---
title: "Electrochemistry and ion transport"
description: "Follow ions from diffusion and double-layer charging to concentration polarization, electro-osmotic flow and impedance, with nine chapters and four experiments."
canonical_url: "https://sandboxphysics.com/stories/electrochemistry/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/electrochemistry/"
---

FR17

# Electrochemistry and ion transport

Sandbox Physics · 2026-10-28

A voltage can make ions accumulate, run short or drag the surrounding liquid. The difference begins at the boundary: can ions cross it, react there or only gather against it? Follow that question from a salt solution between plates to a charged cylinder and an impedance measurement.

## Begin with what the boundary allows

A blocking plate stores charge beside its surface. A selective interface sustains current and depletes the approaching salt. A metal cylinder can induce a charge pattern that drives liquid around it. A reacting electrode divides current between charge storage and electron transfer. Each phenomenon follows from the same questions about ion motion and the boundary it encounters.

The opening chapters introduce chemical potential, flux, conservation and screening as they become useful. The middle chapters derive charging, limiting current, electro-osmotic slip and reaction resistance. The final pair follows a fixed paper calculation and asks what is needed to move from steady transport to instability. Worked examples and explained exercises let you follow the argument even before opening an experiment.

[Shared textbook foundations](https://github.com/yangzichao/sandboxphysics/issues/27) · [FR17 route and scientific scope](https://github.com/yangzichao/sandboxphysics/issues/47)

1. L1 · 1

   ## [Which way do ions move in salt water?](https://sandboxphysics.com/stories/electrochemistry/chemical-potential-flux/)

   Combine diffusion and electric forces into electrochemical potential, derive dilute-ion flux, and work out the sign of an ionic current.

2. L2 · 2

   ## [How ions reshape the field that moves them](https://sandboxphysics.com/stories/electrochemistry/pnp-boundaries/)

   Build Poisson–Nernst–Planck transport from conservation, introduce Debye length, and distinguish blocking, selective and reactive boundaries.

3. L2 · 3

   ## [Why a blocking electrode can still charge](https://sandboxphysics.com/stories/electrochemistry/blocking-double-layer/)

   Connect ion profiles to electrode charge, transient current and charging time, then check the Debye–Hückel and thin-layer limits.

4. L3 · 4

   ## [How ion depletion creates a transport limit](https://sandboxphysics.com/stories/electrochemistry/concentration-polarization/)

   Derive concentration polarization with a Donnan voltage drop, then compare electroneutral diffusion limits with space-charge transport.

5. L3 · 5

   ## [How surface ions drag the surrounding liquid](https://sandboxphysics.com/stories/electrochemistry/electroosmotic-flow/)

   Derive electro-osmotic slip from viscous force balance, then use one cylinder potential to calculate induced charge and quadrupolar flow.

6. L3 · 6

   ## [When ions exchange electrons at an electrode](https://sandboxphysics.com/stories/electrochemistry/butler-volmer/)

   Build Butler–Volmer current from forward and reverse reactions, distinguish local overpotential, and derive charge-transfer resistance.

7. L3 · 7

   ## [What can one impedance arc tell us?](https://sandboxphysics.com/stories/electrochemistry/impedance-identifiability/)

   Introduce complex impedance through sinusoidal response, derive an interfacial arc, and separate parameter uncertainty from non-identifiability.

8. L4 · 8

   ## [Recalculating the induced-charge flow around a cylinder](https://sandboxphysics.com/stories/electrochemistry/reading-iceo-paper/)

   Read the neutral-cylinder figure in Squires and Bazant, then reconstruct a limited weak-field steady result from potential, slip and the Stokes field.

9. L4 · 9

   ## [When a depleted layer starts to move](https://sandboxphysics.com/stories/electrochemistry/electroconvection-frontier/)

   Connect space charge, transverse disturbances and flow feedback, distinguish overlimiting mechanisms, and read electroconvective stability curves.

## Four experiments along the route

- [Watch a blocking cell charge from its ion profiles](https://sandboxphysics.com/labs/double-layer-charging/)
- [Compare ion depletion with a space-charge solution](https://sandboxphysics.com/labs/concentration-polarization/)
- [Turn an electric field around a charged cylinder](https://sandboxphysics.com/labs/induced-charge-electroosmosis/)
- [Acquire an impedance spectrum and fit its saved points](https://sandboxphysics.com/labs/impedance-inference/)

The transport experiments solve ideal dilute-ion models. The cylinder experiment uses a weak-field analytic flow. The impedance experiment generates a labelled synthetic record and fits that saved record. Their theoretical references and simulated data keep their own labels.

For additional fluid intuition, the existing Couette–Poiseuille and capillary-wetting experiments show viscous flow and interfacial forces. They are prerequisite companions to this course. [Viscous flow](https://sandboxphysics.com/labs/couette-poiseuille/) · [Capillary wetting](https://sandboxphysics.com/labs/capillary-wetting/)

## Sources and calculation scope

1. [MIT 10.626, Diffuse charge in electrolytes, 2011 Lecture 24](https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/34aaca3a97887695dd295db7cc0fa3c0_MIT10_626S14_S11lec24.pdf)

   Electrochemical potential, Nernst–Planck flux, charge continuity and dilute-electrolyte screening. Our notation uses number density, a full cell gap and a thermal-voltage scale.

