  We examine statics and dynamics of phase-separated states of dilute binary
mixtures using density functional theory. In our systems, the difference in the
salvation chemical potential $\Delta\mu_s$ between liquid and gas is
considerably larger than the thermal energy $k_BT$ for each solute particle and
the attractive interaction among the solute particles is weaker than that among
the solvent particles. In these conditions, the saturated vapor pressure
increases by an amount equal to the solute density in liquid multiplied by the
large factor $k_BT \exp(\Delta\mu_s/k_BT)$. As a result, phase separation is
induced at low solute densities in liquid and the new phase remains in gaseous
states, while the liquid pressure is outside the coexistence curve of the
solvent. This explains the widely observed formation of stable nanobubbles in
ambient water with a dissolved gas. We calculate the density and stress
profiles across planar and spherical interfaces, where the surface tension
decreases with increasing the interfacial solute adsorption. We realize stable
solute-rich bubbles with radius about 30 nm, which minimize the free energy
functional. We then study dynamics around such a bubble after a decompression
of the surrounding liquid, where the bubble undergoes a damped oscillation. In
addition, we present some exact and approximate expressions for the surface
tension and the interfacial stress tensor.
