Fluctuations of Fluid Membranes

We consider a single membrane with zero spontaneous curvature, described by the bending free energy of Eq. (6.22). In the Monge gauge, in the approximation of small curvatures, the free energy per unit area can be written (see Chapter 1)  

It is only the weak, global conservation of area constraint that gives a term in the free energy proportional to the area. This constraint is negligible for large areas and we thus consider here the thermodynamic limit of the fluctuations of infinitely large, nearly flat membranes. 

The mean-square fluctuation of the height increases algebraically with the system size  

It is this correlation function that defines the curvature of the membrane since it describes how the normal bends as one goes along the membrane a distance r. The correlation function is given by 

The normal-normal correlation function can be used to define the persistence length of the membrane as the distance over which the normal becomes decorrelated via the thermal undulations the distance r at which gnir) is of order unity. The persistence length, is defined as 

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Podgornik, R., Parsegian, V.A. Thermal-mechanical fluctuations of fluid membranes in confined geometries The case of soft confinement. Langmuir 1992, 8, 557-62. 

G. Gompper, J. Goos. Fluctuations and phase behavior of passages in a stack of fluid membranes. J Phys II France 5 621-634, 1995. 

Vesicle shapes are not static entities but show quite pronounced thermal fluctuations due to the extreme softness of fluid membranes [8-13]. Thus, it is necessary to understand the interplay of equilibrium thermal fluctuations with the mean shape determined by the membrane material parameters and vesicle geometry [14,15]. The signatures of several shape transitions in the fluctuation spectrum of a prolate vesicle are described. 

In some sense the physical properties of fluid membranes are unique because they have negligible surface tension. Consequently, their free energy is governed by their geometrical shape and its fluctuations. The rigidity k associated with the restoring force to layer bending is then the important modulus which in many cases will determine the physical state of the membrane. From a biophysical view point the physical nature of a fluid membrane surface may in some cases have a profound influence on the precise mechanism of membrane-membrane interactions which influence processes such as cell-cell contact. 

Soft membranes attracted the attention of physicists in recent years because of novel properties related to their nonplanarity. Typically, thermal energies are sufficent to produce marked deviations from the planar state. In the case of fluid membranes, these fluctuations depend primarily on their bending elasticity. The energy of bending per unit area, g, is usually expressed by a quadratic form in the principal curvatures, C and C2 which are splays in the language of liquid crystals ... 

Spontaneous and tension-induced adhesion of fluid membranes, including the unbinding transition, is a very new field of physics, both experimentally and theoretically. We have tried to give a reveiw of present activities, emphasizing theoretical concepts. The latter are all based on the assumption that the undulations controlled by the bending rigidity are the only out-of-plane fluctuations of these membranes. The theoretical models were developed with a view to electrically neutral biological model membranes. They fail in part when applied to these systems. 

Other possible direct probes are optical experiments similar to studies [113] of vesicles but expanded towards shorter A (20-30 A). Alternatively neutron spin-echo studies of stacked bilayer arrays, which can probe the 10-30 A range [114], might possibly be applicable here. Finally, the x-ray grazing-incidence technique has been shown to be a powerful tool for studying short wavelength fluctuations at fluid interfaces [100]. The application of this technique to the investigation of membrane surface fluctuations can reasonably be expected in the near future [115,116]. 

H. Noguchi and G. Gompper, Dynamics of fluid vesicles in shear flow effect of membrane viscosity and thermal fluctuations, Phys. Rev. E 72, 011901 (2005). 

All of the above considerations have sometimes led to a too rigid picture of the membrane structure. Of course, the mentioned types of fluctuations (protrusions, fluctuations in area per molecule, chain interdigitations) do exist and will turn out to be important. Without these, the membrane would lack any mechanism to, for example, adjust to the environmental conditions or to accommodate additives. Here we come to the central theme of this review. In order to come to predictive models for permeation in, and transport through bilayers, it is necessary to go beyond the surfactant parameter approach and the fluid mosaic model. 

The efficiency of the filtration process should not be significantly affected by the pressure differential across the surface of the membrane or pressure fluctuations produced by the pumping of fluids through it. 

Komura, S. and Seki, K. (1993) Dynamical fluctuations of spherically closed fluid membranes. Physica A, 192,27-46. 

B.-Y. Ha. Modes of counterion density fluctuations and counterion-mediated attractions between like-charged fluid membranes. Preprint (2000). 

Fig. 9 C-DARR spectra of the aliphatic regions of U-[ C, NKwhyfi-PR at 273 (blue, gel phase) and 313 K (red, liquid crystal phase) (a) and the modified homology model of green PR (b). Helical residues influenced by changes in membrane elasticity (labeled in blue) are found in helices C, E, F, and G as well as in loops EC and EF. These residues disappear in the fluid membrane but are visible in the gel phase. This indicates that especially helices C and G but also E and F undergo thermal equilibrium fluctuations in the ground state of PR. Adapted from [41] with permission from the American Chemical Society...

Flexible, solid membranes, are also of interest. However, they are experimentally much less prevalent and are somewhat more complicated to treat since in addition to the membrane shape one must include the effects of shear. Their curvature energy is discussed in the problems at the end of this chapter. Another type of system that has received much theoretical attention is that of a tethered membrane which may describe polymerized, but not crystalline sheets. While a single fluid membrane that is unconstrained by walls or other membranes is strongly affected by thermal fluctuations ( crumpled ), solid membranes, particularly if self-avoidance of the membrane is included, tend to be more weakly affected by fluctuations and are hence flattef . 

Valves in the gas headers control the pressmes on the cells. In the usual case where pressures are close to atmospheric, pressure drops everywhere must be kept very low. Each header is sized generously in order to keep the gas pressure essentially equal on every cell in the line. The two gas pressmes can be controlled independently. However, the differential pressure between the cathode and anode chambers may be more important than the individual header pressures. In a membrane cell, for example, fluctuations in differential pressure cause vibration of the membranes. This can lead to physical damage, sometimes allowing passage of cell fluids, and to early failure. A frequent practice is to control the differential pressure directly, forcing any pressure fluctuations on the two... 

West B, Schmid F (2010) Fluctuations and elastic properties of lipid membranes in the fluid and gel state a coarse-grained Monte Carlo study. Soft Matter 6 1275-1280... 

Fournier JB, Barbetta C (2008) Direct calculation Irom the stress tensor of the lateral surface tension of fluctuating fluid membranes. Phys Rev Lett 100 078103... 

Bending elasticity is a long-standing concept of continuum mechanics and has been used mostly to deal with solid rods and plates. More recently, it has been applied to fluid membranes, especially the lipid bilayers of giant vesicles, to understand their equilibrium shapes and shape fluctuations. For continuum theory to be applicable, the membranes should be reasonably smooth or, in other words, not fluctuate too much. 

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