# Experimental Tetrahedrality

###### tags: `glasses` `tetrahedrality` `local structures` `experimental`

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### Owners (the only one with the permission to edit the main test)

Susana, Frank, Giuseppe

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## Background
Based on the paper of Tetrahedrality Dictates Dynamics in HS Mixtures, we would like to test the predictions in an experimental realization. To do so, the group of Marco Laurati from the University of Florence have been working on experimental HS particles. With those, we would like to compute the $n_{tet}$ and see how well correlated is with dynamics.
<!-->[name=SMA] Specifications from the experiment-->

## To Do
:::success
green is for finished tasks
:::
* Compare the $g(r)$ at different packing fractions with the simulations.

:::success
* $n_{tet}$ Vs $x_L$ of experimental data.
* Check the correct sizes of the particles.
* Fix overlapping of particles.
* Compare results with other packing fractions or size ratios.
* $\tau_D$ Vs $n_{tet}$ of experimental data.
:::

## Results

The data reported by Marco correspond to a system with packing fraction of roughly $\eta\simeq0.575$ and size ratio $q\simeq 0.85$. So far, we have the following compositions $x_L$ 0.2,0.3,0.4,0.5 and 0.6. (These are the ones reported)
In the experimental system, the box is divided in "sub-boxes" (for $x_L=0.2,0.3$ and $0.4$ the volume corresponds to $61.63*61.63*20.38$ $\mu m^3$). We have between 10-12 volumes for each composition.

### Analysis
#### Comparing System Features

The compositions reported and the ones from the snapshots are slightly different:

| $x_L$ Reported | $x_L$ Found |
|:--------------:|:-----------:|
| $0.2$ | $0.172 \pm 0.005$ |
| $0.3$ | $0.251 \pm 0.003$ |
| $0.4$ | $0.315 \pm 0.002$ |
| $0.5$ | $0.407 \pm 0.004$ |
| $0.6$ | $0.482 \pm 0.007$ |
| $0.7$ | $0.597 \pm 0.005$ |

Note that these are the average compositions from each of the volumes $\pm$ standard deviation.

The size of the particles is also slightly different. The ones reported from the MSD are:

| $\sigma_L$ | $\sigma_S$ |
|:----------:|:----------:|
| 1.62 | 1.36 |

For sake of comparison, I calculate the $g(r)$ and assign as particle diameter the value corresponding to the first peak of $g(r)$. This corresponds to $\sigma_L=1.925$ and for the small particles $\sigma_S=1.425$. This leads us to a size ratio of **$q\simeq0.74$**.
In the following picture we show the $g(r)$ for all packing fractions (the values are shifted by a factor of 2 to compare all the $\eta$)

| $\sigma_L$ | $\sigma_S$ |
|:------------------------------------:|:------------------------------------:|
|  |  |
Note that, this $g(r)$ correspond to the averaged one.

<!-- >[name=Frank] Hmm.. most likely they estimated the number fraction based on the mass or total volume of both species of colloids they added. So if the large particles are a bit larger than expected, there would be fewer of them than planned, resulting in a lower $x_L$. (I haven't checked if the numbers for this work out.) -->

<!-- >[name=SMA] For sake of comparison, I've calculated the packing fraction $\eta$ from the experimental data. To do so, I used the radius of the particles obtained from the first peak of $g(r)$.
> In order to calculate the size of the box, I assume that the box origin is at (0,0,0). I found the particle with the largest value of $x$, $y$ and $z$, I assign that value as the lenght of the box. Finally, I take into account all the particles that are at least $\sigma/2$ from the value taken as the edge of the box, and so the whole particle is inside the box. -->

Finally, I've calculated the packing fraction $\eta$ from the experimental data to compare. To do so, I used the radius of the particles obtained from the first peak of the $g(r)$.
In order to get the size of the box, I found the particle with the largest and the smallest value of $x$, $y$ and $z$. I assign those values as the lenght of the box. Finally, I calculate the packing fraction from a "sub-box".
I calculate the packing fraction $\eta$ for two different values of $\sigma_L$: $\sigma_L=1.925$ and $\sigma_L=1.85$. The diameter of the small particle is fixed to $\sigma_S=1.425$.

| $\sigma_L=1.925, \sigma_S=1.425$ | $\sigma_L=1.85, \sigma_S=1.425$ |
|:------------------------------------:|:------------------------------------:|
| |  |

>[name=SMA] These packing fractions correspond to the averaged values over all volumes.

