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REVIEW
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A fresh look at the Laponite phase diagram
Barbara Ruzicka*a and Emanuela Zaccarelli*b
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Published on 20 January 2011 on http://pubs.rsc.org | doi:10.1039/C0SM00590H
Received 28th June 2010, Accepted 18th October 2010
DOI: 10.1039/c0sm00590h
By reviewing experimental and theoretical works, we discuss the phase diagram of Laponite
suspensions, with the aim of clarifying several issues that have caused a long-time controversy on the
system. We claim that, if aging and sample preparation are correctly taken into account, a unifying
picture emerges from different experimental studies. Multiple non-ergodic states are identified,
including a low concentration equilibrium gel and a high concentration Wigner glass in the absence of
salt. These findings are critically discussed also with respect to recent evidence of phase separation at
extremely low densities and to existing theoretical and numerical works.
I.
Introduction
Colloidal clays have recently emerged as complex model systems
with a very rich phase diagram, encompassing fluid, gel and
glassy states.1–3 Often these disordered states interfere with
ordered ones, like nematic and columnar phases.4,5 In this review,
we discuss in detail the multiplicity of states observed in
Laponite, a synthetic smectite clay which has been the subject of
an intense debate after the pioneering work of Thompson and
Butterworth.6
Laponite is nowadays widely used as a rheology-modifier in
many technological applications, such as surface coatings,
ceramic glazes, paints, household cleaners and personal care
products, as well as film former and to build optimized nanocomposites. Laponite platelets are nanometre-sized disks with
1 : 25 aspect ratio and net charges of opposite signs on the faces
(negative) and on the rim (positive).7 The complexity of the single
platelet has its counterpart in a phase diagram which includes
flocculation, disordered (gels and glasses) and ordered (nematic)
states. Despite numerous studies since 1995, this phase diagram
has been very discussed and controversial. The early studies of
Mourchid and coworkers8,9 have reported the formation of gel
states under very dilute conditions, well below the predicted
threshold for the occurrence of an isotropic-nematic transition,
due to the presence of charges. However, soon after these
pioneering works, it was realized that several mechanisms could
be responsible for this scenario. For example, a low-density
non-ergodic state could result from pure electrostatic repulsion
(Wigner glass)10 or from the formation of a bonded ‘house-ofcards’ (HOC)11 network between rim-face charges. Subsequent
experimental studies have tried to elucidate the nature of this
non-ergodic state, but for many years different results and
a
CNR-IPCF and Dipartimento di Fisica, Universit
a di Roma La Sapienza,
P.le A. Moro 2, 00185 Roma, Italy. E-mail: [email protected]
b
CNR-ISC and Dipartimento di Fisica, Universit
a di Roma La Sapienza,
P.le A. Moro 2, 00185 Roma, Italy. E-mail: emanuela.zaccarelli@phys.
uniroma1.it
1268 | Soft Matter, 2011, 7, 1268–1286
interpretations were proposed. This generated a remarkable
scepticism in a part of the scientific community about Laponite,
which has been overcome only recently thanks to three important
observations: (i) the adoption of a standard (and fixed) protocol
of sample preparation which is essential in order to obtain
reproducible data, among which is filtration, which eliminates
the presence of large Laponite aggregates;12,13 (ii) the awareness
that Laponite undergoes extremely subtle aging dynamics,14
which were only recently demonstrated to take place even at very
low clay concentrations,15–18 so that data need to be compared on
similar waiting time scales; (iii) the understanding that different
mechanisms are at hand in different regions of the phase
diagram. Taking into account these issues when comparing
different sets of data, it is now clear that Laponite forms different
non-ergodic states. In pure water, gel states are found at low clay
concentrations and Wigner glasses at higher ones. With
increasing salt concentration, attraction becomes dominant,
leading to the formation of attractive glasses,18 and to macroscopic phase separation (also called flocculation or sedimentation). Very recently, the observation of macroscopic phase
separation at very low densities has been reported,19 even in the
absence of salt, within a time-extended observation window.
These results provide evidence of the formation of empty liquids
and equilibrium gels, similarly to what was predicted for patchy
particles,20 and open new perspectives for using colloidal clays as
suitable anisotropic building blocks for material design and
self-assembly.21
For the reasons discussed above, we believe that a comprehensive work is needed in order to clarify the controversial
aspects and to shed light on the unifying scenario that is now
emerging from the literature. This is precisely the aim of this
review which, by framing the past experimental activity in the
light of recent understanding, will provide a fresh look at the
Laponite phase diagram combining together in a congruous way
previous results. In addition, we complement the experimental
findings with those coming from theory and simulations, trying
to provide guidelines for future theoretical and numerical studies
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in order to better describe the experimental data. In this way, we
believe that this review will also contribute to stimulate a closer
experimental/theoretical activity in the future.
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A. Clarification about phase or state diagrams, definition of
various non-ergodic states
First of all, we want to clarify that in this work we sometimes
refer to ‘phase diagram’ also with respect to those diagrams
containing non-ergodic states. Such non-ergodic states are
metastable states in which the system is not able to reach its
underlying equilibrium configuration (e.g. ordered state) due to
the intervening of an arrest mechanism. The arrest can be driven
by different interparticle interactions, as discussed below, giving
rise to a variety of non-ergodic states, such as gels and glasses of
various nature (Wigner, attractive, repulsive). Hence,
throughout the manuscript, when describing phase diagrams we
are actually referring, for simplicity, to ‘state diagrams’, which in
a broader sense also include out-of-equilibrium states. In
particular, our approach will be to try to identify the state
diagram of Laponite at long enough times, i.e. when the system,
through slow aging dynamics, reaches a long-time stable
(although in many cases out-of-equilibrium) state.
Secondly, since many different non-ergodic states are observed
in colloidal systems1,2 and also in Laponite and other clay
suspensions, we provide here a definition of their relevant
features depending on the microscopic interactions between the
particles.
Gels and glasses. Both are disordered solid states, i.e. they do
not flow if turned upside down, while their structural features
display no long-range order. They can be differentiated, in the
broader sense, by their density. Glasses typically arise in dense
fluids, due to supercooling or compression, and the typical
picture illustrating the mechanism of arrest is the so-called cage
effect: particles are trapped by their nearest-neighbours and
cannot relax to their underlying equilibrium state. Gels on the
other hand are found at much lower densities and typically can
be attributed to the formation of a particle network, where
particles are linked together via attractive bonds. So while
attraction is necessary to form a gel, a glass is typically repulsive
and its colloidal prototype is the hard-sphere glass.
Wigner glasses. However, glasses can also be found at very low
density in the absence of attraction. In this case, particles do not
form a network, but remain spatially disconnected although
arrested in a sort of empty cage. In this sense the localization
length of the particles should be larger, or at least comparable to
the particle size.22 This can be realized by means of a long-range
electrostatic repulsion, which prevents particles to get close and
effectively traps them far apart from each other. An efficient way
to distinguish a Wigner glass from a gel is to perform a dilution
experiment2 in such a way that the resulting nominal particle
concentration is that of a fluid under equilibrium conditions.
Wigner glasses should melt, while gels should remain solid
because particle–particle bonds are not altered by the solvent.23
Attractive glasses. At high densities, glasses can also be of
attractive nature, i.e. stabilized by bonds, but in a locally
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crowded configuration. This occurs when excluded-volume
interactions are complemented by a short-range attraction. The
extrapolation to lower densities of an attractive glass may be
called a gel, if there is a continuous path connecting the two
states. However, for short-ranged attractive spherically
symmetric potentials, gels are obtained only through an intervening phase separation, which gives rise to a denser part of the
system which gets eventually arrested (at low enough temperature).24,25 In this case, gels and attractive glasses are conceptually
very different.
Competing interactions. If particles interact with both attractive and repulsive interactions competing with each other, e.g.
short-range depletion attraction and long-range electrostatic
repulsion, the macroscopic phase separation can be pre-empted
by microphase separation, i.e. the formation of finite-size clusters.26,27 Under these conditions, the system would display
a typical ‘cluster peak’ in the static structure factor,28 arising at
distances much larger than the nearest neighbour particle
distance. These equilibrium clusters in turn become the building
blocks of a non-ergodic state. At low densities, they can give rise
to a Wigner glass of clusters (or cluster glass),29,30 where clusters
are trapped in a disordered way into a metastable state by the
electrostatic repulsion, while with increasing densities clusters
branch and percolate, leading to a gel state.29,31
Patchy models. These primitive models are used to describe
colloidal particles interacting only through localized sticky spots
(patches) or, more generally, having a fixed small valence. In this
case, the phase separation region shrinks to very low packing
fractions and temperatures20,32 with decreasing valence, thereby
generating empty liquids, i.e. liquid states with vanishing density,
but still larger than the critical density. Under these conditions,
gel states can be accessed in equilibrium, because particles are
able to form macroscopic bonded networks with increasing bond
lifetime, without the intervening of phase separation. These are
called equilibrium gels as opposed to those obtained in spherically attractive systems through an irreversible spinodal decomposition process.
II. What is Laponite?
Laponite is a synthetic smectite clay with a structure and
composition closely resembling the natural clay mineral hectorite. It is a layered hydrous magnesium silicate belonging to the
family of (2 : 1) phyllosilicates built up of sheets of octahedrally
coordinated magnesium oxide sandwiched between two parallel
sheets of tetrahedrally coordinated silica, as shown in Fig. 1b.
The empirical formula (Fig. 1a) shows the presence of oxygen
atoms and OH groups, while some magnesium atoms are
substituted by lithium forming a net negative charge that is
balanced by interlayer cations, predominantly sodium ions.
Laponite is produced by Rockwood Additives Ltd as a fine white
powder where Laponite crystals are arranged into stacks held
together electrostatically by sharing of sodium ions in the interlayer region between adjacent crystals. There are different types
of Laponite available, the most studied being Laponite RD and
Laponite XLG, the latter having a lower heavy metals content.
Once dispersed in water, Laponite hydrates and swells. The
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only when particles are close to contact, after the electrostatic
attraction has driven the aggregation process, or at high enough
salt concentration (larger than z101 M 36).
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III.
Fig. 1 (a) Empirical formula of Laponite. (b) Idealised structural
formula of Laponite drawn in perspective. (c) Single Laponite platelet.
release of the Na+ ions leads to a negative charge on the faces
with ions forming double layers around them, while a protonation process of the OH groups, localized where the crystal
structure terminates, forms a positive charge. Therefore,
Laponite in water forms a colloidal dispersion of charged disklike particles, shown in Fig. 1c, with a diameter of 25 nm and
a thickness of 1 nm with both negative charges on the faces and
positive ones on the rims.7 The negative charges have been
recently quantified for different clay concentrations by conductivity measurements.18 Their value is found to be smaller than the
nominal charge (700 e), which would correspond to the release
of all Na+ ions from the particle surface. The situation of the rim
charges is more debated. A number of works in the literature
report that they are positive (typically 10% of the negative
charges) and slightly decreasing with increasing pH for
pH (11.7,33 Tawari and co-workers also suggest that for
pH T 11 the rim charge is neutralized,7 so that we cannot exclude
that above this value it could eventually become negative,
although there is, to our knowledge, no experimental evidence of
this negative regime. For this reason a careful investigation of the
dependence of rim charges on pH would be very valuable,
similarly to what has been done for montmorillonite.34
With regards to pH, Cummins has recently performed
a systematic investigation which has convincingly established
that Laponite solutions are always found at pH x 10.3 Whether
the samples are prepared in water with a starting pH ¼ 10 or
when Laponite powder is added to deionized water without fixed
pH, he observes that the resulting final pH of the solutions is 10.
