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Sessile Nanodroplets on Elliptical Patches of Enhanced LyophilicityClick to copy article linkArticle link copied!
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† Physics of Fluids Group, Department of Applied Physics and J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
‡ Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands
§ Soft Condensed Matter, Debye Institute for Nanomaterials Science, Department of Physics, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands
∥ Soft Matter and Interfaces Group, School of Engineering, RMIT University, Melbourne, VIC 3001, Australia
*E-mail: [email protected]
Abstract
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We theoretically investigate the shape of a nanodroplet on a lyophilic elliptical patch in lyophobic surroundings on a flat substrate. To compute the droplet equilibrium shape, we minimize its interfacial free energy using both Surface Evolver and Monte Carlo calculations, finding good agreement between the two methods. We observe different droplet shapes, which are controlled by the droplet volume and the aspect ratio of the ellipse. In particular, we study the behavior of the nanodroplet contact angle along the three-phase contact line, explaining the different droplet shapes. Although the nanodroplet contact angle is constant and fixed by Young’s law inside and outside the elliptical patch, its value varies along the rim of the elliptical patch. We find that because of the pinning of the nanodroplet contact line at the rim of the elliptical patch, which has a nonconstant curvature, there is a regime of aspect ratios of the elliptical patch in which the nanodroplet starts expanding to the lyophobic part of the substrate, although there is still a finite area of the lyophilic patch free to be wetted.
Copyright © 2017 American Chemical Society
Introduction
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Equilibrium shapes of nanodroplets, which are positioned on a patterned surface, are of great interest to both fundamental research (1-8) and many industrial applications such as printing, (9) microfluidics, (10) and catalysis. (11) If the nanodroplet is deposited on a homogeneous substrate, then it will form a spherical cap, with a contact angle dictated by Young’s law. However, if we pattern the flat substrate with chemical heterogeneities, then the nanodroplet will no longer form a spherical cap but another equilibrium shape with constant mean curvature, which is determined by the Young–Laplace equation. (12) If the length scale of the chemical heterogeneities is much smaller than the length scale of the nanodroplet and if the heterogeneities are regularly and densely distributed over the substrate, then the nanodroplet has the apparent contact angle predicted by the modified Cassie–Baxter law. (13, 14) However, when the chemical heterogeneities are on the same length scale as the nanodroplet itself, a strong coupling of droplet shape and surface heterogeneities emerges. In such a case, we have to numerically minimize the interfacial free energy of the nanodroplet because in three dimensions the minimization problem can often not be solved analytically. (15, 16) The numerical techniques used in calculating the nanodroplet equilibrium shapes include the gradient descent method, (17, 18) metropolis stochastic calculations, (19) hybrid energy minimizations, (20) and lattice Boltzman calculations. (21) Wetting experiments and calculations on isolated chemical defects, such as circular islands (22) and single stripes, (23-26) have shown that the contact angles along the three-phase contact line are determined by the local properties of the substrate, which will be discussed in more detail in the Results section.
In this article, we analyze the morphology of a nanodroplet on a single lyophilic elliptical patch on a flat substrate. The single elliptical patch is an intermediate case between a circular island and a single stripe, but it is qualitatively unique because the rim of the elliptical patch has a nonconstant curvature that affects the nanodroplet equilibrium shape and shape transformations via the Young–Laplace equation. In Figure 1, we show the possible nanodroplet states we expect to find in this system, which will be explained in more detail further on in the article. These states are analogous to those of the nanodroplet positioned on a single finite lyophilic stripe, (24) with the exception of the long cylindrical drops, which cannot be obtained because of the curvature of the rim of the elliptical patch. We expect two possible pathways, through either state B or state C, because the contact line of the nanodroplet has a tendency to pin itself in the region of the sharp transition of wettability (rim of the elliptical patch), which determines the value of one out of the two principal radii of curvature.
Figure 1
Figure 1. Graphical representation of all possible droplet states as seen from the top view.
Theoretical Definition of the Problem
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Because of the small size of a nanodroplet, we can neglect the effect of gravity. Moreover, for simplicity we ignore line tension contributions. Therefore, the shape of the nanodroplet is controlled by the surface tension among the three phases present in the system (27) (s, solid; l, liquid; and v, vapor). If the droplet is in contact with a flat substrate, described by the plane z = 0, with patterned chemical heterogeneities, then we can write the interfacial free energy E as
(1)where γij denotes the surface tension between phases i and j and Aij represents the area of the interface between these two phases. Normalizing the interfacial free energy E with γlv and using Young’s law for the contact angle
(2)we obtain the expression for a reduced interfacial free energy Ẽ
(3)
(1)where γij denotes the surface tension between phases i and j and Aij represents the area of the interface between these two phases. Normalizing the interfacial free energy E with γlv and using Young’s law for the contact angle(2)we obtain the expression for a reduced interfacial free energy Ẽ(3)Numerical Methods and Procedure
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We minimize eq 3 under the volume constraint to obtain the equilibrium shape of the nanodroplet. The geometry of our system is presented in Figure 2, where the origin of the coordinate system (x, y, z) is in the center of the elliptical patch, with large semiaxis a and small semiaxis b. We define ϕ as a viewing angle where the value zero corresponds to the direction of the x axis and R(ϕ) denotes the distance from the center of the elliptical patch, with which we will describe the contact line of the nanodroplet. We define the Young contact angle in our system as
(4)where θ1 is always smaller than θ2, so the surface of the elliptical patch has an enhanced lyophilicity compared to the rest of the flat substrate.
(4)where θ1 is always smaller than θ2, so the surface of the elliptical patch has an enhanced lyophilicity compared to the rest of the flat substrate.Figure 2
Figure 2. Geometry of an elliptical patch defined by semiaxes a and b, with Young’s contact angle θ1 inside the patch and θ2 outside the patch.
We minimize eq 3 numerically with the Surface Evolver (17, 18) and with Monte Carlo (19) calculations. Surface Evolver, developed by Brakke, is a free software package for minimizing the interfacial free energy; it was used with great success to calculate equilibrium wetting morphologies. (13, 23, 24, 28, 29) After setting the initial shape of the droplet, Surface Evolver triangulates the interface of the nanodroplet and moves the points of each triangle with an energy gradient descent method. The contact area Asl is omitted from the Surface Evolver calculation, and we replace it with the integral of the second term of eq 3. In the Monte Carlo calculation, we use a simulated annealing method to calculate the global minimum of the interfacial free energy (eq 3), with the fluid–fluid interface represented by a grid of points. During the Monte Carlo simulation, a random shift in the morphology of the nanodroplet is introduced, and the interfacial free energy values (eq 3) before and after the shift are compared. If the value of the interfacial free energy is smaller after the shift, then we accept the new morphology. If it is larger, then we assign a probability of accepting the new morphology weighted by a temperature-like parameter T. This procedure is repeated continuously while T is gradually lowered, and the simulation ends when T = 0 (Metropolis algorithm). (19) In all of the presented calculations, we use the large semiaxis a as our unit of length, and we consider different aspect ratios b/a by tuning the small semiaxis b and different volumes V/a3 of the nanodroplet. The values of the Young’s contact angles in all calculations are set to θ1 = 30° (lyophilic patch) and θ2 = 60° (lyophobic surface). First, we position the nanodroplet center of mass above the center of the elliptical patch, and we set the nanodroplet initial volume to V = 0.01a3. After calculating the nanodroplet equilibrium shape, i.e., corresponding to the minimum in the interfacial free energy, we increment the nanodroplet volume by ΔV = 0.01a3 and recalculate its equilibrium shape. We repeat this process until we reach V = a3.
Results
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When the nanodroplet is sufficiently small to have the whole three-phase contact line inside the elliptical patch or when it is large enough to have the whole contact line outside the elliptical patch, the nanodroplet has a spherical-cap shape, with the contact angle defined by eq 4. We analytically calculate threshold volumes Vmin and Vmax of the nanodroplet, for which the nanodroplet is, respectively, too large to form the spherical-cap shape inside the patch or sufficiently large to form the spherical-cap shape completely covering the elliptical patch, namely,
(5)
(6)which correspond, using θ1 = 30° and θ2 = 60°, to Vmin ≈ 0.431b3 and Vmax ≈ 1.007a3. For all values of the nanodroplet volume between threshold volumes Vmin and Vmax, the nanodroplet will be in one of the following four possible morphologies:
(5)(6)which correspond, using θ1 = 30° and θ2 = 60°, to Vmin ≈ 0.431b3 and Vmax ≈ 1.007a3. For all values of the nanodroplet volume between threshold volumes Vmin and Vmax, the nanodroplet will be in one of the following four possible morphologies:Droplet state A: the nanodroplet has a part of the three-phase contact line pinned at the rim of the elliptical patch, and the rest of the contact line is located inside the elliptical patch. As we increase its volume, the nanodroplet can evolve to droplet state B or C, depending on the aspect ratio of the elliptical patch.
Droplet state B: the nanodroplet partially spreads outside the elliptical patch, although the elliptical patch (which is more lyophilic) is not fully wetted.
Droplet state C: the whole three-phase contact line of the nanodroplet is pinned at the rim of the elliptical path.
Droplet state D: part of the nanodroplet contact line is still pinned to the rim of the elliptical patch, and the rest of the contact line is outside of the elliptical patch. Both states B and C undergo a morphological transformation to state D once the volume has become sufficiently large.
In Figure 3, we present results for the equilibrium shapes of a nanodroplet on an elliptical patch of aspect ratio b/a = 0.7 as obtained from Monte Carlo simulations for varying droplet volumes V as labeled. The results as obtained from the Surface Evolver are very similar (Figure 4). We find that the shape transformation proceeds via droplet state C to state D as anticipated in Figure 1 upon increasing the droplet volume V. We now focus on the properties of the three-phase contact line and the local contact angle of the nanodroplet. In Figure 4, we show the position of contact line R(ϕ) and local contact angle θ(ϕ) as a function of viewing angle ϕ for different volumes of the nanodroplet, corresponding to different droplet states. For the chosen aspect ratio of elliptical patch b/a = 0.4, the nanodroplet will undergo the transformation from droplet state A to state B, therefore avoiding state C. We notice from these results that along the contact line of the nanodroplet the local contact angle is exactly the Young’s angle predicted from eq 4 if the contact line is locally either inside (θ1) or outside (θ2) the elliptical patch. However, when the contact line is exactly at the rim of the elliptical patch, Young’s law cannot be obeyed (12, 24) and the local contact angle has a value of between θ1 and θ2. Although Young’s law cannot be obeyed, the net force exhibited on the nanodroplet is zero as a result of the inversion symmetry of the system with respect to the x and y axes. When the nanodroplet volume reaches the value of Vmin, the contact line starts touching the rim of the elliptical patch, and at this point, the local contact angle starts increasing and the nanodroplet transforms from state A to state B. As we further increase the volume of the nanodroplet, once the value of the local contact angle reaches θ2, the contact line of the nanodroplet will locally leave the rim of the elliptical patch and will move outside the patch. Because the rim of the elliptical patch has a nonconstant curvature, the nanodroplet cannot have an equilibrium morphology where the local contact angle has a value of θ2 and is constant along the contact line in regions where it is pinned to the rim. This fact determines whether the growing nanodroplet will go through state B or state C. For a given volume, when R(90°) = R(270°) = b and θ(90°) = θ(270°) = θ2, if R(0°) = R(180°) < a, then the nanodroplet will undergo a morphological transition from state A to state B. Instead, if R(0°) = R(180°) = a, then the nanodroplet will be in state C. Both states B and C eventually go through the transition to state D. Once the nanodroplet volume reaches the value of Vmax, the nanodroplet equilibrium shape is a spherical cap with a Young’s contact angle of θ2.
