Effect of Size on Electron Transfer to Silver Clusters - American

13070

J. Phys. Chem. C 2008, 112, 13070–13078

Effect of Size on Electron Transfer to Silver Clusters R. C. Baetzold* 14204 Allen Road, Albion, New York 14411 ReceiVed: January 14, 2008; ReVised Manuscript ReceiVed: May 26, 2008

The electronic properties of silver clusters containing up to eight atoms and adsorbed to a positive kink on the (001) surface of silver bromide in contact with aqueous solution are calculated. The necessary reorganization energies and standard free energies of reaction are computed so that the activation energy for transferring an electron from a reducing molecule in solution to silver clusters could be determined with diabatic Marcus electron transfer theory for different cluster sizes. The activation energy oscillates between odd- and evensized neutral clusters and decreases with increasing size. In the presence of aqueous silver ions, a minimum cluster size of four atoms can be deduced for electron transfer from several reducing agents. These calculated results are discussed in relation to many experimental results. The calculations involve a self-consistent density functional method used to obtain solid state properties of the adsorbed silver clusters and a continuum solvation model (IEF-PCM) used to obtain free energies of solvation at the cluster surface in contact with the aqueous phase at 300 K. I. Introduction The transfer of electrons from a molecule in solution to a small metal cluster attached to the surface of a semiconductor is a fundamental process that is important in numerous proposed or developed technologies. These include electroless deposition,1 metal-catalyzed water splitting at a photocatalyst,2 photocatalytic reactions at TiO2,3 and photographic development.4,5 The specific focus in this paper will deal with the underlying reaction of electron transfer from reducing molecules in aqueous environment to small silver clusters attached to the surface of silver bromide. This reaction represents photographic development, an area where there is extensive experimental literature. The development reaction involving reducing agent (R) in solution with silver ion at the silver cluster (Agn) is written

R + Ag+ + Agn f O + Agn+1

(1)

leading to a silver cluster enlarged by one atom and the oxidized form of the reducing agent (O) in solution. There are two basic schemes6 that categorize the reactions involving electron transfer to the small silver cluster and the process by which silver ions subsequently become reduced and deposit at the cluster. In one process, silver ions dissolved in solution become reduced at the cluster in a reaction that is commonly termed physical development. Alternatively, the source of silver ions may come directly from the silver bromide, where the ions are reduced in essentially a solid state process. Chemical development is the term applied to this process. As explained by James,6 the silver cluster “acts as a catalyst for the reduction of silver ions in the sense that the reaction takes place more rapidly in its presence than absence”. The reactions considered in this report are those in which crystallites of silver bromide containing the silver cluster act independently of each other. Many early experiments have been reviewed6 in order to characterize the development reaction. A simple kinetic rate law is not followed, but the rate of growth of silver has been shown to equal the rate of oxidation of the reducing molecule, giving * Author e-mail: [email protected].

evidence that the overall rate is equal to the rate of reduction of silver ions. Several studies also measured the apparent activation energy for silver deposition. When heats of ionization are accounted for and the silver ion concentration in solution is held constant, values of 7.5-9.0 kcal/mol are typically found for the activation energy. The question of the minimum size of the silver cluster necessary to catalyze the development reaction is longstanding. Some years ago Hamilton and Logel7 outlined much of the early research on this matter. It was long accepted that a minimum critical size for catalysis exists for specific reaction conditions, although there still remains some question whether catalytic ability changes abruptly or more gradually with size. The detailed mechanism by which silver clusters decrease the activation energy for the reaction is unknown, but experiment6 is consistent with the idea that the silver cluster acts as a tiny electrode to accept electrons. Many authors8,9 have noted that, based upon energy levels in the solid state, the silver cluster introduces unfilled energy levels within the band gap of silver bromide that accept electrons more easily than if the electron was required to enter the conduction band. The evidence for a minimum critical size for silver clusters required to catalyze the development reaction comes from several experiments, although each has some limitations in directly specifying sizes of the clusters. In a classic physical development study, Hamilton and Logel7 prepared evaporated silver clusters on silica supports and were able to characterize the distribution of cluster sizes with decoration techniques through the use of electron microscopy. They conducted experiments in which the fractions of growing silver clusters were counted as a function of time in the aqueous reducing solution containing silver ions. Analysis of the data using their nucleation model led to the conclusion that Ag4 was a minimum critical size at which all clusters catalyzed silver ion deposition. Fayet, et al.10 deposited mass-selected silver cluster cations upon AgBr microcrystals that were then used to study the development reaction. Under the conditions of their experiment, Ag4, but not Ag3 or Ag, would catalyze the development reaction. Other analyses have generally agreed with the above findings

10.1021/jp800330f CCC: $40.75  2008 American Chemical Society Published on Web 08/05/2008

