  Polymer microgels exhibit intriguing macroscopic flow properties arising from
their unique microscopic structure. Microgel colloids comprise a crosslinked
polymer network with a radially decaying density profile, resulting in a dense
core surrounded by a fuzzy corona. Notably, microgels synthesized from
poly(N-isopropylacrylamide) (PNIPAM) are thermoresponsive, capable of adjusting
their size and density profile based on temperature. Above the lower critical
solution temperature ($T_\text{LCST}\sim 33$ $^\circ$C), the microgel's polymer
network collapses, leading to the expulsion of water through a reversible
process. Conversely, below $33$ $^\circ$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 yield stress and elastic modulus of the dense
microgel suspensions. Here, we introduce a model for the linear elastic shear
modulus through the minimization of 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.
