Lattice Boltzmann simulations of drop dynamics H. Kusumaatmaja a , A. Dupuis a,b , J. M. Yeomans a a The
Rudolf Peierls Centre for Theoretical Physics, Oxford University, 1 Keble Road, Oxford OX1 3NP, UK
b Institute
of Computational Science, Universit¨atstrasse 6, ETH Zentrum, CH - 8092 Z¨urich, Switzerland
Abstract We present a free energy lattice Boltzmann approach to modelling the dynamics of liquid drops on chemically patterned substrates. We start by describing a choice of free energy that reproduces the bulk behaviour of a liquid-gas system together with the varying contact angles on surfaces with chemical patterning. After showing how the formulation of the free energy fits in to the framework of lattice Boltzmann simulations, numerical results are presented to highlight the applicability of the approach. Key words: Lattice Boltzmann, Wetting, Drop dynamics PACS: 47.11.+j, 68.08.Bc, 47.61.Jd
Our aim in this paper is to present the details of a free energy lattice Boltzmann approach [1–3] to modelling the dynamics of liquid drops moving on surfaces patterned with regions of different wettabilities (contact angles). There has been considerable interest in recent years from both the academic and the industrial communities in studying the morphologies and dynamics of these drops [4–9], driven by the possibility of using chemical patterning for ink-jet or microfluidic applications, or simply as a first step to understand their behaviour on non-homogeneous surfaces. We first write down a suitable choice of free energy and describe how it relates to the various physical parameters of the fluid, in particular the contact angle. The lattice Boltzmann algorithm is summarised, showing how the thermodynamics is coupled to the hydrodynamics of the fluid. Then new results are presented for the dynamics of liquid drops on surfaces patterned with narrow (compared to the drop radius), alternating, hydrophilic and hydrophobic stripes. Email addresses:
[email protected] (H. Kusumaatmaja),
[email protected] (A. Dupuis),
[email protected] (J. M. Yeomans).
Preprint submitted to Elsevier Science
1 November 2005
Thermodynamics of the fluid: The equilibrium properties of a one-component, two-phase fluid can be described by a Landau free energy functional [1] Z
Ψ= V
Z Z κ 2 (ψb (n) + (∂α n) )dV − µb ndV + ψs (ns )dS , 2 V
(1)
S
where the second integral corresponds to a Lagrange multiplier that conserves the total mass of the system. The first term ψb describes the bulk free energy of the system, which we choose for convenience to have the form [2] ψb (n) = pc (νn + 1)2 (νn2 − 2νn + 3 − 2βτw ) = pc (νn2 − βτw )2 + µb n − pb ,
(2)
c where νn = (n − nc )/nc , τw = (Tc − T )/Tc , µb = ∂n ψb |n=nb = 4p (1 − βτw ), nc 2 pb = pc (1 − βτw ) and n, nc (= 3.5), T , Tc (= 4/7) and pc (= 1/8) are the local density, critical density, local temperature, critical temperature and critical pressure of the fluid respectively. β is a constant typically chosen to be 0.1. One can easily see from Eq. (2) that this√choice of free energy will lead to two coexisting bulk phases of density nc (1 ± βτw ).
The second term in Eq. (1) models the free energy associated with any interfaces in the system. The parameter κ is related to two physical quantities [2]: √ the surface tension via γ = (4 2κpc (βτw )3/2 nc )/3, and the interface width via q ξ = (κn2c )/(4βτw pc ). The last term (i.e. the surface term) in Eq. (1) describes the interactions between the fluid and the solid surface. Following Cahn [10] the surface energy density is taken to be ψs (n) = −φ ns , where ns is the value of the fluid density at the surface, so that the strength of interaction is parameterised by the variable φ. Minimising the free energy gives an equilibrium boundary condition κ∂⊥ n =
dψs = −φ . dns
(3)
We should like to stress here that the variable φ is in general a function of position and is related to the contact angle θ by [2] q
φ = 2βτw
r
π α α 2pc κ sign( − θ) cos (1 − cos ) , 2 3 3 2
(4)
where α = cos−1 (sin2 θ) and the function sign returns the sign of its argument. Any variation in the contact angle θ can be modelled easily by constraining the normal derivative ∂⊥ n at different regions of the surface to take the appropriate values given by Eq. (3). Suitable wetting boundary conditions for non-flat surfaces, which are needed to model superhydrophobic surfaces, can be found in [11] Hydrodynamics of the fluid: The hydrodynamics of the fluid is described by the continuity (5) and the NavierStokes (6) equations for a nonideal fluid ∂t n + ∂α (nuα ) = 0 , ∂t (nuα ) + ∂β (nuα uβ ) = −∂β Pαβ + ν∂β [n(∂β uα + ∂α uβ + δαβ ∂γ uγ )] +naα ,
(5) (6)
where u, P, ν, and a are the local velocity, pressure tensor, kinematic viscosity, and acceleration respectively. The thermodynamic properties of the system enter the hydrodynamic description through the pressure tensor P, which satisfies #
"
∂ κ − δαβ (ψb − µb n + (∂γ n)2 ) Pαβ = ∂β n ∂(∂α n) 2 κ Pαβ = (pb (n) − (∂γ n)2 − κn∂γγ n)δαβ + κ(∂α n)(∂β n) , 2 pb (n) = pc (νn + 1)2 (3νn2 − 2νn + 1 − 2βτw ).
