Scanning Electrochemical Microscopy (softpotato.analytical.secm)

The softpotato.analytical.secm module provides normalized approach curve models and feedback expressions for scanning electrochemical microscopy.

secm

Analytical equations for Scanning Electrochemical Microscopy (SECM) approach curves.

This module provides closed-form analytical approximations for steady-state feedback approach curves at an inlaid circular microdisk ultramicroelectrode (UME) tip surrounded by an insulating sheath (insulator-to-electrode radius ratio \(RG = r_g / a\)).

The analytical approximations were developed by Christine Lefrou and Renaud Cornut and describe:

• Pure positive feedback over conductive substrates (diffusion-controlled mediator regeneration).
• Pure negative feedback over insulating substrates (hindered diffusion).
• Approach curves over substrates with finite/irreversible first-order redox kinetics (\(\kappa = \Lambda = k a / D\)), seamlessly interpolating between negative and positive feedback.
• Steady-state diffusion-limited current at infinite distance (\(i_{T,\infty}\)) and dimensional tip current (\(i_T\)) in Amperes.

Functions:

Name	Description
secm_approach_positive_feedback	Normalized steady-state tip current over conductive substrates (positive feedback).
secm_approach_negative_feedback	Normalized steady-state tip current over insulating substrates (negative feedback).
secm_approach_curve	Normalized steady-state tip current for arbitrary substrate kinetics (\(\kappa \ge 0\)).
secm_limiting_current_infinite	Saito steady-state diffusion-limited current at infinite distance (\(i_{T,\infty}\)).
secm_tip_current	Dimensional steady-state tip current in Amperes (\(i_T = I_T \cdot i_{T,\infty}\)).

Conventions & Units

• Normalized distance: \(L = d / a\) (\(L > 0\)), where \(d\) is the tip-substrate separation distance and \(a\) is the electroactive microdisk radius.
• Insulator ratio: \(RG = r_g / a\) (\(RG > 1\)), where \(r_g\) is the total radius including the surrounding glass insulator sheath.
• Normalized current: \(I_T = i_T / i_{T,\infty}\), where \(i_{T,\infty} = 4 n F D c^* a\) is the steady-state diffusion-limited current in bulk solution (\(L \to \infty\)).
• Substrate kinetics parameter: \(\kappa = \Lambda = \frac{k a}{D}\) (\(\kappa \ge 0\)), where \(k\) is the heterogeneous first-order rate constant at the substrate, \(a\) is the microdisk radius, and \(D\) is the diffusion coefficient of the mediator.

Standard IUPAC sign conventions are applied for dimensional current: - Cathodic (reduction) currents are negative (\(I < 0\)). - Anodic (oxidation) currents are positive (\(I > 0\)).

References

• R. Cornut, C. Lefrou, "New analytical approximation of feedback approach curves with a microdisk SECM tip and irreversible kinetic reaction at the substrate", J. Electroanal. Chem., 621 (2008) 178–184.
• R. Cornut, C. Lefrou, "A unified new analytical approximation for negative feedback currents with a microdisk SECM tip", J. Electroanal. Chem., 608 (2007) 59–66.
• C. Lefrou, "A unified new analytical approximation for positive feedback currents with a microdisk SECM tip", J. Electroanal. Chem., 592 (2006) 103–112.
• C. Lefrou, R. Cornut, "Analytical Expressions for Quantitative Scanning Electrochemical Microscopy (SECM)", ChemPhysChem, 11 (2010) 547–556.
• Y. Saito, "A theoretical study on the diffusion current at the stationary spherical and disc electrodes", Rev. Polarogr., 15 (1968) 177–187.

secm_approach_positive_feedback

secm_approach_positive_feedback(L: float | ArrayLike, RG: float | ArrayLike = 10.0) -> float | np.ndarray

Compute normalized steady-state tip current for positive feedback SECM.

