Dear people in Physics Stackexchange,
My question is mostly related to the following papers:
- U. Seifert, Z. Phys. B 97, 299 (1995). "The concept of effective tension for fluctuating vesicles".
- U. Seifert, K. Berndl, and R. Lipowsky, Phys. Rev. A 44, 1182 (1991). "Shape transformations of vesicles: Phase diagrams for spontaneous-curvature and bilayer-coupling models".
In these paper, the Lagrange multipliers $\Sigma$ and $P$ are introduced to take care of effective constraints on area and volume, respectively. If I understood those papers properly, I believe the area constraint and volume constraint originates from the lack of lipid molecules in aqueous solution and osmotic pressure.
Hence, I thought that the free energy $F = \kappa G + \Sigma A + PV$ represents "flaccid" vesicle which has much excess area. In other words, I thought that Lagrange multiplier $\Sigma$ and $P$ are not physical surface tension related to the stretching and osmotic pressure difference, respectively. I imagined them just as a mathematical term to take care of area and volume constraint.
However, Wikipedia(http://en.wikipedia.org/wiki/Elasticity_of_cell_membranes) and Ou-Yang Zhong-can and Helfrich used the free energy $F = \kappa G + \Sigma A + PV$ to determine the shape of the vesicle, and they said explicitly that $\Sigma$ represents tensile stress and and $P$ represents the osmotic pressure difference between the outer and inner medium. (Actually they used the greek alphabet $\lambda$ instead of $\Sigma$ and $\Delta p$ instead of $P$) [Ref. H. J. Deuling and W. Helfrich, J. Physique 37, 1335 (1976). "The curvature elasticity of fluid membranes: a catalogue of vesicle shapes", Ou-Yang Zhong-can and W. Helfrich, Phys. Rev. Lett. 59, 2486 (1987). "Instability and deformation of a spherical vesicle by pressure".]
Then, are these two quantities($\Sigma$ and $P$) measurable or experimentally observable? I thought that the Lagrange multiplier is determined by which curvature model I chose and the prescribed area and volume for a vesicle.
If $\Sigma$ and $P$ represents true physical tensile stress and osmotic pressure difference, then is the phase diagram in above mentioned Seifert et al.'s paper(PRA, 1991) for 'tense' vesicles and not flexible membrane although the membrane has much excess area? I thought that the paper was solely about fluid, flaccid membrane. Which part I am misunderstanding?
Are $\Sigma$ and $P$ vanishing if closed membrane is truly flexible and flaccid due to large excess area? In other words, if I want to find shape of truly flaccid membrane without any tension or osmotic pressure, should I set $\Sigma = 0$ and $P = 0$? If so, which part takes care of the area and volume constraint?
In addition, the fluctuation is strongly related to the area constraints. I thought that the effective tension (which is different from $\Sigma$ because this new effective tension is related to the excess area rather than whole area of the fluid membrane) can take care of it. This effective tension can be considered as entropic suppression of fluctuation due to area constraints. If so, the Lagrange multiplier related to the excess area seems observable while the Lagrange multiplier related to the total area is not. Am I understanding correctly?
