Appendix: Brownian Dynamics Simulations of Active Brownian

Chapter II: Active Matter with Spatially Varying Transport Properties

2.7 Appendix: Brownian Dynamics Simulations of Active Brownian

Langevin equations. For active Brownian particles the equations are given by

0=−𝜁𝑼𝛼+𝑭^{𝑠𝑤𝑖 𝑚}𝛼 +𝑭^𝑐𝛼+𝑭^𝐵𝛼, (2.60)

0=−𝜁_𝑅𝛀𝛼+𝑳^𝐵𝛼, (2.61)

𝜕𝒒𝛼

𝜕 𝑡

=𝛀𝛼×𝒒𝛼, (2.62)

where𝑼𝛼 and 𝛀𝛼 are the translational and angular velocities of particle 𝛼. The resistive medium in which the particles move (e.g. a fluid) exert translational and rotational drag forces and torques: −𝜁𝑼𝛼and−𝜁_𝑅𝛀𝛼, respectively. Each individual

particle 𝛼 has its self-propulsive swim force 𝑭^{𝑠𝑤𝑖 𝑚}_𝛼 = 𝜁 𝑈₀𝒒_𝛼, where 𝑈₀ is the free-space swim velocity and 𝒒𝛼 is the direction of swimming. The interparticle collisional force on particle 𝛼 is 𝑭^𝑐𝛼. In dilute suspensions of ABPs interactions between particles are rare and the interparticle forces can be neglected. The random Brownian force/torque are𝑭^𝐵𝛼 and𝑳𝛼^𝐵, with the white noise statistics:

𝑭^𝐵𝛼(𝑡)=0, 𝑭^𝐵𝛼(𝑡)𝑭^𝐵_𝛽(𝑡^′)=2𝜁²𝐷_𝑇𝑰𝛿_{𝛼 𝛽}𝛿(𝑡−𝑡^′), (2.63) 𝑳𝛼^𝐵(𝑡)=0, 𝑳^𝐵𝛼(𝑡)𝑳^𝐵_𝛽(𝑡^′)=2𝜁²

𝑅/𝜏_𝑅𝑰𝛿_{𝛼 𝛽}𝛿(𝑡−𝑡^′). (2.64) Here, · represents the ensemble average, 𝐷_𝑇 is the translational diffusivity, 𝛿_{𝛼 𝛽}is the Kronecker delta, and𝜏_𝑅 =1/𝐷_𝑅is the reorientation time scale.

The BD simulation is performed by integrating the Langevin equations numerically in time. In chapters 2 and 3 we have used the explicit Euler method with intrinsic error from the numerical method of 𝑂( (Δ𝑡)²). However, errors of 𝑂(Δ𝑡) occur in positions of particles whenever they pass the interface of the two regions with different activities. Therefore, the overall magnitude of error of simulations is𝑂(Δ𝑡).

Our choice of Δ𝑡 ensures that errors in positions of particles from each step is at most an order of 0.1% of the shortest length scale in the system. For suspensions in bounded domains, we use the potential-free algorithm [25] to compute forces exerted by hard walls. The pressure exerted by particles on a wall is obtained by dividing the total force exerted by particles on the wall with the area (or length in 2D) of the wall.

References

[1] I. Buttinoni, G. Volpe, F. Kümmel, G. Volpe, and C. Bechinger, “Active Brownian motion tunable by light,”J. Phys. Condens. Matter, vol. 24, no. 28, p. 284 129, Jul. 2012.

[2] J. M. Walter, D. Greenfield, C. Bustamante, and J. Liphardt, “Light-powering Escherichia coli with proteorhodopsin,”Proc. Natl. Acad. Sci., vol. 104, no. 7, pp. 2408–2412, Feb. 2007.

[3] N. A. Söker, S. Auschra, V. Holubec, K. Kroy, and F. Cichos, “How activity landscapes polarize microswimmers without alignment forces,” Physical Review Letters, vol. 126, no. 22, p. 228 001, 2021.

[4] J. Hasnain, G. Menzl, S. Jungblut, and C. Dellago, “Crystallization and flow in active patch systems,”Soft Matter, vol. 13, no. 5, pp. 930–936, Feb. 2017.

