Microscale Liquid Dynamics and the Effect on Macroscale

LETTER pubs.acs.org/Langmuir

Microscale Liquid Dynamics and the Effect on Macroscale Propagation in Pillar Arrays Rong Xiao and Evelyn N. Wang* Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States

bS Supporting Information ABSTRACT: Liquid dynamics in micropillar arrays have received significant fundamental interest and have offered opportunities for the development of advanced microfluidic, thermal management, and energy-harvesting devices. However, a comprehensive understanding of complex liquid behavior and the effect on macroscopic propagation rates in micropillar arrays is needed. In this work, we investigated the microscopic sweeping behavior of the liquid front along the spreading direction in micropillar arrays where the sweeping distance scales with the one-fifth power of time. We explain the scaling with a simplified model that captures the capillary pressure gradient at the liquid front. Furthermore, we show that such microscopic dynamics is the mechanism that decreases the macroscopic propagation rate. This effect is a result of the reduction in the interfacial energy difference used to generate the capillary pressure, which is explained with an energy-based model and corroborated with experiments. The results indicate the importance of accounting for the microscopic dynamics of the liquid on microstructured surfaces, particularly in sparse geometries.

P

illar arrays with micrometer characteristic dimensions have been studied extensively in the past few decades to achieve extreme static contact angles (close to 180 or 0°) by magnifying the intrinsic hydrophobicity or hydrophilicity.1
5 More recently, liquid dynamics in hydrophilic micropillar arrays have attracted attention because of the significant demand for high-performance microfluidic,6
8 thermal management,9 and energy-harvesting devices.10 On such surfaces, unique liquid behavior with rich fundamental physics such as anisotropic,11 unidirectional,12 and multilayer spreading13 has been demonstrated. Although a complete understanding of such highly complex phenomena is desired, an essential first step is to develop the predictive capability for “macroscopic” liquid dynamics across large arrays of periodic microstructures. For this purpose, the Washburn approach,14 which balances the capillary pressure and the viscous resistance, is widely utilized, where the liquid propagation distance is proportional to t1/2 (i.e., x = Gt1/2, with t being the propagation time). The propagation coefficient, G, is determined by the relative magnitude of the capillary pressure and viscous resistance. Recent investigations have focused on macroscopic liquid propagation in micropillar arrays where simulation tools were used to determine the viscous drag9,15 and capillary pressure accurately.16 In our recent work, we performed detailed experimental investigations of the macroscopic liquid propagation on a large range of micropillar geometries (Figure 1a,b). The propagation behavior follows that of Washburn (Figure 1c) as expected; however, significant overpredictions in the propagation coefficients by our previous model16 were observed in sparse pillar arrays (Figure 1d). Ishino et al. similarly identified these r 2011 American Chemical Society

two distinct propagation behaviors in dense and sparse pillars,17 which were qualitatively attributed to different dominant length scales affecting the viscous resistance. Meanwhile, previous investigations indicated the presence of the complex microscopic behavior of the liquid that occurs between individual pillars as a possible mechanism leading to the disparity in propagation coefficients. One example of such microscopic behavior is “zipping” identified by Sbragaglia et al.18 and Pirat et al.19 Once the liquid front reaches a single pillar in a subsequent column, the liquid immediately propagates perpendicular to the spreading direction at a higher rate compared to the liquid front moving toward the next column. Although these studies provide increased insight, understanding the microscopic behavior of the liquid front along the propagation direction is lacking but needed, which can significantly contribute to the macroscopic liquid dynamics. In this work, we investigated the microscopic dynamics along the spreading direction as well as the effect of such microscopic behavior on macroscopic propagation rates. A series of time lapse images of the microscopic sweeping behavior of the liquid front between columns of pillars are shown in Figure 2a, where the liquid propagates from left to right. “Zipping”18,19 can be clearly observed such that the liquid front first wets the upper pillars and propagates downward to wet lower ones in the column, which leads to a slanted liquid front. However, the sweeping behavior along the spreading direction is Received: June 11, 2011 Revised: July 22, 2011 Published: July 25, 2011 10360

