Two-dimensional Nematic Liquid Crystal Research Articles

Relative kinetic stability of defect patterns in two-dimensional nematic liquid crystals with rectangular confinement.

Guiding and dynamically modulating topological defects are critical challenges in defect engineering of liquid crystals. Here, we employ molecular dynamics simulations to investigate the transition dynamics and relative kinetic stability of defect patterns in two-dimensional nematic Gay-Berne liquid crystals confined within rectangular geometries. We observe the formation of various defect patterns including long-axis, diagonal, X-shaped, composite, and bend configurations under different confinement conditions. The competition between boundary effects and the uniformity of nematic orientation induces the continuous realignment of liquid crystal molecules, facilitating the spatially continuous transformation of defect patterns over time. This transition involves changes in both defect types and their locations, typically initiating from defect regions. Furthermore, we demonstrate that the relative stability of these defect patterns can be effectively controlled by adjusting confinement parameters and external field conditions. Our findings provide fundamental insights into the transition kinetics of defect patterns in confined nematic liquid crystals, thereby enhancing our ability to manipulate topological defects for advanced applications.

Droplet Formation in Dynamic Stratified Liquid–Liquid Systems for Solution-Based Deposition Methods

Abstract The assembly of a two-dimensional (2D) nematic liquid crystal at an interface between two liquids can be exploited to assemble densely packed and highly aligned arrays of rod-like nanoparticles. This method is especially relevant to creating arrays of semiconducting carbon nanotubes (CNTs) for high-performance electronics. When a dense solvent containing CNTs flows over a less dense water subphase in a confined channel, the locally aligned arrays of nanoparticles align globally with the flow direction and can be transferred to the substrate. For large substrates and long channels, the dense solvent tends to slow and create a pool, which then drops through the interface and disturbs the delicate deposition process. Understanding this phenomenon is critical to improving and scaling up similar manufacturing processes. Here, data are collected, and an empirical model is developed to understand and predict the pooling behavior of a suspended fluid flowing over a less dense subphase. The model is demonstrated with two different solvents and proves to be accurate within +/− 15%. With a better understanding of the physics governing the system, the model is then used to suggest methods for minimizing pooling behavior.

Active nematic fluid films

Coupling surface deformations with active stresses in two-dimensional nematic liquid crystal films leads to a rich area of investigation, particularly in biological fluid mechanics across multiple scales from tissue mechanics to cell membrane mechanics. In Al-Izzi & Morris (J. Fluid Mech., vol. 957, 2023, A4), the authors derive the complete set of governing equations for such systems. Their results provide an extended theoretical framework with which active nematic fluid films with in-plane flow and out-of-plane deformation can be analysed. To illustrate the potential applications of this framework, a few specific biologically inspired examples are discussed.

Defect patterns of two-dimensional nematic liquid crystals in confinement.

A two-dimensional or quasi-two-dimensional nematic liquid crystal refers to a surface-confined system. When such a system is further confined by external line boundaries or excluded from internal line boundaries, the nematic directors form a deformed texture that may display defect points or defect lines, for which winding numbers can be clearly defined. Here, a particular attention is paid to the case when the liquid crystal molecules prefer to form a boundary nematic texture in parallel to the wall surface (i.e., following the homogeneous boundary condition). A general theory, based on geometric argument, is presented for the relationship between the sum of all winding numbers in the system (the total winding number) and the type of confinement angles and curved segments. The conclusion is validated by comparing the theoretical defect rule with existing nematic textures observed experimentally and theoretically in recent years.

Aligned 2D carbon nanotube liquid crystals for wafer-scale electronics.

Semiconducting carbon nanotubes promise faster performance and lower power consumption than Si in field-effect transistors (FETs) if they can be aligned in dense arrays. Here, we demonstrate that nanotubes collected at a liquid/liquid interface self-organize to form two-dimensional (2D) nematic liquid crystals that globally align with flow. The 2D liquid crystals are transferred onto substrates in a continuous process generating dense arrays of nanotubes aligned within ±6°, ideal for electronics. Nanotube ordering improves with increasing concentration and decreasing temperature due to the underlying liquid crystal phenomena. The excellent alignment and uniformity of the transferred assemblies enable FETs with exceptional on-state current density averaging 520 μA μm−1at only −0.6 V, and variation of only 19%. FETs with ion gel top gates demonstrate subthreshold swing as low as 60 mV decade−1. Deposition across a 10-cm substrate is achieved, evidencing the promise of 2D nanotube liquid crystals for commercial semiconductor electronics.

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.

Generalized van der Waals theory for phase behavior of two-dimensional nematic liquid crystals. II. Phase coexistence and adsorption.

