Introduction
The surface properties of metal oxides have attracted great attention for years. Especially, TiO2 has been studied intensively owing to its wide-ranging applications such as photocatalysis, energy storage, fuel cells and pollution abatement [1–9]. In catalysis, despite the fact that TiO2 is not itself an efficient catalyst, deposition of metal particles on TiO2 can significantly enhance the catalytic activity. For more active sites, decreasing the size of metal particles with increasing surface-to-volume ratio which maximizes the surface area exposed to the reactant is the most popular way to design the efficient catalytic system. However, metal atoms in smaller particles with higher chemical potential generally exhibit a larger thermodynamic driving force to sinter or agglomerate and thereby the number of active sites decreases at realistic technology process conditions [10–13]. Thus, suppressing sintering of supported particles becomes one of the dominant issues for gaining efficient catalysts.
As TiO2(110) is thermally stable, reducible and nonpolar, TiO2(110) is used as the support in fundamental as well as in industrial research. In metal-TiO2(110) system, sintering of supported metals is frequently evidenced in experiments [14–16]. Take Pd on TiO2(110) for example, by studying the sintering of Pd particle after annealing, researchers suggested that the sintering of Pd particles could be attributed to the particle migration and coalescence [17–19], in which the growth of the particles is by the whole metal particles diffusion across the support surface. However, Howard et al. [20] indicated that the evolution of the size distribution observed in real time by scanning tunnelling microscopy (STM) is consistent with Ostwald ripening mechanism in which larger particles grow at expense of smaller particles through the migration of atoms. Although Su et al. suggested that particle coalescence is possible only for clusters with less than 5 Pd atoms while bigger Pd particles prefer to grow through Ostwald ripening on the CeO2(111) surface [21], the real system may be more complicated in which two mechanisms are both likely in the process of sintering [22]. Obviously, there is still room for discussion regarding sintering mechanism of TiO2(110)-supported Pd. The process of particle migration and coalescence can be directly observed by the modern technology such as in situ transmission electron microscope (TEM). Nevertheless, the process of the migration of atoms of Ostwald ripening is difficult to be captured. Thus, the theoretical study of Ostwald ripening of TiO2(110)-supported Pd particles would be helpful to extend the understanding of the sintering.
Simultaneously, the initial size of particles can largely influence the sintering of the supported particles. Campbell suggested that the metal adsorption energies increase with increasing size of the nanoparticles until their diameter exceeds about 6 nm [23]. It is indicated that the smaller particles could be more unstable than bigger particles. Subsequently, Hu and Li found that the smaller particles have lower onset temperature and shorter half-life time for the ripening of supported particles [24]. Moreover, the sintering of Pd particles is a strong function of the initial conditions such as the initial particle size [24,25]. Recently, it is found that the sintering rates of Al2O3-supported Pd particles with a larger initial particle size were slower than those with smaller ones [26]. However, similar size effect has not yet been quantitatively addressed for Pd on TiO2(110) until now.
During recent years, a kinetic method was developed to study of Ostwald ripening according to an ab initio atomistic thermodynamic theory [27–29]. The Ostwald ripening of late transition metals on TiO2(110) has been studied by the kinetics [30]. However, the surface energy of the metal is used from the experiments [31]. As we know, many experiments have been successfully discussed by first-principles calculations [32]. Here, based on first-principles calculations including the surface energy and the total activation energy, we will extend the kinetic method of Ostwald ripening to provide quantitative information on TiO2(110)-supported Pd particles. It will be helpful to further improve our understanding of the sintering of TiO2(110)-supported metal particles.
Theoretical method
Ripening kinetics
The generalized kinetics rate equation of Ostwald ripening (OR) for supported particles under reaction condition was formulated in the earlier work (and reference therein) [28,33]. The fundamental rate equation is derived from microscopic origin and can be applied to a wide range of temperature, pressure, composition, support and particle distribution. When one particle with a curvature radius (R) on one support, the ripening rate equation for the supported particle is
Three main parts are included in this equation. There are the prefactor part , chemical potential part and total activation energy part (Etot). Prefactor is bound up with the ripening mechanism (diffusion and interface control). It can be largely influenced by the shape and size of supported particles. Detailed description about prefactor can be found in Ref. [29]. The critical curvature radius R” is the size of particles when the monomer attachment and detachment are in dynamic balance. kB is Boltzmann constant and T is temperature.
