Engineering of Submicron Particles Fundamental Concepts and Models

Brings together in one place the fundamental theory and models, and the practical aspects of submicron particle engineering

This book attempts to resolve the tricky aspects of engineering submicron particles by discussing the fundamental theories of frequently used research tools?both theoretical and experimental. The first part covers the Fundamental Models and includes sections on nucleation, growth, inter-molecular and inter-particle forces, colloidal stability, and kinetics. The second part examines the Modelling of a Suspension and features chapters on fundamental concepts of particulate systems, writing the number balance, modelling systems with particle breakage and aggregation, and Monte Carlo simulation. The book also offers plenty of diagrams, software, examples, brief experimental demonstrations, and exercises with answers.

Engineering of Submicron Particles: Fundamental Concepts and Models offers a lengthy discussion of classical nucleation theory, and introduces other nucleation mechanisms like organizer mechanisms. It also looks at older growth models like diffusion controlled or surface nucleation controlled growth, along with new generation models like connected net analysis. Aggregation models and inter-particle potentials are touched upon in a prelude on intermolecular and surface forces. The book also provides analytical and numerical solutions of population balance models so readers can solve basic population balance equations independently. 

• Presents the fundamental theory, practical aspects, and models of submicron particle engineering
• Teaches readers to write number balances for their own system of interest
• Provides software with open code for solution of population balance model through discretization
• Filled with diagrams, examples, demonstrations, and exercises

Engineering of Submicron Particles: Fundamental Concepts and Models will appeal to researchers in chemical engineering, physics, chemistry, engineering, and mathematics concerned with particulate systems. It is also a good text for advanced students taking particle technology courses.

