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Transcript
```Scattering of Electromagnetic Radiation by
Complex Microstructures in the Resonant Regime
Erik Shepard Thiele
A DISSERTATION
Ill
•
Materials Science and Engineering
Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy.
1998
Supervisor of Dissertation
By
Erik Shepard Thiele
1998
ii
Dedication
To my family.
iii
Acknowledgments
I begin by thanking and applauding my research advisor, Roger French, who
first encouraged me to embark on this journey. Roger has helped me to appreciate the
real point of forming a piece of work like this thesis. His support and guidance has
been indispensable, and without these I would have missed out on most of the deep
satisfaction of this process. I am very grateful to my faculty advisor, Peter Davies,
whom I have had the good fortune of knowing since my undergraduate days. I have
always respected Peter for his scientific leadership, but even more for his genuine
concern for all of us who have worked with him. I am also very thankful for the
guidance and encouragement I have received from Professors Dawn Bonnell and
Takeshi Egami throughout the development of this dissertation.
By far the most satisfying aspect of this experience has been the enjoyment of
working (and playing) with so many terrific people along the way. Naming all of you
here might actually prevent my being able to finish what writing I have yet to complete,
so I don’t dare try! You know who you are. I also wish to express my gratitude to the
DuPont Company and to my management, past and present, for supporting this
research. I recognize what a unique opportunity they have provided me.
Finally, I want to give thanks for that handful of very special people of whom life has
seemingly made gifts to me. My family means the world to me, and I appreciate this
today more than ever. Dear friends like Mike, Mark, Claire, Eric, Mike, Tom, Phil, and
Claudio are treasures. Though our paths cross and diverge over time, these people are
iv
like stars in the sky, always with me. Alors, je finis en remerciant avec un coeur bien
joyeux ma femme du désert, le vrai trésor au bout de cette chapitre-ci de ma légende
personnelle. Tu me fait m’épanouir, bellissima, et je t’aime! Vraiment, je serai avec toi
pour toujours.
v
Abstract
A computational optics approach is used to determine the scattering properties
of complex microstructures in the resonant regime. The time-domain finite element
method employed in this research to solve Maxwell’s equations can be used to
determine the scattering properties of scattering systems exhibiting arbitrary, complex
shapes and optical anisotropy. A detailed study of the factors influencing the accuracy
of this numerical method demonstrates that the construction of the finite element model
strongly influences accuracy; error in these computations is confined to <3% for typical
models. The computational optics approach is applied to determine the light scattering
efficiencies of several different white pigment systems. It is demonstrated that the light
scattering efficiency of core-shell white pigment particles is limited by the content of
low-index material and therefore cannot attain the same efficiency as optimized
particles of the high-index material.
Optically anisotropic spheres of rutile titania
exhibit the expected dependence of scattering properties upon the angle of light
incidence and polarization. Under conditions of random illumination, the scattering
efficiencies of these spheres can be approximated to within 10% using models based
upon isotropic materials. The agglomeration of both colloidal titania and latex particles
causes substantial deviation from the light scattering properties of the isolated primary
particles. In the case of rutile titania particles of optimal size, for example, losses in
volume-normalized scattering efficiency of 70% occur for the largest agglomerates
studied. The use of single, equivalent mass spheres to approximate the light scattering
properties of the agglomerate structures succeeds in predicting trends in these properties
vi
but fails as a rigorous description of these properties. For small clusters of either latex
or titania particles with varying interparticle separation distance, optical interactions
between neighboring particles diminish rapidly with increasing particle separation. In
most cases, changes in scattering efficiency with interparticle separation distance could
be well fit using an exponential decay. The decay constant of these spatial variations
varies in the range 0.25-0.45 light wavelengths for these systems, with appreciable
optical interactions not extending beyond 2-3 light wavelengths.
vii
Scattering of Electromagnetic Radiation by ........................................................................ i
Complex Microstructures in the Resonant Regime ............................................................. i
Dedication.......................................................................................................................... iii
Acknowledgments ............................................................................................................. iv
Abstract.............................................................................................................................. vi
List of Tables ..................................................................................................................... xi
List of Illustrations............................................................................................................ xii
Chapter 1. Introduction.......................................................................................................1
1.1 Single Scattering.......................................................................................................1
1.2 Scattering by Ensembles of Scattering Features.......................................................4
1.3 Formulation of the Present Research........................................................................6
1.4 On Vector Notation ..................................................................................................7
Chapter 2. Literature Review..............................................................................................8
2.1 Electromagnetic Radiation Scattering in Materials Science.....................................8
2.2 Light Scattering Properties of Complex Microstructures.........................................9
2.2.1 Light Scattering by White Pigment Particles.....................................................9
2.2.1.1 Application of Mie Theory to Optically Anisotropic Materials ...............11
2.2.1.2 Kubelka-Munk Analysis of Paint Film Scattering Efficiencies ...............12
2.2.1.3 Computational Modeling of Scattering by White Pigments: Previous
Studies....................................................................................................................14
2.2.2 Light Scattering by Concentrated Particle Dispersions ...................................16
2.2.2.1 The Crowding Effect in Paint Films.........................................................16
2.2.2.2 Light Scattering in Stereolithography.......................................................21
2.2.3 Light Scattering by Particle Agglomerates......................................................23
2.2.3.1 Computational Approaches ......................................................................23
2.2.3.2 Computational Results for Particle Agglomerates ...................................25
2.3 Principles of Light Scattering ..................................................................................27
2.3.1 Maxwell’s Equations .......................................................................................27
2.3.2 Exact, Analytical Solutions of Light Scattering ..............................................30
2.3.3 Approximate and Numerical Solutions of Light Scattering ............................33
2.3.3.1 Rayleigh Theory .......................................................................................33
2.3.3.2 Geometrical Optics ...................................................................................35
2.3.3.3 Point Matching..........................................................................................38
2.3.3.4 Perturbation Methods................................................................................38
2.3.3.5 Coupled-Dipole Method ...........................................................................39
2.3.3.6 T-Matrix or Extended Boundary Condition Method ................................41
2.3.3.7 Method of Moments .................................................................................42
2.3.3.8 Finite Difference Time-Domain Method..................................................45
2.2.3.9 Finite Element Method .............................................................................47
Chapter 3. Computational Methods..................................................................................55
viii
3.1 Mie Theory .............................................................................................................55
3.1.1 Computation of the Scattering Coefficients an and bn .....................................55
3.1.2 Far-Field Scattering Parameters.......................................................................59
3.2 Finite Element Time-Domain Method ...................................................................62
3.2.1 Numerical Implementation ..............................................................................63
3.2.2 Use of the EMFlex Software in Practice .........................................................67
3.2.3 Computation of Far-Field Scattering Parameters ............................................68
3.2.4 Simulation of Random Illumination: The Illuminator.....................................69
Chapter 4. Performance of the Time-Domain Finite Element Method ............................75
4.1 Materials System of Interest ...................................................................................75
4.2 Computational Efficiency.......................................................................................82
4.2.1 Memory Requirements For Finite Element Computations..............................82
4.2.2 CPU Time Requirements For Finite Element Computations ..........................83
4.2.3 CPU Time Requirements For Far-Field Extrapolation....................................84
4.3 Computational Accuracy ........................................................................................87
4.3.1 Effects of Finite Element Mesh Density..........................................................87
4.3.2 Effects of Mesh Density on Surface of Far Field Sphere ................................90
4.3.3 Effects of Far-Field Sphere Radius..................................................................91
4.3.4 Effects of Kirchoff Box Placement in the Model ............................................92
4.3.5 Effects of Particle Position Relative to Model Walls ......................................94
4.3.6 Effects of Model Dimensions Compared to Light Wavelength ......................97
4.3.7 Effects of Strong Resonance: High-Index Spheres in Vacuum.....................102
4.4 Guidelines For Building Robust Finite Element Models .....................................105
Chapter 5. A Potpourri of Pigment Particles ..................................................................106
5.1 Core-Shell Pigment Particles................................................................................107
5.1.1 Introduction....................................................................................................107
5.1.2 Scattering Properties of Titania-Coated Core-Shell Pigments ......................108
5.1.2.1 Refractive index of medium at 1.45 .......................................................110
5.1.2.2 Refractive index of medium at 1.0 (air)..................................................118
5.2 Anisotropic Spheres of Rutile Titania ..................................................................120
5.2.1 Introduction....................................................................................................120
5.2.2 Scattering By Anisotropic Spheres of Rutile Titania ....................................125
5.2.3 Angle and Polarization Dependence of Scattering by an Anisotropic Titania
Sphere ......................................................................................................................131
5.2.4 On the Validity of the Average Index and Weighted Sum Approximations .132
5.3 Scattering by a Morphological Rutile Titania ......................................................136
5.3.1 Introduction....................................................................................................136
5.3.2 Scattering by a Morphological Rutile Particle...............................................138
5.3.3 Discussion......................................................................................................141
5.3.3.1 Scattering Coefficients of the Morphological Rutile Particle.................141
5.3.3.2 Mie Theory in Particle Size Analysis by Light Scattering .....................142
Chapter 6. Clusters of Latex Spheres: A Low-Contrast System ....................................145
6.1 Materials System of Interest .................................................................................146
6.2 Light Scattering by Latex Sphere Clusters of Increasing Size .............................150
ix
6.3 Near-Field Interactions Between Neighboring Latex Particles............................157
Chapter 7. Clusters of Titania Particles: A High-Contrast System ................................165
7.1 Materials Systems of Interest ................................................................................165
7.2 Light Scattering by Titania Sphere Clusters of Increasing Size ...........................168
7.3 Near-Field Interactions Between Neighboring Titania Particles..........................173
7.4 Near-Field Interactions Between Two Morphological Rutile Particles ...............176
Chapter 8. Discussion: Light Scattering by Complex Microstructures in the Resonant
Regime.............................................................................................................................184
8.1 Accuracy of the Time-Domain Finite Element Method .......................................184
8.2 Light Scattering by Complex, Resonant Particles ................................................185
8.3 Light Scattering by Densely Packed Particle Aggregates ....................................188
8.4 Near-Field Optical Interactions Between Particles ..............................................195
8.4.1 Titania and Latex Systems.............................................................................195
8.4.2 Extension to Other Materials Systems...........................................................201
Chapter 9. Conclusions...................................................................................................203
Appendix. Typical EMFlex Input Deck .........................................................................207
Bibliography ....................................................................................................................210
x
List of Tables
Table 1. Error in scattering cross section as a function of separation distance between
the bottom of the sphere and the bottom wall of the model. The corresponding
models are shown in Figure 20. .............................................................................. 97
Table 2. Error in scattering cross section of finite element calculations in which the
model edge length was systematically varied. All calculations are for the case of a
40-nm sphere at the center of a cubic model. ....................................................... 101
Table 3. Error in scattering cross section of finite element calculations in which the
model edge length was systematically varied. All calculations are for the case of a
200-nm sphere at the center of a cubic model. ..................................................... 101
Table 4. Error in scattering cross section as a function of sphere diameter for the three
cases indicated on the plot in Figure 22................................................................ 104
Table 5. Scattering coefficients S and σ computed for anisotropic spheres of different
diameter using the finite element approach. ......................................................... 130
Table 6. Values of χ2 (Equation 80) associated with S and σ values for the average
index and weighted sum approximation, compared to anisotropic sphere scattering
data generated using the finite element approach................................................. 135
Table 7. Comparison between the far-field scattering coefficients for the single rutile
particle and the equivalent volume sphere, using both the average index and
weighted sum approximations. ............................................................................. 143
Table 8. Diameters of spheres exhibiting the same far-field scattering coefficients as
the single rutile particle, using both the average index and weighted sum
approximations...................................................................................................... 143
Table 9. Diameters of spheres exhibiting the same far-field scattering coefficients as the
system of two rutile particles for each interparticle separation distance considered,
using both the average index and weighted sum approximations. ....................... 183
Table 10. Results of parametric fits of the six curves shown in Figure 69 and Figure 70
using the expression in Equation 81. .................................................................... 200
xi
List of Illustrations
Figure 1. Schematic diagram of scattering object size versus radiation wavelength
identifying three scattering domains: geometrical optics, the resonant regime, and
the Rayleigh scattering regime. ................................................................................ 4
Figure 2. The ordinary (lower, dashed curve) and extraordinary (upper, dotted curve)
refractive indices of rutile titania versus light wavelength. These data were
compiled from multiple sources by Ribarsky.13 ..................................................... 11
Figure 3. The scattering coefficient S* for rutile titania particles in a typical paint film
as a function of pigment volume concentration. (After Braun25) .......................... 21
Figure 4. Schematic diagram of a stereolithography apparatus for producing complex
ceramic parts using rapid prototyping. (After Liao.) ............................................. 23
Figure 5. Schematic diagram showing reflection and refraction of a light ray using the
geometrical optics approximation. (After Bohren and Huffman1) ........................ 38
Figure 6. Flow chart showing the formulation and evaluation of light scattering
problems using the Illuminator software. ............................................................... 73
Figure 7. A representation of the spherical illumination coordinate system used in
Illuminator runs. In this case, there is an 18o increment in the spherical coordinate
angles phi and theta between adjacent surface area elements. The individual
illumination directions pass through the centers of the surface area elements....... 74
Figure 8. Dependence of the scattering coefficient S upon sphere diameter for the case
of a sphere with n = 2.74 embedded in a resin with n = 1.51. The illumination
wavelength is 560 nm. ............................................................................................ 78
Figure 9. Dependence of the angle-weighted scattering coefficient σ upon sphere
diameter for the case of a sphere with n = 2.74 embedded in a resin with n = 1.51.
The illumination wavelength is 560 nm. ................................................................ 78
Figure 10. Finite element model of a 200-nm sphere with n = 2.74 embedded in a
polymeric resin with n = 1.51. The 560-nm illuminating light propagates in the +x
direction and is polarized in the z direction. This model contains 1,340,000
volume elements. .................................................................................................... 79
Figure 11. Time history of z-polarized electric field amplitude at the center of the
sphere shown in the computation of Figure 10....................................................... 80
Figure 12. Finite element model of a 200-nm sphere with n = 2.74 embedded in a
polymeric resin with n = 1.51. The 560-nm illuminating light propagates in the +x
direction and is polarized in the z direction. This model contains 168,000 volume
elements. ................................................................................................................. 81
Figure 13. Dependence of Cray C-94 memory requirement upon the number of volume
elements in the finite element model for the present series of EMFlex
computations........................................................................................................... 83
Figure 14. Dependence of Cray C-94 CPU time requirement upon the number of
volume elements in the finite element model for the present series of EMFlex
computations........................................................................................................... 84
xii
Figure 15. CPU time required for a single far-field extrapolation is shown in as a
function of the angular discretization on the surface of the far-field sphere. ......... 86
Figure 16. CPU time required for a single far-field extrapolation is shown in as a
function of the number of surface elements on the far-field sphere. ...................... 86
Figure 17. Error in scattering cross section compared to the exact result from Mie
theory as a function of ρ, the number of volume elements in a model per minimum
wavelength of the 560-nm light propagating in the finite element models. ........... 89
Figure 18. Error in scattering cross section compared to the exact result from Mie
theory as a function of as a function of angular discretization on the surface of the
far-field sphere surface. .......................................................................................... 91
Figure 19. Error in scattering cross section compared to the exact result from Mie
theory as a function of as a function of far-field sphere radius. ............................. 92
Figure 20. Cross-sectional slices through the centers of four finite element models in
which the spacing between the 200-nm sphere and the bottom wall of the model
has been systematically varied. Scattered light intensity is shown in each case. .. 96
Figure 21. Finite element model of a 40-nm sphere with n = 2.74 embedded in a
polymeric resin with n = 1.51. The 560-nm illuminating light propagates in the +x
direction and is polarized in the z direction.......................................................... 100
Figure 22. The scattering coefficient S (µm−1) as a function of sphere diameter for the
case of a sphere with n = 3.0 in a medium with n = 1.0. This represents a more
resonant system than can be achieved in practice, since no known, non-lossy
material exhibits such a high refractive index in the visible spectrum................. 104
Figure 23. Schematic representation of titania coated on a core particle of lower
refractive index. .................................................................................................... 109
Figure 24. Scattering coefficient σ (µm−1) versus coated particle diameter and
refractive index of the core material for titania-coated particles with coating
thickness equal to 10% of the particle radius. ...................................................... 112
Figure 25. Scattering coefficient σ (µm−1) versus silica core diameter and titania shell
diameter for a coated particle. The global maximum of this plot corresponds to a
solid titania sphere of 0.2 µm diameter. ............................................................... 112
Figure 26. Scattering coefficient σ for five coated particles with different outside
diameters............................................................................................................... 114
Figure 27. Scattering coefficient σ versus particle diameter for a solid titania sphere
and a titania-coated silica particle with coating thickness equal to 10% of the
Figure 28. Scattering coefficient σ divided by the density of the coated particle. The
global maximum of this plot corresponds to a solid titania sphere with no silica
present................................................................................................................... 117
Figure 29. Scattering coefficient σ divided by the weight fraction of titania present in
each coated particle. The global maximum of this plot corresponds to a solid
titania sphere with no silica present...................................................................... 117
xiii
Figure 30. Scattering coefficient σ for the five particles shown in Figure 26, with the
cellulose surrounding medium replaced by air (refractive index 1.0 instead of
1.45). ..................................................................................................................... 120
Figure 31. Scattering coefficient S versus sphere diameter for an optically isotropic
sphere from Mie theory, using the average index approximation. ....................... 123
Figure 32. Angle-weighted scattering coefficient σ versus sphere diameter for an
optically isotropic sphere from Mie theory, using the average index approximation.
.............................................................................................................................. 123
Figure 33. Scattering coefficient S versus sphere diameter for an optically isotropic
sphere from Mie theory, using the weighted sum approximation. ....................... 124
Figure 34. Angle-weighted scattering coefficient σ versus sphere diameter for an
optically isotropic sphere from Mie theory, using the weighted sum approximation.
.............................................................................................................................. 124
Figure 35. Finite element model and scattered intensities for a 0.2-µm anisotropic
sphere of rutile titania. The 560-nm radiation propagates in the +x direction, with
unit incident intensity. Polarization is parallel to the y direction. ....................... 128
Figure 36. The scattering coefficient S as a function of sphere diameter computed using
the finite element method (solid points) for anisotropic spheres of rutile titania with
n =2.74 embedded in a medium with n = 1.51. The illumination wavelength is 560
nm. The long dashed curve corresponds to the average index approximation, while
the dotted curve corresponds to the weighted sum approximation....................... 129
Figure 37. The angle-weighted scattering coefficient σ as a function of sphere diameter
computed using the finite element method (solid points) for anisotropic spheres of
rutile titania embedded in a medium with n = 1.51. The illumination wavelength is
560 nm. The long dashed curve corresponds to the average index approximation,
while the dotted curve corresponds to the weighted sum approximation............. 129
Figure 38. Scattering coefficient S as a function of the angle of incidence (relative to
the optic axis) for the 0.2-µm anisotropic sphere of rutile titania. Hollow circles
are for polarization perpendicular to the optic axis of the sphere; filled circles are
for polarization coplanar with the optic axis. ....................................................... 132
Figure 39. High resolution scanning electron micrograph of rutile titania particles,
showing their characteristic, morphological shapes. (Micrograph courtesy E.D.
Boyes, DuPont Central Research and Development.) .......................................... 137
Figure 40. Finite element model and near field scattering results for an anisotropic,
morphological rutile particle with a width of 0.175 µm, and a length of 0.35 µm in
n = 1.51 resin; the light wavelength is 560 nm..................................................... 140
Figure 41. The scattering coefficient S as a function of sphere diameter for a latex
particle with n = 1.50 in a water medium with n = 1.33. The illuminating light
wavelength is 488 nm. The 200-nm sphere diameter studied in this chapter is
marked with an “x.” .............................................................................................. 148
Figure 42. The angle-weighted scattering coefficient σ as a function of sphere diameter
for a latex particle with n = 1.50 in a water medium with n = 1.33. The
illuminating light wavelength is 488 nm. The 200-nm sphere diameter studied in
this chapter is marked with an “x.”....................................................................... 148
xiv
Figure 43. Finite element model for a 200-nm latex sphere with n = 1.50 in a water
medium with n = 1.33. The illuminating wavelength is 488 nm, and light
propagates in the +x direction............................................................................... 149
Figure 44. Finite element models of latex sphere clusters containing 3, 7, 13, and 27
spheres. ................................................................................................................. 154
Figure 45. Finite element model of a cluster of seven 200-nm latex spheres n = 1.50 in
a water matrix with n = 1.33. The 488-nm illuminating light propagates in the +x
direction and is polarized in the z direction.......................................................... 155
Figure 46. The scattering coefficient S as a function of the number of 200-nm latex
particles composing a particle cluster under conditions of random illumination. 156
Figure 47. The quantity (1−g) as a function of the number of 200-nm latex spheres
composing a particle cluster under conditions of random illumination................ 156
Figure 48. The scattering coefficient σ as a function of the number of 200-nm latex
spheres composing a particle cluster under conditions of random illumination... 157
Figure 49. Finite element model of a cluster of seven 200-nm latex spheres n = 1.50 in
a water matrix with n = 1.33. The surface-to-surface interparticle spacing is 0.4
µm. The 488-nm illuminating light propagates in the +x direction and is polarized
in the z direction. .................................................................................................. 162
Figure 50. Dependence of the scattering coefficient S upon surface-to-surface
interparticle separation distance for the clusters of seven latex spheres. The dashed
line corresponds to the value associated with a single 200-nm latex primary
particle. ................................................................................................................. 163
Figure 51. Dependence of the quantity (1−g) upon surface-to-surface interparticle
separation distance for the clusters of seven latex spheres. The dashed line
corresponds to the value associated with a single 200-nm latex primary particle.
.............................................................................................................................. 163
Figure 52. Dependence of the angle-weighted scattering coefficient σ upon surface-tosurface interparticle separation distance for the clusters of seven latex spheres. The
dashed line corresponds to the value associated with a single 200-nm latex primary
particle. ................................................................................................................. 164
Figure 53. The scattering coefficient S as a function of sphere diameter for a rutile
particle with n = 2.74 in a polymer medium with n = 1.51 under the average index
approximation. The illuminating light wavelength is 560 nm. The 200-nm sphere
diameter studied in this chapter is marked with an “x.” ....................................... 167
Figure 54. The angle-weighted scattering coefficient σ as a function of sphere diameter
for a rutile particle with n = 2.74 in a polymer medium with n = 1.51 under the
average index approximation. The illuminating light wavelength is 560 nm. The
200-nm sphere diameter studied in this chapter is marked with an “x.” .............. 167
Figure 55. Finite element model of a cluster of seven 200-nm titania spheres with n =
2.74 in a polymer matrix with n = 1.51. The 560-nm illuminating light propagates
in the +x direction and is polarized in the z direction........................................... 171
Figure 56. The scattering coefficient S as a function of the number of 200-nm titania
spheres composing a particle cluster. These results are for the case of random
illumination of each cluster, resulting in orientation-averaged results................. 172
xv
Figure 57. The scattering coefficient σ as a function of the number of 200-nm titania
spheres composing a particle cluster. These results are for the case of random
illumination of each cluster, resulting in orientation-averaged results................. 172
Figure 58. Dependence of the scattering coefficient S upon surface-to-surface
interparticle separation distance for the clusters of seven titania spheres. The
dashed line corresponds to the value associated with a single 200-nm titania
primary particle..................................................................................................... 175
Figure 59. Dependence of the angle-weighted scattering coefficient σ upon surface-tosurface interparticle separation distance for the clusters of seven titania spheres.
The dashed line corresponds to the value associated with a single 200-nm titania
primary particle..................................................................................................... 175
Figure 60. Finite element model cross section and near field scattering results for two
anisotropic morphological rutile particles in n = 1.51 resin for light incident normal
to the cross section shown and with a wavelength of 560 nm. The interparticle
spacing is 0.5 µm. ................................................................................................. 179
Figure 61. Finite element model cross section and near field scattering results for two
anisotropic morphological rutile particles in n = 1.51 resin for light incident normal
to the cross section shown and with a wavelength of 560 nm. The interparticle
spacing is 0.3 µm. ................................................................................................. 180
Figure 62. Finite element model cross section and near field scattering results for two
anisotropic morphological rutile particles in n = 1.51 resin for light incident normal
to the cross section shown and with a wavelength of 560 nm. The interparticle
spacing is 0 µm. .................................................................................................... 181
Figure 63. Scattering coefficient S for two morphological rutile particles as a function
of interparticle separation. The horizontal line shows the results for one of the
morphological rutile particles. .............................................................................. 182
Figure 64. Angle-weighted scattering coefficient σ for two morphological rutile
particles as a function of interparticle separation. The horizontal line shows the
results for one of the morphological rutile particles. ............................................ 182
Figure 65. The scattering coefficient S as a function of the number of 200-nm latex
particles composing a particle cluster (solid curve). For comparison, the results of
Mie theory computations on single, equivalent-mass spheres are shown (dashed
curve). ................................................................................................................... 193
Figure 66. The scattering coefficient σ as a function of the number of 200-nm latex
particles composing a particle cluster (solid curve). For comparison, the results of
Mie theory computations on single, equivalent-volume spheres are shown (dashed
curve). ................................................................................................................... 193
Figure 67. The scattering coefficient S as a function of the number of 200-nm titania
particles composing a particle cluster (solid curve). For comparison, the results of
Mie theory computations on single, equivalent-mass spheres are shown (dashed
curve). ................................................................................................................... 194
Figure 68. The scattering coefficient σ as a function of the number of 200-nm titania
particles composing a particle cluster (solid curve). For comparison, the results of
xvi
Mie theory computations on single, equivalent-mass spheres are shown (dashed
curve). ................................................................................................................... 194
Figure 69. Scattering coefficient S (%) for the three different multiple-particle systems
as a function of the center-to-center interparticle spacing expressed in wavelengths.
The value 100% corresponds to the S value for an isolated primary particle in each
case........................................................................................................................ 200
Figure 70. Angle-weighted scattering coefficient σ (%) for the three multiple-particle
systems as a function of the center-to-center interparticle spacing expressed in
wavelengths. The value 100% corresponds to the σ value for an isolated primary
particle in each case. ............................................................................................. 201
xvii
Chapter 1. Introduction
Scattering phenomena underlie the human perception of the natural world
through the sense of sight, and the scattering properties of physical systems are widely
exploited in both science and technology. Scattering of electromagnetic radiation by
microstructures encompasses a broad range of physical phenomena. Wavelengths in the
electromagnetic spectrum span an infinite range, and those exploited technologically
range from kilometer-long radio signals to sub-angstrom gamma rays. The nature of the
scattering interaction between electromagnetic radiation and materials is determined
primarily by the relationship between radiation wavelength and the physical dimensions
of scattering features in materials systems. With an appreciation of this premise, a
unified view of electromagnetic radiation scattering by microstructures is possible,
spanning the orders of magnitude in size which exist in both the electromagnetic
spectrum and the physical world.
1.1 Single Scattering
The fundamental case in the analysis of electromagnetic radiation scattering is
that of single scattering, in which the interaction between radiation and a single,
isolated scattering feature is considered. Results of single scattering can be extended to
ensembles which contain a sufficiently small number of particles with interparticle
separations large enough that each particle develops its individual scattering pattern
without influence from its neighbors. A qualitative criterion for the assumption of
single scattering is that each particle in an ensemble experiences the same incident
1
electric field, with negligible scattered field from other features in the ensemble. The
treatment of an ensemble of scatterers in the single scattering regime is straightforward:
the scattering properties of the ensemble is simply the sum of contributions from the
individual scatterers.1,2
The degree of mathematical complexity in the treatment of the radiation
scattering by an isolated scattering feature is strongly affected by the relationship
between radiation wavelength and the physical size of a scattering feature, as suggested
above. In general, simplifying approximations can be applied to the separate cases in
which the wavelength is at least an order of magnitude greater or less than the size of a
scattering feature. The former case includes the well-known Rayleigh scattering theory
which explains the basis of the blue color of the daytime sky on a clear day, for
example, while the latter case includes the application of geometrical optics techniques
to the prediction of visible image formation by macroscopic objects. For cases in which
the wavelength of electromagnetic radiation is on the order of the size of a scattering
feature, the mathematical description of scattering is potentially intractable using even
the most sophisticated analytical approaches.
This is especially true for material
systems exhibiting high optical contrast between the scattering feature and its
surrounding medium, since very strong coupling and resonance can occur. Scattering
efficiencies in this resonant regime can be very high. An example of one such system is
the scattering of visible light by an optimized white pigment particle, where near-field
scattered intensities in the vicinity of the particle can be an order of magnitude greater
than that of the incident light and the scattering cross section of a particle can be many
2
times greater than its geometric cross section. These domains of Rayleigh scattering,
the resonant regime, and geometrical optics are illustrated in the schematic diagram in
Figure 1 showing object size mapped versus radiation wavelength.
The potentially high scattering efficiency of systems in which the radiation
wavelength is on the order of the size of scattering features makes them of particular
interest in science and technology. In the scientific community, the use of scattering
techniques to characterize structures is most effective when there is strong coupling
between the features in a structure and the probe radiation. Likewise, technology uses
this resonance to advantage in applications as diverse as paint films and cellular
telephone technology. Unfortunately, the mathematical analysis of scattering by objects
with size on the order of the radiation wavelength is complex.
