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considered. This provides a stringent test for both the classical theory of homogeneous and heterogeneous nu- cleation. For this purpose we propose a new method [16, 17] to measure surface free energies (and hence nu- cleation barriers) of liquid droplets in the bulk and of droplets attached to walls over the experimentally rele- vant range. No “atomistic” identification of which par- ticles belong to the droplet and which to the vapor is required. No bias potential to stabilize droplets of a par- ticular size [8] is needed, and fluctuations of the droplet are not constrained. We independently determine the droplet volume and the chemical potential that charac- terizes the ( stable ) equilibrium between the small droplet and surrounding vapor in the finite system and derive an expression for the line tension. 

We test Turnbull’s formula (5) with Monte Carlo simulations of the standard nearest neighbor (Ising) lattice gas model on the simple cubic lattice: H = − J ∑ S$_{i}$S$_{j}$, S$_{i,j}$ = ± 1. Phase coexistence between sat- urated vapor at density ρ$_{v}$ and liquid ρ$_{ℓ}$ occurs at a (known) chemical potential µ coex . (In magnetic nota- 

tion, the magnetic field H = ( µ − µ coex ) / 2, and ρ$_{ℓ}$ , ρ$_{v}$ are related to the spontaneous magnetization m coex as ρ$_{v}$ = (1 − m coex ) / 2, ρ$_{ℓ}$ = (1 + m coex ) / 2.) Bulk sys- tems are studied in cubic simulation boxes with periodic boundary conditions in all directions. For the heteroge- neous case we choose a L × L × D geometry with pe- riodic boundary conditions in x and y direction. Sur- face fields H 1 and H$_{D}$ = − H 1 act on (the first layer of) the two free L × L surfaces. The temperature is set to k$_{B}$T/J = 3 . 0 to stay away from the bulk critical tem- perature (at k$_{B}$T/J ≈ 4 . 51) and above the roughening transition temperature [18] (at k$_{B}$T/J ≈ 2 . 45). Hence, the correlation length ξ is less than a lattice spacing and the interface free energy does (to a good approximation) not depend on the interface orientation. 

Wetting for the Ising model has been thoroughly studied before [19] and by varying the surface field H 1 we can control the contact angle (Fig. 2). Note that the local surface layer magnetization m 1 = − ( ∂f$_{s}$ ( T, H, H 1 ) /∂H 1 )$_{T,H}$ [20] ( f$_{s}$ is the surface excess free energy per spin). With f$_{s}$ ( T, H → 0 $^{−}$, H 1 ) = γ$_{wv}$ , f$_{s}$ ( T, H → 0 $^{+}$, H 1 ) = γ$_{wℓ}$ and Young’s equation (3), we obtain θ from thermodynamic integration 

Here, we exploit the Ising symmetry f$_{s}$ ( T, H → 0 $^{−}$, H 1 ) = f$_{s}$ ( T, H → 0 $^{+}$, − H 1 ) to use the local mag- netization m$_{D}$ at the other surface at which H$_{D}$ = − H 1 acts. γ$_{vℓ}$ = 0 . 434 was given by the very accurate esti- mates of Hasenbusch and Pinn [21]. 

In the following we describe how to obtain the surface free energy as a function of the droplet radius. First, we measure the chemical potential (in the canonical en- semble) as a function of density (Fig. 3) by applying an 

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