Hence, as discussed above, it appears that under normal experimental conditions Laponite rim charges should be positive.
Laponite dispersions are usually considered monodisperse
suspensions of single platelets. However a grade of polydispersity
has been found by different authors14,35 and atomic force
microscopy measurements on a very diluted Laponite sample
have shown that particles may not be completely delaminated
and a fraction of them (20%) are in dimer configurations.35
To conclude this section, it is important to describe the relevant Laponite–Laponite interactions. In addition to the excluded
volume, there is the (isotropic) van der Waals attraction as well
as the complex (and anisotropic) electrostatic interaction, which
can be either repulsive (face–face, rim–rim) or attractive (rim–
face). We will see in the following that, depending on experimental conditions, the system can be dominated by either
repulsive or attractive interactions. As regards to the role of the
van der Waals attraction, this is expected to become important
1270 | Soft Matter, 2011, 7, 1268–1286
Experiments
In this section we will try to give a global interpretation of
Laponite phase diagram taking into account the results obtained
by different groups on this system. We report results and figures
taken from the original papers where clay concentration is
sometimes reported in weight/weight % (Cw) and sometimes in
weight/volume (g L1; 1 g L1 ¼ 0.1 Cw(%)). Different groups
quantify the molarity by either ionic strength (I) or salt
concentration (Cs). The two quantities are not identical because
when Laponite is dissolved in water, sodium ions are released
from the platelets and the effective ionic strength is higher than
Cs because of counterion contributions. For example, for
samples prepared in free water (without added salt) one has: pH
10,3 salt concentration Cs ¼ 104 M and ionic strength I z 2 104 M.
We start by addressing the aging dynamics and the problem of
reproducibility, in order to clarify the two main issues which have
caused over the years considerable controversy about Laponite.
We then revise the different versions of the phase diagram that
have been proposed in the literature. Finally we conclude the
section by unifying the different measurements and interpretations and illustrating our view about Laponite state diagram.
A.
Aging phenomena
It is nowadays well known that all experimental studies on
Laponite suspensions must take into account the aging evolution
of the samples, which takes place in a very large range of clay and
salt concentrations. Indeed, Laponite samples are found to age
from an initially liquid state up to an arrested state within a time
that, depending both on clay and salt concentrations, can vary
from minutes/hours/months.37 An example of the aging evolution of both dynamic and static structure factors for both a low
and a high concentration sample in salt free water conditions is
shown in Fig. 2. As waiting time tw increases, the intermediate
scattering functions measured by dynamic light scattering (DLS)
decay in a slower manner both for low (a) and high (b) clay
concentrations, until a qualitative change occurs. In the
measured time window a crossover between a complete and an
incomplete decay is observed. This is the signature of a transition
to a non-ergodic, dynamically arrested state. We note that this
transition occurs at waiting times of the order of 103 and 102 h for
the low and high concentration samples respectively. In addition
to the aging dependence of the intermediate scattering functions,
the static structure factors S(Q), measured by small angle X-ray
scattering (SAXS) as the ratio between the intensity and the form
factor of a single platelet, also evolve with increasing tw. In panel
(c) of Fig. 2, S(Q) are reported for two different waiting times
and for both low and high Cw. Right after sample preparation
(tw ¼ 0) S(Q) is flat at low Q for both samples, indicating that the
system is homogeneous. However, when the system becomes
non-ergodic, S(Q) shows dramatic differences between the two
concentrations. With increasing waiting time, the high Cw S(Q)
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Fig. 2 Aging evolution for dynamic and static properties of Laponite
suspensions: measured intensity correlation functions from DLS for low
(a) and high (b) clay concentrations for different waiting times tw, up to
a non-ergodic transition (incomplete decay in the observation time
window); (c) static structure factors S(Q) from SAXS again for a low and
a high clay concentration at two different waiting times: tw ¼ 0 (symbols)
and after they have become non-ergodic (lines). Arrows indicate the
positions of the nearest-neighbor peaks for the two concentrations. Data
taken from Refs. [15,17].
shows only a slight decrease at all Q, while the low Cw S(Q)
shows a spectacular increase at low Q.†
These examples clearly show that aging phenomena need
always to be taken into account for Laponite systems, since most
characteristic features (both static and dynamic) depend on the
time at which observations are made. In the past, neglecting the
waiting time dependence has caused erroneous interpretation of
measurements and has induced controversy in the comparison of
different results.
B. Reproducibility
Since 1992 the pioneering experiments of Thompson and
Butterworth6 reported that considerable care must be taken in
the method of samples preparation, in particular to avoid
Laponite dissolution, which is found to occur in aqueous solutions with pH < 9, as signalled by the magnesium concentration
that increases exponentially with decreasing pH. Subsequently
Mourchid and Levitz38 investigated the long-time behavior
(over one year) for two sets of samples, both corresponding to
Cw ¼ 1% and pH ¼ 10, but one purged with N2 after preparation,
sealed and stored in a glove box under safe atmosphere, while the
second set sealed and stored in ambient atmosphere. The authors
found that the amount of Mg2+ was zero in the first case and
more than 5 104 M in the second one, indicating significant
dissolution in the absence of a safe atmosphere storage of the
samples.
Another crucial point to address is that of filtration of the
samples. Both Bonn12 and Nicolai13 have shown, respectively for
a high concentration sample (Cw ¼ 3.5%, Cs ¼ 104 M at tw ¼
500 s (in the arrested state))12 and for a very low concentration
† Note that this genuine increase should not be confused with that
observed in unfiltered samples, as discussed in the next paragraph.
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sample (Cw ¼ 0.2%, Cs ¼ 104 M at tw ¼ 1 day (in the liquid
state)),13 that there is a dramatic difference between filtered and
unfiltered samples, the latter ones showing a strong upturn at low
Q in the scattered intensity, which is absent in the former ones.
Hence, scattering experiments performed without filtration
cannot be considered reliable because samples also contain
a fraction of very large aggregates, which dominate the scattering
at small wave vectors.
More recently Cummins3 has performed an accurate investigation of samples prepared under different conditions, such as
filtration and pH. The author finds that the aging that takes place
between mixing and filtration is not completely reversed by
filtration. We want to stress this important point that has been
previously neglected. This intervening aging can, indeed,
certainly affect the results and the comparison between different
studies. In practice, samples prepared from a unique stock
Laponite solution but filtered after a long period of time will
undergo different aging evolutions. Several studies have further
shown that properties and phenomenologies of as-prepared
samples (the subject of this review) and rejuvenated ones are
markedly different.39,40 We believe that if one wants to follow the
aging evolution of ‘‘fresh samples’’ (and not of rejuvenated ones)
these should be filtered just after the end of the mixing process
whose duration should remain constant. Cummins concludes
also that the aging behavior of Laponite suspensions is strongly
affected by the sample preparation procedure, making it essentially impossible to compare the results of experiments that
follow different methods of preparation. We partially disagree
with this statement: although a quantitative comparison in terms
of waiting times obtained by experiments based on different
methods of preparation or even on different Laponite batches is
not possible, the basic phenomenologies obtained at different
clay and salt concentrations can be compared and an agreement
between results obtained by different groups is now finally
emerging. This point has been discussed in a recent work by
Jabbari and coworkers,18 who showed that despite the difference
in aging speed, probably due to different types of Laponite or
different procedures of samples preparation, the concentration
dependence of ergodicity-breaking times found by different
groups is very similar throughout a wide range of Laponite
concentration. This is reported in Fig. 3,18 where the results in
salt free water from the studies of three different groups14,15,18 are
Fig. 3 Figure reproduced from18 illustrating the reproducibility of
Laponite results: the ergodicity-breaking time teb from Jabbari et al.18
(squares), Kroon et al.14 (circles) and from Ruzicka et al.15 (triangles,
using tN
w f teb) as a function of Laponite concentration in pure water.
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compared. The consistency is maintained also in the presence of
added salt, as shown in Fig. 7 of Ref. [18], comparing results
from Refs. [18] and [37].
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C. Protocol for sample preparation
From the reasons discussed above we believe that an accurate
and always identical protocol for sample preparation is crucial in
order to obtain reliable and reproducible results. We describe in
a few words the steps necessary for this purpose:
1. Use a glove box to prepare and load samples in closed
vessels in safe atmosphere (e.g. under N2). Keep this condition
throughout the experiment;
2. Disperse Laponite powder in deionized water or at the
desired salt concentration;
3. Stir the system vigorously for a fixed time (e.g. 30 min);
4. Filter the system immediately after (e.g. through a 0.45 mm
filter). This filtration process defines the starting waiting time
(tw ¼ 0).
D.
Phase diagram
The first investigations of Laponite phase diagram were carried
out in 1995 by Mourchid and coworkers,8,9 who reported a rich
phenomenology taking place at different clay concentrations and
ionic strengths, through rheological, osmometric and birefringence measurements. Several regions in the phase diagram
reproduced in Fig. 4 were identified: at low ionic strength an
isotropic liquid (IL) is found for low Laponite concentration,
while an isotropic gel (IG), followed by a nematic gel (NG), are
obtained with increasing Cw. Flocculation (F) is observed at high
ionic strength for all investigated clay concentrations. These
early studies, although neglecting the aging of the samples, which
have led to a misinterpretation of the IL region,15 opened the way
for a large number of subsequent experimental and theoretical
investigations.
We start by briefly discussing the NG region since most of the
debate on Laponite has not been focused on this phase and hence
the available observations are scarce. To our knowledge, apart
from the report of Mourchid et al.,9 there exists another
important piece of evidence of birefringence in Laponite gels by
Fig. 4 State diagram by Mourchid et al., reproduced from.9 F, flocculation; IL, isotropic liquid; IG, isotropic gel; NG, nematic gel. Data
points are obtained by rheological (filled circles), osmometric (open
circles), and birefringence (filled triangles) data.
1272 | Soft Matter, 2011, 7, 1268–1286
Gabriel and co-workers.41,42 These studies report the formation
of ordered nematic phases in a clay concentration range
comparable to that reported in Fig. 4, if one takes into account
the different aspect ratio (1 : 40) of Laponite B samples used in
these experiments. In addition, Shahin and Joshi recently
reported the occurrence of birefringence in rejuvenated Laponite
samples at Cw ¼ 2.8%.43 Hence, it is expected that for Cw T 3.0%
orientational order will interfere with the formation of (isotropic)
non-ergodic states.
