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Adhesion forces measured between
particles and substrates with
nano-roughness
J. Drelich
Associate professor, Department of Materials Science and Engineering,
Michigan Technological University, Houghton, Michigan
Abstract
This report briefly reviews the measurements and interpretation of adhesion forces between microscopic probes
and substrates having micro- and nano-scaled roughness characteristics. The measurements were recently conducted at Michigan Technological University using atomic force microscopy. The laboratory study concentrated
on both rigid and deformable materials, with the probes larger than the size of roughness features of interacting
surfaces. For such systems, a detailed surface characterization, in terms of roughness parameters, asperity size
and shape and spacing between asperities, as well as analysis of deformation of materials, is required. Analysis of
the experimental data can be done using theoretical models suitable for rough surfaces, which would allow one
to normalize the measured forces. Alternatively, loading of the probe during adhesion force measurements can
be managed properly to control deformations of the nano-scaled surface irregularities. Controlling the deformation reduces and even eliminates the effects of nano-roughness on the probe-substrate contact area, promoting
conditions described by the contact mechanics models for the sphere-smooth substrate system.
Key words: Adhesion forces, Nano-scale roughness
Introduction
Adhesion forces play an important role in physical processing and separation of mineral particles. In recent
years they have been measured at a submicrometer scale
using atomic force microscope (AFM). Many parameters
affect an AFM measurement, and surface nano-roughness,
common feature of mineral substrates used in the AFM
experimentation, is one of the most important.
Roughness of particles and/or substrates cannot be
eliminated for real-world materials, and this property
must be taken into consideration in the interpretation of
the measured AFM forces. Asperities of rough surfaces
can either reduce or increase the contact area of interacting
surfaces, and often various contact areas are encountered
between successive measurements for the same system
(Tormoen et al., 2004; Drelich and Mittal, 2005). As a
result, the experimental data presented in the literature
indicate inconsistency and poor reproducibility of adhesion forces measured with the AFM technique (Drelich
et al., 2004a, 2004b; Drelich and Mittal, 2005). For AFM
force measurements to become an accepted technique for
particle-substrate adhesion characterization, problems associated with the effects of surface roughness during force
measurement must be understood and resolved.
The authors recently studied adhesion forces between
microscopic probes and substrates having nano- to microscaled roughness characteristics. This review provides
general guidelines on the measurements and interpretation of the adhesion forces involving particles and
substrates with nano-rough surfaces. This current study
concentrates on both rigid (inorganic) and deformable
(polymeric) materials. The probes used are larger than
the size of roughness features of the interacting surfaces,
and controlled loadings are applied during the adhesion
force measurements. To succeed with interpretation of
such systems, a detailed surface analysis, in terms of
roughness parameters, asperity size and shape and spacing
between asperities as well as analysis of deformation of
materials, is required. Then, analysis of the experimental
data involves theoretical models that are suitable for a
particular system.
In pull-off force measurements, loads are managed
properly to avoid plastic deformation and damaging of
the probe. However, the viscoelastic and plastic deformations of the nano-scaled surface irregularities of compliant
materials are sometimes managed by increased loads to
reduce or eliminate the effects of nano-roughness on the
probe-substrate contact area.
Preprint number 05-059, presented at the SME Annual Meeting, Feb. 28-March 2, 2005, Salt Lake City, Utah. Revised
manuscript received and accepted for publication February 2006. Discussion of this peer-reviewed and approved paper
is invited and must be submitted to SME Publications Dept. prior to May 31, 2007. Copyright 2006, Society for Mining,
Metallurgy, and Exploration, Inc.
November 2006 • Vol. 23 No. 4
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Measurements of pull-off forces using atomic
force microscopy
The principles of the AFM are well described in the literature
(Cappella and Dietler, 1999; Drelich and Mittal, 2005). Briefly,
the AFM measures the deflection of a cantilever spring with a
sharp cantilever tip (tip radius is usually 10 to 100 nm) or an
attached particle (a particle with a diameter ranging from about
2 µm to more than 20 µm is glued to the end of the cantilever)
as a function of displacement from a horizontal position. The
deflection of cantilever with attached probe, monitored by a
laser-photodiode system, relates to the forces acting between
a probing tip and a substrate. Figure 1 shows a schematic of
the cantilever deflection vs. tip-substrate distance curves and
a colloidal probe.
As shown in Fig. 1, the AFM measurement starts at a large
tip-surface separation in the so-called nontouching regime.
