Maximum indentation depth using the AFM in contact mode

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Ben Ohler replied on Fri, Aug 13 2010 12:27 PM

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Ashok,

You can predict the indentation under different conditions using the common DMT/Hertz indentation model:

F = (4/3)*E/(1-v^2)*(R^0.5)*(d^1.5)

where: F=Force, E=Elastic modulus, v= Poisson's ratio, R=Tip radius, d=indentation depth (i.e. deformation). It is often assumed that v=0.5, though in fact it's not usually well known.

An important assumption in this model is that it represents a sphere indenting a flat, homogeneous material. So it is absolutely critical that the indentation depth does not exceed the radius of the indentor. So if you are using a 10 micron particle probe then you will not be able to model the results for indentations greater than 5 microns. You don't state why you would like to indent deeper, but this will definitely pose a problem when trying to interpret the data.

Going back to your question, the first important factor to consider is how high of a force can you apply with that cantilever. If you start at a 0V non-contact deflection, then you might assume that you could measure up to 10V of deflection. Using the deflection sensitivity and the spring constant, you might conclude that F = 10V * 60nm/V * 0.01N/m = 6nN. But because the photodetector response is not perfectly linear over its whole range, best practice is to use a smaller portion of the range for force measurements. There's no absolute rule, but I would suggest 2V as a reasonable value. Using 2V of deflection, you calculate a maximum force of 1.2nN. You could, of course, use a stiffer cantilever to reach higher forces.

The modulus of different tissues can vary by orders of magnitude, but let's assume 100 kPa. Using the model above with R= 5000nm and F= 1.2nN, you calculate a deformation of only 20.9nm. Again, I'm not aware of a hard rule about how much indentation you should have, but it must be less than the tip radius but it should also be some reasonable portion of the radius. For lack of any consensus, let's say 10%.

This suggests that you need both a smaller sphere and a stiffer cantilever. Let's say R=2000nm and we target d=200nm and use a k=0.32N/m cantilever. With E still 100kPa, that would require F=22.5nN, which is a reasonable force for that cantilever. If you plug the equation above into Excel you can run any scenario you like.

I think you may have been approaching the question a little differently, considering the limited Z range of the MultiMode. It's true in the case you propose that this would be a problem. Clearly you can't achieve an indentation depth greater than your Z range even with an infinitely stiff cantilever. Ordinarily though this is not an issue because you would not normally use a particle tip much bigger than 5um radius and you would normally want to limit the indentation to maybe 500nm. So that's well in the range of the Z piezo, but you do need to consider whether you can apply sufficient force for that indentation, as I outlined above.

I hope that gives you some helpful background to help plan your measurements.

Regards,

-Ben

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replied on Fri, Aug 13 2010 2:35 PM

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Ashok,

In a previous study [Microscopy: Science, Technology, Applications and Education A. Méndez-Vilas and J. Díaz (Eds.) Combining atomic force microscopy and live cell imaging to study calcium responses in dorsal root ganglion neurons to a locally applied mechanical stimulus L. Ponce, A. Berquand, M. Petersen, and M. Hafner)] we also used 0.01N/m tipless cantilevers functionalized with a 10um bead to apply a minimal loading force to stimulate living neurons and simultaneously record a fluorescence response. The minimal loading force which could be calculated could be estimated around 150pN if I remember correctly.

Now, applying a maximal loading force into a tissue certainly requires a bead at the cantilever end but I really don’t understand why you want to use supersoft cantilevers… On the contrary, I would use tapping cantilevers having spring constants of 150-300pN. Other than this, the maximum indentation depth you can reach will highly depend on the piezo working distance. At first glance, I would say that the maximum force you can apply is about a couple of hundreds of nN, maximum a few uN, but the maximum indentation depth won’t exceed 20um. Also, at such a degree of piezo extension, you will never be able to control precisely the force you are applying. Of course, if you don’t necessarily want to target a specific location very precisely, you are not restricted to AFM: any nanodip system could make it.

Good luck!

Alex

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ashokr replied on Sun, Aug 15 2010 4:12 PM

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Ben and Alex:

First of all, thank you very much for your responses. I am new to AFM technology and I really appreciate the time you've taken.

Let me clarify a couple of things.

