Origin of Asymmetric Electric Double Layers at Electrified Oxide/Electrolyte Interfaces

The structure of electric double layers (EDLs) dictates the chemistry and the physics of electrified interfaces, and the differential capacitance is the key property for characterizing EDLs. Here we develop a theoretical model for computing the differential Helmholtz capacitance CH of oxide–electrolyte interfaces using density functional theory-based finite-field molecular dynamics simulations. It is found that the dipole of interfacial adsorbed groups (i.e., water molecule, hydroxyl ion, and proton) at the electrified SnO2(110)/NaCl interfaces significantly modulates the double layer potential which leads to the asymmetric distribution of CH. We also find that the dissociative water adsorption prefers the inner sphere binding of counterions, which in turn leads to a higher Helmholtz capacitance, compared with that of the nondissociative case at the interface. This work provides a molecular interpretation of asymmetric EDLs seen experimentally in a range of metal oxides/hydroxides.

S emiconducting oxide−electrolyte interfaces are highly electrified under working conditions. 1,2 The charges of semiconducting oxide surfaces come from two sources. The difference in the electrochemical potentials between oxide and electrolyte leads to the depletion or accumulation of electrons between two phases and, therefore, the formation of the space charge layer. In addition, the adsorption of protons and hydroxyl groups generates positive charges and negative charges, respectively, at these oxide surfaces. 3 The net interfacial proton charge is compensated by counterions from the electrolyte, which builds up the so-called protonic electric double layer (EDL), 4 which spans for just 3−5 Å, that is, the Helmholtz layer, at high ionic strength and under the flat band potential condition. 4,5 EDLs play an important role in electrochemistry, 6,7 photoelectrocatalysis, 8 colloid science, and geochemistry. Some of the fundamental questions therein are the following: What are the surface composition and ions distribution in the EDL? How would adsorbed water molecules respond to the electric field? 9 How would the dissociation and the recombination of interfacial water molecules affect the double layer potential ΔU? What would the differential capacitance look like, that is, the change of the Helmholtz capacitance with respect to the surface charge density? 10 In this regard, experimental methods such as X-ray photoelectron spectroscopy and X-ray standing wave provide the ion adsorption and the surface potential information; 11,12 the sum-frequency generation spectroscopy probes the polarization of water at charged interfaces; 13,14 and titration experiments reveal the change of surface charge density versus pH. 15,16 Nevertheless, the analysis and extraction of this microscopic interfacial information on EDL is rather difficult if not impossible. 17−19 This calls for density functional theory-based molecular dynamics simulation (DFTMD), which is a suitable computational method for describing the microscopic structure and the dynamics of water and ions near an interface. 20,21 Recently, it has been shown that the finite-field DFTMD is a promising technique for modeling EDLs at electrified solid−liquid interfaces. 22−24 Finite-field DFTMD simulations relies on the constant electric displacement D Hamiltonian introduced by Stengel, Spaldin, and Vanderbilt, 25 and the corresponding expression of the average Helmholtz capacitance C H is shown to be 22 where L z is the dimension of the supercell perpendicular to the interface, q is the introduced surface proton charge, A is the area of the x, y cross section, and ⟨P z ⟩ indicates the ensemble averaged supercell polarization. Because ⟨P z ⟩ converges rapidly in the finite-field simulations, this makes C H a new observable within the accessible time-scale of DFTMD. Note that in conjugation with eq 1, two sides of the oxide slab take opposite types but the same amount of proton charges in the simulated supercell.
In the first work of applying finite-field DFTMD for modeling EDLs, 23 it was found that the Helmholtz capacitance at electrified rutile TiO 2 (110)/NaCl interfaces is much higher at high pH than that at low pH for the given surface charge density, and the interfacial proton transfer at low pH increases significantly the capacitance value. Compared with rutile TiO 2 , the isostructural cassiterite SnO 2 has a characteristic dissociative water adsorption, 26 Then, the question naturally arises: how would the proton transfer affect the differential capacitance and vice versa? To answer that, herein the electrified SnO 2 (110)/NaCl electrolyte interfaces at different surface charge densities σ were simulated with finite-field DFTMD, using the CP2K/Quickstep package 28,29 with the PBE functional. 30 Detailed descriptions of the computational setup are given in the Supporting Information.
