Langmuir 2004, 20, 2227-2232
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Adhesion Force of a Wedge P. Schiller,* M. Wahab, and H.-J. Mo¨gel TU-Bergakademie Freiberg, Institut fu¨ r Physikalische Chemie, Leipziger Strasse 29, 09599 Freiberg, Germany Received April 7, 2003. In Final Form: December 17, 2003 The macroscopic theory of capillarity is usually applied to evaluate the adhesion forces produced by fluid bridges between solid particles. A refined mesoscopic description incorporates the disjoining pressure, which results from the long range forces between the fluid and the solid substrates. In the case of simple nonpolar fluids the influence of the disjoining pressure on adhesion is expected to be negligibly small. Water bridges, however, sometimes have very large disjoining pressures on various substrates. Then mesoscopic theory leads to pull-off forces which differ from the predictions of the macroscopic approach.
1. Introduction The interaction of particles with solid surfaces has found renewed interest due to the usage of the atomic force microscope.1-4 It is well-known that water condensing from the atmosphere on hydrophilic surfaces strongly influences the mechanical properties of granular matter. Particularly, water bridges considerably increase the adhesion force between solid surfaces. A very large number of papers refer to the evaluation of these capillary forces for different solid-liquid interface geometries.5-10 Most theoretical studies are based on the classical macroscopic theory of capillarity. The only material parameters entering into the macroscopic description are the surface tension σ of the fluid-air interface and the contact angle θ at the line where the solid-, liquid-, and the gas-phase meet. The macroscopic theory deals with the evaluation of the meniscus shape by solving the Young-Laplace equation.11,12 This differential equation accounts for the constant mean curvature of the liquid-vapor interface in mechanical equilibrium. In narrow slits around the point of contact between two solid surfaces, water bridges already appear at vapor pressures considerably below the liquid-vapor equilibrium. Such a capillary condensation leads to a concave meniscus so that the pressure in the liquid bridge is lower than the gas pressure of the surrounding atmosphere. Then the liquid bridge produces an attraction force between the solid surfaces. Usually, the adhesion is maximal if both solid interfaces touch each other and decreases if the distance between the interfaces is increased. In many cases approximate formulas for the adhesion force can be derived. The contribution of the capillary force K to the total pull-off force of a spherical particle adhering to a flat surface is well described by the * To whom correspondence may be addressed. E-mail:
[email protected]. (1) Eastman, T.; Da-Ming, Z. Langmuir 1996, 12, 2859. (2) Sedin, D. L.; Rowlen, K. L. Anal. Chem. 2000, 72, 2183. (3) Skulason, H.; Frisbie, C. D. Langmuir 2000, 16, 6294. (4) Xiao, X.; Qian, L. Langmuir 2000, 16, 8153. (5) Marmur, A. Langmuir 1993, 9, 1922. (6) de Lazzer, A.; Dreyer, M.; Rath, H. J. Langmuir 1999, 15, 4551. (7) Hauge, E. H. Phys. Rev. A 1992, 46, 4994. (8) Adams, M. J.; Johnson, S. A.; Seville, J. P. K.; Willet, C. D. Langmuir 2002, 18, 6180. (9) Willett, C. D.; Adams, J. M.; Johnson, S. A.; Seville, J. P. K. Langmuir 2000, 16, 9396. (10) Attard, P. Langmuir 2000, 16, 4455. (11) Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces, and Membranes; Addison-Wesley Publishing Company: Reading, MA, 1994. (12) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827.