Now we turn to discuss the IG region of Fig. 4, which has been
widely studied, especially for Cw 3.0%, in salt free water
systems (Cs ¼ 104 M). The work of Kroon et al.,14 by means of
DLS measurements, reported how samples, initially liquid, were
performing aging up to the final non-ergodic state within
a waiting time that depends on clay concentration, as signaled by
a drastic change in the static part of the scattered intensity. Later
on, Bonn and co-workers44 investigated by DLS a sample at
Cw ¼ 3.5%. The combination of (i) a flat, Q-independent,
scattered intensity typical of a homogeneous system and (ii) the
extremely low density of the samples (translated into packing
fraction, f ¼ 0.014) as compared to the typical value of the glassy
density observed in normal (spherical) colloids (f T 0.50), led the
authors to attribute the nature of this non-ergodic state to
a Wigner glass. Taking into account the electrostatic interactions, the authors calculated (assuming a Debye screening
length of about 30 nm at Cs ¼ 104 M) an effective volume
fraction f z 0.43 much larger than that actually occupied by the
platelets and comparable with that of a glassy system.
In favor of these results, subsequent investigations by Levitz
et al.45 have reported the observation of a re-entrant fluid–solid
transition for lower salt concentrations (105 M # Cs # 104 M
obtained by means of ion exchange resins) and quite low clay
concentrations (0.4% # Cw < 2.0%). They performed ultra small
angle X-ray scattering (USAXS) and SAXS measurements and
find that the electrostatic interactions are responsible for the
formation of a solid disordered state (no Bragg peaks are
observed), in agreement with the Wigner glass interpretation.
Scrutinizing their data, the measured structure factors seem to
indicate the presence of so-called microdomains (clusters), rather
than a fractal-like large aggregate (gel), perhaps a hint to the
presence of Wigner glass of clusters, a state that was recently
found in spherical colloids.29,30 However, these early results may
suffer from the absence of filtration and lack of reproducibility.
Nicolai and co-workers were the first to perform static and
dynamic light scattering measurements for low–very low clay
concentration and upon varying salt concentration in the range
103 # C #102 M.46–48 They find that the position of the sol–gel
line in the state diagram depends on waiting time, an issue that
was not taken into account by Mourchid et al.9 Indeed, they
observe that the gel-like behavior does not immediately manifest
after sample preparation, but develops slowly with increasing
waiting time. The gelation time is found to increase strongly with
decreasing salt and Laponite concentration. Therefore, they state
that Mourchid’s state diagram (Fig. 4) depends on the waiting
time used in the experiment and should not be considered as an
‘‘equilibrium’’ phase diagram. In addition, Nicolai et al. observe
sedimentation of the samples at very low clay or high salt
concentrations both by visual inspection and by light scattering
measurements. While Mourchid and coworkers9 reported
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flocculation (sedimentation) only for Cs > 20 mM independently
of Laponite concentration, Nicolai et al.48 observe sedimentation
down to Cs ¼ 1 mM for very low clay concentration (Cw < 0.3%).
This discrepancy could be due to the fact that in Ref. [9] such low
clay concentrations were not investigated or to the fact that
sedimentation for low Cw takes several months, and hence it
would arise at longer waiting times, not monitored in Ref. [9].
Investigating the evolution of the scattered intensity by varying
both salt and clay concentrations (but for Cw always lower than
2.0%) at a certain waiting time, the authors observe for some
(filtered) samples a Q-dependence that can be described by
a power law, indicating that fractal aggregates are formed. From
these studies, on samples with Cw < 2.0% with and without added
salt, the group of Nicolai concludes that the origin of the
formation of a non-ergodic state is gelation, and they extensively
argue against the interpretation of a Wigner glass claimed by
other authors.48
The controversy about gel or glass nature of arrested states in
Laponite is also tackled by Tanaka and co-workers,49 who
attempt to provide a comprehensive interpretation of the results
reported in the literature, trying to reconcile the claim of the
existence of a Wigner glass with that in favour of a gel state.
Combining observations by several groups9,13,44,45,47 they propose
a new phase diagram, reproduced in Fig. 5. The idea is to use as
a reference the state diagram for uncharged spherical colloids,50
and to adapt it to Laponite taking into account the effect of
charges and the results of experimental observations. The low Cw
region is again considered to be in a liquid/sol phase (no matter
the issue of tw), while the high Cw region shows boundaries to
different non-ergodic states. With increasing salt concentration,
at first the system is speculated to pass from Wigner (in salt-free
water and at ultralow salt concentrations) to attractive glass,
because the estimated Debye length would become smaller than
the platelet diameter. Then, upon further addition of salt,
a transition to a gel state before encountering phase separation, is
suggested, following ideas for spherical colloids (which were
recently however contradicted by the scenario of arrested phase
separation25).
In the same year Ruzicka and coworkers15 reported a DLS
study of the aging dynamics of Laponite samples in a large range
of clay concentrations (Cw ¼ (0.3 3.1)%) in salt free water
conditions. For the first time, the issue of waiting time is properly
taken into account even at low clay concentrations and
Fig. 5 State diagram by Tanaka et al.49
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a surprising new picture emerges. Indeed, at variance with
previous determinations indicating a stable liquid phase for
Cw < 1.8%, the authors find that aging towards arrested states
takes place in the whole examined Cw range. Results from this
study are shown in Fig. 2 (a and b panels), demonstrating that
even for very low Laponite samples (down to Cw ¼ 0.3%), the
samples undergo aging up to a final non-ergodic state. The liquid
region reported by Mourchid9 and Tanaka49 (respectively shown
in Fig. 4 and in Fig. 5) is then replaced by a solid one. The very
long waiting time necessary to obtain the arrested state, of the
order of a few months for low Cw, is the reason why previous
studies have interpreted the liquid phase as the stable one in this
region. Furthermore, the analysis of the scattering data allows
the authors to draw another important conclusion. Following
Abou et al.,51 who had previously reported the presence of two
relaxation times in DLS measurements for high concentration
samples, the intermediate scattering function is well reproduced
by the functional form
b
f ðq; tÞ ¼ Aeðt=s1 Þ þ ð1 AÞeððt=s2 ÞÞ
(1)
where an exponential term describes the fast (microscopic)
relaxation with a characteristic time s1, while a stretched exponential term, corresponding to the slow relaxation, is governed
by s2 and the stretching exponent b. Performing fits in the whole
concentration region, Ruzicka and co-workers are able to identify the presence of two distinct arrested states. From the
evidence of two different master curves of the fit parameters,
a low concentration non-ergodic state (IG1), occurring for
Cw < 2.0%, can be differentiated from a high concentration one
(IG2), which is found for Cw $ 2.0%. The fit parameters are
shown in panels (a) and (b) of Fig. 6 where the mean time sm ^
s2G(1/b)/b15 and the stretching exponent b are shown as a function of waiting time rescaled by tN
w , the waiting time of the
Fig. 6 Waiting time dependence of the sm (a) and b (b) as a function of
4
the scaled variable tw/tN
M and for high concentrations (full
w for Cs ¼ 10
symbols, Cw ¼ (2.2, 2.5, 2.8)%) and low concentrations (open symbols,
Cw ¼ (0.3 O 1.5)%). All the data collapse on two master curves, one for
low and one for high clay concentrations. (c) Concentration dependence
of the divergence time tN
w . (d) Concentration dependence of the B
parameter.
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sol / arrested state transition. The latter is estimated from
fitting each sm for all waiting times for each sample with the
N
expression sm ¼ s0 expBðtw =ðtw tw ÞÞ , together with the parameter B,
which quantifies how the arrested state is reached. These two
parameters are also shown respectively in panel (c) and (d) of
Fig. 6. The existence of a finite tN
w for all investigated samples,
changing by more than two orders of magnitude in the investigated concentration regime, demonstrates the instability at long
waiting times of the liquid state, while the behavior of the B
parameter, roughly constant within the two different regions,
allows the authors to distinguish the presence of two different
mechanisms of arrest. A systematic study was also performed in
the presence of salt by the same authors,37 showing that the
evidence of two distinct arrested states is maintained up to
Cs # 3 103 M. For higher Cs a clear transition between two
states is not observed, due to the immediate arrest of the high Cw
samples. For this reason, the nature of the non-ergodic state
observed at low Cw and Cs > 3 103 M remains unclear,
because the evolution of the B parameter can not be clearly
attributed to IG1 or IG2 (see Fig. 4 in Ref. [37]). Based on these
extensive measurements and analysis, Ruzicka et al. have
proposed a new Laponite phase diagram, which is reported in
Fig. 7. The nature of the two non-ergodic states remains to be
fully assessed, but at those early times Ruzicka et al.15 speculated
that they could be both interpreted as Wigner glasses but with an
important difference. At high Cw the authors follow the interpretation of a Wigner glass made of individual particles, as
proposed by Bonn.44 On the other hand, at low Cw the much
slower evolution toward arrest suggests that an intermediate
aggregation could take place. In this picture, equilibrium clusters
could form due to the competing long-range repulsion and shortrange attraction.28 In this case, the residual repulsion between the
clusters could then act as a mechanism for generating a Wigner
glass of clusters.27
The existence of two different non-ergodic states, despite their
(still unknown) nature, is already sufficient to clarify some of the
controversy reported above among the different experimental
results. In particular, the debate between gel and Wigner glass
can be partially solved by observing that all of Nicolai’s
Fig. 7 State diagram by Ruzicka et al.52 The solid-dashed line separates
the two different arrested states IG1 and IG2 respectively for low (open
circles) and high (full circles) clay concentrations as measured in
Ref. [15,37]. For high Cs no clear transition is observed and the nature of
the non-ergodic state is unclear (squares).
1274 | Soft Matter, 2011, 7, 1268–1286
measurements are in the IG1 region, while the Wigner glass
identified by Bonn is located in the IG2 region. The next question
to be addressed is whether it is possible to really discriminate
between the two states. To this end, Ruzicka and co-workers17
carried out a systematic investigation by SAXS of the static
structure factor evolution with waiting time in both concentration regions, following samples aging up to several months.
Although previous SAXS measurements of Laponite could be
found in the literature,8,53 they were performed either on unfiltered samples54 or neglecting the waiting time dependence,
considering indeed the samples as equilibrium states.55 The new
measurements by Ruzicka et al.17 clearly show a different
evolution, also for the structural properties of the system, in the
two distinct non-ergodic regions. This is partially shown in
(c) panel of Fig. 2. With increasing tw, S(Q) for the high
concentration samples show only a slight decrease at all in Q,
with no evidence of formation of larger aggregates. Indeed, the
main peak is found at Q 0.15 nm1 (left arrow in Fig. 2),
corresponding to a length scale of z40 nm, well beyond the
diameter of the platelet, pointing to a disconnected, homogeneous (glass) structure. On the other hand, the low concentration
S(Q) displays a dramatic increase at low Q with tw, signalling the
development of an inhomogeneous structure in the system,
compatible with a gel state. Further evidence of the attractive
bonds that are being established in this regime is the appearance
of a contact peak at Q T 0.4 nm1 (right arrow in Fig. 2), corresponding to a length scale of (15 nm, a value compatible with
T-bonded platelets. Although this is by far not a conclusive
proof, we believe this is a strong hint in favour of the establishment of a ‘house-of-cards’ network in Laponite systems at low
clay concentrations, a matter which has long been predicted11
and still a matter of debate.
