Mathematical Modeling of Electrochemical Promotion and of Metal

Jul 25, 2001 - A surface diffusion-reaction model is developed and solved to describe the effect of electrochemi- cal promotion (also known as non-Far...
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Ind. Eng. Chem. Res. 2001, 40, 4209-4215

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Mathematical Modeling of Electrochemical Promotion and of Metal-Support Interactions C. G. Vayenas* and G. E. Pitselis Department of Chemical Engineering, University of Patras, GR-26500 Patras, Greece

A surface diffusion-reaction model is developed and solved to describe the effect of electrochemical promotion (also known as non-Faradaic electrochemical modification of catalytic activity, NEMCA effect) on porous conductive catalyst films on solid electrolyte supports. The model accounts for the migration (backspillover) of promoting anionic, Oδ-, species from the solid electrolyte onto the catalyst surface. The same type of model is then applied to describe the effect of metal-support interactions for the case of finely dispersed metal nanoparticles on ZrO2and TiO2-based porous supports where the same type of Oδ- backspillover mechanism has recently been shown to be operative. Two basic dimensionless numbers are obtained that dictate the maximum allowable thickness of electrochemically promoted catalysts, or the maximum crystallite size in dispersed supported metal catalysts, to fully utilize the promoting species. Introduction The catalytic activity and selectivity of porous polycrystalline metal catalyst films deposited on solid electrolytes can be altered dramatically and reversibly via application of current or potential ((1 V) between the catalyst film, which also acts as a working electrode, and a counterelectrode also deposited on the solid electrolyte1-8 (Figure 1). This phenomenon is known as electrochemical promotion or non-Faradaic electrochemical modification of catalytic activity (NEMCA effect).2-8 Work in this area has been reviewed in several book chapters9-12 and in a book.13 The importance of electrochemical promotion in catalysis, surface science, and electrochemistry has been discussed by Pritchard,14 Haber,15 Bockris,16 and Wieckowski.17 Two parameters are commonly used to characterize the magnitude of electrochemical promotion: (1) The first is the Faradaic efficiency, Λ, defined by

Λ ) (r - ro)/(I/2F)

(1)

where r is the electrochemically promoted catalytic rate, ro is the open-circuit (unpromoted) rate, I is the applied current, and F is Faraday’s constant. A catalyst is electrochemically promoted when |Λ| > 1. When Λ > 1, the reaction is termed electrophobic,9-12 and when Λ < -1, the reaction is termed electrophilic.9-12 Λ values as high as 3 × 105 and as low as -3 × 104 have been measured.9-13 (2) The second is the rate enhancement ratio, F, defined by

F ) r/ro

(2)

Values of F as high as18 150 or as low as19 zero (i.e., complete catalyst poisoning) have been measured. The NEMCA effect has been studied already with more than 60 catalytic systems9-13 and does not appear to be limited to any type of conductive catalyst, catalytic * To whom correspondence should be addressed. E-mail: [email protected].

Figure 1. Principle and basic experimental setup of electrochemical promotion (NEMCA) investigations.1-13

reaction, or solid electrolyte. Aqueous electrolytes have also been used.20,21 It has been shown, using a variety of surface spectroscopic techniques including X-ray photoelectron spectroscopy (XPS),22,23 ultraviolet photoelectron spectroscopy (UPS),24 temperature-programmed desorption (TPD),25 Scanning tunneling microscopy,26 AC impedance spectroscopy,27,28 and work function measurements,3,29-33 that electrochemical promotion is due to electrochemically controlled migration (backspillover) of promoting species (Oδ- in the case of O2- conductors such as Y2O3-stabilized ZrO2 (YSZ) or mixed electronicionic conductors such as TiO233 or CeO2,34 Naδ+ in the case of Na+ conductors such as β′′-Al2O335) from the solid electrolyte onto the gas-exposed electrode surface. (We use the term backspillover to denote migration of a species from an oxide to a metal and the term spillover for the reverse process.36-38) These backspillover species, which migrate on the gas-exposed catalyst electrode, are accompanied by their compensating (screening) charge in the metal and establish an overall neutral double layer on the gas-exposed catalyst electrode surface12,31-34,39,40 (Figure 2). The exact value of the negative charge δ- of the promoting Oδ- species is not yet fully known. XPS has shown that the Oδ- that migrates on Pt surfaces interfaced with YSZ under anodic (positive) current application is energetically indistinguishable from O2in YSZ.22,32 This suggests that Oδ- is O2-. This is also supported by dipole moment measurements.25 Because,

