pubs.acs.org/Langmuir © 2009 American Chemical Society
Optical Observation of High-Frequency Drop Oscillations by a Spectrum Compression Technique applied to the Capillary Pressure Tensiometry Giuseppe Loglio,*,† Piero Pandolfini,† Reinhard Miller,‡ and Francesca Ravera§ †
University of Florence, Department of Organic Chemistry, Via della Lastruccia, 13, I-50019 Sesto Fiorentino (Firenze), Italy, ‡Max-Planck Institute for Colloids and Interfaces, Am M€ uhlenberg 1, D-14476 Potsdam-Golm, Germany, and §Istituto per l’Energetica e le Interfasi, IENI-CNR, UOS Genova, Via De Marini 6, I-16149 Genoa, Italy Received April 20, 2009. Revised Manuscript Received September 16, 2009 Image acquisition and subsampling of periodic high-frequency drop oscillations is presented as an advantageous metrological procedure in capillary pressure tensiometry (CPT). The observation of a finite sequence of single tone or of multiharmonic cycles, subsampled in an expanded time-scale interval, allows the characteristics of the real oscillations to be well-reconstructed in a frequency-compressed spectrum, where each component is translated toward lower frequencies. The introduced technique is applied to nanoliter-sized water drops, oscillating in a hydrocarbon matrix up to 150 Hz frequency, by using a standard PAL CCD camera provided with an electronic shutter. Application examples show the important role of this technique in data analysis and interpretation of typical high-frequency oscillating drop/bubble experiments. In particular, this technique is effective to check the onset of critical hydrodynamic effects and allows for the determination of the intrinsic elasticity of the liquid/cell system as a function of frequency by comparison of the liquid volume, as displaced by a piezo-actuator, and the actually observed drop volume-amplitude oscillation. The knowledge of this quantity is fundamental for the calculation of the dilational viscoelasticity from the acquired pressure data in the CPT.
1. Introduction The study of the dynamic properties of interfacial layers at fluid interfaces is an important topic due to its relevance for many technological and natural processes involving multiphase systems like liquid films, emulsions, and foams.1-4 Dilational rheology is considered an effective tool for investigating such dynamic aspects, and the experimental studies about this topic are often based on dilational viscoelasticity measurements by drop/bubble tensiometers.5,6 Such experimental techniques have become more and more efficient due to the implementation of advanced instrumentations, which improve the speed of the drop/bubble control and the data acquisition,7-10 and to the application of new theoretical approaches which allow the acquired data to be interpreted also under dynamic conditions.11-13 Moreover, because the relaxation processes responsible for the dynamic behavior of fluid interfaces, like diffusion exchange, surface *
[email protected]. (1) Ivanov, I, B.; Kralchevsky, P. A Colloids Surf., A 1997, 128, 155. (2) Liu, J.; Xu, Z.; Masliyah, J. J. Colloid Interface Sci. 2005, 287, 507. (3) Schramm, L. L.; Stasiuk, E. N.; Turner, D. Fuel Process. Technol. 2003, 80, 101. (4) Koelsch, P.; Motschmann, H. Langmuir 2005, 21, 6265. (5) Liggieri, L.; Attolini, V.; Ferrari, M.; Ravera, F. J. Colloid Interface Sci. 2002, 252, 225. (6) Ravera, F.; Ferrari, M.; Santini, E.; Liggieri, L. Adv. Colloid Interface Sci. 2005, 117, 75. (7) Cabezas, M. G.; Bateni, A.; Montanero, J. M.; Neumann, A. W. Colloids Surf., A 2005, 255, 193. (8) Hoorfar, M.; Neumann, A. W. Adv. Colloid Interface Sci. 2006, 121, 25. (9) Kotsmar, C.; Grigoriev, D. O.; Makievski, A. V.; Ferri, J. K.; Kr€agel, J.; Miller, R.; M€ohwald, H. Colloid Polym. Sci. 2008, 286, 1071. (10) Loglio, G.; Pandolfini, P.; Makievski, A. V.; Miller, R. J. Colloid Interface Sci. 2003, 265, 161. (11) Kovalchuk, V. I.; Kr€agel, J.; Makievski, A. V.; Loglio, G.; Liggieri, L.; Ravera, F.; Miller, R. J. Colloid Interface Sci. 2002, 252, 433. (12) Liggieri, L.; Ferrari, M.; Mondelli, D.; Ravera, F. Fararday Discuss. 2005, 129, 125. (13) Kovalchuk, V. I.; Kr€agel, J.; Makievski, A. V.; Ravera, F.; Liggieri, L.; Loglio, G.; Fainerman, V. B.; Miller, R. J. Colloid Interface Sci. 2004, 280, 498.
