Table Of ContentEngineeringofSubmicronParticles
Engineering of Submicron Particles
FundamentalConceptsandModels
JayantaChakraborty
DepartmentofChemicalEngineering
IndianInstituteofTechnologyKharagpur
India
Thiseditionfirstpublished2019
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LibraryofCongressCataloging-in-PublicationData
Names:Chakraborty,Jayanta,1976-author.
Title:Engineeringofsubmicronparticles:fundamentalconceptsandmodels/
JayantaChakraborty,DepartmentofChemicalEngineering,IndianInstitute
ofTechnologyKharagpur,India.
Description:Hoboken,NJ,USA:JohnWiley&Sons,Inc.,[2019]|Includes
bibliographicalreferencesandindex.|
Identifiers:LCCN2019015637(print)|LCCN2019018492(ebook)|ISBN
9781119296454(AdobePDF)|ISBN9781119296782(ePub)|ISBN9781119296461
(hardcover)
Subjects:LCSH:Nanoparticles.
Classification:LCCTA418.78(ebook)|LCCTA418.78.C4752019(print)|
DDC620.1/15–dc23
LCrecordavailableathttps://lccn.loc.gov/2019015637
CoverDesign:Wiley
CoverImage:CourtesyofJayantaChakraborty
Setin10/12ptWarnockbySPiGlobal,Pondicherry,India
10 9 8 7 6 5 4 3 2 1
Dedicatedtomyparents,whohadthecouragetopushusforhighereducation
againstmanyodds.
vii
Contents
Preface xi
AbouttheCompanionWebsite xv
1 Nucleation 1
1.1 ThermodynamicsofInterfaces 1
1.1.1 TheInterfaceisaSurfaceofHighEnergy 1
1.1.2 TheInterfaceisaSurfaceUnderTension 3
1.1.3 PressureDropAcrossCurvedInterfaces 3
1.1.3.1 CapillaryRise 6
1.1.4 Vapour–LiquidEquilibriumAcrossCurvedInterfaces 7
1.1.4.1 ThomsonEquation 11
1.1.5 StabilityoftheEquilibrium 12
1.2 HomogeneousNucleation 13
1.2.1 EnergeticsofHomogeneousNucleation 13
1.2.1.1 EnergeticsinTermsofNumberofUnits 16
1.2.2 KineticsofHomogeneousNucleation 17
1.2.2.1 ConcentrationofEmbryos/Nuclei 18
1.2.2.2 ChainofReactionsTowardsFormationofNuclei 19
1.2.2.3 AlgebraicManipulationoftheRateExpression 22
1.2.2.4 VariousFormsofHomogeneousNucleationRate 24
1.2.3 ExperimentalAspectsofHomogeneousNucleation 26
1.2.3.1 InvestigationUsingaCloudChamber 26
1.2.3.2 OtherMethods 27
1.3 Non-HomogeneousNucleation 28
1.3.1 HeterogeneousNucleation 28
1.3.2 NucleatingAgentsandOrganizers 30
1.3.3 SecondaryNucleation 30
1.4 Exercises 31
Bibliography 33
viii Contents
2 Growth 35
2.1 TraditionalCrystalGrowthModels 36
2.1.1 DiffusionControlledGrowth 37
2.1.2 SurfaceNucleationControlledGrowth 38
2.1.2.1 RateofMononuclearGrowth 40
2.1.3 SurfaceDislocationControlledGrowth:BCFTheory 41
2.1.3.1 RateofSurfaceDislocationControlledGrowth 42
2.2 FaceGrowthTheories 44
2.2.1 ShapeofaCrystal 45
2.2.2 LawsofFaceGrowth 47
2.2.2.1 LawofBravaisandFriedel 47
2.2.3 Flat,Stepped,andKinkedFaces 47
2.3 MeasurementofParticleSizeandShape 49
2.3.1 OpticalMicroscopy 50
2.3.2 ElectronMicroscopy 51
2.3.3 LightScattering 51
2.3.3.1 RayleighScattering 52
2.3.3.2 StaticandDynamicLight-ScatteringTechniques 55
2.4 Exercises 55
Bibliography 56
3 Inter-ParticleForces 57
3.1 Inter-MolecularForces 58
3.1.1 Charge–ChargeInteractions 58
3.1.2 Charge–DipoleInteractions 59
3.1.3 Dipole–DipoleInteractions 60
3.1.4 Dipole–InducedDipoleInteractions 61
3.1.5 InducedDipole–InducedDipoleInteractions 62
3.1.6 vanderWaalsInteraction 62
3.1.7 RepulsivePotentialandtheNetInteractionEnergy 63
3.2 Inter-ParticleForces 63
3.2.1 Hamaker’sPairwiseAdditivityApproach 64
