Mapping the Influence of Gravity on Pendular Liquid Bridges between

There are four distinct regimes ranging from gravity free to complete draining ..... corresponding Bond numbers for these bridges are 10 and. 5. The d...
1 downloads 0 Views 79KB Size
Download PDF
6180

Langmuir 2002, 18, 6180-6184

Mapping the Influence of Gravity on Pendular Liquid Bridges between Rigid Spheres Michael J. Adams,*,† Simon A. Johnson,† Jonathan P. K. Seville,‡ and Christopher D. Willett† Unilever Research Port Sunlight, Bebington, Wirral, Merseyside, CH63 3JW, United Kingdom, and School of Chemical Engineering, The University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom Received December 18, 2001. In Final Form: April 22, 2002 The influence of gravity on the capillary forces and stabilities of pendular liquid bridges between spherical bodies is examined including the special case of a sphere and a flat surface. Previously, this has only been possible by measurement or calculation for a particular bridge system. In the current work, it is shown that a map involving the dimensionless bridge volume, V*, and the Bond number, Bo, may be used to represent the various regimes of behavior. There are four distinct regimes ranging from gravity free to complete draining in which it is not possible to form stable junctions. The boundaries are defined by contours of constant values of the group V*Bo which may be regarded as the Bond number with the appropriate length scale for this system.

Introduction The capillary forces generated by the presence of pendular liquid bridges between solid surfaces are of considerable scientific and technological importance. Examples include the effects on the statics and dynamics of sandpiles1-3 and the influence of capillary condensation in determining the time and humidity dependence of the static friction of particle assemblies.4 Often it is reasonable to assume that such bridges are sufficiently small that their physical properties are not influenced by gravity or inertial forces. However, there are many technological examples for which this may not be the case. These include liquid drainage in packed particulate beds5 during filtration, for example, and the granulation of particles by a liquid binder using equipment designed to induce collisional breakup of oversized granules.6 Moist granular media are rather complex systems. Although discrete computational methods have been employed to develop a microscopic understanding of their behavior, they require efficient and accurate algorithms for the grain-grain interaction laws.7 While the equilibrium geometries, capillary forces, surface free energies, and stabilities of pendular bridges have been the subject of considerable experimental and theoretical studies (see ref 8), simple and general criteria for predicting the influence of gravity on these properties have proven difficult to establish, even for an axisymmetric geometry like that shown in Figure * Corresponding author. E-mail: [email protected]. † Unilever Research Port Sunlight. ‡ The University of Birmingham. (1) Hornbaker, D. J.; Albert, R.; Albert, I.; Baraba´si, A.-L.; Schiffer, P. Nature 1997, 387, 765. (2) Halsey, T. C.; Levine, A. J. Phys. Rev. Lett. 1998, 80, 3141. (3) Tegzes, P.; Albert, R.; Paskvan, M.; Baraba´si, A.-L.; Vicsek, T.; Schiffer, P. Phys. Rev. E 1999, 60, 5823. (4) Bocquet, L.; Charlaix, E.; Ciliberto, S.; Crassous, J. Nature 1998, 396, 735. (5) Turner, G. A.; Hewitt, G. F. Trans. Inst. Chem. Eng. 1959, 37, 329. (6) Seville, J. P. K.; Tu¨zu¨n, U.; Clift, R. Processing of Particulate Solids; Kluwer Academic: Dordrecht, 1997. (7) Lian, G.; Thornton, C.; Adams, M. J. Chem. Eng. Sci. 1998, 53, 3381. (8) Orr, F. M.; Scriven, L. E.; Rivas, A. P. J. Fluid Mech. 1975, 67, 723.

