.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/polymer/chain_statistics/plot_01_chain_scaling.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_statistics_plot_01_chain_scaling.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_polymer_chain_statistics_plot_01_chain_scaling.py:


Flory's solvent-quality exponents: ideal vs. real chain scaling
===================================================================

Flory's mean-field argument (*Principles of Polymer Chemistry*, 1953)
predicts how excluded volume changes a chain's size scaling.
:class:`~chemistrykit.polymer.systems.chain_statistics.IdealChain` gives
the *exact* random-walk result :math:`\langle R^2\rangle=nb^2`.
:class:`~chemistrykit.polymer.systems.chain_statistics.RealChain` scales
instead as :math:`R\sim bn^\nu`, with the Flory exponent :math:`\nu`
set by solvent quality: a good solvent swells the chain
(:math:`\nu\approx0.6`), a poor solvent collapses it (:math:`\nu=1/3`),
and the theta solvent reproduces the ideal chain exactly
(:math:`\nu=1/2`).

.. GENERATED FROM PYTHON SOURCE LINES 18-34

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.polymer.systems.chain_statistics import IdealChain, RealChain, flory_exponent
    from chemistrykit.polymer.visualizers.polymer_plots import plot_chain_scaling

    b = 0.5  # segment length
    n_values = np.logspace(1, 5, 30)

    ideal = IdealChain()
    theta = RealChain.theta_solvent()
    good = RealChain.good_solvent()
    poor = RealChain.poor_solvent()

    print(f"Flory exponents: theta={flory_exponent('theta')}, good={flory_exponent('good'):.4f}, poor={flory_exponent('poor'):.4f}")





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

 .. code-block:: none

    Flory exponents: theta=0.5, good=0.6000, poor=0.3333




.. GENERATED FROM PYTHON SOURCE LINES 35-36

The ideal chain's <R^2> scales exactly linearly with n.

.. GENERATED FROM PYTHON SOURCE LINES 36-45

.. code-block:: Python

    n1, n2 = 1000.0, 4000.0
    ratio = ideal.mean_square_end_to_end(n2, b) / ideal.mean_square_end_to_end(n1, b)
    print(f"\n<R^2>(4000) / <R^2>(1000) for the ideal chain: {ratio:.6f} (exactly 4.0)")

    # theta-solvent RealChain reproduces the ideal chain's end-to-end distance exactly.
    n_test = 5000.0
    match = np.isclose(theta.end_to_end_distance(n_test, b), ideal.end_to_end_distance(n_test, b))
    print(f"theta-solvent RealChain matches IdealChain exactly: {bool(match)}")





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

 .. code-block:: none


    <R^2>(4000) / <R^2>(1000) for the ideal chain: 4.000000 (exactly 4.0)
    theta-solvent RealChain matches IdealChain exactly: True




.. GENERATED FROM PYTHON SOURCE LINES 46-48

A good solvent swells the chain; a poor solvent collapses it, relative
to the ideal chain of the same length.

.. GENERATED FROM PYTHON SOURCE LINES 48-54

.. code-block:: Python

    n_test = 10000.0
    print(f"\nAt n={n_test:.0f}, b={b}:")
    print(f"  ideal chain R = {ideal.end_to_end_distance(n_test, b):.2f}")
    print(f"  good-solvent R = {good.end_to_end_distance(n_test, b):.2f}  (swollen)")
    print(f"  poor-solvent R = {poor.end_to_end_distance(n_test, b):.2f}  (collapsed)")





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

 .. code-block:: none


    At n=10000, b=0.5:
      ideal chain R = 50.00
      good-solvent R = 125.59  (swollen)
      poor-solvent R = 10.77  (collapsed)




.. GENERATED FROM PYTHON SOURCE LINES 55-59

.. code-block:: Python

    models = {"ideal (theta)": ideal, "good solvent": good, "poor solvent": poor}
    ax = plot_chain_scaling(models, n_values, b)
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/polymer/chain_statistics/images/sphx_glr_plot_01_chain_scaling_001.png
   :alt: Chain-size scaling
   :srcset: /api/gallery/polymer/chain_statistics/images/sphx_glr_plot_01_chain_scaling_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.068 seconds)


.. _sphx_glr_download_api_gallery_polymer_chain_statistics_plot_01_chain_scaling.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_chain_scaling.ipynb <plot_01_chain_scaling.ipynb>`

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

      :download:`Download Python source code: plot_01_chain_scaling.py <plot_01_chain_scaling.py>`

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

      :download:`Download zipped: plot_01_chain_scaling.zip <plot_01_chain_scaling.zip>`


.. only:: html

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

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