Phase behavior of nematic-nanoparticle mixtures

Hölbl, Arbresha; Ranjkesh, Amid; Abina, Andreja; Kralj, Samo; Zidanšek, Aleksander

doi:10.3389/frsfm.2023.1193904

MINI REVIEW article

Front. Soft Matter, 01 June 2023

Sec. Liquid Crystals

Volume 3 - 2023 | https://doi.org/10.3389/frsfm.2023.1193904

Phase behavior of nematic-nanoparticle mixtures

Arbresha Hölbl¹

Amid Ranjkesh²

Andreja Abina³

Samo Kralj^1,2

Aleksander Zidanšek^1,2,3*

¹Faculty of Natural Sciences and Mathematics, University of Maribor, Maribor, Slovenia
²Condensed Matter Department, J. Stefan Institute, Ljubljana, Slovenia
³Jožef Stefan International Postgraduate School, Ljubljana, Slovenia

We study the effects of nanoparticles (NPs) on thermotropic nematic liquid crystals (LCs) in relatively dilute NP–LC mixtures. We are interested in the fundamental generic mechanisms that quantitatively and qualitatively affect the phase behavior of LCs. A simple molecular field analysis shows that a phase transition will likely occur upon entry into the ordered phase. Moreover, the interaction between nematogenic NPs and LCs could force a sergeant–soldier-like behavior, in which only the phase behavior of one component is affected despite the symmetric appearance of the coupling term. When NPs are anisotropic, their influence on LC phase behavior can be qualitatively different depending on the anchoring, even in the absence of the disorder. We illustrate numerically that a random-field-type disorder might impose either short-range, quasi-long-range, or even long-range order, which might survive.

1 Introduction

Mixing soft materials with appropriate nanoparticles (NPs) can yield effective materials exhibiting new or anomalously enhanced properties (Balazs et al., 1979; Hamley, 2003). If soft materials exhibit a kind of orientational or translational order, the richness of the resulting qualitatively different effective materials explodes. Liquid crystalline (LC) materials are an excellent example. They combine a unique combination of liquid character, order, softness, and optical transparency (de Gennes, 1995; Kleman and Lavrentovich, 2003; Oswald and Pieranski, 2019). The liquid character enables relatively simple preparation of mixtures. Order in LC matrices can give rise to long-range forces among immersed NPs (Poulin et al., 1979; Pires et al., 2007). Softness can enable relatively strong responses in the LC matrix in the presence of adequate NPs (Lelidis et al., 1993; Li et al., 2006; Palffy-Muhoray, 2007). Furthermore, optical transparency enables relatively simple observation of NP-driven changes (Poulin et al., 1979; Pires et al., 2007). It should be noted that long-range forces among NPs could trigger the self-assembling of NPs, which can open different structural pathways (Hegmann et al., 2007; Bisoyi and Kumar, 2011; Lagerwall and Scalia, 2012; Lagerwall and Scalia, 2017).

Nematic (de Gennes, 1995; Kleman and Lavrentovich, 2003; Oswald and Pieranski, 2019) orientational order represents the simplest LC phase structure. It is described by the nematic molecular field. In bulk nematic equilibrium, the molecular field is uniaxial and spatially homogeneously aligned along a single symmetry-breaking orientation. Local uniaxial order is commonly described in terms of the nematic director field $\vec{n}$ and the nematic order parameter S. The unit vector points along the local uniaxial direction, where the states $\pm \vec{n}$ are physically equivalent. The amplitude field S describes the amount of ordering along $\vec{n} .$ In thermotropic LCs, one commonly obtains a nematic phase by cooling it from the isotropic (ordinary liquid) phase in which S = 0. Perturbed nematic order could generally exhibit biaxial states, which requires description in terms of the tensor nematic order parameter $\underline{Q}$ (see Supplementary Appendix SA).

