Cholesteric ordering predicted using a coarse-grained polymeric model with helical interactions

Liang Wu and Huai Sun *
School of Chemistry and Chemical Engineering, Key Laboratory of Scientific and Engineering Computing of Ministry of Education, Shanghai Jiao Tong University, 200240, Shanghai, China. E-mail:

Received 20th October 2017 , Accepted 23rd November 2017

First published on 23rd November 2017

The understanding of cholesteric liquid crystals at a molecular level is challenging. Limited insights are available to bridge between molecular structures and macroscopic chiral organization. In the present study, we introduce a novel coarse-grained (CG) molecular model, which is represented by flexible chain particles with helical interactions (FCh), to study the liquid crystalline phase behavior of cholesteric molecules such as double strand DNA and α-helix polypeptides using molecular dynamics (MD) simulations. The isotropic–cholesteric phase transitions of FCh molecules were simulated for varying chain flexibilities. A wall confinement was used to break the periodicity along the cholesteric helix director in order to predict the equilibrium cholesteric pitch. The left-handed cholesteric phase was shown for FCh molecules with right-handed chiral interactions, and a spatially inhomogeneous distribution of the nematic order parameter profile was observed in cholesteric phases. It was found that the chain flexibility plays an important role in determining the macroscopic cholesteric pitch and the structure of the cholesteric liquid crystal phase. The simulations provide insight into the relationship between microscopic molecular characteristics and the macroscopic phase behavior.

1. Introduction

Cholesteric liquid crystals (CLCs) first discovered by Reinitzer in cholesterol derivatives in the late 19th century exhibit fascinating phase behavior between that of liquid and crystalline states.1,2 Compared with a uniaxial nematic phase in which molecules preferentially align along a common orientation (nematic director), the CLC phase, also known as the chiral nematic phase, shows a helical organization of the nematic director field in which the local nematic director is distorted around the helix direction.1,2 The periodicity of rotation of the nematic director field determines macroscopic chirality. Therefore, the CLC phase is described by a characteristic length, the cholesteric pitch (chiral pitch) P. The CLC phase is observed in both thermotropic and lyotropic liquid crystal systems, such as DNA duplexes,3–10 virus suspensions,11–16 cellulose nanocrystals,17,18 and α-helix secondary polypeptides.19–23 The formation and characteristics of CLCs are known to be sensitive to not only the structure of the molecules involved but also the external conditions such the concentration, acidity, ionic strength, pressure and temperature.

Understanding of the CLC behavior at a molecular level has been a major focus in the study of liquid crystals. In addition to numerous experimental works, theoretical and computational studies have been carried out in order to understand the cholesteric LC phase behavior.24–26 The modelling of liquid crystals can be dated back to Onsager's approach27,28 in which the repulsive interaction (excluded volume) alone is found to be responsible for the formation of orientational ordered phases. It has been accepted that a complete description of the LC phase behavior is beyond simple steric repulsive models.29–32 For chiral molecules, such as DNA and α-helix polypeptides, models of hard helix particles,33,34 twisted triangular prisms,35,36 curled rods37 and crescent-shaped (bent) particles with purely repulsive interactions38 have been introduced in publications. The role of long-range interactions in stabilizing the chiral nematic phase has been disputed for cholesteric LCs.39 As experimental studies12,13,16 have shown that the contribution of electrostatic forces is an important factor for the formation of chiral nematic phases, progress has been made by incorporating long-range forces into hard-core particle models. The model of hard spherocylinders equipped with square-well attraction has been employed in reproducing the LC phase diagram of poly-γ-benzyl-L-glutamate (PBLG) solution.40 The Lennard-Jones potential has also been used in simulations and theory to study LCs.41 More sophisticated intermolecular potentials such as Gossens’ model,32,42–44 the patchy rod model,45–47 and the chiral hard Gaussian fluid model48 have been introduced to represent interactions between chiral molecules. However, most models have been used to qualitatively describe the CLC phase behavior with little attention to the correlation with molecular details such as the chain flexibility and helical side interactions.

