Eur. Phys. J. E 16, 253–258 (2005)
PHYSICAL JOURNAL E
Evidence of the hexagonal columnar liquid-crystal phase of hard
colloidal platelets by high-resolution SAXS
D. van der Beek, A.V. Petukhov, S.M. Oversteegen, G.J. Vroege, and H.N.W. Lekkerkerkera
Van ’t Hoff Laboratory for Physical and Colloid Chemistry, Utrecht University Padualaan 8, 3584 CH Utrecht, The Netherlands
Received 4 August 2004 and Received in final form 14 October 2004 /
Published online: 20 January 2005 – c ? EDP Sciences / Societ` a Italiana di Fisica / Springer-Verlag 2005
Abstract. We report Small-Angle X-ray Scattering (SAXS) measurements of the columnar phase of hard
colloidal gibbsite platelets. We have been able to create large oriented domains of the columnar phase
both perpendicular and parallel to the sample wall, varying the volume fraction of platelets and adding
non-adsorbing polymer to the dispersion. In conjunction with the increased resolution of the SAXS setup,
this allowed a detailed analysis of the columnar phase, providing unambiguous evidence for the hexagonal
nature of the phase.
PACS. 61.10.-i X-ray diffraction and scattering – 61.30.-v Liquid crystals – 82.70.-y Disperse systems;
Colloidal suspensions that form periodic self-assembling
structures on sub-micrometre scales are of potential tech-
nological interest. For instance, three-dimensional ar-
rangements of spheres in colloidal crystals  might serve
as photonic materials [2–4], intended to manipulate light.
Colloidal particles with non-spherical shapes (such as rods
and plates) are of particular interest because of their abil-
ity to form liquid crystals that might serve as templates
for ordered porous materials.
It has long been known that dispersions of anisotropic
colloids display liquid-crystal phases. The earliest reports
date back to the 1920s and 1930s, when suspensions of rod-
and platelike colloids were found to exhibit the isotropic
(I) to nematic (N) phase transition. For rodlike colloids,
the I-N transition was found to occur in suspensions of
vanadium pentoxide (V2O5) ribbons  and tobacco mo-
saic virus (TMV) rods . Already in the 1940s, Lars On-
sager proposed an explanation for the I-N transition on a
purely entropic basis : the competition between pack-
ing entropy (which favours the nematic state) and orien-
tational entropy (favouring the isotropic state) determines
the I-N phase behaviour. As the packing entropy becomes
more important at higher volume fractions, the particles
tend to align and form a nematic phase at high enough
concentration. Onsager also showed that the particle
shape alone is enough to induce such behaviour. Moreover,
at a later stage, it was found that rodlike colloids show the
nematic–to–smectic-A phase transition, as in the case of
TMV [8,9], fd virus [10–12] and β-FeOOH . A detailed
description of this phase transition came with theory [14–
18] and computer simulations [19,20], showing that hard-
core interactions are enough to induce such ordering.
As stated, platelike particles also exhibit the I-N tran-
sition. The first ever observation of this kind was done by
Langmuir , who reported in 1938 on sols of California
bentonite clay particles that, after standing for several 100
hours, separated into two distinct phases —the isotropic
and nematic phase. During the last decades, there have
been quite a few studies on natural and synthetic clays, in-
vestigating their phase behaviour [22–26], rheological [27–
29] and structural properties [30–33]. However, in almost
all of the studies, clay suspensions were found to gel rather
than phase separate (I-N), as was the case with Lang-
muir’s sample. Model systems of colloidal platelets that do
show the I-N transition have recently become available [34,
35], confirming both Onsager’s theory and computer sim-
ulations . From computer simulations by Veerman and
Frenkel , and later Zhang et al. , it was found that
hard platelets may also form another liquid-crystal phase,
namely the orientationally and 2D translationally ordered
columnar (C) phase. This phase has also been observed
experimentally in suspensions of sterically stabilised and
charged colloidal plates [39–41]. There is a very interest-
ing issue, raised in one of these reports , which is
related to the inherent size polydispersity in these syn-
thetic suspensions: it is quite surprising that such systems,
with a rather high polydispersity of up to 25%, show the
columnar liquid-crystal phase. In contrast, crystallisation
of hard spheres is suspected to be frustrated due to the so-
called terminal polydispersity, with proposed values from
5% to 12% [42–46].
254The European Physical Journal E
Fig. 1. Panel A shows a Transmission Electron Micrograph of the colloidal gibbsite platelets; the scale bar denotes 500 nm.
