Particle Adhesion and Removal E-Book

K. L. Mittal

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Adhesion and Adhesives - Fundamental and Applied Aspects
• Sprache: Englisch

Seitenzahl: 809

Veröffentlichungsjahr: 2015

Beschreibung

The book provides a comprehensive and easily accessible reference source covering all important aspects of particle adhesion and removal.  The core objective is to cover both fundamental and applied aspects of particle adhesion and removal with emphasis on recent developments.

 Among the topics to be covered include:

1. Fundamentals of surface forces in particle adhesion and removal.
2. Mechanisms of particle adhesion and removal.
3. Experimental methods (e.g. AFM, SFA,SFM,IFM, etc.) to understand  particle-particle and particle-substrate interactions.
4. Mechanics of adhesion of micro- and  nanoscale particles.
5. Various factors affecting particle adhesion to a variety of substrates.
6. Surface modification techniques to modulate particle adhesion.
7. Various cleaning methods (both wet & dry) for particle removal.
8. Relevance of particle adhesion in a host of technologies ranging from simple to ultra-sophisticated.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Preface

Acknowledgements

Part 1: Particle Adhesion: Fundamentals

Chapter 1: Fundamental Forces in Particle Adhesion

1.1 Introduction

1.2 Various Forces in Particle Adhesion

1.3 Summary

References

Chapter 2: Mechanics of Particle Adhesion and Removal

2.1 Introduction

2.2 Models

2.3 Simulations Results

2.4 Summary and Conclusions

Acknowledgements

References

Chapter 3: Microscopic Particle Contact Adhesion Models and Macroscopic Behavior of Surface Modified Particles

3.1 Introduction

3.2 Constitutive Contact Models

3.3 Macroscopic Powder Behavior - Continuum Mechanics Approach

3.4 Surface Modification to Alter the Adhesion Properties

3.5 Experimental Measurements of the Adhesion Forces

3.6 Summary and Conclusions

Acknowledgements

List of Symbols

References

Chapter 4: Characterization of Single Particle Adhesion: A Review of Recent Progress

4.1 Introduction

4.2 Background

4.3 Recent Developments

4.4 Conclusions and Remarks

Acknowledgments

List of Symbols

References

Part 2: Particle Removal Techniques

Chapter 5: High Intensity Ultrasonic Cleaning for Particle Removal

5.1 Introduction

5.2 Ultrasound and Ultrasonics

5.3 Cavitation Phenomenon

5.4 Generation of Ultrasound – Transducers

5.5 Ultrasonic Generators

5.6 Principles of Ultrasonic Cleaning for Particle Removal

5.7 Determination of Residual Particles on Surfaces

5.8 Ultrasonic Aqueous Cleaning Equipment and Process

5.9 Precision Cleaning

5.10 Contaminants

5.11 Ultrasonic Cavitation Forces and Surface Cleaning

5.12 Cleaning Chemistry

5.13 Mechanism of Cleaning

5.14 Cavitation Erosion

5.15 Summary

References

Chapter 6: Megasonic Cleaning for Particle Removal

6.1 Introduction

6.2 Principles of Megasonic Cleaning

6.3 Particle Removal Mechanisms During Megasonic Cleaning

6.4 Types of Megasonic Systems

6.5 Particle Removal and Feature Damage in Megasonic Cleaning

6.6 Summary

References

Chapter 7: High Speed Air Jet Removal of Particles from Solid Surfaces

7.1 Introduction

7.2 Fundamental Characteristics of the Air Jet

7.3 Fundamentals of Air Jet Particle Removal

7.4 New Methods Using Air Jet

7.5 Summary and Prospect

List of Symbols

References

Chapter 8: Droplet Spray Technique for Particle Removal

8.1 Introduction

8.2 Droplet Impact Phenomena

8.3 Cleaning Process Window

8.4 Droplet Spray Technique for Semiconductor Wafer Cleaning

8.5 Summary

References

Chapter 9: Laser-Induced High-Pressure Micro-Spray Process for Nanoscale Particle Removal

9.1 Introduction

9.2 Concept of Droplet Opto-Hydrodynamic Cleaning (DOC)

9.3 Micro-Spray Generation by LIB

9.4 Mechanisms of Particle Removal by Laser-Induced Spray Jet

9.5 Generation of Micro-Spray Jet

9.6 Nanoscale Particle Removal

9.7 Summary

References

Chapter 10: Wiper-Based Cleaning of Particles from Surfaces

10.1 Introduction

10.2 Basic Mechanism of Wiping for Cleaning of Particles and Other Contaminants

10.3 Various Types of Wipers

10.4 Proper Ways to Carry Out Wiping or How to Use Wipers Properly

10.6 Results Obtained Using Wiping

10.7 Future Directions

10.8 Summary

References

Chapter 11: Application of Strippable Coatings for Removal of Particulate Contaminants

11.1 Introduction

11.2 Coating Description

11.3 Types of Strippable Coatings

11.4 Issues with Strippable Coatings

11.5 Precision Cleaning Applications

11.6 Summary

Disclaimer

References

Chapter 12: Cryoaerosol Cleaning of Particles from Surfaces

12.1 Introduction

12.2 History of Cryoaerosol Cleaning

12.3 Thermodynamics of Cryoaerosol Processes

12.4 Cleaning Mechanism

12.5 Factors Affecting Cleaning Performance

12.6 Results Obtained by Cryoaerosol Cleaning

12.7 Summary and Prospects

References

Chapter 13: Supercritical Carbon Dioxide Cleaning: Relevance to Particle Removal

13.1 Introduction

13.2 Surface Cleanliness Levels

13.3 Dense Phase Fluids

13.4 Principles of Supercritical CO2 Cleaning

13.5 Advantages and Disadvantages of Supercritical CO2 Cleaning

13.6 Applications

13.7 Summary and Conclusions

Acknowledgement

Disclaimer

References

Chapter 14: The Use of Surfactants to Enhance Particle Removal from Surfaces

14.1 Introduction

14.2 Solid-Solid Interactions

14.3 Introduction to Surfactants

14.4 Surfactant Adsorption at Solid Surfaces

14.5 Surfactants and Particulate Removal

14.6 Prospects

14.7 Summary

Acknowledgements

References

Index

Particle Adhesion and Removal

Scrivener Publishing100 Cummings Center, Suite 541JBeverly, MA 01915-6106

Adhesion and Adhesives: Fundamental and Applied Aspects

The topics to be covered include, but not limited to, basic and theoretical aspects of adhesion; modeling of adhesion phenomena; mechanisms of adhesion; surface and interfacial analysis and characterization; unraveling of events at interfaces; characterization of interphases; adhesion of thin films and coatings; adhesion aspects in reinforced composites; formation, characterization and durability of adhesive joints; surface preparation methods; polymer surface modification; biological adhesion; particle adhesion; adhesion of metallized plastics; adhesion of diamond-like films; adhesion promoters; contact angle, wettability and adhesion; superhydrophobicity and superhydrophilicity With regards to adhesives, the Series will include, but not limited to, green adhesives; novel and high-performance adhesives; and medical adhesive applications.