Then, with or without a slight deflection depending on the
long-distance tip-surface interactions, the surface approaches
the tip following the horizontal line moving right to left in
the force vs. distance curve. At a certain point, when the tipsubstrate attractive interactions overcome the stiffness of the
cantilever, the transition from nontouching to touching occurs,
and the tip “jumps” onto the sample surface.
Moving the substrate still further causes deflection of the
cantilever equal to the distance that the substrate is moved
(maximum deflection of cantilever in this region relates to
maximum loading that probe experiences in contact with the
substrate). This is referred to as the touching regime or constant
compliance region represented by the diagonal line in the left
part of the force vs. distance curve. For rigid particles in contact
with rigid substrate the curve representing compliance region
has shape of the straight line. However, this line deviates from
a constant slope if deformation of particle or substrate or both
take place, and the degree of material deformation dictates the
shape of the curve (Biggs and Spinks, 1998; Beach and Drelich,
2003; Rutland et al., 2004).
On retracting the surface from the tip, i.e., going towards
the right in the force vs. distance curve, the cantilever again
moves with the surface. The cantilever deflects towards the
surface due to adhesion force before the tip breaks contact
with the surface, going through the lowest point in the force
vs. distance curve. At this “jump-off” point, the tip completely
loses contact with the surface, and the cycle is complete. The
force required to pull the tip off the substrate surface is called
pull-off or adhesion force (F) and is calculated from Hooke’s
law as follows
F = k ∆x (1)
where
k is the spring constant of cantilever and
∆x is the maximum deflection of cantilever during tipsubstrate adhesion.
The accuracy of the F measurement depends on the precision
of determination of both the spring constant of cantilever and
its deflection during pull-off force measurements. The laser
beam-photodiode detector systems of the AFM instruments
are usually capable of recording deflection of the cantilever
to sub-Angstrom precision and the major error in pull-off
force determination is associated with the k value determination. This error, however, is constant and cannot cause the
scatter of F values always reported in the literature, even for
the highest quality and most carefully prepared probes and
substrates. Standard deviation in k value determination at 10%
to 20% of an average value is commonly reported. Errors in
MINERALS & METALLURGICAL PROCESSING 227
Figure 1 — Schematic of a cantilever deflection vs. vertical
position of cantilever curve in the AFM measurement of
surface forces. Photograph is an SEM micrograph of glass
probe glued to a V-frame cantilever.
piezo responses resulting from piezo hysteresis and creep, as
well as fluctuation of light intensity and its interference, may
cause variations in the F value determination. Nevertheless,
the major causes for the scatter of F values are imperfections
in solid surfaces of both substrates and probes and uncontrolled
deformations of the materials posed by uncontrolled and varied
loading conditions.
Characterization of the substrate and the probe size and its
shape, including determination of surface defects and nanoroughness at the tip apex, are important tasks for analysis of the
results from AFM pull-off force measurements. The surfaces
of both probe and substrate can be imaged using field-emission scanning electron microscopy. Such images, however,
very often cannot provide details on the nano-scaled surface
imperfections. Imaging of surfaces with the AFM instrument
at high resolutions is more convenient, although due to small
dimension and spherical shape of the probe, scanning the surface of the probe is more challenging task than scanning the
substrate. Colloidal probes can be imaged either directly with
another (much sharper) AFM cantilever or using the reverse
imaging technique (called blind tip reconstruction technique in
the author’s previous papers) (Drelich et al. 2004a, 2004b).
Figure 2 shows an example of AFM image and corresponding cross section of this image produced by scanning the colloidal glass probe over the array of sharp spikes (TGT grating
from NT-MDT Co., Moscow, Russia). Nano-roughness of the
colloidal probe surface is obvious, particularly from the crosssectional profile. Small asperities such as those located at the
probe apex (Fig. 2) can have a profound effect on the contact
area between the probe and substrate during the pull-off force
measurements. The shape, size and spacing between asperities are important characteristics of rough surfaces and their
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Figure 2 — (a) Three-dimensional AFM image of the ~3 x 3-µm area of the 10-µm colloidal probe (borosilicate glass)
obtained by reverse imaging method (left image) during scanning over the array of sharp spikes separated by 3 µm, and
(b) cross section of this AFM image (from Drelich et al., 2004a).
determination is often crucial for a proper interpretation of the
measured pull-off forces, as will be shown later.
