Modeling Polycrystalline Electrode-electrolyte Interfaces: The Differential Capacitance

The general continuum model relies on universal, i.e. material independent balance equations in the volume domains and on the surface. However, the model equations contain several material dependent constitutive quantities which need appropriate modeling. We summarize here the equilibrium equations that result from the much more complex modeling framework.¹⁹ The key ingredients for the material modeling are the free energy densities^d  ρψ, $\mathop{}\limits_{s}\rho \mathop{}\limits_{s}\psi$ in the bulk and on the volume, respectively. Due to the assumed constant dielectric susceptibility, the free energy densities are of the rather general structure

$$\begin{eqnarray}\begin{array}{rcl}\rho \psi & = & \rho \hat{\psi }(T,{\left({n}_{\alpha }\right)}_{\alpha \in {{ \mathcal I }}_{{\mathtt{K}}}})-\tfrac{1}{2}\chi {\varepsilon }_{0}| {\rm{\nabla }}\varphi {| }^{2},\\ \mathop{}\limits_{s}\rho \mathop{}\limits_{s}\psi & = & \mathop{}\limits_{s}\rho \hat{\mathop{}\limits_{s}\psi }(\mathop{}\limits_{s}T,{\left(\mathop{}\limits_{s}n{}_{\alpha }\right)}_{\alpha \in {{ \mathcal I }}_{{\rm{\Sigma }}}}).\end{array}\end{eqnarray} \tag{ 8 }$$

The chemical potentials are defined as

$$\begin{eqnarray}&&{\mu }_{\alpha }=\displaystyle \frac{\partial \rho \psi }{\partial {n}_{\alpha }},\quad \mathop{}\limits_{s}\mu {}_{\alpha }=\displaystyle \frac{\partial \mathop{}\limits_{s}\rho \mathop{}\limits_{s}\psi }{\partial \mathop{}\limits_{s}n{}_{\alpha }}.\end{eqnarray} \tag{ 9 }$$

By means of Gibbs-Duhem relations, we introduce the material pressure p and surface tension $\mathop{}\limits_{s}\gamma$ as

$$\begin{eqnarray}&&p=-\rho \hat{\psi }+\displaystyle \sum _{\alpha \in {{ \mathcal I }}_{{\mathtt{K}}}}{n}_{\alpha }{\mu }_{\alpha },\quad \mathop{}\limits_{s}\gamma =\mathop{}\limits_{s}\rho \hat{\mathop{}\limits_{s}\psi }-\displaystyle \sum _{\alpha \in {{ \mathcal I }}_{{\rm{\Sigma }}}}\mathop{}\limits_{s}n{}_{\alpha }\mathop{}\limits_{s}\mu {}_{\alpha }.\end{eqnarray} \tag{ 10 }$$

Equilibrium bulk equations. The equilibrium of an incompressible mixture in a bulk domain Ω is characterized by

$$\begin{eqnarray}&&-(1+\chi ){\varepsilon }_{0}{\rm{\Delta }}\varphi =q,\end{eqnarray} \tag{ 11a }$$

$$\begin{eqnarray}&&{\rm{\nabla }}p=-q{\rm{\nabla }}\varphi ,\end{eqnarray} \tag{ 11b }$$

$$\begin{eqnarray}&&0={\rm{\nabla }}({\mu }_{\alpha }+{z}_{\alpha }{e}_{0}\varphi )\quad \ \mathrm{for}\ \alpha \in {{ \mathcal I }}_{{\mathtt{K}}}.\end{eqnarray} \tag{ 11c }$$

The Poisson Eq. 11a determines the electric potential, 11b is the momentum equation, 11c states that the electrochemical potentials of all species are constant in equilibrium. We remark that these equations are not independent, but 11a and 11c together with 10(left) already imply 11b.

Equilibrium surface equations. On the surface Σ between the bulk domains ${{\rm{\Omega }}}^{{\mathtt{K}}}$, for ${\mathtt{K}}={\mathtt{M}},{\mathtt{E}}$, the equilibrium is given by

$$\begin{eqnarray}&&-[[(1+\chi ){\varepsilon }_{0}{\rm{\nabla }}\varphi {\boldsymbol{\nu }}]]=\mathop{}\limits_{s}q,\end{eqnarray} \tag{ 12a }$$

$$\begin{eqnarray}&&[[(p{\bf{1}}+(1+\chi ){\varepsilon }_{0}({\rm{\nabla }}\varphi \otimes {\rm{\nabla }}\varphi -\tfrac{1}{2}| {\rm{\nabla }}\varphi {| }^{2}{\bf{1}})){\boldsymbol{\nu }}]]=0,\end{eqnarray} \tag{ 12b }$$

$$\begin{eqnarray}&&\mathop{}\limits_{{s}}{\mu }{}_{{\alpha }}={{\mu }}_{{\alpha }}{| }_{{\rm{\Sigma }}}^{{\mathtt{K}}}\,\,{\rm{f}}{\rm{o}}{\rm{r}}\,{\alpha }\in {{ \mathcal I }}_{{\mathtt{K}}},{\mathtt{K}}={\mathtt{E}},{\mathtt{M}}.\end{eqnarray} \tag{ 12c }$$

Suitable free energy densities for the volume domains ${{\rm{\Omega }}}^{{\mathtt{E}}}$ of the electrolyte, ${{\rm{\Omega }}}^{{\mathtt{M}}}$ of the metal and for the metal-electrolyte surface Σ have been derived in Refs. 16–18 and are summarized here. The free energy densities account for the entropy of mixing and elastic effects due to the finite size of the molecules. For simplicity, we consider only the incompressible limit case. The rather simplistic assumption of constant surface chemical potential of electrons turns out to be just the appropriate way to capture a central feature of metals. It results in the necessary constraint that excess charge is stored on the metal surface.

Not included into the free energy density of the electrolyte are Debye-type electrostatic interactions between ions. As a consequence, the model cannot be expected to reproduce typical square-root behavior in the limit of strong dilution. For the determination of interface capacitance however, the most relevant regime is the opposite limit of concentrated solutions.

Bulk electrolyte material model in ${{\rm{\Omega }}}^{{\mathtt{E}}}$. The electrolyte in the domain ${{\rm{\Omega }}}^{{\mathtt{E}}}$ is a mixture of several species, one of them being the solvent that we refer to by the index α = 0. The index set of the electrolytic species is denoted by ${{ \mathcal I }}_{{\mathtt{E}}}$. The mole fractions y_α of the constituents are defined as

$$\begin{eqnarray}&&{y}_{\alpha }=\displaystyle \frac{{n}_{\alpha }}{n}\quad \mathrm{with}\quad n=\displaystyle \sum _{\beta \in {{ \mathcal I }}_{{\mathtt{E}}}}{n}_{\beta }.\end{eqnarray} \tag{ 13 }$$

In many solvents, ions are solvated, meaning that a number κ_α of solvent molecules are bound into a solvation shell around a center ion. Thus, the partial volume ${\upsilon }_{\alpha }^{{ref}}$ of a solvated ion in the electrolyte is typically much larger than the partial volume of the solvent ${\upsilon }_{0}^{{ref}}$. Consequently, we consider the electrolyte as an incompressible liquid mixture of free solvent molecules, undissociated species and solvated ions.^e The incompressibility of the mixture is characterized by the constraint

$$\begin{eqnarray}&&1=\displaystyle \sum _{\alpha \in {{ \mathcal I }}_{{\mathtt{E}}}}{\upsilon }_{\alpha }^{{\mathtt{E}}}{n}_{\alpha }.\end{eqnarray} \tag{ 14 }$$

The chemical potentials for an ideal mixture of solvated ions according to Ref. 17 are in the incompressible limit^f

$$\begin{eqnarray}&&{\mu }_{\alpha }={g}_{\alpha }^{{ref}}+{\upsilon }_{\alpha }^{{ref}}(p-{p}^{{\mathtt{E}}})+{k}_{B}T\mathrm{ln}({y}_{\alpha }),\quad \alpha \in {{ \mathcal I }}_{{\mathtt{E}}},\end{eqnarray} \tag{ 15 }$$

with reference Gibbs free energy ${g}_{\alpha }^{{ref}}={\psi }_{\alpha }^{{ref}}+{\upsilon }_{\alpha }^{{ref}}\,{p}^{{\mathtt{E}}}$, with a constant reference energy ${\psi }_{\alpha }^{{ref}}$.