Indeed, rigorous
analytical approaches to solving Maxwell’s equations for scattering in three dimensions
have been historically limited to coordinate systems in which solutions to the scalar
wave equation can be separated in the three spatial variables. In the important case of
the spherical coordinate system, the result is the well-known Mie theory (1908). Mie
theory describes the scattering of electromagnetic radiation of arbitrary wavelength by
an isolated, optically isotropic sphere of arbitrary diameter. An advantage of Mie
theory is that it is easily implemented computationally. A disadvantage of Mie theory is
that it is limited to a spherical particle and is not applicable to the myriad systems in
which the scattering features are irregularly shaped and/or optically anisotropic. In
addition, spherical particles can exhibit surface wave resonances which do not occur
with more irregularly shaped particles. The radiation scattering properties of irregularly
3
shaped objects with size on the order of the radiation wavelength are of growing
importance in the scientific community as the detailed understanding of physical
microstructures increases.
Figure 1. Schematic diagram of scattering object size versus radiation wavelength
identifying three scattering domains: geometrical optics, the resonant regime, and
the Rayleigh scattering regime.
1.2 Scattering by Ensembles of Scattering Features
In ensembles containing sufficiently large numbers of scattering features, the
collective radiation scattering properties of the ensemble can deviate substantially from
the superposed scattering properties of the constituent individual features.
These
deviations can be the result of diffraction or near-field optical interactions between
neighboring particles.
The term multiple scattering refers to the phenomenon of
collective scattering by a large ensemble of individual scattering features and is a wellestablished subject area which encompasses, and expands upon, the study of radiation
4
scattering by isolated scattering features.
Multiple scattering theories describe
mathematically the transfer of radiation through a collection of optically isolated
scattering features, each having known scattering properties.
A criterion for the
application of multiple scattering theories is that no near-field optical interactions exist
between the scattering features in the ensemble. Multiple scattering theories are widely
applied in diverse subject areas, including meteorology, astronomy, colloid science,
optical lithography, materials science, and color science.
An important aspect of
multiple scattering theories is the presumption of optical isolation of the individual
scattering features in an ensemble. This condition requires that the scattered wave front
produced by an individual feature develops independently of neighboring features; that
is, the scattering features are sufficiently spaced physically that there exist no near-field
optical interactions between neighbors. This condition is met only in dilute dispersions
of scattering not exceeding ~5% by volume.
In concentrated dispersions, the scattering properties of individual features are
affected by near-field optical interactions with neighbors; this regime of scattering by
ensembles of isolated features has therefore been described as dependent scattering.
The detailed mathematical description of near-field interactions between individual
scattering features is complex, and the development of understanding of these effects in
systems of practical interest is still in its earliest stages.
The use of large-scale
numerical computation is essential in approaching these problems rigorously, and recent
advances in the scale of available computing power promise substantial advances in the
understanding of dependent scattering effects in the near future.
5
1.3 Formulation of the Present Research
In the present research, the electromagnetic radiation scattering properties of
several different classes of complex microstructures are studied using a numerical
approach. Of particular interest are systems containing scattering features which are on
the order of the radiation wavelength in size, in which near-field interactions occur
among the scattering features. The numerical approach used is a time-domain finite
element formulation of Maxwell’s equations which produces full near-field solutions to
cases of electromagnetic radiation scattering by microstructures having arbitrary
geometry. The method can be applied to isolated scattering features of arbitrary shape
with anisotropic optical properties, overcoming the restrictions of analytical approaches
such as Mie theory. In addition, the finite element approach can be readily applied to
ensembles of multiple scattering features to determine, quantitatively, the effect of nearfield interactions on the radiation scattering properties. The time-domain finite element
method allows precise control of microstructures, facilitating systematic study of the
effects of microstructure on system scattering properties which is not possible
experimentally.
The structure-property relations in light scattering by complex
microstructures is the primary focus of this dissertation.
Chapter 2 reviews electromagnetic radiation scattering studies in materials
science, emphasizing previous work on scattering by systems of multiple particles, and
describes analytical and numerical approaches to the computation of radiation
scattering. Chapter 3 discusses the computational approaches applied in the present
research. In Chapter 4, the factors controlling the accuracy of the time-domain finite
6
element approach employed in the present research are identified. Chapter 5 considers
three diverse cases of light scattering by single, resonant white pigment particles
containing titania. Chapter 6 describes the light scattering properties of multiple-particle
clusters of low-contrast, weakly scattering latex spheres. Chapter 7 presents an
analogous study of the light scattering properties of clusters of high-contrast, strongly
resonant titania particles.
Chapter 8 contains a unifying discussion of the results
presented in Chapters 4-7. Finally, Chapter 9 contains general conclusions drawn from
these collective studies.
1.4 On Vector Notation
Many of the physical quantities of interest in the present work, such as electric
field, are vectors. Throughout this dissertation, vectors are denoted in equations as
italic characters with superscript arrows, E , and in the text as boldface characters, E.
1
C.F. Bohren, D. R. Huffman, Absorption and Scattering of Light By Small Particles, John Wiley, New
York, (1983).
2
H.C. van de Hulst, Light Scattering by Small Particles, Dover, New York (1981).
7
Chapter 2. Literature Review
This chapter reviews previous publications in the scientific literature relevant to
the present research and consists of three main sections.
In Section 2.1 presents
introductory comments on the role of radiation scattering in materials science. Section
2.2 reviews previous studies of electromagnetic radiation scattering by complex
microstructures.
Section 2.3 reviews the mathematical basis for electromagnetic
radiation scattering (Maxwell’s equations), followed by description of diverse analytical
and numerical approaches to computing the scattering properties of microstructures.
2.1 Electromagnetic Radiation Scattering in Materials Science
Electromagnetic radiation scattering plays a critical role in materials science.
Scattering measurements are widely used to gain information about the microstructures
of systems. For example, x-ray diffraction and scattering are widely used as a means to
gain statistical information about the arrangement of scattering features within a
system.1 In addition, visible light scattering is an important experimental tool for
determining the distribution of particle sizes in a sample.2
The scattering properties of a materials system is often a key design
consideration facing its developers. For example, the scattering efficiency of particles
in a pigmented film are affected dramatically by the ability of the formulator to control
particle size and dispersion quality. Likewise, surface roughness affects the optical
appearance, and therefore the aesthetic appeal, of commercial engineering polymers. In
these examples, the formulation of a materials system requires microstructural control
8
to impart the desired light scattering properties. This ability to impart the desired
optical properties requires understanding of how systematic changes to a microstructure
The questions of a) how measurement of radiation scattering properties can be
used to gain microstructural information and b) how microstructure affects radiation
scattering properties are two sides of the same coin. The former is the “hard” or
“inverse” problem; the radiation scattering properties of a particular microstructure are
uniquely determined for any wavelength.
The latter is the “simple” or “forward”
problem; it requires the application of models to recover imputed microstructures from
convoluted data. In the present research, the latter question is of primary interest:
determination of the radiation scattering properties of complex microstructures. By
improving the existing understanding of how complex microstructures scatter radiation,
the dual challenges of building desired optical properties into microstructures and
determination of microstructure from scattering data are simultaneously diminished.
2.2 Light Scattering Properties of Complex Microstructures
2.2.1 Light Scattering by White Pigment Particles
White pigments form one of the most widely used classes of optical materials,
the most important of which is titania. Annual worldwide sales of titania total ~10
billion pounds. Titania pigment is used as an opacifying agent in paint films, plastics,
and paper. Titania is non-toxic, inexpensive, and its stoichiometric phases are virtually
non-absorbing in the visible spectrum. These characteristics, combined with the high
9
refractive indices of both the rutile and anatase titania phases in the visible, have
contributed to its universal acceptance in the marketplace.3,4
The prediction and
optimization of the optical performance of titania in end-use applications has been an
active area of research for decades.
In 1993, Fields, et al., presented a set of
experimental data and approximate modeling which suggest that the scattering
efficiency of commercial rutile titania grades could be improved by as much as 20% by
controlling the particle size distribution and minimizing particle agglomeration.5
Titania occurs in three distinct, naturally occurring polymorphs: rutile, anatase,
and brookite. Of these, both rutile and anatase are commonly manufactured for use as a
white pigment. Both phases exhibit tetragonal crystal structures,6 with a single optic
axis (parallel to the c-axis). Both crystals therefore exhibit two different refractive
indices at a given wavelength: the ordinary no (for radiation polarized perpendicular to
the c-axis) and the extraordinary ne (for radiation polarized parallel to the c-axis ). The
electronic and optical properties of rutile titania in and surrounding the visible spectrum
have been studied both experimentally7,8 and theoretically.9,10,11,12
The optical
properties of rutile over a wide range of wavelengths from multiple investigators have
been compiled by Ribarsky.13 The ordinary and extraordinary refractive indices of
rutile are plotted versus wavelength in Figure 2 over the range 400-700 nm.
Experimental investigations of anatase14,15 have been comparatively limited, apparently
due to the relative scarcity of large, defect-free single crystals.14 As in the case of rutile,
the electronic and optical properties of anatase have been studied theoretically.11 Rutile
titania exhibits both higher ordinary and extraordinary refractive indices throughout the
10
visible spectrum and therefore is superior to anatase in its light scattering performance
as a white pigment, with particle size optimized. At the light wavelength 560 nm, the
center of the visible spectrum, the ordinary and extraordinary refractive indices of rutile
titania are 2.64 and 2.94, respectively.13 At this same wavelength, the ordinary and
extraordinary refractive indices of anatase titania are 2.58 and 2.50, respectively.16
Refractive Index n
3.50
3.25
3.00
2.75
2.50
400
450
500
550
600
650
700
Wavelength (nm)
Figure 2. The ordinary (lower, dashed curve) and extraordinary (upper, dotted
curve) refractive indices of rutile titania versus light wavelength. These data were
compiled from multiple sources by Ribarsky.13
2.2.1.1 Application of Mie Theory to Optically Anisotropic Materials
All previous computational investigations of the light scattering properties of
rutile titania have been based upon Mie theory (Section 2.3.2), which is limited to the
case of an isolated, optically isotropic sphere. This restriction presents the difficulty of
how to address the optical anisotropy of rutile titania using a model which presumes
isotropic materials. Two different approximations have been used. In the first, which is
11
termed the average index approximation, the refractive indices encountered by light
polarized parallel to each of the three principal axes (a, b, and c) of the rutile titania
crystal are averaged. This average refractive index nave is then used in Mie theory
calculations. Since the ordinary refractive index no is encountered by light polarized
parallel to both a and b and the extraordinary refractive index ne is encountered by light
polarized parallel to c only, the expression for nave is:
Equation 1
n ave =
2n o + n e
3
In the second approximation, which is termed the weighted sum approximation, the
scattering properties of a rutile titania sphere are computed using the ordinary and
extraordinary refractive index separately. The two results are then summed together in
a weighted fashion. Since the ordinary refractive index no is encountered by light
polarized parallel to both a and b and the extraordinary refractive index ne is
encountered by light polarized parallel to c only, this weighted sum consists of twothirds the result for the ordinary refractive index plus one-third the result for the
extraordinary refractive index. The weighted sum approximation was used in the work
of Ross.17 The average index approach has been used by Palmer, et al.,18 who suggest
that this method is as equally justifiable as the weighted sum approximation for titania
sphere diameters near that (~0.2 µm) which produces optimal scattering of green light.
2.2.1.2 Kubelka-Munk Analysis of Paint Film Scattering Efficiencies
Experimental characterization of the performance of white pigments in a paint
film is typically based upon reflectance measurements. The resulting data are then
12
analyzed using the Kubelka-Munk theory,19,20,21 an approximate multiple scattering
theory which presumes only two fluxes of light. These fluxes propagate in directions
normal to the film surface: one upward toward the film surface and one downward
toward the substrate.
Kubelka-Munk analysis of scattering by a film yields two
parameters: the scattering coefficient S* and the absorption coefficient K*;a both are
expressed in units of inverse length. The reciprocals of these quantities have units of
length and can be interpreted as mean free path lengths between scattering or absorption
events. In white paint films with no colorants present, K* is very close to zero, and only
S* is considered. The Kubelka-Munk scattering coefficient can be expressed either on a
particle basis, S*particle, or on a film basis, S*film. The latter is obtained by multiplying
the former by the volume fraction φ of white pigment in the film:
Equation 2
S *film = S *particleφ
The quantity S*film depends upon the concentration of pigment in a film and is therefore
an extensive quantity; in the limit of low pigment concentrations in the film, it is
directly proportional to pigment concentration. The quantity S*particle, on the other hand,
is the scattering efficiency of the white pigment itself; in the limit of low pigment
a
It is not standard practice in the literature to denote the Kubelka-Munk coefficients with an asterisk.
This notation has been adopted here to differentiate these experimentally determined scattering
coefficients from the computed coefficients S and σ used in the present research. This conflict arises
from different usage of the symbol S by different researchers. In the present work, the experimentally
determined S* is directly proportional to σ.
13
concentrations in the film, it is independent of pigment concentration.
The
normalization of results on a particle volume basis therefore facilitates comparison of
pigments in different paint formulations by placing the scattering coefficient S* on an
intensive rather than extensive basis.
For this reason, experimentally determined
scattering efficiencies are presented on a particle basis throughout the present chapter.
2.2.1.3
Computational Modeling of Scattering by White Pigments:
Previous Studies
Mie theory computations were used by Ross17 in 1971 to investigate the
theoretically maximum scattering efficiency of rutile titania in paint films, work which
included comparison to the scattering efficiency of air voids in paint films.
The
computations for rutile titania were based upon the weighted sum approximation for a
sphere embedded in a resin having the refractive index of a typical acrylic resin. Ross
then applied corrections for surface reflections by a paint film on a substrate of arbitrary
reflectance to the Mie theory results so that they could be compared directly with
scattering coefficients derived from experiment. These experimentally determined
scattering coefficients were derived from the Kubelka-Munk theory. Mie theory is a
single scattering approach and does not include the effects of near-field optical
interactions between neighboring particles in a paint film. It has therefore been widely
suggested that Mie theory results can be realistically related only to paint films
containing less than 5-10% titania by volume. The computations performed by Ross
showed that the maximum scattering coefficient for rutile titania occurs at a diameter
0.20 µm for 560-nm illuminating light. Ross concludes that actual pigmented films
14
exhibit scattering efficiencies about 40% less than that associated with the optimal rutile
sphere. In addition, his results show that the light scattering by the optimal spherical air
void is only 12% that of the optimal rutile titania sphere.
In 1989, Palmer, et al., used Mie theory to compute the optical performance of
rutile titania spheres compared to different coated particle architectures.18 The average
index approximation was used to address the optical anisotropy of rutile. Palmer, et al.,
performed spectral computations which allowed them to calculate light scattering
properties of a sphere for both tungsten light and sunlight and to correct for the response
of the human eye. In some of the computations, they applied the Kubelka-Munk
multiple scattering theory to estimate the films thickness required for complete opacity
of a film over a black substrate. Their computations yield the result that 0.22 µm is the
optimal sphere diameter for rutile titania, in close agreement with the value cited by
Ross. In addition, Palmer, et al., investigated the light scattering properties of coated
spheres consisting of hollow titania, air-encapsulated titania, titania-coated quartz, and
quartz-coated titania. They conclude that optimized rutile titania is superior to the
optimized form of each of these coated architectures.
In 1993, Hsu, et al., proposed the use of titania-coated silica to achieve opacity
in paper systems, claiming scattering efficiency comparable to rutile titania.22 These
claims, which contradict the conclusions of Palmer, et al., were substantiated by Hsu, et
al., in part using Mie theory calculations on coated particles. In that same year, a
United States Patent was issued to these authors covering a range of coated
compositions.23 Additionally, an international patent application was filed in 1994 by a
15
separate group of inventors covering compositions of coated titania.24 The application
of Mie theory to compute the light scattering efficiency of these coated particles is
performed in detail in Chapter 5.
2.2.2 Light Scattering by Concentrated Particle Dispersions
2.2.2.1 The Crowding Effect in Paint Films
The near-field optical interactions which occur between neighboring particles in
a concentrated dispersion can result in very different scattering properties compared to
the superimposed properties of the isolated, individual particles. In paint films, the
crowding effect manifests itself as a dramatic decrease in scattering efficiency on a
volume-normalized basis as the pigment volume concentration (PVC) increases above
5-10%.
At low PVC, interparticle spacings are sufficiently large that near-field
interactions are very weak. A plot of the volume-normalized scattering coefficient S*
for rutile titania pigment in a typical paint formulation is shown in Figure 3; the
scattering coefficient at 35% PVC is only 40% compared to that of the formulation of
lowest PVC.25
An intuitive explanation for the crowding effect is that closely
neighboring particles in a film shadow each other from the incident light, resulting in
reduced scattering efficiency. An additional consideration is that the scattering cross
sections of optimized pigment particles are several times greater than their geometric
cross sections, meaning that the effective scattering diameters of the particles can
overlap without the particles actually touching.
16
In 1961, Bruehlman, et al., measured the magnitude of the crowding effect in a
series of designed experiments in which both the particle size distribution of the
pigments and the refractive index of the alkyd dispersing media were systematically
varied.26 They conclude that the optimal size of titanium dioxide particles is a function
of PVC, with smaller particle sizes favored at low (0−10%) PVC and larger particle
sizes favored at high (>10%) PVC. This conclusion is one which cannot be justified
using, say, Mie theory, since the conditions of single scattering are not satisfied in
concentrated paint films.
In 1974, Tunstall and Hird measured the crowding effect in an alkyd paint
system and concluded that the loss in particle scattering efficiency associated with the
crowding effect is determined both by particle size and pigment volume concentration.
They developed a simple physical model based upon single scattering theory in which
the loss of scattering efficiency is determined by the surface-to-surface separation of
particles in a film measured in units of light wavelength. The resulting equation fits
their data successfully except at low PVC. This equation is of the form
Equation 3
S* = 1 −
 Zd  X

3

exp−
−
1


φ
 λ  φ

K
where S* is the volume-normalized particle scattering coefficient, K and Z are
constants, φ is the pigment volume fraction, d is the particle diameter, λ is the light
wavelength, and X is the maximum packing fraction of the pigment in the medium. The
second term in the exponential is proportional to the surface-to-surface spacing of
particles and is equal to zero when φ = X. According to Equation 3, S* is unity when φ
17
= 0 and less than unity when φ = X. Stieg had earlier observed that the scattering
coefficient of a pigment is proportional to the surface-to-surface particle spacing
divided by the center-to-center spacing,27 which is to say:
Equation 4
S* ∝ 1 −
1
D
where D is the normalized distance between centers of neighboring particles in particle
diameters. The expression for D in terms of the maximum packing fraction and PVC is:
Equation 5
D=3
X
φ
Stieg’s expression for the scattering efficiency of the particles in a film is therefore of
the form:
Equation 6

φ 

S * = K 1 − 3
X 

where K is a constant. Strictly speaking, this equation incorrectly implies that the
scattering coefficient equals zero if X = φ, when in fact any interstitial resin between the
pigment particles under this condition would produce some light scattering. Air voids
form as φ approaches X in a paint film, and the introduction of this new source of light
scattering is not explicitly accounted for in these models of crowding behavior. It must
be recognized, therefore, that Equation 6 is applicable only in the range where φ does
not approach X. A second problem with Stieg’s model, observed by Fitzwater and
Hook, is that values of φ exceeding unity are sometimes required to fit data, which is
physically unreasonable.28
18
In 1985, Fitzwater and Hook applied an analytical approach to the crowding
effect between spherical particles using Mie theory.28 Their approach is based upon
partial wave analysis of light scattering by a spherical particle. The scattered field from
a particle at steady state can be described as a series summation of partial waves, each
with a definite location in space (unlike the total scattered wave). Partial waves with
low index are located near the particle center, and waves with high index are located far
from the center. Only the first few partial waves are required for an accurate solution;
higher index waves can be discarded, resulting in a finite series summation. Through
analysis of the amplitudes and locations of the scattered partial waves, it can be
determined what fraction of the scattered field lies outside the physical diameter of the
particle. This allows an estimate of the amount of scattered light “shared” between
neighboring particles having a known separation distance and, by extension, an estimate
of the reduced scattering efficiency of the particles. Fitzwater and Hook compute the
crowding effect for both rutile titania in paint and latex in water. The agreement
between their predictions and experiment is qualitatively good, reproducing the major
features in the experimental data. Their model is approximate not only because of the
assumptions about the effective scattering diameter, but also due to its limitation to
spherical particles and a monodisperse particle size distribution. The Fitzwater-Hook
model predicts that the scattering efficiency losses due to the crowding effect are less
severe for larger particles than smaller particles. This is because the scattered partial
waves are more completely contained within the physical diameters of larger particles
than of smaller particles.
This prediction is in agreement with the experimental
19
observations of Bruehlman, et al., cited above. Based upon their model, Fitzwater and
Hook propose a modified version of Stieg’s model of crowding (Equation 6),
Equation 7

m 3φ
S * = K 1 − 3

X





The addition of this constant eliminates the problem in Stieg’s model of packing
fractions exceeding unity.
Proposed approaches to reducing the crowding effect in paints have centered
upon optimizing pigment particle dispersion (that is, maximizing the average distance
between particles in a film). Stieg has emphasized the importance of using large
(several micron) particles of inexpensive material to reduce the demand for binder at a
given PVC, thus providing more free binder for the pigment particles to occupy.29
Inexpensive filler materials are referred to in the paint industry as extenders; the volume
they contribute to a film is considered in computing PVC. Braun, on the other hand,
has suggested that the use of very small (sub-micron) extender particles could provide
effective spacing of titania pigment particles by occupying interstitial volume between
neighboring pigment particles.25 Acknowledging that such an approach creates the
problem of higher specific surface areas of the particles within a film, Braun suggests
that the use of thick coatings on titania pigment particles is a viable approach to enforce
separation between neighboring particles in a film.
20
Figure 3. The scattering coefficient S* for rutile titania particles in a typical paint
film as a function of pigment volume concentration. (After Braun25)
2.2.2.2 Light Scattering in Stereolithography
This discussion of light scattering by concentrated particle dispersions has
focused entirely upon pigmented paint films; it is in this field where the most extensive
characterization of the optics of concentrated dispersions has occurred historically.
Recently, there has been a rapid growth in interest in the behavior of concentrated
ceramics dispersions in photopolymer solutions.
In stereolithography, complex
geometries are formed by ultraviolet laser exposure of a ceramic particle slurry in a
photosensitive solution. Exposure by the laser causes the solution to polymerize, and
the laser position is controlled very accurately by computer. Generation of threedimensional shapes is possible since the laser scans with two degrees of freedom in the
x-y plane and the slurry bath can be translated in the z direction. The geometry of a
21
stereolithography apparatus is shown in Figure 4. Once a three-dimensional geometry
is formed, the part is fired to burn out the polymer and form a dense ceramic.
A critical issue in stereolithography is control of the exposure depth of the laser
into the ceramic particle slurry; prediction of this depth requires an understanding of its
dependence upon such factors as laser power, laser dose, laser wavelength, the
refractive index of the ceramic particles, the volume concentration of ceramic particles,
and ceramic particle size. The relationship between cure depth Dc and the scattering
coefficient S* of the particles (neglecting absorption) is:
Equation 8
Dc =
E 
1
ln 0 
S *φ  Ec 
where φ is the volume fraction of particles, E0 is the exposure energy of the laser at the
surface of the suspension, and Ec is the critical exposure energy required to polymerize
the monomer solution. In 1997, Griffith and Halloran performed a series of model
experiments using alumina and silica particles in UV curable solutions and proposed the
following empirical form of the cure depth in concentrated dispersions:30
Equation 9
2dλ n 02  E 0 
ln 
Dc =
3s ∆n 2  E c 
where d is the average particle diameter, λ is the laser wavelength in the photopolymer
medium, s is the average surface-to-surface interparticle spacing in the medium, n0 is
the refractive index of the medium, and ∆n is the difference in refractive index between
the particles and the medium. Equation 9, when written in the form used above in the
case of paint films, is:
22
Equation 10
S* =
K
φ
3
X
φ
where K is a constant (containing λ, n0 and ∆n). Equation 10 exhibits the obvious
problem of diverging to infinity as φ approaches zero. While Equation 9 apparently
provides a reasonable fit to the limited data of Griffith and Halloran, this relation cannot
apply to systems exhibiting wide variations in the parameters contained in the
expression.
Figure 4. Schematic diagram of a stereolithography apparatus for producing
complex ceramic parts using rapid prototyping. (After Liao.31)
2.2.3 Light Scattering by Particle Agglomerates
2.2.3.1 Computational Approaches
During the past three decades, there has been considerable progress in the
experimental and rigorous theoretical understanding of the light scattering behavior of
23
multiple-particle agglomerates or clusters. Spherical particles have been by far the most
extensively investigated particle shape, owing to the availability of analytical
expressions for radiation scattering by a single sphere from Mie theory. In this case, the
light scattered by one sphere in the cluster is decomposed into a series expansion of
outgoing spherical waves which, in turn, impinge upon the other spheres in the cluster.
The field incident upon any sphere in the cluster is therefore a superposition of the total
incident field and scattered fields from all of the other particles in the cluster.
Coefficients coupling the multipole modes in one sphere to those in every other sphere
in the system appear explicitly in the solution.
Solution of the problem requires
overcoming the challenge of representing the spherical waves about any sphere in the
cluster as an expansion about any other arbitrary origin. The addition theorems for
spherical vector wave functions in integral form which produce the required translation
coefficients were established by Stein32 and Cruzen33 in the early 1960s. In 1971,
Bruning and Lo published the first comprehensive solution for a two-sphere cluster.34,35
In 1988, the order-of-scattering approach was developed by Fuller and Kattawar, who
extended Bruning and Lo’s solution to the case of larger clusters by considering
pairwise interactions between the constituent spheres. In 1991, Mackowski36 developed
a set of recurrence relations for the translation coefficients which provide a significantly
more efficient computational approach than the integral forms. Borghese, et al., in 1984
were the first to produce expressions for the cross sections of multiple sphere clusters.37
In 1995, Xu derived expressions for both the scattering cross sections and the elements
of the Mueller scattering matrix of multiple-sphere clusters.38 (The Mueller scattering
24
matrix describes changes in polarization of the incident light due to the scattering
interaction.) In 1996, Mackowski and Mishchenko described a T-matrix approach to
the problem of light scattering by sphere aggregates, facilitating computation of the
scattering properties under conditions of random illumination. Fuller points out that the
convergence rate of the solution for a multiple-sphere cluster depends upon the
maximum spacing between a pair of spheres in the cluster, resulting from additional
terms required to compute accurately the translation terms required to transform
multipole coefficients from one sphere location in the cluster to another.39
2.2.3.2 Computational Results for Particle Agglomerates
Computational results for multiple-sphere clusters have been limited, with most
of the activity in the literature focusing upon the development of more computationally
efficient methods as opposed to the application of those methods. Borghese, et al.,37
present computed results on the scattering cross sections of two-sphere systems as a
function of illumination direction and sphere separation distance. For even the largest
sphere separation distances studied (200 sphere diameters), the scattering cross section
of the two-sphere system failed to converge to the sum of the cross sections of the two
independent spheres. The explanation for this observation is that the spheres studied in
this work were very small compared to the radiation wavelength, corresponding to the
Rayleigh scattering limit. While the maximum sphere separation distance studied was
very large compared to the sphere diameter, this distance was smaller than the light
wavelength, causing the two particles to behave like a single Rayleigh scatterer with
twice the volume of the individual spheres. It can be stated, in general, that deviations
25
in the optical behavior of an ensemble of spheres from the behavior of the isolated
spheres as a function of interparticle separation (expressed in diameters of the spheres)
will be greatest for small particles.
This is consistent with the experimental
observations of Bruehlman, et al.,26 and the predictions of Stieg27 and Fitzwater and
Hook28 for crowded paint systems.
In their computations on touching, slightly
absorbing two-sphere clusters, Mishchenko, et al.,40 reported that two randomly
oriented spheres with diameters greater than one wavelength can become effectively
independent scatterers at surface-to-surface separations as small as one sphere diameter;
this is not the case for spheres with size smaller than the incident wavelength. Fuller39
has observed that the scattering cross sections of two touching spheres which are each
large compared to the wavelength rapidly approach those of the independent spheres
with increasing separation, provided that one particle is not in the shadow of the other.
If shadowing effects are significant, Fuller reports that the cross sections may deviate
from those of the independent sphere by 5−10% for separations as large as 25
diameters.
Approximate methods which have been applied to the study of light scattering
by particle agglomerates include geometrical optics models (Section 2.3.3.2), which are
limited to the case where the primary particle size is substantially greater than the light
wavelength, and coupled-dipole models (Section 2.3.3.4), which are limited to the case
where the primary particles are substantially smaller than the light wavelength.
Geometrical optics models have been applied to the study of light scattering by ice
crystal agglomerates41 and fractal agglomerates of spheres.42
The coupled-dipole
26
method (Section 2.3.3.5) has been applied to the study of light scattering by fractal
agglomerates.43
2.3 Principles of Light Scattering
The phenomenon of electromagnetic radiation scattering is governed by
Maxwell’s equations, a set of physical laws which provides a rigorous basis for the
mathematical description of classical wave propagation. Their validity appears to be
universal, applying to all interactions between electromagnetic energy and matter. The
historic development of the physics underlying Maxwell’s equations can be found in
virtually any elementary physics text. Maxwell’s equations are described in Section
2.3.1.
Exact, analytical solutions to Maxwell’s equations for different scattering
geometries are described in Section 2.3.2.
Finally, a variety of approximate and
numerical solutions of light scattering are described in Section 2.3.3.
