  Hydrogels are particularly versatile materials that are widely found in both
Nature and industry. One key reason for this versatility is their high water
content, which lets them dramatically change their volume and many of their
mechanical properties -- often by orders of magnitude -- as they swell and dry
out. Currently, we lack techniques that can precisely characterize how these
properties change with water content. To overcome this challenge, here we
develop Gel-Freezing Osmometry (GelFrO): an extension of freezing-point
osmometry. We show how GelFrO can measure a hydrogel's mechanical response to
compression and osmotic pressure, while only using small, $O(100\mu$L$)$
samples. Because the technique allows measurement of properties over an
unusually wide range of water contents, it allows us to accurately test
theoretical predictions. We find simple, power-law behavior for both mechanical
response to compression, and osmotic pressure, while these are not
well-captured by classical Flory-Huggins theory. We interpret this power-law
behavior as a hallmark of a microscopic fractal structure of the gel's polymer
network, and propose a simple way to connect the gel's fractal dimension to its
mechanical and osmotic properties. This connection is supported by observations
of hydrogel microstructures using small-angle x-ray scattering. Finally, our
results motivate us to propose an updated constitutive model describing
hydrogel swelling, and mechanical response.
