Article pubs.acs.org/Langmuir
Which Controls Wetting? Contact Line versus Interfacial Area: Simple Experiments on Capillary Rise C. W. Extrand* and Sung In Moon Entegris, Inc., 101 Peavey Road, Chaska, Minnesota 55318, United States ABSTRACT: Working equations that describe wetting phenomena can be derived in a variety of ways starting from capillary forces, Laplace pressure, or solid surface energies. We examined the relative importance of the contact line and interfacial areas in the capillary rise inside small diameter glass tubes. A series of simple experiments demonstrate that this wetting phenomenon is controlled by interactions in the vicinity of the contact line.
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vertical or z direction, rapidly at first, and then it slows (Figure 1b). Eventually, the liquid reaches a final height, h, where the liquid stops (Figure 1c). If the diameter of the capillary is sufficiently small, such that h ≫ D, then the ultimate height, h, of a liquid rise can be estimated as
INTRODUCTION Even though wetting has been studied for more than three hundred years, there is no universal agreement on the underlying physicochemical mechanisms. One of the most fundamental questions is in regards to the driving mechanism for wetting. Is wetting controlled by interactions within the interfacial areas or at the contact line? Much of the recent discussion around this question has centered on sessile drop measurements and the validity of the Wenzel and Cassie equations,1,2 which were intended to describe wetting behavior on rough and super hydrophobic surfaces. In the past few years, a number of simple, seemingly unequivocal experiments have been performed on sessile drops that suggest that wetting is controlled by interactions in the vicinity of the contact line.3,4 That conclusion met with considerable opposition and resistance.5−7 For example, it has been suggested that disagreement between theory and experiment “does not arise from a fault with Cassie theory but from an incorrect choice of surface area fractions.”5 While this debate has centered on sessile drops, it may be possible to gain a broader perspective by looking at other wetting geometries for clues for the true underpinnings of the physical phenomena. Therefore, in this study, we performed a series of simple experiments involving capillary rise in small tubes that test the roles of contact line and interfacial areas.
h=
ρgD
(1)
where g is the acceleration of gravity. The first careful observations of liquids rising in capillary tubes8 and between glass plates9 were reported by Hauksbee in the early eighteenth century.10 He noted that liquids rose higher in smaller diameter tubes, and that vacuum had no influence on how high a liquid rose.8 The origin of eq 1 is often attributed to Jurin,11 who experimentally determined that the height of liquid rise in a capillary is inversely proportional to its diameter. Today, eq 1 is derived a variety of ways,12−14 by starting with a balance between gravitational forces and capillary forces at the contact line, by examining the interplay between hydrostatic and Laplace pressure at the air−liquid interface, or by examining the change in solid surface energy of the capillary. Examples of the three approaches are summarized below for small diameter capillary tubes. Derivation by Forces. The simplest method to solve for the rise height involves a force balance. The gravitational force ( fg) acting downward on the liquid column of height, h, is
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THEORY Capillary Rise. Consider the capillary tube of internal diameter, D, shown in Figure 1. The capillary is brought into contact with a liquid of surface tension, γl, and density, ρ, such that the bottom just touches the liquid, Figure 1a. A meniscus forms around the inner and outer diameter of the capillary. The meniscus wets the tube with a contact angle of θ. If θ < 90°, the liquid inside the capillary immediately begins to rise in the © 2012 American Chemical Society
−4γl cos θ
⎛π ⎞ fg = ρg ⎜ ⎟D2h ⎝4⎠
(2)
Received: September 21, 2012 Revised: October 15, 2012 Published: October 22, 2012 15629
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w=
∫0
h
⎛π ⎞ ⎛π ⎞ 2 2 2 ⎜ ⎟ρgD (h − z) dz = ⎜ ⎟ρgD h ⎝8⎠ ⎝4⎠
(7)
18
According to Brown, the total energy (Δeb) needed to lift the liquid is the sum of the work (w) and the change in potential energy (Δep),
Δeb = w + Δep
(8)
As these two quantities, w and Δep, are said to be equal, the total bulk energy change is Δeb =
⎛π ⎞ 2 2 ⎜ ⎟ρgD h ⎝4⎠
(9)
On the other hand, the change in the interfacial energy (Δes) that drives the rise of liquid can be estimated from the area of the wetted wall, the difference in the surface energy of the bare solid (γs), and the interfacial energy between the liquid and the wetted solid (γsl),
Δes = πDh(γs − γsl)
(10)
By inserting the Young equation15 into eq 10, γl cos θ = γs − γsl
the change in surface energy is given in terms of the liquid surface tension and contact angle.
Figure 1. A depiction of capillary rise in a small diameter tube. (a) The tube contacts the liquid. (b) Liquid rises vertically inside the tube. (c) The liquid stops at an equilibrium height of h. (d) A close-up view of the meniscus.
