INTERACTION OF NONIDEAL SURFACES IN PARTICULATE SYSTEMS
By
JOSHUA JOHN ADLER
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2001
Copyright 2001
by
Joshua J. Adler
This study is dedicated to my father, who taught me that there is science in all things.
ACKNOWLEDGMENTS
I would like to acknowledge my advisor, Brij Moudgil, for giving me the
opportunity to pursue a broad range of scientific explorations while enhancing my
professional skills. His hard work and attention to detail was an excellent example during
my studies. Yakov Rabinovich additionally deserves acknowledgment for introducing me
to and guiding me through the finer points and rich history of the interaction of surfaces.
Many of the advances achieved during my graduate school career would not have been
possible without his assistance. I would also like to thank the National Science
Foundation's Engineering Research Center for Particle Science and Technology and our
industrial partners for financially supporting this research.
Through my involvement with the Center, I have interacted with a number of
professors who have aided me in my graduate school career. Those which deserve special
recognition include Hassan ElShall, Dinesh O. Shah, Rajiv K. Singh, and E. Dow
Whitney for their useful discussions, involvement in collaborative projects, and
willingness to serve on my advisory committee. Other professors and industry
professionals who deserve special recognition include James Adair, Lennart Bergstr6m,
P. C. Kapur, Richard Klimpel, Hans Lyklema, Pradip, Mitch Willis, and Abbas Zaman.
My time was further enhanced by the support and collaboration with both past
and present students, postdocs, and research scientists including Ali Ata, Bahar Basim,
Scott Brown, Aaron Clapp, Kim Christmas, Ravi Domodaran, Madhavan Esayanur, Mike
Fahey, James FitzGerald, WooHyuk Jang, Christopher Izzo, Raja Kalayanaraman,
James Kanicky, Dhirendra Kumar, Uday Mahaj an, Sharad Mathur, Anders Meurk, Byron
Palla, Alex Patist, Robert Pekrul, Ron Sabo, Sudhir Shrotri, Pankaj Singh, Prakash
Thatavarthy, Steve Truesdale, Rachel Worthen, Joseph Zhu, and all the other students
and friends who made my time at the University of Florida interesting.
Lastly, I would like to thank the staff of the Engineering Research Center for
Particle Science and Technology including Soo Aldrich, Rhonda Blair, Gill Brubaker,
Shelley Burleson, Bret Chen, Anne Donnelly, John Henderson, Lenny Kennedy, Sophie
Leone, Kevin Powers, Byron Salter, Erik Sander, Gary Scheiffele, and Nancy Sorkin.
TABLE OF CONTENTS
page
A C K N O W L E D G M E N T S ................................................................................................. iv
LIST OF TABLES ............... .......... .......... ............ ix
L IST O F F IG U R E S .............................. ............................... ..................................... x
ABSTRACT ......................................................................... ......... xii
1 IN TR O D U C TIO N ................. ........ ............................ .... ........ .................
2 INTERPARTICLE FORCE THEORY.....................................................7
van der Waals Forces ........................................... ........... ....... ........ 7
M icroscopic Approach. .......................... ... ................................. ........... 8
M acroscopic A approach ............................... ......... .......................................... 9
R retardation ..................................................... ................................... 14
S screen in g ............... ... .. ............ .................................... 16
The Derj aguin Approximation ................ ......... ....... .......................... 17
Surface Energy Approach to Adhesion............................ ............................ ..... 19
Surface E energy ............... ................................................................... .................. 19
Hertzian Mechanics ................... ............................ .. 21
DMT Mechanics ............................................... .. 22
JK R M e c h an ic s ......................................................................................................... 2 4
Electrostatic Forces ............. ................. ..... ..... .......... ... ........ .. 26
O rigin of Surface C harge ...................... .. .. ......... .. ....................... .............. 27
Electrical D ouble Layers.......................................................... ......................... 29
Zeta Potential ............................................... ............... 33
Boundary Conditions for Calculation of Electrostatic Force ................................ 35
Linearized PoissonBoltzmann Approach ......................................................... 38
N onL near A approach ........ ............................................................. .............. 40
DLVO and Extended DLVO Theory..................................... ......................... ........ 43
Surface F force Sum m ary.............................................................. .......................... 44
3 ROLE OF ASPERITY GEOMETRY ON THE ADHESION OF SURFACES ............45
In tro d u ctio n ........................................................... ............... 4 5
M modified R um pf M odel ............................................................. ..... ...................... 49
Proposed Model ............................................ 52
S u m m ary ...................................................................................... 6 3
4 ADHESION BETWEEN NANOSCALE ROUGH SURFACES (VALIDATION) ......64
Introduction....................... ............... ..... ............. 64
E x p e rim e n ta l ...................................................................................................... 6 6
M e th o d s ................................................................................ 6 6
M materials ............................................................. ........ ...... 6 7
R e su lts ................................................................................................... . ..................... 7 0
Comparison with the Modified Rumpf Model ................................ ...... 72
Comparison with the Proposed Model (van der Waals Approach) .............................. 74
Comparison with the Proposed Model (Surface Energy Approach) ......................... 80
S u m m ary ....................................................................... 8 5
5 CAPILLARY FORCES BETWEEN NANOSCALE ROUGH SURFACES ................87
Introduction .......................................... 87
Theory ........... ................... ... ................... 90
Adhesion to Surfaces with Nanoscale Roughness .............................. .............. 90
Capillary Adhesion to Surfaces with Nanoscale Roughness ............................. 92
A Critical Relative Humidity for Capillary Forces .............. .......................... 96
E xperim mental ......... .................. .................................... ........................... 99
M materials ........................................................................................... 9 9
M ethods...................... .... .............. 100
Results and Discussion .......................................... 102
Sum m ary ............. ... ...... ... ...... ........ .... ................. 110
6 INFLUENCE OF A TRANSITION LAYER ON INTERPARTICLE FORCE ........112
Intro du action ............................................ .............. 1 12
Long Range Interaction Forces ...................... ................................. .............. 114
Structuring of W after at the Silica Interface ........................................................... 115
GelLayer Formation ..... ........................................... 118
Forces between Surfaces with Transition Layers ................................ ................. ... 120
Homogeneous GelLayer ................. ........... .................................. 121
GelLayer with an Exponential Compositional Decay ................... .................... 124
E x p e rim e n ta l ..................................................................................................... 12 9
M materials ........................................................... ........ ...... 12 9
Methods ......................................... 130
Results ........................ .............. .........131
D iscu ssio n ............................................................................................... 1 3 5
Previous GelLayer M odels ............................................................. ............. 135
Hom ogeneous GelLayer M odel ..................................................................... 135
GelLayer with an Exponential Compositional Decay .......................................... 138
Sum m ary .......................................... ...................... .. ........ .. .. ............. 143
7 SUMMARY AND SUGGESTIONS FOR FUTURE WORK ...................................144
S u m m ary ......................................................................................................... 14 4
Suggestions for Future W ork ............................................................................... . .... 146
L IST O F R E FE R E N C E S ............................................................................................ 149
BIOGRAPHICAL SKETCH ............................................... .......... ...............156
viii
LIST OF TABLES
Table Page
21 van der Waals Attraction between Different Geometries.............. ...............10
22 Components of van der W aals Attraction................. .............................................. 11
23 Results of the Derjaguin Approximation................ ............ ........................19
41 Characteristics of the Model Surfaces.... .................. ..........................70
42 Measured Force of Adhesion .................................. ............... 71
43 Normalized Force of Adhesion: Glass Sphere ................. .... ... ................... ...... 83
44 Normalized Force of Adhesion: AFM Tip ................. ...............84
51 Substrate Characteristics and Adhesion Parameters................................ ............... 107
61 Fitting Parameters for the Relative Effective Hamaker Constant ...........................128
LIST OF FIGURES
Figure Page
11 Relative Magnitude of Forces Acting on an Adhered Particle...............................
21 Calculation of the Derjaguin Approximation. ......... ... ........... ................18
22 H ertzian M mechanics. .......................................................................... ....................23
23 DM T M mechanics .............. ... .. ............ ............................. .......... 24
24 JK R M mechanics .......... ........................................................ ......... .... ..... 26
25 Schematic of the SternGrahame Electrical Double Layer .............. ................... 31
31 Asperity Schematic Using the Rumpf Model..........................................................47
32 Prediction of Adhesion Using the Rumpf Model ....................................................51
33 Asperity Schematic Using the Proposed Model ................... ......................... 53
34 E effective Z one of Interaction .......................................................................... ...... 59
35 C om prison of A dhesion M odels..................................................................... ..... 61
41 Experim ental Configuration ............................................... ............................. 67
4 2 Su rface P rofiles............... .... ................................ .. .......... ........ ..... .... ...... .. 69
43 Comparison of Measured Adhesion with Modified Rumpf Model.............................74
44 Asperity Schematic for Surfaces with Two Roughness Scales .................................76
45 Experimental and Theoretical Adhesion for Sphere/Surface Interaction ..................78
46 Experimental and Theoretical Adhesion for AFM Tip/Surface Interaction ..............80
51 C apillary Schem atic.......... ............................................................... .......... . ....... 93
52 Particle/A sperity C apillaries.............................................. .............................. 94
53 Morphology of CVD Silica Samples............................. ...... ...............101
54 Approaching and Retracting Force Profiles ................................... .................103
55 A dhesion Profiles for Silica................................................ ............ ............... 105
56 Adhesion Force Profile for Other Substrates.................................. ............... 108
61 F ive L ayer Ideal Interaction ........................................................... .....................123
62 Five Layer N onIdeal Interaction ........................................ ......................... 125
63 E effective H am aker C onstant........................................................................ ....... 127
64 Interaction Force Profile between Boiled Surfaces.............................132
65 Interaction Force Profile between Heat Treated Surfaces .............. ... ...............133
66 Interaction Force Profile in Attractive Regime ............................... ............... 134
67 Interaction Force Profile with Electrostatic and DLVO Fitting .............................136
68 Experimental Effective Hamaker Constant ......... ................. ....... ............ 137
69 Theoretical van der W aals Force Comparison .................................. ............... 140
610 Theoretical Interaction Force Profiles ........................ ........................ 141
611 Fitted Interaction Force Profiles ................ ........ ................ ... ............ 142
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
INTERACTION OF NONIDEAL SURFACES IN PARTICULATE SYSTEMS
By
JOSHUA JOHN ADLER
May 2001
Chairman: Dr. Brij M. Moudgil
Major Department: Department of Materials Science and Engineering
Adhesion between surfaces plays a critical role in the macroscopic behavior of
particulate systems such as the flow of cohesive powders, the removal of particulates
from substrates, and the formation of particulate coating on host surfaces. Similarly, the
repulsive force profile between two particles in suspension influences properties such as
theological behavior and polishing efficiency. The majority of theoretical predictions of
surface forces assume that the interface between the particle and surrounding media is
ideal. Namely, that there is an atomically sharp demarcation between the properties of the
bulk material and the properties of the medium. Furthermore, the interface is often
considered perfectly monotonic. While this assumption is likely valid at larger separation
distances, significant deviation from theory is expected when two particles are in a
contact or nearcontact configuration, as they are in adhesion processes or high force
operations.
This study explores the impact of the nonideality of surfaces on the adhesion
between particulates in dry and humid atmospheres as well as in solution. Specifically,
the influence of nanoscale roughness in the range of 0.2 to 20 nm RMS has been
investigated on the van der Waals, polar, and capillary forces that bind particulates
together or prevent particulate removal from surfaces. Additionally, the influence of
transition layers, consisting of chemically modified layers of nanoscale thickness at the
interface, on the interaction of particulates in suspension has been delineated. For each of
these cases, theoretical predictions are developed and validated through direct
measurement of surface forces. The presence of nonideal interfaces is determined to
have a significant impact on the magnitude of the surface force that govern the ensemble
behavior of many particulate systems.
CHAPTER 1
INTRODUCTION
Particleparticle interactions govern phenomena such as dispersion,
agglomeration, coatings, and polishing which play a critical role in materials and product
development. The behavior and properties of a particulate system are controlled by the
balance of forces acting on a given particle. Gravitational forces are proportional to the
radius cubed, whereas, most surface forces are directly proportional to the radius. Hence,
the relative importance of surface forces is increased as particle size is reduced.
The balance of forces acting on a particle is shown schematically in Figure 11.
For a large particle, adhesion behavior, for example, is dominated by gravitational force.
The weight of a particle is the principal mechanism that resists removal from surfaces.
Alternatively, at very small particle sizes, adhesion due to surface forces dominates. In
order to predict behavior, these extremes should be compared to forces that act to remove
particle from surfaces. In many cases, such as water or air jets, the removal force is
proportional to the particle radius squared (projected area) [MAS94]. Hence, three
regimes are often observed in particulate systems as a function of size. Of particular
interest in this study, is the middle region of transition between the dominance of surface
forces and gravitational force. For many systems this transition occurs in the micron size
range.
For particles in this regime, surface forces may be utilized to control the viscosity
of suspensions; to separate particles of different properties through operations such as
108
107 Gravitational
106
105
104
Z 103
S 102
S101
S100 Removal
o 10
0 10 2
t 103
~ 104
S10 Adhesion
106
107
108
109 .
109108107106105104103102101 100 101
Particle Radius (m)
Figure 11: Relative Magnitude of Forces Acting on an Adhered Particle. Shown are the
relative magnitudes and regions of dominance of adhesion, removal, and
gravitational forces for a hypothetical particle adhered to a surface.
filtration, flocculation, or flotation; to polish surfaces with controlled rate and minimal
defects; to enhance the flowability of powders; and to synthesize novel particulates by
crystallization or particulate coatings. In many of these examples, utilizing the inherent
properties of the surface to promote or prevent agglomeration is not sufficient. Rather the
surfaces must be modified by chemical additives. These reagents, such as polymers and
surfactants, may be applied to significantly alter the repulsive or attractive forces between
particles through a variety of mechanisms.
Similar to the balance surface forces and gravitational forces, there is also
competition between surface forces. The net surface force may either be attractive or
repulsive depending on the dominance of a particular mechanism. Interactions of surfaces
in gaseous media tend to be attractive due to contributions from van der Waals forces,
polar forces, plastic deformation, tribocharging, image charging, and capillary
condensation. In a liquid medium, the net force may either be attractive or repulsive due
to the action of van der Waals, electrostatic, solvation, hydrophobic, and steric forces. In
both dry and wet systems, the various forces may not be completely independent of each
other, and may significantly affect individual and ensemble particulate behavior.
There have been a number of methodologies developed to predict the interaction
of surfaces, as will be discussed further in Chapter 2. However, these models are often
based on an ideal model of the surface. The interface between a particle and the medium
is characterized as perfectly smooth and discrete. However, most interfaces, due to the
reaction with the environment during formation or rearrangement of the bonding between
atoms, are compositionally different from the bulk material. The question then arises as
to the contribution of the modified interface to the overall force. Most surfaces also have
some degree of roughness associated with them. Even for surfaces normally considered
smooth, the atoms themselves have an inherent size and thus give rise to roughness that
could affect particulate interactions.
