Engineering of Submicron Particles E-Book

Jayanta Chakraborty

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 262

Veröffentlichungsjahr: 2019

Beschreibung

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.

Leseprobe

Table of Contents

Cover

Preface

About the Companion Website

1 Nucleation

1.1 Thermodynamics of Interfaces

1.2 Homogeneous Nucleation

1.3 Non‐Homogeneous Nucleation

1.4 Exercises

Bibliography

2 Growth

2.1 Traditional Crystal Growth Models

2.2 Face Growth Theories

2.3 Measurement of Particle Size and Shape

2.4 Exercises

Bibliography

3 Inter‐Particle Forces

3.1 Inter‐Molecular Forces

3.2 Inter‐Particle Forces

3.3 Measurement of Inter‐Molecular Forces

3.4 Measurement of Forces between Surfaces

3.5 Exercises

Bibliography

4 Stability

Charged Interface

4.1 Electrostatic Potential Near a Charged Surface

4.2 Solution of the Poisson–Boltzmann Equation

4.3 Repulsive Force between Two Surfaces

4.4 Steric Stabilization

4.5 Kinetics of Stability

4.6 Measurement of Surface Potential

4.7 Exercises

Bibliography

5 Elementary Concepts of Number Balance

5.1 State of a Particle

5.2 State of a Population of Particles

5.3 Number Balance for a Seeded Batch Crystallizer

5.4 Number Balance for Open Systems

5.5 Exercises

Bibliography

6 Breakage and Aggregation

6.1 Breakage Functions

6.2 Number Balance for Breakage

6.3 The Process of Aggregation

6.4 Exercises

Bibliography

7 Solution of the Population Balance Equation

7.1 Operations Involving Moments of the PBE

7.2 Analytical Solutions of the PBE

7.3 Numerical Solution of the PBE

7.4 Exercises

Bibliography

8 Kinetic Monte Carlo Simulation

8.1 Random Variables

8.2 Algorithm for KMC Simulation

8.3 Exercises

Bibliography

Appendix A: Mathematical Topics

A.1 Geometry of a Heterogeneous Drop

A.2 Young's Equation

A.3 Chord Theorem

A.4 Jacobian of Variable Transformation in a Multiple Integral [1]

A.5 Method of Characteristics [2]

Bibliography

Appendix B: Solution of Selected Problems

B.1 General Problem Solving Strategy

B.2 Solutions of Selected Problems

Bibliography

Appendix C: Codes

C.1 Distance‐Dependant Potential

C.2 Solution of Breakage PBE

C.3 Solution of Aggregation PBE

C.4 Sampling of a Discrete Distribution

C.5 Sampling of a Continuous Distribution

C.6 Simulation of Breakage Using KMC

C.7 Simulation of Brainvita Game

Appendix D: Experimental Demonstration

Bibliography

Index

End User License Agreement

List of Tables

Chapter 3

Table 3.1 Relative magnitude of various components of van der Waals forces in

J...

Table 3.2 van der Waals interactions among various macroscopic bodies

Table 3.3 Hamaker constants calculated from Lifshitz's theory

Chapter 4

Table 4.1 Solutions of the Poisson–Boltzmann equation and their domain of applic...

List of Illustrations

Chapter 1

Figure 1.1 Distance‐dependant interaction potential. The negative sign indicat...

Figure 1.2 Difference in coordination number between surface and bulk atoms. I...

Figure 1.3 The geometry of a curved interface: derivation of the Young–Laplace...

Figure 1.4 Surface of zero net curvature.

Figure 1.5 The contact angle and curved interface in a capillary tube: the rea...

Figure 1.6 Vapour–liquid equilibrium in an isothermal system.

Figure 1.7 Formation of a liquid nucleus in a supersaturated vapour.

Figure 1.8

as a function of

for

and

.

Figure 1.9 Embryos crossing the activation barrier to become nuclei.

Figure 1.10 Significant portion of the integrand in eqn 1.14.

Figure 1.11 Experimental determination of the critical supersaturation of homo...

Figure 1.12 Left: homogeneous vs heterogeneous drops. Right: drop volume as a ...

Figure 1.13 Slow formation of a bubble on the tip (Exercise 1.2).

Figure 1.14 A complex cavity (Exercise 1.6).

Chapter 2

Figure 2.1 Build up and exhaustion of supersaturation during the nucleation gr...

Figure 2.2 Transport, adsorption, and integration of the growth unit to the gr...

Figure 2.3 Mass balance in a spherical shell around the growing spherical part...

Figure 2.4 Formation of surface nuclei.

