Molecular field theory with atomistic modeling for the curvature elasticity of nematic liquid...

Molecular field theory with atomistic modeling for the curvature elasticity ofnematic liquid crystalsMirko Cestari, Alessandro Bosco, and Alberta Ferrarini

Citation: J. Chem. Phys. 131, 054104 (2009); doi: 10.1063/1.3193555 View online: http://dx.doi.org/10.1063/1.3193555 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v131/i5 Published by the American Institute of Physics.

Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

Downloaded 08 Apr 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

Molecular field theory with atomistic modeling for the curvature elasticityof nematic liquid crystals

Mirko Cestari,1 Alessandro Bosco,1,2 and Alberta Ferrarini1,a1Dipartimento di Scienze Chimiche, Universit di Padova, via Marzolo 1, 35131 Padova, Italy2International School for Advanced Studies (SISSA), via Beirut 2-4, 34151 Trieste, ItalyReceived 24 February 2009; accepted 8 July 2009; published online 3 August 2009

Liquid crystals oppose a restoring force to distortions of the main alignment axis, the so-calleddirector. For nematics this behavior is characterized by the three elastic moduli associated with thesplay K11, twist K22, and bend K33 modes; in addition, two moduli for mixed splay-bend k13and saddle-splay k24 can be defined. The elastic constants are material properties which depend onthe mesogen structure, but the relation between molecular features and deformations on a muchlonger scale has not been fully elucidated. The prediction of elastic properties is a challenge fortheoretical and computational methods: atomistic simulations require large samples and must beintegrated by statistical thermodynamics models to connect intermolecular correlations and elasticresponse. Here we present a molecular field theory, wherein expressions for the elastic constants ofnematics are derived starting from a simple form of the single molecule orientational distributionfunction; this is parametrized according to the amount of molecular surface aligned to the nematicdirector. Such a model allows a detailed account of the chemical structure; moreover theconformational freedom, which is a common feature of mesogens, can be easily included. Given theatomic coordinates, the elastic constants can be calculated without any adjustable parameter at a lowcomputational cost. The example of 4-n-pentyl,4-cyanobiphenyl 5CB is used to illustrate thecapability of the developed methodology; even for this mesogen, which is usually taken as aprototypal rodlike system, we predict a significant dependence of the elastic moduli on themolecular conformation. We show that good estimates of magnitude and temperature dependence ofthe elastic constants are obtained, provided that the molecular geometry is correctly taken intoaccount. 2009 American Institute of Physics. DOI: 10.1063/1.3193555

I. INTRODUCTION

Liquid crystals LCs are characterized by curvatureelasticity: they oppose a restoring force to distortions of thedirector, i.e., the average molecular alignment axis. Elastic-ity is one of the main properties controlling the LC behavioron the submicrometric length scale and plays a key role forapplications: in LC cells the director profile, which deter-mines the optical behavior, is the result of the tradeoff be-tween the deformation induced by an external electric fieldand the elastic response of the LC medium.1,2

The elastic continuum theory of nematic LCs was devel-oped starting from the early 30s of last century by Oseen,3Zocher,4 and Frank;5 subsequently a microscopic approachwas developed by Nehring and Saupe.6 The general expres-sion of the density of elastic energy for nonchiral nematics iscomprised of five contributions, each characterized by a dif-ferent modulus: in addition to the three deformations sche-matized in Fig. 1, which are denoted as splay, twist, and bendwith moduli K11, K22, and K33, respectively, there are twoother modes called splay-bend with modulus k13 and saddle-splay with modulus k24. These mixed modes are sometimesneglected considering that when calculating the deformation

free energy of a sample, their volume integrals can be re-duced to surface integrals.2 We shall denote the splay, twist,and bend deformations as bulk contributions and the mixeddeformations as surfacelike terms, as in most of the litera-ture.

The elastic moduli of nematics have a magnitude of theorder of some piconewton with values depending on thechemical structure of the constituents.7 The requirement ofstability of the nematic state with uniform director posesconditions on these values: the so-called Ericksen inequali-ties K110, K220, K330, K22 k24, and 2K11K22+k24 were obtained by minimization of the elastic energywithout taking into account the splay-bend term.8 Difficultiesin the minimization arise if this contribution is included. Thisis a controversial issue, which has been widely debated;914 itwas shown that spontaneous deformations induced by the k13and k24 terms can occur in a nematic layer even if the Erick-sen inequalities are satisfied.15

In the case of nematics formed by elongated moleculescalamitics, the sequence K33K11K22 is generally found.These elastic constants generally increase with the degree oforder; but those for splay and twist exhibit a relatively weaktemperature dependence, whereas the bend stiffness canchange more steeply. K11 and K22 are approximately propor-tional to Szz

2, with Szz being the major orientational order

parameter. The bend elastic constant is also the most sensi-aAuthor to whom correspondence should be addressed. Electronic mail:

[email protected].

THE JOURNAL OF CHEMICAL PHYSICS 131, 054104 2009

0021-9606/2009/1315/054104/16/$25.00 2009 American Institute of Physics131, 054104-1


tive to changes in the molecular structure. We can take as anexample 4-n-pentyl,4-cyanobiphenyl 5CB, whose struc-ture is displayed in Fig. 2. Figure 3 shows the temperaturedependence of the bulk elastic constants of 5CB.1619 We cannotice some discrepancies between the data reported in theliterature, which were ascribed to the errors affecting theemployed techniques and to the chemical purity of samplessee Ref. 20 for a survey. As a matter of fact, the experi-mental determination of elastic constants is not trivial andthe results are characterized by non-negligible uncertainty; ingeneral the ratios K22 /K11 and K33 /K11 can be determinedmore accurately than the absolute value of the moduli.

The anisotropy of elasticity defined by the ratios K22 /K11and K33 /K11 is itself an important characteristic of nematics,which can be of utmost relevance for technological applica-tions; for instance, it controls the optical threshold voltageand the steepness of the voltage-transmittance curve intwisted nematic displays.21 The anisotropy of elastic con-stants characterizes the mesoscale behavior of LCs: it deter-mines the anisotropy of the interactions between embeddedcolloidal particles,2224 the structure of topological defects,25and the location of defects in confined nematics.26 The ratiosK22 /K11 and K33 /K11 exhibit a significant yet scarcely under-stood dependence on the molecular structure of theconstituents.7 The latter is particularly sensitive to structuralchanges; the speculation that it would be an increasing func-tion of the length-to-width ratio is contradicted by the obser-vation that it decreases with increasing chain length withinhomologous series27 and that it can change under chemicalmodifications which do not significantly affect the length-to-width ratio.28

The availability of reliable modeling methods would beextremely useful for the synthetic design of LC materialswith tailored elasticity. The prediction of collective proper-ties, which depend on intermolecular correlations, is neces-sarily a complex problem. However, in some cases suitableapproaches have been worked out, which allow to combinean accurate description at the single molecule level, nowa-days feasible by quantum mechanical methods, with an ef-fective representation of the relevant variables of the envi-ronment; a successful example is represented by thecontinuum theory for the dielectric properties of molecularfluids.29 Nothing analogous exists for the elasticity of LCs.

