A Manifesto on the Thermochemistry of Nanoscale Redox Reactions

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A Manifesto on the Thermochemistry of Nanoscale Redox Reactions for Energy Conversion Jennifer Lynn Peper, and James M. Mayer ACS Energy Lett., Just Accepted Manuscript • DOI: 10.1021/acsenergylett.9b00019 • Publication Date (Web): 04 Mar 2019 Downloaded from http://pubs.acs.org on March 4, 2019

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A Manifesto on the Thermochemistry of Nanoscale Redox Reactions for Energy Conversion Jennifer L. Peper* and James M. Mayer* Department of Chemistry, Yale University, New Haven, Connecticut 06520-8107 *[email protected] Abstract Advancements in the use of nanoscale materials for energy conversions will rely on a clear understanding of the reaction thermochemistry. While band edge and Fermi energies can be rigorous thermodynamic descriptors of bulk materials, we show here that their application to nanoscale systems is problematic. A change of one-electron at a thousand-atom nanoparticle or a localized surface site on a catalyst/electrode causes significant nuclear reorganization of the system. Electron transfer is often coupled with solvent reorganization and/or with the transfer of ions, including ion intercalation, surface binding, and movement within the surrounding double layer. These effects can be significant (even for some bulk materials). However, Fermi and band energies, as typically used, do not include the energetics of nuclear reorganizations or cationcoupling. A comprehensive approach to nanoscale charge transfer thermodynamics must explicitly considers these effects in order to have a more complete understanding and greater control of nanoscale redox reactivity.

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Charge transfer reactions at nanoscale solid-solution interfaces are of great technological, environmental, and practical importance. Such processes are crucial steps in the operation of batteries, fuel cells, electrolyzers, photo/electrochemical cells, and other technologies.1 The development of novel nanoscale systems at the intersection of solid-state physics and chemistry research has stimulated advancements in many areas and drawn scientists from diverse backgrounds. While many insightful publications have addressed interfacial charge transfer thermodynamics over the past several decades,1-6 for nanoscale systems these thermochemical treatments are often ill-defined and sometimes misleading. Thermochemistry provides limits on possible reactivity and defines energy efficiency. An improved understanding of solid-solution interface thermodynamics for nanoscale redox reactions is critical to progress in this area, and in the design and analysis of many processes and devices.


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Figure 1. A typical illustration of interfacial redox chemistry that places the conduction (ECB) and valence (EVB) band edge energies of a nanomaterial on the same energy scale as the potentials of proton reduction and water oxidation. We argue here that this ubiquitous practice is incomplete, as the commonly used electronic, single-state ECB, EVB (and Fermi energies) do not capture the nuclear reorganization energies inside and outside of the nanomaterial. Reprinted from reference 7 with slight revision. Electron transfers (ET) involving molecules and materials are typically described in the terms of their respective parent fields, solution chemistry and solid-state physics. Soluble redox couples (Ox/Red, eq 1) are described with standard reduction potentials (E°) and Gibbs free energies (ΔG°), ΔG° = −nFE°. In contrast, studies involving materials traditionally use Fermi energies (EF) or Fermi levels‡ and valence/conduction band edge energies (EVB/ECB) to describe analogous ET processes.8 EF, ECB and EVB are similarly used to describe charge transfer reactions of nanoparticles (NPs), as in eq 2. Figure 1 is a representative example of this widespread practice. The (nonstandard) reduction potential (E) from chemistry and the EF from physics are commonly considered equivalent, so that at equilibrium (eq 3), the nanoparticle EF is taken as equal to the potential of the Ox/Red couple. Ox + e–

