API Reference¶
Welcome to the Soft Potato API Reference. The toolkit is organized into modular subpackages and modules for physical constants, closed-form analytical electrochemistry, and numerical PDE solvers.
Constants Module¶
The softpotato.constants module provides standardized physical and electrochemical
constants defined according to CODATA 2018 / 2019 SI conventions [1].
| Constant / Identifier | Symbol | Value & Units |
|---|---|---|
FARADAY (F) |
\(F\) | \(96485.332 \,\text{C}\cdot\text{mol}^{-1}\) |
GAS_CONSTANT (R) |
\(R\) | \(8.31446\,\text{J}\cdot\text{mol}^{-1}\cdot\text{K}^{-1}\) |
STANDARD_TEMPERATURE (T_STD) |
\(T^\circ\) | \(298.15\,\text{K}\) (\(25\,^\circ\text{C}\)) |
AVOGADRO |
\(N_{\text{A}}\) | \(6.02214 \times 10^{23}\,\text{mol}^{-1}\) |
BOLTZMANN |
\(k_{\text{B}}\) | \(1.38065 \times 10^{-23}\,\text{J}\cdot\text{K}^{-1}\) |
ELEMENTARY_CHARGE |
\(e\) | \(1.60218 \times 10^{-19}\,\text{C}\) |
VACUUM_PERMITTIVITY |
\(\varepsilon_0\) | \(8.85419 \times 10^{-12}\,\text{F}\cdot\text{m}^{-1}\) |
Analytical Module¶
The softpotato.analytical subpackage provides closed-form analytical solutions,
asymptotic approximations, and diagnostic formulations across transient step techniques,
hydrodynamics, interfacial kinetics, voltammetry, microelectrodes, SECM, parameter fitting,
and numerical benchmark suites.
Transient Step Techniques¶
The softpotato.analytical.step module provides exact solutions for potentiostatic
and galvanostatic step experiments, including Cottrell, Anson, and Sand formulations.
| Function | Inputs | Output | Description | References |
|---|---|---|---|---|
cottrell |
\(t, n, D, c^*, A\) | \(I(t)\) | Planar chronoamperometric diffusion-limited current transient. | 2 |
cottrell_step |
\(t, \tau, n, D, c^*, A, D_{\text{Red}}\) | \(I(t)\) | Double potential step chronoamperometry reversal current. | 3 |
cottrell_spherical |
\(t, r, n, D, c^*, A\) | \(I(t)\) | Chronoamperometric current at a spherical electrode. | 3 |
cottrell_cylinder |
\(t, r_0, n, D, c^*, l, A\) | \(I(t)\) | Chronoamperometric current at a cylindrical wire/fiber electrode. | 4, 5 |
anson |
\(t, n, D, c^*, A, Q_{\text{dl}}, \Gamma, \tau\) | \(Q(t)\) | Cumulative charge transient for chronocoulometry. | 6 |
sand |
\(t, I, n, D, c^*, A\) | \(c(0, t)\) | Surface reactant concentration transient under constant current. | 7 |
sand_transition_time |
\(I, n, D, c^*, A\) | \(\tau\) | Transition time for galvanostatic chronopotentiometry. | 7 |
sand_potential |
\(t, \tau, E_{1/2}, n, T\) | \(E(t)\) | Potential-time response curve for chronopotentiometry. | 7 |
step_concentration_profile |
\(x, t, D, c^*\) | \(c(x, t)\) | Exact spatial concentration profile following a potential step. | 8 |
step_flux_profile |
\(x, t, D, c^*\) | \(J(x, t)\) | Exact spatial diffusion flux profile following a potential step. | 8 |
Hydrodynamics & Convection Equations¶
The softpotato.analytical.hydrodynamics module provides models for rotating disk
(RDE) and rotating ring-disk (RRDE) convective electrochemical systems.