Molecular dynamics (MD) simulations49 combined with a coarse-grained (CG) force field50–52 provide a powerful approach to study LC phase behaviors.53–61 By using CG beads to represent groups of atoms, the time and length scales of MD simulations can be significantly extended so that phase separation can be observed at an affordable computational expense. Depending on the scope of prediction, the CG bead can be designed to describe the most critical features of a molecule by using sophisticated potential functions to represent intermolecular and intramolecular interactions.

In this work, we introduce a CG model for representing a general chiral molecule, and use the model with MD simulations to study the CLC phase behavior. The model is a coarse-grained cholesteric molecule in which each bead represents a segment of a real molecule such as DNA or α-helix polypeptides. The model includes descriptions of chain flexibility and internal chirality of the molecule, as well as long range intermolecular interactions. Together with MD simulations, the model can be used to study the CLC phase behavior under various thermodynamic conditions such as different temperatures, pressures and molecular flexibility. The model can be parameterized to represent a real CLC molecule.51,62–64 However, as the first step in a series of studies, we focus here on how the molecular flexibility and pressure impact the CLC phase behavior.

The paper is organized as follows: in Section 2 we introduce a new CG model for chiral molecules, and the simulation procedure and computational method used for analysis are detailed in Section 3. The results and discussion are presented in Section 4, in which the analysis is focused on the isotropic–cholesteric phase transition, the characterization of the cholesteric phase and its correlation with pressure and molecular flexibility. We detail our primary conclusion in Section 5.

2. Coarse-grained model

A chiral macromolecule such as a DNA duplex or a α-helix polypeptide is represented by a chain of directional bead groups as shown in Fig. 1(a). Each bead group consists of two connected beads: a large backbone bead (bb) and a small side interaction bead (ib) as shown in Fig. 1(a). The molecular parameters of the FCh model are shown in Fig. 1(b), in which the dihedral angle ϕ between adjacent side beads describes the internal chirality, and the backbone bond angle θ describes the chain flexibility. The diameters of the backbone bead and side bead are defined relative to a general size parameter σ. The backbone bead is large i.e. σbb = 21/6σ and the side bead is much smaller, σib = 0.1σbb. The chain length is described by Nb backbone beads. In this work, we studied FCh molecules with chain length Nb = 10. The bonded interactions of FCh molecules are described by quadratic functions:
ub,i = Kb,i(lil0,i)2(1a)
ua,j = Ka,j(θjθ0,j)2(1b)
where the subscripts i, j refer to the interaction types, Kb,i and Ka,i are the force constants of bond stretching between backbone beads (i = bb–bb) and backbone-side beads (i = bb–ib) and bending between three backbone beads (j = bb–bb–bb) and the angle formed by one side-interaction bead and two backbone beads (j = ib–bb–bb). li, and θj are the corresponding bond length and bond angle while l0,i and θ0,j are their equilibrium values. As we studied FCh molecules of varying flexibility which is described by the stiffness of the angle bending of backbone beads, Ka,bb−bb−bb is shortened as Kangle for convenience. The dihedral interaction of ib–bb–bb–ib is also described by a quadratic function:
ud = Kd(ϕϕ0)2(1c)
where Kd is the potential constant of torsion, ϕ is the dihedral angle and ϕ0 is the equilibrium value. The bonded parameters of FCh molecules used in this study are summarized in Table 1.

image file: c7sm02077e-f1.tif
Fig. 1 (a) Comparison of DNA duplex, α-helix polypeptides, and a flexible chain with helical interactions (FCh model) of right-handed internal chirality as the coarse-grained (CG) representation for cholesteric liquid crystals. The molecular orientation vector û is depicted along the long axis of a FCh molecule. (b) Molecular parameters of the FCh molecule: molecular internal chiral pitch pint, and backbone bond length lbb−bb, backbone bond angle θbb−bb−bb and dihedral angle ϕ.
Table 1 Bonded parameters between backbone beads (bb) and side interaction beads (ib) in the flexible chain with helical interactions (FCh)
Interaction types l 0 (σ) K b,i (ε−1σ−2)
Bond stretch bb–bb 1.0 100
bb–ib 0.616 100

