Particle and Particle Systems Characterization: Small-Angle Scattering (SAS) Applications

346 Pages·2013·14.76 MB·English

by  Wilfried Gille

Condensed Matter & Materials Science PARTICLE AND
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il PARTICLE SYSTEMS
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e
PARTICLE AND
CHARACTERIZATION
PARTICLE SYSTEMS
CHARACTERIZATION P Small-Angle Scattering
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( )
Small-Angle Scattering R Applications
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( )
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Applications
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Small-angle scattering (SAS) is the premier technique for the
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characterization of disordered nanoscale particle ensembles. SAS
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is produced by the particle as a whole and does not depend in any D
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way on the internal crystal structure of the particle. Since the first
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applications of X-ray scattering in the 1930s, SAS has developed
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into a standard method in the field of materials science. SAS is a
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non-destructive method and can be directly applied for solid and
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liquid samples.
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Particle and Particle Systems Characterization: Small-Angle AC
Scattering (SAS) Applications is geared to any scientist who might TL
want to apply SAS to study tightly packed particle ensembles using IE
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elements of stochastic geometry. After completing the book, the
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reader should be able to demonstrate detailed knowledge of the Y
application of SAS for the characterization of physical and chemical S
materials. T
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K18899
Wilfried Gille
K18899_Cover.indd 1 10/21/13 3:48 PM
PARTICLE AND PARTICLE SYSTEMS
CHARACTERIZATION
K18899_FM.indd 1 9/30/13 1:06 PM
K18899_FM.indd 2 9/30/13 1:06 PM
PARTICLE AND PARTICLE SYSTEMS
CHARACTERIZATION
Small-Angle Scattering (SAS) Applications
Dr. Wilfried Gille
Martin Luther University of Halle-Wittenberg
K18899_FM.indd 3 9/30/13 1:06 PM
CRC Press
Taylor & Francis Group
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Contents
Preface ix
1 Scattering experiment and structure functions; particles and
the correlation function of small-angle scattering 1
1.1 Elastic scattering of a plane wave by a thin sample . . . . . . 3
1.1.1 Guinier approximation and Kaya’s scattering patterns 7
1.1.2 Scattering intensity in terms of structure functions . . 14
1.1.3 Particle description via real-space structure functions . 19
1.2 SAS structure functions and scattering intensity . . . . . . . . 22
1.2.1 Scatteringpattern,SAScorrelationfunctionandchord
length distribution density (CLDD). . . . . . . . . . . 22
1.2.2 Indication of homogeneous particles by i(r ) . . . . . 26
A
1.3 Chord length distributions and SAS . . . . . . . . . . . . . . 30
1.3.1 Sample density, particle models and structure func-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.4 SAS structure functions for a fixed order range L . . . . . . . 34
1.4.1 Correlationfunction in terms of the intensity I (h,L) 38
L
1.4.2 Extension to the realistic experiment I(s), s<s . 39
max
1.5 Aspects of data evaluation for a specific L . . . . . . . . . . . 44
1.5.1 The invariant of the smoothed scattering pattern I . 53
L
1.5.2 How can a suitable order range L for L-smoothing be
selected from an experimental scattering pattern? . . . 55
2 Chord lengthdistributiondensitiesofselectedelementaryge-
ometric figures 59
2.1 The cone case–an instructive example . . . . . . . . . . . . . 60
2.1.1 Geometry of the cone case . . . . . . . . . . . . . . . . 61
2.1.2 Flat, balanced, well-balanced and steep cones . . . . . 66
2.1.3 Summarizing remarks about the CLDD of the cone . . 68
2.2 Establishing and representing CLDDs . . . . . . . . . . . . . 69
2.2.1 Mathematica programs for determining CLDDs? . . . 69
2.3 Parallelepiped and limiting cases . . . . . . . . . . . . . . . . 71
2.3.1 The unit cube . . . . . . . . . . . . . . . . . . . . . . . 73
2.4 Right circular cylinder . . . . . . . . . . . . . . . . . . . . . . 73
2.5 Ellipsoid and limiting cases . . . . . . . . . . . . . . . . . . . 74
2.6 Regular tetrahedron (unit length case a=1) . . . . . . . . . 78
v
vi Contents
2.7 Hemisphere and hemisphere shell . . . . . . . . . . . . . . . . 80
2.7.1 Mean CLDD and size distribution of hemisphere shells 81
2.8 The Large Giza Pyramid as a homogeneous body . . . . . . . 81
2.8.1 Approach for determining γ(r) and A (r) . . . . . . . 82
µ
