Table Of ContentZbigniewH.Stachurski
FundamentalsofAmorphousSolids
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ZbigniewH.Stachurski
Fundamentals of Amorphous Solids
StructureandProperties
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V
Contents
Preface XI
1 Spheres,ClustersandPackingofSpheres 1
1.1 Introduction 1
1.2 GeometryofSpheres 9
1.2.1 ASphereandItsNeighbours 9
1.2.2 NeighboursbyTouching 10
1.2.3 HardandSoftSpheres 12
1.3 GeometryofClusters 15
1.3.1 RegularClusters 15
1.3.2 IrregularClusters 17
1.3.3 Coordinationof(1+k)Clusters 18
1.3.3.1 BlockingModelforClusterFormation 19
1.3.3.2 FürthModelforClusterFormation 22
1.3.4 Configurationof(1+k)Clusters 25
1.3.4.1 RegularClusters 25
1.3.4.2 IrregularClusters 26
1.3.4.3 ClosingVectorBasedonRadialVectorPolygon 27
1.3.4.4 PhysicalMeaningoftheClosingVector,ζ 32
1.3.4.5 SphericalHarmonics 33
1.4 GeometryofSpherePackings 34
1.4.1 FixedandLoosePackings 36
1.4.2 OrderedPacking 36
1.4.3 DisorderedPacking 37
1.4.4 RandomPacking 38
1.4.5 RandomSequentialAdditionofHardSpheres 38
1.4.6 RandomClosedPackingofSpheres 40
1.4.7 NeighboursbyVoronoiTessellation 43
1.4.8 NeighboursbyCoordinationShell 47
1.4.8.1 PairDistributionFunction 49
1.4.8.2 TheProbabilityofContacts 50
1.4.8.3 ContactConfigurationFunction 50
VI Contents
1.5 IdealAmorphousAolid(IAS) 51
1.6 ConstructionofanIdealAmorphousSolidClassI 53
1.7 ElementaryTheoryofAmorphousness 59
1.7.1 Background 59
1.7.2 TheAxioms 60
1.7.3 Conjecture 61
1.7.4 TheRules 62
1.7.5 StatisticalCorrespondence 63
1.8 ClassesofIdealAmorphousSolids 64
1.8.1 IASClassI:RandomClosePackingofIndividualAtoms 64
1.8.2 IASClassII:RandomClosePackingofLinearModelChains 64
1.8.3 IASClassIII:RandomClosePackingofThree-Dimensionally
cross-LinkedChains 65
1.9 ImperfectionsinIAS 66
1.9.1 Geometrical(local)Flaws 67
1.9.2 Statistical(global)Flaws 68
1.9.3 TheEffectofFlawsontheDensityofIAS 69
1.9.4 ShortandMediumRangeOrder 72
References 72
BooksonCrystallography 74
BooksonGlasses 74
BooksonRandomWalks 74
BooksonSpherePackings 74
BooksonCrystalImperfections 74
2 CharacteristicsofSpherePackings 75
2.1 GeometricalProperties 75
2.1.1 TheCoordinationDistributionFunction,Ψ(k) 75
2.1.2 Tetrahedricity 76
2.1.3 VoronoiPolyhedraNotation 78
2.1.4 TopologyofClusters 79
2.1.4.1 OrderedClusters 80
2.1.4.2 IrregularClusters 81
2.1.5 TheConfigurationDistributionFunction,Φ (ζ) 82
k
2.1.6 TheVolumeFraction 83
2.1.6.1 RegularPolyhedra 84
2.1.6.2 IrregularPolyhedra 84
2.1.7 ThePackingFraction 85
2.1.7.1 TheAveragePackingFractionfortheRoundCell 86
2.1.7.2 TheLocalPackingFraction 87
2.1.7.3 TheLimitsofPackingFraction 87
2.1.8 RepresentativeVolumeElement 91
2.1.9 DensityofSinglePhase 92
2.1.9.1 DensityofCrystallineSolid 92
2.1.9.2 DensityofAmorphousSolid 92
Contents VII
2.1.10 DensityofaComposite 93
2.1.11 SolidityofPacking 94
2.2 X-rayScattering 96
2.2.1 Introduction 96
2.2.2 GeometryofDiffractionandScattering 96
2.2.3 IntensityofaScatteredWave 97
2.2.3.1 AmorphousSolid 100
2.2.3.2 EhrenfestFormula 100
2.2.3.3 PolyatomicSolid 101
