## Sample Preparation

A key step in the measurement of particle size distributions is the preparation of powder samples for particle size measurement. Powder dispersion is more important than the measurement methods, and instrument limitations or inaccuracies can be secondary to sample preparation.

Various sample methods include batch, continuous, or in situ measurements as outlined in Fig. 1. The procedures for the collection of representative samples and the preparation of dispersed powders or slurries are critical steps when measuring particle size distributions and estimating errors (as briefly discussed below). More detailed information of sampling and dispersion is discussed in the article "Sample and Classification of Powders" in this Volume.

Fig. 1 General methods of powder sampling and particle size measurements

Sampling Errors. Sampling is distinguished as either a probabilistic or nonprobabilistic process where:

Probabilistic sampling is when all elements of the lot are submitted, along with a probability of being selected.

Nonprobabilistic sampling is deterministic rather than being based on probability, for example, as in grab sampling.

### Probabilistic sampling is the preferred method.

In terms of errors, both sampling and analysis are error-generating steps, with the consequence that the overall estimation error (OE) is the sum of the total sampling error (SE) and analytical (or measurement) error (AE). Sampling and analysis involve several preparation stages, alternated with selection stages, all of which can potentially generate errors. Therefore, total overall estimation error is represented by:

= X (PEn + SEn)=AEn where PEn and SEn are preparation and selection errors, respectively, of each stage, n (n = 1, 2, . . .). Various errors described by Allen and Davies (Ref 6) are shown in Fig. 2. Relative particle sizes and errors depend on specific powders. For submicrometer-sized ceramic powders, dispersion-related errors appear to dominate.

Fig. 2 Dispersion error related to particle size

Dispersion. Powder dispersion includes deagglomeration and formation of a stable suspension. Of the several methods used for deagglomeration (ultrasonic probe/bath, stirring, tumbling), ultrasonication with a probe is the most effective in achieving the separation of particles held together by weak forces. An optimal level of power input to the suspension is required to achieve enhanced data reproducibility. The probe diameter, volume of suspension, power input, time of ultrasonication, and rate of power input are some of the parameters that affect deagglomeration.

The surface chemistry of the solvent-powder interface also controls the preparation of a stable suspension. A repulsive force of sufficient strength to overcome attractive forces between the particles in order to disperse them is required. The major attractive force that hinders dispersion is the van der Waals type. Electrical double-layer forces are the counterforces that can effectively provide repulsive forces. The optimal range of electrical double-layer forces in polar liquids is provided by high surface charge, moderate electrolyte concentration, and an adsorbed layer of polyelectrolyte of an appropriate type and concentration. A combination of electrical double-layer repulsion and steric stabilization has been found to be highly effective for dispersing powders in polar solvents. In nonpolar solvents, steric stabilization alone is often sufficiently effective (Ref 7) and represents the only choice available.

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