# QC and Yielding

###### tags: `QC`, `yielding`, `self-assembly`

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

RM, AP, FS, GF

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## Background

The slow dynamics of QC models have recently been compared to those of glass formers. While the overall dynamics appear similar, the vibrational modes show differences that seem to indicate a distinct elastic response, possibly linked to the presence of phasons in quasicrystals.

Our starting point is the following work:
https://arxiv.org/pdf/2508.18856

but the ideas discussed here may also be relevant:
https://arxiv.org/pdf/2402.10295

## Plans

We plan to investigate the yielding properties of a two-dimensional quasicrystal (QC). In particular, in collaboration with Sri, we have extensively studied the behavior of glass formers under cyclic shear. A useful reference in this context is our PNAS paper:

https://www.pnas.org/doi/10.1073/pnas.2100227118

This work provides a comprehensive study of two different models—the well-established Kob-Andersen (KA) model and a silica model. The supplementary information also contains a wealth of additional details.

In that study, we demonstrated that the degree of annealing—i.e., how deeply the glass is prepared in the energy landscape—significantly influences its yielding properties.

- High-energy (poorly annealed) systems exhibit a continuous transition.

- Low-energy (well-annealed) systems exhibit a discontinuous transition.

These results are illustrated in Fig. 1 and Fig. 2 of the paper. Overall, we found no major differences between the two models. However, in another study we did show that the structural properties behave differently.

https://arxiv.org/pdf/2108.07469

## To Do

### Task 1: Not sheared initial conditions
I think it would be good to start, from the unsheard configuration.

See Fig. 1 of the Supplementary Material of the PNAS paper. There, we examined the inherent structure of the model at different temperatures:

- The system is equilibrated at various temperatures.
- Energy minimization yields the inherent structure energy, $E_{IS}$
- The resulting structures are then compared across the different temperatures. Do we always get a QC? Or only at low temperatures?

### Task 2: equilibration to the steady state.

Start to apply ciclic shear using AQS. It would be interesting to look on how the stady state is reached for different $\gamma_{max}$. See Fig. 2 of the supmat.

### Task 3: look at the final energy as a function of the $\gamma_{max}$

You should produce something like Fig. 1a of the main paper. There we will be able to see if there is an effect of the annealing

### OPEN QUESTIONS

* In all of this the important question is the role of the structures. Defects are basically not present in glasses, but they might play a crucial role in QC.
* Do we get shear-bands?

# Results

## SYSTEM

$k\sigma = 10$ and $\delta = 1.35\sigma$ $\sim$ mollified square shoulder.

We also interpolate at the cutoff to 0:

Simulated in lammps with a table of force and energy. Range: 2000 values equispaced $r/\sigma\in [0.6, 1.95]$, interpolated with splines.


$dt = 0.001\tau$ with $\tau=\sigma\sqrt{m/\varepsilon}$

We (stupidly) define $\phi=\rho\sigma^2=N\sigma^2/L^2$
## TASK 1:

#### Protocol
We thermalize the system at $T_{\textrm{start}}/\epsilon=1$ (fast, $10^6$ steps). We anneal the system from $T_{\textrm{start}}$ to $T$ (slow, $10^7$ steps). We equilibrate the system at $T$ (~ $10^6$ timesteps). We measure its potential energy $\langle U\rangle$. Then, we minimize the energy of this state, this gives the inherent structure at $T$ and measure its potential energy $E_{IS}$ (which is the local groundstate energy).

#### $U$ vs $E_{IS}$

$(U - E_{IS})/T\sim 1\Rightarrow$ equipartition respected (low temperature):

[No phasons contributions.](https://doi.org/10.1002/(SICI)1521-3951(200105)225:1%3C21::AID-PSSB21%3E3.0.CO;2-T)

#### Structural properties

| Article | Me |
| -------- | -------- |
|  ~~~~~~~~~~~ | (top: equilibrium, bottom: equilibrium and then minimization)  |

States:

| where? | Equilibrated ~~~~~~~~~~~~~~~~ | Equilibrated + minimized ~~~~~~~~~~~~~~~ |
| --------------------------------------------------- | ----------------------------------------- | ----------------------------------------- |
|  | q4  | q4  |
|  | q4  | q4  |
|  | q12  | q12  |
|  | q6  | q6  |
| | q12  | q12  |
|| ||
||  | |
|$\phi = 0.966$ $T/\varepsilon=0.07$ |||

## TASK 2

#### Protocol

We take an equilibrated + minimized system at $T$. Then, we perform AQS on it. Small deformation of the box -> minimize -> small deformation of the box -> minimize -> ...

