Molecular dynamics simulation of a nematic liquid crystalarnold/science/papers/ref19_5cb1.pdf ·...

Molecular dynamics simulation of a nematic liquid crystal Andrei V. Komolkin,a) Aatto Laaksonen, and Arnold Maliniakb) Division of Physical Chemistry, Arrhenius Laboratory, Stockholm University S-106 91 Stockholm, Sweden

(Received 7 January 1994; accepted 11 May 1994)

The article describes molecular dynamics simulations of 4-n-pentyl4’-cyanobiphenyl(5CB) in the nematic phase at 300 K using two interaction models. The first model comprises united atoms, while in the second, shorter simulation,- the hydrogen atoms are explicitly included. Liquid crystalline order parameters were calculated using various definitions of molecular frames and were found to be in reasonable agreement with experiments. Distributions of dihedral angles and relative populations of various conformations in the alkyl chain have been determined. Translational and rotational diffusion processes were investigated using time correlation functions, and were compared with experimental results. Local order parameters, relevant for deuterium nuclear magnetic resonance (NMR) spectra, were determined for the segments in the alkyl chain. Proton NMR line shapes were calculated from the trajectory using an approximate method for determination of the dipole-dipole Hamiltonian matrix. These line shapes were found to be very sensitive to conformational distributions and therefore to the force field used in the simulation.

I. INTRODUCTION

Liquid crystalline systems, also called mesophases, have attracted considerable scientific attention during the past 20 years. This attention can be partly ascribed to the many tech- nical applications of mesophases, in particular, liquid crystalline displays (LCD), but also to a desire for fundamental understanding of biological systems such as membranes and nucleic acids. Thermotropic liquid crystalline phases, which is the subject of our investigation, are formed by anisotropic molecules which are either elongated or disklike. The thermotropic liquid crystals formed by elongated molecules are mainly divided into nematic and smectic mesophases. These phases are characterized by the presence of a long range orientational order, whereas the positional order is strongly reduced (smectic phases) or nearly absent (nematics), In the present work, we report a molecular dynamics~(MD) simulation of 4-n-pentyl4’-cyanobiphenyl (5CB) (Fig. 1) in the nematic phase.

During the last few years, a dramatic development of computer power has led to a significantiy increased interest in computer simulations of complex systems using both MD and Monte Carlo techniques. In particular, several investigations of liquid crystalline systems have been performed with various degrees of approximation in the interaction models. These models range from simple lattice descriptions’ to de- tailed atomic interaction models and they have been used for investigations of discotic’ as well as nematic phases.3.4 Re- cently, a review of various simulation techniques and physical models of liquid crystalline systems has been presented.5

5CB has been studied extensively both theoretically and experimentally and therefore a large data set is available for confrontation with the results from computer simulation. One reason for this large number of experimental investigations is the convenient temperature range of the nematic liquid crystalline phase, namely, from 295.6 to 308 K. In par-

*Permanent address: Physics Institute, St. Petersburg State University, St. Petersburg 198904, Russia.

‘IAuthor to whom the correspondence should be addressed.

ticular, many investigations using nuclear magnetic resonance (NMR) on deuterium,6-9 carbon-13,“-t2 and proton 13*t4 nuclei have been performed studying both static and dynamic properties of 5CB. Furthermore, various interaction models were compared in an analysis of molecular conformations, using calculations of deuterium NMR qua- drupolar splittings.‘5.‘6 A preliminary investigation of 5CB using MD simulation has been reported,t7 focusing on the significance of electrostatic interactions for the liquid crystalline properties.

In the current study, we use models where the molecular potential is built up from atom-atom interactions. In the analysis of the trajectories, we focus our attention on static and dynamic parameters that are connected with various NMR techniques. We also make a comparison of united atom and explicit atom models for molecular processes and properties~ of the liquid crystal.

The organization of this paper is as follows: In Sec. II, we describe the interaction model and the details of the simulation. Section III is devoted to the analysis of the liquid crystalline properties, while the discussion on the molecular conformations is presented in Sec. IV The dynamical results are described in Sec. V. Finally, Sec. VI is devoted to simulation of deuterium and proton NMR spectra.

II. SIMULATION AND THE INTERACTION MODEL

In the present study, we have carried out two MD simulations using two types of molecular potentials. In the main simulation, the aliphatic CH, and CH, groups and the aromatic CH group are treated as single interaction centers, i.e., as united atoms (UA). In order to investigate the effect of the united atom approximation, we have also performed a shorter simulation using a full atomic (FA) model. We consider the UA simulation as the main part of the study and therefore unless otherwise stated we refer to the UA simulation.

The molecular dynamics simulations were carried out on a sample consisting of 75 mesogens in a rectangular cell. In

J. Chem. Phys. 101 (5), 1 September 1994 0021-9606/94/101(5)/4103/14/$6.00 Q 1994 American Institute of Physics 4103 Downloaded 09 Jan 2002 to 130.237.185.157. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

4104

Ririg A Ring B

FIG. 1. The 5CB molecule; the internal coordinate frames, dihedral angles, and the skeleton atoms are indicated.

the UA simulation, the molecules consist of 19 sites giving a total of 1425 interaction sites in the computational unit cell. The corresponding numbers in the FA simulation are 38 and 2850. The rectangular periodic boundary conditions, together with minimum image convention and a nonbonded, spherical cutoff of 12.5 A, were applied on the distances between the atoms. The equations of motion were integrated using. the Verlet leapfrog algorithm with a time step of 1 fs. In the FA simulation, a time step of 0.5 fs was used. For the analysis of trajectories, we stored the position of each atom every. 10 fs in both simulations. The simulations were performed in the nematic phase at 300 K and the temperature was kept constant by resealing the velocities at each MD time step.‘8,‘g This procedure essentially creates a canonical (NW) ensemble. In the initial configuration, we arranged the molecules in the simulation box in three layers along the Z axis, creating a structure similar to a s-mectic -A- phase. The molecules were randomly oriented around tbe biphenyl symmetry axis, and a total dipole moment was prevented by alter- nating the .vertical orientations in the box. During the equilibration the system was compressed from a cubic cell with the length of 70 A to a rectangular box with sides 25.5, 25.5, and 48.0 A, which corresponds to an experimental density”’ of 1.02 g cm-s. The system was equilibrated during 75 ps, whereas the trajectory was collected for 300 ps in the UA simulation and for 100 ps in the FA simulation,

Atom

Nl C2 c3

C4G

Cd+ CS c9

clo~cll C12rC13

Cl4

&c,9

H,-A, aromatic* IIs-Hi9 aliphatic*

aFA simulation

The mesogens were treated as flexible and the potential energy could be separated into covalent (the tirst three terms) and nonbonded terms

E total= c h--r,cJ2+ c K,(e- &$” bonds angles

+C cos n(0-- y)] dihedrals

+?J($+j+~] where K, , K,, and V, are force~constants representing bond stretching, bond bending, and torsional motion, respectively. The distance between the interacting sites t and j is rij , and A, B, and 4 are the parameters related to Lennard-Jones and electrostatic terms. The SHAKE21 algorithm was used to constrain tbe bonds lengths with the-convergence parameter carefully calibrated to minimize energy fluctuations. Note

TABLE I. Atomic charges and types used in Tables II-V. Note that all aromatic carbons are denoted Cl, while C2 is the type of carbons in the aliphatic chain.

Atomic type

NO co Cl Cl Cl Cl Cl Cl Cl Cl

C2 HI H2

Charge IeI

-0.450 0.100 0.300

-0.075 -0.070

0.250 0.250

-0.070 -0.060

0.100

0 0 0

Komolkin, Laaksonen, and Maliniak: Simulation of a nematic liquid crystal

that when the SHAKE procedure is used, the contribution from the first term in Eq. (1) is not calculated.

