Orientational Order Parameter Research Articles

Theory of nematic-isotropic phase transitions in solutions of rodlike aggregates

ABSTRACT A mean-field theory is introduced to describe phase behaviours for solutions of rodlike aggregates. By combining the Onsager model for rodlike aggregates and the Flory-Huggins model for polydisperse polymer solutions, we derive the free energy for the solutions of the rodlike aggregates. We calculate the mean aggregation number and orientational order parameter of rodlike aggregates. The mean aggregation number jumps at the first-order nematic–isotropic transition (NIT). We also derive the Landau-de Gennes expansion of the free energy and calculate the isotropic-nematic coexistence curves on the temperature-concentration plane. The NIT curves are qualitatively consistent with the experimental results of rodlike aggregates.

13C NMR studies and orientational order of rod-like mesogens with strongly polar terminal cyano group

ABSTRACT 13C NMR studies of two rod-like mesogens with core units consisting of two and three phenyl rings with strongly polar terminal cyano groups are reported. The two-ring mesogen displayed an enantiotropic smectic A (SmA) mesophase while for the three-ring mesogen, enantiotropic nematic and SmA mesophases were observed. The solution 13C NMR studies of both the mesogens are carried out besides static 13C NMR measurements in the liquid crystalline phase. For the two-ring mesogen, the chemical shift anisotropy (CSA) tensors of the core unit are determined by the 2D-separation of undistorted powder patterns by effortless recoupling (SUPER) experiment. Furthermore, 2D-separated local field (SLF) measurements are realised for both the mesogens in the liquid crystalline phase to determine the 13C-1H dipolar couplings to find the orientational order parameters of the phenyl rings. The main order parameter (Szz) of the two-ring mesogen varies from 0.63 to 0.53 across the SmA phase and for the three-ring mesogen, the Szz values are found to be ~0.73 in the SmA phase and ~0.69 in the nematic phase. The CSA values determined from the SUPER experiment are expected to be a significant utility for molecules with polar terminal nitro/cyano groups as ferroelectric nematic phases are realised in them.

Shearmetry of Fluids with Tunable Rheology by Polarized Luminescence of Rare Earth-Doped Nanorods.

Shear stress plays a critical role in regulating physiological processes within microcirculatory systems. While particle imaging velocimetry is a standard technique for quantifying shear flow, uncertainty near boundaries and low resolution remain severe restrictions. Additionally, shear stress determination is particularly challenging in biofluids due to their significant non-Newtonian behaviors. The present study develops a shearmetry technique in physiological settings using a biomimetic fluid containing rare earth-doped luminescent nanorods acting in two roles. First, they are used as colloidal additives adjusting rheological properties in physiological media. Their anisotropic morphology and interparticle interaction synergistically induce a non-Newtonian shear-thinning effect emulating real biofluids. Second, they can probe shear stress due to the shear-induced alignment. The polarized luminescence of the nanorods allows for quantifying their orientational order parameter and thus correlated shear stress. Using scanning confocal microscopy, we demonstrate the tomographic mapping of the shear stress distribution in microfluidics. High shear stress is evident near the constriction and the cellular periphery, in which non-Newtonian effects can have a significant impact. This emerging shearmetry technique is promising for implementation in physiological and rheological environments of biofluids.

Uniaxial Molecular Alignment in Thin Films Composed of Discrete NDI‐T2 Oligomers Investigated by Variable‐Angle Spectroscopic and Imaging Ellipsometry

The understanding of the optical characteristics and the influence of thermally induced crystallization is vital to evaluate newly synthesized organic semiconductors for their potential applications. Variable‐angle spectroscopic ellipsometry (VASE) is employed to determine the complex dielectric function of two recently developed discrete oligomers based on naphthalene diimide (NDI) and bithiophene (T2), applying an anisotropic optical model. It is shown that the as‐cast films are optically near‐isotropic, while the annealed films exhibit pronounced anisotropies with respect to the orientation of the π−π* transition. Differences between the two distinct molecules are observed and discussed. The degree of optical anisotropy is quantified using the average molecular orientation angle φ, associated with the orientation of the π−π* transition dipole, and the related orientation order parameter S. The molecules are shown to form crystallites of several microns in size upon annealing. Imaging ellipsometry is thus additionally employed to study their optical anisotropy on a microscopic scale, and the agreement with results from macroscopic measurements is evaluated.

