Flow Coupling Alters Topological Phase Transition in Nematic Liquid Crystals

TL;DR

This work investigates how hydrodynamic flow coupling alters defect-driven topological transitions in two-dimensional nematics. Using fluctuating nematohydrodynamic simulations of Beris–Edwards Q-tensor dynamics coupled to incompressible flow, the authors show that the canonical BKT binding–unbinding of $\pm \tfrac{1}{2}$ defects persists for flow-uncoupled or non-aligning nematics ($\lambda=0$), but sharply changes for strain-rate–aligning nematics ($\lambda\neq 0$) where spontaneous bend–splay walls form and drive persistent unbinding, preventing defect recombination. The flow alignment parameter $\lambda$ acts as a fundamental control parameter for topological phase behavior in nematic fluids, suggesting that BKT-like transitions emerge only in the absence of flow alignment and pointing to experimental tests and extensions to three dimensions. The findings provide a mechanistic link between rheology under shear-like fluctuations and topological defect dynamics in active and passive nematics, with potential implications for controlled defect engineering in soft matter systems.

Abstract

We investigate how coupling to fluid flow influences defect-mediated transitions in two-dimensional passive nematic fluids using fluctuating nematohydrodynamic simulations. The system is driven by tuning the fluctuation strength, with increasing (decreasing) fluctuations defining the forward (backward) protocol. In the absence of flow coupling, the transition follows the Berezinskii--Kosterlitz--Thouless (BKT) scenario, governed by reversible binding and unbinding of $\pm 1/2$ defect pairs. When hydrodynamics is included, the outcome is controlled by the flow--alignment parameter. For non-aligning nematics ($λ=0$), the transition remains consistent with BKT. By contrast, for strain-rate--aligning nematics ($λ\neq 0$), bend--splay walls emerge, lowering the defect nucleation threshold and preventing sustained recombination: once created, defects remain unbound across the full range of fluctuation strengths in both forward and backward protocols. These results identify flow alignment as a fundamental control parameter for topological phase behavior and suggest that the canonical BKT transition emerges only in the absence of flow alignment.

Figures (4)

