Three-dimensional atomic structure and local chemical order of medium- and high-entropy nanoalloys

Abstract

Medium- and high-entropy alloys (M/HEAs) mix several principal elements with near-equiatomic composition and represent a model-shift strategy for designing previously unknown materials in metallurgy^{1,2,3,4,5,6,7,8}, catalysis^{9,10,11,12,13,14} and other fields^15,16,17,18. One of the core hypotheses of M/HEAs is lattice distortion^5,19,20, which has been investigated by different numerical and experimental techniques^{21,22,23,24,25,26}. However, determining the three-dimensional (3D) lattice distortion in M/HEAs remains a challenge. Moreover, the presumed random elemental mixing in M/HEAs has been questioned by X-ray and neutron studies²⁷, atomistic simulations^28,29,30, energy dispersive spectroscopy^31,32 and electron diffraction^33,34, which suggest the existence of local chemical order in M/HEAs. However, direct experimental observation of the 3D local chemical order has been difficult because energy dispersive spectroscopy integrates the composition of atomic columns along the zone axes^7,32,34 and diffuse electron reflections may originate from planar defects instead of local chemical order³⁵. Here we determine the 3D atomic positions of M/HEA nanoparticles using atomic electron tomography³⁶ and quantitatively characterize the local lattice distortion, strain tensor, twin boundaries, dislocation cores and chemical short-range order (CSRO). We find that the high-entropy alloys have larger local lattice distortion and more heterogeneous strain than the medium-entropy alloys and that strain is correlated to CSRO. We also observe CSRO-mediated twinning in the medium-entropy alloys, that is, twinning occurs in energetically unfavoured CSRO regions but not in energetically favoured CSRO ones, which represents, to our knowledge, the first experimental observation of correlating local chemical order with structural defects in any material. We expect that this work will not only expand our fundamental understanding of this important class of materials but also provide the foundation for tailoring M/HEA properties through engineering lattice distortion and local chemical order.

Main

Strength and ductility are two important material properties, but they are mutually exclusive in most materials³⁷. Recent experiments have demonstrated that several medium- and high-entropy alloys (M/HEAs) can overcome this strength–ductility trade-off^{3,4,5,6,7,8,38,39,40}. The high strength of M/HEAs stems from the different elemental components acting as solutes⁴¹, local chemical order³⁰ and heterogeneous lattice strain^25,26, which raise the energy barrier of dislocation motion⁴⁰. Although the dislocation glide in face-centred cubic (fcc) metals leads to high ductility with reduced strength³⁷, twinning in M/HEAs provides a distinct plasticity mechanism that obstructs dislocation motion and gains strength while remaining ductile⁴². This twinning-induced, simultaneous increase in strength and ductility in M/HEAs has led to both mechanistically driven and property-discovering investigations for structural applications^{3,7,38,39,40,43}. However, the degree and influence of atomic segregation and chemical short-range order (CSRO) on twin formation remain unclear in both M/HEAs and other alloys. Recent atomistic simulations have indicated a link between CSRO and twinning in medium-entropy alloys (MEAs)^28,30, but there is no experimental evidence.

Furthermore, CSRO, lattice distortion and surface strain strongly affect the catalytic activity of M/HEAs^{9,10,11,12,13,14}. Unlike conventional catalysts, M/HEAs confine different elements to the same lattice, which distorts the lattice structure and induces strain^12,13. The lattice distortion and surface strain along with the chemical diversity of adsorption sites increase the activity, selectivity and durability of M/HEA catalysts^{9,13,44,45,46} However, our current understanding of the lattice distortion, strain and CSRO in M/HEA catalysts remains limited because of the dearth of three-dimensional (3D) space and atomic-scale information from diffraction, spectroscopy, electron microscopy and atomistic simulations^{9,10,11,12,13,14}. Here we used atomic electron tomography (AET) to determine the 3D atomic coordinates of NiPdPt-based M/HEA nanoparticles. We quantified the 3D lattice distortion, strain tensor, dislocations, twin boundaries and CSRO of the M/HEAs at the atomic scale. We observe a correlation between strain and CSRO in the M/HEAs and a direct link between CSRO and twinning in the MEAs.

3D lattice distortion and strain tensor

We chose NiPdPt-based M/HEA nanoparticles as a model in this study as these nanoparticles are not only catalytically active^14,47,48,49 but also provide sufficient image contrast for AET to identify their atomic species and types³⁶. The M/HEA nanoparticles were synthesized by a carbothermal shock method⁵⁰ (Methods and Extended Data Table 1), and the distribution of the eight elements in the high-entropy alloys (HEAs) was confirmed by energy dispersive spectroscopy (EDS) maps (Extended Data Fig. 1). The AET experiments were performed with a scanning transmission electron microscope in the annular dark-field mode (Methods). Tomographic tilt series were acquired from six MEA and four HEA nanoparticles (Extended Data Table 2). These nanoparticles were stable under the electron beam by corroborating the consistency of the images taken before, during and after the data acquisition. After image pre-processing, each tilt series was reconstructed by an advanced tomographic algorithm and the 3D atomic coordinates were traced, classified and refined to produce an experimental atomic model (Methods). The experimental precision of the 3D atomic coordinates was estimated to be 19.5 pm (Extended Data Fig. 2). Although we resolved the atomic species in the MEA nanoparticles as Ni, Pd and Pt, the eight elements in the HEA nanoparticles were classified into three types (Co and Ni as type 1; Ru, Rh, Pd and Ag as type 2; Ir and Pt as type 3) because the atomic numbers of several elements differ by only one and cannot be distinguished by AET³⁶. The number of atoms and the atomic species and types of the M/HEA are shown in Extended Data Table 2.

