# Molecular dynamics of thermotropic liquid crystals: anomalous relaxation dynamics of calamitic and discotic liquid crystals

**ABSTRACT** Recent optical kerr effect (OKE) studies have demonstrated that orientational relaxation of rod-like nematogens exhibits temporal power law decay at intermediate times not only near the isotropic–nematic (I–N) phase boundary but also in the nematic phase. Such behaviour has drawn an intriguing analogy with supercooled liquids. We have investigated both collective and single-particle orientational dynamics of a family of model system of thermotropic liquid crystals using extensive computer simulations. Several remarkable features of glassy dynamics are on display including non-exponential relaxation, dynamical heterogeneity, and non-Arrhenius temperature dependence of the orientational relaxation time. Over a temperature range near the I–N phase boundary, the system behaves remarkably like a fragile glass-forming liquid. Using proper scaling, we construct the usual relaxation time versus inverse temperature plot and explicitly demonstrate that one can successfully define a density dependent fragility of liquid crystals. The fragility of liquid crystals shows a temperature and density dependence which is remarkably similar to the fragility of glass forming supercooled liquids. Energy landscape analysis of inherent structures shows that the breakdown of the Arrhenius temperature dependence of relaxation rate occurs at a temperature that marks the onset of the growth of the depth of the potential energy minima explored by the system. A model liquid crystal, consisting of disk-like molecules, has also been investigated in molecular dynamics simulations for orientational relaxation along two isobars starting from the high temperature isotropic phase. The isobars have been so chosen that the phase sequence isotropic (I)–nematic (N)–columnar (C) appears upon cooling along one of them and the sequence isotropic (I)–columnar (C) along the other. While the orientational relaxation in the isotropic phase near the I–N phase transition shows a power law decay at short to intermediate times, such power law relaxation is not observed in the isotropic phase near the I–C phase boundary. The origin of the power law decay in the single-particle second-rank orientational time correlation function (OTCF) is traced to the growth of the orientational pair distribution functions near the I–N phase boundary. As the system settles into the nematic phase, the decay of the single-particle second-rank orientational OTCF follows a pattern that is similar to what is observed with calamitic liquid crystals and supercooled molecular liquids.

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**ABSTRACT:**Recent Kerr relaxation experiments by Gottke et al. have revealed the existence of a pronounced temporal power law decay in the orientational relaxation near the isotropic-nematic phase transition (INPT) of nematogens of rather small aspect ratio, kappa (kappa approximately 3-4). We have carried out very long (50 ns) molecular dynamics simulations of model (Gay-Berne) prolate ellipsoids with aspect ratio 3 in order to investigate the origin of this power law. The model chosen is known to undergo an isotropic to nematic phase transition for a range of density and temperature. The distance dependence of the calculated angular pair correlation function correctly shows the emergence of a long range correlation as the INPT is approached along the density axis. In the vicinity of INPT, the single particle second rank orientational time correlation function exhibits power law decay, (t(-alpha)) with exponent alpha approximately 2/3. More importantly, we find the sudden appearance of a pronounced power-law decay in the collective part of the second rank orientational time correlation function at short times when the density is very close to the transition density. The power law has an exponent close to unity, that is, the correlation function decays almost linearly with time. At long times, the decay is exponential-like, as predicted by Landau-de Gennes mean field theory. Since Kerr relaxation experiments measure the time derivative of the collective second rank orientational pair correlation function, the simulations recover the near independence of the signal on time observed in experiments. In order to capture the microscopic essence of the dynamics of pseudonematic domains inside the isotropic phase, we introduce and calculate a dynamic orientational pair correlation function (DOPCF) obtained from the coefficients in the expansion of the distinct part of orientational van Hove time correlation function in terms of spherical harmonics. The DOPCF exhibits power law relaxation when the pair separation length is below certain critical length. The orientational relaxation of a local director, defined in terms of the sum of unit vectors of all the ellipsoidal molecules, is also found to show slow power law relaxation over a long time scale. These results have been interpreted in terms of a newly developed mode coupling theory of orientational dynamics near the INPT. In the present case, the difference between the single particle and the collective orientational relaxation is huge which can be explained by the frequency dependence of the memory kernel, calculated from the mode coupling theory. The relationship of this power law with the one observed in a supercooled liquid near its glass transition temperature is explored.The Journal of Chemical Physics 07/2004; 120(23):11256-66. · 3.12 Impact Factor - SourceAvailable from: Suman Chakrabarty[Show abstract] [Hide abstract]

**ABSTRACT:**Orientational dynamics in a liquid crystalline system near the isotropic-nematic (IN) phase transition is studied using molecular dynamics simulations of the well-known Lebwohl-Lasher model. As the IN transition temperature is approached from the isotropic side, we find that the decay of the orientational time correlation functions (OTCF) slows down noticeably, giving rise to a power law decay at intermediate time scales. The angular velocity time correlation function also exhibits a rather pronounced power law decay near the IN boundary. In the mean squared angular displacement at comparable time scales, we observe the emergence of a subdiffusive regime which is followed by a superdiffusive regime before the onset of the long-time diffusive behavior. We observe signature of dynamical heterogeneity through pronounced non-Gaussian behavior in orientational motion particularly at lower temperatures. This behavior closely resembles what is usually observed in supercooled liquids. We obtain the free energy as a function of orientational order parameter by the use of the transition matrix Monte Carlo method. The free energy surface is flat for the system considered here and the barrier between isotropic and nematic phases is vanishingly small for this weakly first-order phase transition, hence allowing large scale, collective, and correlated orientational density fluctuations. This might be responsible for the observed power law decay of the OTCFs.Physical Review E 07/2006; 73(6 Pt 1):061706. · 2.31 Impact Factor - SourceAvailable from: arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**We find in a model system of thermotropic liquid crystals that the translational diffusion coefficient parallel to the director D(parallel) first increases and then decreases as temperature drops through the nematic phase, and this reversal occurs where the smectic order parameter of the underlying inherent structures becomes significant. We argue, based on an energy landscape analysis, that the coupling between orientational and translational order can play a role in inducing the nonmonotonic temperature behavior of D(parallel). Such a view is likely to form the foundation of a theoretical framework to explain the anisotropic translation diffusion.Physical Review E 11/2006; 74(4 Pt 1):041704. · 2.31 Impact Factor

