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Frequency Response of a Quartz Crystal Microbalance Loaded by Liquid Drops 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 Science Park Road, #01-01 Capricorn Science Park II, 117528 Singapore, and Department of Modern Mechanics, UniVersity of Science and Technology of China, Hefei, Anhui 230027, P.R. China ReceiVed December 29, 2006. In Final Form: March 13, 2007 The frequency response of a quartz crystal microbalance (QCM) in contact with a spreading liquid drop is studied in this paper. An improved model describing the frequency change of the QCM with the shape evolution of the liquid drop with time is proposed based on hydrodynamic analysis, which has not been reported in the literature. It is found that the drop spreading shape, including the base radius and height, has a significant influence on the frequency response of the QCM, resulting in an unexpected increase in the resonant frequency of the QCM. The model shows that the combination of the knowledge about the radial sensitivity of the QCM and the dynamic spreading of the liquid drop is potentially important to optimize the interpretation of the experimental results. The predicted results are verified with experimental results obtained with silicone oil.
Introduction The quartz crystal microbalance (QCM) has long been used as a mass sensitive device to monitor mass changes and physical properties of thin layers deposited on surfaces. Early applications of the QCM involved the measurement of minute mass deposited under vacuum based on the Sauerbrey relation between the amount of adsorbed mass and the resonant frequency shift of the oscillating quartz.1 Recently, the QCM has been developed to study liquid properties, such as the density and viscosity of liquids based on the Kanazawa equation,2 while the surface is immersed in liquid. It therefore extends the new biochemical sensor applications of the QCM operating in liquid media to in situ monitoring of changes occurring at the fluid-solid interface. However, certain pharmaceutical and biological fluids either are expensive or are available only in extremely limited amounts. It therefore stimulates the replacement of complete immersion by introducing one liquid drop of microliter volume for accessing the liquid properties measured with the QCM. Since the liquid droplet on the QCM surface is not uniform and spreads with time, both the Sauerbrey and the Kanazawa equations may not be suitable for the interpretation of the frequency response of the QCM loaded by liquid drops. Some works have been done recently for the theoretical and experimental studies on the response of the oscillating quartz crystal in contact with a liquid drop.3-6 In the proposed models, the influence of the radial sensitivity of the QCM on the frequency change due to radial spreading of the liquid drop on the oscillating quartz crystal was considered, while the static contact angle between the droplet and the QCM surface was still assumed for * To whom correspondence should be addressed. E-mail: g0402941@ nus.edu.sg. † National University of Singapore. ‡ Institute of High Performance Computing. § University of Science and Technology of China. (1) Sauerbrey, G. Use of quartz vibrator for weighting thin films on a microbalance. Z. Phys. 1959, 155, 206-212. (2) Kanazawa, K. K.; Gordon, J. G. The oscillation frequency of a quartz resonator in contact with liquid. Anal. Chim. Acta 1985, 175, 99-105. (3) Lin, Z. X.; Hill, R. M.; Davies, H. T.; Ward, M. D. Determination of Wetting Velocities of Surfactant Superspreaders with the Quartz Crystal Microbalance. Langmuir 1994, 10, 4060-4068.