simple approximate relation K ) 4πσr cos θ,13 where r is the particle radius. Surprisingly, this adhesion force does not depend on the vapor pressure of the liquid, which condenses from the surrounding gas phase forming the liquid bridge. Thus the pull-off force of the sphere should be independent of the relative humidity of moist air. However, experimental investigations often reveal that the adhesion forces increase with increasing humidity or show even a more complicated behavior.4,14 In many cases the macroscopic approach seems to be insufficient to account for the capillary forces. The essential material parameter of the macroscopic description, the surface tension, accounts for short-range interactions between the molecules at the fluid-air interface. Thin liquid films are additionally subjected to long-range forces, which originate from the solid substrate or result from specific thin film forces. A possible way to improve the theoretical description is based on the augmented Young-Laplace equation15,16 which includes the disjoining pressure and allows for more specific longrange interactions between solid substrates and liquid films. According to the Lifshitz theory the van der Waals interaction resulting from electric field fluctuations do frequently repel the liquid-air interface away from the solid support so that liquid films tend to thicken.11 For example, water films on hydrophilic surfaces are often rather thick due to long-range repulsion forces. Apart from the van der Waals force, electrostatic repulsion can further increase the film thickness if the solid-liquid interface is charged. Long-range forces influence the size and the shape of water bridges between particles. In many cases the pull-off force between two solid bodies is remarkably modified in comparison to the prediction of the macroscopic theory of capillarity. Numerical results for the capillary force for a cone and a sphere in contact with a flat interface have been published by Gao.17 It was found that the corrections of the adhesion due to long-range van der Waals forces are much larger for a conically shaped solid particle than for a sphere. In this paper we investigate how the prediction for the pull-off force of a wedge is altered when long-range interactions are included in the theoretical description of (13) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1992. (14) Fuji, M.; Machida, K.; Takei, T.; Watanabe, T.; Chikazawa, M. Langmuir 1999, 15, 4584. (15) Frumkin, A. N. Zh. Fiz. Khim 1938, 12, 337. (16) Derjaguin, B. V. Kolloid Zh. 1955, 17, 207. (17) Gao, C. Appl. Phys. Lett. 1997, 71, 1801.
10.1021/la030146m CCC: $27.50 © 2004 American Chemical Society Published on Web 02/13/2004
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Figure 1. Water bridge of a wedge resting on a flat solid surface.
Figure 3. Alteration of the meniscus and the interface areas after lifting an obtuse wedge from the surface. For small distances, H, the meniscus is shifted but does not change its shape. (a) According to the macroscopic theory the solid-liquid interface is reduced by the areas Lz∆l and Lz∆m (on the right side of the wedge), while the meniscus area either remains exactly constant in chemical equilibrium (µ ) constant) or is reduced by a negligibly small area ∆A ∝ H2 if the volume of a nonvolatile fluid bridge does not change. (b) For a wetting fluid in an open system with constant vapor pressure, the mesoscopic theory predicts the occurrence of a thin film on the solid surface even far away from the meniscus. After the wedge is lifted by H, the area of this thin film is changed by Lz∆l and Lz∆m on each side of the wedge. (c) In the case of a wetting fluid in a closed box, the fluid bridge volume is changed by a negligible amount ∆V ∝ LzH2 when the wedge is lifted by a small distance H. The dry solid-vapor interface areas increase by 2Lz∆l and 2Lz∆m at the cost of the moist areas. Figure 2. (a) Cross section of the meniscus, if the macroscopic theory is applicable. (b) According to the mesoscopic theory a thin film condenses from the vapor.
capillary forces. As a compromise between a microscopic and the purely macroscopic theory, the mesoscopic approach based on the augmented Laplace-Young equation is used. The wedge geometry allows deriving analytical results that are applicable to any liquid compounds which wet solid surfaces. We consider the important experimental situation where water vapor condenses on hydrophilic solid surfaces. It will be shown that a large disjoining pressure, which is sometimes observed in thin water films on several substrates, leads to significant correction of the macroscopically evaluated pull-off force.
