Abstract
Torques on interfaces can be described by a divergence-free tensor which is fully encoded in the geometry. This tensor consists of two terms, one originating in the couple of the stress, the other capturing an intrinsic contribution due to curvature. In analogy to the description of forces in terms of a stress tensor, the torque on a particle can be expressed as a line integral along a contour surrounding the particle. Interactions between particles mediated by a fluid membrane are studied within this framework. In particular, torque balance places a strong constraint on the shape of the membrane. Symmetric two-particle configurations admit simple analytical expressions which are valid in the fully nonlinear regime; in particular, the problem may be solved exactly in the case of two membrane-bound parallel cylinders. This apparently simple system provides some flavor of the remarkably subtle nonlinear behavior associated with membrane-mediated interactions.
Highlights
The crucial role of fluid membranes as a component of living cells was recognized early on by biologists [1]
The mechanics of membranes has become a fertile area of research among physicists as well [2]
The energy of the membrane is completely determined by the geometry of the surface
Summary
The crucial role of fluid membranes as a component of living cells was recognized early on by biologists [1]. The link is formed by the surface stress tensor and its torque counterpart This approach has already proven its value when applied to the problem of surface-mediated interactions [13, 14, 15]: particles bound to a membrane interact because they deform its shape [16, 17, 18, 19, 20, 21, 22, 23, 24]. The energy of the surface is evaluated by integration, and differentiated with respect to appropriate placement variables of the bound particles This approach relies heavily on the validity of the linearization.
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