How well do these methods really work?
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I have recently been investigating the Sader and Thermal methods for normal force constant calibration. I see a few papers in the literature that compare these and other methods, most of which seem to give very good agreement if everything is done correctly (which is not necessarily trivial, see the string of corrections to the Thermal method in Cook et al.) A few papers show agreement within +/-20% or so between the methods, such as Cook et al. 2006, Ohler 2007. In your view, can one expect to get this good of agreement between methods (or better), or would you suspect that these papers exclude certain kinds of cantilevers or situations. Do you encounter cases where the results seem unreasonable or where there is larger scatter between these and other methods? I see that gmoeller has said that he generally sees good agreement between the methods, but can you elaborate?
S. M. Cook, T. E. Schaffer, K. M. Chynoweth, M. Wigton, R. W. Simmonds, and K. M. Lang, "Practical implementation of dynamic methods for measuring atomic force microscope cantilever spring constants," Nanotechnology, vol. 17, pp. 2135-45, 2006.
B. Ohler, "Cantilever spring constant calibration using laser Doppler vibrometry," Review of Scientific Instruments, vol. 78, pp. 63701-1, 2007. -
Matt, thank you for your post. These are excellent questions. As a very general comment, it seems that +/- 20\% simply is not satisfying, as we would like to be able to determine and verify smaller effects. With the ultimate limit of single-atom or single-molecule interactions that one can probe with AFM, it would be desirable to reliably resolve smaller differences between different measurements and have day-to-day reproducibility much better than this range. I don't see any fundamental reason why we could not do much better - so what are the sources of error? One thing worth discussing to resolve this are the assumptions in particular models/methods.
For example, the Sader method for rectangular cantilevers assumes:
- cantilever is made of a linear, elastic, homogeneous, isotropic material
- has a perfectly prismatic rectangular cross-section
- ignores any compliance at the base (rigid boundary condition)
- tip mass is ignored
- the excitation of the cantilever when measuring resonance frequency and Q are small (i.e., well within the linear deformation regime)
There are probably more. -
We've recently been working on a round robin comparison with other national metrology institutes. In this work, NIST, KRISS (Korea's national measurement lab) and PTB (
Germany's lab) have all calibrated the stiffness of the same cantilever using their own SI-traceable calibration schemes. The techniques are all based on pressing a tipped cantilever against an electromehcanical balance. Force and displacement are measured. One conclusion we are reaching is that the fricion of the contact seems to produce a variability at the level of 1% that will be very difficult to overcome. That said, it appears that cantilevers can be calibrated with accuracy at levels less than 5% in an absoulte fashion, and that this measurement can be repeated reliably. -
The rectangular cross section is actually very important for the Sader method. You cannot simply work in the moment of inertia for a different cross section into the equation because of the hydrodynamic function. The hydrodynamic function in Sader's method is only for rectangular cross sections. If you wanted to work in a different cross section you'd have to correct for this as well, which can be extremely difficult.
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I've tried to compare thermal/Sader and reference cantilever method for quite high spring constant (higher than 2N/m) and generally the thermal method gives, to my mind, too high values (e.g., 3 instead of 2) whereas the Sader and reference cantilever ones gives quite similar values for rectangular cantilever. The Sader methods yields quite wrong results on arrow shaped cantilever but I expect we have to use an equivalent width.
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In my case the reference cantilever are given and calibrated by Veeco... and they use the thermal method to calibrate them (I know that's quite in contradiction with what I said about the thermal method).
By arrow shape, I mean that there's an arrow at the end of a rectangular cantilever (FMR ARROW) -
I've had the same experience with rectangular cantilevers of a similar plan view. I think it just shows how important the geometry is when calibrating these cantilevers.
I also used Veeco's AFM Calibration Cantilevers that were calibrated by the thermal tune method and was very impressed by the quality of the cantilever. When I compared the spring constant values Veeco provided with what I had calculated using the Sader method I got errors ranging from 0.5% to 10%. -
Jonp, you said above that you've observed calibration within 5\%. I presume that you were talking about electrostic force balance calibrations when you said that, right? What about dynamic methods? Do you have a sense for how accurately those can be implemented? In practice, I guess that most users will have to use a dynamic method or a reference cantilever to perform calibration, because ESBs are not likely to ever be cheap enough to have built in to each AFM, do you agree?