Summary Notes – Force spectroscopy measurements and
processing
Cantilever choice
Spring constant
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Absolutely critical to choose a cantilever with the appropriate spring constant. There is no sense in trying to
extract something with extra processing that was badly measured in the first place.
For elasticity measurements, aim for equal indentation of the sample and deflection of the cantilever.
For single molecule binding/unfolding measurements, the forces are on the small end of the measurement
range, so soft cantilevers are required, but the very softest may have too much thermal noise vibration.
For surface adhesion measurements, the force depends on the contact area – i.e. generally larger for spherical
tips, which may need stiffer cantilevers.
Always make a rough calculation, then make some quick test measurements with quick analysis to check the
force range. Check the measurements are in the right range before making a large range of measurements for
statistics!
Other cantilever considerations
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Sometimes the cantilever shape is important, e.g. triangle shape for lateral stability to keep the tip position
stable during force measurements, or rectangle for better lateral force measurements.
The vertical deflection generally changes a lot more over time for silicon nitride cantilevers with gold coating,
than for uncoated silicon cantilevers. So for silicon nitride cantilevers the force “baseline” will need to be
corrected more often, and perhaps the position of the detector will need to be changed for longer experiments.
For chemical modification of the tip, for example to bind molecules for adhesion or recognition experiments, the
starting surface is critical. So even for “new” cantilevers, it may be necessary to plasma clean the surface
before starting the binding protocol.
Calibration for data analysis
Signals to calibrate
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The final analysis of the force curve is usually force against tip-sample separation. This means the vertical
deflection signal from the detector has to be converted from Volts into Newtons, and the z-position signal needs
to be corrected for the deflection of the AFM tip along the same axis.
Height calibration for the piezo movement is not needed, so long as the height (measured) channel is used.
This is the output from the z-sensor, which is linear and does not change over time. Height (measured) is the
best signal for almost all force measurements.
Sensitivity calibration (nm cantilever deflection per Volt signal of the laser detection system) is straightforward, it
is just important to have a clean, hard surface and to have a reasonable measurement range, e.g. around 1V to
have a reasonable detector signal, but not enough to damage the AFM tip.
JPK Force Spectroscopy Note
May 2009
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Spring constant calibration (N/m = nN/nm, nanonewtons force from the cantilever per nm of vertical deflection)
is the most complicated part of the calibration. There are various different methods, but the thermal noise
analysis is becoming the main standard for AFM experiments.
The dimensions of the cantilevers as they are manufactured are quite well defined in the lateral size, but there is
a significant error range in the thickness of the cantilever. The spring constant depends on the cube of the
thickness, so this translates into a large range of spring constants (typically a factor of 3-5). This depends on
the manufacturing process and cantilever type, so the range quoted in the manufacturers literature gives some
basis for the particular cantilever.
The uncertainty in the spring constant means it is necessary to calibrate each cantilever that is used for force
measurements, so that data from different cantilevers can be combined sensibly
Spring constant calibration - general
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There are various different methods to calibrate the spring constant of an AFM cantilever, and unfortunately all
of them have significant problems. If experiments are compared where different methods are used, differences
of maybe 10-20% can be expected.
The methods are reasonably consistent if they are used carefully, so it is often good to pick one particular
method and calibrate all cantilevers as consistently as possible. This means the data from different cantilevers
can be combined well to give good statistics. Then it is just important to realise that there will be some
systematic difference between the results from different methods.
The thermal noise analysis is becoming the main standard for AFM experiments, because it is available in liquid,
online during the experiment, through a fast, automated software analysis. There are some difficulties in the
theoretical analysis due to cantilever shape, liquid damping, etc., but the convenience and speed means it is
now very widely used.
Spring constant calibration methods
Calculation from dimensions
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One of the oldest and most basic methods for determining the cantilever stiffness is to calculate the theoretical
value, based on the exact geometry and material properties. Of course, the difficulties or inaccuracies in this
method are mostly based on the difficulty of knowing the geometry and material properties of a particular
cantilever.
The simplest formula is for a rectangular cantilever:
k=
t
l
w
E⋅w⎛t ⎞
⎜ ⎟
4 ⎝l ⎠
3
f =
(1.8751) 2 t
⋅ 2
2π
l
E
12 ρ
k spring constant, E Young’s modulus, w width, t thickness, l length
f resonance frequency, ρ mass density
JPK Force Spectroscopy Note
May 2009
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The lateral dimensions of cantilevers are typically 30-40 microns beam width, and 100-500 microns beam length.