Molecular field theories have a long history; they contributedto shed light on general relationships between intermolecularinteractions and elastic constants. However, they were devel-oped for systems of simple particles, such as rods or ellip-soids, and cannot account for the real molecular structure andthe molecular flexibility. In their pioneering work Nehringand Saupe6 found KiiSzz

2, where Szz is the major orienta-

tional order parameter, and K11:K22:K33=5:11:5. Subse-quent investigations showed the subtle dependence of therelative values of the elastic constants on the intermolecularinteractions.30 The correct sequence K33K11K22 was ob-tained for hard axially symmetric particles,3134 although theK33 /K11 ratio was overestimated; moreover, such modelscannot account for the temperature dependence of the elasticconstants. Better agreement with experiment could beachieved by superimposing dispersion and electric dipole/quadrupole interactions to the hard core repulsions betweenparticles.3537 Qualitatively similar results were obtained bymolecular dynamics and Monte Carlo simulations of par-ticles with hard core38 or soft interactions.39,40 Differentroutes from trajectories to elastic moduli were proposed;however, in general theoretical models are required to con-nect the information on the local liquid structure obtainedfrom simulations to the elastic properties.41 Moreover, largesamples are needed for the calculation of elastic constantsfrom atomistic simulations,42 and the description of the fulltemperature dependence for a given system would be an ex-tremely demanding task. Good agreement with experimentwas obtained by Zakharov and Maliniak43 for 5CB at 300 Kfrom molecular dynamics simulations with an atomistic forcefield.

Here we present a molecular theory for the elasticity ofnematics, wherein a realistic representation of the structureof mesogens is introduced through the concept of molecularsurface. We make use of a phenomenological model denotedas surface interaction SI, which rests on the assumption thata molecule in the nematic phase tends to align to the directoras much surface as possible.44 The simplest form, consistentwith the nematic symmetry, is assumed for the orienting mo-

FIG. 1. Splay left, twist middle, and bend right deformations.

FIG. 2. Chemical structure of 4-n-pentyl,4-cyanobiphenyl 5CB.

FIG. 3. Temperature dependence of the bulk elastic constants of 5CB asobtained from different experiments: full symbols Refs. 17 and 19, opensymbols Ref. 18, and dashed lines Ref. 16. TNI is the nematic-isotropictransition temperature.

054104-2 Cestari, Bosco, and Ferrarini J. Chem. Phys. 131, 054104 2009


lecular field acting on each element of the molecular surface;the orienting potential experienced by the molecule is thenobtained by integrating the elementary contributions over thewhole surface. The analogy with the RapiniPapoular45 formof the anchoring free energy on the surface of particles em-bedded in nematics can be recognized. Within the SI model,the single molecule orientational distribution in the nematicphase is related to the anisometry of the molecular shape.The chemical structure can be taken into account by a suit-able representation of the molecular surface on the basis ofthe atomic coordinates. The effects of the molecular flexibil-ity can be easily introduced in terms of different molecularconformations;46 this is another important issue for thermo-tropic LCs, whose constituents are always characterized by acertain conformational freedom. Approaches based on the SImodel have been shown to be able to provide good predic-tions of different properties of nematics: temperature depen-dence of orientational order and thermodynamic functions atthe nematic-isotropic transition,44,46 helical twisting power ofchiral dopants,47 and flexoelectric coefficients.48 The molecu-lar surface is customarily used within implicit solvent repre-sentations to parametrize the short-range intermolecular in-teractions in isotropic solutions.49 The success of thepredictions based on the SI model indicates that the molecu-lar surface is also suitable for modeling the anisotropy ofshort-range intermolecular interactions in thermotropic nem-atics.

Some analogy can be found between the theory for theelasticity of nematics presented here and that proposed a fewyears ago by Marrucci and Greco.50 Both can be consideredas extensions of the MaierSaupe theory51 and introduce thedeformation into the single molecule orientational distribu-tion function through the nonuniformity of the director fieldin the space region occupied by a molecule. Despite the simi-lar origin, the developed methodologies are quite different; anovelty in the present work is represented by the realisticaccount of the molecular geometry, which allows a naturalinclusion of the structural differences and eliminates the ar-bitrariness in the definition of the molecular field for a givenmesogen.

Our paper is organized as follows. In Sec. II and III, thetheoretical method is outlined; molecular expressions for thebulk elastic constants K11, K22, and K33 and the two surface-like moduli k13 and k24 are derived using the SI model. Tokeep plain the presentation, only the main expressions willbe reported; all the lengthy algebra is deferred to AppendicesAG. Then the computational methodology adopted for thecalculation of the elastic constants for flexible mesogens willbe outlined. In Sec. V we shall show the results obtained forthe elasticity of 5CB; this example will allow us to highlightthe features of the present methodology and to compare the-oretical results and experimental data. Conclusions and fu-ture outlooks will be summarized in Sec. VI.

II. THEORY: ELASTIC CONSTANTS BY THE SURFACEINTERACTION METHOD

The density of elastic energy in nematic LCs can beexpressed in the following form:6

fel = k2n n + 12K11 n2 + 12K22n n2

+ 12K33n n2 + k13 nn

+ 12 K22 + k24 n n nn , 1

where n is the director. Here the first term accounts for thespontaneous tendency to twist of the director, which charac-terizes the chiral nematic cholesteric phase, and k2 is gen-erally denoted as the chiral strength. The three subsequentterms represent the contributions for splay, twist, and benddistortion modes, respectively, and Kii are the correspondingelastic constants. The two last contributions in Eq. 1 referto splay-bend and saddle-splay deformations, respectively;using the Gauss theorem the volume integrals of these diver-gence forms can be expressed as surface integrals.

Equation 1 is obtained by truncating at the lowest orderterms the expansion of the free energy density in powers ofthe first and second derivatives of the director field in thelimit of long wavelength deformations. The elastic constantsappearing here are material parameters, which depend on thechemical composition of the LC. In the following, an expres-sion for the free energy will be derived starting from thesingle molecule orientational distribution function using theSI model. Then, molecular expression for the elastic con-stants will be obtained as derivatives of this free energy den-sity with respect to deformations.

The orientational distribution of molecules in the nem-atic phase under the conditions of the canonical ensemble isdescribed by the function p,

p =exp U/kBT

Q , 2

where are the Euler angles specifying the molecular ori-entation in a laboratory frame and Q is the orientational par-tition function,

Q = d exp U/kBT . 3In these expressions U is the potential of mean torquefor which the following form is assumed within the SImodel:44

U = kBTS

dSP2n s , 4

where S is the molecular surface, n and s are unit vectors, theformer parallel to the director and the latter normal to thesurface element dS, and P2 is the second Legendre polyno-mial. The form of the orienting potential acting on a surfaceelement is simply chosen on the basis of symmetry consid-erations: it is nothing else than the first nonvanishing term ofthe expansions on a suitable basis set, with the correct sym-metry, of a function of the angle between normal and direc-tor. The parameter with dimension of inverse square lengthspecifies the orienting strength of the medium; according tomolecular field theories44,52 it is assumed to take the form

054104-3 Atomistic modeling of elastic constants J. Chem. Phys. 131, 054104 2009


= 2

vkBTa , 5

where v is the volume per molecule, is a constant, a isdefined as

a = S

dSP2n s , 6

and the angular brackets denote the orientational average

=pd . 7The average value a can be seen as an order parameter,which vanishes in the isotropic phase and quantifies the de-gree of molecular order in the nematic phase.

To connect the molecular orientational distribution func-tion to the Helmholtz free energy, we shall start from thethermodynamic relationship

f = f iso + u Ts , 8where f iso is the free energy density of the isotropic phaseand u, s are the differences of internal energy and entropydensity between the nematic phase and the isotropic phase,respectively. These differences can be approximated as52

u =1

2vU , 9a

s = kBv

ln p . 9b

Using Eqs. 9a and 9b with Eqs. 26, Eq. 8 can berewritten as

f = f iso +2

2v2a2

kBTv

ln Q . 10

Due to the finite dimension of a molecule, in the presence ofdeformations the director orientation is a function of the po-sition on the molecular surface, n=nR. This introduces adependence on the deformation in the molecular field poten-tial U, and then in the orientational distribution functionp, and in the thermodynamic functions which are derivedfrom them. It might be worth noticing the difference fromother approaches, where the molecular expression for thefree energy density was derived from suitable averages of theinteractions between pairs of molecules.53 In such cases, theorientational distribution function with respect to the localdirector was generally assumed to be the same as in the

undeformed nematic phase, and no rotational entropy changewas associated with the deformation.

