We briefly report on some results concerning the adhesion of lipid membranes on rigid
walls. After a short Introduction, in Section 2 we study a unilateral equilibrium
problem for the energy functional of lipid tubules subject to an external field. These
tubules may form assemblies when they are brought in contact, and so made to adhere to
one another along flat interstices. The contact energy is taken to be proportional to
the area of contact through a constant, which is called the
adhesion potential.
This competes against the external field in determining the stability of patterns with
flat interstices. Though the equilibrium problem is highly nonlinear, we determine
explicitly the stability diagram for the adhesion between tubules. We conclude that
the higher the field, the lower the adhesion potential needed to make flat interstices
energetically favourable, though its critical value depends also on the surface tension
of the interface between tubules and the isotropic fluid around them. Focusing our
attention on thin tubules, in Section 3 we are concerned with the effects of the
geometry of an adhesive wall on the equilibrium condition that holds at the
detachment points, where tubules and wall loose contact. As a result,
this condition depends on the curvature of the wall, which thus plays a central
rôle in determining whether the adhesion of a tubule is possible. This rôle
is clearly illustrated by a comparison between two equilibrium problems, where a
tubule adhere either to a flat or to a curved wall.
In Section 4, we obtain a covariant equation that determines the equilibrium shape of
a free lipid membrane, when a general free-energy functional is employed to
describe its elasticity. Moreover, we study the equilibrium of an open membrane free
to adhere to a rigid wall. The characteristic feature of
an open membrane is that it exhibits a border, which could also be
adhesive. Our main objective is to arrive at the equilibrium
conditions to be imposed on an adhesive border. In particular, these
conditions are applied to a simple problem, which models the adhesion of a
membrane to an axisymmetric profile. For this example we are also able lo
study the stability of the adhesive border.
