Phase Behaviour of Lyotropic Liquid Crystals in External Fields And

Eur. Phys. J. Special Topics 222, 3053–3069 (2013) c EDP Sciences, Springer-Verlag 2013 THE EUROPEAN DOI: 10.1140/epjst/e2013-02075-x PHYSICAL JOURNAL SPECIAL TOPICS

Review

Phase behaviour of lyotropic liquid crystals in external ﬁelds and conﬁnement

A.B.G.M. Leferink op Reininka, E. van den Pol, A.V. Petukhov, G.J. Vroege, and H.N.W. Lekkerkerker Van ’t Hoﬀ Laboratory for Physical and Colloid Chemistry, Debye Institute for Nanomaterials Science, Utrecht University, Padualaan 8, 3584 Utrecht, The Netherlands

Received 6 September 2013 / Received in ﬁnal form 17 September 2013 Published online 25 November 2013

Abstract. This mini-review discusses the influence of external fields on the phase behaviour of lyotropic colloidal liquid crystals. The liquid crystal phases reviewed, formed in suspensions of highly anisotropic particles ranging from rod- to board- to plate-like particles, include nematic, smectic and columnar phases. The external fields considered are the earth gravitational field and electric and magnetic fields. For electric and magnetic fields single particle alignment, collective reorientation behaviour of ordered phases and field-induced liquid crystal phase transitions are discussed. Additionally, liquid crystal phase behaviour in various confining geometries, e.g. slit-pore, circular and spherical confinement will be reviewed.

1 Introduction

The vast majority of studies of colloidal suspensions deal with spherical particles [1–3]. Their behaviour in external fields was studied in detail and one can find ex- tended examples in other mini-reviews in this issue [4–8]. In contrast, here we shall only discuss the phase behaviour of suspensions of highly non-spherical particles with high aspect ratio. Anisometric colloidal particles, such as rods and platelets, have the ability to spontaneously self-organize into various liquid crystalline phases when brought in suspension involving isotropic (I), nematic (N), smectic (S) and columnar (C) phases. In the I phase the particles do not possess order. In the N phase the particles are orientationally ordered, but lack long-range positional order. In the S phase orientationally ordered particles are stacked in layers and thus have long range positional order in one dimension. In the C phase the oriented particles stack into columns, which form a periodic structure in two dimensions. Generally, higher ordered phases (S, C ) are found for higher particle volume fractions. The spontaneous formation of lyotropic liquid crystal phases is an entropy driven process. Already in the 1940s Onsager explained in his seminal work [9]theI-N phase transition for rod-like particles on basis of the particle shape alone: at sufficiently high concentration the loss in orientational entropy is smaller than the gain in excluded volume entropy.

a e-mail: [email protected] 3054 The European Physical Journal Special Topics

Later Frenkel and coworkers predicted the formation of N, S and C phases of hard anisotropic particles using computer simulations [10–13]. The first experimental observation of a lyotropic colloidal liquid crystal was of the N phase, which was found in a suspension of ribbon-like vanadium pentoxide (V2O5) particles [14]. Later the N phase was also found in suspensions of rod-like TMV-virus particles [15,16], board-like goethite particles [17] and clay plate-like [18] particles. The nematic phase found in systems of rod-like particles (which align their long axis) is often referred to as prolate nematic (N+), while plate-like particles display oblate nematic ordering (N−), in which the short particle axes are aligned. For particles with a shape exactly in between rod- and plate-like the rare biaxial nematic phase (NB) is predicted [19–21]. In the NB phase both particle axes are aligned [22]. Nowadays a rich variety of different liquid crystalline phases has been observed in suspensions of anisometric (mineral) particles, which has extensively been reviewed in the papers by Gabriel & Davidson [23,24] and Lekkerkerker & Vroege [25]. These include colloidal mineral particles which are known to display rich phase behaviour [23] such as board-like goethite (α-FeOOH) which is able to form N+, NB, S and C phases and plate-like gibbsite (γ-Al(OH)3), which can form N−, C andinrarecasesevenS [26] phases. Lyotropic liquid crystals are highly susceptible to external fields and in this paper we will review the influence of several external fields, i.e. the earth gravitational field, an electric and a magnetic field, on the phase behaviour of lyotropic liquid crystals. Additionally we will discuss the influence of several types of spatial confinement in- cluding slit-pore, 2D circular and 3D spherical confinement. This paper is organised as follows: in Sect. 2 we will discuss how slow sedimentation of particles in the earth gravitational field can lead to the formation of well-ordered liquid crystals phases, even in polydisperse samples. In Sect. 3 we will describe the influence of both external electric and magnetic fields where we make a distinction between external field-induced particle alignment in the I phase, the collective reorientation of ordered phases (N, S, C ) and field-induced phase transitions. These will be discussed in Sects. 3.1, 3.2 and 3.3 respectively. In Sect. 4 we will review the influence of confining geometries. Additionally we will discuss the experimental difficulties and recent progress in analysing confinement effects on liquid crystals. Finally we will conclude in Sect. 5.

2 Earth gravitational field Generally, the influence of the gravitational field on the formation of mineral liquid crystal phases is very important. At low volume fractions a colloidal dispersion builds up a barometric concentration profile [27,28] proportional to exp(−z/lg) where z is the vertical height and lg the gravitational length. The gravitational length is inversely proportional to the mass density difference Δρ (between the particles and the solvent) and the particle volume Vparticle

kBT lg = (1) gΔρVparticle

where g is the gravitational acceleration and kBT the thermal energy. The gravitational length of larger mineral particles can easily be 1 mm or less. The sedimentation profile for more concentrated dispersions may contain different coexisting liquid crystal phases. However, it should be borne in mind that the osmotic pressure at each interface between different phases is different and Gibbs phase rule predicting the maximum number of coexisting phases does not necessarily hold (even for a monodis- perse system). For a (multi-component) polydisperse system the chemical potential Colloidal Dispersions in External Fields 3055

