Swelling Behavior of Cross-Linked Rubber: Explanation of the Elusive Peak in the Swelling Activity Parameter (Dilational Modulus)

There is a considerable body of work in the literature that addresses the swelling of cross-linked rubber in the framework of the Frenkel-Flory-Rehner (FFR) hypothesis that the swelling is controlled the balance of the 0.026 thermodynamics of mixing and the elasticity of the network, i.e., the mixing and elastic contributions to the free energy are additive. One outcome that contradicts the FFR hypothesis of the experimental aspects of the work is that the swelling activity parameter (or dilational modulus) S = lambda ln(a(c)/a(u)) frequently shows a peak when plotted as a function of the swelling stretch lambda(2)(s). a(c) and a(u) are the chemical potentials of cross-linked and un-cross-linked rubber at fixed polymer volume fraction. The focus of the present work is the behavior of S, which has remained elusive of explanation, and that we propose is an artifact of the data treatment used in the determination of the value of S as a function of lambda(2)(s). This is because the data treatment generally has required curve smoothing of (generally) sparse experimental data for swelling of the rubber as a function of the chemical activity and, then, taking the logarithm of the ratio of a(c)/a(u) at fixed volume of polymer in the system. We find that the peak in S disappears when using a model based smoothing method instead of the empirical or polynomial methods of smoothing the individual volume fraction vs chemical activity curves for the cross-linked and un-cross-linked rubbers. We propose that the peak in S is due to the experimental variability in sparse data and is emphasized in the common methods of data smoothing. We show that addition of less than 1% random error to the model data can lead to the peak in S. Importantly, this work shows that the Frenkel-Flory-Rehner theory is consistent with the swelling data and the elusive peak in S is actually an artifact of normal data smoothing procedures.