various models in broad classes of systems. They will enable significant progress in the understanding of nucle- ation phenomena in diverse branches of physics. creases linearly with R , and hence can be interpreted as a line tension contribution, 2 πrτ . I.e., Eq (4) is mod- ified to F$_{s,het}$ = F$_{s,hom}$ · f ( θ ) + 2 πrτ . This division of surface and line contributions relies on our droplet defi- nition, Eq (8), which uses the fact that for the lattice gas the Tolman length [3–5] is zero. Fig. 4b shows a plot of the line tension τ obtained in this way versus contact an- gle. The line tension is negative, and becomes very small (and presumably vanishes) as θ → 0. The estimates for τ have been used (together with the appropriate esti- mates for R ) in Fig. 2, to show the expected deviation of the contact angle for small droplets from the bulk value. An independent check of the line tension is obtained for θ = 90 ◦ by investigating slab configurations (Fig.5) with varying D. In this case the potential can be written as G/ (2 DLk$_{B}$T ) = γ$_{lv}$ +2 τ/D , and the slope in Fig.5 agrees with the estimate shown in Fig.4b. Hence, Eqs. (4) and (5) hold if a correction due to line tension is considered. Interestingly, an experimental hint for the need of a neg- ative line tension describing bubble formation in water extrusion from mesopores was given in [30]. In summary, we have shown that the classical the- ory of heterogeneous nucleation at flat walls, which pre- dicts a reduction of the free energy barrier by a fac- tor f ( θ ), can describe the actual surface free energies F$_{s}$ ( R, θ ), provided a line tension term is included into the latter. While the importance of line tension effects on small nuclei has been stressed in various other con- texts, e.g. [9, 27, 28], the present paper is the first ap- proach which provides a systematic method to obtain both F$_{s}$ ( R ) for spherical droplets in the bulk and F$_{s}$ ( R, θ ) for wall-attached droplets, as well as for the contact an- gle θ and line tension τ . Thus, for the first time we verify Turnbull’s equation from 1950, which still stands as one of the cornerstones of the theory of heterogeneous nucle- ation. The methods, which in this letter are described for the simple Ising model, can readily be generalized to FIG. 5: Plot of the thermodynamic potential G for slab con- figurations (insert), normalized by temperature and total sur- face area 2 LD , versus 2 /D so that the slope yields the line tension ( τ/k B T = 0 . 26 ± 0 . 001). The intercept of the ordinate was fixed to the literature value of the interface tension 0.434 [21]. 4 [1] M. Volmer and A. Weber, Z. Chem. Phys. 119 , 277 (1926) [2] A.C. Zettlemoyer (ed) Nucleation (M. Dekker, New York, 1969) [3] K. Binder and D. Stauffer, Adv. Phys. 25 , 343 (1976) [4] P. Debenedetti: Metastable Liquids Princeton University Press, Princeton (1997) [5] D. Kashchiev: Nucleation: Basic Theory with Applica- tions (Butterworth-Heinemann, Oxford, 2000) [6] J. Curtius, Compt. Rendus Phys. 7 , 1027 (2006) [7] H. Biloni, in Physical Metallurgy (R.W. Cahn and P. Haasen, eds.) North-Holland, Amsterdam (1983), p. 477 [8] P.R. Ten Wolde and D. Frenkel, J. Chem. Phys. 109 , 9901 (1998) [9] S. Auer and D. Frenkel, Phys. Rev. Lett. 91 , 015703 (2003) [10] L. Maibaum, Phys. Rev. Lett. 101 , 256102 (2008) [11] T. Young, Phil. Trans. Roy. Soc. London 95 , 65 (1805) [12] P.G.de Gennes, F. Brochard-Wyart, and D. Qu´er´e, Cap- illary and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer, Berlin 2004) [13] R.D. Gretz, J. Chem. Phys. 45 , 3160 (1966) [14] D. Turnbull, J. Chem. Phys. 18 , 198 (1950); J. Appl. Phys. 21 , 1022 (1950) [15] G. Navascues and P. Taranzona, J. Chem. Phys. 75 , 2441 (1981) [16] D. Winter, P. Virnau, and K. Binder (in preparation) [17] D. Winter, Diplomarbeit (Johannes Gutenberg- Universit¨ at Mainz, 2009, unpublished) [18] K.K. Mon, S. Wansleben, D.P. Landau, and K. Binder, Phys. Rev. B 39 , 7089 (1989) [19] K. Binder and D. P. Landau, Phys. Rev. B 37 , 1745 (1988); K. Binder, D.P. Landau and S. Wansleben, Phys. Rev. B 40 , 6971 (1989) [20] K. Binder and P.C. Hohenberg, Phys. Rev. B 6 , 3461 (1972) [21] M. Hasenbusch, and K. Pinn, Physica A 192 , 342 (1993); ibid 203 , 189 (1994) [22] B. Widom, J. Chem. Phys. 39 , 2808 (1963) [23] M. Biskup, L. Chayes and R. Kotecky, Europhys. Lett. 60 , 21 (2002) [24] K. Binder, Physica A 319 , 99 (2003) [25] L.G. MacDowell, P. Virnau, M, M¨ uller, and K. Binder, J. Chem. Phys. 120 , 5293 (2004) [26] M. Schrader, P. Virnau, and K. Binder, Physical Review E 79 , 061104 (2009) [27] A. Milchev, and K. Binder, J. Chem. Phys. 114 , 8610 (2001) [28] S. Mechkov, G. Oshanin, M. Rauscher, M. Brinkmann, A.M. Cazabtat and S. Dietrich, Europhys. Lett. 80 , 66002 (2007) [29] J.Crassous, E.Charlaix, J.L.Loubet, Europhys.Lett. 28 , 37 (1994) [30] B. Lefevre, A.Saugey, J.L.Barrat, et al., J.Chem.Phys 120 , 4927 (2004)