for each component must be equal at each interface and also the typical gravitational length (1) can strongly vary for components with different particle volumes. Since this effect couples to the fractionation or partitioning of particles over the different phases, sedimented colloidal systems may display very rich phase behaviour. For the nematic phase e.g. the tendency that long rods order first due to their larger excluded volume [29,30] is reinforced by sedimentation and leads to additional fractionation. Generally, in sedimentation-induced phase-separated samples the more highly ordered liquid crystal phases are formed below the less ordered and disordered phases. However, for polydisperse gibbsite platelets a density inversion for the isotropic-nematic transition was found [31] which could be explained by fractionation effects [32]. Thicker platelets sediment faster but they remain, due to their smaller aspect ratio, in the isotropic phase for volume fractions (φ) at which the thinner platelets already formed the nematic phase. Liquid crystalline phases with partial positional order (S, C ) may be more difficult to form since inherent polydispersity may prevent particles to fit within the positionally ordered structures. Hence, low polydispersity rods form layer-like smectic [33,34] phases while the smectic phase destabilizes at higher polydispersity favouring the nematic phase at low density and the columnar phase at higher density [35–37]. Conversely, polydisperse monolayer α-Zirconium phosphate platelets sometimes form a smectic phase rather than a columnar liquid crystal [38]. These platelets are only polydisperse in diameter, which inhibits long range positional order in the columnar phase, whereas the layered smectic phase can easily accommodate the platelets due to their constant thickness. If sedimentation takes place sufficiently slowly, Brownian motion may permit particles to keep re- arranging [39] and fractionating. For gibbsite platelets sometimes a single columnar phase develops slowly [40,41], whereas fast sedimentation in a centrifugal field of 900 g [42] leads to the formation of many small columnar liquid crystals (the larger particles fractionating towards the bottom and giving larger repeat distances). However, under osmotic compression exerted by free polymer a hexatic columnar phase (without true long range positional order) was identified for gibbsite [43] below a region with small columnar liquid crystals. The electrostatically stabilised goethite system of board-like nanoparticles also showed marked influence of sedimentation and fractionation. This was already sus- pected from the observation of a smectic phase within a 55% polydisperse batch [44], i.e. far above the terminal polydispersity of 18% for the smectic phase of spherocylinders [36]. Careful analysis of the size distributions of coexisting isotropic, nematic, smectic and columnar phases [45] showed an almost threefold increase of the particle length over 5 cm sample height, a strongly reduced polydispersity (<28%) in the smectic phase and the exclusive presence of the longest particles in the columnar phase, which acts as a waste disposal for particles that do not fit the smectic periodicity. In contrast, a similar system of 17% overall polydispersity only displayed a gradual shift (of 10%) in the length distribution without changing its form while a columnar phase was completely absent. SAXS measurements showed an almost constant smectic layer thickness, only slightly compressed when going down in the sample. In another chromium-modified goethite system of 30% polydispersity the phase behaviour was monitored over a timespan of five years [47]. Also this sample displayed, due to sedimentation and fractionation, very rich phase behaviour: I, N, S and even two different columnar phases, rectangular centred (RC ) and rectangular simple (RS) (Fig. 1a, b), were found to coexist. Remarkably, the particles stayed highly mobile over years and the liquid crystal phases kept developing. The nematic phase was found to grow at the cost of a better ordered smectic phase, indicating that, like the columnar phase, the nematic phase adsorbs (in this case small) particles that are expelled from the smectic phase. Sedimentation and fractionation in combination with Brownian motion led to the first observation of fractionated crystallization in 3056 The European Physical Journal Special Topics

W a) T b) c)

d01 d11 d11 d20

d10 85 nm 130 nm 132 nm 101 nm

Fig. 1. Schematic drawings of the RC (a) and RS (b) phase with their characteristic cell distances shown in black. The characteristic distance RC − d20 is equal to the distance RS − d11 c) Schematic representation of the RC − RS martensitic transition. After rotating ◦ the particles in the RS phase by 45 only a little sliding of the layers over each other is needed (emphasized by the dotted lines) to obtain the RC phase. Adapted from Ref. [47]. Copyright (2012) by IOP Publishing.

a) b)

S1 C C

S2 S1

Fig. 2. (a) High resolution SAXS pattern of a coexisting RC and S phase measured in a Cr-goethite sample measured in Oct 2011 (after sedimentation for 5 years) at h = 2.5 mm. Reflections labelled S1 originate from the smectic interlayer distances and the liquid-like interactions within the layers. S2 labelled reflections originate from a different smectic domain. Reflections labelled C originate from the RC · d11 and d20 columnar reflections. The black and red dashed arrows emphasize the split Bragg peaks indicating fractionated crystallization in the smectic and columnar phase respectively. Black bar denotes 0.05 nm−1. (b) Smectic interlayer periodicity (D) determined from SAXS plotted against the distance from the bottom of the sample (h) measured over a time of roughly five years. The vertical dashed lines indicate fractionated crystallization. Reprinted from Ref. [47]. Copyright (2012) by IOP Publishing. the goethite system [47] (Fig. 2), i.e. the presence of liquid crystal domains with different periods at the same height in the sample, as observed earlier for gibbsite [48]. Fractionated crystallization was observed in the smectic phase after three years of sedimentation and after an additional two years in the columnar phase as well. Re- markably, the two columnar phases were found to spontaneously transform into each other, which was explained by the equal density of the RC and RS phase and one shared characteristic distance such that a small transformation is enough to trigger the martensitic transition between the two phases (Fig. 1). In conclusion one can say that sedimentation and fractionation are of significant importance in the mineral liquid crystal formation process. It enables the systems to locally reduce the often high polydispersity, which is inherent for anisometric Colloidal Dispersions in External Fields 3057

colloidal particles, leading to the formation of various positionally ordered liquid crystal phases and very rich phase behaviour. A beautiful illustration of the power of gravity-induced sedimentation and fractionation is the observation of a spontaneously formed large (>10 mm2) single smectic domain in a system of highly polydisperse board-like particles [47].

3 External electric and magnetic ﬁelds

The previous section describes how the influence of the earth gravitational field, through sedimentation and fractionation, leads to phase separation into well-ordered liquid crystals. This section now describes how external magnetic and electric fields can be utilized to influence and engineer liquid crystals. The effect of an external field on suspensions of anisotropic particles can roughly be subdivided into three categories: A) Particle alignment in the isotropic phase. Due to weak interparticle correlations the field-induced torque is mostly exerted onto single particles. Their alignment breaks the symmetry of the isotropic phase: a para-nematic (pN ) phase is formed.

B) Reorientation of ordered phases (N, S, C ). The torque is now applied to a whole (liquid crystal) domain. Domains will reorient and fuse to form single domains.

C) Field-induced phase transitions. The external ﬁeld can inﬂuence the phase diagram by shifting the phase boundaries and by inducing new phases.

The three categories will be discussed in subsections 3.1, 3.2 and 3.3 respectively.

