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Microdroplet Deposition under a Liquid Medium Walter Villanueva,*,† Johan Sjo¨dahl,‡ Mårten Stjernstro¨m,‡ Johan Roeraade,‡ and Gustav Amberg† Linne´ Flow Centre, Department of Mechanics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden, and Department of Analytical Chemistry, Royal Institute of Technology, E-100 44 Stockholm, Sweden ReceiVed September 12, 2006 An experimental and numerical study of the factors affecting the reproducibility of microdroplet depositions performed under a liquid medium is presented. In the deposition procedure, sample solution is dispensed from the end of a capillary by the aid of a pressure pulse onto a substrate with pillar-shaped sample anchors. The deposition was modeled using the convective Cahn-Hilliard equation coupled with the Navier-Stokes equations with added surface tension and gravity forces. To avoid a severe time-step restriction imposed by the fourth-order Cahn-Hilliard equation, a semi-implicit scheme was developed. An axisymmetric model was used, and an adaptive finite element method was implemented. In both the experimental and numerical study it was shown that the deposited volume mainly depends on the capillary-substrate distance and the anchor surface wettability. A critical equilibrium contact angle has been identified below which reproducible depositions are facilitated.
Introduction Techniques for precise deposition of minute volumes of fluid media have become essential in life science. In diagnostics,1,2 drug discovery procedures,3,4 and fundamental studies of system biology,5,6 arraying technologies are extensively employed for the production of matrices of capture-molecules immobilized on surfaces. The microarrays are utilized in the screening for biomolecular ligands or to quantify known biomarkers. Important criteria of the technique for liquid dispensing are robustness, reproducibility, throughput, volume working range, and capability of preserving the biological function of the sample molecules. Microvolume dispensing include piezoelectric deposition,7,8 contact printing,5 and electrical deposition techniques, where fluid is transferred dropwise9 or as a spray10,11 due to a potential difference. A shortcoming of all of the aforementioned techniques is their susceptibility to malfunction due to precipitation of macromol* To whom correspondence should be addressed. E-mail: walter@ mech.kth.se. † Department of Mechanics, Royal Institute of Technology. ‡ Department of Analytical Chemistry, Royal Institute of Technology. (1) Robinson, W. H.; DiGennaro, C.; Hueber, W.; Haab, B. B.; Kamachi, M.; Dean, E. J.; Fournel, S.; Fong, D.; Genovese, M. C.; de Vegvar, H. E.; Skriner, K.; Hirschberg, D. L.; Morris, R. I.; Muller, S.; Pruijn, G. J.; van Venrooij, W. J.; Smolen, J. S.; Brown, P. O.; Steinman, L.; Utz, P. J. Nat. Med. 2002, 8, 295-301. (2) Deinhofer, K.; Sevcik, H.; Balic, N.; Harwanegg, C.; Hiller, R.; Rumpold, H.; Mueller, M. W.; Spitzauer, S. Methods 2004, 32, 249-254. (3) Wick, I.; Hardiman, G. Curr. Opin. Drug DiscoV. DeVel. 2005, 8, 347354. (4) Mattoon, D.; Michaud, G.; Merkel, J.; Schweitzer, B. Expert ReV. Proteomics 2005, 2, 879-889. (5) MacBeath, G.; Schreiber, S. L. Science 2000, 289, 1760-1763. (6) Zhu, H.; Bilgin, M.; Bangham, R.; Hall, D.; Casamayor, A.; Bertone, P.; Lan, N.; Jansen, R.; Bidlingmaier, S.; Houfek, T.; Mitchell, T.; Miller, P.; Dean, R. A.; Gerstein, M.; Snyder, M. Science 2001, 293, 2101-2105. (7) Roda, A.; Guardigli, M.; Russo, C.; Pasini, P.; Baraldini, M. Biotechniques 2000, 28, 492-496. (8) Litborn, E.; Stjernstro¨m, M.; Roeraade, J. Anal. Chem. 1998, 70, 48474852. (9) Ericson, C.; Phung, Q. T.; Horn, D. M.; Peters, E. C.; Fitchett, J. R.; Ficarro, S. B.; Salomon, A. R.; Brill, L. M.; Brock, A. Anal. Chem. 2003, 75, 2309-2315. (10) Morozov, V. N.; Morozova, T. Anal. Chem. 1999, 71, 3110-3117. (11) Moerman, R.; Frank, J.; Marijnissen, J. C.; Schalkhammer, T. G.; van Dedem, G. W. Anal. Chem. 2001, 73, 2183-2189.
