Configurational distribution functions Gaussian chain

The present theoretical approach to rubberlike elasticity is novel in that it utilizes the wealth of information which RiS theory provides on the spatial configurations of chain molecules. Specifically, Monte Carlo calculations based on the RIS approximation are used to simulate spatial configurations, and thus distribution functions for end-to-end separation r of the chains. Results are presented for polyethylene and polydimethylsiloxane chains most of which are quite short, in order to elucidate non-Gaussian effects due to limited chain extensibility. 

Curro and Mark 38) have proposed a new non-Gaussian theory of rubber elasticity based on rotational isomeric state simulations of network chain configurations. Specifically, Monte Carlo calculations were used to determine the distribution functions for end-to-end dimensions of the network chains. The utilization of these distribution functions instead of the Gaussian function yields a large decreases in the entropy of the network chains. 

Therefore, in an attempt to obtain simple analytic expressions for the distribution functions of stiff polymer chains, condition (5.2a) is relaxed. The relaxation of this condition is in the original spirit of the use of Wiener integrals. If this condition were imposed for flexible polymer chains, the Wiener measure would be 2[t s)] exp (—3L/2/) and would give equal weight (measure) to all continuous configurations of the polymer. Thus the use of (5.2a) would not yield the correct gaussian distribution for flexible chains. 

Here the question may arise as to which system does the Gaussian distribution function Eq. (2.10) really belong. There is a formal answer It is associated with a limiting configuration of a segment chain, having an infinite number of segments... 

To establish a useful equation of state for the mechanical behavior of a rubber network, it is necessary to predict the most probable overall dimensions of the molecules under the influence of various externally applied forces. An interesting approach to rubber elasticity consists of simulating network chain configurations (and thus the distribution of end-to-end distances) by the rotational isomeric state technique cited above. Based on the actual chemical structure of the chains, it enables one to circumvent the limitations of the Gaussian distribution function in the high deformation range. Nonetheless, the Gaussian distribution function of the end-to-end distance is very useful. It is obtained from a simple hypothetical model, the so-called freely jointed chain, which can be treated either exactly or at various levels of approximation. 

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