.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/electrochem/butler_volmer/plot_02_butler_volmer_equation.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_electrochem_butler_volmer_plot_02_butler_volmer_equation.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_electrochem_butler_volmer_plot_02_butler_volmer_equation.py:


The Butler-Volmer equation: anodic and cathodic partial currents
==================================================================

The Butler-Volmer equation writes the net electrode current as the
difference of an anodic and a cathodic exponential. At zero overpotential
the two partial currents are equal (each is :math:`i_0`) and cancel -- a
dynamic equilibrium; far from equilibrium one term dominates and the
equation becomes Tafel's straight line. This example plots both partial
currents and
:func:`~chemistrykit.electrochem.systems.butler_volmer.butler_volmer_current_density`,
shows the effect of the transfer coefficient :math:`\alpha`, and checks
numerically how fast the Tafel limit is reached.

.. GENERATED FROM PYTHON SOURCE LINES 17-28

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.constants import FARADAY, STANDARD_TEMPERATURE, R
    from chemistrykit.electrochem.systems.butler_volmer import butler_volmer_current_density, tafel_overpotential
    from chemistrykit.electrochem.visualizers.electrochem_plots import plot_tafel

    i0, n = 1.0e-6, 1
    f = n * FARADAY / (R * STANDARD_TEMPERATURE)
    eta = np.linspace(-0.2, 0.2, 400)








.. GENERATED FROM PYTHON SOURCE LINES 29-30

Partial currents for alpha = 0.5 and the net current.

.. GENERATED FROM PYTHON SOURCE LINES 30-46

.. code-block:: Python

    alpha = 0.5
    i_a = i0 * np.exp(alpha * f * eta)
    i_c = -i0 * np.exp(-(1 - alpha) * f * eta)
    i_net = butler_volmer_current_density(i0, eta, alpha=alpha, n=n)
    print(f"net current at eta = 0: {butler_volmer_current_density(i0, 0.0, alpha=alpha, n=n):.1e} (partial currents +/- {i0:.0e})")

    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4))
    ax1.plot(eta, i_a * 1e6, "--", label="anodic")
    ax1.plot(eta, i_c * 1e6, "--", label="cathodic")
    ax1.plot(eta, i_net * 1e6, "k", label="net (Butler-Volmer)")
    ax1.set_ylim(-50, 50)
    ax1.set_xlabel(r"$\eta$ (V)")
    ax1.set_ylabel(r"$i$ ($\mu$A cm$^{-2}$)")
    ax1.set_title(r"$\alpha$ = 0.5")
    ax1.legend()




.. image-sg:: /api/gallery/electrochem/butler_volmer/images/sphx_glr_plot_02_butler_volmer_equation_001.png
   :alt: $\alpha$ = 0.5
   :srcset: /api/gallery/electrochem/butler_volmer/images/sphx_glr_plot_02_butler_volmer_equation_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    net current at eta = 0: 0.0e+00 (partial currents +/- 1e-06)

    <matplotlib.legend.Legend object at 0x1195acad0>



.. GENERATED FROM PYTHON SOURCE LINES 47-48

The transfer coefficient skews the curve toward one branch.

.. GENERATED FROM PYTHON SOURCE LINES 48-56

.. code-block:: Python

    for a in (0.3, 0.5, 0.7):
        ax2.plot(eta, butler_volmer_current_density(i0, eta, alpha=a, n=n) * 1e6, label=rf"$\alpha$ = {a}")
    ax2.set_ylim(-50, 50)
    ax2.set_xlabel(r"$\eta$ (V)")
    ax2.set_title("Effect of the transfer coefficient")
    ax2.legend()
    fig.tight_layout()








.. GENERATED FROM PYTHON SOURCE LINES 57-59

High-overpotential limit: the Tafel form recovers eta from the full
Butler-Volmer current with rapidly shrinking error.

.. GENERATED FROM PYTHON SOURCE LINES 59-67

.. code-block:: Python

    for e in (0.05, 0.10, 0.20, 0.30):
        i_full = butler_volmer_current_density(i0, e, alpha=alpha, n=n)
        err = abs(tafel_overpotential(i_full, i0, alpha=alpha, n=n) - e) / e
        print(f"eta = {e:.2f} V: Tafel-limit relative error = {err:.2%}")

    ax = plot_tafel(i0=i0, eta_range=(-0.4, 0.4), alpha=alpha, n=n)
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/electrochem/butler_volmer/images/sphx_glr_plot_02_butler_volmer_equation_002.png
   :alt: Tafel plot
   :srcset: /api/gallery/electrochem/butler_volmer/images/sphx_glr_plot_02_butler_volmer_equation_002.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    eta = 0.05 V: Tafel-limit relative error = 15.82%
    eta = 0.10 V: Tafel-limit relative error = 1.04%
    eta = 0.20 V: Tafel-limit relative error = 0.01%
    eta = 0.30 V: Tafel-limit relative error = 0.02%





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.080 seconds)


.. _sphx_glr_download_api_gallery_electrochem_butler_volmer_plot_02_butler_volmer_equation.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_butler_volmer_equation.ipynb <plot_02_butler_volmer_equation.ipynb>`

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      :download:`Download Python source code: plot_02_butler_volmer_equation.py <plot_02_butler_volmer_equation.py>`

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