Almost at the same time of the SAXS measurements performed by Ruzicka,17 Jabbari and coworkers16 have also
provided evidence of two distinct non-ergodic states in salt-free
water Laponite suspensions. Investigating samples in the range
Cw ¼ (0.1 O 3.6)%, they perform ensemble-averaged DLS
measurements even in the non-ergodic regime and find evidence
that the evolution with waiting time of the nonergodicity
parameter, i.e. the non-decaying long-time plateau of the intermediate scattering function, again falls on two distinct master
curves. These observations are accompanied by a measurement
of the short-time diffusion coefficient in the two regions, which is
found to be rather constant—indicating rattling in the cage as in
a glass state—for high Cw, while it decreases with tw for low Cw,
which has been interpreted as due to the establishment of a gel
network. Furthermore, with increasing waiting time the scattered
intensity at low Q shows an increase at low Cw versus a rather
constant value for high Cw. The combination of these results
allows Jabbari and co-workers to identify the two states also as
gels at low Cw and repulsive glasses at high Cw. Furthermore,
these authors find that for an intermediate clay concentration
range, Cw ¼ (1.1 O 2.4)%, located in the region of transition
between the two states, the arrest transition can be either of gel or
glass nature, with ‘‘no way of telling a priori’’ how the sample is
going to evolve. These samples are named ‘‘hesitating’’, since they
are found to hesitate between the two options for a long time and
an initial evolution in one of the two directions can lead to a final
state that is the other one. Jabbari et al. have also extended their
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Fig. 8 State diagram by Jabbari et al., reproduced from Ref. [18].
analysis to different salt concentrations18 and proposed a new
phase diagram reproduced in Fig. 8. Here gels (A) and (repulsive)
glasses (B) are reported in salt free water respectively at low and
high concentrations, while an attractive glass (C), with intermediate behavior between a gel and a repulsive glass, is identified at
high salt concentrations and low/intermediate clay concentrations. The hesitating samples are found only in salt free water
conditions. Despite some differences in the intermediate (hesitating) region, the results of Jabbari and co-workers18 are in good
agreement with those of Ruzicka and co-workers concerning the
existence of distinct gel and glass states at low and high clay
concentrations respectively. This can be seen from the comparison between Fig. 7 and 8, giving a great boost in favor of the
reproducibility of results.
We now turn to the results of two very recent papers by
Ruzicka et al.19,23 which have aimed to clarify the behaviour of
the system in the absence of added salt in the full clay concentration window, prior to the nematic transition. Both works have
presented novel experimental evidence, and the interpretation of
the results has been supported by numerical simulations which
will be discussed in Section IV. In chronological order, the first
paper deals with assessing the nature of the glassy state arising at
high clay concentration. By performing a simple dilution
experiment, the authors show that the high Cw non-ergodic state
is melted back to a liquid state, a scenario that is compatible with
that of a Wigner glass, while for low Cw the arrested state does
not melt due to the presence of long-lived bonds which are not
affected by the addition of water. Hence, attractive and repulsive
interactions are found to be respectively dominant in the
formation and stability of the arrested structure for low- and
high-concentration samples. Furthermore, the comparison
between the S(Q) measured with SAXS and both theoretical and
numerical calculations based on screened Coulomb interactions
only, discussed in the following section and reported in Fig. 15,
shows that the non-ergodic state observed at high concentrations
(Cw $ 2.0%) in the absence of salt is, indeed, a Wigner glass.23
In a subsequent work the attention is focused only on low clay
concentrations (Cs < 2.0%) extending the observation time
window to many years and discovering an entirely new
phenomenology.19 From what was discussed above, in this range
of concentrations, samples were observed to progressively age up
to a gel state, as measured by DLS.15,16 The waiting time necessary to undergo arrest was of the order of a thousand hours.
Now, waiting even longer, it is realized that this arrest is only
apparent on the second timescale while subsequent restructuring
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still occurs on the year timescale. It is observed that all samples
below a well-defined concentration threshold (Cw # 1.0%)
undergo an extremely slow phase separation process into clayrich and clay-poor phases. The phase separation terminates
abruptly at Cw z 1.0%, above which the system remains indefinitely in a gel state. This behaviour is illustrated in Fig. 9a where
photographs of the samples, taken three years after their preparation, show the formation of two distinct phases—an upper
transparent fluid and a lower opaque gel—for Cw # 1.0% and
a homogeneous (transparent) arrested state above this Cw value.
SAXS measurements have also been performed to monitor the
evolution of two samples at Cw ¼ 0.8% and Cw ¼ 1.2%,
respectively inside and outside the phase separation region, for
more than one year. Fig. 9b,c show the behavior of S(Q) with
waiting time: both samples display an initial increase in the low Q
signal with increasing waiting time, indicating the onset of
aggregation which is relevant, as expected, for both studied
concentrations. However with the proceeding of the aging
dynamics, the two samples behave very differently. For the
sample inside phase separation (Fig. 9b), the low Q intensity of
S(Q) continues to increase indefinitely (up to the longest
measured waiting time), signaling the ongoing phase separation
process (as revealed also by the turbidity of the samples). On the
other hand, for the sample outside phase separation (Fig. 9c) the
low Q intensity clearly saturates to a finite value after a long time
(of the order of one year), indicating that the system has reached
its long-time equilibrium structure, i.e. a stable network. We
stress that gel formation according to DLS is obtained after a few
thousand hours, while some restructuring still takes place up to
the long-time saturation, after which the gel ceases to age. Taking
these experimental features together with the anisotropic interactions that govern Laponite behaviour (see also in the following
section IV), we can interpret this novel behavior in the context of
patchy particles. In this view, Laponite offers the first experimental realization of empty liquids (sparse networks of bonded
Fig. 9 (a) Photographs of samples for clay concentration Cw # 1.2% at
very long waiting times (tw z 30000 h). The samples with Cw # 1.0%
show evidence of a phase separation, while the Cw ¼ 1.2% gel sample
remains homogeneous at all times. (b) Evolution of S(Q) with waiting
time for Cw ¼ 0.8%, (c) same for Cw ¼ 1.2%. Curves, from bottom to top,
correspond to tw ¼ 500, 900, 1600, 2700, 3400, 4700, 6000, 8700, 11000 h.
Soft Matter, 2011, 7, 1268–1286 | 1275
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particles) and equilibrium gels (arrested empty liquids) in the
intermediate region 1.0% # Cw < 2.0%.
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E. Discussion and summarizing phase diagram from
experimental results
In the above paragraphs we have provided evidence that the
apparently contradictory results reported in the past about
Laponite phase and state behavior were mostly due to inaccurate
comparison of results. In particular, it is now clear that if
Laponite suspensions are investigated taking into account the
aging phenomenon and if a rigid protocol of samples preparation
is followed, reliable and reproducible results can be obtained, as
shown in Fig. 3. We believe that the congruity of results from
several groups should remove any additional scepticism about
Laponite behavior. Although it is a complex system, its physical
properties are extremely interesting and not affected by spurious
and/or uncontrolled effects. Once salt concentration, clay
concentration and waiting time of the measurements are known,
one can determine the position of the sample in the state diagram
and the expected phenomenology. Obviously, the exact position
of the different transitions can slightly depend on type (XLG,
RD, B), batch of Laponite and samples preparation, but the
phenomenology is robust.
To further stress this point, we want to provide here a unifying
state diagram, based on the attentive comparison between
studies performed by different groups. This state diagram
depends on three control parameters: clay concentration, salt
concentration and waiting time. We consider the latter to be
sufficiently long in order to have no additional (macroscopic)
changes in the samples state behavior, and therefore we report in
Fig. 10 the long-time Laponite phase diagram in the (Cw, Cs)
plane.
In salt free water conditions (Cs ¼ 104 M) we can distinguish
four regions, depending on clay concentration. (i) For Cw # 1.0%
an extremely slow phase separation between a clay-poor and
a clay-rich phase takes place.19 (ii) For 1.0% < Cw < 2.0% a gel
state, originating from the attractive interactions between
Fig. 10 New phase diagram of Laponite suspensions proposed in this
review collecting data from different authors obtained with different
techniques for large enough waiting time. Note that SIM refers to
numerical simulations.
1276 | Soft Matter, 2011, 7, 1268–1286
platelets17,18,48 is observed. The combined evidence of phase
separation, SAXS measurements and numerical simulations has
allowed Ruzicka and co-workers19 to identify the anisotropy of
the attractive interactions as responsible for this low-concentration behavior and to interpret these gels as true equilibrium
gels. (iii) A glassy state is found for Cw $ 2.0%. The latter is
dominated by repulsive interactions, which led to its interpretation in terms of a Wigner glass,23 although Jabbari et al.18 simply
describe it as a repulsive glass. (iv) Finally for Cw T 3.0%
Mourchid et al.9 observed the formation of a nematic phase by
observing birefringence (BF) between cross polarizers.
Upon increasing salt concentration, evidence of two different
non-ergodic states, at low and high clay concentrations, persist
up to Cs ¼ 2 mM. For Cs $ 3 mM, the high clay concentration
has not been extensively investigated due to the immediate arrest
of the samples. For lower clay concentrations in this region,
Jabbari et al.18 find evidence of an attractive glass, because the
light scattering data show mixed features of both gels and glasses,
while the older data of Nicolai et al.46 were attributed to gels for
all studied salt concentrations. The results of Ruzicka and
co-workers37 in this region are not conclusive with respect to
any of these two options. For salt concentrations higher than
Cs ¼ 20 mM phase separation in the form of flocculation or
sedimentation of large aggregates is found both by Mourchid
et al.9 by means of visual inspection (VI) and by Mongondry and
co-workers48 by VI and light scattering (LS) when Laponite is
dispersed directly in salt solutions. For very low clay concentrations (Cw < 0.3%) Mongondry et al.48 find the evidence of
a phase separation region, which develops with waiting time, in
a wide range of salt concentrations down to Cs ¼ 1 mM.