To determine the partial charges, we performed an ab initio HOND022 calculation using second-order M%ller- Plesset calculation of the dispersion energy with the G63”” basis set on the biphenyl fragment of the molecule. We have chosen the aliphatic united atoms to be neutral (4 =0) basing this approximation on the fact that the dipole moment of a CH2 group is small, of the order of 0.3 D. The molecular dipole moment was 3.98 D, which canbe compared with an experimental .value of 4.77 D obtained for cyanobiphenylZ3 in benzene solution. All the potential parameters are summa- rized in Tables I-V. The short range, nonbonded parameters were taken from the optimized potentials for liquid simulations (OPLS)24 force field. The relations between these parameters and A, B in Eq. (1) are given in Table II, together with the standard combination rules used for calculation of the cross terms. Scaling factors of 8.0 and 2.0 were used for the Lennard-Jones and electrostatic terms, respectively, in 1,4 nonbonded interactions. Momany, Carruthers, McGuire, and Scheraga (MCMS) potential paramete# were used for the FA system.

TABLE II. Parameters for nobonded interactions for the UA and FA models. Formulas for the calculation of A,, and Bij from Lennard-Jones parameters, and the standard combination rules are included.

Atomic type

UA simulation FA simulation

eji &.I mol-‘) crti CA) Ori Ni f&i (A)

NO 0.72 3.25 0.87 6.1 3.1 co 0.44 3.75 1.51 5.2 3.4 Cl 0.46 3.75 0.93 5.2 3.4 c2 0.50 3.905 0.93 5.2 3.4 Hl 0.42 0.85 2.4 w 0.42 0.85. 2.4

Formulas for A ij ami Bij

J. Chem. Phys., Vol;lOl, No. 5, 1 September 1994

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TABLE III. Bond lengths used in the SHAKE algorithm. TABLE V. Parameters for torsional angles used in the simulations.

Type of bond

NO-CO CO-Cl Cl-Clb Cl--Cl= Cl-C2 cz-c2 CZ-HZ’ Cl-Hl”

“FA simulation. bin a ring. ‘Between the rings.

rc9 (4

1.200 1.426 1.395 1.510 1.510 1.535 1.10 1.05

UA simulation

Torsional Y “” Y "" angle [degrees) (M mol-‘) n (degrees) (k.J mol-‘) n

X-Cl-Cl-Xbsd 0,180 340 1 0,180 170 1 cl-cl-cl-clc 0 4.2 2 0 4.2 2 Cl-Cl-cz-c2 90 0.04 6 90 0.01 6 Cl-Cl-C2-HZa 90 0.01 6 X-C2-C2-Xd 180 12.0 3 180 1.35 3

“FA simulation. bBond Cl-Cl in a ring. ‘Between the rings. dX=any type of atom.

Two MD programs were used: for the main simulation, a version based on the McMoldyn package,s6 whereas the FA simulation was performed using an optimized code” written at this laboratory. For the UA system, 15 s of central process- ing unit (CPU) time were used for one integration step, whereas the corresponding number in the FA simulation was 12 s. This difference reflects the effect of the careful optimi- zation. Note that the time step used-in the computation of the FA system was 0.5 fs (1 fs in UA). The simulations were carried out on a Convex C220 computer.

Uncertainties given for time and p&ticle averaged properties represent one standard deviation and provide information on the fluctuations, i.e., random errors. The systematic errors, introduced due to the limitation in the interaction model, can be estimated by comparison with experiments.

frame Eintra . We can write these energies using linear (p) and angular (Lj momenta, atomic and molecular masses (m and M), and the moment inertia tensor (I) as follows:

Na IPil” %n=C 2m’

i=l i

E,,=i (I-‘L.L), iw

ill. RESULTS AND DISCUSSION with A. Liquid crystal properties ,,s

. . . . 1. Equilibrium of fhe system

The total energy for the system was calculated to be -6125 kJ mol-‘, and since the simulation was performed at constant temperature, the total kinetic energy E,, was constant. We carried out an analysis of various contributions to the kinetic energy in order to investigate whether the simulation described an equilibrium system, as defined by the classical equipartition theorem. The total kinetic energy can be separated into three parts: (1) translational kinetic energy of the molecular center of mass E, ; (2) rotational energy of a molecule around the center of mass I&; and (3) intramolecular kinetic energy of atoms in the molecular coordinate

n mi Pi =Pi-- Pc.m.-mi(I-lLXri), A4 (24

where N, is the number of atoms in’one molecule, and the three contributions in Eq. (2e) to the intramolecular momentum consist of atomic momentum in the laboratory frame, its part of the linear center of mass momentum, and the angular momentum around the molecular center of mass.

TABLE IV. Parameters for bond angles used in the simulations.

qpe of angle

No--Co-Cl X-cl-xb CI-=C2-C2 c2-C2-C2 C1-C2--H2a c2--C2-H2a H2-C2-H2=

‘FA simul&on. bX=any type of atom.

0, @w-N if, &.I radu2 mol-‘)

180 240 120 210 111 265 111 265 108.9 265 108.9 265 110 265

The model molecule in the UA simulation contained 19 atoms connected by 20 chemical bonds with the bond lengths fixed during the simulation. Such a molecule has NM,=37 degrees of freedom distributed as N,=3 translational, N,,=3 rotational, and Nintra =3 1 internal degrees of freedom. We averaged the kinetic energies using the entire trajectory and found that the translational and the rotational energies are essentially equal (k&/E&-0.99), but their contributions to the total kinetic energy are larger than expected: E,=E,,,-0.095.Ekin, whereas N,=N,,m0.081.Nkin. We believe that the reason for this difference (about 20%) is that using the trajectory, sampled at every tenth step, for the calculations of velocities is not accurate enough. We therefore performed the same analysis in the force calculation routine over the last 1000 steps (0.5 ps) of the PA simulation. A molecule with all hydrogens included consists of 38 atoms connected by 39 bonds and a total of 75 degrees of freedom. We obtained the following .relative contributions: E,~Oe041*Ekiny E,t~0.037.E~, and Bi”,m0.922*EEn.

@b)

Komolkin, Laaksonen, and Maliniak: Simulation of a nematic liquid crystal 4105

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-24

FIG. 2. The number density of molecular center of mass along the Z axis in the simulation box for (a) the initial and (b) the final configurations.

These values are in agreement with expected values of N,=N,,=0.04.N~, , Ninka=0*92*Nen. The small difference between E, and Erot is probably due to insufficient averaging of fluctuations of these energies. The analysis indicates that both systems represent thermodynamic equilibrium.

Positional order is usually absent or very strongly reduced in nematic phases. This order can be monitored by calculating the number density atong various directions of the system. In Fig. 2, we show the center of mass density wave along the Z axis of the box, which nearly coincides with the director. The initial configuration [Fig. 2(a)] shows a typical smectic structure which is a result of locating the mesogens in three distinct layers. The final distribution, shown in Fig. 2(b), is reasonably uniform as expected for a nematic system. Note that the distributions are snapshots and not averages and deviation from an uniform distribution is due to the small number of molecules in the simulation cell.

2. Orientational order

We now turn to the description of the orientational order of the nematic phase and for this purpose, we need to define the director of the liquid crystal. The general form of the ordering tensor is given by’s

where N is the number of molecules and $ are the direction cosines between a set of axes fixed in the molecular (u,u) and laboratory (cr,@ frames. Choosing the long molecular axis as z and u = u = z, we define the ordering matrix Q,, ,

4106 Komolkin, Laaksonen, and Maliniak: Simulation of a nematic liquid crystal

FIG. 3. Temporal development of two components of the nematic director nx and ny and the order parameters. S,rX, S, , and S,, are calculated using the moment of inertia frame. SL is defined by a unit vector between sites N, and C,,.

N 3 a;‘=; & 2 L;,L;,-; &.

J=l

The diagonalization of the ordering matrix gives three eigenvalues and eigenvectors. The nematic director n is the eigen- vector associated with the largest eigenvalue of Q, .

Another, and from the computational point of view, much simpler way of determining the nematic director n is given by

1 lv- n=s F ezjj

I==1 (5)

where ezj is the unit vector of the long molecular axis. In this definition, we make use of the fact that the nematic phase possesses a head-tail symmetry.

