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Generalized Archimedes' principle in active fluids

Nitzan Razin, Raphael Voituriez, Jens Elgeti, and Nir S. Gov
Phys. Rev. E 96, 032606 – Published 15 September 2017
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  • INTRODUCTION
  • ONE-DIMENSIONAL MODEL
  • TWO-DIMENSIONAL MODEL
  • DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We show how a gradient in the motility properties of noninteracting pointlike active particles can cause a pressure gradient that pushes a large inert object. We calculate the force on an object inside a system of active particles with position-dependent motion parameters, in one and two dimensions, and show that a modified Archimedes' principle is satisfied. We characterize the system, both in terms of the model parameters and in terms of experimentally measurable quantities: the spatial profiles of the density, velocity and pressure. This theoretical analysis is motivated by recent experiments, which showed that the nucleus of a mouse oocyte (immature egg cell) moves from the cortex to the center due to a gradient of activity of vesicles propelled by molecular motors; it more generally applies to artificial systems of controlled localized activity.

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    • Received 21 March 2017

    DOI:https://doi.org/10.1103/PhysRevE.96.032606

    ©2017 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Nonequilibrium statistical mechanicsSwimming
    1. Physical Systems
    Active Brownian particlesDry active matterLiving matter & active matter
    1. Techniques
    Fokker–Planck equationLangevin equationMolecular dynamics
    Statistical Physics & ThermodynamicsPolymers & Soft MatterPhysics of Living Systems

    Authors & Affiliations

    Nitzan Razin1, Raphael Voituriez2, Jens Elgeti3, and Nir S. Gov1

    • 1Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
    • 2Laboratoire Jean Perrin and Laboratoire de Physique Théorique de la Matière Condensée, CNRS/Université Pierre et Marie Curie, 75005 Paris, France
    • 3Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, D-52425 Jülich, Germany

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    Vol. 96, Iss. 3 — September 2017

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    • Figure 1
      Figure 1

      (a) The 1D system. (b) For v=d200τ0 and α−1(x)=τ0(|x|/d+1): (i) v, α sketch; (ii) particle density (lines) and number of accumulated particles on the edges (dots), with (colored, solid) and without (gray, dashed) the piston (xp/2d=0.2, wp/2d=0.3). (iii) The force on the piston for varying values of wp. (c) For α−1=2τ0 and v(x)=|x|+d400τ0: (i–iii) as in (b). The force is directed towards the center, and its magnitude saturates when the entire piston is in one side of the system. Note that ℓp=v/α is the same in (b) and (c). ρ0≡N2d, F0≡2dNμtτ0.

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    • Figure 2
      Figure 2

      (a) The 2D system of a disk inside a circular box of active particles used in (b–d). (b) The trajectory of a disk in a Dr gradient, where the line color represents time. The initial and final positions of the disk are marked by dashed and solid gray circles, respectively [v=r030πτ0, Dr−1=τ0rR+1, μtparticle/μtdisk=30]. In (c–d), blue corresponds to v and Dr as in (b), and red to v=r060πτ0(rR+1), Dr−1=2τ0. (c) The particle density as a function of the radial coordinate without a disk (bottom), and the average force on a disk as a function of rd (top). The force is time averaged for a static disk. The validity of this for a moving disk is discussed in Appendix pp7. (d) The normalized angular average over ρ(ϕ)cos(ϕ) in a narrow ring around the disk, as a function of rd. This quantifies the asymmetry in the particle accumulation on the disk surface. Inset: The density in the narrow ring as a function of the angle ϕ (rd=4r03π). For a gradient in Dr, more particles are accumulated on the side with smaller Dr, while for a varying v the accumulation on the disk edge is nearly uniform. R=10r03π, α=0, r0≡πRd, ρ0≡Nr02, F0≡r0Nμtτ0.