The free energy density can be expressed as f= fnJ ,nJK ,nJKL , . . ., where nJ are components of the direc-tor, nJK=nJ /RK are first derivatives of such componentswith respect to the position, nJKM =2nJ /RKRM are secondderivatives, and so on. In the long wavelength limit, the Tay-lor expansion of the free energy density can be truncated atthe first terms,

f f0 +

IJ fnIJ

0nIJ +

IJK fnIJK

0nIJK

+12 IJKM

2fnIJ nKM

0nIJnKM , 11

where the subscript 0 refers to the state of uniform directorwith nJK=0, nJKM =0, and so on.

Expressions for the derivatives of the free energy den-sity, Eq. 10, are reported in Appendix A; by substitution ofsuch expressions into Eq. 11 we obtain

f f0 2

v2a0

IJ

anIJ

0nIJ

2

v2a0

IJK

anIJK

0nIJK

122

v2 IJKM a0 2anIJ nKM0+

2

vkBTa0 anIJ anKM0 anIJ0 anKM0

+ 1 2vkBT

a20 a021 a

nIJ

0

+2

vkBTa0a anIJ0 a0 anIJ0 anKM0

+2

vkBTa0a anKM0 a0 anKM0nIJnKM ,

12

where the angular brackets with zero index denote orienta-tional averages calculated with the orientational distributionfunction in the undistorted nematic phase p0,

0 = dp0 . 13Using for derivatives the expressions reported in Appen-

dix B, Eq. 12 can be rewritten as

f f0 + 3kBTv IJL cJILnL

0nIJ +32

kBTv IJKL cKJILnL

0nIJK +32

kBTv IJKM cMJKI 3cJIL,MKN cJILcMKNnL0nN0

+ 3cJIL bJIL a0cJILcMKN bMKN a0cMKN

a0 + a20 a02

nL0nN

0nIJnKM , 14



where the elements of the tensors cn n=3,4 ,6 and b3appear; these are defined in Eqs. B7aB7d asorientational-conformational averages of integrals over themolecular surface. Here we have used Eq. 5, denoting by the orienting strength in the undeformed nematic phase;hereafter this symbol will be used with the same meaningthroughout the text.

To derive explicit expressions for the elastic constants,let us take a reference frame with the Z axis parallel to n0

=nR0. With this choice, nJ0=JZ; moreover, from the con-

straint n n=1, the relationships nZJ=0 and nZJK=nXJnXKnYJnYK follow. Then, the density of elastic energy fel= f f0 reads

fel 3kBTv K=X,Y J=X,Y,Z cJKZnKJ +

32kBTv K=X,Y J,M=X,Y,Z cMJKZnKJM +

32kBTv IJ=X,Y KM=X,Y,Z cKMIJ IJcKMZZ 3cKIZ,MJZ

cKIZcMJZ + 3cKIZ bKIZ a0cKIZcMJZ bMJZ a0cMJZ

a0 + a20 a02 nIKnJM . 15

As a consequence of the symmetry of undeformed nematics,the terms containing third-rank tensor components cJKZ andbJKZ vanish for nonchiral molecules; for chiral molecules,only the components of these tensors with JKZ surviveEqs. B8a and B8b. So, the last term within curly brack-ets in Eq. 15 is absent in the case of achiral molecules;actually, as shown by simple considerations and confirmedby our calculations, even for highly chiral molecular geom-etries this gives a small contribution, which can be safelyneglected. Thus, from comparison between the molecular ex-pression of the elastic energy, Eq. 15, and the NehringSaupe form, Eq. 1, we can write for the elastic constantssee Appendix C,

k2 = 3kBTv

cXYZ, 16a

k13 = 3kBTv

cXZXZ, 16b

Kii = 3kBTv

cIIXX cIIZZ 2cXZXZi1 i3

3cIXZ,IXZ cIXZ2 i2 , 16c

I = X for i = 1, I = Y for i = 2, I = Z for i = 3,

K22 + k24 =32kBTv

cXXXX + cXXYY cXZXZ

6cXXZ,XXZ cXXZ,YYZ . 16d

It follows from Eqs. B8a and B8b that k2 vanishes forachiral molecules.

III. THEORY: INCLUSION OF MOLECULARFLEXIBILITY

The expressions reported above can be easily extendedto take into account the molecular flexibility. In general, for

flexible molecules the orientational distribution function, Eq.2, should be replaced by the torsional-orientational distri-bution function

p, =exp U,/kBT

Q 17

with the partition function

Q = dd exp U,/kBT , 18where collectively denotes all the torsional degrees offreedom and U , is the torsional-orientational potential.The molecular expression for the orienting strength is stillgiven by Eq. 5 with

a =dd exp U,/kBTa,

Q . 19

If the minima of the torsional potential are separated bylarge enough barriers, the system can be simply treated interms of a finite number of conformers, each correspondingto a given minimum; then, the torsional-orientational parti-tion function, Eq. 18, can be approximated as

Q =

m

exp Vm/kBTQm, 20

where the summation is over all conformers; Vm is the tor-sional potential and Qm is the orientational partition functionfor the mth conformer, whose configuration is specified bythe set of torsional angles m,

Qm = d exp Um/kBT . 21Then, the average value of any arbitrary function g ,can be calculated as

g =

m

wmgm, 22



where wm is the statistical weight of the mth conformer

wm =exp Vm/kBTQm

Q , 23

and gm is the orientational average of the function gm=gm , for the mth conformer

gm =d exp Um/kBTgm

Qm. 24

In the nematic phase the statistical weights of conformerscan be different from those in solution and will be a functionof the degree of order. Namely, depending on their shape,conformers can be more or less accommodated in the nem-atic phase; in general, the most elongated will be stabilizedover the bent ones.

For flexible molecules, an additional torsional contribu-tion must be introduced into the expression for the internalenergy density, Eqs. 9a and 9b,

u =1

2vU +

1vV , 25

where V=V Viso, with Viso being the average torsionalpotential in the isotropic phase. The difference of averagetorsional potential between the nematic and the isotropicphase is derived from the change in conformational distribu-tion between them. Using Eq. 25, the Helmholtz free en-ergy density, Eq. 10, becomes

f = f iso +2

2v2a2

kBTv

ln Q + 1vV . 26

It is worth remarking that the angular brackets in these equa-tions indicate averages taken not only over molecular orien-tations but also over conformations according to Eq. 22.

Following the same procedure presented above for rigidmesogens, the expressions for the elastic constants of flexiblemesogens are obtained see Appendix D,

k2 = 3kBTv

cXYZ, 27a

k13 = 3kBTv

cXZXZ, 27b

Kii = 3kBTv cIIXX cIIZZ 2cXZXZi1 i31 + V0

dIIXX dIIZZ 2dXZXZi1 i3

3cIXZ,IXZ1 + V01 a20a02 dIXZ,IXZ+ 9

2

vcYXZ1 + V01 a20a02 cYXZ dYXZi2 ,

27c

I = X for i = 1, I = Y for i = 2, I = Z for i = 3,

K22 + k24 =32kBTv

cXXXX + cXXYY 2cXXZZ1 + V0

dXXXX + dXXYY 2dXXZZ 6cXXZ,XXZ cXXZ,YYZ1 + V01 a20/a0

2

dXXZ,XXZ dXXZ,YYZ , 27d

where dn n=3,4 ,6 and g3 are the tensors defined in Eqs.E1aE1d and V=V /kBT. If V00, i.e., if theconformational distribution is not significantly changed bythe nematic environment, Eqs. 27a27d reduce to theform of Eqs. 16a16d, where, however, the averages haveto be intended as taken over both orientations and conforma-tions. The same holds, of course, also in the presence ofmultiple conformers with identical torsional potential forwhich V=0. In these cases the elastic constants of flexiblemolecules are simply given by the averages of single con-former contributions with conformer weights which dependon both the intrinsic torsional potential and the orientationalmean field according to Eq. 23.