3.1 Particle alignment in the isotropic phase

Theoretical work [49] conducted in 1969 explained the alignment of anisotropic molecules in an external (AC electric) field in terms of the polarizability tensor, which is generally anisotropic for anisotropic molecules. The field induced dipole moments in the molecules vary in strength along the different directions of the molecule, with the largest induced dipole moment generally along the longest molecular axis. The molecules therefore align their long axis in the field leading to an orientationally ordered liquid similar to a liquid crystal mesophase [49], nowadays known as the para-nematic (pN ) phase. The degree of induced order is expressed in the nematic order parameter, which under certain restrictive conditions, i.e. field strength, can reach values of true nematic phases. Similar behaviour applies to anisotropic colloidal particles [50]. Dielectric colloidal particles will be polarized by an applied external electric field. The largest induced dipole moment generally is along the longest particle axis, although depending on the specific internal (crystal) structure the easy axis can be along another axis. For example rod-like tobacco mosaic virus (TMV-virus) particles were found to display peculiar alignment behaviour; they align, depending on the field strength, both perpendicular and parallel to an external electric field, even though the particles do not possess a permanent dipole moment [51]. Magnetic fields are also suitable candidates for particle alignment; they are clean, contact-free and do not induce electrostatic charges [52]. The alignment principle is analogous to electric field 3058 The European Physical Journal Special Topics

induced alignment: anisotropic particles often have an anisotropic magnetic susceptibility tensor and hence the induced moment results in alignment of a specific particle axis in the field. The external field induced formation of the (para-)nematic liquid crystal phase was supported by Khoklov and Semenov who constructed theoretical phase diagrams for stiff, semi-flexible and persistent semi-flexible rod-like particles in external magnetic and electric fields of various field strengths [53]. Electric-field induced orientational order has been observed experimentally for a variety of anisometric colloids in suspension [24], e.g. in suspensions of various natural clay platelets [54,55] and V2O5 ribbon-like particles [56]. The degree of alignment depends on several factors and the dependency on the AC field frequency was determined from light scattering experiments for ellipsoidal and dumbbell latex as well as for rod-like FeOOH particles [57]. In a system of colloidal rods the dependence of the induced alignment on the field strength was determined through measuring the electro-optical response. A linear dependence on the squared field strength was found for weak fields, while the induced order saturates for sufficiently high field strengths [58]. For magnetic field induced alignment, a significant body of work has been performed by Fraden and coworkers on the alignment of colloidal rod-like virus particles. They described the alignment of TMV virus theoretically based on the Onsager theory [59] and experimentally determined the influence of concentration, polydispersity and ionic strength on the angular alignment of stiff tobacco mosaic virus particles [59,60] as well as of semi-flexible fd-virus particles [61] in the entire isotropic range. Also plate-like gibbsite particles were found to align in an external magnetic field. They align their normals perpendicular to the field indicating a negative susceptibility anisotropy and the nematic order parameter was calculated from SAXS for different field strengths [62].

3.2 Reorientation of ordered phases External fields can also be utilized to realign spontaneously formed liquid crystals. In Sect. 2 we described how liquid crystals often consist of several small domains with different orientations. By applying an external field the different domains will realign and fuse to form a large single domain. The majority of research published on this topic is devoted to external fields applied to nematic phases, and has been studied for a variety of anisotropic colloidal particles in dispersion. E.g. in a very dilute nematic phase of V2O5 ribbons it was found that the magnetic field removes the topological defects and produces a completely aligned nematic single domain [63]. More recently similar behaviour was found in a nematic phase of GdPO4 nanorods [64]. Interestingly, the nanorods migrated to regions of high magnetic field which could be attributed to the paramagnetic nature of the rods. The field-induced formation of single nematic domains was also found in suspensions of gibbsite [65], a synthetic clay, and lath-shaped nontronite [66] and disk-shaped beidellite [67,68], both natural clays. Here the magnetic field was also used as a means to distinguish between a true nematic liquid crystal, which easily aligns in the field, and a (nematic) gel, which is rather unresponsive to external fields [68]. When a strong reorienting field is suddenly applied to a single (nematic) domain, or when the direction of the field in which a single domain is formed is suddenly changed, the system is pushed far out of equilibrium and hydrodynamic instabilities can be observed [66–70]. In a sufficiently strong field the so-called Frederiks transition [70] takes place leading to deformation of the uniform director field. Studies of these phenomena allow one to determine the splay and twist elastic constants [71–73]. Recently, elastic constants were determined from the magnetic-field induced transformation of the director fields in differently sized nematic tactoids (nematic droplets Colloidal Dispersions in External Fields 3059

formed in the isotropic phase). Additionally, the I-N interfacial tension could be calculated from the deformation of the tactoids [74,75]. Usually, for rod-like colloids consisting of magnetically isotropic material it is energetically most favourable to align their long axis parallel to the magnetic field [76]. However various colloids do not display this behaviour due to the material they are composed of and/or the specific crystal structure. Examples are the earlier described beidellite [67] and Co-doped ZnO nanowires [77] which were found to align their long axis perpendicular to the field due to their magnetocrystalline anisotropy. The mineral goethite system, consisting of board-like colloidal particles (with three distinct axes) was found to have unusual and highly interesting magnetic properties. This was already discovered in the very early 1900s by Majorana, Cotton and Mouton who observed magnetic field-induced birefringence with a non-monotonic dependence on the field intensity in dispersions of iron oxides: a positive birefringence was observed in weak fields, which decreased to reach even negative values above a certain field strength [78–80]. This behaviour was not understood until a century later, when Lemaire and coworkers found that this behaviour could be attributed to anti-ferromagnetic goethite particles bearing a permanent magnetic moment along the longest axis (L) of the particles –presumably due to uncompensated surface spins– while the magnetic easy axis is along the shortest dimension of the particles (T )[81]. For this reason the particles align their L-axis parallel to weak magnetic fields, but reorient to align their T -axis parallel to fields above a certain critical magnetic field strength (B∗)[82–84]. In a system of goethite board-like particles, with dimensions L × W × T = 254 × 83 × 28 nm3, a small magnetic field ( B∗) was used to align macrodomains of spontaneously formed biaxial liquid crystal phases Fig. (3(a)). The biaxial liquid crystal structures were revealed by SAXS and the field was applied in different directions to construct a complete picture of the structures [85]. The experiments were performed at different heights in a phase separated sedimented sample and both biaxial nematic (NB, Fig. 3) as well as biaxial smectic (SB, Fig. 4) phases were found. In low magnetic fields the particles align their long axis, hence when the magnetic field is directed parallel to the x-ray beam the reflections of the shortest particle dimensions (W, T ) are revealed (Fig. 3d, e and Fig. 4b), whereas the reflections of L and W (or T ) are revealed for a perpendicular oriented magnetic field (Fig. 3f and Fig. 4c). The reflections are found in directions orthogonal to each other and at q-values of which the qW /qT and qL/qW ratios agree well with the ratios between the different particle dimensions taken into account the Debye length (10 nm at most), proving the biaxiality of the phases [85,86]. In accordance with theoretical predictions, biaxiality appeared in a system with particles that have a shape almost exactly in between rod-like and plate-like particles (L/W = W/T). For this particular system L/W is slightly larger than W/T but the particles have an overall polydispersity of 25% which stabilizes the biaxial phases [87].

