Friction, Adhesion, and Deformation

Friction, Adhesion, and Deformation: Dynamic Measurements with the Atomic Force
Microscope
Phil Attard
Ian Wark Research Institute, University of South Australia, Mawson Lakes SA 5095 Australia
(J. Adhesion Sci. Technol. 16, 753–791 (2002).)
capability of the AFM is the measurement of friction,
which is also called friction force microscopy or lateral
force microscopy. Since the original work of Mate el al.
[7] the fields of friction force mapping, (sometimes called
chemical imaging), and of nanotribology, have grown
greatly, (see, for example, papers in refs [8,9]). This research has been severely limited by the lack of a quantitative calibration method for the AFM. This deficiency
has been rectified quite recently by two techniques that
yield the torsional spring constant of the cantilever and
the voltage response of the lateral photodiode to cantilever twist [10–12]. This review begins by summarising
the limitations of previous calibration technique and by
detailing the procedures involved in the newer quantitative methods, (§II). The results that we have obtained
in our laboratory [13] for the quantitative dependence of
friction on adhesion in a system with electric double layer
interactions are then reviewed (§III).
Running title: Friction, Adhesion, and Deformation
Abstract. A selection of recent experimental and theoretical work involving the atomic force microscope is reviewed,
with the focus being upon dynamic measurements. Four topics are covered: calibration techniques for the friction force
microscope, quantitative measurements of friction and the effect of adhesion, measurement and theory for the deformation
and adhesion of viscoelastic particles, and the interaction and
adhesion of hydrophobic surfaces due to bridging nanobubbles.
I. INTRODUCTION
The atomic force microscope (AFM) [1] is commonly
used to image surfaces and to study the interaction and
adhesion of particles. The wide-spread adoption of the
AFM is due to its ease of use, the molecular-level information that it provides, and the variety of surfaces that
can be studied in a broad range of environments. In addition, the computer interface allows flexible control of the
device and the automated acquisition of large volumes of
data, it facilitates multiple repeat experiments to check
reproducibility and to minimise statistical error, and it
enables detailed data analysis. This computer control
opens up the possibility of real-time monitoring of experiments and the exploration of time-dependent effects.
The AFM is well-suited to studying the latter, whereas
the original surface force apparatus [2] and its variants
[3,4] either lack automated data acquisition or suffer from
inertial and other artefacts that must be accounted in
the quantitative interpretation of dynamic force measurements [5,6].
The distinction between equilibrium and nonequilibrium forces is quite important. To some extent,
the primary concern with the AFM has been, (and should
be), to ensure that the experiments are carried out slowly
enough that equilibrium is established at each instant so
that the measured forces are comparable to those measured statically. Beyond that, an exciting field of research exploits the dynamic capabilities of the AFM to
measure non-equilibrium phenomena in a controlled fashion. We review two examples from our laboratory that
show the utility of dynamic AFM measurements for nonequilibrium systems. Results and quantitative analyses
are presented for the deformation, interaction, and adhesion of viscoelastic droplets, (§IV), and for the interaction
and adhesion of spreading, bridging nanobubbles, (§V).
The most obvious technique that utilises the dynamic
II. CALIBRATION OF THE FRICTION FORCE
MICROSCOPE
A. Critical Review
In order to use the AFM various calibrations have to
be performed. The lateral movement of the piezo is often calibrated using model substrates. The expansion
factor that relates the applied voltage to the distance the
piezo expands in the vertical direction normal to the substrate, ∆z, can be measured from the interference fringes
due to the reflection of the laser from the cantilever and
the substrate. The normal spring constant of the cantilever kx can be obtained gravitationally, thermally, or
by resonance techniques [14–16]. The normal photodiode
sensitivity factor, α0 , relates the measured vertical differential photodiode voltage ∆Vvert to the vertical deflection
of the cantilever, ∆x, which in the constant compliance
regime is equal to the piezo movement, ∆x = ∆z. For
the quantitative measurement of friction, in addition to
these one has to obtain the torsional spring constant of
the cantilever, kθ , and the lateral photodiode sensitivity
factor, β, which relates the measured lateral differential
photodiode voltage, ∆Vlat , to the twist angle of the cantilever, ∆θ.
Unfortunately, almost all lateral calibration techniques
that have been used to date are approximate in one way
or another, and the measurements of friction that utilise
1
them must be regarded as qualitative rather than quantitative. Briefly, a critical review of the literature reveals
that in most cases [17–21] the torsional spring constant
is calculated, not measured, using an analytic approximation [22] that idealises the actual geometry of the
cantilever. In addition it ignores the effects of coatings
and thickness variations, which in the case of the normal
spring constant can alter the value by an order of magnitude. The lateral sensitivity factor, which relates the
photo-diode voltage to the twist angle, has also been obtained by assuming it to be proportional to the vertical
sensitivity [18], by modelling the beam path and profile
[19], and by assuming that the tip is pinned during the
initial part of the friction loop [17,23]. Slippage and deformation makes the latter method inaccurate, and others have attempted to improve the method by invoking
certain simple models of friction and deformation [20,21].
Measurements of friction parallel to the long-axis of the
cantilever using the normal spring constant and sensitivity [24,25], erroneously neglect the bending moment of
the cantilever [6,21]. Toikka et al. [23] attempted to use
gravity acting on an attached lever, but the torque they
applied can be shown to give negligible cantilever twist
[12], and it appears that what they measured was in fact
photo-diode saturation. And finally, the commonly used
calibration method of Ogletree et al. [26] is restricted
by the need for a specialised terraced substrate and an
ultra sharp tip. For the calibration this method makes
two assumptions about the friction law, namely that friction is a linear function of the applied load, and that it
vanishes when the applied load is the negative of the adhesion. Counter-examples showing non-linear behaviour
are known [13,27], and obviously, any fundamental study
of friction should test quantitatively such an assumption
rather than invoke it for the calibration. That none of
the previous calibration methods are satisfactory is confirmed by the fact that many FFM papers give friction in
terms of volts rather than Newtons [27–29]. Almost all
friction force maps are similarly uncalibrated and the images are given in terms of volts rather than the physical
friction coefficient.
Feiler et al. [12] have developed a direct technique that
simultaneously measures the cantilever spring constant
and the lateral sensitivity of the photo-diode. That particular method is discussed in detail below.
Meurk et al. [10] have given a method for directly calibrating the lateral sensitivity of the photo-diode. Basically the angle of a reflective substrate is varied with
respect to the laser beam. In some AFM scanners there
is a stepper motor that facilitates the tilt of the head.
From the geometry and the amount of movement the degree of tilt, ∆θ, can be calculated. The change in the
lateral photo-diode voltage, ∆Vlat , is linear in the tilt
angle and the ratio of the two gives the lateral sensitivity
of the AFM.
The torsional spring constant of the cantilever can be
obtained directly by the technique developed by Bogdanovic et al. [11]. Here a protuberance (eg. an upturned
tipped cantilever) is glued to the substrate and force measurements are performed against it with the protuberance in contact off-set from the central axis of the tipless
force measuring cantilever. The latter consequently simultaneously deflects and twists. Recording the normal
and lateral photo-diode voltages in the constant compliance regime at several different lateral off-sets allows the
spring constant divided by the lateral sensitivity to be
obtained. Combined with the method of Meurk et al.
[10], this allows a full calibration of the AFM. (In principle one can also obtain the lateral sensitivity with this
method. However, the small leverage and high torsional
spring constant, makes it impractical to do so.)
B. Quantitative Calibration Technique
∆x
∆θ
∆z
FIG. 1. Rectangular cantilever with attached fibre and
sphere. When the substrate is moved a distance ∆z, the cantilever deflects a distance ∆x and twists an amount ∆θ. The
corresponding changes in the differential photo-diode voltages, ∆Vvert and ∆Vlat , are measured.
We now describe in detail a one-step method that siFig. 1
multaneously measures both the lateral photo-diode
senAttard
J.
Adh.
Sci.
Technol.
sitivity and the torsional spring constant of the cantilever
that has been developed in our laboratory [12]. A glass
fibre 50-200µm in length is glued perpendicular to the
long-axis of the cantilever and parallel to the substrate.
To ensure that the substrate pushes on the end of the
fibre, a colloid sphere is attached at its tip, (see Fig. 1).
Using the well-known colloid probe attachment procedure of Ducker et al. [30], an epoxy resin is used to attach the sphere and a heat-setting adhesive is used to attach the fibre. This allows the fibre to be later removed
and the cantilever used for friction measurements (ie. the
method is non-destructive). Attaching the sphere is convenient but not essential; other ways to ensure that it is
the end of the fibre that touches the substrate include
gluing the fibre to the cantilever at a slight angle, having
a ledge or colloid probe on the substrate, or performing the measurement with the head or substrate tilted a
small amount, (eg. by using the stepper motor).
The essence of the method is that pushing on the tip
of the fibre with a force F produces a torque τ = F L,
where L is the length of the fibre. The cantilever simultaneously deflects, ∆x = F/kx , and twists, ∆θ = τ /kθ .
The deflection, and hence the force and torque, is obtained from the differential vertical photo-diode voltage
∆x = α0 ∆Vvert , where the bare sensitivity factor, α0 , is
measured in the constant compliance regime without the
attached fibre. The actual sensitivity factor with the at2
tached fibre αL is greater than this because only part of
the piezo movement goes into deflecting the cantilever,
∆x < ∆z, (the rest is soaked up by the twist). The bare
vertical sensitivity factor has to be measured in a separate experiment and depends upon the positions of the
laser, the photo-diode, and the cantilever mount. With
practice, it is possible to obtain better than 10% reproducibility in this quantity between different experiments
and after remounting the cantilever. The best way to
ensure this is to to maximise the total vertical signal and
to minimise the differential lateral signal each time.