Interaction types θ 0 (deg) K a,j (ε−1 rad−2)
Bond angle bb–bb–bb 180 100, 75, 50
ib–bb–bb 90 100

Interaction types ϕ 0 (deg) K d (ε−1 rad−2)
Dihedral ib–bb–bb–ib 30 100

The nonbonded interactions are described by pairwise potentials.65 The backbone beads interact via the soft repulsive Weeks-Chandler-Andersen (WCA) potential,

image file: c7sm02077e-t1.tif(2)
where rbb–bb is the distance between two backbone beads. The intermolecular ib–ib interaction is represented by the shifted Lennard-Jones (LJ) 12-6 potential with the cutoff at 2.5σ,
image file: c7sm02077e-t2.tif(3)

The nonbonded interaction between bb and ib beads is estimated by Lorentz–Berthelot combining rules and the energy strength is ε/kBT = 1.0 in simulations.

3. Simulation details

MD simulations were carried out using a rectangular box containing 7200 FCh molecules. The simulation box has fixed dimensions Lx = 29.5σ, Ly = 30.5σ and variable dimension Lz which was determined upon applying normal pressure in the z-direction. The system was built as the isotropic state, and then relaxed in NPzT simulations using the standard periodic boundary condition (PBC). A series of NPzT simulations were carried out using increased pressures Pz* = 3/ε to study the phase transition at constant temperature. The initial configuration of the simulations was isotropic, and the starting configuration at each pressure point was the last configuration of the previous run. The reduced temperature in simulation is T* = kBT/ε = 1.0 in this paper. For calculation of the chiral pitch of the cholesteric phase, the standard PBC must be removed along the chiral director (here the z-axis).66 Among several methods,35,44,46,47 for each relaxed system using the above simulation protocol, we carried out NVT simulations with the hard wall confinement to estimate the cholesteric pitch P (see Fig. 2).
image file: c7sm02077e-f2.tif
Fig. 2 Typical snapshot of the cholesteric state of FCh molecules with confinement by impenetrable walls in the z-direction. The colour of FCh molecules shows molecular orientations relative to the frame of the simulation box.

The system equilibrium density is defined as ρ* = Nmol/V where Nmol is the total number of molecules and the system volume V = LxLyLz. To calculate the density profile along the z-direction, the simulation box is divided into 200 bins of equal width δz, and the bin density is calculated

image file: c7sm02077e-t3.tif(4)
where Nmol(z) is the number of molecules in a bin and 〈A〉 denotes the ensemble average of a physical quantity A. The orientation nematic order parameter S2 is calculated by diagonalizing the second-rank nematic order tensor
image file: c7sm02077e-t4.tif(5)
where ûi represent the unit vector of the i-th FCh molecule (see Fig. 1), ⊗ denotes the dyadic product, and I is the unit matrix. Diagonalization of the nematic order tensor Q produces three eigenvalues. The largest eigenvalue of Q is defined as the nematic order parameter S2 to quantify global orientational order (S2 > 0) or an isotropic state (S2 ∼ 0). The difference between the other two smaller eigenvalues (Sx, Sy) of the tensor Q, Δb = |SxSy| signifies the degree of the biaxial order.

