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Some years ago, on the basis of the excluded-volume interaction of chains, Hess [49] presented a generalized Rouse model in order to treat consistently the dynamics of entangled polymeric liquids. The theory treats a generalized Langevin equation where the entanglement friction function appears as a kernel... 

Equations (35) and (36) define the entanglement friction function in the generalized Rouse equation (34) which now can be solved by Fourier transformation, yielding the frequency-dependent correlators . In order to calculate the dynamic structure factor following Eq. (32), the time-dependent correlators are needed. 

In addition to the Rouse model, the Hess theory contains two further parameters the critical monomer number Nc and the relative strength of the entanglement friction A (0)/ . Furthermore, the change in the monomeric friction coefficient with molecular mass has to be taken into account. Using results for (M) from viscosity data [47], Fig. 16 displays the results of the data fitting, varying only the two model parameters Nc and A (0)/ for the samples with the molecular masses Mw = 3600 and Mw = 6500 g/mol. 

According to Hess, the relative strength of the entanglement friction can be related to the more microscopic parameter q , describing the range of the true interchain interaction potential. A value of q 1 = 7 A, close to the average interchain distance of about 4.7 A, is obtained. 

Spring-bead models relate frictional force to the relative velocity of the medium at the point of interaction. The entanglement friction coefficient above is defined in terms of the relative velocity of the passing chain. Since the coupling point lies, on the average, midway between the centers of the two molecules involved, the macroscopic shear rate must be doubled when applying the result to a spring-bead model. Substitution of 2 CE for Con in the Rouse expression for viscosity yields... 

Wu, S., Entanglement, friction, and free volume between dissimilar chains in compatible polymer blends, J. Polym. Sci. Polym. Phys., 25, 2511-2529 (1987). 

There are a number of important concepts which emerge in our discussion of viscosity. Most of these will come up again in subsequent chapters as we discuss other mechanical states of polymers. The important concepts include free volume, relaxation time, spectrum of relaxation times, entanglement, the friction factor, and reptation. Special attention should be paid to these terms as they are introduced. 

The segmental friction factor introduced in the derivation of the Debye viscosity equation is an important quantity. It will continue to play a role in the discussion of entanglement effects in the theory of viscoelasticity in the next chapter, and again in Chap. 9 in connection with solution viscosity. Now that we have an idea of the magnitude of this parameter, let us examine the range of values it takes on. 

Whether the beads representing subchains are imbedded in an array of small molecules or one of other polymer chains changes the friction factor in Eq. (2.47), but otherwise makes no difference in the model. This excludes chain entanglement effects and limits applicability to M < M., the threshold molecular weight for entanglements. 

The empirical frictional factor (T fric) is independent of shear rate but increases in poor solvent this permits to account for the dependence of the scission rate constant on solvent quality. The entanglement part (r enl), as given by Graessley s theory which considers the effect of entanglement and disentanglement processes, is a complex function of shear rate ... 

Rouse motion has been best documented for PDMS [38-44], a polymer with little entanglement constraints, high flexibility and low monomeric friction. For this polymer NSE experiments were carried out at T = 100 °C to study both the self- and pair-correlation function. 

Thus at a given temperature, the location of the transition zone of E(t)e on the time scale is determined by the monomeric friction factor, the height of the entanglement plateau by Af, and the width of the plateau by (Ml Mf)3. The time dependence of entanglement slippage. (0 nn describes the rate at which the entanglement plateau will drop to the equilibrium... 

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