Isosbestic points in time resolved SAXS: from spectroscopic analogy to model free structural markers during colloidal gelation

Abstract

Gelation is the transition from a fluid state into a system-spanning, out of equilibrim soft-solid network through a hierarchical process that couples local particle interactions to mesoscopic clustering and global connectivity. In time-resolved small-angle X-ray scattering (SAXS), isosbestic points -- scattering wavevectors where scattering intensity remains invariant -- emerge during this transformation, yet their physical meaning has remained unclear. Here, we show that two isosbestic points, $q_1$ and $q_2$, observed during salt-induced gelation of Ludox colloids, reflect fundamental structural constraints rather than a two-species interconversion. The high-$q$ point $q_2$ is a universal geometric marker, determined by particle contact distances, while the low-$q$ point $q_1$ arises from Porod invariant conservation and separates rapidly arrested local clusters from the growing mesoscopic network. By decomposing the Porod invariant across the reciprocal-space regions defined by these points, we define a dimensionless parameter, $\Phi(t/t_g)$, whose sigmoidal evolution provides a simple, model-free, scale-resolved fingerprint of gelation. Together with the combined evolution of $S(q_{\min},t)$ and $S(q \rightarrow 0,t)$, these results establish a quantitative model free framework linking local structuring, global connectivity, and scattering signatures, clarifying the role of isosbestic points in soft-matter transformations.