dx.doi.org/10.1021/la202206p | Langmuir 2011, 27, 10360–10364

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LETTER

Figure 1. Scanning electron micrographs (SEMs) of micropillar arrays with diameter d = 5 μm, height h = 17 μm, and periods l = 20 μm (left) and l = 10 μm (right). The scale bars are 10 μm. (b) Images of the liquid front (dotted line) during propagation on the two micropillar arrays shown in part a. The scale bars are 5 mm. (c) Comparison of experiments for the two micropillar arrays shown in part a (circles and squares) where the propagation distance, x, is proportional to t1/2, with G as the propagation coefficient. (d) Comparison of the propagation coefficient, G, determined by experiments and previous modeling on various geometries.16 The model and experiments are in good agreement with dense pillars (h > l) (circles), but the previous model overestimates the propagation rates on sparse arrays (h < l) (squares). Error bars were determined from multiple experiments.

unaffected by the zipping perpendicular to the spreading direction. Once the liquid front reaches and wets a single pillar, the liquid front starts to sweep locally from left to right. Meanwhile, the zipping continues toward the bottom of the image where the zipping determines only when the liquid front starts to move. As a result, the sweeping distance of the liquid front can be measured locally and investigated independently of the zipping behavior. Silicon micropillar arrays of two geometries with diameters of d = 8 and 9 μm, periods of l = 25 and 20 μm, and heights of h = 17 μm were investigated with both water and isopropyl alcohol. A 5 μL droplet of liquid was deposited and propagated on the substrate. During this process, images of the microscopic spreading behavior were captured with a white-light microscope (Eclipse LV-100, Nikon) with a magnification of 100 (NA = 0.70) and a high-speed camera (Phantom V7.1, Vision Research) at a frame rate of 2000 fps. The sweeping distance of the liquid, s, was measured as a function of time, t, at a specific y location (Figure 2b,c). Droplets were placed at two different locations on the sample with l = 20 μm. In all cases, the sweeping distance was determined to be proportional to t1/5, which is independent of the geometry, the propagation distance, and the liquid. To understand the t1/5 scaling, the liquid front that determines the local pressure gradient was investigated using interference microscopy. The results suggest that the liquid profile can be approximated by a second-order polynomial function y = ax2 + bx + c . Figure 3a shows a representative interference pattern of the liquid front on a pillar array with d = 5.9 μm, l = 15 μm, and h = 3 μm acquired by a confocal microscope system (VThawk, Visitech International) with a laser source (λ = 532 nm). Each dark fringe corresponds to a λ/2n difference in the thickness of the liquid, where n is the index of refraction of the liquid (n = 1.33

for water). The meniscus profiles along two different locations, AB and A0 B0 in Figure 3a are shown in Figure 3b. To determine coefficients a, b, and c, three boundary conditions are needed. From the measured meniscus profile, we determined that yð0Þ ¼ h

ð1aÞ

yðsÞ ¼ 0

ð1bÞ

where s is the sweeping distance and h is the thickness of the liquid at x = 0. Meanwhile, at the contact line x = s,
dy

¼
tan θ ð1cÞ
dx
x¼s

where θ is the contact angle at the liquid front, which is approximately 10° from the measurement. The low contact angle is due to the enhanced wettability from the oxygen plasma treatment of the sample immediately before the experiment.20 Therefore, the curvature of the profile is given by Λ¼

y00 ð1 þ

y02 Þ3=2

¼

2a ð1 þ

4a2 x2

þ 4abx þ b2 Þ3=2

ð2Þ

According to the Young
Laplace equation, the capillary pressure, Pcap, is proportional to the curvature, Λ. As a result, the pressure gradient is proportional to the gradient of the curvature: dPcap dΛ µ dx dx

ð3Þ

By considering the liquid front at x = s and substituting a, b, and c by their values determined by the boundary conditions, the scaling of 10361

