Maier-Saupe Potential Research Articles

Solution landscape of the Onsager model identifies non-axisymmetric critical points

We investigate critical points of the Onsager free-energy model on a sphere with different potential kernels, including the dipolar potential, the Maier–Saupe potential, the coupled dipolar/Maier–Saupe potential, and the Onsager potential. A uniform sampling method is implemented for the discretization of the Onsager model, and solution landscapes of the Onsager model are constructed using saddle dynamics coupled with downward/upward search algorithms. We first construct the solution landscapes with the dipolar and Maier–Saupe potentials, for which all critical points are axisymmetric. For the coupled dipolar/Maier–Saupe potential, the solution landscape shows a novel non-axisymmetric critical point, named tennis, which exists for a wide range of parameters. We further demonstrate various non-axisymmetric critical points in the Onsager model with the Onsager potential, including square, hexagon, octahedral, cubic, quarter, icosahedral, and dodecahedral states. The bifurcation diagram is presented to show the primary and secondary bifurcations of the isotropic state and reveal the emergence of the critical points. The solution landscape provides an efficient approach to show the global structure of the model system as well as the bifurcations of critical points, which can not only support the previous theoretical conjectures but also propose new conjectures based on the numerical findings.

Axisymmetry of critical points for the Onsager functional.

A simple proof is given of the classical result (Fatkullin I, Slastikov V. 2005 Critical points of the Onsager functional on a sphere. Nonlinearity 18, 2565-2580 (doi:10.1088/0951-7715/18/6/008); Liu H et al. 2005 Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential. Commun. Math. Sci. 3, 201-218 (doi:10.4310/CMS.2005.v3.n2.a7)) that critical points for the Onsager functional with the Maier-Saupe molecular interaction are axisymmetric, including the case of stable critical points with an additional dipole-dipole interaction (Zhou H et al. 2007 Characterization of stable kinetic equilibria of rigid, dipolar rod ensembles for coupled dipole-dipole and Maier-Saupe potentials. Nonlinearity 20, 277-297 (doi:10.1088/0951-7715/20/2/003)). The proof avoids spherical polar coordinates, instead using an integral identity on the sphere S2. For general interactions with absolutely continuous kernels the smoothness of all critical points is established, generalizing a result in (Vollmer MAC. 2017 Critical points and bifurcations of the three-dimensional Onsager model for liquid crystals. Archive for Rational Mechanics and Analysis 226, 851-922 (doi:10.1007/s00205-017-1146-8)) for the Onsager interaction. It is also shown that non-axisymmetric critical points exist for a wide variety of interactions including that of Onsager. This article is part of the theme issue 'Topics in mathematical design of complex materials'.

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.

A molecular field approach to pressure-induced phase transitions in liquid crystals: Smectic–nematic transition

Since a rigorous microscopic treatment of a nematic fluid system based on a pairwise interaction potential is immensely complex, we had introduced a simple mean field potential, which was a modification of the Maier–Saupe potential in a previous paper [S. DasGupta et al., “Pressure-induced phase transitions in liquid crystals: A molecular field approach,” Phys. Rev. E 98, 022701 (2018)]. Building upon that, here we have modified that potential to take into account the various aspects of a smectic A–nematic phase transition. In particular, we have studied the dependence of the phase transition on the coupling coefficient between the nematic and smectic order parameters, which in turn depends on the length of alkyl chain, variation of density, entropy, and specific heat. Detailed investigation on the coupling parameter shows the existence of a smectic A–nematic–isotropic triple point as well as a tricritical point where the smectic–nematic phase transition changes its nature from the second to the first order. It is also seen that the application of pressure can result in the appearance of a nematic phase.

Magnetic Field and Dilution Effects on the Phase Diagrams of Simple Statistical Models for Nematic Biaxial Systems

We use a simple statistical model to investigate the effects of an applied magnetic field and of the dilution of site elements on the phase diagrams of biaxial nematic systems, with an emphasis on the stability of the Landau multicritical point. The statistical lattice model consists of intrinsically biaxial nematogenic units, which interact via a Maier–Saupe potential, and which are characterized by a discrete choice of orientations of the microscopic nematic directors. According to previous calculations at zero field and in the absence of dilution, we regain the well-known sequence of biaxial, uniaxial, and disordered structures as the temperature is increased, and locate the Landau point. We then focus on the topological changes induced in the phase diagram by the application of an external magnetic field, and show that the Landau point is destabilized by the presence of an applied field. On the other hand, in the absence of a field, we show that only a quite strong dilution of nematic sites is capable of destabilizing the Landau point.

