214 pages, 130 figures, 2 pictures, 277 references.
ISBN 5-7692-0248-3
ББК В151.0 Г562 В371.2
УДК 514.11:54-14:54-16:539.2
М42
Publication was supported by Russian Foundation for Basic Researches (RFFI), project 99-03-47007.
The book presents a geometrical approach for studying the structure of non-crystalline systems: computer models of liquids, glasses, packings of balls, and any other systems which can be represented as an ensemble of discrete objects arbitrary distributed in space. The method is based on fundamental mathematical results of G.F.Voronoi and D.N.Delaunay about decomposition of space between discrete centers. Mathematical basis and methodology of the method are discussed in the book in details.
The method is applied to different physical problems. Voronoi polyhedra and Delaunay simplexes are used for analysis of local arrangement of atoms (a problem of short range order). The entire Voronoi-Delaunay tessellation is used for studying spatial correlations both between atoms and interatomic voids. In this case one can trace new features of the structure of liquids and glasses (so called medium range order). Another promising way of application of the method is “a percolation analysis” of the structure. The Voronoi network is a convenient instrument for such work. It lies in the depth between atoms and defines "fairways" through the bottle-necks inside atomic system. Using this network, one can study diffusion and percolation properties of the system.
The book is meant for students, post-graduate students and researchers who are concerned with computer simulation and structure study in physics, physical chemistry and material science. It can be helpful for both who intends to learn the method and who uses it in his work. There is no need in special geometrical knowledge for reading this book. The author tried to avoid a formal style in discussion of mathematical questions. However, presentation of the material is complete enough for readers to understand the base of the method and to apply it in scientific researches.
Contents of the book:
PREFACE 3
INTRODUCTION 5
PART I. BASIS OF THE METHOD 7
1. Mathematical sources of the method 7
1.1. Historical remarks 7
1.2. Fundamental results of Voronoi and Delaunay 9
2. A system of discrete points 12
2.1. Voronoi tessellation 12
2.1.1. Voronoi plane, channel and site 12
2.1.2. Voronoi polyhedron 14
2.1.3. Theorem about the Voronoi tessellation 17
2.2 Delaunay tessellation 19
2.2.1. The Delaunay empty ball method. Delaunay simplex 19
2.2.2. Degeneration 21
2.2.3. Theorem about the Delaunay tessellation 23
2.2.4. Simplicial tessellation (triangulation) 24
2.2.5. Features of mutual arrangement of the Delaunay simplices 25
2.3. Duality of the Voronoi and Delaunay tessellations 27
2.3.1. Voronoi-Delaunay tessellation 27
2.3.2. Correspondence between elements of Voronoi and Delaunay tessellation 28
2.3.3. Graphs and networks 29
3. A system of equal balls 30
3.1. S-constructions of Voronoi and Delaunay 30
3.2. Preliminary geometrical conclusions 31
3.3. Geometry of empty space between balls. 33
3.3.1. Simplicial cavity 33
3.3.2. Composite pores. Presentation of pores on the Voronoi network 35
3.4. About permitted travels of a probe 38
3.5. Peculiarities of overlapping balls system 40
4. A system of balls of different radii 43
4.1. Methods to study systems of non-equal balls 43
4.2. Main notions and definitions 46
4.2.1. Voronoi S-surface (Voronoi hyperboloid) 46
4.2.2. Voronoi S-channel. Configuration of three balls 47
4.2.3. Properties of the Voronoi S-channels 50
4.2.4. Interstitial sphere. Configurations of four balls 52
4.3. Voronoi S-region 53
4.3.1. Types of the Voronoi S-regions 54
4.3.2. Properties of the Voronoi S-regions 56
4.4. Voronoi S-tessellation 57
4.5. Delaunay S-simplex 58
4.6. Voronoi S-network 60
4.7. Navigation map of empty space in ball system 61
5. A system of non-spherical convex objects 64
5.1. Generalisations of the Voronoi tessellation 64
5.1.1. Mathematical view on the generalization 64
5.1.2. Medial axes 65
5.1.3. Jonson-Mehl model 66
5.2. Voronoi tessellation of a system of convex objects 68
5.2.1. Preliminary geometrical remarks 68
5.2.2. Theorem about the Voronoi tessellation of a system of convex objects 70
5.2.3. The Voronoi Network of a system of convex objects 71
5.3. Remarks about systems of non-convex objects 72
6. Ways for calculation of the Voronoi-Delaunay tessellation 72
6.1. Numerical presentation of the Voronoi-Delaunay tessellation 73
6.1.1. Voronoi network 74
6.1.2. Delaunay network 74
6.1.3. Correspondence between Voronoi and Delaunay networks 74
6.1.4. The characteristic numbers 75
6.2. Calculation of individual Voronoi polyhedron (the method of "go around of face"). 76
