\section{Introduction}
\input{intro.tex}
\label{intro}
\section{Problem Formulation and Theory}
\input{theory.tex}
\label{theory}
\section{Results and Discussion}
\input{results.tex}
\label{results}
\section{Summary}
\input{summary.tex}
\label{summary}
\begin{acknowledgments}
This work was sponsored by NSF grant CBET-0853379. 
\end{acknowledgments}

















\subsection{Dimensionless Parameters}
To gain an understanding of the relative dominance of different transport mechanisms in the system, we calculate the values of the Damk\"{o}hler and Rayleigh numbers and the parameter $\beta$ from their respective formulas using appropriate values of the system variables.  In addition, we estimate the parameters using values we extract from the simulations.  Specifically, we calculate approximations to these three dimensionless parameters by numerically integrating the relevant fluxes throughout the system geometry and computing the relevant ratios.  Table \ref{salttab2} shows the results of these calculations.

\input{salttab2.tex}

The total reaction-driven flux of protons is given $J$, defined in (\ref{currcons}).  The total diffusive, convective and electromigration fluxes of protons through the fluid are estimated by integrating the $z$-components of the appropriate local fluxes over the two-dimensional annular disk surrounding the middle of the rod and extending from the rod surface to the boundary of the simulation domain.  The parameters are then numerically estimated by computing the relevant flux ratios: convective to diffusive flux ($Ra_e$), electromigration to diffusive ($\beta$), and total reaction flux to diffusive flux ($Da$).

Although the numerical values of the dimensionless parameters differ significantly between the analytical to the numerical versions, the general trends are the same in both cases.  The Rayleigh number is $O(10^{-4})$ in the analytical case and $O(10^{-1})$ in the numerical case, indicating that electroconvection is dominated by diffusion in transporting mass.  The values of $\beta$ are no larger than 0.01 in either case, showing that electromigration is relatively unimportant compared to diffusion.  The Damk\"{o}hler numbers are on similar orders of magnitude for both cases, suggesting that the majority of the charged species injected into the solution by the reactions are transported by diffusion.

In all cases, the dimensionless parameters become smaller as salt concentration is increased.  In the case of the Rayleigh number, this decrease reflects the decreased swimming speed of the rod, reducing the convective flux of all species.  The reduction in $\beta$ is due to the reduction in the tangential electric field magnitude, which drives electromigration flux.  Finally, the Damk\"{o}hler numbers decrease with salt concentration, indicating that the reaction rate decreases slightly as salt is added.  Overall, we conclude from this analysis that the bulk of transport in the system is due to diffusion (to a greater extent than in the case without salt\cite{moran_electrokinetic_2011}), and electroconvection is relatively unimportant in transporting species.

\subsection{Physical Reasons for Speed Decrease}
Figure \ref{fig2} shows that the rod velocity decreases roughly by a factor of 20 when the conductivity is increased from 8.8 to 100 $\mu$S/cm.  This velocity decrease is due to several factors, each of which cause a partial reduction in the swimming speed.  In descending order of importance, these factors are (i) the decrease in magnitude of the propulsive electric field, (ii) the decrease in magnitude of the area-averaged zeta potential, and (iii) the decrease in overall reaction rate.

We calculated the characteristic electric field $E^*$ of the system as the electric field that would need to be externally applied to drive conventional electrophoresis of the rod having the same surface charge at a speed equal to the measured swimming speed.  As conductivity is increased from 8.8 to 100 $\mu$S/cm, $E^*$ decreases from 415 to 38 V/m, a decrease of roughly 90 \%, i.e. $E^* (\sigma = 8.8) / E^* (\sigma = 100)$ = 10.92.  Since swimming speed is directly proportional to $E^*$, the reduction in $E^*$ with conductivity reduces the swimming speed of the rod by roughly an order of magnitude.

In all cases, the electrolyte added to the solution is at a significantly higher bulk concentration than protons and bicarbonate ions.  While the bulk concentration of protons and bicarbonate ions was kept fixed at 0.9 $\mu$mol/L, the electrolyte concentration was varied from 56.4 to 820 $\mu$mol/L.  The counterions in the electrolyte are attracted to the rod to help screen the surface charge and result in a net decrease in proton concentration in the diffuse layer, as shown in Fig. \ref{fig9} (a).  The reduced proton concentration causes a reduction in the reaction rate because the reaction rate on the gold is dependent on the square of the proton concentration.  When conductivity is increased from its minimum to its maximum value, the total reaction rates on the anode and cathode decrease by roughly 20 \%, such that $J (\sigma = 8.8) / J (\sigma = 100)$ = 1.27.  Due to the proportionality between the speed and surface activity identified by Golestanian, Liverpool, and Ajdari,\cite{golestanian_designing_2007} and also proposed and validated in the scaling analysis of our previous work,\cite{moran_locomotion_2010,moran_electrokinetic_2011} we conclude that the reduction in reaction rate causes a decrease in swimming speed of roughly 20 \% from the minimum to the maximum conductivity.