For many colloidal systems, such as particulate suspensions under normal
conditions, the repulsive force between particulates is large enough that the distance of
closest approach is much greater than the inherent roughness or the thickness of a
modified interface. Hence, the overall contribution of such a nonideal surface is
minimized. However, some aspects of the chemically modified interface are accounted
for in the classical treatment of the interaction of particles. For instance, the prediction of
electrostatic repulsion depends on the number of charges developed after reaction with
the environment.
It has not been necessary, until recently, to explore the effect of nonideal
interfaces on other surface force mechanisms in more detail. However, in the field of
cohesive powder flow, models are being developed through which ensemble behavior is
predicted directly from the interaction and relative movement of particles. This is in
contrast to previous approaches that primarily predicted behavior based on empirical
measurement of macroscopic properties under controlled conditions. During adhesion
processes, particles come into direct contact with one another. In the past, deviations
from ideality could be accounted for in a flexible surface energy parameter. However, to
implement these newer powder flow models, more realistic prediction of the interaction
force is needed.
A more precise description of the interface and its effect on surface force is also
becoming necessary in the control of colloidal suspensions, particularly when processing
in extreme environments is required. An extreme environment such as high electrolyte
concentration, low or high pH, high shear rate, or high normal force may cause particles
to interact at much smaller separation distances than under normal solution conditions.
An example of such a system is chemical mechanical polishing (CMP) of microelectronic
substrates where abrasives under chemically active conditions are pressed into a substrate
by a polymeric pad in order to remove material from the substrate. The nature of the
abrasive and substrate surfaces is critical to the prediction of polishing performance. In
CMP systems, the properties of the surface must be evaluated at the nanometer level in
order to accurately predict forces.
In this study, the influence of two types of nonideal interfaces is explored. The
first is the effect of nanoscale roughness on adhesion forces. It is shown that roughness
on the nanoscale has a dramatic influence on adhesion in dry atmospheres, where van der
Waals and polar forces are dominant, as well as humid environments, where capillary
forces begin to dominate. Theoretical models are developed for both cases and validated
by direct measurement of surface forces.
The influence of a compositional alteration of the surface is explained using the
silica/water system as an example. In the past, there have been some theories proposed to
explain the anomalous shortrange surface forces that have been observed in this system.
However, none of the approaches can predict the full range of silica's colloidal stability.
Through theoretical and experimental methods, a more complete theory of the behavior
of silica in aqueous solution and an improved description of the silica surface is
advanced.
In all these examples, changes in the surface that dramatically affect the
interaction of particles are occurring over length scales of less than 20 nm. Furthermore,
it is demonstrated that the most significant deviations from an ideal system occur over a
length scales of only 1 to 2 nm. As processes are developed where particle interactions on
these same length scales are more common, (e.g. CMP, surface cleaning, coatings), and
theoretical predictions of ensemble behavior increasingly rely on accurate prediction of
surface force (e.g. flow of cohesive powder, viscosity of concentrated suspensions), the
need to more precisely describe surface properties and develop models to predict surface
forces is clear. The following chapters are expected to advance the stateoftheart toward
this goal.
The discussion will begin, in Chapter 2, with a brief overview of surface force
predictions between ideal surfaces relevant to the mechanisms described in later chapters.
Chapter 3 will define a model based on the geometry of a nonideal surface with
nanoscale roughness to predict particulate adhesion. This model will then be
experimentally validated and expanded upon in Chapter 4. Chapter 5 introduces capillary
adhesion and explores effects, due to nanoscale roughness, on the magnitude and onset of
capillary adhesion. In Chapter 6 an alternate type of nonideal interface is discussed. The
effect of a surface layer, differing in composition from the bulk, on the interaction of
particles in suspension will be delineated. Finally, Chapter 7 will summarize the findings
of this investigation and propose possible avenues for further research.
CHAPTER 2
INTERPARTICLE FORCE THEORY
Before a detailed discussion of the interaction of nonideal surfaces, a brief
description of current theory regarding the interaction of ideal surfaces is necessary. The
topics presented in this chapter are intended to introduce the fundamentals of surface
forces relating to interactions in gaseous atmospheres and between particles in aqueous
suspension. The scope and application of surface forces is large. Hence, the following
discussions will be limited to concepts necessary for interpretation of the specific
examples in this study.
van der Waals Forces
In 1873, J. D. van der Waals [VAN73] first suggested that the behaviors of gases
and their deviation from ideality could be expressed as a modification of the ideal gas
law, PV = nRT, to
2
(P + 2)(V nb) = nRT (21)
V
where P is pressure, n is the number of moles of gas, V is the volume, R is the gas
constant, T is temperature, and a and b are constants specific to a particular gas. This
modification is significant because a number of assumptions are inherent in the
interpretation of the constants a and b. The constant b particularly describes the finite
volume of the molecules comprising the gas and the constant a is needed to describe
attractive forces between the molecules. A primary result of this development was a
significantly improved model of the behavior of gases. The various types of attractive
forces between molecules are now collectively termed van der Waals forces. The concept
of an attractive force was also used to describe the properties of condensed matter. The
familiar LennardJones potential [LEN28]
C B
Watom/atom 6 12 (22)
r r
where the net potential energy between atoms, Watom/atom, at distance, r, can be described
by the competition of an attractive London dispersion forces [LON37], characterized by
constant C and a Born repulsion term (arising from the overlap of electron clouds)
characterized by constant B. The London constant, C, is primarily related to the
synchronization of instantaneous dipoles created when the energy fields of neighboring
atoms overlap. Both these expressions of the attraction between atoms becomes evident
when the dispersion constant is related to a and b in the nonideal gas law through
9ab
C (23)
47r2N3a
where Na is Avogadro's number. The characteristic frequencies principally associated
with this constant are in the infrared to ultraviolet range of light and are principally
associated with the opacity of solids. As such, the term dispersion force is often used.
Microscopic Approach
Hamaker [HAM37], Derjaguin [DER34], and de Boer [DEB36] all contributed
significantly to the understanding of the van der Waals interactions between macroscopic
bodies based on the individual interactions of London dispersion forces between atoms.
In the microscopic approach, it is assumed that pairwise addition of all the interactions
between the atoms in one body with another body may be used to derive an expression
for the energy of attraction between them. Following this approach, the interaction energy
,W, between two spherical particles may be expressed in the following manner.
A 2R2 2R2 21r1 4R2
Wsph / sph = [+R + Hn)2 (24)
6 H(4R+H) (2R + H)2 (2R + H
where A is Hamaker's constant, R is the particle radius, and H is the separation distance
between the particle surfaces. Note that this relation does not place a restriction on
particle size. However, if it is considered that the radius of the two particles is much
greater than the separation distance, the expression may be reduced to the more familiar
form represented by Eq. (25) [DER34].
AR
Wsph/sph 1 (25)
12H
Energies of interaction based on different geometries, assuming the extent of the
van der Waals attraction is small compared to the radius of curvature, are given in Table
21. However, the more important outcome of these initial approaches was the separation
of the fundamental nature of the interaction of materials from their geometry. In this
manner the force or energy between two bodies may be expressed as a factor which may
be changed depending on shape and a constant resulting from the material's fundamental
composition. This is a critical development considering that although the total van der
Waals attraction is dominated by the London dispersion component, arising from the
interaction of instantaneous dipoles, there are a number of other mechanisms that should
often be taken into account when considering the total attraction. Table 22 describes a
number of these forces and the corresponding atomic/molecular interactions.
Macroscopic Approach
In real materials there may be many fundamental interactions which contribute to
the total interaction force. Additionally in the microscopic approach, the interaction
between atoms or dipoles is calculated as if a vacuum were intervening. However,
considering an atom at the core of a solid particle interacting with an atom in the core of
another particle, this is clearly not the case. The intervening atoms have a significant
impact on the interaction. Hence, it is desirable to determine an improved method to
predict the attraction between solids that takes into account the different types of
interactions and more accurately sums them. Accounting for the types and orientations of
the many type of species in a solid is difficult and hence a different approach must be
taken. In 1956 Liftshitz [LIF56], based on the assumption that both the static and
Table 21: van der Waals Attraction between Different Geometries. Theoretical formulae
for the prediction of the energy or force of attraction between bodies of
differing geometries where A is Hamaker's constant, H is surface separation
distance, R1 and R2 are particle/cylinder radii, and L is cylinder length. Note
that these were derived for conditions in which the extent of significant van
der Waals attraction is far less than the radius of the particles or cylinders and
that force is the negative differential of energy.
Geometry of Interaction
Two Plates
Sphere/Plate
Sphere/Sphere
Crossed Cylinders
Parallel Cylinders
Energy of Interaction (J)
A
12rnH2
AR
6H
A R1R2
6H (R1 +R2)
A RR2
6H
AL I RR,2
12 2H3/2 R, +R2
Force of Interaction (N)
A
67CH3
AR
6H2
A R1R2
6H2 (R1 +R2)
AR1R2
6H2
AL RI R2
8 V2H5/2 R1 +R2
oscillatory fields produced by the atomic components of solids should directly affect the
absorption of electromagnetic energy by the material, derived a method to calculate the
attraction between materials based on the differences in their dielectric spectra. Thus, the
continuum or macroscopic approach was founded. Although it is clear that this approach
Table 22: Components of van der Waals Attraction. Theoretical predictions of the
possible contributions to the total attraction between particles. Q is the electric
charge (C), u is the electric dipole moment (Cm), a is the electric
polarizability (C2m2/J), r is the distance between interacting atoms or
molecules (m), k is Boltzmann's constant (1.381x1021 J/K), T is absolute
temperature (K), h is Plank's constant (6.626x1014 Js), v is the electronic
absorption (ionization) fre uency (1/s), and co is the dielectric permittivity of
free space (8.854x1012 C /Jm) [ISR92].
Interaction Type
Covalent/Metallic
Charge Charge
Charge Fixed Dipole
Charge Rotating Dipole
Fixed Dipole Rotating Dipole
Two Rotating Dipoles
Charge Nonpolar
Fixed Dipole Nonpolar
Rotating Dipole Nonpolar
Two Nonpolar
Hydrogen Bond
Interaction Energy
Complicated
Q1Q2/(47ror)
Qu cos(O)/(47nror2)
Q2u2/(96i2O2kTr4)
ulU2[2cos(O1)cos(02)
sin(Oi)sin(02)cos( )]/(47enor3)
ul2U22/(4 8_2Eo2kTr6)
Q2c(3 2 2o2r4)
u2a(1+3cos2(0))/(322o2 r6)
u2a/(167r2o2r6)
3hva2/(64 2Eo2r6)
Roughly proportional to r2
Coulomb
Keesom
Debye
London
Dispersion
accounts for the different types of bonding and screening in a body, it is still a very
difficult function to measure due to the wide range of frequencies and types of
experiments needed to determine the entire function.
To overcome this barrier Ninham and Parsegian [NIN70a] in the early seventies
proposed that the principle contributions to the overall attraction come from the regions
of dielectric relaxation. In other words, the regions where a specific atomic or molecular
mechanisms creates a resonant vibration. The characteristic absorption frequencies of
some materials are relatively well characterized, for example the spectra of water has
been characterized in the ultraviolet, infrared, microwave, and static frequency regimes
[GIN72, DAGOO]. However, it is still difficult to extract information for a wide variety of
materials.
To simplify this situation it was demonstrated, by Hough and White [HOU80],
that the majority of the contributions to the overall Hamaker constant come from
dielectric relaxations in the UV and infrared regions. This may be expressed in a
simplification of the NinhamParsegian representation as Eq. (26)
C,, Cr
e(i)= IR + U(26)
1+ (/ aR )2 l+(/CO)2
where e is the dielectric response function as a function of the imaginary frequency i, C
is the spectral constant in the IR and UV regions, o is the characteristic absorption
frequency in the IR and UV regions, and is a constant frequency equal to (47t2kT/h) in
which k is Boltzmann's constant, T is absolute temperature, and h is Plank's constant.
This approximation was found to be valid for materials that are transparent and
exhibit negligible absorption or where (co) = n2, n being the index of refraction. Note
also that for these conditions the UV spectral constant, Cuv, may be related to the index
of refraction in the visible regime, nvis, through
Cu = n 1 (27)
and the spectral constant in the IR regime, CIR, may be approximated by the static
dielectric constant, e(0), and Cuv through Eq. (28).
Cn = e(0) Cv 1 (28)
Using these approximations the Hamaker constant can now be expressed by the
static dielectric constant and the UV absorption parameters. To acquire these UV
parameters the index of refraction in the visible and UV regions may be measured and
according to the Cauchy equation,
n2 ()1= (n2())1 +C, (29)
plotted to extract Cuv and couv. The UV relaxation frequency for such materials is
relatively constant at approximately 3x1015 Hz. Considering a single UV relaxation
frequency, an approximation for the Hamaker constant, A131, of material 1 interacting
with similar material through medium, 3, may be written solely as a function of the
differences in the static dielectric constants, e(0), and indices of refraction in the visible
range, n. The result is the TaborWinterton (TAB69) approximation
3kT r (0) 3(0) + 3hmuv (n2 n)2
1 4 Ei(0)+C3(0) 32r (n2 2 (210)
where kT is the product of Boltzmann's constant and temperature and h is Plank's
constant. Note also that a similar equation may be derived for the interaction of dissimilar
materials, 1 and 2, across medium 3.
3 ,1(0) eF3 (0) Fe2 (0) eF3 (0)
A 132 kT +
132 4 e,(0)+e3(0) F2(0)+F3(0)
(211)
3h, (n 2 n 2 2 2 (2
3hco___ ,n1 n _3,n2 n3 )
16 / n2 /2(n22 21/2[ 1 2 2 1/2 2 /2
167r,[2 (n +n 2n3 +3 +n2 3
Through the methodologies described above, a relatively comprehensive theory of
the interaction of surfaces has been developed. However, there is still a debate as to
which measurements of the spectral constants and which approximations are most valid
[BER96, DAGOO, FEROO]. Hence, as models are developed that more accurately link
nanoscale phenomena with ensemble behavior, it is expected that more accurate
determination of many of the constants proposed in the literature will be necessary.
Retardation
The principle atomic process that controls the magnitude of the van der Waals
forces between surfaces is the synchronous oscillations between instantaneous dipoles in
the two materials, known as London dispersion forces [LON37]. However, the speed at
which the field of one instantaneous dipole may interact with another is limited by the
speed of light. As a result, when two bodies are moved further apart there is more of a
phase lag between the alignment of dipoles, this results in the magnitude of the total van
der Waals force being reduced from that predicted by the nonretarded value. Another
consequence of this phenomenon is that as the magnitude of the contribution to the total
van der Waals force of the instantaneous dipoles decreases, the relative contribution of
the static dipoles (permanent dipoles/charges) increases.