Page 1

Solid State and Structural

Chemistry Unit, Indian

Institute of Science,

Bangalore 560012, India

bbagchi@sscu.iisc.ernet.in

REVIEWS

MolecularDynamicsofThermotropic

LiquidCrystals:Anomalousrelaxation

dynamicsofcalamiticanddiscotic

liquidcrystals

Biman Jana AND Biman Bagchi

Abstract | Recent optical kerr effect (OKE) studies have demonstrated that orientational relaxation of

rod-like nematogens exhibits temporal power law decay at intermediate times not only near the

isotropic–nematic (I–N) phase boundary but also in the nematic phase. Such behaviour has drawn an

intriguing analogy with supercooled liquids. We have investigated both collective and single-particle

orientational dynamics of a family of model system of thermotropic liquid crystals using extensive

computer simulations. Several remarkable features of glassy dynamics are on display including

non-exponential relaxation, dynamical heterogeneity, and non-Arrhenius temperature dependence of

the orientational relaxation time. Over a temperature range near the I–N phase boundary, the system

behaves remarkably like a fragile glass-forming liquid. Using proper scaling, we construct the usual

relaxation time versus inverse temperature plot and explicitly demonstrate that one can successfully

define a density dependent fragility of liquid crystals. The fragility of liquid crystals shows a

temperature and density dependence which is remarkably similar to the fragility of glass forming

supercooled liquids. Energy landscape analysis of inherent structures shows that the breakdown of

the Arrhenius temperature dependence of relaxation rate occurs at a temperature that marks the

onset of the growth of the depth of the potential energy minima explored by the system. A model

liquid crystal, consisting of disk-like molecules, has also been investigated in molecular dynamics

simulations for orientational relaxation along two isobars starting from the high temperature

isotropic phase. The isobars have been so chosen that the phase sequence isotropic (I)–nematic

(N)–columnar (C) appears upon cooling along one of them and the sequence isotropic (I)–columnar

(C) along the other. While the orientational relaxation in the isotropic phase near the I–N phase

transition shows a power law decay at short to intermediate times, such power law relaxation is not

observed in the isotropic phase near the I–C phase boundary. The origin of the power law decay in

the single-particle second-rank orientational time correlation function (OTCF) is traced to the growth

of the orientational pair distribution functions near the I–N phase boundary. As the system settles

into the nematic phase, the decay of the single-particle second-rank orientational OTCF follows a

pattern that is similar to what is observed with calamitic liquid crystals and supercooled molecular

liquids.

Introduction

Thermotropic liquid crystals exhibit exotic phase

behavior upon temperature variation. In the

isotropic phase, a liquid does not exhibit any long

range translational or orientational order. The

nematic phase is endowed with a long-ranged

orientational order but lacks translational order.

Further cooling leads to a more ordered smectic

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Biman Jana and Biman Bagchi

phase where two-dimensional translational order

along with long-ranged orientational order sets

in the system. The isotropic-nematic (I–N) phase

transition, which is believed to be weakly first

order in nature with certain characteristics of

the continuous transition, has been a subject of

immense attention in condensed matter physics and

material sciences.1,2In contrast, the dynamics of

thermotropic liquid crystals have been much less

studied, the focus being mostly on the long-time

behavior of orientational relaxation near the I–N

transition.1A series of OKE measurements have,

however, recently studied collective orientational

relaxation in the isotropic phase near the I–N

transition over a wide range of time scales.3,4The

dynamics have been found to be surprisingly rich,

the most intriguing feature being the power law

decay of the OKE signal at short-to-intermediate

times.3,4The relaxation scenario appears to be

strikingly similar to that of supercooled molecular

liquids5, even though the latter do not undergo any

thermodynamic phase transition.

This work essentially consists of two parts.

In the first part we study dynamics of rod-like

molecules that form calamitic liquid crystal. In

the second part, we study the dynamics of disk-

like molecules that form discotic phase. Despite

such a large difference in aspect ratio and nature

of the nematic liquid crystals that they form,

there are aspects of relaxation that are remarkably

similar. However, a molecular level understanding

of observed anomalous orientational dynamics is

missing.

Relaxation of discotic phases has been less

explored than that of calamitic liquid crystals. The

discovery of discotic liquid crystals, that consist

of disk-like molecules is more recent and dates

back only to the late 19706. Upon cooling from the

high temperature isotropic (I) phase, discotic liquid

crystals typically exhibit a nematic (N) phase and/or

a columnar (C) phase7. The discotic nematic phase

is analogous to the nematic phase formed by rodlike

molecules in that there is a long-range orientational

order without the involvement of any long-range

translational order. In the columnar phase that is

typical of discotic liquid crystals, the molecules

are stacked on top of each other giving rise to a

columnar structure. These columns form a long-

range two dimensional order in the orthogonal

plane with either a hexagonal or a rectangular

symmetry. While the sequence of phases I–N–C has

been observed experimentally with a number of

discoticliquidcrystalsuponcooling,therehavebeen

only a few cases where only I–C or I–N transition

is observed8. Although computer simulations of

model liquid crystals have undergone an upsurge

in recent times, discotic liquid crystals are yet to

be studied in detail. Discotic molecules typically

contain an aromatic core with flexible chains added

in the equatorial plane. While atomistic models

could in principle be undertaken, molecular models,

where mesogens are approximated with particles

withwell-definedanisotropicshape,findtheirutility

in obtaining a rather generalized view. A simple

approach along this line involves consideration of

purely repulsive models involving hard bodies9.