| Function | Inputs | Output | Description | References |
|---|---|---|---|---|
levich |
\(\omega, n, D, c^*, A, \nu\) | \(I_{\text{L}}\) | Convective mass-transfer limiting current at an RDE. | 9 |
levich_constant |
\(n, D, c^*, A, \nu\) | \(B\) | Theoretical Levich slope parameter for RDE rotation series. | 9 |
koutecky_levich |
\(\omega, I_{\text{k}}, k, I_{\text{lim}}, n, D, c^*, A, \nu\) | \(I\) | Current under mixed kinetic and convective mass-transfer control. | 10 |
koutecky_levich_analysis |
\(\omega, I, A, c^*, n, \nu\) | \(k, B\) | Kinetic rate constant and Levich slope extraction from RDE data. | 10 |
nernst_diffusion_layer |
\(\omega, D, \nu\) | \(\delta\) | Steady-state stagnant diffusion layer thickness at an RDE. | 9 |
collection_efficiency |
\(r_1, r_2, r_3\) | \(N\) | Theoretical collection efficiency for an RRDE geometry. | 11 |
shielding_factor |
\(r_1, r_2, r_3, N\) | \(S\) | Theoretical geometric shielding factor for an RRDE. | 11 |
ring_collection_current |
\(I_{\text{disk}}, N, n_{\text{ring}}, n_{\text{disk}}\) | \(I_{\text{ring}}\) | Ring electrode current from disk product collection. | 11 |
ring_limiting_current |
\(\omega, r_1, r_2, r_3, n, D, c^*, \nu\) | \(I_{\text{R,lim}}\) | Convective mass-transfer limiting current at an RRDE ring. | 11 |
rotating_ring_disk |
\(r_1, r_2, r_3, \omega, n, D, c^*, \nu\) | \(N, S, I_{\text{D,lim}}, I_{\text{R,lim}}\) | Comprehensive collection and limiting current metrics for RRDE. | 11 |
rad_s_to_rpm |
\(\omega\) | \(f_{\text{RPM}}\) | Angular velocity to rotational speed conversion. | |
rpm_to_rad_s |
\(f_{\text{RPM}}\) | \(\omega\) | Rotational speed to angular velocity conversion. |
Thermodynamics & Interfacial Kinetics¶
The softpotato.analytical.kinetics module provides thermodynamic equilibrium potentials
and electrode kinetic models, including Nernst, Butler–Volmer, and Tafel formulations.
| Function | Inputs | Output | Description | References |
|---|---|---|---|---|
nernst |
\(E^\circ, c_{\text{Ox}}, c_{\text{Red}}, n, T\) | \(E\) | Reversible equilibrium reduction potential. | 12 |
nernst_potential |
\(E^\circ, c_{\text{Ox}}, c_{\text{Red}}, n, T\) | \(E\) | Nernst equilibrium electrode potential calculation. | 12 |
nernst_ratio |
\(E, E^\circ, n, T\) | \(c_{\text{Ox}} / c_{\text{Red}}\) | Equilibrium concentration ratio from applied potential. | 12 |
nernst_equilibrium_concentrations |
\(E, E^\circ, c_{\text{total}}, n, T\) | \(c_{\text{Ox}}, c_{\text{Red}}\) | Individual equilibrium species concentrations from total pool. | 12 |
butler_volmer |
\(\eta, I_0, \alpha, n, T\) | \(I\) | Faradaic current from activation overpotential. | 13, 14 |
butler_volmer_current_density |
\(\eta, j_0, \alpha, n, T\) | \(j\) | Faradaic current density from activation overpotential. | 13, 14 |
butler_volmer_linear |
\(\eta, I_0, n, T\) | \(I\) | Low-overpotential linearized Butler–Volmer current. | 13, 14 |
charge_transfer_resistance |
\(I_0, n, T\) | \(R_{\text{ct}}\) | Charge-transfer resistance at equilibrium potential. | 13, 14 |
exchange_current |
\(k_0, c_{\text{Ox}}^*, c_{\text{Red}}^*, A, \alpha, n\) | \(I_0\) | Standard exchange current from rate constant and concentrations. | 13, 14 |
exchange_current_density |
\(k_0, c_{\text{Ox}}^*, c_{\text{Red}}^*, \alpha, n\) | \(j_0\) | Exchange current density from rate constant and concentrations. | 13, 14 |
tafel |
\(\eta, I_0, \alpha, n, T\) | \(I\) | High-overpotential unidirectional Tafel current approximation. | 15 |
tafel_overpotential |
\(I, I_0, \alpha, n, T\) | \(\eta\) | Activation overpotential calculated from current. | 15 |
tafel_slope |
\(\alpha, n, T\) | \(b\) | Theoretical Tafel slope in Volts per decade. | 15 |
tafel_analysis |
\(\eta, I, A, c^*, n, T\) | \(I_0, \alpha, b\) | Semi-logarithmic Tafel regression for kinetic parameter extraction. | 15 |
Voltammetry & Kinetic Diagnostics¶
The softpotato.analytical.voltammetry module provides peak current expressions and
reversibility diagnostics for cyclic and linear sweep voltammetry.
