# Stochastic Analysis of Electron Transfer and Mass Transport in Confined Solid/Liquid Interfaces

## Abstract

**:**

^{−2}. We believe that our results can contribute to the comprehension of the physical/chemical properties of nano-interfaces, thereby aiding to a better understanding of the capabilities and limitations of the “dip and pull” method.

## 1. Introduction

^{−2}at most. In particular, and in agreement with the Hubbard’s model, we find that the current density is a linear function of the thickness of the liquid layer atop the solid surface. Our findings are general and valid for one-electron redox couples in both aqueous and non-aqueous media.

## 2. Methods: Stochastic Simulation Details

^{+}A

^{−}) at a concentration (ionic strength, I) of 0.1 M (0.1 mol·L

^{−1}). Under these conditions, the Debye length (k

^{−1}) for the electrolyte (describing the electrostatic screening of the charges in solution) can be calculated as follows (Equation (1)) [40]:

^{−1}= [ε

_{r}·ε

_{0}·RT/(2 × 10

^{3}·F

^{2}·I)]

^{½}

_{r}and ε

_{0}are the relative and vacuum (8.854 × 10

^{−12}F·m

^{−1}) dielectric permittivity, respectively. F and R are the Faraday constant (9.64853 × 10

^{4}C·mol

^{−1}) and the molar gas constant (8.31447 J·mol

^{−1}·K

^{−1}), respectively. T is the temperature (in K). For a 0.1 M aqueous solution at r.t (ε

_{r water =}80.2 [41]), k

^{−1}= 0.96 nm. The quantity 3·k

^{−1}(2.88 nm), which is the distance in the solution at which the electrostatic potential due to the charged electrode is reduced to 5% of its value at the electrode surface, can be thereby taken as the thickness of the electrical double layer (EDL). Within the Gouy–Chapman–Stern model [40], the definition of the EDL given in this study comprises the Stern and the diffuse layer. Please note that within this definition, the thickness of the EDL is also equivalent to the separation at which the electrostatic interaction between the charge density at the electrode surface and an elementary charge is comparable in magnitude to the thermal energy, k

_{B}T (equal to about 25.7 meV (4.12 × 10

^{−21}J) at r.t.).

^{2}. The first element (0) is the working electrode (WE). The elements from 1 to 10 represent the electrolyte solution atop the electrode surface, with element 1 simulating the EDL.

^{−5}cm

^{2}·s

^{−1}for both the oxidized and reduced species. The redox couple obeys the following electrochemical equilibrium at zero overpotential (i.e., at E°) (Equation (2)):

^{−7}cm (2.88 nm) for the element 1 to 1.0 × 10

^{−1}cm for the element 10. The total thickness of the liquid electrolyte d was therefore equal to 1320 µm. This value was chosen to ensure that the liquid layer thickness was larger than the diffusion layer thickness l for all the investigated potential sweep ranges (either in the anodic or cathodic direction starting from the equilibrium potential E° of the redox couple) and scan rates. The values of l for all the different simulated conditions are reported in Table 1, and have been determined using the following relation (Equation (3)) [38]:

^{½}

^{8}(and 1 × 10

^{5}, respectively).

^{0}·[C

_{O}(x = 0, t)·exp [−αηF/RT] − C

_{R}(x = 0, t)·exp [(1 − α) ηF/RT]

^{0}is the standard rate constant (in cm·s

^{−1}). The transfer coefficient α (which is a dimensionless number) represents a measure of the symmetry of the reaction energy barrier (ΔG

^{‡}). For α = ½ the barrier is symmetric. For 0 ≤ α < ½ the activated complex (or transition state) is closer to the product R, whereas when ½ > α ≥ 1 the activated complex is closer to the reagents O + e

^{−}[42].

_{O}(0,t) and C

_{R}(0,t) are the concentrations (in mol·L

^{−1}) of the oxidized and reduced species at the electrode surface (x = 0) at the time t. The simulations were performed setting a concentration of 0.5 mM at x = 0 for both the reduced R and oxidized O species, at the equilibrium potential of the redox couple E° at t = 0. The total concentration of the redox couple in solution at each time t of the simulation was therefore equal to 1.0 mM. Different values and combinations of k

^{0}and α were investigated, as described in the “Results and Discussion” section.

_{⫽}, along the z direction in Figure 1) and orthogonal to the sample surface (DP⫠, along the x direction in Figure 1). It should be realized, however, that the time scale for the diffusion along the DP

_{⫽}path is some orders of magnitude longer compared to that for the DP⫠ path, and therefore it can be ignored. To demonstrate that, we use Fick’s first law of diffusion (Equation (5)) [38]:

^{−1}| ~ |n/Σ·1/D·[ΔC(x, y, z)/Δx, y, z]

^{−1}|

^{−7}cm (30 nm) and 1.0 cm, respectively. Therefore, n = 3 × 10

^{−12}mol (for a redox couple concentration C of 1.0 mM = 1 × 10

^{−6}mol·cm

^{−}

^{3}), and the electrolyte layer cross-section Σxy (Σyz) on the xy (yz) plane is equal to 30 × 10

^{−7}cm × 1.0 cm = 3 × 10

^{−6}cm

^{2}(1.0 cm × 1.0 cm = 1.0 cm

^{2}). We then define ∆t

_{⫽}(∆t⫠) as the time needed to completely deplete the redox couple in the liquid layer having DP

_{⫽}(DP⫠) as the only active diffusion pathway (Δz = 1 cm and Δx = 30 × 10

^{−7}cm, respectively, see Figure 1). We found values for ∆t

_{⫽}and ∆t⫠ of 2 × 10

^{5}s and 1.8 × 10

^{−6}s, respectively, thereby justifying the choice of considering the DP⫠ as the only active pathway for the redox couple diffusion in solution during the simulations.

