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C: Physical Processes in Nanomaterials and Nanostructures
Dual-Phase Molecular-Like Charge Transport in Nanoporous Transition Metal Oxides Cyrille Costentin, and Daniel G. Nocera J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b10948 • Publication Date (Web): 18 Dec 2018 Downloaded from http://pubs.acs.org on December 31, 2018
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The Journal of Physical Chemistry
Dual-phase molecular-like transition metal oxides
charge
transport
in
nanoporous
Cyrille Costentin*a,b and Daniel G. Nocera *a Department of Chemistry and Chemical Biology, Harvard University, 12 Oxford Street, Cambridge, MA 02138, United States. b Université Paris Diderot, Sorbonne Paris Cité, Laboratoire d'Electrochimie Moléculaire, Unité Mixte de Recherche Université - CNRS N° 7591, Bâtiment Lavoisier, 15 rue Jean de Baïf, 75205 Paris Cedex 13, France. a
ABSTRACT: The conductivity of porous films in the presence of a large concentration of electrolyte is modeled considering two behaviors: “molecular-like” corresponding to a single electrical phase with localized redox species surrounded by electrolyte and “dual-phase materiallike” corresponding to two distinct conductive electrical phases. In the first case, both simple redox and ion-coupled redox couples are considered. In the second case a transition from insulating to metallic behavior of the electronic phase is described showing the interplay between diffusion and migration of charge carriers. The effect of electrochemical doping is systematically investigated leading to delineation of experimental criteria to discriminate between these two types of behaviors in porous films. Application of the model to electrodeposited cobalt oxide films shows that these films, despite their molecular electrocatalytic activity, have a dual-phase material-like conductive behavior arising from nanocrystallites of cobalt oxide surrounded by an electrolyte phase, thus revealing the complexity of electrodeposited catalytic metal oxide materials.
Introduction Deposition of a thin film at the surface of a conductive electrode is a common method to design a modified electrode 1 for many applications such as electrocatalysts for electricity to fuels conversion, 2 and as the active component in energy storage, 3 electrochromic, 4 photovoltaic 5 or analytical devices. 6 An important property of such films is their conductivity, which should be appreciable in order to realize efficient transduction. In the broad field of modified electrodes, very different types of films comprising conducting polymers, 7 redox films, 8 transition metal oxides, 9 enzymes, 10 nanoparticles, 11 quantum dots, 12 metal-organic framework films, 13 etc.…, have been studied over the past three decades with little control of the film structure and composition. Consequently, a unified microscopic description of conduction properties of modified electrodes is challenging. As a type of modified electrode, nanoporous transition metal oxides films, are widely studied for charge storage properties, 14 and also for catalytic activity toward water splitting 15 as well as for photovoltaics applications.5 In all cases, the electronic conduction of the designed nanoporous material is deemed to be important though there is not a single unified picture for the conduction properties. The nature of film morphology and supporting electrolyte composition has been recognized to be manifest to function. 16, 17 As such a primary characteristic of electroactive coatings, especially in the area of electrocatalysis, is the film’s porosity in the sense of whether or not it is permeable to supporting electrolyte. 18 Herein, we consider porous films and focus on conductivity properties in “wet” conditions, i.e. in the presence of a solvent and a supporting electrolyte. Despite disparity in films’ composition or structure, the macroscopic conductivity properties may be described by the value of conductivity σ obtained from films in a source-drain
Scheme 1. Schematic representation of a source-drain conductivity measurement of a porous film.
steady-state configuration (Scheme 1). Macroscopic supporting electrolyte motion is absent in such measurements and thus σ depends on the uniform “redox state” of the film at zero sourcedrain voltage, i.e. “doping” level, and the nature of transport of the doped charge carriers. In practice, σ can be evaluated as function of the electrochemical doping in a source-drain/gated measurement performed using interdigitated electrode array, as introduced by Murray and coworkers. 19 Nanoporous electroactive materials are typically described in the framework of “molecular-like” or “dual-phase material-like” films. Using simple models, we show here that the characteristics of electronic conductivity as a function of film doping are different for molecular-like or a dual-phase material-like film behaviors, thus leading to distinction criteria. In the case of molecular-like films, we extend a previously derived model 20 to the situation of an ioncoupled redox couple, a likely situation with transition metal centers supporting proton coupled electron transfer (PCET). In the case of dual-phase material-like films, we investigate the transition from low to high doping levels in the electronic conduction phase, a likely situation in transition metal oxides. Finally, we discuss the properties of an electrodeposited cobalt oxide film, 21 which has been considered as an illustrative example of a nanoporous transition metal oxide film exhibiting molecular-like catalytic properties15 but conduction properties that appear to be that of a dual-phase
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Scheme 2. Schematic representation of A molecular-like and B dualphase material-like films.
