Free-draining polymer

Equations (9.42) and (9.46) reveal that the range of a values in the Mark-Houwink equation is traceable to differences in the permeability of the coil to the flow streamlines. It is apparent that the extremes of the nondraining and free-draining polymer molecule bracket the range of intermediate permeabilities for the coil. In the next section we examine how these ideas can be refined still further. 

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... 

A free-draining polymer molecule, referred to as the free-draining coil, is considered by dividing it into identical segments each of which has the same frictional coeflflcient Since solvent molecules permeate all regions of the polymer coil with equal ease (or difficulty), each segment makes the same contribution to / which therefore is given by... 

Size-based separations of homogeneous polyelectrolytes, such as DNA, are not possible in free solution electrophoresis [159]. This is due to the proportionality of the friction hydrodynamic force and total charge of the molecule to its length. The friction hydrodynamic forces exerted on the free-drained polymer coil while it moves as well as the accelerating electrostatic force both increase proportionally with the addition of a nucleotide to the chain. This is why one must typically use a sieving media, such as a gel or an entangled polymer solution, to obtain size-based separations of DNA using electrophoresis. 

An important determinant of (r ) in real polymer solutions is the quality of the solvent (Fig. 2.6). With a very good solvent the solvent-repeat unit interactions are maximized, resulting in a relatively expanded free-draining polymer chain. Conversely, in a very poor solvent the polymer chains are close to their most compact average conformation, behaving similarly to rigid spheres suspended in solution. Both the expanded and contracted... 

An alternative point of view assumes that each repeat unit of the polymer chain offers hydrodynamic resistance to the flow such that f-the friction factor per repeat unit-is applicable to each of the n units. This situation is called the free-draining coil. The free-draining coil is the model upon which the Debye viscosity equation is based in Chap. 2. Accordingly, we use Eq. (2.53) to give the contribution of a single polymer chain to the rate of energy dissipation ... 

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. 

A plot of A versus r, the calibration curve of OTHdC, is shown in Fig. 22.2. The value of constant C depends on whether the solvent/polymer is free draining (totally permeable), a solid sphere (totally nonpermeable), or in between. In the free-draining model by DiMarzio and Guttman (DG model) (3,4), C has a value of approximately 2.7, whereas in the impermeable hard sphere model by Brenner and Gaydos (BG model) (8), its value is approximately 4.89. 

The non-free draining character of flexible polymer chains was considered in the Zimm model [48], In this model, the effect of hydrodynamic interaction at the location of bead i is taken into account by an additional fluid velocity term vj ... 

The Zimm model predicts correctly the experimental scaling exponent xx ss M3/2 determined in dilute solutions under 0-conditions. In concentrated solution and melts, the hydrodynamic interaction between the polymer segments of the same chain is screened by the host molecules (Eq. 28) and a flexible polymer coil behaves much like a free-draining chain with a Rouse spectrum in the relaxation times. 

For the present we consider the case of very small frictional effects due to the beads i.e., the Stokes law radius a is small. We assume that the effects are so small that the motion of the surrounding medium is only very slightly disturbed by the movement of the polymer molecule relative to the medium. The frictional effects due to the polymer molecule are then comparatively easy to treat, for the velocity of the medium everywhere is approximately the same as though the polymer molecule were not present. The solvent streams through the molecule almost (but not entirely) unperturbed by it hence the term free-draining is appropriate for this case. The velocity difference we require in Eq. (11) is simply defined by the motion of the molecule on the one hand and the unperturbed flow of the medium on the other. 

Fig. 138.—The free-draining molecule during translation through the solvent. Flow vectors of the solvent relative to the polymer chain are indicated.

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... 

A free-draining model with equal bead friction coefficients and W = const, such as a stiff bead-spring polymer, may be efficiently simulated with... 

Free-draining models were among the first to be considered [14-18]. For flexible polymer chains of sufficient length, [77] behaves as if the polymer coil occupied a spherical volume through which the solvent cannot flow. Under these conditions,... 

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