IV. COMPUTATIONAL METHODOLOGY

The calculation of elastic constants for a flexible me-sogen requires the atomic coordinates of all conformers. Foreach conformer, tessellation of the molecular surface isperformed.54 Then the surface integrals, which appear in theexpressions for the elastic constants, are evaluated as sumsover the tesserae. Notice that even the volume integrals ap-pearing in Eqs. B7aB7d can be transformed into surfaceintegrals using the Gauss theorem, as shown in Appendix F.For a given value of the orienting strength , order param-eters and orientational averages for each conformer are cal-culated by integrating over the orientational distributionfunction in undeformed nematics according to Eq. 13. Theintegrals over orientations are conveniently expressed as in-tegrals over the Euler angles , angles by properly ex-ploiting the axial symmetry of undeformed nematics, asshown in Appendix G. Numerical integration on the ,rectangle is carried out by the Gaussian quadrature rule.55Altogether, the calculation of the full temperature depen-dence of the elastic constants for a conformer takes a CPUtime of the order of minutes on a desktop personal computer.Finally, the elastic constants are evaluated as sums of singleconformer contributions; the computation time scales lin-early with the number of conformers.

V. RESULTS: THE ELASTIC CONSTANTS OF 5CB

In the following, we shall present the results obtained for4-n-pentyl,4-cyanobiphenyl 5CB, a typical mesogen TNI=308.2 K for which experimental data are available. Thestructure of 5CB is shown in Fig. 2; several conformers arepossible, which are obtained by rotation around the alkylchain bonds and the phenyl-phenyl bond. In our calculationswe have taken the 14 lowest energy conformers having atmost a single gauche state in the tail; the others have beenneglected in view of the energy increase associated with the



introduction of a gauche, which is of the order of kBT in thetemperature range of the nematic phase.56 An arbitrary con-former is identified by a label such as Ptg+ t; the first letterspecifies the sign of the twist angle between phenyl rings Pand M for angles of about +30 and about 30, respec-tively, whereas the subsequent letters denote the conforma-tional state of the CH2CH2 bonds in their order startingfrom the benzene ring t, g+, and g for trans, gauche+, andgauche, respectively. Figure 4 shows four of the lowestenergy conformers of 5CB with their labels. Atomic coordi-nates of conformers were obtained by geometry optimizationat the DFT /B3LYP /6-31g level.57 Molecular surfaceswere generated with van der Waals radii: rC=0.185 nm, rN=0.15 nm, rH=0.1 nm Ref. 58, and a rolling sphere radiusequal to 0.3 nm;59 a density of points equal to 500 nm2 wastaken.

In calculating the elastic constants, the volume per mol-ecule v was taken equal to 275 3, the value obtained fromthe molecular surface calculation.54 Given the small energydifferences between conformers with a g bond obtained fromthe DFT density functional theory calculations, the sametorsional energy was assumed for them, 2.5 kJ mol1 higherthan the energy of the all-trans conformer.

The elastic constants of 5CB were calculated as a func-tion of the orienting strength or, equivalently, of the orien-tational order parameters. The relationship with temperaturewas then established on the basis of the experimental tem-perature dependence of the major order parameter.60 Figure 5shows the principal elements of the calculated Saupe matrix7as a function of temperature. The Saupe matrix was evalu-ated on the basis of single conformer contributions, all ex-pressed in the same molecular fixed frame.

Figure 6a displays the elastic constants calculated for5CB according to Eqs. 27a27d. These can be comparedto the experimental data shown in Fig. 3: we can see that thesequence of elastic moduli is correctly predicted as well astheir temperature dependence. Figure 6b shows the elasticratios K22 /K11 and K33 /K11, which are known to be espe-cially sensitive to changes in the molecular structure.53 The

calculated values are close to those derived from measure-ments; in keeping with the experiment, similar temperaturedependence is predicted for K22 and K11 weaker than that ofK33. Figure 6a also shows that the results are only slightlyaffected by the neglect in Eqs. 27a27d of all the termscontaining V0, the change in average torsional potentialbetween the nematic phase and the isotropic phase.

In most molecular theories the elastic constants are ex-pressed as a function of order parameters, and contributionsof different ranks are distinguished; the lowest order term isproportional to the square of the second rank order param-eter. In the present model, the dependence on order param-eters is implicitly contained in the orientational averages ap-

FIG. 6. a: Bulk elastic constants calculated for 5CB as a function oftemperature; the dashed lines show the results obtained with V0=0 inEqs. 27c and 27d. b Elastic ratios, calculated line and experimentaltriangles Ref. 16, circles Ref. 18. TNI is the nematic-isotropic transitiontemperature.

Pttt Pttg+ Ptg+t Pg+tt

FIG. 4. Conformers of 5CB considered for the calculations of elasticconstants. The labels P and M are used for positive +30 and negative30 biphenyl twist angles, respectively. The symbols t trans and ggauche denote conformational states of CH2CH2 bonds. Each conformeris identified by a sequence of symbols, listed in order, starting from theCH2CH2 bond closer to the benzene ring.

FIG. 5. Temperature dependence of the principal values of the averageSaupe matrix calculated for 5CB. TNI is the nematic-isotropic transitiontemperature.



pearing in the expressions for elastic constants, Eqs.27a27d. Figure 7 displays the elastic constants of 5CBas a function of Szz

2, the square of the major principal value of

the average Saupe matrix. We can see that for the splay andtwist elastic moduli the relationship is close to linear in al-most the whole range; some more deviation from linearityappears for the bend stiffness. Analogous results were foundby other theories.53

An important and still largely unexplored issue concernsthe effects of the molecular flexibility on the elastic moduliof nematics. New insights can be derived from our approach,which allows us to examine the contribution of individualconformers. Some results obtained for 5CB are shown in Fig.8. We can see a strong dependence on the alkyl chain con-formation understandable because changes in chain confor-mation can bring about changes in the whole molecularshape, as appears in Fig. 4. Small elastic constants are pre-dicted for bent conformers and much higher ones for elon-gated conformers. Remarkably, even negative K33 values areobtained for the most bent structures. It is worth stressingthat this result is not in contrast with the stability of thenematic phase with uniform director for 5CB; the negativecontributions to K33 have the only effect of lowering thevalue obtained after averaging over all the conformers.Medium-high values of the elastic constants are predicted forthe all-trans conformer, which is the most stable and isslightly bent in shape.

Single conformers, being chiral, give nonvanishing con-tributions to the chiral strength k2; this means that theywould promote twist distortions of the director.7 The valueobtained would correspond to helical pitches in the range of0.11 m, i.e., comparable to the pitch measured61 andpredicted62 for alkylcyanobiphenyl mesogens with a chiralcenter in the aliphatic chain. However, for each conformerthe mirror image must also be present in any 5CB sample;since enantiomers give contributions of opposite sign, a van-ishing k2 value is obtained for the racemic mixture of con-formers.

Finally, Fig. 9 displays the temperature dependence ofthe surfacelike elastic constants k13 and k24. We can see thatthe former is negative and weakly changing with tempera-ture, whereas the latter is positive at low temperature andbecomes negative at higher temperature; both are smaller

FIG. 8. Contribution of selected conformers to the elastic constants of 5CB.Conformer labels are reported in Fig. 4. For comparison, also the valuesobtained after averaging over conformers are shown dashed lines. TNI isthe nematic-isotropic transition temperature.

FIG. 9. Surfacelike elastic constants calculated for 5CB as a function oftemperature. TNI is the nematic-isotropic transition temperature.

