dick sandstrom, andrei v. komolkin and arnold maliniak- molecular dynamics simulation of a liquid...

Molecular dynamics simulation of a liquid crystalline mixtureDick SandstromDivision of Physical Chemistry, Arrhenius Laboratory, Stockholm University, S-106 91 Stockholm, Sweden

Andrei V. KomolkinPhysics Institute, St-Petersburg State University, 198904 Saint Petersburg, Russia

Arnold MaliniakDivision of Physical Chemistry, Arrhenius Laboratory, Stockholm University, S-106 91 Stockholm, Sweden

~Received 11 December 1996; accepted 23 January 1997!

We present results from a molecular dynamics simulation of benzene dissolved in the mesogen 4-n-pentyl-48-cyanobiphenyl~5CB!. The computer simulation is based on a realistic atom-atompotential and is performed in the nematic phase. Singlet orientational distribution functions arereconstructed from order parameters employing several methods, and the estimated distributions arecompared with those obtained directly from the trajectory. Transport properties have been studiedby calculating translational diffusion coefficients in directions both parallel and perpendicular to theliquid crystalline director. The simulated diffusion coefficients were found to be of the same orderof magnitude as those measured in experiments. Second rank orientational time correlationfunctions are used to investigate molecular reorientations and significant deviations from the smallstep rotational diffusion model are established. Molecular structure and internal dynamics of 5CBhave been examined by correlating the time dependence of dihedral angles with effective torsionalpotentials. ©1997 American Institute of Physics.@S0021-9606~97!50117-6#

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I. INTRODUCTION

Computer simulation is an increasingly important toolthe study of fluid matter which exhibits long range orientional order.1 For example, liquid crystals belong to this caegory of phases. The explosive development of comppower during the last decade has made it possible to intigate both thermodynamical and molecular properties ofdered systems. When the main purpose is to study bulkhavior such as phase transitions, it is often sufficient tovery simple interaction models. The molecules are in thcases represented as rigid particles with an anisotrshape.2,3 However, when the computer simulation aimsdescribing molecular organization, structure and dynamicgreat detail, realistic atom-atom potentials mustemployed.4,5 It is also possible to use a hybrid between thetwo approaches. Palkeet al.have introduced a computationefficient method to study solutes in a model liquid crystallisolvent by combining the simple Gay–Berne potential wthe more elaborate Lennard–Jones site-site potential.6

Recently,7 we reported a molecular dynamics~MD!simulation of a liquid crystalline mixture consisting of bezene and 4-n-pentyl-48-cyanobiphenyl~5CB!. ~See Fig. 1.!The computer simulation was carried out in the nemaphase and used realistic atom-atom potential functions.purpose was to study orientational order and the structurthe mesophase. In this article, a further analysis of thisjectory is presented. We discuss briefly different waysestimating orientational distribution functions in ordered stems. The main objective here, however, is to examine vous dynamic processes.

The rotational motions present in a real mesophase himportant contributions from overall molecular reorientatiodirector fluctuations, and, in the case of a flexible molecu

7438 J. Chem. Phys. 106 (17), 1 May 1997 0021-9606/97/1

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conformational changes due to internal motions. The schastic molecular tumbling and the internal dynamics taplace on a similar timescale. This fact makes experimestudies of the rotational motion of flexible molecules exceingly difficult.8 Moreover, the experimental data must be iterpreted within a theoretical framework and the resultmotional parameters will, of course, depend on which mowe use in the analysis. These problems are avoided in ansimulation. We can easily investigate internal and ovedynamics separately, and perform tests of models commoemployed in analyses of experimental data.

The outline of this article is as follows. The computtional details are briefly summarized in Sec. II. In Sec. III wcompare two different ways of reconstructing the singlet oentational distribution from the second rank and fourth raorder parameters. Translational motion is treated in Secwhile molecular reorientation is studied in Sec. V. FinalSec. VI contains a discussion of molecular conformation ainternal dynamics of the 5CB nematogen.

II. COMPUTATIONAL DETAILS

The MD simulation of the liquid crystalline mixture waperformed in a rectangular cell containing 110 5CB andbenzene molecules. All carbon-hydrogen fragments wmodeled as single interaction centers~i.e., as united atoms!.The simulation was carried out in the nematic phaat a constant temperature of 290 K, and at a density of 0g cm23. The system was equilibrated for 1.0 ns, followeda 510 ps production run. The full details of the simulatiprocedure, force field parameters, and data analysis hbeen described elsewhere.7

We have also performed a molecular dynamics simution of neat benzene in the isotropic liquid phase at 300

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7439Sandstrom, Komolkin, and Maliniak: Simulation of a liquid crystalline mixture

The cell contained 216 molecules and the density was 0g cm23. Except for the absence of partial charges, all intaction parameters describing benzene are identical to temployed for the aromatic rings in 5CB. The total lengththis simulation was 200 ps, which included a 100 ps equbration period. All calculations were carried out on an IBRISC6000/355 computer using an optimized code writtenthis laboratory.9

