Nanoparticle-Stabilized Lattices of Topological Defects in Liquid Crystals

Abstract

We present a review of nanoparticle (NP) stabilized lattices of topological defects (TDs) in liquid crystals (LCs). We focus on conditions for which chiral liquid-crystalline blue phases and twist-grain boundary A phases might be stable in bulk LCs. These phases exhibit lattices of disclinations and dislocations corresponding to line TDs in orientational and translational LC order, respectively. NPs of appropriate size and surface coating can assemble in the cores of defects and stabilize metastable or increase the temperature range of already stable lattices of TDs. The stabilization is achieved by the universal defect core replacement and adaptive defect core targeting mechanisms. Representative experiments revealing these phenomena and potential applications are discussed.

Appendix: Free Energy Densities-Enforced Behavior

Here, we summarize behaviors enforced by free energy densities given by Eq. 3.

The nematic condensation term $f_{c}^{\left( n \right)}$ enforces finite value of s if orientational order is condensated. For example, the I–N phase transition in non-chiral samples (or I–N^* transformation in chiral LCs) takes place at the critical temperature $T_{IN} = T_{n}^{*} + \frac{{b_{n}^{2} }}{{4a_{n} c_{n} }}$ [27]. For $T < T_{IN}$ the equilibrium nematic order parameter $s \equiv s_{eq}$ minimizing $f_{c}^{\left( n \right)}$ reads $s_{eq} = s_{0} \left( {\frac{{3 + \sqrt {9 - 8r} }}{4}} \right)$. Here $s_{0} = \frac{{b_{n} }}{{2c_{n} }}$ and $r = \frac{{T - T_{n}^{*} }}{{T_{IN} - T_{n}^{*} }}$. Note that in equilibrium it holds $f_{c}^{\left( n \right)} \left( {T \le T_{IN} } \right) < 0$ and $f_{c}^{\left( n \right)} \left( {T > T_{IN} } \right) = 0$.

Next, we consider the nematic elastic term $f_{e}^{\left( n \right)}$. The 1st term in Eq. 3b enforces spatially homogeneous amplitude s. Nematic director field configurations are determined by the remaining splay, twist, bend, and saddle-splay term in Eq. 3b, weighted by the Frank elastic constants $K_{1} ,$$K_{2} ,$$K_{3} ,$ and $K_{24} ,$, respectively.

These term favor nematic (for $q = 0$) or cholesteric (for $q \ne 0$) order for a relatively weak saddle-splay contribution. An equilibrium nematic phase is determined by $\vec{n} \equiv \vec{n}_{eq} = \vec{e}$. Here $\vec{e}$ determines a symmetry-breaking direction, which is spatially homogeneous. In the cholesteric phase, a helical structure is formed, where $\vec{n}$ is always perpendicular to the helix axis. For instance, for the helix axis along the z-axis, a cholesteric order could be described by $\vec{n}_{eq} = \left( {\cos \left( {\text{qz}} \right),\sin \left( {\text{qz}} \right),0} \right)$ [19]. In equilibrium nematic and cholesteric phases it holds $f_{c}^{\left( n \right)} = 0$. In addition, each contribution in Eq. 3b equals to zero.

However, for a large enough value of $K_{24}$ and $q \ne 0$, BPs could be energetically advantageous due to the saddle-splay contribution [14]. Note that in the latter contribution is locally different from zero if $\vec{n}$ twists simultaneously along two orthogonal directions, where more details are given in Ref. [14]. In these structures all Frank elastic term contribute, however it holds $f_{c}^{\left( n \right)} < 0.$

The smectic condensation term $f_{c}^{\left( s \right)}$ (Eq. 3c) enforces translational LC order for low enough temperatures. For example, for a positive value of a_s in non-chiral LCs the N–SmA phase transition is of second order at the critical temperature $T_{NA} = T_{s}^{*} .$ Other scenario are presented in [29].

Finally, we focus on the smectic elastic free energy contribution $f_{e}^{\left( s \right)}$, given in Eq. 3d. In the case of spatially constant smectic amplitude $\eta$ it follows [29]

$$f_{e}^{\left( s \right)} = C_{\parallel } \eta^{2} \left| {q_{0} \vec{n} - \nabla \phi } \right|^{2} + C_{ \bot } \eta^{2} \left| {\vec{n} \times \nabla \phi } \right|^{2} .$$

For positive values of the smectic compressibility and smectic bend elastic constants the second term locally enforces parallel alignment of $\vec{n}$ and $\nabla \phi$ (i.e., smectic layer normal is aligned along $\vec{n}$). Let us assume that this is the case, where we set $\vec{n} = \left( {0,0,1} \right)$ and $\psi = \eta e^{\text{iqz}}$, corresponding to a parallel stack of smectic layers along the z-axis exhibiting the layer distance $d = 2\pi /q$. It follows $f_{e}^{\left( s \right)} = C_{\parallel } \eta^{2} \left( {q - q_{0} } \right)^{2}$, enforcing the equilibrium layer spacing $d_{eq} \equiv \frac{2\pi }{q}_{0} = d.$ Note that chirality favors $\nabla \times \vec{n} \ne 0$ which is incompatible with request $q_{0} \vec{n} = \nabla \phi$ (see Eq. 4). In this case of the compressibility term favors melting of the smectic order.