(2)
where
R is the radius of the particle (probing tip); and
c is a constant (where c = 2 in the DMT model and c = 1.5
in the JKR model).
Contact mechanics models used in analysis of
adhesion forces
In AFM pull-off force measurements, continuum contact
mechanics models are commonly used to describe the probesubstrate system with a high degree of ideality. Two contact
mechanics models derived by Johnson et al. (1971) and
Derjaguin et al. (1975), called the JKR and DMT models,
respectively, are frequently used by researchers to interpret the
pull-off forces measured by the AFM technique. In general, the
JKR and the DMT models apply to particle-substrate systems
where the following assumptions are met:
Thus, by knowing which contacts mechanics model applies
to a particular system under study and setting operation conditions during the AFM measurements that satisfy that particular
model, the WA value can be determined. Note that both models
(JKR and DMT) were derived based on the Hertz theory (Hertz,
1896) and that an analytical solution to the DMT model was
provided by Maugis (1992).
Which of the contact mechanics models should be selected
for interpretation of the AFM pull-off forces is not always
straightforward. In general, the DMT model is more appropriate
for systems with hard materials having low surface energy and
small radii of probe curvature. The JKR model applies better to
softer materials with higher surface energy and larger probes.
This generalization, however, does not bring the researcher any
closer to the selection of the appropriate model and mistakes
are often made (Drelich et al., 2004a, 2004b).
Maugis analyzed both the JKR and DMT models and
suggested that the transition between these models can be
predicted from the dimensionless parameter a defined as
(Maugis, 1992)
• deformations of materials are purely elastic, described
by classical continuum elasticity theory;
• materials are elastically isotropic;
• both Young’s modulus and Poisson’s ratio of materials
remain constant during deformation;
• the contact diameter between particle and substrate is
small compared to the diameter of particle;
• a paraboloid describes the curvature of the particle in
the particle-substrate contact area;
• no chemical bonds are formed during adhesion; and
• contact area significantly exceeds molecular/atomic
dimensions.
The difference between JKR and DMT models occurs in
assuming the nature of forces acting between particle and
substrate. Johnson et al. (1971) assumed in their model that attractive forces act only inside the particle-substrate contact area,
whereas Derjaguin et al. (1975) included long-range surface
forces operating outside the particle-substrate contact area.
The JKR and DMT models describe the correlation between
pull-off force (F) and work of adhesion (WA) through a simple
analytical equation of the following form
November 2006 • Vol. 23 No. 4
F = cπ RWA a=
2.06
z0
3
RWA 2
πK2
(3)
where
z0 is the equilibrium separation distance between the probe
and substrate (zo ≅ 0.2 nm for many systems (Yu and
Polycarpou, 2004),
R is the radius of the probe,
WA is the work of adhesion and
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K is the reduced elastic modulus for the particle-substrate
system:
1 3 1− νp
1 − νs2 
= 
+
K 4  Ep
Es 
2
(4)
where
ν is the Poisson ratio,
E is the Young’s modulus and
p and s stand for probe (particle) and substrate, respectively.
For α → ∞ (α ≥ 5) the JKR model applies, whereas the DMT
model is more appropriate for systems with α → 0 (α ≤ 0.1).
The transition between these two models is described by the
Maugis-Dugdale (MD) theory (Maugis, 1992). Often, for AFM
pull-off force measurements, the MD model seems to be more
appropriate than either of the JKR or DMT models (Drelich et
al., 2004a, 2004b).
In the MD model, two parametric equations must be solved to
describe the transition region between JKR and DMT mechanics (see Maugis, 1992, and Carpick et al., 1999, for details).
A problem with this model is that these two equations must
be solved simultaneously, and, as a result, there is no simple
analytical expression for pull-off force, and iteration is needed
to calculate the c value. In 1999 Carpick et al. (1999) came up
with an approximate (and simple) solution that relates c value
with the Maugis parameter
c=
7 1  4.04a 1.4 − 1 
−
4 4  4.04a 1.4 + 1 
(5)
which applies to 1.5 < c < 2.
Although all of the contact mechanics models discussed
above are useful in analysis of the AFM pull-off forces, it
should be recognized that these models were developed by
assuming a spherical particle in contact with a smooth surface
— two ideal geometries. In the real world, particles are not
perfect spheres and surfaces of both particles and substrates are
almost always rough at either a micro- or nano-scale or both.
