Melting of 2D liquid crystal colloidal structure
Using video microscopy, we investigated melting of a two-dimensional colloidal system, formed by glycerol droplets at the free surface of a nematic liquid crystalline layer. Analyzing different structure correlation functions, we conclude that melting occurs through an intermediate hexatic phase, as...
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irk-123456789-321062012-04-09T12:17:52Z Melting of 2D liquid crystal colloidal structure Brodin, A. Nych, A. Ognysta, U. Lev, B. Nazarenko, V. Škarabot, M. Muševič, I. Using video microscopy, we investigated melting of a two-dimensional colloidal system, formed by glycerol droplets at the free surface of a nematic liquid crystalline layer. Analyzing different structure correlation functions, we conclude that melting occurs through an intermediate hexatic phase, as predicted by the Kosterlitz-Thouless-Halperin-Nelson-Young(KTHNY) theory. However, the temperature range of the intermediate phase is rather narrow, <1°C, and the characteristic critical power law decays of the correlation functions are not fully developed. We conclude that the melting of our 2D systems qualitatively occurs according to KTHNY, although quantitative details of the transition scenario may partly depend on the details of interparticle interaction. 2010 Article Melting of 2D liquid crystal colloidal structure / A. Brodin, A. Nych, U. Ognysta, B. Lev, V. Nazarenko, M. Škarabot, I. Muševič // Condensed Matter Physics. — 2010. — Т. 13, № 3. — С. 33601:1-12. — Бібліогр.: 35 назв. — англ. 1607-324X PACS: 61.30.-v, 64.70.pv, 82.70.Dd http://dspace.nbuv.gov.ua/handle/123456789/32106 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Using video microscopy, we investigated melting of a two-dimensional colloidal system, formed by glycerol droplets at the free surface of a nematic liquid crystalline layer. Analyzing different structure correlation functions, we conclude that melting occurs through an intermediate hexatic phase, as predicted by the Kosterlitz-Thouless-Halperin-Nelson-Young(KTHNY) theory. However, the temperature range of the intermediate phase is rather narrow, <1°C, and the characteristic critical power law decays of the correlation functions are not fully developed. We conclude that the melting of our 2D systems qualitatively occurs according to KTHNY, although quantitative details of the transition scenario may partly depend on the details of interparticle interaction. |
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Brodin, A. Nych, A. Ognysta, U. Lev, B. Nazarenko, V. Škarabot, M. Muševič, I. |
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Brodin, A. Nych, A. Ognysta, U. Lev, B. Nazarenko, V. Škarabot, M. Muševič, I. Melting of 2D liquid crystal colloidal structure Condensed Matter Physics |
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Brodin, A. Nych, A. Ognysta, U. Lev, B. Nazarenko, V. Škarabot, M. Muševič, I. |
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Brodin, A. |
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Melting of 2D liquid crystal colloidal structure |
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Melting of 2D liquid crystal colloidal structure |
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Melting of 2D liquid crystal colloidal structure |
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Melting of 2D liquid crystal colloidal structure |
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Melting of 2D liquid crystal colloidal structure |
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melting of 2d liquid crystal colloidal structure |
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Інститут фізики конденсованих систем НАН України |
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2010 |
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Melting of 2D liquid crystal colloidal structure / A. Brodin, A. Nych, U. Ognysta, B. Lev, V. Nazarenko, M. Škarabot, I. Muševič // Condensed Matter Physics. — 2010. — Т. 13, № 3. — С. 33601:1-12. — Бібліогр.: 35 назв. — англ. |
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Condensed Matter Physics |
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Condensed Matter Physics 2010, Vol. 13, No 3, 33601: 1–12
http://www.icmp.lviv.ua/journal
Melting of 2D liquid crystal colloidal structure
A. Brodin1,2, A. Nych1, U. Ognysta1, B. Lev3, V. Nazarenko1, M. Škarabot4, I. Muševič4
1 Institute of Physics NAS Ukraine, Kyiv, Ukraine
2 National Technical University of Ukraine “Kyiv Polytechnic Institute”, Kyiv, Ukraine
3 Bogolyubov Institute for Theoretical Physics NAS Ukraine, Kyiv, Ukraine
4 J. Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
Received January 4, 2010, in final form May 25, 2010
Using video microscopy, we investigated melting of a two-dimensional colloidal system, formed by glycerol
droplets at the free surface of a nematic liquid crystalline layer. Analyzing different structure correlation func-
tions, we conclude that melting occurs through an intermediate hexatic phase, as predicted by the Kosterlitz-
Thouless-Halperin-Nelson-Young(KTHNY) theory. However, the temperature range of the intermediate phase
is rather narrow, . 1
◦C, and the characteristic critical power law decays of the correlation functions are not
fully developed. We conclude that the melting of our 2D systems qualitatively occurs according to KTHNY,
although quantitative details of the transition scenario may partly depend on the details of interparticle inter-
action.
