How Activity Landscapes Polarize Microswimmers without Alignment Forces

Active-particle suspensions exhibit distinct polarization-density patterns in activity landscapes, even without anisotropic particle interactions. Such polarization without alignment forces is at work in motility-induced phase separation and betrays intrinsic microscopic activity to mesoscale observers. Using stable long-term confinement of a single thermophoretic microswimmer in a dedicated force-free particle trap, we examine the polarized interfacial layer at a motility step and confirm that it does not exert pressure onto the bulk. Our observations are quantitatively explained by an analytical theory that can also guide the analysis of more complex geometries and many-body effects.

Figure 1

Setup and parameter measurement. (a) Sketch of the rectangular arena, in which a $1.5\mu \mathrm{m}$ Janus particle is confined by photon nudging. Position and in-plane orientation $\mathbf{n}$ are observed in dark-field microscopy. (b) Orientational correlation time $\tau$ for the active-passive bulk region as a function of the in-plane propulsion speed $v_0$. The horizontal and linear fits serve to identify ${\tau }_{p,a}$. (c) In-plane propulsion speed $v_0$ as a function of the incident laser intensity, with a fit accounting for the weakly variable particle-wall alignment. (d) Effective diffusion coefficient $D_{\mathrm{eff}}$ in the active bulk, obtained from the slopes of the late-time MSD, and Eq. (1) (line) using ${\tau }_a$ from (b). In (b)–(d) squares and circles correspond to the passive and active region, respectively, and error bars show 95% confidence intervals for Gaussian error propagation.

Figure 2

Particle density and polarization. (a) Particle density in the sample plane for $v_0=2.34\mu \mathrm{m}{\mathrm{s}}^{-1}$ and ${\tau }_a=1.96\mathrm{s}$ with the active-passive interface at $x=0$. (b) Its integral along the $y$ direction (excluding a $0.7\mu \mathrm{m}$ lateral boundary region). The bulk densities ${\rho }_p$, ${\rho }_a$ (upper, lower arrow) in the passive-active region were determined at about one decay length ${\lambda }_{p,a}$ from the passive-active interface. The numerical solution [44] of Eq. (3) (solid line) employs reflecting outer boundaries. The (approximate) analytic solution from Eqs. (4)–(8) (dashed line) was normalized over half of the nominal width of the active and passive regions. (c) Sketch of the processes creating the interfacial polarization and depletion layers in a simplified model with binary particle orientations. (d) The experimental and theoretical (numerical and analytical) particle polarization [legend as in (b)]. The thin dotted vertical lines in (b) and (d) mark the borders to the confining photon-nudging region [cf. Fig. 1]. Similar plots for other laser intensities can be found in the Supplemental Material [34].

Figure 3

Bulk density ratio and interface widths. (a) Measured ratio ${\rho }_p/{\rho }_a$ of passive and active bulk densities as function of the (experimental) Péclet number $v_0^2{\tau }_a/2D$ (circles). Our analytical prediction (8) with $D_r=1/{\tau }_a$ (solid line) improves that of Refs. [19, 45], namely ${\rho }_a/{\rho }_p=D_p/D_a$ (dotted line). (b) Decay lengths ${\lambda }_{a,p}$ in the active (circles) and passive (squares) bulk regions as functions of $v_0$. The solid and dashed lines show the theoretical prediction (6) with $D_r=1/{\tau }_a$, while the dot-dashed line assumes $D_r=1/{\tau }_p$, with ${\tau }_{a,p}$ from the fits in Fig. 1. The solid line for ${\lambda }_a$ improves a prediction of Refs. [19, 45] (dotted line). Error bars indicate the 95% confidence intervals of $\rho$ and the Gaussian propagation of uncertainties for $v_0$, ${\tau }_a$, $D$ from Fig. 1, respectively.