New solution of the non-linear Poisson-Boltzmann differential equation for solid particle dispersions in dissymmetrical electrolytes

Q3 Materials Science JCIS open Pub Date : 2024-02-09 DOI:10.1016/j.jciso.2024.100103

Tayssir Hamieh

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Abstract

A new mathematical solution to the non-linear Poisson-Boltzmann differential equation for solid-liquid dispersions in presence of different dissymmetrical electrolytes was given. The analytical expressions of the surface and charge density of solid particles were given. The variations of electrostatic potential ψ (x) and charge density σ (x) of dispersed particles against the distance x were obtained. For colloidal particles in presence of E(m-n) electrolytes with $m \neq n$ with $m \geq 3, n \geq 3$ and for E(2–3) and E(3-2) electrolytes, the mean electrostatic potential as a function of the distance was numerically integrated by Mathematica program version 13.

The experimental study of silica suspensions in presence with the following electrolytes $N a C l$ , ${N a}_{2} {S O}_{4}$ , $C a {C l}_{2}$ , ${N a}_{3} {P O}_{4}$ , $A l {C l}_{3}$ , ${A l}_{2} {({S O}_{4})}_{3}$ , ${C a}_{3} {({P O}_{4})}_{2}$ , ${N a}_{4} {P_{2} O}_{7}$ and ${N a}_{5} {P_{3} O}_{10}$ led to confirm the theoretical predictions obtained from the analytical solution of Poisson-Boltzmann equation. The results obtained allowed to determine the surface potential as a function of pH of the suspension and the electrostatic potential versus the distance x. The variations of the dissociation coefficient of silica surfaces were determined. An important effect of the anion and cation valences of the dissymmetrical electrolytes on the surface charge density and potential was highlighted.

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不对称电解质中固体颗粒分散的非线性泊松-波尔兹曼微分方程的新解

对存在不同不对称电解质的固液分散体的非线性泊松-波尔兹曼微分方程给出了新的数学解决方案。给出了固体颗粒表面和电荷密度的解析表达式。得到了分散粒子的静电势 ψ (x) 和电荷密度 σ (x) 随距离 x 的变化。对于存在 E（m-n）电解质（m≠n，m≥3,n≥3）以及 E（2-3）和 E（3-2）电解质的胶体粒子，用 Mathematica 程序第 13 版对平均静电势随距离的变化进行了数值积分。通过对存在以下电解质的二氧化硅悬浮液进行实验研究：NaCl、Na2SO4、CaCl2、Na3PO4、AlCl3、Al2(SO4)3、Ca3(PO4)2、Na4P2O7 和 Na5P3O10。根据所获得的结果，可以确定表面电位与悬浮液 pH 值的函数关系，以及静电电位与距离 x 的关系。不对称电解质的阴阳离子价对表面电荷密度和电势的重要影响得到了强调。

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