Scaling law of Brownian rotation in dense hard-rod suspensions.

Self-diffusion in dense rod suspensions are subject to strong geometric constraints because of steric interactions. This topological effect is essentially anisotropic when rods are nematically aligned with their neighbors, raising considerable challenges in understanding and analyzing their impacts on the bulk physical properties. Via a classical Doi-Onsager kinetic model with the Maier-Saupe potential, we characterize the long-time rotational Brownian diffusivity for dense suspensions of hard rods of finite aspect ratios, based on quadratic orientation autocorrelation functions. Furthermore, we show that the computed nonmonotonic scalings of the diffusivity as a function of volume fraction can be accurately predicted by an alternative tube model in the nematic phase.

Mean-Field Limit and Phase Transitions for Nematic Liquid Crystals in the Continuum

We discuss thermotropic nematic liquid crystals in the mean-field regime. In the first part of this article, we rigorously carry out the mean-field limit of a system of N rod-like particles as $$N\rightarrow \infty $$ , which yields an effective ‘one-body’ free energy functional. In the second part, we focus on spatially homogeneous systems, for which we study the associated Euler–Lagrange equation, with a focus on phase transitions for general axisymmetric potentials. We prove that the system is isotropic at high temperature, while anisotropic distributions appear through a transcritical bifurcation as the temperature is lowered. Finally, as the temperature goes to zero we also prove, in the concrete case of the Maier–Saupe potential, that the system converges to perfect nematic order.

Maier-Saupe model of polymer nematics: Comparing free energies calculated with Self Consistent Field theory and Monte Carlo simulations

Self Consistent Field (SCF) theory serves as an efficient tool for studying mesoscale structure and thermodynamics of polymeric liquid crystals (LC). We investigate how some of the intrinsic approximations of SCF affect the description of the thermodynamics of polymeric LC, using a coarse-grained model. Polymer nematics are represented as discrete worm-like chains (WLC) where non-bonded interactions are defined combining an isotropic repulsive and an anisotropic attractive Maier-Saupe (MS) potential. The range of the potentials, σ, controls the strength of correlations due to non-bonded interactions. Increasing σ (which can be seen as an increase of coarse-graining) while preserving the integrated strength of the potentials reduces correlations. The model is studied with particle-based Monte Carlo (MC) simulations and SCF theory which uses partial enumeration to describe discrete WLC. In MC simulations the Helmholtz free energy is calculated as a function of strength of MS interactions to obtain reference thermodynamic data. To calculate the free energy of the nematic branch with respect to the disordered melt, we employ a special thermodynamic integration (TI) scheme invoking an external field to bypass the first-order isotropic-nematic transition. Methodological aspects which have not been discussed in earlier implementations of the TI to LC are considered. Special attention is given to the rotational Goldstone mode. The free-energy landscape in MC and SCF is directly compared. For moderate σ the differences highlight the importance of local non-bonded orientation correlations between segments, which SCF neglects. Simple renormalization of parameters in SCF cannot compensate the missing correlations. Increasing σ reduces correlations and SCF reproduces well the free energy in MC simulations.

An SRLS Study of 2H Methyl-Moiety Relaxation and Related Conformational Entropy in Free and Peptide-Bound PLCγ1C SH2.

