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Effects of Interface Slip and Viscoelasticity on the Dynamic Response of Droplet Quartz Crystal Microbalances Han Zhuang,*,† Pin Lu,‡,§ Siak Piang Lim,† and Heow Pueh Lee†,‡ Department of Mechanical Engineering, National University of Singapore, 1 Engineering Drive 2, 117576 Singapore, Institute of High Performance Computing, 1 Fusionopolis Way, Number 16-16 Connexis, 138632, Singapore, and Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230027, P. R. China In the present paper we first present a derivation based on the time-dependent perturbation theory to develop the dynamical equations which can be applied to model the response of a droplet quartz crystal microbalance (QCM) in contact with a single viscoelastic media. Moreover, the no-slip boundary condition across the device-viscoelastic media interface has been relaxed in the present model by using the Ellis-Hayward slip length approach. The model is then used to illustrate the characteristic changes in the frequency and attenuation of the QCM with and without the boundary slippage due to the changes in viscoelasticity as the coated media varies from Newtonian liquid to solid. To complement the theory, experiments have been conducted with microliter droplets of aqueous glycerol solutions and silicone oils with a viscosity in the range of 50∼10 000 cS. The results have confirmed the Newtonian characteristics of the glycerol solutions. In contrast, the acoustic properties of the silicones oils as reflected in the impedance analysis are different from the glycerol solutions. More importantly, it was found that for the silicone oils the frequency steadily increased for several hours and even exceeded the initial value of the unloaded crystal as reflected in the positive frequency shift. Collaborative effects of interfacial slippage and viscoelasticity have been introduced to qualitatively interpret the measured frequency up-shifts for the silicone oils. The present work shows the potential importance of the combined effects of viscoelasticity and interfacial slippage when using the droplet QCM to investigate the rheological behavior of more complex fluids. In recent years, the quartz crystal microbalance (QCM) has been increasingly served as a promising technique for rheological analysis and nondestructive testing of a variety of liquids and films at megahertz frequencies. This technique has its origin in the pioneering work of Sauerbrey,1 who has derived a relation between the decrease in resonant frequency of the QCM and the mass of a thin film rigidly coupled to the surface. When operating * Corresponding author. E-mail:
[email protected]. † National University of Singapore. ‡ Institute of High Performance Computing. § University of Science and Technology of China. (1) Sauerbrey, G. Z. Phys. 1959, 155, 206–222. 10.1021/ac8010523 CCC: $40.75 2008 American Chemical Society Published on Web 09/04/2008
in liquids, the decrease in the resonant frequency of the QCM can be attributed to an effectively coupled interfacial liquid layer since the oscillation cannot penetrate throughout the bulk liquid but will be evanescently damped within a small distance. As a result of the viscous entrainment of liquids, the well-established Kanazawa equation related the frequency shift and energy loss to the square root of the viscosity-density product.2 Since the pioneering work by Mason and co-workers,3,4 the QCM has evolved as an attractive alternative to the conventional viscometers and has been used extensively for the monitoring of the rheological behavior of various fluids, such as low-viscosity liquids,5,6 oil-based liquids,7,8 polymers and gels,9,10 foodstuff,11 and proteinbased pharmaceuticals.12 However, many studies of the liquidphase QCM have been limited to fixed fluid cells with milliliter volumes. For the rheological analysis of certain pharmaceuticals and biological fluids that may be either expensive or available in small volume limitation, it necessitates the development of the droplet QCM with the advantage of using microliter volumes. There are a few prior reported studies of the applications of the droplet QCM for the assessment of viscosity of various liquids.6,7,12 In addition, the same technique of the droplet QCM has been exploited to study the dynamic wetting behavior of liquids13-15 as well as the evaporation of sessile droplets.16 When using the droplet QCM as a rheological sensor, a number of factors must be taken into account, including the type of liquids, the solid-liquid interaction, and the liquid shape. Particular attention has been paid to the response of the QCM in (2) Kanazawa, K. K.; Gordon, J. G. Anal. Chim. Acta 1985, 175, 99–105. (3) Mason, W. P. Trans. Am. Soc. Mech. Eng. 1947, 68, 359–370. (4) Mason, W. P.; Baker, W. O.; McSkimin, H. J.; Hesis, J. H. Phys. Rev. 1949, 75, 936–946. (5) Bund, A.; Schwitzgebel, G. Anal. Chem. 1998, 70, 2584–2588. (6) Saluja, A.; Kalonia, D. S. AAPS PharmSciTech. 