Topological defects in an unconfined nematic fluid induced by single and double spherical colloidal particles

We present numerical solutions to the Landau-de Gennes free-energy model under the one-constant approximation for systems of single and double spherical colloidal particles immersed in an otherwise uniformly aligned nematic liquid crystal. A perfect homeotropic surface anchoring of liquid-crystal molecules on the spherical surface is considered. A large parameter space is carefully examined, including those in the free-energy model and those describing the dimer configurations and the background liquid-crystal orientation. The stability of the resulting liquid-crystal defects appearing in the neighborhood of the colloidal dimer pair is analyzed in light of the numerical results for their free energies. A number of scenarios are considered: a free dimer pair in a nematic fluid where the free-energy ground states are described in terms of a phase diagram, and a constrained dimer pair where the interparticle distance and the relative orientation of the distance vector to the nematic director can be manipulated. We pay particular attention to the nonsymmetric solutions, which yield several metastable defect states that can be observed in real systems. The high-precision numerical calculations are based on a spectral method, which is an enabling factor that allows us to compare the subtle difference in the free energies of different defect structures.

Figure 1

Defect configurations in a nematic liquid crystal surrounding a spherical particle: (a)–(c) quadrupolar (Saturn-ring) state, (d)–(f) dipolar state, as well as the corresponding regimes in the phase diagram (g) where these states have the lowest free energies. Each state is illustrated by using three views: side views of the $\mathsf{Q}$-tensor element $Q_{xx}$ [in (a) and (d)], the same side views of the biaxiality coefficient $\beta$ [in (b) and (e)], and top view of the defect location, indicated by the black circles produced from plotting the isosurface of $c_l=0.1$. (e) is an enlarged version of the defect area on top of the sphere, where the tensor field (white ellipsoids) indicates a $-1/2$ defect line. The reduced parameters used in producing these configurations are: $[\tau ,{\xi }_R]=[-0.2234,0.07209]$ and $[-0.2234,0.007071]$ ($R\approx 0.1\mu \mathrm{m}$ and $R\approx 1\mu \mathrm{m}$ in 5CB) for the quadrupolar and dipolar states, respectively. The dashed line in (g) represents the stability limit of the dipolar state.

Figure 2

(a) Sketch of the coordinate system for a dimer problem; (b) entangled hyperbolic defect (H) state where $\gamma =\pi /2$; (c) unentangled defect rings (${\mathrm{U}}_{\gamma }$) where $\gamma \ne \pi /2$; (d) parallel dipole-dipole state (${\mathrm{D}}_0$) where $\gamma =0$; and (e) the phase diagram that describes the regimes where these defect states have free-energy minima in terms of a reduced temperature $\tau$ and reduced spherical radius $1/{\xi }_R$. The phase diagram was determined based on the ground-state calculation, after consideration of all other possibilities including minimization with respect to the particle distance $D$ and tilt angle $\gamma$. (b) and (c) are illustrations of the defect lines determined from isosurface $S=0.25$. (d) is the cross-section view of $Q_{xx}$. (b) and (c) are produced from numerical solutions at $[\tau ,{\xi }_R]=[-0.2234,0.07209]$ ($R\approx 0.1\mu \mathrm{m}$ in 5CB). (d) is produced from numerical solutions at $[\tau ,{\xi }_R]=[-0.2234,0.007071]$ ($R\approx 1\mu \mathrm{m}$ in 5CB).

Figure 3

Metastable colloidal dimer configurations in a nematic fluid determined from the Landau-de Gennes free energy in the current work: (a) figure-of-eight (E), (b) figure-of-omega (O), (c) bubble-gum (B), (d) tilted antiparallel dipoles (${\mathrm{D}}_{\gamma }$), (e) axisymmetric dipole-quadrupole (${\mathrm{DQ}}_0$), and (f) asymmetric dipole-quadrupole (${\mathrm{DQ}}_{\gamma }$). For clarity, the defect configurations are shown by using different physical quantities. The black curves in plots (a) and (b) are illustrations of the defect lines determined from the isosurface of $S=0.25$. (c) is a cross-section view $Q_{zz}^2\left(\mathbf{r}\right)$ on the $x-z$ plane where the far-field nematic director is along the $z$ axis. (d), (e), and (f) are cross-section views of $Q_{xx}\left(\mathbf{r}\right)$ on the $x-z$ plane where the far-field director makes an angle $\gamma =38.4^{\circ },0^{\circ }$ and $\gamma =46.8^{\circ }$ from the $z$ axis, respectively. (a) and (b) are produced from numerical solutions at $[\tau ,{\xi }_R]=[-0.2234,0.07209]$ ($R\approx 0.1\mu \mathrm{m}$ in 5CB). (c) and (d) are produced from numerical solutions at $[\tau ,{\xi }_R]=[-0.2234,0.007071]$ ($R\approx 1\mu \mathrm{m}$ in 5CB). (e) and (f) are produced from numerical solutions at $[\tau ,{\xi }_R]=[-0.2234,0.01179]$ ($R\approx 0.6\mu \mathrm{m}$ in 5CB).

