Page 1

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.

Page 3

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)

Page 6

6Tanja Schilling et al.

0

2

4

6

8

10

12

14

0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30

ρ*

W

∆F

ISO

NEM

Fig.3.

transition in a system of soft rods with ? = 2 and L/D = 15. The low density peak

corresponds to the isotropic phase (ISO), the high density peak to the nematic

phase (NEM), and the barrier ∆F to the free energy difference between the two

phases (∆F is given by the average peak height as measured from the minimum

in between the peaks). The above distribution was obtained using box dimensions

Lx = 2.1L and Lz = 8.4L.

Coexistence distribution W = kBT lnP(N) of the isotropic to nematic

where ?N? is the average of the full distribution

?∞

and we assume that P(N) has been normalized to unity

1. The coexistence density of the isotropic phase follows trivially from the

average of P(N) in first peak

??N?

and similarly for the nematic phase

?∞

where the factors of two are a consequence of the normalization of P(N).

The interfacial tension γIN is extracted from the logarithm of the prob-

ability distribution W ≡ kBT lnP(N). Since −W corresponds to the free

energy of the system, the average height ∆F of the peaks in W, measured

?N? =

0

NP(N)dN ,

(8)

?∞

0P(N)dN =

ρiso= (2/V )

0

NP(N)dN ,

(9)

ρnem= (2/V )

?N?

NP(N)dN ,

(10)

Page 7

Isotropic–nematic interface7

L = 12L

z

L = 3L

x

Fig.4. Snapshot of a system of soft spherocylinders at IN coexistence. The sphe-

rocylinders are shaded according to their orientation. On the left side of the dashed

line the system is isotropic, on the right side it is nematic. The second interface

coincides with the boundaries of the box in the elongated direction.

with respect to the minimum in between the peaks, equals the free energy bar-

rier separating the isotropic from the nematic phase. Which configurations

contribute to P(N) in the region where the overall density of the system

is between the peaks ρiso ? ρ ? ρnem? A snapshot of the system in this

regime (Fig. 4) reveals a slab geometry, with one isotropic region, and one

nematic region, separated by an interface (because of the periodic boundary

conditions, there are actually two interfaces). Note that the director of the

nematic phase lies in the plane of the interfaces. This was the typical case for

the snapshots studied by us, and is consistent with the theoretical prediction

that in-plane alignment yields the lowest free energy. From this snapshot it

becomes evident why the distribution has a flat region. If the two interfaces

are sufficiently separated they do not interact. Then it is possible to change

the overall system density simply by growing one phase and shrinking the

other without producing any free energy.

The barrier ∆F in Fig. 3 thus corresponds to the free energy cost of

creating two interfaces in the system. In this work, where the box dimensions

are chosen such that Lx= Lyand Lz≥ Lx, the interfaces will be oriented

perpendicular to the elongated direction, since this minimizes the interfacial

area, and hence the free energy of the system. The total interfacial area in

the system thus equals 2L2

free energy per unit area, we may write

x. As the interfacial tension is simply the excess

γIN(Lx) = ∆F/(2L2

x),

(11)

with γIN(Lx) the interfacial tension in a finite simulation box with lateral

dimension Lx[26].

To obtain the interfacial tension in the thermodynamic limit, one can

perform a finite size scaling analysis [26] to estimate limLx→∞γIN(Lx). Al-

ternatively, away from any critical point, the most dominant finite size effects

will likely stem from interactions between the two interfaces. In this case, it is

feasible to use an elongated simulation box with Lz? Lx, such as in Fig. 4.

Page 8

8Tanja Schilling et al.

This article is intended as an introduction to the simulation method. There-

fore we will focus on the second aspect. A finite size analysis of our results

can be found in [32].

If the free energy barrier ∆F is large, transitions between the isotropic

and the nematic phase become less likely, and the simulation will spend most

of the time in only one of the two phases. A crucial ingredient in our sim-

ulation is therefore the use of a biased sampling technique. We use succes-

sive umbrella sampling [21] to enable accurate sampling in regions where

P(N), due to the free energy barrier separating the phases, is very small.