Figure 1a–d and Extended Data Fig. 3 show the experimental 3D atomic models of the 10 M/HEA nanoparticles, exhibiting a single-phase fcc structure. To quantify the local lattice distortion of the M/HEAs, we compared each atom and its nearest neighbours with a reference fcc lattice to determine the 3D atomic displacement (Methods). Figure 1e–h shows the 3D atomic displacements of two representative MEAs and two HEAs, named MEA-1, MEA-2, HEA-1 and HEA-2, respectively. The mean and standard deviation of the atomic displacements of the four nanoparticles are 0.23 ± 0.11 Å, 0.26 ± 0.12 Å, 0.29 ± 0.12 Å and 0.37 ± 0.12 Å, respectively (Fig. 1i–l). A statistical analysis of these data shows that the HEAs have larger local lattice distortion than the MEAs (Extended Data Table 3). From the experimental 3D atomic coordinates, we also determined the local strain tensor of the M/HEAs using a method described elsewhere⁵¹ (Methods). Figure 2 shows the six components of the local strain tensor for MEA-1, MEA-2, HEA-1 and HEA-2, in which the compressive, tensile and shear strains range from −8% to +8%. To quantify the local heterogeneity of the strain tensor, we calculated the standard deviation and the smoothness energy (Methods). The standard deviation measures the average amount of deviation from the mean, whereas the smoothness energy quantifies the local fluctuation. The larger the smoothness energy, the larger the local fluctuation. Figure 2e,f shows that the HEAs have more heterogeneous strain than the MEAs.

Fig. 1: The 3D atomic structure and lattice distortion of M/HEA nanoparticles.

a–d, Experimental atomic models of two MEAs and two HEAs, named MEA-1 (a), MEA-2 (b), HEA-1 (c) and HEA-2 (d), respectively, in which the yellow circles represent the atoms along the twin boundaries. Scale bar, 1 nm. e–h, Atomic layer-by-layer visualization of the 3D displacements of MEA-1, MEA-2, HEA-1 and HEA-2, in which the arrows point to the twin boundaries. i–l, The distributions of the 3D atomic displacements of MEA-1, MEA-2, HEA-1 and HEA-2 with the mean and standard deviation as 0.23 ± 0.11 Å, 0.26 ± 0.12 Å, 0.29 ± 0.12 Å, and 0.37 ± 0.12 Å, respectively.

Fig. 2: 3D strain tensor measurements of the M/HEA nanoparticles.

a–d, The six components of the local strain tensor of a representative atomic layer in MEA-1 (a), MEA-2 (b), HEA-1 (c) and HEA-2 (d) along the $\left[1\bar{1}0\right]$ direction, in which the dashed lines represent the twin boundaries. Scale bar, 2 nm. e, Standard deviation of strain and CSRO for five MEA and four HEA nanoparticles, which quantifies the average amount of deviation from the mean. MEA-5 is excluded in the calculation because of a grain boundary (Extended Data Fig. 3c). f, Smoothness energy (that is, local fluctuation) of strain and CSRO for the M/HEA nanoparticles, which quantifies the local heterogeneity of strain and CSRO. a.u., arbitrary units.

Next, we characterize the twins and dislocations in the M/HEAs. Among the 10 M/HEAs, 4 are twin-free, 2 have a single twin, 3 have double twins, and 1 has a grain boundary and double twins (Fig. 1a–d, Methods and Extended Data Fig. 3). The abundance of the single and double twins in M/HEAs is different from conventional nanoparticles, which have fewer twins except for decahedral multiply twinned nanoparticles^52,53. Compared with the MEAs, the HEAs have more diffuse twin boundaries with each boundary spreading to the neighbouring atomic layers (Extended Data Fig. 4a–e). We also observe that the HEAs are more prone to having dislocations than the MEAs. Extended Data Fig. 4f–i shows the cores of three Shockley partial dislocations and two screw dislocations in the M/HEAs and the corresponding Burgers vectors.

