Landau-de Gennes Model Research Articles

Modelling and simulation of the cholesteric Landau-de Gennes model

Modelling and simulation of the cholesteric Landau-de Gennes model

Geometric modeling of phase ordering for the isotropic–smectic A phase transition

BackgroundLiquid crystal (LC) mesophases have an orientational and positional order that can be found in both synthetic and biological materials. These orders are maintained until some parameter, mainly the temperature or concentration, is changed, inducing a phase transition. Among these transitions, a special sequence of mesophases has been observed, in which priority is given to the direct smectic liquid crystal transition. The description of these transitions is carried out using the Landau–de Gennes (LdG) model, which correlates the free energy of the system with the orientational and positional order.MethodologyThis work explored the direct isotropic-to-smectic A transition studying the free energy landscape constructed with the LdG model and its relation to three curve families: (I) level-set curves, steepest descent, and critical points; (II) lines of curvature (LOC) and geodesics, which are directly connected to the principal curvatures; and (III) the Casorati curvature and shape coefficient that describe the local surface geometries resemblance (sphere, cylinder, and saddle).ResultsThe experimental data on 12-cyanobiphenyl were used to study the three curve families. The presence of unstable nematic and metastable plastic crystal information was found to add information to the already developed smectic A phase diagram. The lines of curvature and geodesics were calculated and laid out on the energy landscape, which highlighted the energetic pathways connecting critical points. The Casorati curvature and shape coefficient were computed, and in addition to the previous family, they framed a geometric region that describes the phase transition zone.Conclusion and significanceA direct link between the energy landscape’s topological geometry, phase transitions, and relevant critical points was established. The shape coefficient delineates a stability zone in which the phase transition develops. The methodology significantly reduces the impact of unknown parametric data. Symmetry breaking with two order parameters (OPs) may lead to novel phase transformation kinetics and droplets with partially ordered surface structures.

Topological Defects in Tactoid Nematic Phase

The purpose research of the work is to illustrate the role of point surface topological defects in nematics in phase transitions using the example of inorganic sols of vanadium pentoxide in water V2O5/H2O.Methods. Basing on the measurements on polarization light microscopy and theoretically, in frames of the Landau-de Gennes model and the Kibble-Zurek model for defects, the influence of point topological defects at the surface of nematic colloid of lyotropic liquid crystals, in particular, in inorganic sols of the V2O5/H2O system, on their critical thermodynamic, electric, optical, elastic and mechanical properties is estimated. For the system V2O5/H2O, we study the developed models of nematic colloid phases with non-trivial topology composed of spindle-shaped particles, in which their topological defects are classified; geometrical parameters of nematic particles in dynamics and evolution depending on physicochemical conditions, phase stability criteria, conditions for coalescence of particles, including, under external magnetic field, and also, experimental observations confirmed the proposed theories and models, are considered.Results. With a glance of coalescence regime, the critical size of a colloid particle of the system V2O5/H2O, ~ 10 nm in the long axis direction, in which the value of magnetic energy equals to the sum of elastic and surface energy of acolloid particle, is estimated.Conclusion. We confirmed, that the prerequisite of sol particle coalescence in the system V2O5/H2O in the magnetic field is annihilation of the point topological defects at their poles, the geometrical criterion of tactoid coalescence in magnetic field is established basing on the different theories of their surface defects.

Modeling the thermo-responsive behaviors of polydomain and monodomain nematic liquid crystal elastomers

Liquid crystal elastomers (LCEs) are polymeric materials that combine liquid crystal orientation and rubber elasticity. The mechanical responses of LCEs strongly depend on the nematic order and temperature. Developing a thermo-order-mechanical coupling model is important for the engineering applications of LCEs. In this work, we first synthesize both polydomain and monodomain LCEs. The shape change with temperature under a certain stress level is further characterized. In the theoretical part, the Landau–de Gennes model and neo-classical model are combined to construct the free energy density of LCEs. A linear term accounting for the effect of interior stress on nematic–isotropic transition is incorporated into the Landau–de Gennes free energy density, while a macro-order parameter is defined to describe the polydomain–monodomain transition for polydomain LCEs induced by the external force. Comparison between the model predictions and experimental results shows acceptably consistency for both polydomain and monodomain LCEs. Therefore, this work provides an efficient approach to predict the shape change of LCEs in various scenarios.

