Revealing 3D atomic packing in liquid-like solids 1

bipyramids of forming icosahedra, the of pentagonal bipyramids into a novel medium-range order, the pentagonal bipyramid network. Molecular pentagonal bipyramid networks in monatomic amorphous which grow in size and form icosahedra the quench from the state. experimental

show the experimental 3D atomic model 70 7 is quenched from 5200 K to 300 K, θ decreases ( Fig. 4d) but the / and β/ ratios remain 140 relatively unchanged (Fig. 4b). 141 The pentagonal bipyramid network 142 We find that a number of pentagonal bipyramids link to each other by sharing four or five 143 atoms with their neighbours (Fig. 5a and b), which we define as vertex-or edge-sharing 144 of the five-fold skeletons, respectively. Figure 5c and Supplementary Fig. 12 show the 145 fraction of pentagonal bipyramids as a function of the number of vertex-and edge-sharing 146 neighbours. We observe in the three samples that 63.5% of bipyramids do not share any 147 vertex with their neighbours, but the majority of them (72.5%) have at least one edge-148 sharing neighbour. Figure 5d and e shows two pentagonal bipyramid clusters with the 149 most vertex-and edge-sharing neighbours, respectively, where the larger cluster is 150 formed by edge-sharing. These results indicate that edge-sharing of the five-fold 151 skeletons is a more dominant feature in the packing of pentagonal bipyramids. We then 152 investigate if these pentagonal bipyramids form icosahedra. An icosahedron requires the 153 packing of 12 pentagonal bipyramids by edge-sharing. Due to geometric frustration 22,27 , 154 the five-fold skeletons in an icosahedron form a regular dodecahedron with the dihedral 155 angle (φ) of 116.57° (Fig. 5f, inset). But we observe that the dihedral angles (φ) between 156 two edge-sharing pentagonal bipyramids peak at 120.7° in the three amorphous samples 157 ( Fig. 5f), which is close to φ = 120° in the absence of geometric frustration. This 158 observation further confirms that the pentagonal bipyramids only assemble partial 159 icosahedra (Fig. 5g). These results do not contradict that 9.8% of all the Voronoi 160 polyhedra in the three samples are distorted icosahedra because the vast majority of these 161 distorted icosahedra have a large distortion with > 0.255 ( Supplementary Fig. 13). 162 When choosing ≤ 0.255, the total numbers of distorted icosahedra and pentagonal 163 8 bipyramids in the three amorphous materials are 17 and 26262, respectively, showing that 164 the pentagonal bipyramids are far more abundant than the distorted icosahedra in the 165 samples. 166 Instead of assembling icosahedra, most pentagonal bipyramids with edge-sharing 167 skeletons form PBNs in these amorphous samples. Figure 5h shows a representative PBN,168 which consists of five partial icosahedra. To investigate if PBNs are prevalent in other amorphous systems such as liquids 177 and metallic glasses, we employed MD simulations using the large-scale 178 atomic/molecular massively parallel simulator 45 (Methods). A bulk Ta solid was melted 179 at 5200 K, quenched at a cooling rate of 10 13 K/s and brought to equilibrium at 300 K. 180 The PDFs of the Ta structures at varying temperatures are shown in Supplementary Fig.  181 19. At 5200 K, the PDF of the Ta liquid resembles those of experimentally measured 182 amorphous materials in terms of the peak and valley positions (dotted curve in Fig. 1c). 183 At 300 K, the splitting of the 2 nd and 3 rd peaks in the PDF indicates the formation of the 184 Ta metallic glass 18,24 (arrows in Supplementary Fig. 19). By analysing the polytetrahedral 185 packing of these Ta structures with ≤ 0.255, we find that pentagonal bipyramids are 186 the most abundant atomic motifs across the entire temperature range and their population 187 9 dramatically increases with the decrease of the temperature (Fig. 6a). At 5200 K, we 188 observe PBNs and partial icosahedra in the Ta liquid (Fig. 6b, c Table 1). With carefully designed sample preparation and data 460 acquisition protocols, our samples were more stable under the electron beam than some of the previously 461 studied glass samples 54,55 . Images taken at 0⁰ tilt angle before, during and after the tilt series indicate that 462 structural change of the samples throughout the experiment was minimal (Supplementary Fig. 4).

