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Geometry-Driven Wetting Transition Keith D. Humfeld and Stephen Garoff* Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received June 4, 2004. In Final Form: July 23, 2004 Wetting states are quantitatively described by the number of inflection points on the liquid-vapor interface and by the macroscopic contact angle. The number of inflection points required for complete, partial, and pseudopartial wetting is determined for geometries with positive, zero, and negative capillary pressures. The wetting state of a material system is not always independent of the magnitude of the capillary pressure; for example, the wetting state of a fluid inside a capillary tube may depend on the capillary radius. In particular, a fluid that pseudopartially wets the inside of a tube exhibits a transition to partial wetting (or complete wetting) as the capillary radius is decreased.
Introduction The wetting state of a solid-fluid-vapor system is typically described by whether the fluid partially wets or completely wets the solid substrate.1 A fluid that exhibits partial wetting forms a droplet on top of a substrate as shown in Figure 1a. When gravitational forces are negligible compared to surface forces (in the case of a low Bond number,2 the case we will consider here), the shape of the droplet is a spherical cap. The angle between the substrate and the extrapolation of the spherical surface to the substrate is called the macroscopic contact angle, θ.3 The surface tensions of the solid-vapor (γsv), solidliquid (γsl), and liquid-vapor (γlv) interfaces are related to the contact angle (θ) via Young’s equation,4
γsv ) γsl + γlv cos θ
(1)
When the surface tensions are very different from each other such that γsv - γsl - γlv > 0, Young’s equation is not satisfied for any angle θ and the fluid spreads over the substrate. This is the complete wetting state and is depicted in Figure 1b. When we consider only the macroscopic picture of wetting, the three surface tensions are sufficient to determine the wetting state of any solidliquid-vapor system. If we look more closely at the fluid-vapor or solidfluid interfaces, we see that an interface is a continuous density gradient.5 This density gradient has some width normal to the interface. Where the fluid body is thick compared to the width of the interface, the interfacial tension, or surface excess free energy per unit area,3 is equal to its bulk value γsl or γlv. Where the thickness of the fluid body is comparable to the width of the interfaces, the solid-liquid and liquid-vapor interfaces overlap and there is a correction to the surface energy, E(h). The correction to the surface energy depends on the positions of all of the fluid elements and thus on the shape of the interface. Typically the correction is described in terms * Corresponding author. E-mail:
[email protected]. (1) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 828-861. (2) The Bond number is defined as Bd ≡ FgL2/γlv, where F is the fluid density, g is the acceleration of gravity, L is the characteristic length of the fluid body, and γlv is the liquid-vapor surface tension. (3) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: London, 1985; p 321. (4) Young, T. Philos. Trans. R. Soc. 1805, 95, 65. (5) Gibbs, J. W. The Collected Works of J. Willard Gibbs; Yale University Press: New Haven, 1960; pp 96-98.
Figure 1. Fluid exhibiting (a) partial wetting and (b) complete wetting on a horizontal substrate.
of its thickness dependence,1,6 neglecting any contribution of the slope of the intersecting interfaces or their relative curvature. Within the accuracy of this approximation, the Derjaguin approximation,7 the solid-liquid and liquidvapor interfaces have the same area and therefore E(h) can be assigned to either the solid-liquid or the liquidvapor interface.8 We choose to assign the energy correction to the liquid-vapor interface, replacing the liquid-vapor surface tension γlv with the surface tension corrected by the “energy correction,” γlv + E(h), where h is the thickness of the fluid body. The energy correction is related to the disjoining pressure, Π(h), by Π(h) ) -dE(h)/dh. In regions where the fluid body is thin and thus the energy correction is nonzero, the shape of the interface deviates from that of a spherical cap and conforms to the augmented Young’s equation,9
γsv ) γsl + (γlv + E(h)) cos R - hPcap
(2)
where R is the local angle of inclination of the liquidvapor interface relative to the solid-liquid interface and Pcap is the capillary pressure in the drop. A thermodynamic proof has shown that as the fluid body approaches zero (6) Brochard-Wyart, F.; di Meglio, J.-M.; Que´re´, D.; de Gennes, P.-G. Langmuir 1991, 7, 335-338. (7) White, L. R. J. Colloid Interface Sci. 1983, 95, 286-288. (8) Solomentsev, Y.; White, L. R. J. Colloid Interface Sci. 1999, 218, 122-136. (9) Hirasaki, G. Thermodynamics of Thin Films and Three-Phase Contact Regions. In Interfacial Phenomena in Petroleum Recovery; Morrow, N., Ed.; Marcel Dekker: New York, 1991; pp 33, 42.
