VOLUME
86, NUMBER 5 PHYSICAL REVIEW LETTERS 29J
ANUARY
2001
Phase Behavior of Two-Dimensional Colloidal Systems in the Presence of Periodic Light Fields
C. Bechinger, M. Brunner, and P. Leiderer
University of Konstanz, Physics Department, Fach M676, 78457 Konstanz, Germany
(Received 16 August 2000)
We investigate the phase behavior of a two-dimensional suspension of charge stabilized polystyrene
spheres in the presence of a one-dimensional periodic light field. With increasing light intensity we
observe a liquid-solid followed by a solid-liquid transition which are known as laser-induced freezing
and melting, respectively. Here we report on measurements where, in addition to the light intensity, the
single particle density was also systematically varied. As a result, we obtain for the first time the full
thermodynamic information about the system which allows comparison with numerical predictions of
other authors.
DOI: 10.1103/PhysRevLett.86.930 PACS numbers: 82.70.Dd, 64.70.Dv
There has been considerable interest in the freezing and
melting of colloidal particles during the recent years which
has been largely motivated by the fact that colloidal par-
ticles provide ideal model systems for experimental studies
of two-dimensional (2D) melting. Accordingly, such sys-
tems have been intensively investigated by several authors
(see, e.g., [1,2]). While there exist numerous theoretical
and experimental studies on 2D melting on homogeneous
substrates [3–5], only little is known about 2D melting on
corrugated surfaces, although the latter is much more rele-
vant when modeling the substrate potential of a crystalline,
i.e., atomically corrugated surface.
It was the experimental work of Chowdhury, Ackerson,
and Clark which originally demonstrated that one-
dimensional (1D) periodic substrate potentials can cause
interesting effects on the phase behavior of colloidal
particles [6,7]. When they investigated a charge-stabilized
colloidal liquid of polystyrene (PS) particles being con-
fined between two glass plates and additionally exposed
to a standing laser field (with its wave vector q tuned
to the first peak of the direct correlation function), they
observed a phase transition from a liquid to a crystal upon
increasing the laser intensity. This phenomenon which
has been termed laser-induced freezing (LIF) has also
been theoretically analyzed by several authors employing
Monte Carlo (MC) studies [8] and density functional
theory (DFT) [9]. LIF has been explained as an alignment
of particles in the high intensity regions of the interference
pattern which produces a directly stimulated density mode
of particles, followed by a registration of particles between
neighboring lines due to interparticle interactions.
When Chakrabarti et al. [10] theoretically investigated
the phase behavior of such a system as a function of the
laser intensity, in addition to LIF which takes place at
relatively small laser intensities, at higher laser intensities
they found a reentrant laser-induced melting (LIM) transi-
tion, where a remelting of the crystal back into the modu-
lated liquid phase was observed. In order to understand
this — at first glance very surprising — remelting process,
the role of particle fluctuations perpendicular to the inter-
ference fringes has to be taken into account because they
largely contribute to the registration of adjacent lines and
thus to crystallization. Accordingly, upon reducing those
fluctuations by increasing the laser intensity, the crystal
remelts. This effect has also been demonstrated to be in
agreement with experimental studies [11,12].
In previous measurements the phase behavior was
investigated in a regime of fixed particle-particle inter-
action, i.e., constant particle number density and salt
concentration, and only the laser intensity was varied. In
this Letter we present for the first time a systematic study
where in addition to the depth of the light potential the
particle-particle interaction was also varied. Since it is the
counterplay of those two interactions which is the physical
origin of the unusual phase behavior in this situation,
our data provide further insight in the nature of LIF and
LIM and give the full thermodynamic information on the
system. Our data clearly show that reentrant melting from
the crystalline state to a modulated liquid is observed
only if the crystalline state was formed by LIF (and not
by spontaneous crystallization as has been theoretically
suggested [10]). In addition, we observe LIF and LIM
to take place at considerably higher laser fields than
predicted [10] and only in a relatively small region of
particle number densities and salt concentrations.
