Research ArticleAPPLIED PHYSICS

Cross-stream migration of active particles

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Science Advances  26 Jan 2018:
Vol. 4, no. 1, eaao1755
DOI: 10.1126/sciadv.aao1755
  • Fig. 1 Schematic illustration of the model system.

    (A) A particle (white sphere) with axisymmetric coverage by catalyst (black) is driven by an external shear flow (large gold arrows) in the Embedded Image direction near a planar wall (gray). The particle has a height h above the wall and an orientation vector p, which can be specified by the angles θ and φ. When the particle is active, the cap emits solute molecules (green spheres). (B) Schematic illustrating the self-diffusiophoretic and chemiosmotic mechanisms that drive the motion of a chemical microswimmer. The self-generated solute gradient (green spheres) drives flows localized to thin boundary layers on the particle surface (magenta arrows) and on the nearby wall (blue arrows). The particle is shown in the primed frame (red arrows). This frame corotates with the particle around the Embedded Image axis so that the Embedded Image and the particle orientation vector p is always in the plane spanned by Embedded Image and Embedded Image.

  • Fig. 2 Effect of imposed flow on passive and active colloidal particles.

    (A) Passive silica-Pt particles in a flow (R = 1 μm). The plot shows the dependence of the rotation time on flow velocity. V* is the translational velocity of the passive particle and is used to characterize the flow rate. The red dashed line is a theoretical scaling derived in section S4 and fitted to the data. Inset: Time-lapse images of a passive particle rolling in a flow. (B) Tracked trajectories of passive particles in a flow (V* = 14 μm/s). (C) Optical microscopy image capturing the distribution of orientations of active particles (Vp ≈ 6 μm/s) without an imposed flow. (D) Tracked trajectories of active particles without a flow. (E) Same system as (C) with an external flow imposed, V* = 14 μm/s. (F) Tracked trajectories of active particles in a flow (V* = 14 μm/s). (G) Angular probability distributions of active colloidal particles in the absence of an imposed flow, V* = 0 μm/s (green), at V* = 14 μm/s (blue) and at V* = 24 μm/s (red). Inset shows the angular evolution of two different active colloids at imposed flow rates of V* = 14 μm/s (blue) and V* = 24 μm/s (red).

  • Fig. 3 The dependence of cross-stream migration on Vp.

    (A) Probability distributions of α for a particle in the absence of a flow with Vp = 6 μm/s and with an imposed flow corresponding to V* = 14 μm/s for particles of Vp = 6 and 3 μm/s. (B to D) Tracked orientation vectors (line segments) and instantaneous velocities (arrows) for trajectories at different flow and self-propulsion speeds.

  • Fig. 4 Schematic of the contributions to planar alignment steady state.

    (Left) A neutrally buoyant, inactive sphere driven by an external shear flow (gold arrows) near a planar wall. Because the external flow spins the particle around the vorticity axis (that is, the normal to the shear plane), the tip of the particle orientation vector p traces a circular path (magenta). (Middle) An active, heavy sphere in quiescent fluid. Because the wall is uniform and the geometry and activity profile of the particle are axisymmetric, the tip of the orientation vector can only rotate in the Embedded Image direction, that is, directly toward or away from the wall. (Right) Planar alignment steady state, where Embedded Image, but the particle orientation vector has nonzero components Embedded Image and Embedded Image in the flow and vorticity directions, respectively. For planar alignment, the component Embedded Image in the flow direction can be either upstream (Embedded Image) or downstream (Embedded Image), as determined by the function Embedded Image; the downstream case is shown. All contributions to Embedded Image are in the Embedded Image direction (see Eqs. 2 to 5). At a certain angle φ*, all contributions to Embedded Image balance, as shown by the arrows, so that Embedded Image. Note that this fixed point always occurs in pairs related by mirror symmetry across the shear plane; we show the state with py > 0.

  • Fig. 5 Numerically obtained trajectories of inactive and active particles in flow.

