Prediction of Optical and Dielectric Properties of 4-Cyano-4

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Prediction of Optical and Dielectric Properties of 4-Cyano-4-pentylbiphenyl Liquid Crystals by Molecular Dynamics Simulation, Coarse-grained Dynamics Simulation, and Density Functional Theory Calculation Shin-Pon Ju, Sheng-Chieh Huang, Ken-Huang Lin, Hsing-Yin Chen, and Ting-Kai Shen J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b12222 • Publication Date (Web): 17 Jun 2016 Downloaded from http://pubs.acs.org on June 18, 2016

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The Journal of Physical Chemistry

Prediction of Optical and Dielectric Properties of 4-Cyano-4-pentylbiphenyl Liquid Crystals by Molecular Dynamics Simulation, Coarse-grained Dynamics Simulation, and Density Functional Theory Calculation Shin-Pon Ju*a,b, Sheng-Chieh Huanga, Ken-Huang Lina, Hsing-Yin Chenb, and Ting-Kai Shena

a

Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan 804

b

Department of Medicinal and Applied Chemistry, Kaohsiung Medical University, Kaohsiung, Taiwan 807

ACS Paragon Plus Environment

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Abstract The Maier-Meier theory and Vuks theory were used to obtain the dielectric anisotropy and birefringence of the 4-cyano-4-pentylbiphenyl (5CB) liquid crystal system for a benchmark. The molecular dynamics (MD) simulation, coarse-grained molecular dynamics (CGMD) simulation, and the density functional theory (DFT) calculation were used to get the required parameters of the Maier-Meier theory and Vuks theory. The molecular density obtained by the MD simulation is about 1.05 g/cm3, which is very close to the available experimental value of 1.008 g/cm3. In the CGMD simulation, the order parameter is about 0.44 for the 5CB conformation at the temperature that the transformation from the nematic to disorder arrangement occurs. The polarizability (), polarizability anisotropy (∆), and dipole moment () of a 5CB molecule were obtained by the DFT calculation with the functional of B3LYP/6-31+G (2d,p). With these parameters, the birefringence of 5CB by the Vuks theory is very close to the experimental value with the error under 1%, but the dielectric constant and dielectric anisotropy by the Maier-Meier theory display the considerable errors higher than the corresponding experimental data by 218% and 260%, respectively. With the further concern about the Kirkwood’s correlation factor for the effective dipole moment, the dielectric constant and dielectric anisotropy errors decrease from 218% to 12% and from 260% to 2%, respectively. From these results, it shows the dielectric anisotropy and the birefringence of a liquid crystal system can be accurately predicted by the Maier-Meier theory and Vuks theory with the parameters from the MD, CGMD, and DFT calculation as well as considering the effective dipole moment.

Keywords: Molecular dynamics simulation, Coarse-grained molecular dynamics simulation, DFT calculation, Maier-Meier theory, Effective dipole moment


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Introduction Liquid crystals (LCs) have attracted great interest for their potential applications in micro displays 1 and focusing systems 2, and the main aim of them is to increase optical efficiency. Since the orientation of liquid crystal is highly sensitive to the external field and the environment, LCs can be used as a transducing element in various sensors or biosensors 3. In bio-technology, liquid crystals can also be used to reproduce the effects of naturally occurring lens systems 4-6. The focal length of lens is tuned by means of an electric field. As the voltage decreases, the liquid crystal lens display a shorter focal length. This application of liquid crystals could also lead to innovations in contact-len materials. Moreover, the liquid crystal emulsion possesses the stability as well as controlled release and moisturizing properties, and demonstrates better application performance than conventional emulsion systems7. To realize possible applications of liquid crystals, their material properties have been extensively studied by the experimental methods. For example, Holstein et al. 8 investigated the diffusion coefficient of 5CB in the nematic phase by the fundamentals of pulsed field gradient-nuclear magnetic resonance (PFG-NMR). Gwag et al. 9 studied the polar anchoring energy between LC molecules and polyimide films. They found the pre-tilt angle of liquid crystals depends on the orientation and thickness of the polyimide films. Somma et al. 10 explored the orientation dynamical behaviors for oligofluorenes (OFs) by depolarized dynamic light scattering (DLS) and dynamic NMR spectroscopy. Mercuri et al. 11 studied the strain effects on n-hexyl-4'-n-pentyloxy biphenyl- 4-carboxylate , and it was found that the strain in the sample depends on the formation kinetics of sematic phase once the sample is cooled from the isotropic phase. Filpo et al. 12 examined the electro-optical and the morphological properties of reverse-mode polymer-dispersed liquid crystal (PDLC) films. Inoue et al.

13

investigated the microsecond electro-optic response of an

anisotropic-polymer/LC composite. The results showed that the alignment of polymer matrix is retained when an electric field is applied to the nanocomposite. In a space smaller than the wavelength of visible light, the reorientation of nematic liquid crystal molecules results in the scattering-free properties over the entire visible wavelength range and a short decay response time. Although the steady-state properties of liquid crystals can be observed by the experimental approach, there are some difficulties in these experimental methods. For instance, the detailed dynamical behaviors of


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liquid crystal molecules are difficult to inspect because of the resolution limitations of the experimental equipment and the small scale involved. Computer simulation methods can be used to retrieve the information beyond the experimental limitations and provide dynamic behaviors of liquid crystal molecules at the atomic level. The molecular dynamics (MD) simulation is a powerful tool facilitating the investigation of detailed molecular behaviors. As to this aspect, McDonald and Hanna