The two-body (protein and probe) coupled-rotator slowly relaxing local structure (SRLS) approach for NMR relaxation in proteins is extended to derive conformational entropy, Ŝ. This version of SRLS is applied to deuterium relaxation from the C-CDH2 bonds of free and peptide-bound PLCγ1C SH2. Local C-CDH2 motion is described by a correlation time for local diffusion, τ2, and a Maier-Saupe potential, u. On average, τ2, which largely fulfills τ2 ≪ τ1 (τ1 - correlation time for global tumbling), is 270 ± 41 ps and u is 2 ± 0.1 kBT. The PLCγ1C SH2 data were analyzed previously with the model-free (MF) method. SRLS is a generalization of MF, assumed so far to yield the latter for τ2 ≪ τ1 and simple local geometry. Despite these conditions being fulfilled, we find here that τ2 and u differ substantially from their MF counterparts. This is shown to stem from MF (a) disregarding mode-coupling of the first type (see below) and (b) parametrizing the methyl-moiety-related spectral density function (SDF). Our main interest lies in ΔŜ, the conformational entropy difference between the peptide-bound and free PLCγ1C SH2 forms. We find that ΔŜ is rendered inaccurate in MF because factors a and b above impair the accuracy of Saxis, the parameter on which the calculation of ΔŜ is based. Conformational entropy was obtained previously using various simple system-specific models. SRLS is unique in obtaining this important thermodynamic quantity based on a general physically well-defined local potential. It is also unique in its ability to extract the information inherent in 2H relaxation parameters from methyl moieties in protein with accuracy commensurate with data sensitivity.

Bimodal form distribution from modelling biaxial lyotropic liquid crystal solutions through a polydisperse Maier–Saupe model

ABSTRACTBiaxial nematic phases have been the subject of a long list of studies. In particular, they were found for a few lyotropic micellar solutions. There is a debate in the literature on whether the micellar aggregates should be biaxial, or if biaxiality could be the result of perpendicular alignment of uniaxial particles of cylinder-like and disc-like geometry in a mixture. Based on recent studies on the phase stability of such mixtures, we have investigated a polydisperse distribution of uniaxial particles interacting through a Maier–Saupe potential. Our calculations were developed for a general distribution of micellar anisometries. The distribution was obtained from the fitting of our results to the experimental data of Yu and Saupe’s well-known 1980 paper, near the Landau point, yielding a bimodal distribution, with the presence of two quadrupoles referred to objects of opposite symmetry, that is to prolate and oblate micelles. This result lends support to the rationalization of the biaxial phase for lyotropic systems in terms of a polydisperse mixture of rod-like and disc-like micelles.

Deuteron and proton NMR study of D2, p-dichlorobenzene and 1,3,5-trichlorobenzene in bimesogenic liquid crystals with two nematic phases

The solutes dideuterium, 1,3,5-trichlorobenzene and p-dichlorobenzene (pdcb) are co-dissolved in a 61/39wt% mixture of CBC9CB/5CB, a bimesogenic liquid crystal with two nematic phases. NMR spectra are collected for each solute. The local electric field gradient (FZZ) is obtained from the dideuterium spectrum. A double Maier-Saupe potential (MSMS) is used to rationalize the order parameters of pdcb. The liquid-crystal fields G1 and G2 are taken to be due to size and shape interactions and interactions between the solute molecular quadrupole and the mean FZZ of the medium. The FZZ’s obtained from D2 and G2 (from pdcb) are compared and discussed.

Probability distribution function for reorientations in Maier–Saupe potential

Exact analytic solution for the probability distribution function of the non-inertial rotational diffusion equation, i.e., of the Smoluchowski one, in a symmetric Maier–Saupe uniaxial potential of mean torque is obtained via the confluent Heun’s function. Both the ordinary Maier–Saupe potential and the double-well one with variable barrier width are considered. Thus, the present article substantially extends the scope of the potentials amenable to the treatment by reducing Smoluchowski equation to the confluent Heun’s one. The solution is uniformly valid for any barrier height. We use it for the calculation of the mean first passage time. Also the higher eigenvalues for the relaxation decay modes in the case of ordinary Maier–Saupe potential are calculated. The results obtained are in full agreement with those of the approach developed by Coffey, Kalmykov, Déjardin and their coauthors in the whole range of barrier heights.

