A Bidimensional Gay-Berne Calamitic Fluid: Structure and Phase Behavior in Bulk and Strongly Confined Systems

Calderón-Alcaraz, A.; Munguía-Valadez, J.; Hernández, S. I.; Ramírez-Hernández, A.; Sambriski, E. J.; Moreno-Razo, J. A.

doi:10.3389/fphy.2020.622872

ORIGINAL RESEARCH article

Front. Phys., 02 March 2021

Sec. Interdisciplinary Physics

Volume 8 - 2020 | https://doi.org/10.3389/fphy.2020.622872

This article is part of the Research Topic 50 years of Statistical Physics in Mexico: Development, State of the Art and Perspectives View all 17 articles

A Bidimensional Gay-Berne Calamitic Fluid: Structure and Phase Behavior in Bulk and Strongly Confined Systems

A. Calderón-Alcaraz¹

J. Munguía-Valadez¹

S. I. Hernández²

A. Ramírez-Hernández³

E. J. Sambriski⁴

J. A. Moreno-Razo¹*

¹Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, Mexico City, Mexico
²Unidad Multidisciplinaria de Docencia e Investigación-Juriquilla, Facultad de Ciencias, Universidad Nacional Autónoma de México, Juriquilla, Querétaro, Mexico
³Department of Biomedical Engineering and Chemical Engineering, and Department of Physics and Astronomy, The University of Texas at San Antonio, San Antonio, TX, United States
⁴Department of Chemistry, Delaware Valley University, Doylestown, PA, United States

A bidimensional (2D) thermotropic liquid crystal (LC) is investigated with Molecular Dynamics (MD) simulations. The Gay-Berne mesogen with parameterization GB(3, 5, 2, 1) is used to model a calamitic system. Spatial orientation of the LC samples is probed with the nematic order parameter: a sharp isotropic-smectic (I-Sm) transition is observed at lower pressures. At higher pressures, the I-Sm transition involves an intermediate nematic phase. Topology of the orthobaric phase diagram for the 2D case differs from the 3D case in two important respects: 1) the nematic region appears at lower temperatures and slightly lower densities, and 2) the critical point occurs at lower temperature and slightly higher density. The 2D calamitic model is used to probe the structural behavior of LC samples under strong confinement when either planar or homeotropic anchoring prevails. Samples subjected to circular, square, and triangular boundaries are gradually cooled to study how orientational order emerges. Depending on anchoring mode and confining geometry, characteristic topological defects emerge. Textures in these systems are similar to those observed in experiments and simulations of lyotropic LCs.

1 Introduction

Bulk materials exhibit properties imbued by their underlying chemical makeup: the packing of and interactions between atoms and/or molecules impart characteristic traits. On the other hand, metamaterials are synthetically produced and depend more on the relative positioning of building blocks within the structure. Such a trait allows metamaterials to achieve novel properties not exhibited by bulk materials (prominently of an electromagnetic and/or an optical nature). This has facilitated the expansion and miniaturization of existent technologies [1, 2].

Building blocks capable of molecular recognition are essential in the bottom-up production of metamaterials [3–7]. Specifically, complementary moieties can display the ability to “latch” in solution, onto a substrate, or in a combination of scenarios to produce desirable architectures. The threshold concentration of building blocks and formation steps are some issues to consider when optimizing their fabrication [8]. Structural properties in a metamaterial will remain stable provided the interaction strength between units withstands thermal fluctuations in the medium. Bottom-up approaches exploit this feature to circumvent the use of mechanical intervention. Production is scaled by merely increasing the amounts of reactants and relying on system kinetics for product formation. Packing specificity can be modulated by carrying out the assembly under spatial confinement [9–15]. Many fabrication protocols have been optimized by mimicking the strategies latent in biomolecular systems [4, 16, 17].

The solvent is a key component in the production of metamaterials, which must effectively disperse building blocks and stabilize noncovalent interactions holding structures together. Self-assembly reliant on the chemical complementarity embedded in building blocks relegates the solvent to a passive role, serving in large measure as a dispersing agent. A paradigm shift is to screen solvents for an active role in the generation of metamaterials [18–22]. In this scenario, the solvent provides additional bottom-up control that extends the gamut of attainable targets [23–26].

The solvent “paradigm shift” is exemplified in the elastic forces mediated within a liquid crystal (LC) fluid. The intrinsic anisotropy of LCs facilitates spatially ordered mesophases. Solvent order at certain state points can be disturbed in the presence of colloidal inclusions, resulting in topological defects that exert static and dynamic control. Because of their ability to spatially “communicate,” topological defects can couple (obeying topological charge rules) to yield specific colloidal arrangements, including dimers [27–38], wires (i.e., chains) [21, 28, 35, 39–44], and arrays [21, 31–34, 36–38, 43, 45, 46]. Solvents recruited as active agents contributing to the self-assembly of metamaterials enhance a variety of structural possibilities.

