Solid-solid transitions in Pd-Pt nanoalloys

Solid-solid transformations in Pd-Pt nanoalloys in the size range 32–38 atoms and for different compositions are computationally studied by the superposition approximation to the partition function, and by molecular dynamics simulations. A broad spectrum of transition types is shown to take place. These transition types are: (i) one-to-one type, in which the global minimum, which is dominant at low temperatures, transforms into another single isomer with increasing temperature; (ii) one-to-many type, in which the transition is from a single isomer to a family of other isomers; (iii) many-to-many type, in which the transition is between two different families of isomers; (iv) many-to-one type, in which the effect of vibrational entropy is to greatly reduce the number of relevant structures with increasing temperatures. We provide a rationale for these behaviors, which stem from the interplay between energetics and vibrational entropy effects. The vibrational entropy is explained by analyzing the vibrational density of states and the specific features of the normal modes. Quantum effects on the structural transitions are also discussed.

Figure 1

(a) Specific heat, (b) equilibrium probabilities, and (c) dominant structures for ${\text{Pd}}_{32}{\text{Pt}}_6$. In (a) and (b) full and dashed lines correspond to classical and quantum results, respectively. In (b) green and red lines correspond to the cDh an pc6 structures respectively. In c), the cDh structure is on the left and the polyicosahedral pancake (pc6) is on the right. Pd and Pt atoms are shown by small and big spheres, respectively. Note the different shape of the Pt cores in these structures: compact shape for the cDh and hexagonal ring for the pc6.

Figure 2

Ratio between $Z^{\mathrm{vib}}$ of the different minima and $Z^{\mathrm{vib}}$ of the GM in the classical approximation, as a function of the total energy in ${\text{Pd}}_{32}{\text{Pt}}_6$ structures. Red squares correspond to the pc6 cluster and its homotops, green circles to the cDh GM and its homotops, while all others structures are indicated by blue triangles.

Figure 3

Low-temperature phase diagram for ${\text{Pd}}_{n-m}{\text{Pt}}_m$. The $x$ and $y$ axis represent platinum content $m$ and the cluster size $n$, respectively, while the $z$ axis represent the temperature. The two planes correspond to the dominant structural family at 0 K (GM) and 350 K.

Figure 4

Ratio between $Z^{\mathrm{vib}}$ of the different minima and $Z^{\mathrm{vib}}$ of the GM in the classical approximation, as a function of $f_{555}$ in ${\text{Pd}}_{23}{\text{Pt}}_{11}$. The colored shades show the predominant structural family in the region.

Figure 5

(a) specific heat (b) equilibrium probabilities, and (c) dominant structures for ${\text{Pd}}_{23}{\text{Pt}}_{11}$. In (a) and (b), full and dashed lines correspond to classical and quantum results, respectively. In (b) the green lines correspond to the Th family, the red lines to the twinned family and the light blue to the P12 family. In (c), the sizes of the structures are proportional to their equilibrium probabilities at 150 K (left) and 400 K (right).

Figure 6

(a) specific heat, (b) equilibrium probabilities, and (c) dominant structures for ${\text{Pd}}_{19}{\text{Pt}}_{15}$. In (a) full and dotted lines correspond to classical and quantum results for the specific heat, respectively. In (b) only results from the classical approximation are shown: the Dh, twinned and P12 families are marked with red, blue, and light blue lines, respectively; full lines correspond to the total equilibrium probability of a family, while the dashed lines show the equilibrium probability of its dominant cluster. In (c) the sizes of the structures are proportional to their equilibrium probabilities at 100 K (left) and 220 K (right).

Figure 7

Ratio of frequencies ${\omega }_n^{\left(A\right)}/{\omega }_n^{\left(B\right)}$ (see text) of normal modes for different pairs $(A,B)$ of structures. Top: $(+)$ ${\text{Pd}}_{32}{\text{Pt}}_6$ with $A$ and $B$ cDh GM and pc6 structure; $(\times )$ ${\text{Pd}}_{19}{\text{Pt}}_{15}$ with $A$ and $B$ Dh GM and P12 structure; $(*)$ ${\text{Pd}}_{23}{\text{Pt}}_{11}$ with $A$ and $B$ twinned and P12 structure; $(\square )$ ${\text{Pd}}_{23}{\text{Pt}}_{11}$ with $A$ and $B$ Th GM and P12 structure; $(△)$ ${\text{Pd}}_{23}{\text{Pt}}_{11}$ with $A$ and $B$ twinned and Th structure. Bottom: $(△)$ ${\text{Pd}}_{23}{\text{Pt}}_{11}$ with $A$ and $B$ two different P12 structures; $(▽)$ ${\text{Pd}}_{23}{\text{Pt}}_{11}$ with $A$ and $B$ two different twinned structures; $(\diamond )$ ${\text{Pd}}_{17}{\text{Pt}}_{15}$ with $A$ and $B$ two different Dh structures.

Figure 8

VDOS for the cDh GM of ${\text{Pd}}_{32}{\text{Pt}}_6$ and the competing pc6. Frequencies $\nu =\omega /2\pi$ are shown here. The frequency step used to construct the histogram is $\text{d}\nu =0.04\text{THz}$. A small $\left(\sigma =0.04\text{THz}\right)$ Gaussian smearing has been used for a better visualization.

Figure 9

$f_{555}$ as a function of time ${\text{Pd}}_{21}{\text{Pt}}_{15}$ (top) and ${\text{Pd}}_{20}{\text{Pt}}_{15}$ (bottom). $f_{555}$ has been calculated by locally minimizing the structures each 1.6 ns during the molecular dynamics simulations at temperature $T=300$ K.