Reverse Diffusiophoresis in Active Bath

Chapter III: Reverse Osmotic Effects in Active Matter

3.4 Reverse Diffusiophoresis in Active Bath

Diffusiophoresis is a directed motion of a colloidal particle in a solution generated by a concentration gradient of bath or solute particles suspended in the solution. In the classical analysis of diffusiophoresis, a boundary layer analysis is performed near the surface of a phoretic particle using a large separation between the characteristic

length scale of the interactive force 𝑏 and the size of a phoretic particle 𝑅(≫

𝑏). As a leading order result in this ‘thin interfacial limit’, local interactions between the bath particles and the phoretic particle are encapsulated as the ‘slip velocity’ of the suspension at the phoretic particle surface. Then a phoretic motion is generated by the slip velocity to maintain the phoretic particle force- and torque- free. When the interactive force is repulsive, e.g. hard-sphere interactions, the resulted phoretic motion is well-known to be down the concentration gradient — from high concentration to low.

Recently Brady [7] has proposed a new continuum formulation of phoretic motion that is broadly applicable both to a passive and active bath. The aim of the current section is to verify an interesting prediction of a reverse diffusiophoresis in [7] — phoretic motion against the concentration gradient even though the interaction is purely repulsive — which is directly related to the reverse osmotic effect in active matter discussed in the preceding section. Here, we briefly highlight the essence of the new approach.

Instead of performing the boundary layer analysis, Brady utilizes the reciprocal theorem for Stokes flow to obtain a general formula for the phoretic velocity𝑼that is applicable to phoretic particles of arbitrary shape and interactive forces of any range and type. This formulation offers a new alternative perspective, in a nutshell, that the phoretic motion is fundamentally powered by interactive or osmotic force exerted by bath particles but in a reduced amount due to suspending fluid — the osmotic force is not entirely transmitted to the phoretic particle because it is also used to move fluid.

The new perspective is most clearly seen by an expression for the phoretic velocity in a bath of hard-spheres of radius𝑏,

𝑼=−𝑹⁻¹_𝐹𝑈 ·

∫

𝑆𝑐

(𝑰− U𝑈) ·𝒏Π𝑑𝑆+ 𝑹⁻¹_𝐹𝑈· 𝑭^𝑒𝑥𝑡 , (3.2) where𝑹𝐹𝑈is the hydrodynamic resistance tensor coupling the force and the velocity of the phoretic particle,𝑆_𝑐is contact surface between the phoretic and bath particles, U𝑈 is a second-order tensor that gives the fluid velocityU𝑈 ·𝑼 when the phoretic particle translates at a uniform velocity𝑼in a quiescent fluid, 𝒏 is a outward unit normal to the phoretic particle surface, Π = 𝑛 𝑘_𝐵𝑇 is the osmotic pressure of bath particles, and 𝑭^𝑒𝑥𝑡 is the external force on the phoretic particle. Equation (3.2) plainly shows that the amount of force expended by the phoretic motion is reduced from the total osmotic force𝑭^{𝑜 𝑠𝑚 𝑜} =−∫

𝑆_𝑐 𝒏Π 𝑑𝑆because the osmotic force is also

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r U

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n

¹

/ U

₀

(x)

¹

Active bath particle

Phoretic Particle

Figure 3.4: A schematic of reverse diffusiophoresis in a bath of active hard-spheres.

When the swim speed𝑈₀ of highly active (𝑃𝑒 ≫ 1) bath particles varies spatially, the number density is lower in the faster than in the slower region satisfying𝑛𝑈₀ ≈ constant. Due to the reverse osmotic behavior of active particles, the osmotic pressure at the contact surfaceΠ|𝑆𝑐 ∝ ℓ₀is higher on the lower concentration side.

The resulted phoretic motion is from regions of low concentration to high even though interparticle interactions are repulsive: the reverse diffusiophoresis.

consumed to generate a fluid flow especially near the no-slip surface of the phoretic body. It is important to note that eq. (3.2) is generally applicable both to passive and active bath particles.

For a spherical phoretic particle of radius 𝑅, eq. (3.2) becomes 𝑼=−𝐿(𝑅_𝑐)

𝜁_𝑅

∮

𝑟=𝑅𝑐

Π𝒏𝑑𝑆+ 1 𝜁_𝑅

𝑭^𝑒𝑥𝑡 , (3.3)

where 𝑅_𝑐 = 𝑅 +𝑏 is the contact radius, 𝐿(𝑅_𝑐) = 3Δ²(1+2Δ/3)/(2(1+Δ)³) is the hydrodynamic mobility function introduced by Brady in [8], Δ = 𝑏/𝑅, and 𝜁_𝑅 = 6𝜋𝜂 𝑅 is the Stokes drag coefficient of the phoretic particle. Equation (3.3) recovers the classical thin interfacial limit whenΔ≪ 1 and the freely draining model, in which hydrodynamic interactions are neglected and hence the entire osmotic force is transmitted to the phoretic particle, whenΔ≫ 1. In the both cases, the direction of the phoretic motion aligns with the total osmotic force 𝑭^{𝑜 𝑠𝑚 𝑜} = −∮

𝑟=𝑅_𝑐Π𝒏 𝑑𝑆 on the phoretic body.

In a classical diffusiophoresis problem a concentration gradient of passive Brownian bath particles is externally imposed to break the symmetry of the osmotic force density at the contact surface. It can be easily shown that the osmotic pressure of passive bath particles at the contact surface is larger on the high-concentration side.