Supramolecular Arrangement and Conformational and Dynamic Properties of Chiral Smectic Liquid Crystals Obtained through Nuclear Magnetic Resonance: A Brief Review

Ferroelectric and antiferroelectric smectic liquid crystalline (LC) phases are still at the center of investigations and interests for both their fundamental properties and variety of technological applications. This review aims to report the main contributions based on different nuclear magnetic resonance (NMR) techniques to the study of chiral liquid crystalline calamitic mesogens forming smectic phases, such as the SmA, the SmC* (ferroelectric), and the SmC*A (antiferroelectric) phases. 2H NMR and 13C NMR techniques and their combination were of help in clarifying the local orientational properties (i.e., the molecular and fragments’ main orientational order parameters) at the transition between the SmA and the SmC* phases, and in the particular case of de Vries liquid crystals, NMR studies gave important clues regarding the actual models describing the molecular arrangement in these two phases formed by de Vries LCs. Moreover, this review describes how the combination of 2H NMR relaxation times’ analysis, 1H NMR relaxometry, and 1H NMR diffusometry was successfully applied to the study of chiral smectogens forming the SmC* and SmC*A phases, with the determination of relevant parameters describing both rotational molecular and internal motions, collective dynamics, and translational self-diffusion motions. Several cases will be reported concerning NMR investigations of chiral ferroelectric and antiferroelectric phases, underlining the great potential of combined NMR approaches to the study of supramolecular, conformational, and dynamic properties of liquid crystals.

Influence of chalcopyrite nanoplatelets on nematic phases of bend-shaped dimeric molecules: Phase diagram, birefringence, and reorientation transition

We investigated the influence of chalcopyrite (CuFeS2) nanoplatelets on the ordering of uniform and twist-bend nematic phases of achiral bend-shaped dimers, employing polarized optical microscopy, optical and electro-optical measurements. Specifically, aliphatic amine surface-functionalized hydrophobic chalcopyrite nanoplatelets were synthesized and characterized by XRD, TEM, and IR-vis-UV spectroscopy. Nanocomposites of the liquid crystal compound 1”,9”-bis(4-cyanobiphenyl-4'-yl)nonane (CB9CB) with the nanoplatelets were prepared. The study encompasses an assessment of the thermodynamic stability of the nanocomposites, constructing the corresponding phase diagram as a function of temperature and nanoplatelet mass fraction. Large phase temperature shifts were observed for minute mass fractions of NPs. Birefringence measurements were performed to investigate the orientational order parameter and the conical tilt angle's dependence on the mass fraction of the nanoplatelets and the temperature. Additionally, we measured a Fréedericksz-like reorientation transition voltage threshold and switching times as functions of the mass fraction and temperature. A huge increase of the voltage threshold was measured in the twist-bend nematic phase. The dependence of the viscoelastic coefficient ζ on NPls' mass fraction was investigated. The splay elastic constant of the helicoidal structure was estimated, using a coarse grain approximation, to be two orders of magnitude higher than in the uniform nematic phase. Finally, in the twist-bend nematic phase, the impact of temperature and nanoplatelets mass fraction on the hysteresis loop of the reorientation transition was investigated.

Random close packing of semi-flexible polymers in two dimensions: Emergence of local and global order.

Through extensive Monte Carlo simulations, we systematically study the effect of chain stiffness on the packing ability of linear polymers composed of hard spheres in extremely confined monolayers, corresponding effectively to 2D films. First, we explore the limit of random close packing as a function of the equilibrium bending angle and then quantify the local and global order by the degree of crystallinity and the nematic or tetratic orientational order parameter, respectively. A multi-scale wealth of structural behavior is observed, which is inherently absent in the case of athermal individual monomers and is surprisingly richer than its 3D counterpart under bulk conditions. As a general trend, an isotropic to nematic transition is observed at sufficiently high surface coverages, which is followed by the establishment of the tetratic state, which in turn marks the onset of the random close packing. For chains with right-angle bonds, the incompatibility of the imposed bending angle with the neighbor geometry of the triangular crystal leads to a singular intra- and inter-polymer tiling pattern made of squares and triangles with optimal local filling at high surface concentrations. The present study could serve as a first step toward the design of hard colloidal polymers with a tunable structural behavior for 2D applications.

Large-Area Layered Membranes with Precisely Controlled Nano-Confined Channels.

