Che5700 陶瓷粉末處理
粉體粒徑分析
Particle size analysis
• powder, particle (primary, secondary), colloid,
agglomerate (soft, hard), aggregate, granule, crystallite:
slightly different meaning
•Either single particle or a particle system
•Ideal powder: uniform particle size, 0.1 – 1.0 m
(submicron), spherical, no agglomerate (or weak
agglomerate), high purity, batch-to-batch consistency
• Many possible
particle shapes: rod,
fiber, flake, tube
(CNT), flower-like,
etc.
• If not spherical:
need more than
one parameter to
describe the
particle
Che5700 陶瓷粉末處
理
Sampling
• Be representative!! Need theories from statistical theory
• different particles (shape, size, density etc.) – different
motion, should be considered during sampling.
•Golden Rules of Sampling: (a) samples should be in
motion; (b) The whole stream of powder should be taken for
many short increments of time in preference to part of the
stream being taken for the whole time.
• Results highly dependent on sampling techniques!!
Grab sample;
cone & quarter;
Riffling (three
experimental
methods)
From JS Reed,
2nd ed.
Che5700 陶瓷粉末處理
Sampling Accuracy
•Maximum sampling error: E = 2i/P where i = standard
deviation from this sampling; P = weight fraction of material
greater than 44μm (40 ppm here)
•t = (i 2 + n 2) ½ where n = standard deviation from
measurement (total error: substitute i with t); i.e. sampling
error + measurement error
•example: source: 10,000Kg; sample: 10g for analysis, after
classification, particle >44m, Fe impurity (color problem)
40ppm (others = silica particle, 0.5m), estimate sampling
error  i = [P(1-P)/Ws . (P w1 + (1-P) w2) . (1- Ws/Wb)] ½
 Ws =sample weight; Wb = total weight; w1 = impurity
particle weight; w2 = main component particle weight 
result: i = 2.37x10-7; experimental error from sampling E =
1.19%; yet n = 4.0x10-6 (10%) main source of error:
measurement, not sampling  total error: 20%
error
size
From JS Reed
Two-Component Sampling Accuracy
If we count particles (instead of measuring weight),
then sampling error
 σi = [p (1-p)/Ns (1- Ns/Nb)] ½
where p = fraction of particles above a certain size
Ns = number of particles counted
Nb = number of particles in the bulk
This equation can also be used in estimating error from
public opinion poles;
Che5700 陶瓷粉末處理
Different definitions of particle size
•Principal concept: equivalent diameter to a sphere;
•Equivalent items: e.g. volume, surface area, sedimentation
velocity, projected area, (many kinds).
•  dv volume diameter V = /6 dv3 need particle volume
•  ds surface diameter S =  ds2 .need particle surface
•  dvs surface volume diameter dsv = dv3/ds2 ..measure
specific surface area of particle (per unit volume or unit
weight)
•  Stoke’s diameter dStk same sedimentation velocity as a
sphere
Che5700 陶瓷粉末處理
Different definitions of particle size(2)
•  projected area A = /4 da2
•  Sieve diameter: passing opening of a sieve (width
of square opening)
•  Martin diameter: mean chord length of projected
outline of particle
•  Feret’s diameter: mean value of distance between
pairs of parallel tangents to the projected outline of
particles
• We often use software to help with analysis of
projected images (from SEM, TEM)
•Martin diameter: in reality, we can choose
several different directions and average the data
•Feret diameter: distance between parallel
tangents
•Statistical diameter
From TA Ring, 1996.
Coulter counter:
Principle  when each
particle passing through the
aperture, it will displace
same volume of conducting
liquid, resistance then rise 
the frequency and extent of
rise  particle size and
distribution
Problem: two particles
passing at the same time, or
continuous passing of two
particles, particle too large
or too heavy, electrolysis,
aperture blockage etc.
Che5700 陶瓷粉末處理
Microscopic Method
• The basis of all techniques, (seeing is believing)!! OM,
SEM, TEM
•Need standard particles for calibration (e.g. PS
polystyrene monodispersed particle from emulsion
polymerization)
• in association with image analysis software: can handle
large number of images, good statistical results
•ASTM counting requirements: modal size class at least
25 particles, best 10 in each class, total no less than 100
particles; (another suggestion: 700 particles least)
Some Terminology
Rayleigh scattering: particle size much smaller than
wavelength d<<λ  Rθ = Iθ r2/Io = 8 π4α2/λ4 (1+ cos2θ),
where α = polarisability = (no/2π)(dn/dc)(M/L); λ =
wavelength of incident light; no = refractive index of solvent,
dn/dc = change of RI due to concentration, M = molecular
weight (measured value), L = Avogadro No.
 particle size much greater than wavelength: opaque,
only scattering, Fraunhofer diffraction
 Mie scattering: interaction between particle and light
(in general 10λ – λ/10)
Che5700 陶瓷粉末處理
Optical Methods (Optical counters)
•Laser diffraction technique: each particle producing
Fraunhofer diffraction effect when passing parallel laser light,
intensity of diffracted light ~ (size)2; diffracted angle ~ size.