Red

eq 1


3


NP

+ e–

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NP e–

eq 2 NP

+ Red

NP e– + Ox

eq 3 We show here that some of the common assumptions underlying the use of Fermi and band edge energies as thermodynamic descriptors of bulk materials are not appropriate for nanoscale interfacial reactivity. First these assumptions are analyzed, identifying the problems with the use of EF, EVB and ECB as descriptors of outer-sphere electron transfer reactions at nanoscale material, such as eqs 2 and 3. We then emphasize that many interfacial reactions are not pure outer-sphere ET processes but involve the coupled movement of ions, better described as inner-sphere redox reactions. Using classical electrostatic arguments and the example of Li batteries, we show that these effects are significant contributors to the energetics of nanoscale (and sometimes bulk) ET processes. For such inner-sphere processes, EF, EVB and ECB are clearly not complete energetic descriptors. These reactions require the use of a reduction potential (E) and attendant Gibbs free energy (∆G) for which the chemical process is clearly defined. While challenges remain in defining rigorous standard state E° and ∆G° parameters for nanoparticles, as discussed below, the use of well-defined E and ∆G values is a key step towards a more comprehensive and transferable thermodynamic understanding of nanoscale interfacial reactivity.

I.

Single-state parameters versus Gibbs free energies: Contrasting the thermodynamic

treatments of bulk, molecular, and nanoscale systems Fermi energy is defined as the electrochemical potential of electrons in a material, 𝜇𝜇̅ e (eV).2,9-10

The electrochemical potential, not to be confused with reduction potential (E), is the per molar free


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energy change for transferring an electron to a material from vacuum, [𝜕𝜕G/𝜕𝜕ne]j, where all extensive properties of the system are constant (indicated by subscript j) and the amount of electrons added is restricted to an infinitesimal quantity.10,11 More typically, EF is defined, in the context of Fermi-Dirac statistics, as a property of a single material, a single state with a fixed set of electronic levels and electron occupancy.9 In this definition, EF is the energy (E) at which the probability of quantum level occupation, ƒ(E), is 50% in a material (Figure 2A).2,10 Band edge energies are similarly single-state properties, describing the energy of the highest occupied (EVB) and lowest unoccupied (ECB) electronic quantum levels of an ideal semiconductor (Figure 2A).

Figure 2. (A) Band structure of a n-type semiconductor with the electron probability distribution, f(E), from Fermi-Dirac statistics at 0 K (dashed) and finite temperature (solid, red), with EVB, ECB, and EF as properties of this single material. (B) Schematic of an ET equilibrium between a nanoparticle (Mat) and a solution redox couple Ox/Red. At equilibrium, the reduction potential of the semiconductor, E(Mat/Mat•e–), is related to the standard reduction potential, E°(Ox/Red), of the solution-based couple by the bottom equation (dashed, green). Implicit in both definitions of EF is the assumption that the material is large such that the change of 1 electron is an infinitesimal perturbation on the structure and electronic energy levels of the solid.9-11 Both the specific energy levels within the Fermi-Dirac probability function, ƒ(E),9 and the composition of the material (and thus 𝜇𝜇̅ e)10,11 are unchanged upon addition of 1 electron, i.e. ACS Paragon Plus Environment

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the initial and final states (NP and NP•e–, eq 2) are equivalent. For a macroscopic system, this assumption is very accurate, and EF, ECB, and EVB—the properties of a single-state depicted in Figure 2A—are considered thermodynamically rigorous. Thus, the resulting equivalence of the Fermi energy and the reduction potential of a material was developed by the visionary leaders of solid/solution interfacial chemistry for this specific case of outer-sphere ET involving bulk materials.2,3 Unlike the description above of ET at bulk materials, the thermochemical treatments of molecular ET processes explicitly consider the fully relaxed initial and final states of the system. For instance, the Nernst equation for the molecular Ox/Red 1e– half-reaction in Figure 2B is E(Ox/Red) = E°(Ox/Red) – (RT/F)ln([Red]/[Ox]). The need to consider both initial and final states is due to the energetically significant changes in the atomic structure of the molecule and the orientation of the surrounding solvent that occur upon reduction or oxidation of a molecule.12 These changes are the inner- and outer-sphere components of the Marcus reorganization energy (λ = λin + λout), and these are typically large for molecular reactions (~1 eV).12 Due to the substantial reorganization, the ΔG° for ET to a molecule differs substantially (often by ca. λ) from the optical energies. This is illustrated by Hush’s demonstration that intervalence optical transitions of mixed-valent complexes are ~λ larger than ΔG°ET.13,14 In contrast, the Marcus λ of macroscopic electrodes is typically assumed to be zero, 15-17 and the thermodynamic parameters of bulk materials are equated to optical energies.5,6,18 Nanoscale materials, containing only a few thousand atoms, are intermediate between bulk and molecular systems. However, thermochemical analyses of nanomaterial redox reactions almost invariably use bulk treatments, Figure 1. We argue here that ET at a nanoscale material is not an infinitesimal perturbation on the system and that the initial and final states differ significantly due