3.2.2 Lifshitz’sTheory 67
3.3 MeasurementofInter-MolecularForces 68
3.4 MeasurementofForcesbetweenSurfaces 70
3.5 Exercises 73
Bibliography 73
4 Stability 75
ChargedInterface 75
4.1 ElectrostaticPotentialNearaChargedSurface 76
4.2 SolutionofthePoisson–BoltzmannEquation 77
4.3 RepulsiveForcebetweenTwoSurfaces 80
Contents ix
4.4 StericStabilization 85
4.5 KineticsofStability 86
4.5.1 DiffusionofColloidalParticles 87
4.5.2 ParticleAggregationintheAbsenceofPotential 88
4.5.3 ParticleAggregationinthePresenceofaNetPotential 90
4.6 MeasurementofSurfacePotential 92
4.6.1 SurfacePotentialWhenR <<𝜅−1 93
s
4.6.2 SurfacePotentialWhenR >>𝜅−1 95
s
4.7 Exercises 97
Bibliography 99
5 ElementaryConceptsofNumberBalance 101
5.1 StateofaParticle 102
5.2 StateofaPopulationofParticles 105
5.3 NumberBalanceforaSeededBatchCrystallizer 110
5.3.1 CouplingthePBEwithMassBalance 114
5.3.2 ModificationfortheUnseededCase 115
5.4 NumberBalanceforOpenSystems 115
5.5 Exercises 118
Bibliography 120
6 BreakageandAggregation 121
6.1 BreakageFunctions 121
6.2 NumberBalanceforBreakage 126
6.2.1 DiscreteBreakageEquation 129
6.3 TheProcessofAggregation 129
6.3.1 NumberBalanceforAggregation 131
6.3.2 SimplificationoftheAggregationEquation 133
6.3.3 ModelsforAggregationFrequency 136
6.4 Exercises 138
Bibliography 142
7 SolutionofthePopulationBalanceEquation 143
7.1 OperationsInvolvingMomentsofthePBE 143
7.2 AnalyticalSolutionsofthePBE 146
7.2.1 SolutionoftheGrowthEquation:MethodofCharacteristics 146
7.2.2 SolutionoftheAggregationEquation:MethodofLaplace
Transforms 147
7.2.3 SolutionoftheBreakageEquation:SimilaritySolution 148
7.2.3.1 BreakageEquationinTermsofMassFractionUndersize 149
7.2.3.2 SelfSimilarFormoftheBreakageEquation 151
7.3 NumericalSolutionofthePBE 152
7.3.1 DiscretizationUsingFiniteVolume 153
x Contents
7.4 Exercises 155
Bibliography 156
8 KineticMonteCarloSimulation 157
8.1 RandomVariables 157
8.1.1 UniformRandomNumbers 158
8.2 AlgorithmforKMCSimulation 159
8.2.1 SpecificationoftheSystem 160
8.2.2 TimebetweenEvents:IntervalofQuiescence 160
8.2.3 SamplingaDistribution 161
8.2.4 EventsandtheirRegistration 163
8.3 Exercises 166
Bibliography 166
A MathematicalTopics 167
A.1 GeometryofaHeterogeneousDrop 167
A.2 Young’sEquation 168
A.3 ChordTheorem 169
A.4 JacobianofVariableTransformationinaMultipleIntegral 169
A.5 MethodofCharacteristics 171
Bibliography 173
B SolutionofSelectedProblems 175
B.1 GeneralProblemSolvingStrategy 175
B.2 SolutionsofSelectedProblems 176
Bibliography 183
C Codes 185
C.1 Distance-DependantPotential 185
C.2 SolutionofBreakagePBE 186
C.3 SolutionofAggregationPBE 190
C.4 SamplingofaDiscreteDistribution 194
C.5 SamplingofaContinuousDistribution 195
C.6 SimulationofBreakageUsingKMC 196
C.7 SimulationofBrainvitaGame 198
D ExperimentalDemonstration 201
Bibliography 202
Index 203
xi
Preface
Intheprocessindustry,manyproductsandintermediatesexistintheformof
fineparticles.Manynext-generationprocesses,suchascolloidalheattransfer
fluidsforelectroniccooling,alsoinvolvesmallparticles.However,theabilityof
theprocessindustrytodealwithparticulateprocessesinaquantitativewayis
limited.Theprocessindustrymustenhanceitscapabilityintheengineeringof
fineparticles.
