Nematic Tensor Research Articles

Field theory of active chiral hard disks: a first-principles approach to steric interactions

A first-principles approach for active chiral hard disks is presented, that explicitly accounts for steric interactions on the two-body level. We derive an effective one-body equation for the joint probability distribution of positions and angles of the particles. By projecting onto the angular modes, we write a hierarchy for the lowest hydrodynamic modes, i.e. particle density, polarisation, and nematic tensor. Introducing dimensionless variables in the equations, we highlight the assumptions, which—though inherent—are often included implicit in typical closure schemes of the hierarchy. By considering different regimes of the Péclet number, the well-known models in active matter can be obtained through our consideration. Explicitly, we derive an effective diffusive description and by going to higher orders in the closure scheme, we show that this first-principles approach results in the recently introduced Active Model B +, a natural extension of the Model B for active processes. Remarkably, here we find that chirality can change the sign of the phenomenological activity parameters.

Stable Assemblies of Topological Defects in Nematic Orientational Order.

We considered general mechanisms enabling the stabilization of localized assemblies of topological defects (TDs). There is growing evidence that physical fields represent fundamental natural entities, and therefore these features are of interest to all branches of physics. In general, cores of TDs are energetically costly, and consequently, assemblies of TDs are unfavorable. Owing to the richness of universalities in the physics of TDs, it is of interest to identify systems where they are easily experimentally accessible, enabling detailed and well-controlled analysis of their universal behavior, and cross-fertilizing knowledge in different areas of physics. In this respect, thermotropic nematic liquid crystals (NLCs) represent an ideal experiment testbed for such studies. In addition, TDs in NLCs could be exploited in several applications. We present examples that emphasize the importance of curvature imposed on the phase component of the relevant order parameter field. In NLCs, it is represented by the nematic tensor order parameter. Using a simple Landau-type approach, we show how the coupling between chirality and saddle splay elasticity, which can be expressed as a Gaussian curvature contribution, can stabilize Meron TDs. The latter have numerous analogs in other branches of physics. TDs in 2D curved manifolds reveal that the Gaussian curvature dominantly impacts the assembling and stabilization of TDs. Furthermore, a strong enough curvature that serves as an attractor for TDs is a respective field that could be imposed in a fast enough phase transition. Assemblies of created TDs created in such a disordered environment could be stabilized by appropriate impurities.

Core structure of static ferrodark solitons in a spin-1 Bose-Einstein condensate

We develop an analytical description of static ferrodark solitons, the $\mathbb{Z}_2$ topological defects in the magnetic order, in the easy-plane phase of ferromagnetic spin-1 Bose-Einstein condensates. We find that the type-I ferrodark soliton has a single width while the type-II ferrodark soliton exhibits two characteristic length scales. The proposed ansatzes show excellent agreement with numerical results. Spin-singlet amplitudes, nematic tensor densities and nematic currents of ferrodark solitons are also discussed. The $\mathbb{Z}_2$ topological defects in the mass superfluid order, dark-dark-dark vector solitons, are obtained exactly in the parameter regime where exact ferrodark solitons exist. The dark-dark-dark vector soliton has higher excitation energy than ferrodark solitons.

Quantum hydrodynamics of the spinor Bose–Einstein condensate at non-zero temperatures

A finite temperature hydrodynamic model is derived for the spin-1 ultracold bosons by the many-particle quantum hydrodynamic method. It is presented as the two fluid model of the Bose–Einstein condensate (BEC) and normal fluid. The continuity, Euler, spin evolution, and nematic tensor evolution equations are derived for each fluid. The linear and quadratic Zeeman effects are included. Scalar and spin–spin like short-range interactions are considered in the first order by the interaction radius. Obtained hydrodynamic equations are also represented as the set of two nonlinear Pauli equations. The spectrum of the bulk collective excitations is considered for the ferromagnetic phase in the small temperature limit. The spin wave is not affected by the presence of the small temperature in the described minimal coupling model, where the thermal part of the spin-current of the normal fluid is neglected. The two sound waves are affected by the spin evolution in the same way as the change of spectrum of the single sound wave in BEC, where speed of sound is proportional to g1+g2 with gi as the interaction constants.

Curvature Potential Unveiled Topological Defect Attractors

We consider the theoretical and positional assembling of topological defects (TDs) in effectively two-dimensional nematic liquid crystal films. We use a phenomenological Helfrich–Landau–de Gennes-type mesoscopic model in which geometric shapes and nematic orientational order are expressed in terms of a curvature tensor field and a nematic tensor order parameter field. Extrinsic, intrinsic, and total curvature potentials are introduced using the parallel transport concept. These potentials reveal curvature seeded TD attractors. To test ground configurations, we used axially symmetric nematic films exhibiting spherical topology.

