Molecular Dynamics Approach for Predicting Helical Twisting Powers

Article pubs.acs.org/JPCB

Molecular Dynamics Approach for Predicting Helical Twisting Powers of Metal Complex Dopants in Nematic Solvents Go Watanabe*,† and Jun Yoshida*,‡ †

Department of Physics, School of Science, Kitasato University, 1-15-1 Kitasato Minami-ku, Sagamihara, Kanagawa Japan Department of Chemistry, School of Science, Kitasato University, 1-15-1 Kitasato Minami-ku, Sagamihara, Kanagawa Japan

‡

S Supporting Information *

ABSTRACT: Nematic liquid crystals of small molecules are known to transform into chiral nematic liquid crystals with supramolecular helical structures upon doping with enantiomeric compounds. Although this phenomenon is well established, the basic mechanism is still unclear. We have previously examined metal complexes with Δ and Λ chiralities as dopants in nematic liquid crystals and have found that slight differences in the molecular structure determine the handedness of the induced helical structure. In this study, we investigated the microscopic arrangement of liquid crystal molecules around metal complex dopants with the aid of molecular dynamics (MD) simulations. There are several restrictions to performing MD simulations of the chiral nematic system; for example, one pitch of the helix usually exceeds one side of an applicable periodic boundary box (∼102 nm). In view of these simulation problems, we therefore examined racemic systems in which a pair of Δ- and Λ-isomers of the chiral dopant is mixed with liquid crystal molecules. We selected two different octahedral ruthenium complexes as the chiral dopant molecules. As a result, we accurately calculated the ordering matrix that is essential parameter to estimate the helical twisting power of the chiral dopant based on the surface chirality model. Since the microscopic ordering is experimentally hard to be determined, our new approach with using MD simulations accurately deduced the ordering matrix and, with the aid of the surface chirality model, gave reasonable values for the helical twisting powers of each complex.

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value). βM is defined by the equation βM = 1/(x × p) in which x is the molar fraction of the dopant and p is the pitch length of the helix.30−35 Octahedral metal complexes have rigid chiral structures that originate from the propeller-like arrangement of three chelate ligands (blades). The enantiomers are designated as Δ- and Λ-isomers. Our research group has shown that Λruthenium complexes with their elongating direction aligned with the C2 axis (hereafter called Λ-Ru-para, Figure 1) induce left-handed helices, while those with their elongating direction aligned perpendicularly to the C2 axis induce right-handed helices (hereafter called Λ-Ru-per, Figure 1).33−35 We interpreted this phenomenon with the aid of the surface chirality model, a physical theory treating the HTP as the product of the helicity tensor Q and the ordering matrix S,36−40 and were able to successfully reproduce the sign of the HTPs for Δ- and Λ-metal complex dopants assuming the alignment of the elongated ligand with the ordering direction of the liquid crystals. This idea is acceptable by considering the Onsager model which is based on excluded volume effects between rodshaped molecules: however, the microscopic arrangement around the metal complexes was not well understood.

INTRODUCTION Helical supramolecular structures have attracted wide attention due to their optical, electronic, photonic, and chiral properties, which make them suitable for a variety of applications such as circularly polarized light emission,1−6 cholesteric gels,7 and enantioselective recognition/separation of small molecules.8−14 Polymer blends containing low-molecular-weight liquid crystals often form the representative helical supramolecular structures with electro-optical15 and chiroptical properties16 while selfassembly of small molecules can be another trigger of helix formation.13,17−24 Currently, chiral nematic liquid crystals with helical supramolecular structures from nanometer to micrometer pitch are most commonly employed to prepare chiral organic polymers and organic/inorganic hybrid materials.25−29 Since chiral nematic liquid crystals are easily induced by doping enantiomeric compounds with nematic liquid crystals, the chiral nematic template method is advantageous in terms of cost and manipulation. However, the basic mechanism of helix formation in chiral nematics is still not fully understood. Although rational design of chiral nematic liquid crystals is necessary for fine structural control of the resulting helical polymers, it remains difficult even to predict the handedness of the helix formed. Octahedral metal complexes are characteristic dopants in the sense that a clear correlation is observed between the molecular structure and the resultant helical twisting power (HTP, βM © 2016 American Chemical Society