FIG. 7. Bulk elastic constants calculated for 5CB as a function of Szz2 , thesquare of the major principal value of the Saupe matrix. Dashed lines high-light deviations from linearity.



than the bulk moduli and the saddle-splay constant calcu-lated for 5CB satisfies the Ericksen inequalities.8 We havefound a significant dependence of k13 and k24 on the molecu-lar conformation. Unfortunately we cannot evaluate the qual-ity of our predictions by comparison with the experiment forlack of data. Comparison with other theories and simulationsis possible; actually, only a few attempts to evaluate surface-like elastic constants have been reported, all dealing withsimple particle models. Teixeira et al.63 studied a nematicphase formed by GayBerne particles using a generalizedPoniewierskiStecki theory,33 whereas Stelzer et al.64 per-formed molecular dynamics simulations for an analogoussystem. In both cases, k13 and k24, significantly smaller thanthe bulk elastic constants, were obtained in keeping with ourresults for 5CB. However, there is a complete disagreementon the sign of the two constants. Mostly positive k13 and k24values were predicted by Teixeira et al., whereas Stelzer etal.64 found k240 and k130. The discrepancies were as-cribed to different approximations used for evaluating thedirect correlation function, and the strong sensitivity of thesurface constants to details of the latter was inferred.

VI. CONCLUSIONS

We have developed a molecular theory for the elasticityof nematic LCs based on the SI model.44 The relation be-tween molecular orientation and director deformation is in-troduced through the assumption that the nematic directortends to lie perpendicular to the molecular surface at eachpoint. A realistic account of the molecular structure is madepossible by the use of a surface generated from atomic coor-dinates. The mesogen flexibility is easily introduced throughaverages over molecular conformations. The elastic con-stants are expressed in terms of the orientational averages oftensors defined as integrals of suitable functions over themolecular surface; they can be calculated as a function of thedegree of order without any free parameter at low computa-tional cost. This methodology enables us to investigate therole of molecular features and to explore how changes at theatomic level can be conveyed into changes in elastic behav-ior on a quite different length scale. For this reason we thinkthat it can shed light on the origin, still poorly understood, ofthe different elasticity of the nematic phases formed by me-sogens with different structures. The predictive ability of thismethod makes it potentially useful for the synthetic design oftailored mesogens: the elastic constants can be easily calcu-lated once the molecular structure is known.

Promising results have been obtained for the typical caseof 5CB. Several elasticity measurements have been reportedfor this system with some discrepancies; the absolute andrelative magnitude of the elastic constants calculated by ourapproach, as well as their temperature dependence, is in linewith experiments. The calculations show the high sensitivityof the elastic constants, especially that for bending, to themolecular geometry: even for 5CB, which is generally takenas a prototype of a rodlike structure, we have found remark-able differences between the contributions of conformers.The accuracy of our predictions needs now to be checked byinvestigation of other systems; given the approximate form

of the molecular field, we cannot exclude some systematicdiscrepancy between predicted and measured elastic con-stants. However, the value of the proposed approach wouldbe the ability of providing reliable, although not necessarilyextremely accurate, estimates of how the elastic constants areaffected by changes in the molecular structure; presently, noother methodology is available to this purpose.

Within the theoretical framework presented here, alsoexpressions for the surfacelike elastic constants of nematicsare provided. The values obtained for k13 and k24 of 5CBhave opposite signs and are smaller in absolute value thanthose of the bulk elastic moduli. We have found that k13 andk24 are highly sensitive to the molecular geometry; previoustheories and simulations evidenced a strong sensitivity todetails of the direct correlation functions.63,64 The absence ofunambiguous experimental data makes it difficult to assessthe quality of our results. Actually, as discussed by someauthors,13,14 theories for the surface elasticity, as well as theinterpretation of experiments, should be considered within amore general framework, taking into account also the an-choring free energy. So, we think that what we have pre-sented here should be seen as a preliminary exploration ofthe problem of the surface elasticity of nematics, which de-serves further investigation in the future.

Developments of this work along different lines areplanned. First of all, we shall investigate a wider class ofstandard mesogens to assess the capability of the method bycomparison with a larger set of experimental data. Then, weintend to study the elasticity of nonconventional nematicsformed by bent-core mesogens65 and cyanobiphenyl dimers66for which scarce experimental information is available. Ourcalculations have shown the importance of a complete sam-pling of the conformational space, even for a relatively smallmolecule such as 5CB; this issue is expected to become evenmore relevant in the case of molecules with a higher numberof internal degrees of freedom. Thus, the calculation of elas-tic constants will be introduced in a systematic procedure forgenerating and treating conformers of flexible moleculesalong with Monte Carlo sampling of the conformationalspace.67 Another possible development is the extension ofour approach to the elastic properties of biaxial nematics.68

ACKNOWLEDGMENTS

The beginning of this work was supported by ItalianMIUR Grant No. PRIN 2005. The authors gratefully ac-knowledge Dr. Fabio Tombolato for stimulating discussions.

NOMENCLATURE

AbbreviationsLC Liquid crystal

5CB 4-n-pentyl,4-cyanobiphenylSI Surface interaction

TNI Nematic-isotropic transition temperature



APPENDIX A: DERIVATIVES OF THE FREE ENERGY

In this appendix expressions for the derivatives of thefree energy density appearing in Eq. 11 are obtained. FromEq. 10, we can write

fnIJ

=

2

v2a

anIJ

kBTv

1Q

QnIJ

, A1a

2fnIJ nKM

=

2

v2a

2anIJ nKM

+2

v2anIJ

anKM

+kBTv

1Q2

QnIJ

QnKM

kBTv

1Q

2QnIJ nKM

.

A1b

Considering the form of the potential of mean torque, Eq.2, the derivatives of the orientational partition function, Eq.3, are given by

QnIJ

=

2

vkBTQa a

nIJ + a

nIJ , A2a

2QnIJ nKM

= Q 2

vkBT anIJ

anKM

+ a

nIJ anKM

+ a 2anIJ nKM

+ a

2anIJ nKM

2

vkBTa2 a

nIJ

a

nKM

+ a2anIJ

anKM

+ aanIJ

a a

nKM

+ aa anIJ anKM

. A2bBy substitution of Eqs. A2a and A2b into Eqs. A1a andA1b we obtain

fnIJ

=

2

v2a a

nIJ , A3a

2fnIJ nKM

=

2

v2a 2a

nIJ nKM + 2

vkBTa

anIJ

a

nKM a

nIJ a

nKM

2

v2anIJ a

nKM + 2

vkBTaa a

nKM

a anKM 2

v2anKM

anIJ

+2

vkBTaa a

nIJ a a

nIJ

+2

v2anIJ

anKM

1 2vkBT

a2 a2 . A3bFor the first derivative of the average value a, we can write

anIJ

= 1 2vkBT

a2 a21 anIJ + 2

vkBTa

a anIJ a a

nIJ , A4

which when substituted into Eq. A3b leads to

2fnIJ nKM

=

2

v2a 2a

nIJ nKM + 2

vkBTa

anIJ

a

nKM a

nIJ a

nKM

2

v21 2

vkBTa2 a21 a

nIJ

+2

vkBTaa a

nIJ a a

nIJ

anKM

+ 2vkBT

aa anKM

a anKM . A5

An expression analogous in form to Eq. A3a is obtained forthe derivatives of the free energy density with respect tonJKM =

2nJ /RKRM.

APPENDIX B: SURFACE TENSORS

Here we shall make explicit the position dependence ofthe function P2n s appearing in the expression for thepotential of mean torque, Eq. 4, in the presence ofdirector distortions. Let us assume that the origin of the mo-lecular frame is located at the point R0 in the sample; if n0=nR=R0 is the director in this point, the function P2n sat any arbitrary position on the molecular surface, R=R0+r, can be expressed by the Taylor expansion in powers ofthe displacement r,

P2n s P2n0 s + 3

IJK

rKsJsInI0nJK

+32 IJKM rMrKsIsJnI

0nJKM

+32 IJKM rMrKsIsJnIMnJK + . B1

For small displacements on the length scale of the deforma-tion, this expansion can be truncated at the first terms.