6.3. Calculation of the Voronoi network (the method of "the circumscribed sphere"). 79
6.4. Calculation of the Voronoi S-network for system of balls of different radii. 82
6.4.1. General remarks 82
6.5. Solution of the Apollonian problem in three dimensional space 83
6.6. Calculation of the Voronoi network for system of convex objects 86
6.6.1. Distance function 86
6.6.2. Search for a first site 88
6.6.3. Motion along a bond of the Voronoi network 88
6.6.4. Forming of the network 89
PART II. APPLICATION OF THE METHOD. APPLICATION FOR THE AMORPHOGRAPHY 91
7. The structure of non-crystalline systems 91
7.1. Evolution of conceptions about structure of liquids and glasses 91
7.2. The first researches of the structure of liquids 94
7.3. Methods for generation of computer models 96
8. Local order 98
8.1. Voronoi polyhedra analysis 98
8.1.1. Topological characteristics 99
8.1.2. Metrical characteristics 106
8.1.3. Simplified Voronoi polyhedra 109
8.2. The structure of stable and methastable water. Comparison with the tetrahedral network 109
8.2.1. Models 110
8.2.2. Topological analysis 110
8.2.3. Metrical analysis 113
8.3. Analysis of the Delaunay simplices 116
8.3.1. Metric of the Delaunay simplices 116
8.3.2. Measures of shape of the Delaunay simplices 119
8.4. The structure of dense packings of spherical particles 122
9. Intermediate (medium range) order 123
9.1. Approach and terminology of the percolation theory 124
9.2. The structure patterns on the Voronoi network 125
9.2.1. T-colouring 126
9.2.2. O-colouring 128
9.2.3. S-colouring 131
9.3. The structure inhomogeneity in glassy argon. Analysis of the Voronoi network 132
9.3.1. Models 132
9.3.2. Regions of perfect and imperfect structure 133
9.4. The structure patterns on the Delaunay network 136
9.4.1. Universal structure inhomogeneity in a simple glass 136
9.4.2. Free volume distribution in liquid and glass 140
9.5. Geometrical aspect of glass transition 143
APPLICATION FOR THE COMPUTATIONAL POROSIMETRY 147
10. The problem of the structure of pores 147
10.1. Characteristics of pores 148
10.1.1. Radius of the interstitial sphere 148
10.1.2. The simplest and composite pores 150
10.2. The percolation problem of bonds on the navigation map 152
11. Some physical applications 153
11.1. Calculation of accessible volume for test particle 153
11.1.1. Delaunay simplex sampling method 155
11.1.2. Remarks about calculation of the accessible volume in complex systems 157
11.2. Radial distribution function and structure factor of a system of the interstitial spheres 157
11.2.1. Models 159
11.2.2. Analysis of voids 160
11.3. Three dimensional Apollonian packing 162
11.3.1. A procedure to build a packing 163
11.3.2. Models 164
11.3.3. Fractal properties of the packing 165
11.3.4. Super-dense packings of balls 168
11.4. Hierarchy of pores in dense and in loose packings 170
11.4.1. Models 171
11.4.2. Presentation of the pores 173
11.4.3. Analysis of the simplicial cavities 173
11.4.4. Integral analysis of the pores 174
11.4.5. Differential analysis of the pores 175
11.5. Penetrability and diffusion 177
11.5.1. Permeability and percolation 177
11.5.2. Diffusion as a random walking on the Voronoi network 180
11.6. Analysis of voids in complex systems 185
11.6.1. Polydisperse and molecular systems 185
11.6.2. A system of straight lines 186
APPENDIX. Convex polyhedra 189
References 194
Contents 210
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The author thanks co-authors of common publications on this subject: Yu.I.Naberukhin, V.P.Voloshin, V.A.Luchnikov, S.V.Anishchik, V.N.Novikov, W.Brostow, A.Geiger, A.Appelhagen, R.Bieshaar, D.Paschek, L.Oger, P.Richard, J.-P.Truadek, P.Madden, to colleagues for helpful discussions promoted to appearance of the book: G.G.Malenkov, B.R.Gelchinsky, V.P.Il'in, A.D.Medvykh, B.A.Tolkachev, A.Gervuos, and also to academician A.D.Aleksandrov (one of the first students of B.N.Delaunay) for wishing bon voyage from the geometrical school of Delaunay to the author on the beginning of his way to use Voronoi and Delaunay ideas.
Download PDF-copy of the book (in Russian):
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Title, preface, contents |
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Part I. Basis of the method |
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Part II. Application of the method. Application for the amorphography |
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References |
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This copy of the book is free for education and scientific researches. Using any parts of the book (text or figures) in other publications needs citing the book and permission of the author of the book: nikmed at kinetics.nsc.ru.