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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).


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 Soft Matter


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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 Soft Matter

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.


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particles) and equilibrium gels (arrested empty liquids) in the intermediate region 1.0% # Cw < 2.0%.


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.


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 Soft Matter


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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. Soft Matter

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 This journal is ª The Royal Society of Chemistry 2011

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 Soft Matter


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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 Soft Matter

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. This journal is ª The Royal Society of Chemistry 2011

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 Soft Matter


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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 Soft Matter

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 This journal is ª The Royal Society of Chemistry 2011


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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].


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, Soft Matter


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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 Soft Matter

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 This journal is ª The Royal Society of Chemistry 2011

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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 This journal is ª The Royal Society of Chemistry 2011

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. Soft Matter


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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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