# A question about surface tension of membranes and their curvature

I'm reading a review about membranes properties and I have reach a section about fluid membranes. The section discuss the principal curvatures ($c_1, c_2$) and the spontaneous curvatures ($c_0$). After stating some properties of $c_0$, the following two sentences appear:

The membrane also is assumed to be incompressible. Hence, all the contributions to the surface tension vanish.

I'm not sure about the meaning of these two sentences, I think it means that by bending the membrane I can not generate surface tension, is that correct?

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Comment to the question (v1): I interpret it as incompressible here means that the membrane cannot change its area. Since the surface energy is proportional to the area, its contribution to the total energy is a constant. – Qmechanic Jul 5 '11 at 14:08

I wonder which article are you studying. Before I answer the question, I would suggest three references which seem three of the best for beginners.

• S. A. Safran, Statistical Thermodynamics of Surfaces, Interfaces, and Membranes, Addison-Wesley Publishing Company, Massachusetts, 1994.
• S. Leibler, in Statistical Mechanics of Membranes and Surfaces, Proceedings of the Jerusalem Winter School for Theoretical Physics, edited by D. R. Nelson, T. Piran, and S. Weinberg, World Scientific, Singapore, 1989. Ch.3 Equilibrium statistical mechanics of fluctuating films and membranes."
• U. Seifert, Adv. Physics 46, 13 (1997). "Configurations of fluid membranes and vesicles".

Concisely speaking, surface tension of fluid membrane is zero because stretching energy is too large compared to bending energy. Under usual experimental condition, it is about $10^9$ times greater! Thus, membrane is hardly streched from the optimal area. Microscopically, each lipid molecule always keeps its optimal area per molecule. Here optimal area means the area that gives zero stretching energy, i.e. $\frac{\partial f}{\partial a} = 0$ at optimal area $a_0$ where $f$ is the stretching energy per molecule(free energy to increase the area occupied by each lipid molecule) and $a$ is an area occupied by each lipid molecule and $a_0$ is the optimal area per molecule for the system.

Usually, the lipid molecules are very scarce in the solution where vesicle or fluid membranes live in. So we can say that the number of lipids $N$ within fluid membrane is constant. Then, surface tension of fluid membrane is $\sigma = \frac{\partial F}{\partial A} = \frac{\partial (Nf)}{\partial (Na)}$ = $\frac{\partial f}{\partial a} = 0$.

This approach is indeed true for lipid 'monolayers'. However, if you are interested in lipid bilayers, problem requires bit more sophisticated thought. Even a lipid bilayer indeed has zero surface tension. However, this does not mean each monolayer leaflet consisting the bilayer membrane is not stretchable. If a lipid bilayer is bent, one leaflet would be stretched while the other would be compressed. This local compression/stretching of monolayers gives non-local elastic energy so called area-difference elasticity. At first glance, this explanation is very contratdictory, but actually it is not. To clarify this issue. the concept of neutral surface must be introduced.

In real world, membranes are not two-dimensional mathematical surfaces with zero thickness. They do have finite thickness, and this finite thickness has significant role in physics of fluid membranes because membranes are consist of amphiphiles - the molecules have hydrophilic part as well as hydrophobic part. Amphiphilic nature of molecules makes applied stress on each lipid molecule be inhomogeneous when membrane is elastically deformed. (To avoid confusion, I would add a comment: inhomogeneous stress profile means the distribution of the stress along the direction parallel to the lipid molecules. Hydrophilic head and hydrophobic tail would not experience equal amount of stress.) Therefore, we must carefully chose the surface to represent membrane with finite thickness. The interface between hydrophobic chain and hydrophilic head is not usually good one. Usual choice is the neutral surface which decouples stretching energy and bending energy. So far, I have talked about the 'area' occupied by molecule. Such area is specifically the area of "neutral surface", not the other. If you choose some other surface to represent the membrane, the area per molecule would be dependent on curvature of the surface, which is not what we really want.

I think my explanation is not sufficient to fully grasp all physics of lipid membranes. For more information about neutral surface, read Safran's Chapter 6. His introduction is fairly clear and easy. For microscopic model, I suggest you to study Petrov-Derzhanski-Mitov(PDM) model. It is well explained in Leibler's book chapter as well as Appendix A of L. Miao, U. Seifert, M. Wortis, and H.-G. Dobereiner, Phys. Rev. E 49, 5389 (1994). I prefer the latter research article to Leibler's book chapter because Miao et al.'s Appendix A shows more detail steps to follow. Probably, you need to spend some time to follow all the details, but it is quite worth to do.

The concept of surface tension for fluid membrane becomes even more confusing when it comes to fluctuating membranes. All of the sudden, "surface tension" does quite important role to the fluctuating membrane. However, this "surface tension" is not the surface tension we have discussed so far. Such 'effective' tension is irrelevant to the stretching, and rather related to the area conservation constraint. For further study, read following papers:

• W. Helfrich and R.-M. Servuss, Il Nuovo Cimento 3D, 137 (1984). "Undulations, steric interaction and cohesion of fluid membranes". When you read this paper, focus on the concept of 'projected area'.