(Limits in Particle Size Characterization).

pp. 33-47 in: "Science of Whitewares"; eds. V.E. Henkes et al; The American Ceramic Society, Wesrerville, OH (1996)

LIMITS IN PARTICLE SIZE CHARACTERIZATION Herbert Giesche, NYS College of Ceramics at Alfred University ABSTRACT The limitations and specific requirements of different particle size characterization methods are discussed. The theoretical background of each method is shortly summarized and limits in terms of minimum or maximum measurable particle size are discussed together with several experimental parameters which can influence the accuracy or which can cause systematic errors. The paper focuses on sedimentation and optical (light scattering) characterization methods. INTRODUCTION Particle size is one of the most important characteristics in ceramic processing. A precise analysis as well as control throughout all processing steps is essential for high quality production and low loss rates. A large number of publications dealt with specific aspects in sample preparation and the analysis itself. The precise and controlled preparation of a representative sample is the first and essential step in getting reliable analysis results. This sample preparation has to be as closely related as possible to the actual processing procedure or vice versa to the information and conclusions to be drawn from those measurements. Despite the importance of those steps, the present paper will not address those aspects but rather focus on general limitations and possible errors of the various groups of instruments or techniques. DEFINITION OF PARTICLE SIZE In most cases particle size is described as the size of a spherical particle, which would show the same behavior as the actual particle in the specific analysis. As shown below, a variety of size descriptions exist and one has to specify in each presentation of the results, which kind of particle size and how it was determined. Geometric properties: - Diameter, Chord Length (Si ), Feret- (XF ), or Martin- (XM ) Diameter - Surface Equal Sphere X = Surface S

π

- Projected Area X PM = 4 Area π (! non spherical particles are not always in a random position) - Volume Equal Sphere X V = 3 6 Volumeπ 18 η w g Sedimentation Equal Sphere (Stokes): XW = (ρ p − ρ fl ) g

Si

10 0

10 0

80

80

60

60

40

40

20

20

0 1

10

10 0

m ass or volume %

number %

Moreover not only the precision or reliability of the analysis is important but also the way those results are presented. Some are quite obvious, like the difference between a cumulative and differential distribution, or using a linear or logarithmic scale on the size axis. However, there are some hidden differences which are not always indicated or realized. For example, the average particle size will change depending on whether a number-, area-, or volume- (mass-) distribution is shown. There is no general rule, which of those presentations to recommend, this very much depends on the preferences of the operator or the specific requirements of the process. As shown in fig. 1 the sample will have a vastly different mean particle size when plotted as a number- or volumedistribution.

0 10 0 0

Pa rt ic le S iz e / um

Fig. 1: Micrograph (left) and cumulative size distribution (right) The situation gets even more complicated, since most real powders are not spherical. A shape factor can be introduced to account for this. One possible definition was provided by Wadell.1 the shape factor, Ψ, is defined as the surface-ratio of a volume equal sphere 2 XV ⎛ ⎞ over a surface equal sphere. Ψ = ⎜ X S ⎟⎠ ⎝ As can be shown easily the following rank of order is valid for non-spherical particles. XS = XPM ≥ XV ≥ XW IMAGE ANALYSIS Optical- or electron-micrographs can be used to determine the particle size of a sample according to the various definition as mentioned earlier. Even so this technique may provide much more information than just the particle size, there are a number of errors resulting from sample preparation or the analysis set-up. Fig. 2 demonstrates some of those errors, like the under-cut or the random or stable orientation of particles. Especially small particles may not be counted when they are covered by larger particles, or the measured particle size is larger for a sample with powder particles in a stable position compared to randomly oriented particles. In addition computerized image analysis can give faulty results depending on a correct alignment of the threshold value and/or different orientations of particles towards the screen pixels. The latter is especially true for needle like particles as demonstrated in Fig. 3.

under cut, random, or stable

Fig. 2 Errors due to particle position Influence of orientation: a) 22 pixels b) 13 pixels

Fig. 3 Computer assisted image analysis SIEVE ANALYSIS Even with such a simple technique as the sieve analysis a number of errors have to be considered. Some of those errors are quite obvious, like damaged or uneven sieves, a too short analysis time, insufficient tapping, abrasion, breaking of particles, humidity, clogging, or agglomeration of the powder. Yet, in case the tapping frequency becomes too high, the passing-% of the sample likewise decreases. In the latter situation particles do not have enough time to pass through the sieve opening before the movement of the sieve will bounce them back. Thus for a fixed analysis time the amount of sample, remaining on the sieve, passes through a minimum, as shown in fig. 4. Further information about the correct sieve analysis is given in several reference papers.1-5

Fig. 4 Influence of time and vibration frequency on sieve analysis after F. G. Carpenter and V. R. Deitz2 SEDIMENTATION Using Stokes law,6 sedimentation experiments will provide information about the particle size distribution. Stokes law relates the gravitational, the buoyancy, and the drag force acting on the moving (settling) particle in a medium, as shown below. The greatest uncertainty in this equation is related to the drag force. The drag coefficient, Cw , the Reynolds number, Re, and the projected area of the settling particle can not be described sufficiently in all cases. The linear relation between Reynolds number and drag coefficient, Cw = 24/Re, is valid only in the laminar flow regime at Reynold numbers below 0.25. This relation is used for the basic Stokes equation as utilized in most sedimentation experiments. At higher Reynold numbers a transitional region and finally turbulent flow is observed as shown in fig. 5, where spherical particles were assumed.

FG = VP ρ P g =

π

x3 ρP g

buoyancy: FA = VP ρ fl g =

π

x 3 ρ fl g

weight:

drag force: FW = c W (Re) A P

6

6

ρ fl 2

with: VP : volume of particle

ρ pl or ρ fl : density g:

gravity constant

x: particle diameter c W (Re): drag coefficient AP :

projected area of particle

w g : sedimentation velocity

w 2g

1,000

100

transition range for: 0.25 < Cw < 2000 Cw

10

turbulent flow for Cw ≅ 0.45

laminar flow for: 1 Re < 0.25 Cw ≅ 24/Re 0.1 0.1

1

10

100

1000

10000

100000

Reynolds number

Fig. 5 Drag coefficient as a function of Reynolds numbers for spherical particles The limitation in the range of Reynolds numbers thus limits the maximum particle size to be determined for a specific system, since ordinarily the basic Stokes equation, valid for the laminar regime, is used without further corrections. Table 1 summarizes those limitations for a variety of materials. Table 1: Laminar flow restriction, maximum particle diameter at Re=0.25 material

X max = 3

4.5 η (ρ p − ρ fl ) ρ fl g 2

flour quartz alumina lead

density [g/cm3] 1.5 2.65 3.96 11.4

Xmax water 97.2 65.3 53.5 35.3

in

µm

air 43.7 36.2 31.9 22.2

Yet the measurable range can be extended by using various equations to correct for the non linear relation between Cw and Re in the transition range. Table 2 provides a brief overview of equations, the recommended range, and the associated error for Cw for each equation over this range. In order to expand the measurable range a different dispersion media could also be used. Having a higher viscosity liquid expand the range to larger particle sizes. One example is shown in the following diagram (fig. 6).

Table 2: Correction formulae for the transition range7 formula Cw = ...

Re-range

max. error %

(24/Re) +2

< 10

-4/+4

(24/Re) +1

> 1 display Fraunhofer diffraction, whereas Rayleigh-Scattering dominates for small particles (α