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Capillary bridges and capillary forces between two axisymmetric power–law particles
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Capillary bridges and capillary forces between two axisymmetric power–law particles Lefeng Wang, Fengting Su, Huichao Xu, Weibin Rong, Hui Xie ∗ State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150080, China
a r t i c l e
i n f o
Article history: Received 23 April 2015 Received in revised form 21 July 2015 Accepted 18 August 2015 Available online xxx Keywords: Capillary force Power–law profile particles Power–law index Rupture distance
a b s t r a c t Capillary interactions are fundamentally important in many scientific and industrial fields. However, most existing models of the capillary bridges and capillary forces between two solids with a mediated liquid, are based on extremely simple geometrical configurations, such as sphere–plate, sphere–sphere, and plate–plate. The capillary bridge and capillary force between two axisymmetric power–law profile particles with a mediated constant-volume liquid are investigated in this study. A dimensionless method is adopted to calculate the capillary bridge shape between two power–law profile particles based on the Young–Laplace equation. The critical rupture criterion of the liquid bridge is shown in four forms that produce consistent results. It was found that the dimensionless rupture distance changes little when the shape index is larger than 2. The results show that the power–law index has a significant influence on the capillary force between two power–law particles. This is directly attributed to the different shape profiles of power–law particles with different indices. Effects of various other parameters such as ratio of the particle equivalent radii, liquid contact angle, liquid volume, and interparticle distance on the capillary force between two power–law particles are also examined. © 2015 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
Introduction Capillary interactions play a significant role in many scientific and industrial fields. These interactions must be accounted for in studies of particle cohesiveness and wet agglomeration processes, which are related to soil mechanics, granular materials, pharmaceuticals, and civil engineering (Rossetti, Pepin, & Simons, 2003; Sun, Wang, & Hu, 2009). In recent years, capillary interactions between micro- and nano-particles have attracted much attention because of the rapid development of micro and nanotechnology. Capillary interactions can, in principle, be exploited for gripping applications, and several types of capillary grippers have been proposed for microparticle manipulation (Lambert, Seigneur, Koelemeijer, & Jacot, 2006; Fantoni, Hansen, & Santochi, 2013; Fan, Wang, Rong, & Sun, 2015). Capillary forces are also utilized to control self-assembly processes from the millimeter to nanometer scale in microsystem engineering (Mastrangeli et al., 2009). Moreover, capillary interactions greatly affect force measurements and nanomanipulation processes, as revealed in atomic force microscopy (AFM) studies (Butt, Cappella, & Kappl, 2005;
∗ Corresponding author. Tel.: +86 45186404422. E-mail address: [email protected] (H. Xie).
Tabor, Grieser, Dagastine, & Chan, 2012; Kim, Shafiei, Ratchford, & Li, 2011; Onal, Ozcan, & Sitti, 2011). To use them effectively in these related fields, it is necessary to fully understand and accurately model capillary interactions. There has been much work modeling and measuring capillary forces between two particles. Based on thermodynamic equilibrium and non-equilibrium conditions, two main cases are considered when modeling the capillary forces. The first case refers to a liquid bridge that has a constant curvature radius, which is determined from the unsaturated vapor pressure in the environment, as described by the Kelvin equation (Butt, 2008). Pakarinen et al. (2005) investigated the capillary force between a nanosphere and a flat surface and found that the often-used model of the humidity-independent capillary force is reasonable for spherical particles above 1 m. Xiao and Qian (2000) measured the adhesion force between a Si3 N4 AFM tip and SiO2 or n-octadecyltrimethoxysilane (OTE)/SiO2 substrates. For the former, the adhesion force first increases and then decreases because of strong capillary condensation. For the latter, the adhesive force is almost independent of humidity because of weak capillary condensation. The second case corresponds to a constant-volume liquid bridge based on the energy principle. Rabinovich, Esayanur, and Moudgil (2005) calculated the capillary force between two spheres with a
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Please cite this article in press as: Wang, L., et al. Capillary bridges and capillary forces between two axisymmetric power–law particles. Particuology (2015), http://dx.doi.org/10.1016/j.partic.2015.08.005
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When V*Bo < 0.01, the effect of gravity can be neglected (Adams et al., 2002). It is assumed that this criterion is also applicable to the liquid bridge between two power–law profile particles. The profile of the top particle can be written as Z1 =
Fig. 1. Liquid bridge between two axisymmetric power–law profile bodies. The angle ϕ1 is determined by the normal line of the particle profile at point A and the axisymmetric axis of the particle.
