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Langmuir 2008, 24, 1308-1317
Capillary Adhesion in the Limit of Saturation: Thermodynamics, Self-Consistent Field Modeling and Experiment† Joris Sprakel,*,‡,§ Nicolaas A. M. Besseling,‡ Martien A. Cohen Stuart,‡ and Frans A. M. Leermakers‡ Laboratory of Physical Chemistry and Colloid Science, Wageningen UniVersity, Dreijenplein 6, 6703 HB Wageningen, the Netherlands, and Dutch Polymer Institute (DPI), P.O. Box 902, 5600 AX EindhoVen, The Netherlands ReceiVed July 23, 2007. In Final Form: August 29, 2007 We introduce a simple thermodynamic argument for capillary adhesion forces, for various geometries, in the limit of saturation of the bulk phase. For one specific geometry (i.e., the sphere-plate geometry such as that found in the colloidal probe AFM technique), we provide evidence of the validity of our model by comparison with experiment and self-consistent field calculations. With this latter numerical technique, we also discuss deviations from the macroscopic argument both when the system is moved away from saturation and when the capillary bridge becomes so small that macroscopic thermodynamics is no longer accurate.
Introduction The adhesion between particles mediated by liquid bridges, commonly known as capillary adhesion, is of significant technological and scientific interest. In some cases, capillary adhesion is undesired, such as when it causes dry powders to agglomerate when stored in humid environments or when it causes dispersions to phase separate.1 In other cases, the adhesion forces are specifically employed (e.g., for micromanipulation2 or to create larger structures, such as in the granulation of pharmaceutical powder formulations before tabletting3 and in creating the appropriate mixture of sand and water for building sand castles4). Capillary adhesion is even debated to be partially responsible for the strong adhesion of some types of geckos with walls, enabling them to walk upside down.5 These are just a few examples of the wide variety of cases where capillary adhesion is important. Recently, we have shown that it is possible to measure fully equilibrated capillary forces using colloidal probe atomic force microscopy (CP-AFM).6 The absence of hysteresis in these experiments, which is a rare feature in this type of measurement, could be attributed to the combination of having a system with an ultralow interfacial tension (fast nucleation) and a microscopic geometry (limited diffusion lengths). Because of the equilibrium nature of the capillary condensation that leads to these forces, the results can be used to determine thermodynamic properties such as the interfacial tension between the phase that forms the capillary bridge and the outer bulk phase with which it coexists. To do so, one needs to have a relationship that describes the capillary force for that specific geometry as a function of the interfacial tension and the contact angle. In the CP-AFM method, †
Part of the Molecular and Surface Forces special issue. * Corresponding author. Tel: +31-317-485595. E-mail: joris.sprakel@ wur.nl. ‡ Wageningen University. § Dutch Polymer Institute (DPI).
a solid sphere interacts with a planar wall. We will refer to this situation as the sphere-plate geometry. Several options exist for analyzing the capillary forces in the sphere-plate geometry. A well-known solution is that derived by Orr et al.,7,8 who gave a simple expression for the pull-off force (i.e., the adhesion force in contact with the sphere and the flat wall). The result is often stated to be valid only for large spheres and far enough from saturation for the outer bulk phase or in terms of water-vapor coexistence at relative humidities that are not too high. Another solution is that of Willett et al.,9 who predicted the full forceseparation curve for this geometry, again with some significant approximations. In previous work,6 we have shown that this latter model agrees well with our experimental data and, when used to extract the interfacial tension from the force-distance curves, returns approximately the same interfacial tension as measured independently with another technique. In this article, we will introduce a simple thermodynamic approach to find the capillary adhesion force at zero separation (i.e., the pull-off force), which is valid close to saturation (high relative humidity). Although this article will focus mainly on the sphere-plate geometry, we will extend the argument to the geometry of two spheres. The final results of this approach look similar to what can be found in the literature (e.g., refs 7 and 8) but differ fundamentally in their physical background. Subsequently, we will compare the theoretical considerations with a recent experimental result and will show that the experimental data corresponds very well with the thermodynamic model. Finally, we will present numerical results from self-consistent field calculations to provide evidence for our reasoning and to discuss what happens when we move away from saturation and away from the macroscopic limit where the theoretical argument is valid.
