The lattice Boltzmann method for contact line dynamics

The lattice Boltzmann method for contact line dynamics Sudhir Srivastava, J.H.M. ten Thije Boonkkamp, Federico Toschi April 13, 2011

Overview 1 Problem description 2 Huh and Scriven model 3 Lattice Boltzmann method (LBM) LBM for multiphase flow 4 Results 5 Summary

Problem description Contact line g x d A A h U < Uc U = 0 θe B B Dry substrate a θd Figure: 2D schematic for plunging plate problem. Lens Water Wafer Advancing Contact line U = 0 Figure: 2D schematic for immersion lithography. U

Analytical model (Huh and Scriven 1971) Stokes flow approximation For incompressible flow in 2D the velocity field (u r, u φ ) in terms of stream function ψ(r, φ) is given by: u r = 1 r ψ φ, u φ = ψ r, moreover ψ satisfies the biharmonic equation: 4 ψ = 0. Assuming that the velocity is bounded at r = 0 and as r, we have (Michell 1899) ψ = r(a sin φ + b cos φ + cφ sin φ + dφ cos φ).

Boundary and interface conditions Solution applies to both fluid so we have 8 unknowns. P haseb (r, φ) P hasea θ d U a Figure: Model of contact line motion. ψ B / r = 0, at φ = 0, φ = θ d, ψ A / r = 0, at φ = θ d, φ = π, ψ A / φ = ψ B / φ, at φ = θ d, 2 ψ A µ A 2 φ = µ 2 ψ B B 2 φ, at φ = θ d, r 1 ψ B θ r 1 ψ A θ = U, at φ = 0, = U, at φ = π.

50 45 40 35 30 25 20 15 10 5 10 20 30 40 50 60 70 80 90 100 With these boundary condition biharmonic equation has a unique solution: ψ A = r(a A sin φ + b A cos φ + c A φ sin φ + d A φ cos φ) ψ B = r(a B sin φ + b B cos φ + c B φ sin φ + d B φ cos φ). R = 0.16946, θd = 70o, U = 0.01 Figure: Streamline and velocity field for θ d = 70 plate velocity U = 0.01, and viscosity ratio R = 0.1695.

Viscous dissipation Viscous dissipation rate: which gives Ė vis = 2µ A L λ π θ d e ij e ij rdφdr + 2µ B e rr = e φφ = 0, e rφ = r 2 r ( uφ r Ė vis log(l/λ). L θd λ 0 ) + 1 u r 2r φ. e ij e ij rdφdr, Using no-slip boundary conditions to a flow close to contact line leads to energy dissipation (Ė vis ) which is logarithmically diverging (Huh and Scriven 1971). Which is not true in reality.

Use small length scale parameter l s (effective slip length) to remove viscous singularity i.e. u = u wall + l s u n. The slip length parameter l s must be found experimentally because it may depend on plate velocity. No distinction between microscopic and apparent contact angle. In the multiphase lattice Boltzmann method effective slip length can be a consequence of choice of boundary conditions, also the numerical discretization induces a cut off length which removes the viscous singularity.

Lattice Boltzmann method (D2Q9 model) If x is a point in 2D Cartesian coordinates C6 C 2 C 5 C 3 x C 1 C 7 C4 C 8 Figure: D2Q9 lattice

Lattice Boltzmann method (D2Q9 model) If x is a point in 2D Cartesian coordinates C6 C 2 C 5 C 3 x C 1 C 7 C4 C 8 Figure: D2Q9 lattice f i (x + c i t, t + t) f i (x, t) }{{} = fi(x, t) f eq i (ρ, u), }{{ τ } streaming collision where i {0, 1,, 8} is the velocity index, and τ is the relaxation parameter.

