.. _sphx_glr_api_gallery_polymer_chain_growth:

Chain-growth kinetics
------------------------

Free-radical initiation/propagation/termination kinetics, built on
chemistrykit.kinetics's reaction-network engine and integrated
numerically, checked against the closed-form steady-state
approximation. Also combination vs. disproportionation termination, and
the factor-of-two difference in degree of polymerization it produces for
the same kinetic chain length.

Also the Mayo-Lewis copolymer equation and the Poisson distribution of
Szwarc's living polymerization.


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


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

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    <div class="sphx-glr-thumbcontainer" tooltip="free_radical_network builds the lumped initiation/propagation/termination kinetics as a StoichiometricNetwork and integrates it numerically. Because termination is much faster than initiator decomposition, the radical concentration relaxes almost immediately to the steady-state approximation&#x27;s closed-form prediction (~chemistrykit.polymer.systems.chain_growth.steady_state_radical_concentration) -- verified here by comparing the two directly -- while monomer is consumed only slowly, at the classic square-root-of-initiator rate.">

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  .. image:: /api/gallery/polymer/chain_growth/images/thumb/sphx_glr_plot_01_free_radical_kinetics_thumb.png
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  :doc:`/api/gallery/polymer/chain_growth/plot_01_free_radical_kinetics`

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      <div class="sphx-glr-thumbnail-title">Flory's steady-state kinetics of free-radical polymerization</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="free_radical_network&#x27;s mode argument selects between the two ways two growing radical chains can terminate: combination (they fuse into a single dead chain) or disproportionation (one radical abstracts a hydrogen from the other, producing two separate dead chains). Bevington, Melville, and Taylor (&quot;The Termination Reaction in Radical Polymerizations,&quot; J. Polym. Sci. 12 (1954), 449-459) showed how to distinguish the two experimentally, by end-group analysis and by the resulting degree of polymerization&#x27;s relationship to the kinetic chain length (~chemistrykit.polymer.systems.chain_growth.kinetic_chain_length) nu: \bar X_n = 2\nu for combination (two radicals&#x27; worth of monomer per dead chain) but \bar X_n = \nu for disproportionation (one radical&#x27;s worth per dead chain) -- verified here directly against the numerically integrated network.">

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  .. image:: /api/gallery/polymer/chain_growth/images/thumb/sphx_glr_plot_02_combination_vs_disproportionation_thumb.png
    :alt:

  :doc:`/api/gallery/polymer/chain_growth/plot_02_combination_vs_disproportionation`

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      <div class="sphx-glr-thumbnail-title">Combination vs. disproportionation termination</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Mayo and Lewis (1944) showed that, if a growing radical&#x27;s reactivity depends only on its terminal monomer unit, the instantaneous copolymer composition F_1 follows from the feed composition f_1 and two reactivity ratios (~chemistrykit.polymer.systems.copolymerization.mayo_lewis_copolymer_composition). Their own test case, styrene / methyl methacrylate (r_1\approx0.52, r_2\approx0.46), crosses the diagonal at an azeotropic feed (~chemistrykit.polymer.systems.copolymerization.azeotropic_feed_composition) where the copolymer has the same composition as the feed.">

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  .. image:: /api/gallery/polymer/chain_growth/images/thumb/sphx_glr_plot_03_mayo_lewis_copolymerization_thumb.png
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  :doc:`/api/gallery/polymer/chain_growth/plot_03_mayo_lewis_copolymerization`

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      <div class="sphx-glr-thumbnail-title">The Mayo-Lewis copolymer equation: composition vs. feed</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Szwarc (1956) showed that anionic polymerization of styrene initiated by sodium naphthalenide has no termination: every chain starts at once and keeps growing while monomer lasts. Each chain then receives monomers as independent random events, giving a Poisson distribution (~chemistrykit.polymer.systems.living_polymerization.poisson_number_fraction) with \text{PDI}=1+\nu/(1+\nu)^2 (~chemistrykit.polymer.systems.living_polymerization.poisson_pdi) -- close to 1, far narrower than the Flory-Schulz distribution of a step-growth polymer with the same \bar X_n. A seeded stochastic simulation (~chemistrykit.polymer.systems.living_polymerization.simulate_living_polymerization) reproduces the prediction.">

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  .. image:: /api/gallery/polymer/chain_growth/images/thumb/sphx_glr_plot_04_szwarc_living_polymerization_thumb.png
    :alt:

  :doc:`/api/gallery/polymer/chain_growth/plot_04_szwarc_living_polymerization`

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      <div class="sphx-glr-thumbnail-title">Szwarc's living polymerization: the narrow Poisson distribution</div>
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.. toctree::
   :hidden:

   /api/gallery/polymer/chain_growth/plot_01_free_radical_kinetics
   /api/gallery/polymer/chain_growth/plot_02_combination_vs_disproportionation
   /api/gallery/polymer/chain_growth/plot_03_mayo_lewis_copolymerization
   /api/gallery/polymer/chain_growth/plot_04_szwarc_living_polymerization