Coalescents

window.onload = function () {
    //restoreDefaultValues(); //Un-comment this if you want to start with defaults
    Main();
};
const degK=273.1
function Main() {
    //Save settings every time you calculate, so they're always ready on a reload
    saveSettings();
    const inputs = {
        Tgp: sliders.SlideLatexTg.value+degK, //to K
        Tgc: sliders.SlideCTg.value+degK, //to K
        Lrho: sliders.SlideLatexrho.value,
        Crho: sliders.SlideCrho.value,
        K: sliders.SlideK.value,
        S: sliders.SlideSolidsPerc.value/100, //% to fraction
        A: sliders.SlideCoRelPerc.value/100, //% to fraction
        alpha: sliders.Slidealpha.value,
        M: sliders.SlideM.value,
    };

    const result = CalcIt(inputs);
   if (result.plots) {
        for (let i = 0; i < result.plots.length; i++) {
            plotIt(result.plots[i], result.canvas[i]);
        }
    }
}

function CalcIt({Tgp,Tgc,Lrho,Crho,K,S,A,alpha,M}) {
    let pPlot=[], Snow=S, Sinc=(1-Snow)/100,f,Va, invMFFT
    for (let i=0; i<=100;i++){
        f=Snow/(K*(1-Snow)+Snow)
        Va=A*f/Crho/(Snow/Lrho+A*f/Crho)
       invMFFT=Va*(alpha*Tgp-Tgc)/(M*Tgp*Tgc)+1/(M*Tgp)
        pPlot.push({x:Snow*100,y:1/invMFFT-degK})
        Snow+=Sinc
    }
    let plotData = [pPlot], lineLabels = ["MFFT"],  myColors = ['blue']
    const yLabel="MFFT&°C"
    const prmap = {
        plotData: plotData, //An array of 1 or more datasets
        lineLabels: lineLabels, //An array of labels for each dataset
        colors: myColors, //An array of colors for each dataset
        borderWidth: [2, 1], //An array of line widths for each dataset
        hideLegend: true, //Set to true if you don't want to see any labels/legnds.
        xLabel: 'S&%', //Label for the x axis, with an & to separate the units
        yLabel: yLabel, //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: [,100], //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: 'F1', //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 {
        plots: [prmap],
        canvas: ['canvas'],
    };
}

                        

The question

How do you select the right coalescent for a given latex? When you explore this app, the answer is mostly "get the right partition coefficient of coalescent between water and latex". As the Hoy paper shows, the partition coefficient is mostly controlled by, δH, the H-bonding Hansen Solubility Parameter, though presumably for some latices the HSP Distance of the coalescent to the latex plays a role.

Measuring the partition coefficient to common latex types is not so hard, and is something coalescent suppliers could readily do. Maybe you can ask your supplier for some values...

But it's more complicated than just the partition coefficient, so let's see what the theory says.

The theory

Our aim is to find the Minimum Film Forming Temperature, MFFT, for our latex. In the aqueous dispersion we want the MFFT to be high to avoid coalescence of the latex before coating. As the system concentrates on drying we want the MFFT to be reduced to allow the film to form.

We start with a latex with a known Tg_l, along with a coalescent with its own Tg_c which you can set to -100 °C if you have no other information. Initially the latex, l, is present at a % solids, S, with respect to water, and the coalescent, c, is present as % A (input CoRel) relative to the latex. They each have their densities, ρ_l and ρ_c.

Importantly, there is a partition ratio, K, Water/Latex of the coalescent - obviously more water-soluble coalescents have a higher K value.

It's well known that the ratio M (MFFT/Tg) is something like 0.95. This seems to imply that MFFT is only slightly lower than Tg. But the ratio is in Kelvin units, so it's a large reduction. The value changes for different latices, so ideally you would know both Tg and MFFT to input the correct M.

Finally there's a factor α which is perhaps ~ 1.2, described by Hoy as an extra effective plasticisation effect.

From those inputs we can calculate f, V and MFFT

The fraction, f, of coalescent in the latex is given by:

`f=S/(K(100-S)+S)`

The volume fraction, V, of coalescent, which is present at rel % A in the polymer is given by:

`V=(Af/ρ_c)/(S/ρ_l + (Af/ρ_c))`

We can then calculate 1/MFFT via a sort of Fox equation:

`1/"MFFT"=V((αTg_l-Tg_c)/(M.Tg_l. Tg_c))+1/(M.Tg_l)`

This gives us the MFFT at our starting S. But generally the coating is drying, so we repeat the calculation up to "fully dry", with you deciding how "dry" it needs to be before you get film formation via a nicely low MFFT

There's more to coalescence than MFFT

Sure, the rate of coalescence depends on concentration and on whatever energy barrier (e.g. from surfactants) is around the particles. But if the MFFT remains too high, you can't get coalescence.

K values < 1

When K > 1, as the system dries out, you get more coalescent in the latex, so the MFFT decreases. But when K < 1, for numerical reasons I don't understand (and I've found no one who has calculated a drying curve), the MFFT increases. If you can spot the error, or provide an explanation, please let me know. You can see the few lines of calculation in the code, and even a non-coder can work out what is going on.