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Neutron reflectometry is the foremost technique for in situ determination of the volume fraction profiles of polymer brushes at planar interfaces. However, the subtle features in the reflectometry data produced by these diffuse interfaces challenge data interpretation. Historically, data analyses have used least-squares approaches that do not adequately quantify the uncertainty of the modeled profile and ignore the possibility of other structures that also match the collected data (multimodality). Here, a Bayesian statistical approach is used that permits the structural uncertainty and multimodality to be quantified for polymer brush systems. A free-form model is used to describe the volume fraction profile, minimizing assumptions regarding brush structure, while only allowing physically reasonable profiles to be produced. The model allows the total volume of polymer and the profile monotonicity to be constrained. The rigor of the approach is demonstrated via a round-trip analysis of a simulated system, before it is applied to real data examining the well characterized collapse of a thermoresponsive brush. It is shown that, while failure to constrain the interfacial volume and consider multimodality may result in erroneous structures being derived, carefully constraining the model allows for robust determination of polymer brush compositional profiles. This work highlights that an appropriate combination of flexibility and constraint must be used with polymer brush systems to ensure the veracity of the analysis. The code used in this analysis is provided, enabling the reproduction of the results and the application of the method to similar problems.

The refnx Python modules for neutron and X-ray reflectometry data analysis are introduced. A sample analysis illustrates a Bayesian approach using a Markov-chain Monte Carlo algorithm to understand the confidence in the fit parameters., refnx is a model-based neutron and X-ray reflectometry data analysis package written in Python. It is cross platform and has been tested on Linux, macOS and Windows. Its graphical user interface is browser based, through a Jupyter notebook. Model construction is modular, being composed from a series of components that each describe a subset of the interface, parameterized in terms of physically relevant parameters (volume fraction of a polymer, lipid area per molecule etc.). The model and data are used to create an objective, which is used to calculate the residuals, log-likelihood and log-prior probabilities of the system. Objectives are combined to perform co-refinement of multiple data sets and mixed-area models. Prior knowledge of parameter values is encoded as probability distribution functions or bounds on all parameters in the system. Additional prior probability terms can be defined for sets of components, over and above those available from the parameters alone. Algebraic parameter constraints are available. The software offers a choice of fitting approaches, including least-squares (global and gradient-based optimizers) and a Bayesian approach using a Markov-chain Monte Carlo algorithm to investigate the posterior distribution of the model parameters. The Bayesian approach is useful for examining parameter covariances, model selection and variability in the resulting scattering length density profiles. The package is designed to facilitate reproducible research; its use in Jupyter notebooks, and subsequent distribution of those notebooks as supporting information, permits straightforward reproduction of analyses.