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Capillary Forces between Chemically Different Substrates E. J. De Souza,*,†,‡ M. Brinkmann,§ C. Mohrdieck,|,# A. Crosby,⊥ and E. Arzt†,‡ Max Planck Institute for Metals Research, Heisenbergstr. 3, D-70569 Stuttgart, Germany, INM Leibniz Institute for New Materials GmbH, Campus D2 2, 66123 Saarbruecken, Germany, Max Planck Institute for Dynamics and Self-Organisation, Bunsenstr.10, 37073 Go¨ttingen, Germany, Institute of Physical Metallurgy, UniVersity of Stuttgart, Heisenbergstr. 3, D-70569, Stuttgart, Germany, and Polymer Science and Engineering Department, UniVersity of Massachusetts Amherst, 120 GoVernors DriVe, Amherst, Massachusetts ReceiVed March 4, 2008. ReVised Manuscript ReceiVed June 6, 2008 Motivated by experimental results, we present numerical and analytical calculations of the capillary force exerted by a capillary bridge spanning the gap between two parallel flat plates of asymmetric wettabiltiy. Depending on whether the sum of the two contact angles is smaller or larger than 180°, the capillary force is either attractive or repulsive at small separations D between the plates. In either cases the magnitude of the force diverges as D approaches zero. The leading order of this divergence is captured by an analytical expression deduced from the geometry of the meniscus of a flat capillary bridge. The results for substrates with different wettability reveal an interesting behavior: with the sum of the contact angles fixed, the magnitude of the capillary force and the rupture separation decreases as the asymmetry in contact angles is increased. In addition, we present the rupture separation, i.e., the maximal extension of a capillary bridge, as a function of the contact angles. Our results provide an extensive picture of surface wettability effects on capillary adhesion.
1. Introduction The force exerted by a capillary bridge between two substrates was investigated extensively for a variety of conditions in the context of different substrate geometries,1–5 capillary condensation,6–12 and atomic force microscopy.13–15 Close inspection of the literature about capillary force reveals that the effect of contact angle hysteresis and in particular asymmetric wettability has not been investigated in sufficient detail. Recent experimental studies demonstrate that contact angle hysteresis has a major impact on the measurement of forces.16 In this paper, we address the more general problem of stability and force-separation curves of liquid bridges wetting two plane substrates of unlike wettability. * Corresponding author. E-mail:
[email protected]. † Max Planck Institute for Metals Research. ‡ INM Leibniz Institute for New Materials GmbH. § Max Planck Institute for Dynamics and Self-Organisation. | University of Stuttgart. ⊥ University of Massachusetts Amherst. # Present address: EADS Deutschland GmbH, Defence Electronics, Woerthstrasse 85, 89077 Ulm, Germany.
(1) Derjaguin, B. Kolloid-Zeitschrift 1934, 69(2), 155–164. (2) Clark, W. C.; Haynes, J. M.; Mason, G. Chem. Eng. Sci. 1968, 23(7), 810. (3) Orr, F. M.; Scriven, L. E.; Rivas, A. P. J. Fluid Mech. 1975, 67(Feb25), 723–742. (4) Willett, C. D.; Adams, M. J.; Johnson, S. A.; Seville, J. P. K. Langmuir 2000, 16(24), 9396–9405. (5) Farshchi-Tabrizi, M.; Kappl, M.; Cheng, Y. J.; Gutmann, J.; Butt, H. J. Langmuir 2006, 22(5), 2171–2184. (6) Fisher, L. R.; Israelachvili, J. N. Colloids Surf. 1981, 3(4), 303–319. (7) Fisher, L. R.; Israelachvili, J. N. J. Colloid Interface Sci. 1981, 80(2), 528–541. (8) Christenson, H. K. J. Colloid Interface Sci. 1988, 121(1), 170–178. (9) Kohonen, M. M.; Maeda, N.; Christenson, H. K. Phys. ReV. Lett. 1999, 82(23), 4667–4670. (10) Kohonen, M. M.; Christenson, H. K. Langmuir 2000, 16(18), 7285–7288. (11) Maeda, N.; Israelachvili, J. N. J. Phys. Chem. B 2002, 106(14), 3534– 3537. (12) Maeda, N.; Israelachvili, J. N.; Kohonen, M. M. Proc. Natl. Acad. Sci. U.S.A. 2003, 100(3), 803–808. (13) Stifter, T.; Marti, O.; Bhushan, B. Phys. ReV. B 2000, 62(20), 13667– 13673. (14) Malotky, D. L.; Chaudhury, M. K. Langmuir 2001, 17(25), 7823–7829. (15) Zitzler, L.; Herminghaus, S.; Mugele, F. Phys. ReV. B 2002, 66, (15). (16) De Souza, E. J.; Gao, L.; McCarthy, T. J.; Arzt, E.; Crosby, A. J. Langmuir 2008, 24(4), 1391–1396.
