Van der Waals theory: Gibbs–Bogoliubov inequalityThe theoretical framework of the van der Waals theory of Baus et al. begins with the well-known Gibbs–Bogoliubov inequality which reads as follows. Consider a thermodynamic state at temperature T and density ρ for a physical system whose Helmholtz free energy and Hamiltonian are denoted by Ft and Ht, where the subscript t refers to a true system. Suppose at the same T and ρ, we can find a reference system (symbolized by a subscript r) whose Fr, Hr, and related quantities such as the structure, thermodynamic functions, etc. are known, the inequality gives Ft ≤ Fr + 〈Ht − Hr〉rwhich states that the true Ft of a physical system is bounded above by a reference Fr plus a correction term consisting of the difference in Hamiltonians of the true and reference systems. The 〈…〉r in eqn. (1) means an ensemble average taken over the distribution function of the reference system. Note that the perturbation in eqn. (1) is just the difference in potential energies Vt and Vr of the true and reference systems, respectively. Next, we write Vt = VtR + VtA splitting the pair potential into a repulsive part and an attractive part and similarly for the reference system, namely, VrR and VrA. In accord with the van der Waals theory, we take the reference system to be a system of hard spheres, VrR → VrHS, which thus implies VrA = 0 and approximate the VtR to be hard-sphere-like repulsive (R → HS) also. Setting VtR → VtHS = VrHS, simplifies eqn. (1) to where ρHS(r) is the one-particle density associated with the reference system VrHS. Note that in arriving at eqn. (2), we have neglected all two-particle correlations and set the lowest upper bound as a reasonable estimate.[34] The FHS in eqn. (2) is thus the hard sphere free energy which we will utilize below separately for the liquid and solid phases.
Helmholtz free energies of a liquid and a solidTo emphasize the theoretical consistency in the construction of free energies for the liquid and solid, which is essential for an accurate calculation of the phase diagram, we shall present these equations in parallel, stressing the analogy between them.
Since eqn. (2) involves only the attraction, it is convenient to write VtA(r) = −εϕA(r) where ε ≥ 0 is an energy parameter measuring the strength of attraction. Next, we define where the reduced distance x = r/σ, σ being the hard sphere diameter related to the volume fraction η by σ = (6η/πρ)1/3 and φ(x) ≥ 0 describing the range of interaction. For a liquid (solid) which is homogeneous (inhomogeneous), the one-particle density function is ρ → ρHS(r) = , = N/V being the uniform number density for the liquid (solid). Note that ρHS(r) = ∑Ni = 1ψ(r − Ri) where ψ(r − Ri), Ri being Bravais lattice sites in a perfect crystal, is the density profile satisfying the normalized condition ∫drψ(r) = 1. Subject to these choices, ΔF in eqn. (2) can be written in reduced energy units as Similarly, the hard-sphere reduced free energy fHS = FHS/(Nε) can be cast in the form within the spirit of the free-volume approximation. Here δ = /ρ0 (ρ/ρcp) for liquid (solid), T* = kBT/ε is the reduced temperature, ρ0 (ρcp = √2/σ3) is the maximum density of liquid (solid at compact crystal structure) for which a liquid (solid) is stable and Λ = h/(2πmkBT)1/2 is the thermal de Broglie wavelength of an hard sphere. We should stress at this point that, to ensure theoretical consistency in the construction of the free energy f = fHS + Δf, the cell model has been applied on ‘equal footing’ to both the liquid and solid phases. For example, we introduce, as is done in eqn. (5), the free volume (free edge) parameter α which is the fraction of the total volume V that is freely available to hard spheres in a liquid (solid) phase as In this work, we have set ρ0 = (6η0/πσ3)1/3 with an estimated η0 following the quantitative analysis of Baus et al.[31] Also, for the case of solid, δ = 1/x31, x1 = r1/σ being the reduced nearest-neighbor distance for a given crystal structure.
We are now in a position to present our derived expressions for the linear attractive potential shown schematically in Fig. 1. First, Γ for the liquid phase in eqn. (4) can be written as which reduces to the Γ of square-well and linear potentials for λ = 0 and γ = 0, respectively. For the solid phase, we have where in which In eqn. (10), n1 is the number of nearest neighbors and ζσ2 is the inverse width of the Gaussian density profile G(a,x) = ∫x0dta−1e−t is the incomplete gamma function and erf(x) = (2/√π)∫x0dte−t2 is the error function. Given eqns. (5), (8) and (9) for f, it is straightforward to calculate the pressure p = ερ2(∂f/∂ρ) and chemical potential μ = ε(∂/∂ρ)(ρf) from which the corresponding critical points (ηc,T*c) can be determined by ∂p(ρc,T*c)/∂ρ = 0 and ∂2p(ρc,T*c)/∂ρ2 = 0 yielding the critical parameters for the liquid–liquid transition. These predictions for the critical points are reminiscent of the celebrated van der Waals theory for the gas–liquid transition; the parameter Γ now plays the same role as the attraction parameter while η0 simulates the excluded volume factor (see, for example, eqn. (3.49) in .[ref. 35] In fact one can write down the same van der Waals equation of state if the parameters a and b appearing in eqn. (3.49) of [ref. 35] are replaced by εΓ/ρ0 and 1/ρ0, respectively). The critical points for the solid–solid phase transition, however, have to be determined numerically.