III. LONG RANGE ORIENTATIONAL ORDER

The long range orientational order characteristic of luid crystals is traditionally quantified in terms of order prameters. Although useful, these quantities are merely aage values of various geometric functions~see below!.Consequently, they do not fully characterize the molecuorganization in the mesophase. At the single molecule lethe most complete description of the order is instead pvided by the singlet orientational distribution functionf .10

Unfortunately, f is only in exceptional cases accessiblerectly from experiments.11 A relevant question in this contexis, therefore, if it is possible to reconstruct the distributifunction from a small number of measured order parametTo answer this question, different strategies have beenployed to estimatef from the molecular order parameterThe calculated orientational distribution functions~ODFs!are then compared with those obtained directly from the Msimulation. Such considerations are expected to be impornot only in studies of liquid crystals, but also in structurinvestigations of oriented biopolymers.12

We will in this article treat the 5CB molecules as rigiaxially symmetric particles. This assumption is clearly nstrictly valid, but has proven to be a reasonable mode5CB.7 For such objects in a nematic phase,f will only de-pend on the angleb between the unique molecular symmetaxis,z and the director,n. Thez-axis of a solvent moleculewas chosen to coincide with thepara axis, whereas the sixfold symmetry axis defines thez-axis for benzene. Generaprocedures for locating the mesophase director and calcing order parameters in a computer simulation are descrin Ref. 7.

The ODF can formally be expanded in terms of a coplete basis set. If the Legendre polynomialsPL(cosb) areemployed, we may write

f ~b!5 121

52^P2&P2~cosb!1 9

2^P4&P4~cosb!1••• ,~1!

where the normalization condition and the orthogonalitythe Legendre functions have been used.13 Due to the head-tail symmetry of the nematic phase, only even functions

FIG. 1. Chemical structure of 4-n-pentyl-48-cyanobiphenyl~5CB!.

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pear in the expansion. The averages of the Legendre polymials ^PL& (L . 0) constitute the infinite set of ordeparameters. All of them are zero in an isotropic phaswhereas in a system of perfectly aligned molecules theytake the limiting value of one. Explicit expressions for thsecond rank and fourth rank order parameters are listedlow;

^P2&5E 12~3cos

2 b21! f ~b!sin bdb, ~2!

^P4&5E 18~35cos

4 b230cos2 b13! f ~b!sin bdb.

~3!

It is apparent from Eq.~1! that complete knowledge ofthe order in a liquid crystal only can be obtained if a^PL& values are known. The second rank order parame^P2& is relatively easy to determine. For example, NMspectroscopy is a strong candidate for such measuremen14

It is more difficult to obtain the fourth rank term. AlthoughRaman scattering15 and fluorescence depolarization16 maygive ^P4&, results from these techniques are rather cumb

FIG. 2. Singlet orientational distribution functions for~a! 5CB and~b! ben-zene. Solid lines correspond to simulated distributions whereas dashedrepresent estimates obtained using the orthogonal expansion approxim~see the text!.

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7440 Sandstrom, Komolkin, and Maliniak: Simulation of a liquid crystalline mixture

some to interpret. Order parameters of higher rank than fohave proven to be extremely difficult to determinexperimentally.17 We will, therefore, restrict the discussionto ^P2& and ^P4&.

The MD simulation does, of course, provide all ordeparameters. By calculating the lower members of these5CB and benzene at each time step, and then averaging othe whole production run the following result was obtaine^P2

5CB&50.658, ^P45CB&50.292, ^P2

b&520.175, and^P4b&

50.035. The negative second rank order parameter for bzene shows that the disclike solute prefers to orient withC6 axis perpendicular to the liquid crystalline director. Th^PL

i & values were essentially constant during trajectory sapling which indicates a well-equilibrated system.

We started the ODF analysis by assuming that on^P2

i & had been measured. The calculated distribution funtions corresponding to this heavily truncated expansion ashown in Fig. 2~a! ~5CB! and Fig. 2~b! ~benzene!. The dis-tributions obtained directly from the simulation are also included. Clearly, the two term expansion provides a very poestimate of the true ODFs. At some molecular orientationwe even see that the calculated distribution function for 5C

FIG. 3. Singlet orientational distribution functions for~a! 5CB and~b! ben-zene. Solid lines correspond to simulated distributions whereas dashed lrepresent estimates obtained using the maximum-entropy method~see thetext!.


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is negative. This is definitely not an acceptable behavior oprobability distribution. The knowledge of bothP2

i & and^P4

i & improves the ODF estimates. However, the agreembetween calculated and real curves describing 5CB is stillfrom perfect@Fig. 2~a!#. The situation for the solutes, whicare less ordered than the solvent molecules, is much b@Fig. 2~b!#. The three term distribution does in this case dscribe the ordering satisfactory. As expected, this showsthe orthogonal expansion in Eq.~1! converges more rapidlyfor systems of low order.13

Another way of estimatingf (b) is to assume that theODF can be written as

f ~b!5Q21 exp@a2P2~cosb!1a4P4~cosb!1•••#,~4!

whereQ is a normalization constant

Q5E exp@a2P2~cosb!

1a4P4~cosb!1•••#sin bdb. ~5!