2.3.1 Maxwell’s Equations
Maxwell’s equations are two partial differential equations relating the five
vector quantities B, E, D, J, and H. These are magnetic induction, electric field
intensity, electric displacement, current density, and magnetic field intensity,
respectively. These two equations are:
Equation 11
Equation 12
−
∂B
= ∇×E
∂t
∂D
+ J = ∇×H
∂t
27
The set of three constitutive equations which interrelate these five vector quantities
make this pair of differential equations determinate. The constitutive relations for
isotropic materials are:
Equation 13
B = µH
Equation 14
D =εE
Equation 15
J =σE
where µ is the magnetic permeability, ε is the dielectric permeability, and σ is
conductivity. In the case of anisotropic materials, these three scalars are replaced by
tensors. Substituting the constitutive relations into the partial differential equations in
Equation 11 and Equation 12 and eliminating B, D, and J yields the following
determinate pair of Maxwell’s equations in E and H:
−µ
Equation 16
ε
Equation 17
∂H
= ∇×E
∂t
∂E
+σ E = ∇× H
∂t
Eliminating H between these two equations with the assumption that µ is not a function
of position (which is the case in non-magnetic materials) yields the second order partial
differential equation in E:
Equation 18
ε
∂2 E
∂E
1
+σ
= − ∇×∇× E
2
∂t
µ
∂t
This equation is a point form of Maxwell’s equations, meaning that it is valid at a
mathematical point in space, within an arbitrary material. It is the basis for time28
domain solutions to scattering problems using Maxwell’s equations. Below, in the
description of the finite element approach, this point form is elaborated into an integral
form which facilitates numerical solution of this equation throughout an extensive
three-dimensional volume.
In Equation 16 and Equation 17, the time dependence of electric field is
body are of interest; only in cases with time-varying materials properties or fields are
variations in scattering properties of direct interest. It is therefore useful to express the
time dependence of Maxwell’s equations implicitly rather than explicitly.
To
accomplish this, electric and magnetic field are represented as phasors. In the case of
electric field, for example, this representation has the form:
E ( x, y , z, t ) = Re{E ( x, y , z )e iωt }
Equation 19
where ω is the angular frequency of the propagating electromagnetic wave. Equation
18 can therefore be rewritten as:
Equation 20
− εω 2 E + iωσ E = −
1
µ
∇×∇× E .
The vector identity:
Equation 21
( )
∇ × ∇ × E = ∇ ∇ ⋅ E − ∇2 E
in conjunction with the fact that, with no free charges present,
Equation 22
∇⋅ E = 0
can be used to rewrite Equation 20 as:
29
Equation 23
− εω 2 E + iωσ E =
1
µ
∇2 E .
This is the Helmholtz equation for electric field expressed implicitly with respect to
time. This equation is the starting point for frequency-domain solutions to scattering
problems using Maxwell’s equations.
2.3.2 Exact, Analytical Solutions of Light Scattering
Of interest in this dissertation is the study of light scattering by complex
materials microstructures, with most of this work focusing upon the light scattering
behavior of systems containing complex single scattering features or multiple features
which exhibit near-field optical interactions. Even in ensembles of multiple scattering
features where the assumption of single scattering is violated, the scattering properties
of single particles remain the starting point for analysis. The subject of single scattering
has therefore received a great deal of attention, with comprehensive surveys presented
by Bohren and Huffman,1 van de Hulst,2 and Kerker.44
The development of exact, analytical solutions to Maxwell’s equations for
electromagnetic radiation scattering by isolated, three-dimensional bodies is possible in
cases where the surfaces of these bodies correspond to coordinate surfaces of coordinate
systems in which the solution to the scalar wave equation:
Equation 24
∇ 2ψ + k 2ψ = 0
is separable. This has been accomplished in the cases of an infinite circular right
cylinder,45,46 an infinite elliptical right cylinder,47 a sphere,48 and an ellipsoid.49,50,51
30
The solutions for infinite cylinders are of interest in the case of fibers which are very
long compared to their radii. The solution for the case of a sphere is the well-known
Mie theory, named for the German scientist Gustav Mie who presented this solution to
Maxwell’s equations in 1908.48 Mie theory figures centrally in the problem of radiation
scattering by any isolated particle which can be reasonably approximated as a sphere
and has therefore been widely applied in diverse fields where scattering by particles is
of interest.
Owing to its importance, a survey of this theory is presented below.
Solutions for the case of an ellipsoid involve computational effort which is far more
intensive than that required for a sphere, compounded by the necessity of averaging
over all orientations of the particle relative to the incident radiation, and application of
these solutions has therefore not become commonplace.1 It should be noted that
separation of variables can also be applied to systems in which the optical properties
vary with diameter. This has been accomplished for the case of a sphere52,53,54 and an
infinite right cylinder.55,56,57 In this way, systems with coatings or radial segregation of
materials can be rigorously treated.
Clear and detailed developments of the mathematical treatments underlying Mie
theory are presented by Bohren and Huffman,1 van de Hulst,2 and Kerker.44 Given an
isotropic, spherical particle of arbitrary radius and optical properties illuminated by
electromagnetic radiation of arbitrary wavelength, of interest are the scattered fields
from this particle at far field (i.e., at a position located many wavelengths away from the
particle).
The relationship between the electric field vectors associated with the
incident ( Ei ) and scattered ( E s ) electromagnetic waves is required. The first step in this
31
development is to identify a pair of vector harmonics, both of which satisfy the vector
wave equation, generated from the scalar function ψ in Equation 24. These vector
harmonics M and N are of the form:
Equation 25
M = ∇ × ( rψ )
Equation 26
N=
∇ × (M )
k
where r is an arbitrary constant vector. As required for an electromagnetic field, both
M and N are divergence-free and satisfy the vector wave equation:
Equation 27
∇2 A + k 2 A = 0 ,
provided ψ is a solution to the scalar wave equation. The scalar wave equation can be
rewritten in spherical coordinates and ψ represented as the product of three functions,
each a function of just one of the spherical coordinate system variables r, φ, and θ. This
accomplished, the vector spherical harmonics M and N are generated from ψ. The next
step is to represent the incident plane wave as an infinite series with the vector spherical
harmonics as basis functions. Finally, by imposing boundary conditions dictated by
Maxwell’s equations for the tangential components of electric and magnetic field at the
surface of the sphere, the incident and scattered fields outside the sphere can be related
in a series expansion employing a pair of determinate scattering coefficients an and bn.
The forms of these scattering coefficients and the implementation of Mie theory in
computer code are described in Chapter 3.
32
2.3.3 Approximate and Numerical Solutions of Light Scattering
It is often true that the geometries of scattering features whose electromagnetic
radiation scattering properties are of interest do not conform to the geometric limitations
imposed by the exact analytical solutions produced by the separation of variables. For
such cases, which may involve irregular particle shapes or dependent scattering effects,
a host of approximate analytical and numerical approaches have been developed to
determine the radiation scattering properties of the system. Rayleigh scattering theory
and geometrical optics are approximate analytical approaches introduced in Chapter 1
which produce simple expressions for the distribution of scattered light and the
scattering efficiency terms.
The remainder of the techniques discussed below are
numerical approaches which in general require substantial computational resources.
2.3.3.1 Rayleigh Theory
In the case where the wavelength of electromagnetic radiation interacting with a
particle is much greater than the particle diameter, the incident electric field can be
reasonably approximated as constant throughout the volume of the particle.
The
solution of scattering problems under this approximation is termed Rayleigh theory.58
In Rayleigh theory, the particle is presumed to respond to the applied, time-varying
electric field of the radiation as an oscillating dipole which radiates scattered fields in
all directions.
The induced dipole moment of the particle is a vector quantity
proportional to the applied electric field through the polarizability α, a tensor. With the
induced dipole moment known, it is straightforward to compute the resulting electric
33
field at an arbitrary point outside the particle. The induced dipole moment has been
determined analytically for a sphere, a coated sphere,2 an ellipsoid,59 and a coated
ellipsoid.1 In addition, anisotropic polarizability can be incorporated into each of these
particle geometries.60 The polarizability of a material is often isotropic, in which case it
can be treated as a scalar. The angular intensity distribution of scattered light for
unpolarized incident radiation is then given by:
Equation 28
I=
(1 + cos θ )k α
2
2r 2
4
2
I0
where θ is the scattering angle, r is the distance of the observation point from the
particle, I0 is the incident intensity, and
k=
Equation 29
2πn med
λ0
where nmed is the refractive index of the surrounding medium and λ0 is the wavelength
of the incident wavelength in vacuum.
The most fundamental far-field scattering property is the scattering cross
section, defined as the ratio of scattered power to the incident radiation intensity and
expressed in units of area. The scattering cross section of a sphere in the Rayleigh
approximation is given by:
Equation 30
C sca
8πa 2 4 m 2 − 1
x
=
3
m2 + 2
2
34
where a is the particle radius, m is the relative refractive index of the particle (particle
refractive index divided by that of the surrounding medium) and x is the unitless size
parameter:
Equation 31
x = ka .
For fixed particle radius in the Rayleigh regime and with the assumption that m is
independent of wavelength, Equation 30 can be used to explain at least partially the
bluish color of the daytime sky, since scattering cross sections associated with shorter
wavelengths (e.g., blue light) exhibit are greater by a factor (λlong/λshort)4 than scattering
cross sections longer wavelengths (e.g., red light).
The qualitative criterion for validity of the Rayleigh approximation is:
Equation 32
m x << 1 .
The appearance of both m and x in the validity criterion above arises because even in
the case of a particle which is small compared to the radiation wavelength in the
medium surrounding outside the particle, the compressed wavelength of the radiation
inside the particle could be sufficiently short compared to the particle diameter to
the Rayleigh approximation.
2.3.3.2 Geometrical Optics
In the case where the size of a particle is very large compared to the radiation
wavelength, the incident wavefront encountering the particle can be approximated as a
group of independent rays.
In the geometrical optics approximation, the incident
35
wavefront is divided into rays which encounter the particle directly and rays which pass
along its surface. The scattering problem is treated as a separable combination of
reflection, refraction, and diffraction.61,62 The reflected and refracted rays conform to
the laws of Fresnel and Snell, which determine the fraction of light reflected, refracted,
and transmitted at each boundary of the particle, as depicted in Figure 5. These events
depend intimately upon both the geometry and optical properties of the scattering
particle. In the case of irregularly shaped particles, the computational requirements for
geometrical optics can be intensive. The method is best suited to convex and highly
reflective scattering features; the multiple scattering of rays which can occur in the
cases of concave and transparent particles presents computational difficulties.63 Those
incident rays which pass along the surface of the particle undergo Fraunhofer
diffraction. The Fraunhofer diffraction pattern is intense and strongly forward-directed.
This pattern depends only upon the size and shape of the particle and is independent of
the optical properties of the particle. This non-intuitive fact is based in Babinet’s
principle (1837), which states that the diffraction created by a large particle with
arbitrary optical properties is identical to that created by a hole of the same size cut
from a perfectly opaque screen. The diffracted energy is therefore equal to the sum of
the reflected and refracted energies, and it is for this reason that the scattering crosssection of any particle in the geometrical optics limit is equal to exactly twice its
geometric cross-section.64
The light intensity at any observation point is determined in the geometrical
optics approximation by summing the ray contributions from different regions of the
36
particle (or ensemble of multiple particles). If the incident wavefront is coherent, the
scattered rays have a regular phase relationship and amplitudes are summed at the
observation point rather than intensities. In the case of ensembles of multiple particles
having a range of particle sizes, phase relationships can be neglected and intensities of
scattered rays are added at the observation point rather than amplitudes.
The qualitative validity criterion for geometrical optics is that the particle size be
much greater than the radiation wavelength:
x >> 1 .
On a more precise footing, van de Hulst estimates that geometrical optics is valid for
particles which are at least 20 wavelengths in extent.2 For particles smaller than this, it
is difficult to distinguish between rays which are incident upon the particle and those
which pass along its surface. This estimate is in agreement with results comparing the
light scattering properties of hexagonal ice crystals using geometrical optics and a finite
difference time-domain method, for which close agreement is claimed for x > 20.65
37
Figure 5. Schematic diagram showing reflection and refraction of a light ray using
the geometrical optics approximation. (After Bohren and Huffman1)
2.3.3.3 Point Matching
The point matching method66,67 can be used to determine the radiation scattering
properties of non-spherical particles of arbitrary size whose shapes are approximately
spherical. In this method, fields inside and outside the particle are expanded in a
truncated series of vector spherical harmonics and the boundary condition requiring that
tangential field components be continuous at the sphere surface is imposed at a number
of points on the surface.
This number of points corresponding to the number of
unknowns in the series expansions, resulting in a set of simultaneous linear equations.
This method is conceptually simple but suffers from uncertain convergence and its
limitation to nearly spherical particles.63
2.3.3.4 Perturbation Methods
Perturbation methods,68,69,70,71 like the point matching method, attempt to
determine the radiation scattering properties of particles which deviate from a shape
whose scattering properties are known analytically. The shape of any particle can, in
principle, be viewed as a sphere with a perturbed shape. Perturbation methods represent
the scattering properties of the geometrically perturbed particle as an infinite series in
some perturbation parameter of which the results of Mie theory form the first term.
These methods suffer from the same problem of uncertain convergence as the point
matching method, and the computation of a prohibitive number of terms may be
required before the infinite series may be truncated without appreciable error.
It
38
appears that perturbation methods, like the point matching method, are limited to nearly
spherical particles.
2.3.3.5 Coupled-Dipole Method
A scattering object can be approximated as a collection of radiating electric
dipoles which are excited by both the incident electromagnetic radiation and each other.
This is the principle of the coupled-dipole, or discrete-dipole, method. This method was
first developed by DeVoe in 1964 and applied to scattering by molecular aggregates
very small compared to the radiation wavelength.72,73 The method was generalized to
arbitrary size parameters by Purcell and Pennypacker in 1973, who included retardation
effects and applied the method to determine the scattering properties of interstellar
dust;74 the coupled-dipole method is sometimes still referred to as the PurcellPennypacker method. This method has the advantage of ready applicability to arbitrary
geometries without restriction to nearly spherical shapes.
polarizability of a material is a tensor, the coupled dipole method can be used to
compute the scattering properties of anisotropic materials.75 Its accuracy is limited in
principle only by the density of the electric dipole array, which is in turn limited only by
the availability of computational resources.
A critical feature of the coupled-dipole method is the function chosen to relate
the polarizability αj of the discrete dipoles to the dielectric properties of the bulk
material being represented. The Clausius-Mossotti relation provides an exact function
39
for the particular case of an infinite cubic lattice of dipoles in the dc limit, where the
size of the dipoles divided by the radiation wavelength approaches zero:76
α CM =
Equation 33
3(ε − ε med )
n (ε + 2ε med )
where ε is the dielectric function of the scattering material, εmed is the dielectric function
of the surrounding medium, and n is the number of discrete dipoles per unit volume.
Numerous schemes for modifying the Clausius-Mossotti relation have been proposed to
improve the accuracy the coupled-dipole method for realistic, finite-sized scattering
geometries by adding a term on the order of (kd2)77,78 or (kd3),77,79 where k = 2πnmed/λ0
and d is the interdipole spacing.
Recently, Draine and Goodman80 developed an
expression for polarizability of an infinite array of polarizable points so that the array
exhibits the same dispersion relation as a continuum with dielectric constant ε. Draine
and Flatau81 claimed in 1994 that this representation yields the best accuracy to date of
these expressions for dipole polarizability.
The coupled-dipole method features a square matrix with edge length equal to
the number N of discrete dipoles in the scattering geometry being represented. This
NxN matrix expresses the influence of each of the separate dipoles upon each other.
The computational memory required for this method increases as (N3),81 placing a limit
on the number of dipoles which can be used to represent a scattering geometry and,
therefore, upon the accuracy of the method.
40
2.3.3.6 T-Matrix or Extended Boundary Condition Method
Starting with an integral equation formulation of the scattering problem, the Tmatrix or Extended Boundary Condition Method is a promising technique which can be
used to determine the scattering properties of arbitrarily shaped particles.
First
proposed by Waterman in 1965 for the special case of perfectly conducting particles,82
it has continued to be refined and has since been extended to general dielectric
particles,83, including finite cylinders,84 homogeneous ellipsoids,84,85,86 coated
ellipsoids,87 and clusters of dielectric spheres.40,88,89,90
Following here a conceptual development of the T-matrix formulation set forth
by Barber and Yeh,84 the method embodies a series of intuitive steps relying upon the
equivalence theorem, which states that the field scattered by a particle can be
considered the product of initially unknown electric and magnetic surface currents
acting on the surface of the body. The T-matrix approach ultimately defines the fields
internal to the scattering object in terms of the incident field, then the surface currents in
terms of the internal fields, and finally the scattered field in terms of the surface
currents. The incident, internal, and scattered fields are expanded in a series of vector
spherical harmonics with f and g the unknown coefficients of the scattered field. The
result after applying boundary conditions is the linear transition or T-matrix, which
relates the coefficients of the scattered field to the known coefficients a and b of the
incident field:86,91
Equation 34
f
a 
 g  = T b
 
 
41
The elements of the T-matrix are integrals over the surface of the particle which depend
only upon the size, shape and optical properties of the particle. These surface integral
reduce to line integrals for axisymmetric particles, which is why the T-matrix has been
applied almost exclusively to such particles.92 Computationally, application of the Tmatrix method involves linear algebra operations, either matrix inversion or iterative
schemes.88 One advantage of the T-matrix approach is that, once the T matrix is
computed for a particle, it is simple to compute results for arbitrary orientations of that
particle relative to the incident fields. While the T-matrix method can be used to
compute the scattering properties of large volumes, computational requirements
increase dramatically with size parameter and deviation from rotational symmetry.86,93
In addition, numerical instability can arise in T-matrix computations on these lowersymmetry particles.91
2.3.3.7 Method of Moments
Method of moments techniques embody a number of specific methods, each
based upon the solution of an integral form of Maxwell’s equations. The method of
moments was first implemented for Maxwell’s equations by Harrington in 1968.94 Both
surface and volume integration approaches have been described. Like the coupleddipole method, the method of moments is applicable to highly irregularly shaped
objects. Method of moments techniques are well-suited to conductive materials and
homogeneous dielectrics, but the implementation becomes cumbersome for
inhomogeneous bodies and dielectric bodies of large volume.63
The method of
42
moments leads to fully populated matrices and therefore the computer memory required
increases rapidly with increasing object size.
In the surface integral formulation of the moment of methods, as in the T-matrix
method, the problem of radiation scattering by an arbitrary body is recast using the
equivalence principle into unknown electric and magnetic surface currents J and M
acting on the body to create the total E and H fields both inside and outside the body.
Electromagnetic boundary conditions are imposed at the surface of the body to relate
the fields inside and outside the body. Following the notation of Medgyesi-Mitschang,
et al.,95 these equivalent surface currents appear in the form of a coupled pair of integral
equations which have the form:
inc
E (r)
Equation 35
Equation 36
H
inc
(r)
tan
= ( L1 + L2 ) J ( r )
tan = ( K 1 + K 2 ) J ( r )
tan
−( K 1 + K 2 ) M ( r )
tan
 1

1


L
L
+
+
tan 
2 M (r)
2 1
2
η
η
2
 1

tan
where r is the field point on the surface, L and K are known integral operators, the
superscript inc indicates the incident field, and the subscripts 1 and 2 denote regions
outside and inside the body, respectively. The equivalent currents on the surface of the
body are expanded in a set of basis functions fn spanning the surface and substituted into
the pair of equations above. Next, the complex inner product is formed with these
equations, transforming the integral operators L and K into matrix operators. In the
Galerkin form of the method of moments solution, the complex conjugates of fn are
chosen as the testing functions on the surface, and the inner products are formed
43
between the testing functions and the integral equations. The result is a set of linear
equations which are solved to determine J and M. In the determination of the scattering
properties of an arbitrarily shaped body, the surface is represented as a mesh of, say,
triangular elements, and J and M are expanded in terms of a testing function along each
element edge.96
A volume integral formulation of the method of moments described by Hage, et
al., has been applied to inhomogeneous dielectric materials of irregular shape.97 In their
approach, a solid of arbitrary shape is represented as a volumetric mesh of N cubic
elements, with constant electric field assumed in each cell. The mathematical basis of
this approach is to represent Maxwell’s equations in E as the (exact) volume integral
taken over the volume V of the body:
Equation 37
inc
E (r1 ) = E (r1 ) +
k2
4π
∫∫∫ [m
2
]
( r 2 ) − 1 E ( r 2 )G ( r 1 , r 2 )d 3 r 2
V
where r1 is the point of observation, r2 is a point inside the body, m is the complex index
of refraction of the body, and G is the divergence-free Green’s dyadic for the vector
time-independent form of Maxwell’s equations in E. Discretization of the volume V
into a set of N cubic elements makes this integral equation approximate and creates a
system of 3N linear equations with 3N unknowns, the electric field components in each
element. The time required for a computation increases as (N2), which is the chief
disadvantage of this technique. To their credit, Hage, et al., supplement their derivation
of the volume integral form of Maxwell’s equations with microwave scattering
44
experiments on the same arbitrary shaped modeled computationally; good agreement
between calculation and experiment was established.
2.3.3.8 Finite Difference Time-Domain Method
In the finite difference time-domain approach, Maxwell’s equations are solved
numerically in their differential form. This approach is suitable for determining the
radiation scattering properties of arbitrary structures. In the usual finite difference timedomain method, a three-dimensional Cartesian model is constructed which consists of
the scattering structure of interest and the surrounding medium.
It is possible to
implement other coordinate systems with the finite difference approach, but the
Cartesian system affords by far the greatest flexibility in accommodating irregular
geometries. The model is discretized into a large number of rectangular solid mesh
cells, and appropriate optical properties are assigned to each cell.
Absorbing or
radiating (in which only the scattered component of radiation is absorbed) boundary
conditions are assigned to the six outer faces of the model.
propagates through the model in the time domain, and the computation is allowed to
proceed until convergence to steady state is achieved.
The differential equations for E and H solved in the finite difference time
domain approach are the same as those given in Equation 16 and Equation 17:
−µ
Equation 38
Equation 39
ε
∂H
= ∇×E
∂t
∂E
+σ E = ∇× H
∂t
45
The spatial and temporal derivatives appearing in these coupled equations are
represented approximately at any point in the mesh between time steps n and (n+1)
using an approximate difference expression. In the scheme proposed by Kunz and
Luebbers,98 for example, this differencing scheme is explicit and first-order accurate:
n +1
Equation 40
∂E
∂t
≈
E
n +1
−E
∆t
n
Of course, more accurate schemes can be employed, as can more reliably stable,
implicit schemes. Values of E and H are computed at the nodes or centers of the mesh
cells.
Accuracy considerations in the finite difference time domain approach are
analogous to those encountered in the coupled-dipole method and the volume integral
formulation of the method of moments, for example: accuracy is controlled in large part
by the density of the finite difference mesh, which is in turn limited by computational
resources. In the scheme proposed by Kunz and Luebbers, there is no storage of a large
matrix, and the memory requirement increases as (N4/3), where N is the number of mesh
cells in the model. This is a considerable advantage over the couple dipole-method and
the volume integral formulation of the method of moments, which exhibit a (N3)
dependency. Another advantage of the finite difference time-domain approach is that it
produces full near-field solutions through the volume of the model, which can be of
considerable interest; this is not the case in the surface integral formulation of the
method of moments, for example. Disadvantages of the finite difference time domain
approach include the inability to represent exactly any skewed surfaces in a model using
the Cartesian mesh cells, the fact that user intervention is required to identify the steady
46
state solution, the errors introduced by the approximate boundary conditions, and the
necessity of transforming the near-field results to far-field results in an additional
computational step. These same considerations arise in the finite element time-domain
approach used in the present research; many of these issues are addressed in more detail
in Chapter 3.
2.2.3.9 Finite Element Method
Finite element methods in electromagnetics are based upon the solution of an
integral form of Maxwell’s equations. In any finite element approach,99 the governing
differential equation is multiplied by an arbitrary weighting or testing function, and this
product is integrated over the entire volume of the computational space of interest. This
computational space is discretized into a large number of volume elements, the size and
shape of which are dictated by the level of accuracy required and the geometry of the
features contained in the volume. The total field is then approximated within each
element by an arbitrary shape function which interpolates between field values at
element nodes or edges. The only restriction on the weighting and shape functions is
that they be of sufficiently high order to be differentiable according to the integral form
of the governing differential equation. The computational volume is truncated using
boundary conditions which attempt to minimize spurious reflection of radiation back
into the volume. The system of volume elements is assembled into a global, NxN
matrix K which multiplies the vector of unknown field values a at each element node or
edge to form the known vector f derived from boundary conditions:
Equation 41
[K ]{a} = { f }
47
The finite element approach thus yields a system of linear equations which can be
solved for the field throughout each volume element. In the commonly employed
Galerkin formulation, the shape and weighting functions are chosen to be the same
yielding a symmetric K matrix. The matrix K is sparse, yielding a computational
memory requirement which increases as N.100
In frequency-domain finite element approaches to Maxwell’s equations,100,101
the time independent form of Maxwell’s equations is employed. Equation 23, for
example, therefore takes the form:
Equation 42
 1
∫∫∫ − µ ∇
V
2

E ⋅ E ' − εω 2 E ⋅ E ' + iωσ E ⋅ E 'dV = 0

where E' is the weighting or testing function, and this scalar integral equation would be
solved throughout the discretized computational volume V to yield the system of linear
equations in the unknown vector E.
In the present research, the EMFlex software, a versatile finite element timedomain implementation of Maxwell’s equations, is employed. The EMFlex software
has previously been applied to a number of systems of practical interest, including the
determination of light scattering properties of features on silicon wafers102 and
evaluation of the performance of integrated optical devices103 and waveguides.104,105,106
The implementation of this code is described in Chapter 3. Advantages of the timedomain approach used in EMFlex over frequency domain approaches include the use of
an efficient, explicit forward integration scheme which requires minimal computer
memory, a robust and stable path to the steady-state solution, and easy extension to
48
include time-dependent optical properties of materials in the system or transient
phenomena such as pulsed beams.107
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53
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54
Chapter 3. Computational Methods
This chapter describes the two methods used in this research to compute the
electromagnetic radiation scattering properties of microstructures. The first is Mie
theory (Section 3.1), the second is the time-domain finite element approach (Section
3.2).
3.1 Mie Theory
3.1.1 Computation of the Scattering Coefficients an and bn
The separation of variables approach which underlies the derivation of Mie
theory is described in Section 2.3.2. Here, an elaboration of the mathematical forms of
Mie theory is presented, with emphasis on its implementation into computer code which
can be used to calculate the far-field scattering parameters of interest. In the present
research, Bohren and Huffman’s computer codes BHMIE and BHCOAT are employed
to calculate scattering parameters for homogeneous and coated spheres, respectively.1
Both of these codes have been modified to include computation of the asymmetry
parameter (Section 3.1.2).
Given an isotropic, spherical particle of arbitrary radius and optical properties
illuminated by electromagnetic radiation of arbitrary wavelength, of interest are the
scattered fields from this particle at far field. To this end, the relationship between the
electric field vectors associated with the incident ( Ei ) and scattered ( E s )
electromagnetic waves must be established. Equation 43 evokes the 2x2 scattering
55
matrix, a mathematical operator which defines the relationship between scattered and
incident radiation at the arbitrary point (x,y,z) located a distance r from the particle.
The incident radiation is assumed to propagate in the +z direction. In general, the
scattering matrix is fully populated, but for the case of a spherical particle, it is
diagonal.1,2
 E s , par .  e ik ( r − z )  S 2
 =


E
ikr
−
,
.
s
perp
0


Equation 43
0  E i , par . 
.

S1  E i , perp. 
The subscripts of the electric field vectors refer to polarizations parallel (also called p
polarization) and perpendicular (also called s polarization) to the scattering plane, and
k=
Equation 44
2πn med .
λ0
.
In Mie theory computations, the surrounding medium is infinite in extent and therefore
must have zero absorption coefficient; that is, its index of refraction must be a real
number. The values of Sn are given by:
Equation 45
S1 = ∑
2n + 1
(an π n + bn τ n )
n (n + 1)
S2 = ∑
2n + 1
(an τ n + bn π n ) .
n (n + 1)
n
Equation 46
n
The functions τn and πn are polar functions which depend solely upon the scattering
angle θ. They can be written as simple functions of the Legendre function Pn1 :
Equation 47
Pn1
dPn1
πn =
, τn =
sin θ
dθ
56
The functions τn and πn can be conveniently computed by upward recurrence from the
relations:
πn =
2n − 1
(cosθ )π n −1 − n π n −2 ,
n −1
n −1
τ n = n(cosθ )π n − (n + 1)π n −1
Equations 48
beginning with π0 = 0 and π1 = 1.
The terms an and bn are the coefficients of the series expansion for the scattered
fields outside the particle. They, like S1 and S2, are generally complex numbers. The
mathematical expressions for these angle-independent scattering coefficients are
complicated but are computed from only upon materials properties and the spherical
Bessel functions jn ( x ) and hn(1) ( x ) . These expressions are:
Equation 49
Equation 50
µ m2 jn (mx ) xj n (x ) '− µ1 jn (x ) mxj n (mx ) '
an =
µ m2 jn (mx ) xhn(1) (x ) '− µ1hn(1) (x ) mxj n (mx ) '
bn =
µ1 jn (mx ) xj n (x ) '− µ jn (x ) mxj n (mx ) '
µ1 jn (mx ) xhn(1) (x ) '− µ hn(1) (x ) mxj n (mx ) '
where µ and µ1 are the permeabilities of the medium and the particle respectively, and
Equation 51
Equation 52
Equation 53
f (x ) ' =
∂
f (x )
∂x
x = ka
m=
n1
n med
57
where n1 is the real refractive index of the particle. The contrast ratio, m, is the ratio of
the refractive index of the particle to that of the surrounding medium.
The magnitude of the scattered irradiance of a particle as a function of the
scattering angle depends upon the polarization of the incident radiation relative to the
scattering plane. For the case of polarization perpendicular to the scattering plane (i.e.,
i1 as a function of angle is given by the
expression:
Equation 54
i1 = S1 (θ )
2
For the case of polarization parallel to the scattering plane (i.e., the p polarization),
scattered irradiance i2 is given by the expression:
Equation 55
i2 = S 2 (θ )
2
obtained by computing the mean of i1 and i2.