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function of x, the position within the film between the two electrodes. It is important to note that, as opposed to the dual-phase material-like film (vide infra), redox species are highly localized and surrounded by mobile ions of the supporting electrolyte so that only one electrical phase is defined in the film. We consider a conservation condition for both mobile and immobile species in the film and charge displacement occurs under a constraint of electroneutrality. Mobile ion displacements are described by the classical Nernst-Planck equations assuming the validity of the Einstein relationship between the diffusion coefficient and mobility, i.e. without consideration of activity or ion-pairing effects. A modified Nernst-Planck equation is considered for electron hopping between fixed redox sites, also assuming the validity of Einstein relationship. 24 All equations and boundary conditions are given in the supporting information (SI). The film conductivity is defined by, σ = i/(US/L)
(2)
where S is the surface area of the electrodes and L the distance between electrodes. Introducing the dimensionless current, material-like, 22 thus emphasizing the complexity of films prepared by electrodeposition.
j = i/(FSDeC0redox/L)
Results
and defining the electrical potential u = FU/RT, then σ can be written as,
Among the diverse type of nanoporous films studied over the years, two simple situations with distinct conductivity properties can be formally delineated. The so-called redox films8 will be referred to here as displaying the “molecular-like” film behavior. In such a system, electrical transport occurs via electron hopping between localized redox sites surrounded by the mobile electroinactive ions of the supporting electrolyte (Scheme 2a). In “dualphase material-like” film behavior, 23 two distinct conductive electrical phases can be considered in the film: an electronic conductive phase with a high electronic density so that ohmic transport of electrons occurs upon application of a bias between the source and the drain and an ionic conductive phase corresponding to pores filled by the supporting electrolyte (Scheme 2b). We note that the mechanism for electron transport may include various processes such as hopping between conductive regions but it is fundamentally a migration process driven by an electrical potential gradient. Electronic conductivity of molecular-like redox films. We first consider a porous film containing homogeneously distributed immobile redox species Pz/Qz–1 and a mobile supporting electrolyte comprising counter ions, C+ and A– (for the sake of simplicity, only mono-charged ions are considered). Interplay of diffusion and migration of both electrons and ions has been well described for such systems20 and selected characteristics of the conductivity as function of the doping of the film are recalled here. The film is sandwiched between two electrodes (Scheme 2a) and electrochemically doped by application of a potential E (vs. a reference electrode) to the film. A small potential difference U is applied between both electrodes. At the interfaces, electron transfer reactions are assumed to be fast and thus redox equilibria obey Nernst’s law corresponding to, P 𝑧𝑧 + e− ⇌ Q𝑧𝑧−1
(1)
and characterized by a standard potential E0. Upon application of U, a steady-state current (i) flows through the system with the concentrations and electrical potential being a
𝜎𝜎 =
(3)
0
2 𝑗𝑗 F 𝐷𝐷𝑒𝑒 𝐶𝐶redox ∙ R𝑇𝑇 𝑢𝑢
(4)
where F is Faraday’s constant, R the gas constant, T the temperature, C0redox is the total concentration of redox species in the film and De is the apparent diffusion coefficient of electrons in the film. 25 As shown in the SI, the dimensionless conductivity j/u depends on three parameters, the film “redox state” or “doping” level ξ = F(E– E0redox)/RT, the charge number z of P and the excess factor γ = CA0/ C0redox where CA0 is the concentration of mobile electroinactive anions A– in the film. If γ >> z, corresponding to a large concentration of supporting electrolyte in the film, it is shown (see SI) that the dimensionless current (j)-potential (u) relationship is, 𝑗𝑗 exp(𝜉𝜉) 𝑗𝑗 1 + �� + � 1 + exp(𝜉𝜉) 2 1 + exp(𝜉𝜉) 2 𝑢𝑢 = ℓn � � 𝑗𝑗 exp(𝜉𝜉) 𝑗𝑗 1 � − �� − � 1 + exp(𝜉𝜉) 2 1 + exp(𝜉𝜉) 2 �
(5)
thus leading, for small values of u, to, 0 𝐹𝐹 2 𝐷𝐷𝑒𝑒 𝐶𝐶redox ∙ 𝜎𝜎 = R𝑇𝑇
F 0 (𝐸𝐸 − 𝐸𝐸redox )� R𝑇𝑇 2 F 0 )�� �1 + exp �R𝑇𝑇 (𝐸𝐸 − 𝐸𝐸redox exp �
(6)
The classical relationship for the electronic conductivity in redox films is thus recovered. 26 Figure 1a shows that σ strongly depends on the “redox state” of the film. A maximum occurs at ξ =0; σmax = F2DeC0redox/4RT when E = E0redox, as is observed experimentally for several redox film systems.18,26 Current (j)-potential (u) plots also show that the charge transport is not ohmic with the current leveling off for large values of applied voltage (Figure 1b). In the context of large concentration of electroinactive mobile ions (γ >> z), it is worth noting that there is no electrical potential drop in the film and electron transport is a
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The Journal of Physical Chemistry
Figure 1. Electronic conductivity of molecular-like redox films. (a) Conductivity as function of the film “redox state”. (b) Dimensionless current (j) vs. potential (u) for various values of the film “redox state”, ξ = F(E – E0redox)/RT = –2.2 (black line), 0 (red line), 1.1 (brown
line).
diffusional process. More refined models taking into account the interactions between sites have been derived but they lead to similar overall features.26 We also note that the apparent diffusion coefficient of electrons De depends on redox species concentration25 and may be described using percolation modeling. 27 Ion-coupled electronic conductivity of molecular-like redox films. We now extend the above analysis to the case of a redox film in which the redox process is coupled to ion transfer, both at the electrodes and between redox sites. In the case where this counterion is a proton, the system can be referred to as a PCET film. We assume that the counter-ion involved in the redox process is different from the supporting electrolyte ion and its concentration in the film is much less than that of the supporting electrolyte. The porous film thus contains homogeneously distributed immobile redox species Pz/QYz (Y+ being the mobile counter-ion) and a mobile supporting electrolyte, C+ and A–, between the two electrodes in the same configuration as shown in Schemes 1 and 2a. The film is electrochemically doped by application of a potential E (vs. a reference electrode) to the film and then U is applied between both electrodes. At the interfaces, ion-coupled electron transfer reactions are assumed to be fast and thus redox equilibriums obey Nernst law corresponding to, P 𝑧𝑧 + Y + + e− ⇌ QY 𝑧𝑧
(7)
and characterized by a standard potential EY . For the sake of simplicity, we consider that z = 0. Besides, we note that more complicated situations can be considered in which Y+ is involved in fast chemical reactions with the electrolyte (buffer). This would not fundamentally change the derivation provided the chemical equilibria are fast. The same general framework applies as in the preceding case, i.e. one electrical phase is defined in the film, both mobile and immobile species are conserved in the film and charge displacement occurs under the electroneutrality constraint. Mobile ion displacements, including Y+ (whose diffusion coefficient is DY), are again described by classical Nernst-Planck equations, assuming the validity of Einstein relationship between the diffusion coefficient and mobility. Due to its coupling to Y+, electron transport by hopping between adjacent sites is not sensitive to electrical potential since there is no net charge displacement. It is thus described by an equivalent diffusion process on species QY (see SI) character0
Figure 2. Electronic conductivity of molecular-like redox films with an ion-coupled electronic conductivity mechanism as a function of the film “redox state”. From top to bottom DYCY0/DeyC0redox = ∞ (red line), 0.25, 0.025, 0.0025, 0.00025 (black line).
