Abstract
Biological activity is often highly concentrated on surfaces, across the scales from molecular motors and ciliary arrays to sessile and motile organisms. These ‘active carpets’ locally inject energy into their surrounding fluid. Whereas Fick’s laws of diffusion are established near equilibrium, it is unclear how to solve non-equilibrium transport driven by such boundary-actuated fluctuations. Here, we derive the enhanced diffusivity of molecules or passive particles as a function of distance from an active carpet. Following Schnitzer’s telegraph model, we then cast these results into generalised Fick’s laws. Two archetypal problems are solved using these laws: First, considering sedimentation towards an active carpet, we find a self-cleaning effect where surface-driven fluctuations can repel particles. Second, considering diffusion from a source to an active sink, say nutrient capture by suspension feeders, we find a large molecular flux compared to thermal diffusion. Hence, our results could elucidate certain non-equilibrium properties of active coating materials and life at interfaces.
Highlights
Biological activity is often highly concentrated on surfaces, across the scales from molecular motors and ciliary arrays to sessile and motile organisms
We distinguish between various types of carpets that inject energy into the surrounding medium in different ways: We consider actuators that exert a net force on the liquid, a net torque, or a net stress
We consider the diffusion of molecules from a source to an active sink, such as a surface covered with sessile suspension feeders
Summary
The moments of this total flow distribution (the mean, variance, skewness and kurtosis) are found as a function of distance from the surface, for different carpet types (see ‘Methods: Characterising fluctuations: simulation details’). The mean first-passage time is much larger for distant sources, so the flux decays rapidly with gap size (Fig. 4c), which may be important for biology or applications in confined spaces This system can again be solved analytically (see ‘Methods: Diffusion from a source to an active carpet sink: theory details’). Any natural carpet is likely to feature some heterogeneity in its force distribution that can drive local advection flows (see Methods: Advective and di usive transport’) This imposes a constraint on the generalised Fick’s laws we discussed so far: If the particles are stuck in local advection currents, they cannot diffuse around freely. This is the case in many biological and engineered settings, but care should be taken that these conditions are satisfied
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