Sintering of Pt Nanoparticles via Volatile PtO2: Simulation and

Research Article pubs.acs.org/acscatalysis

Sintering of Pt Nanoparticles via Volatile PtO2: Simulation and Comparison with Experiments Philipp N. Plessow†,‡,§ and Frank Abild-Pedersen*,† †

SUNCAT Center for Interface Science and Catalysis, SLAC National Accelerator Laboratory, Menlo Park, California 94025, United States ‡ SUNCAT Center for Interface Science and Catalysis, Department of Chemical Engineering, Stanford University, Stanford, California 94305, United States § Institute of Catalysis Research and Technology (IKFT), Hermann-von-Helmholtz-Platz 1, D-76344 Eggenstein-Leopoldshafen, Germany S Supporting Information *

ABSTRACT: It is a longstanding question whether sintering of platinum under oxidizing conditions is mediated by surface migration of Pt species or through the gas phase, by PtO2(g). Clearly, a rational approach to avoid sintering requires understanding the underlying mechanism. A basic theory for the simulation of ripening through the vapor phase has been derived by Wynblatt and Gjostein. Recent modeling efforts, however, have focused entirely on surface-mediated ripening. In this work, we explicitly model ripening through PtO2(g) and study how oxygen pressure, temperature, and shape of the particle size distribution affect sintering. On the basis of the available data on α-quartz, adsorption of monomeric Pt species on the support is extremely weak and has therefore not been explicitly simulated, while this may be important for more strongly interacting supports. Our simulations clearly show that ripening through the gas phase is predicted to be relevant. Assuming clean Pt particles, sintering is generally overestimated. This can be remedied by explicitly including oxygen coverage effects that lower both surface free energies and the sticking coefficient of PtO2(g). Additionally, masstransport limitations in the gas phase may play a role. Using a parameterization that accounts for these effects, we can quantitatively reproduce a number of experiments from the literature, including pressure and temperature dependence. This substantiates the hypothesis of ripening via PtO2(g) as an alternative to surface-mediated ripening. KEYWORDS: DFT, catalysis, transition state, BEP, adsorbates

1. INTRODUCTION Heterogeneous metal catalysts are generally most efficiently used as dispersed particles, because this increases the catalytically active surface area.1−6 At the same time, the higher surface area means that agglomeration of these particles to fewer, larger particles is thermodynamically favorable. This thermodynamic driving force is the cause of sintering, which is an important factor in determining the long-term performance of real-world catalysts. In particular, sintering of Pt particles under an oxidizing atmosphere is important, due to the relevance of Pt in automotive exhaust cleaning and other important areas of catalysis.7 For the case of Pt, transmission electron microscopy (TEM) experiments have in many cases suggested thatdespite sinteringparticle migration is negligible.8−10 This means that sintering must occur via surface-mediated or gas-phasemediated ripening. It is well-known that Pt forms a stable volatile oxide, PtO2, and it has been hypothesized that this could be responsible for sintering behavior in the presence of O2(g).8,11−16 At the same time, most simulations of sintering phenomena have employed a functional form corresponding to surfacemediated transport of Pt atoms. While atoms are exchanged © XXXX American Chemical Society

through the particle perimeter in surface-mediated ripening, gas-phase ripening would be mediated through the surface of the particle.17 Consequently, the dependence of the flux of atoms on particle size should be different. Furthermore, the sintering rate will depend on oxygen pressure, as observed experimentally.14 Simulations of surface-mediated sintering generally require an estimate of the apparent activation energy required to remove an atom from the particle and let it diffuse over the support. Importantly, this will generally be higher than the energy of the atom in the most stable site of the support due to the diffusion barrier.18 Since these kind of values are hard to access, many simulations employ fitted parameters instead.8,9,13,14,19 The employed apparent activation energies were stated only in a few cases: 3.77 eV on silica14 relative to Pt(bulk), which is about −2.05 eV relative to Pt(g). Reliable data for the diffusion mechanism of platinum atoms on either alumina or silica is scarce. Density functional theory (DFT) calculations gave an adsorption energy for the platinum Received: June 10, 2016 Revised: September 9, 2016

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DOI: 10.1021/acscatal.6b01646 ACS Catal. 2016, 6, 7098−7108