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rearrangement of the adsorbed molecules, and so forth, occur over a broad time scale, an important point for the advancement in this kind of investigation is to access high frequency ranges. Dilational viscoelasticity measurements as a function of the area perturbation frequency can be fruitfully performed by the capillary pressure tensiometry (CPT).14-17 When this kind of tensiometer is used according to the oscillating drop method, the capillary pressure is acquired during the harmonic perturbation of the drop interfacial area, and the dilational viscoelasticity is indirectly obtained from this pressure signal and from the variation of the geometrical characteristics of the drop (radius, surface area). While at low frequency these geometrical characteristics can be directly measured by imaging techniques, the direct acquisition of the drop profile variation becomes more and more unreliable at increasing frequencies, particularly beyond 0.5 Hz oscillations. This limiting frequency requires that the dilational viscoelasticity be calculated on the basis of specific experiment theories accounting for hydrodynamic effects and also considers the influence of the cell compressibility on the drop behavior.11-13 In this work, a technique is proposed for the optical observation of high-frequency drop oscillations, based on the subsampling of a periodic high-frequency function. This technique, denoted as spectrum translation/compression technique and commonly used in other engineering fields,18 is applied here to the image acquisition in capillary pressure tensiometry. Actually, (14) Liggieri, L.; Ravera, F. In Drops and Bubbles in Interfacial Research, M€obius, D., Miller R., Eds.; Elsevier: Amsterdam, 1998, p 239. (15) Wantke, K. D.; Lunkenheimer, K.; Hempt, C. J. Colloid Interface Sci. 1993, 159, 28. (16) Nagarajan, R.; Wasan, D. T. J. Colloid Interface Sci. 1993, 159, 164. (17) Russev, S. C.; Alexandrov, N.; Marinova, K. G.; Danov, K. D.; Denkov, N. D.; Lyutov, L.; Vulchev, V.; Bilke-Krause, C. Rev. Sci. Instrum. 2008, 79, 104102. (18) Yaagoubi, M. E.; Neveux, G.; Barataud, D.; Reveyrand, T.; Nebus, J.-M.; Verbeyst, F.; Gizard, F.; Puech, J. IEEE Transactions on Microwave Theory and Techniques 2008, 56, 1180.
Published on Web 10/02/2009
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Article elements: (1) a mechanical piston for displacing the liquid, contained in the cell, outward from the capillary in an oscillating mode, (2) a pressure transducer, (3) a port connected to an external syringe pump, (4) a valve connected to a vacuum system, and (5) the capillary immersed in an open optical cell.
2.1. Detailed Instrument Components
Figure 1. Block diagram of the oscillating drop instrument.
the spectrum compression technique (SCT) allows oscillatingdrop visualization to be progressively extended toward a wide frequency band and, contemporaneously, maintaining an essential optical resolution for the acquired images. In practice, this article shows how the SCT advantageously permits a physically complete reconstruction of sinusoidal changes of drop volumes in the hundred-hertz band, just acquiring the images by a standard PAL CCD camera. By using CCD cameras with more advanced protocols than the traditional PAL protocol, the SCT functionality grants the optional fruitful features of accessing highest-frequency ranges at basic image resolution, subsampling the oscillations in a prolonged time interval. In addition to expounding the technical aspects and suitability of the spectrum compression procedure to CPT, in subsequent sections some application examples are presented which prove the importance of these techniques in data analysis and interpretation of typical high-frequency oscillating drop/bubble experiments. In particular, this technique is effective to check the onset of critical hydrodynamic effects and allows for the determination of the intrinsic elasticity/compressibility of the liquid/cell system as a function of frequency. The results obtained here can be used to prove the functioning of a particular CPT setup and also for the validation of theoretical approaches specifically used for the CPT data interpretation.
