Surface Tension Measurement at the Microscale by Passive

Feb 15, 2012 - the resonance of capillary waves, which are naturally excited by .... Here, the spectrum of the capillary waves on a finite surface is ...
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Surface Tension Measurement at the Microscale by Passive Resonance of Capillary Waves Christian Pigot† and Akihide Hibara*,‡ †

LIMMS/CNRS-IIS UMI 2820, Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan ‡ Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan S Supporting Information *

ABSTRACT: The properties of fluid interfaces increase in importance as the physical scale decreases and, hence, characterization of surface tension becomes all the more critical. However, there is to date no method to characterize this parameter on microscale surfaces. We propose here a simple method based on the resonance of capillary waves, which are naturally excited by thermal fluctuations, under one-dimensional spatial restrictions using single-beam dynamic light scattering. The principle was verified at methanol/air interfaces in polydimethylsiloxane (PDMS) microchannels having various widths. Characteristic comb-shape power spectra were experimentally obtained. Theoretical analysis showed that the spectral peaks correspond to the first or higher modes of the capillary wave resonance in the restricted space between the parallel channel walls. A useful relation between successive modes was derived to eliminate the effects of damping at the soft PDMS walls. Thus, for methanol, two values were calculated from three successive modes (24.8 and 21.2 mN/m); the literature value is 22.02 mN/m. For acetonitrile, the value obtained was 28.2 ± 5 mN/m, close to the literature value of 28.6 mN/m. Although accuracy and precision require further elucidation, this novel method is expected to become a powerful tool at the micro/nanoscale.

A

is still a challenge. Most macroscale characterization methods are either invasive or rely on gravity and so cannot be downscaled to microflows. Thus, a new method has to be invented. To date, few groups have approached this issue. Hudson et al.22 reported a method for microfluidic interfacial tensiometry based on the deformation of a microdroplet in a diverging or converging flow. Although the method seems dependable and useful, the pressure required for the deformation increases as the droplet size is reduced. Furthermore, the precision of their droplet-shape imaging technique is limited to droplets bigger than a few tens of micrometers across. We present here a new method to measure surface tension for confined fluids. This method relies on the resonance of waves propagating at the interface between two fluids, where their dispersion relation depends on the surface tension. Measuring the resonance mode of these waves thus allows direct access to the surface tension. The practical challenge of this method lies in the method of detecting the resonance. For this, we use an optical spectroscopic method that is specially

recent trend in fluidics research is the study of ever smaller volumes of liquids to probe fundamental phenomena that operate at very small scales.1 For example, a single biological cell, the basic building block of life, has a volume of several picoliters. By studying 10 μm-sized droplets, loss of information by molecular diffusion or averaging can be avoided.2 Furthermore, femtoliter-volume (V = 1 × 10−15 L) droplets with nanomolar concentrations (C = 10−9 M) would contain but several molecules (CV = 10−23 mol) of solutes, allowing one to probe the chemistry and biochemistry of countable numbers of enzymes and proteins.3−5 However, aside from molecular behaviors in restricted volumes, which have been the focus of conventional studies thus far, an important factor to consider is the role of interfaces. This is because size reduction leads to an increase in the surface-to-volume ratio. In particular, the capillary force is one of the important phenomena in microspaces6 and has been well characterized by the so-called capillary rise experiments.7 Indeed, this force plays a major role in microscale devices,8,9 where it is used for driving liquid displacement,8,10 interface shaping,9,11,12 or self-alignment of small objects.12−15 In addition, the force has also recently been used as a means of energy conversion for electrowetting,16,17 mixing,18,19 and mechanical transduction.20,21 However, although the force is often used and discussed in microdevices, precise characterization of the force © 2012 American Chemical Society

Received: January 10, 2012 Accepted: February 15, 2012 Published: February 15, 2012 2557

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designed to meet the requirements in terms of both the spatial resolution and spectral range. In this paper, we elucidate, both theoretically and experimentally, the resonance of capillary waves. The detection principle is precisely described, and the capability of such a resonator to measure surface tension is proven.

