Theoretical bicontinuous models

These fascinating bicontinuous or sponge phases have attracted considerable theoretical interest. Percolation theory [112] is an important component of such models as it can be used to describe conductivity and other physical properties of microemulsions. Topological analysis [113] and geometric models [114] are useful, as are thermodynamic analyses [115-118] balancing curvature elasticity and entropy. Similar elastic modulus considerations enter into models of the properties and stability of droplet phases [119-121] and phase behavior of microemulsions in general [97, 122]. 

Finally, one word about the lattice theories of microemulsions [30 36]. In these models the space is divided into cells in which either water or oil can be found. This reduces the problepi to a kind of lattice gas, for which there is a rich literature in statistical mechanics that could be extended to microemulsions. A predictive treatment of both droplet and bicontinuous microemulsions was developed recently by Nagarajan and Ruckenstein [37], which, in contrast to the previous theoretical approaches, takes into account the molecular structures of the surfactant, cosurfactant, and hydrocarbon molecules. The treatment is similar to that employed by Nagarajan and Ruckenstein for solubilization [38]. 

A theoretical phase diagram for a lyotropic system is shown in Figure 17 and reveals, in addition, the Vj and V2 phases which are bicontinuous cubic phases (normal and reversed, respectively) whose structure can be described by models involving interpenetrating rods or periodic minimal surfaces. Note also that each pair of phases is separated, at least in principle, by a cubic phase (a, b, c, d in Figure 17), and with a biphasic interface (two phases coexisting). 

The result, Eq. (43), can also be used to calculate the elastic constants of interfaces in ternary diblock-copolymer systems [100]. The saddle-splay modulus is found to be always positive, which favors the formation of ordered bicontinuous structures, as observed experimentally [9] and theoretically [77,80] in diblock-copolymer systems. In contrast, molecular models for diblock-copolymer monolayers [68,69], which are applicable to the strong-segregation limit, always give a negative value of k. This result can be understood intuitively [68], as the volume of a saddle-shaped film of constant thickness is smaller than... 

Self-diffusion data from bicontinuous structures contain, in principal, information on the average coordination number of the microstructure. The experimental results can, however, not be compared with the theoretical results directly. There is an additional reduction of D/Dq due to solvation of the surfactant film (a lateral friction felt by the solvent molecules in the solvent layers closest to the film) and an obstruction due to the finite film volume. These effects, which both increases with the surfactant-to-solvent ratio can however be included in the model, as will be discussed for the case of the sponge phase in one of the following sections. Extending the model to include solvation, however, introduces additional parameters which will affect the uncertainty. 

The issue is more difficult when the composition of the quenched material lies close to the center of the miscibility gap. In this case, the final two-phase system exhibits a bicontinuous geometry, both phases occupying about the same volume fraction, and thus the structure cannot be modeled by a set ofisolated nano-objects. For the centtal part of the miscibihty gap, a theoretical model named spinodal decomposition was proposed (Calm, 1965). At advanced stages, even after nearly having reached the equilibrimn concentrations, both phases stiU evolve by a coarsening process. 

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