First Principles Calculations on Site-Dependent Dissolution Potentials

J. Phys. Chem. C 2010, 114, 17557–17568

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First Principles Calculations on Site-Dependent Dissolution Potentials of Supported and Unsupported Pt Particles Ryosuke Jinnouchi,* Eishiro Toyoda, Tatsuya Hatanaka, and Yu Morimoto Toyota Central Research & DeVelopment Laboratories, Inc., Nagakute, Aichi, Japan, 480-1192 ReceiVed: July 15, 2010; ReVised Manuscript ReceiVed: August 25, 2010

Site-dependent redox potentials of the Pt dissolution reaction, Pt f Pt2+(aq) + 2e-, from Pt particles with and without carbon supports were calculated by a first principles method. The calculation result showed a clear site-dependence in which the redox potentials for the Pt atoms at edges are lower than those for Pt atoms at flat surfaces. This site-dependence is roughly correlated with the d-band center of the dissolving Pt atom, and Pt atoms with higher d-band centers dissolve more easily to the solution. The effects of perfect and defected graphenes on the redox potential are less than 0.1 V. For the platinum atom attaching to the carbons, a slightly negative effect (lowering the redox potential) is obtained, because the bond strength between Pt particles and carbons is enhanced by the defect creation after the dissolution reaction. 1. Introduction Highly dispersed Pt nanoparticles on carbons are used as electrocatalysts for oxygen reduction reactions (ORR) in proton exchange membrane fuel cells (PEMFCs) because of their higher durability and activity than those of other materials.1-4 Their outstanding performances have been achieved by enormous efforts including developments of new materials such as alloys,1,5-8 monolayer catalysts,9-11 size-controlled nanoparticles,12 stable carbon supports with high surface areas,1-4,13,14 and optimizations by changing their combinations. However, their performances are still insufficient for a wide distribution of PEMFCs. One of the most critical problems is the dissolution of Pt during the fuel cells operations, the formula of which is likely expressed as,

Ptn f Ptn-1 + Pt2+(aq) + 2e-

(1)

The dissolved Pt ion diffuses and deposits on Pt particles in the catalyst layer or in the proton exchange membrane, and the catalytic surface area decreases by the Ostwald ripening mechanism.15 The reversible potential of the dissolution reaction, which we simply call as dissolution potential in this study, for the bulk Pt is given as 1.188 V (SHE) at the standard condition from the thermodynamic data.16 For plane single crystal surfaces, Komanicky et al. observed that Pt atoms in the (111) terraces started dissolving at 1.15 V (RHE) in a 0.6 M HClO4 solution by in situ atomic force microscopy (AFM) and inductively coupled plasma mass spectrometer (ICP-MS) analysis.17 They also reported that Pt atoms at edges and corners started dissolving at 0.65 V (RHE) and suggested the layer-by-layer dissolution mechanism at this low electrode potential. The concentration of Pt2+(aq) in the electrolyte was measured as about 5 × 10-7 M in their study. The bulk dissolution potential corresponding to this experimental condition is calculated as 1.024 V (RHE) by the Nernst equation with the measured * To whom correspondence should be addressed. Phone: +81-561-717290. Fax: +81-561-61-4120. E-mail: [email protected].

concentration for Pt2+(aq) and the concentration of 0.422 M for H+(aq) estimated from tabulated thermodynamic data for HClO4(aq), ClO4-(aq), and H+(aq).16 Hence, their experimental results indicate that the observed dissolution potential strongly depends on dissolving sites and significantly differs from the bulk dissolution potential derived from the thermodynamic data. In contrast, Wakisaka et al. recently performed in situ scanning tunneling microscopy (STM) measurement on Pt(111) in a 0.01 M HF solution and reported that no significant morphological change was observed at the electrode potential lower than 0.90 V (RHE).18 They found that adsorbed oxygen atoms started covering the surface at 0.90 V (RHE) and filled the surface at 1.12 V (RHE). Then, absorbed oxygen atoms in the subsurface appeared at the electrode potential higher than 1.12 V (RHE). They also performed ICP-MS measurement of dissolved Pt ions and reported the concentration lower than the detection limit of about 5 × 10-10 M. The bulk dissolution potential corresponding to this experimental condition is calculated as 1.030 V (RHE) by the Nernst equation with using 5 × 10-10 M for Pt2+(aq) and 0.01 M for H+(aq). Thus, their experimental results indicate that the surface is passivated by oxides and the Pt dissolution reaction does not occur at the bulk dissolution potential. Accordingly, the atomistic mechanism of the dissolution reaction and its onset potential are not fully clarified for single crystal surfaces. For nanoparticles, lower dissolution potentials are predicted by using the Gibbs-Thomson equation for a homogeneous particle model as follows,

1 bulk Udiss ) Udiss -

1 ∆µPt )

1 ∆µPt 2e

2σPtVPt rSPt

(2)

(3)

where σPt is the surface energy of the particle, VPt and SPt are respectively the volume and surface area of one Pt atom, and r is the particle radius [see Appendix A].19 Darling and Meyers used this simple equation in their macroscopic numerical model

10.1021/jp106593d  2010 American Chemical Society Published on Web 09/28/2010

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representing the catalytic degradation in fuel cells.20 Furthermore, a recent study using in situ STM measurement and density functional theory (DFT) calculations by Tang, et al. supported this simple trend.21 In addition, they suggested that the dissolution pathway for nanoparticles is more likely the direct dissolution reaction described as eq 1 than the electrochemical oxidation of the surface followed by the chemical dissolution of the oxides as follows:

