Fluid coexistence close to criticality: scaling algorithms for precise simulation

Abstract 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 Q L ( T ; 〈 ρ 〉 L ) ≡ 〈 m 2 〉 L 2 / 〈 m 4 〉 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 − ) / Δ ρ ∞ ( T ) universally to 1 2 ( Q m + + Q m − ) , where Δ 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, L 1 , L 2 , … , L n . Starting at a T 0 sufficiently far below T c and suitably choosing intervals Δ T j = T j + 1 − T j > 0 yields Δ ρ ∞ ( T j ) and precisely locates T c . 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 ρ c up to | T − T c | / T c = 10 −4 and 10−3, respectively. Universality of the scaling functions and the exponent β is verified and the ( T c , ρ c ) estimates confirm previous values based on data from above T c . The algorithm extends directly to calculating the diameter, ρ diam ( 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 Δ ρ ∞ ( T ) from limited Q-data when Ising-type criticality may be assumed.