14

used MD

simulations to investigate the appearance of a liquid crystal fluid wetting a surface, which shows the wetting types significantly depend on the interaction energy. Capar and Cebe

15

examined the order parameter,

rotational viscosity coefficient, and rotational diffusion coefficient of liquid crystals for different alkyl chain lengths by the MD simulation. From their simulation results, the calculated properties were compared with the experimental data and the satisfactory results were obtained. Mirantsev and Virga

16

used MD

simulation to study the twisted behaviors of the LC system in the nematic phase, which was confined between two bounding substrates. Although all atom MD simulation is a powerful approach for clearly understanding the dynamic behavior at the atomic level, for an enormous polymeric system, all atom MD simulation has the possibility of becoming trapped at some local energy minima and prevented from further structural evolution. Using this method is also difficult because a sufficient time is needed to allow the system for relaxation of chain motion when the initial polymer molecules were distributed randomly for avoiding the artificial arrangement. Accordingly, several coarse-grained methods have been developed to overcome these limitations of all atom MD simulation. Adopting this technique, Bates

17

adopted the bond fluctuation

model to find the relationship between the phase behavior and the stiffness of large flexible liquid crystals. Cifelli et al.

18

examined the order parameter and the diffusion coefficient of liquid crystals using the

elementary liquid crystal model. Lin et al. 19 employed the rod-like molecular model to investigate the order parameter and phase behavior of lyotropic liquid crystal solution. Zhang et al.

20

established a

coarse-grained model for the 4-cyano-4-pentylbiphenyl (5CB) molecule, and their results demonstrate that both the phase transition temperature from nematic to isotropic phase as well as the diffusion coefficients are in good agreement with the experimental results. In addition, the resultant 5CB CG force field exhibits a good transferability to the extent simulation model

21

. Mukherjee et al.


4

22

investigated the translocation

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mechanisms in sematic LCs for a compound comprised of a stiff azobenzene core with flexible tails. Their results reveal that the molecules can permeate from one sematic layer to the next by two different mechanisms, with and without significant reorientation. In our previous research 23, we have examined the size and chain length effects on structural behavior of a biphenylcyclohexane-based liquid crystal (BCH5H), and carried out the configurational-bias Monte Carlo (CBMC) simulation by the link-vector method. According to the papers reviewed above, it can be seen that the main advantage of CG method is to predict the equilibrium arrangement of macromolecule system by the CG beads, which correspond to a certain groups of atoms. Predictions of dielectric constant, the optical refractive indices, and the anisotropy of LCs have been conducted by adopting empirical equations with some parameters obtained from the quantum chemical calculation. For example, Demus 24 employed the Maier-Meier and Vuks theories to calculate the dielectric anisotropy and the birefringence for a 5CB LC system. The quantum chemical method implemented in the MOPAC package

25

was adopted to obtain the dipole moment, polarizability, and polarizability anisotropy

of 5CB system and those substituted with different functional groups. Although the calculated results of the dielectric anisotropy and the birefringence by Maier-Meier and Vuks theories were in good agreement with the corresponding experimental values, two required parameters for the Maier-Meier theory, system density and order parameter, were taken artificially from the empirical values for different LC molecules. Labidi and Djebaili

26

stated the DFT calculation results are more accurate than those from the semi-empirical

calculation, and the correlation coefficients between the DFT values and the experimental ones were excellent statistically. In Saitoh’s study 27, anisotropy of polarizability, dipole moment and angle between dipole moment and the main axis of polarizability of a liquid crystal molecule were obtained by the empirical quantum chemical calculation. By the Maier-Meier theory with the parameters from the quantum chemical calculation and the empirical order parameter value of 0.7, the dielectric anisotropy is in agreement well with the experimental values. Yokota 28 investigated

the dielectric anisotropies of several

LC systems including 2CB, 4CB, 5CB, 5BP1, 6PA3 by the empirical quantum chemical calculations and the Maier-Meier theory using the empirical value of 0.7 for the order parameter. They found the overestimated values were often obtained in the polarizability calculations. As these studies demonstrate,


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while good agreement with experimental values can sometimes be obtained even when using substituted artificial values for certain density and order parameter into the Maier-Meier equation and Vuks equation, precise calculations of these two parameters without using the empirical values are necessary to improve the agreement with the related experiment results. In the present study, the optical properties and dielectric anisotropies of a LC system were predicted theoretically using the Maier-Meier equation and Vuks equation. The required parameters for these two equations were obtained by several different simulation methods including the MD simulation, CGMD simulation, and the DFT calculation. Beside predicting the system density, the MD simulation was used for preparing the reference bond length and bending angle distributions as well as the radial distribution profiles for the CG potential parametrization. The coarse-grained molecular dynamics (CGMD) simulation was used to get the order parameter of the LC system at different temperatures. Additionally, the polarizability, polarizability anisotropy, and dipole moment were determined by the DFT calculation. The parameters obtained by the abovementioned theories were then substituted into the Maier-Meier theory and the Vuks equation to get the optical anisotropy and the dielectric anisotropy of a LC system. Since 5CB is the earliest synthesized and the most popular liquid crystal molecule, and has much available experimental data, the LC system of 5CB molecule was chosen as the benchmark example for the simulation procedure.