10.1021/ie010001f CCC: $20.00 © 2001 American Chemical Society Published on Web 05/05/2001

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knowledge, this is the first diffusion-reaction modeling involving not a reactant but rather a promoting species. Modeling Physical Considerations and Kinetics. Experiment1-13 has shown that electrochemically promoted kinetics very frequently follow the following simple expression

ln(r/ro) ) R∆Φ/kbT

(3)

where ∆Φ is the change in work function, Φ, of the catalytically active surface induced by the application of a current, I, and concomitant change, ∆UWR, in the potential of the catalyst electrode (working electrode, W) relative to a reference (R) electrode. The latter, ∆UWR, is related to ∆Φ via an important relationship in solid-state electrochemistry3,9-13,29-32,40

∆Φ ) e∆UWR Figure 2. Schematic of a metal catalyst electrode particle supported on Y2O3-doped ZrO2 (YSZ), an O2- conductor, under open-circuit conditions (top) and upon application of an anodic (positive) current showing the backspillover-formed double layer at the metal-gas interface (bottom).31-34

however, the evidence is not yet entirely conclusive, it is safer to use Oδ-, rather than O2-, for the promoting anionic species. Electrochemical Promotion and Metal-Support Interactions It has recently been shown that the same type of Oδbackspillover is the main molecular mechanism of the effect of metal-support interactions (MSI)41,42,43 in commercial dispersed nanocrystalline metal catalysts supported on highly porous ZrO2, TiO2, CeO2, and doped ZrO2 and TiO2 supports13,44,45 (Figure 3). It has also been shown9-13,38 that the promoting backspillover Oδ- species, which increase the catalyst surface work function by up to 1 eV,9-13,44 are significantly less reactive for oxidation reactions than normally chemisorbed O on the same catalyst surface where the two species coexist.22,25 In fact, the ratio of the reactivities of O and Oδ- equals the value of the apparent Faradaic efficiency Λ in electrochemically promoted catalysts,9-13 which provides a direct explanation of the non-Faradaic nature of electrochemical promotion. The recently established13,44,45 mechanistic equivalence between electrochemical promotion and MSI shows that electrochemical promotion is an electrochemically controlled metal-support interaction44 (Figure 3). Conversely a metal-support interaction is mechanistically equivalent with a “wireless” NEMCA configuration13,46 in which gaseous oxygen continuously replaces spent O2- in the support. The present work addresses an important question frequently raised in electrochemical promotion studies: 13 How thick can a porous metal electrode deposited on a solid electrolyte be in order to maintain the electrochemical promotion (NEMCA) effect? The same type of analysis is applicable for the case of nanoparticle catalysts on commercial supports such as ZrO2, TiO2, YSZ, CeO2, and doped ZrO2 or TiO2. To the best of our

(4)

The parameter R in eq 3 is positive for electrophobic reactions (∂r/∂Φ > 0, Λ > 1) and negative for electrophilic ones (∂r/∂Φ < 0, Λ < -1). More complex electrochemical promotion behavior is frequently encountered,13 leading to volcano-type or inverted-volcano-type behavior.13 However, even then, eq 3 is satisfied over relatively wide (0.2-0.3 eV) ∆Φ regions, so we will limit the present analysis to this type of promotional kinetics. It should be noted that eq 3, originally found as an experimental observation,2-12 can be rationalized by rigorous mathematical models that account explicitly for the electrostatic dipole interactions between the adsorbates and the backspillover-formed effective double layer.13 The electrochemically induced change in work function ∆Φ (eq 3) is related to the coverage θi of the promoting species (e.g., Oδ-) on the catalyst surface via the Helmholz equation

