Surface fluctuations bending

Fig. 7.6. Measured power spectrum of the photocurrent fluctuations (heterodyne scheme) wavelength of the surface wave (bending mode) A-30.9 pm, and thickness of the film (Aj + 2A ) —47.S nm. The composition of the solution is the same as in Fig. 7.S. (From Ref. 139, courtesy of North Holland Publishing Company, Amsterdam.)... 

Fluctuations of interfaces are directly relevant to a number of interfacial phenomena. One example, ion transfer across a liquid-liquid interface, will be discussed in Section 6.1. Another example is the behavior of monolayers of surfactants on water surfaces. Surface fluctuations are also fundamental to several processes in water-membrane systems, such as unassisted ion transport across lipid bilayers and the hydration forces acting between two membranes. Here, however, the problem is more complicated because not only capillary waves but also bending motions of the whole bilayer have to be taken into account. Furthermore, the concept of the surface tension is less clear in this case. This topic is discussed in Molecular Dynamics Studies of Lipid Bilayers. 

In the present paper, we will argue that the experimental results of Sentenac and Benattar,19 which apparently contradict sharply the DLVO theory, can be explained in terms of the traditional theory, when the thermal undulations of the film interfaces are also taken into account. The interface will be treated as a membrane subjected to thermal undulations, which can take place at either constant or nonconstant surface area. In the first case the fluctuations generate bending, while in the second they can produce bothbending and change of area. 

In what follows, the two fluctuating interfaces will be replaced by many small, independent surfaces of area S, separated by a distance z (see Figure 4). The (metastable) distribution of the distances between the surfaces, in an ensemble subjected to a constant pressure, will be assumed Boltzmannian. It will be also assumed that the fluctuating interfaces have constant total areas and an elastic bending modulus Kq. Let us first consider that the interfaces interact per unit area through a harmonic potential... 

Bending moduli can in principle be obtained for two types of systems (i) extended, flat surfaces or interfaces, the subject matter of this section, and (ii) surfaces that are already strongly curved, and for which y is zero or extremely low, such as in vesicles or micro-emulsions. For instance such moduli can be inferred from shape fluctuations, from the Kerr effect (sec. 1.7.14] or from polydispersity using some scattering technique. We repeat that this type of measurement is often ambiguous because the bending contributions to the Helmholtz energy can only be estimated when all other contributions are accurately known. 

SDS-l-pentanol-PEO and water-toluene-SDS-l-pentanol-PEO systems and they found an increasing droplet size as a function of the polymer content. Controversial discussions followed [59,60]. If the elastic moduli were the only affected magnitudes one would argue With increasing bending rigidity ( case 1 ) the fluctuations would be diminished, and the radius would decrease due to the opposite surface-to-volume effects. The opposite trends would support case 2 . However, the size could also be changed by polymer incorporation inside the surfactant layer or by polymer-surfactant complexes inside the water domains. These effects would influence the droplet size differently. 

Intensity fluctuation spectroscopy was used in our laboratory to study the dynamic behavior of surface ripples on thin liquid films. Both squeezing and bending modes were examined. To our knowledge one other group of researchers has obtained dynamic light-scattering data from thin soap films but as far as we know, nothing has been published in the official literature. Also some experiments were reported on lipid bilayers in water. ... 

The phenomenon of phase separation naturally gives rise to the presence of interfaces whose study is the focus of the chapters that follow. In this section, we discuss the mathematical definition of an interface or surface (of zero thickness) and show how one can calculate the area and curvature of a general surface. These concepts are particularly useful in the statistical physics of surfaces, interfaces, and membranes since one often has terms in the energy that depend on the area e.g., surface tension) and/or curvature e.g., bending energy). Of course the physical problem is more complicated since often the interface shape and size is not known but is determined self-consistently by the system. Thus one often asks which surfaces have a given area or curvature. An additional complication is that the surfaces of interest are often not deterministic but rather stochastic thermal fluctuations of the surface... 

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