2. [MIT 10.626, Double layer structure, 2011 Lecture 25](https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/b74a3f2744e2c150142cc1eab0d25fdc_MIT10_626S14_S11lec25.pdf)

   Planar diffuse-layer structure and differential capacitance. The chapter derives its small-potential limit and separately states the ideal point-ion approximation.

3. [Bazant, Thornton & Ajdari, Diffuse-Charge Dynamics in Electrochemical Systems, arXiv:cond-mat/0401118v1 (2004)](https://arxiv.org/abs/cond-mat/0401118v1)

   Sections III–V and the thin-layer analysis define the blocking-plane PNP problem, conservation and charging scales; published as Physical Review E 70, 021506. The paper uses plates at plus and minus a half-gap. Our full-gap convention gives a factor one-half in the two-layer RC time. Our browser omits the compact Stern layer.

4. [MIT 10.626, Ion concentration polarization, 2011 Lecture 23](https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/94d59af721c346ed05c99f6359bf05fb_MIT10_626S14_S11lec23.pdf)

   Selective transport, concentration gradients and diffusion-limited current. Our reservoir-to-interface problem states its own cation-activity boundary and explicitly includes the Donnan voltage in the total applied voltage.

5. [MIT 10.626, Electrokinetics, 2011 Lecture 29](https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/resources/mit10_626s14_s11lec29/)

   Thin-double-layer electro-osmosis and viscous force balance. The chapter derives slip with the wall normal, zeta-potential sign and tangential field defined before use.

6. [MIT 10.626, Faradaic reactions in dilute solutions, 2011 Lecture 12](https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/c4b64ab4d5bb9fd4d37209cff799bc78_MIT10_626S14_S11lec12.pdf)

   Local reaction stoichiometry and the transition-state picture behind charge transfer. The course uses anodic current positive and separates local activity from reservoir concentration.

7. [MIT 10.626, Butler–Volmer equation, 2011 Lecture 13](https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/56cfa6e0f28bc8fc1a647cbe679384d1_MIT10_626S14_S11lec13.pdf)

   Forward and reverse activation currents, equilibrium potential and overpotential. Our worked example expands the local fixed-concentration law near equilibrium and uses whole-electrode exchange current in the impedance model.

8. [MIT 10.626, Impedance spectroscopy, 2011 Lecture 5](https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/d5e1b842d6daa3d9dc1991b916639b09_MIT10_626S14_S11lec05.pdf)

   Sinusoidal response, complex amplitudes and circuit impedances. The course introduces positive-exponential phasors before using them and plots negative imaginary impedance upward.

9. [MIT 10.626, Impedance of electrodes, 2011 Lecture 6](https://ocw.mit.edu/courses/10-626-electrochemical-energy-systems-spring-2014/6aacd39911619ff5b669d80a0a0c7bbf_MIT10_626S14_S11lec06.pdf)

   Reaction resistance and double-layer capacitance in a lumped small-signal electrode. Our spectrum uses a series solution resistor and one parallel reaction-resistor/capacitor pair, with no diffusion element.

10. [Squires & Bazant, Induced-charge electro-osmosis, arXiv:physics/0304090v2 (9 February 2004)](https://arxiv.org/abs/physics/0304090v2)

    Frozen preprint of Journal of Fluid Mechanics 509, 217–252: section 3.1, PDF equations (3.4)–(3.9), Figure 3(a) and Table 1. The limited reproduction calculates the steady field, charge, slip and outer Stokes flow for a fixed neutral cylinder. The HTML conversion renumbers equations consecutively.

11. [Bazant & Squires, Induced-charge Electrokinetic Phenomena: Theory and Microfluidic Applications, arXiv:physics/0306100v2 (2004)](https://arxiv.org/abs/physics/0306100v2)

    Physical Review Letters 92, 066101, especially equation (1) and Figure 1(c). This short paper supplies the physical picture; it is distinct from the longer JFM paper. Its original figures are linked, not redistributed.

12. [Sandbox Physics analytic ICEO reference, version 1 (original calculated values)](https://sandboxphysics.com/data/frontier/fr17/methods-iceo-reference.json)

    Original computed surface and velocity samples from the stated cylinder solution, with units and parameters. These are theoretical values, not experimental measurements or digitized points from a published figure.

13. [Rubinstein & Zaltzman, The driving factors of electro-convective instability in concentration polarization, arXiv:1602.00137v1 (2016)](https://arxiv.org/abs/1602.00137v1)

    Sections I–II, equations (3)–(9) and Figures 1–3 distinguish selective conduction, fluid coupling and a transverse stability calculation. The course reads this published research; its one-dimensional PNP experiment does not reproduce the neutral-stability curves.

14. [Dydek et al., Overlimiting current in a microchannel, arXiv:1108.2325v1 (2011)](https://arxiv.org/abs/1108.2325v1)

    Figures 1–3 and the separate surface-conduction, side-wall-flow and instability mechanisms. Their channel geometry and voltage convention differ from the course half-cell; our PNP curve is not a reproduction of those figures.

15. [Kilic, Bazant & Ajdari, Steric effects in the dynamics of electrolytes at large applied voltages I, arXiv:physics/0611030v1 (2006)](https://arxiv.org/abs/physics/0611030v1)

    Sections I–II explain why point-ion dilute theory can overpredict near-surface concentrations and introduce finite-size alternatives. These modifications are reading material, not active equations in the ideal PNP browser solver.