Panel B depicts the normalised diameter distribution of the colloidal gibbsite platelets used in this study, together with a
Gaussian fit. From the fit, we obtain ?D? = 232 nm and σ = 20%, based on 176 platelets measured.
Nevertheless, on the basis of their Small-Angle X-ray
Scattering (SAXS) data, van der Kooij et al.  argue
that the observed high-density liquid-crystal phase is most
likely a hexagonal columnar phase. Their argument for a
hexagonal packing of the columns is founded on the ob-
servation of three low-angle scattering peaks within the
powder pattern. However, they noted the absence of a
fourth low-angle peak that should also be present in the
scattering pattern of a hexagonal structure. Conclusive ev-
idence would require sufficiently large monodomains with
columns perpendicular to the sample wall, revealing scat-
tering patterns with sixfold symmetry corresponding to
hexagonal packing. Usually, oriented samples are prepared
by applying an external field, e.g. magnetic fields [47–51]
and shear flow [52–54]. Here, we use sterically stabilised
colloidal gibbsite platelets with non-adsorbing polymer
as a depleting agent. We have been able to successfully
prepare large crystals of hard colloidal spheres for SAXS
studies [55,56] using such non-adsorbing polymer before.
In addition, in the case of platelets, the depletion inter-
action is expected to favour a perpendicular orientation
of the columns with respect to the sample walls. More-
over, we took advantage of the enhanced resolution of the
SAXS setup  in order to resolve the missing scatter-
ing peak. Our results show that we indeed could prepare
large oriented domains of columnar phase, allowing us to
demonstrate its hexagonal columnar nature.
2 Experimental section
We synthesised hexagonal colloidal gibbsite (Al(OH)3)
platelets  that were subsequently grafted with an end-
functionalised polyisobutene and suspended in toluene to
obtain a model system of hard platelets [34,39]. Transmis-
sion Electron Microscopy (TEM, see Fig. 1A) and Atomic
Force Microscopy (AFM) were used to determine the av-
erage corner-to-corner diameter, D, and thickness, L, of
the dry particle core, although the latter might contain
a contribution of the collapsed steric stabiliser. The di-
ameter distribution is shown in Figure 1B. We found
Table 1. Details of the samples used in this study. Here, age is
the time between sample preparation and measurements of the
SAXS patterns, φ denotes the volume fraction of the platelets
and cpol is the polymer concentration.
?D? = 232 nm and ?L? = 13 nm and a polydispersity
of σ = 20% in both dimensions. In solution the thickness
of the sterically stabilizing polyisobutene brush is esti-
mated as 2–3 nm. This gives us the effective dimensions
Deff= 237 nm and Leff= 18 nm.
We prepared several samples of this dispersion with
effective volume fractions ranging from 0.37 to 0.40 both
with and without non-adsorbing polymer (polydimethyl-
siloxane, Mw≈ 423 kDa and Rg≈ 33 nm ). The sam-
ples were thoroughly homogenised and subsequently put
in flat capillaries (internal dimensions 0.3×3.0 mm2), af-
ter which they were put away to phase separate. Within a
few days, phase separation was complete and yielded mul-
tiple phase equilibria . Each sample at least exhibited
nematic and columnar phases. Furthermore, Bragg reflec-
tions became visible, already hinting at the presence of a
columnar phase. On a timescale of months, these colours
got quite bright and distinct, see Figure 2, although we
were not able to detect a change in the colour. The re-
flections allowed us to get an estimate of the hexagonal
lattice spacing d(100)= 220 ± 15 nm using Bragg’s law.
From the set of samples, three representative samples
(hereafter called A, B, and C) were selected for analy-
sis with SAXS. The sample details are given in Table 1.
We used the recently developed  high-resolution SAXS
setup of the Dutch-Belgian beamline BM-26 DUBBLE
at the European Synchrotron Radiation Facility (ESRF,
Grenoble, France). One of the challenges in the application
of SAXS to our suspensions of colloidal platelets is related
to the existence of two distinctly different spatial scales.
D. van der Beek et al.: Evidence of the hexagonal columnar phase of hard colloidal platelets 255
Fig. 2. Bragg reflections from the columnar phase of a sample with experimental conditions comparable to the ones in this
study (platelets and non-adsorbing polymer). The colour of the reflections (cyan and red) varies with the incident angle. These
Bragg reflections already hint at the presence of a columnar structure, and allow a crude estimate of the inter-columnar spacing.
