Electrical Phenomena at Interfaces and Biointerfaces

Fundamentals and Applications in Nano-, Bio-, and Environmental Sciences

Hiroyuki Ohshima

• Publisher: Wiley
• Year: 2012
• Print ISBN:
  9780470582558
• eBook ISBN:
  9781118135426
• Edition: 1
• Topic: Sciences physiques
• Subtopic: Chimie physique et théorique

This book bridges three different fields: nanoscience, bioscience, and environmental sciences. It starts with fundamental electrostatics at interfaces and includes a detailed description of fundamental theories dealing with electrical double layers around a charged particle, electrokinetics, and electrical double layer interaction between charged particles. The stated fundamentals are provided as the underpinnings of sections two, three, and four, which address electrokinetic phenomena that occur in nanoscience, bioscience, and environmental science. Applications in nanomaterials, fuel cells, electronic materials, biomaterials, stems cells, microbiology, water purificiaion, and humic substances are discussed.

PART I: FUNDAMENTALS

1

POTENTIAL AND CHARGE OF A HARD PARTICLE AND A SOFT PARTICLE

Hiroyuki Ohshima

1.1 INTRODUCTION

When a charged colloidal particle is immersed in an electrolyte solution, mobile electrolyte ions form an ionic cloud around the particle. As a result of electrostatic interaction between electrolyte ions and particle surface charges, in the ionic cloud the concentration of counterions (electrolyte ions with charges of the sign opposite to that of the particle surface charges) becomes very high, while that of coions (electrolyte ions with charges of the same sign as the particle surface charges) is very low. Figure 1.1 schematically shows the distribution of ions around a charged spherical particle of radius a. The ionic cloud together with the particle surface charge forms an electrical double layer. Such an electrical double layer is often called an electrical diffuse double layer since the distribution of electrolyte ions in the ionic cloud takes a diffusive structure due to the thermal motion of ions. The electrostatic interaction between colloidal particles and the motion of colloidal particles in an external field (e.g., electric field and gravitational field) depend strongly on the distributions of electrolyte ions and of the electric potential across the electrical double layer around the particle surface [1–5].

Figure 1.1. Electrical double layer of thickness 1/κ around a spherical charged particle of radius a.

1.2 THE POISSON–BOLTZMANN EQUATION

Consider a uniformly charged particle immersed in a liquid containing N ionic species with valence z_i and bulk concentration (number density)

(i = 1, 2 … N) (in units of cubic meter). From the electroneutrality condition, we have

(1.1)

The electric potential ψ(r) at position r outside the particle, measured relative to the bulk solution phase, where ψ is set equal to zero, is related to the charge density ρ_el(r) at the same point by the Poisson equation, viz.,

(1.2)

where Δ is the Laplacian, ε_r is the relative permittivity of the electrolyte solution, and ε_o is the permittivity of a vacuum. We assume that the distribution of the electrolyte ions n_i(r) obeys Boltzmann’s law, viz.,

(1.3)

where n_i(r) is the concentration (number density) of the ith ionic species at position r, e is the elementary electric charge, k is Boltzmann’s constant, and T is the absolute temperature. The charge density ρ_el(r) at position r is thus given by

(1.4)

Combining Equations 1.2 and 1.4 gives

(1.5)

This is the Poisson–Boltzmann equation for the potential distribution ψ(r), which is subject to the following boundary conditions:

(1.6)

and

(1.7)

If the internal electric fields inside the particle can be neglected, then the surface charge density σ of the particle is related to the potential derivative normal to the particle surface as

(1.8)

where n is the outward normal at the particle surface.

1.3 LOW POTENTIAL CASE

If the potential ψ is low, viz.,

(1.9)

then Equation 1.5 reduces to the following linearized Poisson–Boltzmann equation (Debye–Hückel equation):

(1.10)

with

(1.11)

where κ is called the Debye–Hückel parameter. The reciprocal of κ (i.e., 1/κ), which is called the Debye length, corresponds to the thickness of the double layer. Note that

in Equations 1.5 and 1.10 is given in units of cubic meter. If one uses the units of M (mole per liter), then

must be replaced by 1000

, N_A being Avogadro’s number. Expressions for κ for various types of electrolytes are explicitly given in Table 1.1.

Linearized Equation 1.10 can be solved for particles of various shapes. Table 1.2 gives the potential distribution for a planar surface, a sphere of radius a, and a cylinder of radius a, which can be obtained by solving Equation 1.10 (with Δ = d²/dx² for a planar surface, Δ = d²/dr² + 2/r·d/dr for a sphere, and Δ = d²/dr² + 1/r·d/dr for a cylinder) subject to Equations 1.6 and 1.7, where x is the distance from the planar surface located at x = 0 and r is the distance from the sphere center or the cylinder axis. Table 1.2 also shows the surface potential ψ_o/surface charge density σ relationship, which can be obtained by substituting ψ into Equation 1.8.

TABLE 1.1. Debye–Hückel Parameter for Various Electrolytes

Symmetrical electrolyte of valence z and bulk concentration n

1-1 symmetrical electrolyte of bulk conce...

Table of contents

• Cover
• Title page
• Copyright page
• PREFACE
• CONTRIBUTORS
• PART I: FUNDAMENTALS
• PART II: APPLICATIONS IN NANO- AND ENVIRONMENTAL SCIENCES
• PART III: APPLICATIONS IN BIOSCIENCES
• Index