ecules or salts. To handle this problem we have previously presented a concept, where a liquid lid of an immiscible organic fluid is employed to hinder evaporation from aqueous solutions and circumvent precipitation and concentration biases.12 The strategy based on fluorocarbon liquid lids was utilized in sample preparation schemes for MALDI-TOF mass spectrometry to allow the deposition of picoliter volumes of biomolecules dissolved in a mixture of aqueous and organic solution without any detrimental effects of evaporation.13 In this procedure, the sample solution is transferred from the end of a capillary kept under a layer of liquid fluorocarbon onto a micromachined silicon chip by the action of a pressure pulse. The sample solution suddenly wets the anchor surface upon contact and forms a liquid bridge. Then, the pressure is released, allowing the sample solution to flow back into the capillary, and at some point the liquid bridge snapsoff leaving a droplet of sample solution on the anchor. The deposition process is viewed as a free surface flow problem since it involves motion of liquid-liquid interfaces. For a review of the nonlinear dynamics and breakup of free surface flows, see Eggers.14 One important consideration when dealing with such a problem is how to model the moving interfaces. Either a sharpinterface or a diffuse-interface model can be utilized. For a review of diffuse-interface models in fluid mechanics, see Anderson et al.15 One type of diffuse-interface models of particular interest are phase-field-based models. Phase-field-based models are known to be mass and energy conservative and do not require intervention during simulation. The phase-field model that we have chosen to employ can also handle contact-line dynamics.16 More specifically, the model is a system of partial differential equations composed of the convective Cahn-Hilliard equation coupled with the incompressible Navier-Stokes equations with added surface tension and gravity forces. The Cahn-Hilliard equation models the creation, evolution, and dissolution of (12) Litborn, E.; Roeraade, J. J. Chromatogr. B: Biomed. Sci. Appl. 2000, 745, 137-147. (13) Sjo¨dahl, J.; Kempka, M.; Hermansson, K.; Thorse´n, A.; Roeraade, J. Anal. Chem. 2005, 77, 827-832. (14) Eggers, J. ReV. Mod. Phys. 1997, 69, 865-929. (15) Anderson, D. M.; McFadden, G. B.; Wheeler, A. A. Annu. ReV. Fluid Mech. 1998, 30, 139-165. (16) Jacqmin, D. J. Fluid Mech. 2000, 402, 57-88.
10.1021/la0626712 CCC: $37.00 © 2007 American Chemical Society Published on Web 12/08/2006
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Figure 1. Scheme of a deposition of fluid media under a liquid lid of fluorocarbon.
diffusively controlled phase-field interfaces,17 while the NavierStokes equations describe the motion of the liquids. The dynamics of the liquid-liquid interface in the deposition process are influenced by the interplay of surface tension, viscous, and gravitational forces. In some situations, surface tension tries to resist any deformation caused by viscous and/or gravitational forces. In other cases, surface tension dominates and induces deformation such as liquid column breakups, which usually happens when the liquid column is in an unstable configuration. Many factors affect the contribution of each of the interacting forces on the general behavior of the deposition process, namely, the properties of the liquids involved, the wettability of the surfaces, the height of deposition, etc. Our goal is to investigate experimentally and numerically the factors affecting reproducibility of depositing microdroplets under a liquid medium.
Figure 2. Axisymmetric model. The governing equations are expressed in cylindrical coordinates (r,z). A Poiseuille flow is set as a boundary condition at the top edge of the capillary inlet.
