An Analytical Model for the Spectral Induced Polarization of Metallic Particles Including Stern‐Layer Polarization

Abstract

Spectral induced polarization (SIP) is sensitive to metallic grains in geologic materials. Classical diffuse‐layer theory predicts a weaker dependence of relaxation time on electrolyte conductivity than is commonly measured, whereas surface‐capacitance models can reproduce the stronger dependence but do not explicitly resolve the diffuse layer or the transition between the two regimes. We develop an analytical model for a spherical electronically conductive particle in a symmetric electrolyte by coupling the linearized Poisson–Nernst–Planck equations in the diffuse layer to a self‐consistent Stern‐layer Robin boundary condition. The solution yields a closed‐form dipole reflection coefficient and, through mixing, the effective complex conductivity of a dilute suspension. By explicitly resolving the diffuse layer, the model yields a single relaxation time, , governed by the ratio of the effective Stern length to the Debye length. It interpolates between a diffuse‐layer‐controlled limit, , and a compact‐layer‐controlled limit, . Consequently, the dependence of relaxation time on electrolyte conductivity changes from the weaker square‐root scaling predicted by classical diffuse‐layer theory to the inverse scaling commonly reported in laboratory measurements and obtained by surface‐capacitance models. Comparison with published measurements shows consistency with the compact‐layer‐dominated regime over the experimentally investigated conductivity range, although measurements at sufficiently low salinity are required to test the predicted crossover directly.