Long-length-scale correlations

A detailed analysis of the fluctuation spectrum in eqn [91] shows that the bond-bond correlation function indeed can be approximated by eqn [79]. At long length scales, the spectrum of fluctuations of the semiflexible polyelectrolyte chain is that of a semiflexible chain with the effective chain s bending rigidity Xi equal to the sum of the bare chain s bending constant and the OSF contribution... 

It is important to note that in calculation of the intramolecular structure factors for input into PRISM theory, one can include as much (or as little) chemical detail regarding the molecular architecture as desired. For questions regarding intermolecular packing on relatively long length scales (e.g., the so-called correlation hole regime corresponding... 

Hydrodynamic interactions on a long length scale can be measured with two-point rheology, in which fluorescent or other beads are mixed with a polymer solution, and videomicroscopy is used to measure the Brownian displacements AR, of pairs of beads. The cross-correlations AR, ARy) determine the cross-diffusion tensors as a function of the separation between beads. For beads a fraction of a micron in size in polymer solutions and interbead distances out to 100 xm, measurements of Crocker, et al.(25), Gardel, et a/.(26), and Chen, et al.(21) agree the cross-diffusion tensor falls off with distance as /R, and has at least approximately the magnitude expected for the Oseen interaction in these viscous polymer... 

These models provide a simple means of determining the intramolecular correlation function even for multicomponent systems, but the assumption of a freely jointed chain severely limits their predictive capabilities. Although they perform adequately over long length scales, MD simulations have shown that these models are incapable of capturing structural details on a monomeric length scale [76]. A more accurate... 

At first, one would tend to reconsider conventional crossover due to mean-field criticality associated with long-range interactions in terms of the refined theories. Conventional crossover conforms to the first case mentioned—that is, small u with the correlation length of the critical fluctuations to be larger than 0. However, in the latter case one expects smooth crossover with slowly and monotonously varying critical exponents, as observed in nonionic fluids. Thus, the sharp and nonmonotonous behavior cannot be reconciled with one length scale only. 

Now, when spin-spin correlations are not neglected, there should exist a temperature below which a phase transition to a more organized state will occur. If this state is characterized by order on an infinite length scale (i.e. the length scale of the correlations is essentially the dimension of the sample) this is known as long-range order (LRO). The transition that occurs is analogous to the condensation of a gas to form a crystalline solid. We are not concerned here with a detailed discussion of the voluminous literature on phase transitions of this type (also known as critical phenomena). 

The short chains with MRouse chains with relaxation times on the order of the experimental time-scale. For example, the Rouse times of PSD3 and of PSDIO are approximately 9s and 19s seconds, respectively at 115°C [35,36]. Consequently, the residual orientation at long times for these chains can be attributed to orientational coupling interactions with the long chains of the polymer matrix. Similar orientational correlations have been observed on various systems by NMR spectroscopy studies of stretched elastomers where even dissolved solvent molecules and free chains were shown to possess a very short-length scale local orientation [37]. 

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