The experimental setup has been described elsewhere;
therefore it will be discussed here only in brief. We used
aqueous suspensions of sulfate PS particles of 3 mm
diameter and a polydispersity of 4% (Interfacial Dynamics
Corporation). Because of sulfate-terminated surface
groups which partially dissociate off when in contact with
water, the suspended particles are negatively charged and
experience a screened electrostatic repulsion which can
be described by [13,14]
F共r兲 苷
共Z
ⴱ
e兲
2
4pe
r
e
0
µ
exp共kR兲
1 1kR
∂
exp共2kr兲
r
. (1)
Here Z
ⴱ
e is the renormalized charge of the particles which
has been roughly determined to be Z
ⴱ
艐 20 000 [15], e
r
930 0031-9007兾01兾86(5)兾930(4)$15.00 © 2001 The American Physical Society
First publ. in: Physical Review Letters 86 (2001), 5, pp. 930-933
Konstanzer Online-Publikations-System (KOPS)
URL: http://www.ub.uni-konstanz.de/kops/volltexte/2007/2832/
URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-28329
VOLUME
86, NUMBER 5 PHYSICAL REVIEW LETTERS 29J
ANUARY
2001
is the dielectric constant of water, k is the inverse Debye
screening length, and r is the distance between particle
centers. The experiments are performed in a closed
circuit which is composed of the sample cell, a vessel of
ion exchange resin, and an electrical conductivity probe
to control the ionic strength of the suspension [16]. A
peristaltic pump is used to pump the highly deionized sus-
pension through the circuit. The Debye screening length
was estimated from the interparticle distribution of the col-
loids to be on the order of k
21
艐 400 nm. The sample cell
consists of two horizontally aligned parallel glass surfaces
with a distance of 1 mm. In the course of our experiments
we tested several cells with different spacings between
20 mm and 1 mm and observed no differences. We first
deionized the circuit almost completely as confirmed by
the value of the ionic conductivity s 苷 0.07 mS兾cm.
Then the sample cell was disconnected from the circuit
to allow stable conditions during several hours. This
was determined by measurements of the mean particle
distance which was observed to change only by about 3%
during 5 h. The laser potential was created by two slightly
crossed laser beams of a linearly polarized Nd:YVO
4
laser (l 苷 532 nm, P
max
苷 2W) which overlapped in
the sample plane, thus forming an interference pattern.
Because of the polarizability of the PS spheres this
provides a periodic, one-dimensional potential for the
particles. The periodicity could be adjusted by variation
of the crossing angle. Because of the almost vertical
incidence of the laser beams onto the sample the particles
experienced additionally a vertical light pressure which
pushed them towards the negatively charged bottom silica
plate of our cell. This vertical force is estimated to be in
the range of pN and largely reduces vertical fluctuations
of the particles, thus confining the system effectively
to two dimensions. The sample, which was in addition
illuminated with white light from the top, was imaged
with a microscope objective onto a CCD camera chip
connected to a computer for further analysis. The intense
Nd:YVO
4
laser light was blocked with an optical filter.
In order to compare our experimental measurements
quantitatively to theoretical predictions, one must experi
mentally determine the depth of the laser potential as a
function of the intensity of the interference fringes. This
was achieved by exposing the above-mentioned light pat-
tern to a highly diluted colloidal suspension. From the
probability distribution of the particles perpendicular to
the laser lines, we obtained the laser potential acting on
the particles by employing the Boltzmann statistics. The
shape of the laser potential in the central region of the over-
lapping laser beams is found to be
V 共x兲 苷 2V
0
∑
1 1 cos
µ
2px
d
∂∏
(2)
with V
0
being the potential amplitude and d the period
of the fringe spacing. The amplitude V
0
as obtained
from our experiments increases linearly as a function of
the laser intensity I (symbols in Fig. 1). According to
Loudiyi et al. the colloid-light interaction V
0
can be
calculated by integration over the potential contributions
for each infinitesimal volume element of the colloidal
sphere [17]
V
0
苷 6n
2
W
s
3
0
P
cr
2
0
µ
n
2
2 1
n
2
1 2
∂
√
j
1
共
ps
0
d
兲
2ps
0
d
!
. (3)
Here, P is the laser power, c is the velocity of light in
vacuum, n is the ratio of the refraction indices of poly-
styrene n
P
and water n
W
, s
0
is the colloidal particle
diameter, j
1
the first-order spherical Bessel function, and
r
0
is the waist radius of the Gaussian laser beam. When
comparing the slope of a linear fit through our data points
(as indicated by the dotted line in Fig. 1) with the result
of Eq. (3) after inserting the corresponding values of our
experiment (r
0
苷 220 mm, n
W
苷 1.33, n
P
苷 1.59 [18]),
we found agreement within 10% which is within our ex-
perimental errors.