Theoretical Background The optical anisotropy was derived by the Vuks equation determined by the Maier-Meier theory

30

29

and the dielectric anisotropy was

. Before using these two equations, some required parameters

should be obtained by several simulation methods in advance (i.e., using MD for the density of 5CB, coarse-grained molecular dynamics (CGMD) for the order parameter, and quantum mechanics calculation for molecular polarizability). The following is a brief description of the present simulation procedure. By extending the Onsager theory used originally for an isotropic system

31

, the Maier-Meier equation

was used to calculate the dielectric anisotropy by considering the anisotropy of the molecular polarizability, the orientation of the permanent dipole moment of a molecule, and the order parameter derived from the average angle between the mean director and the long-axial directions of all dipole molecules in the system.


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This equation has been successfully applied to predict the dielectric polarization of the nematic liquid crystal molecules 24, 29. The formula for dielectric constant ε is shown as follows: ℎ

ε − 1 = + 3

(1)

The dielectric anisotropy ∆ε defined by the difference between the dielectric constants parallel and perpendicular to the molecular axis is formulated as:

∆ =

∆ −

!

(1 − 3 (cos &) )( )

(2)

where N is the molecular number density; is the vacuum permittivity, which is defined as 8.854×10-12 (C2/J·m); kB is the Boltzman constant; T is absolute temperature; µ is dipole moment of the dipole molecule; S is the long range orientation order; and β is the dipole moment orientation angle relative to the long principal axis of the molecular frame. In addition to these parameters, ∆α is the polarizability anisotropy and α is the polarizability of a molecule. These two parameters are defined as follows: α=

+,, -+.. -+//

∆α = 11 −

(3)

0

2+.. -+// 3

(4)

where 11 is the molecular polarizability parallel to the molecular long principal axis, and 44 and 55 are the molecular polarizabilities perpendicular to the molecular long principal axis. In Eqs. (3) and (4), the reaction field factor F and cavity factor h are listed in Eq. (5) and Eq. (6), with the internal parameter f also determined in Eq. (5): 7

F = (789+)

(5)

0

h = -7

(6)

As shown in Eq. (7), the formula of f in Eq. (5) is: f = ( = )(-7)

(7)

where a represents the radius of the spherical cavity in the Onsager theory model. In the optical properties of liquid crystal calculations, the optical anisotropy equation relating molecular polarizability to refractive index was proposed by Vuks 29. The effective molecular polarizability


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is assumed by the rotationally averaged polarizability of a single molecule. The Vuks equation is shown as follows: ? 87

? - = 0 @ − ? 87

?C - = 0 @ +

∆+A 0

∆+A 0

(8)

B

B

(9)

where the parameters of DE and D are the principal refractive indices along the extraordinary and ordinary ray, respectively. The birefringence ∆n is defined as the difference between nE and D , and

D , DE , and D can also be correlated by the following equation: D = (DE + 2D )/3

(10)

For the Maier-Meier equation, there are eight required parameters ( , , ℎ, ∆, , , &, ) ) for determining the dielectric anisotropy ∆ε, and five parameters are necessary to derive the birefringence ∆n for the Vuks equation. Table 1 summarizes the approaches of some previous researchers to artificially use some parameters for the Maier-Meier equation, as well as our approaches to calculate these parameters without using any artificial values. In Refs. 22 and 30, the semi-empirical quantum chemistry method for AM1 was used

24, 32

, and by this method the electronic properties of 5CB including the dipole moment, the

polarizability tensors (11 , 44 , 55 ), and the polarizability anisotropy ∆α are not accurate enough when compared to those obtained by DFT calculations using either B3LYP/6-31G(d)

32

or the B3LYP/6-31+G

(2d,p) used in this study. Except for the electronic parameters, previous studies usually used 0.7 and 1000 kg/m3 for the order parameter and the LC molecule density without any experimental or simulation evidence

24

. In the current study, CGMD was used to monitor the variation of 5CB order parameter with

increasing temperature and then obtain a reasonable order parameter when the phase transfers from the nematic state to the disordered phase. Furthermore, the density of 5CB was also predicted by the MD simulation instead of using the artificial value of 1000 kg/m3. The goal of this study is to establish a process for predicting the LC properties by the Maier-Meier equation and the Vuks equation with all parameters from reasonable simulation results.


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Quantum chemical calculation The polarizability tesnsor for obtaining 11 , 44 , 55 as well as the dipole moment µ were calculated by the DFT calculation implemented by GAUSSIAN09 33. The exchange-correlation interaction was treated within the B3LYP using the basis set of 6-31+G(2d,p), which can support the conclusion that such hybrid functionals can help determine vibrational force field, frequencies, and spectra, as well as thermochemical properties. Therefore, B3LYP can predict more accurate results than those from the AM1 semi-empirical calculation used by several previous studies

24, 26-28, 32

. The most stable geometry of 5CB was determined o

first. During the geometry optimization, the convergent conditions were set as 0.001 A for the atomic displacement, 10-6 eV/atom for the energy change, and frequency of 589 nm for the plane-wave cut off energy27.