Series Editor: Dr. K.L. Mittal1983 Route 52,P.O. Box 1280, Hopewell Junction, NY 12533, USAEmail: [email protected]

Publishers at ScrivenerMartin Scrivener([email protected])Phillip Carmical ([email protected])

Copyright © 2015 by Scrivener Publishing LLC. All rights reserved.

Co-published by John Wiley & Sons, Inc. Hoboken, New Jersey and Scrivener Publishing LLC, Salem,Massachusetts.Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

ISBN 978-1-118-83153-3

Preface

The importance of particle adhesion and removal is quite manifest in many areas of human endeavor (ranging from microelectronics to optics, and space to biomedical). A complete catalog of modern precision and sophisticated technologies where removal of particles from surfaces is of cardinal importance will be prohibitively long, but the following eclectic examples should suffice to underscore the concern about particles on a variety of surfaces where particulate contamination is a běte noire. In the semiconductor world of ever-shrinking dimensions, particles which, just a few years ago, were cosmetically undesirable but functionally innocuous, are now “killer” defects. As device sizes get smaller, there will be more and more concern about smaller and smaller particles. In the information storage technology, the gap between the head and the disk is very narrow, and if a particle is trapped in the gap this can have very grave consequences. The implications of particulate contamination on sensitive optical surfaces are all too manifest. So the particulate contamination on surfaces is an anathema from functional, yield, and reliability points of view. With the burgeoning interest in nanotechnologies, the need to remove nano and sub-nano particles will be more and more intense. Apropos, it should be mentioned that in some situations, particle adhesion is a desideratum. For example, in photocopying the toner particles must adhere well to obtain photocopies, but these should not adhere to wrong places otherwise the result will be a dirty photocopy. Here also one can see the importance of particle removal.

One of us (KLM) has edited a series of books called “Particles on Surfaces: Detection, Adhesion and Removal” but the last volume (Volume 9) was published in 2006. Since 2006 there has been an enormous level of research activity, particularly in removing nanosize particles, and thus it was obvious that recent developments needed consolidation and this provided the vindication for the present book. This book was conceived with the core purpose of providing a comprehensive and easily accessible reference source covering important aspects/ramifications of particle adhesion and removal, with emphasis on recent developments in understanding nanoparticle adhesion mechanism(s) and their removal. All signals indicate that R&D activity in the arena of removal of nanometer size particles will continue unabated.

Now coming to this book (containing 14 chapters), it is divided into two parts: Part 1: Particle Adhesion: Fundamentals, and Part 2: Particle Removal Techniques. The topics covered include: Fundamental forces in particle adhesion; mechanics of particle adhesion and removal; microscopic particle adhesion models and surface modified particles; characterization of single particle adhesion; high intensity ultrasonic removal of particles; megasonic cleaning for particle removal; high speed air jet removal of particles; droplet spray technique for particle removal; laser-induced high-pressure micro-spray technique for particle removal; wiper-based cleaning of particles; application of strippable coatings for removal of particulate contaminants; cryogenic cleaning of particles; supercritical carbon dioxide cleaning: relevance to particle removal; and use of surfactants to enhance particle removal.

This book represents the cumulative contribution of many internationally renowned subject matter experts in the domain of particle adhesion and removal. The book reflects the state-of-the-art with special attention to recent and novel developments.

The book containing bountiful information on the fundamental and applied aspects of particle adhesion and removal provides a unified and comprehensive source. It should serve as a portal for the neophyte and a commentary on the recent developments for the veteran. The book should be of interest to researchers in academia and R&D, manufacturing, and quality control personnel in microelectronics, aerospace, automotive, optics, solar panels, pharmaceutical, biomedical, equipment cleaning and wafer reclaiming industries. Essentially, anyone involved in or concerned with removal of particles should find this book of immense value. Also, we hope that this book will serve as a fountainhead for new ideas pertaining to particle removal.

Acknowledgements

Now comes the pleasant task of thanking those who made this book possible. First and foremost, we are deeply thankful to the authors for their interest, enthusiasm, cooperation and contribution without which this book would not have seen the light of day. Also we are much appreciative of Martin Scrivener (Scrivener Publishing) for his steadfast interest in and continued support for this book project.

K.L. MittalHopewell Junction, NY, USAE-mail: [email protected]

Ravi JaiswalVaranasi, UP, IndiaE-mail: [email protected]

October 25, 2014

Part 1

PARTICLE ADHESION: FUNDAMENTALS

Chapter 1

Fundamental Forces in Particle Adhesion

Stephen Beaudoin1, Priyanka Jaiswal2, Aaron Harrison1, Jennifer Laster1, Kathryn Smith1, Melissa Sweat1, and Myles Thomas1

1School of Chemical Engineering, Purdue University, W. Lafayette, IN, USA,

2Department of Applied Chemistry & Polymer Technology, Delhi Technological University (formerly Delhi College of Engineering), New Delhi, India

*Corresponding author: [email protected]

Abstract

van der Waals, capillary, and electrostatic forces acting at the interface between a particle and a surface drive the adhesion behavior of the particles. If one can describe the nature and the strength of these forces as a function of the properties of the two interacting solids and the intervening medium, it is possible to predict and, in many cases, to control particle adhesion. This chapter focuses on the factors that influence the nature and strength of the forces, the fundamental theories that describe them, and the relevant mathematical expressions required to quantify them, with a caveat that the analysis presented is limited to systems with ideal geometry. Specifically, more advanced analysis, which may account for aspects such as roughness, non-uniform shape, deformation, and other complicating aspects, is not treated.

Keywords: Particle adhesion, van der Waals force, Hamaker constant, electrostatic force, double layer, capillary force, surface tension, surface energy.

1.1 Introduction

Particle adhesion influences many areas of science and engineering, including semiconductor fabrication, pharmaceuticals, cosmetics, mining, separations, petroleum production, surface coating, and food processing, to name a few. In the context of this chapter, adhesion is an interfacial phenomenon which appears when two solid bodies, one of which is of colloidal dimensions, approach each other closely. As the two surfaces approach, a complex interplay of van der Waals, electrostatic, and capillary forces drives the resulting behavior. Thorough knowledge of these surface forces is essential to understanding particle adhesion.

