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Phase Transitions Due to Softness of the Potential Core

J. Chem. Phys. 56, 4274 (1972); doi:10.1063/1.1677857

Issue Date: 1 May 1972

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G. Stell
Department of Mechanics, State University of New York, Stony Brook, New York 11790

P. C. Hemmer
Institute of Theoretical Physics, NTH, Trondheim, Norway
This paper extends a previous demonstration [Hemmer and Stell, Phys. Rev. Letters 24, 1284 (1970)] showing that for a system in which the pair potential has a hard core plus a negative part, softening the hard core can produce a second transition if a first already exists. Detailed analytic and numerical results for one-dimensional continuum fluids are given, and our discussion of the lattice gas is further developed. In particular interactions that are repulsive over next-nearest neighbor cells as well as nearest-neighbor cells are considered, and it is rigorously shown that as many as four first-order phase transitions can occur for such potentials, even in one dimension. The relevance of our work to certain features found in real systems (e.g. the possible breakdown of the law of rectilinear diameters, and isostructural solid-solid transitions) is also discussed, as is the novel critical behavior to be expected of certain two- and three-dimensional lattice systems. ©1972 The American Institute of Physics
History: Received 22 January 1971
Permalink: http://link.aip.org/link/?JCPSA6/56/4274/1
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0021-9606 (print)   1089-7690 (online)
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REFERENCES (26)
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  1. P. C. Hemmer and G. Stell, Phys. Rev. Letters 24, 1284 (1970).
  2. B. Widom, J. Chem. Phys. 43, 3098 (1965).
  3. J. L. Lebowitz and O. Penrose, J. Math. Phys. 7, 98 (1966).
  4. The thermodynamics for the potential (2) was first considered by T. Nagamiya, Proc. Phys. Math. Soc. Japan 22, 1033 (1940).
    5. For more general one-dimensional treatments see, for example, the articles by H. Takahasi and L. Van Hove, in Mathematical Physics in One Dimension, edited by E. H. Lieb and D. C. Mattis (Academic, New York, 1966), as well as further references cited in that volume.
  5. One has to solve an integral equation. See the article by Van Hove cited in footnote 4 as well as J. A. Barker, Australian J. Phys., 15, 127 (1962).
  6. This is equivalent to introducing n-body forces (n being the largest integer less than lambda+2). For 1<lambda<2 the three-body force shields completely the forces otherwise found between next-nearest-neighbor particles.
  7. We use the commonly accepted notation for critical exponents: delta refers to the shape of the critical isotherm, beta to the shape of the coexistence curve, gamma to the temperature dependence of the isothermal compressibility KT, and alpha to the temperature dependence of the specific heat CV. See M. E. Fisher, Rept. Progr. Phys. 30, 615 (1967).
  8. There is a good general description in J. Frenkel, Kinetic Theory of Liquids (Dover, New York, 1955), Chap. II, Sec. 3.
  9. H. D. Baehr, Forsch. Ingenieurw. 29, 143 (1963)
    10. and Brennstoff-Wärme-Kraft 15, 514 (1963).
    J. Rowlinson, Liquids and Liquid Mixtures (Butterworths, London, 1969).
    J. M. H. Levelt Sengers, Ind. Eng. Chem. Fundamentals 9, 470 (1970).
  10. J. Rowlinson and B. Widom, J. Chem. Phys. 52, 1670 (1970).
    11. The current experimental picture regarding the possible breakdown of the law of rectilinear diameters is carefully analyzed in J. M. H. Levelt Sengers, J. Straub, and M. Vicentini-Missoni, J. Chem. Phys. 54, 5034 (1971).