Observation of CSRO-mediated twinning

To quantify the local chemical order in the M/HEAs, we compute the CSRO parameters (α_ij) between each atom and its nearest neighbours^30,54 (Methods). For pairs of the same species or types (i = j), a positive α_ii indicates a tendency to segregate, and a negative α_ii the opposite. For pairs of different species or types (i ≠ j), a negative α_ij indicates favourable inter-mixing, and a positive α_ij the opposite. A quantitative analysis of CSRO and strain indicates that the HEAs have more heterogeneous CSRO than the MEAs, and that CSRO is correlated with strain in the M/HEAs (Fig. 2e,f). Figure 3a,b and Extended Data Fig. 5a–d show the 3D distribution of the six CSRO parameters (α_NiNi, α_PdPd, α_PtPt, α_NiPd, α_NiPt and α_PdPt) of twin-free MEA-1. The 3D distribution is heterogeneous with the formation of pockets of local chemical order, indicating that CSRO can propagate from the ångström scale to the nanometre scale. To quantify the local chemical order, we averaged every CSRO parameter for each atomic layer along the [111] direction (Fig. 3c,d and Extended Data Fig. 5e–h). We observe that all the average α_NiPt values are negative and most of the average α_PdPt values are positive, indicating the tendency of the inter-mixing between Ni and Pt atoms and the separation between the Pd and Pt atoms. To validate our experimental observations, we performed density functional theory (DFT)-based calculations to predict energy-favoured CSRO in a twin-free NiPdPt MEA (Methods). Figure 3i shows the histogram of the six average CSRO parameters of the twin-free MEA between the DFT calculations and experimental observations, confirming a favourable bonding between Ni and Pt atoms and an unfavourable bonding between Pd and Pt atoms in twin-free MEAs.

Fig. 3: Experimental observation of the correlation between CSRO and twinning in MEAs.

a,b, The 3D distribution of α_NiPt (a) and α_PdPt (b) in twin-free MEA-1. Scale bar, 1 nm. c,d, Histogram of the average α_NiPt (c) and α_PdPt (d) values for each atomic layer along the [111] direction, indicating the tendency of inter-mixing between the Ni and Pt atoms and the separation between the Pd and Pt atoms. e,f, 3D distribution of α_NiPt (e) and α_PdPt (f) in double-twinned MEA-2 (the twins marked with yellow planes), exhibiting more heterogeneous CSRO than twin-free MEA-1 (a,b). g,h, Histogram of the average α_NiPt (g) and α_PdPt (h) values for each atomic layer along the [111] direction (the yellow bars indicate the twin positions), which are the opposite of α_NiPt and α_PdPt for twin-free MEA-1 (c,d). i, Histogram of the average values for the six CSRO parameters of a DFT-calculated bulk MEA, twin-free MEA-1 and double-twinned MEA-2.

As a comparison, we calculated the six CSRO parameters for double-twinned MEA-2. Figure 3e,f and Extended Data Fig. 5i–l show the 3D distribution of the six parameters, in which the yellow planes represent the twin boundaries. The CSRO parameters of the double-twinned MEA are more heterogeneous than that of the twin-free MEA. Some pockets of local chemical order are connected to each other to form elongated structures that extend over a few nanometres. Figure 3g,h and Extended Data Fig. 5m–p show the histograms of the six average CSRO parameters of each atomic layer parallel to the twin boundaries. Most of the α_NiPt and α_PdPt values are positive and negative, respectively (Fig. 3g,h), which are the opposite of α_NiPt and α_PdPt of twin-free MEA-1 (Fig. 3c,d). The average α_NiPt and α_PdPt values of the double-twinned MEA are the reverse of those of the twin-free MEA as well as the DFT calculations of a twin-free bulk MEA (Fig. 3i). These observations indicate that the separation of the favourable atomic species (Ni and Pt) and the inter-mixing of the unfavourable atomic species (Pd and Pt) facilitate the formation of the twins, which are further corroborated by the analysis of the other double-twinned MEA nanoparticle (Extended Data Fig. 6).

To investigate the impact of CSRO on the twin-formation energy (E_TF) of the MEAs, we used the experimental 3D atomic coordinates and species as a direct input to molecular dynamics simulations and calculated E_TF as a function of the twin position (Methods). We first applied this approach to a single-twinned MEA. Figure 4a–e shows the change of E_TF by moving the twin from the zeroth to the tenth atomic layer along the [111] direction, in which the zeroth layer indicates twin-free MEA. Four representative atomic configurations are shown in Fig. 4a–d with the twin marked in yellow. We observe that E_TF changes from negative to positive when the twin is moved from atomic layers 5 to 6 (Fig. 4e). The experimentally determined twin position is in layer 5 (yellow bar), which is next to the minimum E_TF in layer 4. Next, we used the experimental 3D atomic coordinates of double-twinned MEA-2 as input to molecular dynamics simulations to calculate E_TF as a function of the twin separation. While fixing one twin, we moved the other twin layer-by-layer along the [111] direction and computed the corresponding E_TF (Fig. 4f–i). We find that E_TF changes from negative to positive between a twin separation of atomic layers 5 and 6. The experimentally determined twin separation is five atomic layers (yellow bar in Fig. 4i), which is next to the minimum E_TF with a twin separation of four layers. We analysed the other double-twinned MEA nanoparticle and obtained consistent results (Extended Data Fig. 7). All these observations show a correlation between the CSRO and twinning in the MEAs, that is, energetically unfavourable CSRO lowers the E_TF. As the samples were synthesized by heating the metal precursors to about 2,000 K in approximately 50 ms, followed by rapid cooling at about 10⁵ K s⁻¹ (Methods)⁵⁰, this quenching process essentially freezes the CSRO in the molten alloy because there is not sufficient time for atomic diffusion to appreciably change the CSRO. Therefore, we conclude that the CSRO mediates the formation of twins in the MEA nanoparticles.

