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2.9.3 Donnan Equilibrium Swelling
Consider a porous medium whose solid matrix holds a fixed electrical charge and whose interstitial fluid consists of a solvent and two monovalent counter-ions (such as Na and Cl . The boundaries of the medium are permeable to the solvent and ions. The fixed charge density is denoted by ; it is a measure of the number of fixed charges per volume of the interstitial fluid in the current configuration. This charge density may be either negative or positive, thereby producing an imbalance in the concentration of anions and cations in the interstitial fluid. To determine the osmolarity of the interstitial fluid, it is necessary to equate the mechano-chemical potential of the solvent and the mechano-electrochemical potential of the ions between the porous medium and its surrounding bath. When assuming ideal physicochemical behavior, the interstitial osmolarity (resulting from the interstitial ions) is given by where is the salt concentration in the bath. Alternatively, we note that the osmolarity of the bath is . Though this expression may be equated with eq.(2.9-2), the resulting value of is not constant in this case, since ions may transport into or out of the pore space; therefore that relation is not useful here.
However, since the number of charges fixed to the solid matrix is invariant, we may manipulate eq.(2.9-2) to produce a relation between the fixed charge density in the current configuration, , and the corresponding value in the reference configuration, , Now the osmotic pressure resulting from the difference in osmolarity between the porous medium and its surrounding bath is given by This expression may be substituted into eq.(2.9-3) to evaluate the corresponding elasticity tensor.
When the osmotic pressure results from an imbalance in osmolarity produced by a fixed charge density, it is called a Donnan osmotic pressure. The analysis associated with this relation is called Donnan equilibrium.