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Electron diffraction experiment and analysis of the amorphous Ta film and Pd nanoparticles. The 464 electron diffraction patterns of the Ta film and Pd nanoparticles were acquired using a Thermo Fischer 465 Themis transmission electron microscope equipped with a Ceta 2 camera (insets in Supplementary Fig. 5a 466 and c). The accelerating voltage was 300 kV and a 10 m SA aperture was used to reduce the area, from 467 which the diffraction was collected with the central beam blocked by a beam stop. To calculate the structure 468 factor from each diffraction pattern, a mask was generated to remove the beam stop by properly 469 thresholding the intensity. By fitting the first diffraction ring, the centre of the diffraction pattern was 470 identified and the radially averaged intensity was obtained. A gold sample was used as a reference to 471 calibrate the reciprocal space unit, yielding the intensity distribution as a function of the spatial frequency, 472 I(q). The structure factor, S(q), was computed from I(q) by using the SUePDF software 56 , where the atomic 473 form factor was set by properly selecting the chemical species and electron energy. The background was 474 optimized by specifying the pre-peak and the tail location of I(q), the number of middle reference points 475 and the maximum fitting order. Proper parameters were selected during this step to ensure that the resulting 476 S(q) oscillates and converges to unity at large q ( Supplementary Fig. 5a and c). The PDF was computed by 477 taking the Fourier transform of S(q). For a diffraction pattern with a high signal-to-noise ratio, its reduced 478 PDF has a linear dependence near the origin, from the slope of which the atomic density can be 479 extracted 41,57 . But our samples are very thin ( 10 nm) and their electron diffraction patterns do not have a 480 sufficiently high signal-to-noise ratio, resulting in some oscillations at low spatial frequency. To correct for 481 the oscillations, SUePDF was used to normalize the reduced PDF by fitting a straight line from the origin 482 to the left valley of the first peak, from which the final PDF was obtained ( Supplementary Fig. 5b and d).

483
Image pre-processing. The following four steps were used to perform image pre-processing.

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iii) Background subtraction. For each denoised image, a mask slightly larger than the sample was 502 generated by thresholding. From the background outside the mask, the background level within the mask 503 was estimated using Laplacian interpolation. The estimated background was subtracted from the denoised 504 image. 505 iv) Image alignment. The images in each tilt series were aligned using the following procedure.

506
The tilt series of the two Pd nanoparticles was aligned by the centre of mass and common line method, as 507 described in previous AET experiments 31,32 . For the Ta thin film, we first performed a pre-alignment by 508 using cross-correlation between the images of neighbouring tilt angles. Next, based on reference markers 509 in the sample (in this case we chose an isolated region as the reference marker), we used the common line 510 method and the centre of mass 31,32 to align the thin film along the tilt axis and perpendicular to the tilt axis, 511 respectively. We repeated this alignment process until no further improvement could be made.   iii) The PDF was scaled to approach one at large pair distances.

571
Using this procedure, we calculated the PDFs of all the atoms in the Ta thin film and two Pd nanoparticles.

572
From the PDF of each material, we determined the first valley position, corresponding to the first nearest 573 neighbour shell distance. This distance was used to compute the local bond orientational order (BOO) 574 parameters (see the section below), from which crystal nuclei were identified. After excluding the crystal 575 nuclei, the PDFs of the disordered atoms in the amorphous materials were re-calculated, shown in Fig. 1c.

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The local bond orientational order parameters. We calculated the averaged local BOO parameters (Q4 577 and Q6) to quantify the disorder of the amorphous materials 42,64 . The Q4 and Q6 values were computed based 578 on the procedure published elsewhere 42 , where the first nearest neighbour shell distance (Fig. 1c) was used 579 as a constraint. Q4 and Q6 were used to calculate the normalized BOO parameter, defined as 580 √Q 4 2 + Q 6 2 /√Q 4 fcc 2 + Q 6 fcc 2 , where Q 4 fcc and Q 6 fcc are the reference values of the fcc lattice. We 581 separated crystal nuclei from amorphous structure by setting the normalized BOO parameter larger than or 582 equal to 0.5 40 (red dashed lines in Supplementary Fig. 8a-c) .

583
The Voronoi tessellation, coordination number and local mass density distribution. The Voronoi 584 tessellation of each 3D atomic model was calculated by following procedure published elsewhere 4 . To 585 characterize the nearest neighbour atoms around each centre atom, a regulation was applied to each Voronoi 586 polyhedron, where neighbouring atoms with the facet area less than 1% of the total Voronoi surface area 587 were removed during the analysis 23 . All the Voronoi polyhedra were then indexed by 〈 3 , 4 , 5 , 6 〉 with 588 denoting the number of i-edged faces. The coordination number was calculated by ∑ .

589
The mass density for each atom was calculated by dividing the atomic mass by its atomic volume, 590 which is defined as the volume of its Voronoi polyhedron without regulation. The densities at all atomic 591 positions were interpolated onto a 3D grid and then convolved with a Gaussian kernel. The width of the