Figure 3
Figure 3. Equilibrium shapes (blue grid) of a nanodroplet with increasing volume on an elliptical patch (red area) with aspect ratio b/a = 0.7, as obtained from Monte Carlo calculations. The respective right figures show a top view of the three-phase contact line (blue line).
Figure 4
Figure 4. Position of contact line R(ϕ) and local contact angle θ(ϕ) of the nanodroplet wetting an elliptical patch with aspect ratio b/a = 0.4 as a function of viewing angle ϕ and for varying droplet volumes as labeled. We show R(ϕ) and θ(ϕ) (a, b) for the nanodroplet in state A, (c, d) for the nanodroplet in state B, and (e, f) for the nanodroplet in state D. Solid lines show the results from the Surface Evolver calculation, and the markers present results from the Monte Carlo calculations.
We can study the transitions in a clearer way if we expand the function R(ϕ) into a (truncated) harmonic series as
(7)In Figure 5, we present coefficients from eq 7 as a function of the volume of the nanodroplet for three different aspect ratios, which represent three different scenarios of droplet states. As already mentioned, for the value of the aspect ratio of elliptical patch b/a = 0.4 (Figure 5a), the nanodroplet cannot wet the elliptical patch in state C, whereas for b/a = 0.7 (Figure 5b), the nanodroplet cannot be in state B, which is visible from constant coefficient values for different values of the volume of the nanodroplet. (The contact line has the same position for different values of the volume of the nanodroplet.) Once the value of coefficient c1 reaches its global maximum, the nanodroplet goes through a morphological transition from state A to either state B or C, depending on the values of R(0°) and R(90°) when θ(90°) and θ(270°) reach θ2. As the volume becomes larger, all of the higher harmonics go to zero because the nanodroplet becomes a spherical cap again. (22) We notice that for a circular patch, where b/a = 1 (Figure 5c), all of the higher harmonics remain zero for any volume of the nanodroplet because the nanodroplet has a spherical cap shape for any volume. Small deviations of the harmonics close to nanodroplet volume V = a3 are a numerical artifact from Surface Evolver due to a complicated definition of the interfacial energy on the flat substrate for this particular system. We summarize all of our results in the state diagram presented in Figure 6, where we show the nanodroplet morphological states with respect to nanodroplet volume V and elliptical patch aspect ratio b/a. Note that the presented state diagram holds for inside-patch and outside-patch Young’s contact angles given by θ1 = 30° and θ2 = 60°, respectively, so it can be quantitatively different for other combinations of these two values.
(7)In Figure 5, we present coefficients from eq 7 as a function of the volume of the nanodroplet for three different aspect ratios, which represent three different scenarios of droplet states. As already mentioned, for the value of the aspect ratio of elliptical patch b/a = 0.4 (Figure 5a), the nanodroplet cannot wet the elliptical patch in state C, whereas for b/a = 0.7 (Figure 5b), the nanodroplet cannot be in state B, which is visible from constant coefficient values for different values of the volume of the nanodroplet. (The contact line has the same position for different values of the volume of the nanodroplet.) Once the value of coefficient c1 reaches its global maximum, the nanodroplet goes through a morphological transition from state A to either state B or C, depending on the values of R(0°) and R(90°) when θ(90°) and θ(270°) reach θ2. As the volume becomes larger, all of the higher harmonics go to zero because the nanodroplet becomes a spherical cap again. (22) We notice that for a circular patch, where b/a = 1 (Figure 5c), all of the higher harmonics remain zero for any volume of the nanodroplet because the nanodroplet has a spherical cap shape for any volume. Small deviations of the harmonics close to nanodroplet volume V = a3 are a numerical artifact from Surface Evolver due to a complicated definition of the interfacial energy on the flat substrate for this particular system. We summarize all of our results in the state diagram presented in Figure 6, where we show the nanodroplet morphological states with respect to nanodroplet volume V and elliptical patch aspect ratio b/a. Note that the presented state diagram holds for inside-patch and outside-patch Young’s contact angles given by θ1 = 30° and θ2 = 60°, respectively, so it can be quantitatively different for other combinations of these two values.Figure 5
Figure 5. Expansion of R(ϕ) into a harmonic series as described in eq 7 for different values of the volume of the nanodroplet (V/a3) and different aspect ratios of the elliptical patch: (a) b/a = 0.4, (b) b/a = 0.7, and (c) b/a = 1. The results are presented in the same way as in Figure 4: solid lines correspond to Surface Evolver calculations, and the markers correspond to the Monte Carlo calculations.
Figure 6
Figure 6. State diagram of the droplet morphologies as a function of aspect ratio b/a of the elliptical patch (with inside patch contact angle θ1 = 30 and outside patch contact angle θ2 = 60) and reduced volume V/a3 of the nanodroplet. Solid lines present volume thresholds calculated in the Surface Evolver calculation, and dashed lines (in good agreement with the solid lines) present Monte Carlo results.
Conclusions
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We have calculated the equilibrium shapes of nanodroplets on elliptical patches of enhanced lyophilicity with two different numerical methods and have obtained good agreement between the two methods. With this work, we have connected the equilibrium shapes of a nanodroplet on isolated circular islands and the single stripe with all of the intermediate cases with varying aspect ratio b/a of the elliptical patch. We calculated all of the threshold volumes at which a morphological transition occurs for the given Young’s angles (θ1 = 30° and θ2 = 60°), which are summarized in Figure 6. The droplet states that we observe are similar to those reported on single lyophilic stripes. (23-25) However, because of the curvature of the rim of the elliptical patch, we do not observe long cylindrical drops on the elliptical patch. For practical applications of the elliptical patch, such as in catalysis, the separation of the aspect ratio of the elliptical patch into two regimes, either state B or state C, is an important result. If we think of chemical patterning as an investment in the substrate to be more efficient (isolation of a certain liquid on the patch), then the expansion of the contact line of the nanodroplet outside of the elliptical patch when there is still area available inside the elliptical patch is an inefficient way of patterning.
Author Information
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- Detlef Lohse - Physics of Fluids Group, Department of Applied Physics and J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands;
http://orcid.org/0000-0003-4138-2255;
- Ivan Dević - Physics of Fluids Group, Department of Applied Physics and J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands;http://orcid.org/0000-0003-0977-9973
- Xuehua Zhang - Soft Matter and Interfaces Group, School of Engineering, RMIT University, Melbourne, VIC 3001, Australia; Physics of Fluids Group, Department of Applied Physics and J. M. Burgers Centre for Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands;http://orcid.org/0000-0001-6093-5324
I.D. and G.S. contributed equally to this work.
Acknowledgment
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G.S. and R.v.R. acknowledge financial support by Marie Curie Initial Training Network SOMATAI. This work was supported by The Netherlands Center for Multiscale Catalytic Energy Conversion (MCEC), an NWO Gravitation Programme funded by the Ministry of Education, Culture and Science of the government of The Netherlands. This work is part of the D-ITP consortium, a program of the NWO that is also funded by the Dutch Ministry of Education, Culture and Science (OCW). We are grateful to Ken Brakke for giving us advice in using Surface Evolver. We also thank S. Karpitschka and C. W. Visser for their insights on this work.
References
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This article references 29 other publications.
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- 5Herminghaus, S.; Brinkmann, M.; Seemann, R. Wetting and Dewetting of Complex Surface Geometries Annu. Rev. Mater. Res. 2008, 38, 101– 121 DOI: 10.1146/annurev.matsci.38.060407.130335Google Scholar5Wetting and dewetting of complex surface geometriesHerminghaus, Stephan; Brinkmann, Martin; Seemann, RalfAnnual Review of Materials Research (2008), 38 (), 101-121CODEN: ARMRCU; ISSN:1531-7331. (Annual Reviews Inc.)A review. Surfaces exhibiting complex topogs., such as those encountered in biol., give rise to an enormously rich variety of interfacial morphologies of a liq. to which they are exposed. In the present article, some basic mechanisms involved in the statics and dynamics of such morphologies were elaborated, focusing on a few simple paradigm topogs. Different liq. interface morphologies on the same sample frequently coexist. To exemplify the impact of the dynamics on the final droplet morphol., the shape instability of filamentous liq. structures in wedge geometries are discussed. Some side effects that may dominate on a larger scale, such as contact line pinning and contact angle hysteresis, seem to play a minor role on the microscopic scale under study. This establishes the validity of simple theor. concepts of wetting as a starting point for describing liqs. at substrate surfaces of high complexity.
- 6Rauscher, M.; Dietrich, S. Wetting Phenomena in Nanofluidics Annu. Rev. Mater. Res. 2008, 38, 143– 172 DOI: 10.1146/annurev.matsci.38.060407.132451Google Scholar6Wetting phenomena in nanofluidicsRauscher, M.; Dietrich, S.Annual Review of Materials Research (2008), 38 (), 143-172CODEN: ARMRCU; ISSN:1531-7331. (Annual Reviews Inc.)A review. We focus on the dynamical aspects of wetting phenomena on the nanoscale for which bulk hydrodynamic equations become invalid. At the nanoscale, phenomena that are irrelevant on the micrometer scale and larger, or that can be summarily incorporated in terms of boundary conditions, become important. Among these features are long-ranged mol. interactions such as dispersion forces, thermal fluctuations, hydrodynamic slip, segregation of mixts. and solns. at walls, and elec. double layers.
