Cylindrical Colloids on a Fluid Membrane

We theoretically study the adhesion and membrane-mediated interaction of cylindrical colloids to a flat fluid membrane. There are two ways to approach this problem. The first way, based on energy, requires finding the equilibrium shape of the membrane given the placement of the particle(s). In order to do so, we need to know how the energy of the surface depends on its shape (i.e. the surface Hamiltonian), as well as how the adhered colloid deforms the membrane. The second way to approach this class of problems is “geometrical”, where forces between the membrane-adhered particles are related directly to the geometry of the deformed membrane via the surface stress tensor. The surface Hamiltonian allows finding the stress at any point on the membrane in terms of local geometry. The force acting on the colloid can then be found by integrating this surface stress tensor along any contour enclosing the colloid. In this thesis, using the approach based on free energy calculations, we look into the problem of cylindrical colloids adhering to a membrane with fixed constant adhesion energy between the membrane and the colloids. Angle-arclength parameterization is used in order to treat the problem beyond small gradient approximation. We present three different cases here: single cylinder adhering on a membrane, two cylinders adhering on the same side of the membrane, and two cylinders adhering on different sides of the membrane. For the single cylinder case we present a structural phase diagram to separate no wrapping, partial wrapping and closure states and we compare it to the phase diagram obtained for a related system of spherical colloids. For two cylinders adhered on the same side of the membrane we obtain repulsive interaction and transition from shallow to deep wrapping as the cylinders move apart from each other. We also look into a phase where two cylinders are vertically stacked and discuss its energetics. For two cylinders adhering to the opposite sides of the membrane, attractive interaction is obtained in accordance with previous results and we further show that in that case two cylinders are generally in contact and a first-order transition from shallow to full wrapping is possible. In the last section, we put a framework for the class of problems where the particle is between the membrane and the supporting interface, where adhesion is assumed between the interface and the membrane.