Parametrization of CG potentials It is very time-consuming to obtain the ordered arrangement of liquid molecules from a random initial molecular distribution by the all atom simulation model, so an alternative way is to use the coarse-grained model. The structural-based coarse-grained model was mainly employed to obtain the order parameter of 5CB molecules. The structural-based CG model of a 5CB molecule is shown in Fig. 1, and consists of five coarse-grained beads labeled by CN, C62, C61, C3, and C2. Each CG bead center corresponds to the mass center of its corresponding fragment of the 5CB all atom model. All CG beads were used as the interactive sites for calculating the non-bonded interaction energy of two non-successive beads of the same chain and beads of different chains. The bead charges shown in Fig. 1 were calculated by the atomic charge summation of the corresponding all atom model fragment modelled by the COMPASS force field 34. The bonded potentials for bond stretching and bond bending can be determined by the inverse Boltzmann relation [5], which can be written as: S(T) = −U VD(W(T))

(11)

where z is the bond length or the bending angle, and P(z) indicates the probability distribution of the lengths and angles. The term kB is the Boltzmann constant and T is the system temperature. These


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probability distributions are obtained by the all atom MD simulation with the condensed-phase optimized molecular potentials for atomistic simulation studies (COMPASS) potential. COMPASS possesses the characteristic of high precision and ability to adapt to a larger and more complicated molecular model. The bonded potentials for bond stretching and bond bending obtained by the inverse Boltzmann relation were finally modeled by a harmonic potential with the equilibrium length and angle as follows: Kb ( R − R0 )2 2

(12)

Eb =

Ka (θ − θ 0 ) 2 2

(13)

Ea =

For the non-bonded interaction energies between beads, the iterative Boltzmann inversion (IBI) approach was also used to generate the successive corrections to the initial CG potential based on deviations between the RDFs of the all atom model and the CG model. In the present study, the target RDF is treated as the initial guess energy, which is taken to be the Boltzmann inversion. The Boltzmann inversion can be expressed by

X (Y) = −U ln([∗ (Y))

(14)

The numerical non-bonded potential was iteratively updated for obtaining more realistic potential energy between beads; the procedure was continuously iterated until the reproduced RDF matched the target RDF. In other words, a valid solution can be determined when the algorithm converges. An iterative scheme [31] was used during CGMD simulation, and is shown as the following: Vi +1 ( r ) = Vi ( r ) + k BT ln

(15)

g i (r ) g (r )

where the g(r) is the target RDF from fully atomistic simulation, and gi(r) is the RDF calculated with the potential Vi(r) in the ith iteration during CGMD simulation. The non-bonded potentials obtained by the above iteration were finally modelled by the following 12-6 Lenard-Jones (LJ) forms: E=D0 [ @ 0 B -2 @ 0 B ] R

R

12

R

6

(16)

R


10

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where D0 is the energy parameter of this LJ potential; R is the calculated distance between two CG beads; R0 is the distance, at which two CG beads forms the lowest energy.

Atomistic MD simulation For preparing the reference distributions of bond lengths and bending angles as well as the target RDFs, the fully atomistic MD simulation was carried out by the Forcite package 35 with the COMPASS force field. A system of 0.9 g/cm0 having 100 5CB molecules was established in the nematic liquid crystal phase by the amorphous cell package of Accelrys Materials Studio 5.5. The long axis of each 5CB molecule was randomly arranged along the x direction with the cone angle ranging from 0 to 10 degrees. Then a 2x2x2 supercell with 800 5CB molecules was used for the NPT MD simulation with the initial temperature of 1000 K and the pressure of 1 atm for the simulated-annealing procedure with the cooling rate of 10K per 3 ps. The temperature and the pressure were controlled via the Nosé-Hoover-Langevin (NHL) thermostat

36

and Souza-Martins barostat 37, respectively. When the temperature of simulated-annealing process reached 300 K, another 500 ps was used to relax the system and data were sampled over the following 500 ps.

Coarse-grained Molecular dynamics simulation The CGMD simulation was conducted to find the ordered LC arrangement and the corresponding order parameter. At the beginning of the simulation, all 5CB CG molecules were randomly arranged within the simulation box. The simulation box is 70Å × 67Å × 66Å, and periodic boundary conditions were imposed in the x, y, and z directions. CGMD simulation was carried out by using the Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS)

38

with the time step length of 15 fs. The simulation was carried

out in the Isothermal–isobaric (NPT) ensemble at the temperature at 300 K and the pressure of 1 atm. The Ewald method was used for calculating the long-ranged Columbic interaction.

Results and discussion


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After the CG parametrization procedure for the 5CB, the CG parameters for bond stretching and bending as well as the LJ non-bonding parameters are listed in Tables 2, 3, and 4, respectively. Using these parameters, the predicted bond length and bending angle distributions as well as the RDF profiles by the all atom and CG models can be seen in Figs. 2, 3, and 4. The CG system with 800 5CB molecules was simulated for 2 ns and the last 1000 ps was used for sampling the required data. In Figs. 2 and 3, it can be seen that the CG bond length and angle distributions generally match the corresponding distributions of all atom model. For the RDF profiles, only the pairs of the same CG type are shown in Fig. 4. One can see the characteristic distances of the first peaks of all atom and CG results are basically close to each other. On these comparisons between the all atom and CG results, the CG model taking the parameters from the current parametrization can basically reflect the 5CB geometrical characteristic of all atom model. The optical properties and dielectric anisotropies were determined by the Maier-Meier equation, and the required parameters for Maier-Meier equation were obtained by the all atom MD simulation, CGMD and the quantum calculations. The molecular density was estimated by the same MD simulation for preparing the reference data to parametrize the CG bonded and non-bonded potential parameters. When the system achieved equilibrium, the molecular density approximately equaled 1.05 g/cm0 , close to the experimental result of 1.008

g/cm0 39.