We believe that this state diagram summarizes and combines
all the results obtained with an attentive sample preparation and
considering the waiting time evolution. It is therefore finally clear
that both gel and glassy arrested states are found by simply
changing clay concentration. The origin of these two arrested
states can be attributed to the dominant attractive and repulsive
interactions, which are both present in Laponite suspensions, in
the two concentration regimes. Attraction dominates at low and
intermediate concentrations, finally resulting in a phase separation at long waiting times for very dilute samples. The striking
observation of a low-concentration gel network followed by
a high-concentration disconnected Wigner glass, opposite to
what is commonly found in spherical colloidal suspensions,29,30
can be attributed to the separate time scales controlling the
interactions and the two arrest processes. While repulsion is felt
almost immediately after samples are prepared, attraction, due to
its anisotropic nature and to the presence of an effective repulsive
barrier, develops on a much longer time scale. The relative
importance of attractive and repulsive interactions is also, in our
opinion, crucial to reconcile the ‘house-of-cards’ view with that
of the existence of birefringent states. These states are clearly
incompatible with an underlying HOC network, but this does not
exclude that HOC becomes relevant in other regions of the phase
diagram. Hence, we believe that while attraction-driven HOC
applies to the low concentration gels, a repulsive picture is able to
describe the formation of Wigner glasses and nematic phases at
higher concentrations.
There are still several points of the state diagram that need to
be better investigated. The first one is to understand the fate of
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the phase separation with increasing salt concentration. The
recent results in the absence of salt, obtained over a very long
observation time window, should be extended to higher Cs, to
understand how (and if) this phase separation joins the flocculation reported at very high salt content. Indeed, this portion of
the phase diagram (low Cw, intermediate Cs) has not been studied
for long enough waiting times. Similarly, also the fate of the gel
state is not clear with increasing salt concentration.37,46 It could
merge with an attractive glass18 or at some critical salt concentration, the equilibrium gel may be destroyed in favour of a gel
resulting from an arrested spinodal decomposition, similarly to
what happens in spherical colloids.25 Secondly, the transition
between the two arrested regions, of gel and glassy nature, needs
to be further investigated to understand when the interaction
goes from attraction to repulsion-dominated, also with respect to
the observation of the so-called hesitating samples.16 Thirdly, the
transition from Wigner glass towards another arrested state
(attractive glass?) should be addressed upon increasing salt
concentration, which induces a progressive screening of repulsive
interactions. Finally the behavior at ultralow salt concentrations
Cs < 104 M should also be further investigated, following the
work of Levitz et al.,45 to verify whether the Wigner glass could
be found down to very low concentrations, for example in the
form of a Wigner glass of clusters.
IV. Simulations and theory
The theoretical description of clay suspensions is very challenging due to the double source of anisotropy that is present in
the system. Indeed both the disk shape of the platelets and the
directional (face-rim) interactions place Laponite, with other
clays, among the prototypes of anisotropic particles, those that
are considered to be the future building blocks for novel selfassembled materials.21 In addition, Laponite also displays, as
discussed extensively in the experimental section, a highly nontrivial aging dynamics, which is found at all clay concentrations
with waiting time dependence ranging from hours/days/months.
The complexity of the problem suggests that, for its theoretical
description, it is preferable to start tackling one aspect at a time,
and once a sufficient understanding of this particular aspect is
gained, the essential ingredients for its description should be
identified and later on combined into a more comprehensive
model. For this reason, at present, a unifying approach, which is
able to describe simultaneously the attractive-dominated regime
at low Cw and the repulsive-dominated one at high Cw, as well as
the crossover region between them, is not available. However,
some progress was recently made for describing either the
Wigner glass state occurring for Cw $ 2.0%23 or the phase
separation and gelation occurring at low densities.19
In this section, we start by revising the microscopic models
which have been proposed over the recent years, highlighting the
points where they succeed and fail, especially in the light of more
recent experimental evidence with respect to what was known at
the time. Next, we turn to examine simple effective potentials,
investigated via simulations and theoretical methods such as
liquid state integral equations. We will conclude the section by
making a summary of the ingredients that appear to be necessary
for a model to describe the complex behavior of Laponite
suspensions.
This journal is ª The Royal Society of Chemistry 2011
A.
Models
Models for clay suspensions started to appear soon after the first
experimental results for Laponite were available.8 The main aim
of these studies was to explain the establishment of a gel network
at very low concentrations, i.e. well below a first-order nematic
transition for neutral disks would occur.56,57 The idea that was
put forward to explain this phenomenon was the formation of
a ‘house-of-cards’ structure, kept together by so-called T (edgeto-face) bonds.11 To realize this condition, Djikstra, Hansen and
Madden58 proposed a pioneering model of hard disks carrying
a fixed point quadrupole at their center, thus incorporating the
peculiar edge-to-face attraction of Laponite platelets. The disks
of diameter s are infinitely thin and, to avoid electrostatic
collapse, an additional infinite repulsive barrier is used for centerto-center distances r < s/2, which does not affect the formation
and stability of T-bonds, which are the energetically preferred
local configurations. In the absence of electrostatic interactions,
disks undergo a transition from isotropic to nematic fluid at
rs3 x 4, with r ¼ N/V.56 By using Monte Carlo (MC) simulations, Djikstra and coworkers have established that the addition
of a quadrupolar interaction allows the occurrence of a gel-like
transition at a density lower than the one where a nematic
transition is observed. The authors explain their results by
making an analogy with wormlike micelles and suggest that the
transition is not only a kinetic one, but a thermodynamic phase
separation between a low density cluster phase and a high density
gel, occurring at a critical value of the quadrupolar moment. This
transition is completely analogous to a standard liquid–gas phase
separation and, in essence, a prototype of low-density phase
separation observed in patchy colloids.20 In a subsequent work59
they also perform Gibbs Ensemble MC (GEMC) simulations to
provide an estimate of the location of the critical point and of the
binodal line of this transition (still named sol–gel transition),
which is reported in Fig. 11(a). The role of 3- and 4-coordinated
Fig. 11 (a) Phase diagram in the density, quadrupolar strength plane
reproduced from Dijkstra et al.,59 together with two simulation snaphots.
A sol–gel transition, interpreted as an analogue of a thermodynamic
phase separation, is observed. At low concentrations (b), disks form
finite-size aggregates, while at larger ones (c) they form a ‘house-of-cards’
network. In both cases disks are connected through T-bonds (some being
highlighted with red lines), which are branched by higher-order bonded
configurations (triangles and squares, see panel b).
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rings of disks (configurations in Fig. 11(b)) is highlighted as the
branching points of the network, which is structured by T-bonds
(snapshot in Fig. 11(c)). The critical point is located at a density
rs3 x 0.75 corresponding to approximately 6% in Laponite
weight concentration, while the high density gel would be stable
at roughly 17%, clearly an overestimation with respect to the
experimental values. These results were tentatively compared to
the available experimental data of Mourchid and co-workers,8
trying to map the so-called isotropic gel (IG) observed in
experiments with the phase transition originated by quadrupolar
interactions. However, we have seen in the experimental section
that these early works were wrong both in terms of the location
of the sol/gel transition and of the nature of the arrested state. IG
was later attributed to a Wigner glass, while gels were observed at
lower clay concentrations (Cw < 2.0%). However, in the light of
the recent experimental results of a low-density phase separation
by Ruzicka and coworkers,19 it is now clear that this model,
which belongs to the class of anisotropic potentials, was a real
precursor of its time. Though at the time it could not be realized
because of the lack of coherent experimental results, the model,
despite being very simple, can really capture, at least qualitatively, the essential physics at the basis of the low-density
behaviour of Laponite. On the other hand, the crude treatment
of electrostatic interactions led to the idea, that this model could
not really be relevant for describing Laponite and therefore more
microscopic models were adopted.
To overcome the simplified picture of the quadrupolar disk
model, which is inadequate for describing electrostatic interactions both at very short range (because multipolar expansion
breaks down) and at long range (because of screening effects),
a more realistic model of Laponite was proposed by Kutter,
Hansen and co-workers some years later.60 The Laponite platelet
was represented in terms of a rigid hexagonal disk with discrete
charge sites. The authors focus on two different versions of the
model (respectively named A and B) which should correspond to
different experimental conditions: the first one represents platelets without rim charges so that the interaction potential is purely
repulsive; the second one also takes into account attraction,
originating from the presence of the positive rim charges and
located on the borders of the platelet. In model A, a total
negative charge of 700 e is distributed regularly within the sites
over the total disk surface, taking into account the case where
Laponite dispersed in water releases all its sodium ions. On the
other hand, in model B a 10% positive fraction of the total charge
is added on the outermost shell of sites of the disk, while keeping
the total charge fixed to 700 e. The dependence on a different
number of sites n ¼ 19, 37, 61 was investigated. This multisite
approach is based on linearized Poisson–Boltzmann (PB) theory,
so that the electrostatic potentials are exponentially screened by
the presence of co- and counterions. Therefore the resulting
interaction energy between two platelets is the sum of site–site
screened Coulomb interactions of the Yukawa form,
Va;Y b ¼
n X
n
X
qia qjb
exp riajb =lD
eriajb
i¼1 j¼1
(2)
where riajb ¼ |ria rjb| is the site-to-site distance between sites i,j
respectively located on the two platelets (a s b), 3 ¼ 78 is the
water dielectric constant at room temperature, qia is the electric
1278 | Soft Matter, 2011, 7, 1268–1286
charge allocated to each site and lD is the Debye length. To avoid
that the attraction between two sites of opposite sign leads to an
instability of the system, an ad-hoc short-range repulsion (similar
to the one used by Djikstra) is introduced, which mimics the
excluded-volume interactions. This is modeled as a soft
repulsion,
Va;S b ¼
n X
n
X
C
:
6
r
i¼1 j¼1 iajb
(3)
where C is an arbitrary constant, which is adjusted in such a way
that the total resulting interaction VTOT ¼ VY + VS between
a positive and a negative charge at contact is around kBT.
The choice of a sufficiently large number of sites provides
a good description of the Laponite form factor at relevant lengthscales (QR < 20, where R ¼ s/2 ¼ 12.5). Results from MD
simulations of 32 and 108 platelets are reported in particular to
address the structural and orientational properties of the system
for different weight concentrations and screening lengths. In the
case of purely repulsive platelets (no rim charges, model A)
different phases are identified, ranging from dilute gas to more
concentrated liquid up to fcc crystal and glassy states. Interestingly, the observation of these states occurs with increasing
screening length, rather than with increasing concentration. This
points to the fact that for experimentally relevant weight
concentrations (Cw < 5%) some considerable repulsion—quantified by a Debye length lD of at least a few nanometres—is
necessary in order to observe structuring of the system. The
addition of rim charges, with the parameters studied by the
authors, induces a strong enhancement of platelet bonding,
only when the electrostatic interactions are sufficiently screened
(lD 1nm). In this case, a clear organization of platelets into
T-bonded clusters (and network) is observed. The work of Kutter
and co-workers60 aimed to provide an exploratory overview of
the model behavior within a certain range of parameters that
would be close to the experimental working conditions. A
subsequent work by Odriozola and co-workers61 has revisited the
two models A and B for two different values of the screening
lengths by using Brownian dynamics (BD) simulations. In this
way, a step forward in the treatment of the solvent is made,
although true hydrodynamic interactions, which may be relevant
in Laponite suspensions, are still neglected. The total number of
discrete sites—represented as hard spheres of diameter 1 nm—on
each platelet is increased to n ¼ 469, so that the true aspect ratio
of the platelet (1 : 25) can be realized. In order to compare
directly with previous results,60 only 61 out of the total number of
sites are charged (with the same charge as in Ref. [60]), while the
others are kept neutral. The system has been studied at several
densities at fixed T ¼ 300 K, with two values of the Debye length
respectively equal to 1 and 3 nm. Similarly to Ref. [60], enhanced
structuring is observed for larger Debye length for Model A,
while evidence for T-bonds and house-of-cards arrangement is
found for Model B. However, a surprising new feature is
reported by Odriozola et al.:61 a considerable number of platelet
pairs is found in the so-called parallel, partially overlapped
(PPO) configuration (see (b) panel in Fig. 13), at a distance
between the centers-of-mass of the disks of approximately 21 nm.