Appropriate NPs could influence the nematic order via different generic mechanisms if inserted into nematic LCs. Important controlling parameters are the concentration of NPs, geometrical shape (Bellini et al., 2000; Kyrou et al., 2018; Kyrou et al., 2020; Skarabot et al., 2022), NP surface treatment (Nobili and Durand, 1992; Lavric et al., 2013a; Oswald and Pieranski, 2019; Kyrou et al., 2020), material characteristics of NPs (Mertelj et al., 2013; Kumar, 2014; Moghadas et al., 2015; Poursamad and Hallaji, 2017; Emdadi et al., 2018a; Emdadi et al., 2018b; Drozd-Rzoska et al., 2019; Jahanbakhsh et al., 2019; Poursamad and Emdadi, 2019; Lahiri et al., 2020; Bury et al., 2022; Ma et al., 2022; Sumandra et al., 2022), and LC material properties (de Gennes, 1995; Kleman and Lavrentovich, 2003; Oswald and Pieranski, 2019). Furthermore, NPs could effectively impose qualitatively different disorders on the LC order (Harris et al., 1973; Cleaver et al., 1996; Crawford and Žumer, 1996; Radzihovsky and Toner, 1997; Popa-Nita and Kralj, 2006). Most studies focusing on the impact of the disorder are performed in LC-aerosil mixtures, which can exhibit at least three qualitatively different disorder characteristics (Jin and Finotello, 2001; Bellini et al., 2002; Leon et al., 2004; Rotunno et al., 2005; Buscaglia et al., 2006; Cordoyiannis et al., 2006; Relaix et al., 2011). Furthermore, NPs could enforce LC matrix topological defects (TDs) (Mermin, 1979; Kurik and Lavrentovich, 1988) in the nematic orientational order. TDs correspond to topologically protected, elastically distorted regions in the orientational order. Nematic LCs could host either point or line defects (Schopohl and Sluckin, 1987; Kurik and Lavrentovich, 1988; Lavrentovich, 1998; Kralj and Virga, 2001). Generally, TDs can strongly interact with NPs, opening the door to predetermined and controlled superstructures stabilized by TD–NP interactions (Kikuchi et al., 2002; Yoshida et al., 2009; Karatairi et al., 2010; Liu et al., 2010; Rozic et al., 2011; Coursault et al., 2012; Wang et al., 2012; Lavric et al., 2013a; Lavric et al., 2013b; Liu et al., 2018).

This paper considers different NP-driven generic mechanisms via which NPs impact nematic thermotropic behavior.

2 Results

Mixtures of nematic LCs and NPs can exhibit complex configurations. In the following, we present some qualitatively different scenarios and discuss generic mechanisms.

Supplementary Appendix SB illustrates a simple mean-field Maier–Saupe analysis that reveals the type of coupling terms in a mixture of two nematic mesogens. It yields the following free energy density expression:

f \approx f_{0} + (1 - p) (a_{0} (T - (1 - p) T^{*}) S^{2} - b S^{3} + c S^{4}),

+ p (A_{0} (T - p T_{2}^{*}) S_{2}^{2} - {B S}_{2}^{3} + C S_{2}^{4}) - w S S_{2} . (1)

Here, $S \equiv S_{1}$ and $S_{2}$ label the nematic order parameters of the first and second components. Their volume concentrations are given by $p_{1} = 1 - p$ and $p_{2} = p$ , respectively. Quantities $a_{0},$ b, c, $T^{*} \equiv T_{1}^{*},$ $A_{0}, B,$ C, and $T_{2}^{*}$ are material constants, in which we neglect temperature dependencies in the temperature windows of our interest. Furthermore, $f_{0}$ determines the contributions of the remaining degrees of freedom.

Let us suppose that the first and second components represent LC molecules and NPs, respectively, and the volume concentration of NPs is given by $p \equiv p_{2}$ . The presence of NPs introduces an additional contribution equal to $p (1 - p) a_{0} T^{*} S^{2}$ in the LC condensation term in Eq. 1.

This term resembles the structure of the so-called Flory–Huggins free energy contribution (Onsager, 1949; Flory, 1956; Doi, 1981):

f_{F H} = p (1 - p) χ, (2)

where $χ$ stands for the Flory–Huggins constant. For a positive value of $χ$ , this term enforces phase separation if $χ$ exceeds the critical value $χ_{c} .$ Suppose that in the absence of ordering, it holds ${χ^{(I)} < χ}_{c}$ , where $χ^{(I)}$ denotes the Flory–Huggins constant in the isotropic phase (where S = 0). On entering the nematic phase, the effective Flory–Huggins interaction increases:

χ^{(N)} = χ^{(I)} + a_{0} T^{*} S^{2} . (3)

In most LCs, the second term, “switched-on” in the nematic, is relatively strong. Consequently, on cooling an isotropic mixture to the nematic phase, phase separation is generally likely to occur (Anderson et al., 2001). Figure 1 illustrates a typical phase separation in a mixture of nematic and spherical NPs (i.e., $S_{2} = 0$ ) on entering the nematic phase. In the phase separation regime, the regions exhibiting essentially strong and low nematic LC order coexist. These regions exhibit relatively low and high concentrations of NPs, respectively.

