Simple molecular model for the spontaneous curvature and the bending constants of nonionic surfactant monolayers at the oil/water interface

A simple analytical model for the spontaneous curvature and the bending constants of nonionic surfactant monolayers at an oil/water interface is developed with the goal of allowing correlation and prediction of phase behavior. The surfactant molecules are treated as diblock copolymers grafted to the interface. The change in the free energy of the surfactant monolayer due to the bending of the interface is calculated as a sum of the contributions of the hydrophobic and the hydrophilic blocks. The equilibrium thickness of a spherical and cylindrical surfactant monolayer is found by minimizing the sum of the free energy of mixing and the stretching energy of the hydrophilic and the hydrophobic blocks. Comparison with the Helfrich model gives expressions for the spontaneous curvature H₀, the bending elasticity κ, and the saddle splay modulus κ̄, of the surfactant monolayer. An analytical expression is obtained for the phase inversion temperature, which correlates well with experimental data for a wide series of nonionic surfactants C_iE_j and oils (n-alkanes). The model gives the temperature dependence of the spontaneous curvature and the bending parameters of nonionic surfactant monolayers. Phase boundaries in C₁₂E₅/octane/water micro-emulsions are calculated without use of any adjustable parameters and are in a good agreement with experimental data.