PoissonBoltzmannIPAM2025

Project to develop Modified Poisson-Boltzmann solver for comparing with molecular level simulations.

We are solving a class of Poisson-Boltzmann systems as described below. First, we have poisson equation for charge density. $$ -\nabla \cdot (\varepsilon(\phi, y_\alpha) \nabla \phi) = \rho(\phi, p)\ in\ \Omega $$ with mole fractions of species $\alpha$ $$ y_\alpha(\phi, p)=y^E_\alpha exp(-\frac{z_\alpha e(\phi-\phi^E)+v_\alpha(p-p^E)}{k_B T}) $$ and charge density $$ \rho(\phi, p) = e \sum_{\alpha=0}^N z_\alpha \frac{y_\alpha (\phi, p)}{\sum_{\alpha=0}^N v_\alpha y_\alpha (\phi, p)} $$ We must also solve for pressure to establish mechanical equilibrium with a simple pressure poisson. $$ \nabla p = -q\nabla\phi \longrightarrow \Delta p = -\nabla \cdot (q\nabla\phi) \ in\ \Omega $$

After simulating we get $q(V)$, $C(V)$, $c_\alpha(x)$, $\phi(x)$.

Getting started

First, we'll clone the repository and instantiate the packages. Make sure you have julia 1.11!

git clone https://github.com/IPAM-ECH2025/PoissonBoltzmannIPAM2025.git

julia -e "using Pkg; Pkg.activate(); Pkg.instantiate()"

You can then open one of the notebooks or execute one of the scripts.

Contents

.
├── assets                       # Some data used by the demo code 
│   └── FedorovKornyshev.png
├── LICENSE
├── results                      # Subdirectory with calculation results
├── notebooks
│   └── ...                      # Various notebooks working with JuliaMPBSolver
├── draftnotebooks
│   └── ...                      # Various notebooks working without JuliaMPBSolver
├── src
│   └── PoissonBoltzmannIPAM2025.jl  # Module with shared code for project
├── packages
│   └── JuliaMPBSolver           # Main package
│       ├── Project.toml
│       └── src
│           └── JuliaMPBSolver.jl
├── Project.toml                 # Julia dependency description
├── README.md
└── scripts
    └── ...                      # Various scripts

Naming nomenclature for notebooks etc:

• MPB: modified Poisson-Boltzmann
• IC... : ion conserving ...
• ...P : ... with pressure equation

Workflow hints

For workflow hints, see

• https://modernjuliaworkflows.org/
• https://j-fu.github.io/marginalia/julia/

Acknowledgements

These codes are being developed thanks to IPAM Long Program - Bridging the Gap: Transitioning from Deterministic to Stochastic Interaction Modeling in Electrochemistry