# Introduction to Colloidal and Microfluidic Nematic Microstructures

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## Abstract

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## 1. Introduction

## 2. Mesoscopic Approach to Nematic Complex Fluids

#### 2.1. Landau-de Gennes Free Energy Approach

#### 2.2. Nematodynamics

## 3. Topological Defects and Nematic Colloids

## 4. Nematic Colloidal Assemblies

## 5. Stationary Nematic Microfluidic Structures

#### 5.1. Porous Nematic Microfluidics for Generation of Umbilic Defect Structures

#### 5.2. Stationary Singular Defect Structures in Junctions of Nematic Microfluidic Channels

#### 5.3. Nematic Flow Past Microfluidic Obstacles

## 6. Functionalized Colloids: Ferromagnetic Liquid Crystal Structures

#### 6.1. Suspensions of Magnetic Platelets in Liquid Crystals

#### 6.2. Dense Suspensions of Magnetic Platelets in Isotropic Fluids

## 7. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of complex nematic structures. Orange shows continuum liquid crystal phase consisting from nematic building blocks with characteristic size a. Green indicates general confinement which can be imposed either by different surfaces, such as particles and channel or cell walls. A scheme of a topological defect is shown in black.

**Figure 2.**Topological charge. (

**a**) The simplest nematic point defects: the radial and hyperbolic hedgehog. (

**b**) A colloidal particle with an accompanying defect, forming an elastic dipole and an elastic quadrupole. The image (

**b**) is reprinted and adapted with permission from the reference [38]–Copyright (2006) AAAS.

**Figure 3.**Topology of disclination lines. (

**a**) Disclinations with planar cross sections with winding numbers $+1/2$ and $-1/2$. (

**b**) Disclination lines with $-1/2$ can twist or writhe, accumulating self-linking $\mathrm{Sl}$ when closed into a loop. (

**c**) Disclinations can be reconfigured in three different ways at possible selected rewiring sites, which can emerge in different nematic colloidal assemblies.

**Figure 4.**Colloidal assemblies of spherical microparticles in a nematic under different confinement conditions. Two-dimensional crystals can be formed from (

**a**), (

**b**) dipolar-quadrupolar, or (

**c**) quadrupolar particle and defect units. (

**d**) Assembly entangled with a knotted disclination loop. (

**e**) Three-dimensional nematic colloidal crystal assembled with laser tweezers, with (right) a 3D confocal image. The images are reprinted and adapted with permission from the references: (

**a**) [121]–Copyritht (2007) APS, (

**b**) [39]–Copyright (2008) APS, (

**c**) [122]–Copyright (2008) APS, (

**d**) [109]–Copyright (2015) National Academy of Sciences, (

**e**) [40]–Copyright (2013) Springer Nature.

**Figure 5.**Knotted and linked disclinations in cholesterics and nematics. (

**a**) Transient linked cholesteric disclinations, observed by Yves Bouligand. (

**b**) A stable knotted cholesteric disclination, by Tai et al. (

**c**,

**d**) Linked and knotted disclinations stabilized by silica microspheres in a $\pi $-twisted cell. (

**e**) A complex link stabilized in a $\pi /2$-cell on a nematic 2D colloidal crystal grid. The images are reprinted and adapted with permission from the references: (

**a**) [123]–Copyright (1974) EDP Sciences, (

**b**) [124]–Copyright (2019) AAAS, (

**c**,

**d**) [125]–Copyright (2011) APS. (

**e**) [109]–Copyright (2015) National Academy of Sciences.

**Figure 6.**Nematic colloids with different particle shapes. Structures show different assembly characteristics, mediated by both the defects and the elastic deformation of the director field. Geometry and topology of the particles play a strong role in the behavior of the nematic host and can be tuned to achieve desired goals. Images are reprinted and adapted with permission from the references: (

**a**) [139]–Copyright (2010) National Academy of Sciences, (

**b**) [129]–Copyright (2013) Springer Nature, (

**c**) [133]–Copyright (2013) Taylor & Francis, (

**d**) [53]–Copyright (2009) AAAS, (

**e**) [128]–Copyright (2019) Springer Nature, (

**f**) [52]–Copyright (2017) Springer Nature, (

**g**) [126]–Copyright (2014) Springer Nature.

**Figure 7.**Porous nematic microfluidics as generator for umbilic defect lattice structures. Porous microchannels with cylindrical barriers are arranged into (

**a**,

**b**) triangular, (

**c**) square, and (

**d**) hexagonal lattices, creating: (

**a**) a triangular lattice of $+1$ umbilics and a rectangular lattice of $-1$ umbilics form, (

**b**) a hexagonal lattice of $+1$ umbilics and a Kagome lattice of $-1$ umbilics, (

**c**) a square lattice of both $+1$ and $-1$ umbilics, and (

**d**) a triangular lattice of $+1$ umbilics and Kagome lattice of $-1$ umbilic. The bottom panels show generalization of the observed structures. (

**e**) Generation of umbilic defects of variable (high) umbilic strength by flow peaks and flow saddle points. A local peak in the velocity field generates a +umbilic, and flow saddle generates a −umbilic. The image is reprinted and adapted with permission from the reference [143]–Copyright (2016) Taylor & Francis.

