Particle Proximity Effect in Nanoparticle Electrocatalysis: Surface

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Particle Proximity Effect in Nanoparticle Electrocatalysis: Surface Charging and Electrostatic Interactions Jun Huang, Jianbo Zhang, and Michael H. Eikerling J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b10842 • Publication Date (Web): 08 Feb 2017 Downloaded from http://pubs.acs.org on February 11, 2017

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The Journal of Physical Chemistry

Particle Proximity Effect in Nanoparticle Electrocatalysis: Surface Charging and Electrostatic Interactions Jun Huang1,2, Jianbo Zhang1,3* and Michael H. Eikerling2,* 1

Department of Automotive Engineering, State Key Laboratory of Automotive Safety and Energy, Tsinghua University, Beijing 100084, China 2

Department of Chemistry, Simon Fraser University, Burnaby, BC V5A 1S6, Canada

3 Beijing

Co-innovation Center for Electric Vehicles, Beijing Institute of Technology, Beijing 100081, China

* Corresponding authors: [email protected] (J. Z.) and [email protected] (M.E.)

1

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ABSTRACT Understanding the structure-activity relation of nanostructured electrocatalysts is crucial for advances in emerging electrochemical energy systems, most prominently polymer electrolyte fuel cells and electrolyzers. In this realm, the surface-specific activity of platinum for the oxygen reduction reaction is a peculiar function of particle size and interparticle spacing. Previous attempts to rationalize the particle size and proximity effects focused on geometric and electronic factors and they largely ignored the role of the support. The present study focuses on a consistent treatment of electrostatic reaction conditions around Pt nanoparticles, taking into consideration the specific surface charging properties of Pt and support. The model reveals a double-layer overlap regime in which the surface-specific activity is significantly enhanced by decreasing the interparticle spacing. Relevance of the model in explaining the particle proximity effect is demonstrated.

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INTRODUCTION For polymer electrolyte fuel cells (PEFCs) to make deep incursions into the automotive industry sector, a marked reduction of the platinum loading will be crucial. Efforts in electrocatalysis research focus therefore on the catalytic activity for the oxygen reduction reaction (ORR)1-5 and the effectiveness factor of the catalyst.6-8 Substantial experimental and computational efforts have been

devoted

to this endeavor.1-5

Particularly,

the

structure-activity relation of the catalytic material remains a topic of unfading interest.8-23 The dependence of the ORR activity on the Pt nanoparticle size has been studied extensively.2,3,9 -18 Recent studies reported that the surface specific activity (SSA) of the ORR decreases with the diameter of Pt nanoparticles decreasing from 15 nm to 2 nm, referred to as particle size effect.17,18 Theoretical calculations attribute this trend to the increased binding energy of adsorbed oxygen at Pt, resulting from the greater surface atom fraction of undercoordinated corner and edge sites at smaller Pt nanoparticles.22,23 In supported nanoparticle catalysts, the average interparticle spacing is also related to the catalyst loading. A recent study found that the interparticle spacing decreases from 4 nm to 2 nm to 1.3 nm when the mass loading of Pt was increased from 4.5% to 9.2% to 21%.24 In the literature, the catalyst loading has been varied to examine the dependence of the ORR activity on the interparticle spacing.9,19,20,24 This structural dependence is referred to as the particle proximity effect.3,19,20,24 Investigation of the particle proximity effect requires that the particle size remains constant when varying the catalyst loading. This prerequisite is difficult to fulfill, explaining in part why much fewer reports on the particle proximity effect can be found, compared to the wealth of extensive studies on the particle size effect. In most articles, the particle proximity effect and the particle size effect are interwoven.9,19,20,24 Keeping this observation in mind, it is no surprise that controversial conclusions on the particle proximity effect can be found, as reviewed in Ref.3. Efforts to deconvolute the particle proximity effect and the particle size effect strove to synthesize well-defined, size-controlled Pt nanoclusters on a planar substrate of glassy 3



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carbon19 or they deposited Pt nanoparticles on a high surface area carbon.20 Results suggested that the SSA increases with decreasing interparticle separation.19,20 Independent investigations reproduced this inverse relation between the SSA and the interparticle spacing.24,25 While experimental interest in dense nanoparticle systems has been high, the modeling literature on this topic is sparse. Mean-filed theoretical models were developed to interpret the experimental phenomena.19,20 In them, the potential distribution in the solution phase is calculated by solving the Poisson-Boltzmann (PB) equation. It was conjectured that a more pronounced overlap of electrochemical double layers (EDLs) upon bringing adjacent Pt nanoparticles closer together reduces the potential drop across the Pt-solution interface. This was presumed to reduce the oxide coverage at a given electrode potential, leading to improved ORR activity.19,20 The oxide coverage is co-determined by the potential difference across the interface and the local pH in solution.26,27 The oxide coverage is believed to be invariant with changes in the potential drop across the interface alone, given that the electrode potential and bulk properties of the solution are fixed.27 Notably, a self-consistent treatment of oxidation reactions, surface charging behavior, ion density and potential profiles in solution that are intrinsically interrelated, as shown in Figure 1, is required for model-based understanding. Since surface charging effects play a central role, this treatment should account for the influence of Pt oxidation on the structure and associated properties of the electrified interface that control the surface charge density on Pt, as well as the distinct surface charging relation of the carbon support that exerts a non-trivial effect on the local reaction conditions.5 In Ref.27, we presented a modified structural model of the Pt-solution interface that self-consistently accounts for polarization effects caused by surface-adsorbed oxygen species and interfacial water molecules. Based on this refined EDL model, a non-monotonic charging relation of Pt was derived. Exploiting the refined EDL model and the modified surface charging relation of Pt, the present study aims to quantitatively explain the particle proximity effect. In doing so, the surface charging relation of the carbon support emerges as 4


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an essential influence.

