Abstract
Several aspects of the theory of the coexistence of phases and equilibrium forms are discussed. In section 1, the problem is studied from the point of view of thermodynamics. In section 2, the statistical mechanical theory is introduced. We consider, in particular, the description of the microscopic interface at low temperatures and the existence of a free energy per unit area (surface tension). In section 3, a proof is given of the microscopic validity of the Wulff construction in a simplified model of a two-dimensional interface. Finally, in section 4, the roughening transition and the formation of facets in an equilibrium crystal are studied. Appendices A and B concern, respectively, the first and second points of section 2, mentioned above.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abraham, D.B., Surface Structures and Phase Transitions, in: Critical Phenomena, ed. by C. Domb, J.L. Lebowitz, Academic Press, London, 10, 1–74 (1986).
Andreev, A.F., Faceting Phase Transitions of Crystals, Soy. Phys. JETP 53, 1063 (1981).
Avron, J.E. H. van Beijeren, L.S. Shulman, & R.K.P. Zia, Roughening Transition, Surface Tension and the Equilibrium Droplet Shapes in a Two-Dimensional Ising System, J. Phys. A: Mat. Gen. 15 L 81–86 (1982).
Balibar, S. & B. Castaign, Possible Observation of the Roughening Transition in Helium, J. Physique Lett. 41, L 369–332 (1980).
Balibar, S. & B. Castaign, Helium: Solid-Liquid Interfaces, Surf. Sci. Rep. 5, 87–143 (1985).
Beijeren, H. van, Interface Sharpness in the Ising Model, Commun. Math. Phys. 40, 1-6 (1975).
Beijeren, H. van & I. Nolden, The Roughening Transition, in: Topics in Current Physics, ed. by W. Schommers, P. von Blackenhagen, Springer, Berlin, 43, 259–300 (1987).
Bricmont, J., A. El Mellouki & J. Fröhlich, Random Surfaces in Statistical Mechanics: Roughening, Rounding, Wetting,… J. Stat. Phys. 42, 743–798 (1986).
Bricmont, J., J.R. Fontaine & J.L. Lebowitz, Surface Tension, Percolation and Roughening, J. Stat. Phys. 29, 193–203 (1982).
Bricmont, J., J.L. Lebowitz, C.E. Pfister & E. Olivieri, Non-Translation Invariant Gibbs States with Coexisting Phases I, Commun. Math. Phys. 66, 1–20 (1979).
Bricmont, J., J.L. Lebowitz & C.E. Pfister, Non-Translation Invariant Gibbs States with Coexisting Phases II, Commun. Math. Phys. 66, 21–36 (1979).
Bricmont, J., J.L. Lebowitz & C.E. Pfister, Non-Translation Invariant Gibbs States with Coexisting Phases III, Commun. Math. Phys. 69, 267–291 (1979).
Bricmont, J., J.L. Lebowitz & C.E. Pfister, On the Surface Tension of Lattice Systems, Ann. Acad. Sci. New York 337, 214–223 (1980).
Burago, Yu.D. & V.A. Zalgaller, Geometric Inequalities, Grundiehren der Math. Wissenschaften, Springer, Berlin, 43 (1988).
Curie, P., Sur la Formation des Cristaux et sur les Constantes Capillaires de leurs Differentes Faces, Bull. Soc. Fr. Mineral, 8, 145–150 (1885). Reprinted in Schneer (1977).
DeConinck, J., F. Dunlop & V. Rivasseau, On the Microscopic Validity of the Wulff Construction and of the Generalized Young Equation, Commun. Math. Phys., 121, 401–419 (1989).
Dobrushin, R.L., Gibbs State Describing the Coexistence of Phases for a Three Dimensional Ising Model, Theory Probab. Appl., 17, 582–600 (1972).
Dobrushin, R.L., R. Koteckÿ & S.B. Shlosman, The Wulff Construction: a Global Shape from Local Interactions, American Mathematical Society, Providence (1992).
Dobrushin, R.L., R. Koteckÿ & S.B. Shlosman, A Microscopic Justification of the Wulff Construction, J. Stat. Phys., 72, 1–14 (1993).