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ACS Catalysis atom on α-Al2O3(0001) of −2.02 eV and a diffusion barrier of 0.66 eV.20,21 This gives an apparent activation barrier that is −1.36 eV relative to the gas phase and therefore considerably less stable than what would be required to make Pt atom migration on the support feasible. Similar results were reported for γ-Al2O3(111), with an adsorption energy of −1.99 eV and an activation barrier of 0.33 eV.22 We have recently investigated thermodynamically stable α-quartz(0001) surfaces and have found the interaction with Pt atoms to be weaker than −1 eV.23 While the adsorption in specific sites or defects of amorphous silica may be stronger than on α-quartz(0001), we do not expect that a viable path for migration of a Pt atom exists on a thermodynamically stable silica surface. This is simply because these surfaces only expose OH groups or Si−O−Si bridges which are chemically saturated and do not bind strongly. Generally, the kinetics of Ostwald ripening will depend on the type of support and the diffusing species: e.g., atom or molecule. Therefore, data on any one type of support or diffusing species generally do not allow conclusions regarding all supports and all potential diffusing species. Our discussion is based on all computed adsorption energies of Pt atoms that we are aware of as well as our investigation of PtO2 on quartz (see Results and Discussion). In our simulations, we will neglect Ostwald ripening, which we believe is appropriate for silica but which may not be true for all supports. For gas-phase-mediated sintering, the support always has an indirect effect on the transport in the gas phase, since it controls the relative positions of the particles and may block diffusion paths: for example, due to a porous structure. These effects are not taken into account in the simulation approach that we adopt. In recent work on Pt supported on ceria,24 it has been shown that Pt atoms can be trapped on ceria. The atomically dispersed platinum turned out to be catalytically active and resistant to sintering. Adsorption of Pt atoms on the support is not taken into account in our simulations, which we believe is appropriate for silica but is not for ceria and potentially for other supports. It is also important to keep in mind that at the high temperatures of sintering experiments (>700 K) there is a significant entropic driving force for evaporation of adsorbed species such as a Pt atom. Due to the low vapor pressures of such unstable species, the free energy gain upon evaporation will easily be −3Eh were correlated. A pseudopotential has been used for platinum.44 The D1 diagnostic45 is largest for PtO2 (0.098) but is still in the range considered trustworthy.46 In addition, the HF HOMO− LUMO gap of PtO2 is large at 11.3 eV. 7099


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3. RESULTS AND DISCUSSION 3.1. Ripening Model. Surface-mediated ripening has not been included explicitly in our simulations. As in our study of the adsorption of Pt atoms on α-quartz(0001),23 we have computed the adsorption energy of PtO2 on the thermodynamically stable quartz surfaces, the fully hydroxylated and fully dehydroxylated (reconstructed) surfaces. As for Pt atoms we find only very weak interaction (nonbinding at the RPBE level, ∼−0.2 eV when van der Waals interactions are taken into account using BEEF-vdW).36 The lack of strong interaction is not surprising, since the quartz surface consists of main-group elements (H, O, Si) that are saturated with stable, covalent bonds. Consequently, for SiO2, due to the weak binding of PtO2, gas-phase-mediated ripening can be expected to dominate over surface-mediated ripening. We stress that we currently do not have data for other surfaces, so that the importance of Ostwald ripening mediated by PtO2 cannot be ruled out for all supports. Gas-phase-mediated ripening via volatile PtO2(g) has long been discussed as a sintering mechanism, and a quantitative model for this process has been derived by Wynblatt and Gjostein.17 It is assumed that the chemical potential μ(r) of a particle with radius r can be described using the Gibbs− Thomson equation: μPt (r ) = μPt,bulk +

2γV r

J=A×

(1)

J(r )tot =

pb =

(2)

Here n is the number of Pt atoms, Atot(n) is the total area based on the Wulff construction, and γavg is the average of the surface free energies according to contributions from different facets. Due to the dimensionality (Atot ∝ r2 and n ∝ r3) this has the same functional form as the Gibbs−Thomson equation.13 From now on we will use the particle radius r when we refer to a specific particle size. We could equally well use the number of platinum atoms or the total volume, but for now it does not matter how exactly those quantities are related. The Wulff construction is based on ratios of surface energies obtained using DFT (see the Supporting Inormation for details). We note that this approximation is expected to break down for small particle sizes, because we enter a region where finite size effects and the contribution of edge energies become increasingly important.48−50 If a particle is in equilibrium with PtO2(g), the total flux of PtO2(g) between particle and gas phase is 0. The equilibrium pressure of PtO2(g) is given by the equilibrium constant and the oxygen pressure: p(PtO2 ) = K × p(O2 )

(4)

S × A(r ) × [pb − peq (r )] 2πmkT

(5)

Generally, we assume steady-state conditions, between the gas phase and particles, which means that the total flux between all particles and gas phase is 0. Within this procedure we do not consider any loss of Pt into the gas phase: e.g., the number of Pt atoms in the nanoparticles is conserved. From this, the background pressure pb in eq 5 is defined so that ∑rJ(r)tot = 0:

A tot (n) × γavg n

2πmkT

where A is the exposed area of the particle, which we derive from a Wulff construction (see the Supporting Inormation) and m is the mass of PtO2(g). A part of the molecules impinging on the surface will simply be reflected. The fraction of the flux of PtO2(g) that is actually formed from Pt(bulk) or is dissociated in the reverse reaction is given by the sticking coefficient for dissociative chemisorption S. This “reactive flux” J × S is the maximum flux of PtO2(g) molecules by which the particle can grow or shrink. Here we assume that the flux of O2(g) that is formed or consumed is not limiting. Although O2(g) generally has a small sticking coefficient,51 this is a reasonable assumption since the pressure of O2(g) is always several orders of magnitude higher than that of PtO2(g). Generally, the flux of O2(g) would become limiting for smaller particles which have the highest flux per area. In test simulations, we have found that explicit inclusion of the flux of O2(g) does not influence the results significantly. The next assumption in the model put forward by Wynblatt and Gjostein is that the there is a general, uniform “background” pressure, pb. This pressure is generally different from the pressure peq(r) at which an individual particle with radius r would be in equilibrium with the gas phase (eq 3). If one assumes a constant background pressure, the total flux of PtO2 molecules for a particle with radius r is given by

