Dynamics of Capillary-Driven Flow in Open Microchannels - The

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Dynamics of Capillary-Driven Flow in Open Microchannels Die Yang, Marta Krasowska, Craig Priest, Mihail N. Popescu, and John Ralston* Ian Wark Research Institute, University of South Australia, Mawson Lakes SA 5095, Australia ABSTRACT: The dynamics of capillary-driven flow was studied for water and waterglycerol mixtures in open hydrophilic microchannels (embedded in a hydrophobic matrix). The position of the advancing meniscus was recorded as a function of time using high speed microscopy and compared with the Washburn equation. The square of the position of the liquid front increased linearly with time, as predicted by Washburn. For a channel of the same depth, irrespective of the shape of the channel cross-section (rectangular or curved), the liquid flow was faster with decreasing channel width. A modified Washburn equation, accounting for the different flow profile in the open, noncylindrical channels, was developed. The theoretical prediction was in good agreement with the experimental data for a no-slip boundary condition at the liquidair interface.

1. INTRODUCTION Many technologies, including diagnostic testing,1 DNA manipulation,2 and chemical microreactors,3 rely on the precise handling of minute volumes of liquid in microfluidic channels. For multiphase microfluidics, where droplets, bubbles, or coflowing streams are confined between microchannel walls, interfacial tension can dominate the fluid behavior and is therefore central to the particular technology.46 The same is true for single phase microfluidics where liquid displaces vapor when it first enters the channels. It is well-known that the initial filling of a channel can be carried out using either mechanical pumping or capillarity. The latter is promising in many applications, as it is spontaneous and no external power or expensive equipment is required; it is particularly important in open channels (where conventional pumping methods fail). That is why our study is dedicated to the dynamics of capillary driven flow in open microchannels of two different geometries commonly encountered in microfluidics. Open microchannels have the potential to play an important role in the future of microfluidic technologies. These microchannels offer ease of surface modification, straightforward cleaning, and relatively simple fabrication methods (no chip bonding is required). Moreover, the risk of microchannels clogging is eliminated. While some open microfluidic systems exploit phenomena such as electrowetting7,8 to move liquids, capillarity remains an effective method for low-cost, in-field applications where autonomous operation is required.9,10 Therefore, an interest in developing open microfluidic systems where the capillarity is the driving force has been increasing.1115 Such systems are already applied for micromanipulation of biological macromolecules in microchannels.16 Nonetheless, a thorough understanding of the dynamics of capillary driven flow in open microchannels is very important, especially where the geometry of the channels varies considerably between fabrication methods and for different applications. r 2011 American Chemical Society

For closed cylindrical channels, capillary-driven flow is welldescribed by the classical Washburn equation (also known as the LucasWashburn equation).17,18 The Washburn equation balances the driving force provided by the decrease in free energy as the fluid wets the walls of the channel against the viscous drag of the liquid. Assuming uniform cylindrical channels, negligible gravity (and other external body forces), and a vapor ambient phase, the Washburn equation is given by: x2 ¼

γR cos θ t 2μ

ð1Þ

where R is the channel radius, μ is the dynamic viscosity of the liquid phase, γ is the surface tension of the liquid, and θ is the static contact angle of the liquid on the channel wall.17,18 Thus, the squared position x2 of the traveling meniscus along the channel is proportional to the time t. The kinetics of capillarydriven flow in channels of different shapes and on rough surfaces has also been investigated experimentally and theoretically,1922 showing that the same behavior, x2∼t, holds.1114,23,24 Furthermore, the Washburn equation has been validated in micro-25 and nanocapillaries2629. The studies of capillary flow in nanocapillaries are supported by molecular dynamics simulations, even for non-Newtonian fluids.30,31 Nonetheless, deviations from the Washburn equation have been observed in capillary flow, especially at short time-scales.15 Theoretical studies have shown that in these early stages of capillary flow inertial effects dominate leading to x∼t2 followed by x∼t, while at later stages these effects are negligible and the dependence x2∼t is observed.32 Other potential contributions to deviations include the uncertainties in determining the radius of the capillary, the formation of gas Received: July 11, 2011 Revised: August 15, 2011 Published: August 16, 2011 18761

dx.doi.org/10.1021/jp2065826 | J. Phys. Chem. C 2011, 115, 18761–18769

The Journal of Physical Chemistry C

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Figure 1. (A) Schematic diagram of the fabrication process. (B) Scanning electron microscopy of two channel cross sections resulting from DRIE (left) and HF (right) etching methods, respectively.