Fig. 4: The twin-formation energy (E_TF) calculated from the experimental 3D atomic coordinates and species of the MEAs.

a–d, Calculation of E_TF of a single-twinned MEA by moving the twin (yellow circles) from the zeroth (top) to the tenth atomic layer along the [111] direction, with four representative atomic configurations showing the twin in layer 0 (that is, twin-free) (a), layer 3 (b), layer 5 (c) and layer 8 (d). Scale bar, 1 nm. e, Histogram of E_TF as a function of the twin position, in which E_TF changes from negative to positive from atomic layers 5 to 6. The experimentally determined twin is in layer 5 (yellow bar), which is next to the minimum E_TF in layer 4. f-h, Calculation of E_TF of the double-twinned MEA by fixing one twin (yellow circles in f) and moving the other twin along the [111] direction, in which the three representative atomic configurations show a twin separation by 0 (that is, a single twin) (f), 5 (g), and 10 atomic layers (h). i, Histogram of E_TF as a function of the twin separation, in which E_TF changes from negative to positive between a twin separation of five and six layers. The experimentally determined twin separation is five atomic layers (yellow bar), which is next to the minimum E_TF with a twin separation of four layers.

Compared with the MEAs, the HEAs exhibit greater local chemical fluctuation. Extended Data Figs. 8a–f and 9a–f show the 3D distribution of the six CSRO parameters for a twin-free and a double-twinned HEA, which are more heterogeneous than those of the MEAs (Fig. 2e). The increase in the chemical complexity from the three-element MEAs to the eight-element HEAs leads to a larger distortion of the twin geometry as shown in the formation of atomic steps along the twins of both the MEAs and HEAs (Extended Data Fig. 4b–e), with the latter containing several such steps that disrupt the stacking of the neighbouring atomic planes. To examine the correlation between CSRO and twinning in the HEAs, we calculated the six average CSRO parameters of the atomic layer along the [111] direction for a twin-free HEA (Extended Data Fig. 8g–l). In comparison, the six average CSRO parameters for a double-twinned HEA are shown in Extended Data Fig. 9g–l. We observe the reverse of α₁₃ between the twin-free and double-twinned HEAs, indicating that the separation of the favourable atomic types 1 and 3 facilitates the formation of the twins in the HEAs (Extended Data Fig. 9m). Because our AET experiment can classify only the eight elements in the HEAs into three types, we cannot calculate the E_TF from the experimental 3D atomic coordinates and types of the HEA as in the case of the MEAs.

Conclusions

Lattice distortion and CSRO are two fundamental features that strongly affect the unique properties of M/HEAs. Despite substantial insights from various studies^{21,22,23,24,25,26,27,28,29,30,31,33,34}, direct 3D structure information remains unknown. Here we overcome this limitation by advancing AET to determine the 3D atomic positions of M/HEAs and quantitatively characterize their local lattice distortion, strain tensor, dislocation cores and CSRO in three dimensions. We find a correlation between strain and CSRO in the M/HEAs. We also observe CSRO-mediated twinning in the MEAs, which confirms the DFT calculations of the NiPdPt MEA (Fig. 3i) and the atomistic simulations of the CrCoNi MEA^28,30. Owing to the tunability of CSRO during materials manufacturing, our 3D atomic-scale insights into the correlation between CSRO and twinning could expand the horizon for the design of M/HEAs and other alloys with targeted structure–property relationships. Although we focus on M/HEA nanoparticles in this work, AET could be combined with EDS to determine the 3D atomic positions and local chemical order of bulk M/HEAs by milling them into needles or thin specimens.

Furthermore, M/HEA catalysts have shown performance enhancement over conventional alloys for various multi-step reactions^12,14, including ammonia oxidation and decomposition^9,50, carbon dioxide reduction^44,45 and methane combustion⁵⁵. M/HEA catalysts also possess near-continuum adsorbate binding energies with greater structural stability^12,14,49. Thus, determining the 3D atomic structure of M/HEA catalysts and measuring their 3D local lattice distortion and strain could pave the way for their rational design in a largely untapped range of compositions and structures. The present case study of NiPdPt-based M/HEA nanoparticles provides insights into the heterogeneous distribution of strain and CSRO and represents an important step in this direction. Recently, we have demonstrated that the experimental 3D atomic coordinates from AET can be used as direct input to DFT to obtain more accurate electronic properties⁵⁶ and, together with machine learning, to identify the active sites of Pt alloy nanocatalysts for the oxygen reduction reaction⁵⁷. Immediate extension of this approach to M/HEA nanoparticles could enable the discovery of yet-unknown attributes of this emerging class of catalysts. We expect that the ability to determine the 3D atomic structure and local chemical order of M/HEAs will extend to both the pursuit of superior strength–ductility combinations of the metallurgical community and the quest towards optimized surface adsorption energies of the catalysis community.

