Chain chemically realistic model

Another strong point of the simulation approach is its ability to selectively change parts of the model Hamiltonian. In this way one can compare a chemically realistic model of PB with a freely rotating chain version of the same polymer and does not have to switch to a completely different polymer with some of the same properties like is unavoidable in experiments [33]. With this approach we could establish that identical structure on the two-body correlation function level (single chain and liquid structure factors) does not imply identical dynamics which raises questions on the applicability of the mode-coupling theory of the glass transition to polymer melts. 

An attractive virtue of PRISM theory is the ability to derive analytic solutions for many problems if the most idealized Gaussian thread chain model of polymer structure is adopted. The relation between the analytic results and numerical PRISM predictions for more chemically realistic models provides considerable insight into the question of what aspects of molecular structure are important for particular bulk properties and phenomena. Moreover, it is at the Gaussian thread level that connections between liquid-state theory and scaling and field-theoretic approaches are most naturally established. Thus, throughout the chapter analytic thread PRISM results are presented and discussed in conjunction with the corresponding numerical studies of more realistic polymer models. 

The second contribution to g(r) in Eq. (3.2) is called the correlation hole effect by deGennes and is associated with the longer wavelength universal aspects of chain connectivity and interchain repulsive forces. On intermediate length scales it has a critical power law form due to chain conformation self-similarity, and this simple analytic form remains an excellent representation even for chemically realistic models when intersite separations exceed an intrinsic (/V-independent) distance of the order of 3-5 site diameters... 

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

Coarse-grained polymer models neglect the chemical detail of a specific polymer chain and include only excluded volume and topology (chain connectivity) as the properties determining universal behavior of polymers. They can be formulated for the continuum (off-lattice) as well as for a lattice. For all coarse-grained models, the repeat unit or monomer unit represents a section of a chemically realistic chain. MD techniques are employed to study dynamics with off-lattice models, whereas MC techniques are used for the lattice models and for efficient equilibration of the continuum models.36 2 A tutorial on coarse-grained modeling can be found in this book series.43... 

The chemically realistic simulations we are discussing have been performed using a united atom representation of PB, which leads to the question How does one actually measure a CH vector reorientation for such a model The answer to this question is to use the trick we discussed in the analysis of the pressure dependence of the melt structure factor of PB. Hydrogen atoms are placed on the backbone carbons at their mechanical equilibrium positions for each structure that has been sampled along the MD trajectory. The CH vector dynamics we are showing in Figure 16 is solely from the backbone reorientations of the chain. 

Fig. 5.1. A chemically realistic description of a polymer chain (bisphenol-A-polycarbo-nate [BPA-PC] in the present example) is mapped approximately onto the bond fluctuation model, by using suitable potentials for the length i of the effective bonds and the angles 0 between them. In this example (3 1 mapping) one chemical repeat unit of BPA-PC containing n = 12 covalent bonds along the backbone of the chain is translated into three effective bonds. From [28]...

Fig. 15. Approximate mapping of a chemically realistic polymer (polyethylene in this example) to the bond fluctuation model on the (simple cubic) lattice. In this coarse-graining one integrates n successive chemical monomers (e.g. n = 3) into one effective monomer which blocks 8 adjacent sites on the simple cubic lattice (or 4 on the square lattice in d = 2 dimensions) from occupation by other monomers. The chemical bonds 1, 2, 3 then correspond to effective bond I, bonds 4, 5, 6 to effective bond II. Some information on the chemical structure can be kept indirectly by using suitable distributions P (9) for the angle between subsequent effective bonds, but so far this has been done for homopolymer melts only [94-99]. In the simplest version of the bond fluctuation model [84-88] studied for blends in d = 3 dimensions [88, 91, 92, 99], bond lengths t are allowed to fluctuate freely from i = 2 to t = v/l0, with t = being excluded to maintain that chains do not cut through each other in the course of the random hops of the effective monomers. From Binder [95]...

The local dynamics is naturally strongly dependent on the exact chemical nature and structure of the polymer one studies. The large scale dynamics, however, is largely universal and is described with the Rouse model whereas for longer chains the tube model and reptation concept is believed to describe the chain dynamics [2]. It is easy to see that no single simulation method can capture the physics of polymer dynamics on all these length and time scales [3]. For situations where we can ignore quantum effects (which can, however, be important in polymer crystals [4]) MD simulations with chemically realistic force fields are the method of choice to study local relaxation. 

While we refer to [90] for a thorough analysis of the complex structural situation, in Figures 3-33 and 3-34 we give two examples of the frequency and vibrational displacements for an isolated GG or an isolated isotactic diad defect in a fully syndiotactic planar zig-zag host chain. It must be noted that gap frequencies are different, but that the vibrational displacements associated to these gap modes involve many chemical units ( 20). The complexity of the problem is shown in Figure 3-35 where the density of vibrational states of a realistic model of configurationally disordered PVC is compared with the experimental spectrum. 

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