Metastable Hexagonal Close-Packed Palladium Hydride in Liquid Cell Transmission Electron Microscopy

Metastable phases—kinetically favored structures—are ubiquitous in nature. Rather than forming thermodynamically stable ground-state structures, crystals grown from high-energy precursors often initially adopt metastable structures depending on the initial conditions such as temperature, pressure, or crystal size. As the crystals grow further, they typically undergo a series of transformations from metastable to lower-energy phases and ultimately energetically stable phases, as described by Wilhelm Ostwald. Metastable phases, however, sometimes provide superior chemical and physical properties, and hence, discovering and synthesizing novel metastable phases are promising avenues for achieving innovations in materials science. The most common strategy for synthesizing a metastable material involves manipulating thermodynamic conditions such as temperature and pressure during the course of synthesis. However, the search for metastable materials has mainly been heuristic, on the basis of experiences, intuitions, or even speculative predictions. This limitation necessitates the advent of a new paradigm to discover novel metastable phases, i.e., by “rational design and synthesis” instead of a “rule of thumb,” and based on ab initio methods, i.e., the calculation of thermodynamic and kinetic properties of materials with various compositions, crystal structures, and crystal sizes. The design rule is embodied in the discovery of a metastable hexagonal close-packed (HCP) palladium hydride (PdH x ), synthesized in a liquid cell environment using a transmission electron microscope (TEM). The metastable HCP structure is stabilized through a unique interplay among the precursor concentrations in the solution: a sucient supply of hydrogen (H) favors the HCP structure on the sub-nanometer scale, and an insucient supply of Pd inhibits further growth and subsequent transition toward the thermodynamically stable face-centered cubic (FCC) structure. The crystal structure was modulated (HCP or FCC) by adjusting the H concentration inside the TEM liquid cell, providing strong evidence for the crucial role of the H concentration. Monte Carlo simulations reveal that an unexpected inhomogeneous distribution of interstitial H atoms, distinct from the predominant occupation at the octahedral interstitial sites, is key to stabilizing

of Pd inhibits further growth and subsequent transition toward the thermodynamically stable facecentered cubic (FCC) structure. The crystal structure was modulated (HCP or FCC) by adjusting the H concentration inside the TEM liquid cell, providing strong evidence for the crucial role of the H concentration. Monte Carlo simulations reveal that an unexpected inhomogeneous distribution of interstitial H atoms, distinct from the predominant occupation at the octahedral interstitial sites, is key to stabilizing nanoscale HCP PdH x . Furthermore, insu cient Pd brings a multi-step nucleation and growth pathway, deduced from in situ liquid cell TEM combined with atomic electron tomography, which maintains the metastable phase intact. These ndings provide new thermodynamic insights into metastability-engineering strategy to be deployed in discovering new metastable phases.

Main Text
The advent of liquid cell transmission electron microscopy (TEM) enables the direct imaging and growth of materials in liquids. The electron beam generated inside the microscope transfers high energy to the irradiated suspending liquid; water molecules decompose into radicals and molecules including hydrated electrons (e h -), hydroxyl radicals (OH · ), hydrogen radicals (H · ), and hydrogen molecules (H 2 ). This radiation chemistry of water results in distinct phenomena in liquid cell TEM. Examples include nanocrystal growth via the reaction of precursors and e h -, the most reactive radiolysis product, and bubbles formed by a high H 2 concentration.
Face-centered cubic (FCC) palladium hydride (PdH x ) constitutes the most important model system for fundamental studies of H in metals 6,7 . Generally, the PdH x phase can form upon the adsorption and dissociation of H 2 on the Pd surface followed by surface, subsurface, and interstitial site occupations of hydrogen atoms 6 . Similar to metallic Pd, the PdH x phase has a FCC structure, where the octahedral (O) interstitial sites are occupied by H atoms, providing two distinct phases: a low-H-concentration a-phase and a high-H-concentration b-phase (x £ 0.7) 6 .
