Attractive Forces

There is a fundamental difference between the rheological behaviour of hard sphere or repulsive particle suspensions and attractive particle suspensions due to the attractive bond between particles. The bonds between particles must be broken in order to pull the particles apart to allow flow to occur. The result of the attractive bonds between particles is that an attractive particle network is formed when the suspension is at rest. The attractive bonding produces material behaviour that is characterized by viscoelasticity, a yield stress (minimum stress required for flow) and shear thinning behaviour.

The shear thinning of an attractive particle network is more pronounced than for hard sphere suspensions of the same particles at the same volume fraction

and is caused by a different mechanism. The mechanism for shear thinning is illustrated in Figure 5.18. At rest, the particle network spans the entire volume of the container and resists flow. At low shear rates, the particle network is broken up in to large clusters that flow as units. A large amount of liquid is trapped within the particle clusters and the viscosity is high. As shear rate is increased and hydrodynamic forces overcome interparticle attraction, the particle clusters are broken down into smaller and smaller flow units releasing more and more liquid and reducing the viscosity. At very high shear rates the particle network is completely broken down and the particles flow as individuals again almost as if they were non-interacting.

Greater magnitude interparticle attraction results in increased viscosities at all shear rates. Thus stronger attraction between particles results in higher viscos- ities. Attractive particle networks also exhibit a yield stress (minimum stress required for flow) because there is an attractive force [as shown in Figure 5.2(a) and (b)] which must be exceeded in order to pull two particles apart. The yield stress of the suspension also depends upon the magnitude of the attraction, with stronger attraction resulting in higher yield stresses. Figure 5.19 shows an example of the yield stress of alumina suspensions as a function of pH. The maximum in yield stress corresponds with the IEP of the powder. At low salt concentration, as the pH is adjusted away from the IEP the EDL repulsion increases as the zeta potential increases (see Figure 5.7) thus reducing the overall attraction and decreasing the yield stress. As the salt concentration is increased at pH away from the IEP, the magnitude of the zeta potential decreases (see Figure 5.7) and the EDL repulsion decreases. As such, the resulting overall interaction is attractive and the attraction increases as salt content is increased.

1 10 100 1 000 10 000 100 000 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Volume fraction Relative viscosity φ actual φ eff (a) (b)

Figure 5.17 (a) Illustration of suspension of particles with volume fraction 0.4 (grey circles) with repulsive interaction extending to the dotted line, resulting in an effective volume fraction of 0.57. (b) Relative viscosity of suspensions of repulsive particles (black dots and dotted line) as a function of actual volume fraction. When the rheological results are plotted as a function of effective volume fraction (open dots) the data maps onto the Quemada model (solid line)

0.1 1 10 100 1000 100 00 1000 100 10 1 0.1 0.01 Shear rate (s–1) Suspensi on vi scosi ty ( P a s) shear thinning of hard sphere suspension shear thinning of attractive particle network

Figure 5.18 Comparison of typical shear thinning behaviour of attractive particle net- work with less pronounced shear thinning of hard sphere suspensions. The attractive particle network is broken down into smaller flow units as the shear rate is increased

0 50 100 150 200 250 13 11 9 7 5 pH

Yield stress (Pa)

0.01 M 0.1 M 0.3 M 1.0 M

alumina in KCl solution

Figure 5.19 Yield stress of 25 vol % alumina suspensions (0.3mm diameter) as a function of pH and salt concentration. (Data from Johnsonet al., 1999)

The result presented in Figure 5.19 indicates that the yield stress increases as salt is increased at pH away from the IEP consistent with the force predictions.

It has been pointed out in the previous section that the rheological behaviour of hard sphere suspensions (non-interacting particles) was not influenced by the size of the particles. This is not true of attractive particle networks. When the particles in a suspension are attractive, smaller particle size results in increased rheological properties such as yield stress, viscosity and elastic modulus (described next). The influence of particle size can be determined by considering that the rheological properties of attractive particle networks depend upon the strength of the bond between particles and the number of bonds per unit volume that need to be broken. For example, consider the shear yield stress:

tY/

Number of bonds

Unit volume
Strength of bond ð5:19Þ

The strength of the bond of an attractive particle network increases linearly with particle size as indicated by Equations (5.8) and (5.11).

Strength of bond/ x ð5:20Þ

This would make one think that the larger size particles result in suspensions with greater yield stress, viscosity and elastic modulus. It is the influence of the number of bonds per unit volume that produces the opposite result. The number of bonds that need to be broken per unit volume depends upon the structure of the particle network and the size of the particles. In the first instance we assume that the structure of the particle network does not vary with particle size. (Details of the aggregate and particle network structure are beyond the scope of the present text.) Then the number of bonds per unit volume simply varies with the inverse cube of the particle size:

Number of bonds Unit volume /

1

x3 ð5:21Þ

When the contributions of the strength of the bond and the number of bonds to be broken are considered one finds that the rheological properties such as yield stress, viscosity and elastic modulus vary inversely with the square of the particle size: tY/ 1 x3
x   /_x1₂ ð5:22Þ

Many experimental measurements confirm this result although there is consider- able variation from the inverse square dependence in many other cases. One example of well controlled experiments that confirm the inverse square depen- dence of the yield stress on the particle size is shown in Figure 5.20.

When a stress less than the yield stress is applied to an attractive particle network, the network responds with an elastic-like response. The attractive bonds between the particles are stretched rather than broken and when the stress is removed the particles are pulled back together again by the attractive bonds and the suspension returns to near its original shape. Because the stretching and

breaking of bonds is a statistical phenomenon, pure elasticity is not usually achieved, rather the attractive particle network is a visco-elastic material display- ing behaviour characteristic of both solids and fluids. Some of the energy imparted to deform the material is stored elastically and some of the energy is dissipated by viscous mechanism.

5.8 NANOPARTICLES

Nanoparticles are finding application in many areas of technology due to their unique properties. The very high ratio of surface atoms to bulk atoms is the primary reason for the unique properties of nanoparticles. Due to the large surface area and quantum effects related to very small nanoparticles, the optical, electronic, and other properties are quite different than for larger particles of the same materials. Because of the unusual properties of nanoparticles, there are numerous emerging applications and processes where nanoparticles will be used. Examples of applications of nanoparticles include such diverse topics as: anti-reflective coatings;

fluorescent labels for biotechnology; drug delivery systems;

clear inorganic (ZnO) sunscreens; high performance solar cells; catalysts; 10 100 1000 10 000 1000 100 Particle size (nm)

Yield stress (Pa)

alumina at pH 9 (its IEP)

Figure 5.20 Yield stress of alumina suspensions at their IEP as a function of particle size. The best fit line has a slope of2:01, correlating quite well with the predicted inverse square particle size dependence as per Equation (5.22). (Data from Zhouet al., 2001).

high density magnetic storage media; high energy density batteries;

self cleaning glass; improved LEDs;

high performance fuel cells; nanostructured materials.

The success of these potential applications for nanoparticles relies heavily on the ability to efficiently produce, transport, separate and safely handle nanoparticles. The concepts presented in this chapter on fine particles and colloids provide a starting point for dealing with these issues.

5.9 WORKED EXAMPLES

WORKED EXAMPLE 5.1