Yield Stress & Time-Dependency: Practical Rheology

Building on the core rheological concepts discussed in application note RH146: Core Rheometry: Yield Stress, Time-Dependency, and Rheometry Tips [1], this complementary application note delves deeper into the rheological behavior of time-dependent materials. In particular, this application note focuses on yield stress. It explores the challenges of defining and measuring yield stress in time-dependent materials. In addition to theoretical insights, this note introduces practical test methods that are better suited for industrial environments, balancing scientific rigor with operational efficiency.

RH146 introduced new users to the core principles of rheology, emphasizing the importance of understanding time-dependency and yield stress in complex fluids. It demonstrated how the TA Instruments™ Discovery™ Core Rheometer, paired with RheoGuide™ Software, enables users to move beyond singlepoint measurements and gain deeper insights into material structure, stability, and performance.

A key takeaway of RH146 is that materials appearing similar under a single shear rate can behave very differently across a broader range of conditions [2]. This work also stresses the importance of selecting appropriate test methods and geometries, identifying common artifacts such as wall slip, and leveraging rheological data to enhance product quality and process efficiency.

The foundational work of RH146 sets the stage for more advanced investigations into yield stress behavior, including distinctions between static and dynamic yield stress, the impact of wall slip, and methods for predicting long-term structural stability in timedependent materials using the Discovery Core Rheometer or other rheometers in the TA Instruments portfolio.

Please note that details of the experimental setups are also available in RH146.

The following table summarizes the key rheological tests and findings discussed throughout RH146.

Table 1. Summary of Experiments

Hair gelFewer points (1/decade) saved 70% time with similar results

Test Category	Purpose	Material(s)	Key Findings
Flow Sweep Tests	Viscosity profile, yield stress	Topical ointment, hair gel, after-sun gel	Different materials require distinct equilibration times, especially at low shear rates; wall slip can significantly impact yield stress measurements
Constant Shear Rate Test	Time to reach steady state	Topical ointment	High shear: faster and low shear: slower Steady state periods
Thixotropy Tests	Assess time-dependency and recovery period	Paints, topical ointments	Significant differences in thixotropic behavior and recovery times of different paints and ointments
Optimized Flow Strategy	Accelerate testing without losing accuracy	Hair gel	Fewer points (1/decade) saved 70% time with similar results
Oscillatory Stress Sweep	Yield stress via G′/G″ crossover	Hair gel	Crossover stress increases with frequency
Flow Onset Analysis	Enhanced yield stress assessment	Hair gel	Onset values vary with data selection; consistency is essential

Yield Stress & Wall Slip

When conducting a flow sweep test to create a flow curve, certain materials such as suspensions or emulsions may show a drop in shear stress at low shear rates. Experienced rheologists recognize this as a clear indication of a phenomenon referred to as wall slip (see Figure 1).

A common approach to mitigate wall slip is to use roughened geometries, such as crosshatched or sandblasted plates, or a vane with a grooved cup.

To mitigate wall slip, TA Instruments offers two types of parallel plates with textured surfaces: crosshatched and sandblasted. Note that crosshatched plates have a surface roughness of around 500 μm, while sandblasted plates have a surface roughness of around 1.5 μm. For some materials, the surface roughness of sandblasted geometries may not be sufficient to fully mitigate wall slip. Figure 2 shows that the sandblasted geometry was not enough to fully mitigate wall slip for this body gel.

Additionally, in some cases, to hold the sample between parallel plates, lower gaps may be needed. If both roughness grades work in mitigating wall slip, use sandblasted geometry with lower gaps, as the teeth of crosshatched plates may cause secondary flows, which specifically at higher shear rates, will influence the data to some extent. In both Figure 2 and Figure 9 in RH146, at high shear rates, the values obtained by crosshatched plates are slightly lower than those obtained by smooth and sandblasted plates, which stems from the secondary flows mentioned.

For concentric cylinders, TA Instruments also offers sandblasted and grooved cups with a vane bob. When wall slip is observed, the vane and grooved cup is a suitable geometry to use. However, it should be taken into account that there will be effects of secondary flows at high shear rates between the vane blades, so their use should be limited to low shear rates where wall slip is present; for higher shear rates, sandblasted or smooth concentric cylinders would be the right choice.

However, as shown in Figure 3, even with roughened plates, stress values may still decline at very low shear rates. This behavior is often misinterpreted as wall slip, when in fact it may result from time-dependent effects. Since wall slip is already mitigated using crosshatched geometries [3, 4], the observed stress drop can be more accurately attributed to the material’s time-dependent behavior, which can lead to inaccurate yield stress evaluations if not properly accounted for.

When the maximum equilibrium time set for each point in the flow sweep is shorter than the time required to reach a steady state at low shear rates, the measured shear stress remains below its true steady-state value. This effect is particularly noticeable when the flow curve is measured from low to high shear rates. As the maximum equilibrium time set for the experiment increases, the shear stress values at very low shear rates also increases. This is because the material has more time to reach its steady-state and the shear stress values approach the actual steady-state value.

Static & Dynamic Yield Stress

In principle, yield stress can be measured at the solid-to-liquid or liquid-to-solid transition. The yield stress values obtained for these transitions are commonly referred to as static and dynamic yield stress, respectively [5, 6].

For time-independent materials, the difference between dynamic and static yield stress values is negligible in most cases. However, for time-dependent materials, this difference can be significant, and depending on the industrial process used, measuring dynamic or static yield stress or both will need to be decided [7, 5]. Static yield stress is often estimated from creep tests, while dynamic yield stress usually arises from appropriate model fitting to a steady-state flow curve, as shown in Figure 4.

