Butler-Volmer equation with concentration dependence?
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.