1 Specific components for the present spectrum compression technique involve the following key parts: (a) the objective, that is, a Mitutoyo model M Plan Apo 10 long-working-distance objective, (b) the WV-BP550A B/W Panasonic CCD video camera, featuring a selectable electronic shutter speed, (c) the model P-843.40 Physik Instrumente piezo-actuator, and (d) the model 33220A Agilent arbitrary waveform generator. 2 The experiments were conducted with two capillaries, both having the main stem with a rather large internal diameter, Dint = 3 mm, but terminating at tips with very different characteristics. The tip of capillary A is a stainless steel needle (20 mm length, 0.1 mm internal diameter) and it is externally coated by Acota Certonal FC-734 perfluoropolymer. Capillary B, made of PEEK polymer, has a 2-mm-length tip with a 0.25 mm internal diameter. 3 The capillary is immersed in an open (28 31 mm2 inner section) Hellma optical cell. 4 The area of the optical field of view (FOV) is adjusted at AFOV = 0.93 0.70 mm2. With such a magnification, the horizontal calibration parameter results in cx = 1.45 μm/ pixel, as determined by a 0.25 mm needle. The optical aspect ratio, proper of the image acquisition system (i.e., CCD array and frame-grabber), was previously determined as cx/cy = 0.987 138.7 Possible optical distortion effects were also previously checked by observation of specific objects with known geometrical characteristics.21 5 Coarse volume, Vsyr, displacements of the inner liquid is effected by the syringe pump, driven by a stepping motor (ΔVsyr = 8.3 nL/step). Fine volume, Vpzt, displacements are obtained by the piston, driven by the piezo-actuator, with a full-scale range of ΔVpzt = 44 nL (at an almost continuous flux, ΔVpzt = 7 10-4 nL/step).
The experimental apparatus used in this study is a drop tensiometer where two measurement techniques are combined: the profile analysis and the capillary pressure tensiometry. The essential experimental setup, the design of the measurement cell, and the inherent calibration procedures have been recently described elsewhere.19 Such a tensiometer is commonly used for dilational viscoelasticity measurements of interfacial layers versus frequency according to the oscillating drop method:20 a lowamplitude sinusoidal oscillation of the drop surface area is applied at a given frequency, ν, and from the pressure response, oscillating at the same frequency, the response of the interfacial tension can be derived by means of specific calculation procedures. For the aims of the present study, similar drop oscillations are analyzed at high variable frequency. The CPT cell configuration used here corresponds to the commercial drop profile tensiometer PAT1 equipped with the oscillating drop pressure module ODBA (SINTERFACE Technologies, Berlin) and is schematically described in Figure 1. A thermostatted closed cell holds five
2.2. Experimental Procedure. Before the start of each experimental run, a water drop is generated at the tip of the capillary inside the hydrocarbon matrix, by means of the syringe pump. A proper value of the drop volume is selected in the 10120 nL range. The hydrodynamic stability22 and the contact-line stability of the drop (a quasi-spherical cap) at the capillary tip are first ascertained. These two stability requirements are crucial items in CPT measurements (particularly for oscillating microsized drops) that rely on accurate vacuum filling of the water cell and on hydrophobization of the capillary tip, respectively. For the compressed spectrum technique, the experimental run consists of producing interleaved packets of high-frequency periodic displacements (at various frequencies and amplitudes) of the liquid inside the closed cell and, at the same time, acquiring at a lower proper frequency the images of the ensuing oscillating drop. Actually, the oscillating liquid displacements are actuated by the piston, via the external function generator and the piezodriver, and the oscillating drop images are acquired via the CCD camera at a given rate and with a given exposure time. 2.3. Software. Dedicated software, developed in LabVIEW graphical programming language, controls CPT active devices and acquires the pertinent electronic signals and CCD camera
(19) Del Gaudio, L.; Pandolfini, P.; Ravera, F.; Kr€agel, J.; Santini, E.; Makievski, A. V.; Noskov, B. A.; Liggieri, L.; Miller, R.; Loglio, G. Colloids Surf., A 2008, 323, 3. (20) Kovalchuk, V. I.; Kr€agel, J.; Aksenenko, E. V.; Loglio, G.; Liggieri, L. In Novel Methods to Study Interfacial Layers, M€obius, D.; Miller, R., Eds.; Studies in Interface Science Series, Vol 11; Elsevier: Amsterdam, 2001; pp 485-516.