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to the non-zero nuclear reorganization accompanying ET. Therefore, the single-state energy parameters are not appropriate measures of the thermodynamics of nanoscale systems, e.g., the Fermi energy is not equal to the free energy change for electron transfer. Like the analogous molecular redox couple, the energetics of the nanoparticle half-reaction in Figure 2B include the initial (Mat) and final (Mat•e–) states, E(Mat/Mat•e–). The next section considers the magnitude of the errors introduced by applying the assumption of zero reorganization to nanoscale ET reactions.

II. Nuclear reorganization at nanomaterials and interfaces upon outer-sphere electron transfer Unlike macroscopic materials, the solvation energy of colloidal nanoparticles (NPs) changes (by ΔΔGsolv) upon ET. For a sphere in a dielectric continuum, ΔGsolv varies inversely with the radius, r, according to the simple electrostatic Born equation.19 The ΔΔGsolv of a typical molecular couple (related to λout) is on the order of hundreds of meV; for example, λout = 290 meV for [Ru(bpy)3]3+/2+ (r = 0.7 nm).12 Since small NPs are not so much larger than molecular complexes, the change in their solvation energy upon ET is predicted to be significant.15 Weaver, Murray and coworkers developed a more complete electrostatic model for estimating ΔΔGsolv upon ET to colloid-like structures, shown in Figure 3 as a change in potential (ΔE = ΔΔGsolv/−nF). Weaver’s model predicted the linear quantization of the energy required for sequential reduction of nanoscale spheres or “tiny capacitors” (Figure 3, equation).19-21 This effect was observed experimentally in the classic studies of small ligated gold nanoparticles (r = 0.5−1.5 nm) where the potential for nanoparticle reduction changed by -300 mV per electron.20-21 The relationship between the interfacial potential E and the charge of a colloidal nanoparticle qe is the capacitance C (Figure 3, right axis, C ≡ Δqe/ΔE).19 The large ∆E (sub-attofarad capacitance) ACS Paragon Plus Environment

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observed for small gold nanoparticles demonstrate the significant changes in the solvation of small nanomaterials upon ET.19-21

Figure 3. The quantized ΔE and capacitance (attofarad, aF) of a spherical nanoparticle as a function of radius (r) in an electrolyte solution with a uniform ligand monolayer (or Debye screening length) of thickness d = 0.77 nm and relative permittivity of the surrounding monolayer ϵ = 3, as per hexanethiolate-ligated gold nanoparticles.20 The change in q is 1e– per NP. ΔE is inversely related to C. In the equation, e is the electronic charge and ϵ0 is the permittivity of free space. (Note that unlike the 1/r dependence of ΔΔGsolv in the Born solvation model, the presented equation gives a ΔΔGsolv/nF = ΔE α d/(r2+rd) dependence.)

