Three-dimensional lattice-Boltzmann simulations of critical spinodal decomposition in binary immiscible fluids

We use a modified Shan-Chen, noiseless lattice-BGK model for binary immiscible, incompressible, athermal fluids in three dimensions to simulate the coarsening of domains following a deep quench below the spinodal point from a symmetric and homogeneous mixture into a two-phase configuration. The model is derivable from a continuous-time Boltzmann-BGK equation in the presence of an intercomponent body force. We find the average domain size grows with time as $t^{\gamma },$ where $\gamma$ increases in the range $0.545\pm 0.014<\mathrm{}\gamma <0.717\mathrm{}\pm 0.002,$ consistent with a crossover between diffusive $t^{1/3}$ and hydrodynamic viscous, $t^{1.0},$ behavior. We find good collapse onto a single scaling function, yet the domain growth exponents differ from previous results for similar values of the unique characteristic length $L_0$ and time $T_0$ that can be constructed out of the fluid’s parameters. This rebuts claims of universality for the dynamical scaling hypothesis. For $\mathrm{Re}=2.7\mathrm{}$ and small wave numbers q we also find a $q^2\leftrightarrow q^4$ crossover in the scaled structure function, which disappears when the dynamical scaling reasonably improves at later stages $(\mathrm{Re}=37).\mathrm{}$ This excludes noise as the cause for a $q^2$ behavior, as analytically derived from Yeung and proposed by Appert et al. and Love et al. on the basis of their lattice-gas simulations. We also observe exponential temporal growth of the structure function during the initial stages of the dynamics and for wave numbers less than a threshold value, in accordance with the diffusive Cahn-Hilliard Model B. However, this exponential growth is also present in regimes proscribed by that model. There is no evidence that regions of parameter space for which the scheme is numerically stable become unstable as the simulations proceed, in agreement with finite-difference relaxational models and in contradistinction with an unconditionally unstable lattice-BGK free-energy model previously reported. Those numerical instabilities that do arise in this model are the result of large intercomponent forces which turn the equilibrium distribution negative.