What’s the frequency?

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When I first got into the field of trying to make quantitative nanomechanical measurements with AFM, I was trying to measure the storage modulus (a viscoelastic modulus – see my blog on moduli) of polyethylene.  I went to my friendly rheology colleagues a couple of floors below my lab to discuss the results with them.  They were very puzzled at the moduli numbers we obtained. After much back and forth, it dawned (upon my rheologist friend before it dawned on me) that measuring the modulus at about 800kHz, which is what I was doing, was pretty close to useless in terms of a comparison to moduli that my fellow polymer chemists were measuring.  That is because when my colleagues measure moduli, they typically measure them in the 1-200Hz range, maximum 300Hz. That was at a frequency of over 1,000x lower than the frequency I was measuring.  And guess what, polymers are viscoelastic!  I.e. their moduli depend on frequency.  We were talking apples and oranges. That was back in 2010.

I don’t think enough polymer chemists populated the conferences and were involved in the development of AFM instrumentation, so this rather not-so-subtle problem was pretty much lost on the AFM community.  I am happy to report that in the ensuing 8 years, with more involvement of people who do appreciate this problem, the AFM community has become more aware of this intense frequency mismatch between AFM measurements and bulk mechanical measurements that are typically done with an instrument like a dynamic mechanical analyzer (DMA). 

There is a way around this frequency mismatch called time-temperature superposition (TTS).  This principle is a concept in polymer physics and can be used to collect viscoelastic moduli as a function of temperature that is then converted to viscoelastic moduli as a function of frequency.   Shift factors have to be calculated, ideally by an expert who is well-versed in this field.  But using TTS, the moduli at high frequency can be converted to a lower frequency counterpart to compare with bulk mechanical analysis.   It is quite fun to see all these AFM instrumentation engineers and physicists now refer to time temperature superposition in their talks as if they are discussing cantilever dynamics!

So for the most part, AFM scientists are now aware of this frequency mismatch problem, understanding that resonant AFM methods that occur at the hundreds of kHz or even MHz are in a different frequency regime than commonly reported viscoelastic moduli measured for bulk materials.

What still concerns though is that sometimes in an AFM measurement we think we know the frequency the sample is experiencing, yet still we don’t.  This especially concerns me for tapping mode and force curve measurements.   In a tapping mode measurement, we know the frequency of the cantilever as we tune it to its resonance frequency or very close to its resonance frequency.  For force curves, we set a frequency for the cantilever as it approaches the surface (usually a low frequency in the few kHz or Hz regime).   But that is not necessarily the frequency the sample is experiencing.  The tip makes contact with the sample only in a very small part of its motion and even when it reverses motion, but most of the motion the tip is out of contact with the sample.  It’s akin to hitting the sample with a hammer, where in that case the sample experiences that frequency plus higher order harmonics.  In my experience, trying to match viscoelastic moduli from these kind of measurements with bulk measurements still doesn’t work.  In addition to frequency problem, there is still another problem of remaining in a linear response regime for the material, but that will be the subject of another blog!

Dalia Yablon, PhD


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