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Probing liquid surface waves, liquid properties and liquid films with light diffraction

Surface waves on liquids act as a dynamical phase grating for incident light. In this article, we revisit the classical method of probing such waves (wavelengths of the order of mm) as well as inherent properties of liquids and liquid films on liquids, using optical diffraction. A combination of simulation and experiment is proposed to trace out the surface wave profiles in various situations (\emph{eg.} for one or more vertical, slightly immersed, electrically driven exciters). Subsequently, the surface tension and the spatial damping coefficient (related to viscosity) of a variety of liquids are measured carefully in order to gauge the efficiency of measuring liquid properties using this optical probe. The final set of results deal with liquid films where dispersion relations, surface and interface modes, interfacial tension and related issues are investigated in some detail, both theoretically and experimentally. On the whole, our observations and analyses seem to support the claim that this simple, low--cost apparatus is capable of providing a wealth of information on liquids and liquid surface waves in a non--destructive way.


I. INTRODUCTION
Optical probes are usually non-destructive -they do not change the properties of the medium being probed in any significant way (unless the probe power is large). Diffraction and scattering of light, as is known over centuries, are therefore capable of providing a variety of information. In this article, we revisit a well-known classical method of investigating liquid surface waves using light diffraction. Although this method is well-studied in the context of measurements of surface tension of liquids [1], our motivation for revisiting it is primarily aimed at demonstrating its utility and capability in understanding a wider class of situations/phenomena. In particular, we investigate two aspects -surface wave profiles and liquid properties for waves on a single liquid and also for waves on liquid films.
Before we begin with details, it is worth summarizing some of the earlier work on liquid as well as solid surface waves. Use of laser light scattering is a unique technique for noncontact, noninvasive study of surface acoustic waves and capillary waves [2]. Technological advances directed toward diverse applications of surface acoustic waves on solid surfaces are now well established, and has resulted in important developments in the fabrication of practical devices [3,4,5,6,7,8,9,10,11].
However, relatively few articles deal with surface waves on liquid surfaces [12,13,14,15,16,17,18,19,20]. To mention a few, high frequency capillary waves at the surface of liquid gallium and the mercury liquid-vapor interface have been studied by means of quasielastic light-scattering spectroscopy [21]. The spatial damping coefficient of low-frequency surface waves at air-water interfaces, using a novel heterodyne light-scattering technique, has been measured and reported in [22]. Quasi-elastic light scattering has been used to study thermally excited capillary waves on free liquid surfaces over a considerably wider range of surface wavenumbers [13,16]. Measurements of the decay coefficient of capillary waves in liquids covered with monolayers of stearic acid, oleyl alcohol, and hemicyanine are reported in [23]. Also, the measurement of the pressure-area compression isotherm in Langmuir monolayer films using the laser light diffraction from surface capillary waves is reported in [24].
In a recent article, we have shown how the effect of interfering waves on a liquid surface could be inferred from the nature of the diffraction pattern [15]. Here we begin by proposing a combined method of simulation and experiment, sufficiently general in nature and capable of verifying the nature of the actual liquid surface wave profile. Having known the profile, we then focus on the dispersion relation and discuss experiments to determine the surface tension and viscosity of liquids. Finally, we turn to liquid films where we try to understand the profile through the various modes (surface and interface) and then go on to check the values of interfacial tension.
Section II describes the basic scheme of our experimental arrangement. The methodology of our experiment and typical results pertaining to the surface wave profile are discussed in section III. In section IV, we discuss theory, observations and measurements of surface tension and viscosity for different liquids. Section V first discusses the theoretical platform for surface capillary wave on liquid films. We then verify our theory with experiments.
Finally, in section VI, we summarize our results with some concluding remarks. As shown in Fig. 1, a petridish of diameter about 18.5 cm is filled with the experimental liquid to a depth of ∼ 1 cm. A metal pin with its blunt end glued vertically upright to the diaphragm of a loud speaker (held above the petridish) acts as an exciter. When slightly immersed in the liquid and driven by a low frequency sinusoidal signal generator, this exciter vibrates vertically up and down and generates the desired liquid surface waves . To study the   profile of surface waves we place the petridish-loud-speaker assembly on the prism-table of a   spectrometer in order to choose the diffracting element at any position on the liquid surface   and to record the corresponding position from the prism-table scale (with a least count of 0.1 degree). Light from a 5 mW He-Ne laser (λ = 632.8 nm) having a beam diameter of ∼ 1.8 mm is directed to fall on the liquid surface at an angle of incidence of 77 0 (θ i ), as noted in our experiment. The laser beam incident on the dynamical phase grating formed on the liquid surface, produces Fraunhofer diffraction pattern which is observed on a screen placed at a fairly large distance (3.42 meters in our case) from the diffraction center. The images of the diffraction pattern are recorded using a digital camera (make Sony, DSC -P93, 5.1 mega pixels). Spurious noise in the image is removed using Photoshop software.