the dynamic wetting process. To optimize the interpretation of the experimental results, the theoretical models can be further modified to account for the contribution of the dynamic contact angle during the spreading process. For this purpose, an improved model linking the frequency change of the QCM with the time evolution of a liquid drop spreading spontaneously on the active electrode of the QCM is suggested in this paper by considering the effects of the dynamic contact angle. Dynamic spreading of liquid droplets on solid surfaces has been well studied by many investigators using the hydrodynamic approach.7-14 The spreading behavior including the dynamic contact angle of the liquid drop can be properly modeled with this approach. By combining the coupling expressions of the frequency shift for the QCM and the dynamic spreading for the liquid droplet, a set of modified relations are derived and presented. Based on the proposed theoretical platform, it is possible to quantitatively analyze how the variation of the dynamic spreading influences the frequency change of the QCM loaded by a liquid drop. It therefore reveals that the combination of the knowledge (4) Lin, Z. X.; Stoebe, T.; Hill, R. M.; Davies, H. T.; Ward, M. D. Improved Accuracy in Dynamic Quartz Crystal Microbalance Measurements of Surfactant Enhanced Spreading. Langmuir 1996, 12, 345-347. (5) Stoebe, T.; Hill, R. M.; Ward, M. D.; Davies, H. T. Enhanced Spreading of Aqueous Films Containing Ionic Surfactants on Solid Substrates. Langmuir 1997, 13, 7276-7281. (6) Joyce, M. J.; Todaro, P.; Penfold, R.; Port, S. N.; May, J. A. W.; Barnes, C.; Peyton, A. J. Evaporation of Sessile Drops: Application of the Quartz Crystal Microbalance. Langmuir 2000, 16, 4024-4033. (7) de Gennes, P. G. Wetting: statics and dynamics. ReV. Mod. Phys. 1985, 57, 827-863. (8) Tanner, L. H. The spreading of silicone oil drops on horizontal surfaces. J. Phys. D: Appl. Phys. 1979, 12, 1473-1484. (9) Starov, V. M.; Kalinin, V. V.; Chen, J. D. Spreading of liquid drops over dry surfaces. AdV. Colloid Interface Sci. 1994, 50, 187-221. (10) . Cazabat, A. M; Gerdes, S.; Valignat, M. P.; Villette, S. Dynamics of Wetting: From Theory to Experiment. Interface Sci. 1997, 5, 129-139. (11) Cazabat, A. M. Wetting from macroscopic to microscopic scale. AdV. Colloid Interface Sci. 1992, 42, 65-87. (12) Coninck, J. D.; de Ruijter, M. J.; Voue´, M. Dynamics of wetting. Curr. Opin. Colloid Interface Sci. 2001, 6, 49-53. (13) de Ruijter, M. J.; Coninck, J. D.; Oshanin, G. Droplet Spreading: Partial Wetting Regime Revisited. Langmuir 1999, 15, 2209-2216. (14) Seaver, A.; Berg, J. C. Spreading of a droplet on a solid surface. J. Appl. Polym. Sci. 1994, 52, 431-435.
10.1021/la063767z CCC: $37.00 © 2007 American Chemical Society Published on Web 05/15/2007
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Figure 1. Schematic view of a liquid drop localized on the QCM surface with the contact angle θ and the base radius rb. re is the radius of the active electrode.
about the radial sensitivity of the QCM and the wetting kinetics of the liquid drop may facilitate the interpretation of the experimental results. As an example, comparisons for the results predicted by the theoretical model and obtained by the experiments performed with silicone oil are made.
Theory The Sauerbrey relation first quantified the decrease of the QCM resonant frequency when an ideal mass layer is bound to the quartz surface:1
∆f ) -
2f02∆m AQ(µQFQ)1/2
) S0∆m
(1)
where ∆f is the resonant frequency shift, f0 is the fundamental resonant frequency, ∆m is the surface mass change, AQ is the active electrode area, FQ is the density of the quartz (2651 kg m-3), µQ is the shear modulus of the quartz (2.95 × 1010 N m-2), and S0 is the uniform mass sensitivity defined by
S0 ) -
2f02 AQ(µQFQ)1/2
(2)
When the quartz crystal is immersed in liquid media and oscillates at its resonant frequency, an exponentially damped shear wave is radiated into the liquid. The frequency shift under liquid loading thus depends on the viscosity η and the density F of the liquid, and it can be estimated by the Kanazawa equation:2
(
∆f ) -f03/2
ηF πµQFQ
)
1/2
(3)
This frequency shift corresponds to an effective mass of the liquid within half of the decay length δ. Therefore, the Kanazawa equation can be recovered with the Sauerbrey equation by using the assumption of ∆m ) AFδ/2. The value of the decay length δ can be obtained from
δ ) (η/(πf0F))1/2
(4)
It is known that the mass sensitivity of the QCM is the highest at the center of the sensing surface and vanishes toward the edges.15-17 Under most conditions, the radial mass sensitivity (15) Hiller, A. C.; Ward, M. D. Scanning Electrochemical Mass Sensitivity Mapping of the Quartz Crystal Microbalance in Liquid Media. Anal. Chem. 1992, 64, 2539-2554. (16) Ward, M. D.; Delawski, E. J. Radial Mass Sensitivity of the Quartz Crystal Microbalance in Liquid Media. Anal. Chem. 1991, 63, 886-890. (17) Sekimoto, H. Analysis of Trapped Energy Resonators with Circular Electrodes. IEEE Trans. Sonics Ultrason. 1984, 31, 664-669.
