Solid solution hardening, concentration dependence

[CarlJose]

When you add impurity atoms to a material, the yield strength often increases by a process known as solid solution hardening. This is because the impurity atoms create a barrier to dislocation motion. The literature describing this phenomenon dates back to the 1960s with some famous papers by R.L. Fleischer.

Anyway, this hardening is supposedly dependent on the spacing between the impurity atoms. Since the spacing between impurities should be proportional to the impurity concentration in some way (i.e., more impurities, closer spacing between them), increase in yield strength is often plotted against impurity concentration. Experimentally, this increase has been shown to vary with the square-root of impurity concentration.

*My question is this: Why is it a square-root dependence, and not a cube-root dependence? Atomic concentration is per unit volume, so if you increase impurity concentration, the spacing between these impurity atoms should scale with the cube root of the concentration, right? Every resource I have found states that impurity spacing varies with the square-root of impurity concentration but I have yet to find a good explanation for this. Am I missing something? Can someone explain?

 

[mfb]

You could argue that atoms in a material rarely move along a line. You can have different parts moving against each other, where the whole area is affected. The number of impurity atoms in a specific area goes with $n^{2/3}$. The true dependence could be between those effects.

 

[CarlJose]

You could argue that atoms in a material rarely move along a line. You can have different parts moving against each other, where the whole area is affected. The number of impurity atoms in a specific area goes with $n^{2/3}$. The true dependence could be between those effects.

Thanks for the reply.

Why would the number of impurity atoms in a specific area go with $n^{2/3}$? (and I'm assuming you are using '$n$' to represent atomic concentration?) Is there some sort of geometrical justification for this?

Also, I guess I'd be willing to accept that the $n^{1/2}$ dependence is just an empirical observation (and between two functional limits), but I've found a few sources that state it as if it has some sort of theoretical justification. I'm suspicious that it has something to do with the impurities necessarily being within a crystalline lattice, but I can't figure out why that would make a difference...

 

[mfb]

Sorry, my previous post did not make sense. A surface cutting through a volume of N atoms will have N^2/3 atoms (neglecting prefactors), but the number of impurity atoms within that surface will scale linearly with the total number of impurity atoms in the material.

The mean area without impurities (as part of a larger area we consider) should scale with the inverse square of the average linear distance, which gives n^-2/3, but that is harder to translate to material properties.

 

[alphy]

The square-root dependence of solid solution hardening on impurity concentration has been extensively studied and is well-supported by experimental data. This relationship can be explained by the underlying mechanisms of solid solution hardening.

As you mentioned, the addition of impurity atoms creates a barrier to dislocation motion, which in turn increases the yield strength of the material. This barrier is created by the interaction between dislocations and the impurity atoms. The closer the impurity atoms are to each other, the more effective this barrier will be in hindering dislocation motion.

However, as the impurity concentration increases, the spacing between impurity atoms does not scale linearly with the concentration. This is because the impurity atoms also interact with the host atoms, causing a distortion in the crystal lattice. This distortion creates a strain field around the impurity atoms, which in turn affects the spacing between them.

The square-root dependence arises from the balance between these two effects - the barrier created by the impurity atoms and the distortion of the crystal lattice. At low impurity concentrations, the barrier effect dominates and the yield strength increases at a faster rate. However, as the concentration increases, the distortion effect becomes more significant and the yield strength increases at a slower rate. This results in a square-root dependence.

On the other hand, a cube-root dependence assumes that the spacing between impurity atoms is solely determined by the concentration. In reality, the distortion of the crystal lattice plays a significant role in determining this spacing, leading to a square-root dependence.

In summary, the square-root dependence of solid solution hardening on impurity concentration can be explained by the balance between the barrier effect of impurity atoms and the distortion of the crystal lattice. This relationship has been extensively studied and is well-supported by experimental data.