The site-dependent dissolution potential for the elementary reaction 1 should, instead, be calculated by the binding energy Eb of a Pt atom with other Pt atoms in the particle as follows [see Appendix A]:

Ptn + H2O(aq) f O-Ptn + 2H+(aq) + 2e-,

Eb ) Etot[Ptn(g)] - Etot[Ptn-1(g)] - Etot[Pt(g)]

(4)

O-Ptn + 2H+(aq) f Ptn-1 + Pt2+(aq) + H2O(aq)

(5) Thus, according to Tang et al., surfaces of nanoparticles are not passivated by oxides. Two questions are inspired from these pioneering studies. One is about the site-dependence of the dissolution potential, and the other is about the metal-support interaction. In the following paragraphs, the importance of these two questions is mentioned, and previous studies and their problems are described. Site Dependence. Surfaces of nanoparticles consist of various facets, such as (111) and (100), and edges between these low index facets. Even in the case of well-characterized single crystal surfaces, there are some defect sites. Since surface energies before and after the dissolution reaction depend on dissolving sites, the change in the Gibbs free energy of reaction 1 also depends on dissolving sites. When the surface has an inhomogeneous distribution in the dissolution potential, the dissolution reaction is likely to initiate at the surface site with the lowest dissolution potential. This site-dependence of the dissolution potential is important for the atomistic scale designing of highly durable catalyst nanoparticles, such as Au decorations to Pt nanoparticles,11 but was not included in the simple thermodynamic model described as eqs 2 and 3, which use a single surface energy before and after the dissolution reaction for all dissolving sites. The DFT calculations for Pt particles performed by Tang et al. do not take this site-dependence into consideration either, since their dissolution potential was given by the change in the averaged cohesive energy as follows:

all bulk Udiss ) Udiss +

Ecoh - Ebulk coh 2e

(6)

Etot[Ptn(g)] - nEtot[Pt(g)] n

(7)

Ebulk coh ) Etot[Ptbulk(g)] - Etot[Pt(g)]

(8)

Ecoh )

bulk are cohesive energies per Pt atom for the where Ecoh and Ecoh Pt particle and the bulk crystal, respectively, calculated by total energies of the Pt particle (Etot[Ptn(g)]), the Pt atom (Etot[Pt(g)]) and the Pt bulk crystal (Etot[Ptbulk(g)]) through eqs 7 and 8. It all is the dissolution potential for the should be noted that Udiss overall dissolution reaction of the entire particle described as follows [see Appendix A]:

Ptn f nPt2+(aq) + 2ne-

(9)

1 bulk Udiss ) Udiss +

Eb - Ebulk coh 2e

(10) (11)

In addition to the problem in which the site-dependence is not taken into consideration in their DFT calculation, there is another problem in their choice of the thermodynamic dissolution 1 equation of Udiss described as eqs 2 and 3 for reaction 1 for all comparisons with the DFT-based dissolution equation of Udiss described as eqs 6-8 for reaction 9. For this comparison, a all for reaction better choice is a thermodynamic equation of Udiss 9 as shown below [see Appendix A],

all Udiss

)

bulk Udiss

all ∆µPt )

all ∆µPt 2e

3σPtVPt rSPt

(12)

(13)

Metal-Support Interaction. It is important to know the length, amplitude, and direction of this effect for designing supports giving higher durability to Pt particles. The metalsupport interaction, however, has not been sufficiently understood and was not taken into account by the atomistic model developed by Tang et al. and the macroscopic model developed by Darling and Meyers. There are several studies reporting that power densities of fuel cells were maintained with using graphitized carbons or carbon nanotubes as supports.3,4,22-25 Although these studies suggest that the crystallization of carbon surfaces gives higher durability, the mechanism is not clear. The crystallization would enhance the metal-support binding energy by the increase in the π-site as anchorages inhibiting aggregations or dissolutions of Pt atoms.25 The crystallization would also enhance the resistance to the carbon oxidation which causes growth in the Pt particle through the decrease in the support surface area and the flooding in the catalyst layer through the decrease in the hydrophobicity of support surfaces by the increase in the surface functional groups. Since it is difficult to separately evaluate these effects, it is unclear whether the crystallization of carbon surfaces changes the nature of the metal-support interaction or not. The electronic nature of the metal-support interaction has been investigated by XPS measurements.26-30 It is commonly accepted that Pt particles on carbons have higher core level binding energies than those of the bulk Pt. The cause of this enhancement, however, is not clear. Yu and Ye mentioned that the energy shift is caused by the electron donation from Pt particles to carbons; the electron donation is beneficial for the activity and the durability of electrocatalysts.3 There is another general consideration that the energy shift comes from the slow screening of core holes for small particles.27,29,30 This so-called final state effect depends on the conductivity of substrates and the strength of interactions between metal particles and substrates, and it is difficult to distinguish the final state effect from

First Principles Calculations

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Figure 1. Models of Pt particles with (A) n ) 1, (B) n ) 2, (C) n ) 3, (D) n ) 4, (E) n ) 13, (F) n ) 55, (G) n ) 79, and (H) n ) 201.