Handles gas (aerosol) or liquid samples.
• Sensing volume can be very small such that only one particle
counted at a time.
取自粉粒體粒徑量測
技術, 高立圖書1998
Forward scatter, side scatter, back scatter
Dynamic Light Scattering (DLS)
• Also termed as Photon Correlation Spectroscopy
(PCS); or Quasi-elastic Light Scattering QELS)
•Electric-filed time correlation function obtained
from the scattered light due to motion of particles
was analyzed to evaluate the average decay rate:
Γ= D q2
• D = diffusion coefficient of particles;
•Stokes-Einstein equation: R = kBT/(6πηD) to get
particle size (R)
• q = (4πno/λo) sin(θ/2); no refractive index of
solvent; λo = wavelength of light
PCS or DLS 基本上是量測粒子散射光的相對強度, 運動前
後的差別, 利用correlator來分析數據, complex
mathematics
Uncertainty analysis by dynamic
light scattering
•source: 機械工業, No 288, 94-100, 2007
• factors of uncertainty: Boltzman constant,
wavelength of laser, scattering angle, diffraction
coefficient, viscosity of solution, etc.
• Based on the experiences of the authors: (以PS
standard particles) for 20nm, 100nm & 1000 nm;
their uncertainty 3.2 nm, 6.2 nm & 48 nm
respectively.
Che5700 陶瓷粉末處理
Hydrodynamic Chromatography
•Same as other chromatography technique, particles of
different size will move at different speed through the
column, to become separated and then detected;
smaller particles affected more by wall, move at slower
rate.
disadvantage: low sample recovery (may be trapped),
possible interaction between particle and packing
material, excessive sample dispersion, relative low
resolution etc.
Rf = (time
of passage
of marker)/
(time of
passage of
colloid)
versus
colloid size

calibration
Fluid flow
+
centrifuge
produce
separation
X-ray Line Broadening
•From full width at half maximum of XRD peaks, estimate
of crystallite size (an average number); need complex
mathematical treatment to get size distribution.
•Scherrer equation: d hkl = k /(o cos); k = constant
(mostly taken as 0.9 or 1.0; o = width at half height;  =
x ray wavelength;  = diffraction angle (notice 2 ) [in
theory: βo2=βm2-βi2, βm = measured breadth, βi =
instrument breadth]
•Reasons for broadening: small grain size, strain (or
disorder) inside, instrumental error - use single crystal (Si)
for calibration
From JS Reed, 2nd ed. Good instrument and practice should
obtain similar results, not easy for one method to dominate.