bubbles at the interface in case of very narrow capillaries, or the velocity dependence of the advancing contact angle;33 however, these are not our focus here. Among the studies of capillary-driven flow in open microchannels,1115,19,23,24,3437 the x2∼t dependence was determined in triangular,13 rectangular,34 and skewed U-shaped channels.34 The same dependence was also observed for flow driven by a hydrophilic stripe.15 For triangular channels, the Laplace pressure of the liquid filament varies as a function of the local filling height, and the resulting dynamic instability, which leads to the formation of isolated drops, can be a problem in such channels.36 In microfluidics, a very common cross-section geometry is that of a rectangular or curved rectangular channel resulting from the fabrication methods. Capillary-driven flow in rectangular cross sections has received little attention to date, despite its relevance to microfluidic applications.34,37 An example of modeling the flow behavior in open rectangular microchannels, for the immobilization of individual cells in the channels, has been given by Ryu et al.38 In this paper, we study the dynamics of capillary-driven liquid flow in open, hydrophilic channels embedded in a hydrophobic surface, for two different microchannel geometries most commonly used in microfluidic applications: rectangular or “curved”. We investigate the influence of channel cross-section and liquid viscosity on the rate of liquid flow for water and waterglycerol mixtures. The results are rationalized using a modified Washburn equation.

2. EXPERIMENTAL SECTION Materials. 1-Undecanethiol (98%), ammonium cerium(IV) nitrate (98.5%), and hexadecane (99%) were purchased from Sigma-Aldrich and used in the present study without further purification. Ethanol (analytical reagent (A.R.), 99%), propan-2-ol

(A.R.), acetone (A.R.), glycerol (99.5%), hydrogen peroxide (A.R., 30%), potassium iodide (A.R.), and iodine (A.R.) were obtained from Chem-Supply, Australia. Hydrofluoric acid (50%) was obtained from Univar, and sulfuric acid (A.R. 98%) was obtained from Labscan Asia Co. Ltd., Thailand. Pure gold (99.9%) granules were purchased from PWBECK, Australia, while chromium crystals (99.9%) were obtained from Sigma-Aldrich. Ultrapure water with a pH of 5.6 ( 0.1, a resistivity of 18.2 MΩ 3 cm, and surface tension 72.8 mN/m at 22 °C was obtained from a NANOpore Diamond Barnstead system and used in all experiments. Pyrex discs (1 mm thick, 58 mm in diameter) and quartz rectangular (46.5 mm 38 mm) plates (1 mm thick, polished on both sides) purchased from Herbert A. Groiss & Co. (Australia) were used as substrates. These substrates were cleaned by dedusting in N2, sonication in propan-2-ol (1 min), drying in N2, sonication in 1 M KOH (5 min), rinsing with copious amounts of Milli-Q water (followed by sonication), drying in N2, and plasma treating (Harrick, PDC-OD2) for 60 s in air. Sample Preparation and Characterization. A K975x Turbo evaporator (Emitech, U.K.) under vacuum (∼105 mbar) was used to deposit a 10 nm thick chromium layer (for secure anchoring of the gold), followed by a 20 nm thick layer of gold onto the cleaned Pyrex or quartz substrates. The microchannel and reservoir fabrication on such substrates involved lithography and etching, as shown in Figure 1A. To prepare a rectangular channel, a 15 μm thick layer of negative photoresist (SU8-10, MicroChem Corporation, USA) was spin-coated with an automated spin coater (Karl Suss Delta 80RC) onto the quartz/ Cr/Au substrate and exposed through a Cr-glass mask to UV radiation (mask aligner, EVG 620). Afterward, the photoresist was developed with SU-8 developer (Micro Materials and Research Consumables Pty. Ltd., Australia), leaving only the SU-8 structure behind. The patterned photoresist was hard-baked at 18762