In the absence of gravity, the liquid-air interface of a mechanically stable liquid bridge is a surface of constant mean curvature. This observation allows construction of equilibrium configurations of rotationally symmetric bridges from these classes of surfaces. The appropriate boundary condition on the substrate, e.g., a given contact angle on the surface, selects a number of solution branches. Typically, the stability of the solutions belonging to a certain branch is unknown and has to be inferred from further considerations. A particularly simple wetting geometry of high symmetry is given by two opposed and chemically homogeneous flat plates held at a fixed separation. Vogel17,18 and Langbein19 formulated sufficient conditions for the stability of rotationally symmetric capillary bridges in this geometry for arbitrary but identical contact angles. According to their work, a capillary bridge is stable if (i) the enclosed volume increases with increasing Laplace pressure and (ii) the contour of the bridge has no inflection point. However, these conditions are not necessarily fulfilled for a mechanically stable bridge such that a complete classification of stable and unstable solutions requires the knowledge of the full set of solution branches. For the case of two plates with equal contact angles, this classification has been performed by Fortes,20 Carter,21 and later by Langbein.22 Local stability of the symmetric solution branch is lost either at a turning point corresponding to a maximal separation at fixed liquid volume, or at a bifurcation point involving two branches whose solutions exhibit one inflection point and break the up-down symmetry. The latter scenario is encountered for contact angles larger than 31.2° while the loss of stability at maximal separation applies to contact angles below this value. Assuming a fixed volume, Fortes and Carter determined the range of separations where mechanically stable capillary (17) Vogel, T. I. SIAM J. Appl. Math. 1987, 47(3), 516–525. (18) Vogel, T. I. SIAM J. Appl. Math. 1989, 49(4), 1009–1028. (19) Langbein, D. MicrograV. Sci. Technol. 1992, 1(5), 2–11. (20) Fortes, M. A. J. Colloid Interface Sci. 1982, 88(2), 338–352. (21) Carter, W. C. Acta Metallurgica 1988, 36(8), 2283–2292. (22) Langbein, D., Capillary Surfaces; Springer: Berlin, 2002; pp 15-16, 52-56, 119-129.
10.1021/la800680n CCC: $40.75 2008 American Chemical Society Published on Web 08/13/2008
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Figure 1. Image of a liquid bridge of volume V ≈ 0.5 µL between different substrates with θ1 ) 108° and θ2 ) 60°, recorded during the measurement of a force-displacement curve. This picture illustrates that the surface profile of the bridge cannot easily be approximated by a sphere.
bridges exist and presented, in addition to a stability diagram, force-separation curves for a variety of contact angles. Recent experimental investigations16 of a capillary bridge between two flat plates stress the need for further investigations of the effect of different contact angles on the capillary force. The experimental force-separation curves for plates of unlike wettability show strong deviations from the behavior predicted for identical contact angles. In this work, we present the results of numerical and analytical calculations for this asymmetric situation extending those presented by Fortes and Carter. In order to cross check our results we compared the analytically obtained solutions with numerical minimizations of the interfacial energy. Finally, we compare the calculations with experimental results.