Figure 2.5 Micrographs of two‐dimensional nucleation: polynuclear growth [3].

Figure 2.6 Growth of semiconductor nanocrystals (data for Example 2.1).

Figure 2.7 Screw dislocation.

Figure 2.8 Micrograph of growth spirals [3].

Figure 2.9 Schematic of spiral growth.

Figure 2.10 Faceted crystal.

Figure 2.11 A two‐dimensional faceted crystal. This crystal is defined by the ...

Figure 2.12 Kinetic shape of a crystal. Top: the crystal continues to grow in ...

Figure 2.13 Different lattice planes, their interplaner distances, and their g...

Figure 2.14 Flat, stepped, and kinked crystal faces.

Figure 2.15 The top surface has two periodic bond chains. It grows epitaxially...

Figure 2.16 The periodic bond chain (PBC) for the lattice shown in Figure 2.15...

Figure 2.17 Scattering of light from a particle.

Figure 2.18 Rayleigh scattering.

Chapter 3

Figure 3.1 Charge–dipole interaction.

Figure 3.2 Dipole–dipole interaction.

Figure 3.3 Hamaker's pairwise additivity approach.

Figure 3.4 Hamaker's pairwise additivity approach.

Figure 3.5 Molecule–semi‐infinite body interaction.

Figure 3.6 Sphere–semi‐infinite body interaction.

Figure 3.7 Tracing the force displacement curve using a spring.

Figure 3.8 Surface force apparatus (figure taken from Ref. [2]).

Chapter 4

Figure 4.1 Dissociation of ions at the water–solid interface to form a charged...

Figure 4.2 An elementary volume near a charged surface.

Figure 4.3 Surface pressure.

Figure 4.4 Counter‐ions arranged in a single layer at a distance

from the su...

Figure 4.5 Various possible landscapes for the net potential.

Figure 4.6 Diffusion of particles with radius

towards a particle with radius...

Figure 4.7

potential: the case when

. The surface charge is negative. The v...

Figure 4.8

potential: the case when

. The surface charge is negative. The v...

Figure 4.9 The situation in the EDL when

. The surface charge is negative.

Figure 4.10 Interaction between two spheres.

Chapter 5

Figure 5.1 An MSMPR and a batch crystallizer. In both cases, the vessel is wel...

Figure 5.2 SEM micrograph of copper sulphate pentahydrate crystal.

Figure 5.3 TEM micrograph of copper nanoparticles.

Figure 5.4 Representation of a population of particles using sieve analysis an...

Figure 5.5 Particle size distribution of the same sample obtained with three d...

Figure 5.6 Number densities obtained by dividing the bin population by the bin...

Figure 5.7 Number density function at an instant in a batch crystallizer. The ...

Figure 5.8 Number balance for a pure growth process. The number of particles t...

Figure 5.9 Number balance for a pure growth process. The evolved number densit...

Figure 5.10 An MSMPR crystallizer.

Chapter 6

Figure 6.1 Beta probability density function.

Figure 6.2 Birth and death due to breakage processes.

Figure 6.3 Liquid–liquid dispersion.

Figure 6.4 Birth and death of particles due to pure aggregation.

Figure 6.5 Extensional degradation of polymers.

Figure 6.6 Cell cycle.

Figure 6.7 Making ‘Granola Grin’.

Figure 6.8 A controlled environment chamber.

Chapter 7

Figure 7.1 Change of limit for moment operation. Gray indicates the original o...

Figure 7.2 Division of the domain into one finite and one infinite domain, and...

Figure 7.3 Discretization of space using a geometric grid. The bin boundaries ...

Chapter 8

Figure 8.1 Selection of events for a discrete probability distribution.

Figure 8.2 Selection of IQ: sampling from a continuous distribution.

Figure 8.3 Daughter distribution functions: (a) probability density and (b) cu...

Figure 8.4 Progress of a peg solitaire game.

Guide

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Engineering of Submicron Particles

Fundamental Concepts and Models

Jayanta Chakraborty

Department of Chemical EngineeringIndian Institute of Technology KharagpurIndia

This edition first published 2019© 2019 John Wiley & Sons Ltd

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Library of Congress Cataloging-in-Publication Data

Names: Chakraborty, Jayanta, 1976- author.Title: Engineering of submicron particles : fundamental concepts and models / Jayanta Chakraborty, Department of Chemical Engineering, Indian Institute of Technology Kharagpur, India.Description: Hoboken, NJ, USA : John Wiley & Sons, Inc., [2019] | Includes bibliographical references and index. |Identifiers: LCCN 2019015637 (print) | LCCN 2019018492 (ebook) | ISBN 9781119296454 (Adobe PDF) | ISBN 9781119296782 (ePub) | ISBN 9781119296461 (hardcover)Subjects: LCSH: Nanoparticles.Classification: LCC TA418.78 (ebook) | LCC TA418.78 .C475 2019 (print) | DDC 620.1/15–dc23LC record available at https://lccn.loc.gov/2019015637

Cover Design: WileyCover Image: Courtesy of Jayanta Chakraborty

Dedicated to my parents, who had the courage to push us for higher education against many odds.

