Real polymer molecules

In order to present clear concepts it is necessary that idealized definitions be adopted but it is reeognized that the realities of polymer science must be faced. Deviations from ideality arise with polymers at both molecular and bulk levels in ways that have no parallel with the ordinary small molecules of organic or inorganic chemistry. Although such deviations are not explicitly taken into account in the definitions below, the terms recommended can usefully be applied to the predominant structural features of real polymer molecules, if necessary with self-explanatory, if imprecise, qualifications such as essentially. , almost completely. , or highly. . Although such expressions lack the rigour beloved by the purist, every experienced polymer scientist knows that communication in this discipline is impossible without them. 

The term in brackets on the left-hand side of Eq. (3-37) is called the intrinsic viscosity or limiting viscosity number. It reflects the contribution of the polymeric solute to the difference between the viscosity of the mixture and that of the solvent. The effects of solvent viscosity and polymer concentration have been removed, as outlined earlier. It now remains to be seen how the term on the right-hand side of Eq. (3-37) can be related to an average molecular weight of a real polymer molecule. To do this we first have to express the volume V of the equivalent hydro-dynamic sphere as a function of the molecular weight A/ of a monodisperse solute. Later we substitute an average molecular weight of a polydisperse polymer for M in the monodisperse case. 

Figure 3-13. Bead-and-spring representation of a real polymer molecule in dilute solution.

A real polymer molecule is not equivalent to the above model as it i.s not freely jointed, and a correction must be introduced to account for the fixed angle between monomeric units. Provided that the geometry of the chain is known, it is possible to calculate the effect of the bond angle. 

The result in equation (4.8) for a freely jointed chain leads to the definition of the characteristic ratio C of a real polymer molecule composed of n bonds. This is... 

The bonds of the polymer molecule have been considered thus far as volumeless lines in space. In contradistinction, the bonds of real polymer molecules occupy flnite volumes in space. The repulsion between segments that are far removed from one another along the contour of the chain but spatially contiguous will cause the polymer chain to expand beyond the unperturbed dimensions. The magnitude of this dimensional expansion is measured by the intramolecular expansion factor a... 

Equation 12.67 predicts that the specific viscosity is proportional to the volume of the equivalent hydrodynamic sphere. The Einstein viscosity relation was derived for rigid spherical particles in solution. However, real polymer molecules are neither rigid nor spherical. Instead the spatial form of the polymer molecule in solution is regarded as a random coil. Theories based on this characteristic form of polymer molecules have resulted in the expression... 

Conformational States of Real Polymer Molecules in the Solid State... 

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. 

There are two obvious objections to the application of the freely orienting chain molecule when applied to real polymer molecules. 

In the models for polymer chain conformation that we have considered so far, the polymer chain is allowed to intersect itself, because each link is a vector that takes up no volume. This is clearly unrealistic for real polymer molecules, where the segments occupy a certain volume and the chain cannot cross itself. This leads to excluded volume, which cannot be occupied by other segments. Polymer coils which have excluded volume are said to be perturbed, whereas (r )J gives the unperturbed dimensions of the coil assuming volumeless links. The perturbed dimensions (r ) / are related to the unperturbed dimensions by the expansion factor, a ... 

As the structures of polymers were elucidated and the authors started to report them, the need arose for a structure-based nomenclature. Still, it has to be remembered that in many cases structural representations are idealised. Deviations from ideality have to be ignored in constructing a useful nomenclature which still applies to the predominant structure of real polymer molecules. 

In this section I discuss dynamic Monte Carlo methods for generating SAWs in various ensembles. I emphasize that the stochastic dynamics is merely a numerical algorithm, whose goal is to provide statistically independent samples from the desired distribution tt in the smallest CPU time possible. It need not correspond to any real "physical dynamics Indeed, the most efficient algorithms typically make non-local moves which would be impossible for real polymer molecules. 

Hookean dumbbells are infinitely extensible, and real polymer molecules are not this suggests that finitely extensible springs ought to be used, and consequently the FENE dumbbell model has been much studied. The spring force is taken to be... 

Having obtained the average value of the square of the chain end-to-end distance and the distribution of end-to-end values about this mean, it is worth pausing and again asking if there is any relation between these results and results for real polymer molecules. In other words, how closely do freely jointed chains approximate actual macromolecules If the answer is not very closely, then how do we modify the freely jointed chain results to make them apply to polymers ... 

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