Experimental Section Chemicals. Liquid fluorocarbon (FC-75 Fluorinert, bp ) 100 °C, F ) 1.77 g/mL, dynamic viscosity ) 1.42 × 10-3 kg/ms, diffusivity of FC-75 in water ) 2.7 × 10-5 cm2/s estimated with the WilkeChang method by 3M) was purchased from 3M Company (St. Paul, MN). The surface tension of water in FC-75 is 2.85 × 10-2 N/m. Dimethyldichlorosilane, 2% (w/v) in octamethylcyclotetrasiloxane, (Repel-silane) was acquired from Amersham Biosciences (Uppsala, Sweden). Acetonitrile (99.9%, F ) 0.786 g/mL) was purchased from Merck (Darmstadt, Germany). Deionized water was obtained from a Synergy water system (Millipore, Bedford, MA). Robotic Setup for Liquid Dispensing. The practical experiments on liquid dispensing were conducted with a computer-controlled robotic setup mounted on a vibration-free optical table (model 78451-12, TMC, Peabody, MA). The sample solution was dispensed from the end of a ca. 3 dm fused silica capillary of 20 µm inner diameter (part no. TSP020090, Polymicro Technologies, Phoenix, AZ) by pressurizing a closed sample container connected to the inlet end of the capillary. The capillary outlet was positioned above a silicon chip featuring sample anchoring spots in the form of 3 µm high cylindrical plateaus of 50 µm diameter fabricated by deep reactive ion etching according to a procedure previously described.13 The chip was placed at the bottom of a 4 × 4 × 4 cm3 glass cuvette (Hellma GmbH, Mllheim, Germany) filled with ca. 10 mL of volatile fluorocarbon acting as a liquid lid (Figure 1). The outlet end of the capillary was kept beneath the fluorocarbon surface at a height above the chip that could be adjusted with the aid of a linear stage (model M-ILS50CC, Newport). To facilitate a precise lateral alignment of individual anchors to the capillary end, the cuvette containing the chip was placed on a XY-table (model TIXY 200, Micro-Controle S.A., Evry Cedex, France). The translational stages were maneuvered with a multi-axis motion controller (model MM-4000, Newport, Irvine, CA). A CM-10 microscope (Nikon, Tokyo, Japan) equipped with a CCD camera (model XC-ES30CE, Sony, Tokyo, Japan) and an ultra-long working distance objective with 20× magnification was employed to visualize separate anchors and the capillary end. An additional CCD camera (model C2400-75i, Hamamatsu Pho(17) Bates, P. W.; Fife, P. C. SIAM J. Appl. Math. 1993, 53, 990-1008.
Figure 3. Close-up of the mesh around the anchor at two different moments. The adaptive scheme ensures mesh resolution along the vicinity of the interface. tonics, Japan) with a 5× magnifying objective was positioned at 90° with respect to the microscope. By application of a 1 bar pressure to the sample vessel, a drop of sample solution is formed at the outlet end of the capillary. The pressure is retained until the drop makes contact with the top surface of the anchor positioned beneath. As the pressure is subsequently released, the sample solution flows back and a droplet of sample fluid is left behind on the anchor. To avoid siphoning, care was taken to keep the sample container in level with the capillary outlet. The capillary had been cut with a high-precision cleaver (model OFC-2000, Oxford Fiber Ltd, Rugby, UK) after removal of the polyimide coating, which resulted in a planar 68 µm o.d. end face. To reduce wetting, the whole exterior of the outlet end had been hydrophobized by immersion for 10 s in a 2% dimethyldichlorosilane solution, while flushing the capillary with air. Evaluation. Depositions of deionized water and a 50:50 acetonitrile/water solution (v/v) were made on 50 µm diameter anchors with capillary-anchor distances of 50, 100, 150, and 200 µm, respectively, i.e., eight experimental test series were performed. Each series consisted of 10 depositions carried out under the respective conditions stated above. Videos of the depositions were recorded with the imaging software PCTV Vision 2.50 (Pinnacle Systems, Mountain View, CA). The height of the deposited droplet and its width at the base were determined with the freeware MB-ruler 1.5 for digital measurements on screen (www.download.com). On the basis of the height and the width of the droplet, its volume was calculated as a segment of a sphere. All experimentally determined droplet volumes are given as means based on 10 replicate depositions and with 95% confidence intervals. The equilibrium contact angles were determined with the MB-ruler freeware from images of 50:50 acetonitrile/water drops on an anchor and on the circumference of
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Figure 4. Experimental microdroplet deposition of acetonitrile/water made from a height of 55 µm under liquid fluorocarbon on a 50 µm anchor. The volume deposited is 54 ( 3 pL.