0.8
0.7
γ (pNm/V)
0.6
0.5
0.4
0.3
0.2
0.1
0
70
90
130
150
170
Fig. 3
Attard
J. Adh. Sci. Technol.
FIG. 3. Lateral sensitivity factor for different fibres. The
cantilevers were taken from the same batch. Circles indicate
fibres with an end-attached sphere, diamonds indicate bare
fibres, filled symbols are for approach, and open symbols are
for retraction. The data are from Ref. [12].
1.5
4/21/01 2:31 PM
Calib.xls Fig3
1
Lateral Voltage
110
Length (µm)
2
0.5
0
-0.5
-1
We found that the calibration procedure was straightforward and relatively robust. The method was less successful whenever there was significant adhesion between
the substrate and the tip of the fibre or the attached
sphere. We minimised such adhesion by using silica surfaces and conducting the calibration in water at natural
pH.
It is possible to verify independently the procedure by
obtaining the sensitivity factor that relates the change in
angle to the change in the lateral photo-diode signal, and
comparing this with the value obtained by the method
of Meurk et al. [10]. From the slope of the constant compliance region of the force curve with the attached fibre,
one can obtain the constants
-1.5
-2
-2.5
-3
-3
-2
-1
0
1
2
Vertical Voltage
Fig. 2
Attard
J. Adh. Sci. Technol.
FIG. 2. Lateral differential photo-diode voltage as a function of the vertical voltage. Both were measured for a cantilever with an attached fibre over the whole approach regime
of a single force measurement. The data are from Ref. [12].
4/21/01 2:30 PM
Calib.xls Fig2
The calibration factor of primary interest is the one
that relates the differential lateral photo-diode voltage
to an applied torque, τ = γ∆Vlat . This is given by
τ
∆Vlat
kx ∆xL
=
∆Vlat
∆Vvert
= k x α0 L
.
∆Vlat
αL =
γ=
∆z
∆z
, and βL =
,
∆Vvert
∆Vlat
(2)
for the vertical and lateral deflections, respectively. With
these the lateral sensitivity can be shown to be given by
[12]
(1)
βL (1 − α0 /αL )
∆θ
=
.
∆Vlat
L
This equation predicts a linear relationship between the
two photo-diode signals, which, as can be seen in Fig. 2,
is indeed the case. The slope of this line, combined with
the measured values for the vertical spring constant, the
bare vertical sensitivity factor, and the length of the fibre, gives the factor that converts the differential lateral
photo-diode voltage to the applied torque in general (ie.
independent of the attached fibre). Figure 3 shows the
lateral sensitivity factor obtained using a number of different fibres. That the same value is obtained each time
shows that it is an intrinsic property of the cantilever
and AFM set-up. It also confirms that remounting the
cantilever does not preclude reproducible results being
obtained.
(3)
A value of 3×10−4 rad/V was obtained using our method
[12], compared to 1.7 × 10−4 rad/V using the method of
Meurk et al. [10].
The torsional spring constant itself is given by [12]
kθ =
−kx L2
.
1 − αL /α0
(4)
A value of 2 × 10−9 N m was obtained using our method
[12], compared to 1.2 × 10−9 N m calculated from the
method of Neumeister and Ducker [22].
3
8
III. ADHESION AND FRICTION
10
6
Force (nN)
A. Intrinsic Force
One of the oldest ideas concerning the nature of friction is embodied in Amontons’ law, which states that
the friction force f is proportional to the applied load L,
f = µL, where µ is the coefficient of friction. For the
case of adhesive surfaces, where a negative load needs to
be applied to separate them, it is known that there can
be substantial friction even when the load is zero. Hence
Amontons’ law may be slightly modified
µ(L + A), L ≥ −A
(5)
f=
0,
L < −A,
1
4
0.1
2
0
10
20
30
40
0
-2
-4
0
10
20
30
40
50
Separation, h (nm)
FIG. 4. Force on approach as a function of separation. The
Fig 4
substrate is TiO2 , the 7µm diameter colloid probe is SiO
2,
Attard
J Adh Sci top
Tech
and the background electrolyte is 1mM KNO3 . From
to bottom the curves correspond to pH = 8, 7, 6, 5, and
4. The inset shows constant potential (ψSiO2 = −50mV and
ψTiO2 = −43mV) and constant charge fits to the pH = 8 case
on a log scale [33].
where A > 0 is the adhesion, which is the greatest tension that the surfaces can sustain. This modified version
reflects the plausible idea that friction only occurs when
the surfaces are in contact. Amontons’ law raises several
questions: Is friction a linear function of load? Is the
only role of adhesion to shift the effective load? What
is the law for non-adhesive surfaces? Is friction zero for
surfaces not in contact? And what does contact mean on
a molecular scale?
The AFM is an ideal tool to test the fundamental nature of friction, and we set out to answer quantitatively
these and other questions [13]. We chose a system that
would allow us to change the adhesion in a controlled
manner so that as far as possible all other variables were
kept constant. We used a titanium dioxide substrate,
(root mean square roughness of 0.3nm), and a silicon
dioxide colloid probe, (root mean square roughness of
0.8nm, 7µm diameter). The measurements were carried
out in aqueous electrolyte, (10−3 M KNO3 ), as a function
of pH. The SiO2 is negatively charged at practically all
pH, (its point of zero charge is ≈ pH 2), whereas TiO2 is
positively charged at low pH and negatively charged at
high pH, (its point of zero charge is ≈ pH 4.5). Hence at
low pH the attractive double layer interaction between
the surfaces causes them to be adhesive, and at high pH
they repel each other and do not adhere.
There have been several other AFM studies of friction
between surfaces with electrical double layer interactions
[27,31,32]. In some cases an applied voltage has been
used to modify the adhesion, but the friction coefficients
and force laws have all been qualitative in the sense of the
preceding section. A critical discussion of these results is
given in Ref. [13].
The load, which is the applied normal force, is shown
in Figs 4 and 5 as a function of separation for various
pH. It can be seen that the surfaces do indeed interact
with an electric double layer interaction, and that the
pH controls the sign and the magnitude of the force law.
For pH 4 and 5 the attractive double layer interaction
gives an adhesion of A = 10.5 and 4.4 nN, respectively.
However at higher pH the surfaces do not adhere.
8
Force (nN)
4
0
4
3
2
1
0
-4
-8
0
-12
0
10
20
30
40
5
10
50
15
60
Separation, h (nm)
FIG. 5. Same as preceding figure on retraction. The inset
Fig 5
magnifies the three highest pH at small separations [33]. Attard
J Adh Sci Tech
In view of Eq. (5), we are motivated to define the
detachment force Fdetach as the minimum applied force
necessary to keep the surfaces in contact [13]. For nonadhesive surfaces this is a positive quantity, and for adhesive surfaces it is negative, (in fact it is the negative of
the adhesion). The detachment force at pH = 6, 7, and 8
was Fdetach = 1.4, 2.6, and 3.5 nN, respectively, (Fig. 5).
In view of the close relationship between adhesion and
the detachment force, one may define an intrinsic force,
Fintrinsic = L − Fdetach ,
(6)
which may be thought of as the force in excess of that
when the surfaces are just in contact. In this language, Amontons’ law generalised to non-adhesive surfaces would read f = µFintrinsic .
We measured friction as a function of applied load at
various pH. This was done in the usual fashion [7] by
moving the substrate back and forth in the direction perpendicular to the long axis of the cantilever and recording
friction loops. The length of the scan in each direction
was 0.5µm, and the velocity was 1µm/s. The lateral calibration factor, obtained as detailed above [12], was used
to convert (half) the voltage difference between the two
4
arms of the friction loop to the applied torque τ . The
friction force was obtained as f = τ /2r, where r = 7µm
is the radius of the colloid probe. The applied load was
fixed by using the set-point feature of the AFM (ie. the
vertical deflection signal was held constant during the
friction loop).
These experiments show that for this system friction
is not a linear function of load (ie. the friction coefficient
µ = df /dL is not independent of load). There is a noticeable curvature in the plot, with friction increasing more
rapidly at higher loads. The loads that have been applied here are relatively weak, (the average pressure in
the contact region (see below) is less than about 10MPa,
and the peak pressure is less than about 100MPa [13]),
and it is not clear what will happen at higher loads than
these.
Whilst it is not implausible that the friction should be
zero for negative intrinsic forces in all cases, (this corresponds to the surfaces being out of contact), it is a little
surprising that for positive intrinsic forces the increase
in friction is the same in all cases. After all, not only
is the adhesion and the normal force law different at the
different pH, but also the surface chemistry varies due to
the different amount of ion binding that occurs. The fact
that the latter has almost no effect on friction is perhaps
not unexpected since over the range of pH studied, for
TiO2 only about 1% of the surface sites are converted
from H+ at low pH to OH− at high pH, and for SiO2
the change is about 10% [34]. Nevertheless it is not immediately obvious why surfaces with different adhesions
display quantitatively the same friction for the same intrinsic force.
20
ph4
ph5
ph6
ph7
ph8
18
Friction Force (nN)
16
14
12
10
8
6
4
2
0
0
1
2
3
4
5
6
7
8
F applied (nN)
FIG. 6. Friction force as a function of applied load [33].
Fig 6
Attard
J Adh Sci Tech
Friction is plotted as a function of applied load in Fig 6.
In general friction increases with increasing load. At a
given applied load, friction is also larger the lower the pH.
