Stationary particle currents in sedimenting active matter wetting a wall

Recently it was predicted, on the basis of a lattice gas model, that scalar active matter in a gravitational field would rise against gravity up a confining wall or inside a thin capillary—in spite of repulsive particle-wall interactions [Phys. Rev. Lett. 124, 048001 (2020)]. In this paper we confirm this prediction with sedimenting active Brownian particles (ABPs) in a box numerically and elucidate the mechanism leading to the formation of a meniscus rising above the bulk of the sedimentation region. The height of the meniscus increases with the activity of the system, algebraically with the Péclet number. The formation of the meniscus is determined by a stationary circular particle current, a vortex, centered at the base of the meniscus, whose size and strength increase with the ABP activity. The origin of these vortices can be traced back to the confinement of the ABPs in a box: already the stationary state of ideal (noninteracting) ABPs without gravitation displays circular currents that arrange in a highly symmetric way in the eight octants of the box. Gravitation distorts this vortex configuration downward, leaving two major vortices at the two side walls, with a strong downward flow along the walls. Repulsive interactions between the ABPs change this situation only as soon as motility induced phase separation (MIPS) sets in and forms a dense, sedimented liquid region at the bottom, which pushes the center of the vortex upwards towards the liquid-gas interface. Self-propelled particles therefore represent an impressive realization of scalar active matter that forms stationary particle currents being able to perform visible work against gravity or any other external field, which we predict to be observable experimentally in active colloids under gravitation.

Figure 1

Stationary state of ABPs in a box with reflecting walls. Box dimension is $100\times 400$, particle number is 5000, gravity is in $-\widehat{y}$ direction, $F_0=100, {\mathrm{Pe}}_s=30$, and ${\mathrm{Pe}}_g=6$. Shown quantities are time-averaged. (a) Particle density $\rho (x,y)$. (b) Modulus of the current density $\left|\mathbf{J}\right(x,y\left)\right|$. (c) Curl amplitude $A(x,y)$ together with arrows indicating current orientation ${\varphi }_J(x,y)$. (d) Average particle orientation $\overline{\theta }(x,y)$.

Figure 2

(a) Decay of the isodensity profile $h\left(x\right)$, scaled by ${\mathrm{Pe}}_s^2$, as a function of distance from the wall $x$, for different ${\mathrm{Pe}}_s$ and ${\mathrm{Pe}}_g$. The collapsed curves have been fitted to a double-exponential function, shown in solid line. (b) Density profile $\rho \left(y\right)\sim exp(-y/{\lambda }_{\mathrm{sed}})$ of the dilute layer, for three different sets of (${\mathrm{Pe}}_s,\alpha$). (c) Scaling plot for the sedimentation length extracted from panel (b): ${\lambda }_{\mathrm{sed}}\sim {\mathrm{Pe}}_s^2/{\mathrm{Pe}}_g$, such that ${\lambda }_{\mathrm{sed}}\alpha \sim {\mathrm{Pe}}_s$ (with $\alpha ={\mathrm{Pe}}_s/{\mathrm{Pe}}_g$) is linear in ${\mathrm{Pe}}_s$. The size of the simulation box is $200\times 800$ and the particle density is ${\rho }_0=0.125$. (d) The wall and bulk density profiles $\rho (x_{\mathrm{wall}},y)$ (black), $\rho (x_{\mathrm{bulk}},y)$ (red) are respectively shown for the parameter set $F_0=20, {\mathrm{Pe}}_s=50$, and $\alpha =0.4$. The wetting height $\mathrm{\Delta }{h}^{\max }$ is estimated from the difference curve $\rho (y,x_{\mathrm{wall}})-\rho (y,x_{\mathrm{bulk}})$ (green) from the $y$ values where it decays to a value smaller than 0.01, as indicated schematically in the plot.

Figure 3

(a) Maximal wetting height $\mathrm{\Delta }{h}^{\max }$ as a function of ${\mathrm{Pe}}_g$ for different ${\mathrm{Pe}}_s$. The scaling $\mathrm{\Delta }{h}^{\max }\sim {\mathrm{Pe}}_s^4/{\mathrm{Pe}}_g^{2.1}$ is shown in solid line. (b) Maximal wetting height $\mathrm{\Delta }{h}^{\max }$ as a function of ${\lambda }_{\mathrm{sed}}$ for different ${\mathrm{Pe}}_s$. The scaling $\mathrm{\Delta }{h}^{\max }\sim {\lambda }_{\mathrm{sed}}^{1.8}$ is shown in solid line. ${\lambda }_{\mathrm{sed}}$ for each set of $({\mathrm{Pe}}_s,{\mathrm{Pe}}_g)$ has been estimated separately. Box dimension is $200\times 500$ and particle number is $2\times {10}^4$.

Figure 4

(a) Enstrophy plotted against ${\mathrm{Pe}}_g$ for a fixed ${\mathrm{Pe}}_s=35$ shows a decrease with increasing gravitational force. (b) Enstrophy increases as a function of ${\mathrm{Pe}}_s$ for fixed ${\mathrm{Pe}}_g=8$.

