2.4 Topological Defects
31
as in the above-mentioned computations by Peruani et al (2008). Both active and
passive particles interacted with their nearest neighbors by minimizing the energy
dependent on their alignment.
The resulting phase diagram in the plane spanned by the concentration C and the
relative strength of interactions, measured by the product of the effective interaction
energy ε and the inverse temperature β, is shown in Fig. 2.8a. The equilibrium system
is isotropic at low density and high temperature and nematic at high density and low
temperature; the red solid line marks the locus of the phase transition. Of course, this
is unlike the isotropic/nematic phase transition in liquids, where the density does
not change. In vibrated layers, on the other hand, as in the lattice model, increasing
density enhances the alignment, but there is no direct analogue to temperature, since
the vibration strength is the source, not only of disorder, but of activity itself.
The behavior of the active system is more complicated. As density grows and
temperature decreases, it first transforms from the disordered isotropic (I) state
to the locally ordered inhomogeneous mixed (IM) state. At low temperatures and
densities, this transition takes place through the appearance of dense ordered bands
(BS). At higher densities, the active nematic shows bistability between the mixed
(IM) and the homogeneous globally ordered (HO) states. Typical patterns in these
regimes are shown in Fig. 2.8b; evidently, the HO state is not really highly ordered.
2.4 Topological Defects
Order is hardly ever perfect even in passive nematic liquids, let alone active ones.
The nematic texture may be influenced by boundary conditions that impose a certain
alignment at the confining walls. In the two schemes in Fig. 2.9a, the orientation of
a 2D nematic at the elliptical boundary is either normal or parallel. In both cases,
the alignment direction makes a full revolution by 2π around the confining line.
The only way to resolve it is through the formation of defects, which are said to be
topological since they are determined exclusively by geometric properties preserved
under continuous deformations, independently of any physical interactions. However,
the number of defects does depend on their energy, which is scaled in the common
one-constant approximation as the square of their topological charge, equal to the
circulation along a surrounding contour. Therefore the charges should be minimal,
and more highly charged defects are unstable to splitting1.
For a nematic, the minimal charge is 1/2, which corresponds to circulation by π,
and the circulation around the boundary in Fig. 2.9a is therefore compensated by the
formation of two defects with charge +1/2. If the alignment was polar, a single defect
of unit charge would form, as in the XY model (Fig. 1.8) where the 2D alignment is
vectorial, or as in the experiments by Blair et al (Sect. 2.2) where the 2D alignment
of rods became vectorial due to their freedom to rise vertically and therefore a
single vortex was formed. In the region with two holes with normal alignment on
1 Only in the limit of an infinite ratio of bend to splay energies does splitting a unit charge leave the
energy invariant (Shin et al, 2008).
31
as in the above-mentioned computations by Peruani et al (2008). Both active and
passive particles interacted with their nearest neighbors by minimizing the energy
dependent on their alignment.
The resulting phase diagram in the plane spanned by the concentration C and the
relative strength of interactions, measured by the product of the effective interaction
energy ε and the inverse temperature β, is shown in Fig. 2.8a. The equilibrium system
is isotropic at low density and high temperature and nematic at high density and low
temperature; the red solid line marks the locus of the phase transition. Of course, this
is unlike the isotropic/nematic phase transition in liquids, where the density does
not change. In vibrated layers, on the other hand, as in the lattice model, increasing
density enhances the alignment, but there is no direct analogue to temperature, since
the vibration strength is the source, not only of disorder, but of activity itself.
The behavior of the active system is more complicated. As density grows and
temperature decreases, it first transforms from the disordered isotropic (I) state
to the locally ordered inhomogeneous mixed (IM) state. At low temperatures and
densities, this transition takes place through the appearance of dense ordered bands
(BS). At higher densities, the active nematic shows bistability between the mixed
(IM) and the homogeneous globally ordered (HO) states. Typical patterns in these
regimes are shown in Fig. 2.8b; evidently, the HO state is not really highly ordered.
2.4 Topological Defects
Order is hardly ever perfect even in passive nematic liquids, let alone active ones.
The nematic texture may be influenced by boundary conditions that impose a certain
alignment at the confining walls. In the two schemes in Fig. 2.9a, the orientation of
a 2D nematic at the elliptical boundary is either normal or parallel. In both cases,
the alignment direction makes a full revolution by 2π around the confining line.
The only way to resolve it is through the formation of defects, which are said to be
topological since they are determined exclusively by geometric properties preserved
under continuous deformations, independently of any physical interactions. However,
the number of defects does depend on their energy, which is scaled in the common
one-constant approximation as the square of their topological charge, equal to the
circulation along a surrounding contour. Therefore the charges should be minimal,
and more highly charged defects are unstable to splitting1.
For a nematic, the minimal charge is 1/2, which corresponds to circulation by π,
and the circulation around the boundary in Fig. 2.9a is therefore compensated by the
formation of two defects with charge +1/2. If the alignment was polar, a single defect
of unit charge would form, as in the XY model (Fig. 1.8) where the 2D alignment is
vectorial, or as in the experiments by Blair et al (Sect. 2.2) where the 2D alignment
of rods became vectorial due to their freedom to rise vertically and therefore a
single vortex was formed. In the region with two holes with normal alignment on
1 Only in the limit of an infinite ratio of bend to splay energies does splitting a unit charge leave the
energy invariant (Shin et al, 2008).