Data presentation and interpretation 2.1 Introduction I he behavior - PDF document

Data presentation and interpretation 2.1 Introduction I he behavior and properties of particulate material are, to a large extent, dependent on particle morphology (shape, texture etc.) size and size distribution. Therefore proper measurement, informative data presentation and correct data interpretation are fundamental to an understanding of powder handling and end-use properties. In this chapter the following questions will be addressed: What is meant by particle size? What is meant by particle diameter? For a single particle? For an assembly of particles? How is the average size of an assembly of particles defined? What is meant by particle shape? What is meant by particle size distribution? As well as answering these questions, methods of presenting data will be covered together with data analysis and interpretation. Physical characterization differs from chemical assay in that frequently a unique value does not exist. The determined amount of copper in an ore sample should not depend upon the analytical procedure employed whereas the measured size distribution is method dependent. Only homogeneous, spherical particles have an unambiguous size. The following story illustrates the problem. Some extra terrestrial beings (ETB) were sent to earth to study humans. Their homes were spherical and the more important the ETB the bigger the sphere. The ETB who landed in the Arctic had no problem in defining the shape of the igloos as hemispherical with a single (base) diameter. The ETB who landed in North America classified the wigwams as conical but required two dimensions, height and base diameter, to describe their size. The ETB

Data presentation and interpretation 5 7 who landed in New York classified the skyscrapers as cuboid with three dimensions mutually perpendicular. The one who landed in London gazed about him despairingly before committing suicide. One of the purposes of this chapter is to reduce the possibility of similar tragedies. 2.2 Particle size Ihe size of a spherical homogeneous particle is uniquely defined by its diameter. For regular, compact particles such as cubes or regular tetrahedra, a single dimension can be used to define size. With some regular particles it may be necessary to specify more than one dimension: For a cone the base diameter and height are required whilst for a cuboid three dimensions are needed. Derived diameters are determined by measuring size-dependent properties of particles and relating them to single linear dimensions. The most widely used of these are the equivalent spherical diameters. Thus, a unit cube has the same volume as a sphere of diameter 1.24 units; hence this is the derived volume diameter. The diameter therefore depends upon the measured property. Consider a cube of side 1 cm; its volume V = 1 cm^ and its superficial surface area S ^ 6 cm^, d^ is the diameter of a sphere having the same volume as the cube and d^ is the diameter of a sphere having the same surface area. ^6^'" V^^-dl so that 1.241 d, V 6 ' \TIJ O ]2 5' = 7C(i; so that < ^ 5 = - I =1.382 The surface to volume ratio is of fundamental importance since it controls the rate at which a particle interacts with its surroundings. This is given by: nd] „. S 6J2 . 6 S — = —^ Thus — = — ~ I.e. 5 ^ „ = V {^/6)dl V dl d,. Hence, for a unit cube d^^= 1. Thus a sphere of diameter 1.241 cm has the same volume as the cube, a sphere of diameter 1.382 cm has the same superficial surface area and a sphere of diameter 1cm has the same surface to volume ratio. Definitions of the symbols used are given in Table 2.1.

58 Powder sampling and particle size determination If one were dealing with crystals of known shape it would be more sensible to relate the dimension to that shape, but this is not common practice; for the unit cube this procedure would make all the above-derived diameters equal to unity. A spherical homogeneous particle settling in a fluid rapidly reaches a constant 'terminal' velocity that is uniquely related to the diameter of the sphere. If an irregularly shaped particle is allowed to settle in a fluid, its terminal velocity may be compared with that of a sphere of the same density settling under similar conditions. The size of the particle, defined as its free-falling diameter, is then equated to the diameter of that sphere. In the laminar flow region (i?^<0.25) irregularly shaped particles settle in random orientation and a single particle generates a range of equivalent diameters depending on its orientation. The Stokes diameter is some average of these. Outside the laminar flow region, such particles orientate themselves to give maximum resistance to motion and the free falling diameter that is generated will be the smallest of these (Figure 2.1). Thus the free-falling diameter for a non-spherical particle is smaller in the intermediate region than in the laminar flow region. ''s/^n^ 236 M m ^ 5 . ^ = 277 M m Oo o ^a,max = 252Mm d, = 204Mm d^^^l'25m Fig. 2.1 Stokes diameter for an irregular particle of volume diameter 204 |Lim. With maximum resistance to drag the particle will fall at the same speed as a sphere of diameter 236 jim. With minimum resistance to drag the particle will fall at the same speed as a sphere of diameter 277 \im.

Data presentation and interpretation 59 Table 2.1 Definitions of particle diameters Symbol Diameter Formula Definition d^ Volume Diameter of a sphere having the same volume V^^d^ as the particle d^ Surface Diameter of a sphere having the same s = 71^3 external surface area as the particle -[dll4) d^^, Surface- Diameter of a sphere having the same ratio of volume external surface area to volume as the particle (Sauter) d^ Drag Diameter of a sphere having the same Fij =3;rd^rju resistance to motion as the particle in a fluid of the same viscosity and at the same velocity {d^ approaches d^ when Re is small) df Free- Diameter of a sphere having the same free- falling falling speed as a particle of the same density in a fluid of the same density and viscosity d^f Stokes Free-falling diameter in the laminar flow *st = Wdd region d^ Projected Diameter of a circle having the same area projected area as the particle in stable orientation d Projected Diameter of a circle having the same area projected area as the particle in random orientation [for convex particles, mean value for all orientations d == d^], d^ Perimeter Diameter of a circle having the same perimeter as the projected oufline of the P = ltd,. particle d^ Sieve Width of the minimum square aperture through which the particle will pass • df^ Feret The distance between pairs of parallel tangents to the projected outline of the particle in some fixed direction *^M Martin Chord length, parallel to some fixed direcfion, which divides the particle projected outline into two equal areas y^ Unrolled Chord length through the centroid of the particle oufline statistical diameters, often defined in terms of the mean value for a particular particle.

60 Powder sampling and particle size determination Fig. 2.2 The projected area of a particle is orientation dependent. Martin's diameter {dj^^) is 246 ^im, the Feret diameter {dp) is 312 | L i m and the projected area diameter in stable orientation (climax) * ^ ^52 | L i m (the particle is the same as in Figure 2.1) For irregular particles, the assigned size depends upon the method of measurement, hence the particle sizing technique should, whenever possible, duplicate the process one wishes to control. Thus, for paint pigments the projected area is important since this controls hiding power, whereas for chemical reactants the total surface area should be determined. The projected area diameter may be determined by microscopy for each individual particle, but surface area is usually determined for the powder as a whole. The magnitude of this surface will depend upon the method of measurement, permeametry for example, giving a much lower area than gas adsorption. Further, both of these methods depend upon the size of the gas molecules used in the determinations, since less surface may be accessible for larger molecules. The sieve diameter, for square mesh sieves, is the length of the minimum square aperture through which the particles can pass, though this definition needs modification for sieves which do not have square apertures. Microscopy is the only widely used particle sizing technique in which individual particles are observed and measured. A single particle can have an infinite number of linear dimensions and it is only when these are averaged that a meaningful value results. 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.