Introduction
This application note describes a method for measuring lateral forces as a function of applied load with a Pacific Nanotechnology
Nano-R™ SPM using a LabView interface to the Nano-R™ Software supplied with the AFM. Conventional friction measurements with
realistic, large-area contacts are generally characterized by a friction coefficient μ = FL/FN, the ratio of the lateral force FL to the
normal force FN and, according to Amontons’ law, this parameter is generally relatively independent of the normal force FN. Since
the lateral force depends on the contact area AC and the shear strength of the contacting interface S, as FL = ACS, this implies that
the contact area is proportional to the normal force. When measuring the friction force using an atomic force microscope, the contact
pressures are invariably below the plastic yield strength of the interface, which therefore deforms elastically. In this case, the contact
area does not vary linearly with the normal force. In addition, measurement of a force-distance curve in AFM indicates that there is
adhesion between the tip and substrate that contributes to the resulting normal force experienced by the tip-surface contact, which
must be taken into account.
The elastic contact area in AFM is given either by JKR [1,2] or DMT [3] theory, where JKR theory applies to compliant materials with
large, short-range attractive forces, while DMT theory applies to stiff materials with small, long-range attractive forces. The nondimensional
parameter μ defines which theory is applicable where:
Equation 1:
Where: R is the tip radius
γ is the work per unit area red elastic modulus
z0 represents the equilibrium squired to separate the tip from the surface
E* is the reducepacing for the interaction potential of the surfaces [4].
If μ<0.1, DMT theory should apply, while if μ>5, JKR theory should be valid. For DMT theory, the contact area ADMT is given by:
Equation 2:
Where: Lcorr = L + 2πγR and is the normal load corrected for adhesion between the
tip and substrate [3]
2πγR is the pull-off force measured from the force distance curve.
In the case of JKR theory [1,2], the contact radius a is related to the load L as:
Equation 3:
Where: the pull-off force is now 3πγR
Measurement of an interfacial shear strength in AFM requires a knowledge of the tip radius and the normal and lateral forces. However,
comparative measurements of frictional forces can be made, using the same AFM tip, without calibrating these values.
The method is similar to MFM. Images can be collected in DC (contact mode) by recording the deflection of the cantilever or by AC mode (close contact mode) where the cantilever is oscillated above the surface and either the phase or amplitude of the cantilever is recorded. In this work we chose to do DC imaging. However, there is no reason that AC imaging cannot be performed and by following the MFM application note this can easily be done.
Background
The position of the cantilever is monitored by the quadrant detector, where normal forces cause a vertical deflection of the cantilever
and a corresponding vertical motion of the “red dot” in the Red Dot Alignment screen shown in Figure 1, while a lateral force results in
a lateral motion of the “red dot” in the alignment screen. The corresponding vertical and normal voltages from the quadrant detector
are therefore proportional to the normal and lateral forces respectively. In order to completely decouple the normal and lateral forces
from each other, ideally the quadrants of the detector should be precisely aligned with the normal motion of the cantilever. In practice,
the quadrant detector is aligned at some, generally small, angle Θ with respect to the vertical motion of the cantilever so that pure
vertical motion of the cantilever can give rise to a lateral voltage change, while a purely lateral motion can give rise to a vertical voltage
change. Thus, in order to precisely measure normal and lateral forces using AFM the angle Θ must be determined. The method for
performing this measurement is outlined in reference 5.
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A friction force versus normal force curve is measured by initially bringing the tip into contact with the surface with some initial normal
force (given by some value of the quadrant detector voltage). The frictional force is then measured by moving the cantilever laterally in
one direction at some sliding velocity, and by measuring the lateral deflection
voltage in the quadrant detector. Since the sample is unlikely to be completely
flat, a tilt of the sample with respect to the horizontal trajectory of the tip can
add contribute to the measured forces. The lateral force is therefore also
measured for sliding in the opposite direction resulting in a “friction loop”
as shown in the figure. The friction force for a particular normal force is
taken as the average of the forward and reverse lateral forces. The normal
force is then decreased by some amount and the friction loop collected
once again. This procedure is repeated until the friction force decreases to
zero. The resulting plot yields values of lateral quadrant detector deflection
voltage versus normal quadrant detector voltage. Since the resulting force
at the surface is a sum of the applied force and adhesion between the
tip and substrate, for part of the measurement, for resulting forces less
that the pull-off force, the cantilever is in tension. In this case, the system
is susceptible to vibrations that can cause the tip to some out of contact
resulting in a sudden drop to zero lateral force and the tip moves freely in
space. This effect is minimized by carrying out the experiment as rapidly
as possible. If such effects are found, changing the location of the AFM or
mounting the system on an anti-vibration table may be required.
Precise measurements of contact areas also require a knowledge of the tip radius, the elastic modulus E* of the interface and the
normal and lateral force constants kN and kL respectively. The value of E* is obtained from:
Equation 4:
Where: E1 and E2 are the bulk moduli of the tip and substrate materials and v1 and
v2 their Poisson’s ratios
Tip radii can be measured directly using transmission electron microscopy (TEM). Note that the resolution available in scanning electron
microscopy (SEM) is generally not sufficiently high to allow precise radii to be measured. Alternatively it is possible to reconstruct tip
shapes by scanning a surface of known topology using SPIP software (http://www.nanoscience.com/products/SPIP.html). This latter
strategy suffers from the problem that the measurement itself may result in a change in tip shape or in contamination of the tip. The
normal force constant is determined from the from the resonance behavior of the tip [5], which can be measured using the Nano-R™
Software. The lateral force constant can me measured by collecting a lateral force versus normal force curve on a surface of known
topology. |