Much research has been devoted to three-dimensional (3D) self-assembly [25], though a two-dimensional (2D) variant continues to be of interest from an exploratory perspective [47–55] as well as in applied technologies [1, 56]. Optimal function is achieved via slab geometry in many devices, including optoelectronic/photonic materials [57–61], sensors [60, 62–68], display technologies [69–71], smart glass [72, 73], spatial light modulators [74–77], and tunable filters [78–81]. However, dimensionality plays an essential role in the type and extent of structural order that a condensed phase can maintain [52, 82–88]. When coupling the elastic forces of topological defects in LC media, colloidal ordering induced via a substrate can differ significantly from that observed via topological mediation in the bulk [21, 24, 25, 35, 36]. Slab assembly becomes relevant in 3D colloidal arrangements because it yields intermediates: metamaterials are finalized upon “stacking” slabs in layer-by-layer synthesis to achieve a target 3D structure [89–91].

In this work, we focus on two aspects of a thermotropic calamitic LC fluid relevant to colloidal self-assembly: 1) the changes in topology of the solvent phase diagram due to a reduction in dimensionality from 3D to 2D, and 2) the mesophase behavior of the solvent under strong confinement in slab geometry. The Gay-Berne (GB) model [92] is used here because it captures salient mesogenic features and has a relatively low computational overhead. Prior work with the GB mesogen has focused on different mesophases in bulk 3D systems [93–98]. Additionally, surface-induced ordering (i.e., anchoring) via boundary walls has been studied in thin films [99], droplets [100], and toroidal cavities [101]. Several GB parameterizations have reproduced nematic and smectic phases [102–104]. More recently, a discotic parameterization has been used to explore nematic and columnar phases [105–109], providing insight on structural and dynamic measurements at the molecular level [110–112]. The recognition of specific design principles has stimulated the attainment of novel targets [27].

Despite serving as a point of reference for 3D phenomena, the phase behavior of a strictly 2D thermotropic GB LC system has been limited [113, 114]. On the experimental front, optical microscopy commonly provides information on quasi-2D samples, and in most cases, data merely reflect 2D projections of an underlying 3D system [56, 115]. Renewed interest in the organization of rigid biopolymers as effective 2D systems (in bulk and under confinement) has led to new and interesting textures observed under strong confinement [116–122]. Simple simulation models reproduced the phenomenology observed in 2D [123–126]. Because those efforts focused on lyotropic liquid crystals, we extend the field by considering a thermotropic fluid. Specifically, we explore how shape of the confining area and type of anchoring induced by boundary walls affect mesophase behavior.

2 Model and Methods

The GB model is a generalization of the Lennard-Jones potential defining the interaction between anisotropic molecules. Each molecule i is represented as an ellipsoid having a center-of-mass position $r_{i}$ and a unit vector ${\hat{e}}_{i}$ along the principal (major) axis denoting its orientation (Figure 1).

FIGURE 1

FIGURE 1. Schematic defining the degrees of freedom and parameters associated with the $GB (3,5,2,1)$ mesogens. Molecular axes are defined by unit vectors ${\hat{e}}_{i}$ and ${\hat{e}}_{j}$ . The center-to-center unit vector ${\hat{r}}_{i j}$ tempers the interaction potential relative to mesogen-mesogen orientation, affecting both length [Eq. 2] and energy [Eq. 5] scales. The interaction potential is scaled by a model length $σ_{0}$ defined with respect to the side-side length $σ_{ss}$ . The end-end length $σ_{ee}$ is used to define the molecular aspect ratio; for the parameterization used in this work, $σ_{ee} / σ_{ss} = 3$ .

The intermolecular interaction between the ith and jth mesogens is written as

U_{G B} (r_{i j}, {\hat{e}}_{i}, {\hat{e}}_{j}) = 4 ε ({\hat{r}}_{i j}, {\hat{e}}_{i}, {\hat{e}}_{j}) [Ξ_{i j}^{- 12} - Ξ_{i j}^{- 6}], (1)

where $r_{i j} = r_{i} - r_{j}$ and the scaled distance is given by

Ξ_{i j} = \frac{1}{σ_{0}} [r_{i j} - σ ({\hat{r}}_{i j}, {\hat{e}}_{i}, {\hat{e}}_{j}) + σ_{0}], (2)

where $σ_{0}$ the width of the mesogen (i.e., the minor axis) and $r_{i j} = | r_{i j} |$ is the magnitude of the intermolecular (center-to-center) separation vector. The relative orientation of mesogens within the medium must be taken into account to regulate the strength of the intermesogen interaction. This requires a fully specified function of a general variable ω,

