Fluid-wall interactions in pseudopotential lattice Boltzmann models

Designing proper fluid-wall interaction forces to achieve proper wetting conditions is an important area of interest in pseudopotential lattice Boltzmann models. In this paper, we propose a modified fluid-wall interaction force that applies for pseudopotential models of both single-component fluids and partially miscible multicomponent fluids, such as hydrocarbon mixtures. A reliable correlation that predicts the resulting liquid contact angle on a flat solid surface is also proposed. This correlation works well over a wide variety of pseudopotential lattice Boltzmann models and thermodynamic conditions.

Figure 1

Static contact angle $\theta \approx {60}^{\circ }$ by four tested fluid-wall interaction schemes: (a) Sukop and Thorne's scheme, (b) the scheme of Benzi et al., (c) the scheme of Li et al., and (d) the proposed scheme. The parameters are $G_w=-2.71,{\rho }_w={\rho }_{v,\mathrm{sat}}+0.48({\rho }_{l,\mathrm{sat}}-{\rho }_{v,\mathrm{sat}})\varphi =1.21,\mathrm{\Delta }\rho =0$, and $\phi =1.1$.

Figure 2

Static contact angle $\theta \approx {90}^{\circ }$ by four tested fluid-wall interaction schemes: (a) Sukop and Thorne's scheme, (b) the scheme of Benzi et al., (c) the scheme of Li et al., and (d) the proposed scheme. The parameters are $G_w=-2.14,{\rho }_w={\rho }_{v,\mathrm{sat}}+0.305({\rho }_{l,\mathrm{sat}}-{\rho }_{v,\mathrm{sat}}),\varphi =1.0,\mathrm{\Delta }\rho =0$, and $\phi =1.0$.

Figure 3

Static contact angle $\theta \approx {120}^{\circ }$ by four tested fluid-wall interaction schemes: (a) Sukop and Thorne's scheme, (b) the scheme of Benzi et al., (c) the scheme of Li et al., and (d) the proposed scheme. The parameters are $G_w=-1.58,{\rho }_w={\rho }_{v,\mathrm{sat}}+0.155({\rho }_{l,\mathrm{sat}}-{\rho }_{v,\mathrm{sat}}),\varphi =1.0\mathrm{\Delta }\rho =0.385$, and $\phi =0.9$.

Figure 4

Spurious currents around a static droplet of contact angle $\theta ={60}^{\circ }$ on a flat surface by four tested fluid-wall interaction schemes: (a) Sukop and Thorne's scheme, (b) the scheme of Benzi et al., (c) the scheme of Li et al., and (d) the proposed scheme.

Figure 5

Spurious currents around a static droplet of contact angle $\theta ={90}^{\circ }$ on a flat surface by four tested fluid-wall interaction schemes: (a) Sukop and Thorne's scheme, (b) the scheme of Benzi et al., (c) the scheme of Li et al., and (d) the proposed scheme.

Figure 6

Spurious currents around a static droplet of contact angle $\theta ={120}^{\circ }$ on a flat surface by four tested fluid-wall interaction schemes: (a) Sukop and Thorne's scheme, (b) the scheme of Benzi et al., (c) the scheme of Li et al., and (d) the proposed scheme.

Figure 7

The resulting contact angles with based on different grid meshes. The initial droplet radii are 30,–50 for the grid resolutions of $240\times 180,320\times 240$, and $400\times 300$, respectively.

Figure 8

The comparison between the actual contact angles from PP LB simulations and their expectations for the reduced PR EOS under different temperatures. Parameters (in LB units) used in the correlation: at $T_r=0.9,{\rho }_{l,\mathrm{sat}}=5.908,{\rho }_{v,\mathrm{sat}}=0.5801,{\gamma }_{lg}=0.02880$; at $T_r=0.8,{\rho }_{l,\mathrm{sat}}=7.204,{\rho }_{v,\mathrm{sat}}=0.1971,{\gamma }_{lg}=0.07474$; at $T_r=0.7,{\rho }_{l,\mathrm{sat}}=8.080,{\rho }_{v,\mathrm{sat}}=0.05563,{\gamma }_{lg}=0.1274$; at $T_r=0.6,{\rho }_{l,\mathrm{sat}}=8.725,{\rho }_{v,\mathrm{sat}}=0.01023,{\gamma }_{lg}=0.1844$.

