Capillary Forces between Two Spheres with a Fixed Volume Liquid

The capillary force as a function of the separation distance, for a liquid bridge with a ..... force is valid only at H = 0 and invalid for any finite...
4 downloads 0 Views 196KB Size
Download PDF
10992

Langmuir 2005, 21, 10992-10997

Capillary Forces between Two Spheres with a Fixed Volume Liquid Bridge: Theory and Experiment Yakov I. Rabinovich,† Madhavan S. Esayanur,‡ and Brij M. Moudgil*,†,‡ Particle Engineering Research Center and Department of Materials Science and Engineering, University of Florida, Gainesville, Florida 32611 Received June 30, 2005. In Final Form: August 16, 2005 Capillary forces are commonly encountered in nature because of the spontaneous condensation of liquid from surrounding vapor, leading to the formation of a liquid bridge. In most cases, the advent of capillary forces by condensation leads to undesirable events such as an increase in the strength of granules, which leads to flow problems and/or caking of powder samples. The prediction and control of the magnitude of capillary forces is necessary for eliminating or minimizing these undesirable events. The capillary force as a function of the separation distance, for a liquid bridge with a fixed volume in a sphere/plate geometry, was calculated using different expressions reported previously. These relationships were developed earlier, either on the basis of the total energy of two solid surfaces interacting through the liquid and the ambient vapor or by direct calculation of the force as a result of the differential gas pressure across the liquid bridge. It is shown that the results obtained using these methodologies (total energy or differential pressure) agree, confirming that a total-energy-based approach is applicable, despite the thermodynamic nonequilibrium conditions of a fixed volume bridge rupture process. On the basis of the formulas for the capillary force between a sphere and a plane surface, equations for the calculation of the capillary force between two spheres are derived in this study. Experimental measurements using an atomic force microscope (AFM) validate the formulas developed. The most common approach for transforming interaction force or energy from that of sphere/plate geometry to that of sphere/sphere geometry is the Derjaguin approximation. However, a comparison of the theoretical formulas derived in this study for the interaction of two spheres with those for sphere/plate geometry shows that the Derjaguin approximation is only valid at zero separation distance. This study attempts to explain the inapplicability of the Derjaguin approximation at larger separation distances. In particular, the area of a liquid bridge changes with the separation distance, H, and thereby does not permit the application of the “integral method,” as used in the Derjaguin approximation.

1. Introduction The increase in the strength of powders due to the absorption of moisture from the ambient atmosphere, leading to flow and process-related problems, has been identified and researched since the late 1960s. On the other hand, the applicability of capillary forces in industrially important applications, such as crystallization and agglomeration, is well established, and considerable effort has been directed toward developing analytical expressions to quantify and estimate the force due to the presence of a capillary bridge between two surfaces.1,2 Most studies reported so far have focused on the case of a sphere/plate geometry or sphere/sphere geometry with spheres of equal size. Marmur3 and de Lazzer et al.4 independently developed approximate analytical expressions for the interaction of complex geometries (spherical, paraboloidal, and conical particles) with a flat surface and showed reasonable agreement with the theoretical results obtained using the numerical integration of the Young* Address correspondence to: Dr. Brij M. Moudgil, 205 Particle Science and Technology Building, P.O. Box 116135, Gainesville, FL 32611-6135. E-mail: [email protected]. Phone: (352) 8461194. Fax: (352) 846-1196. † Particle Engineering Research Center. ‡ Department of Materials Science and Engineering. (1) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: San Diego, CA, 1992. (2) Rabinovich, Y. I.; Esayanur, M. S.; Johanson, K. D.; Adler, J. J.; Moudgil, B. M. Measurement of Oil-Mediated Particle Adhesion to a Silica Substrate by Atomic Force Microscopy. J. Adhes. Sci. Technol. 2002, 16 (7), 887-903. (3) Marmur, A. Tip Surface Capillary Interactions. Langmuir 1993, 9 (7), 1922-1926. (4) de Lazzer, A.; Dreyer, M.; Rath, H. R. Particle-Surface Capillary Forces. Langmuir 1999, 15 (13), 4551-4559.