1.2 Various Forces in Particle Adhesion

In most applications of practical interest, the forces that control the adhesion between solid particles and solid surfaces are van der Waals (dipole) forces, electrostatic forces, and forces resulting from any liquid bridges due to capillaries or adsorbed molecular water between the two solids. Depending on the composition of the particle, the solid, and the ambient medium (air of varying relative humidity or aqueous solution are of interest here), the relative importance of these may change. This chapter provides an overview of these varying forces.

1.2.1 Capillary Forces

When a solid particle of characteristic dimension on the order of 100 micrometers or smaller is in contact with a solid surface in a gaseous medium (air), the relative humidity (RH) of the air is a critical factor in the relative importance of the forces that will influence the adhesion between the particle and surface [1,2]. Specifically, water molecules in humid air will minimize their free energy by adsorbing on surfaces at low humidity and by condensing onto surfaces at higher humidity, if the surfaces of interest are sufficiently hydrophilic [3–8]. If condensed moisture forms liquid bridges between a particle and a surface, the capillary forces resulting from these liquid bridges will generally be the controlling forces in the particle adhesion [9]. The behavior of adsorbed water molecules has been studied using gravimetric methods, ellipsometry, nuclear magnetic resonance (NMR), atomic force microscopy (AFM) and the surface force apparatus (SFA), among others [3–8, 10–19].

1.2.1.1 Forces Across a Curved Liquid Interface

When a solid surface comes in contact with a liquid medium, the difference in the magnitude of the net cohesive forces between the liquid molecules (i.e., Fl-l), and the net adhesion force between the liquid and the solid molecules (i.e., Fs-l) initiates the formation of a liquid meniscus at the solid/liquid interface. The nature of the curvature of the liquid meniscus (concave or convex) depends on which force, Fs-l (concave) or Fl-l (convex) is dominant. This leads to the phenomenon of wetting or de-wetting of the surface. Figure 1.1 shows an example of a liquid climbing on a solid plate. In this case, Fs-l > Fl-l. Solid surfaces which have Fs-l > Fl-l are known as high energy surfaces. If the liquid is an aqueous solution, these are known as hydrophilic surfaces. If the liquid is non-aqueous, they are known as lyophilic surfaces. Such surfaces facilitate wetting. Mica, silicon dioxide, metals, and oxidized surfaces in general are typically hydrophilic. Solid surfaces in which Fs-l < Fl-l are known as low energy surfaces. If the liquid is an aqueous solution, these are the hydrophobic surfaces. If the liquid is non-aqueous, they are the lyophobic surfaces. They facilitate de-wetting. Most organic surfaces, including most polymers, are hydrophobic. The surface energy of such materials can be increased by surface modifications (e.g., surface oxidation achieved via ultraviolet radiation, plasma discharge, laser irradiation, etc.) to enhance their hydrophilicity [20].

Figure 1.1 Meniscus formation on a solid plate partially immersed in a wetting liquid.

1.2.1.1.1 Surface Tension Force Acting at a Solid/Liquid Interface

The origin of surface tension is the unbalanced intermolecular force acting on the liquid molecules at the surface. The molecules present in the bulk of the liquid experience no net intermolecular force as they are surrounded by molecules of similar properties and hence are in a low energy state. However, the liquid molecules present at a liquid/solid or liquid/air interface are in an unbalanced or high energy state as they experience a net intermolecular force resulting from the difference in properties of the molecules in the different media. This leads to the development of the surface tension force. The surface tension (γ) is quantified as the net surface tension force acting on a unit length of the liquid/solid or liquid/air interface. Figure 1.2 is a schematic of a spherical particle in contact with a solid surface through a liquid medium. The surface tension force, Fst, acting on the solid/liquid boundary (the dotted line) can be obtained as

(1.1)

where α is the angle of inclination of the liquid meniscus from the vertical, and lwetted is the perimeter of the meniscus boundary on the solid surface.

Figure 1.2 Schematic showing surface tension force acting at the solid/liquid interface.

1.2.1.1.2 Capillary Pressure Force Acting Across a Curved Liquid Interface

The micro-/nano-contacts between two solid surfaces act as active sites for condensation in a humid environment if the RH is above a critical value. When condensed moisture comes in contact with the solid surfaces, a liquid meniscus is formed in the contact region bridging the two solid surfaces, as shown in Figure 1.3.

Figure 1.3 A liquid bridge surrounding a solid particle in contact with a flat substrate.

Menisci form through two methods on solid surfaces: the spontaneous condensation of a vapor in a confined space (otherwise known as capillary condensation) and, for non-volatile liquids, the combination of adsorbed layers (on the two adhering surfaces) merged into a meniscus. A meniscus induces a pressure difference across the liquid-vapor interface, as shown in Figure 1.4, where the pressure on the liquid side of the meniscus is lower than that in the surrounding vapor. This pressure difference is described by the Young-Laplace equation

(1.2)

where ΔP is the pressure difference across the meniscus (the Laplace pressure), γl is the surface tension of the liquid condensate, and rn and rp are the two principal radii of curvature (ROC) of the liquid bridge between the surfaces [21]. The Laplace pressure acts over an area, A, and induces a force that pulls the two surfaces together increasing the total adhesion force [9]. The normal surface tension force around the circumference of the meniscus (Equation 1.1) also contributes to the force, but it is usually small compared to the pressure-induced force and is often not considered for micro-scale particles [9].

Figure 1.4 A spherical particle adhering onto a flat substrate with a liquid bridge formed at the solid-solid interface. The meniscus geometry is shown on the right.

The following relations can be obtained for the geometry shown:

(1.3)

(1.4)

where d is the height of the particle inside the liquid bridge, and D is the separation distance, as shown in Figure 1.4, θ1 and θ2 are the contact angles of the liquid with the sphere (1) and the flat substrate (2), and φ is the half angle subtended at the center of the sphere by the wetted area of the sphere (this is also known as the embracing’ or ‘filling’ angle).

The ROC, rn, can also be obtained from the geometry shown in Figure 1.4:

(1.5)

where R is the particle radius. The equilibrium capillary pressure force, Fcp, is found by multiplying the Laplace pressure by the interaction area using the Young-Laplace equation [22]

(1.6)

where rc is the radius of the contact circle at the solid particle/liquid/air interface, and is given by:

(1.7)

For a large sphere (RD and Rd), the following approximations can be made:

I. The embracing angle, φ, will be very small in comparison to the contact angle, θ1

II. will be very small, hence φ + θ1 ≈ θ1

III.rc can be obtained using the geometry shown in Figure 1.4,

(1.8)

IV.rnrp from Equations 1.4 and 1.5, therefore 1/rn in Equation 1.6 can be neglected.