Nematic tensor evolution in the partially-polarized spin-1 Bose–Einstein condensates

We develop the complete hydrodynamical description of the spin-1 partially polarized Bose–Einstein condensate (BEC), based on the many-particle quantum hydrodynamic formalism. We derive the continuity equation, momentum balance equation, spin evolution and nematicity evolution equations considering the spin–spin dipole long-range interactions and short-range interactions between partially spin-polarized atoms of condensate. The nematic or quadrupolar tensor plays an important role in the experiments of spin-nematic squeezing. We predict the influence of the spin–spin dipole interactions and spin torque force on the dynamics of nematic tensor. The collective modes for phonons and magnons are predicted for the ferromagnetic phase of spinor condensate, where the spin–spin and short-range interactions modify the dispersion equation.

Root shape adaptation to mechanical stress derived from unidirectional vibrations in Populus nigra.

While it is known that plant roots can change their shapes to the stress direction, it remains unclear if the root orientation can change as a means for mechanical reinforcement. When stress in form of a unidirectional vibration is applied to cuttings of Populus nigra for 5 min a day over a period of 20 days, the root system architecture changes. The contribution of roots with a diameter larger than 0.04 cm increases, while the allocation to roots smaller than 0.03 cm decreases. In addition to the root diameter allocation, the root orientation in the stem proximity was analyzed by appearance and with a nematic tensor analysis in an attempt to calculate the average root orientation. The significant different allocation to roots with a larger diameter, and the tendency of roots to align in the vicinity of the stress axis (not significantly different), are indicating a mechanical reinforcement to cope with the received strain. This work indicates an adaptive root system architecture and a possible adaptive root orientation for mechanical reinforcement.

Mechanics of nematic membranes: Euler–Lagrange equations, Noether charges, stress, torque and boundary conditions of the surface Frank’s nematic field

The mechanics of a flexible membrane decorated with a nematic liquid-crystal texture is considered in a variational framework. The variations on the splay, twist and the bend energy are obtained from the local deformations leading to changes in the shape membrane. The Euler–Lagrange derivatives and the Noether charges are identified from the variational equations. The nematic stress tensor is obtained as a consequence of translational invariance. Likewise, the rotational invariance implies the torque nematic tensor. The corresponding boundary conditions are obtained for the free-edges in the open-membrane configuration. These results constitute the basis of a generalized theory of elasticity for anisotropic nematic membranes. Some relevant consequences of the presence of nematic ordering are visualized at revolution surfaces with axial symmetry.

Topological defect enabled formation of nematic domains

ABSTRACTWe study numerically curvature and electric field driven domain formation and reorientations in two-dimensional nematic liquid crystals within square confinement. We use the Landau-de Gennes description in terms of the nematic tensor order parameter. We show that on increasing the amplitude of a sinusoidally shaped confining wall, pairs {defect, antidefect} are formed, which enable insertion of domains displaying significantly different average orientational order with respect to the surrounding nematic body. Furthermore, we study E-driven transformation between two competing degenerate domain-type structures. We show that transformations could be mediated via qualitatively different intermediate structures possessing topological defects.

Computational mean-field modeling of confined active fluids

We present a new framework for the efficient simulation of the dynamics of active fluids in complex two- and three-dimensional microfluidic geometries. Focusing on the case of a suspension of microswimmers such as motile bacteria, we adopt a continuum mean-field model based on partial differential equations for the evolution of the concentration, polarization and nematic tensor fields, which are nonlinearly coupled to the Navier-Stokes equations for the fluid flow driven by internal active stresses. A level set method combined with an adaptive mesh refinement scheme on Quad-/Octree grids is used to capture complex domain shapes while refining the solution near boundaries or in the neighborhood of sharp gradients. A hybrid finite volumes/finite differences method is implemented in which the concentration field is treated using finite volumes to ensure mass conservation, while the polarization and nematic alignment fields are treated using a combination of finite differences and finite volumes for enhanced accuracy. The governing equations for these fields are solved along with the Navier-Stokes equations, which are evolved using an unconditionally stable projection solver. We illustrate the versatility and robustness of our method by analyzing spontaneous active flows in various two- and three-dimensional systems. Our results show excellent agreement with previous models and experiments and pave the way for further developments in active microfluidics.

The impact of mechanical compression on cortical microtubules in Arabidopsis: a quantitative pipeline.

