1. science
2. mathematics
3. geometry

2.2 Cantilever-Sample Force Interaction
2.2.1 Cantilever-sample interaction potential. AFM operation modes
Introduction.
Sample investigation is available thanks to the forces acting between a cantilever and a
surface. They are quite different. One or another force dominate at different tip-sample
separations.

During contact and the surface deformation by the cantilever, the elastic repulsion force
dominates; this approximation is called the Hertz model and is considered in the chapter
"Elastic interactions. The Hertz problem".

At tip-sample separations of the order of several tens of angstrom the major interaction is
the intermolecular interaction called the Van der Waals force (see chapter "The Van der
Waals force").

At the same distance between the tip and the sample and in the presence of liquid films, the
interaction is influenced much by capillary and adhesion forces. The range of capillary forces
considered in the chapter "Capillary forces" is determined by the liquid film thickness.

At larger separations the electrostatic interaction starts to dominate.

At separations of the order of a thousand of angstroms magnetic forces considered in the
chapter "Magnetic force microscopy" prevail.
Probe-surface interaction potential.
"Joining" potentials of forces acting at various tip-sample separations one can construct
the curve shown in Fig. 1 that allows to classify operation modes of an atomic force
microscope.
Fig. 1. Probe-sample interaction potential.
Atomic force microscope operation modes.
Depending on tip-sample separation during scanning, three modes of an atomic force
microscope are available (see Fig. 1): 1) contact, 2) non-contact, 3) "semicontact" which is
intermediate between contact and non-contact.
In the contact mode the probe tip directly touches the sample surface during
scanning. In the non-contact mode the probe is far enough and do not touch the surface.
In the semicontact mode the intermittent contact occurs. The last two AFM modes are
needed to implement modulation (or vibration) techniques.
Each mode is used to solve particular problems, some investigations being conducted
using various techniques in different modes. This gives the researcher a wide scope of
opportunities and allows to operate a microscope in a most appropriate and effective mode
under given experimental conditions.
For example, three AFM modes of the surface relief measurements exist:

contact atomic force microscopy – surface topography imaging in the contact mode.

non-contact atomic force microscopy – surface topography imaging in the non-contact mode
based on vibrating technique.

"semicontact" atomic force microscopy (or intermittent-contact atomic force microscopy) – in
this case the vibrating technique is used; the oscillating tip barely taps the sample surface.
Experimental techniques based on various AFM modes will be reviewed in respective
chapters. The instrument operation in the contact AFM and "semicontact" AFM is the basis for
another AFM techniques. Correct combining of measurements in three modes permits to
acquire new interesting results.
2.2.2 Elastic interactions. The Hertz problem
2.2.2.1 The Hertz problem definition
When the cantilever and the sample are in contact, elastic forces start to act giving rise
to both the sample and tip deformations which can affect the acquired image. To properly
interpret the results and choose the measuring mode one should have a clear idea of elastic
interactions in contact and "semicontact" modes.
Such consideration is necessary in order to:

avoid tip or sample damage during scanning - even at low loading force the pressure
in a contact zone can exceed the strength limit because contact area is very small.

reconstruct properly the sample surface topography basing on the acquired image
profile in case when surface features are of the same size as the tip curvature radius.

analyze forces in the "semicontact" mode at a moment of the tip contact with the
surface which directly affect the cantilever oscillation and are one of the damping
reasons.
Elastic deformations in the contact zone (the Hertz problem).
Let us consider first only the elastic force. The Hertz problem is deformations
determination at local contact of bodies under load
action.
We have to adopt some simplifying assumptions [1].
1.
Suppose that both the cantilever and sample materials are isotropic, i.e. their elastic
properties are described only by two pairs of parameters – Young's moduli
2.
3.
,
and
Poisson ratios
,
. (For the anisotropic materials the number of such independent
elastic characteristics can reach 21).
Assume that in the vicinity of the contact point the undeformed parts of bodies surface
in perpendicular planes orthogonal to the plane in the given point (Fig. 1) are
described by two curvature radii
,
(for the tip) and
sample area).
Deformations are small compared to surfaces curvature radii.
,
(for the studied
Fig. 1. Hertz problem definition.
Summary.

The Hertz problem solution allows to determine deformation parameters at a "point" of two
bodies contact.