(21) Lahooti, S.; Del Rio, O. I.; Neumann, A. W.; Cheng, P. Axisymmetric Drop Shape Analysis (ADSA), In Applied Surface Thermodynamics; Neumann, A. W., Spelt, J. K., Eds.; Surfactant Science Series, Vol 63; Marcel Dekker Inc.: New York, 1996; pp 486-487. (22) Liggieri, L.; Ravera, F.; Passerone, A. J. Colloid Interface Sci. 1990, 140, 436.
2. Experimental Section
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a multiharmonic wave, respectively. From these figures, it is clear that the shape of a single high-frequency cycle can be reconstructed by sampling just one point at definite intervals in a sequence of cycles observed in an expanded time domain. Obviously, in order to downscale a periodic multifrequency oscillation the recording time must be conveniently expanded. In mathematical terms, the sampling operation is expressed by the sifting property of a finite series of equally spaced Dirac delta functions Z
¼Q þ ¥ nX -¥
Figure 2. Reconstruction of the amplitude and shape of a sinusoidal cycle by subsampling the oscillating signal in an expanded time interval.
δðt -nts ÞIðtÞ dt ¼ Iðnts Þ n ¼ 1, 2, ::::i, :::Q
where I(t) = I0 sin(2πνt) and ts is the sampling acquisition period. In order to optimize the effectiveness of this technique for the reconstruction of the periodic real signal, it is important to assume a suitable relationship between ts and the frequency of the real signal. This relation can be easily found for a single-frequency periodic signal, assuming the sampling period as a multiple K of the oscillation period plus a fraction R of it ts ¼
Figure 3. Reconstruction of the amplitude and shape of a biharmonic cycle by subsampling the oscillating signal in an expanded time interval. images, following the action sequence of a pre-established time line.19,23 2.4. Materials. The studied liquid-liquid system is composed by twice-distilled water, forming a droplet as a dispersed phase, and by n-decane (Fluka, purum, CAS 124-18-5) as a continuous phase.
3. Results and Discussion 3.1. Spectrum Compression Technique. The spectrum downscaling (compression) transformation has been proven to be a reliable metrology technique in different fields of engineering.18 In consideration of a strict analogy, the same principle and the same mathematical formalism can be transferred to highfrequency measurements of CPT, provided that we are concerned with steady-state periodic oscillations of the drops. Moreover, when a slow baseline (or amplitude) drift is occurring during the periodic oscillations, this technique is also applicable in CPT as it fruitfully detects the extent and the trend of the drift phenomenon. However, for drifting oscillations, the reconstruction of the signal should be done with caution. Essentially, this measurement technique is based on subsampling an analog signal at a properly adjusted lower rate than the required rate for representing the cycle characteristics, as illustrated in Figures 2 and 3, for a pure sinusoidal oscillation and for (23) Javadi, A.; Javadi, K.; Kr€agel, J.; Miller, R.; Kovalchuk, V. I.; Ferri, J. K.; Bastani D.; Taeibi-Rahni, M. Proceedings of IMECE06, paper no. 15949, 2006 ASME International Mechanical Engineering Congress and Exposition, November 5-10, 2006, Chicago, Illinois, USA, pp 1-9.