Figure 1. Schematic illustrating interrelations between surface oxidation reactions, surface charging behavior and local reaction conditions.

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THE MODEL Different geometrical configurations considered in this work are illustrated in Figure 2. Figure 2 (a) shows a one-dimensional periodic array of stripes of Pt separated by stripes of carbon, designated as 1D array of strips. This system does not mimic a practical Pt-C catalyst in a PEFC. We present it to convey the basic ideas about the EDL overlap between Pt and carbon. Figure 2 (b) displays a regular hexagonal array of circular Pt discs that are embedded into a carbon surface, designated as 2D array of circular disks. An approximate treatment of this case is obtained by creating the Wigner–Seitz cell of a Pt disk in this array and replacing it by a circular area on the support as indicated by the whitened area. The many particle problem is thereby reduced to the problem of a single effective particle with cylindrical symmetry. Figure 2 (c) considers the same particle arrangements as Figure 2 (b); however, the Pt surface is not coplanar with the support, but it involves 3D particle geometries. This case is

designated as 2D array of spherical particles. An offset distance, > 0 , is introduced between the center of spherical Pt particles and the carbon support plane.; the special case

= 0 corresponds to hemispherical particles. For particles with spherical geometry, the

model equations can be solved only numerically, as explained in Appendix A.4. The set of equations for each of the model systems depicted in Figure 2 consists of three parts: (1) the PB equation for the electrolyte phase surrounding nanoparticles; (2) the surface charging relations of Pt and carbon as the boundary conditions of the PB equation; and (3) the Butler-Volmer (BV) equation. Details of the mathematical treatment are provided in Appendix A.1 to A.3. Here we only recapitulate the basic ideas. The PB equation quantifies the electrostatic interactions between the charged electrode surface and ions in the solution, and determines the ion density and potential profiles in the solution phase as a function of the free surface charge densities on Pt and carbon. The surface charge densities on Pt and carbon are self-consistently solved for by imposing the condition that the potential must be continuous from the interior of the Pt/carbon phase to the interior of the bulk solution phase. Then, the ORR current on the Pt surface as a function of interfacial conditions can be

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calculated using the BV equation.

Figure 2. Schematic of the model systems: (a) alternating Pt and carbon stripes; (b) hexagonal array of circular Pt disks immersed in the co-planar carbon substrate; (c) hexagonal array of Pt nanoparticles with an offset distance H.

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RESULTS Surface charging relations The surface charging relations of Pt and carbon, employed in this model, are shown in Figure 3. The EDL model presented in Ref. 27 revealed the non-monotonic charging relation of Pt,

caused by interfacial polarization effects. Upon increasing the electrode potential, , the

free surface charge density of Pt, , exhibits a transition from negative to positive surface

charge density in the range of 0.1~0.6 , corresponding to the normal charging behavior of > 0.6 , decreases with potential, exhibiting a negative metals. At higher potentials,

differential capacitance, and it becomes negative again for > 0.8 . Polarization effects

caused by surface oxide formation and orientational ordering of interfacial water molecules are responsible for this non-trivial charging behavior. The non-monotonic charging relation of Pt is consistent with experimental findings of Frumkin and Petrii28 and Garcia-Araez et al.29 For the surface of the carbon support, we postulate a linear relation between the free surface charge density, , and . Bayram and Ayranci reported the changes in

of an

activated carbon cloth upon polarization.30 The activated carbon cloth was polarized in Na2SO4 or KH2PO4/KOH buffer solutions at potentials from −1.5 to 5.0 V (RHE), resulting in

an increase of

from 0.164 to 0.355 V (RHE). In this regard, we could assume that

is constant in the case of PEFCs where the electrode potential is varied in a narrow window between 0.6 and 0.9 V. This linear relation is parameterized with the potential of zero charge of carbon, . A range of values for

of different carbon materials, such as glassy

carbon and carbon aerogel, have been reported; they are typically found in the range of

0.1-0.4 V (RHE).30,31,3233 We use 0.2 V as a basic value of , implying that is positive in

the potential range of 0.6-0.9 V that is typical for the ORR .

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Figure 3. Surface charging relations of Pt and carbon calculated using parameters in Table 1.

Protonic depletion effect of carbon In what follows, we examine the spatial distributions of the solution phase potential,

, , and the proton concentration, , , for the 1D array of stripes case in Figure 2 (a). Figure 4 shows results for the periodic array of stripes at varying interparticle spacing,

= 2" # " obtained by changing the half width of the carbon stripe, ". The half

width of the Pt stripe is held constant, " = 1 nm. In Figure 4, is fixed at 0.9 V RHE,

corresponding to the potential, at which the SSA of Pt-C catalysts is usually measured. As can be seen in the upper panel of Figure 4, , near the Pt surface is smaller than

that near the carbon support, and overall , is shifted towards more positive values upon increasing

from 0.5 nm to 4 nm. This potential shift clearly manifests the

nonlinear interplay of charging effects at Pt and carbon. Based on the Boltzmann relation, = exp .#

/

01

2 # 34 , the increased , at a larger

decrease of near Pt sites.

leads to a

According to the charging relations in Figure 3, the Pt surface is negatively charged at

= 0.9 V RHE and it attracts protons. On the contrary, at this potential, the carbon

surface is positively charged and it repels protons. Therefore, , is enhanced in the vicinity of Pt and depleted near carbon, as shown in the lower panel of Figure 4. The

distribution of , at the Pt surface is highly nonuniform, with the fringes being 9



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progressively depleted upon expansion of the C surface domain. It is evident that the average value of , and thus the SSA for the ORR is bound to decrease as

grows from

0.5 nm to 4 nm. Note that the distribution of anion concentration possesses an opposite dependence on

compared with (see Figure S1 in supporting information). Since

σ 6 7 between 0 V and 1.0 V, the above analysis shall be applicable at other potentials in

this potential range (see results at 0.6 V in Figure S2).

