Liquid Crystal Phases in Suspensions of Charged Plate-Like Particles

Letter pubs.acs.org/JPCL

Liquid Crystal Phases in Suspensions of Charged Plate-Like Particles Maxime Delhorme,† Christophe Labbez,*,† and Bo Jönsson*,‡ †

Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS, Université de Bourgogne, 21078 Dijon Cedex, France Department of Theoretical Chemistry, Chemical Center, POB 124, S-221 00 Lund, Sweden

‡

ABSTRACT: Anisotropic interactions in colloidal suspensions have recently emerged as a route for the design of new soft materials. Nonisotropic particles can form nematic, smectic, hexatic, and columnar liquid crystals. Although the formation of these phases is well rationalized when excluded volume is solely at play, the role of electrostatic interactions still remains unclear and even less so when particles present a charge heterogeneity, for example, clays. Here, we use Monte Carlo simulations of concentrated suspensions of charged disk-like particles to reveal the role of Coulomb interactions and charge anisotropy underlying liquid crystal formation and structures. We observe a vast zoo of exotic structures, going from hexatic to columnar phases, which are shown to be controlled by the charge anisotropy. The particle volume fraction at which these phases start to form is found to decrease with increasing Coulomb interactions and charge anisotropy, which suggests a route to tune the structure of aqueous liquid crystals. SECTION: Glasses, Colloids, Polymers, and Soft Matter

A

refs 21 and 30), their role is far less well understood at higher particle concentrations, that is, in the concentration range where the lamellar and columnar phases are found. As an example, Lekkerkerker and co-workers21 discovered that suspensions of charged gibbsite particles undergo an isotropic−columnar (I−C) phase transition at volume fractions much lower than those predicted based on simple excluded volume effects. Inorganic disk-like particles often exhibit a charge on the edges of opposite sign to that of the basal planes, for example, in clays and gibbsite, depending on pH. This charge anisotropy may have profound consequences of fundamental as well as industrial interest, in particular, in the gelation process of clays.37−40 Its role in the formation and structure of lamellar and columnar phases has so far not been investigated, except by Morales-Andra et al., who report the formation of interpenetrated hexagonal columns for purely repulsive platelets.41 In the present study, we explore via Monte Carlo (MC) simulations the liquid crystal phases of suspensions of charged disk-like particles over a wide range of particle densities and ionic strengths. We consider three systems that differ by the charge anisotropy of the constituent platelets. With the results of the MC simulations, we address the following questions: (i) How do the volume fraction and ionic strength affect the liquid crystalline phases formed? and (ii) How do anisotropic Coulomb interactions influence the formation of liquid crystals? The model plate-like particles that we are considering are made up of nT = 199 soft spheres distributed on a compact hexagonal lattice forming a disk of diameter D ≈ 15 nm and thickness h = 1 nm. The edge sites are either neutral or carry a

n interesting property of anisotropic particles is their ability to form liquid crystals.1,2 The earliest observation of such phases dates back to the 1920s and 1930s on rod-like vanadium pentoxide,3 tobacco mosaic virus particles,4 and disklike clay particles (but in a gel state)5 that exhibit an isotropic− nematic (I−N) phase transition. In the 1940s, Onsager formulated a simple statistical mechanical approach6,7 for hard rods and disks in which the I−N transition was rationalized as a competition between orientational entropy (favoring the I phase) and translational entropy (favoring the N phase). This theory has been confirmed by Veillard-Baron using Monte Carlo simulations.8 During the last decades, due to a renewed interest in liquid crystals and their applications, numerous experimental and theoretical studies have been performed. Onsager’s theory has been experimentally validated with model systems composed of sterically stabilized platelets of gibbsite9,10 and hydrotalcite.11 New theories12,13 and numerical simulations14−17 have also been presented that give a fairly good description of dispersions of such hard and anisotropic particles. In particular, a columnar phase was predicted from Monte Carlo simulations by Veerman and Frenkel14 and was later confirmed by experiments using sterically stabilized nickel hydroxide,18 gibbsite,10 and copper sulfide.19 Nematic, lamellar, and columnar phases have also been found with charged platelets of various minerals ranging from hydroxides to layered double hydroxides.18,20−27 Recently, Michot and co-workers succeeded to observe a true I−N transition28−31 in suspensions of charged disk-like clay particles, unlike the multiple previous attempts,5,32−36 where the transition was always found to be preempted by the formation of a gel. Although electrostatic interactions appear to be well approximated using ef fective dimensions of the platelets in dilute suspensions (where the I−N transition occurs; see, e.g., © 2012 American Chemical Society

