.. _sphx_glr_api_gallery_polymer_solution_properties:

Solution properties
----------------------

Dilute-solution viscosity as a molar-mass probe (Staudinger's rule and
the Mark-Houwink equation) and the Flory-Huggins lattice theory of
polymer-solution thermodynamics.


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    <div class="sphx-glr-thumbcontainer" tooltip="Staudinger&#x27;s case that rubber and cellulose are genuine covalent macromolecules rested heavily on dilute-solution viscosity: he found the specific viscosity per unit concentration rising in proportion to the chain length, \eta_\text{sp}/c=K_mM (~chemistrykit.polymer.systems.solution_viscosity.staudinger_specific_viscosity). If the chains were colloidal aggregates held together by weak association, the &quot;particle size&quot; would fall apart on dilution or change of solvent; covalent chains keep their length, so a polymer-analogous reaction (e.g. hydrogenating rubber) leaves the degree of polymerization -- and hence \eta_\text{sp}/c -- unchanged. This example contrasts the two pictures.">

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  .. image:: /api/gallery/polymer/solution_properties/images/thumb/sphx_glr_plot_01_staudinger_viscosity_law_thumb.png
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  :doc:`/api/gallery/polymer/solution_properties/plot_01_staudinger_viscosity_law`

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      <div class="sphx-glr-thumbnail-title">Staudinger's macromolecules: solution viscosity grows with chain length</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Mark (1938) and Houwink (1940) generalized Staudinger&#x27;s linear rule to [\eta]=KM^a (~chemistrykit.polymer.systems.solution_viscosity.mark_houwink_intrinsic_viscosity), the relation still used to turn a viscosity measurement into a molar mass. Via Flory-Fox, [\eta]\propto R^3/M with R\propto M^\nu gives a=3\nu-1 (~chemistrykit.polymer.systems.solution_viscosity.mark_houwink_exponent_from_flory): a=0.5 in a theta solvent, a=0.8 in a good solvent, and Staudinger&#x27;s a=1 is recovered only for stiffer chains. Here synthetic &quot;measurements&quot; with 3% seeded noise are fit back to recover K and a (~chemistrykit.polymer.systems.solution_viscosity.fit_mark_houwink).">

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  .. image:: /api/gallery/polymer/solution_properties/images/thumb/sphx_glr_plot_02_mark_houwink_thumb.png
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  :doc:`/api/gallery/polymer/solution_properties/plot_02_mark_houwink`

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      <div class="sphx-glr-thumbnail-title">The Mark-Houwink equation: intrinsic viscosity vs. molar mass</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Flory and Huggins (1941-1942) placed polymer segments and solvent on a lattice to get the mixing free energy per site (~chemistrykit.polymer.systems.flory_huggins.flory_huggins_free_energy)">

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  .. image:: /api/gallery/polymer/solution_properties/images/thumb/sphx_glr_plot_03_flory_huggins_thumb.png
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  :doc:`/api/gallery/polymer/solution_properties/plot_03_flory_huggins`

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      <div class="sphx-glr-thumbnail-title">Flory-Huggins lattice theory: free energy of mixing and the phase diagram</div>
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.. toctree::
   :hidden:

   /api/gallery/polymer/solution_properties/plot_01_staudinger_viscosity_law
   /api/gallery/polymer/solution_properties/plot_02_mark_houwink
   /api/gallery/polymer/solution_properties/plot_03_flory_huggins