Liquid Crystal Phases in Suspensions of Charged Plate-like Particles

Journal: The Journal of Physical Chemistry Letters

Manuscript ID: jz-2012-00380n

Manuscript Type: Letter

Date Submitted by the Author:

28-Mar-2012

Complete List of Authors: Delhorme, Maxime; Lund University, Teoretisk Kemi Labbez, Christophe; Université de Bourgogne, Institut Carnot de Bourgogne, UMR 5209 CNRS Jönsson, Bo; Lund University, Theoretical Chemistry

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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, and Department of Theoretical Chemistry, Chemical Center, POB

124, S-221 00 Lund, Sweden

E-mail: christophe.labbez@u-bourgogne.fr; bo.jonsson@teokem.lu.se

∗To whom correspondence should be addressed†Burgundy University‡Lund University

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Abstract

Anisotropic interactions in colloidal suspensions has recently emerged as a route for the

design of new soft materials of potential technological interest; for example they can self-

assemble into periodic 3D structures on the meso- macro-scale and serve as photonic materials.

Non isotropic particles such as rods and discs can form nematic, smectic, hexatic and columnar

liquid crystals. Although the formation of these phases are 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, e.g. clays. Here we use Monte Carlo

simulations of concentrated suspensions of charged disc-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 strongly decrease with increasing Coulomb interactions and

charge anisotropy. Our results suggest that the structure of aqueous liquid crystals can be tuned

with the anisotropic Coulomb interactions.

Keywords : Liquid crystal; charged discotic particles; Monte Carlo simulation; columnar phase

An interesting property of anisotropic particles is their ability to form liquid crystals.1,2 The

earliest observation of such phases date back to the twenties and thirties on rod-like vanadium pen-

toxide,3 tobacco mosaic virus particles4 and disclike clay particles (but in a gel state)5 that exhibit

an isotropic-nematic (I-N) phase transition. In the forties, Onsager formulated a simple statistical

mechanical approach6,7 for hard rods and discs in which the I-N transition is rationalized as a com-

petition 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 Dur-

ing 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 experimen-

tally 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,

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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 gibbsite10 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 disc-like clay

particles, unlike the multiple previous attempts,5,32–36 where the transition was always found to

be pre-empted by the formation of a gel. Although electrostatic interactions appear to be well

approximated using effective dimensions of the platelets in dilute suspensions (where the I-N tran-

sition occurs) see e.g.,21,30 their role is far less well understood at higher particle concentrations,

i.e. 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 disc-like particles often exhibit a charge on the edges of opposite sign to that of

the basal planes, e.g. 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. that 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 disc-like

particles over a wide range of particle density and ionic strength. We consider three systems which

differ by the charge anisotropy of the constituent platelets. With the results of the MC simulations

we address the following questions: i) What are the liquid crystal formed varying the particle

volume fraction and the ionic strength? ii) Does the Coulomb interaction with the edge charges

favor the formation of liquid crystals?

The model platelike particle, which we are considering, are made up of nT = 199 soft spheres

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distributed on a compact hexagonal lattice forming a disc of diameter D≈ 15 nm and thickness h=

1 nm. The edge sites are either neutral or carry a 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, is 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 both the concentration of salt (cs) and counterion (cc), κ2 = e2(2cs+cc)ε0εrkT where k

is Boltzmann’s constant, T = 298 K the absolute temperature and ε0 is the permittivity of vacuum.

cs was varied from 1 to 100 mM. A shifted Lennard-Jones (uLJ) potential is used to account for the

finite size of particles. In addition to uLJ a screened Coulomb potential uel is employed to describe

the electrostatic contribution. The total interaction between two sites of charge Zi and Z j separated

a distance ri j then reads,

utot(ri j) =ZiZ j exp(−κri j)

4πεrε0ri j+4εLJ((

σLJ

ri j)12− (

σLJ

ri j)6)+ εLJ (1)

with σLJ = 6√

2 nm and εLJ = 0.5kT . The full configurational energy of the N platelets system

them becomes

U =N

∑i=1

N

∑j>i

nT

∑α=1

nT

∑β=1

uel(rαi ,r

β

j )+uLJ(rαi ,r

β

j ) (2)

where indices i, j refer to platelets and indices α,β to sites on these platelets, respectively. The

model system constituted 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 multi-

level coarse graining approach was developed to reduce the computational cost of simulations.