Figure 5. Numerical microdroplet deposition at a height of 50 µm from the anchor. The volume deposited is 37 pL.
Figure 6. Normalized velocity profile before and after snap-off for an acetonitrile/water deposition made from a height of 50 µm. The speed of the snap-off is at least 5 times as high as the inlet centerline flow speed. a dimethyldichlorosiliane-treated fused silica capillary (uncertainty of ∼(8°). Care was taken that the contact lines of the drops on the anchor or capillary were not pinned on edges during the measurements. Mathematical Modeling. Phase-Field Method. Consider the case of an isothermal, viscous, and incompressible binary fluid consisting of two components, A and B in a domain Ω. An order parameter, a phase-field C, analogous to the relative concentration of the two components can be introduced to characterize the two different phases. In each bulk phase, C assumes a distinct constant value that changes rapidly but smoothly in the interfacial region. For example, C assumes the value CA ) -1 in component A while it takes the value CB ) 1 in component B. The transition from CA to CB describes the interfacial region. With the introduction of a free energy density, the system can be modeled by a set of equations: the Cahn-Hilliard equation, modified to account for fluid motion, and the NavierStokes equations with surface tension forcing and forces due to gravity ∂C 1 1 + (u‚∇)C ) ∇2φ ) ∇2(Ψ′(C) - Cn2∇2C) ∂t Pe Pe 1 Bo C∇φ + g(C) (∂u∂t + (u‚∇)u) ) -∇p + ∇ u - Ca‚Cn Ca 2
Re
∇‚u ) 0 where Ψ is a double-well potential and φ is the chemical potential. The dimensionless physical parameters are the Reynolds number, Re, Capillary number, Ca, Bond number, Bo, and Peclet number, Pe, given by Re )
F0UcLc 2x2µUc Ca ) µ 3σ Bo )
x2∆FgL2c 2x2LcUcξ Pe ) 3σ 3κσ
where ∆F, µ, Uc, κ, and σ are the density difference, bulk viscosity, characteristic velocity, mobility, and surface tension, respectively. Details of the nondimensionalization can be seen in ref 18. The Boussinesq approximation has been employed between the two liquids. The Reynolds number is the ratio between the inertial and viscous forces. The Capillary number gives the ratio between the viscous and surface tension forces, while the Bond number is the ratio between the gravitational and surface tension forces. The Peclet number is the ratio between the convective and diffusive mass transport. The Cahn number, Cn ) ξ/Lc, is a dimensionless numerical parameter that provides a measure of the ratio between the meanfield thickness, ξ, and the characteristic length, Lc, which is taken to be the capillary radius. It is shown in ref 18 that the dynamic wetting of a liquid drop on a solid surface is insensitive to the Cahn number, Cn. Following Jacqmin,16 two boundary conditions are set for C. First, the no-flux condition n‚∇φ ) 0. Second, the wetting condition, n‚∇C ) -kg′(C)/Cn, where k is the wetting coefficient which will be discussed later and g(C) is a local surface energy set to 0.75C - 0.25C3. A contact of a liquid-liquid interface to a solid phase gives rise to wetting. The wetting of a liquid on a solid can be classified into two types: total wetting, when the liquid spreads completely, and partial wetting, when the liquid at equilibrium rests on the solid with a contact angle θe.19 Both are characterized by the spreading parameter S ) σSM - (σSL + σ), where the σ’s are surface tensions at the solid/medium (medium is either air or another liquid), solid/liquid, and liquid/medium interfaces, respectively. S > 0 corresponds to total wetting and S < 0 corresponds to partial wetting. High-energy surfaces such as metallic surfaces having higher σSM values thus increases S, compared to low-energy surfaces like plastics. The Young’s relation which is only valid when S < 0 can be defined as cos θe ) k ) S/σ + 1 where θe is the equilibrium contact angle of the liquid-liquid interface at a solid surface. Numerical Treatment. The numerical