The adjustable parametersaL are found by fitting the orderparameters to the right-hand side of Eqs.~2! and ~3!. Thereason for choosing an exponential function is thatmaximum-entropy~ME! principle as well as mean fieldtheory predict this functional form.13,18 Moreover, this typeof ODF fulfils the condition of always being positive. Results from least-squares fits of the known molecular orparameters to the distribution in Eq.~4! are shown in Fig. 3.The coefficientsaL derived from this analysis are given iTable I. In this case, knowledge of^P2

i & alone is quite suf-ficient to obtain an orientational distribution function whicagrees with the ODF extracted directly from the trajectoThis holds for both 5CB and benzene. Including^P4

i & in thefitting procedure yields only a slight improvement betwethe estimated and true distributions. From this we may cclude that the form of ODF in Eq.~4! is superior to theorthogonal expansion in Eq.~1!.

IV. TRANSLATIONAL DIFFUSION

The anisotropic translational diffusion in ordered medcan be described by a second rank tensor. In uniaxial liqcrystals this tensor is diagonal and axially symmetric wprincipal valuesDZZ5D i ~the diffusion coefficient paralleto the director! andDXX5DYY5D' ~the diffusion coefficientperpendicular to the director!. In nematic mesophases it habeen found that the diffusion parallel ton is faster than thatperpendicular ton.19

es

TABLE I. Values of aL parameters obtained from the order parameanalysis.

^Pi& used in fitting

Benzene 5CB

a2 a4 a2 a4

^P2& 21.05 ••• 3.44 •••^P2& and ^P4& 21.01 0.13 3.54 20.10

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We determinedD i andD' using the Green–Kubo rela-tion between the linear velocity time correlation functioCn(t)5^n(0)n(t)& and the diffusion coefficient20

Da51

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^na~0!na~ t !&dt5kBT

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wherena is the linear velocity of a molecule either paralle(a5i! or perpendicular~a5'! to the director. HerekB isthe Boltzmann constant,T is the absolute temperature,m isthe molecular mass, andta is the correlation time defined asthe integral over the normalized time correlation functio~TCF!. The TCFs for 5CB are displayed in Fig. 4~a! and thecorresponding diffusion coefficients areD i 5 3.0310210andD'51.2310210 m2 s21 (D i /D'52.5!. Within error limits,these values are identical to those found in our previous Msimulation of neat 5CB at 300 K.21 The temperature differ-ence~10° lower in the present study! is compensated by thereduced density and the translational diffusion remaintherefore, unchanged. Quasielastic neutron scatterin22

~QENS! has been used to determine the diffusion of 5CBthe nematic phase and these measurements yielded at 29D i55.3310211 andD'54.1310211 m2 s21, indicating thatour simulated diffusion coefficients are slightly to high. It i

FIG. 4. Normalized linear velocity time correlation functions describing~a!5CB and~b! benzene.


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nevertheless encouraging to see that theDa values obtainedfrom the MD simulation are on the same order of magnituas those determined by QENS. It is also important to remeber that measurements of translational diffusion in thermtropic liquid crystals are neither trivial nor free fromartefacts.19

The linear velocity TCFs for benzene in the isotropand nematic phase are shown in Fig. 4~b!. Only the correla-tion function corresponding to theZ-axis of the cubic simu-lation box is displayed for the liquid phase. TheX and Ycomponents are virtually identical to theZ component sinceall three directions in space are equivalent in an isotrosystem. The diffusion coefficient in liquid benzene2.231029 m2 s21 which is in agreement with experimentadata23 and previous computer simulations.24 As reflected bythe TCFs in Fig. 4~b!, the diffusive motion of benzene parallel and perpendicular ton are different from each other inthe nematic mesophase. The diffusion coefficients forsolute in the liquid crystalline phase are as followD i58.0310210 and D'57.0310210 m2 s21 (D i /D'

51.1!. In accordance with other studies of small solutesnematic solvents, the anisotropy of the diffusion of benzeis significantly smaller than that of 5CB.6,19 Zax et al. haveexploited multiple quantum NMR to determine the transtional diffusion of benzene in a nematic liquid crystal.25

These measurements yieldedD i51.1310210 andD'50.9310210 m2 s21 (D i /D'51.2!. The temperature was in thistudy not specified which makes a direct comparisontween experimental and simulated values impossible. Hever, it is clear that the ratioD i /D' for benzene is wellreproduced in the MD simulation.

V. MOLECULAR REORIENTATION

A wide range of spectroscopic methods can give infmation about dynamic processes on a molecular level.example, nuclear spin relaxation is a well-established tenique which provides useful knowledge about molecureorientation.26 Although powerful, this method yields onlyindirect information. To extract motional parameters suchcorrelation times, one has to employ a theoretical mode

TABLE II. Explicit expressions for the second rank correlation functionst 5 0. Initial values offmn(t) obtained directly from the TCFs are comparewith values calculated usingP2& and ^P4& ~in brackets!.

mn fmn(0) Benzenea Benzeneb 5CBb

00 1/512^P2&/7118 P4&/35 0.201 0.166~.168! 0.538~.538!01 1/51^P2&/7212 P4&/35 0.200 0.168~.163! 0.193~.194!02 1/522^P2&/713^P4&/35 0.199 0.250~.253! 0.038~.037!10 1/51^P2&/7212 P4&/35 0.200 0.168~.163! 0.193~.194!11 1/51^P2&/1418^P4&/35 0.202 0.202~.196! 0.291~.314!12 1/52^P2&/722^P4&/35 0.202 0.207~.223! 0.085~.089!20 1/522^P2&/713^P4&/35 0.199 0.250~.253! 0.038~.037!21 1/52^P2&/722^P4&/35 0.202 0.207~.223! 0.085~.089!22 1/512^P2&/71^P4&/70 0.206 0.138~.150! 0.391~.392!

aIn liquid phase.bIn nematic phase.