Surface roughness can significantly alter the true contact area
between the probe and the substrate from that predicted by
contact mechanics models, making analysis of the measured
pull-off forces difficult.
Taking roughness into account in interpretation
of pull-off forces
In measurements with rough substrates the pull-off forces often
change as a result of change in the particle-substrate contact
area. Simple contact mechanics models, discussed above, can
be applied to the systems with rough surfaces if a single-point
contact between the probe and the substrate is established, for
example, through asperities. For such situations, Eq. (2) still
can be applied after the following modification
 RR 
F = cπ  1 2  WA
 R1 + R2 
(6)
where
R1 and R2 are the radii of curvature for asperities, probe and
substrate, respectively, in adhesional contact.
R1 = R and R2 = ∞ for the probe and the substrate, respectively,
if they are free of asperities. Note that Eq. (6) is valid for asperities that can be treated as spherical in their upper section.
MINERALS & METALLURGICAL PROCESSING 229
Figure 3 — Standard deviation (normalized per average
pull-off force) vs. ratio of the probe diameter to the asperityto-asperity distance recorded for glass probes with D = 5,
10 and 40 µm on nano- and micro-rough silicon substrates
(from Tormoen et al., 2005).
As R2 changes for surfaces of random roughness characteristics, the pull-off force changes as predicted by Eq. (6). Equation (6) is useful in interpretation of measured pull-off forces
when single contact between the probe and rough substrate is
established and the shape of asperity in contact with the probe
is known. It also predicts a distribution of measured pull-off
forces instead of one value when measurements are conducted
on random locations of the rough substrate.
Poor reproducibility and scatter of measured pull-off force
are typical problems that every researcher faces when experiments involve rough surfaces. The recent statistical analysis of
the pull-off forces measured on rough substrates indicates the
scatter of the pull-off forces increases with increasing surface
roughness (Tormoen and Drelich, 2005; Tormoen, 2005). As an
example, Fig. 3 illustrates how the scatter in the force values
changes as the probe size to surface roughness ratio changes
(D/λ ≡ probe diameter divided by the asperity-to-asperity
distance; ∆F/F parameter is the standard deviation for the
measured pull-off forces normalized by the mean value). For the
data in Fig. 3, ∆F/F increases from 4.4% to 14.8% calculated
for nano-rough surfaces to 40.4% to 64.3% for micro-rough
surfaces, i.e., ∆F/F increases with increasing roughness. Zero
∆F/F value is expected for the probe-substrate system with
smooth surfaces, if the substrate is free of chemical and/or
mechanical heterogeneity.
For rough substrates with random roughness characteristics and roughness features with much smaller dimensions
than the size of the probe, Eq. (6) has limited application in
interpretation of the measured pull-off forces. This approach
simply neglects the possibility of multiple contacts between the
probe and the substrate (probe in contact with more than one
asperity). In extreme scenarios, when the particle is in contact
with the top of asperity the pull-off force is much lower than
when particle touches the sides of the asperities or entirely
penetrates an inter-asperity valley.
The plot in Fig. 4 shows differences in pull-off force values
as the probe made of a glass sphere with a diameter of 10 μm
is positioned differently over the silicon grid composed of a
regular array of triangular ridges with an approximate spacing of
3 μm and ridge peak radii of <10 nm. Force values approached
a maximum of about 50 nN and a minimum of about 10 nN.
The data follow a pattern that mirrors the structure of the grid
(model micro-rough substrate), as shown by the guideline in
Fig. 4. The largest force values correspond to double contact
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points and the lowest force values correspond to single contact
point scenario. The intermediate values correspond to contact
of the probe with one ridge and long range-interaction with the
second ridge, as schematically illustrated in Fig. 5.
Most of the substrates and probes have random roughness
and quantitative calculations of the adhesion force between
a rigid particle and rigid rough surface are difficult for many
reasons. The size, shape, homogeneity, mechanical properties
and distribution of the asperities (deviations from an ideal planar
surface) influence the actual area of contact, and, therefore,
directly affect the pull-off force (Beach et al., 2002). The particle
can also have an irregular geometry leading to more difficulties
in quantitative analysis of pull-off force data.
Rabinovich et al. (2000) suggested that by measuring the
root mean square (RMS) roughness parameters along with
asperity sizes and distribution for both the probe and substrate,
one could predict the adhesional contact and ultimately account
for different adhesion forces than what is expected from ideal
geometries. The Rabinovich model states
Figure 4 —Averaged pull-off force vs. x-offset of the piezo
(= position of the probe over the grid structure) for the
10 µm probe on TGG grating. A guideline is included in
the plot to aid the eye in following the data trend (from
Tormoen et al., 2005).
