Statistical thermodynamics of liquid-liquid phase separation in ternary systems during complex coacervation

Liquid-liquid phase separation leading to complex coacervation in a ternary system (oppositely charged polyion and macroion in a solvent) is discussed within the framework of a statistical thermodynamics model. The polyion and the macroion in the ternary system interact to form soluble aggregates (complexes) in the solvent, which undergoes liquid-liquid phase separation. Four necessary conditions are shown to drive the phase separation: (i) (σ{23}){3}r/Φ{23c}≥(64/9α{2})(χ{23}Φ{3}){2} , (ii) r≥[64(χ{23}Φ{3}){2}/9α{2}σ{23}{3}]{1/2}, (iii) χ{23}≥(2χ{231}-1)/Φ{23c}Φ{3}, and (iv) (σ{23}){2}/sqrt[I]≥8/3α(2χ{231}-1) (where σ{23} is the surface charge on the complex formed due to binding of the polyelectrolyte and macroion, Φ{23c} is the critical volume fraction of the complex, χ{23} is the Flory interaction parameter between polyelectrolyte and macroion, χ{231} is the same between solvent and the complex, Φ{3} is the volume fraction of the macroions, I is the ionic strength of the solution, α is electrostatic interaction parameter and r is typically of the order of molecular weight of the polyions). It has been shown that coacervation always requires a hydrated medium. In the case of a colloidal macroion and polyelectrolyte coacervation, molecular weight of polyelectrolyte must satisfy the condition r≥10{3} Da to exhibit liquid-liquid phase separation. This model has been successfully applied to study the coacervation phenomenon observed in aqueous Laponite (macroion)-gelatin (polyion) system where it was found that the coacervate volume fraction, δΦ{23}∼χ{231}{2} (where δΦ{23} is the volume fraction of coacervates formed during phase separation). The free energy and entropy of this process have been evaluated, and a free-energy landscape has been drawn for this system that maps the pathway leading to phase separation.