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When Sessile Drops Are No Longer Small: Transitions from Spherical to Fully Flattened C. W. Extrand* and Sung In Moon Entegris, Inc., 3500 Lyman Boulevard, Chaska, Minnesota 55318 Received February 3, 2010. Revised Manuscript Received June 1, 2010 We measured the dimensions and contact angles of sessile drops using three liquids on a variety of polymer and silicon surfaces. Drops ranged in size from a few microliters to several milliliters. With increasing liquid volume, heights of the drops initially rose steeply and then gradually tapered to a constant value. The heights of small, undistorted drops as well as the heights of the largest drops were accurately predicted by well-established models. A recently derived expression for meniscus height was used to estimate the heights of intermediate-size drops. While it was not exact, this expression produced reasonable approximations without having to resort to iterative numerical methods. We also identified transition points where gravity began to distort drop shape and ultimately limited drop height. Relatively simple closed analytical expressions for estimating these transition points were also derived. Predicted values of the height and volume at the onset of distortion agreed fairly well with the measured ones. Contact angles carefully measured by the tangent method were independent of drop size.
Introduction 1,2
Sessile liquid drops are widely used to assess wettability. Usually, a small drop is gently deposited on a horizontal solid surface. A small amount of liquid may be added to or withdrawn from the drop to advance or retract its contact line, and then, a contact angle (θ), depicted in Figure 1, is measured between the solid surface and a tangent line originating from the triple point between the liquid, solid, and surrounding gas. Contact angles can also be measured indirectly from drop dimensions. However, choosing the correct model requires knowledge of the drop shape. *Tel: 952-556-8619. E-mail:
[email protected]. (1) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; Wiley: New York, 1990. (2) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Science, 3rd ed.; CRC Press: New York, 1997. (3) Bashforth, F.; Adams, J. C. An Attempt to Test the Theories of Capillary Action By Comparing the Theoretical and Measured Forms of Drops of Fluid; University Press: Cambridge, England, 1883. (4) Ferguson, A. On the Theoretical Shape of Large Bubbles and Drops, with Other Allied Problems. Phil. Mag. 1913, Ser. 6, 25, 507-520. (5) Ellefson, B. S.; Taylor, N. W. Surface Properties of Fused Salts and Glasses: I. Sessile-drop Method for Determining Surface Tension and Density of Viscous Liquids at High Temperatures. J. Am. Ceram. Soc. 1938, 21, (1), 193-205. (6) Blaisedell, B. E. The Physical Properties of Fluid Interfaces of Large Radius of Curvature. I. Integration of Laplace’s Equation for the Equilibrium Meridian of a Fluid of Axial Symmetry in a Gravitational Field. Numerical Integration and Tables for Sessile Drops of Moderately Large Size. J. Math. Phys. 1940, 19, (1), 186-216. (7) Staicopolus, D. N., The Computation of Surface Tension and of Contact Angle by the Sessile-drop Method. J. Colloid Sci. 1962, 17, (5), 439-447. (8) Butler, J. N.; Bloom, B. H. A Curve-Fitting Method for Calculating Interfacial Tension from the Shape of a Sessile Drop. Surf. Sci. 1966, 4, (1), 1-17. (9) Maze, C.; Burnet, G. A Non-linear Regression Method for Calculating Surface Tension and Contact Angle from the Shape of a Sessile Drop. Surf. Sci. 1969, 13, (2), 451-470. (10) Huh, C.; Scriven, L. E. Shapes of Axisymmetric Fluid Interfaces of Unbounded Extent. J. Colloid Interface Sci. 1969, 30, (3), 323-337. (11) Padday, J. F. Surface Tension. Part II. The Measurement of Surface Tension. In Surface and Colloid Science, Matijevic, E., Ed.; Wiley: New York, 1969; Vol. 1, pp 101-149. (12) Princen, H. M. The Equilibrium Shape of Interfaces, Drops and Bubbles. Rigid and Deformable Particles at Interfaces. In Surface and Colloid Science, Matijevic, E., Ed.; Wiley: New York, 1969; Vol. 2, pp 1-84. (13) Johnson, R. E., Jr.; Dettre, R. H. Wettability and Contact Angles. In Surface and Colloid Science, Matijevic, E., Ed.; Wiley: New York, 1969; Vol. 2, pp 85-153.