Key words: liquid crystal, 2D colloidal structure, phase transitions in 2D structures
PACS: 61.30.-v, 64.70.pv, 82.70.Dd
1. Introduction
Two-dimensional (2D) ordering and the formation of periodic 2D crystal-like structure, although
less familiar than 3D crystallization, is still widespread enough and can even be observed in the
cup of morning coffee as Bénard convection cells [1]. In recent decades, 2D systems and their
order-disorder transition has received considerable attention in computational and experimental
physics. Laboratory 2D systems are usually formed by colloidal particles that are large enough
to be individually observable by optical means, so that the structure and even individual particle
trajectories can be monitored by optical microscopy, thus making the complete structural and
dynamical data readily experimentally accessible [2]. For instance, charged dust particles in “dusty
plasma” readily form the so-called Wiegner or Coulomb crystal, whose melting transition can be
experimentally investigated by optical means [3].
2D lattices are in many respects similar to and considered as convenient models for 3D systems,
although it has been recognized that the analogy is only partial and that, in certain respects, 2D
order has characteristics that are unique to its dimensionality [4]. In particular, it has been shown
that 1D and 2D crystalline lattices would be thermally unstable, and thus spontaneous crystalline
order should not exist in one or two dimensions [5, 6], which is sometimes quoted as Peierls-
Mermin theorem [4]. A more familiar consequence of this property is the absence of true long-range
positional order in the two-dimensional crystal, so that its positional correlation function decays
algebraically to zero with distance, in contrast to the 3D case where a finite value is attained.
This led Kosterlitz and Thouless to suggest [7] that the nature of the melting transition in 2D
is different from 3D and to formulate a theory of melting as a continuous transition mediated by
the dissociation of dislocation pairs. The problem of 2D melting was subsequently worked out in
greater detail by Halperin, Nelson, and Young [8, 9], with the resulting microscopic theory now
commonly referred to as Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory. It has to
be noted that no comparable microscopic theory exists in 3D, which again sets the 2D case apart.
Qualitatively, KTHNY predicts a two-stage 2D melting that occurs from the hexagonal (equilateral
c© A. Brodin, A. Nych, U. Ognysta, B. Lev, V. Nazarenko, M. Škarabot, I. Muševič 33601-1
http://www.icmp.lviv.ua/journal
A. Brodin et al.
triangular) through an intermediate “hexatic”phase, in which the positional and bond-orientational
correlation functions exhibit characteristic algebraic decays, and only via a second transition does
the system become an ordinary fluid where all correlation functions decay exponentially. Since
the formulation of KTHNY theory, the 2D melting transition has been a matter of continuous
debate. Several computer simulations and experiments have been performed (for a review, see [10–
12]), although the results were not always consistent. Overall, it has to be noted that KTHNY
only describes how a 2D crystal could melt, so that other scenarios cannot be excluded. For
instance, it appears that the KTHNY transitions can be preempted by other processes leading
to a single first order transition as in 3D [13]. On the other hand, experiments on paramagnetic
colloidal particles, whose dipolar interaction could be conveniently controlled by magnetic field,
concluded a perfect quantitative agreement with the two-stage KTHNY scenario [14, 15]. Similar
conclusion followed from a recent work on charged millimeter-sized steel balls [20]. Yet another
experimental work on uncharged polymer microspheres [16], while confirming the existence of
hexatic phase, concluded that the liquid-to-hexatic and hexatic-to-solid transitions are strongly first
order transitions. A work on charged colloids [17], while confirming the existence of an intermediate
phase, concluded that this phase was different from the KTHNY expectations. Several works on
dusty plasmas (e. g., [3, 18, 19]) appear to point towards a grain-boundary-induced rather than
KTHNY melting scenario. Overall, it appears that the KTHNY scenario may not be truly universal
but rather depends on the specific properties of the systems, predominantly related to the details of
interparticle interaction, so that further investigation of systems with different interactions is highly
motivated. Recently, there has been discovered a new class of self-ordering 2D colloidal systems,
formed by colloidal particles (droplets) at the free surface of a nematic liquid crystal (NLC) [21–
23]. These colloids, which interact via liquid crystal mediated elastic forces, readily form hexagonal
crystals that exhibit melting transition at temperatures in the range Tmelting = 26÷28◦C, which
are relatively far from the temperature of the nematic-isotropic liquid transition, TNI =36◦C. Such
a colloidal system allows for observation of the 2D melting phenomena over the large area with a
constant colloidal fraction, size of colloids, and interaction between the particles, as the main effect
of the temperature on the system occurs only through the magnitude of thermal fluctuations of the
colloidal particles. In the present contribution, we perform an analysis of the melting transition in
these systems in an attempt to test the applicability of KTHNY and its predictions to this novel
class of 2D lattices.
