We now calculate the apparent slip length $\lambda$ that would be inferred by tracking particle motion in a pressure-driven flow. In the limit $\kappa h \gg 1$, the streaming electric field is given by equation so that the particle velocity becomes, at leading order in $a/d$ and $1/\kappa
h$, $$\label{large1}
U_{\rm P}(z)  = -\frac{h^2}{2\mu}\frac{\d p}{\d
x}\left\{1-\frac{z^2}{h^2} +\frac{2\zeta_w(f(\kappa a
)\zeta_p-\zeta_w)(\e)^2}{\sigma\mu h^2}
\left[1+\frac{8\kappa}{\sigma\mu
  h}\left(\frac{\e k_B
T}{e}\right)^2
\left(\frac{t_w^2}{1-t_w^2}\right)\right]^{-1}\right\} \cdot$$ Comparing with the formula for the velocity in a flow satisfying the partial slip boundary condition , we see that the particle behaves as if it was passively advected by a pressure-driven flow with a finite slip length $\lambda$ given by $$\label{slip1}
\frac{\lambda}{h}=\frac{\zeta_w(f(\kappa a)\zeta_p-\zeta_w)( \e
e)^2}{\sigma\mu (eh)^2+{8\kappa h}\left({\e k_B T}\right)^2
\left(\frac{t_w^2}{1-t_w^2}\right)}\cdot$$ The condition for a positive apparent slip, $\lambda> 0$, is therefore $$\label{condlarge1}
  \zeta_w(f(\kappa a)\zeta_p- \zeta_w)>0.$$ This result can also be understood in the following way: (1) the particle and the wall must have the same charge sign, $\zeta_w\zeta_p>0$; this is usually the case in water where surfaces typically acquire negative charge, for example due to the ionization of sulfate or carboxylic surface groups; (2) the particle zeta potential must be sufficiently large $|\zeta_p|>|\zeta_w|/f(\kappa a)$ (or, equivalently, the wall zeta potential must be sufficiently 