2. Methods Experimental. We used the same experimental setup described in ref16 to measure the force-separation curves of capillary bridges between substrates of different wettability. A drop of deionized water was placed on a lower substrate and moved in the direction of an upper substrate connected to a cantilever. The deflection of the cantilever was measured when a liquid bridge formed. The force-separation curve was obtained from the deflection measurements while pulling the substrates apart and pushing them toward each other. Numerical Minimization. As depicted in Figure 1, we assume two perfectly plane, rigid, inert, and chemically homogeneous substrates with different contact angles θ1 and θ2, a separation D of the substrates and a constant liquid volume V. Under these conditions, we obtain mechanically stable bridges as local minima of the interfacial energy E given by the sum
E ) γlvAlv + (γls1 - γvs1)Als1 + (γls2 - γvs2)Als2
(1)
In eq 1, γij is the interfacial energy per unit area related to the interface between the bulk phases i and j being the wetting liquid (l), the ambient air (v), and the two substrates (s1) and (s2). The respective surface areas are denoted by Aij. For energy minimization we use the software package Surface EVolVer34 that has been successfully applied to analyze axially symmetric bridges between equal plates,16 liquid systems among three particles,23 configurations of the liquid contact line on patterned super hydrophobic substrates,24 and multiple liquid bridges.25 With the help of the energy (23) Hilden, J. L.; Trumble, K. P. J. Colloid Interface Sci. 2003, 267(2), 463– 474. (24) Dorrer, C.; Ruhe, J. Langmuir 2007, 23(6), 3179–3183. (25) De Souza, E. J.; Mohrdieck, C.; Brinkmann, M.; Arzt, E. Enhancement of capillary forces by multiple liquid bridges. LangmuirAccepted for publication, 2008.
minimization, it is possible to calculate the capillary force as a function of separation. To directly compare our results with previous work, we use the same normalized quantities as in refs,21,25 i.e., we introduce a normalized separation d ) D/s, a normalized radius of the liquid-solid contact r(d) ) R(d)/s, a normalized interfacial energy e(d) ) E(d)/(γlvs2), and a normalized capillary force f(d) ) F(d)/ (2πγlvs), where s) (3V/4π)1/3 is the radius of a fictional liquid sphere of the same volume V as the liquid bridge. The capillary force is obtained as the first derivative of the interfacial energy with respect to separation. Analytical Calculation. The shape of an axially symmetric capillary bridge can be mathematically described by a surface of revolution, i.e., by rotating a plane curve. For this particular case, the condition of constant mean curvature can be integrated leading to the periodic two parameter family of Delaunay curves26,27 (see Appendix A1). As shown in a previous publication by Souza et al.,16,25 the capillary forces acting on the plates can be extracted from the Laplace pressure of the bridge, the contact angle, and the radius of the surface being wet by the liquid. Applying this method, one arrives at a parametric representation of the force-separation curves. The more general case of unlike contact angles is treated in an almost identical manner. Breaking the up down symmetry of boundary conditions, however, changes the topology of the solution branches qualitatively. In particular, the bifurcation scenario found for equal contact angles no longer exists (a locally stable branch of symmetric capillary bridges bifurcates into two unstable asymmetric branches and an unstable symmetric branch). Instead, two pairs of solution branches are found. The two branches belonging to a pair merge in a turning point and from an open loop. As corroborated by comparison to the results of the numerical minimizations, only one out of the four solution branches corresponds to locally stable capillary bridges. Small separation allows approximation of the meridional profile of a capillary bridge by a circular arc and calculation of the capillary force from the Laplace pressure and the area of the wet substrate. Details of this procedure are shown in Appendix A2. As the plate to plate separation D approaches zero, the contribution of the Laplace pressure to the capillary forces F dominates over the contribution of surface tension, and we may write in rescaled units
f(θ1, θ2, d) )
2(cos θ1 + cos θ2) 2
3d
+ O(d-1⁄2)
(2)
This equation is valid only for d , 1, i.e., only if the diameter of a capillary bridge is much larger than the separation between the plates.
3. Results Effect of Contact Angle Asymmetry. For later reference, we will begin with a short summary of the findings for the symmetric situation. Figure 2 presents previously published curves of normalized force versus normalized plate separation for identical contact angles (for details, see refs 21,25). In this case, forces are attractive (positive) for contact angles θ e 90°, i.e., hydrophilic surfaces, and increase for decreasing values of θ. The opposite behavior is seen for θ > 90°, on hydrophobic surfaces. Attractive forces can be observed for all contact angles at sufficiently large separations. The maximal separation for stable liquid bridges (point of rupture) is largest for θ ) 90° and decreases monotonically above and below. The results of these forceseparation curves (symbols) from numerical minimizations are in excellent agreement with the analytically calculated ones (solid lines)21 over the entire range of separations and for all contact angles. Let us now turn to the case of unlike contact angles. Figure 3 shows normalized force-separation curves for fixed θ1) 30°, (26) Delaunay, C. E. J. Math. Pure Appl. 1841, 6, 309–315. (27) Roe, R. J. J. Colloid Interface Sci. 1975, 50(1), 70–79.
