Fixman-Alben distribution

Figure 1 Distributions for the end-to-end distance of a PDMS chain having n = 20 skeletal bonds of length / = 1.64 A. The Fixman-Alben distribution (dotted curve) and that from a Monte Carlo simulation (solid curve) are compared with the Gaussian approximation (dashed curve).

Among various representations of w(r) for short chains are the Hermite series [31,32], the Fixman-Alben distribution [33], and Monte Carlo simulations [34], The Fixman-Alben distribution is given by... 

Erman, B., Mark, J.E., 1988. Use of the Fixman-Alben distribution function in the analysis of non-Gaussian rubberlike elasticity. J. Chem. Phys. 89, 3314-3316. 

More quantitative characterization of limited chain extensibility requires a non-Gaussian distribution function for the end-to-end separation, r, of the short network chains. The Fixman-Alben distribution was used to calculate stress-strain isotherms in elongation for bimodal PDMS networks. Good agreement was found between theory and experiment. Other non-Gaussian distribution functions have also been successfully used. The experimental isotherms can also be interpreted... 

For chains having fewer than 50 bonds, such as the short chains in a network having bimodal distribution of lengths, for example, the distribution departs markedly from the Gaussian limit. A distribution that has been employed for short chains is the Fixman-Alben density distribution given by... 

It is also possible to interpret the upturns in modulus in these isotherms using analytic expressions, for example the Fixman-Alben modification [121] of the Gaussian distribution function, combined with the constrained-junction theory and reasonable values of the constraint parameter k [122]. 

Fixman, M. Alben, R., Polymer Conformational Statistics 0.1. Probability Distribution. J. Chem. Phys. 1973,58(4), 1553-1558. 

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