This rather extreme choice is inspired by the idea

that the equilibrium structure of a dense liquid is

essentially determined by the repulsive forces which

fix the molecular shape. Along this line, thin hard

platelets, hard oblate ellipsoids of revolution, and

cut hard spheres have been investigated. Such an

approach is appealing for its simplicity9. However,

temperature plays no direct role in purely repulsive

models on the contrary to what is desired for

thermotropic liquid crystals9. In this respect, the

Gay-Berne pair potential10, which is essentially a

generalization of the Lennard-Jones potential to

incorporate anisotropic interactions, or one of its

variants10, where mesogens are approximated with

soft ellipsoids of revolution, appears to serve as a

more realistic model. In fact, discotic liquid crystals,

modeled by the Gay-Berne family of potentials,

have been found to capture the key features of the

experimentally observed phase behavior11.

In this article, we present results of molecular

dynamics simulations of a family of model

systems consisting of both rod-like and disk-

like molecules across the I–N and I–C transition.

Given the involvement of the phase transition

to an orientationally ordered mesophase upon

lowering the temperature, we choose to probe the

single-particle and collective orientational dynamics

in order to make comparison with relaxation

behaviour observed for supercooled liquids. We

have calculated the non-Gaussian parameter in the

orientational degrees of freedom in order to probe

the heterogeneous dynamics present in the system

near I–N transition. We have defined a fragility

index to quantitatively measure the glassy dynamics

observed in the orientational degrees of freedom.

We have also explored plausible correlation of the

features of the underlying energy landscape with

the observed non-Arrhenius dynamics in analogy

with supercooled liquids. This work follows up our

recent work12, which has reported the emergence

of power law decay regime(s) in orientational

relaxationacrosstheisotropic-nematictransition.In

the spirit of the universal power law in orientational

relaxation in thermotropic liquid crystals suggested

therein12, we compare the orientational dynamics

we observed here with those of calamitic liquid

crystals obtained from recent optical Kerr effect

measurements and molecular dynamics simulations

studies. We further discuss the analogous dynamics

observed in supercooled molecular liquids.

76

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Molecular Dynamics of Thermotropic Liquid Crystals: Anomalous relaxation dynamics of calamitic and discotic liquid crystals

REVIEW

Models and simulation details

A. Rod-like molecules

The systems we have studied consist of ellipsoids of

revolution. The Gay-Berne (GB) pair potential10,

that is well established to serve as a model potential

for systems of thermotropic liquid crystals, has

been employed. The GB pair potential, which

uses a single-site representation for each ellipsoid

of revolution, is an elegant generalization of the

extensively used isotropic Lennard-Jones potential

to incorporate anisotropy in both the attractive

and the repulsive parts of the interaction10,11. In

the GB pair potential, ith ellipsoid of revolution

is represented by the position riof its center of

mass and a unit vector eialong the long axis of the

ellipsoid. The interaction potential between two

ellipsoids of revolution i and j is given by

UGB

ij

=4ε?ˆ rij,ei,ej

??

ρ−12

ij

−ρ−6

ij

?

(1)

where

ρij=rij−σ?ˆ rij,ei,ej

?+σss

σss

(2)

Here σssdefines the thickness or equivalently,

the separation between the two ellipsoids of

revolution in a side-by-side configuration, rijis

the distance between the centers of mass of the

ellipsoids of revolution i and j, and ˆ rij=rij/rijis

a unit vector along the intermolecular separation

vector rij. The molecular shape parameter σ and the

energy parameter ε both depend on the unit vectors

eiand ejas well as on ˆ rijas given by the following

set of equations:

σij

?

ˆ rij,ei,ej

?

=σ0

?

1−χ

2

??

ei·ˆ rij+ej·ˆ rij

1+χ

?2

?

ei·ej

?

−

?

ei·ˆ rij−ej·ˆ rij

1−χ

?2

?

ei·ej

?

??−1/2

(3)

with χ = (κ2+1)/ (κ2−1) and

ε?ˆ rij,ei,ej

where the exponents ν and μ are the adjustable

parameter, and

?=ε0

?ε1

?ei,ej

??ν?ε2

?ˆ rij,ei,ej

??μ(4)

ε1

?ei,ej

?=

?

1−χ2?ei·ej

?2?−1/2

(5)

and

ε2

?

ˆ rij,ei,ej

?

=1−χ?

2

??

ei·ˆ rij+ej·ˆ rij

1+χ??

?

?2

ei·ej

?

+

?

ei·ˆ rij−ej·ˆ rij

1−χ??

?2

ei·ej

?

(6)

with χ?=(κ? 1/μ−1)/(κ? 1/μ+1). Here κ=σee/σss

is the aspect ratio of the ellipsoid of revolution with

σeedenoting the separation between two ellipsoids

of revolution in a end-to-end configuration, and

σss=σ0, and κ?=εss/εee, where εssis the depth of

the minimum of the potential for a pair of ellipsoids

of revolution aligned in a side-by-side configuration,

and εeeis the corresponding depth for the end-to-

endalignment.Here ε0isthedepthoftheminimum

of the pair potential between two ellipsoids of

revolution aligned in cross configuration. The GB

pair potential defines a family of models, each

member of which is characterized by the values

chosen for the set of four parameters κ, κ?, μ, and

ν, and is represented by GB(κ, κ?, μ, ν)11. Systems

consist of 500 ellipsoids of revolution in a cubic

box with periodic boundary conditions at several

temperatures, starting from the high-temperature

isotropic phase down to the nematic phase across

the I–N phase boundary have been simulated. We

have carried out several simulations with different

aspectratios(κ )whereforeachaspectratioisochors

of different densities have been investigated. All

quantities are given in reduced units defined in

terms of the Gay-Berne potential parameters ε0

and σ0: length in units of σ0, temperature in

units ofε0

kB, and time in units of

being the mass of the ellipsoids of revolution.

