Using Exponential Chirp Rheometry to Characterize Time-Evolving Materials

insights from industryGareth McKinleyProfessor of Mechanical Engineering MIT

In this interview, AZoMaterials speaks with Professor Gareth McKinley, Professor of Mechanical Engineering at MIT, about exponential chirp rheometry and how optimally windowed chirps can be used to rapidly probe the linear viscoelastic response of gels and other time-evolving materials.

Can you please introduce yourself and your role at MIT?

I am Gareth McKinley, a mechanical engineer and professor at MIT. A large portion of my work focuses on optimally windowed chirp rheometry.

The idea behind these chirps is that they are a very good way to quickly probe the linear viscoelastic responses of complex materials, such as gels and other time-evolving materials. These might be materials that cross-link or change over time.

This work was done largely at MIT by a number of my graduate students, including Michela Geri, Bavand Keshavarz, and Mohua Das, together with collaborators Thibaut Divoux from the University of Lyon, Christian Clasen from KU Leuven, and Dan Curtis from Swansea University.

Why are time-evolving materials challenging to characterize using rheology?

The key goal is to resolve not just frequency dependence, but also time dependence in rheology.

One way to do that is to probe a single frequency and then examine how the response changes over time. However, we are also interested in the frequency response of a complicated material such as a viscoelastic system.

Gelling systems are of particular interest. More generally, I call them mutating materials, with "mutating" simply meaning that things change over time. They might be gelling, curing, cross-linking, or evaporating.

There have been a number of approaches to this problem. One is the multi-wave method, in which you add multiple sine waves together and use Fourier analysis to separate them. The danger is that when you add several waves together, you can get positive superposition of the strains and end up in a nonlinear state.

Other approaches include short-time Fourier transforms, white noise, and step-strain methods such as iRheo. What we are interested in here is another approach: the chirp.

What is the mutation number, and why is it important when studying these materials?

The mutation number is associated with how unsteady the material itself is, or how much the material is changing.

Henning Winter first defined the mutation number about 30 years ago as the experiment's timescale divided by the timescale over which the material changes.

Our goal is to keep that number small. We want the timescale of the experiment to be very small compared with how fast the material is changing.

If you have ever looked at a sample curing or drying in a rheometer, you might measure a property such as G′ and see it change over time. One way to determine the mutation timescale is to examine the time rate of change of the logarithm of that property. That gives you a rate, and one over that rate gives you a time constant.

For the chirp experiments, the key is to keep the mutation number well below one. In practice, we try to keep it below about 0.1.

What is an exponential chirp, and how does it differ from a conventional oscillatory rheology measurement?

The idea is that we are not going to put in a single frequency. Rather, we will let the frequency vary over time.

A useful way to think about this is frequency modulation; this is the "FM" of FM radio. In large-amplitude oscillatory shear, or LAOS, you might pick a single frequency and change the strain amplitude. With a chirp, we do something different. We stay at small strains in the linear viscoelastic regime and rapidly vary the frequency.

There are different kinds of chirps. The simplest is a linear chirp, but rheologists typically want to span a very wide frequency range, so we normally want something faster than linear.

What we have coded into our software is basically an exponential form. We start from a very low frequency and span to a high frequency. A sigmoidal function is another good possibility, and other functional forms are certainly possible, but generally an exponential or a sigmoidal function is probably the right form for rheometry.

Why do you need to apply a window function to the exponential chirp?

If we use an exponential chirp on its own, we can get spectral leakage. Some of the signal is lost from where we want it to be in the frequency spectrum.

There is actually a nice analogy with bats and dolphins. They change the amplitude and frequency of their signals: they start at a low amplitude, rise to a high amplitude, and then return to a low amplitude.

We do something similar by using a window function. We take the chirp and multiply it by a function that starts at zero, reaches a maximum, remains constant for a while, then returns to zero. Because the signal goes to zero at the beginning and the end, it prevents information from leaking out of the signal.

The mathematical function we use is a Tukey window, a cosine-squared function.

There is an optimization involved. If there is no window, we see wiggles and noisy oscillations in the power spectrum. As we increase the windowing, the signal becomes smoother. But if we increase it too much, we start losing information, particularly at the high-frequency end.