2004, 5, 47. (7) Ash, D. C.; Joyce, M. J.; Barnes, C.; Booth, C. J.; Jefferies, A. C. Meas. Sci. Technol. 2003, 14, 1955–1962. (8) Kuntner, J.; Stangl, G.; Jakoby, B. IEEE Sens. J. 2005, 5, 850–856. (9) Bund, A.; Chmiel, H.; Schwitzgebel, G. Phys. Chem. Chem. Phys. 1999, 1, 3933–3938. (10) Buckin, V.; Kudryashov, E. Adv. Colloid Interface Sci. 2001, 89, 401–422. (11) Dewar, R. J.; Ash, D. C.; German, M. J.; Joyce, M. J. J. Food Eng. 2006, 75, 461–468. (12) Saluja, A.; Kalonia, D. S. J. Pharm. Sci. 2005, 94, 1161–1168. (13) Lin, Z. X.; Hill, R. M.; Davis, H. T.; Ward, M. D. Langmuir 1994, 10, 4060– 4068. (14) Lin, Z. X.; Ward, M. D. Anal. Chem. 1996, 68, 1285–1291. (15) Zhuang, H.; Lu, P.; Lim, S. P.; Lee, H. P. Langmuir 2007, 23, 7392–7397. (16) Joyce, M. J.; Todaro, P.; Penfold, R.; Port, S. N.; May, J. A. W.; Barnes, C.; Peyton, A. J. Langmuir 2000, 16, 4023–4033.
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contact with a fluid media that is neither a rigid solid nor a Newtonian liquid. A number of models based on either the continuum mechanics approach or the equivalent circuit approach have been established to probe the influence of the viscoelasticity on the characteristic response of the QCM on loadings of viscoelastic fluid media.17-19 According to Maxwell’s fluid theory, with increasing molecular size or concentration of fluids, the intermolecular interactions will be more prominent and will result in a relaxation time of the molecular motions approaching to the time scale of the resonant oscillation of the QCM. Consequently, the contribution of the viscoelasticity on the responses of the QCM will increase, which could in principle give rise to various combinations of the frequency and attenuation responses.17 Although the no-slip boundary condition is prevailing in classical hydrodynamics, one aspect of the QCM response that has become a contentious issue is the possible role of the slippage phenomenon at the device-liquid interface, which has its genesis in the experimental observation of the QCM sensor exhibiting the surface energy effects.20 A number of theoretical and experimental studies in support of the slippage phenomenon have been summarized in the recent reviews.21-24 However, Martin et al.25 who have examined the role of surface roughness have argued that the anomalous sensor responses associated with varying surface free energy could be ascribed to the roughness effects. In order to quantify the slippage effect, the slip boundary conditions, either as stress or displacement/ velocity mismatch across the interface, have been incorporated into the usual no-slip models of the acoustic wave resonators.24 Ellis and Hayward26 have recently related the speed mismatch at the solid-liquid interface to the fluid velocity gradient by introducing a slip length b. Although the slippage in both the Newtonian liquid and amorphous solid limits is known to decouple the media from the substrate and thereby reduce both the frequency shift and damping,17 the influence of the slip at the interface between the substrate and the viscoelastic media on the response of the QCM will be more complicated and a rigorous treatment has been rarely reported. In the present paper we first present a derivation based on the time-dependent perturbation theory to develop the dynamical equations which can be applied to model the response of a droplet quartz crystal microbalance in contact with a single viscoelastic media. Equations reported by Hunt et al.27 have been reformulated to characterize the changes in both the QCM frequency and attenuation as the coated media is varied from simple Newtonian liquid to solidlike material. Moreover, the no-slip boundary
condition at the solid-viscoelastic media interface is then relaxed in the model by using the Ellis-Hayward slip length approach. Therefore, it allows the influence of the slippage at the interface on the response of the droplet QCM to be examined. To complement the theory, a series of experiments have been performed which measured either the frequency spectra of the complex impedance or the real-time frequency changes of the QCM by loading microliter droplets of glycerol solutions and silicone oils with a viscosity between 50 and 10 000 cS. The results show that the combined effects of the viscoelasticity and interfacial slippage could lead to specific predictions that possibly interpret the anomalous response observed in the experiments. Thus, it should be taken into account when using the droplet QCM to investigate the rheological behavior of more complex fluids.