The profile of the nematic order parameter S2(z) along the z-direction is calculated using bins. In order to reduce the standard deviation, the number of bins is reduced to 75 so that each bin contains 7200/75 = 96 molecules, which leads to an estimated error of 0.10 in the calculated order parameter S2(z). In each bin the twist angle profile cos[thin space (1/6-em)]α0(z) = |[n with combining circumflex](z[n with combining circumflex](z = 0)| is calculated based on the local nematic director profiles [n with combining circumflex](z). The profile cos[thin space (1/6-em)]α0(z) describes the distortion of the local nematic director along the helix axis with respect to the directors in the first bin [n with combining circumflex](z = 0).

The orientational correlation functions are computed from MD simulations to measure the long-range correlation between molecular orientations of FCh fluids in the cholesteric phase,66

image file: c7sm02077e-t5.tif(6)
image file: c7sm02077e-t6.tif(7)

The MD simulations were carried out using the LAMMPS package67 using the LJ unit and time step Δt = 0.0002. The Nosé–Hoover thermostat68 was used to control the system temperature, and the Nosé–Hoover barostat algorithm was applied to control the pressure. Each NPzT simulation was run for 2–4 × 107 MD steps, and then the resulting configuration was used in NVT simulations with hard walls on the both ends for 1–2 × 108 MD steps. The thermodynamic and structural properties, such as the density profile ρ(z), the local nematic order tensor S2(z), etc., were calculated based on the data collected in the final 2 × 107 MD steps, with an interval of 2000 time steps.

4. Results and discussion

4.1 Isotropic–cholesteric phase transition

We first studied an FCh model with an internal chirality parameter of ϕ0 = 30° and backbone bending constant Kangle = 100. The bulk density and nematic order parameter S2 as a function of applied reduced pressure, obtained from NPzT MD simulations, are shown in Fig. 3. With increased normal pressure Pz*, the system density increases and undergoes a transition from the isotropic to cholesteric (IC) phase. The highest density of the isotropic state is ρI* = 0.0339 corresponding to Pz* = 0.09, and the lowest density of the cholesteric phase is ρC* = 0.0386 under Pz* = 0.1. Hence, the coexistence pressure is estimated by arithmetic average to be PIC* = 0.095. Although free energy calculations would be required for an accurate determination of the phase boundary,69 the estimation of coexistence pressure and density using the averaged values provides a reasonable description of the LC phase transition point.70 It can be seen from Fig. 3b that the nematic order parameter shows a transition at ρI* ∼ 0.035, from S2 < 0.05 corresponding to an isotropic state at low density to S2 ∼ 0.25 for the cholesteric phase.44 The data of the isotropic–cholesteric equation-of-state are available in the ESI.
image file: c7sm02077e-f3.tif
Fig. 3 (a) Isotropic–cholesteric phase behavior of FCh molecules with chain length Nb = 10 and angle bending constant kangle = 100 at temperature T* = 1.0, and the inset is enlargement of the isotropic–cholesteric transition region; (b) density (reduced density ρ* = ρσ3) dependence of the nematic order parameter S2 in both isotropic and cholesteric phases. The statistical error in each figure is smaller than symbols.

At the coexistence pressure Pz* = 0.095 we extended the NPzT simulation up to 3 × 108 steps. As shown by the snapshot of the final configuration (Fig. 4a), the system shows the isotropic state with some local nematic domains. The formation of the pre-transitional state is due to the system size and particularly the very slow transition dynamics. A detailed analysis of the pre-transitional state will be left for future investigations of the dynamics of the system. The isotropic–cholesteric phase transition is observed at pressure Pz* = 0.10. A series of snapshots taken from the NPzT MD trajectory are shown in Fig. 4b. After 4 × 107 MD steps, local nematic clusters are clearly visible. The last configuration of the NPzT simulation was used for the NVT simulation with wall confinements. After 1 × 108 steps the nematic clusters are merged and chiral organization of FCh molecules is seen throughout the box. Additional 1.6 × 108 step simulations lead to the cholesteric phase with a stable bulk chiral pitch. Since the simulation box is sufficiently large, we can clearly see the equilibrium cholesteric state with a full chiral pitch in the simulation box.

image file: c7sm02077e-f4.tif
Fig. 4 (a) Snapshot of the pre-transitional state at Pz* = 0.095. (b) Molecular dynamics trajectory of the formation of the cholesteric phase under normal pressure Pz* = 0.10 of FCh molecules with chain length Nb = 10 and bending constant Kangle = 100 at constant temperature T* = 1.0.