Figure \ref{fig4} shows that the average zeta potential decreases with the addition of salt.  This change in $\zeta$, which is closely linked to the rod potential, is driven by the decrease in reaction rates and the conservation of current requirement.  The reaction rate on the cathode is reduced as salt is added, due to the limited availability of protons (see Fig. \ref{fig9}) to participate in the peroxide reduction reaction.  Since current must be conserved, the rod potential self-adjusts such that the rate of peroxide oxidation on the anode decreases.  The average zeta potential decreases by roughly 45 \%, from $-$67.9 to $-$37.1 mV.  $\bar{\zeta} (\sigma = 8.8) / \bar{\zeta} (\sigma = 100) = 1.83$.  The change in zeta potential should result in a reduction in the swimming speed by roughly a factor of 2.

Since speed is linearly related to $E^*$, zeta potential, and reaction rate, we can multiply the reduction factors together to obtain an estimate of the total velocity reduction factor due to all three effects.  The result is 25.38, which is in reasonably good agreement with the observed speed reduction factor of $U (\sigma = 8.8) / U (\sigma = 100) = 19.86$.

In summary, the conductivity-induced speed decrease originates from the decrease in the characteristic electric field (due to Ohm's law), and to a lesser degree, the reduction of the cathode reaction rate due to exclusion of protons in the diffuse layer by the added nonreactive salt. 



\subsection{Governing Equations and Scaling Analysis}
Following our previous work,\cite{moran_locomotion_2010,moran_electrokinetic_2011} we apply the Poisson-Nernst-Planck-Stokes system of equations to this problem.  In the dilute solution limit, the concentration distributions of all species obey the dimensionless advection-diffusion equation,
\begin{equation}
Ra_e \left( \tilde{\mathbf{u}} \cdot \tilde{\nabla} \tilde{c}_k \right) = \tilde{\nabla}^2 \tilde{c}_k - \beta_k \tilde{\nabla } \cdot \left( \tilde{c}_k \tilde{\mathbf{E}} \right),
\label{AD}
\end{equation}
where $\tilde{\mathbf{u}}$ is the fluid velocity normalized by the electroviscous velocity $U_{ev}$, $\tilde{c}_k$ is the concentration of species $k$ normalized by the background proton and bicarbonate ion concentration, $c_{\pm,\infty}$, $\tilde{\mathbf{E}} = -\tilde{\nabla} \tilde{\phi}$ is the electric field normalized by a characteristic electric field $E_0$,  $Ra_e$ is the electric Rayleigh number,
\begin{equation}
Ra_e = \frac{U_{ev} a }{D_+},
\label{Raedef}
\end{equation}
$D_+$ is the diffusivity of protons, $a$ is a length scale over which the tangential electric field is significant, and $\beta_k$ is a dimensionless parameter that quantifies the relative importance of electromigration and diffusion of ion $k$ given as,
\begin{equation}
\beta_k = \frac{z_k F E_0 a}{RT}.
\label{betadef}
\end{equation}
In this case, all ions are monovalent, and the parameter $\beta$ therefore has the same value for every ion. Throughout the paper, dimensionless variables are indicated with a tilde, while dimensional variables and constants have no tilde.  For oxygen and hydrogen peroxide, which are uncharged, the electromigration term is omitted.