In this study, we synthesized metastable hexagonal close-packed (HCP) PdH x nanoparticles via liquid-cell TEM by employing a graphene liquid cell (GLC) 8 . Speci cally, a Pd precursor solution was encapsulated between two multilayer graphene layers, and e h produced by the electron beam initiated the nucleation and growth of nanoparticles (Movies S1 and S2) 9 , as observed by TEM (Methods). Owing to the free rotation of the nanoparticles in the liquid 10 , diffraction patterns (DPs) with distinct zone axes can be acquired to determine their crystal structures.
Remarkably, DPs obtained from in situ TEM (Figs. 1a, b) and ex situ high-resolution scanning transmission electron microscopy (HR-STEM) (Fig. 1e, Methods) images of the synthesized nanoparticles corresponded to an HCP lattice rather than an FCC lattice, the structure of almost every Pd compound ever reported [11][12][13] , regardless of its growth pathway, i.e., monomer attachment (particle 1, Figs. 1a, c) or To verify the MC-derived formation mechanism of HCP PdH x , we synthesized PdH x nanoparticles by modulating the H concentration via tuning the electron dose rate (EDR) and adopting a different liquid cell system for TEM, a Si 3 N 4 liquid cell (SLC) (Fig. 3). These experiments suggest three distinctive trends.
First, all of the observed nanoparticles exhibit the HCP structure at high EDRs (> 5 × 10 3 e -/Å 2 s) in both liquid cells (Figs. 3a, b), which indicates the effect of pressure in GLC due to van der Waals forces 15,16 and the preferential formation of the HCP structure by epitaxial growth on top of the hexagonal graphene window material 17 can be excluded.
Second, at low EDR (< 5 × 10 2 e -/Å 2 s), both FCC (~30%) and HCP (~70%) particles were observed in GLC ( Fig. 3c), whereas 100% of the nanoparticles observed in SLC were FCC (Fig. 3d), implying that the H concentration plays a crucial role in HCP formation, as suggested by MC simulations. The production of FCC particles around hydrogen bubbles (which form when the radiolysis products exceed their critical concentration 9 ) in GLC under the higher EDR is another piece of evidence for the key role of H in the formation of the HCP structure (Extended Data Figs. 3a, b). These bubbles remove supersaturated H, resulting in a condition favoring the formation of FCC PdH x .
The distinction between GLC and SLC may have been due to their different electron absorption rates (EARs), which are in turn due to their different components with different thicknesses, namely, graphene (~1 nm) and Si 3 N 4 (50 nm) windows. The measurement of the transmitted electron intensity indicated an approximately 10% lower EAR for the GLC than that for the SLC, which can result in a higher H concentration that facilitates the formation of HCP. In addition, Pd atoms adsorbed on the graphene surface can promote the production of atomic H and catalyze Pd-H bonding, further facilitating the formation of HCP PdH x in GLC 18 .
Third, the unit cell volumes of the HCP PdH x nanoparticles at high EDRs were greater than those of the HCP nanoparticles at low EDRs, presumably owing to the higher H concentration (Figs. 3e-g). The H content x in HCP PdH x was estimated by comparing the volume per Pd atom derived from TEM with that from the DFT calculations (Extended Data Fig. 4a). The approximate H content increases with EDR and approaches x ~ 1 at a high EDR, suggesting equivalenced PdH formation. In contrast, the maximum estimated H content in FCC nanoparticles at a low EDR is x ~ 0.2 (Extended Data Fig. 4b).
We also produced nanoparticles without using the TEM by instead employing a newly designed system for electron-beam-induced synthesis (Extended Data Figs. 3c-e, Methods). All of these particles exhibited an FCC structure with lattice constants slightly larger than those of the standard FCC Pd (Extended Data Fig. 3d), with an x value of only approximately 0.002, which may have been due to a low H supply as a consequence of low EDRs. With this information in hand, we conclude that the HCP PdH x formation process is highly dependent on the anomalous H supply originating from radiolysis.