dx.doi.org/10.1021/jp2065826 |J. Phys. Chem. C 2011, 115, 18761–18769


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Figure 2. (A) Schematic diagram (top view) of liquid flow from a reservoir into a hydrophilic microchannel of width W. The hatched and solid gray areas denote the hydrophobic and hydrophilic regions of the substrate, respectively. The liquid reservoir is a circular indentation (depth equal to that of the channel) with a diameter of 4 mm. (B) Snapshots (optical images) of a typical flow experiment (shown for a 64 μm wide curved channel). The left side of the image is aligned with the edge of the reservoir. (C) Profilometer measurement of the experimental liquid profile at a fixed position (∼2 mm away from the reservoir) after a few seconds from the moment of liquid deposition (shown for an 86 μm wide curved channel).

150 °C for 5 min to enhance the structural integrity and the chemical resistance of the microstructures. Au and Cr layers were removed by chemical etching in a KI/I2 solution (4 g of KI, 1 g of I2, 40 mL of H2O) for less than 5 s and a 10% aqueous cerium ammonium nitrate (NH4)2(Ce(NO3)6) solution for 10 s, respectively. The exposed silica was then etched by deep reactive ion etching (DRIE) (ULVAC 570NLD) to form the microchannel. Finally, the photoresist was removed by immersing the sample for 30 min in hot Piranha solution (3:1 mixture of 96% H2SO4/30% H2O2). The depth of the rectangular microchannels was 19.1 ( 0.1 μm for all experiments. Microchannels with a curved cross section were fabricated by wet etching. On cleaned Pyrex/Cr/Au substrates, a 2 μm thick layer of positive photoresist (AZ 1518, Clarient) was spin-coated and exposed through a Cr-glass mask to UV radiation. The substrate was developed in an aqueous solution of AZ developer (Micro Materials and Research Consumables Pty. Ltd., Australia) to remove the exposed photoresist. Au and Cr layers were removed by chemical etching as described above for the rectangular channels. The exposed Pyrex glass was etched in 50% HF, for 2 min. The Pyrex was etched evenly in all directions by HF to obtain “curved” channels with a depth of 18.4 ( 0.1 μm. Since such isotropic etching removes Pyrex from under the photoresist layer to a distance equal to the etch depth, the removal of Cr and Au layers from beneath the overhanging photoresist was essential (with the same etch solutions used earlier, but in reverse order). Finally, the photoresist layer was removed by dissolution in acetone.

For both types of microchannels, the depth was measured using an optical profilometer (WYKO NT9100). Scanning electron microscopy (SEM) (CamScan) was used to image the cross section of the channels (shown in Figure 1B). SEM gives a very accurate image of the channel shapes confirming the channel design (shown/ presented in Figure 1A). The etched channels were uniform and smooth, which makes them ideal for studying the liquid flow. In both cases described above, hydrophilic (SiO2) microchannels embedded in a gold-coated substrate were obtained. To change the wettability of the gold layer, a self-assembled monolayer of alkane thiol was adsorbed overnight from a solution of 1-undecanethiol in ethanol (103 M) at room temperature. Since this thiol adsorbs only at the gold surface, a hydrophilic SiO2 microchannel embedded in a hydrophobic surface was formed. The wettability of thiol-modified gold and SiO2 surfaces were characterized by measuring the water contact angle using the sessile drop technique (OCA 20 instrument, DataPhysics). For the SiO2 channels, the contact angle measurements were carried out on the flat area of the SiO2 reservoir (see Figure 2A) and found to be less than 10°. For the alkane thiol treated gold surface the advancing and receding contact angles were found to be 107 ( 2° and 95 ( 2°, respectively. The value of the advancing contact angle is very close to that previously observed for the same material and the hysteresis is as expected.39 Experimental Setup. The schematic of microchip and liquid for top-view experiments of the flow dynamics is shown in Figure 2A. The sample was placed in a clean glass Petri dish, 18763


