Theory of lateral and flip-flop phase separations in bilayer biomembranes and surfactants

In this work, we consider bilayer biomembranes or surfactants made of two amphiphiles A and B. Under a variation of a suitable parameter, such as temperature or difference of lengths of hydrophobic chains, these systems undergo a phase separation from a homogeneous liquid-phase to two distinct liquid-phases. Two physical situations can be distinguished: (1) The amphiphiles A and B prefer to jump from a monolayer to the other (flip-flop transition), (2) the mixture phase separates on each monolayer, and there is no jump from one sheet towards the second one (lateral transition). To investigate the associated critical phase behavior, we first introduce a field theory, constructed with two order parameters (or fields) φ and ψ, which are nothing else but the composition fluctuations relative to the monolayers. Beside the usual terms proportional to φ2, ψ2, φ4 and ψ4, the free energy contains an extra one, −Cφψ, which describes the lowest order coupling between the two monolayers. The coupling constant C is positive for the lateral phase separation, and negative for the vertical one. We show that its sign results from a competition between the chemical segregation of amphiphiles and the curvature asymmetry. With the help of this free energy, we first identify the liquid-phases, and show the existence of a critical point, Tc, of which the location depends naturally on the value of the coupling constant C. In particular, for those bilayer biomembranes or surfactants made of amphiphiles of the same chemical nature but with different lengths, and at fixed temperature, we show the existence of a critical line in the (Δc0,Δl)-plane, along which the bilayer undergoes a phase separation. Here, Δc0 and Δl account for the curvature gap and the length difference, respectively. Second, we determine the behavior of the composition fluctuations, φ and ψ, and the total one, Φ=φ+ψ, upon temperature, T, and chemical potential difference, Δμ, in the critical region. Third, we determine the critical behavior of the partial compressibilities, κφφ, κψψ and κφψ, and the overall one, κtot=κφφ+κψψ+2κφψ. Finally, we remark that the flip-flop phase separation shows some analogy with the classical para-ferrimagnetic transition of coupled paramagnetic materials of Curie-Weiss type.