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Interpreting Laser Diffraction Results for Non-Spherical Particles

David M. Scott Advanced Particle Sensors [email protected] Horiba Webinar Series – Dec. 10, 2019

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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.

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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

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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

Sieving Dynamic Light Scattering

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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

Frac

tion

(%)

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.

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Particles Are Often Non-Spherical !

10 µm

0.2 µm

Polymer flakes

Sepiolite (clay) Kaolinite (clay)

Crystallized amino acid E 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

≈

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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

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0.0001

0.001

0.01

0.1

1

-0.4 -0.2 0 0.2 0.4

Inte

nsity

(arb

. uni

ts)

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.

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Interpreting PSD Results for

Non-Spherical Particles

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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

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y = 0.8943x!R² = 0.97101!

0!5!

10!15!20!25!30!35!40!45!50!

0! 10! 20! 30! 40! 50!

B fr

om fi

t (un

its o

f λ)!

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

Inte

nsi

ty (

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

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0

1

2

3

4

5

6

0 5 10 15 20

Rat

io o

f Dia

mat

ers

(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

. vol

ume

frac

tion

(%)

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.

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Examples: Rod-Like Particles

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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

Diff

eren

tial V

olum

e Fr

actio

n (%

)

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

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0 1 2 3 4 5 6 7

0.1 1 10 100 1000

Diff

. Vol

ume

Frac

tion

(%)

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

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0

1

2

3

4

5

6

0.1 1 10 100

Diff

. vol

ume

frac

tion

(%)

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 "L

engt

h" (m

icro

ns)

Time (minutes)

The median crystal length decreases about 0.6 microns per minute during recirculation

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Example: Ellipsoidal Particles

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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)

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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

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Examples: Plate-Like Particles

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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

Volu

me

Frac

tion

(%)

Credit: SEM image from USGS

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Deconvolution of Kaolinite PSD

Example of deconvolution of ESD PSD obtained with kaolinite

Volu

me

Frac

tion

(%)

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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.

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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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