Theory of closed-loop liquid-liquid immiscibility in mixtures of molecules with directional attractive forces

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Abstract
The phase equilibria of a binary mixture of equal-sized hard spheres (diameters σ1 = σ2) with mean-field attractive forces between like species (a 11 = a 22) and non between unlike species (a 12 = 0) are determined using an ‘augmented’ van der Waals equation of state. This system is found to have a symmetrical phase diagram exhibiting a large extent of liquid-liquid immiscibility and a tricritical point. Directional attractive forces between the unlike species and then introduced in the form of off-centre, square-well bonding sites, and a perturbation term representing the contribution due to bonding is added to the equation of state. The phase behaviour of systems with varying degrees of association is investigated. As the strength of the site-site bonding interaction is increased relative to that of the mean-field attractions, closed-loop liquid-liquid immiscibility is found with the corresponding lower and upper critical solution points. The region of liquid-liquid coexistence decreases with increasing bonding strength until it disappears completely due to extensive association between the unlike species of the mixture. The fraction of bonded molecules is determined to provide an explanation of the phase behaviour; closed-loop immiscibility is a direct consequence of the dramatic increase in bonding as the temperature is decreased. Existing calculations for lattice models exhibiting closed-loop coexistence are discussed in the context of the results of this continuum approach, a first of its kind. Advantages of the approach in describing the properties of real systems exhibiting closed-loop immiscibility are pointed out.