10.1021/la048614v CCC: $27.50 © 2004 American Chemical Society Published on Web 09/08/2004
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Figure 2. Fluid exhibiting (a) partial wetting and (b) pseudopartial wetting on a horizontal substrate, viewed very close to the end of the capillary phase. The drop in each case extends off the figure to the right. Inflection points are marked on the liquid-vapor interface.
thickness the local angle of inclination must approach 0°,7 or
limR ) 0 hf0
(3)
Three different wetting states can simultaneously satisfy eqs 2 and 3. These are complete wetting (Figure 1b), partial wetting (Figures 1a and 2a), and pseudopartial wetting (Figure 2b). The partial wetting state has a microscopic foot protruding from the edge of the drop,8 and the pseudopartial wetting state has an extended film protruding from the edge of the drop.6,8-12 To determine the wetting state of a solid-liquid-vapor system, all of the following are required: the surface tensions, the energy correction functional, and the capillary pressure. A fluid that partially wets a horizontal substrate (forming a droplet with a positive capillary pressure) will partially wet a vertical plate made of the same material (forming a meniscus with zero capillary pressure) and will partially wet the inside of a tube (forming a meniscus with negative capillary pressure). In this regard, the wetting state seems to be independent of the capillary pressure. However, the equilibrium wetting state of a solidliquid-vapor system may depend on the geometry of the system as well as the material properties of surface tension and energy correction. To show this, we introduce a method of quantitatively describing the wetting state. We show how the wetting state depends on the geometry of the system including its dependence on the sign of the capillary pressure and give a specific instance in which geometric parameters enter into the determining of the wetting state. Further, we predict a transition in wetting state from pseudopartial wetting to partial wetting for a wellcharacterized system. Wetting States Defined by Inflection Points To predict a transition in wetting state, we must clarify the difference between partial and pseudopartial wetting. (10) Hirasaki, G. Shape of Mensicus/Film Transition Region. In Interfacial Phenomena in Petroleum Recovery; Morrow, N., Ed.; Marcel Dekker: New York, 1991; p 78. (11) Churaev, N. Rev. Phys. Appl. 1988, 23, 975-987. (12) Sharma, A. Langmuir 1993, 9, 3580-3586.
Consider a drop partially wetting a substrate as in Figure 2a. The local angle of inclination, R, must approach 0° as the fluid approaches zero thickness, so a microscopic foot protrudes from the end of the capillary body of any partial wetting system. A drop pseudopartially wetting a substrate (Figure 2b) has a film, perhaps of finite length,12 protruding from its tip. It seems arbitrary whether the structure protruding from the end of a drop is a foot or a film. There have been no quantitative criteria for distinguishing partial wetting from pseudopartial wetting. We introduce a criterion here to distinguish these two states. A drop on a horizontal substrate forms a body of rotation, and Figure 2 depicts the liquid-vapor interface along a radial line of this body. Along this radial line, there are points marked that correspond to points where the second derivative of the liquid-vapor interface profile with respect to the radial position is zero. We define these points as inflection points although there is a second, nonzero curvature normal to the radial. Examining the two interface shapes of Figure 2, we see that a drop partially wetting a horizontal substrate has one inflection point, and a drop pseudopartially wetting a horizontal plate has three inflection points. This criterion can be applied in any geometry. The condition for an inflection point along this surface, in terms of the material properties of the system, is found by examining the differential equation governing the shape of the liquid-vapor interface,8
0 ) -Π(h) cos R + (γlv + E(h))K - Pcap
(4)
where K is the curvature of the liquid-vapor interface. At an inflection point, one of the two terms of the curvature is zero. Typically the disjoining pressure term is much larger than the remaining curvature term due to a matter of length scales.13 Assuming that the local angle of inclination is small, we arrive at an approximate condition for an inflection point,10
Π(h) ) -Pcap
(5)
We derive an exact condition for an inflection point in a specific geometry later in this paper. Using this approximate result, we see that a given material system, and thus a given disjoining pressure isotherm, has a different number of inflection points depending on the sign of the capillary pressure but has the same wetting state. Figure 3 shows a sample disjoining pressure isotherm for a partially wetting system with two horizontal lines corresponding to negative and positive capillary pressures. The line of positive capillary pressure satisfies eq 5 once, and the line of negative capillary pressure never satisfies eq 5. The drop shown in Figure 2a could result from Π(h) shown in Figure 3 with Pcap > 0. It has one inflection point. Figure 4 shows a different partial wetting system which could also result from Π(h) shown in Figure 3 with Pcap < 0. This system has no inflection points. Thus eq 5 gives the number of inflection points required for a partial wetting system for either sign of the capillary pressure. We complete the study of the relationship between the number of inflection points and the wetting state by cataloging the number of inflection points required for (13) The disjoining pressure term is on the order of γlv/hΠ, where hΠ, the range of the disjoining pressure, may be as large as tens of nanometers. The energy correction term is on the order of γlv/R2, where R2 is the second, nonzero radius of curvature of the liquid-vapor interface. Since hΠ , R2, the disjoining pressure term is typically much larger than the energy correction term.