The mass as well as the moment of inertia of

each of the ellipsoids of revolution have been

set equal to unity. The intermolecular potential

is truncated at a distance rcutand shifted such

that U(rij= rcut) = 0,rij being the separation

between two ellipsoids of revolution i and j. The

equations of motion have been integrated using

the velocity-verlet algorithm with integration time

step dt =0.0015.12Equilibration has been done by

periodic rescaling of linear and angular velocities

of particles. This has been done for a time period of

tqfollowing which the system has been allowed to

propagatewithaconstantenergyforatimeperiodof

tein order to ensure equilibration upon observation

of no drift of temperature, pressure, and potential

energy. The data collection has been executed in a

microcanonical ensemble. At each state point, local

?σ2

0m

ε0

?1/2

, m

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Biman Jana and Biman Bagchi

Figure 1: The time evolution of the a) single particle OTCF on a log-log

plot for GB(3,5,2,1) along an isochor with density ρ=0.32 across I–N

transition at temperatures T =2.008, 1.697,1.499,1.396, 1.310,1.199

and 1.102 from left to right and b) collective second rank OTCF for the

same system and isochor at temperatures T =2.008, 1.499,1.396,

1.310,1.199 and 1.102 form top to bottom. TI−N is located between

T =1.499 and T =1.396. The portions fitted with straight line

correspond to power law decay regime.

potential energy minimization has been executed

by the conjugate gradient technique for a subset of

200 statistically independent configurations. The

landscape analysis has been done with a system

size of 256 ellipsoids of revolution, which is big

enough for having no qualitative change in the

results due to the system size.14Minimization has

been performed with three position coordinates and

two Euler angles for each particle, the third Euler

angle being redundant for ellipsoids of revolution.

B. Disk-like molecules

The parameterization, that we have employed here,

is κ = 0.345, κ?=0.2,μ =1,ν =211. Molecular

dynamics simulations have been performed with

the model discotic system containing 500 oblate

ellipsoids of revolution in a cubic box with periodic

boundary conditions. All the quantities reported

here are given in reduced units. The intermolecular

potential has been truncated at a distance rcut=1.6

and shifted. The equations of motion have been

integrated following the velocity-Verlet algorithm

with the integration time steps of dt =0.0015 in

the reduced units. Equilibration has been done in

an NPT ensemble. Following this, the system has

been allowed to propagate with a constant energy

and density in order to ensure equilibration. Upon

observation of no drift in temperature, pressure,

and potential energy, the data collection has been

executed in a microcanonical ensemble. The model

discotic system has been melted from an initial

fcc configuration at high temperatures and low

densities, and studied along two isobars at pressures

P =25 and P =10 at several temperatures.

Results and discussion

A. Calamitic liquid Crystals (rod-like molecules)

I. Single particle orientational dynamics

The orientational dynamics of the system at the

single particle level may be described by the first

and second order single particle orientational time

correlation functions (OTCF)Cs

are defined by

??

l(t)(l=1,2), which

Cs

l(t)=

iPl(ei(t).ei(0))?

??

iPl(ei(0).ei(0))?

(7)

where Plis the l-th rank Legendre polynomial

and the angular brackets stand for ensemble

averaging. Figure 1a shows the single particle

second rank OTCF in a log-log plot as the

temperature is lowered from high temperature

isotropic phase to low temperature nematic phase

across the I–N transition. The I–N transition

is marked by a jump in the orientational order

parameter S, defined for an N-particle system as

the largest eigenvalue of the ordering matrix Q:

Qαβ=1

α-component (in the space-fixed frame) of the unit

orientation vector eialong the principal symmetry

axis of the i-th ellipsoid of revolution.15Note the

emergence of the power law decay at short to

intermediate times near the I–N phase boundary. As

the I–N phase boundary is crossed upon cooling,

theadventoftwopowerlawdecayregimesseparated

by an intervening plateau at short-to-intermediate

times imparts a step-like feature to the temporal

behavior of the second rank OTCF. Such power

law relaxation near I–N phase boundary was an

area of great interest in the recent past16–21and

it has been investigated that the scenario is not a

unique property of the model we have studied; it

is a rather universal phenomenon of second rank

OTCF.12Such a feature bears remarkable similarity

to what is observed for supercooled liquids as the

glass transition is approached from the above.22,23

While for the supercooled liquid the emergence of

step-like feature is well understood as a consequence

of β relaxation, the origin of such a feature observed

for liquid crystal defied of reliable explanation.

N

?N

i=1

1

2

?3eiαeiβ−δαβ

?, where eiαis the

78

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Molecular Dynamics of Thermotropic Liquid Crystals: Anomalous relaxation dynamics of calamitic and discotic liquid crystals

REVIEW

Figure 2: a) The orientational correlation time in the logarithmic scale

as function of the inverse of the scaled temperature, the scaling being

done by the isotropic to nematic transition temperature TI−N. For the

insets, the horizontal and the vertical axis labels read same as that of

the main frame and are thus omitted for clarity. Along each isochor, the

solid line is the Arrhenius fit to the subset of the high-temperature data

and the dotted line corresponds to the fit to the data near the

isotropic-nematic phase boundary with the VFT form. b) The fragility

index m shown as a function of density for different aspect ratios. The

dashed lines are guide to the eye to illustrate the fact that the

dependence of the fragility index on the density is becoming stronger as

the aspect ratio becomes smaller.

II. Collective orientational dynamics

In experiments, one can probe orientational

relaxation through the decay of the OKE signal,

which is given by the negative of the time derivative

of the collective second rank OTCF Cc

The later is defined by

2(t).4,24,25

Cc

2(t)=

??

i

?

jP2

?ei(0)·ej(t)??

