Mechanical Description of Active Matter with Spatially Varying Ac-

Chapter III: Reverse Osmotic Effects in Active Matter

3.2 Mechanical Description of Active Matter with Spatially Varying Ac-

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

D

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, D

, U

Region 1 Region 2

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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

volume element with number density𝑛then gives𝑛⟨𝑭^{ℎ 𝑦 𝑑𝑟 𝑜}⟩ =−𝜁𝒋_𝑛+𝜁 𝑈𝒎. The particle flux is 𝒋𝑛 = 𝑛(𝒖𝑝 − 𝒖𝑓), where 𝒖𝑝 = _𝑁¹ Í

𝛼𝑼𝛼 is the average particle velocity. The drag force is proportional to the velocity of a particlerelativeto the fluid,𝑼𝛼−𝒖𝑓, where𝒖𝑓 is the fluid velocity. Only relative motion produces a drag force, a requirement of Galilean invariance.

The fluid momentum balance (3.1) shows that if there is net polar order — a net swim force — in any region this can be balanced by a net particle flux, 𝒋_𝑛 =𝑈𝒎, with ∇𝑝_𝑓 = 0, or if there is no particle flux 𝒋_𝑛 = 0, then there must be a fluid pressure gradient,∇𝑝_𝑓 ≠ 0.

How can we understand the origin of this fluid pressure gradient? When an active particle moves an equal volume of fluid is displaced in the opposite direction — there is no net mass (or volume) flux for force-free motion. (For ‘dry’ active matter, when one takes a step an equivalent amount of ground ‘moves’ in the opposite direction.) Out in the bulk where particles are swimming randomly and there is no polar order, there is no net flux of material across any plane. At a no-flux boundary the active particle’s velocity normal to the boundary is zero; the force from the boundary balances the swim force. However, the particle has not stopped its ‘swim strokes’

and thus is still displacing fluid away from the boundary. Since the fluid does not exit (or enter) the boundary there must be a fluid pressure difference between the wall and the bulk to shut off the flow generated by the active swimming.

This is easily seen from the solution for the concentration and polar order adjacent to a plane wall. From [1]𝑚_𝑧 =−¹

6𝑛^{𝑏𝑢𝑙 𝑘}ℓ𝜆𝑒^{−𝜆 𝑧}, where the accumulation boundary layer thickness 𝜆⁻¹ = 𝛿/√︁

2(1+𝐷^{𝑠𝑤𝑖 𝑚}/𝐷_𝑇). From (3.1) 𝜕 𝑝_𝑓/𝜕 𝑧 = −𝜁 𝑈 𝑚_𝑧 = 𝑛^{𝑏𝑢𝑙 𝑘}𝜁 𝑈 ℓ𝜆𝑒^{−𝜆 𝑧}/6 and thus 𝑝^𝑤

𝑓 − 𝑝^{𝑏𝑢𝑙 𝑘}

𝑓 =−𝑛^{𝑏𝑢𝑙 𝑘}𝜁 𝐷^{𝑠𝑤𝑖 𝑚}. The drop in fluid pressure at the wall is precisely equal to the swim pressure exerted by the active particles on the boundary. The total pressure𝑃= 𝑝^𝑤

𝑓 +𝑛^𝑤𝑘_𝐵𝑇 =𝑝^{𝑏𝑢𝑙 𝑘}

𝑓 +𝑛^{𝑏𝑢𝑙 𝑘}𝑘_𝐵𝑇is constant at each location and there is no net flux of particles or fluid in the system.

A fluid pressure jump occurs not just adjacent to no-flux walls, but also at a point of discontinuity in the swim speed. As seen in Fig. 2.9(b), there is net polar order across the interface pointing into the slower region and a jump in concentration and thus a jump in the osmotic pressure. Since the overall pressure is constant, the jump in fluid pressure Δ𝑝_𝑓 = 𝑝^{𝑏𝑢𝑙 𝑘}

𝑓1 − 𝑝^{𝑏𝑢𝑙 𝑘}

𝑓2 = (𝑛^{𝑏𝑢𝑙 𝑘}

2 −𝑛^{𝑏𝑢𝑙 𝑘}

1 )𝑘_𝐵𝑇 = (𝑛^{𝑏𝑢𝑙 𝑘}

1 +

𝑛^{𝑏𝑢𝑙 𝑘}

2 )𝑘_𝐵𝑇(𝑈₁−𝑈₂)/(𝑈₁+𝑈₂), where we have made use of𝑛𝑈=constant. A fluid pressure difference arises from a difference in swim speeds.

-2 -1 0 1 2 0

0.2 0.4 0.6 0.8 1

Fast Slow

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Total Pressurep=pf + ⇧ = constant

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Fluid Pressurep_f

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Particle Phase Pressure ⇧ =nkBT

Total Pressurep=p_f + ⇧ = constant

Fluid Pressurepf

Particle Phase Pressure ⇧ =nk_BT

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x/L

0 <latexit sha1_base64="lar7Cb4rIaKFEQtzGb4Nlt3lLEU=">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</latexit>

0.5

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0.5

Fast Slow

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p

^w_f

Figure 3.3: (Left) A schematic of the pressure distributions. The pressure exerted by active particlesΠ = 𝑛 𝑘_𝐵𝑇 on a wall is larger in the region with the faster swim speed. Since the total pressure 𝑝 in a force-free active suspension is constant for a mechanical force balance, the fluid pressure 𝑝_𝑓 is lower at the wall in the region with the faster swim speed generating a fluid pressure difference at the two walls Δ𝑝^𝑤

𝑓. (Right) A schematic of a novel pumping device powered by the activity of suspended particles. When two regions with different swim speeds are connected by a tube and the walls are semi-permeable membranes, the fluid pressure difference Δ𝑝^𝑤

𝑓 will generate a flow of fluid from the slower to the faster region — from regions of high concentration to low!

Dalam dokumen Effect and Motility-Induced Phase Separation (Halaman 58-61)