### Abstract

The nonlinear Poisson-Boltzmann (PB) equation is solved using Newton-Krylov iterations coupled with pseudo-transient continuation. The PB potential is used to compute the electrostatic energy and evaluate the force on a user-specified contour. The PB solver is embedded in a existing, 3D, massively parallel, unstructured-grid, finite element code. Either Dirichlet or mixed boundary conditions are allowed. The latter specifies surface charges, approximates far-field conditions, or linearizes conditions "regulating" the surface charge. Stability and robustness are proved using results for backward Euler differencing of diffusion equations. Potentials and energies of charged spheres and plates are computed and results compared to analysis. An approximation to the potential of the nonlinear, spherical charge is derived by combining two analytic formulæ. The potential and force due to a conical probe interacting with a flat plate are computed for two types of boundary conditions: constant potential and constant charge. The second case is compared with direct force measurements by chemical force microscopy. The problem is highly nonlinear - surface potentials of the linear and nonlinear PB equations differ by over an order of magnitude. Comparison of the simulated and experimentally measured forces shows that approximately half of the surface carboxylic acid groups, of density 1/(0.2 nm^{2}), ionize in the electrolyte implying surface charges of 0.4 C/m^{2}, surface potentials of 0.27 V, and a force of 0.6 nN when the probe and plate are 8.7 nm apart.

Original language | English (US) |
---|---|

Pages (from-to) | 62-79 |

Number of pages | 18 |

Journal | Journal of Colloid and Interface Science |

Volume | 247 |

Issue number | 1 |

DOIs | |

State | Published - 2002 |

Externally published | Yes |

### Fingerprint

### Keywords

- Chemical force microscopy
- Finite elements
- Massively parallel
- Nonlinear Poisson-Boltzmann
- Pseudo-transient continuation
- Unstructured grids

### ASJC Scopus subject areas

- Colloid and Surface Chemistry
- Physical and Theoretical Chemistry
- Surfaces and Interfaces

### Cite this

*Journal of Colloid and Interface Science*,

*247*(1), 62-79. https://doi.org/10.1006/jcis.2001.8033

**Solution of the nonlinear Poisson-Boltzmann equation using pseudo-transient continuation and the finite element method.** / Shestakov, A. I.; Milovich, J. L.; Noy, Aleksandr.

Research output: Contribution to journal › Article

*Journal of Colloid and Interface Science*, vol. 247, no. 1, pp. 62-79. https://doi.org/10.1006/jcis.2001.8033

}

TY - JOUR

T1 - Solution of the nonlinear Poisson-Boltzmann equation using pseudo-transient continuation and the finite element method

AU - Shestakov, A. I.

AU - Milovich, J. L.

AU - Noy, Aleksandr

PY - 2002

Y1 - 2002

N2 - The nonlinear Poisson-Boltzmann (PB) equation is solved using Newton-Krylov iterations coupled with pseudo-transient continuation. The PB potential is used to compute the electrostatic energy and evaluate the force on a user-specified contour. The PB solver is embedded in a existing, 3D, massively parallel, unstructured-grid, finite element code. Either Dirichlet or mixed boundary conditions are allowed. The latter specifies surface charges, approximates far-field conditions, or linearizes conditions "regulating" the surface charge. Stability and robustness are proved using results for backward Euler differencing of diffusion equations. Potentials and energies of charged spheres and plates are computed and results compared to analysis. An approximation to the potential of the nonlinear, spherical charge is derived by combining two analytic formulæ. The potential and force due to a conical probe interacting with a flat plate are computed for two types of boundary conditions: constant potential and constant charge. The second case is compared with direct force measurements by chemical force microscopy. The problem is highly nonlinear - surface potentials of the linear and nonlinear PB equations differ by over an order of magnitude. Comparison of the simulated and experimentally measured forces shows that approximately half of the surface carboxylic acid groups, of density 1/(0.2 nm2), ionize in the electrolyte implying surface charges of 0.4 C/m2, surface potentials of 0.27 V, and a force of 0.6 nN when the probe and plate are 8.7 nm apart.

AB - The nonlinear Poisson-Boltzmann (PB) equation is solved using Newton-Krylov iterations coupled with pseudo-transient continuation. The PB potential is used to compute the electrostatic energy and evaluate the force on a user-specified contour. The PB solver is embedded in a existing, 3D, massively parallel, unstructured-grid, finite element code. Either Dirichlet or mixed boundary conditions are allowed. The latter specifies surface charges, approximates far-field conditions, or linearizes conditions "regulating" the surface charge. Stability and robustness are proved using results for backward Euler differencing of diffusion equations. Potentials and energies of charged spheres and plates are computed and results compared to analysis. An approximation to the potential of the nonlinear, spherical charge is derived by combining two analytic formulæ. The potential and force due to a conical probe interacting with a flat plate are computed for two types of boundary conditions: constant potential and constant charge. The second case is compared with direct force measurements by chemical force microscopy. The problem is highly nonlinear - surface potentials of the linear and nonlinear PB equations differ by over an order of magnitude. Comparison of the simulated and experimentally measured forces shows that approximately half of the surface carboxylic acid groups, of density 1/(0.2 nm2), ionize in the electrolyte implying surface charges of 0.4 C/m2, surface potentials of 0.27 V, and a force of 0.6 nN when the probe and plate are 8.7 nm apart.

KW - Chemical force microscopy

KW - Finite elements

KW - Massively parallel

KW - Nonlinear Poisson-Boltzmann

KW - Pseudo-transient continuation

KW - Unstructured grids

UR - http://www.scopus.com/inward/record.url?scp=0036351487&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036351487&partnerID=8YFLogxK

U2 - 10.1006/jcis.2001.8033

DO - 10.1006/jcis.2001.8033

M3 - Article

VL - 247

SP - 62

EP - 79

JO - Journal of Colloid and Interface Science

JF - Journal of Colloid and Interface Science

SN - 0021-9797

IS - 1

ER -