2.2.4 FactorsAffectingIntegratedScatteredIntensity 102
2.2.4.1 IntegratedIntensityofPowderPatternLinesfromCrystalline
Body 102
2.2.4.2 IntegratedScatteredIntensityfromMonoatomicBody 102
2.3 GlassTransitionMeasuredbyCalorimetry 103
References 104
3 GlassyMaterialsandIdealAmorphousSolids 107
3.1 Introduction 107
3.1.1 Solidification 108
3.1.1.1 SolidificationbyMeansofCrystallization 110
3.1.1.2 SolidificationthroughVitrification 111
3.1.2 CognateGroupsofAmorphousMaterials(Glasses) 114
3.1.2.1 MetallicGlasses 114
3.1.2.2 InorganicGlasses 116
3.1.2.3 OrganicGlasses 117
3.1.2.4 AmorphousThinFilms 119
3.2 SummaryofModelsofAmorphousSolids 119
3.2.1 LatticewithAtomicDisorder 120
3.2.2 DisorderedClustersonLattice 121
3.2.3 GeometricModelsforAmorphousNetworks 121
3.2.4 PackingofRegularbutIncongruentClusters 121
3.2.5 IrregularClusters–RandomPacking 123
3.2.6 MolecularDynamics 123
3.2.7 MonteCarloMethod 124
3.3 IASModelofa-Argon 125
3.3.1 IASParameters 126
3.3.2 RoundCellSimulationandAnalysis 127
3.3.2.1 CoordinationDistributionFunction 129
3.3.2.2 VoronoiVolumeandConfigurationDistributionFunctions 129
3.3.2.3 RadialDistributionFunction 129
3.3.2.4 X-rayScatteringfromtheIASModel 131
3.3.2.5 CrystallineandAmorphousCluster 132
3.3.3 Summaryofa-ArIASStructure 132
3.4 IASModelofa-NiNbAlloy 133
3.4.1 Introduction 133
VIII Contents
3.4.2 IASModelofa-NiNbAlloy 133
3.4.2.1 CoordinationDistributionFunctions 133
3.4.2.2 VoronoiVolumeDistribution 135
3.4.2.3 PairDistributionFunction 135
3.4.2.4 ProbabilityofContacts 135
3.4.3 X-rayScatteringfroma-NiNbAlloy 136
3.4.3.1 ExperimentalResults 136
3.4.3.2 TheoreticalResults 138
3.4.4 Densityofa-Ni62-Nb38Alloy 138
3.4.4.1 CrystallineAlloy 138
3.4.4.2 AmorphousAlloy 139
3.4.5 Summaryofa-NiNbIASStructure 140
3.5 IASModelofa-MgCuGdAlloy 140
3.5.1 PhysicalPropertiesoftheElements 140
3.5.2 IASSimulationofa-MgCuGdAlloy 140
3.5.2.1 CoordinationDistributionFunctions 143
3.5.2.2 ConfigurationDistributionFunction 143
3.5.2.3 RadialDistributionFunction 143
3.5.2.4 ProbabilityofContacts 147
3.5.2.5 ClusterCompositionAccordingtoIAS 147
3.5.2.6 ClusterCompositionAccordingtoMD 148
3.5.3 X-rayScatteringfroma-Mg65-Cu25-Gd10Alloy 149
3.5.3.1 FlatPlateX-rayScatteringPattern 149
3.5.3.2 CalibrationbasedonSiPowderPattern 149
3.5.3.3 UncertaintiesandCorrections 153
3.5.4 DensityofMg65-Cu25-Gd10Alloy 155
3.5.4.1 CrystallineAlloy 155
3.5.4.2 AmorphousAlloy 156
3.5.5 Summaryofa-MgCuGdIASStructure 156
3.6 IASModelofa-ZrTiCuNiBeAlloy 157
3.6.1 TransmissionElectronMicroscopy 157
3.6.2 IASSimulationofAmorphousa-ZrTiCuNiBeAlloy 159
3.6.2.1 CoordinationDistributionFunction 159
3.6.2.2 VoronoiVolumeDistribution 161
3.6.2.3 RadialDistributionFunction 161
3.6.3 AtomicProbeofthea-ZrTiCuNiBeAlloy 162
3.6.3.1 ProbabilityofContacts 162
3.6.4 SelectedClustersfromthea-ZrTiCuNiBeAlloy 165
3.6.5 X-rayScatteringfromthea-ZrTiCuNiBeAlloy 165
3.6.6 DensityofZrTiCuNiBeAlloy 167
3.6.6.1 CrystallineAlloy 167
3.6.6.2 AmorphousAlloy 168
3.6.6.3 VitreloyAlloys 169
3.6.7 Summaryofa-ZrTiCuNiBeIASStructure 169
3.7 IASModelofa-Polyethylene(a-PE) 169
