Vesicle adhesion and microemulsion droplet dimerization: Small bending rigidity regime

To study the vesicle-substrate unbinding transition and the onset of microemulsion aggregation, we calculate the curvature free energy of a vesicle adhered to a substrate and of two microemulsion droplets forming a dimer. Analytical expressions are derived in the small bending rigidity regime in which the length (k/σ)1/2, constructed from the rigidity constant of bending k and surface tension σ, is small compared to the typical size of the vesicle (droplet), (k/σ)1/2≪R. The leading contribution to the curvature free energy is shown to be proportional to k1/2. The formulas derived are used to understand the experimentally observed aggregation of microemulsion droplets occurring in the direction of vanishing spontaneous curvature. In this way we intend to bridge the gap between the liquid state theories used to describe aggregation processes in microemulsion systems and the bending energy concept originally introduced by Helfrich to describe vesicles shapes and fluctuations as well as phase diagrams of microemulsion systems.