**Figure 10.** Sketch of a border.

Of course, the molecular splay has an energetic cost: according to Helfrich [13],
it can be given the simple form

where is a positive constitutive parameter which represents a line tension. Here we are primarily concerned with the appropriate equilibrium conditions along a border, which differ from those already obtained for the points where the principal curvatures of a vesicle suffer a jump, as do the admissible deformations of the membrane. Along a border, a fluid membrane can only

**Figure 11.** An open fluid membrane that adheres to a rigid
wall.
is the *free membrane* and
the
*adhering* one;
is the *adhering contour*, and
the *adhesive border* of
.
The principal curvatures of the membrane may
jump along
.
The unit normal
to
is the same as the unit normal to the wall.

On the other hand, an open membrane could also adhere to the wall along the border and be completely detached elsewhere (see Figure 12).

**Figure 12.** An open fluid membrane in contact with a rigid
wall only along its border
.
Here, there is neither an adhering contour
nor an adhering membrane. The angle
that the normal to
makes with the normal to the wall along the border of the
membrane is the *contact angle*.

In this case, the admissible displacements of the border can no longer be tangent to the membrane.

Starting from (1) and (2), one can derive a general equilibrium
equation for a free membrane, which reduces to Ou-Yang and Helfrich's [14]
when their special energy functional is used. This equation was obtained in [15]
through an *intrinsic* method, that is, which does not resort to any specific
system of coordinates to represent the equilibrium shape of the membrane:

In (18)
,
div _{s} and
denote,
respectively the surface Laplacian, the surface divergence, and the surface
gradient;
is the unit normal vector to the membrane and
is a
Lagrange multiplier which accounts for the inextensibility of the membrane.
Moreover,
is the mean curvature and
he Gaussian curvature.

Besides this equation, an open membrane in contact with a wall must also
satisfy other equilibrium conditions on both its possible adhering contours
and its adhesive borders. The former equilibrium condition reads as

where is the outer normal to along the adhering contour , as shown in Figure 11. Equation (19) restricts the possible jumps of the principal curvatures of the membrane across the adhering contour and is a further extension of the condition found by Seifert and Lipowsky [6].

On an adhesive border ,
when the membrane is everywhere tangent to the
wall (see Figure 11) the equilibrium condition reads as

where the line tension is defined as in (17), is the Lagrange multiplier associated with the constraint on the total area of the membrane, while is the

where is the

where is the

When axisymmetric membranes are considered and
is taken as

where

where and are the principal curvatures of the membrane along , while is the unit outer normal to , tangent to (see Figure 12).

As an application of (20), we have studied in [15] the stability
for an adhesive border, when the adhesive body is a cone, as illustrated in
Figures 13*a* & 13*b*

**Figure 13.** The adhesion of a membrane to two axisymmetric
bodies: the cones in *(a) *and *(b)* are reverse to one another, but
with the same
amplitude .
The equilibrium of the border
is unstable
for the former and stable for the latter.

The analysis of (20) allows us to conclude that the equilibrium
configuration shown in Figure 13a is *unstable*, whereas that shown in
Figure 13b is *stable*.