(17) McHale, G.; Lucklum, R.; Newton, M. I.; Cowen, J. A. J. Appl. Phys. 2000, 88, 7304–7312. (18) Arnau, A.; Jime´nez, Y.; Sogorb, T. J. Appl. Phys. 2000, 88, 4498–4506. (19) Voinova, M. V.; Rodahl, M.; Jonson, M.; Kasemo, B. Phys. Scr. 1999, 59, 391–396. (20) Thompson, M.; Nisman, R.; Hayward, G. L.; Sindi, H.; Stevenson, A. C.; Lowe, C. R. Analyst 2000, 125, 1525–1528. (21) Steinem, C.; Janshoff, A. Piezoelectric Sensors; Springer: New York, 2007; pp 123-130. (22) Neto, C.; Evans, D. R.; Bonaccurso, E.; Butt, H. J.; Craig, V. S. J. Rep. Prog. Phys. 2005, 68, 2859–2897. (23) Lauga, E.; Brenner, M. P.; Stone, H. A.; Foss, J.; Tropea, C.; Yarin, A. Handbook of Experimental Fluid Dynamics; Springer: New York, 2005. (24) Ellis, J. S.; Thompson, M. Phys. Chem. Chem. Phys. 2004, 6, 4928–4938. (25) Martin, S. J.; Frye, G. C.; Ricco, A. J.; Senturia, S. D. Anal. Chem. 1993, 65, 2910–2922. (26) Ellis, J. S.; Hayward, G. L. J. Appl. Phys. 2003, 94, 7856–7867. (27) Hunt, W. D.; Stubbs, D. D.; Lee, S. H. Proc. IEEE 2003, 91, 890–901.
Equation 3 is essentially the expression derived by McHale and Newton,28 which can be used to model either a dominantly roughness-induced response or a partially slip-induced response. If further assuming the no-slip boundary condition (the slip length b ) 0), the Kanazawa result2 can be deduced
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THEORY In this section, we first present a theoretical model via timedependent perturbation theory on the characterization of a thicknessshear mode resonator (AT-cut quartz crystal) which is loaded with a single viscoelastic media (See the Supporting Information for the detailed derivation). We begin with the differential equation (eq 1).
t
]{ [ [
˜(1 - i) ωu ˜ δ ∂∆ω +∆ω ˜) ∂t π√Fqµq 4 + 1 + b (1 + i) ˜ δ
[
-ωu Fliq -
iωuη Vf2
] ]}
∂Fliq iωu ∂η - 2 ∂t V ∂t f
+i
(1)
Denotations of all symbols in eq 1 are given in the Supporting Information. As a simple analysis of eq 1, we assume that neither the complex frequency shift ∆ω ˜ , the liquid density Fliq, nor the complex visocity η changes with time. It yields
∆ω ˜)-
[
˜ ωu2 Fliqδ (1 - i) 2 b 1 + (1 + i) π√Fqµq ˜ δ
]
(2)
For Newtonian liquids (η′ ) constant and η′′ ) 0), eq 2) can be further simplified. On the basis of a first-order approximation for small slip length, it yields
∆ω ˜)-
ωu2 Fliqδ 2b 1-i δ π Fµ 2
√
[(
q q
∆ωK ) -
∆ΓK )
) ]
ωu2 Fliqδ π Fµ 2
√
(3)
(4a)
q q
ωu2
Fliqδ 2π √Fqµq 2 2
(28) McHale, G.; Newton, M. I. J. Appl. Phys. 2004, 5, 373–380.