Fig. 3. SAXS patterns obtained in the columnar phase of samples A, B and C, along with the assigned Miller indices. The upper
panels depict the entire SAXS patterns, while the lower panels present the magnified views of the small scattering angle regions
near the beamstop. Sample A yields ringlike diffraction features typical for diffraction from a powder. In contrast, samples B
and C show strong predominant orientation of the columns, either along the vertical direction (in B) or along the X-ray beam
(in C), as shown by the inserted sketches. The hexagonal pattern in C points to the presence of the hexagonal columnar phase.
Due to the small particle thickness, the face-to-face inter-
particle structure leads to scattering at relatively large an-
gles. On the other hand, due to the relatively large particle
diameter, the side-to-side structure results in scattering at
very small angles, requiring a high reciprocal-space resolu-
tion. To achieve the latter, the X-ray beam was carefully
focused at the position of the X-ray detector consisting
of a phosphor screen coupled to a 16-bit CCD camera
(Photonic Science) with a pixel size of 22 µm. In order to
increasethemaximumaccessibleq-range, a relatively high
X-ray energy of 18 keV (λ = 0.69˚ A) and a shorter sample-
detector distance (about 5 m) were used. In addition, the
detector and beamstop were mounted off-centre to max-
imise the q-range even further. These settings allowed us
to achieve a resolution of 0.003 nm−1(the full-width at
half maximum of the instrument function), which is at
least 3 times higher than before . The smallest acces-
sible scattering angle corresponded to qmin= 0.023 nm−1.
The maximum q values were about qmax,h = 0.4 nm−1
in the horizontal plane and qmax,v = 0.24 nm−1in the
256 The European Physical Journal E
3 Results and discussion
Figure 3 displays the obtained SAXS patterns for samples
A to C. In the following we will index the reflections using
Miller indices (hkl). For a hexagonal packing we expect re-
flections perpendicular to the columns with q(hk0)propor-
tional to√h2+ hk + k2, while we use l to indicate (liquid-
like) order within the columns. Sample A shows the char-
acteristics expected for a columnar phase, i.e., four scat-
tering peaks with q-ratios of 1:√3:√4:√7 at small angles
and a much broader, liquidlike peak at large angle. The
small-angle inter-columnar peaks have an apparent width
of 0.003 nm−1, determined by the instrument’s resolution.
The large-angle intra-columnar peak has a width of order
of 0.07 nm−1. The ringlike features are typical for diffrac-
tion from a powder, hence, the domains of columnar phase
must be much smaller than the irradiated volume (300 ×
300 × 300 µm3). We attribute the small domains to the
relatively high volume fraction that causes fast crystalli-
sation yielding small crystallites . Sample B shows the
columnar scattering peaks more clearly than sample A. In
this case the scattering is dominated by a larger single do-
main, likely due to slower crystallisation and higher mobil-
ity at the lower volume fraction involved. The domain has
vertically oriented columns, as illustrated in the inset in
panel B. The liquidlike (001) and (00−1) peaks are located
outside the detector area, but they were visible in similar
samples with skewed orientations. In sample C we find
a single domain with columns directed perpendicularly
to the wall and we observe a hexagonal scattering pat-
tern due to inter-columnar scattering. Here the presence of
non-adsorbing polymer is found to favour the face-to-wall
anchoring of the gibbsite platelets, which can be under-
stood by the stronger depletion attraction in this config-
uration. We also note that in the azimuthal direction the
inter-columnar Bragg peaks are broad, of the order of 30◦,
suggesting a significant spread of the crystal orientations.
From the SAXS patterns, we calculated averaged ra-
dial intensity profiles that are depicted in Figure 4. Due
to the improved resolution of the SAXS setup, the inter-
columnar (100), (110), (200) and (210) reflections are
clearly resolved. Yet, we are not able to resolve the in-
trinsic width of the Bragg peaks. In addition, in samples
B and C we benefit from the predominant orientation of
the columnar domains that enhances the visibility of the
inter-columnar reflections. This is due to an increase of
the intensity of the reflections themselves and to a faster
decay of the background scattering intensity Iscat q val-
ues from q(100)to q(001). The behaviour of the background
can be understood by taking into account the anisotropy
of the scattering of a single particle, i.e., the form fac-
tor F(q). For disklike particles it can be factorised as
F(q) = Fn(qn) ×F||(q||), where qnand q||are the com-
ponents of the scattering vector perpendicular and par-
allel to the disk, respectively. In the q-range of interest
(from q(100)to q(001)) the first factor Fn(qn) does not de-
cay appreciably, while the second one decays as F||(q||)
∝ |q|||−3. Thus, the strongest scattering is observed for
small q||, when the scattering vector q is nearly orthogo-
Fig. 4. Averaged radial intensity profiles of the SAXS pat-
terns. The profiles of sample B were obtained from small (about
5 pixel wide) horizontal (B-hor) and vertical (B-ver) slices,
while A and C are azimuthal averages over the whole available
detector area except for the part covered by the beamstop (the
same mask is used for both A and C). The curves are shifted
vertically for clarity. The dashed lines present the power law
decays (I ∝ qn) with n = −2 and n = −3.