Figure 4

(a) Reduced free-energy branches of ${\mathrm{U}}_{\gamma }$ (circle), ${\mathrm{D}}_0$ (squares), and ${\mathrm{DQ}}_0$ (triangles) as functions of $1/{\xi }_R$ at $\tau =-0.2234$, and (b) reduced free energies of ${\mathrm{D}}_0$ (squares) and ${\mathrm{DQ}}_0$ (triangles) obtained when $D/R$ is fixed as a constraint at $[\tau ,{\xi }_R]=[-0.2234,0.007071]$ ($R\approx 1\mu \mathrm{m}$ in 5CB).

Figure 5

(a)–(d) Three possible dipole-dipole configurations at $\gamma =0$ (parallel, p-p type antiparallel, and h-h type antiparallel) and ${\mathrm{D}}_{\gamma }$ configuration for $D=4R$ shown by $Q_{xx}$, (e) reduced free energies $F$ calculated as functions of $D/R$ for the first three configurations [squares, diamonds, and circles for (a), (b) and (c)], (f) optimal $\gamma$ determined when the free energy is minimized with a $D/R$-constraint in the ${\mathrm{D}}_{\gamma }$ state, and (g) reduced free energies of ${\mathrm{D}}_{\gamma }$ (triangles) and ${\mathrm{D}}_0$ (squares) as functions of $D/R$. The configuration in (d) has $\gamma =83.6^{\circ }$. All figures are produced from numerical solutions at $[\tau ,{\xi }_R]=[-0.2234,0.007071]$ ($R\approx 1\mu \mathrm{m}$ in 5CB); in this example, the free energy at the reduced units $1\times {10}^{-4}$ corresponds to about $8.3\times {10}^3{k}_{B}T$.

Figure 6

(a)–(b) Two possible configurations of dipole-quadrupole pair aligned with $\gamma =0$ for $D=4R$ (shown by a $Q_{xx}$ plot), (c) ${\mathrm{DQ}}_{\gamma }$ configuration shown by a $Q_{xx}$ plot for $D=4R$, (d) reduced free energies $F$ as functions of $D/R$ for the first two configurations [circles for (a) and squares for (b)], (e) the optimal $\gamma$ for ${\mathrm{DQ}}_{\gamma }$ at a given $D/R$, and (f) the reduced free energies as functions of $D/R$ for ${\mathrm{DQ}}_{\gamma }$ [triangles] and ${\mathrm{DQ}}_0$ [circles]. All figures are produced from numerical solutions at $[\tau ,{\xi }_R]=[-0.2234,0.01179]$ ($R\approx 0.6\mu \mathrm{m}$ in 5CB); in this example, the free energy at the reduced units $1\times {10}^{-4}$ corresponds to about $1.8\times {10}^3{k}_{B}T$.

Figure 7

(a)–(f) Illustration of CR, R, ${\mathrm{U}}_{90}$, H, E, and O, (g) reduced free energies $F$ of R, ${\mathrm{U}}_{90}$, H, E, O and ${\mathrm{U}}_{\gamma }$ at $[\tau ,{\xi }_R]=[-0.2234,0.07209]$ ($R\approx 0.1\mu \mathrm{m}$ in 5CB), (h) reduced free energy of CR at $[\tau ,{\xi }_R]=[-0.2234,0.3604]$ ($R\approx 20\mathrm{nm}$ in 5CB), (i) optimal $\gamma$ in ${\mathrm{U}}_{\gamma }$ at $[\tau ,{\xi }_R]=[-0.2234,0.07209]$, and (j) optimal $\gamma$ in ${\mathrm{U}}_{\gamma }$ as a function of $1/{\xi }_R$ for a fixed $D/R=2.1$ at $\tau =-0.2234$. (a) is a $\beta$ plot and (b)–(f) contain defect lines visualized by an isosurface $S=0.25$. The inset shows the details of the minimum region in (g). In (g) and (h), the reduced free energies having a value $1\times {10}^{-2}$ correspond to about $8.3\times {10}^2{k}_{B}T$ and $6.6k_{B}T$, respectively.

