Energetics-based model for a diffusiophoretic motion of a deformable droplet

We construct a mathematical model for a diffusiophoretic motion of a deformable droplet, which is floating at a liquid surface and is driven by the surface tension gradient originating from the surface concentration field of the chemicals that are emitted from the droplet. We define the free energy of the system by including the surface and line energies. From the calculation of the functional of the free energy, we obtain a mathematical model for the diffusiophoretic motion with deformation. By only considering the deformation of the second mode, we explicitly derive the time-evolution equations for the translational motion and the elliptic deformation. There are three stable states: an immobile circular droplet, an immobile elliptically deformed droplet, and a moving droplet with the elliptic deformation in which the minor axis meets the motion direction, and we discuss the transition between these three stable states.