Interpreting Laser Diffraction Results for Non-Spherical Particles

Advanced Particle Sensors LLC
particle characterization for the real world TM
Copyright © 2019 Advanced Particle Sensors LLC
Interpreting Laser Diffraction
Results for Non-Spherical
Particles
David M. Scott
Advanced Particle Sensors
david.scott@particlesci.com
Horiba Webinar Series – Dec. 10, 2019

2
D. M. Scott / Advanced Particle Sensors, “Interpreting Diffraction Results for Non-Spherical Particles”, Horiba Webinar 12/10/19
Abstract
Particle shape is often overlooked in Laser Diffraction measurements,
but it affects the diffraction pattern used to determine particle size
distribution (PSD). As a result, laser diffraction instruments tend to
report bi-modal PSDs for non-spherical particles, even for samples
containing a single size class. Therefore a bi-modal result is ambiguous
unless shape is considered.
Equipped with only qualitative knowledge of particle shape, the particle
analyst can resolve this inherent ambiguity and even use laser diffraction
to measure aspect ratio of non-spherical particles. This webinar explains
the origin of this effect, describes how to interpret PSD data in such
cases, and demonstrates practical applications for measurements of
organic crystals, polymer flakes, bacteria, yeast, and clays.
This presentation is based on a conference paper by David M. Scott and Tatsushi Matsuyama, "Laser
diffraction of acicular particles: practical applications”, at the 2014 International Conference on Optical
Particle Characterization (OPC 2014), published in N. Aya et al., Editors, Proceedings of SPIE Vol. 9232
(SPIE, Bellingham, WA 2014), 923210.

3
Outline
1.  Brief Review of Laser Diffraction for Spherical Particles
2.  Impact of Shape on Laser Diffraction Results
3.  The Measurement Dilemma and Its Resolution
4.  Interpreting Results for Non-Spherical Particles
5.  Application Examples for Real-World Samples
6.  References

4
Benefits of Laser Diffraction
•  Fast
•  Simple
•  Wide range of sizes (less than 100 nm to over 2 mm)
•  Requires very little sample (<0.1 g powder, <0.2 mL dispersion)
•  Versatile: dispersions, droplets in liquids, sprays, solvents, etc.
•  On-line and in-process applications are possible
•  Commonly used throughout industry
10 nm 100 nm 1µm 10 µm 100 µm 1 mm (particle size)
Laser Diffraction
SievingDynamic Light Scattering

5
Laser Diffraction (Spherical Particles)
Laser θ
Flow cell with
transparent windows
Flow
Lens Detector
Light intensity
Physical
Particle
Size
Distribution
Diffraction
Pattern
Mie or Fraunhofer Scattering Model
Data
Inversion
Algorithm
Reported
Particle
Size
Distribution
Detector design and scattering model typically assume that particles are spherical
Detector
Geometry
Scott, Industrial Process Sensors (CRC Press, 2008), Fig. 8.12

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0%
20%
40%
60%
80%
100%
1 10 100 1000
Fraction(%)
Size (µm)
Particle Size Distribution (PSD)
•  Samples of interest usually contain a range (distribution) of
particle sizes.
•  Laser Diffraction size distributions are typically reported on the
basis of particle volume.
•  PSDs frequently (but not always) approximate a log-normal
distribution, which can be described by a median size and a
geometric standard deviation.
•  Log-Normal distributions shown on a semi-log graph resemble
a Normal distribution that is plotted on linear axes.

7
Particles Are Often Non-Spherical !
10 µm
0.2 µm
Polymer flakes
Sepiolite (clay)
Kaolinite (clay)
Crystallized amino acidE Coli
Rods
Ellipsoids
Disks
Credit:USGS

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Sizing a Non-Spherical Particle
Various definitions of particle size :
•  Feret’s diameter (F)
•  Minor axis of the projection (S)
•  Martin’s diameter (M)
•  Equivalent Sphere Diameter, ESD (V)
•  etc.
Many applications
“assume the
spherical cow”, or
the ESD (diameter
of the sphere with
equivalent volume).
Credit: cow photo from USDA.gov
Scott, Industrial Process Sensors (CRC
Press, 2008), Fig. 8.1
≈

9
Diffraction from Rods and Spheroids
Diffraction equations are given by Matsuyama et al. (2000) and Takano et al. (2012)
Rectangle (rods)
Ellipse (spheroids, disks)
Credit: Adapted from an image by Christophe Finot
Sphere
Unlike diffraction from a sphere,
for rods and disks the intensity
depends on the azimuthal angle
as well as the polar angle.