This discussion of Mie theory has focused primarily upon the case of a
homogeneous sphere.
These results can be readily extended to the case of an
inhomogeneous sphere consisting of concentric shells with different optical properties.
This is accomplished by expanding the fields within each coating layer in spherical
harmonic basis functions (just as in the solid core of the coated sphere) and imposing
the electromagnetic boundary conditions at both of the interfaces associated with each
coating layer. The result is a modified pair of expressions for the scattering coefficients
58
an and bn outside of the coated sphere, which are functions of the optical properties and
thicknesses of the coating layers.1
3.1.2 Far-Field Scattering Parameters
The most important angle-independent quantity relating to scattering efficiency
of sphere is the scattering cross section, Csca. Scattering cross section is the total power
scattered by the sphere, divided by the incident intensity. It is expressed in units of area.
If one imagines a spherical, mathematical surface with a radius very large compared to
the radiation wavelength and centered at the particle, the scattering cross section can be
visualized as the following surface integral taken over this far-field, mathematical
surface:
C sca =
Equation 56
∫∫ i(θ )dΣ
Σ
i0
where i(θ) is scattered intensity as a function of scattering angle, dΣ is an infinitesimal
element of surface area, and i0 is the incident intensity. The far-field surface must be a
large distance from the particle to ensure that evanescent fields in the vicinity of the
particle do not reach the far-field surface. The scattering cross section can be computed
from the scattering coefficients as:
Equation 57
C sca =
2π
k2
∞
∑ (2n + 1) ( a
n =1
2
n
2
+ bn )
The scattering cross section can be visualized as the geometric area of the incident plane
wave which is scattered by the particle.
59
In general, the scattering cross section increases with increasing sphere radius.
It is therefore useful to normalize the scattering cross section using the geometric
properties of a particular sphere of interest. One means of normalization is to compute
the ratio of the scattering cross section to the geometric cross section of the sphere. The
result is the unitless scattering efficiency, Qsca:
Equation 58
Q sca =
C sca
πa 2
The scattering efficiency approaches zero as sphere radius approaches zero and
approaches the value 2 as sphere radius approaches infinity. Between these two limits,
Qsca can in principle attain arbitrarily large values for very high-index particles of
optimal size for strong scattering, though in practice for non-absorbing materials of
physically realistic refractive index, peak values in the resonant regime greater than 2
but less than 10 are observed. The scattering efficiency Qsca is an intuitive quantity
which facilitates comparison of the scattering performance of particles with very
different geometric or physical properties.
It is useful to normalize the scattering cross section in terms of the volume of a
scattering particle. It is more natural on physical grounds to normalize scattering
performance by volume, employing a geometric normalization which is threedimensional, than by the two-dimensional cross-sectional area. Paint formulations and
pigment costs, for example, are based upon pigment particle volume rather than upon
particle cross sectional area. Therefore, it useful to evoke a third far-field scattering
60
parameter, the scattering coefficient, S. S is defined as the scattering cross section of a
particle divided by the particle volume:
Equation 59
S=
C sca
 4π a

 3
3



=
3Q sca
4a
S is expressed in units of inverse length. The reader is cautioned that this S is not the
same quantity as the S* defined for particles or films in the discussion of KubelkaMunk theory in Chapter 2.
The asymmetry parameter g is the average cosine of the scattering angle,
weighted by scattered intensity. It is a unitless quantity. The asymmetry parameter
provides important information about the relative importance of forward- and backscattering by the spherical particle. Evoking the far-field spherical surface with the
scattering particle at its center described in the case of the scattering cross section, the
asymmetry parameter is defined as:
Equation 60
g=
∫∫ i(θ ) cosθdΣ
Σ
∫∫ i(θ )dΣ
Σ
The asymmetry parameter can be computed from the scattering coefficients an and bn
as:1
Equation 61
where the asterisk denotes the complex conjugate.
61
In the context of multiple scattering, where radiation encounters a large number
of independently scattering particles while traversing an ensemble of such particles, the
asymmetry parameter strongly effects the redirection of scattered radiation and
therefore the scattering efficiency of the ensemble of particles as a whole.
The
scattering parameter σ combines the asymmetry parameter and the scattering coefficient
S as follows:
Equation 62
σ = S (1 − g )
Like the scattering efficiency S, the angle-weighted scattering coefficient σ is expressed
in units of inverse length. In the case of a semi-infinite film containing a large number
of scattering particles, for example, the scattering coefficient σ provides direct
information about how effectively the ensemble back-scatters incident light through
multiple scattering.17,18,2
While it has been assumed in this discussion of far-field scattering parameters
that the scattering particle is a sphere, the quantities Csca, Qsca, S, g, and σ are all equally
well defined for irregularly shaped particles.
3.2 Finite Element Time-Domain Method
The EMFlex time-domain finite element software for solving Maxwell’s
equations is employed in much of the present research. In Section 3.2.1, the basis of its
numerical implementation basis is presented. In Section 3.2.2, details of its application
to problems of interest are described.
62
3.2.1 Numerical Implementation
The following description of the mathematical basis of the EMFlex code follows
closely an unpublished report by Wojcik107, one of the developers of the code. As
described in Section 2.2.3.9, the finite element method allows piece-wise solution of
Maxwell’s equations throughout a discretized computational volume containing
scattering features. The partial differential equation which is solved is the combined
form of Maxwell equations presented in Section 2.3.1:
ε
Equation 63
∂2 E
∂E
1
+σ
= − ∇×∇× E .
2
∂t
µ
∂t
Forming the inner dot product of this equation with the arbitrary vector weighting
function G and integrating over the volume V of the computational volume yields the
integral equation:
Equation 64
(
)
 ∂2 E
1
∂ E 
ε
G
dV = − ∫∫∫ G ⋅ ∇ × ∇ × E dV .
+σ
⋅
2
∫∫∫
 ∂t
µ V
∂t 
V

Using a vector identity, the integrand of the right-hand side of Equation 64 can be
rewritten as:
Equation 65
(
)
G ⋅ ∇ × ∇ × E = ∇ × E ⋅ ∇ × G − ∇G × ∇ × E .
Substituting Equation 65 into Equation 64 and applying the divergence theorem to the
second term on the right-hand side of Equation 64 gives:
Equation 66
V
(
)
(
)
 ∂ E
1
1
∂ E 
+σ
dV = − ∫∫∫ ∇ × E ⋅ ∇ × G dV − ∫∫ G ⋅ n × ∇ × E dΣ
2
µ V
µ Σ
∂t 
 ∂t
∫∫∫ G ⋅  ε
2
63
where n is the outward unit vector normal to the surface of the computational volume.
The second order spatial operator appearing in Equation 64 is thus replaced in Equation
66 by first order operators, making numerical implementation of this governing
equation simpler.
The fields E and G are assumed to be separable in time and space, so that:
Equations 67
E ( x, t ) = S ( x ) f (t ) ; G ( x, t ) = S ( x ) g (t )
where the same spatial dependence of the shape function S(x) is assumed for both E and
G. This is the Galerkin formulation in which the weighting and shape functions are
taken to be the same and yields a symmetric system of linear equations in the finite
element formulation. Substituting Equations 67 into the integrands of Equation 66
yields the differential equation:
Equation 68

T 
∂f
 ∂2 f
g M 2 + C
−K f −Bf=0
∂t
 ∂t

where the superscript T indicates the transpose and
Equation 69
M ≡ ∫∫∫ (S T εS )dV ; C ≡ ∫∫∫ (S T σS )dV ; K ≡
V
V
1
(∇ × S ) (∇ × S )dV .
µ ∫∫∫
T
V
K, L, and M are symmetric coefficient matrices, and:
Equation 70
B≡
(S
µ ∫∫
1
T
)
n × ∇ × S dΣ
Σ
is the coefficient matrix defined by the surface integral. Since g(t) is an arbitrary
function, Equation 68 can be rewritten as:
64
Equation 71
M
∂f
∂2 f
+C
= (K + B ) f .
2
∂t
∂t
It is this ordinary differential equation which is solved using the finite element approach
by discretizing the computational volume, solving the integral equation within each
element, and constructing the global system of linear equations in the unknown electric
field f(t) at each node in the mesh. Rectangular solid volume elements, with eight
corner nodes, are employed in EMFlex. There are therefore twenty-four unknown
electric field components (three at each node) associated with each element in the finite
element mesh.
The interpolating shape function S(x) varies linearly within each
element between the electric field values at each corner.
In the time dimension,
EMFlex uses the central difference approximation, which is second-order accurate, to
compute the second derivative with respect to time:
Equation 72
∂ 2 f (t ) f (t + ∆t ) − 2 f (t ) + f (t − ∆t )
≈
∂t 2
( ∆t ) 2
This is an explicit forward integration scheme.
In EMFlex finite element models, there are several different boundary
conditions which can be assigned to each of the model walls. The absorbing boundary
condition is intended to absorb all incident radiation at the wall, both scattered and nonscattered.
The reflecting boundary condition assigns the properties of a perfect
conductor to a model wall, resulting in complete reflection of all incident radiation.
Symmetric and asymmetric boundary conditions can be used to impart lateral
periodicity to models. Finally, the illumination boundary condition prescribes that only
65
scattered radiation is absorbed at a model wall, allowing non-scattered radiation to pass
through. The illumination and absorbing boundary conditions can contribute to error at
a model wall by causing a measure of spurious reflection of light back into the model.
The illumination boundary condition is used in the finite element models throughout the
present research.
In practice, an individual EMFlex finite element computation results in the
steady-state properties of full electric field quantities (phase and amplitude) in the nearfield surrounding the structure of interest. These near-field quantities can be expressed
in terms of full, scattered, or reflected components. It is necessary to extrapolate these
near-field quantities to far field in a second computational step to extract the far-field
scattering parameters of interest. For this purpose, a mathematical surface (termed the
Kirchoff box) is defined just inside the walls of the finite element model, on which the
equivalent electric and magnetic currents J and M are computed from the tangential
components of scattered electric and magnetic field.45 From these equivalent currents
the scattered electric field at near field is extrapolated onto the surface of a far-field
sphere having a radius which is much greater than the wavelength of light using the
equivalence theorem.45 Extrapolation to far field eliminates evanescent electric field
components which are present at near field so that only propagating electromagnetic
waves contribute to the far-field result.
The simulation of diffuse illumination is
accomplished by superimposing the far-field scattered intensities computed from a
series of individual computations in which the illumination directions have been varied
66
step-wise over the full range of necessary orientations. The extrapolation of near-field
intensities to far field is performed using the EMFlex post-processing software, Review.
3.2.2 Use of the EMFlex Software in Practice
The methodology used to run the EMFlex code is straightforward. The platform
used for the EMFlex code is a Cray C-94 supercomputer. The platform used for the
Review code is a Silicon Graphics, Inc. (SGI) Power Challenge mainframe computer.
Individual EMFlex jobs are submitted to the Cray from the SGI in the form of text input
decks. A typical EMFlex input deck is included in the Appendix. The EMFlex input
deck contains parameters defining the geometry of the finite element model, the optical
properties of the materials contained in the model, and parameters defining the
illumination conditions. The geometric parameters of the finite element model include
the physical dimensions of the Cartesian computational space (a rectangular solid in
three-dimensional calculations), the number of finite elements along each edge of the
model, and the size, shape, and position of particles embedded within the model. The
illumination conditions include the free-space wavelength of the illuminating plane
wave radiation, its propagation direction, its amplitude, and its polarization, along with
the boundary condition implemented at each wall of the model.
Upon completion of a run on the Cray, a binary data block containing all of the
user-requested near-field quantities (such as electric field amplitude, phase, and
scattered intensity) is returned to the SGI for subsequent post processing. This post-
67
processing typically includes extrapolation of near-field intensities to the surface of a
far-field sphere and computation of the far-field scattering parameters.
3.2.3 Computation of Far-Field Scattering Parameters
The computation of the scattering cross section Csca and the asymmetry
parameter g for a given microstructure using EMFlex is accomplished by numerical
summations of scattered intensities on the surface of the far-field sphere. In practice the
surface of this far-field sphere is discretized into j surface area elements delineated by a
spherical coordinate system with a user-defined mesh density in the two spherical
angles phi and theta. In these computations, there is an established relationship between
the spherical coordinate system of the far-field surface and the Cartesian coordinate
system of the finite element model.
The scattering cross section of a given
microstructure is given by:
Equation 73
C sca =
∑ i(θ
j
) ⋅ ∆Σ j
j
i0
where θ is the scattering angle, i(θj) is the scattered intensity within surface element j,
∆Σj is the surface area of the surface element j, and i0 is the incident radiation intensity
in the finite element computation. The discretized expression in Equation 73 for the
scattering cross section is analogous to the continuum expression given in Equation 56.
The asymmetry parameter g is computed numerically on the surface of the farfield sphere through the summation:
68
Equation 74
g=
∑ i(θ
j
j
) ⋅ cosθ j ⋅ ∆Σ j
∑ i(θ
j
) ⋅ ∆Σ j
.
j
The discretized expression in Equation 74 for the asymmetry parameter is analogous to
the continuum expression given in Equation 60.
With the scattering cross section and asymmetry parameter computed in this
manner for a given finite element computation, the scattering parameters S and σ can be
computed, with knowledge the volume of scattering material contained in the finite
element model.
3.2.4 Simulation of Random Illumination: The Illuminator
The general problem of interest in this research is the random illumination of a
particular microstructure, producing an orientation-averaged result of its light scattering
properties. This condition is appropriate to a film containing pigment particles, for
example, which generally would have no particular orientation in the film relative to the
incident light.
Because the illumination conditions in an individual EMFlex
computation prescribe the wavelength, polarization, and propagation direction of a
plane wave of light, it is not possible to simulate random illumination with a single
computation. Random illumination, however, can be simulated by superimposing farfield intensities from numerous individual EMFlex computations. To facilitate such a
superposition scheme, a computational tools package called the Illuminator has been
developed. The Illuminator creates and submits the individual job files constituting a
random illumination computation to the Cray, performs extrapolation of near-field
69
scattering results to far field, and superimposes the individual far-field results to form
images of scattering signatures. The Illuminator is an extensive script which is run by
the Review software.
The individual steps of an Illuminator run are shown in the flow chart of Figure
6. As this flow chart indicates, the first step is to define the finite element model to be
used in the Illuminator run. This model definition file, called the seed file, specifies the
orientation of the Cartesian coordinate system used to define the model, the model
dimensions, particle position(s) and shape(s), and the optical properties of the
constituent materials of the model.
A second definition file employed by the
Illuminator is the control file. The Illuminator control file specifies the mesh density of
the finite element grid (the number of finite element edge lengths per light wavelength
in the model), the illumination wavelength, illumination directions (expressed in the
spherical coordinate angles theta and phi), and the parameters controlling the
extrapolation of near-field scattered intensities to far field.
The control file also
specifies more advanced options available in the Illuminator software, such as the
ability to specify a series of jobs in which the particle size(s) in a model are
systematically scaled.
There are two distinct coordinate systems employed in each Illuminator run.
The first is the Cartesian coordinate system associated with the three-dimensional finite
element model. The second is the illumination coordinate system. The illumination
directions for an Illuminator run are expressed in spherical coordinates by specifying an
overall range and angular increment for the angles theta and phi. Theta is the polar
70
angle, measured from the positive z axis of the Cartesian coordinate system; phi is the
azimuthal angle measured counter-clockwise in the x-y plane from the positive x axis of
the Cartesian coordinate system. The illumination conditions can be visualized by
evoking the idea of the illumination sphere, an example of which is shown in Figure 7.
The surface of this illumination sphere is divided into area elements determined by a
regular increment (18o, in this case) of the two angles theta and phi. In the spherical
illumination coordinate system, the individual illumination directions lie at the centers
of the patches on the surface of the illumination sphere. One consequence of using
spherical coordinates is that such a scheme leads to a higher density of illumination
directions near the poles of the spherical coordinate system. If the mean scattering
cross-section of the individual illuminations were computed in a non-weighted fashion,
the resulting average scattering cross-section would be inappropriately biased toward
the values associated with those illumination directions near the poles of the
illumination coordinate system. To correct this effect, the Illuminator calculates an
average scattering cross-section where the scattering cross-sections of the individual
illuminations are weighted by the area of the corresponding area element on the surface
of the illumination sphere. The illumination sphere should not be confused with the farfield extrapolation surface; the two are separate.
An important consideration in formulating an Illuminator control file is the
spatial symmetry of the finite element model, since the presence of spatial symmetry
elements can substantially reduce the number of illumination directions required to
simulate random illumination. In the case of a finite element model geometry having
71
three mutually perpendicular mirror planes, for example, illumination over a single
octant of the spherical coordinate system produces the complete scattering results. This
reduces by a factor of eight the computational time required for the Illuminator run
compared to a model requiring illumination over the full spherical coordinate system.
Once the Illuminator seed and control files have been defined, the Illuminator
run is started on the SGI Power Challenge. In the first stage of an Illuminator run, as
Figure 6 indicates, the individual jobs corresponding to the different illumination
directions of a random illumination are created and submitted one by one to the Cray C94. The Cray computes the full near-field scattering results at steady state for each
individual job and returns the results to the SGI Power Challenge. Once all of these
individual jobs have been computed on the Cray, the Illuminator scripts prompts the
Review software to perform the extrapolation of near-field results to far field for each
individual job using the parameters specified in the control file. The completion of
these extrapolations to far field marks the end of an Illuminator run.
72
Illuminator Seed File:
Define Finite Element Model:
Model Dimensions, Materials Properties,
Particle Position(s), Particle Shape(s)
Illuminator Control File:
Define Finite Element Mesh Density,
Illumination Wavelength(s), Illumination Directions,
Extrapolation Conditions
Initiate Illuminator Software
on SGI Challenge Computer
Cray Jobs Submitted Sequentially,
Processed
Full Near-Field Results
Returned from Cray to SGI Challenge
Near-Field Results Extrapolated
to Far Field on SGI Challenge
Complete Extrapolation Results Written to File
(Illuminator Run Complete)
Near-Field Results Reviewed
Graphically or Numerically
on SGI Challenge
Figure 6. Flow chart showing the formulation and evaluation of light scattering
problems using the Illuminator software.
73
Figure 7. A representation of the spherical illumination coordinate system used in
Illuminator runs. In this case, there is an 18o increment in the spherical coordinate
angles phi and theta between adjacent surface area elements. The individual
illumination directions pass through the centers of the surface area elements.
1
W.M. Irvine, “Light Scattering by Spherical Particles: Radiation Pressure, Asymmetry Factor, and
Extinction Cross Section,” J. Opt. Soc. Am., 55 (1), 16-21 (1965).
2
H.C. van de Hulst, Multiple Light Scattering, Academic Press, New York (1980).
74
Chapter 4.
Performance of the Time-Domain Finite Element
Method
This chapter describes the computational efficiency and accuracy of the finite
element method. Section 4.1 presents the dependence of memory requirements and
CPU times on the number of finite elements in EMFlex computations, along with the
dependence of CPU times on the surface element density of the far-field extrapolation
sphere. Section 4.2 considers the dependence of the accuracy of the finite element
method upon selected model properties and upon the far-field extrapolation parameters.
4.1 Materials System of Interest
The materials system chosen for this set of studies is an isotropic sphere
representing rutile titania embedded in a polymeric resin typical of paints, illuminated
with 560-nm light (the center of the visible spectrum). The refractive indices of the
sphere and the resin are 2.74 and 1.51, respectively, with zero absorption coefficient.
These materials have been chosen for these studies because of the large refractive index
mismatch between these two materials (making it a challenging system to model
accurately) and the practical interest in this scattering system. The spherical particle
shape has been chosen because comparison with the analytical solution from Mie theory
can be used to determine the level of accuracy of the finite element computations. As
noted throughout the sections below, most of the computations have been performed
using a sphere diameter of 200 nm, which corresponds to the strongest resonance
between the particle and the incident light. The resonant nature of this particle makes
75
this system especially challenging to model accurately using a numerical method, since
steep gradients in electric field can be expected in the immediate vicinity of the particle.
This resonance is illustrated in both Figure 8, which shows the scattering coefficient S
as a function of sphere diameter, and Figure 9, which shows the scattering coefficient σ
as a function of sphere diameter. Values of these parameters for the case of the 200-nm
diameter sphere are marked with an “x” in each figure.
A finite element model of this 200-nm sphere is shown in Figure 10. The upper
left panel of this figure shows a three-dimensional view of the particle (with the
surrounding medium hidden), and the upper right panel shows a cross-sectional view of
the model. The extra space below the sphere in the model was provided because it was
suspected that interactions between the forward-scattered light and the lower model
wall could produce errors (Section 4.2.4). The lower two panels show the variation in
scattered light intensity with position through these same two views of the model. In
this computation, the 560-nm light propagates in the +x direction (from top to bottom in
Figure 10) and is polarized in the z direction. Fifteen wave cycles were allowed to
propagate through the model in this computation. Shown in Figure 11 is a plot of
electric field amplitude polarized in the z direction at the center of the spherical particle
as a function of time, demonstrating convergence to steady state. This model contains
1.34 million cubic elements, each 3.6 nm on edge. The incident intensity of light in this
model is unity. It is interesting to note that scattered intensities several times higher
than this incident intensity are observed in the immediate vicinity of the particle in the
lower two panels of the figure. This is typical of particles which are resonant with the
76
incident wavelength and larger particles. The distribution of scattered intensity in the
cross-sectional view of the particle in indicates that most of the light is scattered in the
forward direction; this is expected for particles which are not small compared to the
light wavelength (i.e., outside the Rayleigh regime). For comparison with the finite
element computation shown in Figure 10, an analogous result showing the same 200nm sphere size but in a model containing only 168,000 elements is shown in Figure 12
exhibits a more noticeably rough surface than that shown in Figure 10, resulting from
the coarser finite element mesh in the former case. In addition, comparison of the
distribution of scattered light in these two figures indicates that there are quantitative
differences in the maximum intensities in the near-field zone of the sphere.
Using the 200-nm sphere system described above as a basis, different series of
finite element computations are performed in which particular parameters are
systematically varied. These results are presented throughout this chapter. One set of
computations can often serve more than one purpose. In the case of the investigation of
the effect of finite element mesh density in the model, for example, the same set of
results provides data for memory and CPU time requirements as a function of the
number of volume elements in a model, in addition to the effect of finite element mesh
density upon error.
77
30
x
S ( µ m −1)
25
20
15
10
5
0
0
.2
.4
.6
.8
1
Sphere Diameter (Microns)
Figure 8. Dependence of the scattering coefficient S upon sphere diameter for the
case of a sphere with n = 2.74 embedded in a resin with n = 1.51. The illumination
wavelength is 560 nm.
x
12
σ ( µ m −1)
10
8
6
4
2
0
0
.2
.4
.6
.8
1
Sphere Diameter (Microns)
Figure 9. Dependence of the angle-weighted scattering coefficient σ upon sphere
diameter for the case of a sphere with n = 2.74 embedded in a resin with n = 1.51.
The illumination wavelength is 560 nm.
78
Figure 10. Finite element model of a 200-nm sphere with n = 2.74 embedded in a
polymeric resin with n = 1.51. The 560-nm illuminating light propagates in the +x
direction and is polarized in the z direction. This model contains 1,340,000 volume
elements.
79
Figure 11. Time history of z-polarized electric field amplitude at the center of the
sphere shown in the computation of Figure 10.
80
Figure 12. Finite element model of a 200-nm sphere with n = 2.74 embedded in a
polymeric resin with n = 1.51. The 560-nm illuminating light propagates in the +x
direction and is polarized in the z direction. This model contains 168,000 volume
elements.
81
4.2 Computational Efficiency
4.2.1 Memory Requirements For Finite Element Computations
In this series of computations, the number of volume elements used in the
construction of finite element models for the 200-nm titania sphere embedded in resin
has been systematically varied. For the coarsest finite element mesh densities studied
(having the fewest volume elements), the edge lengths of the finite element model have
been extended a greater distance from the surface of the sphere to ensure an adequate
number of elements between the sphere surface and the wall so as not to induce large
errors in the computation (Section 4.3.5). The dependence of the Cray C-94 memory
requirement in megawords (MW) upon the number of volume elements in each model
is shown in Figure 13. (On the Cray C-94, a word is 64 bits, or eight bytes.) The
memory requirement increases linearly with the number of volume elements in the
finite element model; this is an important advantage of the time-domain finite element
approach over many of the alternative numerical schemes described in Chapter 2 which
exhibit higher-order dependence of memory requirement on volume elements. The
memory requirement is an important property of an individual finite element
computation, as the maximum memory available for routine use on this Cray C-94 is 40
MW. This memory limit constrains the maximum number of elements available for a
computation to about 2,500,000, which in turn places an upper limitation upon the
volume of material which can be modeled. The actual upper limit for the volume of
material depends upon the radiation wavelength studied and the required level of
82
accuracy of the computation, since it is the relationship between element size and
radiation wavelength that determines both the physical volume of the model and the
expected level of error in the computation.
Memory (MW)
25
20
15
10
5
0
0.0
0.3
0.6
0.9
1.2
1.5
6
Elements (x10 )
Figure 13. Dependence of Cray C-94 memory requirement upon the number of
volume elements in the finite element model for the present series of EMFlex
computations.
4.2.2 CPU Time Requirements For Finite Element Computations
The dependence of Cray C-94 CPU time upon the number of volume elements
in a finite element model for the series of computations described in Section 4.1.1 is
shown in Figure 14. As in the case of the memory requirement (Figure 13), CPU time
shows a linear dependence upon the number of volume elements in the finite element
model. This linear dependence is likewise an important advantage of the time-domain
finite element approach over many of the alternative numerical approaches described in
Section 2.3.3, such as the method of moments which exhibits a (N2) dependence of CPU
83
time upon the number of elements. It should be noted that the same number of wave
cycles (15) were allowed to propagate in the finite element model in each of the
computations in Figure 14. For fixed finite element model geometry, the Cray C-94
CPU time depends linearly upon the number of cycles allowed to propagate in a given
computation. CPU times of 45-60 minutes are typical of individual finite element
computations performed throughout this research.
The vector nature of the Cray
architecture offers a factor of 3-10 speed-up over conventional scalar workstations with
the EMFlex code.
CPU Time (Min.)
80
60
40
20
0
0.0
0.3
0.6
0.9
1.2
1.5
6
Elements (x10 )
Figure 14. Dependence of Cray C-94 CPU time requirement upon the number of
volume elements in the finite element model for the present series of EMFlex
computations.
4.2.3 CPU Time Requirements For Far-Field Extrapolation
Once steady-state near-field quantities are computed using EMFlex in a finite
element model, these fields are extrapolated to far field using the Review software. As
84
described in Section 3.2.3, the surface of the far-field sphere onto which near-field
scattered intensities are extrapolated is divided into a number of surface elements
according to a spherical coordinate system. In the present series of computations, the
discretization in both angles theta and phi of the spherical coordinate system in the
extrapolation step has been systematically varied.
The angular discretizations
investigated are 0.5, 1, 2, 3, 5, 10, and 15 degrees. The radius of the far-field sphere is
500 µm in all cases; the radius of the far-field sphere has no effect upon CPU time
required for the extrapolation. The CPU time required on the SGI Power Challenge
computer for a single far-field extrapolation is shown in Figure 15 as a function of the
angular discretization on the surface of the far-field sphere. This data indicates that the
required CPU time increases dramatically with increasing angular on the surface of the
far-field sphere. Over six hours of CPU time are required for 0.5-degree angular
discretization, while only about thirty minutes are required for 2-degree angular
discretization.
The data showing CPU time dependence upon angular discretization in Figure
15 has been plotted as a function of the number of surface elements on the far-field
sphere in Figure 16; the number of surface elements is proportional to the inverse
square of the angular discretization. The plot of CPU time as a function of the number
of surface elements shown in Figure 16 indicates that the dependence of CPU time upon
the number of surface elements on the far-field sphere is of higher order than linear.
The selection of angular discretization on the far-field surface affects CPU time
strongly, and this selection must therefore be made with consideration of the effect of
85
angular discretization upon accuracy of the far-field result. This accuracy issue is
described in Section 4.3.2.
CPU Time (Min.)
600
500
400
300
200
100
0
0
2
4
6
8
10
12
14
16
Angular Discretization (Deg.)
Figure 15. CPU time required for a single far-field extrapolation is shown in as a
function of the angular discretization on the surface of the far-field sphere.
CPU Time (Min.)
600
500
400
300
200
100
0
0.0
0.3
0.6
0.9
1.2
1.5
5
Surface Elements (x10 )
Figure 16. CPU time required for a single far-field extrapolation is shown in as a
function of the number of surface elements on the far-field sphere.
86
4.3 Computational Accuracy
4.3.1 Effects of Finite Element Mesh Density
The 200-nm titania sphere in polymeric resin is used here to determine the
influence of finite element mesh density upon the accuracy of the time-domain finite
element method. The size of the cubic volume elements in these computations is
expected to exert a strong influence upon the accuracy of the results, since the ability to
represent the geometries of curved particles such as spheres and to compute gradients in
electric field are both controlled by the size of the rectangular solid volume elements.
The most useful way to parameterize the finite element mesh density is to compare the
edge length of cubic volume elements composing a finite element model to the
wavelength of the electromagnetic radiation propagating in the model.
