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We employ Monte Carlo simulations in a specialized isothermal-isobaric and in the grand canonical ensemble to study structure formation in chiral liquid crystals as a function of molecular chirality. Our model potential consists of a simple Lennard-Jones potential, where the attractive contribution has been modified to represent the orientation dependence of the interaction between a pair of chiral liquid-crystal molecules. The liquid crystal is confined between a pair of planar and atomically smooth substrates onto which molecules are anchored in a hybrid fashion. Hybrid anchoring allows for the formation of helical structures in the direction perpendicular to the substrate plane without exposing the helix to spurious strains. At low chirality, we observe a cholesteric phase, which is transformed into a blue phase at higher chirality. More specifically, by studying the unit cell and the spatial arrangement of disclination lines, this blue phase can be established as blue phase II. If the distance between the confining substrates and molecular chirality are chosen properly, we see a third structure, which may be thought of as a hybrid, exhibiting mixed features of a cholesteric and a blue phase.

Chirality is a symmetry property that is omnipresent, not only in many fields of science [

There is a vast number of examples from all scientific disciplines that comport with Lord Kelvin’s definition of chirality. To name just a few, in mathematics, the well-known Möbius strip (see, for example, Figure 5.6 in [

A particularly interesting class of materials often exhibiting chirality are liquid crystals. Here, it is the combined effect of molecular chirality and the formation of orientationally ordered, but positionally disordered, nematic or smectic mesophases, which gives rise to new structures, such as cholesteric or blue phases, to which the present article is devoted. Both cholesteric and blue phases are characterized by molecular helices in one or more spatial directions. The pitch length characterizing these helical structures ranges typically from a few hundred nanometers to a few microns in length, such that chiral liquid crystals exhibit interesting optical properties over a wide range of wavelengths from the ultraviolet all the way to the visible part of the electromagnetic spectrum [

In particular, blue phases exhibit specular reflections of visible light that can be controlled by external fields, such that these phases may be thought of as tunable photonic materials [

In the context of novel nanomaterials, the fabrication of three-dimensional nanostructures has been achieved recently by polymer templating blue phase I [

On the theory side, quite a bit of work has been invested to understand defects in liquid crystals and their topology [

The interaction potential of a nanocolloid with a

Structural richness in ordered chiral liquid-crystalline phases is also highlighted by Fukuda and Žumer, who observe that highly chiral nematic liquid crystal phases are capable of forming two-dimensional, so-called Skyrmion lattices, as a thermodynamically stable morphology [

Almost all these previous theoretical studies have been based upon mesoscopic approaches, such as the Frank-Oseen elastic equations [

Quite a bit of simulation work had been carried out in the 1990s by Memmer and coworkers. They studied the temperature dependence of the pitch that characterizes cholesteric phases [

The remainder of our paper is organized as follows. In Section 2, we introduce our model system. Section 3 is given as a summary of some key theoretical concepts. In Section 4, we present our results which we summarize in concluding Section 5.

Our model consists of

We introduce shorthand notation, _{1}_{2}_{N}_{1}_{2}_{N}_{1}_{2}_{N}

where _{ij}_{i}_{j}

where _{ff} is the potential well depth, _{ij}_{ij}_{i}_{j}

where
_{1} = 0.04 = −_{2}_{3}, determines the amount of chirality, while its sign defines the handedness of the chiral molecules. The most preferred configuration for two achiral molecules (_{3} = 0) is a side-side configuration (_{i}_{ij}_{j}_{ij}_{3}, the preferred conformation of a pair of mesogens is still the side-by-side one, but now, with slightly tilted orientations, where the preferred tilt angle depends on the magnitude of _{3}. Clearly, Ψ remains unaltered if one changes the sign of _{i}_{j}

The achiral version of the interaction potential was suggested originally by Hess and Su some time ago [

Chiral, as well as achiral liquid crystals are experimentally usually investigated under confinement, often between smooth parallel walls that may prefer a specific orientational alignment of the mesogens with respect to the surface plane. They can serve to manipulate the preferred global orientation, due to specific anchoring at the solid surfaces, which then allows one to fix the orientation of the global director in the nematic phase. “Anchoring” refers to an energetic discrimination of preferred (or undesired) molecular orientations, as we will explain in some more detail shortly. In the present study, we confine the chiral liquid crystal to a nanoscopic slit-pore with atomically smooth substrate surfaces to make contact with the geometry of a typical experimental setup. The fluid-substrate contribution to the total configurational potential energy in

where the fluid-substrate interaction potential is given by the Yukawa-like expression:

Here, _{fs} determines the depth of the attractive well and Δ_{i}_{i}_{z}_{z}_{z}

are unique functions of the screening length, ^{−1}. They are introduced to guarantee that the location of the minimum of the fluid-substrate potential, _{min}, defined by:

remains fixed and that the depth of the attractive well:

is preserved as one varies the range of fluid-substrate attraction. However, in this work, we employ a fixed short-range, but sufficiently strong fluid-substrate attraction characterized by _{fs}_{ff} = 3.0.

Fluid-substrate attractions are “switched on/off” by the anchoring functions, 0 ≤ ^{(}^{k}^{)}( _{i}

where the unit vector, _{α}^{(1)}≠ ^{(2)}, see ^{(1)} = ^{(2)}) is advantageous for reasons to be explained in Section 4.1.

In this work, we consider a liquid crystal composed of _{z}_{z}_{x}_{y}_{α}_{z}_{αα}

where
_{||}_{z}

At the molecular level,

where _{||}_{z}_{B} is Boltzmann’s constant. As discussed elsewhere in detail (see pp. 33–70 in [

where _{B}

where ℐ is the moment of inertia, ^{2}_{B}^{1/2} is the thermal de Broglie wavelength,

is the configuration integral and ^{N}_{i}_{i}

Thus, from _{zz}

In some cases, the simulations have not been carried out in the specialized isothermal-isobaric ensemble introduced above, but in the grand canonical ensemble instead. In this case, the relevant thermodynamic potential is the grand potential:

and

where the grand canonical partition function is given by:

Hence, in the grand canonical ensemble, a thermodynamic equilibrium state is characterized by the set, {_{z}

In this work, we wish to analyze the local orientational order forming in a confined chiral liquid crystal for which we introduced the intermolecular interaction potentials already in Section 2. A suitable quantitative measure of local orientational order is provided through the local alignment tensor, which we define as:

where _{i}

Because _{−} (_{0} (_{+} (_{−} (_{0} (_{+} (_{+} (_{+} (

Another quantity of interest is the pressure tensor,

where _{i}_{xx}_{yy}

where superscript ^{(}^{α}^{)} (_{3}, on the last line of that equation. Monitoring _{xx}_{yy}_{||}_{x}_{y}_{x}_{y}_{x}_{y}_{||}_{xx}_{yy}_{||}_{||}^{3}_{ff} exceeds this deviation by more than two orders of magnitude.

We employ Monte Carlo (MC) simulations in the specialized isothermal-isobaric and in the grand canonical ensembles introduced briefly in Section 3. Because our intermolecular potentials are short range [see _{x}_{y}_{z}, the implementation of hybrid anchoring [see _{z}_{z}

Because we are not interested in investigating any specific chiral liquid crystal, we express our quantities of interest in terms of “reduced” (_{ff}, temperature in units of _{ff}_{B} and pressure in units of _{ff}^{3}.

For the anisotropy parameters chosen throughout this work [see Section 2, _{ff} (_{ij}_{i}_{j}_{i}_{ij}_{i}_{ij}_{i}_{j}

In the isothermal-isobaric ensemble MC simulations, we take _{||}_{3} = 0) is sufficiently deep in the nematic phase, as reflected by the relatively high value of the global nematic order parameter, _{||}

Our results are based upon systems comprising between 5, 000 and 40, 000 mesogens and runs consisting of about 5 ^{4} MC cycles that have been carefully equilibrated using a similar number of cycles prior to sampling any data. In the specialized isothermal-isobaric ensemble, a MC cycle consists of ^{−4}. In both ensembles, we employ the standard generalized Metropolis algorithms described in Chapter 5 of [_{||}^{−1} ln

To save computer time and because our fluid-fluid interaction potential [see _{c} = 3.0. No corrections are applied for neglected interactions beyond _{c}. Moreover, _{ff} (_{c}) remains unshifted with respect to _{ff} = 0. For the fluid-fluid interactions, we utilize a combination of a link-cell and a conventional Verlet neighbor list, as described in the book by Allen and Tildesley [_{n}_{x}_{y}