4.2 Structure of the cholesteric phase

Using the wall confinements, the bulk cholesteric states corresponding to different normal pressures were studied. In Fig. 5, the number density profiles ρ*(z) and the nematic order parameter profiles S2(z) are presented. The density profiles exhibit moderate fluctuations (<3%) around the value of the bulk average indicated by the dashed lines in Fig. 5a. The surface wetting behaviour of FCh molecules near both walls is also observed. The wall effect on the bulk phase behavior is negligible in the system since the simulation box in the z-axis is large enough for a clearly distinct bulk phase formed in the central region of the system.
image file: c7sm02077e-f5.tif
Fig. 5 (a) Molecular number density profiles ρ*(z) of the systems obtained at Pz* = 0.09, 0.1, 0.15 and 0.18. The dashed lines correspond to the averaged bulk densities. (b) Nematic order parameter profiles S2(z) where the z-coordinates are scaled by the box length in the z-axis, and the values of S2(z) for Pz* = 0.15 and 0.18 are offset by 0.2 and 0.4 respectively for data clarity.

The comparison of the nematic order parameter profiles for different pressures is given in Fig. 5b showing different patterns in isotropic and cholesteric phases. For Pz* = 0.09 the highest-density isotropic state, the bulk region of the system shows the low value of S2(z) ∼ 0.17 indicating a disordered state. The nematic order profiles S2(z) for Pz* = 0.15 and 0.18 are shifted by 0.2 and 0.4, respectively, for data clarity. The S2(z) profiles in the cholesteric phase exhibit periodic fluctuations around a mean value throughout the simulation box. The periodicities in S2(z) profiles PS2 vary depending on the normal pressure applied, which is defined as the distance between two neighbouring minima or maxima in S2(z). The periodicity profile PS2 under Pz* = 0.1–0.18, along with the maximum, average and minimum values for S2(z) in the bulk region is shown in Fig. 6. The periodicity PS2 decreases from 32σ to 15σ as Pz* increases. The values of maximum, average and minimum in S2(z) profiles are slightly increased upon increasing normal pressure.

image file: c7sm02077e-f6.tif
Fig. 6 (left axis) The average periodicity of nematic order parameter profiles PS2 (right axis), which is defined as the distance between two minimum or maximum peaks of the S2(z) profile in the bulk.

The inhomogeneous distribution of S2(z) in the bulk suggests the spatial correlation of molecular orientations along the helix axis and the local nematic director [n with combining circumflex](z) is perturbed by the orientational distributions in neighbouring bins. The system in the cholesteric phase shows fluctuations between a less ordered domain (Smin2b) and a higher order region (Smax2b). The evidence of inhomogeneous distributions of S2(z) reveals that local uniaxial nematic approximation adopted in theoretical modelling of cholesteric material should be an effective assumption. A classical DFT based on the Onsager-Straley approach71 provided a detailed analysis of the effect of the density gradient on chiral twist and it was found that a weak density gradient imparted by an external field results in the non-uniform twist profile of the cholesteric phase by density-orientation coupling. Our simulation data show that the orientation inhomogeneity can appear in the bulk cholesteric phase. It is considered that the oscillation in S2(z) is due to the local helical twist of the director field. Further analysis of the oscillatory S2(z) profile is provided in Section 4.3.

