FIG. 3: Chemical potential as a function of density. The den- sity of the sphere-cap shaped droplet at overall density ρ is given by ρ ′ l , the density of the environment by ρ ′ v . Δ F s is ob- tained by integrating the chemical potential over the density from ρ ′ v to ρ . FIG. 4: a) Plot of F s ( R, θ ) /k B T vs. R for k B T/J = 3 . 0 and a broad range of values for θ . Dotted lines are results for L = D = 30, dashes lines for D = L = 40, dash-dotted lines for L = 60, D = 20. For the systems (bulk, H 1 /J = 0 and H 1 /J = 0 . 4 ( θ ≈ 58 $^{o}$)) the classical predictions 4 πR $^{2}$γ vℓ and 4 πR $^{2}$γ vℓ f ( θ ) are included (full curves). b) Plot of τ vs. θ for k B T/J = 3 . 0. Crosses are data for D = L = 30, circles for D = L = 40, diamonds D = 10, L = 60, and squares for D = 10, R = 80. adaptation [16, 17] of the Widom test particle insertion method [22]. In the flat region of the isotherm depicted in Fig. 3, a single droplet is present in the simulation box, which will be used for the determination of the surface free energy. At small densities the ascending branch of µ represents pure vapor (with no single large droplet being present), and the subsequent rapid decrease signifies the system size dependent transition towards a single droplet phase [23–26]. At larger densities, the droplet region is delimited by a transition towards a cylindrical droplet which is stabilized by the boundary conditions. Note that for each system size, only a certain range of droplet sizes can be stabilized and we have to simulate a whole range of box sizes to cover all radii. As indicated in Fig. 3 for each droplet state at den- sity ρ , there exists one vapor state at ρ ′ v and one liquid state at ρ ′ l which have the same chemical potential. Equi- librium conditions require that the chemical potential is constant throughout the box. Therefore, the droplet has density ρ ′ l and is surrounded by an environment with den- sity ρ ′ $_{v}$. The surface free energy of the droplet is simply given by the difference in free energy between the droplet and the vapor state and can be determined by thermo- dynamic integration: F$_{c,bulk}$ ( R ) = 4 πR $^{2}$γ$_{vℓ}$ [16, 26] ( γ$_{vℓ}$ being the interfacial tension for a flat interface.) For R = 5 the difference be- tween the theoretical estimate and our simulation results is already less than 5% (at k$_{B}$T/J = 3 . 0) and quickly vanishes with increasing droplet radius. The validity of the capillarity approximation for the bulk agrees with conclusions drawn from surface force measurements on liquid bridges [29], but contrasts a large body of other work (see [26] for a discussion). We have also determined the surface free energy for a sphere-cap shaped droplet attached to the wall for sev- eral contact angles as a function of the radius of cur- vature. It is important to note that within statistical errors different choices of L and D yield identical re- sults. For θ = 90 o and θ = 58 . 2 o we have also in- cluded the theoretical estimate from the capillarity ap- proximation F$_{c,het}$ = F$_{c,bulk}$ · f ( θ ). When we consider F$_{c,het}$ ( R, θ ) − F$_{s}$ ( R, θ ), we find that the difference in- If we further require that the (average) droplet is spher- ical in the bulk and sphere-cap shaped for the hetero- geneous case, V$_{drop}$ becomes 4 / 3 πR 3 or 4 / 3 πR $^{3}$f ( θ ), re- spectively, and we obtain R ( ρ ) and finally F$_{s}$ ( R ) and F$_{s}$ ( R, θ ). Fig. 4 a highlights the main results of our study. For the bulk we can directly test the classical theory of homogeneous nucleation by comparing our simula- tion results for F$_{s}$ ( R ) with the capillary approximation The number of particles in the box is conserved: F$_{s}$ ( ρ ) = ∫ ρ ρ ′ µ ( ρ $^{′}$) dρ $^{′}$. (7) ρ ′ $_{v}$( V − V$_{drop}$ ) + ρ ′ $_{l}$V$_{drop}$ = ρV. (8) 5 10 15 R 100 200 300 400 F s /kT bulk capillarity  approx. 30 40 50 60 70 80 90 -0.25 -0.2 -0.15 -0.1 30 40 slab 3