Calculates the normalized steady-state tip current \(I_T^{\text{cond}}(L, RG)\) at an inlaid microdisk ultramicroelectrode approaching a conductive substrate under pure diffusion-controlled mediator regeneration, using the analytical approximation of Cornut and Lefrou (2008):

\[ I_T^{\text{cond}}(L, RG) = \alpha(RG) + \frac{\pi}{4 \beta(RG) \arctan(L)} + \left(1 - \alpha(RG) - \frac{1}{2\beta(RG)}\right) \frac{2}{\pi} \arctan(L) \]

where \(\alpha(RG)\) and \(\beta(RG)\) are geometric coefficients defined by:

\[ \theta(RG) &= \frac{2}{\pi} \arccos\left(\frac{1}{RG}\right) \\ \alpha(RG) &= \ln(2) + \ln(2)(1 - \theta) - \ln(2)(1 - \theta^2) \\ \beta(RG) &= 1 + 0.639(1 - \theta) - 0.186(1 - \theta^2) \]

As \(L \to \infty\), \(I_T^{\text{cond}} \to 1.0\) exactly for all \(RG\). As \(L \to 0\), \(I_T^{\text{cond}} \sim \frac{\pi}{4 \beta(RG) L} \to \infty\).

Parameters:

Name	Type	Description	Default
L	float or array - like	Normalized tip-to-substrate distance \(L = d / a\) (\(L > 0\)).	required
RG	float or array - like	Insulator radius ratio \(RG = r_g / a\) (\(RG > 1\)).	10.0

Returns:

Type	Description
float or ndarray	Normalized tip current \(I_T = i_T / i_{T,\infty}\) (dimensionless, \(I_T > 1\)).

Raises:

Type	Description
ValueError	If any element of L is \(\le 0\) or RG is \(\le 1\).

Examples:

>>> from softpotato.analytical.secm import secm_approach_positive_feedback
>>> # Normalized current at L = 1.0, RG = 10.0
>>> round(float(secm_approach_positive_feedback(1.0, RG=10.0)), 4)
1.5628
>>> # Asymptotic approach to bulk current at large L
>>> round(float(secm_approach_positive_feedback(100.0, RG=10.0)), 4)
1.0041

Source code in src/softpotato/analytical/secm.py

def secm_approach_positive_feedback(
    L: float | ArrayLike,
    RG: float | ArrayLike = 10.0,
) -> float | np.ndarray:
    r"""Compute normalized steady-state tip current for positive feedback SECM.

    Calculates the normalized steady-state tip current :math:`I_T^{\text{cond}}(L, RG)`
    at an inlaid microdisk ultramicroelectrode approaching a conductive substrate under
    pure diffusion-controlled mediator regeneration, using the analytical approximation
    of Cornut and Lefrou (2008):

    .. math::

        I_T^{\text{cond}}(L, RG) = \alpha(RG) + \frac{\pi}{4 \beta(RG) \arctan(L)}
        + \left(1 - \alpha(RG) - \frac{1}{2\beta(RG)}\right) \frac{2}{\pi} \arctan(L)

    where :math:`\alpha(RG)` and :math:`\beta(RG)` are geometric coefficients defined by:

    .. math::

        \theta(RG) &= \frac{2}{\pi} \arccos\left(\frac{1}{RG}\right) \\
        \alpha(RG) &= \ln(2) + \ln(2)(1 - \theta) - \ln(2)(1 - \theta^2) \\
        \beta(RG) &= 1 + 0.639(1 - \theta) - 0.186(1 - \theta^2)

    As :math:`L \to \infty`, :math:`I_T^{\text{cond}} \to 1.0` exactly for all :math:`RG`.
    As :math:`L \to 0`, :math:`I_T^{\text{cond}} \sim \frac{\pi}{4 \beta(RG) L} \to \infty`.

    Parameters
    ----------
    L : float or array-like
        Normalized tip-to-substrate distance :math:`L = d / a` (:math:`L > 0`).
    RG : float or array-like, default 10.0
        Insulator radius ratio :math:`RG = r_g / a` (:math:`RG > 1`).

