2058
Langmuir 2006, 22, 2058-2064
Lateral Capillary Forces between Solid Bodies on Liquid Surface: A Lattice Boltzmann Study Hiroyuki Shinto,* Daisuke Komiyama, and Ko Higashitani Department of Chemical Engineering, Kyoto UniVersity, Nishikyo-ku, Kyoto 615-8510, Japan ReceiVed May 13, 2005. In Final Form: NoVember 18, 2005 When two solid bodies are placed on the surface of a dense liquid under gravitation, they deform the liquid surface to experience a lateral capillary force between themselves that can be attractive and repulsive, depending on the wettabilities and weights of the bodies. In the present study, the lateral capillary force between two square bodies at a liquid-vapor interface has been examined using numerical simulations based on a two-dimensional two-phase lattice Boltzmann (LB) method. The particular situations were simulated, where every body was vertically constrained and had the fixed triple points at its upper or lower corners. Here, the triple point indicates the place at which vapor, liquid, and solid phases meet. The interaction force between these two bodies was calculated as a function of the separation distance, the interfacial tension, and the gravitational acceleration. The simulation results agree well with the analytical expression of the lateral capillary interaction, indicating that our LB method can reproduce the interaction force between two bodies of various wettabilities at a liquid-vapor interface in mechanical equilibrium.
1. Introduction When particles are partially immersed in a liquid phase or a liquid film, they deform the liquid surface to experience the lateral capillary force between themselves. This force is utilized to make two-dimensional particle arrays and piles of them, which are expected to form materials and devices with novel functions. 1 The lateral capillary force has been extensively studied by the research groups of Kralchevsky and Nagayama,1-3 where the force between partially immersed bodies of various shapes was experimentally measured and the results were compared with those of the analytical expressions proposed. Their analytical expressions successfully predict the force between the body pairs on a mechanically equilibrated liquid surface (i.e., the two-body hydrostatic force) only when strict conditions are met: the bodies are sufficiently apart from each other, the deformation of the liquid surface is small enough, and so forth.1-3 Moreover, it is rather difficult to take into account the many-body effects4 and the hydrodynamic effects5 on the lateral capillary force using their analytical expressions. Numerical simulation based on the lattice Boltzmann (LB) method is a good tool to reproduce fluid flows in the presence of solid bodies of various shapes.6-9 Ladd proposed a heuristic technique to calculate the force and the torque acting on a solid body in a single-phase flow in which the modified collision rules are applied and then a net momentum transfer between the fluid * To whom correspondence should be addressed. Phone: +81-75-3832672. Fax: +81-75-383-2652. E-mail:
[email protected]. (1) Kralchevsky, P. A.; Nagayama, K. Particles at Fluid Interfaces and Membranes; Elsevier: Amsterdam, The Netherlands, 2001. (2) Kralchevsky, P. A.; Nagayama, K. Langmuir 1994, 10, 23 and references therein. (3) Kralchevsky, P. A.; Nagayama, K. AdV. Colloid Interface Sci. 2000, 85, 145 and references therein. (4) Yamaki, M.; Higo, J.; Nagayama, K. Langmuir 1995, 11, 2975. (5) Danov, K. D.; Pouligny, B.; Kralchevsky, P. A. Langmuir 2001, 17, 6599. (6) Rothman, D. H.; Zaleski, S. Lattice-Gas Cellular Automata: Simple Models of Complex Hydrodynamics; Cambridge University Press: Cambridge, England, 1997. (7) Chopard B.; Droz M. Cellular Automata Modeling of Physical Systems; Cambridge University Press: Cambridge, England, 1998. (8) Succi, S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond; Clarendon Press: Oxford, England, 2001. (9) Chen, S.; Doolen, G. D. Annu. ReV. Fluid Mech. 1998, 30, 329.
