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Uphill transport in competitive drift-diffusion models with volume exclusion

Francesco Casini, Cristian Giardinà, Jacopo Nicolini, Luca Selmi, Cecilia Vernia

TL;DR

The paper addresses uphill transport in crowded drift-diffusion systems with volume exclusion and develops a two-scale framework that links a microscopic hydrodynamic limit (SHDL) to continuum electrostatic models. It shows uphill regimes emerge from the competition between diffusion and field-driven drift, providing phase diagrams and simulation results for a two-species exclusion process and clarifying how these regimes persist when electrostatics are included via mPNP-type formalisms. A self-consistent, electrostatic SHDL variant (P-SHDL) is shown to align closely with the modified Poisson–Nernst–Planck (mPNP) model under near-equilibrium conditions, clarifying the validity of equilibrium-based corrective fluxes. The physical case study of an ion-permeable membrane demonstrates practical relevance to nanoscale electrolytes and iontronic devices, showing how steric effects and electrostatics jointly control uphill transport across realistic boundary conditions. Overall, the work bridges discrete exclusion-based particle models and continuum electrochemical descriptions, highlighting uphill transport as a robust, scale-spanning phenomenon with potential impact on membrane technologies and nanoscale electrolytes.

Abstract

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.

Uphill transport in competitive drift-diffusion models with volume exclusion

TL;DR

The paper addresses uphill transport in crowded drift-diffusion systems with volume exclusion and develops a two-scale framework that links a microscopic hydrodynamic limit (SHDL) to continuum electrostatic models. It shows uphill regimes emerge from the competition between diffusion and field-driven drift, providing phase diagrams and simulation results for a two-species exclusion process and clarifying how these regimes persist when electrostatics are included via mPNP-type formalisms. A self-consistent, electrostatic SHDL variant (P-SHDL) is shown to align closely with the modified Poisson–Nernst–Planck (mPNP) model under near-equilibrium conditions, clarifying the validity of equilibrium-based corrective fluxes. The physical case study of an ion-permeable membrane demonstrates practical relevance to nanoscale electrolytes and iontronic devices, showing how steric effects and electrostatics jointly control uphill transport across realistic boundary conditions. Overall, the work bridges discrete exclusion-based particle models and continuum electrochemical descriptions, highlighting uphill transport as a robust, scale-spanning phenomenon with potential impact on membrane technologies and nanoscale electrolytes.

Abstract

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.
Paper Structure (24 sections, 31 equations, 13 figures, 1 table)

This paper contains 24 sections, 31 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Uphill transport in exclusion models with drift
  3. Multispecies Weakly Asymmetric Simple Exclusion process
  4. Hydrodynamic limit
  5. Fluxes and continuity equations
  6. The steady state (SHDL)
  7. Uphill transport for the SHDL model with constant drift
  8. Definition of uphill transport in the multispecies setup.
  9. The case of a linear potential.
  10. Uphill transport phase diagrams.
  11. Simulation set #1: $\Delta>0$, $\Delta_{1}>0$, $\Delta_{2}<0$ and $|\Delta_{1}|>|\Delta_{2}|$.
  12. Simulation set #2: $\Delta>0$, $\Delta_{1}>0$, $\Delta_{2}>0$ and $|\Delta_{1}|>|\Delta_{2}|$.
  13. Drift-diffusion models for charged particles with volume exclusion
  14. The modified Poisson Nernst Planck model (mPNP)
  15. Normalization of the mPNP model
  16. ...and 9 more sections

Figures (13)

  • Figure 1: Representation of the microscopic Markov model with two particle species (blue and red) hopping across sites with weakly asymmetric probability rates.
  • Figure 2: The plots represent the lines of sign change (symbols), and the regions of uphill transport (shaded areas) for the flux of species 1 (red), species 2 (blue), and for the global flux (green), in the space of the field parameters $a, b$. Notice the presence of regions where more than one uphill regime (partial or global) coexists. The insets show the linear interpolation of the boundary density values.
  • Figure 3: The plots represent the lines of sign change (symbols), and the regions of uphill transport (shaded areas) for the flux of species 1 (red), species 2 (blue), and for the global flux (green), in the space of the field parameters $a, b$. Notice the presence of regions where more than one uphill regime (partial or global) coexist in graph (a). The insets show the linear interpolation of the boundary density values.
  • Figure 4: Sketch of the physical system that is mapped into the SHDL model.
  • Figure 5: Comparison of the mPNP, P-SHDL, and SHDL models at high voltage and high boundary densities. The mPNP and P-SHDL profiles nearly coincide, confirming the validity of the P-SHDL equilibrium approximation in this parameters' range. The SHDL model deviates from the other two models, reflecting the violation from the Poisson equation and the linear potential profile. Nonetheless, the fluxes of species $1$ and $2$ have the same sign for the three models. Boundary conditions: $\Psi^R = 19.34$, $\Psi^L = 0$, $\rho_1^R=0.45$, $\rho_2^R=0.11$, $\rho_1^L=\rho_2^L=0.22$, corresponding to $\varphi^R = 500\,$mV, $\varphi^L = 0\,$mV, $c_1^R=2000\,$mM, $c_2^R=500\,$mM, $c_1^L=c_2^L=1000\,$mM respectively.
  • ...and 8 more figures

Theorems & Definitions (1)

  • Definition 1: Uphill transport