The broadband power spectrum (PBb) is obtained by summing the contributions of each propagating mode by combining eqs 1, 2, and 3 as

THEORY Even at rest, an interface between two immiscible fluids is disturbed by thermal fluctuations. As a consequence, this interface, albeit flat at the macroscale, has an ever-changing roughness of a few angstroms in amplitude. One way to restore equilibrium is to dissipate these perturbations via capillary waves. Thus, waves that are excited by thermal fluctuations propagate on a fluid/fluid interface.23 Previous work on volume acoustic waves showed that waves actuated by thermal fluctuations resonate if their propagation length is greater than the characteristic size of their confinement.24 Indeed, this has been demonstrated in the case of volume acoustic waves in liquids,24,25 solids,26 and gases.27 However, the case of surface waves has not been considered before. Here, the spectrum of the capillary waves on a finite surface is estimated by adaptation of the simple model used by Sandercock for volume acoustic wave resonance.25 This model is a very good approximation of resonance frequencies and gives a satisfactory idea of the shape of the power spectrum. The capillary wave dispersion relation is given by Lamb’s equation28 as

(4)

∞

P Bb(f ) =

■

f=

1 2π

γ q3 ρ1 + ρ2

n=0

η1 + η2 2 q ρ1 + ρ2

P(f ) =

⎛ f − f ⎞2 n 1+⎜ ⎟ ⎝ Δfn ⎠

ΔF 2 2

ΔF + (f − F0)2

P Bb(f )

(5)

Figure 1 shows the simulated spectrum for capillary waves propagating along an infinitely long surface with a width of 70

(1)

Figure 1. Simulation of power spectra of capillary waves for infinite and finite surfaces. The infinite case has a broad single peak, whereas the restricted case has a comb shape. The insets show the cross section of the deformation of the interface (not to scale).

(2)

where ηi represents the viscosity of fluid i. From eqs 1 and 2, the propagation length of capillary waves can be estimated. For a wave in the 100 kHz range, typical of commonly used fluids, the length is below the millimeter range. Resonance thus occurs on surfaces that are hundreds of micrometers across or smaller. To resonate, the propagating waves must satisfy the boundary conditions. For a one-dimensional resonator, assuming that the reflections at the edges are elastic, the boundary conditions are expressed as n π = wq

1

where f n is the central frequency of the nth mode, and Δf n is its degree of broadening. Note that because of the limitations of the means of observation, the accessible spectrum is limited. Thus, the instrumental function has to be taken into account. We model it by a Lorentzian whose central frequency F and HWHM ΔF are assumed. The global spectrum is thus estimated by relating the broadband power spectrum with the instrumental function as

where f is the frequency; γ, the surface tension; ρi, the density of fluid i; and q, the wavenumber of the capillary wave. Because of the viscosities of both fluids, the propagation of the waves is attenuated, leading to a broadening of the power spectrum. This is modeled by a Lorentzian whose central frequency is given by Lamb’s equation and the half width at half-maximum (HWHM):28

1 Δf = 2π

∑

μm (lower curve) and on an infinite surface (upper curve). The central frequency F and the broadening of the instrumental function ΔF are assumed to be equal to 50 kHz. While the infinite surface exhibits a power spectrum with a single large peak, the spectrum of the micrometer-confined surface has a comb shape, with each peak corresponding to a resonant mode. As the mode number n increases, the interval and HWHM increase. The HWHM depends on the square of the wavenumber, whereas the central frequency depends on the 3 /2 power of the wavenumber. The quality factor of the resonator thus decreases with the square root of the wavenumber. As a consequence and in contrast to mechanical resonator sensors, the resolution decreases with the frequency.

(3)

where n is an integer and w is the width of the resonance cavity. In the case of elastic reflection, the energy transmitted by the wave to the wall is null. As a consequence, the pressure exerted by the wave on the wall should also be null,29 resulting in an interface perpendicular to the wall. The boundary conditions are then free. However, assuming clamped boundary conditions leads to the same boundary conditions, hence the same resonance frequencies, the present experiment cannot confirm or invalidate the present hypothesis.