Γ (ω) = 1 - ω [\frac{c_{i}^{2} + c_{j}^{2} - 2 ω c_{i} c_{j} c_{i j}}{1 - ω^{2} c_{i j}^{2}}], (3)

where $c_{i} = {\hat{e}}_{i} \cdot {\hat{r}}_{i j}$ , $c_{j} = {\hat{e}}_{j} \cdot {\hat{r}}_{i j}$ , $c_{i j} = {\hat{e}}_{i} \cdot {\hat{e}}_{j}$ , and ${\hat{r}}_{i j} = r_{i j} / | r_{i j} |$ is the unit (center-to-center) separation vector. The orientation-dependent length scale (range) parameter $σ ({\hat{r}}_{i j}, {\hat{e}}_{i}, {\hat{e}}_{j})$ is computed as

σ ({\hat{r}}_{i j}, {\hat{e}}_{i}, {\hat{e}}_{j}) = σ_{0} {[Γ (χ)]}^{- 1 / 2} (4)

Here, $χ = (κ^{2} - 1) / (κ^{2} + 1)$ , where $κ = σ_{e e} / σ_{0} = σ_{e e} / σ_{s s}$ is the length-to-width (aspect) ratio of the mesogen (Figure 1). The strength anisotropy function $ε ({\hat{r}}_{i j}, {\hat{e}}_{i}, {\hat{e}}_{j})$ is defined by the product

ε ({\hat{r}}_{i j}, {\hat{e}}_{i}, {\hat{e}}_{j}) = ε_{0} {[ε_{1} ({\hat{e}}_{i}, {\hat{e}}_{j})]}^{ν} {[ε_{2} ({\hat{r}}_{i j}, {\hat{e}}_{i}, {\hat{e}}_{j})]}^{μ}, (5)

where the exponents ν and μ are adjustable parameters. The energy anisotropy functions are defined as

ε_{1} ({\hat{e}}_{i}, {\hat{e}}_{j}) = {[1 - χ^{2} {({\hat{e}}_{i} \cdot {\hat{e}}_{j})}^{2}]}^{- 1 / 2} (6)

and

ε_{2} ({\hat{r}}_{i j}, {\hat{e}}_{i}, {\hat{e}}_{j}) = Γ (χ^{'}) . (7)

The parameter $χ^{'}$ depends on the ratio of the potential well depths corresponding to side-side (ss) and end-end (ee) configurations, $κ^{'} = ε_{s s} / ε_{e e}$ . More specifically, $χ^{'} = [{(κ^{'})}^{1 / μ} - 1] / [{(κ^{'})}^{1 / μ} + 1]$ .

The GB model uses four parameters conventionally represented as $G B (κ, κ^{'}, μ, ν)$ . Previous work has shown that specific parameter sets reproduce thermodynamic and structural properties of experimental systems [105, 107, 127]. A complete phase diagram of the 3D GB model is available for $G B (3,5,2,1)$ [104, 128–132], which corresponds to a calamitic mesogen. Moreover, this parameterization has been used to investigate intermolecular interactions in nematic samples [133–135]. Simulations have been previously reported for GB discotic mesogens focused on tracing changes in phase behavior under confinement [99, 130, 136–140] and in droplets [141]. The reader interested in additional parameterizations is referred to previous work [127, 129, 135, 142–146].

The $G B (3,5,2,1)$ parameterization is used in this work to elucidate the role that dimensionality plays on mesophase behavior. Shown in Figure 2 are representative interaction energy curves as a function of intermesogen distance for different relative orientations. For the $G B (3,5,2,1)$ mesogen, the side-side arrangement is preferred over other configurations, a feature that promotes the nematic phase at reasonably accessible temperatures. The $G B (3,5,2,1)$ parameterization was chosen because it has been extensively used as a model for prolate LCs, such as the alkylbiphenyl mesogen family.

FIGURE 2

FIGURE 2. Intermlecular potential for the $GB (3,5,2,1)$ mesogen as a function of intermesogen distance. Representative mesogen arrangements are highlighted to emphasize how $U_{GB} (r)$ is tempered according to relative mesogen-mesogen orientation.

Confined systems were modeled with walls constructed from an array of spherical (i.e., circular in 2D) particles. The mesogen-wall interaction is obtained by taking the limit of Eqs. 1 and 2 when one of the interacting mesogens becomes a sphere (i.e., a wall particle) [147–149]. In that limit, the range parameter and strength anisotropy functions are given by

σ_{w} ({\hat{r}}_{i j}, {\hat{e}}_{j}) = σ_{0} {[1 - χ {({\hat{r}}_{i j} \cdot {\hat{e}}_{j})}^{2}]}^{- 1 / 2} (8)

and

ε_{w} ({\hat{r}}_{i j}, {\hat{e}}_{j}) = ε_{0} [1 - χ'_{w} {({\hat{r}}_{i j} \cdot {\hat{e}}_{j})}^{2}] (9)

where

χ'_{w} = 1 - {(ε_{h} / ε_{p})}^{1 / μ} . (10)

For Eqs. 8–10, the ith molecule denotes a wall-type particle, the jth molecule refers to a mesogen, $ε_{h}$ corresponds to the energy scale for homeotropic anchoring (when ${\hat{e}}_{j}$ is locally perpendicular to the confining wall), and $ε_{p}$ represents the energy scale for planar anchoring (when ${\hat{e}}_{j}$ is locally parallel to the confining wall). Anchoring conditions can thus be controlled by adjusting these two parameters.