Received: March 28, 2012 Accepted: April 24, 2012 Published: April 25, 2012 1315

dx.doi.org/10.1021/jz300380n | J. Phys. Chem. Lett. 2012, 3, 1315−1320

The Journal of Physical Chemistry Letters

Letter

Figure 1. Tentative phase diagrams, ionic strength versus particle volume fraction ϕ, for ne = 38 (left) and 48 (right). Blue |, attractive gel phase with particles mostly in an overlapping coin configuration; blue crosses, attractive gel phase with particles mostly in a house of card configuration; red Z, zigzag columnar phase; red diamonds, interpenetrated hexagonal columnar phase; red squares, interpenetrated rectangular columnar. Liquid crystal phases that are in coexistence with a gel phase (fully percolated network of disordered particles) are indicated as open symbols.

Figure 2. MC simulation snapshots at different state points, (a) at I = 10 mM, ϕ = 21%, and ne = 38; (b) at I = 10 mM, ϕ = 18%, and ne = 48; (c) at I = 10 mM, ϕ = 21%, and ne = 48.

where indices i,j refer to platelets and indices α,β to sites on these platelets, respectively. The model system composed of N = 200 platelets dispersed in a cubic box of volume Vbox filled with a 1−1 salt solution is solved using Monte Carlo simulations in the canonical ensemble (N,V,T). Periodic boundary conditions are applied using the minimum image convention. The platelets are free to move and rotate all through the available space. Collective displacements of platelets were performed in addition to usual single platelet translations and rotations by the use of a cluster move technique42 employing cylinders and infinite layers instead of the usual spheres. Finally, a multilevel coarse-graining approach was developed to reduce the computational cost of simulations. It relies on the idea that for calculating the electrostatic energy between two particles, a detailed charge description is only needed at short separation; see ref 43 for more details. The coarse-graining reduces the computing time by 1 to 2 orders of magnitude. The code was further parallelized and run on eight processors. The results for ne = 38 and 48 are summarized in tentative phase diagrams; see Figure 1 and related snapshots in Figure 2. Irrespective of the charge anisotropy, the suspensions undergo a gel−columnar phase transition. The gel−columnar phase coexistence is observed in some of the simulations; see Figure 2a. One feature of Figure 1 is the strong dependence of the gel−columnar transition with the charge anisotropy and the ionic strength. Indeed, when ne = 0 (not shown), a columnar phase is formed at ϕ as large as 29%, while when ne = 48, a columnar phase coexisting with a gel can already be observed at

positive elementary charge, e. ne of the edge sites are positively charged. The nb basal sites are negatively charged, −e. The net charge of a platelet is always negative and is given by Znet = (ne − nb)e. We used ne = 0, 38, and 48. The solvent, that is, water, was treated as a structureless dielectric continuum characterized by its relative permittivity, εr = 78. Salt and counterions are represented implicitly via the Debye screening length, κ−1, which depends on the concentration of both the salt (cs) and counterion (cc), κ2 = [e2(2cs + cc)]/(ε0εrkT), where k is Boltzmann’s constant, T = 298 K is the absolute temperature, and ε0 is the permittivity of vacuum. cs was varied from 1 to 100 mM. A shifted Lennard-Jones (uLJ) potential was used to account for the finite size of the particles. In addition to uLJ, a screened Coulomb potential uel was employed to describe the electrostatic contribution. The total interaction between two sites of charge Zi and Zj separated by a distance rij then reads tot

u (rij) =

ZiZj exp( −κrij) 4πεrε0rij

⎡⎛ ⎞12 ⎛ ⎞6 ⎤ σLJ σLJ + 4εLJ⎢⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥⎥ r ⎝ rij ⎠ ⎦ ⎣⎝ ij ⎠