Δes = πDhγl cos θ
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RESULTS AND DISCUSSION All three approaches produce the same result for a homogeneous capillary of constant cross-sectional area. In order to correctly solve more complex problems involving different geometries and/or heterogeneous interfaces, the underlying mechanisms must be understood. Therefore, what controls wetting and capillary rise? Is it forces at the contact line? Is it the Laplace pressure at the air−liquid interface? Or, is it the changes in the liquid−solid interface? To answer these questions, we start with a few simple capillary rise experiments shown in Figure 2. The capillary tubes are hydrophilic (θ ∼ 0°) borosilicate glass from Pyrex and
(3)
When the liquid has risen to its maximum height, h, these two forces are equal. Thus, eqs 2 and 3 can be equated and solved for h to yield the capillary rise equation given above. Derivation by Pressures. Alternatively, capillary rise can be estimated from pressures.16,17 The hydrostatic pressure in a capillary depends on the density and height of the liquid Δph = ρgh
(12)
Finally, by combining eqs 9 and 12, we once again arrive at the classic expression for capillary rise, eq 1.
The weight of the liquid is suspended by a capillary force (fc), which depends on the length of the contact line (πD) and the vertical component of the liquid surface tension (γl cosθ),10,15 fc = πDγl cos θ
(11)
(4)
It is assumed that the hydrostatic pressure is supported by the curved air−liquid interface at the top of the liquid column. For sufficiently small diameter tubes, that meniscus is spherically shaped (i.e., not distorted by gravity) and has a Laplace pressure (Δpc) of 2γ Δpc = l (5) r where r is the radius of curvature of the air−liquid interface. The radius of curvature of the air−liquid interface is tied to the capillary diameter via the cosine of the contact angle cos θ =
D/2 r
(6)
Combining eqs 4−6, again yields eq 1. Derivation by Energies. In yet another approach, the capillary rise can be estimated by analyzing energies. The work, w, done to lift the liquid can be calculated as18−21
Figure 2. (a) A 1.2 mm glass capillary tube touching the water surface. (b) The same capillary plunged below the surface to a depth of d = 11 mm. Arrows show the position of the menisci. 15630
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height to depend on the immersion depth, d. Accordingly, eq 10 of the “energy derivation” would include a depth term
have an internal diameter of D = 1.2 mm. In the experiments shown in Figures 2−4, the liquid is water (γl = 72 mN/m and ρ
Δes = πD(h + d) ·(γs − γsl)
(13)
and a working equation for capillary rise would have the following form, ⎛ 4γ cos θ ⎞ 4dγl cos θ =0 h2 − ⎜ l ⎟h − ρgD ⎝ ρgD ⎠
(14)
For the example shown in Figure 2b where d = 11 mm, eq 14 incorrectly overpredicts h to be 39 mm. (Note here: Only the wetted area on the inside of the capillary tube is included; if both inside and outside were accounted for, the overprediction would be even greater.) In Figure 3, the experiment was repeated with an additional variation: the bottom portion (about 10 mm in length) of the plasma treated capillary tube was dipped in a toluene solution of polystyrene and allowed to dry.23 This rendered the lower portion of the tube hydrophobic with an advancing contact angle of θ ∼ 90°. When the coated end of the capillary was lowered a short distance below the surface, water did not rise (Figure 3a). In Figure 3b, the coated capillary was lowered farther until water reached the uncoated hydrophilic portion of the tube; the water rose to the same height as the uncoated tubes. Again, the area of the wetted solid did not influence the rise height. Applying the hydrophobic polystyrene coating to only the inside or outside also produced the same capillary rise as the uncoated tubes.24 We repeated these experiments with glass tubes of different diameters (D = 0.65 and 1.4 mm) and different liquids (ethylene glycol and hexadecane). The results were similar: the rise was independent of how far the capillary tube was immersed into the liquid. Also, rise was not affected by the chemical composition of the immersed portion of the tubes. As an aside, immersion depth and wetted area did not influence the shape or the height of the meniscus around the outer wetted perimeter of the capillary tubes unless it was coated. From these experiments, we conclude that the liquid−solid interfacial area does not control capillary rise. This leaves two possibilities: wetting and rise are controlled by interactions either in the vicinity of the contact line or within the air−liquid interface of the meniscus. One of the well-known properties of liquids is the tendency to minimize the area of its air−liquid interfaces. If the air−liquid interface were to control the wetting, the said interface in capillary tubes would always be flat (or nearly flat except at the periphery where the liquid contacts the wall of the capillary). However, this is generally not true. Unless θ = 90°, the air−liquid interface exhibits curvature. If gravity is not a factor, the interface is spherically shaped. The air−liquid interface minimizes its area within the constraints imposed at the contact line. Or mathematically, forces at the contact line dictate the boundary conditions for the minimization of the air−liquid interface. If gravity is significant, the interface must accommodate this force too.25−28 Therefore, through process of elimination, we can conclude that capillary rise is controlled by neither the liquid−solid nor the air−liquid interface. Rather, capillary forces generated in the vicinity of the contact line drive the liquid rise in small diameter, lyophilic tubes. Liquid rises until the hydrostatic pressure of the liquid column equals the Laplace pressure of the air−liquid interface. (In the Appendix, we show a simple experiment that demonstrates that capillary forces must be
Figure 3. (a) The polystryrene-coated portion of a 1.2 mm capillary contacting a water surface. No rise occurs. (b) The coated portion of the tube is fully immersed such that water contacts the uncoated portion: water rises. Arrows show the position of the menisci.
