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5.2.8 Donnan Equilibrium Swelling
The swelling pressure is described by the equations for ideal Donnan equilibrium, assuming that the material is porous, with a charged solid matrix, and the external bathing environment consists of a salt solution of monovalent counter-ions. Since osmotic swelling must be resisted by a solid material, this material is not stable on its own. It must be combined with an elastic material that resists the swelling.
The Cauchy stress for this material is the stress from the Donnan equilibrium response : where is the osmotic pressure, given by is the bath osmolarity (twice the concentration) and is the fixed charge density in the current configuration, related to the reference configuration via, where is the relative volume, is the universal gas constant and is the absolute temperature.
Note that may be negative or positive. The gel porosity is unitless and must be in the range . The corresponding spatial elasticity tensor is