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Effect of External Field on the Nematic
to Isotropic Transition: An Entropic
Sampling Study
a

b

a

a

Regina Jose , D. Jayasri , Surajit Dhara , K. P. N. Murthy & V.
S. S. SastryaaSchool of Physics, University of Hyderabad, Hyderabad, Indiab

Faculty of Mathematics and Physics, University of Ljubljana,
Ljubljana, Slovenia
Available online: 30 Jun 2011

To cite this article: Regina Jose, D. Jayasri, Surajit Dhara, K. P. N. Murthy & V. S. S. Sastry (2011):
Effect of External Field on the Nematic to Isotropic Transition: An Entropic Sampling Study, Molecular
Crystals and Liquid Crystals, 545:1, 168/[1392]-175/[1399]
To link to this article: http://dx.doi.org/10.1080/15421406.2011.571999

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Mol. Cryst. Liq. Cryst., Vol. 545: pp. 168=[1392]–175=[1399], 2011
Copyright # Taylor & Francis Group, LLC
ISSN: 1542-1406 print=1563-5287 online
DOI: 10.1080/15421406.2011.571999

Effect of External Field on the Nematic
to Isotropic Transition: An Entropic
Sampling Study
REGINA JOSE,1 D. JAYASRI,2 SURAJIT DHARA,1
K. P. N. MURTHY,1 AND V. S. S. SASTRY1
Downloaded by [University of Hyderabad] at 03:13 01 July 20111

School of Physics, University of Hyderabad, Hyderabad, India
Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana,
Slovenia
2

Earlier experiments on the effect of an external field on the nematic-isotropic transition temperature indicate its linear increase with field in the low field regime, incontrast to a quadratic dependence at higher fields. This was explained phenomenologically as due to the effective suppression of longer wavelength order director
modes at low fields. We investigate this effect by carrying out Monte Carlo simulation on a lattice model based on Lebwohl-Lasher potential. We achieve the necessary
high temperature resolution (one part in 104) by performing an entropic sampling on
the system, and our results support this phenomenological interpretation.
Keywords Entropic sampling; Monte Carlo methods; paranematic-nematictransition; Wang-Landau algorithm

1. Introduction
The anisotropic properties of the liquid crystalline phases, like nematics, arise fromlong range orientational order of the constituent anisotropic molecules while retaining fluid like flow behaviour. Coupling of external fields like electric field, to theresulting dielectric anisotropy of the medium leads to controllable molecular orientation. Its manifestation is for example seen as an observable shift in thenematic-isotropic (NI) transition temperature, as well as a weak orientational orderin the high temperature isotropic phase (paranematic phase). It is known that thistransition remains weakly first order for a material with positive dielectric anisotropy and it vanishes at a critical value of the applied field in the medium. The
consequent changes in the (dis)order show up as observable variation in different
physical properties, indicating new phases, and other field induced critical effects[1]. The effect of the applied electric field on the paranematic-nematic transition temperature (TPN) was investigated experimentally in liquid crystals (LC) with both
positive and negative dielectric anisotropies [2–7]. The nature of the transition is
Address correspondence to Regina Jose, School of Physics, University of Hyderabad,
Hyderabad, 500046, India. Tel.: þ91-40 23011033; Fax: þ91-40 23010227; E-mail: reginajose.c@gmail.com

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Field Effect on the Nematic-Isotropic Transition

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affected differently owing to the qualitative difference between the ordering
directions induced by intermolecular interactions on one hand and the coupling of
the molecules with the field on the other [8]. However, in both the cases it was experimentally shown that TPN increases linearly for small applied fields (107 V=m) contrary to the expectation based on the Landau-de-Gennes energy considerations,which in fact predicts a quadratic dependence. The origin for this result (at low
fields) is attributed to quenching of director fluctuation modes by the applied fieldleading to additional dependence on the field which turns out to be linear. Of course
experimental detection of this subtle effect needs application of very low fields andhence require experimental facilities to measure rather very small shifts in temperature [5]. The enhancement in free energy, DF due to loss of entropy resulting fromthe quenching of long wavelength director modes was calculated phenomenologically by introducing two length scales: the electric coherence length n / 1=jEaj andthe cut-off wave vector qc / 1=l (l is of the order of the molecular dimension).
The enumeration of the modes in the presence of these two bounding cut-off wave
vector amplitudes, needed to compute the decrease in the entropy at an applied field
Ea, leads to [5]DF ¼ ajEa j þ dEa2