_{O,R}(x, t)/∂t = D·∂

^{2}C

_{O,R}(x, t)/∂x

^{2}

^{0}= 1.0 cm·s

^{−1}) are reported in Figure 2a. The anodic and cathodic peaks, exhibiting a typical asymmetric line shape, do not shift by varying the scan rate as expected from a purely reversible ET. This is also confirmed by the peak-to-peak separation (Δ

_{pp}) equal to 59.16 ± 0.25 mV, in agreement with the separation expected at r.t. (T = 298 K) for one electron, Nernstian-reversible ET (ln10·RT/F ≈ 2.303·RT/F = 59.13 mV) [43]. The anodic (cathodic) current density at the peak maxima (minima) is proportional to the square root of the scan rate v, as reported in Figure 2b, and in line with the Randles–Sevcik equation for planar, semi-infinite diffusion conditions (Equation (8)) [38]:

_{p}

_{A,C}= 0.4463·(F

^{3}/RT)

^{½}·D

^{½}·C(x = 0)

_{O,R}·v

^{½}

^{0}= 1.0 × 10

^{−}

^{7}cm·s

^{−1}) as a function of the potential scan rate. In contrast to the fast ET, the Δ

_{pp}between the anodic and cathodic branch of the cyclic voltammetry increases by increasing the scan rate, with the Δ

_{pp}significantly exceeding the quantity 2.303·RT/F = 59.13 mV as expected from an irreversible redox couple (Δ

_{pp}passes from 545 mV at a scan rate of 25 mV·s

^{−1}to 650 mV at 200 mV·s

^{−1}). Additionally, in this case, the simulations are in agreement with the experimental results [44,45], showing that the anodic and cathodic peak currents obey Equation (8) as well (Figure 2d).

## 3. Results and Discussion

^{−1}·s at a scan rate of 25 mV·s

^{−1}to 0.66 cm·V

^{−1}·s at 200 mV·s

^{−1}. For the simulations of a “confined interface” with d = 30 nm, H ranges instead from 1.2 × 10

^{−4}cm·V

^{−1}·s at a scan rate of 25 mV·s

^{−1}to 1.5 × 10

^{−5}cm·V

^{−1}·s at 200 mV·s

^{−1}.

^{−1}·s, Equation (8) does not hold and must be substituted by the Hubbard relations. For a reversible (k

^{0}>> 4 × 10

^{−3}cm·s

^{−1}[35]) monoelectronic ET, the oxidation and reduction peaks are centered at the equilibrium potential of the redox couple (η = 0), and are characterized by a full-width at half-maximum (FWHM) equal to 3.530·RT/F = 90.6 mV at r.t. (T = 298 K) [43]. The peak current density j

_{p}is found to be linearly dependent on the thickness of the liquid layer (d), the concentration of reactants at the electrode surface (C(x = 0)

_{O,R}), and the potential scan rate (v) [35] (Equation (9)):

_{p}

_{A,C}

^{Rev}= ± ¼·F

^{2}/(RT)·d·C(x = 0)

_{O,R}·v

_{p}depends also on the transfer coefficient α (Equation (10)):

_{p}

_{A,C}

^{Irr}= ± α/2.718·F

^{2}/(RT)·d·C(x = 0)

_{O,R}·v

_{p A,C}

^{Irr}= ± RT/(αF)·ln [(RTk

^{0})/(αFdv)]

^{0}≤ 1.0 × 10

^{−}

^{5}cm·s

^{−}

^{1}and η

_{p C,A}≥ 100 mV [35].

^{0}, the potential scan rate v, and the transfer coefficient α.

#### 3.1. Voltammetric Response of the “Confined Interface” for a Symmetric (α = ½) ET as a Function of d (v = 100 mV·s^{−}^{1})

^{0}equal to 1.0 cm·s

^{−}

^{1}and 1.0 × 10

^{−}

^{7}cm·s

^{−}

^{1}, respectively). For both kinetics, we simulated a symmetric (α = ½) energy barrier. The CVs for the reversible and irreversible ET simulated at a scan rate of 100 mV·s

^{−}

^{1}are shown in Figure 3a,b, respectively, as a function of d (equal to 10, 20, 30, 40, and 50 nm).

