pubs.acs.org/Langmuir © 2009 American Chemical Society
Capillary Forces during Liquid Nanodispensing Laure Fabie, Hugo Durou, and Thierry Ondarc-uhu* Nanosciences group, CEMES-CNRS, Universit e de Toulouse, 29 rue Jeanne Marvig, 31055 Toulouse cedex 4, France Received July 17, 2009. Revised Manuscript Received September 29, 2009 We present a comprehensive study of the capillary force measured during the liquid nanodispensing of attoliter droplets with an atomic force microscope tip. Due to the presence of a nanochannel drilled at the tip apex and connected to a reservoir droplet deposited on the cantilever, we observe a large variety of force curves during the deposition process. We propose a numerical method which accounts for most of the experimental observations. In particular, we clearly demonstrate the influence of the nanochannel diameter. This study leads to a better understanding of the mechanisms of liquid transfer from the tip to the surface and also provides a real time monitoring of the dispensing. Besides these applications, the method we use, which can handle a large variety of conditions and also complex geometries, may find a wide range of applications.
1. Introduction The capillary force exerted by a liquid meniscus bridging two solid surfaces has been intensively studied because of its implication in many natural, industrial, and everyday life processes. For example, the presence of water in a granular media leads to the formation of liquid menisci between grains which dramatically changes the strength of the powder1 compared to dry state, as experienced by everybody with sand on a beach. Capillary forces are also a technological relevant issue, since they can be a detrimental effect for heads of hard disks.2 On the contrary, one can take advantage of this force to manipulate small objects.3 At the nanometer scale, capillary forces are an important issue in atomic force microscopy (AFM) due to the presence of a meniscus between the tip and surface during imaging.4-6 Since AFM is also used to measure the local mechanical properties or chemical composition of materials, it is essential to understand in detail the contribution of capillarity to the total force experienced by the tip. For these reasons, a considerable effort has been dedicated to develop analytical expressions to quantify the force due to the presence of a capillary bridge between two surfaces. Models were developed to describe a large variety of geometries and conditions.7 Nevertheless, a method allowing treatment of different conditions and more complex geometries than the axisymmetric objects usually considered (mainly planes and spheres) is still lacking. In this Article, we present a study of the capillary forces measured during the deposition of ultrasmall droplets with a recently developed liquid nanodispensing (NADIS) method.8,9 An AFM tip with an aperture at its apex is used to transfer liquid *Corresponding author. E-mail:
[email protected]. (1) Bocquet, L.; Charlaix, E.; Ciliberto, S.; Crassous, J. Nature 1998, 396, 735– 737. (2) Chilamakuri, S. K.; Bhushan, B. J. Appl. Phys. 1999, 86, 4649–4656. (3) Bhushan, B.; Ling, X. J. Colloid Interface Sci. 2009, 329, 196–201. (4) Malotky, D. L.; Chaudhury, M. K. Langmuir 2001, 17, 7823–7829. (5) Zitzler, L.; Herminghaus, S.; Mugele, F. Phys. Rev. B 2002, 66. (6) Bowles, A. P.; Hsia, Y. T.; Jones, P. M.; White, L. R.; Schneider, J. W. Langmuir 2009, 25, 2101–2106. (7) Butt, H. J.; Kappl, M. Adv. Colloid Interface Sci. 2009, 146, 48–60. (8) Meister, A.; Liley, M.; Brugger, J.; Pugin, R.; Heinzelmann, H. Appl. Phys. Lett. 2004, 85, 6260–6262. (9) Fang, A. P.; Dujardin, E.; Ondarc-uhu, T. Nano Lett. 2006, 6, 2368–2374.
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from a reservoir droplet deposited on the top of the cantilever to the surface by simple contact (Figure 1a). This technique combines the resolution of other scanning probe lithography methods, such as dip pen lithography,10 and the flexibility of liquid lithography methods (for example, inkjet or pins arrays used for biochip fabrication) in terms of transferable molecules. Any soluble molecules, nanoparticles, and proteins can be patterned with lateral resolution in the 100 nm range (Figure 1b). The volume of the manipulated droplets ranges from femtoliter (10-15 L) down to attoliter (10-18 L) or smaller, which for standard dilutions contains only a small number of solute molecules. The influence of relevant parameters (aperture diameter, tip surface properties, contact time) on the spot sizes was reported in ref 9. Here, we propose to study in detail the capillary forces measured during the deposition process. The aim of this paper is twofold. The large variety of force curve shapes obtained during nanodispensing requires the development of a method able to consider a wide range of conditions and geometries. We demonstrate in the following that a method based on energy minimization with the “surface evolver” software11 accounts for most of the experimental data. The other objective of this study is to provide information on the mechanism of liquid transfer from tip to surface in liquid nanodispensing. The understanding of the underlying process may provide a way to improve the deposition method. We also aim at getting a simple method to monitor, in real time, the deposition process, based on the capillary force which is the only available information during deposition. The paper is organized as follows. In a first section, we describe the methods used for liquid nanodispensing and measuring force curves, and, after a brief review of the existing methods, we present in detail the simulations we used. We then compare the calculated force curves with the experimental ones. We first study a model situation using standard AFM tips. We then consider hydrophilic NADIS tips and show the influence of several parameters, emphasizing the role of tip aperture size. The case of hydrophobic NADIS tips is finally discussed. (10) Piner, R. D.; Zhu, J.; Xu, F.; Hong, S. H.; Mirkin, C. A. Science 1999, 283, 661–663. (11) Brakke, K. Exp. Math. 1992, 1, 141–165.
Published on Web 10/20/2009
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Figure 1. (a) Schematic representation of liquid dispensing by NADIS tip. (b) Tapping mode AFM image of an array of 250 nm in diameter molecule spots deposited with a hydrophobic NADIS tip with an 80 nm 120 nm oval aperture on a aminefunctionalized silicon dioxide substrate.