[5] J. Stenhammar, R. Wittkowski, D. Marenduzzo, and M. E. Cates, “Light- induced self-assembly of active rectification devices,”Sci. Adv., vol. 2, no. 4, e1501850, Apr. 2016.

[6] G. Frangipane et al., “Dynamic density shaping of photokinetic E. Coli,”

Elife, vol. 7, pp. 1–14, Aug. 2018.

[7] J. Arlt, V. A. Martinez, A. Dawson, T. Pilizota, and W. C. Poon, “Painting with light-powered bacteria,”Nat. Commun., vol. 9, no. 1, p. 768, Dec. 2018.

[8] N. Koumakis, A. T. Brown, J. Arlt, S. E. Griffiths, V. A. Martinez, and W. C.

Poon, “Dynamic optical rectification and delivery of active particles,” Soft Matter, vol. 15, no. 35, pp. 7026–7032, 2019.

[9] T. D. Ross, H. J. Lee, Z. Qu, R. A. Banks, R. Phillips, and M. Thomson,

“Controlling organization and forces in active matter through optically defined boundaries,”Nature, vol. 572, no. 7768, pp. 224–229, 2019.

[10] M. J. Schnitzer, “Theory of continuum random walks and application to chemotaxis,”Phys. Rev. E, vol. 48, no. 4, pp. 2553–2568, 1993.

[11] J. Tailleur and M. E. Cates, “Statistical mechanics of interacting run-and- tumble bacteria,”Phys. Rev. Lett., vol. 100, no. 21, p. 218 103, 2008.

[12] M. E. Cates and J. Tailleur, “When are active Brownian particles and run- and-tumble particles equivalent? Consequences for motility-induced phase separation,”Europhys. Lett., vol. 101, no. 2, p. 20 010, Jan. 2013.

[13] M. E. Cates and J. Tailleur, “Motility-Induced Phase Separation,”Annu. Rev.

Condens. Matter Phys., vol. 6, no. 1, pp. 219–244, Mar. 2015.

[14] N. G. van Kampen, “Diffusion in inhomogeneous media,”Zeitschrift für Phys.

B Condens. Matter, vol. 68, no. 2-3, pp. 135–138, Jun. 1987.

[15] W. Yan and J. F. Brady, “The force on a boundary in active matter,”J. Fluid Mech., vol. 785, R1.1–R1.11, Dec. 2015.

[16] F. Hecht, “New development in freefem++,”Journal of numerical mathemat- ics, vol. 20, no. 3-4, pp. 251–266, 2012.

[17] S. C. Takatori and J. F. Brady, “Forces, stresses and the (thermo?) dynamics of active matter,”Curr. Opin. Colloid Interface Sci., vol. 21, pp. 24–33, Feb.

2016.

[18] A. K. Omar, Y. Wu, Z.-G. Wang, and J. F. Brady, “Swimming to Stability:

Structural and Dynamical Control via Active Doping,” ACS Nano, vol. 13, no. 1, pp. 560–572, Jan. 2019.

[19] S. C. Takatori, R. De Dier, J. Vermant, and J. F. Brady, “Acoustic trapping of active matter,”Nature communications, vol. 7, no. 1, pp. 1–7, 2016.

[20] A. Dulaney and J. Brady, “Waves in active matter: The transition from ballistic to diffusive behavior,”Physical Review E, vol. 101, no. 5, p. 052 609, 2020.

[21] F. J. Sevilla and L. A. G. Nava, “Theory of diffusion of active particles that move at constant speed in two dimensions,”Physical Review E, vol. 90, no. 2, p. 022 130, 2014.

[22] F. J. Sevilla and M. Sandoval, “Smoluchowski diffusion equation for active brownian swimmers,”Physical Review E, vol. 91, no. 5, p. 052 150, 2015.

[23] S. Goldstein, “On diffusion by discontinuous movements, and on the telegraph equation,” The Quarterly Journal of Mechanics and Applied Mathematics, vol. 4, no. 2, pp. 129–156, 1951.

[24] J. Masoliver and G. H. Weiss, “Finite-velocity diffusion,”European Journal of Physics, vol. 17, no. 4, p. 190, 1996.