and the solid sites is integrated.10,11 This technique has been refined by several researchers.12-14 Recently, Inamuro et al. established the robust method, in which integrating stress tensor (and momentum flux) on the surface of a body gives the force and the torque.15 As for two-phase flows (e.g., liquid-vapor and two-component flows), there are several LB methods based on the chromodynamic model,16,17 the pseudopotential model,18,19 and the free-preenergy model.20,21 To our best knowledge, however, the force and the torque on a solid body in two-phase flows have never been studied by LB methods. In the present study, the focus is on two solid bodies at the liquid-vapor interface in mechanical equilibrium, in which the bodies deform the interface to experience the interaction force, namely, the lateral capillary force. Systematic simulations of the system have been carried out by the two-phase LB method based on a free-energy model. The interaction force between two vertically constrained bodies with the fixed triple line at the upper or lower edge is calculated as a function of the separation distance, the interfacial tension, and the gravitational acceleration. Here, the triple line indicates the place at which the vapor, liquid, and solid phases meet. The results are compared with those from the analytical expression. Although it is relatively straightforward to investigate both the many-body effects and the hydrodynamic effects on the lateral capillary force by LB simulations, this investigation is not within the scope of the present study, but will be in our future study. (10) Ladd, A. J. C. J. Fluid Mech. 1994, 271, 285. (11) Ladd, A. J. C. J. Fluid Mech. 1994, 271, 311. (12) Aidun, C. K.; Lu, Y.; Ding, E.-J. J. Fluid Mech. 1998, 373, 287. (13) Heemels, M. W.; Hagen, M. H. J.; Lowe, C. P. J. Comput. Phys. 2000, 164, 48. (14) Nguyen, N.-Q.; Ladd, A. J. C. Phys. ReV. E 2002, 66, 046708. (15) Inamuro, T.; Maeba, K.; Ogino, F. Int. J. Multiphase Flow 2000, 26, 1981. (16) Gunstensen, A. K.; Rothman, D. H.; Zaleski, S.; Zanetti, G. Phys. ReV. A 1991, 43, 4320. (17) Grunau, D.; Chen, S.; Eggert, K. Phys. Fluids A 1993, 5, 2557. (18) Shan, X.; Chen, H. Phys. ReV. E 1993, 47, 1815. (19) Shan, X.; Chen, H. Phys. ReV. E 1994, 49, 2941. (20) Swift, M. R.; Osborn, W. R.; Yeomans, J. M. Phys. ReV. Lett. 1995, 75, 830. (21) Swift, M. R.; Orlandini, E.; Osborn, W. R.; Yeomans, J. M. Phys. ReV. E 1996, 54, 5041.
10.1021/la0512751 CCC: $33.50 © 2006 American Chemical Society Published on Web 01/27/2006
LB Study of Lateral Capillary Forces
Langmuir, Vol. 22, No. 5, 2006 2059
The order parameter φ distinguishing two phases, the flow velocity u, and the pressure p are defined as 9
φ(x,t) ) Figure 1. Sketch of the capillary meniscus around two square bodies of side length l, which are vertically constrained to be partially immersed in a liquid phase and have the fixed triple points at their upper corners; ζ(x) is the shape of the meniscus formed around the bodies, and h1 and h2 denote its elevation at the respective body above the flat liquid surface (i.e., y ) 0) far from it; d t L - l is the gap distance between the body surfaces; PL and PV are the pressures inside the liquid phase and in the upper vapor phase, respectively; γLV is the liquid-vapor interfacial tension; g is the gravitational acceleration.
1 fi(x + ci∆x,t + ∆t) ) fi(x,t) - [fi(x,t) - f eq i (x,t)] τf gi(x + ci∆x,t + ∆t) ) gi(x,t) -
(2.1)
1 [g (x,t) - geq i (x,t)] τg i 3Eiciy(F - FV)g∆x (2.2)
eq where f eq i and gi are the equilibrium distribution functions, τf and τg are the dimensionless single relaxation times, ∆x is the spacing of the square lattice, and ∆t t (U0/c)∆x is the time step during which the particles travel along the lattice spacing. The effect of gravity is considered by the third term in the right-hand side of eq 2.2, where g is the dimensionless gravitational acceleration; F and Ei will be given in eqs 2.6 and 2.9, respectively.22
(22) Inamuro, T. Bussei Kenkyu 2001, 77, 197. (23) Inamuro, T.; Tomita, R.; Ogino, F. Int. J. Mod. Phys. B 2003, 17, 21.
