Quantifying Yield Points and Limits on Rheology Curves
Properties such as yield stress, the onset rate for shear thinning or the limit of linear viscoelasticity are not well-defined values in the same way that viscosity at a defined shear rate can be. Take a look at the curve below which shows a typical viscosity vs shear stress profile where a significant yield is occurring.
Figure 1: An defined yield point is not easy to identify
You can see that the yield could be said to occur at any point within the stress range marked “yield zone”. I am often asked about the best way to deal with this: how do we get some meaningful, reproducible numbers from the curves? Here are some typical approaches:
Method One: Threshold Value
You need to identify the plateau value first; see if your software has the facility to select some plateau data-points and obtain an mean value. Then set your threshold value as a defined percentage below this (see figure 2 – dotted line). Your yield stress is the interpolated stress value at which your curve crosses this threshold.
Figure 2: Dotted line shows "Threshold" value
Method Two: Extrapolated onset
This method works best with Viscosity vs Shear Rate curves where you want to identify the onset of shear-thinning or Power Law behaviour. Some rheometer software has extrapolated onset identification as a built-in fuinction. The software should guide you through it but, in essence, you fit a straight line to both the lower Newtonian plateau and the shear thinning regions of the curves and hit the “identify onset” or similar button. The crossover point of the two lines is then reported and “stamped” on the graph (see fig 3).
Figure 3: Extrapolated onset
Method Three: Threshold Gradient
If your software has the ability to plot the derivative of a curve then you can set a threshold gradient, the crossing of which signifies the plateau end, then identify and interpolate the point at which the gradient of the log-log curve passes, for example, -0.5 as shown in fig 4.
Figure 4: Threshold Gradient and Point-of-Inflection Methods
Method Four: Point of Inflection
This is good for viscosity vs stress curves. Plot the gradient curve and identify the stress value at which the gradient curve shows a negative peak (see fig 4.).
This signifies the point of inflection on the viscosity profile.
As for your choice of method, this should be governed by:
The quality of the correlation of the obtained value to the observed flow behaviour. Sometimes you may want to know at what stresses the yielding process starts. If, for example, you have a product that relies on it’s yield stress to keep it in a certain position or shape for a long period of time then even the first stirrings of yield may be a problem. Here I would probably go for Method 1 and set a fairly close threshold value. Alternatively, if your interest is in more “brutal” initiations of flow (imagine scooping some skin cream from a tub) then the contribution of yield stress to texture would be more closely correlated to the yield at the point of inflection – the point where the yielding process is at it’s greatest.
Ease of execution
Derivative curves exaggerate the noise in the data so you need to ensure you have a good quality curve from the start. This is usually more of a problem for viscosity profiles rather than oscillatory tests, so you may need to put a bit of work into your low-shear data collection or choose a non-gradient method such as Method 1 or 2. For oscillatory “linearity checks” such as stress or strain sweeps you may not get a point of inflection easily – with these tests I usually adopt the Threshold Gradient approach.
The reproducibility and “spread” of results obtained.
You may find that threshold methods - if set too “tightly” - can lead to less-than-desirable reproducibility; in this case you should loosen the criteria or adopt another method.
Hope this helps. As always, if you need some help with this, or any other aspect of rheology, please don’t hesitate to drop me a line.