Figure 4. (a) Hydrophilic glass capillary (D = 1.2 mm) is immersed in water. (b) Same capillary tube is attached to a large polystyrene cup to create a bimodal cross-sectional area. This bimodal tube was raised to expose the wider, lower portion, yet it still supported a water column with the same height as the capillary shown in (a). Arrows show the position of the menisci.
= 998 kg/m3).22 In Figure 2a, the capillary is lowered into contact with the water surface. The end of the tube just touched the water. A spherically shaped meniscus formed. Water initially rose rapidly, then slowed before reaching an equilibrium height of h = 24 mm. The predicted height from eq 1 of 24.5 mm agrees well with the measured one. In Figure 2b, the same experiment is repeated with one variation. Rather than lowering to make contact with the surface of the water, the capillary tube is plunged below the surface to a depth of d = 11 mm. This has no influence on the rise of the water. The water still climbed 24 mm. Thus, the change in liquid−solid interfacial area did not influence capillary rise. If an increase in interfacial energies were to control this phenomenon then we would have expected the rise 15631
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Edlund, J. McDaniel, L. Monson, J. Pillion, and B. Powell for their suggestions on the technical content and text.
balanced against hydrostatic forces rather than the mass of the liquid column.) In his classic textbook on physical chemistry of surfaces, Adamson stated that wetting problems may be tackled “using either the concept of surface tension or the (mathematically) equivalent concept of surface free energy.”12 While there is value in being able to solve problems with different approaches, it is important to understand the limitations of the assumptions used for a particular method. Otherwise, an incorrect assumption may be applied to a problem where the investigator may not have the good fortune to arrive at the correct result.
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(1) Cassie, A. B. D.; Baxter, S. Wettability of Porous Surfaces. Trans. Faraday Soc. 1944, 40 (21), 546−551. (2) Wenzel, R. N. Resistance of Solid Surfaces to Wetting by Water. Ind. Eng. Chem. 1936, 28 (8), 988−994. (3) Extrand, C. W. Contact Angles and Hysteresis on Surfaces with Chemically Heterogeneous Islands. Langmuir 2003, 19 (9), 3793− 3796. (4) Gao, L.; McCarthy, T. J. How Wenzel and Cassie Were Wrong. Langmuir 2007, 23 (26), 3762−3765. (5) Panchagnula, M. V.; Vedantam, S. Comment on How Wenzel and Cassie Were Wrong by Gao and McCarthy. Langmuir 2007, 23 (26), 13242. (6) McHale, G. Cassie and Wenzel: Were They Really So Wrong? Langmuir 2007, 23 (15), 8200−8205. (7) Marmur, A.; Bittoun, E. When Wenzel and Cassie are Right: Reconciling Local and Global Considerations. Langmuir 2009, 25 (3), 1277−1281. (8) Hauksbee, F. An Experiment Made at Gresham-College, Shewing That the Seemingly Spontaneous Ascention of Water in Small Tubes Open at Both Ends is the Same in Vacuo as in the Open Air. Phil. Trans. 1706, 25 (1), 2223−2224. (9) Hauksbee, F. Several Experiments Touching the Seeming Spontaneous Ascent of Water. Phil. Trans. 1708, 26 (1), 258−266. (10) Maxwell, J. C. Capillary Action. In Encyclopaedia Britannica, 9th ed.; Baynes, T. S., Ed. Henry G. Allen & Co.: New York, 1888; Vol. 5th, pp 56−71. (11) Jurin, J. An Account of Some Experiments Shown before the Royal Society; With an Enquiry into the Cause of the Ascent and Suspension of Water in Capillary Tubes. Phil. Trans. 1717−1719, 30 (1), 739−747. (12) Adamson, A. W. Physical Chemistry of Surfaces, 5th ed.; Wiley: New York, 1990. (13) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Science, 3rd ed.; CRC Press: New York, 1997. (14) De Gennes, P.