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Figure 2. Review of results calculated in refs.21,25 Capillary force versus displacement displayed for different but identical contacts angles θ ) θ1 ) θ2. Hydrophilic surfaces (contact angles θ e 90°) lead to attractive forces (positive) while hydrophobic ones (contact angles θ > 90°) to predominantly repulsive forces (negative). Comparison between analytical (solid curves) and numerical (squares) results demonstrates the excellent accuracy of the numerical model.
60°, 120°, and 150° (a, b, c and d). If one substrate is hydrophilic, cf., θ1 ) 60° (Figure 3b), the capillary force can be attractive over the whole range of separations even if the second substrate is hydrophobic in the range 90° e θ2 e 120°. The contact angle asymmetry thus results in an extension of the attractive region in the force-separation diagram up to contact angles θ2 ) 120°. Analogously, one hydrophobic substrate (θ1)120°, Figure 3c) can produce repulsive interactions at small separations even if the second substrate is hydrophilic with a contact angle θ2 > 60°. At large separations close to the point of rupture, however, we observe an attractive capillary force. This asymmetry in the size of the attractive and repulsive regions becomes even more pronounced if the first substrate is either strongly hydrophilic or strongly hydrophobic as shown in Figure 3a,d. The results of force-separation curves obtained from numerical minimizations (symbols) are in excellent agreement with the analytically calculated curves (solid lines) over the entire range of separations and contact angles. A comparison of the analytical results to numerical minimizations showed that the locally stable branch of capillary bridges always corresponds to solutions which do not display an inflection point in their contour at small separations. This branch of solutions is stable up to a turning point as can be seen in force-separation curves: the attractive force decreases shortly before reaching the point of rupture. In the range of contact angles above 30° and for small differences between the contact angles, the infinite negative slope of the curve at the point of rupture can hardly be noticed. The case θ1 + θ2 ) 180°, i.e., cos(θ1) + cos(θ2) ) 0, leads to a singular behavior of the parametric curves which describe the contour of the bridges, as it happens for the symmetric situation at θ1 ) θ2 ) 90°. Therefore, these special cases were solved only numerically (symbols), as, for example, in Figure 3a for θ1 ) 30° and θ2 ) 150°. Complementary to the results shown in Figure 3, Figure 4 displays the analytical results (solid and dashed lines) around the numerically solved singularity (symbols) for θ1 ) 30°, θ2 ) 150° (a) and θ1 ) 60°, θ2 ) 120° (b). The analytical force-separation curves behave similarly to the results for the symmetric situation with contact angles θ ) 89°, 91°, as shown in Figure 2. Forces are attractive over the entire range of separations if at least one of the contact angles is smaller than any of the angles which lead to the singularity, as, for example, θ1 ) 30°, θ2 ) 149° (Figure 4a) and θ1 ) 60°, θ2 ) 119° (Figure
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4b). These situations correspond to θ ) 89° in the case of identical contact angles. Otherwise, the forces are initially repulsive for small separations and become attractive with increasing separation similar to the symmetric case with contact angle θ ) 91°. Asymptotic Behavior. While Figures 3 and 4 focus on the behavior of the force for finite separations up to the point of rupture, the asymptotic behavior for d f 0 is shown in Figure 5. Here, we calculate the force for the same contact angles θ1 as in Figure 3b,c and vary θ2 in steps of ∆θ2 ) 10° for better resolution. The results are displayed as a color map whose axes are the normalized separation d and the second contact angle θ2; the colors correspond to different force values for normalized separations ranging from d ) 0.001 to 0.05. The magnitude of the capillary force approaches infinity if d approaches zero for all combinations of contact angle. The black lines represent isolines of equal force where the black line labeled “0” corresponds to the isoline for f ) 0 that separates the regions of attractive and repulsive forces. Figure 5a clearly demonstrate that for θ1 ) 60° the region of attractive forces dominates over the region of repulsive forces, while the opposite holds for θ1 ) 120°. Using eq 2, we calculated the asymptotic behavior of the force separation curves for similar combinations of contact angles as in Figure 3. We compared the accuracy of the asymptotic expression (2) used to the force at finite separations via the magnitude of the fractional difference between the exact (analytically or numerically) calculated force and the asymptotic