Fig. 5(a) shows the initial arrangement of 800 5CB CG molecules, which are randomly arranged in the simulation box. After the equilibrium process for 2000 ps at the NPT ensemble, the configuration forms an ordered LC arrangement, as shown in Fig. 5(b). In order to understand the molecular arrangement within the 5CB LC, the orientaional order parameter was used. The orientaional order parameter, P2, is expressed by 15

P2 =

3 cos2 θ − 1

(17)

2

where P2 is the second-rank order parameter common used in the analysis of polymers, and the value for cos is written as


12

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cos θ = u ⋅ n

(18)

where u is a unit vector representing the long axis direction of a certain 5CB CG molecule in the CG system and can be calculated from the eigenvector corresponding to the smallest eigenvalue of the moment of the inertia tensor for the specific 5CB CG molecule. The unit vector n stands for the director of the CG system and can be found by diagonalizing a second-rank ordering tensor Q of the system 15. From the definition of the orientational order parameter, Eq. (17), the CG system tends to be isotropic (in a disordered conformation) as P2 approaches zero, whereas it behaves like a crystal (in an ordered-arranged conformation) as P2 approaches unity. During the simulation, the determination of orientaional order parameter of 5CB reveals that the conformation gradually forms an ordered structure from the initial random arrangement, and the orientaional order parameter varies from 0.06 to about 0.6, which can be seen in Fig. 5(c). This indicates that the conformation of liquid crystals forms an ordered structure at the system equilibrium state, and the results also ensure the feasibility of the force field. As we mentioned above, the density and the orientaional order parameter in previous works were artificially taken from the empirical values for different LC molecules. The precision of order parameters often was ignored in those studies

24, 26-28, 32

. However, the orders of LCs configuration are indeed affected

by the system size. The relationship between the order parameter and system size is shown in Fig. 6, and it can be seen that the order parameters decrease with the increasing number of molecules used in the simulation system. The larger error bars explain that the LC configurations are not stable under a system size of 2000 5CB molecules at 300K, because the smaller LC configuration is more affected during the temperature equilibrium process. The minor differences are found obviously in a large system; the order parameters are distributed from 0.57 to 0.59 when the size is in excess of 4800 5CB molecules. Although only a slight error bar occurs with a system size of 13000 5CB molecules, such a large system will require excessive computational time. Therefore, the system with 8000 5CB molecules was adopted. The average density of this CGMD system is about 0.99 g/cm3, which is very close to the experimental value and that from the MD simulation, indicating our CG potential can basically reflect the interactions between the fragments of an all atom model of 5CB.


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To realize the phase behaviors of the 5CB LCs at different temperatures, the system temperature was elevated from 300 K to 400 K. Figure 7 demonstrates the value of order parameter versus system temperature. In Fig. 7, as the temperature is lower than 340 K, the order parameter does not display a significant variation, and the value of order parameter is about 0.58. When the temperature increase from 340 to 390 K, the order parameter displays a parabolic decrease from 0.58 to 0.44, indicating the system conformation remains in an ordered LC arrangement. According to the snapshots for 5CB at 300 and 360 K, one can see the 5CB arranges in a smectic phase within the temperature ranges between 300 and 390 K. At the temperature between 390 and 395, the order parameter almost remains a constant about 0.44 and it can be seen from the snapshot for 5CB at 395 K that the 5CB arrangement turns out to be in the nematic phase. When the temperature is higher than about 395 K, the order parameter has a distinct drop, and decreases from 0.44 to 0.02. This is because the 5CB molecules possess extra kinetic energy at higher temperatures, so the translational and rotational movements of 5CB molecules in the system become more significant, resulting in the decrease of the order parameter. When the temperature exceeds 400 K, the order parameter remains at almost a constant value of 0.01, indicating the 5CB molecules of LC system have transferred into an isotropic arrangement. For accurate investigation on the 5CB phase transition from the smectic to nematic phase, the translational order parameters at different temperatures are also shown in Fig. 7. The translational order parameter, τ is defined as the following equation 40:

τ = a〈exp @e

< f

B gh ∙ j〉h a

(19)

where ui is the mass center position vector of 5CB CG molecule i and the unit vector n stands for the

director of the CG system. The interlayer spacing d is the distance maximizing the translational order parameter, which is not known before the calculation. For a perfect layered structure, this value is equal to 1, whereas this value turns out to be 0 for a system without any translational symmetry. In Fig. 7, it can be seen the translational order parameter displays a distinct drop when the system temperature is higher than 360 K. At 395 K, the orientational order parameter is about 0.44, whereas the translational order parameter is very small and close to those of isotropic phases, indicating the phase at 395 K is in the nematic phase.


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The experimental clearing point of 5CB is about 308K, where there is still a difference of 80 K between the CGMD and experimental value. This is because coarse graining a 5CB benzene fragment with all atoms in the same plane into a spherical CG bead might not correctly reflect the packing of phenyl rings in the 5CB nematic phase at 300K and 1 atm. The same situation can also be seen in the previous 5CB CG study

[18]

