Fig. 2: Dynamic light scattering results for various types of physical gels exhibiting universal hierarchical gel dynamics with unique stretched exponent β = 1/3.

a Cartoon of a general dynamic gel network with physical crosslinks. b Basic principle of DLS technique on a gel network. c–f Correlation function g₁(q, t) measured by DLS at scattering angle 30° for various gels and their corresponding best fitting functions (at the top) for the fitting curve (red line) to the raw data (black squares) with fitting residuals (blue triangles) for poly(acrylamide) gel with 1% chemical crosslinking density, as a reference system (c), poly(acrylamide) physical gel without chemical crosslinkers assembled by hydrogen bonds (d), polyampholyte physical gel of poly (sodium 4-vinylbenzenesulfonate-co-3-(methacryloylamino) propyl-trimethylammonium chloride) with ionic bonds at stoichiometric charge ratio r = 0.52 (number of negatively charged groups to the total number of charged groups) without chemical crosslinkers dialyzed against 2 M NaCl solutions (e), and lysozyme protein physical gel assembled by hydrophobic associations without chemical crosslinkers (f). g Superposition of normalized correlation function g₁(q, t) as a plot against the decay time in units of the characteristic relaxation time τ_β at scattering angle 30° for three types of physical gels in (d–f), all with universal law of stretched exponent β = 1/3. h Correlation between the measured characteristic relaxation time τ_β (filled circles, right axis) and the specific bond binding energy (bars, left axis) obtained from literature^42,43,44,45 for the three types of physical gels in (d–f). Error bars reflect the standard deviation from measurements of three different spatial locations within three replicate gel samples.