## Probe 2 Factor

### Quick Start

The Probe Tack test can now be analyzed via two factors: the *bulk* properties and the *interfacial* ones. So it becomes possible to separate out, and optimize, properties of the PSA system.

The spherical probe used here has larger extensions than from the flat one used in the Probe Tack app.

### Probe 2 Factor

_{max}ref kPa

_{f}@ 5/s & Γ/G=2

_{max}kPa

_{f}

_{A}J/m²

```
//One universal basic required here to get things going once loaded
window.onload = function () {
//restoreDefaultValues(); //Un-comment this if you want to start with defaults
Main();
};
//Any global variables go here
//Main is hard wired as THE place to start calculating when input changes
//It does no calculations itself, it merely sets them up, sends off variables, gets results and, if necessary, plots them.
function Main() {
saveSettings();
//Send all the inputs as a structured object
//If you need to convert to, say, SI units, do it here!
const inputs = {
h: sliders.Slideh.value,
edot: sliders.Slideedot.value,
ef5: sliders.Slideef5.value,
GammaG: sliders.SlideGammaG.value,
sigmaMax5: sliders.SlidesigmaMax5.value,
}
//Send inputs off to CalcIt where the names are instantly available
//Get all the resonses as an object, result
const result = CalcIt(inputs)
//Set all the text box outputs
document.getElementById('Wa').value = result.Wa
document.getElementById('ef').value = result.ef
document.getElementById('sigmaMax').value = result.sigmaMax
document.getElementById('Gamma').value = result.Gamma
document.getElementById('G').value = result.G
//Do all relevant plots by calling plotIt - if there's no plot, nothing happens
//plotIt is part of the app infrastructure in app.new.js
if(result.plots) {
for(let i = 0; i < result.plots.length; i++) {
plotIt(result.plots[i], result.canvas[i]);
}
}
//You might have some other stuff to do here, but for most apps that's it for CalcIt!
}
//Here's the app calculation
//The inputs are just the names provided - their order in the curly brackets is unimportant!
//By convention the input values are provided with the correct units within Main
function CalcIt({h,edot,ef5,GammaG,sigmaMax5}) {
var ef = ef5 / 2 * (1 + edot / 5)
if (ef > 1.5 * ef5) ef = 1.5 * ef5
ef*=GammaG/2
var sigmaMax = sigmaMax5 * Math.pow(edot/5, 0.333) * 25/h //We normalize to 25μm
var sigmaSteady = 0.4 * sigmaMax
var sigmaPlateau = 1.2 *GammaG/2* sigmaSteady
var rmap = new Map()
var sigmaPlot = [], GPlot=[], e, sigma
sigmaPlot.push({ x: 0, y: 0 })
sigmaPlot.push({ x: 0.1, y: sigmaMax })
var inPeak = true
var Wa = sigmaMax/2 * 1e3 * 0.1 * 25 / 1e6 //Half the initial peak
var Wtmp=[],Stmp=[]
Wtmp.push(Wa);Stmp.push(sigmaMax)
for (var e = 0.2; e <= ef; e+=0.1) {
//e = ef * i / 100
if (inPeak) {
sigma = sigmaMax * Math.exp(-(Math.pow((e - 0.1) / 0.25, 2)))
if (sigma < sigmaSteady) {
sigma = sigmaSteady
inPeak = false
}
} else {
sigma = sigmaSteady + Math.max(0, e - 1) / (ef - 1) * (sigmaPlateau - sigmaSteady)
}
sigmaPlot.push({ x: e, y: sigma })
Wa += sigma * 1e3 * 0.1 * 25 / 1e6
Wtmp.push(Wa);Stmp.push(sigma)
}
sigmaPlot.push({ x: e + 0.1, y: 0 })
//Now go back through the loop till we hit G
var WG=Wa/(1+GammaG),Wi=Wtmp.length-1,thee=0,thesigma=0
for (e=ef; e>=0.1;e-=0.1){
if (Wa-Wtmp[Wi]>=WG) {
thee=e;thesigma=Stmp[Wi]
break
}
Wi--
}
GPlot.push({x:thee,y:0})
GPlot.push({x:thee,y:thesigma})
//Now set up all the graphing data detail by detail.
const prmap = {
plotData: [sigmaPlot,GPlot], //An array of 1 or more datasets
lineLabels: ["σ","←J:G→"], //An array of labels for each dataset
xLabel: "ε& ", //Label for the x axis, with an & to separate the units
yLabel: "σ&kPa", //Label for the y axis, with an & to separate the units
y2Label: null, //Label for the y2 axis, null if not needed
yAxisL1R2: [], //Array to say which axis each dataset goes on. Blank=Left=1
logX: false, //Is the x-axis in log form?
xTicks: undefined, //We can define a tick function if we're being fancy
logY: false, //Is the y-axis in log form?
yTicks: undefined, //We can define a tick function if we're being fancy
legendPosition: 'top', //Where we want the legend - top, bottom, left, right
xMinMax: [,], //Set min and max, e.g. [-10,100], leave one or both blank for auto
yMinMax: [,], //Set min and max, e.g. [-10,100], leave one or both blank for auto
y2MinMax: [,], //Set min and max, e.g. [-10,100], leave one or both blank for auto
xSigFigs: 'F1', //These are the sig figs for the Tooltip readout. A wide choice!
ySigFigs: 'F0', //F for Fixed, P for Precision, E for exponential
};
//Now we return everything - text boxes, plot and the name of the canvas, which is 'canvas' for a single plot
return {
sigmaMax : sigmaMax.toFixed(0),
Wa: Wa.toFixed(1),
ef : ef.toFixed(1),
Gamma : (Wa-WG).toFixed(1),
G : WG.toFixed(1),
plots: [prmap],
canvas: ['canvas'],
};
}
```

