The interaction forces between colloidal particles
6.1 Introduction
The interaction forces between charged surfaces and particles is of key relevance for many environmental and industrial systems and processes. These interactions can be between particles on the one hand, and flat or porous materials on the other, as in studies of particle deposition on surfaces, and in studies of particle capture in porous media. In this case the two materials/particles that are interacting across water will be different in their surface structure or chemistry (and thus have a different charge while in contact with the same bulk solution). This topic is called hetero-interaction and refers to interaction between dissimilar materials or surfaces.
On the other hand, homo-interaction refers to the interaction between materials or surfaces that have the same structure and chemistry (i.e., are similar), and therefore also have the same charge density on their surfaces. When they have different shapes and sizes, the interaction is still classified as homo-interaction. Though hetero-interaction may be more relevant in practice, homo-interaction is a more common topic of scientific study, and already presents many challenges.
Indeed, when particles that are the same are brought in contact, the situation can already be breathtakingly complex, with an an attraction changing at some separation to repulsion, again attraction, and again repulsion. This situation is illustrated in Fig. 6.2b for a surface charge density ofΣ =15 mC/m2 (not shown is the region of weak attraction for distances beyond 10 nm). This example illustrates the complexity of the problem of the interaction of electrostatics, surface ionization, and Van der Waals forces, even when the two surfaces are the same.
It is interesting to notice how these curves for the interaction force versus distance depend on pH and salt concentration, 𝑐∞. As we will explain, it is possible that at a certain pH or𝑐∞ two particles are repulsive (the particles are dispersed), but at a different pH or salt concentration, an attraction develops between them (the particles will now ‘coagulate’), but the attraction disappears again when the original pH/salt concentration is reinstituted. This behaviour indicates that an attractive ‘minimum’ can be reversed upon changing solution conditions, i.e., in this case coagulation is reversible. This theoretical prediction is also observed experimentally (‘ repeptization’).
An interesting problem is to what extent dynamic effects must be incorporated in this theory. Particles that approach a surface, or approach one another, do so with a certain (initial) velocity, and the redistribution of ions in the gap between the particles may not be fast enough for equilibrium to remain established during the entire period of encounter, and likewise the surface charge may not adjust fast enough for there to be equilibrium at each
Introduction 133 separation. Thus, during particle collusion the surface charge may remain constant even when according to chemical equilibrium the charge would change.i
This advanced topic of the dynamics of particle interaction will not be discussed here, and we will limit the explanation to the forces between particles, with chemical equilibrium always established. We first explain the interaction of equally charged surfaces, where we first consider surfaces with a fixed, or constant, charge (CC condition), and then analyze the interaction between ionizable surfaces, i.e., surfaces of which the charge changes when particles come closer (charge regulation, CR). After that we address hetero-interaction, i.e., the interaction between different materials, both for CC and CR boundary conditions.
To calculate the electrostatic contribution to the attractive or repulsive forces between colloidal particles, we must solve the Poisson-Boltzmann (PB) equation in the entire space between and around the two charged surfaces. But thanks to the Derjaguin approximation, mathematically we only have to solve a 1D planar geometry to describe the full problem of the interaction between real 3D particles. However, even for a 1:1 salt solution there are no good analytical solutions to the PB equation that work at all separations and charge densities, let alone for asymmetric salts or when ion volume effects are included.
In this chapter we first discuss three approximations to the (1D planar) PB equation:
the Gregory approach valid at low potentials, the more general Ettelaie-Buscall approach, and the Donnan approach for surfaces that are very near. In Fig. 6.3 we also show results obtained from numerical solution of the full PB-equation. To analyze colloidal stability, we combine the electrostatic contribution to the interaction force, with the Van der Waals force (attractive for homo-interaction), to describe the total colloidal interaction force. We first discuss the force between flat surfaces (called disjoining pressure), and then move to curved surfaces (particles) and include the Van der Waals force. For flat surfaces we also discuss hetero-interaction.
It was F.G. Donnan who before 1911 was the first to bring up the idea that the stability of colloids is likely due to a repulsive electrostatic pressure because of an increase in ion concentration in the gap between two particles, and that work must be done to bring particles into close contact against this electrical force (reported in R. Ellis, 1912). The complete
iIn AFM studies of the homo-interaction of two charged materials, an intriguing phenomenon is the ‘snap-off’
force: two surfaces that are repulsive all the way until they are pushed into contact, when they are subsequently pulled apart, they are found to adhere. One explanation is that during approach, the particle charge remains non-zero, leading to repulsion, but during contact, the charge decreases to zero or close to zero in the region of contact (a consequence of the establishment of chemical equilibrium and EDL overlap for ionizable surfaces, see §6.3). When the two particles are now pulled apart, the electrostatic repulsion drops away, and only the attractive Van der Waals force remains, so the particles now stick together, hence the required snap-off force to pull them apart (discussed in Biesheuvel, 2002).
134 The interaction forces between colloidal particles
theory of colloidal interaction is therefore best called DVDW theory, referring to Donnan and Van der Waals.
The disjoining pressure,𝑃, has the unit of Pa (=N/m2) and is the pressure that must be applied to keep two flat surfaces at a certain distance. Thus when the surfaces are attractive, and we actually must pull on them,𝑃is negative. Contributions to𝑃arise from electrostatics (Donnan, ion entropy),𝑃
e, and from the Van der Waals force,𝑃
vdw. Integration of the curve of pressure,𝑃, against separation of the surfaces,𝐷, results in an interaction energy,𝑉(unit J/m2), according to
𝑉=
∫ ∞
𝐷
𝑃d𝐷 . (6.1)
For curved surfaces, for instance spherical particles with a radius 𝑎, this energy of flat surfaces,𝑉, is proportional to the disjoining force, 𝐹, between these curved surfaces (unit N) with the proportionality factor dependent on𝑎. This is the Derjaguin approximation, which is valid when the Debye length is small compared to the curvature of the particles, a condition that is almost always applicable.
We can integrate𝐹once again over separation𝐷to obtain the energy between the curved objects (particles),𝐸(unit J), according to
𝐸 =
∫ ∞
𝐷
𝐹d𝐷 (6.2)
And finally, this energy can be integrated one more time over𝐷(involving an exponential dependence on energy𝐸) to obtain the stability ratio𝑊, a property that can be correlated with experiments on the coagulation of a dispersion of small particles. (Other words for coagulation are: flocculation, agglomeration, aggregation, phase separation.) When such a dispersion does not coagulate, i.e., it is ‘stable,’ the attractive forces are apparently not strong enough to overcome the repulsive forces, or there is an attraction at close distances but there is a repulsive barrier that is large enough.ii
iiA dispersion is in between a solution and a suspension in terms of particle size. A solution consists of solvent and smaller molecules, and macromolecules of perhaps a few kDa. The term suspension refers to a mixture of a fluid and suspended larger particles, with sizes at least 100 nm. In a gravity field these particles will sediment over time. A dispersion is in between these two classes, and particles in a dispersion have sizes from a few nm to perhaps 1𝜇m. They do not sediment due to gravity but remain dispersed.
Colloidal interaction of particles with a fixed surface charge 135