Since the
effective wavelength of radiation propagating in a dielectric material is inversely
proportional to the refractive index of that material, it is the wavelength of a plane wave
traveling through the material with the highest refractive index nmax in the model that
effective should be considered. The minimum effective wavelength λmin in the model
is:
Equation 75
λmin =
λ0
n max
where λ0 is the wavelength of the illuminating radiation in vacuum. In the present
series of computations, the highest index material in the model is the scattering sphere
with n = 2.74. The minimum wavelength of the 560-nm light in these computations is
87
therefore λmin = 560 nm/2.74 = 204 nm. The number of cubic volume elements per
minimum wavelength in the model is given by the unitless quantity ρ:
Equation 76
ρ=
λ min
l edge
where ledge is the edge length of the cubic volume elements in the model. In the case of
the finite element models shown in Figure 10 and Figure 12, the values of ρ are 68 and
20, respectively.
The values of ρ investigated here are 10, 15, 20, 25, 30, 35, 45, 55, and 68. The
sphere diameter of 200 nm was used in each computation, and the edge length of the
finite element model were varied, when necessary, in order to maintain at least ten
elements between the sphere surface and the nearest model walls. In the extrapolation
to far field, the radius of the far-field sphere was 500 µm and the angular discretization
on the far-field sphere surface was 3 degrees in all cases. The scattering cross section
was computed for each case and compared with the exact result from Mie theory. The
resulting error is plotted in Figure 17 as a function of the finite element mesh density ρ.
As expected, the error increases with decreasing ρ, with an error less than 1% for ρ = 68
and an error greater than 8% for ρ = 10. Finite element mesh densities with ρ ≥ 20 are
used in all of the computations described in subsequent chapters of this dissertation, and
ρ ≥ 40 in most cases. These resolutions in the finite element mesh ensure that
maximum expected errors are limited to the values in Figure 17.
88
Evaluations of computational error in many systems the 200-nm titania sphere in
resin one chosen for the data in Figure 17 have confirmed that the values shown in this
figure represent a reasonable upper bound on computational error. Knowledge of these
error levels is critical, since it provides a basis for differentiating the real,
microstructural influences upon computed scattering properties from numerical error.
For example, if a computation on the effects of particle agglomeration with ρ = 40
exhibits a 30% change in scattering cross section relative to the primary particles, this
change can be attributed with confidence to the microstructural influence rather than to
numerical error.
10
Error (%)
8
6
4
2
0
0
10
20
30
40
50
60
70
ρ (Elements/λmin.)
Figure 17. Error in scattering cross section compared to the exact result from Mie
theory as a function of ρ, the number of volume elements in a model per minimum
wavelength of the 560-nm light propagating in the finite element models.
89
4.3.2 Effects of Mesh Density on Surface of Far Field Sphere
In Section 4.2.3, the effects of angular discretization on the surface of the farfield sphere upon the CPU time required for extrapolation of fields from near to far field
were described. Here, the errors in scattering cross section for that same series of
computations are compared to the exact result from Mie theory. These results are
plotted in Figure 18 as a function of angular discretization on the far-field sphere
surface. As expected, computational error increases with decreasing angular resolution.
The error increases non-linearly with angular discretization, with an error of ~0.7% for
all values less than or equal to 3 degrees. For values greater than 3 degrees, error
increases rapidly with increasing angular discretization. As shown in Figure 15, the
CPU time required for extrapolation decreases very rapidly with increasing angular
discretization on the surface of the far-field sphere; there is therefore a trade-off
between increased accuracy and increased CPU time in extrapolation of near-field
results to far field. This trade-off is well-balanced for the angular discretization 2-3
degrees, which is the range of values typically used in this research. Investigation of
error as a function of the illumination angle have confirmed that these values yield good
accuracy while requiring reasonable CPU times for the computation.
90
3.0
Error (%)
2.5
2.0
1.5
1.0
0.5
0.0
0
3
6
9
12
15
Angular Discretization (Deg.)
Figure 18. Error in scattering cross section compared to the exact result from Mie
theory as a function of as a function of angular discretization on the surface of the
far-field sphere surface.
4.3.3 Effects of Far-Field Sphere Radius
In order to determine the effects of the far-field sphere radius upon
computational error, the fields in the finite element model shown in Figure 10 have
been extrapolated to far-field sphere surfaces having radii of 2, 5, 10, 50, and 500 µm.
The angular discretization in the two spherical coordinate system angles was 3 degrees
in all cases. The resulting error compared to the exact solution known from Mie theory
is plotted as a function of far-field sphere radius in Figure 19. These data indicate that
the far-field sphere radius affects error only in the limit of very small extrapolation
radii. In the case of small extrapolation radii, evanescent components of the scattered
light still contribute to intensity on the surface of the extrapolation surface. This is an
important result, for it provides direct information about the length scale over which
91
near-field optical interactions in this system are important. For radii greater than or
equal to 10 µm, there is no dependence of error upon the extrapolation radius. In all
cases, the CPU time required for extrapolation was independent of the extrapolation
radius. Based upon the results shown in Figure 19, a far-field sphere radius of 500 µm
is used in all subsequent far-field extrapolations in this research.
1.0
Error (%)
0.9
0.8
0.7
0.6
0.5
0
1
2
3
Figure 19. Error in scattering cross section compared to the exact result from Mie
theory as a function of as a function of far-field sphere radius.
4.3.4 Effects of Kirchoff Box Placement in the Model
Using the model shown in Figure 10, the position of the Kirchoff box
surrounding the scattering sphere in this model has been systematically varied to
determine its effects upon computational error. The Kirchoff box is the mathematical
surface on which equivalent electric and magnetic currents are computed for subsequent
extrapolation of near-field results to far field (Section 3.2.1). The Kirchoff box surface
92
must be sufficiently displaced from the particle surface (where electric field gradients
are potentially greatest) and from the model walls (where electric field values may be
spuriously influenced by the boundary conditions).
The usual placement of the
Kirchoff box is just inside the walls of the model but with sufficient space between it
and the wall to prevent boundary condition effects from introducing errors into the
extrapolation.
Using the model shown in Figure 10, the Kirchoff box was placed precisely
half-way between each wall of the model and the surface of the sphere. The error in the
scattering cross section for this calculation (compared to the exact solution known from
Mie theory) was 0.76%. In subsequent calculations, the six sides of the Kirchoff box
were progressively moved closer to the surface of the spherical particle. With just two
elements separating the Kirchoff box from the sphere surface on all six sides, the error
in scattering cross section was 0.79%, indicating no deviation from the previously cited
result. With the Kirchoff box just one element away from the sphere surface, however,
the error in scattering cross section increased to 4.3%, suggesting that the presence of
strong electric field gradients immediately at the sphere surface caused significant error
in the extrapolation operation. The Kirchoff box was then moved progressively closer
to the walls of the model from the half-way point between the sphere and the model
walls. With the Kirchoff box positioned two elements away from the model walls on all
six sides, the error in scattering cross section was 0.75%, in good agreement with case
where the Kirchoff box was farther removed from the model wall. With the Kirchoff
box just one element away from the model wall, however, the error jumped to 186%, a
93
result of errors in electric field vectors and gradients produced by the boundary
conditions at the model walls.
The results described in this section indicate that the positioning of the Kirchoff
box can affect the accuracy of these finite element calculations.
No significant
problems are anticipated, however, provided that the sides of the Kirchoff box are
several elements removed from both the surfaces of the scattering particle and the
model walls.
4.3.5 Effects of Particle Position Relative to Model Walls
Illumination boundary conditions (Section 3.2.1), which are intended to absorb
only scattered radiation, are used at each of the six model walls of finite element models
throughout this research. Since scattered light can be spuriously reflected back into the
model at the walls due to the imperfection of the boundary condition, increased error
could be expected as the intensity of scattered radiation encountering the walls
increased. To investigate these effects, the 200-nm titania sphere shown in Figure 10
was displaced from the top edge of the model to the bottom edge over a series of four
computations. Figure 20 shows the steady-state scattered electric field intensities on
cross-sectional slices through the centers of these four models. The 560-nm light
propagates in the +x direction in each case. It is evident in Figure 20 that the forward
scattered light interacts more strongly with the bottom wall of the model as the sphere
moves closer to it, with the extreme case appearing in the bottom right panel of the
94
figure. The distances between the bottom surface of the sphere and the bottom of the
model in these four models are listed in Table 1.
Table 1 summarizes the results of extrapolating the near-field scattering results
shown in Figure 20 to far field. The results of this extrapolation have been compared to
the exact scattering cross section known from Mie theory; the errors associated with
those computations are presented in Table 1. Interestingly, the error in scattering cross
section decreases systematically as the scattering sphere is moved closer to the bottom
wall of the model (or, alternatively, away from the top of the model). This observation
was surprising; it was expected that increased interaction of forward-scattered light with
the bottom wall of the model would cause increased error. The results of Table 1
provide no evidence that such a problem arose in these computations. An important
result the data in Table 1 is that there is no evidence that the interaction between the
forward scattered light and the model wall has caused the computation to fall into
severe error. This result is encouraging, since it suggests that it is not necessary to
provide large volumes of surrounding medium in the forward-scattering direction,
allowing the implementation of models containing fewer volume elements and therefore
requiring shorter CPU times for computation. It is possible that the increased error
associated with longer distances between the sphere and the bottom wall of the model is
a result of resonance between forward-scattered light and the bottom model wall. These
effects could also occur in the back-scattered direction (i.e., between the top of the
sphere and the top model wall in Figure 20), but they would be substantially weaker
since significantly less light is back-scattered than forward-scattered.
95
Figure 20. Cross-sectional slices through the centers of four finite element models
in which the spacing between the 200-nm sphere and the bottom wall of the model
has been systematically varied. Scattered light intensity is shown in each case.
96
Table 1. Error in scattering cross section as a function of
separation distance between the bottom of the sphere and the
bottom wall of the model. The corresponding models are shown in
Figure 20.
Sphere-Bottom Wall Distance (nm)
Error (%)
185
1.9
150
0.74
85
0.21
20
0.02
4.3.6 Effects of Model Dimensions Compared to Light Wavelength
In addition to Kirchoff box and model wall placement, the edge length of a finite
element model compared to the radiation wavelength can have significant impact upon
computational error. Figure 21 shows a finite element model of a 40-nm titania sphere
with refractive index n = 2.74 embedded in the polymeric resin with refractive index n =
1.51. As in all previous computations in this chapter, the wavelength of the incident
light is 560 nm. This represents a non-resonant case in the Rayleigh scattering regime.
Referring to the plot of the scattering coefficient S as a function of sphere diameter
shown in Figure 8, the 40-nm sphere diameter is very weakly scattering compared to the
200-nm sphere diameter. The edge length of the model shown in Figure 21 is 400 nm,
and light propagates in the +x direction. The two panels of Figure 21 showing nearfield scattered intensity indicate that the particle scatters light nearly isotropically rather
than in the forward direction, which is expected in the Rayleigh regime. To investigate
97
the effects of model size upon accuracy of these finite element computations, the edge
length of the cubic model in this series of computations was systematically varied
between 100 and 400 nm, without changing the edge length of the cubic volume
elements composing the model. The scattering sphere was located at the center of the
model in all cases.
The columns of Table 2 show the edge length of the model, the optical length of
the model, and the percent error in scattering cross section versus the known result from
Mie theory for each case. The optical length of the model, which is defined here as the
number of wavelengths of light which would exist along the linear path from the top to
the bottom of the model which passes through the center of the sphere. Since the
refractive indices of the sphere and the surrounding medium are both greater than unity,
the wavelength of radiation propagating in these materials is compressed. The optical
length is the sum of the individual propagation distances through the different media
present in the model multiplied by their respective refractive indices, divided by the free
wavelength of the incident light:
Equation 77
lopt . = (l1n1 + l2 n2 + ...) / λ0
For example, in the case of the finite element model in Figure 21 with 400-nm edge
length, a line passing from the top of the model to the bottom and passing through the
center of the 40-nm sphere encounters 40 nm of material with n = 2.74 (the sphere) and
360 nm with n = 1.51 (the surrounding medium). The optical length is therefore
computed as:
98
Equation 78
lopt . = [(40nm ⋅ 2.74 + 360nm ⋅ 1.514 ) / 560nm] = 1.16 .
As Table 2 indicates, the error in scattering cross section is strongly dependent
upon the edge length of the model. For models smaller than 400 nm on edge, error
increases abruptly from roughly 1% to the range 17-41%. These effects are attributed
to the size of the model relative to the light wavelength. In the case of models which
are so small compared to the light wavelength that the optical length is less than unity,
the problem of not developing a steady-state, propagating wave in the model arises.
Instead of at least one full wavelength of light propagating through the model, electric
field amplitudes in the model can fluctuate unstably in response to the fluctuating
electric field of the incident radiation, to which most of the model is at once subject.
The same analysis shown in Table 2 has also been applied to the problem of the
more resonant 200-nm sphere described in the preceding section. In this series of
experiments, the 200-nm sphere with n = 2.74 has been placed at the center of a cubic
model with a surrounding medium with n = 1.51 and the edge length of the model
systematically decreased until it was in close proximity to the surface of the particle.
The results of this series of experiments, shown in Table 3, support the idea that models
must be sufficiently large to ensure that the optical length of the model is greater than
unity in order to avoid large errors in scattering cross section in these finite element
computations. Comparing results from Table 2 and Table 3, it appears that an optical
length of 1.2 is sufficient to ensure minimal computational errors for the materials
employed in these two series of computations.
99
Figure 21. Finite element model of a 40-nm sphere with n = 2.74 embedded in a
polymeric resin with n = 1.51. The 560-nm illuminating light propagates in the +x
direction and is polarized in the z direction.
100
Table 2. Error in scattering cross section of finite element
calculations in which the model edge length was systematically
varied. All calculations are for the case of a 40-nm sphere at the
center of a cubic model.
Model Edge Length (nm)
Optical Length
% Error
100
0.4
41
200
0.6
26
300
0.9
17
350
1.0
30
400
1.2
0.9
Table 3. Error in scattering cross section of finite element
calculations in which the model edge length was systematically
varied. All calculations are for the case of a 200-nm sphere at the
center of a cubic model.
Model Edge Length (nm)
Optical Length
% Error
234
1.07
4.7
254
1.12
1.9
272
1.17
0.94
300
1.25
0.97
101
4.3.7 Effects of Strong Resonance: High-Index Spheres in Vacuum
As the mismatch between the refractive index of a sphere and the medium in
which it is embedded is increased, the maximum scattering efficiency which is
attainable increases. This is illustrated in the following example calculated using Mie
theory. Figure 8 shows a plot of scattering coefficient S (µm−1) versus diameter for the
case of a sphere with refractive index n = 2.74 embedded in a medium with n = 1.51.
The maximum value observed, representing the most resonant condition, is S ≈ 26 µm−1
for the sphere diameter 200 nm. Figure 22, in comparison, shows a plot of S versus
diameter for the case of a sphere with n = 3.0 embedded in a medium with n = 1.0 (e.g.,
air or vacuum). The maximum value observed in this case is S ≈ 68 µm−1 for the sphere
diameter 177 nm, a substantially greater value than the previous case considered. This
higher value implies much greater scattered intensities and electric fields gradients at
near field.
These make this highly resonant system more susceptible to errors
associated with the mesh density and its ability to describe sharp gradients accurately.
To quantify the computational error of the time-domain finite element approach
in highly resonant cases, the three sphere diameters marked with “x” in Figure 22 have
been investigated. The point corresponding to the first resonant peak in this figure
(diameter 0.177 µm) is the most strongly resonant condition and therefore the most
difficult to model accurately using the finite element approach. The other two points,
corresponding to sphere diameters 0.16 and 0.22 µm, are not as strongly resonant and
lower numerical errors are expected in these cases.
102
The results of these calculations are summarized in Table 4.
The errors
associated with the calculations for the less resonant cases (sphere diameters 0.16 and
0.22 µm) are both less than 1%, indicating very close agreement with the known
solutions from Mie theory. The error associated with the strongly resonant case (sphere
diameter 0.177 µm), on the other hand, is 6.4%, deviating substantially from the exact
solution. This series of calculations demonstrates that error in the time-domain finite
element approach depends upon the degree of resonance in a specific calculation as well
as the factors identified above. It should be noted that the extreme contrast ratio of m =
3.0 between the sphere and the surrounding medium in the case studied here exceeds
any which is encountered in practice; there are no known, non-lossy dielectric materials
with refractive index as high as 3.0. The calculations described in this section are
therefore non-physical but nevertheless highlight the effects of resonance upon
computational accuracy of the finite element method.
103
70
x
S ( µ m −1)
60
50
40
x
30
x
20
10
0
0
.05
.1
.15
.2
.25
.3
.35
Sphere Diameter (Microns)
Figure 22. The scattering coefficient S (µm−1) as a function of sphere diameter for
the case of a sphere with n = 3.0 in a medium with n = 1.0. This represents a more
resonant system than can be achieved in practice, since no known, non-lossy
material exhibits such a high refractive index in the visible spectrum.
Table 4. Error in scattering cross section as a function of
sphere diameter for the three cases indicated on the plot
in Figure 22.
Sphere Diameter (nm)
Error (%)
160
0.64
177
6.4
220
0.85
104
4.4 Guidelines For Building Robust Finite Element Models
The observations of the present chapter may be used to formulate a set of simple
guidelines for the construction of robust finite element models for the EMFlex software.
These guidelines are designed to minimize the error of the finite element. They are:
1. The optical length (Equation 77)of the finite element model edges should be
at least 1.2 wavelengths.
2. The finite element mesh should be as fine as memory limitations allow.
3. The scattering microstructure should be positioned at the center of the
model.
4. The scattering microstructure should be at least eight elements from any
model wall.
5. The Kirchoff surface should be placed at least three elements from any
model wall and the scattering microstructure.
6. The surface of the far-field extrapolation surface should be placed a very
large optical distance from the finite element model.
7. The angular mesh on the surface of the far-field extrapolation surface should
be as fine as possible to ensure minimal error; CPU time restrictions are the
limitation here.
105
Chapter 5. A Potpourri of Pigment Particles
In this chapter, the light scattering properties of three different white pigment
systems are determined using computational modeling. In Section 5.1, Mie theory
computations are performed for the case of core-shell composite pigment architectures
for the case of coated spheres. In this study, the average index approximation for rutile
titania is used; titania is modeled as an optically isotropic material. This assumption is
justified, since the most dramatic changes in scattering behavior arise from the presence
of the second material in the coated particle. In Section 5.2, the time-domain finite
element approach is used to determine the light scattering behavior of optically
anisotropic spheres of rutile titania. These results are compared to the average index
and weighted sum approximations to determine their ability to describe the scattering
efficiency of rutile titania. Finally, in Section 5.3, the light scattering properties of a
morphological rutile titania particle are computed using the time-domain finite element
method. These results are compared to the scattering performance of the equivalentvolume sphere under both the average index and weighted sum approximations. In
addition, the scattering data are used to calculate equivalent spherical diameters in order
to highlight the difficulties in particle size analysis which can arise in modeling
complex particle shapes as simple spheres.
106
5.1 Core-Shell Pigment Particles
5.1.1 Introduction
One of the largest market segments for titania white pigment is the paper
industry. Since titania is more expensive than pulp and the other mineral pigments
which are used in paper formulations, such as kaolin clay and calcium carbonate,
formulators of paper systems attempt to use titania as efficiently as possible. In paper
applications, it is normally most cost-effective to optimize the content of the non-titania
components and then add titania as required to achieve target opacity. Formulation
strategies for optimization of both paper furnish and coating formulations have been
proposed by Houle1 and Kwoka.2,3,4
Recently, it has been proposed that the scattering efficiency of white pigment
can be improved by coating spherical silica particles with a titania shell.22,23,24 The
lower titania content of such particles suggests that they could potentially provide better
value versus pure titania pigments. To address this issue, the light scattering properties
of model core-shell pigments have been determined using a computational approach.
Results are presented below for the general case of a titania shell over cores with
refractive index varying from 1.0 (hollow titania spheres) up to 2.74 (solid titania
spheres). This is followed by a comparison of the scattering performance of titaniacoated silica versus pure titanium dioxide.
107
5.1.2 Scattering Properties of Titania-Coated Core-Shell Pigments
A schematic diagram of the configuration for the titania-coated core-shell
particles is shown in Figure 23. All critical parameters are indicated. The core is a
sphere of diameter DC.
The titanium dioxide coating forms a shell with outside
diameter DT. The reported calculations include particles with shell diameters up to 1
µm and consider the full range of 100% titania (i.e., DC = 0) to 100% core material
(i.e., DC = DT). In the case of a silica core, for example, it is not practical to deposit
more than 40 weight percent titania on the core. Since the volume percent titania is
related to the cube of DC/DT and since titania is more dense than silica, this composition
corresponds to a thin shell around the silica core, with a shell diameter only 10% larger
than the core. For clarity, the schematic diagram in Figure 23 illustrates a particle with
a shell diameter about 20% larger than the core diameter.
This represents an
impractical case with a titania content of 64 wt %. For practical cases, the thickness of
the titania layer would be much thinner than illustrated.
108
Surrounding Medium
Cellulose or
Paper Coating
{
Core
}
n m = 1.45
Core Diameter
DC
n C : Air
1.0
SiO2 1.46
TiO2 Shell
n T = 2.74
DT
TiO2 Shell Diameter
Figure 23. Schematic representation of titania coated on a core particle of lower
refractive index.
The other key parameters in Figure 23 are the refractive indices of the
component materials. The value of 2.74 for titania is the average refractive index at the
center of the visible spectrum. The refractive index of the core is an independent
variable. A core refractive index of 1.0 (air) represents a hollow titania shell, while a
core index of 1.46 represents titania-coated silica. A core with refractive index equal to
that of rutile titania represents a solid rutile particle. The refractive index for the
surrounding medium in a paper system is more difficult to assign. The value of 1.45
used in these calculations is the average refractive index of a representative paper
coating composed of a mixture of clays and calcium carbonate (refractive index 1.6),
starch or latex binders (refractive index 1.5), and air voids (refractive index 1.0).
Uncoated papers are composed of cellulose fiber (refractive index 1.45), clays, calcium
109
carbonate, and air voids. A refractive index lower than 1.45 should be used to represent
the surroundings for titania in filled sheet applications.
5.1.2.1 Refractive index of medium at 1.45
A plot of the angle-weighted scattering coefficient σ for the general case of
rutile titania-coated cores in a paper coating or cellulose matrix is shown in Figure 24.
The results were calculated for the light wavelength 560 nm, the center of the visible
spectrum. This is the wavelength where the human eye is most sensitive and close to
the center of the wavelength bands used for measurement of paper opacity. Absolute
values of scattering coefficients are dependent on wavelength, but results calculated at
560 nm provide a sound representation of the light scattering properties that affect
opacity.
For the particles in Figure 24, the ratio of the core diameter to total diameter,
DC/DT, is fixed at 0.9. The three-dimensional plot shows the scattering efficiency for
particles with outer diameter, DT, from 0.0 to 1 µm. Each contour line represents the
dependence of σ on diameter for a given core refractive index. The refractive index of
the core material is incremented by 0.035 with each contour line and ranges from 1.0
(air) to 2.74 (rutile).
The global maximum of this plot, indicating the optimally
scattering particle structure, corresponds to a solid rutile particle (core and shell
refractive index both equal to 2.74) with of diameter 0.2 µm.
In the case of titania-coated silica, the core index is 1.46. The data in Figure 25
are plotted as a function of the diameters of the silica core and the titania shell
110
diameters, both of which vary in the range 0.0−1.0 µm. Each contour line represents a
0.002 µm increment of outer shell diameter. As in Figure 24, the global maximum
corresponds to a solid rutile sphere (i.e., silica core diameter equal to zero). Variation
of the core and shell diameters results in the undulating series of local minima and
maxima in Figure 25. The greatest local maximum corresponding to a sphere having
substantial silica content occurs at a titania shell diameter of about 0.35 µm, with a
silica core diameter of about 0.2 µm. The σ value associated with this coated silica
particle is only about 64% that of the optimized solid rutile sphere.
111
12
σ ( µ m -1)
10
8
6
4
0
.2
.4
.6
.8
1
1
nd
ex
eI
0
2
1.5
Co
r
2
3
2.5
Sphere Diameter (Microns)
Figure 24. Scattering coefficient σ (µm−1) versus coated particle diameter and
refractive index of the core material for titania-coated particles with coating
thickness equal to 10% of the particle radius.
0
.2
.4
.6
.8
Silica Core Diameter (Microns)
A r b it r a r y Y / A r b it r a r y X
1
on
s)
Ti
O
.2
0
2 S
he
.4
ll
Di
a
.6
er
.8
(M
5
i cr
1
m
et
σ ( µ m -1)
10
Figure 25. Scattering coefficient σ (µm−1) versus silica core diameter and titania
shell diameter for a coated particle. The global maximum of this plot corresponds
to a solid titania sphere of 0.2 µm diameter.
112
For greater clarity, five contours have been extracted from the three-dimensional
plot in Figure 25 and shown in the two-dimensional plot of Figure 26. This figure
shows σ as a function of silica core diameter for spheres having five different titania
shell diameters. The sphere with titania shell diameter 0.20 µm is that which exhibits
the highest σ value associated with solid titania (silica core diameter equal to zero).
There is an obvious decrease in σ with increasing silica core diameter. Many of the
titania-coated silica particles nearly match the scattering efficiency of the pure titania
particles with the same outside diameter, but each is deficient relative to pure titania at
its optimum diameter. For example, in the case of particles with titania shell diameter
0.5 µm, the scattering coefficient is nearly constant as the silica core is increased from 0
to 0.4 µm. The scattering coefficient of this set of particles is 70% below that of the
optimized solid titania particle. A sphere with titania shell diameter 0.35 µm is the
particle which exhibits the highest σ value among those spheres containing a substantial
amount of silica (silica core diameter 0.2 µm). This core-shell particle has DC/DT =
0.2/0.35 = 0.57 and a titania content of 89 weight percent, which means that the titania
coating is impractically thick from a synthesis standpoint. All other silica-containing
particles exhibit even lower σ values than those having titania shell diameters of 0.35
and 0.5 µm.
As noted previously, physically realizable core-shell architectures are limited to
those with a titania shell thickness that is less than or equal to 10% of the outer particle
diameter (DC/DT = 0.9) This limiting case has been plotted in Figure 27, which shows σ
113
as a function of sphere diameter both for solid titania and for coated particles with
DC/DT = 0.9. The results for the two structures are strikingly different; the maximum σ
value achieved by a solid titania sphere is 10.5 µm−1, while the maximum value
achieved by the silica-coated particle is only about 1.7 µm−1. The two curves are
approximately equal for particle diameters greater than ~0.7 µm. The resonant peaks
observed in Figure 27 for the pure rutile particles are an artifact the high spatial
symmetry of a sphere.
In practice, these surface wave resonances are not
experimentally observed even for particles with narrow size distributions.5
Titania shell diameter 0.2 um
10
σ ( µm −1)
8
0.35 um
6
4
0.5 um
0.75 um
2
1.0 um
0
0
r b it r a r y Y / A r b it r a r y X
.2
.4
.6
.8
1
Silica Core Diameter (Microns)
Figure 26. Scattering coefficient σ for five coated particles with different outside
diameters.
114
10
Pure Rutile
σ ( µ m-1)
8
6
4
2
0
10% Rutile on Silica
0
.2
it r a r y Y / A r b it r a r y X
.4
.6
.8
1
Particle Diameter (Microns)
Figure 27. Scattering coefficient σ versus particle diameter for a solid titania
sphere and a titania-coated silica particle with coating thickness equal to 10% of
The Mie theory computations presented here describe the light scattering
properties of an isolated spherical particle and cannot be applied to systems in which the
crowding effect is important. However, crowding can be viewed as agglomeration of
individual particles into clusters. If the scattering properties of these clusters could be
reasonably approximately by computing the scattering properties of equivalent-volume
spheres, the data in Figure 27 at higher diameters suggest that optimized rutile particles
will show more loss due to crowding than particles of titania-coated silica. This effect
will not apply for most paper applications where titania concentrations are below ~15%
by volume.
The incentive for depositing titania coatings on a low density and inexpensive
core material like silica is to attempt to increase the light scattering efficiency per pound
of pigment or to reduce the price of the pigment. The scattering coefficients in Figure
115
24–Figure 27 can be divided by the density of the coated particle, thereby expressing σ
on a weight basis (m2/g). Using specific gravities of 2.2 g/ml for silica and 4.26 g/ml
for rutile titania, the data in Figure 25 were normalized by dividing by the density of the
particle; these results are shown in Figure 28. While this normalization scheme has the
effect of slightly increasing the effective σ values of those particles containing silica,
the global maximum in Figure 28 nevertheless corresponds to a solid titania sphere,
with no silica present, as in Figure 24–Figure 27.
It is also of interest to normalize the light scattering data in Figure 25 by
dividing the σ values by the weight fraction of titania present in the particle. This
normalization scheme reveals the relative scattering efficiency for the limiting case
where the value of the titania-coated silica pigment is assessed only by its titania
content. The results of this normalization scheme are shown in Figure 29. This data
does not include the diverging σ values which occur for those cases where the titania
coating thickness approaches zero. Even with this normalization scheme, the global
maximum in Figure 29 corresponds to a solid titania sphere, with no silica present.
116
2
.4
.2
0
.2
r b it r a r y Y /Silica
A r b it r a
ry X
Core
.4
.6
Diameter (Microns)
.8
i cr
Ti
O
0
(M
ll
Di
am
et
.6
er
.8
1
on
s)
1
2 Sh
e
σ (Density-Normalized)
3
1
.2
0
0
.2
A r b it r a r y Y /Silica
A r b it r a
ry X
Core
.4
.6
Diameter (Microns)
.8
1
m
et
2 S
he
.4
ll
Di
a
.6
er
.8
(M
5
i cr
1
on
s)
10
Ti
O
σ (Normalized by TiO2 Wt. Frac.)
Figure 28. Scattering coefficient σ divided by the density of the coated particle.