2.8.2 Analytic results for small chords r . . . . . . . . . . . 83
2.9 Rhombic prism Y based on the plane rhombus X . . . . . . . 87
2.10 Scattering pattern I(h) and CLDD A(r) of a lens . . . . . . . 88
3 Chord length distributions of infinitely long cylinders 95
3.1 The infinitely long cylinder case . . . . . . . . . . . . . . . . . 96
3.2 Transformation 1: From the right section of a cylinder to a
spatial cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.2.1 Pentagonaland hexagonal rods . . . . . . . . . . . . . 98
3.2.2 Triangle/triangularrod and rectangle/rectangularrod 100
3.2.3 Ellipse/elliptic rod and the elliptic needle . . . . . . . 100
3.2.4 Semicircular rod of radius R . . . . . . . . . . . . . . . 102
3.2.5 Wedge cases and triangular/rectangularrods . . . . . 102
3.2.6 Infinitely long hollow cylinder . . . . . . . . . . . . . . 102
3.3 Recognition analysis of rods with oval right section from the
SAS correlation function . . . . . . . . . . . . . . . . . . . . . 104
3.3.1 Behavior of the cylinder CF for r . . . . . . . . . 104
→∞
3.4 Transformation 2: From spatial cylinder C to the base X of
the cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.5 Specificparticleparametersintermsofchordlengthmoments:
The case of dilated cylinders . . . . . . . . . . . . . . . . . . . 109
3.6 Cylinders of arbitrary height H with oval RS . . . . . . . . . 110
3.7 CLDDs of particle ensembles with size distribution . . . . . . 114
4 Particle-to-particle interference – a useful tool 115
4.1 Particle packing is characterized by the pair correlation func-
tion g(r) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.1.1 Explanationofthefunctiong(r)andRipley’sK func-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.1.2 Different working functions and denotations in differ-
ent fields . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.2 Quasi-diluted and non-touching particles . . . . . . . . . . . . 119
4.3 Correlation function and scattering pattern of two infinitely
long parallel cylinders . . . . . . . . . . . . . . . . . . . . . . 127
4.4 Fundamental connection between γ(r), c and g(r) . . . . . . . 132
4.5 Cylinderarraysandpackagesofparallelinfinitelylongcircular
cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.6 Connections between SAS and WAS . . . . . . . . . . . . . . 155
4.6.1 The function FREQ(r ) describes all distances r . . 155
k k
4.6.2 Scattering pattern of an aggregate of N spheres (AN) 158
Contents vii
4.7 Chord length distributions: An alternative approach to the
pair correlationfunction . . . . . . . . . . . . . . . . . . . . . 162
5 Scattering patterns and structure functions of Boolean mod-
els 169
5.1 Short-order range approachfor orderless systems . . . . . . . 170
5.2 The Boolean model for convex grains – the set Ξ . . . . . . . 171
5.2.1 Connections between the functions γ(r) and γ (r) for
0
arbitrary grains of density N =n . . . . . . . . . . . . 172
5.2.2 The chord length distributions of both phases . . . . . 174
5.2.3 Moments of the CLDD for both phases of the Bm . . 176
5.2.4 The second moments of ϕ(l) and f(m) fix c; 0 c<1 177
≤
5.2.5 Interrelated CLDD moments and scattering patterns . 177
5.3 Inserting spherical grains of constant diameter . . . . . . . . . 179
5.4 Size distribution of spherical grains . . . . . . . . . . . . . . . 184
5.5 Chord length distributions of the Poissonslice model . . . . . 188
5.6 Practical relevance of Boolean models . . . . . . . . . . . . . 192
6 The “Dead Leaves” model 193
6.1 Structure functions and scattering pattern of a PC . . . . . . 195
6.2 The uncovered “Dead Leaves” model . . . . . . . . . . . . . . 205
7 Tessellations, fragment particles and puzzles 207
7.1 Tessellations: original state and destroyed state . . . . . . . . 209
7.2 Puzzle particles resulting from DLm tessellations . . . . . . . 210
7.3 Punch-matrix/particle puzzles . . . . . . . . . . . . . . . . . . 214
7.4 Analysis of nearly arbitrary fragment particles via their CLDD 220
7.5 Predicting the fitting ability of fragments from SAS . . . . . . 227
7.6 Porous materials as “drifted apart tessellations” . . . . . . . . 230
8 Volume fraction of random two-phase samples for a fixed or-
der range L from γ(r,L) 237
8.1 The linear simulation model . . . . . . . . . . . . . . . . . . . 239
8.1.1 LSM for an amorphous state of an AlDyNi alloy . . . 247
8.1.2 LSM analysis of a VYCOR glass of 33% porosity . . . 249
8.1.3 Concluding remarks on the LSM approach . . . . . . . 250
8.2 Analysis of porous materials via ν-chords . . . . . . . . . . . 251
8.2.1 Pore analysis of a silica aerogel from SAS data . . . . 253
8.2.2 Macropore analysis of a controlled porous glass . . . . 255
8.3 The volume fraction depends on the order range L . . . . . . 257
8.4 The Synecek approach for ensembles of spheres . . . . . . . . 259
8.5 Volume fraction investigation of Boolean models . . . . . . . 261
8.6 About the realistic porosity of porous materials . . . . . . . . 262
viii Contents
9 Interrelations between the moments of the chord length dis-
