numerically the maximum apparent slip lengths which could be obtained in an experiment, $\lambda^*$, as a function of the particle zeta potential $\zeta_p$. The results are displayed in Figure \[maximum\] (right). We first note that $\lambda^*$ increases with $|\zeta_p|$. Furthermore, the maximum slip length can take values as low as molecular sizes or below and, in the case of pure water, can be as high as hundreds of nanometers.

The data for the low values of $|\zeta_p|$ display a power-law behavior, which we can analyze as follows. Let us consider formula . The two terms in the denominator will be of the same order of magnitude if $t_w$ is larger than a critical value $\tilde{t}_w$ which is given by $$\label{critical}
\tilde{t}_w\approx \left(\frac{1}{1+ \frac{48\pi \ell \e k_B
T}{e^2 \kappa h}}\right)^{1/2},$$ where we have used to relate the conductivity to the screening length. The smallest value of will be obtained, say, for $\kappa h \approx 10$, in which case we get $\tilde{t}_w \approx 0.86$ which corresponds to a critical wall zeta potential $\tilde{\zeta}_w\approx 135$ mV. Consequently, when $\zeta_w\lesssim \tilde{\zeta}_w$, can be simplified to $$\label{}
\frac{\lambda}{h}=\frac{\zeta_w(f(\kappa
a)\zeta_p-\zeta_w)(\e)^2}{\sigma\mu h^2},$$ for which it is easy to get $$\label{exp}
\zeta_w^*=\frac{f(\kappa a)}{2}\zeta_p \,,\quad
{\lambda^*}=\frac{(\e f(\kappa a)\zeta_p)^2}{4\sigma\mu h}\cdot$$ 