Figure 2. Schematic diagram of the experimental setup.
distribution S(r) of the QCM empirically follows a Gaussian function:3,18-20
S(r) ) K exp(-βr2/re2)
(5)
where K is the maximum mass sensitivity at r ) 0 and is decreased with mass loading, re is the radius of the sensing surface of the QCM, and β is a parameter defining the steepness of the sensitivity dependence on r. The value of the parameter β for various modified sensing surfaces was observed with β ranging from 2.01 to 2.15, and it was chosen to be 2 in the present analysis for simplicity.3 Since the electrode configuration of the QCM used in our experiments is similar to that used in ref 3 where the bottom electrode had a radius about half that of the top electrode, we also chose β ≈ 2 in the modeling. It should be noted that the mass sensitivity distribution S(r) actually depends on the crystal configuration as well as the electrode geometry. Hiller and Ward15 experimentally demonstrated that the crystal contouring for the plano-convex quartz resonator increased the sensitivity toward the resonator center; Josse and co-workers20 studied the influences of the electrode geometries of the quartz resonators on the mass sensitivities; and Lin et al.3 showed that the loading conditions also have an influence on the appearance of S(r). Therefore, the sensitivity parameters K and β in eq 5 vary for crystals having different resonant frequencies, electrode geometries, and working environments, which could be determined by experiments.3 As a liquid drop is introduced onto the QCM surface as shown in Figure 1, it will spread spontaneously, accompanied by a decreased dynamic contact angle and an increased base radius. (18) Martin, B. A.; Hager, H. E. Flow profile above a quartz crystal vibrating in liquid. J. Appl. Phys. 1989, 65, 2627-2629. (19) McKenna, L.; Newton, M. I.; McHale, G.; Lucklum, R.; Schroeder, J. Compressional acoustic wave generation in microdroplets of water in contact with quartz crystal resonators. J. Appl. Phys. 2001, 89, 676-680. (20) Josse, F.; Lee, Y.; Martin, S. J.; Cernosek, R. W. Analysis of the Radial Dependence of Mass Sensitivity for Modified-Electrode Quartz Crystal Resonators. Anal. Chem. 1998, 70, 237-247.
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Figure 3. Measured (continuous curve) and calculated (dotted curve) resonant frequency shifts of a 5 MHz QCM upon adding a 2 µL drop of silicone oil to the center of the gold electrode.
In the present analysis, it is assumed that the drop takes the shape of a spherical cap during the spreading process. The Kanazawa equation modeling the response of the QCM coming into contact with a semi-infinite liquid seems to be oversimplified. Instead, the frequency shift due to radial spreading of the liquid drop on the gold electrode of the QCM, as shown in Figure 1, can be formulated as6
∆f(t) ) 2πF
(δ2∫
rx(t)
0
rS(r) dr +
r rS(r) h(r,t) dr) ∫rx(t) b
(6)
Figure 4. Calculated base radius and dynamic contact angle as a function of time for a 2 µL drop of silicone oil spreading on the QCM surface (circles, dynamic contact angle; triangles, base radius).