3.3 Field-induced phase transitions In addition to (re)alignment and reorientation phenomena, external fields can influence the phase diagram and can induce new phases. Extensive work on electric- field-induced phase transitions in a system of charge-stabilized rod-like virus particles (fd-virus) has been performed by Dhont and coworkers [88–90]. This system forms a chiral nematic (N* ) phase in suspensions with high salt concentrations (∼mM), however this particular research focused on concentrated suspensions with a low salt concentration (0.1 mM). Here the chirality of the particles is screened by the double layer and a regular non-chiral nematic phase (N) is formed in the absence of an external field. Upon application of the electrical field, the alignment of particles is not 3060 The European Physical Journal Special Topics

T L (a) (b) (c)

B = 3 mT B = 40 mT B = 40 mT

(d) (e) (f)

Intensity 150 200 500 (g) (h) hor (i) vert Intensity Intensity Intensity 0.00 0.10 0.20 0.00 0.10 0.20 0.00 0.04 0.08 q (nm-1) q (nm-1) q (nm-1)

Fig. 3. Schematic illustrations of (a) a board-like goethite particle, (b) and (c) differently oriented NB phases corresponding to the SAXS patterns of the NB phase in a magnetic field oriented parallel to the x-ray beam of 3 mT (d) and 40 mT (e) and in a 40 mT magnetic field oriented perpendicular to the x-ray beam (f), respectively. (g), (h) and (i) show the I(q) profiles corresponding to the SAXS-patterns in (d), (e) and (f) respectively. Reprinted from Ref. [85]. Copyright (2009) by The American Physical Society. driven by polarization of the particles itself, but by polarization of the electric double layer. Kang et al. observed various transitions depending on the electric field strength and field frequency [88], e.g. transitions from N to N ∗ and to multidomain N ∗ phases (where the domain size decreases with increasing field strength). Moreover, various dynamic states, in which nematic domains continuously melt and reform, were observed. Above a certain frequency in the kHz range, in contrast to all other phases, a uniform homeotropic phase is formed which is stabilized by field-induced electro- osmotic flow. Due to peculiar magnetic properties of goethite –alignment along L in low magnetic fields, but alignment along T in high magnetic fields– a stable biaxial nematic ∗ phase (NB) is formed when a strong (above B ) magnetic field is applied to the nematic phase. This was first observed in experiments performed by Lemaire and coworkers [81,83,84,91] and biaxiality of the nematic phase was also found by theory Colloidal Dispersions in External Fields 3061

B = 3 mT B = 40 mT

(a) (b) B = 40 mT W L

Fig. 4. SAXS patterns of the SB phase in a magnetic ﬁeld oriented parallel to the x-ray beam of 3 mT (a), 40 mT (b) and in a 40 mT magnetic ﬁeld oriented perpendicular to the x-ray beam (c). (d) Schematic illustration of the SB phase corresponding to (a). Reprinted from Ref. [85]. Copyright (2009) by The American Physical Society.

[92] modelling the particles as spherocylinders. It was found that this property, in combination with the system’s inherent polydispersity, can be utilized to induce nematic- nematic phase separation in a dispersion of slightly elongated goethite boards [93,94]. The strength of the induced magnetic moment scales with particle volume, thus for larger goethite particles the critical magnetic field (B∗) is slightly lower. Applying a magnetic field with a carefully chosen field strength to the N+ phase induces a NB phase for part of the particles. The created excluded volume between the differently oriented particles is sufficient to cause macroscopic phase separation. Particle alignment in dense suspensions in a strong magnetic field is a complicated process which can lead to the formation of positionally ordered phases. It was found e.g. that applying a strong magnetic field to a dense nematic phase can trigger the N → C phase transition [84,91]. SAXS revealed the formation of single rectangular centered columnar (Fig. 1a) domains. Also the positionally ordered smectic phase was found to transform into the columnar phase in a strong (>B∗) magnetic field [44]. Later it was found that this transition strongly depends on polydispersity. Generally, systems with a low polydispersity display an I-N-S phase separation, whereas for systems with a high polydispersity an additional columnar phase is formed (I-N-S- C). Smectic phases found in I-N-S-C phase separated samples transform into the columnar phase at high field strengths, whereas smectic phases formed in dispersions with a low polydispersity, do not [95]. This is illustrated in Fig. 5. The smectic phases in a low magnetic field (Fig. 5a, c) are neatly aligned single domains. Figure 5b clearly shows that the smectic phase found in an I-N-S-C phase separated sedimentation sample is transformed into a columnar domain; the pattern reveals the 11 and 20 reflections of the RC phase. The smectic phase found in a I-N-S phase separated sample on the other hand, reorients to align perpendicular to a large magnetic field (Fig. 5d), but the phase remains smectic. Although the latter reflects single particle behaviour, it was found that the exact reorientation process is complicated. SAXS and x-ray microscopy revealed smectic-C type undulations during the reorientation process and the particles appeared to reorient more or less collectively in small domains [95]. Dispersions of chromium-modified goethite, earlier described in Sect. 2, spontaneously form two different columnar phases: the centered rectangular (RC ) and the 3062 The European Physical Journal Special Topics

B 60 mT 1 T

high polydispersity

a) b) 120 mT 1.4 T

low polydispersity

c) d) Fig. 5. SAXS patterns of smectic phases found in goethite dispersions with high (35%) and low polydispersity (17%) aligned in a weak magnetic field (a, c respectively). The smectic phase found in the highly polydisperse dispersion transforms into the columnar phase in a strong magnetic field (b), whereas the smectic phase found in a system with low polydispersity remains smectic in a strong magnetic field (d). Adapted from Ref. [95] by permission from The Royal Society of Chemistry.

simple rectangular (RS) columnar phase. These phases can spontaneously transform into each other via a simple martensitic transition (Fig. 1), but the RS → RC transition was also found to be induced by a strong magnetic field [96] because the RC phase is energetically favourable in a strong magnetic field. The T axes of the particles are fixed by the field, thereby freezing all particle rotations except the rotation around the T -axis and the RC structure leaves more room for these particular particle rotations than the RS structure. These results led to further investigation of the complex field-induced reorientation behaviour of the columnar phase. SAXS revealed deformed patterns of the transient columnar phase which imply a certain pathway of reorientation. The columnar reflections were broadened in specific directions only, which indicated that alignment proceeds via collective rotation of the domains, during which nanoshear is induced between layers of particles, which then slide over each other. These findings support the shear induced martensitic transition of the RS-RC phases [97]. Clearly, external fields are important tools to engineer liquid crystal phases. Ex- ternal fields can induce order in originally disordered suspensions of anisotropic particles, but can also induce the formation of single liquid crystal domains. Additionally, complete phase transitions to positionally ordered phases can be induced.