Preface

In the process industry, many products and intermediates exist in the form of fine particles. Many next‐generation processes, such as colloidal heat transfer fluids for electronic cooling, also involve small particles. However, the ability of the process industry to deal with particulate processes in a quantitative way is limited. The process industry must enhance its capability in the engineering of fine particles.

Many research laboratories also produce and handle submicron particles. In a broader sense, such particulate systems include powders, polymers, colloids or even human populations. While many engineering textbooks and reference books deal with particles of micron scale and above, submicron particles are discussed mostly under very specialized subtopics and a reference book discussing the fundamental concepts of such systems is missing.

Everyday activities in an industrial or academic research laboratory where particulate systems are involved require application of a number of quantitative relations called models. Even experimental facilities use models to relate the raw data with the quantity of interest and often the user is not aware that the outputs are actually from a model. Most models are not straightforward and no single resource is available to provide understanding of frequently used techniques and concepts. New researchers often find themselves at a loss and tend to trust data blindly. This book attempts to resolve this problem by discussing the fundamental theories behind many frequently encountered particulate processes. A large number of diagrams, software, examples, brief experimental demonstrations, and exercises with answers are included and have been carefully planned to provide good learning.

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 material in the appendices to be also useful to students from other branches of science and engineering.

Models are used at various levels in particle technology. A set of basic models describe the fundamental process of nucleation, growth, and aggregation of particles. In these models, the rate of nucleation of particles from a medium of given supersaturation, the rate of increase of size of a particle of given size under a set of environmental conditions, and the rate of aggregation of given pairs are provided.

Classical nucleation theory is discussed at length in this text. Other nucleation 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 discussed with a brief but useful prelude on inter‐molecular and surface forces.

The basic models alone cannot describe the dynamics of an engineering system containing a large number of particles of varying attributes. For this a number balance equation (population balance) is needed. In this book the emphasis is on formulating the number balance equation (the population balance model) for a given system. Analytical and numerical solutions of population balance models are also discussed briefly. Software with open code is provided for the solution of a population balance model through discretization.

To my knowledge no book serves such a diverse yet unified purpose. This book has been in my mind throughout my career over the past decade, during which I made my journey from an experimental laboratory to two theoretical laboratories and then back to experiments. This book contains useful insights which I acquired over time.

This book is heavily indebted to several books and monographs which helped me in assimilating the content. I kept close to the flow of ideas and concepts of the parent books whenever I felt that was best for the reader. I acknowledge major contributions from the following books and monographs:

Foundations of Colloid Science

by R. J. Hunter and

Kinetics of Precipitation

by A. E. Nielsen for the nucleation and growth chapters.

The chapter on inter‐molecular and inter‐particle force has ideas and contents from

Intermolecular and Surface Forces

by Jacob Israelachvili.

The stability chapter is heavily indebted to Paul C. Hiementz (

Principles of Colloid and Surface Chemistry

).

The particulate system modelling section is indebted to

Population Balances

by D. Ramkrishna and

Theory of Particulate Processes

by A. D. Randolph and M. A. Larson.

Much of the book is also influenced by the lecture notes circulated during my graduate course on modelling at the Indian Institute of Science, Bangalore by Prof. K. S. Gandhi and Prof. Sanjeev Kumar.

Apart from these major resources there are many other books and monographs that helped me to understand, assimilate, and express the ideas. I also acknowledge help from students at IIT Kharagpur who took this course (Fundamentals of Particle Technology, CH60026), asked critical questions, and helped me write this book. I hope this book will be useful to the others. Of course there are multiple errors and omissions which I'm eager to hear from the readers and correct in a future edition.

Jayanta ChakrabortyIIT KharagpurAutumn 2018

About the Companion Website

To access supplementary materials for this book please use the download link shown below:

http://booksupport.wiley.com

Please enter the book title, author name or ISBN to access this material.