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Unfortunately, there is no simple means to predict the transition of drop shape from small and spherical to large and completely flattened. One must generally rely on complex iterative numerical calculations.3-23 Therefore, in this study, we measured drop dimensions and contact angles of progressively larger sessile drops. We compared measured values of small, intermediate, and large drops to those estimated from established and recently published models. We also try to answer the following questions. How small must a sessile drop be so that gravity does not distort its shape? How big must a sessile drop be so that gravity limits its height?
Theory Consider the sessile liquid drop shown in Figure 1. The drop resides on a flat, horizontal solid surface. For a given volume (V ), it exhibits a height of h, base diameter of 2a, and contact angle (14) Graves, D. J.; Merrill, E. W.; Smith, K. A.; Gilliland, E. R. Cinematographic Method for Measurement of Rapidly Changing Surface Tension-area Functions. J. Colloid Interface Sci. 1971, 37, (2), 303-311. (15) Padday, J. F.; Pitt, A. Axisymmetric Meniscus Profiles. J. Colloid Interface Sci. 1972, 38, (2), 323-334. (16) Padday, J. F. Sessile Drop Profiles: Corrected Methods for Surface Tension and Spreading Coefficients. Proc. R. Soc. London, Ser. A 1972, 330, (1583), 561572. (17) Brown, R. A.; Orr, F. M., Jr.; Scriven, L. E., Static Drop on an Inclined Plate: Analysis by the Finite Element Method. J. Colloid Interface Sci. 1980, 73, (1), 76-87. (18) Huh, C.; Reed, R. L. A Method for Estimating Interfacial Tensions and Contact Angles from Sessile and Pendant Drop Shapes. J. Colloid Interface Sci. 1983, 91, (2), 472-484. (19) Dimitrov, A. S.; Kralchevsky, P. A.; Nikolov, A. D.; Noshi, H.; Matsumoto, M. Contact Angle Measurements with Sessile Drops and Bubbles. J. Colloid Interface Sci. 1991, 145, (1), 279-282. (20) Chatterjee, J. Limiting Conditions for Applying the Spherical Section Assumption in Contact Angle Estimation. J. Colloid Interface Sci. 2003, 259, (1), 139-147. (21) Lin, M.-W. Y. a. S.-Y. A Method for Correcting the Contact Angle from the θ/2 Method. Colloids Surf., A 2003, 220, (1-3), 199-210. (22) Shapiro, B.; Moon, H.; Garrell, R. L.; Kim, C.-J. Equilibrium Behavior of Sessile Drops under Surface Tension, Applied External Fields, and Material Variations. J. Appl. Phys. 2003, 93, (9), 5794-5811. (23) Lanchon-Ducauquis, H.; Biguenet, F.; Liraud, T.; Csapo, E.; Guemouri, Y. Modelling Wetting Behavior. In Continuum Thermomechanics, Springer: Netherlands: 2002; Vol. 76, pp 197-208.
Published on Web 06/16/2010
DOI: 10.1021/la1005133
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Article
Extrand and Moon
Figure 1. Illustration of a sessile liquid that depicts its height (h),
a small cylinder using an equation from Ferguson as a starting point.32) Large Drops. If a sessile liquid drop is sufficiently large, gravity will completely flatten it. Here, the balance between capillary and gravitational forces controls the ultimate height (hl). Mathematically, this occurs where the base of the drop is sufficiently large (2a f ¥). Thus, from eq 4 " #1=2 γ ð1 - cos θa Þ hl ¼ 2 Fg
base diameter (2a), and contact angle (θ).
