Self-Consistent Field Theory and Its Applications by M. W. Matsen

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1.1 Gaussian Chain 9 r 2 , the preceding integral can then be expressed in the slightly more convenient form, p m (r) = 2π r ∫ ∞ 0 ∫ r+r1 dr 1 r 1 p m−n (r 1 ) dr 2 r 2 p n (r 2 ) (1.7) |r−r 1| With the recursion relation in Eq. (1.7), it is now a straightforward exercise to march through calculating p m (r) for larger and larger segments. Starting with m =2and n =1, the two-monomer distribution, p 2 (r) = { 1 8πb 2 r , if r
10 1 Self-consistent field theory and its applications 0.3 16 p m (r) a 3 0.2 0.1 8 m=4 0.0 0.0 0.5 1.0 1.5 r/a Figure 1.5: Probability distribution, p m(r), for a coarse-grained segment with m monomers. The dashed curve denotes the asymptotic limit in Eq. (1.11) for m →∞. relation in Eq. (1.5); in other words, if p m−n (r 1 ) and p n (r 2 ) are both Gaussians, then so is p m (r). This universality actually extends beyond freely-jointed chains to a more general class that can be referred to as Markov chains. An equilibrium configuration of a Markov chain can still be generated by starting at one end and attaching monomers one-by-one according to a given probability distribution. However, in this case, the displacement of the i’th monomer, r i , can have a distribution, p 1 (r i ; r i−1 ), that depends on the r i−1 of the preceding monomer. In fact, this can be generalized such that the probability of r i depends on r i−1 , r i−2 ,..., r i−I so long as I remains finite. If this is the case, then sufficiently large coarse-grained segments will still develop a Gaussian distribution with an average chain size obeying R 0 = aN 1/2 . While the universality class of random walks applies to an impressive range of models, it does not include those with self-avoiding interactions. In that case, the position and orientation of a particular monomer is dependent upon all others in the chain. Nevertheless, such polymer configurations can still be generated with random walks, but those configurations containing overlaps must now be disregarded. The smaller compact configurations are more likely to contain overlapping monomers, and therefore self-avoidance favors larger and more open configurations. Consequently, the size of an isolated polymer in solution scales as R 0 ∝ N ν with a larger exponent of ν ≈ 0.6 (de Gennes 1979). However, when the polymer is in a melt with other polymers, it has to avoid them as well as itself. In this case, the advantage of the more open configurations is lost, and the polymer reverts back to random-walk statistics with the Gaussian probability in Eq. (1.11), although with a modified statistical segment length (Wang 1995). Fetters et al. (1994) provide a library of experimental results for R 0 as a function of
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Bibliography Almdal, K., Rosedale,
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BIBLIOGRAPHY 81 Matsen, M. W. and G
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Index action 17 binodal point 46 bl
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Self-Consistent Field Theory and Its Applications by M. W. Matsen

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