    (A) Numerically computed trajectory of a passive bottom-heavy particle near a wall (gray) and driven by an external shear flow (gold arrows) with dimensionless strength Embedded Image (concerning dimensionless parameters, see Materials and Methods). The initial condition of the particle is θ0 = 30° and φ0 = 315°, and the particle height is fixed as h/R = 1.2. The particle rotates and is carried downstream by the external flow with no cross-streamline migration. (B) Numerically computed trajectory of an active Janus particle with the external flow strength, particle materials, and initial conditions as in (A). The particle rotates so that its inert face points largely in the Embedded Image direction, with a slight downstream orientation (px > 0). With this steady orientation, the particle swims across flow streamlines as it moves downstream. (C) Probability density function (pdf) for φ, for a catalytic Janus particle in a shear flow with Embedded Image. (D) Probability density function for the same particle with Embedded Image.

  • Fig. 6 Phase portraits of inactive and active particles in flow.

    (A to C) Phase portraits on the sphere |p| = 1 for an inactive, bottom-heavy particle (A) in shear flow with Embedded Image and for an active, bottom-heavy particle (B and C) in a flow with Embedded Image (B) and Embedded Image (C). The red dashed lines indicate the plane of mirror symmetry py = 0. The cyan circle in (A) indicates the center of oscillatory motion. Green circles in (B) and (C) indicate stable fixed points (“attractors”). For (A) and (B), the magenta circles show the initial conditions for the trajectory in Fig. 5A (A) and that in Fig. 5B (B). The parameters characterizing the particle heaviness and activity are given in Materials and Methods. (D) Vector field on the unit sphere representing the contribution of activity to the motion in (B) and (C). The green dashed line indicates pz = 0. (E) Oscillation in pz with time for the inactive particle in Fig. 5A, also corresponding to the magenta circle in (A). The black dashed line gives the result obtained by numerical integration. The red dashed line gives the analytical solution for the linearized equations. The slight disagreement between them is due to the large amplitude of the oscillation, for which the effect of the nonlinearity in Eqs. 2 to 4 is important. (F) For the active particle in Fig. 5B, with initial conditions corresponding to the magenta circle in (B), the oscillation in pz decays with time, and the orientation p eventually approaches the stable fixed point p*. The analytical solution is shown for the later part of the decay, for which pz has a small amplitude and is well described by linear theory.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/1/eaao1755/DC1

    section S1. Activity-induced reorientation of particles

    section S2. Method to detect the in-plane angle of Janus particles

    section S3. Flow profile in square capillary

    section S4. Estimation of shear rate from particle rotation

    section S5. Interim states in cross-stream migration of active particles

    section S6. Effect of particle size on orientational stability

    section S7. Effect of propulsion velocity on orientational stability

    section S8. Contributions of shear to the equations of motion

    section S9. Fixed points of governing equations

    section S10. Linear stability of planar alignment

    section S11. Steady angle of a particle as a function of flow rate

    section S12. Calculation of gravitational contribution to particle motion

    section S13. Motion of particle in three dimensions

    section S14. Brownian dynamics simulations

    section S15. Legends for movies S1 to S3

    fig. S1. Experimentally measured probability distribution of θ for active particles.

    fig. S2. Illustration of method to detect the in-plane angle of Janus particles.

    fig. S3. Flow profile in square capillary.

    fig. S4. Rotation time τ as a function of speed V* for inactive Janus particles in a flow.

    fig. S5. Interim states in cross-stream migration of active particles.

    fig. S6. Probability distribution function of φ for particles of R = 1 and 2.5 μm at Vp 6 μm/s and V* ≈ 24 μm/s.

    fig. S7. Sample trajectories of R = 2.5 μm particles show very little deviation from their preferred cross-stream orientation at Vp ≈ 6 μm/s and V* ≈ 24 μm/s.

    fig. S8. Fluctuations in φ obtained from five different particles for two values of Vp at V* ≈ 24 μm/s.

