Langmuir 2008, 24, 10565-10568
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Effect of Capillary Pressure and Surface Tension on the Deformation of Elastic Surfaces by Sessile Liquid Microdrops: An Experimental Investigation Ramo´n Pericet-Ca´mara, Andreas Best, Hans-Ju¨rgen Butt, and Elmar Bonaccurso* Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany ReceiVed June 13, 2008. ReVised Manuscript ReceiVed August 11, 2008 Sessile liquid drops are predicted to deform an elastic surface onto which they are placed because of the combined action of the liquid surface tension at the periphery of the drop and the capillary pressure inside the drop. Here, we show for the first time the in situ experimental confirmation of the effect of capillary pressure on this deformation. We demonstrate micrometer-scale deformations made possible by using a low Young’s modulus material as an elastic surface. The experimental profiles of the deformed surfaces fit well the theoretical predictions for surfaces with a Young’s modulus between 25 and 340 kPa.
Introduction When a liquid drop with a volume on the order of 1 nL is deposited on a solid surface, gravity effects can be neglected, and it typically adopts a spherical cap shape. The macroscopic contact angle θ of the droplet with the substrate results from the balance of the interfacial free energies between solid and liquid, γSL, solid and vapor, γSV, and liquid and vapor, γLV. This balance is represented by Young’s equation1
cos θ )
γSV - γSL γLV
(1)
which is one of the most important expressions in the study of wetting phenomena.2 Equation 1 was derived by assuming perfectly smooth, homogeneous, and rigid solid surfaces.3,4 Whereas the influence of roughness and heterogeneity on wetting has been addressed extensively by experiment5 and theory,6-10 the influence of surface deformation has been investigated to a smaller extent. For macroscopic drops on soft surfaces, it has been predicted that Young’s equation remains valid, although it might break down for microscopic drops.11 It is known that an elastic surface bearing a sessile drop undergoes a deformation that depends on the properties of the materials involved and their surface interactions (Figure 1). Indeed, the sessile drop pulls the surface at the three-phase contact line (TPCL) while it pushes at its contact area as a consequence of its capillary or Laplace pressure.12-15 Rusanov13 developed analytical expressions for the complete profile of an elastic surface by the action of a sessile * Corresponding author. E-mail:
[email protected]. (1) Young, T. Philos. Trans. R. Soc., Ser. A 1805, 95, 65. (2) de Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827. (3) Johnson, R. E. J. Phys. Chem. 1959, 63, 1655–1658. (4) McNutt, J. E.; Andes, G. M. J. Chem. Phys. 1959, 30, 1300–1303. (5) Tuteja, A.; Choi, W.; Ma, M. L.; Mabry, J. M.; Mazzella, S. A.; Rutledge, G. C.; McKinley, G. H.; Cohen, R. E. Science 2007, 318, 1618–1622. (6) Cassie, A. B. D. Disc. Faraday Soc. 1948, 3, 11–16. (7) Drelich, J.; Miller, J. D.; Kumar, A.; Whitesides, G. M. Colloids Surf., A 1994, 93, 1–13. (8) Gao, L. C.; McCarthy, T. J. Langmuir 2007, 23, 3762–3765. (9) Huh, C.; Mason, S. G. J. Colloid Interface Sci. 1977, 60, 11–38. (10) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988–994. (11) Shanahan, M. E. R. J. Phys. D: Appl. Phys. 1987, 20, 945–950. (12) Lester, G. R. J. Colloid Sci. 1961, 16, 315–326. (13) Rusanov, A. I. Colloid J. USSR 1975, 37, 614–622. (14) Shanahan, M. E. R.; Degennes, P. G. Compt. Rend. Acad. Sci. 1986, 302, 517–521. (15) White, L. R. J. Colloid Interface Sci. 2003, 258, 82–96.