Figure 5

(a) Representative plot for the mean displacement ${\sigma }_{\mathrm{tracer}}$ of a tracer particle on a vortical loop. From the peak value of ${\sigma }_{\mathrm{tracer}}$ one can read off the size of the loop. The largest peak value among all the oscillatory tracer particle trajectories gives the size of the largest vortex. (b) The largest amplitude in ${\sigma }_{\mathrm{tracer}}$ is plotted as a function of ${\mathrm{Pe}}_s$ for three different ${\mathrm{Pe}}_g$ values. (c) Area integral $W\left(r\right)={\int }_{S}AdS$, where $S$ is the area of a circle of radius $r$ drawn around the center of the largest vortex. $W\left(r\right)$ vs $r$ is plotted for different values of ${\mathrm{Pe}}_s,{\mathrm{Pe}}_g$. (d) The saturation value of $W\left(r\right)$ is plotted against ${\mathrm{Pe}}_s$ for two different values of ${\mathrm{Pe}}_g$.

Figure 6

Areas of the largest curl clusters (a) $S_{\mathrm{wall}}$ and (b) $S_{\mathrm{bulk}}$ plotted against ${\mathrm{Pe}}_s$ for three different values of ${\mathrm{Pe}}_g$. $S_{\mathrm{wall}}$ and $S_{\mathrm{bulk}}$ show power-law behavior $\approx {\mathrm{Pe}}_s^{\nu }$ with $\nu =3$ and $\nu =2$, respectively.

Figure 7

(a) Density ${\rho }_A$ for the curl variable as a function of the distance $r_0$ measured from the lower corners of the box. Since the density profiles are symmetric about $x=L/2$, we average over the left and right halves of the box for measuring ${\rho }_A$. (b) Length scale $r_1$ plotted against ${\mathrm{Pe}}_s$ for three different ${\mathrm{Pe}}_g$ values. (c) Two-point correlation function of the curl amplitude $C_A$ plotted against $r_0$ for five sets of ${\mathrm{Pe}}_s,{\mathrm{Pe}}_g$. (d) Correlation length $\zeta$ estimated for $C_A$ using an exponential fit, as a function of ${\mathrm{Pe}}_s$, for three different values of ${\mathrm{Pe}}_g$.

Figure 8

(a) Maximum wetting height $\mathrm{\Delta }{h}^{\max }$ plotted against $F_0$ indicates that the wetting height decreases as interparticle repulsion is increased and approaches a constant value for large $F_0$. (b) Total number of particles $N(y>y_{\mathrm{iso}})$ elevated above isodensity line scaled by the total number of particles in the system plotted against $F_0$. (c) Width of the meniscus plotted against $F_0$. The width of the meniscus $W_{\mathrm{men}}$ increases as a function of $F_0$. (d) Maximum amplitude of ${\sigma }_{\mathrm{tracer}}$ plotted as a function of $F_0$.

Figure 9

Current lines and curl amplitude for fixed ${\mathrm{Pe}}_s=30$ and ${\mathrm{Pe}}_g=6$ ($\alpha =0.2$), and decreasing interparticle repulsive force: (a) $F_0=100$, (b) $F_0=3$, (c) $F_0=0.3$, and (d) $F_0=0$. Box dimension is $100\times 400$, and particle number is 5000.

Figure 10

Steady-state density, polarization and current-density profiles for noninteracting ABPs with ${\mathrm{Pe}}_s=2$ and $\alpha =0.25$ in a $20\times 20$ box, obtained numerically with $\text{FreeFem}++$. (a) Steady-state density $\rho (x,y)$. The isodensity line $\rho =1$ is shown by a solid black line. (b) Mean orientation $\overline{\theta }(x,y)$ obtained from the steady-state polarization $\mathbf{P}(x,y)$. (c) The current-density lines ${\varphi }_J(x,y)$ are shown by arrows, and the curl amplitude $A(x,y)$ is represented by the color bar. (d) Zoomed-in version of the bottom-left corner of panel (c). Three main vortices are observed: vortex 1 and vortex 3 along the bottom and left walls, respectively, and antivortex 2 at the corner.

Figure 11

(a) Density profile at the center of the box ${\rho }_{\mathrm{bulk}}$ and near the vertical wall ${\rho }_{\mathrm{wall}}$ for noninteracting ABPs with ${\mathrm{Pe}}_s=2$ and $\alpha =0.25$ in a $25\times 25$ box, obtained numerically with $\text{FreeFem}++$. For an isodensity line chosen in the exponential decay regime, the wetting height is always $\mathrm{\Delta }h=6.1$. (b) Sedimentation length ${\lambda }_{\mathrm{sed}}$ as a function of ${\mathrm{Pe}}_s$, for several ${\mathrm{Pe}}_g$. (c) and (d) Wetting height $\mathrm{\Delta }h$ as a function of ${\mathrm{Pe}}_s$ and ${\mathrm{Pe}}_s^2/{\mathrm{Pe}}_g$, respectively, for several ${\mathrm{Pe}}_g$. It is calculated with the isodensity line $\rho =1$ in a $100\times 100$ box. The dotted line represents the fitted curve for all presented data.

Figure 12

Current-density lines ${\varphi }_J(x,y)$ for noninteracting ABPs with ${\mathrm{Pe}}_s=2$ and (a) $\alpha =0$, (b) $\alpha =0.25$, (c) $\alpha =0.5$, and (d) $\alpha =0.75$, numerically calculated with $\text{FreeFEM}++$ in a $10\times 10$ box. The color bar represents the curl amplitude of the current $A(x,y)$.

Figure 13

(a)–(c) Vortex area $S_i$ as a function of ${\mathrm{Pe}}_s$ for noninteracting ABPs with several $\alpha$ and in a $10\times 10$ box. (d)–(f) Circulation $W_i={\int }_{S_i}AdS$ of the corresponding vortices. The labels 1, 2, and 3 correspond to the vortices described on Fig. 10.