A Poisson-Boltzmann Solver for Charge-Regulating Colloids

This repository contains a code that numerically solves the Poisson-Boltzmann equation for a spherical colloid in a cell, coupled to a reservoir at a given pH-value and monovalent salt concentration, with either a constant charge (Neumann) or a self-consistent charge-regulation boundary condition on the colloid surface. It was used to the produce the theoretical results in this preprint. The code also contains functionality to obtain the renormalized (effective) charge of the colloid using the Trizac prescription. Internally, the code uses open-source Python modules, including Numpy and Scipy for numerical operations, Pandas for data processing and Pint for unit conversions.

Dependencies

• Pint
• Pandas
• Numpy
• SciPy

Contents

• checks/: folder with various scripts that check the numerical solver against analytical solutions and results from the literature
• paper/: folder with scripts and numerical results presented in the paper
• LICENSE: license of pb_solver
• pb_solver.py: source code of pb_solver
• requirements.txt: list of required libraries to use pb_solver

Usage

Setting up a virtual environment

To use the Poisson-Boltzmann Solver, first clone this repository locally:

git clone [email protected]:davidbbeyer/pb_solver.git

To handle the dependencies of the solver, it is most convenient to use a Python virtual enviroment. In order to set up a virtual environment, the Python module venv is needed. If venv is not included in your Python distribution, your first need to install it, e.g. on Ubuntu:

sudo apt install python3-venv

To set up a virtual environment and install the Python dependencies, run the following commands:

python3 -m venv pb_solver
source pb_solver/bin/activate
python3 -m pip install -r requirements.txt
deactivate

Running the Checks

The folder checks/ contains various scripts to check the validity of the numerical results against analytical solutions and results from the literature. To run these checks (e.g. checks/check_debye_hueckel.py), simply activate the virtual environment and run the script using Python:

source pb_solver/bin/activate
python3 checks/check_debye_hueckel.py

The included checks are:

• checks/check_alexander.py: Checks that the code is able to reproduce the effective charges reported by Alexander et al. (Fig. 4 of JCP 80 (11), 5776-5781) in the presence of salt.
• checks/check_charge_regulation.py: Compares the degree of ionization of a charge-regulating colloid in the limit of a vanishing Bjerrum length to an independent numerical calculation that couples the ideal Donnan theory and the Henderson-Hasselbalch equation.
• checks/check_charges_ideal: Contains a script to check that the workflow used to produce the data in the preprint gives the expected result in the limit of a vanishing Bjerrum length.
• checks/check_debye_hueckel.py: Compares the numerical solution of the nonlinear Poisson-Boltzmann equation to the linearized analytical solution (Debye-Hückel) in the case of a fixed low surface charge density.
• checks/check_donnan.py: Compares the value of the electrostatic potential at the cell boundary to the analytical ideal Donnan potential in the limit of a vanishing Bjerrum length.

References

Check out the corresponding preprint and references therein to learn more about Poisson-Boltzmann theory, charge regulation, charge renormalization and applications of the solver.

@article{vogel2024co2,
  title={CO2-induced Drastic Decharging of Dielectric Surfaces in Aqueous Suspensions},
  author={Vogel, Peter and Beyer, David and Holm, Christian and Palberg, Thomas},
  journal={arXiv preprint},
  year={2024},
  doi={10.48550/arXiv.2409.03049},
}