Effect of Size on Electron Transfer to Silver Clusters of critical size. Hada and Kawasaki11 analyzed curves of the fraction of silver bromide microcrystals reduced versus light exposure and concluded that Ag4 was the critical size for pure microcrystals as well as those treated with sulfur-containing agents. Hailstone and Hamilton12 developed extensive models for the mechanisms of growth and nucleation of silver clusters formed by the action of light on silver bromide. The analysis of their data of reduced silver versus light exposure yielded minimum developable sizes of 4 atoms for sulfur exposed samples and 2-3 atoms for sulfur plus gold exposed samples. The few examples cited here from a rather vast literature support the concept of a minimum critical size for catalysis; however, it has been acknowledged that the minimum developable size can depend upon the redox potential of the reducing molecule7–9 and conditions of the experiment such as time of reaction. Some of the earliest molecular orbital calculations for silver clusters free or adsorbed to silver bromide substrates13,14 showed an odd-even oscillation in their electronic properties, depending upon whether there is an open or closed shell of electrons. These trends are accepted based upon experimental photoionization experiments15 of gas phase beams containing clusters and improved methods of calculation.16–18 The implications of this odd-even effect for the minimum development size were discussed in terms of electron acceptance at neutral silver clusters.8,19 The ultimate critical size would be determined from even sized neutral clusters where the electron affinity is lowest, making the smallest developable size an odd sized cluster. Kawasaki, et al.20 measured the photon energy to oxidize silver clusters formed on silver bromide and found an oscillation in the threshold photon energy of 1.6, 0.75, and 1.1 eV that was assigned to Ag2, Ag3, and Ag4, respectively and gives support to the odd-even concept. More recently, Tani21 has called attention to the odd-even effects in silver clusters supported on silver bromide microcrystals based upon an analysis using the Kubo effect. These studies used electron spin resonance (ESR), electron microscopy, and temperature dependent rate studies to conclude that even-sized neutral clusters were formed when the silver bromide microcrystals were treated with certain reducing agents but that a distribution containing odd sizes could be formed under light exposure. The study supporting the odd-even concept also showed that the activation energy for development varied with the redox potential of the reducing agent. It was concluded that the rate of electron transfer from reducing agent to silver cluster was rate limiting for the development reaction. A later experimental kinetic study22 has analyzed the developability of silver clusters based upon electron transfer to the electron accepting level of a silver cluster in relation to size, composition, band structure, the number of electrons, and its site on the crystal. These studies found the smallest catalytic centers were Ag4 or Ag2Au when treated with gold salts. One additional background study of interest is the pulse radiolysis experiments of Belloni, et al.23 in which the growth of silver clusters in aqueous solution from silver ions was monitored. This work called attention to the importance of solvation energy changes with cluster size and employed a Born approximation to demonstrate the effect. The studies also called attention to the importance of positively charged silver clusters, as opposed to neutral, and proposed a growth mechanism based upon positively charged silver clusters in photographic systems. The phenomenon of critical cluster size observed in the development reaction is observed more broadly in the field of metal clusters. Early experiments by Hamilton and Logel24 involved preparation of vacuum evaporated distributions of Pt

J. Phys. Chem. C, Vol. 112, No. 34, 2008 13071 and Pd clusters supported on carbon or silica. Through methods such as those discussed above they were able to deduce minimum critical sizes for electroless deposition of metal ions on the catalyst. For Pd clusters, four or more atoms were required to catalyze nickel electroless deposition. In the case of Pt clusters, a rate of deposition dependent upon cluster size was observed that increases with size up to about 50 atoms and then decreases with further size increases. More recently, the impact of nanosized clusters in heterogeneous catalysis has been noted25 and examples of size dependent behavior given. Supersonic beams of Pt clusters in the gas phase have shown size effects for activating methane, where dimers or larger clusters are active.26 The catalytic oxidation of CO on gold clusters deposited size selectively on oxide support in the range up to 20 atoms has been extensively studied.27,28 Neutral Au8 was shown to be the smallest size active for CO oxidation.29 Clusters prepared by decarbonylation have been studied by Li and Gates.30 They showed that the catalytic activity for ethane hydrogenation varies strongly as the size of the Irn cluster changes from n ) 2, 4, 6. These few examples from the literature illustrate the point that size effects, and in many cases a minimum critical size for catalysis in metal clusters, is a phenomena of broad existence in many different applications. The origin of size effects in catalysis is generally traced to the electronic properties of the clusters. Concerning the development reaction, many authors have interpreted the origin of these effects in terms of cluster energy levels. Trautweiler9 was one of the first to do so. Hamilton8 emphasized the electronaccepting level of the silver cluster within the band gap of AgBr since this position is found in the solid state description of the system as deduced by experiment and calculations. Belloni, et al.23 emphasized the ionization potential of silver clusters based upon experiments conducted in solution. The purpose of this report is to perform a computational treatment of silver clusters adsorbed to silver bromide and in contact with an aqueous phase that will allow the study of the reaction with reducing agents. This work will extend our previous solid state treatments of silver clusters adsorbed to silver bromide surfaces.14,16,17 This system will allow an examination of the factors responsible for critical size and will give understanding of the mechanism of silver catalysis under conditions of physical or chemical development. The computational treatment of the supported silver cluster requires that we account for covalent and ionic interactions between cluster and AgBr support, as well as the polarization that occurs in the AgBr phase. In addition, the solvation effects on the silver cluster and reducing molecule in their various charge states are to be accounted for. These are treated with the polarized continuum model (IEF-PCM).31 It is noted that the silver cluster is attached to the silver bromide surface, so only a fraction of the cluster atoms are in contact with the solution phase. One additional consideration is the requirement that the calculation be appropriate for reaction at room temperature. Several approximations will be required in order to include all of these effects. A goal of this work is to understand the development reaction and the critical size phenomena in terms of the quantitative diabatic Marcus electron transfer theory.32 In the case of a small silver cluster with unoccupied energy levels spaced at more than kT, where T is temperature and k is the Boltzmann constant, we consider only one electron-accepting level. Electron transfer from the HOMO of the reducing agent to the LUMO of the silver cluster is considered for this reaction. This analysis reduces the electron transfer problem to that of a two level

13072 J. Phys. Chem. C, Vol. 112, No. 34, 2008

Baetzold

Figure 1. A fragment of the unrelaxed AgBr crystal surface is shown at the positive kink. Blue spheres represent silver and brown represent bromine ions. Lines are a guide to the eye.

system, much like for molecules in homogeneous solution, where the rate constant (K) for electron transfer is given by eq 2,32

K ) (2π ⁄ h)|HDA|2(π ⁄ λkT)1⁄2exp(-(λ + ∆G°)2 ⁄ 4λkT) (2) where h is Planck’s constant, HDA is the matrix element coupling donor and acceptor states, λ is the nuclear reorganization energy parameter, and ∆G° is the standard free energy change for the reaction. A more complete discussion of these terms required in the analysis and how they are calculated will be given in the Methods section. In this first work, the focus will be on the terms in the exponent in order to draw conclusions about the free energy of activation (∆G*), where

