Segment excluded volume

Hence, in a good solvent (T > 6 for. systems with UCST), the repulsion component predominates in the potential y>(r) (Figures 1.40a, 2.34b), the segment excluded volume 3 > / o (Equation 136), which leads to an increase in tlie molecular coil size in dilute solutions (see Equations 157, 113, 153, 115). 

Ill dilute solutions, no compensation of the segment excluded volume by other macro-molecules takes place, so the coil size R determines the upper boundary of As the polymer concentration rises, may, in principle, diminish to the linear sizes of the segment excluded volume... 

In the above considerations, only binary interactions of m2icromoleculc(s) segments were taken into account in the state equations. The measure of the intramolecular pair interactions is the segment excluded volume 0 (the binary cluster integral, Equation 120). 

As was repeatedly emphasized, away from the phase separation region (i.e. at high temperatures for systems with a UCST), the intersegment interaction in a polymer chain reduces to the existence of segment excluded volume, which manifests itself by swelling (expansion) of polymer coils. These interactions also lead to a correlation of the segment density fluctuations as it occurs in general-type systems near the critical point. 

In a very good solvent, a chain s segments are repelled from each other, which is formalized in the existence of segment excluded volume at the intramolecular level and of polymer coil excluded volume at the intermolecular level. In this case, very long chauns obey the universal scaling laws characteristic of the proximity of the critical point in general-type systems. 

First-order perturbation calculations of the mixed second virial coefficients A have been carried out for chains differing in molecular weight but having the same architecture and chemical structure (so that the segment-segment excluded volume parameter is the same for all polymer species), and for some combinations of topologically different chains, e,g. linear chains with rings, stars or combs, stars with stars of different functionality, stars with combs, The simplest result is that for... 

Use of random flight statistics to derive rg for the coil assumes the individual segments exclude no volume from one another. While physically unrealistic, this assumption makes the derivation mathematically manageable. Neglecting this volume exclusion means that coil dimensions are underestimated by the random fight model, but this effect can be offset by applying the result to a solvent in which polymer-polymer contacts are somewhat favored over polymer-solvent contacts. 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

Two segments of a given polymer molecule cannot occupy the same space and, indeed, experience increasing repulsion as they move closer together. Hence the polymer has around it a region into which its segments cannot move or move only reluctantly, this being known as the excluded volume. The actual size of the excluded volume is not fixed but varies with solvent and temperature. 

Proceeding in a manner paralleling the derivation of the excluded volume for a pair of molecules, we consider the polymer molecule to consist of a swarm of segments distributed on the average about the molecular center of gravity in accordance with the Gaussian formula (see Eq. XII-51). This spatial distribution in the unperturbed molecule, as it would exist on the average in the total absence of inter-... 

For a chain with excluded volume the probability of the chain avoiding one segment of volume Vc is Pex(R) — 1—Vc/R3. The total number of combinations of two chain units is n(n—l)/2 nz/2 so the total probability of exclusion is,... 

Fig. 10 Schematic diagram for the explanation of spinodal decomposition due to the orientation fluctuation of stiff segments occurring in the induction period prior to crystallization [19]. On the basis of Doi s kinetic theory [24,25] Vexcl> excluded volume b, nearly equal to the diameter d of the rod or stiff segment L, rod length (9, angle between neighboring rods v, critical stiff segment concentration...

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