^{0}= 1.0 cm·s

^{−1}), the anodic and cathodic peaks are symmetric around the standard potential of the redox couple (η = 0). This is confirmed by the fact that the peaks can be fitted with a single Gaussian function, as reported in Figure 4a for the anodic peak simulated for d = 30 nm and v = 100 mV·s

^{−1}. For the irreversible ET process (k

^{0}= 1.0 × 10

^{−7}cm·s

^{−1}), the voltammetric peaks are instead well-separated and exhibit a pronounced asymmetric tail at low overpotentials. The anodic peak reported in Figure 4a, obtained for the irreversible ET with d = 30 nm and v = 100 mV·s

^{−1}, shows that the line shape can be reproduced by using a Lognormal function. A decrease in the reaction rate not only induces an asymmetry of the peak line shape, but also causes a broadening of the voltammetric features (Figure 4b). A FWHM of 90.6 ± 1.4 mV is found for k

^{0}= 1.0 cm·s

^{−1}, as expected for a Nernstian-reversible ET process at r.t [43]. For k

^{0}= 1.0 × 10

^{−7}cm·s

^{−1}, the peak FWHM for both the anodic and cathodic peaks is instead equal to 120.6 ± 1.2 mV.

_{p}as a function of the electrolyte layer thickness, for the reversible and irreversible ET processes. The trend is linear in both cases, in agreement with the functional dependency outlined by Equations (10) and (11), respectively. Furthermore, the current density slightly decreases when passing from reversible to irreversible ET, with the corresponding ratio equal to 1.32. This is in agreement with the peak current density that results from the Hubbard’s model: Equations (9) and (10) provide, in fact, a ratio equal to 2.718/(4·α) = 1.36, for a symmetric energy barrier (α = ½).

_{p}(Figure 4d) is centered at η = 0 V for all the investigated d values for the reversible ET, whereas it shows a logarithmic dependency from d for k

^{0}= 1.0 × 10

^{−}

^{7}cm·s

^{−}

^{1}, in agreement with Equation (11).

#### 3.2. Voltammetric Response of the “Confined Interface” (d = 30 nm) for a Symmetric ET (α = ½) as a Function of k^{0}

^{0}, with a symmetric energy barrier (α = ½). Figure 5a reports the simulated CVs (at a fixed scan rate of 100 mV·s

^{−1}) for a reversible (A: k

^{0}= 1.0 × 10

^{−}

^{2}cm·s

^{−1}) and three irreversible ET processes (B: k

^{0}= 1.0 × 10

^{−}

^{6}cm·s

^{−1}, C: k

^{0}= 1.0 × 10

^{−}

^{8}cm·s

^{−1}, and D: k

^{0}= 1.0 × 10

^{−}

^{10}cm·s

^{−1}).

^{0}, in agreement with Equation (10). Second, strongly asymmetric anodic/cathodic peaks characterize the irreversible ET, as discussed above. The FWHM increases from 90.6 ± 1.3 mV (3.53·RT/F = 90.6 mV [43]) for the reversible ET to 120.6 ± 1.1 mV for all k

^{0}< 1.0 × 10

^{−}

^{5}cm·s

^{−1}. k

^{0}= 1.0 × 10

^{−}

^{5}cm·s

^{−1}can therefore be taken as a value separating the reversible from the irreversible ET. This is confirmed by the negligible Δ

_{pp}for k

^{0}> 1.0 × 10

^{−}

^{5}cm·s

^{−}

^{1}, whereas for k

^{0}< 1.0 × 10

^{−}

^{5}cm·s

^{−}

^{1}the peak separation increases with decreasing reaction rates (Figure 5b). For values of k

^{0}< 1.0 × 10

^{−}

^{5}cm·s

^{−}

^{1}, the trend of the overpotential η

_{p C,A}

^{Irr}at which the voltammetric peaks are centered is linear with the natural logarithm of k

^{0}, in agreement with Hubbard’s model for an irreversible ET processes occurring in a confined environment (Equation (11)) [35].

#### 3.3. Voltammetric Response of the “Confined Interface” (d = 30 nm) for a Symmetric (α = ½) ET as a Function of v

^{0}= 1.0 cm·s

^{−1}. To simulate the “confined interface”, the thickness of the electrolyte layer was set to 30.0 nm. The corresponding voltammetric response of the interface under the application of the overpotential with different scan rates is reported in Figure 6a. As expected for a reversible ET process, the line shape of the voltammetric peaks is purely Gaussian (with a FWHM of 90.6 ± 1.1 mV) and no separation exists between the anodic and cathodic peaks, which are centered at the equilibrium potential of the redox couple (η = 0 V, E = E°) irrespective of the scan rate. As Figure 6b shows, the current density at the anodic and cathodic peaks increases linearly with the potential scan rate, in line with the Hubbard’s model for a reversible ET (Equation (9)).

^{0}= 1.0 × 10

^{−}

^{7}cm·s

^{−}

^{1}) as a function of the scan rate. The simulated CVs are reported in Figure 7a. The FWHM of both the anodic and cathodic peaks remains constant at around 120.6 ± 1.3 mV, thereby ruling out any effect of the scan rate on the broadening of the voltammetric waves. Finally, in agreement with Equations (10) and (11), the peak current density j

_{p}(Figure 7c) and overpotential η

_{p}(Figure 7d) show a linear and logarithmic dependency on the scan rate, respectively.