Singular perturbation of manifold-valued maps with anisotropic energy

We establish small energy H\"{o}lder bounds for minimizers $u_\varepsilon$ of \[E_\varepsilon (u):=\int_\Omega W(\nabla u)+ \frac{1}{\varepsilon^2} \int_\Omega f(u),\] where $W$ is a positive definite quadratic form and the potential $f$ constrains $u$ to be close to a given manifold $\mathcal N$. This implies that, up to subsequence, $u_\varepsilon$ converges locally uniformly to an $\mathcal N$-valued $W$-harmonic map, away from its singular set. We treat general energies, covering in particular the 3D Landau-de Gennes model for liquid crystals, with three distinct elastic constants. Similar results are known in the isotropic case $W(\nabla u)=\vert \nabla u\vert^2$ and rely on three ingredients: a monotonicity formula for the scale-invariant energy on small balls, a uniform pointwise bound, and a Bochner equation for the energy density. In the level of generality we consider, all of these ingredients are absent. In particular, the lack of monotonicity formula is an important reason why optimal estimates on the singular set of $W$-harmonic maps constitute an open problem. Our novel argument relies on showing appropriate decay for the energy on small balls, separately at scales smaller and larger than $\varepsilon$: the former is obtained from the regularity of solutions to elliptic systems while the latter is inherited from the regularity of $W$-harmonic maps. This also allows us to handle physically relevant boundary conditions for which, even in the isotropic case, uniform convergence up to the boundary was open.

Uniform profile near the point defect of Landau-de Gennes model

For the Landau-de Gennes functional on 3D domains, $$\begin{aligned} I_\varepsilon (Q,\Omega ):=\int _{\Omega }\left\{ \frac{1}{2}|\nabla Q|^2+\frac{1}{\varepsilon ^2}\left( -\frac{a^2}{2}\mathrm {tr}(Q^2)-\frac{b^2}{3}\mathrm {tr}(Q^3)+\frac{c^2}{4}[\mathrm {tr}(Q^2)]^2 \right) \right\} \,dx, \end{aligned}$$ it is well-known that under suitable boundary conditions, the global minimizer $$Q_\varepsilon $$ converges strongly in $$H^1(\Omega )$$ to a uniaxial minimizer $$Q_*=s_+(n_*\otimes n_*-\frac{1}{3}\mathrm {Id})$$ up to some subsequence $$\varepsilon _n\rightarrow \infty $$ , where $$n_*\in H^1(\Omega ,\mathbb {S}^2)$$ is a minimizing harmonic map. In this paper we further investigate the structure of $$Q_\varepsilon $$ near the core of a point defect $$x_0$$ which is a singular point of the map $$n_*$$ . The main strategy is to study the blow-up profile of $$Q_{\varepsilon _n}(x_n+\varepsilon _n y)$$ where $$\{x_n\}$$ are carefully chosen and converge to $$x_0$$ . We prove that $$Q_{\varepsilon _n}(x_n+\varepsilon _n y)$$ converges in $$C^2_{loc}(\mathbb {R}^n)$$ to a tangent map Q(x) which at infinity behaves like a “hedgehog" solution that coincides with the asymptotic profile of $$n_*$$ near $$x_0$$ . Moreover, such convergence result implies that the minimizer $$Q_{\varepsilon _n}$$ can be well approximated by the Oseen-Frank minimizer $$n_*$$ outside the $$O(\varepsilon _n)$$ neighborhood of the point defect.