Figure 4. Distributions of , (upper panel) and , (lower panel) for the model of linear stripes (Figure 2 (a)), with varying stripe separation, , as indicated. The width of the Pt band is fixed at " = 1 nm and the metal potential is held at = 0.9 V RHE. Other parameters are listed in Table 1.

Particle proximity effect and its parametric analysis The SSA of the Pt-C catalysts, defined as the value of the ORR current, 89:: , at =

0.9 V RHE, is shown in Figure 5 as a function of

cf. Figure 2(a). 89:: as a function of

Figure 5(a), the values of >?@ and

for the model of 1D array of stripes,

at 0.6, 0.7 and 0.8 V is shown in Figure S3. In

are varied to examine the effect of the surface

charging relation of carbon on the SSA. Figure 5(b) reveals the influence of the Pt band width, " , referred to as the particle size effect.

All curves in Figure 5 exhibit the same trend of the SSA increasing with decreasing

.

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Note that " is held constant at 1 nm for all cases. As discussed in the preceding section, this particle proximity effect is attributed to the different surface charging properties of Pt and carbon that induce the dependence of the distributions of and on

.

At

0.9 V (vs. RHE), the protophobic carbon, with > 0, depletes , near the

protophilic Pt surface, which has 6 0. At small

,

this protonic depletion effect of

carbon is concentrated on the Pt/C interface, while, it is intensified and extended to a larger area of the Pt stripe with increasing

,

causing the decrease in the SSA.

A larger value of >?@ increases the magnitude of at 0.9 V, enhances the protonic depletion effect, and results in a further decrease of the SSA, as shown in Figure 5(a). On the contrary, increasing

decreases the magnitude of , suppresses the depletion effect,

and improves the SSA. As far as the electrostatic reaction conditions are concerned, a catalyst support with more negative surface charge benefits the ORR performance of supported Pt nanoparticle catalyst systems. Figure 5 (b) examines the influence of Pt size on the particle proximity effect by increasing

" from 1.0 to 1.5 and 2.0 nm. Theoretical studies have reported that the oxygen binding energy increases when the Pt size decreases.22,23 Consequently, the exchange current density,

A 89:: in Eq. (A-22), should be lower for smaller Pt nanoparticles. However, in Figure 5 (b),

A 89:: is assumed constant. Given a certain

,

it is found that the SSA increases with

increasing " , because it diminishes the proton depletion effect exerted by carbon. An

important message is that the particle size effect is essentially interwoven with the particle proximity effect.

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Figure 5. (a) Effect of the surface charging relation of carbon on the particle proximity effect (the unit of >?@ and are Fm C and V, respectively); (b) effect of the Pt size on the particle proximity effect. Other parameters have their basic values as listed in Table 1. Figure 6 (a) compares SSA vs.

curves for the distinct model geometries in Figure 2,

using identical parameters. The curves display the same overall trend that the SSA decreases with increasing

,

while they reveal peculiar differences in the structural sensitivity.

The 2D array of spherical particles (H = 0 nm) case and the 1D array of stripes case show a sharp increase of the SSA with decreasing

in the range below 2 nm. The curve of the

2D array of spherical particles case is almost flat for

> 2 nm, while the other two

cases keep dropping. Figure 6 (b) replots Figure 6 (a) in a logarithmic scale of

cases, an approximately linear relation between the SSA and log GH

GH

.

For all

is revealed for

6 2 nm. However, the slope of this linear relation depends on the model geometry.

The slope of the 1D array of stripes and 2D array of spherical cases are almost identical and larger than that of the 2D array of circular disks. Figure 6 (c) depicts variations in the distribution of in the vicinity of the Pt-solution

interface with varying

for the 2D array of spherical particle case. When

62

impact of the double layer overlap between neighboring particles. However, for

>2

nm, is enhanced in the equatorial region between the particles, due to the pronounced nm the impact of the protophobic carbon surface prevails, shifting the region of high

towards the polar caps of particles. The distribution of remains almost invariant with a 12


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further increase of

above 2 nm. The 2D array of circular disks case possesses a

relatively weak sensitivity compared to the 1D array of stripes and 2D array of spherical particles cases. Another conclusion draw from Figure 6 is that increasing H (relative to H = 0) only slightly improves the SSA. The positive effect of H on the SSA results from the protection of the upper hemisphere from the depletion effect exerted by the carbon support.

Figure 6. (a) Comparison of SSA vs. curves for cases of 1D array of stripes, 2D array of circular disks, and 2D array of spherical nanoparticles (H = 0 or 0.5 nm) using the parameters listed in Table 1; (b) semilog plot of (a); (c) distribution of in the vicinity of the Pt-solution interface for the hemispherical nanoparticle case with varying . The Pt radius is 1 nm in all cases.

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DISCUSSION Double-layer overlap regime Regarding the proximity effect, we can distinguish two regimes: a double-layer overlap regime, in which the SSA is significantly affected by

,

at small particle separation, and a

regime of independent nanoparticles, in which the SSA is insensitive to variations in

and the protophobic carbon effect dominates, at large

.

The characteristic value of

for the transition between these two regimes is referred to as the double-layer

overlap distance, denoted as KL9 . For the case in Figure 6, we can conclude that KL9 ≈ 2 nm.