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the motion. Computer simulation has turned out to be a vaable tool in developing and testing such models.27

A. Basic definitions and limiting values

A convenient way to characterize the rotational motiis by using orientational time correlation functions

fmn;m8n8LL8 ~ t !5^Dmn

L* ~V~0!!Dm8n8L8 ~V~ t !!&, ~7!

whereDmnL (V) is a Wigner rotation matrix element of ran

L, andV5$a,b,g% is a set of time dependent Euler anglrelating the orientation of the director frameD to the mo-lecular fixed axis systemM . In contrast to isotropic liquidsTCFs in ordered systems can couple tensorial propertiedifferent rank~i.e., L may differ fromL8!. We will concen-trate on correlation functions withL5L852 which, from anexperimental point of view, are among the most importones. Second rank TCFs appear, e.g., in the expressionnuclear spin relaxation rates and Raman band shapes. Ifollowing we simplify the notation by dropping the supescripts L and L8 on f. Note that only the real part ofmn;m8n8(t) needs to be considered.

Symmetry arguments involving both the molecules athe phase may be used to reduce the number of indepenTCFs. This issue has been discussed in a series of papeZannoni and co-workers.28–30 The phase symmetry entethrough the indicesm andm8, whereasn andn8 are relatedto the symmetry of the molecule. The simplest possible sation in an ordered system occurs when both moleculemesophase possess axial symmetry, leading to the followselection rule31

fmn;m8n8~ t !5^Dmn2* ~V~0!!Dm8n8

2~V~ t !!&dmm8dnn8 , ~8!

wheredmm8 is the Kroenecker delta. In this case, we hanine independent orientational correlation functionsfmn(t)with m andn equal to 0, 1, or 2.

The initial values of the TCFs ^Dmn2* (V(0))

Dmn2 (V(0))& can be expressed in terms of the orientatio

order parameters.31 Explicit relationships valid for cylindri-cally symmetric particles in a uniaxial phase are givenTable II. Notice that the second rank correlation functionst50 depend on bothP2& and ^P4&. These expressions armodel independent and provide, therefore, a useful checthe previously determined order parameters. In Table IIcomparefmn(0) obtained directly from the correlation functions with initial values calculated using the order paramegiven in Sec. III. The agreement is satisfactory for both bzene and 5CB.

The values of the correlation functions at the other tilimit, i.e., whent→`, are also model independent;

fmn~`!5^Dmn2* &^Dmn

2 &, ~9!

which simply corresponds to a product of two second raorder parameters. In isotropic liquids there is no long ranorientational order and all TCFs decay to zero. For a uniamolecule in a nematic liquid crystal there exists only ononvanishing order parameter of rank two;^D00

2* &5^D002 &


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5^P2&. Consequently, all correlation functions are expecto decay to zero exceptf00(t) which should approach^P2&

2.This is as far as we can go without invoking a dynam

model for the molecular reorientation.

B. The small step rotational diffusion model

Rotational dynamics of molecules in isotropic as wellin anisotropic systems is often described by the small sdiffusion model.32 This model is based on the assumptithat molecular reorientations proceed through a randomquence of infinitesimal angular jumps. For a rigid uniaxmolecule, it is possible to define a molecular fixed framewhich the rotational diffusion tensor is diagonal with princpal componentsRi andR' . Ri describes the spinning motion of the molecule about its symmetry axis whereasR'

refers to the tumbling motion of the symmetry axis.To derive expressions forfmn(t), one must solve the

rotational diffusion equation in the presence of a macscopic restoring potentialU(b). The overall shape ofU(b) is dictated by the symmetry of the phase. The apolaof a nematic mesophase requires, e.g., the ordering poteto be an even function. In general, the correlation functiomay be written as infinite sums of decaying exponentials33

fmn~ t !5fmn~`!1(i51

`

ai exp@2bit#, ~10!

whereai andbi are determined numerically. In practice, E~10! has to be truncated and a single exponential approxition has often been employed8,34

fmn~ t !5fmn~`!1@fmn~0!2fmn~`!#exp@2t/tmn#.~11!

The above equation has a number of attractive properties~i!it is analytically very simple,~ii ! it is exact att50 and att5`, ~iii ! it is exact in the limits of perfect as well as ovanishing orientational order~i.e., in isotropic systems!, and~iv! it is exact to linear order in time for short times. Seveauthors have derived expressions for the correlation ttmn appearing in Eq.~11!. Moro and Nordio

35 used a pertur-bation calculation, while Szabo36 and Kirov et al.37 em-ployed a short time expansion. Interestingly, these twoproaches yield identical results fortmn :

tmn215cmnR'1n2~Ri2R'!. ~12!

