Motility-Induced Phase Separation - Mechanical Theory of Phase Coexistence

Chapter IV: Mechanical Theory of Phase Coexistence

4.4 Motility-Induced Phase Separation

which is an integral condition for no net mass transport between the bulk phases.

By comparing eq.(4.36) and the Maxwell construction (4.7), the integrating factor 𝐸(𝜌) = 𝑎/𝜌² up to a nonzero multiplicative constant, which results 𝑏 = 𝜌 𝑑(𝑎/𝜌)/𝑑𝜌.

As a final note, we emphasize that effects ofat leastthe second order density gradients need to be considered to properly describe microscopic details of even-order tensorial quantities (e.g. free energy, pressure, stress, etc.). Odd-order density gradients do not affect the even-order tensors due to the reflection symmetry. For example, even if first order gradient terms are present, they do not impart information about microscopic details to even-order tensorial quantities. Thus, for both thermodynamic and mechanical descriptions of phase interfaces, of which the central physical quantities are free energy and stress, the mass transport condition should include effects of at least the second order gradients in density. Otherwise, we simply obtain macroscopic description of the system as seen in the local density approximation from the preceding section.

drag forces and torques: −𝜁𝑼𝛼 and −𝜁_𝑅𝛀𝛼, where 𝜁 and 𝜁_𝑅 are translation and rotational drag coefficients. The self-propulsive swim force is 𝑭^{𝑠𝑤𝑖 𝑚}𝛼 = 𝜁 𝑈₀𝒒𝛼, where𝑈₀is the free-space swim velocity and𝒒𝛼is the direction of swimming. The interparticle collisional force on particle𝛼is𝑭^𝑐𝛼. Our theory applies to any types of interaction potentials but the results presented in this section are obtained with the Weeks-Chandler-Andersen (WCA) potential to highlight a unique property of MIPS

— a liquid-gas phase separation with purely repulsive interactions. The random Brownian force and reorientation torque are𝑭^𝐵_𝛼 and 𝑳_𝛼^𝐵 and follow the white noise statistics:

𝑭_𝛼^𝐵(𝑡)=0, 𝑭_𝛼^𝐵(𝑡)𝑭^𝐵_𝛽(𝑡^′)=2𝜁²𝐷_𝑇𝑰𝛿_{𝛼 𝛽}𝛿(𝑡−𝑡^′), (4.40) 𝑳𝛼^𝐵(𝑡)=0, 𝑳𝛼^𝐵(𝑡)𝑳^𝐵_𝛽(𝑡^′)=2𝜁²

𝑅/𝜏_𝑅𝑰𝛿_{𝛼 𝛽}𝛿(𝑡−𝑡^′), (4.41) where 𝐷_𝑇 is the translational diffusivity, 𝛿_{𝛼 𝛽}is the Kronecker delta, and 𝜏_𝑅 is the time scale of random reorientations.

A dimensional analysis shows that there are four governing dimensionless parameters in the problem: 𝜙,𝑈₀𝜏_𝑅/𝐷,𝑈²

0𝜏_𝑅/𝐷_𝑇, and𝜖/(𝐷 𝜁 𝑈₀), where𝜙is the volume (or area in 2D) fraction and 𝐷 and 𝜖 are the length and energy parameter of the interaction potential. Here we consider the hard-sphere limit in which the repulsive interactive force is much larger than the Brownian and swim forces: 𝜖/(𝐷 𝜁 𝑈₀) ≫1 and 𝜖/(𝜁 𝐷_𝑇) ≫ 1. Consequently, the problem is essentially governed by three dimensionless parameters: 𝜙,𝑈₀𝜏_𝑅/𝐷, and𝑈²

0𝜏_𝑅/𝐷_𝑇.