Let ${y}_{\alpha }^{{\mathtt{E}}}$, ${\varphi }^{{\mathtt{E}}}$ and ${p}^{{\mathtt{E}}}$ denote the values of the mole fractions, the electric potential and the pressure at some bulk point ${x}^{{\mathtt{E}}}$ in ${{\rm{\Omega }}}^{{\mathtt{E}}}$. In equilibrium, the constant electrochemical potentials according to 11c and the material model 15 then imply for the bulk mole fractions

$$\begin{eqnarray}\begin{array}{rcl}\underline{\mathrm{equilibrium}:}\ \quad {y}_{\alpha } & = & {y}_{\alpha }^{{\mathtt{E}}}\cdot \exp \left(-\displaystyle \frac{{z}_{\alpha }{e}_{0}}{{k}_{B}T}(\varphi -{\varphi }^{{\mathtt{E}}})\right.\\ & & -\left.\displaystyle \frac{{\upsilon }_{\alpha }^{{ref}}}{{k}_{B}T}\left(p-{p}^{{\mathtt{E}}}\right)\right)\ \mathrm{in}\ {{\rm{\Omega }}}^{{\mathtt{E}}}.\end{array}\end{eqnarray} \tag{ 16 }$$

Note that the pressure p in the electrolytic boundary layer can be expressed as a function of the potential difference φ − ${\varphi }^{{\mathtt{E}}}$ by solving the condition 13 with the representations 16, i.e. the non-linear implicit relation

$$\begin{eqnarray}&&\displaystyle \sum _{\alpha \in {{ \mathcal I }}_{{\mathtt{E}}}}{y}_{\alpha }^{{\mathtt{E}}}\cdot \exp \left(-\displaystyle \frac{{z}_{\alpha }{e}_{0}}{{k}_{B}T}(\varphi -{\varphi }^{{\mathtt{E}}})-\displaystyle \frac{{\upsilon }_{\alpha }^{{ref}}}{{k}_{B}T}\left(p-{p}^{{\mathtt{E}}}\right)\right)-1=0.\end{eqnarray} \tag{ 17 }$$

yields $p=\hat{p}(\varphi -{\varphi }^{{\mathtt{E}}})$.

Bulk metal material model in ${{\rm{\Omega }}}^{{\mathtt{M}}}$. We consider a metal in the domain ${{\rm{\Omega }}}^{{\mathtt{M}}}$ as a binary mixture of positive metal ions M and free electrons e with z_e = −1, i.e. ${{ \mathcal I }}_{{\mathtt{M}}}=\{{\rm{M}},{\rm{e}}\}$. For the metal bulk free energy density, we adopt the Sommerfeld model²² and assume incompressibility, cf.¹⁷ We do not specify the model here, because as a consequence of the following surface material model, it turns out that the metal bulk does not influence any of the results in this paper and therefore can be ignored.

Surface material model on Σ. The surface Σ between the domains ${{\rm{\Omega }}}^{{\mathtt{E}}}$ and ${{\rm{\Omega }}}^{{\mathtt{M}}}$ is considered as mixture of the surface metal ions, surface electrons, electrolytic adsorbates and possibly reaction products of the aforementioned species. The index set of surface constituents is denoted by ${{ \mathcal I }}_{{\rm{\Sigma }}}$ with ${{ \mathcal I }}_{{\mathtt{E}}}\cup {{ \mathcal I }}_{{\mathtt{M}}}\subseteq {{ \mathcal I }}_{{\rm{\Sigma }}}$. Note that we consider on the surface also a solvation effect, whereby each adsorbed ion binds $\mathop{}\limits_{s}\kappa {}_{\alpha }$ solvent molecules which propagates into the partial molar area ${a}_{\alpha }^{{ref}}$.

We adopt the surface free energy model proposed in Ref. 17. In particular, the surface chemical potential of the electrons is assumed to be a constant value depending only on the material and the crystallographic surface orientation. We propose to choose this value related to the electron work function W_Σ of the surface Σ. Analogously to the metal volume, we have an incompressibility constraint on the surface stating ${a}_{M}^{{ref}}\mathop{}\limits_{s}n{}_{M}=1$, where ${a}_{M}^{{ref}}$ is the partial area of surface metal ions. On the electrolyte side, we have to account for adsorption from the volume. Since the surface is not necessarily completely covered with adsorbates, we introduce a number density of surface vacancies via

$$\begin{eqnarray}&&\mathop{}\limits_{s}n{}_{V}=\mathop{}\limits_{s}n{}_{M}-\displaystyle \sum _{\alpha \in {{ \mathcal I }}_{{\rm{\Sigma }}}\setminus {{ \mathcal I }}_{M}}\displaystyle \frac{{a}_{\alpha }^{{ref}}}{{a}_{M}^{{ref}}}\mathop{}\limits_{s}n{}_{\alpha }.\end{eqnarray} \tag{ 18 }$$

Then, we define the surface fractions of vacancies and adsorbates by

$$\begin{eqnarray}\begin{array}{rcl}\mathop{}\limits_{s}n & = & \mathop{}\limits_{s}n{}_{V}+\displaystyle \sum _{\alpha \in {{ \mathcal I }}_{{\rm{\Sigma }}}\setminus {{ \mathcal I }}_{M}}\mathop{}\limits_{s}n{}_{\alpha },\quad \mathop{}\limits_{s}y{}_{V}=\displaystyle \frac{\mathop{}\limits_{s}n{}_{V}}{\mathop{}\limits_{s}n}\quad \ \mathrm{and}\\ \mathop{}\limits_{s}y{}_{\alpha } & = & \displaystyle \frac{\mathop{}\limits_{s}n{}_{\alpha }}{\mathop{}\limits_{s}n}\quad \mathrm{for}\ \alpha \in {{ \mathcal I }}_{{\rm{\Sigma }}}\setminus {{ \mathcal I }}_{M}.\end{array}\end{eqnarray} \tag{ 19 }$$

The chemical potentials of the surface species are

$$\begin{eqnarray}&&\begin{array}{l}\mathop{}\limits_{s}\mu {}_{\alpha }=\mathop{}\limits_{s}\psi {}_{\alpha }^{{ref}}+{k}_{B}T\mathrm{ln}\left(\mathop{}\limits_{s}y{}_{\alpha }\right)-\tfrac{{a}_{\alpha }^{{ref}}}{{a}_{M}^{{ref}}}{k}_{B}T\mathrm{ln}\left(\mathop{}\limits_{s}y{}_{V}\right)\\ \quad \ \ \mathrm{for}\ \alpha \in {{ \mathcal I }}_{{\rm{\Sigma }}}\setminus {{ \mathcal I }}_{M},\end{array}\end{eqnarray} \tag{ 20a }$$

$$\begin{eqnarray}&&\mathop{}\limits_{s}\mu {}_{M}=\mathop{}\limits_{s}\psi {}_{M}^{{ref}}+{k}_{B}T\mathrm{ln}\left(\mathop{}\limits_{s}y{}_{V}\right)-{a}_{M}^{{ref}}\mathop{}\limits_{s}\gamma ,\end{eqnarray} \tag{ 20b }$$

$$\begin{eqnarray}&&\mathop{}\limits_{s}\mu {}_{e}=\mathop{}\limits_{s}\mu {}_{e}^{{ref}}\quad (=\mathrm{const}.).\end{eqnarray} \tag{ 20c }$$

