Abstract
We study equilibrium droplets in two-phase systems at parameter values corresponding to phase coexistence. Specifically, we give a self-contained microscopic derivation of the Gibbs–Thomson formula for the deviation of the pressure and the density away from their equilibrium values which, according to the interpretation of the classical thermodynamics, appears due to the presence of a curved interface. The general—albeit heuristic—reasoning is corroborated by a rigorous proof in the case of the two-dimensional Ising lattice gas.
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REFERENCES
K. Alexander, J. T. Chayes, and L. Chayes, The Wulff construction and asymptotics of the finite cluster distribution for two-dimensional Bernoulli percolation, Comm. Math. Phys. 131:1–51 (1990).
K. Binder, Theory of evaporation/condensation transition of equilibrium droplets in finite volumes, Physica A 319:99–114 (2003).
K. Binder and M. H. Kalos, Critical clusters in a supersaturated vapor: Theory and Monte Carlo simulation, J. Statist. Phys. 22:363–396 (1980).
M. Biskup, L. Chayes, and R. Kotecký, On the formation/dissolution of equilibrium droplets, Europhys. Lett. 60:21–27 (2002).
M. Biskup, L. Chayes, and R. Kotecký, Critical region for droplet formation in the twodimensional Ising model, Comm. Math. Phys. 242:137–183 (2003).
T. Bodineau, The Wulff construction in three and more dimensions, Comm. Math. Phys. 207:197–229 (1999).
T. Bodineau, D. Ioffe, and Y. Velenik, Rigorous probabilistic analysis of equilibrium crystal shapes, J. Math. Phys. 41:1033–1098 (2000).
C. Borgs and R. Kotecký, Surface-induced finite-size effects for first-order phase transitions, J. Stat. Phys. 79:43–115 (1995).
J. Bricmont, J. L. Lebowitz, and C.-E. Pfister, On the local structure of the phase separation line in the two-dimensional Ising system, J. Stat. Phys. 26:313–332 (1981).
R. Cerf, Large deviations for three dimensional supercritical percolation, Astérisque 267: vi+177 (2000).
R. Cerf and A. Pisztora, On the Wulff crystal in the Ising model, Ann. Probab. 28:947–1017 (2000).
J. T. Chayes, L. Chayes, and R. H. Schonmann, Exponential decay of connectivities in the two-dimensional Ising model, J. Stat. Phys. 49:433–445 (1987).
P. Curie, Sur la formation des cristaux et sur les constantes capillaires de leurs différentes faces, Bull. Soc. Fr. Mineral. 8:145 (1885); Reprinted in OEuvres de Pierre Curie(Gauthier-Villars, Paris, 1908), pp. 153-157.
R. L. Dobrushin, R. Kotecký, and S. B. Shlosman, Wulff Construction. A Global Shape from Local Interaction(Amer. Math. Soc., Providence, RI, 1992).
R. L. Dobrushin and S. B. Shlosman, in Probability Contributions to Statistical Mechanics(Amer. Math. Soc., Providence, RI, 1994), pp. 91–219.
H.-O. Georgii, Gibbs Measures and Phase Transitions, de Gruyter Studies in Mathematics, Vol. 9 (Walter de Gruyter, Berlin, 1988).
J. W. Gibbs, On the equilibrium of heterogeneous substances (1876), in Collected Works, Vol. 1 (Longmans, Green, 1928).
R. B. Griffiths, C. A. Hurst, and S. Sherman, Concavity of magnetization of an Ising ferromagnet in a positive external field, J. Math. Phys. 11:790–795 (1970).
D. Ioffe, Large deviations for the 2D Ising model: A lower bound without cluster expansions, J. Stat. Phys. 74:411–432 (1994).
D. Ioffe, Exact large deviation bounds up to Tc for the Ising model in two dimensions, Probab. Theory Related Fields 102:313–330 (1995).
D. Ioffe and R. H. Schonmann, Dobrushin-Kotecký-Shlosman theorem up to the critical temperature, Comm. Math. Phys. 199:117–167 (1998).
B. Krishnamachari, J. McLean, B. Cooper, and J. Sethna, Gibbs-Thomson formula for small island sizes: Corrections for high vapor densities, Phys. Rev. B 54:8899–8907 (1996).
R. Kotecký and D. Preiss, Cluster expansion for abstract polymer models, Comm. Math. Phys. 103:491–498 (1986).
L. D. Landau and E. M. Lifshitz, Course of theoretical physics, in Statistical Physics, Vol. 5 (Pergamon Press, Oxford/Edinburgh/New York, 1968).
T. Neuhaus and J. S. Hager, 2D crystal shapes, droplet condensation and supercritical slowing down in simulations of first order phase transitions, J. Stat. Phys. 113:47–83 (2003).
C.-E. Pfister, Large deviations and phase separation in the two-dimensional Ising model, Helv. Phys. Acta 64:953–1054 (1991).
C.-E. Pfister and Y. Velenik, Large deviations and continuum limit in the 2D Ising model,Probab. Theory Related Fields 109:435–506 (1997).
R. H. Schonmann and S. B. Shlosman, Wulff droplets and the metastable relaxation of kinetic Ising models, Comm. Math. Phys. 194:389–462 (1998).
D. Tabor, Gases, Liquids and Solids: And Other States of Matter,3rd Edn. (Cambridge University Press, Cambridge, 1991).
G. Wulff, Zur Frage des Geschwindigkeit des Wachsturms und der Auflösung der Krystallflachen, Z. Krystallog. Mineral. 34:449–530 (1901).
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Biskup, M., Chayes, L. & Kotecký, R. A Proof of the Gibbs—Thomson Formula in the Droplet Formation Regime. Journal of Statistical Physics 116, 175–203 (2004). https://doi.org/10.1023/B:JOSS.0000037209.36990.eb
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DOI: https://doi.org/10.1023/B:JOSS.0000037209.36990.eb