of θ. Depending on how the drop was deposited on the surface, the contact angle may be an advancing, receding, or intermediate value. Small Drops. It is well-established that, if drops are sufficiently small, their shape can be described as a segment of a sphere. In this case, height (hs) can be estimated from base diameter (2a) and contact angle (θ) as24-26 1 - cos θ sin θ θ ¼ a3 ¼ a 3 tan hs ¼ a 3 sin θ 1 þ cos θ 2
The volume (Vs) of small drops can be calculated as a function of dimensions and/or contact angle. For example, if h and 2a are known, then27,28 Vs ¼
1 πhs ð3a2 þ h2s Þ 6
ð2Þ
Alternatively, if 2a and θ are available, then Vs ¼
1 θ θ π 3 a3 tan 3 þ tan2 6 2 2
hm
Transition from Spherical to Distorted. As drops grow larger and larger, eventually their size reaches a point where gravity begins to distort their shape. The magnitude of the drop dimensions for this transition are often estimated from the capillary length (λ)34,35 1=2 γ λ ¼ ð7Þ Fg
ð3Þ
hs ¼ hm
a2xc þ
1=2 γ γ sin2 θ axc - 2 ¼ 0 Fg Fg 1 - cos θ
ð9Þ
that has a real root of 2axc
(24) Mack, G. L. The Determination of Contact Angles from Measurements of the Dimensions of Small Bubbles and Drops. I. The Spheroidal Segment Method for Acute Angles. J. Phys. Chem. 1936, 40, (2), 159-167. (25) Mack, G. L.; Lee, D. A. The Determination of Contact Angles from Measurements of the Dimensions of Small Bubbles and Drops. II. The Sessile Drop Method for Obtuse Angles. J. Phys. Chem. 1936, 40, (2), 169-176. (26) Bartell, F. E.; Zuidema, H. H. Wetting Characteristics of Solids of Low Surface Tension such as Talc, Waxes and Resins. J. Am. Chem. Soc. 1936, 58, (8), 1449-1454. (27) Young, J. A.; Phillips, R. J. An Indirect Method for the Measurement of Contact Angles. J. Chem. Educ. 1966, 43, (1), 36-37. (28) Beyer, W. H. Standard Mathematical Tables, 27th ed.; CRC Press: Boca Raton, FL, 1984. (29) de Laplace, P. S. Mecanique Celeste; Courier: Paris, 1805; Vol. t. 4, Supplement au X Livre. (30) Poisson, S. D. Nouvelle Theorie de l’Action Capillaire; Bachelier: Paris, 1831. (31) Extrand, C. W.; Moon, S. I. Critical Meniscus Height of Liquids at the Circular Edge of Cylindrical Disks and Rods. Langmuir 2009, 25, (2), 992-996. (32) Ferguson, A. On the Shape of the Capillary Surface Formed by the External Contact of a Liquid with a Cylinder of Large Radius. Phil. Mag. 1912, Ser. 6, 24, (144), 837-844.
ð8Þ
Substituting eqs 1 and 4 into eq 8 and rearranging yields, the following quadratic equation
ð4Þ
where γ is the surface tension of the liquid, F its density, and g is the acceleration due to gravity (9.81 m/s2). (This approximate expression was originally derived for the meniscus height around
11816 DOI: 10.1021/la1005133
ð6Þ
It is also possible to estimate drop height, diameter, and volume at this critical juncture. At this crossover point
Intermediate Drops. The interplay between surface tension and gravity that determines the shape of larger liquid drops was first analyzed by Laplace29 and Poisson30 in the early 19th century. Bashforth and Adams3 gave the first exact numerical solutions of the shape of gravity-distorted sessile drops. In the analysis presented here, we employ an approximate closed analytical solution for the height (hm) of liquid drops of intermediate size31 " #1=2 " 1=2 # - 1=2 γ γ 2 ð1 - cos θÞ ¼ 2 1þ Fg Fg 2a
Equation 5 has also been derived by Quincke33 and Padday16 for very large drops. One can also arrive at this expression by simplifying an 1831 Poisson equation.13,30 The volume of large drops (Vl) can be approximated as a disk of height hl and diameter 2a Vl ¼ πhl a2
ð1Þ
ð5Þ
2 3 !1=2 1=2 2 γ sin θ 4 1þ8 ¼ - 15 Fg 1 - cos θ
ð10Þ
where 2axc is the base diameter of the drop. Combining eqs 1 and 10 yields an expression for estimating drop height (hxc) where gravity begins to cause distortion 2 3 !1=2 1=2 2 1 γ θ sin θ tan 4 1 þ 8 - 15 ð11Þ hxc ¼ 2 Fg 2 1 - cos θ Similarly, combining eqs 3 and 10 yields an expression for estimating drop volume (Vxc) at this transition Vxc
2 33 !1=2 3=2 1 γ θ sin2 θ 2 θ 4 π ¼ tan 3 þ tan - 15 1þ8 48 Fg 2 2 1 - cos θ
ð12Þ (33) Quincke, G. Ueber die Capillarit€ats-constanten geschmolzener chemischer Verbindungen. Ann. Phys. Chimie 1869, 138, (1), 141-155. (34) Batchelor, G. K. An Introduction to Fluid Mechanics. Cambridge University Press: Cambridge, UK, 1967. (35) De Gennes, P.-G.; Brochard-Wyart, F.; Quere, D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves; Springer: New York, 2004.