(2.3)
9
∑cigi(x,t) F(x,t) i)1
u(x,t) )
(2.4)
1 9 p(x,t) ) [ gi(x,t) - F(x,t)] 3 i)1
∑
(2.5)
with the density F
2. Simulation Methods Let us here consider the system illustrated in Figure 1, in which two solid bodies are placed on the surface of a dense liquid under gravitation. Every body is vertically constrained and has the fixed triple points at its upper or lower corners; this results in the deformation of the liquid surface. Consequently, there arises a force between the bodies due to interfacial tension effects. This type of lateral capillary force can be attractive and repulsive, depending on the positions of the triple points that are determined by the wettabilities of the bodies in our modeling, as will be demonstrated in sections 3.2 and 4. It is noted that another two types of lateral capillary forces, flotation and immersion, are driven by the particle weight and the particle wettability, respectively.2 2.1. Two-Phase Liquid-Vapor Flows. Hereafter, we use nondimensional variables, which are defined by using the characteristic quantities of a length L0, a flow speed U0, a particle speed c, a time scale t0 () L0/U0), and a reference density F0. The two-dimensional nine-velocity (D2Q9) model was employed, where the physical space of interest is divided into a square lattice and the evolution of the particle population at each lattice site is computed. The D2Q9 model has velocity vectors ci ) (0, 0), (1, 0), (0, 1), (-1, 0), (0, -1), (1, 1), (-1, 1), (-1, -1), and (1, -1) for i ) 1, 2, ..., and 9, respectively. We adopted the two-phase LB method of Inamuro,22,23 in which the coexistence of the liquid phase of density FL with the vapor phase of density FV is assumed and two particle distribution functions, fi and gi, are used. The distribution functions, fi(x,t) and gi(x,t) with velocity ci at node x and at time t, evolve following
1
fi(x,t) ∑ i)1
F(x,t) )
φ(x,t) - φV (F - FV) + FV φ L - φV L
(2.6)
where φ ranges from the minimum φV to the maximum φL. The two equilibrium distribution functions are expanded as a power series of the local velocity: 2 f eq i ) Hiφ + Fi(p0 - κfφ∇ φ) + 3 9 φ Eiφ 3uRciR - uRuR + uRuβciRciβ + EiκfGRβ ciRciβ (2.7) 2 2
(
)
3 9 geq i ) HiF + Ei 3p + F 3uRciR - uRuR + uRuβciRciβ + 2 2 ∂F F + EiGRβ ciRciβ (2.8) 2FiωguR ∂xR
[
)]
(
with
E1 ) 4/9,
E2 ) ‚‚‚ ) E5 ) 1/9,
E6 ) ‚‚‚ ) E9 ) 1/36
F1 ) -5/3, Fi ) 3Ei for i ) 2, 3, ..., 9 H1 ) 1, φ ) GRβ
(
H2 ) ‚‚‚ ) H9 ) 0
(
9 ∂φ ∂φ 1 ∂φ ∂φ δ 2 ∂xR ∂xβ 2 ∂xγ ∂xγ Rβ
)
(2.9)
)
(2.10)
9 ∂F ∂F 1 ∂F ∂F F ) κg δ + GRβ 2 ∂xR ∂xβ 2 ∂xγ ∂xγ Rβ ∂F ∂F ∂F 9 ω u + uR - uγ δRβ (2.11) 2 g β∂xR ∂xβ ∂xγ
(
ωg )
)
1 1 τ - ∆x 3 g 2
(
)
(2.12)
where κf and κg are the constant parameters determining the interface width and the interfacial tension, respectively.22 In the above equations, subscripts R, β, and γ denote Cartesian coordinates (R, β, γ ) x, y), and δRβ is the Kronecker delta. The summation convection is applied to R and β in eqs 2.7 and 2.8 and to γ in eqs 2.10 and 2.11. In eq 2.7, p0 is given by
p0 )
φT - aφ2 1 - bφ
(2.13)
which is the equation of state of a van der Waals fluid with the parameters a, b, and T determining φL and φV.
2060 Langmuir, Vol. 22, No. 5, 2006
Shinto et al. Table 1. Systems of Lattice Boltzmann Simulations
system
χ1
χ2
L/∆x
κg/(∆x)2 (γLV/10-4∆x)
g/10-6
q(L - l)
number of time steps
A.1 A.2 A.3 B.1 B.2 B.3 C.1 C.2 C.3
1 1 1 -1 -1 -1 1 1 1
1 1 1 -1 -1 -1 -1 -1 -1
40-80 64 64 40-80 64 64 40-80 64 64
0.01 (2.8) 0.01-0.05 (2.8-13.7) 0.01 (2.8) 0.01 (2.8) 0.01-0.05 (2.8-13.7) 0.01 (2.8) 0.01 (2.8) 0.01-0.05 (2.8-13.7) 0.01 (2.8)
5.0 5.0 2.5-7.5 5.0 5.0 2.5-7.5 5.0 5.0 2.5-7.5
1.9-5.4 4.0-1.8 2.8-4.9 1.9-5.4 4.0-1.8 2.8-4.9 1.9-5.4 4.0-1.8 2.8-4.9
340 000-740 000 280 000-560 000 460 000-780 000 180 000-620 000 220 000-480 000 380 000-660 000 1 500 000-2 140 000 1 540 000-1 920 000 1 400 000-3 240 000
The stress tensor σ ) {σRβ} is represented as
σRβ ) -
τg -
1
PRβ -
2τg
τg -
1 2
τg 1 2
6τg
(
9
gi(ciR - uR)(ciβ - uβ) ∑ i)1
)
∂F ∂F ∂F uβ + uR + uγ δRβ ∆x (2.14) ∂xR ∂xβ ∂xγ
surface, and then the halfway bounce-back, no-slip boundary condition8 is applied on the fluid-side lattice nodes that are next to the solid surface. Once the fluid densities at solid-side nodes are given, the fluid near the solid surfaces autonomously evolves to minimize the total free energy of the system.26 In the present study, the surface of a body or a wall was represented by a collection of the continuous lattice nodes with the identical density FS; consequently, these surfaces were chemically homogeneous. The affinity of the solid surface for fluid, χ, is defined as
