Free-draining limit

As discussed in connection with Eq. (9.47), the Kirkwood-Riseman theory predicts that a = 1 in the free-draining limit. This limit is expected for small values of n, however, and does not explain a > 0.5 for high molecular weight polymers. 

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. 

In the free-draining limit of no hydrodynamic interactions, we have... 

Although/ A in both the non-free-draining limit for low salt solutions and the free-draining limit, the terms appearing as prefactors are qualitatively different. 

For K 7 0, a part of Aqo cancels 1 jft exactly in the non-free-draining limit and the remainder is dependent on the structure factor of the polymer and the size exponent v. For large values of KRg, p becomes... 

Kirkwood and Riseman have developed a theory that allows for variable degrees of solvent drainage through the coil domain. We shall not go into this theory in any detail, except to note that it should reduce to Equation (87) in the free-draining limit and to the Einstein equation in the nondraining limit. The Kirkwood-Riseman theory can be written in the form... 

In recent articles1,2 on the dynamics of stiff polymer chains, the Langevin version of Brownian motion theory was used instead of the more common Fokker-Planck approach. These investigations were made, however, only in the free-draining limit. 

Dean P (1967) Atomic vibration in solids. J Inst Math Appl 3 98—165 De Gennes PG (1967) Quasi-elastic scattering of neutrons by dilute polymer solutions I. Free-draining limit. Physics 3(1) 37—45... 

It should be pointed out that in the Zimm, or impermeable coil, limit the specific value of R ff is uninfluential since the friction coefficient C( ) = C/v( ) is independent of R ff, see Eqs. (3.1.7 ) and (3.1.9), v( ) > 1. This is not true in the partial-draining case, in which v(q) is of order unity although larger than the Rouse free-draining limit v(q) = 1, and its full expression must be considered. 

Solutions. Four sets of data indicate that the non-free draining limit of the Zimm theory (29) describes low to moderate frequency dependence of intrinsic complex modulus for linear polymers in 0-solvents. Figure 3.2 shows the result for PIB in benzene at 24° C, the 0-temperature for this system (2,92). In this figure, G R and GR are... 

For the bead-spring model, a/b has to be smaller than 1/2 to avoid the interpenetration of the neighboring spheres. The value of ft described by Eq. (3.1) satisfies this criterion and so is consistent with the model on which the theory is based However, it should be noted that this favorable result is obtained within the framework of the Zimm theory. The value of ft at the non-free draining limit is 1/4 for Gaussian chains but it is different from 1/4 for chains of other distribution. Moreover,... 

The segmental motion of a polymer chain was successfully described by a bead-spring model, discussed by Rouse [17] in the so-called free-draining limit and by Zimm [18] in the hydrodynamic limit, de Gennes [19,20] calculated the coherent and incoherent intermediate scattering functions for both the Rouse and Zimm models. In the low Q and long time limit, the time decay of the intermediate scattering function depends on and and the Q dependence of the... 

So far in this review, we have confined our attention to dense melts, where we found good agreement to the reptation model. For short times, however, not all the data fit to the Rouse model perfectly. One way to examine this in more detail is to study crossover from solution to melt in the free draining limit as a function of density. Experimentally this certainly is not possible, because of the effects of hydrodynamics, which influence the dynamics very strongly. The bond fluctuation algorithm was used because at the relatively low densities of interest the MD is not as suitable. ... 

The Rouse free-draining limit is based on a local response of the monomers to external forces and ignores any long range hydrodynamic interactions. However, it is well known that the motion of each monomer creates a backflow velocity field in the solvent... 

The Rouse model assumes unperturbed chains in the free-draining limit (the limit of no hydrodynamic interactions between monomers) that can be represented as a set of beads connected along the chain contour. The Zimm model, in addition to the Rouse model features, takes into account the hydrodynamic interaction of the beads. 

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