These configurations, which become more important at higher
concentrations in particular toward the possible formation of
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Fig. 12 Waiting time dependence of (a) collective density auto-correlation functions Fc(Q;t,tw) for Q ¼ 0.4 nm1 and (b) corresponding relaxation times scoll, estimated as Fc(Q;scoll,tw) ¼ 1/e, as well as those
extracted from the fits of the self intermediate scattering functions (slow
sc ^ s1, fast sf ^ s2, according to eqn (1)) and related mean relaxation
time sm ¼ scG(1/b)/b, and stretching exponent b (inset) for Q ¼ 0.7 nm1,
reproduced from Ref. [63].
ordered phases,62 are stabilized by the presence of the additional
uncharged sites present in the model,61 which were absent in the
previous version of the model.60 With increasing volume fraction
for lD ¼ 1 nm, a minimum in the energy behavior is observed at
a certain density rmin 2.24 1023/m3. At lower densities,
T-bonds are more abundant in the system with only a minor
contribution of PPO configurations, while at larger densities the
parallel configuration (stacks) starts to dominate due to packing
effects becoming relevant. However, the value of rmin would
correspond to an experimental weight concentration of z28%,
well beyond the experimentally studied range. The larger value of
Debye length studied, lD ¼ 3 nm, does not present evidence of
a minimum in the energy, which is found to be always repulsive
and monotonically increasing with volume fraction. In this case,
no PPO configurations are found, while some evidence of T-like
structure is found at higher densities but with platelets at
a distance larger than the contact distance, so that they can not
be really considered as T-bonds. A further increase of density
(roughly half of that found with lD ¼ 1 nm) again produces the
emergence of parallel stacks. Finally the study of translational
diffusion coefficient provides evidence of a certain slowing down
of the dynamics with increasing volume fraction for both models,
however a detailed study of an eventual dynamic arrest transition
and aging dynamics is not accomplished. Indeed, it seems
possible that, at the large densities studied here, the system could
be found in out-of-equilibrium conditions.
To tackle precisely this issue, a subsequent numerical work by
Mossa and co-workers63 focused only on Model A and carefully
investigated the aging dynamics of the system along a single
isochore, corresponding to z9% in weight concentration, still
larger than the relevant experimental range, but significantly
lower than the ones studied by Odriozola and co-workers.
Interestingly, the model does not show evidence of a nematic
phase, despite this being expected from the experiments at this
volume fraction. The screening length is fixed to lD ¼ 3 nm,
a value coinciding with the more long-ranged case studied in
Ref. [61]. The idea behind this study was to understand whether
This journal is ª The Royal Society of Chemistry 2011
this model would show the clear signatures of a glass transition,
which due to the solely electrostatic repulsive interactions in play,
could be interpreted as a Wigner glass. The system was thermalized at high temperature and then instantaneously quenched
to room temperature, in order to monitor the relaxation of the
system with increasing waiting time. Following the quench, the
system never reaches equilibrium during the whole duration of
the simulation runs, with the energy slowly, but monotonically
decreasing with time. However, despite the strong variation of
the energy and dynamical properties with aging time, the structure of the system appears to change only slightly. This is in
agreement with the experimental findings of Ruzicka for 2.0% <
Cw # 3.0%.23 A significant slowing down of the structural
relaxation time, for both the self and collective density correlation functions, is observed with increasing waiting time, suggesting that the system is strongly out-of-equilibrium and has
become trapped in a non-ergodic state. In Fig. 12(a) the collective density auto-correlation functions Fc(Q;t,tw) are reported as
a function of waiting time for a fixed value of Q. The curves are in
qualitative agreement with the experimental data of Ruzicka,15
reported in Fig. 2. A fit to the data for the self autocorrelation
functions was performed, using the same functional form as for
the experimental measurements (eqn (1)). The increase of the
relaxation times (Fig. 12(b)) exceeds two orders of magnitude,
while the stretching exponent b is found to decrease with tw (inset
of Fig. 12(b)), again in good agreement with experiments. Given
the purely repulsive nature of the electrostatic interactions
involved, this non-ergodic state can be convincingly identified as
a Wigner glass.
A recent work of Jonsson et al.36 has addressed the fundamental question of what are the effective interactions between
two platelets described with Model B, by performing Monte
Carlo simulations of two platelets immersed in an effective
solvent. The two platelets are schematized in Fig. 13 to show
respectively a T-bond (a) and a PPO (b) configuration. Again due
to the high computational cost, the effect of salt is reduced to
a Yukawa-screening of the Coulomb interactions. A more
Fig. 13 Schematic model of two platelets made of charged discrete sites
that are positive on the edge (blue/dark) and negative on the face
(red/light) in typical T-bond (a) and PPO (b) configurations, and effective
potential between them at various salt concentrations (c) and (d),
reproduced from Ref. [36]. The global minimum is always found for the
PPO configurations, while the T-bonded ones are only local minima.
However, in the absence of salt, this model predicts no net attractive
contribution from the T-bonded configurations.
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systematic study of the Debye length has been carried out,
monitoring the system from no added salt (Cs ¼ 104 M, lD 30
nm) to high salt concentration (Cs ¼ 101 M, lD 1 nm). This
work largely confirms the results of the bulk simulations of
Odriozola,61 but with some important differences. First, they
claim that with no added salt (Cs < 5 mM), no net attraction
develops, the effective potential remaining purely repulsive. A
slight addition of salt triggers attraction, but the global minimum
is found to correspond to PPO configurations, rather than to
T-bonded ones. This trend is maintained at all studied salt
concentrations. Interestingly, the PPO preferred configuration is
found for Cs > 10 mM, while the local minimum
corresponding to the T-bond emerges only at higher salt
concentrations Cs > 40 mM. These results are shown in Fig. 13
(c) and (d). However, at finite densities, the T-shaped configuration will better optimize the available space so that it might
become more favorable due to packing. The role of van der
Waals attraction is also estimated in this concentration range,
being found to produce only minor quantitative changes in the
effective potential. For much larger salt concentrations
(Cs T 100 mM), this will eventually become dominant and favor
the stacking configurations.
Finally, we describe the patchy model of Laponite recently
introduced by Ruzicka et al.19 to describe the low-density
behavior of the experimental system in the context of low-valence
systems. While up to this point we have seen that models (with
the exception of the Djikstra58 quadrupolar model) have either
considered only the electrostatic repulsion or have added on top
of this the face-rim attraction, this model approaches the
problem from the opposite point of view. The overall repulsive
electrostatic interaction is neglected, while the electrostatic
attraction between opposite face-rim charges is condensed into
short-ranged attractive sites located on the particle surface.
Hence, the model can be classified as a primitive model, in the
light of previous work on charged molecules such as water.64
Following Kutter et al.,60 a Laponite platelet is still represented
as hard rigid disk but without any charged sites. In addition to
excluded-volume interactions, each disk is decorated with three
sites on the rim and one at the center of each face (five sites in
total) as shown in Fig. 14a, similarly to previous work for lowvalence colloidal spheres.20,32 Since only face-rim bonds can
form, a square-well attraction is active only between face and rim
sites (rim–rim and face–face sites are non-interacting). The shortrange nature of this attraction ensures that each site is involved at
most in one T-bond.65 The idea behind this model is that the
anisotropy of the platelet shape, combined with the directional
face-rim attraction, favors the formation of low-density bonded
networks. Indeed, Laponite forms macroscopic networks at
extremely low densities, so that these must be practically empty,
suggesting that only a few bond per particle are formed and
hence particles experience an effectively low valence.
The phase diagram of this model has been studied by GEMC
simulations and is reported in Fig. 14b. The gas–liquid coexistence region is confined in a narrow window of density and
temperature, consistently with previously studied patchyspheres models.20 Also the percolation line is reported, which
divides the state points where the system is made of finite
(transient) clusters from those where it forms a spanning, but
still transient cluster.
1280 | Soft Matter, 2011, 7, 1268–1286
Fig. 14 Phase diagram of the patchy model of Laponite of Ref. [19]. (a)
A cartoon of a Laponite platelet: a rigid disc composed of 19 sites (red
spheres) with 5 attractive patches (blue small spheres), three located on
the rim and one at the center of each face. (b) Numerical phase diagram:
binodal and percolation lines in the r* T* plane, where r* is the number
density scaled by the close-packing density and T* is the thermal energy
scaled by the bond strength. (c,d) Final gel configurations after a quench
at low T respectively inside (r* x 0.08, c) and outside (r* x 0.16, d) the
phase separation region. All platelets are connected into a single cluster
(gel), which is clearly inhomogeneous (homogeneous) inside (outside) the
binodal region.
Having established the phase diagram, the out-of-equilibrium
dynamics of the system were also studied by performing low-T
quenches inside and outside the phase separation region and
monitoring the evolution of the system with waiting time, to
mimic the experimental protocol. By means of MC simulations,
the static structure factors at different tw were observed to obey
the same behavior previously described for the experimental
S(Q). In particular, two different scenarios occur at long times
after preparation: while for state points inside the phase separation region the low Q increase of S(Q) continues indefinitely,
for points outside this region the growth stops after a finite
waiting time, showing no further evolution. Snapshots of the
final configurations of the simulations are shown in Fig. 14c,d
after a quench respectively inside and outside the phase coexistence region. Independently from the density of the quench, the
final configuration is always characterized by a single spanning
cluster incorporating all particles. The structure of such a cluster
is highly inhomogeneous for quenches inside the coexistence
region (Fig. 14c) and homogeneous for quenches in the empty
liquid region (Fig. 14d). Since at these low T the bond lifetime
becomes much longer than the simulation time, the bonded
network is persistent, i.e. the system forms a gel.