fixed volume liquid bridge, and discussed the applicability of the Derjaguin approximation for the interaction between the spheres. De Souza, Brinkmann, Mohrdieck, Crosby, and Arzt (2008) investigated the capillary force between two plates with chemically different properties, and noted that the capillary force decreases as the asymmetry in contact angles is increased with the fixed sum of contact angles. Yang, Tu, and Fang (2010) proposed a rupture model of a capillary bridge between a micro sphere and a plane, and it was found that the rupture distance increases with increasing spherical radius, sphere hydrophobicity, and environmental humidity. Geometrical configurations considered in capillary force modeling have been mostly limited to sphere–plate (Israelachvili, 1992), sphere–sphere (Willett, Adams, Johnson, & Seville, 2000; Lian, Thornton, & Adams, 1993; Payam & Fathipour, 2011), and plate–plate (De Souza et al., 2008). However, in many real cases, particle shapes are not so ideal and simple. For example, the general shape of an AFM probe tip could be better described by a power–law axisymmetric function as a result of the fabrication method or in-use wear (Grierson, Liu, Carpick, & Turner, 2013). Recently, the capillary force between a power–law probe tip and a spherical particle or a plate under different humidity conditions was investigated (Wang & Régnier, 2015). The effects of relative humidity on the capillary forces were revealed in a dimensionless form which is applicable for equilibrium conditions of the liquid bridge. However, in applications such as micro/nano manipulation, we are more interested in the capillary force between two particles with a constant-volume liquid bridge. In this study, the capillary bridge and capillary force between two power–law profile particles with a constant-volume liquid bridge are investigated. In the Modeling the capillary bridge and capillary force section, modeling processes of the capillary bridge and capillary force between two power–law particles are presented in detail. In the Results and discussion section, the rupture criteria for the liquid bridges are demonstrated, and effects of each parameter on the capillary forces are analyzed. Finally, conclusions are given in the Conclusions section.
n −1
n1 R11
+ D,
(1)
where n1 is the power–law shape index of the top particle. For special cases, it denotes a parabolic shape when n1 = 2. R1 can be treated as the equivalent radius of the top particle, and the equivalent half-filling angle can be defined as ϕ1 = XA /R1 accordingly. D is the distance between the two power–law profile particles. When n1 = 1, it is a conical shape with conical angle of 45◦ (Zheng & Yu, 2007). Similarly, the profile of the bottom particle can be written as X n2
Z2 = −
n −1
n2 R22
,
(2)
where n2 is the power–law shape index of the bottom particle. R2 can be treated as the equivalent radius of the bottom particle, and the equivalent half-filling angle can be defined as ϕ2 = XB /R2 accordingly. The dimensionless method was used to simplify the calculation of the liquid bridge shape. If we let z = Z/R1 , x = X/R1 , and d = D/R1 , then Eqs. (1) and (2) can be nondimensionalized as follows: z1 =
xn1 + d, n1
z2 = −
xn2
n2 R2 /R1
(3)
n2 −1 .
(4)
According to the Young–Laplace equation, the liquid meniscus Z3 = f(X) satisfies the following equation: −
X 1 + X2
3/2 + X
1
P
1 + X2
1/2 = ,
(5)
where X = dX/dZ, X = d2 X/dZ 2 . P is the hydrostatic pressure difference across the air–liquid interface and is the surface tension of the liquid. Eq. (5) can be nondimensionalized as −
x 1 + x 2
3/2 + x
1 1 + x 2
1/2 =
R1 P = p.
(6)
The dimensionless volume v of the liquid bridge is
v = v0 − vt − vb ,
(7)
where v0 , vt , and vb represent the dimensionless volume of the column formed by the meniscus profile, the dimensionless volume of the immersed part of the top power–law particle, and the dimensionless volume of the immersed part of the bottom power–law particle, respectively. These terms can be obtained through integration:
Modeling the capillary bridge and capillary force Fig. 1 shows the liquid bridge between two power–law profile bodies. In this figure, 1 and 2 are the liquid contact angles at the two solid–liquid interfaces. Herein, relatively small liquid bridges are considered and gravitational deformations of the menisci are neglected. For the liquid bridge between two equal spheres with radii R, it was found that a modified Bond number V*Bo can be used to predict the effect of gravity on the pendular liquid bridge, in which V* is the dimensionless volume with V* = V/R3 , and Bo is the Bond number with Bo = gR2 / (Adams, Johnson, Seville, & Willett, 2002). The parameter is the density of the liquid, g is the acceleration due to gravity, and is the surface tension of the liquid.
X n1
zA x2 dz3 ,
v0 =
(8)
zB
zA vt =
x2 dz1 =
xn1 +2 , n1 + 2 A
x2 dz2 =
n2 + 2
(9)
d
0 vb =
R n2 −1 2
R1
xBn2 +2
(10)
zB
Please cite this article in press as: Wang, L., et al. Capillary bridges and capillary forces between two axisymmetric power–law particles. Particuology (2015), http://dx.doi.org/10.1016/j.partic.2015.08.005
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Fig. 2. The relationships between the dimensionless parameters and the dimensionless separation distance: (a) the dimensionless neck radius xN , (b) the dimensionless immersing radius xA on the top particle, (c) the dimensionless mean curvature H, and (d) the dimensionless capillary force f, for n1 = n2 = 2, R2 /R1 = 1, 1 = 2 = 30◦ and for various liquid bridge volumes.
