Swelling of networks and volume phase transitions

Abstract

All synthetic and biological networks swell when exposed to low molecular weight solvents. The degree of swelling at equilibrium depends on factors such as temperature, length of the network chains, size of the solvent molecules, and the strength of thermodynamic interaction between the polymer chains and solvent molecules. As in the previous applications, the thermodynamics of the system may be described by the change in the Gibbs free energy ΔG of the system, which is related to the change in the Helmholtz free energy ΔA by ΔG = ΔA + Δ(pV). At constant pressure, the pressure–volume product does not change significantly in swelling, and ΔG can be replaced by ΔA. The total change results from the change in the elastic free energy ΔA_el of the network upon isotropic dilation with the introduction of the solvent and from the change in the free energy of mixing ΔA_mix of the solvent molecules with the chains constituting the network. It is assumed (Flory, 1953; Treloar, 1975; Erman and Flory, 1985; Erman and Mark, 2005) that the change in the total free energy is the direct sum of the two terms, i.e., ΔA = ΔA_el + ΔA_mix (8.1). Expressions for ΔA_el for an affine network and a phantom network are given by Eqs. (5.10) and (5.21), respectively. The deformation ratios λ₁, λ₂, and λ₃ in these equations must now correspond to the state of isotropic dilation, that is, λ₁ = λ₂ = λ₃ = (V_m/V₀)^(1/3) = (v_2c/v_2m)^(1/3) (8.2), where V_m is the volume of solvent plus polymer, and v_2m is the volume fraction of polymer at equilibrium (maximum) degree of swelling when exposed to excess solvent. Substituting Eq. (8.2) into Eqs. (5.10) and (5.21), respectively, leads to ΔA_el = (3νkT/2)[(v_2c/v_2m)^(2/3) − 1] − μkT ln(v_2c/v_2m) (affine) (8.3) and ΔA_el = (3ξkT/2)[(v_2c/v_2m)^(2/3) − 1] (phantom) (8.4).