pubs.acs.org/Langmuir © 2010 American Chemical Society
Prediction and Optimization of Liquid Propagation in Micropillar Arrays Rong Xiao,† Ryan Enright,†,‡ and Evelyn N. Wang*,† †
Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, and ‡Stokes Institute, University of Limerick, Limerick, Ireland Received May 22, 2010. Revised Manuscript Received August 1, 2010
Prediction and optimization of liquid propagation rates in micropillar arrays are important for various lab-on-a-chip, biomedical, and thermal management applications. In this work, a semianalytical model based on the balance between capillary pressure and viscous resistance was developed to predict liquid propagation rates in micropillar arrays with height-to-period ratios greater than 1 and diameter-to-period ratios less than 0.57. These geometries represent the most useful regimes for practical applications requiring large propagation rates. The capillary pressure was obtained using an energy approach where the meniscus shape was predicted using Surface Evolver simulations and experimentally verified by interference microscopy. The combined viscous resistance of the pillars and the substrate was determined using Brinkman’s equation with a numerically obtained permeability and corroborated with finite element simulations. The model shows excellent agreement with one-dimensional propagation experiments of deionized water in silicon micropillar arrays, highlighting the importance of accurately capturing the details of the meniscus shape and the viscous losses. Furthermore, an effective propagation coefficient was obtained through dimensionless analysis that is functionally dependent only on the micropillar geometry. The work offers design guidelines to obtain optimal liquid propagation rates on micropillar surfaces.
Liquid propagation in superhydrophilic micropillar arrays has received significant interest due to rich interfacial phenomena as well as broad applications in microfluidics for lab-on-a-chip,1 biomedical,2 and thermal management systems.3 Previous studies have demonstrated opportunities to achieve various liquid behavior using three-dimensional micro/nanostructures such as anisotropic,4 unidirectional,5 and multilayer spreading.6 However, utilization of such phenomena in practical systems requires detailed understanding of the complex liquid-solid interactions that determines the wetting and transport behavior. The first step is to obtain quantitative understanding of liquid dynamics on uniformly spaced micropillar arrays. In particular, the prediction and optimization of propagation rates will facilitate the design of high performance heat pipes3 and electrochromatography chips.7 Washburn first proposed a model to predict liquid propagation rates in capillary tubes that balances capillary pressure with viscous resistance.8 Such a model has subsequently been adapted to micro/nanopillar arrays where the key challenges are quantifying the capillary pressure and viscous resistance. Recently, Nam et al. *To whom correspondence should be addressed. E-mail: enwang@ mit.edu. Telephone: þ1 617 324 3311.
(1) Cui, H.-H.; Lim, K.-M. Pillar Array Microtraps with Negative Dielectrophoresis. Langmuir 2009, 25(6), 3336. (2) Nagrath, S.; Sequist, L. V.; Maheswaran, S.; Bell, D. W.; Irimia, D.; Ulkus, L.; Smith, M. R.; Kwak, E. L.; Digumarthy, S.; Muzikansky, A.; Ryan, P.; Balis, U. J.; Tompkins, R. G.; Haber, D. A.; Toner, M. Isolation of rare circulating tumour cells in cancer patients by microchip technology. Nature 2007, 450(7173), 1235. (3) Nam, Y.; Sharratt, S.; Byon, C.; Kim, S. J.; Ju, Y. S. Fabrication and Characterization of the Capillary Performance of Superhydrophilic Cu Micropost Arrays. J. Microelectromech. Syst. 2010, 19(3), 581. (4) Courbin, L.; Denieul, E.; Dressaire, E.; Roper, M.; Ajdari, A.; Stone, H. A. Imbibition by polygonal spreading on microdecorated surfaces. Nat. Mater. 2007, 6(9), 661. (5) Chu, K.-H.; Xiao, R.; Wang, E. N. Uni-directional liquid spreading on asymmetric nanostructured surfaces. Nat. Mater. 2010, 9, 413–417. (6) Xiao, R.; Chu, K.-H.; Wang, E. N. Multilayer liquid spreading on superhydrophilic nanostructured surfaces. Appl. Phys. Lett. 2009, 94(19), 193104. (7) He, B.; Tait, N.; Regnier, F. Fabrication of Nanocolumns for Liquid Chromatography. Anal. Chem. 1998, 70(18), 3790. (8) Washburn, E. W. The Dynamics of Capillary Flow. Phys. Rev. 1921, 17(3), 273.
