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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: 

The number of particles in the box is conserved: 

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 

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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- 