Using the 2D array of spherical particles model, we examine in Figure 7 how KL9 depends

on the value of

that controls the protophobicity, the electrolyte concentration that

determines the Debye length, and the size of Pt particles. By increasing N from 1 nm to 2

nm (the red curve) or increasing

from 0.2 V to 0.4 V (the magenta curve), the SSA vs

curve is uplifted, while the value of KL9 remains unchanged. On the contrary, as

shown in the blue curve, KL9 decreases from 2 nm to ~1 nm when O is changed from 0.1 M to 1 M by adding supporting electrolyte. The Debye length, PK = Q

RS 01

C/ T UVWV

, is the

characteristic length of the diffuse layer of the metal-solution interface, describing the

effective screening effects of the electrified interface; it is closely related to KL9 . In this

regard, O is the key parameter to modulate KL9 . As PK ~HXH increasing HXH .

A.Y

, KL9 decreases with

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Figure 7. Effect of Pt particle size, electrolyte concentration and pzc of carbon on the SSA vs. curves for the case of 2D array of spherical nanoparticles (H = 0 nm) using the parameters listed in Table 1.

Comparison with experiments In Figure 8, we compare model results with experimental data, using the model parameters listed in Table 2. The red pentagrams represent the experimental data of the Pt nanoclusters

(N = 0.3 nm) deposited on a planar glassy carbon surface in 0.1 M HClO4 at 0.85 V vs RHE.19 This experimental system can be well described by the 2D array of circular disks, viz. a hexagonal array of circular Pt disks immersed in the co-planar carbon substrate. Model A results in the solid black curve are calculated using 89:: = 0.3 [ 10 \ A m C (smaller than

the basic value of 1.5 [ 10 \ A m C as a result of the smaller Pt particle size according to the particle size effect) and other parameters listed in Table 1. The blue circles and black squares correspond to Pt nanoparticles with a diameter of 2 nm that were deposited on Vulcan XC72R (235 m2g-1), and Ketjenblack EC-300J (BET surface area: 795 m2g-1), respectively.20 These two sets of data were measured in oxygen saturated 0.1 M HClO4 at 0.9 V and at 20°C. In this case, the 2D array of spherical particles case is chosen for model calculations because it best captures essential aspects of the experimental particle system, viz. particle size, circular shape, and coordination geometry. Model results in

A A the solid blue and black curves are calculated using 89:: = 1.0 [ 10 \ A m C and 89:: =

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1.5 [ 10 \A m C, respectively. It is found that for Pt/Vulcan XC72R data fall into double-layer overlap

regime,

while

for

Pt/Ketjenblack

EC-300J

data

correspond

to

the

independent-nanoparticle regime. The magenta diamonds show the SSA of dispersed Pt crystallites on the support of 584 m2g-1 carbon that was measured in 1.5 M H2SO4 at 60°C. 9

A = 0.1 [ 10 \ A m C . Model results are calculated using 89::

Accurate determination of

is crucial to the interpretation of the proximity effect. In

the literature, there are two methods to estimate

.

One method is to measure the BET

surface area of the carbon support, ^, and then to obtain

according to

=

_^/a # 2NGH , where N is the number of nanoparticles.192024 The other method is to determine the value from the visual analysis of TEM images.24 Proch et al. recently compared

these two approaches, revealing significant experimental uncertainties in

.

24

For

example, the values obtained via TEM image analysis (open blue triangles in Figure 8) are much smaller than those calculated by virtue of the BET surface area of carbon (solid blue triangles in Figure 8).24 This situation calls for more reliable experimental methods and analysis tools in the determination of

.

Figure 8. Comparison of the SSA versus

and experimental data (markers).

curves between model results (solid curves)

A framework bridging nano and macro scales Before the close of this section, we discuss the implications of this study for fuel cell

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electrocatalysis. DFT studies explore the impact of electronic effects and specific surface geometries of the catalyst on adsorption properties of reaction intermediates, reaction pathways,

and

intrinsic

activity,

making

vital

contributions

to

understanding

structure-activity relations in electrocatalysis.2,3,22,23 However, it remains a formidable challenge for DFT studies to self-consistently describe the structure of the electrified interface that extends from the bulk metal to the bulk solution and involves polarizable layers of adsorbed surface oxygen species and interfacial water molecules.5 Moreover, consistently treating the surfacial pH and electrostatic reaction conditions remains problematic for DFT studies.5 The presented work focuses on important mesoscale effects. It provides a framework to interlink results of DFT studies on adsorbate formation, surface polarization effects, reaction pathways and intrinsic activity, with important support and particle proximity effects that are specific for catalytic systems involving supported nanoparticles. Understanding these mesoscale effects provides crucial input for porous electrode models that attempt to rationalize the electrochemical performance, effectiveness factor of catalyst utilization, impedance response and durability of the electrode as a function of its multiphase composition, pore space morphology and the external reaction conditions provided. One must be fully aware that the mean-field approach in this study fails when the characteristic scale is smaller than several ion diameters, say 1 nm. Under such conditions, an accurate treatment of the particle proximity effect calls for explicit molecular simulations, which is beyond the scope of this study.

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CONCLUSION This modeling study explored the proximity effect in nanoparticle electrocatalysis. The model-based analyses focused on rationalizing the local electrostatic reaction conditions around Pt nanoparticles. In this context, it brought to light the important role of the carbon support that possesses distinct surface charging properties in comparison to the Pt catalyst. In the potential regime that matters for the oxygen reduction reaction, the carbon support is positively charged and it thus repels protons; the surface of Pt, on the other hand, exhibits a negative surface charge due to surface polarization effects and it attracts protons. Two competing trends are thus at play in determining the proton density in the interparticle region: proton depletion caused by the proximity to the Carbon surface and proton accumulation caused by the double layer overlap at close proximity of neighboring Pt nanoparticles. A high density of Pt nanoparticles will be beneficial in this regard, utilizing the particle proximity effect; even though the greater proximity clashes with another foremost issue encountered in fuel cell electrodes: catalyst degradation due to particle coagulation. One important message the present study conveys is that the electrostatic effect at the scale of the Debye length (1~100 nm) plays an important role in the understanding of the structure-activity relation of nanoparticle electrocatalysis, in addition to the geometric and electronic factors at the atomic scale (0.1~1 nm). The theoretical model presented in this study should be an important module in a hierarchical modeling framework to rationalize electrocatalytic phenomena in complex electrodes for polymer electrolyte fuel cells and electrolyzers, complementing studies of reaction mechanisms and pathways using DFT calculations at the lower end and porous electrode theory that rationalizes the interplay of transport and reaction at the upper end of the spectrum of scales. Understanding how modifications in nanoscale materials structure and properties affect local reaction conditions in porous electrodes and, thereby, determine their catalytic function is crucial for electrocatalyst design.