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ð1Þ
n ¼0
1 ðK þ RÞ ν
ð2Þ
The fraction of period R is for definition related to the number of sampling per cycle, i.e., R = 1/M. M must be large enough to warrant a good definition of the sinusoidal signal (for the practical case, it is at least assumed M = 25 corresponding to R = 0.04). This subsampling produces a compression of the frequency spectrum, with a translation of the frequency ν to a lower value νt, which from eq 2 reads νt ¼
R ν K þR
ð3Þ
Equations 2 and 3 refer to the easiest case of a sinusoidal oscillation. For a multiharmonic periodic signal, constituted by a fundamental frequency ν0 and N harmonics, we have IðtÞ ¼
N X
an expð -2πinνo tÞ
ð4Þ
n ¼ -N
where the highest harmonic component is, obviously, νmax = Nν0. The operator transforms each of the n components of the periodic signal at frequency ν = nν0 into a component of the new spectrum at frequency νt, n ¼
R νn K þR
ð5Þ
Notice that in this case R is the fraction of the period of the fundamental component and it must be 0 < R < 1/N. This transformation is acceptable if, for each value of n, an integer Zn exists such that the following condition is satisfied 0 < jnν0 -Zn νs je
νs 2
ð6Þ
where νs is the sampling frequency, which in this case is νs = ν0/ (K þ R). The meaning of this condition is clear if the two limiting cases are analyzed. The first case provides νn = Znνs implying Langmuir 2009, 25(21), 12780–12786
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Figure 5. Plot of the sampling comb, pertinent to a standard PAL CCD camera provided with an electronic shutter. Figure 4. Example of a spectrum downscaling transformation obtained for ts = 101 ms.
necessarily R = 0, which means that each sampling always corresponds to the same phase angle of the original oscillation. The second limiting case analogously corresponds to R = 0.5 implying two points per cycle are sampled (Nyquist condition). The obtained compression of the frequency spectrum is clear if the difference between two consecutive harmonics is considered in the original spectrum and compared with the corresponding difference in the transformed spectrum. In fact, in the original spectrum νnþ1 -νn ¼ ν0
ð7Þ
the difference between the two corresponding harmonics in the transformed spectrum is νt, nþ1 -νt, n ¼
R ν0 K þR
ð8Þ
which means that the spectrum is compressed by a factor of R/(K þ R). Figure 4 presents an example of a possible spectrum scaling from a hundred cycles/s frequency down to a few cycles/s. 3.2. Adoption of the Spectrum Compression Technique to CPT Measurements. Specifically in CPT, rather than an electronic signal, an optical continuously changing image is sampled by the CCD devices of the camera, adopting the best compromise between illumination intensity and exposure time (as defined by the electronic shutter setting). The interlaced PAL frames are recorded and saved in a matrix format. The saved images are subsequently off-line split into odd and even fields. Thus, instead of a sequence of scalar numbers, we get a sequence of submatrices in the time domain Iðnts Þ f I½640, 480 ¼ I1 ½640, 240 þ I2 ½640, 240 Finally, by further postprocessing of the acquired drop images, we can determine all the geometrical properties for the oscillating drops (i.e., volume, area, radius, apex coordinates) as a function of time.24 In order to realize the whole procedure of image acquisition, saving in matrix format and postprocessing of data, a dedicated LabVIEW routine has been developed. Figure 5 depicts the sampling comb, together with the particular timing parameters, convenient for a standard PAL CCD
(24) Loglio, G.; Pandolfini, P.; Miller, R.; Makievski, A. V.; Ravera, F.; Ferrari, M.; Liggieri, L. In Novel Methods to Study Interfacial Layers, M€obius, D.; Miller, R., Eds.; Studies in Interface Science Series, Vol 11; Elsevier: Amsterdam, 2001, pp 439-485.