simulations were carried out using FemLego,20 a symbolic tool to solve partial differential equations with an adaptive finite element method. The partial differential equations, boundary conditions, initial conditions, and method of solving each equation are all specified in a Maple worksheet (www.maplesoft.com). We consider an axisymmetric model where the rotational symmetry is around the vertical axis (see Figure 2). The Cahn-Hilliard equation is treated as a system where the chemical potential, φ, is computed separately followed by the calculation of the composition equation C. The right-hand side of the composition equation is the Laplacian of the chemical potential. (18) Villanueva, W.; Amberg, G. Int. J. Multiphase Flow 2006, 32, 10721086. (19) Gennes, P. G.; de Brochard-Wyart, F.; Que´re´, D. Capillarity and wetting phenomena; Springer-Verlag: New York, 2004; pp 16-18. (20) Amberg, G.; To¨nhardt, R.; Winkler, C. Math. Comput. Sim. 1999, 49, 257-274.
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Figure 7. Experimental microdroplet deposition of acetonitrile/water made from a height of 96 µm under liquid fluorocarbon on a 50 µm anchor. The volume deposited is 27 ( 1 pL.
Figure 8. Numerical microdroplet deposition made from a height of 100 µm. The volume deposited is 28 pL. was employed here), and TOL is a given tolerance. If an element satisfies the error criterion, it is marked for derefinement unless it is an original element. At the next refinement step, elements containing hanging nodes are marked for refinement. The refinement/ derefinement stops if and only if no element is marked for refinement/ derefinement. The interface moves slowly, so the adaptivity is implemented at every tenth to twentieth time-step. The use of mesh adaptivity makes the computation at least 18 times more efficient.18 Figure 9. Normalized velocity profile near the moment of pinchoff for an acetonitrile/water deposition made from a height of 100 µm. The speed of the snap-off is at least 5 times as high as the inlet centerline flow speed. The system of equations is solved semi-implicitly with a streamline diffusion method. Both chemical potential and composition equations are discretised in space using piecewise linear functions. The resulting linear systems of both equations are solved using a modified conjugate gradient method (CG), which is described as follows. While solving the composition equation with the standard CG, the chemical potential is solved and updated inside the CG’s iteration loop. The chemical potential is linearized and lumped, and the resulting system is easy to solve, requiring only a few iterations with a diagonal solve. After solving the composition equation, the chemical potential equation is solved again but with a standard CG to get a more accurate solution for the surface tension forcing needed in the Navier-Stokes equations. The Navier-Stokes equations are solved using a projection method by Guermond and Quartapelle21 with an added pressure stabilization term. Piecewise linear functions are used to discretise the NavierStokes equations. The resulting linear systems for the momentum equations are solved using the generalized minimal residual method (GMRES), while CG is used for the pressure equation. The interface is resolved with at least six node points, so the use of mesh adaptivity is essential. An ad hoc error criterion is used to ensure mesh resolution along the vicinity of the interface (see Figure 3). The mesh adaptivity is implemented as follows. At each mesh refinement step, an element, K, is marked for refinement if the element size h > hmin (the minimum h allowed) and the element does not satisfy the following error criterion ||H(a - |C(x,t)|)||L1(K) e TOL
(1)
where H(‚) is the Heaviside step function, a is a constant (a ) 0.99 (21) Guermond, J. L.; Quartapelle, L. In Proceedings of the International Conference on Finite Elements in Fluids; Venezia, 1995; pp 367-374.