Since the adhesion increases with decreasing pH, one may
restate this fact as the higher the adhesion the higher
the friction at a given applied load. Moreover, friction
is non-zero at zero loads for adhesive surfaces. For nonadhesive surfaces, friction is zero for small but non-zero
applied loads.
B. Elastic Deformation
In order to investigate this question further we carried
out elastic deformation calculations of the sphere and
substrate under the experimental conditions [13]. Elastic deformation has long been thought to play a dominant
role in friction of macroscopic bodies, mainly in the context of using contact mechanics to account for asperity
flattening [35,36]. We however were in a position to go
beyond contact theories such as JKR [37] or DMT [38].
We used the soft-contact algorithm of Attard and Parker
[39,40] and invoked the actual experimentally measured
force law, which is of course of non-zero range. The algorithm self-consistently calculates the surface shape of
the elastically deformed bodies due to the local pressure,
which in turn depends upon the local separation of the
deformed bodies. In this way we obtain the actual surface
shape and the actual pressure profile, whereas contact
mechanics assumes simplified and non-physical forms for
both. We fitted a smooth curve to the measured force
law at the different pH, and using the Derjaguin approximation, differentiated this to obtain the pressure as a
function of surface separation. The latter is required by
the algorithm [39,40], as is discussed in the following section. The calculations presented in Ref. [13] are the first
elastic deformation calculations using an actual experimentally measured force law. For the present calculations
there was no hysteresis between the loading and unloading cycles. (The hysteresis observed in the original papers
40
pH4
pH5
pH6
pH7
pH8
pH5.5
Friction Force (nN)
35
30
25
20
15
10
5
0
-5
0
5
10
15
20
25
30
35
F intrinsic (nN)
FIG. 7. Friction force as a function of intrinsic load [33].Fig 7
Attard
J Adh Sci Tech
The quantitative behaviour of friction with pH is not
obvious when plotted as a function of applied load. But
when plotted against intrinsic load, Fig. 7, the utility of
the detachment force becomes evident. The functional
form of the friction force law is fundamentally independent of pH, and all the measurements lie on a single universal curve. In other words, the major role of pH is to
determine the adhesion, (or more precisely the detachment force). Once this parameter has been experimentally determined from a normal force measurement at a
given pH, the friction at that pH may be predicted from
the friction measured at any other pH merely by shifting
the load by the detachment force.
5
1.2
[39,40] for soft adhesive bodies has since been attributed
to a non-equilibrium viscoelastic effect [41,42]; see §IV.)
1.1
1
1.2
0.9
h(r) (nm)
1.1
1
h(r) (nm)
0.9
0.8
0.7
0.6
0.8
0.5
0.7
0.4
0.6
0.3
-0.08
0.5
0.4
0.3
-0.08
-0.04
-0.02
0
0.02
0.04
0.06
0.08
r (µm)
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
FIG. 9. Calculated surface profiles for an intrinsic force of
15nN. From top to bottom, the virtually indistinguishable
curves correspond to a pH of 8, 7, 6, 5, and 4, respectively.
The data are from Ref. [13].
Fig 9
Attard
J Adh Sci Tech
4/21/01 2:44 PM
friction.xls fig9
0.08
r (µm)
Fig 8
Attard
J Adh Sci Tech
FIG. 8. Calculated surface profiles for an applied load
of 5nN. From top to bottom, the pH is 8, 7, 6, 5,
and 4, and in each case the measured force law has
been used in the calculations. Young’s modulus for SiO2 ,
E/(1 − ν 2 ) = 7.7 × 1010 N/m2 , has also been used. The bottom dashed curve is for an applied load of 720nN for the pH4
case. The abscissa is the distance from the central axis in microns and the ordinate is the local separation in nanometres.
The data are from Ref. [13].
friction.xls fig8
4/21/01 2:44 PM
-0.06
Figure 9 shows the surface shapes at the different pH
at an intrinsic load of 15nN, which corresponds to an applied load of 5nN for the pH4 case. The change from
Fig. 8 is quite dramatic, and one can see that the profiles have coalesced. In other words, surfaces at a given
intrinsic load have the same shape and local surface separation. Given that friction is also a universal function
of intrinsic load, (Fig. 7), one may conclude that friction
is a function of the local separation and independent of
the force law. In so far as the short-range interactions
between the atoms on the two surfaces can be expected
to be independent of pH, one can say that these are the
interactions that determine friction. Friction occurs between two bodies when energy can be transferred from
one to another, which means that they have to be close
enough for the interaction between atoms on the two surfaces to be comparable to the thermal energy [13]. One
concludes that the only role of adhesion in friction is to
decrease the amount of applied load that is necessary to
bring the surfaces to a given separation.
Figure 8 shows the resultant surface shape at an applied load of 5nN. This load is greater than all the detachment forces, and in all cases the surfaces showed non-zero
friction. It can be seen that very little surface flattening
has occurred, and that the surfaces at different pH are
effectively displaced parallel to each other.
Also included in Fig. 8 is a high load (720nN) case,
which shows substantial flattening. However there is not
a well defined contact region, and there is certainly not a
sharp change in the surface profile to demark contact despite the fact that these calculations are for the adhesive
pH4 surfaces.
The fitted force law includes a Lennard-Jones soft repulsion with length scale 0.5nm [13], and one could define
contact as local separations smaller than this. Such an arbitrary definition is somewhat problematic, particularly
since the curves at 5nN load, which are not in contact by
the definition, display non-zero friction. In view of this
discussion of the meaning of contact for systems with realistic surface forces of non-zero range, the inapplicability
of simple contact theories such as Hertz, JKR, or DMT
is clear. One might also conclude that the experimental
verification or refutation of Amontons’ second law, (for
a given load friction is independent of contact area), at
the molecular level will be difficult.
IV. VISCOELASTIC DEFORMATION AND
ADHESION
A. Viscoelastic Theory
The shape of the deformed surfaces given above were
obtained by solving the equations of continuum elasticity theory in the semi-infinite half-space approximation
[39,43]
Z
−2
p(h(s))
.
(7)
u(r) =
ds
πE
|r − s|
Here the elasticity parameter E is given in terms of the
Young’s moduli and the Poisson’s ratios of the two bodies, 2/E = (1 − ν12 )/E1 + (1 − ν22 )/E2 , r and s are the lateral distance from the central axis, and p(h) is the pressure between two infinite planar walls at a separation of
6
h. The total deformation normal to the surfaces at each
position is u(r), and hence the local separation between
the two bodies is h(r) = h0 (r)−u(r). Here the local separation of the undeformed surfaces is h0 (r) = h0 + r2 /2R,
where R−1 = R1−1 + R2−1 is the effective radius of the
interacting bodies; in general the Ri are related to the
principle radii of curvature of each body [44].
For contact theories such as Hertz, JKR, or DMT, the
pressure pc (r) that appears in the integrand of Eq. (7) is
a specified function of radius that when integrated gives
u(r) = r 2 /2R, which corresponds to a flat contact region, h(r) = 0. In contrast for realistic force laws that
have non-zero range, such as van der Waals, electric double layer, or the actual measured p(h) discussed above,
the integral must be evaluated numerically. Because in
this case the local separation depends upon the deformation, Eq. (7) represents a non-linear integral equation
that must be solved by iteration for each nominal separation h0 .
An efficient algorithm for the solution of the noncontact elastic equation has been given by the author
[39,41], and it has been used to analyse a variety of force
laws [13,39–42]. Other workers have also calculated the
elastic deformation of the solids using realistic surface
forces of finite range [45–52]. There have of course been
a large number of experimental studies to measure the
interaction of deformable solids. These include AFM
measurements [53–63] , as well as results obtained with
the surface force apparatus and the JKR device [64–73].
These studies in general show that the adhesion and interaction is hysteretic and time-dependent, particularly
for highly deformable solids with high surface energies.
Such behaviour is characteristic of viscoelastic materials.
Maugis and Barquins have given a review of viscoelastic
adhesion experiments, which they attempt to interpret
in quasi-JKR terms, introducing a somewhat ill-defined
time-dependent surface energy [74].
A proper theoretical treatment of the deformation and
adhesion of viscoelastic materials involves replacing the
elasticity parameter, which gives the instantaneous response to the pressure, by the creep compliance function, which gives the response to past pressure changes.
In this way the prior history of the sample is accounted
for. Hence the generalisation of the elastic half-space
equation involves a time convolution integral [75,76],
u(r, t) − u(r, t0 )
Z
Z t
ṗ(h(s, t0 ))
−2
ds
.
=
dt0
0
πE(t − t )
|r − s|
t0
ṗc (s, t), whereas here ṗ(h(s, t)) is determined by the physical force law and the past rate of change of separation.
An algorithm has been developed for solving the full
non-contact problem for the case that the creep compliance function has exponential form [75]
1
1
E∞ − E0 −t/τ
=
+
e
.
E(t)
E∞
E∞ E0
(9)
Here E0 and E∞ are the short- and long-time elasticity
parameters, respectively, and τ is the relaxation time.
The algorithm can be generalised to more complex materials with multiple relaxation times [75]. The present
three parameter model is perhaps the simplest model of
viscoelastic materials, although an alternative three parameter expression, E(t)−1 = C0 + C1 tm , 0 < m < 1,
has also been studied as a model for liquid-like materials
[79–81].
With the exponential creep compliance function, differentiation of the deformation yields [75]
u̇(r, t) =
−1
[u(r, t) − u∞ (r, t)]
τ
Z
ṗ(h(s, t))
2
ds
,
−
πE0
|r − s|
(10)
where u∞ is the static deformation that would occur in
the long-time limit if the pressure profile were fixed at its
current value,
Z
p(h(s, t))
−2
ds
.
