Droplet condensation in the lattice gas with density functional theory

A density functional for the lattice gas with next-neighbor attractions (Ising model) from fundamental measure theory is applied to the problem of droplet states in three-dimensional, finite systems. The density functional is constructed via an auxiliary model with hard lattice gas particles and lattice polymers to incorporate the attractions. Similar to previous simulation studies, the sequence of droplets changing to cylinders and to planar slabs is found upon increasing the average density $\overline{\rho }$ in the system. Owing to the discreteness of the lattice, additional effects in the state curve for the chemical potential $\mu \left(\overline{\rho }\right)$ are seen upon lowering the temperature away from the critical temperature [oscillations in $\mu \left(\overline{\rho }\right)$ in the slab portion and spiky undulations in $\mu \left(\overline{\rho }\right)$ in the cylinder portion as well as an undulatory behavior of the radius of the surface of tension $R_{\mathrm{s}}$ in the droplet region]. This behavior in the cylinder and droplet region is related to washed-out layering transitions at the surface of liquid cylinders and droplets. The analysis of the large-radius behavior of the surface tension $\gamma \left(R_{\mathrm{s}}\right)$ gave a dominant contribution $\propto 1/{R_{\mathrm{s}}}^2$, although the consistency of $\gamma \left(R_{\mathrm{s}}\right)$ with the asymptotic behavior of the radius-dependent Tolman length seems to suggest a weak logarithmic contribution $\propto ln{R}_{\mathrm{s}}/{R_{\mathrm{s}}}^2$ in $\gamma \left(R_{\mathrm{s}}\right)$. The coefficient of this logarithmic term is smaller than a universal value derived with field-theoretic methods.

Figure 1

Binodal for the Highlander functional and the standard mean-field functional. For the Highlander functional, we investigated droplets at the three temperatures $\beta \epsilon =1.0$, 1.2, and 1.4 (horizontal blue solid lines) and for the mean-field functional we chose the three temperatures $\beta \epsilon =0.82$, 0.99, and 1.15 (horizontal green dashed lines). The red cross marks the quasiexact critical point from simulations [25] ${\beta }_{c,\text{sim}}\epsilon \approx 0.887$.

Figure 2

(a) $\mu$ equation of state for the Highlander functional at $\beta \epsilon =1.0$ for three system sizes distinguished by symbols. (b) Density profile for a stable droplet in the midplane of the box at $\overline{\rho }=0.155$ for system size ${45}^3$ with one-dimensional cuts.

Figure 3

$\mu$ equation of state for the Highlander functional at $\beta \epsilon =1.4$ for three system sizes distinguished by symbols. Enlargements show $\mu \left(\overline{\rho }\right)$ for the droplet, cylinder, and slab portions of the $\mu$ equation of state.

Figure 4

Equimolar droplet radius $R_{\mathrm{e}}$ and radius $R_{\mathrm{s}}$ of the surface of tension for (a) the Highlander functional and $\beta \epsilon =1.4$ and (b) the mean-field functional and $\beta \epsilon =1.15$.

Figure 5

Surface tension $\gamma \left(R_{\mathrm{s}}\right)$ (a) and Tolman-length (b) vs $1/R_{\mathrm{s}}$ for the Highlander functional at $\beta \epsilon =1.0$. The fits in panel (a) are according to Eq. (18) with $A$ considered universal (fit 1) or a fit parameter (fit 2) or $A=0$ (fit 3). The resulting fit parameters are shown in Table 1. The dotted, dash-dotted, and dashed lines in panel (b) are according to the GTKB-consistent Eq. (19) with the parameters taken from fits 1, 2, and 3 from panel (a). In contrast to panel (b), fits 2 and 3 are visually indistinguishable in panel (a), therefore only fit 2 is shown in panel (a).

Figure 6

Surface tension $\gamma \left(R_{\mathrm{s}}\right)$ (a, c) and Tolman-length (b, d) vs $1/R_{\mathrm{s}}$ for the Highlander functional at $\beta \epsilon =1.2$ and 1.4. For information on the fits see the caption of Fig. 5. Fits 2 and 3 are indistinguishable in (a, c) and their difference is not very illuminating in (b, d) owing to the oscillatory numerical data, therefore only fits 1 and 2 are shown.

Figure 7

Comparison of droplet, cylindrical and slab density profiles for the Highlander functional at $\beta \epsilon =1.0$. Panels (a) show the heat-map of two-dimensional cross-sections of the corresponding geometries (left: droplet; middle left: cylinder with normal of the plane parallel to the symmetry axis; middle right: cylinder with normal of the plane orthogonal to the symmetry axis; right slab with normal of the plane orthogonal to normal of the slab.). Panel (b) shows the density profile along the corresponding lines in panels (a). The average density of the droplet is $\overline{\rho }=0.155$, the cylinder has $\overline{\rho }=0.3$ and the slab is at $\overline{\rho }=0.4$.

Figure 8

$\mu$ equation of state for the mean-field functional in 2D at $\beta \epsilon =1.5$ for one system size ${64}^2$. Enlargements show $\mu \left(\overline{\rho }\right)$ for the circular droplet and slab portions of the $\mu$ equation of state.

Figure 9

Droplet solutions for the mean-field functional in 2D at $\beta \epsilon =1.5$. (a) $R_{\mathrm{e}}$ and $R_{\mathrm{s}}$ vs $\overline{\rho }$, symbols mark states whose density profiles along the $x$ axis are shown in panel (b) using the same symbols. Arrows mark the two discrete jumps seen in panel (a).