Coupling of surface chemistry and electric double layer at TiO$_2$ electrochemical interfaces

TL;DR

This study uses advanced density functional theory simulations to explore how surface chemistry influences the electric double layer and capacitance at TiO₂ interfaces, revealing pH-dependent microscopic mechanisms affecting electrochemical behavior.

Contribution

It provides the first atomistic-level insights into the coupling of surface chemistry and electric double layer at TiO₂ interfaces using finite-field molecular dynamics.

Findings

Water molecules fluctuate more at high pH, increasing capacitance.

Proton transfers at low pH significantly raise capacitance.

Results match trends observed in titration experiments.

Abstract

Surfaces of metal oxides at working conditions are usually electrified due to the acid-base chemistry. The charged interface compensated with counterions forms the so-called electric double layer. The coupling of surface chemistry and electric double layer is considered to be crucial but poorly understood because of lacking the information at the atomistic scale. Here, we used the latest development in density functional theory based finite-field molecular dynamics simulation to investigate pH-dependence of the Helmholtz capacitance at electrified rutile TiO$_2$ (110)-NaCl electrolyte interfaces. It is found that, due to competing forces from surface adsorption and from electric double layer, water molecules have a stronger structural fluctuation at high pH and this leads to a much larger capacitance. It is also seen that, interfacial proton transfers at low pH increase significantly the capacitance value. These findings elucidate the microscopic origin for the same trend observed in titration experiments.

Tables

Table 1. Table 1: The averaged Helmholtz capacitance C H subscript 𝐶 H C_{\text{H}} , protonic Helmholtz capacitances C H + superscript subscript 𝐶 H C_{\text{H}}^{+} and deprotonic Helmholtz capacitance C H − superscript subscript 𝐶 H C_{\text{H}}^{-} at different surface charge q 𝑞 q . For q = 2 ​ e 𝑞 2 𝑒 q=2e , the corresponding surface charge density for the supercell size used here is about 20 μ 𝜇 \mu C/cm 2 .

$q$	$C_{\text{H}}$ ($\mu$F/cm2)	$C_{\text{H}}^{+}$ ($\mu$F/cm2)	$C_{\text{H}}^{-}$ ($\mu$F/cm2)
2$e$	81	67	101
4$e$	72	59	85

Equations

$$1/C_{\text{EDL}}=1/C_{\text{SC}}+1/C_{\text{H}}+1/C_{\text{GC}}$$

$$C_{\text{H}}={\left(\frac{\partial q}{2\pi L_{z}A\partial ⟨P_z⟩}\right)}_{\bar{D}}$$

Full text

]Department of Chemistry-Ångström Laboratory, Uppsala University, Lägerhyddsvägen 1, BOX 538, 75121, Uppsala, Sweden

]Institut für Chemie, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

]Department of Chemistry, University of Cambridge, Lensfield Rd, Cambridge CB2 1EW, United Kingdom

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The surface chemistry of metal oxides is more involved than that of metals because the surface charge depends on proton exchange driven by pH of the environment 1, 2. The condition in which the surface has no net proton charge is called the point of zero proton charge (PZPC) 3, 4, 5. However, working conditions for most earth-abundant metal oxide based photo-catalysts are not at PZPC and the corresponding solid-liquid interface is highly electrified 6. By attracting counterions from the solution side, the electric double layer (EDL) is formed at the interface in which the electric field can be as high as 109 V/m 7. Therefore, including the coupling of surface chemistry and EDL at metal oxide-electrolyte interfaces is necessary.

The differential capacitance $C_{\text{EDL}}$ of the EDL at oxide materials-electrolyte interfaces has been long regarded as a key probe of its structure. For semiconducting oxides, the capacitance can be resolved into three distinct components connected in series 8, 9:

[TABLE]

The first component is the result of the depletion or accumulation of electrons in the space charge (SC) layer of the semiconductor electrode, which can be 10 to 100 nm thick. The second component $C_{\text{H}}$ is the Helmholtz capacitance due to specific adsorption of hydroxide groups or protons and corresponding counter ions. The last component $C_{\text{GC}}$ called the Gouy-Chapman capacitance, stems from the diffusive electrolyte and depends on the ionic strength.