We have found that the windowing parameter should be a relatively small number, around 10%, and that is what we call our optimal chirp.

What advantages does the exponential chirp offer over other broadband approaches, such as a step-strain experiment?

One advantage is the way the power is distributed across the frequency spectrum.

For an exponential chirp, the power is almost constant across the spectrum. If you perform a step-strain or iRheo-style experiment, the step certainly contains a lot of frequency information, but the power spectrum decays at a much faster rate.

You can quantify that by integrating the amount of power in each decade of the signal. For the exponential chirp, the power is approximately constant per decade. In a step experiment, however, the power per decade decreases as you move to higher frequencies.

Another way of thinking about it is that, with the exponential chirp, I am exciting my material with approximately the same strain-energy input across the entire frequency spectrum.

How can this technique help researchers study rapidly gelling materials?

A good example is casein gelation and the formation of cheese.

There is a long induction time: you have to wait more than an hour for the gelation to happen. But once it starts, the mutation takes only about 120 seconds.

If you really want to measure that process, your experiment has to be much shorter than 120 seconds because that is the rate of change of the elastic modulus. We can perform a chirp that is only 10 or 15 seconds long, so the mutation number is less than 0.1.

Before gelation, the signal is very small and is essentially dominated by noise. But as soon as gelation occurs, you can see the phase angle drop, the moduli increase, and you can monitor the evolution of cheese formation.

That illustrates why chirps are useful for time-evolving systems. You can acquire the frequency-dependent information quickly enough that the material has not changed too much during the measurement.

What factors determine how fast an exponential chirp experiment can be performed?

The length of the signal is really set by the frequency range you want to investigate. We need a sufficiently large time-bandwidth product, and then we need to window the chirp to avoid spectral leakage or signal loss.

There are also questions about how quickly the rheometer itself can respond and how the instrument's bandwidth constrains the chirp.

So, you cannot simply make the experiment arbitrarily short. You need sufficient signal to resolve the frequency information you are interested in, while keeping the experiment short relative to the material's mutation time.

That is why the mutation number is such an important design parameter. If you can design the chirp so that the mutation number remains below about 0.1, then you are in a good regime for studying these evolving materials.

Can exponential chirp rheometry also be used when a material is in the nonlinear regime?

At the moment, there are important limitations that make this challenging.

The analysis we use is based on Fourier analysis, so we assume linear contributions from different frequencies. If you apply pre-stress, for example, you are typically already pulling the material into a nonlinear range. That means there will be interactions between different harmonics.

You could certainly impose a chirp and consider how to deconvolute that signal, but at the moment I would say there is no way to obtain quantitative frequency information for different frequencies when you are in a nonlinear state.

You have to be very careful to distinguish nonlinear interactions, such as between a first and third harmonic, from fundamental contributions at different frequencies.

For the technique I have described here, the way to think about it on a Pipkin diagram is that we are spanning frequency while staying at a low strain amplitude in the linear viscoelastic regime. Work is continuing on how these approaches might eventually be extended into nonlinear rheology.

About Professor Gareth McKinley

Professor Gareth H. McKinley is the School of Engineering Professor of Teaching Innovation and Professor of Mechanical Engineering at the Massachusetts Institute of Technology (MIT). His research focuses on rheology and the behavior of complex fluids, including non-Newtonian fluid dynamics, extensional rheology, gels, and time-dependent materials.

Professor McKinley studied engineering at the University of Cambridge, receiving his BA in 1985 and MEng in 1986. He subsequently attended MIT, where he received his PhD in 1991.

His research has explored the rheology and fluid mechanics of complex materials, with particular interests in understanding how material microstructure influences macroscopic flow behavior and in developing experimental approaches for characterizing challenging fluid systems.

Professor McKinley has received numerous honors for his contributions to rheology and fluid mechanics. These include the Bingham Medal from the Society of Rheology and the Gold Medal of the British Society of Rheology. He was elected to the US National Academy of Engineering and to the Royal Society as a Fellow in 2019.

His work at MIT continues to combine fundamental rheological research with the development of new measurement techniques for complex and time-evolving materials.

This information has been sourced, reviewed, and adapted from materials provided by Waters Materials Sciences (formerly TA Instruments).

For more information on this source, please visit TA Instruments.