The Rheology Handbook. Thomas Mezger

       Figure 3.17: Viscosity functions of dispersions:

      dependence on the particle concentration (with the volume fraction solid Φ)

       Figure 3.18: Viscosity curve of a shear-thickening material showing a “dilatancy peak” at a high shear rate

      Note 1: Shear-thickening, time-dependent and independent of time

      Sometimes, the term “shear-thickening” is used to describe time-dependent flow behavior at a constant shear load (see Figure 3.37: no. 3; and Figure 3.39: left-hand interval). There is a difference between time-dependent shear-thickening behavior (see Chapter 3.4) and shear-thickening behavior which is independent of time (as explained in this section; see also Note 1 in Chapter 3.3.2). If no other information is given, the term should be understood in common usage as the latter one.

      Note 2: Dilatancy peak

      Sometimes with highly concentrated dispersions, shear-thickening does not occur until higher shear rates are reached. If this behavior is presented in a diagram on a logarithmic scale, the viscosity curve often shows initially shear-thinning behavior up to medium shear rates before a “dilatancy peak is occurring at higher shear rates finally (see Figure 3.18, in logarithmic scales).

      Example 1: Plastisols in automotive industry

      A PVC plastisol – as a paste-like micro-suspension – showed shear-thinning behavior in the range of γ ̇ = 1 to 100 s-1, and then a dilatancy peak at γ ̇ = 500 s-1. This may cause problems when it is sprayed as an automotive underbody coating or seam sealing (e. g. by blocking the flatstream spray nozzle).

      Example 2: Paper coatings

      After a certain shear-thinning range, highly concentrated preparations of paper coatings (suspensions showing a pigment content of about 70 weight-%) may display shear-thickening behavior beginning at around γ ̇ = 1000 s-1, often followed by a dilatancy peak. The peak was found to be shifted towards higher shear rate values for coatings showing a lower pigment concentration. Shear-thickening may cause problems during a coating process leading to coating streaks including the danger of tearing the paper [3.23].

      Example 3: Hair shampoos containing surfactant superstructures

      In the range of γ ̇ = 1 to 15 s-1 a shampoo displayed shear-thinning behavior and at γ ̇ = 30 s-1 occurred a dilatancy peak. The shampoo manufacturer wanted the shampoo to show a certain superstructure in this shear rate range, which corresponds to the process when flowing out of the shampoo bottle. Consumers then subconsciously believe that they have bought a liquid showing “body”. In this case, the goal was reached via superstructures, built up by visco-

      elastic surfactants. (VES; see also Chapter 9.1.1d: worm-like micelles).

      Note 3: Temperature-dependent shear thickening of elastomers .

      Different temperatures can lead to fundamentally different kinds of flow behavior of filled and unfilled elastomers and rubber). mixtures.

       Figure 3.19: Viscosity curve of a highly filled suspension showing immediate shear thickening when reaching the critical shear rate γ ̇ crit (“dilatant switch”)

      Example 4: Shear-induced or flow-induced dilatant behavior of elastomers

      A filled elastomer showed shear-thinning behavior at T = +80 °C across the whole shear rate range of γ ̇ = 0 to 100 s-1. At T = +40 °C it first indicated shear-thinning, followed by shear-

      thickening when exceeding γ ̇ = 50 s-1. At T = +23 °C finally, already at around γ ̇ = 10 s-1 pronounced shear-thickening was occurring. Possible reason: At low temperatures, shear-induced or flow- induced crystallization can be expected for this material. A result of this is shear thickening and hardening, [3.16]. See also Chapter 9.2.2: Shear-induced effects.

      Note 4: Composite materials as a “dilatant switch”

      Shear-thickening fluids (STF) were developed with the aim of immediate thickening as soon as a defined limiting value of loading is exceeded.

      Example 5: “Nano-fluids” or “smart fluids” for shock-resistant or bullet-proof materials

      These water-based dispersions are mixtures of PEG (polyethylene glycol) and colloidal silica particles (60 weight-%, with d = 400 nm, mono-dispersely distributed). After initial shear-thinning behavior showing η = 100 Pas at γ ̇ = 0.001 s-1 and η = 2 Pas at γ ̇ = 40 s-1, the viscosity value at the “critical shear rate value” of γ ̇ crit = 50 s-1 immediately steps upwards to η > 500 Pas; see Figure 3.19. At this limiting value of the loading velocity, the silica particles are agglomerating, abruptly forming a rigid “hydro-cluster” due to interparticle interactions.

      Applications: Shock-proof, stab-proof and bullet-proof protective clothing as a combination of this “nano-dispersion” with synthetic technical textile fabrics; reinforced technical polymers for special functions (e. g. as “nano-composite” STF-Kevlar”) [3.17].

      For “Mr. and Ms. Cleverly”

      Note 5: Increased flow resistance due to flow instabilities and turbulence

      Increased flow resistance can also occur due to hydrodynamic flow instabilities which may lead to secondary flow effects and even to turbulent flow behavior showing vortices at high shear rates. In this case, flow curves and viscosity curves will display as well higher values for shear stress and viscosity as well as higher curve slope values compared to curves measured at regular (i. e. laminar) flow conditions, therefore giving at the first glance an impression of shear-thickening behavior.

      When performing tests on liquids using concentric cylinder measuring geometries with a rotating inner cylinder (Searle method, see Chapter 10.2.1.2a) there is a critical upper limit between laminar and turbulent flow conditions in the circular gap. Exceeding this limit, secondary flow effects may occur for the reason of centrifugal forces or inertial forces due to the mass of the fluid. The critical limiting value can be calculated in the form of a Taylor number (Ta). The range of turbulent flow is also reached when the critical Reynolds number (Re) is exceeded. Re numbers represent the ratio between the forces of inertia and flow resistance. (More about Ta number and Re number: see Chapters 10.2.2.4 and 11.3.1.3.)

      Example 6: Turbulent flow of water

      Water was measured at different temperatures using a double-gap measuring geometry. The limiting value of the shear rate range of ideal-viscous flow behavior was found at

       γ ̇ = 1300 s-1 at T = +10 °C showing η = 1.3 mPas

       γ ̇ = 1000 s-1 at T = +20 °C showing η = 1.0 mPas

       γ ̇ = 800 s-1 at T = +30 °C showing η = 0.80 mPas

      In each viscosity curve at the mentioned upper limit of the shear rate a clear bend was observed, followed by a distinctive increase in the slope of the viscosity curve, indicating the begin of the turbulent flow range.

      Note 6: Observation and visualization of turbulent behavior

      Using