The relationship between surface structure and ion complexation of oxide-solution interface is a long-standing topic in colloid chemistry. Experimental interest in these metal oxide systems dates back to early 60s 10, 11. In titration experiments, the variation of surface charges due to adsorption/desorption of H+ is measured against the pH in solution. To interprete experiments, 2-pK model including a phenomenological neutral site (SOH0), dissociate site (SO-) and associate site (SOH${}_{2}^{+}$) was introduced (“S” for the surface) 12. However, the structural information was completely overlooked there until the development of surface complexation model in 1989 13, 14. In the multi-site complexation (MUSIC) model and other types of bond valence model 15, 16, proton affinity (or pKa) is correlated with the Pauling bond valance charge (the charge of a cation divided by its coordination number) and the Me-H distance. To connect with titration experiments, a Gouy-Chapman-Stern type EDL model was adopted to match the surface charges $q$ and the potential generated from surface complexation models $\Delta\phi$. Thus, the capacitance $C_{\text{H}}$ in surface complexation models is a (global) fitting parameter to titration experiments 17, 18.

A more subtle yet important aspect is that proton affinity itself depends on the capacitance of EDL. Indeed, in the original paper of surface complexation model, authors subtracted Nernstian EDL contribution to pKa of the solution monomers but neglected it for the surface protonation reactions 13. This was inevitable because the dielectric constant of bulk water is well known as 78 in ambient conditions while the dielectric constant of EDL at oxide-electrolyte interfaces remains a puzzle 19.

In parallel to the development by colloid chemists, electrochemists have their own interests in electrified metal oxide-electrolyte interfaces. Since the discovery of the Honda-Fujishima effect in 1972 for TiO2 20, metal oxide photo-electrochemistry attracts the perennial interest in water splitting research using solar energy. The focus there was to identify metal oxides having the right band alignment with respect to the water redox potential, for example, the conduction band minimum should be higher than the H+/H2 reduction potential 8, 9, 21. Over the years, experimental characterizations have moved from ultra-high vacuum condition to solid-liquid interface due to the advancement in surface science techniques 22, 23, 24, 25, 26.

As for colloid chemists, the potential distribution at TiO2 aqueous electrolyte interfaces also interested electrochemists. From the slope of Mott-Schottky plots, the Helmholtz capacitance as a function of pH was determined 27. Despite that the reported median Helmholtz capacitance as 50 $\mu$F/cm2 is surprisingly close to the well-quoted value of 64 $\mu$F/cm2 by colloid chemists 28, opposite trends in its pH-dependence were seen 27, 29.

Similar to the approximation colloid chemists took in building surface complexation model, the standard Pourbaix diagram in electrochemistry neglects the non-Nernstian contribution of EDL to the proton affinity 30, 31. Therefore, elucidating the microscopic origin for the pH-dependence of Helmholtz capacitance and quantifying its contribution to the surface pKa is a common challenge in both fields.

Density functional theory based molecular dynamics (DFTMD) is a suitable technique to tackle this challenge which encompasses the electronic, structural and dynamical ingredients on an equal footing. DFTMD modeling of solid-liquid interface has been applied to different areas, such as studying the structure of water (defects) at solid interfaces 32, 33, 34, 35, 36, 37, 38, spectroscopic modeling of surface-sensitive vibrational signals 39, 40, 41, computing the surface acidity 42, 43, 44, 45 and determining the redox potential 46, 47, 48, 49.

In spite of progress, electric properties, such as the dielectric constant and the interfacial capacitance which are at the heart of modeling the electrified solid-liquid interface, were usually thought to be beyond the reach of DFTMD. Thanks to the development of constant electric displacement $\bar{D}$ Hamiltonian for the modeling of ferroelectric nanocapacitors by Vanderbilt and co-workers 50, electric properties became more accessible to DFTMD simulations 51, 52, 53, 54, 55, 56. We are therefore in a position to apply the hybrid constant $\bar{D}$ simulation with PBE functional 57 to the charged rutile TiO2 (110)-NaCl electrolyte interface as implemented in CP2K 58, 59. In Eq. 1, $C_{\text{SC}}$ is the smallest of three capacitances and will normally dominate in the inverse sum. However, its effect and associated depletion layer can been eliminated at the flatband potential condition by an appropriate bias potential 30, 8. At the high ionic strength which is relevant for photoelectrocatalysis, the diffuse ionic layer has a higher capacitance and the inverse $C_{\text{GC}}$ term can therefore be ignored. Based on these considerations, modeling the Helmholtz capacitance $C_{\text{H}}$ is what we focus on in this work.