The concentration distributions are coupled to the electrostatic potential distribution through Poisson's equation,
\begin{equation}
- \frac{\varepsilon E_0}{F a c_{\pm,\infty}} \tilde{\nabla}^2 \tilde{\phi} = \tilde{\rho}_e = \sum_k z_k \tilde{c}_k.
\label{poisson}
\end{equation}
Here $\varepsilon$ is the permittivity of the solution, $c_{\pm,\infty}$ is the bulk ion concentration, $\tilde{\phi}$ is the dimensionless electric potential normalized by $E_0 a$, and the summation is carried out over all ionic species.  The fluid flow is described by the incompressible continuity and Stokes equations:
\begin{equation}
\tilde{\nabla} \cdot \tilde{\mathbf{u}} = 0,
\label{COM}
\end{equation}
\begin{equation}
0 = \frac{1}{Re} \left( - \tilde{\nabla} \tilde{p} + \tilde{\nabla}^2 \tilde{\mathbf{u}} + \tilde{\rho}_e \tilde{\mathbf{E}} \right).
\label{stokes}
\end{equation}
Here $Re = \rho U_{ev} d / \eta$ is the Reynolds number (which is never larger than 10$^{-4}$ in the simulations considered here), $\tilde{p}$ is the pressure normalized by $\eta U_{ev}/d$,  $d$ is a viscous length scale, and $\tilde{\rho}_e \tilde{\mathbf{E}}$ is the electrical body force resulting from the coupling of free charge in the solution and the self-generated electric field.  Equations (\ref{AD}) and (\ref{poisson})-(\ref{stokes}) constitute a coupled, nonlinear system that is difficult to solve in general.  Approximate versions of these equations have been solved analytically to study the self-propulsion of a spherical cell,\cite{lammert_ion_1996} the autonomous fluid circulation near metallic disk electrodes in a peroxide solution,\cite{kline_catalytically_2006} and recently electrokinetic self-propulsion of synthetic particles similar to the case considered here.\cite{yariv_electrokinetic_2011,sabass_nonlinear_2012}  Generally, obtaining analytical solutions to these equations requires one to make simplifying assumptions, e.g. that the reactions cause small perturbations of the field variables from the equilibrium state,\cite{kline_catalytically_2006,yariv_electrokinetic_2011} or that the electrical double layer (EDL) surrounding the particle is negligibly thin compared to the size of the particle.\cite{yariv_electrokinetic_2011,sabass_nonlinear_2012}  We wish to describe in rigorous detail the physical phenomena occurring at the rod/solution interface, and we accordingly make no simplifying assumptions about the size of the EDL or the concentration perturbations. This allows us to study the system for a wide range of parameter values, and obtain numerical solutions to the full nonlinear equations to account for the cylindrical geometry and incorporate all of the important physical phenomena leading to self-propulsion.

The natural velocity scaling in this system is the electroviscous velocity, $U_{ev} = \rho_{e,0} E_0 / (\eta / d^2)$, which was originally introduced by \citet{hoburg_internal_1976} and reflects the balance of viscous and electrical body forces in the system.  We expect that the speed of the rod scales with $U_{ev}$.  In our previous work we derived the general scaling relation for a charged rod with zeta potential $\zeta$ and area-averaged proton flux $j_+$ given as
\begin{equation}
U_{ev} \propto \frac{F L a d^2}{\lambda_D^2 \eta D_+} j_+ \zeta,
\label{UevJFM}
\end{equation}
where $L$ is a length scale for the charge density distribution.  Previously, we assumed that $L$ and $d$ both scale with the Debye length, while $a$ was proportional to the length of the rod, $h$.\cite{moran_locomotion_2010,moran_electrokinetic_2011}  Although we have confirmed that the tangential electric field does increase in magnitude with increasing $h$ (assuming the particle is suspended in an infinite medium and neglecting the role of the particle mass), as shown in the supplementary section, here we are specifically interested in changes in electric field due to changes in solution conductivity.  According to Ohm's law, $i = \sigma E$, the electric field should scale inversely with solution conductivity.  Solution conductivity scales approximately with ionic strength, $I$, and therefore we would expect $E_0 \propto 1 / I$.  Ionic strength is related to the Debye length according to
\begin{equation}
\lambda_D^2 \equiv \frac{\varepsilon RT}{2 F^2 I}.
\end{equation}
From this definition and the inverse relationship between electric field and ionic strength, we see that the electric field can scale with the square of Debye length, $E_0 \propto \lambda_D^2$.\cite{MoranPhD13}  Note that we are not claiming that the electric field scales with the physical thickness of the EDL.  Instead, we concur with previous work\cite{paxton_catalytically_2006} that the electric field scales inversely with the conductivity of the solution, through Ohm's law, which we express in terms of the definition of the Debye thickness.  Making this substitution into equation (\ref{UevJFM}), the scaling result for the swimming speed becomes
\begin{equation}
U_{ev} \propto \frac{F \lambda_D^2}{\eta D_+} j_+ \zeta \propto \frac{\varepsilon \zeta}{\eta} \frac{RT}{FD_+ I} j_+.
\label{scaling}
\end{equation}
Again, this relation resembles the Helmholtz-Smoluchowski-like expression, Eq. (\ref{HSPax}), except with the effective electric field given by $E_0 \propto RT j_+ / FD_+ I$, and the equation is stated in terms of reaction rate $j_+$ instead of current density $i$.  This equation predicts that the swimming speed should scale quadratically with Debye length (or inversely with ionic strength).  A quadratic relationship between speed and Debye length was also asymptotically derived by Sabass and Seifert. \cite{sabass_nonlinear_2012}  The prediction of an inverse dependence on conductivity was also made by \citet{paxton_catalytically_2006} and by Golestanian, Liverpool and Ajdari.\cite{golestanian_designing_2007}  In all three cases, the predicted form for the swimming speed is proportional to the Helmholtz-Smoluchowski expression, except with different forms for the electric field.  In addition, Golestanian's formula and our simulations account for the cylindrical geometry of the particle.

