# ch8 - Advanced Topics in Equilibrium Statistical Mechanics...

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Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson Lecture 14 8. Polymer Statistical Mechanics A. Coarse-Graining Polymer science is a particularly rich area of application of statistical mechan- ics. Before we can talk about the statistical thermo of polymers in solution or in the melt state, we need to discuss some basic models for summing the conformational states of a single polymer chain . Common polymers are: where N , the “degree of polymerization” can range from 10 3 –10 5 . How are we to describe such molecules within a CM framework? 1. An “atomistic” approach . Here we construct pair potentials, usually within an “interaction site” framework for all the atoms composing the macro- molecule. Rotations around backbone C-C bonds are permitted, which leads to large numbers of “conformational states” e N . While in principle , we can apply our MD and MC methods to such models, in practice they are very computationally demanding! For example, a polymer melt has a relaxation time that scales as τ τ 0 N 3 . 4 1

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where τ 0 1 × 10 9 sec is the time for a conformational transition of one monomer. Even for a modest N 10 3 , this gives τ 10 sec, which is nearly 10 15 larger than the timestep ∆ t that we would like to use in an MD simulation! 2. A “coarse-grained” approach . Here, we ignore the atomic details below about 10 A, but simply preserve the connectivity and architecture of the polymer. Think of this as a low-magnification view of a chain, e.g., In the coarse-grained approach, we need to impose some new “effective” interactions among the larger units with which we work. A disadvantage of the CG approach is that we often can’t ask precise ques- tions about properties that depend on monomer shape and interactions , e.g., T g or T m values. An advantage of the CG approach is that if allows one to more easily identify universal scaling relations and properties; i.e. those independent of the specific type of polymer. For example, the scaling law τ τ 0 N 3 . 4 holds for all linear polymers, and while the prefactor τ 0 depends on chemical details, the scaling exponent 3.4 does not! Another advantage is that coarse- grained models will allow us to extract a number of such relations analytically ! B. CG Models of Single-Chain Statistics The crudest single-chain model is the “freely jointed chain” model, where we describe a polymer as a freely-hinged sequence of N segments of fixed length b : We could use { R 1 , . . . , R N +1 } as coordinates, but it is easier to use bond vectors to describe a particular “conformational state” of the chain. r i R i +1 R i 2
Next, we write r i = n i b , where the n i are unit vectors, | n i | = 1. In the FJC model, all bond orientations are equally likely . Thus, we compute conforma- tional averages as: f ( { r } ) N i =1 1 4 π unit sphere d n i f ( { n b } ) Let’s try this out. Notice the end-to-end vector R of a chain is i r i = b i n i : R = sum N i b n i = i b n i = 0 So, on average, the chain is pointing in no particular direction.

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