On standing waves and gradient-flow for the Landau–De Gennes model of nematic liquid crystals

The article treats the existence of standing waves and solutions to gradient-flow equation for the Landau–De Gennes models of liquid crystals, a state of matter intermediate between the solid state and the liquid one. The variables of the general problem are the velocity field of the particles and the Q-tensor, a symmetric traceless matrix which measures the anisotropy of the material. In particular, we consider the system without the velocity field and with an energy functional unbounded from below. At the beginning we focus on the stationary problem. We outline two variational approaches to get a critical point for the relative energy functional: by the Mountain Pass Theorem and by proving the existence of a least energy solution. Next we describe a relationship between these solutions. Finally we consider the evolution problem and provide some Strichartz-type estimates for the linear problem. By several applications of these results to our problem, we prove via contraction arguments the existence of local solutions and, moreover, global existence for initial data with small L^2-norm.

Elastic anisotropy in the reduced Landau-de Gennes model.

We study the effects of elastic anisotropy on Landau–de Gennes critical points, for nematic liquid crystals, on a square domain. The elastic anisotropy is captured by a parameter, , and the critical points are described by 3 d.f. We analytically construct a symmetric critical point for all admissible values of , which is necessarily globally stable for small domains, i.e. when the square edge length, , is small enough. We perform asymptotic analyses and numerical studies to discover at least five classes of these symmetric critical points—the , , and solutions, of which the , and solutions can be stable. Furthermore, we demonstrate that the novel solution is energetically preferable for large and large , and prove associated stability results that corroborate the stabilizing effects of for reduced Landau–de Gennes critical points. We complement our analysis with numerically computed bifurcation diagrams for different values of , which illustrate the interplay of elastic anisotropy and geometry for nematic solution landscapes, at low temperatures.

Accelerated Gradient Descent Methods for the Uniaxially Constrained Landau-de Gennes Model

Accelerated Gradient Descent Methods for the Uniaxially Constrained Landau-de Gennes Model

Solution landscapes of the simplified Ericksen–Leslie model and its comparisonwith the reduced Landau–deGennes model

We investigate the solution landscapes of a simplified Ericksen–Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau–de Gennes (rLdG) models can be viewed as Ginzburg–Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare them with the solution landscapes of the corresponding rLdG model. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviours and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solutionC±with interior defects, high-index ‘fake defect’ solutions, novel tessellating solutions and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics.

The Saturn Ring Effect in Nematic Liquid Crystals with External Field: Effective Energy and Hysteresis

In this work we consider the Landau-de Gennes model for liquid crystals with an external electromagnetic field to model the occurrence of the saturn ring effect under the assumption of rotational equivariance. After a rescaling of the energy, a variational limit is derived. Our analysis relies on precise estimates around the singularities and the study of a radial auxiliary problem in regions, where a continuous director field exists. Studying the limit problem, we explain the transition between the dipole and saturn ring configuration and the occurrence of a hysteresis phenomenon, giving a rigorous explanation of what was conjectured previously by [H. Stark, Eur. Phys. J. B 10, 311-321 (1999)].

Numerical method for the equilibrium configurations of a Maier-Saupe bulk potential in a Q-tensor model of an anisotropic nematic liquid crystal

We present a numerical method, based on a tensor order parameter description of a nematic phase, that allows fully anisotropic elasticity. Our method thus extends the Landau-de Gennes Q-tensor theory to anisotropic phases. A microscopic model of the nematogen is introduced (the Maier-Saupe potential in the case discussed in this paper), combined with a constraint on eigenvalue bounds of Q. This ensures a physically valid order parameter Q (i.e., the eigenvalue bounds are maintained), while allowing for more general gradient energy densities that can include cubic nonlinearities, and therefore elastic anisotropy. We demonstrate the method in two specific two dimensional examples in which the Landau-de Gennes model including elastic anisotropy is known to fail, as well as in three dimensions for the cases of a hedgehog point defect, a disclination line, and a disclination ring. The details of the numerical implementation are also discussed.

