--- title: "How ions reshape the field that moves them" description: "Build Poisson–Nernst–Planck transport from conservation, introduce Debye length, and distinguish blocking, selective and reactive boundaries." canonical_url: "https://sandboxphysics.com/stories/electrochemistry/pnp-boundaries/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/electrochemistry/pnp-boundaries/" last_modified: "2026-10-28" --- [Electrochemistry and ion transport](https://sandboxphysics.com/stories/electrochemistry/) L2 · Chapter 2 # How ions reshape the field that moves them Sandbox Physics · 2026-10-28 Ion motion changes the electric field, and the changed field moves the ions again. Poisson–Nernst–Planck theory closes this feedback by conserving each species and calculating potential from the remaining charge. The boundary decides whether the outcome is charge storage, sustained transport or chemical reaction. ## Before you begin Read chapter one on flux and electrochemical potential. A derivative gives a local rate or slope; an integral adds particle numbers along the channel. You do not need to solve partial differential equations in advance. [Review the preceding chapter](https://sandboxphysics.com/stories/electrochemistry/chemical-potential-flux/) - Explain ion conservation as an imbalance of entering and leaving flux. - Use Debye length to judge whether a charge layer is thin. - Choose appropriate boundaries for blocking plates, a selective interface and a reacting electrode. ## Conserve the ions in a small box Take a short box along $X$ with fixed cross-sectional area. If the entering number flux exceeds the leaving flux, number density $n_\pm$ increases. Divide the flux difference by box length and shrink the box to obtain the spatial derivative. Each species obeys its own conservation equation. For now the liquid is stationary and the bulk contains no ion generation or recombination. Crossing the two faces is the only way to change the number in a box. This is also a strong test of a numerical calculation. $$ \partial_\tau n_\pm=-\partial_XN_\pm $$ Dimensional time $\tau$ is in seconds. Integrating along a closed channel shows that total particle number can change only through end fluxes. Reading: Bazant ## Net charge bends the potential For a monovalent symmetric electrolyte, the ion charges are $+e$ and $-e$. Their number-density difference multiplied by elementary charge gives charge density. Gauss’s law relates charge to the gradient of the field. Replacing the field by the negative potential gradient gives Poisson’s equation. A uniform field gives a straight potential profile. Curvature means net charge is present. A small amount of charge can produce an important voltage change, especially within a thin surface layer. Potential and ion profiles must therefore be solved together. $$ -\varepsilon\partial_X^2\Phi=e(n_+-n_-) $$ The permittivity $\varepsilon$ is in farads per metre. Water is a continuum with constant permittivity in this model; microscopic ionic correlations are averaged out. Reading: Bazant · MIT 10.626 ## Choose scales before comparing solutions Let $L$ be the full gap, $n_0$ the initial density of each ion, and $D$ their common diffusivity. Divide position by gap, time by diffusion time, potential by thermal voltage and densities by their initial value. The resulting dimensionless parameters describe whole families of physical cells. Debye length $\lambda_D$ is the screening distance for a small potential disturbance. Its ratio to gap, $\delta$, determines how much of the cell is occupied by a charge layer. More salt supplies stronger screening and a shorter Debye length. $$ x=\frac XL,\quad t=\frac{D\tau}{L^2},\quad c_\pm=\frac{n_\pm}{n_0},\quad\phi=\frac{\Phi}{V_T},\quad\delta=\frac{\lambda_D}{L} $$ All these variables are dimensionless. We use the full gap. Papers that use a half-gap have corresponding factors of two in charging times. $$ \lambda_D=\sqrt{\frac{\varepsilon k_BT}{2e^2n_0}},\qquad J_\pm=-\left(\partial_x c_\pm\pm c_\pm\partial_x\phi\right) $$ Dimensionless number flux $J_\pm$ has dimensional scale $Dn_0/L$. Both monovalent species contribute to screening, giving the factor two. $$ \partial_t c_\pm=-\partial_xJ_\pm,\qquad -2\delta^2\partial_x^2\phi=c_+-c_- $$ Two conservation equations and Poisson’s equation form the ideal dilute PNP model shared by the two transport experiments. Reading: Bazant · MIT 10.626 ## The boundary changes the physical problem A blocking electrode is a wall to both ions: their normal fluxes vanish. Electrons can still enter the metal through the external circuit and attract counterions in solution, so the cell stores charge. If a compact layer is neglected, the solution surface takes the imposed electrode potential. A compact layer instead divides voltage according to its capacitance. A selective boundary transmits one species. Our concentration-polarization problem has a fixed-concentration reservoir on the left. On the right, cation boundary activity is prescribed and anion flux vanishes. The reservoir replenishes transmitted cations, so total ion number may change. At a reactive electrode, ion flux follows the interfacial reaction rate. Stoichiometry must match consumed or produced ions to transferred electrons. Chapter six introduces Butler–Volmer kinetics. Confusing these boundaries can turn capacitive charging into a spurious sustained reaction current. Reading: Bazant · MIT 10.626 · MIT 10.626 ## Estimate a screening length At room temperature take water permittivity $\varepsilon=6.95\times10^{-10}\;\mathrm{F\,m^{-1}}$. A concentration of $1\;\mathrm{mM}=1\;\mathrm{mol\,m^{-3}}$ per species gives $n_0=6.022\times10^{23}\;\mathrm{m^{-3}}$ after multiplication by Avogadro’s constant. The Debye length is about $9.62\;\mathrm{nm}$. Ten times more salt reduces that length by the square root of ten, to $3.04\;\mathrm{nm}$. In a $1\;\mathrm{\mu m}$ full gap, the first solution has $\delta\approx0.00962$. Most of its interior can be nearly neutral, although its surface charge layers still matter. The browser also explores larger ratios so that the internal layers can be resolved. Electroneutrality is a scale approximation for the bulk. Where ions become depleted, local screening weakens and the local Debye length grows. Chapter four follows the consequences as charge extends farther into the solution. ## Try the prediction ### [Compare thin and overlapping charge layers](https://sandboxphysics.com/labs/double-layer-charging/) 1. Predict how a larger Debye-length ratio will change the ion profiles. 2. Keep voltage fixed and change the ratio. Identify where most of the potential drop occurs. 3. Check total ion counts for the closed cell, then explain how external electrode current can still flow. ## Exercises and explanations 1. The total cation number slowly decreases between blocking electrodes. What should you inspect first? Read the explanation Check that the two boundary number fluxes are exactly zero and that each internal face flux enters its neighbouring cells with opposite signs. Boundary flux controls total number; even a strong internal field cannot destroy cations. 2. Double the full gap at fixed temperature and salt concentration. What happens to Debye length, its ratio to gap, and diffusion time? Read the explanation Debye length stays fixed because it depends on the solution. Its ratio to gap halves. Cross-cell diffusion time grows with gap squared and becomes four times longer. Geometric length and screening length play different roles. [Previous: Which way do ions move in salt water?](https://sandboxphysics.com/stories/electrochemistry/chemical-potential-flux/) [Next: Why a blocking electrode can still charge](https://sandboxphysics.com/stories/electrochemistry/blocking-double-layer/) ## 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, 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.