## G-Values: G', G'' and tanδ

### Quick Start

You really, really, really cannot formulate a PSA^{1} 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! Why does it matter? If your PSA is too elastic or too plastic under given conditions, then it's useless.

### G-Values

Imagine a block of PSA 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 PSA was a "pure solid" so that on release of any stress the PSA would spring (literally) back to its original position. The PSA is, then, a purely elastic solid. Or imagine instead that the PSA was a "pure liquid" so that on release of any stress the PSA would not return at all to its original state. This PSA then 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. Any PSA that was purely elastic would be useless; it would just be another hard bit of polymer. And any PSA that was purely plastic would also be useless; you don't get much useful adhesion between two surfaces with a layer of water.

Clearly, then, a PSA has to be 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 think of a pack of cards, with the bottom of the deck firmly anchored and the top of the deck attached to our oscillating strain. As the stress is applied to the top card some of it is transmitted to the next card, then to the next and so forth, but with a delay. By the time the top card has reached maximum stress (and strain) and starts to return, cards lower down are still moving in the original direction because of the influence of the cards above. It turns out that the response measured at the bottom of the pack is exactly out of phase with the top. By the time the top reaches a maximum stress, the bottom is experiencing a minimum strain, and when the top reaches the minimum stress the bottom experiences a maximum strain.

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 PSA 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)

Given that a PSA works by absorbing energy during an attempt to form a crack, it's clear that a "loss" modulus is of great significance. You cannot formulate a good PSA without a good G''.

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 PSA 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.

Of course the WLF section has already alerted you to the fact that G' and G'' on their own are meaningless. Although we've spoken of measuring them 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 PSA, 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 WLF, the same question can be "At what temperature was this measured at the given frequency?" You cannot understand PSA without G' and G'' and you cannot understand them without knowing temperature and time. So now we are ready to understand the first thing we need for a basic PSA which is to ensure that it meets the "Dahlquist criterion".

Rheology via shear gives the shear modulus G. The tensile modulus, E is related to the shear modulus via the Poisson ratio ν:

E=G.2(1+ν)

The bulk modulus K, i.e. in compression, is given by:

K=E/[3(1-ν)]

For a PSA, ν is effectively 0.5 so E is 3G and K is infinite - i.e. if you try to compress a PSA it simply must squeeze sideways, and if it can't squeeze sideways then you can't compress it.

If you want viscosity (Pa.s) from G'' (Pa) you just divide by the frequency (1/s) at which the G'' was measured.

^{1}A true story. I spent a day lecturing a company about PSA science, including the importance of rheology. At the end of that day they said "Great; here's our rheometer, show us how we can use it for our formulations". I'd not been near a rheometer for many years and was rather nervous about doing rather than merely talking. But after a few errors and some hasty WLF calculations in Excel, it turned out that we could do the right measurements on their rheometer, resulting in a goldmine of useful information about why some formulations were good and others were not.