Some analytical formulas for the equilibrium states of a swollen hydrogel shell

A polymer network can imbibe water, forming an aggregate called hydrogel, and undergo a large and inhomogeneous deformation with external mechanical constraint. Due to the large deformation, nonlinearity plays a crucial role, which also causes the mathematical difficulty for obtaining analytical solutions. Based on an existing model for the equilibrium states of a swollen hydrogel with a core-shell structure, this paper seeks analytical (or semi-analytical) solutions to deformation by perturbation methods for three cases, i.e. free-swelling, nearly free-swelling and general inhomogeneous swelling. Particularly for the general inhomogeneous swelling, we introduce an extended method of matched asymptotics to construct the semi-analytical solution of the governing nonlinear second-order variable-coefficient differential equation. The semi-analytical solution captures the boundary layer behavior of the deformation. Also, analytical formulas for the radial and hoop stretches and stresses are obtained at the two boundary surfaces of the shell, making the influence of the parameters explicit. An interesting finding is that the deformation is characterized by a single material parameter (called the hydrogel deformation constant, which is one-fifth power of the ratio of the shear modulus due to mixing to the shear modulus due to stretching), although the free-energy function for the hydrogel contains two material parameters. Comparisons with numerical solutions are also made and good agreements are found. © 2011 The Royal Society of Chemistry.