Exact solution of Smoluchowski’s equation for reorientational motion in Maier–Saupe potential

The analytic treatment of the non-inertial rotational diffusion equation, i.e., of Smoluchowski’s one (SE), in a symmetric genuinely double-well Maier–Saupe uniaxial potential of mean torque is considered. Such potential may find applications to reorientations of the fragments of structure in polymers and proteins. We obtain the exact solution of SE via the confluent Heun’s function. The solution is uniformly valid for any barrier height. We apply the obtained solution to the calculation of the mean first passage time and the longitudinal correlation time and obtain their precise dependence on the barrier height. In the intermediate to high barrier (low temperature) region the results of our approach are in full agreement with those of the approach developed by Coffey, Kalmykov, Déjardin and their coauthors. In the low barrier (high temperature) region our results noticeably distinguish from the predictions of the literature formula and give appreciably greater values for the transition rates from the potential well. The reason is that the above mentioned formula is obtained in the stationary limit. We conclude that for very small barrier heights the transient dynamics plays a crucial role and has to be taken into account explicitly. When this requirement is satisfied (as, e.g, at the calculation of the longitudinal correlation time) we obtain absolute identity of our results with the literature formula in the whole range of barrier heights. The drawbacks of our approach are its applicability only to the symmetric potential and its inability to yield an analytical expression for the smallest non-vanishing eigenvalue.

Comparative Study of the Thermal Dependence of the Birefringence and Order Parameter of Two nOCB Compounds

The optical properties, birefringence and order parameter of two alkoxycyanobiphenyls (nOCBs, where n = 5 or 8) have been studied as a function of temperature. The variation of orientational order parameter has been determined experimentally. Theoretical order parameter values have also been computed in the smectic range using McMillan potential and in the nematic range using Maier–Saupe potential and compared with the experimentally obtained values. The variations of polarizabilities with temperature have also been evaluated from density variation and birefringence studies.

Stability of equilibria of nematic liquid crystalline polymers

We provide an analytical study on the stability of equilibria of rigid rodlike nematic liquid crystalline polymers (LCPs) governed by the Smoluchowski equation with the Maier-Saupe intermolecular potential. We simplify the expression of the free energy of an orientational distribution function of rodlike LCP molecules by properly selecting a coordinate system and then investigate its stability with respect to perturbations of orientational probability density. By computing the Hessian matrix explicitly, we are able to prove the hysteresis phenomenon of nematic LCPs: when the normalized polymer concentration b is below a critical value b* (6.7314863965), the only equilibrium state is isotropic and it is stable; when b* <b < 15/2, two anisotropic (prolate) equilibrium states occur together with a stable isotropic equilibrium state. Here the more aligned prolate state is stable whereas the less aligned prolate state is unstable. When b > 15/2, there are three equilibrium states: a stable prolate state, an unstable isotropic state and an unstable oblate state.

Multiple branches of ordered states of polymer ensembles with the Onsager excluded volume potential

We study the branches of equilibrium states of rigid polymer rods with the Onsager excluded volume potential in two-dimensional space. Since the probability density and the potential are related by the Boltzmann relation at equilibrium, we represent an equilibrium state using the Fourier coefficients of the Onsager potential. We derive a non-linear system for the Fourier coefficients of the equilibrium state. We describe a procedure for solving the non-linear system. The procedure yields multiple branches of ordered states. This suggests that the phase diagram of rigid polymer rods with the Onsager potential has a more complex structure than that with the Maier–Saupe potential. A study of free energy indicates that the first branch of ordered states is stable while the subsequent branches are unstable. However, the instability of the subsequent branches does not mean they are not interesting. Each of these unstable branches, under certain external potential, can be made metastable, and thus may be observed.

Extendability of Equilibria of Nematic Polymers

The purpose of this paper is to study the extendability of equilibrium states of rodlike nematic polymers with the Maier-Saupe intermolecular potential. We formulate equilibrium states as solutions of a nonlinear system and calculate the determinant of the Jacobian matrix of the nonlinear system. It is found that the Jacobian matrix is nonsingular everywhere except at two equilibrium states. These two special equilibrium states correspond to two points in the phase diagram. One point is the folding point where the stable prolate branch folds into the unstable prolate branch; the other point is the intersection point of the nematic branch and the isotropic branch where the unstable prolate state becomes the unstable oblate state. Our result establishes the existence and uniqueness of equilibrium states in the presence of small perturbations away from these two special equilibrium states.