ð1ÞHere d << a, leading to prominent linear dependence on Ea in the low field regime.
The experimentally observed linear shift in TPN, in the field range up to 107 V=m, iswithin 0.2 C for medium with positive dielectric anisotropy and within 0.6 C for thatwith negative dielectric anisotropy.
In this context, the present work reports Monte Carlo simulation results on thevariation of the paranematic-nematic transition temperature with applied field. The
motivation is to see if the phenomenological interpretation of the experimental
observation (in terms of quenching of certain director modes) is also borne out by
carrying out simulations based on microscopic model. We used the well studied
lattice-model (Lebwohl-Lasher) potential to simulate the results, augmented with adielectric energy density contribution with the usual quadratic dependence on thefield. The interest is to investigate if a linear dependence emerges from such a model
at low enough fields. The challenge of achieving a high precision in the temperature
variation during simulation was met by employing the more recent entropic
sampling method (generally referred to as the non-Boltzmann sampling) [9], thereby
achieving temperature resolution of the simulated results comparable to theexperiments.
In section 2 we provide a brief introduction to the entropic sampling techniqueand details of simulation. Section 3 presents the results and discussion comparing
them with experimental findings. Conclusions are presented in section 4.

2. (a) Entropic Sampling
In standard Monte Carlo simulations, based on given Hamiltonian models, a
random walk is performed in the relevant configuration space seeking the required
statistical distribution (the Boltzmann probability prescription) by the applicationof Metropolis algorithm [10]. The resulting set of microstates, after the equilibration,corresponds to the canonical ensemble appropriate to the set experimental

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conditions, primarily the temperature. This procedure, which is now termed as
Boltzmann sampling, is to be repeated for each specific experimental condition
and the equilibrium physical properties are averages over the respective ensembleof microstates. More recent sampling methods attempt at achieving very high resolution in temperature, while not needing to carry out separate simulation for eachvalue. Besides, they also address with reasonable success certain non-trivial issuesof quasi-ergodicity and critical slowing down, encountered for example near different phase transitions [11–15]. In such cases, a random walk is performed which is
uniform with respect to the energy E of the system through a suitable algorithm.
While the set of microstates collected during this energy-uniform walk are unphysicalfrom experimental view point, they indeed contain all microstates relevant to therange of temperature of interest. Canonical ensembles appropriate to the desired
temperatures are then extracted from the so-called entropic ensemble throughreweighting procedure. We employ one such convenient algorithm [9], modified tosuit LC systems [16] to construct those entropic ensembles. The crucial step in thisprocedure is the accurate estimate of the density of states g(E) of the given system.
Wang-Landau algorithm provides a learning procedure to compute g(E). An
energy-uniform ensemble is then readily obtained by performing a random walk,compensating for the bias in the configuration space due to inherent density distribution of states. The macroscopic value of any physical property, say O, in thermal
equilibrium at temperature T (b ¼ 1=kBT), is then computed from the collection of
microstates in the entropic ensemble fCig by a reweighting procedure as
PN
hOi ¼

i¼1 OðCi ÞgðEðCi ÞÞ exp½bEðCi ÞPN
i¼1 gðEðCi ÞÞ exp½bEðCi Þð2Þ(B) Model Potential and Simulation Details
In the lattice model used here, a cubic lattice (of length L ¼ 20) is considered,imposing periodic boundary conditions to minimize the finite size effects. Each lattice site is occupied by a headless vector (typically representing a small group of uniaxial and apolar liquid crystal molecules), and the interactions among the lattice sites
are restricted to only the nearest neighbours. Since the nematic-isotropic transition is
primarily connected with the orientational degrees of freedom, one such latticemodel (Lebwohl-Lasher) [17] permitting interactions based on the relative orientations of the molecules has been extensively used as it captures the essential featuresof the N-I transition remarkably well. We use this model for the present work,
supplemented by the interaction term (with positive dielectric anisotropy) withthe applied electric field. The total energy thus is expressed as
E ¼ eX
hiji

P2 ðcosðhij ÞÞ  Ea2Xi

P2 ðcosðhi ÞÞð3Þwhere e is the nearest neighbor positive coupling constant, hij is the angle between
unit vectors representing the local directors at ith and jth sites, hi is the angle between
the local director at ith site and the field direction. e sets the energy scale and the temperature T is specified in reduced units given by T ¼ kBt=e, where t is the actual sample temperature. Due to the assumed positive dielectric anisotropy of the system, theField Effect on the Nematic-Isotropic Transition

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applied field favours parallel alignment of the long axes with the field. After obtaining the density of states (DoS) with respect to E at a given field value as describedabove, several independent energy-uniform random walks were performed to collecta number of entropic ensembles each consisting of 3  107 microstates. This wasfound necessary to improve the statistical quality of the data to meet the experimental conditions. Computation of the different physical quantities, averaged over
several canonical ensembles constructed under identical conditions yielded accuracies compatible with the experimental errors in the temperature measurement.
Canonical ensembles so extracted have at least 7  105 microstates near the transition. Transition temperatures are determined from the peak position of specificheat CV plotted as a function of temperature at each field value. The CV values
are computed from the energy fluctuations as

CV ¼

1
ðhE 2 i  hEi2 Þ
KB T 2

ð4Þ

TPN values obtained from different canonical simulations (at a given T and Ea) areaveraged so as to yield an error not exceeding 3  104.

