# Characteristics of a crystal population

## Analytic expressions of particle size distribution

We retain the distribution of both Gauss and Rosin-Rammler.

1) The frequency density f(x) of Gaussian distribution is given by:

о is the distribution’s *standard deviation.*

The *descending fluid* relative to size x includes particles of size smaller than x.

The inverse function of P(x) is (x) = erf"^{1} (P) = x_{p}

On the other hand, Gaussian distribution is such that:
and the *variation coefficient* of the distribution is, by definition,

For Gaussian distribution:

2) According to Rosin-Rammler, the rejected material is *given directly *by:

Hence:

n is the uniformity parameter; x’ is the characteristic size.

We can directly deduce:

Moreover, according to [1.5]:

Indeed, by convention, we have applied the definition of CV given by equation [1.5] to all the distributions.

## Mean size and solubility

For highly soluble crystals, the mean size is in the order of several hundred micrometers, and with the increasing oversaturation, this size becomes independent of the oversaturation a due to the preferential attrition of large crystals.

Of less soluble crystals, typically о » 1, with primary nucleation dominating. Then, aggregation appears, providing the “visible seeds” and particles of several dozen micrometers (however, only where repulsive forces do not intervene). Note that oversaturation must not be too high, since intense nucleation causes it to drop significantly and the agglomerates would only be weakly bonded due to the absence of any strong bonds.

## Coefficient of variation and attrition

Attrition influences the coefficient of variation CV, that is, the spread of the grain size distribution.

Resistant particles, though principally the largest of them, are abraded by agitation. Spread is reduced, becoming more limited at a specific agitation energy in the order of 1 W.kg^{-1} (per kilogram of suspension).

On the other hand, fragile particles are all abraded and eroded with production of particles between 1 and 150 pm, which significantly extends the range of sizes. However, if agitation increases too much, large particles will also be broken and the size distribution will again be narrowed.