The final expression for the capillary pressure force between a large spherical particle and a planar surface, using the above approximations, can be obtained as:

(1.9)

(1.10)

It is apparent from Equation 1.10 that the capillary force for the case of a large spherical particle in contact with a flat substrate is humidity independent (as d, a humidity-dependent parameter which quantifies the height of the liquid bridge, gets canceled out); and hence the capillary force in this case is a function of only the particle size and the surface tension. This has also been shown experimentally [22]. However, the capillary forces for small particles have strong humidity dependence [6, 23].

Most parameters, except d, in Equation 1.9 are usually available to calculate the capillary force between a sphere and a flat plate. The estimation of d requires knowledge of the embracing angle (φ) or the volume of the liquid bridge (V).

a)Relation between d and φ

It is apparent from the geometry shown in Figure 1.4 that

(1.11)

The embracing angle φ will be very small for large spheres or for small liquid bridge volume.

b)Relation between d and V

The following relation exists between d and the liquid bridge volume V[24]

(1.12)

For the case of small separation distance D, d.

For the case of large separation distance D,

The total capillary force acting between a sphere and a flat plate can be determined by combining the capillary pressure force (Equation 1.9) with the surface tension force (Equation 1.1) [25]

(1.13)

where is the angle of the liquid meniscus (at the particle/liquid/air interface) from the vertical as shown in Figure 1.4. Finally,

(1.14)

The filling angle φ is still unknown, but can be estimated by the Kelvin equation, which relates the equilibrium ROC of the meniscus to the ambient relative humidity (RH) [26, 27]

(1.15)

where rk is the so-called ‘Kelvin radius’. Specifically, by substituting Equations 1.4 and 1.5 into Equation 1.15, one may determine φ numerically based on Equation 1.16, and then solve Equations 1.6 and 1.14 to determine Fcp and Ftot

(1.16)

Figure 1.5 Maximum separation distance between two solid surfaces to allow capillary condensation as a function of RH at room temperature.

It is important to note that the Kelvin equation applies to systems containing continuum or bulk liquid water (systems in which the water in the liquid phase has the density and surface tension of bulk liquid water). Adsorbed moisture in molecular form still has an impact on adhesion, but the Kelvin equation is inappropriate for describing the behavior in such systems. Water has an effective diameter of roughly 0.37 nm [21]. At 30% RH, the liquid bridge between two solid surfaces would have a characteristic Kelvin radius corresponding to ~ 3 water molecules. At 50% RH, this number increases to somewhere between 4 and 5 water molecules. In either case, the argument that such a small amount of water would behave like ‘bulk water, exhibiting surface tension effects in the same manner as bulk water, is tenuous.

The Young-Laplace and Kelvin equations are almost universally used to predict the capillary force between macroscopic bodies [1, 6, 11, 28–37]. Butt and Kappl summarize these capillary forces for smooth systems, such as plane-plane, sphere-plane, cone-plane, sphere-sphere, and cone-cone geometries [37]. Furthermore, with only slight modifications, these equations can also model the dependence of the capillary force on the surface roughness [35, 38, 39] and heterogeneity [38].

The conditions in which the surface tension force should be considered when calculating the total capillary force have been demonstrated through numerical simulation in an idealized modeling framework, as shown in Figure 1.6 [40]. This figure shows computational values of the total capillary force, the surface tension force, and the capillary pressure force, for three differently-sized spherical particles interacting with a flat surface. For ease of comparison, the total force is normalized by the particle radius. The relevant region of RH to be considered in this figure is the region with RH > 50%, as this is the condition where condensed moisture can be well-represented as continuum. In this region, it can be seen that the surface tension force (inverted triangles) makes a negligible contribution to the overall force (solid lines) when the particle size is above 1000 nm. When the particle radius is 100 nm, the surface tension force contribution to the overall force is no longer negligible, and when the particle is nano-scale and the RH is high, the surface tension force and the capillary pressure force make comparable contributions to the overall force.

Figure 1.6 Theoretical capillary pressure (Fcp), surface tension (Fst), and total capillary forces (Ftot) between a sphere and flat surface separated by 3Å.

Reproduced with permission [40].

1.2.1.2 Effects of RH on Capillary Forces

Capillary forces resulting from condensed moisture between hydrophilic surfaces generally increase continuously with increasing humidity to a maximum and then decrease, while humidity has little effect on adhesion when one or both of the interacting surfaces are hydrophobic [2, 3, 6–8, 18, 19, 32, 41–43]. Frequently, the increase in adhesion with increasing RH continues until RH attains roughly 50%, after which the force becomes independent of further RH increases or drops with such increases [3, 6, 32, 43]. Figures 1.7 and 1.8 show such results in two AFM-based studies. The magnitude of these changes is influenced by the geometry and composition of the surfaces (or the asperities on the surfaces) in contact [5–7, 15–17].

Figure 1.7 Measured adhesion forces between Si3N4 AFM cantilevers with nanoscale radii of curvature (~20 nm) and stainless steel (hydrophilic) and Perspex® (hydrophobic) surfaces. Each data point is the average of 1200 measurements [42].

These behaviors have been attributed to: 1) variations in the radii of curvature of the menisci of condensed moisture between non-uniform features (asperities) of the interacting surfaces; 2) variations in the thickness of any adsorbed molecular water films; 3) the complexity of the local separation distance between the interacting surfaces due to the complementarity of the asperities/topography of the surfaces (basically how well do the asperities on the two surfaces fit together and how does this influence the accessibility of condensed moisture to elements of the interacting surfaces); 4) changes in the cantilever tip geometry as a result of dulling during multiple tip-substrate interactions (unique to AFM-based studies); 5) composition-driven variations in the contact angles of water on both the probe and substrate, leading to local variations in capillary forces; and 6) dissolution and reaction of surface species on the two interacting surfaces [5]. While these effects do have an influence on the observed behavior, a true description over the entire range of humidity requires a detailed consideration of the form of water at the interface between the particle and surface. Specifically, the Young-Laplace and Kelvin equations are of great utility for equilibrated systems where: 1) moisture is present on a surface in sufficient quantity that it maintains the properties of bulk water, 2) the radius of the particle is much greater than both rn and rp, and 3) the volume of the liquid bridge is approximately constant [37]. However, as the size of the particle approaches the nanoscale, the validity of these criteria is disputable, throwing into question the applicability of these continuum models.

In Figures 1.7 and 1.8, the fact that the dependence of the adhesion force on RH is different for hydrophilic and hydrophobic surfaces is to be expected. However, the maxima in adhesion force observed in the hydrophilic systems are not consistent with existing continuum models. The Kelvin equation typically predicts a monotonic increase in adhesion force with increasing RH. It also predicts radii of curvature on the order of a few molecular diameters for equilibrated menisci at low RH. For example, at 20 °C and 50% RH, the Kelvin radius for water is 0.8 nm [9]. With an effective diameter of approximately 0.37 nm [9], water is thus expected to form a meniscus with less than 3 molecules. Such a constraint considerably stretches the intra- and intermolecular bonds within the liquid. To further illustrate this discrepancy, it has been shown that when the mechanical properties of water dictate the change in surface tension on a nanoscale in water-induced capillary systems, the Kelvin equation is only applicable above 45% RH at best [44]. Below 45% RH, the macroscopic or continuum assumptions in the Young-Laplace and Kelvin equations oversimplify the existence, formation, and magnitude of capillary forces on the nanoscale, especially for polar liquids.