\subsection{Boundary Conditions}
The simulations are conducted in the reference frame of a stationary rod.  We apply the no-slip condition at the rod surface,
\begin{equation}
\tilde{\mathbf{u}} = \mathbf{0}.
\end{equation}
At the domain boundary far from the rod, we prescribe vanishing viscous stress:
\begin{equation}
\left[ \tilde{\nabla} \tilde{\mathbf{u}} + \left( \tilde{\nabla} \tilde{\mathbf{u}} \right)^T \right] = 0.
\end{equation}
Here $\tilde{\nabla} \tilde{\mathbf{u}}$ is the velocity gradient tensor and the superscript $T$ denotes the transpose.  This boundary condition effectively enforces a slip condition at the outer boundary, approximating an infinite medium.  We evaluate the average fluid speed in the axial direction along this boundary to determine the swimming speed of the rod.

Since anions do not react, their boundary condition is zero flux at the rod surface,
\begin{equation}
\mathbf{n} \cdot \tilde{\mathbf{j}}_- = 0,
\end{equation}
where $\mathbf{n}$ is the outward normal vector pointing into the fluid and the dimensionless flux of ion $k$ is defined as $\tilde{\mathbf{j}}_k = -\tilde{\nabla} \tilde{c}_k + \beta_k \tilde{c}_k \tilde{\mathbf{E}} + Ra_e \tilde{c}_k \tilde{\mathbf{u}}.$  The electrochemical reactions generate fluxes of protons leaving the anode surface and entering the cathode surface.  These reactions are represented (dimensionally) in the model by
\begin{equation}
\mathbf{n} \cdot \mathbf{j}_+ = \left\{
                                          \begin{array}{ll}
                                          j_{+,a} = K_{ox} c_{\mathrm{H}_2 \mathrm{O}_2} \exp \left[ \frac{(1-\alpha) m F \Delta \phi_S}{RT} \right], \textrm{   }0 < z < 1 \textrm{ }\mu\textrm{m},\\
                                          j_{+,c} = K_{red} c_{\mathrm{H}_2 \mathrm{O}_2} c_+^2 \exp \left[ - \frac{\alpha m F \Delta \phi_S}{RT} \right], \textrm{  } - 1 \textrm{ }\mu\textrm{m} < z < 0
                                          \end{array}
                                          \right.
\label{fluxbc}
\end{equation}
where the subscript + denotes protons, the subscript $a$ indicates the anodic flux due to the peroxide oxidation reaction, and the subscript $c$ indicates cathodic flux for the peroxide reduction reaction.  Positive values of the axial coordinate $z$ indicate the anode side of the rod (typically Pt), and negative values indicate the cathode (typically Au) side.  Here, $K_{ox}$ and $K_{red}$ are the rate constants for peroxide oxidation and reduction, $\alpha$ is a dimensionless parameter between 0 and 1 (set here to 0.5) that quantifies the asymmetry of the energy barrier for the reaction, $m$ is the number of electrons transferred in the electrochemical reaction ($m=2$ for both reactions considered here), and $\Delta \phi_S$ is the voltage across the compact Stern layer of adsorbed species on the surface of the rod.\cite{bazant_current-voltage_2005}