Consequently, the direction of the total osmotic force, and hence the phoretic motion induced by repulsive hard-sphere interactions, is down the concentration gradient.

When bath particles are active, however, a concentration gradient need not be maintained by external measures. As shown in Chapter 2, the number density at steady state can be inhomogeneous by itself when the swim speed spatially varies in the suspension. Moreover, in this case, the osmotic pressure exerted by the active bath particles is smaller on the high-concentration side — the reverse osmotic effect

— as shown in the preceding section (see Fig. 3.1(a)). The resulted phoretic motion is then against the number density gradient even though interactions are purely repulsive: the reverse diffusiophoresis. Brady [7] predicts that the total osmotic force, and hence the reverse diffusiophoretic speed, is linearly proportional to the gradient of swim speed evaluated at the center of the phoretic particle (𝒙 = 0) when the bath particles are highly active (𝑃𝑒 =𝑈²

0/(𝐷_𝑅𝐷_𝑅) ≫ 1) and the spatial variation occurs slowly (∇ℓ₀ ≪ 1):

𝑭^{𝑜 𝑠𝑚 𝑜}=−

∮

𝑟=𝑅_𝑐

Π_∞^{𝑠𝑤𝑖 𝑚}𝒏𝑑𝑆≈ − 𝜁Ω𝑑𝑅^𝑑

𝑐

𝑑²(𝑑−1)𝑛^∞(0)𝑈₀(0)∇ℓ₀(0), (3.4) where Ω𝑑 is the surface area of the unit 𝑑-sphere (e.g. Ω₂ = 2𝜋) and super- and sub-scripted∞indicates background values in the absence of a phoretic particle.

To test this prediction we performed BD simulations and computed the total osmotic force on the phoretic body inside a bath of purely active particles with linear gradient of swim speed. In the simulations we measure the force on a fixed particle in the active bath instead of letting it move freely. From eq. (3.3) this is equivalent to applying the external force𝐹_𝑒𝑥𝑡 (e.g. optical tweezer) to make the phoretic velocity 𝑼 =0. Here, we assume that the number density distribution of active bath particles is not significantly altered by fixing the phoretic particle. We also assume that a fluid flow generated by the local osmotic force can be neglected. These are standard assumptions in the analysis of phoresis, which are valid when the phoretic speed 𝑈/𝑈₀ ≪1.

Figure 3.5 shows that the BD simulation results agree with Brady’s prediction. The total osmotic force has a sign opposite to the swim speed gradient, which indicates that the force is directed towards regions of slower swim speed where the number

10^-3 10^-2 -10^-2

-10^-3 -10^-4

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Slope = 1

Figure 3.5: The total osmotic force 𝑭^{𝑜 𝑠𝑚 𝑜} = −∮

𝑟=𝑅_𝑐Π𝒏 𝑑𝑆on a circular phoretic body fixed at𝑥=0 in a bath of purely active particles versus the linear swim speed gradient 𝑑𝑈₀/𝑑𝑥 of active bath particles. The osmotic force is scaled with the product of the swim force𝜁 𝑈₀(0), background number density𝑛^∞(0)at the center of the phoretic particle, and area pervaded by the phoretic particle and the swim speed gradient is scaled with the rotational diffusivity. The contact radius𝑅_𝑐 is 10 times longer than the run length at the centerℓ₀(0). The results obtained by the BD simulations confirm Brady’s prediction [7] of the reverse diffusiophoresis that the total osmotic force and hence the phoretic speed is linearly proportional to the swim speed gradient.

density is higher. Consequently the resulting phoretic motion is towards regions of higher bulk bath particle concentration even though the active bulk particles repel the phoretic particles — reverse diffusiophoresis! We also confirm that the magnitude of the total osmotic force is linearly proportional to the swim speed gradient. The phoretic speed then is also linearly proportional to the swim speed gradient from eq. (3.3).

We note that in an active bath a phoretic motion can be achieved even without a concentration gradient. For example, suppose that the reorientation time 𝜏_𝑅 spatially varies while the swim speed𝑈₀ and the translational diffusivity 𝐷_𝑇 are uniform throughout a suspension. Then the background number density of active bath particles is constant since 𝑈₀ does not change but the osmotic pressure at the no-flux contact surface Π|𝑆𝑐 is proportional to the run length ℓ = 𝑈₀𝜏_𝑅 (see Fig. 3.1(b)) and therefore varies. Thus, a phoretic motion in this case is down the gradient of the reorientation time𝜏_𝑅. More generally, a spherical phoretic particle in a bath of active hard-spheres moves down a run length gradient regardless of the

concentration gradient as pointed out by Brady [7].

Indeed active bath particles do not even require any spatial variations in transport properties to generate a directed motion of an immersed body. All that matters is a broken symmetry of the osmotic pressure at the contact surface, which can be also achieved by an asymmetric body shape of the immersed object (e.g. chevron, chiral gear, etc.) in active matter systems. From discussions in this section it is clear that the (classical) diffusiophoresis is just a special case of ‘osmophoresis’ [9], in which a concentration gradient is a source of the broken symmetry to generate nonzero total osmotic force on a phoretic body, from Brady’s new perspective. Thus, the reverse diffusiophoresis in an active bath can be considered as an immediate consequence of the reverse osmotic effect in active matter systems.