We focus on a strictly 2D thermotropic liquid crystal in this work: “flat” ellipsoidal mesogens evolve in a plane. All results reported herein were generated by performing MD simulations in the canonical (NAT, where A is constant area for 2D, analogous to NVT where V is constant volume in 3D) and isothermal-isobaric (NPT) ensembles. Translational and rotational equations of motion were integrated using the velocity-Verlet algorithm [150]. For bulk samples, in either the NAT or NPT ensembles, the time step used was $δ t = 0.001$ . For systems in confined regions, the time step used was $δ t = 0.002$ . The coupling parameters for simulations were as follows: $Q_{t h e r m o s t a t} = 10$ , $Q_{b a r o s t a t} = 1,000$ . In the case of bulk samples, the simulation cell was defined with lateral dimensions $L_{x}$ and $L_{y}$ : periodic boundary conditions were applied in all directions. All simulations were initialized at relatively high temperature (i.e., $T^{*} = k_{B} T / ε_{0} = 1.0$ ). Low-temperature states $(T^{*} = k_{B} T / ε_{0} = 0.1)$ were attained by cooling the system gradually. Velocities were assigned from a Maxwell-Boltzmann distribution and the moments of inertia were set to $I = (σ_{0}^{2} / 20) (κ^{2} + 1)$ [151]. All particles were set to unit mass $(m = 1)$ and intermolecular potentials were truncated at a cutoff length scale $r_{c} = (κ + 1) σ_{0}$ for expediency. Interparticle potentials were shifted to enforce a smoothly vanishing force at $r_{c}$ . Simulations were run for at least $5 \times 10^{6}$ time steps for equilibration and another $5 \times 10^{6}$ time steps for production.

Global orientational order is characterized by the orientational traceless tensor $Q$ [152], specialized for the 2D case [153] as

Q = \frac{1}{N} \sum_{i = 1}^{N} (2 {\hat{e}}_{i} \otimes {\hat{e}}_{i} - I), (11)

where $\otimes$ denotes the tensor product and I is the identity matrix. Diagonalization of Q leads to two eigenvalues $(λ_{+} and λ_{-})$ . The nematic (Maier-Saupe) order parameter S is defined in terms of the highest eigenvalue $λ_{+}$ , so that $S = λ_{+}$ . The parameter S is equal to zero for isotropic configurations and increases as orientational order increases.

3 Results and Discussion

3.1 Orientational Order and Liquid Structure

In this section, we present data for a series of samples of increasing size (mesogen number) to elucidate the orientational order of the LC liquid as a function of temperature. We focus here on characterizing differences due specifically to sample size, considering $N \in {500,1000,4000}$ . Profiles for the Maier-Saupe order parameter are presented in Figure 3.

FIGURE 3

FIGURE 3. Order parameter S as a function of temperature $T^{∗}$ at different pressures $P^{∗}$ $\in$ [5.0 (black), 3.0 (red), 1.0 (orange), 0.5 (blue), 0.1 (green)] in the bulk 2D system. The response in the order parameter S is compared among different ensemble sizes: N $\in$ [500 (triangles), 1,000 (squares), 4,000 (diamonds)]. Finite-size effects become pronounced as pressure increases (and correspondingly at higher temperature). For $P^{∗} = 5.0$ , results are also included for $N \in$ [10,000 (dashed line), 50,000 (circles with error bars)]. Finite-size effects appear to dissipate when N is of $ (10^{4})$ .

The nematic order parameter S displays a state with low orientational order $(S \approx 0)$ when the temperature exceeds a threshold depending on the pressure $P^{∗}$ of the system. The onset of orientational order shows a jump in S, such that $S \approx 1$ when the temperature is sufficiently low: this high value of S indicates the formation of the smectic phase. An isotropic-smectic transition (I-Sm) takes place without an intermediate nematic state when the pressure is sufficiently low (i.e., for $P^{*} < 2$ ). As the pressure increases, the I-Sm transition occurs by passing through a range of nematic state points, corresponding approximately to $0.25 < S < 0.75$ . A finite-size effect in S becomes pronounced at higher pressure (i.e., $P^{*} ≳ 2$ ): the transition appears less sharp as the ensemble size N decreases.