Two-dimensional (2D) nanosheets-based membranes, which have controlled 2D nano-confined channels, are highly desirable for molecular/ionic sieving and confined reactions. However, it is still difficult to develop an efficient method to prepare large-area membranes with high stability, high orientation, and accurately adjustable interlayer spacing. Here, we present a strategy to produce metal ion cross-linked membranes with precisely controlled 2D nano-confined channels and high stability in different solutions using superspreading shear-flow-induced assembly strategy. For example, membranes based on graphene oxide (GO) exhibit interlayer spacing ranging from 8.0±0.1 Å to 10.3±0.2 Å, with a precision of down to 1 Å. At the same time, the value of the orientation order parameter (f) of GO membranes is up to 0.95 and GO membranes exhibit superb stability in different solutions. The strategy we present, which can be generalized to the preparation of 2D nano-confined channels based on a variety of 2D materials, will expand the application scope and provide better performances of membranes.

Large‐Area Layered Membranes with Precisely Controlled Nano‐Confined Channels

Two‐dimensional (2D) nanosheets‐based membranes, which have controlled 2D nano‐confined channels, are highly desirable for molecular/ionic sieving and confined reactions. However, it is still difficult to develop an efficient method to prepare large‐area membranes with high stability, high orientation, and accurately adjustable interlayer spacing. Here, we present a strategy to produce metal ion cross‐linked membranes with precisely controlled 2D nano‐confined channels and high stability in different solutions using superspreading shear‐flow‐induced assembly strategy. For example, membranes based on graphene oxide (GO) exhibit interlayer spacing ranging from 8.0 ± 0.1 Å to 10.3 ± 0.2 Å, with a precision of down to 1 Å. At the same time, the value of the orientation order parameter (f) of GO membranes is up to 0.95 and GO membranes exhibit superb stability in different solutions. The strategy we present, which can be generalized to the preparation of 2D nano‐confined channels based on a variety of 2D materials, will expand the application scope and provide better performances of membranes.

Linear dependence of grain boundary energy on structural orderliness in FCC metals

Establishing the correlation between grain boundary (GB) structure and energy is crucial for advancing the fundamental understanding of polycrystalline materials, yet this correlation remains elusive. In this study, systematic atomic simulations are performed to comprehensively explore numerous GBs in different FCC metals and alloys. The findings unveil a universal linear relationship between the GB energy and GB structural orderliness parameter (GB excess centrosymmetry or excess bond orientational order parameters). Remarkably, the identified linear relationship proves universally applicable across different metal systems, GB types, and length scales, rooted in the linear correlation between atomic potential energy and atomic structural orderliness. Furthermore, the ratio between GB energy and orderliness parameter emerges as a potent indicator for gauging the extent of GB relaxation, thereby providing valuable insights to inform the design of nanocrystalline materials with enhanced GB stability.

Molding Process Retaining Gold Nanoparticle Assembly Structures during Transfer to a Polycarbonate Surface.

The immobilization of gold nanoparticle (AuNP) linear surface assemblies on polycarbonate (PC) melt surface via molding is investigated. The order of the particle assemblies is preserved during the molding process. The assemblies on PC exhibit plasmonic coupling features and dichroic properties. The structure of the assemblies is quantified based on Scanning Electron Microscopy (SEM) and image analysis data using an orientational order parameter. The transfer process from mold to melt shows high structural fidelity. The order parameter of around 0.98 reflects the orientation of the lines and remains unaffected, independent of the injection direction of the melt relative to the particle lines. This is discussed in the frame of fountain flow during injection molding. The particles were permanently fixed and withstood the injection molding process, detachment of the substrate, and extraction in boiling ethanol. The plasmonic particles coupled strongly within the dense nanoparticle lines to produce anisotropic optical properties, as quantified by dichroic ratios of 0.28 and 0.52 using ultraviolet-visible-near-infrared (UV-Vis-NIR) spectroscopy. AuNP line assemblies on a polymer surface may be a basis for plasmonic devices like surface-enhanced Raman scattering (SERS) sensors or a precursor for nanowires. Their embedding via injection molding constitutes an important link between particle-self-assembly approaches for optically functional surfaces and polymer processing techniques.