Using Eq. B1, the integral a defined in Eq. 6 can beapproximated as

a a0 + 3

IJK

TKJInI0nJK +

32 IJKM TMKJInI

0nJKM

+32 IJKM TMKJInIMnJK, B2

where TIJKM are elements of the nth rank Tn tensors



TIJ = S

sIsJdS , B3a

TIJK = S

rIsJsKdS , B3b

TIJKM = S

rIrJsKsMdS . B3c

From the definition, the following symmetry relations arederived:

TIJ = TJI, B4a

TIJK = TIKJ, B4b

TIJKM = TJIKM = TIJMK = TJIKM . B4c

The derivatives appearing in Eq. 12 can then be expressedas

a

nJK= 3

ITKJInI

0 + 3

IM

TMKJInIM , B5a

a

nJKM=

32I TMKJInI

0, B5b

2a

nJK nIM= 3TMKJI. B5c

The Tn tensors defined in Eqs. B3aB3c depend on thelocation of the origin of the molecular frame. Under a shift inthe origin, OO, the coordinates of points in the molecularsurface transform as r=rr0, where the vector r0 specifiesthe position of the new origin in the original frame. It fol-lows that for the components of the Tn tensors we can write

TIJ = TIJ, B6a

TIJK = TIJK r0,ITJK, B6b

TIJKM = TIJKM + r0,Ir0,JTKM r0,ITJKM r0,JTIKM , B6c

where r0,I and r0,J are Cartesian components of the vector r0.The expression for the free energy density, Eq. 12, con-

tains averages of the derivatives, Eqs. B5aB5c, takenover the orientational distribution function in undeformednematics p0 introduced in Eq. 13. After further averagingover all positions of the origin within the molecular volume,as appropriate in a fluid without positional order, we candefine the following tensors:

cIJK = TIJK0 1v

vdrr0,ITJK0, B7a

bIJK = aTIJK0 1v

vdrr0,IaTJK0, B7b

cIJKM = TIJKM0 +1v

vdrr0,Ir0,JTKM0

1v

vdrr0,ITJKM0 1v v drr0,JTIKM0,

B7c

cIJK,LMN = TIJKTLMN0 +1v

vdrr0,Ir0,LTJKTMN0

1v

vdrr0,ITJKTLMN0

1v

vdrr0,LTIJKTMN0, B7d

where v is the molecular volume.Some relationships, which exist between the components

of the cn tensors, by virtue of the uniaxial symmetry of theundistorted nematic phase, are exploited in deriving the ex-pressions for the elastic constants. In a laboratory frame withthe Z axis parallel to the director, after averaging over thedistribution function for undeformed nematics p0, the num-ber of independent components of the cn tensors is reduced.In particular, for the relevant elements of the third-rank ten-sor we can write

TIJZ0 = IJZTIJZ0 =0 for achiral molecules ,B8a

aTIJZ0 = IJZaTIJZ0 =0 for achiral molecules ,B8b

where IJZ is the LeviCivita symbol.69 The components ofthe fourth rank tensors obey the following relationships:

TXXXX0 = TYYYY0, B9a

TXXZZ0 = TYYZZ0, B9b

TXXYY0 = TYYXX0, B9c

TZZXX0 = TZZYY0, B9d

TXZXZ0 = TYZYZ0, B9e

TXZYX0 = TYZXY0 = 0, B9f

TXXYZ0 = TYYXZ0 = TXXXZ0 = TYYYZ0 = TYXXZ0= TXYYZ0 = 0, B9g

TYZXZ0 = TZYXZ0 = 0, B9h

TXYXX0 = TXYYY0 = TXYZZ0 = 0, B9i

TXXXY0 = TYYXY0 = TZZXY0 = 0, B9j



TXXXX0 = TXXYY0 + 2TXYXY0, B9k

and for the components of the sixth rank tensors we have

TXXZTXXZ0 = TYYZTYYZ0, B10a

TXYZTXYZ0 = TYXZTYXZ0, B10b

TZXZTZXZ0 = TZYZTZYZ0, B10c

TXXZTXXZ0 = TXXZTYYZ0 + TXYZTXYZ0 + TXYZTYXZ0.B10d

Analogous symmetry relations can be written for the tensorsdefined in Eqs. B7aB7d.

APPENDIX C: REWRITING EQUATION 15For comparison with Eq. 1, it is convenient to collect

the terms in Eq. 15 as follows:

fel = 3kBTv

cXYZnYX nXY + 3kBTv

cXZXZnXXZ + nYYZ +32kBTv

cXXXX cXXZZ 3cYXZ,YXZnXX2 + nYY

2

+32kBTv cYYXX cYYZZ 3cYXZ,YXZ cYXZcYXZ + 3 cYXZ bYXZ a0cYXZ2a0 + a20 a02 nXY2 + nYX2

+32kBTv

cZZXX cZZZZ 3cZXZ,ZXZnXZ2 + nYZ

2 + 3kBTv

cYXYX 3cXXZ,YYZnXXnYY

+ 3kBTv cYXYX 3cYXZ,XYZ cYXZcXYZ 3 cYXZ bYXZ a0cYXZ2a0 + a20 a02 nXYnYX, C1

where the relationship nKJM =nKMJ has been used in the first line.After some algebra, Eq. C1 can be rewritten as

fel = 3kBTv

cXYZnYX nXY + 3kBTv

cXZXZnXX + nYY2 nXZ2 + nYZ

2 + nXXZ + nYYZ

+32kBTv

cXXXX cXXZZ 2cXZXZ 3cXXZ,XXZnXX + nYY2

+32kBTv cYYXX cYYZZ 3cYXZ,YXZ cYXZ2 + 3 cYXZ bYXZ a0cYXZ2a0 + a20 a02 nXY2 + nYX2

+32kBTv

cZZXX cZZZZ + 2cXZXZ 3cZXZ,ZXZnXZ2 + nYZ

2 +32kBTv

cXXXX + 2cXXZZ cXXYY

+ 6cXXZ,XXZ cXXZ,YYZnXXnYY nXYnYX . C2

APPENDIX D: DERIVATIVES OF THE FREE ENERGYCASE OF MULTIPLE CONFORMERS

In this appendix we shall obtain expressions for the de-rivatives of the free energy density, Eq. 26, with respect todirector deformations. We can write

fnIJ

=

2

v2a

anIJ

kBTv

1Q

QnIJ

+1v

VnIJ

, D1a

2fnIJ nKM

=

2

v2a

2anIJ nKM

+2

v2anIJ

anKM

+kBTv

1Q2

QnIJ

QnKM

kBTv

1Q

2QnIJ nKM

+1v

2VnIJ nKM

. D1b

It is convenient to exploit the following relationships, whichcan be derived from the definitions of orientational-conformational averages and conformer weight, Eqs. 22and 23:

VnIJ

=

1Qm exp Vm/kBT

QmnIJ

Vm V , D2a

2VnIJ nKM

=

1Qm exp Vm/kBT

2QmnIJ nKM

Vm V

1Q2m,n exp Vm + Vn/kBT

QmnIJ

QnnKM

Vm + Vn 2V , D2b

where the index m denotes the mth conformer and summa-tions are extended to all conformers.



The derivatives of the orientational partition functionsfor individual conformers defined in Eq. 21 are given by

QmnIJ

=

2

vkBTQma amnIJ + amanIJ , D3a

2QmnIJ nKM

=

2

vkBTQm anIJ amnKM + amnIJ anKM + a

2amnIJ nKM

+2a

nIJ nKM

+2

vkBTa2am

nIJ

amnKM

+ am2 anIJ

anKM

+ aanIJ

amamnKM

+ aamamnIJ

anKM

,D3b

with the upper bar indicating the orientational average as inEq. 24.

