Based on the conservative phase field model developed by Lowengrub and Truskinovsky [Proc. R. Soc. London A 454, 2617 (1998)] for almost incompressible liquid binary mixtures, we propose an extended scheme for studying immiscible/miscible liquids. Below a critical temperature Tc, the liquids are immiscible with separating interfaces. Above Tc, the interfacial effects vanish, and the liquids become perfectly miscible. The free-energy density of the system depends not only on the phase field variable ϕ (which describes the system composition), but also on the reduced temperature r=(Tc−T)/Tc which measures the distance to the critical point described by Tc. The free energy suffers transformations through Tc in a way to permit a two-phase system in the subcritical (immiscible) regime and a monophase in the supercritical (miscible) regime. Numerical simulations in two spatial dimensions have been performed for isothermal problems (with r as control parameter) as well as for nonisothermal problems with the energy equation describing the temperature distribution. These simulations reveal the behavior of liquid mixtures and droplet coalescence placed in temperature gradients with temperatures continuously varying from T<Tc to T>Tc, problems that could be of large interest in phase transitions in micro- and nanofluidics.Read less <