Geometric phase analysis of magnetic skyrmion lattices in Lorentz transmission electron microscopy images

Magnetic skyrmions are quasi-particles with a swirling spin texture that form two-dimensional lattices. Skyrmion lattices can exhibit defects in response to geometric constraints, variations of temperature or applied magnetic fields. Measuring deformations in skyrmion lattices is important to understand the interplay between the lattice structure and external influences. Geometric phase analysis (GPA) is a Fourier-based image processing method that is used to measure deformation fields in high resolution transmission electron microscopy (TEM) images of crystalline materials. Here, we show that GPA can be applied quantitatively to Lorentz TEM images of two-dimensional skyrmion lattices obtained from a chiral magnet of FeGe. First, GPA is used to map deformation fields around a 5–7 dislocation and the results are compared with the linear theory of elasticity. Second, rotation angles between skyrmion crystal grains are measured and compared with angles calculated from the density of dislocations. Third, an orientational order parameter and the corresponding correlation function are calculated to describe the evolution of the disorder as a function of applied magnetic field. The influence of sources of artifacts such as geometric distortions and large defoci are also discussed.

Cholesteric liquid crystal phase variations within nanotubes: an in-depth analysis of the influences of twist radius and molecular pitch

ABSTRACT This study investigates the self-assembly and phase transitions of cholesteric liquid crystals (CLCs) when confined in nanotubes, specifically focusing on the variations in their molecular structures. A dissipative particle dynamics method was used to investigate how the changes in helix radius ( r LC ) and pitch ( P ) affect the behaviour of the LCs. For pitches of 24, 12, and 6, the orientation order parameter ( P 2 ) showed distinct trends during cooling. Smaller helix radii resulted in isotropic to nematic to smectic phase transitions, whereas larger radii resulted in a novel smectic phase with spiral structures (Sm s ). Circumferential local-order parameters ( P 2 r and P 2 z ) were introduced to better characterise the phases, and a disordered spiral-structure phase (Sm s 2 ) with local perturbations was observed. The transitions between isotropic, nematic, and smectic phases, highlighting the influence of the pitch and helix radius, were observed from the phase diagrams. Smaller pitches and radii prevented the formation of stable structures; however, specific combinations induced characteristic spiral structures in the smectic phase. These results provide new insights into the self-assembly of confined CLCs, which have potential applications in the design of photonic devices and LC-based sensors. This study advances our understanding of phase control in quasi-one-dimensional systems and highlights the importance of molecular variations in liquid crystal behaviour.

Emergent gauge symmetry in active Brownian matter.

We investigate a two-dimensional system of interacting active Brownian particles. Using the Martin-Siggia-Rose-Janssen-de Dominicis formalism, we built up the generating functional for correlation functions. We study in detail the hydrodynamic regime with a constant density stationary state. Our findings reveal that, within a small density fluctuations regime, an emergent U(1) gauge symmetry arises, originated from the conservation of fluid vorticity. Consequently, the interaction between the orientational order parameter and density fluctuations can be cast into a gauge theory, where the concept of "electric charge density" aligns with the local vorticity of the original fluid. We study in detail the case of a microscopic local two-body interaction. We show that, upon integrating out the gauge fields, the stationary states of the rotational degrees of freedom satisfy a nonlocal Frank free energy for a nematic fluid. We give explicit expressions for the splay and bend elastic constants as a function of the Péclet number (Pe) and the diffusion interaction constant (k_{d}).

A confined rod: mean field theory for hard rod-like particles

ABSTRACT In this paper, we model the configurations of a system of hard rods by viewing each rod in a cell formed by its neighbours. By minimising the free energy in the model and performing molecular dynamics, where, in both cases, the shape of the cell is a free parameter, we obtain the equilibrium orientational order parameter, free energy and pressure of the system. Our model enables the calculation of anisotropic stresses exerted on the walls of the cell due to shape change of the rod in photoisomerisation. These results are a key step towards understanding molecular shape change effects in photomechanical systems under illumination.

Neural network molecular dynamics study of LiGe2(PO4)3: Investigation of structure

Machine learning interatomic potential for LiGe2(PO4)3 in a form of a set of neural networks (DeePMD-model) was trained on DFT data. DFT simulations in GGA PBE approximation were performed for ordered and disordered LiGe2(PO4)3 supercell with 432 atoms which allowed the developed potential to describe crystalline, molten and glassy states. The developed DeePMD-potential was verified using literature and our experimental data demonstrating good agreement with it. Neural network molecular dynamic simulations indicate that P atoms have tetrahedral oxygen environment in both crystalline and glassy state. Coordination environment of Ge atoms is more complex. They have octahedral oxygen environment in crystalline state and mixed environment in glassy state: 4-, 5- and 6-coordinated Ge atoms were found. It was shown by calculation of orientational order parameters that in glassy LiGe2(PO4)3 4-coordinated Ge atoms have tetrahedral oxygen environment, while 6-coordinated Ge atoms have octahedral oxygen environment. Analysis of O-Ge-O angle distribution for 5-coordinated Ge atoms in glassy LiGe2(PO4)3 demonstrates that the environment of Ge atoms in this case is represented by various structures: square pyramids, trigonal bipyramids and transitional structures from 5-coordinated Ge atoms to tetrahedrons.