Adsorption of asymmetric particles or molecules into monolayers is important for many biological and technologically relevant physical systems. In-plane ordering can drastically affect adsorption and phase behavior. In this work, a generalized van der Waals theory previously developed [M. V. Zonta and E. R. Soulé, Phys. Rev. E 100, 062703 (2019)10.1103/PhysRevE.100.062703] is used to calculated phase behavior and adsorption isotherms in a system of hard-core rodlike particles with in-plane nematic order, as a function of the model parameters (aspect ratio L/B, isotropic and anisotropic interaction parameters χ and ν,and adsorption constant K_{ads}). For small L/B, isotropic-nematic and/or (depending on χ) isotropic liquid-gas coexistence is observed; as L/B increases, coexistence between two different nematic phases appears at low temperature, and liquid-gas equilibrium ceases to be observed for large enough L/B; this is understood considering that as aspect ratio increases, the range of stability of the nematic phase becomes larger. Adsorption isotherms are found to significantly deviate from Langmuir behavior, and are strongly affected by ordering and interactions (surface density in the adsorbed layer increases as interaction parameters and ordering increase). Phase coexistence is observed as discontinuous transitions in adsorption isotherms, where adsorption-desorption hysteresis cycles are possible.

Asymptotic behavior of two-dimensional stochastic nematic liquid crystal flows with multiplicative noise

Under non-periodic boundary conditions, we consider the long-term behavior of stochastic two-dimensional (2D) nematic liquid crystal flows with multiplicative noise. The structure of the corresponding stochastic model is different from that of the deterministic model. Noise destroys the basic balance law for the nematic liquid crystal flows, so we cannot employ the standard argument to obtain uniform a priori estimates of the solutions to this stochastic model. To overcome this problem, we use logarithmic energy estimates and the Itô formula in a Banach space to obtain uniform estimates that improve the previous result where the orientation field grows exponentially w.r.t. time t. In order to study the existence of a random attractor, we need to show that the solution is a stochastic flow. However, this is not obvious because of the particularly multiplicative noise in the orientation field. To show the flow property of the orientation field, we construct linear stochastic partial differential equations where the solutions are stochastic flows. After determining the relationship between these linear equations and the orientation field equation, we prove that each component of the orientation field is indeed a stochastic flow. The global well-posedness of the stochastic 2D nematic liquid crystal flows is only established for weak solutions, so the existence of the random attractor should be demonstrated in the weak solution space. Thus, using the standard method, we need to derive uniform a priori estimates in the strong solution space. However, the standard method fails because of the ill-posedness of the strong solution. We show that by proving the compactness property of the stochastic flow and using the regularity of the solutions, we can construct a compact absorbing ball in the weak solution space, which implies the existence of the random attractor. Finally, we show the existence of the invariant measure for the corresponding stochastic model, which is the first result for the long-term behavior of stochastic nematic liquid crystals under Dirichlet boundary conditions for the velocity field and Neumann boundary conditions for the orientation field.

Global regularity to the 2D non-isothermal inhomogeneous nematic liquid crystal flows

ABSTRACT In this paper, we prove the global strong solutions for the Cauchy problem of two-dimensional (2D) incompressible non-isothermal nematic liquid crystal flows, if the initial orientation satisfies a geometric condition. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states. When d is a constant vector and , we also extend the corresponding result in Wang Y. [Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier–Stokes flows with vacuum. Discrete Contin Dyn Syst B. doi:10.3934/dcdsb.2020099.] to the whole space , where the global strong solution of 2D inhomogeneous incompressible heat conducting Navier–Stokes flows is established on bounded domain.

Generalized van der Waals theory for phase behavior of two-dimensional nematic liquid crystals: Phase ordering and the equation of state.

Liquid crystalline ordering of anisotropic particles in two dimensions is important in many physical and biological systems and their phase behavior is still a topic of interest. A generalized van der Waals theory is formulated, accounting for repulsive excluded volume and attractive van der Waals and Maier-Saupe interactions, for rectangles confined to two dimensions. The phase ordering transitions and equation of state are analyzed as a function of the model parameters (aspect ratioL/B and isotropic and anisotropic interaction parameters χ and ν). Different phase transitions are observed: continuous isotropic-nematic (high L/B and ν), first-order isotropic-nematic (intermediate L/B and small ν), and continuous isotropic-tetratic (small L/B and ν) followed by a continuous tetratic-nematic transition at higher densities. Increasing L/B decreases the pressure, and this effect is more pronounced in the nematic than in the isotropic phase. Increasing both interaction parameters decreases pressure and can lead to phase separation.

On the existence of skyrmions in planar liquid crystals

The study of topologically nontrivial field configurations is an important topic in many branches of physics and applied sciences. In this paper we are interested to the existence of such structures, the so-called skyrmions, in the context of liquid crystals. More precisely, we consider a two-dimensional nematic or cholesteric liquid crystal. In the nematic case we use a Bogomol'nyi type decomposition in order to get a topological lower bound on the configurations with a given degree for the full Oseen-Frank energy functional, and so we can find a global minimum of degree $\pm 1$ for the energy. Then we consider the cholesteric case in presence of an electric field under the one constant approximation assumption, and, by using the concentration-compactness method, we prove the existence of a minimum again on the configurations of degree $\pm 1$, for sufficiently large electric fields.

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.

Large Deviations for Stochastic Nematic Liquid Crystals Driven by Multiplicative Gaussian Noise

We study a stochastic two-dimensional nematic liquid crystal model with multiplicative Gaussian noise. We prove the Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions using weak convergence method

Pair creation, motion, and annihilation of topological defects in two-dimensional nematic liquid crystals.