∆G * ) (λ + ∆G°)2 ⁄ 4λ

(3)

for the reactions considered at various silver cluster sizes. II. Methods A brief review of the method17 used to calculate the solid state properties of silver clusters attached to the positive kink on the AgBr (001) surface is presented. A neutral hemispherical array containing quantum and classical ions is first constructed. The array has a positive kink site located on the flat side of the hemisphere. A crystal fragment showing ions near the positive kink site is shown in Figure 1. The Ag17Br5 fragment at the kink site is treated quantum mechanically and is directly attached to the closest 194 ions that are represented in the shell model by core and shell charges that are coupled harmonically but can displace relative to one another. The core and shell interact through Coulomb terms and short-range pair potentials with the other crystal ions. This unit is enclosed in 4414 point-charges that are fixed in position at normal lattice sites and provide longrange electrostatic potentials. The positions of the atoms in the silver cluster, the ions of the Ag17Br5 quantum fragment, and the 194 core-shell ions are simultaneously optimized. Neutral and charged silver clusters are considered for each size. Starting positions for silver clusters are derived from the next smaller cluster and are appropriate for a stepwise growth process. Neutral and singly charged clusters are considered up to eight atoms in size. This procedure is contained in the embed5 computer code.33 The same parameters as before were employed including the LANL2DZ basis and pseudopotentials34 for silver and bromine ions, full core pseudopotential for silver,18 and the interatomic potential35 for classical ions. The B3LYP density functional theory36 using the Gaussian-03 code37 was employed for quantum mechanical calculations. The procedure outlined so far provides a total energy of the silver cluster attached to the silver bromide surface that can be used to calculate ionization potentials (IP) and electron affinity (EA) by energy differences of appropriate charge states. The binding energy (BE) is calculated by the energy released upon bringing the separated silver atoms and ions to their equilibrium configuration on the silver bromide surface. These energy terms are appropriate at the temperature 0 K, and it is required to convert these to free energy terms in order to apply the Marcus

Figure 2. The equilibrium structure of PPD in water at 300 K.

theory. The procedure for including the temperature effects is approximate since the cluster is supported on a silver bromide array, but fortunately the corrections are small. The vibrational zero point energy and vibrational entropy components were found by approximating the adsorbed silver cluster with a free silver cluster having the same charge and the same shape at its equilibrium geometry. These components are calculated using the appropriate partition functions. The zero-point energy and TS term, where T is the temperature and S is the entropy, are applied to the internal energy in order to approximate a free energy. There is no suitable means to convert the internal energy to an enthalpy, but this effect is expected to be small for solid state systems. Differences in free energies were employed to obtain the free energy to remove an electron from the neutral cluster (∆GIP) or to attach an electron to the neutral cluster (∆GEA) at the temperature of 300 K. Solvation energy terms are calculated using the polarizable continuum model (IEF-PCM) using atomic and group radii from the united atom model,38 which are contained within the Gaussian-03 code.37 These solvation energy terms are employed to correct the silver cluster solid state IP and EA values to obtain appropriate values when in contact with aqueous solution at 300 K. The model taken for this calculation is the silver cluster attached to a truncated model of the silver bromide, represented by the Ag12Br12 unit that is part of Figure 1. The calculation is applied for different charge states of the silver cluster at each cluster size. The ionization free energy and electron attachment free energy in aqueous solution at 300 K are calculated from the corresponding solid state values. The difference in free energy of solvation for positive and neutral adsorbed silver clusters is added to the solid state ionization potential to obtain the corresponding value in solution. Likewise, the difference in free energy of solvation for neutral versus negative clusters is added to the electron affinity in the solid state to calculate the corresponding value in solution. Various input parameters in the IEF-PCM method were tested. The values determined from UAHF radii38 are reported, but other sets of radii such as those of Bondi39 were tested and were found to have small effects on the solvation energy differences that are needed. The CPCM40 method was also studied, but results did not differ significantly from the IEF-PCM method. The radii scaling factor of 1.2 recommended in the Gaussian handbook was employed.41 Several reducing molecules used in the development reaction4,5 are examined in this report. The prototypical agent N,Ndimethyl-p-phenylenediammine molecule (PPD) is considered extensively. The structure of each reducing molecules is optimized in water at 300 K, and that for PPD is shown in Figure 2. The 6-31G** basis was used with B3LYP density functional theory with the IEF-PCM method. The calculations were used

Effect of Size on Electron Transfer to Silver Clusters

J. Phys. Chem. C, Vol. 112, No. 34, 2008 13073

Figure 3. A sketch of the free energy vs displacement for reactants (right curve) and products (left curve) in an electron transfer reaction. The reorganization energy is the change in reactant curve from its minimum to the position at product minimum.

to give a free energy for ionization in solution (∆GO R ) derived from the difference in total free energy of the positive and neutral clusters at equilibrium. The reorganization energy is the free energy required to change the reactants from their equilibrium geometry and local environment to the corresponding product nuclear configurations. This term includes a sum of inner-sphere and outer-sphere components. The outer-sphere, or solvent reorganization energy, is calculated with the standard equation of Marcus32a

λo ) (e ⁄ 2)(1 ⁄ a1 + 1 ⁄ a2 - 2 ⁄ R)(1 ⁄ ε∞ - 1 ⁄ εo) 2

(4)

where a1, a2, and R are the radius of the donor and acceptor and the distance between them, respectively; e is the electron charge; ε∞ is the high frequency dielectric constant, and εo is the static dielectric constant of the solvent. The inner sphere component of the reorganization energy arises from the nuclear changes in the donor and acceptor. The four-point method42 is used to calculate these components. This calculation involves determining the free energy difference of the donor molecule in its equilibrium reduced geometry (GR) and at the geometry of the oxidized state (GOR).