Dobrushin, R.L. & O. Hryniv, Fluctuations of Shapes of Large Areas under Paths of Random Walks, Probab. Theory Relat. Fields, 105, 423–458 (1996).
Dobrushin, R.L. & S.B. Shlosman, The Problem of Translation Invariance of Gibbs Fields at Low Temperatures, Soy. Sci. Revs., Math. Phys. C, 5, 53–185 (1985).
Dobrushin, R.L. & S.B. Shlosman, Thermodynamic Inequalities and the Geometry of the Wulff Construction, in: Ideas and Methods in Mathematical Analysis, Stochastics and Applications, ed. by S. Albeverio et al., Cambridge University Press, Cambridge, (1992).
Fröhlich, J. & T. Spencer, The Kosterlitz-Thouless Transition in Two-Dimensional Abelian Spin Systems and the Coulomb Gas, Commun. Math. Phys., 81, 527–602 (1981).
Galgani, L., L. Manzoni & A. Scotti, Asymptotic Equivalence of Equilibrium Ensembles of Classical Statistical Mechanics, J. Math. Phys., 12, 933–935 (1971).
Gallavotti, G., The Phase Separation Line in the Two-Dimensional Ising Model, Commun. Math. Phys., 27, 103–136 (1972a).
Gallavotti, G., Instabilities and Phase Transition in the Ising Model. A Review, Riv. Nuov. Cim., 2, 133–169 (1972b).
Gallavotti, G., A. Martin-Lof, & S. Miracle-Sole, Some Problems Connected with the Description of Coexisting Phases at Low Temperatures in Ising Models, in: Mathematical Methods in Statistical Mechanics, ed. by Lenard, A., Lecture Notes in Physics, Springer, Berlin 20, 162–204 (1973).
and 343–524 (1978). Reprinted in The scientific papers of J. W. Gibbs, Longmans, Green & Co., London, 1, 55–353 (1906).
Goldschmidt, V., Atlas der Kristallformen, Universitätsverlag, Heidelberg (1913–1923).
Herring, C., Some Theorems on the Free Energy of a Crystal Surface, Phys. Rev., 82, 87–93 (1951).
Heyraud, J.C. & J.J. Métois, Establishment of the Equilibrium Shape of Metal Crystallites on a Foreign Substrate: Gold on Graphite, Jour. Cryst. Growth, 50, 571–574 (1980).
Heyraud, J.C. & J.J. Métois, Equilibrium Shape and Temperature: Lead on Graphite, Surf. Sci., 128, 334–342 (1983).
Holickÿ, P., R. Koteckÿ, & M. Zahradnik, Rigid Interfaces for Lattice Models at Low Temperature, J. Stat. Phys., 50, 755–812 (1988).
loffe, D., Large Deviations for the 2D Ising Model: A Lower Bound without Cluster Expansions, J. Stat. Phys., 74, 411–432 (1994).
loffe, D., Exact Large Deviations Bounds up to 71 for the Ising Model in Two Dimensions, Probab. Theory Relat. Fields, 102, 313–330 (1995).
Keshishev, K.O., A.Ya. Parshin, & A.V. Babkin, Crystallization Waves in He4, Soy. Phys. JETP, 53, 362–369 (1981).
Kotecky, R. Statistical Mechanics of Interfaces and Equilibrium Crystal Shapes. in: IX International Congress of Mathematical Physics, ed. by B. Simon Adam Hilger, Bristol, 148–163 (1989).
Kotecky, R. & S. Miracle-Sole, A Model with Roughening Transition at Low Temperatures, Phys. Rev. B, 34, 2049–2051 (1986).
Kotecky, R. & S. Miracle-Sole, Roughening Transition for the Ising Model on a Bcc Lattice. A Case in the Theory of Ground States. J. Stat. Phys., 47, 773–799 (1987a).
Kotecky, R. & S. Miracle-Sole, A Roughening Transition Indicated by the Behavior of Ground States, in: VIIIth International Congress on Mathematical Physics, ed. by M. Mebkhout, R. Seneor, World Scientific, Singapore, 331–337 (1987b).
Kotecky, R. & C.E. Pfister, Equilibrium Shapes of Crystals Attached to Walls, J. Stat. Phys., 76, 419–445 (1994).
