We derive analytical and numerical results for the geometric obstruction factor in the case of self-diffusion within model cubic-phase microstructures and apply the results to the analysis of literature self-diffusion data in cubic phases, bicontinuous microemulsions, and L3 (or "L*") phases. Each model microstructure is defined by a dividing surface, which divides the polar regions - surfactant head groups, and usually water - from the nonpolar regions - surfactant tails and possibly oil. The polar-apolar dividing surfaces treated are (1) interconnected cylinders, and (2) smooth surfaces of constant mean curvature, recently computed by one of the authors, which are generalizations of periodic minimal surfaces of identically zero mean curvature. The surfactant self-diffusion can often be modeled as diffusion of a particle confined to the polar-apolar dividing surface, with a constant diffusion coefficient D0. Self-diffusion within the labyrinthine subvolumes created by each dividing surface is solved by a three-dimensional finite element calculation, yielding curves of β = Deff/D0 (the "obstruction factor") versus volume fraction. Then for the case of surface diffusion, i.e., surfactant self-diffusion, the surface diffusion equation is solved over the surface by a two-dimensional finite element method. It is also proven analytically that the effective diffusion coefficient Deff for a particle diffusing over any minimal surface of cubic symmetry is exactly (2/3)D0. Assuming an interconnected-cylinder microstructure leads to an apparently universal relation between volume fraction and obstruction factor. For the constant mean curvature models we find an approximate universal relation between the obstruction factor and mean curvature in the regime of low (dimensionless) mean curvature, which quantifies the observation that higher coordination number leads to a steeper decrease of the obstruction factor with volume fraction, for dividing surfaces of low mean curvature. Application of the results for cubic phases in the DDAB/water/styrene system yields information on the degree of head-group hydration over a range of water contents from 17% to 58%. For L3 phases in the C12E3-water system, results for the "D" family of constant-mean-curvature surfaces provide an excellent fit of the water self-diffusion data. However, or microemulsion systems, including systems based on CnEm surfactants and on Aerosol OT, the slope of the measured obstruction factor as a function of volume fraction is in many cases significantly higher than the theoretical values for families of low or moderate coordination number, and even higher than the value for the randomly decorated Voronoi model of fairly high coordination. This appears to indicate that as the oil:water ratio moves away from unity, obstructing "neck" regions, and perhaps even topological disconnections, develop due to a combination of nonzero spontaneous mean curvature and thermal fluctuation. © 1990 American Chemical Society.