**Figure 8.**Flow stabilized structures in junctions of nematic-filled channels. Defect dynamics are shown in a junction with extensile flow (i.e., two outlet channels indicated by two blue arrows, and four inlet channels). Initial director profile has 2 outward escaping profiles and 4 inward escaping equilibrium configurations with a $-1$ defect at the channel junction center. Flow direction in top and bottom channels is aligned with the direction of director escape. As the nematic undergoes a flow-aligning transition in the left and right channel, a pair of $+1$ defects is created at open channel boundaries. The pair coalesces with the previously residing $-1$ defect, forming a $+1$ defect and, thus, preserving the bulk topological charge. Similarly, undergoing a flow alignment transition, $-1$ defects are created in the up and in the down channel. The defects interact and form a stationary state, consisting of a single $-1$ bulk defect, which is displaced from the center of the junction in the direction of one of the outgoing flows. Time is measured in units of nematic characteristic time scale ${\tau}_{\mathrm{N}}=\frac{{\xi}_{\mathrm{N}}^{2}}{\Gamma L}$. The image is reprinted and adapted with permission from the reference [146]–Copyright (2016) Taylor & Francis.

**Figure 9.**Nematic flow past a cylindrical barrier in nematic microchannel. (

**A**) Morphological evolution of the defect structures in the presence of flow. Defects are drawn in red; black lines show the corresponding director. (

**B**) Extension of the singular loop (measured between the pillar center and the leading end of the defect) shows a non-linear dependence with the Ericksen number Er. Insets show extension of the semi-integer defect loop with increasing the flow speed, observed between crossed-polarizers. Scale bar: 50 mm. (

**C**) Time sequence of polarized micrographs representing the flow-alignment of the nematic director in microchannel. A distinct birefringent domain (green in appearance) with a parabolic boundary is observed upstream of the micro-pillar. The image is reprinted and adapted with permission from the reference [147]–Copyright (2013) RSC.

**Figure 10.**Ferromagnetic suspension of Ba hexaferrite nanoplates in NLC: (

**a**) Magnetic nanoplates (red and blue) orient with their magnetic moments along the average order of liquid-crystal molecules (yellow ellipsoids). (

**b**) Polarized-light microscopy images of two types of antiparallel magnetic domains form with the magnetization along the NLC orientation (denoted by n). P and A indicate the orientation of the polarizer and analyzer, respectively. The upper images show the suspension in the absence of the field; in the right-hand side image, the domain walls are drawn. The bottom images show the response of the domains to a magnetic field. The image is reprinted and adapted with permission from the reference [151]–Copyright (2018) Elsevier B.V.

**Figure 11.**Nematic ferrofluid from suspension of barium ferrite nanoplatelets in n-butanol. (

**a**) Nanoplatelet suspensions viewed in transmitted light with optical polarization conditions indicated (Polarizer: magenta, P; analyzer: cyan, A). Low-volume fraction suspensions are isotropic (Iso), appearing dark between crossed polarizers. The orange/red color is due to optical absorption by the nanoplatelets. At higher concentrations ($\varphi \gtrsim 0.28$), a birefringent ferromagnetic nematic (NF) phase appears in the lower part of the cell. (

**b**) An applied in-plane magnetic field induces birefringence in the isotropic phase, with the principal axes of the optical dielectric tensor along and normal to external magnetic field and the induced macroscopic magnetization density parallel to external magnetic field. (

**c**) The NF phase is separated gravitationally from the isotropic region by a sharp, horizontal interface. Equilibrium Iso and NF structures deduced from birefringence and dichroism measurements are illustrated. (

**d**) The Iso phase is magnetized, and the Iso–NF interface becomes continuous, under applied magnetic field. Samples are sealed in rectangular glass capillaries. The boundaries of the cells are indicated by the solid thin white lines, and the air–liquid interfaces are indicated by the dashed yellow lines. The image is reprinted and adapted with permission from the reference [20]–Copyright (2016) Springer Nature.

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**MDPI and ACS Style**

Čopar, S.; Ravnik, M.; Žumer, S.
Introduction to Colloidal and Microfluidic Nematic Microstructures. *Crystals* **2021**, *11*, 956.
https://doi.org/10.3390/cryst11080956

**AMA Style**

Čopar S, Ravnik M, Žumer S.
Introduction to Colloidal and Microfluidic Nematic Microstructures. *Crystals*. 2021; 11(8):956.
https://doi.org/10.3390/cryst11080956

**Chicago/Turabian Style**

Čopar, Simon, Miha Ravnik, and Slobodan Žumer.
2021. "Introduction to Colloidal and Microfluidic Nematic Microstructures" *Crystals* 11, no. 8: 956.
https://doi.org/10.3390/cryst11080956