2. Experiment
In our experiments we have used hexagonal array of glycerol droplets trapped at the NLC-
air interface and stabilized by the long range elastic-capillary interactions [21, 23]. A cleaned
glass plate of 2 × 2 cm size was covered with the polymer (Elvamide 8061) film and rubbed to
ensure good planar alignment of NLC 5CB (4-n-pentyl-4-cyanobiphenyl, Merck). A layer of 5CB
(clearing temperature TNI =36◦C) of 40÷60 µm thickness was deposited directly onto the substrate
and allowed to equilibrate for 10 minutes. Then the substrate with 5CB film was held upside
down above a petri dish filled with glycerol. The petri dish was heated up to 120◦C to facilitate
transfer of glycerol onto the NLC layer. During this process, the glycerol evaporates from the petri
dish and condenses on the NLC layer forming almost uniform size droplets (see figure 1 (a)). By
adjusting deposition time and distance between the glycerol surface and substrate, the amount of
condensed glycerol can be altered and therefore the number and size of the glycerol droplets can
be effectively controlled. The procedure generally yields 2D crystalline hexagonal lattice. After the
deposition, the substrate with NLC film was placed onto a microscope hot stage (MK1, Instec
Inc.) for observation and temperature control. Figure 1 (b) shows an optical microscopy image of
such hexagonal structure at the free surface of the NLC layer. From the optical images (blurred by
diffraction) of the droplets we estimated their size to be 2R ' 1.0±0.2 µm, and the lattice constant
a ' 1.9 ± 0.1 µm. This estimation is further justified by an observation that the glycerol droplets
exhibit Brownian motion with considerable magnitude, suggesting that their surface-to-surface
separation is ≈ 0.5a, so that the structure is not densely packed [23].
33601-2
Melting of 2D LC colloidal structure
Figure 1. (a) Schematic representation of the preparation procedure of 2D colloidal structure
formed by glycerol droplets trapped at the free surface of a nematic liquid crystal film at 26◦C.
Black lines at the right side show the orientation of LC molecules (NLC director) inside the
LC layer, which changes from planar at the bottom to homeotropic at the top surface. (b)
Microscopic image of the 2D colloidal lattice, formed by glycerol droplets at the LC-air interface.
The structure on the snapshot is not ideal hexagonal due to considerable Brownian motion of
the droplets.
Digital images of the 2D structures, taken at different temperatures, were first analyzed in
terms of the particle positions (coordinates) by using custom-written MatLab programs according
to computerized videomicroscopy algorithms [24]. From the so-obtained data of (x, y) coordinates of
the particles’ centers, we then computed various structural correlation functions, discussed below.
In order to find the structural “bonds” between the nearest neighbors, we performed Delaunay
triangulation procedure on the sets of particle positions, obtained from recorded video-frames.
The resulting structural data in terms of bond positions, lengths, and orientations, were used for
visualization of structural changes during melting (see figure 2), and for further statistical analyses,
such as determination of the number of nearest neighbours of each particle, and computing bond-
orientational correlation functions.
3. Results and discussion
Our 2D hexagonal crystals, figure 1, exhibit “softening” and melting well below the nematic-
to-isotropic transition of 5CB (' 36◦C in our case). In order to visualize the structural changes
upon softening and melting, we show in figure 2 snapshots of the structure at four representative
temperatures, presented as wire frames of interparticle bonds.
Figure 2. Structural changes of the ordered phase of colloidal droplets with changing the tem-
perature. Positions of the glycerol droplets were digitized and Delaunay triangulation mesh was
calculated to represent lattices at different temperatures. Lattice sites with number of neigh-
bors greater or smaller than six are highlighted with red and blue color respectively. Increasing
number of lattice defects as well as disappearance of the long-range ordering indicates melting
of the colloidal crystal.
From visual inspection of figure 2, the close to perfect hexagonal structure observed at 26◦C is
almost completely lost by 27.5◦C, although the structural changes are obviously gradual, i. e. the
33601-3
A. Brodin et al.
loss of crystalline order occurs through continuous increase of the number of defects, as seen at
27◦C and 27.5◦C, rather than through an abrupt transition at any particular temperature. Thus,
melting of our colloidal system is a continuous rather than first-order transition, and therefore it
qualitatively corresponds to the KTHNY scenario.
From the particular examples of figure 2 it may appear that the creation of structural defects
with increasing the temperature is not uniform, but rather occurs through diffusion of the defects
from the outer parts of the structure towards the center, so that the structural disordering and
concomitant melting are rather grain-boundary-induced, similarly to the results of, e. g., [3, 18, 19].