Here μPt,bulk is the chemical potential of bulk Pt at r → ∞, V is the volume per atom (∼15 Å3 for Pt), and γ is the surface free energy. It is important to note that the difference between surface energies and surface free energies will be significant at high temperatures and in the presence of adsorbates. In the following, we will use a similar approach based on particle shapes by Wulff construction:47 μPt (n) = μPt,bulk +

p

∑r A(r ) × peq (r ) ∑r A(r )

(6)

A particle that has peq(r) = pb neither grows nor shrinks and its radius, rcrit, is referred to as the critical radius.52 One can now, to first order in δt, compute the amount of Pt that is transferred between the particles via PtO2(g): δnPt(r ) ≈ J(r )tot × δt

(7)

The accuracy of this procedure depends on the time step δt, or, more precisely, on how much it changes the particle sizes and consequently their fluxes J. We use an adaptive time step and have tested the accuracy of our approach thoroughly (see the Supporting Inormation). This adaptive approach leads to small time steps at fast sintering and larger time steps at slower sintering, and over a simulated time of 24 h, a typical average time step is 20 s. 3.2. Determination of Physical Properties. We will first briefly discuss how the different parameters affect sintering. Both the oxygen partial pressure and the sticking coefficient S lead to a linear increase in the flux J(r)tot and hence an overall increase in sintering (eq 5). An increase in the surface area influences the flux in the same way and in addition it also increases the chemical potential of the particles μPt(r), as seen from eq 2. This variation in μPt(r) is similarly affected by changes to the surface free energy. Hence, a higher surface free

(3)

The number of PtO2 molecules hitting the surface follows from the ideal gas law: 7100


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ACS Catalysis energy or a larger area leads to a steeper slope of μPt(r) with respect to particle size, r. The chemical potential enters the equilibrium constant for formation of PtO2(g), K (eq 3), and hence the flux, J(r)tot, exponentially. Therefore, faster sintering is caused by a higher surface free energy or a higher specific exposed area, which could be caused by less adhesion. 3.2.1. Energetics. Generally, we would like to obtain most parameters from first principles. Unfortunately, DFT is in many cases not accurate enough.53 For the bulk chemical potential, we therefore use the experimental value μPt,bulk = −5.82 eV.54 The formation energy of PtO2(g) from O2 and Pt(g) can be obtained on the CCSD(T)/def2-QZVPP level (−4.13 eV). This gives a formation energy of PtO2(g) of 1.69 eV relative to bulk platinum, which we will use as an approximation to the formation free energy, ΔG. The formation enthalpy ΔH = 1.81 eV reported in ref 29 is in good agreement. The experimental free energy of formation is lower, moderately temperature dependent, and varies in the temperature range of 500−900 °C from 1.69 to 1.64 eV.28 A large uncertainty is the exact state of the particles under reaction conditions. This will determine their shape and surface area and most importantly their stability as a function of size and the sticking coefficient of PtO2. The most relevant sintering experiments have partial O2 pressures ranging from 0.1 to 40 mbar at temperatures of about 800−950 K. We will discuss thermodynamics in terms of the chemical potential of oxygen, μO = μO2/2, relative to its value at 0 K. The relevant chemical potentials range from μO = −1.1 to −1.3 eV. In sintering experiments, the particles show a rounded shape.8,9,55 In thermodynamic equilibrium, a deviation from the faceted Wulff shape of clean particles can only be explained by adsorbates. This is in line with temperature-programmed desorption (TPD) and other experimental techniques that show significant oxygen coverage under the conditions we are interested in.15,55,56 Unfortunately, these experiments do not allow a simple extraction of accurate adsorption or surface free energies of the particles. Pt(111) is probably the best studied system in terms of the atomic adsorption of oxygen on platinum. In the low coverage limit (θO = 1/9), the experimental reaction energy for dissociation of O2 on Pt(111) is −1.05 eV per oxygen.36,57−59 Without entropic correction for the adsorbed oxygen, one would expect Pt(111) to be clean at T > 800 K and p(O2) < 1 mbar. TPD spectra, however, show significant oxygen desorption on Pt(111) into the range of 900 K, with a peak in the range of 700−800 K.60−63 Importantly, TPD experiments operate under vacuum, so that one would expect the oxygen coverage to be even higher in sintering experiments, with p(O2) > 1 mbar. On the basis of computed vibrations64 and the harmonic approximation, adsorption free energies would be lower by about −0.2 eV per adsorbed oxygen. Although anharmonic corrections may be important, this explains the presence of adsorbed oxygen at higher temperatures. For nanoparticles, adsorption energies are less certain. Finite size effects are expected to lead to stronger adsorption only for particles smaller than Pt147 (∼1.6 nm).50 Generally, smaller particles will, however, have a larger concentration of undercoordinated sites such as edges and kinks. Temperatureprogrammed desorption experiments on more open facets, Pt(211),61,65 Pt(533),65,66 and Pt(321),67 show generally more complex desorption peaks, but in the same temperature range