representation for curved interfaces
Wlv ) (2Lz)
1 1
( (du dx ) )
σ1+
2 1/2
dx
(1)
where the factor 2 takes into account that there is also a meniscus on the left side of the wedge. The free energy contributions of the liquid-solid (Wls) and vapor-solid (Wvs) interfaces can be written as
Wls + Wvs ) (4Lz)
x1γls + (L - x1)γvs
(2)
sin (β/2)
Then the free energy is expressed as F ) Wlv + Wls + Wvs. Omitting an nonimportant constant term, we obtain
2. Macroscopic Theory Figure 1 depicts the geometry of the wedge resting on a flat solid interface. It is assumed that the length Lz of the wedge is much larger than its width B ) 2L cos β. In this case we can determine the free energy and the pulloff force per unit length of the wedge length neglecting the disturbances of the meniscus shape at both ends of the wedge. Formally, the free energy includes three material parameters, namely, the interface energies of the vapor-liquid (σ), the vapor-solid (γvs), and the liquidsolid (γls) interfaces. But it turns out that only σ and the difference γsl - γsv appear in relevant results, so that the number of independent material parameters reduces from three to two. It is useful to introduce the x-y Cartesian system shown in Figure 2. In this system the meniscus on the right half of the wedge can be described by a function u of the coordinate x. Symmetry requires that at x ) 0 the condition (du/dx)x)0 ) 0 is satisfied. The interface energy for the liquid-gas interface can be expressed as Wlv ) 2σAlv, where the area Alv can be obtained from the Moivre
∫-xx
F ) (2Lz)
∫-xx
1 1
(
σ(1 + (u′)2)1/2 +
γls - γvs
)
sin (β/2)
dx
(3)
with u′ ) du/dx. When the pull-off force is measured, different experimental conditions can be imposed. In most cases the fluid bridge and the surrounding vapor are in chemical equilibrium and the chemical potential of the particles in the fluid bridge and the vapor phase are equal. In other cases this equilibrium condition may be violated and the time needed to achieve complete chemical equilibrium is much longer than the duration of a typical experiment. Then the volume of the liquid water bridge remains approximately constant. We have two possibilities to evaluate the pull-off force by using a variational method. When the constant volume condition is applied, the adhesion can be obtained from the free energy derivative K ) -(∂F/∂H)V, where the distance H is defined in Figure 3. If the vapor and the liquid bridge are in thermodynamic equilibrium, the chemical potential µ is equal in the whole system and the grand canonical potential Ω is useful to
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Langmuir, Vol. 20, No. 6, 2004 2229
determine the adhesion force, which is obtained from the relation K ) -(∂Ω/∂H)µ. It turns out that in the framework of the macroscopic theory both relations for K lead to the same value of the pull-off force for a wedge in contact with a flat surface (H ) 0). 2.1. Constant Chemical Potential. The particle density F of the liquid phase is assumed to be constant and much larger than the vapor density Fv. If the liquid bridge is in equilibrium with the vapor, the grand canonical potential Ω is more appropriate than the free energy F. Introducing the chemical potential µ, the potential Ω is obtained from the Legendre transformation Ω ) F -µ ∫ F dV, where V denotes the volume of the fluid water bridge. We mention that in this expression the fluid density F should be replaced by the difference F - Fv if the density of the vapor Fv is not negligibly small compared to F.18 Taking into account the symmetry of the meniscus, the volume of the water bridge is expressed as (Figure 2)
V)
∫-xx
1 1
(2Lz)(u - |x| cot(β/2)) dx
(4)
Thus, for constant density F, the equation which determines the shape of the meniscus is obtained from the condition that the functional
