Free draining chains

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 ... 

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. 

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

We may note in passing that the intrinsic viscosity of a fully extended rod molecule, for which is proportional to the square of the length, should depend on the square of the molecular weight, in the free-draining approximation. In a more accurate treatment which avoids this approximation, the simple dependence on is moderated by a factor which depends on the effective thickness of the chain (or bead density along the chain) compared with the chain length. 

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,... 

Notice that the left-hand side is now energy dissipation per unit volume. Comparing this result with Equation (3) shows that the increment in viscosity caused by the free-draining chain is given by the coefficient of (dv/dy)2 and may be written... 

Note that h is proportional to n1/2 in 0-solvents, and thus to N112. For 0 = 0 the flow disturbance is zero, the chain is said to be free draining, and the original Rouse model is recovered. For hP, flow in the coil interior is presumed to be substantially reduced, the chain is frequently said to behave as an impenetrable coil, and the Zimm model is obtained. Equations (4.10-4.12) continue to apply for all values of h, although the distribution of relaxation times depends on h. Some results for the two limiting cases and large N are ... 

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. 

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