Laplace equation. The derived expressions for capillary forces were plotted as a function of the parameter x/R, in which x is one of the radii of the liquid bridge, and R is the radius of the particle. However, the radius of curvature, x, cannot be measured directly, requiring relationships between x and other parameters, such as the separation distance, H, the volume of the liquid bridge, V, and the relative humidity of the environment. The unknown variables such as the radius of curvature and the volume of the liquid bridge have been approximated using either geometric or numerical simplification to provide a reasonable estimate of the capillary force. Formulas for the capillary force between solids with a liquid bridge have been reported for thermodynamic equilibrium5 as well as for nonequilibrium cases1. The first case corresponds to a constant radius of the liquid bridge, as determined from the undersaturated vapor pressure in the atmosphere using the Kelvin equation6. The second (nonequilibrium) case is related to a fixed volume liquid bridge. Previously reported experimental measurements of capillary forces have mainly focused on the condensation of water vapor from the ambient atmosphere forming a liquid bridge.7-9 Studies relating to the use of volatile liquids, such as water, for the (5) Newmann, A. W.; J. K. Spelt Applied Surface Thermodynamics; Marcel Dekker: New York, 1996. (6) Adamson, A. W. Physical Chemistry of Surfaces, 2nd ed.; Interscience Publishers: New York, 1967. (7) Fisher, L. R.; Israelachvili, J. N. Experimental Studies on the Applicability of the Kelvin Equation to Highly Curved Concave Menisci. J. Colloid Interface Sci. 1981, 80 (2), 528-541. (8) Biggs, S.; Cain, R. G.; Dagastine, R. R.; Page, N. W. Direct Measurements of the Adhesion between a Glass Particle and a Glass Surface in a Humid Atmosphere. J. Adhes. Sci. Technol. 2002, 16 (7), 869-885.

10.1021/la0517639 CCC: $30.25 © 2005 American Chemical Society Published on Web 10/05/2005

Capillary Force between Spheres with Liquid Bridges

formation of the liquid bridge would correspond to the constant radius condition. Fewer studies have focused on experiments with nonvolatile liquids.2,10,11 The nonvolatile liquid bridges conform to the constant volume condition. Israelachvili1 developed an expression for calculating the capillary force of a fixed volume liquid bridge on the basis of the total energy of the bridge. However, the validity of the expression for a thermodynamically nonequilibrium process, such as the pulling of a liquid bridge of fixed volume, was not discussed. Rabinovich et al.2 used another approach based on the pressure difference across the liquid bridge to derive a formula for the capillary force. In this paper, a comparison between the theoretical results derived from the formulas developed by Israelachvili1 and Rabinovich et al.2 is provided. One of the most common and simple geometries for the estimation of the capillary force is the sphere/plate interaction. The extrapolation of translating the theoretical expressions for estimating the capillary force from the sphere/plate geometry to the sphere/sphere geometry is not straightforward. The formulas for the interaction of two spheres separated by a liquid bridge are also developed in this study, and their predictions are compared against experimentally measured values using an atomic force microscope (AFM). The results of the calculations based on sphere/sphere interaction formulas are discussed from the point of view of the Derjaguin approximation.12 2. Experimental Section 2.1 Materials. Capillary forces between individual glass spheres 20-50 µm in diameter (from Duke Scientific Corp.) and silica substrates were measured using a Digital Instruments AFM. The root-mean-square (RMS) roughness of the glass spheres was measured to be 0.2 nm by AFM imaging. The silica substrates used were provided by Dr. H. Arwin (Linko¨ping University, Sweden) and were fabricated from 180-nm-thick oxidized silicon wafers with a 0.2-nm RMS roughness. All silica surfaces were cleaned by being rinsed in acetone, methanol, and deionized (DI) water immediately before experimentation. Colloidal glass particles were glued to a tapping mode TESP rectangular AFM cantilever (supplied by Digital Instruments Inc.) using a low-temperature melting glue (Epon R 1004f, Shell Chemical Company). Glass spheres were also glued to a silica substrate using a two-part epoxy adhesive. White mineral oil of “sharpening stone” grade was obtained from Norton Co. (Littleton, NH) and was used to form the liquid layer between the two surfaces. The viscosity of the oil, as measured using a capillary viscometer, was 25 cP. The advancing contact angle of the oil on the silica and glass surfaces ranged from 0 to 10°. 2.2 Methods. Surface forces were measured on a Digital Instruments Nanoscope III AFM according to the methods described by Ducker et al.13 The spring constant k for each cantilever was calibrated by the frequency method suggested by Cleveland et al.14 The average value of the spring constant k was 27 ( 5 N/m. The sphere, which was attached to a cantilever of (9) Quon, R. A.; Ulman, A.; Vanderlick, T. K. Impact of Humidity on Adhesion between Rough Surfaces. Langmuir 2000, 16 (23), 89128916. (10) Willett, C. D.; Adams, M. J.; Johnson, S. A.; Seville, J. P. K. Capillary Bridges between Two Spherical Bodies. Langmuir 2000, 16 (24), 9396-9405. (11) Moudgil, B. M.; Rabinovich, Y. I.; Esayanur, M. S.; Johanson, K. D.; J.; Adler, J. Oil Mediated Particulate Adhesion and Mechanical Properties of Powder. Proceedings of the 4th World Congress on Particle Technology, Sydney, Australia, 2002. (12) Derjaguin, B. V. Untersuchungen ber die Reibung und Adhesion (in German). Kolloid-Z. 1934, 69, 155-164. (13) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Measurement of Forces in Liquids Using a Force Microscope. Langmuir 1992, 8 (7), 1831-1836. (14) Cleveland, J. P.; Manne, S.; Bocek, D.; Hansma, P. K. A Nondestructive Method for Determining the Spring Constant of Cantilevers for Scanning Force Microscopy. Rev. Sci. Instrum. 1993, 64 (2), 403-405.