1.8,9 In the current paper, we develop an approach for this case of two spherical solid bodies in vertical alignment connected by an inviscid liquid junction. It is based on representing the various regimes of behavior as a map. The capillary attractive force as a function of the separation distance between a sphere and a planar surface, which is a special case of two unequal spheres, was also measured as a basis for verifying the calculated values. The capillary forces between two rigid spheres arise from the axial component of the surface tension and the capillary pressure. The first term is always attractive, and the second is also attractive provided that the bridge curvature is negative. The calculation of the curvature is based on the calculus of variations10 or the equivalent Laplace-Young equation which has been solved analytically in terms of elliptic integrals8 and also using numerical procedures (e.g., ref 11). This equation may be written in the following form to include the effect of gravity for spheres in vertical alignment:8

∆p(z) ) ∆p(0) + z∆Fg ) γ

[

]

r¨ (z) 1 (1) 2 1/2 (1 + r˘ (z) ) r(z) (1 + r˘ (z)2)3/2

where the dot notation refers to differentiation with respect to the axial coordinate z, and r(z) is the meridional profile of the bridge (see Figure 1). The parameter ∆p(z) is the local pressure difference across the air-liquid interface, and ∆p(0) is the local pressure difference at the threephase contact line on the upper sphere (see Figure 1). The parameter ∆F is the density difference across the airliquid interface, g is the acceleration due to gravity, and γ is the surface tension of the liquid. The term in the square brackets is the local curvature of the bridge, ξ(z). When the influence of gravity may be neglected, ξ(z) is constant for a given bridge. Experimental data have been obtained under these conditions, for sphere-sphere and sphere-flat systems, that are consistent with the predic(9) Princen, H. M. J. Colloid Interface Sci. 1968, 26, 249. (10) De Bisschop, F. R. E.; Rigole, W. J. L. J. Colloid Interface Sci. 1982, 88, 117. (11) Hotta, K.; Takeda, K.; Iinoya, K. Powder Technol. 1974, 10, 231.

10.1021/la011823k CCC: $22.00 © 2002 American Chemical Society Published on Web 07/11/2002

Influence of Gravity on Pendular Liquid Bridges

Langmuir, Vol. 18, No. 16, 2002 6181

Figure 1. A schematic representation of a capillary liquid bridge, defining the coordinate system.

tions of the Laplace-Young equation at all separation distances.12,13 This is despite the difficulties associated with the accurate measurement of the forces and displacements for the small volumes of the liquid bridges involved. A major problem with some earlier work was the observed maxima in the capillary attractive force at small gaps during separation [e.g., ref 14]. It has now been established that this phenomenon may arise when there is wetting hysteresis even if the equilibrium contact angle is relatively small.15 Under this condition, a bridge is initially pinned at the three-phase contact line until the contact angle corresponds to the receding value. Under “gravity-free” conditions, it has been shown that when the total free energy is expressed as a function of the separation distance, 2S, the Laplace-Young equation has two solution branches such that the stable bridge configuration corresponds to the lower energy branch;16 two corresponding branches are also obtained if the solution is expressed in terms of other bridge parameters such as the force. Moreover, it has been observed that solutions do not exist at separation distances greater than some critical value and this defines the stability limit or rupture distance, 2Sc. This quantity has been related with reasonably high accuracy to the bridge volume, V, by the following expression for equal spheres:16

2Sc* ) (1 + 0.5φ)V*1/3

(2)

where φ is the contact angle (rad), 2Sc* () 2Sc /R) is the dimensionless rupture distance, V* () V/R3) is the dimensionless bridge volume, and R is the sphere radius. The above closed-form relationship is suitable as an approximate particle-particle interaction law of the type described previously for discrete simulation methods. It will be extended in the current paper to account for the effect of gravity. With relatively large bridge volumes, (12) Willett, C. D.; Adams, M. J.; Johnson, S. A.; Seville, J. P. K. Langmuir 2000, 16, 9396. (13) Pitois, O.; Moucheront, P.; Chateau, X. J. Colloid Interface Sci. 2000, 231, 26. (14) Mason, G.; Clark, W. C. Chem. Eng. Sci. 1965, 20, 859. (15) Willett, C. D.; Adams, M. J.; Johnson, S. A.; Seville, J. P. K. Powder Technol., in press. (16) Lian, G.; Thornton, C.; Adams, M. J. J. Colloid Interface Sci. 1993, 161, 138.