[25] D. R. Foss and J. F. Brady, “Brownian Dynamics simulation of hard-sphere colloidal dispersions,”J. Rheol., vol. 44, no. 3, pp. 629–651, May 2000.

C h a p t e r 3

REVERSE OSMOTIC EFFECTS IN ACTIVE MATTER

This Chapter includes content from our previously published article:

[1] H. Row and J. F. Brady, “Reverse osmotic effect in active matter,” Physical Review E, vol. 101, no. 6, p. 062 604, 2020. doi:10.1103/PhysRevE.101.

062604,

In the preceding Chapter we obtained analytical and numerical solutions to describe distributions of active matter when its transport properties abruptly vary in space.

Now we consider the mechanics of the systems and reveal more important consequences of the variation in transport properties. Applying a macroscopic mechanical force balance to active suspensions we find a spatial variation in activity can result a spontaneous reverse osmotic effect in active matter. We show that the reverse osmotic effect can be utilized as a novel way to pump fluid from regions of high concentration to low. We also show that an object in a bath of active particles with spatially varying activity can exhibit a reverse diffusiophoretic motion.

3.1 Particle Phase Pressure on Walls

In order to describe the mechanics of active suspensions, we start by considering the pressure exerted by active particles, or the particle phase pressure. When active matter systems are confined, particles accumulate at the no-flux walls in sharp boundary layers [1] as shown in the preceding Chapter. The number density at a wall 𝑛^𝑤 can be much higher than in the bulk 𝑛^{𝑏𝑢𝑙 𝑘} depending on the activity𝑃𝑒 = 𝑈²/(𝐷_𝑇𝐷_𝑅). Since particles exert a local osmotic pressureΠ^{𝑜 𝑠𝑚 𝑜}=𝑛𝜁 𝐷_𝑇 =𝑛 𝑘_𝐵𝑇 at each point in space [2], the pressure exerted by active particles is higher on the wall (Π^𝑤 =𝑛^𝑤𝑘_𝐵𝑇) than far from it (Π^{𝑏𝑢𝑙 𝑘} = 𝑛^{𝑏𝑢𝑙 𝑘}𝑘_𝐵𝑇). The extra pressure on no-flux boundaries due to the activity is called the swim pressureΠ^{𝑠𝑤𝑖 𝑚} = Π^𝑤−Π^{𝑏𝑢𝑙 𝑘} [1, 3]. (See section 1.3 for more detailed discussions on the swim pressure.)

At high (𝑃𝑒 ≫ 1) activities, the swim pressure is the dominant contribution to the particle phase pressure at a wall. Since𝑛^{𝑏𝑢𝑙 𝑘}𝑈 is nearly constant for highly active particles as shown in the previous Chapter (see Fig. 2.3), the pressure exerted by particles on the wallsΠ^𝑤 ≈ Π^{𝑠𝑤𝑖 𝑚} ∼ 𝑛^{𝑏𝑢𝑙 𝑘}𝜁 𝐷^{𝑠𝑤𝑖 𝑚} ∼ 𝜁(𝑛^{𝑏𝑢𝑙 𝑘}𝑈)ℓ will be different in the two regions if the run lengths are differentℓ₁ ≠ℓ₂.

10⁰ 10² 10⁴

0.5 1 1.5 2 2.5 3

10⁰ 10² 10⁴

10⁰ 10¹

100 120 140 160 180 200

300 320 340 360 380 400

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10²

10³

P e2

100 120 140 160 180 200

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100 120 140 160 180 200

300 320 340 360 380

10⁴

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(b)

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U16=U2

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DR16=DR2

Figure 3.1: The ratio of the pressures exerted by ABPs on wall 1 (𝑧=−𝐿₁) and wall 2 (𝑧=𝐿₂) scaled with the ratio of corresponding run lengths when an active suspension is bounded by the two parallel walls as described in Fig. 2.7. The transport properties in the two regions are the same except for (a) the swim speed (𝑃𝑒₁/𝑃𝑒₂ =𝑈²