Weaver’s experimentally validated electrostatic model (Figure 3) qualitatively predicts the

transition from low to high capacitance when spherical nanomaterials become larger than r ≈ 5 nm.19 For instance, the ΔE for a sphere with r = 100 nm is calculated to be only 35 μV under these experimental conditions (Figure 3).20 This model of the outer-sphere reorganization provides valuable estimates of nanomaterial ΔΔGsolv, ΔE, and C. It does not, however, include other potentially important factors, such as ligand density, ionic strength, the nature of the added charge (localized vs delocalized), or cation-coupling, which can be very important for nanomaterials, vide infra.


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The inner-sphere reorganization energies of nanomaterials are more difficult to estimate a priori, but they are likely reduced relative to those of molecules due to their larger size, delocalized electronic structures, and rigid lattices. However, it should be noted that significant relaxation energies (~200 meV) have been reported for charge transfers at flexible organic photovoltaic materials and for localized trap states, common in many semiconductors.22,23 More studies of λin would be of interest, especially for localized reactive sites of bulk interfaces, where corrosion, heterogeneous catalysis, and electrochemical catalysis typically occur. In sum, ET reactions of small nanoscale systems (r < ~5 nm) typically have significant nuclear reorganizations (λout + λin). The resulting difference in the initial and final states of ET processes is an important aspect of nanoscale energetics. This difference is not accounted for in the application of single-state energy parameters from solid-state physics, such as Fermi and bandedge energies. Because they don’t include nuclear reorganizations, EF, ECB and EVB are more directly connected with optical energies and may be more relevant to femtosecond processes such as plasmonic breathing modes or the photoinduced charge transfer step in a dye-sensitized solar cell or photoelectrosynthesis cell (DSSC or DSPEC). However, the energetic parameters of these ultrafast processes should not be mixed with equilibrated thermochemical values of the slower chemical steps discussed here.

III.

Cation-coupled electron transfers

The discussion above focused on nanoscale redox reactions that involve only outer-sphere electron transfer (ET). Yet many nanoscale redox reactions involve ET that is stoichiometrically accompanied by a charge-balancing cation. This is most evident for lithium-ion batteries, where the fundamental assumption is that every electron transfer is coupled to intercalation of a Li+ ion


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(eq 4).24 This holds both for the anode and the cathode, which typically involve carbon (CLix) or metal oxide (LixMO2) materials. - + Li+ Mat + e Mat + e + Li +

Mat Li++ e-Mat Li e

eq 4

The final state of the material in eq 4 (Mat•Li+•e–) is clearly different from the initial one, by one additional Li+ and e-. The Fermi energy, as defined by Fermi-Dirac statistics or as the electrochemical potential of electrons in the material (𝜇𝜇̅ e = [𝜕𝜕G/𝜕𝜕ne]j), neglects the energies associated with Li+ intercalation: the electrostatic stabilization, nuclear reorganization, and ionic bonding. As such, EF, ECB, and EVB are incomplete in the energetic description of eq 4, even for bulk materials. In fact, these energetic terms are much less prevalent in the battery literature. The thermodynamic description of eq 4 at nanoscale (and bulk) materials must include the contributions of both e– and Li+ transfers. This is illustrated in the representation of eq 4 in a square scheme (Scheme 1) where the Gibbs free energies for the Li+ and e– transfers are respectively represented as the Li+ association constant (KLi, ΔG° = −RTln(KLi)) and the standard reduction potential (E°, ΔG° = −FE°). By Hess’s law, the free energy for the Li+-coupled ET (diagonal arrow in Scheme 1) is equal to the sum of consecutive Li+ transfer and electron transfer steps; ΔG°(Mat/Mat•Li+•e–) = −RTlnKLi(Mat•Li+) − FE°(Mat•Li+). Li+ intercalation accompanies ET only because the reduced material (Mat•e–) has a higher affinity for Li+ than the original material (Mat). In other words, ET is Li+-coupled because KLi(Mat•Li+) < KLi(Mat•Li+•e-).25 For small TiO2 nanoclusters, Friesener et al. calculated a 108 increase in the KLi for Li+ intercalation upon reduction.26 The change in the cation binding constant upon electron addition is a requirement for cation-coupled electron transfer, and it is another clear indication that the properties of the material change upon the addition of a single electron.


