Instability and self-propulsion of active droplets along a wall

Active droplets can swim spontaneously in viscous flows as a result of the nonlinear convective transport of a chemical solute produced at their surface by the Marangoni and/or phoretic flows generated by that solute's inhomogeneous distribution, provided the ratio of convective-to-diffusive solute transport, or Péclet number $\text{Pe}$, is large enough. As the result of their net buoyancy, active drops typically evolve at a small finite distance $d$ from rigid boundaries. Yet, existing models systematically focus on unbounded flows, ignoring the effect of the wall proximity on the intrinsically nonlinear nature of their propulsion mechanism. In contrast, we obtain here a critical insight on the propulsion of active drops near walls by analyzing their stability against nonaxisymmetric perturbations and the resulting emergence of self-propulsion along the wall with no limiting assumption on the wall distance $d$. Dipolar or quadrupolar axisymmetric (levitating) base states are identified depending on $d$ and $\text{Pe}$. Perhaps counterintuitively, a reduction in the drop-wall separation $d$ is observed to destabilize these modes and to promote self-propulsion, as a result of the confinement-induced localization of the chemical gradients driving the motion. In addition, quadrupolar states are more unstable than their dipolar counterparts due to the redistribution of the chemical perturbation by the base flow, favoring the emergence of stronger slip forcing on the drop surface.

Figure 1

An active drop levitating above a rigid wall. The equilibrium distance, $d^{*}$, is dictated by a balance between the external force, ${\mathbf{F}}^{\mathrm{ext}}$, and the “phoretic force,” ${\mathbf{F}}_p$. The isosurfaces of constant $\xi$ and $\mu$ on the left half define the (body-fixed) bispherical grid. Also shown is the origin of the $(\rho ,z)$ cylindrical coordinate system.

Figure 2

The vertical force acting on the (fixed) drop as a function of its separation, $d$, from the wall, for $\text{Pe}=5$. The symbols represent results from the numerical analysis described in Sec. 3 and the dashed line is the analytical approximation for the case when $d\gg 1$ and $\text{Pe}<8$.

Figure 3

Characteristics of the axisymmetric base state. Top: Vertical force, $F_z$, on the drop as a function of the Péclet number, $\text{Pe}$, for different drop-wall separation distances, $d$. Center: Solute distribution and streamlines (isolines of the stream function $\psi$) around the drop for $d=50$ (representative of the unbounded limit $d\to \infty$) for $\text{Pe}=7$ (left), $\text{Pe}=10$ (center), and $\text{Pe}=14$ (right). No streamlines are shown for $\text{Pe}=7$, as the fluid is essentially at rest. Bottom: Solute distribution and streamlines for $\text{Pe}=14$ for $d=5$ (left), $d=2$ (center), and $d=1$ (right).

Figure 4

The steady-state solution branches of the symmetric bifurcation observed beyond $\text{Pe}=8$ in the base state for $d\to \infty$ and $\text{Pe}=10$, as shown in Fig. 3 and discussed in Sec. 3. $F_z>0$ ($F_z<0$) for the solution on the left (right).

Figure 5

Bistability of the base state: the concentration distribution and streamlines are shown for the two different branches of solutions coexisting for $d=5$ and $\text{Pe}=14$, showing respectively dipolar (left) and quadrupolar (right) symmetry. The relative stability of these two branches is analyzed in Sec. 5c and Fig. 8; the quadrupolar branch is more unstable to longitudinal swimming modes.

Figure 6

Evolution of the growth rate of the unstable swimming mode for levitating droplets (forced $F_z\ne 0$, $F_x=F_y=0$) for drop-wall separations $d=5$ and $d=10$. The results for a force-free particle in an unbounded fluid (i.e., $d\to \infty$, $\mathbf{F}=\mathbf{0}$) are shown for comparison.

Figure 7

Instability mechanism for $\text{Pe}<8$ in the far-field interaction limit, $d\gg 1$. Left: An initially isotropic solute distribution becomes asymmetric due to a small translation of the drop from its initial position (dashed). A representative isoline of solute concentration is materialized by the blue circle. The resulting concentration perturbations, $c^{\prime }$, depend on the local sign of ${\mathbf{V}}^{\prime }·\nabla c^b$ [see Eq. (40)]. Right: The fore-aft asymmetry in solute concentration leads to the surface slip ${\mathbf{u}}_s^{\prime }={\nabla }_s{c}^{\prime }$, which is maintained if advection is strong, causing the drop to swim spontaneously [see Eq. (42)].

Figure 8

Higher instability of the quadrupolar base state (right) than the dipolar base state (left), for a horizontally swimming drop (denoted by the black arrow), shown here for $\text{Pe}=14$. The contours are qualitative depictions of the solute distribution around the drop in the self-propelling state, with lighter (darker) colors denoting higher (lower) concentrations.

Figure 9

(a) Evolution with $\text{Pe}$ of the growth rate, ${\sigma }_r$, of the unstable horizontal swimming mode for small-to-moderate wall distance, $d$. (b) Evolution with $d$ of the critical Péclet number for the onset of self-propulsion, ${\text{Pe}}_c$. The log-log inset shows the asymptotic dependence of ${\text{Pe}}_c$ in the lubrication limit $d\to 0$, where the dashed line is a power-law fit, $\sim d^{k_f}$, $k_f\approx 0.46$.

Figure 10

Evolution with $d$ of the perturbation concentration $c^{\prime }$ around the drop normalized by the corresponding swimming speed (color) and normalized auxiliary surface traction $|\mathbf{n}·\widehat{\mathbf{\sigma }}·{\mathbf{e}}_{\theta }|$ (black and white). An $x\text{−}z$ cross section of the distributions is shown. The drop is translating toward the $+{\mathbf{e}}_x$ direction leading to a solute deficiency in its direction of motion. Here, ${\mathbf{e}}_{\theta }$ is the tangent unit vector on the drop's surface with $\theta$ measured counterclockwise from ${\mathbf{e}}_x$. The top row shows these same quantities in the limit $d\to \infty$, and a clear symmetry w.r.t. the direction of motion, ${\mathbf{e}}_x$, is observed.

Figure 11

Coordinate systems and notations used in the asymptotic analysis for evaluation of the force experienced by a fixed, phoretic particle 1 in the presence of an identical phoretic particle 2, placed “far away” at a dimensionless center-to-center distance $2(d+1)$. We show in Appendix pp1 that the leading order force experienced by each particle in this configuration is identical to that experienced if the other particle were replaced by a wall placed along the mid-plane.