Article pubs.acs.org/Langmuir
Vibration Spectroscopy of a Sessile Drop and Its Contact Line S. Mettu† and M. K. Chaudhury* Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015, United States ABSTRACT: Resonance frequencies of small sessile liquid drops (1−20 μL) were estimated from the power spectra of their height fluctuations after subjecting them to white noise vibration. Various resonance modes could be identified with this method as a function of the mass of the drop. Studies with water drops on such supports as polystyrene (θ ≈ 80°) and a superhydrophobic surface of microfibrillar silicone rubber (θ ≈ 162°) demonstrated that the resonant frequency decreases with the contact angle, θ. This trend is in remarkable agreement with the current models of the resonant vibration of sessile drops. A novel aspect of this study is the analysis of the modes of a slipping contact line that indicated that its higher frequency modes are more severely damped than its lower ones. Another case is with the glycerol−water solutions, where the resonance frequency decreases with the concentration of glycerol purely due to the capillary effects. The interface fluctuation, on the other hand, is strongly correlated with the kinematic viscosity of the liquid. Thus, these experiments provide a means to measure the surface tension and the viscosity of very small droplets. identified the first two modes of a drop undergoing vibration parallel to a substrate surface as a function of the wetting angle. Noblin et al.25 used the stage of a loud speaker to excite a drop. Deformations of large flattened drops were measured with a laser beam deflecting from the surface of the drop. When the deflection of the laser beam was Fourier-transformed, multiple resonance frequencies were readily obtained from a single run. While Noblin et al.25 mentioned this experiment in their paper, they did not report any related experimental data, other than pointing out that these were essentially the same as those obtained using frequency sweeps. In recent years, we have been using computer generated Gaussian white noise vibration to induce shape fluctuation in a liquid drop39,40 in the context of generating a rectified motion on a surface. In that connection, we did observe that a drop excited by a white noise exhibits certain shapes that are reminiscent of their resonance modes (Figure 1). While it is perfectly possible to excite multiple modes with a sawtooth wave that is also composed of multiple harmonics, it lacks the advantage of performing the experiment with a white noise, which in principle has equal power at each frequency. In the current study, we analyze the fluctuation of a liquid drop submitted to a random noise in the same spirit as Noblin et al.25 to identify various resonance modes. The specific experiment involves recording the height fluctuation of a drop with a high speed camera and then calculating its power spectrum. Two types of supports were used. In one case (polystyrene), the contact line was pinned to the surface, and in the other case (microfibrillar PDMS or polydimethylsixolane), the drop exhibited substantial slippage. The novel aspect of this work is the analysis of the power spectrum of the fluctuating contact line, thereby identifying its own resonance modes as
1. INTRODUCTION It is well-known that a vibrating liquid drop exhibits multiple resonance modes that depend on its mass and surface tension.1−4 Viscosity has also an effect, but at very low Reynolds numbers. In recent years, the subject has gained considerable vitality because of its importance in various technological applications including crystallization,5,6 spray coating,7 inkjet printing,8 vibrated drop motion,9−13 and microscale heat transfer technologies involving dropwise condensation.12 The oscillations of drops are also useful in estimating the surface tensions of liquids.14−16 Several experimental techniques have been developed for characterizing the resonant modes of liquid drops.5,6,10,11,17−36 One such method involves subjecting a liquid drop to external vibration and recording the resulting dynamics with a high speed camera.5,6,10,11,17−24,36 Resonance modes can be identified either from the shape of the drop at the resonance condition or by analyzing the dynamics of the liquid surface. These direct measurements include reflection25−30 of light from the surface, examining the motion of an optically trapped particle inside the drop,31 and utilization of an AFM microscopy32 that can be used to perturb the surface as well. Recently, significant advances have also taken place in characterizing the internal flow of a vibrating drop using particle velocimetry.37,38 While most methods use mechanical vibrations, some of the newer methods used electrowettinginduced oscillation of a liquid drop.35 An important issue that comes up is how to drive the oscillation of a drop so that multiple modes are excited simultaneously. In this context, while most commonly used methods use a frequency sweep, a few attempts25−30,32−35 have been made to estimate multiple resonance frequencies using a single experimental run. For example, Hill and Eaves33,34 reported multiple resonance modes of a magnetically levitated liquid drop that was disturbed by a jet of air. Sharp27,28 also used a jet of air recently to perturb the drop and thence © 2012 American Chemical Society