nal to the platelets. In sample C this strong contribution
is absent since the scattering vector q is almost parallel
to the platelets, so q ≈ q||. As one can see in Figure 4,
the background intensity indeed closely follows a power
law decay Isc(q) ∝ qnwith exponent n = −3, arising from
the decay of F||. In contrast, in sample A, the particles
have all possible orientations including those with q ≈ qn,
which give the main contribution to the background. Av-
eraging over all possible orientations leads to a power law
decay with a smaller exponent, n = −2. We have ob-
served a similar power law decay with n = −2 in a dilute
isotropic suspension (not shown). In sample B, both phe-
nomena are visible —the background scattering intensity
along the columns (perpendicular to the platelets, vertical
direction) is much stronger and decays more slowly than
when parallel to the platelets (horizontal direction).
We further note that, although the diffraction in sam-
ple B is dominated by a domain with vertically oriented
columns, other orientations are also present, as deduced
from the observation that the columnar reflections form
rings, albeit with well-pronounced maxima in the hori-
zontal direction. Also, the maxima of the (100), (110), and
(210) reflections in the same direction suggest a spread in
the orientations of the hexagons formed by neighbouring
columns. In contrast, sample C shows long-range bond ori-
entational order leading to a well-pronounced hexagonal
pattern. Furthermore, the q−3-decay of the background
also indicates the existence of one single domain within
the irradiated volume in sample C.
From the averaged radial intensity profiles we find
the characteristic spacings as listed in Table 2. The
D. van der Beek et al.: Evidence of the hexagonal columnar phase of hard colloidal platelets257
Table 2. Measured q-values of the scattering in the columnar phase obtained from the radial profiles of the SAXS patterns and
the inter- and intra-columnar spacings aD, respectively, aL, as calculated from the q-values.
inter-columnar spacings are calculated using the relation
columnar distance between the platelets is defined as
aL= 2π/q(001). Although samples A and B differ in overall
platelet volume fraction, due to the first-order character
of the nematic-columnar transition, the volume fractions
of their columnar phases are the same. This explains their
corresponding aD. Sample C, on the other hand, contains
non-adsorbing polymer, which enhances size fractionation
between the coexisting phases [59,60], resulting in a higher
average particle diameter in the columnar phase and hence
a larger columnar spacing.
The formation of columnar crystals in our samples is
actually quite surprising, considering the high polydisper-
sity of our platelets. In general, phase-separating colloidal
systems, of spheres [46,61–63] or platelets [59,60,64,65],
deal with this by fractionation of the particles between
the phases, leading to sub-phases with a (slightly) smaller
polydispersity than the parent suspension. In sample C,
where non-adsorbing polymer enhances fractionation even
more, this lowering of the polydispersity facilitates the for-
mation of the single columnar crystal. In addition to phase
fractionation, there is the possibility of local fractionation,
which is likely to take place in sample A due to its being
a powder. Local fractionation leads to crystals within the
columnar phase having a different average particle diam-
eter (hence slightly different periods) and a slightly lower
polydispersity each. However, in sample C, which contains
a (large) single domain, such local fractionation is much
more difficult as it would require the particles to travel too
large distances (at least 1000 times their own diameter).
The q(hk0)-values for samples A, B, and C correspond
to nearest-neighbour distances of 258 nm, 260 nm, and
268 nm, respectively. From the histogram (of the parent
suspension) it is immediately clear that at least 21% of the
particles do not fit into a columnar phase with these spac-
ings. For sample C, where size fractionation is strongest,
the average particle diameter is larger than the parent’s
average, making accommodation of the particles in the
columnar phase even more difficult.