Received: May 9, 2016 Revised: June 22, 2016 Published: June 22, 2016 6858

DOI: 10.1021/acs.jpcb.6b04669 J. Phys. Chem. B 2016, 120, 6858−6864

Article

The Journal of Physical Chemistry B

used for both MBBA and chiral dopant molecules. The Universal Force Field (UFF)48 parameters are adopted as the energy parameters for the bond stretching and bond-angle bending interactions, including the interatomic bonds between ruthenium and its neighboring oxygen atoms. The partial atomic charges of MBBA, Ru-para, and Ru-per (see Figures S1−S5 in the Supporting Information) were calculated using the restrained electrostatic potential (RESP) methodology,49 based on quantum chemical calculations using the LANL2DZ basis set with the associated effective core potential (ECP) for the ruthenium atoms and the 6-31G* basis set for the other atoms (all calculations were performed with the GAUSSIAN 09 program50). Simulation Systems and Conditions. We performed MD simulations for a pure MBBA system and for a racemic system in which Δ- and Λ-isomers of Ru-para or Ru-per chiral dopants were added to MBBA molecules (hereafter called N− Ru-para and N−Ru-per, respectively). The system of Δ- and Λ-isomers of the chiral dopant and nematic liquid crystal molecules is in a racemic state in which a helical structure is not induced. Although the orientational order of liquid crystal molecules in the cholesteric phase can be analyzed simply by performing MD simulations for the system in which one enantiomer is blended with nematics, the resulting system will exhibit a micrometer-scale helical structure that cannot be easily simulated by atomic MD methods. Here, our approach is to carry out all-atom MD simulations for the racemic system in which both Δ- and Λ-isomers are doped. Since macroscopic helices are not induced in the racemic system, MD simulations with nanometer-scale periodic boundary box can be performed for the accurate and precise prediction of the local orientational order of liquid crystal molecules surrounding the chiral dopants. In the initial structure of the pure MBBA system, 1080 molecules were randomly distributed without overlapping in a cubic periodic boundary box with a 9.15 nm side length by means of the program implemented in the GROMACS package.46 The initial state was not perfectly isotropic but had the orientational order without intention. Then, we carried out the relaxation runs for about 10 ns before equilibration MD runs (see Figure S6 in the Supporting Information).51 We also performed the MD simulations for the different starting configuration in the same way and verified that the order parameter of the system at the equilibrated state reached the same value (around 0.55 as in the last 10 ns of Figure 3) and was not depend on the effect of the initial structure (see Figure S7 in the Supporting Information). For the N−Ru-para system, the Δ- and Λ-isomers were inserted at coordinates (2.28 nm, 2.28 nm, 2.28 nm) and (6.86 nm, 6.86 nm, 6.86 nm) of the MD simulation box of pure MBBA, respectively, followed by the removal of six MBBA molecules that overlapped with chiral molecules. In N−Ru-per, the Δ- and Λ-isomers were positioned at (2.00 nm, 2.28 nm, 2.50 nm) and (7.10 nm, 6.80 nm, 7.00 nm), respectively, and 1076 MBBA molecules remained in the box after removal of the overlapping molecules. The molar fraction of the chiral molecules doped into MBBA molecules is about 0.2 mol %, which is consistent with previous experiments.33,35 After running the steepest descent energy minimization, each system was equilibrated at 1 bar pressure and a constant temperature ranging from 250 to 350 K. The initial relaxation processes at 250 K for 5 ns, 350 K for 1 ns, and 300 K for 5 ns were continuously run using the Berendsen thermostat and barostat with relaxation times of 0.2 and 2.0 ps, respectively.

Figure 1. Molecular structures of the two types of ruthenium complexes simulated in the present study. Both illustrate the Λisomers.

Molecular simulation methodologies such as molecular dynamics (MD) and Monte Carlo have been extensively used to investigate the microscopic arrangements of molecular assemblies, including liquid crystals. Recent studies using molecular simulations have successfully predicted the βM values of several types of chiral dopants.41−45 However, previous theoretical approaches have not utilized a detailed atomistic model for host nematic molecules, nor have they treated all the intermolecular interactions between the chiral solute and nematic solvent. To elucidate the relationship between the HTP and the molecular shapes of the chiral dopant, and to accurately and precisely determine the ordering matrix, MD simulations are required to consider all interatomic interactions, including van der Waals and electrostatic interactions. The treatment of boundary conditions is a problem frequently encountered when performing MD simulations of binary systems of nematic liquid crystal molecules and chiral dopants. Since one pitch of a helix usually exceeds the length of a periodic boundary box (∼102 nm), the simulation of a chiral system under periodic boundary conditions often gives unphysical results. In order to simulate a chiral system reliably by MD methods, a single box system composed of tens of thousands of molecules that form micrometer-scale helical structures would be required, which is not practical given current computational capabilities. In this study, to avoid the boundary condition problem, we focus on nematic liquid crystals doped with racemic metal complexes so that we can perform. MD simulations for a racemic system composed of two different enantiomers and surrounding nematics in which macroscopic helixes are not induced. In estimating the βM value, we need the values of Q and S. Q can be deduced from the molecular structure of the chiral dopant without requiring MD simulations, whereas S can be calculated from MD simulations of the local orientation state of the racemic system, which does not possess macroscopic helical structures.