Now we coarse-grain the microscopic dynamics of ABPs using the standard Irving- Kirkwood-Noll procedure [33, 37, 38] (see appendix 4.7 for the derivations of equations in the current section) and obtain a particle balance equation to describe the mass transport in the active suspension:

𝜕 𝑛

𝜕 𝑡

+ ∇ · 𝒋𝑛 =0, (4.42)

𝒋_𝑛 =𝑈₀𝒎+ 1 𝜁

𝑭^{𝑒𝑥 𝑡}𝑛+ ∇ ·𝝈^𝑐

−𝐷_𝑇∇𝑛 , (4.43)

where 𝑛 = ⟨Í

𝛼𝛿(𝒙 −𝒙𝛼)⟩ is the average number density field, 𝒋_𝑛 is the particle flux, and 𝝈^𝑐 is the collisional stress of the ABPs. The swimming motion of ABPs brings about a unique advective transport of particles 𝑈₀𝒎, where 𝒎 =

⟨Í

𝛼𝛿(𝒙−𝒙𝛼)𝒒𝛼⟩is the polar order or net-orientation vector field. We note that the particle balance (4.42)-(4.43) is exact.

Equation (4.43) is indeed a mechanical force balance on the ABPs. By multiplying the drag coefficient𝜁to eq. (4.43) and using the Stokes-Einstein-Sutherland relation 𝜁 𝐷_𝑇 = 𝑘_𝐵𝑇, we obtain

0=−𝜁𝒋𝑛+𝜁 𝑈₀𝒎+𝑭^𝑒𝑥𝑡𝑛+ ∇ · (𝝈^𝑐+𝝈^{𝑜 𝑠𝑚 𝑜}), (4.44) where𝝈^{𝑜 𝑠𝑚 𝑜} =−𝑛 𝑘_𝐵𝑇𝑰 is the osmotic stress. The suspending medium exerts the drag force−𝜁𝒋𝑛on particles for their relative motion to the medium. The average of the individual swim forces acts as a self-generated ‘body’ force, 𝜁 𝑈₀𝒎, on the particle phase [39] and the divergence of particle stresses balances the forces.

In order to understand how the self-generated body force develops, we obtain an evolution equation for the polar order from the microscopic dynamics:

𝜕𝒎

𝜕 𝑡

+ ∇ · 𝒋𝑚 + 𝑑−1 𝜏_𝑅

𝒎=0, (4.45)

𝒋_𝑚 =𝑈₀𝑈

𝑸+ 1 𝑑

𝑛𝑰

+ 1 𝜁

𝑭^{𝑒𝑥 𝑡}𝒎+ 1 𝜁

∇ · 𝝈 𝒎 𝑛

−𝐷_𝑇∇𝒎, (4.46) where𝑈(𝑛)is the reduction in the swim velocity from its free- space value𝑈₀due to collisions with other particles, 𝑑 is the spatial dimension for reorientation, and 𝑸 = ⟨Í

𝛼𝛿(𝒙− 𝒙𝛼) (𝒒𝛼𝒒𝛼− 𝑰/𝑑)⟩ is the nematic order tensor field. The nematic order satisfies an equation analogous to the polar order equation:

𝜕𝑸

𝜕 𝑡

+ ∇ · 𝒋_𝑄 + 2𝑑 𝜏_𝑅

𝑸 =0, (4.47)

𝒋𝑄 =𝑈₀𝑈𝑩+𝑈₀𝒎· 𝑈

𝑑+2𝜶− 1 𝑑

𝑰 𝑰

! + 1

𝜁

𝑭^𝑒𝑥𝑡𝑸− 1 𝑑𝜁

∇ ·𝝈^𝑐𝑰−𝐷_𝑇∇𝑸, (4.48) where 𝑩 = ⟨Í

𝛼𝛿(𝒙− 𝒙𝛼) (𝒒𝛼𝒒𝛼𝒒𝛼 −𝜶 · 𝒒𝛼/(𝑑 +2)⟩ is the traceless third order orientational moment tensor field and𝜶is the fourth order isotropic tensor.