solution (Figure 6). The intersections between the relative error with the dashed blue line show the normalized separation at which the asymptotic expression reaches the accuracy of 10%. In general, for keeping an accuracy of approximately 10%, d should not exceed the limit of roughly 0.4. For a fixed hydrophilic contact angle, this maximal normalized separation decreases with increasing contact angle and the opposite is true for fixed hydrophobic contact angles. Figure 7a displays the numerically calculated normalized forces (symbols) as a function of θ1 and θ2 for a fixed normalized separation d ) 0.05, θ1 in steps of 30° and θ2 in steps of 10°. As expected from the particular form of expression (2), the function f(θ1, θ2) ) C[cos(θ1)+cos(θ2)] with a suitable constant C (solid lines in Figure 5a) matches the numerical results very well. This relationship was first postulated by O’Brien et al.28 for the case of a liquid bridge between a sphere and plate. The curves in Figure 7a can be better visualized as a color map shown in Figure 7b which displays the normalized force (colors) for all permutations of the contact angles. Here, we vary θ1 and θ2 in steps of ∆θ ) 10° for better visualization. In this map, the diagonal black line marks the force for the condition f ) 0. As already mentioned, the capillary force becomes attractive close to the point of rupture. With the exclusion of the case of complete dewetting on either plate (θ1 ) θ2 ) 180°), this applies to all combinations of contact angles. Figure 8a illustrates the rupture separations for (θ1 ) θ2) calculated via different methods while Figure 8b illustrates the rupture separations for many combinations of contact angles as obtained from our numerical calculations. For a given contact angle on one substrate, the normalized rupture separation is maximum when θ1 equals θ2. With increasing contact angle asymmetry, the rupture separation decreases monotonically in a manner that cannot be fitted with a linear relation, power law function, or other basic functional dependencies. Application of the Model to Experiments. We investigated the effect of contact angle asymmetry by conducting measurements of the capillary force on the force-separation curves in (28) O’Brien, W. J.; Hermann, J. J. J. Adhes. 1973, 5(2), 91–103.
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Figure 3. Four examples of asymmetric configuration, where θ1 is fixed while θ2 is varied. A hydrophilic substrate with θ1 ) 60° (b) can lead to entirely attractive forces even if the second substrate is hydrophobic (θ2 ) 120°). Conversely, a hydrophobic substrate with θ1 ) 120° (c) can lead to predominantly repulsive forces even if the second substrate is hydrophilic (θ2 ) 90°). These effects are more pronounced if the first substrate is superhydrophilic (a) or superhydrophobic (d). In these cases, the force remains attractive or repulsive even for large contact angle asymmetries. Also, here the comparison between our analytical (solid curves calculated via extended Carter) and numerical (squares) results demonstrate the excellent accuracy of the numerical model. Note that singularities occur via analytical calculations for θ1 + θ2 ) 180° and therefore only the numerical curves are displayed (see text for details).
Figure 4. Complementary analytical solutions of the results shown in Figure 3 for force-distance curves (solid and dashed lines) near the numerically calculated singularity (circles).
dynamic experiments. In these experiments,16 the contact angle asymmetry was generated by using different planar, rigid substrates spanned by a liquid bridge. During the dynamic force measurements, the liquid bridge was either continuously stretched or compressed at constant velocity. The advancing and receding contact angles were measured via the evaporation method as described in ref29 and were then used as input parameter for the numerical model. Liquid volumes were determined from the
images of the droplets prior to the formation of a liquid bridge and after its rupture. The velocity used to move the bottom plate relative to the top plate was set to V ) 10 [µm/s], which ensured that evaporation of an aqueous volume of about 0.5 [µL] was negligible at low relative humidity of 18%. (29) Gao, L. C.; McCarthy, T. J. Langmuir 2006, 22(14), 6234–6237.
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the different shapes of the liquid bridge during approach and separation.
4. Discussion
Figure 5. Normalized capillary force for d f 0. The black lines depict lines of equal force, where the label depicts the value of the force. The white region marks the asymptotic behavior of the force f f (∞. In (a), the attractive region of the force dominates the force-map, while in (b), it is the repulsive region that dominates. The singularity occurs in (a) and in (b) for θ1 + θ2 ) 180°.
