On the relationship between the plateau modulus and the threshold frequency in peptide gels.

Relations between static and dynamic viscoelastic responses in gels can be very elucidating and may provide useful tools to study the behavior of bio-materials such as protein hydrogels. An important example comes from the viscoelasticity of semisolid gel-like materials, which is characterized by two regimes: a low-frequency regime where the storage modulus $G^{\prime}(\omega)$ displays a constant value $G_{\text{eq}}$, and a high-frequency power-law stiffening regime, where $G^{\prime}(\omega) \sim \omega^{n}$. Recently, by considering Monte Carlo simulations to study the formation of peptides networks, we found an intriguing and somewhat related power-law relationship between the plateau modulus and the threshold frequency, i.e. $G_{\text{eq}} \sim ( \omega^{*} )^{\Delta}$ with $\Delta = 2/3$. Here we present a simple theoretical approach to describe that relationship and test its validity by using experimental data from a $\beta$-lactoglobulin gel. We show that our approach can be used even in the coarsening regime where the fractal model fails. Remarkably, the very same exponent $\Delta$ is found to describe the experimental data.