Uphill transport in competitive drift-diffusion models with volume exclusion

This paper addresses uphill transport (defined as a regime in which particle flow is opposite to the prescriptions of Fick’s diffusion) in drift-diffusion particle transport constrained by volume exclusion. Firstly, we show that the stationary hydrodynamic limit of a multispecies, weakly asymmetric exclusion process (SHDL) naturally predicts precisely characterized uphill regimes in the space of external drivings. Then, with specific reference to systems of oppositely charged particles, we identify well-defined model hypotheses and extensions whereby the SHDL converges to the modified Poisson-Nernst-Planck model, thus bridging the gap between exclusion-based particle models and continuum descriptions commonly used in engineering. The merits and limitations of the models in describing the particle fluxes and predicting uphill transport conditions are investigated in detail with respect to the adopted approximations and simplifications. The results demonstrate the persistence of uphill transport phenomena across modeling scales, clarify the conditions under which they occur, and suggest that uphill transport may play a significant role in nanoscale electrolytes, confined ionic and iontronic devices, and membrane-based technologies.

💡 Research Summary

This paper investigates the phenomenon of uphill (or reverse) transport—where particle flux runs opposite to the direction prescribed by Fick’s law—in drift‑diffusion systems that incorporate volume exclusion. The authors begin by formulating a multispecies weakly asymmetric simple exclusion process (M‑WASEP) in which particles hop on a one‑dimensional lattice with rates biased by the discrete gradients of two external potentials Ψ₁ and Ψ₂. By performing a diffusive space‑time scaling (N → ∞, t ∼ N²), they derive the hydrodynamic limit (HDL), a set of coupled nonlinear PDEs for the macroscopic densities ρ₁(ξ,t) and ρ₂(ξ,t). The exclusion constraint ρ₁+ρ₂ ≤ 1 gives rise to a mobility matrix that naturally splits the total flux into four contributions: (i) a Fickian diffusion term driven by concentration gradients, (ii) a drift term proportional to the external fields, (iii) a corrective term arising from mutual exclusion (quadratic in the densities), and (iv) the sum of the three, i.e., the total flux.

In the stationary regime (SHDL) the fluxes become spatially constant, and the authors define partial uphill transport (each species’ flux has the same sign as its boundary concentration difference) and global uphill transport (the sum of the fluxes shares the sign of the total concentration difference). By assuming linear potentials Ψᵢ = aᵢ ξ, they obtain explicit conditions on the boundary differences Δ₁, Δ₂ and field strengths a, b that delineate regions of uphill behavior in parameter space.

The second major contribution is the connection of SHDL to a modified Poisson‑Nernst‑Planck (mPNP) model for oppositely charged ions. The electrostatic potential Φ satisfies Poisson’s equation with source term ρ₁ − ρ₂, and the same exclusion‑induced mobility matrix appears in the ionic fluxes. Numerical comparisons show that SHDL and mPNP predict identical uphill regimes, confirming the consistency of the microscopic and continuum descriptions.

Finally, the authors apply the framework to an ion‑selective membrane sandwiched between two electrolytes. By varying boundary concentrations, applied voltages, and steric (exclusion) parameters, they demonstrate that strong concentration gradients combined with significant electric fields can trigger uphill transport, whereas weakening the exclusion effect restores ordinary diffusive behavior. The study highlights that uphill transport is robust across scales and may play a crucial role in nanofluidic electrolytes, iontronic devices, and membrane technologies where high ion densities and confinement make volume exclusion unavoidable.