Figure 8

Phase diagrams of energetically preferred configurations at a fixed $\gamma =\pi /2$ in terms of $1/{\xi }_R$ and $D/R$ for fixed (a) $\tau =0.0625$ and (b) $\tau =-0.2234$. The range $1/{\xi }_R=[0,40]$ approximately corresponds to $R=[0,0.3\mu \mathrm{m}]$ in 5CB at a room temperature.

Figure 9

(a) Illustration of the H-R pathway as the two particles in H are stretched apart, with distances from $D/R=3.1,3.2$, to 3.3, respectively, from left to right. (b) Corresponding reduced free energies of H (diamonds) and R (triangles) branches, and (c) behavior of the two order parameters defined for H (diamonds) and R (triangles). The calculation was performed for $[\tau ,{\xi }_R]=[-0.2234,0.3604]$ ($R\approx 20\mathrm{nm}$ in 5CB); with this choice, a free energy at the reduced units $1\times {10}^{-2}$ corresponds to about 6.6 $k_{B}T$ in 5CB. Shown in color are $\beta$ plots. The defect lines are visualized by an isosurface $c_l=0.05$.

Figure 10

Illustration of a possible two-stage transition when the particles in H is stretched away as $D/R$ increase. The H state can make a first-order transition to jump into a ${\mathrm{U}}_{90}$ state [shown in (a), where $D/R=2.6,2.7$ and 2.8, respectively, from left to right], and the ${\mathrm{U}}_{90}$ state smoothly crosses over to R [shown in (b), where $D/R=3,3.1$ and 3.2, respectively, from left to right]. The H state could be trapped in its metastable form, directly makes an H-R transition described in Fig. 9. These transitions are assessed by the free energies displayed in (c), where diamonds, circles, and triangles represent data points calculated for the H, ${\mathrm{U}}_{90}$, and R branches. The inset of (c) shows $F_{\mathrm{U}}-F_{\mathrm{H}}$ near the transition point. The order parameter for the ${\mathrm{U}}_{90},P_{\mathrm{U}90}$, is displayed in (d). The system considered here has reduced parameters $[\tau ,{\xi }_R]=[-0.2234,0.1768]$ ($R\approx 40\mathrm{nm}$ in 5CB); with this choice, a free energy at the reduced units $1\times {10}^{-2}$ corresponds to about $53k_{B}T$ in 5CB. Shown in color are $\beta$ plots. The defect lines are visualized by an isosurface $S=0.2$.

Figure 11

(a) Illustration of a possible H-E pathway in a stretching experiment starting from H, where $D/R=2.1,2.2$, and 2.3, respectively, from left to right, (b) reduced free energies as functions of $D/R$ for H (diamonds) and E (squares), and (c) the order parameter for E. The inset shows $F_{\mathrm{E}}-F_{\mathrm{H}}$ near the transition point. A jump in the order parameter exists after the two free energy curves cross each other. The calculation was performed for $[\tau ,{\xi }_R]=[-0.2234,0.08839]$ ($R\approx 80\mathrm{nm}$ in 5CB); with this choice, the free energy at the reduced units $1\times {10}^{-2}$ corresponds to about $4.3\times {10}^2{k}_{B}T$ in 5CB. The defect lines are visualized by an isosurface $S=0.25$.

Figure 12

(a) Orientational profiles in the bubble-gum configuration shown by $Q_{zz}^2$ plot at $D=4R$, (b) detailed map between the two spheres at $D=2.2R$ where the white ellipsoids are illustrated according to $\mathsf{Q}$ to display the main directors, and (c) reduced free energy $F$ in B (circles), which can be compared with that of ${\mathrm{D}}_0$ (squares). The example given here corresponds to a system having $[\tau ,{\xi }_R]=[-0.2234,0.007071]$ ($R\approx 1\mu \mathrm{m}$ in 5CB); with this choice, a free energy at the reduced units $1\times {10}^{-4}$ corresponds to about $8.3\times {10}^3{k}_{B}T$.

Figure 13

Relative errors estimated for determination of the free energies in a single-particle problem. In (a), the quadrupolar solution was obtained at $[\tau ,\xi ]=[-0.2234,0.07209]$ ($R\approx 0.1\mu \mathrm{m}$ in 5CB) with grid number up to $N=128$. In (b), the dipolar solution was obtained at $[\tau ,\xi ]=[-0.2234,0.01179]$ ($R\approx 0.6\mu \mathrm{m}$ in 5CB) with the grid number up to $N=256$.