In ref 3, this relation was obtained based on the geometrical analysis of the droplet volume, which is a fairly rough approximation without considering the dynamic spreading of the droplet. For a more accurate analysis, the relation should be determined within the hydrodynamic scheme of wetting kinetics. Considering a Newtonian liquid drop spreading on a smooth and homogeneous solid surface, a rather universal relation between the apparent contact angle θ and the velocity of the contact line U is given by7
where rb is the drop’s base radius and h(r,t) is the height at the local radius r, that is
h(r,t) ) rb((csc2 θ - (r/rb)2)1/2 - cot θ)
(7)
rx(t) denotes the radius where h(r,t) falls within the effective thickness δ/2, so that h(rx(t)) ) δ/2. θ is the dynamic contact angle formed between the moving liquid interface and the solid surface at the contact line. It should be noted that the present analysis is only concerned with the case of hydrophilic wetting (θ < 90°). Following the derivation in Appendix A, it yields the frequency shift ∆f as (A.6):
[
( )]
rb2 re2 ∆f ) πFKδ 1 - cot θ exp -β 2 2β re
(8)
It reflects that both the mass sensitivity distribution of the QCM and the spreading behavior of the liquid drop contribute to the frequency response. Normalizing ∆f by the frequency shift upon liquid immersion, that is ∆fKanazawa ) πFKδ(re2/2β), it yields the dimensionless quantity, ∆fNormalized:
∆f
Normalized
) ∆f/∆f
Kanazawa
( )
) 1 - cot θ exp -
βrb2 re
2
(9)
Equation 9 involves the base radius rb and the dynamic contact angle θ, suggesting that the radial sensitivity of the QCM and the variation in the dynamic contact angle are coupled to affect the frequency change of the QCM. Of necessity, we need to consider an additional relation between the base radius and the dynamic contact angle to assess the frequency change over time.
U)
drb(t) γLV m = θ dt η
(10)
where η and γLV are the viscosity and the surface tension of the liquid, respectively. This relation is consistent with the Tanner law as a result of the zero-order approximation:
θ3 ≈ 9Φ(Ca)
(11)
where Φ is a numerical constant8,21 and Ca is the capillary number and is equal to ηU/γLV. We herein examine a small, thin Newtonian droplet that is rather close in shape to a spherical cap. Its central height hc is much smaller than its base radius rb when its contact angle is assumed to be small (θ , 90°). Following the derivation in ref 7 that is outlined in Appendix B, we can arrive at the spreading law:
rb3m+1 =
γLV m tV η
(12)
where the drop volume V is assumed to be constant due to weak evaporation. Consequently, the spreading rate U can be obtained as
U)
( )
γLV drb 1 ) dt 3m + 1 η
1/(3m+1)
Vm/(3m+1)t-3m/(3m+1) (13)
Using eqs 11 and 13, it yields
θ)
[
( )
9Φ γLV 3m + 1 η
-3m/(3m+1)
Vm/(3m+1)t-3m/(3m+1)
]
1/3
(14)
Incorporating eqs 12 and 14 with eq 8, we develop a relationship between the time evolution of the Newtonian liquid drop, which
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Figure 5. Typical sequence of four video images showing the spreading of a 2 µL droplet of silicone oil on the QCM surface (A-D). The edges of the droplet detected by the image analysis program are shown together with the corresponding times (a-d).
is added on the QCM surface and spreads spontaneously toward its equilibrium configuration, and the resulting frequency change. With this relation, it is possible to quantitatively analyze the undiscovered role of the variation in the dynamic contact angle on the frequency change of the QCM loaded by a liquid drop. Experimental Section Apparatus. A schematic diagram of the experimental setup is shown in Figure 2. The 5 MHz AT-cut quartz crystal (P/N 1492111, model SC-501-1, Maxtek Inc., Santa Fe Springs, CA) was 2.54 cm in diameter and 0.33 mm thick. Polished gold electrodes (∼160 nm thick) were deposited on chromium adhesion layers (∼15 nm thick) on both sides of the crystal. A polished electrode finish is helpful for ensuring surface homogeneity and excluding the roughness effect. An asymmetric configuration was adopted where the upper electrode, for contact with the liquid drop, had a larger radius (re,upper ) 6.45 mm) than the lower electrode (re,lower ) 3.3 mm). The quartz chip was horizontally mounted in the experimental holder (CHC-100, Maxtek Inc.), and it was driven with the research quartz crystal microbalance (RQCM) (P/N 603800, Maxtek Inc., Santa Fe Springs, CA). Frequency measurements were also performed with the RQCM. All the measured data were recorded and processed by a personal computer. The liquid drop was added to the center of the gold electrode using a digital micropipet (model PW10, Witeg, Wertheim, Germany) with a plastic tip to prevent contamination and to ensure accuracy. The volume of the liquid drop employed was 2.0 ( 0.01 µL. The deposition of the drop was handled with care so as to locate the drop at the desired location on the electrode and to prevent any possible splashing when the drop landed on the electrode. The contours of the drop were simultaneously monitored with the video camera from the top. The magnification was up to 14.2×. The image resolution was 550 × 400 pixels; the frame rate was 12 frames per second (fps). Images were analyzed to extract the evolution of the drop’s base radius over time. All the experiments were conducted at a temperature of 23 ( 0.5 °C and at a relative humidity of 60 ( 5% . The slight fluctuations in the temperature and the humidity have a minor influence on the drop spreading behavior. The measurements were repeated several times. The results for the discussion in the following section are shown as mean values from all measurements. Materials. Silicone oil (TSF 451-50, Toshiba Silicone) without further purification was used. Its density and viscosity are 0.96 kg m-3 and 50 cP, respectively. Prior to use, the crystal chips were first sonicated in 18 MΩ water for ∼10 min, then rinsed with pure ethanol for ∼30 min, and eventually dried with air. (21) Gerdes, S.; Cazabat, A. M.; Stro¨m, G. The Spreading of Silicone Oil Droplets on a Surface with Parallel V-Shaped Grooves. Langmuir 1997, 13, 7258-7264.