Conventionally, the Helmholtz capacitance can be computed as $C_{\text{H}}=q/\Delta\phi$ according to the textbook definition, where $q$ is the surface charge and $\Delta\phi$ is the potential drop crossing the Helmholtz layer 60, 61. Instead, by exploring the modern theory of polarization and constant $\bar{D}$ method developed by Vanderbilt and co-worker for treating ferroelectric nanocapacitors 50, we reformulated this problem in terms of the macroscopic polarization $P_{z}$ of the system 53. In our setup, the two sides of the oxide material can be charged at fixed chemical composition of the supercell by bearing same amount but opposite types of proton charges. This scheme leads to a succinct expression of the average Helmholtz capacitance $C_{\text{H}}$ based on the supercell polarization $P_{z}$ 53:

[TABLE]

where $L_{z}$ is the dimension of the model system perpendicular to the interface, $q$ is the imposed proton charge, $A$ is the area of the x, y cross section and $\langle\cdots\rangle$ indicates the ensemble average. The advantage of this formulation is three-fold. First, it does not require additional vacuum slab in the modeling as commonly used in surface science 62, 63 and the oxide is treated as one piece of material. Second, it removes the finite-size dependence of the oxide slab which plagues the computation of the Helmholtz capacitance 53. Third, by switching the electric boundary condition to constant $\bar{D}$, the relaxation of $P_{z}$ is significantly accelerated as predicted by the Debye theory of dielectrics 51, 52.

When the oxide surface is charged by protonation or deprotonation, $P_{z}$ deviates from zero in response. Therefore, $P_{z}$ has to be zero at the PZPC and this serves as a critical test for the convergence of DFTMD simulations 64. As shown in Fig. 1, at the PZPC, $M_{z}$ relaxes to zero rapidly when switching the boundary condition from $\bar{E}=0$ to $\bar{D}=0$. Within 10 picoseconds, its time average is about 0.02 D. For the case of electrified interface with surface charge of 4$e$, $M_{z}$ turns out to be -4.76 D within a similar time-scale. In all cases, classical MD simulations were leveraged to pre-equilibrate the system before applying the hybrid constant $\bar{D}$ DFTMD simulations 55, 56.

Simulation results of the average Helmholtz capacitance $C_{\text{H}}$ at the rutile TiO2 (110)-NaCl interface are shown in Table 1. Following Eq. 2, one obtains the average Helmholtz capacitance $C_{\text{H}}$ which is about 76 $\mu$F/cm2. Although starting from very different initial configurations, $C_{\text{H}}$ at surface charge $q=2e$ and $q=4e$ are in excellence agreement with each other. In order to decompose the overall Helmholtz capacitance into protonic $C_{\text{H}}^{+}$ and deprotonic $C_{\text{H}}^{-}$, we resorted to the macroscopic averaging technique 65, 59 and monitored the shift of the macro-averaged electrostatic potential with respect to the PZPC for the protonic side (Fig. 2 inset). From this decomposition, we found that $C_{\text{H}}^{-}$ is about 50% higher than $C_{\text{H}}^{+}$. At surface charge $q=2e$ (about 20 $\mu$C/cm2 for the supercell used here 59), the surface potential for the deprotonic side is about 200 mV and that for the protonic side is about 300 mV, which can be measured in principle using the surface-sensitive vibrational spectroscopy 66 or the binding energy shift in X-ray photoelectron spectroscopy 67.

The range of experimental estimated capacitance for rutile TiO2-NaCl electrolyte goes from 64 to 160 $\mu$F/cm2 68, 28, 29. The scattered data reflect the nature of the observed capacitance which depends on ionic strength and surface roughness 69. In particular, the asymmetric pH-dependence (a higher Helmholtz capacitance at high pH) that we observed from DFTMD simulations is in accord with titration experiments of rutile at higher ionic strength 29, 70, 71 and has been also seen in other metal oxides, such as ZnO 72.

The Stern layer width (the charge separation distance) for the negatively charged TiO2 surface is about 2Å  in our simulations (Fig. 2b) and this leads to an estimation of the interfacial dielectric constant of about 23. This number can be compared with the commonly assumed value of 26 for rutile TiO2 in geochemistry and colloid science 73. On the other hand, we found the Stern layer width for the positively charge TiO2 surface is about the same ($\sim$2Å, Fig. 2b), thus the smaller capacitance of the protonic side $C_{\text{H}}^{+}$ (Table 1) suggests an interfacial dielectric constant of 15 instead. It is worth noting that the maxima in the radial distribution functions of Na${}^{+}-$O${}_{\textrm{w}}$ and Cl${}^{-}-$H${}_{\textrm{w}}$ in bulk salt solutions are 2.4Å  and 2.9Å  respectively with the PBE functional 74. Therefore, the asymmetry of the interfacial capacitances found here should largely come from the difference in the dielectric screening at the interface.

At the PZPC condition, water molecules are adsorbed to the rutile TiO2 (110) surface as dimers 75. This is evidenced by the angular distribution of water dipole moments with respect to the normal vector of the surface (Fig. 3a). Water dipole moment preferably points out towards the electrolyte solution with primary and secondary peaks at 48∘ and 75∘. For the positively charged side, the electric field of EDL enhances this pattern by shifting primary peak and secondary peaks to 33∘ and 48∘ respectively. The splitting between the two peaks becomes narrower and the overall distribution is less broad in comparison with the neutral surface. The situation is reversed for the negatively charged side. The angular distribution spans almost the whole range of possible values with primary and secondary peaks around 89∘ and 60 ∘.