Ω ) F - FµV
(5)
has a minimum. The condition for a minimum of Ω leads to the Eulerian equation
[
]
µF u′ d ) dx (1 + (u′)2)1/2 σ
(6)
) Ω(H) - Ω(0) if the wedge is shifted away from the flat interface by a distance H. Looking at Figure 3a, it becomes obvious that the area ∆n Lz of the liquid-vapor interface does not alter when the wedge is lifted, but the meniscus volume is reduced by ∆V ) LzH2 cot β. Thus we obtain
∆Ω ) (2Lz)(γsv - γsl)(∆l + ∆m) - LzµFH2 cot β (12) where ∆l ) H cot β and ∆m ) H/sin β correspond to the shift of the three-phase line on the solid interface. Taking into account eq 7, the pull-off force K ) -(∂Ω/∂H)µ for H ) 0 can be transformed into K ) -(2Lz)σ cos θ tan[(R + π)/4]. In the important case of a completely wetted interface (θ ) 0), this formula is simplified to
K/Lz ) -2σ tan[(R + π)/4]
Equation 13 can also be obtained by using a more conventional way. According to the Young-Laplace equation, a pressure jump, ∆p ) σ/R, occurs at the curved meniscus and thus the pressure in the liquid bridge is lower than that in the gas phase. For θ ) 0 the adhesion force is equal to 2A0∆p, where 2A0 is the area on the flat solid surface covered by the liquid bridge. A geometrical consideration results in 2A0 ) 2Lz|R| tan[(R + π)/4] and thus eq 13 is obtained. 2.2. Constant Volume. If the volume of a nonvolatile fluid boundary is kept constant, the shape of the meniscus is still circular, but now the circle radius depends on the fluid bridge volume instead of the chemical potential. When the meniscus is shifted, the free energy alteration can be written as
∆F ) (2Lz)(γsv - γsl)(∆l + ∆m) + (2Lz)σ∆n (14)
and the Young equation
σ cos θ ) γsv - γsl
(7)
for the angle θ at the boundaries x ) x1 and x ) -x1 (Figure 2). The solutions of eq 6 are circles
(u - u0)2 + (x - x0)2 ) R2
Let us consider the alteration of ∆l, ∆m, and ∆n after lifting the wedge away from the flat support by a small distance H (Figure 3a). For constant bridge volume V a geometrical consideration results in
∆l ) H cot β + O(H2)
(8)
∆m ) H/sin β + O(H2)
with radius
R ) σ/µF
(9)
Equation 9 determines the radius of curvature R as a function of the chemical potential µ. On the other hand, R can be obtained from the Kelvin equation
σ R) FkT ln(p/p0)
(13)
(15)
and
∆n ) O(H2) where the symbol O(H2) indicates that small terms proportional to H2 are neglected. Then we evaluate ∆F ) F(H) - F(0)
(10) ∆F ) (2Lz)(γsv - γsl)(cot β + 1/sin β) H + O(H2) (16)
where k is the Boltzmann constant, T the temperature, and p the vapor pressure. The reference value p0 corresponds to the condensation pressure, when bulk liquid and the vapor are in equilibrium. Comparing eqs 9 and 10 we arrive at
and thus, taking into account eq 7, the pull-off force K ) -(∂F/∂H)V for H ) 0 has exactly the same value as in the case of the constant chemical potential (eq 13).
µ ) kT ln(p/p0)
According to the macroscopic theory of capillarity, solid surfaces are assumed to be either dry or covered by thick fluid layers. In many experiments, however, thin films with thicknesses varying between a few and a few hundred nanometers are found on solid surfaces. Let us restrict our attention to the case of a wetting fluid (θ ) 0). Particularly, water films condensing from the vapor are formed on hydrophilic solid surfaces. The stability of these films can be explained by long-range repulsive van der Waals and electrostatic forces exerted by solid substrates.
(11)
The case µ ) 0 corresponds to the liquid-vapor equilibrium for a flat liquid-vapor interface (|R| ) ∞). The center (x0, u0) of the circle (eq 8) must be chosen in such a way that the condition (du/dx)x)0 ) 0 and eq 7 are satisfied. Let us consider the change of the thermodynamic potential ∆Ω (18) Rejmer; K.; Dietrich, S.; Napiorkowski, M. Phys. Rev. E 1999, 60, 4027.
