J. Phys. Chem. B 2004, 108, 8447-8451
8447
Solid-Liquid Phase Transition of the Hard-Core Attractive Yukawa System and Its Colloidal Implication Shiqi Zhou* Research Institute of Modern Statistical Mechanics, Zhuzhou Institute of Technology, Wenhua Road, Zhuzhou City, 412008, P. R. China ReceiVed: February 1, 2004; In Final Form: April 15, 2004
Both a free volume approach for Helmholtz free energy and theoretically based fitted formulas for a radial distribution function of hard-sphere solid are combined to describe the Helmholtz free energy of a solid phase with the hard-core attractive Yukawa (HCAY) potential in the framework of first-order thermodynamic perturbation theory. The corresponding Helmholtz free energy of the fluid phase is calculated from a recently proposed analytical expression based on the inverse temperature expansion and mean spherical approximation. The predicted solid-liquid phase transition for the HCAY system is in far better agreement with simulation than a previous perturbation weighted density approximation. The present formalism is employed as a predictor of protein solidification and metastable liquid-liquid coexistence of crystallizing protein solution, in satisfactory agreement with the available experimental data.
I. Introduction The study of systems consisting of particles with hard-core and short-range attractive tails has received much attention in recent years.1 Potentials with hard-core repulsions and attractive tails represent a large area of practical interest, such as rare gases, proteins, colloids, nanoparticles, fullerenes, and many others. To investigate such matters theoretically, several model potentials2 are proposed, such as Lennard-Jones (LJ) potential, adhesive hard-sphere (AH) potential, square well (SW) potential, hard-core attractive Yukawa (HCAY) potential, etc. Among these model potentials, the SW potential is not continuous in the attractive part, in disagreement with the real potentials, and the AH potential is a special case of SW potential with the well width infinitely small but the well depth still finite. The system of SW potential has analytical solution,3 but only applicable to a small range of potential parameters. The AH system has an analytical solution under the Percus-Yevick approximation;4 however, the AH potential as a representation of a realistic hardcore repulsion plus short-ranged attraction interaction is limited in the fact that it is unable to distinguish two situations with the same adhesiveness and densities, but different interaction strengths and/or ranges, and the system of monodisperse AH potential is also not thermodynamically stable.5 The LJ system has no analytical solution for its structural functions, and this potential also has only a potential parameter relevant to the interaction magnitude; therefore it mainly describes rare gases. The unique potential removing all of the above limitations is the HCAY potential; it has two potential parameters, one () for interaction magnitude and the other (κ) for interaction range. Therefore the HCAY potential can represent various potentials, for example, the LJ potential, a potential dictating a onecomponent plasma system, and short-range potentials dictating, for example, proteins and colloidal systems by adjusting the two potential parameters. Especially the analytical mean spherical approximation (MSA) solution for the HCAY potential has been improved upon by the inverse temperature expansion (ITE) * Corresponding author. E-mail:
[email protected].
technique,6 and the resultant Helmholtz free energy and radial distribution function are in very good agreement with the corresponding simulation data.6 So it will be very convenient and profitable to employ the HCAY potential for investigating areas of practical interest. The main aim of the present paper is to propose a formulation of solid-liquid phase transition of the HCAY potential system and then combine the formalism and the HCAY model potential to describe systems of practical interest. The solid-liquid transition of a system with the HCAY potential or other potentials is investigated theoretically by different authors. One route7 is based on the modified weighted density approximation (MWDA)8 for hard-sphere (HS) freezing. It is well-known that the MWDA is only accurately applicable to the HS fluid and hard-core part of the non-hard-sphere fluid.9 One example is the almost inverse relationship9 between the success of a WDAbased method in describing a hard-sphere system and how generally applicable it is to other interactions with attractive parts. For example, the WDA of Curtin and Ashcroft10 gives very good results for a hard-sphere system and systems with purely repulsive interactions that freeze into close-packed crystals, but the full theory fails when attractive forces are introduced, as in the LJ system.11 Another example is the MWDA, which yields results for the liquid to fcc solid transition for inverse-power and point Yukawa interaction systems that represent a significant improvement over those of the earlier second-order perturbation density functional theory (DFT)12 for freezing. However, it fails to predict any liquid to bcc solid transition, even under conditions where the simulations indicate that this should be the equilibrium solid structure. Especially, one of the most accurate theories for hard-sphere freezing, the generalized effective liquid approximation (GELA),13 gives results almost identical with the simulation data for a hard-sphere system but fails for any other interaction potential.9 Another problem for the case of hard-core + short-range attractive potential is that the melting density becomes very high when the temperature becomes lower, which makes the solid structure highly localized, and thus the calculation of the MWDA
10.1021/jp0495500 CCC: $27.50 © 2004 American Chemical Society Published on Web 05/25/2004
8448 J. Phys. Chem. B, Vol. 108, No. 24, 2004
Zhou
becomes very heavy computationally. Another route is the perturbation WDA due to Mederos et al.,14 which is based on dividing the interaction potential into a repulsive part and an attractive part and then treating the former by WDA and the latter perturbationally; it gives a description of the correlation in the crystal based on the local-compressibility relation. The perturbational WDA reduces to the standard perturbation theory of simple fluids in the uniform density limit. The third route15 is the usual perturbation theory, which extends the uniform fluid first-order thermodynamic perturbation theory (TPT) to uniform solids. Although the fluid and solid are treated on equal footing, TPT is not equally valid for both the solid and fluid phase. For a system with interaction potential βuλ(r) with λ ) 1
βuλ(r) ) ∝
r