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COMPUTATIONAL METHODS Model Details. The MD program GROMACS 4.6.746 was used to carry out the MD simulations. The liquid crystal molecule selected was N-(4-methoxybenzylidene)-4-n-butylaniline (MBBA), and the chiral dopants were Ru-para and Ru-per. MBBA exhibits a nematic liquid crystal phase in the temperature range 294 to 320 K. The molecular model utilized in our simulations is the detailed atomic model, which treats all atoms explicitly. To calculate intra- and intermolecular interactions, generalized Amber force field parameters47 were 6859

DOI: 10.1021/acs.jpcb.6b04669 J. Phys. Chem. B 2016, 120, 6858−6864

Article

The Journal of Physical Chemistry B Then, equilibration MD runs were carried out for up to 60 ns for the pure MBBA system, 75 ns for the N−Ru-para system, and 85 ns for the N−Ru-per systems. Each equilibration run was performed at a constant temperature (300 K) and pressure (1 bar) using the Nosé−Hoover52 thermostat and Parrinello− Rahman53 barostat with relaxation times of 1.0 and 5.0 ps, respectively. The time step was set to 2 fs and all bonds connected to hydrogen atoms were constrained by using the LINCS method.54 The smooth particle-mesh Ewald (PME) method was employed to treat the long-rang electrostatic interactions and the real space cutoff and the grid spacing was 1.4 and 0.30 nm. A snapshot of the MD simulation for N−Rupara after reaching equilibration is shown in Figure 2.

Figure 3. Time profiles of the order parameters of MBBA molecules for pure MBBA, N−Ru-para, and N−Ru-per. All the systems reached equilibration in the final 10 ns. The inset shows the long axis vector li of the MBBA molecule, defined as a vector connecting two end carbon atoms in a molecular core part. ni is the normalized vector of the long axis.

in the orientational order of the host nematics: however, the orientational order of our racemic system is preserved. Radial Distribution Functions and Diffusion Coefficients. Next, to confirm that each simulated system does not have positional order similar to that in the solid state, we analyzed the radial distribution function (RDF) of the liquid crystal molecules, g(r), which is defined as

Figure 2. Snapshot of MD simulation for N−Ru-para at 75 ns. Spheres show Ru-para molecules and MBBA molecules are represented by sticks.

g (r ) =

■

1 N

N

⎛3 1 ⎞⎟ ⎜ n n δαβ iα iβ − 2 2 ⎠

∑⎝ i=1

N

∑ ΔNi(r) i=1

(2)

where ΔNi(r) is the number of molecules in the shell at a distance r from the ith molecule, Δr is the thickness of the shell, V is the volume of the system, and N is the total number of molecules. As shown in Figure 4, the g(r) functions for all

RESULTS AND DISCUSSION Nematic Order Parameters of Simulated Systems. In order to confirm that each system was equilibrated and to analyze its orientational structure, we calculated the order parameters of the MBBA molecules in the pure MBBA system and in the racemic systems of N−Ru-para and N−Ru-per. According to the general definition, the order parameter is calculated from the largest positive eigenvalue of the order parameter tensor P, as expressed below: Pαβ =

