.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/polymer/solution_properties/plot_02_mark_houwink.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_polymer_solution_properties_plot_02_mark_houwink.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_polymer_solution_properties_plot_02_mark_houwink.py:


The Mark-Houwink equation: intrinsic viscosity vs. molar mass
================================================================

Mark (1938) and Houwink (1940) generalized Staudinger's linear rule to
:math:`[\eta]=KM^a`
(:func:`~chemistrykit.polymer.systems.solution_viscosity.mark_houwink_intrinsic_viscosity`),
the relation still used to turn a viscosity measurement into a molar
mass. Via Flory-Fox, :math:`[\eta]\propto R^3/M` with :math:`R\propto
M^\nu` gives :math:`a=3\nu-1`
(:func:`~chemistrykit.polymer.systems.solution_viscosity.mark_houwink_exponent_from_flory`):
:math:`a=0.5` in a theta solvent, :math:`a=0.8` in a good solvent, and
Staudinger's :math:`a=1` is recovered only for stiffer chains. Here
synthetic "measurements" with 3% seeded noise are fit back to recover
:math:`K` and :math:`a`
(:func:`~chemistrykit.polymer.systems.solution_viscosity.fit_mark_houwink`).

.. GENERATED FROM PYTHON SOURCE LINES 20-50



.. image-sg:: /api/gallery/polymer/solution_properties/images/sphx_glr_plot_02_mark_houwink_001.png
   :alt: Mark-Houwink: $[\eta] = K M^a$
   :srcset: /api/gallery/polymer/solution_properties/images/sphx_glr_plot_02_mark_houwink_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    theta solvent (nu=1/2): true a = 0.500, fitted a = 0.496; true K = 0.08, fitted K = 0.0835
    good solvent (nu=3/5): true a = 0.800, fitted a = 0.811; true K = 0.01, fitted K = 0.00872






|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.polymer.systems.solution_viscosity import (
        fit_mark_houwink,
        mark_houwink_exponent_from_flory,
        mark_houwink_intrinsic_viscosity,
    )

    rng = np.random.default_rng(0)
    M = np.logspace(4, 6.5, 12)
    cases = {
        "theta solvent (nu=1/2)": (0.08, mark_houwink_exponent_from_flory(0.5)),
        "good solvent (nu=3/5)": (0.01, mark_houwink_exponent_from_flory(0.6)),
    }

    fig, ax = plt.subplots(figsize=(6.5, 4.5))
    for label, (K, a) in cases.items():
        eta = mark_houwink_intrinsic_viscosity(M, K, a) * (1 + 0.03 * rng.standard_normal(M.size))
        K_fit, a_fit = fit_mark_houwink(M, eta)
        print(f"{label}: true a = {a:.3f}, fitted a = {a_fit:.3f}; true K = {K:.3g}, fitted K = {K_fit:.3g}")
        ax.loglog(M, eta, "o", label=f"{label}: a = {a_fit:.2f}")
        ax.loglog(M, mark_houwink_intrinsic_viscosity(M, K_fit, a_fit), "k-", lw=0.8)

    ax.set_xlabel("molar mass M (g/mol)")
    ax.set_ylabel(r"$[\eta]$ (mL/g)")
    ax.set_title(r"Mark-Houwink: $[\eta] = K M^a$")
    ax.legend()
    plt.tight_layout()
    plt.show()


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

   **Total running time of the script:** (0 minutes 0.064 seconds)


.. _sphx_glr_download_api_gallery_polymer_solution_properties_plot_02_mark_houwink.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_mark_houwink.ipynb <plot_02_mark_houwink.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_mark_houwink.py <plot_02_mark_houwink.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_mark_houwink.zip <plot_02_mark_houwink.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_