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ACKNOWLEDGEMENT J Huang and J Zhang gratefully acknowledge financial support from National Natural Science Foundation of China under the grant number of U1664259, State Key Laboratory of Automotive Safety and Energy under the grant number of KF16042, and SAIC motor under the project number 20152001569. M. Eikerling gratefully acknowledges financial assistance by an Automotive Partnership Canada grant, file number APCPJ417858, that supports the Catalysis Research for Polymer Electrolyte Fuel Cells (CaRPE-FC) network.

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APPENDIX A.1 The general model The Poisson equation is written as, ∇ ∙ d ∇ = #e f gh h , i

(A-1)

where d is the dielectric constant of the solution phase, the solution phase potential,

gh , h , the charge number and concentration of species i, and F the Faraday constant, respectively.

We consider a monovalent electrolyte solution with the following ionic species: C k , A , H k ,

m , , 9m , respectively. OH , of which the bulk concentrations are O # , O # 9

m = no / with the dissociation O is the total concentration of all cations or anions. 9

constant of water, no . The possible anion contamination effect is neglected, which is reasonable for F

and ClOp .

Ion concentrations are related to the solution phase potential by the Boltzmann equation, h = h exp q#

gi e 2 # 3s, Nr

(A-2)

where the bulk ion concentration and bulk potential are given by h and , respectively,

N is the gas constant, r is the temperature. Combining Eqs. (A-1) and (A-2) leads to, ∇ ∙ d ∇ = #e O qexp q#

e e 2 # 3s # exp q 2 # 3ss, Nr Nr

(A-3)

which can be simplified to, ∇ ∙ d ∇ =

2e C O

2 # 3, Nr

(A-4)

if the Debye-Hückel approximation holds.34 This approximation is valid for ≪ 0.12 Cm-2 for a 1M solution.27 A.2 1D array of stripes For the linear stripe case, the dimensionless form of Eq. (A-4) is written as, 20


Page 21 of 32

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u C v u C v + = v , uw C uw C

(A-5)

with dimensionless variables defined as, w =

e , w = , v = 2 # 3, PK PK Nr

(A-6)

where the Debye length, PK , is,

z Nr PK = y C . 2e O

(A-7)

The boundary conditions to close Eq. (A-5) are, (1) symmetry conditions at w = 0, and w = " /PK, uv = 0. uw

(A-8)

v = 0.

(A-9)

uv PK e "

=# q ∗ Θw + # ∗ Θ .w # 4s, uw z Nr

PK

(A-10)

(2) reference potential in the bulk solution phase, w = ∞, (3) free surface charge density at the solid/solution interface, w = 0,

where and represent the free charge density on the Pt and carbon surface,

respectively, Θw is the Heaviside function, " is the width of the Pt stripe in the basic

element of length ", as shown in Figure 2 (a).

Using separation of variables, we solve Eq. (A-5) with boundary conditions Eqs. (A-8) to (A-10) as follows, v w, w = # exp#w ~A # f exp #_1 + C w ~ cos w, ∈

with coefficients, ~A = #

# " # " 2ePK q + s, z Nr "

(A-11)

(A-12)

21



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~ =

Page 22 of 32

2e PCK # "

sin . 4, z Nr " _1 + C PK =

(A-13)

PK . "

(A-14)

As a result, the interfacial potential on the solution side is given by,

=

2PK "

. # # 4 # f z " ∈

#

PK y1 + "

"

sin . 4 cos . 4. " " C

(A-15)

From Eq. (A-15), we can calculate the average over Pt in the Pt-C unit, ,, and

that over the carbon support, ,, which are used in the subsequent expressions for

and , respectively.

The expression for the surface charging relation of Pt electrodes is derived from the refined

structural model of the Pt-solution interface as detailed in Ref. 27. is calculated from the potential continuity condition as follows,

a # # , # z 9 + z O o ∙ tanh 9 = , 9 9 + + z z z 9

9

(A-16)

where is the potential in the bulk Pt, a constant potential drop at the metal

surface due to the electron spillover, z 9 and 9 the dielectric constant and thickness of

the Pt oxide layer, z and the dielectric constant and thickness of the Inner Helmholtz plane (IHP), z9 and 9 the dielectric constant and thickness of the Outer

Helmholtz plane (OHP), a O the Pt atom density, and o the water dipole moment. The variable X, which represents the dimensionless total field-dependent adsorption energy, is given by, q

C 0.6 a o a o

+ a s tanh # = ,

O

C z Nr z Nr

(A-17)

in the two-state water model employed in Ref.27, with Avogadro's number NA. The surfacial oxide dipole density is given by 9 = a O ∙ 9 ∙ ¡ ∙ 9 ,

(A-18)

22


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with the average charge number of ions in the oxide layer, , the elementary charge, ¡, and the oxide coverage, 9 , obtained as the solution of

A e2 # # ¢9 3 9 £9 9 + 2.3 ∙ pH = ln . 4+ , Nr 1 # 9 Nr

(A-19)

the surfacial pH, pH

Using the Boltzmann relation, Eq. (A-19) can be rewritten in terms of bulk properties, as follows, A e2 # ¢9 9 £9 9 3 + 2.3pH = ln . , 4+ Nr 1 # 9 Nr

(A-20)

A with the equilibrium potential of the Pt oxidation reaction at standard state, ¢9 , the bulk

pH, pH , and the Frumkin-type correction term, ξOXθOX, that accounts for deviations from a Langmuir-type adsorption behavior.