The definition of the Hertz problem stipulates that the model of isotropic elastic media is
used and deformations are small.
References.
1.
Landau L.D., Livshits E.M. Theory of elasticity. – Nauka, 1987. – 246 p. (in Russian)
2.2.2.2 The Hertz problem solution
The general solution to this problem is well known (see chapter 2.2.2.3) though it is
written in an implicit form [1]. In order to get the general idea of the deformations in elastic
contact and obtain characteristic numerical values, we will confine to the analysis of two
spherical surfaces interaction – the tip and the small sample area. This means that
,
.
Fig. 2. Relation between contact area
Fig. 1. Hertz problem definition.
radius
and penetration depth
deformed state.
in
Under the load the contacting bodies deform in such a way that instead of a contact
point some contact area arises. Since the problem symmetry is axial, this area is clearly
circular. Denote its radius by
.
Let us introduce the following convenient quantities:
Young's modulus of the given pair of materials:
and effective
(1)
At small deformations (assumption 3 in chapter 2.2.2.1) the following geometric
relation between penetration depth
and contact circle radius
is valid:
(2)
which is clear from Fig. 2.
The Hertz problem solution relates the loading force
and the penetration depth
:
(3)
Accordingly, the pressure is the following function of the force:
(4)
The given solution for the case of two spherical bodies contact includes one important
special case of the flat sample contact with the tip having curvature radius
).
(
,
Let us depict the Hertz problem solution, i.e. the dependence of the penetration depth
(horizontal axis) upon the loading force (vertical axis) for positive
branch corresponds to the Hertz problem solution.
. In Fig. 3, the rising
Fig. 3. Force
depending on penetration depth
(graph of the Hertz problem solution).
As mentioned above, the solution can be obtained in implicit form for any kind of
surfaces (stipulated in condition 2 in chapter 2.2.2.1), however our goal is an exact
numerical result. Nevertheless, this result by the order of magnitude is the same as in our
simplified case. Therefore, we can estimate the characteristic contact pressure from formula
(4).
Results are tabulated in tables which present magnitudes of contact area and pressure
at various Young's modulus of a studied material. The data are calculated for the silicon
cantilever –
– with the curvature radius
magnitudes
and
Sample Young's
modulus, Pa
at two loading force
.
Contact area radius
, nm
Penetration due to
deformation
Contact pressure
, GPa
, nm
108
7.2
16
5.2
24
0.03
0.07
9
3.4
7.2
1.1
5.2
0.14
0.3
10
1.6
3.4
0.25
1.1
0.63
1.4
11
0.9
1.8
0.07
0.3
2.2
4.7
12
0.7
1.4
0.04
0.2
3.7
7.9
50
5
50
10
10
10
10
at loading force
5
50
, nN
5
Table 1. Comparative analysis of contact deformations arising during AFMinvestigation of materials having different elastic properties [2].
Material and its Young's
modulus
Quartz glass,
Kapron,
Contact area
radius
, nm
Penetration due to
deformation
Contact pressure
, GPa
, nm
3.74
8.04
1.04
6.46
0.11
0.25
3.24
6.98
1.05
4.87
0.15
0.33
Copper,
Tungsten,
Diamond,
0.79
1.7
0.062
0.29
2.55
5.51
0.68
1.46
0.046
0.21
3.44
7.47
0.64
1.38
0.041
0.19
3.88
8.36
50
5
50
at loading force
5
50
, nN
5
Table 2. Comparative analysis of contact deformations arising during AFMinvestigation of materials having different elastic properties using silicon
cantilever.
It is clearly seen that the contact pressure is higher for more stiff samples.
The other restriction (assumption 1 in section 2.2.2.1) is the problem solution within the
model of the continuum with isotropic characteristics. Naturally, on the microlevel, the
molecular structure is of great importance, therefore such assumption is rather relative. That
is why the Hertz problem solution with a more exact geometrical characteristics of contacting
surfaces (in contrast to the considered case) makes no sense because assumption 1 in
section 2.2.2.1 itself is a very crude approximation.
Summary.

In a place of a "point" contact of the tip and the surface the contact area is produced.

The Hertz problem solution allows to find the contact area radius and penetration depth as a
function of applied load.