- 7Lohse, D.; Zhang, X. Surface Nanobubbles and Nanodroplets Rev. Mod. Phys. 2015, 87, 981– 1035 DOI: 10.1103/RevModPhys.87.981Google Scholar7Surface nanobubbles and nanodropletsLohse, Detlef; Zhang, XuehuaReviews of Modern Physics (2015), 87 (3), 981-1036CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)Surface nanobubbles are nanoscopic gaseous domains on immersed substrates which can survive for days. They were first speculated to exist about 20 years ago, based on stepwise features in force curves between two hydrophobic surfaces, eventually leading to the first at. force microscopy (AFM) image in 2000. While in the early years it was suspected that they may be an artifact caused by AFM, meanwhile their existence has been confirmed with various other methods, including through direct optical observation. Their existence seems to be paradoxical, as a simple classical est. suggests that they should dissolve in microseconds, due to the large Laplace pressure inside these nanoscopic spherical-cap-shaped objects. Moreover, their contact angle (on the gas side) is much smaller than one would expect from macroscopic counterparts. This review will not only give an overview on surface nanobubbles, but also on surface nanodroplets, which are nanoscopic droplets (e.g., of oil) on (hydrophobic) substrates immersed in water, as they show similar properties and can easily be confused with surface nanobubbles and as they are produced in a similar way, namely, by a solvent exchange process, leading to local oversatn. of the water with gas or oil, resp., and thus to nucleation. The review starts with how surface nanobubbles and nanudroplets can he made, how they can be obsd. (both individually and collectively), and what their properties are. Mol. dynamic simulations and theories to account for the long lifetime of the surface nanobubbles are then reported on. The crucial element contributing to the long lifetime of surface nanobubbles and nanodroplets is pinning of the three-phase contact line at chem. or geometric surface heterogeneities. The dynamical evolution of the surface nanobubbles then follows from the diffusion equation, Laplace's equation, and Henry's law. In particular, one obtains stable surface nanobubbles when the gas influx from the gas-oversatd. water and the outfiux due to Laplace pressure balance. This is only possible for small enough surface bubbles. It is therefore the gas or oil oversatn. ζ that dets. the contact angle of the surface nanobubble or nanodroplet and not the Young equation. The review also covers the potential technol. relevance of surface nanobubbles and nanodroplets, namely, in flotation, in (photo)catalysis and electrolysis, in nanomaterial engineering, for transport in and out of nanofluidic devices, and for plasmonic bubbles, vapor nanobubbles, and energy conversion. Also given is a discussion on surface nanobubbles and nanudroplets in a nutshell, including theor. predictions resulting from it and future directions. Studying the nucleation, growth, and dissoln. dynamics of surface nanobubbles and nanodroplets will shed new light on the problems of contact line pinning and contact angle hysteresis on the submicron scale.
- 8Mendez-Vilas, A.; Jodar-Reyes, A. B.; Gonzalez-Martin, M. L. Ultrasmall Liquid Droplets on Solid Surfaces: Production, Imaging, and Relevance for Current Wetting Research Small 2009, 5, 1366– 1390 DOI: 10.1002/smll.200800819Google Scholar8Ultrasmall liquid droplets on solid surfaces: production, imaging, and relevance for current wetting researchMendez-Vilas, Antonio; Jodar-Reyes, Ana Belen; Gonzalez-Martin, Maria LuisaSmall (2009), 5 (12), 1366-1390CODEN: SMALBC; ISSN:1613-6810. (Wiley-VCH Verlag GmbH & Co. KGaA)A review. The investigation of micro- and nanoscale droplets on solid surfaces offers a wide range of research opportunities both at a fundamental and an applied level. On the fundamental side, advances in the techniques for prodn. and imaging of such ultrasmall droplets will allow wetting theories to be tested down to the nanometer scale, where they predict the significant influence of phenomena such as the contact line tension or evapn., which can be neglected in the case of macroscopic droplets. On the applied side, these advances will pave the way for characterizing a diverse set of industrially important materials such as textile or biomedical micro- and nanofibers, powd. solids, and topog. or chem. nanopatterned surfaces, as well as micro-and nanoscale devices, with relevance in diverse industries from biomedical to petroleum engineering. Here, the basic principles of wetting at the micro- and nanoscales are presented, and the essential characteristics of the main exptl. techniques available for producing and imaging these droplets are described. In addn., the main fundamental and applied results are reviewed. The most problematic aspects of studying such ultrasmall droplets, and the developments that are in progress that are thought to circumvent them in the coming years, are highlighted.
- 9Soltman, D.; Smith, B.; Kang, H.; Morris, S. J. S.; Subramanian, V. Methodology for Inkjet Printing of Partially Wetting Films Langmuir 2010, 26, 15686– 15693 DOI: 10.1021/la102053jGoogle Scholar9Methodology for Ink-Jet Printing of Partially Wetting FilmsSoltman, Dan; Smith, Ben; Kang, Hongki; Morris, S. J. S.; Subramanian, VivekLangmuir (2010), 26 (19), 15686-15693CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)Ink-jet printing of precisely defined structures is crit. for the realization of a range of printed electronics applications. We develop and demonstrate a methodol. to optimize the ink-jet printing of two-dimensional, partially wetting films. When printed inks have a pos. retreating contact angle, we show that any fixed spacing is ineffective for printing two-dimensional features. With fixed spacing, the bead contact angle begins large, leading to a bulging overflow of its intended footprint. Each addnl. line reduces the bead contact angle, eventually leading to sepn. of the bead. We propose a printing scheme that adjusts the line-to-line spacing to maintain a bead's contact angle between its advancing and retreating values as it is printed. Implementing this approach requires an understanding of the two-dimensional bead surface and compensation for evapn. during the print. We derive an analytic equation for the bead's surface with pinned contact lines and use an empirical fit for mass loss due to evapn. Finally, we demonstrate that enhanced contact angle hysteresis, achieved by preprinting a feature's border, leads to better corner definition.
- 10Lee, T.; Charrault, E.; Neto, C. Interfacial slip on rough, patterned and soft surfaces: A review of experiments and simulations Adv. Colloid Interface Sci. 2014, 210, 21– 38 DOI: 10.1016/j.cis.2014.02.015Google Scholar10Interfacial slip on rough, patterned and soft surfaces: A review of experiments and simulationsLee, Thomas; Charrault, Eric; Neto, ChiaraAdvances in Colloid and Interface Science (2014), 210 (), 21-38CODEN: ACISB9; ISSN:0001-8686. (Elsevier B.V.)Advancements in the fabrication of microfluidic and nanofluidic devices and the study of liqs. in confined geometries rely on understanding the boundary conditions for the flow of liqs. at solid surfaces. Over the past ten years, a large no. of research groups have turned to investigating flow boundary conditions, and the occurrence of interfacial slip has become increasingly well-accepted and understood. While the dependence of slip on surface wettability is fairly well understood, the effect of other surface modifications that affect surface roughness, structure and compliance, on interfacial slip is still under intense investigation. In this paper we review investigations published in the past ten years on boundary conditions for flow on complex surfaces, by which we mean rough and structured surfaces, surfaces decorated with chem. patterns, grafted with polymer layers, with adsorbed nanobubbles, and superhydrophobic surfaces. The review is divided in two interconnected parts, the first dedicated to phys. expts. and the second to computational expts. on interfacial slip of simple (Newtonian) liqs. on these complex surfaces. Our work is intended as an entry-level review for researchers moving into the field of interfacial slip, and as an indication of outstanding problems that need to be addressed for the field to reach full maturity.
- 11Clausen, B. S.; Schiøtz, J.; Gråbæk, L.; Ovesen, C. V.; Jacobsen, K. W.; Nørskov, J. K.; Topsøe, H. Wetting/ non-wetting phenomena during catalysis: Evidence from in situ on-line EXAFS studies of Cu-based catalysts Top. Catal. 1994, 1, 367– 376 DOI: 10.1007/BF01492289Google Scholar11Wetting/non-wetting phenomena during catalysis: evidence from in situ online EXAFS studies of Cu-based catalystsClausen, Bjerne S.; Schioetz, Jakob; Graabaek, Lars; Ovesen, Charlotte V.; Jacobsen, Karsten W.; Noerskov, Jens K.; Tops.danisho.e, HenrikTopics in Catalysis (1994), 1 (3,4, Frontiers in Catalysis: Ammonia Synthesis and Beyond), 367-76CODEN: TOCAFI; ISSN:1022-5528. (Baltzer)We have used in situ EXAFS to provide exptl. evidence for a reversible change in the morphol. of the metallic particles in a high surface area, porous catalyst system, Cu/ZnO, contg. small metallic copper particles. By changing the oxidn. potential in the synthesis gas mixt., it is found that the apparent Cu-Cu coordination no. changes in an essentially reversible manner suggesting that the small metallic Cu particles dynamically change morphol. This indicates that a wetting/non-wetting phenomenon takes place in the Cu/ZnO system with changing partial pressures of oxygen. Under similar conditions, these effects are not obsd. when copper is supported on, for example, SiO2. A model based on surface and interface free energies provides a simple explanation for the obsd. results. Since the wetting/non-wetting processes are accompanied by a change in the active surface area, the obsd. behavior has important general implications and such effects must be incorporated into microkinetic models in order to provide a proper description of the catalyst performance.
- 12Lenz, P.; Lipowsky, R. Morphological Transitions of Wetting Layers on Structured Surfaces Phys. Rev. Lett. 1998, 80, 1920– 1923 DOI: 10.1103/PhysRevLett.80.1920Google Scholar12Morphological transitions of wetting layers on structured surfacesLenz, Peter; Lipowsky, ReinhardPhysical Review Letters (1998), 80 (9), 1920-1923CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)The morphol. of wetting layers on structured or imprinted surfaces is detd. by the geometry of the underlying surface domains. Droplets which cover a single domain exhibit contact angles which do not satisfy Young's equation. For surface patterns consisting of many surface domains, the wetting layer exhibits several distinct morphologies (homogeneous droplet patterns, heterogeneous droplet patterns, film states) and may undergo morphol. transitions between these different states. The latter transitions exhibit spontaneous symmetry breaking.
- 13Brandon, S.; Haimovich, N.; Yeger, E.; Marmur, A. Partial wetting of chemically patterned surfaces: The effect of drop size J. Colloid Interface Sci. 2003, 263, 237– 243 DOI: 10.1016/S0021-9797(03)00285-6Google Scholar13Partial wetting of chemically patterned surfaces: The effect of drop sizeBrandon, Simon; Haimovich, Nir; Yeger, Einat; Marmur, AbrahamJournal of Colloid and Interface Science (2003), 263 (1), 237-243CODEN: JCISA5; ISSN:0021-9797. (Elsevier Science)Partial wetting of chem. heterogeneous substrates is simulated. Three-dimensional sessile drops in equil. with smooth surfaces supporting ordered chem. patterns are considered. Significant features are obsd. as a result of changing the drop vol. The no. of equilibrated drops is found either to remain const. or to increase with growing drop vol. The shape of larger drops appears to approach that of a spherical cap and their three-phase contact line seems, on a larger scale, more circular in shape than that of smaller drops. In addn., as the vol. is increased, the av. contact angle of drops whose free energy is lowest among all equil.-shaped drops of the same vol. appears to approach the angle predicted by Cassie. Finally, contrary to results obtained with two-dimensional drops, contact angle hysteresis obsd. in this system is shown to exhibit a degree of vol. dependence in the advancing and receding angles. Qual. differences in the wetting behavior assocd. with the two different chem. patterns considered here, as well as differences between results obtained with two-dimensional and three-dimensional drops, can possibly be attributed to variations in the level of constraint imposed on the drop by the different patterns and by the dimensionality of the system.