Chemical potential of atoms in supported particles with regard to bulk counterpart is approximated here by Gibbs-Thomson (G-T) relation [34].
where and are the volume of the atom and the surface energy of supported metal particles respectively. When the particle size is small than 6 nm diameter, would frequently depend on the size of the supported particles. It originates from that the coordination number of atoms of the surface decreases as the size decreases. Considering modest influence on the result of ripening [35], we ignored the effect of the change of with size. For simplicity, the surface energy of Pd particles (88.1 meV/Å2) originating from the DFT calculation in Section 3.1 was used in the kinetics of ripening.
Etot is the total activation energy of ripening process. It is the sum of the formation energy (Ef) of monomer or metal-reactant complexes and corresponding diffusion barrier (Ed),
where Ema/ox is the total energy of the adatom on the support, Eox is the total energy of the support, and EB is the total energy of the bulk metal. Ema/ox, Eox, EB and Ed can be obtained by density functional theory (DFT). The details of the calculations will be presented in Section 2.2.
DFT calculations
To get the surface energy of Pd and Etot, DFT calculations were performed using the Vienna Ab initio Simulation Program (VASP [36–38]). Surface energy of Pd facets was calculated using the (1×1) slab model, and G-centered k-point meshes of were used for slab calculations. Through a series of comprehensive tests, it was determined that vaccum and slab thicknesses of at least 25 Å separated by a 15 Å vacuum layer were sufficient to ensure necessary convergence. The projector-augmented wave (PAW) method together with Perdew-Burke-Erzernhof (PBE) exchange-correlation functional was used. The kinetic energy cutoff was 400 eV. This system was relaxed by using of conjugate-gradient algorithm until the Hellman-Feynman force [39] on each atom was less than 0.02 eV/Å. The surface energy () is computed using the following expression:where ES is the total energy of the slab, EB is the bulk energy for metal Pd, n is the number of Pd unit cell in the slab, and A is the corresponding surface area of the slab.
We used revised Perdew-Burke-Erzernhof (RPBE) exchange-correlation functional, which is believed better to describe the surface adsorption alleviating the potential over binding [40], to calculate Etot. The kinetic energy cutoff for the plane wave basis set was 400 eV. Here the spin polarization was considered in this paper. The optimized crystal parameters of the rutile TiO2 bulk are a = 4.661 Å, c = 2.968 Å which are comparable with experimental values of a = 4.593 Å, c = 2.958 Å [41]. A (2×4) surface supercell of rutile TiO2(110) with a slab of four Ti-O layers that was separated from its periodic images by vacuum space of 15 Å was used. For Pd atom adsorption on the surface, the top two Ti-O layers and the adsorbates were allowed to relax while the other atoms in the bottom layers were fixed in their positions. By using conjugate-gradient algorithm, this system was relaxed until the Hellman-Feynman force [39] on each atom was less than 0.03 eV/Å. The G point was used to sample the surface Brillouin zone, as has been done in previous studies involving TiO2 [27,42]. The climbing image nudged elastic band (CI-NEB) method [43,44] was utilized to locate the transition states for the diffusion of a single Pd atom on the surface. In this study, at least seven images (including the initial and final states) were calculated and vibrational analysis showing a single imaginary mode was used to confirm the transition states optimized. In all DFT calculations, we neglected zero-point energies and entropy corrections in this paper.