as Pt(111). Calculations of Bray et al.64 using PW91,68 which is among the most accurate functionals for oxygen adsorption on platinum,36 have shown that the adsorption energy of atomic oxygen on Pt(321) can be significantly lower than on Pt(111), −1.46 eV per atomic oxygen. We will now try to estimate by how much this could lower the surface free energy, assuming undercoordinated sites are present. According to the harmonic approximation (see above), the free energy would be about −1.66 eV. At oxygen chemical potentials in the range of −1.1 to −1.3 eV, the formation free energy of adsorbed oxygen is about −0.4 to −0.5 eV. In the case of Pt(321), the coverage corresponding to our conditions is identical with the concentration of the kinked site, θ = 1/5.64 With an area per surface Pt atom of about 7 Å2, this lowers the surface free energy by −10 to −15 meV/Å2. A very high concentration, θ = 1/3, of such sites (at the same binding energy) could lower the surface free energy by −20 to −25 meV/Å2. Stronger adsorption on undercoordinated sites and a higher density of undercoordinated sites can explain the TPD experiments on oxygen-covered Pt nanoparticles. These experiments on SiO2supported Pt nanoparticles have clearly shown a particle size dependence of the peak of oxygen desorption.56 For particles with a size of about 2 nm (3.6 nm), oxygen desorption was detected until about 1000 K (920 K) with peaks at 800−950 K. To obtain the surface free energies of oxygen-covered particles, one first needs the surface free energies of clean particles. Surface energies of clean, crystalline platinum facets are not well-known. Values obtained from DFT for Pt(111) range from 60 to 130 meV/Å2,69 while surface energies deduced from experiments on solid (liquid) Pt give about 13170 (15571,72) meV/Å2. The best values for the Pt(111) surface energy are probably those obtained from RPA53,69,73 and PBEsol:74,75 both are about 130 meV/Å2.69 Since we have no accurate way of calculating the surface free energies of the particles under reaction conditions, we will treat the surface free energy as a fit parameter, starting from the value of the clean surface, which we take to be γ111 = 130 meV/Å2, as did Wynblatt.30 The effective surface free energy is γavg = γ111 + Δγ

(8)

3.2.2. Particle Shape. Another question is as follows: what adhesion energy should be used for the Wulff construction? Generally, higher adhesion will lead to a lower surface free energy at the interface and therefore to less sintering. It will also lead to more wetting, consequently increasing the interface area relative to the exposed area, thus lowering the surface free energy even further. In terms of the Wulff construction, we have considered two cases, which can be solved analytically with moderate effort: the case of no adhesion, where the only effect of the support is to block one 111 facet from the gas phase, and the case of strong adhesion, where the adhesion energy exactly cancels the surface free energy at the interface. In the former case, the particle shape is that of a free, unsupported particle. In the latter case, the particle shows strong wetting and has the shape of the free particle cut in half along the 111 plane. Experiments on both alumina and silica have shown a contact angle certainly larger than 90°13 and somewhere between 90 and 180°.19 For a given set of particles with defined total mass (i.e., total volume, or total number of Pt atoms), the effect of adhesion is large, as described above. For example, a spherical particle with a diameter of 3 nm has about 936 Pt atoms and, given this number of atoms, stronger adhesion would lead to a more 7101


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trapping probability, because of its significant mass and its strong binding to the clean surface. Once PtO2(g) is trapped, it can migrate to a step or kink, attach there, dissociate, and therefore grow the nanoparticle with an average energy gain corresponding to the chemical potential of Pt of that particle. PtO2* could also dissociate first and the formed Pt atom could then migrate and attach to a stepped or kinked site. We have examined the first part of the latter possibility, dissociation of adsorbed PtO2*. Subsequent diffusion of a Pt atom is fast (ΔEa = 0.3 eV), and we also do not expect the attachment of Pt to a stepped or kinked site to have a significant barrier. Since the barriers for dissociation of PtO2* are clearly lower than for desorption (Figure 1), we find S ≈ 1 at θ0O ≤ 2/9. At higher coverages, we expect the sticking coefficient to be significantly smaller. At θ0O = 3/9, the precursor state is weakly bound, −0.25 eV. Rearrangement of adjacent oxygen atoms leads to a more stable state but goes via a transition state close to the desorption energy (E⧧ = −0.09 eV). In Figure 1, we only depict the energy of PtO2* after rearrangement. Dissociation of the first oxygen bond requires a higher barrier (E⧧ = −0.02 eV). A refinement of the sticking coefficient at θ0O = 3/9 (S ≈ 0.5) would obviously require more accurate free energy barriers for desorption and dissociation. It is clear, however, that this is the coverage range in which the sticking coefficient falls off from 1 to very small values since the precursor is only weakly bound at higher coverages and desorption of PtO2* becomes more likely than dissociation. While we cannot quantitatively apply these results to Pt particles, they certainly show the coverage dependence of the sticking coefficient. According to the discussion above, we clearly expect the coverage to depend on particle size, oxygen pressure, and temperature. The coverage in turn changes the sticking coefficient, as shown above. While we will neglect pressure and temperature dependence of the sticking coefficient altogether, we will try to include the size dependence in an effective way. Since we cannot assign an accurate value to the sticking coefficient, or to its particle size dependence, we will have to treat it as an adjustable parameter. As the surface free energy is already fitted, we would like to avoid introducing more adjustable parameters. It is clear that the overall effect of a lower sticking coefficient, i.e. less sintering, can be achieved by an artificially lower surface free energy. Furthermore, it turns out that the main effect of lowering the surface free energy is similar to an introduction of a size-dependent sticking coefficient. A change in surface free energy, Δγ, enters the equilibrium constant (eq 3) by multiplication with a Boltzmann factor