### Comparing with simulations
#### Radial distribution Function
To start the comparisson between simulations and experiments, I compared the $g(r)$ obtained from the simulations and the one given by the experiments.

<!--> [name=Frank] It would be nice to also see $g_{LS}(r)$. Also, it might be nice to do a few runs with (Gaussian) polydispersity on both species, to see how much you need to get a similar broadening of the first peaks...-->

<!--> [name=SMA] Here there are now the $g_{LS}(r)$, I think the normalization of the experimental data has a small bug in somewhere but we can see the comparison between the peaks in sim and experiments. -->

I compared with the simulation at $\eta=0.575$, $q=0.75$ and different size ratios. First by taking $\sigma_L=1.925$:

| $x_L$ (Sim) | $g(r)$ $\sigma_L=1.925$ |
|:----------------------:|:--------------------------------------------:|
| $0.2$ <br> $\simeq0.17$ | |
| $0.3$ $\simeq0.25$ |  |
| $0.4$ $\simeq0.31$ | |
| $0.5$ $\simeq0.4$ | |
| $0.6$ $\simeq0.49$ |  |
| $0.7$ $\simeq0.6$ | |

We compared as well the $g(r)$ when $\sigma_L=1.85$ and the same state point as before.
In general, we can see that the data corresponding to large $x_L$ is almost on top of the simulation one.

| $x_L$ (Sim) | $g(r)$ $\sigma_L=1.85$ |
|:----------------------:|:--------------------------------------------:|
| $0.2$ <br> $\simeq0.18$ | |
| $0.3$ $\simeq0.25$ |  |
| $0.4$ $\simeq0.31$ |  |
| $0.5$ $\simeq0.4$ |  |
| $0.6$ $\simeq0.49$ | |
| $0.7$ $\simeq0.6$ | |

### Tetrahedrality
In order to calculate the number of tetrahedra per particle we follow:
* First, I renormalized the experimental data according to the approximate $\sigma_L$ (I did it also without renormalizing obtaining the same results as the voronoi construction do not need information about the size of the particles).
* Then, due to the non-periodic boundary conditions of the system, I used the modified TCC without periodic boundary conditions.
* Finally, in order to calculate the averaged $n_{tet}$ of the whole system I took into account only the particles whose distance to the edges of the box is greater than $3\sigma_L$.

| Experimental $n_{tet}$ | Comparison Simulation |
|:------------------------------------:|:------------------------------------:|
| | |

It seems that the data of $x_L=0.5$ and $0.6$ approaches quite well to the simulations at a size ratio corresponding to $q=0.7$. However, at lower $x_L$ the $n_{tet}$ is smaller in experiments.

<!-->[name=SMA] Possible reasons: Problem with TCC and the boundary conditions? Are there overlapping particles? Structural data from experiments?-->

<!-->[name=Frank] I suppose missing particles are also an option, as you mentioned earlier. Do you know if the number of missing particles depends on the composition? Could you plot the distribution of $n_{tet}$ as well?-->
>

<!-->[name=SMA] I suspect now that the number of missing particles is not that high, I have re calculated the $\eta$ with the size of the particles obtained from the $g(r)$ and it is really close to 0.575.-->

### Distribution of $n_{tet}$
In order to check from where the possible differences arise, we calculated the distribution of $n_{tet}$ in experiments and simulations. (Note that in simulation corresponds to just one snapshot.)

| $x_L$ (Sim) | Experiments |Simulations at $q=0.7$ and $\eta=0.575$|
|:----------------------:|:--------------------------------------------:|:---:|
| $0.2$ <br> $\simeq0.18$ | ||
| $0.3$ $\simeq0.25$ | ||
| $0.4$ $\simeq0.31$ |||
| $0.5$ $\simeq0.4$ |||
| $0.6$ $\simeq0.49$ |||
| $0.7$ $\simeq0.6$ | ||

### Diffusion time
From the experimental data we have the following diffusion time:

So far, the diffusion time and the $n_{tet}$ stills not following a nice exponential relation.
<!--

<!-- >[name=Frank] True, but I'm not very convinced by the $n_{tet}$ data yet.-->

### Comparison with polydisperse binary mixtures

In order to have a better comparison with the experiments, we simulate polydisperse binary mixtures of hard spheres. To do so, I take each of the mean sizes of the particles (large or small) as the mean value of a normal distribution with different standard deviations (degree of polydispersity). I sampled the sizes of the particles from 15 bins for each normal distribution.

The system now has $N=10000$ particles, a packing fraction of $\eta=0.575$ and I sampled two size ratios ($q=0.7$ and $q=0.75$). So far, I have simulated three different polydispersities ($std=0.04$, $0.05$ and $0.06$).

