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

Abstract: 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.