Results and Discussion Figure 3 shows the time-dependent frequency shift ∆f of the QCM as a 2 µL drop of silicone oil was added to the center of the gold electrode. The resonant frequency shift was continuously recorded throughout the spreading process (∼160 s). The initial abrupt frequency drop due to the addition of the drop was followed by a short decreasing trend until the maximum frequency reduction was reached (∆fmax ≈ -650 Hz). The mechanism of the frequency drop in the initial stage may be explained as follows. Since the thickness of the liquid film originally extends beyond the penetration depth, not all the mass present can be sensed. When the liquid spreads to cover a larger portion of the sensing area, an increasing amount of mass within the penetration depth can be detected and hence results in further frequency reduction. When the initial drop radius reaches ∼2.85 mm (see Figure 6) and the contact angle is estimated to be 14.5°, the initial frequency drop calculated by eq 8 is ∆fcal max ≈ -655 Hz. This is in agreement with the measured value of the initial frequency drop in Figure 3. After the maximum reduction was reached, the resonant frequency surprisingly began to rise, accompanying the increased wetting area. The resonant frequency increased at ∼150 Hz from t ≈ 10 s to t ≈ 16 s. Such an increase in the resonant frequency may be due to the dynamic spreading of the droplet, which has not been reported before. The data clearly indicate that the response of the QCM upon droplet loading cannot be explained in view of the radial sensitivity of the QCM. The special increase in the resonant frequency must then be related to the undiscovered role of the dynamic contact angle as the liquid drop spreads on the QCM surface. Using the present relations given by eqs 8, 12, and 14, the dotted curve in Figure 3 represents the calculated curve with m and Φ fitted to be 3.6 and 18.2, respectively, under the assumption that β ) 2. For the liquid, the surface tension of silicone oil (γ ) 20 mJ m-2) was used. The calculated curve shows the same tendency as the experimental data in the time domain where an increase of the resonant frequency was observed. Given the simplicity of the theory, the fits are fairly good. The fitting values of m and Φ are influenced slightly by errors in the β value. A 10% error in β results in errors of (4% in m and (10% in Φ. In Figure 4, the time evolution of the base radius and the dynamic contact angle for the drop spreading on the QCM surface is shown in accordance with eqs 12 and 14. It highlights that the dynamic contact angle of the drop falls steeply at the beginning, and then it declines gradually toward 9°. For validation, we
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and dynamic contact angle of the liquid drop. While the previous models only cover the influence of the radial sensitivity of the QCM, the present theoretical platform is an attempt to reveal the undiscovered role of the dynamic contact angle on the frequency response of the QCM loaded by a liquid drop. The theoretical results are found to be in good agreement with the experimental data obtained with silicone oil. It is found that the contribution of the dynamic contact angle becomes more pronounced as it decreases dramatically during the spreading process, resulting in an unexpected increase in the resonant frequency of the QCM. It therefore demonstrates that the combination of the knowledge about the radial sensitivity of the QCM and the dynamic contact angle of the liquid drop may potentially improve the interpretation of the experimental results.
Appendix A Figure 6. Base radius of the silicone oil drop spreading on the QCM surface as a function of time. The continuous curve is the experimental data acquired with the video camera, and the dotted curve is the calculated data as shown in Figure 4.