∆Φ )

eNM P ∆θ o i i

(5)

where e ()1.6 × 10-19C) is the unit electric charge, o ) 8.85 × 10-12 C2/(J m), NM is the catalyst surface atom density (atom/m2), and Pi is the dipole moment of the backspillover species (Oδ-) under consideration. Typically, Pi is on the order of 1 D (Debye). The Debye unit, D, equals 3.3 × 10-30 C m. In writing the Helmholz equation in the form of eq 5, we assume that ∆Φ is dictated largely by the change in the coverage of the backspillover species (Oδ-) and not by any concomitant changes in the coverage of coadsorbed reactants or products (e.g., O). This is reasonable in view of the fact that the ionic backspillover species have significantly higher dipole moment, Pi, values than the more covalently bonded reactants, intermediates, and products. Combining eqs 2, 3, and 5 and noting that

θi ) Ci/Ci,max

(6)

where Ci is the surface concentration (mol/m2) of the backspillover species and Ci,max is its maximum possible value, one obtains

ln F )

ReNMPi ∆θi okbT

(7)

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Figure 3. Schematic of a metal particle (∼1 µm) of a metal catalyst film deposited on YSZ or TiO2 under electrochemical promotion conditions (left) and of a metal nanoparticle (∼1 nm) deposited on a porous YSZ or TiO2 support, showing the locations of the classical double layers formed at the metal-support interface and of the effective double layers formed at the metal-gas interface. The energy diagrams (bottom) indicate schematically the spatial constancy of the Fermi level EF (or electrochemical potential µ j e) of the electrons, of the chemical potential of oxygen, and of the electrochemical potential of O2-. Note that, under the application of electrical bias (left), µ j O2remains spatially constant, but µ j e and µ j O2 both bend in the solid electrolyte support. The Fermi level µ j e, and thus the work function Φ, of the metal can be affected by varying UWR (left) or by varying via doping the Fermi level of the support (right).

with kb)1.38 × 10-23 J/K, or equivalently

ln F )

ReNMPi ∆C okbTCi,max i

(8)

Equation 7 can also be written as

ln F ) Π∆θi

(9)

where the dimensionless parameter Π is defined as

Π)

ReNMPi okbT

(10)

For typical experimental1-13 parameter values (R ) 0.5, NM ) 1019 atom/m2, Pi ) 1 D ) 3.3 × 10-30 C m, T ) 673 K), the dimensionless parameter Π equals 32, which implies, in view of eq 9, dramatic rate enhancement ratio F values (e.g., F ) 120) even for moderate (∼15%) changes in the coverage θi of the promoting backspillover species, as experimentally observed.1-13 The promotional kinetics described by eq 3, or by its equivalent eq 9, imply uniform distribution of the backspillover-promoting species on the catalyst surface. This requires fast ion backspillover relative to its desorption or surface reaction. As already noted, the backspillover-promoting species Oδ- can eventually desorb as O222,25 or can react with an oxidizable reactant (e.g., CO or C2H4), albeit at a rate that is Λ times slower than that of a normally chemisorbed O atom.10-13 It is therefore important to examine under what conditions the above criterion is met (i.e., fast ion backspillover relative to its desorption or consumption), for otherwise the promotional process will be “internally diffusion limited” not because of slow diffusion of the

Figure 4. Schematic of an electrochemically promoted metal catalyst film supported on an O2- conductor.

reactants but because of slow diffusion (backspillover) of the promoting species. Mathematical Modeling Electrochemical Promotion. We consider the porous metal catalyst film shown in Figure 4, which is interfaced with an O2- conductor. When a positive current, I, is applied between the catalyst and a counterelectrode, oxide ions, O2-, are supplied from the solid electrolyte to the three-phase boundaries (tpb’s), solid electrolyte-metal-gas, at a rate I/2F. Some of these O2- ions will form O2 at the tpb and desorb