Results and Discussion Depositions of two different solutions were made on anchors with four capillary-anchor distances. We set up a numerical model and the numerical physical parameters consisted of the Capillary number, Ca, Reynolds number, Re, Bond number, Bo, Peclet number, Pe, and the equilibrium contact angles θ1 and θ2, where θ1 corresponds to the equilibrium contact angle of the liquid-liquid interface on the anchor surface while θ2 is the equilibrium contact angle of the liquid-liquid interface at the end surface of the capillary. The actual values of all the properties of the liquids used in the experiments are not known to us. Also, aqueous surfaces are susceptible to impurities or contamination, rendering the values of surface tension uncertain. Therefore, before making an actual comparison or investigation, the numerical simulation has been calibrated to match one particular experimental case that is the deposition of acetonitrile/water solution with a 100 µm capillary/anchor distance. This is done in the following way: the geometry and equilibrium contact angles that correspond to the above-mentioned experimental case have been used. Next, values for the following parameters, Re ) 0.1, Ca ) 0.02, Pe ) 10, were explicitly set and then we used the Bond number to achieve precise agreement and found it to be Bo ) 0.05. The established values of the model parameters, i.e., {Re ) 0.1, Ca ) 0.02, Bo ) 0.05, Pe ) 10, θ1 ) 56°, θ2 ) 90°} were then used to study the effect of deposition height on the dynamics of the deposition. Moreover, the influence of the physical parameters including the wettability of the surfaces on the dynamics of the deposition was also investigated. Influence of Deposition Height. The interplay between surface tension, viscous, and gravitational forces determines the dynamics of the liquid bridge and its snap-off. Depositions made on 50 µm diameter anchors with different capillary-anchor distances have been studied to assess the effect of the deposition height on the resulting droplet size.
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Figure 10. Experimental microdroplet deposition of acetonitrile/water made from a height of 195 µm under liquid fluorocarbon on a 50 µm anchor. The volume deposited is 29 ( 1 pL.
Figure 11. Numerical microdroplet deposition made from a height of 200 µm. The volume deposited is 25 pL.
Figure 12. Graph showing the effect of anchor-capillary distance on the volume of the deposited droplet using 50 µm cylindrical anchors.
When the sample solution makes contact with the anchor, the shape of the liquid bridge that is formed is almost spherical. As the solution flows back into the capillary, the liquid bridge is constricted, and at a certain point the liquid column will resemble a conical frustum. This is seen in Figure 4a, where a deposition of acetonitrile/water is shown. If the extraction of liquid is stopped at this point, the column will keep its configuration and pinchoff is unlikely to occur since the height of the liquid column (55 µm) is comparable to its diameter ((50 + 68)/2 ) 59 µm). The buoyancy (which is proportional to the square of the characteristic length) of the sample solution does not alter the shape of the liquid column since the surface tension will resist any deformation. However, further extraction of solution will dewet the capillary
end surface (see Figure 4b), partly because viscous forces dominate in the inlet region and in addition the 90° equilibrium contact angle on the capillary surface makes the wetting line easy to drag. Note that the upper wetting line did not move symmetrically all the way to the edge of the inlet. The asymmetric behavior is likely due to microscopic surface irregularities. This anomaly will influence the resulting droplet size, and it is therefore concluded that it would be beneficial to conduct the depositions under such conditions that the capillary end face is not dewetted prior to snap-off. A numerical droplet deposition of acetonitrile/water is shown in Figure 5. The initial shape of the liquid bridge is a frustum that matches the profile observed in the experiments (Figure 4a). A Poiseuille flow is set as a boundary condition at the top edge of the capillary inlet with a maximum nondimensional velocity of 1 located at the center of the flow profile. The behavior of the liquid bridge is similar to that observed in the experiments except for the non-axisymmetry seen in the experimental case. It should be noted that the numerical simulation captures the dewetting of the capillary end face and that the profile in the experiment is very similar to the simulated profile. The model predicts that the deposited volume is 37 pL, which is smaller than the experimental value of 54 ( 3 pL. This discrepancy is likely due to the asymmetric dewetting observed in the experiments. The velocity flow profile before and after snap-off is given in Figure 6. Initially, most of the bulk of the liquid bridge is in the inlet region, where viscous forces dominate and therefore the deformation of the liquid column is mainly caused by extraction of fluid. The deformation will lead to an unstable configuration, where surface tension forces prevail, resulting in a rapid snap-off with a speed at least 5 times as high as the inlet centerline flow speed (Figure 6b). Effects of buoyancy are observed in depositions made with capillary-anchor distances of about 100 µm or greater due to
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Figure 13. Simulation showing the effect of employing a low Bond number (negligible gravity). Microdroplet deposition of acetonitrile/ water on a 50 µm cylindrical anchor from a 100 µm deposition height with Bo ) 0.001.