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Figure 2. (a) Time lapse images of the liquid front represented by the red dotted line sweeping between columns of pillars with the white dotted line as the reference location. The scale bar is 25 μm. The sweeping distance of the liquid front, s, scales with t1/5 for both (b) water and (c) isopropyl alcohol, which is independent of the geometry and location. Error bars were determined on the basis of the uncertainty in the imaging.

the pressure gradient with known quantities is given as dPcap dΛ 4 tan θðh
s tan θÞ2 µ 3ð1 þ tan2 θÞ
5=2 µ dx dx s4 ð4Þ For a wetting liquid, tan θ is small so that h
s tan θ ≈ h, and the pressure gradient scales with s
4, dPcap 1 µ 4 s dx

ð5Þ

Moreover, for pressure-driven flow, the velocity of the liquid is proportional to the pressure gradient: U ¼

ds dP µ µ s
4 dt dx

ð6Þ

such that s is proportional to t1/5: s µ t 1=5

ð7Þ

The detailed derivation is given in the Supporting Information. The model results match the experimental results well (Figure 2b,c). Note that this scaling is distinct from that obtained from Tanner’s law.21 Although Tanner also examined the sweeping of the liquid front driven by capillary pressure, the thickness of the liquid, h,

decreases. In micropillar arrays, the liquid pins at the top of the pillars and maintains a constant h.15,17 Therefore, the sweeping distance is proportional to the one-fifth power of time in pillar arrays compared to the one-seventh power in Tanner’s case. With an increased understanding of the microscopic dynamics of the propagating liquid front, we investigated the effect of this microscopic behavior on the macroscopic propagation. From a macroscopic perspective, the propagation distance is proportional to t1/2, which suggests that the macroscale propagation is determined by the balance between the viscous resistance and the capillary pressure. However, the microscopic sweeping behavior affects the propagation coefficient by influencing the capillary pressure, which is generated by wetting a hydrophilic surface. Two time scales are involved in the overall propagation process as shown in Figure 4a, associated with (1) the local sweeping behavior to reach the next column of pillars from state 1 to state 2 (Figure 4a) and (2) after the sweeping, the capillary pressure to bring liquid from the reservoir to fill the unit cell and initiate sweeping in the next unit cell from state 2 to state 3 (Figure 4a). During the sweeping process, the bottom surface wets but this part of the interfacial energy is dissipated immediately by viscous effects (Re ≈ 10
2). Therefore, the amount of the interfacial energy needed to generate the capillary pressure to fill the unit cell is reduced. When the pillars are sparse, the dissipated part of 10362

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Figure 3. (a) Interference microscopy image of the liquid meniscus sweeping between pillars (laser wavelength λ = 532 nm). The diameter, period, and height of the pillar array are d = 5.1 μm, l = 15 μm, and h = 3 μm, respectively. Each dark fringe corresponds to a λ/2n difference in the liquid thickness. The scale bar is 15 μm. (b) Measured meniscus profiles along two locations, AB and A0 B0 . The profiles are well captured by quadratic functions at both locations. Error bars were defined by the maximum possible height difference within one dark fringe (λ/2n).

states 1 and 3,   π ΔE0 ¼ γ l2
d2 þ rf π dh cos θ
γAm 4

ð8Þ

where d, h, and l are the diameter, height, and period of the pillar array, respectively, rf is the roughness factor of the side of the pillars, γ is the surface tension of the liquid, and θ is the intrinsic contact angle on a silicon surface. However, when the timescale associated with the sweeping process is significant, the capillary pressure needed to fill the unit cell is generated only by the interfacial energy difference between states 2 and 3, Z l pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ y02 ðxÞ dx
Am
hl þ γ cos θðrf π dhÞ ΔE ¼ γ½l 0

ð9Þ

Figure 4. (a) Schematics approximating the microscopic sweeping process, which reduces the interfacial energy to generate the capillary pressure. (b) Overestimation factors (OF) determined from previous experiments and the current model represented by the circles and solid line, respectively, on various geometries. The model matches the experiments well, showing that the sweeping behavior explains the overprediction of the capillary pressure and propagation coefficient.