(11)
u∞ (r, t) =
πE∞
|r − s|
The rate of change of the pressure is
h
i
ṗ(h(r, t)) = p0 (h(r, t)) ḣ0 (t) − u̇(r, t) ,
(12)
where ḣ0 (t) is the specified drive trajectory. Accordingly,
Eq. (10) represents a linear integral equation for the rate
of change of deformation. It can be solved using the
same algorithm that has been developed for the elastic
problem [39,41]. It is then a simple matter to solve the
differential equation for the deformation by simple time
stepping along the trajectory, u(r, t + ∆t ) = u(r, t) +
∆t u̇(r, t).
The algorithm has been used to obtain results for an
electric double layer repulsion [75] and for a van der
Waals attraction [76]. The latter is
6
z0
A
−
1
,
(13)
p(h) =
6πh3 h6
(8)
Here ṗ(h(s, t)) is the time rate of change of the pressure. The particles are assumed stationary up to time t0 ,
and, if interacting or in contact, have at that time fixed
deformation corresponding to static elastic equilibrium,
u(r, t0 ) = u∞ (r). This expression is essentially equivalent to that used by a number of authors [77–80], with
the difference being that the latter have treated contact
problems, with ṗ(h(s, t)) replaced by a specified analytic
where A is the Hamaker constant, and z0 characterises
the length scale of the soft-wall repulsion. Fig. 10 shows
the shape of viscoelastic spheres during their interaction.
The total time spent on the loading branch is ten times
the relaxation time, so that one expects to see viscoelastic effects. At the largest separation prior to approach
7
the surfaces are undeformed. Prior to contact on approach they bulge toward each other under the influence
of the van der Waals attraction. There is a relatively
rapid jump into contact, and initially a fast spreading of
the flattened contact region, which continues to grow as
the particles are driven further together. At the edge of
the contact region there is a noticeable rounding of the
surface profiles on the approach branch. Following the
reversal of the motion, (unloading), the surfaces become
extended as they are pulled apart, and there is a sharper
transition between contact and non-contact than on the
loading branch. It should be noted, however, that even
in this case the slopes at the edge of the contact region
are not discontinuous as predicted by the JKR theory.
Following the turning point, the surfaces are effectively
pinned in contact for a time, and then the contact region
begins to recede. After the surfaces jump apart there remains a memory of the stretching that occurred during
unloading, and for a time comparable to the relaxation
time of the material, the deformed separation is smaller
on the unloading branch out of contact than at the corresponding position upon loading.
as one might expect since this corresponds to effectively
stiffer materials.
70
10
Force F(t)/2πR (mN/m)
60
5
50
0
40
-5
30
-10
20
-1
0
1
2
10
0
-10
-20
-30
-10
-8
-6
-4
-2
0
2
4
6
Nominal Separation h0(t) (nm)
FIG. 11. Interaction forces for adhesive viscoelastic
spheres. From inside to outside the hysteresis loops correspond to driving velocities of |ḣ0 | = 1, 2, and 5 µm/s, using
the viscoelastic parameters of Fig. 10. The crosses represent
the static equilibrium elastic result for E∞ = 109 N m−2 . Inset. Loading curves near initial contact. The circles represent
the static equilibrium elastic result for E0 = 1010 N m−2 , and
the bold curve is the force for rigid particles. The data are
from Ref. [76].
4/21/01 3:13 PM
visco.xls Fig11
Fig 11
Attard
J Adh Sci Tech
25
Following the reversal of the direction of motion in
Fig. 11, a small increase in the nominal separation gives
a large decrease in the applied load, which causes the unloading branch to lie beneath the loading branch. This
behaviour is reflected in the surface profiles, (Fig. 10),
where on the loading branch, increasing the load causes
the contact area to grow, whereas immediately following
the turning point, decreasing the load stretches the surfaces at fixed contact area. The hysteresis in the force
curves manifests the fact that a certain energy has to
be put into the system to move the surfaces a nominal
distance on loading, and less energy is recovered from
the system in moving the same distance on unloading.
This is precisely what one expects from a viscoelastic
system. The size of the hysteresis loop increases with
drive speed. As the speed is decreased, both loops appear to coalesce on the long-time elastic result, which
corresponds to static equilibrium, Eq. (7).
Figure 11 also shows that the adhesion, which is the
maximum tension on the force loop, increases with drive
velocity. Because the position is here controlled, we are
able to calculate the trajectory past the force minimum
and beyond the jump out of contact. In an experiment
that controlled the load, the force minimum would be the
last point measured in contact. The position of the minimum force moves to smaller, (more negative), nominal
separations as the velocity is increased. It can be seen
that the adhesion of the viscoelastic particles is significantly greater than that of elastic particles.
The velocity dependence of the adhesion is explored
in more detail in Fig. 12. As the velocity is decreased,
the curves asymptote to the static equilibrium elastic result, calculated from Eq. (7). It should be noted that
h(r,t) (nm)
20
15
10
5
0
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
r (µm)
FIG. 10. Surface profiles for adhesive viscoelastic spheres.
The profiles are plotted every millisecond, or every 2nm from
h0 =10nm (top) to -10nm (bottom). The drive speed is
|ḣ0 | = 2µm/s and the Hamaker constant is A = 10−19 J, with
z0 = 0.5nm and R = 10µm. The viscoelastic parameters are
E0 = 1010 N m−2 , E∞ = 109 N m−2 , and τ = 1ms. The
right hand panel is for loading and the left hand panel is for
unloading. The data are from Ref. [76].
4/21/01 3:12 PM
visco.xls Fig10
Fig 10
Attard
J Adh Sci Tech
This hysteresis in surface shape is reflected in the difference in force versus nominal separation curves on the
loading and unloading branches, Fig. (11). On approach,
prior to contact a given attraction occurs at larger nominal separation, for slower driving speeds. In these cases
there is an increased bulge leading to smaller actual separations, a consequence of the fact that viscoelastic materials soften over longer time-scales. The jump of the surfaces into contact is reflected in a sharp decrease in the
force. Once in contact the force increases and the nominal separation becomes negative, which is a reflection of
the deformation and growth of the flattened contact region under increasing load. The faster the particles are
driven together, the steeper is the slope of the force curve,
8
p
where f (t) ≡ 8πκRP 2 /E02 exp −κ[h0 (t) − u(t)]. and
u∞ (t) = −E0 f (t)/E∞ κ. For a given trajectory h0 (t),
the deformation u(t) is readily obtained from the preceding equation for u̇(t) by simple time-stepping. The
force in this approximation is essentially as given by
Derjaguin, except of course that the actual deformed
separation is used rather than the nominal separation
that would be appropriate for rigid particles. That is,
F (t) = 2πRκ−1 P exp −κ[h0 (t) − u(t)].
This central deformation approximation is tested
against the exact results for the pre-contact deformation
of a viscoelastic sphere being driven toward a substrate in
Fig. 13. The deformation is negative, which corresponds
to flattening of the particles under their mutual repulsion. It may be seen that differential equation is quantitatively accurate for the deformation. It correctly shows
that at a given position h0 , the deformation is greater at
the slower driving speed because the soft component of
the elasticity has more time to take effect. Conversely,
the force is greater at the faster driving speed because
the surface separation of the effectively stiffer material is
smaller at a given position (not shown).
the latter is not given by the JKR prediction, which as a
contact approximation that neglects the range of the van
der Waals interaction, is not exact. It can be seen that
for elastic materials, the JKR approximation is more accurate for particles with larger surface energies. As the
velocity increases, and the system is given less time to
equilibrate, viscoelastic effects become more evident and
the adhesion increases. For the present parameters, at
speeds greater than about 10µm/s, there occurs a noticeable dependence of the normalised adhesion on the
surface energy, with higher energy particles showing less
(normalised) adhesion. The actual adhesion increases
with surface energy at all driving velocities. This suggests that at very high speeds the adhesion will be independent of the surface energy.
6
5.5
5
2F*/3πγR
4.5
4
3.5
3
2.5
2
-0.1
1
0.01
0.1
1
Central Deformation u(t) (nm)
1.5
10
Speed (µm/s)
FIG. 12. Adhesion. The maximum tension normalised by
the JKR elastic adhesion as a function of drive velocity (logarithmic scale). The parameters are as in Fig. 10, except that
the Hamaker constant is A = 1, 5, and 10 ×10−20 J, (the surface energy is γ ≡ A/16πz02 = 0.80, 3.98, and 7.96mN/m), for
the dotted, dashed, and solid curves, respectively. The data
are from Ref. [76].
Fig 12
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J Sci Adh Tech
4/21/01 3:13 PM
visco.xls Fig12
-0.3
-0.7
-1.1
10-4
1
2
3
4
-1.5
0
1
2
3
4
5
Nominal Separation h0(t) (nm)
FIG. 13. Pre-contact flattening for repulsive forces. The
symbols represent the exact calculation, and the solid curves
are the central deformation approximation, Eq. (15). The
parameters are as in Fig. 10, with P = 107 N m−2 and
κ−1 = 1nm being used in the pressure law, Eq. (14). A
constant driving velocity of ḣ0 = 5 (upper) and of 1µm s−1
(lower) is used. The inset shows the corresponding forces normalised by the radius for ḣ0 = 1µm s−1 , with the bold curve
representing the infinitely rigid case (no deformation). The
data are from Ref. [75].
