# Confusion of moduli

A lot of AFM nanomechanical measurements are geared towards measuring modulus. But what is modulus? There are so many different kinds: Youngs modulus, elastic modulus, tensile modulus, compression modulus, bending modulus, shear modulus, storage modulus, loss modulus, and reduced modulus. Which one does AFM measure? Can it measure more than one? What bulk modulus measurement should I be comparing my AFM modulus measurement with? What is even the difference between all these moduli?

Let’s back up for a moment. Modulus is generally associated with a material’s stiffness. It relates the stress (pressure) on a sample with an accompanying strain or displacement by the sample in response to the stress. The modulus relates these two behaviors. Why are there so many different kinds of moduli? Because you can induce stress through a variety of motions.

The first delineation point is how you are applying stress? For many samples you can just pull it in a particular direction. For samples that are viscoelastic - which are materials whose mechanical properties have a frequency dependence such as silly putty or polymers - you might want to exert the stress at a particular frequency.

Let’s start with the easier case of samples that are just elastic. Youngs modulus or elastic modulus is the most general term that relates a material’s stress to its strain. You can pull the material apart so that you measure a tensile modulus or compress the sample to measure a compression modulus. You can even bend the sample or twist it to measure a bending and torsional modulus respectively. These motions are shown in the schematic below.

A common way to measure a tensile modulus is with the Instron tensile tester, an image of which is shown below. The sample is actually the tiny white dogbone in the middle, and the rest is hardware to accurately measure stress and displacement. Some of these testers are several stories high depending on the material they want to measure such as metal or concrete!

**An Instron tensile tester**

Viscoelastic samples are a bit more complicated. As mentioned, for these kinds of sample that exhibit a frequency dependence to their mechanical properties, you drive the sample strain in a sinusoidal motion and then measure the stress response. The stress response typically will lag the strain by a parameter called the phase lag. The response that is in phase with the strain is termed the storage modulus which measures the elastically stored energy. The response that is out of phase with the strain is termed the loss modulus that measures the viscous dissipation of energy. Even viscoelastic moduli come in different forms: you can have viscoelastic strain moduli (denoted by the letter G) or viscoelastic tensile moduli (denoted by the letter E).

**A diagram explaining the different types of moduli**

Finally, there is a reduced modulus which is particularly relevant for AFM and nanoindentation measurements. With these tools, your actual measured observable is a kind of combined modulus of both tip and sample. The reduced modulus is a function of the modulus of your substrate, the modulus of your tip (silicon for AFM and diamond for nanoindentation) and then the poissons ratio of your tip and sample. By knowing the modulus and poissons ratio of your tipand the poissons ratio of your sample, you can then back out the modulus of your sample.

So which modulus does AFM measure? Turns out the answer to that is a bit tricky and depends on which AFM mode you are using. For viscoelastic materials, which are frequency dependent, the answer is even more tricky because AFM is essentially a frequency dependent method and so you have to be careful about all the frequencies at play here. Stay tuned for my next blog to hear more about the moduli we are actually measuring with AFM.

*Dalia*

**Dalia Yablon, Ph.D.**

SurfaceChar LLC

www.surfacechar.com