Interaction of colloidal particles

Hiroyuki Ohshima    Faculty of Pharmaceutical Sciences, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan

Abstract

In this chapter we discuss interactions between colloidal particles in an electrolyte solution such as drug particles of colloidal dimensions (nanometres to micrometres) and biological cells on the basis of the Derjaguin–Landau–Verwey–Overbeek theory of colloid stability. This theory assumes that two fundamental interactions are acting between two approaching colloidal particles, that is, the electrostatic repulsive interaction due to overlapping of the electrical double layers surrounding the particles and the van der Waals attractive interaction and that the balance of these two interactions determines the stability of a suspension of colloidal particles. We treat not only hard particles without surface structures but also soft particles, i.e., polyelectrolyte-coated particles, which serve as a model for biological cells.

Keywords

Interaction of colloidal particles

Electrical double layer interaction

van der Waals interaction

Surface potential

Hamaker constant

DLVO theory

Chapter Contents

1.1 Introduction   2

1.2 Potential Distribution Around a Charged Surface: the Poisson–Boltzmann equation   2

1.2.1 Hard Particle   3

1.2.2 Soft Particles   6

1.3 Electrical Double Layer Interaction Between Two Particles   8

1.3.1 Linear Superposition Approximation   9

1.3.2 Derjaguin’s Approximation   11

1.3.2.1 Two Spheres   12

1.3.2.2 Two Cylinders   13

1.4 van der Waals Interaction Between Two Particles   14

1.4.1 Two Molecules   15

1.4.2 A Molecule and a Plate   16

1.4.3 Two Parallel Plates   17

1.4.4 Two Spheres   18

1.4.5 Two Cylinders   19

1.4.6 Two Particles Immersed in a Medium   20

1.4.7 Two Parallel Plates Covered with Surface Layers   21

1.5 DLVO Theory of Colloid Stability   23

1.5.1 Total Interaction Energy Between Two Spherical Particles   23

1.5.2 Positions of a Potential Maximum and a Secondary Minimum   23

1.5.3 The Height of a Potential Maximum and the Depth of a Secondary Minimum   26

1.5.4 Stability Map   26

1.6 Conclusion   27

References   27

1.1 Introduction

The stability of colloidal systems consisting of charged particles can be essentially explained by the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory [1–12]. According to this theory, the stability of a suspension of colloidal particles is determined by the balance between the electrostatic interaction and the van der Waals interaction between particles. In this chapter we start with the electrical double layer around a charge particle in an electrolyte solution (Figure 1.1). We treat both hard particles and soft particles, i.e., polyelectrolyte-coated particles [8, 10, 13–15] (Figure 1.2). We discuss the electrostatic interaction between two approaching particles due to the overlapping of the electrical double layers around them. We then consider the van der Waals interaction between particles. Finally we discuss the stability of a colloidal suspension on the basis of the total interacting energy (the electrostatic energy and the van der Waals energy) between particles.

1.2 Potential distribution around a charged surface: the Poisson–Boltzmann equation

Around a charged colloidal particle immersed in an electrolyte solution, mobile electrolyte ions form an ionic cloud of thickness 1/κ (called the Debye length), κ being the Debye–Hückel parameter (Figure 1.1). As a result of Coulomb interaction between electrolyte ions and particle surface charges, in the ionic cloud the concentration of counter ions (electrolyte ions with charges of the sign opposite to that of the particle surface charges) becomes much higher than that of coions (electrolyte ions with charges of the same sign as the particle surface charges). The ionic cloud together with the particle surface charge forms an electrical double layer, which is often called an electrical diffuse double layer, since the distribution of electrolyte ions in the ionic cloud takes a diffusive structure due to thermal motion of ions. Electrostatic interactions between colloidal particles depend strongly on the distributions of electrolyte ions and the electric potential across the electrical double layer around the particle surface [1–12].

1.2.1 Hard particle

First we consider a uniformly charged plate-like hard particle immersed in a liquid containing M ionic species with valence zi and bulk concentration (number density) ni∞ (i = 1, 2 … M) (in units of m− 3). We take an x-axis perpendicular to the plate surface with its origin 0 so that the region x > 0 corresponds to the electrolyte solution (Figure 1.3(a)). From the electroneutrality condition, we have

i=1Mzini∞=0

  (1.1)

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

2ψdx2=−ρelxɛrɛ0

  (1.2)

where ɛr is the relative permittivity of the electrolyte solution, and ɛ0 is the permittivity of a vacuum. We assume that the distribution of the electrolyte ions obeys Boltzmann’s low, viz.,

ix=ni∞exp−zieψxkT

  (1.3)

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

elx=∑i=1Mzienix=∑i=1Mzieni∞exp−zieψxkT

  (1.4)

Combining Eqs. (1.2) and (1.4) gives

2ψdx2=−1ɛrɛ0∑i=1Mzieni∞exp−zieψxkT

  (1.5)

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

0=ψ0attheparticlesurface

  (1.6)

x→0asx→∞

  (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 at the particle surface as

ψdxx=0+=−σɛrɛ0

  (1.8)

If the potential ψ is low, viz.,

ieψkT≪1

  (1.9)

then Eq. (1.5) reduces to the following linearised Poisson–Boltzmann equation (Debye–Hückel...