• Figure 1: The transition from defect-free to defect-laden states under temperature fluctuations, with snapshots of equilibrium configurations for passive nematic systems: (a–d) without flow coupling, and (e–l) with hydrodynamic coupling. The latter includes (e–h) non-aligning nematics with $\lambda = 0$ and (i–l) strain-rate–aligning nematics with $\lambda = 1$. Vertical dashed lines indicate the defect creation temperatures $T_{dc}$ for each case. Black lines represent the director field, while $+1/2$ ($-1/2$) defects are marked by green circles (blue triangles). For flow-decoupled as well as non-aligning nematics, increasing the temperature (forward protocol) drives the system through successive transitions: from (b,f) a defect-free state, to (c,g) a state of bound defect pairs, and finally to (d,h) a defect gas with numerous unbound defects. Conversely, decreasing the temperature (backward protocol) from (d,h) results in the (c,g) rebinding and annihilation of defects, eventually restoring (b,f) a defect-free state . In contrast, for strain-rate–aligning nematics ($\lambda = 1$), the system evolves from (j) a defect-free state to (k) a state where defect pairs unbind immediately upon creation, resulting in unbound defects, and finally to (l) a defect-rich state with numerous free defects as the temperature increases. Lowering the temperature retraces the same path, so defects remain unbound whenever present.
• Figure 2: Snapshots at different temperature with +1/2 defects as green circles, $-$1/2 defects as blue triangles, and the elastic energy as a heat map are shown in (a-d) for the non-aligning nematics with $\lambda = 0$ and for (e-h) strain-rate-aligning nematics with $\lambda = 1$. We note that bend walls are only present for $\lambda = 1$ and absent for $\lambda = 0$. (i,j) Averaged director-director correlations $C_{\textup{nn}}(r)$ for (i) $\lambda = 0$ and (j) $\lambda = 1$ for both protocols. For $\lambda=0$, in the forward protocol, quasi-long-range correlations in the director field are present when defects are absent ($T^*=0.030$) or bound ($T^* = 0.045$), whereas they decay exponentially when defects are unbound ($T^* = 0.070$), consistent with the expectation that defect unbinding destroys quasi-long-range order. In the backward case, this long-range order at low temperature is only partly restored, owing to the long relaxation time needed for full equilibration at low temperature. For $\lambda=1$, director correlations decay much more rapidly in comparison. Local maxima in the director correlations evidence the presence of bend wall and their characteristic spacing, which decreases with increasing temperature.
• Figure 3: Defect unbinding for non-aligning ($\lambda=0$) and strain-rate aligning ($\lambda=1$) passive nematics. (a,b) Average nearest-neighbor distance $r_{nn}$, normalized by the average defect spacing $r_{av}$, for $\lambda=0$ and $\lambda=1$. The dashed horizontal line shows the expectation value for free and non-interacting defects, $r_{nn}^{\textup{free}} = 0.5 r_{av}$. In (a), the forward protocol shows that $r_{nn} \ll r_{nn}^{\textup{free}}$ at small temperatures, thus indicating bound defects. At higher temperatures, defects become unbound as $r_{nn} \rightarrow r_{nn}^{\textup{free}}$. For $T^* \geq 0.05$, the backward case follows the same trend, with free defects becoming bound as the temperature is lowered gradually. For $T^* < 0.05$, however, $r_{nn}$ increases, as cooling to such low temperatures decreases defect velocities and increases relaxation times accordingly. This can be seen from (c), which shows that the average NN-distance for each frame for the smallest backward temperatures are only slowly decreasing as the system relaxes. In contrast, (b) reveals that $r_{nn} \approx r_{nn}^{\textup{free}}$ for all temperatures, indicating that defects are unbound whenever present. (d) the NN-distance changes considerably over time, even for the smallest temperatures, as expected from unbound defects. (e,f) Cluster-based measures: the time-averaged neutralization length $l_{\max}$ and percolation length $l_{\text{perc}}$ as functions of effective temperature $T^*$ for the forward protocol. In (e), $l_{\max} < l_{\text{perc}}$ for small temperatures, confirming bound defects. At higher temperatures, $l_{\max} = l_{\text{perc}}$, and defects are unbound. In contrast, (f) reveals that for strain-rate aligning nematics, defects are unbound at all temperatures.
• Figure 4: Defect unbinding and orientational order in passive nematics in the absence of flow. (a) Cluster-based measures: the time-averaged neutralization length $l_{\max}$ and percolation length $l_{\text{perc}}$ as functions of effective temperature $T^*$ for the forward protocol. $l_{\max}$ is an upper bound on the separation length of all defect pairs, from which it follows that defects are bound for $T^* \lessapprox 0.20$. For $T^* \gtrapprox 0.25$, $l_{\max} = l_{\text{perc}}$ within uncertainty, and defects are unbound, consistent with BKT-like unbinding behavior. (b) Director correlation function $C_{\textup{nn}}(r)$ for representative temperatures with solid (dashed) lines indicating the forward (backward) protocol. For $T^*=0.15$, defects are tightly bound, and the director field exhibit quasi--long-range order. For $T^*=0.22$, defects are more loosely bound, and quasi--long-range order is weakened. As for the flow-coupled non-aligning case (Fig. \ref{['Fig2:Dir-corr']}i), nematic order is only partially restored in the backward protocol. For $T^*\geq0.25$, director-director correlations decay rapidly, consistent with defect unbinding destroying nematic order. (c) Average nearest-neighbor distance $r_{nn}$ normalized by the average defect spacing $r_{av}$. The horizontal line marks the expected NN-distance for independently distributed points, $r_{nn}^{\textup{free}}=0.5r_{av}$. For $T^* \lessapprox 0.22$, $r_{nn}$ is considerably less than that expected of free defects, thus indicating bound defects, while at higher temperatures, defects become unbound. For $T^* < 0.17$ in the backward protocol, $r_{nn}$ increases, as cooling to such low temperatures decreases defect velocities considerably and so slows the relaxation toward a defect-free equilibrium state accordingly. This can be seen from (d), which shows that the average NN-distance for each frame against time for the smallest backward temperatures are only slowly decreasing as the system relaxes. The discontinuous jumps indicate the annihilation of a defect pair.