With respect to the same bulk point in the electrolyte as above, i.e. the values ${y}_{\alpha }^{{\mathtt{E}}}$, ${\varphi }^{{\mathtt{E}}}$ and ${p}^{{\mathtt{E}}}$, the surface mole fraction of the vacancies, the electrolytic adsorbates, and the surface reaction products, have the representations in terms of the potential difference $U=\varphi {| }_{{\rm{\Sigma }}}-{\varphi }^{\infty }$,

$$\begin{eqnarray}&&\mathop{}\limits_{s}y{}_{V}=\exp \left(-\displaystyle \frac{{a}_{V}^{{ref}}}{{k}_{B}T}\mathop{}\limits_{s}\gamma \right).\end{eqnarray} \tag{ 21a }$$

$$\begin{eqnarray}&&\mathop{}\limits_{s}y{}_{\alpha }={y}_{\alpha }^{{\mathtt{E}}}\exp \left(-\displaystyle \frac{{\rm{\Delta }}{\widetilde{g}}_{\alpha }}{{k}_{B}T}-\displaystyle \frac{{e}_{0}}{{k}_{B}T}{z}_{\alpha }U-\displaystyle \frac{{a}_{\alpha }^{{ref}}}{{k}_{B}T}\mathop{}\limits_{s}\gamma \right)\ \mathrm{for}\ \alpha \in {{ \mathcal I }}_{{\mathtt{E}}},\end{eqnarray} \tag{ 21b }$$

$$\begin{eqnarray}&&\begin{array}{l}\mathop{}\limits_{s}y{}_{\beta }=\displaystyle \prod _{\alpha \in {{ \mathcal I }}_{E}}{\left({y}_{\alpha }^{{\mathtt{E}}}\right)}^{{\nu }_{\alpha \beta }}\exp \left(-\displaystyle \frac{{\rm{\Delta }}{\widetilde{g}}_{\beta }}{{k}_{B}T}-\displaystyle \frac{{e}_{0}}{{k}_{B}T}\left(\displaystyle \sum _{\delta \in {{ \mathcal I }}_{{\mathtt{E}}}}{\nu }_{\delta \beta }{z}_{\delta }\right)U-\displaystyle \frac{{a}_{\beta }^{{ref}}}{{k}_{B}T}\mathop{}\limits_{s}\gamma \right)\\ \quad \ \mathrm{for}\ \beta \in {{ \mathcal I }}_{{\rm{\Sigma }}}\setminus {{ \mathcal I }}_{E}\setminus {{ \mathcal I }}_{M},\end{array}\end{eqnarray} \tag{ 21c }$$

where the amount of adsorbates and reaction products on the surface is controlled by the corresponding Gibbs energies defined by

$$\begin{eqnarray}&&{\rm{\Delta }}{\widetilde{g}}_{\alpha }=\mathop{}\limits_{s}\psi {}_{\alpha }^{{ref}}-({\psi }_{\alpha }^{{\mathtt{E}}}+{\upsilon }_{\alpha }^{{ref}}{p}^{{\mathtt{E}}})\ \mathrm{for}\ \alpha \in {{ \mathcal I }}_{{\mathtt{E}}},\end{eqnarray} \tag{ 22a }$$

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{\widetilde{g}}_{\beta } & = & \mathop{}\limits_{s}\psi {}_{\beta }^{{ref}}-\displaystyle \sum _{\alpha \in {{ \mathcal I }}_{M}}{\nu }_{\alpha \beta }({\mu }_{\alpha }^{M}+{e}_{0}{z}_{\alpha }{U}^{M})\\ & & -\ \displaystyle \sum _{\alpha \in {{ \mathcal I }}_{M}}{\nu }_{\alpha \beta }({\psi }_{\alpha }^{{\mathtt{E}}}+{\upsilon }_{\alpha }^{{ref}}{p}^{{\mathtt{E}}})\ \mathrm{for}\ \alpha \in {{ \mathcal I }}_{{\mathtt{E}}}.\end{array}\end{eqnarray} \tag{ 22b }$$

The above representations and the definition 19 of the surface mole fractions yield an algebraic equation which determines the surface tension $\mathop{}\limits_{s}\gamma$ as a function of U.

Given the simple geometric setting and assuming the surface to be homogeneous in space, all quantities in the electrolyte can only depend on the spatial coordinate normal to Σ. Then, we define the boundary layer- and the surface charge density by

$$\begin{eqnarray}&&{Q}^{\mathrm{BL}}={\int }_{0}^{{z}^{{\mathtt{E}}}}q\,{\rm{d}}z\mathop{}\limits_{s}Q=-{e}_{0}\displaystyle \sum _{\alpha \in {{ \mathcal I }}_{{\rm{\Sigma }}}\setminus {{ \mathcal I }}_{M}}{z}_{\alpha }\mathop{}\limits_{s}n{}_{\alpha }.\end{eqnarray} \tag{ 24 }$$

In equilibrium, both quantities are completely determined by the potential difference $U=\mathop{}\limits_{s}\varphi -{\varphi }^{{\mathtt{E}}}.$ It is a remarkable feature of the applied material model in Section 3.2 that it is possible to determine the boundary layer charge density ${Q}^{\mathrm{BL}}$ as function of U^g without needing to spatially resolve the bulk Eqs. 11–12, i.e. 

$$\begin{eqnarray}&&{Q}^{\mathrm{BL}}=\mathrm{sgn}(U)\sqrt{2{\varepsilon }_{0}(1+\chi )(\hat{p}(U)-{p}^{{\mathtt{E}}})}=: {\hat{Q}}^{\mathrm{BL}}(U).\end{eqnarray} \tag{ 25 }$$

where $\hat{p}(U)$ is the material pressure determined from 17. Moreover, all surface mole fractions are determined according to 21 and we get for the surface charge density

$$\begin{eqnarray}&&\begin{array}{l}\hat{\mathop{}\limits_{s}Q}(U)\\ \ =\ -\displaystyle \frac{{\displaystyle \sum }_{(\alpha \in {{ \mathcal I }}_{{\mathtt{E}}})}{z}_{\alpha }{e}_{0}\mathop{}\limits_{s}y{}_{\alpha }+{\displaystyle \sum }_{(\alpha \in {{ \mathcal I }}_{{\mathtt{E}}})}{\displaystyle \sum }_{(\beta \in {{ \mathcal I }}_{{\rm{\Sigma }}}\setminus {{ \mathcal I }}_{E,M})}{\nu }_{\alpha \beta }{z}_{\alpha }{e}_{0}\mathop{}\limits_{s}y{}_{\beta }}{{a}_{V}^{{ref}}\mathop{}\limits_{s}y{}_{V}+{\displaystyle \sum }_{(\alpha \in {{ \mathcal I }}_{{\rm{\Sigma }}}\setminus {{ \mathcal I }}_{M})}{a}_{\alpha }^{{ref}}\mathop{}\limits_{s}y{}_{\alpha }}.\end{array}\end{eqnarray} \tag{ 26 }$$

Thus, we can introduce the differential capacity as

$$\begin{eqnarray}&&\hat{C}(U)=\displaystyle \frac{d}{{dU}}\left({Q}^{\mathrm{BL}}(U)+\mathop{}\limits_{s}Q(U)\right)={C}^{\mathrm{BL}}(U)+\mathop{}\limits_{s}C(U)\end{eqnarray} \tag{ 27 }$$