- 14Marmur, A.; Bittoun, E. When Wenzel and Cassie Are Right: Reconciling Local and Global Considerations Langmuir 2009, 25, 1277– 1281 DOI: 10.1021/la802667bGoogle Scholar14When Wenzel and Cassie Are Right: Reconciling Local and Global ConsiderationsMarmur, Abraham; Bittoun, EyalLangmuir (2009), 25 (3), 1277-1281CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)The condition under which the Wenzel or Cassie equation correctly ests. the most stable contact angle is reiterated and demonstrated: these equations do hold when the drop size is sufficiently large compared with the wavelength of roughness or chem. heterogeneity. The numerical demonstrations somewhat mimic recent expts. that seemingly refuted the Wenzel and Cassie equations and show that these expts. were performed only for drops of sizes similar in order of magnitude to the wavelength of roughness or chem. heterogeneity. Under such conditions, the Wenzel and Cassie equations are a priori not expected to be valid. It is also explained that both the local equil. condition at the contact line and the global equil. condition involving the wetted area within the contact line are necessary and complementary.
- 15Gao, L.; McCarthy, T. J. How Wenzel and Cassie Were Wrong Langmuir 2007, 23, 3762– 3765 DOI: 10.1021/la062634aGoogle Scholar15How Wenzel and Cassie Were WrongGao, Lichao; McCarthy, Thomas J.Langmuir (2007), 23 (7), 3762-3765CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)We argue using exptl. data that contact lines and not contact areas are important in detg. wettability. Three types of two-component surfaces were prepd. that contain "spots" in a surrounding field: a hydrophilic spot in a hydrophobic field, a rough spot in a smooth field, and a smooth spot in a rough field. Water contact angles were measured within the spots and with the spot confined to within the contact line of the sessile drop. Spot diam. and contact line diam. were varied. All of the data indicate that contact angle behavior (advancing, receding, and hysteresis) is detd. by interactions of the liq. and the solid at the three-phase contact line alone and that the interfacial area within the contact perimeter is irrelevant. The point is made that Wenzel's and Cassie's equations are valid only to the extent that the structure of the contact area reflects the ground state energies of contact lines and the transition states between them.
- 16McHale, G. Cassie and Wenzel: Were They Really So Wrong? Langmuir 2007, 23, 8200– 8205 DOI: 10.1021/la7011167Google Scholar16Cassie and Wenzel: Were They Really So Wrong?McHale, G.Langmuir (2007), 23 (15), 8200-8205CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)A review of the theor. basis of the Cassie-Baxter and Wenzel equations. The properties of superhydrophobic surfaces are often understood by ref. to the Cassie-Baxter and Wenzel equations. Recently, in a paper deliberately entitled to be provocative, it was suggested that these equations are wrong; a suggestion said to be justified using exptl. data. It is argued that these models are not so much wrong as have assumptions that define the limitations on their applicability and that with suitable generalization they can be used with surfaces possessing some types of spatially varying defect distributions. The relation is discussed of the models to the previously published expts. and using min. energy considerations review the derivations of the equations for surfaces with defect distributions. It is argued that this means the roughness parameter and surface area fractions are quantities local to the droplet perimeter and that the published data can be interpreted within the models. Versions of the Cassie-Baxter and Wenzel equations involving roughness and Cassie-Baxter solid fraction functions local to the three-phase contact line on the assumption that the droplet retains an av. axisymmetry shape are derived. Moreover, it is indicated that, for superhydrophobic surfaces, the definition of droplet perimeter does not necessarily coincide with the three-phase contact line. As a consequence, the three-phase contact lines within the contact perimeter beneath the droplet can be important in detg. the obsd. contact angle on superhydrophobic surfaces.
- 17Brakke, K. A. The Surface Evolver Exp. Math. 1992, 1, 141– 165 DOI: 10.1080/10586458.1992.10504253Google ScholarThere is no corresponding record for this reference.
- 18Brakke, K. A.Surface Evolver Manual, version 2.70; 2013; http://facstaff.susqu.edu/brakke/evolver/downloads/manual270.pdf.Google ScholarThere is no corresponding record for this reference.
- 19Soligno, G.; Dijkstra, M.; van Roij, R. The equilibrium shape of fluid-fluid interfaces: Derivation and a new numerical method for Young’s and Young-Laplace equations J. Chem. Phys. 2014, 141, 244702 DOI: 10.1063/1.4904391Google Scholar19The equilibrium shape of fluid-fluid interfaces: Derivation and a new numerical method for Young's and Young-Laplace equationsSoligno, Giuseppe; Dijkstra, Marjolein; van Roij, ReneJournal of Chemical Physics (2014), 141 (24), 244702/1-244702/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Many phys. problems require explicit knowledge of the equil. shape of the interface between two fluid phases. Here, we present a new numerical method which is simply implementable and easily adaptable for a wide range of problems involving capillary deformations of fluid-fluid interfaces. We apply a simulated annealing algorithm to find the interface shape that minimizes the thermodn. potential of the system. First, for completeness, we provide an anal. proof that minimizing this potential is equiv. to solving the Young-Laplace equation and the Young law. Then, we illustrate our numerical method showing two-dimensional results for fluid-fluid menisci between vertical or inclined walls and curved surfaces, capillary interactions between vertical walls, equil. shapes of sessile heavy droplets on a flat horizontal solid surface, and of droplets pending from flat or curved solid surfaces. Finally, we show illustrative three-dimensional results to point out the applicability of the method to micro- or nano-particles adsorbed at a fluid-fluid interface. (c) 2014 American Institute of Physics.
- 20Matsui, H.; Noda, Y.; Hasegawa, T. Hybrid Energy-Minimization Simulation of Equilibrium Droplet Shapes on Hydrophilic/Hydrophobic Patterned Surfaces Langmuir 2012, 28, 15450– 15453 DOI: 10.1021/la303717nGoogle Scholar20Hybrid Energy-Minimization Simulation of Equilibrium Droplet Shapes on Hydrophilic/Hydrophobic Patterned SurfacesMatsui, Hiroyuki; Noda, Yuki; Hasegawa, TatsuoLangmuir (2012), 28 (44), 15450-15453CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)The authors have developed an efficient algorithm for simulating the equil. shape of a microdroplet placed on a flat substrate that has a fine, discontinuous, and arbitrarily shaped hydrophilic/hydrophobic patterned surface. The method uses a hybrid energy-minimization technique that combines the direct search method to det. the droplet shape around solid/liq. contact lines with the gradient descent method for the other parts of the droplet surface. The method provides high-convergence at a low computational cost with sufficient mesh resoln., providing a useful tool for the optimal design of printed electronic devices.
- 21Dupuis, A.; Léopoldès, J.; Bucknall, D. G.; Yeomans, J. M. Control of drop positioning using chemical patterning Appl. Phys. Lett. 2005, 87, 024103 DOI: 10.1063/1.1984098Google ScholarThere is no corresponding record for this reference.
- 22Extrand, C. W. Contact Angles and Hysteresis on Surfaces with Chemically Heterogeneous Islands Langmuir 2003, 19, 3793– 3796 DOI: 10.1021/la0268350Google Scholar22Contact angles and hysteresis on surfaces with chemically heterogeneous islandsExtrand, C. W.Langmuir (2003), 19 (9), 3793-3796CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)Wetting behavior was studied on surfaces with a single, circular heterogeneous island. Lyophobic islands were created on lyophilic Si wafers using polystyrene. Alternately, lyophobic perfluoroalkoxy fluoropolymer film was etched to make lyophilic domains. Contact angles and hysteresis were measured with water and hexadecane. Small sessile drops were deposited on the center of an island and liq. was sequentially added, eventually forcing the contact line to advance beyond the island perimeter onto the surrounding area. Even though the underlying contact area contained a mixt. of lyophilic and lyophobic domains, the contact angles, both advancing and receding, were equal to the angles exhibited by the homogeneous periphery. Or in other words, if the heterogeneity was completely contained with the contact area and did not intersect the contact line, then no area averaging of the contact angles occurred. These findings suggest that interactions at the contact line, not the contact area, control wetting of heterogeneous surfaces.
- 23Darhuber, A. A.; Troian, S. M.; Wagner, S. Morphology of liquid microstructures on chemically patterned surfaces J. Appl. Phys. 2000, 87, 7768– 7775 DOI: 10.1063/1.373452Google Scholar23Morphology of liquid microstructures on chemically patterned surfacesDarhuber, Anton A.; Troian, Sandra M.; Miller, Scott M.; Wagner, SigurdJournal of Applied Physics (2000), 87 (11), 7768-7775CODEN: JAPIAU; ISSN:0021-8979. (American Institute of Physics)The authors study the equil. conformations of liq. microstructures on flat but chem. heterogeneous substrates using energy minimization computations. The surface patterns, which establish regions of different surface energy, induce deformations of the liq.-solid contact line. Depending on the geometry, these deformations either promote or impede capillary breakup and bulge formation. The contact angles of the liq. on the hydrophilic and hydrophobic regions, as well as the pattern geometry and vol. of liq. deposited, strongly affect the equil. shapes. Moreover, due to the small scale of the liq. features, the presence of chem. or topol. surface defects significantly influence the final liq. shapes. Preliminary expts. with arrays of parallel hydrophilic strips produce shapes resembling the simulated forms. These encouraging results provide a basis for the development of high resoln. lithog. by direct wet printing.
- 24Brinkmann, M.; Lipowsky, R. Wetting morphologies on substrates with striped surface domains J. Appl. Phys. 2002, 92, 4296– 4306 DOI: 10.1063/1.1506003Google Scholar24Wetting morphologies on substrates with striped surface domainsBrinkmann, Martin; Lipowsky, ReinhardJournal of Applied Physics (2002), 92 (8), 4296-4306CODEN: JAPIAU; ISSN:0021-8979. (American Institute of Physics)The wetting and dewetting of chem. structured substrates with striped surface domains is studied theor. The lyophilic stripes and the lyophobic substrate are characterized by different contact angles θγ and θδ, resp. We det. the complete bifurcation diagram for the wetting morphologies (i) on a single lyophilic stripe and (ii) on 2 neighboring stripes sepd. by a lyophobic one. Long channels can only be formed on the lyophilic stripes if the contact angle θγ is smaller than a certain threshold value θch(V) which depends only weakly on the vol. V and attains the finite value θch(∞) in the limit of large V. This asymptotic value is equal to θch(∞) = aρc = g/cm3os(π/4)≃38° for all lyophobic substrates with θδ ≥ π/2. For a given value of θγ<θch(∞), the extended channels spread onto the lyophilic stripes with essentially const. cross section.