Here you will find valuable material designed to enhance your learning, including:

Demo Videos

Source Codes

1Nucleation

Nucleation means the creation of new particles, i.e. the creation of a new phase and associated interface. Hence, in order to understand the nucleation process, we need to learn a few key ideas from physical chemistry. The energy of the interface is vital to the nucleation process, which in turn controls the nucleation rate. Hence, our discussion will involve both thermodynamics and kinetics: the thermodynamics of the interface will provide the magnitude of the driving force and the kinetics will provide the rate of nucleation.

1.1 Thermodynamics of Interfaces

An interface is a surface where one phase ends and another starts. It is a narrow region often in the order of a few angstroms where the properties change from that of one phase to another. For a liquid–vapour interface, the density of the medium undergoes an abrupt change. For a liquid–liquid interface, two dissimilar atoms are in contact at the interface.

1.1.1 The Interface is a Surface of High Energy

An interface is known to contain higher energy than the bulk phase. This difference in energy is key to phenomena relating to many important technical problems. The excess energy of the interface, or interfacial energy, is due to the difference between the energies of atoms on surface and in the bulk. The difference may be due to the change in density between the two phases or to the difference in chemical nature.

Let us consider the former as an example. For a liquid–vapour interface the liquid is in contact with its vapour. Hence, although similar atoms/molecules are present on both sides of the interface, the densities are very different. This leads to a different coordination number of atoms in bulk versus atoms on the surface. Next we show how this leads to interfacial energy.

The energy of interaction of two isolated atoms at infinite separation is zero. If they are brought closer together, they start to interact. At a specific distance, , contact between the two atoms, the energy of interaction, becomes a minimum.

Figure 1.1 Distance‐dependant interaction potential. The negative sign indicates energy release.

Let us denote the amount of energy released by bringing a pair of atoms of same chemical species to this distance by Hence, the energy per atom for constructing a pair is . What is the energy released per atom where all atoms are jam‐packed, as shown in Figure 1.2? A pair of simplifications will be useful in analyzing the case:

only the nearest neighbours of an atom can impart some force on an atom

interaction is pairwise additive.

Figure 1.2 Difference in coordination number between surface and bulk atoms. In this case, the surface atom coordinates with only four other atoms while the bulk atom coordinates with six.

With these assumptions, an atom in the bulk (see Figure 1.2) will release energy corresponding to the pairs A‐1, A‐2, … A‐6 as shown in the figure (the coordination number is 6). If we denote the coordination number in bulk by , the total energy released per atom will be

Now, the coordination number is different for the surface. Hence, the energy released per atom for the surface will be

where the coordination number for the surface is . Because , more energy is released for the atoms in the bulk than on the surface. Hence, the surface atoms retain more energy. Hence the system that has more interface, has more energy. In other words, interface contains energy.

1.1.2 The Interface is a Surface Under Tension

Will the surface resist its extension? It should. More surface will require more atoms to join the surface, leaving the bulk, and hence it goes energy uphill. Hence, interfaces normally act like a stretched membrane.

The extra interface will require extra energy, which will be supplied by external work. If we denote the energy needed for the creation of a differential amount of surface by , the following proportionality can be written:

Inserting the constant of proportionality:

It is clear that the constant is the surface energy per unit area and hence is interpreted as the specific surface energy. If work is done by a constant external force to increase the area and the increase in area can be written as , the above equation becomes

or

which leads to the popular interpretation of specific surface energy as ‘surface tension’ with unit force/length.

1.1.3 Pressure Drop Across Curved Interfaces

The higher energy of the interface leads to difference in pressure across a curved interface [1]. Let us consider a small area, as shown in the Figure 1.3, and perturb the surface by varying the pressure differentially: the surface expands differentially in response to the differential increase in pressure. The increase in area is

This additional area will require additional surface energy, which is

This much energy must be supplied by working against a difference in pressure () between the two sides of the curved interface. If the inside pressure during the perturbation changes only differentially, the pressure difference across the interface remains even after the perturbation. Hence the work is given by . The increase in volume, , in this case is given by and hence the energy balance can be written as

or

Figure 1.3 The geometry of a curved interface: derivation of the Young–Laplace equation

Using the property of similar triangles, we can write

and

These two expressions lead to

and

Hence, the above equation can be written as

(1.1)

This is known as the Young–Laplace equation and gives the pressure difference across a curved interface as a function of its curvature.

Example 1.1

What is the pressure inside a small water droplet of radius 1 μm and one with radius 1 nm? The surface tension of water is 75 mN/m.

