Implementation of contact line motion based on the phase-field lattice Boltzmann method

This paper proposes a strategy to implement the free-energy-based wetting boundary condition within the phase-field lattice Boltzmann method. The greatest advantage of the proposed method is that the implementation of contact line motion can be significantly simplified while still maintaining good accuracy. For this purpose, the liquid-solid free energy is treated as a part of the chemical potential instead of the boundary condition, thus avoiding complicated interpolations with irregular geometries. Several numerical testing cases, including droplet spreading processes on the idea flat, inclined, and curved boundaries, are conducted, and the results demonstrate that the proposed method has good ability and satisfactory accuracy to simulate contact line motions.

Figure 1

Schematic illustration for the lattice nodes around the physical boundary.

Figure 2

Schematic illustration for (a) volume fraction $\epsilon$, (b) the calculated effective surface area $a_v/dx$, and (c) lattice nodes involved in calculating the effective surface area of a curved boundary. The red dots represent boundary nodes and the green ones denotes fluid nodes.

Figure 3

Schematic illustration for the calculation of the gradient of order parameter along the boundary normal direction.

Figure 4

Schematic illustration for the wetting of a droplet on an idea flat surface.

Figure 5

The predicted droplet equilibrium shapes in 2D using the present boundary treatment with wide range of prescribed contact angles, (a) $\theta ={10}^{\circ }$, (b) $\theta ={20}^{\circ }$, (c) $\theta ={30}^{\circ }$, (d) $\theta ={60}^{\circ }$, (e) $\theta ={90}^{\circ }$, (f) $\theta ={120}^{\circ }$, (g) $\theta ={150}^{\circ }$, and (h) $\theta ={160}^{\circ }$.

Figure 6

Equilibrium velocity field for the cases of two-dimensional droplet spreading on the flat ideal wall. (1) Speed vector illustration and (2) Contour plot of velocity $\left|\mathit{u}\right|$.

Figure 7

The predicted droplet equilibrium shapes in three dimensions using the present boundary treatment with wide range of prescribed contact angles (a) $\theta ={10}^{\circ }$, (b) $\theta ={20}^{\circ }$, (c) $\theta ={30}^{\circ }$, (d) $\theta ={60}^{\circ }$, (e) $\theta ={90}^{\circ }$, (f) $\theta ={120}^{\circ }$, (g) $\theta ={150}^{\circ }$, and (h) $\theta ={160}^{\circ }$.

Figure 8

Contour plot of velocity $\left|\mathit{u}\right|$ at equilibrium state for the cases of three-dimensional droplet spreading on the flat ideal wall.

Figure 9

Schematic illustration for the wetting of a droplet on an idea inclined surface, $\theta$ is the contact angle, $\gamma$ represents the inclined angle of the solid boundary.

Figure 10

The predicted droplet equilibrium shapes on the inclined surface in two dimensions using the present boundary treatment with wide range of prescribed contact angles and different inclined angles $\gamma$, (a) $\gamma =arctan\left(1.0\right)$, (b) $\gamma =arctan\left(0.5\right)$, (c) $\gamma =arctan\left(0.25\right)$.

Figure 11

The predicted droplet equilibrium shapes in two dimensions using the wetting boundary treatment with different width of phase interface $W$; contact angle (a) $\theta ={90}^{\circ }$ and (b) $\theta ={120}^{\circ }$.

Figure 12

Contour plot of velocity $\left|\mathit{u}\right|$ at equilibrium state for the cases of two-dimensional droplet spreading on the inclined ideal wall. (1) Contact angle $\theta ={60}^{\circ }$, (2) contact angle $\theta ={120}^{\circ }$.

Figure 13

The predicted droplet equilibrium shapes on the inclined surface in three dimensions using the present boundary treatment with wide range of prescribed contact angles and different inclined angles $\gamma ,$ (a) $\gamma =arctan\left(1.0\right)$, (b) $\gamma =arctan\left(0.5\right)$, (c) $\gamma =arctan\left(0.25\right)$.

Figure 14

Contour plot of velocity $\left|\mathit{u}\right|$ at equilibrium state for the cases of three-dimensional droplet spreading on the inclined ideal wall. (1) contact angle $\theta ={60}^{\circ }$, (2) contact angle $\theta ={120}^{\circ }$.

Figure 15

Schematic illustration for the wetting of a droplet on a cylindrical solid, $\theta$ is the contact angle.

Figure 16

The predicted droplet equilibrium shapes on the cylindrical solid in two dimensions using the present boundary treatment with wide range of prescribed contact angles.

Figure 17

Equilibrium velocity field for the cases of two-dimensional droplet spreading on the cylindrical surface. (1) Speed vector illustration and (2) Contour plot of velocity $\left|\mathit{u}\right|$.

Figure 18

The predicted droplet equilibrium shapes on a 3D solid sphere using the present boundary treatment with wide range of prescribed contact angles. The first row shows the 3D wetting morphology of the droplet, and the second row provides the distribution of order parameters in the central cross section, along with a comparison to the reference solution.

Figure 19

Contour plot of velocity $\left|\mathit{u}\right|$ at equilibrium state for the cases of three-dimensional droplet spreading on the phere surface.

Figure 20

Time evolution of the droplet spreading shapes with different contact angles, (a) $\theta ={60}^{\circ }$, (b) $\theta ={120}^{\circ }$. The results of case 1 with $\chi =-\sqrt{2\beta /\kappa }cos\theta (\varphi -{\varphi }^2)$ are shown with color figure. The contours with $\varphi =0.5$ of case 2 and case 3 are marked with black line and white dashed line, respectively.