Diameter statistical

For this particle, an infinite number of statistical diameters radiating from the center of gravity of the particle exist. The average imrolled diameter of the projected contour is the integrated average defined by [2] 

This calculation is tedious to perform for each particle of a distribution even with the advent of computer aided image analysis. Faster techniques simply measure a linear dimension parallel to some fixed direction and assume that the particles are oriented randomly so that these measurements average out when a sufficiently large population has been sized. Several linear dimensions typically are used. Ferefs diameter is the mean value of the distance between pairs of parallel tangents to the projected outline. Martin s diameter is the mean cord length of the projected outline of the particle. In addition, the maximum 

Martin s Diameter Proiected Area Diameter Maximum Horizontal Intercept 

FIGURE 2.3 Diiferent particle diameters. Taken from Stockham and Fortman [3]. 

A commonly used size dependent property is the equivalent spherical diameter. The equivalent spherical diameter is the diameter of a sphere with the same volume as the particle. For a cube this sphere would have a diameter 1.24 times the edge length of the cube. Another common equivalent spherical diameter is the Stokes diameter. The Stokes diameter is the diameter of a sphere that has the same terminal settling velocity as an irregular particle. (Note Settling has to be under laminar flow [i.e., Re3molds number less than 0.2] in both cases and the density of both the particle and the sphere are assumed to be the same). 

Every laboratory dealing with particulate systems should possess a microscope. A siiq le one is adequate for most purposes and should have the following features. 

1) An eyepiece lens of magnification lOx, into which a graticule may be placed so that a saze conqrarison may be made with the particle images. 

2) Two objective lenses, of lOx and 20x magnification, with as large a numerical aperture as pos le. 

4) The capability of moving the stage in two perpendicular directions in the horizontal plane. 

Martin s diameter Mis the length of line which bisects the image of the particle. The line may be drawn in any direction but once diosai the direction must remain constant for all measurements in the distribution. 

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Avera.ge Particle Size. Average particle size refers to a statistical diameter, the value of which depends to a certain extent on the method of deterniiaation. The average particle size can be calculated from the particle-size distribution (see Size measurement of particles). 

Ferefs diameter (Fig. 20-5) is the perpendicular projection, in a fixed direction, of the tangents to the extremities of the particle profile. Martin s diameter is a hne, parallel to a fixed direction, which divides the particle profile into two equal areas. Since the magnitude of these statistical diameters varies with particle orientation, these diameters have meaning only when a sufficient number of measurements are averaged. 

Several definitions depend on the measurement of a particle in a particular orientation. Thus Feret s statistical diameter is the mean distance apart of two parallel lines which are tangential to the particle in an arbitrarily fixed direction, irrespective of the orientation of each particle coming up for inspection. This is shown in Figure 1.1. 

The computations entailed in determining the average statistical diameters just discussed are given in Table 7. The computations are for a distribution of sizes given in Table 4. It is obvious from this table that values of dav are about a third smaller than. the next largest statistical diameter dw. The use of a correct diameter giving proper weight to the physical property to be measured is apparent. 

With uniform particles the difference between the statistical diameters diminishes. When all particles are the same size, the statistical diameters are the same. 

Table 7—Computation of Statistical Diameters of a Distribution of 245 Particles Measured by Means of an Optical Micrometer...

Suppose we select at random out of all these particles, say 1000 of precisely the same apparent diameter d, and then place them in a box. Then these 1000 particles thus sorted out would form a grade whose statistical diameter is called d. 

Corresponding formulas for other statistical diameters discussed in... 

Surface in Terms of Statistical Diameters—A few aspects of the statistical method have been discussed briefly. Reference to Chapter 3 indicates that there are several formulas for computing approximate particle surface. They are based on the assumption that the particles are spheres having particular average diameters. If it is desired to compute the specific surface of an aggregate (surface per unit-weight), Eq (3-6) may be used, that is,... 

For an assembly of particles, each linear measurement quantifies the particle size in only one direction. If the particles are in random orientation, and if sufficient particles are counted, the size distribution of these measurements reflects the size distribution of the particles perpendicular to the viewing direction. Because of the need to count a large number of particles in order to generate meaningful data these diameters are called statistical diameters. 

For a single particle the expectation of a statistical diameter and its coefficient of variation may be calculated from the following equations [1] (Figure 2.3). 

They define statistical diameters and coefficients of variation according to equations (2.1) to (2.4). 

It is possible to generate more than one sphere that is equivalent to a given irregular particle shape. It is useful to evaluate Feret s or Martin s statistical diameter, " which depends on both the orientation and... 

Fig. 1 Influence of particle orientation on statistical diameters. The change in Feret s diameter is shown by the distances, df. Martin s diameter (dm) corresponds to the dashed lines.

Statistical diameters are often used in microscopy as they can be easily and rapidly measured, but the disadvantage is that they do not give information about the particle properties such as volume, mass, or surface area. However, for quality control applications this information may not be important. Figure 5 illustrates the most commonly used diameters. The USP <776> defines Martin s and Feret s diameter and other less commonly used diameters as (7) ... 

Martin et a , have shown that a volume factor, Ov, and a surface factor, a, can be assigned as a correction factor to the chosen statistical diameter, d, of the irregular particle of interest when estimating its volume and surface (36). One can then write the surface area and volume as ... 

Since the statistical diameter, d, chosen for a perfect sphere is equal to dy and d the volume and surface factor are ... 

Sample preparation must be done carefully because biases in sample preparation will lead to inaccurate results. The statistical diameters that are popular for characterizing particles in microscopy are based on a random orientation thus, biases in orientation due to improper sample preparation will affect the values. Any factors that cause the particles to preferentially orient on the microscope slide will affect the results. For example, spreading the particles out with a spatula may causes a preferential orientation. Another example is particles dispersed in a liquid when the liquid is sprayed or poured onto the microscope slide the particles could orient themselves with the flow lines of the liquid and this could lead to their non random orientation. 

An irregular particle can be described by a number of sizes. There are three groups of definitions the equivalent sphere diameters, the equivalent circle diameters and the statistical diameters. In the first group are the diameters of a sphere which would have the same property as the particle itself (e.g. the same volume, the same settling velocity, etc.) in the second group are the diameters of a circle that would have the same property as the projected outline of the particles (e.g. projected area or perimeter). The third group of sizes are obtained when a linear dimension is measured (usually by microscopy) parallel to a fixed direction. 

These and other diameters are defined in general texts such as Cadle [16] and Orr [1,2]. Martin s and Feret s diameters are statistical diameters associated with microscopy and image analysis. By definition, Martin s diameter is the distance between opposite sides of a particle measured i a line tnsecting the projected area. Feret s diameter is the distance betweoi parallel tangents cm opposite sides of the particle profile. To isure statistical significance, all measuremoits are made in the same directi[Pg.207]

Two other statistical diameters are often encountered, viz. the modal and median diameters both are determined from frequency plots (size interval versus number of particles in each interval). The modal diameter is the diameter at the peak of the frequency curve, whereas the median diameter defines a midpoint in the distribution - half the total number of particles are smaller than the median, half are larger. If the distribution curve obeys the Gaussian or Normal Error law, the median and modal diameters coincide. 

Field scanning Microscope 1-100 Length, projected area statistical diameters... 

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