Intermolecular excluded volume

In practice, the scheme as explained above is not implemented. The consecutive generation of all possible chain conformations is a very expensive step. The reason for this is that there are of the order of ZN number of conformations, where Z is the lattice coordination number. A clever trick is to generate a subset of all possible conformations and to use this set in the SCF scheme. This approach is known in the literature as the single-chain mean-field theory, and has found many applications in surfactant and polymeric systems [96]. The important property of these calculations is that intramolecular excluded-volume correlations are rather accurately accounted for. The intermolecular excluded-volume correlations are of course treated on the mean-field level. The CPU time scales with the size of the set of conformations used. One of the obvious problems of this method is that one should make sure that the relevant conformations are included in the set. Typically, the set of conformations is very large, and, as a consequence, the method remains extremely CPU intensive. 

We begin by formulating the free energy of liquid-crystalline polymer solutions using the wormlike hard spherocylinder model, a cylinder with hemispheres at both ends. This model allows the intermolecular excluded volume to be expressed more simply than a hard cylinder. It is characterized by the length of the cylinder part Lc( 3 L - d), the Kuhn segment number N, and the hard-core diameter d. We assume that the interaction potential between them is given by... 

In dilute solution, all macromolecular chains undergo interactions with each other resulting in the so-called intermolecular excluded volume effect, corresponding to the intermolecular potential. This effect is also observed if one does not assume particular cohesive forces to occur between the macromolecular chains. Under these conditions, the second virial coefficient is calculated from the equation1,2) ... 

There is an intermolecular excluded volume effect in a good" solvent... 

Taking these in order, we will first consider the intermolecular excluded volume effect. Essentially, this can be thought of as the chains repelling one another from... 

Results have been obtained for the density distribution, intermolecular excluded volume effect at the interface, the relation between molecular orientation and bulk density, and the relation between molecular orientation and diameter. Only the first one will be discussed here in detail. 

The shape and size of macromolecules together with the segment distribution within these forms determine the excluded volume of the polymer. Compact molecular shapes such as helices, ellipsoids, and spheres have only an external (intermolecular) excluded volume the space occupied by a given volume in space cannot be occupied by others, and so is an excluded volume for other molecules. Coils, on the other hand, with their loose internal structure, also have, additionally, an internal (intramolecular) excluded volume, since the space occupied by one segment is not available to another segment of the same molecule. 

The free energy is separable into two contributions (1) the intra-molecular interactions associated with the helix-coil transition (2) the intermolecular excluded volume interactions which are responsible for any nematic order. The difference in free energy per chain between a helical and randomly coiled molecule arising from intramolecular interactions is -N n s. The intermolecular excluded volume interactions in the mean field Meier-Saupe like approximation [Eqs. (III.2) - (III.6) with V = N] gives a contribution to the free energy difference, (AF) which is sketched in... 

Exponent in interface theory Intermolecular excluded volume u = Stokes terminal velocity Volume fraction V2 = volume fraction of polymer... 

Note, however, that these authors only consider a reaction to be diftusion controlled , if a system cannot reach an equilibrinm on the typical time scale of reaction and local depletion of reactants occurs. For termination reactions the typical time scale of reaction is severely decreased by intermolecular excluded volume effects, the mean time needed for reaction is larger than typical polymer relaxation times and consequently this reaction is not drSiision controlled. It is important to realize that this definition differs from the one used in this thesis. 

The relationship between scattering and intermolecular excluded volume is likewise more eomplex than in the... 

The PRISM theory of Curro and Schweizer extends RISM theory to polymers by considering the intramolecular structure of flexible polymers [14, 15, 16, 69, 70, 17, 18]. The theory assumes that the Flory ideality concept is valid and polymers exhibit ideal behavior in the melt phase. This is justified by the fact that the intramolecular excluded volume is nearly balanced by intermolecular excluded volume when a chain is surrounded by identical chains, so excluded volume forces can be neglected [3, 71]. They also developed a perturbative scheme to account for chain end effects [15]. 

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