Particle dynamics at fluid interfaces studied by the color gradient lattice Boltzmann method coupled with the smoothed profile method

We suggest a numerical method to describe particle dynamics at the fluid interface. We adopt a coupling strategy by combining the color gradient lattice Boltzmann method (CGLBM) and smoothed profile method (SPM). The proposed scheme correctly resolves the momentum transfer among the solid particles and fluid phases while effectively controlling the wetting condition. To validate the present algorithm (CGLBM-SPM), we perform several simulation tests like wetting a single solid particle and capillary interactions in two solid particles floating at the fluid interface. Simulation results show a good agreement with the analytical solutions available and look qualitatively reasonable. From these analyses, we conclude that the key features of the particle dynamics at the fluid interface are correctly resolved in our simulation method. In addition, we apply the present method for spinodal decomposition of a ternary mixture, which contains two-immiscible fluids with solid particles. By adding solid particles, fluid segregation is much suppressed than in the binary liquid mixture case. Furthermore, it has different morphology, such as with the jamming structure of the particles at the fluid interface, and captured images are similar to bicontinuous interfacially jammed emulsion gels in literature. From these results, we confirm the feasibility of the present method to describe soft matters; in particular, emulsion systems that contain solid particles at the interface.

Figure 1

The schematic diagram of the system.

Figure 2

Density field of fluids $A,B$, and $C$ at the equilibrium state (a) without a solid particle and (b) with a solid particle (the present algorithm). The solid line (inside of fluid $C)$ corresponds to ${\rho }_C=0.5$.

Figure 3

The equilibrium particle position for various wetting parameters. (a) Density field of fluids with a solid particle at equilibrium. (b) (Normalized) equilibrium height of the solid particle versus the wetting parameter.

Figure 4

$L_2$ norm error for the lattice grid resolution.

Figure 5

$L_2$ norm error as a function of parameter ${\beta }_0$.

Figure 6

Schematic diagram of the simulation system: two heavy particles floating at the fluid interface with interparticle distance $d$. ${\psi }_1$ and ${\psi }_2$ denote contact angles around the solid particle.

Figure 7

(a) Simulation results for $\mathrm{Bo}=0.1$ and $0.4$. (b) The equilibrium contact angle of the particle $\left({\psi }^{eq}\right)$ versus the Bond number $\left(\mathrm{Bo}\right)$. The triangle is the simulation result and the grey dashed line is an analytic solution predicted by Eq. (37).

Figure 8

(a) Simulation results for interparticle distances $d/a=8$ and $d/a=4\left(\mathrm{Bo}=0.2\right)$. (b) Capillary interactions for the lateral directions versus interparticle distance (both dimensionless forms). The square symbol is the simulation result and the grey dashed line is the analytic solution obtained by Eqs. (38)$-$(41).

Figure 9

The lateral particle motion induced by capillary interaction. (a) Initial particle configuration. (b) The particles are vertically accelerated in the same direction. (c) The particles are accelerated vertically in the opposite direction. The right side of the figure describes temporal particle separation in the lateral direction.

Figure 10

Spinodal decomposition of fluid mixtures. (a) Binary $\left(\varphi =0\right)$, (b) ternary $\left(\varphi =0.1\right)$, and (c) ternary $\left(\varphi =0.3\right)$ mixtures. Results correspond to $t=0,t=50000$, and $t=600000$ from left to right.