The remaining problem is to decide on the definition of the molecular coordinate system and in particular the long axis z. For a flexible molecule, the choice of coordinate system is not obvious, but one possibility is to use the moment of inertia frame.29 Using this definition, the contributions to the molecular shape from both the rigid, aromatic part and the flexible alkyl chain are included. The inertia tensor can be calculated for all molecules at every time step using proper atomic masses (the mass of united atoms is given as the sum of carbon atom and hydrogens). When the tensor is diagonalized, the three eigenvectors e, , ey, and ez define the principal axis system, whereas the eigenvalues I,, , I),,, , and I,, are the principal moments of inertia.

We can now construct the ordering matrix Q, using z and the axes defined by the simulation box cy, p=X, Y,Z. In Fig. 3, the components of the director n, and rzI, which are the projections on the coordinate system of the simulation cell, are shown as a function of the simulation time. It can be noted that the director is stable during the entire simulation. The director Auctuations of a liquid crystal occur on a time scaIe of milliseconds and require considerably larger systems. These fluctuations can therefore not be observed in our MD simulation. We now define molecular order parameters by

J. Chem. Phys., Vol. 101, No. 5, 1 September 1994


TABLE VI. Order parameters calculated using various methods and different definitions of molecular coordinate systems. Estimated standard deviations are given in parentheses.

Method

Moment of inertia Frame of the ring A Frame of the ring B Biaxiality of tensor of inertia ‘H NMR for SCB-d,, (298.7 K) ‘H NMR for 5CB (300.8 K) Moment of inertia (FA) Exp. (305 K)n

s zc sxx - s,, 0.72(l) -0.04(3) 0.70(l) -0.05/3) 0.70(l) -0.05(s) 0.66(l) -0.09(l) 0.61 . . . 0.59 . . . 0.43(2) O.O3(3) 0.567 0.057

‘Reference 13.

See=; 2 i COS2 Oj-k

t

, j=l i

where e is a unit vector fixed in a molecule and 6’ is the angle between this vector and the nematic director. The order parameters SZ, , S,,, and S,, calculated during 300 ps are included in Fig. 3. Note that the 5CB molecule is not uniaxial and the biaxiality can be defined as (S, - S,,). We cannot be conclusive on the point whether the biaxial@ is induced by the finite size of the system or if it has physical significance. Experimental results,10’13 however, predict a biaxiality of 5CB that is in good agreement with our results. The nematic order parameter S,, and the biaxiality of the phase calculated using molecular axes defined by the inertia tensor are given in Table VI. Note that the sign of the biaxiality obtained from the simulations is opposite to that determined from experiments. This fact is simply a consequence of the definition of molecular coordinate system.

(6)

The order parameters are only unambiguously defined for rigid molecules. Interpretation of experimental results is often dependent on the selection of molecular frame; on the other hand the computer simulation technique provides a unique possibility to use different definitions of this frame. It is possible, e.g., to replace the inertia tensor with the polar- izability tensor, which probably coincides with the polar biphenyl core of the molecule. We have therefore used the coordinate systems fixed in the benzene rings, as shown in Fig. 1, and the order parameters calculated using these two coordinate frames, rings A and B, are included in Table VI.

Yet another method for determination of molecular ordering is provided by the biaxiality of the moment of inertia tensor for calculation of the order parameters. In this scheme,” the order parameters are defined by

Szz= 1 - (A,+A,)/2AZ, S,,= - $+A,/2AZ,

Sxx= _ $+A,/2A,, (7) where ellipsoid semiaxes A, are given by

A,=[(lpp~I,,--1,,)5/2M]“2 (81 and M is the molecular mass. Using this definition, the limits for S,, are 1 and 0 for an infinite rod and an ideal sphere, corresponding to a system with perfect order and complete disorder, respectively. The limits of S, and.S,, are -0.5 and


0, and the ordering matrix is traceless. Note that using this method, we do not perform any transformations between the coordinate systems and only make use of the biaxiality of the inertia tensor, i.e., the averaged shape of the molecules. The order parameters calculated from biaxiality are included in Table VI.

The last source of information about orientational order is proton NMR line shapes and the details of these computations will be given in Sec. VI B. The experimental values in Table VI are derived from proton multiple quantum NMR investigationI performed at 305 K. We therefore expect the experimental order parameter to be lower than the simulated value at 300 K. This is indeed the case in the UA simulation. The nematic order parameter S, for the FA model, on the other hand, is clearly smaller than those derived from experiment and the UA simulation. It is, however, difficult to trace a single reason for the difference between the two simulations, taking into account that the models used are entirely different. A general conclusion is that the agreement between simulated and experimental values is reasonable. Note that different definitions used in our analysis seem to produce results that are within the significance interval. We conclude therefore that several possibilities for a choice of the molecular frame exist, which nearly coincide and essentially give similar nematic order parameters for 5CB.

f(cos 8) =c exp[a,P,(cos 0) + L@,(COS S)

Before cIosing this section, we will discuss the orientational distribution functions f(cos B), where 6’ is the angle between a vector fixed in a molecule and the nematic director. These functions can be calculated from the trajectory for various vectors and analyzed~ using functional form given by3’

fU~P,(COS e>+***-j, (9)

where c is a normalization constant and the coefficients ai are related to (Pj(cos e)), i.e., liquid crystal order parameters.31 Truncation of the expansion after the first term gives an expression that is consistent with the Maier-Saupe theory.“” In Fig. 4(a), the distributions are shown for long molecular axes defined using the inertia tensor and by two unit vectors between the nitrogen and two carbon atoms with the numbers 14 and 16, respectively. All three distributions show a sharp peak around 0=0, which is consistent with the definition of the nematic director. The distributions in Fig. 4(b) are associated with vectors corresponding to the carbon-carbon bonds along the aliphatic chain, with the numbers indicated in Fig. 1. It can be noted that bonds 15-16 and 17- 18 are not parallel with the long molecular axis, and therefore the distributions do not peak at 0=0. Bonds 16-17 and 18-19 nearly coincide with the long axis, and the distributions are clearly broader, but similar to these in Fig. 4(a). The very different shape of various distributions in Fig. 4(b) can be considered as a demonstration of the even-odd effect in the alkyl chain. The quantitative analysis was performed by using a nonlinear least squares fitting of the distributions to the expression in Eq. (9). The coefficients ai derived from this analysis are collected in Table VII. All the distributions that represent the long molecular axis, i.e.,



0.4 a) __ tensor of inertia 0.3 _. _ LIIs!.EC!_6 -.-..-.. . . .._.....”

-----Nl-Cl4 8 go.2 ~ .___. l._-- ,__ I.-._” _-~ _--, -

G=r i

0 T.--i---‘ 0 0.25 0.5 0.75 1 case

FIG. 4. Orientational distribution functions flcos 8) for vectors fixed in the molecular frame. (a) The long molecular axis using three definitions-the tensor of inertia and two unit vectors N,-C,, and Nt-Cts. (b) Bonds between the carbons in the alkyl chain.

these in Fig. 4(a) are sufficiently determined using the two leading terms in Eq. (9), whereas the other distributions require three parameters for a satisfactory fit. The even-odd effect is clearly reflected in the values of (P,(cos 0)) along the alkyl chain. Note that a perfect uniaxial phase of rigid molecules is completely determined by S=(P,(cos 0)) and therefore the first term of the expansion should be sufficient.

3. Radial distribution functions Positional order in the nematic phase is reflected in the

radial distribution functions g(rij), where rij is the distance between sites i and j on different molecules. Figure 5(a) shows the distribution functions for nitrogen atoms gNN and methyl groups ga. The nitrogen distribution is only slightly structured, with a weak maximum around 10 fi. This is expected since the nitrogen atoms have high partial charges, and therefore the repulsive electrostatic forces are rather strong. The methyl distribution, on the other hand, shows a relatively sharp first maximum at 5 8, and a weak

TABLE VII. The coefficients aj derived from the fitting of f(cos 6).

Vector P,bs m a2 a4

I; 0.725 4.67 -0.25 NI-C,, 0.720 4.44 -0.32 W-014 0.708 4.51 -0.18 c15-c,6 -0.103 -0.79 -1.22 0.35 %-cl7 0.621 2.87 0.37 c17-G3 -0.077 -0.54 -1.11 0.42 %-Cl9 0.467 1.62 0.98

is

ob---“r-7-7 1 ~I ’ ’ t * 4

distance (htp 1.5.