drop. He considered a stress vector P to be a combination of the surface tension at the TPCL and the capillary pressure at the liquid-solid interface. With this, he calculated the vector z(x, y) of the vertical displacement at all points of the surface by using the theory of elasticity.16 In a 1D representation (Figure 1), Rusanov calculated the surface profile z(r) for three surface sections: (i) the deformation underneath the drop (r e a); (ii) the deformation at the TPCL, where the TPCL is assumed to be a thin line of finite thickness t (a e r e a + t); and (iii) the deformation of the surface not covered by the drop (r g a + t)
{ ()
γLV sin θ r r 4(1 - ν2) ∆PaE + aE πE a t a r π(1 - 2ν) (a + t)E + γ ∆ cos θ a+t 4(1 - ν) LV rea (2a) 2 γLV sin θ a a 4(1 - ν ) ∆PrG + rG z(r) ) πE r t r π(1 - 2ν)γLV∆ cos θ r (a + t)E + (a + t - r) a+t 4(1 - ν)t aerea+t (2b) 2) γ sin θ ( a LV a a+t 4 1-ν ∆PrG + z(r) ) r G -G πE r t r r rga+t (2c) z(r) )
[ ()
( )]
{ () ( )]
{ ()
[ ()
}
}
[ ( ) ( )]}
where G(k t E(k) - (1 - k2)K(k), and E(k) and K(k) are the total normal elliptical Legendre integrals of the first and second kind, respectively, ∆ cos θ is the difference between the macroscopic contact angle and the equilibrium contact angle given by the Young’s equation, t is the so-called “thickness of the liquid-gas interfacial layer”, V is the Poisson’s ratio of the material, a is the contact radius of the drop, E is the Young’s modulus of the surface material, and ∆P is the Laplace pressure of the sessile drop. White’s model15 extended the latter theory by addressing the transmission of the surface tension stress to the substrate at the contact line on the microscopic scale and predicted a deviation of the microscopic contact angle from the macroscopic. The surface deformation according to these models is negligible for (16) Landau, L. D.; Lifshitz, E. M. Theory of Elasticity, 3rd ed; ButterworthHeinemann: Oxford, 1999; Vol. 7.
10.1021/la801862m CCC: $40.75 2008 American Chemical Society Published on Web 08/23/2008
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Figure 1. Scheme of the deformation of an elastic surface by a sessile droplet. (a) L, S, and V stand for liquid, solid, and gas. The dotted line displays the solid surface without deformation. θ is the equilibrium contact angle of the droplet, h is the maximum height of the ridge, and d is the maximum depth of the depression under the drop. (b) Magnified scheme of the contact point at the TPCL. The surface tension acts over a surface layer of thickness t.
hard materials with Young’s moduli above 1 GPa. However, for materials with Young’s moduli on the order of 100 kPa the deformation may reach several micrometers. For soft materials such as gels or biological tissues, the deformation might even be larger because the Young’s modulus is usually lower than 100 kPa. The presence of an elevated rim at the contact line of a sessile drop on an elastic surface has already been confirmed experimentally in situ17 and ex situ.18-20 This “wetting ridge” can modify the spreading dynamics of a liquid.17 However, the effect of the capillary pressure of the drop on the surface deformation still has not been measured, to the best of our knowledge, in any of the experiments in the literature. Herein, we investigate the deformation profile of an elastic surface by a sessile drop using laser scanning confocal microscopy (LSCM). Both the solid/vapor interface in the proximity of the contact line and the liquid/solid interface between the drop and the substrate are imaged and analyzed. The effect of the softness of the surface on the deformation is studied and compared with the calculations realized by Rusanov.13
Experimental Section In our model system, drops of 1-butyl-3-methylimidazolium hexafluorophosphate ionic liquid (Fluka, Switzerland) dyed with the fluorophore Nile red (Sigma-Aldrich, Germany) are deposited with a Nano-Plotter NP 2.0 (GeSiM GmbH, Germany) ink-jet dispenser. The ionic liquid is diluted in dimethylformamide (DMF) 30% w/w to decrease its viscosity. The dispenser, a mobile piezoelectric-driven microdosage pipet, cannot nominally emit liquids with a viscosity higher than 5 mPa · s. The drops are deposited on previously fabricated surfaces made of Sylgard 184 (Dow Corning), which is a silicone elastomeric polymer. The base product has to be cross-linked using a curing agent before use. The Sylgard is spin coated onto a thin round glass substrate (L ) 25 mm), and it is then cured in a 100 °C preheated oven for 45 min. We produced films with thicknesses from 23.7 to 27.9 µm for mixtures from 20:1 to 50:1 by weight of the base/curing agent. Within this thickness range, the influence of the stiff underlying substrate is negligible21,22 because the maximum vertical deformations that we expect are below 3 µm. Tensile tests were made on differently cross-linked Sylgard samples to evaluate the Young’s modulus E as a function of the cross-linking ratio. The tests showed that the cross-linked PDMS behaves purely elastically. Additional proof is that after the removal of the drops (17) Carre, A.; Gastel, J. C.; Shanahan, M. E. R. Nature 1996, 379, 432–434. (18) Extrand, C. W.; Kumagai, Y. J. Colloid Interface Sci. 1996, 184, 191– 200. (19) Pu, G.; Guo, J.; Gwin, L. E.; Severtson, S. J. Langmuir 2007, 23, 12142– 12146. (20) Saiz, E.; Tomsia, A. P.; Cannon, R. M. Acta Mater. 1998, 46, 2349–2361. (21) Tsukruk, V. V.; Sidorenko, A.; Gorbunov, V. V.; Chizhik, S. A. Langmuir 2001, 17, 6715–6719. (22) Cappella, B.; Silbernagl, D. Langmuir 2007, 23, 10779–10787.