3. Mesoscopic Theory
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Long-range forces should also influence the shape of liquid bridges and can be incorporated into the theoretical description. If electric surface charges do not occur, the thickness of an adsorbed liquid film depends on the chemical potential and the effective Hamaker constant, which characterizes the layered system consisting of the solid, the liquid film, and the gaseous phase (Figure 2b). For such a system the free energy (3) is replaced by the extended expression16
F ) (2Lz)
∫-ll
[
σ(1 + (u′)2)1/2 +
γsl sin(β/2)
+
]
W(h) dx sin(β/2) (17)
where
h ) u sin(β/2) - |x| cos(β/2)
(18)
is the local film thickness (Figure 2b) and the potential of the long-range forces
W(h) )
∫h∞ Π(h) dh
(19)
is obtained from the disjoining pressure Π(h).13 The length l, which characterizes the size of the wedge, is assumed to be large enough so that extended thin film regions with almost constant thicknesses exist. In the regions where the film thickness alteration is small (|dh/dx| , 1) the disjoining pressure of many nonpolar fluids can be expressed as13,11
Π(h) )
|A| 6πh3
(20)
where A < 0 is the effective Hamaker constant. Unfortunately, a simple formula for the disjoining pressure in the strongly curved meniscus region does not exist. However, in this paper we do not need this expression for evaluating the pull-off force of a wedge. Inserting eq 17 into eq 5, we obtain from δΩ/δu ) 0 the Eulerian equation17,18
[
]
Figure 4. Different experimental conditions for a fluid adsorbed on a solid surface. (a) Liquid droplet on a plane solid interface. (b) Open system with constant chemical potential. The mesoscopic theory predicts that a thin film with defined thickness h0 condenses from the vapor phase on the solid surface (S > 0 and A < 0). (c) If the solid substrate is confined in a box, and if there is not enough vapor confined in the system, a pancake can be formed. Beyond the line, where the solid, the vapor, and the liquid meet, the pancake thickness is obtained from eq 27. Outside the pancake region the solid surface area is dry, although the spreading condition S > 0 is satisfied.11
corresponds to the free energy density of adsorbed films. Since the area of the solid-liquid interface remains constant, the product of σ + W(h0) and 2Lz(∆l + ∆m) is equal to the change of the free energy resulting from the increase of the liquid film area. Using eq 5 the alteration of the grand canonical potential after lifting the wedge by a distance H is found to be
Ω(H) - Ω(0) ) (2Lz)[σ + W(h0)](cot β + 1/sin β) H - µF[V(H) - V(0)] (23) where
V(H) - V(0) ) (2Lz)(cot β + 1/sin β)h0H - LzH2 cot β (24)
(21)
is the change of the fluid volume. Using eq 23, the geometric relation cot β + 1/sin β ) tan[(R + π)/4] and the formula K ) -(∂Ω/∂H)µ for H ) 0, we obtain the pull-off force
which is called the augmented Young-Laplace equation. Considering the plane film far away from the curved part of the meniscus (u′ ) 0), this equation is reduced to
K ) -(2Lz)[σ + W(h0) - µFh0] tan[(R + π)/4] (25)
µF + Π(h) u′ d ) dx (1 + (u′)2)1/2 σ
Π (h0) ) -µF
(22)
where h0 denotes the equilibrium thickness of the liquid film. 3.1. Open System. The chemical potential of the open system is fixed by a constant vapor pressure. As in the macroscopic approach, it is not necessary to solve the Eulerian equation for predicting the pull-off force. We consider an obtuse wedge assuming that the apex of the meniscus is far away from the wedge tip (Figure 3b). Beyond the curved part of the meniscus, extended plane film regions with constant thickness h0 occur. If the obtuse wedge is lifted from the plane solid surface by a small distance H, the meniscus shifts toward the tip of the wedge but preserves its initial shape. The essential difference between the configurations for H ) 0 and H * 0 is the increase of the thin liquid film areas by 2Lz∆l and 2Lz∆m and a small alteration of the liquid volume by h0(2Lz∆l + 2Lz∆m) - LzH2 cot β. The expression γsl + σ + W(h0)
Replacing the chemical potential µ by using eq 22 leads to
K/Lz ) -2[σ + W(h0) + h0Π(h0)] tan[(R + π)/4] (26) Apart from the surface tension σ we need the disjoining pressure Π(h) as a function of the film thickness h to obtain K. After evaluating the film thickness h0 by solving the equation Π(h0) + FkT ln(p/p0) ) 0, the pull-off force in dependence on the relative vapor pressure p/p0 is obtained by eqs 26 and 19. When the liquid bridge consists of water, the pressure ratio p/p0 coincides with the relative humidity of the air. 3.2. Closed Box. Finally, let us consider a closed box containing the plane substrate and the wedge (Figure 4). If A < 0 and if there is not a sufficient amount of material in the box to cover completely the solid surfaces, dry regions which form pancakes with thickness