BIG. 5. Intermolecular radial distribution functions. (a) Nitrogen gNN and methyl groups g,, . (b) Centers of mass-molecular gee , rings A and B gas, and two A rings gAA .

broad second maximum at 12 fi. An analysis of x-ray diffraction measurements33 suggests molecular association due to high dipole moment of 5CB. The local structure proposed in this analysis predicts antiparallel orientation of neighboring aromatic cores and a tendency to local layer formation. The distance between nitrogen atoms derived from this structure is 10 A and is in fact in good agreement with the value predicted by our gNN. In addition, from scanning tunneling microscope (STM) experiments, an average value of 5.5 A was reported for separation between aliphatic chains of aCB molecules adsorbed on a solid surface of MoS~.~~ In Fig. 5(b), the radial distribution functions are displayed for different centers of mass-molecular gee, aromatic rings A gAA , and between A and B aromatic rings gAB . The distributions are similar and show a broad first maximum at 7 A. This maximum is clearly smaller for gee, due to the flexibility of the aliphatic chain which shifts the center of mass.

The radial distribution functions provide information on distances only, and it is therefore difficult to draw any conclusions concerning molecular orientations. We note that the distributions of aromatic rings and of the center of mass have their first maximum at 7 A, whereas the nitrogen distribution gNN peaks at 10 A, which may indicate that neighboring molecules prefer an antiparallel orientation. An approximate intermolecular spacing derived from x-ray diffraction is close to 5 8, which is shorter compared with the radial distribution functions in Fig. 5(b). This fact is a possible indication of a lower degree of local correlation in the MD simulation. We have calculated the angle between unit vectors parallel to the molecular dipole moments and found no strong correlation or indication of antiparallel dimers. Mo- lecular mechanics calculations35 predict a very small differ-



‘: 2 75

g 2 50 0

; 25

the (PS)

FIG. 6. Temporal development of translational diffusion coefficients calculated during the simulation. Note that D, is an average of X and Y components.

ence between parallel and antiparallel orientations of a dimer, whereas & antiparallel dimer formed due to the electrostatic interactions is discussed in several experimental studies.33p36*37 The radial distribution functions show close similarity to those obtained for many isotropic, disordered systems. This is in fact not surprising since a nematic liquid crystal exhibits very limited positional order.

4. Translational diffusion

Translational diffusion coefficients can be obtained from the mean square displacement (MSD) according to

D,,=lim(1/2t) (

2 [R&(t)-&0)12 , i

(10) t--t- j-1

where (Y=X, Y, and Z detine the coordinates of the simulation box and R,(t) is the position of ‘a molecular center of mass at time t.

In a liquid crystal, the diffusion is anisotropic, and for a uniaxial or nearly uniaxial phase, we can distinguish two components of the diffusion tensor D,=Dll and D,= Dyy=D, , referring to the nematic director, or the Z axis of the simulation box. Figure 6 displays the temporal development of the diffusion constants during 300 ps of sam- pling. The perpendicular component is an average of X and Y, which is also reflected in a strongly reduced noise level.

The diffusion constants can a1s.o be evaluated from the linear velocity time correlation functions according to


time (ps)

FIG. 7. Normalized time correlation functions of linear velocities of molecular center of mass.

cage effect, the integral is a factor of 3 larger, which is associated with a faster translational diffusion along this axis.

In Table VIII, the diffusion constants obtained from the mean-square displacement and from the correlation functions are given, together with experimental results. In the FA calculation, the diffusion constants determined from MSD did not reach equilibrium values and therefore only an upper limit is given in Table VIII. The diffusion constants derived from correlation functions for the two models are very similar. The experimental diffusion coefficients have been determined at 296 K from quasielastic neutron scattering (QENS) measurements.37 Taking into account that the simulation was performed at a higher temperature, the agreement between calculated and experimental values is reasonable. In a recent preliminary investigation, a value of D,=72XlO-” m’? s-l obtained from NMR using the Fourier transform pulsed- gradient spin-echo (FT-PGSE) techniqueI has been reported; this value seems, however, to be much too high. There are various experimental problems connected with measurements of diffusion coefficients in therrnotropic liquid crystal~.~s The neutron scattering method is not direct, i.e., it requires a separation of contributions from other dynamical processes. The F’I-PGSE method, on the other hand, is completely insensitive to other types of molecular motions, but is strongly limited by short Spin-spin relaxation times, which cause a dramatic reduction of the signaL3* In order to com- pare the diffusional and molecular shape anisotropies, the following relation is used37*39:

Da+ ];(v,(O).v,(t))dt=!$ Q-,, TABLE VIII. Translational diffusion coefficients. Estimated standard deviations are given in parentheses.

where the brackets indicate an ensemble average, v is the linear velocity of a molecule, M is the molecular mass, k, is the Boltzmann constant, and 7cr is the correlation time defined as an integral of the time correlation function. Normal- ized velocity time correlation functions are shown in Fig. 7. The correlation functions for X and Y components are essentially equal, as expected from D,== D yy == D, and these differ clearly from the parallel component D,= D/l . The inte- grals 7, are 0.1 and 0.3 ps for perpendicular and parallel components, respectively. Note that in spite of a strongly negative region of Z correlation function, often denoted as a

Determined from

Simulation MSD (UA) C,(t) (U-4 MSD (FA) C,(t) (F-Q

D,XlO” (m’s-‘)

10(l) 25(2) W) 30(2)

<3 <3 C9 26(2)

Simulation MSD (290 K) 140 250 Experimental QENS (296.5 K)b 4.1 5.3

FT-PGSE (308 K) 72 . . .

‘Reference 17. bReference 37. “Reference 14.


4110

a)

o,(degrees)

T-----T 4 I SC o-j,l& ...I ~~~~o

-90 0 $0 @,(degrees)

FIG. 8. Distributions of dihedral angles 8, and 8, in (a) UA simulation and (b) FA simulation.

3 -2j(l-s)+2S+ 1 - DL y(S+2)+l-S ’

where y=41/rd, and 1 and d are the molecular length and diameter, respectively. Using DIIID, =3 and the nematic order parameter S=O.7 gives l/d=4, which is a reasonable result. From experimental diffusional anisotropy, If d = 1.3 was obtained,37 an unexpectedly low value for the 5CB molecule. This small shape anisotropy has been discussed in terms of possible formation of dimers due to electrostatic attraction between antiparaIle1 oriented molecules. We note that the diffusional anisotropy in our simulation is in agreement with that of other nematic phases of rodlike molecules, which usually have this ratio in the range between 1.5 and 4.38

IV. MOLECULAR STRUCTURE

A. Dihedral angle distributions

Dihedral angles and therefore molecular conformations are determined by the explicit torsional potential represented by the third term of Eq. (1) and by intramolecular nonbonded electrostatic and Lennard-Jones interactions. In addition, the dihedral angles are affected by intermolecular interactions and therefore depend on molecular orientation. In Fig. 1, five torsional angles 8,-0s are indicated and these were chosen to be monitored during the analysis. The first angle 8, determines the relative orientation of phenyl rings in the biphenyl fragment and & is the torsional angle between a phenyl ring and the aliphatic chain. The three dihedral angles of the alkyl chain are denoted 8,-8,. Figures 8 and 9 show the distributions of the five torsional angles obtained from UA and FA simulations. The distributions of the inter-ring tor-

2.5 h-. a)

8, (degrees)

2.5~~b)

A 1.54 I a- E 1 n Qs

Komolkin, Laaksonen, and Maliniak: Simulation of a nematic liquid crystal

“$JQ&j+q #q;, [l, ei( degrees)

FIG. 9. Distributions of dihedral angles e3, 8,, and C& in (a) UA simulation and (b) FA simulation.

