Ab initio morphology prediction of Pd, Ag, Au, and Pt nanoparticles on (0001) sapphire substrates

Nanoparticles exhibit unique properties that differ from those of bulk metals, and extensive research on their properties has been conducted to synthesize them with specific characteristics.^1–3 Previous studies have established that the properties of nanoparticles depend on their size and shape.^4–8 However, accurate experimental investigation of the morphology of these nanoparticles is challenging, especially when synthesized on substrates. In our previous studies, we introduced a novel real-time method based on acoustic techniques for evaluating the morphology⁹ and growth of nanoparticles on sapphire substrates.^10,11 We analyzed the morphology of Pd-based core–shell nanoparticles by examining the time-dependent variations in the full width at half maximum (FWHM) of the resonant peak in relation with the surface energy of each species. The shape and internal structure of the core–shell nanoparticles were clarified considering the surface energy of each metal. However, according to the well-known Young–Dupre equation,¹² the interfacial effect between nanoparticles also influences the morphology of nanoparticles. Thus, these effects must be considered to gain more comprehensive insights into the morphology.

In this study, to complement our previous research, we used energy principles to predict the morphology of Pd, Ag, Au, and Pt nanoparticles on (0001) sapphire substrates. To this end, density functional theory (DFT) atomistic simulations were performed, incorporating the Young–Dupre equation. The computational results were compared with our previous experimental observations to validate both the previously proposed experimental methods and $abinitio$ morphology prediction using DFT simulations, as introduced in this study.

DFT atomistic simulations were performed using the Vienna Ab initio Simulation Package.¹⁴ The electron–ion interaction was described using the projector-augmented wave method.¹⁵ The exchange-correlation between electrons was treated using the Perdew–Burke–Ernzerhof generalized gradient approximation,¹⁶ with an energy cutoff of 520 eV for the planewave basis set. Energy convergence criteria for the electronic and ionic structure relaxations were set as $1.0× 10 − 6$ and $1.0× 10 − 3$ eV, respectively. A $5×5×1$ k-point mesh was used for all atomic models shown in Fig. 1.

Table II provides the calculated values for $E X$⁠, $E X - A l 2 O 3$⁠, $A A l 2 O 3$⁠, and $W X - A l 2 O 3$⁠. Notably, $W X - A l 2 O 3$ for Ag and Au are lower than those for Pd and Pt, suggesting a more spherical shape for Ag and Au nanoparticles, as implied by Eq. (1). Table III presents the calculated $γ X$ and contact angle $θ$ for each nanoparticle size. For Ag, the calculated surface energy is similar to those reported previously.^23–25 The plots of $γ X$ for different nanoparticle sizes are shown in Fig. 3. The contact angles for all nanoparticles are larger than $90 °$⁠, indicating their spherical nature. In particular, Au nanoparticles exhibit a higher contact angle compared with other nanoparticles. The spherical nucleation of Ag nanoparticles on the Si substrate has been experimentally observed.²⁶ This outcome is consistent with our result, even though the substrate is different. The difference in contact angle between each species is not substantial, although $W X - A l 2 O 3$ for Ag and Au are lower than those of Pd and Pt. This phenomenon is attributable to the lower surface energy of Ag and Au, resulting in the nanoparticles being flattened on the substrate. Thus, although we discussed the morphology of nanoparticles on the substrate only using the surface energy of nanoparticle in our previous study,¹¹ not only the surface energy, but the interfacial effect between the nanoparticle and substrate is also important; the competition between adhesion energy $W X - A l 2 O 3$ and surface energy $γ X$ is crucial for understanding the morphology of nanoparticles on the substrate, with these values being highly species-dependent.