Langmuir, Vol. 21, No. 24, 2005 10993

Figure 1. Illustration of a sphere/plate geometry interaction with a liquid bridge. a known spring constant, was positioned close to the flat substrate. Several larger drops (millimeters in diameter) of oil were deposited on the flat silica substrate using a microsyringe, and these drops were disrupted by a gas jet, leaving many small droplets (micrometers in diameter). The substrate was moved toward and away from the colloidal probe by a piezoelectric scanner, and the deflection of the cantilever was monitored by a laser reflected from the top of the cantilever onto a positionsensitive photodiode. The force (given by the cantilever deflection times the spring constant) between the two surfaces was measured as a function of the separation distance. The force/ distance profiles were normalized with respect to the radius of the sphere. In other words, the data are presented in terms of energy per unit area of the flat surface. This allows for the determination of the force for different particle geometries and sizes, as long as the distance range, where the forces act, is less than the radius of curvature of the particles. To predict the adhesion force for either of the boundary conditions (constant radius or constant volume), the initial volume of the liquid bridge must be known. For the condensation of vapor in a capillary, this can be derived from the geometry of the bridge and the Kelvin equation. For nonvolatile liquids (e.g., oil) the volume is dependent on the size of the oil droplet present on the surface and cannot be predicted from theory.

3. Theoretical Calculation of Capillary Adhesion Force between a Sphere and a Plate for a Liquid Bridge of Fixed Volume An expression for the calculation of the capillary force between a sphere and a plate separated by a liquid bridge of small fixed volume, V, as illustrated in Figure 1, is given by

Fsp/pl ) -

4πγR cos θ - 2πγR sin R sin(θ + R) 1 + H/dsp/pl

(1)

in which γ is the liquid surface tension, θ is the contact angle, R is the “embracing angle”, and H is the shortest distance between the sphere and the plate. The above expression is similar to the derivation by Israelachvili1 except that the second term on the right-hand side of eq 1 was not included. This term is the force due to the vertical component of the surface tension of the liquid bridge.3,4 This force does not play a significant role in the case of a small volume liquid bridge (i.e., R , 1). Rabinovich et al.11 derived an expression for determining dsp/pl:

dsp/pl ) -H + xH2 + V/(πR)

(2)

On the other hand, it appears from the geometry, as shown in Figure 1, that

10994

Langmuir, Vol. 21, No. 24, 2005

Rabinovich et al.

dsp/pl ) RR2/2

(3)

Comparing eqs 2 and 3, we obtain the following relationship between the embracing angle, R, and the liquid bridge volume, V:

R2sp/pl )

( x

2H -1 + R

1+

)

V πRH2

(4)

Equation 1 can be further simplified for small and large separation distances; namely, for small H, V/πRH2 . 1. Combining eqs 1 and 2 results in (see eq A10 in Appendix II)

4πγR cos θ - 2πγR sin R sin(θ + R) Fsp/pl ) 1 + HxπR/V

(5)

For a large separation distance, H, the expression is modified as (see eq A7 in Appendix II)