Note also that phase coexistence is only observed if the chemical potential

µ is set equal to its coexistence value. This value is in general not known

at the start of the simulation, but it may easily be obtained by using the

equation P(N|µ1) = P(N|µ0)eβ(µ1−µ0)N, with P(N|µα) the probability dis-

tribution P(N) at chemical potential µα. In the simulations, we typically set

the chemical potential to zero and use successive umbrella sampling to obtain

the corresponding probability distribution. We then use the above equation

to obtain the desired coexistence distribution, in which the area under both

peaks is equal.

4 Results

4.1Profiles

First we estimated the minimum box size necessary to contain two indepen-

dent interfaces. We ran an NVT Monte Carlo simulation of hard spherocylin-

ders at coexistence in a very elongated box (Lz = 10L) and measured the

interfacial width. Fig. 5 shows the density (top) and the nematic order pa-

rameter (bottom) perpendicular to the interface. We do not see any trace

of non-monotonous behaviour. In agreement with Onsager theory, the inter-

faces are shifted with respect to one another. Moving from the nematic to the

isotropic region, first ρ drops and then S2. Measured from the point where

the density is 0.9ρisoto the point, where it is 0.9ρnemthe interface is roughly

4L wide. This means that very large simulation boxes will be necessary to

properly decouple two interfaces.

4.2Phase diagram

First we used our grand canonical Monte Carlo scheme to determine the IN

phase diagram of the soft spherocylinder system of eqn. 3 using ? = 2. For

several rod elongations L/D, we measured the distribution P(N), from which

ρisoand ρnemwere obtained. The system size used in these simulations is typ-

ically Lx= Ly= 2.1L and Lz= 4.2L. In Fig. 6, we plot the reduced density

of the isotropic and the nematic phase as function of L/D. We observe that

the phase diagram is qualitatively similar to that of hard spherocylinders

Page 9

Isotropic–nematic interface9

0.19

0.2

0.21

0.22

0.23

ρ∗

024

6

810

z [L/D]

0

0.2

0.4

0.6

S2

Fig.5. Profiles of density and nematic order perpendicular to the interface for

L/D = 15. In agreement to Onsager theory, the profiles are shifted by roughly

0.3L/D with respect to one another.

[12]. The quantitative difference being that, for soft rods, the IN transition

is shifted towards higher density. The inset of Fig. 6 shows the concentration

variable c = πDL2ρ/4 as a function of D/L. For hard spherocylinders, On-

sager theory predicts that cISO= 3.29 and cNEM= 4.19 in the limit of infinite

rod length, or equivalently D/L → 0. In case of the soft potential of eqn. 3,

these values must be multiplied by (1−e−β?)−1≈ 1.16 for ? = 2. In the inset

of Fig. 6, the corresponding limits are marked with arrows. As in [12], we ob-

serve that the simulation data for the isotropic phase smoothly approach the

Onsager limit, while the nematic branch of the binodal seems to overshoot

the Onsager limit. This we attribute to equilibration problems. To simulate

the IN transition in the limit D/L → 0, large system sizes are required, and

it becomes increasingly difficult to obtain accurate results. To quantify the

uncertainty in our measurements, additional independent simulations for rod

elongation L/D = 25, 30, and 35 were performed. The corresponding data

are also shown in Fig. 6. For L/D ≥ 30, we observe significant scatter, while

for L/D ≤ 25, the uncertainty is typically smaller than the symbol size used

in the plots.

4.3Interfacial tension

Next, the interfacial tension γINis determined for L/D = 10 and L/D = 15.

Unfortunately, the system size used to compute the phase diagram in the

previous section, was insufficient to accurately extract the interfacial tension

Page 10

10Tanja Schilling et al.