The relationship between the actual volume V of the liquid bridge and its dimensionless volume v is v = V/R1 3 . If the volume of the liquid bridge is constant, the final meniscus profile can be derived through Eq. (6) and related constraint conditions. Here we use an iterative method based on the given contact angles and constant liquid volume. By first assuming an immersing liquid radius xA on the top power–law profile particle, the contact angle on the bottom power–law particle can be calculated based on Eq. (6). The calculated contact angle is compared with the given one. If they are not equal, another immersing liquid radius is selected. If they are equal, the volume of the liquid bridge is calculated and compared with the given volume. The process is realized with a bisection method. When the final immersing liquid radius on the top power–law profile body xA is obtained, the profile of the liquid bridge can be determined. The method was also adopted to investigate the capillary force between a microsphere and a flat surface (Lambert & Delchambre, 2005). The total downward force acting on the upper particle F is the sum of the capillary pressure force Fc , the surface tension force Fs , and the buoyancy force Fb : F = Fc + Fs + Fb
(11)
Eq. (11) can be nondimensionalized as f = fc + fs + fb
(12)
These dimensionless forces can be expressed as (Lian et al., 1993): fc = −pxA2
fs = 2xA sin 1 + 1 fb =
xA n1 + 2
R12 g n1 +2
(13) (14) (15)
In this study, the liquid volume is sufficiently small, and the buoyancy force is neglected. The relationship between the actual capillary force F and the dimensionless capillary force f can be calculated through the following equation: F = fR1
(16)
Results and discussion In this section, the rupture criteria of the capillary bridge between two power–law axisymmetric particles will be discussed first, and then effects of various parameters on the capillary force between two power–law particles will be investigated in detail. Rupture of the capillary bridge between two power–law particles Several criteria have been adopted to predict the critical rupture of the liquid bridge between two spherical particles, such as the neck diameter, the half-filling angle, the mean curvature, and the capillary force. The applicability of the corresponding criteria for the axisymmetric power–law profile particles will be studied here. According to Eq. (6), the dimensionless mean curvature H can be expressed as H=
R1 p P = 2 2
(17)
Fig. 2(a)–(d) shows the relationship of these four variables as a function of separation distance in a dimensionless form. There are two branches of the solution based on the numerical method in the Modeling the capillary bridge and capillary force section, and they converge at the maximum value of the dimensionless separation distance. It can be seen that the critical dimensionless separation distance is identical for each criterion. This indicates that these criteria are also applicable for axisymmetric power–law particles. We focus our discussion on the rupture behavior and the capillary force of the liquid bridge. From Fig. 2(b), there is a minimum in the half-filling angle on the top power–law particle. The half-filling angle is also equivalent to the dimensionless immersing radius. The rupture distance will be underestimated if the minimum of the equivalent half-filling angle is adopted as a rupture criterion. This is similar to the capillary interaction between two spherical particles (Lian et al., 1993). Fig. 3 shows the relationship between the dimensionless rupture distance and dimensionless volume for different particle shapes. It is clear that the rupture distance increases with the liquid volume for various power–law particles. When n = 2, the power–law
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Fig. 3. The dimensionless rupture distance as a function of dimensionless volume for power–law particles with various shape indices of n1 = n2 = n, for R2 /R1 = 1, 1 = 2 = 30◦ .
Fig. 4. The dimensionless capillary force as a function of dimensionless separation distance with different shape indices of n1 = n2 = n, for R2 /R1 = 1, 1 = 2 = 30◦ , v = 0.01.
particles become paraboloids, and the rupture distance between them is almost equal to that between two spherical particles. This indicates that the capillary interaction between two spheres can be used to describe that between two paraboloids especially when the liquid volume is sufficiently small. However, when the shape index n ≥ 2, the dimensionless rupture distance decreases very gently with the increased shape index. Also, when the shape index increases gradually from n = 2, it tends to approach the special case of a plane/plane configuration. The data reveals that the dimensionless rupture distance is smaller when the shape index n = 1.5 than when n = 2. This can be attributed to the sharper particle profile when n < 2, and because the capillary bridge tends to become unstable at a large separation distances. Accordingly, the rupture distance of the capillary bridge for the cone/cone configuration for n = 1 is much lower than other configurations. Influences of parameters on capillary forces between two power–law particles Fig. 4 shows the relationship between the dimensionless capillary force and the dimensionless separation distance for power–law profile particles with different power–law indices n1 = n2 = 2, 3, and 6. The dimensionless capillary force increases gradually as the power–law index increases at the same dimensionless separation distance. This is attributed to the effects of the indices on the particle profiles based on Eqs. (3) and (4). The tip of the particle becomes flatter with increasing power–law index, as shown in Fig. 5. In other words, the flatter the particle tip becomes, the larger the dimensionless capillary force will be for a fixed dimensionless volume. At the zero separation distance, the dimensionless capillary forces vary largely as the power–law index n increases. As mentioned, when the shape index tends to infinity, the plate–plate
Fig. 5. The liquid bridge between power–law particles with different shape indices of n1 = n2 = n. The thinner lines denote the particle profiles, and the thick lines denote the liquid bridge profiles.
Fig. 6. The dimensionless capillary force as a function of dimensionless separation distance with different equivalent radius ratios for n1 = n2 = 2, 1 = 2 = 30◦ , v = 0.01.
