Flow Coupling Alters Topological Phase Transition in Nematic Liquid Crystals

This study demonstrates that while passive nematic fluids without flow alignment exhibit a standard Berezinskii--Kosterlitz--Thouless transition, coupling to fluid flow in strain-rate-aligning nematics fundamentally alters the topological phase transition by promoting bend-splay walls that lower the defect nucleation threshold and prevent defect recombination, thereby suppressing the canonical BKT scenario.

The Gist

Imagine a crowd of people at a concert, all trying to face the same direction. In physics, this is like a nematic liquid crystal—a material where molecules are free to move around but generally point the same way, like a school of fish or a crowd of fans.

Sometimes, things go wrong. Two people in the crowd might face opposite directions, creating a "knot" or a defect. In the world of physics, these are called topological defects (specifically, +1/2 and -1/2 defects).

For decades, scientists believed these defects behaved like a predictable game of magnetic attraction.

• The Old Rule (BKT Transition): At low energy (cold), the "knots" would pair up and hold hands, keeping the crowd orderly. As you heated things up (added energy), the pairs would let go, scatter, and the crowd would become chaotic. This was a reversible dance: cool it down, and they would hold hands again.

This paper asks a simple question: What happens if the crowd isn't just standing still, but is also flowing like a river? Does the current change how the knots behave?

The Big Discovery: The "River" Changes the Rules

The researchers ran massive computer simulations to see what happens when these liquid crystals flow. They found that the answer depends entirely on how the molecules react to the flow.

1. The "Floaters" (Non-Aligning Nematics)

Imagine a crowd of people wearing life jackets who don't care which way the water is flowing; they just float.

• Result: Even with the water moving, the old rules apply. The knots still pair up when it's cold and scatter when it's hot. The flow didn't break the dance.

2. The "Surfers" (Strain-Rate Aligning Nematics)

Now, imagine a crowd of surfers. When the water ripples or shears, they instinctively turn to ride the wave.

• Result: Everything breaks.
• The "Wall" Effect: Instead of just floating, the flow creates invisible "walls" or ridges in the water. These walls act like fences.
• The Broken Dance: When a knot (defect) is born, it gets stuck on these walls. The flow pushes it along the wall, but the wall prevents it from finding its partner to hold hands.
• The Consequence: Once the knots appear, they never pair up again, no matter how much you cool the system down. They remain scattered and chaotic forever. The "reversible" dance is gone; the system is stuck in a state of permanent disorder.

The Active Twist: The "Self-Powered" Crowd

The paper also looked at Active Nematics—materials where the molecules are alive and generate their own energy (like bacteria swimming).

• Result: It doesn't matter if they are "floaters" or "surfers." Because they are constantly swimming and creating their own currents, they generate their own "walls" and turbulence.
• Outcome: The knots are always unbound. They are constantly being pushed apart by their own energy. The orderly pairing never happens.

The Takeaway in Plain English

Think of the BKT transition (the old rule) as a game of Musical Chairs where the music stops, and everyone finds a partner.

• Without Flow: When the music stops (cooling), everyone finds a partner. When the music starts (heating), they let go.
• With Flow (The Surfers): The music is playing, but there are conveyor belts (the flow walls) running through the room. Even when the music stops, the conveyor belts keep pushing people apart. They can't find their partners because the floor itself is moving them away.

Why does this matter?
This study shows that the "textbook" rules of how these materials change from order to chaos are only true if the material is perfectly still. In the real world, where things flow, move, or are active (like in our bodies or in new robotic materials), the rules are totally different. The flow doesn't just speed things up; it fundamentally rewrites the laws of physics for how these materials organize themselves.

In short: If you want to control these materials (for better screens, medical devices, or soft robots), you can't just look at the temperature. You have to control the flow, because the flow decides whether the pieces can ever stick together again.

Technical Summary

1. Problem Statement

The paper investigates how coupling the orientational order of nematic liquid crystals to hydrodynamic flow influences topological phase transitions, specifically the defect-mediated transition between ordered and disordered states.

• Context: In purely relaxational (non-hydrodynamic) 2D nematics, the transition follows the Berezinskii–Kosterlitz–Thouless (BKT) scenario. Here, $\pm 1/2$ topological defects form bound pairs at low temperatures (quasi-long-range order) and unbind into a gas of free defects at high temperatures.
• The Gap: While it is known that hydrodynamic backflow creates an asymmetry between $+1/2$ and $-1/2$ defects, it remains unclear whether flow coupling merely renormalizes the BKT transition temperature or fundamentally alters the nature of the transition (i.e., whether bound pairs can still exist).
• Key Question: Does the flow-alignment parameter ($\lambda$), which dictates how the nematic director responds to shear/strain, preserve the BKT binding-unbinding mechanism, or does it destabilize bound pairs entirely?

2. Methodology

The authors employed large-scale two-dimensional fluctuating nematohydrodynamic simulations using a hybrid Lattice Boltzmann method.

