effect of particle size and shape on powder...

Dr James Elliott

Effect of particle size and shape on powder properties

PARTICLE TECHNOLOGYPart III Materials SciencePart IIB Chemical Engineering

MSM III M2/CET IIB – Lecture 5

23/10/2006

Handouts will available online from http://www.msm.cam.ac.uk/Teaching/PtIII/M2/

Copyright © 2002 University of Cambridge. Not to be quoted or copied without permission.

A.1.1 The physical nature of granular materials

Granular materials are rather unusual state of matterThey have both similarities and differences to more familiar states, such as solids and liquids:

– When densely packed, capable of resisting a shear stress(similar to solid, different to a liquid)

– Usually weak in tension and very inhomogeneous stress distribution in compression (force chains)

– Generally have no long-range positional order (similar to liquids and glassy solids)

– Non-thermodynamic (average K.E. of particles » kBT) systems, in other words there is no well-defined granular “temperature”

– corollary: most properties of interest are not a function of state (beware!) and therefore powders are highly process-sensitive


A.1.2 The physical nature of granular materials

Consider a sand-filled hour glass…

The reasons for some of these unusual types of behaviour will be explained in following lectures, but for now we must start with some basic definitions…

For a liquid, flow rate proportional to height of liquid above the neck

For a granular material, flow rate is independent of height, and related only to the diameter of the orifice


2.1 Some basic definitions

Solid density, ρs– The density of solid particles (i.e. bulk material)

Bulk density, ρb– The density of the particles and voids (i.e. bulk powder)

Packing fraction, η– The ratio of solid volume to bulk powder volume

Void fraction (or “voidage”), ε = 1 – η– The ratio of void volume to bulk powder volume

Note that the bulk density depends on both the present and past treatment history of the powder (an example of a non-state dependent powder property)

( )b s s gρ ρ η ρ (1 ε) ρ ε, ususally ignored= = − +


2.2 Other variables characterising powdersParticle size distribution (PSD)

– Discussed at length in lectures 1-4

Particle shape, and shape distribution– Sphericity [ψ = (S.A. of equiv. sphere)/(S.A. of particle)]– Angularity [spectrum of spatial frequencies]– Concavity/convexity

Forces acting between particles– Cohesive or non-cohesive

Dry/Wet [lectures 6-7/10-12, respectively]Charged/uncharged

Single particle (material) properties– Mechanical properties (Young’s modulus, Poisson’s ratio, etc.)– Porous/non-porous


2.3.1 Methods for characterising PSDs

Sieving– Measure fraction of particles passing through stack of sieves of

decreasing mesh size[1]

– Results hard to interpret for low ψ

[1] J.P.K. Seville, U. Tüzün, R. Clift in Processing of Particulate solids. (1981).



Microscopy– Direct observation of particles (in 2D projection!) using optical or

preferably scanning electron microscopy (SEM). Use automated image analysis software to extract PSD and shape distribution.

Laser diffraction– Mentioned in lectures 1-4

– Fraunhofer diffraction instruments good for in situ measurement of PSD but can be unreliable for small, transparent particles[1]

[1] J.P.K. Seville, U. Tüzün, R. Clift in Processing of Particulate solids. (1997).



X-ray microtomography (XMT)– A relatively new technique, used to require synchrotron X-ray

source but now can be done with desktop equipment

SampleX-ray Shadow

Images

Raw DataExperiment 3D Image

1. Acquisition

1.

2. Reconstruction

2.

Stage:

Axis of rotationAxis of rotationAxis of rotation

2D Detector

2D Detector

2D Detector

Incident X-ray beam

Incident X-ray beam



X-ray microtomography (XMT)– Comparison of results with SEM (left, as cumulative distribution)

and laser diffraction methods (right, as volume PSD) show that there are discrepancies between ‘direct’ and ‘indirect’ methods

– Be aware!

180 200 220 240 260 280 300 3200

20

40

60

80

100

SEM CT

Acc

umul

ativ

e vo

l/wt(%

)

Particle Size (µm)

100 200 300 400 5000

1

2

3

4

5

6

Vol

umn

(%)

Particle size (µm)

200-300 um d(0.5)=248.257

150 200 250 300 3500

2

4

6

8

10

12

14

16

18

20

Vol

ume

(%)

Particle Size (µm)

CT 2000 Particles


2.4 Methods for characterising particle shape

Can parameterise particles by using– Simple parameters (see lectures 1-3)

volumesphericityangularity

– Fourier or spherical harmonic expansion (rarely used)

[1] D. Huller & R. Wiechart in Particle Size Analysis. (1981).

Two dimensional map of particles of varying angularity, and sphericity[1]

Sphericity (decreasing)

Angularity (increasing)


2.5.1 Filling space with geometric solids

Maximise the proportion of space occupied by particles– Increase stiffness and strength of powder compacts– Decrease permeability of powder compacts– Try not to affect the “flowability” of uncompacted powder

“Apollonian” packing

First known multifractal (ca. 200 BCE)Fractal dimension D ~ 1.3058.[1]

asymptotic PSD is power-law (slope -2.25)[2]

Log(size) →

Log(

num

ber)

[1] B.B. Mandelbrot. “The fractal geometry of nature”. (1982).[2] Y. Rouault Powder Technol. 102, 274-280 (1999).



Lattice packings of monosized spheres can be thought of in terms of stacking sequences of close-packed layers…ABABAB… hexagonal close-packed (h.c.p.)

…ABCABC… cubic close-packed (c.c.p.)