(4b)
Figure 2. Calculated correction factor of the frequency shift χf changes as a function of the inverse loss factor η′′/η′ without and with the slip boundary condition: (a) b/δ ) 0; (b) b/δ ) 0.1. Figure 1. Calculated correction factors of (a) the frequency shift χf and (b) the bandwidth χR change as a function of the ratio of the slip length to the decay length b/δ and the inverse loss factor η′′/η′.
where ∆ωK and ∆ΓK are the Kanazawa frequency shift and bandwidth shift, respectively. Applying the Kanazawa result (eq 4a), we can rewrite eq 2 as ∆ω ) χf∆ωK
(5a)
∆Γ ) χR∆ΓK
(5b)
where
[( [(
χR ) Im
)] )( ) ]
)(
2b η'' -i 1-i ˜ η δ
1⁄2
2b η'' -1+i 1-i ˜ η δ
1⁄2
χf ) Re 1 -
(6a) Figure 3. Schematic diagram of the experimental setup.
(6b)
χf and χR are the correction factors for the Kanazawa frequency shift and bandwidth shift, respectively. Assuming the no-slip boundary condition (b ) 0), χf and χR can be reduced to the expressions derived by Arnau et al.29 in which an extended Butterworth-Van Dyke (EBVD) model was extracted for the QCM in contact with a viscoelastic fluid media. However, the derivation in this article is more comprehensive in that both the χf and χR account for the combined effects of viscoelasticity and slippage at the device-liquid interface. For access to such combined effects, the calculated correction factors of the frequency shift χf and the bandwidth χR as a function of the ratio of the slip length to the decay length b/δ and the inverse loss factor η′′/η′ are shown in Figure 1. Clearly, when either b/δ or η′′/η′ increases, both the frequency and attenuation of the QCM calculated from our model (eq 5a) exhibit higher values than the counterparts calculated from the Kanazawa model (eq 4a). It is also worth noting that the extreme combination of b/δ and η′′/η′ may even give rise to a resonant frequency higher than the unloaded QCM as shown by the change in sign in Figure 1. Furthermore, as shown in Figure 2, we choose b/δ ) 0 and 0.1 to illustrate the change in the correction factor of the frequency shift χf to the inverse loss factor η′′/η′ without and with a slip boundary condition. It is (29) Arnau, A.; Jime´nez, Y.; Sogorb, T. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 2001, 48, 1367–1382.