sional angle 0, are displayed in Figs. 8(a) and 8(b). We note the different shapes of these distributions, apparently reflect- ing a difference in the effective torsional potential of the two models. The UA distribution shows a broad, but single maximum at O”, indicating an essentially planar structure of the two rings, whereas two degenerate maxima at zL2.5” are ex- hibited by the FA distribution. Using multiple quantum proton NMR,13 this dihedral angle has been determined to be 32”. We conclude therefore that the effective torsional potential in the UA simulation fails to correctly predict the inter- ring structure. An important contribution that determines this angle is the repulsion between adjacent ortho protons on the two rings; obviously this repulsion is absent in the UA model. In the FA simulation, these protons are explicitly included, and the repulsive interactions are clearly effective. It can be noted that this dihedral angle is strongly dependent on the state of aggregation, and in the case of unsubstituted biphenyl, it ranges from 45” in the gas phase to 0” in the solid state.40 The distribution of the dihedral angle between the biphenyl part and the chain & shows very little structure in both simulations and essentially has the shape of a cosine function. In the rotation of the chain around the bond con- necting it to the biphenyl part, the C2 symmetry of the ring is combined with the C, symmetry of the first aliphatic carbon, giving a resultant C6 symmetry. Usually torsional potential for a rotation around a sixfold axis has low energy barrier. In condensed matter, however, the energy barrier can increase due to complicated inter- and intramolecular interactions. We note that the population difference in these distributions is less than for other dihedral angles, in agreement with previous conclusions6 about a low energy barrier for this rotational potential. Structural determinations of alkyl deriva-



TABLE IX. Relative populations of chain conformations. 1 I

Ref. 42 State UA 6) FA (46) (8) Multiplicity

rtt 44.7 15.4 33.2 I ttg 2 8.1 11.5 7.17 2 tgrt 15.6 12.6 11.59 2 tgzzgz 0.8 5.4 2.48 2 tg+gT 0.0 2.7 0.05 2 gktt 1.6 3.5 4.85 2 gCtgk 0.0 3.2 1.82 2 g-ctg+ 0.6 1.9 2.64 2 g’g+t 1.0 1.3 1.74 2 g‘cg-cgT 0.0 0.0 0.01 2 grgrt 0.0 0.0 0.04 2 gzg+gr 0.0 0.0 0.00 2 gzg+gc 0.0 0.0 0.02 2 g-tg2g-e 0.0 0.2 0.95 2

’ I 2 0.5

‘1 c I, 16 ’ I :! ,’ 5 C 17

FIG. 10. Intramolecular radial distribution functions between the nitrogen atom and the carbon atoms in the alkyl &tin C, .

B. Intramolecular distribution functions

tives of benzene predict41 a minimum of the potential at 90”. This prediction is indeed confirmed in our simulations since the @ distributions peak at 90”.

We now turn to a discussion of chain conformations. In Fig. 9(a), the three distributions of dihedral angles e3-F)s are shown for the UA simulation, while the corresponding distributions obtained in the FA simulation are displayed in Fig. 9(b). All the distributions in the two simulations show two sharp maxima at t 180”, corresponding to the tram (t) conformation, and smaller peaks at 567” which are due to the two gazlche (g) conformations. Note the even-odd effect, which in particular is clearly demonstrated in the UA simulation. The distributions derived from the FA simulation are much broader, indicating that the chain is clearly more disordered than in the UA model. Relative statistical weights for the different conformations in the alkyl chain are given in Table IX. The conformers are labeled according to the se- quences along the chain and g-C indicates an average of the two gauche states (g ’ and g -). It is to be expected that the alkyl chain prefers the all-truns state. The relative populations of this state are clearly different for the two models. Such populations determined from deuterium NMR experimental data interpreted using the rotameric state model42 are included in Table IX. These predict the all-truns state to account for 335, which is in between the values derived from the FA and UA models; the corresponding value in the isotropic state42 is 22.7%. Also an analysis of proton NMR spectra based on the rotameric state model for the chain conformations43 predicts the statistical weight of the all-trans state to be between 25.7% and 325%. In a Monte Carlo computer simulation of n-pentane in isotropic phase at 25 0C,24*44 nearly 50% of the molecules were found in the all-trans state. A distribution of conformations similar to that predicted by the UA model has been recently obtained from a MD simulation of tram-4-(tram-4-n- pentylcyclohexyl)cyclohexylcarbonitrile (CCHS) in nematic and isotropic phases.30 We will return to the discussion of the conformational distributions in Sec. VI, in connection with deuterium and proton Nh4R spectra.


14 distaA:e (A)

18 20

Figure 10 displays intramolecular radial distribution functions which show the probability of finding an united atom (methylene or methyl group) C, of the alkyl chain at a distance r from the nitrogen atom. We note that the distribution corresponding to the first united atom shows. only one sharp peak, which is consistent with the fact that this group does not undergo any isomerization processes. The trans- gauche isomerization of the second unit does not change the interatomic distance to the nitrogen and the distribution function therefore again consists of a single peak. The last three distributions Ct7-Ct9 show a sharp maximum and a second, significantly weaker maximum at a shorter distance; these peaks correspond to the trans and gauche conformations, respectively. Assuming that the relative populations obey the Boltzmann law, we find that the truns-guuche isomerization energy is about 6 kJ mol-‘, which is similar to the energy difference derived from the effective torsional potential of CCHS in the nematic phase.“’ We can also note that the maximum of the methyl distribution is at a distance of 18 A from the nitrogen atom; the same distance has been used in the discussion of the molecular length37 of 5CB. The ellipsoid axis 2.A, defined by Eq. (8), which corresponds to the long molecular axis, has a length of 17.9 A.

V. DYNAMICS

A. Molecular reorientations

Molecular reorientations are conveniently investigated using time correlation functions (TCFS) C(t) defined by

(13)

where eru is a unit vector fixed in the molecule and Pi(x) is the Ith order Legendre polynomial. Different spectroscopic methods monitor correlation functions of Legendre polynomials with different values of I. In the following, we will focus our attention on a comparison with nuclear spin relaxation results, which are sensitive to I=2 functions with P2(x)= 1/2(3x”- 1). Figure 11 shows TCFs describing reorientations of some unit vectors during 100 ps. These vectors are defined using the coordinate system of the principal moment of inertia and the local frames fixed in the phenyl rings A and B (Fig. l), respectively. Note that the TCFs corresponding to the x and y components of the local frames



O-j.---,--- 0 20 40

time (i$ 80 100

FIG. 11. Normalized reorientational correlation functions of unit vectors defining molecular frames. Tensor of inertia frame a,, a,, , and a,. Local frames of the phenyl rings e,, , eA, , e,, , eax , eRy , and e,, .

and of the inertia tensor are identical, and it can therefore be concluded that the 5CB molecule is nearly uniaxial. The z components of the two rings coincide and the para axis is considered here as the molecular symmetry axis. In the no- tation used for symmetrical rotors, the molecular reorientations can be decomposed into two components-the reorien- tation of the long axis, or the molecular tumbling, and the motion around molecular axis, the spinning motion. These reorientations are characterized as perpendicular and parallel referring to the molecular symmetry axis, and the corresponding rotational diffusion constants are R, and RII, respectively. The molecular tumbling is described by the TCFs (Ftg. 11) of vectors e,, , e,, , and e,, , which represent the moment of inertia and the two para axes of the rings (Fig. 1). It can be noted that the initial decay of the phenyl frame TCFs is somewhat faster than that defined by the moment of inertia. This more rapid initial loss of correlation can be ascribed to fast atomic fluctuations and small deformations of the aromatic rings!5 The TCFs associated with the spinning motion show an opposite trend; the inertia tensor function decays slower than the vectors fixed in the phenyl frames. This observation can be explained in the following way: the elements are defined at each time step by Z$-=-lY,> Ii,, but because Z,, and ZyY are nearly equal, the local motion in the alkyl chain may interchange the two axes. In other words, in two consecutive time steps, the molecule passes through an uniaxial state. This process implies that the correlation of the unit vectors associated with these axes will be very rapidly lost, which explains the fast initial decay. We were in fact able to monitor this effect by a careful analysis at every time step. After a rapid initial loss of correlation, the TCFs corresponding to the molecular tumbling show slow decays which are essentially identical for all three vectors. Assuming that the long-time decays are exponential, i.e., that the motion is in the diffisional limit, we fitted the last 60 ps of the TCFs to an exponential function. The correlation time estimated from this fit was found to be 1.1 ns. The decay of the spinning motion TCFs is considerably faster, and we estimated this correlation time to be 0.14 ns. We have also calculated TCFs for Legendre polynomials with 1 =l (not shown) and found the correlation times for the tumbling and spinning motions to be 2.6 and 0.36 ns. In the diffusional limit, the ratio ~t/7~ (the subscripts refer to the order of the Legendre polynomi-

TABLE X. Rotational diffusion constants. Estimated standard deviations are given in parentheses.

