We explore the pressure of active particles on curved surfaces and its relation to other interfacial properties. We use both direct simulations of the active systems as well as simulations of an equilibrium system with effective (pair) interactions designed to capture the effects of activity. Comparing the active and effective passive systems in terms of their bulk pressure, we elaborate that the most useful theoretical route to this quantity is via the density profile at a flat wall. This is corroborated by extending the study to curved surfaces and establishing a connection to the particle adsorption and integrated surface excess pressure (surface tension). In the ideal-gas limit, the effect of curvature on the mechanical properties can be calculated analytically in the passive system with effective interactions and shows good (but not exact) agreement with simulations of the active models. It turns out that even the linear correction to the pressure is model specific and equals the planar adsorption in each case, which means that a known equilibrium sum rule can be extended to a regime at small but nonzero activity. In turn, the relation between the planar adsorption and the surface tension is reminiscent of the Gibbs adsorption theorem at an effective temperature. At finite densities, where particle interactions play a role, the presented effective-potential approximation captures the effect of density on the dependence of the pressure on curvature.
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Although σ, as defined in Eq. (12), is usually referred to as surface tension, it differs from the actual surface tension, defined as an integral of the pressure tensor anisotropy, by an additional wall term. This can be understood by rewriting the integral over p Θ(∓(r − R)) in terms of the normal pressure, which vanishes inside the wall and equals p in the bulk, and using the condition ∇·p = −ρ∇v for hydrostatic stability. The additional wall term does, however, only play a role if the wall potential is soft, which means that at a hard wall σ equals the surface tension. Due to this minor difference, we shortly call σ a total surface tension or just surface tension if there is no risk of confusion with the true surface tension. For a passive system, the true surface tension is a pure many-body term, which is always zero for an ideal gas, whereas the total surface tension, i.e., the excess grand potential per surface area σ, determines surface thermodynamics; compare Ref. 46.