gravity causes geometric distortions of a bridge so that the curvature, ξ(z), is no longer constant. In this case, the capillary forces are affected at all separation distances and there is a change in the rupture distance.17 In addition, the buoyancy forces become significant compared with the total capillary force.8 A dimensionless group known as the Bond number is used to estimate the importance of gravitational shape distortions for free droplets of a liquid. It is conventionally given by ∆FgD2/γ, where D is the droplet diameter, which corresponds to the ratio of the gravitational and surface tension forces.18 The Bond number, Bo, has also been adopted for axisymmetric pendular bridges between two equal spheres19 and between a sphere and a flat surface8 with the sphere radius assumed to be the characteristic length; that is, Bo ) ∆FgR2/γ. It was argued8 that the influence of gravity is negligible if Bo/|2ξR| , 1, viz., if the Bond number is small compared with the dimensionless curvature. An alternative criterion |∆p| . h∆Fg, where h is the difference in height between the highest and lowest points in a bridge (see Figure 1), was also proposed. This states that the capillary pressure is large compared with the hydrostatic head caused by the gravitational field.9 These criteria were found to be conservative, and exact calculations were required to evaluate the influence of gravity for a particular bridge system. The limitation of formulating a gravity criterion in terms of the capillary pressure alone may be appreciated by noting that the pressure difference is zero for a bridge with a catenoidal meridional profile. In principle, the problem reduces to identifying a more appropriate length scale, L, in the Bond number than the sphere radius; that is, Bo ) ∆FgL2/γ. It is this approach that was adopted in the current work. Calculation of the Capillary Forces The bridge geometry for equal or unequal spheres was calculated in the current work by a numerical integration of eq 1 using a method described previously;11 the liquidsolid contact angle was set to zero for these calculations. The total attractive force acting on each sphere, Fi, was obtained from

Fi 1 ) sin2 βi - ξiRi sin2 βi 2πRiγ 2 ∆FgRi2 2 1 - cos βi + cos3 βi (3) 2γ 3 3

[

]

where i ) u or l such that u and l refer to the upper and lower spheres and the upper sign refers to the upper sphere, βi is the half-filling angle (see Figure 1), and ξi is the curvature at the three-phase contact line for each sphere; that is, ξu ) ξ(0) and ξl ) ξ(h). The three contributions to the force arise respectively from the surface tension, capillary pressure, and buoyancy. The term in square brackets is equal to Vb /πRi3, where Vb is the partial volume of the sphere that is submerged. An overall force balance shows that the difference in the forces between the upper and lower solid bodies arises from the weight of the bridge () V∆Fg).19 This provided an independent verification of the accuracy of the calculations. If one of the solid bodies is actually a flat surface, the associated buoyancy term is zero and it is also more (17) Bayramli, E.; Abou-Obeid, A.; van de Ven, T. G. M. J. Colloid Interface Sci. 1987, 116, 490. (18) Clift, R.; Grace, J. R.; Weber, M. E. Bubbles, Drops and Particles; Academic Press: New York, 1978. (19) Mazzone, D. N.; Tardos, G. I.; Pfeffer, R. J. Colloid Interface Sci. 1986, 113, 544.

6182

Langmuir, Vol. 18, No. 16, 2002

Adams et al.