SummaryExogenous mechanical perturbations on living tissues are commonly used to investigate whether cell effectors can respond to mechanical cues. However, in most of these experiments, the applied mechanical stress and/or the biological response are described only qualitatively. We developed a quantitative pipeline based on microindentation and image analysis to investigate the impact of a controlled and prolonged compression on microtubule behaviour in the Arabidopsis shoot apical meristem, using microtubule fluorescent marker lines. We found that a compressive stress, in the order of magnitude of turgor pressure, induced apparent microtubule bundling. Importantly, that response could be reversed several hours after the release of compression. Next, we tested the contribution of microtubule severing to compression‐induced bundling: microtubule bundling seemed less pronounced in the katanin mutant, in which microtubule severing is dramatically reduced. Conversely, some microtubule bundles could still be observed 16 h after the release of compression in the spiral2 mutant, in which severing rate is instead increased. To quantify the impact of mechanical stress on anisotropy and orientation of microtubule arrays, we used the nematic tensor based FibrilTool ImageJ/Fiji plugin. To assess the degree of apparent bundling of the network, we developed several methods, some of which were borrowed from geostatistics. The final microtubule bundling response could notably be related to tissue growth velocity that was recorded by the indenter during compression. Because both input and output are quantified, this pipeline is an initial step towards correlating more precisely the cytoskeleton response to mechanical stress in living tissues.

Spin squeezing and entanglement for an arbitrary spin

A complete set of generalized spin-squeezing inequalities is derived for an ensemble of particles with an arbitrary spin. Our conditions are formulated with the first and second moments of the collective angular momentum coordinates. A method for mapping the spin-squeezing inequalities for spin-1/2 particles to entanglement conditions for spin-j particles is also presented. We apply our mapping to obtain a generalization of the original spin-squeezing inequality to higher spins. We show that, for large particle numbers, a spin-squeezing parameter for entanglement detection based on one of our inequalities is strictly stronger than the original spin-squeezing parameter defined in [A. Sorensen et al., Nature 409, 63 (2001)]. We present a coordinate system independent form of our inequalities that contains, besides the correlation and covariance tensors of the collective angular momentum operators, the nematic tensor appearing in the theory of spin nematics. Finally, we discuss how to measure the quantities appearing in our inequalities in experiments.

FibrilTool, an ImageJ plug-in to quantify fibrillar structures in raw microscopy images

Cell biology heavily relies on the behavior of fibrillar structures, such as the cytoskeleton, yet the analysis of their behavior in tissues often remains qualitative. Image analysis tools have been developed to quantify this behavior, but they often involve an image pre-processing stage that may bias the output and/or they require specific software. Here we describe FibrilTool, an ImageJ plug-in based on the concept of nematic tensor, which can provide a quantitative description of the anisotropy of fiber arrays and their average orientation in cells, directly from raw images obtained by any form of microscopy. FibrilTool has been validated on microtubules, actin and cellulose microfibrils, but it may also help analyze other fibrillar structures, such as collagen, or the texture of various materials. The tool is ImageJ-based, and it is therefore freely accessible to the scientific community and does not require specific computational setup. The tool provides the average orientation and anisotropy of fiber arrays in a given region of interest (ROI) in a few seconds.

Spin turbulence with small spin magnitude in spin-1 spinor Bose-Einstein condensates

We theoretically and numerically study spin turbulence (ST) with small spin magnitude in spin-1 spinor Bose-Einstein condensates by using the spin-1 spinor Gross-Pitaevskii (GP)equations. This kind of ST is realized in two cases: (i) with antiferromagnetic interaction and (ii) with ferromagnetic interaction under a static magnetic field. The ST with small spin magnitude can exhibit two characteristic power laws in the spectrum of the spin-dependent interaction energy: -1 and -7/3 power laws in the low- and high-wave-number regions, respectively. These power laws are derived from a Kolmogorov-type dimensional scaling analysis for the equations of motion of the spin vector and nematic tensor. To confirm these power laws, we perform a numerical calculation of the spin-1 spinor GP equations in a two-dimensional uniform system. In case (i), the -7/3 power law appears in the high-wave-number region, but the spectrum in the low-wave-number region deviates from the -1 power law. In contrast, both -1 and -7/3 power laws are found to clearly appear in case (ii).

Tensorial conservation law for nematic polymers

We derive the "conservation law" for nematic polymers in tensorial form valid for quadrupolar orientational order, in contradistinction to the conservation law in the case of polar orientational order. Due to microscopic differences in the coupling between the orientational field deformations and the density variations for polar and quadrupolar order, we find that the respective order parameters satisfy fundamentally distinct constraints. Being necessarily scalar in its form, the tensorial conservation law is obtained straightforwardly from the gradients of the polymer nematic tensor field and connects the spatial variation of this tensor field with density variations. We analyze the differences between the polar and the tensorial forms of the conservation law, present some explicit orientational fields that satisfy the tensorial constraint, and discuss the role of singular "hairpins," which do not affect the local quadrupolar order of polymer nematics, but nevertheless influence its gradients.