Methods

Sample synthesis

The M/HEA samples were prepared by a carbothermal shock method⁵⁰. Metal precursors in ethanol (0.05 M) were mixed into the desired multi-element composition (Extended Data Table 1) and dipped onto a carbon heater for Joule heating, in which reduced graphene oxide was used as the substrate for dispersion and stabilization. After drying at room temperature, metal precursors on the reduced graphene oxide were rapidly heated to a high temperature (about 2,000 K for approximately 50 ms) by Joule heating followed by rapid cooling (about 10⁵ K s⁻¹). This sample synthesis procedure enabled us to fully mix various elements into M/HEA nanoparticles^14,50.

AET data acquisition

The AET experiments of 10 M/HEA nanoparticles were performed using the TEAM 0.5 microscope with the TEAM stage at the National Center for Electron Microscopy, with the scanning transmission electron microscope operated in the annular dark-field (ADF-STEM) mode at 200 kV or 300 kV (Extended Data Table 2). At each tilt angle, a nearby nanoparticle or a sample feature was used as a fiducial to align and focus the image, thereby reducing the unnecessary exposure of the region of interest to the electron beam⁵⁸. To minimize the drift distortion and electron dose at each tilt angle, three to four sequential images were taken with a dwell time of 3 µs. The total electron dose of each tilt series was optimized to be between 4.7 × 10⁵ e⁻ Å⁻² and 1.1 × 10⁶ e⁻ Å⁻² to reduce the beam damage (Extended Data Table 2). For each sample, we confirmed the structural stability on beam exposure by comparing images before, during and after the acquisition of the tilt series.

Image pre-processing

A multi-step image pre-processing protocol was performed on each AET dataset as outlined below.

• 1. Drift correction: To compensate for sample drift during data acquisition, we collected three to four images at each tilt angle, computed the cross-correlation coefficient between the images and identified the relative drift vectors by the maximum cross-correlation. We used a step size of 0.1 pixels as the drift in typical ADF-STEM images is smaller than 1 pixel. We applied the drift correction to each image along the slow scan direction and corrected for it by interpolating the raw images with drift-corrected pixel positions. The drift-corrected images were then averaged to form a single image at each tilt angle.

• 2. Image denoising: To remove the Poisson and Gaussian noise from the drift-corrected images, we used the block-matching and 3D filtering algorithm⁵⁹, which has been successfully used to denoise experimental AET datasets^60,61. To optimize the block-matching and 3D filtering denoising parameters for each dataset, we first estimated the level of each noise type (Poisson and Gaussian) in the image stack. Then, these noise levels were added to several simulated ADF-STEM images of model nanoparticles with similar size and elemental distribution as the experimental data. The denoising parameters leading to the maximum cross-correlation between the simulated images and the experimental images were applied to the experimental data.

• 3. Background subtraction: For each denoised image, we used Otsu thresholding in MATLAB to generate a mask of the nanoparticle, which is slightly larger than its boundary. Using the background outside the masked region, we performed a Laplacian interpolation to estimate the background level inside the masked zone and subtracted it from the denoised image⁶⁰.

• 4. Image alignment: The background-subtracted images were aligned by the common line method along the tilt axis and the centre of mass method perpendicular to the tilt axis, which can align experimental tilt series with a sub-pixel accuracy^52,58.

Tomographic reconstruction

Each pre-processed dataset was reconstructed by the Real Space Iterative Reconstruction (RESIRE) algorithm⁶². RESIRE iteratively minimizes the difference between the experimental and calculated images of a sample using gradient descent. RESIRE is superior to other tomographic reconstruction algorithms by incorporating angular refinement and spatial alignment⁶². A typical RESIRE reconstruction converges after about 200 iterations. Following an initial reconstruction and an iterative process of angular refinement and spatial alignment, the background of the new set of images was re-evaluated and re-subtracted. The resulting images, after undergoing further angular refinement and spatial realignment, were used to generate the final 3D reconstruction by RESIRE.

Determining the 3D atomic coordinates and species or types

From the final 3D reconstructions, we determined the 3D atomic coordinates and species or types using the following steps.

• 1. Each reconstruction underwent spline interpolation by a factor of 3 to produce a finer grid, from which all the local maxima were identified. Using a polynomial fitting method⁶⁰, we identified the positions of all the peaks (that is, potential atoms) from a 0.8 Å × 0.8 Å × 0.8 Å volume around each peak. An initial list of the potential atoms was obtained by searching through all the fit peak positions with the constraint that the minimum distance between neighbouring peaks is 2 Å, given that all the interatomic distances in our samples are larger than this value.

• 2. To remove the non-atoms from this list, we performed K-means clustering of the integrated intensity of the local volume around each potential atom position^63,64.

• 3. By overlaying the 3D atomic positions with the 3D reconstructions, we manually checked the atomic positions and corrected a small fraction (<1%) of unidentified or misidentified atoms. This manual correction is routine for atom tracing and refinement in macromolecular crystallography⁶⁵.

• 4. The distribution of the integrated intensity of each atom is affected by several factors, including the atomic number, the electron probe size, the inner and outer angles of the detector, sample drift, angular errors, the missing wedge and noise. To accurately classify the Ni, Pd and Pt atoms in MEAs and the three types of atom in HEAs, we used K-means clustering^63,64, which has been demonstrated as a robust method to obtain consistent results from different independent runs^60,66.