An abundant supply of H may also modulate the growth kinetics. According to DFT-calculations, adding another layer of Pd onto the (111) surface of FCC PdH requires more energy than adding onto the (0001) surface of HCP PdH. This higher energy cost is compensated by a more exothermic reaction with H atoms in the FCC structure (Extended Data Fig. 5). This result can be interpreted as the passivation of FCC PdH particles by H radicals, which in turn stabilizes the surface and disfavors the further incorporation of Pd, resulting in slower growth. In addition, more rapidly growing HCP particles would quickly deplete the Pd precursors, further suppressing the growth of FCC particles.
As a metastable solid phase initially forms, it dissolves and is replaced by a more stable phase via several thermodynamically permissible pathways until the most stable phase is formed, as described by Wilhelm Ostwald 3 . Hence, a remaining question is how an HCP nanoparticle maintains its crystal structure instead of transforming into FCC, even when it enters a size regime that is unstable for HCP as it grows. Generally, the synthesis of a nanocrystal in an aqueous solution is based on there being a su cient supply of precursors to facilitate the formation of a thermodynamically stable phase.
In contrast, in liquid cell TEM, the initially grown metastable phases form a region of supersaturated precursors with much higher concentrations than those of these precursors in the as-made solutions 19 ; at the same time, the initially grown metastable phases are rapidly depleting the precursors in the surrounding area 20 , a process that can delay further growth and transformation into the stable phase, eventually leaving the metastable phase intact. We heated PdH x nanoparticle at 500 °C for 2 h in an in situ heating system (Methods, Extended Data Fig. 6) to induce phase transformation. Remarkably, the HCP nanoparticle held its structure after annealing, indicating that HCP PdH x is indeed a metastable phase and that the energy barrier between this phase and the stable phase is quite high.
To gain deeper insights into the growth mechanism of HCP PdH x , the detailed three-dimensional (3D) atomic arrangement of a PdH x nanoparticle was investigated by performing atomic electron tomography (AET) 21-23 . A STEM tomography dataset was acquired at 33 different tilt angles ranging from -70° to 70°, and an atomic-resolution 3D tomogram was obtained using the GENFIRE algorithm (Methods) 24 . Individual atoms were clearly resolved in the reconstruction as local intensity maxima (Figs. 4a-c). The full 3D coordinates of individual atoms were traced by 3D Gaussian tting at each local maximum, resulting in a 3D atomic model of the nanoparticle (Methods). To verify whether the nanoparticle indeed forms an HCP structure, local hexagonal stacking types were determined for each atom. As shown in Fig.  4d, each atom in an HCP crystal can have two different types of nearest neighbor arrangements, namely, type A and type B. A perfect HCP crystal is expected to show alternating layers of type-A and type-B atoms. Interestingly, the measured nanoparticle was not a perfect HCP single crystal, but it exhibited a complex domain structure with different regions of ABAB and BABA stacking, as shown in Figs. 4a-c, e. Nevertheless, the 3D connectivity of these different types of domains (blue and red regions in Fig. 4e, exhibiting ABAB or BABA stacking, respectively) and their excellent agreement with the ideal HCP structure (Extended Data Fig. 7a) suggested that the nanoparticle indeed possessed HCP ordering.
Remarkably, the boundary between the two anti-phase HCP stacking domains (Fig. 4e) appears to be severely indented and irregularly shaped. This observation cannot be easily explained by coalescence events (aggregation or oriented attachment) because they usually result in sharp planar boundaries 8,10,[25][26][27] . Instead, the presence of a complex boundary structure (Fig. 4f) suggests that the nanoparticle probably formed as a result of a multi-step crystallization process that is more energetically favorable than direct crystallization 28-30 , as described in Extended Data Fig. 8h. Note that no diffraction peaks were observed at the early stage of growth (< ~1 s), indicating an amorphous phase, which further supports the multi-step crystallization mechanism (Extended Data Figs. 8a-g).
