Transient Behavior of Microswimmers with Abrupt Variation in Ac-

Chapter II: Active Matter with Spatially Varying Transport Properties

2.5 Transient Behavior of Microswimmers with Abrupt Variation in Ac-

speed. We once again note that the slight discrepancy in the number density at high 𝑃𝑒 = 10² results from the 𝑸=0-closure used to describe the boundary layers not from the singular perturbation analysis. The singular perturbation solution is valid as long as the sizes of the two regions are larger than the boundary-layer thickness:

Δ𝑅 ≫ 𝐷_𝑇₁/𝑈₁and(1−Δ)𝑅 ≫ 𝐷_𝑇₂/𝑈₂. When the activity is high (𝑃𝑒 > 10²) the accuracy can be improved by accounting for the nematic field associated with rapid changes of the number density and polar order inside the boundary layers [15].

2.5 Transient Behavior of Microswimmers with Abrupt Variation in Activity

in the properties in section 2.2 (see Fig. 2.1) in order to study the effect of an abrupt change in transport properties. Since the translational diffusivity just smears out spatial gradients, purely active particles (𝐷_𝑇 =0) are considered in order to understand the essential physics. The moment equations closed by𝑸=0are

𝜕 𝑛_𝑖

𝜕 𝑡

+ ∇ · 𝒋𝑛,𝑖 =0, 𝒋𝑛,𝑖 =𝑈_𝑖𝒎𝑖 (2.50)

𝜕𝒎𝑖

𝜕 𝑡

+ ∇ · 𝒋𝑚,𝑖 + (𝑑−1)𝐷_𝑅,𝑖𝒎𝑖 =0, 𝒋𝑚,𝑖 = 1 𝑑

𝑈_𝑖𝑛_𝑖𝑰, (2.51) where subscripts𝑖(=1 or 2)represents the index of regions. We discuss the validity of the assumption 𝑸 = 0 later in this section. For mathematical simplicity we consider an instantaneous infinite line source of particles with random orientations.

The particles are initially located at at the interface of the two regions (𝑥 =0) and released at 𝑡 = 0. The corresponding initial conditions are 𝑛₁ = 𝑛₂ = 𝛿(𝑥) and 𝒎₁= 𝒎₂ =0at𝑡=0. Also at the interface the fluxes are continuous: 𝑛₁𝑈₁ =𝑛₂𝑈₂ and 𝒎₁𝑈₁ = 𝒎₂𝑈₂ at 𝑥=0. Finally particles swim into an infinite free space so 𝑛_𝑖→0 and 𝒎𝑖→0as|𝑥| → ∞. The associated telegraph equation for each region is

𝜕²𝑛_𝑖

𝜕 𝑡²

+ (𝑑−1)𝐷_𝑅,𝑖

𝜕 𝑛_𝑖

𝜕 𝑡

= 𝑈²

𝑖

𝑑

∇²𝑛_𝑖 . (2.52)

For the telegraph equation (2.52) the initial and boundary conditions for the polar order should be rewritten in terms of the number density. From eq. (2.50) the initial condition for the polar order can be expressed as an initial condition for the number density𝜕 𝑛_𝑖/𝜕 𝑡=0 at𝑡=0. Also by integrating eq. (2.50) over each region and using the boundary condition for the polar order, we obtain an integral condition for the particle conservation 𝑁₁+𝑁₂ =1, where 𝑁₁ =∫0

−∞𝑛₁ 𝑑𝑥 and 𝑁₂ = ∫∞

0 𝑛₂ 𝑑𝑥 are the fractions of particles in regions 1 (𝑥 <0) and 2 (𝑥 ≥ 0).

Laplace transforming the telegraph equation (2.52) in time we obtain 𝑠²𝑛ˆ_𝑖 + (𝑑−1)𝐷_𝑅,𝑖𝑠𝑛ˆ_𝑖 =

𝑈²

𝑖

𝑑

𝜕²𝑛ˆ_𝑖

𝜕 𝑥²

, (2.53)

where ˆ𝑛_𝑖(𝑥 , 𝑠) is the transformed number density. The equation is easily solved to give

𝑛_𝑖(𝑥 , 𝑠) =𝑛ˆ_𝑖(0, 𝑠)exp 1

𝑈_𝑖

√︃

𝑑(𝑠²+ (𝑑−1)𝐷_𝑅,𝑖𝑠)

. (2.54)

In order to check the partitioning of particles we integrate the number density in each region and use the convolution theorem to obtain

𝑁_𝑖(𝑡) = 1

√ 𝑑

∫ ^𝑡

𝑈_𝑖𝑛_𝑖(0, 𝑡^′)exp

−𝑑−1

2 𝐷_𝑅,𝑖(𝑡−𝑡^′))

𝐼₀

𝑑−1

2 𝐷_𝑅,𝑖(𝑡−𝑡^′)

𝑑 𝑡^′. (2.55)

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⌧R,1=⌧R,2

<latexit sha1_base64="XUHwMqyyOxG67Q0O2SyG17G8k/M=">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</latexit>

U1= 2U2

<latexit sha1_base64="OZNkhLpjvozWzSEQBOMBqHRbXqY=">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</latexit>

U1=U2

<latexit sha1_base64="aYDreq5la6mSgzmVqxgv9++SKjk=">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</latexit>

⌧R,1= 2⌧R,2

<latexit sha1_base64="BlFbKLR4gggStB7FGOiexdNV0CM=">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</latexit> t=⌧R,1