-G.; Brochard-Wyart, F.; Quéré, D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves; Springer: New York, 2004. (15) Young, T. An Essay on the Cohesion of Fluids. Phil. Trans. R. Soc. London 1805, 95 (1), 65−87. (16) Laplace, P. S. Supplément au Xe Livre Mécanique Celeste; Courier: Paris, 1805; Vol. t. 4. (17) Poisson, S. D. Nouvelle Théorie de l’Action Capillaire; Bachelier: Paris, 1831. (18) Brown, R. C. Note on the Energy Associated with Capillary Rise. Proc. Phys. Soc., London 1941, 53 (3), 233−234. (19) Markworth, A. J. Liquid Rise in a Capillary Tube. A Treatment on the Basis of Energy. J. Chem. Educ. 1971, 48 (8), 528. (20) Walton, A. J. Surface Tension and Capillary Rise. Phys. Educ. 1972, 7 (8), 491−498. (21) Henriksson, U.; Eriksson, J. C. Thermodynamics of Capillary Rise: Why Is the Meniscus Curved? J. Chem. Educ. 2004, 81 (1), 150− 154. (22) Weast, R. C. Handbook of Chemistry and Physics, 73rd ed.; CRC: Boca Raton, FL, 1992. (23) Polystyrene (PS) from Aldrich with a molecular weight of ∼192,000 g/mol was dissolved in toluene (>99.5%, Aldrich) to make a 1 wt % solution. To coat only the inside, one end of a tube was touched to the PS solution; the solution rose, and then was removed by wicking with a tissue. The remaining layer of PS solution was allowed to dry. To coat the outside, the top of the tube was plugged, and then the bottom end was immersed in the PS solution. These sequences were combined to coat both sides. (24) For tubes with the hydrophobic coating on the outside, rise happened as soon as the capillary tip touched the water. For tubes with
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CONCLUSIONS Simple experiments using capillary tubes demonstrate that liquid−solid interfacial areas are relatively unimportant in wetting. The shape of the air−liquid interface is subjugated to solid geometry and boundary conditions dictated by the contact angle where the liquid and solid meet. Thus, capillary forces acting in the vicinity of the contact line control wetting and thus the rise of liquid in the capillary tubes.
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APPENDIX In the body of this work, we performed experiments and analysis for capillary tubes with a constant diameter and crosssectional area, where one could derive the correct form of eq 1 by balancing the capillary forces against forces associated with either the mass of the liquid or its hydrostatic pressure. However, this approach is not valid for tubes with nonuniform cross sections. We show an example in Figure 4 that demonstrates that the contact line and the air−liquid interface are supporting a hydrostatic pressure, not the mass of the liquid column. The same hydrophilic glass capillary used in early experiments (D = 1.2 mm) is shown in Figure 4a. It is immersed in water, which has risen to a height of 24 mm. In Figure 4b, the same capillary tube has been attached to a large polystyrene cup with a silicone adhesive. This “capillary”, which had a bimodal crosssectional area, was immersed to completely wet the lower, much wider portion, and then immersed a bit more to initiate rise in the upper, narrower portion. With the wide portion completely immersed, capillary rise in the upper tube was the same as the simple tube shown in Figure 4a. Next, if the bimodal tube was raised to expose the wider, lower portion, as shown in Figure 4b, the height of the water relative to the horizontal air−liquid interface did not change. The mass of water in the column shown in Figure 4b is more than 10000 times greater than the column in Figure 4a, yet they are the same height. Thus, we conclude that a capillary force and Laplace pressure are balancing a hydrostatic pressure, not the mass of liquid in the “column”. This outcome was anticipated by Jurin,11 but to the best of our knowledge had not been experimentally tested.
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REFERENCES
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Tel: (952) 556-8619. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank Entegris management for supporting this work and allowing publication. Also, thanks to B. Arriola, M. Amari, T. 15632
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the coating on the inside or both sides, the coated portion of the tube was pushed beneath the water surface to initiate rise. (25) 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. (26) Rayleigh, L. On the Theory of Capillary Tube. Proc. R. Soc. Lond., Ser. A 1916, 92 (1), 184−195. (27) Princen, H. M. The Equilibrium Shape of Interfaces, Drops and Bubbles. Rigid and Deformable Particles at Interfaces. In Surface and Colloid Science; Matijević, E., Ed; Wiley: New York, 1969; Vol. 2, pp 1−84. (28) Padday, J. F. Theory of Surface Tension. In Surface and Colloid Science; Matijević, E., Ed; Wiley: New York, 1969; Vol. 1, pp 39−248.
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