??

i

?

jP2

?ei(0)·ej(0)??.

(8)

Calculation of this correlation function is

computationally demanding, particularly at longer

times. In order to set a direct link with experimental

results, we show the temporal behavior of the OKE

signal in the log-log plot for the system across the

I–N phase transition in figure 1b. The short-to-

intermediate-time power law regime is evident in

the OKE signal for the system studied here. Like

single particle second rank OTCF, it is also verified

tobeauniversalphenomenonnearI–Ntransition.12

III. Fragility of liquid crystals

We estimate the orientational correlation time

τ as the time taken for Cs

90%, i.e.,Cs

in the logarithmic scale as a function of the

inverse temperature along the three isochors for

each of the three systems considered. We have

scaled the temperature by TI−Nin the spirit of

Angell’s plot, that displays the shear viscosity

(or the structural relaxation time, the inverse

diffusivity, etc.) of glass-forming liquids as a

function of the inverse of the scaled temperature,

the scaling being done in the latter case by the

glass transition temperature Tg.26,27For all the

three systems, two distinct features are common:

(i) in the isotropic phase far away from the I–N

transition, the orientational correlation time τ

exhibits the Arrhenius temperature dependence,

i.e., τ(T)=τ0exp(E/kBT), where the activation

energy E and the pre-factor τ0are both independent

of temperature; (ii) in the isotropic phase near

the I–N transition, the temperature dependence

of τ shows marked deviation from the Arrhenius

behavior and can be well described by the

Vogel-Fulcher-Tammann (VFT) equationτ(T)=

τ0exp[B/(T−TVFT)], where τ0, B, and TVFT

are constants, independent of temperature. Again

these features bear remarkable similarity with

those observed for fragile glass-forming liquid.

A non-Arrhenius temperature behavior is taken

to be the signature of fragile liquids. For fragile

liquids, the temperature dependence of the shear

viscosity follows the Arrhenius behavior far above

Tgand can be fitted to the VFT functional form in

the deeply supercooled regime near Tg.26,27The

striking resemblance in the dynamical behavior

described above between the isotropic phase of

thermotropic liquid crystals near the I–N transition

and supercooled liquids near the glass transition

has prompted us to attempt a quantitative measure

of glassy behavior near the I–N transition. For

supercooled liquids, one quantifies the dynamics by

a parameter called fragility index which measures

the rapidity at which the liquid’s properties (such as

viscosity) change as the glassy state is approached. In

thesamespirit28thatoffersaquantitativeestimation

of the fragile behavior of supercooled liquids, we

here define the fragility index m of a thermotropic

liquid crystalline system as21

2(t) to decay by

2(t = τ) = 0.1. Figure 2(a) shows τ

m=dlog10τ(T)

dTI−N/T

????T=TI−N

.

(9)

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Biman Jana and Biman Bagchi

Figure 3: Time evolution of the rotational non-Gaussian parameter

αR

dependence is shown at several temperatures (T =3.5, 3.25, 3.0, 2.75,

2.5, 2.25, 2.0, 1.88, 1.82, 1.78, and 1.5) across the isotropic-nematic

(I–N) transition along an isochor at density = 0.33. a) On a different

scale along the vertical axis (appearing on the right), time evolution of

the mean square angular deviation??φ2(t)?is shown in a log-log plot

phase (T =1.5) and the other two temperatures (T =3.0 and 2.75) that

are closest to the I–N transition on either side along with the time

evolution of αR

(appearing on the right), the time evolution of the single-particle second

rank orientational time correlation function Cs

plot for the two temperatures (T =3.0 and 2.75) that are closest to the

I–N transition on either side along with the time evolution of αR

solid lines denote the curves for the high temperature isotropic phase

and the dashed lines for the low temperature nematic phase.

2(t) in a semi-log plot for the system with aspect ratio κ=3. The time

for three temperatures: the highest temperature studied in the isotropic

2(t), and b) On a different scale along the vertical axis

2(t) is shown in a log-log

2(t).The

It is clear from the above equation that if τ(T)

follows Arrhenius temperature dependence, m will

be constant throughout the whole temperature

range. Figure 2(b) shows the density dependence

of the fragility index for the three systems with

different aspect ratios. For a given aspect ratio, the

fragility index increases with increasing density,

the numerical values of the fragility index m

being comparable to those of supercooled liquids.

The density dependence observed in the present

work is remarkably similar to those observed for

supercooled liquids. For the range of aspect ratios

studied here, the dependence of the fragility index

on the density is becoming stronger as the aspect

ratio becomes smaller.

IV. Heterogeneous dynamics

Another hallmark of fragile glass-forming liquids

is spatially heterogeneous dynamics29reflected

in non-Gaussian dynamical behavior.30It is

intuitive that the growth of the pseudo-nematic

domains, characterized by local nematic order,

in the isotropic phase near the I–N transition

would result in heterogeneous dynamics in liquid

crystals. We have, therefore, monitored the time

evolution of the rotational non-Gaussian parameter

(NGP),31,32

αR

2(t), which in the present case is

defined as

??φ4(t)?