(7)
The lattice Boltzmann algorithm: The basic idea of a lattice Boltzmann algorithm [12] is to associate distribution functions {fi (x, t)}, discrete in time and space, to a set of velocity directions i. For a 3-D system, one needs to take at least fifteen velocity vectors vi = (0, 0, 0), (±1, 0, 0), (0, ±1, 0), (0, 0, ±1), (±1, ±1, ±1). The distributions are related to the physical variables by X i
fi = n,
X
fi viα = nuα .
(8)
i
Taking a single-time relaxation approximation, the evolution equation for a given distribution function fi takes the form fieq (r, t) − fi (r, t) + nwσ viα aα . fi (r + vi ∆t, t + ∆t) = fi (r, t) + τ where wσ = w1 = 1/3 if |v| = 1 and wσ = w2 = 1/24 if |v| = 3
√
(9)
3. The relaxation
time τ tunes the kinematic viscosity, ν = ((∆x)2 (τ − 1/2))/(3∆t) [1]. ∆x and ∆t represent the discretisation in space and time respectively. It can be shown [1,4] that Eq. (9) reproduces Eqs. (5) and (6) in the continuum limit if the correct thermodynamic and hydrodynamic information is input to the simulation by a suitable choice of local equilibrium functions, i.e. if the following constraints are satisfied X eq
fi = n ,
i
X eq
X eq
fi viα = nuα ,
(10)
i
fi viα viβ = Pαβ + nuα uβ + ν[uα ∂β n + uβ ∂α n + δαβ uγ ∂γ n] .
i
A possible choice for fieq is a power series expansion in the velocity [4] fieq = Aσ + Bσ uα viα + Cσ u2 + Dσ uα uβ viα viβ + Gσαβ viα viβ , wσ κ Aσ = 2 (pb (n) − (∂γ n)2 − κn∂γγ n + νuγ ∂γ n) , c 2 wσ n 3wσ n wσ n Cσ = − 2 , Dσ = , Bσ = 2 , c 2c 2c4 1 G1αα = 4 (κ(∂α n)2 + 2νuα ∂α n) , G2αα = 0 , 2c 1 G2αβ = (κ(∂α n)(∂β n) + νuα ∂β n + νuβ ∂α n) . 16c4
(11)
In addition to the wetting boundary conditions discussed above, we also impose a no-slip boundary condition u = 0 on the surface [4]. We now show how this lattice Boltzmann algorithm can be used to simulate the dynamics of liquid drops on chemically heterogeneous surface. In general, the surfaces can be patterned with regions of different sizes, shapes and wettabilities: the limitation is merely the spatial discretisation taken in the simulation. Here we shall study the case where the surface is patterned with narrow, alternating, hydrophilic and hydrophobic stripes along the y-direction, perpendicular to the motion of the drop which we take to be along x. We take θ1 = 60o , θ2 = 110o , and δ1 /R = δ2 /R = 10/25, where θ is the contact angle, δ is the stripe width, R is the drop radius, and the hydrophilic and hydrophobic stripes are labelled 1 and 2 respectively. Fig. 1 shows snapshots of the drop as it moves across the surface. The crenellated edges of the drop show immediately that the drop prefers to wet the hydrophilic rather than the hydrophobic stripes. Note that as the drop reaches a new hydrophilic stripe (Figs. 1(b) – 1(d)), it prefers to wet the stripe in the y-direction, instead of going forward in the x-direction, until the decrease in surface free energy due 4
t = 155000
t = 160000
t = 163000
t = 170000
Fig. 1. Snapshots of the drop morphologies as it moves across the patterned surface. Simulation parameters: (Lx , Ly , Lz ) = (140, 100, 80), θ1 = 60o (dark grey), θ2 = 110o (light grey), δ1 /R = δ2 /R = 10/25, ax = 2.5 10−7 , κ = 0.004, pc = 1/8, nc = 3.5, T = 0.4, Tc = 4/7, and ν = 0.1. t = 150000
t = 154000
