--- title: "Concentration Polarization and Space Charge Interactive Lab | Sandbox Physics" description: "What changes when a membrane draws ions faster than diffusion can replace them?" canonical_url: "https://sandboxphysics.com/labs/concentration-polarization/" language: "en" page_type: "lab" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/labs/concentration-polarization/" --- Electrochemistry · A selective interface runs short of salt # Concentration Polarization and Space Charge **What changes when a membrane draws ions faster than diffusion can replace them?** A selective membrane removes one ion species and leaves a depleted layer in front of it. The electroneutral approximation approaches a diffusion-limited current. When depletion becomes strong, net space charge can develop and the full transport model departs from that limit. Pull cations toward an ideal selective membrane. Watch the depleted region grow, compare its final profile with the electroneutral approximation, and increase the voltage until space charge carries the current beyond the classical limit. > Interactive simulation: open this page in a web browser to change the parameters and run the model. Model assumptions This is a one-dimensional dilute ideal electrolyte with equal ion diffusivities, a fixed-concentration reservoir and a cation-selective interface. The interface fixes cation concentration and voltage and blocks anions. Applied voltage includes the interfacial Donnan drop. The two-micrometre gap and enlarged screening length are teaching scales. The calculation has no fluid flow, electroconvective instability, surface conduction, water splitting or real membrane microstructure. Simulated transport record ## Follow the ions and the current together ### Ion concentration across the cell ### Potential across the cell ### Current after the voltage step Cations Anions Potential / total current Dashed: electroneutral steady limit The neutral reference approaches twice the current-density scale. The full model includes the charged interface layer and its transient. Both are simulated or analytic model results. Equations and primary sources $$ J_\pm=-\partial_xc_\pm\mp c_\pm\partial_x\phi,\quad -2\delta^2\partial_x^2\phi=c_+-c_-,\quad j_{\rm tot}=J_+-J_-+2\delta^2\partial_tE $$ [Bazant, Thornton and Ajdari (2004), blocking-cell charge dynamics](https://arxiv.org/abs/cond-mat/0401118v1) [Rubinstein and Zaltzman (2016), concentration polarization and electroconvection](https://arxiv.org/abs/1602.00137v1) [Learn how diffusion, charge and flow connect](https://sandboxphysics.com/stories/electrochemistry/) Physics tutorial ## Concentration Polarization and Space Charge **Background** The reservoir can supply both ions, but the membrane takes only cations. Anions cannot pass through it. Their zero steady flux links the local concentration to the potential, and salt becomes scarce just in front of the membrane. **Why it matters** The classical limiting current follows from a nearly neutral liquid. Strong depletion is precisely where this approximation becomes fragile, so it is useful to keep Poisson’s equation and inspect the local charge. Start with the essentials - **Focus question**: What changes when a membrane draws ions faster than diffusion can replace them? - **One-sentence intuition**: The depleted concentration and the space-charge field must be solved together. A one-dimensional current above the neutral diffusion limit does not predict the onset of fluid motion. ### Core mathematical model #### Diffusion and electrical drift $$ J_\pm=-\partial_xc_\pm\mp c_\pm\partial_x\phi,\qquad \partial_tc_\pm=-\partial_xJ_\pm $$ Distance is scaled with the full gap, concentration with the reservoir concentration, potential with the thermal voltage, and time with gap squared divided by diffusivity. Positive flux points from left to right. Both ions have the same diffusion coefficient. #### Charge changes the potential $$ -2\delta^2\partial_x^2\phi=c_+-c_-,\qquad \delta=\lambda_{\rm D}/L $$ The concentration difference is the net charge in these units. A small screening ratio lets appreciable charge collect in narrow layers. The numerical mesh resolves these layers, with its convergence checked separately. #### The electroneutral steady comparison $$ c(x)=1-\frac{j x}{2},\quad \phi(x)=\ln c(x),\quad j=2(1-e^{-V/2}) $$ The left reservoir has unit concentration and zero potential. The right membrane fixes cation concentration at unity and potential at minus the applied voltage while blocking anions. The voltage includes the Donnan drop; the neutral outer profile approaches a limiting current of two. ### Common difficulties #### A neutral limit is an approximation **Typical misconception** The classical limiting current is an absolute ceiling for every electrolyte system. **Better mental model** The neutral diffusion limit assumes local electroneutrality and no other transport mechanisms. Space charge changes the electrical migration; fluid flow and surface transport are separate extensions. ### Run the experiment 1. 01 #### Start where diffusion keeps up Choose Diffusion keeps up and let the evolution settle. Predict where the two concentration curves should almost coincide, then compare them with the dashed neutral reference. **What to observe:** Away from the membrane charge layer, the two ion concentrations nearly coincide and decline toward the interface. The reference is the steady outer solution, so early transient curves need not match it. 2. 02 #### Follow the depleted region Choose A depleted layer. Drag the time marker in the cell to compare an early frame with a late one. Then increase Applied voltage. **What to observe:** Salt is depleted near the membrane and the potential drop becomes concentrated there. The ionic flux and current are computed from these same concentration and potential arrays. 3. 03 #### Inspect space charge Choose Space charge spreads. Compare the full current with the classical limit and inspect the separation of the positive and negative concentration curves. **What to observe:** The full one-dimensional model can carry current above the neutral limit through a charged region. Flow, surface conduction and water splitting would require additional physics. Physical conclusion Identify the approximation behind a limiting current before extending it to a membrane experiment. ## Related experiments and articles Electrochemistry and ion transport · Reading path ### From ion diffusion to charge, flow and impedance Articles on this path that use this experiment, followed by the other experiments they suggest. 1. [Article **How ion depletion creates a transport limit**](https://sandboxphysics.com/stories/electrochemistry/concentration-polarization/) 2. [Article **When a depleted layer starts to move**](https://sandboxphysics.com/stories/electrochemistry/electroconvection-frontier/) 3. [Lab **Induced-Charge Electro-osmosis** Distinguish imposed electro-osmotic flow from instability](https://sandboxphysics.com/labs/induced-charge-electroosmosis/) Fields & matter · Starter path ## From a field to an electron microscope Map an electric field and steer a charged particle. 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