Magneto-optic dynamics in a ferromagnetic nematic liquid crystal

We investigate dynamic magneto-optic effects in a ferromagnetic nematic liquid crystal experimentally and theoretically. Experimentally we measure the magnetization and the phase difference of the transmitted light when an external magnetic field is applied. As a model we study the coupled dynamics of the magnetization, $\mathbf{M}$, and the director field, $\mathbf{n}$, associated with the liquid crystalline orientational order. We demonstrate that the experimentally studied macroscopic dynamic behavior reveals the importance of a dynamic cross-coupling between $\mathbf{M}$ and $\mathbf{n}$. The experimental data are used to extract the value of the dissipative cross-coupling coefficient. We also make concrete predictions about how reversible cross-coupling terms between the magnetization and the director could be detected experimentally by measurements of the transmitted light intensity as well as by analyzing the azimuthal angle of the magnetization and the director out of the plane spanned by the anchoring axis and the external magnetic field. We derive the eigenmodes of the coupled system and study their relaxation rates. We show that in the usual experimental setup used for measuring the relaxation rates of the splay-bend or twist-bend eigenmodes of a nematic liquid crystal one expects for a ferromagnetic nematic liquid crystal a mixture of at least two eigenmodes.

Figure 1

Top: Sketch of the experimental setup and definition of coordinate axes [30]. The thick yellow arrows indicate the direction of the light passing through the polarizer and the analyzer. In the absence of an applied magnetic field ($\mathbf{H}$, $z$ direction), the equilibrium director ($\mathbf{n}$) and magnetization ($\mathbf{M}$) fields are only slightly pretilted from the $x$ direction. Inset: Distortion of the NLC director (ellipsoids, schematic) prevents flocculation of the suspended nanoplatelets carrying a magnetic moment ${\mathbf{p}}_m$ parallel to $\mathbf{n}$ in equilibrium. Bottom: Ferromagnetic E7 nematic 20-$\mu \mathrm{m}$ sample placed between crossed polarizers, with the director at an angle of ${45}^{\circ }$. Polarizing optical microscopy image width corresponds to 700 $\mu \mathrm{m}$. The spheres are cell spacers.

Figure 2

Comparison of experimental and theoretical static results. Top: Normalized phase difference $r\left(H\right)$. Bottom: Magnetization component $M_z$ as functions of the magnetic field ${\mu }_{0}H$.

Figure 3

For low magnetic fields, the numerically calculated polar angle of the director is in agreement with Eq. (50).

Figure 4

For large magnetic fields, the numerically calculated polar angle of the director is in agreement with Eq. (57).

Figure 5

Comparison of numeric and analytic results at low and high values of the applied magnetic field. Top: Magnetic field dependence of the normalized phase difference for small magnetic fields is in agreement with Eq. (52) below 0.5 mT, whereas the approximation for large magnetic fields, Eq. (61), is within 1% of the numerical value already when above 0.8 mT. Bottom: Magnetic field dependence of the $z$ component of the magnetization for small magnetic fields is in agreement with Eq. (55) below 0.5 mT, whereas the approximation for large magnetic fields, Eq. (60), is within 1% of the numerical value already when above 0.8 mT.

Figure 6

Top: Time evolution of the measured normalized phase difference, $r\left(H\right)$, fitted by the dynamic model Eqs. (1)–(11). The linear-quadratic onset of $r\left(H\right)$ is in accord with the analytic result given in Eq. (79). Bottom: The corresponding theoretical time evolution of $M_z/M_0$, initially growing linearly as given in Eq. (85).

Figure 7

The overall relaxation rate, $1/\tau \left(H\right)$, as a function of the magnetic field ${\mu }_{0}H$, extracted from the experimental data and the theoretical results using the fitting function Eq. (68). Inset: Without the dynamic cross-coupling, the relaxation rate levels off already at low fields (dashed).

Figure 8

Inverse of the time of the minimum determined from the measured normalized phase difference as a function of the magnetic field. The linear behavior in magnetic field is in agreement with Eq. (80).

Figure 9

Pretilt, determined from experimental data using Eq. (82).

Figure 10

The coefficient $k$ of Eq. (79) as a function of the magnetic field ${\mu }_{0}H$. The straight line fits are used to extract ${\chi }_2^D$.

Figure 11

The coefficient $p$ of Eq. (79) as a function of the magnetic field ${\mu }_{0}H$. The straight line fit is used to extract ${\phi }_s$.

Figure 12

Normalized phase difference at different values of the dissipative cross-coupling parameter ${\chi }_1^D$, ${\chi }_2^D=0$, ${\mu }_{0}H=50$ mT. Inset: The initial behavior is not affected.

Figure 13

Normalized phase difference at different values of the dissipative cross-coupling parameter ${\chi }_2^D$, ${\chi }_1^D=0$, ${\mu }_{0}H=50$ mT. Inset: The initial behavior is strongly affected as well.

Figure 14

The time dependencies of the azimuthal angles (degrees) of the director ($\phi$) and the magnetization ($\delta$) at $z=d/2$ for different values of ${\chi }^R$, ${\chi }_1^D={\chi }_2^D=0$, ${\mu }_{0}H=10$ mT.

Figure 15

Time dependence of the normalized intensity of transmitted light for zero and nonzero values of the reversible cross-coupling coefficient ${\chi }^R$; ${\mu }_{0}H=5$ mT.

Figure 16

Experimental and numerical normalized phase difference as a function of time at different values of the applied magnetic field.

Figure 17

Normalized phase difference as a function of time at 5 mT, calculated with ${\chi }_2^D=0$ and ${\chi }_2^D=4.0$ (Pa ${\mathrm{s})}^{-1}$.

Figure 18

Experimental switch-off relaxation rate of the normalized phase difference as a function of the applied magnetic field.

Figure 19

Relaxation rate of the normalized phase difference and $z$ component of the magnetization after switching off the magnetic field of strength ${\mu }_{0}H$ at ${\chi }_2^D=4$ (Pa ${\mathrm{s})}^{-1}$ and ${\phi }_s=0$.

Figure 20

Relaxation rate at ${\mu }_{0}H=50$ mT as a function of the dissipative coefficient ${\chi }_2^D$ for different values of the director rotational viscosity ${\gamma }_1$.

Figure 21

The normalized coefficients Eq. (103) of the eigenvectors ${\mathbf{t}}_1$ and ${\mathbf{t}}_2$ as a function of the applied magnetic field with $q_x=0$ and $q_{\perp }=\pi /2$; $K_1=K_2$.

Figure 22

The normalized coefficients Eq. (104) of the eigenvectors ${\mathbf{p}}_1$ and ${\mathbf{p}}_2$ as a function of the applied magnetic field with $q_x=0$ and $q_{\perp }=\pi /2$; $K_1=K_2$.

Figure 23

Relaxation rates of almost pure bend fluctuations ($q_x\gg q_{\perp }$) and the corresponding dynamic eigenmodes as a function of the applied magnetic field. The dashed lines represent the limiting behavior of the relaxation rates, described by Eqs. (132) and (133). For clarity, a smaller value of the rotational viscosity was used to make the asymptotic behavior set in sooner.