+ εLJ

(1)

with σLJ = 21/6 nm and εLJ = 0.5kT. The full configurational energy of the N platelets system then becomes N

U=

N

nT

nT

∑ ∑ ∑ ∑ uel(riα , r jβ) + u LJ(riα , r jβ) i=1 j>i α=1 β=1

(2) 1316



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Figure 3. Bidimensional distribution function, that is, radial and angular, of suspensions of discotic particles under various conditions accompanied by an illustration of the corresponding phase. (a) ne = +48, cs = 1 mM, and ϕ = 18%; (b) ne = +38, cs = 10 mM, and ϕ = 21%; (c) ne = +38, cs = 10 mM, and ϕ = 25%.

ϕ = 14%. Conversely, the gel−columnar transition is favored when decreasing the ionic strength of the solution as revealed by the positive slope of the boundary between the gel and columnar areas. On the same line as our previous work on the sol−gel transition,43 this comes as further evidence that a positive slope of the transition line between two phases in a cs versus ϕ phase diagram (of anisotropic particles) does not necessarily mean that interparticular interactions are dominated by repulsive interactions, contrary to suspensions of charged isotropic particles. In agreement with MC simulations of Frenkel et al,14 on neutral platelets, a N−C phase transition is found above ϕ = 36%. Note that in the case of ne = 0, the columnar phase is also found to be preempted with a nematic phase (ND type), in line with experiments of Lekkerkerker et al.21,23 Interestingly enough, the experimental N−C transitions fall in the same range of particle volume fractions as those found in our simulations. These results demonstrate and clarify the crucial role played by the charge anisotropy and, more generally, by the Coulomb interactions in the formation of columnar liquid crystals. Figure 3 displays the typical two-dimensional pair distribution functions, that is, angular (θ) and radial (r), for particles in

(a) an interpenetrated hexagonal columnar phase (Chin), (b) a zigzag columnar phase (Chz), and (c) a rectangular columnar phase (Crin). θ is defined as the angle between the normals of the particles. In the Chin columnar phase, we observe sharp peaks at h and 2h, where h ≈ 38 Å is the intracolumnar distance, followed by more broad peaks at l, l√3, and l√4, where l is the intercolumnar distance, characteristic of a six-fold symmetry. The intercolumnar spacing, l ≈ 120 Å, is significantly lower than the particle diameter, D = 150 Å, which signals an intercalation of the columns favored by the strong edge−basal attraction. All of the peaks in the g(r,θ) are found with θ < π/30, which is the signature of a parallel arrangement of the platelets, that is, a long-range angular correlation, within the columnar phase. In the Chz phase (Figure 3b), on the other hand, not all peaks are found at such low angle, but instead, two distinct angles are found, that is, θ1 ≈ 0 and θ2 ≈ π/6, illustrating the zigzag configuration of the particles. The whole structure can be described as a parallel superposition of planes, separated by a distance h/2, with h ≈ 80 Å, that encompass the center of mass (cm) of the particles. In these planes, the cm's are hexagonally organized. This is shown by the sharp peaks at l and √3l, with l 1317



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Figure 4. Bidimensional distribution function, that is, radial and angular, of a suspension of neutral discotic particles at high volume fraction, ϕ = 36%. The result is virtually the same for charged particles at all salt concentrations and ne values.

