Rotating Disk Electrodes beyond the Levich Approximation: Physics-Informed Neural Networks Reveal and Quantify Edge Effects

Abstract

Physics-informed neural networks are used to characterize the mass transport to the rotating disk electrode (RDE), the most widely employed hydrodynamic electrode in electroanalysis. The PINN approach was first quantitatively verified via 1D simulations under the Levich approximation for cyclic voltammetry and chronoamperometry, allowing comparison of the results with finite difference simulations and analytical equations. However, the Levich approximation is only accurate for high Schmidt numbers (Sc > 1000). The PINN approach allowed consideration of smaller Sc, achieving an analytical level of accuracy (error <0.1%) comparable with independent numerical evaluation and confirming that the errors in the Levich equation can be as high as 3% when Sc = 1000 for rapidly diffusing species in aqueous solution. Entirely novel, the PINNs permit the solution of the 2D diffusion equation under cylindrical geometry incorporating radial diffusion and reveal the rotating disk electrode edge effect as a consequence of the nonuniform accessibility of the disc with greater currents flowing near the extremities. The contribution to the total current is quantified as a function of the rotation speed, disk radius, and analyte diffusion coefficient. The success in extending the theory for the rotating disk electrode beyond the Levich equation shows that PINNs can be an easier and more powerful substitute for conventional methods, both analytical and simulation based.

Near Equilibration Times in the Diffusion-Controlled Adsorption

“…A certain surface active substance that is dissolved in very low concentration and that is adsorbing to the surface of a rotating disk is assumed: A ↔ A ads If the edge effect is not negligible, the mass transport at the rotating disk must be calculated in a cylindrical coordinate system, but if the disk surface is considered as infinite large, the transport within a diffusion layer and the adsorption equilibrium can be defined by the following equations: ∂ c / ∂ t = D ∂ 2 c / ∂ x 2 − V ∂ c / ∂ x t = 0 , 0.25em x ≥ 0 : 0.25em 0.25em 0.25em c = c * , 0.25em θ = 0 t > 0 , 0.25em x → ∞ : 0.25em 0.25em 0.25em c → c * x = 0 : 0.25em 0.25em 0.25em β c x = 0 exp false( a θ false) = θ / false( 1 − θ false) D false( ∂ c / ∂ x ) x = 0 = Γ max d θ / d t V = pr...…”

“…A certain surface active substance that is dissolved in very low concentration and that is adsorbing to the surface of a rotating disk is assumed: If the edge effect is not negligible, the mass transport at the rotating disk must be calculated in a cylindrical coordinate system, but if the disk surface is considered as infinite large, the transport within a diffusion layer and the adsorption equilibrium can be defined by the following equations: Here, θ = Γ/Γ max is the surface coverage, Γ max is the maximum surface concentration of the adsorbed substance, V is the flow rate of solution in the axial direction, a = 2 g / RT and g is the interaction energy, R is gas constant, T is temperature, and the meanings of other symbols are explained in Table . Equation was solved by the digital simulation …”

“…An adsorption is the process of accumulation of molecules at the interface. − If the boundary between two phases is created suddenly, the adsorption will result in the diminished concentration of these molecules in the vicinity of the interface. This is the reason for the diffusion-controlled adsorption. , If the rates of adsorption and desorption are much higher than the rate of the interface creation, the former processes appear equilibrated at any measurement time. , This is local equilibrium, and both the amount of adsorbate and the concentration of dissolved surfactant at the interface increase under the influence of flux of adsorptive from the bulk of solution. , When the concentrations of later at the interface and in the bulk become equal, the flux vanishes and the general equilibrium is established. − It is expressed by the adsorption isotherm, that is, the equilibrium relation between the quantity of the adsorbed material and the concentration in the bulk solution at constant temperature. − Usually, the rate of equilibration is enhanced by convection so that the diffusion is restricted to the thin layer at the interface. − The term kinetics of adsorption frequently includes this transport component. − The rotating disk is a good solid–liquid interface because of precisely defined convection and diffusion fluxes. , The adsorption at this type of interface can depend on interactions between adsorbed molecules, which is described by Fowler–Guggenheim or Frumkin isotherm. − These isotherms are developed for the monolayer adsorption using the kinetic principle that the rates of adsorption and desorption are equal in the equilibrium. The first rate is proportional to the product of bulk concentration of dissolved surfactant and the free fraction of the surface of the adsorbent, while the second rate is proportional to the occupied fraction of the surface. If there is lateral interaction of the adsorbed molecules, the isotherm includes the term that corresponds to the energy of interaction. , The parameters of the isotherm are thermodynamic quantities and must be measured under bulk equilibrium conditions.…”

Redox-detecting deep learning for mechanism discernment in multi-redox cyclic voltammograms

“…[8][9][10] Machine-learning techniques have been applied to mechanistic classification of single-redox voltammograms, [11][12][13] and numerical fitting of voltammogram data under a pre-determined mechanistic assignment. [14][15][16] It is proposed that machine learning's expertise in pattern recognition and feature extraction 17 is complementary if not substitutive to manual inspection of electrochemical data. 11, 12, 18 For example, our recent work reported a deep-learning (DL) model based on the architecture of ResNet (Residual Neural Network) 19 that automatically analyzes cyclic voltammograms (Fig.…”

Removing 65 Years of Approximation in Rotating Ring Disk Electrode Theory with Physics-Informed Neural Networks

“…The recent introduction of PINN, as a novel discretization-free partial differential equation (PDE) solver, has proven effective and accurate, to solve complicated PDEs in various domains, from modeling and reconstructing fluid mechanics flow fields, , to material fatigue prediction and solid mechanics, , and to blood pressure and hemodynamics estimation in healthcare. , In the field of electrochemistry, PINN has re-educated hydrodynamic electrochemistry simulation in areas ranging from single and double microband channel electrodes to the rotating disk electrode with analytical levels of accuracy. , In 2024, PINN is no longer at its infancy, or is complementary to traditional finite difference and finite element methods . The Electrochemistry-Informed Neural Netwok (ECINN) embedded electrochemical kinetic laws with mass transport equations, achieving simultaneous discovery of electrochemical rate constants, transfer coefficients, and diffusion coefficients .…”