6.2.2 Poisson-Boltzmann Model with Sheet Charge

Looking at the region containing the charged macromolecules linked to the surface and not being dispensed in a gel, one recognizes that its height is typically in the deca nanometer regime. Thus, it is extremely thin compared to the rest of the device dimensions. Therefore, to save mesh points, one introduces the average charge density replacing this region with an equivalent sheet charge $\sigma_{\mathrm{Sheet}}(x)$ at the surface $y_{0}$.

$$\varepsilon_{0} \nabla \cdot (\varepsilon_{\mathrm{sol}} \nabla \... ...}{k_{B}T}(\psi(x,y)-\psi_{\mu})} + \sigma_{\mathrm{Sheet}}(x)\, \delta(y-y_{0})$$ (6.2)