Abstract
Amongst the theoretical approaches towards the dynamics and phase behaviour of suspensions of active Brownian particles (ABPs), no attempt has been made to specify the motility-induced inter-particle correlations as quantified by the pair-correlation function. Here, we derive expressions for the pair-correlation function for ABPs with very short-ranged direct interactions for small and large swimming velocities and low concentrations. The pair-correlation function is the solution of a differential equation that is obtained from the Fokker-Planck equation for the probability density function of the positions and orientations of the ABPs. For large swimming Peclet numbers, λ, the pair-correlation function is highly asymmetric. The pair-correlation function attains a large value, ∼λ, within a small region of spatial extent, ∼1/λ, near contact of the ABPs when the ABPs approach each other. The pair-correlation function is small within a large region of spatial extent, ∼λ1/3, when the ABPs move apart, with a contact value that is essentially zero. From the explicit expressions for the pair-correlation function, Fick's diffusion equation is generalized to include motility. It is shown that mass transport, in case of large swimming velocities, is dominated by a preferred swimming direction that is induced by concentration gradients. The expression for the pair-correlation function derived in this paper could serve as a starting point to obtain approximate results for high concentrations, which could then be employed in a first-principles analysis of the dynamics and phase behaviour of ABPs at higher concentrations.
Highlights
Active matter consisting of suspensions of self-propelling particles ranges from synthetic colloidal particles to cells and bacteria
It is important to notice that inside the boundary layer, (R À 1) B 1/l, while outside the boundary layer, g = 1, where the exponential in eqn (73) is essentially zero. This justifies the neglect of the derivatives with respect to X and a in eqn (65), and shows that the rotational contribution can be neglected within the frontsector
The interpretation of the latter is that, for the large relative velocities under consideration, the time that the particles spent within the narrow boundary layer is sufficiently small that rotation does not occur
Summary
Active matter consisting of suspensions of self-propelling particles ranges from synthetic colloidal particles to cells and bacteria. 36 and 37, the analysis of MIPS for non-chiral and chiral ABPs in 2D is based on the Smoluchowki equation, where inter-particle correlations are accounted for by numerical input obtained from simulations.[38] An independent calculation of the pair-correlation function is not pursued in these studies It is the purpose of the present study to derive explicit expressions for the 3D pair-correlation function, for small concentrations where binary interactions are dominant.
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