2.2 Rod-like Particles
27
Fig. 2.5 Phase diagram of the regimes of self-propelled rod-like particles corresponding to the
snapshots shown in the insets. The particle orientations are color-coded as shown in the upper left
corner (Abkenar et al, 2013)
filaments in motility assays (Sect. 5.4) where intersections were not excluded. The
detailed phase diagram of regimes dependent on the scaled density and velocity (or
Péclet number) is shown in Fig. 2.5.
A statistical theory of “dry” active nematics has been developed (Bertin at al,
2013) along the same lines as the corresponding theory of polar particles, based
on Boltzmann’s hypothesis of “molecular chaos” (Bertin at al, 2006) and leading
to hydrodynamic equations with coefficients derivable from the microscopic model.
It was amended by Shi et al (2014) to account for particle diffusion. Pure nematic
symmetry can be ensured in this model by a high direction-reversal rate. The limitations mentioned in Sect. 1.4 remain in place, but the advantage of having continuous
equations lies in the ease in carrying out linear stability analysis. In this way, Shi et
al computed both the critical density, below which the disordered solution is stable,
and the stability limit of the ordered solution. The latter, naturally, stabilizes as the
density increases, but the lower limit depends on the relative magnitudes of the
effective longitudinal and transverse diffusivities D 0 = (D − D ⊥ )/(D + D ⊥ ) and
turns out to be different when derived from hydrodynamic or kinetic equations, as
shown in the master panel of Fig. 2.6.
The patterns obtained in computations by Shi et al (2014), based on the kinetic
model, show nematic bands forming when a slightly perturbed isotropic homogeneous state breaks into pieces at a later time. The same is observed in very long
27
Fig. 2.5 Phase diagram of the regimes of self-propelled rod-like particles corresponding to the
snapshots shown in the insets. The particle orientations are color-coded as shown in the upper left
corner (Abkenar et al, 2013)
filaments in motility assays (Sect. 5.4) where intersections were not excluded. The
detailed phase diagram of regimes dependent on the scaled density and velocity (or
Péclet number) is shown in Fig. 2.5.
A statistical theory of “dry” active nematics has been developed (Bertin at al,
2013) along the same lines as the corresponding theory of polar particles, based
on Boltzmann’s hypothesis of “molecular chaos” (Bertin at al, 2006) and leading
to hydrodynamic equations with coefficients derivable from the microscopic model.
It was amended by Shi et al (2014) to account for particle diffusion. Pure nematic
symmetry can be ensured in this model by a high direction-reversal rate. The limitations mentioned in Sect. 1.4 remain in place, but the advantage of having continuous
equations lies in the ease in carrying out linear stability analysis. In this way, Shi et
al computed both the critical density, below which the disordered solution is stable,
and the stability limit of the ordered solution. The latter, naturally, stabilizes as the
density increases, but the lower limit depends on the relative magnitudes of the
effective longitudinal and transverse diffusivities D 0 = (D − D ⊥ )/(D + D ⊥ ) and
turns out to be different when derived from hydrodynamic or kinetic equations, as
shown in the master panel of Fig. 2.6.
The patterns obtained in computations by Shi et al (2014), based on the kinetic
model, show nematic bands forming when a slightly perturbed isotropic homogeneous state breaks into pieces at a later time. The same is observed in very long