These dimensions can be manufactured with a reasonable degree of accuracy, and also measured reasonably
accurately for a particular cantilever using optical or scanning electron microscopy.
The thickness of the cantilever beam ranges from around 5 microns for stiff cantilevers (for dynamic modes in
air, for example, 40 N/m spring constant), down to a couple of hundred nanometers for soft cantilevers (contact
mode in liquid, 0.01 N/m).
The small size of the cantilever thickness means that the relative error is higher, this dimension is most difficult
to control during manufacturing, and it is quite difficult to measure accurately after manufacture, even using
scanning electron microscopy.
For triangular cantilevers, more complex equations are needed to calculate the spring constant from the
dimensions, or some groups use finite element modeling calculations.
For cantilevers with metal coatings, the different materials could affect the calculated spring constant.
This method is not regularly used for soft cantilevers, it may be more useful for stiff cantilevers.
Dimensions plus frequency and/or Q
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Various more advanced calculations have been developed to get around the difficulty of determining the
thickness by measuring related values.
The frequency equation above also depends on the thickness of the cantilever, and the resonance frequency of
the cantilever is relatively easy to measure. So this opens up one approach to getting a more accurate spring
constant value.
One method is to use the lateral dimensions plus the resonant frequency of the free fluctuations of the
cantilever (Sader 1995).
A related method is to use the dimensions plus the frequency and quality factor (Q) of the resonance of the
cantilever (Sader 1999)
The method is relatively easy for rectangular cantilevers, if the nominal dimensions from the manufacturer are
used then only a frequency sweep and fit are required. The error in the nominal lateral dimensions is usually
reasonable, of course the method is more time-consuming if the dimensions have to be measured more
accurately for each cantilever.
These methods generally depend on the shape of the cantilever, so may be more complicated for V-shaped
cantilevers.
These methods are sometimes useful for stiff cantilevers, where the thermal noise method is not so appropriate,
and the shape is often a simple rectangle. For soft cantilevers, and particularly in liquid, this is not typically
used.
Added mass
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If the mass of the cantilever could be known, then this would be a way to reach the spring constant value.
Measuring the mass of the cantilever is not generally practical, but by changing the mass, the value can be
deduced.
This method generally involves adding small, known, masses to the end of the cantilever, and looking at the
change in the resonance frequency.
One problem is the accuracy – it is important to have well-characterised masses of the particles that are added
(usually metal spheres) and to position them at the very end of the cantilever. Both the mass and position
uncertainty give errors in the spring constant.
JPK Force Spectroscopy Note
May 2009
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The other problem is the time-consuming nature of the measurements (including characterisation of the
particles), which means that this technique is not generally used for routine calibration.
Reference cantilever
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The spring constant for a new cantilever can be measured by comparing it with a known cantilever.
Usually reference cantilevers are bought from cantilever manufacturers, and a single reference cantilever chip
(usually with several cantilever arms that have different spring constants) is used again and again as a
reference.
This method is most accurate when the reference cantilever has roughly the same spring constant as the test
cantilever.
The test cantilever force curve against a hard surface is measured to give a sensitivity calibration (using the
slope of the indentation part of the force curve). The test cantilever is then pressed against the reference
cantilever to get a spring constant comparison using the slope of this force curve.
⎞
⎛s
k c = k r ⎜⎜ s − 1⎟⎟
⎠
⎝ sr
sr is the slope measured on the reference cantilever
ss is the slope measured on a solid support
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The measurement is relatively straightforward, although it is important to align the cantilevers carefully so that
the meet only at the very end of the cantilever arm.
As the measurement is off-line and made separately, it is still more time-consuming than the thermal noise
method.
In general, this method should be just as accurate or straightforward for cantilevers with different V-shapes as
for simple rectangular cantilevers, because a direct comparison measurement is made rather than a theoretical
calculation.
Thermal noise analysis
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The position of the end of the cantilever is constantly fluctuating because of the thermal vibrations from the
environment, this can be thought of as a kind of diffusion restricted or balanced by the restoring force from the
spring constant.
The thermal environment of the cantilever is known, and the deflection of the cantilever can be measured
accurately, so the balance between them can be used to calculate the spring constant. The details will be
described in the next section.
This method is based on measuring the free fluctuations of the cantilever, so the main advantages are because
it is a passive measurement and can be made in liquid and actually in-situ during an experiment.
