Free-draining molecule

Equation (9.44) treats the free-draining molecule as an assembly of independent hydrodynamic units and shows that in this limit [r ] (nf/r o)(rJ/n). 

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

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... 

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. 

If the motion of the molecule is one of translation, as it is during sedimentation in a centrifugal field, the velocity of every bead is the same, and in the free-draining case the difference in velocity Aw, for each bead relative to the solvent is the same as the (relative) translational velocity u of the molecule as a whole. Fig. 138 is illustrative of this case. The total force on the molecule is then... 

In the free-draining case sy/2 is also the relative velocity of the medium in the vicinity of a bead at a distance s from the center. Hence the frictional force acting on the bead is sy/2, and the rate of energy dissipation by the action of the bead is the product of the force and the velocity, or sy/2y. The total energy dissipated per unit time by the molecule will be given by the sum of such terms for each bead, or... 

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. 

The peculiarity of this expression, however, is that it does not make sense for dilute solutions of Gaussian coil molecules. In fact, the free-draining case is characterized by the limit of infinetely smaE friction coefficient . For this case, the contributions of the chain molecules to the viscosity of the solution becomes zero. 

With a finite value of necessarily some intramolecular hydrodynamic interaction or shielding must occur. The importance of eq. (3.53) lies at the present time, in the fact that it can be adapted for concentrated, solvent free systems like polymer melts. As Bueche (13) pointed out, in these systems every chain molecule is surrounded by chain molecules of the same sort. As all these molecules are necessarily equivalent, one cannot speak of a hydrodynamic shielding effect. This would imply that certain chains are permanently immobilized within the coils of other chains. The contrary is expected, viz. that the centre of gravity of each chain wiH independently foHow, in the average, the affine deformation of the medium as a continuum. From this reasoning Bueche deduces that the free-draining case should be applicable to polymer melts. Eq. (3.53) is then used (after omission of rj0) for the evaluation of an apparent friction factor . After introduction of this apparent friction factor into eq. (3.50), the set of relaxation times reads ... 

In these eqs. index 2 stands for two branches, i.e. for the linear molecule. is the mean square radius of gyration which is equal to < h2 >/6 for linear chains of sufficient length. As an example, the values for a star molecule of four equal branches are quoted. One obtains for the free-draining case ... 

Kuhn for statistically coiled molecules. The two dotted lines denoted by F and N stand for the free-draining and the non-draining case of Zimm s theoty for Gaussian coils. The hatched area indicates the area where the experimental points obtained on solutions of anionic polystyrenes are located (See Fig. 3.1). 

The non-free-draining water attached to a polysaccharide molecule and the free-draining water surrounding it travel at different velocities across a boundary whose location is a function of fc and, therefore, a function of Rg and ti0 [Eq. (4.1) Flory, 1953 Tanford, 1961] ... 

Cellulose Molecules in Solution are Partially Free Draining Chains Evidence 1... 

CD molecules are partially free draining chains and the negative temperature dependence of the limiting viscosity number fn] can be attributed to the temperature dependence of the unperturbed chain dimension A. 

In a real situation, the motion of the segments of a chain relative to the molecules of the solvent environment will exert a force in the liquid, and as a consequence the velocity distribution of the liquid medium in the vicinity of the moving segments will be altered. This effect, in turn, will affect the motion of the segments of the chain. To simplify the problem, the so-called free-draining approximation is often used. This approximation assumes that hydrodynamic interactions are negligible so that the velocity of the liquid medium is unaffected by the moving polymer molecules. This assumption was used in the model developed by Rouse (5) to describe the dynamics of polymers in dilute solutions. 

---