Statistics, non-Gaussian

For a polyelectrolyte chain that has non-Gaussian statistics, exact analytical expression for B is not feasible. To get some insight, we notice that the static structure factor has the limiting behavior. 

MODIFICATIONS TO SIMPLE STATISTICAL THEORY— NON-GAUSSIAN STATISTICS... 

In this section some general considerations of interest regarding non-Gaussian statistical theory are made with the aim of bringing the simple network model discussed in Section 3,2 closer to a real network (2,4). 

The expression of the probability density function, P r), in logarithmic form corresponding to non-Gaussian statistics furnished by Kuhn and Griin in 1942 for a chain with n links of length / is given by... 

Strain hardening at high deformations A can be explained by the non-Gaussian statistics of strongly deformed chains. Recall that the Gaussian approximation for a freely jointed chain model is valid for end-to-end distances much shorter than that for a fully stretched state R < / max = bN. In Section 2.6.2, the Langevin functional dependence of normalized end-to-end distance R/Nb on the normalized force Jb/ kT) for a freely jointed chain [Eq. (2.112)] was derived ... 

Modifications to these classical statistical models can also be made, such as by the incorporation of loose chains (non-load-bearing chains), physical crosslinks (temporary or permanent) and intramolecular crosslinks (loops). At higher deformations (strains) or increasing crosslink densities, it may be necessary to use a non-Gaussian statistical treatment that considers the finite extensibility of the chain. Non-Gaussian models are reviewed extensively by Treloar (1975). 

Figure 7.10). This occurs only because the two curves are superimposed at low strains. The modulus of real networks is higher than theoretical because of entanglement effects. Use of non-Gaussian statistics only lessens the discrepancy between the calculated and the experimental curves but the reasons for this discrepancy are not understood. 

The full derivation of the nominal stress (/)-strain relationship for a chain obeying non-Gaussian statistics is complicated. The final equation as derived by Kuhn is ... 

It is also important to note that a whole family of non-Gaussian statistical treatments is reported in the literature. Treloar (1975, section 3.10) presents an excellent review on this topic. 

The discrepancy between high-elasticity classical theory and experimental curves o-e or o-X for elastoplastics indicated above is due to two factors firstly, by essentially non-Gaussian statistics of real polymer networks and, secondly, by the lack of coordination of two main postulates, lying in the basis of entropic high-elasticity classical theory - Gaussian statistics and elastoplastics incompressibility. The last condition is characterised by the criterion v = 0.5, where v is Poisson s ratio [46]. 

The Gaussian theory considers the number of possible conformations of a chain having a specified end-to-end distance. A more accurate non-Gaussian statistical treatment of the random chain is based on the distribution of sin j, i.e. of the angle between the direction of a random link and of the end-to-end vector. From the probability of finding n links in the range AGj, ri2 in A 2 and so on, the entropy of a single chain is derived [2b] as... 

Finally some of the eoneepts were brought together in a section on the modeling of reliability data using non-Gaussian statistics. The next chapter will continue with the analysis of data with random errors but will expand flic concepts in this chapter to examples where measured data sets depend on one or more independent variables. 

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