Figure 2. Fluorescent drop on an elastic surface imaged by laser scanning confocal microscopy (LSCM). Red displays the fluorescent droplet, and the green line is the reflecting surface. (a) Schematic representation of a droplet containing a fluorescent dye on an elastic surface. The green and red arrows symbolize the reflected laser beam and the emitted fluorescence, respectively. (b) Experimental image of an ionic liquid droplet labeled with Nile red on a soft PDMS surface obtained with LSCM. The white dashed line displays the theoretical profile as calculated from Rusanov’s model in eq 2. Please note that the x/z length scale is 15:1.
the stresses are relieved and the surfaces flatten again. The nominal Poisson’s ratio V of our elastic samples is 0.5. We used simultaneous fluorescence and reflection laser scanning confocal microscopy (LSCM) (Carl Zeiss AG, Germany) using a He-Ne excitation laser at λ ) 543 nm to attain 3D images of the fluorescing ionic liquid droplet and the reflecting elastic surface (Figure 2a).
Results and Discussion Figure 2b shows the lateral image of a fluorescing drop with a contact radius of 33.5 µm and a contact angle of θ ) 60° sitting on PDMS with a Young’s modulus of E ) 25 kPa. The macroscopic contact angle is considered to be equal to the equilibrium contact angle because we did not observe a significant change in the contact angle of the ionic liquid drops on the four substrates that we considered. The surface tension of the ionic liquid is γL ) 48.8 mN/m.23 The confocal scan was realized around the solid surface to maximize resolution in the region of interest. From this image, the profile of the deformed surface was obtained. The thin green line shows the Gaussian distribution center of the light reflected at the PDMS surface that is not covered by the droplet. The reflected light at the liquid-solid interface between the drop and the PDMS surface is not visible because of their very similar refractive indexes. Nevertheless, the fluorescence of the dyed ionic liquid, shown here in red, in contrast with the nonfluorescing elastomer allows the measurement of the deformation underneath the droplet. The threshold (23) Huddleston, J. G.; Visser, A. E.; Reichert, W. M.; Willauer, H. D.; Broker, G. A.; Rogers, R. D. Green Chem. 2001, 3, 156–164. (24) Ridler, T. W.; Calvard, S. IEEE Trans. Syst. Man Cybernetics 1978, 8, 630–632.
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Figure 3. Maximum depth of a PDMS surface deformed by a sessile ionic liquid droplet versus the Young’s modulus of the bulk elastomer. The circles represent the maximum experimental depth d of the depression under the droplet. The solid circles represent the depression depths under droplets with a contact radius a of 33.0 ( 0.7 µm, and the open circles represent the depression depths under droplets with a contact radius a of 52.6 ( 1.2 µm. The error bars on the x axis are the maximum and minimum values of four different measurements of PDMS elasticity. Those on the y axis are the size of a pixel in the confocal scan images. The curve corresponds to values of the maximum depression depth calculated according to Rusanov’s model.
of the fluorescent image has been defined using the Isodata algorithm24 implemented in ImageJ analysis software (Wayne Rasband, National Institutes of Health) to allow us to locate the correct position of the liquid-solid interface. It shows that the bottom of the ionic liquid droplet is below the base level of the elastomer. This experimentally confirms for the first time that the elastic surface is deformed downward by the action of the capillary pressure of the droplet. The vertical distance between the bottom of the drop and the flat undeformed PDMS surface is 2.3 µm, which is established as the maximum depth of the depression. In the same graph (dashed line in Figure 2b), a plot of the theoretical deformation profile calculated from eq 2 is displayed for comparison. We fitted the experimental curves with Rusanov’s analytical model instead of using White’s model. In fact, with our experimental technique we cannot resolve the deformation profile in the close proximity of the three-phase contact line, and thus we cannot measure the microscopic contact angle. We used the above-mentioned experimental parameters of the PDMS and of the liquid in the calculation. We had only one free parameter, the thickness of the surface layer t. The best fits to the four experimental profiles yielded all t ) 16 nm. This value agrees in order of magnitude with h0 for electrolyte liquids in White’s theory, h0 ) 10 nm.15 h0 scales the extent of a precursor film in the microscopic region at the three-phase contact line. In this region, the disjoining pressure plays a role in the stress of the drop transmitted to the elastic surface. Nevertheless, the effect of the disjoining pressure on the surface deformation cannot be resolved by our experimental technique, and it has been neglected It is observed that the experimental profile of the deformed PDMS surface reproduces the theoretical one predicted by Rusanov fairly well. As a next step, the deformation of the PDMS surface as a function of the elastic properties was studied. The experimental maximum depth d at the center of the depressions decreased upon increasing the Young’s modulus of the PDMS as observed in Figure 3. The theoretical values of the depression depth in the center of the deformation are also plotted for comparison and show similar behavior. Experimental data agree well with those calculated with Rusanov’s model. One must bear in mind that the confocal scanning volume has a z extension of about 1 µm, which is in the range of the lowest attainable with LSCM. The
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Figure 4. Theoretical and experimental profiles of the deformed PDMS surface close to the TPCL for four different Young’s moduli E. The open red circles represent the experimental profiles obtained from a white-light confocal profilometer. The solid lines are the calculated profiles according to Rusanov’s model (eq 2).
