Applications area and advantages of the capillary waves method

Transcription

1 Applications area and advantages of the capillary waves method Surface waves at the liquid-gas interface (mainly capillary waves) provide a convenient probe of the bulk and surface properties of liquids. The consequent application of this experimental method and corresponding theory of surface wave damping can yield essentially new information about visco-elastic properties in various two-dimensional systems containing surfactants and polymers. The experimental technique applied in our group is based on the reflection of a laser beam from the oscillating liquid surface and gives the possibility of fast and sufficiently precise measurements of the capillary wave characteristics, especially as compared with earlier techniques where the wave probes touch the liquid surface. The main advantages of the capillary wave method can be summarized as follows: 1. broad frequencies range from about 30 Hz to about 3000 Hz, which can be easily enlarged when using longitudinal surface waves or Faraday ripples 2. contact-less method, which can be used for remote sensing of the liquid surface 3. absolute experimental technique which does not require any preliminary calibration 4. unlike many other methods for the investigation of non-equilibrium surface phenomena, measurements of capillary wave characteristics are possible at really small deviations from equilibrium. For example, the wave characteristics can be easily measured even when the ratio of the wave amplitude to the wavelength is less than 0.1%. This allows us to study the complicated influence of non-linear hydrodynamics processes using the theory of linear waves 5. the influence of the three-phase contact (contact angles) on the measurements of surface properties, usually impossible to be estimated independently, especially at nonequilibrium conditions, is entirely excluded 6. theory of capillary waves is one of the most developed sections of hydrodynamics, and corresponding literature is very comprehensive; all theoretical results can be used for interpretation of experimental data on surface wave characteristics 7. linear surface waves (cf 4.) are one of the simplest hydrodynamics subjects; for example, the hydrodynamic description of liquid jets and bubbles requires more sophisticated approaches even in a linear approximation. 1

2 Theoretical basis The most general equation connecting the characteristics of capillary waves, the parameters of the coexisting bulk phases as well as the dynamic and equilibrium properties of the interfacial layer is the dispersion equation. To derive this equation for capillary waves one has to solve the boundary problem for the hydrodynamic equations of a viscous fluid. The problem can be essentially simplified by neglecting all non-linear terms in the hydrodynamic equations because the amplitude of capillary waves is much less than the wavelength. The following set of functions represents the problem solution for the gas-liquid interface under the condition that the liquid motion fades away from the interface v x = ( ikae kz mbe mz ) e i(kx+ωt) (1) v z = ( kae kz + ikbe mz ) e i(kx+ωt) (2) p = p kz i(kx+ωt) 0 ρgz + iωρae e (3) Here the waves propagate in x-direction, the z-axis is directed towards the depth of the liquid and the xy plane coincides with the unperturbed interface, v x and v z are the liquid velocity components along the corresponding axes, p and p 0 are the pressure and hydrostatic pressure at z = 0, respectively, ρ and µ are density and viscosity of the liquid, respectively, g is acceleration of gravity, t is the time, ω is angular frequency, k = 2π/λ +iα is the complex wave number, λ is the wavelength, α is damping coefficient, m 2 = k 2 + iωρ/µ, Re[m] > 0, A and B are the wave amplitudes, which must be determined from the boundary conditions. The relationships follow from the balance of the tangential and normal components of the forces at the gas-liquid interface, taking into account the surface tension γ. Substitution of expressions (1)-(3) into the boundary conditions leads to a system of two homogeneous algebraic equations. The determinant of the system must be equal to zero if a non-trivial solution exists. This condition yields the dispersion equation of surface waves ( ρω γk ρgk)( ρω mk ε) εk ( γk + ρgk) + 4iρµω k + 4µ ω k (m k) = 0 (4) where ε = ε r +iε i is the complex dilational dynamic surface elasticity. This complex equation is equivalent to two real equations. If the surface tension and wave characteristics are known (for example, determined from experiment) equation (4) allows to evaluate the real and imaginary parts of the dynamic surface elasticity. 2

3 Principles of measurements The damping coefficient and the wavelength can be determined from measuring the wave amplitude and phase at different distances from the wave generator. Because the transverse capillary waves disturb the shape of the liquid surface the wave amplitude can be determined by measuring the local deviation of the surface from the mean position (surface elevation). The shape of the liquid surface is described by a sinusoidal function (cf. Eq. 2) ζ = ζ e cos( ωt α x 0 kx) where ζ is the surface elevation, ζ 0 is the amplitude of elevation at x = 0. The non-contacting capillary wave excitation proves to be more convenient from the point of view of physico-chemical applications. Acoustic or alternating electrical fields can be used for this purpose. In the latter case, an alternating voltage is supplied to a razor blade placed close to the liquid surface. Due to electrocapillarity, the liquid tends to rise up to the region of the intense electric field at the tip of the razor blade. Surface tension and gravitation oppose this displacement. Because the strength of electrocapillarity is proportional to the square of applied voltage, the capillary waves are generated at a frequency twice that of the applied wave. The most precise measurement of the surface oscillation of the type given by Eq. (5) is the specular reflection of a laser beam from the liquid surface. The reflection angle from the liquid surface is proportional to the local slope of surface and thus to the wave amplitude in case of linear waves. Usually, the reflected beam is directed to the optical position sensitive detector (PSD). The light spot oscillates along the surface of the PSD leading to an alternating electric signal at the PSD output with the same frequency as that of the capillary wave. The amplitude of the signal is proportional to the deflection of the beam from the central position and, consequently, to the wave amplitude. The movement of the optical system parallel to the surface relative to the wave generator allows to measure the signal at different distances. According to Eq. (5) the wave amplitude and, consequently, signal amplitude I depend exponentially on the distance. Therefore, the measurement of I as a function of x yields the damping coefficient α = ( ln I ln I ) ( x ) x1 where I 1 and I 2 are the signals corresponding to the distances x 1 and x 2. In practice one usually measures the value of I at several values of x and the slope of the linear dependence ln I = f(x) yields the damping coefficient. The phase of the electrical signal (5) 3