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Scheme 1: Thermochemical square scheme showing the addition of Li+ (horizontal arrows), e(vertical arrows) or both Li+ and e− (Li+-coupled ET or Li atom transfer, diagonal arrows), to a material (Mat).

+ Li+

Mat

KLi(Mat•Li+) + + Li + e–

+ e– E °(Mat) e–

Mat•

+ Li+ KLi(Mat•Li+•e–)

Mat•Li

+

+ e– + E °(Mat•Li ) + Mat•Li •e–

While the stoichiometric coupling of Li+ and e– is so ingrained as to be unstated in lithium ion batteries, the coupling of other cation transfers to nanoscale ET processes is typically ignored. Yet examples of cation-coupling are increasingly well established in the literature.26-30 In particular, reductive proton intercalation has been identified in many oxide systems.30-36 The ~60 mV per pH unit dependence of the ET energetics at metal oxide-aqueous interfaces is increasingly understood to indicate proton-coupled electron transfer30 (though this is not the textbook explanation of the shifts of the band edge energies with pH37). From their extensive work with nanoporous TiO2, ZnO and SnO2 electrodes, Hupp et al. concluded that “charge-compensating cation intercalation is a general mode of reactivity for metal oxide semiconductors.”30,34-36 Studies of colloidal ZnO NPs in aprotic solvents have allowed comparisons of the stabilizing effects of different cations on the potential of ZnO-based conduction band electrons.29,38-39 Consistent with Hupp’s emphasis on charge compensation, roughly twice as many electrons were added to ZnO NPs treated with divalent cations (Mg2+ and Ca2+) than NPs treated with Na+ or H+ at similar potentials.29 Cation-coupled ET to NPs could involve intercalation of the cation, as in lithium ion batteries, or binding of the cation to the surface. Even in the latter case, the electrostatic stabilization of a


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reduced material from cation adsorption at the material’s surface is significant for nanoscale systems. In an extension of the electrostatic model discussed in Section II above, Liu, Gamelin, Li et al. predicted that the interfacial potential of a conducting sphere is stabilized by up to several hundred meV by a unit positive charge at the surface, following ΔE’∝ 1/(r+δ), where r is radius and δ is the cation distance from the surface.40 Cation surface binding is, in practice, very similar to the accumulation of cations in the inner Helmholtz layer surrounding a nanomaterial. Cation binding is thus connected to the capacitance and λout discussed above, which include contributions from cation movement (or transfer) within the double layer in addition to the reorganization of the solvent dipoles and the ligand shell. These arguments suggest that under typical experimental conditions, most ET reactions at nanoscale solid/solution interfaces are coupled to charge compensating cations in some fashion. This need not always be in the simple 1:1 stoichiometry of lithium ion batteries described above. This includes reactions of colloidal, isolated NPs as well as NPs on solid supports and porous NP films. The origin of the coupling is the change in the association constant of the cation upon material reduction (Scheme 1). In these cases, the free energy of cation-coupled ET depends on the nature and concentration of the cation, not just on the intrinsic properties of the material. Fermi and band edge energies, which focus exclusively on the energetics of the electron, are clearly not appropriate descriptors of cation-coupled processes.