configuration arises. For the plate–plate configuration, the radius of the liquid bridge tends to be infinity when the separation distance of the two plates is ideally zero, so the force tends to be infinity. However, the actual separation distance between two plates cannot be absolutely zero due to exponentially-increasing close-range repulsive intermolecular interactions and surface roughness effects (Rabinovich, Adler, Ata, Singh, & Moudgil, 2000). Fig. 6 shows the influence of the equivalent radius ratio R2 /R1 of power–law profile bodies on the capillary force. The stable dimensionless capillary force decreases as the equivalent radius ratio decreases at the same dimensionless separation distance. The bottom particle becomes sharper soon with decreasing R2 /R1 , and the liquid bridge radius on the top or bottom particle will decrease gradually; thus, the dimensionless force at a given dimensionless separation distance decreases with decreasing values of R2 /R1 . It can also be seen that the curve of the stable capillary force becomes flatter when the equivalent radius ratio decreases. Additionally, the dimensionless rupture distance of the liquid bridge decreases gradually with decreasing equivalent radius ratio. Fig. 7 shows the influence of the liquid contact angle on the capillary force between two axisymmetric power–law particles. The dimensionless capillary force decreases when the contact angle increases at small distances, and they tend to approach each other before the liquid bridge ruptures. The maximal stable capillary force that occurs at zero separation distance decreases when the contact angle becomes larger. However, the rupture distance increases with increased contact angle. Fig. 8 shows the influence of the dimensionless liquid volume on the dimensionless capillary force when n1 = n2 = 3. This parameter dependence when n1 = n2 = 2 has been demonstrated in Fig. 2(d). From Fig. 2(d), the dimensionless force at zero separation changes little when n = 2. This is consistent with capillary
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dimensionless capillary force increases with the increased power–law index, which is attributed to the flatter power–law particle profile. There may be a distinct variation of the dimensionless capillary force at zero separation distance because of the increased power–law index. The dimensionless force decreases as the ratio of the equivalent radii of the bottom and top particles decreases, and it is affected by the contact angle and liquid bridge volume. These results show more general cases for the capillary interaction between two axisymmetric particles. Acknowledgements
Fig. 7. The dimensionless capillary force as a function of dimensionless separation with different contact angles for n1 = n2 = 2, R2 /R1 = 1, v = 0.01.
This work was supported by the National Natural Science Foundation of China under grant Nos. 51175130 and 61222311, Self-Planned Task of State Key Laboratory of Robotics and System (HIT) under grant Nos. SKLRS201501A04 and SKLRS201301A02, and the Fundamental Research Funds for the Central University under grant No. HIT.BRETIV.201309. References
Fig. 8. The dimensionless capillary force as a function of dimensionless separation with different dimensionless volumes for n1 = n2 = 3, R2 /R1 = 1, 1 = 2 = 30◦ .
interactions between two spheres (Lian et al., 1993). From Fig. 8, the maximum dimensionless capillary force at zero distance decreases quickly with increasing liquid bridge volume when n = 3. This is because the particles become flatter when the power–law index is 3. Accordingly, the change of liquid volume will induce a distinct change in the half-filling angle for the particles compared to n = 2. As a result, there is a distinct variation of the capillary force between particles at zero separation distance. The dimensionless force–distance curve also becomes steeper at the beginning stage when the liquid volume decreases. This result is similar to the capillary force effects seen between two spherical particles. This implies a decreased dimensionless rupture distance of the liquid bridge between two power–law axisymmetric particles as the liquid volume decreases. Conclusions In this study, an analytical method and process to calculate the capillary force between two power–law profile particles is described. Two solutions of the immersing radii on the top particles were obtained according to the Young–Laplace equation when the liquid bridge volume and separation distance were known, allowing two branches of capillary bridge shapes and capillary forces to be derived. It is shown that the two capillary force solutions converge to the critical point which is the rupture distance. The rupture distance of the liquid bridge between two power–law particles is consistent with the neck radius, immersing radius on the top particle or equivalent half-filling angle, average curvature radius, and capillary force. For the special case of a paraboloid–paraboloid configuration, the capillary interaction is similar that between two spheres. When the gravity of the liquid can be neglected, the
Adams, M. J., Johnson, S. A., Seville, J. P. K., & Willett, C. D. (2002). Mapping the influence of gravity on pendular liquid bridges between rigid spheres. Langmuir, 18(16), 6180–6184. Butt, H.-J., Cappella, B., & Kappl, M. (2005). Force measurements with the atomic force microscope: Technique, interpretation and application. Surface Science Reports, 59(1–6), 1–152. Butt, H.