At low pH, the SnO 2 (110) surfaces are positively charged by adsorbing protons to Sn 2 O br sites, and the direction of dissociative proton transfer is opposite to that of the electric field E in the EDL (Figure 1a, I and II). At high pH, the SnO 2 (110) surfaces are negatively charged by the desorption proton from terminal Sn 5c O w H 2 , and the direction of the dissociative proton transfer is the same as that of the electric field E in the EDL (Figure 1a, III and IV). Because of this contrast, the degree of interfacial water dissociation (reaction 2) α increases with the pH in electrolyte (Figure 1b), when averaging the number of Sn 5c O w H 2 , Sn 5c O w H − , Sn 2 O br H + , and Sn 2 O br sites (16 sites in total on each surface) over DFTMD trajectories (see Supporting Information). Meanwhile, we find the free energy of water dissociation (Reaction 2) ΔA diss decreases with the pH (see Supporting Information).
When looking at the orientation of interfacial water, it is found that the adsorbed water molecules point toward the electrolyte solution ( Figure 1c), with the angle θ between the water dipole and surface normal of about 60°at the PZC. This value is downshifted to 30°−40°and up-shifted to 80°−90°, in response to the electric field in EDL. As shown in Figure 1d, the average angle of adsorbed water molecules shows a monotonic increment as a function of σ in spite of a strong presence of the water dissociation.
It has long been known that the dipole of interfacial hydroxyl groups at oxide−electrolyte interfaces strongly affect the potential offset. 31−33 As shown in Figure 2a (see Supporting Information), the band edge is shifted upward with reference to the standard hydrogen electrode (SHE) scale 34 when going from the vacuum surface, to the water monolayer (ML) adsorption and the fully solvated SnO 2 (110)/H 2 O interface. This effect is also manifested when restraining water molecules adsorbed on the left side of the SnO 2 slab not undergoing dissociation reactions at the PZC. In this case, interfacial water molecules would have different orientational distributions on the two sides of the SnO 2 slab (Figure 2b), and the average θ of the left side (restrained) is about 10°larger than that of the right side (free). This leads to a total net dipole pointing to the opposite of the surface normal ( Figure 2c). As a consequence and illustrated in Figure 2d, the left side (restrained) of the slab has a higher potential than that of the right side. On the basis of these observations made in Figure 1 and 2, we have formulated a theoretical model of differential Helmholtz capacitance at oxide-electrolyte interfaces to take the water dipole contribution into account explicitly. We begin with the textbook definition of the capacitance C H where U is the potential drop across the Helmholtz layer, and σ is the surface charge density.
Then, applying the fundamental relation D = E + 4πP and using the field expressions instead of potential and charge density, we can rewrite the above expression as where l H is the width of the Helmholtz layer, ϵ H°i s the corresponding interfacial dielectric constant. P PZC is the total polarization at the PZC, and ΔP = P − P PZC , which contains the difference in the polarization with respect to PZC.
From the equation above, it is clear that we will have a constant term C H°, which is independent of D, and a water polarization term ΔP(D), which is a function of D.
In case we are using the polarization of adsorbed surface O−H groups ΔP w as a descriptor, the ΔP w includes the contribution of adsorbed water molecules in both molecular and dissociative forms. One may scale ΔP w by the dielectric constant of the double layer ϵ H°t o account for the electrostatic screening. This recasts eq 7 into the following form: where ΔM w (D) = Ω · ΔP w (D), the volume Ω = A · l H . Here ΔM w (D) is the total dipole moment difference of interfacial groups (including the molecular and dissociative forms of adsorbed water) with respect to PZC. We can get . M w , M dis , N w , and N dis are the dipole moment and number of adsorbed water molecule and dissociated (OH − +H + ) groups (see Figure S5a). Note that in the model introduced here, there is only one free parameter ϵ H°i n eq 9. Instead, both l H and ΔM w (D) can be determined from finite-field DFTMD simulations. This allows us to obtain the differential capacitance rather than applying the one-shot estimator (eq 1).