4 Conﬁning geometries

In the first sections of this paper the influence of the earth gravitational field as well as of external electric and magnetic fields on liquid crystal phase behaviour was reviewed. The discussed phenomena all concerned bulk properties. Anisometric colloidal particles have, due to particle-wall interactions, a favourable orientation near a wall or a surface. For example plate-like particles prefer to align their normal perpendicular to a flat wall (homeotropic anchoring), whereas rod-like particles often align their longest axis parallel to a flat wall (homogenous anchoring). For bulk samples the effect of the wall-induced alignment decays away from the wall and can often be neglected. For sufficiently confined samples though, the wall anchoring can have a significant Colloidal Dispersions in External Fields 3063

influence on the liquid crystal phase behaviour. E.g. theoretical work on suspensions of rod- [98] and plate-like particles [99] confined between two planar parallel walls (e.g. slit-pore confinement) revealed that sufficiently strong confinement lowers the I-N binodal; alignment imposed by the walls induces formation of the N phase which is now found at volume fractions for which the bulk is isotropic. This phenomenon, known as capillary nematization, vanishes again below a critical wall separation distance of some particle lengths. In another system of biaxial micellar aggregates with weak shape anisotropy the nematic phase was confined between parallel walls. Wall anchoring suppresses fluctuations around the long particle axis forcing particles to align their shortest axis perpendicular to the wall leading to a N+ → N− phase transition [100]. Regarding positionally ordered phases in confinement, extensive theoretical research was performed by de las Heras and coworkers [101–103]. They focused on smectic phase formation of hard rods in slit-pore confinement. The complete phase diagram was mapped out as a function of wall separation and chemical potential. Like capillary nematization, they predicted capillary smectization for specific wall separation distances [101]. For even smaller distances frustrated smectic ordering was predicted. Later, also the phase diagram was mapped out for a binary mixture of rods with identical diameters but dissimilar lengths. Here theory showed that confinement can induce S−S demixing [102,103]. Other theoretical work showed that the presence of hard walls inhibits the second order N-S transition; instead a first order layering transition is predicted [104]. Little experimental work has been published on positionally ordered liquid crystals of hard rod/plate-like particles in confinement. The systems that are predominantly studied are lyotropic lamellar systems of surfactant particles. The formation of a monodomain ordered structure was found for the lamellar system in slit-pore confinement, which was attributed to the surface-assisted nucleation and corresponding suppression of nucleation in the bulk [105]. A similar system in a wedge-shaped cell suffered from geometrical frustration due to the spatial gradient of the cell thickness. This resulted in the formation of edge-dislocations (the partial stacking of a layer in between two other layers) near the tip of the wedge. The individual edge dislocations could be directly observed with optical microscopy, and the coarsening of the edge dislocation was followed over time [106]. Various work has been published on wetting and filling behaviour of nematic liquid crystals of rod-like particles in patterned surfaces. Patricio and coworkers focused on three classes of periodic surfaces: triangular, sinusoidal and rectangular; their theoretical work showed that the nematic director field strongly depends on the geometry of the surface [107]. Experimental work on this topic has been performed by Dammone and coworkers who studied the director fields of nematic liquid crystals of fd-virus rod-like particles in channels with wedge structured walls for various wedge angles [108]. They found a splay-to-bend transition with increasing wedge angle. Interest- ingly, the defect formed where splay configuration meets bend configuration could be observed with laser scanning confocal microscopy (LSCM). Nematic liquid crystals confined in 2D circular nanocavities of only a few particle lengths in diameter were theoretically studied by de las Heras et al. [109,110]. Homeotropic anchoring of the rods at the surface was assumed to be energetically favorable. It was found that for small cavity radii, the cavity is free of defects; the nematic liquid crystals have a homogeneous director field at the cost of increasing the surface free energy. Conversely, for larger cavities, the director fields are inhomogeneous and point defects are formed. Experimentally, similar observations have been done on nematic (3D) tactoids in suspensions of plate-like gibbsite particles. The shape and director field of these nematic tactoids were studied as a function of tactoid size. The plate-like particles entropically prefer homeotropic anchoring whereas the elastic free energy prefers a homogenous director field and it is therefore the competition between interfacial and elastic free 3064 The European Physical Journal Special Topics

energies that determines the properties of the tactoids. Also here the director field is found to be homogeneous in small tactoids, but the tactoids are deformed and have an oblate shape. The larger tactoids remain spherical with a radial director field with a point-defect in the center [111]. Interestingly, in a similar system columnar tactoids were found to form in the nematic phase. These droplets consist of hexagonally arranged stacks of platelets leading to disk-shaped droplets [112]. Generally, tactoids reveal a mirror-symmetry non-chiral structure. However Lavrentovich and coworkers reported on chiral symmetry breaking in nematic tactoids formed in polymer-crowded solutions of elongated molecular aggregates [113]. They explained this chiral induc- tion as a replacement of the non-chiral but energetically costly splay packing with chiral twisted packing. In general, the effects of several types of confinement on suspensions of anisotropic particles have been studied extensively theoretically. Experimental research has been performed, but not to the same extent and it mainly focused on nematic liquid crystal phases of surfactant molecules. Barely any experimental research has been performed on confined positionally ordered liquid crystals of (hard) colloidal particles. The chal- lenge lies in the characterization techniques, e.g. SAXS is an excellent technique to characterize positionally ordered liquid crystals (S, C), but since confinement samples are often micrometer sized the scattered intensity is low. Difficulty also lies in characterizing the structures locally. However, colloids are on the verge of becoming accessible with the improved optical techniques. For example Lettinga and Grelet were able to directly visualize single particle transport between smectic layers of fd- virus using laser scanning confocal microscopy (LSCM) [39]. Research has been performed on adapting the plate-like gibbsite system for LSCM by increasing the particle size and by fluorescent labeling of the platelets. There are two possible ways to grow the platelets to larger sizes: by stepwise growing the seed particles to the desired size [114] or in a one-step synthesis [115]. The platelets can be made fluorescent by growing a fluorescent silica shell around the platelets [116]. Although the shortest dimension of the resulting platelets remains too small to be observed, single particles could be resolved by adding non-fluorescent platelets which increases the average distance between the fluorescently labeled particles and conse- quently liquid crystal structures could be determined from LSCM (to be published). By increasing the particle size the sedimentation speed will increase and inherently the formation of well-ordered liquid crystals can be hindered. E.g. for large gibbsite platelets in aqueous solution it was found that phase separation only took place after the main part of the sample had sedimented into an amorphous phase [114]. For sterically stabilized gibbsite in an organic solvent the hexagonal columnar phase was found, although SAXS revealed the presence of orientational fluctuations in the columnar structure; the average platelet orientation was decoupled from the column axis. These effects could be ascribed to gravitational compaction [118]. Another liquid crystal phase frequently found in fast sedimenting systems is the columnar nematic phase (NC ). In this phase where true hexagonal columnar order is inhibited the particles are stacked into short columns which in turn display nematic ordering. In the NC phase the overall particle orientation is constant, however due to strong fluctuations the stacks of particles often have different orientations. During SAXS measurements the side-to-side correlation peak therefore shifts to larger Q-values when patterns are taken while the NC sample is rotated around its vertical axis (Fig. 6). Conversely, for a well-ordered hexagonal columnar phase these side-to-side correlations remain at fixed Q [46]. It is believed though, that sufficiently strong confinement can suppress the fluctuations found in these distorted (columnar) phases. In conclusion one can say that while theory predicts confinement effects to be able to induce the formation of (well-ordered) liquid crystals, it remains challenging Colloidal Dispersions in External Fields 3065

a) b)

0°

D cos 30°

Fig. 6. (a) Schematic representations of the NC structure where short particle stacks are formed where the particle normal (ζ) and stack axis (Ξ) are decoupled due to the rotation of Ξ over an angle α causing the side-to-side correlation to be shorter according to D cos α and (b) schematic representation of the interstack correlation peak in the reciprocal space and its broadening due to δΞ variations (green lines). By rotating the sample by 30◦ the correlation peak is observed at a larger scattering vector (blue arrows). Reprinted from Ref. [46]. Copyright (2011) by IOP Publishing.

to study these eﬀects experimentally. However, recent progress with characterization techniques [119,120] and the development of colloidal systems [121,122] will hopefully enable such research.

