# How to measure a solid-solid surface energy?

Many techniques exist to measure the surface energy between a liquid and a liquid or a liquid and a gas (see e.g. the wiki page).

Methods to measure the surface energy between a solid and a fluid are rare, but still there is a method developed by Zisman (see e.g. here) that allows you to at least estimate it by extrapolation for solid/gas or solid/liquid, depending on the environment that you use in your experiment

What I wonder: is there a method to measure the surface energy between two non-elastic solids?

One option I could think of is that you could melt one of the solids and then use the technique of Zisman, but this will limit your knowledge to high temperature surface energy, whereas the ones at low temperature are the thing you are typically interested in.

EDIT: just for future reference, this is a study on surface energies between 2 solids, but with 1 being highly elastic

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– Luboš Motl Feb 27 at 7:24
That is actually a good point. I normally use the terms somewhat interchangeable, but I agree that in case of solids it doesn't make much sense to talk about surface tension – michielm Feb 27 at 7:59

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Think about high-temperature creep experiments using very fine-grained polycrystalline samples to measure the thermal activation energy of pure diffusion creep. You might be able to estimate the average inter-granular surface energy using a diffusion-creep law. – Mark Rovetta Feb 28 at 0:28
I don't really understand how that would work. Could you expand a bit?! And doesn't this have the same issue as melting the material, i.e. you get the surface energy at way to high temperatures? – michielm Mar 1 at 9:00
@MarkRovetta could you explain this creep experiment?! – michielm Mar 2 at 7:53
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## 1 Answer

I have done some searching and found out that there is a technique that has been around for roughly 10 years already and it is surprisingly simple (if you have the right, expensive, equipment). It can be found in this JCIS paper (which is also freely available here).

The technique works as follows: an atomic force microscope (AFM) with a well-defined spherical tip made out of solid 1 is brought into contact with solid 2. Then the tip is pulled of the surface again and the work of adhesion is measured. Based on the pull-off force and theoretical contact mechanics models (for details see the paper) you can calculate the surface energy $\gamma$ between the two solids from the following equation: $$\gamma = \frac{F}{2\pi c R}$$ where $F$ is the pull-off force, $R$ is the tip radius and $c$ is a constant between 1.5 and 2 depending on the details of the contact model. The paper explains how to choose which model is appropriate for the type of measurement you do.

Some conditions (assumptions) for the theoretical models apply:

1. deformations of materials are purely elastic, described by classical continuum elasticity theory
2. materials are elastically isotropic
3. both Young’s modulus and Poisson’s ratio ofmaterials remain constant during deformation
4. the contact diameter between particle and substrate is small compared to the diameter of particle
5. a paraboloid describes the curvature of the particle in the particle–substrate contact area
6. no chemical bonds are formed during adhesion
7. contact area significantly exceeds molecular/atomic dimensions

The paper explains in quite some details how deviations from these conditions are often source of error, but also how they can be met to get an appropriate measurement.

So to conclude: the surface energy of a solid-solid system can be measured using AFM when taking into account that the assumptions of models used in data processing are thoroughly met.

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