# Quasicrystals

###### tags: `masters projects`

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### Owners (the only one with the permission to edit the main text, others can comment)

Bas, Laura

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We are studying core-corona particles. These particles are characterized by a shoulder potential as shown below (taken from Harini Pattabhiraman's thesis, titled *Quasi-periodic and periodic photonic crystals: A simulation study of theirself-assembly, stability and photonic properties*, page 15).


It turns out that under the right circumstances these particles will order themselves as a quasiqrystal.
From Harini Pattabhiraman's thesis, titled *Quasi-periodic and periodic photonic crystals: A simulation study of theirself-assembly, stability and photonic properties*, page 36, I read that we can produce a quasicrystal in the following manner. Start with a crystal in hexagonal phase at reduced density $\rho^* = 0.98$ at reduced temperature $T^* = 1.0$. Let the system evolve and slowly let the temperature cool to $T^* = 0.278$. The result will be a quasicrystal.

### Hexacrystal
This is the starting configuration of the particles. All particles are nicely arranged in a hexacrystal at density $\rho^*=0.98$.
### Attempts at quasicrystals
Below are some of the resulting particle systems.
#### Structure factor
Below are the structure factors of the resulting systems, in the same order.
> [name=laura]Testing adding a comment

# Change of plan
## New method
The previous method does not seem to yield a quasicrystal, even after a lot of time. Therefore I change my approach. We switch from NVT to an NPT ensemble. We consider the following phase diagram (special thanks to Emanuele Boattini), which is correct for core corona particles with $\sigma_{HD}=1.4$:
Particularly, I am going to look at the specified red region. So I take reduced pressure $P^* = 21.0$ and reduced temperature $T^* = 0.37$.

Also, I reduced the number of particles, as the simulations take a lot of time. I let the simulation start in a fluid phase at a packing fraction $\eta = 0.2$.

## Result

Corresponding structure factor S(k):
> [Laura] This looks like it might be heading towards a quasicrystal, but still not there yet. What was the initial config? Fluid or Hex?
> [Bas] The initial config was randomly placed particles at packing fraction $\eta = 0.2$.

### Volume change
After some searching, I found that there was an error in my volume change step. In the acceptance rule for a volume change, I only took the hard core interactions into account and forgot about the soft interactions.

I got the code working. I produced the following quasicrystal:
The structure factor looks like this.
This still could be beter.

If we look at the bonds of this structure we get the following picture (here I think I used a cutoff value of $r_{\text{cut}} = 1.35$.
We see many trian1gles and squares. However, there are also many defects. In the picture below I indicated all defects in blue.

## Defects
### What are defects?
We are going to invesigate defects more thoroughly. First I discuss what a defect in a quasicrystal entails.
We can create a graph, by taking the centers of all particles to be nodes and then drawing an edge between to nodes if the particles are closer than a certain cutoff value. This is how the previous figure was created.

Now if we look at our quasicrystal, we notice that the bonds form squares and triangles. Ideally, the whole box should be filled with only squares and equilateral triangles. That means that anything that departs from this is part of a defect.

We can now look at what shapes each node is a part of.
Now obviously, every angle in a perfect square is 90 degrees and every angle in a perfect equilateral triangle is 60 degrees. Also, if for all figures that a certain node is part of, we sum the angle at this node, we clearly have to arrive at 360 degrees. It follows that if a node is part of only perfect squares and equilateral triangles, there are three options:
1. The node is part of 4 squares,
2. The node is part of 2 squares and 3 triangles,
3. The node is part of 6 triangles.

These options are illustrated below (taken from *Defect-mediated relaxation in the random tiling phase of a binary mixture: Birth, death and mobility of an atomic zipper* by Elisabeth Tondl, Malcolm Ramsay, Peter Harrowell, and Asaph Widmer-Cooper, https://doi.org/10.1063/1.4867388)

### Implementation
Since there are a lot of defects in the self assembled quasicrystal, we will take a different route.
Using code by Frank, I am now able to create perfect quasicrystals. E.g.

with bonds:
The structure factor:

In this perfect quasicrystal, every particle is exactly a distance 1 away from its direct neighbors. Since the diameter of the hard core of a particle is also 1, this means that this configuration is allowed. However, since computers have finite precision, some neighbors have ever so slightly smaller distance than 1. Therefore, I scale the system slightly such that the minimum distance between all particles is slightly greater than 1.

From this point I ran the simulation at $P^* = 21.0$ and $T^* = 0.37$, to see if defects would be spontaneously created. This does not seem to be the case (I might run the simulation for longer later on to more thoroughly check this). Now we are going to introduce a defect in this crystal, so that we can study its properties.