• 5. We performed a local reclassification of all the atomic species or types. Each atom was defined to be at the centre of a sphere of radius 10  Å. The average intensity distribution of the three atomic species or types was computed within this sphere. We then computed the L₂ norm of the intensity distribution between the central atom and the average Ni, Pd and Pt or type-1, -2 and -3 atoms. The atom was assigned to the species or type with the smallest L₂ norm. After repeating this step for all the atoms, an initial experimental 3D atomic model of the sample was obtained.

• 6. The 3D atomic coordinates of the initial model were refined by minimizing the error between the experimental and calculated images using a gradient descent as described elsewhere⁶⁴. The convergence of the iterative process was verified by monitoring the L₂ norm error.

Local lattice distortion and strain tensor measurements

The local lattice distortion is defined as the deviation of the experimental atomic positions in the M/HEA nanoparticles from those in a perfect fcc lattice. For each atom, its local lattice distortion (Δd_i) was calculated by

$${\Delta d}_{i}=\frac{1}{N}\sum _{j}\left|{{\bf{r}}}_{ji}-{{\bf{r}}}_{ji}^{0}\right|$$

(1)

where N is the number of the nearest neighbours of atom i, r_ji is the experimental 3D coordinates of the jth nearest-neighbour atom, and ${{\bf{r}}}_{{ji}}^{0}$ is the perfect fcc lattice structure that is aligned to r_ji based on the three Euler angles determined by a breadth-first search algorithm⁶⁰. A cutoff of a quarter of the nearest-neighbour bond length was applied to eliminate the contribution from some surface atoms with a large deviation. The strain tensor measurements follow a procedure described elsewhere⁵¹. Briefly, the atomic displacements were calculated from the measured atomic positions and a perfect fcc lattice structure. The atomic displacements were convolved with a 2-Å-wide 3D Gaussian kernel to increase the signal-to-noise ratio and precision, but reduce the 3D resolution to about 4 Å. The strain tensor was obtained by numerical differentiation of the 3D atomic displacements, in which a mask was applied to remove the edge effect⁵¹.

The CSRO parameters

After identifying the nearest neighbours of each atom, we computed the CSRO parameters (α_ij) between the central atom (i) and its nearest neighbours (j) by^30,54

$${\alpha }_{{ij}}=\frac{{p}_{{ij}}-{C}_{j}}{{\delta }_{{ij}}-{C}_{j}}$$

(2)

where p_ij denotes the average probability that a j-type atom is the nearest neighbour to an i-type atom, c_j indicates the average concentration of j-type atoms and δ_ij is the Kronecker delta function. To eliminate the boundary effect, we removed the surface atoms of each M/HEA nanoparticle from the calculation of the CSRO parameters. Following this procedure, we computed the six CSRO parameters, which are bounded between −1 and +1, for every atom in the M/HEA nanoparticles. To obtain a local CSRO distribution such as those in Fig. 3 and Extended Data Figs. 5, 6, 8 and 9, we interpolated the CSRO parameters onto 3D grids and convolved them with a Gaussian kernel. The width of the Gaussian kernel was determined by the first valley of the pair distribution function of the nanoparticle, corresponding to the first nearest-neighbour shell distance.

The twin order parameter

From the experimental 3D atomic coordinates, we fit the nearest-neighbour atoms around each atom to a perfect fcc and hexagonal close-packed (hcp) lattice by the breadth-first search algorithm⁶⁰. If a nearest-neighbour atom has a deviation larger than a cutoff of 0.75 Å, we set its deviation to be the cutoff value. This step was to eliminate the effect of some surface atoms with a large deviation. The twin order parameter (η) was calculated by⁶⁷

$$\eta =\frac{{d}_{{\rm{fcc}}}-{d}_{{\rm{hcp}}}}{{d}_{\max }}$$

(3)

where η = 1 and −1 represent the hcp and fcc structures, respectively, d_fcc and d_hcp are the sum of the deviation of the nearest-neighbour atoms from a perfect fcc lattice and hcp lattice, respectively, and d_max is the maximum deviation. In this study, we also calculated η using different cutoff values and obtained consistent results. With η, we can identify intrinsic and extrinsic stacking faults and twin boundaries by calculating the separation between two hcp layers: 0 for intrinsic stacking faults, 1 for extrinsic stacking faults and at least 2 for twin boundaries. In our experimental data, the separation between two hcp layers is at least 2, indicating that these are twin boundaries instead of intrinsic or extrinsic stacking faults.

Correlation between strain tensor and CSRO

To quantify the local heterogeneity of the strain tensor and CSRO, we calculated the standard deviation and the smoothness energy⁶⁸. The standard deviation (σ) of strain or CSRO quantifies the average amount of deviation from the mean, defined as

$$\sigma =\sqrt{\frac{1}{6}{\sum }_{\beta }{\sigma }_{\beta }^{2}}$$

(4)

where σ_β is the standard deviation of component β of the strain tensor or CSRO. The smoothness energy (S_E) is defined as⁶⁸

$${S}_{{\rm{E}}}=\frac{1}{6N}\sum _{n,i,\beta }{({\varDelta }_{i}{f}_{\beta }^{n})}^{2}$$

(5)

where ${f}_{\beta }^{n}$ is component β of the strain tensor or CSRO, n is the nth voxel of a volume containing N voxels and Δ_i is the ith component of Laplacian with i equal to x, y or z. S_E averages the square of the second derivative of the local strain tensor or CSRO and quantifies the local fluctuation in each dataset. Larger S_E means larger local fluctuation (that is, more heterogeneous). Figure 2e,f shows the standard deviation and the smoothness energy of the strain tensor and CSRO for the M/HEA nanoparticles, in which a mask was applied to remove the edge effect. This quantitative analysis shows that (1) the HEAs have more heterogeneous strain and CSRO than the MEAs and (2) strain is correlated to CSRO in the M/HEAs.