Equilibrium distribution function The equilibrium distribution function f eq i { f eq i (ρ, u) = W i ρ 1 + 1 c 2 (c i u) + 1 s 2c 2 s given by [ 1 c 2 s ]} (c i u) 2 u u, where c s is the sound speed in LBM, for D2Q9 model c s = 1/ 3. f eq i is second order approximation of the Maxwell-Boltzmann distribution function, W i are the weights corresponding to D2Q9 model. Hydrodynamic variables: ρ(x, t) = 8 f i (x, t), (ρu)(x, t) = i=0 8 c i f i (x, t). i=0

Multiphase flow Shan-Chen model 1 Inter-particle potential : If x, x in 2D Cartesian coordinate system V(x, x ) := G(x, x )ψ(x)ψ(x ), where G is the interaction parameter, ψ(ρ) = 1 exp( ρ) is the effective density. x x 1 PRE Shan-Chen 1993-94

Multiphase flow Shan-Chen model (Shan-Chen 1993-94) Inter-particle potential : If x, x in 2D Cartesian coordinate system V(x, x ) := G(x, x )ψ(x)ψ(x ), where G is the interaction parameter, ψ(ρ) = {1 exp( ρ)} is the effective density. Restrict to nearest neighbor interaction { G(x, x G for x x ) = = c, 0 otherwise. x

Multiphase flow Net momentum at x is given by F(x) = Gψ(x) i W i ψ(x + c i t)c i Shift in momentum in f eq lim F(x) = V(x) c 0 ρu (x) = ρu(x) + τf(x). Under sufficient low Mach number assumption the Chapman-Enskog expansion(wolf-gladrow, Dieter 2000) gives us the continuity and momentum balance equations for multiphase flow.

Phase diagram This change in the velocity modifies the pressure p. The modified pressure is given by p = ρc 2 s + G 2 c2 s ψ 2. 0.7 0.6 0.5 G = -4 G = -5 G = -6 G = -7 0.4 Pressure(P) 0.3 0.2 0.1 0-0.1-0.2-0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 Density (ρ) Figure: Phase diagram showing that below the critical value of G there is no phase separation.

Numerical Simulation and Results Open boundary SIZEX SIZEY = 1024 1024, τ = 1, ν = 0.1667, γ = 0.0266, G = 4.5, ρ l = 1.493, ρ v = 0.253, R = 0.17, ρ w = 1.1(θ e = 45 ± 5 o ), Body Force(g) = 6.4778 10 7. Hydrostatic pressure Body force No flux No flux θ e Wettable wall + No slip Figure: Schematic for the boundary conditions used for the simulation of plunging plate problem. Boundary Conditions: Wettable bottom wall θ e 50 o, Body force in -x direction, pressure gradient in x-direction, No slip at south boundary, open-boundary (fluid is at equilibrium) north boundary.

Results Y 50 45 40 35 30 25 20 15 10 5 0 HS velocity field LBM density contour LBM velocity field time =3900000 U plate = 0.03 520 530 540 550 560 570 580 590 600 610 X Figure: Comparison of Huh-Scriven velocity field and the velocity field obtained from lattice Boltzmann multiphase simulations

Results 100 100 15 15 80 20 80 20 60 25 60 25 40 30 40 30 20 35 20 35 0 480 500 520 540 560 580 600 620 640 660 40 0 480 500 520 540 560 580 600 620 640 660 Figure: Local viscous dissipation rate calculated form the velocity field obtained form the Huh-Scriven velocity field (right figure) and lattice Boltzmann multiphase simulations (left figure). 40

Results 8 x time =3900000,λ = 30,U = 0.03 plate 10 4 300 HS velocity field 1000 LB velocity field 800 7 250 600 Viscous Dissipation Rate 6 5 4 3 200 150 100 400 200 200 400 600 800 1000 2 50 1 10 1 10 2 10 3 L 0 500 520 540 560 580 600 620 640 660 680 700 Figure: Viscous dissipation rate versus L (radial distance form contact line) for plate velocity 0.03(left), Interface position (right).

Conclusion and Future Work Various boundary conditions for multiphase LBM has been implemented and tested. LBM is in good agreement with the dissipation model in the outer region. Study of capillary number for plunging plate problem depending on various density ratios and contact angles needs to be done. Implementation of multi-range Shan-Chen model.

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