The coefficientscmn , which depend onP2& and ^P4&, aretabulated in Ref. 37. In the isotropic limit,cmn reduces to 6for all combinations ofm andn. Hence, the rotational dif-fusion model predicts in this casem-independent TCFs.

The correlation functions for benzene in the isotropliquid phase are shown in Fig. 5. All TCFs are displayednormalized form, that is

fmn~ t ![^Dmn2* ~V~0!!Dmn

2 ~V~ t !!&/

^Dmn2* ~V~0!!Dmn

2 ~V~0!!&. ~13!

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The anisotropic motion of benzene manifests itself byclearn dependence. As expected on symmetry grounds,correlation functions do not depend on the director indm. The short time behavior~t,0.5 ps! is nonexponential in-dicating that the nature of the motion is not diffusive.relatively free molecular rotation is possible at these shtimes and Gaussian-shaped TCFs, characteristic ofinertial-type of motion, are observed.38 After this first inertial

FIG. 5. Normalized second rank orientational time correlation functionsbenzene in the isotropic liquid phase. Note the difference between theaxis in ~a! as compared to those in~b! and ~c!.


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effect, the correlation functions withn50 or 1 decay ap-proximately as single exponentials. On the other hand,simulated TCFs withn52 deviate significantly from the dif-fusion model in the entire time interval. A possible explantion could be that the spinning motion of a benzene molecis dominated by inertial and not by diffusive reorientatioThis complicated behavior makes it impossible to extrbothRi andR' from the orientational time correlation func

reFIG. 6. Normalized second rank orientational time correlation functionsbenzene in the liquid crystalline mixture. Note the difference betweentime axis in~a! as compared to those in~b! and ~c!.

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tions. We have nevertheless fittedfm0(t) to single exponen-tials in the time interval 1–5 ps, which resulted in an averacorrelation timetm05t'51.5 ps. Similar values oft' havepreviously been obtained from MD simulations,6,24 and fromNMR and Raman experiments.39 For example, Do¨lle et al.have performed nuclear spin relaxation measurements inuid benzene and found a value oft'51.6 ps at 293 K.40

The simulated TCFs for benzene in the liquid crystallimixture are shown in Fig. 6. The correlation functions are

FIG. 7. Normalized second rank orientational time correlation functions5CB.


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this case dependent on bothm andn. These curves exhibiseveral characteristic features which we also observedisotropic benzene;~i! a Gaussian beginning signifying initiafree rotation,~ii ! fm0 can, after a rapid initial decay, bapproximated by single exponentials, and~iii ! fm2 arestrongly nonexponential. To facilitate a direct compariswith the rotational motion of benzene in the isotropic phathe three decays in Fig. 6~a! were fitted to an exponentiabetween 2 and 12 ps. The correlation times, correspondinthese exponentials, were found to be;t00532 ps,t10517 ps, andt20516 ps. By using Eq.~12! together withthe second rank and fourth rank order parameters from SIII, we obtained an estimate oft''20 ps (t'51/6R'!.Comparison of this correlation time with the one calculatin the liquid phase shows thatt' increases by an order omagnitude from an isotropic liquid to a liquid crystal.should, however, be remembered that the simulation ofbinary mixture was performed at 290 K, while isotropic bezene was simulated at 300 K. To preserve reasonable sttical significance, the correlation functions were only calclated during 12 ps. The TCFs withn50 decay rather slowlyand the infinite time limits are clearly not reached within thtime duration.

Orientational time correlation functions for 5CB are diplayed in Fig. 7. These TCFs were calculated for a coonate system attached to the cyano ring, with thez-axis in thedirection of thepara axis. By this choice, the influence frominternal bond rotations on the simulated correlation functiois avoided. The TCFs depend on bothm andn, and have arapid initial decay. A similar short time behavior was oserved in an MD simulation of a one-component mesomphic system composed of rodlike molecules.41 In that study,the authors also reported that the second rank correlafunctions can be well described by single exponentials a10 ps. This is in sharp contrast to what we observe in FigEven if the first 50 ps of the TCFs are ignored, Eq.~11! doesstill provide an inadequate description of the rotational dnamics of the solvent molecules. We tried to account formultiexponential decays of the 5CB correlation functionsusing the numerical solutions of the rotational diffusioequation provided by Vold and Vold.33 However, we couldnot find any combination ofRi andR' which is consistentwith the curves in Fig. 7.

The above analysis is based on the approximationthe cyano ring fragment is cylindrically symmetric about tpara axis. This assumption is clearly not strictly valid sincthe local biaxiality of a phenyl ring in 5CB is arounA6(^D022

2 &1^D022 &)/2'0.05.42 Tarroni and Zannoni have

investigated the influence of molecular biaxiality on corretion functions.28 Their calculations indicate that the effect foTCFs with n5n8 ~i.e., the type of correlation functionwhich we consider! is small. Therefore, we do not believthat the failure of the diffusion model in describing the rottional dynamics of 5CB is a consequence of molecular asmetry.