For the derivatives of the average torsional potential wecan write

VnIJ

=

2

vkBT1Qm exp Vm/kBTQm

a amnIJ

+ amanIJ

Vm V=

2

vkBTaV a

nIJ V a

nIJ

+anIJ

Va aV , D4a

2VnIJ nKM

=

2

vkBT anIJ

V a

nKM + a

nKM

V a

nIJ + aV 2a

nIJ nKM + 2

vkBTa2V a

nIJ

a

nKM + Va2a

nIJ

anKM

+ aanIJ

Va a

nKM + a a

nKM

Va a

nIJ 2

vkBTV a

nIJ

anKM

+ anKM

anIJ + a 2a

nIJ nKM

+2

vkBTa2 a

nIJ

a

nKM + a2a

nIJ

anKM

+ aanIJ

a a

nKM + a a

nKM

a a

nIJ 2

vkBT2aa

V anIJ + Vaa

nIJ a

nKM + a

nKM 2

vkBT2aaV a

nKM + Va a

nKM a

nIJ + a

nIJ

+ 2 2vkBT

2Va2 anKM

+ anKM

anIJ + a

nIJ . D4b

By substitution of Eqs. D3a and D3b and Eqs. D4a and D4b into Eqs. D1a and D1b, using Eq. A4, we obtain

fnIJ

=

2

v2a a

nIJ + 2

v2kBTaV anIJ V anIJ+

anIJ 2

vkBTaa a

nIJ a a

nIJ

1 2vkBT

a2 a2Va aV , D5a

2fnIJ nKM

=

2

v2anIJ

anKM1 2

vkBTa2 a2

2

v2

anIJ 2

vkBTaa a

nIJ a a

nIJ a

nKM 2

vkBTaa a

nKM a a

nKM

1 2vkBT

a2 a2

1v 2vkBT

2Va2 Va2 + 2aVa Va



anIJ 2

vkBTaa a

nIJ a a

nIJ a

nKM 2

vkBTaa a

nKM a a

nKM

1 2vkBT

a2 a22+

2

v2kBTV a

nKM V a

nKM + 2

vkBTaVa a

nKM Va a

nKM + a2aV a

nKM

Va anKM

aV anKM

a

nIJ 2

vkBTaa a

nIJ a a

nIJ

1 2vkBT

a2 a2+

2

v2kBTV a

nIJ

V anIJ + 2

vkBTaVa a

nIJ Va a

nIJ + a2aV a

nIJ Va a

nIJ a

V anIJ

a

nKM 2

vkBTaa a

nKM a a

nKM

1 2vkBT

a2 a2

2

v2a 2a

nIJ nKM + 2

vkBTa

anIJ

a

nKM a

nIJ a

nKM + 2

v2kBTaV 2a

nIJ nKM aV 2a

nIJ nKM

+2

vkBTa2V a

nIJ

a

nKM a2V a

nIJ

a

nKM + a22V a

nIJ a

nKM V a

nIJ a

nKM

anIJV a

nKM . D5b

An expression analogous in form to Eq. D5a is obtained forthe derivatives of the free energy density with respect tonJKM =

2nJ /RKRM.

APPENDIX E: SURFACE TENSORS CASE OFMULTIPLE CONFORMERS

If the mesogen flexibility is taken into account, in addi-tion to those defined by Eqs. B7aB7d, some other ten-sors appear in the molecular expressions for the elastic con-stants. Their components are defined as

dIJK = VTIJK0 1v

vdrrIVTJK0, E1a

gIJK = VaTIJK0 1v

vdrrIVaTJK0, E1b

dIJKM = VTIJKM0 +1v

vdrrIrJVTKM0

1v

vdrrIVTJKM0 1v v drrJVTIKM0,

E1c

gIJK,LMN = VTIJKTLMN0 +1v

vdrrIrLVTJKTMN0

1v

vdrrIVTJKTLMN0

1v

vdrrLVTIJKTMN0, E1d

where the angular brackets denote orientational-conformational averages, defined by Eq. 22. The index 0 isused for averages performed over the orientational distribu-tion in the undistorted nematic phase.

APPENDIX F: TRANSFORMING VOLUME INTOSURFACE INTEGRALS

For computational purposes it is convenient to transformthe volume integrals in Eqs. B7aB7d into integrals overthe molecular surface using the Gauss theorem.69 For ex-ample we can write



v

drrI = v dSrIsK I K1/2v

dSrI2sK I = K , F1a

v

drrIrJ =v

dSrIrJsK I K,J K

1/2v

dSrI2rJsK I J,I = K or J = K

1/3v

dSrI3sK I = J = K .

F1b

So, if we introduce the tensors

GIJK = S

dSrIrJsK, F2a

GIJKM = S

dSrIrJrKsM , F2b

the average tensor components cIJL and bIJL appearing in theexpressions for elastic constants and defined by Eqs.B7aB7d can be expressed in a compact form as

cIJL = TIJL0 GIKTJL0 I K1/2GIIKTJL0 I = K , F3abIJL = TIJL0 aGIKTJL0 I K1/2aGIIKTJL0 I = K , F3b

cIJLM = TIJLM0 + GIJKTLM0 GIKTJLM0 GJKTILM0 I K,J K

1/2GIIJKTLM0 1/2GIIKTJLM0 GJKTILM0 I J,I = K1/2GIIJKTLM0 GIKTJLM0 1/2GJJKTILM0 I J,J = K

1/3GIIIKTLM0 GJJKTILM0 I = J = K , F3c

cIJL,KMN = TIJLTKMN0 + GIJKKTJLTMN0 GIKTJLM0 GJKTILM0 I K,J K

1/2GIIJKTLM0 1/2GIIKTJLM0 GJKTILM0 I J,I = K1/2GIIJKTLM0 GIKTJLM 1/2GJJKTILM0 I J,J = K

1/3GIIIKTLM0 GJJKTILM0 I = J = K . F3d

APPENDIX G: CALCULATION OF AVERAGESURFACE TENSORS

The tensor components defined in Appendix B dependon the three Euler angles , ,; however, exploitation ofthe axial symmetry of undeformed nematic phase allows usto calculate their orientational averages as double integralsover the , Euler angles. For example, we can calculateTXZXZ0 and TZXZTZXZ0 as

TXZXZ0 = TXZXZ0 + TYZYZ0 , G1a

TZXZTZXZ0 = TZXZTZXZ0 + TZYZTZYZ0 , G1b

where

0 = sin d d p0, , G2with p0 being the orientational partition function in unde-formed nematics.

The integrals TXXXX0, TXXYY0, and TXYXY0 can beobtained by solving the set of algebraic equations

TXXXX0 + TYYYY0 = TXXXX0 + TYYYY0+ TXXYY0 + TYYXX0

TXXYY0 TYYXX0 = TXXYY0 + TYYXX0 2TXYXY0 G3

together with Eq. B9k.Expressions analogous to Eqs. G1a and G1b or Eq.

G3 can be used for all average tensor components appear-ing in the expressions for the elastic constants.

1 P. G. de Gennes, The Physics of Liquid Crystals Clarendon, Oxford,1974.

2 P. Oswald and P. Pieranski, Nematic and Cholesteric Liquid CrystalsTaylor & Francis, London, 2005.

3 C. W. Oseen, Trans. Faraday Soc. 29, 883 1933.4 H. Zocher, Trans. Faraday Soc. 29, 945 1933.5 F. C. Frank, Discuss. Faraday Soc. 25, 19 1958.6 J. Nehring and A. Saupe, J. Chem. Phys. 54, 337 1971; J. Chem. Phys.

