The linearised Poisson-Boltzmann (Debye-Huckel) equation for the electrostatic…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Plasma Physics · The plasma state and single-particle motion

Conditions of Validity, Ch Green functions: inverting the operator

Problem

The linearised Poisson-Boltzmann (Debye-Huckel) equation for the electrostatic potential around a point ion in an electrolyte is −(∇2−κ2)ϕ=qϵ δ(3)(r),-(\nabla^2-\kappa^2)\phi = \frac{q}{\epsilon}\,\delta^{(3)}(\mathbf r), where qq is the ion's charge, ϵ\epsilon the permittivity of the solvent, and κ−1=λD\kappa^{-1}=\lambda_D the Debye screening length set by the solution's ionic strength. This is the Helmholtz operator of this section with k2k^2 replaced by −κ2-\kappa^2. ((a)) Following the same Fourier-transform-and-residue route used for −(∇2+k2)G=δ(3)(r)-(\nabla^2+k^2)G=\delta^{(3)}(\mathbf r), derive the free-space Green function G(r)G(r) of the screened operator −(∇2−κ2)-(\nabla^2-\kappa^2). State, in one sentence, why here, unlike the Helmholtz case, the choice of contour and which pole it encloses is forced by convergence alone, with no second physical argument needed to pick between two candidate kernels. ((b)) A monovalent ion of charge q=1.602×10−19 Cq=1.602\times10^{-19}\,\mathrm C sits in water (ϵ=ϵrϵ0\epsilon=\epsilon_r\epsilon_0, ϵr=78\epsilon_r=78, ϵ0=8.854×10−12 F m−1\epsilon_0=8.854\times10^{-12}\,\mathrm{F\,m^{-1}}) in a solution giving a Debye length λD=0.70 nm\lambda_D=0.70\,\mathrm{nm}. Find the potential ϕ=qG(r)/ϵ\phi=qG(r)/\epsilon at r=2.0 nmr=2.0\,\mathrm{nm} from the ion, and the ratio of this screened potential to the bare Coulomb potential q/4πϵrq/4\pi\epsilon r at the same distance. ((c)) Find the distance at which the screened potential has fallen to 1%1\% of the bare Coulomb potential at that distance.

Browse the problem library