Results
DFT calculation
DFT calculations were used to obtain the average surface energy of Pd. We considered various orientations of facet-center cubic Pd including (111), (100), (210), (221), (311) and (322). After the surface energies of Pd surfaces considered were calculated, the result is shown in Table 1. Subsequently, Fig. 1 presents the equilibrium morphology of Pd particle which can be obtained according to the Wulff construction. Table 1 shows the exposed facets, corresponding surface energies and ratio. It is clear that (111), (100), (322) and (221) facets cover 49.73%, 15.99%, 14.26% and 14.12% region exposed, respectively. Furthermore, it is found that the average surface energy of 88.1 meV/Å2 for Pd particle. The difference between the average surface energy and the surface energy of (111) is less than 6 meV/Å2. The slight difference is in that (111) covers about 50% of the equilibrium morphology of Pd particle. Our calculated average surface energy is close to the measured surface energy of liquid Pd (94 meV/Å2) [45].
Subsequently, we calculated the formation energy (Ef) of Pd on TiO2(110). Various adsorption sites were considered and only the most stable position is indicated below. The most favorable position for Pd on TiO2(110) is on the hollow site (Fig. 2(a)). Pd atom prefers to bond with a protruded twofold coordinated O atom (O2c) and a fivefold coordinated Ti atom (Ti5c), in line with the previous calculations of Pd [46,47]. Here, the formation energy (Ef) of the most stable position as shown in Fig. 2(a) is 2.05 eV. Transition state (TS) was searched on every possible diffusion path by CI-NEB method. The diffusion barrier (Ed) considered is the least one of the values of the possible diffusion paths. In Fig. 2(b), transition state is on the migration of Pd atom along [001] direction. Ed of Pd along [001] (0.17 eV) is 0.04 eV smaller than the migration along [10]. Since Ef is much larger than Ed, it is indicated that the ripening of TiO2(110)-supported Pd particles are in the interface control limit in which the most of the energy for ripening should be consumed during the detachment/attachment process rather than the diffusion process.
Tab.1 Calculated surface energies (γ, in meV/Å2), surface area proportion (ƒ) of the facets exposed on faced-centered cubic (FCC) Pd from Wulff constructions |
| Pd facets | fi/% | gi/(meV·Å−2) |
|---|---|---|
| (111) | 49.73 | 82.6 |
| (100) | 15.99 | 95.8 |
| (322) | 14.26 | 90.1 |
| (221) | 14.12 | 91.8 |
| (210) | 3.58 | 101 |
| (311) | 2.32 | 98.8 |
| Ʃ | 100 | 88.1 |
Fig.2 Side view (top) and top view (bottom) for (a) the most stable atom position and (b) corresponding transition state position of Pd on TiO2(110). Gray and blue balls represent titanium and palladium atoms. Magenta and red balls represent the upmost bridging oxygens and other oxygens respectively |
Ripening of Pd on TiO2(110)
Fig.3 (a) PSD corresponding to different temperatures; (b) evolution of the normalized volume V, dispersion D and particle number N, and average diameter〈d〉in right y-axis versus ramping temperature for Pd on TiO2(110). Star represents onset temperature Ton. The supported particle ensemble responds to a linear temperature ramp process starting from 200 K at a rate of 1 K/s. The contact angle (a) was set as 90o. The surface energy of Pd is 88.1 meV/Å2.〈d0〉= 3 nm, rsd = 10% |
Firstly, DFT-resulted γ= 88.1 meV/Å2 and Etot = 2.22 eV were used to investigate the ripening kinetics under temperature programed condition. The initial particle size distribution for Pd particles was set as a normalized Gaussion distribution. For this distribution, the initial main particle size (〈d0〉) is 3 nm while the relative standard deviation (rsd) is 10%. The supported particle ensemble responds to a linear temperature ramp process (temperature programed condition). In the temperature process, the starting temperature is 200 K and subsequent it increases with a rate of 1 K/s. Figure 3 shows the details evolution of TiO2-supported Pd particles under this temperature programed condition. The evolution of particle size distribution (PSD) is shown in Fig. 3(a). As the ramping temperature is less than 750 K, the change in PSD shape is very slow. When the temperature is at 750 K, it is found that the peak height decreases about 7% and the peak position shifts about 0.1 nm from left to right. When the temperature is higher than 750 K, TiO2-supported Pd particles ripen faster than before. Moreover, it is obvious that there is a long tail toward the small particles for PSD shape at 850 K in Fig. 3(a). This style of PSD shape is namely Lifshitz-Slyozov-Wagner (LSW)-type distribution. This is because one larger particle needs expense a couple of smaller particles together. In line with above finding in PSD, the normalized dispersion and total particle number plotted in Fig. 3(b) starts to decrease only when the ramping temperature increases up to a threshold. After this, both dispersion and particle number decrease rapidly along with the increase of the average size. Here, onset temperature Ton, corresponding 10% decreases of the particle number for ripening under temperature programed condition, is defined to compare with experimental results. When 〈d0〉 and rsd of TiO2(110)-supported Pd particles are 3 nm and 10% respectively, resulted Ton is 777 K. In comparison with the previous study, the surface energy of Pd used in this paper is lower than that used in the literature [30]. This is understandable that the calculated Ton in this paper is about 25 K higher than the previous reported result [30] since particles with lower surface energy would have stronger ripening resistance [29].
