Curvature spontaneous

The spontaneous curvature plays a very important role in surfactant science. For molecules with a large polar head and small tail, the molecules have a distinct wedge shape. This means that the monolayer, if aUowed to do whatever it wants, tends to curve towards oil (// 0). Similarly, for molecules with a bulky tail and a small polar head, the monolayer curves towards water (Ho 0). Of course, the nature of the spontaneous curvature goes beyond this simple cartoon. Not only is the architecture of the polar head and alkyl chain important, but the nature of the oil and aqueous solvents, the presence of electrolytes, and temperature also play a role. These effects can be more clearly seen when one considers the distribution of the pressure in the interfacial region. It is well known that the interfacial tension between two phases can be represented as the zeroth moment of the excess pressure in the interfacial region  

The spontaneous curvature of surfactant layers can be controlled in many ways see Table 7.1. For ionic surfactants, one can control the contribution coming from the counter-ions by adjusting the concentration of salt. As the electrolyte concentration increases, the ionic atmosphere approaches the oil-water interface (Debye length decreases), the transverse pressiue moment decreases and the spontaneous curvature faUs. Increasing the temperature does the opposite, because the osmotic pressxne (jt) of the counter-ions is proportional to temperature due to the osmotic pressure ideal gas law n = cRT. 

Factor Effect on Ho (taking the sign into account)  

Branching of surfactant chain at a given number of carbon atoms i i 

Desorbing neutral species, e.g. sugars Nature of hydrocarbon oil 1 i 

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Fig. 60 Schematic illustration for formation of cylindrical morphology in a blend of slightly asymmetric lower molecular weight PS-b-PI (/3-chain) with large symmetric PS-fc-PI (a-chain). a Molecule of /S-chain with non-zero spontaneous curvature, b Cylindrical morphology formed by neat /3 chains shown in a. Here mean curvature of cylinder (solid line) is larger than spontaneous curvature of /3-chain (dashed lines). c Cylindrical morphology formed by binary blend of /3-chains shown in a and large symmetric copolymers (a-chain). In this case, mean curvature of cylinder closely fits to spontaneous curvature of /3-chain. From [180]. Copyright 2001 American Chemical Society...

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. 

In biological systems, one often observes membrane structures with nonzero spontaneous curvatures, e.g. in mitochondria. This type of bilayer structure is also essential in various transport related processes such as endo- and exocy-tosis (see Chapter 8 of this volume). These curved membrane systems may be stabilised by protein aggregation in the bilayer, or may be the result of the fact that biological membranes are constantly kept off-equilibrium by lipid transport and/or by (active) transport processes across the bilayer. These interesting... 

The most frequent emulsiflcation using phase inversion is known as the PIT (Phase Inversion Temperature) method [81-83] and occurs through a temperature quench. This method is based on the phase behavior of nonionic surfactants and the correlation existing between the so-called surfactant spontaneous curvature and the type of emulsion obtained. 

Figure 1.4. For a nonionic surfactant, influence of the temperature on (a) the surfactant morphology and hence the spontaneous curvature, (b) the type of self-assembly, (c) the phase diagram, the number of coexisting phases is indicated (d) the coexisting phases at equilibrium, and (e) the corresponding emulsions.

Emulsilication through phase inversion is based on a change in the surfactant spontaneous curvature induced by temperature. This concept can be generalized considering any parameter influencing the spontaneous curvature of a surfactant, for example, salinity, pH, presence of a cosurfactant, and nature of the oil. The concept of inversion has often been reported in the literature by means of a formulation-composition map. In the following, we shall sum up this empirical approach which can be useful for formulators. 

Figure 5.3. Scheme explaining the influence of the spontaneous curvature on the activation energy for coalescence in a W/O emulsion. 

It is probable that numerous interfacial parameters are involved (surface tension, spontaneous curvature, Gibbs elasticity, surface forces) and differ from one system to the other, according the nature of the surfactants and of the dispersed phase. Only systematic measurements of > will allow going beyond empirics. Besides the numerous fundamental questions, it is also necessary to measure practical reason, which is predicting the emulsion lifetime. This remains a serious challenge for anyone working in the field of emulsions because of the polydisperse and complex evolution of the droplet size distribution. Finally, it is clear that the mean-field approaches adopted to measure > are acceptable as long as the droplet polydispersity remains quite low (P < 50%) and that more elaborate models are required for very polydisperse systems to account for the spatial fiuctuations in the droplet distribution. 

As indicated above, miscibiUty gaps are small and intermediate lamellar liquid crystalline phases dissolve rapidly into the aqueous phase if the surfactant or surfactant mixture is rather hydrophihc with a high spontaneous curvature (low (v/la)), for instance at temperatures below Tq for pure nonionic surfactants. In this case dissolution, which converts lamellae of zero curvature to aggregates with significant curvature as surfactant concentration decreases, occurs spontaneously because it reduces system free energy. 

Various structures of DNA tubes have been devised based on the main building blocks exploited in 2D crystals four-arm junctions [73] and different kinds of DX [68] and TX tiles [101]. The spontaneous curvature of DNA constructs is in this case exploited rather than minimized or compensated as in 2D crystals. Specifically, the curvature is the consequence of the symmetry of the different tiles and of the nature, the number, and the spatial arrangement of the crossover points, whose ultimate origin is an implementation of the approach proposed in [100] and shown in Fig. 22 (see [102] for a detailed discussion). 

Co is called the spontaneous curvature. The spontaneous curvature is a more general parameter than the surfactant parameter Ns, defined by Eq. (12.4). It makes it easier to discuss the phase behavior of microemulsions because we get away from the simple geometric picture. 

Example 12.9. We consider a film with zero spontaneous curvature (6 0 = 0). What is the elastic energy for bending such a film to a sphere of radius R1 With C = 1/R and C2 = 1/R we obtain... 

The spontaneous curvature can vary in the range of -0.5 nm-1 to +0.5 nm, depending on the polar head group, the length and number of apolar chains, and the nature of the oil. Most microemulsions that have been studied contain four or even more components. In addition to water, oil, and surfactant usually a cosurfactant is added. Typical cosurfactants are alcohols. With their small head group and relatively large hydrophobic tail they tend to decrease the spontaneous curvature of surfactants. 

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