Segmental friction

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. 

To the extent that the segmental friction factor f is independent of M, then Eq. (2.56) predicts a first-power dependence of viscosity on the molecular weight of the polymer in agreement with experiment. A more detailed analysis of f shows that segmental motion is easier in the neighborhood of a chain end because the wagging chain end tends to open up the structure of the melt and... 

Equation (2.56) not only enables us to understand the basis for the first-power dependence of rj on M, but also presents us with a new and important theoretical parameter, the segmental friction factor. We shall see in the next chapter that it is a quantity which can also be extracted from measurements of the viscoelasticity of polymers. 

Table 2.3 Segmental Friction Factors Ranked in Order of Decreasing Values for Polymers Compared 100°C Above Their Respective Glass Transition Temperatures...

Although we still need to explain the use of this theory, Eq. (3.98) shows that segmental friction factors are accessible through viscoelastic studies. This fact was anticipated in the list of f values given in Table 2.3. 

Rather than discuss the penetration of the flow streamlines into the molecular domain of a polymer in terms of viscosity, we shall do this for the overall friction factor of the molecule instead. The latter is a similar but somewhat simpler situation to examine. For a free-draining polymer molecule, the net friction factor f is related to the segmental friction factor by... 

Random coils. Equation (9.53) gives the Kirkwood-Riseman expression for the friction factor of a random coil. In the free-draining limit, the segmental friction factor can, in turn, be evaluated from f. In the nondraining limit the radius of gyration can be determined. We have already discussed f in Chap. 2 and (rg ) in this chapter and again in Chapter 10, so we shall not examine the information provided by D for the random coil any further. 

To guarantee the transformational invariance of DG = kBT/(NQ (see Table 4) in the case of Rouse relaxation, the replacement of N by N/A, requires the simultaneous replacement of the segmental friction coefficient by X which is natural, since friction is proportional to the number of segments involved. 

This treatment assumes that the forces between molecules in relative motion are related directly to the thermodynamic properties of the solution. The excluded volume does indeed exert an indirect effect on transport properties in dilute solutions through its influence on chain dimensions. Also, there is probably a close relationship between such thermodynamic properties as isothermal compressibility and the free volume parameters which control segmental friction. However, there is no evidence to support a direct connection between solution thermodynamics and the frictional forces associated with large scale molecular structure at any level of polymer concentration. 

Step 5 Calculate pipe straight segment friction pressure loss APf. 

The friction fector f per chain group is an important parameter obtained in the empirical analysis. We have tabulated the inherent segmental friction factor Co (the frequency factor of the rate process of mass transport) in Table 2. These values are principally calculated from by Eq. (2.7) ... 

We now ask which of the two friction factors has been used in the expression for zp equation (3-88). The segmental friction factor ps has been eliminated from this expression through its dependence on 77, the steady-state... 

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