-J. (2008). Capillary forces: Influence of roughness and heterogeneity. Langmuir, 24(9), 4715–4721. De Souza, E. J., Brinkmann, M., Mohrdieck, C., Crosby, A., & Arzt, E. (2008). Capillary forces between chemically different substrates. Langmuir, 24(18), 10161–10168. Fan, Z., Wang, L., Rong, W., & Sun, L. (2015). Dropwise condensation on a hydrophobic probe-tip for manipulating micro-objects. Applied Physics Letters, 106(8), 084105. Fantoni, G., Hansen, H. N., & Santochi, M. (2013). A new capillary gripper for mini and micro parts. CIRP Annals—Manufacturing Technology, 62(1), 17–20. Grierson, D. S., Liu, J., Carpick, R. W., & Turner, K. T. (2013). Adhesion of nanoscale asperities with power–law profiles. Journal of the Mechanics and Physics of Solids, 61(2), 597–610. Israelachvili, J. N. (1992). Intermolecular and surface forces. London: Academic Press. Kim, S., Shafiei, F., Ratchford, D., & Li, X. (2011). Controlled AFM manipulation of small nanoparticles and assembly of hybrid nanostructures. Nanotechnology, 22(11), 115301. Lambert, P., & Delchambre, A. (2005). Parameters ruling capillary forces at the submillimetric scale. Langmuir, 21(21), 9537–9543. Lambert, P., Seigneur, F., Koelemeijer, S., & Jacot, J. (2006). A case study of surface tension gripping: The watch bearing. Journal of Micromechanics and Microengineering, 16(7), 1267–1276. Lian, G., Thornton, C., & Adams, M. J. (1993). A theoretical study of the liquid bridge forces between two rigid spherical bodies. Journal of Colloid and Interface Science, 161(1), 138–147. Mastrangeli, M., Abbasi, S., Varel, C., Van Hoof, C., Celis, J.-P., & Böhringer, K. F. (2009). Self-assembly from milli- to nanoscales: Methods and applications. Journal of Micromechanics and Microengineering, 19(8), 083001. Onal, C. D., Ozcan, O., & Sitti, M. (2011). Automated 2-D nanoparticle manipulation using atomic force microscopy. IEEE Transactions on Nanotechnology, 10(3), 472–481. Pakarinen, O. H., Foster, A. S., Paajanen, M., Kalinainen, T., Katainen, J., Makkonen, I., et al. (2005). Towards an accurate description of the capillary force in nanoparticle-surface interactions. Modelling and Simulation in Materials Science and Engineering, 13(7), 1175–1186. Payam, A. F., & Fathipour, M. (2011). A capillary force model for interactions between two spheres. Particuology, 9(4), 381–386. Rabinovich, Y. I., Adler, J. J., Ata, A., Singh, R. K., & Moudgil, B. M. (2000). Adhesion between nanoscale rough surfaces. I. Role of asperity geometry. Journal of Colloid and Interface Science, 232(1), 10–16. Rossetti, D., Pepin, X., & Simons, S. J. R. (2003). Rupture energy and wetting behavior of pendular liquid bridges in relation to the spherical agglomeration process. Journal of Colloid and Interface Science, 261(1), 161–169. Sun, Q., Wang, G., & Hu, K. (2009). Some open problems in granular matter mechanics. Process in Natural Science, 19(5), 523–529. Tabor, R. F., Grieser, F., Dagastine, R. R., & Chan, D. Y. C. (2012). Measurement and analysis of forces in bubble and droplet systems using AFM. Journal of Colloid and Interface Science, 371(1), 1–14. Wang, L., & Régnier, S. (2015). Capillary force between a probe tip with a power–law profile and a surface or a nanoparticle. Modelling and Simulation in Materials Science and Engineering, 23(1), 015001. Willett, C. D., Adams, M. J., Johnson, S. A., & Seville, J. P. K. (2000). Capillary bridges between two spherical bodies. Langmuir, 16(24), 9396–9405.
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ARTICLE IN PRESS
PARTIC-847; No. of Pages 6
Particuology xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Particuology journal homepage: www.elsevier.com/locate/partic
Capillary bridges and capillary forces between two axisymmetric power–law particles Lefeng Wang, Fengting Su, Huichao Xu, Weibin Rong, Hui Xie ∗ State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150080, China
a r t i c l e
i n f o
Article history: Received 23 April 2015 Received in revised form 21 July 2015 Accepted 18 August 2015 Available online xxx Keywords: Capillary force Power–law profile particles Power–law index Rupture distance
a b s t r a c t Capillary interactions are fundamentally important in many scientific and industrial fields. However, most existing models of the capillary bridges and capillary forces between two solids with a mediated liquid, are based on extremely simple geometrical configurations, such as sphere–plate, sphere–sphere, and plate–plate. The capillary bridge and capillary force between two axisymmetric power–law profile particles with a mediated constant-volume liquid are investigated in this study. A dimensionless method is adopted to calculate the capillary bridge shape between two power–law profile particles based on the Young–Laplace equation. The critical rupture criterion of the liquid bridge is shown in four forms that produce consistent results. It was found that the dimensionless rupture distance changes little when the shape index is larger than 2. The results show that the power–law index has a significant influence on the capillary force between two power–law particles. This is directly attributed to the different shape profiles of power–law particles with different indices. Effects of various other parameters such as ratio of the particle equivalent radii, liquid contact angle, liquid volume, and interparticle distance on the capillary force between two power–law particles are also examined. © 2015 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
Introduction Capillary interactions play a significant role in many scientific and industrial fields. These interactions must be accounted for in studies of particle cohesiveness and wet agglomeration processes, which are related to soil mechanics, granular materials, pharmaceuticals, and civil engineering (Rossetti, Pepin, & Simons, 2003; Sun, Wang, & Hu, 2009). In recent years, capillary interactions between micro- and nano-particles have attracted much attention because of the rapid development of micro and nanotechnology. Capillary interactions can, in principle, be exploited for gripping applications, and several types of capillary grippers have been proposed for microparticle manipulation (Lambert, Seigneur, Koelemeijer, & Jacot, 2006; Fantoni, Hansen, & Santochi, 2013; Fan, Wang, Rong, & Sun, 2015). Capillary forces are also utilized to control self-assembly processes from the millimeter to nanometer scale in microsystem engineering (Mastrangeli et al., 2009). Moreover, capillary interactions greatly affect force measurements and nanomanipulation processes, as revealed in atomic force microscopy (AFM) studies (Butt, Cappella, & Kappl, 2005;
∗ Corresponding author. Tel.: +86 45186404422. E-mail address: [email protected] (H. Xie).