Received: July 21, 2012 Revised: September 7, 2012 Published: September 10, 2012 14100
dx.doi.org/10.1021/la302958m | Langmuir 2012, 28, 14100−14106
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accelerations of the stage was Gaussian that was white up to a practical bandwidth of 5 kHz. With these random accelerations (γ), the nominal strength of the noise was estimated as K = ⟨γ2(t)⟩τc, where τc is the time duration (40 μs) of the pulse. This estimation of noise strength is based on the condition that the noise is strictly white and Gaussian. This is not exactly the case for the type of noise that we generate experimentally with a mechanical oscillator, which has a tendency to spring back after each excitation, as the autocorrelation of the output noise exhibits a negative peak; thus the noise strength needs to be corrected when it is used for quantitative treatment of the data. For most of this Article, we use the nominal definition of K. However, toward the end of this Article (i.e., section 5), this value of K was corrected by multiplying it with a numerical factor (∼0.03) as in ref 41. For most experiments, white noise of (nominal) strength K = 0.17 m2/s3 was used. The fluctuating liquid drops (Figure 1c) were recorded with a high speed camera (Redlake, MotionPro, model 2000) operating at 1000−2000 frames/s, which were analyzed later using a motion analysis software MIDAS (Xcitex Inc., U.S.) for automatic tracking of the height and the contact line of the drops. To study the effects of viscosity on the resonance spectra, we carried out experiments with 10 μL liquid drops of various glycerol−water solutions as per the previous studies of Sharp.28 The viscosities of these solutions as measured using a rotating spindle viscometer (model: Anton Paar Physica MCR301) were 1, 2, 3.9, 6.2, and 10.9 cP (centipoise), respectively, for pure water, and 20, 40, 50, and 60 vol % of glycerol in water, respectively. The corresponding surface tensions (du Nouy ring method, FDS Dataphysics, Germany) were found to be 72.5, 70.8, 69.4, 69.1, and 67.9 mN/m, respectively. Although there is a small change in the surface tension of the solution with the increase amount of the glycerol concentration, the change in viscosity is more pronounced as it varies from 1 cP (water) to 10.9 cP (60% glycerol).
Figure 1. (a) Schematic of a liquid drop on a solid substrate subjected to white noise vibration perpendicular to the plate. (b) Gaussian distribution of acceleration of plate. (c) A few randomly selected frames of a 20 μL water drop vibrating on a microfibrillar PDMS surface show some of the resonance modes.
well. Comparison of the contact line fluctuation with that of the height provided additional insights into the degree of slippage that a drop undergoes on a surface in relation to the mode number. The first objective of this study is the simultaneous identification of resonance modes of small drops that are dominated by surface tension, and then to examine the applicability of a simple model to predict these modes. The second objective is to study how these modes are affected by the wettability of a surface and the ensuing slippage at the contact line. In all cases, we restricted our studies to the small drops (1−20 μL), because of their relevance to drop fluidics.
3. RESULTS AND DISCUSSION It is well-known that the power spectrum of an ideal white noise has equal power at all frequencies. Hence, it is theoretically possible to excite all of the resonance modes of a liquid drop when it is subjected to a true white noise vibration. However, in a laboratory setting, a white noise vibration can be generated with only a finite bandwidth. The number of modes that can be excited is thus limited by this bandwidth as well as the resolution of the camera used to record the data. In our experiment, the mechanical oscillator had a practical bandwidth of 5 kHz, with the highest frame speed of 2000 fps for the video camera. With these limitations, the highest frequency that could be measured was 1 kHz. The side view (Figure 1c) of the recorded image of a vibrating drop was analyzed with an automatic motion tracking MIDAS software (Xcitex Inc., U.S.). This fluctuation was subsequently fast Fourier transformed (FFT) using OriginLab Software to identify resonance modes. The noisy power spectrum was further denoised with wavelet transform using MATLAB that enabled one-dimensional automatic denoising of the power using a heuristic threshold of level 4, symmetric wavelet decomposition. Several power spectra (five or more) were added together and averaged to improve the signal-to-noise ratio. The averaged power spectrum was denoised and despiked to improve the clarity in the position of resonance peaks (Figure 2). We observe some satellite peaks apart from the main resonance peaks (Figure 2a), the origin of which is not understood at present.
