Undulation free energy

There is a discussion in the literature about the effect of undulation entropy on the equilibrium membrane tension [14,15], Formally, undulations are included in the surface tension, and thus we need not worry about this. However, if in some model the two are artificially decoupled, one may allow for a very small (positive) surface tension as the equilibrium structure. In other words, the entropy (per unit area) from undulations should compensate for the tension (excess free energy per unit area). 

It should be noted that the interaction free energy of the bilayers reported in Figure 25 is an intrinsic interaction. For a complete picture, one needs to add the van der Waals contribution, but also, more importantly, the contributions due to the undulation. To some extent, these two contributions cancel each other out, and it is expected that the intrinsic effects as discussed above are relevant for the interaction of bilayers at rather close proximity. 

Thermodynamic Equations. We briefly describe the thermodynamic formalism of Ruckenstein.10 Let us consider a liquid crystal of volume V per unit area, in equilibrium, and denote by <5i the average thickness of the water layer and by d2 the average thickness of the oil layer. The Helmholtz free energy of the system can be written as the sum of a free energy F0 of a hypothetical system in which the lamellae are treated as bulk, planar, phases and a free energy Pi, which accounts for the smallness of the lamellae, the interactions between them, and their thermal undulations. One can write... 

The free energy per unit area for a pair of successive lamellae, which includes the effect of thermal undulations, is computed as follows. Let us denote by U(x, X2) the interaction potential between planar water and oil lamellae of thicknesses X and x2, which include the van der Waals, hydration, steric, and electrostatic interactions... 

A similar procedure can be used to calculate the interaction free energy between two undulating (or rough) surfaces. Assumingthat the separation distance between surfaces is normally distributed around H with a dispersion o, the interaction free energy is given by... 

When the system is subjected to an external pressure pcxi, the undulations of the membranes occur in such a manna- to minimize the total Gibbs free energy of the system, which is the sum between the enthalpy of the system and the entropic contribution provided by Eq. (10) (see Appendix B) ... 

For an external pressure pext, the membranes undulations minimize the Gibbs free energy, based on the enthalpy ... 

Minimization of the Gibbs free energy with respect to s provides the value so as a function of the external pressure, which in turn provides the average distance between undulating membranes, for each value of the external pressure [25] ... 

Consider the fluctuations of a surface defined in the Monge representation as z = h(x, y). The area of the flat surface is denoted by A. For slowly varying fluctuations of this surface about a flat shape (h = ho, where is a constant) the additional surface free energy of the undulated interface over that of the flat one (AFg = — yA) is approximately... 

In the case of capillary waves, the flat surface or interface is basically stable and the waves are the result of thermal fluctuations. However, there are interface geometries where a simple interface shape — such as a cylindrical tube of one phase in other or the surface of a cylinder of fluid or solid — is intrinsically unstable because there are other geometries of lower surface areas and hence lower interfacial free energies. A simple energetic/thermodynamic argument (see the next example) can be used to show that the free energy of an undulated cylinder with undulations whose wavelength exceeds a critical value proportional to the cylinder radius, is lower than that of the perfect cylinder. These undulations eventually lead to the breakup of the cylinder into... 

While the Rayleigh instability of a cylindrical surface is driven by the surface energy, the flat surface of a thin film can become unstable under the influence of van der Waals forces (see Chapter 5). In this case, the surface tension provides a restoring force which stabilizes short-wavelength undulations. Again, the critical wavelength is determined by a balance of the free energies (see the problems at the end of this chapter). 

To determine the stabiiity, we consider the interfaciai free energy of Eq. (3.1) and determine whether a smaii perturbation of the interface raises or iowers the free energy, if the free energy is increased by the perturbation, the surface is stabie — aithough the entropy gained by unduiations results in a finite ampiitude to these undulations (see Fig. 3.1 and Fig. 3.2) via thermal fluctuations. If the free energy is lowered by the perturbation, the... 

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