Particles bound to an interface interact because they deform its shape. The
stresses that result are fully encoded in the geometry and described by a
divergence-free surface stress tensor. This stress tensor can be used to
express the force on a particle as a line integral along any conveniently
chosen closed contour that surrounds the particle. The resulting expression is
exact (i.e., free of any "smallness" assumptions) and independent of the chosen
surface parametrization. Additional surface degrees of freedom, such as vector
fields describing lipid tilt, are readily included in this formalism. As an
illustration, we derive the exact force for several important surface
Hamiltonians in various symmetric two-particle configurations in terms of the
midplane geometry; its sign is evident in certain interesting limits.
Specializing to the linear regime, where the shape can be analytically
determined, these general expressions yield force-distance relations, several
of which have originally been derived by using an energy based approach.Comment: 18 pages, 7 figures, REVTeX4 style; final version, as appeared in
Phys. Rev. E. Compared to v2 several minor mistakes, as well as one important
minus sign in Eqn. (18a) have been cured. Compared to v1, this version is
significantly extended: Lipid tilt degrees of freedom for membranes are
included in the stress framework, more technical details are given, estimates
for the magnitude of forces are mad