A smart paper^{1} from the lab of Prof Kenneth Shull at Northwestern U and a team at Dow Chemical has given us deep insights into how we can properly engineer PSAs. *The authors have asked me to mention that the theory always assumes adhesive rather than cohesive failure.*

The key insight is that you can, in principle, engineer the bulk properties independently from the interfacial properties, with predictable results in the ensuing work of adhesion, W_{A}. OK, so this applies specifically to the probe tack test and translating it to peel may be tricky, but the core ideas are profound

As you pull the probe up from the sample there is a rapid increase in the stress for a small increase in strain. The maximum, σ_{max} depends, of course, on the adhesive, but the paper shows that it increases linearly in rate of pull and decreases on thickness (the reference value you enter is defined at 5/s and 25μm). Faster and thinner gives a bigger peak. Then the adhesive starts to cavitate and the stress decreases to σ_{steady}, which also follows the speed/thickness rule. Generally there is some strain hardening as the sample stretches further to reach σ_{plateau} before it detaches at a failure strain ε_{f}. The core behaviour depends on bulk properties, but the detachment depends on interfacial properties

### Detachment

The adhesive detaches at a critical elastic energy G_{c} which depends on the interfacial interactions. It happens at an elongation ε_{c} which, for the purposes of the app, is the same as the failure strain ε_{f}. Most of the useful absorption of crack energy has been done by the bulk dissipation, Γ. It turns out that different PSA materials have different ratios of dissipation to elastic energy and Γ/G captures the essence of a material, with higher values being generally better.

Where things get tricky is if an adhesive with a high G_{c} has a low Γ/G. That low Γ/G means that you reach G_{c} at lower strains, even though that actual G_{c} is higher.

If, as in the paper, you can make the bulk have a large Γ/G but with some trick (e.g. a thin extra coating) you achieve a large G_{c} then adhesion is improved.

### Appifying these insights

The paper cleverly analyses a lot of experimental data. Here we are showing the general ideas. We enter the thickness and extension rate, expecting (and finding) that the calculated W_{A} increases if either is increased. We also specify the failure strain, ε_{f} defined at 5/s and at a good Γ/G=2. Now as you change the extension rate and/or Γ/G you see how ε_{f} changes. The ←J:G→ line in the graph shows how much of Wa comes from G (the area to the right) and from Γ (to the left). Of course, ε_{f} depends on the surface. In the paper the difference between adhesion to glass and to PE were entirely due to changes in ε_{f}. And, if you've been reading what I've said about changing surfaces, this is nothing to do with surface energy. The G_{c} values are in the multi J/m² range, not the 40 mJ/m² typical of surface energies. It's a potent mix of chemistry and polymer relaxation that so far has not been properly investigated, but the equipment used in the paper is nicely set up to explore these issues.

Given that W_{A} is the area under the curve, if everything else stays equal (which it never does!) then decreasing ε_{f} reduces adhesion.

We've all seen these curves change in complicated ways, but till this paper there wasn't a way to capture the subtle combination of bulk and interfacial effects.

### Cavitation and fingering

With probe tests you can usually see an initial cavitation event occuring in the centre, then fingers start to come in from the side as you pull harder. It seems natural to assume that the exact form of cavitation and fingering should have a big effect on the adhesion. If you go to extremes (where the assumptions behind this paper fall down) then, yes, it makes a difference. But the paper shows that nothing special happens to their data when they go from cavity to fingers. This is good news: it means that we don't have to worry too much about the niceties of cavitation/fingering theory!

### Putting this into practice

Presumably σ_{max}, as I've argued elsewhere, should be obtained by conventional tensile testing at high strain rates, and Γ/G can be derived from the same tensile tests or via conventional rheology as some sort of mix of G'' and G'. Influencing G_{c} via chemistry or physics should start to become rational.

The dilemma we all face is that a "weak" PSA gives a high Γ/G but such a small σ_{max} that the overall adhesion is poor. A "strong" PSA gives a large σ_{max} but it might be strong because it's highly elastic, so Γ/G is low.

I'm not saying that sorting out the tradeoffs will be easy. But I am saying that after this paper we have a much better methodology for thinking about and formulating around the tradeoffs.

### Acknowledgement

I am most grateful to two of the authors, Dr William Griffith and Dr Qifeng Wang for their expert help in refining this page. All responsibility for errors is mine.

^{1}Qifeng Wang, William B. Griffith, Melinda Einsla, Sipei Zhang, Michaeleen L. Pacholski,and Kenneth R. Shull, *Bulk and Interfacial Contributions to the Adhesion of Acrylic Emulsion-Based Pressure-Sensitive Adhesives*, Macromolecules 2020, 53, 6975−6983