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VI. MOLECULAR STRUCTURE AND INTERNALROTATIONAL MOTION

It is well known that molecular structure and internbond rotations greatly influence the physical properties oliquid crystal.43 The determination of conformational distrbutions usually involves NMR experiments in whicthrough-space magnetic dipole-dipole~DD! couplings aremeasured.44,45 The extracted set of DD couplings must binterpreted within a theoretical framework, and the estimaprobability distributions will depend on the model employeFortunately, this model-dependence is often rather small46

It is much more difficult to study therate of rotationsabout molecular bonds. A careful analysis of NMR linshapes can, under favorable conditions, provide informaabout the internal dynamics. These motions are, howeusually so fast in molecules confined in liquid crystallineisotropic phases so that the dynamical information is lostthis fast-motion regime one can instead measure nuclearrelaxation rates which are sensitive to the rate of conformtional changes. The main problem associated with thisproach is that the spin relaxation may be dominated by omotions such as overall molecular tumbling and direcfluctuations.47

Computer simulations are ideally suited for investigtions of molecular structure and internal dynamics sincehave complete knowledge of the temporal developmeneach atom coordinate in the simulation cell. This leads tunique opportunity to disentangle the internal motion frothe overall molecular reorientation, and hence to a possibto study bond rotations directly.

We will concentrate on the inter-ring angle and the fidihedral angle in the chain~Q1 , andQ2 in Fig. 1! since theyhave been studied experimentally. The motions of segmlocated further out in the aliphatic chain of 5CB were briediscussed in our previous paper.21

FIG. 8. Effective torsional potentials describing the inter-ringQ1 and thering-chainQ2 dihedral angles in 5CB.


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A. Effective torsional potentials

The internal rotational motions of an isolated molecuare in an MD simulation modeled using bonded as wellnonbonded interactions. The first contribution consists of‘‘naked’’ torsional potential whereas the latter includes thelectrostatic and Lennard–Jones terms. Moreover, the mlecular dihedral angles may be affected by anisotropic intemolecular interactions.

A convenient way to examine dihedral angles is to caculate an effective torsional potentialVeff(Q) from

p~Q!5C exp@2Veff~Q!/RT#, ~14!

wherep(Q) is the torsional angle distribution function,C isa normalization constant, andR is the molar gas constant.Effective torsional potentials forQ1 andQ2 are shown inFig. 8. The functions are not exactly symmetric abouQ590°, as they should be for 5CB. This is because the Msimulation does not sample the whole conformational spaproperly. The noisy region atQ 5 90° for the inter-ring angleis another consequence of insufficient dihedral angle sapling.

The structure of a biphenyl molecule is very sensitivethe aggregation state: The angle of twist is; 45° in the gas

FIG. 9. Temporal developments of~a! the inter-ringQ1 and ~b! the ring-chainQ2 dihedral angles in 5CB.

, No. 17, 1 May 1997

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phase, close to 0° in the solid phase, and; 35° when dis-solved in isotropic or anisotropic liquids.46,48NMR has beenused to determine the conformation of the cyanobiphefragment in neat 5CB.42,44,49,50These measurements, whicwere performed in the nematic phase, yielded a torsioangle value of; 35°. The inter-ring angle depicted in Fig.has a minimum energy conformation atQ1 5 39°, which cor-relates well with the experimental results discussed aboThis confirms that our force field parameters adequatelyscribe the main structural features of the cyanobiphenyl mety. It is more difficult to assess the correctness of the potial barrier heights. An additive potential~AP! analysis50 ofproton DD couplings obtained in neat 5CB indicates tVeff(Q150°)'2andVeff(Q1590°)'5 kJ mol21, i.e., some-what lower than the values found in the MD simulatioVeff(Q1 5 0°)'7 andVeff(Q1590°)'14 kJ mol21. On theother hand, a structural investigation of 4-methox48-cyanobiphenyl yielded Veff(Q150°)'5 and Veff(Q1

5 90°)' 12 kJ mol21 using the APmethod.51

The effective torsional potential for the ring-chain boexhibits a broad minimum when the chain is oriented ppendicular to the ring plane~i.e., atQ25690°!, with a bar-rier to rotation of around 5 kJ mol21. The conformation ofthis 5CB fragment was recently studied by Emsleyet al.ex-ploring proton NMR spectroscopy.52 The rotational potentiaobtained in that investigation has a minimum at690°, inagreement with our result. The barrier height determinfrom the NMR experiments~.22 kJ mol21) is, however,larger than that observed in the MD simulation. This iscourse a reflection of our particular choice of molecular fofield.

B. Dihedral dynamics

The torsional angle history of a representative 5CB mecule is plotted in Fig. 9. Temporal developments are shofor the 510 ps long production run. We start with discussthe internal dynamics of the inter-ring angleQ1 . Three dis-tinct motions are easily observed in Fig. 9~a!: ~1! A very fasttorsional libration within the four equivalent potential wellocated aboutQ15639° andQ156141° ~cf. Fig. 8!. Thismotion occurs on a subpicosecond timescale and has amean-square~RMS! fluctuation of 11°.~2! Transitions whichinvolve a passage over barriers atQ150° andQ15180°.We will refer to this as a low barrier crossing~LBC!. ~3!Transitions over the potential barriers atQ15690° whichwill be called high barrier crossings~HBC!. The latter typeof transition occurs only twice for the particular 5CB moecule described in Fig. 9~a! ~one at;210 ps and the other a;420 ps!. We have calculated transition rates for LBCs aHBCs, and found the following; kLBC584 andkHBC53 ns21. These numbers represent averages oversolvent molecules. The rates were calculated by countingtotal number of transitions and dividing by the productitime duration. A transition was counted when the torsioangle crossed a potential barrier and reached the bottomthe well corresponding to the new minimum energy confmation.