For the case of elastic particles, a relatively accurate
analytic approximation for the elastic integral has been
developed to treat the pre-contact situation [39]. The
elastic central deformation approximation (CDA) consists of replacing the deformation u(r) everywhere by its
value on the central axis, u(0). An analogous approximation can be made for the viscoelastic case, and results
in the form of an analytic differential equation have been
presented for the van der Waals attraction used above
[76], and for an electric double layer repulsion [75]. The
latter has the form
Fig 13
Attard
J Sci Adh Tech
The inset of Fig. 13 compares the load on a viscoelastic
sphere to that on an undeformable one at a given position. It can be seen that the load required to move the deformable particle a nominal amount (the drive distance)
is less than that required for a rigid particle because the
surface separation between deformed particles is greater
than that between undeformed particles. The agreement
between the central deformation approximation Eq. (15)
and the exact calculations in the inset confirms the validity of the elastic Derjaguin approximation. As the latter
shows, the major effect of deformation on the force arises
from the change in surface separation rather than from
(14)
In this case, the analytic approximation for the central
deformation u(t) ≡ u(0, t) is [75]
f (t)ḣ0 (t) − [u(t) − u∞ (t)] /τ
,
1 + f (t)
h0(t) (nm)
-1.3
0
B. Central Deformation Approximation
u̇(t) =
10-3
-0.9
4/21/01 3:14 PM
visco.xls Fig13
p(h) = P e−κh .
10-2
Force
F/2πR
(N/m)
-0.5
(15)
9
any increase in contact area due to flattening.
separations the deformation is always negligible because
here the force is weak. In practical terms of course it
is a matter of whether or not one has the instrumental
resolution to measure weak enough forces, and this is determined by the ratio of the cantilever spring constant to
the deformability of the substrate or particle. Assuming
that this regime is accessible, then at large separations
the measured force must equal that between rigid particles. If the latter is known, then this fact can be used to
shift the experimental data so that it coincides with the
known force law at large separations. When this is done,
the drive distance, which has arbitrary zero, is converted
to a nominal separation, which is the separation between
rigid particles. This procedure is now illustrated, as is the
method of calculating the deformation of the particles,
which allows the conversion of the nominal separation to
the actual separation.
C. Deformation and Adhesion Measurements
The AFM is an ideal tool for the study of viscoelastic
effects because of its real-time acquisition of data during
controlled dynamic measurements. The data that are
directly obtainable are the force as a function of drive
distance for both loading and unloading, and the adhesion. Detailed analysis of this data using the elastic and
viscoelastic theories described above should allow the extraction of the amount of deformation, and the values of
the elastic parameters and relaxation times.
In our laboratory we have recently commenced a research program of quantitative AFM measurement and
analysis of the interaction, deformation, and adhesion of
viscoelastic particles [82]. We use an emulsion polymerisation process to make poly-dimethylsiloxane (PDMS)
droplets [83,84]. The deformability ranges from liquidto solid-like, and is controlled by the ratio of trimer to
monomer cross-linker used in the synthesis. Depending
upon the conditions, micron-sized droplets form, and are
transferred to the AFM on a hydrophobic glass slide
to which they are allowed to adhere. A 7µm silica
colloid probe is attached to the cantilever; the welldefined and known geometry and surface chemistry of
the probe enables a quantitative analysis of the measurement. The zeta potential of the droplets is measured
by electrophoresis [85]. The surface chemistry of the
droplets is very similar to that of the silica probe; at
pH9.6 the zeta potentials are -46 and -62 mV, respectively.
There have been a number of previous AFM studies of
deformable solid surfaces [53–63]. In addition, the AFM
has been applied to air bubbles [86–89] and to oil droplets
[90–93]. Measurements of such systems raise two immediate issues: the determination of the normal sensitivity
factor, which relates the measured vertical photo-diode
voltage to the deflection of the cantilever, and the determination of the zero of separation. Two further issues of
analysis arise: the conversion of the nominal separation
to the actual separation, (ie. the determination of the deformation), and the relation of the material and surface
properties of the substrate to the measured interaction.
One can perform the vertical calibration by a prior
measurement on a hard substrate in the constant compliance regime. We performed this calibration in situ by
simply moving off the droplet and pressing against the
substrate [82]. If this is not possible, (because either
the drop is macroscopic or because a deformable probe
is attached to the cantilever), then one can perform the
calibration on another cantilever provided that one takes
care with the remounting and alignment of the laser, as
described in §II above and in Ref. [12].
The matter of determination of the zero of separation
can only be done if the force law is known. At large
100
10
Force
(nN)
80
1
Force (nN)
60
0.1
40
20
Nominal Separation (nm)
0.01
-100
-50
0
50
0
-20
-40
100
200
300
400
500
600
700
Drive Distance (nm)
FIG. 14. AFM measurement of the force between a PDMS
droplet (-46mV) and a silica sphere (-62mV) in 1mM KNO3 at
pH9.8. The drive speed is 1.2 µm/s, and the drive distance is
with respect to an arbitrary zero. The flat force extrema arise
from photo-diode saturation. Inset. Force on a logarithmic
plot. The zero of the nominal separation is determined by
shifting the data to coincide with the electric double layer
force at large separation calculated using the measured zeta
potentials. The straight line is the linear Poisson-Boltzmann
law for rigid particles, and the partly obscured curve is the
elastic central deformation approximation, Eq. (18), with a
fitted elasticity parameter, E∞ = 7 × 105 J m−3 . The CDA
is shown dashed for h0 < −19nm, which, for a pure double
layer interaction, is the point of actual contact, h = 0. The
data are from Ref. [82].
4/21/01 3:14 PM
visco.xls Fig14
Fig 14
Attard
J Sci Adh Tech
Figure 14 shows the force between a silica sphere, (diameter 7µm), and a solid-like PDMS droplet, (diameter
1.2µm, 50% trimer), measured as a function of the drive
distance [82]. After the initial zero force regime, one can
see the electric double layer repulsion due to the interaction of the two negatively charged surfaces. At a force
of around 20nN there is a jump into contact due to a
van der Waals attraction, followed by a soft compliance
regime. The latter is characterised by a finite slope and
a non-zero curvature. Upon reversing the direction, (ignoring the instrumental saturation at about 35nN force),
the soft compliance is again evident, with the change in
10
slope indicating hysteresis. The adhesion of the surfaces
contributes to this hysteresis, and they do not jump apart
until being driven a distance of several hundred nanometres from the point of maximum load. (Again the instrumental saturation at about -35nN is ignored.)
The analysis of the data is illustrated in the inset to
Fig. 14. The zero of separation is established by shifting
the measured data horizontally to coincide with the linear Poisson-Boltzmann law at large separations. It can
be seen that over a limited regime the data is indeed
linear on the log plot, with a slope corresponding to the
expected Debye length. The relatively short range of this
regime is due to a combination of the large deformability
of the PDMS droplet and the stiffness of the cantilever,
k = 0.58N/m, chosen in order to measure larger applied
loads and as much of the adhesion as possible. The data
at the largest separations are only just significant compared to the resolution of the AFM; that the data apparently begins to decay faster than the Debye length at the
extremity of the range exhibited is due to contamination
by interference fringes.
The linear Poisson-Boltzmann law used here is given
−κD h0
, where κ−1
by F (h0 ) = 2πRκ−1
D = 9.6nm is
D P0 e
the Debye screening length, h0 is the nominal separation
(between rigid particles), and R = 0.6µm is the radius of
the PDMS droplet. In linear Poisson-Boltzmann theory,
the pre-factor in the pressure law, Eq. (14), is given by
P0 =
20 r κ2D ψ1 ψ2 ,
The inset of Fig. (14) compares this elastic CDA with
the measured data using a fitted elasticity of E∞ = 7 ×
105 N/m2 . At large separations in the weak force regime
it coincides with the rigid particle result, but due to the
extreme softness of the particles, the force increases much
less rapidly than the linear Poisson-Boltzmann predicts.
The CDA predicts that the surfaces come into actual contact, (h = 0), at a nominal separation of h0 = −19nm,
and the theory is continued past this point as a dashed
line. There is a noticeable increase in the steepness of
the data beyond this point, which suggests that the force
is no longer a pure double layer interaction. The agreement between the approximation and the measurements
is quite good, which confirms the utility of the former
and the role of deformation in the latter.
The CDA shows becomes relatively linear on the log
plot at negative nominal separations, as do the measurements. Effectively, the Debye length has been renormalised due to the elasticity of the substrate. It is
straightforward to obtain from Eq. (18) an expression
for the CDA decay length in this regime. The limiting
force is given by
0 −κh0
,
F (h0 ) = 2πRκ−1
D P0 e
where the decay length is
κ=
(16)
eqψ/2kB T − 1
.
eqψ/2kB T + 1
P00 = P0 e−κω .
(20)
(21)
The length ω was defined above and the regime of validity
of this result is −ω < h0 κ−1
D .
The amount of deformation is substantial, being of the
order of 100nm at the largest applied loads, compared to
a particle diameter of 1200nm. It is possible that the turn
up in the force just prior to the van der Waals jump could
be due to the contribution from the underlying rigid substrate at these large deformations. Alternatively, there
is some evidence that this is instead due to a steric repulsion due to extended polymer chains; see above and
below.
The viscoelastic nature of the PDMS droplet is clearly
exhibited in Fig. 15, which shows the velocity dependence of the interaction. (The hydrodynamic drainage
force is negligible here.) In general the repulsive force at
a given drive position increases with increasing drive velocity. This is consistent with the notions that underlie
the creep compliance function, namely that viscoelastic
materials are initially stiff and soften over time. One
may conclude from the data that relaxation processes
decrease the force at a given nominal separation for particles that are being more slowly loaded. The physical
mechanism by which this occurs is the flattening of the
particle, which increases the actual separation and consequently decreases the force. Driving more slowly allows
time for this deformation to occur.