2. Material and Methods Liquid Nanodispensing. NADIS tips were fabricated by modifying commercially available tips (OMCL-RC, Olympus) by focused ion beam (FIB). A nanochannel was milled at the tip apex in a two step process described in detail in ref 9: to overcome the limitation in hole aspect ratio inherent to FIB machining and to reach small aperture sizes, the tip wall was first thinned from the inner side; the cantilever was then turned over, and the final aperture was milled from the tip side for optimal centering at the apex. This technique gave a precise control of the shape and size of tip apertures in the 100 nm range down to a diameter of 35 nm (Figure 2a,b). The surface properties of the tip were then eventually modified by thiolate chemistry of the gold layer coating the tip outer wall. The NADIS tips were loaded with glycerol with a micropipet connected to a microinjector (Narishige) and controlled by a micromanipulator (The Micromanipulator Inc.). Droplets with a diameter of about 20 μm injected on the cantilever were used as the reservoir (Figure 2c). A Multimode PicoForce atomic force microscope (Veeco) was operated in force spectroscopy mode to transfer the liquid by contact of the loaded tip with the surface. We demonstrated that femto- to sub-attoliter volumes could be successfully deposited, corresponding to lateral dimensions of droplets ranging from micrometers down to 75 nm. The main parameters controlling droplet sizes are tip and substrate wettability, nanochannel diameter, and contact time.9 On the contrary, contact force and tip velocity have no significant influence on the deposition process. Capillary Force Measurements. Since the dispensing was realized in force curve mode, we recorded, for each spot, the force exerted on the tip during the liquid transfer. The capillary force responsible for the deflection of the cantilever was recorded as a function of the tip-surface distance z obtained by subtracting the cantilever deflection from the displacement of the tip by the piezoelectric element. During the tip approach, a jump to contact was observed when the tip apex (or the liquid at the tip apex) touched the surface. The tip was then pushed on the surface until a given value was obtained. This approach curve which contained no information about the liquid transfer was used, in the contact regime, to determine the deflection sensitivity which is required to calibrate cantilever deflection. In this paper, we concentrate on the retraction curves obtained when the tip is pulled out of the surface. During this stage, the tip experiences a large attractive force extending over large tip separation distances. Such a behavior, resulting from the liquid meniscus bridging tip and surface, is not observed when using a standard AFM tip on a dry surface. At a certain Langmuir 2010, 26(3), 1870–1878
Figure 2. (a) SEM image of the apex of a NADIS tip with a rectangular aperture of 200 nm 250 nm; (b) SEM image of a NADIS tip with 35 nm diameter nanochannel; (c) optical micrograph of a NADIS tip loaded with a reservoir droplet of about 30 μm.
Figure 3. Force curves measured during the retraction of NADIS tips in different experimental conditions: (blue) tip with a 400 nm aperture with a hydrophilic outer wall (cleaned in a piranha solution); (green) same tip made hydrophobic by surface modification by dodecanethiol; (red) hydrophobic tip with a 35 nm diameter aperture. tip-surface separation, the meniscus then breaks and the force abruptly drops to zero. Interestingly, the shape of the curves obtained with NADIS tips is greatly influenced by the experimental conditions as illustrated in Figure 3 gathering the three main observed behaviors. Force curves recorded during a deposit with a hydrophilic tip with large aperture (in blue) present a minimum. The magnitude of the (attractive) force slowly increases and then decreases before the snapoff. Functionalization of the same tip with dodecanethiol in order to make its outer wall hydrophobic results in the disappearance of the minimum: the force magnitude (green) decreases regularly, and the contact breaks earlier. Curves obtained with a tip with very small aperture show a totally different shape (red). The concavity of the curve is opposite from the ones of the previous examples. The retraction curves contain a lot of information about the mechanism of liquid transfer from the tip to the surface. In order to get insights into the processes involved in the deposition, we performed simulations of these curves. The method used is developed in the next section. Note that, in Figure 3, the deflection was converted into a force using the nominal cantilever spring constant k given by the manufacturer. Nevertheless, the actual spring constant may deviate to a large extent from this value. In the following, for comparison with calculated curves, we therefore considered k as an adjustable parameter. DOI: 10.1021/la902614s
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Figure 4. (a) Schematic representation of the studied configuration; (b,c) shape of the meniscus between a conical tip and a surface before and after optimization using “surface evolver” software, respectively. It is important to note that when the force gradient is higher than the cantilever stiffness, the tip abruptly jumps to contact or off contact depending on the direction of motion. This is the case at the very end of the separation process when the meniscus breaks. The snapoff may therefore happen due to this mechanical instability rather than from the meniscus instability. In order to keep contact as long as possible and come as close as possible from the meniscus capillary instability, we used cantilevers with a spring constant of 82 mN/m which is higher than the estimated value of the stiffness of a liquid meniscus of the order of the surface tension12 (63 mN/m for glycerol). Simulation of Force Curves. The capillary force exerted by a liquid meniscus bridging two surfaces has been studied in great extent. All models considered an axisymmetric geometry and negligible gravity effects (meniscus characteristic sizes are much smaller than the capillary length). The force is calculated either by a derivation of the total interface energy of the system or by a direct computation of the force from the meniscus shape. Lambert et al.13 recently demonstrated the equivalence between the two methods provided one considers, in the second method, both the Laplace pressure term resulting from the pressure inside the meniscus and the tension term due to the liquid surface tension force applied on the contact line. In applications involving capillary condensation between surfaces with a large radius of curvature compared to the Kelvin length or in experiments with a surface force apparatus, the radius of curvature of the surfaces is large compared with surface separation. In this case, the tension term can be neglected, and, in a first approximation, the capillary force is given by F=4πRγ cos θ where R=[(1/R1) þ 1/R2)]-1 and cos θ=1/2(cos θ1 þ cos θ2), with R1 and R2 being the radii of curvature of solid surfaces and θ1 and θ2 being the liquid contact angles on the surfaces.14 Refinements of this model were proposed in order to take into account tension terms,15 precise meniscus shape,16 or surface roughness17,18 which may play an important role. When capillary forces acting on an AFM tip are concerned, the radius of curvature of the solid surfaces cannot be neglected anymore. Contrary to the previous case, the exact shape of the liquid interface has to be computed using the Laplace equation and defining boundary conditions at contact lines. A classical description consists of approximating the profile of the interface (12) Jai, C.; Aime, J. P.; Boisgard, R. Europhys. Lett. 2008, 81, 34003. (13) Lambert, P.; Chau, A.; Delchambre, A.; Regnier, S. Langmuir 2008, 24, 3157–3163. (14) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: New York, 1992. (15) Rabinovich, Y. I.; Esayanur, M. S.; Moudgil, B. M. Langmuir 2005, 21, 10992–10997. (16) de Lazzer, A.; Dreyer, M.; Rath, H. J. Langmuir 1999, 15, 4551–4559. (17) Butt, H. J. Langmuir 2008, 24, 4715–4721. (18) de Boer, M. P.; de Boer, P. C. T. J. Colloid Interface Sci. 2007, 311, 171–185.
