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Determining equilibrium osmolarity in Poly(ethylene glycol) / Chondrotin sulfate gels mimicking articular cartilage

We present an experimentally guided, multi-phase, multi-species polyelectrolyte gel model to make quantitative predictions on the electro-chemical properties of articular cartilage. The mixture theory consists of two different types of polymers: Poly(ethylene gylcol) (PEG), Chondrotin sulfate (ChS), water (acting as solvent) and several different ions: H$^+$, Na$^+$, Cl$^-$. The polymer chains have covalent cross-links modeled using Doi rubber elasticity theory. Numerical studies on polymer volume fraction and net osmolarity (difference in the solute concentration across the gel) show the interplay between ionic bath concentrations, pH, polymer mass in the solvent and the average charge per monomer; governing the equilibrium swelled / de-swelled state of the gel. We conclude that swelling is aided due to a higher average charge per monomer (or a higher percentage of charged ChS component of the polymer), low solute concentration in the bath, a high pH or a low cross-link fraction. However, the swelling-deswelling transitions could be continuous or discontinuous depending upon the relative influence of the various competing forces.