Figure 9a-c presents measured force-separation curves (red lines) for three different sets of plates. In each of these experiments, the branch of the curve marked “advancing curve” corresponds to a compression of the liquid bridge, i.e., the plates are moved toward one another. In this situation, the measured equilibrium contact angles at the two plates correspond to the advancing angles. The branch marked “receding curve” represents a liquid bridge that is stretched by pulling the two plates apart from each other and the corresponding contact angles are receding angles. Four different angles characterize one dynamic experiment: two advancing and two receding angles for each of the two plates. The black and blue curves in each figure represent exact calculated force-separation curves that have been fitted to the advancing and receding branches of the measured curve, respectively, by adjusting the values of the contact angles θ1 and θ2 in each fit. The fit curves in Figure 9a show that both substrates are hydrophilic and their advancing contact angles correspond to 50° and 75°, while their receding contact angles correspond to 30° and 40°, which results in a weak hysteresis of the measured force-distance curve. A similar effect occurs in Figure 9b. However, a more pronounced hysteresis can be seen in Figure 9c, where the advancing contact angles are approximated by 60° and 125° and the receding contact angles by 40° and 110°. Recorded images during the experiments are inserted to illustrate
Capillary Force for Substrates with Different Wettability. From the results presented in section 3.2, the question arises, why is the meniscus force attractive for certain combinations of contact angles? In general, a dominantly attractive or repulsive force between the plates at small separations arises from the negative or positive Laplace pressure, respectively. However, the contribution to the force due to Laplace pressure becomes weak at large separations while the attractive contribution of surface tension becomes dominant. This phenomenon is wellknown and has been explained for the case of identical plates in refs.20,21,25 In the case of two plates of unlike wettability, this explanation is still valid: the sign of the Laplace pressure in the limit of small separations, i.e., flat capillary bridges, is negative if the sum of the two contact angles is smaller than 180°. This can be seen from the shape of the liquid meniscus, which is curved inward. In this case, the capillary force is always attractive. If the sum of the contact angles is larger than 180°, the meniscus of a flat capillary bridge bends outward and the positive Laplace pressure inside the bridge leads to a repulsive force. However, at larger separation the attractive contribution of surface tension comes into play and changes the sign of the force to attractive close to the point of rupture. Furthermore, the force-separation curve for substrates with different contact angles cannot be simply obtained by averaging both angles and applying the force-separation curve for equal contact angles, as has been done for example in ref.30 Comparison between the force-separation curve for θ ) 90° (displayed in Figure 2) with the curves for the same average (displayed in Figure 4a,b) clearly shows that the magnitude and the rupture point of these force-separation curves are different. The separation of rupture attains its maximum for the symmetric case and decreases with an increasing degree of asymmetry. A similar observation can be made for the capillary force in the limit of small separations: the absolute value of the force decreases as the asymmetry in the contact angles is increased. In general, the complete force-separation curves cannot be easily anticipated for arbitrarily given contact angles. The arcapproximation proposed previously in ref28 to calculate the complete force-separation curve between a sphere and a plate was used in later studies, as, for example, in refs 30,31 without verifying its applicability. Our results for two parallel plates demonstrate that eq 2 derived from the arc-approximation correctly predicted the asymptotic behavior for d f 0. However, expression 2 becomes inaccurate for large separations. The solution of these cases is otherwise given by a parametric curve that cannot be solved for f as a function of d. Comparison between Calculations and Experimental Results. The results in Figures 3 are useful in determining the magnitude contact angle hystereses. Direct measurement of contact angles can be difficult, especially for surfaces with large contact angles. A drop of water on such a surface does not adhere but tends to stick to the needle of the dispenser or run down the inclined surface. Therefore, the method proposed in ref16 to measure the contact angle hysteresis from the hysteresis of the capillary force can be extended to arbitrary combinations of substrates and is not limited to situations where they are identical. (30) Rabinovich, Y. I.; Adler, J. J.; Esayanur, M. S.; Ata, A.; Singh, R. K.; Moudgil, B. M. AdV. Colloid Interface Sci. 2002, 96(1-3), 213–230. (31) Gao, C. Appl. Phys. Lett. 1997, 71(13), 1801–1803.
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Figure 6. Fractional difference between asymptotic and analytically calculated force-distance curves near d goes to 0 for different combinations of contact angles (a-d). The curves show the percentage of discrepancy depending on the normalized distance between the substrates. The dashed blue lines at 10% indicate a limit of accuracy of the asymptotic solution.