To investigate how finite-size effects are pronounced at higher pressures, simulations were performed for $P^{*} = 5.0$ . At this pressure, finite-size effects are accentuated. We performed simulations for $N \in {10000,50000}$ . As can be gleaned from Figure 3, results for the larger two systems are close to one another. Hence, finite-size effects seemingly dissipate when the ensemble size is at least of $ (10^{4})$ .

To gain insight into the local structure, we analyzed MD snapshots for $N = 50,000$ when $P^{*} = 5.0$ via a temperature sweep shown in Figure 4. An eightfold magnification of a portion of the ensemble is shown to aid discerning local mesophase order: the entire ensemble for each temperature is provided in the Supplementary Material. At high temperatures, translational entropy overwhelms the cohesive energy of the LC medium and a disordered phase is the most stable state accessible to the system. At intermediate temperatures, the ensemble displays small clusters of mesogens with correlated orientation. Clustering grows in spatial extent with decreasing temperature. At even lower temperatures, the cohesive energy overtakes the decreasing translational entropy: a liquid phase with smectic order ensues at these temperatures. The smectic mesophase is stabilized (i.e., structural order increases in spatial extent) upon further cooling.

FIGURE 4

FIGURE 4. Snapshots of the bulk 2D system when P^∗ $P^{*} = 5.0$ and $N = 50,000$ . Shown is an eightfold-magnified region of the ensemble to discern the local order of mesogens. Configurations are arranged as a temperature sweep (arrows) with each configuration showing its temperature $T^{∗}$ . Mesogens are colored according to orientation, as indicated by the color bar (bottom).

3.2 Phase Diagram Topology and Mesophases

A major contribution in this study is the orthobaric $[i .e ., (ρ^{*}, T^{*})]$ phase diagram for the 2D system shown in Figure 5. The 3D case for the same GB mesogen is overlaid with gray and red shadows: such a comparison enables us to appreciate how dimensionality affects topology. The phase diagram for the 3D case was previously reported [128, 129]. The 2D phase diagram reported here was obtained from MD simulations performed in the isothermal-isobaric (NPT) ensemble. The pressure $P^{*}$ was controlled with a Nosé-Hoover barostat; samples consisted of $N = 1,000$ mesogens. Phase regions were outlined by acquiring data for $P^{*} \in {0.1, 0.5, 1.0, 2.0, 3.0, 5.0}$ . For each pressure, the system was initialized at a high temperature and gradually cooled in steps of $Δ T^{*} = 0.02$ for at least $5 \times 10^{6}$ time steps.

FIGURE 5

FIGURE 5. The orthobaric phase diagram for the $GB (3,5,2,1)$ mesogen in 2D (lines) and 3D (gray shadow). The 2D critical point (diamond) appears when $ρ_{c}^{∗} = 0.159 \pm 0.002$ and $T_{c}^{∗} = 0.202 \pm 0.007$ . The nematic phase region is highlighted in both cases: 2D (blue shadow) and 3D (red shadow). For 2D, the nematic region is stable over a wider $T^{∗}$ -range for a slightly narrower $ρ^{∗}$ -range. Three specific isobars (dotted lines, labeled with $P^{∗}$ ) are shown for comparison. Only the top-right region of the phase diagram is most sensitive to finite-size effects as shown in the response of the order parameter S in Figure 3.

When compared to the 3D case, the 2D system displays an evident shift in its phase boundaries. This behavior is justified by the fact that thermal fluctuations are stronger when the dimensionality of the system is reduced [154–156]. The 2D system shows that the nematic phase emerges over a wider (nearly double) range in temperature at slightly lower densities. Additionally, the isotropic phase occupies a larger area of stability in the $(ρ^{*}, T^{*})$ -plane, extending to lower temperatures and higher densities in 2D. The critical point appears at a lower temperature $(T_{c}^{*} = 0.202 \pm 0.007)$ and slightly higher density $(ρ_{c}^{*} = 0.159 \pm 0.002)$ when compared to the 3D system. Our result for the 2D critical point compares very well with previous work [157, 158].

It is instructional to consider how finite-size effects influence the topology of the phase diagram. For this purpose, three isobars are included: $P^{*} \in {0.1, 3.0, 5.0}$ . For the $(ρ^{*}, T^{*})$ region shown, only slight deviations from the boundaries in the top-right corner would be expected. Boundaries shown on the phase diagram serve as a guide to the eye based on the available data extracted from the isothermal-isobaric simulations. The slim regions conveyed by solid lines are best estimates that outline the limits of phase stability and do not represent true coexistence lines.

We note that isothermal-isobaric simulations can probe metastable regions that elude canonical simulations without yielding coexistence (i.e., phase separation). The complete mapping of such phase boundaries would require free energy calculations, such as Gibbs ensembles [159–161], Gibbs-Duhem integration [162], histogram reweighting [163], or the Frenkel-Ladd method [164], among others. The coexistence of mesophases, however, was verified by independent canonical simulations. A sample cooling routine highlighting the coexistence of different mesophases is shown in Figure 6. The snapshots trim out sparsely populated regions of the full simulation cell observed at lower temperatures.