where P ) {PRβ} is the pressure tensor
(
)
κg ∂F ∂F PRβ ) p - |∇F|2 δRβ + κg 2 ∂xR ∂xβ
χ≡ (2.15)
The kinematic viscosity ν and the interfacial tension γLV are respectively given by
ν)
1 1 τ - ∆x 3 g 2
(
)
dξ ∫-∞∞ (∂F ∂ξ)
γLV ) κg
2
(2.16) (2.17)
where ξ is the coordinate normal to the interface.22,24 The derivatives in eqs 2.7, 2.8, 2.10, 2.11, 2.14, and 2.15 are calculated using the finite-difference approximations
∂λ ∂xR ∇2λ ≈
9
1
≈
∑ 6∆x i)2
1
9
3∆x
[
ciRλ(x + ci∆x)
λ(x + ci∆x) - 8λ(x)] ∑ i)2
(2.18)
(2.19)
where λ indicates φ or F. The parameters in eq 2.13 were chosen as a ) 1.0, b ) 6.7, and T ) 3.5 × 10-2, which resulted in φL ) 9.714 × 10-2 and φV ) 1.134 × 10-2 because of the requirements for the coexistence chemical potential at φ ) φL and φV.25 The density ratio FL/FV was set at 5 (i.e., FL ) 0.5 and FV ) 0.1). The other parameters used are τf ) 1.0, τg ) 1.1, and κf ) 0.8(∆x)2 throughout the present simulations, whereas several values of κg and g were employed, as summarized in Table 1. 2.2. Solid Surfaces with Wettability. The solid body and the solid wall of a fluid-wettable surface were modeled following our previous study:26 The fluid densities, ranging from FV to FL, are assigned to the solid-side lattice nodes that are on the solid (24) Inamuro, T.; Konishi, N.; Ogino, F. Comput. Phys. Commun. 2000, 129, 32. (25) Yang, A. J. M.; Fleming, P. D., III; Gibbs, J. H. J. Chem. Phys. 1976, 64, 3732. (26) Iwahara, D.; Shinto, H.; Miyahara, M.; Higashitani, K. Langmuir 2003, 19, 9086.
FS - Fd FL - Fd
(2.20)
where FV e FS e FL, and Fd ≡ (FL + FV)/2. The parameter χ satisfies -1 e χ e 1 and is referred to as the surface affinity in the present study. The solid surface becomes more wettable with increasing χ.26 It is noted that all the solid-side nodes have the same density FL or FV when χ ) 1 or -1, which corresponds to the complete wettable or the complete unwettable surface. 2.3. Force on Solid Body. Because a circular body is represented by a collection of lattice nodes in LB simulations, the outer edge is not smooth but rather rough stepwise.10-15 For the sake of simplicity, a square body embedded in a part of the lattice grid was considered in the present study, as illustrated in Figure 2. Following Inamuro et al.,15 the force acting on square body i, Fi, was calculated by
Fi )
∫S {σ ‚ n - Fu[(u - ui) ‚ n]}dS - dtd ∫V FudV
(2.21)
where S is a closed surface with outward unit normal vector n, and V denotes the region that is inside the closed surface and outside the square body. Because the square bodies were immobile, the velocities were ui ≡ 0. The interaction force between the bodies 1 and 2, respectively centered at x1 and x2, was given by
x1 - x2 1 F(L) ) (F1 - F2) ‚ 2 |x1 - x2|
(2.22)
where L ≡ |x1 - x2| is the center-to-center distance between the bodies. The interaction force is repulsive when F > 0, whereas it is attractive when F < 0. 2.4. Simulation Details. The simulation region was represented by the lattice nodes of Lx × Ly ) 255∆x × 63∆x and confined within the solid walls. The wettable bottom wall of χ ) 1 (FS ) FL) was placed at y ) 0 in the range of ∆x e x e Lx, whereas the unwettable upper wall of χ ) -1 (FS ) FV) was located at y ) Ly + ∆x in the range of ∆x e x e Lx. The sidewalls of χ ) 0 (FS ) Fd), which are between the wettable and unwettable walls, were positioned at x ) 0 and Lx + ∆x. The square bodies 1 and 2 of side length l ) 17∆x were fixed at x1 ) (128∆x L/2, 32∆x) and x2 ) (128∆x + L/2, 32∆x), respectively. Three
LB Study of Lateral Capillary Forces
Langmuir, Vol. 22, No. 5, 2006 2061
one when the liquid phase has sufficiently large thickness. In this case, one obtains
PL(ζ) ) PL(0) - FLgζ
(3.2)
PV(ζ) ) PL(0) - FVgζ
(3.3)
When the slope of the meniscus around the bodies is small enough (i.e., |dζ/dx| , 1), combining eqs 3.1-3.3 results in a linearized form of the Laplace equation Figure 2. Square body of side length l embedded in a part of the lattice grid and the fluid-side nodes next to the body surface; S represents a closed surface with outward unit normal vector n; V denotes the region that is inside the closed surface and outside the square body; ∆x is the spacing of the square lattice.
