Self-organised structures in mixed active-passive suspensions due to hydrodynamic interactions

This study utilizes three-dimensional Stokesian dynamics to demonstrate that hydrodynamic interactions in mixed suspensions of bottom-heavy squirmers and passive spheres can induce novel self-organised structures, such as fibrillar and lamellar phase separations, while generally disrupting orientational order.

The Gist

Imagine a crowded dance floor. On one side, you have a group of energetic dancers who can move on their own (the active swimmers). On the other side, you have a group of people who just stand there or shuffle along passively, unable to dance on their own (the passive obstacles).

This paper is a scientific investigation into what happens when you mix these two groups together in a 3D space, like a giant, invisible ballroom filled with water. The researchers wanted to know: Do the dancers organize themselves? Do they push the passive people around? And does the type of dance move matter?

Here is a simple breakdown of their findings, using everyday analogies.

1. The Dancers: Three Types of Moves

The researchers studied three types of "dancers" (microswimmers), defined by how they push against the water:

• Pushers (The "Back-Propellers"): Like a swimmer pushing water backward to go forward. They push things away from their sides.
• Pullers (The "Front-Propellers"): Like a swimmer pulling water toward them to go forward. They pull things toward their sides.
• Neutrals: They don't push or pull much; they just glide.

2. The Problem: The "Crowded Room" Effect

In a room full of only dancers, they often form beautiful, synchronized patterns (like a flash mob). But when you throw in a bunch of passive people (obstacles), it gets messy.

• The Finding: The passive people act like traffic cones. They disrupt the dancers' ability to sync up. Unless there are very few passive people, the dancers struggle to form a coordinated group. The "traffic" breaks the rhythm.

3. The Twist: Gravity as a "Conductor"

The researchers added a special rule: Bottom-Heaviness. Imagine the dancers are wearing heavy boots on their feet. This makes them naturally want to stand upright (align with gravity). This acts like a conductor telling the dancers which way to face.

When this "conductor" is introduced, things get interesting:

A. The "Fiber" Formation (Medium Density)

When the room is moderately crowded and the dancers are "bottom-heavy," they don't just mix randomly. They sort themselves out!

• The Analogy: Imagine the dancers forming lanes or highways to run down, while the passive people get stuck in the "shoulder" areas between the lanes.
• The Result: The active swimmers form long, thin streams (fibers) moving in one direction, leaving the passive particles behind in a nearly stationary zone. It's like a highway with fast cars and a slow-moving shoulder.

B. The "Sandwich" Effect (High Density + Strong Pullers)

This is the most surprising discovery. When the room is very crowded, the "Puller" dancers (who pull things toward them) and the passive particles create a strange, layered structure.

• The Analogy: Imagine a club sandwich.
  • Layer 1 (Top): A solid wall of passive particles.
  • Layer 2 (Middle): A layer of Puller swimmers pushing against that wall.
  • Layer 3 (Bottom): An empty gap of water.
• How it works: The Pullers swim forward and pull the passive particles with them, creating a solid "wall" of passive stuff. But because the room is so packed, the Pullers can't push through the wall. So, they get stuck behind it, creating a gap. The passive particles are essentially being "herded" into a solid block by the swimmers.

4. Why Does This Matter?

You might wonder, "Who cares about microscopic balls in water?"

• Real World Connection: This isn't just about balls. It's about bacteria, algae, and cells living in complex environments like soil, your intestines, or mucus.
• The Takeaway: If you want to move things around in a complex fluid (like delivering medicine through the body), you can't just rely on the bacteria swimming. You have to understand how they interact with the "traffic" (other particles) and how external forces (like gravity or magnetic fields) change their behavior.
• The Lesson: The type of swimmer matters. Pushers, pullers, and neutrals behave completely differently when mixed with obstacles. If you want to control them, you need to know which "dance move" they are doing.

Summary

In a nutshell:

1. Without help: Mixing active swimmers with passive obstacles usually creates chaos and stops them from organizing.
2. With gravity (the conductor): The swimmers can organize into lanes or "sandwiches," depending on how crowded it is and what kind of swimmer they are.
3. The Surprise: Passive particles can get "herded" into solid blocks or lanes, changing how they move and how the whole system flows.

The study shows that in the microscopic world, hydrodynamics (how fluids move around objects) is the invisible hand that arranges the furniture, creating structures we wouldn't expect just by looking at the particles themselves.

Technical Summary

1. Problem Statement

Microswimmers (active particles) in suspension exhibit diverse collective behaviors, such as motility-induced phase separation (MIPS), bacterial turbulence, and bioconvection. While the dynamics of pure active suspensions and the "active doping" effect (few active particles in a passive bath) are well-studied, the behavior of dense mixed suspensions containing significant proportions of both active and passive particles remains poorly understood.

The primary challenge lies in accurately modeling many-body hydrodynamic interactions (HIs). Previous studies often relied on Active Brownian Particle (ABP) models which neglect HIs, or focused on quasi-2D geometries. The formation of self-organized structures in 3D mixed suspensions, specifically how passive obstacles influence the collective dynamics of different swimmer types (pushers, pullers, neutrals) under varying densities and external torques, has remained unexplored due to computational complexity.

2. Methodology

The authors employed Stokesian Dynamics to simulate the 3D dynamics of a suspension containing $N=216$ spherical particles in a periodic domain.