#### 3.4. Voltammetric Response of the “Confined Interface” (d = 30 nm) for an Irreversible ET (k^{0} = 1.0 × 10^{−}^{7} cm·s^{−}^{1}) as a Function of α

^{0}= 1.0 × 10

^{−}

^{7}cm·s

^{−}

^{1}) with α spanning a range of 0.1–1.0 and a fixed potential scan rate of 100 mV·s

^{−}

^{1}. The simulated CVs are reported in Figure 8a. It is clear that α has a strong influence on the FWHM, the magnitude, and the potential at which the voltammetric waves are centered. The FWHM of both the anodic and cathodic peaks decreases when α increases (Figure 8b). We found that the trend can be fitted with the following function (Equation (12)):

_{p A,C}= RT/(3.926·F·α)·ln [½α/(10

^{−8}·RT)]

_{p}is directly proportional to α, in agreement with the predictions of the Hubbard’s model (Equation (10)). Similarly, the trend of the peak overpotential η

_{p}as a function of α is well described by Equation (11), as shown by Figure 8d.

## 4. Conclusions

- We show that the electrochemical response of nanometer-sized interfaces can be described with the thin-layer voltammetry theory originally elaborated by Hubbard for liquid layer thicknesses spanning from tens to hundreds of micrometers. For both space scales, the common dominant factor is that the diffusion of reactants is confined to the direction orthogonal to the interface. This is in agreement with our recent experimental work where the oxygen evolution reaction (OER) was investigated on a poly crystalline Pt surface immersed in 1.0M KOH aqueous solution [32]. Under those conditions, the hydroxyl anions present in the solution were depleted from the thin liquid layer due to the ongoing oxidation to molecular oxygen, eventually causing the loss of potential control at the interface [32]. We concluded that the applied overpotential (~700 mV with respect to the thermodynamic water oxidation potential, +1.23 V vs. RHE) sustained an electrolyte consumption rate in the thin electrolyte layer that was not counterbalanced on the same time scale by the diffusion rate from the macroscopic liquid meniscus.
- We investigated the confined interface by simulating reversible and irreversible electron transfer processes as a function of the liquid layer thickness. For irreversible electron transfers, we find that the current density and the line shape of the voltammetric features are strongly dependent on the symmetry of the reactant and product free energy curves around the energy barrier. In addition, for both types of electron transfers, we observe that the current density is a linear function of the liquid layer thickness, and that the peak current density values are on the order of hundreds of nA·cm
^{−}^{2}at most. It is noteworthy to compare this value with the one experimentally retrieved using two different working electrode configurations, as reported in ref. [13]: the first preserved the usual “dip and pull” geometry [7,13], while in the second one the bottom part of the sample immersed in the electrolyte was masked to approximate the current density reached at the “confined interface” [13]. We determined a current density ratio between the two experimental configurations of about 3, with the current density for the “masked” working electrode reaching some hundreds of μA·cm^{−}^{2}at most [13]. The discrepancy between this value and the one obtained in this work from the stochastic simulations can be easily explained in terms of the macroscopic liquid meniscus still present on the “masked” electrode, ensuring the necessary electrochemical continuity between the electrolyte layer on the sample and the bulk solution [13]. We use the capillary length as a “yardstick” to characterize the curvature of the meniscus, finding a value of about 4 mm for the liquid water/water vapor interface at r.t [47]. Therefore, although showing an expected decreasing trend when passing from a “bulk” to a “confined interface”, the current density values found in ref. [13] are dominated by the presence of the meniscus and do not capture the true properties of electrolyte layers with thicknesses limited to few tens of nanometers.

