Conventional Shape Factors

Particle shape is a fundamental characteristic of powder particles and thus influences the properties of particulate systems. Various shape terms have been proposed to quantitatively represent particle shape. Early systems tended to measure one specific feature of a particle.

Table 3 (Ref 23, 24, 25, 26, 27, 28, 29, 30) lists some of the most frequently used shape terms. The applicability and/or limitations of various shape factors, also referred to as shape parameters, are discussed in the following section of this article on stereological characterization of shape. For discussion purposes, the following method developed by Hausner (Ref 28) is used to demonstrate the applicability of shape factors in quantitative analysis of particle shape.

Table 3 Shape terms and their definitions

Volume specific surface (Sv)

V: Particle volume

S: Particle surface area

Heywood ratios (Ref 24)

Elongation ratio (n)

Bh: Breadth--the minimum distance between two parallel planes that are perpendicular to planes defining Th_

Th: Thickness--minimum distance between two parallel planes tangential to the particle maximum stability plane_

Lh: Length—distance between two parallel planes that are perpendicular to both Th and Bh planes

: Jh = f/k da: Diameter of a circle having the same projected area where/=

k-VJdl

S: Particle surface area

V: Particle volume

Wadell

-J4.84 (k/f)m dv: Diameter of the sphere that has the same volume of the particle f k: Diameter of the sphere having the same surface area as the particle (defined in Heywood ratios)

r: ith radius of curvature along the particle profile which is smaller than the radius of the largest inscribed circle_

N: Total number of radii of curvature smaller than the inscribed circle_

R: Radius of the largest inscribed circle

Krumbein (Ref 28)

Sphericity ( v

Lk: Longest dimension of the particle

Sphericity ( v

Tk: Particle thickness

Bk: Breadth--measured perpendicular to L

Tk: Particle thickness

Hausner (Ref 29)

Elongation ratio (x)

a: Length of the enveloping rectangle that has the minimum area b: Width of the rectangle

A: Projected area of the particle c: Perimeter of the projected profile

Church shape factor (Vc) (Ref 30)

dM: Martin's diameter--length of the chord which divides the profile into two equal areas with respect to a fixed direction_

E(dM): Expectation of Martin's diameter_

dF: Feret's diameter, the distance between a pair of parallel tangents of the particle profile with respect to a fixed direction_

E(dF): Expectation of Feret's diameter_

Centroid aspect ratio (CAR) (Ref 31)

dm: Longest chord passing through the centroid

A rectangle of minimum area is drawn around the cross section of a particle (particle projection) as it is observed under the microscope (Fig. 25). The ratio of the rectangle side lengths permits calculation of particle elongation:

Elongating factor, x = -

Fig. 25 Determination of particle size characterization. (a) Length of the enveloping rectangle, which has a minimum area. (b) Width of the rectangle. (c) Circumference of the projected particle. A, surface area of the projected particle

The ratio of the area (4) of the projected particle to the area of the enveloping rectangle of minimum area (a*b) indicates the bulkiness of the particle:

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