Figure 10. Sketch of a border.
Of course, the molecular splay has an energetic cost: according to Helfrich ,
it can be given the simple form
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  when their special energy functional is used. This equation was obtained in  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
On an adhesive border ,
when the membrane is everywhere tangent to the
wall (see Figure 11) the equilibrium condition reads as
When axisymmetric membranes are considered and
is taken as
As an application of (20), we have studied in  the stability for an adhesive border, when the adhesive body is a cone, as illustrated in Figures 13a & 13b
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.