| Function | Inputs | Output | Description | References |
|---|---|---|---|---|
randles_sevcik |
\(v, c^*, D, A, n, T\) | \(I_{\text{p}}\) | Reversible electron transfer peak voltammetric current. | 16, 17 |
randles_sevcik_irreversible |
\(v, \alpha, n_\alpha, c^*, D, A, n, T\) | \(I_{\text{p}}\) | Totally irreversible electron transfer peak current. | 18, 19 |
randles_sevcik_quasi |
\(v, k_0, \Lambda, c^*, D, A, \alpha, n, T\) | \(I_{\text{p}}\) | Quasi-reversible voltammetric peak current and wave shape. | 19 |
peak_potential_irreversible |
\(v, k_0, E^{\circ\prime}, D, \alpha, n_\alpha, T\) | \(E_{\text{p}}\) | Peak potential shift for totally irreversible kinetics. | 18, 19 |
nicholson_delta_ep |
\(\psi, n\) | \(\Delta E_{\text{p}}\) | Peak potential separation from Nicholson parameter. | 20, 21 |
nicholson_psi |
\(\Delta E_{\text{p}}, k_0, v, D, n, T\) | \(\psi\) | Nicholson dimensionless kinetic parameter calculation. | 20, 21 |
nicholson_rate_constant |
\(\Delta E_{\text{p}}, v, D, n, T\) | \(k_0\) | Standard heterogeneous rate constant extraction from peak split. | 20, 21 |
matsuda_ayabe_lambda |
\(k_0, v, D, n, T\) | \(\Lambda\) | Dimensionless Matsuda–Ayabe reversibility parameter. | 19 |
matsuda_ayabe |
\(k_0, v, D, n, \alpha, T\) | \(\Lambda, \text{zone}\) | Reversibility classification (reversible, quasi, irreversible). | 19 |
Microelectrode Geometries¶
The softpotato.analytical.microelectrodes module provides steady-state and transient
diffusion-limited current expressions for ultramicroelectrode (UME) geometries.
| Function | Inputs | Output | Description | References |
|---|---|---|---|---|
microdisc_limiting_current |
\(a, n, D, c^*\) | \(I_{\text{ss}}\) | Saito steady-state limiting current at an inlaid circular microdisc. | 22, 23 |
microdisc_transient |
\(t, a, n, D, c^*\) | \(I(t)\) | Shoup–Szabo full-time chronoamperometric transient at a microdisc. | 24 |
mahon_oldham_transient |
\(t, a, n, D, c^*\) | \(I(t)\) | High-precision Mahon–Oldham chronoamperometric transient. | 25 |
microsphere_limiting_current |
\(r, n, D, c^*\) | \(I_{\text{ss}}\) | Steady-state spherical diffusion-limited current. | 22 |
microhemisphere_limiting_current |
\(r, n, D, c^*\) | \(I_{\text{ss}}\) | Steady-state hemispherical diffusion-limited current. | 22 |
microband_limiting_current |
\(t, w, l, n, D, c^*\) | \(I(t)\) | Quasi-steady-state diffusion-limited current at an inlaid microband. | 26, 27 |
Scanning Electrochemical Microscopy¶
The softpotato.analytical.secm module provides normalized approach curve models
and feedback current expressions for scanning electrochemical microscopy.
| Function | Inputs | Output | Description | References |
|---|---|---|---|---|
secm_limiting_current_infinite |
\(a, n, D, c^*\) | \(I_{\text{T},\infty}\) | Unhindered bulk steady-state current at microelectrode tip. | 22 |
secm_approach_curve |
\(L, R_{\text{g}}, \kappa\) | \(I_{\text{T}}^{\text{norm}}\) | Normalized tip approach curve over substrates of arbitrary kinetics. | 28, 29 |
secm_approach_positive_feedback |
\(L, R_{\text{g}}\) | \(I_{\text{T}}^{\text{norm}}\) | Pure positive feedback approach curve over conductive substrate. | 28, 29 |
secm_approach_negative_feedback |
\(L, R_{\text{g}}\) | \(I_{\text{T}}^{\text{norm}}\) | Pure negative feedback approach curve over insulating substrate. | 28, 29 |
secm_tip_current |
\(L, R_{\text{g}}, \kappa, a, n, D, c^*\) | \(I_{\text{T}}\) | Dimensional steady-state tip current in Amperes. | 28, 29 |
Parameter Fitting & Regression¶
The softpotato.analytical.fitting module provides non-linear regression and parameter
extraction routines for experimental electrochemical data.
