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Active phase separation by turning towards regions of higher density

Nature Physics volume 17, pages 961–967 (2021)Cite this article

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Abstract

Studies of active matter, from molecular assemblies to animal groups, have revealed two broad classes of behaviour: a tendency to align yields orientational order and collective motion, whereas particle repulsion leads to self-trapping and motility-induced phase separation. Here we report a third class of behaviour: orientational interactions that produce active phase separation. Combining theory and experiments on self-propelled Janus colloids, we show that stronger repulsion on the rear than on the front of these particles produces non-reciprocal torques that reorient particle motion towards high-density regions. Particles thus self-propel towards crowded areas, which leads to phase separation. Clusters remain fluid and exhibit fast particle turnover, in contrast to the jammed clusters that typically arise from self-trapping, and interfaces are sufficiently wide that they span entire clusters. Overall, our work identifies a torque-based mechanism for phase separation in active fluids, and our theory predicts that these orientational interactions yield coexisting phases that lack internal orientational order.

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Fig. 1: Active phase separation in metal–dielectric Janus colloids.
Fig. 2: Particle turnover dynamics of non-jammed clusters whose population remains nearly constant.
Fig. 3: Particle orientation, velocity and correlations in non-jammed clusters.
Fig. 4: State diagram of torque-based active phase separation.

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Data availability

Source data are provided with this paper. All other data that support the plots and findings of this study are available from the authors upon request.

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All codes are available from the authors upon request.

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Acknowledgements

J.Z. and S.G. were supported by the taxpayers of South Korea through the Institute of Basic Science, project code IBS-R020-D1. R.A. thanks J. Tailleur for insightful discussions, and acknowledges discussions with the participants of the virtual ‘Active 20’ KITP programme, supported in part by the National Science Foundation under grant no. NSF PHY-1748958. R.A. acknowledges support from the Human Frontier Science Program (LT000475/2018-C). J.Y. holds a Career Award at the Scientific Interface from the Burroughs Wellcome Fund. N.S.W. acknowledges support from the National Science Foundation, through the Center for the Physics of Biological Function (PHY-1734030).

Author information

Author notes

  1. These authors contributed equally: Jie Zhang, Ricard Alert.

Authors and Affiliations

  1. Department of Physics, University of California at Santa Barbara, Santa Barbara, CA, USA

    Jie Zhang

  2. Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ, USA

    Ricard Alert & Ned S. Wingreen

  3. Princeton Center for Theoretical Science, Princeton University, Princeton, NJ, USA

    Ricard Alert

  4. Department of Molecular, Cellular and Developmental Biology, Yale University, New Haven, CT, USA

    Jing Yan

  5. Department of Molecular Biology, Princeton University, Princeton, NJ, USA

    Ned S. Wingreen

  6. Center for Soft and Living Matter, Institute for Basic Science (IBS), Ulsan, South Korea

    Steve Granick

  7. Departments of Chemistry and Physics, Ulsan National Institute of Science and Technology (UNIST), Ulsan, South Korea

    Steve Granick

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  1. Jie Zhang
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Contributions

J.Z., J.Y. and S.G. conceived the experiment. J.Z. performed the experiments and analysed data with help from J.Y. R.A. conceived and developed the theory and analysed data. N.S.W. supervised the theory. All authors discussed and interpreted the results. J.Z., R.A., N.S.W. and S.G. wrote the manuscript.

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Correspondence to Steve Granick.

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Extended data

Extended Data Fig. 1 Ostwald ripening of active colloids.

Series of snapshots showing a small cluster shrinking and disappearing as part of the coarsening process.

Extended Data Fig. 2 Clusters with and without stuck particles.

a, A bright field microscopy image of active phase separation in a late stage with yellow stars labelling particles that, stuck to the coverslip, remain in the same location throughout the experiment. b, The number (magenta) and fraction (blue) of clusters with and without (solid and dotted curves, respectively) at least one stuck particle. In the beginning of the experiment, more clusters are formed without than with stuck particles. With elapsed time, the number of clusters forming both with and without stuck particles decreases (magenta), but clusters with stuck particles become the most abundant.

