The hard-hexagon model (particles with nearest-neighbor exclusion on the triangular lattice) is exactly solved, and Strobl and Harrington (2013) combined the exact activity a(θ)a(\theta) with the exact combinatorial identity ΩN+1=Tav/(N+1)\Omega_{N+1} = T_{\mathrm{av}}/(N+1) to obtain a closed piecewise-algebraic expression for the fraction θav\theta_{\mathrm{av}} of available (adsorbable) sites as a function of the occupied fraction θ\theta. One issue was explicitly left open: the paper reports a small non-monotonicity of this expression near the phase transition at θc=(5−5)/10≈0.27639\theta_c = (5-\sqrt{5})/10 \approx 0.27639 (an increase of about 0.0002 between θ=0.2734\theta = 0.2734 and θ=0.2792\theta = 0.2792), an effect far below the resolution of its 33×33 Monte Carlo (agreement ±0.008) and 6×6 exact enumeration (agreement ±0.02). Because the analytical expression is exact in the thermodynamic limit (it is θ/a(θ)\theta/a(\theta) with Joyce's exact activity), the paper's note that this 'warrants further investigation' poses an analytical question: to determine the fine structure that the exact function actually implies. This problem asks for a completion of the available-site characterization: an exact analysis of the fine structure of θav(θ)\theta_{\mathrm{av}}(\theta) — its monotonicity intervals, local extrema, and one-sided critical slopes.

Background

The hard-hexagon model — hard particles with nearest-neighbor exclusion on the triangular lattice (coordination number 6, maximum coverage 1/3) — was exactly solved by Baxter (1980, 1981), who computed the partition function per site κ(z)\kappa(z) (the grand-canonical free energy per site, with zz the activity, i.e. the fugacity) and the critical activity zc=(11+55)/2≈11.09z_c = (11+5\sqrt{5})/2 \approx 11.09; the transition is second order, in the universality class of the three-state Potts model (the q=3q = 3 generalization of the Ising model), with specific-heat exponent α=1/3\alpha = 1/3. Joyce (1988) recast the solution in algebraic form, giving the activity a(θ)=za(\theta) = z as an explicit function of the coverage θ\theta, separately below and above θc\theta_c, with the coverage-related singularity governed by α\alpha. Strobl and Harrington (2013), motivated by the need for analytic lattice-gas models of electrochemical adsorption and voltammetry (current–potential curves recorded during adsorption and desorption, where mean-field Frumkin-type isotherms are known to misplace phase transitions), studied available sites: unoccupied sites with no occupied nearest neighbor, i.e. sites at which an additional particle could be placed — available is not the same as merely unoccupied, since an occupied site has no occupied nearest neighbor but cannot accept another particle. Using the exact identity ΩN+1=Tav/(N+1)\Omega_{N+1} = T_{\mathrm{av}}/(N+1) relating the number ΩN+1\Omega_{N+1} of (N+1)(N+1)-particle configurations to the number TavT_{\mathrm{av}} of particle-available-site pairs, they proved a(θ)=θ/θava(\theta) = \theta/\theta_{\mathrm{av}} exactly (its combinatorial content verified to zero residual for all N≤12N \le 12 on the 6×6 system), and thence obtained θav(θ)\theta_{\mathrm{av}}(\theta) in closed piecewise-algebraic form from Joyce's activity, e.g. θav(θc)=(95−20)/5≈0.02492\theta_{\mathrm{av}}(\theta_c) = (9\sqrt{5}-20)/5 \approx 0.02492. The exact expression is internally consistent: (95−20)/5=θc/zc(9\sqrt{5}-20)/5 = \theta_c/z_c identically. Their mixed-adsorption application combines the hexagons with a second, non-interacting Langmuir species and replaces the distribution of the number of available sites by its mean (the delta-function or self-averaging approximation), under which the mixed-system equations become exact only in the limit where that distribution is a delta function. The activity–available-site relation has since been applied in electrochemistry (Kibler and Al-Shakran 2016; Rinaldo et al. 2014), and further exact-solution results for the hard-hexagon partition function appear in Assis et al. (2013).