Typical magnitudes for the AFM are as follows
a.
contact area radius – up to 10 nm;
b.
penetration depth – up to 20 nm;
c.
contact pressure – up to 10 GPa.
References.
1.
2.
Landau L.D., Livshits E.M. Theory of elasticity. – Nauka, 1987. – 246 p. (in Russian)
Gallyamov M.O., Yaminsky I.V. Scanning probe microscopy: basic principles,
distortions analysis (218 kB).
2.2.2.3 Exact Hertz problem definition and its solution in a general form
Let two solids be in a point contact (Fig. 1). We have to adopt the following simplifying
assumptions [1]:
1.
Bodies are filled with uniform isotropic linearly elastic media characterized by Young's
2.
3.
moduli
,
and Poisson ratios
,
.
The surfaces curvature weakly affects the mode of deformation.
Boundary surfaces are interchanged by the elliptic paraboloid.
4.
The point of contact is not the singular point, the contact area is the simply connected
domain and its contour is ellipse.
Fig. 1. Two bodies contact before
deformation.
Fig. 2. Deformation two bodies. Surfaces
before deformation are shown by dotted
line, and squeezed surfaces – full line. The
characters
and
denote lengths, which
are determined by equations (1) and (2).
The equation of surface near the point of contact is as follows
(1)
where summation is conducted over doubly recurring indices
characterizes the surface curvature and its principal values are
principal radii of contacting surfaces in point О).
,
. The tensor
and
(here
,
–
Similarly, for the second body:
(2)
Suppose the bodies are compressed by some force
and as a result they become
deformed and approach each other within small distance
(Fig. 2). Then, the contact area
will not be a point but surface portion having area
for an ellipse). Let
(
and
be
the components (along the
and
axes respectively) of displacement vectors of both
bodies surface points at compression (Fig. 2).
From the picture it is seen that for the points of the contact area the following equation
is valid:
(3)
or
(4)
For points outside the contact area the following is true:
(5)
Choose the axes
it. Denoting by
and
and
directions so that tensor
principal axes diagonalize
the principal values of this tensor, rewrite it as (4):
(6)
Quantities
and
are related to curvature radii
following formulas given here without derivation:
,
and
,
of both surfaces by
(7)
where
– angle between those normal sections of surfaces which have curvature radii
and
. Sings of curvature radii are considered to be positive if corresponding centers of
curvature are located inside the corresponding body and negative in the opposite case.
Denote by
the pressure between compressed bodies in a point of their contact.
The pressure outside the contact area is evidently
of normal forces
to be plain):
. Displacement
under the action
is determined by the following expression (surfaces are considered
(8)
Notice that from (8) it follows that ratio
is constant and is equal to:
(9)
Relations (7) and (9) directly determine the deformations
the contact area. Substituting expressions (8) into (7) we get:
and
distribution across
(10)
This integral equation describes pressure
distribution across the contact area.
Its solution can be found by computing technique used in the potential theory.
That is why we must consider the problem from the potential theory.
Let the charge with density
be uniformly distributed over the triaxial ellipsoid
(11)
Then the potential inside the ellipsoid is determined by the following expression:
(12)
In the extreme case of almost plane (in the
the potential is:
-direction) ellipsoid, i.e. when
,
(13)
(
-coordinates inside the ellipsoid are supposed zero).
The expression for the potential can be written in the other way:
(14)
where integration is performed over the ellipsoid volume. Assuming
in the radicand
and integrating by
within
, supposing
, we get:
(15)
where integration is performed over the ellipse
expressions for
area. Equating both
, we get the following:
(16)
Compare the integral equation (16) and equation (10). It is seen that the right sides of
equations contain similar quadratic functions of
and
while the left sides contain
integrals of the same type. Therefore, it is clear that the contact zone (which is the region of
integration in integral (10)) is limited by ellipse of the following type:
(17)
and that function
should be as follows:
(18)
The const is chosen so that integral
over the contact area is equal to force
of bodies compression. The result is:
(19)
This formula determines the pressure distribution over the contact area. Notice that
pressure at the center is half as much again the average pressure
.
Substitute (19) into (10) and replace the resulting integral by its expression in
accordance with (16):
(20)
where
– effective Young's modulus:
(21)
Equating coefficients at
and
as well as absolute terms of both sides, we get:
(22)
(23)
(24)
Equations (22), (23) define semi-axes
and
and
of the contact area at given force
(
are known quantities for given bodies). Next, using expression (22), we can
obtain the relationship between force
and bodies penetration
the right sides of equations are elliptical.
caused by it. Integrals in
Applying the obtained formulas to the case of two spheres with radii
we can write:
and
contact,
(25)
From the case symmetry it follows that
radius can be calculated from (23), (24) as:
, i.e. the contact area is circle which
(26)
in this case is the difference between the sum
center distance. From (18) the following relationship between
and the spheres center-toand
can be obtained:
(27)
So
and correspondingly
.
Dependence of the
,
type is valid not only for spheres
but for another bodies of finite dimensions. It can be easily proven from the similarity
consideration. If we substitute
,
,
, where
– arbitrary
constant, equations (23), (24) will not change. The right side of equation (22) will be
multiplied by
follows that
, therefore, it will be unchanged if
should be proportional to
is substituted for
. From this it
.
Summary.

The Hertz problem allows to determine parameters of deformation in a "point" of two bodies
contact.

Definition of the Hertz problem implies the use of uniform isotropic linearly elastic media
model and the assumption of deformations smallness.

In a place of the tip-sample "point" contact the contact area arises.

The Hertz problem solution relates the deformation and applied load. Penetration
proportional to the compressing force as
is
.
References.
1.
Landau L.D., Livshits E.M. Theory of elasticity – Nauka, 1987. – 246 p. (in Russian)
2.2.2.4 The effect of elastic deformations during experiment
Materials destruction during scanning.
Once the contact pressure is estimated according to formula (4) in chapter 2.2.2.2, it is
easy to determine which material can be damaged during scanning. For that it is enough to
compare materials ultimate strength (measured in Pa) and arising stress (pressure
Appendix 1.
). See
However, even if this strength is exceeded, no probe or sample material destruction can
occur during scanning. The point is that the overcritical pressure should act longer than the
damage process duration (relaxation time of elastic deformations is about
rather fast scanning of large areas this condition may fail. See Appendix 2.
s). During
Reconstruction of the surface feature shape from the scan line profile.
The change in the probe vertical position during scanning in the contact mode produces
profile which can differ much from the real surface topography. One of the reasons for that is
the elastic deformation of the tip and the sample. For example, the decrease in the organic
molecules vertical dimensions was experimentally established. Because these materials are
very soft, the probe "indents" protrusions on their surfaces (See Appendix 3 and chapter
2.5.1).
The second reason for the difference between scan profile and real surface geometry is
the tip-sample convolution. Its consideration is important when studying small (of the order
of the tip curvature radius) surface features. A finite tip dimension results in the lack of the
ability to probe narrow cavities on the sample surface thus decreasing their depth and width.
Similarly, convex features image appears wider. The convolution phenomenon is best
understood from Fig. 1.
Fig. 1. Tip convolution during scanning. The scan profile can differ much from real
surface geometry.
It can be seen that for an object having in reality radius
the measured
dimensions are larger depending on the tip radius (see details in the chapter 2.5.2).
If one assumes the simultaneous effect of convolution and deformation, it becomes
clear how much the image profile can differ from the real topography. In Appendix 4 it is
demonstrated that the acquired image needs to be analyzed and even computer processed in
order to obtain the sample real topography.
Non-elastic conservative contact forces.
As tip makes contact with the sample, some other forces arise besides the elastic one.
For example, the Van der Waals interaction (revealed not only when two bodies touch but
within some distance between them) leads to the contact pressure decrease because Van der
Waals forces in contrast to elastic ones are attractive but not repulsive.
Fig. 2. Plots of force
vs. penetration depth
. Shown are the Hertz problem solution
as well as solution with a hysteresis loop accounting for the nonconservative forces.
This alongwith other attractive microscopic interactions (not discussed here) result in
the downward shift of the plot (Fig. 3 in chapter 2.2.2.2) which illustrates the Hertz problem
solution. As can be seen, at
the force is negative. This means that as the tip slightly
touches the sample, an attractive force acts.
Nonconservative effects.
Besides elastic and Van der Waals forces, some other nonconservative forces exist:
from friction to energy dissipation by arising elastic waves ѕ phonones. These forces modify
the Hertz problem solution even greater. Let us consider the experimental consequences of
such contact interactions.
Particularly, due to the nonconservative forces, the tip adhesion (sticking to the surface)
takes place. In this case the touch and separation happen differently, i.e. the hysteresis
occurs.
The tip adhered to the surface carries a small "stuck" portion of the sample (during the
upward move) which, for some time before separation, goes up producing a neck (Fig. 3).
Fig. 3. "Sticking" of the sample surface portion to the tip is due to
nonconservative forces and results in hysteresis.
Such deformations will be treated as having negative penetration depth
. This means
that during the cantilever retraction upward, the mentioned plot (Fig. 2) can shift to the left
from the vertical axis until the stepwise separation occurs. In Fig. 2 arrows show the path in
coordinates when the tip moves down and up.
To describe the left part of the plot one should use more complex (compared to the
Hertz model) analytical models. They are examined, e.g. in [1].
The hysteresis loop in the plot means that in order to press the tip into the sample
surface and then separate it and bring it back, some work must be done. In other words, if
one hits the sample surface with the cantilever, the collision will be non-elastic. In
semicontact vibration mode such non-elastic sticking is one of the damping factors.
Summary.