The first step is to compare the radial distribution functions and then, the number of tetrahedra per particle.

| $x_L$ (Sim) | $g(r)$ $\sigma_L=1.925$ |
|:----------------------:|:--------------------------------------------:|
| $0.2$ <br> $\simeq0.17$ ||
| $0.3$ $\simeq0.25$ | |
| $0.4$ $\simeq0.31$ | |
| $0.5$ $\simeq0.4$ ||
| $0.6$ $\simeq0.49$ | |
| $0.7$ $\simeq0.6$ ||

| $x_L$ (Sim) | $g(r)$ $\sigma_L=1.85$ |
|:----------------------:|:--------------------------------------------:|
| $0.2$ <br> $\simeq0.17$ ||
| $0.3$ $\simeq0.25$ ||
| $0.4$ $\simeq0.31$ | |
| $0.5$ $\simeq0.4$ ||
| $0.6$ $\simeq0.49$ ||
| $0.7$ $\simeq0.6$ ||

>[name=SMA] I only compared the $g(r)$ with 0.05 and 0.06 because at the end 0.04 has a larger first peak. I could imagine that for the large particles taking the $std=0.05$ or $0.06$ it is agood approach. And because, for the small particles the first peak from the expertiments is larger than those of the simulations it might be that the polydispersity in that case is closer to $0.04$.
>
Finally we compare the number of tetrahedra per particle:
>[name=SMA] The trends of the number of tetrahedra didn't extremely changed, it is true that at larger polydispersity the $n_{tet}$ slightly decreases.


>[name=SMA] I compared as well the diffusion time of the polydisperse systems and the experimental data. As we can see both of them are really close to each other. Note that this corresponds to a $q=0.7$. Then the question is why at low compositions we are not capturing the tet, is it a problem with the coordinates?

### Packing Fraction?
Playing a little bit with the data, I plot all the packing fractions we explored from simulations. And on top I add (in closed symbols) the experimental points to check at what packing fraction the values we get from experiments correspond. The three points that we have found that are closer to the simulation values are the ones corresponging to $x_L=0.3$, $0.5$ and $0.6$. However $x_L=0.3$ seems to correspond to a smaller packing fraction.

One thing important to remark is that the variation of the $n_{tet}$ at larger compositions is not that radical and they grow simillarly. However, the systems with small compositions suchs as $x_L=0.2$ and $x_L=0.25$ have a larger variation with packing fraction which can lead to have mismatches between the simulation and experiments if the packing fractions is slightly different.


A possible route we can take (if it is not too difficult in experiments) to focus ourselves into the systems with compositions close to $50-50$ and explore different packing fractions. We will still capture the effect of tetrahedrality and it might be that the packing fraction can be better controled.

### Modifying the packing fraction
Now, we will focus on following the changes on the structure and dynamics by changing the packing fraction. To do so, we fix a size ratio and composition and we follow the behaviour as it is shown in the previous image.

So far, we have from the experiments one set of systems with different packing fractions. These systems correspond to the same size ratio $q\simeq 0.72$, with size of the large particles corresponding to $\sigma_L\simeq 1.975$ and $\sigma_S\simeq 1.425$ and a composition of $x_L\simeq0.4$. Note that, from the previous results this composition has values far from the ones predicted from simulation.

Once more the results of this composition and size ratio for the different packing fractions are considerably smaller than the ones from the simulation.


>[name=SMA]I think with the composition near to $x_L=0.5$ or $x_L=0.6$ where we found the best agreement we can follow better the changes with packing fraction.

## Fixed size ratio and composition, changing packing fraction

In order to better follow the changes in $<n_{tet}>$ with packing fraction, this time from the experiments the size ratio and the composition was fixed and different packing fractions were explored.

Same as before, first we check the packing fraction values, the radial distribution function and the number of tetrahedra.

The composition is $x_L=0.750$ and the size ratio is $q\simeq 0.7-0.75$.
First we check the packing fraction. In the following plot we have the packing fraction in the x-axis the one reported from experiments and in the y-axis the one calculated.
>[name=SMA] The following plot was calculated by assuming the size of the particle equal to the place where the first maximum of the $g(r)$ is.