The Journal of Physical Chemistry C and a Milli-Q water droplet of 5 μL was deposited by a needle connected to the syringe in the syringe pump onto the reservoir area. Once the droplet contacted the inlet of the channel, the liquid started flowing in the microchannel due to the capillary force. A high speed camera (Photron) (4000 frames per second) mounted on a microscope (Olympus, U-LH50MH) allowed us to track the position of the meniscus as a function of time. The field of view for a single measurement was approximately 1.8 mm. The recorded frames were analyzed using ImageJ software. A typical series of top-view images is shown in Figure 2B. After each experiment, the sample was rinsed thoroughly with ethanol and pure water and stored under water for further use. For each channel and liquid combination, three series of independent measurements were conducted, showing good agreement between experiments. To measure the liquid filling height in the channel, the side-view of the liquid profile was recorded using an optical profilometer (WYKO NT9100) at a fixed position (∼2 mm away from the reservoir, close to the observation range in the experiment), as shown in Figure 2C. Since the liquid flows very fast, the measured profile represents the final condition of the liquid in the channel. The measurements were carried out ∼10 s after the liquid droplet was deposited in the reservoir. Characterization of Liquid Properties. The surface tension and the viscosity of the glycerol/water mixtures (against air) were measured using an OCA 20 instrument (Data Physics) and a Cannon-Manning semimicro viscometer (Cannon Instruments Company), respectively, at room temperature.40 Their values are listed in Table 1 and are in the range of the values reported in the literature.41

3. RESULTS AND DISCUSSION When a droplet is deposited onto the reservoir area, the contact line expands until it makes contact with the inlet of the microchannel.42 At this moment, liquid starts flowing along the hydrophilic microchannel, driven by the capillary force. Profilometer measurements show that the liquid fills the channel to the top with a very small curvature of the top liquidair interface (see Figure 2C). Although such profiles cannot be recorded dynamically during the filling, the fact that at short times and far from the reservoir (droplet) the channel is completely filled supports our assumption that during the whole flow the liquid fills the channel almost to the top. In Figures 3 and 4, the squared position of the meniscus, x2, is shown against time for the two different channel cross sections, various liquids, and various microchannel widths, W. The dynamics of the advancing meniscus show Washburn-like behavior; that is, x2 increases linearly with time t, the time t being measured from the moment corresponding to the first frame in which the liquid is seen in the channel (in some cases a weak oscillatory behavior around x2∼t, which is due to the oscillatory motion of the reservoir droplet, can be observed). Note that this linear relationship starts only after a short time t0 (t0 is the transient time as the hydrodynamic flow field in the channel relaxes to the quasisteady state characteristic of a Washburn-like dynamics). It is difficult to precisely determine t0 from experiments (including in this study), and it is usually neglected in most interpretations of experimental data, as t/t0 is typically quite large. However, in this present study, due to the high rate of liquid flow and to the short spatial range of observation along the channel (∼1.8 mm), the ratio t/t0 must be considered. To estimate this ratio, we take the intercept of the time axis based on a linear fit to our data

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Table 1. Surface Tension and Viscosity of Water/Glycerol Mixtures at Room Temperature glycerol (%)a

a

surface tension γ (mN/m)b

viscosity μ (mPa 3 s)

0

72.8

0.89

20 30

68.3 66.7

1.99 2.98

Volume percentage. b Precision of measurement was (0.5 mN/m.

Figure 3. Progress of the liquid meniscus, x2, with time, t, for pure water flowing in rectangular and curved open microchannels. The solid lines show the linear fit, x2 = k(t t0), to the data in the linear regime. The vertical broken line shows tc, which is the time required for all data sets to enter Washburn behavior for each type of channel.

(to an R2 value >0.9997, achieved by progressively excluding the earliest data points in the fit), as shown in Figures 3 and 4. Typical values for t0 are within 210 ms for water flowing along the rectangular and curved channels, respectively, and within 5 20 ms for a 30% glycerolwater mixture. As we follow the moving meniscus for at least several tens of milliseconds, a regime of linear dependence of x2 with t appears to be well-established in our experiments.43 This linear behavior is observed to be independent of the exact shape of the cross section, that is, irrespective of whether the channels were rectangular or curved; the shape of the channel shows solely in a prefactor that determines the flow rate (the slope of the fits shown in Figure 3, see also eq 1). 18764