We begin the structural analysis with a discussion of the local director for a system at relatively low chirality, _{3} = 0.14. Unfortunately, in general,

A mesogen located in that bulk-like region is surrounded by nearest neighbors in all three spatial directions. The orientation of these neighbors is tilted with respect to the reference mesogen, because at _{3} ≠ 0, this configuration is energetically favorable. If, on account of thermal fluctuations, the tilt angle deviates from its optimum value (at

Looking next at the associated local director,

where _{z}_{z} perfectly. Therefore, the plots of _{x}_{y}_{z}_{z}

These structural features are further corroborated by the plot of a typical “snapshot” of a configuration in the cholesteric phase taken from our MC simulations (see _{z}

In fact, closer scrutiny reveals that the helical structures formed by the liquid crystal depend on the chirality coupling parameter, _{3}. Simulations in the specialized isothermal-isobaric ensemble were performed with 15,000 mesogens and a substrate distance of _{z}_{3} = 0.08, several simulations are performed, where the chirality is slightly increased by Δ_{3} = 0.02 between subsequent simulations; in each of these simulations, the nematic director is computed as a function of _{3} can be attributed to the stronger twist between neighboring mesogens. For very high chiralities, we observe the pitch length to reach a plateau. For increasingly larger values of _{3}, highly twisted structures are energetically favored. However, the formation of these structures eventually causes even nearest neighbor molecules to deviate strongly from a side-side conformation favored energetically by the terms proportional to _{1} and _{2} in

Moreover, one notices that the local nematic order decreases with enhanced chirality. This can be seen from plots in _{3} = 0.90 decays to a value characteristic of an isotropic phase. However, the liquid crystal is not isotropic locally, but does indeed form a highly structured, morphologically distinct phase under these conditions, as we shall demonstrate shortly.

As we showed above in the cholesteric phase, mesogens are aligning with a local nematic director pointing in a direction somewhere in the _{3} is sufficiently small, nematic order can be preserved

Now, with increasing _{3}, in-plane, twisted conformations of mesogens become energetically more favorable, which, in turn, destroys the local, in-plane nematic order of the cholesteric phase described before. In fact, under the present conditions, the liquid crystal exhibits the complex structure reminiscent of a blue phase. This can be seen from the snapshots presented in _{3}.

In particular, one notices that in each of the three plots, regions of blue colored mesogens at the center exist. As one moves out of these regions from the center in any radial direction, the color of the mesogens changes from blue to green and, eventually, to red, as one reaches the circumference of a region in which the mesogens are aligned with the respective line of vision. This change in color reflects a change in orientation, where mesogens at the center of each region are aligned with the line of vision, whereas along the circumference, they are oriented in an orthogonal fashion with that respective line. Hence, this orientational change is characteristic of a double-twist alignment of the mesogens. Because the global topology of the three structures depicted in

It seems somewhat surprising that the blue phase, which is isotropic _{x}_{y}_{x}_{x}_{x}_{3} = 0.90 is optically isotropic in a global sense, as is expected for the blue phase [

The statistical accuracy of our data can be rationalized as follows. Taking

where cos

As indicated by snapshots in _{z}_{z}_{x}_{z}

We note in passing that observing the complex three-dimensional double-twist structure of a blue phase is not only a theoretical challenge, but also an experimental one. Indeed, the typical platelet structure of the blue phases seen experimentally is a result of a non-homogeneous crystallization [

Nevertheless, we stress that the results obtained in this study clearly show that MC simulations of fairly large systems—if carried out with care—are capable of reproducing three-dimensional blue-phase structures comprising a number of pitches in each direction. However, up to this point, our results do not permit us to identify the specific blue phase (I, II or III) forming in our simulations. While blue phase I and II are characterized by a regular lattice of double-twist helices, blue phase III exhibits an amorphous structure. We, therefore, speculate that the presented structure might represent a blue phase III, which also is seen experimentally for systems with higher chirality [_{3} = 0.30, causing a larger pitch length and containing only

This interpretation is further corroborated by an inspection of the associated disclination lines, which we define as regions in space with a nematic order parameter,