This would mean a grain-boundary-induced transition [13], which is different from the KTHNY
scenario. As far as we cannot exclude the occurrence of grain-boundary-induced melting in some
of our colloidal structures, we note that in most cases the structural changes were rather uniform,
qualitatively in accord with KTHNY.
We now proceed to quantitative analysis of the structural changes upon softening and melting,
and compute the radial distribution function g(|r|) of the particles according to
g(|r|) =
1
N
〈
N
∑
j 6=k
δ(r + rj − rk)
〉
, (1)
where N is the number of particles and 〈. . .〉 denotes ensemble average over a series of structures,
photographed at the same temperature. Figure 3 shows g(r) at several selected temperatures.
The peaks at 27.5◦C are smeared out in comparison with 26◦C – in particular, the characteristic
splitting of the second peak that corresponds to the second (
√
3a) and third (2a) neighbor distances
in the second coordination shell is no longer recognizable. The disappearance of the splitting is
obviously associated with crystal melting, although it is not possible to use it as a quantitative
indicator, since the peak broadening occurs not only because of structural changes but also due to
increased fluctuations of particle positions at higher temperatures. The first peak, whose area gives
the coordination number, appears to somewhat decrease in magnitude, which can be attributed
to the growing number of structural defects with increasing temperature. At large distances, the
peaks broaden, merge with one another, and become lost in the continuum background g(r) → 1.
Overall, it does not appear possible to conclude on the details of structural changes from the pair
correlation function. Indeed, it is well known that the pair correlation function cannot sensitively
reveal the presence or absence of long-range order, the liquid g(r) being virtually indistinguishable
from the one of a “hot solid”, whose atoms are displaced from the equilibrium lattice sites due to
thermal motion [4].
r (µm)
g(
r)
2 4 6 8 10 12 14
0
1
2
3
4
5
6
7
8
9
26.0 °C
26.5 °C
27.0 °C
27.5 °C
28.0 °C
Figure 3. Radial distribution function g(r) of the 2D colloidal lattice at indicated temperatures.
Note that the splitting of the second peak, clearly visible at 26◦C, disappears at higher temper-
atures, indicating the loss of hexagonal order. Plots are evenly shifted vertically for clarity.
33601-4
Melting of 2D LC colloidal structure
Next, we analyze the structure factor S(q)
S(q) =
1
N
N
∑
j,k=1
e−iq·(rj−rk), (2)
where N is the number of particles and rj , rk their position vectors. For a perfect 3D crystal with
its infinite Bragg planes, S(q) consists simply of an array of delta functions at the points of the
reciprocal lattice q = G, superimposed on a background that is related to the thermal fluctuations
of the atoms about their equilibrium positions and that is expressed through the Debye-Waller
factor. These Bragg peaks remain infinitely sharp at any temperature, as long as the crystal has
not melted, since the atoms occupy, on average, equilibrium lattice sites (long range order). In 2D,
as mentioned, the long range order is not expected to occur, and thus the infinite “Bragg lines”
do not exist, so that the peaks in S(q) are progressively smeared out with increasing q. Figure 4
shows S(q) for a set of about 700 particles.
qx (µm−1)
q y (µ
m
−1
)
a) 26.0 °C
−10 −5 0 5 10
−10
−5
0
5
10
qx (µm−1)
q y (µ
m
−1
)
b) 26.5 °C
−10 −5 0 5 10
−10
−5
0
5
10
qx (µm−1)
q y (µ
m
−1
)
c) 27.0 °C
−10 −5 0 5 10
−10
−5
0
5
10
qx (µm−1)
q y (µ
m
−1
)
d) 27.5 °C
−10 −5 0 5 10
−10
−5
0
5
10
Figure 4. Static structure factor (interference function) S(q) of a system of about 700 particles
at selected temperatures. With increase of temperature high-order peaks vanish while six first-
order peaks are blurred into ring, which clearly indicates loss of both positional and angular
ordering. Darker shading corresponds to higher magnitude of S(q).
The “diffraction peaks”, seen in figure 4 (a), are indeed rather broad and have low intensity,
only about a few percent compared to the central peak. Furthermore, the peaks at q & 5µm−1
beyond the first Brillouin zone are only barely distinguishable, as really expected in the case of
only a relatively short translational order. It is also seen that the lattice is not exactly hexagonal
but is somewhat deformed.
When the temperature of the system increases, the first-order peaks of S(q) are smeared out,
and by 27.5◦C almost completely disappear, transforming into a nearly isotropic ring (see fig-
ure 4 (d)). Higher-order peaks vanish already at 27◦C, see figure 4 (c). Such changes of S(q)
indicate that our system melts at about 27◦C.