h1 ) (|A|/4πS)1/2
(27)
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are expected to be stable,11,12 where S ) γsv - γsl - σ > 0 denotes the spreading coefficient. The pancake shape is the result of an equilibrium between long-range repulsion forces and short-range forces incorporated in the surface tension. If the value of S is positive but small, the pancake thickness can be rather large. Figure 3c demonstrates the expected shape of the meniscus around the wedge for the pancake regime. When the wedge is lifted, the threephase lines are shifted toward the tip of the wedge. Then the dry areas on the solid surfaces increase and areas covered by the fluid film decrease (Figure 3c). Neglecting a possible small reduction of the bridge volume, which produces a term proportional to H2, the alteration of the free energy 17 can be expressed as
∆F ) (2Lz)(γsv - γsl)(cot β + 1/sin β)H + O(H2) (28) The corresponding pull-off force per wedge length Lz is
K/Lz ) -2(γsv - γsl) tan[(R + π)/4]
(29)
In this expression the factor (γsv - γsl) cannot replaced by σ, because the Young equation (7) is not necessarily valid. Because of S > 0 the condition γsv - γsl > σ is satisfied, and thus eq 29 predicts a stronger pull-off force than the macroscopic theory (eq 13).
Summarizing the previously obtained eqs 13, 26, and 29 for wetting fluids (θ ) 0), we find that the pull-off force for the wedge can be written as
(30)
where the factor ω is equal to σ when the macroscopic theory of capillarity is applied. Experimental investigations mostly refer to open systems with constant relative air humidity, which corresponds to a fixed chemical potential µ. For this case we compare the results of the macroscopic and the mesoscopic theory of capillary forces. According to the mesoscopic theory (eq 26), the factor ω can be written as ω ) σ + ∆ω, where
∆ω ) h0Π(h0) +
∫h∞ Π(h) dh 0
(31)
corresponds to the correction of the macroscopic result. Let us first compare the predictions of the macroscopic and mesoscopic description for a nonpolar simple liquid. In this case eq 20 for the disjoining pressure is applicable and the correction term for the pull-off force is
∆ω ) |A|/4πh02
()
ln
4. Discussion
K ) -2ωLz tan[(R + π)/4]
value. Probably, ∆ω is generally negligibly small for simple nonpolar liquids. The most important fluid, which can strongly enhance adhesion between hydrophilic surfaces, is water. Water condensing from the atmosphere forms films on hydrophilic solid surfaces. For many solid substrates the water film is much thicker than that evaluated by the Lifshitz theory,20 which accounts for the van der Waals force. In the case of water on quartz, the disjoining pressure is 3 orders of magnitude larger than expected. Even if the electrostatic contribution produced by surface charges is taken into account, the water film is much thicker than the value estimated by considering a combination of van der Waals and electrostatic repulsion forces.21 A possible explanation of this discrepancy is based on strong structural forces resulting from the alignment of water molecules on solid surfaces. Thus we expect that in the case of water bridges the mesoscopic and the macroscopic theory lead to different results. A complete evaluation of the pull-off force by using eqs 30 and 31 requires knowledge of the disjoining pressure Π(h) for water film thicknesses h in the range between a few and a few hundred nanometers. Accurate and complete data necessary to evaluate ∆ω seem to be not available yet. Pashley et al.22 found that the relation
(32)
As an example we assume that the solid substrate is quartz and the vapor consists of octane. In this case the disjoining pressure is obtained from the relation Π(h) ) (0.9 × 10-21)J/h3.19 The concept attributed to the disjoining pressure and the mesoscopic theory is no longer valid if the thickness of the wetting film becomes comparable to the particle size. Thus the film thickness should be comparable to or larger than 1 nm. For h0 ) 1 nm, eq 31 leads to ∆ω = 1.5 × 10-3 N/m. Compared to the surface tension of octane (σ = 20 10-3 N/m), the value of ∆ω is rather small. The correction to the macroscopically evaluated expression for the pull-off force only amounts a few percent of its total (19) Davis, T. D. Statistical Mechanics of Phases, Interfaces and Thin Films; VCH Publishers: New York, 1996.