Fsp/pl ) -

2γV cos θ - 2πγR sin R sin(θ + R) (6) H2

Figure 2. Theoretical force/distance profiles of sphere/plate interactions. The two sets of curves, curves 1 and 3 and curves 2 and 4, are theoretical estimates based on eqs 8-10 and eqs 1, 2, and 4, respectively. The following parameters are used: for curves 1 and 2, the radius of the sphere, R, is 12 µm, the volume of the liquid bridge, V, is 7 × 108 nm3, and the contact angle θ is 0°; for curves 3 and 4, R ) 25 µm, V ) 170 × 108 nm3, and θ ) 10°. The surface tension of the oil γ is 26 mN/m. The agreement of the two sets of curves confirms the validity of either approach (total energy and the pressure difference across the bridge) for the estimation of the capillary force.

Note that, in eq 6, the first term on the right-hand side is independent of the radius of the sphere, R. The derivation by Israelachvili, which is based on the total energy of the liquid bridge, is given by

Wtot,sp/pl ) -2πγR2R2 cos θ

(7)

In this equation, only the energy of the solid surface under the liquid bridge is taken into account, and the energy of the surface of the meniscus itself is ignored. As a result, Israelachvili1 obtained an expression for Fsp/pl with only the first term on the right-hand side of eq 1. However, to check the validity of eq 1 for a thermodynamically nonequilibrium process (the separation of a fixed volume liquid bridge), Rabinovich et al.2 developed the following formulas based on the pressure difference inside and outside the liquid bridge:

Fsp/pl/R )

{

πγx′ 1 -

x′[cos(θp + R) + cos θf)] H′ + 1 - x1 - x′

2

- 2 sin(θp + R)

}

(8)

R ) arctan[x′/(1 - x′2)]

(9)

x′ ) x-2H′ + 2xH′2 + V/πR3

(10)

in which θp and θf are the contact angles of liquid on the spherical particle and the flat surface, respectively; x′ ) x/R, in which x is one of the liquid bridge radii (as shown in Figure 1); and H′ ) H/R. The theoretical capillary force calculated using eqs 1, 2, 4, and 8-10, are compared in Figure 2. The two pairs of curves relate to two different liquid bridge volumes. The two curves in each set agree, suggesting that either of the two approaches (that of Israelachvili or that of Rabinovich et al.) is applicable for the prediction of the capillary force. At zero separation distance (H ) 0), the difference between the forces calculated using the two sets of equations, ∆F(H), is given by

∆F(H ) 0) ) -πγRR(1 + 2 sin θ + R cos θ) (11)

Figure 3. Geometry of the sphere/sphere interaction with a liquid bridge, in which AA is the plane of symmetry.

in which (for H ) 0)

R2 ) 2xV/(πR3) , 1 For a small R and a low contact angle θ, the force difference given by eq 11 is small compared with the total capillary force. However, for θ > 90°, the total capillary force is small, and, as a result, the magnitude of ∆F can be comparable to the total force. In our opinion, the difference between the forces calculated using the two sets of formulas is the result of the inherent assumptions made in the development of eqs 1 and 8, rather than the result of a different approach. For an annulus of fixed volume, the reasonable agreement between these equations (as shown in Figure 2) proves that eq 7 is correct for the total energy of the annulus for sphere/plate geometry. 4. Theoretical Calculation of the Capillary Force between Two Spheres The Derjaguin approximation12 can be used to calculate the force of adhesion between two spheres (as shown in Figure 3) or for sphere/plate geometry (Figure 1), and is given by

F(H) ) kRU(H)

(12)

Capillary Force between Spheres with Liquid Bridges

in which k ) π or 2π for the interaction of two spheres or a sphere/plate geometry, respectively, and U(H) is the specific (per unit area) energy of interaction for two flat surfaces at a separation distance of H. However, the application of the Derjaguin approximation for two different surface energies (i.e., the energy of interaction through the inside of the liquid bridge and the dry interaction outside) is not clear. Taking into account the specific energy, U, acting through a liquid layer at the point of contact,

U ) -2γ cos θ

(13)