• Particle Models:
  • Active Swimmers: Modeled as "squirmers" with a surface velocity profile defined by the squirmer parameter $\beta$ (determining pusher, neutral, or puller behavior) and swimming speed $U_0$.
  • Passive Particles: Modeled as inert, non-Brownian spheres with no-slip boundary conditions.
  • Bottom-Heaviness: An external orienting torque ($G_{bh}$) was applied to simulate gravitational alignment (gravitaxis), ranging from none ($G_{bh}=0$) to weak ($10$) and strong ($100$).
• Hydrodynamics:
  • The simulation accounts for both far-field interactions (via Ewald summation) and near-field interactions (via lubrication theory and a pre-computed database).
  • Steric repulsion forces prevent particle overlap.
• Parameters:
  • Volume Fraction ($\phi$): Ranged from 0.1 to 0.5 (high density).
  • Active Fraction ($\alpha$): Varied from 1/6 to 1.
  • Swimmer Types: Pushers ($\beta = -1, -3$), Neutrals ($\beta = 0$), and Pullers ($\beta = 1$).
  • Initial Conditions: Both initially mixed and initially phase-separated configurations were tested.
• Metrics:
  • Coherence: Measured by mean orientation vector $|\langle \mathbf{e} \rangle|$ and mean velocity.
  • Transport: Measured by mean-squared displacement (MSD) and dispersion tensors.
  • Phase Separation: Quantified using a spatial cross-correlation metric ($S$) derived from binned particle densities.

3. Key Contributions

• First 3D Analysis of Dense Mixed Suspensions: This study provides the first comprehensive 3D investigation of how passive obstacles alter the self-organization of active squirmers, explicitly including full hydrodynamic interactions.
• Discovery of Novel Structures: The identification of a unique "sandwich-like" lamellar phase separation in strongly bottom-heavy puller suspensions, a structure not observed in pure active suspensions.
• Mechanism of Order Disruption: Demonstrating that passive particles generally disrupt orientational order in mixed suspensions by halving the effective alignment interactions (superposition of swimmer-passive vs. swimmer-swimmer interactions).
• Role of Bottom-Heaviness: Systematically mapping how external torques (simulating gravity) can dynamically induce phase separation and specific transport mechanisms that are absent in isotropic systems.

4. Key Results

A. No Bottom-Heaviness ($G_{bh} = 0$)

• Disruption of Order: In initially mixed suspensions, the presence of passive spheres generally prevents the formation of coherent orientational order for pushers and neutral/puller swimmers, unless the active fraction is very high ($\alpha \ge 5/6$).
• Metastable Phase Separation: For neutral and puller squirmers at very high densities ($\phi=0.5$) and low active fractions ($\alpha \le 0.5$), an initially phase-separated state is metastable. The passive layer acts as a quasi-2D confinement, allowing a coherent state to form within the active layer. However, this state does not emerge dynamically from a mixed state; it relies on initial conditions.
• Transport: Passive particle transport is negligible unless a coherent active state exists, in which case passive particles exhibit ballistic motion at high densities.

B. Weak Bottom-Heaviness ($G_{bh} = 10$)

• Dynamic Phase Separation (Fibrillar): The external torque breaks symmetry, leading to dynamic phase separation into "lanes" or fibrillar structures.
  • Neutral/Pullers: Form aligned jets of swimmers separated by nearly stationary passive regions.
  • Pushers: Strong pushers struggle to maintain order due to trapping by obstacles, leading to suppressed alignment compared to neutral/puller types.
• Transport Efficiency: Pushers are surprisingly efficient at transporting passive particles at low-to-medium densities due to lateral hydrodynamic attraction, despite their tendency to destabilize order.

C. Strong Bottom-Heaviness ($G_{bh} = 100$)

• Coherent Transport: Strong alignment enforces order across all swimmer types.
• Sandwich-like Lamellar Structure (Novel Finding): In puller suspensions at high density ($\phi \ge 0.4$) and moderate-to-high active fractions, a unique 3-layer structure emerges:
  1. A layer of active swimmers.
  2. A layer of passive particles pushed by the swimmers.
  3. A fluid gap.
  • Mechanism: The dipolar flow of pullers attracts other pullers horizontally and directs passive particles to accumulate in front/behind. Strong bottom-heaviness prevents reorientation, stabilizing this layered "sandwich" against the breakup instabilities seen in pure puller suspensions.
• Pusher Transport: Strong pushers maintain coherent formations and transport passive particles via strong lateral hydrodynamic attraction.

5. Significance and Implications

• Biological Relevance: The findings offer insights into how microorganisms (e.g., Volvox, Chlamydomonas, magnetotactic bacteria) behave in complex, crowded environments like soil, the gut, or polymer solutions where non-motile cells or debris are present.
• Control of Active Matter: The study highlights that hydrodynamic interactions are the dominant factor in determining microstructure. It suggests that external torques (gravity or magnetic fields) can be used to control the transport and separation of active and passive components in complex fluids.
• Theoretical Advancement: The results challenge the assumption that MIPS or turbulence in active suspensions behave similarly in mixed systems. The presence of passive particles fundamentally alters the stability of collective states, often suppressing order unless specific conditions (high density, strong external torque) are met.
• Future Directions: The study identifies the need for larger computational domains to fully analyze the stability of the observed lamellar bands, as the current domain size may be insufficient to capture long-wavelength instabilities.

In conclusion, this paper establishes that the interplay between swimmer type, passive obstruction, and external alignment creates a rich phase space of self-organized structures, with hydrodynamic interactions playing a critical role in both the formation and stability of these states.