Chain real /self-interacting

Besides this basic aspect of the heterogeneous character of the chain, real polymers are complex objects on their own, typically fluctuating in a solvent and interacting with the molecules of the solvent as well as with other portion of themselves, the so called self-interaction or excluded volume interaction (two classical references on polymers and polymer models are [Flory (1953)] and [de Gennes (1979)]). However, in simplified models, polymers are often reduced to random walk paths on a lattice and the self-interaction is modeled by the self-avoiding constraint a random walk path, self-... 

Interaction ofthe electrons in the framework of the self-consistent field approximation is accounted for by considering the induced density fluctuations as a response of independent particles to Oext + Poissons equation [2], This means, physically, that collective excitations of the electrons can occur, taken into account via a chain of electron-holeexcitations. These collective excitations show up in S(q, ) as a distinct energy loss feature. Figure 2 shows the shape of the real and imaginary parts of the dielectric function in RPA (er(q, ), Si(q, )) and the resulting dielectric response... 

The upper part of the phase diagram (Fig. 5.1) corresponds to good solvents. At low concentrations, polymer coils are far from each other and behave as isolated real chains (see Section 3.3.1.1). At temperatures for which the excluded volume interaction within each chain exceeds the thermal energy kT, they begin to swell. The Flory theory prediction for the size of swollen real chains with excluded volume v > /y/N is ihs, same as the result for a self-avoiding walk of thermal blobs [Eq. (3.77)] ... 

A number of theoretical models use a single-chain approach to simulate topological constraints in real polymer networks. The basic idea is that one starts from the statistical mechanics of a single network chain which is subjected to a spatial domain of constraints. The constraining potential is introduced in a heuristic manner and cannot be calculated within the frame of the chosen model self-consistently. Hence, the strength of the topological interaction must be characterized by best-fit parameters of the model. 

Note that the sharpness of the transition in the change on going from the depletion zone to the parabolic zone is due to limitations in the analytical function and does not reflect real transition behaviour. The more gentle transitions indicated in the theoretical SCF profile are more realistic. In the self-consistent field calculation a lattice model is not presumed the volume fraction of the tethered chains is calculated from a diffusion equation that involves polymer propagators and a (z-dependent) potential function that includes enthalpic interactions between the two copolymer blocks and between each block and the solvent. Initially the potential function is set to zero, the pol)nner propagators are calculated and then the volume fraction variation of the tethered block. A new potential is calculated from this volume fraction profile and the process reiterated imtil the difference in volume fraction profiles calculated by sequential iterations is smaller than some defined tolerance. The approach bears similarities to the SCF approach of Shull (1991) but makes no allowance for the dry brush case, i.e. that in which the relative molecular mass of the solvent approaches that of the tethered polymer molecule. 

Single-chain-in-mean-field (SCMF) simulation [40-42, 86] is an approximate, computational method that retains the computational advantage of self-consistent field theory but additionally includes fluctuation effects because, in contrast to self-consistent theory, SCMF simulations aim at preserving the instantaneous description of the fluctuating interactions of a segment with its environment. In this partide-based simulation technique, one studies an ensemble of molecules in fluctuating, real, external fields. The explicit particle coordinates are the degrees of freedom and not the collective variables, densities and fields. 

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