The force for the sphere/plate interaction based on eq 12 is given by

Fsp/pl(H ) 0) ) -4πγR cos θ

(14)

which corresponds to the first term of eq 1 at H ) 0. However, at H * 0, the Derjaguin approximation for capillary force is invalid, which is evident from the comparison of eqs 12 and 1. Unlike eq 12, eq 1 shows that Fsp/pl is not proportional to the sphere radius R because the value of dsp/pl depends on the sphere radius. Moreover, for the limiting case of a large separation distance (given by eq 6), the first term of the total capillary force is independent of the radius of the sphere. The validity of the Derjaguin approximation for capillary force was also studied by Willett et al.,10 who concluded that the approximation is valid for H ) 0 and very large distances and inapplicable at intermediate distances. This conclusion (at least for H ) 0) was based on the proportionality of the capillary force to the sphere radius. However, as made clear from the discussion above, the proportionality of the capillary force to the radius is a necessary, but not sufficient, condition for the validity of the Derjaguin approximation. Moreover, at large separation distances (eq 6), the force is not proportional to the radius. Therefore, we believe that the Derjaguin approximation for capillary force is valid only at H ) 0 and invalid for any finite separation distance. One of the possible reasons for the invalidity is the fact that the capillary force acts in a restricted area under the liquid bridge, and this area changes with the separation distance, H. As a result, the integral method used in the Derjaguin approximation is not applicable for the capillary liquid bridge. An alternative method to derive the formula for the capillary force between two spherical particles is discussed below. For two spheres, of radius R and contact angle θ, the total energy of the liquid bridge is given by eq 7. The force between the two spheres with a liquid bridge derived from the total energy is

Langmuir, Vol. 21, No. 24, 2005 10995

Fsp/sp(H,V) ) -

2πRγ cos θ 1 + [H/2dsp/sp(H,V)]

(18)

When the attraction force due to the vertical component of the liquid bridge is taken into account, the complete formula for the capillary force can be expressed as

Fsp/sp(H,V) ) 2πRγ cos θ - 2πγR sin R sin(θ + R) (19) 1 + [H/2dsp/sp(H,V)] The value of dsp/sp(H,V) for the interaction of two spheres can be obtained from eq 16 as

dsp/sp(H,V) ) (H/2) × [-1 + x1 + 2V/(πRH2)]

(20)

A comparison of experimental data and the theoretical force calculated using eq 19 is presented in the following section. 5. Experimental Results for Sphere/Sphere Capillary Force A comparison of the experimentally measured capillary force between two spheres and the theoretically calculated force is shown in Figure 4. The experimental force/distance curves were obtained using an AFM for spherical particles with radii in the range of 19-35 µm. The theoretical curves plotted in Figure 4 are based on eqs 19 and 20 with the volume of the liquid bridge, V, as the fitting parameter. There are maxima in the absolute value of attractive (detachment) force present at small separation distances (H ≈10 nm), which are not clearly seen in Figure 4 because of the large scale of the abscissa. These maxima are believed to be due to contact angle hysteresis, as described by Willett et al.15 Because of the lack of theoretical formulas for the capillary force between two spherical particles of unequal size, an effective particle radius was used:12

Reff )

2R1R2 R1 + R2

(21)

(17)

Although the effective particle radius is based on the Derjaguin approximation, it is used as a first approximation for estimating the capillary force between two spheres of unequal size. To account for the variation in the experimentally measured forces due to the use of a different set of cantilevers and particles, the surface tension γ was also used as a fitting parameter. The experimentally measured value for the surface tension was 27.5 mN/m. The theoretical value of the capillary force F is directly proportional to the value of γ. The use of a fitting value for the surface tension is believed to account for the errors in the calibration of the cantilever stiffness, the deviations from the nonspherical shape of the particles, and their exact size. The fitting values of γ ) 24-28 mN/m (instead of 27.5 mN/m) indicate a total error of (10%. The values of Vfit were based on a least-squares fitting method, comparing the theoretical force distance curve with the experimental data. As seen from Figure 4, the theoretical and experimental force curves (for three different liquid bridge volumes) agree, thereby validating eqs 19 and 20. On the other

Combining eqs 15 and 17, the following equation for the force between the two spheres is obtained:

(15) Willett, C. D.; Adams, M. J.; Johnson, S. A.; Seville, J. P. K. Effects of Wetting Hysteresis on Pendular Liquid Bridges between Rigid Spheres. Powder Technol. 2003, 130 (1), 63-69.