Determined from R,XlO* R,XlO'

(s-9 cs-‘1

Simulation UA FA

Experiment ‘H NMR (300 K) 13C NMR (300 K)b

M(2) Wl) 0.8(l) 5.0(5) 0.5-1.5 12-28 3.5 5.7

‘References 6 and 7. bReference 46.

als) should be 3, and we conclude therefore that the motion is indeed diffusional. The rotational diffusion constants are then given by R,= l/65-, and RII= 11671. The constants obtained from. the simulation along with experimental values are collected in Table X. We note that the motional anisotropies RIIIR, derived from experiments using deuterium and carbon-13 NMR are very different. In order to determine rotational diffusion constants from carbon-13 experiments, a separation of different relaxation mechanisms (contributions from dipole-dipole and chemical shift anisotropy) to the measured relaxation rates is required.“’ We can therefore ar- gue that the deuterium analysis677 provides a more convinc- ing description of the molecular dynamics of the rigid aromatic part. The rotational diffusion constants obtained from deuterium data are well reproduced by our simulation. Fi- nally, we note that the rotational diffusion constants derived from the FA simulation differ by a factor of 2 compared with the UA model, but are within the range of experimental results.

B. Local motions in the alkyl chain

The problem of local dynamics in flexible molecules in isotropic and ordered phases has attracted considerable attention during last few years.40,47-49 In particular, several dynamic models have been proposed for interpretation of nuclear spin relaxation in alkyl chains.6T7’50951 A general com- plication, which makes the theoretical analysis of local dynamics very intricate, is the coupling between overall and local motions. This problem is especiahy difficult to over- come when the overall dynamics is highly anisotropic, as it is in the case of a rodlike molecule such as 5CB. Figure 12 shows the reorientational correlation functions of various C-H vectors of the alkyl chain for both UA and FA simulations. In the UA simulation, positions of various hydrogen atoms were reconstructed from the trajectory -of the carbon skeleton neglecting C-H vibrations and assuming tetrahedral symmetry of all methylenes and the methyl group. Note that the TCF for the methyl group in the UA model does not include the most important dynamical process, namely, the explicit rotation around the Cts-Cl9 bond. In fact, the torsional angle defined by Ct-,-Cts-C,,-H is fixed in the re- construction algorithm. The flexibility along the alkyl chain is expected to increase; and the reorientations of a C-H vector in the methyl group-should therefore lose the correlation more rapidly than the methylenes in the chain. We note in Fig. 12 that TCFs indeed show this behavior, i.e., the internal




oj-7-m r --‘- ---‘-- --I 0 10

tine (p3Sq 40 50

oj--, ( ..2’6 .-,-.-.,-. ri 0 10 30 50

time (ps)

FIG. 12. Normalized reorientational correlation functions for C,-H vectors in the alkyl chain in (a) UA simulation and (b) FA simulation.

motion increases along the chain. In particular, this is found in the FA simulation, where a significant difference between segmental motion in the first carbon and the methyl group is observed. Assuming an exponential form of the last part of the TCFs, we can estimate characteristic decays for these functions. Accordingly, the correlation times for the C-H vectors related to the five carbons along the alkyl chain are 132, 108, 98, 68, and 40 ps. We find that the correlation times associated with reorientations of the two first methylene groups are similar to the overall spinning motion of the molecule (140 ps); these motions are probably strongly coupled. In the UA simulation, the difference .between the correlation times for various segments of the chain is very small, and we suspect that the united atom model of the aliphatic chain does not predict correct dynamic behavior. The origin of this failure may be traced to the isotropic na- ture of the model, where the interactions of methylene groups do not depend on their mutual orientations. The problem of anisotropy of the interactions is clearly solved by using the FA model, and it can also be treated using an anisotropic united atom models2

VI. CALCULATION OF NMR SPECTRA

A. Deuterium NMR

Investigations of deuterium NMR line shapes are used extensively to study dynamic processes and structures in liquid crystalline systems.53 These studies are particularly convenient when the director of the mesophase orients in the magnetic field of an NMR spectrometer. In most of conven- tional, rod-like, nematic phases such as XB, the director is parallel to the field. This situation is in fact analogous with our simulation, where the sample is “oriented” and the di-

E m

-+I- deuterium -&---UA - --6-- Carbon-13 -cFA

----r- I I I

1 3 4 5 position in the chain

FIG. 13. Order parameters in the alkyl chain. The experimental values were taken from Ref. 5.5 for deuterium NMR and Ref. 10 for carbon-13 NMR.

rector is parallel to the Z axis of the simulation box. We can assume that the quadrupole interaction tensor for deuterons is cylindrically symmetric, with the symmetry axis parallel to the C-D bond. A deuterium NMR spectrum then consists of a pair of lines centered on the resonance frequency with a spacing of54

Av= ;qBcDScD, (14)

where qcn is the quadrupole coupling constant e’qQlh, and Sco is the component of the ordering matrix along the C-D bond. The relation between this component and the nematic order parameter S,, is given by

SC-= +((3 cos2 8- l)s,,) , (15)

where 8 is the angle between the C-D vector and the long molecular axis. The nematic order parameter can be determined from deuterons which are rigidly attached to the molecular frame and for which 8 is consequently fixed. In the case of 5CB, this order parameter can be determined from the aromatic deuterons, and Scn for the various methylene groups can be evaluated from the other splittings. In the computer simulation, the order parameters S,, can be evaluated directly from the trajectory by calculating the scalar product of a C-D vector and the nematic director. The C-D vectors in the UA simulation were obtained from the carbon skeleton as described in the previous section. In Fig. 13, the order parameters SC,, calculated from FA and UA simulations are plotted together with experimental values55 derived from deuterium NMR. In this figure we have also included order parameters derived from C-H dipolar couplings using carbon-13 2D NMR.” The values obtained from the simulations are negative, since the C-D vectors are almost orthogo- nal to the long molecular axis: The sign is, however, not available directly from deuterium NMR experiments, and we have therefore defined all the order parameters as positive. Both experimental and simulated values indicate that the alkyl chain does not exist in an all-trans conformation, since this state requires all order parameters to be equal.

First, we note that the UA simulation gives order parameters that are clearly higher than the experimental. This difference corresponds almost exactly to the higher molecular order in the simulated liquid crystal and is reflected in the values of Szz (Table VI). The FA simulation shows much better agreement with experimental results, which we, how-

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ever, consider to be somewhat accidental. In fact, two effects cancel apparently each other-low nematic order parameter (Table VI) and highly disordered alkyl chain. The disorder of the chain increases obviously the numerical value of the first transformation in Eq. (15). Both the simulations and the experiments predict an increase of the order parameter at the third carbon of the chain. Such an increase was also obtained from theoretical calculations56 based on the rotational iso- merit state model.

spectra have been proposed58-60; and to our knowledge, the largest direct calculation was performed on a 1Zspin system.“’

B. Proton NMR in liquid crystals

1. Nuclear spin Hamiltonian and the computational procedure

In this work, we have calculated the spectra of 5CB using an approximate method. The details of this method have been presented previously62 and only a few of the most important approximations will be discussed here. The 19- spin system of 5CB is divided into several subsystems according to the strength of the dipolar interaction. We assume that the kth subsystem contains Nk spins, and accordingly, the dimension of the Hamiltonian is 2Nk. The remaining E, spins (Ek= N- NJ are denoted as external and create an additional magnetic field Bext,i,

The proton NMR spectrum of molecules in liquid crystalline systems is mainly determined by intramolecular dipolar interactions. In analogy with solids, and in contrast to isotropic liquids, these interactions do not average to zero. Assuming the magnetic field to be parallel to the 2 axis in the laboratory frame, and using the high field approximation, the Hamiltonian for dipole-dipole interactions is given byfl

Hi=-2$ Bij 1ziIzj-i Ii*Ij -2 B,~~iyf~i* !

