Equilibrium Onions ?

– We demonstrate the possibility of a stable equilibrium multi-lamellar (“onion”) phase in pure lamellar systems (no excess solvent) due to a sufficiently negative Gaussian curvature modulus. The onion phase is stabilized by non-linear elastic moduli coupled to a polydisperse size distribution (Apollonian packing) to allow space-filling without appreciable elastic distortion. This model is compared to experiments on copolymer-decorated lamellar surfactant systems, with reasonable qualitative agreement. Introduction. – The existence of vesicles at thermal equilibrium is a long-standing, and still controversial, problem [1]. Unilamellar and multi-lamellar vesicles (MLVs) were first induced in lyotropic lamellar phases with excess water by adding energy (e.g. shear flow, ultrasound, electric field) [2]. In a very elegant work, using membranes generated by a chemical reaction, Hoffman and co-workers demonstrated that any of lamellae, unilamellar vesicles, or MLVs can be prepared in the same system, depending on the mechanical path chosen [3]. However in some special cases [4, 5] equilibrium unilamellar vesicles have been demonstrated. These systems are all in the dilute regime (large excess of water). Some experiments have suggested that MLVs (or onions) can be stabilized in the semi-dilute regime (excess water) [5]. Theories to explain the stability of dilute unilamellar vesicles either describe a competition between the entropy of mixing and the curvature energy of the vesicles [5–7] or a symmetrybreaking instability leading to a spontaneous curvature [8]. Onions are also predicted to be stabilized in the dilute and semi-dilute regimes due to an unstable curvature energy; in these cases, a transition towards unilamellar vesicles is avoided by imposing either a core energy [9] or a cutoff in the entropy of the Helfrich interactions [10]. Indeed, Simons and Cates [10] have discussed the stability of unilamellar vesicles and the transition to onions as the concentration increases even when the curvature energy is unfavorable, due to entropic reasons. In the concentrated regime (homogeneous lamellar phase with no excess water), very monodisperse onions can be prepared under shear [11–13]. The vesicles fill space and remain in © EDP Sciences 2 EUROPHYSICS LETTERS the one phase region without expelling solvent, by distorting into space-filling polyhedra [14]. The resulting texture is a lattice of disclinations which can be either disordered [11] or ordered [13]. Moreover many experimental systems “spontaneously” exhibit onions in regions of the phase diagrams that seem to be continuously linked to the lamellar phase [15]. This evolution from a texture of polydisperse vesicles to a texture of focal conics upon tuning a parameter, such as a co-surfactant concentration or the volume fraction of solvent, seems to be generic. A striking signature of this evolution is the change in the rheology, from a viscous gel in the “onion” phase to a more fluid phase in the “lamellar” focal conic phase. Thus, whether an onion texture is at equilibrium or not seems yet unclear. In this work we propose a model based on a non-quadratic elastic energy, which shows that a pure lamellar phase, or one of space-filling deformed monodisperse onions, is unstable with respect to a polydisperse space-filling packing (we study an Apollonian packing [16]). We compare these predictions to experiments on copolymer-doped lamellar phases, originally introduced by Ligoure et al. [17]. Model. – The bending free energy per area F/A for membranes is usually taken, following Helfrich, as the simplest quadratic function of the mean and Gaussian curvatures, respectively 2H = r 1 +r −1 2 and G = 1/(r1r2), where r1 and r2 are the two principal radii of curvature [18]. This suffices for large curvature radii r ≫ δ, where δ is the layer thickness. However, if onions exist, because of the high concentration of membrane, high curvatures (small r) are expected near the core of the onions. We will thus use the free energy, relative to a flat state, of a symmetric bilayer expanded to quartic order in the curvature radii [19]: