Damping coefficient of capillary waves

Insoluble monolayers on an aqueous substrate have been investigated by means of the capillary wave method for many years. Lucassen and Hansen (1966) in their pioneering work neglected the surface viscosity and considered only pure elastic films. Subsequent studies showed that the surface elasticity of real surface films is a complex quantity, and both the equilibrium surface properties and the kinetic coefficients of relaxation processes in the films influence the characteristics of surface waves. However, it has been discovered recently that the real situation is even more complicated and the macroscopic structure of surface films influences the dependency of the damping coefficient of capillary waves on the area per molecule (Miyano and Tamada 1992, 1993, Noskov and Zubkova 1995, Noskov et al. 1997, Chou and Nelson 1994, Chou et al. 1995, Noskov 1991, 1998, Huhnerfuss et al. this issue). Some peculiarities of the experimental data can be explained, if one takes into account the capillary wave scattering by two-dimensional particles (Noskov et al. 1997). 

Fig. 5.18. Dependence of the damping coefficient of capillary waves on SDS concentration at a frequency of 200 Hz [96] curves 1 and 2 are calculated according to Eq. (5.256) for a diffusion - controlled adsorption mechanism (1) and for the barrier -controlled adsorption mechanism (2), 3 corresponds to the experimental data.

Surfactant Results of calculations on the basis of experimental data on the damping coefficient of capillary waves Results of relaxation methods of the bulk phase ... 

Figure 1.5 shows the growth rate of the capillary instability for different liquid viscosities. Viscosity dampens the instability with a damping coefficient of hpl lp and shifts the fastest growing perturbations toward longer waves. For p = 0, Rayleigh solution is obtained, whereas for very viscous jets with (3pk /2p) al2pa , a> = a/ pd) — k a ). The breakup length for a viscous jet is found as ...

Fig. 4 General solution for the dispersion equation on water at 25 °C. The damping coefficient a vs. the real capillary wave frequency o> , for isopleths of constant dynamic dilation elasticity ed (solid radial curves), and dilational viscosity k (dashed circular curves). The plot was generated for a reference subphase at k = 32431 m 1, ad = 71.97 mN m-1, /i = 0mNsm 1, p = 997.0kgm 3, jj = 0.894mPas and g = 9.80ms 2. The limits correspond to I = Pure Liquid Limit, II = Maximum Velocity Limit for a Purely Elastic Surface Film, III = Maximum Damping Coefficient for the same, IV = Minimum Velocity Limit, V = Surface Film with an Infinite Lateral Modulus and VI = Maximum Damping Coefficient for a Perfectly Viscous Surface Film...

Fig. 6 The effect of transverse viscosity on the polar plot of Fig. 4. The damping coefficient, a, is plotted vs. the real capillary wave frequency, 0> for several different transverse viscosities (/x in the figure has units of 10 5 mNsm ). Only the isopleths for Sd = 0 and k = 0 are shown to give the outermost loop of Fig. 4. The plot was generated using the same condition as in Fig. 4, k = 32 431 m, ad = 71.97mN nr1, p = 997.0 kg nr3, r) = 0.894 mPa s and g = 9.80 m s 2...

In the general case transverse surface waves have a vortical component and Eq. (8) is assumed to be an approximation only. This means that Eq. (13) does not allow to calculate the damping coefficient. However, the real part of the complex wave number K for slightly damped capillary waves (ReK > > ImK) can be estimated. 

First, we consider the influence of the liquid s viscosity on the damping of plane capillary waves on deep water. Suppose the liquid has low viscosity so that viscous effects only manifest themselves inside a thin boundary layer near the interface. Hence, outside the boundary layer, the liquid flow is potential, and the potential is described by the Laplace equation, while the liquid flow near the surface is described by the boundary layer equations with the accompanying condition that the tangent viscous stress at the free interface must be zero. The solution of this problem can be found in [2]. The main difference from the case of a non-viscous liquid is the appearance of a coefficient of the form exp(—jSjt) in the... 

A longitudinal wave propagation approach has been adopted in view of the problems of applying capillary wave techniques to liquid/liquid interfaces. These arise from two quarters. Firstly the variation of damping coefficient with elastic modulus can be shown (12) to be negligible at sufficiently high and low elastic moduli. 

The measurements of the propagation characteristics of the capillary wave, e.g., the propagation velocity and the damping coefficient, are effective for the study of the dynamic properties of materials existing on the gas-liquid interface. The theoretical studies for the insoluble monolayers have been performed by Dorrestein, Mayer and Eliassen", and Mann and Du, while those for the soluble monolayer have been performed by van den Tempel and van de Riet, Hansen and Mann, and Lucassen and Hansen. The former has developed their theories taking account of the surface rheologies, and the latter with the assumption that the rate-determining step of surfactant transfer between the surface and the bulk phase is the diffusion process. 

The apparatus used was a modification of that described by Davies and Vose the schematic diagram is shown in Fig. 1. The capillary wave was generated by a vibrator attached to a drive-unit of the trumpet speaker. The vibrator was made of Teflon, which has a weak affinity to all solutions. The flash of the stroboscope was synchronized with the signal of the oscillator, and the stationary image of focus was observed with a microscope. The propagation velocity and the damping coefficient were obtained as has been described in Brownes paper . The frequency range of the apparatus was from 25 Hz to 4 kHz. 

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