Page 57 - ILL Annual Report 2019

SCIENTIFIC HIGHLIGHTS
54-55
This approach directly addresses an issue as fundamental as the choice of the most grounded model for the time- dependence of ISFs, based on a probabilistic inference of the number and form of individual decay processes contributing to it.
The analysis presented does not resort to a preselected interpretative framework and can thus be considered model-free. In fact, the reversible jump technique
allows, by switching between different dimensions in
the model hyperparameter space, one to choose from
a large class of models—i.e. any model that might be considered a particular case of a sum of stretched or simple exponentials or a combination of them, here including the Rouse and Zimm models—and to treat the number k of decay channels as a parameter itself to be estimated conditionally on the available data. As a result, comparison of the ISFs of either PEG-coated or PEG2000 nanoparticle suspensions enabled us to disentangle the translational diffusion of the nanoparticles from the internal dynamics of the polymer grafted to them.
Our findings also indicate that the relaxation of the polymeric corona takes the form of a simple exponential decay—as it should once this particular dynamics
is singled out—and is still perfectly consistent with a
Figure 2
An example of a posterior distribution for the number k of possible relaxation channel decays of PEG2000 polymer solution in D2O, as derived by measurements on IN11 and IN15. The maximum at k = 2 of the P(k|y) immediately suggests a two-exponentials model for the decays of the relative ISFs.
Figure 3
The three faces of Bayes (© Slackprop.Wordpress).
Rouse description of a tethered polymer brush. Most importantly, this inferential method clearly favours a multi-exponential profile over more involved ansätze based on the Kohlrausch–Williams–Watts distribution
of relaxation times. The latter modelling, often argued
to be the most appropriate for complex materials, can
at best provide an approximated phenomenological description of ISFs, even when supported by theoretical predictions. Its use, besides being often poorly justified by experimental evidence, leads to relevant physical insight derived from ISFs being overlooked while delivering confusing and fuzzy portrayals of relaxation dynamics. Conversely, implementing a multi-exponential model into the Bayesian inference approach gives a better account of the dynamical complexity of the system under scrutiny by unveiling the fine structure of ISFs without the risk of over-parameterisation, since the Bayes theorem naturally integrates an elegant and effective implementation of the Occam’s razor principle. This Bayesian approach not only provides a quantitative evaluation of the statistically most supported model (figure 2), it also draws an entire distribution for each model parameter (figure 3). Instead of delivering apodictic or conclusive statements on the physics under study, Bayesian methods assist investigators in the search of an evidence-based, probabilistic interpretation of experimental data.
Figure 3
Joint posterior distribution at four selected momentum transfer Q values, and marginalised with respect to the other model parameters, for the two relaxation times in the decay model recognised as the most statistically supported by the experimental data for a PEG2000 AuNP water solution at T = 318 K and λ = 10 Å. From the shape of such a joint distribution the absence of correlation between the two parameters and how the precision in the determination of the parameters changes with Q can be observed.
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