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Transport Dynamics in Open Microfluidic Grooves Jean-Christophe Baret† and Michel M. J. Decre´ Philips Research Laboratories EindhoVen, Prof. Holstlaan 4, 5656 AA EindhoVen, The Netherlands
Stephan Herminghaus and Ralf Seemann* Max Planck Institute for Dynamics and Self-Organization, D-37018 Go¨ttingen, Germany ReceiVed December 11, 2006. In Final Form: January 16, 2007 In microscopic rectangular grooves various liquid wetting morphologies can be found, depending on the wettability and details of the geometry. When these morphologies are combined with a method to vary the apparent contact angle reversibly, transitions between droplike objects and elongated liquid filaments can be induced. Liquid can thus be transported on demand along the grooves. The dynamics of liquid filaments advancing into grooves as well as receding from grooves has been studied, varying the contact angle using the electrowetting effect. The dynamics of the receding filament is purely capillarity driven and depends only on the contact angle, the viscosity of the liquid, and the geometry of the groove. The length and the dynamics of the advancing filaments, on the other hand, are strongly dependent on the ionic content of the liquid and the applied ac voltage.
Introduction Various open microfluidic concepts with freely accessible liquid surfaces have been developed as an alternative to standard microfluidic devices, where microchannels are surrounded by a solid matrix.1-3 Generally, there are two strategies to guide liquid by open microfluidic structures. The first one is to prepare wettability patterns on planar substrates.4-10 Here, the Laplace pressure of the liquid morphologies is always positive, such that liquid will never spread spontaneously along these patterns when brought in contact with a large reservoir. The second strategy is to offer an appropriate surface topography to the liquid.11,12 This exploits the fact that the liquid rather wets wedges and grooves than planar surfacessprovided that the intrinsic contact angle is sufficiently small. Depending on control parameters such as contact angle, liquid volume, and geometry of the surface topography, a rich variety of liquid morphologies can be found at steps13 and in rectangular grooves.14 Let us consider a planar substrate with straight grooves etched into it. The wettability (i.e., the contact angle) shall be the same on the walls of the groove as on the ridges between the grooves. For very large contact angles the liquid has no reason to enter the grooves and will form droplike morphologies, regardless of the cross section of the grooves (cf. Figure 1a). If the contact angle is below a critical value, however, which depends on the geometry of the groove, the liquid enters the grooves. Here, the † Present address: ISIS-ULP, 8 alle ´ e Gaspard Monge, 67083 Strasbourg, France.
(1) Thorsen, T.; Maerkl, S. J.; Quake, S. R. Science 2002, 298, 580. (2) Squires, T. M.; Quake, S. R. ReV. Mod. Phys. 2005, 77, 977. (3) Weigl, B. H.; Bardell, R. L.; Cabera, C. R. AdV. Drug Del. ReV. 2003, 55, 349. (4) Darhuber, A. A.; Troian, S. M. Annu. ReV. Fluid Mech. 2005, 37, 425. (5) Gau, H.; Herminghaus, S.; Lenz, P.; Lipowsky, R. Science 1999, 283, 46. (6) Kataoka, D.; Troian, S. Nature 1999, 402, 794. (7) Wang, J., et al. Nat. Mater. 2004, 3, 171. (8) Zhao, B., et al. Science 2001, 291, 1023. (9) Daniel, S.; Chaudhury, M. K.; Chan, J. C. Science 2001, 291, 633. (10) Hayes, R. A.; Feenstra, B. J. Nature 2003, 425, 383; Annu. ReV. Fluid Mech. 2005, 37, 425. (11) Concus, P.; Finn, R. Proc. Natl. Acad. Sci. U.S.A. 1969, 63, 292. (12) Rasco´n, C.; Parry, A. O. Nature 2000, 407, 986. (13) Brinkmann, M.; Blossey, R. Eur. Phys. J. E 2004, 14, 79. (14) Seemann, R.; Brinkmann, M.; Kramer, E. J.; Lange, F. F.; Lipowsky, R. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 1848.