A linear relation parameterized by the potential of zero charge is employed to describe the surface charging behavior of the carbon support, = >?@ ∙ 2 # , # 3,

(A-21)

where >?@ is the double layer capacitance of the carbon-solution interface and

the

pzc of carbon. Eqs. (A-15), (A-16) and (A-21) constitute a closed set of equations to determine ,

and . After solving for these variables, the ORR current density, 89:: 2, 3, can be

calculated using the Butler-Volmer equation, 89:: 2x, 3

=

∙ exp q#

A 89::

∙q

9 T 9A T

s

¤¥T

¤¦

∙q A s

§ e A 2 # # ¢9:: 3s , Nr

(A-22)

where 9 is the surfacial oxygen concentration at the reaction plane, the T

surfacial proton concentration, obtained from using the Boltzmann relation

A in Eq. (11), ¨9T , ¨ are the reaction orders of oxygen and proton, 9A T and are

the concentrations of oxygen and proton under reference conditions that correspond 23



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Page 24 of 32

A A to 89:: , ¢9:: the equilibrium potential of the ORR, and αc the effective transfer

coefficient. The surface specific current density of the Pt-C unit in Figure 2(a) is defined as, 89::

ª«

1 = ∙ © 89:: 2, 3 ∙ , "

A

(A-23)

and the SSA is the value of 89:: at = 0.9 V RHE.

Model parameters used in the simulation are listed in Table 1 with references and notes. The parameters of the Pt-solution interface are taken from our previous study.27 The ORR properties are adopted from widely-cited studies.26,35 The structural and operating parameters match the experimental system and conditions in Ref. 20. A.3 2D array of circular disks For the hexagonal array of circular Pt disks, the PB equations are written as, 1 u uv u C v = v , q¬̃ s+ ¬̃ u¬̃ u¬̃ ug̃ C

(A-24)

with the dimensionless variables defined as, ¬̃ =

¬ g e , g̃ = , v = 2 # 3. PK PK Nr

(A-25)

In addition to the symmetrical conditions at ¬̃ = 0, and ¬̃ = N /PK , the key boundary

condition at g̃ = 0 is,

uv ¬̃ , 0 PK e N

=# ® ∗ Θ¬̃ + # ∗ Θ .¬̃ # 4¯. ug̃ z Nr

N

(A-26)

The solution to Eq. (A-24) is obtained using the variable separation method, ¬, g = °A ∗ exp .#

¶

g ¬ g 4 + f ° ∗ ±A .² 4 ∗ exp .#ρ´ 4, PK N Pµ ·¸

(A-27)

with the coefficients expressed as,

24


Page 25 of 32

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°A =

PK N

® + # ∗ .1 # 4¯, z

N

² N

2PK # N ±¸ N ° = , z N ² ±A ² C _C # 1

N = ² , PK

±¸ ² = 0.

(A-28)

(A-29)

(A-30) (A-31)

where ± is the modified Bessel function of the first kind of nth order, and ² is the nth root of ±¸ = 0.

Based on Eq. (A-27), the surfacial potential in the solution phase, ¬, and its average value over Pt, ,, and that over the carbon surface, ,, can be calculated. Then, ,

, and 89:: 2¬, 3 can be solved.

In this scenario, Eq. (A-23) is rewritten as, 0«

1 89:: 2 3= C ∙ © 89:: 2¬, 3 ∙ 2¬ ∙ ¬. N

A

(A-23)

A.4 Numerical simulation for the 3D array of sphericalparticles

The Poisson-Nernst-Planck equations are solved in COMSOL, coupled with the metal surface charging model solved in MATLAB. In the COMSOL model, a 2D axisymmetric geometry according to Figure 2 (c) is built. The ‘electrostatic’ module is used to describe the Poisson equation, and the ‘transport of dilute species’ module is employed to formulate the Nernst-Plank equations. The space charge density in the

‘electrostatic’ module is # A# , and the surface charge density is given by a Matlab function to solve Eqs. (A-16), (A-17), (A-19), (A-21), and (A-22). Eq. (A-16) is simplified below to reduce the complexity. The inputs required from the COMSOL model to feed the MATLAB function are and .

A.5 Simplification of Eq. (A-16) in numerical simulations. Based on the magnitude analysis of Eq.(A-17), we have, 25



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q

C 0.6 a o a o

+ a tanh ≈ , s

O

C z Nr z Nr

Page 26 of 32

(A-24)

Then, Eq.(A-16) can be rearranged as, =

# # , # z 9

9 9 z 9 + z9 + z

9

1 C a O

1+ 0.6

.

(A-25)

26


Page 27 of 32

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REFERENCES 1.

Debe, M.K. Electrocatalyst approaches and challenges for automotive fuel cells. Nature 2012, 486, 43-51.

2.

Stephens, I.E.L.; Bondarenko, A.S.; Grønbjerg, U.; Rossmeisl, J.; Chorkendorff, I. Understanding the electrocatalysis of oxygen reduction on platinum and its alloys. Energy Environ. Sci. 2012, 5, 6744-6762.

3.

Antolini, E. Structural parameters of supported fuel cell catalysts: The effect of particle size, inter-particle distance and metal loading on catalytic activity and fuel cell performance. Appl. Catal. B Environ. 2016, 181, 298-313.