, which used the same CG mapping scheme as the current study. The main aim of CGMD

simulation in the current study is to provide a fast way to obtain the reasonable order parameter value of 5CB in the nematic arrangement for the further calculations by Maier-Meier and Vuks equations. Once the reliable order parameter for 5CB can be obtained by the current CG model, the inconsistency of thermodynamic behaviors (such as the phase transition temperature) between our CGMD predictions and the experimental measurements has less influence on the calculation results of Maier-Meier and Vuks equations. In Fig. 7, although the variation of order parameter with the system temperature do not correspond to the experimental observation, the averaged order parameter value of 0.44 between 390 and 395 K for the 5CB nematic phase is very close to those from previous 5CB studies (0.48 by CGMD20 and 0.5 by atomistic simulation41). This value is more reliable than the artificial value of 0.7 used in previous Maier-Meier and Vuks calculations. The polarizability (), polarizability anisotropy (∆), and dipole moment ( ) of a 5CB molecule were obtained by the DFT calculation and the calculation results can be seen in Table 5. The dipole moment value is larger than those in previous studies by AM1 calculation for 5CB 24, 27-28 and the polarizability and polarizability anisotropy are also larger than previous theoretically predicted values. Since the DFT calculation does not consider the velocities of atoms, the temperature of the DFT system corresponds to 0 K. Consequently, the temperature effects on dielectric constant ε and dielectric anisotropy ∆ε were considered by Eqs. (1) and (2). Because only one 5CB molecule (in the gas phase) was considered for the DFT calculation, the orientational effect among 5CB molecules on the dielectric anisotropy ∆ε cannot be directly considered by the DFT calculation. The correction on ∆ε, which considers the orientational effect among molecules, was conducted by introducing the orientational order parameter S in Eq. (2). For the current study, it should be noted that 300 K was used in Eqs. (1) and (2) for considering the temperature


15


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Page 16 of 43

effects on dielectric constant ε and dielectric anisotropy ∆ε. The orientational order parameter S predicted

by the current CG model at 395 K was used for considering the orientational effect among 5CB molecules on the dielectric anisotropy ∆ε. After determining the required parameters, the dielectric constant (l), dielectric anisotropy (∆l), the

refraction parallel to the light axis (DE ), the refraction perpendicular to the light axis (Dm ), and birefringence

(∆D) can be derived by the Maier-Meier and Vuks theories, as listed in Table 5. In order to demonstrate the reliability of our method, these results are compared to the 5CB experimental values

42

and it can be seen

that the values of DE , Dm , and ∆D are very close to the experiment results. For and ∆ , however, these two parameters display large differences from experiment results. According to Chakraborty’s study 43, the influence of molecular–molecular correlation on the dipole moment should also be considered in order for the Maier-Meier theory to obtain more accurate dielectric constant and dielectric anisotropy values. Therefore, the correlation factor ‘g’ was introduced by the Kirkwood–Frolich theory 24, in which the short range dipole–dipole interaction was included explicitly. In general, the dielectric constant and the anisotropy of a LC system derived by the Maier-Meier theory are much larger than the corresponding experimental values if the dipole moment determined by DFT calculation for an isolated LC molecule was used. Instead of adopting the dipole moment directly from the DFT calculation, the effective dipole moments µeff was used to modify the values of and the ∆

listed in Table 5. The effective dipole moments µeff can be calculated by Kirkwood’s correlation factor[:

[ = μopp /μ

(20)

For [, the relations hold: dimers with parallel dipoles is expressed by [ > 1, dimers with antiparallel

dipoles is expressed by [ < 1, and no association is expressed by [ = 1. In order to accurately predict the

[ factor of a LC system, the Dunmur and Palffy-Muhoray theory 24 was used, and this theory is shown as

the following equations:

g∥ = 1 −

tuvw [yz (7-{)-y| (78{)]

(21)

t~(t )


16

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g = 1 −

tuvw [y| (7-{)-yz (78{)]

(22)

t~(t )

7 = 3( − )/( + 2)

(23)

where = & and = eD&, and β is the dipole moment orientation angle relative to the long

principal axis of the molecular frame. Consequently, and are the longitudinal and transverse components of the molecular dipole, respectively. kB is the Boltzmann constant and T is absolute temperature. The value of 7 depends on the length L and the breadth B of a molecule, and also depends

on the length-to-breadth ratio. The positive and negative values of 7 represent a rod-like molecule and a

discotic molecule, respectively. Terms g ∥ and g are the corresponding Kirkwood’s correlation factor for

the longitudinal and transverse directions of the molecular dipole. The values of [ , [∥ , and [ can also be correlated by the following equation:

[ = (g ∥ + 2g )/3 .

(24)

Figure 8 shows the schematic diagram for the length and the breadth of a 5CB molecule. The ellipsoid volume surrounding a 5CB molecule is equal to the volume enclosed by the Connolly surface created by the Materials Studio 5.5. The semi-major axis of this ellipsoid is the half of the longest distance between two atoms of a 5CB molecule coordinates projected into the principal axis. Then two equal semi-minor axes can be calculated by the ellipsoid volume and the length of the semi-major axis. Then the length ratio of semi-major axis to semi-minor axis is regarded as the L/B ratio for obtaining the 7 value in Eq. 23.

The 7 distribution for 8000 5CB molecules by CGMD simulation is shown in Fig. 9 and the average

value of 7 is about 1.82. By using Eqs. (21) and (22), the required parameters for determining the g factor can be determined and were listed in Table 6. It can be seen that the effective dipole moment (3.52 D) is much smaller than the dipole moment (6.32 D) obtained directly by the DFT calculation. The effective dipole moment was used again in the Maier-Meier equation to predict the dielectric constant and the dielectric anisotropy of the 5CB LC system. Table 7 lists the values of and ∆ after the effective dipole moment was used for the Maier-Meier equation. Comparing the calculated values with the experiment ones,


17


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Page 18 of 43

the errors of the dielectric constant and the dielectric anisotropy between the predicted values and the experimental data become 12% and 2%, which significantly improve the previous results. Consequently, this result indicates the dielectric properties of 5CB LCs are considerably influenced by the dipole-dipole interaction.