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Fabi e et al. to a circle. In this so-called toroidal approximation, the meniscus shape can be determined analytically.19,20 The Laplace equation can also be integrated numerically,16 using a double iteration scheme in the case of a liquid with given volume.21 Recently, De Souza et al. used the “surface evolver” software11 to simulate the meniscus shape and capillary force resulting from a liquid volume bridging two identical substrates with contact angle hysteresis22 or two chemically different substrates.23 Even if this method was used for flat substrates and volume constraints only, it can in principle be applied to surfaces with more complex shapes such as pyramidal AFM tips and can handle pressure or volume constraints. For that reason, we chose this method which is the only one able to describe all the different situations encountered when using NADIS tips. We considered the situation schematized on Figure 4a of a liquid volume bridging a conical tip and a flat surface. The capillary force modeling was realized with the “surface evolver” software which determines liquid shape by energy minimization. First, an initial shape of the system, including the meniscus connecting the tip and the surface, was created by fixing its vertex coordinates (Figure 4b). The mesh was then sequentially refined and the vertex coordinates optimized (using a gradient method) until an equilibrium shape made of approximately 30 000 facets was reached (Figure 4c). The optimization of the meniscus shape was performed with given boundary conditions and constraints that we detail in the following. The boundary conditions describe the behavior of the contact lines, on both the tip and the surface. We defined as Rtip (Rsurf) and θtip (θsurf) the radius of the contact line and the contact angle on the tip (and on the surface), respectively. Depending on the situations, we used as boundary conditions either a constant radius when the contact line did not move on the corresponding solid, or a constant contact angle condition in the cases where the contact line could move. We used tips with various geometries such as cones or pyramids. Although NADIS tips were pyramidal, we used, for the sake of simplicity, in most of the cases, a conical tip. We chose a cone angle which gave, for an identical height, the same volume as the pyramidal tip. Given the geometry of the NADIS tips, this led to a conical tip with half angle R=38.3°. Constraints applied to the liquid body during the minimization were of two kinds: we imposed either the liquid volume or the liquid pressure. As examples, the shape of a liquid metal between a solder and a surface corresponds to a volume constraint, whereas, for capillary condensation, the pressure is defined by humidity conditions. In the case of NADIS, the situation is more complex due to the nanochannel connecting the meniscus to the reservoir which may influence the constraints to be applied. As a result of the simulations for a tip-surface separation z, we obtained the shape of the meniscus and the corresponding values of Rtip, Rsurf, θtip, and θsurf, its volume V, pressure P, and the total interfacial energy of the system W. The capillary force F applied by the liquid on the tip was obtained as the derivative of the energy versus z. We computed the changes of total energy associated with small variations of height Δz around the equilibrium height and deduced F(z) using F(z) = -[ΔW(z)/Δz] = -[W(z þ Δz) W(z - Δz)/2Δz]. The similar finite differences method was used to compute the horizontal force exerted by a defect on a contact line.24 By iterating the whole procedure increasing z, we simulated the whole force curve. At some threshold value Zrupt, the minimization did not converge any more and led to instability of the meniscus. This corresponds to the rupture of the liquid bridge. (19) Orr, F. M.; Scriven, L. E.; Rivas, A. P. J. Fluid Mech. 1975, 67, 723–742. (20) Farshchi-Tabrizi, M.; Kappl, M.; Cheng, Y. J.; Gutmann, J.; Butt, H. J. Langmuir 2006, 22, 2171–2184. (21) Lambert, P.; Delchambre, A. Langmuir 2005, 21, 9537–9543. (22) De Souza, E. J.; Gao, L. C.; McCarthy, T. J.; Arzt, E.; Crosby, A. J. Langmuir 2008, 24, 1391–1396. (23) De Souza, E. J.; Brinkmann, M.; Mohrdieck, C.; Crosby, A.; Arzt, E. Langmuir 2008, 24, 10161–10168. (24) Checco, A. Europhys. Lett. 2009, 85, 16002.
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In order to assess the accuracy of the method and optimize the calculations while keeping reasonable computing time, we considered a model situation leading to an analytical solution. With Rsurf=1 μm, R=θtip=45°, and pressure P=γ/Rsurf=63 kPa, one expects a cylindrical meniscus. The resulting capillary force is equal to F=-2πRsurfγ þ πRsurfγ=-πRsurfγ, where the first term corresponds to the surface tension force on the contact line and the second one to the force resulting from the Laplace pressure. We performed calculations with “surface evolver”, changing the number of refinement steps of the mesh during the surface optimization stage, and the Δz value used in force calculation. A relative error smaller than 10-6 (with respect to the analytical value) was obtained provided one considered more than 20 000 facets and a variation of height Δz smaller than 10-3R. This validated our method and defined conditions to be used in all the calculations. It is important to note that all the methods discussed above only consider an equilibrium situation and do not take into account any viscous force which may play an important role when the meniscus is dynamic. An estimate of the viscous force leads to Fv ≈ 6πηR2/ts, where η is the viscosity of the liquid, R is the radius of the contact zone, and ts is the time required to separate surfaces.25 In the conditions used in NADIS deposition, this formula leads to viscous forces of the order of 10-13 N which are negligible compared to the capillary forces in the range of 10-8 N. Viscous effects may nevertheless play a role at the very end of the separation process when interface velocity becomes larger.
