:orphan:

Examples
========

This gallery walks through every public feature of
``chemistrykit.polymer``: ideal random-walk chain statistics and the
Flory exponent for real chains under different solvent conditions;
molecular-weight-distribution statistics and the closed-form Flory-Schulz
distribution; step-growth kinetics via the Carothers equation; and
chain-growth/free-radical polymerization kinetics built on
``chemistrykit.kinetics``'s reaction-network engine.

Each script in this gallery is self-contained and can be run directly
with ``python examples/polymer/<section>/<script>.py``. Every script
also carries an RST module docstring as its title/description and uses
``# %%`` markers to split narrative text from code, which is exactly what
Sphinx-Gallery renders into the pages below -- the script *is* the source
of truth for what you see, not a copy of it.

Sections
--------

- **chain_statistics** -- ideal-chain end-to-end distance and radius of
  gyration, and the Flory-exponent scaling of real chains under
  theta/good/poor solvent conditions; sampled freely jointed chains and
  the Kratky-Porod worm-like chain.
- **molecular_weight_distribution** -- Mn/Mw/PDI, and the closed-form
  Flory-Schulz (most-probable) chain-length distribution, cross-checked
  by direct numerical summation; Debye light scattering vs. osmometry.
- **step_growth** -- the Carothers equation relating degree of
  polymerization to extent of reaction, plain and with a stoichiometric
  imbalance; Flory-Stockmayer gelation.
- **chain_growth** -- free-radical initiation/propagation/termination
  kinetics, integrated numerically and checked against the steady-state
  approximation's closed form; combination vs. disproportionation; the
  Mayo-Lewis copolymer equation; and Szwarc's living polymerization.
- **solution_properties** -- Staudinger's viscosity rule, the
  Mark-Houwink equation, and Flory-Huggins solution thermodynamics.
- **stereochemistry** -- Ziegler-Natta stereocontrol and Bernoullian
  tacticity statistics.


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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 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
    :alt:

  :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
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  :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>
    </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
    :alt:

  :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>
    </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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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.


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    <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).">

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  .. 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`

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


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    <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.">

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  .. image:: /api/gallery/polymer/chain_statistics/images/thumb/sphx_glr_plot_02_flory_vs_de_gennes_thumb.png
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  :doc:`/api/gallery/polymer/chain_statistics/plot_02_flory_vs_de_gennes`

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


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    <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.">

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  .. image:: /api/gallery/polymer/chain_statistics/images/thumb/sphx_glr_plot_03_kuhn_freely_jointed_chain_thumb.png
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  :doc:`/api/gallery/polymer/chain_statistics/plot_03_kuhn_freely_jointed_chain`

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


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    <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)">

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  .. image:: /api/gallery/polymer/chain_statistics/images/thumb/sphx_glr_plot_04_kratky_porod_worm_like_chain_thumb.png
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  :doc:`/api/gallery/polymer/chain_statistics/plot_04_kratky_porod_worm_like_chain`

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

Mn/Mw/PDI summary statistics, and the closed-form Flory-Schulz
(most-probable) chain-length distribution for an ideal step-growth
polymerization, with its PDI approaching exactly 2 at full conversion.

Also Debye light scattering (Mw) vs. osmometry (Mn).


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    <div class="sphx-glr-thumbcontainer" tooltip="For an ideal step-growth polymerization at extent of reaction p, the Flory-Schulz (most-probable) distribution gives the number and weight fraction of chains of each length in exact closed form. Its number-/weight-average degrees of polymerization, \bar X_n=1/(1-p) and \bar X_w=(1+p)/(1-p), are cross-checked here against direct numerical summation of the distribution, and the PDI =1+p is shown approaching exactly 2 as p approaches 1.">

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  .. image:: /api/gallery/polymer/molecular_weight_distribution/images/thumb/sphx_glr_plot_01_flory_schulz_thumb.png
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  :doc:`/api/gallery/polymer/molecular_weight_distribution/plot_01_flory_schulz`

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      <div class="sphx-glr-thumbnail-title">The Flory-Schulz molecular-weight distribution</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="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 (~chemistrykit.polymer.systems.light_scattering.rayleigh_ratio_dilute_mixture). Osmotic pressure counts molecules instead and gives the number average (~chemistrykit.polymer.systems.light_scattering.osmotic_pressure_dilute_mixture). Here a blend of two narrow fractions is &quot;measured&quot; both ways, and a Debye plot Kc/R_0=1/M_w+2A_2c (~chemistrykit.polymer.systems.light_scattering.debye_Kc_over_R) is extrapolated to c=0.">