    Returns
    ----------
    float or np.ndarray
        Normalized tip current :math:`I_T = i_T / i_{T,\infty}` (dimensionless, :math:`I_T > 1`).

    Raises
    ------
    ValueError
        If any element of ``L`` is :math:`\le 0` or ``RG`` is :math:`\le 1`.

    Examples
    --------
    >>> from softpotato.analytical.secm import secm_approach_positive_feedback
    >>> # Normalized current at L = 1.0, RG = 10.0
    >>> round(float(secm_approach_positive_feedback(1.0, RG=10.0)), 4)
    1.5628
    >>> # Asymptotic approach to bulk current at large L
    >>> round(float(secm_approach_positive_feedback(100.0, RG=10.0)), 4)
    1.0041
    """
    L_arr = np.asarray(L, dtype=float)
    RG_arr = np.asarray(RG, dtype=float)

    is_scalar = (L_arr.ndim == 0) and (RG_arr.ndim == 0)

    _validate_positive(L_arr, "Normalized distance L")
    _validate_rg(RG_arr)

    L_bc, RG_bc = np.broadcast_arrays(L_arr, RG_arr)

    alpha, beta = _compute_alpha_beta(RG_bc)
    atan_L = np.arctan(L_bc)

    # I_T^cond = alpha + pi / (4 * beta * arctan(L)) + (1 - alpha - 1 / (2 * beta)) * (2 / pi) * arctan(L)
    term1 = alpha
    term2 = np.pi / (4.0 * beta * atan_L)
    term3 = (1.0 - alpha - 0.5 / beta) * (2.0 / np.pi) * atan_L

    res = term1 + term2 + term3
    return _format_output(res, is_scalar)

secm_approach_negative_feedback

secm_approach_negative_feedback(L: float | ArrayLike, RG: float | ArrayLike = 10.0) -> float | np.ndarray

Compute normalized steady-state tip current for negative feedback SECM.

Calculates the normalized steady-state tip current \(I_T^{\text{ins}}(L, RG)\) at an inlaid microdisk ultramicroelectrode approaching an inert insulating substrate under hindered diffusion, using the analytical approximation of Cornut and Lefrou (2007, 2008):

\[ I_T^{\text{ins}}(L, RG) = \frac{\frac{2.08}{RG^{0.358}} \left(L - \frac{0.145}{RG}\right) + 1.585} {\frac{2.08}{RG^{0.358}} (L + 0.0023 RG) + 1.57 + \frac{\ln(RG)}{L} + \frac{2}{\pi RG} \ln\left(1 + \frac{\pi RG}{2 L}\right)} \]

As \(L \to \infty\), \(I_T^{\text{ins}} \to 1.0\) for all \(RG\). As \(L \to 0\), \(I_T^{\text{ins}} \to 0.0\).

Parameters:

Name	Type	Description	Default
L	float or array - like	Normalized tip-to-substrate distance \(L = d / a\) (\(L > 0\)).	required
RG	float or array - like	Insulator radius ratio \(RG = r_g / a\) (\(RG > 1\)).	10.0

Returns:

Type	Description
float or ndarray	Normalized tip current \(I_T = i_T / i_{T,\infty}\) (dimensionless, \(0 < I_T < 1\)).

Raises:

Type	Description
ValueError	If any element of L is \(\le 0\) or RG is \(\le 1\).

Examples:

>>> from softpotato.analytical.secm import secm_approach_negative_feedback
>>> # Normalized current at L = 1.0, RG = 10.0
>>> round(float(secm_approach_negative_feedback(1.0, RG=10.0)), 4)
0.4983
>>> # Strong hindered diffusion near substrate (L = 0.05)
>>> round(float(secm_approach_negative_feedback(0.05, RG=10.0)), 4)
0.0337

Source code in src/softpotato/analytical/secm.py

def secm_approach_negative_feedback(
    L: float | ArrayLike,
    RG: float | ArrayLike = 10.0,
) -> float | np.ndarray:
    r"""Compute normalized steady-state tip current for negative feedback SECM.