3. Results and Discussion As evidenced in Figure 3, the capillary force exerted on a NADIS tip during liquid dispensing presents characteristic features which depend on the experimental conditions used. We used the simulation method exposed above to interpret these experimental data. We first present the results of a model experiment using standard AFM tips with the aim to validate the calculation method on an experimental situation. In a second step, we discuss the case of hydrophilic NADIS tips, emphasizing the role of the nanochannel size in the process. Finally, hydrophobic tips are considered. a. Standard AFM Tips. In order to assess the validity of the simulation method, we first realized an experiment using a model situation where constraints and boundary conditions are both unambiguously defined. With that aim, we deposited with a NADIS tip glycerol droplets of different sizes ranging from 1 to a few μm on a Si02 surface. A standard AFM tip (i.e., with no aperture) similar to the ones used for NADIS fabrication was mounted on the atomic force microscope and approached from the deposited droplets. Force curves were then recorded while dipping the tip into the droplets until contact with the solid surface underneath and then retracting until separation of the tip from the liquid meniscus. Considering the fact that the time required for the acquisition (less than 1 s) is much shorter than the evaporation time of micrometer size glycerol droplets (several minutes),26 the volume of the liquid was considered as constant. This defined the type of constraint used in the simulations. Moreover, both tip and surface were cleaned with piranha solution (a 3:1 mixture of concentrated sulfuric acid with 30% hydrogen peroxide, v/v; caution: hazard) in order to make them hydrophilic, so that the liquid did not dewet during the retraction process. The contact lines on tip and surface were therefore considered as fixed. Since the shape of the contact line depends on the geometry of the tip, we considered four cases: (i) circular (25) Cai, S.; Bhushan, B. Mater. Sci. Eng. R 2008, 61, 78–106. (26) Arcamone, J.; Dujardin, E.; Rius, G.; Perez-Murano, F.; Ondarc-uhu, T. J. Phys. Chem. B 2007, 111, 13020–13027.
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Figure 5. Experimental force curves (dots) and calculated ones (color lines) for a standard AFM tip detaching from droplets of different sizes in the micrometer range. Inset: schematic representation of the system showing the origin of the zshift parameter.
contact line on a conical tip; (ii) elliptical one on a tilted conical tip (in the experiments, the tip is tilted by about 10° with respect to the vertical); (iii) square contact line on a pyramidal tip; (iv) contact line geometry optimized with a contact angle on a pyramidal tip (see inset of Figure 12) and kept fixed during the retraction process. The force curves computed in the last three configurations showed very little deviation with respect to the simple conical tip case. The maximum relative deviation on one force curve was of 2%, 1%, and 2%, for cases (ii)-(iv), respectively. For sake of simplicity, we therefore considered a conical tip for all situations with a hydrophilic tip. The model experimental conditions used therefore defined the boundary conditions and constraints to be used in the calculations: we simulated force curves using fixed Rtip and Rsurf parameters and a constant volume constraint V. In Figure 5 are reported with symbols the experimental retraction curves obtained in three droplets of different sizes. Note that the deflection of the cantilever was converted into a force using a cantilever spring constant k used as an adjustable parameter. In the present case, the spring constant of 2 N/m used for the three curves was 2.45 times larger than the nominal value given by the manufacturer. Such a discrepancy is usual for cantilevers in particular because small changes in dimensions lead to large variations of the spring constant. The results of the modelization using a conical tip with half angle 38.3° are reported on Figure 5. In addition to the Rtip, Rsurf, and V parameters used in the calculations, we shifted the calculated curves by a distance zshift taking into account the deviation of the tip shape from a perfectly conical tip which may lead to a translation of the tip-surface contact point. A positive zshift value corresponds to a probe with a overgrown tip, whereas a negative one indicates a truncated tip such as a NADIS tip. We observe, in the three cases, a very good agreement with experimental curves except at the very end where the separation occurred earlier in the simulations. This may come from the precise tip shape or from a dynamical process which may play a role during the thinning and rupture of the meniscus. Except for this point, the shape of the curves, with maximum amplitude for a short separation distance followed by a slow decrease of the capillary force, is very well reproduced. The adjustable parameters used in the calculations were Rtip =1, 0.89, and 0.63 μm, Rsurf=2, 1.8 , and 1.26 μm, V=8, 5.64, and 2 fL, and zshift = 52, 31 and 57 nm, respectively, from the bigger to smaller droplets. Among these parameters, the value of zshift could be determined DOI: 10.1021/la902614s
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Figure 6. Experimental (black dots) and calculated (red line) force curve measured during the retraction of a NADIS tip with a 280 nm aperture leading to droplets with a diameter of 800 nm.
