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.
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 . 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
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.
2 Tanja Schilling et al.
the two contributions flips and the system changes from the isotropic to the
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
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  to 0.34 , 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 .
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.
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 , 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
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
PN(z) − PT(z) dz ,
where h(q) is the Fourier transformed of h(x,y) . In  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
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.
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
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
(3uiαuiβ− δαβ) .
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
Fig.2. Definition of the quantities used in eqn. 3
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.
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
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 . Coexistence is defined as the situation in which the areas under
the peaks are equal:
A(N → N + 1) = min1,
N + 1exp(−β∆E + βµ)
A(N → N − 1) = min1,N
Vexp(−β∆E − βµ)
6Tanja Schilling et al.
0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30
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
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
ρiso= (2/V )
ρnem= (2/V )
L = 12L
L = 3L
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
with γIN(Lx) the interfacial tension in a finite simulation box with lateral
To obtain the interfacial tension in the thermodynamic limit, one can
perform a finite size scaling analysis  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.
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 .
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  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.
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.
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
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.
. 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 , 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.
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
10Tanja Schilling et al.
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
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 . 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.
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
12 Tanja Schilling et al.
0.35 0.36 0.370.38
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
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.
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
10 0.363 0.3880.3970.424
15 0.2440.267 0.2800.307
0.0022 ± 0.0003 0.033
0.0053 ± 0.0001 0.080
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 , 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.
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
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.
1. L. Onsager: Ann. N. Y. Acad. Sci. 51, 627 (1949)
2. M. Doi, N. Kuzuu: Appl. Polym. Symp. 41, 65 (1985)
3. W.E. McMullen: Phys. Rev. A 38, 6384 (1988)
4. Z.Y. Chen, J. Noolandi: Phys. Rev. A 45, 2389 (1992)
5. Z.Y. Chen: Phys. Rev. E 47, 3765 (1993)
6. D.L. Koch, O.G. Harlen: Macromolecules 32, 219 (1999)
7. K. Shundyak, R. van Roij: J. Phys.: Condens. Matter 13, 4789 (2001)
8. E. Velasco, L. Mederos, D.E. Sullivan: Phys. Rev. E 66, 021708 (2002)
9. P. van der Schoot: J. Phys. Chem. B 103, 8804 (1999)
10. W. Chen, D.G. Gray: Langmuir 18, 663 (2002)
11. M. Dijkstra, R. van Roij, R. Evans: Phys. Rev. E 63, 051703 (2001)
12. P. Bolhuis, D. Frenkel: J. Chem. Phys. 106, 666 (1997)
13. M.A. Bates, C. Zannoni: Chem. Phys. Lett. 280, 40 (1997)
14. M.A. Bates, C. Zannoni: Chem. Phys. Lett. 288, 209 (1998)
15. M.P. Allen: J. Chem. Phys. 112, 5447 (2000)
16. M.P. Allen: Chem. Phys. Lett. 331, 513 (2000)
17. M.S. Al-Barwani, M.P. Allen: Phys. Rev. E 62, 6706 (2000)
18. A.J. McDonald, M.P. Allen, F. Schmid: Phys. Rev. E 63, 010701 (2000)
19. N. Akino, F. Schmid, M.P. Allen: Phys. Rev. E 63, 041706 (2001)
20. Q. Yan, J.J. de Pablo, J. Chem. Phys 113, 1276 (2000)
21. P. Virnau, M. M¨ uller: J. Chem. Phys. 120, 10925 (2004)
22. J. Potoff, A. Panagiotopoulos: J. Chem. Phys. 112, 6411 (2000)
23. W. G´ o´ zd´ z: J. Chem. Phys. 119, 3309 (2003)
24. P. Virnau, M. M¨ uller, L.G. MacDowell, K. Binder: J. Chem. Phys. 121, 2169
25. R.L.C. Vink, J. Horbach: J. Chem. Phys. 121, 3253 (2004)
26. K. Binder: Phys. Rev. A 25, 1699 (1982)
27. D.P. Landau, K. Binder: A Guide to Monte Carlo Simulations in Statistical
Physics (Cambridge University Press, Cambridge, 2000)
28. A.D. Bruce, N.B. Wilding: Phys. Rev. Lett. 68, 193 (1992)
29. Y.C. Kim, M.E. Fisher, E. Luijten: Phys. Rev. Lett. 91, 065701 (2003)
30. D. Frenkel, B. Smit: Understanding Molecular Simulation(Academic Press, San
31. M. M¨ uller, N.B. Wilding: Phys. Rev. E 51, 2079 (1995)
32. R. Vink, T. Schilling: cond-mat/0502444 (2005)
33. N.B. Wilding: Am. J. Phys. 69, 10 (2001)