1. I am using a soft cantilever because to get good force-displacement curves, the stiffness of the cantilever should match the stiffness of the tissue being indented. In my application, the tissue that I am indenting (embryonic chicken heart) has a stiffness of about 0.002 N/m. Ideally I would like a cantilever that is about the same stiffness, but the lowest I could find was 0.01 N/m, which is about five times stiffer than the tissue - not ideal, but will work. Remember: if the cantilever is too stiff compared to the tissue, then it will hardly deflect and force measurements will become susceptible to noise (imagine indenting rubber with a thick steel cantilever). On the other hand, if the cantilever is too soft compared to the tissue, then it will deflect a whole lot and cantilever deflection and piezo displacement will be about the same (imagine indenting steel with a rubber cantilever). Indeed this is the procedure used in the AFM analysis to get the deflection sensitivity parameter where we indent a "hard" sample.

2. I am using 10 um bead because, I want to get somewhat "global" material properties.The diameter of my sample is 200 microns and if the tip is too small, then the properties will be too localized. Furthermore, It has been shown that if the tip is too sharp, we will overestimate the stifness of the tissue by as much as a factor of four (see "Elasticity measurement of living cells with an atomic force microscope: data acquisition and processing" by Carl and Shillers)

3. The material that I am indenting is hyperelastic and hence the modulus varies with the indentation depth. This is the reason I wanted to go in as much as 20 microns. The force-displacement is classic Fung-type exponential and the stiffness (and modulus) increases with the indentation depth.

Ben: could you give me a citation for the DMT/Hertz indentation model that you refer to ?

Alex: Usually I am applying between 0 to 20 nN of force. I did try indenting with a super stiff cantilever (RTESP, 40 N/m) and I was able to push the force to a 100 nN, but like I mentioned earlier, the cantilever deflection was too small (only about 3 nM) and I am not sure I trust the data.

And so here are two more questions:

1. The 10v span for maximum deflection - is that a fixed quantity for the Veeco scope ?

2. A recommendation that I use of a stiffer cantilever seems to be a recurrent motif. My concern is that with a soft tissue, the deflection of the cantilever will be too small if the cantilever is too stiff. How reliable are AFM measurements when the cantilever deflects only a few nanometers ?

Thank you again for your time.

Ashok

ps. I would like to attach some pictures of F-d curves. I am still figuring out how to do this.

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Ben Ohler replied on Mon, Aug 16 2010 2:29 PM

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You will find the Hertz model used in almost all papers that analyze AFM nanoindentation. It exists in two common forms, one modeling the indentation of a flat surface by a sphere (what I present above) and the other modeling the indentation of a flat surface by a cone (often referred to a Sneddon model). Manfred Radmacher at University of Bremen is one of the key leaders in applying this approach to biological samples. You can find a list of his publications here. One of the most commonly cited ones is the 1997 paper in IEEE Medicine and Engineering Biology. Note that there is no intrinsic problem with using sharp tips, but from a practical standpoint it is harder to stay within the valid range of the model and a cone may not approximate an AFM tip as well as a sphere approximates a microsphere on an AFM tip.

A stiffness or spring constant is probably not the best way to characterize tissue because clearly that will vary with contact area. So it's hard to infer what AFM cantilever spring constant will be appropriate for your measurement. You may need to adjust your approach to ensure that, 1) your indentation model remains valid (i.e. it closely approximates the real situation) and 2) That the force-distance curve is measured accurately.

Measuring the force-distance curves accurately is mostly just a matter of picking an appropriate cantilever and then calibrating its spring constant. The 10V deflection limit I mentioned is based on the instrumentation. The full deflection range on Veeco AFMs is typically 20 or 24V (+/- 10 or 12V), but we normally start in the center of the range, near 0V. That leaves 10V for repulsive forces and 10V for adhesive forces. But as I mentioned, the detection method becomes more non-linear for larger deflection because the laser spot is roughly circular or elliptical. That's why I suggest keeping the deflection in a smaller, more linear range (e.g. +/- 2V). This isn't an absolute limit of course and neither is there a clear limit on the low end. You mention measuring deflections of a few nanometers, which is certainly getting on the small end but isn't unreasonable. In fact, it's routine for very low force measurements (<100 pN) where the softest cantilevers typically have spring constants of 10-60 pN/nm. But if you are measuring forces more like 20 nN then there is no need to use super soft or super stiff cantilevers. There are a lot of available spring constants in between. The 0.35 N/m (350 pN/nm) cantilever I mention is a common one. So for that one, 20nN of force would result in 62.5nm of deflection, which would be about 1V. Where possible it is just good practice to keep the measurement well above the noise floor of the deflection measurement and well below the point where non-linearity becomes an issue.

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Regards,

-Ben