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Summary
Summary
Small-angle scattering (SAS) is the premier technique for the characterization of disordered nanoscale particle ensembles. SAS is produced by the particle as a whole and does not depend in any way on the internal crystal structure of the particle. Since the first applications of X-ray scattering in the 1930s, SAS has developed into a standard method in the field of materials science. SAS is a non-destructive method and can be directly applied for solid and liquid samples.
Particle and Particle Systems Characterization: Small-Angle Scattering (SAS) Applications is geared to any scientist who might want to apply SAS to study tightly packed particle ensembles using elements of stochastic geometry. After completing the book, the reader should be able to demonstrate detailed knowledge of the application of SAS for the characterization of physical and chemical materials.
Table of Contents
Preface | p. ix |
1 Scattering experiment and structure functions; particles and the correlation function of small-angle scattering | p. 1 |
1.1 Elastic scattering of a plane wave by a thin sample | p. 3 |
1.1.1 Guinier approximation and Kaya's scattering patterns | p. 7 |
1.1.2 Scattering intensity in terms of structure functions | p. 14 |
1.1.3 Particle description via real-space structure functions | p. 19 |
1.2 SAS structure functions and scattering intensity | p. 22 |
1.2.1 Scattering pattern, SAS correlation function and chord length distribution density (CLDD) | p. 22 |
1.2.2 Indication of homogeneous particles by i(r A ) | p. 26 |
1.3 Chord length distributions and SAS | p. 30 |
1.3.1 Sample density, particle models and structure functions | p. 32 |
1.4 SAS structure functions for a fixed order range L | p. 34 |
1.4.1 Correlation function in terms of the intensity I L (h, L) | p. 38 |
1.4.2 Extension to the realistic experiment I(s), s | p. 39 |
1.5 Aspects of data evaluation for a specific L | p. 44 |
1.5.1 The invariant of the smoothed scattering pattern I L | p. 53 |
1.5.2 How can a suitable order range L for L-smoothing be selected from an experimental scattering pattern? | p. 55 |
2 Chord length distribution densities of selected elementary geometric figures | p. 59 |
2.1 The cone case-an instructive example | p. 60 |
2.1.1 Geometry of the cone case | p. 61 |
2.1.2 Flat, balanced, well-balanced and steep cones | p. 66 |
2.1.3 Summarizing remarks about the CLDD of the cone | p. 68 |
2.2 Establishing and representing CLDDs | p. 69 |
2.2.1 Mathematica programs for determining CLDDs? | p. 69 |
2.3 Parallelepiped and limiting cases | p. 71 |
2.3.1 The unit cube | p. 73 |
2.4 Right circular cylinder | p. 73 |
2.5 Ellipsoid and limiting cases | p. 74 |
2.6 Regular tetrahedron (unit length case a = 1) | p. 78 |
2.7 Hemisphere and hemisphere shell | p. 80 |
2.7.1 Mean CLDD and size distribution of hemisphere shells | p. 81 |
2.8 The Large Giza Pyramid as a homogeneous body | p. 81 |
2.8.1 Approach for determining ¿(r) and A ¿ (r) | p. 82 |
2.8.2 Analytic results for small chords r | p. 83 |
2.9 Rhombic prism Y based on the plane rhombus X | p. 87 |
2.10 Scattering pattern I(h) and CLDD A(r) of a lens | p. 88 |
3 Chord length distributions of infinitely long cylinders | p. 95 |
3.1 The infinitely long cylinder case | p. 96 |
3.2 Transformation 1: From the right section of a cylinder to a spatial cylinder | p. 97 |
3.2.1 Pentagonal and hexagonal rods | p. 98 |
3.2.2 Triangle/triangular rod and rectangle/rectangular rod | p. 100 |
3.2.3 Ellipse/elliptic rod and the elliptic needle | p. 100 |
3.2.4 Semicircular rod of radius | p. 102 |
3.2.5 Wedge cases and triangular/rectangular rods | p. 102 |
3.2.6 Infinitely long hollow cylinder | p. 102 |
3.3 Recognition analysis of rods with oval right section from the SAS correlation function | p. 104 |
3.3.1 Behavior of the cylinder CF for r → ∞ | p. 104 |
3.4 Transformation 2: From spatial cylinder C to the base X of the cylinder | p. 107 |
3.5 Specific particle parameters in terms of chord length moments: The case of dilated cylinders | p. 109 |
3.6 Cylinders of arbitrary height H with oval RS | p. 110 |
3.7 CLDDs of particle ensembles with size distribution | p. 114 |
4 Particle-to-particle interference - a useful tool | p. 115 |
4.1 Particle packing is characterized by the pair correlation function g(r) | p. 116 |
4.1.1 Explanation of the function g(r) and Ripley's K function | p. 116 |
4.1.2 Different working functions and denotations in different fields | p. 118 |
4.2 Quasi-diluted and non-touching particles | p. 119 |
4.3 Correlation function and scattering pattern of two infinitely long parallel cylinders | p. 127 |
4.4 Fundamental connection between ¿(r), c and g(r) | p. 132 |