According to Table III and Fig. 3, the surface energy decreases as the nanoparticle size increases, consistent with findings of prior studies.^23–25 Eventually, as nanoparticles grow, the contact angle decreases, and nanoparticles assume a penny-like flattened shape. This is attributable to the decreasing fraction of edge in the total surface area. For sufficiently large nanoparticles, the effect of edges becomes negligible, and surface energy can be accurately described using only the surface energy of specific Miller index planes, outlined in Table I. Considering the shape of adequately large nanoparticles (⁠ $n=∞$⁠) as truncated octahedrons with $(111)$ hexagons and $(100)$ squares,²¹ we estimated their surface energy, as indicated in Table III and Fig. 3. Generally, $γ X$ of the smallest nanoparticles (⁠ $n=55$⁠) is 50% larger than that for a nanoparticle with $n=∞$⁠. Owing to the inherently higher surface energy of Pd and Pt compared with Ag and Au, the edge effect on nanoparticles is more significant for Pd and Pt, resulting in a larger variation in contact angle $θ$ with respect to $n$⁠, compared with that for Ag and Au. Through linear interpolation of the plots, we determined that the edge effect becomes negligible at $n=390$⁠, 470, 440, and 360 for Pd, Ag, Au, and Pt, respectively. Considering that the radius of conventional nanoparticles is larger than 10 nm,² the edge effect of the nanoparticles may be negligible in such conditions.

It is worth mentioning that the shapes of the nanoparticles predicted using Wulff construction tends to deviate from the optimal ones if the nanoparticle size is enough small because Wulff construction neglects contributions from edges and vertices and the discrete nature of atomic plane becomes important for the small-size nanoparticles.²⁷ Actually, in the size range discussed in our study, previous studies reported non-crystalline structures such as icosahedral and decahedral nanoparticles, asymmetric structures and complex structures due to surface reconstruction, that we cannot make using Wulff construction, become optimal.^28–32 To discuss this shape effect of nanoparticle to the contact angle, we employed potential energy (per atom) of $n=55$ icosahedral and $n=144$ decahedral Au nanoparticles from previous DFT calculation, which reported as optimal shapes at the nanoparticle size by Lambie et al.,³⁰ and use them as $E Xnp$ in Eq. (4) and re-calculated the contact angle of $n=55$ and $n=141$ Au nanoparticles roughly. Note we use the DFT data of $n=144$ decahedral Au nanoparticle alternatively for $n=141$ nanoparticle because there are no DFT data of $n=141$ nanoparticle in the literature of Lambie et al.³⁰ The result is $151 °$ for both $n=55$ and $n=141$ nanoparticles, which is almost the same as the original values in Table I (⁠ $152 °$⁠). We think this is because the energy difference between original FCC and optimal complex shapes are not large; only 0.001 eV/atom order (or lower) for Au and Pt though the complex shapes are optimal.³⁰ Eventually, we concluded that the shape effect to the contact angle is not large and negligible.

To validate our $ab$ $initio$ morphology prediction, we compared our results with our previous experimental data, specifically, the FWHM timings for Pd, Ag, Au, and Pt. The film thickness vs FWHM curves for pure Pd, Ag, Au, and Pt atoms, obtained during sputtering onto the sapphire substrate in our previous work,^10,11 were re-plotted in Fig. 4. In our previous work, we suggested that the FWHM peak indicates the time at which the nanoparticles of sputtered atoms come into contact. Thus, an earlier (later) appearance of the peak indicates that the nanoparticles on the substrate are flatter (spherical). This interpretation could successfully explain the formation dynamics of core–shell nanoparticles synthesized by sputtering different metals onto nanoparticles.^10,11 Applying this interpretation to the present results for single-element nanoparticles, Fig. 4 indicates that Pd and Pt nanoparticles may be flatter than Ag and Au nanoparticles, with Ag and Au having higher contact angles. Although the quantitative $θ$ cannot be directly determined from our experimental result, the indication of a higher contact angle of Au nanoparticles compared with Pd and Pt aligns with our prediction. However, the suggestion for Ag nanoparticles contradicts our predictions.

