which the net charge density is essentially zero.

However, is restricted to low surface potentials, typically 20mV, which is a severe approximation. Let us consider for simplicity the case of a two-dimensional channel of height $2h$ in the $z$-direction and let us instead derive the solution to for any value of the zeta potential at the wall $\zeta_w$ but in the limit where the channel dimensions are much larger than the double layers $\kappa h\gg 1$. This limit is appropriate for channel sizes down to $h\approx 5$ $\mu$m in the case of pure water, or even $h\approx 50$ nm in the case of tap water.

Let us define the dimensionless potential $\phi=e\psi/k_BT$ and the dimensionless vertical coordinate $\bar{z}=z/h$. In this case, becomes $$\label{newPB}
\frac{1}{(\kappa h )^2} \frac{\d^2 \phi}{ \d\z^2} =\sinh \phi,$$ with the boundary conditions $\phi(\z=\pm
1)=\phi_w=e\zeta_w/k_BT$.

Since $1/\kappa h\ll 1$, the solution to equation involves boundary layers near $z=\pm 1$. The outer solution $\phi_{\rm \,out}$ is found by taking the limit $1/\kappa h=0$ in and we find $\phi_{\rm \,out}=0$. The inner solution $\phi_{\rm\, in}$ is valid near the boundaries for $\kappa
h(1-|\z|)={\cal O}(1)$, in which case reduces to the Poisson-Boltzmann equation near an infinite plane, whose solution is [@hunter] 