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Figure 6. Effect of the internal cell elasticity as a function of frequency: relative difference between amplitude of drop oscillation and of piston displacement, for two capillaries with different geometrical characteristics.
camera. In this case, the sampling period is constrained to be either ts = 20 ms or a multiple of this time interval. 3.3. Application Examples: Observation of Dynamic Effects. 3.3.1. Effect of the Intrinsic Compressibility of the Experimental Cell. As already mentioned above, the oscillation of the drop volume in a CPT during oscillating drop experiments is obtained by applying a sinusoidal variation of the piezo-piston volume at a given frequency and amplitude. In case of incompressible laminar oscillatory flow, the value of the drop volume change, ΔVdrop, exactly matches the value of the displaced liquid volume by the piston, ΔVpzt. Actually, in real systems, the interplay action of the Poiseuille pressure (due to the viscous flow) and of the intrinsic compressibility/elasticity of the system liquidphase/cell makes the sinusoidal variation of the drop volume different in amplitude and phase shift. As stated in ref 6, this effect can be explained by assuming a simplified model of the experimental systems, where a bubble entrapped in the closed uncompressible liquid phase changes its volume according to the sinusoidal variation of the pressure (see Figure 6). The observation of the actual volume variation during the drop oscillations, by means of the spectrum compression techniques, allows for a quantitative evaluation of the effect of such intrinsic elasticity, otherwise very difficult to predict by hydrodynamic modeling. As an example, Table 1 reports a selection of experiment parameters we adopted for the experimental runs, conducted with a drop of pure water in pure n-decane. The values of the useful oscillating frequencies for the drop are constrained by the sampling period of the PAL CCD camera, such a period being a multiple of Δt = 20 ms, that is, the time interval between two consecutive odd and even fields. For all experiments, we used a convenient record length, that is tr = 12 s, in order to observe a DOI: 10.1021/la901391y
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Table 1. Real and Translated Frequencies, Obtained for a Standard PAL CCD Camera drop-oscillation frequency (Hz)
subsampling frequency (Hz)
translated frequency (Hz)
10.5 13.1 17.5 26.0 51.0 101.0 151.0
10 12.5 16.67 25 50 50 50
0.5 0.6 0.83 1 1 1 1
Figure 8. Volume of drops, oscillating at frequency ν = 51.0 Hz (50.0 Hz subsampling) about two different equilibrium states, namely V = 0.106 mm3 (upper curve) and V = 0.033 mm3 (lower curve). The drop volume is determined by numerical integration of the 1-pixel-height trapezoids observed in the meridian section of an axially symmetric object.
Figure 7. Simple qualitative model explaining the observed differences between drop oscillation amplitude and displaced volume in terms of volume changes of a single bubble of air.