It is instructive to compare characteristic changes of the 2D structure factor on melting with
those in the 3D case. One of the well-known criteria of freezing of a monatomic 3D liquid asserts
that the amplitude S(q0) of the main peak of the liquid structure factor reaches a characteristic
value of 2.85 at freezing [26], which is known as Hansen-Verlet criterion [12]. Furthermore, the
amplitude S(q0) can be considered as an order parameter in a first-principles theory of freezing
[27]. Meanwhile, the corresponding amplitude of the main structure factor peak in 2D reaches
significantly higher values than 2.85 [27–29]. Otherwise, the general applicability of the Hansen-
Verlet criterion in 2D is not clear, nor is the corresponding characteristic value S(q0). The order-
parameter theory of Ramakrishnan and Yussouff [27] suggests the critical value S(q0) ≈ 7 in 2D,
whereas molecular dynamics simulations rather indicate the value of 5.5. Empirical evidence seems
to suggests ≈ 4 [12].
To investigate the temperature behavior of S(q0), we computed S(q), which is the angular
average of the structure factor S(q), from the radial distribution functions g(r) of figure 3 through
the (two-dimensional) Fourier transform [4],
S(q) = 1 + ρ
∫
[g(r) − 1]e−iq·r dr, (3)
33601-5
A. Brodin et al.
where ρ is the number density. Performing the angular integration in equation (3) leads to the
following expression for S(q) through the Hankel transform,
S(q) = 1 + 2πρ
∞
∫
0
[g(r) − 1]J0(qr) rdr, (4)
where J0(qr) is the zero-order Bessel function of the first kind.
q (µm−1)
S
(q
)
0 5 10 15
0
2
4
6
8
10
12
14
26.0 °C
26.5 °C
27.0 °C
27.5 °C
28.0 °C
Figure 5. Angle-averaged structure factor S(q) at indicated temperatures (traces for tempera-
tures below 28.0 ◦C are evenly shifted vertically).
In figure 5 we present temperature-dependent structure factors S(q) of our systems, numerically
computed from the radial distribution functions g(r) of figure 3 with the help of equation (4). It
is immediately obvious from the figure that the structure factors, and thus the structure, exhibit
a dramatic, almost abrupt change upon the temperature change from 27.0◦C to 27.5◦C. Firstly,
the characteristic splitting of the second peak, clearly seen at T 6 27.0◦C, disappears above that
temperature, and secondly, the amplitude of the first peak decreases almost abruptly from & 7
to ≈ 3. The splitting of the second peak of S(q), analogous to the corresponding splitting in g(r)
discussed above, is evidently much more sensitive to the presence of hexagonal order than the
second peak of g(r). From its behavior alone one concludes that our 2D structure melts somewhere
between 27.0◦C and 27.5◦C. The concomitant drop of the first peak amplitude also appears to be
a sensitive indicator of melting, with the typical amplitude in the molten state of ∼ 3. The highest
amplitude, detected at 27.5◦C just prior to freezing, amounts to 3.5, which is rather close to the
empirical value of ≈ 4 [12], but significantly less than the theoretical value of ∼ 7 [27].
Next we turn to analyzing other correlation functions that are more sensitive to the structural
order than the pair correlation function, and for which specific behavior is expected within the
KTHNY theory. Firstly, in table 1 we summarize the expected translational and orientational
order in different phases, according to the KTHNY theory.
Truly long range order is only expected for bond orientations in the crystalline phase, whereas
translational correlation is then only quasi-long range, so that the corresponding correlation func-
tion gG(r) (defined later in equation equation (5)) decays algebraically with distance. In the in-
termediate hexatic phase, bond-orientational correlation function g6(r) (see later equation (6) for
definition) acquires critical algebraic decay. In all cases of short range order, the corresponding
correlation functions decay exponentially with distance. Specifically, in the solid phase one expects
gG(r) ∝ r−η with the exponent η in the range 1/4 6 η(Tm) 6 1/3 at the “melting” temperature
Tm, where transition to the hexatic phase starts to occur. Up to a higher temperature Ti, at which
complete isotropization occurs, the system remains in the hexatic phase with g6(r) ∝ r−η
6 with
the exponent that assumes the value of 1/4 exactly at Ti, i. e. η6(Ti) = 1/4. Thus, one expects
33601-6
Melting of 2D LC colloidal structure
r (µm)
g G
(r
)
a)
0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
26.0 °C
r (µm)
g G
(r
)
e−r
b)
0 5 10 15 20 25
0.1
1
26.0 °C
26.5 °C
27.0 °C
27.5 °C
28.0 °C
r (µm)
g G
(r
)
r−1/3
r−1/4
c)
1 10
0.1
1
26.0 °C
26.5 °C
27.0 °C
27.5 °C
28.0 °C
Figure 6. (a) Translational (positional) correlation function gG(r) at 26◦C. (b), (c) Smoothed
gG(r) (envelopes of the oscillating functions) at selected temperatures, on the semi-log and log-
log scale, respectively. Dashed lines represent characteristic exponential and power-law decays.
