There are five most widely used contact angle schemes in the pseudopotential lattice Boltzmann (LB) model for simulating the wetting phenomenon: The pseudopotential-based scheme (PB scheme), the improved virtual-density scheme (IVD scheme), the modified pseudopotential-based scheme with a ghost fluid layer constructed by using the fluid layer density above the wall (MPB-C scheme), the modified pseudopotential-based scheme with a ghost fluid layer constructed by using the weighted average density of surrounding fluid nodes (MPB-W scheme) and the geometric formulation scheme (GF scheme). But the numerical stability and accuracy of the schemes for wetting simulation remain unclear in the past. In this paper, the numerical stability and accuracy of these schemes are clarified for the first time, by applying the five widely used contact angle schemes to simulate a two-dimensional (2D) sessile droplet on wall and capillary imbibition in a 2D channel as the examples of static wetting and dynamic wetting simulations respectively. (i) It is shown that the simulated contact angles by the GF scheme are consistent at different density ratios for the same prescribed contact angle, but the simulated contact angles by the PB scheme, IVD scheme, MPB-C scheme and MPB-W scheme change with density ratios for the same fluid-solid interaction strength. The PB scheme is found to be the most unstable scheme for simulating static wetting at increased density ratios. (ii) Although the spurious velocity increases with the increased liquid/vapor density ratio for all the contact angle schemes, the magnitude of the spurious velocity in the PB scheme, IVD scheme and GF scheme are smaller than that in the MPB-C scheme and MPB-W scheme. (iii) The fluid density variation near the wall in the PB scheme is the most significant, and the variation can be diminished in the IVD scheme, MPB-C scheme and MPB-W scheme. The variation totally disappeared in the GF scheme. (iv) For the simulation of capillary imbibition, the MPB-C scheme, MPB-W scheme and GF scheme simulate the dynamics of the liquid-vapor interface well, with the GF scheme being the most accurate. The accuracy of the IVD scheme is low at a small contact angle (44 degrees) but gets high at a large contact angle (60 degrees). However, the PB scheme is the most inaccurate in simulating the dynamics of the liquid-vapor interface. As a whole, it is most suggested to apply the GF scheme to simulate static wetting or dynamic wetting, while it is the least suggested to use the PB scheme to simulate static wetting or dynamic wetting.

The wetting phenomenon is not only widespread in nature but also plays an important role in the industry [_{2} storage and etc.

Lattice Boltzmann method (LBM) is a mesoscopic numerical method based on kinetic theory, which preserves microscopic kinetic principles to recover macroscopic hydrodynamics. Fluid dynamics are described by microfluid particle movement governed by the LB equation, which two processes can solve: collision and streaming process [

There are mainly five kinds of widely used contact angle schemes in the pseudopotential LB model. Sukop et al. [

However, implementation of the contact angle schemes in the pseudopotential LB model was demonstrated to bring about many unphysical or inaccurate simulation results in the past. Spurious current is a common problem in simulating multiphase flow in LBM, which mostly occurs adjacent to the multiphase interface [

To clarify the numerical accuracy and stability of different contact angle schemes in simulating static wetting and dynamic wetting phenomenon, the five widely used contact angle schemes as listed in

Scheme No. | Description | Authors |
---|---|---|

Scheme #1 | Pseudopotential-based scheme | Sukop [ |

Scheme #2 | Improved virtual-density scheme | Li et al. [ |

Scheme #3 | Modified pseudopotential-based scheme with a ghost fluid layer of fluid density nearest above wall | Wu et al. [ |

Scheme #4 | Modified pseudopotential-based scheme with a ghost fluid layer of weighted average density of surrounding fluid nodes | Yang et al. [ |

Scheme #5 | Geometric formulation scheme | Ding et al. [ |

The fluid flow dynamics are governed by the macroscopic continuity equation and the Navier-Stokes equation as follows:
_{i} is the discrete velocity, the index _{i}^{eq}_{i}^{eq}_{s}_{i}

In this study, the internal forces or external forces are incorporated into _{int} is the fluid-fluid interaction force among fluid particles and _{ads} is the fluid-solid interaction force between bulk fluid and solid wall. The real macro velocity of fluid particles

The kinetic viscosity

In Shan-Chen pseudopotential LB model, a “effective mass” at every fluid node is introduced to mimic interaction forces among fluid particles (_{int}) [_{int} is calculated following the force approximation approach given by Kupershtokh et al. [_{c} and

The static wetting of a 2D sessile droplet on a plate solid wall is simulated by using different contact angle schemes in the following sections. The schematic of a 2D sessile droplet on a plate solid wall is shown in _{c}; _{c}) in lattice units. The kinetic viscosities of liquid and vapor are set as 0.17 and 2.08. With the evolution of simulation time, the droplet finally reaches equilibrium status. Then, the static contact angle, spurious velocities and droplet liquid density variation near the plate solid wall can be measured. All the algorithm of the LB simulation in this study is coded by the C program language. The units of all the variables are presented in lattice units in our simulation. Conversion from lattice units to physical units is realized by using the reduced properties, with the reduced properties expressed as

The four different contact angle schemes used in this paper as listed in

In this contact angle scheme, the fluid-solid interaction forces _{ads} near the plate solid wall are given by Sukop et al. [_{s} represents the fluid-solid interaction strength, and _{i}) is a switch function which equals to 1 for a solid node and 0 for a fluid node.