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Salmeron, M.; Schlögl, R. Ambient pressure photoelectron spectroscopy: A new tool for surface science and nanotechnology. Surf. Sci. Rep.
**2008**, 63, 169–199. [Google Scholar] [CrossRef] [Green Version] - Crumlin, E.J.; Bluhm, H.; Liu, Z. In situ investigation of electrochemical devices using ambient pressure photoelectron spectroscopy. J. Electr. Spectr. Relat. Phenom.
**2013**, 190, 84–92. [Google Scholar] [CrossRef] - Liu, X.; Yang, W.; Liu, Z. Recent progress on synchrotron-based in-situ soft X-ray spectroscopy for energy materials. Adv. Mater.
**2014**, 26, 7710–7729. [Google Scholar] [CrossRef] [PubMed] - Crumlin, E.J.; Liu, Z.; Bluhm, H.; Yang, W.; Guo, J.; Hussain, Z. X-ray spectroscopy of energy materials under in situ/operando conditions. J. Elect. Spectr. Relat. Phenom.
**2015**, 200, 264–273. [Google Scholar] [CrossRef] [Green Version] - Roy, K.; Artiglia, L.; van Bokhoven, J.A. Ambient Pressure Photoelectron Spectroscopy: Opportunities in Catalysis from Solids to Liquids and Introducing Time Resolution. Chem. Cat. Chem.
**2018**, 10, 666–682. [Google Scholar] [CrossRef] - Lewerenz, H.-J.; Lichterman, M.F.; Richter, M.H.; Crumlin, E.J.; Hu, S.; Axnanda, S.; Favaro, M.; Drisdell, W.; Hussain, Z.; Brunschwig, B.S.; et al. Operando analyses of solar fuels light absorbers and catalysts. Electrochim. Acta
**2016**, 211, 711–719. [Google Scholar] [CrossRef] [Green Version] - Favaro, M.; Abdi, F.F.; Crumlin, E.J.; Liu, Z.; van de Krol, R.; Starr, D.E. Interface science using ambient pressure hard X-ray photoelectron spectroscopy. Surfaces
**2019**, 2, 8. [Google Scholar] [CrossRef] [Green Version] - Liu, Z.; Bluhm, H. Liquid/solid interfaces studied by ambient pressure HAXPES. In Hard X-ray Photoelectron Spectroscopy (HAXPES), 1st ed.; Woicik, J., Ed.; Springer: Cham, Switzerland, 2016; pp. 447–466. ISBN 978-3-319-24043-5. [Google Scholar]
- Axnanda, S.; Crumlin, E.J.; Mao, B.; Rani, S.; Chang, R.; Karlsson, P.G.; Edwards, M.O.M.; Lundqvist, M.; Moberg, R.; Ross, P.N.; et al. Using “tender” X-ray ambient pressure X-ray photoelectron spectroscopy as a direct probe of solid-liquid interface. Sci. Rep.
**2015**, 5, 9788. [Google Scholar] [CrossRef] - Karslıoğlu, O.; Nemšák, S.; Zegkinoglou, I.; Shavorskiy, A.; Hartl, M.; Salmassi, F.; Gullikson, E.M.; Ng, M.L.; Rameshan, C.; Rude, B.; et al. Aqueous solution/metal interfaces investigated in operando by photoelectron spectroscopy. Faraday Discuss.
**2015**, 180, 35–53. [Google Scholar] [CrossRef] [Green Version] - Favaro, M.; Jeong, B.; Ross, P.N.; Yano, J.; Hussain, Z.; Liu, Z.; Crumlin, E.J. Unravelling the electrochemical double layer by direct probing of the solid/liquid interface. Nat. Commun.
**2016**, 7, 12695. [Google Scholar] [CrossRef] - Favaro, M.; Drisdell, W.S.; Marcus, M.A.; Gregoire, J.M.; Crumlin, E.J.; Haber, J.A.; Yano, J. An operando investigation of (Ni–Fe–Co–Ce)O
_{x}system as highly efficient electrocatalyst for oxygen evolution reaction. ACS Catal.**2017**, 7, 1248–1258. [Google Scholar] [CrossRef] [Green Version] - Favaro, M.; Valero-Vidal, C.; Eichhorn, J.; Toma, F.M.; Ross, P.N.; Yano, J.; Liu, Z.; Crumlin, E.J. Elucidating the alkaline oxygen evolution reaction mechanism on platinum. J. Mater. Chem. A
**2017**, 5, 11634–11643. [Google Scholar] [CrossRef] [Green Version] - Favaro, M.; Yang, J.; Nappini, S.; Magnano, E.; Toma, F.M.; Crumlin, E.J.; Yano, J.; Sharp, I.D. Understanding the oxygen evolution reaction mechanism on CoO
_{x}using operando ambient-pressure X-ray photoelectron spectroscopy. J. Am. Chem. Soc.**2017**, 139, 8960–8970. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shavorskiy, A.; Ye, X.; Karslıoğlu, O.; Poletayev, A.D.; Hartl, M.; Zegkinoglou, I.; Trotochaud, L.; Nemšák, S.; Schneider, C.M.; Crumlin, E.J.; et al. Direct mapping of band positions in doped and undoped hematite during photoelectrochemical water splitting. J. Phys. Chem. Lett.
**2017**, 8, 5579–5586. [Google Scholar] [CrossRef] [Green Version] - Calvillo, L.; Fittipaldi, D.; Rüdiger, C.; Agnoli, S.; Favaro, M.; Valero-Vidal, C.; Di Valentin, C.; Vittadini, A.; Bozzolo, N.; Jacomet, S.; et al. Carbothermal Transformation of TiO
_{2}into TiO_{x}C_{y}in UHV: Tracking Intrinsic Chemical Stabilities. J. Phys. Chem. C**2014**, 118, 22601–22610. [Google Scholar] [CrossRef] - Stoerzinger, K.A.; Favaro, M.; Ross, P.N.; Yano, J.; Liu, Z.; Hussain, Z.; Crumlin, E.J. Probing the Surface of Platinum during the Hydrogen Evolution Reaction in Alkaline Electrolyte. J. Phys. Chem. B
**2018**, 122, 864–870. [Google Scholar] [CrossRef] [Green Version] - Gokturk, P.A.; Camci, M.T.; Suzer, S. Lab-based operando x-ray photoelectron spectroscopy for probing low-volatile liquids and their interfaces across a variety of electrosystems. J. Vac. Sci. Technol. A
**2020**, 38, 040805. [Google Scholar] [CrossRef] - Novotny, Z.; Aegerter, D.; Comini, N.; Tobler, B.; Artiglia, L.; Maier, U.; Moehl, T.; Fabbri, E.; Huthwelker, T.; Schmidt, T.J.; et al. Probing the solid–liquid interface with tender x rays: A new ambient-pressure x-ray photoelectron spectroscopy endstation at the Swiss Light Source. Rev. Sci. Instrum.
**2020**, 91, 023103. [Google Scholar] [CrossRef] - Starr, D.E.; Favaro, M.; Abdi, F.F.; Bluhm, D.; Crumlin, E.J.; van de Krol, R. Combined soft and hard X-ray ambient pressure photoelectron spectroscopy studies of semiconductor/electrolyte interfaces. J. Elect. Spectr. Relat. Phenom.
**2017**, 221, 106–115. [Google Scholar] [CrossRef] [Green Version] - Velasco-Vélez, J.J.; Pfeifer, V.; Hävecker, M.; Wang, R.; Centeno, A.; Zurutuza, A.; Algara-Siller, G.; Stotz, E.; Skorupska, K.; Teschner, D.; et al. Atmospheric pressure X-ray photoelectron spectroscopy apparatus: Bridging the pressure gap. Rev. Sci. Instrum.
**2016**, 87, 053121. [Google Scholar] [CrossRef] - Velasco-Vélez, J.J.; Pfeifer, V.; Hävecker, M.; Weatherup, R.S.; Arrigo, R.; Chuang, C.