Source data

Extended Data Fig. 3 Dynamics of cluster growth.

a, Snapshots of cluster growth upon a.c. electric field application (Supplementary Movie 2). Scale bar, 100 μm. The yellow circumference indicates the outline of the cluster in the final snapshot. b, Density profile evolution in the growing cluster shown in a. Averages are over 20 frames (1.2 s) centered at each designated time point. Error bars are s.d.

Source data

Extended Data Fig. 4 Particle-number fluctuations in a quasi-steady state cluster.

The number of particles in this cluster, the same cluster also shown in Figs. 1 and 2, remains roughly constant during the time when averages are taken.

Source data

Extended Data Fig. 5 Particle turnover time depends on cluster size and particle speed.

The turnover time τ, defined in Fig. 2b, depends on the number of particles N in the cluster, and on particle speed and interactions. The number of particles is averaged over the time used to calculate the turnover time τ as in Fig. 2b. This averaging time varies from 30 to 180 s in different clusters. Error bars are s.d. Blue points correspond to an applied electric field with amplitude 66 V/mm and frequency 30 kHz, producing an average single-particle speed of 12 μm/s within clusters. Red points correspond to a field with the same frequency but amplitude 83 V/mm, giving an average single-particle speed of 22 μm/s within clusters. Particles with higher speeds and stronger interactions turn over more quickly.

Source data

Extended Data Fig. 6 Mean square displacements.

Mean square displacements of particle position (a) and angle (b) for particles at different local area fractions ϕ. The particle speed and effective rotational diffusivity shown in Fig. 2c are obtained from the first 0.5 s of these data by fitting MSD = (vt)2 and MSAD = D\({}_{{\rm{r}}}^{{\rm{eff}}}t\), respectively.

Source data

Extended Data Fig. 7 Orientation-velocity coincidence prior to clustering.

Joint probability distribution function (PDF) of the particle orientation and velocity angles, as defined in Fig. 3d, prior to cluster formation. In contrast to the mismatch found in clusters (Figs. 3e and 3f), the orientation and velocity directions coincide before clusters form.

Extended Data Fig. 8 Dynamics and statistics of two-particle interaction events.

a,b, Two examples of interaction events between two particles, which we define by the condition r12 < 3R, with R the particle radius. We also require the minimal interparticle distance to be min r12 < 2.2R. We analyzed 2061 of such interaction events. In a, both particles turn clockwise. Therefore, the interaction torque has the same sign on both particles, showing that torques are non-reciprocal (Γ12 ≠ -Γ21). This type of interaction with particles initially pointing in opposite directions, defined by the condition ∣θ1(t = 0) + θ2(t = 0)∣ < 0.2 rad, occurred in ~33% of the analyzed events. In b, particle 1 changes its orientation very little compared to particle 2, showing another example of non-reciprocal torques. The particles end up aligned in a chain. This type of interaction, defined by the condition \(\min ({\theta }_{1}({\rm{t}})-{\theta }_{2}({\rm{t}}))<0.5\) rad during the interaction event, occurred in ~2.8% of the analyzed events. c,d, Evolution of the interparticle distance, r12, and the change in angle of each particle, Δθ1,2, for the interaction events in a and b, respectively. e,f, Joint probability distribution functions of the angle changes of each particle in an interacting pair. The pre-collision (e) and post-collision (f) phases respectively correspond to the times before and after the particles reach their minimal distance. These histograms show that, statistically, both particles in the interacting pair tend to turn in the same direction, showing that interaction torques are non-reciprocal. Reciprocal torques would lead to particles rotating in opposite directions and by the same magnitude, as indicated by the dashed lines.

