Models for Polymer Chains

If we turn from phenomenological thermodynamics to statistical thermodynamics, then we can interpret the second virial coefficient in terms of molecular parameters via a model. We pursue this approach for two different models, namely, the excluded-volume model for solute molecules with rigid structures and the Flory-Huggins model for polymer chains, in Section 3.4. 

MSI) that uses the same time-dependent Ginzburg Landau kinetic equation as CDS, but starts from (arbitrary) bead models for polymer chains. The methods have been summarized elsewhere. Examples of recent applications include LB simulations of viscoelastic effects in complex fluids under oscillatory shear,DPD simulations of microphase separation in block copoly-mers ° and mesophase formation in amphiphiles, and cell dynamics simulations applied to block copolymers under shear. - DPD is able to reproduce many features of analytical mean field theory but in addition it is possible to study effects such as hydrodynamic interactions. The use of cell dynamics simulations to model non-linear rheology (especially the effect of large amplitude oscillatory shear) in block copolymer miscrostructures is currently being investigated. ... 

The reason why the random flight model has proved so popular theoretically stems from its simplicity, which offers hope for the development of analytic solutions. The problem can usually be cast in the form of a diffusionlike or a Schrodinger-wave-equation-like differential equation, the solutions of which are reasonably well explored. A tendency has developed in recent times to apply extremely sophisticated mathematical procedures to what are really very primitive models for polymer chains (see, e.g. Levine et al., 1978). Whether the ends merit the means in such instances cannot yet be assessed objectively. A strategy that might be more productive in terms of the development of a practical theory for steric stabilization is to aim for a simpler mathematical description of more complex models of polymer chains. It should also be borne in mind in developing ab initio theories that a simple model that may well suffice in polymer solution thermodynamics may be quite inadequate for the simulation of the conformational properties of polymers. Polymer solution thermodynamics seem to be relatively insensitive to molecular architecture per se whereas the conformation of a polymer chain is extremely sensitive to it. 

Models for polymer chains The early theories of steric stabilization 10.5.1 Loss of configurational entropy theories... 

In the first part of this article the review of various theoretical models for polymer chains is given. The models of freely jointed chains, freely rotating chains (including wormlike chains), and chains with fixed bond angles and independent rotational potentials and with interdependent potentials, including rotatimial isomeric state approximation, are presented. 

Thus, the comprehensive cycle of investigations considered can be interpreted in terms of the reptation motion model for polymer chains. The large scale and systematic character of these investigations must play an important part in the final elucidation of the dynamic properties of polymer systems. 

THE "KATZ EFFECT ON THE RANDOM COH. MODEL FOR POLYMER CHAINS... 

The x-ray patterns of natural rubber in the relaxed and extended states led J.R. Katz and others to develop a random coil model for polymer chains. The "Katz Effect" which was repeated by H. Mark helped to establish a relationship between mechanical deformation and concomitant molecular events in all macromolecules. 

Why is there a need for these many different algorithms at all Of course, there would be no need for these algorithms if one wanted to simulate only the simple random walk of Fig. la. But in fact, this is not a useful model for polymer chains in most cases, since it allows arbitrarily many effective monomers to sit on top of each other, ignoring excluded volume interactions completely. A slightly better variant of the RW is the NRRW, Fig. lb, where immediate reversals are forbidden, but it is possible that loops are formed where the NRRW crosses itself. The NRRW is still straightforward to program—at each step there are z — 1 choices to proceed, from which one... 

Bassler (23) hous discussed an alternative model for polymer chain propagation. The approach is based on the esq>erimental fact that the activation energy for TS for the addition of a monomer increases as the length of the radical, oligomer intermediates increases (24). 

Fig. 6. Schematic representation of the spring model for polymer chain The mean end-to-end distance is...

In the models for polymer chain conformation that we have considered so far, the polymer chain is allowed to intersect itself, because each link is a vector that takes up no volume. This is clearly unrealistic for real polymer molecules, where the segments occupy a certain volume and the chain cannot cross itself. This leads to excluded volume, which cannot be occupied by other segments. Polymer coils which have excluded volume are said to be perturbed, whereas (r )J gives the unperturbed dimensions of the coil assuming volumeless links. The perturbed dimensions (r ) / are related to the unperturbed dimensions by the expansion factor, a ... 

There is no unique way to construct coarse-grained models of polymer systems. In fact, the choice of model very much depends on the physical problems that one may wish to address, and also many details are fixed from the desire to construct computationally efficient simulation algo-rithms. Thus many variants of models for polymer chains exist, and there may still be the need to invent new ones This section is also not intended to be exhaustive, but rather restricts attention only to the most common models which are widely used in various contexts, as will become apparent in later chapters of this book. 

---