    Calculates the normalized steady-state tip current :math:`I_T^{\text{ins}}(L, RG)`
    at an inlaid microdisk ultramicroelectrode approaching an inert insulating substrate
    under hindered diffusion, using the analytical approximation of Cornut and Lefrou (2007, 2008):

    .. math::

        I_T^{\text{ins}}(L, RG) = \frac{\frac{2.08}{RG^{0.358}} \left(L - \frac{0.145}{RG}\right) + 1.585}
        {\frac{2.08}{RG^{0.358}} (L + 0.0023 RG) + 1.57 + \frac{\ln(RG)}{L} + \frac{2}{\pi RG} \ln\left(1 + \frac{\pi RG}{2 L}\right)}

    As :math:`L \to \infty`, :math:`I_T^{\text{ins}} \to 1.0` for all :math:`RG`.
    As :math:`L \to 0`, :math:`I_T^{\text{ins}} \to 0.0`.

    Parameters
    ----------
    L : float or array-like
        Normalized tip-to-substrate distance :math:`L = d / a` (:math:`L > 0`).
    RG : float or array-like, default 10.0
        Insulator radius ratio :math:`RG = r_g / a` (:math:`RG > 1`).

    Returns
    ----------
    float or np.ndarray
        Normalized tip current :math:`I_T = i_T / i_{T,\infty}` (dimensionless, :math:`0 < I_T < 1`).

    Raises
    ------
    ValueError
        If any element of ``L`` is :math:`\le 0` or ``RG`` is :math:`\le 1`.

    Examples
    --------
    >>> from softpotato.analytical.secm import secm_approach_negative_feedback
    >>> # Normalized current at L = 1.0, RG = 10.0
    >>> round(float(secm_approach_negative_feedback(1.0, RG=10.0)), 4)
    0.4983
    >>> # Strong hindered diffusion near substrate (L = 0.05)
    >>> round(float(secm_approach_negative_feedback(0.05, RG=10.0)), 4)
    0.0337
    """
    L_arr = np.asarray(L, dtype=float)
    RG_arr = np.asarray(RG, dtype=float)

    is_scalar = (L_arr.ndim == 0) and (RG_arr.ndim == 0)

    _validate_positive(L_arr, "Normalized distance L")
    _validate_rg(RG_arr)

    L_bc, RG_bc = np.broadcast_arrays(L_arr, RG_arr)

    c_rg = 2.08 / (RG_bc**0.358)
    num = c_rg * (L_bc - 0.145 / RG_bc) + 1.585
    den = (
        c_rg * (L_bc + 0.0023 * RG_bc)
        + 1.57
        + np.log(RG_bc) / L_bc
        + (2.0 / (np.pi * RG_bc)) * np.log(1.0 + (np.pi * RG_bc) / (2.0 * L_bc))
    )

    res = num / den
    return _format_output(res, is_scalar)

secm_approach_curve

secm_approach_curve(L: float | ArrayLike, RG: float | ArrayLike = 10.0, kappa: float | ArrayLike = np.inf) -> float | np.ndarray

Compute normalized steady-state tip current for arbitrary substrate kinetics.

Calculates the normalized steady-state tip current \(I_T(L, RG, \kappa)\) at an inlaid microdisk ultramicroelectrode for first-order substrate reaction kinetics characterized by the dimensionless kinetic parameter \(\kappa = \Lambda = \frac{k a}{D}\), using the unified analytical approximation of Cornut and Lefrou (2008, Eq. 22):

\[ I_T(L, RG, \kappa) = I_T^{\text{cond}}\left(L + \frac{1}{\kappa}, RG\right) + \frac{I_T^{\text{ins}}(L, RG) - 1}{\left(1 + 2.47 RG^{0.31} L \kappa\right) \left(1 + L^{0.006 RG + 0.113} \kappa^{0.0236 RG + 0.91}\right)} \]

Special limiting cases:

• Pure positive feedback (\(\kappa \to \infty\)): reduces to secm_approach_positive_feedback (\(I_T^{\text{cond}}\)).
• Pure negative feedback (\(\kappa = 0\)): reduces to secm_approach_negative_feedback (\(I_T^{\text{ins}}\)).
• Finite substrate kinetics (\(0 < \kappa < \infty\)): smoothly interpolates between the negative and positive feedback curves.