overall XPS results for this complex system. Our group applied XPS to measure the density of states of Pt particles on carbons and encountered similar difficulties for the interpretation of obtained results.31 Although several authors performed DFT calculations on Pt particles on carbons, they did not sufficiently estimate the effect of the metal-support interaction on the dissolution property. Calculations of cohesive energies of Ptn (n e 32) particles on graphenes with and without vacancy or B-dope defects were performed, and these calculations showed that graphenes do not give large increase in the cohesive energy.32-35 Taking eq 6 into consideration, these results seem to indicate that the effect all is small. However, of carbons on the dissolution potential Udiss these studies did not give site-dependent dissolution potential 1 Udiss . This Work. In this work, the site-dependence of the dissolution potential was derived by DFT calculations for supported and unsupported Pt particles on graphenes with and without vacancy defects. Following the suggestion by Tang et al.,21 we investigated the redox potential of the direct reaction pathway described as eq 1 without any surface oxides. Correlations between the dissolution potential and the electronic structure of these Pt particles were also investigated and discussed. In section 2, we summarize our models and computational methods. In section 3, we present results and discussion. A summary of this work is presented in section 4. 2. Computational Methods DFT calculations were performed on Pt particles with and without graphenes as a model of carbon supports. The numbers of Pt atoms were 1, 2, 3, 4, 13, 55, 79, and 201 for unsupported particles and 1, 2, 3, 4, 13, 55 for supported particles. The shapes of the particles are shown in Figure 1. For the unsupported Pt particles, calculations with and without a 2-dimensional periodic boundary condition (2D-PBC) were performed. For the supported Pt particles, the calculation with the same 2D-PBC was performed. The applied 2D-PBC corresponds to the 7 × 43 periodic structure shown in Figure 2. Two types of graphenes were used as the models of carbon supports. One is a perfect graphene shown in Figure 2(A), and the other is a graphene with a vacancy defect per unit cell shown in Figure 2(B). We denote the perfect and defected graphenes as C and C-d, respectively. We used a DFT calculation code with localized pseudoatomic orbitals for basis sets and norm-conserving pseudopotentials for

Figure 2. Graphenes (A) without and (B) with a vacancy defect per unit cell. Unit cells are shown as squares.

the interaction between valence electrons and effective cores including atomic nucleus and inner-core electrons.36,37 We adopted basis sets with double-ζ plus polarization (DZP) optimized for the bulk graphite and the bulk Pt. The cutoff radii for these basis sets were 2.9 Å for C and 4.3 Å for Pt. The size of the basis sets is one of the most important empirical parameters in our calculation. We checked the convergence of the binding energy of a Pt atom with other Pt atoms in the Pt55 particle with and without the perfect graphene using the basis sets of DZP and triple-ζ plus double polarizations (TZDP). Although the difference in the absolute value of the binding energies was as much as 0.13 eV, the difference in the relative values among different sites was only 0.04 eV at most, which does not change the conclusion given in this study.

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All calculations were performed using a generalized gradient approximation (GGA) developed by Hammer et al., which we denote as GGA-RPBE in this study.38 Brillouine zone integrations were performed with a 2 × 2 uniform mesh and a Gaussian smearing with the energy width of 0.1 eV.39,40 All total energies were extrapolated to kBT ) 0 eV. Full structural optimizations were carried out to achieve maximum forces of 0.05 eV · Å-1 by a quasi-Newton optimizer.41 3. Results and Discussion 3.1. Parameter Setting. Before discussing the DFT results, parameters used in calculations and discussion are introduced in this section. The particle radius r is defined as follows:

( )

(14)

3 LPt VPt ) 4

(15)

3nVPt r) 4π

1/3

where LPt is the lattice constant for the bulk Pt. We used a theoretically obtained LPt of 4.03 Å, which is larger than the experimental value of 3.92 Å42 but close to another theoretical result of 4.02 Å given by Nørskov, et al.43 The cohesive energy Ecoh and binding energy Eb were all 1 and Udiss through eqs 6 converted to dissolution potentials Udiss and 10, respectively. In these conversions, the theoretically obtained cohesive energy of 5.073 eV · atom-1 for the bulk Pt bulk . Our GGA-RPBE calculation underestimates was used as Ecoh bulk , which was experimentally obtained as 5.84 eV · atom-1.42 Ediss We considered that this underestimation is caused by the inaccuracy in the bulk properties calculated by GGA-RPBE, because using GGA-PBE functional44 gives a closer cohesive energy of 5.681 eV · atom-1 to the experimental value and a theoretical value of 5.59 eV · atom-1 given by a FLAPW method with using the same functional.45 The surface area per Pt atom for the Pt(111) surface calculated as follows was used as SPt in eqs 3 and 13:

SPt )

√3 2 L 4 Pt

TABLE 1: The Number of Atom in a Particle n, the Radius r, the Cohesive Energy Ecoh and the Dissolution Potential Udissall for Unsupported Pt Particles n

r (nm)

Ecoh (eV · atom-1)

Udissall (V (SHE))

1 2 3 4 13 55 79 201 bulk

0.158 0.198 0.227 0.250 0.370 0.600 0.676 0.923 ∞

0.000 1.446 2.106 2.433 3.211 4.003 4.182 4.435 5.073

-1.349 -0.626 -0.296 -0.132 0.257 0.653 0.742 0.869 1.188

DFT calculation results with the Gibbs-Thomson equation described as eqs 2 and 3.21 Kumar and Kawazoe calculated Ecoh of unsupported Pt clusters with more varieties in the shape and size and obtained a similar dependence.47 Hence, we consider that the dependence shown in Figure 3 can be generalized to Pt particles with wide varieties in the shape and size. Table 2 and Figure 4 summarize the site-dependent binding 1 energies Eb calculated by eq 11 and dissolution potentials Udiss of the reaction 1 calculated using eq 10. We performed DFT calculations for Ptn-1 particles with various vacancy defects as shown in Figure 5. Calculation results by the Gibbs-Thomson equation described as eqs 2 and 3 are also shown in Figure 4. In this calculation, previously described values for VPt, SPt and σPt were used. For comparisons, results of similar calculations for single crystal surfaces of Pt(111), Pt(221), and Pt islands on Pt(111) shown in Figure 6 are included in Table 2 and Figure 4. The overall trend in the results by DFT calculations is the same as that from the Gibbs-Thomson equation [i.e., smaller 1 for binding energies Eb and lower reversible potentials Udiss smaller particles]. There are two distinctive deviations, however, between results given by these two methods. The first is that 1 to DFT calculation results show the reversible potential Udiss

