The Free-draining Molecule

The rigid sphere is not an accurate picture of how a polymer molecule affects the flow of the fluid in which it is dissolved, because the fluid can penetrate within the molecule. This recognition led to the development of models in which the molecule is represented as a chain of beads, which contain all the mass of the molecule, connected by springs. In the free-draining model of Rouse [45], there is no effect of one bead on the flow pattern around other beads. This model starts from Stokes law, which gives the drag force f on a sphere in a Newtonian fluid flowing past it at the velocity U as proportional to the radius a of the sphere. In terms of the coefficient of friction f =Fj /U), Stokes law for flow past a sphere is  

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 

The motion occurring during viscous flow consists of a shear in which different layers of the solution move with different velocities. The large polymer molecule finds it impossible to adjust its motions so as to coincide with the velocities of the different layers of the liquid through which it extends. Its situation is depicted in Fig. 139, where vectors representing the unperturbed velocity of the liquid relative to the position of the center of gravity of the molecule are shown. Let the velocity gradient be 7  

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 

The factor of 100 has been introduced in the denominator in order to convert to the usual units (g./lOO ml.) for the intrinsic viscosity. For a linear polymer molecule is proportional to Hence, 

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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.

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] ... 

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. 

Rouse s theory is the simplest molecular theory of polymer relaxation. A later theory of Zimm [28] does not assume that the velocity of the liquid solvent is unaffected by the movement of the polymer molecules (the free draining approximation ). The hydrodynamic interaction between the moving submolecules is taken into account and this gives a modified relaxation spectrum. 

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