λ1 ) GOR - GR

(5)

The reorganization energy component due to the acceptor silver cluster is the free energy difference of the supported silver cluster in the geometry after it accepts an electron at equilibrium / (GAg ) and the silver cluster in its equilibrium geometry before it accepts an electron (GAg), where the corresponding silver bromide ion positions in the local environment are included in the calculation.

λ2 ) G/Ag - GAg

(6)

The total reorganization energy is taken as the sum of inner and outer sphere terms, as in a recent application,43 using eqs 4–6. It is noted that eq 6 can be applied for neutral or positively charged silver clusters. A short discussion of the well-known Marcus formalism32 is included. Consider the familiar concept of a configuration coordinate diagram in Figure 3. The curves represent free energies of the reactants including adsorbed silver cluster and reducing molecule in solution and corresponding products where an electron has been transferred from reducing agent to silver cluster as a function of reaction coordinate or displacement. It is assumed that the free energy is a harmonic function of displacement from equilibrium for both reactants and products. The reorganization energy is the free energy change for reactants moving from the equilibrium position to the nuclear position of the products at equilibrium. The simple relationship in eq 3 for the curve crossing point is obtained based upon the

Figure 4. The equilibrium structure of adsorbed Ag6 cluster and the nearest silver and bromine ions at the positive kink. Red spheres represent cluster atoms.

Figure 5. The equilibrium structure of adsorbed Ag7 cluster and the nearest silver and bromine ions at the positive kink.

reorganization energy and the free energy change from equilibrium reactants to products indicated in the figure. This procedure has been widely applied to homogeneous and heterogeneous reactions with good results and should apply equally to the development reaction. III. Results The previous calculations17 have shown that the positive kink site on the AgBr (001) surface is a unique and favorable site for the photolytic formation of silver clusters where alternate electron trapping and then interstitial silver ion trapping is favorable. The adsorbed neutral silver clusters at the positive kink up to five atoms in size took a near planar equilibrium geometry with deviations from planarity due to the relaxation of the underlying silver bromide model and different bonding to the underlying silver or bromide ions. The present work extends the prior studies to six, seven, and eight atom adsorbed silver clusters in neutral, anionic, and cationic charge states. The equilibrium geometry of the six atom cluster is roughly a planar shape, as shown in Figure 4. The equilibrium structure of the seven and eight atom clusters shows significant deviations from the planar equilibrium structure. Several isomers were studied by starting with different initial coordinates for the next one or two atoms that were added to the six atom cluster. A three-dimensional structure in which the last silver atom is adsorbed above the six atom cluster rather than next to silver bromide was found to be most stable for the seven atom cluster. This pattern continues for the next atom in the eight atom cluster. The charged clusters were found to possess equilibrium geometries similar in shape to the neutral clusters. These neutral clusters are shown in Figures 5 and 6. Table 1 presents the properties of these adsorbed clusters. Note that the oscillatory trends in the ionization potential and electron affinity continue up to the eight atom cluster. The shapes of these clusters are rather similar to those reported44 for the equilibrium gas phase structures. In the gas phase, a planar structure is maintained through the hexamer, and at the heptamer it changes to three-


Baetzold TABLE 2: Free Energy to Add (∆GEA) or Remove (∆GIP) an Electron from Neutral Silver Clusters Adsorbed to the Silver Bromide Positive Kink and in Contact With Water at 300 Ka solvent temperature cluster ∆GIP ∆GEA size ∆IP (eV) ∆EA (eV) ∆IP (eV) ∆EA (eV) (eV) (eV)

Figure 6. The equilibrium structure of adsorbed Ag8 cluster and the nearest silver and bromine ions at the positive kink.

TABLE 1: Ionization Potential (IP), Electron Affinity (EA), and Binding Energy (BE) of Neutral and Charged Silver Clusters Adsorbed to Silver Bromide at the Positive Kink Sitea cluster size 1 2 3 4 5 6 7 8

IP (eV)

EA (eV)

BE (eV) cation

BE (eV) neutral

BE (eV) anion

7.64 6.70 5.30 6.36 5.91 6.72 5.68 6.50

5.34 4.16 5.04 3.98 4.76 4.23 4.76 4.47

0.40 3.80 6.06 6.97 8.43 10.28 12.08 13.03

0.58 3.02 3.89 5.79 6.86 9.52 10.28 12.06

4.56 5.81 7.56 8.38 10.26 12.22 13.69 15.14

a Binding energy is relative to separated silver atoms and ions and the AgBr model.

dimensional. The gas phase ionization potentials reported for the clusters were 7.78, 5.77, 6.46, 5.89, 6.73, 5.57, and 6.22 for n ) 2-8. The ionization potentials found for these adsorbed clusters are surprisingly close to the values reported44 for the gas phase clusters for four atoms and larger. Calculations using the properties for the larger adsorbed clusters show that the photolytic mechanism based upon stepwise alternate electron and then interstitial silver ion capture in order to enlarge the silver cluster by one atom has exothermic energy changes for each cluster size up to eight atoms. When the supported silver clusters are placed in aqueous solvent, the ionization free energy and electron attachment free energy are changed because of the added polarization effects of the solvent. It is assumed that the polarization energy due to the presence of water is additive to the solid state polarization energies already included in the IP and EA data of Table 1. The free energy of solvation enhances the stability of charged silver clusters versus neutral clusters. We used the IEF-PCM model to calculate the free energy of solvation in water at 300 K and attached to silver bromide. To accomplish the IEF-PCM calculation, it was necessary to truncate the overall silver cluster-AgBr model to include the silver cluster and the nearest Ag12Br12 ions. This unit that is part of Figure 1 is used to calculate the differences in solvation energy for silver clusters in each charge state. In addition, the correction terms due to zero-point vibrations and entropy are applied in order to convert the solid state IP and EA values to a free energy. Table 2 shows the correction terms due to temperature dependent terms and the solvation free energy correction terms that are applied to the solid state IP and EA data in Table 1 and which lead to the final ionization free energy and electron attachment free energy in Table 2. It is noted that the vibrational and entropy correction terms are small, whereas the solvation free energy terms are much larger. The presence of water in contact with the silver cluster attached to silver bromide decreases the ionization free energy