Macroscopic Model for Capillary Adhesion in the Limit of Saturation
(1) Olsson, M.; Joabsson, F.; Picullel, L. Langmuir 2005, 21, 1560. (2) Rollot, Y.; Regnier, S.; Guinot, J. C. Int. J. Adhes. Adhes. 1999, 19, 35. (3) Thiel, W. J.; Nguyen, L. T.; Stephenson, P. L. Powder Technol. 1983, 34,
In this section, we will discuss some thermodynamic arguments that can be used to predict capillary adhesion forces on the basis
(4) Nowak, S.; Samadani, A.; Kudrolli, A. Nat. Phys. 2005, 1, 50. (5) Huber, G.; Mantz, H.; Spolenak, R.; Mecke, K.; Jacobs, K.; Gorb, S. N.; Arzt. E. Proc. Natl. Acad. Sci. U.S.A. 2005, 102, 16293. (6) Sprakel, J.; Besseling, N. A. M.; Leermakers, F. A. M.; Cohen Stuart, M. A. Phys. ReV. Lett. 2007, 99, 104504.
(7) Orr, F. M.; Scriven, L. E.; Rivas, A. P. J. Fluid Mech. 1975, 67, 723. (8) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: London, 1985. (9) Willet, C. D.; Adams, M. J.; Johnson, S. A.; Seville, J. P. K. Langmuir 2000, 16, 9396.
75.
10.1021/la702222f CCC: $40.75 © 2008 American Chemical Society Published on Web 11/17/2007
Capillary Adhesion in the Limit of Saturation
Langmuir, Vol. 24, No. 4, 2008 1309
Figure 1. Schematic representation of a capillary bridge between a sphere of radius R and a flat wall indicating the principal radii of curvature r1 and r2, chosen at the waist of the capillary bridge, and the surface separation h. The entire system is rotationally symmetric around the central axis. The situation shown here is for the case of complete wetting (i.e., the contact angle θ ) 0 and an undersaturated bulk phase).
of macroscopic considerations. Hence, the width of the interface is negligible compared to the size of the capillary bridge; therefore, the position of the interface is unambiguously defined. This also implies that the interfacial tension is a constant and that curvature corrections to the interfacial tension are negligible. Self-consistent field calculations, using a technique similar to that used later in this article, have revealed that curvature corrections to the interfacial tension are very small, both for binary systems of simple liquids10 and for ternary polymer/polymer/solvent mixtures.11 We consider two solid surfaces, one of which is a flat wall and the other a solid sphere with radius R, separated by a distance h (Figure 1). When these surfaces are immersed in a fluid mixture having a miscibility gap, a new phase can be formed between the two surfaces that is thermodynamically stable, even when the liquid mixture is undersaturated. When the bulk phase is not saturated, the condensate cannot be stable in the absence of these surfaces. This process of forming a new stable phase between two surfaces is called capillary condensation. Capillary condensation occurs only when the condensing phase preferentially wets the surfaces (i.e., a contact angle θ measured through the condensed phase of less than 90°) and when the separation distance between the surfaces h is small enough. In essence, the phenomenon of capillary condensation is captured by the Kelvin equation12
µ - µ* ) Vm∆P
(1)
which relates the relative under- or oversaturation of the outer bulk phase, given by the difference in the actual chemical potential µ and the chemical potential at saturation µ*, to the pressure difference between the bulk phase and the condensate, ∆P. The coefficient Vm is the molecular volume. The difference in pressure between the condensate and the outer bulk phase is given by the Young-Laplace equation12
∆P ) γJ ) γ
(
1 1 + r1 r2
)
(2)
where γ is the interfacial tension between the condensate and -1 the bulk phase and J ) (r-1 1 + r2 ) is the mean curvature defined by the two principal radii of curvature r1 and r2, as indicated in (10) Oversteegen, S. M.; Blokhuis, E. M. J. Chem. Phys. 2000, 112, 2980. (11) Leermakers, F. A. M.; Barneveld, P. A.; Sprakel, J.; Besseling, N. A. M. Phys. ReV. Lett. 2006, 97, 066103.