(16)

σPt in eq 13 was calculated as the theoretical surface free energy of Pt(111) at kBT ) 0 eV with using a ten layer slab of Pt(111) with a 2 × 2 unit cell as follows:

σPt )

Etot[Pt(111)] - 40Etot[Pt(Bulk)] 8

(17)

In the calculation of the Pt(111) slab, the structural optimization explained in section 2 was also performed. Calculated σPt was 0.955 eV · atom-1, which well-agrees with the experimental value of 1.03 eV · atom-1 by Tyson and Miller.46 3.2. Dissolution Potentials of Unsupported Pt Particles. Cohesive energies Ecoh calculated by eq 7 and dissolution all potentials Udiss calculated by eq 6 of unsupported Pt particles are summarized in Table 1 and Figure 3 (A). Figure 3 shows all with the increase in the monotonic increase in Ecoh and Udiss particle size. This trend can be explained by the simple thermodynamic expression of eqs 12 and 13. This agreement is already shown by Tang et al., although they compared their

Figure 3. (A) The cohesive energy Ecoh and (B) the dissolution all potential Udiss for unsupported Pt particles as functions of the particle radius r. DFT results: b; the thermodynamic equation described as eq 12: sold line.



TABLE 2: The Binding Energy Eb and the Dissolution Potential Udiss1 for Unsupported Pt Particles site num.

Eb (eV)

Udiss1 (V (SHE))

1 2 3 4 5 6 7 8 9 10 11 12 13 14

0.000 2.892 3.424 3.415 2.269 4.602 4.821 5.149 4.820 5.180 5.636 5.109 5.174 5.338

-1.349 0.098 0.363 0.359 -0.214 0.952 1.062 1.226 1.061 1.241 1.469 1.206 1.239 1.320

site num.

Eb (eV)

Udiss1 (V (SHE))

15 16 17 18 19 20 21 22 23 24 25 26 27

5.355 5.711 5.982 6.122 4.028 5.482 6.047 6.072 4.979 5.334 5.966 6.163 6.354

1.329 1.507 1.642 1.712 0.665 1.393 1.675 1.688 1.141 1.318 1.635 1.733 1.823

1 be sensitive to the dissolving site. The second is higher Udiss given by DFT calculations than those given by the Gibbs-Thomson equation for larger particles. These deviations are exemplified with the dissolution reaction energies from the sites 18 and 19 1 for the site 18 is 1.712 V in Figure 6, parts (A) and (B). Udiss (SHE), and that for the site 19 is 0.665 V (SHE). Since both the cases are regarded as the dissolution and deposition reaction from/to the Pt(111) surface, the Gibbs-Thomson equation cannot distinguish them and gives a single dissolution potential of 1.188 V (SHE). The difference in the dissolution potentials between these two sites or between results by the DFT calculations and the Gibbs-Thomson equation stems from defect-dependent surface energies before and after the reactions. The surface energy for the Pt(111) with a vacancy defect or with a Pt adatom is higher than that for the flat plane Pt(111) surface. Thus, the dissolution reaction creating a vacancy defect destabilizes the surface and has a high dissolution potential, and, in contrast, the one removing a Pt adatom stabilizes the surface and has a low dissolution potential. For Pt particles of Pt55, Pt79, and Pt201, which consist of smooth flat surfaces, surface energies

Figure 5. Positions of dissolving sites in Pt particles with (A) n ) 1, (B) n ) 2, (C) n ) 3, (D) n ) 4, (E) n ) 13, (F) n ) 55, (G) n ) 79, and (H) n ) 201. Light colored spheres (yellow colored spheres in web version of this article) show the dissolving atoms.

Figure 6. Positions of dissolving sites in Pt single crystal surfaces of (A) Pt(111), (B) Pt adatom on Pt(111), (C) Pt(221), (D) Pt island forming one row on Pt(111), (E) Pt island forming two rows on Pt(111), (F) Pt island forming three rows on Pt(111), (G) Pt island forming four rows on Pt(111) and (H) Pt island forming five rows on Pt(111). Light colored spheres (yellow colored spheres in web version of this article) show the dissolving atoms.

after the dissolution reactions are higher than those before the dissolution reactions. Hence, the DFT-calculated dissolution potentials for the particles with flat surfaces are higher than those given by the Gibbs-Thomson equation. The explanation given in the previous paragraph can be more rigorously described by the thermodynamic formula including the effect of the defect-dependent surface energy on U1diss shown as follows:

Figure 4. (A) The binding energy Eb and (B) the dissolution potential 1 for unsupported Pt particles as functions of the particle radius r. Udiss DFT results: b; the Gibbs-Thomson equation: sold line.