Che5700 陶瓷粉末處理
Shape Factor
•Surface or volume shape factor: V = v d3; or S = s d2;
 = shape factor, dependent on size measurement; αv
=π/6; αs = π
• shpericity 球形度: = (surface area of a sphere having
same volume)/(actual surface area of particle)  = (d NV/d
2
NS)
• similar definition: circularity = (perimeter of a circle
having same area)/(actual perimeter)
• aspect ratio (長軸比): for fibers, = (length/ diameter) or
(longest to shortest dimension)
All particles
are hydrous
zinc oxide
(from
different
precipitation
conditions):
Shapes are
different
Different name for
different shapes
ψA/ψV  index of
angularity
(shape factor based on
area/shape factor
based on volume)
Acicular: slender and
pointed, needle-like 常用於
描述葉子的形狀
* 取自TA Ring, 1996; S/V = St/V . Dav ; 其中St/V =
surface area/unit volume (specific surface area, similar to
based on unit weight), an easy to measure item (BET data)
More Shape Factors
•Dynamic shape factor  = (d NV/ d st)2 ; same motion
resistance as a shpere; d NV & d st = volume equivalent
diameter based on number & Stoke’s diameter
respectively;
•=
-½
( = sphericity)
• Simple way to quantify shape and can be related to
other properties or processing variables
Che5700 陶瓷粉末處理
Fractal Shapes
• texture like a broccoli or cauliflower, the particle is fractal;
then use fractal shapes, C = circumference (週線)
estimated with a ruler of size Cx ~ x 1-D , where D = fractal
dimension of particle
• In a particle microscopic picture: number of particles in a
circle of radius r  plot log-log figure (N vs r), slope 
fractal dimension
• Fractal shapes traditionally produced by agglomeration
from sol-gel, or flame, plasma synthesis of ceramic
powders
• particle property related to fractal dimension, e.g.  ~ R
D-3; A ~ (r )-1 R D; (r = radius, R agglomerate size)
o
o
Taken from TA
Ring, 1996
Che5700 陶瓷粉末處理
Size Distributions
•expressions: (a) as a figure – (i) cumulative (oversize
or undersize); or (ii) frequency – based on number,
weight or volume, etc. (b) proper mathematical
equations
•CNPF: cumulative distribution based on number,
percentage finer; CNPL (L = larger than this size)
•CMPF: (M for weight) (based on weight)
Size interval: linear or geometric (幾何級數or log scale),
e.g. 2 ½
Che5700 陶瓷粉末處理
Mathematical Equations
•Two most popular equations:
•Normal distribution: f(x) = 1/(2) exp[-(x – x)2/22]
(two adjustable parameters): x &  (average and standard
deviation)  ∫f(x) dx [from -∞ to ∞] = 1; σ=x(84.13) –
x(50) = x(50) – x(15.87)
•Log-normal distribution: f(z) = 1/(z2) exp[-(z – z)2/
2z2] ; z = ln d (similar two parameters) or as f(d) =
1/(lng 2) exp[- (ln(d/dg))2/ 2 (lng)2]  dg g =
geometric mean size & standard deviation; σg = d 84.13/d 50
= d 50/d 15.87
Che5700 陶瓷粉末處理
More Equations
•Rosin – Rammler distribution: f(x) = n b x (n-1) exp( - b
xn) ; n & b adjustable parameters, related to particle
characteristics, after integration, we get: F(x) cumulative
distribution = exp( - b xn) …a simple equation
•Gaudin – Schulmann model: w(d) = a n d
based on weight
(n-1)
; w(d)
• Most equations have two parameters, similar results in
fitting the true distribution, important question = what is
the meaning of the parameters, any physical meaning.
From TA Ring, 1996;
bimodal distribution
Obtained from mixing
of two particles with
different size
distributions, or one
type particle from
two different
formation
mechanisms
Che5700 陶瓷粉末處理
Mean diameters
•Can use “mean”, “modal”
(most populous) or median
(in the middle or50%)
•Mean (or average): several
different ways (equations)
to calculate it
fN(a): number distribution of size a; we can
also have number mean diameter
Number frequency distribution (n) will be very much
different from mass frequency distribution (nd3)
* From TA Ring, 1996; e.g. if expect 1% error, for a size
interval having 20wt%, we need to count about 400
particles; total error = sampling error + sizing (analysis)
error
Determine Error of Size
Distribution (previous table)
• 16-22 μm size interval: Wj = 13%, Nj = 110; Ej = error
= Wj/(√Nj)= 1.2%
• largest error: 60-84 size interval, only 1 particle counted,
Wj = 6.5%, (from figure)  Ej = 5% (?)
• total error = ET = ΣEj Wj ~ 2% for this case
Che5700 陶瓷粉末處理
Characteristics of Agglomerates
•Binding between particles: electrostatic, magnetic,
van der Waals, capillary adhesion, hydrogen bonding,
solid bridge (from atom diffusion) due to sintering,
chemical reaction, dissolution-evaporation etc.
• strength: may be estimated by methods such as
compaction, ultrasonic vibration (indirect);  size
distribution before and after treatment  get an
estimation, in theory can be obtained from: primary
particle size, coordination number, agglomerate
porosity etc.
In general:  = o exp( - b ); /tho = 1 - 
From Am. Cer.
Soc. Bull., 65,
1591, 1986.
conclusion: weak
agglomerate
provides better
sintered density
Soft agglomerate
vs strong
agglomerate
Che5700 陶瓷粉末處理
Some methods to break agglomerates
• For example:
•
•
•
•




研磨 milling
超音波震盪: ultrasonic treatment
分散劑 dispersion agent (chemical method)
高的成型壓力 high forming pressure
 From Vantage Tech. Corp.