.. 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_01_staudinger_viscosity_law.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_01_staudinger_viscosity_law.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_polymer_solution_properties_plot_01_staudinger_viscosity_law.py:


Staudinger's macromolecules: solution viscosity grows with chain length
==========================================================================

Staudinger's case that rubber and cellulose are genuine covalent
macromolecules rested heavily on dilute-solution viscosity: he found the
specific viscosity per unit concentration rising in proportion to the
chain length, :math:`\eta_\text{sp}/c=K_mM`
(:func:`~chemistrykit.polymer.systems.solution_viscosity.staudinger_specific_viscosity`).
If the chains were colloidal aggregates held together by weak
association, the "particle size" would fall apart on dilution or change
of solvent; covalent chains keep their length, so a polymer-analogous
reaction (e.g. hydrogenating rubber) leaves the degree of polymerization
-- and hence :math:`\eta_\text{sp}/c` -- unchanged. This example contrasts
the two pictures.

.. GENERATED FROM PYTHON SOURCE LINES 19-32

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.polymer.systems.solution_viscosity import staudinger_specific_viscosity

    Km = 1.0e-4  # Staudinger constant (per molar-mass unit, per concentration unit)
    M = np.array([1e4, 3e4, 1e5, 3e5])
    c = np.linspace(0.001, 0.01, 10)

    for Mi in M:
        ratio = staudinger_specific_viscosity(c, Mi, Km) / c
        print(f"M = {Mi:8.0f}:  eta_sp/c = {ratio[0]:.2f} at every concentration (constant: {np.allclose(ratio, ratio[0])})")





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

 .. code-block:: none

    M =    10000:  eta_sp/c = 1.00 at every concentration (constant: True)
    M =    30000:  eta_sp/c = 3.00 at every concentration (constant: True)
    M =   100000:  eta_sp/c = 10.00 at every concentration (constant: True)
    M =   300000:  eta_sp/c = 30.00 at every concentration (constant: True)




.. GENERATED FROM PYTHON SOURCE LINES 33-37

Colloidal-aggregate picture: particles of 1e5 units dissociate on
dilution (a simple association equilibrium, apparent size ~ sqrt(c)),
so eta_sp/c would *fall* as the solution is diluted, unlike a real
macromolecule.

.. GENERATED FROM PYTHON SOURCE LINES 37-41

.. code-block:: Python

    M_apparent = 1e5 * np.sqrt(c / c[-1])
    eta_colloid = staudinger_specific_viscosity(c, M_apparent, Km) / c
    eta_macro = staudinger_specific_viscosity(c, 1e5, Km) / c








.. GENERATED FROM PYTHON SOURCE LINES 42-58

.. code-block:: Python

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.2))
    M_scan = np.logspace(3.5, 6, 50)
    axes[0].loglog(M_scan, staudinger_specific_viscosity(1.0, M_scan, Km), "k-")
    axes[0].loglog(M, staudinger_specific_viscosity(1.0, M, Km), "o")
    axes[0].set_xlabel("molar mass M")
    axes[0].set_ylabel(r"$\eta_{sp}/c$")
    axes[0].set_title(r"Staudinger's rule: $\eta_{sp}/c = K_m M$")

    axes[1].plot(c, eta_macro, label="covalent macromolecule (Staudinger)")
    axes[1].plot(c, eta_colloid, "--", label="dissociating colloidal aggregate")
    axes[1].set_xlabel("concentration c")
    axes[1].set_ylabel(r"$\eta_{sp}/c$")
    axes[1].set_title("Dilution test: real molecules keep their size")
    axes[1].legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/polymer/solution_properties/images/sphx_glr_plot_01_staudinger_viscosity_law_001.png
   :alt: Staudinger's rule: $\eta_{sp}/c = K_m M$, Dilution test: real molecules keep their size
   :srcset: /api/gallery/polymer/solution_properties/images/sphx_glr_plot_01_staudinger_viscosity_law_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.092 seconds)


.. _sphx_glr_download_api_gallery_polymer_solution_properties_plot_01_staudinger_viscosity_law.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_staudinger_viscosity_law.ipynb <plot_01_staudinger_viscosity_law.ipynb>`

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

      :download:`Download Python source code: plot_01_staudinger_viscosity_law.py <plot_01_staudinger_viscosity_law.py>`

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

      :download:`Download zipped: plot_01_staudinger_viscosity_law.zip <plot_01_staudinger_viscosity_law.zip>`


.. only:: html

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

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