The equilibrium cholesteric pitch P is estimated by fitting the twist angle profile to the function cos[thin space (1/6-em)]α0(z) = |[n with combining circumflex](z[n with combining circumflex](z = 0)| = |cos(2πz/P)|. The method has been shown to be reliable and convenient for calculation of the microscopic chiral pitch.35 We fit |cos(2πz/P)| to the averaged profiles cos[thin space (1/6-em)]α0(z) = |[n with combining circumflex](z[n with combining circumflex](z = 0)| based on 15[thin space (1/6-em)]000 configurations from the last 107 steps of NVT simulations of three independent MD runs to obtain the average cholesteric pitch P as shown in Fig. 7a. It indicates that the molecules at both ends of the simulation box adopt different orientations, and the local nematic director exhibits periodic oscillations along the z-axis, which correspond to the helical organization. The molecular orientational correlation functions (eqn (6) and (7)) are also plotted in Fig. 7b to identify the cholesteric structure. The system shows different patterns of orientational correlation under varying normal pressure. In an isotropic state of Pz* = 0.09, both S220û(z) and S221û(z) reduce to vanishing and the system shows no chiral ordering. In the uniaxial nematic phase, S220û(z) is a constant due to the orientational alignment between molecules, while in the cholesteric state of FCh fluids, the distortion of molecular alignments along with the helix axis (z-direction) is responsible for the oscillations in S220û(z). S221û(z) in Fig. 7a confirms that the cholesteric structure of FCh molecules has left-handedness.

image file: c7sm02077e-f7.tif
Fig. 7 (a) Twist angle profile cos[thin space (1/6-em)]α0(z) = |[n with combining circumflex](z) = [n with combining circumflex](z = 0)|. Symbols represent three independent MD trajectories and the red continuous line is the fitted function |cos(2πz/P)| used to estimate the macroscopic cholesteric pitch P. (b) Orientational correlation functions (eqn (6) and (7)) for the FCh molecules under varying normal pressure.

The equilibrium cholesteric pitch is plotted with density in Fig. 8. In the density range of interest, the pitch monotonically decreases with the density and the fitting to the scaling relation P ∼ (ρ*)β gives β = −1.37 which is qualitatively consistent with exponents obtained in theoretical predictions (β = −1),24,43 flexible helix (β = −1.67),24 virus suspension (β = −1.45)12 and PBLG (β = −1.8).19 Because of the density range considered in the study, ρ* = 0.04–0.06 corresponding to packing fractions 0.21–0.31, the chiral sense inversion is not observed in our simulations.46,47,72,73

image file: c7sm02077e-f8.tif
Fig. 8 Density dependence of cholesteric pitch P fitted to bulk density using P ∼ (ρ*)β, the resulting exponent β = −1.37.

4.3 Molecular flexibility

We studied the effect of molecular flexibility on the isotropic–cholesteric phase behavior of FCh molecules. In the FCh CG model, the molecular flexibility is described by the force constant in the bond angle term of backbone CG beads (See Fig. 1 and eqn (1)). FCh models with the reduced force constants, Kangle = 75, 50, were simulated with the rest of the model parameters (chain length Nb = 10, internal chirality ϕ = 30°) being identical to the previous section.

In Fig. 9a, the isotropic–cholesteric equations of state of FCh molecules with varying flexibilities (Kangle = 100, 75, 50) are compared. With the increasing chain flexibilities, the transition from the isotropic to cholesteric state is slightly postponed to higher density. The shifting implies that the local orientational ordering is destabilized by increasing molecular flexibilities which results in less anisotropy in the excluded volume of FCh molecules.41 Such an observation is further confirmed by the density dependence of the system nematic order parameter S2 as shown in Fig. 9b. The state of FCh molecules with Kangle = 50 at Pz* = 0.1 remains isotropic and the onset of isotropic–cholesteric transition is located between Pz* = 0.11–0.12. In addition, the biaxiality of the cholesteric phase is quantitatively related to molecular chain flexibilities, as shown in Fig. 9c. The biaxiality increases with increased density (pressure), while FCh molecules with a comparatively rigid backbone (Kangle = 100) exhibit higher biaxiality. This demonstrates the molecular flexibility affects the isotropic–cholesteric phase transition and the local orientational order as well.

