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Monte Carlo study of the isotropic–nematic

interface in suspensions of spherocylinders

Tanja Schilling1, Richard Vink1, and Stefan Wolfsheimer1

Johannes Gutenberg Universit¨ at Mainz, 55099 Mainz, Germany

Abstract. The isotropic to nematic transition in suspensions of anisotropic col-

loids is studied by means of grand canonical Monte Carlo simulation. From mea-

surements of the grand canonical probability distribution of the particle density,

the coexistence densities of the isotropic and the nematic phase are determined, as

well as the interfacial tension.

1Introduction

On change of density, suspensions of rod-like particles undergo a phase tran-

sition between an isotropic fluid phase, where the particle orientations are

evenly distributed, and an anisotropic fluid phase (called “nematic” phase),

where the particle orientations are on average aligned. Fig. 1 shows a sketch

of these phases.

In the 1940s, this phenomenon was explained by Lars Onsager in a theory

based on infinitely elongated hard spherocylinders [1]. Onsager showed that

the basic mechanism of the transition is the interplay between positional

and orientational entropy. The size of the excluded volume – i.e. the volume

around one particle, which another particle cannot enter, because it would

produce an overlap – depends on the angle between the two particles’ axis. If

the particles lie parallel, the excluded volume is minimized. Hence particles

which are aligned, gain accessible volume and therefore positional entropy,

but they loose orientational entropy. At a certain density the balance between

isotropicnematic

Fig.1. Sketches of the isotropic phase (left) and nematic phase (right): In the

isotropic phase, particle positions and orientations are disordered. In the nematic

phase, particle orientations are aligned.

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2 Tanja Schilling et al.

the two contributions flips and the system changes from the isotropic to the

nematic phase.

Onsager theory has been remarkably successful at describing the isotropic

to nematic (IN) transition, and still serves as the basis for many theoretical

investigations of the properties of liquid crystals. Over the last twenty years,

for instance, several groups have investigated the properties of the interface

between the two phases using Onsager–type density functional approaches

[2–8].

The IN interface is an interesting model system, because both phases are

almost incompressible and their densities are similar. Therefore pressure and

density are not important parameters. The only parameters which determine

the properties of the interface are the particle aspect ratio and the orientation

of the director (i.e. the axis of average orientation of the particles in the

nematic phase) with respect to the plane of the interface.

An important finding of the theoretical studies cited above is that the

interfacial tension γINof the IN interface is minimized when the director lies

in the plane of the interface. In this case γINis predicted to be very low, but

the precise value varies considerably between different authors [9,10]. Theo-

retical estimates for γIN typically range from 0.156 [7] to 0.34 [3], in units

of kBT/LD, with L the rod length, D the rod diameter, T the temperature,

and kBthe Boltzmann constant.

To test the accuracy of the theoretical estimates of γIN, one might wish

to make a direct comparison to experimental data. Unfortunately, this is not

straightforward. The models used in theoretical treatments of the IN inter-

face are typically rather simplistic, usually based on a hard or short-ranged

pair potential in a system of monodisperse spherocylinders. Using these mod-

els, it is not reasonable to expect quantitative agreement with experiments,

because the interactions in the experimental system will be much more com-

plex. For example, polydispersity may be an important factor, and it is not

clear to what extent long-range interactions play a role. Also many experi-

mental systems display chirality. And even the experimental determination

of the rod dimensions L and D, required if a comparison to theory is to be

made, presents complications [10].

In order to validate the assumptions made by the various approaches, it is

nevertheless important to test the accuracy of the theoretical predictions. To

this end, computer simulations are ideal, because they, in principle, probe the

phase behavior of the model system without resorting to approximations. In

recent years, several groups have investigated the IN transition by means of

simulations [11–18]. However, the interfacial tension γIN was not measured

in these studies.

To obtain γIN in simulations rather elaborate simulation techniques are

required. One possibility is to measure the anisotropy of the pressure tensor.

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Isotropic–nematic interface3

The interfacial tension is obtained from the difference between the normal

and the transversal pressure tensor components:

?∞

where the interface lies in the xy-plane. In [18], this method is applied to

suspensions of ellipsoids with axial ratio κ = A/B = 15, where A is the

length of the symmetry axis, and B that of the transverse axis. The mea-

sured interfacial tension is 0.006 ± 0.005kBT/B2≈ 0.09kBT/AB if a hard

interaction potential is used, and 0.011±0.004kBT/B2≈ 0.165kBT/AB for

a soft potential. The anisotropy of the pressure tensor is very small, and thus

difficult to measure accurately. Therefore the error bars of these results are

large.

Another approach is via the capillary wave spectrum. The basic idea is

the following: the interface will fluctuate for entropic reasons. As enlargement

of the interfacial area costs energy, the spectrum of the fluctutions is related

to γIN. If the interface is described by a function h(x,y), then one can show

that

γIN=

−∞

PN(z) − PT(z) dz ,

(1)

?|h(q)|2? =

kBT

γINq2,

(2)

where h(q) is the Fourier transformed of h(x,y) [19]. In [19] this approach

was applied to soft ellipsoids with κ = 15. γIN = 0.016 ± 0.002kBT/B2≈

0.24kBT/AB is reported. However, capillary wave theory is only valid in the

long wavelength limit. Therefore very large system sizes are required. More-

over, if periodic boundary conditions are used, two interfaces will be present

in the simulation box. Since γINis very small, large capillary fluctuations can

occur, and one needs to be aware of interactions between the two interfaces.

