Shapes of a liquid droplet in a periodic box

Within the coexistence region between liquid and vapor the equilibrium pressure of a simulated fluid exhibits characteristic jumps and plateaus when plotted as a function of density at constant temperature. These features exclusively pertain to a finite-size sample in a periodic box, as they are washed out in the bulk limit. Below the critical density, at each pressure jump the shape of the liquid drop undergoes a morphological transition, changing from spherical to cylindrical to slablike as the density is increased. We formulate a simple theory of these shape transitions, which is adapted from a calculation originally developed by Binder and coworkers [L. G. MacDowell, P. Virnau, M. Muller, and K. Binder, J. Chem. Phys. 120, 5293 (2004)]. Our focus is on the pressure equation of state (rather than on the chemical potential, as in the original work) and includes an extension to elongated boxes. Predictions based on this theory well agree with extensive Monte Carlo data for the cut-and-shifted Lennard-Jones fluid. We further discuss the thermodynamic stability of liquid drops with shapes other than the three mentioned above, like those found deep inside the liquid-vapor region in simulations starting from scratch. Our theory classifies these more elaborate shapes as metastable.

Figure 1

Comparison between theory and MC data [4] from sequential simulations of the cut-and-shifted LJ model in a periodic cubic box ($N=4000$ and $T=0.90$). In order to equilibrate the system ${10}^6$ MC cycles were produced at each state point, followed by other $4\times {10}^6$ cycles over which equilibrium averages were computed. Top: $\mathrm{\Delta }F/N-P_{\mathrm{v}}(v_{\mathrm{v}}-v)$ for the various competing phases (green, “hom”; cyan, “sph”; blue, “cyl”; black, “slab”). The lower envelope of all the free-energy curves (thick red line) marks the equilibrium curve. The vertical lines mark the location of the shape transitions [see Eqs. (2.12)]. Middle: system pressure relative to $P_{\mathrm{v}}$. The dots are the MC data from Ref. [4] (open dots, forward trajectory; full dots, backward trajectory) while the thick red line is the equilibrium pressure according to our theory. Bottom: the equilibrium drop size [thick red line, see Eqs. (2.13)] vs half box size (black line).

Figure 2

MC pressure data for the cut-and-shifted LJ model in a periodic cubic box with $N=4000$ and $T=0.90$ (only the forward trajectory is shown). The horizontal line lies at the level of $P_{\mathrm{v}}$. The vertical lines mark the liquid-vapor coexistence densities, ${\rho }_{\mathrm{v}}$ and ${\rho }_{\mathrm{l}}$.

Figure 3

Comparison between theory and MC data from sequential simulations of the cut-and-shifted LJ model in a periodic cubic box ($N=4000$ and $T=0.90$). Same as Fig. 1 but the high-density region is shown here.

Figure 4

Comparison between theory for $a=3$ and MC data from sequential simulations of the cut-and-shifted LJ model in a periodic cuboidal box ($N=1500,T=0.90$, and $\beta \gamma =0.175$). In stark contrast with the $a=1$ case, the spherical drop would never be stable for $a=3$. The cylindrical drop is only observed in a narrow range of densities close to 0.1 whereas the slablike drop is stable over a much wider density interval.

Figure 5

MC pressure data for the cut-and-shifted LJ model in a periodic cubic box ($N=4000$ and $T=0.72$) vs theory (for this case, $P_{\mathrm{v}}=0.0062,{\rho }_{\mathrm{v}}=0.0086,{\rho }_{\mathrm{l}}=0.7722,\beta \gamma =0.78$ [20], whereas the value of $K_{\mathrm{v}}$, 173.62, resulted from a long $NPT$ simulation of the liquid at $P_{\mathrm{v}}$). The data points (full dots in the middle panel) were obtained independently from one another, by starting the simulation from scratch at each density. We see that each point lies on a theoretical line, except for the points around $\rho =0.25$ which appear to lie on a different plateau (dashed line). In the corresponding range of densities the liquid drop resembles either a DC or a PS.

Figure 6

Cut-and-shifted LJ model for $N=4000$ and $T=0.72$: results from MC simulations started from scratch. (a) A system snapshot taken at the density $\rho =0.24$. (b) Another snapshot taken at $\rho =0.26$. In these pictures, each particle was given an effective diameter $\sigma$. The symbol colors were chosen according to the number of nearest neighbors each particle has, in turn defined as the number of particles within a distance of $1.45\sigma$ from the given particle ($1.45\sigma$ being the position of the first nonzero minimum of the radial distribution function of the liquid at ${\rho }_{\mathrm{l}}$). While the liquid drop in (a) resembles a DC, the drop in (b) looks like a PS.

Figure 7

Same as in Fig. 6, but after the particle coordinates have been transformed as explained in the main text. The outcome was that a DC became a PS, and vice versa.

Figure 8

Shape transitions for the cut-and-shifted LJ model in a periodic cubic box $(N=4000$ and $T=0.72$): theoretical results (see parameters in Fig. 5 caption). Top: $\mathrm{\Delta }F/N-P_{\mathrm{v}}(v_{\mathrm{v}}-v)$ for various competing phases (blue full line, “cyl”; black full line, “slab”; green long-dashed line, type-1 PS; cyan long-dashed line, DC; magenta long-dashed line, type-2 PS—notice that the free-energy curves for “hom” and “sph” are not shown). The thick red line marks the equilibrium curve (that is, the lower envelope of all the free-energy curves, including “hom” and “sph”). The vertical lines mark the location of the shape transitions. Bottom: the equilibrium drop size (thick red line) vs half box size (black full line). The dotted and dashed lines, respectively, refer to the optimal $d$ and $r$ values for the two types of PS and the DC.