Capillary waves oscillations

Capillary waves may be generated mechanically by means of an oscillating bar, and for this case one writes the solutions to Eqs. IV-25 and IV-26 in the form... 

Ultrasonic atomization is sometimes also termed capillary-wave atomization. In its most common form, 142 a thin film of a molten metal is atomized by the vibrations of the surface on which it flows. Standing waves are induced in the thin film by an oscillator that vibrates vertically to the film surface at ultrasonic frequencies. The liquid metal film is broken up at the antinodes along the surface into fine droplets once the amplitude of the capillary wave exceeds a certain value. The most-frequent diameter of the droplets generated is approximately one fourth of the wavelength of the capillary wave,1 421 and thus decreases with increasing frequency. 

Several possible mechanisms have been proposed to explain the influence of US energy on droplet formation and disruption. One assumes the formation of droplets as a consequence of unstable oscillations at the liquid-liquid interface. Such oscillations contribute to droplet disruption only if the diameter of droplets is considerably larger than the oscillation wavelength, which is about 10 xm for oil-water systems. Therefore, this mechanism, which is known as the capillary waves mechanism and rarely used to explain US-assisted emulsification, is only valid for droplets larger than 10 (xm (first steps of the process). 

One mechanism similar to that of capillary waves is based on the oscillation and subsequent disruption of droplets under US action. The corresponding resonance radius at a frequency of 20 kHz (common for ultrasonic sources) is about 10 xm. This mechanism must be considered as one source of US-assisted emulsification, but can only be applied to immiscible liquid-liquid systems with a diameter within the established range for most of the droplets. In fact, most immiscible liquid-liquid systems are formed by droplets with... 

The experimental methods for the determination of liquid viscosity are similar to those used for gases ( 8.VII F) (i) transpiration, through capillaries, (ii) torque on rotating cylinders, or the damping of oscillating solid discs or spheres, in the liquid, (iii) fall of solid spheres through the liquid, (iv) flow of liquid through an aperture in a plate, (v) capillary waves. Methods (i) and (ii) are mostly used for absolute, the others for comparative, measurements. 

Henceforth we shall use the term capillary waves, or capillary ripples for waves that are so small that interfacial tension contributes significantly to their properties. Two types of such waves can be distinguished spontaneous, or thermal waves and those externally applied. The former type is always present they are caused by spontaneous fluctuations cind have a stochastic nature. In secs. 1.10 and 1.15 it was shown how from these fluctuations interfacial tensions and bending moduli could be obtained. Now the second type will be considered. Transverse or longitudinal perturbations can be applied to the interface, for example by bringing in a mechanically driven oscillator (see sec. 3.7). Such waves are damped, meaning that the amplitude Is attenuated. Damping takes place by viscous friction in the... 

As mentioned in Sect. 2.2.2, the effective interfacial width wD characterizing the bilayer structure may be broadened beyond its intrinsic value w, yielded by a mean field theory (Eqs. 10 and 12). This is due to the capillary wave excitations causing the lateral fluctuation of the depth Ie(x,y) corresponding to the midpoint of the internal interface between coexisting phases. This fluctuation is opposed by the forces due to external interfaces, which try to stabilize the position Ie(x,y) in the center of the bilayer [6, 224, 225]. It was suggested recently [121] that the spectrum of capillary waves for a soft mode phase should be cut off by qb and y. This leads to the conclusion that the effective interfacial width wD should depend on the film thickness D as (wD/2)2= b2+ bD/4. Experimental data [121] obtained for olefinic blends (at T close to Tc) indeed show remarkable increase of the measured interfacial width from wd(D=160 nm)=14.4(3) nm to wd=45(12) nm for thickness D-660 nm, where wD levels off (because is comparable with lateral sample dimensions). This trend is in qualitative agreement with the formula due to capillary oscillations in the soft mode phase . However... 

In the present chapter current relaxation theories will be described first both damping of harmonically generated disturbances and relaxations to transient perturbations. Thereafter, experiments are described, based on the damping of capillary and longitudinal waves, oscillation behaviour of bubbles. Also transient relaxations with pendent drop and drop and bubble pressure measurements are shown. Finally, applications to different interfaces, using surfactants, surfactant mixtures, polymers and polymer/surfactant mixtures are discussed. 

Beside the capillary wave techniques, the oscillating bubble method belongs to the first experiments for measuring the surface dilational elasticity (Lunkenheimer Kretzschmar 1975, Wantke et al. 1980, 1993). For soluble adsorption layers it allows of the exchange of matter at a harmonically deformed bubble surface to be determined. 

The equation derived for the transport of surfactant ions through the DL describes the adsorption kinetics as a reversible process. The qualitatively new result in the theory of ionic adsorption kinetics is the incorporation of electrostatic retardation for both the adsorption and desorption process, which is of essential importance for processes close to equilibrium. Such a situation exists at harmonically disturbed surfaces, used in investigations of adsorption dynamics like the damping of capillary waves or oscillating bubbles. At sufficiently high frequencies the diffusion layer becomes very thin and the adsorption-desorption exchange is controlled only by the ion transport through the DL, i.e. by the electrostatic retardation. At... 

Examination of Fig. 10.4.2A shows that in the breakup of the jet before the drops become spherical they undergo an oscillation about a spherical shape. This oscillation is associated with capillary waves on the drop surface and from dimensional considerations the characteristic oscillation frequency must be alpd ) with d the drop diameter. Rayleigh (1894) (see also Levich 1962) showed this estimate to be exactly the minimum natural oscillation frequency from which the length to form the uniformly spaced spherical drops can be estimated. 

The phenomena of uniform drop formation from a stream of liquid issuing from on orifice were noted as early as 1833 by Savart [2] and described mathematically by Lord Raleigh [3,4] and Weber [5]. In this type of system that is based on their observations, fluid under pressure issues from an orifice, typically 40-80 pm in diameter, and breaks up into uniform drops by the amplification of capillary waves induced onto the jet, usually by an electromechanical device that causes the pressure oscillations to propagate through the fluid. The drops break off from the jet in the presence of an electrostatic field called the charging filed, and thus acquire an electrostatic charge. The charged drops are directed to their... 

To complete the mathematical problem a relationship r(c), a so-called adsorption isotherm, is needed. For the simple case of bubble or drop oscillations (with the surfactant only outside the drop) a solution was derived in Ref. 189 in analogy to the capillary wave theory (183, 184). 

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