Parameters:

Name	Type	Description	Default
L	float or array - like	Normalized tip-to-substrate distance \(L = d / a\) (\(L > 0\)).	required
RG	float or array - like	Insulator radius ratio \(RG = r_g / a\) (\(RG > 1\)).	10.0
kappa	float or array - like	Dimensionless heterogeneous first-order substrate rate constant \(\kappa = \Lambda = \frac{k a}{D}\) (\(\kappa \ge 0\)). Set to np.inf for diffusion-controlled positive feedback, and 0.0 for an inert insulating substrate.	np.inf

Returns:

Type	Description
float or ndarray	Normalized tip current \(I_T = i_T / i_{T,\infty}\) (dimensionless).

Raises:

Type	Description
ValueError	If any element of L is \(\le 0\), RG is \(\le 1\), or kappa is \(< 0\).

Examples:

>>> from softpotato.analytical.secm import secm_approach_curve
>>> import numpy as np
>>> # Positive feedback (kappa = inf)
>>> round(float(secm_approach_curve(1.0, RG=10.0, kappa=np.inf)), 4)
1.5628
>>> # Negative feedback (kappa = 0)
>>> round(float(secm_approach_curve(1.0, RG=10.0, kappa=0.0)), 4)
0.4983
>>> # Intermediate substrate kinetics (kappa = 1.0)
>>> round(float(secm_approach_curve(1.0, RG=10.0, kappa=1.0)), 4)
1.2064

Source code in src/softpotato/analytical/secm.py

def secm_approach_curve(
    L: float | ArrayLike,
    RG: float | ArrayLike = 10.0,
    kappa: float | ArrayLike = np.inf,
) -> float | np.ndarray:
    r"""Compute normalized steady-state tip current for arbitrary substrate kinetics.

    Calculates the normalized steady-state tip current :math:`I_T(L, RG, \kappa)`
    at an inlaid microdisk ultramicroelectrode for first-order substrate reaction kinetics
    characterized by the dimensionless kinetic parameter :math:`\kappa = \Lambda = \frac{k a}{D}`,
    using the unified analytical approximation of Cornut and Lefrou (2008, Eq. 22):

    .. math::

        I_T(L, RG, \kappa) = I_T^{\text{cond}}\left(L + \frac{1}{\kappa}, RG\right)
        + \frac{I_T^{\text{ins}}(L, RG) - 1}{\left(1 + 2.47 RG^{0.31} L \kappa\right)
        \left(1 + L^{0.006 RG + 0.113} \kappa^{0.0236 RG + 0.91}\right)}

    Special limiting cases:

    - Pure positive feedback (:math:`\kappa \to \infty`): reduces to
      :func:`secm_approach_positive_feedback` (:math:`I_T^{\text{cond}}`).
    - Pure negative feedback (:math:`\kappa = 0`): reduces to
      :func:`secm_approach_negative_feedback` (:math:`I_T^{\text{ins}}`).
    - Finite substrate kinetics (:math:`0 < \kappa < \infty`): smoothly interpolates
      between the negative and positive feedback curves.

    Parameters
    ----------
    L : float or array-like
        Normalized tip-to-substrate distance :math:`L = d / a` (:math:`L > 0`).
    RG : float or array-like, default 10.0
        Insulator radius ratio :math:`RG = r_g / a` (:math:`RG > 1`).
    kappa : float or array-like, default np.inf
        Dimensionless heterogeneous first-order substrate rate constant
        :math:`\kappa = \Lambda = \frac{k a}{D}` (:math:`\kappa \ge 0`).
        Set to ``np.inf`` for diffusion-controlled positive feedback, and ``0.0``
        for an inert insulating substrate.