There are several methods to account for the decreased efficiency of interaction
between the instantaneous diploes. It has been demonstrated that the full numerical
solution to the Liftshitz equations do account for retardation [MAH76, PAS77].
However, these calculations are tedious and the full dielectric spectra for many materials
is not described adequately. Hence, for separation distances where the contribution of
instantaneous dipoles is still significant, approximately 5 to 15 nm in water, Gregory
[GRE81] has proposed an analytical correction, given by Eq. (212),
A t =A 15.32H In 1+ (212)
ret 1 5.32H/k1
where Aret is the retarded Hamaker constant, A is the nonretarded Hamaker constant, H
is surface separation distance, and ) is the London wavelength (characteristic wavelength
for instantaneous dipole oscillation: approximately 100 nm).
At larger separation distances the contribution of instantaneous dipoles becomes
small. Under this assumption Casimir and Polder [CAS48a] derived an expression for
van der Waals energy between two plates, Wplt/plt, of the form
B131
Wplt/pit (213)
3H
where B (Jm) is a constant analogous to that of Hamaker equal to
B ch ((0) 3 (0))2(214)
B131 (214)
480 30 ((0) + 3(0))2
where e(0) is the static dielectric constant of material 1 or medium 3, c is the speed of
light, h is Plank's constant, and ( is 0.24 for dielectrics and 1 for conductors.
Note that at large separation distances the magnitude of the van der Waals forces
under these assumptions decays as a function of H3 instead of H2 as predicted by a non
retarded interaction. Hence, even for separation distances in the 515 nm regime the
measured forces may be less than approximately half that predicted by nonretarded
theory. At larger distances in water, the absolute magnitude of the interaction energy due
to van der Waals attraction approaches that of Brownian motion (1 kT) and as such the
long range formulas are not often applied. However, for noncontact interactions in
gaseous atmosphere, they may need to be considered.
Screening
Screening is a phenomena similar to retardation in that it acts to reduce the total
van der Waals interaction between surfaces. However, screening primarily acts to reduce
the contribution of the static component of the total interaction energy and occurs
primarily in aqueous solutions. The van der Waals interaction is essentially an
electromagnetic effect. As such, free charges such as ions in solution will act to reduce
the field strength. In real systems, this effect may act to reduce the static attraction term,
see Eq. (210) or (211), over very short separation distances.
For example in a 0.1 M aqueous solution at 1 nm separation distance the static
contribution to the van der Waals force will be approximately 10% of its contribution at
zero separation distance. Hence at larger separations in high ionic strength solutions the
interaction is primarily controlled by London dispersion forces. This effect has been
described mathematically by Mahanty and Ninham [MAH76] for nonretarded van der
Waals forces through Eq. (215).
Asc.een = A=0 (2dKH)e2" + AcD (215)
where Ascreen is the screened Hamaker constant, Am=o is the zero frequency Hamaker
constant component, Amo is the sum of all the other components, H is separation
distance, and K is a characteristic inverse length, called the DebyeHickel parameter (the
inverse is commonly called the Debye length) that describes the distribution of ions near
a charged interface according to
I = e2 (216)
ErekT
where n is the number concentration of ions, Z is ion valency, e is the fundamental charge
on an electron, r is the relative dielectric constant of water (static), co is the permittivity
of free space, and kT is the product of Boltzmann's constant and absolute temperature.
The Derj aguin Approximation
In the previous section, a methodology to determine the interaction between
bodies was discussed. Using the microscopic approach, Hamaker was able to separate the
critical interaction components into a material constant and a factor dependent on the
geometry of the interacting surfaces [HAM37]. While the macroscopic approach
delineated more clearly the value of the Hamaker constant, changes in the geometry of
the interacting bodies is still a difficult multidimensional integration process.
An approach to simplify this problem was proposed by Derjaguin in 1934
[DER34]. It was proposed that the energy of interaction between two bodies could be
calculated based on the summation of the interaction force as separation distance, H',
goes from H to infinity for infinitesimally thin concentric rings of radius x and thickness
dx interacting with the projection of that ring on the opposing body as if it were two
plates of area 27rdx separated by H', as shown in Figure 21. If the assumption is then
made that the range of the force between the interacting bodies is small compared to their
radii and that at large separation distances the magnitude of the interaction approaches
zero, relatively simple relations for the energy of interaction between flat plates and the
force of interaction between bodies of differing shapes may be extracted.
H'
H
dx
Figure 21: Calculation of the Derjaguin Approximation. Integration of the force between
the flat surface created by a ring of thickness dx and radius x with its
projection on the sphere as separation distance goes from H to o yields the
equivalent energy between flat surfaces.
This approximation is extremely important for the measurement of surface forces
because it simplifies the derivation of analytical formulae and allows comparison
between bodies of different geometries. Although van der Waals attractive force has a
relatively simple form, other forces are considerably more complicated and derivation or
integration may not yield analytical solutions. Additionally, many computations, such as
the distribution of ions from a charged interface, are simplified if the interface is flat. As
illustrated in Table 23, multiplying the interaction energy between flat surfaces by a
factor, the force between different geometry bodies may be calculated.
An even more important result of this approximation is that all of the forces or
energies of interaction between curved bodies are seen to be directly proportional to the
radius or mean radius. Hence, forces measured between spheres, cylinders, and plates
may be directly related to each other. Moreover, experiments measured between surfaces
of differing radii may also be compared to a single theoretical prediction. Surfaces forces
for this reason are often presented normalized by the radius of the interacting bodies.
Note that in this form, force divided by radius is equivalent to energy per unit area.
Surface Energy Approach to Adhesion
Surface Energy
The application of the above models, based on van der Waals attraction, are
necessary to describe the distant dependent nature of the interaction of surfaces.
However, for the specific case when two surfaces are in contact, a different approach to
quantify the magnitude of the adhesion force may be considered. The work of adhesion
(cohesion for similar materials) describes the energy needed to bring two half spaces (flat
plates) from contact to infinite separation distance. This work, W132, may be further
described, Eq. (217), as the sum of the surface energies, 713 or 723, associated with the
newly created interfaces.
Table 23: Results of the Derj aguin Approximation. The interaction force, F, between
surfaces of different geometries, according to the Derjaguin approximation, is
the energy of interaction between flat plates, Wplt/plt, multiplied by a simple
prefactor.
Geometry of Interaction Force Derjaguin Approximation
Sphere/Plate Fsph/plt (27nR)Wplt/plt
Sphere/Sphere Fsph/sph (7tR)Wplt/plt
Crossed Cylinders Fcyl/cyl (2nR)Wplt/plt
W132 + y23 (217)
For liquid interfaces the surface energy is normally referred to as surface tension.
For solid materials in very close contact, this quantity may be referred to as fracture
energy. Since the contact separation distance at which surface energy is defined is not
specified, different methods of evaluating surface energy are expected to provide
different values. For example between two liquid interfaces, separation distance may be
taken as the intermolecular distance (minimum in the LennardJones potential) which is
approximately 0.16 nm [ISR92]. However, between two adhered solid surfaces the atoms
are not free to arrange themselves and separation distance is often larger (approximately
0.3 to 0.4 nm for many materials). Additionally, different types of bonding may occur in
different systems. Considering two adhered silica surfaces, surface energy measurements
based on adhesion result in values of Y13 of 25 to 40 mJ/m2, depending on the degree of
hydroxylation [BRA32, YAM75, YAM98]. However, surface energy based on fracture is
approximately 5000 mJ/m2 [ASH91]. This enormous discrepancy is reasonable if it is
considered, that in the case of fracture, covalent bonds must be overcome to separate the
surfaces. Consequently, surface energy values are often derived from a number of
different sources and may not always incorporate the true fundamental mechanisms of
adhesion relevant to a specific system.
The surface of a material is often different from the bulk. For example hydroxyl
groups or other polar groups often satisfy the broken bonds at the interface. These species
normally have a dipole moment, or may form hydrogen bonds (acid/base reactions) with
similar species. These dipoles at the interface are not accounted for in the determination
of van der Waals forces between surfaces in which the interface is treated as a sharp
transition between the properties of the body and medium. On the other hand, these polar
groups may add significantly to the measured adhesion. For a surface with a high
concentration of hydroxyl groups, such as silica, the contribution of van der Waals
adhesion, assuming a 0.3 nm minimum separation distance, results in a contribution of
only 9.5 mJ/m2 to the total measured surface energy (2540 mJ/m2). For more
hydrophobic materials this contribution is significantly less but should still be considered.
It is known that contributions to adhesion from polar forces decay sharply with
distance and essentially are only critical for surfaces in contact in gaseous atmospheres.
Unfortunately, adequate characterization of the orientation, concentration, and
contribution of the individual polar groups is not available. Because of the importance of
these polar forces, many investigators chose to work with the surface energy approach to
describe adhesion phenomena. Considering that surface energies are dependent on
surface history and are generally poorly characterized, this parameter is often used to fit
experimental data. Despite this limitation, there is substantial theory to describe the
interaction of particles based on the surface energy approach. Since contact is assumed,
these approaches are generally valid only after contact has been achieved.
Hertzian Mechanics
When a particle is in contact with a surface, for example, there are generally two
classes of forces acting between the bodies. The forces mentioned in the preceding
sections tend to attract the particle towards the surface. If the particle were a liquid
droplet, after contact, these forces would begin to deform the droplet until the surface was
wetted. However, for solid materials this deformation is resisted by the force needed to
elastically deform the particle. This force of resistance to elastic deformation, Fsph/plt, was
described mathematically by Hertz [HER81] and may be written as
Fsphpt Ka (218)
R
where a is the radius of the contacting area between the particle and plate, R is the radius
of the particle, and K is the reduced elastic modulus. Note that
K = f v+ v2 (219)
3 E E 1
where v is Poisson's ratio and E is the elastic modulus for materials 1 and 2. From these
relations the indentation depth, 8, relative to initial contact (8 = 0) may also be
determined.
2
a = (220)
R
The above forces are purely repulsive and act to keep the particle from deforming.
In fact, with no externally applied force, the model predicts zero deformation, zero radius
of contact, and zero surface force, Figure 22. In systems that behave according to the
Hertzian indentation model, attractive surface forces should be negligible. For particles in
the micronsize regime, this assumption is not generally valid. Instead the attractive
forces act to pull the particle towards the plate. This action is the resisted by Hertzian
repulsion. An account of net area in contact based on this model was proposed by two
groups for different limiting cases and is described below.
DMT Mechanics
Derjaguin, Muller, and Toporov [DER75] proposed, in the limiting case of high
elastic modulus, small particle size, low attractive forces or appropriate combination of
the above, that any deformation that occurs will not significantly affect the force of
pressure
equilibrium
Figure 22: Hertzian Mechanics. Under pressure, the particle resists being pushed into the
surface by elastic deformation. At equilibrium, the particle is neither attracted
nor repelled from the surface.
attraction between the sphere and flat plate. Hence the resulting interaction may be
written as
Ka3
Fsph /plt = 4Ry,31 (221)
R
where the first term is Hertzian repulsion and the second is attraction. As indicated by
Fig. 23, The indentation depth for this model is the same as for the Hertzian model, Eq.
(220). In DMT mechanics, when the particle detaches from the surface the area in
contact is zero and the particle is truly not deformed. In other words, the force applied to
remove the particle exactly equals the adhesion term and Hertzian repulsion is zero.
Hence, the force of adhesion may be written as Eq. (222).
Fadhesion sph/plt = 4CR7131 (222)
equilibrium
pull off
Figure 23: DMT Mechanics. At equilibrium, attractive surface forces are balanced by
Hertzian repulsion arising from elastic deformation. At pulloff, removal
force equals surface force and the particle is not deformed.
It should be noted that Eq. (222) is really equivalent to the Derjaguin
approximation, Fsph/plt = 27rRW131. The difference is that in this methodology the work of
adhesion is a specific value independent of separation distance. The simplicity of these
mechanics is due to the assumption of a rigid spherical body. However, many materials
such as latex, do not fit the assumptions made in the DMT model.
JKR Mechanics
To extend the model to cases where deformation of the particle should be
accounted for, Johnson, Kendall, and Roberts [JOH71], in the limiting case of low elastic
modulus, large particle size, high attractive forces or appropriate combination of the
above, derived an expression for the attraction between a particle and a plate as a function
of the radius of the contact area, a,
Ka
r Kah C2Ka,1 (223)
sph plt =, 2n 1a 3,,
where K is the reduced elastic modulus, R is the particle radius, and 7131 is surface
energy. The depth of indentation, 6, also has a very different functional form as shown in
Eq. (224).
a 2712 ,
8 2=12a 31 (224)
R 3 K
In the JKR scheme, because the surfaces are deformable, detachment of the
particles does not occur when the penetration depth is zero as in the DMT model, as
shown in Fig. 24. Instead as the particle is pulled from the surface a neck forms between
the two surfaces as they try to remain adhered. Detachment occurs when the neck
becomes too thin to maintain contact (dF/da = 0). In this case, the force of adhesion may
be written as, Eq. (225).
Fadhesion sph/plt = 37RT131 (225)
There are alternative descriptions of the interaction of surfaces, such as Maugis
mechanics for spheres and plates, that more accurately predict adhesion between the
DMT or JKR limits based on material properties [MAU92]. However, for practical
application, the magnitude of the force of adhesion predicted by JKR mechanics is 75%
of that predicted by the DMT model. This difference is relatively small compared to the
uncertainty in the magnitude of the surface energy. Another complication arises from the
comparison of bodies of differing geometry. In the literature, the Derjaguin
approximations are commonly applied to both JKR and DMT mechanics. However, the
validity of these transformations has not been adequately tested.
equilibrium
pull off
Figure 24: JKR Mechanics. At equilibrium, attractive surface forces (although with a
different functional form than in DMT mechanics) are balanced by Hertzian
repulsion arising from elastic deformation. At pulloff, JKR mechanics
predicts neck formation. The maximum force of adhesion occurs when the
neck breaks.
Electrostatic Forces
The forces described above, with the exception of polar forces, operate at some
level whether the particles are in gaseous atmospheres or in aqueous suspension.
Additionally, the forces mentioned above are usually attractive. However, when particles
are introduced to an aqueous medium they often develop charge at the solid/water
interface. If the charges are of the same sign the particles will repel each other. This force
that is often utilized to prevent coagulation in suspensions is called electrostatic
repulsion.
Origin of Surface Charge
Charge on the surface of particles may develop through a variety of mechanisms.
When new surface area is created either through precipitation, fracture, or other
mechanisms the atoms at the interface seek to lower their free energy by combining with
molecules from their environment. In the case of oxides, nitrides, carbides, phosphates, or
sulfides this often takes the form of surface hydroxyl, nitride, carboxylate, phosphate, or
sulfate groups. These groups are essentially Bronsted acids and, as such, upon immersion
in aqueous solution may dissociate. This dissociation is often characterized by a reaction
rate equation such as
MOH M+ + OH (226)
where MOH represents a metal hydroxide, for example, and K is a rate constant equal to
[M ][OH]
K = [M][OH] (227)
[MOH]
with the brackets indicating surface concentrations. Using this terminology it is clear that
the relative magnitude of the charge at the interface is highly dependent on the
concentration of hydroxyl, hydronium, and metal ions at the interface. Dissolved species,
such as these, that may directly affect surface charge are called potential determining
ions.