2O2- f O2(g) + 4e-

(11)

and some will form backspillover Oδ- species according to the reaction

O2- f (Oδ--δ+) + 2e-

(12)

where (Oδ--δ+) denotes the backspillover Oδ- species

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in order to emphasize that it is a dipole (i.e., is overall neutral), as it is accompanied by the compensating (screening) charge δ+ in the metal. We assume that the fraction, f, of O2- arriving at the tpb that follows path 12 is proportional to (1 - θi), where θi is the coverage of (Oδ--δ+) at the tpb (z ) 0). We also assume that the rate of consumption of (Oδ-δ+) on the catalyst surface (due to desorption or reaction) is first-order in θi (or Ci) and denote by Ds the effective surface diffusivity (m2/s) of the backspillover species on the catalyst surface. One thus obtains the following differential steadystate mass balance for the surface concentration, Ci, of the promoting species

d2Ci dz2

- (k/Ds)Ci ) 0

(13)

where k is the first-order rate constant for backspillover ion consumption (due to desorption and/or reaction). One also has the boundary conditions (Figure 4)

z ) 0:

dCi I )(1 - Ci/Ci,max) dz 2FDsltpb

z ) L:dCi/dz ) 0

Figure 5. Dependence of the promotional effectiveness factor, ηp, on the Thiele modulus Φp and dimensionless current J.

and expresses the fraction of the total catalyst surface covered by the promotional backspillover species. Substituting eq 21 into eq 22 and integrating, one obtains

(14)

1/ηP ) 1/J + Φp/(tanh Φp)

(15)

The dependence of ηP on J and Φp is shown in Figure 5. As expected, for large J and small Φp values, ηP approaches unity. This implies maximum promotion of the catalyst surface. For large (>10) J values, eq 23 becomes

where ltpb(m) is the lenght of the three-phase boundary. Recalling that θi ) Ci/Ci,max and defining ξ ) z/L, where L is the thickness of the catalyst film, one can write eqs 13-15 in dimensionless form

(23)

ηP ≈ (tanh Φp)/Φp

(24)

ηP ≈ 1/Φp

(25)

2

d θi dξ2 ξ ) 0:

- Φp2θi ) 0

dθi ) -JΦp2(1 - θi) dξ

ξ ) 1:dθi/dξ ) 0

(16) (17) (18)

which reduces to

for large (>10) Φp values. For small J and large Φp values, one obtains from eq 23

ηP ) J

where the promotional Thiele modulus Φp is defined as

Φp ) Lxk/Ds

(19)

Also, in view of eq 9, one has

ln F )

and the dimensionless current J is defined as

J ) I/(2FkCi,maxLltpb) ) I/(2FkCi,maxAc)

(20)

where, in the last equality, we have expressed the total gas exposed catalyst electrode surface area, Ac, as Lltpb by assuming uniform film porosity and local geometry. Solution of eq 16 with boundary conditions 17 and 18 gives

θi(ξ) )

cosh[Φp(1 - ξ)] sinh Φp/(ΦpJ) + cosh Φp

(21)

Note that, for small Φp and large J values, θi(ξ) approaches 1 for all J. The promotional effectiveness factor, ηP, is defined as

ηP )

∫01θi(ξ) dξ

(22)

(26)

∫01Πθi(ξ) dξ

(27)

and thus

ln F ) ΠηP

(28)

which shows the practical usefulness of the promotional effectiveness factor ηP. Metal-Support Interactions. The above analysis (eqs 13-28) remains valid when one considers the geometry shown in Figure 6 to model a (not necessarily cylindrical) metal crystallite on a high-surface-area support. The only physical difference is that, here, the current, I, can not be directly measured and thus the dimensionless current density, J, can not be directly computed. This difficulty can, however, be overcome if the ratio of the reactivities, Λ, of normally adsorbed and backspillover oxygen is known (e.g., from electrochemical promotion experiments, where Λ, as already noted, also expresses the Faradaic efficiency). Thus, in this case,