Figure 1. Sketches of wetting morphologies: (a) droplike morphology, which can be found for large contact angles; (b) below a certain threshold, the liquid will fill the groove completely with either a negative or positive mean curvature of the liquid/air interface.
liquid either fills the entire cross section of the groove and forms elongated filaments, as shown in Figure 1b, or spreads along the bottom corners and forms liquid wedges.14 These liquid filaments or wedges have either a positive or negative Laplace pressure, depending on the contact angle, which allows filling the grooves even from a reservoir with zero Laplace pressure. Therefore, if a grooved substrate is combined with a technique to vary the wettability, transitions between the liquid morphologies can be induced, resulting in a net flow of liquid along the grooves.14,15 To optimize the design and the possible application of a topographically structured, open microfluidic system, it is beneficial to understand the transporting time of the liquid precisely. The spreading dynamics of wetting liquids from a reservoir or a drop into grooves with triangular cross section has been studied extensively in the past.16-21 However, due to a dynamic instability, which leads to the formation of isolated drops, elongated liquid filaments can never be drained from triangular grooves by capillary forces, even if the wettability can (15) Baret, J.-C.; Decre´, M.; Herminghaus, S.; Seemann, R. Langmuir 2005, 21, 12218. (16) Rye, R. R.; Mann, J. A., Jr.; Yost, F. G. Langmuir 1996, 12, 555. (17) Yost, F. G.; Rye, R. R.; Mann, J. A., Jr. Acta Mater. 1997, 45, 5537. (18) Rye, R. R.; Yost, F. G.; O’Toole, E. J. Langmuir 1998, 14, 3937. (19) Weislogel, M. M.; Lichter, S. J. Fluid. Mech. 1998, 373, 349. (20) Warren, P. B. Phys. ReV. E 2004, 69, 041601. (21) Dussaud, A. D.; Adler, P. M.; Lips, A. Langmuir 2003, 19, 7341
10.1021/la063584c CCC: $37.00 © 2007 American Chemical Society Published on Web 03/23/2007
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Figure 2. Electrowetting curves for advancing contact angles on a plane substrate for various frequencies of the applied voltage. The solid and dashed lines are fits to the data using the Lippmann equation (solid line) and eq 1 (dashed line).
be varied.22 The driving force of this dynamic instability is the Laplace pressure of the liquid filament, which varies locally as a function of the filling height. For liquid filaments in grooves with parallel side walls, however, the Laplace pressure is independent of the local filling height and the grooves can be reversibly filled. In this article we will therefore focus on the dynamics of the advancing and receding liquid filaments in rectangular grooves when the apparent contact angle is varied by electrowetting.23-27 Experimental Section For electrowetting, an electrically conducting substrate is needed which bears a dielectric coating. When a conducting liquid is placed on such a substrate, a capacitor results. When a voltage is now applied to it, the field energy stored in it leads to a decrease of the apparent contact angle. This can be a dramatic effect, spanning several tens of degrees23-27 (cf. Figure 2). Using electrowetting, one can reversibly lower the apparent contact angle, whereas the materials contact angle θY without any voltage applied is the maximum contact angle. Thus, it is advantageous to use hydrophobized substrates, allowing for a large range of accessible contact angles. To minimize electrochemical effects, we use sinusoidal ac voltage for all experiments presented here. Using standard photolithographic methods combined with pulsed reactive ion gas etching, rectangular grooves have been fabricated in conductive, arsenic-doped silicon wafers. Subsequently, an insulating oxide layer with thickness d ) 1.15 ( 0.15 µm was thermally grown into the structured silicon sample. The resulting grooves have a width, W, in the range (15-40) ( 1 µm and a fixed depth, D, of 20 ( 1 µm, yielding aspect ratios, A ) D/W, between 0.50 and 1.33. To increase the materials contact angle, θY, of the liquid, the samples were hydrophobized by grafting a selfassembled monolayer of OTS molecules (octadecyltrichlorosilane) on top, following standard procedures.28 As the wetting liquid we used a mixture of 17 ( 3% water, 80 ( 3% glycerol, and 3 ( 1% NaCl by weight. The volume fractions were chosen such that the mixture was hygroscopically stable under our typical laboratory conditions. The corresponding conductivity of the mixture, σ, is 0.11 ( 0.04 S/m, as measured by a HA8733 Hana conductometer. The liquid/air interfacial tension taken from tables is γ ≈ 65 mN/m and the viscosity η ≈ 80 mPa s. The apparent contact angle, θ, of (22) Khare, K.; Brinkmann, M.; Law, B. M.; Gurevich, E.; Herminghaus, S.; Seemann, R. Submitted for publication. (23) Quilliet, C.; Berge, B. Europhys. Lett. 2002, 60, 99. (24) Someya, T., et al. Langmuir 2002, 18, 5299. (25) Prins, M. W. J.; Welters, W. J. J.; Weekamp, J. W. Science 2001, 291, 277. (26) Mugele, F.; Baret, J.-C. J. Phys.: Condens. Matter 2005, 17, 705. (27) Lippmann, G. Ann. Chim. Phys. 1875, 5, 494. (28) Sagiv, J. J. Am. Chem. Soc. 1980, 102, 92.