-H.; Stotz, E.; Weinberg, G.; Salmeron, M.; Schlögl, R.; et al. Photoelectron spectroscopy at the graphene-liquid interface reveals the electronic structure of an electrodeposited cobalt/graphene electrocatalyst. Angew. Chem. Int. Ed.
**2015**, 54, 14554–14558. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rik, M.; Frevel, L.; Velasco-Vélez, J.J.; Plodinec, M.; Knop-Gericke, A.; Schlögl, R. The Oxidation of Platinum under Wet Conditions Observed by Electrochemical X-ray Photoelectron Spectroscopy. J. Am. Chem. Soc.
**2019**, 141, 6537–6544. [Google Scholar] [CrossRef] [PubMed] - Velasco-Velez, J.-J.; Mom, R.V.; Sandoval-Diaz, L.-E.; Falling, L.J.; Chuang, C.-H.; Gao, D.; Jones, T.E.; Zhu, Q.; Arrigo, R.; Roldan Cuenya, B.; et al. Revealing the Active Phase of Copper during the Electroreduction of CO
_{2}in Aqueous Electrolyte by Correlating In Situ X-ray Spectroscopy and In Situ Electron Microscopy. ACS Energy Lett.**2020**, 5, 2106–2111. [Google Scholar] [CrossRef] [PubMed] - Lichterman, M.F.; Hu, S.; Richter, M.H.; Crumlin, E.J.; Axnanda, S.; Favaro, M.; Drisdell, W.; Hussain, Z.; Mayer, T.; Brunschwig, B.S.; et al. Direct observation of the energetics at a semiconductor/liquid junction by operando X-ray photoelectron spectroscopy. Energy Environ. Sci.
**2015**, 8, 2409–2416. [Google Scholar] [CrossRef] [Green Version] - Kolb, D.M.; Hansen, W.N. Electroreflectance spectra of emersed metal electrodes. Surf. Sci.
**1979**, 79, 205–211. [Google Scholar] [CrossRef] - Hansen, W.N. Electrode resistance and the emersed double layer. Surf. Sci.
**1980**, 101, 109–122. [Google Scholar] [CrossRef] - Kolb, D.M.; Rath, D.L.; Wille, R.; Hansen, W.N. An ESCA study on the electrochemical double layer of emersed electrodes. Ber. Bunsenges. Phys. Chem.
**1983**, 87, 1108. [Google Scholar] [CrossRef] - Siegbahn, H. Electron spectroscopy for chemical analysis of liquids and solutions. J. Phys. Chem.
**1985**, 89, 897–909. [Google Scholar] [CrossRef] - Weingarth, D.; Foelske-Schmitz, A.; Wokaun, A.; Kötz, R. In situ electrochemical XPS study of the Pt/[EMIM][BF
_{4}] system. Electrochem. Comm.**2011**, 13, 619–622. [Google Scholar] [CrossRef] - Booth, S.G.; Tripathi, A.M.; Strashnov, I.; Dryfe, R.A.W.; Walton, A.S. The offset droplet: A new methodology for studying the solid/water interface using X-ray photoelectron spectroscopy. J. Phys. Condens. Matter
**2017**, 29, 454001. [Google Scholar] [CrossRef] [Green Version] - Stoerzinger, K.A.; Favaro, M.; Ross, P.N.; Hussain, Z.; Liu, Z.; Yano, J.; Crumlin, E.J. Stabilizing the meniscus for operando characterization of platinum during the electrolyte-consuming alkaline oxygen evolution reaction. Top. Catal.
**2018**, 61, 2152–2160. [Google Scholar] [CrossRef] [Green Version] - Favaro, M.; Abdi, F.F.; Lamers, M.; Crumlin, E.J.; Liu, Z.; van de Krol, R.; Starr, D.E. Light-induced surface reactions at the bismuth vanadate/potassium phosphate interface. J. Phys. Chem. B
**2018**, 122, 801–809. [Google Scholar] [CrossRef] [PubMed] - Kinetiscope Web Site. Available online: http://hinsberg.net/kinetiscope/index.html (accessed on 7 August 2020).
- Hubbard, A.T. Study of the kinetics of electrochemical reactions by thin-layer voltammetry: I. Theory. J. Electroanal. Chem.
**1969**, 22, 165–174. [Google Scholar] [CrossRef] - Houle, F.A.; Hinsberg, W.D.; Morrison, M.; Sanchez, M.I.; Wallraff, G.; Larson, C.; Hoffnagle, J. Determination of coupled acid catalysis-diffusion processes in a positive-tone chemically amplified photoresist. J. Vac. Sci. Technol. B Microelectron. Nanometer Struct. Process. Meas. Phenom.
**2000**, 18, 1874. [Google Scholar] [CrossRef] - Houle, F.A.; Hinsberg, W.D.; Sanchez, M.I. Kinetic Model for Positive Tone Resist Dissolution and Roughening. Macromolecules
**2002**, 35, 8591–8600. [Google Scholar] [CrossRef] - Bard, A.J.; Faulkner, L.R. Electrochemical Methods, 2nd ed.; John Wiley & Sons Inc.: Hoboken, NJ, USA, 2001; Appendix B; ISBN 13 978-0-471-04372-0. [Google Scholar]
- Wiegel, A.A.; Wilson, K.R.; Hinsberg, W.D.; Houle, F.A. Stochastic methods for aerosol chemistry: A compact molecular description of functionalization and fragmentation in the heterogeneous oxidation of squalane aerosol by OH radicals. Phys. Chem. Chem. Phys.
**2015**, 17, 4398–4411. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lee, L.L. Molecular Thermodynamics of Electrolyte Solutions; World Scientific Publishing Co. Pte. Ltd.: Singapore, 2008; Chapter 11; ISBN 13 978-981-281-418-0. [Google Scholar]
- Archer, G.G.; Wang, P. The Dielectric Constant of Water and Debye-Hückel Limiting Law Slopes. J. Phys. Chem. Ref. Data
**1990**, 19, 371–411. [Google Scholar] [CrossRef] [Green Version] - Bauer, H.H. The electrochemical transfer-coefficient. J. Electroanal. Chem. Interf. Electrochem.
**1968**, 16, 419–432. [Google Scholar] [CrossRef] - Salbeck, J. An electrochemical cell for simultaneous electrochemical and spectroelectrochemical measurements under semi-infinite diffusion conditions and thin-layer conditions. J. Electroanal. Chem.
**1992**, 340, 169–195. [Google Scholar] [CrossRef] - Favaro, M.; Perini, L.; Agnoli, S.; Durante, C.; Granozzi, G.; Gennaro, G. Electrochemical behavior of N and Ar implanted highly oriented pyrolytic graphite substrates and activity toward oxygen reduction reaction. Electrochim. Acta
**2013**, 88, 477–487. [Google Scholar] [CrossRef] - Favaro, M.; Agnoli, S.; Cattelan, M.; Moretto, A.; Durante, C.; Leonardi, S.; Kunze-Liebhäuser, J.; Schneider, O.; Gennaro, A.; Granozzi, G. Shaping graphene oxide by electrochemistry: From foams to self-assembled molecular materials. Carbon
**2014**, 77, 405–415. [Google Scholar] [CrossRef] - Botasini, S.; Mendez, E. Limited Diffusion and Cell Dimensions in a Micrometer Layer of Solution: An Electrochemical Impedance Spectroscopy Study. Chem. Electro. Chem.
**2017**, 4, 1891–1895. [Google Scholar] [CrossRef] - Boucher, E.A. Capillary phenomena: Properties of systems with fluid/fluid interfaces. Rep. Prog. Phys.
**1980**, 43, 497–546. [Google Scholar] [CrossRef]