The global maximum of this plot corresponds to a solid titania sphere with no
silica present.
Figure 29. Scattering coefficient σ divided by the weight fraction of titania present
in each coated particle. The global maximum of this plot corresponds to a solid
titania sphere with no silica present.
117
Finally, the light scattering performance of the silica-titania core-shell pigments
can be compared to a heterogeneous mixture of the two pure pigments to determine
whether any synergy results for the coated particles.
For the titania-coated silica
particles in Figure 27, DC/DT is 0.9 which corresponds to 27.1 volume percent titania.
The highest σ value of 1.70 µm−1 occurs for the titania-coated silica at the diameter
0.687 µm. For pure rutile, the optimal σ value of 10.574 µm−1 occurs at the diameter
0.20 µm. For pure silica, the optimal σ value for pure silica is very low because the
refractive index is close to that of the medium; the optimum occurs at the diameter 0.20
µm but is only σ = 0.0005 µm−1. For a heterogeneous mixture of optimized rutile and
silica spheres, the scattering coefficient σmix is calculated using volume fraction, φ of
each component:
Equation 79
σ mix = φ TiO 2σ TiO 2 + φ SiO 2σ SiO 2
For the core-shell particle containing 27.1 volume % titania, φTiO2 is 0.271 and φSiO2 is
0.729. Therefore, σmix for the mixture of titania and silica spheres computed from
Equation 79 is 2.866 µm−1, which is 1.7 times greater than the scattering coefficient for
the same volumes of titania and silica in the optimized core-shell structure. Similar
evaluations of other combinations of DC and DT for silica and titania also reveal no
evidence of synergy in the scattering properties of the core-shell structures.
5.1.2.2 Refractive index of medium at 1.0 (air)
White pigment particles in filled paper applications are not surrounded only by
cellulose (refractive index 1.45) but instead by both cellulose and air. To address this
118
point, these calculations have been modified by changing the index of refraction of the
surrounding medium from 1.45 to 1.0, that of air. With a surrounding matrix of air, the
difference in σ between the optimized titania sphere and titania-coated silica spheres is
even more pronounced. Figure 30 shows a plot of σ versus silica core diameter for the
five core-shell particles shown in Figure 26. The σ value for a solid titania sphere with
diameter 0.20 µm in Figure 30 is σ ≈ 28 µm−1, while the value for a titania-coated silica
sphere with outer diameter 0.35 µm and silica core diameter 0.3 µm is σ ≈ 10 µm−1.
This difference is much more pronounced than the values observed in Figure 26. Since
pure rutile particles outperform titania-coated silica particles in both a cellulose matrix
and an air matrix, it can be concluded that rutile particles are also more efficient in
porous paper substrates, in which the medium surrounding the particles is a mixture of
cellulose and air.
119
30
Titania shell diameter 0.2 um
σ ( µm−1)
20
0.35 um
10
0.5 um
0.75 um
1.0 um
0
0
it r a r y Y / A r b it r a r y X
.2
.4
.6
.8
1
Silica Core Diameter (Microns)
Figure 30. Scattering coefficient σ for the five particles shown in Figure 26, with
the cellulose surrounding medium replaced by air (refractive index 1.0 instead of
1.45).
5.2 Anisotropic Spheres of Rutile Titania
5.2.1 Introduction
Attempts to use Mie theory to predict the light scattering properties of a rutile
titania particle require approximation to address the optical anisotropy of the crystal.
Rutile titania is uniaxial, exhibiting its ordinary refractive index for light polarized
parallel to either the a or b axis of the crystal and its extraordinary refractive index for
light polarized parallel to the c axis. In this section, the light scattering properties of
optically anisotropic spheres of rutile titania have been computed directly for the first
time using the time-domain finite element approach.
120
As described in Section 2.2.1, two methods for approximating rutile as an
isotropic material have been proposed in the literature. One approach has been to
compute the mean of the refractive indices along the three crystal axes at the light
wavelength of interest18 and subsequently apply Mie theory to compute far-field
scattering parameters.
This is the average index approximation.
For the light
wavelength 560 nm, the average refractive index of rutile titania is 2.74. In these
computations, the refractive index of the surrounding medium is 1.51, representing a
typical paint resin. The results of Mie theory computations for the scattering coefficient
S using the average index approximation are shown in Figure 31 as a plot of S versus
sphere diameter. The greatest S values occur for sphere diameters in the range 0.180.30 µm, with a maximum value of S = 27.74 µm−1 for the diameter 0.272 µm.
Likewise, the results of Mie theory computations for the angle-weighted scattering
coefficient σ using the average index approximation are shown in Figure 32. The
maximum value of the angle-weighted scattering coefficient σ = 12.17 µm−1 occurs for
the sphere diameter 0.20 µm.
A second method for approximating the scattering behavior of rutile titania
using isotropic materials has been to compute separately far-field scattering parameters
for an isotropic sphere having the ordinary and extraordinary refractive indices of rutile
titania, and subsequently compute the weighted sum of these separate far-field
results.17,18
The weighted sum consists of two-thirds the result for the ordinary
refractive index plus one-third the result for the extraordinary refractive index. This is
the weighted sum approximation. The results of Mie theory computations for the
121
scattering coefficient S using the weighted sum approximation are shown in Figure 33
with S plotted as a function of sphere diameter. This result exhibits less pronounced
resonant peaks in S than the analogous plot generated using the average index
approximation shown in Figure 31. This is expected, since the resonant peaks in the
two individual curves used to form the weighted sum are systematically offset from one
another. The maximum S values for the weighted sum approximation occur in the
range 0.18−0.3 µm, as with the average index approximation. The maximum value S =
25.08 µm−1 occurs for the diameter 0.270 µm, about 10% lower than the maximum
value S = 27.74 µm−1 observed in the case of the average index approximation. The
angle-weighted scattering coefficient σ using the weighted sum approximation is shown
in Figure 34. This curve is in close agreement with that computed using the average
index approximation, shown in Figure 32.
In the case of the weighted sum
approximation, the maximum value of the angle-weighted scattering coefficient σ =
11.71 µm−1 occurs for the sphere diameter 0.204 µm, about 4% lower than the
maximum value of the angle-weighted scattering coefficient σ=12.17 µm−1 observed in
the case of the average index approximation.
The average index and weighted sum approximation results shown in Figure
31−Figure 34 will be used as bases for comparison with results for anisotropic spheres
of rutile titania computed using the time-domain finite element method.
Results
obtained using the finite element method contain the effects of the optical activity of the
uniaxial crystal, unlike the results of the average index or weighted sum approximations
based upon scattering by isotropic spheres.
122
30
S ( µ m −1)
25
20
15
10
5
0
0
.2
.4
.6
.8
1
Sphere Diameter (Microns)
Figure 31. Scattering coefficient S versus sphere diameter for an optically
isotropic sphere from Mie theory, using the average index approximation.
12
σ ( µm −1)
10
8
6
4
2
0
0
.2
.4
.6
.8
1
Sphere Diameter (Microns)
Figure 32. Angle-weighted scattering coefficient σ versus sphere diameter for an
optically isotropic sphere from Mie theory, using the average index approximation.
123
30
S ( µ m −1)
25
20
15
10
5
0
0
.2
.4
.6
.8
1
Sphere Diameter (Microns)
Figure 33. Scattering coefficient S versus sphere diameter for an optically
isotropic sphere from Mie theory, using the weighted sum approximation.
12
σ ( µm −1)
10
8
6
4
2
0
0
.2
.4
.6
.8
1
Sphere Diameter (Microns)
Figure 34. Angle-weighted scattering coefficient σ versus sphere diameter for an
optically isotropic sphere from Mie theory, using the weighted sum approximation.
124
5.2.2 Scattering By Anisotropic Spheres of Rutile Titania
The light scattering properties of anisotropic, high-index spheres assigned the
optical constants of rutile titania with diameters in the range 50−350 nm have been
computed using the finite element method.
The light wavelength used in these
computations is 560 nm. At this wavelength, the ordinary refractive index of rutile
titania is 2.64, while the extraordinary index is 2.94.13 The spatial symmetry elements
of an anisotropic, uniaxial sphere include an equatorial mirror plane and a cylindrical
rotation axis perpendicular to this mirror plane. (In this description, the optic axis is
parallel to the cylindrical rotation axis.) The complete light scattering behavior of such
a system can therefore be determined through a series of illuminations along any arc
connecting one pole of the sphere to the equator. Six different illumination directions
are used for each sphere, at the equally spaced angles 7.5°, 22.5°, 37.5°, 52.5°, 67.5°,
and 82.5° relative to the optic axis. Two mutually perpendicular polarizations (parallel
and perpendicular to the optic axis of the sphere) are used for each illumination
direction, resulting in a total of 12 individual illuminations per sphere. The far-field
scattering parameters S and σ are computed for each sphere by computing the averages
of these two quantities, weighting the individual results by the probabilities of their
occurrence under conditions of random illumination (Section 3.2.4). As a result of the
cylindrical rotation axis parallel to the optic axis of these spheres, illumination
directions nearly perpendicular to the optic axis are significantly more probable than
those nearly parallel to the optic axis.
125
The finite element models for each of the spheres studied consist of a spherical
particle at the center of a cubic finite element model. In each case, the sphere is
surrounded by an optically isotropic medium with refractive index n = 1.51,
representative of an acrylic, alkyd paint resin. The finite element mesh density differs
from model to model but is typically ρ = 50-60 elements per wavelength in the highest
index material in the model. In the case of the 0.2-µm sphere, for example, the cubic
element edge length is 3.87 nm. Accuracy analysis of the time-domain finite element
approach (Chapter 4) suggests that the numerical error associated with each of the
calculations in this study should be less than 2%.
The full near-field results of a finite element computation performed on the
anisotropic titania sphere with diameter 0.2 µm are shown in Figure 35. The upper left
panel of the figure shows a three dimensional view of the sphere (with the surrounding
medium removed for clarity), and the upper right panel shows a cross-sectional slice
through the center of the finite element model. The lower two panels show steady-state
scattered intensities in the views shown in the two panels above. In this computation,
the incident light propagates in the +x direction, and its polarization is parallel to the y
direction. The incident intensity of the light is equal to unity. The scattered intensities
observed in the model are as much as 4.3 times greater than the incident intensity.
Using the methodology described above, the far-field parameters S and σ have
been computed for a series of anisotropic spheres of rutile titania with diameters in the
range 0.05-0.35 µm. Results for the scattering coefficient S are summarized in Table 5
and plotted versus sphere diameter in Figure 36 (solid circles). These data exhibit a
126
sharp increase in S as the sphere diameter is increased through the Rayleigh scattering
regime from 0.05-0.2 µm, beyond which resonant peaks are observed with increasing
diameter.
Results for the angle-weighted scattering coefficient σ are likewise
summarized in Table 5 and plotted versus sphere diameter in Figure 37 (solid circles).
The σ data exhibit the same sharp increase the sphere diameter increases from 0.05-0.2.
For sphere diameters greater than 0.2 µm, the σ data in Figure 37 decrease without the
same pronounced resonant peaks observed in the S data in Figure 36.
127
Figure 35. Finite element model and scattered intensities for a 0.2-µm anisotropic
sphere of rutile titania. The 560-nm radiation propagates in the +x direction, with
unit incident intensity. Polarization is parallel to the y direction.
128
30
-1
S (µm )
25
20
15
10
5
0
0.0
0.1
0.2
0.3
0.4
Sphere Diameter (µm)
Figure 36. The scattering coefficient S as a function of sphere diameter computed
using the finite element method (solid points) for anisotropic spheres of rutile
titania with n =2.74 embedded in a medium with n = 1.51. The illumination
wavelength is 560 nm. The long dashed curve corresponds to the average index
approximation, while the dotted curve corresponds to the weighted sum
approximation.
14
12
-1
σ (µm )
10
8
6
4
2
0
0.0
0.1
0.2
0.3
0.4
Sphere Diameter (µm)
Figure 37. The angle-weighted scattering coefficient σ as a function of sphere
diameter computed using the finite element method (solid points) for anisotropic
spheres of rutile titania embedded in a medium with n = 1.51. The illumination
wavelength is 560 nm. The long dashed curve corresponds to the average index
approximation, while the dotted curve corresponds to the weighted sum
approximation.
129
Table 5. Scattering coefficients S and σ computed for anisotropic
spheres of different diameter using the finite element approach.
Diameter (µm)
S (µm−1)
σ (µm−1)
0.050
0.587
0.564
0.100
4.52
3.79
0.150
12.50
7.03
0.175
20.06
8.82
0.187
23.94
11.73
0.200
25.62
12.34
0.212
22.27
11.30
0.225
21.01
10.37
0.250
22.17
9.41
0.275
25.83
9.68
0.300
22.10
9.07
0.325
18.89
9.43
0.350
18.91
8.44
130
5.2.3
Angle and Polarization Dependence of Scattering by an Anisotropic
Titania Sphere
An essential difference between radiation scattering by an optically anisotropic
sphere versus an optically isotropic sphere is the dependence of the far-field scattering
properties upon illumination direction and polarization. This orientation dependence
results from the optical activity of the optically anisotropic sphere. In addition, an
optically anisotropic sphere exhibits polarization dependence in its far-field scattering
parameters for a given illumination direction, with the exception of the special case
where the incident radiation propagates parallel to the optic axis of the sphere. These
effects have been observed in the scattering properties of each of the anisotropic spheres
studied here. The combined effects of the direction of light incidence and polarization
upon scattering properties for the case of the 0.2-µm sphere are shown in Figure 38.
Plotted in this figure is the scattering coefficient S versus the angle of light incidence for
polarization both parallel to (filled circles) and perpendicular to (hollow circles) the
optic axis of the sphere; the angle of incidence is measured between the propagation
direction of the incident light and the optic axis. The curves in Figure 38 converge to
the same values at low angles of incidence; this is expected, since both polarizations of
light incident at low angles encounter primarily the ordinary refractive index of the
crystal. As the angle of incidence increases, the scattering coefficient associated with
polarization parallel to the sphere optic axis increases markedly as the light increasingly
encounters the higher, extraordinary index of the rutile crystal.
For polarization
131
perpendicular to the optic axis, however, no component of the incident light is exposed
to the extraordinary refractive index of the crystal, resulting in no dependence of S upon
the angle of incidence. The scattering coefficient S for unpolarized light incident for
any of the angles shown in Figure 38 can be computed by averaging the S values
associated with the two orthogonal polarizations.
34
32
-1
S (µm )
30
28
26
24
22
20
0
10 20 30 40 50 60 70 80 90
Angle of Incidence (degrees)
Figure 38. Scattering coefficient S as a function of the angle of incidence (relative
to the optic axis) for the 0.2-µm anisotropic sphere of rutile titania. Hollow circles
are for polarization perpendicular to the optic axis of the sphere; filled circles are
for polarization coplanar with the optic axis.
5.2.4 On the Validity of the Average Index and Weighted Sum Approximations
A question of central interest in this study is to what degree the average index
and weighted sum approximations described in Section 5.2.1 succeed in predicting the
far-field scattering parameters associated with the anisotropic spheres investigated here.
132
Agreement in the scattering coefficient S between the results computed using the
finite element method (solid circles) and both the average index (dashed curve) and the
weighted sum (dotted curve) approximations is shown in Figure 36. In both cases,
agreement between the finite element method results for anisotropic spheres and the
approximate approaches is very good for diameters 0.18 µm and smaller. The weighted
sum approximation underestimates the first resonant peak in the finite element results
(0.2–µm sphere), though the average index approximation accurately matches this peak
value. For diameters greater than 0.2 µm, differences of up to 10% occur between the
two approximations and the finite element results. The three data sets in Figure 36
suggest that the periodicity of the resonant peaks in the finite element results is longer
than in either of the two approximations. It has been suggested that such resonant peaks
arise from resonant surface waves on a sphere;2 it is therefore reasonable that the
different symmetry of an anisotropic sphere compared to an isotropic should result in a
quantitatively different behavior of such resonances. It is not clear why the period of
the resonant peaks associated with anisotropic spheres are longer than either of the two
approximations based upon isotropic spheres. Comparison of the average index and
weighted sum approximation data sets in Figure 36 indicate that the main difference
between the two is that the resonant peaks in the weighted sum approximation are less
sharp. That approximation is a weighted superposition of two curves which are slightly
offset with respect to sphere diameter, resulting in averaging of the resonant peaks.
The agreement in the angle-weighted scattering coefficient σ between the finite
element method (solid circles) and both the average index (dashed curve) and weighted
133
sum (dotted curve) approximations are shown in Figure 37. In analogy with results for
the scattering coefficient S in Figure 36, agreement between the finite element data and
both approximations is very good for diameters less than 0.18 µm.
For sphere
diameters greater than this value, the data sets contain the same resonant features, with
the period of the resonant peaks for the finite element data again being greater than that
of the average index approximation. In both S and σ, the maximum values in the
average index approximation (sphere diameter ~0.2 µm) are in very good agreement
with the finite element data. The most significant disagreement between the weighted
sum approximation and the finite element data occurs around the primary resonant
peak, around 0.2 µm. The average index approximation more accurately fits this first
resonant peak than the weighted sum approximation in both S and σ.
The statistical quantity χ2 has been computed as a means to quantify the ability
of both the average index and weighted sum approximations to predict the results for S
and σ of anisotropic spheres obtained using the finite element method. The complete
data sets in Figure 36 and Figure 37 were used. The quantity χ2 has been computed as
Equation 80
n
(Ok − E k )2
k =1
Ek
χ =∑
2
where n is the number of data points (thirteen, in this case), Ok is the value of S or σ
computed for a given sphere diameter using either the average index or weighted sum
approximations, and Ek is the value of S or σ computed for the sphere diameter using
the finite element method. The values of χ2 computed for the average index and
134
weighted sum approximations are summarized in Table 6 for both quantities S and σ.
The value of χ2 is significantly smaller than the number of data points for each data set,
indicating a very good statistical fit in each case. For both S and σ, the weighted sum
approximation exhibits slightly lower χ2 values than the weighted sum approach, but
this can be reversed by selectively omitting certain discrete data points from the data
sets, indicating no clear advantage to either approach from the statistical standpoint.
While both the average index and weighted sum approximations are in good
statistical agreement with the results of finite element computations for anisotropic
spheres, there are possible advantages associated with each for a given circumstance.
The average index approximation offers the advantage of explicitly defining a single
refractive index for the material, therefore requiring only a single set of computations to
obtain far-field results.
The weighted sum approximation offers the advantage of
smoothing out the sharp resonant peaks associated with particular sphere diameters in
Mie theory computations, artifacts of the high symmetry of a sphere which in practice
are not observed in light scattering by real systems of multiple particles.
Table 6. Values of χ2 (Equation 80) associated with S and σ values for the
average index and weighted sum approximation, compared to anisotropic
sphere scattering data generated using the finite element approach.
S
σ
Average Index
Approximation
0.71
0.44
Weighted Sum
Approximation
0.60
0.33
135
5.3 Scattering by a Morphological Rutile Titania
5.3.1 Introduction
A fundamental goal of both consumers and manufacturers of rutile titania
pigment is to maximize the light scattering efficiency of the pigment in end use. The
achievement of this goal requires quantitatively identifying, and subsequently
controlling, the factors which impact the light scattering efficiency of rutile titania
powders. This light scattering efficiency is controlled by a number of factors, including
the particle size distribution, the degree of particle agglomeration, and the degree of
optical interaction between neighboring particles in a pigmented microstructure.
However, the quantitative influence of each of these factors upon the scattering
efficiency of rutile titania is not fully understood, due in large part to the difficulty of
controlling laboratory experiments to the degree that the effect of each can be studied
independently.5,6
Due to these inherent experimental challenges, theoretical modeling attempts
have been made to understand light scattering properties of rutile titania. These efforts
have typically relied upon Mie theory, which provides an exact treatment of the light
scattering properties of a single, optically isotropic sphere.1,2,48 Examples of such
studies include investigation of the theoretical scattering efficiency of rutile titania
versus that of air voids,17 the crowding effect in highly loaded films,25,26,28,7 and the
application of a multiple scattering model to predict the light scattering efficiency of
136
paint films.18 In reality, individual rutile titania particles exhibit both optical anisotropy
and complex shapes. In addition, pigmented films typically contain multiple-particle
aggregates in which near-field optical interactions between neighboring particles play a
critical role. Mie theory is therefore not capable of rigorously describing the light
scattering properties of realistic rutile titania particle shapes and microstructures. To
illustrate this point, a high-resolution scanning electron micrograph of a commercial
rutile titania pigment powder is shown in Figure 39. Both the characteristic, nonspherical shape of the primary particles and particle aggregation are apparent in the
micrograph.
Figure 39. High resolution scanning electron micrograph of rutile titania particles,
showing their characteristic, morphological shapes. (Micrograph courtesy E.D.
Boyes, DuPont Central Research and Development.)
137
5.3.2 Scattering by a Morphological Rutile Particle
Here, the time-domain finite element approach has been applied to the case of
560-nm light scattering by a single, morphological rutile titania particle . This particle,
shown in the upper panels of Figure 40, has a width 0.175 µm and a tip-to-tip length of
0.35 µm. The shape and size of this particle is representative of rutile titania pigment
and has been chosen to illustrate the scattering properties of a typical pigment particle.
In contrast to the finite element computations performed for the case of a single sphere
in the Section 3 of this study, this rutile particle has been assigned the appropriate
anisotropic optical constants. As in the calculations described above, the particle is
surrounded by a medium with refractive index n = 1.51, representative of an acrylic
resin.
This model contains ~1.6 million cubic elements, 3.6 nm on edge.
This
corresponds to a finite element mesh density of ρ = 50 elements per light wavelength in
the model. The upper left panel of Figure 40 shows a three-dimensional view of the
sphere with the surrounding medium hidden; the cubic elements which compose the
particle are visible. The upper right panel shows a cross-sectional view of the finite
element model, including the particle and the surrounding medium. In this case, light is
propagating in the finite element model with components in both the +x and +z
directions with unit incident intensity.
The lower two panels of Figure 40 show
scattered intensities in the geometric domains displayed in the corresponding upper
panels. As in the case of the spherical particle shown in Figure 10, scattered intensities
in the near field can be several times greater than the incident intensity.
138
In general, the rutile titania particles in a paint film are randomly oriented. In
order to determine the light scattering properties of the morphological, anisotropic rutile
particle shown in Figure 40 under conditions of diffuse illumination, a set of individual
finite element computations with different angles of light incidence are performed.
Two individual plane waves having mutually perpendicular polarizations are used for
each illumination direction.
To determine far-field scattering properties under
conditions of random illumination, far-field intensities for the individual illuminations
are superimposed. The presence of four different mirror planes in the spatial symmetry
of the rutile particle shown in Figure 40 requires that only one half-octant of the particle
be illuminated to determine the full light scattering properties of the particle. In the
present case, fifteen different illumination directions were employed over one halfoctant of the particle, which is equivalent to one hundred eighty illumination directions
over the entire particle.
Using the finite element method, the far-field scattering
parameters S = 23.97 µm−1 and σ = 10.07 µm−1 have been computed for the rutile
particle shown in Figure 40.
139
Figure 40. Finite element model and near field scattering results for an
anisotropic, morphological rutile particle with a width of 0.175 µm, and a length of
0.35 µm in n = 1.51 resin; the light wavelength is 560 nm.
140
5.3.3 Discussion
5.3.3.1 Scattering Coefficients of the Morphological Rutile Particle
It is of interest to compare the far-field scattering parameters S and σ computed
for the rutile particle to those from Mie theory for the sphere with equivalent volume,
using both the average index and weighted sum approximations. The volume of the
rutile particle shown in Figure 40 is 7.77x10-3 µm3, corresponding to an equivalent
spherical diameter of 0.246 µm. Such comparisons provide a sense of the adequacy of
Mie theory to describe the optics of realistic rutile particles. Table 7 shows the results
of this comparison, with S and σ values shown for the finite element computation for
the rutile particle and for the equivalent volume sphere computed using both the
average index and weighted sum approximations. For the average index approximation,
the S result is 1.4% lower than the finite element result, and the angle-weighted
scattering coefficient σ result is 4.6% higher than the finite element result. For the
weighted sum approximation, the S result is 6.4% lower than the finite element result,
and the angle-weighted scattering coefficient σ is in agreement with the finite element
result.
For this particular rutile particle, at least, the scattering properties of the
equivalent volume sphere computed using the two approximations and Mie theory are
within ~5% of the finite element result.
141
5.3.3.2 Mie Theory in Particle Size Analysis by Light Scattering
It is common practice in particle sizing methods based upon light scattering to
use Mie theory to compute particle size distributions for particles which are in reality
non-spherical.2 In these methods, the light scattering properties of a system are used to
determine equivalent spherical particle sizes by fitting scattering data to a size
distribution. It is therefore of considerable interest to compare the results of the finite
element calculations on systems containing realistically shaped rutile particles to the
results of Mie theory. This can be achieved by back-calculating equivalent spherical
diameters corresponding to the morphological rutile particle based upon the scattering
parameters S and σ rather than upon particle volume (as in Table 7).
Using the results S = 23.97 µm−1 and σ = 10.07 µm−1 for the isolated rutile
particle shown in Figure 40, the Mie theory results shown in Figure 31−Figure 34 have
been used to compute the spherical diameters exhibiting these same far-field scattering
parameters. These results are shown in Table 8. For both the average index and
weighted sum approximations, four different spherical diameters correspond to
matching S values, and two different spherical diameters correspond to matching angleweighted scattering coefficient σ values. The multiple sphere diameters corresponding
to the single S and σ values associated with this particle result from multiple
occurrences of these values in the curves shown in Figure 31−Figure 34. These results
highlight the potential ambiguity in using Mie theory to impute equivalent spherical
diameters to non-spherical particles based upon far-field scattering properties.
142
Table 7. Comparison between the far-field scattering coefficients for the single
rutile particle and the equivalent volume sphere, using both the average index
and weighted sum approximations.
S (µm−1)
σ (µm−1)
23.97
10.07
Equivalent-Volume Sphere
n = 2.74
22.44
10.07
Equivalent-Volume Sphere
Weighted Sum of n=2.64 and
n = 2.94 Results
23.64
10.54
Anisotropic Rutile Particle
w=0.175 µm, Aspect Ratio 2
Table 8. Diameters of spheres exhibiting the same far-field scattering
coefficients as the single rutile particle, using both the average index and
weighted sum approximations.
Equivalent Spherical Diameter
n = 2.74
S Basis
σ Basis
0.186 µm
or 0.212 µm
0.183 µm
or 0.254 µm
or 0.247 µm
or 0.293 µm
Equivalent Spherical Diameter
Weighted Sum of n = 2.64
and n = 2.94 Results
0.190 µm
or 0.204 µm
or 0.248 µm
0.179 µm
or 0.261 µm
or 0.287 µm
143
1
L.L Houle, J. Boudreau, and G. Shukla, TAPPI 1990 Coating Conference Proceedings, TAPPI Press,
Atlanta (1990).
2
R.A. Kwoka, TAPPI 1990 Dyes, Fillers and Pigments Short Course, TAPPI Press, Atlanta (1990).
3
R.A. Kwoka, TAPPI 1993 Coating Conference Proceedings, TAPPI Press, Atlanta (1993).
4
R.A. Kwoka, and T.W. Logan, TAPPI Journal, 77 (10), 136 (1994).
5
Y. Wang, “Local Field Effect in Small Semiconductor Clusters and Spheres,” J. Phys. Chem., 95 (3),
1119-1124 (1991).
6
J.G. Balfour, “Opacity and Titanium Dioxide Pigments,” J. Oil Colour Chem. Assoc., 73, 225 (1990)
7
D.F. Tunstall and M.J. Hird, “Effect of Particle Crowding on Scattering Power of TiO2 Pigments,” J.
Paint Technol., 46, 33-40 (1974).
144
Chapter 6. Clusters of Latex Spheres: A Low-Contrast System
This chapter describes the light scattering properties of clusters of latex particles
in water studied using the time-domain finite element method. Using synthesis methods
such as emulsion polymerization, polymeric materials can be formed as sub-micron
latex particles which are spherical in shape and can be nearly monodisperse in size. The
use of latex particles is widespread in polymer processing, since these materials can be
readily dispersed in liquids and subsequently formed into complex shapes.
Latex
particles can be stabilized in water, allowing the elimination of potentially flammable
and toxic volatile organic solvents from the processing of the sol.
This valuable
attribute of latices has created a very large market for latex-based paint systems. The
near monodispersity and controllable dispersion characteristics of latex systems makes
them ideal for model studies of particle agglomeration dynamics in liquid suspensions.
These agglomeration characteristics can be studied by microscopy, rheological
measurements, and light scattering techniques.
In Section 6.1, the general scattering properties of the materials system of
interest are described. In Section 6.2, the effects of latex particle agglomeration upon
the scattering properties of the system are described. In Section 6.3, the length scale
over which near-field optical interactions between neighboring latex particles are
determined. The computational results presented in this chapter are placed within the
context of scattering efficiency and hiding power. The latex system studied here is of
particular interest because it is a widely used system with low optical contrast,
145
providing a valuable basis for comparison with the results for clusters of white pigment
particles with high optical contrast described in Chapter 7.
6.1 Materials System of Interest
The system of interest in the present work is a latex material with refractive
index n = 1.50 in a water medium with refractive index n = 1.33. The illuminating
wavelength is 488 nm, a commonly used argon ion laser wavelength. A plot of the
scattering coefficient S versus diameter for a single sphere calculated using Mie theory
for these conditions is shown in Figure 41.
maximum S value is 1.4 µm.
The sphere diameter producing the
The low optical contrast of this system produces
significantly lower S values (one order of magnitude) than the high-contrast pigment
particle systems studied in the previous chapter. A plot of the angle-weighted scattering
coefficient σ versus sphere diameter calculated using Mie theory for the latex system is
shown in Figure 42. The sphere diameter producing the maximum σ value is 0.18 µm.