0.1

0.2

0.3

0.4

0.5

10 2030

L/D

40 50

ρ*

NEM

ISO

3 3

4 4

5 5

0.000.000.05

D/LD/L

0.100.10

cc

0.05

Fig.6. Soft spherocylinder phase diagram of the IN transition using ? = 2. Shown

is the reduced density ρ?of the isotropic phase (closed circles) and of the nematic

phase (open circles) as function of L/D. The inset shows the concentration variable

c as function of D/L for both the isotropic and the nematic phase. The lower and

upper arrow in the inset mark the Onsager limit D/L → 0 for the isotropic and

the nematic phase, respectively. The lines connecting the points serve as a guide to

the eye.

because no flat region between the peaks in P(N) could be distinguished.

To properly extract the interfacial tension, much larger systems turned out

to be required. In this case, care must be taken in the sampling procedure.

Many sampling schemes, especially the ones that are easy to implement such

as successive umbrella sampling, put a bias on the density only. Such schemes

tend to “get stuck” in meta–stable droplet states when the system size be-

comes large [24]. As a result, one may have difficulty reaching the state with

two parallel interfaces, in which case eqn. 11 cannot be used.

Therefore, for large systems, one must carefully check the validity of the

simulation results. We performed a number of additional grand canonical

simulations using a biased Hamiltonian of the form H = H0+ W, with H0

the Hamiltonian of the real system defined by eqn. 3 and W = −kBT lnP(N).

If the measured P(N) is indeed the equilibrium coexistence distribution of

the real system, a simulation using the biased Hamiltonian should visit the

isotropic and the nematic phase equally often on average [33,24]. This is

illustrated in the top frame of Fig. 7, which shows the S2order parameter as

a function of the elapsed simulation time during one such biased simulation.

Page 11

Isotropic–nematic interface11

0.00

0.20

0.40

0.60

0.80

02040 60

time

80100120

S2

0.24

0.25

0.26

0.27

0.28

0204060

time

80 100120

ρ*

Fig.7. Monte Carlo time series of a biased grand canonical simulation. The top

frame shows the S2 order parameter as a function of the invested CPU time, the

lower frame the reduced density, with CPU time expressed in hours on a 2.6 GHz

Pentium. During the simulation, the reduced density was confined to the interval

0.245 < ρ?< 0.275, as indicated by the horizontal lines in the lower figure. The

data were obtained using L/D = 15, ? = 2, Lx = 2.1L and Lz = 8.4L, which are

the same parameters as used in Fig. 3.

Indeed, we observe frequent transitions between the isotropic (S2∼ 0) and

the nematic phase (S2∼ 1). Also shown in Fig. 7 is the corresponding time

series of the reduced density. In case that a perfect estimate for P(N) could

be provided, the measured distribution in the biased simulation will become

flat in the limit of long simulation time. The deviation from a flat distribution

can be used to estimate the error in P(N), or alternatively, to construct a

better estimate for P(N). The latter approach was adopted by us. First,

successive umbrella sampling is used to obtain an initial estimate for P(N).

This estimate is then used as input for a number of biased simulations using

the modified Hamiltonian, and improved iteratively each time.

To obtain the interfacial tension, the most straightforward approach is

to fix the lateral box dimensions at Lx= Ly, and to increase the elongated

dimension Lz? Lxuntil a flat region between the peaks in the distribution

P(N) appears. For soft spherocylinders of elongation L/D = 10, the results of

Page 12

12 Tanja Schilling et al.

0

1

2

3

4

5

6

0.35 0.36 0.370.38

ρ*

0.390.40 0.41

W

a

b

c

Fig.8. Coexistence distributions W = kBT lnP(N) of soft spherocylinders with

L/D = 10 and ? = 2 for various system sizes. In each of the above distributions,

the lateral box dimension was fixed at Lx = Ly = 2.3L, while the perpendicular

dimension was varied: (a) Lz = 2.3L; (b) Lz = 10.35L; (c) Lz = 13.8L. The

corresponding free energy barriers ∆F are: (a) 1.52; (b) 2.68; (c) 2.33, in units of

kBT.

this procedure are shown in Fig. 8. Indeed, we observe that the region between

the peaks becomes flatter as the elongation of the simulation box is increased.