The intensity across the diffraction spots are measured using a photo-detector operating in the linear dynamic range of the photo-currents' response to optical power. During our experiment, the room temperature recorded is 25 0 C.

B. Theoretical background
We discuss here, briefly, the diffraction mechanism which lies at the heart of all our experiments. When monochromatic light of wavelength λ is incident on a circular surface wave of frequency ω and wavevector K, the phase modulation produced by the surface wave is given as where, the factor of cos θ i appears due to oblique incidence of the incident monochromatic light [11,14,15,25]. The field strength E of the diffraction pattern can be estimated from the Fourier transform of the aperture function (in this case the surface wave produced by single exciter). The intensity of the diffraction patterns, obtained from EE * , is given by [25,26] I where z is the horizontal distance between the location of the laser spot on the liquid surface and the screen and Λ = 2π K , is the wavelength of the surface wave. h is the amplitude of the surface wave. x ′ is the coordinate which measures the distance of the diffraction spots from a reference point (central spot) on the observation plane. J n is the Bessel function of order When we generate capillary waves by means of a single pin exciter, the amplitude of the wave at the point of oscillation is almost the same as the amplitude of the oscillating pin in a low viscous liquid [15]. The intensities of different orders depend only on the amplitude of the capillary wave (h), if other parameters like θ i and λ are kept constant (Eqn. 2).
Theoretical plots of wave amplitude versus the intensity ratios, I 0 /I 1 , I 1 /I 0 and I 0 /I 2 , using Eqn. 2 (keeping θ i and λ fixed) are shown in Fig. 2. In our experiment, for a particular ω, the intensity of different order spots in the diffraction pattern are measured by a photodiode detector when the laser beam is focused along a radial line on the liquid surface. The intensity ratios like I 0 /I 1 and I 0 /I 2 are then evaluated. By comparing our experimentally measured ratios with those obtained from the theoretical plot (Fig. 2), we estimate the average value of h. It is to be noted that, for I 0 << I 1 , the value of this ratio I 0 /I 1 becomes very small and could yield erroneous results. In such cases, we used the ratio I 1 /I 0 for estimation of h (however, we were always below the limit for which this ratio becomes very large). Again, by measuring x ′ , we determined the wavelength (Λ) of the surface capillary waves to be 2.1 mm [for more details see [15]]. For several exciters placed at different types of geometric configurations (eg, along a line, or a polygon etc.), the surface wave patterns change, and hence Eqn. 2 must also change.

III. THE SURFACE WAVE PROFILE
A. Simulation of the profile Low frequency surface waves are not visible to the naked eye. One may appreciate this when we simulate the surface wave profile using realistic dimensions and parameters. In a previous study [15], we measured the surface-wave parameters (wavelength, amplitude at the point of excitation) at a given oscillation frequency (220 Hz). Here, we use similar typical values to generate theoretically the distribution of waves on the liquid surface. The equations, which we use to simulate the wave profile for the cases of single and double-pin exciters are (at a fixed time 't', say t=0) and respectively. Here (x 1 , y 1 ) and (x 2 , y 2 ) are the centers of oscillations and δ is the spatial damping coefficient of the liquid. To mimic the intended experiments, we assume that each wave has the same frequency ω, wavelength Λ, and amplitude h (at the center of oscillation).
In our simulations, we use the typical values of Λ = 2.1 mm, h =1.0 mm and D = 8.4 mm (separation between pin-exciters) [15]. δ is chosen to be 0.235 cm −1 (for water at 220 Hz, we obtain this value from experiments discussed later in section IV). The corresponding surface-wave profiles estimated for the two cases of interest are shown in Fig. 3(a) and  These results are shown in Table-I this behavior theoretically, we use the interference treatment for n-oscillating sources. We consider n (≥ 2) sources lying in a straight line. Representing the wave generated by each oscillating source as h exp(i(K.r n − ωt)), where h is the amplitude and r n is the position vector from the n-th source to the point of observation, we find the resultant field ψ P at a certain point P as [26], where R = 1 2 D(n − 1)Cos(φ) + r 1 , is the distance from the centre of the line of oscillators to the point P and β = KDCos(φ), is the phase difference between adjacent sources. Again φ is the angle between R and the line joining the sources of oscillation. For minima at P, the , where m is an integer but not equal to or multiple of n. If we choose D Λ = 4, then Cos(φ) has only 16 allowed values, i.e. 16 hyperbolic minima lines are possible. The angular ditribution of these asymptotic lines perfectly match with the observed pattern. We have carried out this comparison successfully for higher order n-values upto 5. From the above calculations we have seen that the number (N o ) of asymptotic minima lines in the region above (or below) the line joining the oscillation sources is directly proportional to the ratio D Λ and also the number of the interval (n − 1) between the exciters. Taking care of the ± sign of Cos(φ), we can express N o as which gives the same result (for different n nalues) as obtained before. This relationship can, just by measuring the number of minima lines for given D and n, estimate Λ of the capillary waves.