stable particle with less exposed surface area and hence less activity in sintering. From experiment, however, one generally only knows the particle diameter. Therefore, comparison of experiment and theory requires the conversion of radius to particle size. We do this transformation according to the adhesion energy we assume in our Wulff construction, from sphere (hemisphere) in the case of no (strong) adhesion. Therefore, assuming no adhesion would mean the 3 nm particle consists of 936 atoms, whereas assuming strong adhesion, it consists of only 468 atoms. Since smaller particles generally sinter more quickly, this effect largely cancels the effect that adhesion has on a given particle size. In fact, we have found that simulations using strong and weak adhesion give very similar results (see the Supporting Inormation). Again, this is because these simulations use the same initial radii but different particle sizes. In other words, for different adhesion, particles with the same radii behave in a similar way much more than particles with same total mass. Since the effect of adhesion is small, at least in comparison to other parameters, and since for different supports it is not exactly clear how the adhesion energy varies, we will use the Wulff construction in the limit of no adhesion for our simulations. This means that in our simulations different supports such as silica and alumina are treated in exactly the same way. 3.2.3. Sticking Coefficient. In order to estimate the sticking coefficient, we have calculated an energy profile for the adsorption and dissociation of PtO2(g) on Pt(111) (Figure 1).

Figure 1. Energy profile for adsorption of PtO2(g) and activation barriers for two successive Pt−O bond cleavages at four different initial coverages of oxygen, θ0O. All calculations were carried out in a 3 × 3 supercell containing θ0O × 9 oxygen atoms in addition to the adsorbed species “X*”. Diffusion of O* will lead to equilibration according to the equilibrium coverage θ0O. This is labeled Oθ*, and the integral adsorption energy is not significantly coverage dependent for θ0O ≤ 2/ 9. An inset shows the transition state for cleavage of the first Pt−O bond (θ0O = 0), where the adsorbed Pt atom is highlighted in green. Another inset shows how the sticking coefficient becomes smaller with increasing coverage.

⎡ −Δγ Veff ⎤ exp⎢ ⎥ ⎣ kBT r ⎦

that depends on the particle radius, r. While this slightly changes the background pressure, the main effect is the same as introducing this factor as the size-dependent sticking coefficient S(r). We have tested numerically that results using such a sticking coefficient or simply the changed surface free energy give similar results (see the Supporting Inormation for details). A sticking coefficient S(r) of this form has generally the right behavior in that it is smaller for smaller particles and ranges from 0 to 1. 3.2.4. Parameterization. Another concern is how efficiently the gas phase mediates transport in between the particles. The Wynblatt−Gjostein approach assumes a constant background

We have considered initial coverages of oxygen of θ0O = 0, 1/9, 2/9, 3/9. We will first consider low coverages, in which the precursor state, adsorbed PtO2*, is strongly bound: −2.13 eV at θ0O = 0, −2.03 eV at θ0O = 1/9, and −1.63 eV at θ0O = 2/9. According to the Baule estimate,76,77 we therefore expect the probability for trapping into the precursor state to be ∼1. Generally, PtO2(g) is expected to have a relatively high 7102