Tabor, Grieser, Dagastine, & Chan, 2012; Kim, Shafiei, Ratchford, & Li, 2011; Onal, Ozcan, & Sitti, 2011). To use them effectively in these related fields, it is necessary to fully understand and accurately model capillary interactions. There has been much work modeling and measuring capillary forces between two particles. Based on thermodynamic equilibrium and non-equilibrium conditions, two main cases are considered when modeling the capillary forces. The first case refers to a liquid bridge that has a constant curvature radius, which is determined from the unsaturated vapor pressure in the environment, as described by the Kelvin equation (Butt, 2008). Pakarinen et al. (2005) investigated the capillary force between a nanosphere and a flat surface and found that the often-used model of the humidity-independent capillary force is reasonable for spherical particles above 1 m. Xiao and Qian (2000) measured the adhesion force between a Si3 N4 AFM tip and SiO2 or n-octadecyltrimethoxysilane (OTE)/SiO2 substrates. For the former, the adhesion force first increases and then decreases because of strong capillary condensation. For the latter, the adhesive force is almost independent of humidity because of weak capillary condensation. The second case corresponds to a constant-volume liquid bridge based on the energy principle. Rabinovich, Esayanur, and Moudgil (2005) calculated the capillary force between two spheres with a
http://dx.doi.org/10.1016/j.partic.2015.08.005 1674-2001/© 2015 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.
Please cite this article in press as: Wang, L., et al. Capillary bridges and capillary forces between two axisymmetric power–law particles. Particuology (2015), http://dx.doi.org/10.1016/j.partic.2015.08.005
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When V*Bo < 0.01, the effect of gravity can be neglected (Adams et al., 2002). It is assumed that this criterion is also applicable to the liquid bridge between two power–law profile particles. The profile of the top particle can be written as Z1 =
Fig. 1. Liquid bridge between two axisymmetric power–law profile bodies. The angle ϕ1 is determined by the normal line of the particle profile at point A and the axisymmetric axis of the particle.
fixed volume liquid bridge, and discussed the applicability of the Derjaguin approximation for the interaction between the spheres. De Souza, Brinkmann, Mohrdieck, Crosby, and Arzt (2008) investigated the capillary force between two plates with chemically different properties, and noted that the capillary force decreases as the asymmetry in contact angles is increased with the fixed sum of contact angles. Yang, Tu, and Fang (2010) proposed a rupture model of a capillary bridge between a micro sphere and a plane, and it was found that the rupture distance increases with increasing spherical radius, sphere hydrophobicity, and environmental humidity. Geometrical configurations considered in capillary force modeling have been mostly limited to sphere–plate (Israelachvili, 1992), sphere–sphere (Willett, Adams, Johnson, & Seville, 2000; Lian, Thornton, & Adams, 1993; Payam & Fathipour, 2011), and plate–plate (De Souza et al., 2008). However, in many real cases, particle shapes are not so ideal and simple. For example, the general shape of an AFM probe tip could be better described by a power–law axisymmetric function as a result of the fabrication method or in-use wear (Grierson, Liu, Carpick, & Turner, 2013). Recently, the capillary force between a power–law probe tip and a spherical particle or a plate under different humidity conditions was investigated (Wang & Régnier, 2015). The effects of relative humidity on the capillary forces were revealed in a dimensionless form which is applicable for equilibrium conditions of the liquid bridge. However, in applications such as micro/nano manipulation, we are more interested in the capillary force between two particles with a constant-volume liquid bridge. In this study, the capillary bridge and capillary force between two power–law profile particles with a constant-volume liquid bridge are investigated. In the Modeling the capillary bridge and capillary force section, modeling processes of the capillary bridge and capillary force between two power–law particles are presented in detail. In the Results and discussion section, the rupture criteria for the liquid bridges are demonstrated, and effects of each parameter on the capillary forces are analyzed. Finally, conclusions are given in the Conclusions section.
n −1
n1 R11
+ D,
(1)
where n1 is the power–law shape index of the top particle. For special cases, it denotes a parabolic shape when n1 = 2. R1 can be treated as the equivalent radius of the top particle, and the equivalent half-filling angle can be defined as ϕ1 = XA /R1 accordingly. D is the distance between the two power–law profile particles. When n1 = 1, it is a conical shape with conical angle of 45◦ (Zheng & Yu, 2007). Similarly, the profile of the bottom particle can be written as X n2
Z2 = −
n −1
n2 R22
,
(2)
where n2 is the power–law shape index of the bottom particle. R2 can be treated as the equivalent radius of the bottom particle, and the equivalent half-filling angle can be defined as ϕ2 = XB /R2 accordingly. The dimensionless method was used to simplify the calculation of the liquid bridge shape. If we let z = Z/R1 , x = X/R1 , and d = D/R1 , then Eqs. (1) and (2) can be nondimensionalized as follows: z1 =
xn1 + d, n1
z2 = −
xn2
n2 R2 /R1
(3)
n2 −1 .
(4)
According to the Young–Laplace equation, the liquid meniscus Z3 = f(X) satisfies the following equation: −
X 1 + X2
3/2 + X
1
P
1 + X2
1/2 = ,
(5)
where X = dX/dZ, X = d2 X/dZ 2 . P is the hydrostatic pressure difference across the air–liquid interface and is the surface tension of the liquid. Eq. (5) can be nondimensionalized as −
x 1 + x 2
3/2 + x
1 1 + x 2
1/2 =
R1 P = p.
(6)
The dimensionless volume v of the liquid bridge is
v = v0 − vt − vb ,
(7)
where v0 , vt , and vb represent the dimensionless volume of the column formed by the meniscus profile, the dimensionless volume of the immersed part of the top power–law particle, and the dimensionless volume of the immersed part of the bottom power–law particle, respectively. These terms can be obtained through integration:
Modeling the capillary bridge and capillary force Fig. 1 shows the liquid bridge between two power–law profile bodies. In this figure, 1 and 2 are the liquid contact angles at the two solid–liquid interfaces. Herein, relatively small liquid bridges are considered and gravitational deformations of the menisci are neglected. For the liquid bridge between two equal spheres with radii R, it was found that a modified Bond number V*Bo can be used to predict the effect of gravity on the pendular liquid bridge, in which V* is the dimensionless volume with V* = V/R3 , and Bo is the Bond number with Bo = gR2 / (Adams, Johnson, Seville, & Willett, 2002). The parameter is the density of the liquid, g is the acceleration due to gravity, and is the surface tension of the liquid.