4 is changed too at the movement of the optical system relative to the wave generator. If the distance changes from x to the x + λ, the phase changes by 2π. Thus, measurements of the phase allow us to obtain the wavelength. The phase of the electrical signal is usually compared with the constant phase of a standard electric generator by means of Lissajoux figures on the oscilloscope screen or by means of the output signal of a phase lock-in amplifier. Because of high precision at the determination of the beam position at the surface (micrometrical screw!), the measurements of the wavelength are usually much more precise than the determination of the damping coefficient (precision of α is a few percents while that of λ is about ± 0.1%). Experimental set-up A schematic diagram of the experimental set-up and the corresponding photographic picture are shown in Figs. 1 and 2, respectively. Fig. 1 Schematic set-up for measurements of the capillary wave characteristics 4

5 A stainless steel razor blade 1 (60 mm wide) is placed above the centre of the liquid surface 2. It is attached to a 3D-translation stage, which can be moved in directions parallel and perpendicular to the liquid surface. Moreover, a special device allows a smooth change of the angle between the blade and the surface so that the blade can be fixed parallel to the surface with a precision better than ± 0.05 degree. The function generator MXG-9810 (Conrad Electronics) 3 produces a square-type voltage (up to approximately 10 V) of a certain constant frequency. This voltage is amplified by the high-voltage amplifier HOPP-0.6B5 (Wulf- Mueller, Hamburg) 4 up to about 300 V and is supplied to the blade. The liquid under investigation is grounded by means of an auxiliary platinum electrode 5. Capillary waves excited by electrocapillarity propagate along the liquid surface in a direction perpendicular to the blade. Because the negative part of the output signal at the function generator is absent, the wave frequency coincides with the generator frequency. All optical devices are attached to a separate plate. The light beam from the cylindrical laser (He-Ne , Linos Photonics) 6 goes through the beam splitter cube 7, reflects from the reflection prism 8 mounted on a linear translation stage 9, and then reflects from the liquid surface 2. The reflected beam returns to the reflection prism 8 and beam splitter cube 7, changes direction, reflects from the second reflection prism 10 and reaches finally the PSD 11. Fig. 2 Photograph of the experimental set-up. Because the power of the laser is not very important, it is used with the lowest power. The only requirement is that the power of the laser beam at the PSD must exceed the power of other beams incident to the PSD from all other light sources by at least an order of magnitude. 5

6 Another important condition relates to the diameter of the laser beam at the liquid surface d (d < 0.5 λ). If d > λ, the oscillations of the surface do not lead to oscillations of the beam. This condition is especially important at high frequencies of capillary waves. In this case, a special optical devices is sued to decrease the diameter of the light spot. However, if the range of high frequencies is out of the scope, one can measure the capillary wave characteristics without any additional optical devices. The experimental accuracy is determined also by the mechanical system moving the reflection prism 8. The movement of the translation stage must be parallel to the direction of the capillary wave propagation. This can be achieved by means of screws regulating the orientation of the upper surface of the tables supporting the translation stage. Moreover, the movement of the stage has to be smooth enough to not excite additional oscillations of the liquid surface. Therefore, the isolation of the vessel with the liquid under investigation from chaotic vibrations is of crucial importance. This isolation can be achieved by using a piezoelectric vibration free unit. After a preliminary amplification by an embedded device the alternating signal from PSD is passed to a lock-in amplifier (LIA- MVD-200L Dual phase Lock-In-Amplifier Module, FEMTO Messtechnik GmbH, Berlin) 12. Fig. 3 Typical measured signal as a function of the distance between reflection point and blade. 6

7 The reference signal supplied to the lock-in amplifier from the frequency generator 3 is divided into two orthogonal signals with a phase difference of 90 deg. Then both are compared with the measured signal. The two resulting signals are filtered and transformed into DC-signals, which correspond to the X (without additional phase shift relative to the measured signal) and Y (phase shift 90 deg) components of the output signal. These signals depend on the amplitude of the initial input signal and on the phase differences between the latter and the corresponding components of the reference signal. Additionally, both of these DC-signals go through a vector summing amplifier unit (embedded in the lock-in amplifier) in order to obtain only the magnitude oscillation at each point of the investigated surface. The signals from the lock-in amplifier output pass to the oscilloscopic card of the PC 13, which provides their graphic presentation and evaluation. The magnitude output allows to determine only the damping coefficient α. The simultaneous determination of both capillary wave characteristics is possible if one of X- or Y-outputs is used. In this case, Eq. (5), taken at t = 0, describes the dependence of the signal on the distance between the blade and beam reflection point at the surface, from which α and λ can be evaluated. An example of this dependence and principles of its processing are schematically shown in Fig.3. 7