Another test to determine yield stress, specifically in QA/QC, is the oscillatory stress amplitude sweep test. Since this oscillatory sweep test is conducted from low to high stress amplitude values, the yield stress obtained will be closer to the static yield stress value than the dynamic one (See Figure 5). Therefore, some discrepancy is expected between this value and the yield stress obtained from the flow curve—typically, the static yield stress is higher than the dynamic yield stress. As discussed in RH146, the results of this test are frequency-dependent: higher frequencies lead to a higher crossover point and G’ onset value. It’s also worth noting that the crossover point—where viscous behavior overtakes elastic behavior—occurs at a stress slightly above the yield stress. To estimate a value closer to the true yield stress, identify the onset of flow where G’ begins to decline, as shown in Figure 5.

Stress ramps are another tool to determine yield stress. However, since ramps are conducted under non-steady-state conditions, they are not ideal for characterizing time-dependent materials unless replicating the industrial process is required [8].

Note that the static yield stress of time-dependent materials depends on shear history, so a constant initial condition is necessary for a fair and reliable comparison as discussed earlier in RH146.

Creep tests can be employed to identify static yield stress. These tests involve imposing a constant shear stress for a fixed period on a fresh, pristine sample under constant initial conditions. The output is the time evolution of the shear rate or strain. This test is repeated for different shear stress values, starting from a relatively low value and increasing in small increments. When the shear rate approaches zero monotonically, the imposed stress is below the static yield stress. The tests will continue until the shear rate tends to a non-zero steady-state value, indicating that the imposed stress is above the static yield stress [5, 7, 9]. However, this method is extremely time-consuming and may not be popular in industry.

As an alternative, creep-recovery tests help determine how much deformation is recoverable (elastic) and how much is permanent (viscous). It provides insights in shorter periods into whether the applied stress is close to, below, or above the (static) yield stress. During the creep, stress is applied to the sample for a set period. In the recovery, the stress is removed, and the sample’s recovery is observed. The deformation curve is influenced by both the applied stress and the sample’s microstructure.

For purely viscous materials, deformation increases almost linearly during the creep phase and remains constant once the stress is removed, showing no elastic response. In contrast, purely elastic materials exhibit instantaneous deformation when stress is applied and return to zero during the recovery phase. Most materials exhibit both viscous and elastic properties, with the degree of each depending on the sample’s characteristics and the magnitude of the applied stress [10].

In Figure 6, at σ = 20 Pa, the hair gel shows instant elastic response when stress is applied and recovers almost fully when the stress is removed. When the stress is raised to σ = 60 Pa, the creep test curve changes from a step elastic shape to a curved viscoelastic shape, indicating a transition in the material’s behavior.

This transition signifies that the material is moving from a purely elastic response to a viscoelastic response, where it can no longer recover instantaneously and completely. The curved viscoelastic profile indicates that the material is approaching its yield stress.

As the stress increases to σ = 80 Pa in Figure 7, three distinct regions are observed in the strain response over time during the creep phase. Initially, there is an instantaneous step increase in strain, representing the immediate elastic deformation of the material. Following this, the material exhibits a viscoelastic response, characterized by a curved strain-time relationship where the strain increases at a decreasing rate. This phase reflects the material’s time-dependent elastic behavior. Finally, the material enters a linear viscous response phase, where the strain increases linearly with time, indicating a steady-state flow and the material’s viscous behavior [11].

When a material exhibits a linear viscous response at higher stresses, it indicates that the elastic component is no longer significant. In this scenario, the material behaves like a fluid, continuously deforming under stress. Once the stress is removed, the deformation remains constant, signifying that the applied stress is above the yield stress. For time-dependent materials, if the length of the creep and recovery periods changes, the observed response could also change. Therefore, choose recovery periods that are representative of the application or process.

Long-Term Stability

Many industries prefer to maintain their materials in a gel state during storage to ensure homogeneity. Additionally, a higher yield stress can prevent phase separation and sedimentation, maintaining the material’s structure [8]. This is particularly critical for products like pharmaceuticals, cosmetics, and food, where consistency is essential for performance and safety.

Frequency sweeps are quick and simple tests for predicting the long-term stability of materials and ensuring they remain in their desired state during storage and transport. At very low frequencies (i.e., long periods), the test can predict the long-term state of the material. In Figure 8, a frequency sweep test was conducted at a strain within the linear viscoelastic region for both the hair gel and a shampoo. Observe that for the hair gel, even at low frequencies, the storage modulus G’ stays above the loss modulus G’’), indicating it remains in a gel state. However, for the shampoo, there is a crossover, and at low frequencies, G’’ exceeds G’, indicating that the shampoo transitions out of the gel state.

In oscillatory tests, the phase angle (δ) measures the lag between the applied sinusoidal strain and the resulting stress, reflecting the material’s viscoelastic nature. A phase angle of 0° indicates purely elastic behavior, while 90° indicates purely viscous behavior. For viscoelastic materials, the phase angle (δ) falls between 0° and 90°.

In Figure 9, the frequency sweep of the same shampoo sample is presented in terms of phase angle. The phase angle directly indicates the balance between elastic and viscous behavior, simplifying the communication of the material’s properties. Observe that within the frequency range of 10 to 20 rad/s (around 3 Hz), the phase angle crosses 45°, indicating a transition from an elastically dominated response to a viscously dominated response.

In Figure 6, the hair gel shows an elastic dominant response at σ = 20 Pa stress. In contrast, the shampoo exhibits a viscousdominant behavior in the creep-recovery tests at this same stress (Figure 10). This indicates a much lower yield stress value for the shampoo, corroborating the results in Figure 8.