Viscoelastic Stress Relaxation, Fast and Slow

Abstract

Soft materials such as colloids, pastes, and polymer liquids are defined rheologically by how they build and relax stress during flow and deformation. Their internal connectivity manifests in a broad spectrum of viscoelastic eigenmodes, with relaxation ranging from fast to slow and contributions that vary from weak to strong. The interplay of these modes determines how the material deforms under shear, compression, or stretching across different processing timescales. Traditional measures of viscoelasticity, such as the Deborah number (De) and the Weissenberg number (Wi), condense this complexity into single scalar values. While useful for certain purposes, these scalar measures mask the fast/slow interplay of relaxation processes that shape the distinct responses of soft matter. To overcome this limitation, we introduce the “Spectral Classification of Processes and Eigenmodes” (SCOPE) framework. SCOPE explicitly accounts for the distributed nature of both process times and material relaxation times. It generalizes the classical De and Wi into their functional counterparts—the Deborah Function and the Weissenberg Function—which connect applied stress and strain to the full spectrum of relaxation times (0 < τ <τ_max), thereby covering the entire range of process timescales and types of deformation. By doing so, SCOPE provides a spectral perspective on viscoelasticity that integrates fast and slow dynamics within a single, unified rheological framework. SCOPE provides criteria that separate viscous from elastic eigenmodes, and modes below or above the onset of nonlinearity. In what follows, we introduce the SCOPE framework in detail and demonstrate its functions for viscoelastic liquids.