In order to define critical experiment parameters at which sample or cantilever damage can
take place, one should consider elastic properties.

While scanning, the sample is indented, therefore, to reconstruct the real topography one
should account for an elastic deformation.

At the onset of the tip-sample contact, the other forces besides elastic ones arise including
nonconservative such as adhesion.
References.
1.
Handbook of Micro/Nanotribology / Ed. by Bhushan Bharat. – 2d ed. – Boca Raton
etc.: CRC press, 1999. – 859 p.
2.2.2.5 Appendices
Appendix 1.
An interesting fact is that if one sets equal scan parameters and uses the same probe
when studying hard and soft samples, the first one can be damaged while the second can
stay undamaged.
Consider two cases of scanning flat samples from mica and pyrolytic graphite. A silicon
probe with known characteristics is used. Let us calculate the contact pressure developed
under the same load force
respective materials.
and compare it with the ultimate strength of
The following values are used:
tip curvature radius:
load force:
moduli of elasticity:
[1], [2]
[2]
[2]
ultimate strength:
[1], [2]
[2]
[2]
To find the "effective elasticity"
we use formula (1) in chapter 2.2.2.2 where for the
sake of simplicity Poisson's ratios are ignored.
Substitution of values into formula (4) in chapter 2.2.2.2 yields:
It is clear that the softer sample can not be damaged. This, as pointed out before,
arises from the fact that for hard materials the contact area is very small so the pressure is
much larger as compared to softer materials.
Appendix 2.
The possibility of the sample or the tip destruction depends on the scan speed in the
contact mode. If during static measurements or slow scanning the load can exceed the
critical value, the destruction can occur not at high cantilever speed.
The reason is as follows: though at high probe speed the deformations at every point
are "overcritical", their duration is small so the sample has no time to be destroyed.
Because the scan speed depends on the scan area, the effect can suddenly manifest
itself while changing the image size when its decreasing results in the sample damage.
Let the pyrolitic graphite sample be imaged by the silicon cantilever. The goal is to
choose such scanning parameters that materials in contact are not damaged.
The following values are used:
tip curvature radius:
load force:
moduli of elasticity:
[1], [2]
[2]
ultimate strength:
[1], [2]
[2]
elastic relaxation time:
To find the "effective elasticity"
we use formula (1) in chapter 2.2.2.2 where for the
sake of simplicity Poisson's ratios are ignored:
The radius of the contact area arising from the force
formula (3) in chapter 2.2.2.2:
action can be expressed from
Let cantilever be moved with horizontal speed
. The time of the tip action upon the
given point (i.e. the time of tip travel across the contact area diameter) should be less than
the relaxation time:
The speed, in turn, is the line length
multiplied by the line scan frequency
–
. Therefore, materials will not be destroyed if:
т.е.
For example, if line scan frequency is
dimension is
, the minimum permissible scan area
.
Appendix 3.
Due to the elastic penetration of the tip into the sample the scan line profile differs from
the real geometry.
Let us determine the silicon cantilever penetration into the large organic molecule.
The following values are used:
tip curvature radius:
molecule dimension:
load force:
moduli of elasticity:
[1], [2]
[2]
To find the "effective elasticity"
we use formula (1) in chapter 2.2.2.2 where for the
sake of simplicity Poisson's ratios are neglected:
Using formula (3) in chapter 2.2.2.2 the penetration depth
can be expressed as:
where
Calculation gives
which is more than 10% of the molecule dimension.
Appendix 4.
While studying microobjects placed onto the substrate, one should notice that the tip
penetration results in a height lowering of small particles. It was experimentally proved that
this lowering can reach tens percent of the undeformed molecule dimension.
In order to calculate the profile change it is necessary to know not only the tip
penetration into the microparticle
depth
(see Appendix 3) but "particle-substrate" penetration
and "tip-substrate" penetration
. As seen in Fig. 1, the height lowering is:
Fig. 1. On the calculation of height lowering when scanning large molecules.
Let the large organic molecule placed onto the flat graphite substrate be scanned by the
silicon cantilever. The height lowering will be calculated using the following values:
tip curvature radius:
molecule dimension:
load force:
moduli of elasticity:
[1], [2]
[2]
[2]
Calculation of the penetration for every pair of materials is performed like in Appendix
3:
Total:
.
References.
1.
2.
Physical magnitudes. Reference book/ Ed. Grigor'ev I.S., Meilikhova I.Z.. – Moscow:
Energoatomizdat Publ., 1991. – 1231 pp.
Gallyamov M.O., Yaminsky I.V. Scanning probe microscopy: basic principles,
distortions analysis (218 kB).
2.2.3 Capillary forces
2.2.3.1 Basic principles of the surface tension theory
In most cases, the sample under investigation contains on its surface a microscopic
liquid film which affects much the cantilever interaction with the surface because the surface
tension force is of great importance on a microscale.
It is generally known that any interface is characterized by the free energy which is
proportional to its surface
:
(1)
where
– coefficient of surface tension (dyne/cm). It is clear that expressions like (1)
should be written for all surfaces of the system. At equilibrium, the free energy is minimal so
interfaces reshape in order to minimize the total
value.
Consider two consequences of this principle which will give us not only the intuitive
understanding of phenomena but provide with necessary formulas.
Fig. 1. Contact angle in case when one of surfaces is solid.
The boundary line of three media is characterized by the so called contact angle
1). If one of the surfaces is solid, the Neumann relation [1] is applied:
(Fig.
(2)
Subscripts denote the media separated by the surface with a given coefficient of surface
tension (CST). Actually, formula (2) arises from the demand of the free energy minimum.
One can verify that any deviation of
change that
in relation (2) will result in such an interface area
will increase.
The second consequence is the curved surface pressure appearance. Consider a local
flat surface area. It is clear that any deformation gives rise to its square and, hence,
increase. In order to minimize the free energy the curvature will tend to decrease which is
the evidence of the curved surface pressure appearance. The corresponding Laplace formula
for the difference between inside and outside pressure of a liquid is rather simple [1]:
(3)
where
and
– surface curvature radii in orthogonal planes (Fig. 2). If center of the
curvature is positioned outside liquid,
is taken negative.
Fig. 2. Liquid surface element.
Summary.