F
cπλ2 2WA
=
+B
R 58 R( RMS2 )
where
B=
(7)
A
1
2
6 zo 2 
58 R( RMS1 )   1.82( RMS2 )  (8)
1
+
1
+

 

λ12
zo
r
RMS =
32 ∫ y 2 rdr
0
λ2
kp
(9)
c is a constant (JKR c = 1.5, DMT c = 2),
kp is the surface packing density for closed-packed spheres
(kp = 0.907),
A is the Hamaker constant,
z0 is the distance of closest approach between the two
surfaces,
λ is the asperity-to-asperity distance,
r is the radius of curvature of the asperity apex and
R is the radius of the probe.
In the Rabinovich model, surfaces that exhibit two scales
of roughness, both smaller than the size of spherical probe,
are considered. The first type of roughness, called RMS1, is
associated with a longer peak-to-peak distance, λ1. A second
smaller roughness, called RMS2, is associated with a smaller
peak-to-peak distance, λ2. For many systems, B is small and
Eq. (7) can be reduced to
cπλ2 2
WA
58 RMS2
(10)
The Rabinovich model was tested in this study by analyzing
the results of pull-off forces measured for colloidal probes in
contact with rough substrates and found to provide a reasonable
estimate of the measured forces (Beach et al., 2002; Drelich et
al., 2004a, 204b; Tormoen et al. 2005). Discrepancies between
experimental and theoretical force values were noted as well,
although these discrepancies are believed to be mainly caused
by a broad variation in size, shape and distribution of surface
asperities, i.e., random nature of the roughness versus symmetrical one assumed in the model.
Figure 5 — Schematic of possible contact scenarios for a
probe on a rough substrate: (a) a single asperity contact,
(b) a single asperity contact with a noncontact attraction
imposed by a second asperity and (c) a double asperity
contact scenario.
November 2006 • Vol. 23 No. 4
F=
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Elimination of roughness effects in adhesion
force measurements
As discussed above, substrate roughness, for which the asperities’ dimensions are comparable or smaller to the dimensions
of the probe, as well as any surface roughness at the probe
apex, can affect the sphere-substrate contact area to a degree
that is not taken into account by simple contact mechanics
models. If this happens, interpretation of the pull-off forces is
difficult or even impossible. However, both microscopic and
submicroscopic roughness effects can be eliminated in many
experiments by appropriate selection and preparation of the
substrates. Specifically, nanoscale asperities of deformable
materials can be squeezed and flattened out during the pull-off
force measurements to such a degree that the probe-substrate
contact area is the same as predicted by one of the contact
mechanics models for ideal sphere-flat geometry. To reach
such experimental conditions, stiff cantilevers and higher
loads need to be used in the AFM pull-off force measurements
(Biggs and Spinks, 1998; Nalaskowski et al., 2003; Drelich,
2004b; Tormoen and Drelich, 2005). As an example, Fig. 6
shows pull-off force data recorded for ~14-µm polyethylene
probe, having nano-rough surface, in contact with a silicon
wafer. The polyethylene probe was mounted on ~17 N/m
cantilever.
Figure 6 presents the pull-off forces measured for varying loading regimes. The loads varied from ~330 to ~3,900
nN and the recorded forces varied from ~6,200 to 7,150 nN.
Figure 6 shows that at low loads, force data are more widely
scattered, from ~6,200 to 7,150 nN, whereas the data are much
more tightly grouped at the high loads, ~6,770 to 6,930 nN.
Decrease in standard deviation of measured pull-off force
reflects the progressing deformation of nano-scaled asperities
of the polymeric probe.
The asperities deform under applied load (Reitsma et al.,
2000; Beach and Drelich, 2003; Rutland et al., 2004). Plastic
deformation can be analyzed using the model proposed by
Maugis and Pollock (1984). According to this model, load P is
needed to initiate plastic deformation of asperity with the radius
of curvature of Ra or smaller (Drelich et al., 2004a, 2004b)
Ra ≤
PE 2
10800π Y 3 Figure 7 — Plastically deformed asperity size vs. applied
load for polyethylene probe calculated based on Eq. (11)
using E = 0.14 GPa and Y = 6.9 MPa (based on data from
Tormoen and Drelich, 2005).