The equations for the polar order (4.45)-(4.46) and nematic order (4.47)-(4.48) show the well-known hierarchical structure of orientational moments due to the coupling between the moments of neighboring orders [39, 40]. Due to the sink terms in the equations, a higher order moment has an order of the gradient of a lower order moment (e.g. 𝒎 ∼ ∇𝑛, 𝑸 ∼ ∇𝒎 ∼ ∇∇𝑛, etc.) for high activities (ℓ²

0/(𝐷_𝑇𝜏_𝑅) ≫ 1) that are required to achieve MIPS. Thus, for active suspensions the hierarchy of orientational moments naturally introduces microscopic nonlocal effects via orientational moments even without a density-gradient expansion. As discussed at the end of the preceding section, a mechanical description of interfaces

requires effects of at least the second order density gradients. We close the hierarchy by keeping all terms up to the order𝑂(𝑘²) in Fourier space assuming reasonably slow variations in the number density and investigate results of the minimal closed set of equations.1

It is notable that at steady state the polar order equation (4.45) allows the self- generated ‘body’ force to be written in a form of divergence: 𝜁 𝑈₀𝒎 = ∇ ·𝝈^{𝑠𝑤𝑖 𝑚}, where𝝈^{𝑠𝑤𝑖 𝑚} ≡ 𝜁 ℓ₀𝒋_𝑚/(𝑑−1)is the swim stress [41] andℓ₀=𝑈₀𝜏_𝑅is the run length in a free space. The swim stress is an effective stress arising from the self-propulsive forces. While the swim stress is not a true stress [36, 42] but a ‘body’ force recast as a form of a stress — analogous to the equivalent (or dynamic) pressure in fluid mechanics [17] or Maxwell stress in electrostatics [43] —2it is useful to determine the force exerted by ABPs on no-flux boundaries at steady state [39]. We show below that this steady-state property of active matter systems is indeed important in determining the phase coexistence criteria of MIPS.

Now let us consider a planar MIPS interface with the normal in𝑧direction at steady state when there is no net mass transport 𝑗_𝑛,𝑧 =0. We label the coexisting phases 𝛼 (𝑧→ −∞) and 𝛽 (𝑧→ +∞). Here, we focus on the effects of the self-propelling motion of ABPs to discover the underlying physics of MIPS by considering athermal limit (𝐷_𝑇 = 0), but it is straightforward to include the thermal Brownian motion.

Now, the problem is described by two dimensionless parameters: 𝜙andℓ₀/𝐷. The mechanical force balance and the polar and nematic order equations become

0=𝜁 𝑈₀𝑚_𝑧− 𝑑Π^𝑐 𝑑 𝑧

=0, (4.49)

𝑑 𝑑 𝑧

𝑈₀𝑈

𝑄_{𝑧 𝑧}+ 1 𝑑

𝑛 + 𝑑−1 𝜏_𝑅

𝑚_𝑧 =0, (4.50)

𝑑 𝑑 𝑧

3𝑈 𝑑+2 − 1

𝑑

𝑈₀𝑚_𝑧 + 1 𝑑𝜁

𝑑Π^𝑐 𝑑 𝑧

! + 2𝑑

𝜏_𝑅

𝑄_{𝑧 𝑧} =0, (4.51) where Π^𝑐 is the collisional pressure. Combining and integrating eqs. (4.49) and (4.50), we obtain

𝜁 𝑈₀ℓ₀

𝑑−1𝑈 𝑄_{𝑧 𝑧}+Π^{𝑠𝑤𝑖 𝑚} +Π^𝑐 =const., (4.52)

1The effects of the correlation between the collisional stress and orientation∇ · (𝝈 𝒎/𝑛)/𝜁 (∼

𝑂(𝑘²))in the polar order flux (4.46) is investigated in appendix 4.8.

2Treating the swim stress as a true stress leads to extremely negative interfacial tension of MIPS which contradicts a stable interfaces between coexisting phases [36, 44, 45].

whereΠ^{𝑠𝑤𝑖 𝑚} =𝜁 𝑈₀ℓ₀𝑈 𝑛/(𝑑(𝑑−1))is the swim pressure [8, 41]. From the nematic order equation (4.51),𝑄_{𝑧 𝑧} =0 deep inside phases where spatial gradients vanish due to the homogeneity. The sum of the swim and collisional pressures defines anactive pressure which is identical in the bulk part of coexisting phases. This motivates us to name the undetermined integration constant in eq. (4.49) as the coexistence active pressureΠ_{𝑐𝑜 𝑒𝑥}^{𝑎 𝑐𝑡} :

Π_{𝑐𝑜 𝑒𝑥}^{𝑎 𝑐𝑡} = Π_𝛼^{𝑎 𝑐𝑡} = Π^{𝑎 𝑐𝑡}_𝛽 . (4.53) The active pressure represents the pressure exerted by ABPs on a no-flux boundary at steady-state and acts as an observable steady-state pressure. For the mechanical stability𝜕Π^{𝑎 𝑐𝑡}/𝜕 𝑛 ≥ 0. Thus, the spinodal is determined by𝜕Π^{𝑎 𝑐𝑡}/𝜕 𝑛=0 and the critical point determined by𝜕²Π^{𝑎 𝑐𝑡}/𝜕 𝑛²=0.

