Gaussian chain, models

The root mean square centroid separation is /, the Kuhn step length. 5 (r) is the three-dimensional Dirac delta function. The distribution Gq can be evaluated to be 

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The Gaussian chain model yields a spring constant even for a single bond k=3k T/ , where k is the Boltzmann constant. From Eq. 3.3 the chain extension between arbitrary points along the chain may be computed to R n) -R(m)y)= n - m ... 

The guiding principle in writing down the self-repelling Gaussian chain model is mathematical simplicity, not microscopic faithfulness. Do we have, any idea why such a primitive model properly can explain the experiments To investigate this question we consider how a realistic microscopic description could be reduced to our model. 

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. 

To derive the results quoted in Sect. 3.1 for our Gaussian chain model, we here calculate the unperturbed (since there is no interaction) Greensfunction defined as... 

The interacting Gaussian chain model is defined by Eqs. (2.2)-(2.4). As before we consider the generalization to d-dimensional space, and we evaluate expectation values by expanding in powers of 3e. We demonstrate the method with the example of the endpoint Greensfunction G(p n) of a single chain system, which generalizes Gh p n) (Eq, 3.33). 

In this section, we describe a theory for calculating observables resulting from incoherent excited state transport among chromophores randomly distributed in low concentration on isolated, flexible polymer chains. The pair correlation function used to describe the distribution of the chromophores is based on a Gaussian chain model. The method for calculating the excitation transfer dynamics is an extension to finite, inhomogeneous systems of a truncated cumulant expansion method developed by Huber for infinite, homogeneous systems (25.26). 

In Chapter 3, we used the Rouse model for a polymer chain to study the diffusion motion and the time-correlation function of the end-to-end vector. The Rouse model was first developed to describe polymer viscoelastic behavior in a dilute solution. In spite of its original intention, the theory successfully interprets the viscoelastic behavior of the entanglement-free poljuner melt or blend-solution system. The Rouse theory, developed on the Gaussian chain model, effectively simplifies the complexity associated with the large number of intra-molecular degrees of freedom and describes the slow dynamic viscoelastic behavior — slower than the motion of a single Rouse segment. 

This equation also implies that each entanglement strand is sufficiently long to be described by a Gaussian chain model, i.e. 

Doi and Edwards treated the polymer chain as a Gaussian chain in the time region t < T q. They assumed that the equilibration process of segmental redistribution took place before T q. Even though they did not deal with the dynamic behavior of the equilibration process, they used the Gaussian chain model to obtain the stress at the end of the equilibration process (Eq. (8.32)). To be consistent, the dynamic aspect of the Gaussian chain picture needs to be included for t > Teq. 

The phantom network can account qualitatively for many properties of crosslinked elastomers, but the quantitative explanation of basic properties is wrong. For example, stress-strain properties, especially in simple extension, show departures from the phantom network results even at extension ratios covered by the Gaussian chain model. The explanation of these departures, phenomenologically described by the famous Mooney-Rivlin Eq. (1)... 

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