Laue, M. von, Der Wulffsche Satz für die Gleichgewichtsform von Kristallen, Z. Kristallogr., 105, 124–133 (1943). Reprinted in Schneer (1977).
Lebowitz, J.L. & C.E. Pfister, Surface Tension and Phase Coexistence, Phys. Rev. Lett., 46, 1031–1033 (1981).
Messager, A., S. Miracle-Sole & S. Ruiz, Convexity Properties of the Surface Tension and Equilibrium Crystals, J. Stat. Phys., 67, 449–470 (1992).
Miracle-Sole, S., Theorems on Phase Transitions with a Treatment for the Ising Model, in: Critical Phenomena, ed. by J. Brey, R.B. Jones, Lecture Notes in Physics, Springer, Berlin. 80, 190–215 (1976).
Miracle-Sole, S., Surface Tension, Step Free Energy and Facets in the Equilibrium Crystal Shape, J. Stat. Phys., 79, 183–214 (1995a).
Miracle-Sole, S., On the Microscopic Theory of Phase Coexistence, in: 25 Years of Non-Equilibrium Statistical Mechanics, ed. by J.J. Brey, Springer, Lecture Notes in Physics 445, Berlin, 312–321 (1995b).
Miracle-Sole, S. & J. Ruiz, On the Wulff Construction as a Problem of Equivalence of Ensembles, in: On Three Levels: Micro, Meso and Macroscopic Approaches in Physics, ed. by M. Fannes, Plenum Press, New York, 295–302 (1994).
Pavlovska, A. & D. Nenov, Experimental Investigation of the Surface Melting of Equilibrium Form Faces of Diphenil, Surf. Sci., 27, 211–217 (1971).
Pavlovska, A. & D. Nenov, Les Surfaces Non-Singulières sur la Forme d’Équilibre du Naphtalène, Jour. Cryst. Growth, 12, 9–12 (1971).
Pavlovska, A. & D. Nenov, Experimental Study of the Surface Melting of Tetrabrommethane, Jour. Cryst. Growth, 39, 346–352 (1977).
Pfister, C.E., Large Deviations and Phase Separation in the Two-Dimensional Ising Model, Heiv. Phys. Acta, 64, 953–1054 (1991).
Rottman, C. & M. Wortis, Statistical Mechanics of Equilibrium Crystal Shapes: Interfacial Phase Diagrams and Phase Transitions, Phys. Rep., 103, 59–79 (1984).
Ruelle, D., Statistical Mechanics, Rigorous Results, Benjamin, New York (1969).
Schneer, C.J., ed., Morphological Basis for the Reticular Hypothesis Am. Mineralogist, 55, 1466–1488 (1970). Reprinted in Schneer (1977).
Schneer, C.J., ed., Crystal Form and Structure, Benchmark papers in Geology, 34, Dowden, Hutchinson & Ross, Stroudsbourg (Penn) (1977).
Taylor, J.E., Some Crystalline Variational Techniques and Results, Asterisque, 154–155, 307–320 (1987).
Weeks, J.D., G.H. Gilmer & H.J. Leamy, Structural Transition in the Ising Model Interface, Phys. Rev. Lett., 31, 549–551 (1973).
Winterbottom, W.L., Equilibrium Shape of a Small Particle in Contact with a Foreign Substrate, Acta Metallurgica, 15, 303–310 (1967).
Wolf, P.E., S. Balibar & F. Gallet, Experimental Observations of a Third Roughening Transition in hcp 4He Crystals, Phys. Rev. Lett., 51, 1366–1369 (1983).
Wulff, G., Zur Frage der Geschwindigheit des Wachstums und der Auflosung der Kristallflachen, Z. Kristallog., 34, 449–530 (1901). Reprinted in Schneer (1977).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Miracle-Sole, S. (1999). Interfaces. In: Goles, E., Martínez, S. (eds) Cellular Automata and Complex Systems. Nonlinear Phenomena and Complex Systems, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9223-9_5
Download citation
DOI: https://doi.org/10.1007/978-94-015-9223-9_5
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5154-7
Online ISBN: 978-94-015-9223-9
eBook Packages: Springer Book Archive