FIGURE 6

FIGURE 6. A representative cooling sweep in the NAT ensemble. Configurations are shown for an ensemble with $N = 2,000$ and $ρ^{∗} = 0.10$ . The temperature $T^{∗}$ decreases going from left to right over a narrow temperature window as shown. Snapshots of the simulation box have been trimmed to improve the visual clarity of the more densely populated regions observed at lower temperature.

3.3 Confinement: Point Defects and Domain Walls

The extent to which mesophase structure is affected by strong confinement was also explored in this study. Inspiration for this lies in the rich structures and topological defects observed in lyotropic systems: the similarity in the textures observed in our thermotropic system highlights certain universal traits of topological defects. From an applications standpoint, this is of interest because topological defects can be recruited for the self-ordering of colloidal particles. In the case of a 2D system, this arrangement has the potential to yield monolayers of colloidal particles with specific positional constraints.

The 2D LC samples were confined within walls consisting of an array of fixed Lennard-Jones particles. Three different confinement scenarios were considered in this study: circular, square, and triangular. The mesogen packing fraction was kept approximately at $η = 0.75$ in all cases to ensure a nematic state point consistent with the bulk 2D phase diagram. As a point of reference, the packing fraction for a 2D hexagonal lattice composed of circular units is $η \approx 0.907$ [165].

To characterize the way the confining walls exert a structuring effect on the mesogenic liquid, two anchoring conditions were considered: homeotropic $(ε_{h} / ε_{p} = 5.0)$ and planar $(ε_{h} / ε_{p} = 0.2)$ cases (Eq. 10). The nematic field emerging from the sample in the bulk region displays a dominant direction. However, the boundaries defining the confined area disrupt any such dominant alignment. The resulting director field persists in response to a delicate balance between anchoring conditions imposed by the confining walls and the strong tendency of neighboring mesogens to mutually align. As a result, this synergy has the effect of stabilizing topological defects within the confined region.

Data for circular confinement are shown in Figure 7. Topological defects are sharply sensitive to the type of anchoring. For homeotropic anchoring, the confinement radius in this work affords a low-temperature director field giving rise to two defects (with topological charge $+ 1 / 2$ ), localized away from the wall but separated in relation to one another. This behavior is consistent with density functional theory predictions [119]. As temperature increases, the defects move away from one another until they approach the wall: at sufficiently high temperature $(T^{*} \sim 1.80)$ , the defects dissipate to yield a single, isotropic configuration. Such an outcome is possible because the anchoring energies at the confining surfaces are of finite strength. As can be seen in the configuration snapshots, thermal fluctuations are sufficiently strong to overcome the orientation induced by anchoring.

FIGURE 7

FIGURE 7. The 2D $GB (3,5,2,1)$ mesogen system confined by a circular boundary. The series from top to bottom represents a cooling sweep: temperature is indicated by the centered temperature bar. Renditions are shown for homeotropic ( $ε_{h} / ε_{p} = 5.0$ , two leftmost columns) and planar ( $ε_{h} / ε_{p} = 0.2$ , two rightmost columns) anchoring. Each set of columns on either side of the temperature bar correspond as follows: configuration snapshots (left) and scalar field of the local order parameter (right).

The situation changes for planar anchoring: at low temperature, two defects are present, but they are located at opposite poles of the confining circle. As a result, the so-called polar nematic configuration is observed. This state is distinguished by a layered mesophase similar to the smectic-like state that dominates all but two thin surface shells on opposite ends $(T^{*} \sim 0.20)$ . Another configuration also observed at the lowest temperature possesses boundary disclinations, but the main topological defect consists of point defects appearing on opposite poles of the circular boundary (refer to the discussion on circular boundary confinement in Section 3.4). Radially oriented domains, like those observed in the case of homeotropic anchoring, are absent in planar anchoring. Nematic-like order emerges as temperature increases for both anchoring modes under circular confinement. Although two polar defects persist at high temperature in the two systems $(T^{*} \sim 1.20)$ , the type of anchoring can be distinguished by probing the relative distance between defects: the separation between the two defects is always larger (nearing the boundary wall) for planar anchoring. As expected, internal order is mostly lost in both types of anchoring at sufficiently high temperatures. Our observations are consistent with those documented for a 2D fluid of hard rods in the high-density regime, with a sufficiently small aspect ratio [124].