To construct the binodal of MIPS, one more condition is required to determine the coexistence active pressure. Now we need to utilize the microscopic information in the continuum-balance equation. From eqs. (4.49), (4.51) and (4.52) we obtain

Π^{𝑎 𝑐𝑡}−Π_{𝑐𝑜 𝑒𝑥}^{𝑎 𝑐𝑡} = 3ℓ²

2𝑑(𝑑−1) (𝑑+2)𝑈 𝑑 𝑑 𝑧

𝑈

𝑑Π^𝑐 𝑑 𝑧

. (4.54)

Using the chain rule eq. (4.54) can be expressed as Π^{𝑎 𝑐𝑡}−Π_{𝑐𝑜 𝑒𝑥}^{𝑎 𝑐𝑡} = 3ℓ²

2𝑑(𝑑−1) (𝑑+2)𝑈

𝜕

𝜕 𝑛

𝑈

𝜕Π^𝑐

𝜕 𝑛

𝑑𝑛 𝑑 𝑧

+𝑈

𝜕Π^𝑐

𝜕 𝑛 𝑑𝑛² 𝑑 𝑧²

, (4.55) which has the the same mathematical form with eq. (4.30).3 Thus, Eifantis and Serrin’s analysis [31] applies and there exists an integrating factor (4.35) such that the integration of eq. (4.55) from phase 𝛼 to 𝛽 makes the second order gradient terms on the right-hand side disappear. Indeed the existence of such integrating factor is more clearly seen by using the collisional pressure gradient𝑑Π^𝑐/𝑑 𝑧for the original equation (4.54):

∫ Π^𝑐_𝛽

Π^𝑐𝛼

Π^{𝑎 𝑐𝑡} −Π_{𝑐𝑜 𝑒𝑥}^{𝑎 𝑐𝑡}

𝑑Π^𝑐 = 3ℓ²

2𝑑(𝑑−1) (𝑑+2)

∫ ∞

−∞

𝑈 𝑑Π^𝑐

𝑑 𝑧 𝑑 𝑑 𝑧

𝑈

𝑑Π^𝑐 𝑑 𝑧

𝑑 𝑧

= 3ℓ²

4𝑑(𝑑−1) (𝑑+2)

𝑈

𝑑Π^𝑐 𝑑 𝑧

2#^𝛽

𝛼

=0, (4.56) where the last equation is obtained by the homogeneity in each bulk phase.

3We note that the gradient terms are all multiplied byℓ²

0and thus are unique to active suspensions.

Gradient terms similar to those present in equilibrium systems could be included in (4.54) but are not necessary. Furthermore, these terms are directly connected to the surface tension of an interface, and MIPS interface has been shown to have essentially zero surface tension [36].

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Mechanical Theory Maxwell Construction

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0 5 10

8 10 12

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⇧

8 10 12

⇧

act

= ⇧

+ ⇧

swim

0 4 10

⇧

^coex

Mechanical Theory Maxwell Construction

(a)

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(b)

Figure 4.2: (a) Generalized equal-area construction from the mechanical theory (eqs. (4.53) and (4.56)) in the (Π^{𝑎 𝑐𝑡} = [Π^𝑐 +Π^{𝑠𝑤𝑖 𝑚}], Π^𝑐) plane and (b) Maxwell construction forℓ₀/𝐷 =35×2^−1/6in 2D. Pressures are normalized by𝜁 𝑈₀/(𝜋 𝐷). The shaded areas are visual guides for equal-area constructions.