2. EXPERIMENT Figure 1a shows the schematic of the experimental procedure. Two types of supports were used for the sessile drop of water. One was a polystyrene-coated silicon wafer, and the other was a microfibrillar PDMS (polydimethylsiloxane or silicone) rubber, the details of which have been described in a previous publication.13 The microfibrillar PDMS rubber had square fibrils of 10 μm in size and 25 μm in height with a fibrillar spacing of 50 μm. The advancing and receding angles on the polystyrene surface were 91° and 68° (hysteresis of 23°), respectively. This surface, due to high contact angle hysteresis, pins a liquid drop on the surface. By contrast, the advancing and receding contact angles on the microfibrillar surface were 170° and 157°, respectively, with a cosine average contact angle of ∼162°. When vibrated, the contact line contracted and expanded rather freely on this surface. The experimental details were already described elsewhere.39−42 Below, we briefly outline the procedure for completeness. The test substrates were placed on an aluminum stage, which was attached to a mechanical oscillator (Pasco Scientific, model no. SF-9324). After deionized water drops of volumes ranging from 1 to 20 μL were placed onto the test substrate, the stage was subjected to a random vibration generated using a signal generator (Agilent, model 33120A) and amplified with a power amplifier (Sherwood, model no. RX-4105). The whole experimental setup was isolated from the spurious ground vibration by placing it on a vibration isolation table (Micro-g, TMC). The acceleration of the supporting aluminum plate was estimated with a calibrated accelerometer (PCB Peizotronics, model no. 353B17) driven by a Signal Conditioner (PCB Peizotronics, model no. 482) that was connected to an oscilloscope (Tektronix, model no. TDS 3012B). The pdf (probability distribution function) of the
4. RESONANCE MODES A few randomly selected frames of the vibrating drop (Figure 1c) show some of the resonance modes. Figures 2 and 3 demonstrate the typical vibration spectra that can be obtained from the measurements of the height fluctuations. We note that 14101
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Figure 4. Resonance frequencies of water drops as a function of mass on a polystyrene-coated silicon wafer (a) and a microfibrillar PDMS surface (b). Open symbols are the experimental data, whereas the solid lines are the linear fits through the data by forcing the lines to meet at (0, 0). The slopes of these lines are compared to the theoretical predictions in Tables 1 and 2.
Figure 2. (a) Comparison of denoised power spectra of a 10 μL water drop vibrating on a fibrillar PDMS and a polystyrene surface. The power spectrum on fibrillar PDMS surface is shifted up by 1 order of magnitude (y-axis shift) to facilitate the clarity of presentation. The mode numbers are shown on the spectra. Gaussian probability distribution of height fluctuations of the same size drop on a fibrillar PDMS (b) and a polystyrene surface (c) corresponding to an observation time of 0.001 s. Here, x̃ is the nondimensional height fluctuation x̃ = (x − xp)/σx expressed in terms of the position of the peak (xp) and the standard deviation (σx) of the height fluctuations. White noise of strength K = 0.17 m2/s3 was used to vibrate the drop, which was recorded with a video camera at 2000 frames/s.
γlvω̃j2
f j2 =
(1)
6πm
where γlv is the surface tension of the drop and m is mass, and ω̃ j are roots of the following equation: ⎛ (2l − 1) !! ⎞ l(4l + 1) ⎜ ⎟ =0 2 ⎠ ω̃j − 4l(2l − 1)(l + 1) ⎝ 2ll! 2
∞
F(ωj̃ ) =
∑ l=1
(2)
To compare the experimental data with the predictions of eq 1, we numerically solved eq 2 to find its roots as follows: ω̃ 2 = 4.4268, ω̃ 3 = 10.5920, ω̃ 4 = 18.0377, ω̃ 5 = 26.5954, ω̃ 6 = 36.1426, with which the resonance frequencies were calculated from eq 1. The data summarized in Figure 4a and the corresponding Table 1 comparing the slopes of the lines suggest that the agreement between the theory and experiment is very good. Figure 3. Effect of the strength of white noise vibration on resonance frequencies of a 10 μL water drop on polystyrene (a) and microfibrillar PDMS surface (b). The strength of vibration has no effect on the peak positions.