Denote csc ) csc θ and cot ) cot θ. Substituting eq 7 into the formula h(rx(t)) ) δ/2, it yields
δ/2 ) rb((csc2 - (rx/rb)2)1/2 - cot)
(A.1)
Solving eq A.1, we can find the expression for rx: simultaneously monitored the contours of the silicone oil drop spreading on the QCM surface with the video camera from the top. Figure 5 shows the typical sequence of four video images when a 2 µL droplet of silicone oil spreads on the QCM surface, and the edges of the droplet detected by the image analysis program are also shown together with the corresponding times. In Figure 6, the calculated value of the base radius agrees fairly well with the tendency of the actual value obtained with the video images. It can be deduced for the above results that the observed increment in the resonant frequency is a consequence of the variation in the dynamic spreading of the drop. As the dynamic contact angle of the liquid drop decreases dramatically during the spreading process, the effect of the cotangent term in eq 8 becomes more pronounced. Consequently, the resonant frequency of the QCM may rise. In this case, to quantitatively analyze the experimental data of the QCM loaded by a spreading liquid drop, it is essential to consider the coupled effects caused by the radial sensitivity of the QCM and the variation of the dynamic contact angle. However, there are a few issues that need to be addressed in the future for improving the present analysis. First, the spherical shape assumption for the droplet may not be valid, since the spreading of the drop may be asymmetric due to surface heterogeneity. Second, the effective thickness of the drop detectable with the QCM is likely to be underestimated because the shear wave actually propagates beyond δ/2.3 Third, the possible contribution due to the compressional wave coupling with the shear wave is ignored.19 Furthermore, Φ in the Tanner law is likely to be dependent on the spreading velocities. The extent of the difference between the low- and high-velocity spreading regimes can be up to 40%.21
Conclusion We have derived an improved model describing the frequency change of the QCM with the time evolution of a Newtonian liquid drop that spreads spontaneously on the electrode of the QCM. The spreading behavior of the liquid drop is modeled with simple hydrodynamic relations that characterize the base radius
[
rx2 ) rb2 1 -
( )]
δ δ cot rb rb
2
(A.2)
Since δ/rb , 1, we can approximate eq A.2 as
[
rx2 ≈ rb2 1 -
δ cot rb
]
(A.3)
Substituting eq A.3 into eq 6, it yields
{
re2 xπ rb cot -a csc2 ∆f ) πFKδ e 1 + 1/2 erfi(xay) 2β δ a
|} y2 y1
(A.4)
where a ) β(rb2/re2), y2 ) cot, and y1 ) cot[1 + δ/(rb cot)]1/2 ≈ cot + (1/2)(δ/rb). The “imaginary error function” erfi(z) is defined as
erfi(z) )
-2i xπ
∫0iz e-x
2
dx
Accordingly, we can simplify eq A.4 as
{
( )
re2 xπ rb cot -a csc2 2xa δ 1 + 1/2 × e ∆f ) πFKδ 2β δ 2r a xπ b exp(-ayj2) re2 ) πFKδ {1 - cot exp[-a(csc2 - yj2)]} 2β
}
(A.5)
where yj ∈ [cot, cot + δ/2rb]. Since δ/2rb , cot, yj can be approximated as a constant equal to cot θ within the interval. Accordingly, we arrive at
re2 ∆f ) πFKδ (1 - cot θ e-a) 2β
[
( )]
re2 rb2 ) πFKδ 1 - cot θ exp -β 2 2β re
(A.6)
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Appendix B
Substituting eq B.3 into eq 10, it yields
We examine a small, thin Newtonian droplet that is rather close to a spherical cap. Its central height hc is much smaller than its base radius rb when its contact angle is assumed to be small (θ , 90°). Under these assumptions, we expect
π V ) hcrb2 2
(B.1)
1 hc ) rbθ 2
(B.2)
()
drb(t) γLV V = dt η r3 b
V rb3
(B.4)
and the spreading law becomes
rb3m+1 =
Using eqs B.1 and B.2 to eliminate hc, we can obtain
θ≈
m
(B.3) LA063767Z
γLV m tV η
(B.5)