Conclusion Two empirical theories, Maier-Meier theory and Vuks theory, have been used to obtain the dielectric anisotropy and birefringence of the 5CB liquid crystal system. All required parameters were obtained by the theoretical approaches including MD, CGMD, and the DFT calculation. By the COMPASS force field, the MD simulation was carried out for obtaining the 5CB density. When the system achieves equilibrium, the density is about 1.05 g/cm3, which is close to experimental value of 1.008 g/cm3. By the CGMD simulation, the average order parameter value of 0.44 for 5CB in the nematic arrangement is very close to those predicted by previous 5CB studies. With the system density from MD simulation and the order parameter from the CGMD simulation as well as the polarizability (), polarizability anisotropy (∆), and dipole

moment ( ) from the DFT calculation, the birefringence of 5CB by the Vuks theory is very close to the experimental value with the error under 1%. However, the dielectric constant and dielectric anisotropy by the Maier-Meier theory display the considerable errors higher than the corresponding experimental data by 218% and 260%. By the further concern about the Kirkwood’s correlation factor, the Maier-Meier theory with the effective dipole moment can accurately predict the dielectric constant and dielectric anisotropy of 5CB system with the errors smaller than 12% and 2%, respectively. From these results, it shows the dielectric anisotropy and the birefringence of a liquid crystal system can be accurately predicted by the Maier-Meier theory and Vuks theory with the parameters from the MD, CGMD, and DFT calculation by considering the effective dipole moment.

Acknowledgements The authors would like to thank the Ministry of Science and Technology, R.O.C. under Grant No. MOST 104-2221-E-110-008 and NSYSU-KMU JOINT RESEARCH PROJECT (#NSYSUKMU105-P024)


18

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37. Souza, I.; Martins, J., Metric Tensor as the Dynamical Variable for Variable-Cell-Shape Molecular Dynamics. Physical Review B 1997, 55, 8733-8742. 38. Plimpton, S.; Crozier, P.; Thompson, A., Lammps-Large-Scale Atomic/Molecular Massively Parallel Simulator. Sandia National Laboratories 2007, 18. 39. Karat,

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-N-Alkyl-4-Cyanobiphenyls. Molecular Crystals and Liquid Crystals 1976, 36, 51-64. 40. Ji, Q.; Lefort, R.; Busselez, R.; Morineau, D., Structure and Dynamics of a Gay–Berne Liquid Crystal Confined in Cylindrical Nanopores. The Journal of Chemical Physics 2009, 130, 234501. 41. Cacelli, I.; De Gaetani, L.; Prampolini, G.; Tani, A., Liquid Crystal Properties of the N-Alkyl-Cyanobiphenyl Series from Atomistic Simulations with Ab Initio Derived Force Fields. The Journal of Physical Chemistry B 2007, 111, 2130-2137. 42. Cummins, P. G.; Dunmur, D. A.; Laidler, D. A., The Dielectric Properties of Nematic 44 ′ N-Pentylcyanobiphenyi. Molecular Crystals and Liquid Crystals 1975, 30, 109-123. 43. Chakraborty, S.; Mukhopadhyay, A., Optical and Dielectric Properties of the Mesogen 4-Cyano-4′ -N–Octyl Biphenyl as a Function of Temperature. Phase Transitions 2006, 79, 201-212.


22

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Table and figure captions Table 1 Comparisons of correlation factor (F) in the Maier-Meier equation, orientaional order parameter, and density between different 5CB LCs studies. Table 2 Parameters of bond stretching potential in the CGMD simulation. Table 3 Parameters of bond bending potential in the CGMD simulation. Table 4 Parameters of 12-6 Lenard-Jones potential. Table 5 Dielectric constant (l), dielectric anisotropy (∆l), polarizability (), polarizability anisotropy (∆), and dipole moment ( ), the refraction parallel to the light axis (DE ), the refraction perpendicular to

the light axis (Dm ), and birefringence (∆D) of 5CB molecule at 300K.

Table 6 Required parameters for obtaining the g factor and the effective dipole moment of 5CB system at 300 K. Table 7 Optical and dielectric properties of 5CB system modified by the effective dipole moment at 300K.

Figure 1 The coarse-grained model of a 5CB molecule mapped from the all atom model. C2, C3, C61, C62 and CN are the types of CG beads and the partial charges are derived by the summation of the atomic charges from the COMPASS force field. Figure 2 Probability distributions of bond lengths between the bonded beads for all atom and CG models. (a) C2-C3; (b) C3-C61; (c) C61-C62; (d) C62-CN. Figure 3 Probability distributions of bending angles for all atom and CG models at (a) C2-C3-C61; (b) C3-C61-C62; (c) C61-C62-CN. Figure 4 Radial distribution functions for all atom and CG models. (a) C2-C2；(b) C3-C3；(c) C61-C61； (d) C62-C62；(e) CN-CN.


23


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Page 24 of 43

Figure 5 (a) Initial structure of 800 5CB CG chains; (b) The equilibrium structure of 800 5CB CG chains at 300K; (c) orientaional order parameter variation of 800 5CB CG chains at 300 K during the simulation. Figure 6 Orientaional order parameters of the 5CB CG systems with different chain numbers at 300 K. Figure 7 Orientational and translational order parameter variations of 8000 5CB CG chains at different temperatures. The snapshots at 300, 360, 395, and 450 K are shown in the inserts. Figure 8 The length and the breadth of a 5CB molecule. Figure 9 Probability distribution of K1 value for 8000 5CB molecules.


