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Unformatted text preview: Advanced Topics in Equilibrium Statistical Mechanics Glenn Fredrickson Lecture 14 8. Polymer Statistical Mechanics A. CoarseGraining 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 CC 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 N 3 . 4 1 where τ ∼ 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 “coarsegrained” 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 lowmagnification view of a chain, e.g., In the coarsegrained 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 N 3 . 4 holds for all linear polymers, and while the prefactor τ 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 SingleChain Statistics The crudest singlechain model is the “freely jointed chain” model, where we describe a polymer as a freelyhinged 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: h f ( { r } ) i ≡ N Y i =1 1 4 π Z unit sphere d n i f ( { n b } ) Let’s try this out. Notice the endtoendLet’s try this out....
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This document was uploaded on 05/18/2010.
 Spring '09

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