Ternary free-energy lattice Boltzmann model with tunable surface tensions and contact angles

Ciro Semprebon, Timm Krüger, and Halim Kusumaatmaja

Phys. Rev. E 93, 033305 – Published 11 March 2016

Abstract

We present a ternary free-energy lattice Boltzmann model. The distinguishing feature of our model is that we are able to analytically derive and independently vary all fluid-fluid surface tensions and the solid surface contact angles. We carry out a number of benchmark tests: (i) double emulsions and liquid lenses to validate the surface tensions, (ii) ternary fluids in contact with a square well to compare the contact angles against analytical predictions, and (iii) ternary phase separation to verify that the multicomponent fluid dynamics is accurately captured. Additionally we also describe how the model presented here can be extended to include an arbitrary number of fluid components.

Received 4 November 2015
Revised 15 February 2016

DOI:https://doi.org/10.1103/PhysRevE.93.033305

Published by the American Physical Society

Physics Subject Headings (PhySH)

Contact line dynamics

Liquid-liquid interfaces

Lattice-Boltzmann methods

Fluid Dynamics

Authors & Affiliations

Ciro Semprebon¹, Timm Krüger², and Halim Kusumaatmaja^1,*

¹Department of Physics, Durham University, Durham DH1 3LE, United Kingdom
²School of Engineering, The University of Edinburgh, Edinburgh EH9 3JL, United Kingdom

^*halim.kusumaatmaja@durham.ac.uk

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Vol. 93, Iss. 3 — March 2016

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Figure 1
(a) Comparison between analytical predictions for the Neumann angles $θ_{1}, θ_{2}$ , and $θ_{3}$ and the simulation results for a liquid lens [panels (b)–(d)] and a double emulsion [panels (e)–(g)] at equilibrium. In the simulation, $α = 1, κ_{1} = 0.01$ , and $κ_{2} = 0.02$ are fixed, while $κ_{3}$ is varied. Specifically, $κ_{3} = 0.05$ in panels (b) and (e), $κ_{3} = 0.15$ in panels (c) and (f), and $κ_{3} = 0.001$ in panels (d) and (g). We have also used $τ_{ϕ} = 1.0, τ_{ψ} = 2 / 3$ , and $Γ_{ϕ} = Γ_{ψ} = 1.0$ . Panels (c) and (f) show the isosurfaces $C = 0.5$ for three-dimensional simulation results. All remaining panels are for two-dimensional systems.
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Figure 2
Validation of the wetting boundary conditions at solid walls: simulations in both panels are carried out with $α = 1, τ_{ρ} = 1.0, τ_{ϕ} = 1.0, τ_{ψ} = 2 / 3$ , and $Γ_{ϕ} = Γ_{ψ} = 1.0$ . (a) Three fluid phases at equilibrium confined to a square well, showing simultaneously the three Neumann angles and the three different pairs of contact angles. Here $κ_{1} = 0.01, κ_{2} = 0.02, κ_{3} = 0.03, h_{1} = - 0.002, h_{2} = 0.002$ , and $h_{3} = 0.0$ . (b) Comparison between the predicted and simulated contact angles as the parameters $h_{1}, h_{2}$ , and $h_{3}$ are varied. Here $κ_{1} = κ_{2} = κ_{3} = 0.01$ , The agreement departs for very small and very large angles. The minimum and maximum values shown are $θ = 14^{\circ}$ and $θ = 166^{\circ}$ , respectively. Outside this range, the deviation in contact angle is $> 4^{\circ}$ between prediction and measurement.
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Figure 3
Ternary phase separation for three different sets of fluid compositions: (i) $C_{1} = C_{2} = 1 / 4$ and $C_{3} = 1 / 2$ [panels (a)–(d)], (ii) $C_{1} = C_{2} = C_{3} = 1 / 3$ [panels (e)–(h)], and (iii) $C_{1} = C_{2} = 2 / 5$ and $C_{3} = 1 / 5$ [panels (i)–(l)]. Here we use $κ_{1} = κ_{2} = κ_{3} = 0.01, τ_{ρ} = 1.0, τ_{ϕ} = 1.0, τ_{ψ} = 2 / 3, Γ_{ϕ} = Γ_{ψ} = 1.0$ and the fluid compositions are initialized with random perturbations of amplitude $δ = 0.01$ . Panels (d), (h), and (l) show the time evolution of the quantity $χ_{m} = 〈 C_{m}^{2} {(1 - C_{m})}^{2} 〉 6 L / α$ , where $L = 200 Δ x$ is the size of the square domain in lattice units and $m = (1, 2, 3)$ . The rapid decay of $χ_{m}$ indicates the segregation of the phase $C_{m}$ . Once the phase is well separated, its value represents the total length of the boundary between $C_{m}$ and the other two phases. The black circles and vertical dashed lines indicate the time of snapshots reproduced in the upper panels.
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Figure 4
Scaling of the typical domain size $L = A / χ$ , during phase separation in two dimensions. (a) Diffuse regime. (b) Inertial regime. The area of the domain size is $400 \times 400$ lattice spacings. Ternary and binary mixtures are introduced by setting $C_{1} = C_{2} = C_{3} = 1 / 3$ and $C_{1} = C_{2} = 1 / 2$ respectively. The parameters $τ_{ρ} = 1.0, τ_{ϕ} = 1.0, τ_{ψ} = 2 / 3, Γ_{ϕ} = Γ_{ψ} = 1.0$ are the same in all cases. To achieve the diffusive regime we remove the advection term in the Cahn-Hilliard equation. For the diffuse regime we set $κ_{1} = κ_{2} = κ_{3} = 0.01$ , while for the inertial regime we set $κ_{1} = κ_{2} = κ_{3} = 0.04$ . Each curve is shifted by its time constant $t_{0}$ obtained from a fitting routine.
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