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It relies on the idea that for calculating the electrostatic energy between two particles a detailed

charge description is only needed at short separation, see43 for more details. The coarse graining

reduces the computing time by one to two orders of magnitude. The code was further parallelized

and run on eight processors.

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

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

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 2-a. One feature of Figure 1 is the strong dependence of the gel-columnar

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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 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 bound-

ary between the gel and columnar areas. On the same line as our previous work on the sol-gel

transition,43 this comes as a 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 al14 on neutral

platelets a nematic-columar 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.

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Figure 3: Bi-dimensional distribution function, i.e. radial and angular, of suspensions of discotic particles undervarious conditions accompanied with an illustration of the corresponding phase. a) ne =+48, cs = 1mM and φ = 18%;b) ne =+38 cs = 10mM and φ =21 %; c) ne =+38 cs = 10mM and φ =25%

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Figure 3 displays the typical two-dimensional pair distribution functions, i.e. angular (θ ) and

radial (r), for particles in a) an interpenetrated hexagonal columnar phase (Chin), b) a zig-zag

columnar phase (Chz) and c) a rectangular columnar phase (Crin). θ is defined as the angle be-

tween the normals of the particles. In the Chin columnar phase we observe sharp peaks at h and 2h,

where h ∼ 38Å, is the intra columnar distance, followed by more broad peaks at l, l√

3 and l√

4,

where l is the inter columnar distance, characteristic of a sixfold symmetry. The inter-columnar

spacing, l ∼ 120Å, 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 the peaks in the g(r,θ)

are found with θ < π/30 which is the signature of a parallel arrangement of the platelets, i.e. a

long range angular correlation, within the columnar phase.

In the Chz phase (Figure 3-b), on the other hand, not all peaks are found at such low angle,

but, instead, two distinct angles are found, i.e. θ1 ∼ 0 and θ2 ∼ π/6, illustrating the zig− zag

configuration of the particles. The whole structure can be described as a parallel superposition of

planes, separated a distance h/2 with h ∼ 80 Å, that encompass the center of mass (cm) of the

particles. In these planes the cm are hexagonally organized. This is shown by the sharp peaks at

l and√

3l, with l ∼ 120 Å. Within a plane the particles are parallel to each others, i.e. θ1 = 0,

but are tilted with respect to the plane by θ2/2. Each of those planes are 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 an 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 ∼ 60Å. 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 3-c), all particles lie parallel to each other as shown by the virtually

zero value of g(r,θ 6= 0). The first two peaks arise from the intra-columnar spacing positioned at

h and 2h with h∼ 30 Å. The three last and broad peaks correspond to the inter-columnar spacing

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located at l ∼ 160Å, lin =√(2 ∗ (l/2)2 +(h/2)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) (1/2,1/2,1/2).

Figure 4: Bi-dimensional distribution function, i.e. radial and angular, of a suspension of neutral discotic particlesat high volume fraction, φ = 36%. The result is virtually the same for charged particles at all salt concentrations andne values.

At volume fractions where the columnar phases are formed the Debye screening length is dom-

inated 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 non-trivial interplay of repulsive and attractive interactions. These complex non-

isotropic 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 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 elec-

trostatic 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

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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.

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

A more detailed view on the variation of the inter- and intra-columnar distances as well as the

intra-columnar particle angle with the salt concentration, the charge anisotropy and the particle vol-

ume fraction are 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, i.e. Chz→ Crin and

Ch→Crin. In the same qualitative way Mourad et al23 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 de-

tailed balance of the repulsive and attractive electrostatic interactions. It is worth noting that at

φ =29% no distinction between the structures and interparticle distances can be made as ne is de-

creased from 48 to 38 and even to 0 (not shown), although, with uncharged particles, an hexagonal

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columnar phase is formed at larger φ (∼36%).

To conclude, we have shown through Monte Carlo simulations of concentrated dispersions of

charged plate-like particles with varying charge anisotropy the important role played by electro-

static 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 par-

ticles, 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.

Acknowledgement

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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