Spontaneous curvature of the

More recently, Smith et al. have developed another model based on spontaneous curvature.163 Their analysis is motivated by a remarkable experimental study of the elastic properties of individual helical ribbons formed in model biles. As mentioned in Section 5.2, they measure the change in pitch angle and radius for helical ribbons stretched between a rigid rod and a movable cantilever. They find that the results are inconsistent with the following set of three assumptions (a) The helix is in equilibrium, so that the number of helical turns between the contacts is free to relax, (b) The tilt direction is uniform, as will be discussed below in Section 6.3. (c) The free energy is given by the chiral model of Eq. (5). For that reason, they eliminate assumption (c) and consider an alternative model in which the curvature is favored not by a chiral asymmetry but by an asymmetry between the two sides of the bilayer membrane, that is, by a spontaneous curvature of the bilayer. With this assumption, they are able to explain the measurements of elastic properties. 

Fig. 1.10 The lipid domain of one phospholipid component (e.g. DPPA) forms a protrusion within the two-dimensional structure of the second component (e.g. DPPC). The protrusion is caused by differences in spontaneous curvature of the two phospholipids. (Reprinted from ref. 94 with permission from Wiley-VCH.)...

Here, cu and C are the local mean curvatures of each of the two membrane monolayers, and nm denotes the bending rigidity of a single monolayer that is here assumed to be the same for each leaflet and for both lipid species. The spontaneous curvatures of the two leaflets, and cj are described as sums of the spontaneous curvatures of the pure lipid constituents weighted by their local compositions. This approximation has been previously validated [36,48]. 

The type of structure observed is closely related to the spontaneous curvature Co of the surfactant assemblies [7]. By using an analogy with liquid crystals, which can also adopt layered structures, Helfrich [8] introduced the concept of the elastic-free energy associated with thermally excited deviations from the spontaneous curvature of the microstructures. This elastic-free energy per unit area is given by... 

Malmsten, M. andLindman, B. (1989) Ellipsometric studies of cleaning of hard surface. Relation to spontaneous curvature of the surfactant monolayer. Langmuir, 5, 1105. 

The self-diffusion approach relies on the fact that molecular displacements over macroscopic distances are very sensitive to confinement and thus to microstructure. For example, we found that at the same composition (water, oil, surfactant), the ratio between water and oil self-diffusion coefficients could differ by a factor of 100 000. This also illustrates that the microstructure is primarily determined by the spontaneous curvature of the surfactant film and not by the oil-to-water ratio. Contributions to a better understanding of microemulsion structures with FT spin-echo NMR self-diffusion starting with Stilbs, included also Nilsson, Olsson, Soderman, Khan, Guering, Monduzzi, Ceglie, Das and many others in Lund. In this work [49-63], the access to suitable systems was very important. Here, the contacts with Friberg, Shinoda, Strey and Langevin played a central role. 

Finally similar transition can be induced by using UV-sensitive groups such as poly(ethylene oxide)-poly(methylphenylsilane) (PEO-PMPS) and azobenzene-containing poly(methacrylate)-poly(acrylic acid) (PAA-PAzoMA) [59]. More recently Mabrouk et al. [60] have shown an asymmetric polymersomes whose membrane has only one leaflet composed of UV-sensitive liquid-crystalline copolymer Poly(ethylene glycol)-poly(4-butyloxy-2-(4-(methacryloyloxy)butyloxy)-4-(4-butyloxybenzoyloxy)azobenzene) (PEG-fe-PMAazo444). Once exposed to UV light these polymersome burst due to changes in the spontaneous curvature of the membrane. 

This form for the free energy per unit area was discussed by Helfirich and states that the mean curvature which minimizes the free energy has a value Co, termed the spontaneous curvature of the membrane. The energy cost of deviating from the spontaneous curvature is the bending or curvature modulus, k. The parameter k, known as the saddle-splay modulus, measures the energy cost of saddlelike deformations. 

The quantity cq is the spontaneous curvature of the membrane, which this model endows with a simple physical meaning When the imposed head area, Lq is larger than the optimal area, vo/tg, dictated by the chain packing, the preferred curvature is negative the system prefers to pack with the heads on the outside . Note that the free energy of the curved interface is lower than that of the flat interface the system accommodates part of the strain induced by the mismatch between the heads and chains by bending. 

Calculate the interfacial tension between a dispersion of water-in-oil mi-croemulsion droplets that coexist with a phase of excess water as a function of the bending modulus and spontaneous curvature of the drops. Use the fact that at the interface there is a monolayer of surfactant that is constrained to be flat, while the spontaneous curvature implies that the lowest energy state is a curved interface. 

In this technique, a transition in the affinity is obtained by changing the water volume fraction, instead of changing the temperature. By successively adding water into oil, initially water droplets are formed in a continuous oil phase. Increasing the water volume fraction changes the spontaneous curvature of the surfactant from initially stabilizing a w/o microemulsion to an o/w microemulsion at the inversion locus. This transition is referred to as PIC. PIC method of emulsification involves... 

Moreover Porte et al. [52] have shown that in bilayer phases, k is related to the spontaneous curvature of the monolayer and the elastic constants of the monolayer ... 

We note that bicontinuity results from a particular spontaneous curvature of the surfactant films rather than from a certain solvent volume fraction, which is a secondary factor in determining microstructure. Note that for nonionic surfactants it was shown that the diffusion behavior was determined by temperature and not by solvent composition. For different systems at the same composition, we may have either water droplets, oil droplets, or a bicontinuous structure. An example is given in Fig. 17. Furthermore, one could argue that, to be consistent, all surfactant structures of infinite aggregates (including lamellar and hexagonal) should be described as percolated. 

Figure 19 Double-oil diffusion experiment with nonionic surfactant, (a) Self-diffusion coefficients and (b) diffusion coefficient ratio A" as a function of temperature in a water-rich microemulsion with nonionic surfactant. A transition from oil-in-water droplets to a bicontinuous microstructure occurs with increasing temperature (decreasing spontaneous curvature of the C12E5 surfactant film). The maximum in K indicates that an attractive interaction between the micelles is operating prior to the formation of a bicontinuous structure. Kq = 1.69 is the diffusion coefficient ratio in the pure oil mixture and is indicated as a broken line in (b). Note that the initial decrease of the self-diffusion coefficients shows that the droplets grow in size before the bicontinuous transition. The phase boundary at 25.7 C is indicated as a vertical broken line. (Data from Ref 43.)...

Clearly, the siuface tension has a minimum when the spontaneous curvature of the surfactant film equals the mean ciuvature of the interface. The mean curvature for a flat interface is zero, larger than zero for an interface curving towards the oil (oil-in-water emulsions), and smaller than zero for a water-in-oil emulsion. Hence, a large positive spontaneous monolayer curvature, as for a strongly hydro-philic surfactant, favors oil-in-water emulsions and vice versa. The Kabalnov-Wennerstrom model also allows the thickness of the film to vary in order to minimize the free energy of hole formation, i.e., the mean curvature of... 

---