Effects of the bulk volume fraction on solutions of modified Poisson–Boltzmann equations

To investigate the effect of the bulk volume fraction ν on the structure of the electrical double layer (EDL) in electrolyte solutions, we revisit modified Poisson–Boltzmann (MPB) equations (Borukhov et al. (1997) [5]) over a bounded smooth domain. Under Robin boundary conditions, we establish the exact first two order terms of the asymptotic expansions (as 0<ν≪10<ν≪1) for solutions. In particular, such results demonstrate the effect of the bulk volume fraction ν on the slope of the solution curve (corresponding to the profile of the electrostatic potential) at boundary points. An interesting outcome from our result is that in the direction of the outward normal to the boundary, the solution curve is getting steeper when the bulk volume fraction is getting smaller. Besides, in the case of small Debye screening length, we go further to establish the stability property for solutions with respect to ν   in both L∞L∞- and H1H1-norms, respectively. These results provide a basic understanding of the effect of the bulk volume fraction on the EDL structure at the charged surface and on the behavior of the electrostatic potential in the bulk of dilute electrolyte solutions, both at a theoretical level and for practical applications.