experimentally. The positive values of zshift correspond to a tip which is longer than the pyramidal one. This is indeed the case with these tips which are sharpened at the tip apex (see sketch in the inset of Figure 5). The estimation of the additional height by scanning electron microscopy (SEM) imaging gives, for new tips, a value of about 100 nm which compares well with the value of zshift used in the simulations (about 50 nm), in particular if one considers that the tip used in the experiments may be damaged by repeated contact with the surface. This model experiment demonstrates the validity of our simulations to describe experimental situations. In the following, we describe the application of this method to account for more complex situations corresponding to liquid dispensing by NADIS tips. b. Hydrophilic NADIS Tips. We consider here the case of NADIS tips cleaned using a piranha solution for 15 min prior to loading and dispensing. This treatment led to a hydrophilic tip outer wall as confirmed by the spreading of the liquid on the cantilever observed during the loading of the reservoir. The deposition was realized on a SiO2 substrate functionalized by aminopropyltriethoxysilane, a surface treatment which is relevant in many applications. This surface exhibited a strong hysteresis with an advancing contact angle of 50° and a receding contact angle of 12°. This hysteresis led to a strong pinning of the contact line. Due to the small value of the receding contact angle, we considered that the liquid could not dewet during the retraction process and therefore assumed a fixed contact line. The surface properties of both tip and substrate led us to use in the calculations, as in the model experiment presented above, boundary conditions defined by fixed Rsurf and Rtip parameters. However, contrary to the previous case, in the case of NADIS tips, the value of Rsurf was not considered as an adjustable parameter. It was measured experimentally as the size of the corresponding molecule spot deposited on the surface after evaporation of the solvent. In the following, we compare experimental and simulated force curves obtained under these conditions. As shown in Figure 3, besides surface chemistry, the size of the nanochannel at the tip apex plays an important role in the measured capillary force. We therefore present separately the cases of large and small apertures. i. NADIS Tips with Large Apertures (>200 nm). An example of force curve obtained with a NADIS tip with a 280 nm square aperture on a SiO2 substrate functionalized by aminopropyltriethoxysilane is reported Figure 6. We observe a large attractive force due to the presence of the liquid meniscus between the tip and surface. During the retraction, the magnitude of the capillary force increased slowly and after a smooth 1874 DOI: 10.1021/la902614s
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Figure 7. Experimental (black dots) and calculated (red line) force curve obtained with a NADIS tip with a 35 nm aperture.
maximum decreased until the point where, for a tip surface separation of about 300 nm, the meniscus suddenly broke. These features were very reproducible and appeared for all the measurements performed in similar conditions. The channel at the apex of NADIS constitutes the main difference compared to standard AFM tips. In the case of tips with large apertures, that is, tips with a channel diameter larger than 200 nm, we cannot consider the volume of the liquid bridge as constant since liquid can flow in the channel and feed the droplet. On the contrary, if the liquid flow is fast enough, the pressure in the meniscus is in permanent equilibrium with the pressure in the reservoir. We therefore assumed a constant pressure during the entire separation process. As a result, in the modelizations reported in the following, we imposed a constant pressure constraint. The value of the pressure was fixed at P = 6400 Pa which is the Laplace pressure associated with a droplet with a 20 μm radius of curvature, as can be estimated from optical imaging of reservoir droplets on cantilevers. This value may change from experiment to experiment, but we checked that a variation of pressure by a factor of 2 only modified the capillary force by less than 2%. As explained above, in the case of hydrophilic surfaces, we considered that the radii of the meniscus on the tip and surface Rtip and Rsurf, respectively, remained constant. The value of Rtip which was unknown experimentally was used as an adjustable parameter, whereas the value of Rsurf was measured experimentally from the size of the molecule spots after evaporation of the solvent. In Figure 6, we compare the experimental capillary force with the simulated ones. We note an excellent agreement between both curves. The smooth maximum of the force magnitude is very well reproduced. The adjustable parameters used in the calculations were Rtip=250 nm, k=1.96 N/m, and zshift=-146 nm, while the value of Rsurf measured experimentally was 400 nm. The value of the cantilever stiffness was comparable to the one found in the previous experiment. As for zshift, we found a negative value which is compatible with the fact that the NADIS tip was a pyramid truncated by about 160 nm at its apex. The value of Rtip corresponds to a wetted part of the tip of 500 nm in diameter which is significantly larger than the size of the aperture. This is coherent with a fact that the tip is hydrophilic and therefore partially wetted by the liquid. The very good agreement between experimental and calculated curves, together with the realistic values of the parameters which could be assessed experimentally, are good indications of the validity of our approach. In particular, it validates the constant pressure assumption and shows that the meniscus volume does Langmuir 2010, 26(3), 1870–1878
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not remain constant and is fed by the reservoir during the retraction process. ii. NADIS Tips with Small Aperture. In the case of NADIS tips with small apertures, typically smaller than 50 nm in diameter, the shape of force curves is totally different from the ones discussed above. A typical curve obtained with a 35 nm aperture is reported on Figure 7. The capillary force decreases rapidly at the beginning of the separation process and then more slowly until the liquid bridge breaks. Such features cannot be reproduced in the calculations using the constraints and boundary conditions used in the experiments reported above made with larger apertures. The main assumption in the case of large aperture was that the liquid could flow rapidly in the channel so that a hydrostatic equilibrium established at every time between the meniscus and the reservoir droplet. This is probably no longer the case when decreasing the size of the aperture. For ultimate aperture sizes, such as the one reported on Figure 7, we made the assumption that the liquid flow was negligible in the nanochannel. The liquid could no longer feed the liquid meniscus whose volume remained constant during the separation process. We therefore performed simulations using a volume constraint. The volume V was used as an adjustable parameter, contrary to the case of pressure constraint for which the P parameter could be estimated experimentally. Despite the fact that in the experiments the tip was hydrophobic, we used constant Rtip and Rsurf boundary conditions. This is due to the fact that, close to the tip apex, the gold layer coating the outer wall is damaged on a region of about 120 nm around the aperture (see SEM image Figure 2b). The resulting silicon nitride region was not modified by thiol chemistry and therefore remained hydrophilic. Consequently, we assumed that the liquid wetted only this part without spreading on the hydrophobic tip. We used as boundary conditions Rtip =65 nm and Rsurf = 95 nm. The result of the calculation reported in Figure 11 remarkably reproduces the experimental data with the following set of parameters: V=4 aL, zshift=-28 nm, and k= 2.4 N/m. Note that the zshift value is significantly smaller than the truncated part of the tip measured by SEM which leads to an estimated value of 55 nm. Nevertheless, the precise shape of the tip apex is difficult to determine and may play a role in the determination of the separation at contact. The force curves obtained with a small aperture could be reproduced remarkably well with our method, using a volume constraint. In the following, we discuss more in detail the origin of the capillary force in the cases of large and small aperture, and justify the constraints used in both cases. iii. Origin of Capillary Force. In order to compare in greater detail the two situations reported above when varying aperture sizes, we report in Figure 8 the calculated curves extracted from Figures 6 and 7, after normalization by 2πγRtip for the force and Rtip for the separation. The large aperture case (constant pressure) is plotted in blue, whereas the small aperture (constant volume) is plotted in red. Note that, in order to compare on a larger range of separations, we did not take into account the tip but simply considered a liquid bridge between two disks of radii Rtip and Rsurf separated by a distance z. Even in the normalized representation, we observe a large difference of behavior: in the case of constant pressure, the force evolves rather slowly and separation occurs for a large separation, whereas for constant volume the tip experiences a much larger force over a shorter distance, leading to a sharp maximum. The separation process is therefore shorter in the case of small aperture but involves higher forces which may become of importance for dispensing on soft materials. As mentioned above, the capillary force has two components: the Laplace term due to the pressure P inside the meniscus is given Langmuir 2010, 26(3), 1870–1878
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Figure 8. Normalized calculated force curves in the cases of constant pressure (blue) or constant volume (red) constraints. In the latter case, the tension and Laplace contributions are plotted in purple and green, respectively.