The stabilization of the sub-nanometer-sized particles of HCP PdH predicted by MC simulations corroborates the multi-step crystallization process suggested by the AET experiment. Overall, the experimentally resolved atomic structure shows a domain size of approximately 1-2 nm (Extended Data Fig. 7a), similar to the nucleus diameter of 1.2 nm that favors the formation of HCP PdH (Figs. 2b-c).
The transition to HCP crystal nuclei from the amorphous phase and subsequent growth of the crystalline parts via merging enables the formation of multi-domain HCP PdH nanoparticles, which can grow larger than about a nanometer, the size limit for HCP PdH single crystals being more stable than FCC PdH.
In this study, we discovered a new metastable HCP PdH x for the rst time. The unique environment in the liquid cell-namely, the limited quantity of Pd precursors and continuous supply of H-results in the formation of the HCP phase owing to the nonuniform H distribution, and the transformation to the stable FCC phase was inhibited because of the multi-step growth pathway. The metastable HCP PdH x exhibited a superior H capacity and stronger H binding compared with its FCC counterpart. This insight may provide a new route for designing and synthesizing novel energy storage materials. For example, in contrast with the conventional fabrication procedure based on a phase transformation driven by intercalation and dissolution, the procedure involving synthesizing materials for H storage or Li-ion battery electrodes under a high concentration of H or Li could result in a new metastable phase. Such synthesis of novel metastable phases by the proposed approach will be pursued in future work. 16 Yang, X., Li, H., Ahuja, R., Kang, T. & Luo, W. Formation and electronic properties of palladium hydrides and palladium-rhodium dihydride alloys under pressure. Sci. Rep. 7, 3520 (2017).

Methods
Precursor solution. The palladium precursor solution was prepared by dissolving sodium tetrachloropalladate(II) (Na 2 PdCl 4 , Aldrich) and isopropyl alcohol (Junsei) in distilled water 31 . The Pd concentration varied from 3.67 mM to 18 mM, but it did not affect the phase and molar volume of nucleated particles. Most experiments were performed with a 3.67 mM precursor solution. Isopropanol was used as a hydroxyl radical scavenger to minimize the probability of dissolving palladium nanoparticles. The molar concentration of isopropanol in the aqueous solution was 0.1 M 9,32 .
Liquid TEM cell. The graphene liquid cell (GLC) was fabricated following the previous reports 8 . Multilayer graphene, which was purchased from Graphene Square Incorporation, was adopted to increase durability of GLC under bombardment of electrons accelerated at a few hundred kilovolts. Silicon nitride liquid cell (SLC) experiments were conducted with a commercial in situ liquid holder, Protochips Poseidon 210, the window of which was 50 nm thick.

TEM. Electron-beam-induced PdH x nanoparticle growth in liquid was traced in real time with a Titan TEM
(FEI Titan 80-300) operated at 300 kV, which was equipped with a OneView CCD camera (Gatan) and a monochromator. We used this TEM to irradiate electrons on a liquid cell to generate particles for ex situ analysis, which was conducted after fully drying the liquid, i.e. by irradiating the liquid in the GLC for ~30 min or exposing the disassembled SLC to vacuum for 48 h. HR-TEM and HR-STEM images as well as raw images for AET of the particles generated in GLC under a high electron dose rate were acquired with a double Cs-corrected Titan Themis TEM (FEI) operated at 300 kV. HR-TEM images of the particles generated at lower dose rates or in SLCs were obtained using the aforementioned Titan TEM. Monochromated electron energy loss (EEL) spectra of the HCP particles formed inside the SLC were acquired using the Titan TEM operated at 80 kV 6,33 . Energy dispersive X-ray spectroscopy (EDS) analysis was conducted with a Talos TEM (FEI Talos F200X) operated at 200 kV, equipped with a Super-X EDS system that utilizes four silicon drift detectors (SDD).
