Colloidal nanoparticles in liquid crystals: Bulk properties, biaxiality and untwisting in cholesterics

TL;DR

This work develops a homogenized Landau–de Gennes framework to model dilute suspensions of colloidal nanoparticles in nematic and cholesteric liquid crystals. By integrating NP–LC surface energies into a bulk term $f_{hom}$ and forming a modified bulk energy $f_m$, the authors show that NPs can eliminate the first-order isotropic–nematic transition, stabilize biaxial phases, and impose symmetry through the NP boundary tensor $\mathbf{X}$. In spatially inhomogeneous settings, particularly cholesteric-filled channels, NP-induced symmetry breaking drives temperature-dependent untwisting transitions with reversible hysteresis in large domains, enabling tunable optical responses. The analysis combines explicit critical-point calculations, asymptotics for limiting NP strengths, and numerical simulations (including gradient flows) to illustrate how NP properties control bulk and confined LC configurations. The results suggest broad applicability to diverse NP geometries and surface energies, offering a route to engineer novel LC phases and responsive metamaterials.

Abstract

We study the effects of colloidal nanoparticles (NPs) in liquid crystal samples in the dilute limit, in a Landau--de Gennes theoretical framework. The effects of the suspended NPs are captured by a homogenized energy, as outlined in~\cite{canevari2020design}. For spatially homogeneous samples, we explicitly compute the critical points and minimizers of the modified Landau--de Gennes energy and show that the presence of NP eliminates the first-order isotropic-nematic phase transition, stabilises elusive biaxial phases over some temperature ranges and that the symmetry of the NP boundary conditions or surface treatments dictates the bulk equilibrium phase at high temperatures. We also numerically demonstrate structural transitions from twisted helical director profiles to untwisted director profiles in cholesteric-filled channel geometries, driven by the collective effects of the NPs and increasing temperature. These transitions are reversible upon lowering the temperature in sufficiently large domains, where thermal hysteresis can also be observed. This behaviour opens interesting avenues for tuning the optical properties of confined, nano-doped cholesteric systems.

Figures (8)

• Figure 1: Critical points of $f_m$ for prolate uniaxial $\mathbf{X}$ (about $x$-axis) with $\pazocal{B}=1.5$, $\Sigma_s=4\pi$, $W=0.5$ and $(s_*,r_*)=(1,0)$. Recall that $\pazocal{A} = \tilde{\pazocal{A}} - \Sigma_s W$. In all plots, solid lines indicate stable critical points and dashed lines indicate unstable critical points. Similarly, blue lines indicate prolate symmetry and red ones oblate symmetry. (a) Three uniaxial solutions about $x$-axis, which as $\pazocal{A}\to-\infty$ approach the two uniaxial solutions about $x$-axis and the isotropic branch. (b) Biaxial solutions, which as $\pazocal{A}\to-\infty$ approach the uniaxial solutions about $y$-axis. (c) Biaxial solutions, which as $\pazocal{A}\to-\infty$ approach the uniaxial solutions about $z$-axis. (d) Biaxiality parameter $\beta$ pertaining to plot (b) and (c).
• Figure 2: Critical points of $f_m$ for oblate uniaxial $\mathbf{X}$ (about $x$-axis) with $\pazocal{B}=1.5$, $\Sigma_s=4\pi$, $W=0.5$ and $(s_*,r_*)=(-1,0)$. In all plots, solid lines indicate stable critical points and dashed lines indicate unstable critical points. Similarly, blue lines indicate prolate symmetry and the red ones oblate symmetry. (a) Three uniaxial solutions about the $x$-axis. (b) Two biaxial solutions about the $y$-axis. (c) Two biaxial solutions about the $z$-axis. (d) Biaxiality parameter $\beta$ pertaining to plot (b) and (c).
• Figure 3: The critical points of $f_m$ for prolate biaxial $\mathbf{X}$ (about $x$-axis) with $\pazocal{B}=1.5$, $\Sigma_s=4\pi$, $W=0.5$ and $(s_*,r_*)=(1,1/4)$; the biaxiality parameter of $\mathbf{X}$ is $\beta_*=0.44$. In all plots, solid lines indicate stable critical points and dashed lines indicate unstable critical points. Similarly, blue lines indicate prolate symmetry and the red ones oblate symmetry. (a) Three biaxial solutions about the $x$-axis. (b) Two biaxial solutions about the $y$-axis. (c) Two biaxial solutions about the $z$-axis. (d) The biaxiality parameter $\beta$ of the seven solutions.
• Figure 4: The critical points of $f_m$ for oblate biaxial $\mathbf{X}$ (about $z$-axis) with $\pazocal{B}=1.5$, $\Sigma_s=4\pi$, $W=0.5$ and $(s_*,r_*)=(1,3/4)$; the biaxiality parameter of $\mathbf{X}$ is $\beta_*=0.44$. In all plots, solid lines indicate stable critical points and dashed lines indicate unstable critical points. Similarly, blue lines indicate prolate symmetry and the red ones oblate symmetry. (a) Three biaxial solutions about the $x$-axis. (b) Two biaxial solutions about the $y$-axis. (c) Two biaxial solutions about the $z$-axis. (d) The biaxiality parameter $\beta$ of the seven critical points.
• Figure 5: Numerical solutions of \ref{['q1']}-\ref{['q5']} subject to the Dirichlet conditions in \ref{['dirichlet']} that demonstrate the transition from the uniaxial helical $\mathbf{Q}$-field to an uniaxial (with thin biaxial boundary layers at $z=0,1$) untwisted $\mathbf{Q}$-field, as $\pazocal{A}$ is increased. The parameter values are $\lambda=1000$ with $W=0.5$, $\pazocal{B}=1.5$, $\eta=4.5$, $\sigma=4\pi$, $\Sigma_s=4\pi$ (same parametric values are used in subsequent figures) and $\mathbf{X} = \left(\mathbf{e}_x\otimes\mathbf{e}_x -\frac{1}{3}\mathbf{I}\right)$. The two snippets on the left in each subfigure shows the dominant director field of $\mathbf{Q}$, projected onto $xz$- and $yz$-planes. The top row pertain to $\pi$-twisting helices and the bottom row to $2\pi$-twisting helices. In both cases, the helical $\mathbf{Q}$-fields disappear roughly around $\pazocal{A}=-5$ and this transition occurs at lower temperatures for larger $\lambda$.