shown in Figure 2 that the frequency increases toward the unloaded resonant frequency as the elastic contribution of semiinfinite media varying from liquid to solid increases, whether the interfacial slippage is present or not. However, Figure 2b indicates that the slippage will decouple the loaded media from the device substrate and thereby increase the resonant frequency. Again, it suggests that the slippage be possibly responsible for the anomalously positive frequency shift of the QCM when operating in fluids. EXPERIMENTS The diagram of the experimental setup is shown in Figure 3. AT-cut quartz crystals of 5 MHz resonant frequency (diameter 25.4 mm, thickness 0.33 mm) were supplied by SRS Inc. Polished gold electrodes (160 nm) were deposited on chromium adhesion layers (15 nm) on both sides. An asymmetric pattern was adopted, where the upper electrode in contact with the fluid media had a larger radius (rEUpper ) 6.45 mm) than the lower one (rELower ) 3.3 mm). The crystal mounted in the standard holder was driven and monitored in real-time by the RQCM (Maxtek Inc.). The RQCM was linked to a PC, which was used to record and process the time courses of frequency changes of the loaded crystals. Alternatively, an Agilent 4294A precision impedance analyzer was used to analyze the frequency spectra of the impedance of the unloaded and loaded crystals. The crystal holder was enclosed in a chamber, where the temperature and the relative humidity were monitored and recorded as 23 ± 0.5 °C and 60 ± 5%. An optical microscope was used with the acoustic measurements, and this allowed observations of the changes in droplet shape. Analytical Chemistry, Vol. 80, No. 19, October 1, 2008
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Table 1. Measured and Normalized Frequency Change of the Droplet QCM Loaded with the Aqueous Glycerol Solutions at 23 ((0.5) °Ca wt % glycerol
10
50
70
90
95
97
98
99
100
-∆f (Hz) -∆fnorm b (kHz) FL (g cm-3) ηL (cP) νL (cS)
110 0.11 1.026 0.903 0.880
275 0.27 1.131 5.105 4.516
450 0.53 1.183 18.3 15.5
954 1.61 1.235 164 133
1060 2.48 1.248 380 304
1025 3.02 1.253 553 441
948 3.35 1.256 674 537
870 3.64 1.258 825 656
776 4.11 1.261 1011 802
a FL, ηL, and νL are the density, dynamic viscosity, and kinetic viscosity of the aqueous glycerol solutions, respectively. b Measured frequency changes normalized to the area covered by the droplet.
Measurements were performed on two different groups of liquids: (a) glycerol-water mixtures; (b) KF96 silicone oils with the viscosity ranging from 50 to 10 000 cS (Shin-Estu Chemical, Japan). For each experiment, one droplet of liquid of approximate volume 2 µL was introduced to the top electrode of the quartz crystal by a digital adjustable pipet. A microscope with a homemade crosshair reticule was used to center the droplet. During the experiments using the RQCM, the frequency changes were recorded in real time until the steady loaded frequencies were measured. Typically, the measurements for glycerol solutions were followed for 2∼3 min whereas those for silicone oils would be for at least 6 h for the frequency shift exhibiting the long-term trend. During the experiments using the impedance analyzer, the magnitude and the phase angle of the impedance of the loaded crystals at the series resonant frequency (defined herein as the frequency of the minimum impedance) would be measured every 15 min for 2 h. Crystals used for glycerol solutions were washed throughly in pure water followed by ethanol and dried in air for at least 1 h. In contrast, crystals used for silicone oils were washed in xylene (Fisher Scientific, U.K.) and dried in air until the initial values of the frequency and resistance under the unloaded conditions were obtained. RESULTS AND DISCUSSION We have first measured the resonant frequency changes of the quartz resonator with RQCM upon loading of the droplets of different glycerol solutions to complement the exemplary analysis of the silicone oils presented below. The area covered by the droplets of the glycerol solutions which were extracted from the microscope images ranged from 2.33 to 5.54 mm2, varying with the glycerol concentration. As the mass sensitivity of the quartz crystal varies locally,30 it is necessary to normalize the measured frequency change to the fractional contact area so as to illustrate the role of the viscosity-density product for different liquids. The normalized frequency shift ∆fnorm is calculated through multiplying the frequency shift ∆f by the factor A/AC, where AC is the contact area and A is the area of the smaller electrode (∼34.2 mm2). Table 1 shows the measured and normalized frequency changes of the droplet QCM upon aqueous glycerol solutions with varying density and viscosity. From Figure 4, in which ∆fnormare plotted against (Fη)1/2, it shows that the normalized frequency shifts are linear with increasing density-viscosity product for different glycerol solutions. Figure 4 also shows the trend predicted by the Kanazawa equation (eq 4a). The data are consistent with the predicted trend within the experimental error, (30) Hillier, A. C.; Ward, M. D. Anal. Chem. 1992, 64, 2539–2554.