XRD. X-ray diffraction (XRD) analysis was performed at room temperature in the 2θ range of 10-90° on a D8 Advance (Bruker AXS, Germany) diffractometer equipped with a LynxEye line detector using Cu-Kα radiation (λ = 1.5418 Å) at 40 kV and 40 mA and a scanning rate of 0.5 °/min with a step size of 0.02°.
In order to prove that the lattice of Pd/C was expanded relative to that of pristine Pd, an internal standard of Si powder (NIST SRM 640e) was used. The lattice parameters of Pd/C were re ned with the Le Bail method 34 using TOPAS software (Bruker AXS, Germany) 35 ; the lattice parameter of Si was xed to a = 5.431179 Å.
Lattice constant calculation. The lattice constants of the nanoparticles were calculated from the diffraction patterns of the HR-TEM and HR-STEM images of nanoparticles along the [1-21-3] and [-2110] zone axes. We used the crystallographic relationship formula of the HCP structure below to derive the lattice constants 36 .
The lattice constant a was directly calculated from the (1-100) planes, and the lattice constant c was calculated from the (1-101) planes and the aforementioned (1-100) planes for the particle along the [1-21-3] zone axis. As for the particle along the [-2110] zone axis, the lattice constants a and c were directly calculated from the (0-110) and (0002) planes, respectively. Notably, some particles had quite different lattice constants from the majority of particles. Speci cally, some particles had far higher c values, while others had far lower values. These differences were attributed to the deviation in the number of H atoms residing in the particles. As the HCP palladium hydride nanoparticles formed explosively, some particles might have been formed in out-of-average surroundings, i.e., some might have been exposed to an excess of H radicals, while others might have not.
In situ d-spacing tracking. The plane index of diffraction spot, and corresponding d-spacing, acquired when the particle was aligned along the zone axis was obvious, as plane information is manifested in the zone axis pattern. When the particle was not along the zone axis, the diffraction spot of the particle was assigned to a plane index considering the continuity of fringes, and the degree of proximity between the aforementioned d-spacing value in the video and average interplanar distance measured ex situ.
Bulk scale PdH x /C preparation through e-beam irradiation. To synthesize PdH x nanoparticles via irradiation with an electron beam in a batch reactor outside TEM, 1 L of 3.67 mM Na 2 PdCl 4 in a 0.1 M isopropyl alcohol aqueous solution was prepared. Then, carbon black was added to the reaction solution, which was homogeneously mixed under ultrasonication for 10 min at room temperature. Thereafter, the reaction vessel was placed under electron beam irradiation at a dose rate of 1.9 × 10 -3 e -/Å 2 s for several minutes. The resulting solution was ltered with distilled water several times, and black precipitates were obtained. Data acquisition. A tomographic tilt series of a PdH x nanoparticle was obtained using the Titan Themis TEM. In HAADF-STEM mode, 33 tilt series images of 1,024 × 1,024 pixels were obtained between -70.0°a nd +70.0° with a 35.50 pm pixel size. Ten images were consecutively acquired for each tilt angle with a 3 μs dwell time. A linear drift constant was estimated from consecutive images and corrected for each angle. Then, scan distortion correction was applied to correct the image distortions owing to slight misalignment between the x and y directions of the scan coil, based on a reference image of the Si (110) zone axis taken under the same measurement conditions. Image denoising. HAADF-STEM images suffer from Poisson-Gaussian noise. The Poisson and Gaussian noise parameters were estimated from the image statistics of the ten consecutively acquired images, and a sparsity-based transform domain denoising (BM3D) was applied, as described in Refs. 10-14. 21-24,37 GENFIRE reconstruction. After denoising, the projections were aligned based on the common line and center of mass methods for alignment along the tilt axis direction and perpendicular to the tilt axis direction, respectively 23,24,37-39 . Tomographic reconstruction was performed using GENFIRE for the aligned tilt-series images 23,24,37-40 with an oversampling ratio of 4 and interpolation radius of 0.1 pixel. To improve the quality of the reconstruction, an angular re nement process was applied to minimize the discrepancy between the forward-projected projections of the reconstruction volume and the experimental projections 40 . The nal reconstruction obtained after the re nement process showed a clear atomicresolution internal structure.