The much wider angular distribution of water molecules at the negatively charged side comes from two competing factors: chemisorption of water molecules at rutile (110) would orientate water dipole towards to the electrolyte while the electric field in EDL tends to flip it. Indeed, adsorption and desorption can happen dynamically at the negatively charged side in contrast to the positively charged side because of this competition, as seen in Fig. 3b. Since the extent of dielectric screening is proportional to the magnitude of dipole moment fluctuation as formulated in Kirkwood-Onsager theory 76, 52, it is not a surprise that we found the interfacial dielectric constant at the deprotonic side (therefore $C_{\text{H}}^{-}$) is much higher than that of protonic one.

This dynamical adsorption-desorption process of water molecules observed at the negatively charged side may play a role for the alternative mechanism of O2 production in alkaline solution (high pH) 77, since adsorbed water molecules at the TiO2 surface not only provide the raw material for OH$\cdot$ production but may also block surface sites 22.

The dissociative adsorption of water molecules at the bare rutile TiO2 (110) surface was extensively discussed in literature 32, 33, 34. Indeed, this is also observed for the neutral surface mediated by the neighboring water molecule (Ref. 36 and Fig. 4 a). One would expect the electric field due to the EDL will magnify this effect. However, instead of the dissociative adsorption, the hydrolysis of adsorbed water molecules releases ions to the electrolyte solution. In our simulations, this dominantly happens for the positively charged rutile TiO2 (110) surface (Fig. 4 b), in accord with a much higher electric field generated in the protonic side.

The pKa of the protonic rutile TiO2 (110) was determined to be about -1.0 in DFTMD 42. Therefore, the estimated reaction free energy for the proton transfer between the charged surface groups Ti2OH+ and solvating water molecules is comparable to thermal fluctuation energy at room temperature. Indeed, spontaneous proton exchange between them was observed for the protonic side of rutile TiO2 (110), as shown in Fig. 4 c. On the contrary, the exchange of OH- between TiOH- and solvating water molecules is endothermic and no corresponding OH- transfer event was observed at the deprotonic side. Instead, the resonance of charged group TiOH- and surface groups with adsorbed water molecules TiH2O happens (Fig. 4 d).

To quantify the effect of surface acid-base chemistry on the Helmholtz capacitance, we also carried out additional simulations by constraining adsorbed water molecules and charged surface groups Ti2OH+ to not undergo reaction. When only constraining adsorbed water molecules, the average Helmholtz capacitance $C_{\text{H}}$ at $q=4e$ turns out to be about 80$\mu$F/cm2. This value is very similar to the one about 72$\mu$F/cm2 in Table 1 without constraints. However, when both adsorbed water molecules and charged surface groups are constrained, $C_{\text{H}}$ becomes about 44$\mu$F/cm2 which is close to results obtained previously 61. This significant reduction of capacitance can be understood, because the proton exchange between Ti2OH+ and solvating water molecules will shorten the charge separation distance in the EDL. Therefore, the surface acidity of the oxide is another determining factor for the Helmholtz capacitance.

In summary, using finite-field DFTMD simulations, we discovered the microscopic origin for the pH-dependent of Helmholtz capacitance at rutile TiO2 (110) seen in titration experiments. At high pH, water molecules have a stronger structural fluctuation and this lead to a much larger capacitance. At low pH, proton transfer increases the capacitance value by reducing the charge seperation distance. These observations for rutile TiO2 (110) and the finite-field DFTMD modeling approach presented in this study pave the way to investigate electrochemical reactivity at electrified metal oxide-electrolyte interfaces, e.g. the Non-Nernstian contribution of the capacitance to the Pourbaix diagram and the hole trapping in alkaline solutions.

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CZ thanks Deutsche Forschungsgemeinschaft (DFG) for a research fellowship (No. ZH 477/1-1) during his stay in Cambridge. CZ is grateful to Uppsala University for a start-up grant and to Åforsk Foundation for a research grant (Ref. nr. 18-460). Funding from the Swedish National Strategic e-Science program eSSENCE is also gratefully acknowledged. Computational resources were provided by the UK Car-Parrinello (UKCP) consortium funded by the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom. CZ also thanks J. Cheng for helpful discussions and T. Sayer for reading the manuscript.

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The hybrid constant $\bar{D}$ Hamiltonian and its corresponding CP2K input syntax. The details of CP2K simulations of electrified rutile TiO2 (110)-NaCl electrolyte interfaces.