A Free Energy Model for the Plateau Shear Modulus in Thermosensitive Microgel Suspensions

Polymer microgels exhibit intriguing macroscopic flow properties arising from their unique microscopic structure. Microgel colloids usually comprise a cross-linked polymer network with a radially decaying density profile, resulting in a dense core surrounded by a fuzzy corona. Notably, microgels synthesized from poly(<i>N</i>-isopropylacrylamide) (PNIPAM) are thermoresponsive and capable of adjusting their size and density profile based on temperature. Above the lower critical solution temperature (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>LCST</mml:mi></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:mn>33</mml:mn></mml:math> °C), the microgel's polymer network collapses, expulsing water through a reversible process. Conversely, below 33 °C, the microgel's network swells, becoming highly compressible and allowing overpacking to effective volume fractions exceeding one. Under conditions of dense packing, microgels undergo deformation in distinct stages: corona compression and faceting, interpenetration, and finally, isotropic compression. Each stage exhibits a characteristic signature in the dense microgel suspensions' yield stress and elastic modulus. Here, we introduce a model for the linear elastic shear modulus by minimizing a quasi-equilibrium free energy, encompassing all relevant energetic contributions. We validate our model by comparing its predictions to experimental results from oscillatory shear rheology tests on microgel suspensions at different densities and temperatures. Our findings demonstrate that combining macroscopic rheological measurements with the model allows for temperature-dependent characterization of polymer interaction parameters.