Manyresearchlaboratoriesalsoproduceandhandlesubmicronparticles.In
abroadersense,suchparticulatesystemsincludepowders,polymers,colloids
orevenhumanpopulations.Whilemanyengineeringtextbooksandreference
books deal with particles of micron scale and above, submicron particles are
discussed mostly under very specialized subtopics and a reference book dis-
cussingthefundamentalconceptsofsuchsystemsismissing.
Everyday activities in an industrial or academic research laboratory where
particulatesystemsareinvolvedrequireapplicationofanumberofquantita-
tiverelationscalledmodels.Evenexperimentalfacilitiesusemodelstorelate
therawdatawiththequantityofinterestandoftentheuserisnotawarethat
theoutputsareactuallyfromamodel.Mostmodelsarenotstraightforwardand
nosingleresourceisavailabletoprovideunderstandingoffrequentlyusedtech-
niquesandconcepts.Newresearchersoftenfindthemselvesatalossandtend
totrustdatablindly.Thisbookattemptstoresolvethisproblembydiscussing
thefundamentaltheoriesbehindmanyfrequentlyencounteredparticulatepro-
cesses. A large number of diagrams, software, examples, brief experimental
demonstrations,andexerciseswithanswersareincludedandhavebeencare-
fullyplannedtoprovidegoodlearning.
Particulate systems are used by physicists, chemists, mathematicians, and
engineers. It is difficult to provide fundamental knowledge to the degree
demanded by all. This book is mainly aimed at senior undergraduate or
graduate chemical engineering students but provides enough background
materialintheappendicestobealsousefultostudentsfromotherbranchesof
scienceandengineering.
xii Preface
Modelsareusedatvariouslevelsinparticletechnology.Asetofbasicmodels
describe the fundamental process of nucleation, growth, and aggregation of
particles. In these models, the rate of nucleation of particles from a medium
ofgivensupersaturation,therateofincreaseofsizeofaparticleofgivensize
underasetofenvironmentalconditions,andtherateofaggregationofgiven
pairsareprovided.
Classical nucleation theory is discussed at length in this text. Other nucle-
ation mechanisms, e.g. the organizer mechanism, are also introduced. For
growth, the classical growth models such as diffusion controlled and surface
nucleation controlled growth are discussed, along with newer models like
connected net analysis. Aggregation models and inter-particle potentials are
discussedwithabriefbutusefulpreludeoninter-molecularandsurfaceforces.
Thebasicmodelsalonecannotdescribethedynamicsofanengineeringsys-
temcontainingalargenumberofparticlesofvaryingattributes.Forthisanum-
berbalanceequation(populationbalance)isneeded.Inthisbooktheemphasis
isonformulatingthenumberbalanceequation(thepopulationbalancemodel)
for a given system. Analytical and numerical solutions of population balance
modelsarealsodiscussedbriefly.Softwarewithopencodeisprovidedforthe
solutionofapopulationbalancemodelthroughdiscretization.
To my knowledge no book serves such a diverse yet unified purpose. This
bookhasbeeninmymindthroughoutmycareeroverthepastdecade,during
whichImademyjourneyfromanexperimentallaboratorytotwotheoretical
laboratoriesandthenbacktoexperiments.Thisbookcontainsusefulinsights
whichIacquiredovertime.
Thisbookisheavilyindebtedtoseveralbooksandmonographswhichhelped
meinassimilatingthecontent.Ikeptclosetotheflowofideasandconceptsof
the parent books whenever I felt that was best for the reader. I acknowledge
majorcontributionsfromthefollowingbooksandmonographs:
• FoundationsofColloidSciencebyR.J.HunterandKineticsofPrecipitation
byA.E.Nielsenforthenucleationandgrowthchapters.
• Thechapteroninter-molecularandinter-particleforcehasideasandcon-
tentsfromIntermolecularandSurfaceForcesbyJacobIsraelachvili.
• ThestabilitychapterisheavilyindebtedtoPaulC.Hiementz(Principlesof
ColloidandSurfaceChemistry).
• TheparticulatesystemmodellingsectionisindebtedtoPopulationBalances
byD.RamkrishnaandTheoryofParticulateProcessesbyA.D.Randolphand
M.A.Larson.
• Muchofthebookisalsoinfluencedbythelecturenotescirculatedduringmy
graduatecourseonmodellingattheIndianInstituteofScience,Bangaloreby
Prof.K.S.GandhiandProf.SanjeevKumar.
Apart from these major resources there are many other books and mono-
graphsthathelpedmetounderstand,assimilate,andexpresstheideas.Ialso