- 25Ferraro, D.; Semprebon, C.; Tóth, T.; Locatelli, E.; Pierno, M.; Mistura, G.; Brinkmann, M. Morphological Transitions of Droplets Wetting Rectangular Domains Langmuir 2012, 28, 13919– 13923 DOI: 10.1021/la302854tGoogle Scholar25Morphological Transitions of Droplets Wetting Rectangular DomainsFerraro, Davide; Semprebon, Ciro; Toth, Tamara; Locatelli, Emanuele; Pierno, Matteo; Mistura, Giampaolo; Brinkmann, MartinLangmuir (2012), 28 (39), 13919-13923CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)The authors report the results of comprehensive expts. and numerical calcns. of interfacial morphologies of H2O confined to the hydrophilic top face of rectangular posts of width W = 500 μm and lengths between L = 5W and 30W. A continuous evolution of the interfacial shape from a homogeneous liq. filament to a bulged filament and back is obsd. during changes in the liq. vol. Above a certain threshold length of L* = 16.0 W, the transition between the 2 morphologies is discontinuous and a bistability of interfacial shapes is obsd. in a certain interval of the reduced liq. vol. V/W3.
- 26Jansen, H. P.; Bliznyuk, O.; Kooij, E. S.; Poelsema, B.; Zandvliet, H. J. W. Simulating Anisotropic Droplet Shapes on Chemically Striped Patterned Surfaces Langmuir 2012, 28, 499– 505 DOI: 10.1021/la2039625Google Scholar26Simulating Anisotropic Droplet Shapes on Chemically Striped Patterned SurfacesJansen, H. Patrick; Bliznyuk, Olesya; Kooij, E. Stefan; Poelsema, Bene; Zandvliet, Harold J. W.Langmuir (2012), 28 (1), 499-505CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)The equil. shape of droplets on surfaces, functionalized with stripes of alternating wettability, have been investigated using simulations employing a finite element method. Expts. show that a droplet deposited on a surface with relatively narrow hydrophobic stripes compared to the hydrophilic stripes adopts a strongly elongated shape. The aspect ratio, the length of the droplet divided by the width, decreases toward unity when a droplet is deposited on a surface with relatively narrow hydrophilic stripes. The aspect ratio and the contact angle parallel to the stripes show unique scaling behavior as a function of the ratio between the widths of the hydrophobic and hydrophilic stripes. For a small ratio, the contact angle parallel to the stripes is low and the aspect ratio high, while for a large ratio, the contact angle parallel is high and the aspect ratio low. The simulations exhibit similar scaling behavior, both for the aspect ratio of the droplets and for the contact angles in the direction parallel to the stripes. Two liqs. with different surface tensions have been investigated both exptl. and in simulations; similarities and differences between the findings are discussed. Generally, three parameters are needed to describe the droplet geometry: (i) the equil. contact angles on the hydrophilic and (ii) hydrophobic areas and (iii) the ratio of the widths of these chem. defined stripes. Furthermore, we derive a simple anal. expression that proves to be a good approxn. in the quant. description of the droplet aspect ratio.
- 27Woodward, J. T.; Gwin, H.; Schwartz, D. Contact Angles on Surfaces with Mesoscopic Chemical Heterogeneity Langmuir 2000, 16, 2957– 2961 DOI: 10.1021/la991068zGoogle Scholar27Contact angles on surfaces with mesoscopic chemical heterogeneityWoodward, J. T.; Gwin, H.; Schwartz, D. K.Langmuir (2000), 16 (6), 2957-2961CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)The contact angle of water was measured on surfaces composed of random hydrophilic and hydrophobic patches with typical length scales of 10-100 nm. By quenching self-assembled monolayers at various stages of growth, the fractional surface coverage of the hydrophobic patches was varied in the range 0.04-0.97 as detd. by AFM. The cosine of the contact angle of water cos θ was systematically lower than the prediction of the mean field Cassie equation cos θC. The deviation from this prediction cos θC - cos θ had an approx. linear dependence on the total contour length between hydrophobic and hydrophilic patches (a measure of the degree of heterogeneity). The contact angle was insensitive to droplet size, suggesting that line tension effects were minimal. Also, contact angles of hexadecane were in good agreement with the Cassie prediction. Two possible explanations for the obsd. behavior are proposed. Long-range (approx. 5 nm) hydrophobic interactions may result in a relatively hydrophobic boundary region around each hydrophobic patch which effectively increases the coverage of the hydrophobic phase altering the equil. contact angle. Alternatively, an increased d. of "pinning" sites may prevent the contact angle from relaxing to the equil. value.
- 28Seemann, R.; Brinkmann, M.; Kramer, E. J.; Lange, F. F.; Lipowsky, R. Wetting morphologies at microstructured surfaces Proc. Natl. Acad. Sci. U. S. A. 2005, 102, 1848– 1852 DOI: 10.1073/pnas.0407721102Google Scholar28Wetting morphologies at microstructured surfacesSeemann, Ralf; Brinkmann, Martin; Kramer, Edward J.; Lange, Frederick F.; Lipowsky, ReinhardProceedings of the National Academy of Sciences of the United States of America (2005), 102 (6), 1848-1852CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)The wetting of microstructured surfaces is studied both exptl. and theor. Even relatively simple surface topogs. such as grooves with rectangular cross section exhibit a large variety of different wetting morphologies as obsd. by at. force microscopy. This polymorphism arises from liq. wedge formation along the groove corners and from contact line pinning along the groove edges. A global morphol. diagram is derived that depends only on two system parameters: (i) the aspect ratio of the groove geometry and (ii) The contact angle of the underlying substrate material. For microfluidics, the most interesting shape regimes involve extended liq. filaments, which can grow and shrink in length while their cross section stays essentially const. Thus, any method by which one can vary the contact angle can be used to switch the length of the filament, as is demonstrated in the context of electrowetting.
- 29Chen, Y.; He, B.; Lee, J.; Patankar, N. A. Anisotropy in the wetting of rough surfaces J. Colloid Interface Sci. 2005, 281, 458– 464 DOI: 10.1016/j.jcis.2004.07.038Google Scholar29Anisotropy in the wetting of rough surfacesChen, Yong; He, Bo; Lee, Junghoon; Patankar, Neelesh A.Journal of Colloid and Interface Science (2005), 281 (2), 458-464CODEN: JCISA5; ISSN:0021-9797. (Elsevier)Surface roughness amplifies the water-repellency of hydrophobic materials. If the roughness geometry is, on av., isotropic then the shape of a sessile drop is almost spherical and the apparent contact angle of the drop on the rough surface is nearly uniform along the contact line. If the roughness geometry is not isotropic, e.g., parallel grooves, then the apparent contact angle is no longer uniform along the contact line. The apparent contact angles obsd. perpendicular and parallel to the direction of the grooves are different. A better understanding of this problem is crit. in designing rough superhydrophobic surfaces. The primary objective of this work is to det. the mechanism of anisotropic wetting and to propose a methodol. to quantify the apparent contact angles and the drop shape. A theor. and an exptl. study of wetting of surfaces with parallel groove geometry is reported.
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Abstract
Figure 1
Figure 1. Graphical representation of all possible droplet states as seen from the top view.
Figure 2
Figure 2. Geometry of an elliptical patch defined by semiaxes a and b, with Young’s contact angle θ1 inside the patch and θ2 outside the patch.
Figure 3
Figure 3. Equilibrium shapes (blue grid) of a nanodroplet with increasing volume on an elliptical patch (red area) with aspect ratio b/a = 0.7, as obtained from Monte Carlo calculations. The respective right figures show a top view of the three-phase contact line (blue line).
Figure 4
Figure 4. Position of contact line R(ϕ) and local contact angle θ(ϕ) of the nanodroplet wetting an elliptical patch with aspect ratio b/a = 0.4 as a function of viewing angle ϕ and for varying droplet volumes as labeled. We show R(ϕ) and θ(ϕ) (a, b) for the nanodroplet in state A, (c, d) for the nanodroplet in state B, and (e, f) for the nanodroplet in state D. Solid lines show the results from the Surface Evolver calculation, and the markers present results from the Monte Carlo calculations.
Figure 5
Figure 5. Expansion of R(ϕ) into a harmonic series as described in eq 7 for different values of the volume of the nanodroplet (V/a3) and different aspect ratios of the elliptical patch: (a) b/a = 0.4, (b) b/a = 0.7, and (c) b/a = 1. The results are presented in the same way as in Figure 4: solid lines correspond to Surface Evolver calculations, and the markers correspond to the Monte Carlo calculations.
Figure 6
Figure 6. State diagram of the droplet morphologies as a function of aspect ratio b/a of the elliptical patch (with inside patch contact angle θ1 = 30 and outside patch contact angle θ2 = 60) and reduced volume V/a3 of the nanodroplet. Solid lines present volume thresholds calculated in the Surface Evolver calculation, and dashed lines (in good agreement with the solid lines) present Monte Carlo results.
References
This article references 29 other publications.
- 1Marmur, A. Contact Angle Hysteresis on Heterogeneous Smooth Surfaces J. Colloid Interface Sci. 1994, 168, 40– 46 DOI: 10.1006/jcis.1994.13911Contact angle hysteresis on heterogeneous smooth surfacesMarmur, AbrahamJournal of Colloid and Interface Science (1994), 168 (1), 40-6CODEN: JCISA5; ISSN:0021-9797.An oscillatory dependence of the highest and lowest possible contact angles on the vol. of a drop is demonstrated for a simple model of a 2-dimensional (cylindrical) drop placed on a periodically heterogeneous but smooth solid surface. This type of dependence may explain the macroscopic jumps obsd. during measurements of contact angle hysteresis. In addn., the present calcns. reveal the possibility of dependence of the advancing and receding contact angles on the drop vol., in qual. agreement with available exptl. data. The effect of asymmetry in placing the drop relative to the surface heterogeneity pattern is also discussed and is an important factor in the interpretation of contact angle hysteresis.
- 2de Gennes, P. G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves; Springer: New York, 2004.There is no corresponding record for this reference.
- 3Quere, D. Wetting and roughness Annu. Rev. Mater. Res. 2008, 38, 71– 99 DOI: 10.1146/annurev.matsci.38.060407.1324343Wetting and roughnessQuere, DavidAnnual Review of Materials Research (2008), 38 (), 71-99CODEN: ARMRCU; ISSN:1531-7331. (Annual Reviews Inc.)We discuss in this review how the roughness of a solid impacts its wettability. We see in particular that both the apparent contact angle and the contact angle hysteresis can be dramatically affected by the presence of roughness. Owing to the development of refined method for setting very well-controlled micro- or nanotextures on a solid, these effects are being exploited to induce novel wetting properties, such as spontaneous filmification, superhydrophobicity, superoleophobicity, and interfacial slip, that could not be achieved without roughness.