Figure 3. Time series of optical micrographs showing a liquid filament advancing into a groove and receding from a groove with W ) 40 µm and H ) 20 µm: (a) switching the applied voltage up to U0 ) 80 V and f ) 20 kHz; (b) switching the voltage down to zero. this experimental system can be tuned from about 90°, without any voltage applied, down to about 50° for the applied rms voltage U0 ≈ 90 V, as shown in Figure 2. Due to contact angle hysteresis, the receding contact angle is always 10-15° smaller than the apparent advancing contact angle. The apparent contact angle is independent of the frequency of the applied voltage within the tested range of f ) 5-30 kHz, as well as of the position on the wafer, indicating a good lateral uniformity of the OTS coating and the oxide layer. For small voltages, here up to ∼45 V, the apparent contact angle, θ, is described by the Lippmann equation, cos θ ) cos θY + U02/(2dγ), where is the dielectric permittivity of the insulating silicon oxide layer.26 The Lippmann equation is shown as a solid line in Figure 2. For larger voltages a deviation from the Lippmann equation is observed, which is wellknown but not yet understood. In our experiments, the voltage of interest ranges from 40 to 90 V. In this range, the corresponding apparent contact angle θ deviates from the expectation of the Lippmann equation and is approximately described by the linear relation cos θ ) cos θY +
U0 UL
(1)
shown as a dashed line in Figure 2. UL was determined to 122 V by fitting eq 1 to the electrowetting data. Hence, in the following we will use this equation to describe the dependence of the apparent contact angle as function of the applied voltage.
Results and Discussion When the applied voltage is switched up to a value above the filling threshold UT, respectively θT, a liquid filament advances into the groove, as shown in Figure 3a. The length of the liquid filament, L, increases quickly for early times and plateaus after some time. When the voltage is switched back to zero, the liquid filament reversibly recedes to its feeding drop (cf. Figure 3b). The receding velocity starts slowly and accelerates for smaller filament length, L. The time scales for the groove filling and draining are similar and depend on the groove dimension, geometry, and wettability. The maximum length and the dynamics of advancing liquid filaments are furthermore dependent on the
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Figure 4. Dynamics of advancing and receding (insets) liquid filaments in grooves with W ) D ) 20 µm for (a) various frequencies, f ) 10, 15, 30, 60 kHz, of the applied voltage U0 ) 80 V and for (b) various voltages, U0 ) 55, 60, 71, 80 V, at fixed frequency f ) 20 kHz. The dashed lines are fits to the early time dynamics of the advancing filaments and the dynamics of receding filaments, respectively.