Fsp/sp(H,V) ) -

dW dR ) 4πR2Rγ cos θ dH dH

(15)

The volume of the liquid bridge between the two spheres is given by

Vsp/sp ) πR2R2H + 0.5πR3R4

(16)

For a fixed volume of the liquid bridge, dV/dH ) 0, and from eq 16,

-1 dR ) dH (2H/R) + 2RR

10996

Langmuir, Vol. 21, No. 24, 2005

Rabinovich et al.

Figure 4. Experimental (points) and theoretical (curves) data for the capillary force between two silica spheres with radii R1 ) 19 µm and R2 ) 35 (curve 1), 32.5 (curve 2), and 27.5 (curve 3) µm. The theoretical curves are estimates based on eqs 19 and 20 with surface tension, γ ) 27, 24, and 28 mN/m, and the volume of the liquid bridge, V, is 2, 12, and 36 × 108 nm3 (for curves 1, 2, and 3, respectively). The contact angle between the oil and the spheres was θ )10°. The good agreement between the theoretical and the experimental data proves the validity of eqs 19 and 20 for the sphere/sphere capillary interaction.

Figure 5. Comparison of the simplified set of formulas for the capillary force between sphere/plate and sphere/sphere geometries. Curves 1 and 3 show the capillary force for sphere/sphere geometry on the basis of eqs 18 and A11, respectively. Curves 2, 4, and 5 are for sphere/plate geometry and are based on eqs 1, A10, and A7, respectively. The values of the parameters in the experiment are θ ) 6°, V ) 1010 nm3, R ) 20 µm, and γ ) 28 mN/m.

hand, this verification is semiquantitative because of the use of a fitting value for the liquid bridge volume. An independent estimation of the volume of the bridge can be obtained from either the point of contact of the liquid film on the two interacting surfaces or the point of rupture of the liquid bridge on the experimental force/distance curves. An independent estimation of the capillary liquid bridge volume was performed using the following formula:2

Vexp )

πHc3 1 1+ 12 1 - cos θ

(

)

(22)

in which Hc is the critical separation distance at which the two oil-coated spheres touch each other. This distance was measured from the extending (approaching) portion of the force distance curve. For example, for the experimental force/distance curve shown in Figure 7 from the study by Rabinovich et al.2, Vexp ) 3 × 109 nm3, and Vfit ) 8 × 109 nm3. One cannot expect better agreement between the two values because eq 22 is very sensitive to the value of the contact angle. Thus, the measurements of V in the study by Rabinovich et al.2 prove that the fitting values of the bridge volume are within a reasonable range. However, in this paper, for the capillary interaction between two spheres, there is a substantial difference between the values of Vfit and Vexp, which is unexplainable. In Appendix II, several simplified equations are developed on the basis of eqs 1 and 18. These equations should be valid for the calculation of Fsp/sp and Fsp/pl in the range of large and small separation distances. The comparison of eqs A7, A10, and A11 with the more precise eqs 1 and 18 is shown in Figure 5. Apparently, this set of equations is valid in the corresponding range of distances, that is, 0-200 nm for short-range forces and above 1 µm for long-range forces. Another test of the applicability of the Derjaguin approximation for the capillary force can be performed on the basis of eq A7 and Figure 5. One of the conslusions of the Derjaguin approximation (eq 12) is that

Fsp/pl ) 2Fsp/sp

(23)

However, we find from eq A7 that this condition is not fulfilled, which confirms the conclusion made in this paper regarding the inapplicability of the Derjaguin approximation for the capillary force at a distance other than H ) 0.

Figure 6. Ratio of Fsp/pl to Fsp/sp vs separation distance H. Curve 1 is based on eq 24. V ) 1010 nm3; R ) 20 µm. Curve 2 is based on the Derjaguin approximation (eq 23).

To further clarify the applicability, the theoretical ratio of Fsp/pl/Fsp/sp was calculated as a function of the separation distance H. Using the first terms of eq 1 and eq 18, the following formula is obtained:

Fsp/pl 1 - 1/(1 - x1 + 2V/(πRH2) )2 Fsp/sp 1 - 1/(1 - x1 + V/(πRH2)

(24)