    Returns
    ----------
    float or np.ndarray
        Normalized tip current :math:`I_T = i_T / i_{T,\infty}` (dimensionless).

    Raises
    ------
    ValueError
        If any element of ``L`` is :math:`\le 0`, ``RG`` is :math:`\le 1`, or ``kappa`` is :math:`< 0`.

    Examples
    --------
    >>> from softpotato.analytical.secm import secm_approach_curve
    >>> import numpy as np
    >>> # Positive feedback (kappa = inf)
    >>> round(float(secm_approach_curve(1.0, RG=10.0, kappa=np.inf)), 4)
    1.5628
    >>> # Negative feedback (kappa = 0)
    >>> round(float(secm_approach_curve(1.0, RG=10.0, kappa=0.0)), 4)
    0.4983
    >>> # Intermediate substrate kinetics (kappa = 1.0)
    >>> round(float(secm_approach_curve(1.0, RG=10.0, kappa=1.0)), 4)
    1.2064
    """
    L_arr = np.asarray(L, dtype=float)
    RG_arr = np.asarray(RG, dtype=float)
    kappa_arr = np.asarray(kappa, dtype=float)

    is_scalar = (L_arr.ndim == 0) and (RG_arr.ndim == 0) and (kappa_arr.ndim == 0)

    _validate_positive(L_arr, "Normalized distance L")
    _validate_rg(RG_arr)
    _validate_non_negative(kappa_arr, "Substrate kinetics parameter kappa")

    L_bc, RG_bc, k_bc = np.broadcast_arrays(L_arr, RG_arr, kappa_arr)

    # Base feedback bounds
    i_ins = secm_approach_negative_feedback(L_bc, RG=RG_bc)
    i_cond = secm_approach_positive_feedback(L_bc, RG=RG_bc)

    # Initialize output array matching broadcast shape
    res = np.zeros_like(L_bc, dtype=float)

    # Mask regimes: pure positive, pure negative, finite kinetics
    is_pos = np.isinf(k_bc)
    is_neg = k_bc == 0.0
    is_finite = ~is_pos & ~is_neg

    if np.any(is_pos):
        res[is_pos] = np.asarray(i_cond)[is_pos]

    if np.any(is_neg):
        res[is_neg] = np.asarray(i_ins)[is_neg]

    if np.any(is_finite):
        l_fin = L_bc[is_finite]
        rg_fin = RG_bc[is_finite]
        k_fin = k_bc[is_finite]
        ins_fin = np.asarray(i_ins)[is_finite]

        # Effective distance for conductive part: L + 1 / kappa
        l_eff = l_fin + 1.0 / k_fin
        cond_eff = secm_approach_positive_feedback(l_eff, RG=rg_fin)

        term_den1 = 1.0 + 2.47 * (rg_fin**0.31) * l_fin * k_fin
        pow_l = 0.006 * rg_fin + 0.113
        pow_k = 0.0236 * rg_fin + 0.91
        term_den2 = 1.0 + (l_fin**pow_l) * (k_fin**pow_k)

        kinetic_correction = (ins_fin - 1.0) / (term_den1 * term_den2)
        res[is_finite] = cond_eff + kinetic_correction

    return _format_output(res, is_scalar)

secm_limiting_current_infinite

secm_limiting_current_infinite(radius: float | ArrayLike, n: float = 1, D: float = 1e-05, c_bulk: float = 0.001, F: float = FARADAY) -> float | np.ndarray

Compute steady-state diffusion-limited current at infinite distance from substrate.