Figure 1. With these two basic thermodynamic equations, we can derive the main premises of capillary condensation. If we have a condensed liquid in a bulk phase below its saturation point, then we can see from the Kelvin equation that this would require a negative Laplace pressure to meet equilibrium. From the Laplace equation, we can deduce that a spherical droplet of the condensed phase, in which r1 is equal to r2 by definition, cannot have a negative Laplace pressure. In other words, this situation will not be thermodynamically stable. For a capillary condensate however, which in sphere-plate geometry has a characteristic hyperboloid shape, the two radii of curvature are not equal. By definition, r1 is always positive and r2 is negative, hence when -1/r2 is larger than then 1/r1 a net negative curvature results and the condensed phase can be thermodynamically stable. Note that, because the Laplace pressure must be homogeneous within the capillary condensate, the mean curvature J must be constant over the entire surface of the capillary bridge. r1 and r2, however, may vary from point to point along the interface. In the following discussion, we conveniently choose r1 and r2 at the waist of the capillary bridge (i.e., the point where the bridge has its smallest diameter) because this is the only position where the interfacial tension acts in the same direction as the total adhesion force between the two surfaces. In a grand canonical ensemble, that is, when the total system volume V, the temperature T, and the chemical potentials µ are fixed, the interaction energy between the two surfaces (Gint) in presence of a capillary condensate is given by
Gint (h) ) γAc(h) - ∆PVc(h)
(3)
where Ac and Vc are the surface area and the volume of the capillary bridge, respectively. The interaction force between the surfaces is found by differentiating with respect to the separation distance h
F(h) )
-∂Gint(h) ) -γLb(h) + ∆PAb(h) ∂h
(4)
where Lb is the circumference of the capillary bridge at its waist and Ab is the surface area of the cross section at this same position. Because the capillary bridge is rotationally symmetric, we find Lb ) 2πr1 and Ab ) πr12. With eq 2, we can rewrite eq 4 as
F ) -2πr1γ + πr12
(
)
1 1 + γ r1 r2
(5)
In capillary condensation, in full thermodynamic equilibrium, the Laplace pressure is smaller than or equal to zero, hence both terms in the above equation give a zero or negative contribution to the force; in other words, capillary condensation necessarily leads to attraction between the two surfaces. Although the result in eq 5 for the adhesion force seems fairly simple, the difficulty is in predicting the values of r1 and r2 for a given situation. For certain specific situations, this can be dealt with exactly, as we will show below. One frequently used solution has been given by Orr et al.,7 who derived the adhesion force between and sphere and a flat wall at contact (h ) 0) due to the presence of a capillary bridge. The simple result that they give, F(h ) 0) ) -4πRγ cos θ,8 is found after some severe approximations. Their derivation starts by neglecting the γA term in the interaction energy (given in eq 3), leaving only the ∆PV term. The second main step in their derivation is a geometric approximation (i.e., -r2 , r1) to (12) Lyklema, J. Fundamentals of Interface and Colloid Science; Academic Press: London, 1991; Vol. 1.
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Sprakel et al.
R cos θ, which follows from simple geometric arguments, as illustrated in Figure 2. Combining this result with eq 8, we find an exact expression for the capillary adhesion force between a sphere and a plate in the limit of saturation:
lim F(h ) 0) ) -2πRγ cos θ
µfµ*
Figure 2. Schematic representation of a capillary bridge between a sphere of radius R and a flat wall at contact (h ) 0) at saturation of the surrounding bulk phase and with a finite contact angle θ. Under these specific conditions, the Laplace pressure must be zero (eq 6), hence r1 ) -r2. From geometric arguments, we derive r1 ) R cos θ.