In order to keep the total light intensity independent of
V
0
in the following experiments, one of the laser beams
was directed through a l兾2 plate to rotate its polarization
with respect to the other. Accordingly, the variation of V
0
was achieved by adjusting the angle of the l兾2 plate while
keeping the total laser intensity constant.
To distinguish the different thermodynamic phases we
first determined the particle center coordinates by means of
a particle-recognition algorithm. From those data the time
averaged single particle density r共x, y兲 and the pair cor-
relation function g共x, y兲 were calculated, the latter being
particularly useful for differentiating the modulated liquid
from the crystalline phase. Here, x and y denote the di-
rection perpendicular and along the interference fringes,
0.0 0.2 0.4 0.6 0.8 1.0
0
1
2
3
4
5
32 36 40 44 48
0
2
4
6
2V
0
V
0
/k
B
T
laser intensity [W]
laser potential [k
B
T]
x-position [µm]
FIG. 1. Amplitude of light potential acting on colloidal par-
ticles of 3 mm diameter as a function of the incident laser inten-
sity. The symbols correspond to measured data and the dashed
line corresponds to a theoretical prediction. The inset shows the
spatially modulated laser potential experienced by the particles
at the center region of the periodic light field (symbols). The
line corresponds to a cosine function.
931
VOLUME
86, NUMBER 5 PHYSICAL REVIEW LETTERS 29J
ANUARY
2001
respectively. Since in the modulated liquid phase no reg-
istration between adjacent lines occurs, no correlations
among particles in neighboring lines are observed. This
leads to a g共x, y兲 plot with smeared-out lines along the
interference fringes. In contrast, adjacent lines in the crys-
talline phase are interlocked due to particle excursions per-
pendicular to the laser lines which leads to well-defined,
nonoverlapping patches in the vicinity of the reference
point in g共x, y兲 [11]. In addition, when fitting the decays
g共y兲 of the different phases, we find that the modulated
liquid phases always have short range order and an expo-
nential decay, whereas the decay in the crystal extends over
a much longer range and can be described with an algebraic
function g共y兲 2 1 ~ y
2h
. It has been mentioned that un-
like conventional 2D melting, in the presence of a modu-
lated light field, h is universal at the melting transition
and should be equal to 1 at the melting temperature [19].
This is also consistent with our previous experiments [11].
In contrast to earlier measurements, where the particle
number density was held constant, here we measured sys-
tematically the phase behavior for different particle num-
ber densities as a function of the light potential amplitude
V
0
. Particular attention was paid to the fact that the pe-
riodicity of the laser potential d was adjusted properly to
obtain a hexagonal crystal, i.e., d 苷
p
3
2
a. Otherwise a
distorted lattice would have been observed. The range of
d was between 6 and 8 mm. Only the central region of the
interference pattern (corresponding to about 400 particles)
was used for the data analysis to ensure V
0
to be constant
within about 5%. To obtain sufficient statistics, g共x, y兲
was averaged over more than one thousand pictures with a
time interval of one second each. The result of more than
one hundred single measurements is shown in Fig. 2. We
plotted the vertical axis in units of 共ka兲
21
with a being
0 2 4 6 8 1012141618
0.044
0.048
0.052
0.056
0.060
0.064
modulated liquid
crystal
(
κ
a)
-1
V
0
/k
B
T
FIG. 2. Experimentally determined phase diagram as a func-
tion of 共ka兲
21
vs
V
0
kT
. The open symbols denote the modulated
liquid and the closed symbols denote the crystalline phase, re-
spectively. For clarity, error bars are plotted only for a few
data points.
the mean distance of next neighbor particles which was
measured for each particle concentration in the absence of
the laser field. As can be seen, the value of 共ka兲
21
where
the transition towards the crystal occurs decreases at small
laser intensities as a function of V
0
. This is the charac-
teristic feature of LIF and is in agreement with numeri-
cal calculations [9]. For larger values of V
0
, however, the
separation line between the crystalline and the modulated
liquid region is shifted back to higher 共ka兲
21
values and
starts to saturate at the highest values which could be ob-
tained with our setup. It is this up bending which gives rise
to the LIM phenomenon. If 共ka兲
21
is in a range between
0.045 and 0.048, with increasing V
0
one observes the fol-
lowing sequence of states: isotropic liquid — modulated
liquid —crystal — modulated liquid, which is in agreement
with earlier results [11,12].