Table 1. Comparisons of the Slopes of Linear Fits through the Experimental Data of the Drop Vibrating on the Polystyrene Surface with Those Calculated from Eq 1 (Lyubimov et al.43) and Eq 3 (Noblin et al.25)a slope (kg0.5 s−1) theory
the positions of the resonant peaks on either the polystyrene (j = 2−6) or the fibrillar PDMS surface (j = 2−5) are rather independent of the strength of the noise (Figure 3). The resonance peaks corresponding to j = 6, 7, and 8 on fibrillar PDMS are damped out (Figure 3b) at low power of K ≈ 0.006 m2/s3 as compared to higher powers. However, the position of peaks corresponding to j = 2−5 on fibrillar PDMS is independent of the strength of the noise. The observation that resonance frequencies are independent of strength of noise coupled with the fact that the height fluctuations are Gaussian (Figure 2b and c) suggest that there are no overwhelming nonlinearities39,40 associated with these drop dynamics. For each mode, the resonance frequency of the drop vibration varies inversely with the square root of its mass (Figure 4), which is in agreement with the theories of drop oscillation.1−4 The most exact treatment of the vibration of a sessile drop is due to Lyubimov et al.,43 which has been discussed by others in the recent past.32,36 Using the notations of ref 32, the resonance frequency ( f j) corresponding to a mode j, as given by Lyubimov et al.,43 is:
mode (j) 2 3 4 5 6 a
experiment 0.28 0.66 1.06 1.65 2.29
± ± ± ± ±
0.01 0.01 0.02 0.02 0.02
Noblin et al.
25
Lyubimov et al.43
0.32 0.69 1.15 1.68 2.27
0.27 0.66 1.12 1.65 2.24
All of the lines were forced to pass through the origin.
For the drop on the fibrillar surface, the equation of Lyubimov et al.43 could not be used as it was not meant to apply for contact angles larger than 90°. Here, we explored the approximate analysis of Noblin et al., 25 which is an improvisation of a one-dimensional gravity-capillary wave vector relation of Landau and Lifshitz.44 The Landau−Lifshitz equation predicts the resonance frequency (f j) for a thin layer of liquid on a surface as follows: f j2 = 14102
γ ⎞ ⎛ 1 ⎞⎛ ⎜ ⎟⎜gq + lv q 3⎟ tanh(q h) j ⎝ 4π 2 ⎠⎝ j ρ j⎠
(3)
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where g is gravitational acceleration, qj is the wave vector for the jth mode, γlv and ρ are surface tension and density of liquid, and h is the height of the film. Noblin et al.25 applied eq 3 to liquid drops by approximating h = V/πR2 as the mean height of drop with volume V. The first term of eq 3 accounts for the effect of gravitational acceleration (which is negligible for the small drops), whereas the second term accounts for the effect of surface tension to the normal vibration modes. Noblin et al.25 intuitively improvised this equation for liquid drops by modifying the wave vector relation to its meridian (side view) perimeter. Here, the wave vector for the jth mode is given by qj = π(j − 1/2)/p for a pinned contact line and qj = π(j − 1)/p for a moving contact line. These wave vector relations are derived by expressing the meridian perimeter of a drop in terms of the wavelength of the normal modes. Here, p = Rθ is one-half of the meridian perimeter of liquid drop of volume V with R (eq 4) being the radius of the circle of the spherical cap that subtends an angle of contact θ with the solid surface. ⎞1/3 ⎛ 3V 1 R=⎜ ⎟ ⎝ π (2 − 3 cos θ + cos3 θ ) ⎠
line to slip. On the other hand, the agreement is good for the higher modes with the model of no slip. Even though the predictions of the approximate model of Noblin et al.25 may not be taken too seriously in trying to explain the above discrepancy, we were, nevertheless, persuaded to find out the nature of the fluctuation of the contact line of the drop in relation to its height. Fluctuation of the contact line, in a sense, was studied previously by Noblin et al.45 by studying the oscillation of the contour of a drop when it was subjected to high amplitude vibration. In this context, certain interesting connections were made between contact line slippage and the transition of the drop from a circular (harmonic mode) to a noncircular shape (subharmonic modes). In our studies, the fluctuations of the contact diameter of the drop on a polystyrene and a fibrillar PDMS surface were measured directly when it was subjected to a white noise vibration. The contact line on polystyrene surface showed negligible slippage when examined visually. However, as shown in Figure 5a, two
(4)
For the polystyrene surface, we used θ = 90° in calculating the resonance modes assuming that the drop is deposited onto the surface in an advancing mode. Numerical calculation, however, shows that even if we use the average contact angle of 80°, instead of 90°, the calculated resonance frequencies increase by