DFT calculations

DFT-based lattice Monte Carlo approach²⁸ was used to show the characteristic CSRO for the NiPdPt MEA and NiCoRuRhPdAgIrPt HEA. The 256-atom configurations were generated as a special quasi-random structure⁶⁹. The Monte Carlo simulations were run for 3,000 steps, corresponding to about 12 swap trials per atom, at a Monte Carlo temperature of 600 K. Energy calculations were performed using the Vienna ab initio simulation package^70,71. A plane wave cutoff energy was chosen at 380 eV, and the Brillouin zone integrations were performed using Monkhorst–Pack meshes⁷¹ with a single k-point (Γ). Projector augmented wave potentials⁷² with the Perdew–Burke–Ernzerhof generalized-gradient approximation⁷³ were adopted for the exchange-correlation functional. From the DFT results, the lattice distortion parameter²³ was calculated as 0.089 Å and 0.11 Å for the NiPtPd MEA and NiCoRuRhPdAgIrPt HEA, respectively, which confirm our experimental observations (Fig. 1i–l).

Twin-formation energy of the MEAs

We calculated the twin-formation energy of the NiPdPt MEAs from the experimentally measured 3D atomic coordinates using the LAMMPS software package⁷⁴ with the empirical embedded-atom method potential⁷⁵. The experimental 3D atomic coordinates of the MEA nanoparticles were obtained from the AET experiments. To eliminate the boundary effect, the surface atoms of each MEA nanoparticle were not used in the molecular dynamics calculations. For the supercell, the shrink-wrapped non-periodic boundary conditions were imposed along all three directions. As shown in Fig. 4 and Extended Data Fig. 7, the migration of the twin boundary was carried out by gradually shifting the close-packed (111) atomic planes along the $\left\langle 112\right\rangle$ direction by the Burgers vector of the Shockley partial ${b}_{s}=\frac{a}{6}\langle 11\bar{2}\rangle$. For the twin boundary at different positions, the twin-formation energy was calculated by the energy difference between the twinned and twin-free configurations after energy minimization.

Extended data figures and tables

Extended Data Fig. 1 EDS maps of the HEA nanoparticles.

a, Low-resolution ADF-STEM image of the nanoparticles. Scale bars, 20 nm. b-i, EDS maps showing the distribution of Co (b), Ni (c), Ru (d), Rh (e), Pd (f), Ag (g), Ir (h) and Pt (i) in the nanoparticles.

Extended Data Fig. 2 3D precision estimation.

a, b, Comparison between a representative experimental (after denoising) (a) and a multi-slice calculated image (b) of MEA-3. The multi-slice images were convolved with a Gaussian function to match the incoherence effects in the experimental images. Scale bar, 2 nm. c, Histogram of the deviation of the 3D atomic positions between the experimental atomic model and that obtained from 55 multi-slice images. The root-mean-square deviation of the histogram is 19.5 pm.

Extended Data Fig. 3 Experimental 3D atomic models of the other six M/HEA nanoparticles.

a-f, Experimental atomic models of four MEAs and two HEAs, named MEA-3 (a), MEA-4 (b), MEA-5 (c), MEA-6 (d), HEA-3 (e) and HEA-4 (f), in which the yellow circles represent the atoms along the twin boundaries in (a-c) and grey circles indicate the atoms on the grain boundary in (c). Scale bar, 1 nm.

Extended Data Fig. 4 Twin boundaries and dislocations in the M/HEAs.

a-e, The twin boundaries of three representative MEAs (a, twin-free; b, single twin; c, double twins) and two HEAs (d, single twin; e, double twins), showing more diffuse twin boundaries in the HEAs with each boundary spreading to the neighbouring atomic layers. The twin order parameter of 1 and −1 represents a hcp (i.e., twinning) and fcc structure, respectively. f, Two Shockley partial dislocations in MEA-5 with opposite Burgers vectors $\frac{a}{6}\left[121\right]$ and $\frac{a}{6}[\bar{1}\,\bar{2}\bar{1}]$, as the gliding process was frozen near the boundary during the rapid cooling process of the nanoparticle. g, A screw dislocation in MEA-2 with the Burgers vector $\frac{a}{2}[110]$, where the zigzag lines in light blue show the characteristic feature of the screw dislocation. h, A Shockley partial dislocation in HEA-4 with the Burgers vector $\frac{a}{6}\left[121\right]$, which exists near the boundary of the nanoparticle. i, A screw dislocation in HEA-3 with the Burgers vector $\frac{a}{2}[110]$. Scale bars, 1 nm (a); and 5 Å (f).

