Dipolar colloids in nematostatics: Tensorial structure, symmetry, different types, and their interaction

In spite of the analogy to the electrostatics, the three-dimensional colloidal nematostatics is substantially different in both its mathematical structure and its physical implications. The general tensorial structure of elastic multipoles derived in V. M. Pergamenshchik and V. O. Uzunova [Eur. Phys. J. E 23, 161 (2007); Phys. Rev. E 76, 011707 (2007)] allows for a classification of different types of colloids in the nematostatics. In comparison to their electrostatic counterparts, the elastic multipoles have one extra tensorial index. Based on this structure, we identify possible types of elastic dipoles. An elastic dipole is characterized by three coefficients—isotropic strength, anisotropy, and chirality—and a two-component vector along the unperturbed director. The relationship between the dipole type and symmetry groups is established and sketches of various representative types of dipolar colloids are given. Instead of a single electric dipole, in the nematostatics there are four different pure types (dipolar singlets) and eight mixed types of elastic dipoles (one quintet, one quartet, two triplets, and four doublets). It is shown that the full symmetry of the colloid-induced director field and the colloid’s shape (body) symmetry determine different dipole components. For instance, a helicoidal component of the anchoring easy axes can make a chiral elastic dipole of a colloid with the quadrupolar shape symmetry. The interaction potentials for different singlet and doublet dipoles are derived and illustrated in terms of the dipolar dyads and elastic Coulomb law. We argue that multipole parameters must be found by pure numerical means, as from ansatz director distributions one can find only orders of their magnitudes.

Figure 1

A quadrupole has a HMP and infinite number of VMPs. A finite helical component $n_{\varphi }$ removes all of the mirror planes and makes it a uniaxial helicoid (pure $C$ dipole) with the quadrupolar body symmetry.

Figure 2

Illustration of the elastic Coulomb law. The dipoles of the two upper pairs of particles attract one another as the directors at their neighboring sides are tilted same way (“parallel,” the $n_x$ components are of the same sign). The dipoles of the two lower pairs of particles repel one another as the directors at their neighboring sides are tilted opposite ways (“crossed,” the $n_x$ components are of different sign). The cone like colloids in the left diagram can be qualitatively represented by two point defects, but the chiral dipoles in the right diagram cannot be represented by a combination of point defects.

Figure 3

Two interacting particles situated in the laboratory RF $O(x,y,z)$. The particles’ orientation is determined by the orientation of their eigenframes in which their dipole dyads have the form (16). The eigenframe $O_0(x_{01},y_{01},z_{01})$ of particle 1 has the $z_{01}$ axis parallel to the $z$ axis, and its $x_{01}$ makes an angle ${\varphi }_1$ with the $\mathit{xz}$ plane. The eigenframe $O_0(x_{02},y_{02},z_{02})$ of particle 2 has the $z_{02}$ axis parallel to the $z$ axis, and its $x_{02}$ axis makes an angle ${\varphi }_2$ with the $\mathit{xz}$ plane. The mutual location of the particles is determined by the vector $R\mathbf{u}$, where $R$ is the distance and $\mathbf{u}$ is the unit vector directed from particle 2 to particle 1. This vector makes an angle $\theta$ with the $z$ axis. The interparticle distance $R$ and angles ${\varphi }_1$, ${\varphi }_2$, and $\theta$ fully determine the dipole-dipole interaction of the two particles.

Figure 4

Conventional schematic of a pure $d$ type (uniaxial) dipole and its dyad in the eigenframe. This form is equivalent to a circular cone (see text for details).

Figure 5

Two configurations of the maximum attraction of two pure $d$ dipoles. The black arrows are the ${\mathbf{d}}_x$ vectors of the $d$ dyads. The ${\mathbf{d}}_y$ vectors (not shown) are along the $y$ axis which is normal to the figure plane; $\left|{\mathbf{d}}_x\right|=\left|{\mathbf{d}}_y\right|=d.$

Figure 6

Conventional schematic of a pure $\Delta$ type (anisotropic or mirror) dipole and its dyad in its eigenframe.

Figure 7

The two configurations of maximum attraction of two pure $\Delta$ dipoles. The solid arrows are the ${\mathbf{d}}_x$ vectors; the dashed arrows are the ${\mathbf{d}}_y$ vectors of the $\Delta$ dyads (in the laboratory RF).

Figure 8

Conventional schematic of a pure $z$-type (longitudinal) dipole and its dyad in its eigenframe.

Figure 9

The two configurations of maximum attraction of two pure $z$ dipoles. The solid arrows are the ${\mathbf{d}}_x$ vectors, ${\mathbf{d}}_y=0.$

Figure 10

Conventional schematic of a pure $C$-type dipole (uniaxial helicoid) and its dyad in its eigenframe. This form is equivalent to a circular one (see text for details).

Figure 11

The configurations of maximum attraction of two pure $C$ dipoles. Two helicoids of the same handedness attract one another along the director, whereas those of opposite handedness attract one another across the director. The solid arrows are the ${\mathbf{d}}_x$ vectors; the dashed arrows are the ${\mathbf{d}}_y$ vectors of the $C$ dyads.

Figure 12

The two configurations of maximum attraction of two (nonchiral biaxial) $d\Delta$ dipoles. Solid arrows are the ${\mathbf{d}}_x$ vectors; dashed arrows are the ${\mathbf{d}}_y$ vectors of the $d\Delta$ dyads.

Figure 13

The single configuration of maximum attraction of $d$ and $\Delta$ dipoles. Solid arrows are the ${\mathbf{d}}_x$ vectors; dashed arrows are the ${\mathbf{d}}_y$ vectors of the $d\Delta$ dyads (shown in the laboratory RF).

Figure 14

The configurations of maximum attraction of $C$ dipole to $\Delta$ dipole lie in the horizontal plane $\theta =\pi /2$. There are two opposite directions for each handedness. Arrows indicate the director at the colloids to demonstrate that the attraction directions can be understood from the attraction of “parallel directors.”

Figure 15

The configurations of maximum attraction of the $C$ dipole to the $z$ dipole (with $\gamma <0$) lie in the plane $\varphi =\pi /2$ along $\theta =\pi /4$ or $\theta =3\pi /4$. There are two opposite directions for each handedness. Arrows indicate the director at the colloids to demonstrate the elastic Coulomb law.

Figure 16

The configurations of maximum attraction of $z$ dipole to $d$ dipole lie on the cones $\theta =\pi /4$ and $\theta =3\pi /4$.

Figure 17

The configurations of maximum attraction of $z$ dipole to $\Delta$ dipole lie in the planes $\varphi =0$ and $\varphi =\pi /2$ along $\theta =\pi /4$ or $\theta =3\pi /4$ of the eigenframe of the $\Delta$ dipole.