Infinite-dilution limit of cylindrical counterions

In a neutral counterion-only cylindrical cell, send the outer radius to infinity at fixed cylinder charge. Below the Manning parameter threshold, number density tends to zero at each fixed radius. Above threshold, the cylindrical counterion-only Poisson-Boltzmann profile retains line charge $\lambda -q/{\ell }_B$ locally, while the rest recedes to infinity. Therefore the limit of the full finite-cell charge integral is not the charge integral of the pointwise limiting density.