image file: c7sm02077e-f9.tif
Fig. 9 (a) Isotropic–cholesteric phase behavior of FCh molecules with varying molecular flexibilities. The results belonging to FCh with Kangle = 75, 50 are offset by 0.05 and 0.1 in the Pz*-axis respectively for clarity. (b) Orientational order parameter for FCh molecules with varying molecular flexibilities. The data belonging to FCh with Kangle = 75, 50 are offset by 0.1 and 0.2 in the S2-axis, respectively, for clarity. (c) Density dependence of the biaxiality parameter (Δb) of the cholesteric phase of FCh molecules with varying molecular flexibilities.

We also examined the effect of molecular flexibility on the structure of the cholesteric phase formed using the FCh model. Snapshots of FCh molecules of varying flexibilities under the normal pressure Pz* = 0.18 are illustrated in Fig. 10a. The equilibrium cholesteric pitches P are extracted by fitting |cos(2πz/P)| to the average twist angle profile |cos[thin space (1/6-em)]α0(z)| as shown in Fig. 10b. It is clear from the snapshots and twist angle profiles that high molecular flexibility produces a larger chiral pitch. It indicates that the backbone flexibility of FCh molecules weakens the local twist which leads to a large chiral pitch. In the cholesteric phase, FCh molecules with varying flexibilities also show the inhomogeneous distribution of the nematic order parameter S2(z) along the helix director, as shown in Fig. 10c. The amplitude of the S2(z) oscillation becomes narrow when the molecular flexibility (Kangle) increases, and it shows that the oscillatory S2(z) profile is affected by the local twist of the director field (see Fig. S3 in the ESI). The periodicity PS2 is progressively expanded with increased molecular flexibility which is consistent with the trend that the cholesteric pitch is increased with higher molecular flexibility. The molecular flexibility then becomes an important factor which can be used to fine tune the detailed structure of cholesteric phases.

image file: c7sm02077e-f10.tif
Fig. 10 (a) Snapshots of FCh molecules of varying flexibilities Kangle = 100, 75, 50 under normal pressure Pz* = 0.18, with molecules that are colour coded corresponding to their orientations. (b) Twist angle profile |cos[thin space (1/6-em)]α0(z)| = |[n with combining circumflex](z[n with combining circumflex](z = 0)|. (c) Nematic order parameter S2(z).

The density dependence of the cholesteric pitch of flexible FCh systems is plotted in Fig. 11 to examine the scaling behavior of the chiral pitch. The obtained chiral pitches for different densities are correlated with density using the power law P ∼ (ρ*)β. As indicated in Fig. 11, the exponent β decreases with the increasing molecular flexibilities. The obtained exponent is β = −1.56 for the FCh molecules of Kangle = 50, β = −1.47 for the system of Kangle = 75, and β = −1.37 for the most rigid FCh molecules considered. The trend shows that the density dependence of cholesteric pitch does not obey a simple power law in which the exponent is quasi-universal.24,34 We also notice that the statistical error for more flexible FCh systems becomes large, particularly in the lower density cholesteric states in which longer relaxation times are required, and the density dependence of the cholesteric pitch P varies upon molecular flexibility. Our simulations support that molecular flexibility plays an important role in determining the detailed structure of the cholesteric states of FCh molecules.

image file: c7sm02077e-f11.tif
Fig. 11 Density dependence of the cholesteric pitch for FCh molecules with varying molecular flexibilities, Kangle = 100, 75 and 50. Symbols correspond to simulation results and dashed lines are fitted to P ∼ (ρ*)β. The fitting parameter β is plotted against molecular flexibilities Kangle in the inset.

