Self-phoretic Brownian dynamics simulations

Abstract

A realistic and effective model to simulate phoretic Brownian dynamics swimmers based on the general form of the thermophoretic force is here presented. The collective behavior of self-phoretic dimers is investigated with this model and compared with two simpler versions, allowing the understanding of the subtle interplay of steric interactions, propulsion, and phoretic effects. The phoretic Brownian dynamics method has control parameters which can be tuned to closely map the properties of experiments or simulations with explicit solvent, in particular those performed with multiparticle collision dynamics. The combination of the phoretic Brownian method and multiparticle collision dynamics is a powerful tool to precisely identify the importance of hydrodynamic interactions in systems of self-phoretic swimmers.

Fig. 1

Sketches of the propulsion direction of self-phoretic asymmetric dimers and phoretic interaction between dimer pairs, which is: a attractive for the case of colloids drifting up gradient, and b repulsive in the opposite case. c Sketch of the implemented forces in the Ph-BD model

Fig. 2

Self-propulsion velocity $v_s$ of single dimers simulated with MPC as a function of the temperature gradient $⟨\mathrm{\nabla }T⟩$ felt by the phoretic bead, for various dimer types. Results for dimers with phoretic bead $s_p=6$. Circles (in blue) correspond to thermophobic dimers; triangles (in red) correspond to thermophilic dimers. Full symbols correspond to asymmetric dimers ($\gamma =3$); empty symbols to symmetric dimers ($\gamma =1$). Lines relate to linear fits to Eq. (9) for small gradients

Fig. 3

Dynamical quantities measured with single-dimer Ph-BD simulations as a function of the phoretic bead size and normalized by the values of the hydrodynamic simulations. a Rotational diffusion coefficient $D_r$, b self-propulsion velocity $v_s$, c resulting Péclet number Pe. All quantities are normalized by the reference values obtained for hydrodynamic simulations with $s_p^{\mathit{hi}}=6$, in Table 1. The dashed lines at unity indicate perfect agreement between Ph-BD and MPC simulations. Thick vertical gray line corresponds to the case with optimal agreement for both $v_s$ and Pe, which occurs for $s_p^{\mathit{bd}}=8$

Fig. 4

Snapshots of ensembles of 200 self-thermophilic swimmers at times around $300{\tau }_b$ for density $\varphi =0.2$ simulated with the Ph-BD method. a Asymmetric dimers ($\gamma =3$), showing a few large stable clusters; b symmetric dimers ($\gamma =1$), showing a number of small transient clusters

Fig. 5

Bounding time ${\tau }_c$ calculated as a time average and shown here as a function of simulation time normalized with the dimer ballistic times ${\tau }_b$. Results for simulations with the three Brownian dynamics methods for 200 thermophilic dimers distinguished by the inset labels. Results at densities $\varphi =0.2$ are shown with light colors, while results with $\varphi =0.3$ are displayed with darker colors. a Asymmetric dimers and b symmetric dimers