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Influence of Young's modulus and geometrical shapes on the 2D simulation of pH-sensitive hydrogels by the meshless random differential quadrature method

In this paper, 2D simulation of a pH-sensitive hydrogel is performed by a novel strong-form meshless method called the random differential quadrature (RDQ) method. So far the simulations of pH-responsive hydrogels have been performed over 1D hydrogel domains by simplification, where the hydrogel is allowed to deform in one direction only with a constant axis-symmetric cross-section. However, for an irregular cross-section, in which the hydrogel swells unevenly in different directions, it truly becomes the 2D problem. The RDQ method is a novel meshless technique based on the fixed reproducing kernel particle method and the differential quadrature method. The diffusion of mobile ionic species between the hydrogel and solution is simulated by the system of the Poisson–Nernst–Planck equations, and the hydrogel swelling is captured by mechanical equilibrium equations. The analytical expressions of displacements in the x and y directions are derived for a constant osmotic pressure at field nodes located along the interface between the hydrogel and solution domains. The numerical values of the displacements are verified with the corresponding analytical values obtained from the derived expressions for a hydrogel with square geometry. It is shown from the simulation results that the RDQ method is capable of capturing the jumps in the values of field variables across the interface between the multiple domains. The hydrogel swelling is studied by changing Young's modulus and the geometrical shape, and the simulation results are found qualitatively in good agreement with the physics of the problem.