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
