Effective segment

To close this section we calculate the effective segment size... 

Now such terms also result from one-body potentials. Recall the general decomposition (2.9) of the effective segment potential ... 

It is clear a priori that it does not make sense to choose r > Rg, since then the coil would be smaller than an effective segment. To leading order of bare perturbation theory Rg is given by the expression for a noninteracting chain R 2N (Eq. (3,32) d = 3), where we neglect polydispersity effects. Renormalizing this expression we find... 

Here the exponential weighting factors represent gaussian statistics (gyration radii = bf N / 6, where we allow different effective segment lengths bA, bB for... 

It is noteworthy that Eq. (3.1), as well as the other results of this section, can be applied not only to the model of freely jointed segments but also to any other model of semiflexible macromolecules it is necessary only to replace p in all equations by the ratio of the effective segment length to its width. In fact, the translational entropy... 

Expression (5.14) gives the operator g for the persistent model. A similar operator has already been used for the analysis of the persistent coils in Ref.35-37 1Z. In particular, in these references it was shown that the effective segment of the persistent chain described by the operator (5.14) is equal to... 

It does not affect the exponents in the equation (i.e.. the dependence of on N), however, but simply introduces a prefactor. This suggests a different approach, where we consider the number of adjacent bonds whose combinations of allowed rotations essentially behave like a freely jointed unit when taken collectively. We would then have Nx effective segments each of length lp known as the Kuhn segment length (Figure 8-36), defined in Equation 8-12 ... 

In the vicinity of the phase coexistence, p = Pcoex(T), the distribution P p) exhibits two pronounced peaks that are separated by a deep valley . The corresponding free energy profile, F p) = —kBT x P p), for polymers that are comprised of At = 10 effective segments and temperature ksT/e = 1.68, is presented in Fig. 13. 

Fig. 22. (a) Snapshot of an interface between two coexisting phases in a binary polymer blend in the bond fluctuation model (invariant polymerization index // = 91, incompatibihty 17, linear box dimension L 7.5iJe, or number of effective segments N = 32, interaction e/ksT = 0.1, monomer number density po = 1/16.0). (b) Cartoon of the configuration illustrating loops of a chain into the domain of opposite type, fluctuations of the local interface position (capillary waves) and composition fluctuations in the bulk and the shrinking of the chains in the minority phase. Prom Miiller [109]... 

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