1 2 3 4 5 6 7 8

-1.52 -3.21 -2.01 -2.02 -1.73 -1.56 -1.69 -1.41

0.35 0.42 0.64 0.53 0.83 0.47 0.77 0.60

0.00 -0.01 0.00 -0.03 0.05 0.06 0.07 0.07

0.00 0.01 0.04 0.01 0.02 0.09 0.02 0.15

6.12 3.48 3.29 4.31 4.23 5.22 4.06 5.16

5.69 4.59 5.72 4.52 5.61 4.79 5.55 5.22

a Corrections to the solid state values due to solvent and temperature effects are shown.

and increases the electron attachment free energy versus values appropriate to the solid state due to the greater solvation free energy for charged versus neutral clusters. Odd-even effects found in the solid state are also found for the electron attachment free energy and to a lesser extent for the ionization free energy in the presence of water. It is noted that the ionization free energy (∆GIP) becomes less than the electron attachment free energy (∆GEA) for many cluster sizes, unlike the reverse in the solid state. On a band picture framework, these ∆GIP and ∆GEA values for clusters of silver two atoms and larger are located in the band gap region (3.6-6.2 eV) of silver bromide. Thus, electron transfer from a molecule in solution to either neutral or positively charged silver clusters is energetically more favorable than electron transfer to the conduction band of silver bromide in the presence of solvent. This point is in accord with much previous analysis of photographic development. However, electron transfer to Ag2+ or Ag3+ is not favorable according to this calculation because of the small ∆GIP value for the dimer and trimer, which is quite different than in the solid state. Consider two possible reactions for electron transfer from a reducing agent in aqueous solution to the silver cluster attached to silver bromide that is in contact with water. The neutral (Agn) or positive silver cluster (Agn+) is considered.

R + Agn f O + Agn-

(7)

R + Agn+ f O + Agn

(8)

Using the ∆GIP and ∆GEA values from Table 2 as well as the free energy change for oxidation of the reducing agent, the standard free energy change (∆G°) for reactions 7 and 8 may be computed in eqs 9 and 10, respectively.

∆GO R

∆G° ) ∆GEA - ∆GR°

(9)

∆G° ) GIP - ∆GR°

(10)

Here, is the standard free energy of ionization of the reducing agent in solution. This value is computed at 300 K using the difference of the total free energies for the reduced and oxidized molecules at their equilibrium geometry in water at 300 K. Table 3 shows these values for several typical reducing agents used in the development reaction. In addition, the inner sphere reorganization energy computed from the free energy difference in eq 5 is shown in Table 3. The free energy of electron transfer from reducing agent to silver cluster may be evaluated using eqs 9 and 10. The prototypical reducing agent PPD was evaluated to give values of the free energy changes reported in Table 4. The reaction at


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TABLE 3: Standard Free Energy to Remove an Electron (∆G°R), Reorganization Energy (λ1) and Activation Energy for Electron Transfer to Ag4(∆G*4) for Reducing Agents in Water at 300 K reducing Agent

∆GRO (eV)

λ1 (eV)

∆G/4 (eV)

o-hydroquinone p-hydroquinone phenylhydrazine p-aminophenol p-phenylenediammine PPD p-hydroquinone anion

5.37 5.11 4.52 4.35 4.18 4.09 3.73

0.25 0.27 0.69 0.20 0.35 0.42 0.15

1.29 0.91 0.69 0.48 0.44 0.42 0.23

TABLE 4: Standard Free Energy Change (eV) for Electron Transfer from PPD to Silver Clusters Adsorbed to the Silver Bromide Positive Kink and in Contact With Water at 300 K cluster size 1 2 3 4 5 6 7 8

positive silver cluster

neutral silver cluster

-2.03 0.61 0.80 -0.22 -0.14 -1.13 0.03 -1.07

-1.60 -0.50 -1.63 -0.43 -1.52 -0.70 -1.46 -1.13

cluster size 1 2 3 4 5 6 7 8

(11)

is calculated to be -2.4, -2.1,-1.4,-0.3,-1.6, -0.3, and -0.3 eV for n ) 1-7, respectively. At sufficient silver ion concentration in solution, all of the neutral silver clusters would be converted to positively charged clusters. The driving force is particularly strong to form Ag2+, Ag3+, Ag4+, and Ag6+. This situation is rather different than in the absence of aqueous phase,17 where the positively clusters are less stable in the solid state than the neutral cluster plus the silver ion at an interstitial position. In the presence of solution phase containing sufficient silver ion, reaction 11 can occur once the concentration of interstitial silver ion is saturated. This can be a very complicated process since the space charge layer exists at the surface of silver bromide. Nevertheless, it follows from this consideration that in physical development, where silver ion is present in contact with the silver cluster, that reaction 11 will be an important factor in controlling the development reaction. The outer sphere reorganization energies are evaluated with eq 3. A sphere radius that encloses the silver cluster or the reducing molecule is calculated using atomic or united atom radii of 1.574, 1.725, 1.590, 1.86, and 1.77 Å for Ag, C, N, CH3, and NH2, from the united atom model.39 The radii calculated for silver clusters are shown in Table 5, and the value 5.29 Å was calculated for the PPD molecule. The outer sphere

cluster radius (Å)

λ0 (eV)

