# Using the Power Law Model to Quantify Shear Thickening Behavior

While shear thinning takes place in most polymer structured materials and suspensions, certain materials also show shear thickening behavior where there is an increase in viscosity with increasing shear stress or shear rate. This can also be referred to as dilatancy, and even though this refers to a particular shear thickening mechanism, the terms are used interchangeably most of the time. In most cases, shear thickening takes place over several years of shear rates, and there can be a region of shear thinning at higher and lower shear rates.

Typically, particulate suspensions or dispersions containing a very high concentration of pastes, solid particles or associative polymers such as HEUR, HASE polymers display shear thickening. Materials that display shear thickening are less common in industrial applications compared to materials that exhibit shear thinning. However, shear thickening materials can pose severe processing problems. Materials that are subjected to orientation or micro-structural changes on shear application will show shear thickening due to an increase in flow resistance.

In the case of suspensions, this normally occurs in materials that show shear thinning behaviour at lower shear stresses and shear rates. At a critical shear rate or shear stress, the ordered flow that causes shear thinning is interrupted, and the so-called jamming or hydro-cluster formation can take place. This causes an increase in the observed viscosity and a transient solid-like response. Shear thickening can also take place in polymers, especially in amphiphilic polymers, which may stretch or open up at high shear rates, exposing sections of the chain that can form temporary intermolecular links.

## Power-Law Model

The power-law model is used for mathematically defining the shear thickening behavior.

Where:
k is the consistency
η is the power law index
σ is the shear stress
ý is the shear rate

For shear thickening fluids, η is greater than 1.

It is important to note that an increase in viscosity at high shear rates may take place through a phenomenon like fluid turbulence. This effect occurs with lower viscosity fluids and can be determined using Reynolds number calculations.

## Experimental Procedure

By creating a table of shear rate tests and studying the curve obtained by fitting a power law model, the shear thickening behavior of a 75 % w/w corn starch/water suspension mixture was studied. A Kinexus rotational rheometer, equipped with a Peltier plate cartridge and a roughened parallel plate measuring system, was employed to perform rotational rheometer measurements at 25 °C, using previously configured sequences in the rSpace software.

In order to make sure that both the samples were subjected to a controllable and consistent loading protocol, a standard loading sequence was used. Using an equilibrium table of shear rates test between 0.1 and 100 s-1, the flow curve was produced. A power law model was then fitted to a manually selected part of this curve.

## Results and Discussion

The viscosity-shear rate profile for the corn starch dispersion is shown in Figure 1. At low shear rates, the sample displays shear thinning behavior. In contrast, a sharp increase in viscosity, characteristic of shear thickening behavior, is seen at a critical shear rate of around 8 s-1.

By fitting a power law model to the data between 0.15 and 6.5 s-1, the obtained value for the power law index η is 0.57, which confirms the shear thinning behavior (η< 1). When the same model is fitted to the data between 10 and 20 s-1, the value of η is 3.01, which shows considerable shear thickening (η>1).

It is important to note that cylindrical or cone-plate geometry can also be used. If the material is likely to show wall slip effects, a sand-blasted geometry must be considered. Larger geometries are beneficial for measurements at low torques that are more likely to be encountered at low shear stresses and rates. Deploying a solvent trap is also suggested for these tests, as solvent evaporation around the measuring system edges can invalidate the test, especially while working at high temperatures.

Figure 1. Viscosity-Shear rate dependence of a corn starch-water mixture.

## Conclusion

The evaluated corn starch-water mixture showed strong shear-thickening behavior beyond 8 s-1 as corroborated by the power law index (η), which provided a value of 3 for data between 10 and 20 s-1.

This information has been sourced, reviewed and adapted from materials provided by Malvern Panalytical.

## Citations

• APA

Malvern Panalytical. (2019, September 03). Using the Power Law Model to Quantify Shear Thickening Behavior. AZoM. Retrieved on October 19, 2019 from https://www.azom.com/article.aspx?ArticleID=12171.

• MLA

Malvern Panalytical. "Using the Power Law Model to Quantify Shear Thickening Behavior". AZoM. 19 October 2019. <https://www.azom.com/article.aspx?ArticleID=12171>.

• Chicago

Malvern Panalytical. "Using the Power Law Model to Quantify Shear Thickening Behavior". AZoM. https://www.azom.com/article.aspx?ArticleID=12171. (accessed October 19, 2019).

• Harvard

Malvern Panalytical. 2019. Using the Power Law Model to Quantify Shear Thickening Behavior. AZoM, viewed 19 October 2019, https://www.azom.com/article.aspx?ArticleID=12171.