56, 5527 1972.7 G. Vertogen and W. H. de Jeu, Thermotropic Liquid Crystals: Fundamen-tals Springer, New York, 1988.

8 J. L. Ericksen, Phys. Fluids 9, 1205 1966.9 C. Oldano and G. Barbero, Phys. Lett. 110A, 213 1985.

10 H. P. Hinov, Mol. Cryst. Liq. Cryst. 148, 197 1987; 178, 53 1990.11 O. D. Lavrentovich and V. M. Pergamenschik, Int. J. Mod. Phys. B 9,

2389 1995.12 S. Faetti and M. Riccardi, J. Phys. II 5, 1165 1995.13 H. Yokoyama, Phys. Rev. E 55, 2938 1997.14 V. M. Pergamenshchik and S. Zumer, Phys. Rev. E 59, R2531 1999.15 V. M. Pergamenshchik, Phys. Rev. E 61, 3936 2000.



16 N. V. Madhusudana and R. Pratibha, Mol. Cryst. Liq. Cryst. 89, 2491982.

17 M. J. Bradshav, E. P. Raynes, J. D. Bunning, and T. E. Faber, J. Phys.France 46, 1513 1985.

18 G.-P. Chen, H. Takezoe, and A. Fukuda, Liq. Cryst. 5, 341 1989.19 H. J. Coles, in The Optics of Thermotropic Liquid Crystals, edited by S.

Elston and R. Sambles Taylor & Francis, London, 1998.20 A. Bogi and S. Faetti, Liq. Cryst. 28, 729 2001.21 M. Schadt, Annu. Rev. Mater. Sci. 27, 305 1997.22 M. Ravnik, M. karabot, S. umer, U. Tkalec, I. Poberaj, D. Babi, N.

Osterman, and I. Muevi, Phys. Rev. Lett. 99, 247801 2007.23 K. Takahashi, M. Ichikawa, and Y. Kimura, Phys. Rev. E 77, 020703

2008.24 C. Lapointe, N. Cappallo, D. H. Reich, and R. L. Leheny, J. Appl. Phys.

97, 10Q304 2005.25 S. D. Hudson and E. L. Thomas, Phys. Rev. Lett. 62, 1993 1989.26 H. Shin, M. J. Bowick, and X. Xing, Phys. Rev. Lett. 101, 037802

2008.27 W. H. de Jeu and W. A. P. Claassen, J. Chem. Phys. 67, 3705 1977.28 M. Schadt and P. R. Gerber, Z. Naturforsch. C 37a, 165 1982.29 Continuum Solvation Models in Chemical Physics. From Theory to Ap-

plications, edited by B. Mennucci and R. Cammi Wiley, New York,2007.

30 G. Barbero and L. R. Evangelista, Phys. Rev. E 56, 6189 1997; L. R.Evangelista, I. Hibler, and H. Mukai, ibid. 58, 3245 1998.

31 J. P. Straley, Phys. Rev. A 8, 2181 1973.32 R. G. Priest, Phys. Rev. A 7, 720 1973.33 A. Poniewierski and J. Stecki, Mol. Phys. 38, 1931 1979.34 A. M. Somoza and P. Tarazona, Mol. Phys. 72, 991 1991.35 J. Stecki and A. Poniewierski, Mol. Phys. 41, 1451 1980.36 W. M. Gelbart and A. Ben Shaul, J. Chem. Phys. 77, 916 1982.37 A. Srivastava and S. Singh, J. Phys.: Condens. Matter 16, 7169 2004.38 M. P. Allen and D. Frenkel, Phys. Rev. A 37, 1813 1988; 42, 3641

1990.39 M. P. Allen, M. A. Warren, M. R. Wilson, A. Sauron, and W. Smith, J.

Chem. Phys. 105, 2850 1996.40 N. H. Phuong, G. Germano, and F. Schmid, J. Chem. Phys. 115, 7227

2001.41 M. P. Allen and A. J. Masters, J. Mater. Chem. 11, 2678 2001.42 M. R. Wilson, Int. Rev. Phys. Chem. 24, 421 2005.43 A. V. Zakharov and A. Maliniak, Eur. Phys. J. E 4, 85 2001.44 A. Ferrarini, G. J. Moro, P. L. Nordio, and G. R. Luckhurst, Mol. Phys.

77, 1 1992.45 A. Rapini and M. Papoular, J. Phys. Paris, Colloq. 30, C4-54 1969.46 A. Ferrarini, G. R. Luckhurst, P. L. Nordio, and S. J. Roskilly, J. Chem.

Phys. 100, 1460 1994.47 A. Ferrarini, G. J. Moro, and P. L. Nordio, Phys. Rev. E 53, 681 1996;

Mol. Phys. 87, 485 1996.48 A. Ferrarini, Phys. Rev. E 64, 021710 2001.49 A. R. Leach, Molecular Modelling: Principles and Applications Dorset,

Dorchester, 2001.50 G. Marrucci and F. Greco, Mol. Cryst. Liq. Cryst. 206, 17 1991.51 W. Maier and A. Saupe, Z. Naturforsch. A 14A, 882 1959; 15A, 287

1960.52 G. R. Luckhurst and G. W. Gray, The Molecular Physics of Liquid Crys-

tals Academic, New York, 1979.53 S. Singh, Phys. Rep. 277, 283 1996.54 M. F. Sanner, J. C. Spehner, and A. J. Olson, Biopolymers 38, 305

1996.55 W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Nu-

merical Recipes Cambridge University Press, Cambridge, England,1992.

56 C. J. Adam, A. Ferrarini, M. R. Wilson, G. J. Ackland, and J. Crain, Mol.Phys. 97, 541 1999.

57 M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., GAUSSIAN03, RevisionC.02, Gaussian, Inc., Wallingford, CT, 2004.

58 Handbook of Chemistry and Physics, edited by D. R. Lide CRC, BocaRaton, 1996.

59 A. Ferrarini, F. Janssen, G. J. Moro, and P. L. Nordio, Liq. Cryst. 26, 2011999.

60 L. G. P. Dalmolen, S. J. Picken, A. F. de Jong, and W. H. de Jeu, J. Phys.France 46, 1443 1985.

61 G. W. Gray and D. G. McDonnell, Mol. Cryst. Liq. Cryst. 37, 1891976; Mol. Cryst. Liq. Cryst. Lett. 34, 211 1977.

62 S. M. Todd, A. Ferrarini, and G. J. Moro, Phys. Chem. Chem. Phys. 3,5535 2001.

63 P. I. C. Teixeira, V. M. Pergamenshchik, and T. J. Sluckin, Mol. Phys. 80,1339 1993.

64 J. Stelzer, H.-R. Trebin, and L. Longa, J. Chem. Phys. 103, 3098 1995;107, 1295 1997.

65 D. Wiant, J. T. Gleeson, N. Eber, K. Fodor-Csorba, A. Jakli, and T.Toth-Katona, Phys. Rev. E 72, 041712 2005.

66 N. V. Tsvetkov, V. V. Zuev, I. V. Ksenofontov, and V. N. Tsvetkov, Mol.Cryst. Liq. Cryst. 331, 1901 1999.

67 M. Cestari, Ph.D. thesis, Universit di Padova, 2009.68 Y. Singh, K. Rajesh, V. J. Menon, and S. Singh, Phys. Rev. E 49, 501

1994; L. Longa, J. Stelzer, and D. Dunmur, J. Chem. Phys. 109, 15551998; A. Kapanowski, Phys. Rev. E 77, 052702 2008.

69 G. B. Arfken and H. J. Weber, Mathematical Methods for PhysicistsAcademic, New York, 2001.



Molecular field theory with atomistic modeling for the curvature elasticity of nematic liquid...

Documents

Transcript of Molecular field theory with atomistic modeling for the curvature elasticity of nematic liquid...