Vibrational Properties of Pd Nanocubes

Abstract

The atomic disorder and the vibrational properties of Pd nanocubes have been studied through a combined use of X-ray diffraction and molecular dynamics simulations. The latter show that the trend of the mean square relative displacement as a function of the radius of the coordination shells is characteristic of the nanoparticle shape and can be described by a combined model: A correlated Debye model for the thermal displacement and a parametric expression for the static disorder. This combined model, supplemented by results of line profile analysis of the diffraction patterns collected at different temperatures (100, 200, and 300 K) can explain the observed increase in the Debye-Waller coefficient, and shed light on the effect of the finite domain size and of the atomic disorder on the vibrational properties of metal nanocrystals.

Keywords: Debye–Waller coefficient; Vibrational properties of nanocrystals; X-ray diffraction; mean square displacement; metal nanocrystals; molecular dynamics; nanocubes.

Figure A1

Graphical explanation of Equation (A9). The function is a simplification of the static component trend calculated from MD simulations.

Figure A2

Static MSRD results from Figure 3 in the main text with static model of Equation (A9). Best fit of ${\sigma }_{0,max}^2$ are, respectively, 0.00594 (D = 41.84 Å), 0.00640 (D = 62.26 Å), and 0.00703 (D = 87.54 Å).

Figure A3

Static MSRD results for nanoparticle of roughly the same number of atoms of a cube of edge 6 nm (15,000 atoms), but with different degrees of truncation (0% is a cube, 100% an octahedron). The overall shape is always similar to the expression A9, except for the case of more “spherical” shapes (truncation 40% and 60%) that exhibit a trend closer to the sphere (Figure 3 in main text). Compared to Figure A3, the data have been presented with larger bins for a better visualization of the trends.

Figure A4

Graphical depiction of the common volume function along direction [111] for a cube.

Figure 1

Transmission Electron Microscopy (TEM) images of the Pd nanocubes (see Supporting Information in [23]) From (a) to (d) progressive magnification to the nanoparticles. Image (a) and (b) show the limited size dispersion and an example in (a) of the non-crystallographic fraction (the multiply-twinned particle on the right of the image). In (c) and (d) a magnification of a single nanoparticle, showing details of the truncated cube shape. (Images adapted from [23], with permission from iucr journals, 2019).

Figure 2

Trends of the Mean Square Relative Displacement (MSRD) for a spherical (a) and truncated cube (b) crystal, N = 6986 and N = 4923 Pd atoms, respectively; corresponding number of atom pairs (N_r) is shown by histograms in (c) and (d). MSRD is shown for the total (blue), static (orange), and thermal (green) components, as of Equation (3), (4), and (5), respectively. Insets in (c) and (d) show images of the nanoparticle shapes.

Figure 3

Static MSRD component from Molecular Dynamics (MD) simulations for three spheres (a) and three truncated cubes (b) of increasing size. The dotted line is the trend according to the parametric model of Equation (7).

Figure 4

MSRD trends for a spherical Pd nanocrystal with 7 nm diameter (N = 12161 Pd atoms): total (blue), static (orange), and thermal (green) components of MSRD, with the best fit of the Correlated Debye (CD) model (line) (a). The sum of CD model and static component (dark blue) is compared with the total MSRD (blue) (b).

Figure 5

Whole powder pattern model (WPPM) results. X-Ray Diffraction (XRD) patterns (circle), model (line), and difference (residual, line below) at 300 (a), 200 (b), and 100 K (c). Insets: in (a), refined shape of the truncated nanocube; in (b), Thermal Diffuse Scattering (TDS) component at 100 (blue), 200 (red), and 300 K (black); in (c), intensity log scale plot.

Figure 6

WPPM results. Lognormal distribution of cube edges, D (a) and microstrain distribution along three crystallographic directions: [100] (black), [110] (red), [111] (green) (b).

Figure 7

Left axis: Debye-Waller coefficient, B_ISO, from the XRD patterns collected at 100, 200, and 300 K (♦) for the Pd nanocubes and corresponding room temperature (RT) value for bulk Pd (◊) [36]. Red line is the combined model of Equation (8), black line is the extrapolation according to the Debye model for the literature bulk value. The arrow highlights the increase in the static component. Right axis: Unit cell parameter measured from the XRD patterns (▪). See text for details.