**Figure 1.**Solid/liquid interface parametrization used in this work. The elements from 1 to 10 represent the electrolyte solution atop the WE surface (element 0), with element 1 simulating the EDL. The colored arrows indicate electron transfer (ET) and Fick diffusion (FD) between the different elements. The right inset reports the schematization of the two possible diffusion pathways (DP), parallel and orthogonal to the interface.

**Figure 2.**Simulation results of a symmetric (α = 0.5), Nernstian-reversible (k

^{0}= 1.0 cm·s

^{−1}(

**a**,

**b**) and irreversible ET (k

^{0}= 1.0 × 10

^{−7}cm·s

^{−1}(

**c**,

**d**) for a “bulk interface” (d = 1320 µm). For both set of simulations, the initial concentration of the oxidized and reduced species in solution was set to 0.5 mM, for a total concentration of the redox couple equal to 1.0 mM. (

**a**,

**c**) and (

**b**,

**d**) report the simulated cyclic voltammograms and the corresponding peak current densities as a function of the potential scan rate v. For both ET processes, the simulations show that the peak current density obeys the Randles–Sevcik equation.

**Figure 3.**Simulation results for a symmetric (α = ½) ET between the redox couple in solution and the electrode surface, as a function of d (at a fixed scan rate v of 100 mV·s

^{−1}); (

**a**) reversible ET (k

^{0}= 1.0 cm·s

^{−1}); (

**b**) irreversible ET (k

^{0}= 1.0 × 10

^{−7}cm·s

^{−1}). For both sets of simulations, the initial concentration of the oxidized and reduced species in solution was set to 0.5 mM, for a total concentration of the redox couple equal to 1.0 mM.

**Figure 4.**(

**a**): Anodic peaks and corresponding Gaussian and Lognormal fits for a reversible (k

^{0}= 1.0 cm·s

^{−1}) and irreversible ET (k

^{0}= 1.0 × 10

^{−7}cm·s

^{−1}), respectively. The simulations were carried out for a symmetric energy barrier (α = ½) and setting d = 30 nm and v = 100 mV·s

^{−1}. Trends of the peak FWHM (

**b**), current density j

_{p}(

**c**), and overpotential η

_{p}(

**d**) as a function of the electrolyte layer thickness d (symbols). The figures also report the corresponding predictions of the Hubbard’s model (red lines).