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Figure 4. Normalized resonant frequency changes of the QCM for theaqueousglycerolsolutionsasafunctionofincreasingdensity-viscosity product. Open symbols represent the experimental data. The line represents the predicted response (Kanazawa equation).
indicating Newtonian characteristics of the glycerol solutions. In contrast, the glycerol solutions have been shown to exhibit the viscoelastic characteristics at high glycerol concentration.5 It can be understood that as Bund and co-worker5 utilized the 10 MHz quartz crystals, the onset of the viscoelastic characteristics at high glycerol concentration could be detectable due to the comparatively higher oscillatory frequency. In order to further investigate the viscosity (viscoelasticity) response of the QCM, the measurements have been conducted on silicone oils with a viscosity ranging from 50 to 10 000 cS. The shear-wave penetration depth δ for different silicone oils, which changes with the kinematic viscosity, is greatest for 10 000 cS oil (FL ) 0.975 g cm-3) at 25 µm for a 5 MHz crystal see eq S-2 in the Supporting Information. The penetration depth for other oils (FL ) 0.960∼0.970 g cm-3) is lower than that of 10 000 cS oil due to the lower viscosity. However, the droplet height can be easily in excess of this penetration depth. Typically, for the 1 000 cS sample the droplet height (∼0.1 mm) and the contact area (∼42.0 mm2) were measured at t ≈ 4.5 h postinjection. Because the silicone oils tended to have a complete wetting of the gold electrode, enough volume of the oils was added to ensure the complete coverage of the electrode by different oils which can minimize the influence of the contact area. On the other hand, a small volume (2 µL) of the droplet avoids any spillage of the liquid beyond the electrode portion of the QCM which could lead to the field fringing effect.30
Figure 5. Experimental plots of the frequency spectra of the impedance magnitude (blue curve) and the phase angle (red curve) for the unloaded and loaded quartz crystals as recorded by the impedance analyzer. Triangles mark the minimum impedance and the corresponding phase angle at the series resonant frequency. (a) Unloaded quartz crystal, (b) 99.8% glycerol, (c) 50 cS silicone oil measured at t ) 90 min, and (d) 500 cS silicone oil measured at t ) 90 min.
Figure 5 compares the representative frequency spectra of the impedance magnitude |Z| and phase angle θ for the crystal unloaded or loaded with the liquids on the surface. The triangles in each plot mark the minimum impedance |Z|fs and the corresponding phase angle θfs at the series resonant frequency. It can be seen that the presence of the liquids covering the crystal surface (compared to the blank quartz crystal) gives rise to a significant influence on both |Z| and θ, which is primarily expressed in a diminution and broadening of the peaks as shown in Figure 5. Such diminution and broadening of the peaks arises from an increase in energy dissipation due to the replacement of the crystal-air interface with the crystal-liquid interface. Additionally, the impedance results show that θfs for 99.8% glycerol was -62.6° whereas θfs for silicones oils were measured at levels between -82.8° and -85.1°; |Z|fs for 99.8% glycerol was 99.7 Ω whereas |Z|fs for silicones oils were at levels between 143.0 and 148.4 Ω. The experimental results imply that silicone oil samples may lag more behind the driven crystal surface compared with glycerol, albeit the density-viscosity product Fη for 99.8% glycerol
(∼10.4 g2 cm4/s) is much larger than those for silicone samples (Fη are 0.48 and 4.9 g2 cm4/s for 50 and 500 cS silicone samples, respectively). Figure 6 compares the measured real-time changes in the resonant frequency ∆f of the QCM versus the time t upon the droplet loading of silicone oils with increasing viscosity. Similar frequency responses have been measured with all the silicone oils. Initially, the frequency decreased dramatically during the first few minutes (∆f ) -200 to ∼-550 Hz). We attributed the initial frequency decrease to the replacement of the crystal-air interface with the crystal-liquid interface when the droplets of the silicone oils cover the top electrode on the crystal. Thereafter, we found that the silicone oils would exhibit a quite complex long-term behavior and the resonant frequency of the crystal started to increase steadily in all cases, albeit the magnitude and kinetics of the increase in frequency were primarily dependent on the viscosity of the silicone oils. Except for the 50 cS sample, the final frequency changes for other samples exceeded that of the unloaded crystal. Positive frequency shifts between 1250 and 1400 Analytical Chemistry, Vol. 80, No. 19, October 1, 2008