αR

2(t)=

2??φ2(t)?2−1

(10)

where

??φ2n(t)?=1

Here φiis the rotation vector like the position

vector riappears in the case of translational NGP of

ith ellipsoid of revolution, the change of which is

defined by ?φi(t)=φi(t)−φi(0)=?t

and N is the number of ellipsoids of revolution in

the system. NGP will have value equal to zero when

system dynamics is spatially homogeneous and will

have a non-zero value when the system dynamics

is spatially heterogeneous. As a typical behavior,

Fig. 3(a) and (b) show the time dependence of the

rotational NGP for one of the systems at several

temperatures across the I–N transition along an

isochor. On approaching the I–N transition upon

cooling, a bimodal feature starts appearing with

the growth of a second peak, which eventually

becomes the dominant one, at longer times.21,31We

further investigate the appearance of this bimodal

feature in NGP plot. To this end we calculate

mean square angular deviation (MSAD) of the

system at different temperatures starting from high

temperature isotropic phase to low temperature

nematic phase. The appearance of the bimodal

feature in the rotational NGP is accompanied by a

signature of a sub-diffusive regime in the temporal

evolution of the MSAD, the time scale of the short-

time peak and that of the onset of the sub-diffusive

regime being comparable, as shown in Fig. 3(a).21,31

We note that the dominant peak appears on a time

scale which is comparable to that of onset of the

diffusive motion in orientational degrees of freedom

N

N

?

i=1

?|φi(t)−φi(0)|2n?.

(11)

0dt?ω?t??,

ωibeing the corresponding angular velocity,22,23

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Figure 4: a) The temperature dependence of the average inherent structure energy per particle eIS along three isochors at

densities ρ=0.31, 0.32, and 0.33 for κ=3. The inset shows the root mean square fluctuation in inherent structure energyσeIS,

computed from a subset of 200 configurations for each state point, as a function of temperature T at the same three densities.

b) The evolution of the average order parameter S with temperature both for the inherent structures (filled) and the

corresponding pre-quenched ones (empty). The inset shows the temperature dependence of the average potential energy Epot at a

state point obtained from averaging over the molecular dynamics trajectory. For clarity, Epot is shown for the state points along

only one isochor corresponding to the density ρ=0.32. The state points considered in our simulations correspond to (i) the

isotropic (I) phase for T ? 1.297 and the smectic-B (Sm–B) phase for T ? 0.595 along the isochor at ρ=0.31; (ii) I for T ? 1.495

and Sm-B for T ? 0.706 at ρ=0.32; (iii) I for T ? 2.089 and Sm-B for T ? 0.803 at ρ=0.33. c) The inverse temperature

dependence of the single-particle orientational relaxation times τs

straight lines are the Arrhenius fits for the subsets of data points, each set corresponding to a fixed density: ρ=0.31 (circle),

ρ=0.32 (square), ρ=0.33 (triangle up).

l, l =1 (filled) and l =2 (empty), in the logarithmic scale. The

(ODOF) as evident in Fig. 3(a).21,31Similar feature

has been observed recently for supercooled water.31

We further find that the time scale at which the

long-time peak appears is also comparable to the

time scale of onset of the plateau that is observed

in the time evolution of Cs

3(b).21,31

2(t), as shown in Fig.

V. Energy landscape analysis

Several studies have attempted to interpret the

dynamics of glass-forming liquids in terms of the

features of the underlying energy landscapes.33–38

Energy landscape analysis gives the potential energy,

which devoid of any kind of thermal motions, of

inherent structures of the parent liquid and hence

provides a better understanding of the structure and

dynamics of the parent liquid. Figure 4a displays the

average inherent structure energy as the change in

temperaturedrivesthesystemacrossthemesophases

along three different isochors. Figure 4b shows the

concomitant evolution of the average orientational

order parameter S both for the inherent structures

and the corresponding pre-quenched ones. It is

evident that the average inherent structure energy

remains fairly insensitive to temperature in the

isotropic phase before it starts undergoing a steady

fall below a certain temperature that corresponds

to the onset of the growth of the orientational

order14. As the orientational order grows through

the nematic phase, the system continues to explore

deeper potential energy minima until a plateau is

reached on arrival at the smectic phase14. In the

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Biman Jana and Biman Bagchi

Figure 5: The average second-rank orientational order parameter ?P2? as a function of temperature

along two isobars. The circles correspond to the data for the pressure P =25 and the squares for

P =10. The phase boundaries are shown by vertical dotted lines for P =25 and by a vertical solid line

for P =10.

1

0.8

0.6

0.4

0.2

02

2.53

T

<P2>

3.54

inset of Fig. 4a, the location of the maximum of the

mean square fluctuation in the inherent structure

energy shows that the system explores potential

energy minima spanning over a broader energy

range as it settles into the nematic phase. This

suggests the critical role of fluctuation effects in the

nematic phase. The average potential energy for a

state point obtained from the molecular dynamics

trajectory, however, decreases rather smoothly in

all three phases with decrease in temperature as

illustrated in the inset of Fig. 4b. It is evident that

the signature of the I–N transition is quite weak here

in contrast to that of the nematic-smectic transition.

We have repeated the same analysis for a larger

system size to check the effect of finite system size,

but qualitatively ended up with same conclusions as

the smaller one. Note that this has been observed for

a glassy system,36where the average IS energy also

falls over a temperature range.34Like supercooled

liquid, we have also observed a Gaussian form of

number density of IS with eIS.21

Figure 4c illustrates the correlation of the

energy landscape behaviour with the dynamics

the system exhibits. Here, we define relaxation

times τs

dramatic slow down of orientational dynamics with

decreasing temperature near the I–N transition

manifests in the temperature dependence of

these relaxation times. Figure 4c shows that in

the isotropic phase far from the I–N transition

region τs

l(T) exhibits the Arrhenius behavior, i.e.,

τs

l(T)=τ0,lexp[El/(kBT)], where the activation

energy Eland the infinite temperature relaxation

time τ0,lare independent of temperature. We find

l(T) as the time when Cs

l(t)=e−1.14The

that the breakdown of the Arrhenius behavior

occurs at a temperature that marks the onset of

the growth of the depth of the potential energy

minima explored by the system.14Such correlations

of different other properties with the landscape

have been investigated in several other studies for

both supercooled liquids and thermotropic liquid

crystals.39,40

B. Discotic liquid crystals (disk-like molecules)

We first need to characterize the phases that appear

along the isobars studied here. To this end, we have

monitored the average second-rank orientational

order parameter ?P2?. ?P2? tends to zero in the

isotropic phase but retains a non-zero value because

of the finite size of the system. In the nematic phase,

?P2? has a value above 0.4. For the columnar phase,

?P2? is above 0.9. In the present case, we observe

the I–N–C phase sequence along the isobar at the

higher pressure and the sequence I–C along the

other isobar. The temperature dependence of ?P2?

has been shown in Figure 5.