The expressions for the proton fluxes $j_+$ on the rod segments are given by Butler-Volmer equations with Frumkin's correction,\cite{frumkin_hydrogen_1933} and reflect the dependence of the kinetics of the electrochemical reactions on the local reactant concentrations and on the voltage across the compact Stern layer.\cite{moran_electrokinetic_2011,frumkin_hydrogen_1933,delahay_double_1965,bard_electrochemical_2000,bazant_current-voltage_2005}  In equation (\ref{fluxbc}), we have implemented the Tafel approximation, meaning that the reactions are assumed to proceed in one direction only and the backward components of each reaction are considered negligible.  For a derivation of Eq. (\ref{fluxbc}) starting from the full Butler-Volmer equation, the reader is referred to our previous paper.\cite{moran_electrokinetic_2011}

The reaction rates for species other than protons are related to the proton fluxes according to the stoichiometry of the reactions.  On the anode, one peroxide molecule is consumed for every two protons released into the solution:
\begin{equation}
\mathbf{n} \cdot \mathbf{j}_{\textrm{H}_2\textrm{O}_2,a} = - \frac{j_{+,a}}{2},
\end{equation}
and one oxygen molecule is generated for every two protons generated:
\begin{equation}
\mathbf{n} \cdot \mathbf{j}_{\textrm{O}_2,a} = \frac{{j}_{+,a}}{2}.
\end{equation}
On the cathode, one peroxide molecule is consumed for every two protons consumed:
\begin{equation}
\mathbf{n} \cdot \mathbf{j}_{\textrm{H}_2\textrm{O}_2,c} = \frac{j_{+,c}}{2}.
\end{equation}
It has been suggested by Wang \textit{et al.},\cite{wang_bipolar_2006} among others, that the four-electron reduction of O$_2$ may also occur on the cathode end, perhaps even as the dominant reaction.  While this is possible, O$_2$ reduction is not likely the dominant reduction reaction on the cathode side, since our previous work shows that the rods move faster in solutions purged of O$_2$ and slower in O$_2$-rich solutions.\cite{calvo-marzal_electrochemically-triggered_2009}  We therefore assume that peroxide reduction is the only reaction occurring on cathode, and that oxygen is nonreactive on the cathode end:
\begin{equation}
\mathbf{n} \cdot \mathbf{j}_{\textrm{O}_2,c} = 0.
\end{equation}
Far from the rod, the concentrations of all chemical species approach their bulk values,
\begin{equation}
\tilde{c}_k \rightarrow 1 \textrm{ as } \left| \tilde{\mathbf{r}} \right| \rightarrow \infty.
\end{equation}
The boundary conditions for proton flux can also be stated in dimensionless form.  Since the proton flux takes a different form on the anode and cathode, here the Damk\"{o}hler number is defined differently for each metal.  The Damk\"{o}hler numbers can be defined in terms of the reaction kinetic expressions, equation (\ref{fluxbc}).  On the anode, the dimensionless boundary condition takes the form
\begin{equation}
\mathbf{n} \cdot \tilde{\mathbf{j}}_+ = Da_{anode} \tilde{c}_{\textrm{H}_2 \textrm{O}_2},
\end{equation}
where
\begin{equation}
Da_{anode} = \frac{K_{ox} a c_{\textrm{H}_2 \textrm{O}_2,\infty}}{D_+ c_{+,\infty}}.
\label{Da_anode_def}
\end{equation}
On the cathode, the dimensionless boundary condition reads
\begin{equation}
\mathbf{n} \cdot \tilde{\mathbf{j}}_+ = Da_{cathode} \tilde{c}_{\textrm{H}_2 \textrm{O}_2} \tilde{c}_+^2,
\end{equation}
where
\begin{equation}
Da_{cathode} = \frac{K_{red} a c_{\textrm{H}_2 \textrm{O}_2,\infty} c_{+,\infty}}{D_+}.
\label{Da_cathode_def}
\end{equation}
Although the Damk\"{o}hler numbers are defined differently on the anode and cathode, the rate constants in each definition have different units, so that the Damk\"{o}hler number is dimensionless in each case.  In stating the definitions of the Damk\"{o}hler numbers we have ignored the exponential terms in the kinetic expressions (i.e., we have assumed these terms to be equal to unity).  Although these terms could be included, they would not significantly alter the magnitude of the Damk\"{o}hler numbers.  In this work, the exponential terms range in magnitude from 0.97 to 1.03 in all cases studied.