1 ∆µPt 4πr2∆σPt + 2e 2eSPt

(18)

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Jinnouchi et al. Nb

εd )

Nb

/ ckikσSjkk ∑ ∑ ∑ fikσεikσ ∑ ∑ cjikσ k

σ

i

j)1,j∈d k)1 Nb Nb / fikσ cjikσ ckikσSjkk i∈d j)1,j∈d k)1

∑∑∑ σ

k

(21)

∑ ∑

where fikσ, εikσ, and cjikσ are respectively the occupation number, eigenenergy and eigenvector for an eigenstate specified with spin index σ and band index i at a k-point in the Brillouin zone, Sjkk is the overlap matrix between j-th and k-th localized basis sets, and the summation for j was performed with respect to d-components in the specific atom. This figure shows a trend of higher U1diss for Pt atoms with lower εd. Therefore, the spacial distribution of εd shown in Figure 8 also indicates the charac1 . Atoms at edges have higher εd teristic site-dependence of Udiss and, therefore, are easier to dissolve than atoms in flat surfaces. Potentials for Pt dissolution from the particles with r ) 0.600 and 0.923 nm are respectively 0.952 and 1.206 V (SHE) for the first Pt atoms on the edges, but are 0.767 and 0.941 V (SHE) for the averages of all the outermost shell atoms. In the calculations of averaged potentials, equations summarized below were used: 1 Figure 7. Dissolution potentials Udiss as functions of (A): the number of nearest neighbor Pt atoms; (B): the d-band center of dissolving Pt atom. Particles: b; single crystal surfaces: 2; fitted functions: solid line.

∆σPt ) σPt - σ′Pt

(19)

shell bulk Udiss ) Udiss +

Eshell coh )

bulk Eshell coh - Ecoh 2e

(22)

Etot[Ptn(g)] - Etot[Ptn-nshell(g)] - nshellE[Pt(g)] nshell

(23) where σ′Pt and σPt are surface energies before and after the dissolution reaction, respectively [see Appendix B]. Since ∆σPt is positive for the dissolution of a Pt atom from Pt(111), Pt55, 1 Pt79, and Pt201, this reaction has higher Udiss than that estimated by the Gibbs-Thomson equation. Accordingly, ∆σPt strongly 1 . affects Udiss How does ∆σPt depend on geometrical and electronic natures of the dissolving site? Atoms with a larger number of nearest neighbor atoms are expected to have higher ∆σPt because of larger bonding energy Eb with surrounding atoms, which linearly correlates with ∆σPt as follows [see Appendix B],

Ecoh )

2

1 Eb = Ebulk coh - ∆µPt +

4πr ∆σPt 2eSPt

where n ) 55 and 201, and nshell ) 42 and 122 for particles with r ) 0.600 and 0.923 nm, respectively. This result suggests that the dissolution reaction occurs through the layer-by-layer mechanism, in which the reaction is initiated at edges of particles. 3.3. Dissolution Potentials of Supported Pt Particles. Before discussing the DFT results of supported Pt particles, parameters used in calculations and discussion are introduced in this paragraph. The cohesive energy Ecoh of the supported Pt particle is defined as follows:

nEtot[Pt(g)] + Etot[C(g)] - Etot[Ptn /C(g)] n

(24)

(20)

This expectation is met by our calculation results as shown in 1 Figure 7 (A), which shows Udiss as a function of the number nc of nearest neighbor atoms for the unsupported Pt particles and single crystal surfaces. The linear correlation shown in Figure 7(A) is similar to the ones obtained by DFT-calculations on copper nanoparticles performed by Taylor et al.48 and on metal-supported metal adstructures performed by Greeley.49 These linear correlations indicate that the metal-metal bond interactions in nanoparticles can be roughly described as additive pair potentials. The electronic structure also has an important role in the 1 1 . Figure 7(B) shows Udiss as a function of determination of Udiss the d-band center εd calculated by using a following equation:

all by using eq 6. To clarify the effect Ecoh was converted to Udiss of the support, we introduce the support interaction energy Eint defined as the following equation:

Eint )

Etot[Ptn(g)] + Etot[C(g)] - Etot[Ptn /C(g)] n

(25) Then, Ecoh is rewritten as follows: n(g) Ecoh ) EPt coh + Eint

(26)

Ptn(g) where Ecoh is the cohesive energy of unsupported Pt particles calculated by using eq 7. Equation 26 indicates that Eint



Figure 8. A color map of the d-band center for the Pt201 particle.

TABLE 3: The Number of Atoms in a Particle n, the Radius r, the Cohesive Energy Ecoh, the Support Interaction Energy Eint, and the Dissolution Potential Udissall for Supported Pt Particles or the Pt Tri-Layer on the Perfect Graphenea r (nm) Ecoh (eV · atom-1) Eint(eV · atom-1) Udissall(V (SHE))

n 1 2 3 4 13 55 Pt skin

0.158 0.198 0.227 0.250 0.370 0.600

1.150 (6.019) 1.634 (4.572) 2.321 (4.145) 2.587 (3.323) 3.300 (3.762) 3.994 (4.050) 4.569

1.150 (6.019) 0.188 (3.126) 0.216 (2.040) 0.154 (0.891) 0.089 (0.551) -0.009 (0.047) 0.005

-1.349 (-0.774) -0.626 (-0.532) -0.296 (-0.188) -0.132 (-0.055) 0.257 (0.301) 0.653 (0.648) 0.936

a Values for Ecoh, Eint, and Udissall in the parentheses correspond to results for Pt particles on the defected graphene.

all characterizes the effects of support on Ecoh and Udiss . The sitedependent binding energy Eb for the supported Pt particle was calculated by the following equation:

Eb ) Etot[Ptn-1 /C(g)] + Etot[Pt(g)] - Etot[Ptn /C(g)]

Figure 9. (A) The Cohesive energy Ecoh, (B) the dissolution potential all Udiss and (C) the interaction energy Eint for unsupported and supported Pt particles as functions of the particle radius r.