$\delta \gamma= 0.0001$ (typical yield strain: $\gamma_{\text{yield}}\sim 0.15$)
#### Playing a bit
Small system:


-------

Large system:


BOOP 12:
{%youtube i9txakXpW5g %}
angle q12:

shear stress:
{%youtube Vno7byPtgPM %}

---------

(stress way less homogeneous)
| | square crystals | Hexagonal crystals |
| --- | ----------------------------------------------------------------------------------- | --------------------------------------------- |
| STRESS |   |  |
| BOOP |   |  |

#### Yielding

$\phi = 0.933$, influence of various starting point at different $T/\varepsilon$:


| Param: $\phi = 0.933$ | Tiling | Stress | $q_{12}$ angle |
| --------------------- | ------------------------------------ | --- | ------------------------------------ |
| $T = 0.05$ |  |  |  |
| $T = 0.11$ |  |  |  |
| $T = 0.16$ |  |  |  |
| $T = 0.22$ |  |  |  |
| $T = 0.27$ |  |  | |

#### Cyclic shear

##### Single things



| Stress colored | $q_{12}$ angle colored |
| -------- | -------- |
|  |  |



##### Comparison QC with liquidlike and crystalike structure

| SQUARE | LIQUID LIKE ($\phi = 0.7$), badly equilibrated :) |
| ------------------------------------ | ----------- |
| stress  | stress  |
|  |  |
|| |

##### Quasicrystal formed by shearing.
|Initial config|Final config|
|------------|----------------|
|||

| Tiling | Angle $q_{12}$ |
| -------- | -------- |
|  |  |

UNFORTUNATELY FOR ME, IT SEEMS THAT THIS PACKING FRACTION CORRESPONDS TO A SQUARE QC COEXISTENCE. At equilibrium it does not seem to be the case? This density self-assembles a perfect QC. At intermediate temperature at least, perhaps this density at low temperature is in coexistence.

Indeed, the squares partially melt if we use the last cyclically sheared snapshot as an initial condition for a langevin dynamics:

| Temperature | Short time | Larger time | Tiling |
| --- | -------- | -------- | -------- |
| $T/\epsilon = 0.13$ |  |  |  |
| $T/\epsilon = 0.02$ | ... |  | .... |

Plausible phase diagram at low $T$:

##### First cluster results:

| $\gamma_{max}=0.04$ | $\gamma_{max}=0.06$ | $\gamma_{max}=0.08$ | $\gamma_{max}=0.11$ | $\gamma_{max}=0.14$ |
| --------------------------------------------------- | --------------------------------------------------- | --------------------------------------------------- | --------------------------------------------------- | --- |
|  |  |  |  |  |

Shittiest plot of the year: Energy vs cycles for two different temperatures (Solid lines are one temperature and the dashed line are an other temperature). Minimum of energy around yielding (I think i saw that on some thesis and or talk of sri? The nonmonotonoicity of $U$ with $\gamma_{max}$)

I find it interesting, that, at yielding ($\gamma = 0.7$) the system has to go first into a state of high energy before going back to the low energy state.

---------------------
Final $q_4$ after a lot of cycles (min 100, except for small T, for which the simulations havent run):


| | Initial | Final |
| -------- | ---------------------------------- | ---------------------------------- |
| $q_{4}$ | |  |
| $q_{12}$ |  |  |

$q_{12}$ remains more or less constant (altough it increases a bit). $q_{4}$ grows like crazy. The angle averaged $q_{12}$ however, grows, that is, we anneal the polycrystallinity.


**CONCLUSION**: THIS STATE POINT IS PROBABLY TOO DILUTE AS IT FAVORS SQUARES

___________________________________

#### Higher densities:

$\phi = 0.966$

| Initial condition equilibrated at $T/\varepsilon=0.05$ ($\gamma_{max} = 0$) | $\gamma_{max}=0.04$ | $\gamma_{max}=0.053$ | $\gamma_{max}=0.067$ (around yielding) | $\gamma_{max}=0.12$ |
| -------------------------------------- | ------------------------------------- | -------------------- | --- | --- |
|  |  |  |  |  |

SHEAR; DESTROYER OF HEXAGONS (note the change of average angle via nucleation). This state is a hexagon liquid coexistence equilibrated at $T/\varepsilon=0.27$. Sheared at $\gamma_{max}\simeq 0.1$:

| Angle $q_{12}$ | Tiling |
| -------- | -------- |
|  |  |

Frame by frame:

| Initial condition | 4th cycle | 14th cycle | 38th cycle |
| --------------------------------------------------- | --------------------------------------------------- | --------------------------------------------------- | --- |
|  |  |  |  |


General:

$\phi = 0.95$

$\phi = 0.966$

In both case, highest $q_{12}$ at yielding

-----------------------

Steady state:

|Thermal phase diagram| Cyclically sheared $\phi = 0.933$ | Cyclically sheared $\phi = 0.95$ | Cyclically sheared $\phi = 0.9666$ |
|----| -------- | -------- | -------- |
||  |  ||

____________________

Reversible to irreversible properties:

Note the timescale difference.
| | Below yielding | At yielding | Above yielding |
| ------------- | --------------------------------------------------- | --------------------------------------------------- | --------------------------------------------------- |
| $\phi = 0.95$ |  |  |  |
| $\phi = 0.966$ | | |  |

Solid lines are $T/\varepsilon = 0.21$, dashed lines are $T/\varepsilon=0.05$ (not that lower temperature does not always lead to lower energy due to the large number of hexagons):

| $\phi = 0.95$ | $\phi = 0.966$ |
| -------- | -------- |
|  |  |



### Phase stability at $T=0$

Mixed tiling.Comming edge length $s$. Area of triangle: $A_T=\tfrac{\sqrt3}{4}s^2$ and of square $A_T=s^2$. Area constraint:

$$\rho_SA_S+\rho_TA_T = 1$$

with $\rho_S$ the number of square by unit of area. We have on average this many number of edge by unit of area: $e=\dfrac{3\rho_T+4\rho_S}{2}$. Euler identity is: $V-E+F=0$ with $F=(\rho_T+\rho_S)A$. The vertex density is: $v=e-f=(\rho_T+2\rho_s)/2$. We want one vertex per unit of area (to obtain the fundamental tile). $v = \rho\rightarrow \rho = (\rho_T+2\rho_S)/2$. We get:

$$s= \sqrt{\dfrac 1 \rho}\sqrt{\dfrac{1}{\sqrt{3}/2+x_S\left(1-\sqrt{3}/2\right)}}$$

with $x_S$ the fraction of square tiles. We see that it interpolates between the square lattice and the hexagonal lattice.


### Shear bands

#### Single shear

I dumped stupidly the data, so I have almost no data :) :) :)

________________________


| | $\gamma = 0$ | $\gamma = 0.1$ | $\gamma = 0.2$ |
| --- | ------------ | -------------- | -------------- |
| $N = 25\times10^3$ |  |  |  |
|$N = 5\times 10^3$||||

Seems that we have shear bands around defects. The simulations are faster than expected, so now, I'm running cyclic with 2

#### Cyclic shear

|$\gamma_{max}$|Displacement map|$q_{12}$ angle|
| -------- | -------- | -------- |
|0.07 (yield)| |  |
|0.09|||
|0.12|  | |
|0.13| ||
|0.14|||
|0.16| ||
|0.18|||
|0.20|||

Shear bands form in large systems. So, the idea of the perfect QC at yielding breaks down.


### Small system, best QC?

Best ones around $\phi = 0.97-0.975$

Above yielding
| $\phi = 0.93$ | $\phi = 0.94$ | $\phi = 0.96$ | $\phi = 0.98$ | $\phi = 1$ |
| --- | ---------------------------------- | ---------------------------------- | ---------------------------------- | ---------------------------------- |
|  |  |  |  |  |

### Shear annealing

#### Too high $\Delta \gamma$

I'm starting way above yielding: $\gamma_{max}=2\gamma_{yield}$, and slowly decreasing the $\gamma_{max}$ at each full cycle:

| (global) $\vert\langle q_{12}\rangle\vert$ | (local) $\langle\vert q_{4}\vert\rangle$ | (local) $\langle\vert q_6\vert\rangle$ |
| -------- | -------- | -------- |
|  |  |  |

I was a bit too enthusiastic and decided to use a large $\Delta\gamma$ ($\Delta\gamma=0.01$ instead of $0.0001$), I think it's suppressing the shear bands.

Weird step like behavior due to large $\Delta \gamma$

$f_{anneal}$ is how much i decrease $\gamma_{max}$ after each strobo

| $p$ as strain is decreased | $U$ as strain is decreased |$\sigma_{xy}$ as strain is decreased |
| -------- | -------- | -------- |
| |  |  |

BOOP as strain is decreased


#### Smaller $\Delta\gamma$

We now take a smaller $\Delta\gamma(=0.0001)$.

We now see shear bands:

|Strobo displacement| (global) $\vert\langle q_{12}\rangle\vert$ | (local) $\langle\vert q_{4}\vert\rangle$ | (local) $\langle\vert q_6\vert\rangle$ |
| ------ | -------- | -------- | -------- |
|  |  |  | |

| $p$ as strain is decreased | $U$ as strain is decreased |$\sigma_{xy}$ as strain is decreased |
| -------- | -------- | -------- |
| |  | |


Without annealing:

### Finite rate

We now solve: $\dfrac{d\gamma}{dt}=\dot\gamma$ coupled to $\Gamma \dfrac{dx_i}{dt}=F_i$. The typical timescale (relaxational) is:$\tau = \dfrac{\Gamma \sigma^{2}}{\varepsilon}$. Therefore, at fixed radius and $\varepsilon$, the only interesting quantity is $\dot\gamma\tau$.

We still observe the same kind of thing ($\gamma\dot\tau = 0.001$):