an increased liquid column volume. The gravitational effect in combination with the continuous extraction of solution will deform the liquid column into a sinusoidally shaped pillar with the neck near the anchor (Figure 7a-c). Note that the capillary end face is not dewetted prior to snap-off. Since the snap-off occurs closer to the anchor than in the depositions made from 50 µm height, a droplet of smaller volume is left behind. The wetting line of the droplet is fixed on the edge of the anchor due to the strong contact force (θ1 ) 56°). The numerical simulations capture the deformation of the liquid column and the pinning of the wetting lines on the edges of both surfaces (Figure 8a-c). The volume of the deposited droplet in the simulation is 28 pL compared to the experimental value of 27 ( 1 pL. Similar to the results from the simulations of depositions made from 50 µm height, the maximum velocity in the pinch-off region is at least 5 times the velocity at the capillary inlet (Figure 9). In depositions made from a height of 200 µm, the shape of the liquid column and the snap-off behavior resemble the observations of the depositions made from 100 µm (Figure 11). The numerically estimated droplet volume is 25pL (Figure 11) as compared to the experimental value 29 ( 1 pL (acetonitrile/ water). For all the geometrical cases tested, the dynamics of the depositions made with pure water were similar to those observed with acetonitrile/water solution. We can identify two stages in the deposition process. The first stage is the deformation of the liquid column. The second stage starts when the liquid column becomes unstable, that is, when the aspect ratio between the liquid column’s length and its neck radius exceeds a critical value. At this point, surface tension starts to dominate and the liquid column snaps off. This is known in the literature as the Plateau-Rayleigh instability.19 This instability states that a liquid column becomes unstable when the aspect ratio between the cylinder’s length and its original radius exceeds 2π. The effect of the deposition height on the resulting droplet size is shown in Figure 12. The two kinds of liquids used in the experiments gave comparable results. Except for the data from depositions made from a height of 50 µm, a good agreement both qualitatively and quantitatively between the experimental and numerical results was achieved. The discrepancy in the data obtained with 50 µm height is explained by the inhomogeneous capillary end face, which disturbs the droplet formation at short capillary-anchor distances. It is concluded that the highest droplet volume reproducibility is obtained with capillary-anchor distances of about 100 µm or greater. Influence of Physical Parameters. In this section, the effect of the model parameters on the dynamics of depositions is discussed. Depositions made from 100 µm are used as a model case in the discussion. Influence of the Dimensionless Physical Parameters. To study the influence of the Ca, Bo, Re, and Pe numbers on the outcome
Figure 14. Graph showing the effect of the wetting properties of the anchor and capillary end face on the volume of the deposited droplet for the case with a 50 µm diameter anchor and 100 µm deposition height.