Table 1. Translational and orientational order in different phases, according to the KTHNY
theory.
solid hexatic liquid
Translational quasi- short- short-
order long- range range
range
Bond- long- quasi- short-
angle range long- range
order range
characteristic power law decays of the correlation functions at the characteristic temperatures Tm
and Ti.
We first consider the translational (positional) correlation function gG(r), defined as follows [8]:
gG(|r − r′|) = 〈exp[iG · (h(r) − h′(r′))]〉
= 〈exp[iG · (r − r′)]〉 ,
(5)
where G denotes a reciprocal lattice vector, h(r) = r − R is the displacement of the particle
at r with respect to the ideal lattice site R, and the last equality follows from the fact that
exp[iG · (R′ − R)] = 1 for any R′, R. The translational correlation function gG(r) measures the
decay of translational order with distance. In a 3D solid, it would decay to a finite value at large
r (long range order), whereas in the 2D case we expect it to decay algebraically to zero.
Figure 6 (a) shows the translational correlation function gG(r) at 26◦C, computed with the re-
ciprocal lattice vectors, obtained from figure 4. The correlation function exhibits strong oscillation,
reminiscent of the corresponding oscillation of the radial distribution function g(r) in figure 3 and
reflecting the local order. We are mainly interested in the loss of correlation with distance which
is reflected in the behavior of the upper envelope function, rather than fast oscillations due to the
local order. We therefore extract only the peak maxima from the oscillating gG(r), computed at
different temperatures, and plot those envelopes in figure 6 (b) and (c) on a semi-log and log-log
scale, respectively.
Surprisingly, inspecting figure 6 (b) it appears that at the lowest temperatures 26◦C and 26.5◦C,
which correspond to a solid hexagonal structure, the correlation decay is close to exponential, see
dashed line in figure 6 (b), with a correlation length ξ = 9.3 µm = 5.3a. At higher temperatures
the correlation length decreases sharply, although the decay, rather than becoming/remaining ex-
ponential, appears to turn algebraic, see figure 6 (c). We cautiously note that such an “anomalous”
behavior, while probably being genuine for our systems, may also be an artifact due to the limited
33601-7
A. Brodin et al.
structure size and numerical inaccuracies. One of the possible sources of such inaccuracies is in
the determination of the reciprocal lattice vector G, which enters equation (5). Obviously, the
reciprocal vector cannot be exactly defined – in fact, none exists in the true sense, due to the lack
of long range order.
At distances shorter than the correlation length ξ, the decay of gG(r) at T 6 26.5◦C can be
approximated with a power law, whose exponent η is close to 1/3 or 1/4, see the log-log plot of
figure 6 (c). At 27◦C the decay is already somewhat faster than r−1/3, whereas at T > 27.5◦C,
when the structure must have melted according to the preceding analysis, the decay, albeit not
truly exponential, is clearly much faster than r−1/3. Thus, it appears that the hexatic-like phase
only occurs in the narrow temperature range of ≈26.5◦C to 27◦C, i. e. over a mere 1◦C or less.
Now we turn the attention to the bond-orientational correlation function g6(r), defined as
follows [10]:
g6(|rb − r′b|) = 〈exp[6iθ(rb, r
′
b)]〉 , (6)
where θ(rb, r
′
b) is the angle between the bonds at rb and r′b. The bond-orientational correlation
function measures the probability that two bonds a distance r apart are hexagonally oriented
with respect to each other. It is expected that, although 2D lattices lack long range translational
order, they do possess long range orientational order, so that g6(r) decays algebraically to a finite
temperature-dependent level in the ordered state, whereas the decay goes over to exponential in the
melt. Figure 7 shows the orientational correlation function g6(r) at several selected temperatures
together with the guide-lines of characteristic r−1/4 power law, expected at the hexatic-isotropic
transition.
r (µm)
g 6(r
)
1 10
0.01
0.1
1
26.0 °C
26.5 °C
27.0 °C
27.5 °C
28.0 °C
Figure 7. Bond-orientational correlation functions g6(r) of 2D colloidal structures at indicated
temperatures. Slanted dashed lines are power laws r−1/4, presented as guides to the eye.
The correlation functions g6(r) in figure 7 exhibit considerable oscillations due to local order
analogous to, yet smaller than the corresponding oscillations of gG(r) in figure 6 (a). Similarly to
the case of gG(r), we are only interested in the (smooth) behavior of the corresponding envelopes.