There are several problems due to the rather simple assumptions of the original version, and there are several
different correction factors that are needed to take care of special effects from the shape of the cantilever,
details of the hydrodynamic damping etc. Most of these factors just give a consistent shift of the values by 5-
JPK Force Spectroscopy Note
May 2009
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10%, however, and are not so important if the main focus is only to have consistent force measurements. This
is particularly true if the same type of cantilever is used for all the measurements.
For cases where results from differently shaped cantilevers should be combined, or where an accurate absolute
value of force is needed to compare with another technique, then these correction factors should all be
considered, and this can be a confusing process to begin with.
The method is most suited to soft cantilevers, where the free fluctuations are more significant, and where other
methods have significant problems. This is typical for the case of single molecule or single cell force
measurements.
The method is becoming established as the standard technique for these kind of cantilevers and experiments,
mainly because of the convenience.
Thermal noise analysis details
General description of the method
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The energy in the fluctuations comes from the natural thermal environment of the cantilever, for instance at
room temperature or physiological temperature.
The size of the fluctuations is measured by the AFM system. The vertical deflection is measured over some
time, and the collection of different vertical deflection values is analyzed. In theory, the data could be analyzed
as a histogram of vertical deflection values.
The measurement should improve with time, because of better statistics. However, the low-frequency
components would dominate over longer times, for instance because of cantilever deflection drift.
Therefore, the measurements are most often analysed by looking at the frequency dependence of the
fluctuations. This allows a more focused analysis of the data at the actual resonance, so that for instance low
frequencies or specific noise sources are excluded.
A Lorenz fit is made to the resonance peak (free fluctuations plotted against frequency), and the area under the
curve is used as a measure of the energy in the resonance.
Equipartition theory says that the energy in any free mode of the system has to be equal to the thermal energy
due to the absolute temperature of the system, ½ kB T, where kB is the Boltzmann constant (not related to the
spring constant!). The measured energy in the resonance is given by the spring constant and the average
value of the vertical deflection of the cantilever, here q.
1
1
k BT = k q 2
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The value of q2 is what is measured from the Lorentz fit to the frequency spectrum. This assumes, however,
that the movement of the cantilever is completely harmonic. In fact, there are various correction factors that are
needed to get a more accurate value from the fit.
JPK Force Spectroscopy Note
May 2009
Thermal noise in the JPK online SPM software
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The online SPM software from JPK Instruments is equipped with automatic thermal noise analysis for cantilever
calibration, several corrections are included and there is space for user input of specific correction factors,
depending on the type of cantilever and resonance peak.
The whole method is based on the simple harmonic oscillator equations. Consider first of all the amplitude –
frequency dependence of a simple harmonic oscillator:
2
A 2 ( f ) = η 2 + ADC
f 04
f 02 f 2
(f − f ) +
Q2
2
ADC is the D.C. amplitude (A in the software)
2 2
0
η
is the white noise background
f0
resonance frequency (f in software)
Q
Quality factor, width of resonance
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The values here are fitted from the thermal noise curve in the software, and the area under the fit curve (note
the original data) is used for the thermal noise calculation.
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The parameters to set in the software include the temperature, for the calculation of the thermal energy. In
general, the actual room temperature is not very critical, since it is the absolute temperature (normally
measured temperature plus 273) which is important. So a couple of degrees difference is fairly insignificant. If
the temperature is particularly high or low, this can be an important number to set.
There is a correction factor for the angle of the cantilever mount. This is included to take account of the
difference between the force or deflection normal to the cantilever, and in the vertical plane. The sensitivity
calibration using the piezo movement uses a vertical movement that is not perpendicular to the plane of the
mounted cantilever. This value is always 10 degrees for standard cantilevers and mounting, if there is a special
mount then this value can be changed. Both the normal and vertical spring constants are displayed in the
software, the vertical one is relevant for the experiment, and the normal value for comparison with the
manufacturer’s value.
There is another text field in the settings panel of the thermal noise calibration software to enter user-defined
correction factors. These factors depend on the type of cantilever, and whether the first or higher resonance
peaks are used for the calculation. Any factor entered here will be multiplied with the results of the basic
calculation to give the actual value used by the software for the measurements and display.
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Correction factors for the thermal noise analysis
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The rather simple assumptions in the basic thermal noise analysis cause a few systematic errors in the
measurements. There are various differences with the real measurement system.
One source of error is that the sensitivity measured by the force curve on a hard surface is for a relatively large,
static deflection of the tip. The cantilever bending shape during dynamic fluctuations is rather different, and
since the detection system is primarily sensitive to angular deflections it has a slightly different sensitivity for the
measurements of the thermal noise.