1 V 1 4πr 2Δr N N

(1)

where N is the total number of molecules, subscripts α and β represent the coordinates x, y, and z, and ni is the normalized vector of the ith molecular long axis. The long axis of a liquid crystal molecule is defined as the vector connecting the two end carbon atoms in the rigid core part. Figure 3 shows the nematic order parameters of the pure MBBA system and the racemic systems of N−Ru-para and N−Ru-per. For the pure MBBA system, the average order parameter value for the final 10 ns of the simulation was 0.54 ± 0.01. The characteristics of the system obtained from the MD simulations might approach those of bulk nematics, since the calculated order parameter is consistent with the experimental value for MBBA (∼0.55).55 For the final 10 ns of the simulation, the average order parameter of the liquid crystal molecules in the N−Ru-para and N−Ru-per systems were 0.47 ± 0.01 and 0.48 ± 0.01, respectively. These results indicate that both racemic systems possessed the same degree of orientational order as that of the pure MBBA system. Doping different types of chemical compounds into host nematic liquid crystals causes a decrease

Figure 4. RDFs of the center of mass of MBBA molecules for pure MBBA, N−Ru-para, and N−Ru-per. Each function is the average of the final 5 ns of the simulation.

systems show quite similar curves, do not exhibit discrete sharp peaks, and asymptotically converge to 1. These characteristics of g(r) are typical of those of liquid-like phases. Thus, we confirmed that the MBBA molecules in our system were arranged in a disorderly manner, similar to molecules in the liquid or glass phase. In order to examine the fluidity of the system, we analyzed the mean square displacement (MSD) of the liquid crystal molecules for all the systems. The resulting profiles are shown in Figure 5. The MSD for each system 6860

DOI: 10.1021/acs.jpcb.6b04669 J. Phys. Chem. B 2016, 120, 6858−6864

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The Journal of Physical Chemistry B

Figure 6. Schematic figures of Λ-Ru-para and Λ-Ru-per represented by a pair of thin triangular and rectangular panels. The elongated rectangular panel is planar with the y-z plane for Λ-Ru-para and the xz plane for Λ-Ru-per.

Q xx = 0, Q yy = −

Figure 5. MSDs of the center of mass of MBBA molecules for pure MBBA, N−Ru-para, and N−Ru-per. Each function is the average of the final 5 ns of the simulation.

N

∑ |ri(t ) − ri(0)|2 i=1

= lim 6Dt t →∞

(3)

Q xx =

the self-diffusion coefficient D of the system can be calculated from the MSD of the simulation. The resulting D values were 2.38 × 10−11, 1.97 × 10−11 m2/s, and 2.04 × 10−11 m2/s of the MBBA molecules for pure MBBA, N−Ru-para, and N−Ru-per, respectively. These D values are consistent with those reported for MBBA in previous experimental studies, which are on the order of 10−11 m2/s.56−58 Theoretical Estimations of HTPs of Chiral Dopants. We now discuss the theoretical estimation of the HTP (βM) of the ruthenium chiral dopants in MBBA solvent, based on the surface chirality model proposed by Ferrarini et al.36,37 The surface chirality model takes into account the molecular interactions between the dopant and nematic solvent molecules. It has been confirmed that this model can successfully predict βM for several types of chiral molecules. Using the surface chirality model, βM can be obtained simply by multiplying physical properties of the nematic solvent and the chiral parameter of a solute molecule as follows: ⎛ Nϵ ⎞ βM = ⎜ A an ⎟χ ⎝ 2πK 22vm ⎠

2 (Q xxSxx + Q yySyy + Q zzSzz) 3

(6)

3 3 3 3 a , Q yy = 0, Q zz = − a 8 8

(7)

For the Δ-isomer, the signs of Qxx, Qyy, and Qzz should be inverted, while the sign of each element of S should not be inverted. Therefore, the sign of the chiral parameter should be changed from positive to negative, and vice versa. In the above calculation, the elongated part of each complex (rectangular panel in Figure 6) was supposed to be planar with the central diketone moiety (triangular panel in Figure 6). On this tentative treatment, the contribution of the rectangular panel is negligible in the calculation of Q.35 This treatment has been successful for estimating the HTPs of metal complex dopants, in particular to predict the sign. The calculation of Q for the case of Λ-Ru-para with twisted rectangular panel is described in the Supporting Information and discussed in the subsequent section. S can be calculated from the local orientational structure of the nematic hosts around the chiral dopant. The unit vectors along the y- and z-axes are denoted as ey(t) and ez(t) for the simplified model of N−Ru-para (Figure 7: for the N−Ru-para system, the unit vector in the direction of the x-axis is not defined, because eq 6 implies that Sxx is not required to obtain the chiral parameter). The unit vectors ex(t) and ez(t) for the N−Ru-per system are shown in Figure 7. The elements of S

(4)