X n1
zA x2 dz3 ,
v0 =
(8)
zB
zA vt =
x2 dz1 =
xn1 +2 , n1 + 2 A
x2 dz2 =
n2 + 2
(9)
d
0 vb =
R n2 −1 2
R1
xBn2 +2
(10)
zB
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Fig. 2. The relationships between the dimensionless parameters and the dimensionless separation distance: (a) the dimensionless neck radius xN , (b) the dimensionless immersing radius xA on the top particle, (c) the dimensionless mean curvature H, and (d) the dimensionless capillary force f, for n1 = n2 = 2, R2 /R1 = 1, 1 = 2 = 30◦ and for various liquid bridge volumes.
The relationship between the actual volume V of the liquid bridge and its dimensionless volume v is v = V/R1 3 . If the volume of the liquid bridge is constant, the final meniscus profile can be derived through Eq. (6) and related constraint conditions. Here we use an iterative method based on the given contact angles and constant liquid volume. By first assuming an immersing liquid radius xA on the top power–law profile particle, the contact angle on the bottom power–law particle can be calculated based on Eq. (6). The calculated contact angle is compared with the given one. If they are not equal, another immersing liquid radius is selected. If they are equal, the volume of the liquid bridge is calculated and compared with the given volume. The process is realized with a bisection method. When the final immersing liquid radius on the top power–law profile body xA is obtained, the profile of the liquid bridge can be determined. The method was also adopted to investigate the capillary force between a microsphere and a flat surface (Lambert & Delchambre, 2005). The total downward force acting on the upper particle F is the sum of the capillary pressure force Fc , the surface tension force Fs , and the buoyancy force Fb : F = Fc + Fs + Fb
(11)
Eq. (11) can be nondimensionalized as f = fc + fs + fb
(12)
These dimensionless forces can be expressed as (Lian et al., 1993): fc = −pxA2
fs = 2xA sin 1 + 1 fb =
xA n1 + 2
R12 g n1 +2
(13) (14) (15)
In this study, the liquid volume is sufficiently small, and the buoyancy force is neglected. The relationship between the actual capillary force F and the dimensionless capillary force f can be calculated through the following equation: F = fR1
(16)
Results and discussion In this section, the rupture criteria of the capillary bridge between two power–law axisymmetric particles will be discussed first, and then effects of various parameters on the capillary force between two power–law particles will be investigated in detail. Rupture of the capillary bridge between two power–law particles Several criteria have been adopted to predict the critical rupture of the liquid bridge between two spherical particles, such as the neck diameter, the half-filling angle, the mean curvature, and the capillary force. The applicability of the corresponding criteria for the axisymmetric power–law profile particles will be studied here. According to Eq. (6), the dimensionless mean curvature H can be expressed as H=
R1 p P = 2 2
(17)
Fig. 2(a)–(d) shows the relationship of these four variables as a function of separation distance in a dimensionless form. There are two branches of the solution based on the numerical method in the Modeling the capillary bridge and capillary force section, and they converge at the maximum value of the dimensionless separation distance. It can be seen that the critical dimensionless separation distance is identical for each criterion. This indicates that these criteria are also applicable for axisymmetric power–law particles. We focus our discussion on the rupture behavior and the capillary force of the liquid bridge. From Fig. 2(b), there is a minimum in the half-filling angle on the top power–law particle. The half-filling angle is also equivalent to the dimensionless immersing radius. The rupture distance will be underestimated if the minimum of the equivalent half-filling angle is adopted as a rupture criterion. This is similar to the capillary interaction between two spherical particles (Lian et al., 1993). Fig. 3 shows the relationship between the dimensionless rupture distance and dimensionless volume for different particle shapes. It is clear that the rupture distance increases with the liquid volume for various power–law particles. When n = 2, the power–law
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Fig. 3. The dimensionless rupture distance as a function of dimensionless volume for power–law particles with various shape indices of n1 = n2 = n, for R2 /R1 = 1, 1 = 2 = 30◦ .
Fig. 4. The dimensionless capillary force as a function of dimensionless separation distance with different shape indices of n1 = n2 = n, for R2 /R1 = 1, 1 = 2 = 30◦ , v = 0.01.