p 0.3 nm )p0 h
(33)
fits fairly accurately the experimental data for the film thickness h for water on quartz at humidities p/p0 varying between 0.9 and 0.98. It should be mentioned, however, that results of other authors somewhat differ from eq 33, but the general behavior is similar.23 Combining the equation Π(h0) ) -FkT ln(p/p0) and eq 33 leads to
Π(h) )
(0.3hnm) FkT
(34)
Unfortunately, inserting this disjoining pressure into eq 31 leads to a divergent integral. Hence eq 34 cannot be accurate for very thick films (h > 30 nm), and thus we must replace the upper integration constant ∞ by a finite value h1, e.g., h1 ) 30 nm. Then, for T = 300 K and FkT = 1.33 × 108 N/m2, eq 31 can be written as
∆ω ) -40 × 10-3
(mN) ln(nmh ) + ω
(35)
0
where ω0 is an integration constant. In principle, this integration constant could be evaluated by using accurate data of the disjoining pressure for very thick films (h > 30 nm). Combining eqs 33 and 35 the humidity dependence of the factor ω ) σ + ∆ω is expressed as
ω ) 40 × 10-3
(mN) ln[-ln(p/p )] + ω 0
1
(36)
where ω1 is an unknown constant. For very thick films close to the saturation limit of the vapor pressure, the macroscopic approach is always valid (ω ) σ). In this case the radius of meniscus curvature is still much larger than the film thickness, but the potential W(h) is negligibly small so that long-range forces do not influence the (20) Mu¨ller, H. J. Langmuir 1998, 14, 6789. (21) Pashley, R. M.; Kitchener, J. A. J. Colloid Interface Sci. 1979, 71, 491. (22) Pashley, R. M. J. Colloid Interface Sci. 1980, 78, 246. (23) Gee, L. M.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1990, 140, 450.
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Figure 5. Factor ω of the adhesion force versus the relative humidity for different models. (a) Macroscopic theory for water (σ ) ω ) 73 × 10-3 N/m). (b) Mesoscopic approach with constant chemical potential using the analytical formula for the disjoining pressure of water on quartz proposed in ref 22. (c) Constant volume (pancake regime with A < 0 and S > 0).
capillary force. As eq 36 fails for humidities close to the condensation point (p/p0 > 0.98), eq 34 cannot be extrapolated into the region of very thick films. Assuming that ω = σ for a thick film at humidity p/p0 ) 0.98, the constant term ω1 can be chosen as ω1 = 0.23 N/m. In Figure 5 the dependence of ω on humidity is compared with the macroscopically obtained result. The adhesion force increases with decreasing film thickness or decreasing humidity of the atmosphere. Obviously, water films with thicknesses exceeding a few nanometers produce essential correction of the macroscopically evaluated results for capillary forces. It should be mentioned, however, that for low humidities and microscopically thin water films the pull-off force can oscillate13 and should be described by a suitable microscopic theory. Finally, let us compare recent experiments on the humidity dependence of the adhesion force and predictions of the mesoscopic description. In experimental investigations adhesion forces can be obtained by using the atomic force microscope (AFM) or the surface force apparatus. The pull-off force between flat surfaces and AFM tips or microspheres of different sizes has been extensively studied as a function of relative humidity. Most experiments can be described by a sphere-on-flat model. In this case, instead of eq 13, the macroscopic theory leads to the pull-off force K ) 4πrσ,13 where r is the radius of the sphere. This equation for K is applicable when the minimal curvature radius |R| of the meniscus is small compared to the radius of the sphere (|R| , r). On condition that |R| , r, an approximate expression for the pull-off force can be obtained from the mesoscopic approach by using similar geometrical assumptions as utilized in the macroscopic description.13 Then the mesoscopic theory for the sphereon-flat geometry leads to the approximate formula K ) 4πr(σ + ∆ω),24 which replaces eq 26 for the wedge geometry. Accordingly, the general behavior of the pulloff force in the high humidity region should be the same for both geometries. The pull-off force decreases with increasing humidity as expected for water on quartz (Figure 5). We emphasize that this result is only relevant for hydrophilic substrates with extraordinary high disjoining pressure. If substrates obey the Lifshitz theory, we usually obtain ∆ω , σ and thus the reduction of (24) Schiller, P. In preparation.