I. Single particle orientational dynamics

We have investigated orientational dynamics

at the single-particle level by monitoring the

temporal evolution of the corresponding second-

rank orientational time correlation functions

(OTCF). In Fig. 6, we show the time evolution

of the single-particle second-rank OTCF at several

temperatures in log-log plots. The emergence of a

power law decay at short-to-intermediate times near

the I–N phase boundary is notable in Fig. 6(a). It

follows from Fig. 6(a) that as the system transits

82

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Figure 6: Time evolution of the single-particle second-rank OTCF in log-log plots for the discotic

system at several temperatures. The dashed lines are the simulation data corresponding to increasing

orientational order parameter from the bottom to the top. The solid lines are the linear fits to the

data, showing the power law decay regimes. (a) Along the isobar at P =25.0 at several temperatures:

T =2.991, 2.693, 2.646, and 2.594; (b) Along the isobar at P =10.0 at all the temperatures studied for

the isotropic phase.

across the I–N phase boundary, two power law

relaxation regimes, separated by a plateau, appear

giving rise to a step-like feature. However, the decay

of the single-particle second-rank OTCF in the

isotropic phase near the isotropic-columnar phase

boundary does not follow any power law as evident

in 6(b).

II. Collective orientational dynamics

In optical heterodyne detected optical Kerr effect

measurements (OHD-OKE), one probes collective

orientational relaxation41. In recent OHD-OKE

experiments with calamitic liquid crystals, the decay

of the OKE signal has been found to follow a

complex pattern.3,4The most intriguing feature

has been the power law decay regimes at short-to-

intermediate times.4,5We have therefore monitored

the time evolution of the collective second-rank

OTCF. In the present case, the negative of the

time derivative of the collective second-rank OTCF

provides a measure of the experimentally observable

OHD-OKE signal. As monitoring the time evolution

of collective orientational correlation function

is computationally quite demanding, we have

restricted ourselves to the short-to-intermediate

time dynamics that would suffice to compare

the most intriguing aspect of the experimental

observations. In Fig. 7, we show in log-log plots the

temporal behavior of the OKE signal derived from

present system at several temperatures. A short-to-

intermediate-time power law regime is evident in

the decay of the OKE signal on either side of the I–N

transition as illustrated in Fig. 7(a). In consistency

with the single-particle dynamics, such a power law

decay regime is not observed for the OKE signal in

the isotropic phase near the I–C phase boundary

as apparent in Fig. 7(b). It follows from the time

evolution of the single-particle second-rank OTCF

shown in Fig. 6(a) that as the system settles into

the nematic phase, two power law decay regimes,

that are separated by a plateau, emerge. Such a

feature bears a close resemblance with what has been

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Biman Jana and Biman Bagchi

Figure 7: The short-to-intermediate time decay of the OKE signal in

log-log plots for the discotic system. The dashed lines are the simulation

data and the solid lines show the linear fits to the data showing the

power law decay regimes: ∼ t−α. The values of the power law exponent

α are given below in the parenthesis. (a) Along the isobar at P =25.0 at

several temperatures: T =2.991, T =2.693 (α=0.208), T =2.646 (α=0.194),

and T =2.594 (α=0.178). (b) Along the isobar at P =10.0 at several

temperatures: T =2.298, 2.196, and 2.143. Temperature decreases from

the top to the bottom at the left of the figure in each case.

observed recently for a model system of calamitic

liquid crystals21. The decay pattern is also similar

to those observed for models supercooled molecular

liquids. In fact, based on a series of OHD-OKE

measurements Fayers and coworkers have recently

drawn an analogy in the orientational dynamics

between calamitic liquid crystals in their isotropic

phase near the I–N transition and supercooled

molecular liquids. The analogous dynamics could

be captured in a subsequent molecular dynamics

simulation study of model systems of these two

classes of soft condensed matter. The short-to-

intermediate time power law decay of the OKE

signal observed therein bears a close similarity

with what is found in the present discotic system

across the I–N transition. The contrasting behavior

observed in orientational relaxation in the isotropic

phase near the I–N and the I–C phase boundaries is

noteworthy. Such an observation may throw new

light on the origin of the power law relaxation

in the isotropic phase near the I–N transition.

While the I–C transition is strongly first order

in nature, the I–N transition is only weakly first

order with certain characteristics of the continuous

transition. This is reflected in the present case in

a much larger change in the density marking the

I–C transition as compared to the I–N transition

(data not shown). The weakly first order nature

of the I–N transition appears to play a role in the

short-to-intermediate time power law relaxation.

It seems fair to trace the origin of the power law

decay in orientational relaxation to the growth in

the orientational correlation length in the isotropic

phase near the I–N transition.

III. Theoretical analysis

The I–N phase transition is weakly first order

both in calamitic and discotic systems. This is

manifested in the growing orientational pair

correlation length as the I–N phase boundary is

approached from the high temperature isotropic

phase. Apparently, a second order phase transition

at a temperature only slightly lower (by ≈ 1 K),

where the orientational correlation length would

have diverged, is preempted by the weakly first order

phase transition. Nevertheless, even this weakly

first order phase transition is driven by the growing

correlation length. The temperature dependent

growth of this correlation length ξ(T) can be given

by the following expression1

ξ(T)=A?T∗−T?−ν

(12)

where ν is 0.5 in the Landau mean-field theory.