Under the standard assumption that the potential difference between an ideally non-polarizable reference electrode and the bulk electrolyte remains constant under potential variations, it is possible to show¹⁷ that

$$\begin{eqnarray}&&U=\mathop{}\limits_{s}\varphi -{\varphi }^{{\mathtt{E}}}=E+\displaystyle \frac{1}{{e}_{0}}\mathop{}\limits_{s}\mu {}_{e}-{E}^{{ref}}\quad \mathrm{with}\quad {E}^{{ref}}=\mathrm{const}.,\end{eqnarray} \tag{ 28 }$$

where E is the applied voltage in a three electrode setup, $\mathop{}\limits_{s}\mu {}_{e}$ the surface chemical potential of electrons, and ${E}^{{ref}}$ a constant which depends on the actual reference electrode. For a specific surface Σ, the differential capacity in the absolute scale E is given by an appropriate shift of 27, i.e. 

$$\begin{eqnarray}&&C(E)=\hat{C}\left(E+\tfrac{1}{{e}_{0}}\mathop{}\limits_{s}\mu {}_{e}-{E}^{{ref}}\right).\end{eqnarray} \tag{ 29 }$$

We consider now a planar surface Σ which is composed of N facets Σ_i, i.e. ${\rm{\Sigma }}={\cup }_{i=1}^{N}{{\rm{\Sigma }}}_{i}$ as sketched in Fig. 2. We assume finite size surfaces patterns that can be periodically extended to the whole plane. We denote by $| {{\rm{\Sigma }}}_{i}|$ the area of the facet Σ_i whereby the total area of Σ is $| {\rm{\Sigma }}| ={\sum }_{i=1}^{N}| {{\rm{\Sigma }}}_{i}|$ and surface fraction ${s}_{i}:= \tfrac{| {{\rm{\Sigma }}}_{i}| }{| {\rm{\Sigma }}| }$. We make the assumption

• The facet boundaries $\partial {{\rm{\Sigma }}}_{i}\cap \partial {{\rm{\Sigma }}}_{j}$ for $i\ne j$ constitute no independent thermodynamic entities.

Thus, each facet Σ_i can be treated as a finite size planar single crystal surface, and we can apply the material model of Section 3.2 on each facet Σ_i separately. Since the surface density of metal ions depends on the cystallographic orientation of the surface, the material parameter a_M is different for each individual facet Σ_i. As a consequence, in the absence of adsorption to the surface, the surface tension $\mathop{}\limits_{s}\gamma$ differs between the facets Σ_i.^h In addition, we have to assign to each facet Σ_i a constant value of $\mathop{}\limits_{s}\mu {}_{e}$, which has to be related to the electron work function on a surface of the respective cystallographic orientation. Because all facets are connected by electric conductors, the electrochemical potential of the electrons is constant over all grains in the metal ${{\rm{\Omega }}}_{{\mathtt{M}}}$, i.e. 

$$\begin{eqnarray}&&(\mathop{}\limits_{s}\mu {}_{e}-{e}_{0}\mathop{}\limits_{s}\varphi ){| }_{{{\rm{\Sigma }}}_{i}}=({\mu }_{e}-{e}_{0}\varphi ){| }_{{{\rm{\Omega }}}_{{\mathtt{M}}}}=(\mathop{}\limits_{s}\mu {}_{e}-{e}_{0}\mathop{}\limits_{s}\varphi ){| }_{{{\rm{\Sigma }}}_{j}}.\end{eqnarray} \tag{ 30 }$$

Between each two surfaces, we thus have the pairwise potential difference

$$\begin{eqnarray}&&\varphi {| }_{{{\rm{\Sigma }}}_{j}}-\varphi {| }_{{{\rm{\Sigma }}}_{i}}=\displaystyle \frac{1}{{e}_{0}}(\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{j}}-\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}}).\end{eqnarray} \tag{ 31 }$$

Patterned planar surface. As in the single crystal case before, we can expect boundary layers that decay exponentially from the surface Σ into the electrolyte bulk. The length scale of the boundary layer is given by the Debye length according to ²³ and the electrolyte bulk can be characterized by certain constant far field values ${y}_{\alpha }^{{\mathtt{E}}}$, ${\varphi }^{{\mathtt{E}}}$ and ${p}^{{\mathtt{E}}}$ in a distance from the surface that is large compared to $1[\mathrm{nm}$. We assume that the electrolyte in ${{\rm{\Omega }}}^{{\mathtt{E}}}$ is not influenced by any other boundary layer than the one at Σ. Given the applied potential E − ${E}^{{ref}}$ we set ${\varphi }^{{\mathtt{E}}}=0$, which corresponds to an ideally polarizable reference electrode. Then, the state in the electrolyte in ${{\rm{\Omega }}}^{{\mathtt{E}}}$ is given by the the generalized Poisson-Boltzmann boundary value problem similar to Refs. 16, 17, 19

$$\begin{eqnarray}&&-(1+\chi ){\varepsilon }_{0}{\rm{\Delta }}\varphi =q(\varphi ,p)\quad \ \mathrm{in}\ {{\rm{\Omega }}}^{{\mathtt{E}}},\end{eqnarray} \tag{ 32a }$$

$$\begin{eqnarray}&&1=\displaystyle \sum _{\alpha \in {{ \mathcal I }}_{{\mathtt{E}}}}{y}_{\alpha }(\varphi ,p)\quad \quad \ \mathrm{in}\ {{\rm{\Omega }}}^{{\mathtt{E}}},\end{eqnarray} \tag{ 32b }$$

$$\begin{eqnarray}&&\varphi {| }_{{{\rm{\Sigma }}}_{i}}=\tfrac{1}{{e}_{0}}(\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}}-(E-{E}^{{ref}}))\quad \ \mathrm{on}\ {\rm{\Sigma }},\end{eqnarray} \tag{ 32c }$$

$$\begin{eqnarray}&&{\rm{\nabla }}\varphi \to 0\quad \quad \ \mathrm{far}\ \mathrm{away}\ \mathrm{from}\ {\rm{\Sigma }},\end{eqnarray} \tag{ 32d }$$

where y_α(φ, p) is given by 16 for $\alpha \in {{ \mathcal I }}_{{\mathtt{E}}}$.

Given a solution of 32 in ${{\rm{\Omega }}}^{{\mathtt{E}}}$, the boundary layer charge q is determined by y_α according to 7 and 14. The surface charge density can be calculated by 26 on each facet Σ_i separately and we get

$$\begin{eqnarray}&&\mathop{}\limits_{s}Q{| }_{{{\rm{\Sigma }}}_{i}}=\mathop{}\limits_{s}\hat{Q}\left(E-{E}^{{ref}}+\tfrac{1}{{e}_{0}}\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}}\right)\end{eqnarray} \tag{ 33 }$$

Thus for the surface Σ, we conclude

$$\begin{eqnarray}&&\mathop{}\limits_{s}Q{}_{\mathrm{poly}}(E)=\displaystyle \sum _{i=1}^{N}{s}_{i}\mathop{}\limits_{s}\hat{Q}\left(E-{E}^{{ref}}+\tfrac{1}{{e}_{0}}\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}}\right).\end{eqnarray} \tag{ 34 }$$

Upon solution of the boundary value problem, the boundary layer charge density is computed from

$$\begin{eqnarray}&&{Q}_{\mathrm{poly}}^{\mathrm{BL}}(E)=-\displaystyle \frac{1}{| {\rm{\Sigma }}| }{\int }_{{{\rm{\Omega }}}^{{\mathtt{E}}}}q(x)\,{dx},\end{eqnarray} \tag{ 35 }$$

Finally, the total surface charge is ${Q}_{\mathrm{poly}}(E)={Q}_{\mathrm{poly}}^{\mathrm{BL}}(E)+\mathop{}\limits_{s}Q{}_{\mathrm{poly}}(E)$ and the differential capacity of the patterned surface is computed by taking the derivative with respect to the applied potential E.