>[name=SMA] From the experiments the sizes of the particles are $\sigma_L=1.78 \mu m$ and $\sigma_S=1.26 \mu m$. By taking this sizes then we obtain:
>

| Experimental | Simulation | Error |
|:------------:| ---------- | ----- |
| 0.47 | 0.472 | 0.007 |
| 0.49 | 0.495 | 0.004 |
| 0.51 | 0.512 | 0.013 |
| 0.52 | 0.516 | 0.013 |
| 0.57 | 0.515 | 0.004 |
| 0.58 | 0.579 | 0.021 |
| 0.59 | 0.586 | 0.022 |
| 0.60 | 0.596 | 0.005 |

### Radial distribution function $g(r)$
I compared the radial distribution functions from the experiments and simulations:


>[name=SMA] The best agreement of both $g(r)$ is at high packing fractions. At low packing fractions the first peak of the polydisperse simulations is understimated.

### Tetrahedrality
After, we calculate the number of tetrahedra per particle. There are no big variations at low packing fractions, however at higher packing fractions differently from expected the $<n_{tet}>$ decreases.
>[name=SMA] I will recheck the calculations with the refined coordinates to see if at higher packing fractions it gets better.

|Experiment | Simulation and Experiment |
| -------- | -------- |
|  | |

>[name=SMA] In the simulations the number of tetrahedra increases as expected. However, in the experiments it seems it decreases.

## Polydisperse system one component (November 2021)

In order to remove some of the sources of discrepancy between experiments and simulations, we decided to go to single component polydisperse systems. These systems avoid crystallization and also present a glassy dynamics that it is well captured by the tetrahedra as we saw from the PRL paper.
From the experiments, there are samples at different packing fractions ranging from $\eta=0.50$ to $0.60$ and the polydispersity of the particles is about $5\%$.

In order to compare with the simulations, we simulate as well a single component system, with two polydispersities $5\%$ and $6\%$, both of them giving similar results. As before, we compare the $g(r)$, the $<n_{tet}>$ and the MSD.

### Experimental Results

|$g(r)$ | $MSD$ |
| -------- | -------- |
|  |  |

### Comparison Radial Distribution Function: Simulation and experiments
The first thing we compare is the overall structure between the simulations and the experimental data.
To match better with the $g(r)$ from the experiments I took a value close to the peak of the experimental $g(r)$ which match best the first peak of the simulation $g(r)$. For $\eta=0.50,0.52$ this corresponds to $1.725\mu m$ for the rest of them is $1.7 \mu m$.

In general, the overall structure is quite similar, for some of the packing fractions we find almost a perfect match between them.
| | | |
| ------------------------------------ | ------------------------------------ | ------------------------------------ |
|  |  |  |
|  | |  |

### Comparison $n_{tet}$: Simulation and experiments

We follow the same procedure than before to count the number of tetrahedron. The results in this case are still quite bellow compared to the simulations. However, now we are finding the right trend in experiments, where the $<n_{tet}>$ is growing as a function of $\eta$. It might be that by comparing with the diffusion time, now we would fine a better trend than before.


### Comparison $n_{tet}$: Histograms

In order to check what is happening in more detail, we calculate the histogram of the $<n_{tet}>$. Something that is quite promising is that in overall the trends are more similar this time. The average value is still bellow, but now we don't find that many particles with 0 tetrahedron, which could have been a problem when we had the binary mixture and the tracking.

| $\eta$ | Experiments |Simulations at $std=0.06$|
|:----------------------:|:--------------------------------------------:|:---:|
| $0.50$ | ||
|$0.52$|||
|$0.54$|||
|$0.56$|||
|$0.58$|||
|$0.60$|||

>[name=SMA] The histograms from the simulations are calculated with just one configuration. There were some problems with the cluster and the memory, hence it will be better to run them again.

To Do: We could calculate now the diffusion time of the experiments to compare with the ones of simulations and also compare the relation between $<n_{tet}>$ and the $\tau_D$.
Check other types of clusters, as the icosahedral to see if we find roughly the same numbers. Do other polydispersities? Compare with simulations with charges?

### Tetrahedra as a function of time (Feb-March 2022)

In order to understand if the local structure of the systems is still evolving with time, we follow now in the experiments the evolution of the number of tetrahedra as a function of the experimental time. This was performed for the large packing fraction $\eta=0.58$ where stronger changes in the local structure are expected.

>[name=SMA] We have the idea that the number of tetrahedra were still evolving to reach larger values comparable to the ones in the simulations. Preliminary, the previous systems were showing an increasing-noisy trend. Which lead to the supposition that the local structure was still evolving, even though the $g(r)$ of the simulations and the experiments were almost on top pf each other.

In general, we observe that the local structure is still evolving in the experimental systems. However, not expected the data show a decreasing trend, instead of an increasing as it was expected.