Figure 1

(Color online) Chemical potential difference $\widehat{\mu }=\mu \left(N\right)-{\mu }_{\text{coex}}$ plotted vs density $\rho$ for a cubic $L\times L\times L$ box of size $L=15.8\sigma$ ($\sigma$ is the Lennard-Jones diameter) at $T=0.68T_{\text{c}}$. The states at $\widehat{\mu }=0$ with density ${\rho }_{\text{v}}$ (vapor) and ${\rho }_{\ell }$ (liquid) approximate very accurately the homogeneous phases that can coexist in the thermodynamic limit. In the finite box, the homogeneous vapor (indicated by the snapshot picture denoted as “hom”) also exists for densities exceeding ${\rho }_{\text{v}}$, up to about the density ${\rho }_{\text{t}}$ (dotted vertical line) where the droplet evaporation/condensation transition occurs. For densities from about ${\rho }_{\text{t}}$ to about ${\rho }_{\text{t}2}$ the system is in an inhomogeneous state. A spherical (sph) droplet coexists with surrounding supersaturated vapor. At ${\rho }_{\text{t}2}$ the shape of the liquid droplet changes from spherical to cylindrical (cyl), which is the stable two-phase configuration for ${\rho }_{\text{t}2}<\rho <{\rho }_{\text{t}3}$, while in the central region around ${\rho }_{\text{d}}=\left({\rho }_{\ell }+{\rho }_{\text{v}}\right)/2$ a slablike two-phase configuration (slb) is stable $\left({\rho }_{\text{t}3}<\rho <{\rho }_{\text{t}4}\right)$. At ${\rho }_{\text{t}4}$ the character of two-phase coexistence in the box changes again. The vapor changes from a slablike region to a cylindrical domain (cyl2), while for ${\rho }_{\text{t}5}<\rho <{\rho }_{\text{t}6}$ one has a coexistence of a spherical bubble (sph2) with surrounding liquid. Inset: this graph illustrates that for $\widehat{\mu }>0$ there exists a regime of $\widehat{\mu }$ where three phases correspond to the same value of $\widehat{\mu }$ and hence can coexist with each other—a homogeneous vapor at density ${\rho }_{\text{v}}^{\prime }>{\rho }_{\text{v}}$, a homogeneous liquid droplet coexists with surrounding vapor, at average density $\rho$ (in the regime ${\rho }_{\text{t}}<\rho <{\rho }_{\text{t}2}$) in the simulation box, and a homogeneous liquid at density ${\rho }_{\ell }^{\prime }$. Note that the occurrence of all these phases was already discussed in [56, 57].Reuse & Permissions

Figure 2

(Color online) Typical snapshot picture of a spherical droplet in a box of linear dimension $L=33.7\sigma$ at average density $\rho =0.1{\sigma }^{-3}$ (total particle number $N=3827$ and $N=3427$ particles in the droplet).Reuse & Permissions

Figure 3

(Color online) Normalized free-energy density $f_L\left(\rho ,T\right)\equiv L^{-3}F\left(N,V,T\right)/k_{\text{B}}T$ plotted vs density $\rho$ for six choices of $L$ at $T=0.68T_{\text{c}}$. Only for the three smallest sizes the functions are shown over the full density range. The free-energy branches describing homogeneous vapor and liquid are fitted to simple polynomials of the density.Reuse & Permissions

Figure 4

(Color online) Radial distribution of the excess chemical potential ${\mu }_{\text{ex}}\left(r\right)$ (broken curves), the ideal gas part ${\mu }_{\text{id}}\left(r\right)$ (dotted curves) and their sum (full curves) for boxes of linear dimension $L=15.8\sigma$ (upper part) and $L=33.7\sigma$ (lower part) at density $\rho =0.1{\sigma }^{-3}$ and $T=0.68T_{\text{c}}$ (central part of the figure). The small pictures on the right and left sides show corresponding results for ${\mu }_{\text{ex}},{\mu }_{\text{id}}$ and $\mu \left(N\right)$ in bulk vapor and liquid phases at the densities ${\rho }_{\text{v}}^{\prime },{\rho }_{\ell }^{\prime }$ as defined in Fig. 1. Note the different abscissa scales for $r$ in the upper and lower part, respectively.Reuse & Permissions