Figure 2. The capillary forces as functions of the separation distance between an upper sphere of radius 2.381 mm and a lower planar surface. The experimental data for the sphere have been measured for bridge volumes of 0.620 (O), 2.261 (0), and 5.736 µL (4). The corresponding calculated forces are shown which include (full lines) and neglect (dashed lines) the influence of gravity.

appropriate to express the total attractive force in terms of the bridge radius at the three-phase contact line, ri () Ri sin βi ). For the case of a zero contact angle, the surface tension term associated with the flat surface is also zero and the solution simplifies to Fi ) -πri2γξi for 1/Ri ) 0, where ru ) r(0) or rl ) r(h) depending on whether the flat surface is above or below the sphere. Experimental Section The details of the experimental equipment and methodology used to measure the stability and also the capillary forces as a function of the separation distance have been described previously.12 This earlier work was aimed at microscopic bridges for which the influence of gravity was negligible. Here, pendular bridges were formed between an upper sphere of radius 2.381 mm and a lower planar surface. A precision synthetic sapphire sphere (Dejay Distribution Ltd., Wokingham, U.K.) and a sapphire disk (Melles Griot Ltd., Cambridge, U.K.) were used for this purpose. They were cleaned with alcohol (AnalaR grade, BDH Laboratory Supplies, Poole, U.K.). This was sufficient to ensure that the silicone fluid (DC200, Fluka Chemicals, Gillingham, U.K.), which was used to form the pendular bridges, perfectly wetted the sapphire surfaces (φ ) 0°). The measured surface tension and density of this fluid were 20.6 mN/m and 960 kg/m3, respectively. The equipment for monitoring the capillary forces as a function of separation distance was based on a relatively stiff microbalance (MK2 vacuum head with Multicard II controller, CI Electronics Ltd., Salisbury, U.K.) and a piezoelectric actuator (17 PAZ 015 and 001 actuator and controller, Melles Griot Ltd.) with feedback for accurate displacement control. A separation velocity of 1 µm/s was imposed which was sufficiently slow to eliminate a viscous contribution to the total force except for separation distances 2S < 5 µm. All data within this limit were discarded.

Experimental Results There is a close agreement between the calculated and measured data, as is shown in Figure 2, which correspond to bridge volumes of 0.620, 2.261, and 5.736 µL between a sphere and a flat surface (Ru ) 2.381 mm and 1/Rl ) 0). The curves calculated from the theory when the influence of gravity is not taken into account are also shown in Figure 2. There is an increasing deviation of the measured data from these calculated values with increasing bridge volume. The experimental data clearly demonstrate that

Figure 3. Dimensionless plots of the calculated forces between equal spheres as functions of the half-separation distance for the cases when gravity is included (full lines) and neglected (dashed lines): (a) Bo ) 10, V* ) 0.01; (b) Bo ) 5, V* ) 0.1. For the case when gravity is included, the two solutions refer to the force associated with the upper and lower spheres.

rupture occurs at the maximum separation distance for which a solution to eq 1 exists; the calculated curves shown in Figure 2 correspond to the lower energy branches referred to previously and only continue partially into the higher energy branches. Discussion Figure 3 shows dimensionless plots of the forces, Fi* ) Fi/2πRiγ, calculated for upper and lower equal spheres as a function of the half-separation distance S* () S/Ri) for two dimensionless bridge volumes of 0.01 and 0.1. The corresponding Bond numbers for these bridges are 10 and 5. The divergence of the forces from the gravity-free data, which are also shown in the figure, and the reduction in the rupture distances are greater for the smaller Bond number. This clearly arises from the larger bridge volume and suggests that the Bond number alone is not a reliable indicator of the influence of gravity as was discussed previously. The simplest scaling of the volume leads to an alternative dimensionless group, V*Bo. The values of V*Bo in Figure 3a,b are 0.1 and 0.5. A higher energy solution branch does not exist for the larger value of V*Bo. This is indicative of a draining limit that has been observed in previous calculations17 and will be considered later. Essentially, the influence of gravity causes the liquid filling

Influence of Gravity on Pendular Liquid Bridges

Langmuir, Vol. 18, No. 16, 2002 6183

Figure 4. Dimensionless plots of the half-rupture distances for equal spheres as functions of the bridge volume where the data points correspond to the calculated stability limits. The lines are the best-fit polynomial curves passing through the origin where the values of V*Bo are 0.025 (b), 0.05 (]), 0.1 (1), 0.25 (4), 0.5 (9), 1.0 (O), and 1.5 (0).

