Cover image for Particle and particle systems characterization : small-angle scattering (SAS) applications
Title:
Particle and particle systems characterization : small-angle scattering (SAS) applications
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Publication Information:
Boca Raton : CRC Press/Taylor & Francis, 2014
Physical Description:
xii, 336 pages : illustrations ; 24 cm.
ISBN:
9781466581777

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30000010330280 TA418.8 G55 2014 Open Access Book Book
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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

Prefacep. ix
1 Scattering experiment and structure functions; particles and the correlation function of small-angle scatteringp. 1
1.1 Elastic scattering of a plane wave by a thin samplep. 3
1.1.1 Guinier approximation and Kaya's scattering patternsp. 7
1.1.2 Scattering intensity in terms of structure functionsp. 14
1.1.3 Particle description via real-space structure functionsp. 19
1.2 SAS structure functions and scattering intensityp. 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 SASp. 30
1.3.1 Sample density, particle models and structure functionsp. 32
1.4 SAS structure functions for a fixed order range Lp. 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), sp. 39
1.5 Aspects of data evaluation for a specific Lp. 44
1.5.1 The invariant of the smoothed scattering pattern I Lp. 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 figuresp. 59
2.1 The cone case-an instructive examplep. 60
2.1.1 Geometry of the cone casep. 61
2.1.2 Flat, balanced, well-balanced and steep conesp. 66
2.1.3 Summarizing remarks about the CLDD of the conep. 68
2.2 Establishing and representing CLDDsp. 69
2.2.1 Mathematica programs for determining CLDDs?p. 69
2.3 Parallelepiped and limiting casesp. 71
2.3.1 The unit cubep. 73
2.4 Right circular cylinderp. 73
2.5 Ellipsoid and limiting casesp. 74
2.6 Regular tetrahedron (unit length case a = 1)p. 78
2.7 Hemisphere and hemisphere shellp. 80
2.7.1 Mean CLDD and size distribution of hemisphere shellsp. 81
2.8 The Large Giza Pyramid as a homogeneous bodyp. 81
2.8.1 Approach for determining ¿(r) and A ¿ (r)p. 82
2.8.2 Analytic results for small chords rp. 83
2.9 Rhombic prism Y based on the plane rhombus Xp. 87
2.10 Scattering pattern I(h) and CLDD A(r) of a lensp. 88
3 Chord length distributions of infinitely long cylindersp. 95
3.1 The infinitely long cylinder casep. 96
3.2 Transformation 1: From the right section of a cylinder to a spatial cylinderp. 97
3.2.1 Pentagonal and hexagonal rodsp. 98
3.2.2 Triangle/triangular rod and rectangle/rectangular rodp. 100
3.2.3 Ellipse/elliptic rod and the elliptic needlep. 100
3.2.4 Semicircular rod of radiusp. 102
3.2.5 Wedge cases and triangular/rectangular rodsp. 102
3.2.6 Infinitely long hollow cylinderp. 102
3.3 Recognition analysis of rods with oval right section from the SAS correlation functionp. 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 cylinderp. 107
3.5 Specific particle parameters in terms of chord length moments: The case of dilated cylindersp. 109
3.6 Cylinders of arbitrary height H with oval RSp. 110
3.7 CLDDs of particle ensembles with size distributionp. 114
4 Particle-to-particle interference - a useful toolp. 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 functionp. 116
4.1.2 Different working functions and denotations in different fieldsp. 118
4.2 Quasi-diluted and non-touching particlesp. 119
4.3 Correlation function and scattering pattern of two infinitely long parallel cylindersp. 127
4.4 Fundamental connection between ¿(r), c and g(r)p. 132
4.5 Cylinder arrays and packages of parallel infinitely long circular cylindersp. 148
4.6 Connections between SAS and WASp. 155
4.6.1 The function FREQ(¿ k ) describes all distances ¿ kp. 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 functionp. 162
5 Scattering patterns and structure functions of Boolean modelsp. 169
5.1 Short-order range approach for order less systemsp. 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 = np. 172
5.2.2 The chord length distributions of both phasesp. 174
5.2.3 Moments of the CLDD for both phases of the Bmp. 176
5.2.4 The second moments of ¿(l) and f(m) fix c; 0 ≤ cp. 177
5.2.5 Interrelated CLDD moments and scattering patternsp. 177
5.3 Inserting spherical grains of constant diameterp. 179
5.4 Size distribution of spherical grainsp. 184
5.5 Chord length distributions of the Poisson slice modelp. 188
5.6 Practical relevance of Boolean modelsp. 192
6 The "Dead Leaves" modelp. 193
6.1 Structure functions and scattering pattern of a PCp. 195
6.2 The uncovered "Dead Leaves" modelp. 205
7 Tessellations, fragment particles and puzzlesp. 207
7.1 Tessellations: original state and destroyed statep. 209
7.2 Puzzle particles resulting from DLm tessellationsp. 210
7.3 Punch-matrix/particle puzzlesp. 214
7.4 Analysis of nearly arbitrary fragment particles via their CLDDp. 220
7.5 Predicting the fitting ability of fragments from SASp. 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 modelp. 239
8.1.1 LSM for an amorphous state of an AlDyNi alloyp. 247
8.1.2 LSM analysis of a VYCOR glass of 33% porosityp. 249
8.1.3 Concluding remarks on the LSM approachp. 250
8.2 Analysis of porous materials via ¿-chordsp. 251
8.2.1 Pore analysis of a silica aerogel from SAS datap. 253
8.2.2 Macropore analysis of a controlled porous glassp. 255
8.3 The volume fraction depends on the order range Lp. 257
8.4 The Synecek approach for ensembles of spheresp. 259
8.5 Volume fraction investigation of Boolean modelsp. 261
8.6 About the realistic porosity of porous materialsp. 262
9 Interrelations between the moments of the chord length distributions of random two-phase systemsp. 269
9.1 Single particle case and particle ensemblesp. 270
9.2 Interrelations between CLD moments of random particle ensemblesp. 272
9.2.1 Connection between the three functions g, ¿ and fp. 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 conclusionsp. 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 fractionp. 288
10 Exercises on problems of particle characterization: examplesp. 289
10.1 The phase difference in a point of observation Pp. 290
10.2 Scattering pattern, CF and CLDD of single particlesp. 292
10.2.1 Determination of particle size distributions for a fixed known particle shapep. 292
10.2.2 About the Pi plot of a scattering patternp. 293
10.2.3 Comparing single particle correlation functionsp. 294
10.2.4 Scattering pattern of a hemisphere of radius Rp. 295
10.2.5 The "butterfly cylinder" and its scattering patternp. 296
10.2.6 Scattering equivalence of (widely separated) particlesp. 299
10.2.7 About the significance of the SAS correlation functionp. 299
10.2.8 The mean chord length of an elliptic needlep. 301
10.3 Structure functions parameters of special modelsp. 303
10.3.1 The first zero of the SAS CFp. 303
10.3.2 Different models, different scattering patternsp. 304
10.3.3 Boolean model contra hard single particles and quasi-diluted particle ensemblesp. 306
10.3.4 A special relation for detecting the c of a Bmp. 307
10.3.5 Properties of the SAS CF of a Bmp. 308
10.3.6 DLm from Poisson polyhedral grainsp. 309
10.4 Moments of g(r), integral parameters and cp. 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 parametersp. 311
10.4.3 Volume fraction and integral parametersp. 312
10.4.4 Application of Eq. (9.16) for a ceramic micropowderp. 313
Referencesp. 315
Indexp. 333