Frictional Properties of Polymer Molecules in Dilute Solution

FRICTIONAL PROPERTIES OF POLYMER MOLECULES IN DILUTE SOLUTION 

When a polymer molecule moves in a dilute solution it undergoes frictional interactions with solvent molecules. The nature and effect of these frictional interactions depend upon the size and shape of the polymer molecule. Thus, the chain dimensions of polymer molecules can be evaluated from measurements of their frictional properties [25]. 

In a laminar flow at a definite shear rate, different parts of the polymer molecule move at different rates depending on whether they are in the zone of rapid or relatively slow flow, and as a result the polymer molecule is under the action of a couple of forces which makes it rotate in the flow. Rotation and translational movement of polymer molecules causes friction between their chain segments and the solvent molecules. This is manifested in an increase in viscosity of the solution compared to the viscosity of the pure solvent. 

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 

A nondraining polymer molecule, also referred to as the impermeable coil, can be represented by an equivalent impermeable hydrodynamic sphere of radius R. The frictional coefficient of this sphere which represents the frictional coefficient of the non-draining polymer coil can thus be written, 

6 Frictional Properties of Polymer Molecules in Dilute Solution 

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. 

4 Frictional properties of polymer molecules in dilute solution 

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. 

A non-draining polymer molecule can be represented by an equivalent impermeable hydrodynamic particle, i.e. one which has the same frictional coefficient as the polymer molecule. Thus a non-draining random coil can be represented by an equivalent impermeable hydrodynamic sphere of radius Rh- From Stokes Law 

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The viscosity method makes use of the fact that the exponent, a, in the Mark-Houwink equation (see Frictional Properties of Polymer Molecules in Dilute Solution), rj = KM° , is equal to 0.5 for a random coil in a theta-solvent. A series of polymers of the same type with widely different known molecular weights is used to determine intrinsic viscosities [t ] at different temperatures and hence a at different temperatures. The theta-temperature can thus be determined either by direct experiment or, if it is not in the measurable range, by calculation. 

In the present chapter we shall be concerned with quantitative treatment of the swelling action of the solvent on the polymer molecule in infinitely dilute solution, and in particular with the factor a by which the linear dimensions of the molecule are altered as a consequence thereof. The frictional characteristics of polymer molecules in dilute solution, as manifested in solution viscosities, sedimentation velocities, and diffusion rates, depend directly on the size of the molecular domain. Hence these properties are intimately related to the molecular configuration, including the factor a. It is for this reason that treatment of intramolecular thermodynamic interaction has been reserved for the present chapter, where it may be presented in conjunction with the discussion of intrinsic viscosity and related subjects. 

The various physical methods in use at present involve measurements, respectively, of osmotic pressure, light scattering, sedimentation equilibrium, sedimentation velocity in conjunction with diffusion, or solution viscosity. All except the last mentioned are absolute methods. Each requires extrapolation to infinite dilution for rigorous fulfillment of the requirements of theory. These various physical methods depend basically on evaluation of the thermodynamic properties of the solution (i.e., the change in free energy due to the presence of polymer molecules) or of the kinetic behavior (i.e., frictional coefficient or viscosity increment), or of a combination of the two. Polymer solutions usually exhibit deviations from their limiting infinite dilution behavior at remarkably low concentrations. Hence one is obliged not only to conduct the experiments at low concentrations but also to extrapolate to infinite dilution from measurements made at the lowest experimentally feasible concentrations. 

This treatment assumes that the forces between molecules in relative motion are related directly to the thermodynamic properties of the solution. The excluded volume does indeed exert an indirect effect on transport properties in dilute solutions through its influence on chain dimensions. Also, there is probably a close relationship between such thermodynamic properties as isothermal compressibility and the free volume parameters which control segmental friction. However, there is no evidence to support a direct connection between solution thermodynamics and the frictional forces associated with large scale molecular structure at any level of polymer concentration. 

Hydrodynamic properties, such as the translational diffusion coefficient, or the shear viscosity, are very useful in the conformational study of chain molecules, and are routinely employed to characterize different types of polymers [15,20, 21]. One can consider the translational friction coefficient, fi, related to a transport property, the translational diffusion coefficient, D, through the Einstein equation, applicable for infinitely dilute solutions ... 

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