# Evaluating Product Spreading Characteristics on Rotational Rheometer Using the Power Law Model

#### Introduction

The rheological properties of a product can influence how it is visually and texturally perceived by a consumer and how it is likely to behave during product use. For example, very Shear ThinningThe most common type of non-Newtonian behavior is shear thinning or pseudoplastic flow, where the fluid vis­cosity decreases with increasing shear.shear thinning materials will be highly responsive to changes in applied StressStress is defined as a level of force applied on a sample with a well-defined cross section. (Stress = force/area). Samples having a circular or rectangular cross section can be compressed or stretched. Elastic materials like rubber can be stretched up to 5 to 10 times their original length.stress while Newtonian materials will show much lesser dependence. Such a response is important when considering the ease of spreading or ‘spread-ability’.

The process of spreading causes a consequent reduction in the layer thickness as it is distributed over a wider surface area as shown in Figure 1. Since shear rate is equal to the applied velocity divided by the layer thickness, spreading can therefore not be attributed to a single shear rate.

A better way of assessing spread-ability is by characterizing the change in viscosity over a range of shear rates as shown in Figure 2. The region of interest is the Shear ThinningThe most common type of non-Newtonian behavior is shear thinning or pseudoplastic flow, where the fluid vis­cosity decreases with increasing shear.shear thinning or power law region since this describes how easily the material structure breaks down with applied shear. This region appears linear on a log-log plot of viscosity vs. shear rate with a constant gradient, but shows power law dependence when plotted on a linear scale.

Mathematically this region of the flow curve can be described using the Power Law or Ostwald de Waele Model given by Equation 1.

k is the consistency
n is the power law index
σ is the shear rate

Consistency has the units of Pasn but is numerically equal to the viscosity measured at 1 s-1. The power law index ranges from 0 for very Shear ThinningThe most common type of non-Newtonian behavior is shear thinning or pseudoplastic flow, where the fluid vis­cosity decreases with increasing shear.shear thinning materials to 1 for Newtonian materials.

This information can be represented on a chart similar to that shown in Figure 3. Materials with low k values and/or low n values should be easiest to spread.

#### Results and Discussion

Figure 4 shows the viscosity-shear rate curve for a number of commercial products and their corresponding fitting parameters, with graphical presentation of the latter in Figure 5.

Although toothpaste and hand cream have similar k values, hand cream has a much lower n value making it more Shear ThinningThe most common type of non-Newtonian behavior is shear thinning or pseudoplastic flow, where the fluid vis­cosity decreases with increasing shear.shear thinning and easier to spread. Conversely, syrup and chocolate sauce have much lower k values but are not Shear ThinningThe most common type of non-Newtonian behavior is shear thinning or pseudoplastic flow, where the fluid vis­cosity decreases with increasing shear.shear thinning hence they appear thick and sticky during application. The body lotion has both a relatively low k and n value making it much easier to apply. To quantitatively compare the StressStress is defined as a level of force applied on a sample with a well-defined cross section. (Stress = force/area). Samples having a circular or rectangular cross section can be compressed or stretched. Elastic materials like rubber can be stretched up to 5 to 10 times their original length.stress requirement for the spreading of a hand cream and syrup respectively at equivalent shear rates the values of n and k can be substituted into Equation 1. Considering a single shear rate of 1 s-1 which may equivocate with a thicker product layer than the StressStress is defined as a level of force applied on a sample with a well-defined cross section. (Stress = force/area). Samples having a circular or rectangular cross section can be compressed or stretched. Elastic materials like rubber can be stretched up to 5 to 10 times their original length.stress required to maintain flow at this shear rate is 279 Pa for the hand cream and 10 P a for syrup (σ = k at 1 s-1). At a shear rate of 1000 s-1 which would relate to a thinner layer of material resulting from the spreading process the StressStress is defined as a level of force applied on a sample with a well-defined cross section. (Stress = force/area). Samples having a circular or rectangular cross section can be compressed or stretched. Elastic materials like rubber can be stretched up to 5 to 10 times their original length.stress requirement increases to 734 Pa for the hand cream and 10,000 Pa for the syrup. This highlights importance of Non-NewtonianA non-Newtonian fluid is one that exhibits a viscosity that varies as a function of the applied shear rate or shear stress.non-Newtonian behavior in the spreading process.

#### Conclusion

A shear rate ramp test with a Power Law ModelThe power law model is a common rheological model to quantify (typically) the shear thinning nature of a sample, with the value closer to zero indicating a more shear thinning material.power law model fit was used to characterize the spread-ability of various commercial products using the power law fitting parameters k and n. Low values of k and n indicate lower viscosity and a greater degree of Shear ThinningThe most common type of non-Newtonian behavior is shear thinning or pseudoplastic flow, where the fluid vis­cosity decreases with increasing shear.shear thinning respectably, which will contribute to easier spreading.