Ensemble of planar single crystal surfaces. The total boundary layer charge density can be decomposed into the contributions adjacent to the different facets Σ_i, viz. 

$$\begin{eqnarray}&&{Q}^{\mathrm{BL}}=\displaystyle \sum _{i=1}^{N}{s}_{i}{Q}_{{{\rm{\Sigma }}}_{i}}^{\mathrm{BL}}\quad \ \mathrm{with}\ \quad {Q}_{{{\rm{\Sigma }}}_{i}}^{\mathrm{BL}}=-\displaystyle \frac{1}{| {{\rm{\Sigma }}}_{i}| }{\int }_{{{\rm{\Sigma }}}_{i}\times [0,\infty ]}q(z)\,{dz}.\end{eqnarray} \tag{ 36 }$$

Since boundary effects propagate into the interior with a length scale in the order of the chosen reference length of the electrolyte, boundary effects due to the finite size of Σ_i can be neglected once the characteristic facet diameter is sufficiently large, i.e. $1[\mathrm{nm}]\ll {d}^{\mathrm{facet}}$. Then, the charge density in front of Σ_i is well approximated by the boundary layer charge 25 at a planar single crystal with the same material parameters as Σ_i, i.e. 

$$\begin{eqnarray}&&{Q}_{{{\rm{\Sigma }}}_{i}}^{\mathrm{BL}}-{\hat{Q}}^{\mathrm{BL}}\left(E-{E}^{{ref}}+\tfrac{1}{{e}_{0}}\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}}\right)\to 0\quad \ \mathrm{for}\ \displaystyle \frac{1[\mathrm{nm}]}{{d}^{\mathrm{facet}}}\to 0.\end{eqnarray} \tag{ 37 }$$

Thus, in the limit of large facets, the boundary layer charge density ${Q}_{\mathrm{poly}}^{\mathrm{BL}}$ of the polycrystal is given by the according density of an ensemble of non-interacting (infinitely large) planar single crystal surfaces, all in contact with the same electrolyte, viz.

$$\begin{eqnarray}&&{Q}_{\mathrm{poly}}^{\mathrm{BL}}(E)=\displaystyle \sum _{i=1}^{N}{s}_{i}\ {\hat{Q}}^{\mathrm{BL}}\left(E-{E}^{{ref}}+\tfrac{1}{{e}_{0}}\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}}\right),\end{eqnarray} \tag{ 38 }$$

The boundary layer capacity related to 38 is given by 3. As 33 can be applied independent of the size of ${d}^{\mathrm{facet}}$, we conclude

$$\begin{eqnarray}&&{C}_{\mathrm{poly}}(E)=\displaystyle \sum _{i=1}^{N}{s}_{i}\ \hat{C}\left(E-{E}^{{ref}}+\tfrac{1}{{e}_{0}}\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}}\right).\end{eqnarray} \tag{ 39 }$$

The potential of zero charge E^PZC is determined from Eq. 37, i.e. ${Q}_{\mathrm{poly}}^{\mathrm{BL}}(E)\mathop{=}\limits^{!}0$. Intuitively one would expect that the potential of zero charge E^PZC is determined as $\tfrac{1}{{e}_{0}}{\sum }_{i=1}^{N}{s}_{i}\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}}+{E}^{{ref}}$. But since the functions ${\hat{Q}}^{\mathrm{BL}}$ are non-linear with respect to their argument, this does not hold, i.e. in general

$$\begin{eqnarray}&&{E}^{\mathrm{PZC}}-{E}^{{ref}}\ne \tfrac{1}{{e}_{0}}\displaystyle \sum _{i=1}^{N}{s}_{i}\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}}=: \mathop{}\limits_{s}\bar{\mu }{}_{e}\end{eqnarray} \tag{ 40 }$$

but

$$\begin{eqnarray}&&{Q}_{\mathrm{poly}}^{\mathrm{BL}}(E)\left|{}_{E={E}^{\mathrm{PZC}}}\right.=0.\end{eqnarray} \tag{ 41 }$$

In Section 6 we provide some exemplary calculations showing the difference between ${E}^{\mathrm{PZC}}-{E}^{{ref}}$ and $\mathop{}\limits_{s}\bar{\mu }{}_{e}$ for several parameter variations.

We consider first the most simple configuration of a symmetric bi-crystalline surface, where N = 2, ${s}_{1}={s}_{2}=1/2$. We choose an average potential $\mathop{}\limits_{s}\bar{\mu }{}_{e}$ such that $\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{1}}-\mathop{}\limits_{s}\bar{\mu }{}_{e}=-(\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{2}}-\mathop{}\limits_{s}\bar{\mu }{}_{e})=-0.1\,\mathrm{eV}$. On the surface, we thus have the boundary values $\mathop{}\limits_{s}\varphi {| }_{{{\rm{\Sigma }}}_{i}}={{\rm{\Phi }}}_{i}$ for i = 1, 2 with ${{\rm{\Phi }}}_{1}=-{{\rm{\Phi }}}_{2}=-0.1{\rm{V}}$. The simplest realization of this configuration consists of a pattern of parallel stripes with alternating prescribed potential. Here, we let ${d}^{\mathrm{facet}}$ denote the width of the stripes. With an appropriate choice of the coordinate system, the boundary values can be described by a 1D-function such that

$$\begin{eqnarray}\mathop{}\limits_{s}\varphi (x)=\left\{\begin{array}{ll}{{\rm{\Phi }}}_{1} & \ \mathrm{for}\ 2\,| x| \lt {d}^{\mathrm{facet}},\\ {{\rm{\Phi }}}_{2} & \ \mathrm{for}\ 2\,| x-{d}^{\mathrm{facet}}| \lt {d}^{\mathrm{facet}},\end{array}\right.\end{eqnarray} \tag{ 42 }$$

and periodic continuation. Accordingly, the potential and the charge density in the electrolyte domain z > 0 can be determined from a 2D-FEM computation by integration of the free charge q in space according to 35. The choice of the boundary values implies that in the far field the potential approaches $\varphi \to {\varphi }^{{\mathtt{E}}}=-(E-{E}^{{ref}})-\tfrac{1}{{e}_{0}}\mathop{}\limits_{s}\bar{\mu }{}_{e}$. Plots of the electric potential for $E={E}^{{ref}}-\tfrac{1}{{e}_{0}}\mathop{}\limits_{s}\bar{\mu }{}_{e}$ and an electrolyte with a bulk concentration of $0.1\,\mathrm{mol}\,{{\rm{l}}}^{-1}$ and the remaining parameters according to Table I are given in Fig. 6 (left). In the upper plot, where the facet size was chosen as ${d}^{\mathrm{facet}}=10{L}^{\mathrm{Debye}}\approx 12.28\,\mathrm{nm}$, one can observe that the piece-wise constant boundary data to a large extend propagates into the electrolyte and decays with increasing distance to the boundary. To the contrary, in the lower plot, where ${d}^{\mathrm{facet}}=\tfrac{5}{2}{L}^{\mathrm{Debye}}\approx 3.07\,\mathrm{nm}$, the profile of the potential is dominated by facet boundary effects such that the regions of parallel iso-lines almost disappear.