(17)
As discussed above following Eq. (15), the central
deformation approximation (CDA) for elastic particles
gives for the pre-contact deformation [39]
p
u = − 8πR/κD E 2 P0 e−κD [h0 −u]
≡ −ωe−κD [h0 −u] .
κD
,
1 + ωκD
and the renormalised pressure coefficient is
where 0 = 8.854 × 10−12 is the permittivity of free
space, r = 78 is the dielectric constant of water, and
ψ1 = −46mV and ψ2 = −62mV are the surface potentials
of the PDMS and the silica sphere, respectively, which
are measured independently by electrophoresis [85]. In
practice an effective surface potential is used, which
essentially converts this into the non-linear PoissonBoltzmann law in the asymptotic regime [94,95]. One replaces ψ by 4γkB T /q, where q = 1.6×10−19 is the charge
on the monovalent electrolyte ions, kB = 1.38 × 10−23 is
Boltzmann’s constant, T = 300K is the temperature,
and
γ=
(19)
(18)
Although this can be solved by iteration to obtain the
deformation u for any nominal separation h0 , for the purposes of plotting it is easier to specify h and to calculate
directly the corresponding u and h0 . The resultant force
−κD h
is F (h0 ) = 2πRκ−1
, where the actual separation
D P0 e
is h = h0 − u.
11
2.5
measured data likely indicate actual contact of a diffuse
polymeric steric layer. (Miklavcic and Marčelja have used
mean-field theory to model the interaction of polyelectrolytes and obtained a similar initial softening of the
double layer repulsion followed by a steeper steric interaction [96].) That this kink occurs at a substantially
lower load than the putative van der Waals jump identified in Fig. 14, and is of different character, supports a
model of the PDMS droplet as a dense core surrounded
by a diffuse corona of polymer tails.
10
F (nN)
Force (nN)
2
1
1.5
1
h0 (nm)
0.1
-50
0.5
0
0
-30
-20
-10
0
10
20
30
40
50
20
Nominal Separation (nm)
FIG. 15. Velocity dependence of the PDMS loading curve.
From top to bottom the velocities are 3, 1, and 0.5 µm/s.
The curves are the viscoelastic central deformation approximation using fitted parameters E0 = 5 × 106 J m−3 ,
E∞ = 5 × 105 J m−3 , and τ = 0.03s. The bold curve is
the double layer force between rigid particles. Inset. Force
on a logarithmic scale. The data are from Ref. [82].
Fig 15
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J Sci Adh Tech
15
10
Force (nN)
4/21/01 3:14 PM
visco.xls Fig15
5
0
-5
The viscoelastic CDA has been fitted to the data in
Fig. 15. The long-time elasticity, E∞ = 5 × 105 N m−2 ,
is a little less than that used in the elastic CDA fitted in Fig. 14; evidently the latter incorporates some
of the initial stiffness. The fitted short-time elasticity,
E0 = 5 × 106 N m−2 , is substantially greater than the
short time one. At the fastest driving velocity shown
the loading curve approaches that between rigid surfaces. The relaxation time used in the approximation
is τ = 0.03s, and it is sufficient to describe the transition from short- to long-time behaviour observed in the
experiments.
The viscoelastic CDA may be described as semiquantitative. There are a number of reasons for the evident discrepancies between the theory and the experiments. First is the obvious fact that the CDA is an approximation to the full viscoelastic theory. In particular
it is not accurate when there is substantial surface flattening, as occurs, for example, in the post-contact regime.
Second of course is the simplicity of the three-parameter
viscoelastic model. Doubtless there are multiple relaxation modes in the PDMS droplet, and the model is only
useful in so far as one dominates the experiment. Third is
the use of the purely exponential double layer force law.
Close to actual contact this is not correct, (due for example, to the non-linear nature of the Poisson-Boltzmann
equation and also to charge regulation effects, such as
constant potential boundary conditions). Despite these
simplifications the CDA represents a viable approximate
theory that can be used to extract the material parameters of viscoelastic materials.
An additional consideration is that close to contact
other forces will start to contribute, as discussed in connection with the CDA prediction of contact in Fig. 14.
In particular, the kink in the data in Fig. 15 at a load of
1.5–2nN. is evidence of such a non-electric double layer
force. This and the subsequent steeper gradient of the
-10
-50
0
50
100
150
Nominal Separation (nm)
FIG. 16. Hysteresis and adhesion of the PDMS particle.
The velocities are |ḣ0 | = 4, 2, and 0.5 µm/s, from top to
bottom at the point of reversal. The data are from Ref. [82].
4/21/01 3:14 PM
visco.xls Fig16
Fig 16
Attard
J Sci Adh Tech
Figure 16 shows the velocity dependence of the hysteresis and and the adhesion of the PDMS droplet. The
area of the hysteresis loops, which gives the amount of
energy dissipation, increases with drive speed, which is
what one would expect for a viscous system. The maximum load drops with decreasing speed, as predicted by
the viscoelastic theory, Fig. 11. The difference between
Fig. 11 and Fig. 16 is that in the former the turning point
is at a fixed nominal separation, whereas in the latter it is
at a fixed drive distance; the nominal separation at fixed
drive distance decreases with speed due to the decreased
cantilever deflection.
The adhesion, which is the minimum load, or, equivalently, the maximum tension, also increases with drive
speed. What is also noticeable on the retraction curves
are the long-range attractions that increase with separation and that appear as discrete steps. These may be
attributed to individual bridging polymers, with the flat
regions corresponding to the peeling of the polymer from
the silica sphere segment by segment, and the regions
of increasing force corresponding to the stretching of the
individual polymer chains. Such forces between individual bridging polymers have been explored in other AFM
measurements [63,97–101]. Between one and three bridging chains can be seen in the individual force curves in
Fig. 16. The force due to the longest bridging polymer
is remarkably independent of velocity.
12
a rapid jump into contact with no data available between
the onset of the attraction and the jump; but see below).
These steps in the data provided the key to understanding the physical origin of the force. It was proposed that
there were sub-microscopic bubbles present on the hydrophobic surfaces, and that each step represented the
instant of attachment of a bubble on one surface to the
other surface [107,126]. These bridging bubbles spread
along the surfaces and give rise to the measured force.
An attractive feature of the ‘nanobubble’ theory is that
the range of the interaction between hydrophobic surfaces
is set by the height of the bubbles on the isolated surface,
and there is no need to invoke any new long-range force
to account for the data. The fact that calculations of the
force due to multiple bridging bubbles were in quantitative agreement with the measured data provided strong
support for the proposed physical origin [107].
V. BRIDGING NANOBUBBLE DYNAMICS
A. Experimental Evidence
In 1972 Blake and Kitchener [102] found that bubbles ruptured at inexplicably large separations from hydrophobic surfaces, but it took a decade before the existence of a long range attraction between such surfaces
was confirmed by direct force measurements [103–105].
The force appeared to be universally present between
hydrophobic surfaces, (ie. those on which water droplets
had a high contact angle), and was much stronger than
the van der Waals attraction, which was the only other
known attractive force between identical surfaces. It produced extremely large adhesions, and it had a measurable range of hundreds of nanometres [106,107], which is
orders of magnitude larger than most surface forces.
The broad features of this unusual force were reproduced in a number of laboratories and many efforts were
made to explain its origin. The earliest attempt at a
quantitative theory suggested that the surfaces coupled
by correlated electrostatic fluctuations, with the consequence that the decay length of the attraction should
be half the Debye length [108]. This idea was subsequently taken up and developed by a number of authors [109–112]. Although several experiments appear
to show the predicted dependence on the electrolyte concentration, [104,105,113], the vast majority are insensitive to the concentration or valence of the electrolyte
[107,114–117]. One must conclude that the proposed
electrostatic mechanism is not in general the origin for
the measured hydrophobic attraction. It had also been
proposed that surface induced structure in the water was
responsible for the long range interaction [118]. This
poly-structural theory is contradicted by the evidence
from computer simulations, which show that the structure induced by surfaces propagates less than about 1nm
into the water [119,120]. Further, the fact that the solvophobic force measured in non-hydrogen bonding organic
liquids is almost identical to that measured in water has
also been taken as evidence against the theory [121]. Finally, vapour cavities had been observed between the
hydrophobic surfaces when they were in contact [122],
and a theory for the force in terms of separation-induced
spinodal cavitation has been developed [123–125]. It is
difficult to design an experimental test of this theory.
In 1994 Parker [107] explored the phenomenon with
his MASIF device [3,4], a type of AFM that uses macroscopic surfaces, (radii 2mm), and like that instrument
electronically collects large volumes of data at high resolution. Some of this data is reproduced in Fig. 17, where
the extreme range and strength of the attraction is evident. The steps in the force at large separations had not
previously been seen with the surface forces apparatus,
because of its low resolution and few data points. (They
are also difficult to see with the AFM because the low inertia and weak spring constant of the cantilever leads to
0
50
100
150
200
0
-2
Separation (nm)
F/R (mN/m)
0
-4
-6
-8
-10
-12
-0.2
-0.4
-0.6
-0.8
-1
100
150
200
250
FIG. 17. Force measured between hydrophobic glass surfaces in water, (MASIF, R = 2.1mm). Three separate approach curves are shown. Inset. Magnification at large
separations showing steps in the data. The data are from
Ref. [107].