To better study the thermal resistance against ripening for Pd on TiO2(110), OR evolution is performed under isothermal condition. The initial PSD under isothermal condition is the same as those under temperature programed condition. Under isothermal condition (600 K), the evolution of the PSD is shown in Fig. 4(a). First 25% decrease of the particle number of supported Pd particles occurs at t = 5 days. At this time, the peak height decreases about 40% and the peak position right shifts 0.3 nm. When another 25% of the initial particle number decreases, it occurs at t = 14 days and the peak position is at 3.9 nm. While third 25% decrease of the particle number occurs at much late t = 56 days and the corresponding peak at 4.9 nm. At this moment, PSD shape is clearly LSW-type distribution. There is a long tail from 4.9 nm to the smaller size. This PSD shape is similar with that at 850 K under temperature programed condition in Fig. 3(a). It is obvious that the ripening rate gradually decreases with time, accompanying with an increase of the average particle size. In Fig. 4(b), the normalized dispersion and particle number decrease but the average diameter increases with time. It is found that the evolution of Pd dispersion becomes smooth after about 20 days. The variation trend of the Pd particle number is similar to that of Pd dispersion. However, the decreasing rate of Pd particle number is even larger than that of the Pd distribution. Accordingly, the average size increases gradually. To better evaluate the long-time behavior of ripening, the half-life time (t1/2), which represents the time necessarily for half decrease of the particle number, is used as measurement of lifetime of Pd particles. When Pd particles (〈d0〉= 3 nm and rsd = 10%) on TiO2(110) is at 600 K, resulted t1/2 is about 14 days.
Fig.4 (a) PSD corresponding to different time; (b) evolution of the normalized volume V, dispersion D and particle number N, and average diameter 〈d〉 in right y-axis versus ripening time for Pd on TiO2(110). Star represents half-life time t1/2. The contact angle (a) was set as 90o. The surface energy of Pd is 88.1 meV/Å2. 〈d0〉= 3 nm, rsd = 10%, T=600 K |
Figures 5(a) and 5(b) show the dependence of Ton and t1/2 on size respectively. It can be found that both Ton and t1/2 have the linear function with the size of Pd particles. Moreover, both Ton and t1/2 decrease faster along with the decrease of size. In Fig. 5(a), Ton decreases about 300 K with the decrease of 〈d0〉 from 6 to 1 nm. By STM, Howard et al. found Ostwald ripening of TiO2(110)-supported Pd particles at 750 K. Although the particle diameter in the experiment is from about 3 nm to about 11 nm, the experimental temperature is in our calculated range of Tons. It is found that our calculated Ton for particles smaller than 6 nm is lower than the so-called Tamman temperature (914 K, typically half of the bulk melting point of Pd [48]). The origin is that the melting point for smaller particles of Pd is smaller than that for the bulk [49]. When the size is higher than 6 nm, the calculated Ton could be close to Tamman temperature. Moreover, since Tamman temperature ignores the size and support effect, our calculated results could provide more information to quantitively elaborate the ripening for Pd on TiO2(110). While 〈d0〉 increases every 1 nm, Ton increases more than 45% from 1 to 6 nm but less than 16% from 6 to 16 nm. The increment ratio of Ton for increasing every 1 nm is even less than 1% for particles larger than 16 nm. Similar size effect is also found between t1/2 and 〈d0〉 as shown in Fig. 5(b). Along with the decrease of 〈d0〉 from 6 to 1 nm, t1/2 decreases 5–8 orders of magnitude for 300, 450, 600 and 750 K respectively. However, the increment of t1/2 for increasing every 1 nm is less than 0.4 orders of magnitude for particles larger than 6 nm. It is concluded the small size would dramatically worsen the resistance against ripening for its lower onset temperature and shorter half-life time. The reason is the smaller particle has higher chemical potential (as indicated in Eq. (2)) to promote ripening. The size effect is more concrete with the small particle size (less than 6 nm). This effect of size dependence on ripening coincides with that obtained by Campbell groups [23,50]. To better show the picture of the size effect, Fig. 5 just provides the value of Ton and t1/2 for particles less than 6 nm. Moreover, along with the increase of temperature from 300 to 750 K, the decrement of t1/2 from 1 to 6 nm decreases from about 8 orders of magnitude for 300 K to 5 orders of magnitude for 750 K. It is indicated that higher temperature can mitigate the different performance on ripening originating from the particle size.
