.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/polymer/molecular_weight_distribution/plot_02_debye_light_scattering.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_molecular_weight_distribution_plot_02_debye_light_scattering.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_polymer_molecular_weight_distribution_plot_02_debye_light_scattering.py:


Debye light scattering measures Mw; osmometry measures Mn
============================================================

Debye (1944) showed that the light scattered by a dilute polymer
solution, extrapolated to zero angle and zero concentration, gives an
absolute molar mass -- the *weight* average, because each chain scatters
in proportion to the square of its mass
(:func:`~chemistrykit.polymer.systems.light_scattering.rayleigh_ratio_dilute_mixture`).
Osmotic pressure counts molecules instead and gives the *number* average
(:func:`~chemistrykit.polymer.systems.light_scattering.osmotic_pressure_dilute_mixture`).
Here a blend of two narrow fractions is "measured" both ways, and a Debye
plot :math:`Kc/R_0=1/M_w+2A_2c`
(:func:`~chemistrykit.polymer.systems.light_scattering.debye_Kc_over_R`) is
extrapolated to :math:`c=0`.

.. GENERATED FROM PYTHON SOURCE LINES 19-40

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.polymer.systems.light_scattering import (
        debye_Kc_over_R,
        osmotic_pressure_dilute_mixture,
        rayleigh_ratio_dilute_mixture,
    )
    from chemistrykit.polymer.systems.molecular_weight_distribution import (
        number_average_molar_mass,
        polydispersity_index,
        weight_average_molar_mass,
    )

    # Equal masses of a 20 kg/mol and a 500 kg/mol fraction.
    M_i = np.array([20.0, 500.0])  # kg/mol
    w_i = np.array([0.5, 0.5])  # mass fractions
    Mn = number_average_molar_mass(w_i / M_i, M_i)
    Mw = weight_average_molar_mass(w_i / M_i, M_i)
    print(f"True Mn = {Mn:.1f} kg/mol, Mw = {Mw:.1f} kg/mol, PDI = {polydispersity_index(Mn, Mw):.2f}")





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

 .. code-block:: none

    True Mn = 38.5 kg/mol, Mw = 260.0 kg/mol, PDI = 6.76




.. GENERATED FROM PYTHON SOURCE LINES 41-42

"Measure" each at a dilute total concentration c (kg/m^3).

.. GENERATED FROM PYTHON SOURCE LINES 42-51

.. code-block:: Python

    c_total = 1.0
    K, T = 1.0, 298.15
    R0 = rayleigh_ratio_dilute_mixture(c_total * w_i, M_i, K)
    Pi = osmotic_pressure_dilute_mixture(c_total * w_i, M_i, T)
    M_light = R0 / (K * c_total)
    M_osmo = 8.314462618 * T * c_total / Pi
    print(f"Light scattering apparent M = {M_light:.1f} kg/mol  (= Mw)")
    print(f"Osmometry apparent M        = {M_osmo:.1f} kg/mol  (= Mn)")





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

 .. code-block:: none

    Light scattering apparent M = 260.0 kg/mol  (= Mw)
    Osmometry apparent M        = 38.5 kg/mol  (= Mn)




.. GENERATED FROM PYTHON SOURCE LINES 52-53

Debye plot with a second virial coefficient; the intercept is 1/Mw.

.. GENERATED FROM PYTHON SOURCE LINES 53-75

.. code-block:: Python

    A2 = 2.0e-4
    c = np.linspace(0.5, 5.0, 8)
    rng = np.random.default_rng(1)
    y = debye_Kc_over_R(c, Mw, A2) * (1 + 0.005 * rng.standard_normal(c.size))
    slope, intercept = np.polyfit(c, y, 1)
    print(f"Debye-plot intercept -> Mw = {1 / intercept:.1f} kg/mol; slope -> A2 = {slope / 2:.2e}")

    fig, axes = plt.subplots(1, 2, figsize=(11, 4.2))
    cc = np.linspace(0, 5.0, 50)
    axes[0].plot(c, y, "o", label="simulated data")
    axes[0].plot(cc, intercept + slope * cc, "k-", lw=0.8, label="linear extrapolation")
    axes[0].axhline(1 / Mn, color="crimson", ls="--", label="1/Mn (would be wrong)")
    axes[0].set_xlabel("c (kg/m$^3$)")
    axes[0].set_ylabel(r"$Kc/R_0$ (mol/kg)")
    axes[0].set_title("Debye plot: intercept = 1/Mw")
    axes[0].legend()

    axes[1].bar(["Mn (osmometry)", "Mw (light scattering)"], [M_osmo, M_light], color=["crimson", "steelblue"])
    axes[1].set_ylabel("apparent M (kg/mol)")
    axes[1].set_title("Same blend, two different averages")
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/polymer/molecular_weight_distribution/images/sphx_glr_plot_02_debye_light_scattering_001.png
   :alt: Debye plot: intercept = 1/Mw, Same blend, two different averages
   :srcset: /api/gallery/polymer/molecular_weight_distribution/images/sphx_glr_plot_02_debye_light_scattering_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    Debye-plot intercept -> Mw = 259.6 kg/mol; slope -> A2 = 2.00e-04





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

   **Total running time of the script:** (0 minutes 0.061 seconds)


.. _sphx_glr_download_api_gallery_polymer_molecular_weight_distribution_plot_02_debye_light_scattering.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_debye_light_scattering.ipynb <plot_02_debye_light_scattering.ipynb>`

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

      :download:`Download Python source code: plot_02_debye_light_scattering.py <plot_02_debye_light_scattering.py>`

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

      :download:`Download zipped: plot_02_debye_light_scattering.zip <plot_02_debye_light_scattering.zip>`


.. only:: html

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

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