λ2 (eV) positive cluster

λ2 (eV) neutral cluster

1.57 2.50 2.93 3.94 4.33 4.72 5.38 5.38

2.12 1.32 1.14 0.90 0.84 0.80 0.75 0.75

0.54 1.44 1.46 1.13 0.92 1.08 1.16 1.37

1.58 1.33 2.05 1.16 1.18 1.26 0.92 1.15

TABLE 6: Total Reorganization Energy (λ) and Free Energy of Activation (∆G*) for Electron Transfer from PPD to Positively and Neutral Charged Silver Clusters Adsorbed to the Silver Bromide Positive Kink Site and in Contact with Water at 300 K cluster size

neutral clusters (reaction 7) is exothermic for all cluster sizes. The reaction at positively charged clusters is exothermic for many larger clusters. The free energy change oscillates between odd- and even-sized clusters where the trends are reversed for the neutral versus positively charged silver cluster reactions. Values for the other reducing agents are shifted depending upon their ∆GOR value, but the trends are similar and will be discussed later. Values for the single silver atom are presented in Table 4 and several succeeding tables for completeness. It is known from experiment that the single silver atom formed by light is not sufficiently stable to participate in these processes. Consider the charge of silver clusters adsorbed to the silver bromide surface that are present when in contact with solutions containing silver ion. The free energy change for the reaction

Ag+ + Agn f Agn+1+

TABLE 5: Reorganization Energy Components for Silver Clusters Adsorbed to the Silver Bromide Positive Kink Site and in Contact with Water at 300 K

1 2 3 4 5 6 7 8

positive

neutral

λ (eV)

∆G* (eV)

λ (eV)

∆G* (eV)

3.08 3.18 3.02 2.45 2.18 2.30 2.33 2.54

0.09 1.13 1.21 0.51 0.48 0.15 0.60 0.36

4.12 3.07 3.61 2.48 2.44 2.48 2.10 2.32

0.37 0.54 0.27 0.42 0.09 0.32 0.05 0.15

reorganization energies calculated from the radii using eq 4 are shown in Table 5 and decrease as the silver cluster size increases. The inner sphere components are calculated for various reducing molecules and are presented in Table 3. The inner sphere components for silver clusters are calculated, as explained in the Methods section, for electron acceptance at either positive or neutral silver clusters. In these cases the total energy of the embedded clusters with surrounding relaxed ions is compared in order to arrive at the reorganization energy component. Due to the complex nature of this relaxation, there are no clear trends in the λ2 term with cluster size shown in Table 5. The free energy of activation is computed for electron transfer using the Marcus formula in eq 3 for electron transfer to the neutral or positive cluster according to eqs 9 and 10, respectively. These values are shown in Table 6 along with the total free energy change and total reorganization energy where the reducing molecule PPD is considered. There are lower barriers for electron transfer at the neutral silver clusters than cationic clusters. The activation energies generally decrease with increasing cluster size. There is an odd-even oscillation in the activation energy values that is larger for even-sized neutral clusters and odd-sized positive clusters. In looking for a reason why larger clusters would be better catalysts for electron transfer, the trends to generally smaller activation energies with increasing size is very important. The decreasing values of the reorganization energy as the cluster size increases appear to be a very important parameter in determining the corresponding decline in activation energy with size. To further explore the effects of cluster size, silver clusters free of silver bromide support are examined in solution. The structure was fixed at the equilibrium geometries calculated for the adsorbed clusters in order to facilitate comparison with adsorbed clusters. All of the techniques used to calculate the standard free energy of reaction and reorganization energy are applied to determine the relevant properties for this system as


Baetzold

TABLE 7: Standard Free Energy Change (∆G°) and Activation Energy (∆G*) Values for Electron Transfer from PPD to Positively and Neutral Charged Silver Clusters in Water at 300 K cluster size 1 2 3 4 5 6 7 8

positive

neutral

∆G° (eV)

∆G* (eV)

∆G° (eV)

∆G* (eV)

0.64 -0.58 1.05 0.27 0.64 -0.35 0.59 -0.07

1.00 0.19 1.09 0.48 0.72 0.16 0.66 0.26

0.63 1.48 1.34 1.17 1.18 1.52 0.89 0.83

0.99 1.49 1.35 1.17 1.18 1.54 0.91 0.85

presented in Table 7. The PPD reducing molecule is also considered for electron transfer according to reactions 7 and 8. The activation energies for electron transfer were larger at the neutral clusters. In the case where excess silver ions are present in solution there is a strong driving force to adsorb at the neutral silver cluster, converting it to a positive charged cluster. Exothermic reaction energies are computed to be 3 eV or greater for each cluster up to eight atoms in size. This factor in addition to activation energy values show that reaction at the positively charged cluster is favored compared to the neutral cluster. Decreases in activation energy are noted as the size increases. IV. Discussion The trend in solid state properties of Ag6, Ag7, and Ag8 adsorbed to the silver bromide positive kink continues the trend observed for the smaller clusters. The values of the IP and EA oscillate between even and odd sizes. These clusters can accept an electron from the conduction band of silver bromide since their EA is greater than the conduction band edge at 3.6 eV. The odd-sized silver clusters can capture a photohole, but the even-sized clusters cannot, based upon the IP values versus the valence band edge at 6.2 eV. The reaction energies are exothermic for the alternate electron trapping

Agn + e- f Agn-

(12)

and then interstitial silver ion capture

Agn- + Ag+ f Agn+1

(13)

for all cluster sizes up to eight atoms. This result extends earlier findings17 to eight atoms and supports the nucleation and growth mechanism that Hamilton and Hailstone have discussed8,12,19 for the photolysis mechanism of silver cluster formation in silver bromide. The effects of aqueous solvent on the silver clusters adsorbed to silver bromide are evaluated. It was found that the presence of water decreased the ionization free energy and increased the electron attachment free energy of the silver clusters. For many cluster sizes ∆GIP < ∆GEA, unlike the solid state. The odd-even oscillations with cluster size are generally present with some exceptions at the smallest sizes. The shifts due to aqueous solvent are significant, and it is clear that these must be considered in evaluating the development reaction. The silver clusters are more easily oxidized in aqueous water than in the solid state. The even sized clusters have IP values greater than the valence band edge in the solid state but not in water where they can react with holes. The values of the standard free energy changes are related to the standard electrochemical redox potentials (E°) by the equation

