Modeling flexible polymers with constraints

In the following, we first want to consider linear, flexible polymers with stiff bonds of unit length (ra+ = 1) and pairwise interactions among nonbonded monomers are modeled by a standard U potential. Thus, the energy of a conformation X reads 

Since we are interested in classifying conformational pseudophases of polymers with respect to their thickness, it is useful to introduce the restricted conformational space 7 = X rge(X) p of all conformations X with a global radius of curvature larger 

The canonical partition function of the restricted conformational space thus reads 

---

We now consider the case where the beads are subject to rigid constraints. This is necessary to deal with the problems of suspensions of a rigid body, or polymers with rigid constraints (such as the rodlike polymer, or the freely jointed model), but the reader who is interested only in flexible polymers can omit this section. 

For displacement updates, the use of spherical coordinates is rather overdone, but spherical updates that serve to satisfy geometric constraints in the model are necessary and will be described in the following. It is an obvious limitation of the displacement update that it can only be used in connection with polymer models that allow for changes of the bond length of bonded monomers. The widely used elastic flexible polymer model with FENE bonds (see Sect. 1.6.4) is an example. If the bond is rather stiff, as is cpiite natural for protein models, the displacement update is not applicable at all. In such cases, the most elementary updates are rotational updates. 

The formulation of the foundations of the statistical mechanics of polymers in bulk is just in its infancy. A number of questions of principle still require careful attention. This review can then only be of a state-of-the-art nature, and only a few very simple applications can be considered. In the next section, a simple model is considered which serves as a zeroth-order model for a system of polymers in bulk. This model, which employs the Wiener integral formulation for flexible chains, has great pedagogical value since it introduces some of the formidable problems to be encountered in any statistical mechanical description of polymers in bulk. This model then naturally leads to a discussion of the nature of statistical mechanics for systems with internal constraints (Section IX). 

While the flow and creep of a polymer are necessarily associated with translational motion, the internal rotational shape change is taking place at the same time. So the total movement is a combination of the two. As before, when polymers approach the ideal flexible model with copious internal rotation, then their behaviour ceases to be sensitive to local conformational constraints and can... 

It is important to observe that only the first term in equation (137) depends on chain architecture and the lattice coordination number, and that both it and the second term cancel out of the entropy of mixing, leaving only the third term equal to Q/Qq. Consequently the theory would apply alike to solutions of flexible and stiff-chain polymers (except that above a certain concentration the latter would respond to packing constraints and minimize the free energy by separation of an ordered phase). The factorization of the combinatorial factors in equation (137) is the fundamental reason why the lattice calculation works at all, despite the extreme artificiality of picturing the chain as fitting a sequence of regular lattice sites with a definite coordination number. These aspects of the model simply disappear in the final result. The independence of the intermolecular factor also implies that the chain conformation should be independent of dilution Rq should be the same in pure liquid polymer as in solution. Naturally this rationale would not hold for dilute solutions, for which the intermolecular factor in equation (137) is not valid. 

---