eliminate one of the two radii of curvature from the equations, which holds only for large enough spheres. Finally, the result as stated above is obtained. However, if we consider Kelvin’s law (eq 1), then we must realize that this approach is invalid in the limit of saturation, where the chemical potential of the component that wets the surfaces µ becomes equal to the chemical potential of that component at saturation µ* because the Laplace pressure should vanish exactly at saturation:
lim ∆P ) 0
µfµ*
(6)
Hence, the assumption that the γA term becomes negligible compared to the Laplace pressure term is fundamentally incorrect when the bulk phase is close to saturation. Of course, this is recognized by other researchers in the field.13 Here we will introduce a simple thermodynamic approach to describe the capillary adhesion force at contact in the limit of saturation, which does not contain severe approximations. Starting from eqs 2 and 6, we easily find that in the limit of saturation the mean curvature must be zero, hence
lim r1 ) - r2
µfµ*
(7)
For the capillary bridge, the condition in eq 7 can be satisfied by the hyperboloid shape of the condensate. In this limit, the interaction force between the surfaces can be simplified to
lim F ) -2πr1γ
µfµ*
(8)
because the Laplace pressure term in eq 5 must vanish at saturation. Now let us focus on the adhesion force, which is the force that holds the sphere and the wall together when they are in contact (i.e., at h ) 0). For this condition in the limit of saturation, a schematic representation is given in Figure 2. For the case of complete wetting in the limit of saturation, the minimal width of the neck is found at the meridian of the sphere because this is the only solution for which r1 equals -r2 while satisfying the boundary condition θ ) 0. This means that in the limit of saturation and for the case of complete wetting r1 ) -r2 ) R. For nonzero contact angles smaller that 90°, r1 ) -r2 ) (13) Pakarinen, O. H.; Foster, A. S.; Paajanen, M.; Kalinainen, T.; Katainen, J.; Makkonen, I.; Lahtinen, J.; Nieminen, R. M. Model. Simul. Mater. Sci. Eng. 2005, 13, 1175.
sphere-plate
(9)
Note that our result differs only by a factor of 2 from the wellknown result of Orr et al., as discussed above, yet the derivation of our result is significantly different and contains no severe approximations. The result in eq 9 is analogous to the expression used in the Wilhelmy plate method for measuring interfacial tension.14 However, the underlying physical arguments are different. In the Wilhelmy plate equation, the hydrostatic pressure in the capillary bridge balances the finite Laplace pressure, whereas here we neglect gravitational effects and have a vanishing Laplace pressure. An assessment of the effect of gravity on these types of capillary bridges is given in ref 15. One way of estimating the relative importance of gravity is by means of the dimensionless Bond number, which is the ratio of the body forces (gravity) to the interfacial tension forces:
Bo )
∆Fgl2 γ
(10)
For our problem, ∆F is the difference in density between the phase forming the capillary bridge and the surrounding bulk phase, g is gravitational acceleration, and l is the characteristic dimension of the capillary bridge. For large Bond numbers (.1), the body forces dominate, and for small Bond numbers (,1), the surface tension forces are most important. In the CP-AFM setup, where the sphere size is typically on the order of micrometers, l will also be on the order of micrometers or smaller. For a capillary bridge of water, in coexistence with its vapor, the Bond number is on the order of 10-7, and for the system discussed in the experimental part of this article, which demixes into two liquid phases with only minor density differences and ultralow interfacial tension, we find a Bond number of around 10-4. This means that in both cases gravitational effects are completely negligible. For complete wetting (θ ) 0) and at saturation, the wetting layers on the flat surface can grow infinitely large, and a macroscopic bulk phase that swallows the particle will be formed instead of the mesoscopic capillary bridge between the two surfaces. For systems with a nonzero contact angle, this is not the case because the adsorption layers will always have finite microscopic dimensions, even at saturation. This implies that defining capillary forces at saturation for the latter type of wetting is useful but has no meaning when we are dealing with complete wetting. As a consequence, we formulate our results in the limit of saturation such that the analysis is valid both for complete and partial wetting situations. In the Introduction, we mentioned the model of Willett and co-workers,9 who gave a numerical solution of the capillary force as a function of the separation distance h in the toroidal approximation (i.e., assuming that the shape of the capillary bridge matches the void in the center of a torus and by assuming that the volume of the liquid bridge remains constant at all values of h). If we numerically take the limit to zero separation of the adhesion force predicted with the model of Willett et al., then (14) Heertjes, P. M.; Smet, E. C. D.; Witvoet, W. C. Chem. Eng. Sci. 1971, 26, 1479. (15) Adams, M. J.; Johnson, S. A.; Seville, J. P. K.; Willett, C. D. Langmuir 2002, 18, 6180.