Phase behaviour of n-CB liquid crystals confined to controlled pore glasses

In this review, we present the phase behaviour of liquid crystals (LCs) confined to Controlled-Pore Glasses (CPG) of various pore sizes. We consider either non-treated or silane-treated CPG pore surfaces, which impose parallel and homeotropic alignment to n-cyanobiphenyl (n-CB) LCs, respectively. The resulting nematic or smectic ordering depends on the characteristic void radii R and is well described in terms of the Landau-de Gennes free energy. Experimental data for different n-CB LCs, 5CB, 8CB, and 12CB, demonstrate phase transitions in orientation and translational degrees of freedom. In the bulk, 5CB and 8CB exhibit on cooling from the isotropic (I) phase the 1st order isotropic-nematic (I−N) phase transitions. In addition, 8CB also exhibits on further cooling a 2nd order nematic-smectic A (N−SmA) phase transition. 12CB, on the other hand, only has a 1st order isotropic-smectic A (I−SmA) phase transition. The bulk LCs exhibit a variety of different phase structures. When LCs are confined to porous media, their phase structures can be modified and even glass transition is possible.For LCs confined to CPG, the phase transition temperatures are shifted, depending on the pore size and the pore surface treatment. The Landau-de Gennes model predicts that the present phase transition shifts display inversely linear and inversely quadratic contributions in terms of the characteristic pore sizes. Furthermore, for strong enough confinement the phase transitions are replaced by gradual evolution of LCs order. These predictions are in line with nuclear magnetic resonance, small- angle X-ray scattering, and high-resolution calorimetry experimental observations.

Manipulation of m = 1 topological disclination line core structure

Topological defects (TDs) arise in phases reached via symmetry-breaking phase transitions. Their behavior displays several universalities and is of interest to all branches of physics, including particle physics, soft-condensed matter, and cosmology. Ideal testing beds to study the physics of TDs are liquid crystals (LCs), where TDs are relatively easily approachable experimentally. In the paper, we perform a numerical investigation of a locally perturbed m = 1 line defect core structure in the nematic LC phase confined within a plane-parallel cell. We use the mesoscopic Landau-de Gennes model in terms of the tensor nematic order parameter. Relatively thin LC cells are considered, where a m = 1 disclination is enforced by appropriately patterned confining surfaces. We demonstrate that an imposed local melting, which breaks local cylindrical symmetry, could trigger splitting of the disclination into two m = 1/2 disclination lines. Such manipulations could enable transformations between qualitatively different defect core structures. Cores of TDs could be also manipulated by nanoparticles which could be exploited in various applications.

Solution landscape of a reduced Landau–de Gennes model on a hexagon

We investigate the solution landscape of a reduced Landau–de Gennes model for nematic liquid crystals (NLCs) on a two-dimensional hexagon at a fixed temperature, as a function of λ—the edge length. This is a generic example for reduced approaches on regular polygons. We apply the high-index optimization-based shrinking dimer method to systematically construct the solution landscape consisting of multiple solutions, with different defect configurations, and relationships between them. We report a new stable T state with index-0 that has an interior −1/2 defect; new classes of high-index saddle points with multiple interior defects referred to as H-class and TD-class saddle points; changes in the Morse index of saddle points as λ 2 increases and novel pathways mediated by high-index saddle points that can control and steer dynamical pathways on the solution landscape. The range of topological degrees, locations and multiplicity of defects offered by these saddle points can be used to navigate the complex solution landscapes of NLCs and other related soft matter systems.