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The dihedral angle history for the ring-chain bondQ2 isshown in Fig. 9~b!. In this case, the internal dynamics cosists of two types of motion:~1! Fast torsional librations withan RMS fluctuation of 28°. This value is considerably highthan that observed for the inter-ring angle. The reasonthis behavior is that the wells of the rotational potential fQ2 are much broader than those forQ1 ~see Fig. 8!. ~2!Passages over barriers atQ250° andQ25180°. Thetran-sition rate for this type of motion is 88 ns21, i.e., close towhat we found forkLBC . The similarity between these twrates stems from the fact that both transitions involve a brier crossing of height;6 kJ mol21.

VII. SUMMARY

We have in this article presented results from an Msimulation of benzene dissolved in the nematic liquid crys5CB. A large number of static and dynamic molecular chacteristics of both the solute and solvent were extracted fthe trajectory. Many of the orientation and time dependproperties obtained are essentially not accessible by expmental means.

The singlet orientational distribution functions were rconstructed from the order parameters using two differapproaches; a truncated orthogonal expansion andmaximum-entropy function. It was found that the latter wsuperior for both benzene and 5CB. In fact, knowledge^P2& alone provides, using the ME method, a probabildistribution in remarkably good agreement with the simlated one~cf. Fig. 3!.

The transport properties discussed in Sec. IV confithat translational diffusion along the director is faster thdiffusion perpendicular to the director in a nematic liqucrystal. The anisotropy ratioD i /D' is 2.5 for 5CB and 1.1for benzene which is a reasonable result given the differein molecular shape.

Molecular reorientation was examined in terms of orietational time correlation functions. The short time behavof the TCFs for benzene exhibit characteristic features ofinertial component. We analyzed the correlation functioemploying the small step rotational diffusion model and tconclusions may be summarized as follows:~i! The reorien-tation of theC6 axis in liquid benzene is rather well described by the diffusion model. The correlation time of thmotion increases by an order of magnitude from the isotroliquid to the liquid crystalline phase.~ii ! Benzene correlationfunctions withn52 ~i.e., those which depend strongly on thspinning motion! are not adequately described by small stdiffusion. This holds true for both the isotropic and the aisotropic system. It is not easy to assess the reliabilitythese findings since we exploited united atoms for the solHowever, a previous MD simulation of liquid benzene usia twelve-siteab initio potential indicated also departure fropure rotational diffusion.24 ~iii ! The time correlation func-tions for 5CB do not exhibit a single exponential decaylong times—an assumption often made in experimenanalyses. A more rigorous treatment, based on numericalutions of the Smoluchowski equation,33 did also fail in ac-

, No. 17, 1 May 1997

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counting for the observed molecular reorientation. It is byond the scope of this article to establish what sortmotional model that best describes the set of orientatiocorrelations in Fig. 7. We merely note that a routine useEq. ~11! in experimental data analysis may lead to a dracally erroneous picture of molecular motions.

Computer simulations offer a unique possibility to stumolecular structure and internal dynamics in a model inpendent fashion. It proved very useful to correlate effecttorsional potentials with the dihedral angle history. For eample, one may easily estimate transition rates and thetial extent of torsional librations.

ACKNOWLEDGMENTS

This work was supported by the Swedish Natural Sence Research Council and the Carl Trygger Foundation

1M. P. Allen and M. R. Wilson, J. Comp. Aid. Molec. Des.3, 335 ~1989!.2D. Frenkel, Mol. Phys.60, 1 ~1987!.3M. P. Allen, Liq. Cryst.8, 499 ~1990!.4S. J. Picken, W. F. van Gunsteren, P. T. Van Duijnen, and W. H. DeLiq. Cryst.6, 357 ~1989!.

5C. W. Cross and B. M. Fung, J. Chem. Phys.101, 6839~1994!.6W. E. Palke, J. W. Emsley, and D. J. Tildesley, Mol. Phys.82, 177~1994!.7D. Sandstro¨m, A. V. Komolkin, and A. Maliniak, J. Chem. Phys.104,9620 ~1996!.

8The Molecular Dynamics of Liquid Crystals, edited by G. R. Luckhurstand C. A. Veracini~Kluwer Academic, Dordrecht, 1994!.

9A. V. Komolkin ~unpublished!.10G. R. Luckhurst,Nuclear Magnetic Resonance of Liquid Crystals, editedby J. W. Emsley~Reidel, Dordrecht, 1985!.

11G. S. Harbison, V. D. Vogt, and H. W. Spiess, J. Chem. Phys.86, 1206~1987!.

12Z. Song, O. N. Antzutkin, A. Rupprecht, and M. H. Levitt, Chem. PhLett. 253, 349 ~1996!.