Figure 3 shows the phase diagram as obtained by
means of MC simulations from Chakrabarti
et al. [9].
The phase separation line has a pronounced minimum at
nonzero laser potentials which is in agreement with our
data. Besides this qualitative agreement, however, there
are several important differences between Figs. 2 and 3.
The experimental data suggest that LIF and LIM occur at
considerably higher V
0
than in the MC simulations.
Possibly some deviations between theory and experiment
stem from the fact that the latter were performed in finite
size systems, whereas the simulation results were obtained
by extrapolation to the thermodynamic limit. We believe,
however, that this is not sufficient to explain such a
large difference because our values for LIF are consistent
with experimental and numerical results of other authors
[6,8,17]. The second, and even more significant, differ-
ence between Figs. 2 and 3 is the qualitative behavior of
the phase separation line at high V
0
. The MC simulations
suggest that 共ka兲
21
at V
0
苷 0 is below the corresponding
value at very high V
0
. This implies that LIM is not
restricted to crystals formed by LIF but may also appear
in systems where the particle concentration is above that
for spontaneous crystallization. This is in contrast to our
experiments, where LIM was only observed for light-
induced crystals. The latter is also supported by a recent
theoretical work by Frey et al. where the qualitative
phase behavior of a 2D system of charged colloids in the
presence of a periodic light potential was reinvestigated
[19]. By using the same concepts developed in the context
of dislocation mediated melting theory, their data also
support the existence of LIF and LIM. In contrast to
Ref. [9], however, the melting temperature of the reentrant
modulated liquid is found to be higher than that of the
modulated liquid at small V
0
. This directly corresponds
to our finding that reentrant melting is only possible in
a range of 共ka兲
21
values which allows for LIF and thus
demonstrates the unique properties of the light-induced
crystalline phase. Therefore, this result is more than
a quantitative correction to the above-mentioned MC
simulations since our data provide strong evidence for
932
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86, NUMBER 5 PHYSICAL REVIEW LETTERS 29J
ANUARY
2001
FIG. 3. Phase diagram in the same parameter space as Fig. 2
as obtained by MC simulations. Filled squares denote first-order
transition points, whereas open circles correspond to continuous
transition points. The data are taken from Ref. [10].
the fluctuation-induced nature of the LIM transition.
In additon, because the region of 共ka兲
21
values where
reentrance occurs is rather small, this might explain why
such an effect has experimentally not been observed
before although such behavior should also be found in
other 2D systems with periodic 1D potentials.
From our experiments, it is difficult to resolve the con-
troversially discussed nature of LIF and LIM phase tran-
sitions conclusively at this point. When investigating the
region near the LIF and LIM transition in detail, we never
observed any hysteresis effects within our experimental
errors. This observation is consistent with a continu-
ous phase transition for LIF and LIM as predicted by
Frey et al. [19], but at this point not sufficient to make
any final statements yet.
So far, we have investigated only the influence of peri-
odic 1D potentials on the phase behavior of colloidal par-
ticles. By interfering more than two laser beams, however,
we can also produce 2D potentials of hexagonal, quadratic,
or rhombic geometry which would then mimic a more re-
alistic surface potential compared to the case studied here.
Since the surface potential depth can be continously var-
ied by the laser intensity, investigations on colloidal model
systems might help to understand the details of the phase
behavior of atomic adsorbates on crystalline sufaces. Ex-
periments are in progress to study the phase behavior under
such conditions.
In summary, we have studied the phase behavior of a
2D colloidal suspension subjected to a periodic 1D light
potential. We have varied both the mean particle distance
as well as the light potential amplitude. As a result we
obtained the phase diagram which shows a minimum in the
共ka兲
21
vs V
0
plane which is the characteristic feature of
the LIF and LIM transitions. While at the present moment
theoretical results obtained by different techniques are not
yet consistent, our data show good agreement with recent
calculations of Frey et al. [19].
We acknowledge fruitful discussions with P. Nielaba,
W. Strepp, and E. Frey.
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![FIG. 3. Phase diagram in the same parameter space as Fig. 2 as obtained by MC simulations. Filled squares denote first-order transition points, whereas open circles correspond to continuous transition points. The data are taken from Ref. [10].](/figures/fig-3-phase-diagram-in-the-same-parameter-space-as-fig-2-as-1c66lb0y.webp)