√3√h2+ hk + k2/q(hk0), while the average intra-
To summarise, in this work we present clear evidence of
the formation of hexagonal columnar liquid crystals in sus-
pensions of polydisperse hard colloidal platelets. Apart
from a powder of small columnar crystals, we find macro-
scopically large single-domain crystals. Our results suggest
that addition of non-adsorbing polymer promotes the for-
mation of single-domain crystals with unique orientation
and no sign of disordered areas, as we have observed before
in a suspension of hard colloidal spheres . The macro-
scopically large crystals open up possibilities to fabricate
nanostructured materials with a sub-micron periodicity
that are of potential interest as, e.g., photonic materials.
The authors thank Patrick Davidson for enlightening dis-
cussions. Igor Dolbnya and the other crew at the BM-26
DUBBLE-beamline (ESRF, Grenoble) are kindly thanked for
their assistance during the measurements. The work of SMO is
part of the SoftLink research programme of the “Stichting voor
Fundamenteel Onderzoek der Materie (FOM)”, which is finan-
cially supported by the “Nederlandse Organisatie voor Weten-
schappelijk Onderzoek (NWO)”.
1. P.N. Pusey, W. van Megen, Nature 320, 340 (1986).
2. A. Imhof, D.J. Pine, Nature 389, 948 (1997).
3. J.E.G.J. Wijnhoven, W.L. Vos, Science 281, 802 (1998).
4. A. Blanco, E. Chomski, S. Grabtchak, M. Ibisate, S. John,
S.W. Leonard, C. Lopez, F. Meseguer, H. Miguez, J.P.
Mondia, G.A. Ozin, O. Toader, H.M. van Driel, Nature
405, 437 (2000).
5. H. Zocher, Z. Anorg. Chem. 147, 91 (1925).
6. F.C. Bawden, N.W. Pirie, J.D. Bernal, I. Fankuchen, Na-
ture 138, 1051 (1936).
7. L. Onsager, Phys. Rev. 62, 558 (1942); Ann. N. Y. Acad.
Sci. 51, 627 (1949).
8. G. Oster, J. Gen. Physiol. 33, 445 (1950).
9. U. Kreibig, C. Wetter, Z. Naturforsch. C 35, 750 (1980).
10. J. Lapointe, D.A. Marvin, Mol. Cryst. Liq. Cryst. 19, 269
11. F.P. Booy, A.G. Fowler, Int. J. Biol. Macromol. 7, 327
12. Z. Dogic, S. Fraden, Phys. Rev. Lett. 78, 2417 (1997).
13. Y. Maeda, S. Hachisu, Colloids Surf. 6, 1 (1983); 7, 357
14. M. Wadati, A. Isihara, Mol. Cryst. Liq. Cryst. 17, 95
15. M. Hosino, H. Nakano, H. Kimura, J. Phys. Soc. Jpn. 46,
1709 (1979); 47, 740 (1979); 51, 741 (1982).
16. B. Mulder, Phys. Rev. A 35, 3095 (1987).
17. X. Wen, R.B. Meyer, Phys. Rev. Lett. 59, 1325 (1987).
258The European Physical Journal E Download full-text
18. A. Poniewierski, R. Holyst, Phys. Rev. Lett. 61, 2461
19. A. Stroobants, H.N.W. Lekkerkerker, D. Frenkel, Phys.
Rev. Lett. 57, 1452 (1986); Phys. Rev. A 36, 2929 (1987).
20. D. Frenkel, H.N.W. Lekkerkerker, A. Stroobants, Nature
332, 822 (1988).
21. I. Langmuir, J. Chem. Phys. 6, 873 (1938).
22. A. Mourchid, A. Delville, J. Lambard, E. L´ ecolier, P.
Levitz, Langmuir 11, 1942 (1995).
23. M. Kroon, G.H. Wegdam, R. Sprik, Phys. Rev. E 54, 6541
24. A. Mourchid, E. L´ ecolier, H. Van Damme, P. Levitz, Lang-
muir 14, 4718 (1998).
25. P. Levitz, E. L´ ecolier, A. Mourchid, A. Delville, S. Lyon-
nard, Europhys. Lett. 49, 672 (2000).
26. J.C.P. Gabriel, C. Sanchez, P. Davidson, J. Phys. Chem.
100, 11139 (1996).
27. H. van Olphen, J. Colloid Interface Sci. 19, 313 (1964).
28. E. L´ ecolier, A. Mourchid, P. Levitz, Prog. Colloid Polym.
Sci. 110, 16 (1998).
29. P.F. Luckham, S. Rossi, Adv. Colloid Interface Sci. 89, 43
30. J.D.F. Ramsay, S.W. Swanton, J. Bunce, J. Chem. Soc.
Faraday Trans. 86, 3919 (1990).