Here NA is Avogadro’s number, ϵan is the anchoring strength, K22 is the twist Frank elastic constant, and vm is the molar volume of the host nematic. By using Q and S, the chiral parameter χ is written as shown in eq 5. χ=−

3 3 a 8

where a is the side length of the triangular acetylacetonate part. For Ru-per, the dibenzoylmethanate was considered as a pair of triangular and rectangular panels in which the longest side of the rectangular panel was connected to the triangular panel, as shown in Figure 6. Taking the z-axis as parallel to the direction of the longest side of the rectangular panel, the diagonal elements of Q for Λ-Ru-per were calculated as follows:

increases linearly with increasing MD simulation time, as is typical for fluids. By using the Einstein relation given by 1 N

3 3 a , Q zz = 8

(5)

A useful method for evaluating Q simply and accurately is to regard the chiral metal complexes as a pair of thin triangular and rectangular panels.33,35 For Ru-para, the triangular and rectangular panels correspond to the acetylacetonate and phenyl acetylene moieties, respectively, as shown in Figure 6. The longest side of the elongated rectangular panel is set parallel to the z-axis. Details of the calculation have been reported in previous studies.35 Following our previous calculations, the diagonal elements of Q for Λ-Ru-para were calculated as follows:

Figure 7. Unit vectors defined to calculate S for the N−Ru-para and N−Ru-per systems: ey and ez for Ru-para and ex and ez for Ru-per. The isomer type does not affect the definitions of the unit vectors. 6861

DOI: 10.1021/acs.jpcb.6b04669 J. Phys. Chem. B 2016, 120, 6858−6864

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The Journal of Physical Chemistry B

Table 1. Diagonal Elements of the Ordering Matrix S Averaged for the Final 2 ns in MD Simulations and the Helicity Tensor Q and the Chirality Order Parameter χ Calculated for Δ- and Λ-Isomers of Ru-para and Ru-per Sxx Syy Szz Qxx Qyy Qzz χ

Δ-Ru-para

Λ-Ru-para

Δ-Ru-per

Λ-Ru-per

− −0.10 ± 0.08 0.20 ± 0.09 − 77 −77 19

− −0.09 ± 0.08 0.28 ± 0.09 − −77 77 −23

−0.09 ± 0.07 − 0.27 ± 0.08 −77 − 77 −23

−0.16 ± 0.02 − 0.35 ± 0.05 77 − −77 32

experiments. For Ru-para, the theoretical values (βM = 46 and −56 μm−1 for Δ- and Λ-isomers respectively) are very close to the experimental values (βM = 60 and −64 μm−1 for Δand Λ-isomers respectively).35 Our previous X-ray study and DFT calculation demonstrate that the phenylene unit is almost planar with the central 1,3-diketonate unit.35 Moreover, the rectangular panel is negligible in the calculation of βM for the case of Ru-para (see Supporting Information). Consistency of experimental and theoretical βM values supports the rationality of our procedure combining MD simulation and surface chirality model. In contrast, for Ru-per, there are slight differences between the theoretical (βM = −55 and 78 μm−1 for Δ- and Λ-isomers respectively) and experimental (βM = −146 and 130 μm−1 for Δ- and Λ-isomers respectively) values.33 We regarded the dibenzoylmethanate of the Ru-per molecule as the combined shape of one thin isosceles right triangular panel and one thin rectangular panel located on the same plane; nevertheless, strictly speaking the phenyl groups are supposed to be slightly twisted toward the central 1,3-diketonate unit. Moreover, the two phenylene groups may independently twist from the central 1,3-diketonate unit. The slight twist of the rectangular panel produces new diagonal components in Qii and affect βM in the case that the twist is asymmetrically fixed (see Supporting Information). Our conclusion in the Supporting Information for the treatment of the rectangular panel is that neglecting the twist is reasonable for Ru-para but not fully applicable to Ru-per. The simple treatment of the dibenzoylmethanate as a rectangular panel may be insufficient to explain the βM values precisely and accurately for Ru-per. In general, chiral dopants with various conformations are still difficult to predict βM, because it is hard to judge the contribution of each conformation to βM. In this sense, our approach using MD simulations still has the regulation that each ligand should be planar like the case of Ru-para. Although it is not applicable to all systems, our approach is promising for evaluating βM of the chiral dopants with rigid or high symmetric configuration. Investigation of such chiral nematics is interesting and important for understating the fundamental mechanism of helix formation in the chiral nematics. However, it is generally difficult to experimentally measure the βM values for the chiral nematics containing a very small amount of dopants.