Extended Data Fig. 5 3D distribution of the other four CSRO parameters of twin-free MEA-1 and double-twinned MEA-2.

a-d, 3D distribution of ${\alpha }_{{\rm{NiNi}}}$, ${\alpha }_{{\rm{PdPd}}}$, ${\alpha }_{{\rm{PtPt}}}$, and ${\alpha }_{{\rm{NiPd}}}$ in MEA-1, showing the formation of local chemical-order pockets. e-h, Histogram of the average ${\alpha }_{{\rm{NiNi}}}$, ${\alpha }_{{\rm{PdPd}}}$, ${\alpha }_{{\rm{PtPt}}}$, and ${\alpha }_{{\rm{NiPd}}}$ values for each atomic layer along the [111] direction in MEA-1. i-l, 3D distribution of ${\alpha }_{{\rm{NiNi}}}$, ${\alpha }_{{\rm{PdPd}}}$, ${\alpha }_{{\rm{PtPt}}}$, and ${\alpha }_{{\rm{NiPd}}}$ in MEA-2 (the twins labelled as yellow planes), exhibiting more heterogeneous CSRO than twin-free MEA-1 (a-d). m-p, Histogram of the average ${\alpha }_{{\rm{NiNi}}}$, ${\alpha }_{{\rm{PdPd}}}$, ${\alpha }_{{\rm{PtPt}}}$, and ${\alpha }_{{\rm{NiPd}}}$ values for each atomic layer along the [111] direction in MEA-2. Scale bar, 1 nm.

Extended Data Fig. 6 3D distribution of the six CSRO parameters in double-twinned MEA-3.

a-f, 3D distribution of ${\alpha }_{{\rm{NiNi}}}$, ${\alpha }_{{\rm{PdPd}}}$, ${\alpha }_{{\rm{PtPt}}}$, ${\alpha }_{{\rm{NiPd}}}$, ${\alpha }_{{\rm{NiPt}}}$, and ${\alpha }_{{\rm{PdPt}}}$, where the twins are labelled as yellow planes. g-l, Histogram of the average ${\alpha }_{{\rm{NiNi}}}$, ${\alpha }_{{\rm{PdPd}}}$, ${\alpha }_{{\rm{PtPt}}}$, ${\alpha }_{{\rm{NiPd}}}$, ${\alpha }_{{\rm{NiPt}}}$, and ${\alpha }_{{\rm{PdPt}}}$ values for each atomic layer along the [111] direction, where the yellow bars indicate the twin positions. Scale bar, 1 nm.

Extended Data Fig. 7 Twin formation energy (E_TF) calculated from experimental 3D atomic coordinates of double-twinned MEA-3.

a-c, Calculation of E_TF of the double-twinned MEA by fixing one twin (top yellow circles) and moving the other twin along the [111] direction, in which the three representative atomic configurations show a twin separation by 0 (i.e., a single twin) (a), 9 (b), and 13 atomic layers (c). d, Histogram of E_TF as a function of the twin separation. The experimentally determined twin separation is 9 atomic layers (yellow bar), which is next to the minimum E_TF with a twin separation of 10 layers. Scale bar, 1 nm.

Extended Data Fig. 8 3D distribution of the six CSRO parameters in twin-free HEA-1.

a-f, 3D distribution of ${\alpha }_{11}$, ${\alpha }_{22}$, ${\alpha }_{33}$, ${\alpha }_{12}$, ${\alpha }_{13}$, and ${\alpha }_{23}$, which are more heterogeneous than those of the twin-free MEAs (Fig. 3a,b, Extended Data Fig. 5a–d). g-l, Histograms of the average ${\alpha }_{11}$, ${\alpha }_{22}$, ${\alpha }_{33}$, ${\alpha }_{12}$, ${\alpha }_{13}$, and ${\alpha }_{23}$ values for the atomic layer along the [111] direction. Scale bar, 1 nm.

Extended Data Fig. 9 3D distribution of the six CSRO parameters in double-twinned HEA-2.

a-f, 3D distribution of ${\alpha }_{11}$, ${\alpha }_{22}$, ${\alpha }_{33}$, ${\alpha }_{12}$, ${\alpha }_{13}$, and ${\alpha }_{23}$, exhibiting greater local chemical fluctuations than the double-twinned MEA (Fig. 3e,f, Extended Data Figs. 5i–l and 6a–f). g-l, Histograms of the average ${\alpha }_{11}$, ${\alpha }_{22}$, ${\alpha }_{33}$, ${\alpha }_{12}$, ${\alpha }_{13}$, and ${\alpha }_{23}$ values of the atomic layer along the [111] direction. m, Histogram of the average values for the six CSRO parameters of a DFT-calculated bulk HEA, twin-free HEA-1, and double-twinned HEA-2. Scale bar, 1 nm.

Extended Data Table 1 Metal precursors used for the synthesis of M/HEA nanoparticles

Extended Data Table 2 AET data collection, processing, reconstruction, refinement and statistics of the ten M/HEA nanoparticles

Extended Data Table 3 Unpaired t-test on the 3D atomic displacements of MEA-1, MEA-2, HEA-1 and HEA-2