.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/polymer/chain_growth/plot_04_szwarc_living_polymerization.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_chain_growth_plot_04_szwarc_living_polymerization.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_polymer_chain_growth_plot_04_szwarc_living_polymerization.py:


Szwarc's living polymerization: the narrow Poisson distribution
==================================================================

Szwarc (1956) showed that anionic polymerization of styrene initiated by
sodium naphthalenide has no termination: every chain starts at once and
keeps growing while monomer lasts. Each chain then receives monomers as
independent random events, giving a Poisson distribution
(:func:`~chemistrykit.polymer.systems.living_polymerization.poisson_number_fraction`)
with :math:`\text{PDI}=1+\nu/(1+\nu)^2`
(:func:`~chemistrykit.polymer.systems.living_polymerization.poisson_pdi`) -- close to 1,
far narrower than the Flory-Schulz distribution of a step-growth
polymer with the same :math:`\bar X_n`. A seeded stochastic simulation
(:func:`~chemistrykit.polymer.systems.living_polymerization.simulate_living_polymerization`)
reproduces the prediction.

.. GENERATED FROM PYTHON SOURCE LINES 19-44

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.polymer.systems.living_polymerization import (
        poisson_number_average_DP,
        poisson_number_fraction,
        poisson_pdi,
        simulate_living_polymerization,
    )
    from chemistrykit.polymer.systems.molecular_weight_distribution import (
        flory_schulz_number_fraction,
        flory_schulz_pdi,
    )

    n_chains, n_monomers = 5000, 250000
    nu = n_monomers / n_chains
    x_sim = simulate_living_polymerization(n_chains, n_monomers, rng=0)
    pdi_sim = np.mean(x_sim**2) / np.mean(x_sim) ** 2
    print(f"nu = {nu:.0f}: Xn simulated = {x_sim.mean():.2f}, Poisson = {poisson_number_average_DP(nu):.2f}")
    print(f"PDI simulated = {pdi_sim:.4f}, Poisson = {poisson_pdi(nu):.4f}")

    # Step-growth polymer with the same Xn = 1/(1-p)
    p = 1 - 1 / poisson_number_average_DP(nu)
    print(f"Flory-Schulz PDI at the same Xn: {flory_schulz_pdi(p):.3f}")





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

 .. code-block:: none

    nu = 50: Xn simulated = 51.00, Poisson = 51.00
    PDI simulated = 1.0197, Poisson = 1.0192
    Flory-Schulz PDI at the same Xn: 1.980




.. GENERATED FROM PYTHON SOURCE LINES 45-64

.. code-block:: Python

    x = np.arange(1, 160)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4.2))
    axes[0].hist(x_sim, bins=np.arange(0.5, 160.5, 2), density=True, alpha=0.5, label="stochastic simulation")
    axes[0].plot(x, poisson_number_fraction(x, nu), "k-", label="Poisson (living)")
    axes[0].plot(x, flory_schulz_number_fraction(x, p), "r--", label="Flory-Schulz, same Xn")
    axes[0].set_xlabel("chain length x")
    axes[0].set_ylabel("number fraction")
    axes[0].set_title(f"Living chains, nu = {nu:.0f}")
    axes[0].legend()

    nus = np.logspace(0, 3, 100)
    axes[1].semilogx(nus, poisson_pdi(nus), label="living (Poisson)")
    axes[1].axhline(2.0, color="r", ls="--", label="step growth limit (2)")
    axes[1].set_xlabel(r"$\nu$ (monomers per initiator)")
    axes[1].set_ylabel("PDI")
    axes[1].set_title("Polydispersity")
    axes[1].legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/polymer/chain_growth/images/sphx_glr_plot_04_szwarc_living_polymerization_001.png
   :alt: Living chains, nu = 50, Polydispersity
   :srcset: /api/gallery/polymer/chain_growth/images/sphx_glr_plot_04_szwarc_living_polymerization_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.105 seconds)


.. _sphx_glr_download_api_gallery_polymer_chain_growth_plot_04_szwarc_living_polymerization.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_04_szwarc_living_polymerization.ipynb <plot_04_szwarc_living_polymerization.ipynb>`

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

      :download:`Download Python source code: plot_04_szwarc_living_polymerization.py <plot_04_szwarc_living_polymerization.py>`

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

      :download:`Download zipped: plot_04_szwarc_living_polymerization.zip <plot_04_szwarc_living_polymerization.zip>`


.. only:: html

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

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