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conditions can easily be explored under otherwise the exact same conditions. We will first examine how sintering depends on p(O2) and temperature. The effect of p(O2) is very transparent in the simulations, since the flux of PtO2(g) through the boundary of the particles is proportional to the pressure of PtO2(g), which in turn is proportional to the pressure of O2(g). In terms of temperature dependence, we expect ripening mediated by PtO2(g) to behave similarly to ripening mediated by Pt atoms on the support surface. This is because the prefactor in eq 4 does not vary significantly. Therefore, the temperature dependence of PtO2(g)-mediated ripening mainly results from the equilibrium constant in eq 3. Consequently, the flux of ripening via both Pt on the surface and PtO2(g) is proportional to a Boltzmann factor. We have investigated the effect of temperature and p(O2) for a PSD (dinitial = 3.5 nm, σ = 1.0 nm) and a sintering time of 6 h, which is in the typical range of experiments.8,9,13,78 The extent of sintering is computationally best discussed in terms of the fraction of particles that have disappeared, which gives 0 for no sintering and 1 for full sintering. As can be seen in Figure 2c, the fraction of sintered particles is highly correlated with the shift of the mean diameter that is usually discussed in sintering experiments. It is therefore always possible to transform between these two representations of sintering. As one would expect, higher temperature and higher p(O2) lead to more sintering (Figure 2a). Within reasonable oxygen partial pressure (100 nm). One can therefore expect that PtO2(g) molecules that are emitted from a particle in the direction of other particles will typically hit these particles without prior collision in the gas phase. This may lead to local correlation effects, as have been proposed for surface-mediated ripening.8 We expect that our procedure of adjusting the surface free energy will, in addition to the actual effect of oxygen coverage on the surface free energy, also effectively include a sizedependent sticking coefficient, as discussed above, and potentially mass-transport limitations. In fact, agreement with experiments13,78 requires a surface free energy of about 75 meV/Å2. The stabilization (Δγ = −55 meV/Å2) is certainly lower than the expected effect of oxygen coverage on surface free energies according to our considerations above. Another set of experiments8 requires a slightly different value of Δγ but can be equally well reproduced (see the Supporting Inormation for details). This is not surprising, because the results can hardly be explained on the basis of the data used for simulation (mean initial diameter, oxygen pressure, temperature, and simulation time). The only significant difference is the higher temperature (ΔT = 50 K) in ref 8, which should lead to more sintering, while the final diameter is actually slightly lower by 0.1 nm in these experiments. We believe that these discrepancies are most likely due to details in the experimental setup, such as the spatial distribution of the particles and gas flow. Since in their insight in terms of comparison to simulations the experiments of Simonsen et al.8 are similar to those of Tabib Zadeh Adibi et al.,13 we have chosen to discuss the latter in detail, as experiments by the same group78 can be reproduced with the exact same parameterization. We note in passing that simulations on surface-mediated ripening have employed adjusted surface free energies that are significantly below the value of clean Pt(111): 31−35 and 93 meV/Å2.8,14 Our reasoning above may also explain why surface free energies that low have worked in practice. While it would be desirable to have more accurately determined parameters, we wish to point out that our model does not include more empiricism than any of the surface-mediated ripening simulations that we are aware of. In Table 1 we give the parameters that we employed in the simulations. 3.2.5. Extent of Sintering as a Function of Conditions. The main advantage of theory over experiment is that different Table 1. Chosen Parameters param

value

S μPt,bulk ΔGform(PtO2) γ111/γ100 γ111 γadh rexptl ↔ Vtheor Δγ(fit)

1 −5.82 eV 1.69 eV 0.77 130 meV/Å2 0 Vtheor = (4/3)πrexptl3 −55 meV/Å2 7103


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Figure 3. Time dependence of the number of particles, the mean diameter, and the standard deviation of the diameter as shown for a few simulations taken from Figure 2. In each part of the graph, trajectories with σinitial = 0.05 (0.25) nm are marked in red (blue) and ⟨dinitial⟩ is marked with different line styles.

Figure 2. Extent of sintering as measured by the fraction of particles that have disappeared due to sintering is shown (a) as a function of temperature and the partial pressure of oxygen for a given PSD and (b) as a function of the mean initial particle diameter and the standard deviation relative to ⟨dinitial⟩ for 40 mbar of O2(g), 600 °C, and a sintering time of 6 h. Using the data from (b), the correlation between the fraction of sintered particles and the shift in mean diameter, Δ⟨d⟩ = ⟨dfinal⟩ − ⟨dinitial⟩ is shown in (c).

To make a quantitative comparison, we will use the mean diameter of the PSD as a function of time to compare experiment and theory. We will focus here on published experiments where we were able to obtain detailed data for analysis and where that data can be compared best with simulations. First, we will study the course of sintering over time in comparison to the experiments of Tabib Zadeh Adibi et al.13 Second, we investigate the effect of temperature and oxygen pressure, again comparing with experiments.78 3.3.1. Sintering Kinetics. Tabib Zadeh Adibi et al. have studied sintering of Pt nanoparticles on three different supports (alumina, plasma-enhanced chemical vapor deposition (PECVD) silica, and sputtered silica).13 Simulations were started from a normal distribution that was derived from the PSD of PECVD silica after 10 min of sintering (⟨d⟩ = 3.5 nm, σ = 1.0 nm). This is because initial data of all three supports are not available and the PSDs of the three supports after 10 min are very similar (⟨d⟩ = 3.2, 3.5, 3.6 nm and σ = 1.1, 1.0, 1.1 nm). Experiments were run at a temperature of 600 °C and a partial pressure of oxygen of p(O2) = 40 mbar. Over a course of 24 h, PSDs have been determined, from which we have extracted the mean diameter, ⟨d⟩. The evolution of ⟨d⟩ over time (Figure 4) shows a steady increase, clearly indicating

particles have sintered. It is clear that in the limit of σ → 0, there is no driving force for sintering and therefore no shift in the mean diameter. In order to suppress sintering, it is therefore useful to employ particles with a narrow PSD. From Figure 2 it is not clear what the ideal initial diameter would be. In Figure 3, we show the time dependence for a few PSDs from Figure 2. We find that for a given narrow PSD, σinitial = 0.05 nma larger initial mean diameter ⟨dinitial⟩ leads to slower sintering. As expected, the number of particles shrinks faster for particles with ⟨dinitial⟩ ≈ 3.5 nm in comparison to those with ⟨dinitial⟩ ≈ 4.2 nm. However, less intuitively, when the PSD with ⟨dinitial⟩ ≈ 3.5 nm reaches ⟨d⟩ ≈ 4.2 nm, its width has increased more than that of the PSD with ⟨dinitial⟩ ≈ 4.2 nm. Consequently, in this case, the PSD with lower ⟨dinitial⟩ sinters quickly and the trajectories of ⟨d⟩(t) cross. 3.3. Comparison of Theory and Experiment. There are numerous experimental studies on sintering of Pt nanoparticles.8,9,13,78 The most easily accessibleand therefore commonly reportedresult is the PSD as a function of time. 7104