Then, we applied the eq 9 to fit U − U PZC , that is, the potential drop crossing the Helmholtz layer with respect to that at the The fitting result of U(σ) − U PZC is given in Figure 3a, where the dashed line indicates the first constant term in eq 9. The free parameter ϵ H°t urns out to be 24, which is quite close to the commonly assumed values for rutile structures. 36 This agreement further justifies the theoretical model we formulated for the differential Helmholtz capacitance at oxide−electrolyte interfaces.
By taking the analytical derivative of U(σ) − U PZC , we obtain the differential capacitance C H of the Helmholtz layer as shown in Figure 3c. We find that C H shows a maximum of ∼151 μF/ cm 2 at the negatively surface charge density −10 μC/cm 2 , and it decreases to ∼100 and ∼60 μF/cm 2 when the surface charge density moves away to −40 and 40 μC/cm 2 , respectively. In addition, we find that C H at the negatively charged surface is about 50% higher than that at the positively charged surface for the same value of |σ|, and this finding of asymmetric distribution of the differential C H is in accord with what has been seen at electrified TiO 2 (110)/NaCl interfaces from finite-field DFTMD simulations 23 and in agreement with titration experiments of SnO 2 at higher ionic strength. 15,37 In fact, the asymmetric Helmholtz capacitance has been also seen in a range of metal oxides/hydroxides, such as ZnO, 38 TiO 2 , 16 α-Al 2 O 3 39 and γ-FeOOH. 40 This common feature observed in different oxides suggests that there should be a fundamental reason behind its cause.
When taking the derivative of eq 10 with respect to D and combining it with eq 7, this leads to the expression of the differential capacitance C H at the oxide surface ( Figure 4b) as Ä It is worth stressing that D PZC is not due to the proton charge as in D but an intrinsic property of metal oxide surfaces. Moreover, D PZC is positive, as in the same direction of the water dipole at the PZC (Figure 1c). This offset D PZC due to the specific orientation of adsorbed water at the PZC is the origin for the The final question is how the chemical specificity of surfaces comes into play. Clearly, the magnitude and the sign of D PZC is system-dependent. More importantly, in the case of SnO 2 (110), the interaction between dissociated OH − , H + and counterions in electrolyte is direct and strong, forming the inner sphere coordination. For example, at the positively charged interfaces, as shown in Figure 5a,e, the counterion Cl − is stabilized by two neighboring Sn 2 O br H + and two Sn 5c O w H − groups. Similarly, at the negatively charged interfaces, the counterion Na + is stabilized by two neighboring Sn 5c O w H − or two Sn 2 O br groups as shown in Figure 5b,f.
In contrast, the outer sphere coordination is preferred when restraining adsorbed water molecules not undergoing the dissociation (Figure 5c,d). Consequently, the average C H of SnO 2 with dissociatively adsorbed water molecules is about 109 μF/cm 2 , which is 45% larger than that of SnO 2 with water molecules restrained to the molecular adsorption ∼61 μF/cm 2 ( Figure S7c). Overall, they indicate that theoretical models connecting the macroscopic property and the microscopy information are essential for interpreting the differential capacitance C H of oxide−electrolyte interfaces.
In summary, through a combination of finite−finite DFTMD simulations and a theoretical model, the quantitative relationship between interfacial water orientation, proton transfer and differential capacitance at the charged water interface of SnO 2 (110) was established. It has been shown that the general phenomenon of asymmetric electric double layers seen in a range of metal oxides/hydroxides can be explained by the specific orientation of chemisorbed water molecules at the point of zero charge.
Detailed descriptions of computational setup, the degree of water dissociation α, the free energy of water dissociation ΔA diss , band edge alignment, double layer potentials and differential capacitance at electrified SnO 2 (110)/NaCl interfaces (PDF)