Optimized equation of the state of the square-well fluid of variable range based on a fourth-order free-energy expansion

The free energy of square-well (SW) systems of hard-core diameter sigma with ranges 1 <=lambda <= 3 is expanded in a perturbation series. This interval covers most ranges of interest, from short-ranged SW fluids (lambda similar or equal to 1.2) used in modeling colloids to long ranges (lambda similar or equal to 3) where the van der Waals classic approximation holds. The first four terms are evaluated by means of extensive Monte Carlo simulations. The calculations are corrected for the thermodynamic limit and care is taken to evaluate and to control the various sources of error. The results for the first two terms in the series confirm well-known independent results but have an increased estimated accuracy and cover a wider set of well ranges. The results for the third- and fourth-order terms are novel. The free-energy expansion for systems with short and intermediate ranges, 1 <=lambda <= 2, is seen to have properties similar to those of systems with longer ranges, 2 <=lambda <= 3. An equation of state (EOS) is built to represent the free-energy data. The thermodynamics given by this EOS, confronted against independent computer simulations, is shown to predict accurately the internal energy, pressure, specific heat, and chemical potential of the SW fluids considered and for densities 0 <=rho sigma(3)<= 0.9 including subcritical temperatures. This fourth-order theory is estimated to be accurate except for a small region at high density, rho sigma(3)approximate to 0.9, and low temperature where terms of still higher order might be needed.