4.

Kongkanand, A.; Mathias, M.F. The priority and challenge of high-power performance of low-platinum proton-exchange membrane fuel cells. J. Phys. Chem. Lett. 2016, 7, 1127-1137.

5.

Eslamibidgoli, M.; Huang, J.; Kadyk, T.; Malek, A.; Eikerling, M. How theory and simulation can drive fuel cell electrocatalysis. Nano Energy 2016, doi:10.1016/j.nanoen.2016.06.004.

6.

Xia, Z.; Wang, Q.; Eikerling, M.; Liu, Z. Effectiveness factor of Pt utilization in cathode catalyst layer of polymer electrolyte fuel cells. Can. J. Chem. 2008, 86, 657-667.

7.

Wang, Q.; Eikerling, M.; Song, D.; Liu, Z. Structure and performance of different types of agglomerates in cathode catalyst layers of PEM fuel cells. J. Electroanal. Chem. 2004, 573, 61-69.

8.

Huang, J.; Zhang, J.; Eikerling, M. Theory of electrostatic phenomena in water-filled Pt nanopores. Faraday Discuss. 2016, 193, 427-446.

9.

Watanabe, M.; Saegusa, S.; Stonehart, P. Electro-catalytic activity on supported platinum crystallites for oxygen reduction in sulphuric acid. Chem. Lett. 1988, 17, 1487-1490.

10. Kinoshita, K. Particle size effects for oxygen reduction on highly dispersed platinum in acid electrolytes. J. Electrochem. Soc. 1990, 137, 845-848. 11. Takasu, Y.; Ohashi, N.; Zhang, X.G.; Murakami, Y.; Minagawa, H.; Sato, S.; Yahikozawa, K. Size effects of platinum particles on the electroreduction of oxygen. Electrochim. Acta 1996, 41, 2595-2600. 12. Gamez, A.; Richard, D.; Gallezot, P.; Gloaguen, F.; Faure, R.; Durand, R. Oxygen reduction on well-defined platinum nanoparticles inside recast ionomer. Electrochim. Acta 1996, 41, 307-314. 13. Mukerjee, S.; McBreen, J. Effect of particle size on the electrocatalysis by carbon-supported Pt electrocatalysts: an in situ XAS investigation. J. Electroanal. Chem. 1998, 448, 163-171. 14. Mayrhofer, K.J.J.; Blizanac, B.B.; Arenz, M.; Stamenkovic, V.R.; Ross, P.N.; Markovic, N.M. 27



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The impact of geometric and surface electronic properties of Pt-catalysts on the particle size effect in electrocatalysis. J. Phys. Chem. B 2005, 109, 14433-14440. 15. Mayrhofer, K.J.J.; Strmcnik, D.; Blizanac, B.B.; Stamenkovic, V.; Arenz, M.; Markovic, N.M. Measurement of oxygen reduction activities via the rotating disc electrode method: From Pt model surfaces to carbon-supported high surface area catalysts. Electrochim. Acta 2008, 53, 3181-3188. 16. Nesselberger, M.; Ashton, S.; Meier, J.C.; Katsounaros, I.; Mayrhofer, K.J.J.; Arenz, M. The particle size effect on the oxygen reduction reaction activity of Pt catalysts: influence of electrolyte and relation to single crystal models. J. Am. Chem. Soc. 2011, 133, 17428-17433. 17. Shao, M.H.; Peles, A.; Shoemaker, K. Electrocatalysis on platinum nanoparticles: particle size effect on oxygen reduction reaction activity. Nano lett. 2011, 11, 3714-3719. 18. Perez-Alonso, F.J.; McCarthy, D.N.; Nierhoff, A.; Hernandez-Fernandez, P.; Strebel, C.; Stephens, I.E.L.; Nielsen, J.H.; Chorkendorff, I. The effect of size on the oxygen electroreduction activity of mass-selected platinum nanoparticles. Angew. Chem., Int. Ed. 2012, 51, 4641-4643. 19. Nesselberger, M.; Roefzaad, M.; Hamou, R.F.; Biedermann, P.U.; Schweinberger, F.F.; Kunz, S.; Schloegl, K.; Wiberg, G.K.; Ashton, S.; Heiz, U.; Mayrhofer, K.J. The effect of particle proximity on the oxygen reduction rate of size-selected platinum clusters. Nat. Mater. 2013, 12, 919-924. 20. Speder, J.; Altmann, L.; Bäumer, M.; Kirkensgaard, J.J.K.; Mortensen, K.; Arenz, M. The particle proximity effect: from model to high surface area fuel cell catalysts. RSC Adv. 2014, 4, 14971-14978. 21. Park, Y.C.; Tokiwa, H.; Kakinuma, K.; Watanabe, M.; Uchida, M. Effects of carbon supports on Pt distribution, ionomer coverage and cathode performance for polymer electrolyte fuel cells. J. Power Sources 2016, 315, 179-191. 22. Tritsaris, G.A.; Greeley, J.; Rossmeisl, J.; Nørskov, J.K. Atomic-scale modeling of particle size effects for the oxygen reduction reaction on Pt. Catal. Lett. 2011, 141, 909-913. 23. Tripković, V.; Cerri, I.; Bligaard, T.; Rossmeisl, J. The influence of particle shape and size on the activity of platinum nanoparticles for oxygen reduction reaction: a density functional theory study. Catal. Lett. 2014, 144, 380-388. 24. Proch, S.; Kodama, K.; Inaba, M.; Oishi, K.; Takahashi, N.; Morimoto, Y. The “particle proximity effect” in three dimensions: a case study on Vulcan XC 72R. Electrocatalysis, 2016, 7, 249-261. 25. Taylor, S.; Fabbri, E.; Levecque, P.; Schmidt, T.J.; Conrad, O. The effect of platinum loading and surface morphology on oxygen reduction activity. Electrocatalysis, 2016, 7, 287– 296. 26. Sepa, D.B.; Vojnovic, M.V.; Damjanovic, A. Reaction intermediates as a controlling factor 28