In the context of patchy particles, these results can be easily
interpreted. At low densities the system prefers to phase separate
in order to reach a denser phase where most of the possible
bonds, according to the fixed valence, are satisfied. However,
above a certain finite density, the system does not need to phase
separate in order to reach its ground state in energy. Indeed, the
low valence favours the establishment of a fully-connected
network. In this optimal network-forming region, the system at
sufficiently low T will not further change its structure, because
the number of bonds cannot increase any further.66 For this
reason, this state is called an equilibrium gel. The
particular choice of the valence determines only the location of
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the gas–liquid phase separation, but does not affect the topology
of the phase diagram. In the studied Laponite model, the coexisting liquid density is still much larger than the experimental
values (by a factor of 8), but this could in principle be improved
by adjusting the valence and the aspect ratio (the current 19 sites
hexagon gives 1 : 5 rather than 1 : 25), as well by re-introducing
at some level the neglected electrostatic repulsion.
Despite these problems, the study of this simple model,
combined with the experimental results, has allowed the identification of the crucial anisotropy of Laponite interactions, which
is responsible for its low-density peculiar behavior. This was
made possible thanks to a proper study of the out-of-equilibrium
dynamics of the system, an issue often neglected in previous
studies (with the exception of the work of Mossa et al.63 for the
Wigner glass regime). Indeed, to really compare with experiments, simulations need to focus on the limit where attraction
strength is very large9 by quenching at extremely low temperatures. Under these conditions the bond lifetime becomes very
large (and may exceed the simulation time window), hence
a careful study of the waiting time dependence of the system
properties becomes crucial because the system undergoes a slow
aging dynamics both in simulations and in experiments.
B. Effective potentials
While microscopic models have the advantage of taking into
account the full anisotropic nature of the platelet shape and
interactions, it is sometimes difficult to understand the main
ingredients that should be included in these models. Indeed,
starting from the chemical formula for the platelets and coarsegraining the irrelevant degrees of freedom (e.g. water, co- and
counter-ions), one should be left with the right parameters
modeling the behavior of Laponite in bulk conditions. For
example, instead of a negative charge z 700 e for each
Laponite platelet, a much smaller charge should be used in order
to take into account counter-ion condensation, which limits the
release of sodium ions in solution as indicated by recent
conductivity measurements.18 The (average) residual charge onto
a single platelet may then vary depending of salt, pH and
concentration. Accordingly, also the value of the Debye length
will vary. Therefore, together with the microscopic models, it is
important that the theoretical investigation of Laponite also
aims to establish the effective parameters governing the electrostatic interactions in solution. Hence, studies of effective potentials and their direct comparison with experiments can be very
crucial to ensure that more realistic parameters are considered
when going back to microscopic models. Moreover, the use of
integral equation theories can be of great help, allowing us to
obtain useful information on the structure in a rather straightforward manner. For these reasons, some investigations of this
kind have proceeded in parallel with the simulations of more
microscopic models discussed in the previous section.
To start with, Trizac and coworkers67–69 have applied standard
DLVO theory to disk-shaped particles, incorporating also the
non-linear effects of counter-ion condensation (charge renormalization), in order to derive an effective anisotropic pair
potential between platelets of arbitrary relative orientations. In
particular, by solving linearized PB theory for two discs at large
distances (r/lD [ 1), they have shown that an effective Yukawa
This journal is ª The Royal Society of Chemistry 2011
potential still describes (to leading order) the effective interactions, with a prefactor which depends both on the renormalized
charge and on the orientations of the two platelets.69 At fixed
centre-to-centre distance, the repulsive energy is maximized for
co-planar discs (maximum overlap of electric double layers) and
minimized when the discs are co-axial and parallel. A situation of
intermediate electrostatic energy is that of T-shaped perpendicular discs.
In the case of Laponite, the large value of the bare charge
should not allow linearization of PB, but at sufficient distance
(order of the Debye length) away from the platelet this may still
be valid with an effective charge much smaller than the bare one.
This can be calculated, and for conditions where lD z R (where
R is the platelet radius), this upper bound is estimated to 100,
i.e. considerably smaller than the bare charge. This value of the
effective charge is in agreement with that estimated by Meyer and
coworkers,70 who were able to quantify the effect of counter-ion
condensation, by means of MC simulations of two charged
platelets where co- and counter-ions are taken into account in
terms of the primitive model. The huge reduction of the residual
effective charge provides an effective force that is two orders of
magnitude smaller than what would be obtained using the bare
charge (as done in the microscopic models described above).
Using an average of the two-body platelet potential over angular
degrees of freedom, Trizac and co-workers69 also provided the
location of the sol–gel transition observed in early Laponite
experiments8,9 on a tentative phase diagram. The competition
between the increasing strength of repulsion and the decrease of
the Debye length with increasing clay concentration provides
a re-entrant solid transition, differently from what would be
observed for spheres. This approach yields a gel transition at
a number density corresponding to about 8% in weight concentration (smaller than that for uncharged disks), still larger than
the experimentally observed threshold but closer to it with
respect to all models reported in the previous paragraph.
A generalization of polymer reference interaction site model
(PRISM) theory71 to platelets has been reported by Harnau and
co-workers,72 providing a very good description of the equation
of state of uncharged platelets. Hence, it was also applied to the
case of Laponite, modeling it via the discrete-site representation
of Kutter et al. discussed above,60 but considering only effective
repulsive Yukawa interactions between sites. Again, to match
with experimental data (and to ensure the linearized PB
approach is valid), an effective charge much lower than the
nominal one is used (Zeff ¼ 50) due to counter-ion condensation.
At sufficiently large salt concentration, PRISM describes very
well the experimental data of Mourchid,8 but a considerable
overestimation is found in the absence of salt. This was attributed by the authors to the interference of gel/glass formation or
incipient nematic ordering. In the light of more recent
measurements, it seems plausible that those earlier results are
affected by waiting time dependence. Nonetheless the importance of PRISM as a powerful theoretical tool for describing the
thermodynamic behavior of platelet suspensions has emerged
and its investigation should be further pursued73,74 in the light of
the recent experimental evidences reported previously.
Another important work trying to describe Laponite solutions
in terms of an effective potential was carried out by Li55 a few
years ago, comparing PRISM theoretical predictions for the
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static structure factor to SAXS measurements at very low clay
concentrations (Cw x 0.04 0.4%) for Cs ¼ 103 M. We note
that these authors claim that they are monitoring equilibrium
states but, as discussed previously, several experiments have
shown that this is not the case,13,37 because aggregation takes
place albeit on a very slow time-scale. So, in our view, they are
essentially fitting the data at tw ¼ 0 i.e. at the beginning of the
aggregation process. At first they analyze the form factor and fit
it to that of a polydisperse distribution of platelets, finding
a considerable size polydispersity. The measured S(Q) are
compared to PRISM calculations corresponding to different
interaction potentials. They start by considering only an effective
Yukawa repulsion between the platelets, which provides a very
poor agreement with the experimental data, whatever the effective parameters (surface charge, Debye length, polydispersity
distribution) used. The large S(0) values observed in Laponite
experiments as compared to other purely repulsive systems,
suggests the use of a more complex potential, which takes into
account also a short-range attraction, but whose analytic form is
not provided. In this way, the authors obtain quite a good
description of the experimental data but the number of effective
parameters involved, and their change with Laponite concentration, remains unclear. Moreover, the issue of absence of
waiting time dependence calls for further investigation of the low
Cw region to elucidate the nature of the effective interactions and
the relative strength of attractive and repulsive terms.
On the other hand, the situation occurring at larger clay
concentrations, i.e. 2.0% # Cw # 3.0%, has been clarified in
a very recent study by Ruzicka and coworkers.23 In this work, the
S(Q) measured by SAXS has been compared to integral equation
predictions for purely repulsive objects,23 as shown in Fig. 15. In
this simplified framework, the discotic shape of the platelets was
not taken into account, and only an effective interaction between
the centers of mass of the scattering objects was considered,
following similar ideas for repulsive-interacting particle clusters.29 In this context, interactions are spherical and not very
strong, so that S(Q) obtained within a simple closure such as
Percus–Yevick (PY) or Hypernetted Chain (HNC) is virtually
identical to that obtained from direct simulation for the same
Yukawa potential. Two main parameters were fixed throughout
Fig. 15 S(Q) from SAXS (symbols) and theory (lines) for Laponite in
salt-free water conditions. Inset: comparison with MC simulations of
charged disks. From Ref. [23].
1282 | Soft Matter, 2011, 7, 1268–1286
the studied concentration range, in order to maximize the
agreement with experimental data: the effective charge (Zeff ¼
60), which similarly to previous works69,72 is found to be much
smaller than the bare charge, and the number density of the
scattering objects, which turns out to be smaller than that corresponding to the nominal weight concentration by a factor
0.4. This can be attributed to the fact that platelets may be
found within a distribution of clusters, mainly monomers and
dimers.35 The Debye length is calculated from the parameters
above and found to be varying between 8 and 10 nm with
decreasing concentration in the studied range, while the repulsion strength is found to increase by a factor of z2, compatibly
with the behavior of other charged systems.75 To validate these
results based on spherical interactions, MC simulations of clay
disks have also been carried out in the same work,23 in order to
show that the measured S(Q) is consistent with that of charged
platelets in the absence of attractive interactions (see inset of
Fig. 15). To this aim Model A60 with 19 discrete sites has been
used, the important difference with respect to the original model
being a much lower Zeff. Very good agreement is observed
between theory, simulations and experiments at the same
Laponite concentration as that determined by the theoretical fits,
effective charge of about 70 e and Debye length of approximately
5 nm. This study reinforces the idea that simple effective treatments are very powerful in detecting the relevant effective
parameters, later to be incorporated in more microscopic
models. Moreover, combined with the dilution experiments23
discussed in the experimental section, this study has clarified the
nature of the observed non-ergodic state as a Wigner glass,
revealing that the electrostatic repulsion is sufficient for
describing the structure of the system at high concentration.
C.
Discussion on the theoretical and numerical results
We now want to draw some conclusions on several aspects that
have emerged from the analysis of theoretical and numerical
results, highlighting the points that should be taken into account
in future work.
First of all, it is important to clarify one point that, in our
opinion, has not been realized up to now. The two models (A and
B) proposed by Kutter et al.60 were thought at the time to be
relevant for different experimental conditions. Indeed, while
model A, without rim charges, was designed to describe Laponite
at high pH, Model B was thought to be relevant for describing
the system at lower pH, where the release of OH ions would be
favored and positive rim charges would appear.60,61 However, it
has become clear from recent experimental measurements that
pH remains always close to 10,3 and that, under these normal
conditions, face-rim attraction is very important.17 Therefore, it
would appear that only Model B is relevant for the experimental
system. However, the real distinction between the models should
not be done with respect to different pH conditions, but rather to
distinguish the dominant interactions in the two regions of the
phase diagram (at low salt content): attractive-dominated (at low
Cw) and repulsive-dominated (at high Cw).23 Therefore, the two
models have validity beyond their original purpose. Model B,
with adjusted parameters, should be used to describe the system
at low clay concentrations, where rim charges are dominant. On
the other hand, Model A can, as a first approximation, be used,
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again with adjusted parameters, to describe the relevant (repulsive) interactions at high clay concentrations, thus neglecting the
presence of rim charges.