This value is significantly smaller than the 1.4 µm diameter producing the maximum S
value in Figure 41. This result indicates that the quantity (1−g), where the asymmetry
parameter g is the average cosine of the scattering angle, decreases sharply with
increasing sphere diameter in the sub-micron size range. That is, there is a rapid
transition from isotropic to forward scattering as the latex particle size increases beyond
the Rayleigh regime. The primary particle size studied here is 200 nm, a representative
latex particle diameter. This sphere diameter is indicated in the plots of Figure 41 and
Figure 42 with an “x”; this particle size provides an especially interesting case since it is
146
sub-optimal for the scattering coefficient S and very nearly optimal for the angle-weight
scattering coefficient σ.
The light scattering properties of a single 200-nm latex sphere in water has been
computed using the time-domain finite element method. Shown in Figure 43 are the
model geometry and near field scattered intensities for this computation. The 488-nm
light propagates in the +x direction and is polarized in the z direction. The top two
panels of this figure show a three-dimensional view of the scattering sphere and a crosssectional slice through the center of the model, respectively. The bottom two panels of
this figure show the near-field scattered intensities as a function of position in the
geometries shown in the top two panels. In contrast with the titania sphere in resin
shown in Figure 10, the latex sphere shown in Figure 43 exhibits scattered intensities in
the near field which are much smaller than the unit intensity of the incident light. This
is a result of the relatively low optical contrast of the latex system. The distribution of
scattered light in Figure 43 shows that most of the light is scattered in the forward
direction, with two lobes of concentrated light on the forward-scattering hemisphere of
the particle. The computed values of the scattering coefficients S and σ from the results
shown in Figure 43 are within 0.8% and 0.9% of the exact results computed from Mie
theory, respectively, showing close agreement.
147
3.5
3
S ( µ m −1)
2.5
2
1.5
1
x
.5
0
0
.5
1
1.5
2
2.5
3
3.5
4
Sphere Diameter (Microns)
Figure 41. The scattering coefficient S as a function of sphere diameter for a latex
particle with n = 1.50 in a water medium with n = 1.33. The illuminating light
wavelength is 488 nm. The 200-nm sphere diameter studied in this chapter is
marked with an “x.”
.25
x
σ ( µ m −1)
.2
.15
.1
.05
0
0
.5
1
1.5
2
2.5
3
3.5
4
Sphere Diameter (Microns)
Figure 42. The angle-weighted scattering coefficient σ as a function of sphere
diameter for a latex particle with n = 1.50 in a water medium with n = 1.33. The
illuminating light wavelength is 488 nm. The 200-nm sphere diameter studied in
this chapter is marked with an “x.”
148
Figure 43. Finite element model for a 200-nm latex sphere with n = 1.50 in a water
medium with n = 1.33. The illuminating wavelength is 488 nm, and light
propagates in the +x direction.
149
6.2 Light Scattering by Latex Sphere Clusters of Increasing Size
A series of computations has been performed using the time-domain finite
element method to determine the light scattering properties of particle clusters
containing increasing numbers of the 200-nm primary particle shown in Figure 43.
Cluster sizes investigated consisted of 2, 3, 6, 7, 8, 13, 27 and 64 primary particles. The
packings of these particles are not identical in each case; instead, the highest-symmetry
arrangements have been selected. For example, the clusters of 8, 27, and 64 spheres
share a simple cubic packing, while the cluster of 13 spheres has a face-centered cubic
packing. The geometries of the clusters containing 3, 7, 13, and 27 spheres are shown
in Figure 44 as examples of the cluster packings.
In each computation, random
illumination conditions are simulated using the Illuminator software, yielding
orientation-averaged scattering coefficients along with the individual scattering results
for each illumination direction and polarization.
Near-field optical interactions between the latex spheres in these clusters can be
expected to occur and thus produce light scattering properties which differ from those
of the isolated, 200-nm latex primary particle which composes each cluster. The finite
element method permits direct visualization of these near-field interactions. Shown in
Figure 45 are geometric views of the cluster containing seven 200-nm latex particles
and the distribution of scattered light intensity in the near field. The top two panels of
this figure show a three-dimensional view of the spheres and a cross-sectional slice
through the center of the model, while the bottom two panels show the distribution of
150
scattered light through these two geometric views. The geometry of this model consists
of a central sphere surrounded by two neighboring spheres along each of the three axes
of the Cartesian coordinate system. As in all of the sphere clusters constructed in this
study, the spheres are in hard contact with their neighbors.
The 488-nm light
propagates in the +x direction in the computation shown in Figure 45. The lower two
panels of Figure 45 demonstrate clearly that near-field scattered light in the sevensphere cluster is not distributed evenly among the seven primary particles in the cluster
as would be the case in the limit of single scattering. Instead, the greatest intensity of
scattered light is in the forward-scattering hemisphere of the cluster. This result is
significant and illustrates a phenomenon which is observed in all computations
performed on these latex sphere clusters: a cluster of latex spheres scatters light as a
single entity rather than as a collection of independent particles. This fact lies at the
root of the dramatic effects of sphere clustering upon light scattering which are
described in this and the following sections of this chapter.
The effects of latex particle clustering upon the scattering coefficient S are
shown in Figure 46 with S plotted as a function of the number of spheres contained in a
A discontinuity occurs for the case of the seven-sphere cluster; this is likely due to the
relatively low packing density of this cluster (see Figure 45) compared to the others.
The S results shown in Figure 46 indicate that the efficiency of light scattering by these
latex clusters increases on a volume-normalized basis with increasing numbers of
primary particles in a cluster. The value of S for a six-sphere cluster is two times that
151
associated with a single latex primary, for example, and the value for a sixty-foursphere cluster is nearly four times that associated with the isolated primary.
The quantity (1−g) is plotted in Figure 47 as a function of the number of latex
spheres in a cluster. The data indicate a sharp decrease in this term as cluster size
increases, corresponding to a decrease in the fraction of light deflected away from the
forward-scattered direction. This decrease in (1−g) is the opposite of the trend observed
in the scattering coefficient S. That is, while clustering of greater numbers of the 200nm latex spheres causes increased scattered intensity on a volume-normalized basis, this
additional scattered intensity is directed more strongly in the forward direction. The
opacifying efficiency of these clusters in an aqueous dispersion depends upon the angleweighted scattering coefficient σ, which is the product of S and (1−g). The opposite
trends observed in S and (1−g) indicate that the dependence of σ upon the number of
spheres in a cluster is less strong than the case of either S or (1−g).
The effects of latex particle clustering upon the scattering coefficient σ are
shown in Figure 48 with σ plotted as a function of the number of spheres contained in a
cluster. In contrast to the results of Figure 46 for the scattering coefficient S, the plot in
Figure 48 in general shows a steady, but only slight, decrease in σ with increasing
numbers of latex spheres in a cluster. The global maximum in the data corresponds to
the case of a two-sphere cluster, exhibiting a σ value which is 5% greater than that of
the isolated latex primary. For larger clusters, there is little change in the data between
the cases of 13-, 27-, and 64-sphere clusters. The cluster of thirteen particles exhibits a
152
σ value which is 25% lower than that associated with the isolated latex primary, and
the cluster of sixty-four primary particles exhibits a value which is about 15% lower
than the isolated latex primary. It can be concluded that, in general, clustering of these
latex spheres over the size range investigated here results in poorer scattering efficiency
for a dispersion of these particles compared to the case where no particle agglomeration
is present.
153
Figure 44. Finite element models of latex sphere clusters containing 3, 7, 13, and
27 spheres.
154
Figure 45. Finite element model of a cluster of seven 200-nm latex spheres n = 1.50
in a water matrix with n = 1.33. The 488-nm illuminating light propagates in the
+x direction and is polarized in the z direction.
155
2.0
-1
S (µm )
1.5
1.0
0.5
0.0
0
10
20
30
40
50
60
70
Number of Spheres
Figure 46. The scattering coefficient S as a function of the number of 200-nm latex
particles composing a particle cluster under conditions of random illumination.
0.5
(1-g)
0.4
0.3
0.2
0.1
0.0
0
10
20
30
40
50
60
70
Number of Spheres
Figure 47. The quantity (1−g) as a function of the number of 200-nm latex spheres
composing a particle cluster under conditions of random illumination.
156
-1
σ (µm )
0.3
0.2
0.1
0.0
0
10
20
30
40
50
60
70
Number of Spheres
Figure 48. The scattering coefficient σ as a function of the number of 200-nm latex
spheres composing a particle cluster under conditions of random illumination.
6.3 Near-Field Interactions Between Neighboring Latex Particles
In the previous section, it has been demonstrated that clusters of 200-nm latex
spheres can exhibit substantially different scattering efficiencies than the individual
primary particles composing them. That result calls into question of the length scale of
interparticle separation over which near-field interactions between the particles are
important. In this section, this question is investigated in model systems containing a
small number of 200-nm latex particles illuminated with 488-nm light. These model
systems are based upon the geometry of the seven-sphere cluster shown in Figure 45;
the spacing between the central sphere and the six surrounding spheres are uniformly
varied from one computation to the next.
In the case of a 0.4-µm interparticle
separation, for example, each of the six surrounding spheres is positioned to have a 0.4-
157
µm surface-to-surface interparticle separation from the central sphere. The resulting
shape of the cluster is therefore an octahedron for each interparticle separation distance
investigated. Conditions of random illumination are simulated for each separation
distance using the Illuminator software, and scattering efficiencies are computed on an
orientation-averaged basis as in Section 6.2.
The geometry of the model with 0.4-µm surface-to-surface interparticle
separation distance is shown in Figure 49. The upper two panels of this figure show a
three-dimensional view of the latex spheres and a cross-sectional slice through the
center of the finite element model. Five of the spheres in the seven-sphere cluster are
bisected by this cross-sectional slice; the other two particles are positioned in the +z and
–z directions in the model (out of the page, and into the page, respectively). Shown in
the lower two panels are the distributions of near-field scattered light intensity on the
surfaces of the seven spheres and through the cross-sectional slice through the center of
the model. The incident 488-nm light is polarized in the z-direction in this computation
and propagates in the +x direction in the model. There are several interesting features
in the distributions of scattered light intensity in the near field for this computation.
First, it is clear in both the surface and cross-sectional views that each of the seven
individual latex spheres do not exhibit the same distribution of scattered light on their
surfaces or through their cross-sections. Instead, the two spheres positioned in the
forward scattering (+x) direction appear to have greater scattered intensities in their
immediate vicinity than the other spheres in the cluster. This suggests that at this 0.4µm separation, the seven particle are interacting optically in the near field rather than
158
scattering independently.
This interaction is evident in the scattered intensity
distribution in the cross-sectional slice through Figure 49, in which faint scattered
intensity is visible in the spaces between the five particles visible in that view. It is also
interesting to compare the results for 0.4-µm interparticle separation distance in Figure
49 with those for the touching spheres in Figure 45. It is evident that while the seven
spheres separated by 0.4 µm do not scatter light independently, the distribution of
scattered light between those spheres is more even than the distribution of scattered
light in the cluster of touching spheres. That is, these near-field results suggest that the
light scattering properties of the cluster with 0.4-µm interparticle separation should be
much closer to the result for seven isolated spheres than the case of the cluster of seven
touching spheres.
The dependence of the scattering coefficient S upon surface-to-surface
interparticle separation distance for the clusters of seven latex spheres is shown in
Figure 50. The dashed line in this plot corresponds to the S value associated with a
single, isolated 200-nm latex primary particle. Deviations from this dashed line are the
result of near-field interactions between neighboring spheres in the clusters. At the
maximum interparticle separation distance studied, 0.6 µm, the S value exhibited by the
seven-sphere cluster is only 5% greater than that of a single 200-nm latex primary
particle. The data in Figure 50 indicate a steady increase in S as the interparticle
spacing between the spheres in the cluster decreases to zero. For the case of zero
interparticle separation distance (touching spheres, Figure 45), the value of S is 0.88
µm-1, which is 80% greater than the value associated with a single primary particle.
159
(This case is part of the data set shown in Figure 46.) The interparticle separation
distance at which the seven-sphere cluster exhibits an S value more than 10% greater
than that of an isolated 200-nm primary particle is 0.35 µm, which is very close to the
effective wavelength of the 488-nm light in water with n = 1.33 refractive index.
The quantity (1−g) is plotted in Figure 51 as a function of interparticle
separation distance between the seven-sphere clusters. The dashed line in the plot
indicates the value associated with a single 200-nm latex primary particle. As in the
case of particle agglomeration investigated in Section 6.2, the quantity (1−g) exhibits
the opposite trend from the scattering coefficient S, decreasing sharply as the
interparticle separation distance approaches zero. That is, as the seven spheres in these
clusters approach each other and interact more strongly optically, the total amount of
light scattering increases but this scattered intensity is directed more strongly in the
forward direction. The value of (1−g) for the case of touching spheres is 46% lower
than that of a single 200-nm latex primary particle.
The dependence of the angle-weighted scattering coefficient σ upon interparticle
separation distance in the seven-sphere clusters is shown in Figure 52. As in Figure 50
and Figure 51, the dashed line in this figure corresponds to the value associated with a
single 200-nm latex primary particle. The data indicate a relatively weak dependence of
σ upon interparticle separation distance, compared to either S or (1−g). The value
associated with zero interparticle separation is just 5% less than that associated with a
single 200-nm latex primary particle. In contrast to the data for S and (1−g), the σ data
160
oscillate about the value associated with the single latex primary particle, exceeding it
by about 3% with a 0.25-µm interparticle separation distance. The data appears to have
an oscillatory character, but the amplitude of this oscillation is not large compared to
the 1-3% error level expected in these computations.
161
Figure 49. Finite element model of a cluster of seven 200-nm latex spheres n = 1.50
in a water matrix with n = 1.33. The surface-to-surface interparticle spacing is 0.4
µm. The 488-nm illuminating light propagates in the +x direction and is polarized
in the z direction.
162
0.9
S (µm-1)
0.8
0.7
0.6
0.5
0.4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Separation (µm)
Figure 50. Dependence of the scattering coefficient S upon surface-to-surface
interparticle separation distance for the clusters of seven latex spheres. The
dashed line corresponds to the value associated with a single 200-nm latex primary
particle.
0.6
(1-g)
0.5
0.4
0.3
0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Separation (µm)
Figure 51. Dependence of the quantity (1−g) upon surface-to-surface interparticle
separation distance for the clusters of seven latex spheres. The dashed line
corresponds to the value associated with a single 200-nm latex primary particle.
163
0.26
σ (µm-1)
0.25
0.24
0.23
0.22
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Separation (µm)
Figure 52. Dependence of the angle-weighted scattering coefficient σ upon surfaceto-surface interparticle separation distance for the clusters of seven latex spheres.
The dashed line corresponds to the value associated with a single 200-nm latex
primary particle.
164
Chapter 7.
Clusters of Titania Particles: A High-Contrast
System
Chapters 2 and 5 have emphasized the technological importance of the light
scattering properties of high-index white pigment particles.
The present chapter
describes the effects of near-field interactions between white pigment particles upon
scattering efficiency in several different systems.
As stated previously, optical
interactions between neighboring white pigment particles are known to have profound
impact in end-use applications, yet the fundamental aspects of these phenomena have
not been directly studied successfully. In Section 7.1, the general scattering properties
of the materials system of interest are established.
In Section 7.2, the effects of
clustering primary pigment particles which are optimized for scattering efficiency is
described. In Sections 7.3 and 7.4, the magnitude of near-field interactions between
separated particles is investigated for the cases of a seven-sphere cluster and of two
morphological rutile particles, respectively.
7.1 Materials Systems of Interest
In Sections 7.2 and 7.4, the results of studies upon the light scattering properties
of systems of multiple 200-nm spheres of high-index material are described. The
system investigated is that of a high-index material (n = 2.74) embedded in a medium
with index of refraction n = 1.51. The light wavelength studied is 560 nm. This system
is representative of rutile titania spheres using the average index approximation. The
sphere diameter studied here, as in the case of the latex system in the previous chapter,
165
is 200 nm. A plot of the scattering coefficient S versus diameter for a single sphere
calculated using Mie theory is shown in Figure 53. The sphere diameter producing the
maximum S value of 28 µm-1 is 0.3 µm. A plot of the angle-weighted scattering
coefficient σ versus diameter for a single sphere calculated using Mie theory is shown
in Figure 54. The sphere diameter producing the maximum σ value of 12.1 µm-1 is 0.2
µm. The 200-nm rutile sphere diameter studied here is indicated in each of these two
plots with an “x”. The 200-nm diameter studied here is that corresponding to the
maximum of the first resonant peak in S in the plot shown in Figure 53. In addition, this
sphere diameter produces the maximum σ value for this materials system. This particle
size may therefore be expected to produce qualitatively different behaviors upon
clustering than the 200-nm latex sphere studied in the previous chapter.
The light scattering properties of the single 200-nm rutile sphere under these
conditions has been computed using the time-domain finite element method; these
results are shown in Figure 10. A comparison of the near-field scattered intensities in
Figure 10 with those of the single latex sphere shown in Figure 43 indicates that the
main difference between these two systems is the much higher scattered intensities in
the vicinity of the titania sphere in Figure 10.
Indeed, these near-field scattered
intensities are about 50 times higher than in the case of the latex particle. It is the
greater optical contrast between the titania particle and its surrounding medium,
combined with the resonant diameter of the titania sphere, which produces this stronger
scattering interaction.
166
30
x
S ( µ m −1)
25
20
15
10
5
0
0
.2
.4
.6
.8
1
Sphere Diameter (Microns)
Figure 53. The scattering coefficient S as a function of sphere diameter for a rutile
particle with n = 2.74 in a polymer medium with n = 1.51 under the average index
approximation. The illuminating light wavelength is 560 nm. The 200-nm sphere
diameter studied in this chapter is marked with an “x.”
x
12
σ ( µ m −1)
10
8
6
4
2
0
0
.2
.4
.6
.8
1
Sphere Diameter (Microns)
Figure 54. The angle-weighted scattering coefficient σ as a function of sphere
diameter for a rutile particle with n = 2.74 in a polymer medium with n = 1.51
under the average index approximation. The illuminating light wavelength is 560
nm. The 200-nm sphere diameter studied in this chapter is marked with an “x.”
167
7.2 Light Scattering by Titania Sphere Clusters of Increasing Size
As in the case of the latex particles in the previous chapter, a series of
computations has been performed using the time-domain finite element method to
determine the light scattering properties of particle clusters containing increasing
numbers of the 200-nm rutile sphere shown in Figure 10. Cluster sizes investigated
consisted of 2, 3, 4, 6, 7, 8, 13, 20, 27 and 64 primary particles. The same geometrical
packings used in the case of the latex particles have been used here. In each case,
random illumination has been simulated using the Illuminator software, yielding
orientation-averaged scattering coefficients along with the individual scattering results
for each illumination direction and polarization.
Shown in Figure 55 are geometric views of the cluster containing seven titania
particles and the distribution of scattered light in the near field. (Figure 45 shows the
analogous case for latex particles). The top two panels show a three dimensional view
of the spheres and a cross-sectional slice through the center of the model, while the
bottom two panels show the distribution of scattered light at steady state through these
two geometric views. The 560-nm light propagates in the +x direction in this case. As
in the case of the clustered latex spheres, the results in Figure 55 show that the
distribution of scattered light through the seven-sphere cluster is not uniform. Instead
the greatest concentration of scattered light is in the forward-scattering hemisphere of
the cluster. The incident light intensity for the computation shown in Figure 55 is unity.
In the lower two panels of this figure, the maximum scattered intensities observed are
therefore about 9 times higher than the incident intensity, indicating that the titania
168
cluster focuses scattered light in the forward direction. A comparison of near-field
scattered intensities observed for the seven-sphere cluster of titania spheres with those
observed for the single primary particle (Figure 10) indicate scattered intensities which
are about 3 times higher for the cluster.
The effects of titania particle clustering upon the scattering coefficient S are
shown in Figure 56 with S plotted as a function of the number of spheres contained in a
cluster. There is an increase in S by about 15% as a second sphere is added to the first,
and then a dramatic decrease in general as more particles are added to the cluster. The S
value associated with the thirteen-sphere cluster is 50% less than that associated with an
isolated primary particle, and the value associated with the sixty-four sphere cluster is
about 75% less than that of the isolated primary. In general, clustering of 200-nm
titania spheres results in markedly reduced total scattering on a volume-normalized
basis compared to the case of the isolated primary.
The effects of titania particle clustering upon the angle-weighted scattering
coefficient σ are shown in Figure 57 with σ plotted as a function of the number of
spheres contained in a cluster. These σ results share the same characteristic as the S
data in Figure 56, which is a pronounced decrease with increasing cluster size.
Contrary to the S results, the σ results in Figure 57 do not show an increase as the
cluster size is increased from one to two spheres. In addition, the decrease in σ
observed for small cluster sizes (<10 spheres) in Figure 57 with increasing cluster size
is more pronounced than the decrease in S in Figure 56 over the same size range. The
loss in σ for a four-sphere cluster is about 30% compared to the isolated primary
169
particle, while a loss in S of only about 10% occurs for this cluster. Similar losses in the
two coefficients S and σ occur for clusters larger than 10 spheres. The loss in σ for the
thirteen-sphere cluster is 60% compared to the isolated primary, and the loss associated
with the sixty-four sphere cluster is 70%. These results for titania spheres indicate that
the quantity (1−g) decreases rapidly with increasing cluster size for small clusters and
changes more slowly for large clusters.
170
Figure 55. Finite element model of a cluster of seven 200-nm titania spheres with n
= 2.74 in a polymer matrix with n = 1.51. The 560-nm illuminating light
propagates in the +x direction and is polarized in the z direction.
171
35
30
S (µm-1)
25
20
15
10
5
0
0
10
20
30
40
50
60
70
Number of Spheres
Figure 56. The scattering coefficient S as a function of the number of 200-nm
titania spheres composing a particle cluster. These results are for the case of
random illumination of each cluster, resulting in orientation-averaged results.
12
σ (µm-1)
10
8
6
4
2
0
0
10
20
30
40
50
60
70
Number of Spheres
Figure 57. The scattering coefficient σ as a function of the number of 200-nm
titania spheres composing a particle cluster. These results are for the case of
random illumination of each cluster, resulting in orientation-averaged results.
172
7.3 Near-Field Interactions Between Neighboring Titania Particles
In the previous section, the effects of cluster sizes of 200-nm titania spheres
upon scattering efficiency has been studied using the time-domain finite element
method. In the present study, the effects of crowding are investigated for the case of the
seven-sphere cluster shown in Figure 55 by varying the interparticle separation distance
between the center sphere and the six surrounding spheres in the cluster. The surfaceto-surface interparticle separation distance is allowed to vary from zero to 0.5 µm. This
study is analogous to the study of crowding of latex spheres with this same geometry in
Section 6.3.
The dependence of the scattering coefficient S upon interparticle separation
distance in these seven-sphere clusters of titania is shown in Figure 58. The horizontal,
dashed line in this figure indicates the S value associated with a singe, isolated 200-nm
titania sphere of which the cluster is composed. The plot in Figure 58 shows that the S
values for the seven-sphere clusters is nearly equal to that of the isolated titania sphere
(within ~3%) for interparticle separation distances greater than 0.2 µm. The S values
for those clusters with interparticle separation distances greater than 0.4 µm in face
exceed the value associated with an isolated titania sphere by about 2%; this difference
is within the expected error for those computations and may be a result of numerical
error. For interparticle separation distances less than 0.2 µm, the scattering coefficient S
decreases very sharply with decreasing interparticle separation distance. For the case of
touching spheres (zero interparticle separation distance), the value of S is 16% lower
than the value associated with an isolated titania sphere. This 16% loss in the total
173
scattering by the cluster on a volume-normalized basis is the same value indicated in the
Figure 56 for the seven-sphere cluster.
The dependence of the angle-weighted scattering coefficient σ upon interparticle
separation distance in the seven-sphere clusters of titania is shown in Figure 59. The
horizontal, dashed line corresponds to the σ value associated with an isolated 200-nm
titania sphere of which the cluster is composed. The plot in Figure 59 shows that the
value of σ for the seven-sphere cluster with 0.5-µm surface-to-surface interparticle
separation distance is very nearly equal to that associated with an isolated 200-nm
titania sphere. The σ value at this separation distance is about 1.5% less than that of the
isolated primary particle, which is within the range of expected numerical error for this
computation. For interparticle separation distances less than 0.5 µm, the value of σ for
these seven-sphere clusters of titania decreases steadily as the separation distance
approaches zero. For a surface-to-surface interparticle separation distance of 0.2 µm,
the value of σ is 10% less than the value associated with an isolated sphere. For the
cluster of seven touching spheres, the value of σ is 35% less than the value associated
with an isolated sphere, indicating a substantial loss in scattering efficiency a result of
particle crowding and agglomeration in this system.
174
27
S (µm-1)
26
25
24
23
22
21
0.0
0.1
0.2
0.3
0.4
0.5
Separation (µm)
Figure 58. Dependence of the scattering coefficient S upon surface-to-surface
interparticle separation distance for the clusters of seven titania spheres. The
dashed line corresponds to the value associated with a single 200-nm titania
primary particle.
13
σ (µm-1)
12
11
10
9
8
7
0.0
0.1
0.2
0.3
0.4
0.5
Separation (µm)
Figure 59. Dependence of the angle-weighted scattering coefficient σ upon surfaceto-surface interparticle separation distance for the clusters of seven titania
spheres. The dashed line corresponds to the value associated with a single 200-nm
titania primary particle.
175
7.4 Near-Field Interactions Between Two Morphological Rutile Particles
Time domain finite element computations have been performed for the case of
two morphological rutile titania particles which approach each other in a resin. In this
case, two particles with the same geometry and anisotropic optical properties as the
particle shown in Figure 40 are positioned within a finite element model with
interparticle separations ranging from 0.5 µm to 0 µm (touching). Cross-sectional
views of the finite element models are shown in upper panels of Figure 60−Figure 62,
and the logarithm of scattered intensity through the same cross section are shown in the
lower panels. In each of these figures, the illumination direction is perpendicular to the
plane of the cross sections shown, and the light polarization is parallel to the long axes
of the particles. Asymmetry in the near-field scattered intensities around each of the
particles is apparent in each of these figures, indicating that the two particles are
interacting at these separations.
Random illumination conditions have been simulated for each of the
interparticle separations investigated using the Illuminator software. The presence of
three mutually perpendicular mirror planes in the geometry of the finite element models
limited the number of illumination directions to twenty-five in one octant of the finite
element model. The scattering parameter S has been computed for this two-particle
system as a function of surface-to-surface interparticle separation distance. This result
is shown in Figure 63; the horizontal line indicates the value of the scattering coefficient
for a single morphological rutile particle (Figure 40) for comparison with the twoparticle results.
The scattering coefficient S decreases with decreasing separation
176
between the two particles. For an interparticle separation of 0.5 µm, the scattering
coefficient is very nearly equal to that associated with the isolated particle. For the case
of zero interparticle separation, the value of the scattering coefficient S is 8% lower than
that associated with a single, isolated particle. That is, the two-particle aggregate
scatters 8% less total light than the isolated particle.
Results for the angle-weighted scattering coefficient σ for the two-particle
system are shown in Figure 64 as a function of the interparticle separation distance.
The horizontal line in the figure indicates the σ value associated with a single
morphological rutile particle (Figure 40). The results Figure 64 show that the value of
the angle-weighted scattering coefficient σ for the interparticle separation of 0.5 µm is
nearly equal to that associated with the isolated particle. For the case of the two
particles touching, the value is 20% lower, indicating that the two-particle aggregate is
significantly less effective in the limit of multiple scattering than the isolated particle.
As in the case of the isolated rutile particle, it is possible to determine the
diameters of spheres with equivalent far-field scattering properties for the case of two
interacting rutile particles described above. This case is especially interesting since the
total volume of scattering material remains constant as the two rutile particles approach
each other, yet the far-field scattering properties of the system change markedly as the
interparticle distance changes. These results are presented in Table 9, with the sphere
diameters producing the same values of the scattering coefficients S and σ as the twoparticle system for each of the interparticle separation distances considered. As in the
177
case of the single rutile particle, multiple values of S and σ occur for each of the
interparticle separation distances.
178
Figure 60. Finite element model cross section and near field scattering results for
two anisotropic morphological rutile particles in n = 1.51 resin for light incident
normal to the cross section shown and with a wavelength of 560 nm. The
interparticle spacing is 0.5 µm.
179
Figure 61. Finite element model cross section and near field scattering results for
two anisotropic morphological rutile particles in n = 1.51 resin for light incident
normal to the cross section shown and with a wavelength of 560 nm. The
interparticle spacing is 0.3 µm.
180
Figure 62. Finite element model cross section and near field scattering results for
two anisotropic morphological rutile particles in n = 1.51 resin for light incident
normal to the cross section shown and with a wavelength of 560 nm. The
interparticle spacing is 0 µm.
181
S (µm-1)
24
23
22
0.0
0.1
0.2
0.3
0.4
0.5
Separation (µm)
Figure 63. Scattering coefficient S for two morphological rutile particles as a
function of interparticle separation. The horizontal line shows the results for one
of the morphological rutile particles.
σ (µm-1)
10
9
8
0.0
0.1
0.2
0.3
0.4
0.5
Separation (µm)
Figure 64. Angle-weighted scattering coefficient σ for two morphological rutile
particles as a function of interparticle separation. The horizontal line shows the
results for one of the morphological rutile particles.
182
Table 9. Diameters of spheres exhibiting the same far-field scattering
coefficients as the system of two rutile particles for each interparticle
separation distance considered, using both the average index and weighted
sum approximations.