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in the kinetics and mechanism of oxygen reduction at platinum electrodes. Electrochim. Acta 1981, 26, 781-793. 27. Huang, J.; Malek, A.; Zhang, J.; Eikerling, M. Non-monotonic surface charging behavior of Platinum: a paradigm change. J. Phys. Chem. C 2016, 120, 13587–13595. 28. Frumkin, A.N.; Petrii, O.A. Potentials of zero total and zero free charge of platinum group metals. Electrochim. Acta. 1975, 20, 347-359. 29. Garcia-Araez, N.; Climent, V.; Feliu, J. Potential-dependent water orientation on Pt (111), Pt (100), and Pt (110), as inferred from laser-pulsed experiments. Electrostatic and chemical effects. J. Phys. Chem. C 2009, 113, 9290-9304. 30. Bayram, E.; Ayranci, E. A systematic study on the changes in properties of an activated carbon cloth upon polarization. Electrochim. Acta 2011, 56, 2184-2189. 31. Gao, X.; Omosebi, A.; Landon, J.; Liu, K. Surface charge enhanced carbon electrodes for stable and efficient capacitive deionization using inverted adsorption–desorption behavior. Energy Environ. Sci. 2015, 8, 897-909. 32. Zebardast, H.R.; Rogak, S.; Asselin, E. Potential of zero charge of glassy carbon at elevated temperatures. J. Electroanal. Chem. 2014, 724, 36-42. 33. Shao, L.H.; Biener, J.; Kramer, D.; Viswanath, R.N.; Baumann, T.F.; Hamza, A.V.; Weissmüller, J. Electrocapillary maximum and potential of zero charge of carbon aerogel. Phys. Chem. Chem. Phys. 2010, 12, 7580-7587. 34. Debye, P.; Hückel, E. Z. Phys. 1923, 9, 185-206. 35. Parthasarathy, A.; Srinivasan, S.; Appleby, A.J.; Martin, C.R. Pressure dependence of the oxygen reduction reaction at the platinum microelectrode/Nafion interface: electrode kinetics and mass transport. J. Electrochem. Soc. 1992, 139, 2856-2862.

29



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Page 30 of 32

Table 1. A list of model parameters Category

General

Item

Value

Gas constant, R

8.314 J K-1 mol-1

Constant

Faraday constant, F

96485 C mol-1

Constant

Elementary charge, e

1.60[10-19

C

Constant

Water dipole moment, μo

6.02[1023

mol-1

Constant

Vacuum permittivity, εA

1.5[1019

Avogadro’s number, N

3.1 D

Dissociation constant of water, K o

1 [ 10

Pt atom density, N O

Structure parameters Oxide formation

Pt-solution interface

Equilibrium potential of Pt oxidation,

0.716 V

A E9

lateral interaction parameter of oxide adsorption, ξ9

Operating conditions

64.2(1-θOX) kJ mol-1

6.0 ε0 30.0 ε0

Permittivity of bulk water, ε

78.5 ε0

Thickness of the oxide layer, δ 9

0.2 nm

Charge number of oxide dipole, ς

Ref. 27

0.8

Thickness of the IHP, δ

0.275 nm

Potential drop at Pt surface, Δϕ

0.515 nm

Oxygen reaction order, γ9T

1.0

Exchange current density,

1.5[10-3 A mPt-2

0.3 V

A ¢9::

Proton reaction order, γ

1.203 V

Potential of zero charge,

Double

layer

Ref. 26

1.0

A 89::

capacitance

carbon-solution interface, >?@

1.5 A 9C

0.985 mol

A k

of

Surfacial oxygen concentration, 9 T

m-3

Adjustable Ref. 35

1250 mol m-3 0.2 V (RHE) the

0.1 F m-2

Ref. 30,31,32,33

1.3 mol m-3

Bulk solution concentration, O

100 mol m-3

Temperature, T

293.15 K

Bulk pH, pH

Ref. 27

30.0 ε0

Permittivity of the OHP, ε9

Reference proton concentration,

support

Constant

Variable

Reference oxygen concentration,

properties

C 4/√3¿

with a = 3.92Å

According to Pt size in Ref. 20

Transfer coefficient, α

Carbon

Constant

0.2-20 nm

Equilibrium potential,

parameters

m-2

1 nm

Thickness of the OHP, δ9

ORR

mol2 m-6

Length/radius of Pt in the unit, L , R

Interparticle spacing,

Permittivity of the IHP, ε

of the

Typical value

¼

8.85[10-11 F m-1

Permittivity of the oxide layer, ε 9

Parameters

Note

1

Experimental conditions 20

30


Page 31 of 32

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Table 2. Models and parameters used to generate Figure 8 Experimental data Pt

nanoclusters

planar glass carbon19

Test conditions on

0.1 M HClO4 at 0.85 V vs RHE

and

at

room

temperature

A 89:: / A m C

Model 2D

array

of

circular

disks

Pt/Vulcan XC72R20 Pt/Ketjenblack

0.1 M HClO4 at 0.9 V and

EC-300J20 Pt/Carbon9

1.5 M H2SO4 at 0.85 V

2D array of spherical particles

and at 60°C

0.3 [ 10

\

1.0 [ 10

\

0.1 [ 10

\

1.5 [ 10

\

31



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TOC Graphic

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Particle Proximity Effect in Nanoparticle Electrocatalysis: Surface

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