Calculates the Saito steady-state diffusion-limited Faradaic current at an inlaid circular microdisk electrode of radius \(a\) in bulk solution (\(L \to \infty\)):

\[ i_{T,\infty} = 4 n F D c^* a \]

Parameters:

Name	Type	Description	Default
radius	float or array - like	Radius of the microdisk tip \(a\) in meters or cm (\(a > 0\)).	required
n	int or float	Number of electrons transferred per molecule (\(n > 0\)).	1
D	float	Diffusion coefficient \(D\) in \(\text{m}^2/\text{s}\) or \(\text{cm}^2/\text{s}\) (\(D > 0\)).	1e-5
c_bulk	float	Bulk concentration of mediator \(c^*\) in \(\text{mol}/\text{m}^3\) or \(\text{mol}/\text{cm}^3\) (\(c^* \ge 0\)).	1e-3
F	float	Faraday constant in \(\text{C}/\text{mol}\) (\(F > 0\)).	FARADAY

Returns:

Type	Description
float or ndarray	Steady-state limiting current \(i_{T,\infty}\) in Amperes (\(\text{A}\)).

Raises:

Type	Description
ValueError	If radius, D, n, or F is \(\le 0\), or c_bulk is \(< 0\).

Source code in src/softpotato/analytical/secm.py

def secm_limiting_current_infinite(
    radius: float | ArrayLike,
    n: float = 1,
    D: float = 1e-5,
    c_bulk: float = 1e-3,
    F: float = FARADAY,
) -> float | np.ndarray:
    r"""Compute steady-state diffusion-limited current at infinite distance from substrate.

    Calculates the Saito steady-state diffusion-limited Faradaic current at an inlaid
    circular microdisk electrode of radius :math:`a` in bulk solution (:math:`L \to \infty`):

    .. math::

        i_{T,\infty} = 4 n F D c^* a

    Parameters
    ----------
    radius : float or array-like
        Radius of the microdisk tip :math:`a` in meters or cm (:math:`a > 0`).
    n : int or float, default 1
        Number of electrons transferred per molecule (:math:`n > 0`).
    D : float, default 1e-5
        Diffusion coefficient :math:`D` in :math:`\text{m}^2/\text{s}` or
        :math:`\text{cm}^2/\text{s}` (:math:`D > 0`).
    c_bulk : float, default 1e-3
        Bulk concentration of mediator :math:`c^*` in :math:`\text{mol}/\text{m}^3`
        or :math:`\text{mol}/\text{cm}^3` (:math:`c^* \ge 0`).
    F : float, default FARADAY
        Faraday constant in :math:`\text{C}/\text{mol}` (:math:`F > 0`).

    Returns
    ----------
    float or np.ndarray
        Steady-state limiting current :math:`i_{T,\infty}` in Amperes (:math:`\text{A}`).

    Raises
    ------
    ValueError
        If ``radius``, ``D``, ``n``, or ``F`` is :math:`\le 0`, or ``c_bulk`` is :math:`< 0`.
    """
    _validate_common_params(n=n, D=D, c_bulk=c_bulk, F=F)
    _validate_positive(radius, "Electrode radius")

    r_arr = np.asarray(radius, dtype=float)
    is_scalar = r_arr.ndim == 0

    i_inf = 4.0 * n * F * D * c_bulk * r_arr
    return _format_output(i_inf, is_scalar)

secm_tip_current

secm_tip_current(L: float | ArrayLike, RG: float | ArrayLike = 10.0, kappa: float | ArrayLike = np.inf, radius: float = 1e-05, n: float = 1, D: float = 1e-05, c_bulk: float = 0.001, F: float = FARADAY, reduction: bool = True) -> float | np.ndarray

Compute dimensional steady-state tip current in Amperes.

Scales the normalized SECM approach curve \(I_T(L, RG, \kappa)\) by the Saito steady-state limiting current at infinite distance \(i_{T,\infty} = 4 n F D c^* a\):

\[ i_T = I_T(L, RG, \kappa) \cdot 4 n F D c^* a \]

Parameters:

Name	Type	Description	Default
L	float or array - like	Normalized tip-to-substrate distance \(L = d / a\) (\(L > 0\)).	required
RG	float or array - like	Insulator radius ratio \(RG = r_g / a\) (\(RG > 1\)).	10.0
kappa	float or array - like	Dimensionless heterogeneous first-order substrate rate constant \(\kappa = \Lambda = \frac{k a}{D}\) (\(\kappa \ge 0\)).	np.inf
radius	float	Active microdisk radius \(a\) in meters or cm (\(a > 0\)).	1e-5
n	int or float	Number of electrons transferred per molecule (\(n > 0\)).	1
D	float	Diffusion coefficient \(D\) in \(\text{m}^2/\text{s}\) or \(\text{cm}^2/\text{s}\) (\(D > 0\)).	1e-5
c_bulk	float	Bulk concentration of mediator \(c^*\) in \(\text{mol}/\text{m}^3\) or \(\text{mol}/\text{cm}^3\) (\(c^* \ge 0\)).	1e-3
F	float	Faraday constant in \(\text{C}/\text{mol}\) (\(F > 0\)).	FARADAY
reduction	bool	Direction of reaction. Cathodic reduction (True) yields negative dimensional tip current (\(i_T < 0\)), while anodic oxidation (False) yields positive current (\(i_T > 0\)).	True

Returns:

Type	Description
float or ndarray	Dimensional tip current \(i_T\) in Amperes (\(\text{A}\)).

Raises:

Type	Description
ValueError	If any input parameter violates physical bounds.

Source code in src/softpotato/analytical/secm.py

def secm_tip_current(
    L: float | ArrayLike,
    RG: float | ArrayLike = 10.0,
    kappa: float | ArrayLike = np.inf,
    radius: float = 1e-5,
    n: float = 1,
    D: float = 1e-5,
    c_bulk: float = 1e-3,
    F: float = FARADAY,
    reduction: bool = True,
) -> float | np.ndarray:
    r"""Compute dimensional steady-state tip current in Amperes.

    Scales the normalized SECM approach curve :math:`I_T(L, RG, \kappa)` by the Saito
    steady-state limiting current at infinite distance :math:`i_{T,\infty} = 4 n F D c^* a`:

    .. math::

        i_T = I_T(L, RG, \kappa) \cdot 4 n F D c^* a

    Parameters
    ----------
    L : float or array-like
        Normalized tip-to-substrate distance :math:`L = d / a` (:math:`L > 0`).
    RG : float or array-like, default 10.0
        Insulator radius ratio :math:`RG = r_g / a` (:math:`RG > 1`).
    kappa : float or array-like, default np.inf
        Dimensionless heterogeneous first-order substrate rate constant
        :math:`\kappa = \Lambda = \frac{k a}{D}` (:math:`\kappa \ge 0`).
    radius : float, default 1e-5
        Active microdisk radius :math:`a` in meters or cm (:math:`a > 0`).
    n : int or float, default 1
        Number of electrons transferred per molecule (:math:`n > 0`).
    D : float, default 1e-5
        Diffusion coefficient :math:`D` in :math:`\text{m}^2/\text{s}` or
        :math:`\text{cm}^2/\text{s}` (:math:`D > 0`).
    c_bulk : float, default 1e-3
        Bulk concentration of mediator :math:`c^*` in :math:`\text{mol}/\text{m}^3`
        or :math:`\text{mol}/\text{cm}^3` (:math:`c^* \ge 0`).
    F : float, default FARADAY
        Faraday constant in :math:`\text{C}/\text{mol}` (:math:`F > 0`).
    reduction : bool, default True
        Direction of reaction. Cathodic reduction (True) yields negative dimensional tip current
        (:math:`i_T < 0`), while anodic oxidation (False) yields positive current (:math:`i_T > 0`).

    Returns
    ----------
    float or np.ndarray
        Dimensional tip current :math:`i_T` in Amperes (:math:`\text{A}`).

    Raises
    ------
    ValueError
        If any input parameter violates physical bounds.
    """
    i_inf = secm_limiting_current_infinite(radius=radius, n=n, D=D, c_bulk=c_bulk, F=F)
    i_norm = secm_approach_curve(L=L, RG=RG, kappa=kappa)

    sign = -1.0 if reduction else 1.0
    res = i_norm * i_inf * sign

    if isinstance(res, np.ndarray) and res.ndim == 0:
        return float(res.item())
    return res