Square confinement leads to more interesting textures as shown in Figure 8. The reduced symmetry of the boundary frustrates global mesophase order. For both anchoring conditions, domain walls (i.e., boundaries between different orientationally ordered domains) appear at sufficiently low temperature $(T^{*} \sim 0.60)$ . For homeotropic anchoring, domain walls define three regions: a large region with a local director rotated $π / 2$ radians in relation to the local directors characterizing two small regions oriented in the same direction. Mesogens in the large region are highly oriented and form smectic-like layers. These domain walls signal domains possessing different orientations: the free energy is minimized in the system when curved interfaces develop. Moreover, two equivalent states are possible by symmetry: one shown in the snapshots and another obtained by rotating the snapshot $π / 2$ radians. In this way, the system displays two-fold degeneracy. This effect was previously observed using a density-functional approach [166]. An external field can lead to an interchange between the two states dynamically, as previously reported [167].

FIGURE 8

FIGURE 8. The 2D $GB (3,5,2,1)$ mesogen system confined by a square boundary. Data are arranged as in Figure 7.

When planar anchoring is operative under square confinement, four domain walls (i.e., five regions) appear. Four small regions display an orientation aligned with the confining walls and one interior region with a local director tilted slightly in relation to adjacent lateral domains. This arrangement is strikingly similar to the $W \sim 40 L$ system studied by Cortes et al. [117]. It is plausible for the interior region to eventually reorient to create a single region where the local director matches that of two adjacent lateral domains. As temperature increases, those domain walls disappear and two point defects arise close to the corners of the square. The structural behavior of the LC sample under square confinement agrees with previous theoretical models [121].

The most severe confining geometry in this study is the triangular boundary, the results of which are shown in Figure 9. For either type of anchoring, three orientationally ordered regions are discernible at low temperature: the resulting topological defects are driven by an order that permeates from the boundary wall toward the center of the confining region. This effect promotes the formation of a near-centered defect in either anchoring case at fairly high temperature $(T^{*} \sim 2.00)$ . Streak disclinations radiate from the point defect, signaling the partition between distinctly oriented regions giving rise to three domain walls only when the temperature is sufficiently low $(T^{*} \sim 0.60)$ .

FIGURE 9

FIGURE 9. The 2D $GB (3,5,2,1)$ mesogen system confined by a triangular boundary. Data are arranged as in Figure 7. A set of approximate local directors highlighted for the lowest temperature (red arrows) enables differentiating between the two anchoring modes, as discussed in the main text.

Although the scalar fields of the order parameter under triangular confinement for both anchoring cases are similar, the two samples can still be differentiated when accounting for local order. The approximate local directors corresponding to the three oriented domains highlighted in Figure 9 are rotated $π / 6$ radians with respect to one another. This difference arises from the coupling of the anisotropic shape of the calamitic mesogen and the underlying confining geometry. When placed at the center of the triangular region, the set of directors for the homeotropic sample point at the corners of the triangular boundary; the analogous set for planar anchoring results in the bisection of all sides of the triangular region. As expected, the orientationally ordered regions and the streak disclinations dissipate with increasing temperature, though the point defect persists even at high temperature $(T^{*} \sim 1.40)$ .

3.4 Dynamics of Confined Samples

Ancillary data from this work are ensemble trajectories given that systems were evolved with MD simulations. Although static information obtained from simulation snapshots is useful in characterizing orientational order and topological defects, it is equally important to probe the temporal behavior of mesophases. To this end, trajectories were leveraged to probe dynamical fluctuations in the fluid structure and the scalar field of the order parameter. The Supplementary Material includes trajectory visualizations for the systems in Figures 7–9. Each visualization is labeled by temperature and anchoring mode. The timescale in each case corresponds to approximately 3% of an entire simulation run.

The system under circular confinement with planar anchoring at reasonably high temperatures already exhibits features reminiscent of the polar defects stabilized at low temperatures. However, such defects are accompanied by strong fluctuations in intensity and positional alignment. As one point defect vanishes another emerges in the same pole. For homeotropic anchoring, point defects fluctuate in number, intensity, and position at high temperatures. Upon further cooling, the homeotropic sample displays two point defects with minimal fluctuations in position and intensity, immediately after two radially oriented domains form. Before the central smectic-like region sets in, the separation between the two point defects reaches a minimum. Fluctuations in intensity are minimal at the lowest temperature studied: the separation between point defects stabilizes at a slightly larger distance $(T^{*} \sim 0.20)$ than before $(T^{*} \sim 0.80)$ . In the case of planar anchoring, point defects appear on opposite poles of the circular boundary. Unlike the low-temperature configuration in Figure 7, textures in the bulk region of the confined area become richer when the smectic phase sets in: boundary defects of fleeting intensity appear upon further cooling for planar anchoring. However, polar defects persist prominently.

The behavior of topological defects upon sample cooling is similar for both anchoring modes under square confinement. In both cases, a highly fluctuating cross pattern with approximately two point-like defects on opposite corners of the square is observed. As soon as the sample reaches a temperature where the smectic mesophase becomes favorable, the fleeting point-like defects vanish: each anchoring case becomes distinguishable at this point $(T^{*} \sim 0.60)$ . For homeotropic anchoring, two prominent domain walls persist, giving rise to three distinct regions. For planar anchoring, four domain walls become stable, demarcating five distinct regions. The domain walls, although subdued when compared to the homeotropic case, yield a rhomboidal pattern.