Figure 9

The comparison between the actual contact angles from PP LB simulations and their expectations for different values of $a$ in LB units. Parameters (in LB units) used in the correlation are ${\rho }_{l,\mathrm{sat}}=7.204,{\rho }_{v,\mathrm{sat}}=0.1971,{\gamma }_{lg}=0.05169$, 0.074 74, and 0.1095 when $a=1/98, 1/49$, and 2/49, respectively.

Figure 10

The comparison between the actual contact angles from PP LB simulations and their expectations for different choices of cubic EOS: (a) vdW EOS, parameters (in LB units) used in the correlation: at $T_r=0.9,{\rho }_{l,\mathrm{sat}}=5.800,{\rho }_{v,\mathrm{sat}}=1.490,{\gamma }_{lg}=0.01602$; at $T_r=0.8,{\rho }_{l,\mathrm{sat}}=6.764,{\rho }_{v,\mathrm{sat}}=0.8388,{\gamma }_{lg}=0.04481$; at $T_r=0.7,{\rho }_{l,\mathrm{sat}}=7.492,{\rho }_{v,\mathrm{sat}}=0.4481,{\gamma }_{lg}=0.08139$; at $T_r=0.6,{\rho }_{l,\mathrm{sat}}=8.090,{\rho }_{v,\mathrm{sat}}=0.2092,{\gamma }_{lg}=0.1242$ in the LB units. (b) CS EOS, parameters used in the correlation: at $T_r=0.9,{\rho }_{l,\mathrm{sat}}=0.2481,{\rho }_{v,\mathrm{sat}}=0.04543,{\gamma }_{lg}=8.616\times {10}^{-4}$; at $T_r=0.8,{\rho }_{l,\mathrm{sat}}=0.3072,{\rho }_{v,\mathrm{sat}}=0.02173,{\gamma }_{lg}=2.463\times {10}^{-3}$; at $T_r=0.7,{\rho }_{l,\mathrm{sat}}=0.3581,{\rho }_{v,\mathrm{sat}}=9.297\times {10}^{-3},{\gamma }_{lg}=4.585\times {10}^{-3}$; at $T_r=0.6,{\rho }_{l,\mathrm{sat}}=0.4062,{\rho }_{v,\mathrm{sat}}=3.082\times {10}^{-3},{\gamma }_{lg}=7.192\times {10}^{-3}$; at $T_r=0.5,{\rho }_{l,\mathrm{sat}}=0.4541,{\rho }_{v,\mathrm{sat}}=6.268\times {10}^{-4},{\gamma }_{lg}=0.01031$.

Figure 11

The comparison between the actual contact angles from PP LB simulations with the modified definition of the interaction forces of Kupershtokh et al.. Parameters used in the correlation are as same as those in Fig. 8.

Figure 12

The comparison between the actual contact angles from PP LB simulations with partially miscible MC hydrocarbon fluids. Parameters used in the correlation (in LB units), case 1: ${\rho }_{l,\mathrm{sat}}=8.297,{\rho }_{v,\mathrm{sat}}=0.8024,{\gamma }_{lg}=0.03951$; case 2: ${\rho }_{l,\mathrm{sat}}=10.26,{\rho }_{v,\mathrm{sat}}=0.2608,{\gamma }_{lg}=0.1098$; case 3: ${\rho }_{l,\mathrm{sat}}=7.677,{\rho }_{v,\mathrm{sat}}=0.1908,{\gamma }_{lg}=0.08186$; case 4: ${\rho }_{l,\mathrm{sat}}=8.262,{\rho }_{v,\mathrm{sat}}=0.08036,{\gamma }_{lg}=0.1133$.