  11. Defining a higher-order critical state by the requirement that four isothermal density derivatives vanish, one may, following Gibbs, calculate the degree of freedom f of this critical state in a n-component system. Subtraction of the number of equations (four) from the number of independent state variables (n+l) yields f = n−3. Hence one expects a higher-order critical point in a three-component system, a critical line in a four-component system, etc. The fact that such a higher-order point occurs in our one-component system may be ascribed to the fact that we really are considering an ensemble of systems with two dimensionless potential parameters lambda, C that come in addition to the two state variables. One might therefore be tempted to argue that higher-order critical points do not occur in a real one-component system. However, as vigorously stressed by C. Truesdell [The Elements of Continuum Mechanics (Springer, Berlin, 1966), p. 117], this type of argument should be taken cum grano salis. For the standard two-dimensional Ising models, for instance, the first 14 density derivatives vanish at the critical point!
  12. When the hard core extends to m neighbors, one shows easily that the one-phase equation of state is given by p = kTln[1+rho(1−mrho)−1]−arho2, from which the critical parameters follows easily. In particular, rhoc = m−1[2−m−1+(1−m−1+m−2)1/2]−1.
  13. W. K. Theumann and J. S. Høye, J. Chem. Phys. 55, 4159 (1971), have investigated the corresponding spin system.
  14. The lattice gas on a two-dimensional square lattice with an interaction that is infinite between nearest neighbors, otherwise finite and nonnegative, has a transition. A rigorous proof has been given by R. L. Dobruskin [Theoret. Prob. Appl. 13, 201 (1968);
    15. Funct. Anal. Appl. 2, 31, 41 (1968)].
    Other systems in which the existence of transitions can be proven are the three-dimensional spherical model [R. M. Mazo, J. Chem. Phys. 39, 2196 (1963)],
    and the two-dimensional infinite repulsion model of R. Baxter [J. Math. Phys. 11, 3116 (1960)].The literature on approximate methods for studying such transitions is large; in recent years two fruitful methods have been the series-extrapolation method and the transfer-matrix method. We refer to both methods below.
  15. A convenient review of the properties of the hard-sphere fluid is given by H. L. Frisch, Advan. Chem. Phys. 6, 229 (1964).
  16. Pertinent references include D. S. Gaunt and M. E. Fisher, J. Chem. Phys. 43, 2840 (1965);
    17. L. K. Runnels and L. L. Combs, ibid. 45, 2462 (1966);
    F. H. Ree and D. A. Chestnut, ibid. 45, 3983 (1966);
    and N. Karayinis, C. A. Morrison, and D. E. Wortman, J. Math. Phys. 7, 1458 (1966)
    for y = 2 and D. S. Gaunt, J. Chem. Phys. 46, 3237 (1967) for = 3.
  17. See article by D. S. Gaunt, Ref. 16.
  18. L. K. Runnels, J. P. Solvant, and H. R. Streiffer, J. Chem. Phys. 52, 2352 (1970).
  19. L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon, London 1958) p. 260.
  20. J. F. Nagle, Phys. Rev. A 2, 2124 (1970). We are grateful to Dr. Nagle for receiving preprints before publication.
  21. R. B. Griffiths, Phys. Rev. Letters 24, 1479 (1970)
    22. and, in Critical Phenomena in Alloys, Magnets and Superconductors, edited by R. E. Mills, E. Aschev, and R. I. Jaffee (McGraw-Hill, New York, 1970).
  22. C. Domb and D. L. Hunter, Proc. Phys. Soc. (London) 86, 1147 (1965).
  23. However, Domb (preprint) has concluded that in the Ising model one should expect M1/betaTheta rather than M2Theta in (52), which makes (53) and (54) seem less likely in the fluid case. Moreover, N. D. Mermin [Phys. Rev. Letters 26, 169 (1971)]has recently discussed an exactly soluble model in which 1−alpha[prime] rather than 2beta appears in (56). Mermin's result lends support to the conjectures in Refs. 1 and 10 that when 1−alpha[prime] differs from 2beta, the diameter of the coexistence curve behaves like (Tc−T)1−alpha[prime] near the critical point.
  24. See, for example, A. Jayaraman, Phys. Rev. 137, A179 (1965).
  25. A. W. Lawson and T. Y. Tang, Phys. Rev. 76, 301 (1949);
    26. M. K. Wilkinson, ibid. 122, 1409 (1961).
  26. N. W. Ashcroft and J. Lekner, Phys. Rev. 145, 83 (1966).

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