IV. Outlook, recommendations, and conclusions For a small nanoparticle, a change of one electron is a significant perturbation. It can change the affinity of a NP for a cation so strongly that every ET event is explicitly coupled to cation intercalation. Even when cation intercalation is not occurring, the energetics of nanoscale ETs are


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affected by the ligand shell (density, thickness, and permittivity), the ions in solution (ion charge, concentration, and size), and other aspects of Marcus-type inner- and outer-sphere reorganization energies. We have shown here that Fermi and band edge energies, electron focused properties of a singlestate, do not encompass a full description of nanoscale interfacial thermodynamics. Instead, we recommend rooting these discussions in Gibbs free energies and reduction potentials. The use of these typical molecular parameters emphasizes the difference in the initial and final states of the nano-system and conveys the intrinsic sensitivity of nanoscale reactivity to the surrounding medium (as per their similarity to molecular ET reactions). Mat + Red

Mat e

-

+ Ox

eq 5 [Red]

E(Mat/Mat•e–) = E°(Ox/Red) – 59 mV log [Ox]

eq 6

In an equilibrium outer-sphere ET reaction, the reduction potentials of a NP (Mat/Mat•e–) and a parallel solution-based redox couple (Red/Ox) are equal (Figure 2B, eqs 5 and 6 at 298 K).11 Leveraging these equilibrations to evaluate nanoscale thermodynamics is common in the literature; however, the solution of eq 6 is usually equated to the Fermi energy of the material.41-44 Because this equilibrium is a reflection of the initial and final states of the material and the specific experimental conditions, we advocate that the reduction potential, E(Mat/Mat•e–)be used instead. Mat + Red + Li+

Mat e- Li+

+ Ox

eq 7

E(Mat/Mat•Li+•e–) = E1MLi+(Mat/Mat•Li+•e–) + 59 mV log[Li+]

eq 8 [Red]

E1MLi+(Mat/Mat•Li+•e–) + 59 mV log[Li+] = E°(Ox/Red) – 59 mV log [Ox]

eq 9

In the case of explicit cation-coupled electron transfer reactions, the preference for Gibbs energies over Fermi and band energies is incontrovertible (even for bulk materials). The


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thermochemical description of reactions that have a 1 electron : 1 cation stoichiometry (eq 7) should, of course, include the solution concentration (activity) of the cation. Referenced to 1M Li+, E1MLi(Mat/Mat•Li+•e–), the half-reaction potential for cation-coupled electron transfer to the material, E(Mat/Mat•Li+•e–) (eq 4), shifts by 59 mV per magnitude change in [Li+] (eq 8).25 This Nernstian dependence is also reflected in the experimental measurements of E(Mat/Mat•Li+•e–) with a soluble redox couple (eq 9).45 Grätzel used an analogous equation (to eq 9) for the description of 60 mV per pH unit dependence of TiO2 reactivity with methyl viologen.45 Like the lithium ion battery example, the 60 mV per pH unit dependence of the potential to reduce TiO2 thin film electrodes indicates 1:1 proton-coupled electron transfer reactivity.30 Thermodynamic parameters which explicitly include the contributions of cations and/or other species are more robust, predictive, and transferable. We encourage researchers to test for cation-coupling in their redox reactions, to determine the best form of the thermodynamic parameters (eq 6 versus eqs 8 and 9). Additional studies are needed to determine how to convert these measured E and ∆G values into true standard reduction potentials E° and ΔG°. One challenge is defining standard states for nanomaterials. In the limit that a colloid of small NPs is similar to a molecular system, the full equilibrium expressions of equations 5 and 7 should include the ‘concentrations’ of the oxidized and reduced NPs in some form. For example, Keq(eq 5) = ‘[Mat•e–]’[Ox]/‘[Mat]’[Red]. In this limit, the ‘concentrations’ of Mat and Mat•e–, perhaps represented by the concentration of NPs in solution (NP mol L-1) or the ensemble density of unoccupied conduction band states, will also influence the observed nanomaterial equilibria. At the other limit, bulk materials are not included in equilibrium expressions. Because nanomaterials are intermediate between molecules and bulk solids, it is not yet evident how to account for the mass-action effects (how to define