5 Conclusions

In this paper the influence of the earth gravitational field, external electric and magnetic fields and spatial confinement on (the formation of) mineral liquid crystal phases is reviewed. From the body of work published in these fields it can be concluded that these external influences are powerful tools to engineer liquid crystal phases. They enable producing single liquid crystal domains, reducing polydispersity, reducing or inducing defects, directing liquid crystals to desired orientations and inducing the formation of (new) liquid crystal phases. The use of (different types of) external fields enables control of mineral liquid crystals in a way that might be sufficient for applications like displays (in which currently thermotropic liquid crystals are used). E.g. for application in reflective color displays the electric-field induced switching of rod-like pigment particles is currently studied [123,124]. Other published work [125], where spatial confinement is combined with an external magnetic field, reports on liquid crystal alignment with an electro-optical response up to ten times faster than in currently used systems. Also Bubenhofer and coworkers [126] report on magnetic switching of optical reflectivity in a colloidal suspension containing nanomag- nets/micromirrors as a potential alternative for liquid crystal displays. This review hopes to have shown that mineral liquid crystals in combination with external fields display phenomena of fundamental importance, which may lead to interesting future applications.

This work was supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is ﬁnancially supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) within the SFB-TR6 (project D6) research programme. The scattering data shown here are obtained at the Dutch-Belgian beamline BM-26 DUBBLE of the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. 3066 The European Physical Journal Special Topics

References

1. A. Yethiraj, A. van Blaaderen, Nature 421, 513 (2003) 2. H. Löwen, J. Phys.: Condens. Matter 13, R415 (2001) 3. I. Kretzschmar, J.H. Song, Curr. Opin. Colloid In. 16, 84 (2011) 4. T. Palberg, H. Schweinfurth, T. Köller,H. Müller,H.J. Schöpe, A. Reinmüller, Eur. Phys. J. Special Topics 222(11), 2835 (2013) 5. J. Zhao, P. Papadopoulos, M. Roth, C. Dobbrow, E. Roeben, A. Schmidt, H.-J. Butt, G.K. Auernhammer, D. Vollmer, Eur. Phys. J. Special Topics 222(11), 2881 (2013) 6. P. Dillmann, G. Maret, P. Keim, Eur. Phys. J. Special Topics 222(11), 2941 (2013) 7. H. Löwen, T. Horn, T. Neuhaus, B. ten Hagen, Eur. Phys. J. Special Topics 222(11), 2961 (2013) 8. A. Reinmüller,E.C. Oguz, R. Messina, H. Löwen, H.J. Schöpe, T. Palberg, Eur. Phys. J. Special Topics 222(11), 3011 (2013) 9. L. Onsager, Ann. N. Y. Acad. Sci. 51, 627 (1949) 10. R. Eppenga, D. Frenkel, Mol. Phys. 52, 1303 (1984) 11. A. Stroobants, H.N.W. Lekkerkerker, D. Frenkel, Phys. Rev. Lett. 57, 1452 (1986) 12. D. Frenkel, H.N.W. Lekkerkerker, A. Stroobants, Nature 332, 822 (1988) 13. J. Veerman, D. Frenkel, Phys. Rev. A 45, 5632 (1992) 14.H.Zocher,Z.Anorg.Allg.Chem.147, 91 (1925) 15. F. Bawden, N. Pirie, J. Bernal, I. Fankuchen, Nature 138, 1051 (1936) 16. J. Bernal, I. Fankuchen, J. Gen. Physiol. 25, 111 (1941) 17. K. Coper, H. Freundlich, T. Faraday Soc. 33, 0348 (1937) 18. I. Langmuir, J. Chem. Phys. 6, 873 (1938) 19. M. Freiser, Phys. Rev. Lett. 24, 1041 (1970) 20. R. Alben, Phys. Rev. Lett. 30, 778 (1973) 21. J. Straley, Phys. Rev. A. 10, 1881 (1974) 22. L. Yu, A. Saupe, Phys. Rev. Lett. 45, 1000 (1980) 23. P. Davidson, J. Gabriel, Curr. Opin. Colloid In. 9, 377 (2005) 24. J. Gabriel, P. Davidson, Adv. Mater. 12, 9 (2000) 25. H.N.W. Lekkerkerker, G.J. Vroege, Philos. Trans. R. Soc. A 371, 20120263 (2013) 26. D. Kleshchanok, P. Holmqvist, J.M. Meijer, H.N.W. Lekkerkerker, J. Am. Chem. Soc. 134, 5985 (2012) 27. J. Perrin, Les Atomes (Alcan, Paris, 1913) 28. J. Perrin, Ann. Chim. Phys. 18, 5 (1909) 29. H.N.W. Lekkerkerker, P. Coulon, R. Vanderhaegen, R. Deblieck, J. Chem. Phys. 80, 3427 (1984) 30. A. Speranza, P. Sollich, J. Chem. Phys. 117, 5421 (2002) 31. F. van der Kooij, D. van der Beek, H.N.W. Lekkerkerker, J. Phys. Chem. B 105, 1696 (2001) 32. H.H. Wensink, G.J. Vroege, H.N.W. Lekkerkerker, J. Phys. Chem. B 105, 10610 (2001) 33. Z. Dogic, S. Fraden, Phys. Rev. Lett. 78, 2417 (1997) 34. Z. Dogic, S. Fraden, Curr. Opin. Colloid In. 11, 47 (2006) 35. F. Livolant, A. Levelut, J. Doucet, J. Benoit, Nature 339, 724 (1989) 36. M. Bates, D. Frenkel, J. Chem. Phys. 109, 6193 (1998) 37. Y. Martinez-Raton, J.A. Cuesta, Mol. Phys. 107, 415 (2009) 38. D. Sun, H. Sue, Z. Cheng, Y. Martinez-Ration, E. Velasco, Phys. Rev. E 80, 041704 (2009) 39. M.P. Lettinga, E. Grelet, Phys. Rev. Lett. 99, 197802 (2007) 40. F. van der Kooij, K. Kassapidou, H.N.W. Lekkerkerker, Nature 406, 868 (2000) 41. D. van der Beek, T. Schilling, H.N.W. Lekkerkerker, J. Chem. Phys. 121, 5423 (2004) 42. D. van der Beek, P.B. Radstake, A.V. Petukhov, H.N.W. Lekkerkerker, Langmuir 23, 11343 (2007) 43. A.V. Petukhov, D. van der Beek, R.P.A. Dullens, I.P. Dolbnya, G.J. Vroege, H.N.W. Lekkerkerker, Phys. Rev. Lett. 95, 077801 (2005) Colloidal Dispersions in External Fields 3067

