Butler-Volmer equation with concentration dependence? For a simple elementary reaction on electrode

\begin{equation}
O+e^-\rightleftharpoons R
\end{equation}

We can derive the Butler-Volmer equation. But it seems that the formula found on John Newman's Electrochemical System seems to be different from the one found on Allen Bard's Electrochemical Methods, both of which are considered classics. Below is my understanding, if not incorrect.

In the most general case, Newman writes

\begin{equation}
i = i_0\biggl[\exp\biggl(\frac{(1-\beta)nF}{RT}\eta_s\biggr)-\exp\biggl(-\frac{\beta nF}{RT}\eta_s\biggr) \biggr]
\end{equation}

where $\eta_s=V-U$. $U$ is the equilibrium which depends on the surface concentration, and so do $i_0$, the exchange current.

Bard, on the other hand, writes, in the most general case, the current-overpotential equation
\begin{equation}
i_n = i_0\biggl[\frac{C_O}{C_O^*}\exp\biggl(-\frac{\alpha nF}{RT}\eta_s\biggr)-\frac{C_R}{C_R^*}\exp\biggl(\frac{(1-\alpha) nF}{RT}\eta_s\biggr) \biggr]
\end{equation}
which only leads to Butler-Volmer equation if mass-transfer is not concerned. Here $\eta_s=E-E_{eq}$ (with its notation). $E_{eq}$ is the equilibrium potential established on bulk concentration, which is a constant taken to be initial condition. $i_0$ is dependent on the bulk concentration.

Aside from the anodic, cathodic sign convention different, what Bard has written explicitly depends on the surface concentration, which in a sense suggest that even if the over-potential is negative (which, in Bard's convention, leads to positive cathodic current), anodic reaction can still be established if $C_R$ is sufficiently high.

The more widely known equation (also found in Bockris's Modern Electrochemistry vol.2) is the one Newman wrote. But the equation will only lead to (in Newman's convention) positive anodic current if the over-potential $\eta_s$ is positive, even if $C_O$ dominates (which leads to cathodic current in Bard's formula).

It seems that the derivation differ on the reference potential: Newman sets equilibrium potential $U$ to be dependent on the surface concentration while Bard let it be referenced to the open-circuit potential with bulk concentration. What's right and wrong?
I just figure it out that both are right (of course!). It's just that the definition of over-potential $\eta_s$ is different. In the case of Newman's, positive $\eta_s$ will guarantee positive (anodic) current. Here $\eta_s=V-U=\phi_{metal}-\phi_{solution}-U$ is positive in the sense that $V&gt;U$, where $U$ is the equilibrium potential corresponding to the surface concentration. Whereas in the case of Bard's, sometimes positive $\eta_s$ will lead to positive (cathodic) current if $C_O$ is sufficiently large (possibly with convention or precipitation). But here $\eta_s=V-E_{eq}$, and $E_{eq}$ is defined as equilibrium potential corresponding to bulk (initial) condition. So the value of $V$ here will result in negative $\eta_s$ in the case of Newman's

In short, it’s just that the sign of over-potential is different with their corresponding definition of equilibrium potential.