Phase transitions in phospholipid monolayers: Statistical model at the pair approximation

A Langmuir film, consisting of a phospholipid monolayer at the air-water interface, was modeled as a two-dimensional lattice gas corresponding to a ternary mixture of water molecules (w), ordered-chain lipids (o), and disordered-chain lipids (d). The statistical problem is formulated in terms of a spin-1 model, in which the disordered-chain lipid states possess a high degenerescence omega >> 1, and was termed Doniach lattice gas (DLG). Motivated by some open questions in the analysis of the DLG model at the mean-field approximation (MFA) [phys. Rev. E 90, 052705 (2014)], we have reconsidered it at the pair-approximation level by solving the model on a Cayley tree of coordination z. The attractors of the corresponding discrete-map problem are associated with the thermodynamic solutions on the Bethe lattice (the central region of an asymptotically infinite Cayley tree). To check the thermodynamic stability of the possible phases, the grand-potential density was obtained by the method proposed by Gujrati [Phys. Rev. Lett. 74, 809 (1995)]. In general, the previous MFA results are confirmed at the pair-approximation level, but a novel staggered phase, overlooked in the MFA analysis, was found when the condition epsilon(wd) > 1/2(epsilon(ww) + epsilon(dd)) is satisfied, where epsilon(xy) represents the nearest-neighbor intermolecular interactions between single-site states x and y. Model parameters obtained by fitting to experimental data for the two most commonly studied zwitterionic phospholipids, 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) and 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC), yield phase diagrams consistent with the phase transitions observed on Langmuir films of the same lipids under isothermal compression, which present a liquid-condensed to a liquid-expanded first-order transition line ending at a critical point.