Ind. Eng. Chem. Res., Vol. 40, No. 20, 2001 4213 Table 2. Typical Operating Parameters in a Supported Catalyst r/Ac ) 10-6 mol/(s cm2) Λ ) 102 k ) 10-2 s-1 Ci,max ) 10-7 mol/cm2 L ) 3 nm computed parameters J ) 10 Φp ) 4.8‚10-3 ηP ) 0.91 condition for Φp < 1 Φp < 1 w L < 0.6 µm

Figure 6. Schematic of cylindrical or, more generally, fixed-crosssection nanoparticles deposited on an O2- conducting support. Table 1. Typical Operating Parameters in Electrochemical Promotion Studiesa I ) 100 µA k ) 10-2 s-1 AcCi,max ) 10-7 mol L ) 3 µm Ds ) 4 × 10-11 cm2/s computed parameters J ) 0.5 Φp ) 4.8 ηP ) 0.15 condition for Φp < 1 Φp < 1 w L < 0.6 µm a

Electrolyte surface area AE ) 1 cm2, T ) 400 °C.

upon combining eqs 1 and 20, one obtains for J

J)

(r/Ac) ΛkCi,max

(29)

where r is the measured promoted rate of the catalytic reaction and Ac is the total catalyst surface area. Numerical Examples Electrochemically Promoted Films. To estimate ηP in actual electrochemical promotion experiments, we use here typical values1-13 of the operating parameters (Table 1) to calculate J and Φp. The value of k is estimated on the basis of typical NEMCA galvanostatic transients,1-13 which show that the lifetime of the promoting Oδ- species on the catalyst surface is typically 102 s at temperatures of 350-400 °C. The surface diffusivity, Ds, is computed (conservatively) from the diffusivity measurements of Lewis and Gomer47 for O on Pt(111) and Pt(110) near 400 °C. They described their data with the equation 2

Ds ) δ ν exp(∆S/R) exp(-E/RT)

(30)

with δ ) 3 Å, ∆S ) 17 cal/(mol K), ν ) 1012 s-1, and E ) 34.1 kcal/mol. At 400 °C, this gives Ds ) 4 × 10-11 cm2/s. Thus, at 400 °C, an Oδ- backspillover ion can move approximately 1 µm per s. The computed J, Φp, and ηP results (Table 1) show the significance of the present analysis. It is very likely that many of the published electrochemical promotion data1-13 have been obtained under promoter-diffusioncontrolled conditions, i.e., the actual measured (quite large1-13) F values might not correspond to full utilization of the promoting species (ηP ) 1), so that one could

Table 3. Summary of Results. Dimensionless Numbers Dictating the Magnitude of Electrochemical Promotion and Metal-Support Interactions Π ) (ReNMPi/okbT) J ) I/(2FkCi,maxLltpd) ) I/(2FkCi,maxAc) ) (r/Ac)/(ΛkCi,max) Φp ) Lxk/Ds ηP ) 1/[1/J + Φp/(tanh Φp)] F ) exp(ΠηP)