Concentrating first on the lowest temperatures 26◦C and 26.5◦C, where one expects the highest
long-range orientational order, the correlation indeed stays almost constant with the distance,
and thus indeed reveals a long-range orientational order. The decay is neither exponential, nor
algebraic, although it is clearly slower than r−1/4. At 27◦C, the dependence is rather algebraic and
closely follows r−1/4. At 27.5◦C the decay is clearly faster than r−1/4, whereas at 28.0◦C it is still
faster and close to exponential, indicating that melting is complete.
Comparing now the results of figure 6 and figure 7, they are qualitatively consistent with
the KTHNY expectations. Firstly, the translational functions in figure 6 decay faster than the
corresponding orientational ones in figure 7, at least at T 6 27.5◦C. It appears that the system
enters the intermediate hexatic phase at a temperature around 27◦C, where the orientational
function g6(r) exhibits a characteristic algebraic r−1/4 behavior, whereas the translational function
33601-8
Melting of 2D LC colloidal structure
decay is close to r−1/3, and the translational correlation length starts to drop. At T > 27.5◦C, both
g6(r) and gG(r) decay rapidly, signaling the transition to isotropic state. Overall, it thus appears
that the transition happens in a finite but narrow range around 27◦C.
Investigating colloidal systems with varying droplet size, we found that the 2D melting tran-
sition and the temperature range of intermediate phase depend on the size of glycerol droplets.
In the case of smaller droplets of a diameter 2R ' 1 µm, the transition occurs approximately
at 27◦C. For larger droplets, the transition temperature increases, and for droplets of a diame-
ter 2R ' 2.5 µm there was no melting observed within the nematic phase range of 5CB liquid
crystal. Such difference is due to the fact that the NLC-mediated interactions, which stabilize the
2D lattice, are size-dependent and their magnitude increases with the droplet size. Thus, larger
droplets are more strongly bounded, and thus their positions exhibit less thermal fluctuations. On
the contrary, 2D lattices composed of smaller droplets stronger “feel” the changes in temperature.
4. Lindemann parameter
Lindemann condition [30] is a well established empirical melting criterion for 3D solids. In
current interpretation, it states that the crystal melts when r.m.s. (root-mean-square) displacement
of particles from their equilibrium positions due to thermal motion
√
〈∆r2〉 reaches 15% of the
lattice period a. Accordingly, one introduces the Lindemann parameter L =
√
〈∆r2〉/a, whose
value is 0.15 at melting. It turns out that this criterion is also valid in higher dimensions, at
least up to D = 50 [31]. In 2D crystals, however, the situation is different: the particle deviations
diverge at long times (for the same reason that 2D crystals do not possess long range order), so
that the Lindemann parameter, as defined above, appears to have no meaning. The definition can
however be easily adapted to a form suitable to 2D cases, either through the relative displacement
of nearest neighbors i and i + 1 [14, 32], with the “modified” Lm =
√
〈(∆ri − ∆ri+1)2〉/a, or else
by evaluating the displacements ∆r in local coordinate systems of the particles [33]. In both cases,
the diverging long-wavelength fluctuations are excluded, so that the result is finite and essentially
equivalent to the usual definition. Numerically, however, Lm and L are not identical – indeed,
assuming uncorrelated motion of neighboring particles, as in the Einstein solid, then evidently
L2
m = 2L2. Furthermore, some authors use L2 in place of L and/or introduce additional numerical
pre-factors into the definition. Due care has therefore to be taken when comparing data from the
literature. In the following, we assume the classical definition of the Lindemann parameter, and
accordingly reduce the literature data for comparisons, when needed.
Temperature (°C)
L
26 26.5 27 27.5 28
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
Figure 8. Temperature dependence of the Lindemann parameter, estimated from the standard
deviation of the distribution of colloidal bond lengths.
In order to estimate L in our case, we observe that in the case when |∆r| � a and assum-
ing uncorrelated motion of neighboring particles, L2
m essentially gives the bond length variance,
33601-9
A. Brodin et al.
L2
m ≈ 〈(|b|−a)2〉, where b is the bond vector between the nearest neighbors i and i+1. We therefore
use r.m.s. bond length fluctuation to estimate the Lindemann parameter, L =
√
〈(|b| − a)2〉/
√
2a.
Evidently, the r.m.s. bond length fluctuation overestimates L, since it also reflects the static bond-
length disorder, which is significant in the intermediate phase. Yet we believe that, since the
dynamic fluctuations are relatively large at the transition, it gives an acceptable estimate. Temper-
ature dependence of the so-determined Lindemann parameter is presented in figure 8. One observes
that L = 0.09 in the crystal and starts to rise linearly with the temperature at 26.5◦C, i. e. where,
according to the previous analyses, the system enters the intermediate phase. At melting, which
occurs somewhat above 27◦C, it reaches the value L ≈ 0.12 or 0.13, which is thus rather close to
the 3D condition of L = 0.15.
