A Gram–Charlier Analysis of Scattering to Describe Nonideal Polymer Conformations

Theories of interpreting polymer physics and rheology at the molecular level from experiments, including small-angle scattering, typically rely on the assumption that polymer chains possess a Gaussian configuration distribution. This assumption frequently fails to describe features of real polymer molecules both at equilibrium (when polymers have nonlinear topology or heterogeneous chemistry) and out of equilibrium (when they are subjected to nonlinear deformations). To better describe non-Gaussian polymer conformation distributions, we propose a moments analysis based on the Gram-Charlier expansion as a natural framework for describing structure and scattering from non-Gaussian polymers. The expansion describes the conformation distribution in terms of cumulants (equivalent to moments of the distribution) of the underlying segment density distribution function, providing low-dimensional descriptors that can be inferred directly from measured scattering in a way that is agnostic to a polymer's topology, chemistry, or state of deformation. We use this framework to show that cumulants can be used to "fingerprint" non-Gaussian conformation distributions of polymers either at equilibrium (applied to sequence-defined heteropolymers) or out of equilibrium (applied to polymers experiencing nonlinear deformation due to flow). We anticipate that this new analysis method will provide a general framework for examining nonideal polymer configurations and the properties that arise from them.