FIGURE 1

FIGURE 1. Typical phase separation pattern on entering the nematic LC phase in a nematic-NP mixture. In the case shown, NPs are spherical. Regions are exhibiting i) relatively strong nematic and ii) essentially isotropic (or paranematic) order, which are i) poor and ii) rich in NP content, respectively.

Let us suppose that the mixture remains homogeneous. A simple Maier–Saupe-type analysis (Humphries et al., 1972) presented in Supplementary Appendix SB suggests that in a mixture of two nematogenic components, a free energy density $f^{(c o u p l)}$ term coupling ordering of both ingredients arises; see the last term in Eq. 1. It should be noted that despite its symmetric structure (e.g., in Eq. 1), the coupling term reads $f^{(c o u p l)} = - w S_{1} S_{2}$ ); such terms have significantly different impacts on the two components. Here, $S_{i}$ determines the amplitude of nematic ordering in the ith component. Furthermore, in real samples, different structures of coupling terms arise depending on the type of NPs and their surface treatment. Figure 2 illustrates the “sergeant–soldier” type of behavior, where we consider the simplest possible term (Holbl et al., 2022),

f^{(c o u p l)} = - w S_{1}^{2} S_{2}^{2}, (4)

which does not affect the qualitative order-disorder behavior of individual components. For simplicity, we assume that both components have identical condensation material constants with the exceptions of bulk phase transition temperatures of isolated components and w > 0 (i.e., this term tends to increase the degree of order in both components). We assume that an isolated ith component exhibits an isotropic-nematic phase transition at $T_{i}^{(c)}$ , where $T_{2}^{(c)} > T_{1}^{(c)}$ . Thus, the coupling term is absent in the regime ${T > T}_{2}^{(c)}$ , where both order parameters are melted. However, in the temperature regime $T < T_{2}^{(c)}$ , the second component condensates. For this reason, the first mesogen experiences the effective free energy density ordering field contribution

f^{(c o u p l)} = - w_{e f f} S_{1}^{2}, (5)

where ${w_{e f f} = w S}_{2}^{2} > 0 .$ This term renormalizes the phase transition of the first component. It holds

T_{e f f}^{(c)} = T_{1}^{(c)} + \frac{w_{e f f}}{a_{1}}, (6)

where $T_{e f f}^{(c)}$ determines the enhanced phase transition of the first component. Therefore, increasing the value of the coupling strength raises the phase transition temperature of the first component, while the phase transition of the second component is unaffected. The resulting temperature behavior of the mixture is shown in Figure 2. It should be noted that in the case of bilinear coupling (see the last term in Eq. 1), the first component also exhibits pre-transitional effects in the temperature interval $T \in [T_{e f f}^{(c)}, T_{2}^{(c)}]$ if the effective coupling strength is below some critical value. If the latter value is exceeded, the degree of order $S_{1}$ exhibits gradual (noncritical) increase with decreasing temperature below $T_{2}^{(c)}$ .

FIGURE 2

FIGURE 2. “Sergeant–soldier” behavior. For the dimensionless coupling strength $σ = w / w_{c}$ , below the critical value (corresponding to $w_{c}$ ), on increasing $σ$ , the phase transition temperature of “soldier” ( $S_{1})$ monotonously increases(see figures A–D). On the other hand, the phase transition of “sergeant” ( $S_{2})$ is in this regime independent of $σ$ . All graphs from figures (A–D) and presented together in figure (E). In the simulation, we used two identical nematogens, which for $w = 0$ exhibit 1st order transition and dimensionless temperatures $T_{1}^{(c)} = 100$ and $T_{2}^{(c)} = 105 .$

