The Definition of the Mean Value in Particle Characterization

The mean value is a certain type of average taken over the entire distribution.

For example, if there are ten spheres of diameters from 1 to 10, the sum of their diameters from the distribution can be represented by ten spheres each of diameter 5.5; but the sum of their volumes has to be represented by ten spheres each having a diameter of 6.72. Due to their nature, each particle characterization technology will “see” the same sample differently. In the language of statistics, different technologies see particles through different “weighting factors.” For example, when using TEM, one measures particles based on their number, but when using laser diffraction, one detects the light scattering intensity of particles based on their volume. Thus, if one calculates the mean value of a distribution using the “native weighting” of the measurement, i.e., without converting it to the same base, no valid comparison can be made from results of two different measurement technologies, even for spheres. Different mean values thus must be defined and used.

In the so-called moment-ratio notation, mean values are expressed as the ratio between two moments of the number density distribution of parameter x. Here, x is a characteristic parameter, most commonly the diameter d (or x) but it can also be some other parameter. The term Dp,q is used to designate a mean value obtained from summing discrete individual x values to the power of p (xp), which represents the relationship between the measured signal and the particle parameter x. The value of the xp summation is then normalized by a summation of x values to the power of q, which represents the relation between the weighting of each particle to its parameter x in the measurement:

For a particle size distribution, when the exponential term of dq is equal to 1, 2, or 3, the term dq is length, surface, or volume weighted, respectively. The same is true for dp-q. In the above definitions, p and q are restricted to integer values. The last equality (Eq. 3) is particularly useful when a specific mean value cannot be measured directly but the other two mean values are measurable.

In practice, if one uses an electron microscope to measure particles, one will measure the diameters with a graticule, add them up and divide the sum by the number of particles to get an average result. This will then be D1,0; i.e., the number average. If one then does an image analysis the area of particles is what most important. D2,0 will be generated from the image analysis by adding all projected areas and dividing the sum by the total number of particles analyzed. Likewise, in an electric sensing zone measurement, one would get D3,0, and in a laser scattering experiment, one would get D4,3. The table below illustrates one example of how large the difference in the Dp,q values can be, even for a very simple system that consists of four spherical particles with diameters of 1, 2, 3, and 10, respectively (the unit is irrelevant here). In size measurements their corresponding Dp,q values are as follows:

Table 1. Dp,q values of a simple system

D0,0 = 2.78 D1,1 = 5.65 D3,2 = 9.08 D6,3 = 9.88
D1,0 = 4.00 D2,1 = 7.13 D4,2 = 9.41 D4,4 = 9.87
D2,0 = 5.34 D3,1 = 8.05 D5,2 = 9.58 D5,4 = 9.93
D3,0 = 6.37 D4,1 = 8.57 D6,2 = 9.67 D6,4 = 9.95
D4,0 = 7.08 D5,1 = 8.90 D3,3 = 9.55 D5,5 = 9.96
D5,0 = 7.58 D6,1 = 9.10 D4,3 = 9.74 D6,5 = 9.98
D6,0 = 7.94 D2,2 = 8.42 D5,3 = 9.83 D6,6 = 9.99

We can see from the above table that the mean sizes can be quite different if one uses an electron microscope (D1,0 = 4.00), or a laser diffraction instrument (D4,3 = 9.74) to measure this four-particle system. The difference between a number-averaged mean (q = 0) and a mass-averaged mean (q = 3) resides in the fact that in the number average the mean value represents values from particles with the largest population while in the mass average the mean value represents more from particles with the largest size. The same is true for a real distribution. A number distribution may be completely different from its corresponding mass distribution. Fig. 1 shows a real differential volume distribution (the dashed line) with the differential number distribution of the same sample (the solid line). Table 2 lists the values for a few statistical parameters of the sample in Fig.1.

Figure 1. Number (solid line) and volume (dashed line) distributions of the same sample.

Table 2. Statistics of the distribution in Fig. 1 (all values are in mm)

  Volume Weighted Number Weighted
  Arithmetic Geometric Arithmetic Geometric
Mean D4,3 = 1.56 D3,3 = 1.24 D1,0 = 0.68 D0,0 = 0.65
Median 1.37 1.37 0.60 0.60
Mode 0.58 0.58 0.56 0.56
SD 1.09 1.86 0.28 1.30
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