We define an overestimation factor that compares the capillary pressure excluding the sweeping process to that including the sweeping process. Meanwhile, the capillary pressure is defined as the change in interfacial energy per unit volume.16 The volume of liquid filling one unit cell is fixed by the geometry, with or without the sweeping process. Therefore, the ratio of capillary pressures can be determined as the ratio of the interfacial energy differences: OF ¼

ΔE0 ΔE



 π 2 l
d þ rf π dh cos θ
Am 4 ¼ Z l pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ y02 ðxÞ dx
Am
hl þ cos θðrf π dhÞ ½l 2

the interfacial energy becomes more significant, resulting in the deviation between the model prediction and experiments in sparse pillar arrays observed previously. To validate this theory, we developed an energy-based model to determine the capillary pressure in micropillar arrays and compared the results to experiments. The model uses the second-order polynomial determined previously as the meniscus profile of the liquid front. When the timescale associated with the sweeping process is short, the capillary pressure needed to fill the unit cell is generated by the interfacial energy difference between

0

ð10Þ A detailed derivation is included in the Supporting Information. The overestimation factor as a function of the height to distance between pillars is shown as the solid red line in Figure 4b. Although the actual meniscus shape during the sweeping process is complicated such that calculating the exact surface energy is 10363

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Langmuir challenging, the overestimation factor facilitates a comparison with the experiments, as described below. In the experiments, micropillar arrays with d = 5 μm, l = 12.5
30 μm, and h = 17 and 25 μm were investigated. The viscous resistances are calculated from the known geometries,16 and the actual propagation rates are measured experimentally. Therefore the actual capillary pressure can be determined. To make a reasonable comparison with the model results, the overestimation factor for the experiments is defined by the ratio of the capillary pressure determined from the previous model,16 where the sweeping process is not considered, to the experimentally determined capillary pressure, where the sweeping process is present, as shown by the blue circles. Note that as the pillars become sparse, the overestimation factor from the experiments becomes significantly larger than 1, which is also well captured by the model. The results indicate that the reduction in interfacial energy by the microscopic sweeping is the mechanism for the overestimation of the propagation rates. Therefore, to predict the macroscopic propagation rates, the capillary pressure should be modified according to the overestimation factor. In summary, we investigated the microscopic sweeping behavior of liquids in micropillar arrays and determined that the sweeping distance of the liquid front scales with t1/5, which is explained with a simplified model capturing the capillary pressure gradient at the liquid front. We also show through modeling and experiments that the sweeping behavior reduces the capillary pressure, which leads to the decrease in macroscopic propagation rate. This work offers important insights into explaining the role of microscopic dynamics in the prediction of macroscopic liquid propagation rates. Moreover, the local sweeping process can be utilized to achieve complex spreading behavior in microstructures, which promises exciting opportunities for various applications such as ultrasmall volume liquid dispensing or nanoscale thermal management.

LETTER

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’ ASSOCIATED CONTENT

bS

Supporting Information. Detailed derivation of the sweeping distance, s, scaling with time, t. Detailed derivation of the overestimation factor as a function of geometries. This material is available free of charge via the Internet at http:// pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Tel: +1 617 324 3311.

’ ACKNOWLEDGMENT We gratefully acknowledge funding support from the Office of Naval Research (ONR) with Dr. Mark Spector as program manager. We also acknowledge Intel for a generous donation of computers and the MIT Microsystems Technology Lab for fabrication staff support, help, and the use of equipment. ’ REFERENCES (1) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988–994. (2) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546–551. (3) Martines, E.; Seunarine, K.; Morgan, H.; Gadegaard, N.; Wilkinson, C. D. W.; Riehle, M. O. Nano Lett. 2005, 5, 2097–2103. 10364

dx.doi.org/10.1021/la202206p |Langmuir 2011, 27, 10360–10364