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  .. image:: /api/gallery/polymer/molecular_weight_distribution/images/thumb/sphx_glr_plot_02_debye_light_scattering_thumb.png
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  :doc:`/api/gallery/polymer/molecular_weight_distribution/plot_02_debye_light_scattering`

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      <div class="sphx-glr-thumbnail-title">Debye light scattering measures Mw; osmometry measures Mn</div>
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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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Step-growth kinetics
-----------------------

The Carothers equation relating the number-average degree of
polymerization to the extent of reaction, and its generalization to a
stoichiometric imbalance between functional groups.

Also the Flory-Stockmayer and Carothers gel points of branching
polymerizations.


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    <div class="sphx-glr-thumbcontainer" tooltip="Step-growth polymerization&#x27;s number-average degree of polymerization follows the Carothers equation \bar X_n=1/(1-p), which diverges as the extent of reaction p approaches 1 -- reaching a useful high molecular weight requires very high conversion. A stoichiometric imbalance between the two functional groups caps \bar X_n even at complete conversion of the limiting group.">

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  .. image:: /api/gallery/polymer/step_growth/images/thumb/sphx_glr_plot_01_carothers_thumb.png
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  :doc:`/api/gallery/polymer/step_growth/plot_01_carothers`

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      <div class="sphx-glr-thumbnail-title">The Carothers equation</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="With trifunctional or higher monomers, step-growth polymerization builds branched molecules that link into a sample-spanning network at a sharp gel point. For self-condensation of an A_f monomer, Flory (1941) and Stockmayer (1943) found the weight-average degree of polymerization \bar X_w=(1+p)/(1-(f-1)p) (~chemistrykit.polymer.systems.gelation.branching_weight_average_DP) diverges at p_c=1/(f-1) (~chemistrykit.polymer.systems.gelation.flory_stockmayer_gel_point) while the number average (~chemistrykit.polymer.systems.gelation.branching_number_average_DP) stays finite. Carothers&#x27;s criterion \bar X_n\to\infty, p_c=2/f (~chemistrykit.polymer.systems.gelation.carothers_gel_point), overestimates the gel point; experiments fall between the two.">

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  .. image:: /api/gallery/polymer/step_growth/images/thumb/sphx_glr_plot_02_flory_stockmayer_gelation_thumb.png
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  :doc:`/api/gallery/polymer/step_growth/plot_02_flory_stockmayer_gelation`

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      <div class="sphx-glr-thumbnail-title">Flory-Stockmayer gelation: the gel point of a branching polymerization</div>
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Stereochemistry
------------------

Tacticity of vinyl polymers: meso/racemo dyad and triad statistics
contrasting stereospecific (Ziegler-Natta) and random (free-radical)
monomer placement.


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    <div class="sphx-glr-thumbcontainer" tooltip="Natta&#x27;s Ziegler-type catalysts insert each propylene with the same orientation, giving isotactic polypropylene (every methyl on the same side, nearly all meso dyads), whereas free-radical placement is close to random (atactic). With meso-placement probability P_m (Bernoullian statistics), the NMR-observable triad fractions are [mm]=P_m^2, [mr]=2P_m(1-P_m), [rr]=(1-P_m)^2 (~chemistrykit.polymer.systems.tacticity.bernoullian_triad_fractions), and isotactic runs average 1/(1-P_m) dyads (~chemistrykit.polymer.systems.tacticity.mean_isotactic_run_length). Seeded dyad sequences (~chemistrykit.polymer.systems.tacticity.sample_dyad_sequence) make the difference visible.">

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  .. image:: /api/gallery/polymer/stereochemistry/images/thumb/sphx_glr_plot_01_ziegler_natta_tacticity_thumb.png
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  :doc:`/api/gallery/polymer/stereochemistry/plot_01_ziegler_natta_tacticity`

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      <div class="sphx-glr-thumbnail-title">Ziegler-Natta stereocontrol: isotactic vs. atactic polypropylene</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/polymer/chain_growth/index.rst
   /api/gallery/polymer/chain_statistics/index.rst
   /api/gallery/polymer/molecular_weight_distribution/index.rst
   /api/gallery/polymer/solution_properties/index.rst
   /api/gallery/polymer/step_growth/index.rst
   /api/gallery/polymer/stereochemistry/index.rst



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