44. G.J. Vroege, D.M.E. Thies-Weesie, A.V. Petukhov, B.J. Lemaire, P. Davidson, Adv. Mater. 18, 2565 (2006) 45. E. van den Pol, D.M.E. Thies-Weesie, A.V. Petukhov, G.J. Vroege, K. Kvashnina, J. Chem. Phys. 129, 164715 (2008) 46. A.B.G.M. Leferink op Reinink, J.M. Meijer, D. Kleshchanok, D.V. Byelov, G.J. Vroege, A.V. Petukhov, H.N.W. Lekkerkerker, J. Phys.: Condens. Matter 23, 194110 (2011) 47. A.B.G.M. Leferink op Reinink, E. van den Pol, D.V. Byelov, A.V. Petukhov, G.J. Vroege, J. Phys.: Condens. Matter 24, 464127 (2012) 48. D.V. Byelov, M.C.D. Mourad, I. Snigireva, A. Snigirev, A.V. Petukhov, H.N.W. Lekkerkerker, Langmuir 26, 6898 (2010) 49. J. Hanus, Phys. Rev. 178, 420 (1969) 50. A. Gast, C. Zukoski, Adv. Colloid Interface Sci. 30, 153 (1989) 51. H. Asai, N. Watanabe, Biopolymers 15, 383 (1976) 52. G. Maret, K. Dransfeld, Strong and Ultrastrong Magnetic Fields and their Applications (Springer, New York 1985), p. 143 53. A. Khokhlov, A. Semenov, Macromolecules 15, 1272 (1982) 54. I. Dozov, E. Paineau, P. Davidson, K. Antonova, C. Baravian, I. Bihannic, L.J. Michot, J. Phys. Chem. B 115, 7751 (2011) 55. Z. Rozynek, B. Wang, J.O. Fossum, K.D. Knudsen, Eur. Phys. J. E 35, 8 (2012) 56. S. Lamarque-Forget, O. Pelletier, I. Dozov, P. Davidson, P. Martinot-Lagarde, J. Livage, Adv. Mater. 12, 1267 (2000) 57. T. van de Ven, M. Baloch, J. Colloid Interface Sci. 136, 494 (1990) 58. B. van der Zande, G. Koper, H.N.W. Lekkerkerker, J. Phys. Chem. B 103, 5754 (1999) 59.S.Fraden,G.Maret,D.Caspar,Phys.Rev.E48, 2816 (1993) 60.S.Fraden,G.Maret,D.Caspar,R.Meyer,Phys.Rev.Lett.63, 2068 (1989) 61. J. Tang, S. Fraden, Phys. Rev. Lett 71, 3509 (1993) 62. D. van der Beek, A.V. Petukhov, P. Davidson, J. Ferré, J.P. Jamet, H.H. Wensink, G.J. Vroege, W. Bras, H.N.W. Lekkerkerker, Phys. Rev. E 73, 041402 (2006) 63. X. Commeinhes, P. Davidson, C. Bourgaux, J. Livage, Adv. Mater. 9, 900 (1997) 64. B. Abécassis, F. Lerouge, F. Bouquet, S. Kachbi, M. Monteil, P. Davidson, J. Phys. Chem. B 116, 7590 (2012) 65. D. van der Beek, P. Davidson, H.H. Wensink, G.J. Vroege, H.N.W. Lekkerkerker, Phys. Rev. E 77, 031708 (2008) 66. L. Michot, I. Bihannic, S. Maddi, S.S. Funari, C. Baravian, P. Levitz, P. Davidson, Proc. Natl. Acad. Sci. U.S.A. 103, 16101 (2006) 67. E. Paineau, I. Dozov, K. Antonova, P. Davidson, M. Impéror, F. Meneau, I. Bihannic, C. Baravian, A.M. Philippe, P. Levitz, L.J. Michot, Mater. Sci. Eng. 18, 062005 (2011) 68. E. Paineau, I. Dozov, A.M. Philippe, I. Bihannic, F. Meneau, C. Baravian, L.J. Michot, P. Davidson, J. Phys. Chem. B 116, 13516 (2012) 69. S. Fraden, A. Hurd, R. Meyer, M. Cahoon, D. Caspar, J. Phys.-Paris 46, 85 (1985) 70. V. Freedericksz, A. Repiewa, Z. Phys. 42, 532 (1927) 71. A. Hurd, S. Fraden, F. Lonberg, R. Meyer, J. Phys.-Paris 46, 905 (1985) 72. F. Lonberg, S. Fraden, A. Hurd, R. Meyer, Phys. Rev. Lett. 52, 1903 (1984) 73. G. Srajer, S. Fraden, R. Meyer, Phys. Rev. A 39, 4828 (1989) 74. A.A. Verhoeff, R.H.J. Otten, P. van der Schoot, H.N.W. Lekkerkerker, J. Chem. Phys. 134, 044904 (2011) 75. A.A. Verhoeff, R.H.J. Otten, P. van der Schoot, H.N.W. Lekkerkerker, J. Phys. Chem. B 113, 3704 (2009) 76. E.P. Furlani, Permanent Magnet and Electromechanical Devices: Materials, Analysis and Applications (Academic Press: San Diego, CA 2001), p. 518 77. S. Zhang, C.I. Pelligra, G. Keskar, P.W. Majewski, F. Ren, L.D. Pfefferle, C.O. Osuji, ACS Nano 5, 8357 (2011) 78. Q. Majorana, C.R. Acad. Sci. 135, 159 (1902) 79. A. Cotton, Q. Majorana, C.R. Acad. Sci. 141, 317 (1905) 80. Q. Majorana, Rend. Accad. Lincei 11, 374 (1902) 3068 The European Physical Journal Special Topics

