The elastic free energy and the elastic equation of state: Elongation and swelling of polydimethylsiloxane networks

The elastic free energy for a tetrafunctional network is here expressed as where λ₁, etc., are the principal extension ratios relative to the state of reference (wherein 〈r²〉 = 〈r²〉₀), and C₁, C₂, and m are arbitrary parameters. The free energy of a swollen system is taken to be the sum G = G_mix + G_el of G_el and the free energy of mixing \documentclass{article}\pagestyle{empty}\begin{document}$ G_{{\rm mix}} = RT\left {n_1 \ln v_1 + n_2 \ln v_2 + \bar \chi n_1 v_2 } \right] $\end{document}. Stress-strain relations and the chemical potential μ₁ are derived from G as functions of the elasticity parameters C₁, C₂, and m, and of the thermodynamic interaction parameter χ or of \documentclass{article}\pagestyle{empty}\begin{document}$\chi = \bar \chi + n_1 v_2^{ - 1} \partial \bar \chi /\partial n_1 $\end{document}. The dilation of the semi-open system subject to deformation when exposed to diluent at fixed activity is derived as the sum of the dilation at fixed composition and the dilation due to absorption of diluent. Experiments are reported on the dependence of the equilibrium retractive force on elongation for cross-linked polydimethylsiloxanes (PDMS) exposed to benzene or hexamethyldisiloxane vapor at regulated activities. Volume fractions of samples covered the range v₂ = 1.00 to v₂ ≈ 0.30. With the choice of m = ½ the elastic behavior of a given polymer is well represented by one combination of values for C₁ and C₂ at all dilutions by either diluent. The dependence of the Mooney-Rivlin (C₂) term on volume is thus established, at least for PDMS, and the scope of the semi-empirical free energy expression and its consequents is greatly enlarged. Values of χ deduced from the equilibrium swelling of the unstrained networks exposed to benzene at various activities are in excellent agreement with those obtained previously from vapor pressures and osmotic pressures on linear PDMS. The results of Allen, Kirkham, Padget, and Price on the elastic behavior of natural rubber are discussed, with particular reference to the coefficients of dilation with elongation which they determined at fixed composition. The present results lend strong support to the principle of additivity of the free energies attributable to the network and to the bulk liquid system, respectively. This principle is fundamental to the analysis of rubber elasticity.