Next, we consider a dilute mixture where the second component consists of anisotropic particles. For illustration, we assume that the free energy F consists of LC condensation, LC nematic elastic, and NP–LC interfacial free energy contributions, where details are given in Supplementary Appendix SA. Of our interest is the impact of NPs on the LC order–disorder phase transition. We assume that NPs relatively weakly disrupt the nematic director field, as illustrated in Figure 3. In the nematic phase, $\vec{n}$ is essentially homogeneously aligned along a single symmetry-breaking direction. We assume that a nanoparticle locally enforces orientation along the unit vector $\vec{e}$ , which is allowed to fluctuate. We also set S to be essentially spatially homogeneous, i.e., $S \sim \bar{S}$ , where $\bar{(\dots)}$ indicates the spatial averaging over the system’s volume V. Furthermore, we neglect spatial variations in the nematic director field. With this in mind, it follows (see Supplementary Appendix SA)

\frac{F}{V} \sim a_{0} (T - T^{*}) S^{2} - b S^{3} + c S^{4} - p \frac{a_{N P} w S}{v_{N P}} S_{2}, (7)

where $S_{2} \equiv {\bar{P}}_{2} = \bar{\frac{(3 {(\vec{e} ∙ \vec{n})}^{2} - 1)}{2}}$ measures the orientational order of anisotropic NPs. Quantities $N_{N P}$ , $a_{N P}$ , and $v_{N P}$ stand for the number of NPs, NP’s surface area, and NP’s volume, respectively. For weakly interacting NPs, it roughly holds (van der Schoot et al., 2008)

S_{2} \sim \frac{\int P_{2} e^{α x} d x}{\int e^{α x} d x}, (8)

where $\vec{e} ∙ \vec{n} = x \in [- 1,1]$ , $α = a_{N P} w S / (k_{B} T)$ , k_B, and we imposed cylindrical symmetry.

FIGURE 3

FIGURE 3. Dilute nematic LC—anisotropic NP mixture, where NPs relatively weakly interact with the nematic LC host. Consequently, the nematic LC orientational order, described by the nematic director field $\vec{n}$ , is essentially spatially homogeneous. Local anisotropic NP orientation is determined by the unit vector $\vec{e}$ .

In the limit $α ≫ 1$ , it holds $S_{2} \sim 1$ . Eq. 7 yields

\tilde{f} \sim r s^{2} - 2 s^{3} + s^{4} - σ s, (9)

where $r = \frac{T - T^{*}}{T_{I N} - T^{*}}$ is the reduced temperature, $\tilde{f} = f / (a_{0} (T_{I N} - T^{*}) S_{0}^{2})$ , s = S/S₀ stands for the scaled nematic order parameter, the scaling unit $S_{0} = \frac{b}{2 c}$ measures the nematic order at $T_{I N}$ in bulk equilibrium, and $σ \sim p \frac{a_{N P} ξ_{I N}^{2}}{v_{N P} d_{e}}$ is the effective dimensionless field conjugated to the order parameter. Quantities $ξ_{I N}$ and $d_{e}$ stand for the nematic correlation length at $T_{I N}$ and the surface extrapolation length (see Eq. (A6)). The latter length describes the effective strength of LC–NP coupling.

The phase behavior of this effective system is as follows. Typical nematic order parameter temperature variations are depicted in Figure 4. For $σ < σ_{c}$ = 0.5, the system exhibits a first-order transition at $r_{c} = 1 + σ$ , where the bulk I–N phase transition corresponds to $r_{I N} = 1$ . On the other hand, for $σ \geq σ_{c}$ , the system exhibits a gradual evolution of nematic order with decreasing temperature.

FIGURE 4

FIGURE 4. Nematic LC order $s = S / S_{0}$ variations on increasing dimensionless temperature $r = \frac{T - T^{*}}{T_{I N} - T^{*}}$ for different values of the effective field $σ$ . The latter arises due to the effective LC–NP coupling. The upper curve corresponds to $σ = 0.75$ , and the curves below refer to cases $σ = 0.5$ , $σ = 0.25$ , and $σ = 0,$ respectively. In the regime $σ$ < $σ_{c} \equiv 0.5$ , systems exhibit first-order transition between the isotropic ( $σ = 0$ ) or paranematic ( $σ > 0$ ) and nematic phase, which takes place at the dimensionless critical temperature $r_{c} = 1 + σ$ . The dashed curves describe the degree of nematic order at the free energy density maximum separating paranematic (or isotropic) and nematic (meta) stable states. It should be noted that for $σ = 0$ ( $σ = 0.25)$ , the global nematic minimum persists in the regime $r < r_{c} = 1$ ( $r < r_{c}$ = 1.25). In the remaining regime, the isotropic (paranematic) phase corresponds to the global minimum.