(19) i<j i

The perturbational field Bext,i is calculated for all possible states of the external spins, giving 2Ek contributions according to the expression

N Ek

Hd=-2C Bij i

.lci.lzj-i Ii.Ij , 1

(16) Bext,i- -c 2; Bijy-‘,

i<j j

where N is the number of protons in a molecule. The dipole- dipole interaction tensor Bij is defined by

where indices i and j relate to the subsystem and external spins, respectively. The sign in Eq. (20) is related to m, , i.e., the orientation of the jth spin with respect to the magnetic field. To obtain the spectrum for a subsystem, we average over contributions from all possible values of B,,, . The final NMR spectrum of the entire system is the sum of the spectra of the subsystems. The discrete spectral lines are broadened (using a Gaussian function) by a common factor, which car- responds to experimental broadening due to inhomogeneous magnetic field and other spin interactions.

Bij-3 r”fi((3 COZ? ~~ij- l)~~?~)+ (17)

Here ~ij is the angle between the internuclear vector rij and the direction of the magnetic field, and the other symbols have their usual meaning. In the molecular coordinate frame xyz, the interaction tensor is given by29

Bij=t yzh[S,((3 COS2 aij,-l)r,j3)+(S,-SSyy)

X((COS' aijx-C0S2 crij,)rii3)], (18)

where S,, , S, , and S,, are the order parameters of the nematics, ff+ are the angles between the internuclear vector rij and the axes of the molecular frame u =x, y, and z.

Typically, an overall width of a proton spectrum in the liquid crystalline phase is 15-30 kHz. The dipole-dipole interaction energy (in frequency units) between or&protons in a benzene molecule is 24 kHz, whereas this interaction is -30 kHz for methyl protons when the interspin vector is parallel to the magnetic field.

The first step in the calculation of a proton NMR spectrum involves construction of the matrix for the dipolar Hamiltonian (size 2N*2N), in an arbitrary basis of orthonor- ma1 functions. Second, eigenvalues and eigenvectors need to be evaluated by a diagonalization procedure, and finally the frequencies and the intensities of the spectral transitions can be calculated. In principle, a direct computation requires diagonalization of a matrix of order 2N. Constructing the Hamiltonian matrix in the basis of 1, eigenstates results in a block structure, and the computations reduce to N+ 1 problems of diagonalizing smaller matrices. Several methods for simplification of numerical calculations of proton NMR


GQ

2. ‘H NMR spectra of the 5CB molecule

The calculation of the matrix for dipolar interactions was performed using positions of the protons from MD trajectories and Eq. (17). In the FA simulation, these positions are given explicitly, whereas in the UA simulation, the positions were reconstructed from the carbon skeleton (cf. Sec. V B).

Figure 14 shows experimental proton NMR spectra63 of 5CB together with line shapes caIculated from the UA simulation. Spectra for fully protonated (19 spins) system are shown in Fig. 14(a) whereas the spectra of aromatic protons (eight spins) are displayed in Fig. 14(b). The latter corresponds to a molecule with a deuterated alkyl chain SCB-d,, . We note that the calculated spectra differ very clearly from the experimental. This disagreement is caused mainly by the failure in description of the torsional angle between the phenyl rings 8, as discussed in previous sections. We have corrected this angle (in the analysis of the trajectory) for every molecule at each time step by turning ring A through 3 1” around its paru axis. The “corrected” spectra are included in Fig. 14, and the agreement with the experimental line shapes is in fact satisfactory. In particular, the line shape corresponding to the SCB-d,, molecule shows very close agreement with the experimental spectrum. This indicates that the




I -30 -20 -10 0 10 20 30

Frequency (kHz)

-30 -20 -10 0 10’ 20 30 Frequency (kHz)

FCIG. 14. Proton NMR line shapes in the UA simulation (a) a 19 spin system and (b) an eight spin system. The spectra are (---) calculated from the simulation and i---) “corrected” (the details are given in the text). The experimental spectra (-) were taken from Ref. 63.

correction of 8, is sufficient and that the conformational distributions in the alkyl chain are reasonably predicted by the UA model.

Proton line shapes calculated using the FA model trajectory are displayed in Fig. 15. In this simulation, the rigid, aromatic part of the molecule seems to be modeled correctly, which can be observed in Fig. 15(b), where the experimental spectrum of SCB-d,, is compared with the calculated line shape. The total simulated spectrum (the 19-spin system) shown in Fig. 15(a) differs very strongly from the experimental trace. This difference can be mainly attributed to conformational distributions in the alkyl chain, which we also pointed out in the dihedral angles section and in the discussion of order parameters in connection with deuterium NMR.

In Figs. 14 and 15, only the shapes of the spectra are compared; the frequency ranges of experimental and simulated spectra are different. This deviation corresponds to the difference between simulated and experimental nematic order parameters according to Eq. (18). The “corrected” simulated spectra are therefore scaled by the ratio between these parameters. From the scaling we derived order parameters for the simulated liquid crystal; these values have been included in Table VI. In fact, we find very good agreement between the experimental order parameter and that derived from proton NMR spectra.

The final conclusion of this section is that proton NMR line shapes are very sensitive to molecular conformational distributions. Note that compared with.th& values based on experimental results,42 the length of the chain, in terms of all-frans probabilities, is overestimated in the UA and under- estimated in the FA models. Yet the proton spectra calculated

-2; I I

-10 0 10 20 Frequency &Hz)

b)

-;0 -iO 0 10 Frequency (kHz)

FIG. 15. Proton NMR line shapes (---) in the FA simuIation (a) a 19 spin system and (b) an eight spin system. The experimental spectra (-) were taken from Ref. 63.

from these simulations are entirely different, and only the UA model reasonably agrees with the experiments. The calculation of proton line shapes from a MD trajectory therefore provides a powerful test of the description of molecular conformations by the force field used in the simulation.

VII. CONCLUSIONS

Molecular dynamics simulations of 4-n-pentyl-4’- cyanobiphenyl (SCB) have been performed in the nematic phase using two interaction models. In the first model (which we also consider to be the main part of this work), all the CH, CH,, and CH, groups were treated as single interaction centers, i.e., as united atoms (UA). In the second, shorter simulation, the hydrogen atoms were explicitly included, i.e., a full atom (FA) model was used. The simulations were carried out for, respectively, 300 and 100 ps.

The energy of the system remained constant and the nematic director did not move with respect to the simulation cell on the time scale of our simulations. We believe therefore that the system was well equilibrated. Liquid crystalline order parameters were derived using different definitions of molecular frames. The calculated order parameters are in reasonable agreement with experimental values. Simulated translational diffusion constants predicted a somewhat higher anisotropy than shown by experiments. Low experimental anisotropy also requires low anisotropy in the molecular shape, which has been tentatively explained by invoking the concept of molecular aggregation. However, we have not



made any observations which would support such an expla- nation. In particular, the radial distribution functions do not indicate any molecular aggregation.

The UA model failed to predict a correct distribution of the torsional angle between the phenyl rings in the biphenyl fragment. This angle has been experimentally determined to be 30”, whereas the simulation predicts a planar conformation (OO). We believe that this failure is caused by insufficient repulsion between aromatic protons on adjacent rings. In the FA simulation, the distribution peaks at 25”, which is a satisfactory result. This model, however, did not correctly predict relative conformations of the alkyl chain.

Molecular reorientations were determined from time correlation functions for various vectors tixed in the molecule. The rotational diffusion constants for the overall motions were in good agreement with experimental values. We noted that the local mobility, i.e., the flexibility along the alkyl chain, increased using the FA model; this effect was clearly smaller in UA simulation.

16D. J. Photinos, E. T. Samulski, and H. Toriumi, J. Chem. Phys. 94,2758 (1991).

