Probing length scale of dynamic correlation by particle tracking

Dynamic heterogeneity in complex fluids is often inferred from broad relaxation spectra or non-Gaussian probe motion, but these signatures do not by themselves identify the relevant spatial length scales or distinguish heterogeneity from inter-particle dynamic correlation. Here, we develop an analytical framework for multiparticle tracking microrheology that separates these two aspects of dynamics. The framework compares single-particle and multi-particle non-Gaussian parameters αsp2 and αmp2 to infer the accessible length-scale range of diffusivity heterogeneity, and introduces an overlap-based cross-covariance quantity, χ_cross, to quantify inter-particle dynamic correlation. Brownian-dynamics simulations validate the expected behavior of these estimators under simplified heterogeneous-diffusivity and correlated-motion models. The framework is then applied to two-component tetra-PEG hydrogels with symmetric and asymmetric stoichiometric ratios. In post-gel samples, αmp2 > 0 while αsp2 ≈ 0, indicating dynamic heterogeneity on length scales larger than the single-trajectory span and within the experimental field of view, on the order of tens of microns. The two gel systems exhibit distinct dynamical signatures: in the post-gel regime, the symmetric-ratio gel shows slightly smaller αmp2 and little detectable cross-correlation, whereas the asymmetric-ratio gel shows slightly larger αmp2 and significant cross-correlation. These results demonstrate that dynamic heterogeneity and dynamic correlation are not equivalent observables in particle-tracking experiments, and that their combined analysis provides a more resolved description of gelling complex fluids.