1. 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).

  2. 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).

  3. 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 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 (1995).

    J. L. Hutter and J. Bechhoefer, “Calibration Of Atomic-Force Microscope Tips,” Review of Scientific Instruments 64 (7), 1868-1873 (1993).

    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 (1993).


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.

first, measure 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.)

second, 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 its mass.

third, take 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!)

fourth, 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 Sarid's 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 & Mulvaney method.

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.