If $α \sim 1$ , the following roughly holds: $S_{2} \equiv {\bar{P}}_{2} \sim \frac{a_{N P} w \bar{S}}{k_{b} T}$ . In this case, NPs only renormalize the I–N phase transition temperature. It should be noted that we have neglected the nematic director field distortions in the previously considered estimates.

In the following, we discuss cases where NPs introduce a certain degree of randomness. The I–N phase transition is extremely susceptible to disorder due to the existence of Goldstone modes in the nematic director field in the nematic phase (Kleman and Lavrentovich, 2003; Palffy-Muhoray, 2007). These are the consequences of continuous symmetry breaking, via which bulk nematic equilibrium is established. According to the Imry–Ma theorem (Larkin, 1970; Imry and Ma, 1975), even infinitesimally weak random-field-type disorder is sufficient to break the long-range order (LRO) of the pure bulk phase, which is reached via continuous symmetry breaking. The resulting phase should possess short-range order (SRO), and the resulting domain-type pattern is predicted to be dominated by a single characteristic domain size $ξ_{d} \propto w_{R F}^{- \frac{2}{4 - d}}$ . Here, $w_{R F}$ measures the disorder strength, and d is the space-dimensionality. Therefore, in effectively d = 2 or d = 3 systems, one expects $ξ_{d} \propto w_{R F}^{- 1}$ and $ξ_{d} \propto w_{R F}^{- 2}$ , respectively. However, several later experimental and theoretical systems reveal that the resulting behavior is much more complex. For appropriate conditions, configurations exhibiting quasi-long-range order (QLRO) (Denholm and Sluckin, 1993; Chakrabarti, 1998; Feldman, 2000; Cvetko et al., 2009; Ranjkesh et al., 2014) or even LRO (Chakrabarti, 1998; Bisoyi and Kumar, 2011; Ranjkesh et al., 2014) are reported.

A convenient model to study the impact of NP-imposed randomness on the I–N phase transition is the Lebwohl–Lasher lattice model (Lebwohl and Lasher, 1972). Its key ingredients are presented in Supplementary Appendix SC. Using it, we probe the impact of NP-imposed random-type behavior on the LC order–disorder phase transition in the orientational order. Despite its simplicity, this approach well captures the essential features of LCs. In modeling, we vary the concentration p of sites by imposing a randomly selected orientation with finite disorder strength W. In typical studies focusing on a system’s range of orientational order, one commonly measures or calculates the orientational order correlation function G(r). This correlation function measures how orientational correlations decay with relative distance r. However, several studies (Denholm and Sluckin, 1993; Cvetko et al., 2009; Ranjkesh et al., 2014) reveal that G(r) dependence relatively poorly distinguishes between SRO and QLRO. Numerically, it is computationally more effective to extract the range of order using finite-size analysis (Eppenga and Frenkel, 1984; Ranjkesh et al., 2014). Namely, according to the central limit theorem, one expects that ${S \equiv \bar{P}}_{2}$ in the case of SRO exhibits scaling behavior

{\bar{P}}_{2} \propto L^{- γ}, (10)

where $γ = 3 / 2$ in d = 3. If QLRO replaces SRO, it follows $0 < γ < 3 / 2$ . On the other hand, LRO is fingerprinted by $γ = 0$ .

Our simulations used this finite-scaling approach to determine the range of order on the varying concentration of NPs, their imposed anchoring strength W, and temperature. Furthermore, in the presence of disorder, one expects history-dependent behavior. For this reason, we probed systems’ behavior for three different histories, which we refer to as i) annealed history (AH), ii) temperature-quenched history (TQH), and (iii) field-quenched history (FQH). In AH, we gradually decreased temperature stepwise and calculated the structure at a given T by using as the initial “seed” structure the fixed-point configuration calculated at the previous T. In QH, we calculated the structure at a given T by originating from the isotropic phase. Finally, in FQH, the “seed” structure was spatially homogeneously aligned along a single symmetry-breaking direction. In Figures 5, 6, we plot $γ$ for varying several control parameters. In Figure 5, we focus on W-driven behavior for different concentrations of NPs using AH deep in the nematic phase. In Figure 6, we present a more detailed impact of W on LC configurations at one concentration of NPs where we vary the history of samples and probe two different temperatures below the bulk I–N phase transition temperature. One sees that SRO (i.e., configurations characterized by $γ = 3 / 2$ can be reached only for relatively high values of W, especially in diluted samples. The control parameters T and W are given in the dimensionless scaled form. The references are as follows: bulk I–N phase transition is realized at $T \sim 1.1$ , and W is measured with respect to LC–LC neighboring molecular interactions. It should be noted that for low enough values of W and p, one observes LRO.