Additionally, the estimate of the rupture of liquid bridges shown in Figure 8b is crucial for capillary force experiments, because it helps to define the suitable range of separations necessary for a complete force-separation curve. Also, the stability of n-liquid bridges25 of fixed total volume is determined by the point of rupture of a single bridge. Our results in Figure 9a,b,c shows that we can reproduce the measured force-separation curve analytically and by numerical minimization of the interfacial energy with high precision using the liquid volume and both advancing angles and both receding angles of the substrates as input parameter. This demonstrates that the computed force separation curves can accurately assess situations in which the contact angles are (i) different and (ii) at the same time display a hysteresis. Although this is the case in most experimental situations, there are not many studies that address both issues simultaneously. Lambert and Delchambre,32 for example, compare the results of their computer simulation with experimental data for the very specific case of polydimethylsiloxane-oil between a sphere and a plate with contact angles θ1 ) 16° and θ2 ) 12°, but they do not discuss the contact angle hysteresis nor the stability of the bridge. Similar to the experiments performed in ref,16 we emphasize that comparisons between capillary force hysteresis and contact angle hysteresis in our current approach can only be obtained (32) Lambert, P.; Delchambre, A. Langmuir 2005, 21(21), 9537–9543.
if the contact line of a liquid bridge, or drop, spreads and contracts in a radially symmetric way on the substrate. This occurs when the chemistry and topography of the surface are both homogeneous. Contact angles measured via two-dimensional images of a drop will not correspond to the contact angles extracted from force measurements if the shape of the contact line is not spherical on a randomly structured or contaminated surface. Under these conditions, the contact line can pin for a moment in one place leading to more than one advancing or receding contact angle which fits the force-separation curve. For these materials, the kinetics becomes essential for a complete description of the capillary bridge force-separation curve and consequently full characterization of the substrates. It is advisable to use one substrate with well-known hysteresis, which can be easily measured if the plates are identical.16 This procedure allows the advancing and receding contact angles and thus the contact angle hysteresis to be determined for any arbitrary planar substrates. In a biological context, the large magnitude of the capillary force for small distances may be the reason why some animals33 that apply wet adhesion develop an adhesion force that is usually much smaller than predicted by theoretical models. These animals seem to favor relatively large distances between their attachment organs and the substrate, which results in a moderatelysstill (33) Barnes, W. J. P.; Oines, C.; Smith, J. M. J. Comp. Physiol., A: 2006, 192(11), 1179–1191. (34) Brakke, K. The Surface EVolVer, version 2.14; 1999.
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Figure 8. (a) Comparison of rupture distances of a liquid bridge between substrates with equal contact angles obtained by four different methods. (b) Rupture distances of a liquid bridge between two plates with different contact angles obtained numerically with Surface EVolVer. Figure 7. Behavior of the normalized force for a fixed displacement d ) 0.05 and arbitrary contact angles θ1 and θ2. In (a), θ1 and θ2 vary in steps of 30° and 10°, respectively, while in (b) they both vary in steps of 10°. The black line in (b) represents the isoline f ) 0.
sufficientlyslarge adhesion force and at the same time ensures efficient detachment from the substrate. In these cases, the optimum distance is therefore the result of the conflicting interests of tight attachment to and fast detachment from the substrate.
• A strongly hydrophilic or hydrophobic substrate biases the capillary force toward attractive or repulsive values, respectively, even for large contact angle asymmetry. • A simple asymptotic expression exists that matches all calculated force-separation curves for normalized separation d < 0.4 within 10% accuracy and very well for df0. • Good agreement exists between our analytically predicted force-separation cuves and experimental results.
5. Conclusions In this work, we extend the calculations of capillary forces and rupture separations of liquid bridges between two chemically homogeneous and parallel plates to the case of unlike contact angles. The comparison between the results obtained via analytical calculations and numerical minimizations demonstrates that both procedures give identical results. The circular approximation used for the asymptotic regime has shown that this approximation holds only for very small separations. Therefore, it should be used to estimate the capillary force of plates in perfect contact and not for the prediction of capillary forces over the whole range of separations, as sometimes reported in the literature. The capillary force is attractive for any distance whenever the sum of the contact angles is smaller than or equal to 180°. If the sum is larger than 180°, the capillary force is repulsive for sufficiently small separations and is always attractive close to the point of rupture. Furthermore, we showed the following: • Capillary bridges between plates of equal contact angle are mechanically more stable than in asymmetric configurations.