averaged over a sufficiently long equilibration time are defined as Sαα =

1 M

M

⎡3 1⎤ (ni(t ) ·eα(t ))2 − ⎥ 2 2⎦

∑ ⎢⎣ i=1

(8)

t

where subscript α represents the coordinates x, y, and z and M is the number of MBBA molecules within a distance r (nm) from a ruthenium atom of the chiral dopant. We calculated the average value of each element of S for MBBA molecules within 2.0 nm of the ruthenium atom. In order to analyze the sufficient number of MBBA molecules and calculate Syy and Szz for N− Ru-para and Sxx and Szz for N−Ru-per, we compare the time profiles of the elements of S for the molecules in different size of the region (see Figure S8 to S15 in the Supporting Information). Each figure indicates that the time profiles analyzed for the molecules within the distance r > 1.8 nm are similar and liquid crystalline behavior possessing the orientational order is observed in the range of about 2.0 nm from the chiral dopant. Since the distance between the two isomers was greater than 4.0 nm for both N−Ru-para and N−Ru-per, there was no direct interaction between the isomers, and the target areas for analysis of the local orientational order of the nematic solvent did not overlap. The average Syy and Szz of the final 2 ns of the simulation for MBBA molecules around Δ- and Λ-Rupara are listed in Table 1. The average Sxx and Szz in the final 2 ns for Δ- and Λ-Ru-per were also calculated. The larger value of Szz relative to Sxx and Syy, and the observation that host liquid crystal molecules tend to align preferentially with the z-axis, (to which the ligands also elongate), are consistent with previous studies. In order to determine Qxx, Qyy, and Qzz, the side length a of the isosceles triangular panel representing the acetylacetonate was set to 5 Å. According to eq 8, the chiral parameters for each of the enantiomers can be obtained from the ordering matrix and helicity tensor determined above (Table 1). Moreover, by using the anchoring strength, (ϵan = 5kBT N/ nm at T = 300 K and K22 = 3.2 × 10−12 N, reported in a previous study37), and vm = 2.5 × 10−4 m3 calculated from the current results, we estimated βM for the Δ- and Λ-isomers of Ru-para and Ru-per (Table 2). Table 2 clearly indicates that the values of βM obtained in the present simulations show good qualitative agreement with those measured in previous

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CONCLUDING REMARKS In this paper, we presented an MD simulation approach based on the surface chirality model to calculate values of βM for chiral dopants. The MD simulations were performed for the racemic systems in which both Δ- and Λ-isomers of chiral ruthenium complexes were doped with nematic liquid crystals. By analyzing the orientational order of the host liquid crystal molecules surrounding each isomer, we characterized the

Table 2. βM of Δ- and Λ-Isomers of Ru-para and Ru-per Obtained from the MD Simulation Results and Previous Experiments

MD expt

Δ-Ru-para

Λ-Ru-para

Δ-Ru-per

Λ-Ru-per

46 60

−56 −64

−55 −146

78 130 6862

DOI: 10.1021/acs.jpcb.6b04669 J. Phys. Chem. B 2016, 120, 6858−6864

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The Journal of Physical Chemistry B ordering matrix and estimated the values of βM for two types of chiral metal complexes. The results confirmed that the liquid crystal molecules prefer to align along the elongated direction of the metal complexes. Moreover, our approach proved useful for qualitatively determining βM values for chiral dopants that have rigid chiral linkages, and may also be an appropriate method for predicting the helical twisting powers of chiral metal complexes for a variety of host nematics.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.6b04669. Molecular models and partial atomic charges for MBBA, Δ-, Λ-Ru-para, and Δ-, Λ-Ru-per, time profiles of the order parameters of MBBA molecules for pure MBBA, N−Ru-para, and N−Ru-per during the relaxation runs, time profiles of the order parameters of MBBA molecules for another pure MBBA system during the relaxation and equilibration runs, time profiles of the ordering matrices for MBBA molecules within 1.0, 1.4, 1.8, and 2.0 nm of the ruthenium atom of the chiral dopants simulated in the present study, and detailed description of calculation of the helicity tensor (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*(G.W.) E-mail: [email protected]. Telephone: +81(42)-778-9540. *(J.Y.) E-mail: [email protected]. Telephone: +81-(42)778-7980. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

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