Diffuse-interface modeling of liquid-vapor phase separation in a van der Waals fluid

We present simulations of isothermal liquid-vapor spinodal decomposition of a van der Waals fluid, occurring after quenching it from a single-phase equilibrium state to the unsteady two-phase region of its phase diagram, showing that the process is governed by the density gradient-driven, non-equilibrium Korteweg body force. Experimentally, most studies on spinodal decomposition have been performed using light-scattering techniques in microgravity. In fact, under normal gravity conditions, the density difference between the evolving phases in the two-phase fluid causes sedimentation and formation of layered structures, which makes growth measurements of single-phase microdomains hard to perform. That is why most experiments (conducted under normal gravity conditions) on spinodal decomposition have been carried out using nearly isopycnic binary liquids. There, the mixtures start to separate by diffusion and coalescence right after the temperature has crossed that of the miscibility curve, leading to the formation of dendritic, interconnected domains for critical systems, or pseudo-spherical drops of the minority phase surrounded by the majority phase for the off-critical systems. In particular, it was shown (Gupta et al. 1999; Vladimirova et al. 1999a; Mauri et al. 2003; Califano & Mauri 2004; Poesio et al. 2006) that diffusion alone cannot explain the segregation process in low-viscosity liquid mixtures, as the late stage of phase separation is dominated by convection, leading to a linear growth law for the characteristic size of single-phase microdomains. Although experiments and simulations on liquid-vapor phase separation have been carried out in the past (Osborn et al. 1995; Yamamoto & Nakanishi 1996; Warren 2001; Beysens et al. 2002; Sofonea et al. 2004; Borcia & Bestehorn 2007; Oprisan et al. 2008), none of these previous works has focused on spinodal decomposition systematically. In this work, applying the diffuse-interface model (Felderhof 1970; Langer & Turski 1973; Antanovskii 1996; Jasnow & Viñals 1996; Anderson et al. 1998), we investigate liquid-vapor spinodal decomposition in 2-D and 3-D for critical and off-critical van der Waals fluids as a function of a convection parameter expressing the relative magnitude of capillary-toviscous forces. We show that, at the late stages of the process, the mechanism of growth is convection-driven coalescence with a two-thirds power-law scaling for the characteristic size of single-phase microdomains, in agreement with dimensional analysis (Siggia 1979; Furukawa 1994) and experimental measurements (Beysens et al. 2002). Inertial scaling of convection-driven coalescence has also been reported in lattice Boltzmann simulations of binary fluid spinodal decomposition (Appert et al. 1995; Osborn et al. 1995; Kendon et al. 2001; Chin & Coveney 2002). This paper is organized as follows. In Sec. 2 we describe the diffuse-interface model,