E ° ) -∆G° ⁄ F

(14)

for a one electron transfer where F is equal to the unit charge e when free energies are expressed in eV per atom or molecule, as in the present work. This equation may be applied to the ∆GO R values in Table 3 for various reducing agents in order to obtain the standard electrochemical for the half-cell in which the reducing agent is converted to its oxidized form plus an electron. Likewise, the standard free energy change for reactions 7 or 8 given in eq 9 or 10, respectively, as presented in Table 4 may also be converted to a standard redox potential through eq 14. The standard free energy of reaction 8 may be expressed as

∆G° ) -RT ln[(Agn-)(O) ⁄ (Agn)(R)]

(15)

-),

where R is the ideal gas constant and (Agn (Agn), (O), and (R) represent the activities of the corresponding chemical species in eq 8. These equations serve as the link of free energy changes to redox potentials or concentrations of reactants and products. It is noted that the reducing agents in Table 3 often undergo further reaction in which a proton may be ionized or a dismutation reaction takes place. The resulting equilibria are well documented45 and can be included in eq 15 along with reagent activities and variables such as pH and pAg in order to treat a particular reaction condition. The free energy of activation for electron transfer from PPD to silver clusters is plotted in Figure 7. The activation energy is smaller at the neutral silver clusters than the corresponding positively charged silver clusters from two to five atoms in size and correlates with a more exothermic reaction at these neutral clusters. This more favorable reaction at neutral clusters would be unexpected just based upon considerations in the solid state but becomes possible due to the greater solvation energy of charged clusters rather than neutral clusters. Thus, the electron attachment free energy is larger than or roughly comparable to the ionization free energy in solution, and this favors reaction at the neutral clusters. The small silver clusters Ag2+ and Ag3+ are ineffective in accepting an electron. Consider how the computed reaction energies and activation energies could be related to a minimum critical size. Under conditions of strict chemical development, all of the adsorbed silver clusters in contact with solution will have neutral charge. Table 4 shows that for all cluster sizes the development reaction with PPD is exothermic. The activation energy in Table 6 for this reaction oscillates between odd and even sizes but declines as size increases. The even-sized clusters have the higher activation energies and would present the greatest obstacle to reaction. This situation fits the picture discussed earlier by Hamilton and Hailstone8,12,19 that is based upon solid state

Figure 7. A plot of the calculated free energy of activation vs cluster size for neutral (left side) and positively charged (right side) silver clusters to accept an electron from PPD.


Figure 8. A plot of the calculated free energy of activation versus cluster size for neutral silver clusters to accept an electron from p-hydroquinone, p-aminophenol, and p-hydroquinone anion, from top to bottom.

properties of the cluster where an odd-sized cluster represents the minimum critical size. This result is not consistent with several experiments that deduced the even-sized cluster Ag4 to be the minimum developable cluster size. The declining activation energy with increasing cluster size, however, is consistent with the experimental observation that larger cluster sizes should be more active. Consider the development reaction when silver ions are present in solution. The silver ions will adsorb exothermically to the neutral cluster by reaction 11. For the small cluster sizes this reaction is strongly exothermic and Ag2 and Ag3 should be converted to Ag3+ and Ag4+, respectively. The Ag4 cluster has a smaller driving force to convert to Ag5+ and could be present in either form depending upon conditions. The data in Table 6 show that the activation energy for the development reaction is smaller at Ag4 compared to Ag4+ or Ag5+ or any of the smaller clusters. There are pathways at larger clusters with lower activation energies for the successive steps at the clusters Ag5 or Ag6+, Ag6, Ag7 or Ag8+ and Ag8. Thus, Ag4 or possibly Ag5+ is consistent with the minimum size for electron acceptance, above which the activation energy is smaller. This result is consistent with many experiments. The experiments of Hamilton and Logel7 are ones in which aqueous silver ion is present and participates in the development reaction. The experimental reported minimum developable size of Ag4 is consistent with the above analysis. The experiments of Fayet et al.,10 Hada and Kawasaki,11 and Hailstone and Hamilton12 gave a critical size of Ag4, but in these cases it is not clear whether silver ion is present in solution in sufficient concentration to fit this analysis. It is often true that some soluble silver salts are present within silver bromide that could give aqueous silver ion. It is clear, however, that when only neutral silver clusters are present an odd sized minimum critical size is predicted, and this would not be consistent with the above interpretations of some experiments. The activation energy for electron transfer from the other reducing agents follows curves similar to PPD as shown in Figure 8. This plot shows data for p-hydroquinone, p-aminophenol, and p-hydroquinone anion in decreasing order. In each case the activation energy oscillates at the neutral clusters and decreases with size. The shapes of the curves are similar for each reducing agent, and in each case the activation energy is smaller at Ag4 than Ag4+ or Ag5+ and is consistent with this minimum critical size. The results also show that when Ag3 is converted to Ag4+, the activation energy for the reaction is only 0.09 eV greater than Ag4 in reaction with PPD. With stronger reducing agents or modifications to the silver cluster, the minimum critical size of three atoms might be realized. It would also be possible to achieve the three atom critical size with neutral clusters, but a two atom critical size seems most unlikely due to the higher activation energies. The calculations for bare silver clusters in aqueous environment present a different picture for silver cluster growth. In this