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Figure 6. Measurements of shift in the resonant frequency of the QCM versus time for the droplet deposition of silicone oils with the viscosity ranging from 50 to 10 000 cS (a-h). 7352
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Hz were measured after 6 h with silicone oils in the region of 102∼103 cS, while∆f ) +800, +600, and +160 Hz were reached after 6 h with the 3 000, 6 000, and 10 000 cS silicone oils, respectively. In contrast, ∆f ) -110 Hz was measured after 7 h with the 50 cS silicone oil, albeit this negative frequency shift was found much greater than the predicted value (∼-5 kHz) by the Kanazawa equation using the known density (FL ) 0.96 g cm-3) and viscosity. The underlying nature of the hundreds of hertz frequency upshift was not likely due to the evaporation of the silicone oils, because the supplier’s technical document for the silicone fluids (http://www.silicone.jp/e/contact/download.html) shows that KF96 silicone oils are stable for the gaskets and the holder and practically nonvolatile at room temperature due to their extremely low vapor pressure. Rather, it might be an indication of the breakdown of the Kanazawa equation (eq 4a) where only the density-viscosity effect leads to a ∆f change. Thereby, it is suspected that the measured frequency up-shifts could be due to either fluid viscoelasticity or interfacial slippage or both. It is known that viscoelastic behavior can be observed when the silicone oil is subjected to a relatively high oscillation frequency.31,32 As a result of the complex structural composition, the highly viscous silicone oil is likely to take on non-Newtonian character in the region of the megahertz oscillation frequency so that the enhanced effect of viscoelasticity could lead to increasing resonant frequency. Alternatively, because the droplet QCM oscillating at the megahertz frequency experiences high shear rates at the device-liquid interface, it is suspected that interfacial slippage could occur to decouple the silicone oil from the crystal and possibly account for the measured positive frequency changes. (31) McHale, G.; Newton, M. I.; Banerjee, M. K.; Cowen, J. A. Mater. Sci. Eng., C 2000, 12, 17–22. (32) Raimbault, V.; Rebiere, D.; Dejous, C.; Guirardel, M.; Conedera, V. Sensor Actuat. A 2008, 142, 160–165.
In summary, the experimental findings have been qualitatively discussed based on the model proposed in the article for the viscoelasticity and interfacial slip effects on the droplet QCM responses and may stimulate the ongoing argument on the related issues in the QCM community. However, further investigations on both the frequency shift and the dissipation to extract the quantitative information like the complex shear modulus or slip length are expected to confirm our hypothesis. Moreover, in Figure 6, it is worth noting that cyclical resonant peaks in frequency were found during the measurements, albeit the bandwidth of the resonant peaks became wider with increasing viscosity. Since the droplet height is much greater than the shear wave decay length, the shear wave is not likely to be responsible for the rapid changes in frequency. Instead, a possible explanation is that the droplet may act as an acoustic cavity for the compressional wave coupled with the shear wave. As the compressional wave is nearly lossless in most fluids, it may travel to the free surface of the droplet, to reflect, and to interfere constructively.33-35 We are currently performing more experimental studies on this phenomenon, and the results will be reported in the near future. SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.
Received for review May 23, 2008. Accepted July 29, 2008. AC8010523 (33) Schneider, T. W.; Martin, S. J. Anal. Chem. 1995, 67, 3324–3335. (34) McKenna, L.; Newton, M. I.; McHale, G.; Lucklum, R.; Schroeder, J. J. Appl. Phys. 2001, 89, 676–680. (35) Couturier, G.; Boisgard, R.; Jai, C.; Aime´, J. P. J. Appl. Phys. 2007, 101, 093510.
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