## 2 Materials and Methods
Constraining the core-shell thickness to be the same for (i) and (ii) differs from the assumption made in our earlier study [17], necessitated by the higher background scattering masking the more complete observation of core-shell features possible previously. The 3 rd component assumed to be present is also low-density high-defect KDP, dissolved and re-precipitated between sintered KDP powder grains. This component is assumed to be nanoscale in thickness, no  
Figure 3: (a) In situ slit-smeared USAXS/SAXS line-trace data versus q for KDP CSP using p = 125 MPa at 60 \({}^{\circ}\)C. Point-to-point data uncertainties are indicated by scatter in the line traces. (b) Corresponding in situ WAXS (XRD) data with selected XRD peaks indexed. longer associated necessarily with porosity, hence likely to be water-free. All 3 of these components were needed to fit the data. For convenience, all 3 components were assumed to have a lognormal volume-fraction size distribution (a commonly used, analytically accessible, size distribution assumption in small-angle scattering); the pore populations, (i) and (ii), were assumed globular in nature with a core-shell structure (water-filled pore body, low-density KDP around the perimeter), and the standalone low-density KDP was assumed cylindrical in nature with both needle-like or sheet-like shapes allowed during fitting. Table 1 summarizes these morphology assumptions with the X-ray scattering length (atomic form-factor) density (SLD) derived from published X-ray form-factor tables [21]. The X-ray SLD values shown in Table 1 assume the medium outside of all scattering features to be fully dense KDP, and the pores (both (i) and (ii)) to have water-filled cores. The solid shell around each pore is assumed to have an SLD value mid-way between water and fully dense KDP, based on the shell phase having many vacant defects potentially occupied by H\({}_{2}\)O molecules. The low-density KDP phase (iii), not associated with pores, is assumed to be 50% dense KDP, with vacant defects not occupied by H\({}_{2}\)O molecules. Some evidence for the formation of a low-density high-defect solid phase during CSP has been revealed in transmission electron microscopy (TEM) studies of various CSP systems [22]. Here, while inevitably over-simplifying the true situation somewhat, these overall assumptions provide a caliper for quantifying changes in the evolving pore morphology during CSP. Typical 3-component fits to the combined USAXS/SAXS data are shown in Figure S1 of the Supplementary Materials. These are shown both for the slit-smeared data and, equivalently, after desmearing the data, which amplifies the statistical scatter. Figure S2 of the Supplementary Materials shows how each component typically contributes to the overall model fit, and also presents the 3 lognormal size distribution components, themselves, from which the results presented below have been derived. The applicability of using lognormal size distribution analysis to small angle scattering measurements is well established, e.g., [23]. Fits to the smoother slit-smeared data have been conducted for the remainder of this study. The in situ slit-smeared USAXS/SAXS datasets, used for model fitting, are presented in Figures S3-S5 of the Supplementary Materials. Finally, scanning electron microscopy (SEM) was conducted on densified samples following CSP under the two conditions presented in Figure 1 (50 MPa, 60 \({}^{\circ}\)C, 100 min; and 150 MPa, 60 \({}^{\circ}\)C, 10 min). SEM was performed on both the external (uncut) surfaces and fracture surfaces of the densified KDP samples using a JEOL 7100-FT FE-SEM (JEOL Peabody, MA, USA). The main objective of the SEM studies was to evaluate the degree of grain growth occurring during CSP.