Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 2 / Section A(iii) / 16

Let $\Phi (x)$ be the electrostatic potential in volts, relative to the bulk value, and $x$ the distance normal to the surface. The positive quantity $e$ is the elementary charge; the monovalent ions carry $\pm e$. The bulk number density $n_0$ refers to each ionic species in a symmetric electrolyte, so the total bulk ion density is $2n_0$. ${ϵ}_0$ is vacuum permittivity, ${ϵ}_r$ relative permittivity, $k_B$ the Boltzmann constant and $T$ absolute temperature. The hyperbolic sine acts on the dimensionless potential $u=e\Phi /(k_{B}T)$.

The printed sign is inconsistent with physical Poisson–Boltzmann screening under these usual definitions. Indeed,and Poisson's equation gives ${\Phi }^{\prime \prime }=-\rho /({ϵ}_0{ϵ}_r)=+2e{n}_0\sinh u/({ϵ}_0{ϵ}_r)$, with a plus sign. The literal printed equation and the physical screened solution must therefore be distinguished.

The dimensionally meaningful small-potential condition is $|e\Phi |\ll k_{B}T$, not a comparison of volts directly with energy. Expanding $\sinh u=u+O(u^3)$ gives the physical linearized equationFor a single plane with bulk boundary condition $\Phi (\infty )=0$, $\Phi (x)={\Phi }_0{e}^{-x/{\lambda }_D}$. At fixed temperature and permittivity, ${\lambda }_D\propto n_0^{-1/2}$: the curve decreases with increasing density, diverging in the ideal zero-salt limit.

Literal expansion of the printed minus-sign equation instead gives ${\Phi }^{\prime \prime }=-{\kappa }_D^2\Phi$, with sine/cosine solutions. A nonzero such solution cannot decay to zero at infinity. That literal equation does not describe a Debye length. The figure and following ion profiles use the physically derived plus-sign equation, with this correction explicit.

Take a positive surface potential ${\Phi }_0>0$, so positive ions are co-ions and negative ions counterions. With $u_0=e{\Phi }_0/(k_{B}T)\ll 1$,The counterion density is slightly enhanced at the surface, the co-ion density slightly depleted, and both approach $n_0$ over a Debye length. To first order their sum is $2n_0$, while their difference supplies the screening charge. Reverse the surface sign and the ionic roles interchange. These are number-density profiles, not trajectories of individual ions.

For large positive dimensionless surface potential $u_0=e{\Phi }_0/(k_{B}T)\gtrsim 1$, use the nonlinear Boltzmann distribution rather than their linear expansion:The counterions become strongly enriched near the wall and co-ions strongly depleted. A planar mean-field solution isIt approaches the linear screening form far from the wall. Nonlinearity preserves positive densities and produces a much stronger near-wall asymmetry; an extrapolated linear co-ion profile could incorrectly become negative. The total local ion density is $2n_0\cosh u$, no longer constant.

The original figure compares weak- and strong-potential profiles. At very large local concentrations, finite ion size, specific adsorption and ion correlations limit this ideal point-ion mean-field model; it does not predict unlimited physically realizable counterion packing.