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‘concentrations’) of nanomaterials. Still, the advantages of a reduction potential formalism, as in eqs 6, 8 and 9, are clear. When the nanoscale reaction is properly defined, reduction potentials capture all of the energetically important components of the ET process, and, in the case of cationcoupled ET, provide quantitative insight into the effects of cation concentration on reactivity (eqs 8, 9). In sum, while Fermi and band edge energies can be thermodynamically rigorous properties for infinite, bulk solids, these terms are not appropriate for the thermochemical description of nanomaterial solid/solution interfacial charge transfer reactions. The energetics of nanoscale reactions depend on the significant reorganization energies (λ), the coupling of cation transfer with electron transfers, the ionic strength, the ligand density, and other effects. These effects likely contribute to the scatter in measured thermodynamic values from one system to another. For example, a compendium of reported band energies for TiO2 by Finklea shows variations of ±0.5 V at each pH.46 We advocate a more holistic approach to nanoscale charge transfer thermodynamics, which explicitly considers the many nuclear reorganizations beyond the transfer of an electron. The use of reduction potentials and Gibbs free energies are needed to obtain a more complete understanding and greater control of nanoscale redox reactivity. This is especially important in instances of explicit cation-coupled ET, where Fermi and band edge energies are insufficient even for the description of bulk solid. We believe that this effort to place the thermochemistry of nanoscale redox reactions on firmer footing, bringing the molecular intuition implied by the use of Gibbs free energy terms, will stimulate new insights and approaches to critical challenges in energy conversion and other areas.


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AUTHOR INFORMATION [email protected], https://chem.yale.edu/people/james-mayer Notes The authors declare no competing financial interest.

Biographies Jennifer L. Peper Jennifer Peper was born and raised in Indiana, and attended Indiana University (Bloomington) where she worked with Prof. Mu-Hyun Baik. She began her graduate work at the University of Washington (Seattle), moving with the Mayer group to Yale, where she received her PhD in 2019.

James M. Mayer James Mayer was born and raised in New York City. Research with Profs. Edwin Abbott (Hunter College) and William Klemperer (Harvard) led to a PhD from Caltech with Professor John Bercaw. After two years at DuPont, he was a professor at the University of Washington, moving to Yale in 2014.

ACKNOWLEDGMENT This work was supported as part of the Argonne-Northwestern Solar Energy Research Center, an Energy Frontier Research Center funded by the U. S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0001059, and by the U. S. National Science Foundation 1151726 and 1609434.

REFERENCES ‡

For our purposes here, Fermi energy and Fermi level are conceptually equivalent. See Ashcroft

and Girault for more complete discussions.9,10


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1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11.

12. 13. 14.