1.2.1.2.1 Effects of High RH on Adhesion Forces

At sufficiently high humidity (>80%), particle adhesion forces are observed to decrease, as can be seen in Figures 1.7 and 1.8. This is attributed to a combination of factors. First, strongly adsorbed water on a hydrophilic substrate prevents the close approach of the particle and surface. At the higher RH, the liquid bridge can wet a larger fraction of the particle, as the condensed moisture can span a greater distance between the two surfaces. The liquid bridge can also be extended to a greater distance along the substrate. As the extent of the condensed liquid around the particle grows, the values of rp and rn in Equation 1.6 become large which drives the capillary force down. This makes it easier for the adsorbed water to hold the particle away from the surface. When the RH attains 100%, a continuous water layer forms at the solid/solid interface. This completely eliminates the liquid neck and hence the capillary force at the solid contacts, and it screens the van der Waals interactions between the adhering solid surfaces.

1.2.1.2.2 Effects of Low RH on Adhesion Forces

At humidity levels below 45%, molecular-scale representations of the behavior of water are required to explain the role of water in particle adhesion. For this purpose, several approaches have been applied successfully. First, a combination of coarse lattice-gas (LG) models, grand canonical Monte Carlo (GCMC) simulations, and thermodynamic integration techniques has been used to predict the effects of adsorbed moisture between an AFM probe and a hydrophilic surface [8, 45–51]. In these studies, vapor molecules (ci) are allowed to occupy sites on a 3D lattice spanning the interstitial space between the two bodies, as shown schematically (in 2D) in Figure 1.9. The lattice spacing is generally one molecular diameter, each site may be either fully occupied or unoccupied, and molecules are only allowed to interact with their nearest neighbors (NN). Each molecule has its own chemical potential, μ, and the interaction between a molecule and its NN is described using an intermolecular attraction energy, ε. If a molecule’s NN is either of the two solid surfaces in consideration, the top or the bottom surface (particle or substrate), binding energies of bT or bs are imposed.

The total energy of such a system, H, is given by:

(1.17)

where N is the total number of molecules in the system or the total number of occupied sites [52]. The chemical potential is related to RH by

(1.18)

Figure 1.9 Schematic of grand canonical Monte Carlo lattice-gas model.

To determine whether or not a molecule is removed from, added to, or moved to a lattice site, one calculates the change in the total energy of the system (ΔH) resulting from the proposed molecular change. If ΔH is negative (i.e., the change is energetically favorable), the removal, addition or relocation of the molecule is accepted. If ΔH is positive, the removal, addition or relocation of the molecule occurs with a given probability. This Monte Carlo process is repeated hundreds of thousands of times, which allows the system to reach ‘equilibrium’ [52]. The capillary force resulting from this molecularly adsorbed, non-continuum water is calculated by integrating the partial derivative of the excess number of molecules relative to the bulk system (Nex) with respect to changes in the separation distance between the two surfaces (h) for a fixed μ and T [45]:

(1.19)

Many of the effects of molecularly adsorbed water between an AFM probe and a substrate have been predicted with this LG GCMC model (Equations 1.17–1.19) [54]. Additionally, the effects of RH on the pull-off force between AFM cantilevers and substrates of varying hydrophilicity can be predicted qualitatively with this method. Specifically, a maximum in the adhesion force is predicted around 30% RH for a strongly hydrophilic tip; a plateau above 34% is predicted for a hydrophobic tip, due to the interaction of two confined layers of water; and a monotonic increase is predicted for a slightly hydrophilic tip [46, 55]. All of these behaviors have been verified experimentally [3, 11, 30, 32, 56, 57]. The LG GCMC model has also demonstrated that nanoscale roughness on either the tip or the substrate dramatically influences the force-RH curve [48, 51]. For example, for a smooth tip and a smooth surface, a single maximum is seen in the plot of the adhesion (pull-off) force as a function of RH. However, when a rough tip and rough surface are simulated, several local maxima are predicted [51, 52]. The LG GCMC model is computationally simple, and its ability to provide molecular insight into the onset of true capillary forces makes it very attractive [52].

A drawback of the LG GCMC model is that it does not account for molecular shape, dipole moments, or long-range electrostatic interactions. To consider these effects, adhesion forces between AFM probes and surfaces at low RH have also been modeled using molecular dynamics (MD) techniques, which can incorporate these more realistic conditions [58]. The formation and breakage of a true liquid meniscus between an AFM probe and a surface at high (70%) RH was predicted in this manner based on the density profile of water molecules from MD snapshots on a 3D lattice [58]. In other work with a hydrophilic AFM probe tip against a hydrophilic surface, an oscillatory force was predicted as the tip was withdrawn from the surface, indicative of the role of confined layers of molecular water on the adhesion. This simulation also recreated a global maximum in the pull-off force around 20% RH, which correlates with experimental values for gold and mica surfaces [59]. These recent studies are promising approaches to understanding the molecular origin of capillary forces at low RH.

In addition to LG GCMC and MD simulations, density functional theory (DFT) [46, 60, 61] and computational fluid dynamics (CFD) [62, 63] simulations have been used to describe the effect of RH on adhesion. Like the MD simulations, the DFT simulations are computationally less demanding than the LG GCMC model. However, they do not account for strongly adsorbed layers or fluctuations in the meniscus at high RH (the region of transition from water with continuum-like density to vapor with gas-like density) [61].

A final approach to modeling capillary forces at low RH stems from an observation that adsorbed water can form ice-like, monolayer structures on smooth, hydrophilic materials (e.g., mica and silicon dioxide) [10]. This phenomenon has been attributed to the surface having an isosteric heat of adsorption greater than the latent heat of condensation for water [57]. Since the number of hydrogen bonds per water molecule is greater in ice than in the liquid (a monolayer of ice-like molecules is expected to have a higher surface energy than liquid water), the surface tension of liquid water, therefore, should not be expected to account for the total adhesion force between a particle and an ice-inducing surface. By considering van der Waals forces (described later) to account for the surface energy of ice-like molecular water, and the capillary forces (predicted using the Young-Laplace and Kelvin adhesion models), the experimentally observed adhesion behavior of ultra-smooth hydrophilic surfaces against AFM probes as a function of RH has been modeled effectively, including the prediction of an adhesion maximum at roughly 30% RH as shown in Figure 1.7 [30, 64].

