- Theoretical methods
These methods rely on knowledge of cantilever material properties
E, the Young's elastic modulus and n,
Poisson's ratio. The main papers on the subject are by John Sader
& Lee White from Melbourne University. There is also a good
book for those who are Mathematica savvy: "Exploring SPM
with Mathematica" by Sarid. The
book contains routines which calculate k for a number
of different geometries, and employs a variational method described
by Chen et al.
J. E. Sader and E. White, Theoretical analysis of the static
deflection of plates for atomic force microscope applications,
Journal of Applied Physics 74 (1), 1-9 (1994).
J. E. Sader, Parallel Beam Approximation For V-Shaped Atomic
Force Microscope Cantilevers, Review of Scientific Instruments
66 (9), 4583-4587 (1995).
G Chen, R Warmack, T Thundat et al.,
Resonance Response of Scanning Force Microscopy Cantilevers,
Rev. Sci. Instrum. 65 (8), 2532-2537 (1994).
G. Y. Chen, R. J. Warmack, A. Huang et al., Harmonic Response
Of Near-Contact Scanning Force Microscopy, Journal of Applied
Physics 78 (3), 1465-1469 (1995).
- Static Methods
These methods utilise the deflection experienced by a cantilever
when a constant known force is applied to the cantilever. Relevant
papers include those by Senden, Torii, and Gibson. Senden measured
the deflection due to an added mass (his technique requires that
your AFM is invertible - so you can measure the deflection in
opposing directions). Torii and Gibson both have proposed methods
which rely on using reference cantilevers (which in turn must
also be calibrated).
A. Tori, S. Minoru, K. Hane et al., A method for determining
the spring constant of cantilevers for atomic force microscopy,
Meas. Sci. Technol. 7, 179-184 (1996).
T. J. Senden and W. A. Ducker, Experimental Determination
Of Spring Constants In Atomic Force Microscopy, Langmuir
10 (4), 1003-1004 (1994).
C. T. Gibson, G. S. Watson, and S. Myhra, Determination
Of The Spring Constants Of Probes For Force Microscopy/Spectroscopy,
Nanotechnology 7 (3), 259-262 (1996).
- Dynamic Methods
These methods focus on the dynamic behaviour of a cantilever.
Hutter and Bechhoefer describe a method based on measurement
of the thermal response of the cantilever, whereas others such
as Cleveland measured the change of resonant frequency caused
by the addition of known masses to the cantilever. Sader, White
& Mulvaney proposed a method based on measurement of the
unloaded cantilever resonant frequency combined with measurement
of the cantilever dimensions, so that k could be
determined with reference to a length scale invariant quantity
they call the 'normalised effective mass'.
J. E. Sader, I. Larson, P. Mulvaney
et al., Method For The Calibration Of Atomic Force Microscope
Cantilevers, Review of Scientific Instruments 66 (7), 3789-3798
J. L. Hutter and J. Bechhoefer, Calibration Of Atomic-Force
Microscope Tips, Review of Scientific Instruments 64 (7),
J Cleveland and S Manne, A nondestructive
method for determining the spring constant of cantilevers for
scanning force microscopy, Rev. Sci. Instrum. 64 (2), 403-405
Which way is the best?
All of these methods have their drawbacks, and the choice of
method is pretty much up to you. Current opinion indicates that
at least two mechanical methods should be employed, and a theoretical
calculation wouldn't hurt either.
How do I measure k?
I'm using a method that is suited to
what I'm setting out to do - that is, to measure force interactions.
I'm attaching spheres to the cantilevers, Au coating them, then
chemically coating the gold later on. So when I add a sphere its
there to stay...this may not suit everyone.
the resonance response of the cantilever using an AFM (if you
are used to working in an AC mode such as tapping, you already
know how to do this.)
attach a spherical particle to the cantilever. Make sure you know
what the particle is made of, since you want to be able to calculate
an SEM image of the whole cantilever top down to measure the dimensions.
In the same session take an image of a calibration grid so you
really know. Take an image of the sphere and measure its diameter.
Take a side view and measure the thickness of the cantilever (This
is critical as k is proportional to the cube of the thickness!)
remeasure the resonance response of the cantilever.
Now you have heaps of data with which
to determine k.
With the dimensions of the cantilever
and a knowledge of its material properties you can use the Chen method (via
routines as mentioned above). With the two frequency measurements
and the diameter of your particle and its density, i.e. you know
its mass, you can use the Cleveland method. With the unloaded frequency response
and the dimensions and density of the cantilever material you
can use the Sader, White &
Pitfalls and Oddnesses
When you don't check the calibrations of your instruments you
should not be surprised if your experimental results don't match
your calculations. Don't believe tech staff that the machine is
OK. Assume it isn't and prove that it is.
Attaching particles is not for the fainthearted or the impatient.
If you want to try it, write
me and I'll tell you where to get the spheres and resin.
Finding the 1st resonance can be difficult, especially on an
older machine. The clearest resonance may not be the one you want.