Self-motion of a growing single crystal of azobenzene chromophore in triacrylate solution (TA) is investigated in relation to the solid-liquid phase diagram bound by the solidus and liquidus lines. Upon thermal quenching from the isotropic melt to the crystal + liquid gap, various single crystals develop in a manner dependent on concentration and supercooling depth. During the crystal growth, TA solvent is rejected from the growing faceted fronts, enriching with TA in close proximity to the crystal-solution interface. The concentration gradient that formed as the result of TA expulsion induces convective flows in the solution and generates spatial variability of surface tension usually responsible for Marangoni effect. Either or both of these phenomena may have contributed to the observed self-motion including swimming, sinking, and floating of the azobenzene rhomboidal crystal in TA solution. A stationary rhomboidal crystal is also shown to swim upon irradiation with the UV light because of a mechanical torque generated by the trans-cis isomerization. Judging from the sinking or floating behavior of the azobenzene crystal, it may be inferred that the nucleation occurs at the solution-air interface.