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

TL;DR

This work develops an energetics-based description for a deformable, diffusiophoretic droplet at a liquid surface, where motion arises from a surface-tension gradient generated by emitted chemicals. Using a free-energy functional $E=E_s+E_l$ with $\gamma(u)=\gamma_0-\Gamma u$, the authors describe the droplet boundary via a Fourier expansion and couple it to a reaction-diffusion equation for the concentration, yielding gradient-flow dynamics for translation and deformation. They perform numerical PDE simulations for a 2-mode deformation and then reduce the system to a 4D ODE in polar coordinates, extracting a rich bifurcation structure including supercritical and subcritical pitchforks and drift-pitchforks that produce three stable states: IC, ID, and MD moving along the elliptic minor axis. The results establish a universal, two-dimensional framework connecting deformation and self-propulsion, with phase diagrams that align between PDE and ODE analyses and offer insights beyond earlier Ohta–Ohkuma-type models by enabling stable deformed stationary states and bistabilities.

Abstract

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.

Figures (4)

• Figure 1: Numerical simulation results obtained from the PDE model in Eqs. \ref{['eq31']} to \ref{['eq39']}. Snapshots at $t = 1000$ (left panels) and time series (right panels) of the speed $v = \left|d\bm{r}_c/dt\right|$ (red) and the deformation magnitude $s = \sqrt{{a_2}^2+{b_2}^2}$ (blue) are shown for the three typical cases. (a) Immobile circular (IC) droplet at $\kappa_2 = 0.12$ and $\eta_t = 0.085$. (b) Immobile deformed (ID) droplet at $\kappa_2 = 0.116$ and $\eta_t = 0.085$. (c) Mobile deformed (MD) droplet at $\kappa_2 = 0.112$ and $\eta_t = 0.085$. The droplet moves downward as the orange arrow indicates. Grayscale tone displays the concentration field $u$. The initial condition is set as $d\bm{r}_c/dt = \bm{e}_x + 0.1 \bm{e}_y$, $a_2 = 0.1$, $b_2 = 0.01$.
• Figure 2: Numerical simulation results obtained from the PDE model in Eqs. \ref{['eq31']} to \ref{['eq39']}. (a) Two-dimensional phase diagram on the $\eta_t$-$\kappa_2$ plane. The other parameters are set to be $\eta_2 = 0.1$ and $R=1$. The red, green, and dark blue points correspond to an immobile circular (IC) droplet, an immobile deformed (ID) droplet, and a mobile deformed (MD) droplet. The cyan points show the bistability between IC and MD droplets, the yellow points show that between IC and ID droplets, and the gray points show that between IC and mobile circular (MC) droplet. The brown points represent the case that we can classify into none of the above (Three-or-more final steady states due to the slow convergence of the system). (b) One-dimensional bifurcation diagrams with constant $\eta_t$ or $\kappa_2$, which are indicated as arrows at the top and right sides of the plot in panel (a). Speed $v$ and deformation $s$ are plotted. (i,ii) $\kappa_2$ is varied at (i) $\eta_t = 0.082$ and (ii) $\eta_t = 0.090$. (iii-vi) $\kappa_2$ is varied at (iii) $\kappa_2 = 0.160$, (iv) $\kappa_2 = 0.130$, (v) $\kappa_2 = 0.117$, and (vi) $\kappa_2 = 0.100$. The red, green, and dark blue lines represent immobile circular (IC), immobile deformed (ID), and mobile deformed (MD) droplets. For the MD droplet, it moves in its minor-axis direction. The vertical dotted lines are drawn to indicate the transition points and correspondence between $v$ and $s$.
• Figure 3: Analysis of the ODE model in Eqs. \ref{['eq58']} and \ref{['eq59']}. (a) Phase diagram obtained by the ODE model. The boundary curves are given as $\eta_t = f_1$, $\kappa_2 = \tilde{g}_0$, Eq. \ref{['funcboundary']} with Eq. \ref{['Hs']}, and Eq. \ref{['idmd']}. IC, ID, and MD indicate an immobile circular droplet, an immobile deformed droplet, and a mobile deformed droplet, respectively. IC/MD and ID/MD denote the bistable states of IC and MD droplets and ID and MD droplets, respectively. (b) One-dimensional bifurcation diagrams with constant $\eta_t$ or $\kappa_2$, which are indicated as arrows at the top and right sides of the plot in panel (a). Speed $v$ and deformation $s$ are plotted. (i,ii) $\kappa_2$ is varied at (i) $\eta_t = 0.085$ and (ii) $\eta_t = 0.095$. (iii-vi) $\kappa_2$ is varied at (iii) $\kappa_2 = 0.160$, (iv) $\kappa_2 = 0.130$, (v) $\kappa_2 = 0.118$, and (vi) $\kappa_2 = 0.105$. The red, green, dark blue, and brown lines represent immobile circular (IC), immobile deformed (ID), mobile deformed (MD) droplets moving in the minor-axis direction, and MD droplets moving in the major-axis direction. The thick and thin lines indicate the stable and unstable solutions, respectively. The characters "P" and "SN" mean the pitchfork and saddle-node bifurcations. The vertical dotted lines are drawn to indicate the transition points and correspondence between $v$ and $s$.
• Figure 4: Magnitude $s$ of deformation depending on $\kappa_2$ obtained by the numerical simulation only considering the deformation dynamics. The simulation results are shown with red dots. The result of the fitting to a function $\kappa_2 = p_0 - p_2 s^2$ is shown with a blue line. Here, $p_0$ and $p_2$ are obtained to be $p_0 = 0.1184$ and $p_2 = 0.1436$ from the fitting. The green broken line shows the curve using the analytically estimated value for $\tilde{g}_0$.