A thermodynamically consistent free energy model for fluid flows comprised of one gas and two liquid components is presented and implemented using the entropic lattice Boltzmann scheme. The model allows a high density ratio, up to the order of O(10^{3}), between the liquid and gas phases, and a broad range of surface tension ratios, covering partial wetting states where Neumann triangles are formed, and full wetting states where complete encapsulation of one of the fluid components is observed. We further demonstrate that we can capture the bouncing, adhesive, and insertive regimes for the binary collisions between immiscible droplets suspended in air. Our approach opens up a vast range of multiphase flow applications involving one gas and several liquid components.