Preface xi

About the Companion Website xv

1 Nucleation 1

1.1 Thermodynamics of Interfaces 1

1.1.1 The Interface is a Surface of High Energy 1

1.1.2 The Interface is a Surface Under Tension 3

1.1.3 Pressure Drop Across Curved Interfaces 3

1.1.3.1 Capillary Rise 6

1.1.4 Vapour–Liquid Equilibrium Across Curved Interfaces 7

1.1.4.1 Thomson Equation 11

1.1.5 Stability of the Equilibrium 12

1.2 Homogeneous Nucleation 13

1.2.1 Energetics of Homogeneous Nucleation 13

1.2.1.1 Energetics in Terms of Number of Units 16

1.2.2 Kinetics of Homogeneous Nucleation 17

1.2.2.1 Concentration of Embryos/Nuclei 18

1.2.2.2 Chain of Reactions Towards Formation of Nuclei 19

1.2.2.3 Algebraic Manipulation of the Rate Expression 22

1.2.2.4 Various Forms of Homogeneous Nucleation Rate 24

1.2.3 Experimental Aspects of Homogeneous Nucleation 26

1.2.3.1 Investigation Using a Cloud Chamber 26

1.2.3.2 Other Methods 27

1.3 Non-Homogeneous Nucleation 28

1.3.1 Heterogeneous Nucleation 28

1.3.2 Nucleating Agents and Organizers 30

1.3.3 Secondary Nucleation 30

1.4 Exercises 31

Bibliography 33

2 Growth 35

2.1 Traditional Crystal Growth Models 36

2.1.1 Diffusion Controlled Growth 37

2.1.2 Surface Nucleation Controlled Growth 38

2.1.2.1 Rate of Mononuclear Growth 40

2.1.3 Surface Dislocation Controlled Growth: BCF Theory 41

2.1.3.1 Rate of Surface Dislocation Controlled Growth 42

2.2 Face Growth Theories 44

2.2.1 Shape of a Crystal 45

2.2.2 Laws of Face Growth 47

2.2.2.1 Law of Bravais and Friedel 47

2.2.3 Flat, Stepped, and Kinked Faces 47

2.3 Measurement of Particle Size and Shape 49

2.3.1 Optical Microscopy 50

2.3.2 Electron Microscopy 51

2.3.3 Light Scattering 51

2.3.3.1 Rayleigh Scattering 52

2.3.3.2 Static and Dynamic Light-Scattering Techniques 55

2.4 Exercises 55

Bibliography 56

3 Inter-Particle Forces 57

3.1 Inter-Molecular Forces 58

3.1.1 Charge–Charge Interactions 58

3.1.2 Charge–Dipole Interactions 59

3.1.3 Dipole–Dipole Interactions 60

3.1.4 Dipole–Induced Dipole Interactions 61

3.1.5 Induced Dipole–Induced Dipole Interactions 62

3.1.6 van der Waals Interaction 62

3.1.7 Repulsive Potential and the Net Interaction Energy 63

3.2 Inter-Particle Forces 63

3.2.1 Hamaker’s Pairwise Additivity Approach 64

3.2.2 Lifshitz’s Theory 67

3.3 Measurement of Inter-Molecular Forces 68

3.4 Measurement of Forces between Surfaces 70

3.5 Exercises 73

Bibliography 73

4 Stability 75

Charged Interface 75

4.1 Electrostatic Potential Near a Charged Surface 76

4.2 Solution of the Poisson–Boltzmann Equation 77

4.3 Repulsive Force between Two Surfaces 80

4.4 Steric Stabilization 85

4.5 Kinetics of Stability 86

4.5.1 Diffusion of Colloidal Particles 87

4.5.2 Particle Aggregation in the Absence of Potential 88

4.5.3 Particle Aggregation in the Presence of a Net Potential 90

4.6 Measurement of Surface Potential 92

4.6.1 Surface Potential When R_s<< 𝜅⁻¹ 93

4.6.2 Surface Potential When R_s >> 𝜅⁻¹ 95

4.7 Exercises 97

Bibliography 99

5 Elementary Concepts of Number Balance 101

5.1 State of a Particle 102

5.2 State of a Population of Particles 105

5.3 Number Balance for a Seeded Batch Crystallizer 110

5.3.1 Coupling the PBE with Mass Balance 114

5.3.2 Modification for the Unseeded Case 115

5.4 Number Balance for Open Systems 115

5.5 Exercises 118

Bibliography 120

6 Breakage and Aggregation 121

6.1 Breakage Functions 121

6.2 Number Balance for Breakage 126

6.2.1 Discrete Breakage Equation 129

6.3 The Process of Aggregation 129

6.3.1 Number Balance for Aggregation 131

6.3.2 Simplification of the Aggregation Equation 133

6.3.3 Models for Aggregation Frequency 136

6.4 Exercises 138

Bibliography 142

7 Solution of the Population Balance Equation 143

7.1 Operations InvolvingMoments of the PBE 143

7.2 Analytical Solutions of the PBE 146

7.2.1 Solution of the Growth Equation: Method of Characteristics 146

7.2.2 Solution of the Aggregation Equation: Method of Laplace Transforms 147

7.2.3 Solution of the Breakage Equation: Similarity Solution 148

7.2.3.1 Breakage Equation in Terms of Mass Fraction Undersize 149

7.2.3.2 Self Similar Form of the Breakage Equation 151

7.3 Numerical Solution of the PBE 152

7.3.1 Discretization Using Finite Volume 153

7.4 Exercises 155

Bibliography 156

8 Kinetic Monte Carlo Simulation 157

8.1 Random Variables 157

8.1.1 Uniform Random Numbers 158

8.2 Algorithm for KMC Simulation 159

8.2.1 Specification of the System 160

8.2.2 Time between Events: Interval of Quiescence 160

8.2.3 Sampling a Distribution 161

8.2.4 Events and their Registration 163

8.3 Exercises 166

Bibliography 166

A Mathematical Topics 167

A.1 Geometry of a Heterogeneous Drop 167

A.2 Young’s Equation 168

A.3 ChordTheorem 169

A.4 Jacobian of Variable Transformation in a Multiple Integral 169

A.5 Method of Characteristics 171

Bibliography 173

B Solution of Selected Problems 175

B.1 General Problem Solving Strategy 175

B.2 Solutions of Selected Problems 176

Bibliography 183

C Codes 185

C.1 Distance-Dependant Potential 185

C.2 Solution of Breakage PBE 186

C.3 Solution of Aggregation PBE 190

C.4 Sampling of a Discrete Distribution 194

C.5 Sampling of a Continuous Distribution 195

C.6 Simulation of Breakage Using KMC 196

C.7 Simulation of Brainvita Game 198

D Experimental Demonstration 201

Bibliography 202

Index 203

JAYANTA CHAKRABORTY, PHD, is Assistant Professor in the Department of Chemical Engineering at Indian Institute of Technology Kharagpur, India.