Mixtures of active and passive colloids show an intriguing dynamics of self-assembling, which is driven by the active component. Self-phoretic active colloids generate sinks in a chemical concentration field that cause passive colloids to drift toward active colloids by diffusiophoresis. The strength of this effective attraction is governed by the diffusiophoretic parameter, which determines the drift velocity. Simulating the Langevin dynamics of the colloids, we determine the state diagram for increasing diffusiophoretic strength and fixed active velocity. Three main states are distinguished. For weak attraction, passive particles are first scattered in the simulation box and then form a colloidal cloud around its center. Increasing the diffusiophoretic parameter further, passive particles oscillate between the cloud and a compact cluster, which embeds active colloids. Ultimately, in the third state, all particles collapse into a single stable cluster. In the collapse regime, the clustering dynamics of the largest cluster follows a logistic function and the mean cluster velocity vs cluster size decays with a power law. Throughout this article, we discuss our simulation results with regard to the experiments of Singh et al., Adv. Mater. 29(32), 1701328 (2017).
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The swimming velocity depends on the fuel concentration c0, which we introduce below and which we do not vary in our treatment. To concentrate on exploring the principal effect of self-propulsion on the behavior of the binary mixture, we assume here that the presence of other particles does not strongly affect .
According to Ref. 41, the most general form of the diffusiophoretic drift velocity is D = [⟨b⟩1 −⟨(3n ⊗ n −1)b⟩/2]∇c, where n is the normal unit vector on the particle surface, ⟨…⟩ means average over the particle surface, and c is the chemical concentration at the position of the particle. The slip-velocity coefficient b connects the slip velocity to the chemical gradient at a specific location of the particle surface. It depends on the interaction potential of the chemical solute with the particle surface. For the passive colloids, b is constant over the colloid surface. Thus, the second term of D vanishes and we arrive at the expression in the main text with ζtr = −b.
Strictly speaking, the active Janus colloid with its catalytic cap produces an anisotropic concentration field c. We here restrict ourselves to the leading monopole contribution of the chemical sink field resulting from the overall consumption of the fuel with rate k. It is given in Eq. (5). To take into account the anisotropic surface of the Janus colloid, one can add derivatives of the delta function in the sum of Eq. (4). They will produce dipolar and higher multipole contributions to Eq. (5), which decay faster than the monopole. The overall prefactor in front of the sum with the total consumption rate k will not change in this approach. However, in a more elaborate model with reactants and products that decribes reactions at the particle surface and in the bulk fluid, the total consumption rate depends on how well the catalytic cap is approximated by an expansion into Legendre polynomials.43
The value of can be estimated by using the thermal values of the diffusion coefficients that follow from the Stokes-Einstein relations. In this way, one obtains , which is below the value of 2.33a that we specify here. Since the colloids move close to the bottom wall in the experimental setup of Ref. 9, the value is also increased in experiments. In fact, in Ref. 6, a ratio of was measured. According to Refs. 24 and 25, we expect that the qualitative behavior of our system does not change drastically, if we adjust the ratio.