FIGURE 5

FIGURE 5. Variation of $γ$ on increasing the dimensionless anchoring strength W for different concentrations of NPs, which enforce random-field-like disorder for AH. It should be noted that for p = 0.1, one does not achieve SRO, which is anticipated according to the Imry–Ma theorem.

FIGURE 6

FIGURE 6. Variation of $γ$ on increasing the dimensionless anchoring strength W for different dimensionless temperatures T and histories of samples. For W > 1, relatively strong memory effects are observed. Furthermore, in the regime W < 0.2, simulations predict LRO.

3 Conclusion

We illustrated several mechanisms via which NPs could quantitatively or qualitatively modify the phase behavior of the bulk nematic LC phase. The simplicity of nematic ordering explodes into a rich palette of behaviors enabled by varying control parameters. In the study, we addressed only a few of them: concentration of NPs, nature and strength of NP–LC interactions, and anisotropy of NPs. The diversity of phenomena becomes even wider if the ferromagnetic, ferroelectric, or multiferroic properties of NPs are included. The ferromagnetic nanoparticles dispersed within the liquid crystals provide a new way to develop magneto-optic devices (Ji et al., 2019) that can also be used for THz filtering and modulation, in which sensing applications are becoming increasingly widespread (Abina et al., 2022).

In this paper, we focused only on phase behavior. It should be noted that structural behavior could be even richer. For instance, recent technological advances enable the stabilization of nematic structures incorporating diverse configurations of TDs, i.e., lattices of disclinations (Culbreath et al., 2011; Ackerman et al., 2012a; Ackerman et al., 2012b; Evans et al., 2013; Murray et al., 2014; Glazar et al., 2015; Guo et al., 2016; Peng et al., 2017; Wang et al., 2017; Yu et al., 2019; Sohn et al., 2020; Guo et al., 2021) or other assemblies (Harkai et al., 2020). Such structures could be further modified by appropriate NPs, which could make them more robust or introduce additional functionalities in the system. For example, conductive NPs assembled within a line defect can form a conductive wire (Coursault et al., 2012). Furthermore, recent experiments demonstrated (Harkai et al., 2020) that one could efficiently switch among competing line-defect structures using external fields. If these well-controllable defects would drag trapped NPs with them, one could construct rewritable networks of NPs, which could open doors to diverse applications.

Author contributions

AH, SK, AA, and AZ contributed to the conception and design of the study. AR organized and performed the experimental part. AH wrote the first draft of the manuscript. All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Funding

The authors acknowledge support from the Slovenian Research Agency grants P1-0099, P2-0348, and J1-2457.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/frsfm.2023.1193904/full#supplementary-material

References

Abina, A., Puc, U., and Zidanšek, A. (2022). Challenges and opportunities of terahertz technology in construction and demolition waste management. J. Environ. Manage 315, 115118. doi:10.1016/j.jenvman.2022.115118

PubMed Abstract | CrossRef Full Text | Google Scholar

Ackerman, P. J., Qi, Z., Lin, Y., Twombly, C. W., Laviada, M. J., Lansac, Y., et al. (2012). Laser-directed hierarchical assembly of liquid crystal defects and control of optical phase singularities. Sci. Rep. 2, 414. doi:10.1038/srep00414

PubMed Abstract | CrossRef Full Text | Google Scholar

Ackerman, P. J., Qi, Z., and Smalyukh, . Optical generation of crystalline, quasicrystalline, and arbitrary arrays of torons in confined cholesteric liquid crystals for patterning of optical vortices in laser beams. Phys. Rev. E. 2012;86, 021703, doi:10.1103/physreve.86.0217032, 1).

PubMed Abstract | CrossRef Full Text | Google Scholar

Anderson, V. J., Terentjev, E. M., Meeker, S. P., Crain, J., and Poon, W. C. K. (2001). Cellular solid behaviour of liquid crystal colloids - 1. Phase separation and morphology. Eur. Phys. J. E 4 (1), 11–20. doi:10.1007/pl00013680