Next we study n (≥ 3) oscillation sources forming a closed loop shape of an equilateral triangle, square, pentagon, and a hexagon. The number (N c ) of minima lines (which are lying on ether side of a symmetric axis) in this case can be similarly expressed as, The number of lines for the above stated closed geometrical figures at the same values of D and Λ are 12, 16, 20 and 24 respectively. We have not been able to calculate exactly the angular distribution of the minima lines. However, using the simulation mentioned above, etc., are well-known and gives accurate results [27,28]. Here, we focus on a single optical set up which can measure both surface tension and viscosity of low-viscous liquids with reasonable accuracy. Though the measurement of surface tension of liquids using Fraunhofer diffraction of laser light by surface waves is well known in the literature [1], to the best of our knowledge the estimation of the viscosity of a liquid using this probe and, in particular, the simple method which we have followed, is certainly new. In the following we elaborate on the above mentioned measurements in some detail.

A. Surface tension
Consider the well-known dispersion relation for surface capillary wave on liquids [29,30,31] Here ω and K are the angular frequency and wavevector of the surface capillary wave, respectively. α and ρ are the surface tension and density of the liquid and g is the acceleration due to gravity. The above relation shows that waves at the surface of the liquid are dominated by gravity as well as surface tension. Neglecting the gravitational effect and taking the logarithm on both sides of the Eqn. 8 we get which is an equation of a straight line for ln ω vs. ln K with a slope of 3 2 and y-intercept 1 2 ln α ρ , which contains the surface tension α. In our experiments, we have generated the surface capillary wave on the liquid surface with a single exciter. As we discussed before, the expected surface wave profile [shown in Fig. 3(a)] acts as a reflection phase grating for the incident laser light and the intensity distribution satisfies Eqn. 2. If we take the separation between the central order and the first order (n=1) diffraction spots to be d (see Fig. 7), then from the delta function in Eqn.
2 we get the expression for the wavenumber K as where θ = π 2 − θ i is the grazing angle of incidence of the laser beam. For small grazing angle, we can approximate K as where l is the perpendicular height of the central order spot from the liquid surface level (see Fig. 7). As ω changes, the distance between bright spots of the diffraction pattern also of d are obtained through averaging these data. For each frequency ω, we have repeated the experiment at least four times. Hence, using Eqn. 11, we estimate the value of K for a particular ω. In addition, it is reasonable to assume that the frequency, ω, of capillary wave is the same as that of the exciter (driven by a function generator) [15]. Thus, the values of ω are found directly from the frequency readings of the function generator driving the exciter. Following these measurements, we plot a ln -ln graph for K vs. ω and evaluate the slope and y-intercept of this curve. The slope of the curve ( 3 2 ) should verify the power law predicted by the dispersion relation while the y-intercept should give the surface tension.
Our experimental results on three different liquids (water, 1-hexanol and kerosene -a mixture of petrolium hydrocarbons) are shown in Fig. 8. The non-linear least square fit (shown by lines in Fig. 8) to the data with Eqn. 9 verifies the 1.5 exponent of the dispersion relation within experimental uncertainty (1.5±0.03). Using the known value of ρ for each liquid, we estimate the value of α from the measured y-intercept of the straight lines. It is to be noted that the maximum percentage error in measuring α is limited to 6% -8% (for different liquids) by our measurement procedure. We have tabulated the measured values of surface tension for different liquids in Table-II. These results match quite well with the standard values of surface tension for the corresponding liquids [32].