7405.023

η ≅= π



Size and co-ordination of interstices in close-packed structures: tetrahedral and octahedral

Descartes’ circle theorem (DCT) or Soddy’s formula

∑∑+

=

+

=

=⎭⎬⎫

⎩⎨⎧ 2

1

222

1

n

ii

n

ii bnb

Define a bend b = 1/r for sphere of radius rin n dimensions.

Then DCT links the bends of an assembly of n+2 spheres in contact

[1] G.G. Szpiro in “Kepler’s conjecture” Wiley and Sons (2003).


A.2 Filling space with geometric solids

Soddy’s “bowl of integers” [1,2]

[1] F. Soddy, Nature, 77 (1937).[2] http://mathworld.wolfram.com/BowlofIntegers.html

Place two solid spheres of radius 1/2 inside a hollow sphere of radius 1 so that the two smaller spheres touch each other at the centre of the large sphere and are tangent to the large sphere on the extremities of one of its diameters

This arrangement is called the “bowl of integers” since the bend of each of the infinite chain of spheres that can be packed into it such that each successive sphere is tangent to its neighbours is an integer.



The static packing fraction of an N-component mixture ηNis defined by

Accessed experimentally by measuring the bulk volume Vb of a particle assembly Can use a digital camera connected to an image analyser, having previously determined the solid particle volume Vs gravimetricallyThe particle densities must be known (measured using Archimedes’ principle or porosimetry)

∑=

==N

ibsi VV

1N /ηη


2.6.1 Monodisperse diameter spheres (expt.)

For spheres of a single diameter, d, the only factor affecting the packing is the finite size of the container, DThe packing fraction was measured as a function of d/Dfor containers with cylindrical symmetry.

( )Dd07.045.0004.0605.0η ±−±=

Extrapolated to infinite container size (D→∞), this represents 0.605/0.741 = 82% of maximum close-packed density

Why the discrepancy?


2.6.2 Monodisperse diameter spheres (expt.)

Why is close-packing generally not achieved, even for monodisperse size spheres?

Diameter ratio 1:4.47 for same container size reveals that crystallisation is inhibited by presence of container wallsRatio must be at least 1:10 to be free from finite size effects


2.7.1 Binary sphere mixtures: diameter ratio

Examine influence of size on packing fraction: vary diameter ratio d1/d2 of binary sphere mixture at fixed relative concentrations of spheres (1:1 in this case)

Need to carry out experiments in regime where finite size of container is negligible (or extrapolate D → ∞)

d1/d2 = 4.468d1/d2 = 2.333d1/d2 = 1.263


2.7.2 Binary sphere mixtures: diameter ratio

Results from a binary mixture with number fractions of components x1 = x2 = 0.5 shows that minimum in packing fraction occurs when spheres are of equal size

This is an interstitial filling effect, and is therefore fundamentally limited when smaller component fills interstices of larger component to the close-packed limit


2.7.3 A binary sphere mixture: composition ratio

Can also change the composition ratio for fixed particle size. Shows a maximum in the packing fraction around x1/x2 = 0.7 for large size ratios. Plot below is for d1/d2= 30.

This is an interstitial filling effect, and is therefore fundamentally limited when smaller component fills interstices of larger component to the close-packed limit


2.7.3 A binary sphere mixture: composition ratio

Also known as ‘Furnas’ curves in soil mechanics, where εis the void fraction and w is proportion of larger particles

k is the ratio of size of smaller particle to larger particlesCurve tends to asymptotic limit as k → 0


2.8 Beyond binary mixtures…

McGeary[1] showed that packing fractions of up to 0.95 can be reached by using quaternary mixtures of spheres.This is really the maximum number of components it is worth having, as smallest particles begin to be dominated by non-geometric interactions (more in lecture 6)Optimal size ratios and concentrations quoted were 316:38:7:1 and 60.7:23.0:10.2:6.1% by volume, respectively.These mixtures flowed almost freely, displaying almost no ‘viscosity’ whatever (powder flow dealt with in lectures 7-8)

[1] R.K. McGeary J. Amer. Cer. Soc. 44, 513-522 (1961).


2.8.1 Recursive filling model: multi-modal mixtures

Can account for the maximum packing fractions achieved by invoking simple recursive filling of the interstices.

1 0.625 0.625

2 0.840 0.600

3 0.900 0.536

4 0.951 0.530

n ηn x

[1] J.A. Elliott, A. Kelly, and A.H. Windle, J. Mat. Sci. Lett. 21, 1249 (2002).[2] N. Ouchiyama and T. Tanaka, Ind. Eng. Chem. Res. 28, 1530 (1989).

( )1η η 1 ηn n n x+ = + −

( )η 1 1 nn x= − − ( )1/1 1 η nnx = − −

n

i

i

nii

n

iin

xxxc

c

)1(1)1(

η/η ηη

11

−−−

=

==

−

=∑


2.8.2 Recursive filling model: multi-modal mixtures

Predicts relative concentrations for binary, ternary and quaternary sphere mixtures in excellent agreement with McGeary’s experimental data

n c1 c2 c3 c4

Calc. Exp. Calc. Exp. Calc. Exp. Calc. Exp.

2 0.73 0.73 0.27 0.27 - - - -

3 0.66 0.67 0.25 0.23 0.09 0.10 - -

4 0.64 0.61 0.24 0.23 0.09 0.10 0.03 0.06


Summary of Lecture 5

Granular materials are an usual type of material, with characteristics of both liquids and solidsWe began by reviewing some basic definitions of measures used to characterise particle packingWe discussed some methods for characterising PSD and particle shapesWe discussed filling space with geometric solids, in both in random and ordered (crystalline) mannerWe saw the effects of size polydispersity and varying concentrations on the particle packing in binary, ternary and quaternary mixtures of spheresNext lecture, we look at interparticle forces in more detail

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