.. 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_03_flory_huggins.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_03_flory_huggins.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_polymer_solution_properties_plot_03_flory_huggins.py:


Flory-Huggins lattice theory: free energy of mixing and the phase diagram
============================================================================

Flory and Huggins (1941-1942) placed polymer segments and solvent on a
lattice to get the mixing free energy per site
(:func:`~chemistrykit.polymer.systems.flory_huggins.flory_huggins_free_energy`)

.. math::

    \frac{\Delta F_\text{mix}}{k_BT} = \frac{\phi}{N}\ln\phi
    + (1-\phi)\ln(1-\phi) + \chi\phi(1-\phi)

Below the critical :math:`\chi_c` the curve is convex and the solution is
stable; above it a concave region appears and the solution phase
separates. The spinodal
(:func:`~chemistrykit.polymer.systems.flory_huggins.flory_huggins_spinodal_chi`)
has its minimum at the critical point
(:func:`~chemistrykit.polymer.systems.flory_huggins.flory_huggins_critical_point`),
which moves to :math:`\phi_c\to0`, :math:`\chi_c\to1/2` (the theta point)
as the chains get longer.

.. GENERATED FROM PYTHON SOURCE LINES 25-41

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.polymer.systems.flory_huggins import (
        flory_huggins_critical_point,
        flory_huggins_free_energy,
        flory_huggins_spinodal_chi,
    )

    N = 100
    phi_c, chi_c = flory_huggins_critical_point(N)
    print(f"N = {N}: phi_c = {phi_c:.4f}, chi_c = {chi_c:.4f}")
    for Nv in (1, 10, 100, 1000, 10**6):
        pc, cc = flory_huggins_critical_point(Nv)
        print(f"  N = {Nv:>7}: phi_c = {pc:.4f}, chi_c = {cc:.4f}")





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

 .. code-block:: none

    N = 100: phi_c = 0.0909, chi_c = 0.6050
      N =       1: phi_c = 0.5000, chi_c = 2.0000
      N =      10: phi_c = 0.2403, chi_c = 0.8662
      N =     100: phi_c = 0.0909, chi_c = 0.6050
      N =    1000: phi_c = 0.0307, chi_c = 0.5321
      N = 1000000: phi_c = 0.0010, chi_c = 0.5010




.. GENERATED FROM PYTHON SOURCE LINES 42-65

.. code-block:: Python

    phi = np.linspace(1e-4, 0.6, 600)
    fig, axes = plt.subplots(1, 2, figsize=(11, 4.2))
    for chi in (0.4, chi_c, 0.75):
        F = flory_huggins_free_energy(phi, N, chi)
        # subtract the straight chord from phi=0 to phi=0.6 so the curvature is visible
        axes[0].plot(phi, F - F[-1] * phi / phi[-1], label=rf"$\chi$ = {chi:.3f}")
    axes[0].set_xlabel(r"polymer volume fraction $\phi$")
    axes[0].set_ylabel(r"$\Delta F_{mix}/k_BT$ (minus linear term)")
    axes[0].set_title(f"Free energy of mixing, N = {N}")
    axes[0].legend()

    phi_s = np.linspace(1e-3, 0.95, 800)
    for Nv in (10, 100, 1000):
        axes[1].plot(phi_s, flory_huggins_spinodal_chi(phi_s, Nv), label=f"N = {Nv}")
        axes[1].plot(*flory_huggins_critical_point(Nv), "ko", ms=4)
    axes[1].axhline(0.5, color="gray", ls="--", lw=0.8, label=r"theta, $\chi$ = 1/2")
    axes[1].set_ylim(0.3, 1.5)
    axes[1].set_xlabel(r"$\phi$")
    axes[1].set_ylabel(r"spinodal $\chi_s$")
    axes[1].set_title("Spinodals and critical points (dots)")
    axes[1].legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/polymer/solution_properties/images/sphx_glr_plot_03_flory_huggins_001.png
   :alt: Free energy of mixing, N = 100, Spinodals and critical points (dots)
   :srcset: /api/gallery/polymer/solution_properties/images/sphx_glr_plot_03_flory_huggins_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.079 seconds)


.. _sphx_glr_download_api_gallery_polymer_solution_properties_plot_03_flory_huggins.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_03_flory_huggins.ipynb <plot_03_flory_huggins.ipynb>`

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

      :download:`Download Python source code: plot_03_flory_huggins.py <plot_03_flory_huggins.py>`

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

      :download:`Download zipped: plot_03_flory_huggins.zip <plot_03_flory_huggins.zip>`


.. only:: html

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

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