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Figure 3. Disjoining pressure isotherm for a partially wetting system, along with lines corresponding to negative and positive capillary pressures. The point for which eq 5 is satisfied, which will correspond to an inflection point in the liquid-vapor interface, is marked.
Figure 5. Fluid exhibiting partial wetting inside of a tube, with a cylindrical surface bounding the material body. The x-direction is defined as vertical, and h is the thickness of the fluid at the bottom of the surface.
Figure 5 shows a fluid partially wetting the inside of a tube. A surface in the shape of a cylinder defines a material body. For this system to be in static equilibrium, the forces on the top and bottom of this material body must be equal. The components of those forces in the x-direction are equal when
2πaγsv ) 2πaγsl + 2π(a + h)(γlv + E(h)) cos R Pcapπ((a + h)2 - a2) (6)
Figure 4. Fluid exhibiting partial wetting in a system with negative capillary pressure, that is, inside of a slot or a tube. The axis of rotation (or symmetry) is off the figure to the left. Table 1. Number of Inflection Points Required for Complete, Partial, and Pseudopartial Wetting in Geometries with Positive Capillary Pressure and in Geometries with Negative or Zero Capillary Pressure complete wetting partial wetting pseudopartial wetting
Pcap > 0
Pcap e 0
1 1 3
0 0 1
complete, partial, and pseudopartial wetting in geometries with positive and negative (or zero) capillary pressure. Table 1 shows these results. Note that complete wetting and partial wetting require the same number of inflection points and that they are differentiated by the macroscopic contact angle being zero or nonzero. Tube or Slot Geometry Equation 5 is a well-accepted approximate condition for an inflection point, based on the assumptions that the second curvature term is negligible and that the local angle of inclination is very small. To make a quantitative prediction of a transition in wetting state, we derive an exact condition for an inflection point in a specific geometry.
where h is the thickness at the bottom of the Gaussian surface, Pcap ) -2γlv cos θ/a, and a is the capillary radius of the tube. Using Young’s equation, this force balance simplifies to
1 0 ) (γlv + E(h)) cos R + Pcap(a - h) 2
(7)
Since this equation must hold for any position of the bottom of the Gaussian surface, we take the derivative of eq 7 with respect to x and recombine the resulting equation with eq 7 to remove the capillary-pressure-dependent term. This results in
[
0 ) -Π(h) cos R + (γlv + E(h)) -h′′(1 - h′2)-3/2 + cos R (8) a-h
]
where the primes indicate derivatives with respect to x. We have defined an inflection point as a position where h′′ ) 0, so the condition for an inflection point is
ξa(h) ≡
(a - h)Π(h) )1 γlv + E(h)
(9)
Although the derivations differ slightly, eq 9 holds for complete wetting and pseudopartial wetting and also applies to all wetting states within a slot. In all of these cases, the wetting state is a function of the number of inflection points, which is a function of the material properties (γlv, E(h)) and the capillary radius a. Thus inside
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of a slot or a tube the wetting state is a function of the capillary radius. Example System Exhibiting Transition To predict a transition in wetting state caused by a change in capillary radius for an experimental system, we choose a system with a known disjoining pressure isotherm. The Derjaguin-Landau-Verway-Overbeek (DLVO) potential14,15 is a well-studied isotherm that describes a system that exhibits this transition in wetting behavior. The DLVO potential describes the energy correction isotherm for a system with a van der Waals component and an electrostatic component, such as an aqueous ionic-surfactant solution and an ionic substrate. The van der Waals contribution to the energy correction in such a system arises from the London dispersion interaction of a solid and a vapor across a thin fluid layer. The electrostatic contribution to the energy correction arises from the electric potential difference between the two charged surfaces and the screening effect of the ions in solution. For such a system, the DLVO energy correction isotherm takes the form16
E(h) ) A1h-2 + A2e-κh
(10)
κ-1