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damped oscillations of the reservoir drop; as previously discussed, this is based on the quasi-linearity observed in the data in Figure 3A,B), and it is opposed by the hydrodynamic viscous friction. After the initial transient regime (as the liquid accelerates near the entrance of the channel), these two forces balance one another. The result of this force balance, the derivation of which is presented in the Appendix, indeed predicts a linear dependence, as observed in the experiment, with the mobility parameter being given by the equations: k¼

2γD ½2 cos θ ð1 cos θÞp gðpÞ, rectangular μ p2 ð3Þ

k=

2γD ½π cos θ 2 ð1 cos θÞp gðpÞ, curved μ p2 ð4Þ

Figure 4. Progress of the liquid meniscus, x2, with time, t, for 30% glycerolwater mixtures flowing in rectangular and curved open microchannels. The solid lines show the linear fit, x2 = k(t t0), to the data in the linear range. The vertical broken line shows tc, which is the time required for all data sets to enter Washburn behavior for each type of channel.

For the open microchannels studied we observed slower meniscus velocities (slower flow) in wider channels.44 Furthermore, increasing the viscosity μ of the liquid by adding glycerol to the water slows down the velocity of the meniscus but gives the same trend—slower flow in wider channels, as shown in Figure 4. In summary, for all of the channelliquid combinations studied after a transient behavior at very short time-scales t0 (less than a few milliseconds) the flow dynamics obeys the linear dependence: x2 ≈ k t

ð2Þ

similar to eq 1, where k is a “mobility parameter” dependent on the details of the particular system (including the channel cross-section and dimensions, and the viscosity and surface tension of the liquid). Due to the similarity between the capillary-driven flow from a large reservoir into a tube, described by the classic Washburn equation,17,18 and the flow dynamics observed in our microchannel system, we employ the Washburn modeling method to calculate the mobility parameter, k. We assume that the flow is driven by the change in the free energy of the system as the liquid advances along the channel (thus we neglect any effect due to the

In eqs 3 and 4 p = W/D denotes the aspect ratio of the channel width W (measured at the flat bottom) to the depth D, while the function g(p) (see eq A8 in the Appendix) describes the dependence of the average flow velocity, thus of the viscous friction, on the aspect ratio of the cross section of the channel. θ is the equilibrium or Young's contact angle. Note that, as the aspect ratios increase (channels become wider) to a large extent, the k values may become negative for both geometries. This means that for any aspect ratio p there is a critical contact angle θc = arccos [p/(p + 2)] (rectangular), θc = arccos [(p + 2)/(p + π)] (curved), respectively, above which such a Washburn-like filling of the channel is no longer possible and the liquid does not flow into the channels (or at least does not fill the channel). Equations 3 and 4 predict that for fixed D (which is the case of our experiment, where the depth of the channels is fixed and the liquid fills the channel completely; see Figure 2C) the product (μ/γ)k is a function of the aspect ratio p only. Figure 5A,B shows that indeed when plotted in this way there is a good collapse of the experimental data (shown in the corresponding insets), extracted from the linear fits of x2 versus t, onto a master curve. Note that, as the surface tension and viscosity of the liquid were measured independently (see Table 1), eqs 3 and 4 do not contain any free fitting parameters. This result is somewhat surprising because usually it is assumed that at a liquidair interface a full-slip boundary condition holds.45 It has been argued in the past46,47 that the presence of minor contamination interferes with the tangential flow at an interface and can immobilize it. The immobilization of waterair interface can result from the presence of surfactants, but it can also result from the presence of small gradients in surface tension caused by one or more mechanisms that can operate in pure water, as discussed by Yaminsky et al.48 Furthermore, immobile boundary conditions were also observed for other fluidfluid interfaces; examples include the mercuryelectrolyte interface investigated by Connor and Horn, using the surface forces apparatus (SFA)49,50 and the glycerolsilicone interface, investigated by Klaseboer and co-workers, who studied the drainage of the thin liquid film between two deformable drops approaching each other at constant velocity.51 (Note, however, that a lack of slip at the waterair interface has been observed experimentally even in very clean systems, for example, the liquidair interface in the proximity of solid wall.52,53) 18765



Figure 5. Comparison between the product (kμ/γ) calculated from eqs 3 and 4, lines, and the values obtained from fitting our experimental data, symbols, for channels of various aspect ratios, p. The insets show the corresponding values k as a function of p for various liquids, which when plotted as (kμ/γ) show a very good data collapse onto one curve.