A simple mode coupling theory, based on time

dependent density functional theory, shows that

this growing correlation length can give rise to a

power-law decay of the type observed in simulations.

This approach uses the the generalized Debye–

Stokes–Einstein relation between the correlation

time, diffusion, and friction42

C2(z)=

1

(z+6ADR(z))

(13)

and

DR(z)=

kBT

I(z+ς(z))

(14)

where A is equal to 1 for the single-particle

relaxation, but is related to orientational caging

for collective dynamics. It was shown elsewhere,

the growing correlation length can give rise to a

singularfrequencydependenceof ς overafrequency

range ς(z)∼A/zαwith α=0.5. This power law

dependence in the frequency dependence of friction

in turn gives rise to a power law decay in the

orientational time correlation function.

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Figure 8: The orientational pair distribution function g220 (r) for the

model discotic system at state points along two isobars: (a) the one at

P =25 and (b) the other at P =10. The temperature decreases from the

bottom to the top at the position of the dominant peak of the curves

starting from high temperature isotropic phase down to the temperature

which is just above the temperature at which columnar phase appears.

IV. Orientational pair correlation function

Thus, in the above mentioned theory, the origin

of the power law decay is essentially the same as

observed near the critical phenomena. However,

one may not expect a universal behavior since there

is no true divergence. The absence of power law

decay near the I–C phase boundary could then be

due to the absence of any growing correlation length.

The I–C phase transition is strongly first order in

nature where both orientational and positional

order set in at the same time. Since the growth of

orientational correlation is small, a power law decay

is not expected. To verify our assertion, we have

calculated the distance dependent orientational

pair distribution functiongll?m(r)43for the system

studied here along both the isobars and presented

in Fig. 8(a) and Fig. 8(b), respectively. While the

growth of orientational correlation length is clearly

evident across the I–N transition, such a growth

is found to be totally absent in the isotropic phase

near the I–C phase boundary.

Conclusion

We have presented theoretical and computer

simulation studies of dynamics of calamitic and

discotic liquid crystals, both near the I–N phase

boundaryandalsointherespectiveliquidcrystalline

phases. Computer simulation studies of single

particle and collective orientational dynamics of

thermotropic liquid crystals near the isotropic-

nematic (I–N) transition are presented and

compared with the dynamics of supercooled liquids

near glass transition. The short-to-intermediate

time scale power law decay in the orientational

relaxation appeared to be the most intriguing

feature. In analogy with the supercooled liquids,

a fragility index of liquid crystals is introduced to

quantify the glassiness of orientational dynamics

near the I–N transition. Our investigation of

spatially heterogeneous dynamics strengthens the

analogy further. The striking resemblance in

the correspondence between the manner of the

exploration of the potential energy landscape

and the onset of the non-Arrhenius temperature

dependence of the relaxation time might imply a

unique underlying landscape mechanism for slow

dynamics in soft condensed matter.

In the second part of study, of disk-like

molecules, the system has been studied along two

isobars so chosen that the phase sequence I–N-

C appears upon cooling along the one and the

sequence I–C along the other. We have investigated

temperature dependent orientational relaxation

across the I–N transition and in the isotropic

phase near the I–C phase boundary with a focus

on the short-to-intermediate time decay behavior.

While the orientational relaxation across the I–N

phase boundary again shows a power law decay

at short-to-intermediate times, such power law

relaxation is not observed in the isotropic phase

near the I–C phase boundary. Study of orientational

pair distribution function shows that there is a

growth of orientational pair correlation near the

I–N transition whereas such a growth is absent in

the isotropic phase near the I–C phase boundary. As

the system settles into the nematic phase, the decay

of the single-particle second-rank orientational time

correlation function follows a pattern that is similar

to what is observed with calamitic liquid crystals

and supercooled molecular liquids.

The present study brings out the role of

intermolecular correlations in giving rise to the

power law, in a way quite similar to the emergence

of such effects in supercooled liquids, except that

here the fluctuations due to a weakly first order

phase transition makes the effects much more

pronounced, as evident from experiments and

simulations. Energy landscape analysis provides a

convincing testimony to this observation.

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Biman Jana and Biman Bagchi

Future work may look into the relaxation

dynamics of discotic liquid crystals more extensively.

As already mentioned, this system has not

been studied in adequate detail. Another greatly

interesting system is calamitic liquid crystals of

dipolar rod-like molecules because many real

molecular systems are dipolar. Such a system can

exhibit dynamics distinctive of the system. Only a

few studies exist along this line44.

Acknowledgement

It is a great pleasure to thank Dr. Dwaipayan

Chakrabarti for helpful suggestions and discussions

during the preparation of the manuscript. This work

was supported in part by a grant from DST, India.

BJ thanks CSIR, India for providing SRF.

Received 15 April 2009.

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Biman Jana did his undergraduate

study from Calcutta University, West

Bengal and master study from Indian

Institute of Technology, Kanpur. He

is currently pursuing his doctoral

degree at Indian Institute of Science,

Bangalore under the supervision of Prof.

Biman Bagchi. His research focuses on

understanding the dynamics of complex

systems which includes liquid crystals

and biomolecular hydration.

Biman Bagchi obtained his Ph.D. degree

from Brown University in 1981 with

Professor Julian H. Gibbs. He was

Research Associate at University of

Chicago (1981–1983), where he worked

with Professors David W. Oxtoby,

Graham Fleming, and Stuart Rice, and

at University of Maryland (with Robert

Zwanzig) before returning to India in

1984 to join as faculty in Indian Institute

of Science, Bangalore. He is a Fellow of the Indian Academy of

Sciences, Indian National Academy of Science and also of the

Third World Academy, Trieste. His research interests include

statistical mechanics, relaxation phenomenon, chemical reaction

dynamics, phase transitions, protein folding and enzyme kinetics.

86

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