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Figure 5. Mean diameter as a function of time for experiments involving different oxygen pressures (red lines).78 Simulated mean diameters are also shown for the same pressures with matching symbols.

reproduced for oxygen pressures of 5, 2, and 1 mbar and, qualitatively, for 0.5 mbar with the same single parameter Δγ. We are aware of two studies where sintering was investigated in the same experimental setup at different temperatures.13,14 As expected, higher temperatures generally lead to more sintering. Tabib Zadeh Adibi et al. found that Pt particles essentially do not sinter until 500 °C and then sinter significantly at 550 °C.13 Further increasing the temperature to 600 °C leads to a systematically increased sintering rate. A similarly abrupt change in stability has been observed by Porsgaard et al.12 Pt particles were reported to be stable under oxidizing conditions at 400 °C and to significantly shrink when the temperature was increased beyond 450 °C. Our simulations do not predict such a discontinuous temperature dependence. An explanation could be a variation in oxygen coverage (or a surface oxide), essentially a phase change, that could give rise to different surface free energies as well as potentially different sticking coefficients for PtO2(g). This is in agreement with the TPD experiments on Pt nanoparticles, where the hightemperature peak is seen at T > 750 K (∼480 °C).56 Furthermore, decomposition of solid PtO2 in air occurs around 650 °C.79 From experiments,25 Johns et al. have deduced that around T > 1020 K a crossover occurs from surface- to vapormediated ripening, both via PtO2. This temperature range coincides with the point beyond which no significant oxygen desorption signal is observed in TPD experiments, even on small particles.56 Another explanation would therefore be a transition to very low oxygen coverage that would lead to sticking coefficients of PtO2(g) ≈ 1 and therefore to fast sintering. We will compare to the experiments of Tabib Zadeh Adibi et al.,13 which were run in 40 mbar of O2 and three different temperatures for 12 h. The results are discussed in terms of the shift of the mean diameter, Δ⟨d⟩, which is plotted in Figure 6. One can clearly see that at 500 °C (∼773 K), experimentally, Δ⟨d⟩ ≈ 0. The simulations were run using normal distributions that were based on the experimental PSDs (⟨d⟩ = 3.0 nm, σ = 0.7 nm). While the simulations reproduce the high-temperature behavior well, they show a smooth continuous decrease of sintering toward lower temperatures. A plausible explanation for the deviation is that our simulation uses the same surface

Figure 4. (a−c) PSDs from experiment (PECVD silica)13 (red line) and simulations at p = 40 mbar. The mean particle diameter as a function of time for experiment and simulations is shown in (d) for different oxygen pressures.

sintering. On alumina and PECVD silica the final PSDs are not significantly different with final ⟨d⟩ values of 10.0 and 10.8 nm. The authors note, however, that the particle density on alumina is higher.13 Sintering on sputtered silica is slightly slower with a final mean diameter of 8.1 nm. In Figure 4 we show the evolution of the mean diameter obtained experimentally on PECVD silica and simulated with different partial pressures of oxygen. It can be seen that, within the chosen parameterization, the sintering kinetics agree quantitatively. The largest deviation occurs at long sintering times: >6 h. This may partially be due to poorer statistics in the simulations, since the initial number of particles (2000) is significantly reduced to about 50. In both experiment and simulation we see a steady monotonous decline of the sintering rate as measured by d⟨d⟩/dt from an initially high value. Both experimental and simulated PSDs are compared in Figure 4a−c. The PSDs agree reasonably well, in that we do not see significant deviations or distinctive features that would appear to be characteristic. 3.3.2. Effect of Oxygen Pressure and Temperature. Our simulations in Figures 2 and 4 show a pronounced effect due to changes in oxygen pressure. Tabib Zadeh Adibi et al. have also carried out sintering experiments for silica-supported Pt particles with varying oxygen pressure.78 The temperatures are identical (600 °C), and the initial PSDs are very similar. For simulations, the initial distribution from ref 78 was approximated using a normal distribution (⟨d⟩ = 3.0 nm, σ = 0.8 nm). The pressures in these experiment and simulations are significantly lower: p(O2) = 5, 2, 1, 0.5 mbar. Consequently, sintering is less pronounced (Figure 5) with mean diameters increasing from 3 to at most 5 nm, whereas at 40 mbar of O2 the final mean diameter was about 10 nm. It is noteworthy that the trend in terms of oxygen pressures is quantitatively 7105


