## G-Values: G', G'' and tanδ

### Quick Start

You really, really, really cannot get through much of formulation without understanding G' and G''. Most of us have avoided them because they seemed so difficult. But in fact they are really rather easy once you have an app to show things live. All you have to do is tell the app how closely (or not) the response to an oscillating force follows the stimulus. If it follows it closely then the sample (at this temperature and speed) is elastic, if it lags behind then it is plastic or viscous. It's as easy as that

### G-Values

Imagine a sample trapped between two discs. Apply a stress (force) that twists the top disc back and forth in a sinusoidal motion. Measure the strain (% stretch) induced in the sample via that stress, noting that the strain varies sinusoidally with time. You can equally apply an oscillating strain and measure the stress, the two modes are equivalent in theory (though there are practical reasons for choosing one or the other).

Imagine that the sample was a "pure solid" so that on release of any stress the sample would spring (literally) back to its original position. The material is, then, a purely elastic solid. Or imagine instead that it was a "pure liquid" so that on release of any stress the sample would not return at all to its original state. This would be a purely "plastic" liquid. In other words, at one extreme we talk about "elastic" deformations that return to their original position and "plastic" deformations that are permanent.

In reality, most solid samples are a mixture of both. That's why we need G' (which measures the elastic component) and G'' (which measures the plastic component). Going back to our thought experiment, the strain response of a pure elastic is instantaneous - as the stress increases so does the strain. To understand the pure plastic deformation remember that viscous stress is proportional to strain rate. At the top and bottom of the sine curve, the oscillation velocity is near-zero so the rate is zero so the stress is zero. Near the cross-over points, the angular velocity is maximum so the stress is maximum: the responses are exactly 90° out of phase.

As we all know, modulus is Stress/Strain. So in this test set-up we can always measure "modulus" as Maximum_Stress/Maximum_Strain. Why the two "Maximums"? Well for the plastic case at maximum stress the strain is zero so a "modulus" based on Stress/Strain would be infinite at that point.

Our thought experiment therefore gives us two bits of information: the "phase" angle difference δ between the stimulus (stress) and response (strain) and the modulus, G* from Maximum_Stress/Maximum_Strain. What it doesn't seem to tell us is how "elastic" or "plastic" the sample is. This can be done by splitting G* (the "complex" modulus) into two components, plus a useful third value:

- G'=G*cos(δ) - this is the "storage" or "elastic" modulus
- G''=G*sin(δ) - this is the "loss" or "plastic" modulus
- tanδ=G''/G' - a measure of how elastic (tanδ<1) or plastic (tanδ>1)

The app does virtual experiments and derives G*, G', G'' (relative to some arbitrary maximum value=1) and tanδ.

Although this is an artificial graph with an arbitrary definition of the modulus, because you now understand G', G'' and tanδ a lot of things about your sample will start to make more sense. How you measure them is a matter of practicality. Typically you can choose between a rheometer and a DMA (Dynamic Mechanical Analyser) though these days the distinctions between them are rather blurred.

Although we've spoken of measuring G' and G''' via an oscillation, no mention has been made of the frequency. This brings us to a biblical prophetess, Deborah, who said "The mountains flowed before the Lord" and who has thus been honoured with the Deborah Number:

D_{e}=τ/t

D_{e} is the ratio of the relaxation time τ of the system (say a mountain) and the timescale, t (say billions of years), of the measurement. If t>>τ (D_{e}<<1) then the mountain will indeed flow and is plastic. If t<<τ (D_{e}>>1) then even water becomes a very tough elastic solid; indeed ultra-high speed measurements of the modulus of water show that it is comparable to steel. To return to our sample, if D_{e}<1 then G'' wins, if D_{e}>1 then G' wins.

So your first question when shown data on G' and G'' should be "At what frequency was this measured at the given temperature?" or, because of the magic of time-temperature superposition, the same question can be "At what temperature was this measured at the given frequency?" You cannot understand material properties without G' and G'' and you cannot understand them without knowing temperature and time. So you next need the WLF app to see how temperature equals time.