A Time-Dependent Hierarchical Model for Elastic and Inelastic Scattering Data Analysis of Aerogels and Similar Soft Materials

Soft nanomaterials like aerogels are subject to thermal fluctuations, so that their structure randomly fluctuates with time. Neutron elastic and inelastic scattering experiments provide unique structural and dynamic information on such systems with nanometer and nanosecond resolution. The data, however, come in the form of space- and time-correlation functions, and models are required to convert them into time-dependent structures. We present here a general time-dependent stochastic model of hierarchical structures, with scale-invariant fractals as a particular case, which enables one to jointly analyze elastic and inelastic scattering data. In order to describe thermal fluctuations, the model builds on time-dependent generalisations of the Boolean model of penetrable spheres, whereby each sphere is allowed to move either ballistically or diffusively. Analytical expressions are obtained for the correlation functions, which can be used for data fitting. The model is then used to jointly analyze previously published small-angle neutron scattering (SANS) and neutron spin-echo (NSE) data measured on silica aerogels. In addition to structural differences, the approach provides insight into the different scale-dependent mobility of the aggregates that make up the aerogels, in relation with their different connectivities.