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EXPERIMENTAL SECTION Previous work explored the resonance of millimetric capillary waves excited in a zero gravity environment.30 However, no demonstration of such a phenomenon has previously been achieved under normal conditions without excitation. To 2558

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demonstrate the resonance, a simple prototype was designed comprising of a straight open-air microchannel with a width of 70 μm and depth of 30 μm. Here, the interface between the flowing fluid and air was characterized in a chip made of polydimethylsiloxane (PDMS). To minimize any swelling, polar solvents, methanol and acetonitrile, were used in this study. The liquids were introduced into the channel by a syringe-pump. The inlet flow rate was tuned so as to reach a steady state between the volume of fluid introduced and that lost via evaporation. The sensing method is the key feature of this method. Here, capillary waves were observed with a dynamic light scattering method, which is an optical angular-resolved method. Briefly, as a laser beam crosses the interface, the propagating capillary waves scatter some of the incident photons (Figure 2). Along

Figure 3. Schematic of single-beam dynamic light scattering method adapted to the microscale. For simplicity, we used an open-air microchannel flow.

Figure 2. Schematic explaining the principle of dynamic light scattering by using the divergence of a single beam.

with the change in direction, the photons experience a frequency change that is equal to the frequency of the scattering wave (eq 1). However, the number of scattered photons is small and so is the frequency shift of interest (a few tens of kilohertz, as can be seen in Figure 1) with regard to the initial frequency of the photons (1015 Hz). Thus, amplification and downconversion are required for the signal to be sensible to a conventional photodiode. Previous work used a second laser that optically mixes with the scattered photons on the photodiode.31 However, in this method, there is a trade-off between spatial resolution and angular range. Consequently, the achievable frequency range for micrometer spatial resolution is higher than the observable resonance frequency. This limitation is overcome by a single-beam setup, where the photodiode is placed in the incident diverging laser beam32 so that mixing occurs between the low-angle-scattered and the incident (transmitted) beams. The required spatial resolution and frequency range are then achieved. Prior to entering the focusing lens, the laser beam is shaped by optical slits. From circular, the section becomes linear, parallel to the propagation direction of the waves of interest (Figure 3). Thus, only these waves are observed. A 532 nm laser is used (Coherent, Verdi), and the optical signal is converted into an electrical signal by an avalanche photodiode (Hamamatsu, C5331), then amplified by a lownoise preamplifier (NF, SA-430F5), and analyzed by a spectrum analyzer (National Instruments, PXI 5660).

Figure 4. Power spectrum of capillary waves propagating perpendicularly to a microchannel compared with a signal obtained for an infinite surface. The best fitting curve is also shown for the constrained surface case. For clarity, the fitting curve has been down shifted.

power spectrum of the infinite surface has a broad single peak due to instrumental broadening (upper panel) but has the expected comb shape on a constrained surface (lower panel). Comparison of the power spectrum obtained with the optical slits parallel or perpendicular to the microchannel shows the similar difference. When the laser beam is perpendicular to the walls, the capillary waves have a comb shape power spectrum. Oppositely, in the parallel position, the power spectrum exhibits a broad single peak (Figure S2 in the Supporting Information). To deduce the central frequency of each peak, the whole spectrum is fitted by a sum of three Lorentzians (eq 4) using MATLAB. The harmonic of each mode is calculated from eqs 1 and 3 (Table 1). From the results, we note two salient points: the harmonics are not integers and their difference is unity. This discretization is a characteristic of resonance. The first point is explained by the assumption of elastic reflection. Since PDMS is a soft material and its acoustic impedance is low, wave

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RESULTS AND DISCUSSION The power spectra measured on an infinite methanol surface and on a 70 μm wide channel are compared in Figure 4. The 2559