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Article
Transition from Distorted to Flattened. If drops are sufficiently large, gravity limits their height. At the crossover point where their height begins to plateau hm ¼ hl
hm þ ε = hl
ð14Þ
hm = hl - ε
ð15Þ
or
Incorporating eqs 4 and 5 into eq 15 and rearranging produces an expression for the base diameter (2axg) of a sessile drop at the transition point where gravity fully distorts it, such that the central portion of the top surface is almost flat 2axg
ð16Þ
Merging eqs 5 and 15 gives the approximate height at the transition from distorted to fully flattened sessile drops " #1=2 γ ð1 - cos θa Þ hxg =ð1 - εÞhl ¼ ð1 - εÞ 2 Fg
ð17Þ
Finally, combining eqs 6, 16, and 17 yields an expression for estimating drop volume (Vxg) at the point where height plateaus " #1=2 " #2 γ 3 1 ð18Þ ð1 - cos θa Þ Vxg =πð1 - εÞ 2 Fg ð1 - εÞ - 2 - 1
Experimental Details The liquids used in the experiments were 18 MΩ 3 cm deionized (DI) water, ethylene glycol (Sigma-Aldrich, anhydrous 99.8%), and diiodomethane (also commonly known as methylene iodide, Alfa Aesar 97þ% assay). Density (F), surface tension (γ), and capillary length (λ) of the liquids are summarized in Table 1.1,36,37 The solid surfaces were PFA (a PTFE copolymer obtained from DuPont as extruded sheet, 1000 LP, 0.25 mm thick), PC (Sabic Lexan 8010 film, 0.25 mm thick), and silicon wafers (100 mm wafers: Æ100æ, P/Boron, 1-50 Ω cm, Silicon Resource Co., Reno, NV, USA; 200 mm wafers: Æ100æ, P/Boron, 1-50 Ω cm, Siltronic AG, Munich, Germany). Prior to use, the polymer surfaces were rinsed with isopropanol (Brenntag Co.) and 18 MΩ 3 cm DI water, then blown-dry with clean, filtered air. Wettability of the silicon was varied by pretreatment. The smaller Si wafers (Si 1), which were moderately hydrophilic, were used as received. The larger wafers (Si 2) were cleaned and then aged in a container (Entegris Crystalpak) for a month before use, which produced a more hydrophilic surface than the smaller, untreated wafers. Drops having small to intermediate volumes were extruded from a 1 mL glass syringe (M-S, Tokyo, Japan). The syringe plunger was displaced by turning a micrometer. The number of turns (50 gradations per turn) was converted to liquid volume, V, (36) Weast, R. C. Handbook of Chemistry and Physics, 73rd ed.; CRC: Boca Raton, FL, 1992. (37) Wu, S. Polymer Interface and Adhesion; Marcel Dekker: New York, 1982.
Langmuir 2010, 26(14), 11815–11822
γ (mN/m)
F (kg/m3)
λ (mm)
solid
θ (deg)
water
72
998
2.7
ethylene glycol diiodomethane
48 51
1110 3320
2.1 1.2
PFA PC Si 1 Si 2 PFA PFA
108 80 40 15 90 85
liquid
ð13Þ
From a mathematical perspective, drops must attain infinite diameter to reach their limiting height. However, a practical limit is reached when hm approaches hl within some value ε associated with the precision or uncertainty of the estimate
1=2 γ 1 ¼ 2 Fg ð1 - εÞ - 2 - 1
Table 1. Properties of the Liquids and Solid Substrates
via a calibration curve. Larger drops (2a > 10 mm) were deposited with a plastic pipet (3 mL transfer pipet, Samco Scientific Corp.). The mass of the larger drops was measured and converted into volume. Drops were observed, and images were captured using a drop shape analyzer (Kr€ uss DSA10). Drop heights and base diameters were measured using Image-Pro Plus software, except for the larger base diameters, which extended beyond the field of view of the drop shape analyzer. These were measured from above with a supported ruler. Most contact angles (θ) were measured by the tangent method using the resident software of the drop shape analyzer. Alternatively, with the aid of image analysis software, tangent, and base lines were drawn manually and a tangent θ value was measured. In a few instances noted within the text, contact angles were estimated by the height-diameter method, eq 1. Unless stated otherwise, contact angles were advancing values. All measurements were performed at 25 ( 1 C. Error of the dispensed liquid volume from the syringe was usually