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experiments. For both surface-mediated and gas-phase-mediated ripening, it is unclear to what extent oxygen coverage and adhesion modify the chemical potential of the particles. Additionally, the oxygen coverage is expected to change the sticking coefficient of PtO2(g). Lowering the surface free energy relative to its value for clean Pt(111), we can explain sintering kinetics and the dependence of temperature and pressure. On the basis of the currently available information, sintering via PtO2(g) provides the best explanation for the experimental observations. In particular, it explains the surprisingly small influence of the support on sintering kinetics in the same experimental setup,13 in comparison to variations between different experiments on the same type of support.8,13 To explain these findings with surface-mediated ripening requires the energetics for Ostwald ripening on these different supports to beaccidentallyvery similar. Abrupt changes in sintering kinetics at around 450 °C12,13 and 750 °C25 happen to coincide with the onset and end of oxygen desorption peaks in TPD experiments.15,55,56,60−63,65−67 Lower oxygen coverage will, as we have demonstrated, lead to a higher sticking coefficient of PtO2(g) and therefore to observed increased sintering. We believe that more effort should be put into the investigation of gas-phase-mediated ripening and that future simulations on sintering of supported Pt particles under an oxidizing atmosphere should take this mechanism into account. Experiments could probe the importance of PtO2(g)-mediated ripening by systematically varying the gas flow in the vicinity of the surface under otherwise constant conditions.25 It is has been proposed that adsorbed PtO2* acts as the active species at lower temperatures. Since determined adsorption energies on alumina are very weak (−0.8 eV),25 this would require very low barriers for migration on the support and from the particle onto the support to be more favorable than ripening via the gas phase. Determination of the barriers from DFT could therefore help to clarify the relative importance of ripening via volatile or adsorbed PtO2. As for surface-mediated ripening, more accurate surface free energies in the presence of O2(g) as well as adhesion energies and information about particle shape would improve the description of the chemical potential of Pt particles. Additionally, a simulation approach without adjustable parameters would require the calculation of the sticking coefficient of PtO2(g) as a function of oxygen coverage and particle size. Since our study has demonstrated that modeling of gas-phasemediated ripening is viable, it may also be worthwhile to use this approach for other systems where similar mechanisms may be important.

Figure 6. Temperature dependence of the shift of the mean diameter, Δ⟨d⟩, relative to the initial diameter of 3.0 nm, after 12 h of sintering. Experiments were run in 40 mbar of O2.

free energies and sticking coefficients at all temperatures. As noted above, surface free energies and sticking coefficients may be lower due to higher oxygen coverage at lower temperatures.

4. SUMMARY We have carried out simulations of gas-phase-mediated ripening, by explicit calculation of the flux of PtO2(g). We have presented a thorough analysis of how this affects sintering kinetics under relevant conditions and have compared the results to those of published experiments. In agreement with the results of Wynblatt,30 simulations of gas-phase-mediated ripening together with the surface energy of clean Pt(111) and a sticking coefficient of PtO2(g) of 1 overestimate sintering, in comparison to experimental results. The overestimated sintering can be explained by a significant oxygen coverage that lowers the surface free energy of Pt and leads to a lower sticking coefficient. An additional factor could be mass transport limitations in the gas phase. In agreement with experiment,56 smaller particles are expected to bind oxygen more strongly due to a higher density of undercoordinated sites and finite size effects. Therefore, the oxygen coverage, and as a result of that also the sticking coefficient, are expected to be size dependent. We have shown that lowering the surface free energy can also capture the effect of a sizedependent sticking coefficient. Using an adjusted surface free energy allows us to quantitatively reproduce experimental sintering kinetics. Furthermore, the dependence of sintering on oxygen pressure is reproduced over five different oxygen pressures with the same parameters. While the sudden onset of sintering with increasing temperature could not be reproduced, the temperature dependence of the simulation agrees well with the available experimental data for Pt on silica in the hightemperature limit.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acscatal.6b01646. More details of the simulations and convergence tests (PDF)

5. CONCLUSION AND OUTLOOK Ripening of Pt nanoparticles under an oxidizing atmosphere has long been suspected to be mediated by volatile PtO2(g). However, most simulations of this process were based on kinetics for surface-mediated Ostwald ripening. At this point, simulations of both surface-mediated and gasphase-mediated ripening require assumptions regarding their parameters. For surface-mediated ripening of Pt on either alumina or silica, the assumed energetics of the diffusion mechanisms are generally not fully supported by calculations or

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AUTHOR INFORMATION

Corresponding Author

*E-mail for F.A.-P.: [email protected]. Notes

The authors declare no competing financial interest. 7106


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ACKNOWLEDGMENTS We acknowledge support from BASF SE and from the Office of Basic Energy Sciences of the U.S. Department of Energy to the SUNCAT Center for Interface Science and Catalysis at SLAC/ Stanford. We thank Dr. Stig Helveg at Haldor Topsøe A/S for providing the experimental data from refs 8 and 9, Prof. Dr. Henrik Grönbeck for providing the data from ref 13, and Prof. Dr. Christoph Langhammer for providing the data from ref 78. We also thank Dr. Alan C. Luntz for helpful discussions about sticking coefficients.

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