Up to this point we introduced the confining substrate merely as a technical means that allows us to determine the pitch length reliably and without interference from periodic boundary conditions. However, in parallel experiments, confining solid substrates are part of the standard setup. This prompted us to investigate the impact of the presence of these substrates on structural properties of the liquid crystal by varying _{z}_{z}_{3} = 0.24, which is slightly bigger than the one for which the system undergoes a transformation from a cholesteric to a blue phase, according to

According to the plot in _{3}. Thus, fixing the latter causes helices to form that are characterized by a fixed pitch length regardless of whether these helices are part of a cholesteric or a blue-phase structure. From the plot in _{z}_{z}_{z}

A more interesting case is observed at a substrate separation, _{z}_{3} = 0.24. At both substrates, molecules are anchored directionally [see _{3} = 0.24 slightly favors the formation of a blue phase and the directional anchoring would favor formation of a cholesteric phase in which the director rotates around the

A more quantitative analysis of this peculiar new structure is presented in _{x}_{x}_{z}_{z}_{z}_{z}_{z}_{z}

The mixed new structure illustrated by plots in

The uniqueness of the new structure observed under special confinement, chirality and anchoring conditions is also illustrated by the spatial variation of disclination lines, which we define according to the same criterion already introduced above with respect to our analysis of blue phase II. The plot of disclination lines in

We investigated the formation of cholesteric and blue phase II in a new model for chiral liquid crystals by means of MC simulations in a specialized isothermal-isobaric and a grand canonical ensemble. Through specific anchoring conditions at the planar substrates of a slit-pore, we are able to determine the pitch length as a function of the chirality parameter, _{3}, via the nematic director. The latter rotates along one or several axes of the Cartesian coordinate system in the cholesteric and blue phase, respectively. We determine the pitch length over a wide range of chiralities ranging from _{3} = 0.08 to _{3} = 0.9. Knowing the pitch length, we can set up systems that are not exposed to spurious strains, due to periodic boundary conditions, because the side lengths of our simulation cell can be taken as half-integer multiples of the pitch length. Because of this setup, we are able to observe the undisturbed helical structure of blue phases in all three spatial dimensions and observe the same pitch length for each as one would expect. If the pitch length is large enough, we can also visualize the disclination lines characteristic of blue phase II. We emphasize that in our model, chirality has to be sufficiently large to obtain a well-defined blue phase. According to typical snapshots presented in

By choosing a certain chirality parameter, _{3} = 0.24, we also investigated the interplay between confinement, chirality and anchoring conditions at the solid substrates. We could show that even for just a single chirality value, three different phases are possible, the nematic, the cholesteric and a novel confined phase, which may be perceived as a result of a competition between ordinary cholesteric and blue phases. Unlike the latter two, the new phase is characterized by the formation of a helical structure in two dimensions rather than one (cholesteric phase) or three spatial directions (blue phase). Moreover, the new phase in confinement is characterized by a spatial variation of disclination lines that have not been observed before.

We are grateful to the International Graduate Research Training Group “Self-assembled soft matter nanostructures at interfaces” for financial support. This work was also supported in part by NSF’s Research Triangle MRSEC (DMR–1121107).

The authors declare no conflict of interest.

Plots of Cartesian components of the local director field, _{z}_{x}_{y}_{z}

Plot of the sum, _{z}_{z}_{3} = 0.14 (see, also,

Side view of a “snapshot” of a typical configuration in the cholesteric phase, where the _{i}_{x}_{3} = 0.14 with

Pitch length as a function of the chiral coupling constant, _{3}. The red line is intended to guide the eye. The transition from the cholesteric to the blue phase occurs at _{3} = 0.2.

Components _{x}_{3} = 0.14 (cf., _{3} = 0.90. In addition, the local nematic order parameter, _{3} = 0.14 (
_{3} = 0.90 (
_{3} = 0.90, mesogens are directionally anchored at both substrate surfaces [see _{z}_{3} = 0.14, hybrid anchoring is employed [see

“Snapshot” of a typical configuration characteristic of a blue phase. Plots in (_{3} = 0.90.

As _{3} = 0.90 in the blue phase.

(

“Snapshots” of configurations of confined systems where both substrates cause directional alignment. The phase determining parameter is the substrate-substrate distance, which is 8.2 for the nematic phase in (_{3} = 0.24.

As _{x}_{3} = 0.24.

As _{3} = 0.24. One can see the periodic structure in the

As