Of course, most surfaces have more than one type of group either due to a multi
atomic composition, the propensity to form multiple hydroxylated (or other) species, or
changes in the local surface molecular architecture. Similarly, dissolved ions from the
surface, especially metal oxides in aqueous solution, may form a number of species in
solution. All these factors can affect the actual dissociation constant. However, for most
dissociating materials in water, there is a general trend that at low pH surface charge is
positive and negative at high pH. The pH at which there is a net zero charge on the
surface is called the point of zero charge (PZC) and is generally characteristic of a
particular material.
A second method of generating charges at the interface of a particle is through
selective dissolution or adsorption of ions. Many salts including AgI, BaSO4, CaF2, NaC1,
and KC1 generate charge through this mechanism. The origin lies in the more favorable
solubility of one of the ionic constituents over the other and may be predicted through a
combination of hydration energy and surface hydration [VEE97]. The balance between
the ions in this case may be quite delicate as exemplified by the fact that NaCl has a net
positive charge due to the enhanced dissolution of chlorine, whereas KC1 has a net
negative charge due to enhanced dissolution of potassium [MIL92]. These systems may
also approach equilibrium (saturation) as in the case of AgI where the difference in
concentration of the potential determining ions (Ag+ or I) determines surface charge.
Several clay minerals, such as mica, may also develop charge via this mechanism. If
charges between layers are satisfied by metal cations such sodium, potassium, calcium, or
magnesium, when these surfaces are exposed to solution a proportion of these ions may
dissolve leaving a negatively charged site.
The third principle way in which surfaces may acquire charge is through
isomorphic substitution. In this mechanism, an ion may be incorporated into the lattice
that has a different valence than the surrounding structure. An example is the substitution
of aluminum for a silicon atoms in the structure of kaolin clay which leads to a net
negative charge.
In many materials, several mechanisms of surface charge may be operating
simultaneously. Prediction of the surface charge, therefore, is difficult and experimental
techniques become necessary to characterize the surface. In the following sections the
influence of surface charge on the structure of the surrounding solution and the
interaction of surfaces will be discussed in more detail.
Electrical Double Layers
Well over 100 years ago Helmholtz modeled charged surfaces in aqueous solution
as being neutralized by an equal number of oppositely charged ions (counterions),
adsorbed to the interface. This description formed the basis for what is now called the
electrical double layer. That is to say, one layer of charges at the interface being
neutralized by a second layer of counterions in solution. With this description, electrical
potential was suggested to drop linearly from the magnitude at the interface, Wo, to zero at
the midpoint of the counterion layer. Unfortunately, this description was not able to
adequately describe many commonly observed colloidal properties, such as
electrophoresis.
To overcome this discrepancy, Gouy [GOU10] and Chapman [CHA13] assumed,
due to thermal energy of the counterions in the double layer, that not all of them would
necessarily be adsorbed to the interface. Instead, there would likely be a concentration
gradient near the surface. Under this assumption, the local concentration of counterions
near the interface would be greater than in the bulk, whereas the local concentration of
similarly charged ions (coions) would be depleted. If the ions are assumed to be point
charges that do not interact with each other, then according to the principles of
electrostatics the distribution of charge may be described by the PoisonBoltzmann
equation,
d2 2Zen Ze (228)
X2= snh (228)
dx2 r 0 kT
where y is potential, x is distance from the interface, Z is counterion valence, e is the
charge on an electron, n is the electrolyte number density, r is the relative dielectric
constant of the medium, Fo is the permittivity of free space, and kT is the product of
Boltzmann's constant and absolute temperature. This relation, however, is only assumed
to be valid starting at some distance away from the interface.
A more complete description of the electrical double layer was developed by
Stem [STE24] and Grahame [GRA47]. The SternGrahame model utilizes concepts from
both Helmholtz and GouyChapman to describe the overall properties of the electrical
double layer. However there is a primary difference in that the number of counterions
adsorbed to the interface is assumed to be proportional to surface potential and bulk
electrolyte concentration. A schematic illustration of the SternGrahame model of the
electrical double layer is shown in Fig. 25. This figure indicates a net negatively charged
surface that may be described by either a surface potential, Wo, or surface charge, Go. Due
to adsorbed counterions, the potential drops from Wo to y according to
30o6
y 0 (229)
where 8 is the radius of the adsorbed hydrated counterions, Go is surface charge and Fs is
the relative dielectric constant of the Stern layer. Note that this layer may also be referred
to as the Grahame layer and associated potentials or charges are refereed to by either
Stern or Grahame as well as the outer Helmholtz potential (OHP).
To apply Eq. (229), the number of adsorbed counterions needs to be known as a
function of surface potential and bulk electrolyte concentration. Using the formalism of
Grahame, the Stem charge, ao, which is nothing more than the difference in the number
+
o
S coion
counterion
i I
I 'lilIs a c
I ,
I \ I
I I
shear plane
Stern plane: os
Figure 25: Schematic of the StemGrahame Electrical Double Layer. At a net negatively
charged interface, cations (counterions: hatched) will adsorb in the Stern
plane and accumulate near the surface while anions (coions: solid) will be
depleted. At large distances from the surface the concentration of both
counter and coions approaches the bulk concentration. This double layer
structure leads to a linear decay of potential and charge across the Stem plane
and a roughly exponential decay thereafter. The zeta potential, , is measured
at the shear plane.
of charge groups per unit area between the surface and Stern layer multiplied by the
charge on an electron, may be written as the Grahame equation.
8c = 8 onZ2kT sinh( e2 ) (230)
2kT )
Beyond distance 8 the potential in the double layer decays according to Gouy
Chapman theory. Using an approximate solution, as outlined by Israelachvili [ISR92], the
number density of ions at distance x from an interface, nxi, may be written as
1 1 2kT dxJ
where ni is the bulk number concentration of ionic specie i, Er is the relative dielectric
constant of the medium, o is the permittivity of free space, k is Boltzmann's constant, T
is absolute temperature, and y is potential at distance x. For a 11 electrolyte this
expression simplifies to Eq. (232).
dw 8kTn ( et
dx sinh (232)
dx y,c0 2kT
This relation may then be integrated using the integral Jcsch(X)dX = ln[tanh(X/2)]
to yield
1+ tanh e 
S= In 4k 4kT tanh e (233)
e l etanh e e 4kT
S4kT
where W is the Stem potential and K is the inverse Debye length. Note that for low
potentials Eq. (233) reduces to the DebyeHiickel equation.
S= Woei (234)
These approaches have proven successful in describing the distribution of ions
and potential emanating from a charged interface in aqueous solution and form the
foundation for calculation of the electrostatic forces between two such interfaces.
However, most of the above relations are dependent on knowing the potential at the Stern
plane. Through potentiometric techniques the adsorption of potential determining ions
may be used to measure the charge developed at the interface under given ionic strength
conditions. Unfortunately, these measurements are rather tedious and require the
assumptions inherent in Eq. (229) to convert to the Stern potential. An alternative
approach is to measure the potential on the particle directly. Zeta potential is such a
technique.
Zeta Potential
When charged particles in colloidal suspension are exposed to an applied electric
field they are attracted and diffuse toward the electrode of opposite charge in a process
known as electrophoresis. This may seem unusual at first because the excess of counter
ions in the electrical double layer negates the charge on the surface and should result in a
net neutral particle. However, the counterions are attracted to the electrode of similar
charge to the particle. Hence, the propensity of a given counterion to move with the
particle is a competition between the electrostatic attraction for the similarly charged
electrode and hydrodynamic shear on the ion versus the attraction of that ion for the
surface. The boundary at which the counterions cease to move with the particle is shown
in Figure 21 as the shear plane. Since the structure of the double layer is disturbed at the
shear plane, while electrophoresis is occurring, electrolyte concentration falls rapidly to
that of the bulk and a potential difference between the shear surface and ground is
created. In electrophoretic phenomena this potential is known as zeta potential, . It
should be noted that the position of the shear plane is usually quite small (0.4 nm)
compared to the thickness of the double layer or Debye length ( 130 nm) [ATTOO].
Smoluchowski [SMO21] treated this phenomena mathematically by equating the
force of attraction due to the particle's interaction with the electric field with the
resistance of the particle's movement through liquid due to Stokesian flow, Eq. (235),
Fhyd = 6rlRv (235)
where Fhyd is the force resisting the movement of a spherical particle radius, R, moving at
velocity, v, through liquid of viscosity, fr. Assuming that the extent of the electrical
double layer is much smaller than the radius of the particle and that conduction within the
shear plane is negligible, the velocity of the particle may be related to the zeta potential,
, and electric field strength, E, by
1rgit
v = E (236)
where r is the relative dielectric constant of the medium, o is the permittivity of free
space and the electrophoretic mobility, PE, may be expressed as Eq. (237).
PE = (237)
Note that these relations are independent of particle radius as both Stokesian
movement and electrostatic attraction are proportional to radius. Additionally, although
originally calculated for particles of zero dielectric constant the approach has been
validated for particles of arbitrary dielectric constant and shape as reviewed by Hunter
[HUN01]. For particles in the limiting case of a large double layer compared to particle
radius, as may be the case for nanoparticles, Hickel [HUC24] derived an expression
similar to that of Smoluchowski,
v = E(1+ KR) 2 E (238)
31j 31r
where R is particle radius and K is the inverse Debye Length. The discrepancy between
these two cases was resolved by Henry [HEN31] who proposed an equation of the form
v = E f, (KR) (239)
3r1
where fi(KR) is a function ranging from 1 in the Hickel limit of low KR to 3/2 in the
Smoluchowski limit of high KR. Recently Oshima (OSH94) has proposed a single
analytical formula for fi(KR).
S(R) =+ 2.5/(1+ 2e") 3(240)
f (iciR) = 1 + R (240)
2 KcR
Due to the ions within the shear plane the zeta potential is always lower in
magnitude than the Stern potential. However, for relatively low ionic strength conditions
(approximately less than 0.03 M) and low surface potential (approximately less than 30
mV) the zeta potential may be considered a good approximation of the Stem potential.
For electrolyte concentrations and potentials where this approximation is not valid, there
are several methods to calculate the Stem potential based on the distribution of ions
between the shear and Stern planes [ATTOO]. However, to more accurately characterize
the interaction of surfaces direct measurement of electrostatic force, which depends
primarily on the Stern potential and ionic strength, may be necessary.
Boundary Conditions for Calculation of Electrostatic Force
When two electrical double layers of similar charge begin to overlap, electrostatic
repulsion results. If these surfaces are opposite in charge, they will attract. In the above
sections, surfaces were considered to be isolated. However, when surfaces approach one
another the increased electric field strength and higher concentration of counter and
potential determining ions may affect the charge on the surfaces. Hence, before
electrostatic forces may be calculated a brief description of how the surfaces behave in
the presence of each other is necessary.
There are two boundary conditions that act as upper and lower limits to the
electrostatic force magnitude. Under a constant potential boundary condition, the surface
charge decays from its value at infinite separation distance to zero as the two surfaces
approach. In more descriptive terminology, this implies that as the two electrical double
layers overlap the local concentration of both counterions and potential determining
ions increases. Especially for materials that acquire charge by the selective dissolution
mechanism this higher concentration of ions tends to reduce the surface charge. At small
separation distances this effect becomes more pronounced due to the exponential nature
of the ion concentration profile at an interface. Because the charge is reduced at any
given separation distance using this approach, this boundary provides the lower
magnitude limit of electrostatic force.
The boundary condition that predicts the maximum repulsion between surfaces is
referred to as constant charge. Under this assumption, as surfaces approach no change in
surface charge occurs. Hence, at small separation distances surface potential must
approach zero. Physically, this indicates that the electrolyte concentration between the
surfaces must increase more strongly than in the constant potential case in order to
compensate the charge. Surfaces that acquire charge through dissociation are more likely
to approach the constant charge limit as the concentration of counterions does relatively
little to change the surface charge.
Unfortunately, real surfaces cannot be characterized as constant potential or
constant charge. Instead both processes are likely to occur to some degree. To model the
true behavior of surfaces, a charge regulation boundary condition may be employed.
Under this assumption, reactions that account for both the dissociation of surface groups
and the binding of counterions to the interface are proposed. These reactions must
depend on the local concentrations of counter and potential determining ions.
Additionally, for a given concentration of counter and potential determining ions, the
total number of each type of site should be known.
Even for a relatively simple single site solid such as silica or mica this means that
four additional variables need to be considered when predicting the interaction force
profile. For an amphoteric surface such as alumina the relations become significantly
more complicated. Such constants are usually obtained experimentally by fitting directly
measured forces between surfaces. Solving for the unknown charge regulation parameters
(as well as Stern potential and Hamaker constants) is difficult and requires measurement
over a wide range of solution conditions. Hence, determination of the charge regulation
parameters has only been attempted for a few materials such as mica [PAS82] and silica
[GRA93, ZHUOO].
It should be noted, however, that the practical effects of charge regulation
processes only come into effect at relatively short separation distances (less than 30 nm).
In fact for larger separation distances, the two limiting conditions approach one another
and a single analytical expression may be used. However, as ensemble models are refined
and processing conditions become more severe, more detailed knowledge of the behavior
of particles at small separation distances will become necessary.
The variation in surface charge or potential as interfaces approach is important
because these boundary conditions allow the PoissonBoltzmann distribution for
overlapping electrical double layers to be solved and electrostatic forces calculated. In the
following two sections these conditions will be utilized to derive electrostatic force under
the linearized PoissonBoltzmann approximation, which is generally valid for lower
potentials and larger separation distances, as well as the full solution.
Linearized PoissonBoltzmann Approach
To calculate the electrostatic force between surfaces, the full solution to the
PoissonBoltzmann equation must be performed numerically. However, in order to
develop an analytical model, a linear approximation may be employed. If the scaled
potential, y(), between two plates is represented as
Ze
y() = W (241)
kT
where Z is counterion valence, e is the charge on an electron, kT is the product of
Boltzmann's constant and absolute temperature, V6 is the Stem potential, and 4 is a
dimensionless scaled length parameter,
4 = KX (242)
where K is the inverse Debye length and x is distance perpendicular to the interface. The
PoissonBoltzmann equation, Eq. (228) may be written as Eq. (243).
d2y( )
2 = sinh y(4) (243)
d2
Expanding the right side of Eq. (243) in a power series of y(4) yields the
following.
dy() Y()+ y( + y( ... (245)
d42 6 120
If only the first term on the right side of the equation is considered to adequately
describe the PoissonBoltzmann distribution then it becomes possible to analytically
solve for the electrostatic repulsive energy between two flat surfaces. For constant
potential surfaces, these calculations were performed by Hogg et al. and the resultant
equation is often referred to as the HoggHealyFuerstenau (HHF) equation [HOG66].