The equivalence of the active pressures (4.53) and the new integral condition (4.56) allows a new generalized equal-area construction for MIPS rather than the con- ventional Maxwell construction (4.6) for thermodynamic systems. In fig. 4.2(a) we show the equal-area construction of the active pressureΠ^{𝑎 𝑐𝑡} = Π^𝑐+Π^{𝑠𝑤𝑖 𝑚} vs.

Π^𝑐 that determines phase coexistence. From the equal-area construction the collisional pressure in the two phases is determined and then the corresponding number densities are found from the equation of state (EOS) for the collisional pressure.

Figure 4.2(b) shows what the corresponding Maxwell construction would predict.

The predicted coexistence pressure is larger and correspondingly the dilute phase

Homogeneous

Phase-separated

0.2 0.4 0.6 0.8

0 0 20 40 60

80

Mechanical Theory

Maxwell Construction

Simulation Critical point (theory) Critical point (simulation)

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Area fraction

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Ac ti v it y `

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0 0.2 0.4 0.6 0.8 0

20 40 60 80

Figure 4.3: Mechanical theory prediction for the phase behavior of active suspensions (solid line) compared with simulations (open circles). Here,ℓ₀/𝐷 =𝑈₀𝜏_𝑅/𝐷 and𝜙=𝜋 𝐷²𝑛/4, is the area fraction of active particles of diameter 𝐷. Also shown is the phase boundary from using a Maxwell construction (dotted line).

density is higher.

In order to construct aΠ^{𝑎 𝑐𝑡} vs. Π^𝑐 plot we need expressions for the collisional and swim pressures as a function of the number density𝑛. For the equal-area construction the equaion of state (EOS) is required. In the following, we use the EOS of ABPs in 2D from the work by Mallory, Omar, and Brady [8], which includes a careful discussion of how to extrapolate homogeneous system behavior into the two-phase region. The EOS in 3D can be obtained from Brownian dynamics simulations by following the procedures in [8] as

Π^𝑐

𝜁 𝑈₀/(𝜋 𝐷²) =6×2^−7/6 𝜙²

√︁1−𝜙/𝜙₀

, (4.57)

Π^{𝑠𝑤𝑖 𝑚}

𝜁 𝑈₀/(𝜋 𝐷²) =𝜙 ℓ₀

𝐷

𝑈 =𝜙 ℓ₀

𝐷 1+

1−exp

−2^7/6 ℓ₀

𝐷

𝜙 1−𝜙/𝜙₀

−1

(4.58) where𝜙₀ =0.645 is the maximum random packing density achieved in 3D from the simulations. The extrapolation can be challenging, but is made easier here because the collisional pressure is a monotonically increasing function of concentration and, while the swim pressure displays a maximum, the reduction in the swim speed 𝑈(𝑛) is a monotonically decreasing function of concentration; their sum displays the van-der Waals-like loop shown in figs. 4.2(a) and 4.2(b).

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z/D

Figure 4.4: Interfacial profiles of the number density, polar order, and nematic order scaled by the volume of a particle𝑉_𝑝 =𝜋 𝐷³/6 at the interface forℓ₀/𝐷 =50 in 3D obtained by the a) Brownian Dynamics (BD) simulation and (b) mechanical theory.

Figure 4.3 shows a comparison of this new mechanical theory for the motility- induced phase separation (MIPS) of ABPs in 2D. Also shown for comparison is the phase coexistence predicted via the a Maxwell construction, which over predicts the concentration in the dilute phase. The same Π^𝑐 are Π^{𝑠𝑤𝑖 𝑚} are used in both constructions. There are no adjustable parameters in the comparison.

The mechanical theory not only predicts phase coexistence, but, by its very nature, makes predictions about the structure of the interface that can be compared with simulation. A solution of eq. (4.55) is shown in fig. 4.4, where the interfacial profile, polar order,𝑚_𝑧and nematic order,𝑄_{𝑧 𝑧}, are shown in comparison to simulation in this case for 3D in the two-phase region at an activityℓ₀/𝐷 =50. From the particle force balance (4.49), the difference in the collisional pressures between the two phases is balanced by the integral of the swim force: ΔΠ^𝑐 =∫

𝜁 𝑈₀𝑚_𝑧𝑑 𝑧. Particles at the

Dalam dokumen Effect and Motility-Induced Phase Separation (Halaman 82-91)