Color-gradient lattice Boltzmann model for simulating droplet motion with contact-angle hysteresis

Lattice Boltzmann method (LBM) is an effective tool for simulating the contact-line motion due to the nature of its microscopic dynamics. In contact-line motion, contact-angle hysteresis is an inherent phenomenon, but it is neglected in most existing color-gradient based LBMs. In this paper, a color-gradient based multiphase LBM is developed to simulate the contact-line motion, particularly with the hysteresis of contact angle involved. In this model, the perturbation operator based on the continuum surface force concept is introduced to model the interfacial tension, and the recoloring operator proposed by Latva-Kokko and Rothman is used to produce phase segregation and resolve the lattice pinning problem. At the solid surface, the color-conserving wetting boundary condition [Hollis et al., IMA J. Appl. Math. 76, 726 (2011)] is applied to improve the accuracy of simulations and suppress spurious currents at the contact line. In particular, we present a numerical algorithm to allow for the effect of the contact-angle hysteresis, in which an iterative procedure is used to determine the dynamic contact angle. Numerical simulations are conducted to verify the developed model, including the droplet partial wetting process and droplet dynamical behavior in a simple shear flow. The obtained results are compared with theoretical solutions and experimental data, indicating that the model is able to predict the equilibrium droplet shape as well as the dynamic process of partial wetting and thus permits accurate prediction of contact-line motion with the consideration of contact-angle hysteresis.

Figure 1

Illustration of a D2Q9 lattice node on the bottom boundary of a 2D domain (at propagation step).

Figure 2

Equilibrium droplet shapes for ${\theta }_s={20}^{\circ },{50}^{\circ },{90}^{\circ },{120}^{\circ },{150}^{\circ }$. Dashed lines: theoretical shapes. Solid lines: equilibrium shapes by the present LBM.

Figure 3

Dimensionless wet length $L/R_0$ and height $e/R_0$ of the droplet at equilibrium as a function of the static contact angle ${\theta }_s$ at different grid resolutions.

Figure 4

Dimensionless pressure difference $R_0\Delta P/\sigma$ of the droplet at equilibrium as a function of the static contact angle ${\theta }_s$ at different grid resolutions.

Figure 5

Evolution of droplet shape. Time ${\tau }_0$: initial shape. Time ${\tau }_{\infty }$: equilibrium shape. Dimensionless time is taken as $\tau =0$, $4.2$, $14.1$, $28.8$, and 1700.

Figure 6

Time evolution of (a) the contact-line velocity $u_{\mathrm{cl}}$ and (b) the advancing contact point position $x_{\mathrm{cl}}$.

Figure 7

The dimensionless wet length $L^{*}$ as a function of the dimensionless time $\tau$.

Figure 8

Schematic diagram of a droplet meniscus subject to a simple shear flow.

Figure 9

Time evolution of droplet shape for $\mathrm{Ca}=0.10$. Time ${\tau }_0$: initial shape. Time ${\tau }_{\infty }$: equilibrium shape. The dimensionless time is taken as $\tau =0$, $11.34$, $17.01$, 200.

Figure 10

Equilibrium velocity field and streamlines for $\mathrm{Ca}=0.10$. The droplet interface is represented by the black solid line.

Figure 11

Comparison of the simulated equilibrium droplet shapes with the shear flow results of Schleizer and Bonnecaze [34] at $\mathrm{Ca}=0.10$ for $A_d^{*}=0.125$ and 1. The solid lines are the simulated contours at ${\rho }^N=\left\{-0.9,0,0.9\right\}$, while the dashed lines are the results of Schleizer and Bonnecaze [34].

Figure 12

Evolution of droplet shapes at $\left({\theta }_R,{\theta }_A\right)=\left({60}^{\circ },{120}^{\circ }\right)$ for (a) stationary mode and (b) slipping mode. In (a), ${\tau }_0$ denotes initial droplet shape and ${\tau }_{\infty }$ denotes equilibrium shape. In (b), ${\tau }_0$ denotes initial droplet shape, ${\tau }_1$ denotes the shape at $\tau =10.2$ when the droplet starts to move, and ${\tau }_2$ is the droplet shape at $\tau =76.6$ when the simulation terminates.

Figure 13

Evolution of the droplet shape for $\left({\theta }_R,{\theta }_A\right)=\left({60}^{\circ },{120}^{\circ }\right)$ and $\mathrm{Ca}=0.3$ in breakup mode with the times taken as (a) $\tau =0$, (b) $\tau =7.7$, (c) $\tau =48.5$, and (d) $\tau =69$.

Figure 14

Droplet velocity as a function of Capillary number for various hysteresis windows.

Figure 15

Receding and advancing contact angles as a function of Capillary number for $\left({\theta }_R,{\theta }_A\right)=\left({80}^{\circ },{100}^{\circ }\right)$.

Figure 16

Relationship between $\left(\cos {\theta }_a-\cos {\theta }_A\right)$ and contact-line Capillary number for various hysteresis windows.

Figure 17

Critical shapes when the droplet starts to move for different hysteresis windows.