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Next, the applied potential is varied in positive and in negative direction and the boundary value problem 32 is solved. From this solution, the free charge density q and ${Q}^{\mathrm{BL}}(E)$ are computed. Figure 6 (right) shows the resulting capacity for different electrolyte concentrations in dependence of the facet size ${d}^{\mathrm{facet}}$. We observe for increasing facet size ${d}^{\mathrm{facet}}$ a convergence of the computed capacity curves to the algebraic solution 3. In particular, in the potential range outside of the lowest and the largest position of the maxima, the algebraic expression 3 is always a good characterization of the polycrystalline surface, independent of the facet size. Figure 6 also shows that the convergence depends on the electrolyte bulk concentration. We thus repeat the numerical experiment over a larger range of parameter variation, such that the bulk concentration is varied between ${10}^{-4}\,\mathrm{mol}\,{{\rm{l}}}^{-1}$ and $1\,\mathrm{mol}\,{{\rm{l}}}^{-1}$ and the facet diameter is changed between $0.2\,\mathrm{nm}$ and $120\,\mathrm{nm}$. We observe in Fig. 7 that while the behavior with respect to the bulk concentration is not monotonous, in any case the approximation error of 3 in the maximum norm can be bounded by some term proportional to the inverse of ${d}^{\mathrm{facet}}$.

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In addition, 3D-FEM simulations with a checkerboard pattern of the surface were performed where the boundary values are given by periodic continuation of

$$\begin{eqnarray}\mathop{}\limits_{s}\varphi (x,y)=\left\{\begin{array}{ll}{{\rm{\Phi }}}_{1} & \ \mathrm{for}\ \max (| x| ,| y| )\lt {d}^{\mathrm{facet}},\quad \mathrm{sgn}(x\,y)\gt 0,\\ {{\rm{\Phi }}}_{2} & \ \mathrm{for}\ \max (| x| ,| y| )\lt {d}^{\mathrm{facet}},\quad \mathrm{sgn}(x\,y)\lt 0.\end{array}\right.\end{eqnarray} \tag{ 43 }$$

The observations from the 3D computations are analogous as for the 2D case. Since for given ${d}^{\mathrm{facet}}$, the length of the contact lines in the checkerboard pattern is double compared to the striped pattern represented by the 2D case, convergence for increasing facet size is a bit slower here. Together, the 2D and the 3D results indicate that the shape of the facets is not relevant as long as the facet diameter ${d}^{\mathrm{facet}}$ is sufficiently large compared to a reference length in the order of $1[\mathrm{nm}]$. We conclude that already the covered surface fraction of the facets and the corresponding single crystal capacities fully determine the capacity of a polycrystalline surface of sufficiently large facet size.

As FEM simulations in 3D are computationally expensive, we used anisotropic grids with a mesh grading with respect to the a-priori known structure of the potential, see Fig. 8. To check that the grid is sufficiently fine, we compared the resulting capacity curves from numerical solutions of the boundary value problem 32 on different refinement levels of the initial mesh. Refined meshes on the refinement levels 3, 4 and 5 consisted of 2 601, 18 513 and 139 425 nodes, respectively. Figure 8 shows the convergence of the computed capacity curves when refining the mesh.

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In the previous section, we did not discuss the PZC for the patterned polycrystalline surface. The reason is that due to the symmetry, i.e. ${s}_{1}={s}_{2}$, it always coincides with the chosen reference potential. In order to introduce some asymmetry, we slightly modify the configuration to N = 2, ${s}_{1}=1/3$, ${s}_{2}=2/3$, and again set ${{\rm{\Phi }}}_{1}=-{{\rm{\Phi }}}_{2}=-0.1{\rm{V}}$. For a 2D-FEM simulation of this configuration, we choose

$$\begin{eqnarray}\mathop{}\limits_{s}\varphi (x)=\left\{\begin{array}{ll}{{\rm{\Phi }}}_{1} & \ \mathrm{for}\ 2| x| \lt {d}^{\mathrm{facet}},\\ {{\rm{\Phi }}}_{2} & \ \mathrm{for}\ | x-\displaystyle \frac{3}{2}\,{d}^{\mathrm{facet}}| \lt {d}^{\mathrm{facet}},\end{array}\right.\end{eqnarray} \tag{ 44 }$$

with periodic continuation of the boundary data. As in the previous example, we again observe that the capacity of the patterned surface approaches 3 for large facet sizes ${d}^{\mathrm{facet}}$, see Fig. 9 (left). From the computed boundary charge as a function of the applied potential according to 35, we can now determine the potential of zero charge of the patterned surface with finite ${d}^{\mathrm{facet}}$. We observe that for a fixed facet size ${d}^{\mathrm{facet}}$, the PZC is not constant but increases with the bulk electrolyte solution, cf. Fig. 9 (right). Moreover, the potential range, in which the PZC varies with respect to the bulk concentration, gets wider for increasing ${d}^{\mathrm{facet}}$. As a limit for ${d}^{\mathrm{facet}}\to \infty$, the PZC reaches the values we get from the surface fraction weighted averaging of single crystals according to 3. Since the capacity of a planar single crystal is approximately quadratic at its PZC, the boundary layer charge behaves almost cubic. Thus, one should not expect, that the linear combination of single crystal capacities would attain its PZC at the respective linear combination of the individual PZCs.

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In addition, another limit is of interest, however more from a theoretical point of view. Let us consider ${d}^{\mathrm{facet}}\to 0$, although a pattern at the atomic length scale or even blow that scale cannot be realized in practice. We observe, that for ${d}^{\mathrm{facet}}\to 0$, the capacity curve changes from the three-maxima shape according to 3 into the typical two-maxima or camel shape that is well known from the planar single crystal. In fact, for ${d}^{\mathrm{facet}}\to 0$, the capacity converges to the capacity of a planar single crystal surface where the PZC is determined as the sum of the PZCs on the contributing facets weighted by their respective surface fraction ${s}_{i}$ according to 5. Moreover, this limit of the PZC is independent of the bulk electrolyte concentration.

Because of surface roughness, the actual surface area of real surfaces is larger compared to their ideally plain counterparts which are characterized by their so called visible surface area. As a measure of this effect, the surface roughness factor W is introduced as the quotient of the real surface area over the visible. Surface roughness appears on a large range of different length scales, from "physical inhomogeneity" on the atomic length scale to "geometric roughness" on a macroscopic scale of several $\mu {\rm{m}}$. A lot of attention was devoted to the analysis of "geometric" surface roughness which in experiments may be caused e.g. by electrode pre-treatment like polishing. In the context of electrochemical impedance analysis, roughness is then often analyzed in the framework of fractal geometry, where on the larger scales, an interaction of the boundary layer with bulk transport properties of electrolyte has to be expected.

We want to concentrate on the effect of surface roughness on the boundary layer capacitance and therefore consider only deformations of an ideally plain surface that are on a length scale of the facet size ${d}^{\mathrm{facet}}$. We return to the setting of the symmetric bi-crystal of Section 5.1 and let the surface be given by

$$\begin{eqnarray}\tfrac{{d}^{\mathrm{facet}}}{{h}^{\max }}\ h(x)=\left\{\begin{array}{ll}x & \ \mathrm{for}\ 2| x| \lt {d}^{\mathrm{facet}},\\ ({d}^{\mathrm{facet}}-x) & \ \mathrm{for}\ 2| x-{d}^{\mathrm{facet}}| \lt {d}^{\mathrm{facet}},\end{array}\right.\end{eqnarray} \tag{ 45 }$$

where the parameter ${h}^{\max }$ is the maximal elevation of the surface over the supporting pane, see Fig. 10. This results in a surface roughness factor of

$$\begin{eqnarray}&&W=\displaystyle \frac{\sqrt{{\left({d}^{\mathrm{facet}}\right)}^{2}+{\left({h}^{\max }\right)}^{2}}}{{d}^{\mathrm{facet}}}.\end{eqnarray} \tag{ 46 }$$

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We performed 2d FEM simulations of the boundary layer in dependence of the applied potential and the electrolyte concentration varying the surface roughness parameter relative to the Debye length as ${h}^{\max }/{L}^{\mathrm{Debye}}=\tfrac{5}{8}$, $\tfrac{5}{4}$, $\tfrac{5}{2}$. The respective surface roughness factors according to 46 are $W=1.002$, 1.0078, 1.031. We observe that with increasing ${h}^{\max }$, the capacity curves deviate stronger from the calculated capacity of the perfectly smooth surface, see Fig. 11 left. Moreover, Fig. 11 right indicates that surface roughness can be very accurately taken into account by multiplying the surface roughness factor W to the capacity of the perfectly plain surface, i.e.