4/21/01 3:26 PM
dynbbub.xls Fig17
Fig. 17
Attard
J Adh Sci Tech
Further support for the notion that nanobubbles preexisted on the hydrophobic surfaces and that their bridging were responsible for the measured attractions subsequently came from de-aeration experiments, which
showed that the force tends to be more short-ranged
when measured in de-aerated water [116,127]. Wood and
Sharma [127] showed that the force was also of shorter
range when measured between surfaces that had never
been exposed to the atmosphere, which suggests that the
bubbles attach to defects in the surfaces when they were
taken through the air-water interface.
In 1998 Carambassis et al. [117] obtained AFM results
that, by virtue of the detail of the force curves, provided
significant support for nanobubbles as the origin of the
long-range attraction. By using a colloid sphere attached
to the cantilever, they were able to obtain the force due
to a single nanobubble in the contact region, and their
results were more readily interpretable than the multiple
bubble results of Parker et al. [107]. Perhaps the most
striking new feature that appears in Fig. 18 is the short
range repulsion that appears prior to the jump into contact. The data suggests that prior to interaction there is
on one surface in one case a nanobubble of height about
60nm, and in the other case a nanobubble of height about
13
150nm. The evident repulsion prior to the jump toward
contact is in part a double layer interaction between the
liquid-vapour interface and the approaching solid surface.
A quantitative theory for the data following the jump
has been made by Attard [128] and is discussed in more
detail below. According to the theory, the jump into contact following the initial repulsion is due to the bridging
of the bubble between both surfaces, and the extended
soft-contact, varying-complience region is a dynamic effect due to its lateral spreading. The results of Carambassis et al. [117], have been confirmed by a number of
similar AFM measurements [129–132]. These later papers include measurements of forces in de-aerated water,
and concur with Sharma and Woods’ earlier conclusion
that the force was on average shorter ranged in this case
[127]. Finally, infra-red spectroscopy has been used to
show the presence of gaseous CO2 between aggregated
hydrophobic colloids [133].
assuming diffusive equilibrium of the gas with the atmosphere leads to the prediction that all bubbles are unstable [107,126,128]. The constrained Gibbs free energy for
an arbitrary bubble profile z(r) is
G([z]|X, h0 )=
p0 V − N kB T ln V + γAlv − ∆γAsv ,
where kB is Boltzmann’s constant, V [z] is the volume
of the bubble, Alv [z] is the liquid-vapour surface area,
Asv [z] is the solid-vapour surface area, X represents the
fixed variables listed above, and h0 is the separation of
the solid surfaces.
The equilibrium bubble profile, z(r), may be obtained
by functional differentiation, which results in the EularLagrange equations and which was the original procedure
used to obtain the force due to a bridging bubble [107].
Alternatively, the profile may be parameterised by a suitable polynomial expansion and the optimisation may be
carried out with respect to the coefficients, which procedure has certain numerical advantages [128]. If the coefficients are denoted by ai , then the dependence of the profile on them and upon the separation may by symbolised
as z(r; a, h0 ). The equilibrium profile z(r) = z(r; a, h0 ),
is the one that minimises the constrained potential and
hence the equilibrium coefficients satisfy
∂G([z]|X, h0 ) (23)
= 0.
∂ai
a
20
Force (nN)
10
15
1
10
0.1
0.01
5
50
100
150
200
100
120
250
0
-5
-10
-15
0
20
40
60
80
140
160
The thermodynamic potential is the minimum value of
the constrained potential, G(X, h0 ) ≡ G([z]|X, h0 ). The
force between the solids is [128]
∂G(X, h0 )
F (h0 ) = −
∂h0
X
∂G([z]|X, h0 )
=−
∂h0
a,X
∂Alv
∂V
−γ
.
(24)
= ∆p
∂h0 a
∂h0 a
Separation (nm)
FIG. 18. Force between a silica colloid (R = 10.3µm) and
glass surface. Both surfaces were hydrophobed by exposure
to silane vapour, and the AFM measurements were performed
in 9.5mM (crosses) and 0.19mM (triangles) NaCl at a drive
velocity of 4.5µ/s. Inset. Large separation repulsion on a
logarithmic scale. The curve is the calculated hydrodynamic
drainage force. The data are from Ref. [117]
Fig. 18
Attard
J Adh Sci Tech
dynbbub.xls Fig18
4/21/01 3:27 PM
(22)
Taken in total, the evidence in support of the existence
of nanobubbles is overwhelming. There is now general
consensus that they are responsible for the long range
attractions measured between hydrophobic surfaces, as
originally proposed by Attard and co-workers [107,126].
Even though the ai depend upon h0 , the second line follows from the variational nature of the constrained thermodynamic potential, as manifest in the preceding equation [134,135].
One advantage of the constrained thermodynamic potential approach is that the approach to equilibrium
can be explored by holding particular variables constant.
This is illustrated in Fig. 19 where the potential is plotted as a function of the contact radius. Minima in the
potential correspond to equilibrium values. Whether
these minima are local or global determines whether
that particular size is stable or metastable. It can be
seen that there are deep minima at microscopic radii,
and more shallow minima at sub-microscopic radii. Microscopic bubbles are absolutely stable at small separations and sub-microscopic bubbles are absolutely stable
B. Theory for Bridging Bubbles
In order to calculate the force due to a bridging bubble one must first calculate the bubble shape. This is
done by optimising the appropriate constrained thermodynamic potential [134,135]. In this case the external
atmospheric pressure, p0 , the temperature, T , the liquidvapour surface energy γ, and the difference in solid surface energies, ∆γ > 0, (the contact angle at equilibrium
is θ = cos−1 [−∆γ/γ]), are fixed, as is the number of
gas molecules, N . The last condition is important, as
14
at large separations, and that there is an overlapping
regime at intermediate separations where one branch is
metastable with respect to the other. (All the bridging
bubbles are stable with respect to the hemispherical bubble on the isolated surface, which has Gibbs free energy
of 50.35pJ.) Hence the bridging bubble is hysteretic; approaching from large separations the bubble is initially
sub-microscopic before jumping to microscopic dimensions, and conversely upon retraction, with the reverse
jump occurring at larger separations.
Figure 20 shows the equilibrium shape of the bridging
bubble. In accord with the constrained thermodynamic
potential calculations of the preceding figure, one can see
that at small separations the equilibrium bridging bubble
has a microscopic lateral radius, whereas at larger separations it is sub-microscopic. There is a marked distinction
between the two sizes. On the isolated surface this bubble sits as a hemisphere of radius 50nm, height 41.3nm,
and contact radius 49.2nm. Hence it can be seen that at
small separations the bubble has expanded laterally by
more than a factor of twenty. In general the bubbles are
concave or saddle shaped, which indicates that the internal gas pressure is less than the external atmospheric
pressure. However, the departure from cylindrical shape
is relatively small, and it will be shown below that approximating the bubble as a cylinder provides simple but
accurate results for the force due to the bridging bubble.
40
30
20
31
10
30
0
0 29
0
-10
0.05
-0.2
0.1
-0.4
0
0.5
1
1.5
2
Force (µN)
Constrained Gibbs Potential (pJ)
50
Contact Radius (µm)
FIG. 19. Gibbs potential for a bridging bubble as a function of the constrained contact radius. The surface separations are, from bottom to top, h0 = 30, 40, 50, 60, 70, 80 and
90 nm. The equilibrium radius, which is given by the minimum in the potential, is microscopic at small separations,
and sub-microscopic at large separations. The liquid-vapour
surface tension is γ = 0.072 Nm−1 , the external pressure is
p0 = 105 Nm−2 , the hydrophobic surfaces are both of radius
R = 20µm and have an equilibrium contact angle of θ = 100◦ ,
and the number of gas molecules is fixed at N = 1.4×105 . Inset. Magnification of the minimum at sub-microscopic radii.
The data are from Ref. [41].
Fig. 19
Attard
J Adh Sci Tech
dynbbub.xls Fig19
4/21/01 3:27 PM
z(r;h0) (nm)
-40
-50
20
40
200
400
600
800
1000
1200
1400
r (nm)
FIG. 20. Equilibrium shape of a bridging bubble. The bubble shrinks as the separation increases, from right to left the
microscopic bubbles occur at separations of h0 = 0, 10, 20, 30,
40, 50, 60, and 70 nm. The other parameters are as in the preceding figure. Inset. Magnification of the large separation,
sub-microscopic bubbles, with, from right to left, h0 = 60,
70, 80, 90, and 100 nm. The first two profiles are metastable
with respect to their microscopic counterparts at the same
separation. The data are from Ref. [41].
4/21/01 3:27 PM
dynbbub.xls Fig20
-0.05
50
20
40
60
100
80
100
Fig. 21
Attard
J Adh Sci Tech
The hysteresis due to the local minima in the constrained thermodynamic potential appears clearly in the
force plot, Fig. 21. The force due to the bridging bubble is attractive and monotonically increasing with separation. It is weak on the sub-microscopic branch and
much stronger on the microscopic branch. The jump on
approach occurs at smaller separations than that on retraction.
Also shown in Fig. 21 is the force due to a cylindrical
bridging bubble. In this approximation the optimum radius of the cylinder, r(h0 ), is obtained by minimising the
constrained thermodynamic potential given above. For
microscopic cylinders, the pressure inside the bubble may
60
-60
0
-1.4
dynbbub.xls Fig21
4/21/01 3:28 PM
10
-30
-0.03
FIG. 21. The interaction force due to an unconstrained
bridging bubble, (parameters as in Fig. 19). The attraction is large at small separations where the bubble is microscopic, and it is weak at large separations where the bubble is sub-microscopic. Note that the jump between the
two branches occurs at smaller separations on approach,
h0 = 52 nm, than on retraction, h0 = 80 nm, which gives
rise to hysteresis in the force. The dotted curve that terminates at h0 = 76nm is the bridging cylinder approximation, Eq. (25). The horizontal arrow is the classical capillary
adhesion, Eq. (26). Inset. Expansion of the force on the
sub-microscopic branch. No bridging bubble with these parameters is stable beyond h0 = 112 nm. The data are from
Ref. [41].