**Figure 5.**Simulation results for a symmetric ET (α = ½) between the redox couple in solution and the electrode surface, with d = 30.0 nm. (

**a**) CVs simulated at a scan rate v of 100 mV·s

^{−1}, as a function of the standard rate constant k

^{0}(A: k

^{0}= 1.0 × 10

^{−2}cm·s

^{−1}, B: k

^{0}= 1.0 × 10

^{−6}cm·s

^{−1}, C: k

^{0}= 1.0 × 10

^{−8}cm·s

^{−1}, D: k

^{0}= 1.0 × 10

^{−10}cm·s

^{−1}). (

**b**) Overpotentials η

_{p A}and η

_{p C}at which the anodic and cathodic waves are centered as a function of k

^{0}and its natural logarithm (full symbols). The solid red lines describe the η

_{p A}

^{Irr}and η

_{p C}

^{Irr}trends as predicted by the Hubbard’s model (Equation (11)), whereas η

_{p}

^{Rev}= 0 V (E = E°) for a Nernstian-reversible ET (k

^{0}> 1.0 × 10

^{−5}cm·s

^{−1}). The initial concentration of the oxidized and reduced species in solution was set to 0.5 mM, for a total concentration of the redox couple equal to 1.0 mM.

**Figure 6.**Simulation results of a symmetric (α = 0.5), Nernstian-reversible ET (k

^{0}= 1.0 cm·s

^{−1}) between the redox couple in solution and the electrode surface, for a “confined interface” (d = 30 nm). (

**a**,

**b**) Simulated voltammetric response and corresponding peak current densities as a function of the potential scan rate v. The initial concentration of the oxidized and reduced species in solution was set to 0.5 mM, for a total concentration of the redox couple equal to 1.0 mM.

**Figure 7.**Simulation results for a symmetric (α = ½), irreversible ET (k

^{0}= 1.0 × 10

^{−7}cm·s

^{−1}) between the redox couple in solution and the electrode surface, with d = 30.0 nm. (

**a**) Simulated CVs as a function of the scan rate v. The trends of the peak FWHM, current density j

_{p}, and overpotential η

_{p}as a function of v (full symbols) are reported in (

**b**–

**d**), respectively, together with the corresponding predictions of the Hubbard’s model (solid red lines). The initial concentration of the oxidized and reduced species in solution was set to 0.5 mM, for a total concentration of the redox couple equal to 1.0 mM.

**Figure 8.**Simulation results for an irreversible ET (k

^{0}= 1.0 × 10

^{−7}cm·s

^{−1}) between the redox couple in solution and the electrode surface, with d = 30 nm. (

**a**) CVs simulated at a scan rate v of 100 mV·s

^{−1}, as a function of the transfer coefficient α. The trends of the peak FWHM, current density j

_{p}, and overpotential η

_{p}as a function of α (full symbols) are reported in (

**b**–

**d**), respectively, together with the corresponding predictions of the Hubbard’s model (solid red lines). The initial concentration of the oxidized and reduced species in solution was set to 0.5 mM, for a total concentration of the redox couple equal to 1.0 mM.

**Table 1.**Diffusion layer thickness l for all the investigated potential sweep ranges and scan rates. The values of l have been determined using Equation (3) and the diffusion coefficient D of the redox couple, equal to 0.5 × 10

^{−5}cm

^{2}·s

^{−1}. The electrochemical “bulk” conditions are guaranteed by the total thickness of the liquid layer atop the electrode surface (d = 1320 µm), which exceeds l for all the investigated parameters.

Potential Scan Rate v (mV·s^{−1}) | Potential Sweep Duration Δt (s) for ΔV = 1.2 V | Diffusion Layer Thickness l (µm) for ΔV = 1.2 V |
---|---|---|

25 | 48 | 930 |

50 | 24 | 660 |

100 | 12 | 465 |

200 | 6 | 330 |

**Table 2.**Δx dimensions for the liquid elements used to simulate a total electrolyte layer thickness of 10, 20, 30, 40, and 50 nm.

Liquid Electrolyte Layer Thickness d (nm) | Δx Element 1 (EDL) (nm) ^{1} | Δx Elements 2–10 (nm) |
---|---|---|

10 | 2.88 | 0.79 |

20 | 2.88 | 1.90 |

30 | 2.88 | 3.01 |

40 | 2.88 | 4.12 |

50 | 2.88 | 5.24 |

^{1}EDL thickness (defined as 3·k

^{−1}) for a supporting electrolyte concentration of 0.1 M (see text for details).

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Favaro, M.
Stochastic Analysis of Electron Transfer and Mass Transport in Confined Solid/Liquid Interfaces. *Surfaces* **2020**, *3*, 392-407.
https://doi.org/10.3390/surfaces3030029

**AMA Style**

Favaro M.
Stochastic Analysis of Electron Transfer and Mass Transport in Confined Solid/Liquid Interfaces. *Surfaces*. 2020; 3(3):392-407.
https://doi.org/10.3390/surfaces3030029

**Chicago/Turabian Style**

Favaro, Marco.
2020. "Stochastic Analysis of Electron Transfer and Mass Transport in Confined Solid/Liquid Interfaces" *Surfaces* 3, no. 3: 392-407.
https://doi.org/10.3390/surfaces3030029