electrical properties of the system, as we will discuss in more detail in the following. In Figure 4 the dynamics of advancing liquid filaments are shown for various applied frequencies and voltages. The maximum length of a filament for a given applied voltage and frequency is finite, due to the finite conductivity of the liquid and the resulting voltage drop along the filament.15 At equilibrium, the voltage at the tip of the liquid filament equals the critical voltage for the groove filling. Accordingly, the wetting properties and thus the driving forces vary locally along the liquid filament.15 Hence the advancing dynamics slows down as soon as the voltage drop along the filament becomes significant. For the early time dynamics of advancing filaments, the wetting forces are constant and are determined by the applied voltage via the electrowetting effect. As indicated by the dashed lines in Figure 4, the early time dynamics of advancing filaments follow the power-law behavior L2 ) c × t, c being a function of the applied voltage and geometry of the grooves and being independent of the frequency of the applied voltage. With increasing frequency, the maximum filament length, as well as the length up to which L2 ) c × t is valid, decreases. The dynamics of receding liquid filaments are independent of filament length, voltage, and frequency, as shown in the insets of Figure 4, and are described by the same power-law behavior, L2 ) c′ × t. This behavior corresponds to the early time dynamics of Washburn’s law for liquid imbibing a capillary from a large reservoir with viscous dissipation driven by capillary forces.29,30 The
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Figure 5. (a) Coefficients c derived from the early spreading behavior as a function of the wettability for two aspect ratios A ) 1 (circles) and A ) 0.5 (squares), showing the linear dependence c ) κ(cos θ - cos θT). (b) Comparison of the calculated mobility parameter κ (black solid line) with experimentally derived values for various aspect ratio, A. The experimental data were obtained from the early time dynamics of advancing liquid filaments (open squares), complete dynamics of advancing filaments (open circles), and receding filaments (solid diamonds).
coefficients, c and c′, for the early time dynamics of advancing filaments and receding filaments, respectively, are determined from a fit to the experimental data. The results for the early time dynamics of advancing filaments are reported in Figure 5a for two aspect ratios, A, as a function of wettability. Here, the applied voltages were converted to the apparent contact angle using eq 1. The coefficient c is proportional to the deviation from the filling threshold, c ) κ × (cos θ - cos θT), where κ is the mobility parameter of the system depending on the geometry of the grooves. The experimentally derived values of the mobility parameter κ are displayed in Figure 5b. Values derived from the early time dynamics of the advancing filaments are shown as open squares, whereas values derived from receding filaments are shown as solid diamonds. Hence, the early time dynamics of advancing filaments and the dynamics of receding filaments seem to be based on a simple electrocapillary model where the apparent contact angle, θ, is given by the applied voltage, U0. To validate this assumption, we will compare the calculated mobility parameter κ with the experimental values. To calculate the mobility parameter κ, we have to balance the driving force with the viscous flow resistance of the liquid in the grooves. The driving force is given by the excess of energy required to increase the length L of the filament Fdriv ) dE/dL. (29) Oron, A.; Davis, S. H.; Bankoff, S. G. ReV. Mod. Phys. 1997, 69, 931. (30) Washburn, E. W. Phys. ReV. 1921, 17, 273.
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The excess of energy per length is given by the sum of the interfacial energies involved, multiplied by the respective surface area, dE/dL ) γsgΣsg - γslΣsl + γΣ, where γ and Σ denote the interfacial energies and interfacial areas per length of the solid/ gas, solid/liquid, and liquid/gas interfaces, respectively. Using the fact Σsl ) Σsg and the equation of Young and Dupre´, γsg γsl ) γ cos θ, results in Fdriv ) γΣsl(cos θ - Σ/Σsl). For simplicity we assume the liquid/air interface to be flat and parallel to the bottom of the groove. This assumption is reasonable because groove filling occurs for liquid morphologies with about zero Laplace pressure.14,15 The ratio of the surfaces per unit length Σ/Σsl ) 1/(1 + 2A) can be identified as cos θT, for which filling occurs (dE/dL ) 0), where A is again the aspect ratio of the groove. Thus, the driving force can be expressed as
Fdriv ) γW(2A + 1) (cos θ - cos θT)
(2)
To derive the flow resistance, Fvisc ) WD∂p/∂y, we have to solve the Stokes equation:
∂2Vy ∂x2
+
∂2Vy ∂z2
)
1 ∂p ) const η ∂y
(3)