(11)
where
E is the Young’s modulus of material and
Y is the material’s yield strength.
To see how this varies for the polyethylene probe in the
range of loads used in the above-described AFM experiment
(<4 μN), Fig. 7 shows a plot of size of the plastically deformed
asperity (Rasp) vs. the applied load (P). For example, at a load
of 4 μN, asperities as large as with radius Ra ~170 nm should
deform plastically.
The plastic deformation of asperity is not necessary condition
for the probe to establish a full contact with the substrate without
roughness interference. Such contact can already be managed
in probe-substrate elastic deformation regime. Although the
elastic deformation of the probe recovers soon after the load is
reduced or removed or the probe-substrate contact is broken,
this recovery is often a slow process for polymers (if used as
material for the probe or substrate or both). The delay in full
recovery of the original probe shape (or substrate) is caused by
viscoelastic properties of polymers (viscoelastic deformation
exhibits the mechanical characteristics of elastic deformation
coupled with a viscous flow).
MINERALS & METALLURGICAL PROCESSING Figure 6 — Force data collected for polyethylene probe on
a silicon wafer (from Tormoen and Drelich, 2005). Trend
lines are included to demonstrate how the scatter in the
pull-off forces diminishes with increasing maximum applied loads.
231
It is possible to estimate the size of roughness irregularities,
which can be squeezed out during the probe’s elastic/viscoelastic deformation, using the contact mechanics models. For
example, according to the JKR model, the elastic/viscoelastic
deformation of the probe (δ) can be calculated by
δ=
a 2 2 6π aWA
(12)
−
R 3
K
where
a=
3
R
P + 3π RWA + 6π RWA P + ( 3π RWA )2  (13)

K
Figure 8 shows a dependence of δ on P for the polyethylene
probe with R = 7 µm.
As shown in Fig. 8, δ changes from about 6 to 9 nm when
the applied load increases from 0 to 4 µN. It is expected that
all asperities present on the surface of the colloidal probe with
a height equal or smaller than δ should be squeezed out completely during deformation of the probe. Because most of the
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PROOF COPY
Acknowledgments
This paper briefly reviews the research results generated by
two former graduate students, Elvin Beach and Garth Tormoen,
and the author would like to express his appreciation for their
contribution to the program. Financial support provided by the
Petroleum Research Fund and administrated by the American
Chemical Society is gratefully acknowledged.
References
Beach, E.R., and Drelich, J., 2003, "AFM pull-off force measurements with polystyrene (deformable) colloidal probes," in Functional Fillers and Nanoscale
Minerals, J.J. Kellar, M.A. Herpfer and B.M. Moudgil, eds., Society for Mining,
Metallurgy, and Exploration, Inc., Littleton, CO, pp. 177-193.
Beach, E.R., Tormoen, G.W., Drelich, J., and Han, R., 2002, "Pull-off force measurements between rough surfaces by atomic force microscopy," J. Colloid
Interface Sci., Vol. 247, pp. 84-99.
Biggs, S., and Spinks, G., 1998, "Atomic force microscopy investigation of the
adhesion between a single polymer sphere and a flat surface," J. Adhesion
Sci. Technol., Vol. 12, pp. 461-478.
Cappella, B., and Dietler, G., 1999, "Force-distance curves by atomic force
microscopy," Surface Sci. Rep., Vol. 34, pp. 1-104.
Carpick, R.W., Ogletree, D.F., and Salmeron, M., 1999, "A general equation for
fitting contact area and friction vs. load measurements," J. Colloid Interface
Sci., Vol. 211, pp. 395-400.
Derjaguin, B.V., Muller, V.M., and Toporov, Y.P., 1975, "Effect of contact deformations on the adhesion of particles," J. Colloid Interface Sci., Vol. 53, pp.
314-326.
Drelich, J., and Mittal, K.L., eds., 2005, Atomic Force Microscopy in Adhesion
Studies, VSP, Utrecht-Boston.
Drelich, J., Tormoen, G.W., and Beach, E.R., 2004a, "Determination of solid
surface tension from particle-substrate pull-off forces measured with the
atomic force microscope," J. Colloid Interface Sci., in press.
Drelich, J., Tormoen, G.W., and Beach, E.R., 2004b, "Determination of solid
surface tension at the nanoscale using atomic force microscopy," in Contact
Angle, Wettability and Adhesion, Vol. 4, K.L. Mittal, ed.
