Single-chain mean-field theory

In practice, the scheme as explained above is not implemented. The consecutive generation of all possible chain conformations is a very expensive step. The reason for this is that there are of the order of ZN number of conformations, where Z is the lattice coordination number. A clever trick is to generate a subset of all possible conformations and to use this set in the SCF scheme. This approach is known in the literature as the single-chain mean-field theory, and has found many applications in surfactant and polymeric systems [96]. The important property of these calculations is that intramolecular excluded-volume correlations are rather accurately accounted for. The intermolecular excluded-volume correlations are of course treated on the mean-field level. The CPU time scales with the size of the set of conformations used. One of the obvious problems of this method is that one should make sure that the relevant conformations are included in the set. Typically, the set of conformations is very large, and, as a consequence, the method remains extremely CPU intensive. 

Some very powerful theories that have been applied to study grafted polymer layers are the self-consistent field (SCF) approaches, " and the single-chain mean-field (SCMF) theory.Both approaches are based on the same ideas (even though they have been originally derived in very different ways) and the main difference between them is in the degree of accuracy with which the chain configurations are treated (see below). A recent detailed review of the SCMF theory and the differences and similarities with the SCF approaches can be found in Ref. 12. 

In what follows, we use simple mean-field theories to predict polymer phase diagrams and then use numerical simulations to study the kinetics of polymer crystallization behaviors and the morphologies of the resulting polymer crystals. More specifically, in the molecular driving forces for the crystallization of statistical copolymers, the distinction of comonomer sequences from monomer sequences can be represented by the absence (presence) of parallel attractions. We also devote considerable attention to the study of the free-energy landscape of single-chain homopolymer crystallites. For readers interested in the computational techniques that we used, we provide a detailed description in the Appendix. ... 

The most essential step in a mean-field theory is the reduction of the many-body problem to a scheme that treats just a small number of molecules in an external field. The external field is chosen such that it mimics the effect of the other molecules in the system as accurately as possible. In this review we will discuss the Bragg Williams approach. Here the problem is reduced to behaviour of a single chain (molecule) in an external field. Higher order models (e.g. Quasi-chemical or Bethe approximations) are possible but we do not know applications of this for bilayer membranes. 

We now discuss the case where xab is positive but small and start again horn the dilute end. When we reach the overlap concentration the factor Xab is not strong enough to induce segregation. We may still increase the concentration, keeping a single phase, which is a semi-dilute mixture of A and B chains. The interaction Xab >s then a weak perturbation superimposed on a familiar problem—the problem of Ch q>ter HI. As seen in this chapter, the probability of contact between two monomers is much smaller than predicted by mean field theory the reduction amounts to a factor of where d> is the total concentration of monomers (4> = A + b)> This applies in particular to the AB contacts. Their number is reduced by Thus we are led to use the mean field theory with a renormalized segregation factor ... 

In a more detailed calculation the self-consistent mean field theory reduces the problem of calculating the interactions among polymer chains to that of a single noninteracting polymer chain placed in an external field self-consistent with the composition profiles (26). Again, the primary objective is to compute the free energy and polymer distribution functions near the order-disorder transition. 

In the case of general chain architectures, however, the mean fleld problem of a single chain in an external fleld cannot be cast in the form of a modified diflusion equation, and the density that a single chain creates in the external field and the concomitant single-chain partition function have to be estimated by partial enumeration [30-34], This methodology has been successfully applied to study the packing of short hydrocarbon chains in the hydrophobic interior of lipid bilayers [31,32,34] and polymer brushes [33] and to quantitatively compare the results of Monte Carlo simulations to the predictions of the mean field theory without adjustable parameters [30], The latter application is illustrated in Figure 5.2. 

However, there are cases where fluctuations in the local monomer concentration become unimportant. The adsorption behavior of a single polymer chain is then obtained using simple mean-field theory arguments. Mean-field theory is a very useful approximation applicable in many branches of physics, including polymer physics. In a nutshell, each monomer is placed in a field , generated by the averaged interaction with all the other monomers. 

Even with all the simplifying assumptions, the emerging physical picture is quite rich and robust. Adsorption of polymers from dilute solutions can be understood in terms of a single-chain adsorption on the substrate. Mean-field theory is quite successful, but in some cases fluctations in the local monomer concentration play an important role. Adsorption from more concentrated solutions offers rather complex and rich density profiles, with several regimes (proximal, central, distal). 

To find the region of validity of the mean field theory we have to use the mean field exponents rather than the exact ones, so that we have to use the values in equation (16). The correlation length for vulcanization is certainly proportional to the typical size of a single chain of which the... 

Consider a single crosslinked cluster, where the crosslinks can be everywhere along the chains. The mean field theory of Flory and Stockmayer calculates for the size of the cluster the typical N law, where N is the total amount of monomers in the cluster. This law is true in ideal noninteracting systems for any kind of branched molecule or lattice animal. " The cluster of size R has then... 

Logarithmic corrections to the mean-field theory of single chains are known, for example, for the finite-chain Boyle temperature Tb(N), where the second virial coefficient vanishes. The scaling of the deviation of Tb(N) from T reads [124] ... 

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