angle for the upper sphere to decrease continuously with increasing separation distance until eventually the bridge becomes unstable. If V*Bo provides the correct scaling, it should lead to a superposition of the liquid bridge parameters under the influence of gravity. Figure 4 shows dimensionless plots of the calculated rupture distance as a function of the bridge volume for different values of this group. These data may be represented by a polynomial expression in V* which has been used to calculate the curves in the figure. The leading order terms of this polynomial are given by

2Sc* ≈ (1 - 0.48V*Bo)V*1/3

(4)

which reduces to eq 2 in the gravity-free limit for a contact angle of zero. The maximum error associated with this expression is ca. 10%. It may also be seen in Figure 3 that the upper and lower spheres share the weight of the bridge equally except near the rupture points. Thus the half-weight represents a measure of the deviation of the total capillary force, ∆F* () V∆Fg/4πRγ), from the gravity-free result which we will specify arbitrarily as having an upper limit of 0.001 to be significant. Alternatively, this criterion may be written as V*Bo < 0.01, which we will define as the gravity-free region since, according to eq 4, the criterion also provides a sensible limit below which gravity does not significantly influence the rupture distance. The geometric distortion due to gravity may be characterized by the upper and lower filling angles as exemplified in Figure 5 for two dimensionless bridge volumes and for a range of Bond numbers. In all cases, βl > βu and the value of βl initially decreases with increasing separation followed by an increase which continues until bridge rupture. This minimum in the filling angle has been observed previously for calculations made both under gravity-free conditions10 and when the influence of gravity is significant.17 The upper filling angle shows similar behavior for Bond numbers less than some critical value which in parts a and b of Figure 5 may be estimated as being ca. 15 and 1.5, respectively. Thus a critical condition may be defined, to a first approximation, by V*Bo ≈ 0.15 for all bridge volumes although more calculations are required in order to refine this value, which could have

Figure 5. The half-filling angles as functions of the dimensionless half-separation distance between equal spheres for a range of Bond numbers and dimensionless bridge volumes of (a) 0.01 and (b) 0.1. Data are shown for the upper (full lines) and lower (dashed lines) spheres. The vertical lines correspond to the rupture distances.

a small sensitivity to the bridge volume. For V*Bo > 0.15, βu decreases continuously for all separation distances, which we will term draining. Another important aspect of the data shown in Figure 5 is that the rupture distance decreases with increasing Bond number. Eventually, it is not possible to form a stable bridge even when the spheres are in contact; we will term this condition complete draining. In reality, when such draining occurs, a stable bridge will be formed with a smaller volume. Setting Sc* ) 0 in eq 4 allows an estimate to be made of the critical value of V*Bo which is ca. 2. That is, the maximum volume that may be retained between touching spheres is a monotonically decreasing function of the Bond number, which is consistent with previous theoretical work.20 Moreover, the relationship V*Bo ) 2 provides a close approximation to reported experimental data5 of maximum bridge volumes for touching spheres measured in the ranges 0.06 < V * < 0.4 and 2.5 < Bo < 50. On the above basis, we will define a gravity-controlled region (0.15 < V*Bo < 2) such that draining occurs. It is also useful to distinguish a region in which the capillary forces and the rupture distances are significantly affected by gravity but rupture does not occur by the draining (20) Sa´ez, A. E.; Carbonell, R. G. J. Colloid Interface Sci. 1990, 140, 408.

6184

Langmuir, Vol. 18, No. 16, 2002

Adams et al.

