Intramolecular excluded volume theory

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. 

Before proceeding to a review of both scaled particle theory and fuzzy cylinder model theory, it would be useful to mention briefly the unperturbed wormlike (sphero)cylinder model which is the basis of these theories. Usually the intramolecular excluded volume effect can be ignored in stiff-chain polymers even in good solvents, because the distant segments of such polymers have little chance of collision. Therefore, in the subsequent reference to wormlike chains, we always mean that they are unperturbed . 

These two theories neglect the intramolecular excluded volume effect, the multiple contacts between a pair of chains [41], and the reptation-like motion so that their applicability is essentially limited to stiff-polymers. Many favorable comparisons of their predictions with experimental results seem to justify these approximations to stiff polymer systems. 

Improvements upon the theories require more detailed treatment of intramolecular as well as intermolecular interactions. As we mentioned in the previous section, use of Mayer /-functions has been made by Yamakawa and others to take intramolecular excluded volume effects into consid ation. However, in their calculation, parts of macromolecules between two consecutive contact points of two molecules are replaced always by Gaussian-free chains. While this approximation may be correct for a small number of contact points for very long molecules, it certainly invalidates the /-function expansion itself for higher orders. On the other hand, our results in the previous section indicate clearly that collective interactions of two macromolecules play important roles in explaining the molecular weight dependence of Ag. 

Close examination of the radii of ration from the molecular dynamirs simulations reveals that the chains are not completely ideal Overall the chains exhibit nearly ideal scaling behavior for whidi R oc Locally, however, the chains are found to be expanded relative to a freely jointed chain of the same length. This local expansion is a result of local intramolecular excluded volume which has not been completdy screened out in the melt. Thus in order to predict accurately the intermolecular structure one needs to correct for local deviations from Flory s ideality hypothesis. We were able to make this correction by employing the ra(k) directly computed from the molecular dynamics simulation. When the PRISM theory is then used to calculate g(r) from the actual, simulated d>(kX excellent agreement is seen in Fig. 3 between the theory and the simulation... 

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]. 

According to the aforementioned two-parameter theory, intramolecular excluded-volume effects in non theta solvents vanish only at the limit of N = 0. However, this cannot be the case for actual polymers because the probability of monomer-monomer collision in a single molecule should become essentially zero at a certain nonzero N, and thus the theory is inapplicable to stiff chains and short flexible chains. 

Formation of short branches in LDPE is investigated. It is assumed that the probability of an intramolecular rearrangement of the Roedel type is proportional to the probability that the reacting groups are separated by a distance r r and adhere sufficiently closely to "three in a line" geometry. Excluded volume effects are ignored. The calculations rationalize many of the structural features observed in LDPE. For the present purpose, probabilities must be evaluated by using branched-molecule RIS theory. 

A complication arising from the extension of the theory to flexible macromolecules is that in general, the intermolecular and intramolecular radial distribution functions depend on each other.In modeling the bulk of a one-phase polymer melt, however, the situation resolves itself because the excluded volume effect is insignificant under these conditions the polymer chains assume unperturbed dimensions (see also the section on Monte Carlo simulations by Corradini, as described originally in Ref. 99). One may therefore calculate the structure of the unperturbed single chain and employ the result as input to the PRISM theory to calculate the intermolecular correlation functions in the melt. 

Recent theory suggests that cyclization phenomena (1) are much more sensitive to excluded volume effects than other properties of polymer chains. Intramolecular fluorescence quenching processes in molecules containing appropriate end groups permit one to study both the dynamics and thermodynamics of end-to-end cyclization. As a consequence, the sensitivity of polymer cyclization to excluded volume can be examined. 

In several theories, the effects of excluded volume are combined into a static expansion factor a relating the total, experimentally measured mean square radius of gyration <5 > to the value with no long-range intramolecular forces accounted for,... 

Theories of chain expansion use to account for intramolecular segment-segment volume exclusion and relate expansion parameters to through the excluded volume parameter z. These theories show that 1 and Og 1 as 0-... 

An analysis of the intramolecular dynamics in terms of the Rouse modes yields non-exponentially decaying autocorrelation functions of the mode amphmdes. At very short times, a fast decay is found, which turns into a slower exponential decay which is well fitted by Ap exp(-f/Tp), see Fig. 13. Within the accuracy of these calculations, the correlation functions exhibit universal behavior. Zimm theory predicts the dependence Tp for the relaxation times on the mode number for polymers with excluded-volume interactions [6]. With v = 0.62, the exponent a for the polymer of length Am = 40 is found to be in excellent agreement with the theoretical prediction. The exponent for the polymers with Am = 20 is slightly larger. 

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