Eppur si muove: Shape of topological defects -- and consequent motion -- in active nematics

1. Problem Statement

Topological defects in liquid-crystalline systems are fundamental to their large-scale properties. In active nematics (systems driven out of equilibrium by internal energy consumption, e.g., cytoskeletal filaments or bacterial suspensions), defects exhibit unique behaviors not found in equilibrium, most notably the self-propulsion of $+1/2$ charge defects.

A critical gap in existing theoretical literature is the assumption that the shape of the defect core and its far-field distortion remain unperturbed by activity. Previous calculations of defect self-propulsion speeds (e.g., using the Halperin-Mazenko method or perturbative approaches) often assumed the defect retains the shape of a passive system with a single Frank elastic constant.

• The Conflict: This assumption leads to inconsistent predictions. For instance, some theories predict that $+1/2$ defects should not self-propel if certain activity coefficients ($L_2$) are zero, or conversely, that passive systems with two Frank constants should exhibit spontaneous motion (violating energy conservation).
• The Core Question: Does activity modify the shape of topological defects, and if so, how does this shape modification dictate their self-propulsion speed and interactions?

2. Methodology

The authors employ a combination of analytical perturbation theory and numerical simulations to address the problem.

• Theoretical Model: They utilize a generic "dry" active nematic model (ignoring hydrodynamic flow for the primary analysis, though extensions are discussed). The dynamics are governed by a time-dependent Ginzburg-Landau equation for the nematic order parameter tensor $Q_{ij}$:
  $$\partial_t Q_{ij} = G_{ij}$$
  where $G_{ij}$ includes standard Landau-de Gennes terms and activity-induced terms characterized by coefficients $L_1, L_2, K_1, K_2$.

  • The model reduces to a passive system with a single Frank constant when activity coefficients vanish.
  • It reduces to a passive system with two Frank constants (splay and bend having different elastic costs) when activity coefficients are equal ($L_i = K_i = K$).

• Analytical Approach:

  • Perturbative Expansion: They perform asymptotic expansions in the defect core ($r \to 0$) and the far-field ($r \to \infty$).
  • Ansatz: They propose specific functional forms for the order parameter magnitude $S(r, \phi)$ and the angle field $\theta(r, \phi)$, introducing angular-dependent corrections ($A(\phi), B(\phi), a(\phi), b(\phi)$) to account for activity-induced shape changes.
  • Moving Frame: For $+1/2$ defects, they transform to a co-moving reference frame to solve for the steady-state shape and velocity simultaneously.
  • Solvability Conditions: They use the Fredholm alternative (matching asymptotic expansions) to derive the self-propulsion speed $v_d$ without assuming a fixed shape.

• Numerical Simulations:

  • They implemented a Finite Element Method (FEM) solver in MATLAB to solve the non-linear partial differential equations on a circular domain.
  • Boundary conditions were set to induce $\pm 1/2$ defects in the bulk.
  • They tracked the defect position over time to measure self-propulsion speeds and extracted the angular dependence of the order parameter to verify the predicted shapes.

3. Key Contributions

1. First Analytical Derivation of Active Defect Shapes: The paper provides the first closed-form analytical expressions for the shape of both $-1/2$ and $+1/2$ defects in active nematics, explicitly showing how activity deforms the defect profile.
2. Refutation of the "Rigid Shape" Assumption: The authors demonstrate that assuming a defect shape identical to that of a passive single-Frank-constant system leads to qualitatively incorrect predictions, even at arbitrarily small activity levels.
3. Resolution of Paradoxes: They resolve the contradiction where previous theories suggested $+1/2$ defects might not self-propel in certain active regimes or that passive systems might self-propel. By using the correct, activity-modified shape, they show that $+1/2$ defects always self-propel in active nematics, and passive systems correctly yield zero velocity.
4. Link Between Shape and Speed: They establish a direct analytical relationship between the angular deformation of the defect core and its self-propulsion speed.

4. Key Results

A. $-1/2$ Defects

• Shape Modification: Unlike $+1/2$ defects, $-1/2$ defects do not self-propel. However, their shape is significantly modified by activity.
• Far-Field: The authors derived analytical expressions for the far-field corrections to the order parameter magnitude $S$ and angle $\theta$. The corrections depend on the activity coefficients ($L_1, L_2, K_1, K_2$) and exhibit a 3-fold symmetry ($\sin 3\phi, \cos 3\phi$).
• Core: Surprisingly, the core structure of the $-1/2$ defect (specifically the angular field correction $b(\phi)$) remains identical to that of a passive single-Frank-constant system ($b(\phi)=0$), even in active systems.
• Validation: Numerical simulations perfectly matched the analytical predictions for the far-field shape functions $A(\phi)$ and $B(\phi)$.

B. $+1/2$ Defects

• Universal Self-Propulsion: Contrary to previous claims that self-propulsion depends on specific activity terms (like $L_2$), the authors show that $+1/2$ defects always self-propel in active nematics, regardless of the specific nature of the activity ($L_1, L_2, K_1, K_2$).
• Speed-Shape Relation: The self-propulsion speed $v_d$ is intimately linked to the distortion of the angle field at the core. The authors derived the relation:
  $$b(\phi) = \frac{1}{4}(v_d + a_0 L_2) \sin \phi$$
  (where $b(\phi)$ is the angular correction and $a_0$ is a numerical constant). This equation explicitly ties the defect's shape to its velocity.
• Correction of Previous Theories:
  • The "Halperin-Mazenko" method applied with a fixed passive shape yields $v_d = -L_2 a_0$, predicting zero speed if $L_2=0$.
  • The correct perturbative calculation (accounting for shape change) yields a speed proportional to a combination of all activity coefficients:
    $$v_d \log\left(\frac{3.29}{v_d}\right) \approx 0.07(K_1 + K_2) - 0.78 L_2 + 0.64 L_1$$
  • This new formula correctly predicts $v_d = 0$ in the passive limit ($L_i = K_i$), satisfying energy conservation.

C. Numerical Verification

• Simulations confirmed that $+1/2$ defects move with a constant speed independent of system size (until they hit the boundary), whereas in passive systems, any drift is a finite-size boundary effect that vanishes as $R \to \infty$.
• The measured defect shapes in simulations matched the analytical predictions for both $+1/2$ and $-1/2$ defects across various activity regimes.

5. Significance and Implications

• Theoretical Correction: This work fundamentally corrects the theoretical framework for active nematics. It proves that neglecting defect shape deformation leads to erroneous conclusions about defect dynamics, even in the linear (small activity) regime.
• Biological Relevance: Since topological defects drive morphogenesis (e.g., in Hydra), cell extrusion, and bacterial colony growth, accurate modeling of defect motion is crucial. The finding that defects always self-propel in active media suggests a robust mechanism for tissue remodeling and flow generation.
• Future Directions: The authors suggest that defect shape modifications are likely the source of non-reciprocal interactions between defects (where defect A exerts a different force on B than B on A). They also note that these methods can be extended to "wet" active nematics (with hydrodynamics) and other ordered phases (hexatics, smectics), potentially explaining the formation of defect lines and clusters observed in experiments.

In summary, the paper demonstrates that "Eppur si muove" (And yet it moves): the motion of active defects is inextricably linked to their deformation. Ignoring this deformation renders theoretical predictions inconsistent with physical laws and experimental reality.