Analytical Solutions for the Equilibrium states of a Swollen Hydrogel Shell and an Extended Method of Matched Asymptotics
Abstract
A polymer network can imbibe water, forming an aggregate called hydrogel, and undergo 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 equilibrium states of a swollen hydrogel with a coreshell structure, this paper seeks analytical solutions of the deformations by perturbation methods for three cases, i.e. freeswelling, nearly freeswelling and general inhomogeneous swelling. Particularly for the general inhomogeneous swelling, we introduce an extended method of matched asymptotics to construct the analytical solution of the governing nonlinear secondorder variablecoefficient differential equation. The 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), although the freeenergy function for the hydrogel contains two material parameters. Comparisons with numerical solutions are also made and good agreements are found.
 Publication:

arXiv eprints
 Pub Date:
 March 2011
 arXiv:
 arXiv:1103.0622
 Bibcode:
 2011arXiv1103.0622D
 Keywords:

 Physics  Biological Physics;
 Condensed Matter  Soft Condensed Matter;
 Mathematical Physics
 EPrint:
 20 pages, 4 figures