Asymptotic solutions and new insights for cylinder and core–shell polymer gels in a solvent

Abstract

A polymer gel in a solvent can swell to a large extent, and the large deformation caused by the swelling makes studies on the behaviors of gels difficult due to the high nonlinearity. In this paper, we study three homogeneous deformations of a cylinder gel in the cases of free-swelling, swellings under uniaxial constraint and under equal biaxial constraint and the inhomogeneous deformation of a spherical core–shell gel. The aim is to deduce new insights from the asymptotic solutions. In particular, the roles of the chemical potential are fully explored through appropriate parameters. For homogeneous deformations, the governing nonlinear algebraic equations are solved using perturbation methods. Analytical formulae for the stretch and stress are obtained and the key controlling parameters are found, which lead to some interesting insightful information. Also, according to some critical values of this parameter, the plane of the polymer network parameter and the solvent parameter can be divided into five domains, in which the gel has different behaviors. The inhomogeneous deformation of the core–shell gel (which can be used as a drug-carrier in applications) is governed by a complicated variable-coefficient nonlinear differential equation. It is not possible to obtain the exact solution. However, when the above-mentioned parameter is small or large, we manage to construct asymptotic solutions. In the first case, two key transformations are introduced, which make the leading-order equation twice integrable. In the second case, we use an extended method of matched asymptotics to construct the leading-order solution. For a core–shell gel, debonding can happen due to the swelling-induced stress. Our analytical solutions suggest that the best strategy for preventing this is to increase the initial stretch, since such an increase has no influence on the volume ratio but leads to the reduction of the stress. Also, how the geometry of the shell reduces the volume ratio is explicitly revealed. All the analytical results are also validated by numerical solutions.