Fractional Viscoelasticity in Transient Unentangled Polymer Networks
Stress relaxation in transient polymer networks often shows extended power-law behavior, $G(t) \sim t^{-β}$, where the exponent $β$ frequently departs from the value $1/2$ predicted by the sticky Rouse model and its variants. We introduce the fractional inhomogeneous Rouse model (FIRM), which uses a generalized Langevin equation driven by fractional Gaussian noise of exponent $α$, while retaining heterogeneous bead friction to represent sticky cross-links. Thus, FIRM unifies subdiffusive sticker dynamics and chain heterogeneity within a single framework. We show that the relaxation modulus $G(t)$ can be represented as a linear combination of Mittag-Leffler functions. For homogeneous chains, it recovers two power-law regimes, $t^{-α/2}$ and $t^{-2α}$, on either side of the terminal relaxation time. Fitting FIRM to stress relaxation data for an imine-based polystyrene vitrimer shows that $α< 1$ is required to capture the shape of the terminal relaxation. It also accommodates both Arrhenius and non-Arrhenius temperature dependence in the rheological activation energy. We derive expressions for dynamic properties such as mean-squared displacement and dielectric response and outline how generalized memory kernels extend the framework to real materials. Together, these results suggest novel ways in which data from rheology, dielectric spectroscopy, scattering, and other experimental methods may be incorporated into a chemistry-specific molecular model.