A mixed hybrid finite element framework for the simulation of swelling ionized hydrogels

Abstract

Ionized hydrogels, as osmoelastic media, swell enormously (1000 times its original volume in unionized water) due to the osmotic pressure difference caused by the presence of the negatively charged ion groups attached to the solid matrix (polymer chains). The coupling between the extremely large deformations (induced by swelling) and fluid permeation is a field of application that regular poroelasticity formulations cannot handle. In this work, we present a mixed hybrid finite element (MHFE) computational framework featuring a three-field (deformation-chemical potential-flux) formulation. This formulation guarantees that mass conservation is preserved both locally and globally. The impact of such a property on the swelling simulations is demonstrated by four numerical examples in 2D. This paper focuses on the implementation aspects of the MHFE model and shows that it stays robust and accurate for a volume increase of more than 3000%.

The Importance of the Mixing Energy in Ionized Superabsorbent Polymer Swelling Models

“…This method calculates the fluid flux across each element as an independent variable rather than through the numerical differentiation of the chemical potential field, hence separating their dependency on each other. This models efficacy in better replicating large deforming ionized gels has been previously reported [12].…”

“…Successful recent work on reaching large deformation with the finite element method have excelled with the addition of the Terzaghi decomposition to split the stress in the system into the effective stress acting on the solid matrix and the pressure of the fluid in the pores [10,11]. Three-field finite element formulations which solve for position, chemical potential and fluid flux independently, have shown greater stability than two-field formulations [12][13][14]. The assumption of perfect separability of the energies of FR theory has been questioned in several studies.…”

“…Experiments on swelling sodium polyacrylate gels showed that the mixing energy can be assumed negligible when swelling equilibrium is reached [20][21][22][23]. Several studies have used this assumption to develop swelling models of ionized gels [10,12,24,25] as well as biological tissue [26][27][28]. However, the effect of the mixing energy during transient swelling is not documented experimentally due to the difficulty of dynamically splitting the swelling pressures.…”

Comparing mixed hybrid finite element method with standard FEM in swelling simulations involving extremely large deformations

“…As a matter of fact, using proper numerical methods to solve Darcy flow problem is a well-established topic in porous media studies [15,33]. Inspired by the success of mixed formulations in Darcy flow problems, a mixed hybrid finite element method (MHFEM) was implemented to simulate the transient swelling of a hydrogel in one of our previous works [45].…”

“…In this work, we aim to evaluate the effectiveness of MHFEM comparing with standard FEM in terms of robustness, accuracy and efficiency the solution method when applied to simulate the finite transient swelling of hydrogels. As the theoretical basis of the MHFEM is elaborated in [45], in this work, the focus lies on comparing and furthermore demonstrating the advantages of MHFEM over standard FEM in swelling simulations involving large deformations. Results and discussions of numerical examples form the key part of this work.…”