ized by an apparent diffusion coefficient Dey. Note however that mobile ion transport are required to sustain ion coupled-electron transfer at the electrode surfaces. Introducing the dimensionless current (j = i/(FSDeyC0redox/L) and potential (u = FU/RT), similar to eq (4), the conductivity σ can be written as, 𝜎𝜎 =
0 𝑗𝑗 F 2 𝐷𝐷𝑒𝑒𝑒𝑒 𝐶𝐶redox ∙ R𝑇𝑇 u
(8)
As shown in the SI, the dimensionless conductivity j/u depends on four parameters: the film “redox state” or “doping” level ξy = F(E – Ey0)/RT – ℓn CY0 where CY0 is the concentration of mobile counterions Y+ in the film, the supporting electrolyte excess factor γ = C0A / C0redox, the counter-ion excess factor γY = CY0 / C0redox and the ratio of the diffusion coefficients γY = DY / Dey. As previously mentioned, we consider γ >> γY (i.e. C0A >> C0Y), and therefore the dimensionless current (j)-potential (u) relationship is (see SI), 𝑗𝑗 𝑗𝑗 𝑗𝑗 exp(𝜉𝜉𝑌𝑌 ) 1 + �� + � �𝛾𝛾𝑌𝑌 + � 𝛿𝛿𝑌𝑌 1 + exp(𝜉𝜉𝑌𝑌 ) 2 1 + exp(𝜉𝜉𝑌𝑌 ) 2 𝑢𝑢 ≈ ℓn � � (9) 𝑗𝑗 𝑗𝑗 𝑗𝑗 exp(𝜉𝜉𝑌𝑌 ) 1 � − 2� � − 2� �𝛾𝛾𝑌𝑌 − � 𝛿𝛿𝑌𝑌 1 + exp(𝜉𝜉𝑌𝑌 ) 1 + exp(𝜉𝜉𝑌𝑌 ) �
For small values of u we obtain, 1 1 1 = + 𝜎𝜎 𝜎𝜎𝑒𝑒𝑒𝑒 𝜎𝜎𝑌𝑌
(10)
with 𝜎𝜎𝑒𝑒𝑒𝑒 and
0 F 2 𝐷𝐷𝑒𝑒𝑒𝑒 𝐶𝐶redox = ∙ R𝑇𝑇
𝜎𝜎Y =
𝐹𝐹 2 𝐷𝐷Y 𝐶𝐶Y0 R𝑇𝑇
F − 𝐸𝐸y0 �� − ℓn𝐶𝐶Y0 R𝑇𝑇 �𝐸𝐸 2 (11) F �1 + exp �R𝑇𝑇 �𝐸𝐸 − 𝐸𝐸y0 � − ℓn𝐶𝐶Y0 �� exp �
(12)
The ion-coupled electronic conductivity of molecular-like redox film σ is thus controlled by the slowest of the two-components, the ion-coupled conductivity σey, which strongly depends on the “redox state” of the film. σey, goes through a maximum,
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𝜎𝜎𝑒𝑒𝑒𝑒,𝑚𝑚𝑚𝑚𝑚𝑚 when
0 𝐹𝐹 2 𝐷𝐷𝑒𝑒𝑒𝑒 𝐶𝐶redox = 4R𝑇𝑇
𝐸𝐸 = 𝐸𝐸y0 +
R𝑇𝑇 𝐹𝐹
ℓn𝐶𝐶Y0
(13)
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Scheme 3. A Electrochemical n-type doping of dual-phase material-like films. B Concentration of charge carrier in the electronic phase as function of the doping level.