- 4de Gennes, P. G. Wetting: statics and dynamics Rev. Mod. Phys. 1985, 57, 827 DOI: 10.1103/RevModPhys.57.8274Wetting: statics and dynamicsDe Gennes, P. G.Reviews of Modern Physics (1985), 57 (3, Pt. 1), 827-63CODEN: RMPHAT; ISSN:0034-6861.The wetting of solids by liqs. is connected to phys. chem. (wettability), to statistical physics (pinning of the contact line, wetting transitions, etc.), to long-range forces (van der Waals, double layers), and to fluid dynamics. The present review represents an attempt towards a unified picture with special emphasis on certain features of "dry spreading": (1) the final state of a spreading droplet need not be a monomol. film; (2) the spreading drop is surrounded by a precursor film, where most of the available free energy is spent; and (3) polymer melts may slip on the solid and belong to a sep. dynamical class, conceptually related to the spreading of superfluids. Many refs.
- 5Herminghaus, S.; Brinkmann, M.; Seemann, R. Wetting and Dewetting of Complex Surface Geometries Annu. Rev. Mater. Res. 2008, 38, 101– 121 DOI: 10.1146/annurev.matsci.38.060407.1303355Wetting and dewetting of complex surface geometriesHerminghaus, Stephan; Brinkmann, Martin; Seemann, RalfAnnual Review of Materials Research (2008), 38 (), 101-121CODEN: ARMRCU; ISSN:1531-7331. (Annual Reviews Inc.)A review. Surfaces exhibiting complex topogs., such as those encountered in biol., give rise to an enormously rich variety of interfacial morphologies of a liq. to which they are exposed. In the present article, some basic mechanisms involved in the statics and dynamics of such morphologies were elaborated, focusing on a few simple paradigm topogs. Different liq. interface morphologies on the same sample frequently coexist. To exemplify the impact of the dynamics on the final droplet morphol., the shape instability of filamentous liq. structures in wedge geometries are discussed. Some side effects that may dominate on a larger scale, such as contact line pinning and contact angle hysteresis, seem to play a minor role on the microscopic scale under study. This establishes the validity of simple theor. concepts of wetting as a starting point for describing liqs. at substrate surfaces of high complexity.
- 6Rauscher, M.; Dietrich, S. Wetting Phenomena in Nanofluidics Annu. Rev. Mater. Res. 2008, 38, 143– 172 DOI: 10.1146/annurev.matsci.38.060407.1324516Wetting phenomena in nanofluidicsRauscher, M.; Dietrich, S.Annual Review of Materials Research (2008), 38 (), 143-172CODEN: ARMRCU; ISSN:1531-7331. (Annual Reviews Inc.)A review. We focus on the dynamical aspects of wetting phenomena on the nanoscale for which bulk hydrodynamic equations become invalid. At the nanoscale, phenomena that are irrelevant on the micrometer scale and larger, or that can be summarily incorporated in terms of boundary conditions, become important. Among these features are long-ranged mol. interactions such as dispersion forces, thermal fluctuations, hydrodynamic slip, segregation of mixts. and solns. at walls, and elec. double layers.
- 7Lohse, D.; Zhang, X. Surface Nanobubbles and Nanodroplets Rev. Mod. Phys. 2015, 87, 981– 1035 DOI: 10.1103/RevModPhys.87.9817Surface nanobubbles and nanodropletsLohse, Detlef; Zhang, XuehuaReviews of Modern Physics (2015), 87 (3), 981-1036CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)Surface nanobubbles are nanoscopic gaseous domains on immersed substrates which can survive for days. They were first speculated to exist about 20 years ago, based on stepwise features in force curves between two hydrophobic surfaces, eventually leading to the first at. force microscopy (AFM) image in 2000. While in the early years it was suspected that they may be an artifact caused by AFM, meanwhile their existence has been confirmed with various other methods, including through direct optical observation. Their existence seems to be paradoxical, as a simple classical est. suggests that they should dissolve in microseconds, due to the large Laplace pressure inside these nanoscopic spherical-cap-shaped objects. Moreover, their contact angle (on the gas side) is much smaller than one would expect from macroscopic counterparts. This review will not only give an overview on surface nanobubbles, but also on surface nanodroplets, which are nanoscopic droplets (e.g., of oil) on (hydrophobic) substrates immersed in water, as they show similar properties and can easily be confused with surface nanobubbles and as they are produced in a similar way, namely, by a solvent exchange process, leading to local oversatn. of the water with gas or oil, resp., and thus to nucleation. The review starts with how surface nanobubbles and nanudroplets can he made, how they can be obsd. (both individually and collectively), and what their properties are. Mol. dynamic simulations and theories to account for the long lifetime of the surface nanobubbles are then reported on. The crucial element contributing to the long lifetime of surface nanobubbles and nanodroplets is pinning of the three-phase contact line at chem. or geometric surface heterogeneities. The dynamical evolution of the surface nanobubbles then follows from the diffusion equation, Laplace's equation, and Henry's law. In particular, one obtains stable surface nanobubbles when the gas influx from the gas-oversatd. water and the outfiux due to Laplace pressure balance. This is only possible for small enough surface bubbles. It is therefore the gas or oil oversatn. ζ that dets. the contact angle of the surface nanobubble or nanodroplet and not the Young equation. The review also covers the potential technol. relevance of surface nanobubbles and nanodroplets, namely, in flotation, in (photo)catalysis and electrolysis, in nanomaterial engineering, for transport in and out of nanofluidic devices, and for plasmonic bubbles, vapor nanobubbles, and energy conversion. Also given is a discussion on surface nanobubbles and nanudroplets in a nutshell, including theor. predictions resulting from it and future directions. Studying the nucleation, growth, and dissoln. dynamics of surface nanobubbles and nanodroplets will shed new light on the problems of contact line pinning and contact angle hysteresis on the submicron scale.
- 8Mendez-Vilas, A.; Jodar-Reyes, A. B.; Gonzalez-Martin, M. L. Ultrasmall Liquid Droplets on Solid Surfaces: Production, Imaging, and Relevance for Current Wetting Research Small 2009, 5, 1366– 1390 DOI: 10.1002/smll.2008008198Ultrasmall liquid droplets on solid surfaces: production, imaging, and relevance for current wetting researchMendez-Vilas, Antonio; Jodar-Reyes, Ana Belen; Gonzalez-Martin, Maria LuisaSmall (2009), 5 (12), 1366-1390CODEN: SMALBC; ISSN:1613-6810. (Wiley-VCH Verlag GmbH & Co. KGaA)A review. The investigation of micro- and nanoscale droplets on solid surfaces offers a wide range of research opportunities both at a fundamental and an applied level. On the fundamental side, advances in the techniques for prodn. and imaging of such ultrasmall droplets will allow wetting theories to be tested down to the nanometer scale, where they predict the significant influence of phenomena such as the contact line tension or evapn., which can be neglected in the case of macroscopic droplets. On the applied side, these advances will pave the way for characterizing a diverse set of industrially important materials such as textile or biomedical micro- and nanofibers, powd. solids, and topog. or chem. nanopatterned surfaces, as well as micro-and nanoscale devices, with relevance in diverse industries from biomedical to petroleum engineering. Here, the basic principles of wetting at the micro- and nanoscales are presented, and the essential characteristics of the main exptl. techniques available for producing and imaging these droplets are described. In addn., the main fundamental and applied results are reviewed. The most problematic aspects of studying such ultrasmall droplets, and the developments that are in progress that are thought to circumvent them in the coming years, are highlighted.
- 9Soltman, D.; Smith, B.; Kang, H.; Morris, S. J. S.; Subramanian, V. Methodology for Inkjet Printing of Partially Wetting Films Langmuir 2010, 26, 15686– 15693 DOI: 10.1021/la102053j9Methodology for Ink-Jet Printing of Partially Wetting FilmsSoltman, Dan; Smith, Ben; Kang, Hongki; Morris, S. J. S.; Subramanian, VivekLangmuir (2010), 26 (19), 15686-15693CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)Ink-jet printing of precisely defined structures is crit. for the realization of a range of printed electronics applications. We develop and demonstrate a methodol. to optimize the ink-jet printing of two-dimensional, partially wetting films. When printed inks have a pos. retreating contact angle, we show that any fixed spacing is ineffective for printing two-dimensional features. With fixed spacing, the bead contact angle begins large, leading to a bulging overflow of its intended footprint. Each addnl. line reduces the bead contact angle, eventually leading to sepn. of the bead. We propose a printing scheme that adjusts the line-to-line spacing to maintain a bead's contact angle between its advancing and retreating values as it is printed. Implementing this approach requires an understanding of the two-dimensional bead surface and compensation for evapn. during the print. We derive an analytic equation for the bead's surface with pinned contact lines and use an empirical fit for mass loss due to evapn. Finally, we demonstrate that enhanced contact angle hysteresis, achieved by preprinting a feature's border, leads to better corner definition.
- 10Lee, T.; Charrault, E.; Neto, C. Interfacial slip on rough, patterned and soft surfaces: A review of experiments and simulations Adv. Colloid Interface Sci. 2014, 210, 21– 38 DOI: 10.1016/j.cis.2014.02.01510Interfacial slip on rough, patterned and soft surfaces: A review of experiments and simulationsLee, Thomas; Charrault, Eric; Neto, ChiaraAdvances in Colloid and Interface Science (2014), 210 (), 21-38CODEN: ACISB9; ISSN:0001-8686. (Elsevier B.V.)Advancements in the fabrication of microfluidic and nanofluidic devices and the study of liqs. in confined geometries rely on understanding the boundary conditions for the flow of liqs. at solid surfaces. Over the past ten years, a large no. of research groups have turned to investigating flow boundary conditions, and the occurrence of interfacial slip has become increasingly well-accepted and understood. While the dependence of slip on surface wettability is fairly well understood, the effect of other surface modifications that affect surface roughness, structure and compliance, on interfacial slip is still under intense investigation. In this paper we review investigations published in the past ten years on boundary conditions for flow on complex surfaces, by which we mean rough and structured surfaces, surfaces decorated with chem. patterns, grafted with polymer layers, with adsorbed nanobubbles, and superhydrophobic surfaces. The review is divided in two interconnected parts, the first dedicated to phys. expts. and the second to computational expts. on interfacial slip of simple (Newtonian) liqs. on these complex surfaces. Our work is intended as an entry-level review for researchers moving into the field of interfacial slip, and as an indication of outstanding problems that need to be addressed for the field to reach full maturity.
