A full quantitative description of the swelling of smart microgels is still problematic in many cases. The original approach of Flory and Huggins for the monomer–solvent interaction parameter χ cannot be applied to some microgels. The reason for this obviously is that the cross-linking enhances the cooperativity of the volume phase transitions, since all meshes of the network are mechanically coupled. This was ignored in previous approaches, arguing with distinct transition temperatures for different meshes to describe the continuous character of the transition of microgels. Here, we adjust the swelling curves of a series of smart microgels using the Flory–Rehner description, where the polymer–solvent interaction parameter χ is modeled by a Hill-like equation for a cooperative thermotropic transition. This leads to a very good description of all measured microgel swelling curves and yields the physically meaningful Hill parameter ν. A linear decrease of ν is found with increasing concentration of the cross-linker N,N′-methylenebisacrylamide in the microgel particles p(NIPAM), p(NNPAM), and p(NIPMAM). The linearity suggests that the Hill parameter ν corresponds to the number of water molecules per network chain that cooperatively leave the chain at the volume phase transition. Driven by entropy, ν water molecules of the solvate become cooperatively “free” and leave the polymer network.

The description of gel swelling by Flory and Rehner using the original Flory–Huggins interaction parameter for the polymer–solvent interaction cannot be applied to most smart microgels. Here, we compare descriptions of the swelling curves of such microgels using series expansions of the Flory–Huggins parameter $$\chi$$ χ with the results of Hill-like equation for $$\chi$$ χ . We study N-isopropyl-acrylamide particles at different concentrations of the cross-linker N,N-methylenebisacrylamide. The hydrodynamic radius $$R_{\mathrm {H}}$$ R H of the microgel particles is determined using photon correlation spectroscopy. The fits with the series expansion of $$\chi$$ χ nicely follow the experimental data. However, already with the first-order series expansion, the computed $$\Theta$$ Θ temperatures are not physically reasonable. Moreover, the physical meaning of the parameters of the series expansion is not clear. The Hill-like equation, which we recently introduced, yields a good description of all measured microgel swelling curves and provides physically meaningful parameters. For instance, the Hill parameter $$\nu$$ ν corresponds to the number of water molecules per network chain cooperatively leaving the chain at the volume phase transition. Graphical abstract Different approaches to model the Flory-Huggins interaction parameter are explored and compared with respect to the quality of the fit of microgel swelling curves.

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In the present article, we use an improved Flory–Rehner theory to describe the swelling behavior of copolymer microgels, where the interaction parameter is modeled by a Hill-like equation for a cooperative thermotropic transition. This description leads to very good fits of the swelling curves of the copolymer microgels at different comonomer contents (30 mol%, 50 mol% and 70 mol%) obtained by photon correlation spectroscopy. Fixed parameters, which are universally applicable for the respective monomers given in our previous work, are used to fit the swelling curves. The analysis of the swelling curves yields physically reasonable and meaningful results for the remaining adjustable parameters. The comonomer content of the statistical copolymer microgels poly(NNPAM-co-NIPAM), poly(NIPAM-co-NIPMAM) and poly(NIPMAM-co-NNPAM) is determined by nuclear magnetic resonance spectroscopy and is in agreement with the nominal comonomer feed used in the synthesis. To investigate the volume phase transition at a molecular level, swelling curves are also measured by Fourier transformation infrared spectroscopy. The obtained swelling curves are also fitted using the Hill-like model. The fits provide physically reasonable parameters too, consistent with the results from photon correlation spectroscopy.

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