Figure 15. Simulation showing the effect of a less hydrophilic anchor surface. The equilibrium contact angle on the anchor surface is θ1 ) 75°.
of the numerical simulations, one parameter at a time was varied within a certain interval. The effect of the Ca number was examined in the interval 0.015-1.0. The dynamics of the interface and the wetting lines change a little with Ca below 0.1. The snap-off is a bit slower, and slightly larger volumes are deposited as Ca approaches 0.1. When Ca > 0.1, the neck gets closer to the center of the liquid column and the wetting line on the capillary surface moves toward the center of the capillary prior to snapoff. This is not surprising since the wetting line on the capillary surface is close to the region where fluid is being extracted and in addition the equilibrium contact angle on the capillary surface of 90° makes the wetting line easily slip by viscous drag. The Bond number was varied between 0.001 and 0.5, and the surrounding liquid remains denser than the liquid column. If a value of 0.001 is employed, the dynamics of the deposition will change distinctly. The capillary surface dewets prior to snap-off, and a larger volume is deposited, i.e., 120 pL compared to only 26 pL in the Bo ) 0.05 case. The numerical investigation of the effects of the Reynolds number (Re ) {0.001 - 10}) and the Peclet number (Pe ) {1 - 200}) on the dynamics of the deposition and the deposited volume did not reveal any significant differences within the examined ranges. Wettability/Hydrophilicity. To achieve reproducible depositions, the anchor surface should preferably be highly wettable, i.e., the equilibrium contact angle, θe, should be close to 0°. It is well known that the contact angle θe depends on the chemical constitution of the solid and the liquids involved. Solid surfaces can be categorized into hard solids which have high-energy surfaces and molecular solids which have low-energy surfaces. Experimentally we employed a molecular solid, i.e., a silicon chip with a silicon dioxide surface. Directly after chip fabrication,
Microdroplet Deposition under a Liquid Medium
the anchor surface is very hydrophilic, but due to contamination from the surrounding environment, the wettability can vary over the chip surface and was observed to diminish with time. Therefore, the equilibrium contact angles were determined in conjunction with the experiments. The influence of the wettability of the anchor surface and the capillary end face on the volume of the deposited droplet was studied numerically. Simulations were performed with the same geometry and physical parameters as used in the previous section, i.e., {Re ) 0.1, Ca ) 0.02, Bo ) 0.05, Pe ) 10}, while varying either θ1 or θ2. Figure 14 shows the results of the simulations. In all cases, the upper wetting line stays on the edge of the capillary before the pinch-off, which occurs close to the anchor. In other words, with this set of parameters, we did not observe any significant effect of the wetting property of the capillary on the overall behavior of the deposition and on the resulting droplet size. However, the equilibrium contact angle on the anchor surface strongly influences the size of the resulting droplet. Figure 15 shows the dewetting of an anchor surface with θ1 ) 75°. The liquid column dewets the area 10 µm from the anchor edge before it snaps off and then wets back the anchor surface with a radial distance of about 3 µm after it snaps off. Thus, the final droplet base radius is only 18 µm compared to the anchor radius of 25 µm. As θ1 is decreased, see Figure 14, the contact force on the anchor increases and will hence counterbalance the viscous stress acting close to the wetting line in an inward direction. In conclusion, a critical equilibrium contact angle θ1,c ≈ 60° on the anchor surface was identified. Below this threshold value, the wetting line will be pinned to the edge of
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the anchor, and thus reproducible droplet depositions are more easily obtained.
Conclusion We have presented an experimental and numerical study of the factors affecting the reproducibility of microdroplet depositions onto cylindrical anchors under a liquid medium. In the numerical model, the time-step restrictive nature of the fourthorder Cahn-Hilliard equation is avoided by implementing a semi-implicit scheme. A qualitative and quantitative agreement was established between the experimental and numerical results. In both the experimental and numerical study, it was shown that the deposited volume depends significantly on the capillaryanchor distance. This is due to the interplay between the surface tension, viscous, and gravitational forces, and depending on the deposition height, different forces will dominate. Furthermore, we have shown that the wettability of the anchor has a strong influence on the resulting droplet size. In this context, a critical equilibrium contact angle that facilitates reproducible droplet depositions has been determined numerically. Acknowledgment. The authors gratefully acknowledge the financial support from the Computational Phase Transformation program and the Nanochemistry program both funded by the Swedish Foundation for Strategic Research (SSF), the European Community (Contract No. NMP4-CT-2003-505311) and the Swedish Research Council. Part of the work (WV and GA) was carried out within the Linne´ Flow Centre. LA0626712