    fig. S9. The functions f(h/R) and g(h/R) as obtained by Goldman et al. and with the boundary element method.

    fig. S10. Schematic illustration of the three fixed point solutions to Eqs. 2 to 5 in the main text.

    fig. S11. The mean px = |cos (φ*)| plotted as a function of V*.

    fig. S12. Probability distribution of |φ| plotted for two different flow velocities shows a clear shift of the peak position φ* toward 90° at higher flow rates.

    fig. S13. 3D trajectory of a bottom-heavy, chemically active particle in a shear flow.

    fig. S14. Probability distribution functions for components of the particle orientation vector p, obtained from stochastic numerical simulations.

    table S1. Comparison of f(h/R) and g(h/R) as calculated by Goldman et al. and in this work using the boundary element method.

    movie S1. Silica-Pt active Janus particles in the absence of any external shear flow.

    movie S2. Silica-Pt Janus inactive particles in a shear flow (no hydrogen peroxide).

    movie S3. Silica-Pt Janus active particles in a shear flow.

  • Supplementary Materials

    This PDF file includes:

    • section S1. Activity-induced reorientation of particles
    • section S2. Method to detect the in-plane angle of Janus particles
    • section S3. Flow profile in square capillary
    • section S4. Estimation of shear rate from particle rotation
    • section S5. Interim states in cross-stream migration of active particles
    • section S6. Effect of particle size on orientational stability
    • section S7. Effect of propulsion velocity on orientational stability
    • section S8. Contributions of shear to the equations of motion
    • section S9. Fixed points of governing equations
    • section S10. Linear stability of planar alignment
    • section S11. Steady angle of a particle as a function of flow rate
    • section S12. Calculation of gravitational contribution to particle motion
    • section S13. Motion of particle in three dimensions
    • section S14. Brownian dynamics simulations
    • section S15. Legends for movies S1 to S3
    • fig. S1. Experimentally measured probability distribution of θ for active particles.
    • fig. S2. Illustration of method to detect the in-plane angle of Janus particles.
    • fig. S3. Flow profile of in square capillary.
    • fig. S4. Rotation time τ as a function of speed V* for inactive Janus particles in a flow.
    • fig. S5. Interim states in cross-stream migration of active particles.
    • fig. S6. Probability distribution function of ϕ for particles of R = 1 and 2.5 μm at Vp ≈ 6 μm/s and V* ≈ 24 μm/s.
    • fig. S7. Sample trajectories of R = 2.5 μm particles show very little deviation from their preferred cross-stream orientation at Vp ≈ 6 μm/s and V* ≈ 24 μm/s.
    • fig. S8. Fluctuations in ϕ obtained from five different particles for two values of Vp at V* ≈ 24 μm/s.
    • fig. S9. The functions f(h/R) and g(h/R) as obtained by Goldman et al. and with the boundary element method.
    • fig. S10. Schematic illustration of the three fixed point solutions to Eqs. 2 to 5 in the main text.
    • fig. S11. The mean px = |cos (ϕ*)| plotted as a function of V*.
    • fig. S12. Probability distribution of |ϕ| plotted for two different flow velocities shows a clear shift of the peak position ϕ* toward 90° at higher flow rates.
    • fig. S13. 3D trajectory of a bottom-heavy, chemically active particle in a shear flow.
    • fig. S14. Probability distribution functions for components of the particle orientation vector p, obtained from stochastic numerical simulations.
    • table S1. Comparison of f(h/R) and g(h/R) as calculated by Goldman et al. and in this work using the boundary element method.

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    Other Supplementary Material for this manuscript includes the following:

    • movie S1 (.mp4 format). Silica-Pt active Janus particles in the absence of any external shear flow.
    • movie S2 (.mp4 format). Silica-Pt Janus inactive particles in a shear flow (no hydrogen peroxide).
    • movie S3 (.mp4 format). Silica-Pt Janus active particles in a shear flow.

    Files in this Data Supplement:

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