where Vy is the velocity of the liquid in the direction of the groove. x and z are the directions perpendicular to the groove, 0 e z e D, and -W/2 e x e +W/2. ∂p/∂y is the pressure gradient along the liquid filament. Assuming no slip boundary conditions at the solid/liquid interfaces and full slip boundary conditions at the liquid/gas interface, we solve eq 3 using a Fourier series.31 For rectangular grooves with aspect ratio A, the solution for the flow velocity, Vy, reads
Vy(A, xj, jz) )
-1(n - 1)/2 cos(nπxj) ηπ3 ∂y n∈2N+1 n3 cosh(nπ(zj - A))
4W2 ∂p
∑
[
]
- 1 (4)
Here, the coordinates xj ) x/W and jz ) z/W are rescaled by the groove width. The flow rate, Q, is obtained by integrating the flow velocity (eq 4) over the cross-sectional area of the groove 4
A W V (xj,zj) dxj dzj ) ∫xj1/2 )-1/2∫jz)0 y η
∂p ‚G(A) ∂y
(5)
where G(A) is a purely geometry-dependent factor integrating the flow field
G(A) )
∫
∫xj1/2 )-1/2
Vy(xj,zj) dxj dzj V0 jz)0 A
(6)
with V0 = W2η-1 δp/δy. Using the relation Q ) WD‚dL/dt and the result of eq 5, the viscous friction force for a liquid filament with length L can be written as
A2η dL Fvisc ) G(A) dt
(7)
Balancing this with the constant driving force (eq 2) and solving for L gives the time-dependent length of the liquid filaments in grooves with rectangular cross sections: (31) White, F. M. Viscous Fluid Flow; McGraw-Hill: New York, 1974.
(8)
where the mobility parameter is defined as
κ)
2γD 2A + 1 G(A) η A3
(9)
The result of eq 9 is plotted as a function of the groove aspect ratio, A, in Figure 5b together with the experimentally derived mobility parameter. Note that there is no free parameter used to fit the data. All parameters that enter the calculation of κ are determined otherwise. The mobility parameter κ calculated so far from our model describes the groove-filling and drainage dynamics for constant contact angle and thus for constant driving force. In our experiments this condition is fulfilled for all times in case of receding filaments and for the early time dynamics of advancing filaments using electrowetting. It is also valid for the advancing dynamics in the limiting case of dc electrowetting. However, as already mentioned earlier in this text, using ac voltage the voltage varies along extended liquid filaments, which results in a maximum filament length for a given applied voltage. Hence, it is obvious that the simple assumption of constant driving force is not justified for extended liquid filaments advancing into grooves. To understand the dynamics of advancing liquid filaments in more detail, let us first repeat what determines the maximum length of the filaments: due to the finite conductivity of the liquid, and the thin dielectric layer, a liquid filament is electrically equivalent to a free ended coaxial cable. Hence, when an ac voltage is applied, the voltage decays along the liquid filament, as derived in an earlier article:15
U(y˜ , L˜ ) ) UT xcosh2(L˜ - y˜ ) - sin2 (L˜ - y˜ )
cosh(nπA)
Q ) W2
L2(t) ) κ × (cos θ - cos θT)t
(10)
where U(y˜ , L˜ ) is the voltage at distance y˜ ) y/λ for a liquid filament of length L˜ ) L/λ. UT is the threshold voltage for the filling, and λ is the characteristic penetration length of the voltage along the liquid filament:
λ)
dσ DW ‚ xfπ W + 2D
(11)
λ is composed of two characteristic length scales of our system. The first term represents the electrical properties, whereas the second term represents the geometrical properties of the groove. When the length of the filament becomes on the order of λ, the driving force decays according to the voltage loss. Thus, the apparent contact angle, θ, in eq 8 has to be replaced by a voltagedependent contact angle varying along the liquid filament according to eq 10. Because the local contact angle at the tip of the advancing filament determines the driving force, we consider the voltage right at the tip only, U(L˜ ) ) U(y˜ ) L˜ , L˜ ). In terms of the voltage applied at the feeding drop, U0 ) U(y˜ ) 0, L˜ ), we can express U(L˜ ) in the form
U(L˜ )2 )
U02 cosh2(L˜ ) - sin2(L˜ )
(12)
At equilibrium the voltage at the tip equals the threshold voltage for the groove filling so that U(L ˜ eq) ) UT. Remember that for the experimental system and the geometries used in our electrowetting experiments the threshold voltage ranges from 40 to 90 V, so that the apparent contact angle, θ, is well described
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Figure 6. (a) Dynamics of advancing liquid filaments for various applied voltages. The solid lines are numerical fits of eq 13 to the experimental data, whereas κ is used as a fitting parameter. (b) Values of the best fits for κ for grooves with aspect ratio A ) 1.33, 1.0, and 0.66. κ is constant as soon as U0 is sufficiently large compared to UT. The short dashed vertical lines represent the filling thresholds for the various aspect ratios. The short dotted horizontal lines represent the expectation from our model.