To find the reason of the above contradictions between the calculation and experimental observation, considering that the diffusivity of sputtered atoms onto the substrate also affects the nanoparticle morphology and thus the timing of FWHM peaks,^10,11 we investigated the activation energy of the diffusion of single X atoms on a sapphire (0001) surface using the drag method, while detecting the minimum energy path (MEP).³³ One X atom was placed at the FCC A stacking position on the surface of the Model (b) unit cell shown in Fig. 1. The energy change induced by the movement of the X atom from the initial to final A stacking position along the MEP was calculated to estimate the activation energy of diffusion. The initial estimates for the diffusion path of X atom were obtained through linear interpolation between the initial and final A positions, and nine atomic models of intermediate states were produced with equal intervals along the path for the drag method. Details of the energy and atomic structure changes along the path can be found in the  Appendix and supplementary material. The activation energies are calculated to be 0.20, 0.098, 0.055, and 0.37 eV for Pd, Ag, Au, and Pt, respectively. Compared with those of Pd and Pt atom, the activation energy values for the diffusion of Au and Ag atoms are low (less than 0.1 eV), indicating that these atoms can move more freely on the sapphire surface without significant thermal obstacles. This finding is qualitatively consistent with the timing of FWHM peaks shown in Fig. 4. The atomic diffusion process of sputtered atoms on the surface is well captured by the FWHM peaks: a faster surface diffusion results in a later timing of the peak. In this context, we argue that the size of nanoparticles can be discussed using our FWHM peaks. Small nanoparticles grow separately owing to the high activation energy of diffusion for Pd and Pt sputtering, resulting in the earlier appearance of FWHM peak. In contrast, atoms gather to form one large nanoparticle for Ag and Au sputtering, reducing the surface energy and delaying the timing of FWHM peaks. To confirm this interpretation, we investigated the size of nanoparticles resulting from experimental Pd and Au sputtering using scanning electron microscopy (SEM). The SEM images shown in Fig. 5 reveal large Au nanoparticles, whereas the Pd nanoparticles are small.

This paper proposes a novel strategy for the morphology prediction of Pd, Ag, Au, and Pt nanoparticles on (0001) sapphire substrates, based on DFT simulations and the well-known Young–Dupre equation. For all nanoparticles, the contact angles are larger than $90 °$⁠, indicating their spherical nature. Notably, Au nanoparticles demonstrate a higher contact angle compared with the other metal nanoparticles. As the nanoparticle size increases, the contact angles decrease, leading to a more penny-like shape on the substrates. The validity of the proposed $abinitio$ morphology prediction of method is demonstrated through a comparison with previous experimental results. The diffusivities of single Pd, Ag, Au, and Pt atoms on the substrate are evaluated by calculating the activation energy to discuss the inherent physics governing the timing of FWHM peaks. The results demonstrate the higher diffusivity of Au and Ag compared with that of Pd and Pt. From the comparison of DFT and experiment result, although there appears to be no significant correlation between the contact angles and timing of FWHM peaks, the diffusivity of sputtered atoms may indeed be linked to the timing of FWHM peaks. The timing can likely provide insights into the size of the nanoparticles rather than the shape. The proposed method is applicable for the morphology prediction of nanoparticles on other types of substrates, such as silicon, as well as for predicting the morphologies of core–shell nanoparticles. These aspects will be considered in future work.

See the supplementary material for the atomic structure changes along the MEP for the surface diffusion of an X atom on a sapphire (0001) substrate, which we explained in the  Appendix.

This study was partially supported by Japan Society for the Promotion of Science KAKENHI Grant No. JP21K03771. The authors would like to thank Koji Matsuura for his technical assistance.

The authors have no conflicts to disclose.

Akio Ishii: Conceptualization (lead); Formal analysis (equal); Funding acquisition (lead); Investigation (lead); Methodology (lead); Software (lead); Writing – original draft (lead); Writing – review & editing (equal). Nobutomo Nakamura: Formal analysis (equal); Investigation (equal); Writing – original draft (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The calculated changes in the potential energy and atomic structure along the minimum energy path (MEP) for the surface diffusion of an X atom (X = Pd, Ag, Au, Pt) on a sapphire (0001) substrate are shown in Fig. 6. The asymmetrical shape of the potential energy changes with respect to the middle reaction coordinate (denoted as 5) is because the Al atoms around X atoms are not all on the same $(0001)$ plane in the $α$- $A l 2 O 3$ structure: there are certain deviations in the position along the $[0001]$ direction between them. Notably, the atomic structure changes along the MEP do not exhibit significant dependence on the X species, consistent with the observations in our supplementary material. Conversely, the shape of the potential energy surface for diffusion appears to vary with the species. Notably, Pt exhibits a local minimum at the middle reaction coordinate 5, and the stable state for Ag atoms differs from that of other species, occurring at reaction coordinate 8. As the investigation into the underlying reasons for these species-specific variations is not the primary focus of this study, we refrain from delving into further discussion.