Active Colloidal Suspensions Exhibit Polar Order under Gravity

Recently, the steady sedimentation profile of a dilute suspension of chemically powered colloids was studied experimentally [J. Palacci et al., Phys. Rev. Lett. 105, 088304 (2010)]. It was found that the sedimentation length increases quadratically with the swimming speed of the active Brownian particles. Here we investigate theoretically the sedimentation of self-propelled particles undergoing translational and rotational diffusion. We find that the measured increase of the sedimentation length is coupled to a partial alignment of the suspension with the mean swimming direction oriented against the gravitational field. We suggest realistic parameter values to observe this polar order. Furthermore, we find that the dynamics of the active suspension can be derived from a generalized free energy functional.

Figure 1

Dependence of polar order and reduced sedimentation length on $\mathrm{Pe}=v_{0}R/D$ and $\alpha =R/{\delta }_0$: (a),(b) $⟨\cos \theta ⟩$ and $\xi$ as a function of Pe for $\alpha =0.1$; (c),(d) $⟨\cos \theta ⟩$ and $\xi$ as a function of $\alpha$ for $\mathrm{Pe}=2$. Numerical data are computed from the steady solution of Eq. (5) by ${\delta }_{\mathrm{eff}}/{\delta }_0=-\alpha \Omega /{\Omega }_z$ and the definitions of $\Omega$ and $⟨\cos \theta ⟩$ (see text).

Figure 2

(a) Altitude dependence of the particle distribution $\rho (z,{\theta }^{*})$ for different orientations ${\theta }^{*}$. (b) Angular dependence of $\rho (z^{*},\theta )$ at different heights $z^{*}$. (c) Logarithmic plot of the normalized total density $\Omega (z)$ [see (d) for legend]. (d) Rescaled density profiles after discarding the first micrometers. Parameter values: $\alpha =1$, $\mathrm{Pe}=2$ (a),(b) and $\alpha =0.1$ (c),(d).