15070 DOI: 10.1021/la102645u
simulated the meniscus shape on hexagonal pillar arrays using Surface Evolver (SE) to predict the capillary pressure.3 Hasimoto approximated the viscous resistance of pillar arrays by idealizing them as infinitely long cylinders.9 Sangani and Acrivos studied the flow in planar (two-dimensional) circular pillar arrays using numerical method and determined the permeability of such planar arrays.10 Srivastava et al. provided scaling analysis and performed finite element simulations to obtain the viscous resistance in micropillar arrays.11 In addition, Ishino et al. also carried out scaling analysis and determined that the viscous resistance may be dominated by either the bottom surface or the pillars, depending on the pillar height compared to the period of the array.12 While these studies have used various methods to quantify the capillary pressure and viscous resistance in micropillar arrays, a predictive model for both capillarity and viscous resistance is needed. In this work, we developed a semianalytical model to predict liquid propagation rates based on the diameter, height, and period of the micropillar arrays. The model follows the Washburn approach by balancing the capillary pressure and viscous resistance. In particular, we considered the actual meniscus shape to determine the driving capillary pressure for pillar arrays with a height-to-period ratio greater than 1. The viscous resistance was obtained by solving Brinkman’s equation13 for pillar arrays with a diameter-to-period ratio less than 0.57. Experiments were performed on various microfabricated silicon pillars to validate the developed model within the defined geometric range, which is most useful for large propagation rates. (9) Hasimoto, H. On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 1959, 5(02), 317–328. (10) Sangani, A. S.; Acrivos, A. Slow flow past periodic arrays of cylinders with application to heat transfer. Int. J. Multiphase Flow 1982, 8(3), 193. (11) Srivastava, N.; Din, C.; Judson, A.; MacDonald, N. C.; Meinhart, C. D. A unified scaling model for flow through a lattice of microfabricated posts. Lab Chip 2010, 10, 1148–1152. (12) Ishino, C.; Reyssat, M.; Reyssat, E.; Okumura, K.; Quere, D. Wicking within forests of micropillars. Europhys. Lett. 2007, 79(5), 56005. (13) Brinkman, H. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1949, 1(1), 27.
Published on Web 09/01/2010
Langmuir 2010, 26(19), 15070–15075
Xiao et al.
Letter
Figure 1. Scanning electron micrograph (SEM) of a representative micropillar array. (a) Micropillars with diameter d = 2.3 μm, period l = 4.5 μm, and height h = 8.3 μm. (b) Magnified image of a single micropillar from (a) showing scallop features on the sides as a result of deep reactive ion etching (DRIE).
Typical micropillar arrays for microfluidic applications consist of a uniform square array of cylindrical pillars. An example scanning electron micrograph (SEM) of such a surface with diameter d = 2.3 μm, period (pillar center-center distance) l = 4.5 μm, and height h = 8.3 μm is shown in Figure 1a. The pillars were defined by photolithography and etched by deep reactive ion etching (DRIE) in silicon. Due to the alternating etch and passivation steps characteristic of DRIE, scallop features were formed on the sides of the pillars (Figure 1b). We determined the capillary pressure in the micropillar array using a thermodynamic approach minimizing interfacial free energy. The pressure is defined as the change in surface energy per unit volume, Pcap ¼ ΔE=ΔV
ð1Þ
where ΔE is the decrease in surface energy as the liquid fills one unit cell and ΔV is the corresponding volume of the liquid filling one unit cell. To accurately predict the change in surface energy and volume, Surface Evolver (SE)14 was used to simulate the shape of the meniscus with minimum surface energy. A unit cell consisting of four quarter-pillars filled with water was defined as shown in Figure 2a. The effect of the scalloped features was included in the simulations as an effective surface energy where the scallops were approximated as a series of semicircles with a roughness factor, rf = π/2.15 The intrinsic contact angle of deionized (DI) water on the silicon surface was measured via a goniometer to be θ = 38. The model construction process for the SE simulations is discussed in more detail in the Supporting Information. Geometries with diameters, d, and periods, l, ranging from 2 to 8 μm and from 8 to 15 μm, respectively, were considered. Gravitational effects were neglected due to the fact that, for such length scales, the Bond number, defined as Bo = Fgl2/γ, is on the order of 10-5. Simultaneously, we used interference microscopy to examine the shape of the water meniscus in the microfabricated pillar arrays with diameters d ranging from 2.5 to 6.4 μm and periods l ranging from 8 to 30 μm. Fringes were generated by the interference of laser light (λ = 405 nm) reflected at the liquid surface and at the substrate (Figure 2b). The shape of the water meniscus in the pillar arrays was obtained by examining the dark fringes, where each dark fringe corresponds to a λ/2n difference in the relative thickness of the liquid film, where n = 1.3 is the refractive index of water. (14) Brakke, K. A. The Surface Evolver. Exp. Math. 1992, 1(2), 141–165. (15) Wenzel, R. N. Resistance of Solid Surfaces to Wetting by Water. Ind. Eng. Chem. 1936, 28(8), 988–994.
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Figure 2 compares the simulated meniscus shape to experimental interference measurements on a micropillar array with diameter d = 6.4 μm and period l =15 μm. Figure 2a shows the three-dimensional meniscus shape obtained by SE, and Figure 2b shows the top-down view of the experimentally obtained interference patterns on the same pillar geometry. Close to the pillar sidewalls (