Optimization of the Domain Size in Stressed Liquid Crystals

The distributions of orientational and translational order parameters in liquid crystal polymer composites are studied using the methods of molecular-statistical physics. A model is considered in which the composite consists of liquid crystal (LC) domains separated by flat polymer sheets. The perturbation of the orientational and translational order parameters, which is caused by an arbitrary flat polymer sheet in a planar LC layer, is estimated. The effective distance over which this disturbance propagates is estimated. It is shown that it is proportional to the square root of the splay elastic constant of LC. It has been shown that in certain temperature ranges the polymer can induce a more ordered liquid crystal phase near itself (smectic phase inside of nematic phase and nematic phase – inside of isotropic liquid phase). In this case, the orientational order parameter of LC near the polymer can be significantly higher than far from it, and the “perturbation” of the order parameters can propagate over a distance of up to several micrometers with a sharp peak at a distance of the order of one micrometer. Based on the original theory, the methods have been proposed for optimizing the sizes of domains in stressed liquid crystals obtained by mechanical treatment in liquid crystal polymer composites.

Photosynthetically-powered phototactic active nematic liquid crystal fluids and gels

One of the most ancient forms of life dating to ~3.5 billion years ago, cyanobacteria are highly abundant organisms that convert light into energy and motion, often within conjoined filaments and larger colonies that attract a great deal of interest but their active nematic behavior remained unexplored. Here we demonstrate how light causes a spontaneous self-assembly of two- and three-dimensional active nematic states of cyanobacterial filaments, with a plethora of topological defects. We quantify light-controlled evolutions of orientational and velocity order parameters during the transition between disordered and orientationally ordered states of photosynthetic active matter, as well as the subsequent active nematic’s fluid-gel transformation. Patterned illumination and foreign inclusions with different shapes interact with cyanobacterial active nematics in nontrivial ways while inducing interfacial boundary conditions and fractional boojum defects. Our phototactic model system promises opportunities to systematically explore fundamental properties and technological utility of the liquid crystalline active matter.

Manifestation of the hexatic phase in confined two-dimensional systems with circular symmetry

Quasi-two-dimensional systems play an important role in the manufacture of various devices for the needs of nanoelectronics. Obviously, the functional efficiency of such systems depends on their structure, which can change during phase transitions under the influence of external conditions (for example, temperature). Until now, the main attention has been focused on the search for signals of phase transitions in continuous two-dimensional systems. One of the central issues is the analysis of the conditions for the nucleation of the hexatic phase in such systems, which is accompanied by the appearance of defects in the Wigner crystalline phase at a certain temperature. However, both practical and fundamental questions arise about the critical number of electrons at which the symmetry of the crystal lattice in the system under consideration will begin to break and, consequently, the nucleation of defects will start. The dependences of the orientational order parameter and the correlation function, which characterize topological phase transitions, as functions of the number of particles at zero temperature have been studied. The calculation results allows us to establish the precursors of the phase transition from the hexagonal phase to the hexatic one for N = 92, 136, 187, considered as an example.

Theoretical models for the nematic twist-bend phase and the Landau de Gennes theory of Longa and Tomczyk

ABSTRACT We present a critical analysis of the existing phenomenological theories of the nematic twist-bend phase including the models proposed by Dozov and Selinger and the more general Landau de Gennes theory of L. Longa and W. Tomczyk. The role of the flexoelectric polarisation is considered in detail as well as the stabilisation of the nanoscale helical structure of the twist-bend phase. In particular we focus into the mechanism of the reduction of the bend elastic constant determined by negative flexoelectric corrections and discuss why the splay elastic constant is not reduced at all. A molecular theory of elasticity of bent-core nematics, which also enables one to describe the reduction of the bend elastic constant, is considered in detail as well. Finally we consider two different tensor orientational order parameters of the twist-bend phase which describe all kinds of uniaxial and local biaxial ordering of bent-core molecules in the twist-bend phase.