where is the Debye screening length, and A1 and A2 are coefficients dependent on the Hamaker constant of the materials, the salt valence and concentration, the surfactant concentration, and the electric potential of the liquid-vapor surface relative to the solid-liquid surface. A detailed explanation of DLVO theory and eq 10 is found in the text by Adamson and Gast.16 In order for a transition from pseudopartial wetting to partial (or complete) wetting to occur, the energy correction isotherm must have at least two competing terms. The van der Waals force is almost always attractive, leading to a negative sign for the coefficient A1, so we choose a system with a repulsive electrostatic contribution. An aqueous salt-surfactant solution will have a repulsive electrostatic contribution if both the surfactant and surface have the same sign charge. For numerical study, we assume a Hamaker constant of 10-20 J, a potential difference between the solid-liquid and liquid-vapor surfaces of 30 mV, and a liquid-vapor surface tension of 36 mN/m. These values model a solution of hexadecyltrimethylammonium bromide (CTAB) at or above its critical micelle concentration on a cationic surface.16,17 For a given concentration of electrolytes in the solution, for example, 10-3 M, we can plot the ξa(h) of eq 9 for a variety of capillary radii a. Figure 6 shows ξa(h) plotted against h (normalized to the capillary length) for tube radii of 3.8, 4.74, 5.7, and 6.64 µm. As the capillary radius increases, the maximum value of ξa(h) increases. For some transition radius, the maximum of ξa(h) is 1. Below the transition radius, ξa(h) < 1 for all h so there are no inflection points along the liquidvapor interface and the system is partially or completely wetting (see Table 1). Above the transition radius, ξa(h) ) 1 for two thicknesses h, so there are two inflection points and the system is pseudopartially wetting. The transition radius for this 10-3 M electrolyte-surfactant solution is approximately 5.7 µm, and the transition radius decreases with increasing electrolyte concentration. (14) Derjaguin, B.; Landau, L. Acta Physicochim. 1941, 14, 633662. (15) Verwey, E. J. W.; Overbeek, J. T. G. Theory and Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (16) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, 6th ed.; Wiley: New York, 1997; pp 240-242. (17) Velegol, S. B. Tuning Interfacial and Bulk Self-Assembly of Ionic Surfactants via Counterions and Similarly-Charged Polyelectrolytes. Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, 2000.
Figure 6. ξa(h) plotted against the thickness normalized to the capillary length, for tube radii of 3.8, 4.74, 5.7, and 6.64 µm. Larger radii correspond to the taller ξa(h) curves.
To bring this phenomenon into a regime which might be more easily tested experimentally, we can change the signs of both the van der Waals and the electrostatic interactions. The van der Waals interaction is attractive if the index of refraction of the liquid is between that of the solid and the vapor: ns > nl > nv.3 This is the case for many common wetting systems, as the index of refraction of the vapor phase is nearly unity and the index of refraction of most solids is much larger than for most liquids. By choosing a system with a high index of refraction liquid, a low index of refraction solid, or a liquid third phase, it may be possible to achieve {nl > ns and nl > n3} or {nl < ns and nl < n3}, either of which will yield a repulsive van der Waals interaction. So long as the solidliquid surface is charged, an anionic or cationic surfactant can be added to the solution to provide the attractive electrostatic interaction. With this sign of the interactions and the same magnitude of Hamaker constant, electrostatic potential, and surface tension as listed before, the transition radius occurs at a much larger radius. For an electrolyte concentration of 10-3 M the transition radius is 569 mm, for 10-2 M the transition radius is 12 mm, and for 10-1 M the transition radius is 237 µm. Of course, the low Bond number assumption no longer holds for the large radii in the first two of these cases. Conclusions Wetting states are quantitatively described by the number of inflection points on their liquid-vapor interface and their macroscopic contact angle. The number of inflection points for complete, partial, and pseudopartial wetting are cataloged in Table 1 for both positive and negative (or zero) capillary pressures. The same geometric conditions that cause the capillary pressure to be positive, negative, or zero determine the number of inflection points required for each wetting state. Thus a material system that exhibits a particular wetting state in one geometry exhibits the same wetting state in all geometries. The exception to this conclusion is found for a fluid wetting the inside of a slot or tube. In such a system, the equilibrium wetting state can depend on the magnitude of the capillary pressure. For a salt-surfactant solution whose disjoining pressure isotherm is correctly described by DLVO theory, there is a certain capillary radius at which the system transitions from pseudopartial to partial (or complete) wetting. Acknowledgment. We acknowledge the partial support of the National Science Foundation under Grant DMR980220. LA048614V