types of square body pairs were considered, as summarized in Table 1: (A) two wettable bodies of χ1 ) χ2 ) 1; (B) two unwettable bodies of χ1 ) χ2 ) -1; (C) the wettable body of χ1 ) 1 and the unwettable body of χ2 ) -1. As an initial condition, the densities FL and FV were respectively given to the lower half (∆x e y e 32∆x) and the upper half (33∆x e y e Ly) of the simulation cell, except the area of the solid bodies. The halfway bounce-back, no-slip boundary condition8 was imposed on the fluid-side nodes of the walls and the body surfaces. The system was allowed to evolve until it reached mechanical equilibrium, when the separation L, the interfacial tension γLV determined by κg, and the gravitational acceleration g were given as in Table 1. The criterion for judging the mechanical equilibrium was
|
|
F(tn+1) - F(tn) F(tn+1)
< 10-5
(2.23)
where F is the force between the solid bodies calculated by eq 2.22, and tn ) (20 000n)∆t with positive integer n. The number of time steps required for equilibrating each system is listed in Table 1.
3. Analytical Expressions
(x
dζ/dx
)
1 + (dζ/dx)2
)
PV(ζ) - PL(ζ) γLV
(3.1)
where y ) ζ(x) is the equation of the deformed fluid interface. The upper surface of the liquid phase is flat far from the body; this line is chosen as the level y ) 0 (see Figure 1). The pressures PV and PL on the vapor and the liquid sides of the interface depend on the effects of hydrostatic pressure and disjoining pressure. The latter effect is negligible compared to the former
(3.4)
x
(3.5)
(FL - FV)g γLV
where q-1 is called the capillary length. The boundary conditions
{
ζ(-∞) ) 0 ζ[-(d/2 + l)] ) ζ(-d/2) ) h1 ζ(d/2) ) ζ(d/2 + l) ) h2 ζ(∞) ) 0
(3.6)
bring about the solution of eq 3.4 in five distinct regions:
{
ζ(x) ) h1 exp[q{x + (d/2 + l)}] h1
x e - (d/2 + l) -(d/2 + l) < x < -d/2
(h2eqd/2 - h1e-qd/2)eqx + (h1eqd/2 - h2e-qd/2)e-qx eqd - e-qd
-d/2 e x e d/2 d/2 < x < d/2 + l x g d/2 + l
h2 h2 exp[-q{x - (d/2 + l)}]
(3.7)
Two particular situations of eq 3.7 are considered to give the interface profile in the gap between two bodies (i.e., -d/2 e x e d/2):
ζ(x) )
The lateral capillary force between two partially immersed bodies is caused by the overlap of the deformed liquid surfaces around them. In this section, an analytical expression based on the Laplace equation is derived for the lateral capillary force between two square bodies, which are vertically constrained and have the fixed triple points at their corners,2 as shown in Figure 1. 3.1. Slightly Deformed Fluid Interface. Two square bodies of side length l and gap distance d (≡ L - l) between the surfaces are considered, as illustrated in Figure 1. The bodies deform the fluid interface, the shape of which obeys the Laplace equation of capillary:
d dx
q)
d 2ζ ) q2ζ dx2
ζ(x) ) -
h(eqx + e-qx) eqd/2 + e-qd/2 h(eqx - e-qx) eqd/2 - e-qd/2
for h1 ) h2 ) h (* 0)
(3.8)
for h1 ) - h2 ) h (* 0) (3.9)
Equation 3.8 represents the case of two square bodies with the triple points stuck to their upper corners (h1 ) h2 > 0) or their lower corners (h1 ) h2 < 0). Equation 3.9 represents the case of two square bodies in which one has the fixed triple points at the upper corners and the other has them at the lower corners. 3.2. Lateral Capillary Force. The grand thermodynamic potential of the system illustrated in Figure 1 can be written in the form
Ω(x1,x2) ) Wg + Ww + Wm + const
(3.10)
where Wg, Ww, and Wm are the gravitational, wetting, and meniscus contributions to the grand potential Ω, respectively.1-3 The wetting contribution is
Ww ≡ const
(3.11)
because the contact points are immobilized. The gravitational
2062 Langmuir, Vol. 22, No. 5, 2006
Shinto et al.
