Extracting interface correlations from the pair distribution function of composite materials

Using a non-negative matrix factorisation (NMF) approach, we show how the pair distribution function (PDF) of complex mixtures can be deconvolved into the contributions from the individual phase components and also the interface between phases. Our focus is on the model system Fe∥Fe3O4. We establish proof-of-concept using idealised PDF data generated from established theory-driven models of the Fe∥Fe3O4 interface. Using X-ray total scattering measurements for corroded Fe samples, and employing our newly-developed NMF analysis, we extract the experimental interface PDF (‘iPDF’) for this same system. We find excellent agreement between theory and experiment. The implications of our results in the broader context of interface characterisation for complex functional materials are discussed.


Corrosion of Fe
10.0 ml of distilled H 2 O was added to 5.0 g of Fe powder in a Petri dish and the sample was then left under ambient conditions to dry before being mixed. This cycle was repeated so that a total of 30.0 ml of H 2 O was added each week for a total of ten weeks, with a small fraction removed each week for PDF analysis. In addition a separate 5.0 g of Fe was prepared every week to ensure that a backup sample was available in case of problems with the first Fe batch over the corrosion process, but was never used. The corrosion process was repeated in an entirely separate experiment following the same experimental procedure, with the only major difference being the experiment was carried out over a shorter length of time (2 weeks compared to 10 weeks). We measured X-ray total scattering data for both controlled oxidation experiments and the results from the first experiment were reproduced closely. The iPDFs determined from the two experiments are shown in Fig. S1. Figure S1: Comparison of the NMF-derived iPDFs two separate experimental studies of controlled decomposition of Fe. Note that the higher resolution of the iPDF from experiment 1 is because X-ray PDF data were measured using an Ag anode for experiment 1 and a Mo anode for experiment 2.

Total Scattering Measurements
X-ray total scattering patterns were measured using a PANalytical Empyrean X-ray diffractometer fitted with an Ag anode (Q max = 20Å −1 ), a capillary spinner sample stage and a GaliPIX3D detector. These data were processed using GUDRUNX S4, S5 in order to correct for background scattering, Compton scattering, multiple scattering and beam attenuation by the sample container. The resulting X-ray total scattering functions were transformed to PDFs; again we use the normalisation referred to as G (r) in Ref. S5. For comparison, samples from the second Fe corrosion experiment were measured using the same diffractometer fitted with a Mo Anode (Q max = 17Å −1 ). All other instrument and data processing variables were kept as similar as possible to the first Fe corrosion. Figure S2: F (Q) data from X-ray diffraction measurements from the first controlled decomposition of Fe. Grey to orange colour gradient corresponds to increasing volume of water added. Figure S3: X-ray PDFs from the first controlled decomposition of Fe; grey to orange colour gradient corresponds to increasing volume of water added. The same data are presented in  Figure S4: X-ray PDFs from the second controlled decomposition of Fe; grey to orange colour gradient corresponds to increasing volume of water added. Note the lower resolution here is because a Mo anode (Q max = 17Å −1 ) was used compared to an Ag anode in experiment 1.
In order to determine which iron oxide phases were present in partially-decomposed Fe, the PDF of the sample was compared with those measured from pure

Scanning Electron Microscopy
Powdered samples of Fe were scattered onto Cu tape affixed onto aluminium 12.5 mm stubs.
Carbon was evaporated onto the surface of the prepared stubs to a thickness of 7.5 nm; this is to suppress charging caused by the electron beam. Samples were observed using a Zeiss Merlin scanning electron microscope, at an accelerating voltage of 3 kV and probe current 100 pA. energy-dispersive X-ray analysis was carried out under the same conditions, using an Oxford instruments X-max 80 mm detector and Aztec software version 3.0.

2 Computational Modelling of Fe||Fe 3 O 4 Model Building
To study the Fe||Fe 3 O 4 interface we built atomistic models with varying Fe/Fe 3 O 4 fractions and layer thicknesses. We used the DFT-relaxed structures of Ref. S1 and the Fe 3 O 4 unit cell length in the [001] direction was reduced (from 8.396Å to 8.108Å) in order to align with the Fe lattice parameter and guarantee structural registry. We also built atomistic models by aligning the cell dimensions the other way (i.e. stretching the Fe unit cell to fit Fe 3 O 4 ). The iPDF from NMF of X-ray PDFs from these models agree less well with the experimental iPDF. The models described in the main text are outlined in Table S1; the volumes of the various models are different because the layer thickness of Fe and Fe 3 O 4 are different. Figure S6 shows the structure at the Fe||Fe 3 O 4 interface in the atomistic models. Figure   S6 Figure S6(c).

Pawley refinements of X-ray diffraction patterns
Pawley refinements were carried out using TOPAS Academic (version 4.1). S3 The measured scattering patterns of corroded Fe samples were fitted with the diffraction patterns of Fe and Fe 3 O 4 present in the sample using Pawley refinement. Scale factors and the crystallite size were also refined. A representative Pawley refinement is illustrated in large format in Fig. S8, and the corresponding fits for all the room-temperature powder diffraction data for the corroded Fe samples are shown in Fig. S9. The key point here is that the experimental reciprocal space data can be reasonably well accounted for in terms of a conventional two-phase refinement, in spite of the fact that we know the interface contribution to be significant.

Non-Negative Matrix Factorisation
We use the Metropolis Monte Carlo NMF implementation developed in Ref. S6  In this study, the task for NMF was to identify the three fundamental components G * i (r) (i = 1, 2, 3) and weights w ij (j = 1, 2, . . . , 9 for synthetic data; j = 1, 2, . . . , 12 for experimental PDF data) so as to minimise |G calc (r are the elements of G calc (r). We applied the additional constraints that G * i (r) is positive for all i and r, and that 3 i=1 w ij G * i (r) = 1 for all j.

Simulated PDF Data
In the main text we use NMF to deconvolve our synthetic PDF data into 2 components, and

Experimental PDF data
In the main text we use NMF to deconvolve experimental X-ray PDF data of Fe that has undergone controlled decomposition into 3 components; the PDFs of Fe and Fe 3 O 4 were fixed during the NMF. The NMF fits for the results presented in the main text are shown in Fig S14. To test our Metropolis Monte Carlo NMF implementation, we used a similar approach to the one described in the main text but the PDFs of Fe and Fe 3 O 4 were not fixed. The results of NMF with only the PDF of Fe fixed (i.e. Fe 3 O 4 PDF allowed to vary from random initial values), and with no components fixed are given in Fig S15 and Fig S16 respectively. In both cases the iPDFs derived have similar features to the experimental iPDF from the main text with slight changes in relative peak intensities. There are also small changes in the NMF weights for the three different NMF analyses.

Reciprocal space data from Fe||Fe 3 O 4 models
While our focus is on the interpretation of X-ray total scattering data in terms of its real-space transforms, we include for completeness an additional NMF analysis here that uses the reciprocal space data. Our starting point was to calculate the total scattering function for the DFT-driven synthetic models. We used a variant of the total scattering function which we call

Experimental reciprocal space data
We used the same approach to carry out a three-component NMF refinement of the experimental total scattering data, normalised in the same way as described above. The corresponding interface F (Q) function contains a mixture of Bragglike and diffuse features [ Fig. S19(a)], as expected, and the evolution of different phase components tracks oxidation in qualitatively the same way as determined using the real-space transforms: cf Fig. S19(b) and Fig. 3(b) of the main text. Note that we do not expect quantitative agreement here in phase fractions, since the meaning of the iPDF is linked to the real-space range over which the NMF analysis is carried out.