Global orientation dynamics for liquid crystalline polymers

We study global orientation dynamics of the Doi–Smoluchowski equation with the Maier–Saupe potential on the sphere, which arises in the modeling of rigid rod-like molecules of polymers. Using the orientation tensor we first reconfirm the structure and number of equilibrium solutions established in [H.L. Liu, H. Zhang, P.W. Zhang, Axial symmetry and classification of stationary solutions of Doi–Onsager equation on the sphere with Maier–Saupe potential, Comm. Math. Sci. 3 (2) (2005) 201–218]. We then examine global orientation dynamics in terms of eigenvalues of the orientation tensor via the Doi closure approximation. It is shown that for small intensity 0 < α < 4 , all states will evolve into the isotropic phase; for large intensity α > 4.5 , all states will evolve into the nematic prolate phase; and for the intermediate intensity 4 < α < 4.5 , an initial state will evolve into either the isotropic phase or the stable phase of two nematic prolate phases, depending on whether such an initial configuration crosses a critical threshold. Moreover, the uniaxial symmetry structure is shown to be preserved in time.

Characterization of stable kinetic equilibria of rigid, dipolar rod ensembles for coupled dipole–dipole and Maier–Saupe potentials

We study equilibria of the Smoluchowski equation for rigid, dipolar rod ensembles where the intermolecular potential couples the dipole–dipole interaction and the Maier–Saupe interaction. We thereby extend previous analytical results for the decoupled case of the dipolar potential only (Fatkullin and Slastikov 2005 Nonlinearity 18 2565–80; Ji et al Phys. Fluids at press; Wang et al 2005 Commun. Math. Sci. 3 605–20) or the Maier–Saupe potential only (Constantin et al 2004 Arch. Ration. Mech. Anal. 174 365–84; Constantin et al 2004 Discrete Contin. Dyn. Syst. 11 101–12; Constantin and Vukadinovic 2005 Nonlinearity 18 441–3; Constantin 2005 Commun. Math. Sci. 3 531–44; Fatkullin and Slastikov 2005 Commun. Math. Sci. 3 21–6; Liu et al 2005 Commun. Math. Sci. 3 201–18; Luo et al 2005 Nonlinearity 18 379–89; Zhou et al 2005 Nonlinearity 18 2815–25; Zhou and Wang Commun. Math. Sci. at press), and prove certain numerical observations for equilibria of coupled potentials (Ji et al Phys. Fluids at press). We first derive stability conditions, on the magnitude of the polarity vector (the first moment of the orientational probability distribution function) and on the direction of the polarity. We then prove that all stable equilibria of rigid, dipolar rod dispersions are either isotropic or prolate uniaxial. In particular, all stable anisotropic equilibrium distributions admit the following remarkable symmetry: the peak axis of orientation is aligned with both the polarity vector (first moment) and the distinguished director of the uniaxial second moment tensor. The stability is essential in establishing the axisymmetry. To demonstrate that the stability is indeed required, we show that there exist unstable non-axisymmetric equilibria.

Enhancement factor sign inversion triggered by the variation of the incident direction of light in the azo-dye doped liquid crystals

The optical reorientation process of the azo-dye doped liquid crystals (LCs) is studied and the dependence of the enhancement factor on the incident light direction is explained. By analysing the relation between the order parameter and the cis isomer concentration in the azo-dye doped LC system, an analytical expression that describes the dependence of the order parameters on the direction of the incident light is obtained. It is found that, since the order parameters of the guest–host LC system depend on the direction of the incident light, the intermolecular orientational interaction potentials are also related with the incident light direction. In order to describe the interaction of the cis isomer with the liquid crystalline molecules, a revised Maier–Saupe potential expression that allows for a higher-order interaction is used. A microscopic formula of the enhancement factor for the azo-dye doped LC system is derived on the basis of a simplified two-level model. From the microscopic formula, the mechanism behind the dependence of the enhancement factor on the incident direction of light is revealed. The comparison of our computational results with the existent experimental data verified our enhancement factor's microscopic form.