Therefore this method requires very large system sizes.

Clearly, in order to obtain γIN more accurately, much more computer

power or different simulation techniques are required. In this article we present

a method, which allows to reduce statistical errors considerably and therefore

makes an analysis of the finite-size effects possible. Recent advances in grand

canonical sampling methods [20,21] have enabled accurate measurements of

the interfacial tension in simple fluids [22,23] and colloid–polymer suspensions

[24,25]. The aim of this paper is to apply these techniques to the IN tran-

sition in a system of soft spherocylinders, and to extract the corresponding

phase diagram and the interfacial tension. Simulations in the grand canonical

ensemble offer a number of advantages over the more conventional methods

discussed previously. More precisely, in grand canonical simulations, both the

coexistence properties can be probed, as well as the interfacial properties –

where as the methods described above require an independent estimate of the

coexistence densities. Additionally, finite–size scaling methods are available

which can be used to extrapolate simulation data to the thermodynamic limit

[26–29].

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4Tanja Schilling et al.

This article is structured as follows: First, we introduce the soft sphe-

rocylinder model used in this work. Next, we describe the grand canonical

Monte Carlo method, and explain how the coexistence properties, and the

interfacial tension are obtained. Finally, we present our results.

2Model

For numerical reasons, which will be explained in Sect. 3, we do not model

the particles as hard rods, but as repulsive soft rods. Two spherocylinders of

elongation L and diameter D interact via a pair potential of the form

??

where r1, r2, u1and u2are definied in Fig. 2 and r is the distance between

the particles’ axis. The total energy is thus proportional to the number of

overlaps in the system. In this article, the rod diameter D is taken as unit

of length, and kBT as unit of energy. The strength of the potential is set to

? = 2kBT. Note that in the limit ? → ∞, this model approaches a system of

infinitely hard rods. The effect of the particle softness is only a shift in the

coexistence densities to higher values. The transition mechanism remains the

same.

To study the IN transition, we use the density and the rod alignment as

order parameters. Both the isotropic and the nematic phase are fluid phases,

in the sense that long-range positional order of the centers of mass is absent.

Orientational order is measured by the S2 order parameter, defined as the

maximum eigenvalue of the thermal average of the orientational tensor Q:

V (r1,r2,u1,u2) =

:

:

r < D

otherwise0

,

(3)

Qαβ=

1

2N

N

?

i=1

(3uiαuiβ− δαβ) .

(4)

Here, uiα is the α component (α = x,y,z) of the orientation vector ui of

rod i (normalized to unity), and δαβ is the Kronecker delta. In the case of

orientational order S2assumes a value close to one, while in the disordered

isotropic phase, S2is close to zero.

Since the density of the nematic phase is slightly higher than that of the

isotropic phase, we may also use the particle number density ρ = N/V to

r2

r1

1

u2

u

Fig.2. Definition of the quantities used in eqn. 3

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Isotropic–nematic interface5

distinguish between the phases, with N the number of rods in the system, and

V the volume of the simulation box. Following convention, we also introduce

the reduced density ρ?= ρ/ρcp, with ρcp= 2/(√2 + (L/D)√3) the density

of regular close packing of hard spherocylinders.

3Simulation method

The simulations are performed in the grand canonical ensemble. In this en-

semble, the volume V , the temperature T, and the chemical potential µ of

the rods are fixed, while the number of rods N inside the simulation box fluc-

tuates. Insertion and removal of rods are attempted with equal probability,

and accepted with the standard grand canonical Metropolis rules, given by

?

and

?

where ∆E is the energy difference between initial and final state, and β =

1/kBT [27,30]. Here it becomes evident, why this method is difficult to apply

to a system of hard objects – insertion moves will become extremely unlikely,

if overlaps are forbidden. Therefore we introduced a finite energy cost instead.

The simulations are performed in a three dimensional box of size Lx×

Ly×Lzusing periodic boundary conditions in all directions. In this work, we

fix Lx= Ly, but we allow for elongation Lz≥ Lx. Moreover, to avoid double

interactions between rods through the periodic boundaries, we set Lx> 2L.

During the simulations, we measure the probability distribution P(N),

defined as the probability of observing a system containing N rods. The

shape of the distribution will depend on the following parameters:

• the aspect ratio L/D

• the temperature T (in a trivial way, because it just sets the energy scale.)

• the chemical potential µ

• the box dimensions Lxand Lz, because there will be finite-size effects.

At phase coexistence, the distribution P(N) becomes bimodal, with two

peaks of equal area, one located at small values of N corresponding to the

isotropic phase, and one located at high values of N corresponding to the

nematic phase. A typical coexistence distribution is shown in Fig. 3, where

the logarithm of P(N) is plotted.

In order to find the chemical potential of coexistence, we use the equal

area rule [31]. Coexistence is defined as the situation in which the areas under

the peaks are equal:

??N?

A(N → N + 1) = min1,

V

N + 1exp(−β∆E + βµ)

?

(5)

A(N → N − 1) = min1,N

Vexp(−β∆E − βµ)

?

,

(6)

0

P(N)dN =

?∞

?N?

P(N)dN ,

(7)