Figure 5

(Color online) Graphical illustration of the equations used to locate the droplet evaporation/condensation transition by an approximate analytical calculation. At the transition density ${\rho }_{\text{t}}$, the free energy of the state containing a droplet plus surrounding vapor (Fig. 1) and of the state with the denser vapor without a droplet are equal (upper part). The density ${\rho }_{\text{v},\text{t}}^{\prime }$ of the vapor surrounding the droplet at the transition point can be found from the equality of the chemical potentials $\left({\widehat{\mu }}_{\text{t}}\right)$, lower part. Using Eq. (21), ${\widehat{\mu }}_{\text{t}}$ yields the corresponding value of $R$, and Eq. (17) then yields the relation between ${\rho }_{\text{t}}$ and $L$. Both in the plot of $f_L\left(\rho ,T\right)$ vs $\rho$ (upper part) and in the plot of $\widehat{\mu }$ vs $\rho$ (lower part) metastable parts of the branches are shown as broken curves, and the finite size rounding seen in Figs. 1, 3 due to the fact that in a finite simulation box the system can jump back and forth between both states is disregarded.Reuse & Permissions

Figure 6

(Color online) Probability distributions $P_{\text{NVT}}\left(U\right)$ plotted versus energy per particle $-U/\left(N{k}_{\text{B}}T\right)$ for seven values of $N$, for $T=0.68T_{\text{c}}$, and a cubic box with linear dimension $L=50\sigma$ (upper part). Cumulative distribution function $P_{\text{cum},\text{NVT}}\left(U\right)$ for the same cases (lower part).Reuse & Permissions

Figure 7

(Color online) Log-log plot of the density difference ${\rho }_{\text{t}}-{\rho }_{\text{v}}$ versus the system size $L$ for $T/T_{\text{c}}=0.68$. Straight lines exhibit power-law fits through three successive values (slopes being indicated in the figure). Inset: plot of the “effective exponent,” as estimated before versus the inverse system size $L^{-1}$.Reuse & Permissions

Figure 8

(Color online) Chemical potential difference $\widehat{\mu }\left(\rho \right)$ plotted vs the density $\rho$ at $T=0.68T_{\text{c}}$ including data for six choices of $L$. The part of the curves describing pure homogeneous vapor and pure homogeneous liquid that are not affected by finite-size effects due to two-phase coexistence are fitted by functions ${\rho }_{\text{hom}}^{\text{fit}}\left(\widehat{\mu }\right)=-0.823+110.04\widehat{\mu }-2518.1{\widehat{\mu }}^2+20910{\widehat{\mu }}^3$ and ${\rho }_{\text{hom}2}^{\text{fit}}\left(\widehat{\mu }\right)=702.08-2628.4\widehat{\mu }+3255.8{\widehat{\mu }}^2-1333{\widehat{\mu }}^3$, respectively. The inset shows a magnified plot of the $\widehat{\mu }\left(\rho \right)$ vs $\rho$ curves in the region of interest, where two-phase coexistence between a droplet and surrounding vapor always occurs. In this region the data were also fitted by polynomials ${\rho }_{\text{sph}}^{\text{fit}}\left(\widehat{\mu },L\right)$, which now distinctly depend on $L$.Reuse & Permissions

Figure 9

(Color online) Surface free energy $F_{\text{s}}\left(R\right)=4\pi R^2\gamma \left(R\right)$ plotted vs the droplet radius $R$ for $T=0.68T_{\text{c}}$. The thin full curve shows the result of the classical capillarity approximation $4\pi R^2{\gamma }_{\text{v}\ell }$, with ${\gamma }_{\text{v}\ell }$ taken from the independent estimation given in Eq. (13). The insets show magnified plots near $R=10\sigma$ (right) and from $R=2.5\sigma$ to $R=4.5\sigma$ (left). The curves labeled “m” are the results where $R$ was computed from Eqs. (15, 16, 17), while the curves labeled “c” refer to estimations of $R$ from the Stillinger [40] cluster criterion. For $T=0.85T_{\text{c}}$ analogous data are shown for small radii up to $R=5.25\sigma$ (only for method “m”).Reuse & Permissions

Figure 10

(Color online) Plot of $\gamma \left(R\right)/{\gamma }_{\text{v}\ell }$ vs the droplet radius $R$ for $T=0.68T_{\text{c}}$ (full curves) and for $T=0.85T_{\text{c}}$ (broken curve). The curve labeled by “m” has been obtained using “macroscopic measurements” of ${\rho }_{\text{v}}^{\prime },{\rho }_{\ell }^{\prime }$ and applying Eqs. (15, 16, 17) to estimate $R$, while the curve labeled by “c” has been obtained from observations of the droplets identified via the Stillinger connectivity criterion [40]. Inset: the plotted data vs the inverse radius reveals that the deviation from the classical capillarity approximation cannot be described by a Tolman correction, at least not for $T=0.68T_{\text{c}}$. Otherwise the curve would be a straight line through the origin with slope $2\delta$.Reuse & Permissions