particles become paraboloids, and the rupture distance between them is almost equal to that between two spherical particles. This indicates that the capillary interaction between two spheres can be used to describe that between two paraboloids especially when the liquid volume is sufficiently small. However, when the shape index n ≥ 2, the dimensionless rupture distance decreases very gently with the increased shape index. Also, when the shape index increases gradually from n = 2, it tends to approach the special case of a plane/plane configuration. The data reveals that the dimensionless rupture distance is smaller when the shape index n = 1.5 than when n = 2. This can be attributed to the sharper particle profile when n < 2, and because the capillary bridge tends to become unstable at a large separation distances. Accordingly, the rupture distance of the capillary bridge for the cone/cone configuration for n = 1 is much lower than other configurations. Influences of parameters on capillary forces between two power–law particles Fig. 4 shows the relationship between the dimensionless capillary force and the dimensionless separation distance for power–law profile particles with different power–law indices n1 = n2 = 2, 3, and 6. The dimensionless capillary force increases gradually as the power–law index increases at the same dimensionless separation distance. This is attributed to the effects of the indices on the particle profiles based on Eqs. (3) and (4). The tip of the particle becomes flatter with increasing power–law index, as shown in Fig. 5. In other words, the flatter the particle tip becomes, the larger the dimensionless capillary force will be for a fixed dimensionless volume. At the zero separation distance, the dimensionless capillary forces vary largely as the power–law index n increases. As mentioned, when the shape index tends to infinity, the plate–plate
Fig. 5. The liquid bridge between power–law particles with different shape indices of n1 = n2 = n. The thinner lines denote the particle profiles, and the thick lines denote the liquid bridge profiles.
Fig. 6. The dimensionless capillary force as a function of dimensionless separation distance with different equivalent radius ratios for n1 = n2 = 2, 1 = 2 = 30◦ , v = 0.01.
configuration arises. For the plate–plate configuration, the radius of the liquid bridge tends to be infinity when the separation distance of the two plates is ideally zero, so the force tends to be infinity. However, the actual separation distance between two plates cannot be absolutely zero due to exponentially-increasing close-range repulsive intermolecular interactions and surface roughness effects (Rabinovich, Adler, Ata, Singh, & Moudgil, 2000). Fig. 6 shows the influence of the equivalent radius ratio R2 /R1 of power–law profile bodies on the capillary force. The stable dimensionless capillary force decreases as the equivalent radius ratio decreases at the same dimensionless separation distance. The bottom particle becomes sharper soon with decreasing R2 /R1 , and the liquid bridge radius on the top or bottom particle will decrease gradually; thus, the dimensionless force at a given dimensionless separation distance decreases with decreasing values of R2 /R1 . It can also be seen that the curve of the stable capillary force becomes flatter when the equivalent radius ratio decreases. Additionally, the dimensionless rupture distance of the liquid bridge decreases gradually with decreasing equivalent radius ratio. Fig. 7 shows the influence of the liquid contact angle on the capillary force between two axisymmetric power–law particles. The dimensionless capillary force decreases when the contact angle increases at small distances, and they tend to approach each other before the liquid bridge ruptures. The maximal stable capillary force that occurs at zero separation distance decreases when the contact angle becomes larger. However, the rupture distance increases with increased contact angle. Fig. 8 shows the influence of the dimensionless liquid volume on the dimensionless capillary force when n1 = n2 = 3. This parameter dependence when n1 = n2 = 2 has been demonstrated in Fig. 2(d). From Fig. 2(d), the dimensionless force at zero separation changes little when n = 2. This is consistent with capillary
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dimensionless capillary force increases with the increased power–law index, which is attributed to the flatter power–law particle profile. There may be a distinct variation of the dimensionless capillary force at zero separation distance because of the increased power–law index. The dimensionless force decreases as the ratio of the equivalent radii of the bottom and top particles decreases, and it is affected by the contact angle and liquid bridge volume. These results show more general cases for the capillary interaction between two axisymmetric particles. Acknowledgements
Fig. 7. The dimensionless capillary force as a function of dimensionless separation with different contact angles for n1 = n2 = 2, R2 /R1 = 1, v = 0.01.
This work was supported by the National Natural Science Foundation of China under grant Nos. 51175130 and 61222311, Self-Planned Task of State Key Laboratory of Robotics and System (HIT) under grant Nos. SKLRS201501A04 and SKLRS201301A02, and the Fundamental Research Funds for the Central University under grant No. HIT.BRETIV.201309. References
Fig. 8. The dimensionless capillary force as a function of dimensionless separation with different dimensionless volumes for n1 = n2 = 3, R2 /R1 = 1, 1 = 2 = 30◦ .
interactions between two spheres (Lian et al., 1993). From Fig. 8, the maximum dimensionless capillary force at zero distance decreases quickly with increasing liquid bridge volume when n = 3. This is because the particles become flatter when the power–law index is 3. Accordingly, the change of liquid volume will induce a distinct change in the half-filling angle for the particles compared to n = 2. As a result, there is a distinct variation of the capillary force between particles at zero separation distance. The dimensionless force–distance curve also becomes steeper at the beginning stage when the liquid volume decreases. This result is similar to the capillary force effects seen between two spherical particles. This implies a decreased dimensionless rupture distance of the liquid bridge between two power–law axisymmetric particles as the liquid volume decreases. Conclusions In this study, an analytical method and process to calculate the capillary force between two power–law profile particles is described. Two solutions of the immersing radii on the top particles were obtained according to the Young–Laplace equation when the liquid bridge volume and separation distance were known, allowing two branches of capillary bridge shapes and capillary forces to be derived. It is shown that the two capillary force solutions converge to the critical point which is the rupture distance. The rupture distance of the liquid bridge between two power–law particles is consistent with the neck radius, immersing radius on the top particle or equivalent half-filling angle, average curvature radius, and capillary force. For the special case of a paraboloid–paraboloid configuration, the capillary interaction is similar that between two spheres. When the gravity of the liquid can be neglected, the
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Please cite this article in press as: Wang, L., et al. Capillary bridges and capillary forces between two axisymmetric power–law particles. Particuology (2015), http://dx.doi.org/10.1016/j.partic.2015.08.005