Interparticle Separation
(µm)
0.5
0.4
0.2
0.1
0
S Basis
σ Basis
0.186
0.211
0.254
0.292
0.186
0.212
0.254
0.293
0.184
0.215
0.252
0.295
0.182
0.22
0.249
0.297
0.180
0.228
0.242
0.301
0.183
0.278
0.183
0.279
0.181
0.285
0.337
0.349
0.178
0.295
0.328
0.354
0.171
0.360
183
Chapter
8.
Discussion:
Light
Scattering
by
Complex
Microstructures in the Resonant Regime
The light scattering properties of several previously unknown systems have been
determined computationally in this dissertation research. The results of these studies,
presented in Chapters 4−7, provide a basis for the elaboration of new insights about the
light scattering properties of complex microstructures.
This chapter contains a
discussion of these learnings on a general footing.
8.1 Accuracy of the Time-Domain Finite Element Method
Studies have been presented in Chapter 4 upon the accuracy of the time-domain
finite element method with different finite element model configurations and far-field
extrapolation conditions. The methodology used in this work has been to systematically
vary computational conditions in the finite element method and compare the numerical
results to known, analytical results from Mie theory.
The materials systems
investigated using Mie theory were as closely related to the more complex systems of
interest in subsequent research as possible. In each of the variations in computational
conditions studied, wide variations in the accuracy of the finite element method were
observed. It is significant that those computations which were grossly in error showed
no outward signs of any problem, such as crashes or error messages. Without the
foundations of understanding developed in the error analysis work, it would be
impossible to apply the time-domain finite element method to unknown cases with
confidence.
184
The explosion in computational power which has occurred in the late 1990’s has
created an unprecedented environment for successful application of computational
modeling to complex physical phenomena. This opening has been accompanied by the
development of diverse numerical methods in a given field, putting at one’s disposal
numerous approaches to the modeling of a given system. Each approach has its unique
advantages and disadvantages for the solution of a particular problem. In the field of
computational modeling and simulation (and in experimental work, for that matter),
there is an unfortunate tendency for researchers to vaunt the particular numerical
methods they develop or apply, at the expense of alternative approaches, and to avoid
discussion of their shortcomings. Careful and detailed studies of the accuracy of any
numerical method under the full range of computational conditions is critical, for it is
only through a sound understanding of the limitations of a numerical method that its
strengths can be appreciated.
8.2 Light Scattering by Complex, Resonant Particles
The light scattering properties of core-shell pigment particles illustrate important
principles about the design of complex particle architectures in the resonant regime. It
is well-known that the scattering of light arises from dielectric discontinuities between
different materials. It could therefore be argued that increased light scattering could be
produced by creating a large number of interfaces in an engineered composite particle.
One could imagine a complex particle consisting of a series of alternating, concentric
shell layers of high- and low-index materials; each interface in the particle would create
a source of light scattering, resulting in a particle with light scattering properties which
185
are superior to either of the constituent materials alone. In the case of the core-shell
particles studied in Chapter 5, which consist of a high-index titania shell surrounding
lower-index core materials, it has been demonstrated that this sort of approach is not a
viable means of producing pigment particles with superior light scattering properties in
the resonant regime. Instead, the core-shell architecture produces the same trends in
behavior as reducing the effective refractive index of the composite particle. Core-shell
particles with optimized diameter exhibit substantially lower scattering coefficients than
particles of the high-index material with optimized diameter. An explanation for these
observations lies in the relationship between the core-shell particle size and the
wavelength of the illuminating light. Since the core-shell particle diameters producing
optimal light scattering are on the order of the light wavelength in size, the potential for
increased light scattering by a multi-layer particle architecture is limited by the
resolution of the illuminating light. In simple terms, the light does not “see” the internal
particle interfaces so much as it “sees” a particle with reduced effective refractive index.
The strategy for producing white pigment particles with maximum light scattering
power is to produce optimally sized, solid particles of the highest refractive index, nonabsorbing material available.
The effects of optical anisotropy upon light scattering in the resonant regime
have been investigated for the first time in the study presented in Chapter 5. The results
of that study revealed the expected phenomena of orientation- and polarizationdependent light scattering properties in the anisotropic spheres studied. Changes in the
scattering coefficients S by as much as 50% were demonstrated for different
186
illumination directions and light polarizations in the case of the highly resonant 0.2-µm
sphere. This orientation and polarization dependence is an essential difference between
the anisotropic spheres and the isotropic spheres considered in Mie theory. Calculation
of the scattering parameters S and σ on an orientation-averaged basis, on the other hand,
revealed that the scattering properties of anisotropic spheres can be approximated
reasonably, within 10%, by isotropic scattering models. The approximate formulations
do not describe the orientation-averaged scattering behavior of the anisotropic spheres
rigorously, but the errors were not dramatic. These results suggest that dilute ensembles
of optically anisotropic particles which are truly randomly oriented may be well
approximated using isotropic models which can be far less complex to analyze. They
also highlight the fact that there are potential opportunities to improve the scattering
properties of ensembles of anisotropic particles forcing the particles to adopt preferred
orientations during processing. For example, it is possible to deliberately align particles
with high aspect ratios during processing, which may be advantageous for light
scattering performance in certain systems.
Finite element computations on the light scattering properties of a single,
morphological rutile titania particle demonstrate the ready applicability of the method to
even the most complex particle shapes. This resonant particle is faceted and exhibits
anisotropic optical properties. The absence of cylindrical symmetry in the rutile particle
makes its scattering properties especially difficult to model using techniques like the Tmatrix method which are not based upon piece-wise solution of Maxwell’s equations
through a large system of volume elements. The computed light scattering properties of
187
the rutile particle have been compared to those of the equivalent-volume sphere using
two approximations based upon scattering by an isotropic sphere. As in the case of
anisotropic spheres, the isotropic scattering models were in reasonable statistical
agreement with the computed scattering coefficients of the rutile particle, with errors
less than 10% in all cases. It can be concluded that equivalent-volume, isotropic
spheres approximate reasonably the light scattering coefficients of complex single
particles in the resonant regime, but rigorous agreement is not achieved.
The computed coefficients S and σ of the morphological rutile particle were also
used to identify those isotropic sphere diameters producing the same scattering
coefficients. In this case, there were multiple sphere diameters in each case which
exhibited the same scattering coefficients.
This study highlights the ambiguity of
assigning equivalent spherical diameters to particles with complex shapes and therefore
more degrees of freedom in their geometry than a simple sphere.
Particle sizing
methods based upon light scattering and the assignment of equivalent spherical
diameters should be regarded as effective engineering control tools rather than absolute
measures of true particle size.
8.3 Light Scattering by Densely Packed Particle Aggregates
The light scattering coefficients of multiple-sphere clusters containing different
numbers of primary particles have been determined using the time-domain finite
element approach in two different materials systems. The first system, representing
clusters of 200-nm latex spheres in water (Chapter 6), is a low-contrast system in which
188
the refractive index ratio of the two materials is 1.13. The second system, representing
clusters of 200-nm titania spheres in a polymeric resin (Chapter 7), is a high-contrast
system in which the refractive index ratio of the two materials is 1.81. It is interesting
to note that, while 488-nm light was used in the case of the latex-water system and 560nm was used in the titania-resin system, the effective light wavelengths in the two
systems were nearly identical due to the different refractive indices of the water and the
resin. In the case of the latex-water system, the effective light wavelength is 488nm/1.33 = 367 nm.
In the case of the titania-resin system, the effective light
wavelength is 560-nm/1.51 = 371 nm. The large difference in contrast ratio between
these two systems result in very different far-field scattering coefficients for the isolated
200-nm primary particle used to build the clusters in both cases (compare Figure 41Figure 42 with Figure 53-Figure 54.)
The results of computing the scattering coefficients of the packed particle
clusters studied in these two systems illustrate the fact that the multiple-particle clusters
exhibit very different scattering properties than the isolated primaries composing them.
In addition, the trends in the scattering coefficients S and σ as a function of the number
of 200-nm primaries in a cluster were in some cases opposite in these two materials
systems. That is, there exists no simple function which can be used to predict how the
scattering coefficients will change for clustering of arbitrarily sized primary particles in
an arbitrary materials system.
For the formation of multiple-particle clusters from a given primary particle size
and optical constants, an important consideration is the relationship between the
189
scattering coefficient S or σ of the isolated primary particle compared to the diameter
associated with the optimal value of that scattering coefficient. In other words, a critical
question is where the primary particle diameter is positioned on the computed scattering
coefficient curve from Mie theory. This knowledge provides the ability to predict in a
semi-quantitative way how the scattering coefficient of clusters of the primary particle
will trend with increasing cluster size. For example, for a primary particle diameter
which is much smaller than the diameter producing the maximum value of the angleweighted scattering coefficient σ in a given materials system, the formation of multipleparticle clusters of increasing size will result in increased σ values. These increased σ
values, however, will not attain the value of the solid particle which is of optimal size in
the system. Likewise, for a primary particle diameter which is equal to that producing
the maximum σ value in a system, the formation of clusters of increasing size from that
primary particle will result in decreased σ values.
It appears that this sort of
comparison is the only reliable means of predicting trends in scattering coefficients as a
function of the degree of particle clustering.
Comparisons are made here between the scattering properties of clusters
composed of arbitrary numbers of primary particles and the results of Mie theory
computations for single, equivalent-mass spheres. Shown in Figure 65 is a comparison
between values of the scattering coefficient S in the latex-water system as a function of
the number of spheres in a cluster (from Figure 46) and the results of Mie theory
computations for equivalent-mass spherical particles (dashed curve). The Mie theory
computations correctly predict the upward trend in S with increasing cluster size. In a
190
quantitative sense, however, the agreement between the two sets of data is statistically
poor for the largest cluster sizes investigated. For the cluster consisting of sixty-four
200-nm latex spheres, for example, the predicted value of S is 45% greater for the
equivalent-mass sphere than the value computed for the actual cluster. Differences
between the two sets of data arise from the different microstructures of the solid sphere
and the more complex, porous cluster of primary particles. Shown in Figure 66 is an
analogous comparison between values of the angle-weighted scattering coefficient σ in
the latex-water system (from Figure 48) and the results of Mie theory computations for
equivalent-mass spherical particles (dashed curve). In this case, the agreement between
the two sets of data is much better than in the case of the S data in Figure 65. The two
exhibit the same trend with increasing cluster size. For the case of the sixty-four sphere
cluster, the predicted value of σ is 20% less for the equivalent-mass sphere than the
value computed for the actual cluster.
For the titania-resin system, the comparison between S values for the multipleparticle clusters (from Figure 56) and equivalent-mass spheres (dashed curve) is shown
in Figure 67. As in the case of the latex-water system, the equivalent-mass Mie theory
computations correctly predict the trend in S as a function of the number of spheres in a
cluster. The statistical agreement between the two sets of data is poor for the larger
cluster sizes investigated. For cluster sizes greater than eight spheres, the equivalentmass results underestimate the value of S by as much as a factor of 2.5. Shown in
Figure 68 is the analogous comparison between values of the angle-weighted scattering
coefficient σ in the titania-resin system (from Figure 57) and the results of Mie theory
191
computations for equivalent-mass spherical particles (dashed curve). As in the case of
the latex-water system, agreement between the two sets of data is much better than in
the case of the scattering coefficient S. For the largest clusters investigated (twentyseven and sixty-four spheres), deviations between the two sets of data are the greatest,
with the results for clusters greater than the predicted values from Mie theory by 50%
and ~100%, respectively.
It is concluded that semi-quantitative trends in the effects of particle clustering
upon the light scattering coefficients S and σ can be predicted using Mie theory
calculations for equivalent-mass spheres. For a particular materials system, these trends
are determined by the relationship between the size of the primary particle composing
the clusters and the particle size corresponding to the maximum value of the scattering
coefficient. For example, increasing cluster sizes containing particles of sub-optimal
diameter primary particles result in increased scattering coefficient, while increasing
cluster sizes containing optimal diameter primary particles result in decreased scattering
coefficient.
Far better statistical agreement between the scattering properties of
multiple-particle clusters and Mie theory calculations for equivalent-mass spheres has
been found in the angle-weighted scattering coefficient σ than in S. The reasons for this
difference are not understood. This result suggests that opposite trends exist, with
increasing cluster size, in the scattering coefficient S and the factor (1−g), where g is the
asymmetry parameter.
192
3.0
-1
S (µm )
2.5
2.0
1.5
1.0
0.5
0.0
0
10
20
30
40
50
60
70
Number of Spheres
Figure 65. The scattering coefficient S as a function of the number of 200-nm latex
particles composing a particle cluster (solid curve). For comparison, the results of
Mie theory computations on single, equivalent-mass spheres are shown (dashed
curve).
-1
σ (µm )
0.3
0.2
0.1
0.0
0
10
20
30
40
50
60
70
Number of Spheres
Figure 66. The scattering coefficient σ as a function of the number of 200-nm latex
particles composing a particle cluster (solid curve). For comparison, the results of
Mie theory computations on single, equivalent-volume spheres are shown (dashed
curve).
193
35
30
-1
S (µm )
25
20
15
10
5
0
5
10
15
20
25
30
Number of Spheres
Figure 67. The scattering coefficient S as a function of the number of 200-nm
titania particles composing a particle cluster (solid curve). For comparison, the
results of Mie theory computations on single, equivalent-mass spheres are shown
(dashed curve).
12
-1
σ (µm )
10
8
6
4
2
0
0
5
10
15
20
25
30
Number of Spheres
Figure 68. The scattering coefficient σ as a function of the number of 200-nm
titania particles composing a particle cluster (solid curve). For comparison, the
results of Mie theory computations on single, equivalent-mass spheres are shown
(dashed curve).
194
8.4 Near-Field Optical Interactions Between Particles
8.4.1 Titania and Latex Systems
Changes in the number of primary particles composing densely packed clusters
of spheres produce dramatic changes in the far-field scattering properties of both the
latex-water and titania-resin systems compared to the isolated primary particles. Nearfield optical interaction between the neighboring particles in these clusters is the cause
of these deviations in scattering properties. Of interest is the length scale over which
these near-field interactions are important, a question which has not been addressed
comprehensively in the literature.
Computations using the time-domain finite element approach for the case of
multiple particle approaching each other have been performed for three different
systems in this thesis: seven-sphere clusters of both latex particles in water (Section
6.3) and titania particles in resin (Section 7.3), and two morphological rutile particles in
resin (Section 7.4).
Those results were presented in the form of the scattering
coefficients S and σ as functions of the distance (in microns) between neighboring
particles. In order to place these diverse systems on a common basis, these same results
are presented here as a function of the number of effective light wavelengths between
the centers of neighboring particles. The center-to-center spacing is adopted here for
the sake of general applicability to different particles sizes. In the case of the latexwater system, the light wavelength studied was 488 nm, corresponding to an effective
light wavelength of 488-nm/1.33 = 367 nm. In both the titania-resin systems, the light
195
wavelength studied was 560 nm, corresponding to an effective light wavelength of 560nm/1.51 = 371 nm.
In addition, the values of S and σ are expressed here as a
percentage of the values associated with the isolated primary particle in each system;
this normalization scheme makes it convenient to plot the latex-water and titania resin
systems, which exhibited very different magnitudes of S and σ, on the same axis.
The results for the percent-normalized scattering coefficient S are plotted as a
function of the center-to-center interparticle separation distance (in effective light
wavelengths) in Figure 69. The upper curve (solid triangles) corresponds to the sevensphere latex-water system, while the lower curves correspond to the seven-sphere titania
system (solid circles) and the two morphological rutile system (hollow squares). The S
values in the two titania-resin systems converge more rapidly to the value associated
with the isolated primary particle with increasing particle separation than the latexwater system. The S values in both the titania systems is within 10% of the value
associated with the isolated primary particles in ~0.8 light wavelengths, while the latex
system converges to within 10% of the isolated primary particle value in ~1.6 light
wavelengths.
Even for the center-to-center interparticle separation of 2.2 light
wavelengths, the latex system deviates from the value of the isolated primary by several
percent. While this value is on the order of the expected error level in the finite element
method for that finite element mesh density, there is no evidence of numerical noise in
the smoothly decaying curve.
The corresponding plot of the percent-normalized scattering coefficient σ as a
function of the center-to-center interparticle separation distance (in effective light
196
wavelengths) is shown in Figure 70. In this case, a comparison of the interparticle
separation distances for which there is a 10% deviation in each system from the σ
values associated with the isolated primary particles indicates similar length scales
determined for S in Figure 69. While the latex system does not deviate by this much for
any of the separation distances studied, there are deviations of 10% relative to the
isolated primary particles for the seven-sphere titania and the two morphological rutile
systems at 1.4 and 0.8 wavelengths, respectively. All three systems converge to within
3% of the σ values associated with the isolated primary particles for an interparticle
separation distance of 2.0 light wavelengths.
Examining the data in Figure 69 and Figure 70 collectively, it can be concluded
that appreciable near-field optical interactions between neighboring particles with sizes
in the resonant regime occur over approximately two light wavelengths. Beyond this
separation, these optical interactions are very weak compared to those which occur
between neighboring particles which are nearly in contact.
In this context, it is
interesting to consider additionally the data presented in Section 4.3.3 on the effects of
the far-field sphere radius upon error in the extrapolation of near-field scattering results
to far field. In that case, it was determined that placement of the far-field sphere surface
>10 µm from the scattering particle was sufficient to ensure that evanescent
components of the scattered light not effect the accuracy of extrapolation.
For
extrapolation surfaces placed 2 and 5 µm from the spherical titania particle, deviations
of just 0.02% and 0.09%, respectively, were observed in the computed scattering cross
section, (see Figure 19). For the 560 nm light studied in those cases, these distances
197
expressed in light wavelengths are 3.6 and 8.9 wavelengths, respectively.
These
vanishingly small errors confirm that near-field optical interactions operating in these
resonant systems are extremely weak at separation distances greater than 3 light
wavelengths.
The intensity of the evanescent electric field components which exist in the
vicinity of a scattering particle decay exponentially with increasing distance from the
particle. Based upon this physical argument, the three curves shown in Figure 69 and
Figure 70 have been fit using a parametric, exponential curve of the following form:
Equation 81
 ( x − b )
S (%) = 100 + a exp 
 c 
where the left-hand argument can be either S or σ, and x is the separation distance
between the centers of neighboring particles expressed in units of light wavelength in
the medium. The second term on the right-hand side of Equation 81 is the deviation
with separation distance in the scattering efficiency of a multiple-particle cluster
relative to an isolated primary particle. The parameter b is equal to the center-to-center
spacing of touching particles expressed in units of light wavelength, equal to 0.47 for
both the 200-nm titania and latex spheres and 0.47 for the two morphological rutile
particles. The parameters a and c are variable fitting parameters. The parameter a is
approximately equal to the percent deviation in the left-hand argument for touching
particles, relative to the 100% value associated with an isolated primary particle. The
parameter c is a decay constant equal to the distance, expressed in light wavelengths,
198
over which the left-hand argument in Equation 81 decreases by the factor e
(2.718282…).
The results of these curve fits are presented in Table 10, which shows the values
of the fitting parameters a and c for each case and the value of the quality of fit
parameter R2. This quantity is equal to unity for perfect curve fits and less than one for
curve fits which do not intersect all of the points in a data set. Wide variations in the
value of the fitting parameter a are observed in Table 10 for the six different cases
studied, ranging from −34.2 to 74.3; these are consistent with the variations observed in
the scattering coefficients S and σ for touching particles compared to an isolated
primary particle.. The value of the fitting parameter c varies in the range 0.24-0.46.
The variation in the parameter c reflects differences in the percent change in scattering
efficiency relative to an isolated primary particle in the different systems studied.
Larger c values indicate slower convergence of the scattering coefficients to the value
100% with increasing interparticle separation distance. The R2 values in Table 10 are
greater than or equal to 0.93 in all but one case, indicating that the exponential curve
defined in Equation 81 is a sound basis for describing the changes in the scattering
coefficients observed in Figure 69 and Figure 70.
In the case of the scattering
coefficient σ for the seven latex spheres, the R2 value of 0.66 indicates a poor fit. This
result is not surprising considering the erratic behavior of σ in this case.
199
Table 10. Results of parametric fits of the six curves shown in Figure
69 and Figure 70 using the expression in Equation 81.
Case
a
c
R2 value
S, Seven TiO2
-15.9
0.26
0.93
S, Seven Latex
74.3
0.40
0.97
S, Two Morph. Rutiles
-8.29
0.37
0.97
σ, Seven TiO2
-34.2
0.46
0.99
σ, Seven Latex
-6.09
0.24
0.66
σ, Two Morph. Rutiles
-19.2
0.43
0.99
200
Seven TiO2 in Resin
Seven Latex in Water
Two Morph. Rutiles in Resin
Single Particle
S (% Sprimary)
180
160
140
120
100
80
60
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
Separation (λ)
Figure 69. Scattering coefficient S (%) for the three different multiple-particle
systems as a function of the center-to-center interparticle spacing expressed in
wavelengths. The value 100% corresponds to the S value for an isolated primary
particle in each case.
200
110
σ (% σprimary)
100
90
80
Seven TiO2 in Resin
Seven Latex in Water
Two Morph. Rutiles in Resin
Single Particle
70
60
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
Separation (λ)
Figure 70. Angle-weighted scattering coefficient σ (%) for the three multipleparticle systems as a function of the center-to-center interparticle spacing
expressed in wavelengths. The value 100% corresponds to the σ value for an
isolated primary particle in each case.
8.4.2 Extension to Other Materials Systems
The combined use of equivalent-mass, single spheres to approximate the light
scattering properties of particle agglomerates and the curve-fitting methodology for
particle clusters described in Section 8.4.1 suggests a means of approximating the light
scattering properties of systems other than titania and latex. As described in Section
8.3, it is possible to estimate the scattering properties of particle agglomerates through
Mie theory calculations on equivalent-mass spheres. In addition, the results described
in Section 8.4.1 indicate that the scattering coefficient for a particle cluster as a function
of interparticle separation varies between the values associated with the agglomerate
(touching particles) to that of an isolated primary particle (with very large separation
distances). The exponential curve defined in Equation 81 can be used to compute
201
approximately the dependence of S or σ on interparticle separation distance by setting
the parameter a equal to the percent change in scattering coefficient for the agglomerate
(estimated from Mie theory) and setting the parameter c equal to a value in the range
0.25-0.45.
The difficulties associated with this generalization to unknown materials
systems lie in the inaccuracy of using Mie theory to estimate the scattering properties of
agglomerated primary particles and the range of possible values in the exponential
decay length c in Equation 81. The two systems studied, titania in resin and latex in
water, represent two extreme cases in optical contrast, which is useful in defining
bounds on the value of the parameter c. On the other hand, the 200-nm titania primary
particle is the optimal size for scattering efficiency while the 200-nm latex sphere is
smaller than optimal. It would be interesting to determine the scattering properties of
sphere clusters for sub-optimal titania particles and optimal latex particles to complete
the space of primary particle sizes investigated. It should also be pointed out that the
simple exponential form of the expressing in Equation 81 may not be an adequate
description of more complex cases, such as a case where the scattering coefficient S or
σ associated with agglomerates of primary particles passes through a maximum with
changing interparticle separation distance.
202
Chapter 9. Conclusions
In the present study, a computational optics approach using a time-domain finite
element technique has been applied to determine the light scattering properties of
complex microstructures in the resonant regime. The scattering behavior such systems
have been largely unknown due to the difficulties associated with performing the largescale computations required for numerical solutions to Maxwell’s equations. Especially
difficult in this respect are individual scattering features with highly asymmetric
geometries and ensembles containing many scattering features with near−field optical
interactions occurring between them. In this chapter, general conclusions are drawn
from the details of the work presented in earlier chapters.
In the application of any numerical code to simulate physical phenomena, it is
critical that the accuracy of the code be tested against case with a known, analytical
solution. The analytical test case should be as closely related as possible to the physical
phenomena being simulated. The results of numerical methods always contain some
degree of error, and systematic quantification of this error under various simulation
conditions provides an important basis for applying the numerical method with
appropriate confidence in cases having no known solution. In the present work on light
scattering by three-dimensional objects, the analytical results of Mie theory are very
well suited for comparison with the time-domain finite element approach. It is clear
from the accuracy testing in the present study that the construction of the finite element
models bears great influence upon the accuracy of the numerical method.
203
It has been demonstrated in the computation of the light scattering properties of
core-shell pigment particles that the light scattering efficiency of composite particles
containing high- and low-index materials is limited by the content of the lower
refractive index material. In the case of particles whose sizes are on the order of the
illuminating light wavelength, the presence of a lower refractive index material creates
a particle having a reduced overall refractive index, which in turn shifts the optimal
diameter for scattering to larger sizes and reduced the maximum scattering efficiency
possible. While it may appear attractive to create a scattering particle containing
multiple interfaces between two index mismatched materials, these interfaces do not
produce the desired additional refraction and reflection in the particle since these
inhomogeneities are small compared to the light wavelength and effectively invisible.
In the case of light scattering by anisotropic spheres, it has been demonstrated
that, while the effects of optical anisotropy are pronounced when particular angles of
light incidence and polarizations are considered, these effects are effectively washed out
under conditions of random illumination. Reasonable agreement was found between
the results of random illumination of anisotropic spheres using the time-domain finite
element approach and two approximate models based upon scattering by isotropic
spheres. This implies that ensembles of randomly oriented particles which are optically
anisotropic or irregularly shaped may be modeled accurately using optically isotropic
materials. This will certainly not be true in cases involving particles with high aspect
ratios under processing conditions which cause the particles to adopt preferred
orientations.
204
Computation of the light scattering properties of a morphological rutile titania
particle highlighted the inherent difficulty in fitting scattering data by complex particle
shapes to equivalent spherical diameters. As in the case of the anisotropic spheres,
approximating the shaped, optically anisotropic rutile particle as an equivalent volume,
optically isotropic sphere predicted reasonably well the orientation-averaged scattering
properties of the particle. The inverse problem of fitting a spherical particle size based
upon the scattering properties of the complex shaped particle, however, presents the
difficulty of representing particles which require multiple geometrical parameters in
their description as spherical particles with just geometrical parameter. Indeed, strictly
speaking, equivalent spherical diameters computed in the inverse problem are not
meaningful.
In the case of irregularly shaped particles, particle sizing by light
scattering methods may provide a sound measure of engineering control, but the results
of such particle sizing should not be taken in an absolute sense.
The light scattering properties of clusters of multiple spheres of both high and
low refractive index have been studied systematically using the time-domain finite
element approach.
These computations demonstrated that the near-field optical
interactions which act between neighboring particles in wavelength-sized, hard contact
clusters produce light scattering properties which deviate substantially from those of the
individual primary particles composing the cluster. The effects of particle clustering
upon scattering efficiency depends upon the relationship of the size of the primary
particles in the cluster compared to the optimal particle size for scattering in that
materials system. In the case of primary particles which are smaller in size than the
205
optimal size, clustering will produce improved light scattering efficiency compared to
the primary particles. This improvement will be less dramatic than producing fully
dense, single particles of the same mass since the illuminating light samples both the
particles in the cluster and the interstitial material. In the case of primary particles
which are nearly optimal in size for light scattering efficiency, clustering causes a
reduction in scattering efficiency which can be dramatic. In the case of rutile titania
particles, for example, losses in volume-normalized scattering efficiency of 70% occur
for the largest agglomerates studied. This reduction is more pronounced in highcontrast systems, which exhibit strong resonances with the illuminating light at
particular sizes and produce very high scattering efficiencies.
For separated clusters of either latex or titania particles, near-field optical
interactions between neighboring particles diminish rapidly with increasing particle
separation, with appreciable near-field interactions not extending beyond separations of
2-3 light wavelengths. In most cases, changes in scattering efficiency with interparticle
separation distance could be well fit using an exponential decay. The decay constant of
these spatial variations varies in the range 0.25-0.45 light wavelengths for these
systems, with appreciable optical interactions not extending beyond 2-3 light
wavelengths.
206
Appendix. Typical EMFlex Input Deck
grid 111 83 83
geom
xcrd -0.15e-6 0.25e-6
ycrd -0.15e-6 0.15e-6
zcrd -0.15e-6 0.15e-6
end
matr
prop poly
prop rutl
20.30e-12 0.0 1.2566e-6 /* m = 1.51 + 0.0i
66.47e-12 0.0 1.2566e-6 /* m = 2.74 + 0.0i
end
topo
regn poly
sphr rutl 0. 0. 0. 0.1e-6
/* 0.2-um diameter sphere centered at (0,0,0)
end
ilum
symb c0 = 2.998e8
symb wavl = 0.560e-6
symb freq = \$c0 / \$wavl
/* vertically propagating, z-polarized plane wave:
pwav wav1 \$freq 1. 0. 1. 0. 0. 0. 0. 1.
ffld 1 1
207
boun
/* illumination boundary conditions on all six sides
side 1 ilum
side 2 ilum
side 3 ilum
side 4 ilum
side 5 ilum
side 6 ilum
calc
wndo 1 * 1 * 1 * /* calculate amp & phase over entire model
esst scat
intn
/* calculate scattered quantities
/* calculate intensity
pout rate 1
hist ez 20 60 20 20 60 20 20 60 20
/*Collect time histories in the model
rest no
prcs
data out modl
exec cycl 15 /* execute to steady-state
calc on /* calculate amplitude & phase
end
exec cycl /* execute 1 cycle
kirc
/* Kirchoff box definition
ref cntr 0. 0. 0.
208
defn srf1 cntr in
node 5 5 5 77 5 77
node 105 105 5 77 5 77
node 5 105 5 5 5 77
node 5 105 77 77 5 77
node 5 105 5 77 5 5
node 5 105 5 77 77 77
writ srf1
data /* export data for plotting with Review
out insc
stop
209
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