contribution is described as 2
Wg )
migyi + (FL - FV)g∫V ydxdy ∑ i)1
(3.12)
m
where mi is the mass of body i, yi is the vertical position of its center, and Vm is the region comprised between the meniscus surface y ) ζ(x) and its projection on the x axis. Because two bodies are vertically fixed, eq 3.12 reduces to
Wg ) const + (FL - FV)g
∫-∞∞dx ∫0ζydy )
const +
(FL - FV)g 2
∫-∞∞ ζ2dx
(3.13)
The meniscus contribution is given by
Wm ) γLV∆A ) γLV
∫-∞∞[x1 + (dζ/dx)2 - 1]dx ≈ γLV ∞ dζ 2 dx ∫ 2 -∞(dx)
Figure 3. Instantaneous forces between two square bodies as a function of time t: (b) system A, χ1 ) χ2 ) 1; (9) system B, χ1 ) χ2 ) -1; (O) system C, χ1 ) -χ2 ) 1. The parameters used are L ) 64∆x, γLV ) 2.8 × 10-4∆x, and g ) 5.0 × 10-6. The dashed line indicates the zero forces.
(3.14)
where ∆A is the difference between the area of the meniscus and that of its projection on the x axis, and the assumption of |dζ/dx| , 1 is employed again. Using eq 3.4, one obtains the relation 2
(dζdx) ) dxd (ζdζdx) - ζddxζ ) dxd (ζdζdx) - q ζ 2
2 2
(3.15)
2
Equation 3.14 with eq 3.15 becomes
Wm )
γLV 2
γLVq2 ∞ 2 d dζ ζ dx ζ dx ) -∞ dx dx -∞ 2 (FL - FV)g ∞ 2 γLV dζ ∞ ζ ζ dx (3.16) -∞ 2 dx -∞ 2
∫
∞
∫
( ) [ ]
∫
Consequently, eq 3.10, combined with eqs 3.11, 3,13, and 3.16 as well as eq 3.7, reduces to
Ω)
γLV dζ ∞ ζ + const ) 2 dx -∞ γLVq[(h12 + h22)eqd - 2h1h2]
noted that eq 3.20 yields the attractive force, whereas eqs 3.21 gives the repulsive force.
[ ]
eqd - e-qd
+ const (3.17)
γLVq[2h1h2 - (h1 + h2 )e 2
∆Ω ≡ Ω(L) - Ω(∞) ) -
e -e qd
2
-qd
]
-qd
(3.18)
where d ≡ L - l. The lateral capillary force is then described as
d∆Ω ) dL 2(FL - FV)g[h1h2(eqd + e-qd) - (h12 + h22)]
F(L) ) -
(eqd - e-qd)2
(3.19)
Like eq 3.7, two particular cases of eq 3.19 are considered:
F(L) ) -
F(L) )
2(FL - FV)gh2e-qd (1 + e-qd)2
2(FL - FV)gh2e-qd (1 - e-qd)2
Figure 4. Snapshots of the fluid densities for the same systems described in Figure 3 after mechanical equilibration, that is, at t ) 560 000∆t, 480 000∆t, and 1 920 000∆t for systems A, B, and C, respectively. Light gray indicates the vapor phase, dark gray represents the liquid phase, and the black squares are the solid bodies. The top, the bottom, and the side walls are depicted as the nodes with assigned fluid densities FV, FL, and Fd t (FL + FV)/2, respectively.
for h1 ) h2 ) h (* 0) (3.20)
for h1 ) -h2 ) h (* 0)
(3.21)
which correspond to eqs 3.8 and 3.9, respectively. It should be
4. Results and Discussion A typical result of the lateral capillary forces between two bodies as a function of time is depicted in Figure 3, where L ) 64∆x, κg ) 0.01(∆x)2 (i.e., γLV ) 2.8 × 10-4∆x), and g ) 5.0 × 10-6. We determined by eq 2.23 that the forces for systems A (χ1 ) χ2 ) 1), B (χ1 ) χ2 ) -1), and C (χ1 ) -χ2 ) 1) reach the constant values after 560 000, 480 000, and 1 920 000 time steps, respectively. The forces after mechanical equilibration are attractive for systems A and B, whereas that for system C is repulsive. Figure 4 illustrates that the liquid surface deforms to attain the mechanical equilibrium, depending on the wettabilities of the body surfaces: the wettable body of χi ) 1 pulls up the liquid surface (i.e., hi > 0), while the unwettable body of χi ) -1 pushes it down (i.e., hi < 0). Thus, the wettability of the body surface determines the positions of the triple points at the body surface in our modeling. The meniscus positions at two bodies, h1 and h2, explain the direction of the lateral capillary force exerted on the bodies, as expected from eq 3.20 for systems A and B and eq 3.21 for system C. Henceforth, the results of LB simulations after achieving mechanical equilibrium are presented for the systems listed in Table 1. We investigate how the lateral capillary force depends on L, γLV, and g in sections 4.1, 4.2, and 4.3, respectively, where
LB Study of Lateral Capillary Forces
Figure 5. Forces between two square bodies as a function of the separation L: (b) system A.1, χ1 ) χ2 ) 1; (9) system B.1, χ1 ) χ2 ) -1; (O) system C.1, χ1 ) -χ2 ) 1. The parameters used are γLV ) 2.8 × 10-4∆x and g ) 5.0 × 10-6. The solid and the dash lines represent the theoretical force-distance profiles of eqs 3.20 and 3.21, respectively, with γLV ) 2.8 × 10-4∆x, g ) 5.0 × 10-6, FL - FV ) 0.4, and |h1| ) |h2| ) 8.5∆x.