JPK Force Spectroscopy Note
May 2009
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Correction factors have been calculated for instance by Butt and Jaschke (Nanotechnology 1995) to take
account of the difference between z-deflection and angular deflection for the different bending modes of the
cantilever.
Usually the first resonance is used, as this has the largest amplitude, and therefore the best signal to noise for
accurate measurements. For very soft cantilevers in liquid, however, the first resonance is at frequencies
around 1kHz where it is affected by low frequency problems and environmental/acoustic noise. Therefore in
this case the second resonance can give more reliable results. The second and higher resonances have
different relations between z-deflection and angular deflection at the tip, and so different correction factors are
needed.
Peak
Correction factor
Comments
1
0.817
Generally used
2
0.251
Used when first resonance
frequency is too low
3
0.0863
Not generally used
Example correction factors from
Butt and Jaschke,
Nanotechnology 1995
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The shape of the cantilever is important for thermal noise analysis, because the deviations from a simple
harmonic oscillator depend on the shape of the cantilever. Factors have been calculated for rectangular
cantilevers (Butt and Jaschke), and computed using finite element analysis for a particular example of a
triangular cantilever (Stark, Drobek, Heckl), both giving values around 0.97 for the first mode/peak.
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For a fully accurate absolute force measurement, there are probably other minor correction factors that are
required for particular hydrodynamic drag functions, or other simplifying assumptions in the model. However, at
some point the significant errors from other parts of the measurement become more important. It is realistic to
expect errors in the range of 10-20% when comparing different cantilever calibrations, depending on the tip
shape and spring constant, and no method stands out as a real standard reference.
The speed and convenience of the thermal noise method means it is becoming established as the standard. It
is very valuable to be able to check the spring constant in liquid, and this enables the basic calibration of
cantilevers as they are used. As long as the calibration method is consistent and carefully done, the results are
reasonably reliable. For better consistency (translating into narrower force histograms) it is best to combine
results from force curves using the same type of cantilever, where the differences are minimized.
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JPK Force Spectroscopy Note
May 2009
Selected references
General cantilever calibration
C.A. Clifford, M.P. Seah,
"The determination of atomic force microscope cantilever spring constants via
dimensional methods for nanomechanical analysis"
Nanotechnology 16 (2005) 1666-1680.
J.E. Sader, I. Larson, P. Mulvaney, L.R. White
“Method for the calibration of atomic force microscope cantilevers”
Rev. Sci. Instrum. 66 (1995) 3789-3798
Spring constant calculation from
dimensions
Spring constant calculation from
dimensions and frequency
J.E. Sader, J.W.M. Chon, P. Mulvaney
"Calibration of rectangular atomic force microscope cantilevers”
Rev. Sci. Instrum. 70 (1999) 3967-3969.
Spring constant calculation from
dimensions, frequency and Q
J.P. Cleveland, S. Manne, D. Bocek, P.K. Hansma
"A nondestructive method for determining the spring constant of cantilevers for
scanning force microscopy"
Rev. Sci. Instrum. 64 (1993) 403-405.
Spring constant determination from
adding masses
Thermal noise calibration
J.L. Hutter, J. Bechhoefer
"Calibration of atomic-force microscope tips"
Rev. Sci. Instrum. 64 (1993) 1868-1873.
Original paper where thermal noise
analysis is described
H.-J. Butt, M. Jaschke
"Calculation of thermal noise in atomic force microscopy"
Nanotechnology 6 (1995) 1-7
Correction factors for rectangular
cantilevers, higher harmonics
R.W. Stark, T. Drobek, W.M. Heckl,
"Thermomechanical noise of a free v-shaped cantilever for atomic-force
microscopy",
Ultramicroscopy 86 (2001) 207-215
Correction factors for triangular
MLCT cantilevers
R. Levy, M. Maaloum
"Measuring the spring constant of atomic force microscope cantilevers: thermal
fluctuations and other methods"
Nanotechnology 13 (2002) 33-37
A. Maali, C. Hurth, R. Boisgard, C. Jai, T. Cohen-Bouhacina, J.P. Aimé
“Hydrodynamics of oscillating atomic force microscopy cantilevers in viscous fluids”
J. App. Phys. 97 (2005) 074907
JPK Force Spectroscopy Note
May 2009
Comparison of thermal noise with
other dynamic methods for triangular
cantilevers
Correction factors for rectangular
cantilevers, hydrodnamics and
modes