3D identi cation of atomic coordinates (tracing). The 3D atomic coordinates of all atoms in the nanoparticle were determined by tting a 3D Gaussian function on a 5 × 5 × 5 voxel volume near each local maximum. The ttings were performed in descending order from the highest intensity local maximum, and a minimum distance constraint of 2.4 Å was enforced during the process. However, owing to the atom elongation effect resulting from the missing wedge and slight imperfectness of the reconstruction, several connected intensity blobs were not properly traced owing to the failure to properly identify the local maxima. To locate the untraced atoms, the reconstruction volume was sliced along the hexagonal c axis at each atomic layer, and the local maxima in each slice were identi ed. Then, the same Gaussian tting procedure was repeated using the new local maxima in descending order, still enforcing the same minimum distance constraint. After this process, a nal 3D atomic model of 9,868 atoms was obtained.
Assignment of HCP lattice sites. To analyze the 3D atomic structure, proper HCP lattice sites must be assigned for each atom. To begin the assignment procedure, the HCP lattice vectors for the nanoparticle were roughly estimated from the peak positions in the 3D Fourier transformed reconstruction volume. Then, an atom closest to the mean position of the given 3D atomic coordinates was selected. The rst selected atom was assigned to the origin of the HCP lattice. From the starting atom, the positions of the 12 nearest neighbor (n.n.) atoms and corresponding n.n. HCP lattice sites were calculated. At each n.n. position, a sphere was drawn with a radius of ¼ n.n. distance. If an atom was encompassed by the sphere, the atom was assigned to the lattice site corresponding to the n.n. position. This process was continued for all newly assigned HCP lattice sites and repeated until there were no newly assigned sites.
Note that an atom in an HCP structure can have two different types of n.n. structure, as described in Fig.   4d. To determine which n.n. structure the starting atom belonged to, the process above was repeated twice for two different n.n. structures, and the one that resulted in a larger number of assigned HCP sites was chosen. After assigning HCP lattice sites for all available atoms, the HCP lattice vectors were tted to minimize the error between the measured atomic positions and the corresponding lattice positions of the tted HCP lattice. The entire process described above was continuously repeated with the newly obtained HCP lattice vectors until there the lattice vectors stopped changing.
Domain identi cation. For the initial trial, HCP lattice assignment and lattice parameter tting were performed assuming that the entire nanoparticle is an HCP single crystal, resulting in initial HCP lattice vectors. However, the nanoparticle is not guaranteed to form a single crystal. Therefore, local lattice assignments and ttings described above were individually performed for all the atoms within the nanoparticle to check their n.n. ordering. For the local tting, only the atoms within the distance of the mean of n.n. and next-nearest-neighbor (n.n.n.) distance, which was determined from the HCP tting of the entire nanoparticle, were used. From the local ttings, the local lattice constant and n.n. structure type (Fig. 4d, named as type A or type B here) can be obtained for each atom. After the local tting, all the atoms were assigned to the closest atomic layer perpendicular to the hexagonal c axis (Fig. 4e). Then, type-A atoms in odd-numbered layers and type-B atoms in even numbered layers were classi ed as ABAB stacking, and the opposite cases were classi ed as BABA stacking. As shown in Fig. 4e, atoms with the same stacking type form clusters.