Fig. 2. Snapshots of the drop morphologies for δ1 /R = δ2 /R = 7/25. The other parameters are identical to Fig. 1
to spreading in the y-direction is overcome by the surface tension penalty as the drop’s total surface area is increased. Note also that the drop must dewet the back hydrophilic stripe before it can move forward and wet another hydrophilic stripe. For comparison, we also show in Fig. 2, two snapshots of the drop if the stripes’ width δ1 /R = δ2 /R = 7/25. These results illustrate that the lattice Boltzmann algorithm is a powerful way of studying the behaviour of drops on chemically patterned surfaces. The algorithm has a natural length scale ∼ 1 − 100µm which is ideal for studying drop on lithographically patterned surfaces and for microfluidic applications. The input parameters to the simulations are thermodynamic variables, such as surface tension, density, drop sizes and contact angles, and bulk transport coefficients, such as the viscosity. Although microscopic simulation approaches can start from the more funda5
mental standpoint of intermolecular interactions they can only look at much smaller drops so that fluctuations can make the interpretation of the data very difficult. It is important to point out that a problem with mesoscale simulations of liquid–gas systems is that interface widths (including the length over which the density varies near the surface) are too large compared to other length scales and the density difference between liquid and gas is too small. The result of this is that time scales are too fast: indeed it is this that makes the simulations feasible. However, comparisons with experiments so far have shown that the dynamic pathways are correct [4–6]. Similar lattice Boltzmann approaches have been applied to many problems, for example phase ordering [13], the breakup of fluid filaments [14], and drops moving on superhydrophobic surfaces [11].
References [1] M.R. Swift, E. Orlandini, W.R. Osborn and J.M. Yeomans, Phys. Rev. E 54, 5041 (1996). [2] A.J. Briant, A.J. Wagner and J.M. Yeomans, Phys. Rev. E 69, 031602 (2004); A.J. Briant and J.M. Yeomans, Phys. Rev. E 69, 031603 (2004). [3] A different form of free energy has been used in J. Zhang, B. Li, and D. Y. Kwok, Phys. Rev. E 69, 032602 (2004). [4] J. L´eopold`es, A. Dupuis, D.G. Bucknall, and J.M. Yeomans, Langmuir 19, 9818 (2003); A. Dupuis and J.M. Yeomans, Fut. Gen. Comp. Sys. 20, 993 (2004). [5] A. Dupuis, J. L´eopold`es, D.G. Bucknall, and J.M. Yeomans, Appl. Phys. Lett. 87, 024103 (2005). [6] H. Kusumaatmaja, J. L´eopold`es, A. Dupuis, and J.M. Yeomans, in preparation. [7] G. Gau, H. Hermingaus, P. Lenz, and R. Lipowsky, Science 283, 46 (1999); P. Lenz and R. Lipowsky, Phys. Rev. Lett. 80, 1920 (1998). [8] M. Brinkmann and R. Lipowsky, J. Appl. Phys. 92, 4296 (2002). [9] A.A. Darhuber, S.M. Troian, S.M. Miller, and S. Wagner, J. Appl. Phys. 87, 7768 (2000). [10] J.W. Cahn, J. Chem. Phys. 66, 3667 (1977). [11] A. Dupuis and J.M. Yeomans, Langmuir 21, 2624 (2005). [12] S. Succi, The Lattice Boltzmann Equation, For Fluid Dynamics and Beyond, OUP (2001). [13] V.M. Kendon, J.C. Desplat, P. Bladon and M.E. Cates, Phys. Rev. Lett. 83, 576 (1999). [14] J.G. Hagedorn, N.S. Martys, and J.F. Douglas, Phys. Rev. E 69, 056312 (2004).
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