obtain even larger F values (exp Π, eq 28) if thinner catalyst films were used. This is corroborated by the fact that the highest F value so far for C2H4 oxidation on Pt supported on YSZ (F ≈ 60) was reported4,48 for a Pt film with a surface area corresponding to 4.2 × 10-9 mol of Pt4 (L < 0.1 µm), whereas significantly smaller F values (∼10-20) were reported48 for the same reaction on Pt films with surface areas corresponding to 10-7 mol of Pt (L ≈ 3 µm).48 Dispersed Supported Catalysts. To estimate ηP in actual fully dispersed Pt, Rh, and Pd catalysts deposited on highly porous Y2O3-doped ZrO2, WO3-doped TiO2, TiO2, and γ-Al2O3 supports38,41 used for CO and light hydrocarbon oxidation, we use typical44,49 operating parameter values (Table 2) similar to those in Table 1 and assume Λ ) 100, which is a rather conservative1-13,44,49 value, to compute J via eq 29 and Φp via eq 19. The results give ηP ≈ 0.91, which implies that the O2- backspillover mechanism is fully operative under oxidation reaction conditions on nanoparticle metal crystallites supported on ionic or mixed ionicelectronic supports, such as YSZ, TiO2, and CeO2. This is quite reasonable in view of the fact that, as already mentioned, an adsorbed O atom can migrate 1 µm/s on Pt at 400 °C. Thus, unless the oxidation reaction turnover frequency is higher than 103 s-1, which is practically never the case, the Oδ- backspillover double layer is present on the supported nanocrystalline catalyst particles. Conclusions A classical reaction engineering approach50 has been used to model electrochemical promotion and metalsupport interactions. The two effects are mechanistically similar13,44,45 and thus can be described by the same type of equations. The analysis shows that the magnitude of the promotional or metal-support interaction effect depends on three dimensionless numbers, Π, J, and Φp (Table 3), which dictate the actual value of the promotional effectiveness factor. Two qualitative conclusions can be drawn from the numerical examples presented here. First, slow surface diffusion of the promoting species might have limited the maximum F value measured in many of the published electrochemical promotion studies.2-12 Second,

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nanocrystalline metal particles supported on ZrO2, TiO2, CeO2, and doped ZrO2 and TiO2 supports13,41-45 are almost certainly covered by a high coverage of promoting Oδ- species originating from the support, so that diffusion of Oδ- is the dominant mechanism inducing metalsupport interactions on these supports. It will be interesting and important to compare the present model with detailed experimental data. To this end, electrochemical promotion experiments with catalyst films of varying thicknesses are currently underway, together with detailed kinetic studies on supported dispersed catalysts with varying catalyst crystallite sizes. Acknowledgment We thank BASF and DuPont for financial support. C.G.V. also thanks Professor Jimmy Wei for his kind scientific encouragement, advice, and collaboration at MIT in the period 1977-1981. This paper is dedicated to Professor Jimmy Wei on the occasion of his 70th birthday. List of Symbols Ac ) catalyst surface area, m2 AE ) electrolyte surface area, m2 Ci ) surface concentration of species i, mol‚m-2 Ci,max ) maximum surface concentration of species i, mol m-2 Ds ) effective surface diffusivity, m2 s-1 e ) unit charge, 1.6 × 10-19 C E ) activation energy, J mol-1 F ) Faraday’s constant, 96487 C/mol I ) current, A J ) dimensionless current, defined in eq 20 k ) rate constant, s-1 kb ) Boltzmann’s constant, 1.38 × 10-23 J/K L ) catalyst film thickness, m ltpb ) length of the three-phase boundary, m NM ) surface atom density, atom m-2 Pi ) dipole moment of species i, C m r ) reaction rate, mol s-1 ro ) unpromoted (open-circuit) reaction rate, mol s-1 R ) gas constant, 8.314 J/(mol K) S ) entropy, J mol-1 K-1 T ) temperature, K UWR ) potential difference between the working (catalyst) and reference electrodes, V z ) distance, m Greek Symbols R ) NEMCA coefficient, defined in eq 3 o ) dielectric constant, 8.85 × 10-12 C2/(J m) ηp ) promotional effectiveness factor, defined in eq 22 θi ) degree of coverage of species i, Ci/Ci,max Λ ) Faradaic efficiency, defined in eq 1 ξ ) dimensionless distance, z/L Π ) dimensionless work function or dipole moment, defined in eq 10 F ) rate enhancement ratio, r/ro Φ ) work function, eV/atom Φp ) Thiele modulus, defined in eq 19

Note Added after ASAP Posting This article was released ASAP on 5/5/01 with an error in eq 21. The correct version was posted on 7/25/01.

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Received for review January 2, 2001 Revised manuscript received February 15, 2001 Accepted February 20, 2001 IE010001F