31. J.D.F. Ramsay, P. Lindner, J. Chem. Soc. Faraday Trans.
89, 4207 (1993).
32. F. Pignon, J.M. Piau, A. Magnin, Phys. Rev. Lett. 76,
33. B.J. Lemaire, P. Panine, J.C.P. Gabriel, P. Davidson, Eu-
rophys. Lett. 59, 55 (2002).
34. F.M. van der Kooij, H.N.W. Lekkerkerker, J. Phys. Chem.
B 102, 7829 (1998).
35. D. van der Beek, H.N.W. Lekkerkerker, Europhys. Lett.
61, 702 (2003).
36. D. Frenkel, R. Eppenga, Phys. Rev. Lett. 49, 1089 (1982).
37. J.A.C. Veerman, D. Frenkel, Phys. Rev. A 45, 5632 (1992).
38. S.-D. Zhang, P.A. Reynolds, J.S. van Duijneveldt, J.
Chem. Phys. 117, 9947 (2002).
39. F.M. van der Kooij, K. Kassapidou, H.N.W. Lekkerkerker,
Nature 406, 868 (2000).
40. A.B.D. Brown, S.M. Clarke, A.R. Rennie, Langmuir 14,
41. D. van der Beek, H.N.W. Lekkerkerker, Langmuir 20, 8582
42. J.L. Barrat, J.P. Hansen, J. Phys. (Paris) 46, 1547 (1986).
43. P.N. Pusey, J. Phys. (Paris) 48, 709 (1987).
44. R. McRae, A.D.J. Haymet, J. Chem. Phys. 88, 1114
45. P.N. Pusey, in Liquids, Freezing and Glass Transition, Les
Houches Session 51, NATO Advanced Study Institute, Se-
ries B: Physics, edited by J.P. Hansen, D. Levesque, J.
Zinn-Justin (North-Holland, Amsterdam, 1991) pp. 763.
46. P.G. Bolhuis, D.A. Kofke, Phys. Rev. E 54, 634 (1996).
47. J. Torbet, G. Maret, J. Mol. Biol. 134, 843 (1979).
48. E. Senechal, G. Maret, K. Dransfeld, Int. J. Biol. Macro-
mol. 2, 256 (1980).
49. J. Torbet, J.M. Freyssinet, G. Hudry-Clergeon, Nature
289, 91 (1981).
50. J.M. Freyssinet, J. Torbet, G. Hudry-Clergeon, G. Maret,
Proc. Natl. Acad. Sci. USA 80, 1616 (1983).
51. R. Oldenbourg, X. Wen, R.B. Meyer, D.L.D. Caspar, Phys.
Rev. Lett. 61, 1851 (1988).
52. J. Gregory, K.C. Holmes, J. Mol. Biol. 13, 796 (1965).
53. M. Imp´ eror-Clerc, P. Davidson, Eur. Phys. J. B 9, 93
54. A.B.D. Brown, A.R. Rennie, Phys. Rev. E 62, 851 (2000).
55. A.V. Petukhov, D.G.A.L. Aarts, I.P. Dolbnya, E.H.A.
de Hoog, K. Kassapidou, G.J. Vroege, W. Bras, H.N.W.
Lekkerkerker, Phys. Rev. Lett. 88, 208301 (2002).
56. A.V. Petukhov, I.P. Dolbnya, D.G.A.L. Aarts, G.J.
Vroege, H.N.W. Lekkerkerker, Phys. Rev. Lett. 90, 028304
57. A.V. Petukhov, I.P. Dolbnya, D.G.A.L. Aarts, G.J.
Vroege, Phys. Rev. E 69, 031405 (2004).
58. A.M. Wierenga, T.A.J. Lenstra, A.P. Philipse, Colloids
Surf. A 134, 359 (1998).
59. F.M. van der Kooij, M. Vogel, H.N.W. Lekkerkerker, Phys.
Rev. E 62, 5397 (2000).
60. F.M. van der Kooij, H.N.W. Lekkerkerker, Langmuir 16,
61. D.A. Kofke, P.G. Bolhuis, Phys. Rev. E 59, 618 (1999).
62. S.R. Williams, I.K. Snook, W. van Megen, Phys. Rev. E
64, 021506 (2001).
63. N.B. Wilding, P. Sollich, Europhys. Lett. 67, 219 (2004).
64. M.A. Bates, D. Frenkel, J. Chem. Phys. 110, 6553 (1999).
65. F.M. van der Kooij, H.N.W. Lekkerkerker, Phys. Rev.
Lett. 84, 781 (2000).