Spin Turbulence in Spin-1 Spinor Bose-Einstein Condensate with Antiferromagnetic Interaction

We theoretically and numerically study turbulence in spin-1 spinor Bose-Einstein condensates (BECs) with antiferromagnetic (AF) interaction by using the spinor Gross-Pitaevskii equations. In this system, the dynamical instability is caused by the counterflow, leading to the disturbed distribution of the spin density vector, which we call spin turbulence (ST). Previously, we investigated the ST in spin-1 spinor BEC with ferromagnetic interaction, finding that the spectrum of spin-dependent interaction energy exhibited the −7/3 power law. In this study, we investigate the ST in the spin-1 spinor BEC with AF interaction, pointing out the −1 and −7/3 power laws in the low and high wave number regions, respectively. These power laws can be obtained by the Kolmogorov-type dimensional scaling analysis for the equations of spin vector and nematic tensor. However, our numerical calculation finds the −7/3 power law in the high wave number region, while the spectrum in the low wave number region deviates from the −1 power law.

Transformation from walls to disclination lines: Statics and dynamics of the pincement transition

We present an experimental and theoretical study of the pincement phenomenon-transformation of a wall associated with the Fréedericksz transition into a pair of disclination lines. We measure the velocity of the boundary (front) between the two states as a function of the voltage. Experimental results are recovered by numerical simulations based on the nematic tensor order parameter, which also reveal the detailed three-dimensional structure of the front. By introducing reduced models we obtain approximate expressions for the two-state coexistence voltage and the front velocity. We find a bifurcation scenario incorporating a pair of saddle nodes at which the wall and disclination solutions appear or disappear.

Texture formation under phase ordering and phase separation in polymer-liquid crystal mixtures

Computational modeling of texture formation in coupled phase separation-phase ordering processes in polymer/liquid crystal mixtures is performed using a unified model based on the nematic tensor order parameter and gradient orientation elasticity. The computational methods are able to resolve defect nucleation, defect-defect interactions, and defect-particle interactions, as well as global and local morphological features in the concentration and order parameter spatiotemporal behavior. Biphasic structures corresponding to polymer dispersed liquid crystals (PDLCs), crystalline filled nematic (CFNs), and random filled nematics (RFNs) are captured and analyzed using liquid crystal defect physics and structure factors. Under spinodal decomposition due to concentration fluctuations, the PDLC structure emerges, and the nucleation and repulsive interaction of defects within nematic droplets leads to bipolar nematic droplets. Under spinodal decomposition due to ordering fluctuations, the CFNs structure emerges, and the stable polymer droplet crystal is pinned by a lattice of topological defects. For intermediate cases, where the mixture is unstable to both concentration and nematic order fluctuations, the RFN structure emerges, and polymer droplets and fibrils are pinned by a defect network, whose density increases with the curvature of the polymer-liquid crystal interface. The simulations provide an information of the role of topological defects on phase separation-phase ordering processes in polymer-liquid crystal mixtures.

Fluctuations of topological disclination lines in nematic liquid crystals: Renormalization of the string model

Using the tensorial Landau-de Gennes theory, we study the fluctuations of a disclination line in a nematic liquid crystal. By analyzing the structure and the spectrum of the eigenmodes of a line of strength +/-1/2 , we reassess the concept of line tension used in the simple string model of the disclination showing that it does not include the complete set of eigenmodes and must be renormalized. In general, the line tension considerably underestimates the thermal amplitude of a disclination and we find that it is only applicable to severely confined disclinations.

Interfacial Thermodynamics of Polymeric Mesophases

AbstractSummary:A complete mechanical‐thermodynamical formulation for multicomponent nematic polymer‐isotropic fluid interfaces is derived, validated, and used to derive the structure and shape equations for these soft anisotropic polymer interfaces. The fundamental role of liquid crystalline order and long range effects in coupling bulk and interfacial effects, and in coupling thermodynamical/liquid crystalline order/geometrical variables is demonstrated, discussed, and validated. The Gibbs‐Duhem nemato‐thermodynamics equation emerges from an interfacial tension γ = γ(Θ, μ,Q, ∇sQ,k) that depends on temperature (Θ), chemical potential (μ), nematic tensor order parameterQ, surface gradients ofQ, and geometryk, and leads to new couplings in these enhanced phase spaces. The role of entropy and adsorption, and long range effects on interfacial shape and structure selection is revealed. For flat interfaces the preferred structure emerges from a competition between energy, entropy, and adsorption.