A Gram–Charlier\nAnalysis of Scattering to Describe\nNonideal Polymer Conformations

Theories of interpreting polymer physics and rheology\nat the molecular\nlevel from experiments, including small-angle scattering, typically\nrely on the assumption that polymer chains possess a Gaussian configuration\ndistribution. This assumption frequently fails to describe features\nof real polymer molecules both at equilibrium (when polymers have\nnonlinear topology or heterogeneous chemistry) and out of equilibrium\n(when they are subjected to nonlinear deformations). To better describe\nnon-Gaussian polymer conformation distributions, we propose a moments\nanalysis based on the Gram–Charlier expansion as a natural\nframework for describing structure and scattering from non-Gaussian\npolymers. The expansion describes the conformation distribution in\nterms of cumulants (equivalent to moments of the distribution) of\nthe underlying segment density distribution function, providing low-dimensional\ndescriptors that can be inferred directly from measured scattering\nin a way that is agnostic to a polymer’s topology, chemistry,\nor state of deformation. We use this framework to show that cumulants\ncan be used to “fingerprint” non-Gaussian conformation\ndistributions of polymers either at equilibrium (applied to sequence-defined\nheteropolymers) or out of equilibrium (applied to polymers experiencing\nnonlinear deformation due to flow). We anticipate that this new analysis\nmethod will provide a general framework for examining nonideal polymer\nconfigurations and the properties that arise from them.