13C. Zannoni,The Molecular Dynamics of Liquid Crystals, edited by G. R.Luckhurst and C. A. Veracini~Kluwer Academic, Dordrecht, 1994!.

14Nuclear Magnetic Resonance of Liquid Crystals, edited by J. W. Emsley~Reidel, Dordrecht, 1985!.

15S. Jen, N. A. Clark, P. S. Pershan, and E. B. Priestley, J. Chem. Phys66,4635 ~1977!.

16A. Arcioni, R. Tarroni, and C. Zannoni, J. Chem. Soc. Faraday Trans.87,2457 ~1991!.

17I. W. Hamley, S. Garnett, G. R. Luckhurst, S. J. Roskilly, J. Skov Pedsen, R. M. Richardson, and J. M. Seddon, J. Chem. Phys.104, 10 046~1996!.

18R. L. Humphries, P. G. James, and G. R. Luckhurst, J. Chem. Soc. Fday Trans. 268, 1031~1972!.

19G. J. Kruger, Phys. Rep.82, 229 ~1982!.


-falfi-

-e-a-

-

u,

-

ra-

20D. A. McQuarrie, Statistical Mechanics~Harper & Row, New York,1976!.

21A. V. Komolkin, A. Laaksonen, and A. Maliniak, J. Chem. Phys.101,4103 ~1994!.

22A. J. Leadbetter, F. P. Temme, A. Heidemann, and W. S. Howells, ChPhys. Lett.34, 363 ~1975!.

23P. Stilbs and M. E. Moseley, Chem. Script.15, 176 ~1980!.24P. Linse, S. Engstro¨m, and B. Jo¨nsson, Chem. Phys. Lett.115, 95 ~1985!.25D. Zax and A. Pines, J. Chem. Phys.78, 6333~1983!.26R. L. Vold and R. R. Vold,The Molecular Dynamics of Liquid Crystals,edited by G. R. Luckhurst and C. A. Veracini~Kluwer Academic, Dor-drecht, 1994!.

27R. W. Pastor, R. M. Venable, M. Karplus, and A. Szabo, J. Chem. P89, 1128~1988!.

28R. Tarroni and C. Zannoni, J. Chem. Phys.95, 4550~1991!.29E. Berggren, R. Tarroni, and C. Zannoni, J. Chem. Phys.99, 6180~1993!.30E. Berggren and C. Zannoni, Mol. Phys.85, 299 ~1995!.31C. Zannoni and M. Guerra, Mol. Phys.44, 849 ~1981!.32R. Y. Dong,Nuclear Magnetic Resonance of Liquid Crystals~Springer,New York, 1994!.

33R. R. Vold and R. L. Vold, J. Chem. Phys.88, 1443~1988!.34S. Y. Yakovenko, A. A. Minko, B. Arnscheidt, and J. Pelzl, Liq. Cryst.19,449 ~1995!.

35G. Moro and P. L. Nordio, Chem. Phys. Lett.96, 192 ~1983!.36A. Szabo, J. Chem. Phys.81, 150 ~1984!.37N. Kirov, I. Dozov, and M. P. Fontana, J. Chem. Phys.83, 5267~1985!.38A. Maliniak, A. Laaksonen, J. Kowalewski, and P. Stilbs, J. Chem. Ph89, 6434~1988!.

39W. A. Steele, Adv. Chem. Phys.34, 1 ~1976!.40A. Dolle, M. A. Suhm, and H. Weinga¨rtner, J. Chem. Phys.94, 3361

~1991!.41S. Y. Yakovenko, A. A. Muravski, G. Kro¨mer, and A. Geiger, Mol. Phys86, 1099~1995!.

42D. Sandstro¨m and M. H. Levitt, J. Am. Chem. Soc.118, 6966~1996!.43Liquid Crystals, edited by H. Stegemeyer~Springer, New York, 1994!.44S. Sinton and A. Pines, Chem. Phys. Lett.76, 263 ~1980!.45D. Sandstro¨m, K. T. Summanen, and M. H. Levitt, J. Am. Chem. So116, 9357~1994!.

46W. E. Palke, D. Catalano, G. Celebre, and J. W. Emsley, J. Chem. P105, 7026~1996!.

47F. Noack and K. H. Schweikert,The Molecular Dynamics of Liquid Crystals, edited by G. R. Luckhurst and C. A. Veracini~Kluwer Academic,Dordrecht, 1994!.

48M. Akiyama, T. Watanabe, and M. Kakihana, J. Phys. Chem.90, 1752~1986!.

49D. Catalano, L. Di Bari, C. A. Veracini, G. N. Shilstone, and C. ZannoJ. Chem. Phys.94, 3928~1991!.

50G. Celebre, M. Longeri, E. Sicilia, and J. W. Emsley, Liq. Cryst.7, 731~1990!.

51J. W. Emsley, G. Celebre, G. De Luca, M. Longeri, and F. LucchesLiq. Cryst.16, 1037~1994!.

52J. W. Emsley, G. De Luca, G. Celebre, and M. Longeri, Liq. Cryst.20,569 ~1996!.

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dick sandstrom, andrei v. komolkin and arnold maliniak- molecular dynamics simulation of a liquid...

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