Figures 6(a) and 6(b) show the change of Ton and t1/2 along with rsd. When the monodispersity is the poorest (rsd = 50%), Ton and t1/2 are 641 K and 6 days (600 K) respectively. On the contrary, when the monodispersity is excellent (rsd = 10-3%), Ton and t1/2 are 857 K and about 2.5×104 days (600 K), respectively. It is evident that both Ton and t1/2 increase along with the size of Pd particles. Furthermore, it is found that Ton decreases about 216 K from rsd = 10-3% to rsd = 50% while t1/2 for rsd = 10-3% is about 4166 times of that for rsd = 50%. These results tell us that the monodispersity of TiO2(110)-supported Pd particles is helpful to mitigate the ripening, particularly for the smaller particles. Simultaneously, t1/2 decreases with the increase of temperature. The increment of t1/2s for 300 and 450 K, 450 and 600 K as well as 600 and 750 K are about 11, 6 and 4 orders respectively. It is found that the increment of t1/2 between every 150 K could decrease along with the increase of the temperature. The difference of t1/2 between the poorest monodispersity (rsd = 50%) and the highest monodispersity considered (rsd = 10-3%) are about 3.4, 3.5, 3.6 and 3.7 orders of magnitude for 300, 450, 600 to 750 K respectively. The different performance of the ripening originating from different rsds is more obvious at higher temperature than lower temperature. This temperature effect for rsds is the inverse of that for 〈d0〉. It is indicated that the influence of rsd on ripening may be more sensitive at higher temperature.
Conclusions
We calculated the surface energy of some exposed facets of face-centered cubic Pd. It is found that (111) covers most of the exposed area of Wulff construction of Pd. The average surface energy of the equilibrium morphology is close to the surface energy of (111). The most stable adsorption of Pd on TiO2(110) is found that Pd prefers to occupy hollow site. Moreover, it diffuses easier along with [001] than with [10] for its lower diffusion barrier. Subsequently, according to the average surface energy of Pd and the total activation energy (the formation energy and diffusion barrier) calculated from the density functional theory, the ripening of TiO2(110)-supported Pd particles is simulated by a kinetic method. The result concluded that smaller Pd particles would worsen the performance of ripening-resistance according to its lower onset temperature and shorter half-life time. This size effect could be mitigated along with the increase of temperature. Furthermore, it is verified that the monodispersity can improve ripening resistance especially for the smaller Pd particles. However, the different performance of the ripening originating from difference of the relative standard deviation is more obvious at higher temperature than lower temperature. This temperature effect for the relative standard deviation is the inverse of that for the initial main particle size. It is indicated that the influence of dispersity of TiO2(110)-supported Pd particles on ripening may be more sensitive at higher temperature. In this contribution, we extend the first principle kinetics to elaborate the ripening of Pd on TiO2(110). It is expected that the information would be helpful to the study on sintering for metals on oxide in experiments.