We present a framework for the study of disclinations in two-dimensional active nematic liquid crystals and topological defects in general. The order tensor formalism is used to calculate exact multiparticle solutions of the linearized static equations inside a planar uniformly aligned state so that the total charge has to vanish. Topological charge conservation then requires that there is always an equal number of q=1/2 and q=-1/2 charges. Starting from a set of hydrodynamic equations, we derive a low-dimensional dynamical system for the parameters of the static solutions, which describes the motion of a half-disclination pair or of several pairs. Within this formalism, we model defect production and annihilation, as observed in experiments. Our dynamics also provide an estimate for the critical density at which production and annihilation rates are balanced.

Defect driven shapes in nematic droplets: analogies with cell division.

Building on the striking similarity between the structure of the spindle during mitosis in living cells and nematic textures in confined liquid crystals, we use a continuum model of two-dimensional nematic liquid crystal droplets to examine the physical aspects of cell division. The model investigates the interplay between bulk elasticity of the microtubule assembly, described as a nematic liquid crystal, and surface elasticity of the cell cortex, modeled as a bounding flexible membrane, in controlling cell shape and division. The centrosomes at the spindle poles correspond to the cores of the topological defects required to accommodate nematic order in a closed geometry. We map out the progression of both healthy bipolar and faulty multi-polar division as a function of an effective parameter that incorporates active processes and controls centrosome separation. A robust prediction, independent of energetic considerations, is that the transition from a single cell to daughters cells occurs at critical value of this parameter. Our model additionally suggests that microtubule anchoring at the cell cortex may play an important role for successful bipolar division. This can be tested experimentally by regulating microtubule anchoring.

Stability of point defects of degree $$\pm \frac{1}{2}$$ ± 1 2 in a two-dimensional nematic liquid crystal model

We study k-radially symmetric solutions corresponding to topological defects of charge $$\frac{k}{2}$$ for integer $$k\not = 0$$ in the Landau-de Gennes model describing liquid crystals in two-dimensional domains. We show that the solutions whose radial profiles satisfy a natural sign invariance are stable when $$|k| = 1$$ (unlike the case $$|k| > 1$$ which we treated before). The proof crucially uses the monotonicity of the suitable components, obtained by making use of the cooperative character of the system. A uniqueness result for the radial profiles is also established.

Instability of point defects in a two-dimensional nematic liquid crystal model

We study a class of symmetric critical points in a variational 2D Landau–de Gennes model where the state of nematic liquid crystals is described by symmetric traceless 3×3 matrices. These critical points play the role of topological point defects carrying a degree k2 for a nonzero integer k. We prove existence and study the qualitative behavior of these symmetric solutions. Our main result is the instability of critical points when |k|≥2.

Equilibrium configurations of nematic liquid crystals on a torus.

The topology and the geometry of a surface play a fundamental role in determining the equilibrium configurations of thin films of liquid crystals. We propose here a theoretical analysis of a recently introduced surface Frank energy, in the case of two-dimensional nematic liquid crystals coating a toroidal particle. Our aim is to show how a different modeling of the effect of extrinsic curvature acts as a selection principle among equilibria of the classical energy and how new configurations emerge. In particular, our analysis predicts the existence of stable equilibria with complex windings.

Numerical solution of the Ericksen–Leslie dynamic equations for two-dimensional nematic liquid crystal flows

A finite difference method for solving nematic liquid crystal flows under the effect of a magnetic field is developed. The dynamic equations of nematic liquid crystals, given by the Ericksen–Leslie dynamic theory, are employed. These are expressed in terms of primitive variables and solved employing the ideas behind the GENSMAC methodology (Tomé and McKee, 1994; Tomé et al., 2002) [38,41]. These equations are nonlinear partial differential equations consisting of the mass conservation equation and the balance laws of linear and angular momentum. By employing fully developed flow assumptions an analytic solution for steady 2D-channel flow is found. The resulting numerical technique was then, in part, validated by comparing numerical solutions against this analytic solution. Convergence results are presented. To demonstrate the capabilities of the numerical method, the flow of a nematic liquid crystal through various complex geometries are then simulated. Results are obtained for L-shaped channels and planar 4:1 contraction for several values of Reynolds and Ericksen numbers.

Curvature-induced ordering in cylindrical nematic shells

In this paper we study theoretically the equilibrium configurations of two-dimensional nematic liquid crystals on a cylindrical surface. Configurations are described through a tensor-order parameter that provides information about the tangential optical axis and the dispersion of the molecules around it. Equilibrium equations are obtained by the surface Landau–de Gennes energy recently proposed in Napoli and Vergori [Physical Review E 061701 (2012)]. We show that, whenever free boundary conditions are applied, the extrinsic curvature induces the alignment of the optical axis along the cylinder axis. On the contrary, when strong planar anchoring is imposed on the boundaries, the curvature triggers a sort of Freédricksz-like transition in the alignment. We provide the asymptotic solution of the non-linear problem with strong planar anchoring boundary conditions for elongated cylindrical shells.