B. Viscosity
Using the same experimental set-up, we now move on towards estimating the viscosity of a given liquid. We mentioned before that for capillary waves generated by means of a single pin exciter, the amplitude of the wave at the point of oscillation is almost the same as the amplitude of the oscillating pin in a low viscous liquid [15]. However, if we imagine a radial straight line from the source of excitation (on the liquid surface), then along this line the amplitude of surface wave is gradually damped due to the viscosity of the liquid (see Fig. 7). The laser beam is focused along a radial line at different distances on the liquid surface from the point of excitation (x in Fig. 7). We have seen in Sec. II that the intensities of the diffraction spots of different orders depend only on the amplitude of the capillary wave, if other parameters like θ i and λ are kept constant (Eqn. 2). By comparing our experimentally measured intensity ratios of different orders in the diffraction pattern with those obtained from the theoretical plot (Fig. 2), we estimate the average value of h at a particular location on the liquid surface. In Fig. 9, we plot x, the distance from the point of oscillation vs. h, the capillary wave amplitude for three experimental liquids (water, kerosene, and 1-hexanol). We fit the data points with an exponential decay function, where we kept the decay constant (also known as spatial damping coefficient of the liquid), δ, as varying parameter. The values of δ for three different liquids are tabulated in Table-II. The wavevector for a wave of fixed real frequency, ω, has a small imaginary part, which contributes to the spatial damping. Substituting K = K 0 + iδ (where δ << K 0 ) in the wellknown Navier-Stokes equation for waves on a liquid-air interface, one gets [22,33], where η is the viscosity of the liquid. Experimentally, we have obtained the values of α and δ for the above mentioned liquids. The estimated value of viscosity of different liquids, using Eqn. 12, are shown in Table-II. The experimental plots in Fig. 9 are for frequency 220 Hz.
We have also checked our results for two other frequencies (260 Hz and 300 Hz) and have obtained similar results (see Table-II where, here too, we neglect a term due to gravity. Typically, if we choose y = 1 cm and a range of K from 20 to 80 cm −1 (as used in our earlier experiment), tanh (Ky) is nearly unity, and the dispersion relation reduces to the one given by Eqn. 8. The interesting problem we focus on now is that of wave propagation on a liquid film spread out on another liquid. If the thickness of the film is very low, then not only the surface tension but also the interfacial tension across the liquid-film boundary, play a role in the propagation mechanism of the waves. We first work out a general dispersion relation where the surface and interfacial tensions as well as the film thickness appear explicitly in our mathematical expressions. with H being very large. The film is located above y = 0 with a thickness y ′ << H. Under these assumptions, the general expressions for the surface capillary waves in the region (I): -H ≤ y ≤ 0 and in region II : 0 ≤ y ≤ y ′ are given by [30], and respectively. Here A, B and C are arbitrary constants. The above two waves must obey the following boundary conditions arising from the continuity of the velocity component(v y ) and that of pressure at the surface of separation of the two liquids, i.e., Evidently, the first two conditions apply at y = 0 while the third one is valid at the top surface at y = y ′ . In the boundary conditions above, we neglect the effect of gravity in all calculations below. Using Eqn. 14 and Eqn. 15 in Eqn. 16 and Eqn. 17 we obtain and Substituting Eqn. 19 and Eqn. 20 in Eqn. 18 with y = y ′ , we construct a quadratic equation for ω 2 given by: where, we have used the notations ρ ′ ρ = r, α ′ K 3 ρ ′ = ω 2 s and αK 3 ρ = ω 2 i for simplicity. The two roots of Eqn. 21 are the new dispersion relations for our given problem. The two roots are where, P = rω 2 s + ω 2 s tanh Ky Consider two limiting cases of the above dispersion relation. Let us first assume y ′ → ∞.
Then ω 2 + → ω 2 s and ω 2 − → ω 2 i 1+r = αK 3 ρ+ρ ′ . Thus, when y ′ is very large, the ω + mode corresponds to the waves propagating on the upper liquid surface (thus called surface mode). On the other hand, the ω − mode involves the interfacial tension [30] as well as the density of the lower liquid and may be termed as an interface mode. In such a situation (y ′ large), reflecting the probe laser beam off the surface wave profile on the liquid film, we can obtain information In order to avoid any decrease in thickness due to evaporation, we have taken observations within 5 minutes of the formation of the uniformly spread film. The measurements are of the same type as those mentioned while studying surface tension (discussed in the earlier section) except that, in the present case, we do the experiments for varying film thicknesses.
Here, we show our experimental results only for o-xylene films of different thicknesses on water. For films of the other liquids mentioned above, on water, we have observed similar behavior.
To find out the mode which is present on the film surface for a liquid-film of o-xylene on water, we first plot the experimental ω ∼ K behavior with film thickness as a parameter.
22 [32]. It is evident that for a film of thickness as low as 0.016 cm, the measured data corresponds to ω + mode [ Fig. 11 [32].