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

A. I. Shestakov, J. L. Milovich, Aleksandr Noy

Research output: Contribution to journalArticle

59 Citations (Scopus)

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 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.

Original languageEnglish (US)
Pages (from-to)62-79
Number of pages18
JournalJournal of Colloid and Interface Science
Volume247
Issue number1
DOIs
StatePublished - 2002
Externally publishedYes

Fingerprint

Boltzmann equation
Surface charge
finite element method
Surface potential
Finite element method
Boundary conditions
Force measurement
Carboxylic Acids
Carboxylic acids
Electrolytes
Electrostatics
Microscopic examination
boundary conditions
probes
flat plates
carboxylic acids
newton
iteration
far fields
electrolytes

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

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

In: Journal of Colloid and Interface Science, Vol. 247, No. 1, 2002, p. 62-79.

Research output: Contribution to journalArticle

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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.

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