effect of droplet size on the depression depth was also investigated. In Figure 3, measurements of the depression depth realized on the same substrate under drops of contact radii a of 33.0 ( 0.7 and 52.6 ( 1.2 µm are shown for comparison. For all of the investigated Young’s moduli, no effect of the droplet size on the depression depth was observed. The depression depth under drops of contact radii a between 33 and 117 µm at 25 kPa surfaces showed no dependence either, as predicted by Rusanov’s model. Furthermore, the rim outer profile close to the TPCL is studied as a function of the PDMS elasticity (E ) 25-338 kPa). Solid-vapor surface profiles of PDMS were acquired with a µsurf white-light confocal profilometer (Nanofocus AG, Germany) (Figure 4). The lower the Young’s modulus of PDMS, the larger the deformation and the higher the rim protrudes from the mean surface. The theoretical profile of the deformed surface calculated from eq 2 agrees with the measured profiles. Only the experimental peak heights were lower than the theoretical ones, which is a direct consequence of the limitation of reflection microscopy for imaging steep surfaces. Elements steeper than a certain threshold angle return the light beam out of the field of vision of the detector. The white-light confocal profilometer employed here is able to detect the specular reflection of surface slopes up to 36° as stated by the manufacturer, whereas reflection LSCM detected angles only up to about 5° in our measurements. Except for the hardest surface, the experimental gradients next to the top of the rims in Figure 3 are equal to the resolution threshold of the profilometer. Considering that the deformed surface slope theoretically increases with proximity to the TPCL, we can ensure that for these conditions the highest sections of the rim cannot be imaged with this apparatus. However, this drawback can be overcome by extrapolating the experimental values along the theoretical profile until reaching the contact line. The results of this study demonstrate that the stresses of a sessile droplet on an elastic soft surface, namely, its capillary pressure and its surface tension applied at the TPCL, generate a characteristic deformation profile with a craterlike shape. This profile is imaged and quantified here, and it fits with that calculated by continuum elastic theory. It is observed that the surface deformation increases for lower values of the Young’s modulus of bulk PDMS. Furthermore, it is confirmed that the depth of the depression is independent of drop size in the range of our analysis.
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The deformation of soft surfaces certainly influences wetting and dewetting on hydrogels, most biological tissues, and other soft materials. The softness of a surface enhances the hysteresis between the contact angles of an advancing and a receding wetting front,18 and it also slows down the spreading of a liquid on an elastic solid due to viscoelastic energy dissipation.17,25,26 These results are the first step in understanding these phenomena, and we intend to extend the experiments to dynamic systems. Furthermore, the method presented here may be a potential (25) Carre, A.; Shanahan, M. E. R. Langmuir 1995, 11, 24–26. (26) Carre, A.; Shanahan, M. E. R. J. Colloid Interface Sci. 1997, 191, 141– 145.
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technique for measuring the elasticity properties of such materials. This could be of special interest because the elasticity analysis of soft tissues is an important concern in medical diagnosis.27 Acknowledgment. We thank Kaloian Koynov for useful discussions and suggestions. E.B. acknowledges financial support from the Max-Planck Society (MPG), and R.P.-C. acknowledges the Swiss National Science Foundation (SNF) for funding (project PBGE2-112884). LA801862M (27) Samani, A.; Bishop, J.; Luginbuhl, C.; Plewes, D. B. Phys. Med. Biol. 2003, 48, 2183–2198.
