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Journal of Physics A Mathematical and General

The shape of the van der Waals loop and universal critical amplitude ratios

Michael Ellis Fisher, Shun-yong Zinn

Published Sep 18, 1998• 107 citations
Publisher

Abstract

Assuming that a van der Waals loop for a fluid, ferromagnet, etc, can be defined, its shape is analysed on the basis of universal critical amplitude ratios for three-dimensional Ising-type systems. Selected estimates for these ratios are tabulated. Near criticality, the loop attains a universal scaled form given, optimally, by a new parametric extended sine model for the bulk asymptotic equation of state; but a (particular) low-order interpolating Padé approximant provides a reasonable fit. Among other universal features, the reduced spinodal magnetization/density and conjugate field are found to be in contrast to the van der Waals values : thus the spinodal lies appreciably closer to the binodal (or coexistence curve) than predicted classically.

DOI
10.1088/0305-4470/31/37/002

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Assuming that a van der Waals loop for a fluid, ferromagnet, etc, can be defined, its shape is analysed on the basis of universal critical amplitude ratios for three-dimensional Ising-type systems. Selected estimates for these ratios are tabulated. Near criticality, the loop attains a universal scaled form given, optimally, by a new parametric extended sine model for the bulk asymptotic equation of state; but a (particular) low-order interpolating Padé approximant provides a reasonable fit. Among other universal features, the reduced spinodal magnetization/density and conjugate field are found to be in contrast to the van der Waals values : thus the spinodal lies appreciably closer to the binodal (or coexistence curve) than predicted classically.
Parametric scaling representations of the asymptotic equation of state, correlation lengths, and singular part of the Helmholtz free energy in the critical region of a system with a scalar order parameter are considered with the aim of fitting all 15 available independent universal amplitude ratios and describing naturally van der Waals loops. Defects in previous linear and cubic models and their extensions are described: they cannot represent analytically connected van der Waals loops or approximate satisfactorily all the known amplitude ratios. However, novel trigonometric parametric models for the free energy (and correlations) meet all the desiderata as demonstrated by the explicit numerical data presented.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.
We extend our previous determinations of nonasymptotic critical behavior of Phys. Rev. B 32, 7209 (1985) and 35, 3585 (1987) to accurate expressions of the complete classical-to-critical crossover (in three-dimensional field theory) in terms of the temperaturelike scaling field (i.e., along the critical isochore) for (1) the correlation length, the susceptibility, and the specific heat in the homogeneous phase for the n-vector model (n=1 to 3) and (2) the spontaneous magnetization (coexistence curve), the susceptibility, and the specific heat in the inhomogeneous phase for the Ising model (n=1). The present calculations include the seventh-loop order of Murray and Nickel and closely account for the up-to-date estimates of universal asymptotic critical quantities (exponents and amplitude combinations) provided by Guida and Zinn-Justin [J. Phys. A 31, 8103 (1998)].
Thermodynamics in the vicinity of a critical endpoint with nonclassical exponents α, β, γ, δ, … , is analyzed in terms of density variables (mole fractions, magnetizations, etc.). The shapes of the isothermal binodals or two-phase coexistence curves are found at and near the endpoint for symmetric and nonsymmetric situations. The spectator- (or noncritical-) phase binodal at T=Te is characterized by an exponent (δ+1)/δ (≃1.21) with leading corrections of relative order 1/δ (≃0.21), θ4/βδ (≃0.34) and 1−(βδ)−1 (≃0.36); in contrast to classical (van der Waals, mean field, etc.) theory, the critical endpoint binodal is singular with a leading exponent (1−α)/β (≃2.73) and corrections which are elucidated; the remaining, λ-line binodals also display the “renormalized exponent,” (1−α)/β but with more singular corrections. [The numerical values quoted here pertain to (d=3)-dimensional-fluid or Ising-type systems.]
Thermodynamics in the vicinity of a critical endpoint with nonclassical exponents α, β, γ, δ, … , is analyzed in terms of density variables (mole fractions, magnetizations, etc.). The shapes of the isothermal binodals or two-phase coexistence curves are found at and near the endpoint for symmetric and nonsymmetric situations. The spectator- (or noncritical-) phase binodal at T=Te is characterized by an exponent (δ+1)/δ (≃1.21) with leading corrections of relative order 1/δ (≃0.21), θ4/βδ (≃0.34) and 1−(βδ)−1 (≃0.36); in contrast to classical (van der Waals, mean field, etc.) theory, the critical endpoint binodal is singular with a leading exponent (1−α)/β (≃2.73) and corrections which are elucidated; the remaining, λ-line binodals also display the “renormalized exponent,” (1−α)/β but with more singular corrections. [The numerical values quoted here pertain to (d=3)-dimensional-fluid or Ising-type systems.]
We address the problem of determining the critical equation of state of three-dimensional XY\mathrm{XY}XY systems. For this purpose we first consider the small-field expansion of the effective potential (Helmholtz free energy) in the high-temperature phase. We compute the first few nontrivial zero-momentum n-point renormalized couplings, which parametrize such expansion, by analyzing the high-temperature expansion of an improved lattice Hamiltonian with suppressed leading scaling corrections. These results are then used to construct parametric representations of the critical equation of state which are valid in the whole critical regime, satisfy the correct analytic properties (Griffith's analyticity), and take into account the Goldstone singularities at the coexistence curve. A systematic approximation scheme is introduced, which is limited essentially by the number of known terms in the small-field expansion of the effective potential. From our approximate representations of the equation of state, we derive estimates of universal ratios of amplitudes. For the specific-heat amplitude ratio we obtain ${A}^{+}{/A}^{\ensuremath{-}}=1.055(3),$ to be compared with the best experimental estimate ${A}^{+}{/A}^{\ensuremath{-}}=1.054(1).$