(27) Eb for supported Pt particles was converted to the reversible 1 for the elementary dissolution reaction of eq 1 potential Udiss by using eq 10. In the same manner as the introduction of Eint, we introduce the change in the support interaction energy ∆Eint defined as the following equation:

∆Eint ) (Etot[Ptn(g)] + Etot[C(g)] - Etot[Ptn /C(g)]) (Etot[Ptn-1(g)] + Etot[C(g)] - Etot[Ptn-1 /C(g)]) (28) This equation shows that ∆Eint corresponds to the difference in Eint between Ptn and Ptn-1. By using ∆Eint, Eb can be simplified as follows:

Eb )

n(g) EPt b

+ ∆Eint

(29)

where EbPtn(g) is the binding energy of the dissolving Pt atom in the unsupported Pt particle calculated by eq 11. Equation 29 shows that ∆Eint characterizes the effect of the support on Eb 1 and Udiss . all are summarized in Table 3 Results for Ecoh, Eint, and Udiss and Figure 9. For comparison, we also performed calculations for extended Pt skins on the perfect graphene as shown in Figure 10, which may be a proper model for infinitely large Pt particles on the graphene. The periodic structure proposed by Okamoto was applied in this calculation.50 Table 3 also includes this result.

Figure 10. The trilayer skin of Pt on the perfect graphene with a 2 × 2 periodic structure.

TABLE 4: The Number of Atoms n in a Particle, the Radius r, and the Net Valence Charge Q of a Pt Atom in Supported Pt Particles or the Pt Tri-Layer on the Perfect Graphene Obtained by Mulliken Population Analysisa

a

n

r (nm)

Q (e · atom-1)

1 2 3 4 13 55 Pt skin

0.158 0.198 0.227 0.250 0.370 0.600 Pt skin

9.852 (9.632) 9.885 (9.793) 9.963 (9.870) 9.945 (9.903) 9.976 (9.953) 9.996 (9.990) 10.002

Values for Q in parentheses are for Pt particles on the defected graphene.

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Figure 11. Positions of dissolving sites in Pt particles on perfect and defected graphenes with (A) n ) 1, (B) n ) 2, (C) n ) 3, (D) n ) 4, (E) n ) 13, (F) n ) 55. Light colored spheres (yellow colored spheres in web version of this article) show the dissolving atoms.

Figure 9 (C) clearly shows that Eint decreases by increasing particle radius r, and Eint is nearly zero for the largest Pt particle, n ) 55, in this study. Eint for the Pt skin on the graphene is also all nearly zero. Accordingly, effects of graphenes on Ecoh and Udiss are limited to the very small particles whose radius is smaller than 0.6 nm. The disappearance of the effects for large particles is due to the decrease in the ratio of the number of edge sites directly binding with graphenes to the total number of atoms in the particle. The large effect of edge sites on the binding energy between the particle and the graphene was also suggested by Okazaki-Maeda et al.35 They performed DFT calculations for Pt clusters with n e 13 on the perfect and defected graphenes and discovered a sensitive dependence of the cluster shape on the particle size [i.e., clusters with 3 e n e 7 adsorbed as vertical planar structures, clusters with 8 e n e 9 adsorbed as deformed planar structures, and clusters with n g 10 adsorbed as threedimensional structures]. Then, they suggested that the interactions between edges of the Pt clusters and the graphenes have an essential role affecting the size-dependent preferential structures. Pt atoms in small clusters with n e 9 cannot obtain large stabilization energy by the cohesion among them because of their small coordination number and take vertical or deformed planar structures which give a larger number of attachments between the edges and graphene. Pt atoms in large clusters with n g 10 can obtain large stabilization energy by the cohesion among them and take three-dimensional structures, such as a spherical shape. The growth in the size of the spherical shape decreases the ratio of the number of edges binding with the graphene to the total number of atoms in the particles. Hence, Eint converges to zero for these particles. As a consequence,

these DFT calculations show small binding energies between Pt nanoparticles and carbons and disagree with the experimental suggestion that the π-sites on the carbons cause the large stabilization of the Pt nanoparticles.3,25 The small interaction between Pt nanoparticles and carbons means that the number of valence electrons donated from Pt nanoparticles to carbons is small; this also disagrees with the experimental suggestion that the energy shift observed in XPS measurements is caused by the electron donation.3 Table 4 summarizes the net valence charges per Pt atom in Pt particles on perfect and defected graphenes obtained by the Mulliken population analysis.51 The net valence charge per Pt atom in Pt55 particles or the Pt skin on graphenes is very close to the valence electron number of 10 without any electron donation. Hence, our DFT results indicate that the energy shift is likely caused by the other effect, i.e., the slow screening of the core holes.27,29,30 1 Results for Eb, Udiss , and ∆Eint on dissolving sites shown in Figure 11 are summarized in Table 5 and Figure 12. The overall trend is similar to the one for Eint shown in Figure 9 [i.e., smaller ∆Eint for larger particles], and graphenes do not seem to have 1 for Pt particles effect larger than 0.2 eV in Eb and 0.1 V in Udiss with r g 0.6 nm. However, the negative shifts of 0.12-0.19 eV in Eb and 0.06-0.09 V in U1diss for Pt55 particles on graphenes in Table 5 are not negligible. To clarify the reason for this negative effect, we further investigated the site-dependence of ∆Eint for Pt55 particles on graphenes. Figure 13 shows ∆Eint as functions of the height z of the dissolving site shown in Figure 14. For both the perfect and defected graphenes, the Pt atom attached to the graphene has the largest negative shift. This