Figure 5. Mean inter- (black lines) and intracolumnar (red lines) particle distances at various salt concentrations and as a function of the volume fraction of particles with (a) ne = +38 and (b) ne = +48. (c) Mean angle between particles within columns as a function of the volume fraction of particles with ne = +38. The salt concentrations are 1 (circles) and 100 mM (crosses).

≈ 120 Å. Within a plane, the particles are parallel to each others, that is, θ1 = 0, but are tilted with respect to the plane by θ2/2. Each of those planes is sandwiched by two parallel planes, where the particles are tilted by −θ2/2. In other words, the axes of symmetry of two particles belonging to two consecutive planes make angles of θ2 and are parallel when they belong to planes separated by an odd number of planes. The overall g(r,θ) thus displays peaks at positions (a,θ2), (h,θ1), (2a,θ2), (l,θ1), (2h,θ1), (√3l,θ1), and (3a,θ2), where a is the distance of the nearest neighbors that belong to the nearest planes with a ≈ 60 Å. The broadening in θ and r and the weakness of the peaks corresponding to the particles in the nearest layers demonstrate the liquid-like state of the structure. In the Crin phase (Figure 3c), all particles lie parallel to each other, as shown by the virtually zero value of g(r,θ ≠ 0). The first two peaks arise from the intracolumnar spacing positioned at h and 2h, with h ≈ 30 Å. The last three broad peaks correspond to the intercolumnar spacing located at l ≈ 160 Å, lin = [(2(l/2)2 + (h/2)2)]1/2 ≈ 115 Å, and √2l, where l > D and lin < D are the neighbor positions in the plane of the platelet diameter and in the intercalated columns, respectively. These positions reveal the centered tetragonal ordering of the columns for which the structure can be reproduced with a parallelepiped primitive cell of dimensions l·l·h, where the particles occupy the reduced positions (0,0,0) and (1/2,1/2,1/ 2). At volume fractions where the columnar phases are formed, the Debye screening length is dominated by the counterion concentration, and, in general, κ−1 ≪ D. Thus, electrostatic interactions are short-range and greatly affected by the details of the local structure and charge anisotropy, which lead to a nontrivial interplay of repulsive and attractive interactions. These complex nonisotropic interactions eventually result in a vast zoo of exotic liquid crystal phases tuned via the charge anisotropy, ϕ and κ−1, as best seen in the phase diagrams,

Figure 1. As an illustration, the Chin phase is only observed at high charge anisotropy; for example, by decreasing ne from 48 to 38 at ϕ = 21%, the Chz columnar phase is formed. When increasing ϕ above 25%, both the Chin and Chz melt into a Crin phase, in which the intercalated column structure shows that the anisotropic electrostatic interactions are still at play even at such small κ−1 (∼5 − 10 Å). Upon further increasing the particle volume fraction, ϕ > 36%, the electrostatic interactions are completely screened, and the system is mainly driven by the anisotropic excluded volume effects, which lead to the common hexagonal columnar phase, Ch. The Ch phase is found to be virtually identical to that formed from the same system but where only the soft interactions are at play; see Figure 4. A more detailed view of the variation of the inter- and intracolumnar distances as well as the intracolumnar particle angle with the salt concentration, the charge anisotropy, and the particle volume fraction is given in Figure 5. One can see that these distances and orientations are governed by the particle concentration and that the salt is essentially not playing any role. Furthermore, the most important changes are found at the boundaries of the phase transitions, that is, Chz → Crin and Ch → Crin. In the same qualitative way, Mourad et al.23 observed that the salt concentration was not influencing the interparticle spacing between the faces and edges of charged gibbsite particles in columnar phases. As the screening length for such high ϕ is mainly governed by the counterion concentration, this should not come as a surprise. The electrostatic interactions through the charge anisotropy are, on the other hand, an essential ingredient in the formation of the columnar phases, as best illustrated in the phase diagrams in Figure 1, which may appear as contradictory. These results simply show that the overall separation between the particles is dictated by the excluded volume effects, while their specific arrangement within the liquid crystals is governed by the detailed balance of the repulsive and attractive electrostatic 1318



