Diffuse-interface modeling of liquid-vapor coexistence in equilibrium drops using smoothed particle hydrodynamics

We study numerically liquid-vapor phase separation in two-dimensional, nonisothermal, van der Waals (vdW) liquid drops using the method of smoothed particle hydrodynamics (SPH). In contrast to previous SPH simulations of drop formation, our approach is fully adaptive and follows the diffuse-interface model for a single-component fluid, where a reversible, capillary (Korteweg) force is added to the equations of motion to model the rapid but smooth transition of physical quantities through the interface separating the bulk phases. Surface tension arises naturally from the cohesive part of the vdW equation of state and the capillary forces. The drop models all start from a square-shaped liquid and spinodal decomposition is investigated for a range of initial densities and temperatures. The simulations predict the formation of stable, subcritical liquid drops with a vapor atmosphere, with the densities and temperatures of coexisting liquid and vapor in the vdW phase diagram closely matching the binodal curve. We find that the values of surface tension, as determined from the Young-Laplace equation, are in good agreement with the results of independent numerical simulations and experimental data. The models also predict the increase of the vapor pressure with temperature and the fitting to the numerical data reproduces very well the Clausius-Clapeyron relation, thus allowing for the calculation of the vaporization pressure for this vdW fluid.

Figure 1

(a) Phase diagram showing the theoretically calculated binodal and spinodal curves for a van der Waals fluid. The overlayed symbols denote the initial state of the models. The initial density is varied by varying the mass of the particles: $m=0.4$ (dots), 0.5 (squares), 0.7 (triangles), 0.8 (diamonds), and 1 (asterisks). (b) Densities and temperatures of the coexisting phases as obtained from the numerical simulations (dots) compared to the binodal curve for a vdW fluid (solid line).

Figure 2

Stable liquid drops in a vapor atmosphere. The liquid and the vapor phases are in thermal equilibrium with coexistence temperatures from top to bottom of $\approx 0.46$, 0.64, 0.66, and 0.75. The plots show the normalized distribution of SPH particle densities overlayed on the phase diagram (left), the particle positions showing the liquid drop and ambient vapor (middle), and the spatially rendered mass density field using the visualization tool splash (right). The color-scale bar and numbers on the right border indicate the density contrast in terms of our reduced units.

Figure 3

(a) SPH-averaged radial density, (b) pressure, and (c) temperature profiles for an equilibrium drop with a temperature $T\approx 0.64$. The vertical line in (a) marks the position of the equimolar radius. All quantities are expressed in terms of our reduced units.

Figure 4

(a) SPH-averaged radial density, (b) pressure, and (c) temperature profiles for an equilibrium drop with a temperature $T\approx 0.66$. The profiles obtained with $K=0.1$ (solid lines) are compared to those calculated using $K=1$ (dashed line) for an identical model. The vertical line in (a) marks the position of the equimolar radius. All quantities are expressed in terms of reduced units.

Figure 5

Dependence of the surface tension $\sigma$ on temperature for the vdW drops. The SPH results (filled dots) are compared with those of Nugent and Posch [22] (open dots). The open triangles and open squares refer to molecular dynamics [45] and Monte Carlo simulations [46] for a Lennard-Jones fluid, respectively, while the continuous lines correspond to available experimental data for liquid argon [47] (solid line), kripton (dashed line), ethane (dotted line), and R152a (dot-dashed line) [48].

Figure 6

Dependence of the equimolar radius $R$ on temperature. The same symbols in Fig. 1 are used to identify the initial state for each sequence of data. The squares and diamonds represent models with $m=0.5$ and 0.8, respectively, for $K=0.1$ (filled symbols) and $K=1$ (open symbols).

Figure 7

Natural logarithm of the vapor pressure as a function of the inverse of the temperature for all models. The same symbols in Fig. 1 are used to identify the initial state for each sequence of data. The solid line fits the numerical data to the Clausius-Clapeyron equation (29) and the dashed line fits the same data by excluding the scattered points, corresponding to liquid drops with essentially no vapor atmosphere or not fully in equilibrium. The slope of the former linear fit is $\Delta H_{\mathrm{vap}}/{\overline{k}}_B\approx 2.08$, while that of the latter is $\Delta H_{\mathrm{vap}}/{\overline{k}}_B\approx 3.4$, which gives a better representation of the numerical data and defines the vaporization pressure in reduced units for this vdW fluid.

Figure 8

Dependence of the vapor pressure $p_v$ on temperature for all models. The solid line represents the best exponential fit to all numerical data: $p_v\approx 0.19exp(-2.10/T)$, yielding $\Delta H_{\mathrm{vap}}/{\overline{k}}_B\approx 2.10$. The dashed line represents the same fit when the scattered data, corresponding to liquid drops with essentially no vapor atmosphere or not fully in equilibrium, are omitted. It satisfies the relation $p_v\approx 1.69exp(-3.42/T)$ and provides a better representation of the Clausius-Clapeyron relation for the 2D vdW drops.