| Function | Inputs | Output | Description | References |
|---|---|---|---|---|
fit_linear |
\(x, y\) | \(m, c, R^2\) | Ordinary linear regression with complete error diagnostics. | |
fit_curve |
\(f(x, p), x, y, p_0, \sigma\) | \(p, \sigma_p, R^2\) | Physically bounded non-linear least-squares curve fitting. | |
fit_cottrell |
\(t, I, n, A, c^*\) | \(D, I_{\text{dl}}\) | Chronoamperometric Cottrell regression for diffusivity extraction. | 2 |
fit_levich |
\(\omega, I, n, A, c^*, \nu\) | \(D, B\) | RDE Levich analysis for diffusion coefficient determination. | 9 |
fit_koutecky_levich |
\(\omega, I, n, A, c^*, \nu\) | \(I_{\text{k}}, k_0, D\) | Koutecký–Levich analysis for kinetic and transport parameters. | 10 |
fit_tafel |
\(\eta, I, n, T\) | \(I_0, \alpha, b\) | Tafel plot regression for exchange current and transfer coefficient. | 15 |
fit_randles_sevcik |
\(v, I_{\text{p}}, n, A, c^*, T\) | \(D\) | Voltammetric peak current regression for diffusion coefficient. | 16, 17 |
fit_microdisc_transient |
\(t, I, r, c^*, n\) | \(D\) | Microdisc transient current fitting to extract diffusivity or radius. | 24, 25 |
fit_secm_approach |
\(d, I_{\text{norm}}, a\) | \(R_{\text{g}}, d_{\text{offset}}\) | SECM approach curve fitting for sheath ratio and distance offset. | 28, 29 |
Simulation Benchmarking Suite¶
The softpotato.analytical.benchmark module provides validation harnesses, exact
analytical benchmarks, and spatial/temporal order of convergence testing utilities.
| Function | Inputs | Output | Description | References |
|---|---|---|---|---|
compute_error_metrics |
\(u_{\text{num}}, u_{\text{exact}}\) | \(e_{\max}, \text{RMSE}, L_2\) | Standardized error norms between numerical and exact profiles. | 31 |
estimate_convergence_order |
\(\Delta x, e\) | \(p, R^2\) | Empirical discretization order calculation via log-log regression. | 31 |
list_benchmark_cases |
\(\varnothing\) | \(\text{cases}\) | Names of available standard benchmark cases. | |
run_benchmark |
\(\text{case}, \text{solver}\) | \(\text{result}\) | Unified runner executing a specified validation benchmark. | |
run_cottrell_benchmark |
\(\text{solver}, D, c^*, A\) | \(\text{result}\) | Chronoamperometric potential step solver accuracy benchmark. | 2 |
run_fourier_decay_benchmark |
\(\text{solver}, D, L, t_{\text{end}}\) | \(\text{result}\) | 1D Fourier sine diffusion series analytical validation. | 30 |
run_mass_conservation_benchmark |
\(\text{solver}, D, L, t\) | \(\text{result}\) | Zero-flux total mass conservation test across simulation time. | |
run_benchmark_suite |
\(\text{solvers}, \text{cases}\) | \(\text{suite}\) | Comprehensive benchmark suite execution across multiple solvers. | |
verify_spatial_convergence |
\(\text{solver}, \Delta x_i, D, L\) | \(p_x\) | Spatial grid refinement convergence order verification. | 31 |
verify_temporal_convergence |
\(\text{solver}, \Delta t_j, D, L\) | \(p_t\) | Time-step refinement convergence order verification. | 31 |
Solver Module¶
The softpotato.solver subpackage provides numerical partial differential equation (PDE)
solvers and boundary condition primitives for 1D multi-species diffusion and reaction-diffusion systems.
| Class / Function | Inputs | Output | Description | References |
|---|---|---|---|---|
get_solver |
\(\text{name}\) | \(\text{solver}\) | Factory retrieving an instantiated numerical solver by alias. | |
list_solvers |
\(\varnothing\) | \(\text{names}\) | Returns all registered solver names and aliases. | |
register_solver |
\(\text{name}, \text{cls}\) | \(\varnothing\) | Extends solver registry with a custom solver class. | |
DirichletBC |
\(c_{\text{bc}}\) | \(\text{BC}\) | Fixed concentration boundary condition specification. | |
NeumannBC |
\(J_{\text{bc}}\) | \(\text{BC}\) | Fixed diffusion flux boundary condition specification. | |
DiffusionProblem |
\(x, D_i, \text{BC}_i, c_{i,0}, R_i\) | \(\text{Problem}\) | Data container defining multi-species diffusion-reaction problem. | |
ExplicitFiniteDifference |
\(\Delta t\) | \(\text{solver}\) | Forward-Time Central-Space explicit finite difference solver. | |
ImplicitFiniteDifference |
\(\Delta t\) | \(\text{solver}\) | Backward-Time Central-Space unconditionally stable implicit solver. | |
CrankNicolson |
\(\Delta t\) | \(\text{solver}\) | Second-order in space and time Crank–Nicolson solver. | 32 |
ScipyIVPSolver |
\(\text{method}, \text{rtol}, \text{atol}\) | \(\text{solver}\) | Method of Lines solver wrapping SciPy adaptive ODE integrators. | 33 |
solve |
\(\text{Problem}, (t_0, t_1)\) | \(c_i(x, t)\) | Integrates problem across time span to compute concentration fields. |
References¶
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