1.2.1.3 Effects of Bulk Liquid Water on Capillary Forces in Idealized Systems

In systems where the RH is high enough to assure the presence of liquid water, but in which the adhesion between the water and the interacting solid surfaces is not so strong as to limit the closeness of approach of the two surfaces, a number of idealized continuum models may be used to obtain analytical expressions for capillary forces and to describe the effect of the condensed moisture. This section shows derivations of the analytical expressions for capillary forces for systems with ideal geometries.

1.2.1.3.1 Parallel Plates

As a starting point, Figure 1.10 shows two parallel plates of different materials separated by a thin liquid film of thickness, d. The meniscus of the film is cylindrical in shape with the primary ROC, rp, and the secondary ROC, rn (= ∞). The capillary pressure force for this system can be obtained from a modified version of Equation 1.6:

(1.20)

where Axy is the wetted area of the plate.

The ROC, rp of the film meniscus can be related to the film thickness, d, using geometry shown in Figure 1.10b:

(1.21)

where θ1 and θ2 are the contact angles of the liquid against the two plates. Finally, the capillary pressure force for this system can be determined as:

(1.22)

Figure 1.10 (a) Two parallel plates made of different materials separated by a thin sheet of liquid film, (b) expanded view of the geometry at the liquid film meniscus.

(1.23)

Figure 1.11 shows a schematic of two plates of the same material, separated by a liquid column of height, d, and radius, R. The contact angle of the liquid with each plate is θ. The capillary pressure force between these plates can be obtained using Equations 1.6 and 1.23:

(1.24)

where Axy is the wetted area of the plate, and can be approximated to πR2[22].

Figure 1.11 Two parallel plates of same material linked by a columnar liquid bridge.

If the radius of the liquid column is much larger than the column height (i.e., Rd),

(1.25)

The surface tension force acting on the plate in this case can be obtained using Equation 1.26:

(1.26)

The total capillary force acting between the plates can be determined by combining the capillary pressure force with the surface tension force:

(1.27)

1.2.1.3.2 Two Spherical Particles Linked by a Liquid Bridge

Figure 1.12 shows a schematic of two spherical particles of radii R1 and R2, separated by distance D and linked by a liquid column with radii of curvature rp and rn and height (D+d1 +d2). The ROC, rp, can be determined using geometry shown in Figure 1.12.

Figure 1.12 Two spherical particles linked by a liquid bridge. Detail of the meniscus is at the right.

(1.28)

where d1 and d2 are the heights of the two spherical particles inside the liquid bridge, θ1 and θ2 are the contact angles of the liquid with the spheres, and φ1 and φ2 are the ‘embracing’ angles for the spheres.

The ROC, rn, can also be obtained from the geometry shown:

(1.29)

Now the general expression for the capillary force between two spherical particles can be obtained using the Young-Laplace equation as:

(1.30)

where rc is the radius of the contact circle at the solid particle/liquid/air interface.

The capillary force in Equation 1.30 can be calculated if either the particle depths into the liquid bridge (d1 and d2) or the embracing angles(φ1 and φ2) are known as they both are related according to:

(1.31)

When the volume of the liquid bridge, V, is known, the capillary force can be calculated. First, the expression for the volume of the liquid bridge is [65,66]:

(1.32)

where

Next, the following apparent geometric relation can be obtained from Figure 1.12.

(1.33)

The embracing angles can now be calculated using Equations 1.29–1.30, and then the total force can be predicted using Equation 1.27.

For large spheres (R1 and R2D, and R1 and R2d), the following approximations can be made:

I. The embracing angles, and will be very small, and hence φ1 + θ ≈ θ1 and φ2 + θ2 ≈ θ2

II.rnrp from Equations 1.25 and 1.26, therefore 1/rn in Equation 1.30 can be neglected.

The final expression for the capillary pressure force between two large spherical particles linked by a liquid bridge can be obtained using the above approximations in Equation 1.30

(1.34)

(1.35)

The parameter d can be determined from the known embracing angle using Equation 1.31. It can also be estimated if the volume of the liquid bridge, V, is given using the following relation [67]:

(1.36)

For the case of small separation distance D,

For the case of large separation D,

1.2.1.3.3 Non-spherical Particles Against a Flat Surface

Particle shape can influence the capillary forces by changing the geometrical parameters of the liquid meniscus generated at the solid/solid interface in a humid environment. Figure 1.13 (top) shows simulated geometries of five idealized AFM cantilever probes, and Figure 1.13 (bottom) shows predicted capillary forces, based on the Kelvin-Laplace equation, between a flat substrate and these AFM probes [40]. As noted above, the RH levels at which the Kelvin-Laplace equation may be applied are strictly limited by the ability of the two surfaces to approach each other. In this case, where the surface is a theoretical, atomically flat surface and the cantilever tip is assumed to have perfect geometry and no roughness, the Kelvin-Laplace predictions are appropriate at lower RH levels than would be appropriate for realistic systems. The shapes of the simulated AFM probes are assumed to be (a) spherical, (b) polynomial with flat tip, (c) conical, (d) truncated conical, and (e) polynomial with curved tip. As can be seen in Figure 1.13 (bottom), particle geometry has a strong impact on the capillary forces, emphasizing the importance of adequate modeling of the particle geometry for reliable prediction of these forces.

1.2.1.4 Effects of Bulk Liquid Water on Capillary Forces Using Non-ideal Meniscus Shapes

The prediction of capillary forces for non-ideal solid/solid contacts in a humid environment requires a proper accounting of the irregularity in geometry, roughness and material properties of the adhering objects, and the resulting shape of the liquid meniscus formed between them. Even for cases when interacting solid objects are ideally shaped, it is not necessarily true that the meniscus has an ideal shape as was assumed in the calculations considered above. Most theoretical models assume a circular profile for the liquid meniscus, i.e., the ROC (rp) in Fig. 1.4 is assumed to be the same at every point on the liquid meniscus. However, this assumption is rarely true and can give significant deviation, especially for small particles or nano-contacts at high humidity [40]. For such systems, precise predictions of capillary forces can be made by computing in a sequential, point-by-point manner across the liquid meniscus. This can be done by expressing rp and rn as [68]

(1.37)

The expression for the capillary force can now be written using the general form of the Young-Laplace equation:

(1.38)

Equation 1.38 can be solved with proper boundary conditions for the given solid/solid system to obtain the meniscus profile x(y) and the capillary force, Fcp.

For the case of a spheroidal particle in contact with a flat plate, as shown in Figure 1.14, the following boundary conditions can be used to solve Equation 1.38

Figure 1.14 Schematic of a sphere adhered onto a flat plate with a liquid bridge.

