# Effective Field Theory of Particle Interactions Mediated by Fluid Surfaces

Fluid interfaces and membranes can mediate forces between particles bound to them. Bound objects impart local deformations of the surface geometry and modify its thermal fluctuation spectrum, the effects of which spread to distant regions where other objects can respond to it. Such surface-mediated interactions play an important role in aggregation and structure formation of both colloids at fluid interfaces, relevant for many technological applications, and protein inclusions in biological membranes, which are believed to assist in important cellular remodeling processes like endocytosis and exocytosis. While the physical characteristics of these geometric interactions are conceptually straightforward, the corresponding calculations are unfortunately far from trivial. The challenge is that one must enforce conditions at the particle–surface boundaries of finite-sized objects, which themselves may also be subject to thermal fluctuations. In this thesis we develop an Effective Field Theory (EFT) formalism which disentangles the particle boundary conditions from the calculation of the interaction free energy by constructing an equivalent point-particle description. We first motivate the intuition and key steps needed to construct an EFT through a familiar electrostatics problem, which will serve as a guiding analogy for capturing finite-size information in point-like “polarizabilities” and determining their values through a suitable “matching” procedure. We then apply the formalism to construct complete effective energy functionals for flat, curved, and asymmetric rigid particles bound to tension dominated and bending-elasticity dominated surfaces. The interaction potential emerges as a systematic cumulant expansion, for which we provide a powerful diagrammatic technique and derive series expansions for pairwise and multibody interactions, as well as corrections due to thermal fluctuations. In particular, we calculate to high orders—and in some cases to all orders—the elastic and entropic interactions between particles with various fixed and free boundary conditions, and analyze the corresponding energy landscapes to determine the preferred configurations and orientations of anisotropic and multibody systems.