Elasticity of Polymer Networks

Rubbers and gels are three-dimensional networks composed of mutually cross-linked polymers. They behave like solids, but they still have high internal degrees of freedom that are free from constraints of external force the random coils connecting the cross-links are free in thermal Brownian motion. The characteristic elasticity of polymeric materials appears from the conformational entropy of these random coils. In this chapter, we study the structures and mechanical properties of rubbers on the basis of the statistical-mechanical models of polymer networks. 

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Flory.P.J. Elasticity of polymer networks cross-linked in state of strain. Trans. Faraday Soc. 56,722-743 (1960). 

Erman B, Flory PJ (1978) Theory of elasticity of polymer networks. II The effect of geometric constraints on junctions. J Chem Phys 68 5363—5369 Erukhimovich IYa, Irzhak VI, Rostiashvili VG (1976) On concentration dependence of swelling coefficient of weakly non-Gaussian macromolecules. Polym Sci USSR 18 1682-1689... 

Flory PJ (1977) Theory of elasticity of polymer networks. The effect of local constraints on junctions. J Chem Phys 66(12) 5720-5729... 

Heinrich, G., Straube, E. and Helmis, G. Ruber Elasticity of Polymer Networks Theories. Vol. 84, pp. 33-87. 

Therefore, the principal difficulty connected with the application of Eq. (12) is due to the incompleteness of the Gauss invariant. So, the use of the Gauss invariant for adequate classification of topologically different states in many-chain systems is very problematic. Nevertheless, that approach was used repeatedly for consideration of such physically important question as the high-elasticity of polymer networks with topological constraints [15]. Unfortunately,... 

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c> a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

The polymer chain in a tube model and its modifications are widely used for investigation of the rubber elasticity of polymer networks with topological constraints. The detailed review on that problem one can find in the monograph... 

It is well known that the elasticity of polymer networks with constrained chains in the rubbery state is proportional to the number of elastically active chains. The statistical (topological) model of epoxy-aromatic amine networks (see Sect. 2) allows to calculate the number of elastically active chains1 and finally the equilibrium modulus of elasticity Eca,c for a network of given topological structure 9 10). The following Equation 9) was used for the calculations of E, c ... 

A particularly interesting molecular model for the nonlinear elasticity of polymer networks was proposed by Rubinstein and Panyukov [50], the slip-tube model. In this model the confining potential acting on the network chains depends on the deformation and is modeled by virtual chains attached to the network... 

Heinrich, G., Kaliske, M., 1997. Theoretical and numerical formulation of a molecular based constitutive tube-model of rubber elasticity. CompuL Theor. Polym. Sci. 7 (3-4), 227-241. Heinrich, G., Straube, E., Helmis, G., 1988. Rubber Elasticity of Polymer Networks Theories Polymer Physics. Springer, Berlin/Heidelberg, pp. 33-87. 

Until now, all we have deduced from (7.30) was the dependence a(A) at constant temperature. However, it also contains the dependence on T. So, let s analyze how the elasticity of polymer networks is affected by the temperature. 

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