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In Table 1, the surface tension values based on eq 7 are summarized, assuming the specific gravity of the liquid to be 0.791. The calculated values agree well with the literature value (22.07 mN/m33). The dependence of the capillary wave resonance on the surface tension is validated by characterizing the interface between air and acetonitrile. In a 43 μm-wide channel, the first seven resonant modes were observed. With the same method as the above, the surface tension was found to be 28.2 ± 5 mN/m. This value also agrees well with the literature value (28.6 mN/ m). Note that the measurements are sensitive to our accuracy in measuring the resonance frequencies. Both theory and experiments show that the lower the mode, the higher is the quality factor of the resonance and, therefore, the precision of the measurement. Further, a material harder than PDMS would have smaller losses at reflection, which would also improve the quality factor and, consequently, the resolution. Moreover, like gravitational waves, the capillary waves are affected by the bottom of the channel.34 This dependence results in a noticeable deterioration of the quality factor when the depth of the fluid is smaller than the wavelength. In the present case, the depth is of 30 μm while the wavelength ranges from 35 to 80 μm. The quality factor is then decreased, the lower harmonics being more affected than the higher harmonics. We estimate that an order of magnitude improvement should be achievable with deep channels on glass substrate.

Table 1. Calculation of Air/Methanol Harmonic Modes and Surface Tension from the Resonance of Capillary Wavesa peak

frequency (kHz)

harmonic

surface tension (mN/m)

1 2 3

18.6 37.8 59.7

2.7 3.7 4.8

24.8 21.2

a

The surface tension value in the second row was calculated from the resonant frequencies of peaks 1 and 2 using eq 7. Similarly, that in the third row was from peaks 2 and 3.

reflections at the edges are accompanied by energy losses. In such inelastic reflection, eq 3 must be modified as (n + c)π = wq

(6)

where c is a constant, which, in this case, was empirically determined to be equal to 0.7. The frequency dependence of the resonance with respect to the capillary wave’s dispersion relation can be validated by measuring the spectrum of waves in channels of different widths. Thus, the resonance in the channels with widths of 70, 43, and 27 μm was studied here. In each case, a comb-shaped power spectrum was obtained and fitted by a sum of Lorentzians. Using eqs 1 and 3, we plotted the frequency of each mode against the channel width (Figure 5). With the

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CONCLUSIONS We developed a novel method to measure interfacial tension at the microscale that relies on passive resonance of capillary waves. This resonance was demonstrated for the first time under normal laboratory conditions in a one-dimensional case, specifically for several channels with widths of the order of tens of micrometers. The frequency of each mode agrees well with the dispersion relation. Thus, the dependence of the power spectrum on surface tension was validated. Further, the capability of the microfluidic resonator to measure surface tension was demonstrated. This method is label-free, simple, and noninvasive. It thus opens the way to the study of interfaces at the microscale. The measurement principle relies on an optical method, and thus spatial resolutions down to the diffraction limit can be expected.

Figure 5. Power frequency of the resonance modes of an air/methanol interface as a function of the channel width. Squares, experimental data; dotted line, dispersion relation of each mode.

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assumption of c = 0.7 in the boundary conditions, the experimental resonance frequencies fit the dispersion relation well. This adequacy completely validates the resonance of capillary waves. Moreover, the boundary constant is independent of the channel size. It is thus attributed to an interaction between the capillary waves and the PDMS walls of the microchannel and not to the geometrical properties of the device. The phenomenon of resonance is dependent on the surface tension. However, in order to extract the surface tension value from the resonance peak frequency directly, knowledge of the mode number is required. Since the reflections of the waves at the boundaries are not elastic, the constant that must be added to model for inelastic reflection is not known beforehand. However, this problem with the unknown can be overcome by relying on the frequency difference between successive modes as γ=

4ρw3 2/3 (fn + 1 − fn2/3 )3 π

ASSOCIATED CONTENT

* Supporting Information S

Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone: +81-3-5452-6341. Fax: +81-3-5452-6339. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Prof. Keiji Sakai of the Institute of Industrial Science, The University of Tokyo, and Prof. Voltz and Dr. Jalabert for the fruitful discussions. C.P. acknowledges the JSPS Postdoctral Fellowship for Foreign Researchers (Grant

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2200727). A part of this study was supported by the Asahi Glass Foundation and KAKENHI (Grant 23655064).

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