Making the connection - particle size, size distribution and rheology

Jamie Fletcher, Applications Specialist
Adrian Hill, Rheometry Technical Specialist
Malvern Instruments Ltd, Enigma Business Park, Grovewood Road, Malvern, Worcestershire, UK, WR14 1XZ

A number of factors influence the rheology of a suspension, including particle size, particle size distribution, and the volume fraction of solids present. Here we examine the relationship between rheology and particle size parameters. Commonly used rheological terms are described and we present data from example systems to illustrate key points.

Terminology

Viscosity (‘thickness’) is the term that describes resistance to flow. High viscosity liquids are relatively immobile when subjected to shear (a force applied to make them move), whereas low viscosity fluids flow relatively easily. Measurement of viscosity, and other rheological properties, can be made using either capillary or rotational rheometers, the choice of system depending on the properties of the material being tested and the data required.

‘Shear rate’ defines the speed with which a material is deformed. In some processes (spraying for example), materials are subjected to high shear rates (>105 s^-1); in others, (such as pumping or levelling), the associated shear rate is low (10^-1 – 10¹ s^-1). High shear rates tend to occur when a material is being forced rapidly through a narrow gap.

If viscosity remains constant as shear rate increases, a fluid is described as being Newtonian. Non-Newtonian fluids, which fail to exhibit this behaviour, fall into one of two categories – shear thinning or shear thickening. With shear thinning materials viscosity decreases as shear rate increases: application of shear leads to a breakdown of the material’s structure so that it flows more readily. Most fluids and semi-solids fall into this group. Conversely, the viscosity of shear thickening materials increases at rising shear rates.

With regard to suspensions, the volume fraction and the maximum volume fraction can also be influential. It is possible to think of the maximum volume fraction (highest volume of particles that can be added to a fluid) as the amount of free space the particles have in which to move around, and the implications on viscosity are discussed below.

Effect of particle size

Maintaining a constant mass of particles in a suspension while reducing the particle size of the solid phase leads to an increase in the number of particles in the system. The effect of this change on the viscosity of the system across a range of shear rates is shown in figure 1. These data are for latex particles in a pressure-sensitive adhesive and the shape of the graph indicates:

• the fluid is shear thinning (viscosity decreases at higher shear rates)
• viscosity tends to be greater with smaller particles

Fig. 1: The impact of particle size on viscosity.

A higher number of smaller particles results in more particle-particle interactions and an increased resistance to flow. Clearly as shear rate increases, this effect becomes less marked, suggesting that any particle-particle interactions are relatively weak and broken down at high shear rates.

Figure 2 shows data for a talc/epoxy system. In the absence of talc, the epoxy system is Newtonian; adding coarse talc leads to an increase in viscosity, but still the system is Newtonian. The addition o finer talc results in a further, more significant, increase in viscosity, particularly at low shear rates. Colloidal repulsion between a relatively large number of particles gives structure to the fluid, increasing resistance to flow. As in the previous example, this relatively weak structure is broken down at high shear rates. The fluid has become shear thinning.

Fig. 2: The impact of particle size on flow behavior.

Volume fraction

The effects of volume fraction and maximum volume fraction on viscosity are described using the Krieger-Dougherty equation:

where η is the viscosity of the suspension, ηmedium is the viscosity of the base medium, φ is the volume fraction of solids in the suspension, φm is the maximum volume fraction of solids in the suspension and [η] in the intrinsic viscosity of the medium, which is 2.5 for spheres.

This correlation indicates an increase in viscosity with increasing volume fraction. As the volume fraction of solids in the system goes up: the particles become more closely packed together; it becomes more difficult for them to move freely; particle-particle interactions increase; and resistance to flow (viscosity) rises. As the volume fraction nears maximum for the sample, viscosity rises very steeply.

As well as influencing the absolute value of viscosity, volume fraction also affects the nature of the relationship between shear rate and viscosity for the system - flow behaviour. Suspensions with relatively low volume fraction tend to behave as Newtonian fluids, with viscosity independent of shear rate. Increasing volume fraction leads to shear-thinning behaviour. The transition is illustrated in Figure 3 for a latex/pressure-sensitive adhesive system.

Fig. 3: Viscosity as a function of shear rate for different volume fractions.

At the lowest volume fraction the system is almost Newtonian. As volume fraction increases, shear-thinning behaviour becomes evident. Increased volume fraction results in more particle-particle interaction, and resistance to flow increases. The forces between particles are, however, broken down at high shear rates.

A further transition in flow behaviour occurs as volume fraction increases to more than ≈50% of maximum volume fraction. At these solids loadings the free movement of particles is significantly hindered as collisions between particles increase and the system becomes more congested. As shear rate increases, the particles are trying to move more rapidly and the effect becomes more pronounced. Viscosity therefore increases with shear rate; the system is shear thickening at very high shear rates.

Distribution

Particle size distribution (PSD) influences particle packing: a polydisperse population with a broad size distribution packs more closely than a monodisperse sample. The effects on viscosity can be explained with reference to the Krieger-Dougherty equation (see above). For a monodisperse sample the maximum volume fraction is around 62%. With a polydisperse sample smaller particles can fill gaps between larger ones and the maximum volume fraction is greater – around 74%. Increasing the PSD for any given volume fraction of solids will reduce the viscosity of the system. PSD can be a valuable tool for manipulating the viscosity of a system that has a fixed volume fraction.

Viscosity as a function of fraction of large or small talc particles is shown for an epoxy in figure 4. In this example a synergistic effect is seen when particles of both sizes are present at a certain concentration. The resulting viscosity is lower than that achieved using a monodisperse sample of either sized talc.

Fig. 4: Viscosity as a function of polydispersity.

These results show how particle size distribution can be used to manipulate viscosity. If the requirement is for a higher solids loading but the same viscosity, then this can be achieved by broadening the particle size distribution. Conversely, viscosity can be increased by using particles with a narrower size distribution.

In conclusion

It is clear that particle size and size distribution data can be valuable when developing products with specific rheological properties. Clear relationships between particle size, particle size distribution and volume fraction, and viscosity, allow key physical parameters of the suspension to be tuned to meet product specifications.