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    • Figure 3
      Figure 3

      The effect of the persistence length and disk edge modulation. (a) Sketch of simulated system in b–d: A disk with a modulated edge inside a rectangular 2D system with periodic boundary conditions, and ∇ℓp∥x̂. (b) The force on a modulated disk (n=12) as a function of ℓp(0), divided into three components, for a varying modulation amplitude: R1=0 (solid), 0.038r0 (dashed), 0.075r0 (dotted), and S= an area concentric with the disk with edge at distance r=Rd(ϕ)+0.017r0 from the disk center. The modulated disk area r02 is kept constant. Simulation results: the total force: black, FI1: red, FI2: yellow, FJ: blue. The theoretical calculation of FI1 [Eq. (B4)] is plotted in green. As the modulation amplitude increases, the total force on the disk is increased due to a reduction in magnitude of FI2 and an increase in FJ (F axis is transformed using a log-modulus of 105F/F0). (c, d) The current density around the disk for a modulation amplitude of 0 (c) and 0.075r0 (d) for ℓp(0)=2r03π. Lx=20r03π, Ly=10r03π, xd=5r03π, F0≡r0Nμtτ0, v=r0/τ0, Dr−1=τ0ℓp(0)r0(|xLx−0.5|+0.5), α=0. Similar results are obtained for α−1=τ0ℓp(0)r0(|xLx−0.5|+0.5), Dr=0, as shown in Appendix pp5.

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    • Figure 4
      Figure 4

      (a) System sketch of a 2D channel with walls in the x̂ direction, and periodic boundary condition in the y coordinate. The persistence length is a function of x only. (b) The force on the left wall divided into terms according to Eq. (5) (The total force: black, FI1: red, FI2: yellow, FJ: blue) for the system in (a), for S that is a narrow strip around the wall: S={−ε≤x≤0.5x0,0≤y≤Ly}, for some positive ε. FJ=0 due to the translation symmetry in the y direction. FI2 is nonzero, contrary to the similar 1D system. (Lx=50x0, Ly=100x0, v=x0/τ0, Dr−1=τ0(Bx/x0+1), for varying B=∇ℓp, N=100, F0≡x0Nμtτ0).

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    • Figure 5
      Figure 5

      Scaling of the three terms of force on a disk, for (a) ℓp changes at constant Rd and ∇ℓp (b) Rd changes at constant ℓp and ∇ℓp (The total force: black, FI1: red, |FI2|: yellow, |FJ|: blue). Calculated in a rectangular system with periodic boundary conditions, as in Fig. 3, with v=2r030πτ0, in (a) Dr−1=τ0(|x/Lx−1/2|+0.1B) for various B values, (b) Dr−1=τ0(5|x/Lx−1/2|+2.5).

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    • Figure 6
      Figure 6

      Force terms for Dr(x) vs α(x): (a) Same as Fig. 3, shows the force terms for a system with Dr−1=τ0ℓp(0)r0(|xLx−0.5|+0.5) and α=0 (b) The same plot as (a), for α−1=τ0ℓp(0)r0|xLx−0.5|+0.5 and Dr=0. The persistence time τ=(Dr+α)−1 is the same in (a) and (b).

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    • Figure 7
      Figure 7

      Force on a disk as a function of its position, for (1) a static disk held in place (black), (2) a disk that can move due to the forces applied to it by the ABPs, for various ratios of particle and disk motilities μtparticle/μtdisk=30,300,3000 (colored lines). F0≡r0Nμtτ0.

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    • Figure 8
      Figure 8

      Velocity-velocity correlation, in (a) linear scale, (b) log scale. The second point at Δt≈1/2s is negative, but later a slight exponential decay can be observed. In (b) a linear fit (dashed red line) was performed for all points after the second which have a value of more than 1% of the initial correlation, out of which the parameters of an ABP model were extracted: Dr≈0.4s−1, v≈0.09μm/s, D≈0.009μm2/s.

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    • Figure 9
      Figure 9

      ABP model parameter values from fit to binned velocity-velocity correlation. Error bars are 95% confidence bounds of the fit.

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