TABLE 5: The Number of Atoms in a Particle n, the Radius r, the Binding Energy Eb, the Change in the Support Interaction Energy ∆Eint, and the Reversible Potential Udiss1 for Supported Pt Particlesa n

r (nm)

Eb (eV)

∆Eint (eV)

Udiss1 (V (SHE))

1 2 3 4 13 55

0.158 0.198 0.227 0.250 0.370 0.600

1.150 (6.019) 2.118 (3.140) 3.696 (3.291) 3.384 (0.859) 2.914 (3.692) 4.483 (4.419)

1.150 (6.019) -0.774 (0.247) 0.272 (-0.133) -0.030 (-2.556) 0.645 (1.423) -0.119 (-0.183)

-0.774 (1.661) -0.290 (0.221) 0.499 (0.297) 0.343 (-0.919) 0.108 (0.497) 0.893 (0.861)

a Values in the parentheses correspond to results for Pt particles on the defected graphene. 1 Figure 13. Dissolution potentials Udiss of Pt55 particles on perfect and defected graphenes as functions of the distance z between the dissolving site and the graphene. The position of the graphene was determined as the averaged height of all carbon atoms.

Figure 14. Positions of dissolving sites used to obtain data in Figure 13. Light colored spheres (yellow colored spheres in web version of this article) show the dissolving atoms.

Figure 12. (A) The binding energy Eb, (B) the dissolution potential U1diss and (C) the change in the interaction energy ∆Eint for unsupported and supported Pt particles as functions of the particle radius r.

seems unlikely because atoms with a larger number of nearest neighbor atoms have larger binding energies. However, the mechanism of the negative ∆Eint can be understood by taking into consideration of the large effect of defects in Pt particles on the binding energy. As described in the previous section, Pt atoms at edges have a role of an anchorage enhancing the binding energy between the particle and the graphene. Removing the interfacial Pt atom from the particle gives an additional defect site at the interface, which, in turn, increases the binding energy between the Pt particle and the graphene. This is why the calculated dissolution potential for the interfacial Pt atom is lower than that for Pt atoms positioning at upper sites. We found that this site-dependence of the support effect cannot be 1 explained by the correlation between d-band center εd and Udiss as described in Figure 7(B), The spacial distribution of εd caused

Figure 15. Distributions of d-band centers in Pt55 particles on (A) perfect and (B) defected graphenes. Carbon atoms are shown as black and small spheres.

by the support shows a clear lowering of εd for Pt atoms attaching to the graphenes as shown in Figure 15. Accordingly, the correlation in Figure 7(B) cannot be used to predict this negative shift. In summary, our DFT calculations indicate that graphenes all and do not have the influence larger than 0.1 V on both Udiss 1 Udiss for the Pt particle with r g 0.6 nm. Negative shift less 1 is introduced by the defect-induced than 0.1 V in Udiss enhancement of the metal-support binding energy after the dissolution reaction.

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4. Conclusions We have performed DFT calculations on redox potentials of the direct dissolution reaction described as eq 1 for Pt particles with and without carbon supports described as graphenes and clarified the site-dependent properties, considering effects of the particle size and the supports. The results are summarized as follows: (i) Pt atoms at edges are easier to dissolve than those in flat surfaces. This calculation result indicates the layer-by-layer dissolution mechanism in which the reaction is initiated at edges of particles. (ii) Pt atoms with a smaller number of nearest neighbor atoms and with higher d-band centers have lower dissolution potentials. This rough correlation indicates that lowering the d-band center is important not only for the high activity for the oxygen reduction reaction52 but also for the high resistance to the dissolution reaction. (iii) The increase in the particle size decreases the binding energy per Pt atom between the Pt particle and graphene, and the effect of graphene on the dissolution potential is less than 0.1 V for the Pt particle with the radius of 0.6 nm. The small interaction between nanoparticles and graphene results in the small amount of the net charge transfer from Pt particles to graphene. These results suggests that that the π-sites in carbon surfaces do not stabilize Pt nanoparticles. It is also questionable to interpret the energy shift in XPS measurements as a result of the electron donation from Pt nanoparticles to carbons. (iv) Graphene lowers the dissolution potentials of Pt atoms attached to them by 0.06-0.09 V. This unexpected negative effect is caused by the defect-induced enhancement in the binding energy between the Pt particle and the support after the dissolution reaction.

I ) G[Pt2+(g)] - G[Pt(g)]

(A.3)

Gsolv ) G[Pt2+(g)] - G[Pt2+(aq)]

(A.4)

By using eqs A.2-A.4, eq A.1 can be converted to the following equation:

∆G1 ) G[Ptn-1(g)] + G[Pt(g)] + I - Gsolv + 2e φSHE - U - G[Ptn(g)] (A.5) e

(

)

In a similar way, the change in the Gibbs free energy for the reaction 9 is derived as follows:

(

∆Gall ) nG[Pt(g)] + nI - nGsolv + 2ne

The change in the Gibbs free energy is zero at the reversible 1 all and Urev satisfy following equations: potential; therefore, Urev

G[Ptn-1(g)] + G[Pt(g)] + I - Gsolv + φSHE 2e - U1rev - G[Ptn(g)] ) 0 (A.7) e

(

)