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Alignment in Laminar Flow
•  Flow Cells typically operate in laminar flow regime (with
Re << 2000)
•  Velocity across the flow cell has parabolic profile
•  The flow aligns the particles (Jeffery 1922) perpendicular
to the laser beam, but not necessarily in the same
direction.
Laser beam

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Integration over Many Particles
The resulting diffraction pattern
(light intensity versus scattering
angle θ) is a mixture of
contributions from the major
and minor axes of the particle.
θ
Ring detector

12
0.0001
0.001
0.01
0.1
1
-0.4 -0.2 0 0.2 0.4
Intensity(arb.units)
Scattering Angle (radians)
5λ x 10λ
rod
6λ sphere
The Result
The observed diffraction signal (intensity versus scattering angle)
cannot be described by diffraction from a single sphere.
In general, the instrument reports a bi-modal distribution:
Size
Vol.%
Diffraction
pattern
PSD

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The Measurement Dilemma
Given a bi-modal or multi-modal result for an unknown material:
•  Does the PSD signify that there are different size classes (e.g.
primary and aggregate) of quasi-spherical particles?
•  Or, does the apparent PSD indicate that non-spherical particles
are present?
•  How do we interpret PSD results for non-spherical particles?
? ?

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Resolving Ambiguity
To solve the dilemma, we only need to know whether or not the
particles are approximately spherical.
This information may be known a priori (e.g. from crystal habit or
particle formation considerations), but it is most often determined
by imaging:
•  Optical microscopy
•  Dynamic Image Analysis
•  Scanning Electron Microscopy
If the particles are approximately spherical, the PSD may be
reported as usual.

15
Interpreting PSD Results for
Non-Spherical Particles

16
The Approach – Diffraction Equivalence
Since the diffraction signal cannot be described by diffraction from a
single sphere, try a mixture of 2 sizes!
One size (a) represents the particle diameter, and the other (b)
represents the length (scaling to wavelength simplifies the math).
5 λ x 10 λ rectangle
The diffraction pattern of a
5 λ x 10 λ rod is nearly the
same as the pattern from
a mixture of 6.1λ and
10.4λ spheres.
aλ
bλ

17
y = 0.8943x!
R² = 0.97101!
0!
5!
10!
15!
20!
25!
30!
35!
40!
45!
50!
0! 10! 20! 30! 40! 50!
Bfromﬁt(unitsofλ)!
Rod length b (units of λ)!
a=5!
a=10!
Interpreting Particle Length
The previous calculation of equivalent diffraction has been
repeated for a number of rods of various lengths (bλ).
Numerical results show the diameter of the large sphere (B)
is about 10% smaller than the actual rod length (b).
aλ
bλ
B

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1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
Intensity(arb.)
Scattering angle (rad)
Spheroid (25 x 15) diffraction Fit (2 spheres)
Diffraction Equivalence for Spheroids
The diffraction pattern from a spheroid can also be represented
as the combined diffraction from two spheres.
Note: Side lobes cannot be fit accurately with only 2 spheres.
Matsuyama et al. (2000) showed a size distribution is needed .
Prolate spheroid

19
0
1
2
3
4
5
6
0 5 10 15 20
RatioofDiamaters(B/A)
Particle Aspect Ratio (b/a)
Spheroid, a=15
Spheroid, a=5
Rod, a=10
Rod, a=5
Power Law
Estimating Aspect Ratio
Calculations for rods and spheroids show that the aspect ratio (b/a)
and the corresponding ratio (B/A) of spheres giving equivalent
diffraction appear to be related by a universal curve:
Given (B/A), the aspect ratio is estimated as
(B / A) =1.12(b / a)0.563
(b / a) = 0.893(B / A)1.776

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Deconvolution (Fitting)
•  Diffraction from mono-sized rods and spheroids can be
approximated by combining two or more spheres.
•  A range of sphere sizes is required to approximate
diffraction from polydisperse samples.
•  In most cases, bi-lognormal distributions can be fitted
to the observed data, thus deconvolving the length and
width contributions.
•  Median sizes (a and b) of the two component
distributions are related to the median values of the
minor and major axes (A and B) of the subject
particles.

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0
1
2
3
4
5
6
0.1 1 10 100
Diff.volumefraction(%)
Particle size (microns)
Raw PSD bimodal fit Mode A Mode B
A
B
Bi-Log-Normal Distribution
Bi-Log-Normal distribution is characterized by TWO logarithmic
median sizes, TWO logarithmic standard deviations, and the
relative contributions of both size components.
This example shows a bi-log-normal distribution fitted to data:
Note: Median size A and B can be approximated directly (without
fitting) from the positions of the peaks.

