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In the ion-exchange chromatography literature, a term "degree of crosslinking" appears frequently. Ion exchange resins are typically made from divinylbenzene and styrene. One of the stated characteristics is the "degree of crosslink" which is often stated as 5% to 55% crosslink with DVB.

The term degree of crosslink does not appear in the IUPAC's glossary; yet it is a very well established terminology. What is the physical significance of, say, 55% crosslink when the manufacturers quote this number? Is it simply the mole percent of DVB present in the polymer?

I am looking for its exact definition from a reliable source but it seems to be elusive.

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Normally, in practice the amount fraction of DVB in the feed is used as a nominal number for the degree of crosslinking. IUPAC provides a definition in a somewhat unexpected place [1, p. 2352]:

Crosslinking: Property of a solid support prepared from polymeric materials with interconnected strands. Often results from the inclusion of multifunctional monomers in the polymerization reaction, e.g. divinylbenzene in polystyrene production. In such cases, the degree of crosslinking is often quoted as the proportion of the multifunctional monomer in the reaction mixture. The extent of crosslinking is important for physical properties of the solid support, such as the propensity to swell in different solvents [18].

Following reference [18], one can find further clarification [2, p. 2278]:

Care is therefore required in defining or interpreting the DVB content of crosslinked polymers since this might be quoted as a percentage of technical DVB used to make the polymer, or the figure can be adjusted to reflect only the content of actual DVB isomers present. Since for convenience the nominal crosslink ratio or degree of crosslinking of a polymer network is often quoted as the mol% of crosslinker used to prepare the network, defining the actual percentage of DVB isomers employed is more informative since this equates with the nominal crosslink ratio (Fig. 4). Bearing in mind that there is no unambiguous method, and certainly no simple and rapid method, for determining the real crosslink ratio in a polymer network, the nominal figure based on the actual DVB feed is a very useful parameter.

polymerisation of styrene and divinylbenzene
Fig. 4 Polymerisation of styrene and divinylbenzene to form an infinite polymer network.

Further, the paper lists several reasons as to why the real level of crosslinking is always accompanied with a level of uncertainty:

  • The effective crosslink ratio significantly decreases with greater quantity of DVB being fed due to growing number of unreacted vinyl groups.
  • Entanglement crosslinking occurs at higher rates of polymerization in slowly-agitated systems and results in additional mobile crosslinking.

    permanent entanglement crosslink
    Fig. 5 Permanent entanglement crosslink

  • Additional crosslinks can also be created as the result of intramolecular side-reactions.

These factors are tricky to estimate quantitatively; a method allowing to do so would be a single pulse excitation (SPE) 13C solid state NMR.

References

  1. Maclean, D.; Baldwin, J. J.; Ivanov, V. T.; Kato, Y.; Shaw, A.; Schneider, P.; Gordon, E. M. Glossary of Terms Used in Combinatorial Chemistry. Pure and Applied Chemistry 1999, 71 (12), 2349–2365. DOI: 10.1351/pac199971122349.
  2. Sherrington, D. C. Preparation, Structure and Morphology of Polymer Supports. Chem. Commun. 1998, No. 21, 2275–2286. DOI: 10.1039/a803757d.
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    $\begingroup$ Sherrington is missing out: NMR is a very straightforward way to determine the actual degree of crosslinking. Sufficiently swollen networks give useful spectra as-is, otherwise there is CPMAS. Of course the unavoidable (and extremely hard to determine) network heterogenity makes the number nearly as ambigous as the molar ratio of crosslinker in the educt. ;) $\endgroup$
    – Karl
    Commented Dec 28, 2020 at 9:22
  • $\begingroup$ Does the paper give a reference for those "entanglement crosslinks"? I'm sure they exist, but can they be shown to exist, i.e. do they have any effect on the network? $\endgroup$
    – Karl
    Commented Dec 28, 2020 at 16:58
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    $\begingroup$ @Karl Why, entanglement crosslinks is a well-known phenomenon, I guess, since 1950s. Also studied e.g. with NMR (doi.org/10.1021/ma960650g). One of the latest studies on contribution of entanglement crosslinking to mechanical properties: doi.org/10.1021/acs.macromol.0c00682. $\endgroup$
    – andselisk ♦
    Commented Dec 28, 2020 at 17:07
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    $\begingroup$ Thanks! I knew they are relevant for post-crosslinked material (vulcanisation etc.), but found it hard to imagine how they occur in a solution polymerisation. On second thought, emulsion polymerisation of PS with not too high DVB content should be a nice model system to study that. $\endgroup$
    – Karl
    Commented Dec 28, 2020 at 19:25
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    $\begingroup$ @Karl Entanglement cross links occur frequently in DNA, and there is a dedicated enzyme (topoisomerase II) to deal with them. $\endgroup$
    – Karsten ♦
    Commented Feb 28, 2021 at 17:59

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