At the interface between three media the contact angle arises which is determined by
coefficients of surface tension and is calculated according to the Neumann formula (2).

The curved liquid surface produces additional pressure calculated according to the
Laplace formula (3).
References.
1.
Sivukhin D.V. General physics course: thermodynamics and molecular physics. –
Moscow. Nauka Publ., 1983 – 551 pp.
2.2.3.2 Capillary force acting on the probe
Let us examine the effect of the surface tension on AFM measurements. [1]At the
moment of a cantilever contact with a liquid film on a flat surface, the film surface reshapes
producing the "neck". The water wets the cantilever surface (Fig. 1) because the watercantilever contact (if it is hydrophilic) is energetically advantageous as compared to the
water-air contact. Notice that in such cases the contact angle is always less than
.
Fig. 1. "Neck" formation.
It is intuitively clear that the neck curved surface will tend to flatten that is available
only at the expense of the cantilever pulling down. This means that the cantilever attracts to
the sample.
Calculation of this attraction force is a simple task. Let the tip curvature radius be much
larger than other characteristic dimensions of the case. In Fig. 2 the following designations
are introduced:
– tip-sample separation,
lesser curvature radius of the liquid surface,
– "immersion depth",
– film thickness,
– tip-liquid contact area radius.
–
Fig. 2. On the calculation of the
capillary force.
Fig. 3. Explanation of the Laplace formula.
According to the Laplace formula the pressure in liquid will be less than the atmospheric
pressure by the following value:
(1)
,
This pressure is applied to the tip-liquid interface having square
of the cantilever attraction to the sample due to the capillary effect is equal to:
. The force
(2)
It is easy to find that
contact angles
(for the sake of simplicity we assume
for the tip and the sample to be equal). Then we get:
(3)
If we neglect the liquid film thickness (
valid and formula (6) is simplified to [1]:
), then the equation
is
(4)
We will not concentrate attention on
value determination. For the estimation purpose
we will use the maximum value of the capillary attraction force that takes place at
this case the unknown parameter
. In
vanishes:
(5)
Taking into account that cantilever radius
is equal to
is
and the contact angle is small i.e.
, water surface tension at
is close to 1, we get the
following estimation:
. Thus, the capillary force by the order of
magnitude is the same as the Van der Waals interaction and the electrostatic force.
During the cantilever approach-retraction cycle, the hysteresis arises. At the upward
move the neck stays longer because the cantilever surface is already wetted and the liquid
neck goes with the tip. As bonds break, the capillary force stops to act and the cantilever
suddenly returns into its undeflected state.
The cantilever deflection
vs. the distance to the sample
is shown in Fig. 4. At large
distances there is no deflection and the plot is horizontal. The left portion of the plot is linear
with a slope of
. This corresponds to the contact with the surface i.e. the tip is pressed
into the sample and the cantilever deflects towards the surface linearly.
As the cantilever is retracted away from the surface, the tip remains stuck to the
sample: the capillary force holds it and the linear deflection-distance dependence extends
below the horizontal axis until the neck disappears. After that the cantilever sharply flips off
and becomes undeflected.
Summary.