by FL =πRtip2P and the tension term resulting from the surface tension force applied by the liquid interface on the contact line which is equal to FT = -2πRtipγ cos(φ), where φ is the angle between the interface at the contact line and the vertical (Figure 4a). The Laplace pressure was imposed in the case of constant pressure and could be determined numerically using “surface evolver” during the separation at constant volume. We could therefore estimate the Laplace force term, and we deduced the tension term as the difference between total force and Laplace force. In the case of large aperture, the normalized Laplace force is 0.05, showing that its contribution is negligible. The capillary force is therefore mainly due to the tension term. The maximum magnitude is obtained when the liquid interface is vertical on the tip (φ = 0), which leads to a total force of F = -2πRtipγ and therefore a normalized force equal to -1 as observed in Figure 8. For small apertures, the Laplace pressure is no longer negligible and may contribute to the total force. Both tension and Laplace contributions are plotted in Figure 8 with a purple dashed line for FT and green dotted line for FL. We observe that the normalized tension contribution has a maximum magnitude equal to -1, in accordance with the expression given above. The φ=0 situation occurs for a separation much shorter than that in the constant pressure mode. Contrary to the latter case, the Laplace pressure term plays an important role here. A negative pressure27 builds in the meniscus and then decreases to zero. The magnitude of the Laplace force is comparable to the one due to tension and is predominant during a large part of the separation process. This explains that the maximum of the normalized total force is about twice the one of constant pressure case. This discussion shows that the modelization we developed, through the calculation of the various contributions, is useful to give a precise picture of the capillary force measured during NADIS deposition. iv. Transition between Regimes. We showed that a transition between constant pressure and constant volume regimes occurs for nanochannel diameters in the 100 nm range. Above such a value, the pressure can be considered as constant whereas for 35 nm apertures a constant volume model reproduces well the data. In between these two extreme situations, one expects a mixed regime where neither pressure nor volume can be considered as constant. In this case, dynamical parameters such as the flow in (27) Tas, N. R.; Mela, P.; Kramer, T.; Berenschot, J. W.; van den Berg, A. Nano Lett. 2003, 3, 1537–1540.
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the channel have to be computed, which is not possible with our approach. In order to understand the value of the transition diameter, we estimated the flow in the nanochannel during the deposition process. We assumed that the flow is due to a pressure difference between the reservoir (approximately 6400 Pa) and the meniscus. Even if the pressure inside the meniscus may vary during the process, for a rough approximation, we considered a zero pressure inside the liquid bridge. The flow rate was estimated using a simple Poiseuille flow which gives Q = ΔP(πR4/8ηL), where Q is the flow rate, ΔP is the pressure difference, R is the radius of the nanochannel, L is its length, and η is the liquid viscosity. Using a channel length of 200 nm, we obtain flow rates of 3 fL/s and 0.7 aL/s for the two extreme cases we considered, that is, channel diameters of 280 and 35 nm, respectively. We then compared these values with the flow rate required to maintain a constant pressure inside the meniscus. We computed the volume of a liquid bridge during the separation process at constant pressure. We found that the volume increased during the main part of the detachment and decreased by about 30% before the instability of the meniscus. We measured the flow rate as the slope of the V(t) curve during the first stage. In the case of the large channel, we found Q = 0.4 fL/s, which is 1 order of magnitude smaller than the flow rate which can be provided using a Poiseuille flow. This validates the use of a constant pressure constraint in the calculations. As for the 35 nm nanochannels, a flow rate Q=5 aL/s is required to maintain a constant pressure. This value is significantly higher than the one obtained with a Poiseuille flow. Even if these rough estimates would require more sophisticated calculations, they justify the assumptions of the model. These assumptions could be checked experimentally changing the rate of liquid dispensing by changing the tip velocity during retraction of the tip. For large aperture, one could increase the velocity until the constant pressure condition is no longer valid. Unfortunately, this would require tip velocities which are not available with our setup. In the case of the ultimate nanochannels, a decrease of the velocity may lead to a transition from a constant volume to a constant pressure condition by allowing the flow in the channel to feed the liquid bridge. However, this was impossible to achieve because of liquid evaporation. An extrapolation of the data on glycerol sessile drop evaporation we reported recently26 gives a 1 s evaporation time for a 1 aL droplet. In the experiment reported above, the dispensing is performed in about 0.1 s. We therefore consider that evaporation is negligible, but this would not be the case for longer times required to observe a transition to a constant pressure regime. v. Real Time Monitoring of Dispensing. We used the very good agreement between experimental and simulated curves to provide a simple method for real time monitoring of the deposition. The idea was to use the capillary force in order to monitor, during the deposition procedure, the size of the droplets and, if necessary, adjust this size by changing relevant parameters. In order to get a simple correlation between the force curve and the droplet size, we considered one characteristic point of the force curve, the tip-surface separation at meniscus rupture Zrupt, and measured its value as a function of the radius of the droplet Rsurf. We used a NADIS tip with a large aperture to realize different depositions of glycerol solutions, changing the tip-substrate contact time. It was shown previously that, in the regime involving femtoliter droplets, the contact time plays an important role in the size of the spots.9 A pattern with droplets radii ranging from 247 to 500 nm as measured with AFM was obtained (inset of Figure 9). For each deposition, the Zrupt value was determined 1876 DOI: 10.1021/la902614s
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Figure 9. Experimental (dots) and calculated (solid line) values of the tip-surface separation at meniscus failure Zrupt as a function of the meniscus radius Rsurf, measured from a series of deposits obtained over varying contact times. Inset: AFM image of the resulting spots (scale bar: 2 μm).
