Comment on "Exact results for the lower critical solution in the asymmetric model of an interacting binary mixture"

VOLUME 79, NUMBER 18 PHYSICAL REVIEW LETTERS 3NOVEMBER 1997 Comment on “Exact Results for the Lower Critical Solution in the Asymmetric Model of an Interacting Binary Mixture” In a recent Letter, Lin and Taylor (L-T) [1] reported exact results for a square lattice gas model of an interacting binary mixture. In this model, each cell of the lattice can be occupied by a square (particle A) or by four triangles (particles B). The nearest-neighbor coupling between the Aparticles is eAA, while the coupling between Aand Bparticles, eAB is introduced for each edge contact between the squares and triangles. This model can be considered as a generalization of the “venerable” decorated lattice model introduced three decades ago by Widom [2]. With an appropriate transformation, the partition function of this model maps onto the partition function of the twodimensional Ising model. L-T found that the necessary condition for the occurrence of a lower critical point (LCP) in addition to the upper critical point (UCP) is that not only eAA ,0 and eAB ,0but also the ratio s2eAAys2eAB 2e AAd must exceed 1. The aim of this Comment is to show that it is possible to obtain LCP for eAA ,0, eAB , 0, and eAA $2eAB, which implies s.1and s,0. All the lines of critical points for eAA ,0end at the same point. This point corresponds to the critical pressure peand critical temperature of the pure Asystem. In Fig. 1, the dashed line corresponds to the coexistence line of the Apure system. The zone between this line and the line of critical points (continuous line) is where the coexistence of the mixture occurs. FIG. 1. Lines of critical points (continuous lines) (a) eAB  0.45eAAss.1d,(b)e AB 0.49eAAss.1d, (c) eAB 0.6eAA, (d) eAB eAA, and (e) eAB 2`. The reduced units are defined as pp 22psyeAA and Tp 22kBTyeAA where s is the area of the lattice cell. FIG. 2. Coexistence curves at constant pressure for the case (d) of Fig. 1. Continuous line p,pe, dotted line p1.pe, and dot-dashed line p2.p1.pe. The crosses correspond to the location of critical points. For values of s,0, the line of critical points can present a nonmonotonic behavior with a maximum and then the system exhibits a LCP and an UCP for values of the pressure between this maximum and pe. For values smaller than pe, the UCP disappears and the system shows a coexistence with LCP and ending in the pure system (see Fig. 2). Decreasing eAB, the line of critical points becomes monotonic and consequently, the UCP does not appear and the system shows a single LCP for p,pe. The limit eAB2` presents a straight line of critical points. One of the authors (L.F.R.) has benefited from a stimulating correspondence with Professor Michael E. Fisher. José M. Romero-Enrique, Inmaculada Rodrı ´guez-Ponce, and Luis F. Rull Departamento de Fı ´sica Atómica, Molecular y Nuclear Area de Fı ´sica Teórica, Universidad de Sevilla Aptdo 1065, Sevilla, 41080, Spain Umberto Marini Bettolo Marconi Dipartimento di Matematica e Fisica and INFM via Madonna delle Carceri, Università di Camerino 62032 Camerino, Italy Received 16 June 1997 [S0031-9007(97)04401-3] PACS numbers: 61.25.Hq, 05.50.+q, 64.75.+g, 82.60.Lf [1] J.C. Lin and P.L. Taylor, Phys. Rev. Lett. 73, 2863 (1994). [2] B. Widom J. Chem. Phys. 46, 3324 (1967). 0031-9007y97y79(18)y3543(1)$10.00 © 1997 The American Physical Society 3543