To identify the connected clusters, we performed a local connectivity analysis. First, we selected a starting atom, around which we drew a sphere, with the mean of the n.n. and n.n.n. distance for the radius. Then, among the atoms within the sphere, we counted how many atoms share the same stacking type with the selected atom. If the number was larger than a certain threshold, the selected atom was considered to be within a connected domain, and a speci c domain identi er was assigned to it. In this case, we added atoms with the same stacking type as the selected atom to the potential domain atom list, and we repeated the process for all the newly added atoms in the list. For all the atoms that ful lled the number threshold, the same domain identi er was assigned. For atoms which could not meet the number threshold, we checked whether all of the atoms in their local sphere had the same atom type. If that was the case, we assigned the same domain identi er to the atom. Otherwise, the atom was marked as a non-domain atom (not belonging to any domain). We repeated the procedure above until there were no newly added atoms in the potential domain atom list. Finally, a collection of atoms with the same domain identi er could be identi ed, which formed a domain. We repeated this entire process with each other atom as a starting atom with a different domain identi er. After running the process through all available atoms, all connected domains in the nanoparticle could be identi ed. Individual HCP lattices could be separately assigned and tted to each identi ed domain to correctly determine their lattice constants.
We tested several different number thresholds for this analysis, and using a number threshold of 7 yielded the largest number of atoms successfully assigned to HCP lattice sites after separate domain ttings. Using this threshold, we found ve connected domains in the nanoparticle, which contained To understand the surface effect in a PdH x nanoparticle, a 2D slab model, in nite in the a-b plane and nite along the c axis, was employed because the full description of a 3D nanoparticle is computationally demanding. To suppress the interaction between periodic images along the c axis, a vacuum region of at least 8 Å was inserted. Owing to the symmetric arrangement of the atoms, no net dipole moment developed, and the error caused by the periodic images was minimal. The energy changes upon the layerby-layer growth of the FCC and HCP PdH slabs were compared (Extended Data Fig. 5). Alternating Pd and H layers were added, and the formation energy was calculated with respect to FCC Pd and H 2 .
Monte Carlo (MC) simulation using embedded atom method (EAM) potential model. The stability of bulk PdH x crystals with different crystal packing was analyzed using the classical interatomic potential for Pd-H based on the embedded atom method (EAM), which predicts the lattice constants and elastic properties of PdH x 14 . The total energy of the bulk PdH x crystals was calculated using the EAM potential at the DFT-optimized geometry for HCP/FCC PdH x . The DFT-optimized lattice constants were scaled to match the experiments. The H atoms were placed exactly at the octahedral (O) or tetrahedral (T) interstitial sites of the Pd atom unit cells. For the T occupation, H atoms in the equivalenced PdH crystal would ll half of all tetrahedral sites, which were modelled as zinc-blende and wurtzite structures for FCC PdH and HCP PdH, respectively.
An MC simulation of nite-sized PdH nanoparticles was performed to understand the effects of particle size and crystal packing on the relative thermodynamic stability and the internal structure of the nanoparticles. The radii of the nanoparticles were varied from 6 to 30 Å, wherein the Pd atoms were prepared with two different crystal packing (FCC/HCP). The Pd atoms in the FCC/HCP nanoparticles were positioned with ideal FCC and HCC packing using the lattice constants of FCC/HCP PdH obtained from DFT calculations (with scaling to match the experimental lattice constant). The lattice constant of partially hydrogenated PdH x was obtained by linearly scaling the lattice constants of PdH and pure Pd crystals with FCC/HCP packing. In addition, MC simulations were performed to construct a local H concentration map in the experimentally obtained HCP PdH nanoparticle, which used the 3D atomic coordinates of Pd atoms obtained from the AET experiment.
Each MC move involved the random displacement of H atoms in the xed Pd nanocrystals. Initially, all H atoms were randomly placed inside the nanoparticles. Each new H atom position was accepted or rejected via Metropolis criteria in the NVT ensemble, where the system temperature was set at 300 K. To ensure that H atoms remained in the nanoparticle, new H positions that were not within a cut-off distance of 2.5 Å from any Pd atoms were rejected. The total energy of the system was equilibrated after a half million accepted MC moves for the smallest nanoparticles. Each nanoparticle was simulated until 5 million MC moves were accepted. The trajectory of the H atom positions was saved for every 100 successful MC moves. The total energy of the nanoparticles and the H occupancy at different interstitial sites (Fig. 2