Capillary waves thermally excited

Here we briefly present the relevant theory of capillary waves. The thermally excited displacement (r, t) of the free surface of a liquid from the equilibrium position normal to the surface can be Fourier-decomposed into a complete set of surface modes as... 

Recently, the newly developed time-resolved quasielastic laser scattering (QELS) has been applied to follow the changes in the surface tension of the nonpolarized water nitrobenzene interface upon the injection of cetyltrimethylammonium bromide [34] and sodium dodecyl sulfate [35] around or beyond their critical micelle concentrations. As a matter of fact, the method is based on the determination of the frequency of the thermally excited capillary waves at liquid-liquid interfaces. Since the capillary wave frequency is a function of the surface tension, and the change in the surface tension reflects the ion surface concentration, the QELS method allows us to observe the dynamic changes of the ITIES, such as the formation of monolayers of various surfactants [34]. 

There are several experimental techniques suitable for studying e. Some of them are Relaxation after a sudden compression of the monolayer Electrocapillary waves An oscillatory barrier Light Scattering by thermally excited capillary waves. The first two techniques are used in the low - frequency range, below 1 Hz. The last one in the kilohertz range. 

Interface between two liquid solvents — Two liquid solvents can be miscible (e.g., water and ethanol) partially miscible (e.g., water and propylene carbonate), or immiscible (e.g., water and nitrobenzene). Mutual miscibility of the two solvents is connected with the energy of interaction between the solvent molecules, which also determines the width of the phase boundary where the composition varies (Figure) [i]. Molecular dynamic simulation [ii], neutron reflection [iii], vibrational sum frequency spectroscopy [iv], and synchrotron X-ray reflectivity [v] studies have demonstrated that the width of the boundary between two immiscible solvents comprises a contribution from thermally excited capillary waves and intrinsic interfacial structure. Computer calculations and experimental data support the view that the interface between two solvents of very low miscibility is molecularly sharp but with rough protrusions of one solvent into the other (capillary waves), while increasing solvent miscibility leads to the formation of a mixed solvent layer (Figure). In the presence of an electrolyte in both solvent phases, an electrical potential difference can be established at the interface. In the case of two electrolytes with different but constant composition and dissolved in the same solvent, a liquid junction potential is temporarily formed. Equilibrium partition of ions at the - interface between two immiscible electrolyte solutions gives rise to the ion transfer potential, or to the distribution potential, which can be described by the equivalent two-phase Nernst relationship. See also - ion transfer at liquid-liquid interfaces. 

At any finite temperature, the oil/water interface is not flat but is roughened due to the presence of thermally excited capillary waves. The spectrum of capillary waves can be calculated by considering small fluctuations of the order parameter field, (I>(r) = d>(r) -f f/(r), around the planar interface. By an expansion of the free energy functional to second order in r, one finds the energy [42,96]... 

Let us first consider a quiescent emulsion film, say the film between two drops within a floe or cream. At a given sufficiently small thickness of the film, termed the critical thickness (92—97, 115), the attractive surface forces prevails and causes growth of the thermally excited capillary waves. This may lead to either film rupture or transition to a thinner secondary film. Two modes of film undulation have been distinguished symmetric (squeezing, peristaltic) and antisymmetric (bending) modes it is the symmetric mode which is related to the film breakage/transition. The critical thickness, h = h, of a film having area nR can be estimated from the equation (94) ... 

Experimental and theoretical results show evidence that the capillary-wave mechanism is the most fi equent reason for the coalescence of both deformed and spherical emulsion drops. For a certain critieal thiekness (widdi), of the film (gap) between two emulsion drops the amplitude of the thermally excited fluctuation capillary waves begins to grow, promoted by the surface forces, and causes film rupture. The capillary waves can bring about coalescence of two spherical emulsion drops, when flic distance between fliem becomes smaller than a certain critical value, which is estimated to be about 10—50 nm (see Sec. 

In fact, thermally excited fluctuation capillary waves are always present on the film surfaces. With the decrease of the average film thickness, K the attractive surface force... 

Trojanek, A., P. Krtil, and Z. Samec (2001). Quasi-elastic laser light scattering from thermally excited capillary waves on polarised liquid/liquid interfaces Part 1 Effects of adsorption of hexadecyltrimethylammonium chloride at the water/1,2-dichloroethane interface. J. Electroanal. Chem. 517, 77-84. 

One plausible explanation for this discrepancy is that the bare interface whose structure is predicted by self-consistent field theory is broadened by thermally excited capillary waves (Shull et al. 1993, Semenov 1994). The... 

Figure 4.14. (a) An interface without capillary waves, with some intrinsic interface width wi. In (b) the interface is roughened by thermally excited capillary waves. A technique that averages over a distance L will measiue a larger apparent interfacial width w L). 

The interfacial width obtained from fitting the dPS-LDPE NR profiles is given in Table II. This measured interfacial width is described by a self-consistent mean-field theory, which is broadened by thermally excited capillary waves [2]. The measured interfacial width is given by Gaussian quadrature addition of the intrinsic (unbroadened by capillary waves) interfacial width, Ao, and the interfacial width associated with the capillary waves, Ac. This gives -... 

A liquid surface is continually roughened by thermal excitations, which give rise to the hydrodynamic modes known as capillary waves. The r.m.s. amplitudes of the waves are small ( 2 A) but they are efficient light scatterers. The displacement of the liquid surface from its equilibrium position by a wave propagating in the X direction is ... 

Finally, we consider the hydrodynamic theory of thin liquid film rupture. The stability of the liquid films to a great extent is ensured by the property of the adsorbed surfactant to damp the thermally excited fluctuation capillary waves representing peristaltic variations in the film thickness [6]. In addition to the theory of stability of free foam and emulsion films, we consider also the drainage and stability of wetting films, which find application in various coating technologies [7]. 

The surfactant molecules in adsorption monolayers or lamellar bilayers are involved in a thermally excited motion which brings about the appearance of fluctuation capillary waves. The latter also cause a steric interaction (although a short-range one) when two thermally corrugated interfaces approach each other see Sec. VI.E. 

Another property of surfactants is that they influence the capillary waves. This property is employed by several methods for determining the rheological properties of surfactant adsorption monolayers. Moreover, the breakage of liquid films (and the destruction of foams and emulsions) is often related to the growth of the amplitude of thermally excited capillary waves facilitated by attractive surface forces. The theoretical description of these effects was considered in Sec. VIII. 

The paper is organized as follows. First we recall and discuss SnelVs and FresneFs laws for X-ray optics. We then derive the general relation of the density profile across the surface to specular reflectivity (Fig. 1.1a) and to the Qz-variation in grazing incidence diffraction (Fig. 1.1b). Specular reflectivity is illustrated by two examples. The first is reflection from a bare water surface and the determination of the diffuseness of the air-water interface due to thermally excited capillary waves. In the second example we consider a monomolecular film of an amphiphilic molecule, arachidic acid, floating on water, as the area per molecule is varied by a moveable barrier in a Langmuir trough. ... 

The application of X-ray scattering methods to the study of the liquid-vapour interface has been developed theoretically and illustrated by experimental examples. The surfaces of simple liquids are rough due to thermally excited capillary waves. The interface can be characterized by one parameter, the rms. diffuseness, o, which can be determined by X-ray reflectivity measurements, as illustrated for the case of water. 

Capillary wave theory considers the density variation at the interface to be the result of the superposition of thermally excited density fluctuations on a bare intrinsic profile. Mathematically, the instantaneous local density at a... 

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