Capillary Adhesion in the Limit of Saturation
Figure 3. Schematic representation of a capillary bridge between two spheres of equal radius R at contact (h ) 0) and at saturation of the surrounding bulk phase. At saturation, r1 ) -r2 because the Laplace pressure must be zero (eq 6). Using Pythagoras’ theorem, we find that r1 ) 2/3 R cos θ.
we find a pull-off force that approximately, within 5%, equals -2πγR, which is in accordance with eq 9. We must note that Willett’s model does not account at all for the relative undersaturation under which the capillary condensation occurs. The fact that the limit of Willett’s model to h ) 0 gives approximately the same result as eq 9 seems to indicate that it is valid in the limit of saturation, although this does not become apparent from the derivation of their model. We can extend the model for the adhesion force in the limit of saturation to the case of two spheres of equal radius R. For this case, eqs 7 and 8 must again apply. However, the geometry of the capillary bridge is different (Figure 3). With the triangle drawn between the center of one of the solid spheres, the point of contact between the spheres, and the center of the circle that describes the outer radius of curvature, we find that r1 ) -r2 ) 2/ R using Pythagoras’ theorem. This is valid for complete wetting, 3 and for the same reasons as above, we can extend this to partial wetting by correcting with the cosine of the contact angle. The adhesion force between two spheres of equal size in the limit of saturation is thus given by
4 lim F(h ) 0) ) - πRγ cos θ 3
µfµ*
equal spheres (11)
Also, for this geometry an approximate argument is given in ref 8 that is based on the same reasoning as explained above (i.e., neglecting the γA term in the interaction energy and with significant geometric approximations). As in the previous case for the sphere-plate geometry, the result again differs from our eq 11 by a numerical constant; in this case, the difference is a factor of 2/3. We can also derive a more general geometric argument that is not limited to the two specific cases (sphere-plate and equal spheres) discussed above. For the case of two spheres with arbitrary radii R1 and R2, we can construct the geometry, an example of which is shown in Figure 4. The following considerations are valid only for the case of complete wetting (i.e., at zero contact angle). Again, we start with the fact that r1 ) -r2 (eq 7). Below, we express all radii of curvature in terms of r1. First we must define the quantity f that represents the fraction of the vertical line that connects the two centers of the spheres belonging to the upper triangle. Hence, (1 - f) is the fraction of that same vertical line that belongs to the lower of the two adjacent triangles. The total length of this vertical line is R1 + R2.
Langmuir, Vol. 24, No. 4, 2008 1311
Figure 4. Schematic representation of a capillary bridge between two spheres with arbitrary radii R1 and R2, at zero separation (h ) 0), at saturation of the surrounding bulk phase (µ ) µ*), and for complete wetting (θ ) 0). Equation 7 dictates that r1 ) -r2 because the Laplace pressure must be zero at saturation (eq 6). The quantity f is the fraction of the vertical line between the centers of the spheres that is part of the upper triangle, hence (1 - f) is the fraction of the line that is part of the lower adjacent triangle.