In the calculation of the mobility parameter k, a constant driving force and complete filling by liquid (flat liquidair interface to the top) have been assumed. Although a drop oscillation was observed during the experiment, which may jeopardize our assumption of constant driving force, this was observed to have a negligible effect on the flow dynamics, as shown by the linear relationship for the flow behavior. Moreover, such an oscillatory behavior cannot explain an overall significantly slower flow. The assumption of a constant driving force also refers to a constant contact angle preserving the equilibrium value at all times. This has been recently questioned in the light of the dependence of the dynamic contact angle on the wetting-line velocity. However, in a high velocity of capillary flow, which applies to our case, one can safely neglect the effect of a dynamic contact angle.54,55 An incomplete filling of the channel would be compatible with a slower than predicted flow. However, the sideview of the liquid profile recorded and shown in Figure 2C strongly suggests, as discussed, that during flow the liquid fills the channel to the top. The small curvature of the liquidair interface proves that the assumption of a flat liquidair interface is reasonable. These observations rule out the incomplete filling of the channel as being the source of the observed behavior. Therefore, a plausible cause for this no-slip boundary condition at the liquidair interface is the possible presence of traces of surface active impurities at the liquidair interface.

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The morphology of the meniscus during flow deserves further discussion. In the theoretical development we have assumed that the contact angle and the shape of the meniscus do not change during the flow of liquid in the channel. In our experiments, this condition was fulfilled—at least qualitatively—for capillary-driven flow in narrow and medium width channels (W/D from 0.8 to 2.7). However, for wider channels (W/D from 3.9 to 5.4), we have noticed liquid “fingers” extending ahead of the meniscus along the sides of the channel. A similar phenomenon has been reported by Seemann et al.56 for static liquids in rectangular microchannels. In this latter case the existence of liquid fingers is determined by the specific features of the system, including the (static) contact angle and the aspect ratio of the channel. Depending on the nature of the system, a rich variety of static liquid morphologies can be observed in open rectangular channels,56 but little is known, yet, about their existence and behavior when the liquid flows. Thus, the assumption of an unchanging liquid morphology is not obeyed for channels with large aspect ratios (note that for large aspect ratios the critical contact angles for Washburn-like flow into the channels are very low, which can explain the occurrence of such morphological instabilities). The fact that the theoretical predictions are, nevertheless, close to the experimental data, with the bottom of the meniscus as the reference for monitoring liquid motion, implies that these fingers do not alter the capillary driving force significantly compared with the main contribution of the channel, over the time scales of our experiments. Although this morphology change seems to have a relatively minor effect on the flows observed in this study, such changes in the meniscus shape, or those due to a velocity dependent dynamic contact angle33 (which decreases as the velocity of the advancing liquid slows), could play an important role if the liquid flow is observed over time scales significantly longer than the ∼100 ms of the present experiments.

4. CONCLUSION A systematic experimental investigation of the dynamics of capillary-driven flow for water and waterglycerol mixtures in open microchannels has been performed. The results show that, for two channel cross sections commonly used in microfluidic devices, the square of the position of the liquid front (with respect to the inlet of the channel) increases linearly with time, in agreement with Washburn's theory. The flow velocity decreases with increasing channel width due to the different flow profile in an open channel geometry. The experimental observations can be explained by accounting for the noncylindrical geometry of the channels (rectangular or “curved”) in a modified Washburn equation. The theory shows very good agreement with the experimental results, provided that a no-slip boundary condition is invoked at the liquidair interface. Small deviations from the theory are discussed in terms of the assumptions employed, including the dynamic morphology of the meniscus during the capillary-driven flow. As capillarity is an important consideration in the function of microfluidic devices, especially for open channels where pumping is not applicable, our description of capillary-driven flow dynamics in these channel geometries will provide a suitable platform for their future design. ’ APPENDIX To calculate the mobility parameter k, the flow rate of liquid in open microchannels is modeled using a similar force balance argument as that employed in the derivation of the classic 18766



