Hydrochemical interactions in dilute phoretic suspensions: From individual particle properties to collective organization

Janus phoretic colloids (JPs) self-propel as a result of self-generated chemical gradients and exhibit spontaneous nontrivial dynamics within phoretic suspensions, on length scales much larger than the microscopic swimmer size. Such collective dynamics arise from the competition of (i) the self-propulsion velocity of the particles, (ii) the attractive/repulsive chemically mediated interactions between particles, and (iii) the flow disturbance they introduce in the surrounding medium. These three ingredients are directly determined by the shape and physicochemical properties of the colloids' surface. Owing to such link, we adapt a recent and popular kinetic model for dilute suspensions of chemically active JPs where the particle's far-field hydrodynamic and chemical signatures are intrinsically linked and explicitly determined by the design properties. Using linear stability analysis, we show that self-propulsion can induce a wave-selective mechanism for certain particles' configurations consistent with experimental observations. Numerical simulations of the complete kinetic model are further performed to analyze the relative importance of chemical and hydrodynamic interactions in the nonlinear dynamics. Our results show that regular patterns in the particle density are promoted by chemical signaling but prevented by the strong fluid flows generated collectively by the polarized particles, regardless of their chemotactic or antichemotactic nature, i.e., for both puller and pusher swimmers.

Figure 1

Phoretic limit. Left: Evolution of the growth rate of the least stable mode as obtained numerically from Eq. (40) in the phoretic limit (${\xi }_r=0$ and ${\xi }_t=0.375$) for different values of $u_0$ with $\beta =2\pi$ and $d_x=0.05$. The inset shows the same quantity with different axes' ranges to include the case where self-propulsion is absent [$u_0=0$, Eq. (41)]. Right: Schematic representation of the competition between externally induced drift and self-propulsion in the phoretic limit.

Figure 2

Chemotactic limit. Left: Evolution of the growth rate of the least stable mode as obtained numerically from Eq. (40) in the chemotactic limit (${\xi }_r/\phi =0.6, {\xi }_t=-0.375$) for different values of $u_0$ with $\beta =2\pi$ and $d_x=0.05$. Right: Schematic representation of the chemotactic mechanism and the stabilizing effect of self-propulsion.

Figure 3

Antichemotactic limit. Left: Evolution of the growth rate and frequency (inset) of unstable modes of wave number $k$ as obtained from the chemical dispersion relation (40) in the antichemotactic (delay) limit with ${\xi }_r/\phi =-3, {\xi }_t=-0.375, \beta =0.628, d_x=0.05$. Right: Illustration of the antichemotactic instability mechanism: (a) particles orient and swim towards regions of low chemical concentration; (b) particles have reached the region of low solute content and start releasing solute there, but the solute concentration takes time to equilibrate, leading to more particles' aggregation; and (c) in response to the increased particle accumulation, the chemical concentration overshoots [in comparison with (a)], and particles rotate and swim away from the solute-rich region.

Figure 4

Emergence of nonlinear dynamics in the chemotactic limit (${\xi }_r/\phi =1, {\xi }_t=-0.375, u_0=0.5, \beta =2\pi$). Particle density $\mathrm{\Phi }$ (color) and mean direction field $\mathbf{n}$ (arrows) forming asters. Left: With hydrodynamic interactions (${\alpha }_s=-3.1169, {\alpha }_i=0.3142$). Right: Without hydrodynamic interactions (${\alpha }_s={\alpha }_i=0$).

Figure 5

Time series of relevant quantities in the chemotactic limit (${\xi }_r/\phi =1, {\xi }_t=-0.375, u_0=0.5, \beta =2\pi$); $\langle •\rangle$ indicates spatial averages: (a) spatial correlation of the chemical gradient and the mean director, (b) variance of the flow velocity, (c) spatial correlation of the flow velocity and the mean director, and (d) variance of the particle density.

Figure 6

Nonlinear evolution of the particle density $\mathrm{\Phi }$ (color) and fluid velocity $\mathbf{u}$ (arrows) in the chemotactic limit (${\xi }_r/\phi =1, {\xi }_t=-0.375, u_0=0.5, \beta =2\pi$). The entire domain is depicted as in Fig. 4.

Figure 7

Time series of the second central moment of the normalized energy density ${\mu }_{2,2}\left(\mathcal{E}\right)$ in the chemotactic limit (${\xi }_r/\phi =1, {\xi }_t=-0.375, u_0=0.5, \beta =2\pi$) with and without hydrodynamic interactions.

Figure 8

Contour plots of the particle density $\mathrm{\Phi }$ in the antichemotactic limit (${\xi }_r/\phi =-3, {\xi }_t=-0.5, u_0=2, \beta =2\pi /3$). Left: Without hydrodynamic interactions (${\alpha }_s, {\alpha }_i=0$). Center and right: With hydrodynamic interactions (${\alpha }_s=28.05, {\alpha }_i=-0.94$).

Figure 9

Time series of relevant quantities in the antichemotactic limit (${\xi }_r/\phi =-3, {\xi }_t=-0.5, u_0=2, \beta =2\pi /3$). Left: Variance of the flow velocity (top) and spatial correlation of the chemical gradient and the mean director (bottom). Right: Variance of the particle density $\langle {\mathrm{\Phi }}^2\rangle$ with and without hydrodynamic interactions.