Adams, D. M.; Brus, L.; Chidsey, C. E. D.; Creager, S.; Creutz, C.; Kagan, C. R.; Kamat, P. V.; Lieberman, M.; Lindsay, S.; Marcus, R. A., et al. Charge Transfer on the Nanoscale: Current Status. J. Phys. Chem. B 2003, 107, 6668-6697. Reiss, H. The Fermi Level and the Redox Potential. J. Phys. Chem. 1985, 89, 3783-3791. Gerischer, H.; Ekardt, W. Fermi Levels in Electrolytes and the Absolute Scale of Redox Potentials. Appl. Phys. Lett. 1983, 43, 393-395. Bockris, J. O. M.; Khan, S. U. M. Fermi Levels in Solution. Appl. Phys. Lett. 1983, 42, 124125. Cahen, D.; Kahn, A. Electron Energetics at Surfaces and Interfaces: Concepts and Experiments. Adv. Mater. 2003, 15, 271-277. Zhu, X.-Y. How to Draw Energy Level Diagrams in Excitonic Solar Cells. J. Phys. Chem. Lett. 2014, 5, 2283-2288. Maeda, K.; Domen, K. Photocatalytic Water Splitting: Recent Progress and Future Challenges. J. Phys. Chem. Lett. 2010, 1, 2655-2661. Bard, A. J.; Memming, R.; Miller, B. Terminology in Semiconductor Electrochemistry and Photoelectrochemical Energy Conversion (Recommendations 1991). Pure Appl. Chem. 1991, 63, 569. Ashcroft, N. W.; Mermin, N. D. The Sommerfield Theory of Metals. In Solid State Physics, Brooks/Cole Cengage Learning: Singapore, 2011; pp 40-43. Girault, H. H. Electrochemical Potential. In Analytical and Physical Electrochemistry, Marcel Dekker: New York, 2004; pp 1-31. Page 21: "In regard to the electrochemical potential, implicitly, we are making the hypothesis that the phase is large enough so that adding charges does not modify the inner potential (or μ�e) significantly." Castellan, G. W. Systems of Variable Composition: Chemical Equilibrium. In Physical Chemistry, 2d ed ed.; Addison-Wesley Pub. Co: Reading, Mass, 1971; pp 227-263. “For any substance i in a mixture, the value of µi is the increase in free energy which attends the addition of an infinitesimal number of moles of that substance to the mixture per molar of the substance added. (The amount added is restricted to an infinitesimal quantity so that the composition of the mixture, and therefore the value of µi, does not change).” As pointed out by a reviewer, the situation is more complex for an ideal, large-bandgap bulk semiconductor. Such a material would have no states in the band gap and effectively zero population of e– and h+ in the CB and VB due to the very small Boltzmann factor (exp[– Eband gap/kT] = 10–51 for rutile at 298 K). This is really an insulator, not a semiconductor. For essentially any real macroscopic semiconductor, the presence of intrinsic and extrinsic trap states within the band gap will remove this effect. Marcus, R. A.; Sutin, N. Electron Transfers in Chemistry and Biology. Biochim. Biophys. Acta, Bioenergetics 1985, 811, 265-322. D'Alessandro, D. M.; Keene, F. R. Current Trends and Future Challenges in the Experimental, Theoretical and Computational Analysis of Intervalence Charge Transfer (IVCT) Transitions. Chem. Soc. Rev. 2006, 35, 424-440. Hush, N. S. Intervalence-Transfer Absorption. Part 2. Theoretical Considerations and Spectroscopic Data. Prog. Inorg. Chem. 1967.


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ACS Energy Letters

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Quotes requested by the Editorial Office These are not direct quotes from the Perspective but have been adjusted to account for being taken out of context. "While Fermi and band edge energies are thermodynamically rigorous properties for infinite, bulk solids, these terms are not appropriate for the thermochemical description of nanomaterial solid/solution interfacial charge transfer reactions. The energetics of nanoscale reactions depend on the significant reorganization energies (λ), the coupling of cation transfer with electron transfers, the ionic strength, the ligand density, and other effects." "Typical illustrations of interfacial redox chemistry place the Fermi energy (EF), conduction (ECB) and valence (EVB) band edge energies of a nanomaterial on the same energy scale as solution redox couples. We argue here that this ubiquitous practice is incomplete, as the commonly used EF, ECB, and EVB energies do not capture the energies of solvent reorganization and cation-coupling inside and outside of the nanomaterial." "Nanoscale materials, containing only a few thousand atoms, are intermediates between bulk and molecular systems. However, thermochemical analyses of nanomaterial redox reactions almost invariably use bulk treatments, which assume that addition of one electron is an infinitesimal perturbation. We argue here that ET at a nanoscale material is not an infinitesimal perturbation on the system and that the initial and final states of an ET reaction differ significantly due to the non-zero nuclear reorganization accompanying ET.” "Under most experimental conditions, ET reactions at nanoscale solid/solution interfaces are likely coupled to charge compensating cations.... The origin of the coupling is the change in the association constant of the cation upon material reduction. In these cases, the free energy of ET depends on the nature and concentration of the cation, not just on the intrinsic properties of the material.”


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A Manifesto on the Thermochemistry of Nanoscale Redox Reactions

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