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Washburn equation;17,18 however, the symmetry of the cylindrical capillary is lost, and the hydrodynamic boundary conditions (BC) at the wall and at the liquidair interface may be different. We thus assume that the fluid flow in the channel is driven by capillarity and opposed by viscous friction and that after a very short transient regime these two balance each other at every moment. Furthermore, we assume that the liquid flow is confined to the channel and that the depth filled by liquid is constant from entry up to the meniscus, that the top liquidair interface is flat (or of very small curvature), and that the contact angle with the channel walls, as well as the shape of the advancing meniscus, is constant during spreading. The reservoir droplet is large, and thus the changes in volume and shape due to the liquid filling the channel are negligible; moreover, since its radius of curvature is in the millimeter range, its Laplace pressure is very small. Therefore, the drop plays solely the role of a liquid reservoir for the liquid filling the channel. In what follows, we choose a coordinate system with the origin O located at the channel's entry in the midpoint of the bottom wall, such that the planar bottom wall of the channel corresponds to the Oxy plane, the axis Ox is oriented along the channel in the direction of flow, and the Oz axis is oriented into the channel, normal to the bottom wall (see also Figure 1A). The cross section of the rectangular channel thus corresponds to (W/2) e y e +(W/2), and 0 e z e D. The capillary driving force is calculated from the change in the free energy of the system due to the liquid flow in the channel. The free energy change E due to the liquid filling a portion x of the initially empty channel is given by

and assuming that for Newtonian fluids the hydrodynamic flow occurs at low Reynolds number, is laminar and at every moment is well-described by a quasi-steady-state Poiseuille type flow B u= (υ(y,z), 0, 0) induced by the pressure gradient ΔP/x = [Fγ/(area)]/x stemming from the capillary forces. Thus the flow obeys the Stokes equations ∇2 υ ¼

Fμ ¼ 8πμαðpÞx

dE ðA2Þ dx For a rectangular channel with width of W and depth D, the areas in eq A1 are given by Asl = (2 + p)Dx and Al = Dpx, where p = W/D is the aspect ratio. Thus one obtains: dE ðA3Þ ¼ ½2 cos θ ð1 cos θÞpDγl dx For a curved channel of depth D, for which the shape consists of the flat bottom of width W continued by a quarter of a circle of radius D, the areas are given by Asl = (π + p)Dx and Al = (2 + p)Dx, therefore: Fγ ¼

dE ðA4Þ ¼ ½π cos θ 2 ð1 cos θÞpDγl Fγ ¼ dx The viscous resistance force Fμ is obtained by neglecting end effects (at the entry of the capillary and at the meniscus region)

ðA5Þ

dx dt

ðA6Þ

where dx/dt is the (cross-sectional) averaged velocity of the moving liquid and α(p) is a geometrical factor given by αðpÞ ¼

1 p2 8π gðpÞ

with

Z gðpÞ ¼

ðA1Þ

Fγ ¼

∇ uB ¼ 0

(Note that the continuity equation is automatically satisfied by u = (υ(y,z),0,0) and that eq A5 should be solved for a given x; B i.e., the pressure gradient is considered a constant when solving for B u .) The flow field B u is subject to the boundary conditions of no-slip at the solid walls and slip at the liquidair interface (assumed, as we mentioned, to be located at z = D). Then the viscous friction can be written in a form similar to the one corresponding to a closed cylindrical tube, that is,

E ¼ Al γl þ ðγsl γs ÞAsl þ Em ¼ γl ðAl Asl cos θÞ þ Em

where γsl, γs, and γl are the interfacial tensions for the solidliquid, solidair, and liquidair interfaces, respectively, θ is the equilibrium contact angle (from Young's equation, γsl γs = γl cos θ), Asl is the area of the liquidsolid interface, Al is the liquidair surface area, the loss of solidair surface is equal to Al, and Em is the free energy associated with the region from the bottom to the top of the meniscus (which is assumed to be a constant with respect to x, as the shape of the meniscus does not change during the flow). The driving force (the effect of the capillary pressure) Fγ is then obtained as the negative rate of change in free energy with respect to x while keeping all of the other parameters (temperature, etc.) constant, that is,