4.5 Cylinder arrays and packages of parallel infinitely long circular cylinders | p. 148 |
4.6 Connections between SAS and WAS | p. 155 |
4.6.1 The function FREQ(¿ k ) describes all distances ¿ k | p. 155 |
4.6.2 Scattering pattern of an aggregate of N spheres (AN) | p. 158 |
4.7 Chord length distributions: An alternative approach to the pair correlation function | p. 162 |
5 Scattering patterns and structure functions of Boolean models | p. 169 |
5.1 Short-order range approach for order less systems | p. 170 |
5.2 The Boolean model for convex grains - the set ¿ | p. 171 |
5.2.1 Connections between the functions ¿(r) and ¿ 0 (r) for arbitrary grains of density N = n | p. 172 |
5.2.2 The chord length distributions of both phases | p. 174 |
5.2.3 Moments of the CLDD for both phases of the Bm | p. 176 |
5.2.4 The second moments of ¿(l) and f(m) fix c; 0 ≤ c | p. 177 |
5.2.5 Interrelated CLDD moments and scattering patterns | p. 177 |
5.3 Inserting spherical grains of constant diameter | p. 179 |
5.4 Size distribution of spherical grains | p. 184 |
5.5 Chord length distributions of the Poisson slice model | p. 188 |
5.6 Practical relevance of Boolean models | p. 192 |
6 The "Dead Leaves" model | p. 193 |
6.1 Structure functions and scattering pattern of a PC | p. 195 |
6.2 The uncovered "Dead Leaves" model | p. 205 |
7 Tessellations, fragment particles and puzzles | p. 207 |
7.1 Tessellations: original state and destroyed state | p. 209 |
7.2 Puzzle particles resulting from DLm tessellations | p. 210 |
7.3 Punch-matrix/particle puzzles | p. 214 |
7.4 Analysis of nearly arbitrary fragment particles via their CLDD | p. 220 |
7.5 Predicting the fitting ability of fragments from SAS | p. 227 |
7.6 Porous materials as "drifted apart tessellations" | p. 230 |
8 Volume fraction of random two-phase samples for a fixed order range L from ¿(r, L) | p. 237 |
8.1 The linear simulation model | p. 239 |
8.1.1 LSM for an amorphous state of an AlDyNi alloy | p. 247 |
8.1.2 LSM analysis of a VYCOR glass of 33% porosity | p. 249 |
8.1.3 Concluding remarks on the LSM approach | p. 250 |
8.2 Analysis of porous materials via ¿-chords | p. 251 |
8.2.1 Pore analysis of a silica aerogel from SAS data | p. 253 |
8.2.2 Macropore analysis of a controlled porous glass | p. 255 |
8.3 The volume fraction depends on the order range L | p. 257 |
8.4 The Synecek approach for ensembles of spheres | p. 259 |
8.5 Volume fraction investigation of Boolean models | p. 261 |
8.6 About the realistic porosity of porous materials | p. 262 |
9 Interrelations between the moments of the chord length distributions of random two-phase systems | p. 269 |
9.1 Single particle case and particle ensembles | p. 270 |
9.2 Interrelations between CLD moments of random particle ensembles | p. 272 |
9.2.1 Connection between the three functions g, ¿ and f | p. 279 |
9.2.2 The moments M i , l i , m i in terms of Q(t), p(t), q(t) | p. 280 |
9.2.3 Analysis of the second moment M 2 = -Qö(0) | p. 280 |
9.3 CLD concept and data evaluation: Some conclusions | p. 284 |
9.3.1 Taylor series of Q(t) in terms of the moments M n of the function g(r) | p. 286 |
9.3.2 Sampling theorem, the number of independent SAS parameters, CLD moments and volume fraction | p. 288 |
10 Exercises on problems of particle characterization: examples | p. 289 |
10.1 The phase difference in a point of observation P | p. 290 |
10.2 Scattering pattern, CF and CLDD of single particles | p. 292 |
10.2.1 Determination of particle size distributions for a fixed known particle shape | p. 292 |
10.2.2 About the Pi plot of a scattering pattern | p. 293 |
10.2.3 Comparing single particle correlation functions | p. 294 |
10.2.4 Scattering pattern of a hemisphere of radius R | p. 295 |
10.2.5 The "butterfly cylinder" and its scattering pattern | p. 296 |
10.2.6 Scattering equivalence of (widely separated) particles | p. 299 |
10.2.7 About the significance of the SAS correlation function | p. 299 |
10.2.8 The mean chord length of an elliptic needle | p. 301 |
10.3 Structure functions parameters of special models | p. 303 |
10.3.1 The first zero of the SAS CF | p. 303 |
10.3.2 Different models, different scattering patterns | p. 304 |
10.3.3 Boolean model contra hard single particles and quasi-diluted particle ensembles | p. 306 |
10.3.4 A special relation for detecting the c of a Bm | p. 307 |
10.3.5 Properties of the SAS CF of a Bm | p. 308 |
10.3.6 DLm from Poisson polyhedral grains | p. 309 |
10.4 Moments of g(r), integral parameters and c | p. 310 |
10.4.1 Properties of the moment M 2 of g(r) | p. 310 |
10.4.2 Tests of properties of M 2 special model parameters | p. 311 |
10.4.3 Volume fraction and integral parameters | p. 312 |
10.4.4 Application of Eq. (9.16) for a ceramic micropowder | p. 313 |
References | p. 315 |
Index | p. 333 |