Coil molecules free draining

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. 

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. 

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 ... 

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). 

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. 

Two extremes of the frictional behaviour of polymer molecules can be identified, namely free-draining and non-draining. A polymer molecule is said to be free-draining when solvent molecules are able to flow past each segment of the chain, and non-draining when solvent molecules within the coiled polymer chain move with it. These two extremes of behaviour lead to different dependences of the frictional coefficient, of a polymer molecule upon chain length. 

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