Thermodynamic Investigation of Droplet-Droplet and Bubble-Droplet Equilibrium in an Immiscible Medium

In the absence of external fields, interfacial tensions between different phases dictate the equilibrium morphology of a multiphase system. Depending on the relative magnitudes of these interfacial tensions, a composite system made up of immiscible fluids in contact with one another can exhibit contrasting behavior: the formation of lenses in one case and complete encapsulation in another. Relatively simple concepts such as the spreading coefficient (SC) have been extensively used by many researchers to make predictions. However, these qualitative methods are limited to determining the nature of the equilibrium states and do not provide enough information to calculate the exact equilibrium geometries. Moreover, due to the assumptions made, their validity is questionable at smaller scales where pressure forces due to curvature of the interfaces become significant or in systems where a compressible gas phase is present. Here we investigate equilibrium configurations of two fluid drops suspended in another fluid, which can be seen as a simple building block of more complicated systems. We use Gibbsian composite-system thermodynamics to derive equilibrium conditions and the equation acting as the free energy (thermodynamic potential) for this system. These equations are then numerically solved for an example system consisting of a dodecane drop and an air bubble surrounded by water, and the relative stability of distinct equilibrium shapes is investigated based on free-energy comparisons. Quantitative effects of system parameters such as interfacial tensions, volumes, and the scale of the system on geometry and stability are further explored. Multiphase systems similar to the ones analyzed here have broad applications in microfluidics, atmospheric physics, soft photonics, froth flotation, oil recovery, and some biological phenomena.