$$\begin{eqnarray}&&C(E)=W({h}^{\max })\,{C}_{\mathrm{poly}}^{\mathrm{BL}}(E).\end{eqnarray} \tag{ 47 }$$

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In the previous section, we kept the potential difference between the two considered facets constant (viz. $0.2\,{\rm{V}}$), However, this difference is a material dependent characteristic quantity that varies between different materials or different crystallographic orientations. While differences of the PZC, respective $\mathop{}\limits_{s}\mu {}_{e}$, between differently oriented surfaces are considered minor for some metals, e.g. "low melting metals" like $\mathrm{Cd}$, Sn, Pb, Zn and Bi, cf.,^5,10 considerable differences between the low index facets are reported for e.g. Ag, Au, Cu, Pt and W, cf.²⁴ Thus, we examine the situation of an bi-crystal with equal surface fractions ${s}_{1}={s}_{2}=1/2$ and vary the difference of the surface chemical potential of the electrons ${\rm{\Delta }}\mathop{}\limits_{s}\mu {}_{e}=\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{1}}-\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{2}}$ between $0\,\mathrm{eV}$ and $0.8\,\mathrm{eV}$. This corresponds to varying the boundary values such that $0{\rm{V}}\leqslant {{\rm{\Phi }}}_{1}-{{\rm{\Phi }}}_{2}\leqslant 0.8{\rm{V}}$. We observe, that for ${{\rm{\Phi }}}_{1}-{{\rm{\Phi }}}_{2}\lt 0.115{\rm{V}}$ the capacity retains the typical camel shape. For $0.115{\rm{V}}\lt {{\rm{\Phi }}}_{1}-{{\rm{\Phi }}}_{2}\lt 0.375{\rm{V}}$ there is a three maximum shape, whereas for larger potential differences between the facets, we have a four-maximum configuration that again allows to identify the two local extrema of the individual facets, see Fig. 12. Once more, we stress that in experiments one should be aware of the fact that for a polycrystalline surface the PZC in general is not found at a local capacity minimum! For a sufficiently large potential difference Φ₁ − Φ₂, the PZC is in general found between the inner two local maxima of the capacity, and only for specific configurations of the surface fractions ${s}_{i}$ the PZC is located at a local capacity minimum. Even more, for moderate ${{\rm{\Phi }}}_{1}-{{\rm{\Phi }}}_{2}$, the PZC is located near a local capacity maximum!

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As stated in Section 4.2, the potential of zero charge

$$\begin{eqnarray}&&{E}^{\mathrm{PZC}}-{E}^{{ref}}\ne \mathop{}\limits_{s}\bar{\mu }{}_{e}\end{eqnarray} \tag{ 48 }$$

but, it has to be determined from a solution of the non-linear equation ${Q}_{\mathrm{poly}}^{\mathrm{BL}}(E)=0$. This solution is dependent on the salt concentration c, the the surface fractions s_i, and the $\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}}$. For a bi-crystal, Fig. 13 displays the difference $\mathop{}\limits_{s}\bar{\mu }{}_{e}-({E}^{\mathrm{PZC}}-{E}^{{ref}})$.

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To study the effect of adsorption, we return to the simple configuration of a symmetric bi-crystal in Section 5.1, i.e. we choose N = 2, ${s}_{1}={s}_{2}=1/2$. We assume that the adsorption energies Δ g_α do not differ between the different facets. In particular, by the choice of

$$\begin{eqnarray}&&{\rm{\Delta }}{g}_{A}=-0.25\,\mathrm{eV},\quad {\rm{\Delta }}{g}_{C}=+1\,\mathrm{eV},\quad {\rm{\Delta }}{g}_{S}=-0.0735\,\mathrm{eV}.\end{eqnarray} \tag{ 49 }$$

we assume, that cations effectively do not adsorb while anions and the solvent are allowed to adsorb to the surface. For simplicity and clarity of presentation, we do not consider surface reactions like solvation shell stripping or electron transfer reactions. An extension is straightforward by applying 21c for the reaction products and 26 for the surface charge. In the numerical computation, we use for the radius and the partial area of the adsorbed species

$$\begin{eqnarray}&&{r}_{\alpha }^{{ref}}={\left(\tfrac{3}{4\pi }{\upsilon }_{\alpha }^{{ref}}\right)}^{\displaystyle \frac{1}{3}},\quad {a}_{\alpha }^{{ref}}=\tfrac{8}{\sqrt{3}}\pi {\left({r}_{\alpha }^{{ref}}\right)}^{2}.\end{eqnarray} \tag{ 50 }$$

According to 34, the surface capacity of a patterned surface only depends on the material parameters of the contributing facets and their surface fractions ${s}_{i}$, but not on the facet geometry or the facet size parameter ${d}^{\mathrm{facet}}$. Thus, we can combine 3 and 34 to conclude that for large facet size, the capacity of a perfectly plain polycrystal surface is

$$\begin{eqnarray}&&{C}_{\mathrm{poly}}=\displaystyle \sum _{i=1}^{N}{s}_{i}\,\hat{C}\left(E-{E}^{{ref}}+\tfrac{1}{{e}_{0}}\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}}\right).\end{eqnarray} \tag{ 51 }$$

Figure 14 shows the capacity curves of a single crystal surface and illustrates the construction for a bi-crystal according to 51. As already known for single crystal surfaces, if there is adsorption from the electrolyte, the PZC depends on the electrolyte and its bulk concentration, cf. e.g.¹⁷ The same also holds true for the polycrystal and we conclude from 3 and 34 thatⁱ for $1[\mathrm{nm}]\lt {d}^{\mathrm{facet}}\to \infty$, the determination of ${E}_{\mathrm{poly}}^{\mathrm{PZC}}$ requires the solution of the non-linear algebraic equation

$$\begin{eqnarray}&&0=\displaystyle \sum _{i=1}^{N}{s}_{i}\ \hat{Q}\left({E}_{\mathrm{poly}}^{\mathrm{PZC}}-{E}^{{ref}}+\tfrac{1}{{e}_{0}}\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}})\right).\end{eqnarray} \tag{ 52 }$$

The above results can easily be extended to more complex polycrystalline surfaces with many facets of different type. We consider the case of N = 4 different types of facets Σ₁, Σ₂, Σ₃, Σ₄ with the respective surface fractions ${s}_{1}=4/9$, ${s}_{2}={s}_{3}=2/9$, ${s}_{4}=1/9$. Also, we want to have junction points where more than two different facets meet. One possible simple realization on a 2D surface is the periodic symmetric continuation of the pattern sketched in Fig. 15. The boundary data is given as $\mathop{}\limits_{s}\varphi ={{\rm{\Phi }}}_{i}$ on Σ_i, for i = 1, 2, 3, 4. We choose ${{\rm{\Phi }}}_{1}=-{{\rm{\Phi }}}_{2}=-0.1\,{\rm{V}}$. First, we let ${{\rm{\Phi }}}_{4}={{\rm{\Phi }}}_{1}$ and Φ₃ = Φ₂ and get slightly non-symmetric capacity curves^j , see Fig. 15. Next, we change Σ₃ such that ${{\rm{\Phi }}}_{3}=-.061\,{\rm{V}}$ while still Φ₄ = Φ₁. Finally, set ${{\rm{\Phi }}}_{4}=.085\,{\rm{V}}$ and keep the remaining boundary values as before, see Fig. 15 bottom line.