30
-20
-1.2
Separation (nm)
20
-10
-0.01
-1
0
50
0
-0.8
-1.6
60
40
-0.6
Fig. 20
Attard
J Adh Sci Tech
15
be neglected. The inverse formula for the separation as
a function of radius can be given explicitly [128]
h0 = 2
p
R2 − r2 − 2R +
2Rr∆γ − 2r 2 γ
√
.
(rp0 + γ) R2 − r2
and that the thermodynamic driving force towards equilibrium is small compared to dissipative forces, (see the
discussion of viscoelasticity in §III). Similar contact angle hysteresis occurs for a hemispherical bubble in contact with a substrate. Hence for the present problem of
a bridging bubble, one expects hysteresis and velocitydependent effects as the bubble spreads or recedes.
Of course in order to have hysteresis one must have
dissipation, and the simplest model is to invoke a drag
force that is proportional to the velocity and length of
the contact line,
(25)
The force is F = −πr 2 p0 − 2πrγ. It can be seen in
Fig. 21 that the bridging cylinder approximation is quite
accurate for the force on the microscopic branch.
The adhesion or capillary force due to the bridging
bubble is also of interest. The largest radius occurs at
contact, h0 = 0, and in the
cylinder approximai
h bridging
p
tion it is r ∗ = (−3γ/2p0 ) 1 − 1 + 8Rp0 ∆γ/9γ 2 [128].
Fd = −2πarc ṙc .
The capillary adhesion given by F ∗ = −πr ∗2 p0 − 2πr ∗ γ.
As can be seen in Fig. 21, this result is more accurate for
small colloids than the classical result
F ∗ = 2πRγ cos θ.
(27)
Here rc is the contact radius, ṙc is its velocity, and a is
the drag coefficient. The physical origin of the contact
line friction is not clear, although two likely contributing
mechanisms are viscous dissipation due to hydrodynamic
flow in the contact region [137], and jumping of the contact line between asperities [136,138]. In the state of
steady motion of the contact line, the thermodynamic
driving force must exactly balance the drag force,
(26)
(Both results agree in the limit of large R.)
C. Spreading Bubble
−
The calculated force in Fig. 21 appears qualitatively
different to the measured forces shown in Fig. 18. Although the experiments show a definite jump toward
contact, the attraction is about two orders of magnitude weaker than the calculated adhesion. In addition,
the pre-jump repulsion and the soft-contact, varyingcompliance region are missing from the calculations.
Obviously, the calculated force due to the bridging
bubble is only relevant after attachment of the bubble to
the approaching surface, and no attempt has been made
to describe the force curve prior to this point. The large
separation repulsion evident in the inset of Fig. 18 is in
part due to the hydrodynamic drainage force between
the colloid and the substrate, F = −6πηR2 ḣ0 /h, where
η = 10−3 kg m−1 s−1 is the viscosity of water. The sharp
increase in the repulsion immediately prior to the jump is
probably a combination of deformation plus an electrical
double layer repulsion. The decay length of the measured
force was observed to decrease with increasing electrolyte
concentration, but was about one fifth the Debye length
in pure water, and about twice the Debye length in 10mM
monovalent electrolyte [117].
The soft-contact, varying-compliance region prior to
the colloid probe coming into hard contact with the substrate appears to be a dynamic effect due to the spreading of the bubble (ie. surface drying). For the case of a
liquid drop on a surface, it is well-known that a growing drop makes a greater angle of contact with the substrate than a shrinking one, and that the gap between
the advancing and receding angles increasing with increasing velocity [136–138]. The existence of hysteresis
and dynamic effects indicates that the equilibration of
three phase contact occurs over macroscopic time scales,
∂G(rc |X, h0 )
− 2πarc ṙc = 0.
∂rc
(28)
The first term is the derivative of the constrained thermodynamic potential of a bridging bubble of fixed contact
radius rc but otherwise of optimum shape (cf. Fig. 19).
This differential equation for the contact radius may
be solved for a given trajectory h0 (t) by simple timestepping [128]. The force between the probe and the
substrate was taken to be given by Eq. (24).
20
15
Force (nN)
10
5
0
-5
-10
-15
0
20
40
60
80
100
120
140
160
Separation (nm)
FIG. 22. Dynamic force due to a spreading bridging bubble. The AFM data is that of Fig. 18 [117] and the curves are
Eq. (28) using a fitted drag parameter of a = 3.2kN s m−2
[41]. The curve passing through the crosses is for N such that
on the isolated substrate the hemispherical bubble has radius
Rb = 75nm and height zb = 62nm, and the curve passing
through the triangles is for N such that Rb = 200nm and
zb = 165nm. The other parameters are as in Fig. 19.
dynbbub.xls Fig22
4/21/01 3:28 PM
Fig. 22
Attard
J Adh Sci Tech
Figure 22 shows that this model of contact line motion
is able to quantitatively describe the measured data in
the soft contact regime. The rapid jump toward contact
upon bubble attachment, the minimum in the force, and
16
the ever-steepening repulsion are all present in the theoretical calculations. The origin of the repulsion is that
the drag on the contact line prevents the bubble growing
to its optimum size at a given separation. As the colloid
particle is driven toward the substrate, the consequent
compression of the bubble leads to the repulsive force.
Several simplifications have been made in the model
calculations. The calculations are for two identical
spheres of radius 20µm, whereas the experimental data
is for a sphere of radius 10.3µm interacting with a flat
substrate. Similarly, the calculations are for a symmetric bridging bubble, which is likely a poor approximation
to reality immediately following attachment to the approaching surface. Additionally, in the latter regime the
velocity of the contact line is almost certainly changing
rapidly, and assuming steady state conditions likely introduces errors here. Finally, no attempt has been made
to include the pre-attachment forces in the calculations.
The bubble was taken to attach when the separation
equalled its height on the isolated surface, which was fitted to the data, and the initial contact radius was chosen
to give zero normal force at this point.
Because of the variability in the measured data, and
because of the limited number of force curves analysed,
one cannot yet claim to have confirmed the drag law (27).
Nevertheless it is of interest to compare the fitted drag
coefficient, a = 3.2 × 103 N m−1 s−1 with the value of
6 × 10−2 N m−1 s−1 estimated by de Ruijter et al. [138]
from molecular dynamics simulations of a spreading hexadecane droplet. The large discrepancy between the two
may be due in part to the low viscosity of the simulated
liquid, (two orders of magnitude less than that of water),
to the low surface tension, (about one fifth that of water), and to a low level of coupling between the substrate
and the liquid in the simulations. The average speed of
the contact line in the simulations is about 1m s−1 [138],
whereas in the experiments [117] and in the theory [128]
the bubble spreads at about 10µm s−1 . In both simulations and theory the product of drag coefficient and
velocity is 3–6×10−2 N m−1 , is of the same order of magnitude as the surface tension.
Despite the caveats outlined above, the agreement between theory and experiment supports the notions that
bridging bubbles are responsible for the measured forces,
and that it is the motion of the contact line that gives rise
to the details of the force curve. Accordingly, the theory
combined with the dynamic force measurements allows
the phenomenon of dynamic wetting to be followed with
molecular resolution.
viscoelasticity, and wetting.
In the case of friction a quantitative method of calibrating the torsional spring constant and the lateral
photo-diode response was described [12]. The method
is direct, non-destructive, and single step. The friction
between metal oxide surfaces in aqueous electrolyte was
measured as a function of applied load using the pH to
control the adhesion [13]. It was found that with the
detachment force used to shift the applied load, friction
became a universal function of the intrinsic load independent of the pH. Elastic deformation calculations further
revealed that surfaces with the same intrinsic load were
at the same local separation, which suggests that friction
is mediated by the short-range interactions between the
atoms.
A theory for the deformation and adhesion of viscoelastic particles interacting with realistic surface forces of
non-zero range was summarised [75,76]. A triangular
drive trajectory led to hysteretic force loops, with the
hysteresis and the adhesion increasing with velocity. A
central deformation approximation was introduced that
gave accurate analytic results in the pre-contact regime,
and that allowed the zero of separation in AFM force
measurements to be established. AFM measurements on
PDMS droplets were shown to be qualitatively in accord
with the theory, and the viscoelastic material parameters
were extracted from the data by fitting the theory to it
[82].
The force between hydrophobic surfaces has been ascribed to bridging nanobubbles [107], and the softcontact, varying-compliance region observed in AFM
measurements has been attributed to the drying of the
surface as the bubble spreads laterally [117]. This is a dynamic effect that depends upon the drive velocity. The
thermodynamic force due to a bridging bubble has been
calculated, and, assuming steady state conditions and a
simple model of contact line friction, a quantitative account of the measured data has been obtained [128].
Acknowledgement. It has been a privilege to work with
Archie Carrambassis, Adam Feiler, Graeme Gillies, Ian
Larson, John Parker, Mark Rutland, and James Tyrrell,
and I thank them for their very significant contributions
to the experimental work reviewed here. Discussions with
Kristen Bremmel, Sonja Engels, and Clive Prestidge have
also been helpful.
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[6] P. Attard, A. Carambassis, and M. W. Rutland, Langmuir 15, 553 (1999).
VI. SUMMARY AND CONCLUSION
The atomic force microscope is ideally suited to carrying out dynamic measurements that can elucidate a
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17
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