Figure 6. A map of the influence of gravity on the stabilities of pendular bridges between equal spheres showing the following regions: gravity free (GF, V*Bo < 0.01), transitional (TR, 0.01 < V*Bo < 0.15), gravity controlled (GC, 0.15 < V*Bo < 2), and complete draining (CD, V*Bo > 2). Contours of constant half-rupture distance are also shown. Table 1. Regimes of Pendular Bridge Behavior under the Influence of Gravity modified Bond number V*Bo 2.0

regime

description

gravity free transitional

insignificant effect of gravity significant decrease in rupture distance and change in the force by half the bridge weight gravity controlled bridge rupture by a draining mechanism complete draining stable bridges do not exist

mechanism. We will term this the transitional region which is defined by the limits 0.01 < V*Bo < 0.15. The various regions of behavior for the different ranges of V*Bo are summarized in Table 1. They are represented as a map in Figure 6 which includes the rupture contours for constant values of Sc* calculated using eq 4. The contours clearly demonstrate that for a given Bond number, there is an upper bound on the stable bridge volume corresponding to the gravity controlled/complete draining transition. For a given Bond number and separation distance, an initially stable bridge may rupture if the volume is either increased or decreased. It is in the gravity-controlled region that the stable bridge volumes show the greatest sensitivity to the Bond number and asymptotically approach the complete draining limit. This limit has been defined for bridges with a zero separation distance. If separation is imposed on such a bridge, draining would occur and the residual bridge would be in the gravity-controlled region. In the gravity-free regime, it has been shown previously that the total capillary force for a liquid bridge between a pair of unequal spheres, with radii Ru and Rl, was generally close to that calculated for a pair of equal spheres having the harmonic mean radius of the pair, Rul () 2(Ru-1

+ Rl-1)-1).12 Further work would be necessary to determine the value of analogously expressing the modified Bond number in terms of the harmonic mean radius (i.e., V*Bo ) V∆Fg/γRul) to map the effects of gravity on liquid bridges between unequal spheres. In principle, the current approach may be extended to characterize the effect of gravity on the behavior of pendular bridges with nonzero contact angles. However, there are complications associated with wetting hysteresis as mentioned previously. This allows a range of metastable geometries to exist for a particular bridge, due to contact line pinning, in which the contact angle can vary between the advancing and receding values.15 The development of a gravity map to include the angle of elevation of two spheres is a significantly more difficult exercise since it would require a numerical analysis scheme such as the “Surface Evolver”.21 However, an experimental study of the maximum volume of liquid that may be contained in a capillary bridge between two touching spheres as a function of the angle of elevation has been reported.5 Conclusions We have shown that a modified Bond number, V*Bo, may be used to predict and map the effect of gravity on pendular liquid bridges between equal spheres. In dimensional form, this group is written as V∆Fg/γR which may be interpreted as a ratio of the bridge weight and a quantity that scales the total capillary force. This is consistent with the definition of the Bond number for liquid droplets, D2∆Fg/γ, which scales as the ratio Vd∆Fg/γD where Vd and D are the volume and diameter of a droplet. An equivalent interpretation is that the characteristic length scale is (V/R)1/2 in the conventional definition of the Bond number. Although this is less physically meaningful, the same length scale has independently been found to allow gravity-free dimensionless force-separation curves for different bridge volumes to be approximately superimposed.12 The main value of the current work is that it provides simple criteria for assessing the influence of gravity on a capillary bridge for a given pair of spherical particles. The radii of particles are typically in the range from 1 µm to 1 mm. For bridges formed by water, this would correspond approximately to Bond numbers in the range 10-7 < Bo < 10-1. The dimensionless volume of a cylindrical bridge formed between two touching equal spheres with a radius equal to that of the spheres is 2π/3. In this case, for particles with R ) 200 µm the maximum value of V*Bo is therefore ∼0.01. This value is the upper limit of the gravity-free regime. Hence, the influence of gravity may be significant for particles having a greater radius depending on the size of the liquid bridge. For example, for particles with R ) 1 mm the corresponding maximum value of V*Bo is ∼0.3 which is in the gravity-controlled regime. The influence of gravity could therefore be important for the drainage of a packed bed of highly saturated particles of this size. LA011823K (21) Brakke, K. A. Exp. Math. 1992, 1, 141.