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As in Section 5.1, we checked convergence of the capacity calculated from 3D-FEM simulation for finite facet size ${d}^{\mathrm{facet}}$ to the limit Eq. 3 for ${d}^{\mathrm{facet}}\to \infty$ and grid convergence of the numerical solutions.

A precise measurement of all surface facets and their corresponding work functions is extremely expensive, if possible, and even measurements on very similar facets may contain some scatter. Hence, the input values for the mathematical model of the foregoing section can only be determined to a certain precision, which has to be taken into account when realistic polycrystalline surfaces are to be described. We sketch therefore a stochastic description of the polycrystalline surface based on the derived deterministic model and provide some numerical examples of the double layer capacity.

Consider a surface Σ of N > 1 facets Σ_i. We define the set ${{ \mathcal M }}_{{\rm{\Sigma }}}=\{\mathop{}\limits_{s}\mu {}_{e}^{1},\ldots ,\mathop{}\limits_{s}\mu {}_{e}^{J}\}$ of all different values of $\mathop{}\limits_{s}\mu {}_{e}$ on Σ. Moreover we introduce the sets ${{ \mathcal I }}_{\mu }=\{1\leqslant i\leqslant N:\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}}=\mu \}$. Then we denote with Σ_μ all facets which have the value μ for $\mathop{}\limits_{s}\mu {}_{e}$ and the according surface fraction with ${s}_{\mu }$, i.e. 

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mu }={\cup }_{i\in {{ \mathcal I }}_{\mu }}\,{{\rm{\Sigma }}}_{i},\quad {s}_{\mu }=\displaystyle \sum _{i\in {{ \mathcal I }}_{\mu }}\,{s}_{i}.\end{eqnarray} \tag{ 53 }$$

Doing so, we switch the facet labeling from a consecutive numbering of all facets to values of $\mathop{}\limits_{s}\mu {}_{e}$. Since ${\sum }_{\mu \in {{ \mathcal M }}_{{\rm{\Sigma }}}}{s}_{\mu }=1$, we can introduce the discrete probability density

$$\begin{eqnarray}&&f(\omega ):= \displaystyle \sum _{\mu \in {{ \mathcal M }}_{{\rm{\Sigma }}}}{s}_{\mu }\delta (\omega -\mu ),\end{eqnarray} \tag{ 54 }$$

where δ is the Dirac-distribution. The charge ${Q}_{\mathrm{poly}}^{\mathrm{BL}}$ stored in the polycrystalline boundary layer thus rewrites as

$$\begin{eqnarray}\begin{array}{rcl}{Q}_{\mathrm{poly}}^{\mathrm{BL}} & = & \displaystyle \sum _{i=1}^{N}{s}_{i}{\hat{Q}}^{\mathrm{BL}}\left(E-{E}^{{ref}}+\tfrac{1}{{e}_{0}}\mathop{}\limits_{s}\mu {}_{e}{| }_{{{\rm{\Sigma }}}_{i}}\right)\\ & = & \ \displaystyle \sum _{\mu \in {{ \mathcal M }}_{{\rm{\Sigma }}}}{s}_{\mu }{\displaystyle \int }_{-\infty }^{\infty }\delta (\omega -\mu ){\hat{Q}}^{\mathrm{BL}}\left(E-{E}^{{ref}}+\tfrac{1}{{e}_{0}}\omega \right)d\omega \\ & = & \ {\displaystyle \int }_{-\infty }^{\infty }f(\omega ){\hat{Q}}^{\mathrm{BL}}\left(E-{E}^{{ref}}+\tfrac{1}{{e}_{0}}\omega \right)d\omega \\ & = & \ (f\,\ast \,{\hat{Q}}^{\mathrm{BL}})(E-{E}^{{ref}}).\end{array}\end{eqnarray} \tag{ 55 }$$

With the convolution operator $*$ we obtain thus a very compact expression for ${Q}_{\mathrm{poly}}^{\mathrm{BL}}$ in terms of the distribution function f(ω), which describes the facet density parametrized over the surface potential $\mathop{}\limits_{s}\mu {}_{e}$. The potential of zero charge ${E}_{\mathrm{poly}}^{\mathrm{PZC}}$ is thus (implicitly) defined via $(f\,\ast \,{\hat{Q}}^{\mathrm{BL}})(E-{E}^{{ref}})\mathop{=}\limits^{!}0$ and the double layer capacity is

$$\begin{eqnarray}\begin{array}{rcl}{C}_{\mathrm{poly}}^{\mathrm{BL}} & = & \displaystyle \sum _{i=1}^{N}{s}_{i}{\hat{C}}^{\mathrm{BL}}\left(E-{E}^{{ref}}+\tfrac{1}{{e}_{0}}\mathop{}\limits_{s}\mu {}_{e}^{i}\right)\\ & = & \ {\displaystyle \int }_{-\infty }^{\infty }f(\omega ){\hat{C}}^{\mathrm{BL}}\left(E-{E}^{{ref}}+\tfrac{1}{{e}_{0}}\omega \right)d\omega \\ & = & \ (f\,\ast \,{\hat{C}}^{\mathrm{BL}})(E-{E}^{{ref}}).\end{array}\end{eqnarray} \tag{ 56 }$$

Note that ${\hat{Q}}^{\mathrm{BL}}$ and ${\hat{C}}^{\mathrm{BL}}$ remain the non-linear functions described in Section 4.

For realistic polycrystalline electrodes, the Dirac-distribution might smear, whereby f becomes a continuous function. In order to show the consequences of the smearing of the convolution operator * on the non-linear functions of ${C}^{\mathrm{BL}}$ we consider the following example. Let the density f be given by a normal distribution of facets with respect to its mean value $(\mathop{}\limits_{s}\mu {}_{e}-\mathop{}\limits_{s}\bar{\mu }{}_{e})=0$ and with the standard deviation σ, see Fig. 4. For small standard deviation like σ = 0.025, the resulting capacity according to 4 is close to the single crystal capacity, see Fig. 4. However, for σ = 0.2, the local capacity minimum has disappeared and there is only a single capacity maximum. Due to the symmetry of the chosen density f(ω), the PZC is in either case located at the maximum of the density f. We want to stress, that this particular configuration of facets with a broad scatter of the individual facet properties is very different from the limit related to 5 where on each finite size part of the surface a large number of accordingly small facets are considered.

Next, we want to investigate the effect of a polycrystal surface that is mainly covered by facets near to low index surfaces like (100), (110) and (111). For the low index facets of Ag we chose values of $\mathop{}\limits_{s}\mu {}_{e}$ related to the (negative) work function as given in Table II. To describe a non-ideal Ag polycrystal, we consider a probability density that consists of a superposition of scaled normal distributions around the potentials of the low index facets and a standard deviation small enough, such that the peaks do not overlap, see Fig. 16 where σ = 0.025 is chosen. The resulting capacity curve remains close to the capacity of an ideal polycrystal with equal surface fractions ${s}_{i}=\tfrac{1}{3}$ for i = 1, 2, 3. Moreover, we perturb the configuration such that an equi-distribution in a $1\,{\rm{V}}$ potential range is superimposed in the probability density. As a result, we obtain a much more smooth capacity curve. The pure equi-distribution for ω(u) results in a capacity similar to the normal distribution with large standard deviation.

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Table II.  Recommended values for the electron work function on Ag surfaces according to Ref. 24.

surface	(110)	(100)	(111)
work function	$4.10\,\mathrm{eV}$	$4.36\,\mathrm{eV}$	$4.53\,\mathrm{eV}$
$\mathop{}\limits_{s}\mu {}_{e}-\mathop{}\limits_{s}\bar{\mu }{}_{e}$	$+0.2\,\mathrm{eV}$	$-0.06\,\mathrm{eV}$	$-0.23\,\mathrm{eV}$