.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/polymer/step_growth/plot_02_flory_stockmayer_gelation.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_step_growth_plot_02_flory_stockmayer_gelation.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_polymer_step_growth_plot_02_flory_stockmayer_gelation.py:


Flory-Stockmayer gelation: the gel point of a branching polymerization
=========================================================================

With trifunctional or higher monomers, step-growth polymerization builds
branched molecules that link into a sample-spanning network at a sharp
gel point. For self-condensation of an :math:`A_f` monomer, Flory (1941)
and Stockmayer (1943) found the weight-average degree of polymerization
:math:`\bar X_w=(1+p)/(1-(f-1)p)`
(:func:`~chemistrykit.polymer.systems.gelation.branching_weight_average_DP`)
diverges at :math:`p_c=1/(f-1)`
(:func:`~chemistrykit.polymer.systems.gelation.flory_stockmayer_gel_point`)
while the number average
(:func:`~chemistrykit.polymer.systems.gelation.branching_number_average_DP`)
stays finite. Carothers's criterion :math:`\bar X_n\to\infty`,
:math:`p_c=2/f` (:func:`~chemistrykit.polymer.systems.gelation.carothers_gel_point`),
overestimates the gel point; experiments fall between the two.

.. GENERATED FROM PYTHON SOURCE LINES 21-36

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.polymer.systems.gelation import (
        branching_number_average_DP,
        branching_weight_average_DP,
        carothers_gel_point,
        flory_stockmayer_gel_point,
    )

    for f in (2, 3, 4, 6):
        pc = flory_stockmayer_gel_point(f)
        Xn_gel = branching_number_average_DP(min(pc, 0.999), f)
        print(f"f = {f}: Flory-Stockmayer p_c = {pc:.3f}, Carothers p_c = {carothers_gel_point(f):.3f}, Xn at p_c = {Xn_gel:.2f}")





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

 .. code-block:: none

    f = 2: Flory-Stockmayer p_c = 1.000, Carothers p_c = 1.000, Xn at p_c = 1000.00
    f = 3: Flory-Stockmayer p_c = 0.500, Carothers p_c = 0.667, Xn at p_c = 4.00
    f = 4: Flory-Stockmayer p_c = 0.333, Carothers p_c = 0.500, Xn at p_c = 3.00
    f = 6: Flory-Stockmayer p_c = 0.200, Carothers p_c = 0.333, Xn at p_c = 2.50




.. GENERATED FROM PYTHON SOURCE LINES 37-59

.. code-block:: Python

    f = 3
    pc = flory_stockmayer_gel_point(f)
    p = np.linspace(0, pc - 1e-3, 400)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4.2))
    axes[0].semilogy(p, branching_weight_average_DP(p, f), label=r"$\bar X_w$ (diverges)")
    axes[0].semilogy(p, branching_number_average_DP(p, f), label=r"$\bar X_n$ (finite)")
    axes[0].axvline(pc, color="k", ls="--", lw=0.8, label=f"Flory-Stockmayer $p_c$ = {pc:.2f}")
    axes[0].axvline(carothers_gel_point(f), color="crimson", ls=":", label=f"Carothers $p_c$ = {carothers_gel_point(f):.2f}")
    axes[0].set_xlabel("extent of reaction p")
    axes[0].set_ylabel("degree of polymerization")
    axes[0].set_title("A$_3$ self-condensation")
    axes[0].legend()

    fs = np.linspace(2.05, 8, 100)
    axes[1].plot(fs, flory_stockmayer_gel_point(fs), label="Flory-Stockmayer 1/(f-1)")
    axes[1].plot(fs, carothers_gel_point(fs), "--", label="Carothers 2/f")
    axes[1].set_xlabel("functionality f")
    axes[1].set_ylabel("gel point $p_c$")
    axes[1].set_title("Gel point vs. functionality")
    axes[1].legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/polymer/step_growth/images/sphx_glr_plot_02_flory_stockmayer_gelation_001.png
   :alt: A$_3$ self-condensation, Gel point vs. functionality
   :srcset: /api/gallery/polymer/step_growth/images/sphx_glr_plot_02_flory_stockmayer_gelation_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.093 seconds)


.. _sphx_glr_download_api_gallery_polymer_step_growth_plot_02_flory_stockmayer_gelation.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_flory_stockmayer_gelation.ipynb <plot_02_flory_stockmayer_gelation.ipynb>`

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

      :download:`Download Python source code: plot_02_flory_stockmayer_gelation.py <plot_02_flory_stockmayer_gelation.py>`

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

      :download:`Download zipped: plot_02_flory_stockmayer_gelation.zip <plot_02_flory_stockmayer_gelation.zip>`


.. only:: html

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

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