redundant number of cycles. Measurements are conducted at room temperature. The experimental runs are grouped into two categories, in relation to the geometrical properties of the used capillary tip, here denoted as capillary A and capillary B. The thinner capillary A allows the formation of a smaller semispherical drop, in respect to capillary B, hence a drop oscillation around a smaller working volume. Figure 6 illustrates the observable effects of the cell elasticity/ compressibility. As seen in the plot, the viscous resistance, inside the narrow bore of capillary A, decreases to a large extent the drop oscillation amplitude and most of the displaced liquid volume is held by the capacity of the internal elasticity. The opposite effect occurs for the large bore of capillary B, where the relaxation of the internal elasticity delivers additional liquid to the drop. In both cases A and B, the observed behavior as a function of frequency, expressed as a fraction of the displaced volume, is independent of the average drop volume and displaced volume. The experimental findings of Figure 6 can be qualitatively interpreted by the simple model of Figure 7, where all the components of the internal elasticity are assumed to be represented as a single bubble of air. Thus, the positive or negative difference between liquid volumes is attributed to the volume changes of the bubble. The assumption of this simplified model is extensively explained in refs 5, 6. where it has also been shown how the effect of the internal compressibility/elasticity on the drop volume variation, being dependent on the pressure, can be strongly influenced by the frequency and capillary radius. As a final note, the behavior of the data reported in Figure 6 is directly related to the actual pressure amplitude inside the drop phase, which in turn is governed by different involved phenomena, concerning also the dilational viscoelasticity of the interfacial layer. Giving a quantitative interpretation of the frequency behavior of the data obtained is beyond the purposes of the 12784 DOI: 10.1021/la901391y
present work. What is important is showing the effectiveness of such optical techniques in detecting the true amplitude of the oscillating drop volume. 3.3.2. Nonlinear Volume Oscillation Behavior. Figure 8 illustrates a 12 s record of two typical runs of drop oscillations, obtained with the capillary B. In this experiment, the piezo-piston, excited by the wave generator, forces 500 sinusoidal cycles of liquid displacement at 51.0 Hz frequency. As seen in Figure 8, the subsampling technique allows transfer to low frequency of the real oscillations, and hence, 10-cycle oscillations at 1.0 Hz become apparent in an expanded time scale. All of the physical information is maintained in the apparent cycles. The issuing drop volume response, to the forced sinusoidal excitation, behaves in a good linear mode in the case of oscillation about a large equilibrium volume value (upper curve), while the oscillations become more distorted above the hemispherical drop size (lower curve). The observable nonlinearity is caused by the interaction between the capillary pressure and the internal elasticity (in the limit, the nonlinearity degenerates in the hydrodynamic instability phenomenon22). The nonlinearity extent here is quantitatively expressed in terms of the THD parameter (total harmonic distortion), as determined by expansion in a Fourier series of the observed oscillations.25 3.3.3. Viscosity Effects on the Drop Shape. Capillary pressure tensiometry is based on the assumption that the drop shape obeys the Laplace equation. However, such a Laplacian shape could not always be guaranteed to increase oscillation frequency. A visual observation of the drop profile at high frequency is then very important also to verify the required profile. Actually, under dynamic conditions, both inertial and viscous forces can influence the drop shape. Hydrodynamic considerations allow dimensionless quantities to be determined, defining the threshold between negligible viscous effects and the onset of oblate/prolate distortions of the drop profile. Nevertheless, viscous forces cannot be safely neglected for drop oscillation at high frequency with time-varying interfacial tension and, in addition, the theoretical criterion is apparatus-dependent.26,27 To get experimental support for the Laplacian-shape assumption, qualitatively at first glance, we can visually inspect the acquired and saved images. The example of Figure 9 shows two drop images at the maximum and minimum volume oscillation, (25) Loglio, G.; Pandolfini, P.; Miller, R.; Makievski, A.; Kr€agel, J.; Ravera, F. Phys. Chem. Chem. Phys. 2004, 6, 1375. (26) Liao, Y.-C.; Basaran, O. A.; Franses, E. I. Colloids Surf., A 2004, 250, 367. (27) Freer, E. M.; Wong, H.; Radke, C. J. J. Colloid Interface Sci. 2005, 282, 128.
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Figure 11. Standard deviation of the residuals for the fit of the Figure 9. Example of typical images of an oscillating drop at frequency ν = 51.0 Hz, subsampled at the instants of maximum (left panel) and of minimum (right panel) amplitude. Exposure time is Δt = 2 ms. Maximum and minimum volume is Vmax = 0.140 mm3 and Vmin = 0.074 mm3, respectively.
drop profile to the Laplace equation for the sequence of drop images, subsampled between the maximum and minimum volume amplitude during half-cycle oscillation at ν = 51 Hz.