from the force curve and plotted as a function of the corresponding value of Rsurf (Figure 9). We observe an increase of Zrupt with Rsurf, with the value of Zrupt varying by 50% whereas the value of Rsurf changed by a factor 2. In order to model this curve, we simulated force curves at constant pressure with a given Rtip and varying Rsurf, all other parameters remaining unchanged. We extracted the Zrupt value as the tip separation distance at rupture. We found a nonlinear relationship between the Zrupt and Rsurf values, in excellent agreement with experiments, with Rtip as the only adjustable parameter. The value of Rtip used in the calculations was Rtip =200 nm. The demonstrated relationship between Zrupt and Rsurf can be used to monitor liquid nanodispensing. Since the Zrupt value is very easily and quickly determined, it can be used to monitor in real time the size of the deposited droplets. This is an important point for applications for which one cannot wait for droplet evaporation and AFM imaging of deposited spots to check the result of the dispensing process. vi. Deposition on Hydrophobic Surfaces. In the previous cases, we considered deposition on hydrophilic surfaces (or surfaces with very small receding contact angle), leading to a pinned contact line on the substrate. On hydrophobic substrates, this is not the case. It was shown experimentally that, on highly hydrophobic fluorinated SiO2 substrates (receding contact angle 98°), the size of the droplets could be significantly smaller than the aperture size.9 This effect was also demonstrated at larger scale using pipets.28,29 The force curves measured in the NADIS experiments showed no difference compared to the case of a dry tip. During the retraction, the tip remains in contact with the surface and then suddenly detaches. The force drops to zero with no evidence of the formation of a liquid bridge. We computed force curves corresponding to this situation by using a pressure constraint (large aperture) and as boundary conditions a fixed contact line radius Rtip on the (hydrophilic) tip and a fixed contact angle θsurf on the substrate. This was achieved by attributing to the wetted part of the surface an excess surface energy W=γSL - γSV, which can be expressed using the Young-Laplace equation as W=γ cosθsurf. Note that we did not take into account the tip shape and simply considered a disk of radius Rtip at a z distance from the surface. In addition to the capillary (28) Uemura, S.; Stjernstrom, M.; Sjodahl, J.; Roeraade, J. Langmuir 2006, 22, 10272–10276. (29) Qian, B.; Loureiro, M.; Gagnon, D. A.; Tripathi, A.; Breuer, K. S. Phys. Rev. Lett. 2009, 102, 164502.
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Figure 10. (top) Normalized values of Rsurf as a function of the normalized tip-surface separation, calculated for five values of surface contact angle and (bottom) corresponding normalized force curves.
force, we also extracted from the calculations the radius of the contact line on the substrate Rsurf. The normalized force curves obtained for various values of the surface contact angle are reported in Figure 10 (bottom). We observe that the more hydrophobic the surface, the smaller the tip-separation distance at meniscus rupture. For contact angles larger than 90°, the separation at rupture is smaller than 0.7Rtip. This distance is not accessible experimentally due to the presence of the tip which prevents the contact line on the tip from coming close to the surface (even with a truncated tip). This explains why we could not observe experimentally, on the force curves, the formation of a meniscus bridging tip and surface. When decreasing the contact angle, the meniscus remains stable for larger values of z. The maximum magnitude of the force is always close to F= -2πRtipγ which corresponds to the maximum tension force when the liquid interface on the tip is vertical (φ = 0). The small difference between the capillary force and the tension force FT = -2πRtipγ cos φ comes from the positive Laplace pressure contribution which was used in the simulations to account for the pressure in the reservoir. The radius Rsurf of the wetted part of the substrate is also largely influenced by the contact angle as evidenced on Figure 10 (top). For contact angles smaller than 60°, this radius increases during the main part of the separation process and only slightly decreases at the very end of the process. For more hydrophobic surfaces (θsurf > 60°), on the contrary, the wetted area shrinks and can be significantly smaller than Rtip. This is coherent with the experimental observations, but no quantitative comparison is possible. Moreover, in the ultimate stage of the meniscus failure, dynamic effects may play an important role.28,29 c. Hydrophobic Tips. In order to decrease the size of the droplets and reach ultimate dimensions interesting for nanoscale patterning or single molecule deposition, we functionalized a tip via dodecanethiol surface chemistry. This treatment makes the outer wall hydrophobic, confining the liquid at the tip apex by limiting the spreading of liquid on the tip. The use of such tips radically changed the shape of the force curves. Examples of curves obtained with the same tip, before and after dodecanethiol treatment, on the same APTES surface are reported in Figure 3. Contrary to the case of the hydrophilic tips, hydrophobic tips show a monotonous decrease of the force and a greatly reduced rupture point. In order to model this situation, given the size of the tip aperture (diameter 400 nm), we considered a constant pressure Langmuir 2010, 26(3), 1870–1878
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Figure 11. Calculated force curves for hydrophobic NADIS tips when varying tip contact angle.