The solution of Equation 1.38 with these boundary conditions will lead to the derivation of the meniscus profile =f(y, φ, Fcp, R, γt, θ1, θ2). Finally, the expression for the capillary force between a sphere and a flat plate, accounting for the non-uniform ROC, can be obtained using the following condition in the derived meniscus profile:

(1.39)

The value of φ needs to be known to calculate the capillary force. If φ is not given but the volume of the liquid bridge (V) is known, the following relation between V and φ can be obtained.

(1.40)

Equations 1.38 and 1.40 can now be solved together to calculate φ and the capillary force.

1.2.1.5 Concluding Remarks

a. The formation of liquid bridges (due to moisture condensation) at the solid/solid interface leads to the existence of capillary forces.

b. The total capillary force is the combination of the surface tension force and the capillary pressure force. The capillary pressure force, which originates due to the pressure difference across the liquid/vapor meniscus developed at the solid/solid interface, dominates over the surface tension force.

c. The Kelvin diameter (2rk) is defined as the maximum separation distance between the adhering solid surfaces over which the moisture condensation takes place. Since the Kelvin diameter for even the most strongly wetting surfaces is < 2 nm (which is on the order of the roughness height for most substrates) for RH < 50%, any adhesion enhancement with RH at levels below 50% cannot be attributed to continuum moisture effects.

d. The strength of the capillary force depends on the RH, the nature of the adhering substrates, and the temperature. The capillary forces resulting from the condensed moisture between hydrophilic surfaces generally increase with increasing RH, while humidity has little effect on adhesion when one or both of the interacting surfaces are hydrophobic.

e. The magnitude of the capillary force also depends on the geometries of the adhering solid bodies. Relevant analytical expressions to predict capillary forces for the systems with ideal geometries (plate-plate, sphere-plate, and sphere-sphere) have been derived. The approach to determine capillary forces for non-ideal geometries has also been discussed.

1.2.2 van der Waals Forces

van der Waals forces result when dipoles in the surface regions of two interacting bodies respond to electromagnetic radiation propagating between the surfaces of the bodies. These forces are significant over separation distances up to 40 nm, depending on the properties of the medium between the bodies. There are two approaches used to describe these forces, Hamaker’s pairwise additive approach and the Lifshitz continuum approach.

1.2.2.1 Hamaker’s Pairwise Additive Method

Hamaker’s approach for determining van der Waals forces between particles and surfaces begins with the energy of interaction between two particles (of radii R1 and R2) containing ρ1 and ρ2 atoms per cm3, as shown in Figure 1.15. The interaction energy between the two spheres, or any spherically symmetric pair of bodies, is [69–75]

Figure 1.15 Diagram of two spherical particles experiencing mutual vdW interaction [76].

Reprinted with permission.

(1.41)

where dV1, dV2, V1, and V2 are the volume elements and total volumes, respectively, of the two particles; |r| is the separation distance between dV1 and dV2; and ζ12 is the vdW or Hamaker constant, a purely material-dependent quantity. To evaluate the double integral in Equation 1.41, one considers the interaction energy experienced at point P in Figure 1.16. In this case, the surface ABC is given by

(1.42)

where θ0 is found by using the Law of Cosines,

(1.43)

Figure 1.16 Diagram illustrating the geometry associated with the vdW interaction energy experienced at point P [76].

Adapted with permission.

After substituting Equation 1.43 into Equation 1.42, one obtains

(1.44)

Thus, a volume element (i.e., dV1 or dV2) is

(1.45)

Finally, the potential energy of an atom or molecule located at point P is

(1.46)

To incorporate a second sphere of radius R2 and centered around point P, this geometric procedure is repeated and the double integration in Equation 1.42 is repeated between opposing volume elements (in a pairwise fashion). The resulting expression for the interaction energy between two spheres is

(1.47)

(1.48)

When the separation distance between the particles is much smaller than the size of either particle (i.e., |r| Rparticle), Equation 1.48 is readily transformed to describe the vdW interaction between a sphere and an infinite flat plate (R2R1, y → ∞):

(1.49)

To obtain the force of interaction between a sphere and a flat plate, differentiate Equation 1.49 with respect to separation distance d:

(1.50)

When x 1, which is the case for a particle in contact with a flat plate, this expression simplifies even further to [9]

(1.51)

Similarly, the expression for the vdW forces for other regular geometries can also be derived using Hamaker’s pairwise additivity. The vdW force between unit areas of the two opposing parallel plates is derived as

(1.52)

The vdW force between two spherical particles of radii R1 and R2 is given as

(1.53)

These limiting-case results owe their simplicity to Hamaker’s assumption of pairwise additivity of vdW interactions. This means each volume element dV1,i interacts with a second volume element dV2,j over a distance rij, and there is no accounting for the effect of many-body interactions—e.g. reflected electromagnetic (EM) waves caused by the presence of neighboring atoms or molecules, known as the polarizability effect—nor is the retardation effect (i.e., the phase lag induced during transmission of the electric field between interacting elements) considered.

1.2.2.1.1 Calculating the Hamaker Constant: Approximate Forms Using Pairwise Schemes

When a pairwise approach is taken to describing van der Waals forces, the Hamaker constant is often expressed in terms of a combination of Keesom, Debye, and London interactions. Keesom interactions are interactions between polar molecules with permanent dipoles (dipole-dipole interactions). Debye interactions are interactions between polar molecules and non-polar molecules (dipole-induced dipole interactions) or between pairs of polar molecules when the molecules induce dipoles in each other. London interactions are between pairs of non-polar molecules and non-polar molecules (induced dipole-induced dipole interactions; dispersion interactions).

The Keesom interaction energy is described by

(1.54)

The Debye interaction energy between a polar molecule (molecule 1) and a non-polar molecule (molecule 2) is described by

(1.55)

(1.56)

The London dispersion interaction energy between two similar molecules is

(1.57)

(1.58)

When Hamaker constants are determined in this manner, they are calculated analogously to the form in Equation 1.41

(1.59)

It should be noted that when Hamaker constants are determined using this pairwise additive approach, it is assumed that the electromagnetic field propagating between the two interacting objects moves quickly relative to the time that dipoles reorient in the objects. When the separation distance between the interacting bodies is larger than roughly 5 nm in gaseous environments, this assumption begins to fail and dipoles within the interacting surfaces can recover before the reflected field from the opposing surface arrives. This assumption fails at closer separation distances in condensed media, where the propagation of electric fields is slower. In such cases, the van der Waals forces are considered to be ‘retarded’, and the extent of the retardation is related to the properties of the medium between the bodies [9, 77]. When vdW forces are retarded, the various components of the force (Keesom, Debye, London dispersion) are affected differently. Specifically, the dispersion forces, which are the dominant forces at close separations, tend to fade, becoming proportional to 1/r7 rather than 1/r6. In such cases, the overall vdW force begins to follow the 1/r7 pattern. As separation distance continues to increase to the point where the dispersion forces are no longer dominant, one finds that the forces driven by permanent dipoles (Keesom and Debye), assume a dominant role. Their electronic configuration does not change, and as such their dependence on the propagation of electric field through the medium does not change. This causes the vdW force to resume a 1/r6 behavior [9].

