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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 

various models in broad classes of systems. They will enable significant progress in the understanding of nucle- ation phenomena in diverse branches of physics. 

[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) 

