The Noninteracting Gaussian Chain

Ignoring the excluded volume interaction we deal with independent chains. It thus is sufficient to consider a single chain system. The spatial dimension is found to play an important role in the theory, and we therefore work with general dimension d. Wo write the configurational probability for a noninteracting Gaussian chain as... 

Since the internal configurations of disjoint parts of a noninteracting Gaussian chain are uncorrelated, this distribution is identical to the endpoint distribution of a chain of length k > - k ... 

A significant parameter for model behavior is the reduced density p (Eq.(8)) of the mers. For p > 1, the model exhibits a glass transition at sufficiently low temperature this transition is absent for the classic affine network model composed of noninteracting Gaussian chains. 

The simple class of models just discussed is of interest because it is possible to characterize the decay of correlations rather completely. However, these models are rather far from reality since they take no account of interparticle forces. A next step in our examination of the decay of initial correlations is to find an interacting system of comparable simplicity whose dynamics permit us to calculate at least some of the quantities that were calculated for the noninteracting systems. One model for which reasonably complete results can be derived is that of an infinite chain of harmonic oscillators in which initial correlations in momentum are imposed. Since the dynamics of the system can be calculated exactly, one can, in principle, study the decay of correlations due solely to internal interactions (as opposed to interactions with an external heat bath). We will not discuss the most general form of initial correlations but restrict our attention to those in which the initial positions and momenta have a Gaussian distribution so that two-particle correlations characterize the initial distribution completely. Let the displacement of oscillator j from its equilibrium position be denoted by qj and let the momentum of oscillator j be pj. On the assumption that the mass of each oscillator is equal to 1, the momentum is related to displacement by pj =. We shall study... 

Thus all segment correlations of our noninteracting model take the form of Gaussian functions. We stress that for our Gaussian chain model all these results are valid rigorously for all n > 1 or k2 — k] > 1, respectively. 

The index zero indicates that the average is taken by choosing the Gaussian weight °W for the configurations. The quantity ° is the free energy of the noninteracting system. Then let °Z be the partition function of an isolated chain. We have... 

To determine Am> we use the fact that for noninteracting polymers (v = 0), (cftC-t) can be calculated exactly as a sum of the correlation Wctions of independent Gaussian chains. 

Fig. 6.10 Phase diagrams calculated using generalized free volume theory for spherocylinders with L/D = 20 plus interacting polymer chains in a good solvent dashed curves) for size ratios q = 1 left) and q = 2.5 right) in the reservoir representation. Full curves are FVT results for (noninteracting) penetrable hard spheres as depletants as identical to those in Fig. 6.7. As in Fig. 6.7 the Gaussian form for the ODF was used...

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