Due to the film of liquid frequently presented on the sample surface, the capillary force
arises which attracts the probe to the surface.

To quantify the capillary force it is enough to know the tip geometry and the surface
tension coefficient of the liquid.

When the cantilever is retracted from the surface the hysteresis is observed due to the
liquid necking.
References.
1.
Israelachvili J.N. Intermolecular and Surface Forces. – Academic Press, 1998. – 450
p.
2.2.4 The Van der Waals force
2.2.4.1 Intermolecular Van der Waals force
The Van der Waals force or the intermolecular attractive force has three components
of slightly different physical nature but having the same potential dependence on the
intermolecular distance –
. This lucky circumstance allows to compare directly constants
of interaction that correspond to three Van der Waals force components because proportions
between them will be held constant at different
will differ for various materials.
magnitudes. Constants at
multiplier
(1)
All three Van der Waals force components are based on dipoles interaction, therefore we
should remember two basic formulas:
the energy of dipole
placed in field
is [1]:
and the electric field produced by the dipole
[1]:
(2)
is
(3)
where
– unit vector directed from the point at
which the energy is determined to the dipole.
The orientational interaction (or the Casimir force) arises between two polar
molecules each of which has the electric dipole moment. In accordance with (2), (3) the
interaction energy of dipoles
and
separated by distance
(4)
depends sufficiently upon the molecules relative position. Here
directed along the line between molecules.
is the unit vector
In order to reach the potential minimum, dipoles tend to align along the common axis
(Fig. 1). The thermal motion, however, breaks this order. To determine the "resulting"
orientation potential
one should average statistically interactions over all possible
orientations of molecules pair. Notice that in accordance with the Gibbs distribution
, which gives the probability of the system being in the state with energy
at temperature
, the energetically advantageous orientations are preferable. That is why
despite the isotropy of possible mutual orientations, the average result will be nonzero.
Averaging with the use of the Gibbs distribution is performed in accordance with the
following formula:
(5)
where, for the sake of normalization, the denominator is the statistical sum and is the
integration parameter providing enumeration of all the system possible states (a pair of
dipoles mutual orientations).
If
the exponent can be approximated by the series expansion:
(6)
so the energy of orientation interaction is approximated as:
(7)
On performing integration it can be shown that
Introducing constant
, thus,
.
in accordance with (4), we finally have:
(8)
The induction interaction (or the Debye force) arises between polar and nonpolar
molecules. Electric field
generated by dipole
polarizes the other molecule (Fig. 2). The
induced moment calculated in the first order of the quantum perturbation theory is equal to
where
stands for the molecule polarizability.
Then, the potential of induction interaction is computed as follows:
(9)
Thus,
this
kind
of
interaction
also
"universally"
depends
on
though having the other reason and the other constant.
It should be noted that in liquids and solids the polarized molecule experiences the
symmetric influence of many neighbor molecules, the induction interaction being strongly
compensated by their action. The result is that the real induction interaction is estimated as:
(10)
The dispersion interaction (or the London force) is a prevailing one because it
involves nonpolar molecules as well. This third term in (1) is always presented that is why it
is the major one.
In a system of nonpolar molecules the electrons wave function
values of dipole moments in any state
is such that average
are equal to zero:
. However,
nondiagonal matrix elements
are nonzero. Moreover, the second quantum
mechanical correction to the interaction energy calculated as is known [2] according to the
formula below, is nonzero too:
(11)
where perturbation
is given by (4),
molecules in arbitrary states
and
.
,
– energies of the system of two
In a certain sense, "momentary" magnitudes of dipole moments (at zero average value)
are nonzero and they interact (Fig. 3). In the second order of smallness the averaged
magnitude of such "momentary" potential is not already vanished and namely this is the
potential of dispersion interaction.
Correction (11) as is seen, is proportional to the square of perturbation
it is clear that
. From this
(12)
,
Constant
potentials,
is called the Hamaker constant (here
,
,
– ionization
– molecules polarizability).
The classical interpretation of this interaction is as follows. The dipole moment of one
molecule arisen from fluctuations, generates field which, in turn, polarizes the second
molecule. The already nonzero field of the second molecule polarizes the first one. The
potential of this peculiar system with a "positive feedback" is calculated similarly to the
induction interaction.
Relative values of Van der Waals force components are presented in Table 1 [3].
Substance
0.667
0
13.6
0
0
6.3
1.57
0
13.6
0
0
41.3
1.74
0
15.8
0
0
59.3
1.6
0
15.8
0
0
48
0.2
0
24.7
0
0
1.2
1.99
0.12
14.3
0.0034
0.057
67.5
2.63
1.03
13.7
18.6
5.4
105
1.48
1.84
18.0
197
10
48.8
2.24
1.5
11.7
87
10
72.6
Table 1. Magnitudes of polarizability, dipole moment, ionization potential and
energies of different weak interactions between various atoms and molecules.
Obviously, the force is determined by
(13)
,
Estimations of the Van der Waals attraction for AFM studies in the contact mode give:
.
Summary.

The Van der Waals force which arises from the electrostatic interaction of molecular
shells has three components: orientation, induction and dispersion interactions.