Figure 1. Simulation results on the liquid crystal system. (a) density of states, (b) energy
histogram of the entropic ensemble, (c) temperature variation of equilibrium energy (dashedline) and specific heat (solid line), (d) orientational order (dashed line) and its susceptibility(solid line).

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3. Results and Discussion
The results of Wang-Landau simulation without applied field are shown in Figure 1.
Here, (a) and (b) show respectively, the density of states (DoS) with respect toenergy, and the histogram recorded while collecting the microstates in the entropic
ensemble. The temperature evolution of average energy, specific heat CV, averageorientational order and its susceptibility v are shown in Figure 1(c) and
Figure 1(d). For the temperature resolution achieved, the transition appears quite
sharp as seen from the jump in the energy and in the order parameter near TPN. Thisis also evident from the sharply peaked CV and v profiles.
DTPN ¼ TPN(Ea)  TPN(0) (i.e., shift in TPN) determined from the position of theCV peak is shown as a function of Ea in Figure 2. As may be seen, there are two distinct regions of variation of this shift with field. For low values of the field (.0.04 inarbitrary units), the temperature shift is linear, tending to become quadraticallydependent for higher values. As is shown in the inset of the figure, a linear fit seems
to be appropriate in the low field region. The higher field regime fits very well to apolynomial in Ea of order 2, confirming the expected variation from the dielectricenergy.
The variation of CV with temperature for various fields is shown in Figure 3(a).
It is observed that near and above Ea  0.15, the CV peak flattens considerably indicating softening of the transition in the presence of high enough fields. Of course thisalso makes the determination of the TPN (at those high values) relatively more prone
to errors. In fact in a positive dielectric liquid crystal medium, the trajectory of thetransition with field is expected to end in a critical point. Our results are in agreementwith reports on a system of 1000 particles wherein the critical value of Ea2 wasFigure 2. Variation of DTPN with field (solid line is a linear fit and dashed line is a quadratic
fit to the data in the low and high field regimes respectively). The inset focuses on the variationin the low field regime.

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Field Effect on the Nematic-Isotropic Transition

Figure 3. (a) Temperature variation of specific heat at various fields. (b) Variation of thewidths of nematic susceptibility (FWHM) with applied field. (Figure appears in color online.)

reported to be well below 0.2 [6]. The softening of the transition with increase of
applied field is also seen from the variation of the width i.e., full width at half
maximum (FWHM) of the nematic susceptibility (computed from reweightingDownloaded by [University of Hyderabad] at 03:13 01 July 2011

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Figure 4. Energy per lattice site (solid circles) and the corresponding representative entropy(open triangles) versus applied field at T ¼ 1.1239 (just below TPN(Ea ¼ 0)).

results of a single entropic ensemble) and is shown in Figure 3(b). This width remains
almost constant in the lower field regime (consistent with linear temperature shift)and increases for the higher field values. This seems to indicate that the thermally
driven fluctuations in the order near the PN transition are effectively quenched at
the lower values of Ea.
The field dependence of energy at a temperature just below the PN transition isshown in Figure 4 (solid circles). The corresponding values of entropy computedfrom the DoS data i.e., log(g(E)) is also shown in this figure (open triangles). It
may be noted that the values of the entropy shown are representative being accurate
to within an additive constant, and are obtained by appropriate normalization of
DoS over the chosen energy range. It is remarkable that despite the quadratic dependence of the energy on the applied field introduced through the Hamiltonian, themacroscopic energy and consequently the associated entropy, show striking linear
behaviour at low fields conforming both the experimental observations, as well as
lending support to the phenomenological view point of partial quenching of director
modes over a length scale determined by the interplay of the applied field strengthand the elastic constant.

3. Conclusion
We determine the shift in the paranematic-nematic transition temperature due to theapplication of an electric field in a liquid crystal system with positive dielectric anisotropy. We used recent Monte Carlo simulation methods which permits high accuracy and resolution in computing the phase transition temperatures. Our results
agree with the experimental observation of the linear temperature shift at the very
low fields. Further, the observed variation of energy and the representative entropy,
with applied field, also exhibit a linear behavior in the corresponding field regime
thus lending support to the phenomenological interpretation of the observed datain terms of partial quenching of director modes in the presence of applied fields.

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Acknowledgment
The authors acknowledge Centre for Modeling Simulation and Design, University ofHyderabad for providing computational facilities. RJ thanks CSIR, New Delhi forthe award of Research Fellowship.

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