• Governing Equations:
  • Orientational Field ($Q$): Evolved via the Beris–Edwards equations, incorporating a molecular field derived from the Landau–de Gennes free energy and elastic Oseen–Frank terms.
  • Flow Field ($\mathbf{u}$): Evolved via generalized Navier–Stokes equations for incompressible fluids, including viscous, elastic (backflow), and active stress contributions.
  • Fluctuations: Thermal noise ($\xi_Q, \xi_u$) was added as Gaussian random variables to satisfy detailed balance, controlled by an effective dimensionless temperature $T^*$.
• Control Parameters:
  • Flow-Alignment Parameter ($\lambda$):
    • $\lambda = 0$: Non-aligning nematics (director responds only to vorticity, not strain).
    • $\lambda = 1$: Strain-rate-aligning nematics (director aligns with shear flow).
  • Activity ($\zeta$): Simulations were run for both passive (thermal fluctuations only) and active (self-generated stress $\Pi_{active} = -\zeta Q$) systems.
• Protocol: The system was driven through forward (increasing $T^*$) and backward (decreasing $T^*$) protocols to observe hysteresis and reversibility. System sizes ranged from $L=128$ to $512$.

3. Key Contributions & Results

A. Passive Nematics: The Role of Flow Alignment

The study reveals that the presence of flow coupling does not uniformly preserve the BKT scenario; the outcome is strictly controlled by $\lambda$.

1. Non-Aligning Nematics ($\lambda = 0$):

   • Result: The system retains the BKT scenario.
   • Behavior: Defects form bound pairs at low $T^*$ and unbind at high $T^*$. The transition is reversible; cooling the system allows defects to recombine and annihilate, restoring order.
   • Note: The defect creation threshold is slightly lower than in the flow-decoupled case, but the mechanism (binding/unbinding) remains intact.

2. Strain-Rate-Aligning Nematics ($\lambda \neq 0$):

   • Result: The BKT scenario is fundamentally broken.
   • Mechanism: The coupling between orientation and strain rates induces elastic backflow, leading to the spontaneous formation of bend-splay walls (extended orientational distortions).
   • Consequences:
     • These walls lower the defect nucleation threshold significantly ($T^*_{dc} \approx 0.004$ vs $0.13$ in decoupled systems).
     • Persistent Unbinding: Once defects are created, they never form stable bound pairs, regardless of temperature or protocol direction.
     • Wall-Mediated Dynamics: Walls act as barriers that screen the logarithmic ("Coulombic") attraction between opposite charges. Defects are advected along these walls and toward junctions, promoting mixing but preventing recombination.
     • Irreversibility: In the backward protocol, defects do not rebind efficiently; the system remains in a disordered state due to the screening effect of the walls.

B. Active Nematics

• Result: In active nematics (where activity injects energy continuously), defects remain unbound across all activity strengths.
• Independence of $\lambda$: Unlike the passive case, activity generates persistent flow structures even when $\lambda = 0$. Consequently, the binding mechanism is suppressed regardless of the alignment parameter. The system enters a regime of sustained defect unbinding and turbulent dynamics.

C. Quantitative Validation

The authors quantified these findings using:

• Nearest-Neighbor Distance ($r_{nn}$): In $\lambda \neq 0$ and active systems, $r_{nn}$ remains close to the value expected for uncorrelated, free defects ($0.5 r_{av}$) at all temperatures.
• Clustering Analysis: Using neutralization length ($l_{max}$) and percolation length ($l_{perc}$), they showed that for $\lambda \neq 0$, $l_{max} = l_{perc}$ at all temperatures, confirming the absence of stable bound clusters.
• Correlation Functions: Director-director correlations decay rapidly in aligning systems, indicating finite-size domains rather than quasi-long-range order.

4. Significance and Implications

• Fundamental Physics: The paper demonstrates that the canonical BKT transition is a specific case that emerges only in the absence of flow coupling (or specifically when $\lambda=0$). In most experimental nematic systems where $\lambda \neq 0$, the topological phase transition is fundamentally different, characterized by wall-mediated persistent unbinding.
• Control Parameter: The flow-alignment parameter $\lambda$ is identified as a decisive control parameter for defect-mediated transitions, determining whether bound states are even viable.
• Active Matter: The findings explain why active nematics exhibit chaotic, turbulent defect dynamics rather than equilibrium-like ordering; self-generated flows inherently suppress defect binding.
• Experimental Outlook: The results suggest that experimentalists should look for "bend-splay walls" and persistent defect unbinding in sheared or strained nematic films, challenging the standard interpretation of defect dynamics in these materials.

Conclusion

The coupling of nematic order to fluid flow, mediated by the flow-alignment parameter, fundamentally reshapes topological phase transitions. While non-aligning nematics follow the classic BKT binding-unbinding mechanism, strain-rate-aligning nematics and active systems exhibit a new regime where bend-splay walls prevent defect recombination, leading to a state of persistent disorder and unbound defects. This work redefines the understanding of topological transitions in soft matter by highlighting the critical role of hydrodynamic feedback.