J. Phys. Chem. C, Vol. 112, No. 34, 2008 13077 case the strong adsorption of silver ion would convert all clusters to their positive charge. Only processes involving positively charged silver clusters become important, and this deduction agrees well with interpretations from pulse radiolysis experiments23 that pointed to the importance of positively charged clusters. This agreement supports our overall procedure. The trends in properties with silver cluster sizes should be emphasized more than absolute values. This is because of the complex nature of the system studied that required several approximations. Previously,17 many approximations in the solidstate calculation were mentioned. The solvation energy is an additional component of the present work that is calculated by continuum methods that have limitations.31c Despite the approximations inherit in this calculation, the final activation energies in Table 6 agree well with the experimental activation energies6 for the development reaction in the range of 7.5 9.0 kcal/mol, or about 0.4 eV. This indicates that the calculations are realistic. The trends in activation energy with ionization potential of the developer molecule are likewise realistic. These values reported in Table 3 show that the developer standard free energy of ionization in solution scales with the free energy of activation for electron transfer to the Ag4 cluster and, due to eq 14, correlates well with the statements in the literature that reduction potential plays a role in determining the minimum critical size for catalysis by silver clusters. V. Conclusions The electron transfer reaction from aqueous reducing agent to silver clusters on the surface of silver bromide was examined. The presence of water stabilizes charged states of the cluster and leads to more favorable electron transfer to neutral, rather than positively charged, silver clusters for most sizes up to eight atoms. This result is unlike that calculated in the absence of water where the cluster ionization potential exceeds its electron affinity. The free energies of electron transfer were calculated, as well as the reorganization energies of the reactants involved in electron transfer. The Marcus electron transfer theory is applied to this system in order to determine the activation energy for electron transfer at various silver cluster sizes. In the presence of aqueous silver ion, the Ag4 cluster can act as a catalyst for the electron transfer reaction. Smaller clusters have much larger activation energies for accepting electrons, and beyond this size the activation energies decline. The decline in activation energies with increasing cluster size supports the concept of a minimum critical size for accepting electrons from reducing agent, as is commonly observed in these reactions. Several reducing molecules were studied, and the correlation of ionization potential and activation energy for electron transfer was established. Appendix A check on the method of calculating solvation energy changes for silver clusters of various charges attached to the Ag12Br12 fragment was undertaken. In the alternate method a bare silver cluster was considered, and the calculated solvation energy is multiplied by the fraction of solvent-accessible surface available at the kink site in order to compute changes in solvation energy. These values are used to compute the free energy changes for electron transfer. Using these values and the reorganization energy explained in the text, the free energy of activation is calculated and compared for the two methods in the first two columns of Table A1 for the electron transfer from PPD to silver cluster. Data are also calculated for the bare cluster using the CPCM method with Bondi radii and are presented in the third column. There is good agreement between the three sets of data


Baetzold

TABLE A1: Calculated Free Energy of Activation (eV) for Electron Transfer from PPD to Silver Cluster ∆G*, Ag12Br12a

∆G*, barea

∆G*, bareb

8 7 6 5 4 3 2

0.15 0.05 0.32 0.09 0.42 0.27 0.54

0.17 0.07 0.30 0.12 0.38 0.23 0.42

0.20 0.09 0.20 0.12 0.36 0.12 0.44

a

IEF-PCM, UAHF radii, R ) 1.2.

size

b

CPCM, Bondi radii, R )

1.2.

in absolute value and trend with cluster size that supports the methods employed in this work. Acknowledgment. This material is based upon work supported by the National Science Foundation under the following NSF programs: Partnerships for Advanced Computational Infrastructure, Distributed Terascale Facility (DTF), and Terascale Extensions: Enhancements to the Extensible Terascale Facility. References and Notes (1) (a) Brenner, A.; Riddell, G. J. Res. Nat. Bur. Stand. 1946, 37, 31. (b) Yin, X.; Hong, L.; Chen, B.-H. J. Phys. Chem. B 2004, 108, 10919. (2) (a) Maeda, K.; Domen, K. J. Phys. Chem. C 2007, 111, 7851. (b) Kamat, P. V. J. Phys. Chem. C 2007, 111, 2834. (3) Tachikawa, T.; Fujitsuka, M.; Majima, T. J. Phys. Chem. C 2007, 111, 5259. (4) James, T. H. The Theory of the Photographic Process, 4th ed.; Macmillan Publishing Co.: New York., 1977. (5) Tani, T. Photographic SensitiVity; Oxford University Press: New York., 1995. (6) Chapter 13 of ref 4. (7) Hamilton, J. F.; Logel, P. C. Photogr. Sci. Eng. 1974, 18, 507. (8) Hamilton, J. F. AdV. Phys. 1988, 37, 359. (9) Trautweiler, F. Photogr. Sci. Eng. 1968, 12, 138. (10) Fayet, P.; Granzer, F.; Hegenbart, G.; Moisar, E.; Pischel, B.; Woste, L. Phys. ReV. Lett. 1985, 55, 3002. (11) Hada, H.; Kawasaki, M. J. Imaging. Sci. 1985, 29, 51. (12) Hailstone, R. L.; Hamilton, J. F. J. Imaging. Sci. 1985, 29, 125. (13) Baetzold, R. C. J. Chem. Phys. 1971, 55, 4363. (14) (a) Baetzold, R. C. J. Sol. St. Chem. 1973, 6, 352. (b) Hamilton, J. F.; Baetzold, R. C. Photogr. Sci. Eng. 1981, 25, 189. (c) Hamilton, J. F.; Baetzold, R. C. Science 1979, 205, 1213. (15) Taylor, K. J.; Pettiette-Hall, C. L.; Cheshnovsky, O.; Smalley, R. E. J. Phys. Chem. 1991, 96, 3319. (16) Baetzold, R. C. J. Phys. Chem. B 2001, 105, 3577. (17) Baetzold, R. C. J. Phys. Chem. C 2007, 111, 1385. (18) Bonacic-Koutecky, V.; Cespiva, L.; Fantucci, P.; Koutecky, J. J. Chem. Phys. 1993, 98, 7981. (19) Hailstone, R. K.; Hamilton, J. F. J. Imaging. Sci. 1987, 31, 229. (20) Kawasaki, M.; Tsiymuria, Y.; Hada, H. Phys. ReV. Lett. 1986, 57, 2796. (21) Tani, T. J. Appl. Phys. 2002, 91, 4595.

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Effect of Size on Electron Transfer to Silver Clusters - American

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