Appendix A1. Analytical Computation. Provided effects of gravity are negligible, the liquid-gas interface of a liquid bridge is a surface of constant mean curvature. Given the contour r(z) of an axially symmetric liquid bridge in cylindrical coordinates, the mean curvature H of its liquid-gas interface is given by
H)
[
rzz 1 1 + 2 (1 + r 2)3⁄2 r(1 + r 2)1⁄2 z z
]
(A1)
Solutions of eq A1 are periodic into the z-direction and can be expressed in terms of complete elliptic integrals.27 These Delaunay curves are either curly and self-intersecting (nodoids) or wavy and nonintersecting (undoloids). Both classes of Delaunay curves are fixed by two parameters, the minimal radius rmin and the maximal radius rmax of the contour. The absolute value of the mean curvature of the corresponding surface of revolution is given by
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H)
1 rmax ( rmin
De Souza et al.
(A2)
where the positive sign refers to undoloids and the negative sign to nodoids. The sign of the mean curvature depends on the local orientation of the surface.22 In order to obtain the contour of liquid bridges between two plane and parallel plates of different wettability, one has to choose suitable segments of undoloids and nodoids which match the prescribed contact angles on either plates. Because the boundary condition of a fixed contact angle is scaling-invariant one may fulfill the subsidiary constraint of a given volume or a given separation between the plates by a rescaling of all dimensions. Displayed in the force-distance plane, the solution branches form loops whose upper and lower branches diverge to positive infinity or negative infinity if the sum of the contact angles is smaller or larger than 180°. Comparison with numerical minimizations shows that the upper branch represents locally or globally stable solutions while the lower branch refers to unstable solutions. A2. Asymptotic. In the following, we consider a flat capillary bridge held between parallel plates at separation D with contact
Figure 10. Sketch of the circular meniscus of a flat bridge between two parallel plates with contact angles θ1 and θ2.
angles θ1 and θ2, cf. Figure 10. The diameter of the wetted area is large compared to the plate separation D. Because the curvature of the liquid-air interface perpendicular to the symmetry axis is small compared to the curvature parallel to the symmetry axis, the meridional contour of the bridge is approximated well by a circular arc. Since the auxiliary angles ε1 and ε2 are equal (ε1 ) ε2 ) ε), we find three relations between the contact angles θ1, θ2, and the auxiliary angles, φ1, φ2, and ε:
π ) φ1 + ε + θ1
(A3)
π ) φ2 + ε + θ2
(A4)
π ) φ1 + φ2
(A5)
From the three equations above, it follows that
π (θ1 - θ2) 2 2 π (θ2 - θ1) φ2 ) 2 2 π (θ1 + θ2) ε) 2 2 φ1 )
(A6) (A7) (A8)
The radius of curvature of the meniscus contour, R, becomes
R)
D D D ) ) 2 sin φ1 sin ε 2 sin φ2 sin ε (cos θ1 + cos θ2) (A9)
In the asymptotic limit of small distances D f 0, the contribution of surface tension to the total capillary force scales as D-1/2 and can be neglected compared to the contribution of the Laplace pressure. The capillary force is given by the product of the Laplace pressure
PL ) 2Hγlv )
γlv R
(A10)
and the area of the substrate being wet by the liquid, Als and reads:
F ) PLAls )
Figure 9. Measured hysteresis of the capillary force (red curves) and numerically calculated force-distance curves (black and blue curves). By fitting the calculated curves to the advancing and receding branches of the measured curve, the advancing and receding angles of the substrates can be determined. This is demonstrated for three substrates with a small (a and b) and large (c) contact angle hysteresis. The corresponding images beside the graphs illustrate the shape of the bridge for different separations. The red marked scale bar at the bottom left of each image 1 corresponds to 500 µm.
γlvV(cos θ1 + cos θ2) D2
+ O(D-1⁄2) (A11)
Acknowledgment. We thank Dr. K. Kalaitzidou, Dr. E. Chan, Dr. L. Gao, and Professor T. J. McCarthy for their technical support and helpful discussions. Professor B. Bhushan of Ohio State University, Columbus, Ohio, and Prof. S. Herminghaus of the Max Planck Institute of Dynamics and Self-Organization, Goettingen, Germany, have contributed with fruitful discussions. A.J.C. gratefully acknowledges financial support from NSFDMR CAREER Award 0 34 9078 LA702188T. LA800680N
