## 2 Materials and Methods
### Neutron Far-Field Interferometry Imaging at NIST  
In the first application of neutron far-field interferometry to batteries, neutron grating-based interferometry imaging was done at the NG6 cold neutron imaging beamline using the same set of Li-po batteries (NCNR NIST, Gaithersburg) [35]. Batteries were charged, imaged, discharged, and imaged as described in Section 2.1.  
The phase gratings, G1 and G2, were fabricated at the NIST Center for Nanoscale Science and Technology on 0.5 mm thick, 10 cm diameter Si wafers. The linear features of the phase gratings (a rectangular cross-section with a duty cycle 50%) were fabricated with a period of 2.4 \(\upmu\)m, corresponding to a neutron phase shift of \(\pi/2\) at a wavelength of 0.54 nm. The intergrating distance, (\(D\)), between the G2 and G1 gratings was varied between 7 mm and 35 mm with a linear translation stage. The G2-to-detector distance (\(L_{2}\)) was 4.355 m as was the sum of the slit-to-G1 distance (\(L_{1}\)) and intergrating distance (\(D\)), hence the interferometer was in a symmetric geometry (Figure 1). The interferometer was operated in \(D\)-scan mode, where a fixed sample position was used and the G1-G2 distance was varied in 1 mm increments. The autocorrelation lengths, \(\xi\), probed for this neutron experiment were 788 nm to 3.94 \(\upmu\)m based on (\(\lambda\) = 0.54 nm), \(D\) ranging from 7 mm to 35 mm, and \(z/(L_{1}+D+L_{2})\approx 1/2\). The beam defining slits were set to approximately 0.5 mm horizontal and 50 mm vertical which resulted in the images being blurred primarily along the vertical direction. To reduce sharp vertical guide features, a pellet of silicon powder, 6 mm thick, was placed directly downstream of the aperture. The interferometry image sharpness, as defined by the geometric \(L/D\) ratio, is excellent along the horizontal direction with \((L_{1}+D+L_{2})/(\text{slit horizontal})\approx 17,000\) while only 170 along the vertical direction. Recently, deep learning methods have been evaluated to improve image resolution [37].  
The sample was mounted on a rotation stage and positioned 5 cm downstream of G2. This position yields an optical magnification \(\approx\) 2. A 150 \(\upmu\)m thick \({}^{6}\)LiF/ZnS scintillator was imaged with an Andor NEO sCMOS detector optically coupled with Nikkor 50 mm lens (F-stop set to 1.2). The detector was 2560 pixels \(\times\) 2160 pixels and each pixel was 6.5 \(\upmu\)m \(\times\) 6.5 \(\upmu\)m; the lens had a reproduction ratio approximately 7.9 giving effective 51.4 \(\upmu\)m pixels. A geometric magnification of 2 gives an effective pixel size of 25.7 \(\upmu\)m pixels. Median (radius of 2 pixels) and mean (radius of 3 pixels) filtering were used to remove gamma streaks from the images.  
The interferometer was operated in stepped-grating mode, typically with 9 or 12 step positions evenly spaced over 3 \(\upmu\)m. Two software packages were used for reduction of the interferograms to attenuation, differential phase contrast, and dark-field projections. The NIH software was used for initial processing at the beamline [38]. The LSU software was used for production work off-line [36; 39]. The small angle scattering signal was extracted from the dark-field images as a function of the \(D\)-scans and analyzed for the autocorrelation lengths, \(\xi\).