Source data

Extended Data Fig. 9 Pair distribution function measured in experiments.

a, Full g(x, y) used in the calculation of the collective force and torque coefficients ζ0, ζ1, and τ1 (see text and Supplementary Eq. (37)). b, Zoomed-in region, which allows to more clearly appreciate that the pair distribution function is anisotropic, indicating that it is more likely to find another particle in front than behind a reference self-propelled particle. In each panel, the arrow indicates the direction of self-propulsion of the reference particle, dashed lines indicate the coordinate axes, and the white circle indicates the region r < 2R of volume exclusion between two particles.

Extended Data Fig. 10 Effective thermodynamics of torque-based MIPS.

Effective free energy density (a, Supplementary Eq. (56)), chemical potential (b, Supplementary Eq. (55)), and thermodynamic pressure (c–d, Supplementary Eq. (57)) of the active Janus suspension as a function of the area fraction of particles, for v0 = 10 μm/s. The remaining parameter values are evaluated using the estimates in Table I in the Supplementary Note. To better visualize the double-well shape of the free energy βf(ϕ), we added a linear term − 16.5ϕ, which does not affect phase coexistence. Thin dashed lines indicate the densities of the coexisting phases. These densities are obtained from the common-tangent construction on the free energy density (a), which corresponds to equality of both chemical potential (b) and pressure (c), as indicated by thick dashed lines. Alternatively, the common-tangent construction also corresponds to the Maxwell construction on the curve p(1/ϕ), as indicated by the thick dashed line in panel d.

Supplementary information

Supplementary Information

Supplementary Notes, Discussion, Fig. 1 and Table 1.

Supplementary Video 1

Coarsening dynamics during active phase separation. This video, taken with a ×5 objective, shows the phase-separation process of a quasi-two-dimensional Janus particle system (ϕ ≈ 0.12) when a.c. electric field (30 kHz, 83 V mm−1) perpendicular to the imaging plane is applied. The video is played at 10× real speed.

Supplementary Video 2

Growth of a representative active cluster. This video, from which the snapshots in Extended Data Fig. 3 are taken, is obtained using a perpendicular a.c. electric field (30 kHz, 83 V mm−1), and is played at 20× real speed.

Supplementary Video 3

Particle turnover dynamics for an active cluster whose population remains nearly constant. The video is taken after the perpendicular a.c. electric field (30 kHz, 67 V mm−1) has been applied for a few minutes and the clusters are coarsening. Particles are colour-coded according to their distance from the cluster centroid in the first frame, which is an arbitrary time point. Particles joining the cluster after the first frame are not labelled. This video is played at 2× real speed and associated with Fig. 2.

Supplementary Video 4

Particle turnover dynamics for two clusters of different sizes. The plots on the right refer to the videos on the left. Native particles are defined as those particles within the frame at the beginning of the video. N0(t) is the number of native particles remaining in the frame after time t. Both videos are taken after the perpendicular a.c. electric field (30 kHz, 67 V mm−1) has been applied for a few minutes. Particles are colour-coded according to their distance from the cluster centroid in the first frame, which is an arbitrary time point. Particles joining the cluster after the first frame are not labelled. This video is played at 2× real speed.

Supplementary Video 5

Flickering chains in an active cluster. Lines are drawn to connect pairs of particles whose centres are separated by less than 3R (R = 1.5 μm), colour-coded by the degree of pair alignment, with red indicating alignment and blue anti-alignment. This video is played at 0.2× real speed. A snapshot of this video is presented in Fig. 3a.

Supplementary Video 6

High-density uniform phase with transient particle chains. This video, taken with a ×20 objective, shows the high-density uniform phase (ϕ ≈ 0.35) with transient particle chains when an a.c. electric field (30 kHz, 83 V mm−1) perpendicular to the imaging plane is applied.

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Zhang, J., Alert, R., Yan, J. et al. Active phase separation by turning towards regions of higher density. Nat. Phys. 17, 961–967 (2021). https://doi.org/10.1038/s41567-021-01238-8

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