Free-draining polymer molecule

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. 

The frictional behavior of real polymer molecules is made of contributions of both free-draining and non-draining polymer molecules represented by Eqs. (3.136) and (3.139), respectively. The free-draining contribution dominates for very short chain or elongated rodlike molecules. 

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.

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 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. 

A more realistic representation of polymer chains is the Rouse model [59], which considers a polymer molecule to be a linear chain of N free-draining beads interconnected by springs (each of time constant Xu = il4H), and predicts Eq. (16) to apply. 

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