Interparticle force in nematic colloids: Comparison between experiment and theory

We have studied the interparticle force between two colloidal particles in a nematic liquid crystal experimentally and theoretically. The force $F$ was directly measured using dual-beam optical tweezers and was numerically calculated from the equilibrium tensor field around the particles. The dependence of $F$ on the center-to-center distance $R$ between the particles was studied not only for equal-sized particles but also for different-sized ones in various kinds of configurations and arrangements. The magnitude of $F$ between different-sized particles in the dipole configuration depends on their relative arrangement. Both experimental and theoretical force curves are found to be in good agreement with each other. At large $R$, they also make agreement with those predicted by an electrostatic analogy of nematic field.

Figure 1

Polarizing microscope images (cross-Nicoles) of particle-defect pairs. (a) Dipole configuration; (b) Saturn-ring configuration. In the black region, the direction of the director is parallel or perpendicular to the director far from the particle shown as an arrow. In the white region, the direction of the director is distorted from the uniform director field. Although an accompanying defect exists at the right-hand side of the particle, the defect cannot be observed in (a) due to small optical anisotropy of the NLC used in this study. The white regions around the particle are asymmetric about the vertical centerline of the particle. (b) A disclination ring appears as a blurred black line perpendicular to the uniform director field. The four white regions around the particle are almost symmetric.

Figure 2

Various configurations of two particle-defect pairs investigated in this study. (a) Parallel dipole arrangement, (b) antiparallel dipole arrangement, (c) parallel dipole arrangement with different-sized particles (type I), (d) parallel dipole arrangement with different-sized particles (type II), (e) parallel dipole and Saturn-ring arrangement, and (f) antiparallel dipole and Saturn-ring arrangement. An accompanying hyperbolic defect is represented by a point. A disclination ring is represented by a thin line.

Figure 3

Images of two particles in a dipolar configuration with parallel arrangement. (a) Polarizing microscope images (Cross-Nicoles). (b) Intensity profile of the nematic field obtained by numerical calculation shown by grayscale plots of $Q_{\mathrm{zz}}^2$. In the black regions ($Q_{\mathrm{zz}}^2$ $=$ 1), the director aligns along the $z$ axis, while $Q_{\mathrm{zz}}^2$ $=$ 0 in the white region.

Figure 4

Dependence of the interparticle force on the reduced center-to-center interparticle distance $D$. Open circles (red), experimental data $F$ for the particles of radius $a=$ 2.55$\mu$m. Solid line (blue), theoretically calculated data $f$. The distance $R$ is normalized by the radius of the particle $a$. The dependence of $F$ and $f$ on $D$ are also plotted in logarithmic scale in the inset. The best-fitted curve of the empirical formula, $F=C/D^4+A{(D-D_r)}^{-\delta }$, is drawn as a dotted line (green). The broken line (pink) represents the dependence of $D^{-4}$ as a guide of eye. The shadowed region in the inset indicates the region where $F$ deviates from $f$ due to the confinement effect of the cell.

Figure 5

Dependence of the interparticle potential $U\left(D\right)$ on the reduced interparticle distance $D$ for the particles of radius $a$ $=$ 2.55$\mu$m. Experimental data are shown as open circles (red). The best-fitted curve of the power law $U\left(D\right)\sim D^{-\alpha }$ with $\alpha =3.09$ is drawn as a solid line (blue). The inset picture shows a chainlike aggregate formed in an NLC.

Figure 6

Dependence of the interparticle force on the reduced interparticle distance $D$. Open circles (red), experimental data $F$ for the particles of radius $a$ $=$ 2.55$\mu$m. Solid line (blue), theoretically calculated data $f$. The distance is normalized by the radius of the particle $a$. The dependence of $F$ on $D$ is also plotted in logarithmic scale in the inset. The solid line (green) is the best-fitted curve of $F=C_1/D^4+C_2/D^6$, and the broken line (black) is the best-fitted curve of $F=C/{(D+D_0)}^4$ with $D_0=0.893$.

Figure 7

Images of two different-sized particles in a dipolar configuration with parallel arrangement. Upper pictures, polarizing microscope images (Cross-Nicoles). Lower pictures, intensity profile of the nematic field obtained by numerical calculation shown by grayscale plots of $Q_{\mathrm{zz}}^2$. Type I (left) corresponds to the arrangement shown in Fig. 2c. Type II (right) corresponds to the arrangement shown in Fig. 2d.

Figure 8

Dependence of the interparticle force on the reduced interparticle distance $D$. The distance is normalized by the radius of the larger particle. Open squares (red), type I arrangement (for particles of radius 2.55$\mu$m and 2.0$\mu$m). Open circles (blue), type II arrangement. The solid line (red) is the theoretical force $f$ for the particles whose ratio in size is 1:0.8 in the arrangement of type I. The broken line (blue) is the theoretical one for the particles in the arrangement of type II. The dependence of $F$ and $f$ on $D$ is also plotted in logarithmic scale in the inset. The best-fitted curves of Eq. (2) to experimental data are shown in solid lines (black).

Figure 9

Dependence of the interparticle force $f$ obtained theoretically on the reduced interparticle distance $D$. (a) Particles with various size ratios in the arrangement of type I. (b) Particles with various size ratios in the arrangement of type II.

Figure 10

Dependence of the difference in the interparticle force between the two arrangements $\delta F$ and $\delta f$ on the reduced interparticle distance $D$ in a logarithmic scale. Force difference is obtained by subtracting the force in the arrangement of type II from that in the arrangement of type I. The best-fitted curve of $\delta f=C_1^{\prime }/D^5+C_2^{\prime }/D^7$ to $\delta f$ for $D>$2.7 is drawn in a solid line (black). The dotted line (black) exhibits the dependence of $D^{-5}$, which is expected by the electrostatic analogy.

Figure 11

Dependence of the interparticle force on the reduced interparticle distance $D$. Open circles (red), experimental data $F$ for the particles of radius $a$ $=$ 2.55$\mu$m. Solid line (blue), theoretically calculated data $f$. The distance is normalized by the radius of the particle $a$. The dependence of $F$ and $f$ on $D$ are also plotted in a logarithmic scale in the inset. The broken line (green) represents the dependence of $D^{-5}$.

Figure 12

Images of the particles in a dipole and a Saturn-ring configuration. (Left) Polarizing microscope images (cross-Nicoles). (Right) Intensity profiles of the nematic field calculated by simulation. Upper, parallel arrangement [Fig. 2e]. Lower, antiparallel arrangement [Fig. 2f]. Intensity profile of the nematic field obtained by numerical calculation are shown by grayscale plots of $Q_{\mathrm{zz}}^2$.

Figure 13

Dependence of the interparticle force on the reduced interparticle distance $D$ in a logarithmic scale. Open circles (red), experimental data $F$ for the particles of radius $a=$ 2.55$\mu$m. Solid line (blue), theoretically calculated data $f$. The distance is normalized by the radius of the particle $a$. The broken line (black) is the best-fitted line to experimental data (3.2 $<D<$ 8.1), $F\sim D^{-3.67}$. The solid line (red) is the best-fitted line of $F=C_1^{\prime \prime }/D^5+C_2^{\prime \prime }/D^6$ to experimental data (2.3 $<D<$ 7.0). The solid line (blue) is the best-fitted line to theoretical data (2.3 $<D<$ 7.0), $f\sim D^{-4.92}$.