<latexit sha1_base64="kOoPf1A9TFiJQ6LTutVV1pIBMuY=">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</latexit> t=2⌧R,1

<latexit sha1_base64="ZtD4zDZECCTUUT5YXks5V+DEhvw=">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</latexit> t=4⌧R,1 <latexit sha1_base64="xx4xpk/QhdqcwgStkJKxuhiDosM=">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</latexit>

Low <latexit sha1_base64="JwcFbS5zQAGGaWUYQhMwMfhrQvU=">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</latexit>

High

⌧R,1=⌧R,2

U1= 2U2

U1=U2

⌧R,1= 2⌧R,2

Figure 2.12: (Left) Snapshots from BD simulations of purely active particles released at 𝑡 = 0 from the center (white dashed line) where the reorientation time or swim speed abruptly changes. The colormap shows the local number density of particles. The length of the simulation box is 5 times longer than the run length in region 1 (𝑥 <0). (Right) Transient number densities, polar orders, and nematic orders multiplied by the run length in region 1ℓ₁obtained by BD simulations when purely active particles released at𝑡 = 0 from𝑥 =0 where the reorientation time or swim speed abruptly changes. The probability density is normalized so that∫∞

−∞𝑛 𝑑𝑥=1 and the position is scaled with ℓ₁ as well. The shape of number densities shows transition from wave-like (𝑡 ≤ 𝜏_𝑅) to diffusion-like (𝑡 > 𝜏_𝑅) dynamics. When a step change in the swim speed occurs the even-ordered orientational moments (e.g. 𝑛, 𝑸, etc.) are discontinuous at the jump for the continuity of the translational flux

𝒋𝑇 =𝑈 𝑃𝒒.

Here, 𝐼₀ is a modified Bessel function of the first kind. Since𝑈₁𝑛₁=𝑈₂𝑛₂ at the interface (𝑥=0) for the continuity of the polar order flux, eq. (2.55) implies that the particles are partitioned evenly in each region (𝑁₁=𝑁₂) at all time regardless of the swim speeds when the rotational diffusivities in the two regions are the same. The fraction of particles in each region at (≫ 𝜏_𝑅) times is approximated as

𝑁_𝑖(𝑡) ≈ 1

√︁

𝑑(𝑑−1)𝜋 𝐷_𝑅,𝑖

∫ ^𝑡

𝑈_𝑖𝑛_𝑖(0, 𝑡^′)

√ 𝑡−𝑡^′

𝑑 𝑡^′, (2.56)

10⁰ 10¹ 10²

10⁰ 10¹

100 120 140 160 180 200

300 320 340 360 380

100 120 140 160 180 200

300 320 340 360 380 400

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Analytical <latexit sha1_base64="aJ330Jh0cPFiyudhgFJnsik1ZmM=">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</latexit>

100 120 140 160 180 200

300 320 340 360 380

100 120 140 160 180 200

300 320 340 360 380

U1=U2

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⌧R,1=⌧R,2

Figure 2.13: The ratio of numbers of purely active particles in the two regions after randomly oriented particles are released at the interface of the two regions with different transport properties. Black and blue represent a step change in the reorientation time (ℓ₁/ℓ₂ = 𝜏_𝑅,₁/𝜏_𝑅,₂) and the swim speed (ℓ₁/ℓ₂ = 𝑈₁/𝑈₂), respectively. Markers are from BD simulations and lines are the analytical prediction 𝑁₁/𝑁₂=√︁

𝜏_𝑅,₁/𝜏_𝑅,₂. In the BD simulations the numbers of particles are measured

∼3𝜏_𝑅,₁after the release of particles to ensure the values have reached constant.

where we used 𝐼₀(𝑥) ∼𝑒^𝑥/√

2𝜋𝑥 as 𝑥 → ∞. Thus, at long times the ratio of the fractions (or total numbers) of particles in each region is 𝑁₁/𝑁₂ = √︁

𝜏_𝑅,₁/𝜏_𝑅,₂ — the partitioning of particles is governed by the reorientation time.

This also can be explained by the long-time diffusive behavior of ABPs and the steady state solutions from previous sections. When𝑡 ≫ 𝜏_𝑅ABPs effectively diffuse with the swim diffusivity 𝐷^{𝑠𝑤𝑖 𝑚} ∼ 𝑈²𝜏_𝑅 and the average (root-mean-square) distance reached by the released particles scales as

√

𝐷^{𝑠𝑤𝑖 𝑚}𝑡. Since the number density of particles approaches the steady state value specified by𝑛𝑈=constant, the ratio of the number of particles in each region𝑁₁/𝑁₂ ∼

√︃

𝐷^{𝑠𝑤𝑖 𝑚}

1 𝑡𝑈₂/

√︃

𝐷^{𝑠𝑤𝑖 𝑚}

2 𝑡𝑈₁ =√︁

𝜏_𝑅,₁/𝜏_𝑅,₂. We perform BD simulations to verify this prediction. Figure 2.12 shows general shapes of the Green’s function when the swim speed or reorientation time abruptly changes at the position where particles are released. Even in transient cases, the number density is an order of magnitude larger than the nematic order allwoing the 𝑸=0-closure for simple predictions at a long (> 𝜏_𝑅) time scale. At a short (< 𝜏_𝑅) time scale different closure methods including the effect of the nematic order are known to give better accuracy [20]. The ratio of number of particles obtained by BD simulations is shown Fig. 2.13 and excellently agrees with the prediction that

Dalam dokumen Effect and Motility-Induced Phase Separation (Halaman 47-52)