Theory of the lamellar-hexagonal transformation: Tilted mesophases in lyotropic systems

Abstract

A phenomenological model of the lamellar-hexagonal transformation in lyotropic systems is described. A classification is given of the mesophase symmetries that may arise in the transformation process. The approach is shown to apply in an analogous but different way to the transformation between tilted mesophases. ͓S1063-651X͑97͒13211-1͔PACS number͑s͒: 61.30. Gd, 64.60.Ϫi Two-dimensional ͑2D͒ hexagonal mesophases are often found in the phase diagrams of complex fluid systems. In lyotropic mixtures ͓1͔, one of the most common mesophase configurations is that of the ''middle soap'' phase ͑labeled E in Ekwall's notation ͓2͔͒, which consists of parallel amphiphilic rods in hexagonal array, composed of radially disposed molecules of amphiphiles, with the hydrocarbon parts directed inwards and the hydrated polar groups facing outwards. In the phase diagrams of many ternary systems ͓2͔ the E phase is found with the hexagonal F phase, which is composed by aggregates of the inverted type having outwards hydrocarbon chains and inwards polar groups: The E and F phases occupy generally a region of the phase diagram located around a concentration c ϭ 1/2 of surfactant in water, at low and high concentrations of cosurfactant, respectively ͓2͔. Other mesophases displaying hexagonal lattices have been identified in lyotropics, such as the ''complex'' hexagonal phase (H c ) 3 , or the rhombohedral (R3m) phase, in which the planar hexagonal networks of short rods are stacked regularly in a three-dimensional lattice ͓4͔.The most general sequence of phases in which the 2D hexagonal structures appear usually involves the lamellar mesophase from which the E and F phases are separated either by a two-region phase, i.e., by a first-order reconstructive transition, or by intermediate mesophases ͑e.g., rectangular, monoclinic, cubic, etc.͒. The first aim of this article is to give a phenomenological description of the lamellarhexagonal transformation, and of the possible types of related intermediate mesophases. The model, which has been recently introduced in the case of the lamellar tetragonal transformation ͓5͔, makes use of a single symmetry breaking mechanism, consisting in the undulation of the interfaces between the molecular aggregates and the solvent. Another motivation of the present work is to show that when assuming oriented interfaces, the model provides a basis for the description of the various types of tilted mesophases that have been found in lyotropic systems ͓6͔, namely, the ''ripple'' ( P ␤ Ј ) and lamellar hexatic (L ␤ Ј ) phases found, for example, in lecithin ͓7͔. Figure 1͑a͒ shows a two-dimensional view of one of the sequences of mesophases assumed in our approach. It de-composes the successive steps of the transformation between the lamellar (L) and hexagonal (H) phases in the plane perpendicular to the lamellae. One can see that the reorganization of the mesophases results from a periodic undulation of the interfaces separating the molecular aggregates and the solvent. Such a type of periodic distortion has bee...

Elastic energy of tilt and bending of fluid membranes

“…Defining tilt as deviation of the average orientation of the hydrocarbon chains from the normal to the membrane surface, we consider three factors contributing independently to the elastic energy: i) the molecular tilt, ii) the gradient of the tilt accounting for the variation of the tilt along the membrane plane, and iii) bending. In contrast to the previous studies, where independent elastic moduli were introduced for the tilt, tilt gradient, and bending deformations [10][11][12][13][14][15][16][17], we demonstrate that the variation of tilt and bending are additive and that the contributions of both of them to the elastic energy are determined by the elastic moduli known from measurements of pure monolayer bending. The energetic contribution of tilt against the surface normal is determined by an independent elastic coefficient, of which we estimate the value.…”

“…This term has been introduced phenomenologically in all precedent models of tilt deformations [9][10][11][12][13][14][15][16][17] and is related in ours to the transversal shear rigidity profile λ T (z).…”

“…One relates these phases to a 324 The European Physical Journal E particular internal structure of bilayers characterized by a systematic tilt of the stretched hydrocarbon chains with respect to the membrane plane [2][3][4][5][6][7][8], which is seen as an order parameter of the membrane core. Thus, the treatment of the energetics of lipid membranes in the lowtemperature states was performed by combining the effects of bending and those of tilt, the two kinds of contributions having different physical origin [9][10][11][12][13][14][15][16][17].…”

“…Consequently these deformations are related to same first-and second-order expansion coefficients in the free energy: J s , κ, andκ. This bears a substantial simplification for many existing continuum elastic models for lipid layers combining tilt and bending deformations [10][11][12][13][14][15][16][17] and will hopefully simplify future work.…”

Rheological and phase behaviour of amphiphilic lipids

“…A further model has been proposed recently, suggesting that the lamellar mesophase progressively shifts to the hexagonal mesophase by means of symmetrical ruptures along a plane perpendicular to the lamellar bilayer through lamellar mesophases of different symmetry (Bourgoyne et al, 1995), (Mettout et al, 1997), (Holmes, 1998). Obviously, these intermediate mesophases are defective liquid crystals, which may be metastable and likely to yield complex rheological behaviour.…”