tributions of random two-phase systems 269
9.1 Single particle case and particle ensembles . . . . . . . . . . . 270
9.2 Interrelations between CLD moments of random particle en-
sembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
9.2.1 Connection between the three functions g, ϕ and f . . 279
9.2.2 The moments M , l , m in terms of Q(t), p(t), q(t) . . 280
i i i
9.2.3 Analysis of the second moment M = Q (0) . . . . . 280
2 ′′
−
9.3 CLD concept and data evaluation: Some conclusions . . . . . 284
9.3.1 Taylor series of Q(t) in terms of the moments M of
n
the function g(r) . . . . . . . . . . . . . . . . . . . . . 286
9.3.2 Sampling theorem, the number of independent SAS
parameters,CLD moments and volume fraction . . . . 288
10 Exercises on problems of particle characterization: examples289
10.1 The phase difference in a point of observation P . . . . . . . 290
10.2 Scattering pattern, CF and CLDD of single particles . . . . . 292
10.2.1 Determinationofparticlesizedistributions forafixed
known particle shape . . . . . . . . . . . . . . . . . . . 292
10.2.2 About the P plot of a scattering pattern . . . . . . . 293
1
10.2.3 Comparing single particle correlationfunctions . . . . 294
10.2.4 Scattering pattern of a hemisphere of radius R . . . . 295
10.2.5 The “butterfly cylinder” and its scattering pattern . . 296
10.2.6 Scattering equivalence of (widely separated) particles . 299
10.2.7 About the significance of the SAS correlation function 299
10.2.8 The mean chord length of an elliptic needle . . . . . . 301
10.3 Structure functions parameters of special models . . . . . . . 303
10.3.1 The first zero of the SAS CF . . . . . . . . . . . . . . 303
10.3.2 Different models, different scattering patterns . . . . . 304
10.3.3 Booleanmodel contrahardsingleparticlesandquasi-
diluted particle ensembles . . . . . . . . . . . . . . . . 306
10.3.4 A special relation for detecting the c of a Bm . . . . . 307
10.3.5 Properties of the SAS CF of a Bm . . . . . . . . . . . 308
10.3.6 DLm from Poisson polyhedral grains . . . . . . . . . . 309
10.4 Moments of g(r), integral parameters and c . . . . . . . . . . 310
10.4.1 Properties of the moment M of g(r) . . . . . . . . . . 310
2
10.4.2 Tests of properties of M special model parameters . . 311
2
10.4.3 Volume fraction and integral parameters . . . . . . . . 312
10.4.4 Application of Eq. (9.16) for a ceramic micropowder . 313
References 315
Index 333
Preface
The aim of this book is to demonstrate basic knowledge of the application of
small-anglescattering(SAS)forthecharacterizationofphysicalandchemical
materials in various fields to beginners and advanced scientists in a compre-
hensive way. The subject is divided into 10 chapters consisting of 336 pages,
including 164 figures and 237 references.
SAS is a technique of spatial density measurement; however, it does not
produce an image or hologram. A monochromatic neutron, X-ray or light
beam directed at a sample material containing small zones of non-uniform
density allows a scattering effect to be recorded. This phenomenon is based
on the interference property of electromagnetic waves.
The designation particle is a significant one in this book. Particles are en-
sembles of many atoms, which possess a (nearly) homogeneous composition.
Therefore, the general characterization of homogeneous regions of a hetero-
geneous body is referred to as particle characterization for short. SAS can
detect (yields) geometric parameters describing ensembles of small particles
downtothenanometerregion,i.e.,ofmicroscopicregionspossessingadensity
differentfromthatoftheirsurroundings. SASdoesnotdependinanywayon
the internal crystal structure of the particle and is produced by the particle
as a whole.
In contrast to the electron microscope, SAS experiments do not require a
specialsamplepreparationprocedure,areessentiallynon-destructiveandcan
be directly applied for liquid and solid samples.
While anelectronmicroscopeyieldsdirectreal-spaceimages,SASprovides
data in reciprocal space, which yields a scattering pattern. Compared with
a typical sample micrograph of well-selected magnification, the information
content of an interference intensity can be interpreted in many ways. This
interference intensity is calledthe scattering intensity or scattering pattern of
the sample for short. An interpretation of such data always requires a model
as well as initial assumptions about the sample material: For a certain order
range (i.e., on a certain length scale interval), the spatial density of many
materials can be approximated by a two-phase model. Such a model implies
two different densities. For the density contrast of particles, possessing a
constant density inside and a surrounding matrix, there exists a variety of
geometric approachesand applications, modeling sizes,shapes and geometric
arrangements. In the ideal case, both the size and shape of the particles
involvedinamodelcanbedeterminedfromthescatteringintensityexpressed
as a function of the scattering angle.
ix