Fluid coexistence close to criticality: scaling algorithms for precise simulation

A novel algorithm is presented that yields precise estimates of coexisting liquid and gas densities, ρ(T ), from grand canonical Monte Carlo simulations of model fluids near criticality. The algorithm utilizes data for the isothermal minima of the moment ratio QL(T ; 〈ρ〉L) ≡ 〈m2〉2L/〈m 〉L in L× · · ·×L boxes, where m = ρ − 〈ρ〉L. When L→∞ the minima, Q±m(T ;L), tend to zero while their locations, ρ ± m(T ;L), approach ρ (T ) and ρ(T ). Finite-size scaling relates the ratio Y=(ρ+m−ρ − m)/∆ρ∞(T ) universally to 1 2 (Q+m+Q − m), where ∆ρ∞ = ρ (T )−ρ(T ) is the desired width of the coexistence curve. Utilizing the exact limiting (L→∞) form, the corresponding scaling function can be generated in recursive steps by fitting overlapping data for three or more box sizes, L1, L2, · · ·, Ln. Starting at a T0 sufficiently far below Tc and suitably choosing intervals ∆Tj =Tj+1 − Tj > 0 yields ∆ρ∞(Tj) and precisely locates Tc. The algorithm has been applied to simulation data for a hard-core square-well fluid and the restricted primitive model electrolyte for sizes up to L/a = 8-12 (where a is the hard-core diameter): the coexistence curves can be computed to a precision of ±1-2% of ρc up to |T − Tc|/Tc = 10 and 10, respectively. Universality of the scaling functions and the exponent β is verified and the (Tc, ρc) estimates confirm previous values based on data from above Tc. The algorithm extends directly to calculating the diameter, ρdiam(T ) ≡ 1 2 (ρ + ρ), and can lead to estimates of the Yang-Yang ratio. Furthermore, a new, explicit approximant for the basic scaling function Y permits straightforward estimates of ∆ρ∞(T ) from limited Q-data when Ising-type criticality may be assumed.