The resulting curve of Fsp/pl/Fsp/sp versus H, which is calculated with eq 24, is shown in Figure 6. For comparison, the Derjaguin ratio (from eq 23; i.e., a constant value equal to 2) is also shown in Figure 6. This figure confirms that the Derjaguin approximation for the capillary force is applicable only at H ) 0. 6. Conclusions In this study, theoretical formulas reported for the estimation of the capillary force due to a liquid bridge were compared. The distant dependent capillary force can be derived theoretically on the basis of two approaches: (1) the total liquid bridge energy and (2) the pressure difference across the liquid bridge (Laplace equation). To corroborate the application of the two different approaches, experimental measurements of the capillary force using an AFM were compared with the theoretical estimates. The experimental and theoretical results were found to agree. Most of the theoretical expressions for capillary forces that have been reported are based on the sphere/plate geometry for the interaction of the two surfaces. The extension of these expressions to predict the force between

Capillary Force between Spheres with Liquid Bridges

two spheres is based on the Derjaguin approximation. The validity of the Derjaguin approximation for the estimation of the capillary force was studied, and it was shown to be applicable only at zero separation distance. An alternative framework for the theoretical estimation of the capillary force between two spheres was developed. The theoretical formulas developed were validated with experimental data. The prediction of the distance-dependent capillary force of adhesion is important in estimating the total adhesion energy required to control and modify the flow behavior of powder systems and in avoiding segregation (enhance binding) in key industrial processes, such as mixing. Acknowledgment. The authors acknowledge the financial support of the Particle Engineering Research Center (PERC) at the University of Florida, the National Science Foundation (NSF Grant EEC-94-02989), and the Industrial Partners of the PERC Any opinions, findings, conclusions, and/or recommendations expressed in this material are those of the author(s) and do not necessarily reflect those of the National Science Foundation.

Langmuir, Vol. 21, No. 24, 2005 10997

Fsp/pl(H,V,θp,θf) ) -

dsp/pl(H,2V) ) 2dsp/sp(H,V)

(A1)

dsp/pl(H/2,V/2) ) dsp/sp(H,V)

(A2)

Equations for Sphere/Plate and Sphere/Sphere Capillary Force at Large and Small Separation Distances. This section provides simplified versions of the formulas for the adhesion between sphere/plate and sphere/sphere geometries in the two limiting cases of large and small separation (H) distances. At large separation distances, V , πRH2. Both eqs 2 and 20 for dsp/pl and dsp/sp, respectively, yield

1+

)

2V πRH2

(A3)

The force/distance dependence for two spheres based on eq 18 can be derived from another simple method. Introducing a plane of symmetry between two interacting spheres (Figure 3) and comparing the annular geometry in Figures 1 and 3 results in

Fsp/sp(H,V,θp) ) Fsp/pl(H/2,V/2,θp,θf ) π/2) (A4) Now, the first term of eq 1 can be modified for different contact angles θp and θf as follows:

(A6)

As a result, the first terms of eqs 1 and 18 yield

Fsp/pl ) Fsp/sp ) - 2γV cos θ/H2

(A7)

At small separation distances, V . πRH2. In this case, the following formulas are valid:

dsp/pl ) xV/(πR)

(A8)

dsp/sp ) xV/(2πR)

(A9)

Fsp/pl ) -4πγR cos θ(1 - HxπR/V)

(A10)

Fsp/sp ) -2πγR cos θ[1 - HxπR/(2V)]

(A11)

When eqs 3 and 20 are compared, the following equation for Rsp/sp is obtained:

( x

(A5)

Appendix II

dsp/pl ) dsp/sp ) V/(2πRH)

Additional Formulas for the Calculation of Capillary Forces. This section provides some additional formulas for the calculation of capillary forces for sphere/ plate and sphere/sphere geometries. A comparison of eq 2 for dsp/pl with eq 20 yields two relationships between dsp/pl and dsp/sp:

H -1 + R

1 + H/dsp/pl(H,V)

When the Fsp/pl in eq A5 is replaced with eq A4 and eqs 2 and 20 are taken into account, the same eq 18 is obtained. When this method is used for eqs 8-10 (without accounting for the contribution of the vertical component of the surface tension), equations for the force between two spheres are obtained, which provide the same results that eq 18 provides.

Appendix I

Rsp/sp2 )

2πγR(cos θp + cos θf)

Note, that eqs A10 and A11 account only for the contribution of the pressure difference across the liquid bridge and do not account for the vertical component of the surface tension. A comparison of the simplified expressions (eqs A7, A10, and A11) with the more precise expressions (eqs 1 and 18) is discussed in the main text of this paper. LA0517639