1 ΔP μ x

p=2 p=2

0

dy

ðA7Þ

0

Z

1

dz 0 0

y ¼ y=D,

0

0

0

υðy , z Þ , υ0

z ¼ z=D,

ðA8Þ

and υ0 = (1/μ)(ΔP/x)D2. To compute the cross-sectional average in eq A8, one has to explicitly calculate the flow field, which requires that eq A5 is solved subject to appropriate boundary conditions. We discuss the cases of no-slip and full-slip boundary conditions at the top liquidair interface, respectively, because for any partial slip boundary condition the result will be bounded by these two. For no-slip boundary conditions at both solidliquid and liquidair interfaces, the solution is identical to that of flow in closed rectangular channels.57,58 For the case of a full-slip boundary condition at the liquidair interface, the solution of eq A5 is easily obtained from the known result for Poiseuille flow with a no-slip boundary condition in rectangular closed channels by noting that the midheight plane in such a channel is a plane of zero tangential shear stress (full slip). Therefore, one formally replaces the depth D of the channel by 2D in the formulas corresponding to the closed channel. The result then is: 0

0

υðy , z Þ 8 > ΔP > > > > > > < μL ¼ > > ΔP > > > > > : μL

" # 0 4D2 1 coshðnπy Þ 0 1 sinðnπz Þ ðno-slip BCÞ π3 n g 0, nodd n3 cosh nπp=2 2 0 3 ! 0 cosh nπy =2 7 16D2 16 nπz sin 1 ðfull-slip BCÞ 4 5 π3 n g 0, nodd n3 2 cosh nπp=4

∑

∑

ðA9Þ By combining eqs A6A9 to compute Fμ, equating the result to the driving force Fγ (eq A3), and integrating the resulting 18767



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differential equation, one obtains a linear time dependency x2 ¼ k t

ðA10Þ

with k¼

2γD ½2 cos θ ð1 cos θÞp gðpÞ μ p2

ðA11Þ

and 8 16 1 nπ nπ > > ðno-slip BCÞ > < π5 n g 0, nodd n5 2 p tanh 2 p gðpÞ ¼ 128 1 nπ nπ > > p tanh p ðfull-slip BCÞ > 5 4 : π5 n 4 n g 0, nodd

∑

∑

ðA12Þ Because of the complex geometry of the cross section in a curved channel, an exact analytical solution is not possible for this case. However, one can immediately bound the viscous resistance from below by that of a rectangular channel of width W and from above by that corresponding to a rectangular channel of width (W + 2D); for wide channels, the two bounds obviously converge to the same limit. Intuitively, the lower bound is a better approximation for the exact value of the viscous friction because it simply shifts the zero velocity lateral walls into vertical ones slightly inside the channel (but in regions where the velocity would be small anyway), while the upper bound involves adding liquid and liquid flow resistance, over a significant additional cross-section. We therefore approximate the viscous friction in the case of curved channels by that corresponding to a fictitious rectangular channel with depth D and a width W equal to that of the bottom part of the curved channel. For wide channels, this is expected to be quite accurate, but for very small width channels (p close to 1) it may lead to relatively large underestimates of the viscous friction. Following the same reasoning as above and using the corresponding expression, eq A4, for the driving force Fγ, we again obtain the linear dependency in eq A10 but with k=

2γD ½π cos θ 2 ð1 cos θÞp gðpÞ μ p2

ðA13Þ

with the same g(p) as given by eq A12.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Fax: + 61 (0)8 8302 3683. Tel.: + 61 (0)8 8302 3066.

’ ACKNOWLEDGMENT Discussions with Dr. Rossen Sedev are warmly acknowledged. Photolithography masks and microchannels were prepared at the Optofab and University of South Australia nodes of the Australian National Fabrication Facility (ANFF) using Federal and State Government funding, under the National Collaborative Research Infrastructure Strategy to provide nano- and microfabrication facilities for Australia's researchers. The authors acknowledge the financial support from the Australian Research Council Linkage and Discovery Project schemes, Grants AMSRI and DP1094337.

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Dynamics of Capillary-Driven Flow in Open Microchannels - The

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