When comparing the two anchoring modes for triangular confinement, the cooling history is very similar in both cases. A point defect is characteristic of either case upon the slightest hint of ordering. Three domain walls weakly form at high temperature $(T^{∗} \sim 1.00)$ , although they are characterized by strong fluctuations in position and intensity. Upon further cooling, the domain walls assert their presence and a point defect becomes prominent in the sample $(T^{∗} \sim 0.60)$ . As discussed in Section 3.3, due to the similarity between the two anchoring cases, it is only possible to distinguish the two samples by inspecting the relative arrangement of local directors in each sample.

4 Conclusion

Bulk and confined 2D samples were explored for the Gay-Berne mesogen with parameterization $G B (3,5,2,1)$ . This model calamitic exhibits a sharp isotropic-smectic (I-Sm) transition at lower pressures $(P^{*} < 2.0)$ ; at higher pressures $(P^{*} ≳ 2.0)$ , the I-Sm transition involves an intermediate nematic phase. Clusters of locally ordered mesogens reach a threshold size, at sufficiently low temperatures, before the LC sample becomes smectic. The nematic phase shows an extended region of stability, nearly doubling in the temperature range at slightly lower densities. The critical point shifts to a lower temperature and a higher density compared to the analogous 3D system.

Confined samples were subjected to three boundary geometries: circular, square, and triangular. In circular geometry, two point defects emerge: for homeotropic anchoring, point defects are stable in the bulk region of the boundary and remain at a nearly constant separation. For planar anchoring, point defects gravitate toward opposite poles of the boundary. In square geometry, no stable point defects are observed at low temperatures. Instead, two distinct domain walls give rise to three regions under homeotropic anchoring; the structure under planar anchoring results in four interconnecting domain walls, rhomboidal in form, producing five regions. In triangular geometry, confinement yields similar defects when comparing anchoring modes: three domain walls “emanate” from a nearly centered point defect yielding three regions. In this case, local directors must be accounted for to differentiate between anchoring modes.

All systems were studied with MD simulations. The resulting trajectories of confined 2D LC samples were visualized, revealing a complex evolution of textures originating from topological defects. Ordered domains at low temperatures are prefaced with strong thermal fluctuations that cause spatial variations in the mesophase at sufficiently high temperatures. Within the mesophase, flickering in position and intensity of topological defects is minimized at sufficiently low temperatures. Both the confining geometry and anchoring mode contribute to the type of defects observed.

Confined 2D LC systems provide a rich and exciting outlook. An outstanding matter with an eye toward 3D metamaterials is how disclinations couple when colloidal slabs are stacked. Practicable systems could extend layer-by-layer protocols [7, 89, 90, 168–180], thus expanding the gamut of metamaterials attainable by conventional 3D-based methods. Studies on the switching mechanics by applying external fields (as opposed to thermal tempering) would be of interest in the production of devices and associated technologies. A characterization of relevant timescales would offer an important perspective on design principles. Structured colloidal assemblies via topological defects could be exploited to yield colloidal assemblies with screw/twist properties, thus amplifying the availability of chiral materials.

Data Availability Statement

The numerical model simulations upon which this study is based are unwieldy to archive or to transfer. Instead, all information needed to replicate the simulations is provided.

Author Contributions

AC-A and JM-V performed bulk simulations, wrote post-simulation analysis codes, and implemented data analysis. AC-A carried out simulations for small-scale bulk samples as well as strongly confined systems and visualized trajectories. JM-V performed large-scale simulations of bulk samples via parallelization. SH and AR-H contributed to the interpretation of results and drafted relevant sections of the analysis. ES verified results for bulk systems, interpreted dynamical trajectories of strongly confined systems, and completed writing the manuscript. JM-R conceived the study, wrote the Gay-Berne simulation code, drafted initial versions of the manuscript, and coordinated the direction of the project.

Funding

JM-V and JM-R are thankful for the computing time generously provided under grant LANCAD-UNAM-DGTI-344. SH is grateful for funding provided by projects UNAM-DGAPA-PAPIIT IA104319, LANCAD-UNAM-DGTIC-276, and LANCAD-UNAM-DGTIC-087.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2020.622872/full#supplementary-material.

Acknowledgments

JM-R gratefully acknowledges the computing resources provided by the Laboratorio de Supercómputo y Visualización en Paralelo (LSVP) at UAM-I. SH gratefully acknowledges the technical support provided by Carlos Sair Flores Bautista, Alejandro de León Cuevas, and Luis Alberto Aguilar Bautista from the Laboratorio Nacional de Visualización Científica Avanzada (LAVIS-UNAM), as well as by Beatriz Marcela Millán Malo from CFATA-UNAM.

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