W,(H) ptt 2Z2nkT 1 tanh H (246)
( kT K 2
W,(H)plt/plt is the energy of interaction between flat plates under a constant
potential boundary condition, Z is counterion valance, n is the number density of
counterions, e is the charge on an electron, Va is the Stern potential, k is Boltzman's
constant, T is absolute temperature, K is the inverse Debye length, and H is the distance
between the two surfaces. Similarly, the interaction energy between flat plates under a
constant charge boundary condition, W((H)plt/plt, may be written as Eq. (247).
eV )2 KH
W (H)t = 2Z2nkT J coth 1 (247)
(kT K ( 2
As mentioned earlier, at low potentials (generally less than 30 mV), low ionic
strength (generally less than .03 M), and small degrees of double layer overlap (generally
H greater than 30 nm), the constant charge and potential boundary conditions begin to
coincide. Under these conditions the surface charge, Go, may be related to the surface
potential, Wo, which is also equivalent to the Stem potential,
CTo = eoKo0 (248)
and where the electrostatic interaction between flat plates may be written as, Eq. (249).
2
W (H)plt/plt = W (H)plt/pt = 2r 0e = (249)
ErE0K
To extend the range of applicability of Eqs. (246) and (247) especially to higher
potentials and shorter separation distances as well as account for the interaction of
dissimilar surfaces, some authors have extended the power series approximation, Eq. (2
45) to the second and third terms [OSH82]. While immensely useful because the
solutions retain the form of analytical equation, they are very long, over 50 terms, and
still fail to accurately describe the electrostatic force at potentials greater than
approximately 60 mV, especially at short separation distances. Since the principal
discussion in this study is the interaction at relatively small separation distances,
electrostatic force calculation based on the full numerical solution to the Poisson
Boltzmann distribution is more appropriate.
NonLinear Approach
The full nonlinear solution to the PoissonBoltzmann equation must be
performed to accurately predict the electrostatic interaction of surfaces at high surface
potentials or short separation distances. Chan et al. [CHA80] using such an approach
have developed as relatively simple algorithm for these calculations. If the Poisson
Boltzmann equation, Eq. (228) is written in terms of the scaled potential, y, defined by
Eq. (241) to yield Eq. (243) a first integration may be written as
dy= QSgn(ym) (250)
where Sgn(ym) indicates the sign of the potential at the midpoint between the surfaces
and Q is the following.
Q = 2(cosh y coshym) (251)
Note also that at the midpoint between the surfaces, = 0, dy/d must be zero and
hence, Q = 0. With this boundary condition dQ/dy may be calculated as the following.
dQ sinhy Sgn(ym) Q2 (252)
+coshym 1 (252)
dy Q Q 2
From Eqs. (250) and (252) it is now possible to derive d//dQ, Eq. (253).
2+ cos
= +coshyQ1 (253)
dQ 2 m2
If at the midpoint between the surfaces Q = 0 and = 0, and at the interface Q =
Q6 and 5 = KH/2 where K is the inverse Debye length and H is separation distance, then
for selected a value of the midplane potential, ym, numerical integration from Q = 0 to Q6
will yield a specific value of g. Repeating this procedure for ym = 0, or a suitably low
potential, to ym = ys will the define a set of corresponding values for separation distances
and midplane potentials. The electrostatic force per unit area, Fplt/plt, may be written as a
function of midplane potential as
Fplt/plt = 2nkT(cosh ym 1) (254)
where n is the number density of counterions and kT is the product of Boltzmann's
constant and absolute temperature. The interaction energy between plates may then be
calculated by a second numerical integration represented by Eq. (255).
Wplt/pt = F(H) pl/pt dH (255)
H
The above equations were derived under the assumption of a constant surface
potential, y6, which yields a constant value of Q6 at any separation distance. Hence, Q6
has the same form as Q, Eq. (251).
Q6 = 12(coshy coshym) (256)
However, for a constant charge boundary condition the integration limit Q6 must
be determined in terms of charge, a By definition at the interface
dy 4re 1
d = 'T (257)
d4 ErEkT K
which is equivalent to Eq. (250). Therefore, the integration limit may be written as Eq.
(258).
4re 1
Q 47e I (258)
8C, kT K
Although the mathematical description of the electrical double layer and the
calculation of electrostatic force is extensive, there are still many assumptions assumed in
the basic theory. As delineated by Lyklema [LYK95]:
i) Ions are assumed to be point charges.
ii) NonColumbic (specific) adsorption between counterions and coions and
the surface is neglected.
iii) Permittivity of the medium is assumed to be a constant and independent of
electrolyte concentration.
iv) Incomplete dissociation of the electrolyte is ignored.
v) Solvent is considered to be homogeneous.
vi) Polarization of the solvent is not accounted for.
vii) Surface charge is assumed to be smeared out and uniform.
viii) Image forces between ions and the surface are neglected.
ix) Average potential and potential of the mean force are assumed to be equal.
x) Surfaces interact in an infinite volume of solution.
xi) Surfaces are considered ideally smooth.
For most interactions these assumptions are not violated to a significant degree.
However, for interactions in which separation distance is small or ionic strength is high,
models which account for some of these issues may need to be developed. Specific
examples include, if hydrated ions are sufficiently large and in large enough
concentration they may physically resist surfaces from approaching each other [PAS83].
These hydration forces violate the description of ions as point charges. Similarly, for
highly charged interfaces in solutions of relatively high ionic strength, the permittivity of
the medium may be altered, and could continue to change as a function of separation
distance [NIN97]. Under similar conditions, image forces have been suspected to
increase the viscosity of concentrated suspensions [COL98].
DLVO and Extended DLVO Theory
The teams of Derjaguin and Landau in Russia [DER41] as well as Verwey and
Overbeek [VER48] in the Netherlands independently developed the fundamental criteria
for colloidal stability known as DLVO theory. Quite simply, both groups came to the
conclusion that the stability of similar particles in suspension was a competition between
van der Waals attraction and electrostatic repulsion. This may be expressed as the total
energy, Wot, of interaction being equal to the sum of van der Waals attractive energy,
Wvdw, and electrostatic repulsive energy, Weiect, as expressed in Eq. (259).
Wtot = Welect WdW (259)
This seemingly simple relation has proven extremely powerful in predicting the
behavior of nearly ideal particulates and surfaces in solution and has been validated for a
wide variety of conditions and surfaces. Israelachvili [ISR92], Lyklema [LYK95], and
Hunter [HUN01] have reviewed many of the successful applications of DLVO theory.
There are a number examples, however, where DLVO theory does not adequately
describe measured interaction force profiles. These may range from the relatively
complicated, as in examples where the increased concentration of ions in the double layer
aids in the screening of van der Waals attraction such that the two quantities are not
independent [NIN97], to more obvious situations where another mechanism of force is
active. For example, if in a good solvent a polymeric reagent (nonionic) is adsorbed to
the surfaces, a strong steric repulsive force may be observed due to the physical and
osmotic pressure developed by the overlap of soluble polymer chains. However, too
much free polymer in solution, when excluded from the gap between surfaces, may cause
an attractive depletion force, due to osmotic pressure. Additionally, hydration and other
solvation forces arise due to the structuring of hydrated ions or solvent, respectively, in
the gap between surfaces. Alternatively, attraction may occur due to the unfavorable
structure of water near a hydrophobic interface. It has become common practice to also
assume that, as a first approximation, these other force mechanisms are also independent
and additive. Hence, extended DLVO theory may be written as Eq. (260).
Wtot = Welect WvdW + Wsterni Wdepletion + Whydration + (2
(260)
Wsolvation Whydrophobi c + Wother
Certainly not all of these forces can occur simultaneously or are strictly
independent of each other. Many also have different magnitudes and signs such that as a
function of separation distance different forces may become dominant. Delineation of
these forces, their interactions, and fundamental mechanisms is currently being actively
pursued. However, in this study conditions and materials are avoided where these
additional forces are expected to have significant impact. Hence, they will not be
discussed in detail here.
Surface Force Summary
The previous sections have described in some detail the basic concepts of
attraction and repulsion between surfaces under gaseous atmospheric conditions and in a
solvent. Although these models are not necessarily complete, they do provide a sound
theoretical basis for the fundamental interaction of surfaces. However, in all the above
models, the interfaces in contact are assumed to be ideal (smooth and discrete). In the
remainder of this study, these basic concepts will be extended for two specific types of
nonidealities. Namely, surface roughness and chemically impure surfaces. Emphasis will
be placed on the nanoscale characterization of these interfaces and impact of these
deviations on ensemble behavior of particulate systems.
CHAPTER 3
ROLE OF ASPERITY GEOMETRY ON THE ADHESION OF SURFACES
Introduction
As discussed in the previous chapter, there are a wide variety of theoretical
approaches to describe the adhesion between surfaces in a gaseous medium. These
theories are based on the assumption that the geometry of the interacting bodies is known
and that they may be treated at the macroscopic level. This precludes the discussion of
interfaces on which nanoscale roughness or a transition in composition exists. This is
unrealistic because every surface due to the physical size of atoms or crystal structure
exhibits some order of roughness. Similarly, there are very few materials that do not form
some sort of discontinuous layer, such as an oxide, at the interface. The question then
arises as to how significant are these nonidealities in determining the overall interaction
between surfaces. In this chapter, a description of one type of nonideal surface, a surface
on which roughness at the nanoscale is present, will be developed. Through an analysis of
such a surface, models of the interaction force of adhesion between two interfaces will be
developed. This concept will be further validated and expanded in Chapters 4 and 5.
Most particulate surfaces, regardless of preparation method, possess some finite
surface roughness. For many modern materials, particularly in the microelectronics
industry, highly polished surfaces with roughness on the nanometer scale are becoming
increasingly common. Particulate adhesion to these surfaces is of critical importance due
to the possibility of defects caused by small imperfections. Hence, to develop enhanced
cleaning procedures, the strength of adhesion between particles and the substrate needs to
be determined. The existence of nanoscale roughness is known to dramatically reduce
adhesion between surfaces due to a decrease in the real area in contact and an increase in
the distance between the bulk surfaces [TAB77, BRI92, KRU67, JOH71, MIZ96].
However, the magnitude of these effects is less well understood.
Quantitative calculation of adhesion force was performed by lida et al. [IID93] by
modeling the interaction between a smooth glass particle and glass substrate possessing
roughness ranging from 50 400 nm. In this investigation, adhesion force was estimated
by the average force needed to retain 50% of particles on a substrate after bombardment
by fluid or gas. A factor often decrease in adhesion force was initially observed as
roughness increased but was then followed by a region of more gradually decreasing
adhesion force. However, in developing a theoretical model, these investigators only
considered the interaction between the particles and asperities (contact forces) and
ignored the contribution arising from the interaction of the particles with the underlying
surface (noncontact forces). Due to the proximity of the surface at lower root mean
square (RMS) roughness, this omission resulted in underestimated adhesion forces.
One of the most commonly used models of adhesion force in the nanoscale
roughness regime is Rumpf's model [RUM90], which is based on contact of a single
hemispherical asperity, centered at the surface and interacting with a much larger
spherical particle along a line normal to the surface connecting their centers, as shown
schematically in Figure 31. The model consists of two terms that describe the total van
der Waals interaction. The first represents the interaction of the adhering particle in
/ /
I
/ I
/i
i' '/
Figure 31: Asperity Schematic Using the Rumpf Model. Illustration of the geometry
proposed by Rumpf [RUM90] for the interaction of an adhering particle
with a rough surface. Diagram depicts a hemispherical asperity of radius r
and origin at the surface interacting along the ordinate axis with a
spherical adhering particle.
contact with the asperity, while the second describes the "noncontact" force between the
adhering particle and flat surface separated by the height (radius) of the asperity.
Applying Derjaguin's approximation [DER34] for both interactions, Rumpf
obtained the following formula.
Fad= A [ rR R (31)
a 6H r+R (1+r/Ho)2
Where A is the Hamaker constant, R and r are the radii of the adhering particle
and asperity, respectively, and Ho is the distance of closest approach between surfaces
(approximately 0.3 nm). A primary limitation of this model is that the center of the
hemispherical asperity is required to be at the surface. As described in detail later, in real
systems, especially those with low roughness, this assumption may not accurately
describe the surface asperities.
Greenwood and Williamson [GRE66] introduced a roughness model considering
hemispherical asperities of equal radii, but whose origin was offset from the average
surface plane according to Gaussian probability. However, as Fuller and Tabor surmised
[FUL75], the application of a Gaussian distribution to model asperities on a surface may
produce errors because large asperities significantly affect the interaction even though
their number may be low or even singular. Czamecki and Dabros [CZA80] modeled
particles with surface roughness as spherical cores with a rough outer shell. They then
proposed a correction factor, based on asperity height, to predict the van der Waals
attractive force. This approach is valid only for separation distances much larger than the
asperity height. Therefore, it is only applicable for noncontact dispersion forces and not
adhesion force.
Recently, Xie [XIE97] also carried out a theoretical study to determine the effect
of surface roughness on adhesion. A modified van der Waals force depending only on the
radius of asperities was proposed and applied through two geometrical models. The first
model is similar to Rumpf s but ignored the interaction of the particle with the asperity
on the surface. The second assumes the asperity to be a small particle positioned between
the two larger surfaces (sandwich model). It was concluded that if the radii of surface
asperities are smaller than 10 nm, the surface could be treated as smooth.
The investigations described above propose methodologies to predict adhesion
force between surfaces of known asperity geometry. Unfortunately, little has been said
about how well those geometries correlate with known surface roughness profiles,
especially at the nanoscale. In this chapter, a more realistic model of surface roughness
will be developed and the effect on adhesion force explored through parametric
investigation. Chapter 4 will focus on the application of this model to predict the
adhesion force between surfaces with nanoscale roughness. Chapter 5 will extended the
application of this basic approach to the interaction of surfaces in a humid environment
where capillary adhesion forces may become significant.
Modified Rumpf Model
A schematic of the Rumpf model is shown in Figure 31. This model and Eq. (3
1) were proposed by Rumpf to calculate the force of adhesion between a spherical
particle and spherical surface asperity [RUM90]. The radius of the asperity, r, however, is
not easily measured, while the value of RMS (root mean square) roughness is. Hence, in
order to apply the Rumpf model, a relationship between the radius of the asperity and
RMS roughness must be developed.
Considering a spherical coordinate system, with the origin at the center of the
spherical asperity, the horizontal projection, ri, of any radii of the asperity, r, at angle, a,
formed between any radii and vertical axis, may be related to r and a as follows:
r, = r cos a (32)
Similarly, the vertical distance to the average surface plane, y, may be written as
y = r sin a (33)
The general definition of RMS for hemispherical asperities is
r
32 y2 rdrl (34)
RMSS = k
k P
where k = 4r is the peaktopeak distance and kp is the surface packing density for close
packed spheres, 0.907. Note that a negative deviation from the average plane in term of
RMS roughness is equivalent to an asperity. Hence, as long as it is assumed that the
surface contains an equal number of asperities and inverse asperities (pits) the close
packing assumption is valid. Substituting Eqs. (32) and (33) in (34) and integrating, the
following equations are obtained.