Unfortunately, even for the largest system that we could handle, the region

between the peaks still displays some curvature. In other words, the interfaces

are still interacting, indicating that even more extreme box elongations are

required. Ignoring this effect, and applying eqn. 11 to the largest system of

Fig. 8, we obtain for the interfacial tension γIN = 0.0022 kBT/D2. For rod

elongation L/D = 15, the distribution of the largest system that we could

handle is shown in Fig. 3. The height of the barrier reads ∆F = 10.59 kBT,

and the corresponding interfacial tension γIN= 0.0053 kBT/D2.

The advantage of the present simulation approach is that the statistical

errors are small, and that finite size effects are clearly visible as a result. In

contrast, if the pressure tensor or capillary broadening are used to obtain

γIN, the statistical errors will likely obscure any finite size dependence.

Page 13

Isotropic–nematic interface 13

Table 1. Bulk properties of the coexisting isotropic and nematic phase in a system

of soft spherocylinders with ? = 2 and L/D = 10 and 15. Listed are the reduced

density ρ?of the isotropic and the nematic phase, the normalized number density

ρLD2and the interfacial tension γIN, expressed in two types units to facilitate the

comparison to other work.

L/D isotropic phase nematic phase interfacial tension γIN

ρ?

ρLD2

ρ?

ρLD2

10 0.363 0.3880.3970.424

15 0.2440.267 0.2800.307

kBT

D2

kBT

LD

0.0022 ± 0.0003 0.033

0.0053 ± 0.0001 0.080

5Discussion

It is clear from the phase diagram of Fig. 6 that the Onsager limit is not re-

covered until for very large rod elongation, exceeding at least L/D = 40. As a

result, our estimates for the interfacial tension differ profoundly from Onsager

predictions. Typically, γIN in our simulations is four times lower than On-

sager estimates. Note that our simulations also show that γINincreases with

L/D, towards the Onsager result, so there seems to be qualitative agreement.

However, to properly access the Onsager regime, additional simulations for

large elongation L/D are required. Unfortunately, as indicated by the scatter

in the data of Fig. 6, and also in [12], such simulations are very complicated.

It is questionable if present simulation techniques are sufficiently powerful to

extract γIN with any meaningful accuracy in the Onsager regime.

As mentioned in the introduction, computer simulations of soft ellipsoids

with κ = 15 yield interfacial tensions of γIN = 0.011 ± 0.004 kBT/B2and

γIN = 0.016 ± 0.002 kBT/B2[18,19]. For L/D = 15, our result for soft

spherocylinders is considerably lower. Obviously, spherocylinders are not el-

lipsoids, and this may well be the source of the discrepancy. Note also that

the shape of the potential used by us is different from that of Refs. [18,19].

In summary, we have performed grand canonical Monte Carlo simula-

tions of the IN transition in a system of soft spherocylinders. By measuring

the grand canonical order parameter distribution, the coexistence densities

as well as the interfacial tension were obtained. In agreement with theoret-

ical expectations and other simulations, ultra–low values for the interfacial

tension γIN are found. Our results confirm that for short rods, the interfa-

cial tension, as well as the coexistence densities, are considerably lower than

the Onsager predictions. This demonstrates the need for improved theory

to describe the limit of shorter rods, which is required if the connection to

experiments is ever to be made.

5.1Acknowledgement

We are grateful to the Deutsche Forschungsgemeinschaft (DFG) for support

(TR6/A5) and to K. Binder, M. M¨ uller, P. van der Schoot, and R. van Roij

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14Tanja Schilling et al.

for stimulating discussions. We also thank G. T. Barkema for suggesting some

of the numerical optimizations used in this work. T. S. was supported by the

Emmy Noether program of the DFG.

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