## WLF - tTE - tTS

### Quick Start

You often find awesome rheology datasets containing data from -90°C to +120°C and tested at frequencies from 10^{-10} to 10^{4}Hz. How do they get such data? No one can do it directly. Instead they gather data at a practical range of temperatures and frequencies and combine them into a single graph via WLF shifting. The technique is powerful and universal.

### WLF Fitting to a -80°C elastomer

_{g}°C

_{1}

_{2}

Temperature and time (the same as speed) are strongly interrelated and are a key part of the "systems" thinking necessary to understand adhesion. This relationship can be described with three acronyms, all meaning the same thing:

**WLF**Williams-Landell-Ferry**tTE**time-Temperature Equivalence**tTS**time-Temperature Superposition

If you slowly pull a polymer you can imagine that it will give the impression of being weak because it will slowly stretch. A sharp pull will make it appear strong as it doesn't have time to respond. Similarly, the polymer at high temperature will easily flow and at low temperature will be rather rigid. In fact the two sets of scenario are exactly equivalent. Therefore:

.### Time is equivalent to Temperature

Let us do a test for modulus (G') at a two different speeds at a given temperature. Because time is equivalent to temperature we can calculate that the high speed value will be obtained at low speed at a lower temperature and that the low speed result will be obtained at high speed at a higher temperature. Similarly, if we do a test at a given speed at two temperatures we know that we can obtain the high temperature result at a lower speed and the low temperature result at a higher speed. The good news is that there is a (relatively) simple equation to convert temperature into time. It was developed by Williams, Landel and Ferry and is called WLF. It says:

`Log(a_t)=-(C_1(T-T_r))/(C_2+T-T_r)`

Here we have two constants, C_{1} and C_{2}, we have the temperature of interest, T and a reference temperature T_{r}. From these we calculated a_{T} which is the amount by which the time has to be shifted to get the same result at T as you would get at T_{r}.

Before providing the means for you to calculate this, let's see what this actually means. On the left is a famous dataset from Ahagon et al which shows how Log(W) (work of adhesion) depends on Log (Peel Rate) at different temperatures. This is not a rheology test, but the principles are the same. The data are all very interesting, but you can imagine that it wasn't much fun to collect them all.

Now let's look at the graph on the right (the colour codes for the different temperatures are the same) to see the data time shifted via WLF. All the data on the left can be made to fall (fairly) neatly into a single curve. A lot of rheology plots are actually WLF-shifted plots, but you often have to read captions carefully to spot what's happening, because the trick is so common. Note that the time shifting is quite severe. The high peel rates (0.01mm/s) at 80°C get shifted down to ~10^{-15} m/s to get the same values at the T_{g} of -90°C - not the sorts of speeds relevant to a real-world experiment!

Let's see how those datasets really are the same thing. Change the three parameters, T_{g}, C_{1} and C_{2} till the datapoints fall on an adequately single curve. The default value of T_{g} the first time you run the app is deliberately set very high (-25°) so you can see the dramatic effect of T_{g} on the WLF shift. The meanings of C_{1} and C_{2} are discussed below.

As you will see, you can't get a perfect straight line, and there is quite a lot of latitude in the various settings. But you get the general idea that points measured at very different temperatures can be plotted in a single curve when shifted by WLF

I have a popular YouTube video that describes WLF and mention a spreadsheet of some real-world data that I happened to gather one day (it is not important what it is!) and had to hastily analyze without access to the software on the rheometer. Lots of people have asked for access to the spreadsheet, so here is WLF Example.xls, slightly tidied up with some simple instructions.

### The meaning of C_{1} and C_{2}

What do C_{1} and C_{2} mean? C_{1} is a measure of the range of times/frequencies spanned by that process over the entire relevant temperature range. To say that C_{1} is, say, 17.5 is to say that there are 17.5 orders of magnitude in the process. C_{2} is the temperature range that changes the process by half C_{1}; in other words if C_{2} is 51.6 then over 51.6 °C the process changes by ~8.5 orders of magnitude

### Limitations

WLF themselves were clear that their ideas applied only to a subset of temperature/time effects in their area of expertise which was polymers. It does *not* apply to classical Arhenius systems where (typically) there is a doubling of some factor with every 10K of temperature. And in rheology it does *not* apply to thixotropic systems where "time" has two roles. There will be plenty of other cases where it doesn't apply. But still the take-home message is that just about any statement in rheology is invalid if you don't specify both the temperature and the timescale, or, conversely, you should always be aware of both factors before making any confident statement about the rheological behaviour of a system.

### The physics of caramel

Here is a delightful and insightful paper, Simplicity in complexity – towards a soft matter physics of caramel by Simon Weir, Keith M. Bromley, Alex Lips and Wilson C. K. Poon, which is Open Access so freely downloadable. In the paper the tTS principles are used to help understand caramel. But, tTS isn't the only game in town. There is also tQS (Time-Cure Superposition, or Time-Crosslinking Superposition) and tCS (Time-Composition Superposition). It turns out that both tQS and tCS are "well-known principles" but until I read the caramel paper, they were not well-known to me. tQS is especially important for PSA because the (small) degree of crosslinking in a PSA is especially important for functionality.

### And the physics of dough

An equally delightful and insightful paper Pressure-Sensitive Adhesive Properties of Wheat Flour Dough and the Influence of Temperature, Separation Rate, and Moisture Content, by S. S. Heddleson, D. D. Hamann, D. R. Lineback, and L. Slade, freely downloadable via that link, shows that bread dough can be considered as a PSA, and that not only is tTS applicable to the stickiness of dough, but also tWS (Time-Water Superposition). Arguably tWS is the same as tCS in the Caramel paper. But the point is that these large superposition effects are a general aspect of physics, not some rarified concept that applies just to PSA.