- 11Clausen, B. S.; Schiøtz, J.; Gråbæk, L.; Ovesen, C. V.; Jacobsen, K. W.; Nørskov, J. K.; Topsøe, H. Wetting/ non-wetting phenomena during catalysis: Evidence from in situ on-line EXAFS studies of Cu-based catalysts Top. Catal. 1994, 1, 367– 376 DOI: 10.1007/BF0149228911Wetting/non-wetting phenomena during catalysis: evidence from in situ online EXAFS studies of Cu-based catalystsClausen, Bjerne S.; Schioetz, Jakob; Graabaek, Lars; Ovesen, Charlotte V.; Jacobsen, Karsten W.; Noerskov, Jens K.; Tops.danisho.e, HenrikTopics in Catalysis (1994), 1 (3,4, Frontiers in Catalysis: Ammonia Synthesis and Beyond), 367-76CODEN: TOCAFI; ISSN:1022-5528. (Baltzer)We have used in situ EXAFS to provide exptl. evidence for a reversible change in the morphol. of the metallic particles in a high surface area, porous catalyst system, Cu/ZnO, contg. small metallic copper particles. By changing the oxidn. potential in the synthesis gas mixt., it is found that the apparent Cu-Cu coordination no. changes in an essentially reversible manner suggesting that the small metallic Cu particles dynamically change morphol. This indicates that a wetting/non-wetting phenomenon takes place in the Cu/ZnO system with changing partial pressures of oxygen. Under similar conditions, these effects are not obsd. when copper is supported on, for example, SiO2. A model based on surface and interface free energies provides a simple explanation for the obsd. results. Since the wetting/non-wetting processes are accompanied by a change in the active surface area, the obsd. behavior has important general implications and such effects must be incorporated into microkinetic models in order to provide a proper description of the catalyst performance.
- 12Lenz, P.; Lipowsky, R. Morphological Transitions of Wetting Layers on Structured Surfaces Phys. Rev. Lett. 1998, 80, 1920– 1923 DOI: 10.1103/PhysRevLett.80.192012Morphological transitions of wetting layers on structured surfacesLenz, Peter; Lipowsky, ReinhardPhysical Review Letters (1998), 80 (9), 1920-1923CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)The morphol. of wetting layers on structured or imprinted surfaces is detd. by the geometry of the underlying surface domains. Droplets which cover a single domain exhibit contact angles which do not satisfy Young's equation. For surface patterns consisting of many surface domains, the wetting layer exhibits several distinct morphologies (homogeneous droplet patterns, heterogeneous droplet patterns, film states) and may undergo morphol. transitions between these different states. The latter transitions exhibit spontaneous symmetry breaking.
- 13Brandon, S.; Haimovich, N.; Yeger, E.; Marmur, A. Partial wetting of chemically patterned surfaces: The effect of drop size J. Colloid Interface Sci. 2003, 263, 237– 243 DOI: 10.1016/S0021-9797(03)00285-613Partial wetting of chemically patterned surfaces: The effect of drop sizeBrandon, Simon; Haimovich, Nir; Yeger, Einat; Marmur, AbrahamJournal of Colloid and Interface Science (2003), 263 (1), 237-243CODEN: JCISA5; ISSN:0021-9797. (Elsevier Science)Partial wetting of chem. heterogeneous substrates is simulated. Three-dimensional sessile drops in equil. with smooth surfaces supporting ordered chem. patterns are considered. Significant features are obsd. as a result of changing the drop vol. The no. of equilibrated drops is found either to remain const. or to increase with growing drop vol. The shape of larger drops appears to approach that of a spherical cap and their three-phase contact line seems, on a larger scale, more circular in shape than that of smaller drops. In addn., as the vol. is increased, the av. contact angle of drops whose free energy is lowest among all equil.-shaped drops of the same vol. appears to approach the angle predicted by Cassie. Finally, contrary to results obtained with two-dimensional drops, contact angle hysteresis obsd. in this system is shown to exhibit a degree of vol. dependence in the advancing and receding angles. Qual. differences in the wetting behavior assocd. with the two different chem. patterns considered here, as well as differences between results obtained with two-dimensional and three-dimensional drops, can possibly be attributed to variations in the level of constraint imposed on the drop by the different patterns and by the dimensionality of the system.
- 14Marmur, A.; Bittoun, E. When Wenzel and Cassie Are Right: Reconciling Local and Global Considerations Langmuir 2009, 25, 1277– 1281 DOI: 10.1021/la802667b14When Wenzel and Cassie Are Right: Reconciling Local and Global ConsiderationsMarmur, Abraham; Bittoun, EyalLangmuir (2009), 25 (3), 1277-1281CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)The condition under which the Wenzel or Cassie equation correctly ests. the most stable contact angle is reiterated and demonstrated: these equations do hold when the drop size is sufficiently large compared with the wavelength of roughness or chem. heterogeneity. The numerical demonstrations somewhat mimic recent expts. that seemingly refuted the Wenzel and Cassie equations and show that these expts. were performed only for drops of sizes similar in order of magnitude to the wavelength of roughness or chem. heterogeneity. Under such conditions, the Wenzel and Cassie equations are a priori not expected to be valid. It is also explained that both the local equil. condition at the contact line and the global equil. condition involving the wetted area within the contact line are necessary and complementary.
- 15Gao, L.; McCarthy, T. J. How Wenzel and Cassie Were Wrong Langmuir 2007, 23, 3762– 3765 DOI: 10.1021/la062634a15How Wenzel and Cassie Were WrongGao, Lichao; McCarthy, Thomas J.Langmuir (2007), 23 (7), 3762-3765CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)We argue using exptl. data that contact lines and not contact areas are important in detg. wettability. Three types of two-component surfaces were prepd. that contain "spots" in a surrounding field: a hydrophilic spot in a hydrophobic field, a rough spot in a smooth field, and a smooth spot in a rough field. Water contact angles were measured within the spots and with the spot confined to within the contact line of the sessile drop. Spot diam. and contact line diam. were varied. All of the data indicate that contact angle behavior (advancing, receding, and hysteresis) is detd. by interactions of the liq. and the solid at the three-phase contact line alone and that the interfacial area within the contact perimeter is irrelevant. The point is made that Wenzel's and Cassie's equations are valid only to the extent that the structure of the contact area reflects the ground state energies of contact lines and the transition states between them.
- 16McHale, G. Cassie and Wenzel: Were They Really So Wrong? Langmuir 2007, 23, 8200– 8205 DOI: 10.1021/la701116716Cassie and Wenzel: Were They Really So Wrong?McHale, G.Langmuir (2007), 23 (15), 8200-8205CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)A review of the theor. basis of the Cassie-Baxter and Wenzel equations. The properties of superhydrophobic surfaces are often understood by ref. to the Cassie-Baxter and Wenzel equations. Recently, in a paper deliberately entitled to be provocative, it was suggested that these equations are wrong; a suggestion said to be justified using exptl. data. It is argued that these models are not so much wrong as have assumptions that define the limitations on their applicability and that with suitable generalization they can be used with surfaces possessing some types of spatially varying defect distributions. The relation is discussed of the models to the previously published expts. and using min. energy considerations review the derivations of the equations for surfaces with defect distributions. It is argued that this means the roughness parameter and surface area fractions are quantities local to the droplet perimeter and that the published data can be interpreted within the models. Versions of the Cassie-Baxter and Wenzel equations involving roughness and Cassie-Baxter solid fraction functions local to the three-phase contact line on the assumption that the droplet retains an av. axisymmetry shape are derived. Moreover, it is indicated that, for superhydrophobic surfaces, the definition of droplet perimeter does not necessarily coincide with the three-phase contact line. As a consequence, the three-phase contact lines within the contact perimeter beneath the droplet can be important in detg. the obsd. contact angle on superhydrophobic surfaces.
- 17Brakke, K. A. The Surface Evolver Exp. Math. 1992, 1, 141– 165 DOI: 10.1080/10586458.1992.10504253There is no corresponding record for this reference.
- 18Brakke, K. A.Surface Evolver Manual, version 2.70; 2013; http://facstaff.susqu.edu/brakke/evolver/downloads/manual270.pdf.There is no corresponding record for this reference.
- 19Soligno, G.; Dijkstra, M.; van Roij, R. The equilibrium shape of fluid-fluid interfaces: Derivation and a new numerical method for Young’s and Young-Laplace equations J. Chem. Phys. 2014, 141, 244702 DOI: 10.1063/1.490439119The equilibrium shape of fluid-fluid interfaces: Derivation and a new numerical method for Young's and Young-Laplace equationsSoligno, Giuseppe; Dijkstra, Marjolein; van Roij, ReneJournal of Chemical Physics (2014), 141 (24), 244702/1-244702/13CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Many phys. problems require explicit knowledge of the equil. shape of the interface between two fluid phases. Here, we present a new numerical method which is simply implementable and easily adaptable for a wide range of problems involving capillary deformations of fluid-fluid interfaces. We apply a simulated annealing algorithm to find the interface shape that minimizes the thermodn. potential of the system. First, for completeness, we provide an anal. proof that minimizing this potential is equiv. to solving the Young-Laplace equation and the Young law. Then, we illustrate our numerical method showing two-dimensional results for fluid-fluid menisci between vertical or inclined walls and curved surfaces, capillary interactions between vertical walls, equil. shapes of sessile heavy droplets on a flat horizontal solid surface, and of droplets pending from flat or curved solid surfaces. Finally, we show illustrative three-dimensional results to point out the applicability of the method to micro- or nano-particles adsorbed at a fluid-fluid interface. (c) 2014 American Institute of Physics.
- 20Matsui, H.; Noda, Y.; Hasegawa, T. Hybrid Energy-Minimization Simulation of Equilibrium Droplet Shapes on Hydrophilic/Hydrophobic Patterned Surfaces Langmuir 2012, 28, 15450– 15453 DOI: 10.1021/la303717n20Hybrid Energy-Minimization Simulation of Equilibrium Droplet Shapes on Hydrophilic/Hydrophobic Patterned SurfacesMatsui, Hiroyuki; Noda, Yuki; Hasegawa, TatsuoLangmuir (2012), 28 (44), 15450-15453CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)The authors have developed an efficient algorithm for simulating the equil. shape of a microdroplet placed on a flat substrate that has a fine, discontinuous, and arbitrarily shaped hydrophilic/hydrophobic patterned surface. The method uses a hybrid energy-minimization technique that combines the direct search method to det. the droplet shape around solid/liq. contact lines with the gradient descent method for the other parts of the droplet surface. The method provides high-convergence at a low computational cost with sufficient mesh resoln., providing a useful tool for the optimal design of printed electronic devices.
- 21Dupuis, A.; Léopoldès, J.; Bucknall, D. G.; Yeomans, J. M. Control of drop positioning using chemical patterning Appl. Phys. Lett. 2005, 87, 024103 DOI: 10.1063/1.1984098There is no corresponding record for this reference.