24

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Table 1 Comparisons of correlation factor (F) in the Maier-Meier equation, orientaional order parameter, and density between different 5CB LCs studies.

Method (predicted value) Orientaional order

Quantum chemical method

Density (kg/m3)

F parameter

AM1a

Artificial (1)

Artificial (0.7)

Artificial (1000)

AM1b

Artificial(fitted)

Artificial (0.7)

Artificial (1000)

B3LYP/6-31G(d)a

Artificial (1)

Artificial (0.7)

Artificial (1000)

B3LYP/6-31+G (2d,p)(This work)

Eq. 5 (1.42)

CGMD(0.44)

MD (1050)

a32

b24

(a, b:293 K, This work :300 K)


25


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Page 26 of 43

Table 2 Parameters of bond stretching potential in the CGMD simulation. Bond stretching Kb(Kcal/mol/Å2) R0(Å)

bead1

bead2

C2

C3

4.23

3.26

C3

C61

3.41

4.03

C61

C62

145.6

4.29

C62

CN

197.08

3.47


26

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Table 3 Parameters of bond bending potential in the CGMD simulation. Angle bend bead1

bead2

bead3

Ka(Kcal/mol/°2)

θ0 (°)

C2

C3

C61

2.43

152.88

C3

C61

C62

16.22

165.48

C61

C62

CN

30.16

175.04


27


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table 4 Parameters of 12-6 Lenard-Jones potential. Van der Waals D0(kcal/mol) R0(Å)

bead1

bead2

C2

C2

0.28

4.68

C2

C3

0.97

4.95

C2

C61

0.28

5.85

C2

C62

0.28

5.65

C2

CN

0.2

4.25

C3

C3

0.1

5.66

C3

C61

0.9

5.75

C3

C62

0.41

5.75

C3

CN

0.24

4.65

C61

C61

0.01

7.71

C61

C62

0.97

5.15

C61

CN

0.91

5.25

C62

C62

0.01

6.85

C62

CN

1.25

5.15

CN

CN

0.27

3.65


28

Page 28 of 43

Page 29 of 43

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Table 5 Dielectric constant (l), dielectric anisotropy (∆l), polarizability (), polarizability anisotropy (∆), and dipole moment ( ), the refraction parallel to the light axis (DE ), the refraction perpendicular to the light axis (Dm ), and birefringence (∆D) of 5CB molecule at 300K.

l

∆l

α(. . )

∆α(. . )

Simulation

33.39

41.35

220.06

183.58

4.76

Experiment

10.5

11.5

239.35

216.6

Error

218%

260%

8%

15%


29

j

∆j

1.689

1.512

0.177

6.32

1.71

1.53

0.18

25%

1%

1%

2%

µ()

j


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 43

Table 6 Required parameters for obtaining the g factor and the effective dipole moment of 5CB system at 300 K.

K1

∥

,∥

,

1.82

0.30

0.77

3.43(D)

0.77(D)

3.52(D)


30

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Table 7 Optical and dielectric properties of 5CB system modified by the effective dipole moment at 300K.

l

∆l

11.83

11.76

Experiment

10.5a

11.5a

Error

12%

2%

a. 42


31


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Page 32 of 43

CN C3 C61 C62 C2 0.0034 0.0134 0.0286 0.0286 -0.074

Figure 1 The coarse-grained model of a 5CB molecule mapped from the all atom model. C2, C3, C61, C62 and CN are the types of CG beads and the partial charges are derived by the summation of the atomic charges from the COMPASS force field.


32

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(a)

(b)

(c)

(d)

Figure 2 Probability distributions of bond lengths between the bonded beads for all atom and CG models. (a) C2-C3; (b) C3-C61; (c) C61-C62; (d) C62-CN.


33


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(a)

Page 34 of 43

(b)

(c) Figure 3 Probability distributions of bending angles for all atom and CG models at (a) C2-C3-C61; (b) C3-C61-C62; (c) C61-C62-CN.


34

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(a)

(b)

(c)

(d)


35


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(e) Figure 4 Radial distribution functions for all atom and CG models. (a) C2-C2；(b) C3-C3；(c) C61-C61； (d) C62-C62；(e) CN-CN.


36

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(a)

(b)


37


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(c) Figure 5 (a) Initial structure of 800 5CB CG chains; (b) The equilibrium structure of 800 5CB CG chains at 300K; (c) orientaional order parameter variation of 800 5CB CG chains at 300 K during the simulation.


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Figure 6 Orientaional order parameters of the 5CB CG systems with different chain numbers at 300 K.


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0.8

Orientational order parameter Translational order parameter

360K 0.6

Order parameter

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395 K 0.4

300K

450 K

0.2

Experimental 5CB clear point at 308 K 0 280

320

360

400

440

480

Temperature

Figure 7 Orientational and translational order parameter variations of 8000 5CB CG chains at different temperatures. The snapshots at 300, 360, 395, and 450 K are shown in the inserts.


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Figure 8 The length and the breadth of a 5CB molecule.


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 9 Probability distribution of K1 value for 8000 5CB molecules.


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TOC

Coarse-graining mapping


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Prediction of Optical and Dielectric Properties of 4-Cyano-4

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