RMS = 0.673r (35)
or
r= 1.485 RMS (36)
Using this approach, the adhesion force predicted by the Rumpf model may be
calculated by a simple substitution of Eq. (36) into Eq. (31) yielding the following.
AR 1 1
ad 6H +R/(1.48.RMS) (1+1.48.RMS/Ho)2
Adhesion force calculated using Eq. (37) will be termed the modified Rumpf
model in the remainder of the study.
Figure 32 depicts the variation of the total adhesion force between a 10 jPm
particle and a rough surface normalized by the radius of curvature of the adhering particle
as predicted by the modified Rumpf model. In the limit of both very small and very large
surface roughness the predicted adhesion force approaches that of a sphere and smooth
flat plate (185 mN/m). Note that when normalized by the radius of curvature this limiting
value is independent of the size of the adhering particle. In the nanoscale roughness
regime a dramatic decrease in adhesion force is predicted for micron sized adhering
particles. To elucidate the mechanisms responsible for this decrease, the contact
(particle/asperity) and noncontact (particle/surface) components of the total adhesion
51
..................... .......
I '' .I.. '
= V
I
II. .I I
I I )* '
Figure 32: Prediction of Adhesion Using the Rumpf Model. Prediction of the total,
contact, and noncontact force of adhesion, normalized by the radius of the
adhering particle, between a 10 pm smooth sphere and a surface with
nanoscale roughness using the modified Rumpf model, Eq. (37). Hamaker
constant, A, is 1019 J and distance of closest approach, Ho, is 0.3 nm.
force are also plotted in Figure 32. At larger RMS roughness, the contact adhesion force
between the sphere and asperity is seen to dominate the interaction. However, as the
radius of the asperity and hence roughness decreases, the noncontact adhesion force
primarily contributes to the interaction.
It should be noted that even though the scale of adhesion force is normalized by
the radius of curvature of the adhering particle, in the area of nanoscale roughness this
normalization is not entirely accurate. As expected in this model, the noncontact force is
independent of particle size but the contact adhesion force between the particle
and asperity is not. As the particle size increases, the normalized force of adhesion in the
regime affected by nanoscale roughness decreases. This results in a shift in the contact
component towards higher RMS roughness as particle size increases. Hence, the
minimum in the total adhesion force, as predicted by this theory, further decreases in
magnitude and occurs at greater roughness values as the size of the adhering particle
increases.
Proposed Model
Techniques such as atomic force microscopy have enabled detailed investigation
of the morphology of surfaces with small roughness. Through these investigations it has
been determined that the geometry of the Rumpf model does not accurately describe the
surface in the nanoscale regime. More realistically, one can imagine that as the surface
roughness decreases, the radius of the asperity must really increase. Hence, as the surface
roughness approaches zero the radius of the asperity should approach infinity resulting
locally in a flat surface. For this situation to be valid, if the asperity is still modeled as a
sphere, the center of the asperity can not be at the surface. Instead the center must be
located some distance below the surface such that the observed asperity height is
equivalent to the radius in the Rumpf geometric model. In the proposed model of the
geometry of the surface roughness, a new parameter is introduced. Not only should the
height of the asperity used but its breadth must also be considered.
Figure 33 depicts surface roughness as closepacked hemispherical asperity caps
and troughs. As in the Rumpf model a single particle interacting with a single asperity
1 I
\ f
Figure 33: Asperity Schematic Using the Proposed Model. Illustration of the geometric
model used to calculate adhesion force between a spherical adhering particle
and a surface in the proposed model. Diagram depicts a hemispherical
asperity of radius r, peaktopeak distance ), and origin below the average
surface interacting along the vertical axis with a spherical adhering particle.
along a line connecting their centers and perpendicular to an average surface plane will
be considered. However, the asperity will be characterized by radius, r, and peaktopeak
distance, ). Note that in this model the height of the asperity, above the average surface
plane, is not equal to the radius of the asperity. The origin of the asperity, 0, is located
below the average surface plane and is coincident with a spherical coordinate system.
As before, if a is defined to be any angle between the vertical direction and a
radius of the asperity, r, that terminates above the average surface plane, the projection,
ri, on a horizontal plane is the following.
r, = r sin a (38)
The maximum value of the angle, ao, is limited to the intersection of the sphere
describing the asperity and the average surface plane, ao = arcsin (k/4r). Hence, the
vertical distance between any point on the asperity and the horizontal plane may be
written as
y = r(cos a cos co) (39)
For small angles, a and ao << 1, as is the case for nanoscale roughness, Eqs. (38)
and (39) may be simplified, as follows
r = ra (310)
and
y=(r/2)(a~a2) (311)
where ao = k/4r, and hence 0o << 1, and ) << 4r where all angles are in radians.
The radius of the asperity and the maximum value of the peak height, ymax, are not
measured parameters and should be expressed as a function of RMS roughness and peak
topeak distance, k, through a geometric model of the roughness profile. For the
geometry defined above, the value of the peak height, ymax, can be obtained from
ra2 2
Ymax 2 32r (312)
Introducing a relationship between ymax and RMS where kl is a coefficient
yma = kRMS (313)
and substituting in Eq. (312), Eq. (314) is obtained.
r = (314)
32kiRMS
Once again applying the general definition of RMS roughness for close packed
hemispherical asperity caps or troughs
k/4
32 Jy2rdrl (315)
RMS = o k
h2 P
and substituting Eqs. (310) and (311) in (315), Eq. (316) is obtained.
RMS = 0.0172'2 /r (316)
Note that if Eq. (316) is combined with the restriction ) << 4r, the following
condition is imposed.
RMS << k/14.5 (317)
This condition indicates that the proposed adhesion model is valid only for
surfaces with small asperity amplitude and large peaktopeak distances (not a high
aspect ratio or jagged surface). At first, this restriction may seem to severely limit the
systems to which this model is applicable. However, as discussed in Chapter 4, the
roughness on the nanoscale is found to control the theoretical adhesion between surfaces.
Even for surfaces with larger RMS roughness, it is the nanoscale asperities often found
on larger asperities that dominate the adhesion interaction. Hence, the proposed model
may be extended for many types of surfaces with greater RMS roughness if multiple
scales of roughness are considered.
Using Eqs. (314) and (316), allows the coefficient relating RMS roughness and
the maximum peak height to be determined as,
k = 1.817 (318)
To calculate the adhesion force in the framework of this model, we can use the
following equation by considering the contact and noncontact van der Waals forces,
similar to the method proposed by Rumpf [RUM90].
AR r 1
Fad = + Y (319)
6H2 r+R Ym
H0
Note that Eq. (319) becomes Eq. (31) when the origin of the spherical asperity is
coincident with the average surface plane (i.e. when ) = 4r and ymax = r). Substituting
Eqs. (313) and (314) for ymaxand r in Eq. (319), the adhesion force between an
adhering particle and a surface with nanoscale roughness is the following.
AR 1 1
Fad = + (320)
6H 1 32RkRMS 1 kRMS
The first terms in brackets of Eqs. (319) and (320) represent the contact
interaction of an adhering spherical particle with an asperity on the surface. The second
terms account for the noncontact interaction of adhering particle with the average
surface plane.
It should be noted that Eqs. (319) and (320) only account for the van der Waals
attraction between surfaces. However, as described in Chapter 2, it is known that other
forces, such as polar forces [ISR92], may significantly contribute to the adhesion between
contacting surfaces. Furthermore, the proposed models also do not consider elastic
deformation of the surfaces. As a result, the proposed model may underestimate
adhesion. To account for any additional attractive forces and possible elastic deformation
upon detachment from the surface, adhesion force is often calculated through models
considering surface energy such as the JohnsonKendallRoberts [JOH71] or Derjaguin
MullerToporov [DER75] models. The application of these models to the proposed
geometries is discussed further in Chapter 4.
Another assumption inherent in the proposed model and in Rumpf s original work
is that the region outside the contacting asperity may be approximated by a smooth flat
plane. To justify this approximation, the series approximation to the ratio of noncontact
van der Waals attraction between rough planes and smooth planes may be considered
[BRE74, RAB89].
3(RMS2 +RMS) 5(RMS, + RMS2)4
Urough / Usmooth =2+ 2 +4RMS + (321)
H2 H4
Urough and Usmooth are the specific van der Waals energy of interaction between
two rough or smooth planes, respectively. H is the separation distance between the
surfaces or average surface planes. RMS1 and RMS2 are root mean square roughness of
the first and the second interacting planes of the rough surfaces. Although this relation
was originally derived for surfaces far from contact (H >> RMS1 or RMS2), within this
study it is applied at a separation distance corresponding to the asperity height, ymax, as
related to RMS1 through Eqs. (313) and (318). Additionally, the energy between planes
may be related to the force through the Derjaguin approximation for a smooth adhering
sphere (RMS2=0). At contact the ratio becomes independent of RMS1, and approaches a
value of approximately 2.4.
Although this ratio is greater than unity, it should be noted, that it includes the
contribution of the central (contacting) asperity on the plane, which is excluded from the
second terms in Eqs. (319) and (320). More important is the fact that the overall
contribution of the plane is significantly less than the contribution of the contact
interaction between the surfaces. For example, even at extremely small RMS roughness,
where the contact and noncontact terms are closest in magnitude, the noncontact term is
only half the particle/asperity contribution. Above 2 nm RMS roughness, in some cases,
it may even be appropriate to ignore the noncontact interaction between the particle and
the average surface plane.
As indicated earlier, Eqs. (319) and (320) are only applicable over certain ranges
of RMS, ) and r values. As RMS roughness decreases towards an atomically flat surface,
Eq. (320) predicts a force of adhesion that is twice that predicted for the van der Waals
interaction between a smooth surface and sphere. In this limit, the radius of the asperity
approaches infinity and effectively equals the contribution of the planar noncontacting
surface. This can be avoided if we restrict the range of RMS or r. These values may be
evaluated if we introduce an "effective zone", Seff, i.e., the area of the surface that
principally contributes to the interaction between the adhering particle and flat surface or
spherical asperity.
An interacting sphere and flat surface separated by gap H are shown in Figure 3
4. Since van der Waals forces decrease rapidly as a function of distance, the effective
zone for sphere may be determined by the projected area of the sphere on the flat plate
where the separation distance is less than 2H. Beyond 2H, the contribution of noncontact
forces is less than 12% of the force at distance H and may be considered negligible. For a
given separation distance, H, the following formulae are obtained.
AB = 2R sin P (322)
I )
I I
Figure 34: Effective Zone of Interaction. Illustration the concept of an effective zone of
interaction between a flat surface and adhering spherical particle.
RH
cos I (323)
R
For a small angle 13 (H << R), the following expression for the effective zone is
obtained.
Seff = 21rHR (324)
2
The effective zone, Sef, is equal to 4IlrRHl/(r+R) for contact interaction between
two spheres with radii R and r (i.e., adhering particle and spherical asperity), separated by
the minimum intermolecular distance H0, 0.3 nm. For noncontact interaction between the
adhering sphere, with radius R, and the average surface plane separated by gap
(Ho+kiRMS), the effective zone Seft2 is equal to 27rR(Ho+kiRMS). Therefore, Eqs. (319)
and (320) are valid if Sem<< Seff2, meaning that the area of the average surface plane that
contributes to the contact interaction between the particle and asperity is small compared
to the area of interaction between the particle and the average surface plane. Hence, the
noncontact interaction between the particle and average surface plane may be calculated
including the volume in common with that of the asperity. The radius, r, of the asperity
increases when RMS decreases (see Eq. (314)), the following simple limit of
applicability for Eqs. (319) and (320) may be obtained by comparing Sem and Sef2 for
small roughness where RMS is approximately < Ho/kl and r << R. Using Eq. (314) and
this condition, the range of ), where the equations are valid may be obtained from.
k << /32kRRMS (325)
Eq. (325) may be used to determine the lower boundary for small values of
roughness. The upper limit of roughness as described above yields, Eq. (326).
>> 14.5 RMS (326)
The dependence of the total normalized adhesion force as well as the contact and
noncontact components as a function of RMS roughness calculated with Eq. (320), are
presented in Figure 35 for an adhering particle of 10 jpm radius, a Hamaker constant of
1019 J, a distance of closest approach of 0.3 nm, and a peaktopeak distance of 250 nm.
Also replotted for comparison is the total normalized adhesion force as predicted by the
modified Rumpf model, Eq. (37), at equivalent conditions. Unlike the modified Rumpf
model, no minimum in the total normalized adhesion force predicted by the proposed
model is observed. Instead, both the contact and noncontact terms contribute to the total
predicted force of adhesion at small roughness. Also unlike the modified Rumpf model, it
is now the contact term that provides the primary contribution to the total adhesion force.
However, the noncontact term is not negligible in the nanoscale roughness regime.
Hence, the adhesion force predicted by the proposed model may be a full order of
61
I '' lb I. '
i 7
'II.
.. ... .. .. ..
Figure 35: Comparison of Adhesion Models. Prediction of the total, contact, and non
contact force of adhesion, normalized by the radius of the adhering particle,
between a 10 jpm smooth sphere and a surface with nanoscale roughness
using the proposed model, Eq. (320). Hamaker constant, A, is 1019 J, peak
topeak distance, k, is 250 nm, and distance of closest approach, Ho, is 0.3
nm. Also plotted for comparison is the predicted adhesion force from the
modified Rumpf model, Eq. (37), using the same parameters except peakto
peak distance which is not accounted for in the model.
magnitude greater than predicted by the modified Rumpf model. As the radius of the
adhering sphere decreases, the contact contribution to the total predicted force increases,
while the noncontact term remains constant. Hence, with decreasing particle size the
interaction behavior approaches that of a perfect sphere/flat plate interaction.
Using the boundary conditions described above, the proposed model is valid in
the range where 0.1 << RMS << 20 nm, for the given peaktopeak distance. For
roughness of greater magnitude, a correspondingly greater peaktopeak distance would
be present and hence would necessarily increase the upper RMS limit. While the
modified Rumpf model does not theoretically have these limits, its predictive power in
the extremes of roughness may still be limited. Specifically on surfaces with greater
roughness, the asperities are rarely smooth. Hence, the increase in adhesion force
predicted by the Rumpf model may not actually occur. Instead a second order of
roughness, with a smaller radius or peaktopeak distance, superimposed on the first may
need to be considered. This concept will be explored further in Chapter 4.
Both the modified Rumpf and proposed models have other limitations in addition
to those described above. These models were developed for a fixed particles adhering to a
surface. Hence, the particle is not free to find an equilibrium position that would most
likely consist of more than one contact with the surface. This increased number of
contacts holds the possibility of increasing the total adhesion experienced by a particle.
Note however, that during experimental determination of adhesion force by a method
such as atomic force microscopy, the particle is fixed to the cantilever.