1/𝑈²

2) or (b) the rotational diffusivity (𝑃𝑒₁/𝑃𝑒₂ = 𝐷_𝑅₂/𝐷_𝑅₁). The two regions have the same length (𝐿₁ = 𝐿₂) and the distance between the walls 𝐿 =𝐿₁+𝐿₂is 100 times longer than the microscopic length scale𝛿₂=√︁

𝐷_𝑇₂/𝐷_𝑅₂ in region 2. Markers are from pressures directly measured from BD simulations and lines are from the analytical solution (2.33)-(2.36). In the both cases the ratio of pressures exerted by particles on the walls isℓ₁/ℓ₂when activity is high (𝑃𝑒≫ 1)..

To test this prediction, we consider an active suspension bounded by parallel planar walls in section 2.3 (see Fig. 2.7 for a schematic description) and directly measure the pressure exerted by particles on the walls Π^𝑤. In BD simulations we monitor the force transmitted when each APB collides with a wall to compute the pressure.

We compare pressures obtained by BD simulations with the pressures computed by Π^𝑤 =𝑛^𝑤𝑘_𝐵𝑇 using the analytical solution (2.33)-(2.36). As shown in Fig. 3.1, the ratio of pressures exerted by ABPs on the two walls is predicted very accurately. At large𝑃𝑒 the pressure is predicted to be linear in the run length, which is borne out precisely as shown in Fig. 3.1.

3.2 Mechanical Description of Active Matter with Spatially Varying Activity Now we take the whole suspension as a control volume as shown in Fig. 3.2 and consider a macroscopic mechanical force balance on the control volume. If the run length of ABPs differs in the two regions, the pressure exerted the ABPs on the walls is also different. The difference in the particle phase pressures at the walls, however, must be balanced by a difference in pressure in the suspending fluid for otherwise there would be a net force exerted on the container. Active motion is a force-free

D

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, D

, U

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, U

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x

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0

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L

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Figure 3.2: A schematic of a suspension of ABPs with an abrupt change in activity confined between two parallel planar walls (see Fig. 2.7 for detailed description of the system). The whole active suspension enclosed by the red dashed box is taken as a control volume. Since ABPs undergo a force-free motion, the net pressure on the left and right (or top and bottom) sides of the control volume must be identical to satisfy a macroscopic mechanical force balance.

motion: particles push off the suspending fluid and their motion is balanced by the fluid drag. A container of active matter cannot exert any net force on its boundary.

This is clear for so-called ‘wet’ active matter — particles immersed in a fluid — but it also applies to ‘dry’ active matter such as crawling particles who push off a substrate; the combination of particles and substrate is force-free.

The force-free condition requires the total pressure in the suspension to be constant (when there is no flow). The total pressure 𝑃 is the sum of the local osmotic pressure Π^{𝑜 𝑠𝑚 𝑜} = 𝑛 𝑘_𝐵𝑇 exerted by particles and the fluid pressure 𝑝_𝑓. Since 𝑃 = 𝑝_𝑓 +𝑛 𝑘_𝐵𝑇 = const., the fluid pressure distribution is the opposite that of the number density. From the number density balance (1.4), the gradient of the fluid pressure is directly related to the polar order and the flux of the active particles:

∇𝑝_𝑓 =−𝜁 𝑈𝒎+𝜁𝒋𝑛. (3.1)

Equation (3.1) is a momentum balance on the fluid. The right-hand side is the net hydrodynamic force exerted by active particles on the fluid [4]. The total hydrodynamic force on an individual swimmer 𝑭^{ℎ 𝑦 𝑑𝑟 𝑜}𝛼 = −𝜁(𝑼𝛼 − 𝒖𝑓) + 𝜁 𝑈𝒒𝛼

consists of the propulsive swim force 𝑭^{𝑠𝑤𝑖 𝑚}_𝛼 = 𝜁 𝑈𝒒_𝛼 and the fluid drag 𝑭^{𝑑𝑟 𝑎𝑔}𝛼 =

−𝜁(𝑼𝛼−𝒖𝑓). Here,𝛼is the index of each swimmer. Averaging over a continuum

Dalam dokumen Effect and Motility-Induced Phase Separation (Halaman 53-57)