5. Conclusions

In this work, we introduce the FCh molecule as a CG representation of cholesteric LCs. In FCh molecules, the repulsive polymeric backbone chain is combined with side interaction beads which interact via LJ potential, and the molecular flexibility and internal chirality are described by bonded interactions of the FCh molecules. We perform MD simulations of cholesteric liquid crystalline ordering of the FCh model. The isotropic–cholesteric phase behavior of FCh molecules, with varying normal pressures and molecular flexibilities, is studied.

The LC phase behavior of FCh molecules under various pressures shows that high pressure facilitates the formation of a cholesteric state, which is in line with previous studies.46,47 The structure of the bulk cholesteric phase is analysed using a wall confined simulation box in order to break the symmetry along the helix director. FCh molecules show left-handed cholesteric phases in the system when the FCh molecules have right-handed internal chirality. The bulk cholesteric state of FCh molecules shows the inhomogeneous distribution of the nematic ordering parameter profile S2(z), which is thought to be related to local density fluctuations. In developing theoretical descriptions for cholesteric LCs, a local uniaxiality assumption is commonly adopted that the system is locally approximated as the uniaxial nematic state so that the local orientational distribution is decoupled from the coordinates along the cholesteric helix director. The simulation of the FCh model shows that the spatial inhomogeneity of the local orientational distribution exists in the cholesteric LC system. This implies that a feasible method to consider the inhomogeneous orientation distribution is to express the orientational distribution as a function of not only the relative orientation between two LC molecules and also the spatial variation of the local orientational distribution.

The FCh model allows us to study the effect of molecular flexibility on the LC phase behavior and the structure of the cholesteric phase. Simulation results show that high molecular flexibility destabilizes the local orientational order and leads to a shift of the isotropic–cholesteric transition to a high-density region in the system of flexible FCh models. The simulation provides evidence that the equilibrium chiral pitch becomes increased for systems of flexible FCh molecules. We also show that the density dependence of FCh systems does not obey a scaling law P ∼ (ρ*)β with a universal exponent β. The exponent is dependent on molecular flexibility. This idea will provide a guide for the rational design of cholesteric LC materials to control the structure of the cholesteric state by introducing flexible segments such as alkyl groups into the molecular structure.

The FCh model and the simulation in this study place an emphasis on understanding the cholesteric phase behavior from the molecular perspective and the efforts here are devoted to correlate macroscopic chirality with molecular flexibility. It should be stressed that the adoption of the LJ-type potential does not restrict the applications of the FCh model. The model is highly versatile for both thermotropic and lyotropic cholesteric LC when other interactions are taken into consideration, such as columbic forces and dipolar interactions. Such chemical modifications that interaction beads are decorated on a segment of the FCh model would lead to even more rich LC phase behavior.

In the future work, we will focus on the elucidation of the inhomogeneity of the nematic order parameter profile by careful measurement of the twist elastic constant and torque along the helix director, as well as the investigation of the structure of cholesteric phases in high densities.47 It is of particular interest to study the influence of molecular flexibility on chiral sense inversion.72 Since chiral sense inversion occurs at high densities in which the wall confinement results in a dramatic change in the orientational and positional order, some alternatives such as a torque-based technique26,47 are needed. We also plan to theorize the FCh molecular model in the framework of the classical density functional approach,34,36,43,45,48,71,74 by which a wide range of molecular parameters can be explored to present the global phase behavior of the FCh model and sensitivity of the cholesteric pitch.

Conflicts of interest

There are no conflicts to declare.


This work was funded by the National Natural Science Foundation of China [grant number 21473112], [grant number 21403138], [grant number 21673138], [grant number 21603141], [grant number 21604054]; National Basic Research Program of China (973 Program) [grant number 2014CB239702], and China Postdoctoral Science Foundation [No. 2017M611541]. This paper is partially supported by the Materials Genome Initiative Centre, Shanghai Jiao Tong University. The computational resources are supported by the High-Performance Computing Centre of Shanghai Jiao Tong University.

Notes and references

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Electronic supplementary information (ESI) available. See DOI: 10.1039/c7sm02077e

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