Sylvio May and Avinoam Ben-Shaul, 'Spontaneous curvature and thermodynamic stability of mixed amphiphilic layers', J.Chem.Phys. 109, 3839 (1995)

The elastic behavior of mixed bilayers composed of two amphiphilic components with different chain length (and identical head groups) is studied using two molecular level models. In both, the bilayer free energy, is expressed as a sum of chain, head group and interfacial contributions as well as a mixing entropy term. The head group and interfacial terms are modeled using simple phenomenological but general expressions. In one model the chain conformational free energy is calculated using a detailed mean-field molecular theory. The other approach is based on a simple 'compression' model (generalized for mixed layers). Both models lead to similar qualitative conclusions as well as to similar numerical results. The bilayer free energy is expressed as a sum of free energies of its constituents monolayers. A thermodynamic stability analysis is performed to examine the possibility of spontaneous vesicle formation. To this end we expand the bilayer free energy as a power series (up to second order) in terms of the monolayer curvatures, their amphiphilic compositions and the average cross sectional areas per molecule. All variables are coupled with the molecular composition and areas treated as degrees of freedom which are allowed to relax during bending so as to minimize the curvature free energy. Using reasonable molecular interaction parameters we find that a second order transition from a planar to a curved (vesicle) geometry in a mixed bilayer composed of amphiphiles of different chain length is an unlikely event, at least as long as the mixing in each monolayer is ideal. Most of our analysis is devoted to calculating of the spontaneous curvature and the bending rigidity of the bilayer as a function of its amphiphile chain composition. The calculations show that adding short chain amphiphiles to a layer of long chain molecules reduces considerably its bending rigidity, as already known from calculations involving only the chain contributions. However, we find that inclusion of the head group and interfacial interactions moderates the effects of the added short chains. We also find that the bending rigidity of pure monolayers is approximately linear in chain length, as compared to a nearly cubic dependence implied by the chain free energy alone (at constant head group area). One main result involves the calculation of the spontaneous curvature as a function of composition. We find, for different chain mixtures, that upon adding short chains to long chain monolayers, the spontaneous curvature first varies (increases) nearly linearly with composition and then (beyond a mole fraction of 0.5) begins to saturate towards the spontaneous curvature of a pure short chain layer. Qualitative arguments are provided to explain this behavior.