Solution:

Because the drop is spherical, both the radii are equal in this case. Hence, the Young–Laplace equation reduces to

Hence for a 1 μm drop:

If the drop size is 1 nm,

It can be seen that for the first case the pressure difference is merely 1.5 atm whereas for the later it is huge: 1500 atm. Usually, nuclei are very small, of the order of nanometres, and hence they experience huge pressure due to the curved interface.

Example 1.2

What is the pressure inside a small soap bubble of radius 1 cm?

Solution:

Because the soap bubble has two interfaces, the Young–Laplace equation should be written for both interfaces. Denoting as inside pressure, as film pressure, and as outside pressure, and applying the Young–Laplace equation for both interfaces,

Neglecting the film thickness :

Note that we have used the surface tension value of water instead of the surfactant solution in order to obtain an approximate value. The true surface tension of the surfactant solution is dependent on the nature and concentration of the surfactant and should be used for an accurate value.

Example 1.3

A soap bubble can be stabilized on a pair of open tubes, as shown in the Figure 1.4. The pressure is 1 atm on both sides of the bubble yet it has a curved interface. Does it contradict the Young–Laplace equation?

Figure 1.4 Surface of zero net curvature.

Solution:

For any curved surface, the curvatures in two mutually perpendicular directions are related by the Young–Laplace equation. For the above case the pressure difference is zero and hence the Young–Laplace equation reduces to

For the surface shown in Figure 1.4, one curvature is convex while the other is concave to the observer from either side. Hence, one curvature is positive and the other is negative, and their values should be such that the Young–Laplace equation is exactly satisfied. Such surfaces are called surfaces of zero average curvature.

1.1.3.1 Capillary Rise

One of the interesting manifestations of the pressure difference across a curved interface is capillary rise. Figure 1.5 shows a capillary tube and the liquid meniscus. The liquid–vapour interface is a curved surface in the capillary because of a phenomena known as the contact angle. Every liquid–solid–vapour interface maintains a definite angle of contact, as shown in the figure, and hence the interface cannot remain flat. If the tube has a larger diameter the interface is nearly flat except for the edges and if the tube diameter is small (a capillary tube) the interface becomes nearly spherical.

Figure 1.5 The contact angle and curved interface in a capillary tube: the reason for the elevated liquid level in a capillary tube.

In our previous example, we dealt with a liquid drop and showed that the pressure is higher inside the drop. For the liquid meniscus in a capillary, which side has the higher pressure? It is determined by the sign of the curvature. For a spherical surface this boils down to the simple rule that the side which contains the centre has higher pressure. Hence, for the case of a liquid meniscus in a capillary, the pressure in the gas phase is higher.

If the meniscus to co‐inside with the liquid level, the pressure difference cannot be accounted for because the pressure at that level is 1 atm in both the liquid and the gas. But if the liquid level is raised, the pressure on the gas side is 1 atm but the pressure in the liquid side at that raised level is less than 1 atm. It reaches 1 atm at the surface of the liquid pool by addition of the gravity head of the liquid column. A force balance yields:

where is the capillary rise.

Example 1.4

What will be the elevation of the liquid level in a capillary whose radius is 100 μm? The surface tension of water is 75 mN/m.

Solution:

In the above example, it can be seen that significant capillary rise is possible. Moisture is often drawn into narrow inter‐particle spaces through capillary action and it is difficult to drive away such trapped moisture.

1.1.4 Vapour–Liquid Equilibrium Across Curved Interfaces

One of the reasons for our interest in the pressure difference in a small drop is that it alters the chemical potential of the drop. The chemical potential of a phase is a function of pressure, temperature, and composition, and hence excess pressure inside a drop alters its chemical potential. In other words, for a pure substance under isothermal conditions, droplets of different sizes will have different chemical potentials. The quantitive description of size‐dependent chemical potential is given by Kelvin's equation.

Let us consider a closed isothermal vessel containing a pure vapour maintained at temperature and pressure , as shown in Figure 1.6. The chemical potential of the vapour phase is . Now we ask the following question:

Figure 1.6 Vapour–liquid equilibrium in an isothermal system.

What is the Radius of a Drop Which is in Equilibrium with this Vapour?

Let us denote the chemical potential and pressure of the condensed phase by and , respectively. The pressure inside the drop is given by the Young–Laplace equation:

Notice that we have used the condition of mechanical equilibrium, but the condition for chemical equilibrium remained unused. Hence the desired relation may be obtained by using chemical equilibrium. Chemical equilibrium enforces

or

(1.2)

Now, the relation between chemical potential (partial molar Gibbs energy) and other thermodynamic variables is given by the Gibbs–Duhem equation:

Substituting