As stated earlier, the materials in these systems are also considered non
deformable under the applied loads. The effect of plastic deformation is neglected. In the
Rumpf model at nanoscale roughness, the asperity size is necessarily small and so plastic
deformation may appear to play a critical role. However, in the proposed model, the
asperities are modeled as larger spheres with their centers below the surface, hence, the
resistance to deformation is considerably increased. Additionally, other forces known to
act between some surfaces in dry contact such as polar interactions [ISR92] and contact
electrification [HOR92] are also not considered. If the contributions of these forces were
found to be significant, both would result in greater measured adhesion force.
Summary
To this point, a model has been developed to more accurately predict the force of
adhesion between a particle and surface with nanoscale roughness. By modeling the
surface roughness in a manner that more closely describes the true geometry of the
surfaces a larger asperity radius is predicted that significantly impacts the relative
contributions of contact and noncontact adhesion forces. For example, a predicted force
of adhesion nearly two orders of magnitude greater, at 10 nm RMS roughness, than a
similar model using a smaller radius was calculated. As this model more accurately
simulates real surface geometries, it is expected that it will more accurately predict the
measured force of adhesion described in Chapter 4.
CHAPTER 4
ADHESION BETWEEN NANOSCALE ROUGH SURFACES (VALIDATION)
Introduction
Few surfaces are smooth at the atomic level, and even highly polished surfaces
possess some finite surface roughness. Such roughness is known to reduce adhesion
between surfaces or between a particle and a surface due to a decrease in real area of
contact and an increase in the distance between the bulk surfaces [TAB77, BRI92,
KRU67, JOH71, MIZ96]. In Chapter 3, a theoretical framework was developed in order
to more accurately account for nanoscale asperities at interfaces. This analysis resulted in
a new model, based on a geometry, that considers both the height and breadth of
asperities and yields an increased asperity radius compared to previous approaches. In
this chapter, the proposed model will be validated by direct measurement of adhesion by
atomic force microscopy (AFM) and extended to include polar forces and for surfaces
with various scales of roughness.
One of the first systematic investigations of roughness effects on adhesion was
conducted by Fuller and Tabor [FUL75] between a rubber surface and a
polymethylmethacrylate (PMMA) surface with RMS roughness in the range of 120 
1500 nm. They reported that a 1,000 nm increase in roughness reduced the adhesion force
to approximately 10% of that predicted by van der Waals force between smooth surfaces.
This study also indicated that adhesion force on rough surfaces was primarily determined
by the distribution of asperity heights.
lida et al. [IID93] experimentally determined the adhesion forces between smooth
glass particles and glass substrate with roughness of 50 to 400 nm. The adhesion force
was estimated by the average force needed to retain 50% of the particles after
bombardment by gas or liquid. Similar to other investigations, they initially observed a
factor of ten decrease in adhesion force followed by a region of more gradual decrease.
Recently, Shaefer et al. [SCH95], using AFM, have measured adhesion between
8 jpm particles with roughness of 10 20 nm. In addition to deviation from the average
center line, roughness was characterized by the estimated mean radius of curvature
profile obtained by AFM. Experimental values of adhesion force were found to be fifty
times less than the theoretical values for smooth glass particles based on JohnsonKendal
Roberts, JKR, theory [JOH71]. When the estimated radius of the contacting asperity was
used to calculate adhesion rather than the radius of spheres, the experimental data was
only three times lower than predicted for smooth particles.
To understand the effect of surface roughness in the nanometer scale, the change
in adhesion force must be accurately measured for very small deviations from ideally
smooth surfaces. However, to the best of our knowledge, there has been no detailed
investigation of nanoscale surface roughness effects on the adhesion force. As described
previously, experimental studies have been performed for surface roughness values
greater than ten nanometers [MIZ96, FUL75, IID93, SCH95]. However, both theoretical
and experimental difficulties have resulted in a lack of experimental data in the
nanometer surface roughness regime. Experimentally, it is difficult to obtain surfaces
having regular and reproducible roughness of this scale and theoretical complications
arise from an inability to characterize or properly define geometric models based on
experimentally measurable parameters. Due to a lack of experimental data, models
developed to predict adhesion force as a function of surface roughness have not been
accurately verified.
Experimental
Methods
Adhesion force was measured experimentally by atomic force microscopy (AFM)
(Nanoscope III, Digital Instruments Inc.). Details of the technique have been described
elsewhere [DUC92, RAB94]. Adhesion force measurements were made with either a
glass sphere or an AFM silicon nitride tip attached to the cantilever. The experimental
configuration for the two different measurements is shown in Figure 41. Since the radius
of the glass sphere is much larger than the scale of the roughness, it is expected that
contributions from both the contact and noncontact forces could be significant. In the
case of the AFM tip, the radius of curvature is smaller than the scale of the surface
roughness. As a result, the interaction is expected to be dominated by the contact forces
and adhesion values are expected to be much closer to those predicted for the interaction
of smooth surfaces as described in Chapter 3. Comparing these two geometries is
intended to highlight the importance of the relative magnitude of the roughness compared
to the size of the adhering particle.
Systematic errors, appearing due to the cantilever calibration, were found to
produce variation in experimentally measured forces of approximately 10%. Small local
variations of surface topography may also produce significant changes in adhesion force.
As a result, the statistical scattering of adhesion force for a given data point was also
"[ ".1 I 1 I'/ .lL lr. .' L, i
Figure 41: Experimental Configuration. Schematic of the experimental configuration for
spheresurface and tipsurface interactions. The radius of the AFM cantilever
tip is small compared to the surface roughness, whereas the glass sphere is
significantly larger.
approximately 10%. Presented data points are the average of at least 50 measurements at
different locations on the surface.
Materials
Plates with controlled roughness were fabricated in the present study by
deposition of titanium thin films (10 100 nm thick) on a silicon wafer substrate.
Compositional homogeneity of the deposited films was verified by Auger spectroscopy.
"I I I I L. I',' / I I : I I L' I I I L" 1', 1 L ': i ; I I "..
In this system, the roughness of the surface is proportional to the thickness of the
deposited film. In this manner, the RMS roughness was varied from 1.6 to 10.5 nm. The
silicon wafer had an RMS roughness of 0.17 nm. Directly before each measurement
surfaces were cleaned by rinsing with ethanol, methanol, and deionized water.
AFM surface roughness profiles for the plates used in this investigation are shown
in Figure 42(a) through 42(d). Note that sample A is the silicon wafer without a coating
of titanium. Upon initial inspection, the asperities seem approximately semispherical
with their origins at the surface. However, since the vertical scale is greatly exaggerated
the asperities should more appropriately be modeled, in contrast to previously proposed
models of asperities, as spheres with the origin below the surface instead of semispheres.
This allows the asperity to exhibit both the measured height and breadth, whereas, if the
origin was restricted to the surface only, the height parameter is utilized. An important
difference in this approach is that with the origin of the asperity located beneath the
surface, the radius needed to produce a given height necessarily becomes larger.
The surfaces used in this investigation exhibited two types of roughness profiles.
The first roughness to be defined, RMSI, is associated with the longer peaktopeak
distance, )i, fluctuations (approximately 1000 nm) that occur on samples C and D
(Figure 42(c) and 42(d)). The second, RMS2, occurs on all samples and has a peakto
peak distance, k2 of approximately 250 nm. Note that on samples C and D, RMS2 is
superimposed on RMS1. The experimentally measured RMS and k of the four samples
are given in Table 41. Roughness in terms of RMS was determined by the analysis
software of the AFM and the peaktopeak distance taken in different directions from the
images. Methods to calculate the radius of an asperity are detailed in Chapter 3.
" A
arr4
EMn
1iM
J.Ifi
" "
M .35
0.2s
i. is
4k*
1.5
1.HI
I.0o
I.25
Figure 42: Surface Profiles. Surface roughness profiles, as determined by atomic force
microscopy, of titanium deposited on the silicon substrates in order of
increasing thickness and RMS roughness. Characterization parameters are
summarized in Table 41.
\,
1*.'a.
sai
p~
m .2a
El !"'
El
El
Glass spheres were obtained from Duke Scientific Inc. and cantilevers from
Digital Instruments Inc. The spheres possess a radius of approximately 10 jPm and an
RMS roughness of less than 0.2 nm. The measured radius of the AFM tip was
approximately 50 nm.
Results
The force of adhesion, normalized by the radius of the adhering particle, between
either the glass sphere, or AFM tip, and the samples of controlled roughness is presented
in Table 42. For both the particle and tip a relatively sharp decrease in the normalized
force of adhesion is detected upon a surface roughness increase of 0.17 to 1.6 nm RMS.
For the 10 jpm particle, the normalized force of adhesion decreases to 27% of the
adhesion at 0.17 nm RMS. However, upon further increase in roughness to 10.5 nm RMS
the adhesion is still 19% of the adhesion of sample A. Similar trends are observed when
Table 41: Characteristics of the Model Surfaces. Measured roughness parameters, RMS
and ), for the model surfaces. Values of radii of peaks rl and r2 are calculated
from Eq. (416) in Chapter 3. Note that sample A is the uncoated silicon
wafer.
RMSi RMS2 1i 2 rl r2
Sample
(nm) (nm) (nm) (nm) (jPm) (Pjm)
A 0 0.17 250 o, 6.3
B 0 1.64 249 o, 0.75
C 4.3 1.64 1180 237 5.6 0.69
D 10.5 1.64 1046 260 1.8 0.81
the particle size is less than the peaktopeak distance of the roughness. For the
interaction of the AFM tip and sample B the normalized force of adhesion is 83% of the
interaction between the tip and sample A. However, even for a 7 fold increase in surface
roughness to 10.5 nm (sample D) RMS, the adhesion is still nearly 60% of the interaction
between that of the tip and sample A.
The absolute adhesion magnitude between the two samples is also significantly
different, indicating that when normalized by the radius of the adhering particle the
smaller particle (AFM tip) has relatively more intimate contact with the surface. This data
indicates a rather large decrease in the normalized adhesion force with small deviations
from an ideally smooth surface and that this decrease becomes less significant with
further increase in surface roughness. In the following sections these experimental data
will be compared to both the modified Rumpf model and the model developed in this
study (Chapter 3).
Table 42: Measured Force of Adhesion. Measured normalized force of adhesion between
the sphere or tip and the model surfaces normalized by the radius of the
adhering particle.
Sample RMS1 RMS2 Normalized Force of Adhesion (mN/m)
(nm) (nm) Glass Sphere/Plate AFM Tip/Plate
A 0 0.17 100 600
B 0 1.64 27 500
C 4.3 1.64 23 370
D 10.5 1.64 19 350
Comparison with the Modified Rumpf Model
As described in Chapter 3, the model originally proposed by Rumpf [RUM90]
modified to predict adhesion as a function of RMS roughness instead of asperity radius is
given as Eq. (41).
Fad =R + +/ 2 (41)
6H2 1++R/(1.48 RMS) (1+1.8RMS/H )
Adhesion forces calculated using Eq. (41) will be called the modified Rumpf
approach in the remainder of the study. As noted above, samples C and D exhibit more
than one peaktopeak distance of roughness. Since AFM averages the smaller
superimposed roughness to zero this secondary roughness is not accounted for. Therefore
to apply the modified Rumpf model for the surfaces used in this investigation the total
roughness of the surfaces, as calculated by Eq. (42), will be used.
RMS= RMSz = RMS + RMS2 (42)
where RMS1 and RMS2 are average root mean square roughness of the long and short
peaktopeak distances, respectively. Eq. (42) is applicable ifRMS2 is much less than or
much greater than RMS1.
Hamaker constants for the various interactions are needed to apply Eq. (41) and
the proposed model developed below. The Hamaker constant, Anl, is approximately
0.65x1019 J for amorphous silica [ISR92] and 1.72x1019 for silicon nitride [MEU97].
All for silicon was calculated, using the optical data of Visser [VIS72], and determined to
be 2.65x1019 J [FIE96]. The Hamaker constant for metals are expected to be 2 to 4x1019
[ISR92], so for simplicity AI for titanium is considered to be equal to that of silicon.
To estimate the Hamaker constant between dissimilar materials a combining rule
approximation [ISR92]
A12 = JAl A22
(43)
results in A12 = 1.31 1019 J for silica/ silicon and silica/titanium, and A12 = 2.13x1019 J
for silicon nitride/silicon and silicon nitride/titanium.
Utilizing this approach, the theoretical and experimentally measured normalized
adhesion force between the smooth sphere and substrates as a function of the total root
mean square roughness, RMSx, as plotted in Figure 43. Theoretical values were
calculated by the modified Rumpf model, Eq. (41), for a minimum contact distance, Ho,
of 0.3 nm, and a particle diameter, R, of either 10 jam for the glass sphere or 50 nm for
the AFM tip. For both the large and small particles, it is clear that the adhesion forces
calculated using this model underestimate the experimental values by more than an order
of magnitude. This discrepancy is primarily attributed to the underestimation of the
radius of the asperity in Rumpf s model as described in Chapter 3.
The increase in adhesion force predicted for the AFM tip/surface interaction,
especially at roughness greater than approximately 1 nm RMS is also to be noted. This
trend is not mirrored in the experimental data suggesting that the secondary roughness
may actually play a more significant role in controlling the adhesion force. Even though
the radius of the asperity begins to increase to where the adhering particle/asperity forces
theoretically dominate, the secondary roughness reduces the real area of contact, in effect
once again reducing the roughness to the nanoscale. The phenomena of most surfaces
with apparently larger scale roughness comprised of multiple roughness scales may also
help to explain some of the experimentally measured differences described earlier.
Comparison with the Proposed Model (van der Waals Approach)
The roughness of the model surfaces is inadequately described by the geometry
proposed in the Rumpf model. A new model, described in Chapter 3, that more
74
i I I i
1 1  I12
,III I
\ 1". : II ." l., I'ii'i i )
Figure 43: Comparison of Measured Adhesion with Modified Rumpf Model.
Experimentally measured normalized adhesion force between the glass sphere
or AFM tip and the model substrates as a function of increasing nanoscale
roughness, RMSz. The theoretical adhesion calculated with the modified
Rumpf model (Eq. [1]) with A12 =1.31 1019 J for silica/silicon and
silica/titanium, A12 = 2.13x1019 J for silicon nitride/silicon and silicon
nitride/titanium, Ho = 0.3 nm, and Rsphere = 10 jpm or Rtip = 50 nm
underestimates the measured adhesion force by more than an order of
magnitude.
accurately describes the surface roughness must be employed. However, the model must
be additionally modified to account for the second order of roughness on samples C and
D. The geometry of this model is shown schematically in Figure 44. In this model
roughness is described by asperities with heights yl max and y2 max as well as peaktopeak
distances 1i and k2.
To calculate the adhesion force in the framework of this model, Eqs. (314), (3
16), and (318), are considered valid for each superimposed roughness scale. As a result,
similar to derivation of Eq. (319), but also accounting for the contact interaction of
sphere with RMS2 and noncontact interaction with the RMS1 and flat substrate, the
following formulas are obtained.
