When ions are crowded, the effect of steric repulsion between ions (which can produce oscillations in charge density profiles) becomes significant and the conventional Poisson–Boltzmann (PB) equation should be modified. Several modified PB equations were developed but the associated total ionic charge density has no oscillation. This motivates us to derive a general model of PB equations called the PB-steric equations with a parameter \(\Lambda\) , which not only include the conventional and modified PB equations but also have oscillatory total ionic charge density under different assumptions of steric effects and chemical potentials. As \(\Lambda=0\) , the PB-steric equation becomes the conventional PB equation, but as \(\Lambda \gt 0\) , the concentrations of ions and solvent molecules are determined by the Lambert type functions. To approach the modified PB equations, we study the asymptotic limit of PB-steric equations with the Robin boundary condition as \(\Lambda\) goes to infinity. Our theoretical results show that the PB-steric equations (for \(0\leq \Lambda \leq \infty\) ) may include the conventional and modified PB equations. On the other hand, we use the PB-steric equations to find oscillatory total ionic charge density which cannot be obtained in the conventional and modified PB equations.


  1. PB-steric equations
  2. total ionic charge density
  3. oscillation

MSC codes

  1. 35C20
  2. 35J60
  3. 35Q92

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The authors express their thanks to the reviewers for the careful and insightful review.


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Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 1603 - 1622
ISSN (online): 1095-712X


Submitted: 22 August 2022
Accepted: 6 March 2023
Published online: 3 August 2023


  1. PB-steric equations
  2. total ionic charge density
  3. oscillation

MSC codes

  1. 35C20
  2. 35J60
  3. 35Q92



Jhih-Hong Lyu
Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan.
Tai-Chia Lin Contact the author
Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan; National Center for Theoretical Sciences, Mathematics Division, Taipei 10617, Taiwan.

Funding Information

Funding: The research of the second author is partially supported by the National Center for Theoretical Sciences (NCTS) and MOST grant 109-2115M-002-003MY3.

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