## Distribution Calculations

### Quick Start

Two emulsions with the same measured "size" might have very different drop distributions, first, because they might have different definitions of "size" and second, because the same size can be attained via different distributions.

Use the app to get a feel for what this means. Create your own emulsion with different blends of two types of particles (changing their respective Radii, peak Heights and Widths), then read out the different Diameter values, the meaning of which are described below.

### Distribution

_{1}nm

_{1}

_{1}

_{2}nm

_{2}

_{2}

What is the size of your emulsion drops? Neither the question nor the answer makes much sense unless you can simulate some emulsion distributions and look at the cumulative number, area and mass distributions. In this app you can create two different emulsion size distributions with different peak positions, r, heights, h, and widths, w, then look at the cumulative number, area and mass distributions (Cum.N, Cum.A and Cum.M). The peaks are "gamma distribution functions" to give a characteristic tail shape. The effect of w is rather obscure on such functions - just play with w values to get something that looks OK. Once you start to play you will find why those who want to make their distributions look good like number distributions - they can say things like '90% < 300nm'. At the same time users tend to prefer mass distributions because something that is '90% < 300nm' in number terms might be '90% > 300nm' in mass terms, because a few large drops contain more mass than lots of small drops.

Finally the app calculates the different diameters that are often quoted. D[1,0] is the
*number* average (or mean), D[3,2] is the *volume/surface* or
*Sauter* average and D[4,3] is the *mean diameter over volume* or
*DeBroukere* mean. Note that these are sometimes shown as D̅10, D̅32 and D̅43, where the ̅ symbol means "average". Unfortunately these short-hands with the bar can be confused with common symbols such as D50 which is also calculated. D50 means the diameter where half the particles have a volume less and half have a volume more.

Why are the inputs in terms of radius and outputs in terms of diameter? Because that seems to be the normal way of doing these things. It is a reminder that we have to be alert to whether data are in radii or diameters because the literature is inconsistent.