In general, the flux expressions for the anode and cathode are not equal at the junction between them, $z = 0$.  To avoid unphysical discontinuities in the reaction flux and flux gradient, we multiply the flux profile along the length of the rod by a dimensionless sigmoidal weighting function, $\xi (z)$, defined as
\begin{equation}
\xi (z) = \left| \frac{2}{1 + e^{- \gamma z}} - 1 \right|,
\end{equation}
where $\gamma = 10^7$~m$^{-1}$ and $z$ is evaluated in meters.  The function $\xi$ is defined to be roughly equal to 1 at the end of the anode segment, 1 at the end of the cathode segment, and zero at the anode/cathode boundary.  The use of this weighting function reflects a diffuse interface which would result in reduced density of available reaction sites and reaction rate near the junction between anode and cathode.

The surface of the rod is theorized to contain an immobile layer of charged and uncharged adsorbed species, often referred to as the \textit{Stern layer}.  Together with the diffuse layer of ions in the solution adjacent to the rod, these two layers constitute the electrical double layer (EDL).  According to the Stern model of the EDL, the immobile (Stern) layer acts as a linear capacitor in series with the diffuse layer.\cite{bard_electrochemical_2000,bazant_current-voltage_2005}  The electric potential gradient is extrapolated across the Stern layer, from the outer Helmholtz plane to the metal.  Thus, the Stern voltage is linearly related to the normal electric field at the rod surface.  Following the Stern model, we treat this layer as a linear capacitor which leads to the (dimensional) mixed boundary condition\cite{bard_electrochemical_2000,bazant_current-voltage_2005,moran_electrokinetic_2011}
\begin{equation}
\Phi_{rod} + \lambda_S \left( \mathbf{n} \cdot \nabla \phi \right)_{\textrm{OHP}} = \phi_{\textrm{OHP}} \equiv \zeta,
\label{potentialBC}
\end{equation}
where $\Phi_{rod}$ is the electrical potential of the interior of the rod with respect to the bulk solution, $\lambda_S$ is an effective thickness of the Stern layer (set here to 2 {\AA} for all cases), and the subscript OHP indicates that the quantity is evaluated at the outer edge of the Stern layer, often termed the outer Helmholtz plane (OHP).\cite{bard_electrochemical_2000}  Since the rod is conducting, the potential $\Phi_{rod}$ is assumed uniform everywhere in its interior.  The zeta potential $\zeta$ is defined here as the potential at the OHP versus the bulk solution, and in general varies with position on the rod surface.  The voltage across the Stern layer is generally defined as the internal rod potential minus the potential at the OHP, i.e. $\Delta \phi_S \equiv \Phi_{rod} - \zeta (z)$, and therefor depends on the normal electric field at the OHP through the above boundary condition on the potential, (\ref{potentialBC}).

Far from the rod, the electric potential approaches zero,
\begin{equation}
\tilde{\phi} \rightarrow 0\textrm{ as }\left| \tilde{\mathbf{r}} \right| \rightarrow \infty.
\end{equation}

\subsection{Current Conservation}
At steady state, the total charge in the rod must be conserved, implying that the net current into or out of the rod must be zero.  We require that
\begin{equation}
\int_{anode} j_{+,a} dA = - \int_{cathode} j_{+,c} dA \equiv J
\label{currcons}
\end{equation}
at steady state, where the reaction fluxes $j_{+,a}$ and $j_{+,c}$ are given by (\ref{fluxbc}).  The system of equations (\ref{AD})-(\ref{stokes}) is solved concurrently and is closed by iterative determination of the rod potential $\Phi_{rod}$ that produces reaction fluxes that satisfy (\ref{currcons}).  The value of $\Phi_{rod}$ directly affects the reaction rates on both the anode and cathode.  On the cathode, a more negative rod potential would result in a more negative zeta potential, which means that the surface attracts more protons electrostatically to the surface to screen the surface charge.  The elevated proton concentration results in faster reaction rates on the cathode, according to (\ref{fluxbc}).  On the anode, a more negative potential decreases the reaction rate, since this would alter the overpotential bias to favor reduction more and oxidation less.  The value of $\Phi_{rod}$ that satisfies (\ref{currcons}) is observed to vary with salt concentration.  Table \ref{salttab} shows the values and units of the constants used in the simulations.\cite{lide_crc_2004}  The ion mobilities $\nu_k$ are determined from the Nernst-Einstein relation, $D_k = \nu_k RT$.

\input{salttab.tex} 