Ordering behavior and electric response of a ferroelectric nano-doped liquid crystal with ion impurity effects

We present a theoretical study of the simultaneous effects of ferroelectric nanoparticles and ion impurities on the order parameter of liquid crystals. We have focused on the effect of ion impurities on the ordering behavior and electrical response of a ferroelectric nano-doped liquid crystal near the nematic-isotropic phase transition. In this study, a statistical theory based on the orientational distribution of the nanoparticle dopants is used. We have performed the calculations using the Poisson–Boltzmann equation and a modified Landau-De Gennes model. The total density of free energy of the system is obtained within the framework of this theory, and the changes in the concentration of mobile ions due to the presence of ferroelectric nanoparticles are investigated. It is shown that the screening effect of the ion impurities plays a dominant role in the ordering behavior and electrical response of a ferroelectric nano-doped liquid crystal. The ion generation and the consequent changes are also described in some physical properties of a nano-doped system under an applied electric field.

Cubic microlattices embedded in nematic liquid crystals: a Landau-de Gennes study

We consider a Landau-de Gennes model for a connected cubic lattice scaffold in a nematic host, in a dilute regime. We analyse the homogenised limit for both cases in which the lattice of embedded particles presents or not cubic symmetry and then we compute the free effective energy of the composite material. In the cubic symmetry case, we impose different types of surface anchoring energy densities, such as quartic, Rapini-Papoular or more general versions, and, in this case, we show that we can tune any coefficient from the corresponding bulk potential, especially the phase transition temperature. In the case with loss of cubic symmetry, we prove similar results in which the effective free energy functional has now an additional term, which describes a change in the preferred alignment of the liquid crystal particles inside the domain. Moreover, we compute the rate of convergence for how fast the surface energies converge to the homogenised one and also for how fast the minimisers of the free energies tend to the minimiser of the homogenised free energy.

A structure-preserving FEM for the uniaxially constrained $$\mathbf{Q}$$-tensor model of nematic liquid crystals

We consider the one-constant Landau-de Gennes model for nematic liquid crystals. The order parameter is a traceless tensor field $$\mathbf{Q}$$ , which is constrained to be uniaxial: $$\mathbf{Q}= s (\mathbf{n}\otimes \mathbf{n}- d^{-1}\mathbf{I})$$ where $$\mathbf{n}$$ is a director field, $$s\in \mathbb {R}$$ is the degree of orientation, and $$d\ge 2$$ is the dimension. Building on similarities with the one-constant Ericksen energy, we propose a structure-preserving finite element method for the computation of equilibrium configurations. We prove stability and consistency of the method without regularization, and $$\Gamma $$ -convergence of the discrete energies towards the continuous one as the mesh size goes to zero. We design an alternating direction gradient flow algorithm for the solution of the discrete problems, and we show that such a scheme decreases the energy monotonically. Finally, we illustrate the method’s capabilities by presenting some numerical simulations in two and three dimensions including non-orientable line fields.

A domain decomposition approach to accelerate simulations of structure preserving nematic liquid crystal models

We present a method for speeding up equilibrium simulations of liquid crystals for two related models that are degenerate with respect to defects: the Ericksen model and the uniaxially constrained Landau-de Gennes model. The degeneracy induces a non-linear/non-smooth coupling between the two order parameters in both models that makes the discrete problems very stiff to solve. The technique described in this paper uses an alternating Schwarz domain decomposition method that isolates the degenerate region and alleviates some of the stiffness, and is easy to implement. We present numerical results illustrating the speed-up and how it is affected by the size of the sub-domains and the number of sub-iterations used within the degenerate region.

Electro-optical Kerr effect of 6CHBT liquid crystal doped with MgO nanoparticles in different concentration

The Kerr constant and pre-transition temperature of pure liquid crystal, 6CHBT, and doped with different concentrations of MgO nanoparticles were investigated at temperatures above the nematic-isotropic transition temperature by using the nulled intensity method. The MgO nanoparticles were added to liquid crystal at three different concentrations of 0.1, 0.3 and 0.6 (wt.). The results showed that the Kerr constant increase by adding nanoparticles mass percentages up to a certain concentration and then remains constant. The linear dependence of (T-T*) on the Kerr constant is found to be in good agreement with predictions of Landau-De Gennes model.