Using Pythagoras’ theorem, we find the following relations
f 2(R1 + R2)2 + (2r1)2 ) (R1 + r1)2
(12)
for the upper triangle that is linked to the sphere with radius R1 and
(1 - f)2(R1 + R2)2 + (2r1)2 ) (R2 + r1)2
(13)
for the lower triangle that is linked to the sphere with radius R2. Combining these two equations leads to the following expression for f:
(
2 2 1 (R1 + r1) - (R2 + r1) f) +1 2 (R + R )2 1
2
)
(14)
For the two specific geometries that we have discussed above, we can find numerical values for f. For the case of two spheres of equal radius, f ) 1/2. This value reflects the symmetry of the system with respect to the plane of contact between the two spheres. For the sphere-plate geometry in which R2 f ∞, we find that f ) 0. This latter case of course is the stronger case of asymmetry because one of the surfaces has lost its curvature. To use these considerations in predicting the capillary adhesion force on the basis of eq 8, we need to express r1 as a function of f, R1, and R2. We can do so by solving eq 12. This gives a quadratic equation in terms of r1, which logically leads to two possible solutions. With the boundary condition r1 > 0, we find the proper solution:
1 1 r1 ) R1 + x4R12 - 3f 2(R1 + R2)2 3 3
(15)
For the geometry of two equal spheres (R1 ) R2 ) R) for which f ) 1/2, we can show that eq 15 returns r1 ) 2/3R, which is the same result that we have derived above, leading to eq 11. For the sphere-plate geometry (R1 ) R, R2 f ∞ and f ) 0), eq 15 returns r1 ) R, which is again exactly the same as the result leading to eq 9.
Capillary Forces Measured with Colloidal Probe AFM In a recent paper,6 we have shown that it is possible to measure fully equilibrated forces using the colloidal probe AFM technique
1312 Langmuir, Vol. 24, No. 4, 2008
developed by Ducker et al.16 in systems with ultralow interfacial tension. Here we show one example of such an experiment in a system for which an independent experimental result for interfacial tension is available. Silica spheres with a radius of 3 µm (a gift from Philips Laboratories, The Netherlands) are attached to large narrowlegged AFM contact-mode cantilevers, using the method of Giesbers,17 and oxidized silicon wafers (WaferNet, Germany) are used as the flat substrates. The attached probes and substrates are washed with water and ethanol and cleaned in a plasma cleaner (Harrick PDC-32G). The spring constant of the cantilever was determined to be 0.07 ( 0.005 N/m, using the method of Hutter and Bechhoefer.18 The force measurements are performed on a Nanoscope 4 instrument equipped with with a PicoForce scanner (Digital Instruments). The separation between the surfaces is found by determining the zero-distance position (i.e., where the slope of the cantilever deflection versus vertical piezo movement reaches unity (constant compliance regime)). It is known that there is some uncertainty in this method19 that is generally on the order of O(1 nm), which is not significant in the context of the longranged forces discussed here. High-molecular-weight fish gelatin (Norland Products Inc.) and dextran (500 kDa, Sigma) are dissolved in 50 mM KCl (to suppress electrostatic interactions between the silica surfaces and to aid dissolution) at an overall polymer concentration of 100 g/L. Because of the miscibility gap between these biopolymers, two aqueous macroscopic phases are formed, one enriched in dextran and the other enriched in gelatin. Microscopy has revealed that when these two-phase systems are brought into contact with silica surfaces the gelatin-rich phase completely wets the surface (θ ≈ 0),20 hence it is expected that when we bring the dextran-rich phase between two silica surfaces together at close separation, a capillary bridge of the gelatin-enriched phase is formed. We fill the liquid cell of the atomic force miscroscope with the dextran-rich phase at its binodal composition, meaning that this phase is at its saturation point: φbgelatin/ φ/gelatin ) 1. The force-distance curve that results from bringing together the two surfaces in the atomic force microscope is shown in Figure 5. We see a long- ranged, weak (≈ 100 pN) attractive force due to the capillary condensation of a gelatin-rich phase between the surfaces. van der Waals forces can be excluded as a possible origin of the attraction because the force measured here decays much slower than what is predicted for van der Waals attraction between a sphere and a flat surface (i.e., FvdW ∝ h-2). We should also point out that van der Waals forces are almost never found in CP-AFM measurements between silica surfaces in aqueous systems (see ref 21), although theory predicts that it should be possible to measure them in the CP-AFM setup. At small separation distances (