(14)
and the purely ionic conductivity is σY. Figure 2 shows variation of σ with the “redox state” of the film (i.e., E) for various values of DYCY0 / DeyC0redox. σ exhibits a maximum, which transitions to a broad plateau as DYCY0 / DeyC0redox is decreased. Nonetheless, as E is distant enough from Ey0 + (RT/F)ℓnCY0, the conductivity eventually decreases due to the saturation of redox sites by either oxidized or reduced species. As in the preceding case, the presence of a large concentration of supporting electrolyte screens any potential drop so that charge transport is a purely a diffusive process driven either by Y+ ion gradient or Pz/QYz gradients. Dual-phase material-like electronic conductivity. Electrode coatings often contain an electronic conductive phase introduced intentionally to lead to a two-phase system, as sketched on Scheme 2b.23 However, the bulk material of the coating may itself be electronically conductive or semiconductive, leading to an analysis of charge transport based on a band model with partial delocalization of charge carriers resulting from interactions and overlap between redox centers. 28 We now consider such a description in which electronically conductive areas are connected, forming a phase not permeated by ions and another phase corresponding to ionic pores. Note that the electronically conductive phase may results from equilibration of conductive and insulating domains via charge percolation. Intrinsic parameters of the electronic phase are the dielectric constant ε and the apparent diffusion coefficient of charge carriers D. The latter might reflect charge transfer between electronically conductive areas through a percolation process.27,29 We assume the validity of Einstein relationship between the diffusion coefficient and mobility. The electronically conductive phase is characterized by its electrical potential ϕ as well as the concentration of free charge carriers Ce0, a measure of the doping level of the film that is set by the applied potential E (Scheme 3a). We consider here n-type doping where the charge carriers in the electronic conductive phase are electrons (transposition to p-type doped materials is straightforward). Electrochemical doping of the film upon cathodic shifting of E leads to an increase of Ce0 with concomitant formation of immobile positive charge in the film and/or at the interface with the ionic porous phase. This doping process is commonly referred to as a chemical capacitance. 30 Ce0 reaches a maximal value, Cemax, when the “doping” level of the film is large enough so that the electronic phase is fully conductive with a metallic behavior (Scheme 3b). Although the purpose of the present work is not to describe the dynamical doping process, 31 we note that a cathodic shift of the applied potential E induces a Helmholtztype capacitive process corresponding to a shift of ϕ (often referred to as band unpinning) and thus to accumulation of charge at the phase interfaces (electron in the electronic conductive phase side and ions in the pore side). 32 The latter process becomes exclusive as Ce0 has reached its maximal value Cemax (metallic behavior). Importantly, the additional injected charges corresponding to the Helmholtz capacitance charging process do not contribute to electronic conductivity.
At a given doping level, a small bias U is applied between the two electrodes sandwiching the film in a source-drain configuration (Scheme 2b). A steady-state current flows through the system corresponding to free charge carriers that diffuse and/or migrate in the electronically conductive phase according to the Nernst-Planck equation. The local electroneutrality constraint does not hold in each phase and charge density is related to the electric potential through the Poisson equation. Pseudo-local electroneutrality of the film is assured by ionic adjustment in the pores (local concentrations of the mobile ions and electrical potential) relative to the local charge density in the conductive phase. We assume that the concentration of ions in the pores is large enough to prevent ionic transport limitations. As detailed in the SI, derivation of the conductivity in the general case requires numerical calculation due to the interplay of diffusion and migration, leading to, 𝜎𝜎 =
𝑖𝑖 F 2 𝐷𝐷𝐶𝐶e0 ≈ 𝑈𝑈𝑈𝑈/𝐿𝐿 R𝑇𝑇
(15)
The electronic conductivity is therefore a measure of the film doping level provided that the apparent diffusion coefficient is independent of the doping level. 33 The interplay between diffusion and migration depends on the dimensionless parameter θ = λD/L, i.e. the ratio of the Debye characteristic screening distance, 𝜆𝜆D = �
𝜀𝜀R𝑇𝑇 F 2 𝐶𝐶e0
(16)
and the film thickness L. Two limiting behaviors are worth mentioning. When θ → 0, corresponding to a situation where the conductive phase has metallic behavior, i.e. Ce0 = Cemax and its Fermi level is in a conduction band; the Debye characteristic screening distance is much smaller than the film thickness so that an electrical potential gradient can build up in the film. The charge transport is then a pure migration process as expected for a dual ohmic material-like film. The conductivity,
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Figure 3. Electronic conductivity of dual-phase material-like films as function of the n-type doping level.
𝜎𝜎max =
F 2 𝐷𝐷𝐶𝐶emax R𝑇𝑇
(17)
is independent of the applied gated potential E. Alternatively, when θ → ∞, corresponding to the situation of low doping where Ce0
