Elementary and More Advanced Molecular Theories

The basic postulate of elementary molecular theories of rubber elasticity states that the elastic free energy of a network is equal to the sum of the elastic free energies of the individual chains. In this section, the elasticity of the single chain is discussed first, followed by the elementary theory of elasticity of a network. Corrections to the theory coming from intermolecular correlations, which are not accounted for in the elementary theory, are discussed separately. 

The chemical structure of a polymer chain determines its statistical properties, such as its average dimensions in space and its flexibility. These parameters, in turn, affect various properties of a network consisting of these chains. A detailed understanding of the single chain is therefore important in this regard. 

The vector r joining the two ends of the chain takes different values resulting from rotations about the individual bonds. For chains with more than about 50 skeletal bonds, the probability W(r)dxdydz that one end of r is at the origin and the other end is in an infinitesimal volume dV — dxdydz is satisfactorily represented by the Gaussian function [31,32] 

r2)o represents the average of the squared end-to-end vectors, and the subscript zero indicates that the chain is in the unperturbed or so-called theta state [21]. It is now well established that chains in the bulk undiluted state are in the unperturbed state. Equation (4a) represents the probability distribution of the vectorial quantity r. A less detailed form of representation is the distribution zt (r) showing the probability that the magnitude r of r has a certain value irrespective of direction. Thus, the probability that the chain end-to-end length is in the range r to r+dr irrespective of its direction is 

For chains having fewer than 50 bonds, such as the short chains in a bimodal network, for example, the distribution departs markedly from the Gaussian limit. 

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