1.2.2.2 Lifshitz Continuum Approach to van der Waals Forces

In 1954, Lifshitz developed a rigorous method, derived on the basis of quantum electrodynamics, for predicting vdW interactions between bulk condensed-phase media by treating these as continua and relating the fluctuating EM fields of the approaching bodies to their complex dielectric permittivity functions (dielectric functions). These functions are defined according to

(1.60)

where ε’ is the real component of the dielectric permittivity ε” is the imaginary component of the dielectric permittivity and ω is the frequency of the propagating electromagnetic field between the surfaces of the interacting bodies. It has been shown that the imaginary part of the permittivity is always positive and determines the energy dissipation of a wave propagating through a medium [78–89].

The geometry used as the basis for the development of this approach is illustrated in Figure 1.17.

Figure 1.17 Two semi-infinite media separated by distance D, interacting across intervening medium m.

Dzyaloshinskii, Lifshitz, Pitaevskii and Hamermesh (DLPH) used quantum field theory methods in statistical physics to validate Lifshitz’ original approach and prove that vdW interactions can be quantified based on purely macroscopic considerations [79]. The resulting DLPH theory formulae for the vdW energy of interaction between two condensed-phase, semi-infinite, planar media, separated by a small distance D, across an intervening medium m are

(1.61)

(1.62)

(1.63)

(1.64)

To derive the formula for the sphere-plane system geometry, the Derjaguin Transformation is applied to Equation 1.61 [90]. The Derjaguin Transformation relates the interaction force between two spheres to the energy per unit area of two flat plates at the same separation distance

(1.65)

where F(D) is the sphere-sphere interaction force at separation distance D, R1 and R2 are the radii of the spheres, and W(D) is the interaction energy per unit area of flat plates at the same separation distance D. Per this transform, if oppositely curved surfaces are at a separation distance D that is small in comparison to the radii R1, R2 of the bodies, the interaction force between the curved surfaces can be approximated as an interaction between parallel flat plates. This transform is appropriate for micro-scale spheroidal contact within the separation distance from a surface where vdW forces are important. Applying this transform to Equation 1.61 yields the interaction energy between two spheres:

(1.66)

(1.67)

(1.68)

Note that the variable of integration, which was x in Equation 1.61, is now p in Equation 1.66. For very small values of rn, which arise for situations involving bodies in contact, the argument of the natural logarithm in Equation 1.61 approaches zero, and the function is no longer analytical. To avoid this difficulty, the variable transformation x → p has been introduced, and the range of integration has changed correspondingly from [rn, ∞), for which rn can be arbitrarily small, to [1, ∞). Note also that the functions εi and μi are complex. In order to evaluate Equation 1.61, the integrand must be complex-differentiable. Complex-valued functions often contain points, zeroes, poles, asymptotes, and other features at which they are not complex-differentiable in Cartesian coordinates. To avoid difficulties associated with this situation, the second infinite summation, the second Riemann zeta function, sometimes denoted ζ(2), is used in Equation 1.66. This allows the integrand in Equation 1.61 to be mapped onto a space over which it is complex-differentiable over the range of integration. The effect of introducing ζ(2) is that integrand of Equation 1.66 no longer involves a natural logarithm.

For a spherical particle in contact with a planar surface, R2 → ∞ and Equation 1.66 becomes

(1.69)

For non-permanently magnetic materials, the μi’s are all simply unity. This means the magnitudes of the Δji’s depend only on the difference between si and sj. When all of the materials comprising the system have relatively similar dielectric characteristics, the difference between si. and sj. is small and is made even smaller by the Riemannian exponent q. The net effect is that the (ΔLmΔRm)q term in Equation 1.69 can often be omitted, so that the interaction energy becomes

(1.70)

The vdW interaction force of a sphere-plane system may be found by differentiating Equation (1.70) with respect to separation distance D as

(1.71)

In this expression, the Hamaker constant, which is actually a coefficient that varies with the separation distance between the two bodies, is

(1.72)

Before these formulae can be utilized to calculate the vdW force, the complex dielectric permittivities ε(iξ) of L, R, and m must be known. Unfortunately, the exact functional forms of ε(iξ) are rarely known, and in general one must resort to approximation techniques or other simplifications to evaluate their integral relations [9, 78, 79, 83, 85, 91–103]. It has been shown that only the imaginary part ε” of the dielectric permittivity is needed for the determination of ε, via the Kramers-Kronig formula [79, 104].

(1.73)

Thus, if experimental optical data are available over a sufficient range of angular frequencies ω, then the interaction force can be calculated beginning with numerical evaluation of Equation 1.73.

(1.74)

These sampling frequencies are known as Matsubara frequencies. Figure 1.18 illustrates the Matsubara frequency distribution across the EM spectrum. As can be seen, there is an increase in sampling frequency with photon energy. This allows certain spectral regions (specifically, the UV) to make greater contributions to the overall energy or force of interaction for a system. Table 1.1 provides an example of the breakdown of these contributions per spectral region [105].

Figure 1.18 Distribution of Matsubara frequencies across the EM spectrum (based on [88]).

Reprinted with permission.

Table 1.1 Relative Contributions of EM Spectral Regions to Overall Interaction*

Spectral Region

Value of n

Percentage Contribution

Static

0

23

Infrared

1–11

13

Visible

12–24

15

Ultraviolet

25-∞

49

* For Polystyrene-Water-Polystyrene

As mentioned above, the dielectric function of a material is a complex function

(1.75)

where the real part of ε(ω), ε1(ω), describes the speed of light within a medium, and the imaginary part, ε2(ω), describes the absorption of light in the medium. When describing how radiation propagates through a medium, it is more common to use the refractive index (n) because this quantity can be measured experimentally. ε(ω) is related to n(ω) via

(1.76)

which makes n a complex function, according to

(1.77)

where (ω) is the real part of n, and k(Ω) is the imaginary part. By combining Equations 1.75–1.77, one obtains expressions for the real and imaginary components of ε

(1.78)

(1.79)

Because ε(ω) is a complex function, it can be recast in terms of complex frequency, i.e.,

(1.80)

where both ωR and ξ are strictly real variables. ε(ω) can be expanded in terms of these variables and an appropriate function of time as [88]

(1.81)