Figure 12. Plot of the drop profiles for the drops of Figure 9. Figure 10. Drop of Figure 9 (left panel): random distribution of the residuals for the fit of the drop profile to the Laplace equation. For this specific drop, the standard deviation of the fit is σ = 0.52 μm.
picked in the fifth cycle of the experiment of Figure 8, run 1. A further reliable quantitative examination involves the following offline processing steps: (a) detection of the drop profiles for the whole set of saved images by a standard algorithm.28,29 (b) fitting the Laplace equation to the drop profile.24 (c) evaluating the statistical analysis of the fitting residuals, that is, the normal distance between observed and calculated profile points.19 As an example obtained with the used experiment parameters, Figure 10 shows the plot of the fitting residuals for the drop in Figure 9 (left panel). Most values of the residual population fall within the optical resolution. The random distribution, together with the standard deviation parameter, definitely confirms the Laplacian-shape hypothesis. Note that the Laplace equation is even valid in the limit of small microsized drops, which approach a spherical shape. Actually, for the capillary pressure technique, the radius of curvature is only needed (substantially, in the fitting procedure the apex radius approaches the sphere radius and the inherent shape factor approaches zero24). Figure 11 reports the standard deviation, σ, for the succession of drops in between the maximum and minimum amplitude of volume oscillation. Albeit in the central oscillation amplitude the (28) Canny, J. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol PAMI-8, No. 6, 1986; pp 679-698. (29) Zholob, S. A.; Makievski, A. V.; Miller R.; Fainerman, V. B. Adv. Colloid Interface Sci. 2007, 134-135 , 322.
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σ-value moderately increases, likely caused by some blurring of the acquired image, the overall behavior also validates the established Laplacian shape for the specific drop of Figure 9. Note that the pattern of the half-cycle in Figure 10, completed with its mirror image of the preceding half-cycle, is effectively repeated in all observed cycles of the experiment. 3.3.4. Possible Slip of the Droplet at the Capillary Edge. During the oscillation time, the liquid/solid contact line of the drop should be maintained stable at a definite circular line on the capillary tip (in particular at the inner capillary diameter). Possible slip of the oscillating droplets at the capillary edge can be quantitatively ascertained by the coincidence of the (x, z) coordinate values of the uppermost profile points. Figure 12 represents the profiles of the two drops in the example of Figure 9. As seen, the profiles practically concur to the same points and hence attest to the stability of the contact line during the peak-topeak oscillation amplitude. In the case of instability of the drop contact line, due to unwanted wettability, one would have to change the material of the capillary or, optionally, coat the capillary tip surface with a hydrophobic or oleophobic film, depending on the chemical nature of the biphasic liquid system.
4. Conclusions The adoption of the spectrum translation technique in capillary pressure tensiometry allows reconstruction of the oscillation characteristics (i.e., amplitude and shape) for high-frequency periodic steady-state oscillations of micrometer-sized droplets. This measurement technique can be advantageously adopted for CCD cameras with a simple PAL protocol, since the highfrequency oscillation spectrum can be down-transferred by 1 or DOI: 10.1021/la901391y
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2 orders of magnitude. In perspective, the same principle may also be a benefit for CCD cameras with a faster CameraLink protocol, in view of the higher resolution at a lower frame rate and the protocol flexibility of the image-sampling frequency. Moreover, the observation of the actual properties and behavior of high-frequency oscillating drops is an imperative metrological item in capillary pressure tensiometry, giving confidence about the reliability of the obtained results. As a matter of fact, the presented experimental results disclose a number of hydrodynamic effects, which are not easily forecasted by hydrodynamic models. Specifically, the determination of the cell compressibility/ elasticity effect is very important by drop image acquisition, because it can modify in a complex manner, depending on the
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capillary used, the drop oscillation amplitude as a function of frequency. In conclusion, the image acquisition of oscillating drops at high frequency allows various possible phenomena to be detected and error sources to be assessed, namely, (a) nonlinearity of the volume oscillation behavior around the equilibrium state, (b) drop shape deformation, (c) slip of droplet at the capillary edge, and uncontrolled wettability effects. Acknowledgment. This work was performed within the framework of “’MAP AO-99-052, Fundamental and Applied Studies of Emulsion Stability”, FASES project (ESTEC Contract Number 14291/00/NL/SH).
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