The Spectral Viscoelasticity of Gels, Fast and Slow

Gels uniquely combine elasticity and viscosity, making them effective for damping, impact absorption, and noise control. In adhesives, this duality provides both structural strength and vibration management. Viscoelastic behavior arises from two contributions: permanent elasticity, described by the equilibrium modulus G₀, and a spectrum of transient relaxation modes (eigenmodes) with relaxation times that range from milliseconds to the longest relaxation time τ_max extending hours or longer. The fast/slow interplay of eigenmodes governs applications of viscoelastic materials yet often remains obscured in classical rheology. We address this limitation with the SCOPE framework (Spectral Characterization of Process and Eigenmodes). To each eigenmode, SCOPE assigns a spectral Deborah function D and spectral Weissenberg function W, which provide natural, physically transparent criteria for gel classification.

The Deborah function (Winter, Rheologica Acta 2025) compares an eigenmode’s relaxation time with the characteristic timescales of a process. When D<1, the eigenmode relaxes before the process is completed, giving rise to a predominantly viscous response. When D>1, relaxation is incomplete, and the eigenmode contributes predominantly to elastic behavior.

The Weissenberg function, by contrast, spectroscopically measures the strain accumulated during a relaxation time τ. It quantifies how much deformation an eigenmode “remembers” before relaxing. For W>1, deformation grows more rapidly than relaxation can erase memory, separating eigenmodes that generate secondary flow effects (e.g., rod climbing in shear, strain hardening in extension) from those that do not. In addition, W provides a spectroscopic criterion for the onset of nonlinearity: when W exceeds a critical strain γ_c, the specific eigenmode leaves the linear regime. The condition W/γ_c=1 therefore marks the threshold between linear and nonlinear eigenmodes.

Interactions of G₀, D, and W within SCOPE offer complementary spectroscopic perspectives on gel behavior. This will be demonstrated using IRIS Rheo-Hub, a machine learning tool for rheology data analysis, modeling, and visualization.