Figure 6. Same as Figure 5 but recasted in a semilogarithmic graph. The solid and the dash lines represent the theoretical force-distance profiles of eqs 3.20 and 3.21, respectively. See also the caption of Figure 5.
the simulation results are compared with the theoretical predictions by eqs 3.20 and 3.21. 4.1. Dependency on Separation Distance. Figure 5 displays the forces as a function of the separation L for systems A.1, B.1, and C.1, where γLV ) 2.8 × 10-4∆x, and g ) 5.0 × 10-6. The forces are attractive for systems A and B, whereas those for system C are repulsive. For every system, the force becomes stronger with decreasing L. The force-distance profiles for systems A.1 and C.1 almost coincide with the theoretical profiles of eqs 3.20 and 3.21 with γLV ) 2.8 × 10-4∆x, g ) 5.0 × 10-6, FL - FV ) 0.4, and |h1| ) |h2| ) 8.5∆x, although the agreement is relatively poor for system B.1. Figure 5 is recasted in the semilogarithmic graph of Figure 6. The forces exhibit an exponential decrease with the decay length of λ ) 10.1∆x, 17.7∆x, and 11.1∆x for systems A.1, B.1, and C.1, respectively, which were obtained by fitting the relation |F| ∼ exp(-L/λ) to the simulation data of L/∆x ) 56-80; in this region, eqs 3.20 and 3.21 respectively approximate to
F(L) ≈ -2(FL - FV)gh2 exp[-q(L - l)] for h1 ) h2 ) h (* 0) (3.20′) F(L) ≈ 2(FL - FV)gh2 exp[-q(L - l)] for h1 ) -h2 ) h (* 0) (3.21′) because e-qd ) exp[-q(L - l)] , 1 (see also Table 1). The decay lengths are in good agreement with the capillary length q-1 ) 11.8∆x obtained by eq 3.5, except for system B.1.
Langmuir, Vol. 22, No. 5, 2006 2063
Figure 7. Forces between two square bodies as a function of the interfacial tension γLV: (b) system A.2, χ1 ) χ2 ) 1; (9) system B.2, χ1 ) χ2 ) -1; (O) system C.2, χ1 ) -χ2 ) 1. The parameters used are L ) 64∆x and g ) 5.0 × 10-6. The solid and the dash lines represent the theoretical force-interfacial tension profiles of eqs 3.20 and 3.21, respectively, with L ) 64∆x, g ) 5.0 × 10-6, FL FV ) 0.4, and |h1| ) |h2| ) 8.5∆x.
Figure 8. Snapshots of the fluid densities for system A.2 (χ1 ) χ2 ) 1) of different interfacial tensions: (a) γLV ) 2.8 × 10-4∆x; (b) γLV ) 8.3 × 10-4∆x; (c) γLV ) 13.7 × 10-4∆x. See also the caption of Figure 4.
These results demonstrate that our LB simulations can capture the effects of the separation on the lateral capillary forces between two bodies. Nonetheless, further LB simulations are necessary to understand the discrepancy in system B.1 between the decay length and the capillary length. This will be our future study. 4.2. Dependency on Interfacial Tension. Figure 7 shows the forces as a function of interfacial tension γLV for systems A.2, B.2, and C.2, where L ) 64∆x, and g ) 5.0 × 10-6. The force becomes stronger with γLV for every system. The profiles for systems A.2 and C.2 are in quantitative agreement with the theoretical profiles of eqs 3.20 and 3.21 with L ) 64∆x, g ) 5.0 × 10-6, FL - FV ) 0.4, and |h1| ) |h2| ) 8.5∆x. The agreement is rather poor for system B.2. This is also the case for the forcedistance profiles of Figures 5 and 6, as mentioned in section 4.1. We confirmed that eqs 1.24 and 1.25 in ref 2 for three-dimensional systems predict the increase of |F| with γLV like eq 3.20 does in the present study. Figure 8 displays snapshots of the fluid densities for system A.2, where the liquid level at the meniscus center between the wettable bodies is positive and becomes higher with increasing γLV. In contrast, the liquid level at the meniscus center between the unwettable bodies for system B.2 is negative and becomes lower with γLV (snapshots not shown). In other words, the larger interfacial tension causes the more significant oVerlapping of two deformed liquid surfaces between two similar bodies at a given separation, as expected from eq 3.8 with eq 3.5 (i.e., ζ(0) ) h/cosh(qd/2) and q ∼ γLV-1/2); this leads to a stronger attractive force between the bodies for systems A.2 and B.2, as shown in
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theoretical predictions is significant. Again, this is the case for the force-distance profiles of Figures 5 and 6 and the forceinterfacial tension profiles of Figure 7, as mentioned in sections 4.1 and 4.2, respectively. It should be noted that the theoretical profile of eq 3.20 in Figure 10 has a minimum at g ) 1.82 × 10-6 (i.e., qd/2 ) 1.1996...). Likewise, we found that eqs 1.24 and 1.25 in ref 2 for three-dimensional systems give a single-well profile of F as a function of g.