Of course, in reality, rim charges should always be present but,
to zero-th order, they can be neglected for describing the Wigner
glass state. This was demonstrated by the favourable comparison
of S(Q) between theory, simulations and experiments23 within
a purely repulsive description. However, the study of the
dynamical properties needs further investigation. The BD
simulations by Mossa and coworkers63 already provided indications that the purely repulsive potential could describe the
experimental observations, by reproducing the fact that S(Q)
does not show a marked change with increasing waiting time,
while the dynamical behavior continues to age. Also the dependence of the relaxation time and of the stretching exponent with
increasing waiting time is in good qualitative agreement with
experimental results (see for a comparison Fig. 6 and Fig. 12).
We note that a much faster relaxation towards a non-ergodic
state, occurring at much smaller waiting times, is observed on
absolute timescales in the simulations, which could be due to the
simulated density being considerably larger than the experimental one. However, assuming the Q2 scaling of the relaxation
time, one can deduce that the numerical relaxation times are not
too far off the experimental ones, suggesting that perhaps
a combination of the effective potential parameters which have
been found able to describe the static structure of Laponite
solutions23 with BD simulations of Model A could be important
to gain further knowledge about this concentration regime. We
emphasize that it will be crucial to take into account correctly the
effective charge, which most theoretical studies23,69,70 have found
to converge to about 10% of the bare charge, providing
a screening length of the order of 5–10 nm slightly depending on
concentration in salt-free water conditions. Finally, the role of
attractive interactions, which are not needed to correctly describe
the statics, should be better understood in relation to the
dynamic behavior. Indeed, some experimental evidences of
attractive interactions coming into play, at waiting times larger
than those of arrest,23,43 have been reported.
Now turning to lower concentrations, where attraction plays
the leading role, it is now clear that the various versions of Model
B studied so far36,60,61 do not appear to be able to reproduce the
right phenomenology. In particular, the so-called PPO configurations, minimizing the energy for this model, have been
observed in experiments only at very large clay concentrations.62
In the light of the recent evidence provided by Ruzicka et al.,19 it
appears that the essential ingredient that is necessary to consider
in order to produce a low-density phase separation is that of an
anisotropic and localized attraction. Indeed, the phase separation is a result of the limited number of bonds that each platelet is
able to form. Hence the only descriptions that are able to
reproduce this physical behavior are the pioneering model of
Dijkstra and coworkers58 and the recently introduced patchy
Laponite model.19 An important aspect that is still not well
captured by theory is the number density at which the gel states
are formed, all models predicting the formation of a network at
a considerably larger density than the experimental one. To this
end, a proper combination of short-range patchy attraction with
the electrostatic repulsion could be important to correctly
reproduce not only the crossover between gels and Wigner
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glasses taking place around Cw 2%, but also to reach a quantitative agreement with the experimental phase diagram.
V. Conclusions and perspectives
In this review, we have provided evidence that Laponite can be
considered as a model system for investigating several physical
mechanisms often interfering with each other. Indeed, once taken
into account correctly the issues of reproducibility and aging,
a converging picture indicating the presence of multiple nonergodic states, as well as phase separation, is obtained. Moreover, the fact that different timescales govern respectively
attractive and repulsive interactions brings the problem to
a higher level of complexity, which adds onto the anisotropic
shape and the directionality of face-rim interactions. All these
reasons make Laponite an interesting candidate for elucidating
different aspects of soft matter physics. For example,
Laponite has been actively investigated by different experimental
groups76–80 to study the violation of the fluctuation-dissipation
theorem (FDT). Along these lines, a recent numerical study81
suggests that this type of measurements could help clarifying the
different nature of dynamic arrest (gel or glass) in this system.
Most importantly, the recent discovery of Laponite behavior as
a patchy particle system opens new perspectives for exploiting
colloidal clays (at least those showing regimes where attraction
becomes dominant) as suitable candidates for bottom-up selfassembly approaches.21
Coming back to examining the current status of Laponite
phase diagram, there remain numerous open issues to be investigated, as discussed extensively above both in relation to
experiments and to theory and simulations. The ultimate goal of
these efforts would be to obtain a unified description of both
attractive and repulsive-dominated regimes. In order to achieve
such a description, a better understanding of the different
interactions separately is needed. To this aim, a route that was
already proposed by Mongondry82 some years ago but not
systematically studied since then, is the addition of pyrophosphate or polyethylene oxide (PEO) to Laponite suspensions. By
adsorbing onto the positively charged rim and inhibiting rim-face
bond formation, pyrophosphate was found to retard or even
prevent gel formation at low clay concentrations. On the other
hand, PEO can have a complex interaction with Laponite. For
low molar mass, it also slows down the aggregation process, even
inhibiting it under certain conditions, due to steric hindrance of
chains adsorbed onto Laponite platelets. Oppositely for high
molar mass, it leads to the formation of clusters or a weak gel
immediately after mixing,82 since it makes bridges between
platelets. These modifications of the system behavior can be used
to isolate the two competing contributions, either attraction or
repulsion, in order to pave the way for a more accurate
description of Laponite suspensions within a unified model. This
will be important in order to tackle those questions, such as the
fate of the Wigner glass with respect to the gel or with respect to
the attractive glass (or phase separation, or even arrested phase
separation25) at large Cw and Cs, that need to take into account
attractive and repulsive interactions simultaneously. Finally
a more clear determination of the boundaries between disordered
and ordered states is needed at large Cw. This would be an
interesting topic per se, as the crucial parameters controlling the
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occurrence of the isotropic/nematic phase transition with respect
to gel formation are still not completely understood.83,84
We believe that these questions are very important not only for
the variety of technological applications in which Laponite is
used, which need to be based on long-term stable states
(for example equilibrium gels), but also from a fundamental
point of view, because the underlying physical mechanisms at
hand in the complex behavior of Laponite can be of help for the
study of other colloidal clays as well as for systems with anisotropic shape and interactions, including the recently synthesized
patchy particles21 and globular proteins.85,86
VI. Acknowledgements
We thank S. Mossa for providing Fig. 12 and M. Dijkstra and
J. Russo for fruitful discussions. We are grateful to our
colleagues L. Zulian, G. Ruocco, R. Angelini, M. Sztucki, A.
Moussaid, T. Narayanan and F. Sciortino for the ongoing
collaboration in the study of Laponite suspensions.
EZ acknowledges support from ITN-234810-COMPLOIDS,
ERC-226207-PATCHYCOLLOIDS and MIUR-Prin.
Fig. 16 State diagram of Na Cloisite reproduced from Ref. [91] at fixed
waiting times (three months). The different regions (a–e) are pictorially
represented: (a) Wigner glass of clusters; (b) gel of individual platelets; (c)
cluster fluid; (d) phase separation; (e) gel of stacked platelets.
VII. Appendix: comparison with Cloisite
In this Appendix we try to make connection between Laponite
phase diagram with that of another clay suspension, sodium
montmorillonite (Na Cloisite). While many clays have shown
a behavior that can be interpreted only in terms of repulsive
interactions,87–89 Na Cloisite appears to show clustering and
a rather complex behavior originated by the interplay of both
attractive and repulsive interactions, similarly to Laponite. For
this reason, it is of interest to compare the two systems and the
different phases that they can form in the various ranges of salt
and clay concentration.
Sodium montmorillonite is made of platelets with a diameter
of 100 nm and a much larger aspect ratio (1 : 100) with respect to
Laponite. The other main structural difference is the organization of the octahedral layer, which induces substantial heterogeneity in particle growth and a considerably larger size
polydispersity.90 Another important aspect is that, while
Laponite solutions are stable at pH 10 only, montmorillonite
exists down to pH 4.34 Under these much more acidic conditions, the patch-wise charge heterogeneity, i.e. oppositely
charged surface parts of layers, is greatly enhanced.34
A careful characterization of the phase diagram of Na Cloisite
has been recently reported by Shalkevich and co-workers91 by
means of a combination of various experimental techniques for
different weight and salt concentrations. Different from the
available data for Laponite, a systematic study of the aging
dynamics has not been performed and the data refer to a fixed
waiting time corresponding to three months after preparation.
The phase diagram, reported in Fig. 16, does show similarities
to that of Laponite suspensions, considering the latter at
a comparable tw. In the absence of salt, the system is reported91 to
form a Wigner glass state (a), where the building blocks are small
clusters, rather than single platelets. At very large salt concentration, phase separation/flocculation (d) and percolation of
stacks of platelets (e) are observed. In the intermediate salt
concentration range, an equilibrium fluid of clusters (c) and a gel
1284 | Soft Matter, 2011, 7, 1268–1286
state, originating from percolation of individual platelets, (b) are
reported respectively for small and large Cw.
Some notable differences with respect to Laponite can be
identified from this study. First, looking at the effective structure factors in the low-salt regime, a significant growth at low
Q is observed, a feature not present in Laponite SAXS data at
a comparable Cw, e.g. of 2%. This could be explained by the
stronger attraction strength (due to lower pH) in Cloisite,
manifesting in the cluster character of the Wigner glass, or to
some differences in the sample preparation procotol (relatively
large 5 mm filter used for Cloisite).91 Another signature of
stronger attraction comes from the cluster formation at low Cw,
the cluster radius inferred from neutron scattering data being
Rg 400–600 nm, while Laponite Wigner glass appears to be
mostly made by individual particles (or very small clusters,
e.g. dimers).23 Second, as soon as some salt is added, a
re-entrant melting to a cluster fluid phase is observed at low
Cw, with clusters being much larger (order of a few microns) in
this regime. It is plausible that this liquid pocket has some
correspondence with the ‘‘isotropic fluid’’ observed by
Mourchid9 before the gel transition observed by Ruzicka takes
place (order of a few thousand hours).15 In support of the
interpretation of Shalkevich et al.91 comes the characterization
of SAXS data for Laponite, at tw 0 only, in terms of an
effective potential based on the competition between a shortrange attraction and a long-range repulsion.55 However, the
more recent study of Ruzicka et al.19 has revealed that in this
low-salt, low-concentration region the underlying mechanism is
that of an extremely slow phase separation process, which takes
place even after a (transient) gel is formed. Therefore, although
a competition between short-range attraction and long-range
repulsion is at hand, it is the patchy nature of Laponite
attractive interactions which controls the phase behavior and
the long-term stability of the system. It will be interesting to
connect these findings also to Cloisite.
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Finally the transition/crossover between Wigner glass at low
Cs and gel (or attractive glass) at high Cs needs a more systematic
investigation for both clay suspensions. In addition, for Cloisite
the important role of stacked platelets is elucidated in the high
salt concentration regime, while a similar analysis for Laponite
has not been provided.
Notwithstanding these interesting differences, both Na Cloisite and Laponite suspensions show robust evidence of the existence of multiple non-ergodic states upon variation of Cw and Cs.
For this reason, it will be important in the future that the
investigation of both systems proceeds in parallel in order to gain
better understanding of the physical mechanisms behind the
formation of clay low-density gels or glasses.
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