Metastable and unstable hydrodynamics in multiphase lattice Boltzmann

Metastability in liquids is at the foundation of complex phase transformation dynamics such as nucleation and cavitation. Intermolecular interaction details, beyond the equation of state, and thermal hydrodynamic fluctuations play a crucial role. However, most numerical approaches suffer from a slow time and space convergence, thus hindering the convergence to the hydrodynamic limit. This work shows that the Shan-Chen lattice Boltzmann model has the unique capability of simulating the hydrodynamics of the metastable state. The structure factor of density fluctuations is theoretically obtained and numerically verified to a high precision, for all simulated wave vectors, reduced temperatures, and pressures, deep into the metastable region. Such remarkable agreement between the theory and simulations leverages the exact implementation at the lattice level of the mechanical equilibrium condition. The static structure factor is found to consistently diverge as the temperature approaches the critical point or the density approaches the spinodal line at a subcritical temperature. Theoretically predicted critical exponents are observed in both cases. Finally, the phase separation in the unstable branch follows the same pattern, i.e., the generation of interfaces with different topology, as observed in molecular dynamics simulations.

Figure 1

Lower panel: Isotherms at the critical temperature ($\widehat{T}=1$) and subcritical temperature of $\widehat{T}=0.8$. Dashed blue and red (darker and lighter shades) lines are respectively the binodal and spinodal curves, and the blue- and red-shaded areas (darker and lighter shades) the metastable and unstable regions. The five points on the unstable part of the isotherm (red solid line) correspond to the initial densities which phase separate immediately with final interface topologies of droplet (d), droplet-cylinder (dc), flat (f), bubble-cylinder (bc), and bubble (b), all shown in the upper panel.

Figure 2

Density structure factor $S\left(k\right)$ as a function of $k$ at varying pressure along the $\widehat{T}=0.8$ isotherm (top row) and at varying temperature along the binodal (bottom row). The left and right columns show the cases starting from the gas and liquid phase respectively. The color scale, from green to orange (lighter to darker shade), indicates the distance in pressure from the spinodal line or in temperature from the critical point, with the values reported near the corresponding curve. In dashed lines we report Eq. (16) truncated at $O\left(k^2\right)$.

Figure 3

$S\left(k\right)$ as functions of the relative pressure from the spinodal line $|1-{\widehat{\pi }}_{\text{s}}|$ (top), and relative distance from the critical point $1-\widehat{T}$ (bottom). Solid lines indicate the analytical prediction, while the dashed lines the truncation to $O\left(k^2\right)$. The color scale goes from $k\mathrm{\Delta }x=0$ in red to $k\mathrm{\Delta }x=\pi$ in blue (lighter to darker shade). Upon approaching the spinodal line or the critical point the large scale ($k\to 0$ in dot-dashed red) converges to a power-law behavior (in dashed black).

Figure 4

(a) Flat interface bulk pressure profile $P_b=n\left(x\right)T-{\psi }^2\left(x\right)/2$, the bulk pressure value $P_0$ is consistently obtained in both phases away from the interface. (b) Lattice pressure tensor normal component, displaying fluctuations of the order of ${10}^{-13}$, i.e., constant to machine precision. Results are expressed in lattice units [11].

Figure 5

Normalized static density structure function $S\left(k\right)$ for the Carnahan-Starling equation of state implemented with the Shan-Chen multiphase approach. The forcing stencil has been tuned in such a way that at low temperature $S\left(k\right)$ displays similar features to that of realistic fluids: (1) $\overline{S}\left(0\right)<1$, (2) $d\overline{S}{/dk|}_{k=0}>0$, and (3) oscillations around 1 as the limit $k\to \pi$ is approached. The peak of the decreases as the temperature increases as indicated by the arrow. This procedure can be understood as the result of the coarse graining of the molecular properties of a liquid at a given scale. Results are expressed in lattice units [11].