AR r2 ri 1
Fad = + 1 (44)
1+
1+ 58R RMS ( 58R.RMS, 1 82RMS2 2
Fad = 2 (45)
6HO
(1+1.82(RMS + RMS2))2
In Eqs. (44) and (45), the first, second and third terms in brackets correspond to the
interaction of the adhering particle with RMS2, RMS1, and the average surface plane,
respectively. These equations are valid if the radius of the asperity, ri or r2 and the peak
topeak distance, )i or k2, associated with both superimposed roughness are much less
than the radius of the adhering particle. As )1 becomes comparable to R, there is less and
less material in the average surface plane that is not already contained in the asperity.
Hence, for such a situation, the third term in Eqs. (44) and (45) is largely redundant and
must be dropped, yielding the following.
..
\7
Figure 44: Asperity Schematic for Surfaces with Two Roughness Scales. Schematic
illustration of the geometric model used to calculate adhesion force in the
proposed model. The surface is proposed to consist of an array of spherical
asperities and troughs with the origin of the asperities positioned below the
average surface. In this model two superimposed roughness profiles are
needed to model surfaces C and D.
AR 1 1
F = 1 I + (46)
ad 6H2 58R*RMS2
0 6 1+ 5858R.RMS, 1.82RMS2
2 1 Ho
Eq. (46) is applicable at any value of RMS1. In the limit of small values of
RMSI, when RMS1 << k2/58R, Eq. (46) reduces to Eq. (320).
Because the rougher surfaces in this study exhibited multiple roughness
components, theoretical and experimental data are plotted as a function of the two
primary variables, RMS1 and RMS2 at )1 of 1110 nm and k2 of 250 nm, taken as an
average of the values measured on the model surfaces. The Hamaker constant, minimum
contact separation distance, and radius of the sphere were kept identical to those used in
the modified Rumpf model prediction. Figure 45 compares the normalized adhesion
force measured between the glass sphere and model surfaces with theoretically predicted
values using Eq. (46). Comparison of experimental and theoretical data suggests that a
principle contribution to the adhesion force is van der Waals attraction. Note, however,
that a discrepancy still exists between the theoretical and measured values. While these
predictions represent a significant improvement over the modified Rumpf model (Fig.
4.3), the experimentally measured values of adhesion force at RMS2 = 1.64 nm are
consistently larger by almost a factor of two than the theoretically predicted values.
Experimental error alone ( 10%) can not account for this difference. One
remaining factor not taken into account in the proposed model is polar interactions
[ISR92]. Polar forces are known to be significant in contact interactions and, therefore,
may be especially important in the proposed model at low RMS roughness values. To
account for these additional forces a surface energy approach is necessary for the contact
interactions. This may be important because polar forces, primarily hydrogen bonding,
that operate between contacting surfaces may be equal to or greater in magnitude than the
van der Waals component of surface energy [ISR92]. For example, for glass surfaces the
van der Waals component of the work of adhesion at contact is approximately 19 mJ/m2,
while experimentally measured values range from 50 to 80 mJ/m2 depending on the
degree of hydroxylation [BRA32, YAM75, YAM98]. The utility of such an approach
will be explored in the next section.
Contact electrification has also been reported to enhance the adhesion between
dry surfaces [HOR92]. This electrostatic adhesion, related to transfer of charge during
1000 .. ..
1000
2 1
RMS
Figure 45: Experimental and Theoretical Adhesion for Sphere/Surface Interaction.
Experimentally measured normalized adhesion and theoretical predictions for
the interaction of the glass sphere with the model surfaces as a function of
RMS1 and RMS2. Theoretical predictions were made using Eq. (46) with A12
= 1.31 1019 J, Ho = 0.3 nm, R = 10 m, i = 250 nm, and k2 = 1110 nm.
separation of surfaces, may produce adhesion forces two orders of magnitude greater than
the ideal van der Waals forces [HOR92] but was not observed during the present
investigation. Possible reasons include a thin oxidation layer, observed by Auger
spectroscopy, that may have increased the work function of the titanium surfaces such
that charge transfer was not possible or that the measurements were conducted in air at
approximately 25% relatively humidity, as opposed to the experiments of Horn and
Smith [HOR92] where a dry nitrogen atmosphere was used. It is possible that the water
adsorbed under these conditions was enough to dissipate any charge that accumulated on
the surfaces.
Experimental results and theoretical predictions, calculated by Eq. (46), for the
interaction of the AFM tip with the model surfaces are shown in Figure 46. The values
of the variables used are identical to those in Figure 45 with the exception of the smaller
radius of the adhering particle (R = 50 nm) and the different Hamaker constant (A2 =
2.13x1019 J). The same AFM cantilever was used in all experiments to eliminate
variability in spring constant and tip dimensions. Furthermore, the tip required a "break
in" period before reproducible measurements were achieved. This period is assumed to
correspond to the wearing of the tip to a stable radius of curvature.
The radius of tip is smaller than either of the surface roughness peaktopeak
distances. Hence, the tip penetrates between asperities and contact forces dominate. As a
result, the decrease of adhesion force upon increase of the surface roughness from RMS =
0.2 nm to 1.6 nm is only 1.5 times, whereas for the glass particle the same conditions
produced a fivefold decrease in the force of adhesion. Once again the predicted
interaction using Eq. (46) results in an underestimated value of adhesion force as
compared with experimental data. This may be attributed to contributions from polar
forces that are larger in magnitude in the present case because of the greater normalized
surface area in contact. To more accurately evaluate the magnitude of these additional
adhesion forces, the contact force between the adhering particle and an asperity may
alternatively be modeled using the surface energy approach with the proposed description
of the asperity radius.
,, : :
Figure 46: Experimental and Theoretical Adhesion for AFM Tip/Surface Interaction.
Experimentally measured normalized adhesion and theoretical predictions for
the interaction of the AFM cantilever tip with the model surfaces as a
function of RMS1 and RMS2. Theoretical predictions were made using Eq. (4
1110 nm.
Comparison with the Proposed Model (Surface Energy Approach)
It should be noted that all equations for calculation of adhesion force based on
Rumpf s model described above are developed for nondeformable surfaces. To
incorporate this effect and account for the total surface energy, two limiting cases of
adhesion as described in Chapter 2, both of which imply elastically deformable surfaces,
are often applied. DerjaguinMullerToporov (DMT) mechanics [DER75] is often applied
for harder materials where at the point of detachment there is no deformation. Similarly,
JohnsonKendallRoberts (JKR) mechanics [JOH71] is often determined to be the
limiting case for softer materials. Johnson and Greenwood [JOH97] suggested a
parameter, [L, which if greater than 0.01 indicated an interaction more closely
approximated by JKR mechanics as opposed to the DMT model. This parameter may be
written as
=rW2 1/3
P = 23 >0.01 (47)
where W is the work of adhesion between the two interacting surfaces, r is the radius of
the asperity, Ho is the minimum separation distance between contacting surfaces (0.3
nm), and K is the reduced combined elastic modulus [BUR22].
K = 1 2 (48)
3 E, E2
El, E2, vi, and v2 are the elastic moduli and Poisson ratios for materials 1 and 2,
respectively. For asperity or tip radii used in this investigation and W = 91.5 mJ/m2
(silicon nitride/tungsten interaction) the minimum value of P is approximately 0.08,
indicating that the JKR model may be used to evaluate the contact interaction between
the adhering particles and asperities. Using this approach Eq. (46), may be rewritten by
replacing the first term in this equation by adhesion force between two spheres as
determined by JKR theory.
AR
Fad +rWRr 2 (49)
2(r + R) ( 58R RMS, 1.82RMS2
I Ho
As a final criteria to evaluate the applicability of the proposed model, the size of
any plastically deformed zone should be small compared to the size of the asperity. If the
two are of similar magnitudes the asperity will essentially be flattened by the adhering
particle and the effect of surface roughness will be negligible. The model of Pollock and
Maugis [MAU84] assumes that all of the applied force produces plastic deformation of
the surface and so yields the upper limit of the plastically deformed zone. Their equation
has the form
3 Rr
Fad = 3EW = 3ra2plY (410)
2 (R + r)
in which the applied force is set equal to the force of adhesion, Fad, determined by JKR
mechanics with the work of adhesion, W, determined from the fitted values described
below. R is the radius of the adhering particle (either the glass sphere or AFM tip), r is
the radius of the asperity, api is the radius of plastically deformed zone, and Y is the yield
stress of the plastically deformed material (derived from the assumption that hardness is
approximately 3Y). Using this approach, the largest calculated plastic area radius
between glass particle with Y = 3.6 GPa [ASH91], and titanium asperity with Y = 0.18
GPa [ASH91] and W = 69.5 mJ/m2 is approximately 16 nm. Interactions between the
AFM tip with Y = 8.0 GPa [ASH91], and asperities yielded values of approximately 5
nm. This radius overestimates the radius of the plastically deformed zone and is still
much smaller than the radius of the smallest asperity (0.7 jam) or peaktopeak distance
(250 nm). Hence it is appropriate to assume that the asperities are not significantly
plastically flattened by the adhering particles. This example also reinforces the
importance of considering the correct radius of the surface asperities in other types of
surface interactions.
Tables 43 and 44 compare the experimentally measured adhesion between the
glass sphere or AFM tip and the model surfaces to those predicted by Eq. (49). The
principle drawback of the surface energy approach, however, is that while good
estimations of van der Waals attraction for a wide variety of surfaces may be made, no
such methodologies exist for the work of adhesion. To overcome this barrier a best fit
approach was utilized for each pair of interacting surfaces. The results of this fit and the
van der Waals component are also given in Table 43 and 44. It appears that, the van der
Waals component represents approximately 50 70% of the total fitted interaction
energy.
It should be emphasized that the majority of the refinements to the model only
attempt to further increase the accuracy beyond the +50% of the measured value offered
by the model developed in Chapter 3, compared to the nearly 10 to 50 times
Table 43: Normalized Force of Adhesion: Glass Sphere (Surface Energy Approach).
Predicted normalized force of adhesion between the glass sphere and model
surfaces calculated by Eq. (49) and compared with measurement. The fitted
total work of adhesion and the van der Waals contribution to the overall work
of adhesion are also shown.
Work of Adhesion: Fitted Work of Normalized Force
Sample Calculated by Adhesion: Total of Adhesion (mN/m)
van der Waals (mJ/m2) (mJ/m2) Experimental Predicted
A 39 55.0 100 100
B 39 69.5 27 23
C 39 69.5 23 22
D 39 69.5 19 25
underestimation resulting from other models which incorporate asperities with radii in the
nanometer size range. This underestimation of the adhesion occurs not only with the van
der Waals based Rumpf approach as detailed above, but also with more complex models
considering surface energy and elastic deformation such as JKR theory. In fact, exact
prediction remains difficult, even with these more rigorous analyses, due to the increased
number of unknown parameters needed to apply the models. Hence, the principle finding
of the present investigation is that regardless of the exact model used (Rumpf, DMT,
JKR) the predicted adhesion values is primarily dependent on deriving a realistic value of
the radius of the asperities on the surface.
Table 44: Normalized Force of Adhesion: AFM Tip (Surface Energy Approach).
Predicted normalized force of adhesion between the AFM tip and model
surfaces calculated by Eq. (49) and compared with measurement. The fitted
total work of adhesion and the van der Waals contribution to the overall work
of adhesion are also shown.
Work of Adhesion: Fitted Work of Normalized Force
Sample Calculated by Adhesion: Total of Adhesion (mN/m)
van der Waals (mJ/m2) (mJ/m2) Experimental Predicted
A 63 128 600 598
B 63 91.5 500 404
C 63 91.5 370 405
D 63 91.5 350 409
Summary
In this chapter, the force of adhesion between surfaces with nanoscale roughness
was measured. The results validated previous investigations, at larger roughness, where a
decrease of adhesion force as a function of roughness was reported [IID93, SCH95].
Measurement of adhesion was also performed at lower roughness than previously
reported. It was experimentally determined that a roughness value as small as 1.6 nm
RMS was significant enough to decrease the force of adhesion by nearly a factor of five.
The magnitude of this decrease was additionally found to have a strong dependence on
the radius of the sphere contacting the surface. When the adhesion force was probed with
a sphere whose radius was of the same order or smaller than the radius or peaktopeak
distance of the roughness, the decrease of adhesion upon increase of surface roughness to
1.6 nm RMS was only 1.5 times, thus, more closely approximating the contact of an
ideally smooth sphere and flat surface.
As described in Chapter 3 and 4, the key to predicting the interactions is to model
the roughness in a manner that more closely describes the true geometry of the surfaces.
In the proposed model, through an analysis of the surface topography by atomic force
microscopy, a mathematical description of both the height and breadth of the asperities
was determined, which are expressed as RMS roughness and peaktopeak distance,
respectively. The radius of the surface asperities as determined by this method was much
larger than considered in previous models. As a result, the experimental force of adhesion
was predicted within 50% of experimental values using a van der Waals force based
approach. Previous models underestimated adhesion by 10 to 50 times. Application of the
proposed geometries with JKR mechanics was found to further reduce the error, albeit
with fitted (yet realistic) values of the work of adhesion. Using a realistic value of the
radius of asperities on the nanoscale was determined to be the most important parameter
in accurately predicting adhesion between surface with nanoscale roughness.
In the previous chapters, adhesion models have been derived for a variety of
surface geometries. However, only van der Waals and polar forces have so far been
considered. It may be expected that roughness on the nanoscale may also influence other
mechanisms of adhesion. Chapter 5, will explore the effect of the nonideal interface
(surface roughness) in humid atmospheres where capillary condensation may
dramatically increase observed adhesion forces.
CHAPTER 5
CAPILLARY FORCES BETWEEN NANOSCALE ROUGH SURFACES
Introduction
The flow and adhesion behavior of fine powders (approximately less than 10 jam)
is significantly affected by the magnitude of attractive interparticle forces. Hence, the
relative humidity at which capillaries form and the magnitude of capillary forces are
critical parameters in the processing of these materials. Using the general models of non
ideal surfaces developed in previous chapters, approximate theoretical formulae will be
developed to predict the magnitude and onset of capillary adhesion between a smooth
adhering particle and a surface with roughness on the nanometer scale. The theoretical
predictions are validated using experimental adhesion values between a variety of
surfaces.
In industrial operations ranging from the transport and handling of fine powders
to the removal of abrasive particles after a polishing operation, the magnitude of forces
between two surfaces is critical. As described in previous chapters, there are a number of
approaches to predict these interactions based on van der Waals adhesion [HAM37] or
through surface energy based approaches such as JohnsonKendallRoberts (JKR)
[JOH71] or DerjaguinMullerToporov (DMT) [DER75] adhesion theory. With real
surfaces, however, there are a number of difficulties associated with applying these
models. One principal concern is the effect nanoscale roughness has on the real area of
contact and, subsequently, the magnitude of the force of adhesion. In Chapter 4, it has