17S. J. Picken, W. F. van Gunsteren, P. T. Van Duijnen, and W. H. De Jeu, Liq. Cryst..6, 357 (1989).

“L. V. Woodcock, Chem. Phys. Lett. 10, 257 (1971). “D. Brown and J. H. R. Clarke, Mol. Phys. 51, 1243 (1984). *‘J. S. Lewis, E. Tomchuk, and E. Bock, Can. J. Phys. 65, 115 (1987). “J-P Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comput. Phys. 23,

327 (1977). “M. Dupuis, J. Rys, and H. F. King, J. Phys. Chem. 65, 111 (1976). mC. W. N. Cumper, S. K. Dev, and S. R. Landor, J. Chem. Sot. Perkins

Trans. II 46, 537 (1973). a“W. L. Jorgensen, J. D. Madura, and C. J. Swenson, J. Am. Chem. Sot.

106, 6638 (1984). 2SF. A. Momany, L. M. Carruthers, R. F. McGuire, and H. A. Scheraga, J.

Phys. Chem. 78, 1595 (1974).

Order parameters relevant for deuterium NMR spectra were determined for the segments in the alkyl chain. These were in reasonable agreement with experiments. Proton NMR line shapes were calculated from the trajectory using an approximate method for determination of the dipole- dipole Hamiltonian matrix. The spectra proved to be highly sensitive to molecular conformations and molecular order in the liquid crystalline phase.

Finally, we conclude that the extension from a united atom to a full atomic model is very expensive (computer time) and in fact unnecessary. We believe that the UA model may be improved, in particular, the dihedral angle in the biphenyl part and the local dynamics in the alkyl chain. Such an improvement has been proposed by introducing an anisotropic united atom mode1.52

%A. Laaksonen, Comp. Phys. Commun. 42, 271 (1986). “A V. Komolkin (unpublished results). SRI Hashim. G. R. Luckhurst, and S. Romano, Mol. Phys. 56,1217 (1985). “E. E. Bumell and C. A. De Lange, J. Magn. Reson. 39, 461 (1980). 30M. R. Wilson and M. P. Allen. Lia. Crvst. 12. 157 (1992). “R. Eppenga and D. Frenkel, Mol. whys: 52, 1303 (1984).’ 32G. R. Luckhurst, The Molecular Physics of Liquid Crystals (Academic,

New York 1979) p. 85. _ -_ .

33A. J. Leadbetter, R. M. Richardson, and C. N. Colling, J. Phys. 36, Cl-37 (1975).

34Y. Iwakabe, M. Hara, K. Kondo, S. Oh-ham, A. Mukoh, and H. Sasabe, Jpn. J. Appl. Phys. 31, L1771 (1992).

35M. R. Wilson and D. A. Dunmur, Liq. Cryst. 5, 987 (1989). 3aA. 3. Leadbetter, J. Frost, J. Gaughan, G. Gray, and A. MosIey, J. Phys. 40,

375 (1979). 37A. J. Leadbetter, F. P. Temme, A. Heidemann, and W. S. Howells, Chem.

Phys. Lett. 34, 363 (1975). 38G. J. Kriiger, Phys. Rep. 82, 229 (1982). 39K. S. Chu and D. S. Mori, I. Phys. 36, Cl-99 (1975). 40A Ferrarini and P. L. Nordio, J. Chem. Sot. Farad. Trans. 88, 1733

(1992). 4’C. J. R. Counsell, J. W. Emsley, G. R. Luckhurst, and H. S. Sachdev, Mol.

Phys. 63, 33 (1988). 42J. W. Emsley, G. R. Luckhurst, and C. l? Stockley, Proc. R. Sot. London

Ser. A 381, 117 (1982). 43J W. Emsley, J. C. Lindon, and G. R. Luckhurst, Mol. Phys. 30, 1913

(i975). ACKNOWLEDGMENTS

We thank Dick Sandstrom for valuable comments on the manuscript. This work has been supported by the Swedish Natural Science Research Council and Magnus BergvalIs Stiftelse. One of us (A.V.K.) was supported by the Swedish Institute.

‘G. R. Luckhurst and P. Simpson, Mol. Phys. 47, 251 (1982). ‘I. Ono and S. Rondo, Bull. Chem. Sot. Jpn. 65, 1057 (1992). 3M. R. Wilson and M. P Allen, Mol. Cryst. Liq. Cryst. 198, 465 (1991). 4A. V. Komolkin, Y. V. Molchanov, and P. P. Yakutseni, Liq. Cryst. 6, 39

(1989). 5 M. P. Allen and M. R. Wilson, J. Comp. Aid. Mol. Des. 3, 335 (1989). 6R. Y. Dong, J. Chem. Phys. 88, 3962 (1988). 7R. Y. Dong and G. M. Richards, Chem. Phys. Lett. 171, 389 (1990). ‘P. A. Beckmann, .I. W. Emsley, G. R. Luckhurst, and D. L. Turner, Mol.

Phys. 50, 699 (1983). ‘I? A. Beckmann, J. W. Emsley, G. R. Luckhurst, and D. L. Turner, Mol.

Phys. 59, 97 (1986). “‘C B. French, B. M. Fung, and M. Schadt, Liq. Cryst. Chem. Phys. Appl.

l&O, 215 (1!289). “J. S. Lewis, E. Tomchuk, and E. Bock, Liq. Cryst. 5, 1033 (1989). “J S Lewis, E, Tomchuk, H. M. Hutton, and E. Bock, J. Chem. Phys. 78,

632’(1983). 13S. Simon and A. Pines, Chem. Phys. Lett. 76, 263 (1980). 14R. Kiillner, K. H. Schweikert, E Noack, and H. Zimmermann, Liq. Cryst.

13,483 (1993). “E. T. Samulski and R. Y Dong, J. Chem. Phys. 77, 5090 (1982).

44N. G. Almarza, E. Encisco, and E 3. Bermejo, J. Chem. Phys. 96.4625 (1992).

45R. M. Levy, M. Karplus, and J. A. McCammon, J. Am. Chem. Sot. 103, 994 (1981).

&J. S. Lewis, E. Tomchuk, and E. Bock, Liq. Cryst. 14, 1507 (1993). 47R. M. Levy, M. Karplus, and P. Wolynes, J. Am. Chem. Sot. 103, 5998

(1981). 48G. Lipari and A. Szabo, J. Am. Chem. Sot. 104,4546 (1982). 49R. M. Levy and M. Karplus, Adv. Chem. Ser. 204,445 (1983). ‘OR. Y. Dong, Mol. Cryst. Liq. Cryst. 141, 349 (1986). “R. Y. Dong, Phys. Rev. A43,4310 (1991). 52S.‘Toxvaerd, J. Chem. Phys. 93,429O (1990). 53 K. Miiller, P. Meier, and G. Kothe, Progr. NMR Spectrosc. 17,211 (1985). 54C. A. Veracini, in NMR of Liquid Crystals, edited by J. W. Emsley (Reidel,

Dodrecht, 1985). “J. W. Emsley, G. R. Luckhurst, and C. P. Stockley, Mol. Phys. 44, 565

(1981). 56C. J. R. Counsell, J. W. Emsley, N. J. Heaton, and G. R. Luckhurst, Mol.

Phys. 54, 847 (1985). s7M Mehring, Principles of Highs Resolution NMR in Solids (Springer, Ber-

lin, 1983). 58Z. Luz and R. Naor, Mol. Phys. 46, 891 (1982). “H. Schmiedel, B. Hillner, S. Grande, A. Liische, and St. Limmer, J. Magn.

Reson. 40, 369 (1980). 60A. F. Martins, J. B. Ferreira, F. Volino, A. Blumstein, and R. B. Blumstein,

Macromolecules 16, 279 (1983). ‘*D Gailand H. Gerard, J. A. Ratto, F. Volino, and J. B. Ferreira, Macro-

molecules &, 4519 (1992). ,mA. V. Komolkin and Y. V. Molchanov, Liq. Cryst. 4, 117 (1989). fi3 Y. V. Molchanov .(unpublished results).

J. Chem. Phys., Vol. iO1, No. 5, 1 September 1994


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