by eq 1. Thus, combining eqs 1 and 12, and inserting the resulting formulas for cos(θ(U, L˜ )) and cos θT into eq 8, the resulting differential equation for the dynamics of advancing liquid filaments using ac electrowetting can be written as
κ UT dL˜ ) 2L˜ dt λ2 UL
(x
cosh2(L˜ eq) - sin2(L˜ eq) cosh2(L˜ ) - sin2(L˜ )
)
-1
(13)
Equation 13 is solved numerically and fitted to the experimental data, as shown in Figure 6 for various applied voltages, κ being used as a fitting parameter. The fitting curves and the experimental data are in perfect agreement, indicating that the model used correctly describes the complete filling dynamics. Again, it is worth mentioning that for each fitting of κ there is only one unknown parameter, the geometrical factor, G(A). All other experimental parameters in eq 9 are otherwise determined. The electrical length scale λ is determined experimentally from the equilibrium filament length, Leq. As a result λ × (1 + 2A)1/2 )
1.15 mm is used for all the fits together with the corresponding value of A. The variation of the mobility parameter κ is plotted as a function of the applied voltage in Figure 6. According to our model, κ is expected to be independent of the applied voltage. Indeed, κ is observed to become constant as soon as the applied voltage U0 is sufficiently large compared to the threshold voltage UT (see Figure 6). This is reasonable, considering that we assumed a constant cross section of the liquid filament, which might not be completely true for a locally varying apparent contact angle and which is in particular not true for very short filaments where the changing cross section at the tip cannot be neglected. Moreover, the solution for the flow velocity Vy(xj, jz) (eq 4) is only valid for fully developed and constant flow profiles, which is of course not fulfilled for short filaments. An additional effect leading to deviations from our model might be that close to the threshold the driving force has become smaller than the pinning forces: the equilibrium length is thus not reached, leading to an error in determining κ. However, if we average the data for κ in the plateau regimes, we can compare these groove-filling data with the expectations from our model, the results for groove drainage, and the early time dynamics of the groove filling (cf. Figure 5). Considering the fact that there is no free parameter and that some assumptions made to calculate the flow profile are crude, the agreement is satisfactory. The dynamics of the groove filling and draining observed by switching the wettability using electrowetting can thus be described with good quantitative agreement.
Conclusion In summary, we have used electrowetting to study the advancing and receding dynamics of liquid filaments in rectangular, open microgrooves. We have observed that the equilibrium length and dynamics are controlled by the frequency and the amplitude of the applied voltage, offering easy means for active control. We have shown that the dynamics are determined by the characteristics of a capillary flow, provided that the electrical loss along the filament is negligible. This applies for receding liquid filaments, where the dynamics are controlled by the material wetting properties of the system. Furthermore, it applies for the early time dynamics of the groove filling, when the length of the liquid filament is well below the characteristic penetration length of the ac voltage, λ. For larger advancing filament length, the voltage drop along the filament needs to be included as a spatially varying contact angle leading to a spatially varying driving force, which is reduced to zero at the equilibrium filament length. Acknowledgment. We thank Anton Kemmeren, Emil van Thiel, and Dirk Burdinski for help with sample preparation; fruitful discussions with Martin Brinkmann and Frieder Mugele are gratefully acknowledged. The project was supported by DFG priority program 1164 under Grant No. SE1118/2 and by the Marie Curie Industry Host Fellowship (No. IST-1999-80004). LA063584C
