Models of blockage-induced selectivity

Abstract

We examine blockage-induced selectivity in a particulate stream flowing through a channel. Each component of the mixture is characterized by the transit time, τ, necessary to pass through the channel. The model is motivated by filtration and other processes involving blockage. The transit time distribution of exiting particles depends on the entering particle distribution, ψ(τ ), the intensity, λ, of the entering stream, and the blocking rule. With the simple rule that a blockage occurs whenever two particles are present in the channel, the properties of the exiting stream are directly related to the Laplace transform of the entering distribution, ψ(λ). For any entering distribution, the exiting stream is enriched in faster moving components. The selectivity of a species in a binary mixture can be mapped to a thermodynamic system, namely a hard rod mixture at a given pressure and temperature that can model the adsorption of gas mixtures in nanopores. We also examine an alternative rule according to which blocking only occurs if a faster moving particle catches up to a slower one in the channel. The selectivity is quantitatively dierent compared to the simple blocking rule. In a binary mixture the majority component in the entering stream is further enhanced in the exiting stream, independently of the transit times.

Secular dynamics of long-range interacting particles on a sphere in the axisymmetric limit

“…In present context, Eq. (8) generalizes the stability criteria put forward in [7,8] (see Appendix B). Following Eq.…”

“…In this Appendix, we consider the case where the system's interaction is limited to only one harmonic ℓ, so that U (x) = −α ℓ P ℓ (x). This is of particular relevance for classical Heisenberg spins (ℓ = 1) [7,8], and the Maier-Saupe model for liquid crystals (ℓ = 2) [9,10]. In that limit, the dressed susceptibility coefficients from Eq.…”

“…As such, in Eq. (B3), we immediately recover the susceptibility coefficients obtained by a more complex method in [8] (see Eq. (33) therein).…”

“…The Hamiltonian from Eq. (1) encompasses a wide class of long-range interacting systems: (i) U = −α 1 P 1 describes globally coupled classical Heisenberg spins [7,8], (ii) U = −α 2 P 2 is the Maier-Saupe model for liquid crystals [9,10], (iii) U = − ℓ α 2ℓ P 2ℓ captures the process of vector resonant relaxation in galactic nuclei [11][12][13] (up to additional conserved quantities).…”

“…u ′ = u, leading to zero flux and ∂F/∂t = 0. In that case, the system cannot relax under 1/N effects [8,[16][17][18]. It undergoes a so-called kinetic blocking [19], and can only relax under weaker finite-N effects associated with higher-order correlations.…”