81. B. Lemaire, P. Davidson, J. Ferré, J.P. Jamet, P. Panine, I. Dozov, J.P. Jolivet, Phys. Rev. Lett. 88, 125507 (2002) 82. B.J. Lemaire, P. Davidson, J. Ferré, J.P. Jamet, D. Petermann, P. Panine, I. Dozov, J.P. Jolivet, Eur. Phys. J. E 13, 291 (2004) 83. B.J. Lemaire, P. Davidson, D. Petermann, P. Panine, I. Dozov, D. Stoenescu, J.P. Jolivet, Eur. Phys. J. E 13, 309 (2004) 84. B.J. Lemaire, P. Davidson, J. Ferré, J.P. Jamet, D. Petermann, P. Panine, I. Dozov, D. Stoenescuc, J.P. Jolivet, Faraday Discuss. 128, 271 (2005) 85. E. van den Pol, A.V. Petukhov, D.M.E. Thies-Weesie, D.V. Byelov, G.J. Vroege, Phys. Rev. Lett. 103, 258301 (2009) 86. E. van den Pol, D.M.E. Thies-Weesie, A.V. Petukhov, D.V. Byelov, G.J. Vroege, Liq. Cryst. 37, 641 (2010) 87. S. Belli, A. Patti, M. Dijkstra, R. van Roij, Phys. Rev. Lett. 107, 148303 (2011) 88. K. Kang, J.K.G. Dhont, Eur. Phys. Lett. 84, 14005 (2008) 89. K. Kang, J.K.G. Dhont, Soft Matter. 6, 273 (2010) 90. K. Kang, J.K.G. Dhont, Eur. Phys. J. E 30, 333 (2009) 91. B.J. Lemaire, P. Davidson, P. Panine, J.P. Jolivet, Phys. Rev. Lett. 93, 267801 (2004) 92. H.H. Wensink, G.J. Vroege, Phys. Rev. E 72, 031708 (2005) 93. E. van den Pol, A. Lupascu, M.A. Diaconeasa, A.V. Petukhov, D.V. Byelov, G.J. Vroege, J. Phys. Chem. Lett. 1, 2174 (2010) 94. E. van den Pol, A.A. Verhoeff, A. Lupascu, M.A. Diaconeasa, P. Davidson, I. Dozov, B.W.M. Kuipers, D.M.E. Thies-Weesie, G.J. Vroege, J. Phys.: Condens. Matter 23, 194108 (2011) 95. E. van den Pol, A.V. Petukhov, D.V. Byelov, D.M.E. Thies-Weesie, A. Snigirev, I. Snigireva, G.J. Vroege, Soft Matter 6, 4895 (2010) 96. E. van den Pol, A.V. Petukhov, D.M.E. Thies-Weesie, G.J. Vroege, Langmuir 26, 1579 (2010) 97. A.B.G.M. Leferink op Reinink, E. van den Pol, G.J. Vroege, A.V. Petukhov, J. Appl. Crystallogr. (accepted) 98.M.Dijkstra,R.vanRoij,R.Evans,Phys.Rev.E63, 051703 (2001) 99. H. Reich, M. Schmidt, J. Phys.: Condens. Matter 19, 326103 (2007) 100. J. Bonvent, I. Bechtold, M. Vega, E. Oliveira, Phys. Rev. E 62, 3775 (2000) 101. D. de las Heras, E. Velasco, L. Mederos, Phys. Rev. Lett. 94, 017801 (2005) 102. D. de las Heras, Y. Martinez-Raton, E. Velasco, Phys. Chem. Chem. Phys. 12, 10831 (2010) 103. D. de las Heras, Y. Martinez-Raton, E. Velasco, Phys. Rev. E 81, 021706 (2010) 104. A. Malijevsky, S. Varga, J. Phys.: Condens. Matter 22, 175002 (2010) 105. Y. Iwashita, H. Tanaka, Phys. Rev. Lett. 95, 047801 (2005) 106. Y. Iwashita, H. Tanaka, Phys. Rev. E 77, 041706 (2008) 107. P. Patricio, J.M. Romero-Enrique, N.M. Silvestre, N.R. Bernardino, M.M. Telo da Gama, Mol. Phys. 109, 1067 (2011) 108. O.J. Dammone, I. Zacharoudiou, R.P.A. Dullens, J.M. Yeomans, M.P. Lettinga, D.G.A.L. Aarts, Phys. Rev. Lett. 109, 108303 (2012) 109. D. de las Heras, L. Mederos, E. Velasco, Liq. Cryst. 37, 45 (2010) 110. D. de las Heras, E. Velasco, L. Mederos, Phys. Rev. E 79, 061703 (2009) 111. A.A. Verhoeff, I.A. Bakelaar, R.H.J. Otten, P. van der Schoot, H.N.W. Lekkerkerker, Langmuir. 27, 116 (2011) 112. A.A. Verhoeff, H.N.W. Lekkerkerker, Soft Matter 8, 4865 (2012) 113. L. Tortora, O.D. Lavrentovich, Proc. Natl. Acad. Sci. U.S.A. 108, 5163 (2011) 114. J. Wijnhoven, J. Colloid Interface Sci. 292, 403 (2005) 115. S. Shen, P.S. Chow, F. Chen, S. Feng, R.B.H. Tan, J. Cryst. Growth. 292, 136 (2006) 116. C. Vonk, S. Oversteegen, J. Wijnhoven, J. Colloid Interface Sci. 287, 521 (2005) 117. J. Wijnhoven, D. van ’t Zand, D. van der Beek, H.N.W. Lekkerkerker, Langmuir 21, 10422 (2005) Colloidal Dispersions in External Fields 3069

118. M.C.D. Mourad, A.V. Petukhov, G.J. Vroege, H.N.W. Lekkerkerker, Langmuir 26, 14182 (2010) 119. J. F¨olling,S. Polyakova, V. Belov, A. van Blaaderen, M.L. Bossi, S.W. Hell, Small 4, 134 (2008) 120. D.V. Byelov, J.M. Meijer, I. Snigireva, A. Snigirev, L. Rossi, E. van den Pol, A. Kuijk, A.P. Philipse, A. Imhof, A. van Blaaderen, G.J. Vroege, A.V. Petukhov, RSC Advances (in press) 121. A. Kuijk, D.V. Byelov, A.V. Petukhov, A.V. van Blaaderen, A. Imhof, Faraday Discuss. 159, 181 (2012) 122. A. Kuijk, A.V. van Blaaderen, A. Imhof, J. Am. Chem. Soc. 133, 2346 (2011) 123. R.J. Greasty, R.M. Richardson, S. Klein, D. Cherns, M.R. Thomas, C. Pizzey, N. Terrill, C. Rochas, Philos. Trans. R. Soc. A 371, 20120257 (2013) 124. A. Eremin, R. Stannarius, S. Klein, J. Heuer, R.M. Richardson, Adv. Funct. Mater. 21, 556 (2011) 125. M. Boamfa, S. Lazarenko, E. Vermolen, A. Kirilyuk, T. Rasing, Adv. Mater. 17, 610 (2005) 126. S.B. Bubenhofer, E.K. Athanassiou, R.N. Grass, F.M.Koehler, M. Rossier, W.J. Stark, Nanotechnology. 20, 485302 (2009)