In this contact angle scheme, a virtual density _{ads} near wall is calculated as follows [

In this contact angle scheme, the fluid-solid interaction force _{ads} near wall is given by Li et al. [

A ghost fluid layer is constructed at the plate solid wall according to Wu et al. [_{int} on the fluid node at position

In this contact angle scheme, the fluid-solid interaction force _{ads} near wall is calculated following the same

The fluid-fluid interaction force _{int} on the fluid node at position

In the geometric formulation scheme [

In this study, the fluid density ratio in simulation equals to the saturated fluid density ratio at the system temperature. If the system temperature is assigned as 0.9_{c}, the corresponding density ratio is _{c}, the corresponding density ratio is _{s} or _{s} or

The fluid flow velocities in LB simulation of a sessile droplet on a plate wall by using different contact angle schemes at two density ratios (10 and 36) are shown in

Spurious velocities appear mostly adjacent to the multiphase interface, especially at the droplet triple-phase contact line. The comparison of fluid flow velocities around the triple-phase contact line of a sessile droplet on a plate solid wall simulated by Schemes #2 and #3 at the same density ratio of 10 is shown in

The maximum spurious velocities in LB simulation of a sessile droplet on a plate solid wall at different density ratios

The fluid density variation near the solid wall having acute contact angle

The fluid density variation near wall having obtuse contact angle

The liquid density variations near wall (

To compare numerical accuracy of different contact angle schemes in simulating dynamic wetting of solid wall, the five widely used contact angle schemes are applied to simulate capillary imbibition in a two-dimensional (2D) channel formed by two separated parallel plates as shown in _{c} in lattice unit. Kinetic viscosities of liquid and vapor are set as 0.17. The liquid-vapor interface moves forward in the channel with time, driven by the capillary force. It is noted that contact angle hysteresis effects are neglected in this study, in such situation the liquid-vapor interface movement follows the well-known Washburn law [

Three groups of grids (

The comparisons of LB simulation by using different contact angle schemes with the Washburn law of

Five widely used contact angle schemes based on the pseudopotential LB model are applied to simulate the problems of a 2D sessile droplet on a plate solid wall and capillary imbibition in a 2D channel, for the purpose of clarifying numerical stability and accuracy of the contact angle schemes in the simulation of static wetting and dynamic wetting. The main conclusions are summarized as follows:

For contact angle schemes using fluid-solid interaction force to simulate solid wall wetting (PB scheme, IVD scheme, MPB-C scheme and MPB-W scheme) the simulated static contact angles of wall are different at different density ratios, even with the same liquid-solid interaction strength. While GF scheme simulates solid wall wetting based on geometric formulation with a prescribed contact angle, the simulated static contact angles are consistent at different density ratios.

The IVD scheme, MPB-C scheme, MPB-W scheme and GF scheme are more stable than the PB scheme so that the former schemes can simulate a wide range of contact angles in static wetting simulations at large density ratios.

The increased density ratio enhances spurious velocity in all the contact angle schemes, which results in the simulated fluid flow velocity in errors. The maximum spurious velocities in the GF scheme are the smallest at any density ratio so that the GF scheme is supposed to simulate fluid flow velocities more accurately.

The liquid density deviation near wall in the GF scheme is the smallest, and the liquid density deviation near wall in the PB scheme is the largest. So that the thickness of the unphysical fluid density variation layer in the GF scheme is the smallest and the variation is the most significant in the PB scheme.

By constructing a ghost fluid layer on wall with the average density of surrounding fluid nodes (MPB-W) rather than by directly using the fluid layer density nearest the wall (MPB-C), the spurious velocity and fluid density deviation near wall both diminish in static wetting simulation.

The MPB-C scheme, MPB-W scheme and GF scheme can simulate capillary imbibition accurately, with GF scheme having the most accuracy. The IVD scheme has low accuracy in simulating capillary imbibition at low contact angle

This study was sponsored by the

The authors declare that they have no conflicts of interest to report regarding the present study.

_{2}storage: Key impact factors and characterization approaches