Pairing-induced motion of source and inert particles driven by surface tension

We experimentally and theoretically investigate systems with a pair of source and inert particles that interact through a concentration field. The experimental system comprises a camphor disk as the source particle and a metal washer as the inert particle. Both are floated on an aqueous solution of glycerol at various concentrations, where the glycerol modifies the viscosity of the aqueous phase. The particles form a pair owing to the attractive lateral capillary force. As the camphor disk spreads surface-active molecules at the aqueous surface, the camphor disk and metal washer move together, driven by the surface tension gradient. The washer is situated in the front of the camphor disk, keeping the distance constant during their motion, which we call a pairing-induced motion. The pairing-induced motion exhibited a transition between circular and straight motions as the glycerol concentration in the aqueous phase changed. Numerical calculations using a model that considers forces caused by the surface tension gradient and lateral capillary interaction reproduced the observed transition in the pairing-induced motion. Moreover, this transition agrees with the result of the linear stability analysis on the reduced dynamical system obtained by the expansion with respect to the particle velocity. Our results reveal that the effect of the particle velocity cannot be overlooked to describe the interaction through the concentration field.

Figure 1

(a) Experimental setup. A camphor disk and a metal washer were floated at the surface of the glycerol aqueous solution in a square-shaped container (245 mm $\times$ 245 mm $\times$ 25 mm). (b) Definition of the variables obtained from the experimental results.

Figure 2

(a), (b) Superimposed images of the camphor disk and metal washer, whose trajectories are also denoted. Scale bar is $30\mathrm{mm}$. (a) Circular and (b) straight motions observed on the glycerol aqueous solutions at $C$ = (a) 0 and (b) 0.5 $\mathrm{vol}\%$. (c), (d) Time series of the drift angle $\psi$ (orange) and angular velocity $\omega$ (blue) for (c) circular and (d) straight motions. The time $t$ corresponds to that in (a) and (b). (e), (f) Relationship between the drift angle $\psi$ and the angular velocity $\omega$ for (e) circular ($C=0\mathrm{vol}\%)$ and (f) straight motions ($C=0.5\mathrm{vol}\%)$ for the period from $t=0$ to 184 s. Corresponding videos are available in Supplemental Material [31].

Figure 3

(a)–(e) Trajectories of the pair represented by those of the washer. (f)–(j) Distribution $p$ of unsigned curvature $\kappa$ corresponding to (a)–(e). The trajectories in the domain surrounded by red squares in (a)–(e) are used to obtain $p$. The concentrations of glycerol, C, were (a), (f) 0, (b), (g) 0.125, (c), (h) 0.25, (d), (i) 0.375, and (e), (j) 0.5 $\mathrm{vol}\%$. The distribution $p$ shifts gradually to the lower $\kappa$ as $C$ increased, indicating that the circular motion gradually changed into the straight one.

Figure 4

Force applied to the particle. (a) Force originating from the concentration field. (b) Profile of attractive lateral capillary force and repulsive force owing to the short-range exclusive volume effect. The graph shows the force applied to the particle at the origin from another particle at distance $l=\left|\mathit{l}\right|$.

Figure 5

(a), (b) Superimposed images obtained using numerical calculations, in which the color map shows the profile of $u$. The green and pink disks represent the source and inert particles, respectively. The solid and dotted lines exhibit trajectories of the source and inert particles, respectively. The white arrows indicate the direction of motion. (a) Circular motion for $\mathrm{\Gamma }=5.75$ and $\eta =0.4$. (b) Straight motion for $\mathrm{\Gamma }=5.75$ and $\eta =0.425$. (c) Phase diagram that classifies circular and straight motions based on the mean value of the unsigned curvature, $\overline{\kappa }$, on the $\mathrm{\Gamma }\text{−}\eta$ plane. The source and inert particles exhibited the circular motion for higher $\mathrm{\Gamma }$ and lower $\eta$, while they exhibited the straight motion for lower $\mathrm{\Gamma }$ and higher $\eta$. (d) Dependence of $\overline{\kappa }$ on $\eta$ for fixed $\mathrm{\Gamma }=5.75$ on the red line in (c). (e) Dependence of $\left|\psi \right|$ on $\eta$ for fixed $\mathrm{\Gamma }=5.75$ on the red line in (c). The yellow and light blue areas in (d) and (e) represent the parameter regions where circular and straight motions appeared, respectively. (f) Relationship between the drift angle $\psi$ and angular velocity $\omega$ for the data corresponding to all parameters in (c).

Figure 6

Schematic illustration of forces acting on each particle in the circular motion. Arrows indicate the direction and the relative magnitude of these forces based on the numerically obtained data for the parameters corresponding to Fig. 5. The gray curves indicate the trajectories of the particles. The black arrows indicate the viscous resistance forces (${\mathit{F}}_{\mathrm{vis}}^{\mathrm{s}}$, ${\mathit{F}}_{\mathrm{vis}}^{\mathrm{i}}$), the lateral capillary forces (${\mathit{F}}_{\mathrm{cap}}^{\mathrm{s}}$, ${\mathit{F}}_{\mathrm{cap}}^{\mathrm{i}}$), and the driving forces (${\mathit{F}}_{\mathrm{conc}}^{\mathrm{s}}$, ${\mathit{F}}_{\mathrm{conc}}^{\mathrm{i}}$) from the concentration field. The pink and green disks indicate the inert and source particles, respectively. The pink and green arrows show the net force acting on the inert and source particles. The scale of the net forces is enhanced by a factor of 5. The inset represents the wider view to illustrate the relation between trajectories and net forces. Here the scale of the net forces is further enhanced by a factor of 50.

Figure 7

(a) Coordinate system and straight pairing-induced motion. (b) Representative profile of $u_v(\mathit{r},{\mathit{v}}_{\mathrm{s}})$, where ${\mathit{v}}_{\mathrm{s}}={\mathit{e}}_x$. The contours at every interval of 0.2 are shown. (c) $v_0$ against $\eta$ for $\mathrm{\Gamma }=5.75$. (d) ${\ell }_0$ against $\eta$ for $\mathrm{\Gamma }=5.75$. (e) Real parts of eigenvalues $\lambda$ of $A_x$ calculated for each $\eta$. The eigenvalue around $\mathrm{Re}\lambda \simeq -0.05$ (plotted with red) is real, while other eigenvalues around $\mathrm{Re}\lambda \simeq -0.25$ (plotted with blue) are complex. (f) Real parts of the eigenvalues $\lambda$ of $A_y$ calculated for each $\eta$. The zero eigenvalue is not plotted.

Figure 8

(a) Phase diagram that represents the single source particle motion based on the steady velocity $v_s$. The source particle exhibited self-propelled motion for the higher $\mathrm{\Gamma }$ and the lower $\eta$, while it stopped for the lower $\mathrm{\Gamma }$ and the higher $\eta$. (b) Dependence of $v_{\mathrm{s}}$ on $\eta$ for fixed $\mathrm{\Gamma }$ = 5.75 indicated by the red line in (a).