constraint. As a boundary condition on the APTES surface, as in previous cases, we used a constant Rsurf radius. The main change we brought in the modelization compared to hydrophilic tips was the boundary condition on the tip. Due to the hydrophobic character of the tip, the contact line was not assumed as fixed: the liquid could dewet on the tip with a fixed contact angle (the receding contact angle on the dodecanethiol modified gold surface). The same procedure as that for the hydrophobic surfaces was applied to maintain a constant contact angle θtip on the tip. Computed force curves when changing tip contact angle for Rsurf = 500 nm are reported in Figure 11. We observe that the magnitude of the force decreases regularly during the retraction process. This is totally different from the force curves obtained with hydrophilic tips which exhibited a maximum of force amplitude. This result is in qualitative agreement with the experimental observations discussed above. The decrease of the total force results from a shrinking of the wetted part of the tip when it is whithdrawn from the liquid. We observe that the liquid dewets from the tip wall during retraction, with the contact line moving toward the tip apex. Note that the contact line never reaches the end of the tip, with the meniscus instability occurring for a finite value of Rtip. The decrease of Rtip leads to a decrease of both tension and Laplace contributions to the total capillary force which can be estimated in a rough approximation. The simulations show that the height h of the meniscus only changes slightly during the separation process. In the first half of the process, it remains nearly constant and then slowly decreases by about 15% at the end of the process. If one assumes that the meniscus height h is constant, the radius of the contact line Rtip, which can be expressed as Rtip = (h - z)tan R, decreases linearly with the tip-surface separation z. Since the main contribution to the capillary force comes from the tension term which is proportional to Rtip, one expects, in this rough approximation, a linear decrease of the capillary force when the tip is withdrawn from the meniscus. Even if the model assuming a constant contact angle on the tip gives results which are in good qualitative agreement with the experiments, a quantitative comparison is difficult. One can fit portions of the force curve, but it is difficult to find conditions leading to a perfect adjustment of the entire force curve, with realistic parameters. There may be several reasons for this fact. Contrary to the case of hydrophilic tips, in the hydrophobic regime, the precise shape of the tip plays a role. In particular, in the calculations presented above, we considered a conical tip whereas the ones used in the experiments are pyramidal. The DOI: 10.1021/la902614s
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Ga (ions used for milling) which can decrease the efficiency of the dodecanethiol surface treatment. As a result, the motion of the contact line on the hydrophobic tip probably follows a rather complex dynamics which makes a good fit of the whole curve difficult. In the case of the hydrophilic tip, since contact lines are pinned on both the tip and substrate, the situation is more favorable for comparison with calculated curves.
Figure 12. Comparison of force curves obtained with conical and pyramidal tips.
advantage of the method we developed compared to analytical calculations is that the “surface evolver” software can handle more complex boundary conditions than a pure axisymmetric geometry. We therefore compared the results obtained with a conical shape and more realistic pyramidal ones. The angle of the pyramid was given by the manufacturer, whereas for the conical tip we used an half angle R=38.3° which gives, for a same height, the same volume as the pyramid. Examples of meniscus shapes computed using the same constraints (P = 6400 Pa) and tip contact angle (θtip = 30°) for conical and pyramidal tips are reported in Figure 12. We observe that, with the pyramidal tip, the contact line on the tip has a more complex geometry, with liquid spreading more on the facets than on the edges (see inset of Figure 12). Despite the differences in meniscus shapes, both geometries give comparable values of the capillary force. Nevertheless, a noticeable difference appears since the meniscus loses stability earlier on pyramidal tips. As in the case of conical tips, force curves computed with pyramidal tips did not allow a quantitative fit of a whole experimental curve with realistic parameters. Another reason which may explain the difficulty to fit the whole force curve is contact angle hysteresis. It is likely that, during retraction, the contact line does not dewet regularly the tip but rather has a more complex dynamics with different modes. At the beginning of the retraction, the contact angle is given by the advancing contact angle. Depending on the contact angle hysteresis, the contact line may remain pinned before receding on the tip. A pinning on the edges at the apex of the truncated NADIS tips may also change the contact line dynamics.30,31 The overall process is therefore a mixture of several modes. Moreover, as shown in Figure 2, the gold layer coating the tip outer wall is damaged during the FIB imaging. This leads to a surface with heterogeneous properties which changes the local contact angle on the tip and makes a precise comparison with experiments difficult. The region around the tip apex is also doped with (30) Oliver, J. F.; Huh, C.; Mason, S. G. J. Colloid Interface Sci. 1977, 59, 568– 581. (31) Ondarc- uhu, T.; Piednoir, A. Nano Lett. 2005, 5, 1744–1750.
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4. Conclusion We have reported a comprehensive study of capillary force experienced by an AFM tip during liquid nanodispensing. We showed a large variety of behaviors depending on the experimental conditions. A numerical method based on energy minimization has been developed and accounts very well for most of the experimental measurements. Since it can describe many different situations and surfaces with more complex shapes than the usual axisymmetric ones, it is a general method which can be useful for many applications. The very good agreement between experimental and calculated force curves gives a precise description of the process of liquid nanodispensing. In particular, the influence of channel diameter on the transfer mechanism was demonstrated. Through the computation of the various contributions to the capillary force, we could propose a precise picture of the total force acting on the tip. The method can also be used to optimize the NADIS technique. It can help in sizing the aperture size of the tip and substrate surface properties for a given spot size. The simple relationship between spot diameter and tip-surface separation at meniscus rupture also provides a useful real time monitoring of the dispensing process. Since the method considers a system at equilibrium at each point, it cannot describe the viscous forces or situations where the liquid flow in the nanochannel plays an important role. We limited our discussion to the two asymptotic situations observed as a function of the aperture size. To get a more complete description, it would be useful to go further and propose methods to describe the whole dynamics of the system. Given the nanometric sizes of the liquid bridges in the experiments, it would also be interesting to study the influence of line tension32 which may play a role in the sub-100 nm range. More generally, the NADIS technique, besides its application as a surface nanopatterning method, may provide an interesting way to probe wetting properties at the nanometer scale.33 Acknowledgment. The authors wish to thank Erik Dujardin, Aiping Fang, and Virginie Orfila for fruitful collaboration on the NADIS technique and Andre Meister (CSEM, Neuch^atel, Switzerland) for stimulating discussions. The partial support of the EC-funded project NaPa (Contract No. NMP4-CT-2003500120) is gratefully acknowledged. (32) Amirfazli, A.; Neumann, A. W. Adv. Colloid Interface Sci. 2004, 110, 121– 141. (33) Mendez-Vilas, A.; Jodar-Reyes, A. B.; Gonzalez-Martin, M. L. Small 2009, 5, 1366–1390.
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