- 22Extrand, C. W. Contact Angles and Hysteresis on Surfaces with Chemically Heterogeneous Islands Langmuir 2003, 19, 3793– 3796 DOI: 10.1021/la026835022Contact angles and hysteresis on surfaces with chemically heterogeneous islandsExtrand, C. W.Langmuir (2003), 19 (9), 3793-3796CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)Wetting behavior was studied on surfaces with a single, circular heterogeneous island. Lyophobic islands were created on lyophilic Si wafers using polystyrene. Alternately, lyophobic perfluoroalkoxy fluoropolymer film was etched to make lyophilic domains. Contact angles and hysteresis were measured with water and hexadecane. Small sessile drops were deposited on the center of an island and liq. was sequentially added, eventually forcing the contact line to advance beyond the island perimeter onto the surrounding area. Even though the underlying contact area contained a mixt. of lyophilic and lyophobic domains, the contact angles, both advancing and receding, were equal to the angles exhibited by the homogeneous periphery. Or in other words, if the heterogeneity was completely contained with the contact area and did not intersect the contact line, then no area averaging of the contact angles occurred. These findings suggest that interactions at the contact line, not the contact area, control wetting of heterogeneous surfaces.
- 23Darhuber, A. A.; Troian, S. M.; Wagner, S. Morphology of liquid microstructures on chemically patterned surfaces J. Appl. Phys. 2000, 87, 7768– 7775 DOI: 10.1063/1.37345223Morphology of liquid microstructures on chemically patterned surfacesDarhuber, Anton A.; Troian, Sandra M.; Miller, Scott M.; Wagner, SigurdJournal of Applied Physics (2000), 87 (11), 7768-7775CODEN: JAPIAU; ISSN:0021-8979. (American Institute of Physics)The authors study the equil. conformations of liq. microstructures on flat but chem. heterogeneous substrates using energy minimization computations. The surface patterns, which establish regions of different surface energy, induce deformations of the liq.-solid contact line. Depending on the geometry, these deformations either promote or impede capillary breakup and bulge formation. The contact angles of the liq. on the hydrophilic and hydrophobic regions, as well as the pattern geometry and vol. of liq. deposited, strongly affect the equil. shapes. Moreover, due to the small scale of the liq. features, the presence of chem. or topol. surface defects significantly influence the final liq. shapes. Preliminary expts. with arrays of parallel hydrophilic strips produce shapes resembling the simulated forms. These encouraging results provide a basis for the development of high resoln. lithog. by direct wet printing.
- 24Brinkmann, M.; Lipowsky, R. Wetting morphologies on substrates with striped surface domains J. Appl. Phys. 2002, 92, 4296– 4306 DOI: 10.1063/1.150600324Wetting morphologies on substrates with striped surface domainsBrinkmann, Martin; Lipowsky, ReinhardJournal of Applied Physics (2002), 92 (8), 4296-4306CODEN: JAPIAU; ISSN:0021-8979. (American Institute of Physics)The wetting and dewetting of chem. structured substrates with striped surface domains is studied theor. The lyophilic stripes and the lyophobic substrate are characterized by different contact angles θγ and θδ, resp. We det. the complete bifurcation diagram for the wetting morphologies (i) on a single lyophilic stripe and (ii) on 2 neighboring stripes sepd. by a lyophobic one. Long channels can only be formed on the lyophilic stripes if the contact angle θγ is smaller than a certain threshold value θch(V) which depends only weakly on the vol. V and attains the finite value θch(∞) in the limit of large V. This asymptotic value is equal to θch(∞) = aρc = g/cm3os(π/4)≃38° for all lyophobic substrates with θδ ≥ π/2. For a given value of θγ<θch(∞), the extended channels spread onto the lyophilic stripes with essentially const. cross section.
- 25Ferraro, D.; Semprebon, C.; Tóth, T.; Locatelli, E.; Pierno, M.; Mistura, G.; Brinkmann, M. Morphological Transitions of Droplets Wetting Rectangular Domains Langmuir 2012, 28, 13919– 13923 DOI: 10.1021/la302854t25Morphological Transitions of Droplets Wetting Rectangular DomainsFerraro, Davide; Semprebon, Ciro; Toth, Tamara; Locatelli, Emanuele; Pierno, Matteo; Mistura, Giampaolo; Brinkmann, MartinLangmuir (2012), 28 (39), 13919-13923CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)The authors report the results of comprehensive expts. and numerical calcns. of interfacial morphologies of H2O confined to the hydrophilic top face of rectangular posts of width W = 500 μm and lengths between L = 5W and 30W. A continuous evolution of the interfacial shape from a homogeneous liq. filament to a bulged filament and back is obsd. during changes in the liq. vol. Above a certain threshold length of L* = 16.0 W, the transition between the 2 morphologies is discontinuous and a bistability of interfacial shapes is obsd. in a certain interval of the reduced liq. vol. V/W3.
- 26Jansen, H. P.; Bliznyuk, O.; Kooij, E. S.; Poelsema, B.; Zandvliet, H. J. W. Simulating Anisotropic Droplet Shapes on Chemically Striped Patterned Surfaces Langmuir 2012, 28, 499– 505 DOI: 10.1021/la203962526Simulating Anisotropic Droplet Shapes on Chemically Striped Patterned SurfacesJansen, H. Patrick; Bliznyuk, Olesya; Kooij, E. Stefan; Poelsema, Bene; Zandvliet, Harold J. W.Langmuir (2012), 28 (1), 499-505CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)The equil. shape of droplets on surfaces, functionalized with stripes of alternating wettability, have been investigated using simulations employing a finite element method. Expts. show that a droplet deposited on a surface with relatively narrow hydrophobic stripes compared to the hydrophilic stripes adopts a strongly elongated shape. The aspect ratio, the length of the droplet divided by the width, decreases toward unity when a droplet is deposited on a surface with relatively narrow hydrophilic stripes. The aspect ratio and the contact angle parallel to the stripes show unique scaling behavior as a function of the ratio between the widths of the hydrophobic and hydrophilic stripes. For a small ratio, the contact angle parallel to the stripes is low and the aspect ratio high, while for a large ratio, the contact angle parallel is high and the aspect ratio low. The simulations exhibit similar scaling behavior, both for the aspect ratio of the droplets and for the contact angles in the direction parallel to the stripes. Two liqs. with different surface tensions have been investigated both exptl. and in simulations; similarities and differences between the findings are discussed. Generally, three parameters are needed to describe the droplet geometry: (i) the equil. contact angles on the hydrophilic and (ii) hydrophobic areas and (iii) the ratio of the widths of these chem. defined stripes. Furthermore, we derive a simple anal. expression that proves to be a good approxn. in the quant. description of the droplet aspect ratio.
- 27Woodward, J. T.; Gwin, H.; Schwartz, D. Contact Angles on Surfaces with Mesoscopic Chemical Heterogeneity Langmuir 2000, 16, 2957– 2961 DOI: 10.1021/la991068z27Contact angles on surfaces with mesoscopic chemical heterogeneityWoodward, J. T.; Gwin, H.; Schwartz, D. K.Langmuir (2000), 16 (6), 2957-2961CODEN: LANGD5; ISSN:0743-7463. (American Chemical Society)The contact angle of water was measured on surfaces composed of random hydrophilic and hydrophobic patches with typical length scales of 10-100 nm. By quenching self-assembled monolayers at various stages of growth, the fractional surface coverage of the hydrophobic patches was varied in the range 0.04-0.97 as detd. by AFM. The cosine of the contact angle of water cos θ was systematically lower than the prediction of the mean field Cassie equation cos θC. The deviation from this prediction cos θC - cos θ had an approx. linear dependence on the total contour length between hydrophobic and hydrophilic patches (a measure of the degree of heterogeneity). The contact angle was insensitive to droplet size, suggesting that line tension effects were minimal. Also, contact angles of hexadecane were in good agreement with the Cassie prediction. Two possible explanations for the obsd. behavior are proposed. Long-range (approx. 5 nm) hydrophobic interactions may result in a relatively hydrophobic boundary region around each hydrophobic patch which effectively increases the coverage of the hydrophobic phase altering the equil. contact angle. Alternatively, an increased d. of "pinning" sites may prevent the contact angle from relaxing to the equil. value.
- 28Seemann, R.; Brinkmann, M.; Kramer, E. J.; Lange, F. F.; Lipowsky, R. Wetting morphologies at microstructured surfaces Proc. Natl. Acad. Sci. U. S. A. 2005, 102, 1848– 1852 DOI: 10.1073/pnas.040772110228Wetting morphologies at microstructured surfacesSeemann, Ralf; Brinkmann, Martin; Kramer, Edward J.; Lange, Frederick F.; Lipowsky, ReinhardProceedings of the National Academy of Sciences of the United States of America (2005), 102 (6), 1848-1852CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)The wetting of microstructured surfaces is studied both exptl. and theor. Even relatively simple surface topogs. such as grooves with rectangular cross section exhibit a large variety of different wetting morphologies as obsd. by at. force microscopy. This polymorphism arises from liq. wedge formation along the groove corners and from contact line pinning along the groove edges. A global morphol. diagram is derived that depends only on two system parameters: (i) the aspect ratio of the groove geometry and (ii) The contact angle of the underlying substrate material. For microfluidics, the most interesting shape regimes involve extended liq. filaments, which can grow and shrink in length while their cross section stays essentially const. Thus, any method by which one can vary the contact angle can be used to switch the length of the filament, as is demonstrated in the context of electrowetting.
- 29Chen, Y.; He, B.; Lee, J.; Patankar, N. A. Anisotropy in the wetting of rough surfaces J. Colloid Interface Sci. 2005, 281, 458– 464 DOI: 10.1016/j.jcis.2004.07.03829Anisotropy in the wetting of rough surfacesChen, Yong; He, Bo; Lee, Junghoon; Patankar, Neelesh A.Journal of Colloid and Interface Science (2005), 281 (2), 458-464CODEN: JCISA5; ISSN:0021-9797. (Elsevier)Surface roughness amplifies the water-repellency of hydrophobic materials. If the roughness geometry is, on av., isotropic then the shape of a sessile drop is almost spherical and the apparent contact angle of the drop on the rough surface is nearly uniform along the contact line. If the roughness geometry is not isotropic, e.g., parallel grooves, then the apparent contact angle is no longer uniform along the contact line. The apparent contact angles obsd. perpendicular and parallel to the direction of the grooves are different. A better understanding of this problem is crit. in designing rough superhydrophobic surfaces. The primary objective of this work is to det. the mechanism of anisotropic wetting and to propose a methodol. to quantify the apparent contact angles and the drop shape. A theor. and an exptl. study of wetting of surfaces with parallel groove geometry is reported.