.. _sphx_glr_api_gallery_polymer_chain_statistics:

Chain statistics
------------------

Ideal random-walk chain statistics (exact end-to-end distance and radius
of gyration) and the Flory-exponent scaling of real chains under
theta/good/poor solvent conditions.

Also sampled freely jointed (Kuhn) chains and the Kratky-Porod
worm-like chain.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Flory&#x27;s mean-field argument (Principles of Polymer Chemistry, 1953) predicts how excluded volume changes a chain&#x27;s size scaling. IdealChain gives the exact random-walk result \langle R^2\rangle=nb^2. RealChain scales instead as R\sim bn^\nu, with the Flory exponent \nu set by solvent quality: a good solvent swells the chain (\nu\approx0.6), a poor solvent collapses it (\nu=1/3), and the theta solvent reproduces the ideal chain exactly (\nu=1/2).">

.. only:: html

  .. image:: /api/gallery/polymer/chain_statistics/images/thumb/sphx_glr_plot_01_chain_scaling_thumb.png
    :alt:

  :doc:`/api/gallery/polymer/chain_statistics/plot_01_chain_scaling`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Flory's solvent-quality exponents: ideal vs. real chain scaling</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="good_solvent uses Flory&#x27;s original 1953 mean-field good-solvent exponent (\nu=3/5); good_solvent_renormalization_group uses de Gennes&#x27; 1979 renormalization-group-refined value (\nu\approx0.588). The two exponents differ by only about 2%, but because R\sim bn^\nu is a power law, that small an exponent difference compounds with chain length: the two chain-size curves are nearly indistinguishable on a log-log plot for short chains, yet the relative difference between the predicted R values grows steadily as n increases -- exactly why precise critical exponents matter most for long-chain, asymptotic behavior.">

.. only:: html

  .. image:: /api/gallery/polymer/chain_statistics/images/thumb/sphx_glr_plot_02_flory_vs_de_gennes_thumb.png
    :alt:

  :doc:`/api/gallery/polymer/chain_statistics/plot_02_flory_vs_de_gennes`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Flory's mean-field exponent vs. the renormalization-group value</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Kuhn (1934) modeled a flexible polymer as a random walk of n freely jointed segments of length b. freely_jointed_chain samples such conformations directly; averaging their squared end-to-end distance reproduces the exact result \langle R^2\rangle=nb^2 (~chemistrykit.polymer.systems.chain_statistics.IdealChain), and the end-to-end vector&#x27;s components become Gaussian with variance nb^2/3, as the central limit theorem predicts.">

.. only:: html

  .. image:: /api/gallery/polymer/chain_statistics/images/thumb/sphx_glr_plot_03_kuhn_freely_jointed_chain_thumb.png
    :alt:

  :doc:`/api/gallery/polymer/chain_statistics/plot_03_kuhn_freely_jointed_chain`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Kuhn's random-walk chain: sampled freely jointed chains</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A stiff polymer (DNA, many polysaccharides) bends smoothly rather than at free joints. Kratky and Porod&#x27;s (1949) worm-like chain, with persistence length P, gives (~chemistrykit.polymer.systems.chain_statistics.worm_like_chain_mean_square_end_to_end)">

.. only:: html

  .. image:: /api/gallery/polymer/chain_statistics/images/thumb/sphx_glr_plot_04_kratky_porod_worm_like_chain_thumb.png
    :alt:

  :doc:`/api/gallery/polymer/chain_statistics/plot_04_kratky_porod_worm_like_chain`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Kratky-Porod worm-like chain: from rigid rod to random coil</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/polymer/chain_statistics/plot_01_chain_scaling
   /api/gallery/polymer/chain_statistics/plot_02_flory_vs_de_gennes
   /api/gallery/polymer/chain_statistics/plot_03_kuhn_freely_jointed_chain
   /api/gallery/polymer/chain_statistics/plot_04_kratky_porod_worm_like_chain