## 2 Materials and Methods  
For the present experiments, high-purity potassium di-phosphate KH\({}_{2}\)PO\({}_{4}\) (KDP) feedstock material was obtained from Sigma Aldrich (ACS reagent, \(\geq\)99.0%, CAS Number: 7778-77-0, Sigma Aldrich, St. Louis, MO, USA) (Certain commercial products are identified in this paper to specify the materials used and the procedures employed. In no case does such identification imply endorsement or recommendation by the National Institute of Standards and Technology, nor does it indicate that the products are necessarily the best available for the purpose.). The as-received powder was milled dry without a solvent using a high energy planetary mill. While the resulting powder retains a small fraction of coarse macroscale particles within the mix, most of the powder particle volume is sub-micrometer in size. The fine particle size, combined with the large volume fraction of water mixed with the powder prior to cold sintering (20% mass, 46.8% volume of deionized water) results in a smooth CSP densification process, with water or water vapor progressively released from each end of the sample capillary as densification progresses. This feedstock differs from that used in our earlier studies (KDP powder catalog no. P285, \(>\)95% mass pure from Thermo Fisher Scientific Inc., Fair Lawn, NJ, USA, used as received for CSP tests with addition of deionized water at 5% mass). The previous material, used with a much smaller water addition, gave rise to a lower X-ray scattering background and allowed direct observation of X-ray scattering features associated with core-shell effects in fine pores between the KDP grains [17]. Here, the higher water fraction and associated scattering background [18], especially when increased by presence of dissolved or low-density KDP, precluded such direct observation but allowed for a better controlled and smoother CSP process. Core-shell effects could be accommodated in the model fitting to the scattering data by assuming a single time-dependent core shell thickness for all pores in the microstructure. Prior to each CSP measurement, deionized water was added to KDP powder at 20% mass to form a small sample pellet. The pellet was pushed into a 0.75 mm internal diameter quartz capillary, which was then mounted between two pistons within the customized, heated cold sintering stage. Using calibrated heating coils and thermocouple, motor and load cell, force was applied to one of the pistons to generate controlled uniaxial compressive stresses at pre-set temperatures. Load cell accuracy was rated as within \(\pm\)1 N (\(\pm\)2.3 MPa uniaxial stress) and the temperature was calibrated to within \(\pm\)1 \({}^{\circ}\)C. Figure 1 shows the visual change in the sample during CSP under two different uniaxial stress and temperature conditions. The time dependence of the piston movement is also shown in each case. Following a rapid initial powder consolidation step, a regime of linear densification is revealed, followed by an extended period of slow incremental densification, associated with forcing residual porosity and aqueous fluid out of the system. The samples were found to be well-sintered at the end of each run, despite the different time scales involved. Figure 2 shows schematically the arrangement of the CSP cell in the X-ray beam for X-ray scattering and diffraction measurements. In situ X-ray studies of CSP were carried out at the USAXS facility, sector 9-ID of the Advanced Photon Source (APS), Argonne National Laboratory, Argonne, IL. As described elsewhere [15,16], complete USAXS, SAXS and WAXS (XRD) measurements can be made in less than 5 min at this facility without disturbing the sample geometry. The scattering and diffracted intensity data, I(q), were measured as a function of q = (4\(\pi\)/\(\lambda\)) sin\(\theta\), where \(\lambda\) is the X-ray wavelength (0.05904 nm for the 21 keV X-rays used here) and \(\theta\) is one half of the scattering or diffraction angle.