Despite the different nature of the Van der Waals force components, their dependence
on distance is of the same character –
.
References.
1.
Sivukhin D.V. General physics course: Electricity – Moscow. Nauka Publ., 1983. –
687 pp (in Russian)
2.
3.
Landau L.D. Quantum mechanics: Nonrelativistic theory. – Moscow.: Nauka Publ.,
1989. – 767 pp (in Russian)
Rubin A.B. Biophysics: Theoretical biophysics. – Moscow.: Knizny dom Universitet,
1999. – 448 pp (in Russian)
2.2.4.2 Van der Waals probe-sample attraction
As it is shown in the chapter on Van der Waals (VdW) forces, the potential of the
molecules pairwise interaction depends on the distance as
equal to its derivative with respect to distance
:
. The corresponding force is
(1)
where
– Hamaker constant.
Basing on this microscopic description we can determine the attraction force between
the probe and the whole sample. It is equal to the sum of all pairwise interactions between
cantilever molecules and the studied surface:
(2)
It is clear that the result will depend sufficiently on the problem geometry.
Ignoring the discrete distribution of interacting centers (molecules) one can easily
proceed from the pair summation (2) to the double integral:
(3)
where
and
– concentrations of tip and sample molecules (density).
Let us calculate the inner integral denoting it as
. Its physical meaning is the
interaction force between a single molecule and a plane. The attraction force (1) decays
sharply with distance (
) therefore, the distant parts of the system do not contribute
substantially into the integral. Thanks to this, the integration can be performed over the
whole half-space as if it was a uniform sample.
Fig. 1. The single atom – flat sample system.
Introducing the cylindrical coordinates as shown on the Fig. 1, consider the origin of
coordinates to be our molecule. From the problem symmetry it is clear that the resulting
force is directed downward vertically. In this case the horizontal components of the attraction
force between a molecule and two symmetrical with respect to the
-axis molecules are
compensated. That is why it is convenient to consider only vertical force component:
(4)
This force is the same for all points of the ring with radius
around the
get:
axis just comes to multiplication by
, so integration over the angle
. Further calculations are easy so we
(5)
To take the outer integral in (3) we need to integrate over the tip volume:
(6)
Therefore, further calculations should be performed for the specific cantilever tip (see
Appendices).
Summary.

To find the probe-sample interaction force one needs to integrate the pairwise Van der
Waals interaction of cantilever and sample molecules.

The force of Van der Waals interaction between a separate molecule and a flat sample
decreases with a distance as

.
The probe-sample interaction force can be found by integrating the attraction of all
molecules to the sample surface.

2.2.4.3 Appendices


1. Parabolic or spherical probe.
The model applies to probes with a spherical tip at small tip-sample separations
. The same result is obtained generally for the parabolic tip.
(1)

In AFM measurements with silicon probe and sample at
:

.

2. Conical probe.
,
,

The model applies in case when tip curvature radius can be neglected as compared
to tip-sample separation (
).
(2)

In AFM measurements with silicon probe and sample at
:


,
,
.

3. Pyramidal probe.
The model applies in case when tip curvature radius can be neglected as compared
to tip-sample separation (
).
(3)

In AFM measurements with silicon probe and sample at
:


,
,
.

4. Conical probe with a rounded tip.
The model is a generalization of (1) and (2) at arbitrary ratio between
and
.
(4)

where
. At
(semispherical probe) formula (4) transforms into formula
(1) while at

(conical probe) – into formula (2).
In AFM measurements with silicon probe and sample at
:
2.2.7 Adhesion forces
2.2.7.1 The nature of adhesion
.
,
,
In Introduction we discussed two cases of cantilever-sample interaction within the range
of molecular forces action: the Van der Waals attraction if tip is out of contact with the
sample and elastic interaction if they are touching. In the middle range where attraction
forces between some of the probe-sample molecule pairs act (potential is proportional to
) and repulsive forces between some other pairs act too (potential is proportional to
), it is impossible to find the interaction force between the whole probe and the
sample.
Moreover, in the transition region a qualitatively new phenomenon – adhesion arises.
It originates from the short-range molecular forces. The character of adhesion affects the
force curve parts "fitting". The fitting function is called the adhesion interaction.
We have to distinguish between two types of adhesion: probe-liquid film on a surface
and probe-solid sample. If the first case turns into the capillary interaction considered in
chapter 2.2.3.2 "Capillary forces", the origin of adhesion forces between the probe and the
solid is the molecular electrostatic interaction.
Adhesion is a nonconservative process. Forces acting during the cantilever-to-sample
approach differ from the forces during the probe retraction (Fig. 1). Such an operation
requires some work to be done which is called the work of adhesion. This work has the
following components [1]:
(1)
The
subscripts
denote:
–
London
dispersion
interaction,
–
dipole-dipole
(orientation) interaction, – induction interaction,
– hydrogen bond,
–
bond,
–
donor-acceptor bond, – electrostatic interaction. Notice that the first three items represent
the work of Van der Waals forces.
The reason of adhesion – electrostatic forces at two bodies interface arising from the
formation in a contact zone of the electric double layer. Its origin is different for materials of
different type. For metals it is determined by the contact potential, states of outer electrons
of a surface layer atoms as well as by lattice defects; for semiconductors – by surface states
and impurity atoms; for dielectrics – by dipole moment of molecules groups at the phase
boundary.
Adhesion is the irreversible process. For example, if contact potential exists, electrons
start to drift resulting, as is well known, in the entropy increase. That is why forces differ
(see below) during the cantilever approach and retraction and the process is thereby
nonconservative.
To describe the adhesion quantitatively some approximating models are used. For solids
these are various corrections to the Hertz problem solution.
Summary.