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interactions. It is worth noting that at ϕ = 29%, no distinction between the structures and interparticle distances can be made as ne is decreased from 48 to 38 and even to 0 (not shown), although, with uncharged particles, a hexagonal columnar phase is formed at larger ϕ (∼36%). To conclude, we have shown through MC simulations of concentrated dispersions of charged plate-like particles with varying charge anisotropy the important role played by electrostatic interactions. Remarkably, a columnar phase is formed at a volume fraction as low as 14% when strong anisotropic electrostatic interactions are at play, while for the same uncharged particles, a columnar phase is only observed at ϕ > 36%. Furthermore, the detailed structure of the columnar phases has been shown to be greatly affected by the charge anisotropy. In particular, the interplay of repulsive and attractive interactions, tuned by the charge anisotropy and overall screening, were found to lead to exotic intercalated columnar phases: interpenetrated rectangular and hexagonal columnar phase as well as a zigzag columnar phase. These findings give further insights into the relevance of anisotropic interactions in the design of new soft materials.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (C.L.); bo. [email protected] (B.J.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the Region Bourgogne and computational support from CRI, Université de Bourgogne are gratefully acknowledged. The authors would like to thank B. Cabane and P. Davidson for many interesting discussions.

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REFERENCES

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(33) Michot, L. J.; Bihannic, I.; Porsch, K.; Maddi, S.; Baravian, C.; Mougel, J.; Levitz, P. Phase Diagram of Wyoming Na-Montmorillonite Clay. Influence of Particle Anisotropy. Langmuir 2004, 20, 10829− 10837. (34) Gabriel, J. C. P.; Sanchez, C.; Davidson, P. Observation of Nematic Liquid-Crystal Textures in Aqueous Gels of Smectic Clays. J. Phys. Chem. 1996, 100, 11139−11143. (35) Saunders, A. E.; Goodwin, J. W.; Richardson, R. M.; Vincent, B. A Small-Angle X-ray Scattering Study of the Structure of Aqueous Laponite Dispersions. J. Phys. Chem. B 1999, 103, 9211−9218. (36) Lemaire, B. J.; Panine, P.; Gabriel, J. C. P.; Davidson, P. The Measurement by SAXS of the Nematic Order Parameter of Laponite Gels. Europhys. Lett. 2002, 59, 55−61. (37) Dijkstra, M.; Hansen, J. P.; Madden, P. A. Gelation of a Clay Suspension. Phys. Rev. Lett. 1995, 75, 2236−2239. (38) Odriozola, G.; Romero-Bastida, M.; de J. Guevara-Rodriguez, F. Brownian Dynamics Simulations of Laponite Colloid Suspensions. Phys. Rev. E 2004, 70, 021405−001420. (39) Jönsson, B.; Labbez, C.; Cabane, B. Interaction of Nanometric Clay Platelets. Langmuir 2008, 24, 11406−11413. (40) Ruzicka, B.; Zaccarelli, E.; Zulian, L.; Angelini, R.; Sztucki, M.; Moussaid, A.; Narayanan, T.; Sciortino, F. Observation of Empty Liquid and Equilibrium Gels in Colloidal Clays. Nat. Mater. 2011, 10, 56−60. (41) Morales-Andra, L.; Wensink, H. H.; Galindo, A.; Gil-Villegas, A. Anomalous Columnar Order of Charged Colloidal Platelets. J. Chem. Phys. 2012, 136, 034901. (42) Swendsen, R.; Wang, J. S. Nonuniversal Critical Dynamics in Monte Carlo Simulations. Phys. Rev. Lett. 1987, 58, 86−88. (43) Delhorme, M.Thermodynamics and Structure of Plate-Like Particle Dispersions. Ph.D. Thesis, Lund University, Lund, 2012.

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Liquid Crystal Phases in Suspensions of Charged Plate-Like Particles

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