Schiller et al.
adhesion force becomes negligibly small. According to the macroscopic theory of capillarity the pull-off force should be independent of humidity. However, most experiments revealed that the adhesion force between hydrophilic interfaces increases with increasing humidity.25 A possible explanation of this behavior is based on the roughness of solid surfaces. The adhesion forces for large contacts, e.g., contacts between micrometer spheres, falls well below the predictions of the theory. Interpreting AFM studies, Shaeffer et al.26 concluded that this discrepancy is due to the existence of many microasperities between two solid interfaces. Capillary bridge effects only approach their theoretical magnitude when meniscus radii and the thicknesses of adsorbed water films begin to exceed the average asperity size.25 Then the prediction of the macroscopic theory for the adhesion force is valid at very high relative humidity close to the saturation point of the vapor. At moderate humidities the capillary force is lower, because a single water bridge which incorporates all asperities of the contact region does not exist. However, if the curvature radius of the AFM tip is small, say 100 nm, only one or a few asperities exist in the contact region. Then the humidity dependence of the adhesion force is expected to behave differently. Recently, He et al.27 published a generic sketch of the functional relationship between the pull-off force and the relative humidity for nanoasperity contacts. Above a threshold below which the adhesion force remains constant, the pull-off force strongly increases with increasing relative humidity, reaches a maximum, and then gradually decreases at high relative humidity. Monte Carlo simulations for the sphere-on-flat geometry display the same behavior of the pull-off force.28 Actually, a maximum in the plot of the pull-off force versus relative humidity was found by several groups, which investigated the adhesion of sharp hydrophilic AFM tips (tip curvature radii between 20 and 100 nm) on hydrophilic surfaces.4,27,25,29 The decrease of the pull-off force with humidity seems to be in agreement with the mesoscopic theory used in this paper (curve b in Figure 5). The rapid grow of the pull-off force at low humidities, however, is not explained by this theoretical approach. We suspect that only a microscopic theory could be able to explain the increase of the adhesion force in the region of low humidity, where the thickness of adsorbed films does not exceed one or a few molecular layers. Further experimental and theoretical investigations are needed to explain the adhesion force between small nanometer contacts. In conclusion, for nonpolar simple fluids the pull-off force is well described by the macroscopic theory of capillarity. Water condensing from the atmosphere on hydrophilic surfaces is expected to behave differently in several cases. When the value of the disjoining pressure is very high, the mesoscopic theory predicts large deviations from the values evaluated by the macroscopic approach. Then the pull-off force of small nanometer contacts is expected to decrease with increasing humidity. Acknowledgment. Financial support of the Deutsche Forschungsgemeinschaft (SFB 285 and Project SCHI 368/ 3-1) is gratefully acknowledged. LA030146M (25) Jones, R.; Pollock, H. M.; Cleaver, J. A. S. Hodges, C. S. Langmuir 2002, 18, 9045. (26) Shaeffer; D. M.; Carpenter, M.; Gady, B.; Reifenberger, R.; Demejo, L. P.; Rimai, D. S. J. Adhes. Sci. Technol. 1995, 9, 1049. (27) He, M.; Blum, A. S.; Aston, D. E.; Buenviaje, C.; Rene, M. O.; Luginbu¨hl, R. J. Chem. Phys. 2001, 114, 1355. (28) Shinto, H.; Uranishi, K.; Miyahara, M.; Higashitani, K. J. Chem. Phys. 2002, 116, 9500. (29) Xu, L.; Lio, A.; Hu, Jun; Ogletree, D. F.; Salmeron, M. J. Phys. Chem. B 1998, 102, 540.