(

nG[Pt(g)] + nI - nGsolv + 2ne

Appendix A The dissolution potential for reaction 1 is different from that for reaction 9. To show this difference, we derive thermodynamic equations for these two dissolution reactions. The change in the Gibbs free energy for the reaction 1 is written as follows:

(

)

φSHE ∆G1 ) G[Ptn-1(g)] + G[Pt2+(aq)] + 2e -U e G[Ptn(g)] (A.1)

)

φSHE -U e G[Ptn(g)] (A.6)

)

φSHE - Uall rev e G[Ptn(g)] ) 0 (A.8)

In the same manner, the equation satisfied with the bulk bulk dissolution potential Urev can be derived as follows:

(

-Gbulk coh - I + Gsolv + 2e

)

φSHE )0 - Ubulk rev e

(A.9)

bulk is the cohesive free energy defined as follows: where Gcoh

where G is the Gibbs free energy of the system shown in the square bracket, φSHE is the thermodynamic work function for the SHE, and U is the electrode potential in SHE. For simplicity, effects of the solvation by the electric double layer on the Gibbs free energies of Pt particles are neglected in this equation. These effects have nonlinear dependence on the electrode potential U, and the evaluation of these effects requires a full selfconsistent solution of the system including long-range electric double layer.39,40 Calculations for Pt particles using this technique need large computational resources, and we could not perform them in this work. The Gibbs free energy for Pt2+(aq) is related to the Gibbs free energy of an isolated Pt atom in the gas phase as follows:

G[Pt2+(aq)] ) G[Pt(g)] + I - Gsolv

Gbulk coh ) G[Pt(g)] - G[Pt(bulk)]

(A.10)

By using eqs A.7-A.10, U1rev and Uall rev can be simply represented by following equations:

Gb - Gbulk coh 2e

(A.11)

Gcoh-Gbulk coh + 2e

(A.12)

U1rev ) Ubulk rev +

Uall rev

)

Ubull rev

(A.2)

where I is the ionization free energy, and Gsolv is the solvation free energy for the Pt cation. These are defined as the following equations:

Gb and Gcoh are defined as follows:

Gb ) G[Ptn-1(g)] + G[Pt(g)] - G[Ptn(g)]

(A.13)


Gcoh )


nG[Pt(g)] - G[Ptn(g)] n

By neglecting the zero point energies and vibration entropies, eqs A.11 and A.12 are turned into eqs 10 and 6, respectively. These equations correspond to the application of the linear Gibbs energy relationship developed by Anderson’s group to the dissolution reactions.53 Greeley and Nørskov proposed the same equation and applied it to the prediction of the dissolution potential of Pt alloys.54 Thermodynamic equations relating U1rev and Uall rev to the particle radius r are obtained by introducing a homogeneous particle model described below. By using the surface free energy σPt bulk and the bulk cohesive free energy Gcoh , the free energies of Ptn and Ptn-1 particles are written as follows:

G[Ptn(g)] ) nG[Pt(g)] - nGbulk coh +

4πr2 σ SPt Pt

(A.15)

G[Ptn-1(g)] ) (n -1 )G[Pt(g)] - (n -

1)Gbulk coh

4πr'2 + σ SPt Pt (A.16)

where r and r′ are the radii of these particles described as follows:

r)

( 4π3 nV )

1/3

(A.17)

Pt

1/3 3 r' ) (n - 1)VPt 4π

[

]

(A.18)

By substituting eqs A.15-A.18 into eqs A.13 and A.14, Gb and Gcoh are simplified as follows:

2σPtVPt rSPt

Gb = Gbulk coh -

Gcoh ) Gbulk coh -

3σPtVPt rSPt

2/3

=1-

21 3n

(A.20)

(A.21)

Substituting eqs A.19 and A.20, respectively, into eqs A.11 and A.12 gives respective dissolution potentials shown as follows,


1 2σPtVPt 2e rSPt

1 3σPtVPt 2e rSPt

(A.23)

Comparisons of eqs A.22 and A.23, respectively, with eqs 2 and 12 gives respective free energy equations described as eqs 3 and 13. Accordingly, the dissolution potential for the reaction 9 is different from that given by the Gibbs-Thomson equation for reaction 1. Appendix B It is possible to derive a thermodynamic equation of the site1 dependent dissolution potential Urev including the effect of the difference in the surface free energy before and after the dissolution reaction. We define surface free energies of Ptn and Ptn-1 particles as σ′Pt and σPt respectively. The free energies of the particles are written as follows:

G[Ptn(g)] ) nG[Pt(g)] - nGbulk coh +

4πr2 σ′ SPt Pt

G[Ptn-1(g)] ) (n - 1)G[Pt(g)] - (n - 1)Gbulk coh +

(B.1)

4πr'2 σ SPt Pt (B.2)

By substituting eqs A.17, A.18, B.1, and B.2 into eq A.13, we obtain Gb as follows:

Gb =

Gbulk coh

-

1 ∆µPt

4πr2∆σPt + 2eSPt

(B.3)

where the definition of ∆σPt is shown as eq 19. From eq B.3, eq 20 is derived by neglecting the zero point energies and vibration entropies. Substituting eq 20 into eq 10 gives eq 18. Acknowledgment. This work was partially financially supported by NEDO (New Energy and Industrial Technology Development Organization). References and Notes

(A.19)

In the derivation of (A.19), we applied the following approximation:

(1 - n1 )

all bulk Udiss ) Udiss -

(A.14)

(A.22)

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JP106593D

First Principles Calculations on Site-Dependent Dissolution Potentials

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