Problem Statement

Complete the exact characterization of the fine structure of the available-site fraction θav(θ)\theta_{\mathrm{av}}(\theta) in the hard-hexagon model. Using the exact identity θav=θ/a(θ)\theta_{\mathrm{av}} = \theta/a(\theta) with the Baxter–Joyce activity, determine rigorously (i) the monotonicity intervals of θav\theta_{\mathrm{av}}, (ii) the locations and values of any local extrema, and (iii) the one-sided derivatives of θav\theta_{\mathrm{av}} at θ=0\theta = 0, θc=(5−5)/10\theta_c = (5-\sqrt{5})/10, and θ=1/3\theta = 1/3; all three parts are required.

References

  1. J. R. Strobl and D. A. Harrington, 'The role of available sites in the activity of lattice gases with geometric constraints', The Journal of Chemical Physics 139, 104104 (2013). DOI 10.1063/1.4820400, https://doi.org/10.1063/1.4820400. Open accepted-manuscript copy with supplementary material: UVic DSpace, http://hdl.handle.net/1828/10867.
  2. R. J. Baxter, 'Hard hexagons: exact solution', Journal of Physics A: Mathematical and General 13, L61–L70 (1980). DOI 10.1088/0305-4470/13/3/007, https://doi.org/10.1088/0305-4470/13/3/007.
  3. R. J. Baxter, 'Rogers-Ramanujan identities in the hard hexagon model', Journal of Statistical Physics 26, 427–452 (1981). DOI 10.1007/BF01011427, https://doi.org/10.1007/BF01011427. (Companion to the 1980 solution; both are cited as refs. 11 and 12 of Strobl–Harrington. The solution is also presented in book form in R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London (1982), ISBN 0-12-083180-5, Chapter 14.)
  4. G. S. Joyce, 'Exact results for the activity and isothermal compressibility of the hard-hexagon model', Journal of Physics A: Mathematical and General 21, L983–L988 (1988). DOI 10.1088/0305-4470/21/20/005, https://doi.org/10.1088/0305-4470/21/20/005.
  5. G. S. Joyce, 'On the hard hexagon model and the theory of modular functions', Philosophical Transactions of the Royal Society of London A 325, 643–702 (1988). DOI 10.1098/rsta.1988.0077, https://doi.org/10.1098/rsta.1988.0077.
  6. B. Widom, 'Some topics in the theory of fluids', The Journal of Chemical Physics 39, 2808–2812 (1963). DOI 10.1063/1.1734110, https://doi.org/10.1063/1.1734110.
  7. M. Assis, J. L. Jacobsen, I. Jensen, J.-M. Maillard, B. M. McCoy, 'The hard hexagon partition function for complex fugacity', Journal of Physics A: Mathematical and Theoretical 46, 445202 (2013). DOI 10.1088/1751-8113/46/44/445202, https://doi.org/10.1088/1751-8113/46/44/445202; arXiv:1306.6389, https://arxiv.org/abs/1306.6389.
  8. L. A. Kibler and M. Al-Shakran, 'Adsorption of Formate on Au(111) in Acid Solution: Relevance for Electro-Oxidation of Formic Acid', The Journal of Physical Chemistry C 120, 16238–16245 (2016). DOI 10.1021/acs.jpcc.6b02044, https://doi.org/10.1021/acs.jpcc.6b02044.
  9. S. G. Rinaldo, W. Lee, J. Stumper, M. Eikerling, 'Mechanistic Principles of Platinum Oxide Formation and Reduction', Electrocatalysis 5, 262–272 (2014). DOI 10.1007/s12678-014-0189-y, https://doi.org/10.1007/s12678-014-0189-y.