Adhesion is sticking of two surfaces in contact due to electrostatic forces having
different nature for different materials.

Adhesion is a nonconservative process, therefore, to separate surfaces one needs to
expend an additional work. In the contact zone a "neck" arises.
References.
1.
Zimon A.D. Fluid adhesion and wetting – Moscow. Chimiya Publ., 1974. – 416 pp.(in
Russian).
2.2.7.2 The DMT model of solids adhesion
The DMT model [1, 2] (Derjagin, Muller, Toropov – 1975) is applied to tips with small
curvature radius and high stiffness. It is assumed that deformed surfaces geometry differs
not much from that given by the Hertz problem solution. Consideration of the Van der Waals
forces acting along the contact area perimeter results in an additional probe-sample
attraction which weakens forces of elastic repulsion.
Fig. 1. Plot of the force vs. the penetration depth.
The relation between the pressure and the penetration depth is as follows:
(1)
where
modulus
– tip curvature radius,
,
– contact area radius,
– effective Young's
– work of adhesion (see chapter 2.2.7.1).
References.
1.
Derjaguin B.V., Muller V.M., Toropov Yu.P., J. Colloid. Interface Sci. 53, 314 (1975).
2.
Derjuguin B.V., Churayev N.V., Muller V.M. Surface forces. – Moscow: Nauka Publ.
1985 (in Russian)
2.2.7.3 The JKR model of solids adhesion
The JKR model [1] (Johnson, Kendall, Roberts – 1964-1971) applies to tips with large
curvature radius (most likely to macroscopic bodies) and small stiffness. Such systems are
called strongly adhesive. The model accounts for the influence of Van der Waals forces
within the contact zone.
Fig. 1. Plot of the force vs. the penetration depth.
Due to these, the attraction arises which not only weakens the force of elastic repulsion
(the right part of Fig. 1b) but results in the neck creation (
and in the negative force. The plot is described by formulas:
, the left part of Fig. 1b)
(1)
where
– tip curvature radius,
– contact area radius,
– effective Young's
modulus
,
– work of adhesion (see chapter 2.2.7.1). An
example of the system to which the model applies is an eraser and paper.
References.
1.
Johnson K.L. Mechanics of contact interaction – Moscow: Mir, 1987 (in Russian).
2.2.7.4 The Maugis model of solids adhesion
The Maugis mechanics [1] (1992) is the most composite and accurate approach. It
can be applied to any system (any materials) with both high and low adhesion. The amount
of adhesion is determined by parameter
:
(1)
where
– interatomic distance.
DMT and JKR models are extreme cases of the Maugis mechanics corresponding to
different parameters
. For the stiff materials (DMT)
, for compliant materials (JKR)
.
Fig. 1. Plot of the force vs. the penetration depth.
The Maugis model assumes that the molecular attraction force acts within a ring zone at
the contact area border. The Maugis correction to the Hertz problem solution is expressed
implicitly via parameter
:
(2)
where
modulus
– tip curvature radius,
– contact area radius,
,
– effective Young's
– work of adhesion (see chapter 2.2.7.1).
Both JKR model and Maugis mechanics adopt originally the existence of hysteresis
during the approach-retraction cycle. It is assumed that during the cantilever approach the
attraction force arises sharply at the moment of touching, then the system proceeds from
point 0 into point 1 (Fig. 2 at web page). During the cantilever retraction the system
"describes" the other path 1-2 until the jump out of the contact occurs 2-3.
The loop 0-1-2-3 in the plot means that to separate the probe from the sample some
work must be done which is equal to the loop square. This is the work of adhesion
.
References.
1.
Maugis D.J., J. Colloid. Interface Sci. 150, 243 (1992).
2.2.7.5 Comparison of DMT, JKR and Maugis models
To compare the foregoing models we introduce the normalized radius of the contact
area
, force
and penetration depth
:
(1)
In Table 1 are collected major assumptions and restrictions of every theory while in
Table 2 – corresponding normalized equations [1].
Model
Assumptions
Restrictions
Hertz
No surface forces
Not applied to small loads in the
presence of surface forces
DMT
Long-range surface forces act only outside Contact area can be decreased due to
the contact area. Model geometry is as in the limited geometry. Applied only to
the Hertz model
small
JKR
Short-range surface forces act only within
the contact area
Force magnitude can be decreased
due to surface forces.Applied only to
large
Maugis
Tip-sample interface is modeled as a ring.
The solution is analytical but
equations are parametric. Applied to
all
values.
Table 1. Comparison of quantitative adhesion models.
Model
Normalized equations
Hertz
DMT
JKR
Maugi
s
Table 2. Normalized equations of quantitative adhesion models.
Fig. 1 shows plots of normalized force vs. normalized penetration depth for DMT, JKR
and Maugis models at different
the DMT model while at large
. As can be seen, at small
– the JKR model.
the Maugis model approaches
Fig. 1. Plot of the force vs. the penetration depth for the DMT, JKR and Maugis
models
for the different
values.
Summary.

Several theoretical models of adhesion having different ranges of application are
proposed. The most accurate one is the Maugis model.
References.
Handbook of Micro/Nanotribology / Ed. by Bhushan Bharat. - 2d ed. - Boca Raton etc.: CRC
press, 1999. – 859 p.