Analytical solutions of the Poisson-Boltzmann equation within an interstitial electrical double layer in various geometries

Saboorian‐Jooybari, Hadi; Chen, Zhangxin

doi:10.1016/j.chemphys.2019.01.026

Cited by 11 publications

(4 citation statements)

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“…For planar geometry, exact solutions to the nonlinear PB equations also exist for symmetric and asymmetric electrolyte solutions [Andrietti et al, 1976, Chapman, 1913, Gouy, 1910, Grahame, 1953. A few studies have presented analytical solutions to describe electrostatic potential between two similarly charged particles in terms of complex integrals [Coussy, 2010a, Polat and Polat, 2010, Saboorian-Jooybari and Chen, 2019, Zhang et al, 2018. However, approximate solutions are generally only sufficiently accurate for surface separations beyond about one Debye length (please see [Israelachvili, 2011] for the definition and interpretation of Debye length), and one needs to seek numerical solutions [Anandarajah and Chen, 1994, Bohinc et al, 2016, Brumleve and Buck, 1978, Gross and Osterle, 1968, Olivares and McQuarrie, 1975, Ramanathan, 1983 for problems with complex geometry and high surface charge densities.…”

Section: Validation and Discussionmentioning

confidence: 99%

The role of disjoining pressure on the drying shrinkage of cementitious materials

Rahman,

Grasley

2023

Open Geomechanics

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Drying induced shrinkage is often attributed to two major mechanisms-capillary pressure in the bulk pore solution and disjoining pressure in the liquid film separating the vapor phase from the pore wall or separating solid surfaces in nanometric pores. There is sufficient ambiguity in literature regarding the relative contribution of these two mechanisms, as well as the means to quantify their contributions. The objective of this manuscript is to evaluate the contribution of disjoining pressure in the drying shrinkage of cementitious materials. An unconventional approach to determining disjoining pressure within the framework of continuum mechanics is presented. This approach utilizes the conservation of linear momentum to derive a generalized expression of the disjoining pressure from the Lorentz force vector. The expression suggests that disjoining pressure is essentially an osmotic pressure at the contact surfaces that counters the electrostatic contribution to linear momentum. The proposed theory accurately predicts measurements of osmotic pressure found in the literature for the swelling of charged bilayers in a dilute salt solution. Applied to the shrinkage problem, the theory suggests that shrinkage stress is induced by the reduction in the potential gradient between the liquid film and bulk solution from the reference (fully saturated) state. The reduction in the potential gradient is caused by an increase in the concentration of the solutes in the pore solution when liquid water is removed as the relative humidity decreases.

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Section: Validation and Discussionmentioning

confidence: 99%

The role of disjoining pressure on the drying shrinkage of cementitious materials

Rahman,

Grasley

2023

Open Geomechanics

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“…One goal in this work is to calculate the apparent surface charge density. In fact, we place the answer to this question into the more general context of characterizing the degree of counterion release for slab-like planar geometries, and we carry out our analysis using the Poisson–Boltzmann model, which is a simple approach to describe ion concentrations near macroions on the mean-field level. − Analytic solutions of the nonlinear Poisson–Boltzmann equation exist for only a few geometries − and surface charge density distributions or for some limiting cases. , Yet, when available, these solutions provide a conceptual understanding of observable quantities such as the osmotic pressure between macroions or in polyelectrolyte solutions, − the condensation of counterions, − the influence of electrostatics on the phase behavior of charged colloids and nanoparticles, − and on the stability , and rigidity of soft aggregates. , Our present work adds the number of released counterions to this list.…”

Section: Introductionmentioning

confidence: 99%

“…16−19 Analytic solutions of the nonlinear Poisson−Boltzmann equation exist for only a few geometries 20−22 and surface charge density distributions 23 or for some limiting cases. 24,25 Yet, when available, these solutions provide a conceptual understanding of observable quantities such as the osmotic pressure between macroions or in polyelectrolyte solutions, 26−28 the condensation of counterions, 29−32 the influence of electrostatics on the phase behavior of charged colloids and nanoparticles, 33−36 and on the stability 37,38 and rigidity of soft aggregates. 39,40 Our present work adds the number of released counterions to this list.…”

Section: Introductionmentioning

confidence: 99%

Counterion Release from Macroion Assemblies of Planar Geometry

Volpe Bossa,

Hobbie,

May

2024

J. Phys. Chem. B

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Macroions such as nanoparticles, polyelectrolytes, ionic gels, and amphiphiles can form condensed, often selfassembled, phases that are embedded in a solvent region. The condensed phase contains not only the partially or fully immobile charges of their macroions but also corresponding counterions that are mobile and thus free to migrate out of their confinement into the solvent region where they benefit from high translational entropy. Based on the nonlinear Poisson−Boltzmann model for monovalent ions, we quantify the corresponding fraction of released counterions for a planar slab geometry of the macroion phase. Slab thickness, extension of the solvent phase, fixed background charge density provided by the macroions, and dielectric constants inside slab and solvent combine into three dimensionless parameters that the fraction of released counterions depends on. We calculate that fraction and analyze the limits of a thin macroion phase, a large solvent phase, and linearized theory, where simple analytic results become available. Of particular interest is the presence of a single-planar interface that separates a bulk macroion phase from an extended solvent region. We calculate the apparent surface charge density that emerges due to the released counterions. Our model yields a comprehensive description of counterion partitioning between a planar macroion phase and a solvent region on the level of mean-field electrostatics in the absence of added salt ions.

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“…These solutions are explicit in form, but with no exception involve some unknown parameters which have to be determined numerically prior to the evaluation of electric potential. Most recently, Saboorian-Jooybari and Chen [19] presented the exact solution of the P-B equation in the parallel-plate geometry under the assumption of a zero electric potential at the center of geometry. This underlying assumption corresponds to the non-overlapping EDLs which are valid and consistent with the electroneutrality condition in natural settings.…”

Section: Introductionmentioning

confidence: 99%

Perturbation solutions for the nonlinear Poisson–Boltzmann equation with a high-order-accuracy Debye–Hückel approximation

Zhao

Wang

Zeng

2020

Z. Angew. Math. Phys.

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The Poisson-Boltzmann (P-B) equation is of fundamental importance in understanding solid-liquid electrolyte interfaces that are present in many fields. Due to the nonlinearity, it is usually challenging to find the explicit exact solutions of the P-B equation. The present work reports several perturbation solutions for the nonlinear P-B equation in the Cartesian and spherical coordinates. The new solutions contain a perturbation parameter from the high-order-accuracy approximation of the hyperbolic sine function and thus can apply to high zeta potential conditions. The comparison of the perturbation solutions with the traditional Debye-Hückel solutions and the full numerical solutions validates the robustness and accuracy of the perturbation solutions. The perturbation solutions are explicit and analytical and then can be used for a fast calculation of the EDL potential and interaction energy in versatile applications.

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Analytical solutions of the Poisson-Boltzmann equation within an interstitial electrical double layer in various geometries

Cited by 11 publications

References 50 publications

The role of disjoining pressure on the drying shrinkage of cementitious materials

The role of disjoining pressure on the drying shrinkage of cementitious materials

Counterion Release from Macroion Assemblies of Planar Geometry

Perturbation solutions for the nonlinear Poisson–Boltzmann equation with a high-order-accuracy Debye–Hückel approximation

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