It has been suggested in the literature [14, 15, 32] that Lc = 0.13 is the critical value, at which
2D solid melts (the quoted papers actually used L2
m as the modified Lindemann parameter, from
which L =
√
L2
m/2). We confirm that our system melts at, or close to this value, which thus seems
to be universal, although somewhat different estimates of Lc were also suggested, such as Lc ∼ 0.1
in an MD simulation work [33]. Overall, it appears that Lc does exist in 2D, and is close to the
corresponding 3D value of 0.15. We note that the existence of the Lindemann criterion in 2D,
although not explicitly discussed in the KTHNY theory, is closely related to the critical elastic
moduli of the theory, e. g., to Young’s modulus K, whose value at the solid-hexatic transition
is such that Ka2/ kBT = 16π [10, 34]. Approximating the colloidal system with the 2D Einstein
solid, where each particle vibrates in a harmonic potential U = kr2/2 created by its surrounding, it
follows from mean-field-like considerations that the modulus K is proportional to the corresponding
spring constant k. The latter, however, is directly related to the Lindemann factor. Indeed, mean
square displacement 〈r2〉 of a 2D harmonic oscillator is given by its thermal energy kBT such that
〈r2〉 = 2kBT/k [35], which, introducing the Lindemann parameter, leads to ka2/kBT = 2/L2, so
that the elastic constant (and thus the modulus) is related to L. Consequently, the existence of a
critical elastic modulus implies the existence of Lc.
Finally, we try to estimate the depth of the potential wells of lattice sites. We approximate
the potential, which is periodic with the lattice period a, with the trigonometric function U(x) =
U0 sin2 πx/a, so that the corresponding elastic constant is k = 2π2U0/a2. From the reasoning of
the previous paragraph it then follows that U0 = kBT/π2L2. Thus, the depth of the potential wells
decreases from U0 ' 12kBT at 26.0◦C to U0 ' 6kBT at 28.0◦C.
5. Conclusions
In conclusion, we have shown that melting of a two-dimensional colloidal system, formed by glyc-
erol droplets at the free surface of a nematic liquid crystalline layer, occurs qualitatively and even
semi-quantitatively in accordance with the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY)
theory. In particular, melting is obviously a continuous transition through an intermediate hexatic
phase, characterized by quick rise of the number of structural defects. However, the temperature
range of the intermediate phase is rather narrow, perhaps .1◦C. Furthermore, decay of the cor-
relation functions with distance can only roughly be approximated with the characteristic critical
power laws and exponential decays in different phases, suggesting that the details of the transi-
tion scenario do not fully comply with the KTHNY expectations, and reflect peculiarities of the
interparticle interaction.
Acknowledgements
The work was supported by National Academy of Sciences of Ukraine AS of Ukraine via Grants
VC-134 and #5M/09–16; and Fundamental Research State Fund Project UU24/018.
33601-10
Melting of 2D LC colloidal structure
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33601-11
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Плавлення двовимiрної рiдкокристалiчної колоїдної
структури
О. Бродин1,2, А. Нич1, У. Огниста1, Б. Лев3, В. Назаренко1, М. Шкаработ4
I. Мушевич4,
1 Iнститут фiзики НАН України, Київ, Україна
2 Нацiональний технiчний унiверситет України “Київський полiтехнiчний iнститут”, Київ, Україна
3 Iнститут теоретичної фiзики iм. М.М. Боголюбова НАН України, Київ, Україна
4 Iнститут Й. Стефана, Любляна, Словенiя
Використовуючи вiдеомiкроскопiю, ми дослiдили плавлення двовимiрної колоїдної системи, сфор-
мованої глiцериновими краплями на вiльнiй поверхнi шару рiдкого кристала. Аналiзуючи рiзнi струк-
турнi кореляцiйнi функцiї, ми показуємо, що плавлення вiдбувається через промiжну гексатичну фа-
зу, як передбачено теорiєю Костерлiца-Таулесса-Альперiна-Нельсона-Янга (КТАНЯ). Проте темпе-
ратурний дiапазон промiжної фази є дуже вузьким . 1
◦C i характеристичний критичний степiнь
спадання кореляцiйних функцiй не повнiстю встановлюється. Ми зробили висновок, що плавлення
таких двовимiрних систем якiсно описується теорiєю КТАНЯ, хоча кiлькiснi деталi сценарiю переходу
можуть частково залежати вiд деталей взаємодiї мiж частинками.
Ключовi слова: рiдкi кристали, двовимiрнi колоїднi структури, фазовi переходи в двовимiрних
структурах
33601-12
Introduction
Experiment
Results and discussion
Lindemann parameter
Conclusions
|