5. Conclusions
Figure 9. Same as Figure 8 but for system C.2 (χ1 ) -χ2 ) 1).
Figure 10. Forces between two square bodies as a function of the gravitational acceleration g: (b) system A.3, χ1 ) χ2 ) 1; (9) system B.3, χ1 ) χ2 ) -1; (O) system C.3, χ1 ) -χ2 ) 1. The parameters used are L ) 64∆x and γLV ) 2.8 × 10-4∆x. The solid and the dash lines represent the theoretical force-gravitational acceleration profiles of eqs 3.20 and 3.21, respectively, with L ) 64∆x, γLV ) 2.8 × 10-4∆x, FL - FV ) 0.4, and |h1| ) |h2| ) 8.5∆x.
Figure 7. On the other hand, the snapshots of system C.2 shown in Figure 9 demonstrate that the larger interfacial tension sharpens the shape of the meniscus between two dissimilar bodies at a given separation (i.e., [dζ(x)/dx]x)0 ) -(2h/d)[(qd/2)/sinh(qd/ 2)] and q ∼ γLV-1/2) to enhance the anti-oVerlapping of two deformed liquid surfaces between them; this gives rise to the stronger repulsive force for system C.2, as shown in Figure 7. 4.3. Dependency on Gravitational Acceleration. The deformability of the liquid surface, which influences the strength of the lateral capillary force, is varied not only by the interfacial tension γLV, but also by the gravitational acceleration g, as expected from eq 3.7 with eq 3.5. The forces as a function of g are plotted in Figure 10 for system A.3, B.3, and C.3, where L ) 64∆x and γLV ) 2.8 × 10-4∆x. The force becomes weaker with increasing g for every system under our simulation conditions. The simulation results for systems A.3 and C.3 are in quantitative agreement with the theoretical predictions from eqs 3.20 and 3.21 with L ) 64∆x, γLV ) 2.8 × 10-4∆x, FL FV ) 0.4, and |h1| ) |h2| ) 8.5∆x, as shown in Figure 10. For system B.3, the discrepancy between the simulation data and the
We have examined the lateral capillary forces between two square bodies at the liquid-vapor interface in mechanical equilibrium, using numerical simulations based on the twodimensional two-phase LB method of Inamuro.22,23 The particular situations were simulated, in which every body was vertically constrained and had the fixed triple points at its upper or lower corners. In our modeling, the positions of the triple points at the body surface were determined by its wettability. The lateral capillary force between these two bodies was calculated as a function of the separation distance, the interfacial tension, and the gravitational acceleration. As a result, it was found that the lateral capillary force between two square bodies is attractive when both bodies have the fixed triple points at their upper or lower corners (i.e., both bodies are wettable or unwettable in our modeling); in contrast, the force is repulsive when one has the fixed triple points at the upper corners and the other has them at the lower corners (i.e., one is wettable and the other is unwettable). The force-distance profile exhibits an exponential decrease for every body pair. The force at a given separation becomes stronger with the interfacial tension, while it becomes weaker with the gravitational acceleration under our simulation conditions. These simulation results are in quantitative agreement with the results from the analytical expression of the lateral capillary interaction, except for the unwettable body pair with the fixed triple points at their lower corners. The quantitative agreement between the simulation results and the theoretical predictions indicates that our LB method can reproduce the interaction force between two bodies of various wettabilities at a liquid-vapor interface in mechanical equilibrium, namely, one of the hydrostatic forces. Nonetheless, further examination is necessary to understand the discrepancy between the simulations and the theories of the capillary interaction for the unwettable body pair. This will be our future study. Investigation of the hydrodynamic forces acting on the bodies in the liquid-vapor flows will be our future study as well. Acknowledgment. This work was partly supported by the Grants-in-Aid (No. 14750603/15206085) for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology in Japan, and by the Hosokawa Powder Technology Foundation, Japan. LA0512751
