Abstract
Phase separation is induced in the one-dimensional Ising chain (or lattice-gas model of a fluid) by means of an external field that changes sign in the middle of the chain. The magnetization profile (or density profile of the analogous fluid) is obtained analytically. It is found to decay exponentially rapidly to the bulk-phase magnetizations (or densities), the exponential decay parameter being the correlation length in the bulk phases in the presence of the field. This is in accord with earlier theoretical ideas. The interfacial tension is also obtained analytically. In an appropriately defined limit of large neighboring-site spin-spin interactions and small external field the interface becomes infinitely broad while the amplitude of the profile and the interfacial tension both vanish, in close imitation of the approach to a critical point in a real fluid. In this asymptotic limit the interfacial tension is related to the amplitude of the profile in the way that is predicted by earlier theories of interfaces near critical points, with critical-point exponents now those appropriate to one dimension. The exact interfacial profile and tension are used to test several approximations, including a corrected form of the “barometric law” and local (square-gradient) and nonlocal forms of the van der Waals theory.
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References
M. Robert,Helv. Phys. Acta 53:319 (1978), and Ph.D. thesis, École Polytechnique Fédérale de Lausanne (Suisse) 1980.
D. B. Abraham and M. Robert,J. Phys. A 12:L129 (1979),13:2229 (1980).
D. B. Clayton and G. W. Woodbury, Jr.,J. Chem. Phys. 55:3895 (1971).
A. Compagner,Phys. Lett. 21:627 (1966).
J. K. Percus,J. Stat. Phys. 15:505 (1976),28:67 (1982);
A. Robledo,J. Chem. Phys. 72:1701 (1980);
A. Robledo and C. Varea,J. Stat. Phys. 26:513 (1981).
J. M. J. van Leeuwen and H. J. Hilhorst,Physica 107A:319 (1981).
C. Ebner, M. A. Lee, and W. F. Saam,Phys. Rev. A 21:959 (1980); C. Ebner and M. A. Lee,Phys. Rev. A 25:2721 (1982).
W. Kinzel,J. Phys. A 15:L413 (1982); one of us (B.W.) is grateful to P. Nightingale for calling this paper to his attention.
B. McCoy and T. T. Wu,The Two-Dimensional Ising Model (Harvard University Press, Cambridge, Massachusetts, 1973);
E. Ising,Z. Phys. 31:253 (1925).
J. S. Rowlinson and B. Widom,Molecular Theory of Capillarity (Oxford University Press, Oxford, 1982).
M. E. Fisher,J. Math. Phys. 5:944 (1964).
D. R. Nelson and M. E. Fisher,Ann. Phys. (N.Y.) 91:226 (1975).
K. Binder,Phys. Rev. A 25:1699 (1982).
B. Widom,J. Stat. Phys. 19:563 (1978).
B. Widom,J. Chem. Phys. 39:2808 (1963); J. L. Jackson and L. S. Klein,Phys. Fluids 7:279 (1964).
R. B. Griffiths,J. Math. Phys. 8:484 (1967), Section VII. One of us (M.R.) thanks C. Pfister for this observation.
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Robert, M., Widom, B. Field-induced phase separation in one dimension. J Stat Phys 37, 419–437 (1984). https://doi.org/10.1007/BF01011842
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DOI: https://doi.org/10.1007/BF01011842