Abstract
We discuss equilibrium shapes of crystals attached to walls. Optimal shapes for different configurations of walls are found and the minimality of the overall surface tension is proven with the help of a simple geometrical argument based on the isoperimetric inequality and monotonicity. Stability results in the form of Bonnesen inequalities are obtained in the two-dimensional case.
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References
P. Curie, Sur la formation des crystaux et les constantes de capillarité de leur differente phase.Bull. Soc. Fr. Minéral.8:145–150 (1885).
J. W. Cahn, Critical point wetting,J. Chem. Phys. 66:3667–3672 (1977).
J. De Coninck, J. Fruttero, and A. Ziermann, Non-typical Wulff shapes in a corner: A microscopic derivation; a 2D Ising drop in a corner, Mons preprint (1993).
J. De Coninck, A. Messager, and A. Ziermann, Double Wulff construction for asymmetric media, Mons preprint (1993).
A. Dinghas, Über einen geometrischen Satz von Wulff für die Gleichgewichtsform von Kristallen,Z. Kristallog. 105:304–314 (1944).
B. Dacorogna and C.-E. Pfister, Wulff theorem and best constant in Sobolev inequality,J. Math. Pures Appl. 71:97–118 (1992).
R. L. Dobrushin and S. Shlosman, Thermodynamic inequalities for the surface tension and the geometry of the Wulff construction, inIdeas and Methods in Mathematical Analysis, Stochastics, and Applications, Vol. 2, S. Albeverio, J. F. Fenstad, H. Holden, and T. Lindstrom, eds. (Cambridge University Press, Cambridge, 1992).
R. L. Dobrushin, R. Kotecký, and S. Shlosman,The Wulff Construction: A Global Shape from Local Interactions (AMS, Providence, Rhode Island, 1992).
H. Federer,Geometric Measure Theory (Springer, Berlin, 1969).
I. Fonseca, The Wulff theorem revisited,Proc. R. Soc. Lond. A 432:125–145 (1991).
J. Fröhlich and C.-E. Pfister, The wetting and layering transitions in the half-infinite Ising model,Europhys. Lett. 3:845–852 (1987).
J. Fröhlich and C.-E. Pfister, Semi-infinite Ising model. I. Thermodynamic functions and phase diagram in absence of magnetic field,Commun. Math. Phys. 109:493–523 (1987).
J. Fröhlich and C.-E. Pfister, Semi-infinite Ising model. II. The wetting and layering transitions,Commun. Math. Phys. 112:51–74 (1987).
J. W. Gibbs, On the equilibrium of heterogeneous substances, inThe Scientific Papers of J. Williard Gibbs, Vol. 1,Thermodynamics (Longmans, Green & Co. 1906), pp. 315–326.
C. Herring, Some theorems on the free energies of crystal surfaces,Phys. Rev. 82:87–93 (1951).
A. Messager, S. Miracle-Sole, and J. Ruiz, Convexity properties of the surface tension and equilibrium crystal,J. Stat. Phys. 67:449–470 (1992).
C. E. Pfister, Large deviations and phase separation in the two-dimensional Ising model,Helv. Phys. Acta 64:953–1054 (1991).
C. Rottman and M. Wortis, Statistical mechanics of equilibrium crystal shapes: Interfacial phase diagrams and phase transitions,Phys. Rep. 103:59–79 (1984).
J. E. Taylor, Some crystalline variational techniques and results,Astérisque,154–155:307–320 (1987).
G. Wulff, Zur Frage der Geschwindigkeit des Wachstums und der Auflösung der Krystallflachen,Z. Krystallog. Mineral. 34:449 (1901).
W. L. Winterbottom, Equilibrium shape of a small particle in contact with a foreign substrate,Acta Metallurgica 15:303–310 (1967).
R. K. P. Zia, Anisotropic surface tension and equilibrium crystal shapes, inProgress in Statistical Mechanics, C. K. Hu, ed. (World Scientific, Singapore, 1988), pp. 303–357.
A. Ziermann, Partial wetting in a three-phase system with a wall (1993), to appear; Interfaces and grain boundaries, finite size effects, refraction and global shapes (1992), Ph.D. Thesis, preprint CTS-92-06.
R. K. P. Zia, J. E. Avron, and J. Taylor, The summertop construction: Crystals in a corner,J. Stat. Phys. 50:727–736 (1988).
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Kotecký, R., Pfister, C.E. Equilibrium shapes of crystals attached to walls. J Stat Phys 76, 419–445 (1994). https://doi.org/10.1007/BF02188669
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DOI: https://doi.org/10.1007/BF02188669