Hertz, H., 1896, Miscellaneous Papers, Macmillan, London.
Johnson, K.L., Kendall, K., and Roberts, A.D., 1971, "Surface energy and the
contact of elastic solids," Proc. Roy. Soc., London, A324, pp. 301-313.
Maugis, D., 1992, "Adhesion of spheres: the JKR-DMT transition using a Dugdale
model," J. Colloid Interface Sci., Vol. 150, pp. 243-269.
Maugis, D., and Pollock, H.M., 1984, "Surface forces, deformation and adherence
at metal microcontacts," Acta Metall., Vol. 9, pp. 1323-1334.
Nalaskowski, J., Drelich, J., Hupka, J., and Miller, J.D., 2003, "Adhesion between hydrocarbon particles and silica surfaces with different degrees of
hydration as determined by the AFM colloidal probe technique," Langmuir
19, pp. 5311-5317.
Rabinovich, Y.I., Adler, J.J., Ata, A., Singh, R.K., and Moudgil, B.M., 2000, "Adhesion between nanoscale rough surfaces I. Role of asperity geometry," J.
Colloid Interface Sci., Vol. 232, pp. 17-24.
Reitsma, M., Craig, V., and Biggs, S., 2000, "Elasto-plastic and visco-elastic deformations of a polymer sphere measured using colloid probe and scanning
electron microscopy," Int. J. Adhesion Adhesives, Vol. 20, pp. 445-448.
Rutland, M.W., Tyrrell, J.W.G., and Attard, P., 2004, "Analysis of atomic force
microscopy data for deformable materials," J. Adhesion Sci. Technol., Vol.
18, pp. 1199-1215.
Tormoen, G.W., and Drelich, J., 2005, "Deformation of soft colloidal probes during
AFM pull-off force measurements: elimination of nano-roughness effects,"
J. Adhesion Sci. Technol., Vol. 19, pp. 181-198.
Tormoen, G.W., Drelich, J., and Beach, E.R., 2004, "Analysis of atomic force
microscope pull-off forces for gold surfaces portraying nanoscale roughness and specific chemical functionality," J. Adhesion Sci. Technol., Vol.
18, pp. 1-18.
Tormoen, G.W., Drelich, J., and Nalaskowski, J., 2005, "A distribution of AFM
pull-off forces for glass microspheres on a symmetrically structured rough
surface," J. Adhesion Sci. Technol., Vol. 19, pp. 215-234.
Yu, N., and Polycarpou, A.A., 2004, "Adhesive contact based on the Lennard-Jones
potential: a correction to the value of the equilibrium distance as used in the
potential," J. Colloid Interface Sci., Vol. 278, pp. 428-435.
Figure 8 — Vertical deformation of the polyethylene probe
(δ) with a diameter of D = 14 µm in contact with a silicon
wafer under different load P. The deformation was calculated
using the JKR model assuming WA = 200 mJ/m2 and K = 4
GPa (based on data from Tormoen and Drelich, 2005).
irregularities that have been observed on polymeric probes are
several nanometers tall, they most likely are deformed during
the pull-off force measurements if loads at a level of a few
micronewtons are applied. If such experimental conditions
are imposed in adhesion force measurements, asperities have a
limited effect on contact area between the probe and the substrate
and the measured force values are more reproducible.
Final remarks
The contact area between a probe and a substrate is a fundamental parameter for analyzing pull-off force data generated
by atomic force microscopy. Roughness, present at some
scale for all real materials, complicates this task by introducing probe-substrate limited contact through asperities and
causes both change in magnitude and distribution of pull-off
forces. The results from the author’s laboratory indicate that
the scatter in the data decreases with increasing diameter of
the probe as compared to the dimension of surface irregularities, but the magnitude of the pull-off force is more severely
altered by roughness.
The analysis of pull-off forces measured for rough (rigid)
substrates and probes is still possible if the probe diameter is
much larger than the size of roughness features on interacting
surfaces. To succeed in interpreting such complex systems, a
detailed surface analysis in terms of roughness parameters,
asperity size and shape and spacing between asperities is required. Then, analysis of the experimental data should be done
with a theoretical model applicable to rough surfaces.
It is also possible to overcome the roughness effect at nanoscale in measurements of pull-off forces for soft probe and/or
substrates. Operating at adequate loads promotes deformation of asperities to such extend that a saturated contact area
between a probe and a substrate, at different locations of the
surface, is established.
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