Engineering & Science
Nernst-Planck Electrodiffusion Equation: Fickian Diffusion + Electric Migration Flux
Calculate total ionic molar flux J_i = -D_i * (dC/dx + ((z_i * F) / (R * T)) * C * (dPhi/dx)) in umol/(m^2*s) across PEM fuel cells, Li-ion electrolytes, and biological ion channels.
Model coupled concentration diffusion and electric-field drift in batteries & neurons!
Fick's First Law J = -D * (dC/dx) works ONLY for uncharged molecules! Inside a Li-ion battery separator, Nafion fuel-cell proton membrane, electrodialysis desalination stack, or neuronal potassium channel, charged ions of valence z_i experience BOTH random thermal diffusion down concentration gradient dC/dx AND electrophoretic drift down electric potential gradient dPhi/dx (where E = -dPhi/dx)!
Walther Nernst and Max Planck combined Fickian diffusion with the Einstein-Smoluchowski mobility relation u_i = D_i / (R * T) (with F = 96,485.33 C/mol and R = 8.31446 J/(mol*K)) to derive the Nernst-Planck flux equation (without bulk convection): J_i = -D_i * (dC/dx + ((z_i * F) / (R * T)) * C * (dPhi/dx))! Look at how dramatically even a modest electric field dominates diffusion for a monovalent cation (z = +1, D = 1.0e-9 m^2/s, C = 10 mol/m^3, dC/dx = -100 mol/m^4 at T = 298.15 K): pure Fickian diffusion (dPhi/dx = 0) yields just +0.100 umol/(m^2*s), whereas applying dPhi/dx = -50 V/m adds +19.461 umol/(m^2*s) of forward migration (total +19.561 umol/(m^2*s)), and reversing the field to +50 V/m drives ions BACKWARD uphill against concentration at -19.361 umol/(m^2*s)!
| Potential Gradient dPhi/dx | Fickian Diffusion Term | Electric Migration Term | Net Cation Flux J (umol/m^2*s) |
|---|---|---|---|
| Zero Field (dPhi/dx = 0 V/m) | +0.100 umol/(m^2*s) | 0.000 umol/(m^2*s) | +0.100 (Pure Fickian diffusion) |
| Assisting Field (-50 V/m, E = +50) | +0.100 umol/(m^2*s) | +19.461 umol/(m^2*s) | +19.561 (195x stronger flux!) |
| Opposing Field (+50 V/m, E = -50) | +0.100 umol/(m^2*s) | -19.461 umol/(m^2*s) | -19.361 (Pumped uphill vs dC/dx!) |
Total Nernst-Planck Ionic Molar Flux J_i in umol/(m^2*s)
Evaluates -D * (dC/dx + ((z * 96485.33) / (8.31446 * T)) * C * dPhi/dx) and converts mol/(m^2*s) to umol/(m^2*s) (* 10^6).
=IF(OR(B2 <= 0, B4 < 0, B6 <= 0), "Invalid Electrolyte Inputs", ROUND(-1000000 * B2 * (B5 + (((B3 * 96485.33) / (8.31446 * B6)) * B4 * B7)), 3))Pure Migration Component J_mig Only (umol/(m^2*s))
Isolates the electrophoretic drift contribution -D * ((z * F) / (R * T)) * C * (dPhi/dx) * 10^6.
=IF(OR(B2 <= 0, B4 < 0, B6 <= 0), "Invalid Electrolyte Inputs", ROUND(-1000000 * B2 * (((B3 * 96485.33) / (8.31446 * B6)) * B4 * B7), 3))Try the example
Use the original values or change them to test your understanding.