22
Examples: Rod-Like Particles

23
Sepiolite
•  Sepiolite is a magnesium silicate clay with fibrous structure, used
to reinforce materials (e.g. nanocomposites)
•  High power sonication disperses the fibers
•  Peak associated with the fiber diameter becomes more prevalent
as fibers become more dispersed

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0
2
4
6
8
10
12
14
16
18
0.01 0.1 1 10 100 1000
DifferentialVolumeFraction(%)
Size (um)
Sepiolite (Dispersed)
PSD of well-dispersed sepiolite is dominated by the
peak associated with diameter
Diameter
Length
Note: Diffraction theory predicts this effect for
infinitely long rods (Bohren & Huffman 1998)
0.2 µm
Sonication energy:
300 J/g

25
0
1
2
3
4
5
6
7
0.1 1 10 100 1000
Diff.VolumeFraction(%)
Size (microns)
Initial PSD
After 1 min.
After 2 min.
Tyrosine Crystals
~10 μm
Crystals are broken by the pump during recirculation
(note the reduction in the coarse tail of the distribution
over time). No sonication was used in this example.
Tyrosine crystals in methanol
Tyrosine is an amino acid
used in protein synthesis

26
0
1
2
3
4
5
6
0.1 1 10 100
Diff.volumefraction(%)
Particle size (microns)
Raw PSD bimodal fit Mode A Mode B
Deconvolution of Crystal Data
Bi-modal deconvolution is used to determine the approximate
crystal length as a function of time:
y = -0.61x + 9.53
0
2
4
6
8
10
12
0 1 2
"Length"(microns)
Time (minutes)
The median crystal length decreases about
0.6 microns per minute during recirculation

27
Example: Ellipsoidal Particles

28
Polymer Flakes
•  Median of distribution corresponding to width is 143 µm
•  Median of distribution corresponding to length is 406 µm

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E Coli
Best fit to PSD is obtained
with two size modes
Ratio of the two modes is
1.5, consistent with direct
microscopic observation
RMS
error
0.128
RMS
error
0.267
RI = 1.397+0.01i (Balaev et al. 2002)

30
Yeast
Deconvolution of coarse peak yields 5.2
x 7.8 µm, close to observed size
Origin of PSD peak at 1 µm is unclear
(scattering from cell wall??)
Assume RI=1.52+0.01i

31
Examples: Plate-Like Particles

32
Dispersion of Kaolinite
•  Kaolinite was dispersed in an organic solvent with a high shear mixer
•  Samples were drawn at regular intervals of time
•  Power number of the rotor was used to estimate the total specific
energy for each sample
VolumeFraction(%)
Credit: SEM image from USGS

33
Deconvolution of Kaolinite PSD
Example of deconvolution of ESD PSD obtained with kaolinite
VolumeFraction(%)

34
Dispersion of Kaolinite
•  Results of deconvolving the ESD PSD data at each specific energy
•  Component due to diameter decreases in size: plates are breaking
•  Component due to thickness remains constant: there appears to be
little exfoliation of the plates above 100 J/g

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Conclusions
•  Laser Diffraction typically gives bi-modal results for non-
spherical particles, especially for (b/a)>1.5.
•  Interpretation of multi-modal PSDs requires a priori knowledge
of particle shape, but detailed information is not necessary to
monitor particle size and aspect ratio.
•  Bi-lognormal distributions can be fit to PSD data for a variety
of industrial particles, yielding approximate distributions of
particle length and width (or width and thickness).
•  A universal curve has been discovered that shows the true
aspect ratio varies as a power of the ratio of diameters (B/A).
•  Finally, deconvolution of apparent PSD allows us to estimate
aspect ratio and true dimensions of non-spherical particles.

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References
•  A.E. Balaev, K.N. Dvoretski, and V.A. Doubrovski, "Refractive index of escherichia coli
cells", Proc. SPIE 4707, Saratov Fall Meeting 2001: Optical Technologies in Biophysics
and Medicine III, (16 July 2002).
•  C.F. Bohren and D.R. Huffman, Absorption and Scattering of Light by Small
Particles (Wiley 1998), p.211.
•  G. B. Jeffery, “The motion of ellipsoidal particles immersed in a viscous fluid,” Proc. R.
Soc. A 102 (1922) 161-179.
•  T. Matsuyama, H. Yamamoto, and B. Scarlett, “Transformation of Diffraction Pattern due
to Ellipsoids into Equivalent Diameter Distribution for Spheres”, Part. Part. Syst.
Charact. 17 (2000) 41-46.
•  D.M. Scott, Industrial Process Sensors (CRC Press, 2008), Chapter 8.
•  D.M. Scott and T. Matsuyama, "Laser diffraction of acicular particles: practical
applications”, at the 2014 International Conference on Optical Particle Characterization
(OPC 2014), published in N. Aya et al., Editors, Proceedings of SPIE Vol. 9232 (SPIE,
Bellingham, WA 2014), 923210.
•  Y. Takano, K.N. Liou, and P. Yang, “Diffraction by rectangular parallelepiped, hexagonal
cylinder, and three-axis ellipsoid: Some analytic solutions and numerical results”,
Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 1836–1843.

37
Credits
•  Cow image from USDA.gov is in the public domain.
•  Rectangular diffraction pattern is adapted from an image by Christophe
Finot and used under license CC-BY-SA shown at
https://creativecommons.org/licenses/by-sa/1.0/legalcode
•  Kaolinite image from from United States Geological Survey (Bulletin 1614,
1985) is in the public domain

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