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# System prepended metadata

title: '**polydisperse hard disks**'

---



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# **polydisperse hard disks**



**N=10000**


![](https://i.imgur.com/216VNZo.png)



the results that I have obtained for now show the presence of an interface between hexatic and liquid even at polydispersity > 7%. The intefacial region actually increases for higher polydispersity, so the phase transition between hexatic and liquid remains first order. This contradicts the results of Ruiz et al. Simulations with higher Number of disks are necessary to understand if this is only a  size effect. 

**equation of states**

these are the equation of state obtained for N=10000 particles, both pressure in the monte carlo and in the EDMD coincide. For now I see no presence of Maxwell's loop. 
I still have no equation of state for poly=0% because of the cluster crash.

![](https://i.imgur.com/6maWyIQ.png)


poly=9%
![](https://i.imgur.com/dzbc7xt.png)

![](https://i.imgur.com/xWTCy8Y.png)

![](https://i.imgur.com/mjXrIkU.png)



poly=8%
![](https://i.imgur.com/DTFfeSM.png)

![](https://i.imgur.com/WG8Ipk7.png)

![](https://i.imgur.com/N7JU3ZH.png)



poly=7%
![](https://i.imgur.com/y3lJb9P.png)

![](https://i.imgur.com/hVsSYBn.png)

![](https://i.imgur.com/O9nfDlh.png)



poly=6%
![](https://i.imgur.com/SoOSHy0.png)

![](https://i.imgur.com/oKvIaJf.png)

![](https://i.imgur.com/c1x1dN1.png)



poly=5%
![](https://i.imgur.com/tMTW6aq.png)

![](https://i.imgur.com/9bbv6dh.png)

![](https://i.imgur.com/rSL35Ed.png)



poly=4%

![](https://i.imgur.com/Oe7phVP.png)

![](https://i.imgur.com/YgPpXH5.png)


poly=3%

![](https://i.imgur.com/UpFT5C1.png)


![](https://i.imgur.com/NL2cm7Z.png)

![](https://i.imgur.com/zKFCySz.png)



poly=2%

![](https://i.imgur.com/CyRR8Pc.png)


![](https://i.imgur.com/pHyN6ew.png)

![](https://i.imgur.com/n0Pql6N.png)




poly=1%

![](https://i.imgur.com/wshQjj4.png)

![](https://i.imgur.com/Qqyp3Qq.png)

![](https://i.imgur.com/opglYnx.png)



poly=0%

![](https://i.imgur.com/RtOMsOp.png)


![](https://i.imgur.com/TjUDD4X.png)

![](https://i.imgur.com/1Vh4h7M.png)



![](https://i.imgur.com/Lu7NJkk.png)

![](https://i.imgur.com/A9LQ6ev.png)


![](https://i.imgur.com/wYRl4jK.png)

![](https://i.imgur.com/KuoIu4r.png)



**liquid**

![](https://i.imgur.com/JybswOC.png)



![](https://i.imgur.com/sRizWCB.jpg)

![](https://i.imgur.com/5OafaQ1.png)



**hexatic-liquid**

![](https://i.imgur.com/dvpC2P0.png)

avercorr>0.6 yellow
0.6>avercorr>0.3 blue
0.3>avercorr>0.1 cyan
0.1>avercorr>-0.1 green
-0.1>avercorr red

![](https://i.imgur.com/VuxO8aB.png)

![](https://i.imgur.com/eDoAMbt.png)




To be more consistent we have computed the different correlation of the particle using SANN algorithm and we have run simulation longer trying to see phase separation.


![](https://i.imgur.com/NLsBsXA.png)

![](https://i.imgur.com/2OwEw83.png)

unfortunately the crashes of the cluster does not allow us to have better results so for now we cannot see a full phase separation. for now we have only an hint with these results that the phase transition is first order, we need maxwell loop and phase separation. 



**Hexatic-Crystal**

![](https://i.imgur.com/tDd12nn.png)


![](https://i.imgur.com/ASYQKpI.png)

![](https://i.imgur.com/TtkqIm1.png)

Currently I cannot fully distinguish the hexatic and crystal state points. My data analisys program can in principle distinguish between the two of them and show an hint of the two different phases' behaviour. However more particles are necessary to be able to fully distinguish between the two phases. 


result for N=160000

![](https://i.imgur.com/t8ml5NJ.png)

![](https://i.imgur.com/tOH92ym.png)


***Fractionation effects at high packing fraction***

At high packing fraction the system seems to slowly moves toward a fractionation phase before reaching the crystal state.




**Poly = 7%, p.f. = 0.825888**


![](https://i.imgur.com/epm1R3a.png)

![](https://i.imgur.com/uND0X9z.png)



this image looks like a quasi-crystal at first, I will prove that it is not the case.


![](https://i.imgur.com/Ahetu1S.png)

![](https://i.imgur.com/b609F9O.png)

this figure shows the distributions of radii in the green and yellow region (distribution is the green and distribution2 is the yellow).

we also have computed the packing fraction for the green and yellow region: 
green region = 0.815725
yellow region = 0.825127

so the region at interface is slightly less dense compared to the average/yellow region.


![](https://i.imgur.com/DreBu5P.png)

we have splitted the particles into two main clusters depending on the correlations between their neighbours. as you can  see the clusters coincide  with the regions splitted by the stripes in the first image. 

now we can compute the g(x,y) for one of the two clusters:

![](https://i.imgur.com/PJtmYCH.png)


for the other cluster we have: 

![](https://i.imgur.com/VSw2dxl.png)

combined together the two g(x,y) form the first figure.


we can also compute the distributions of radii for the two different clusters:

![](https://i.imgur.com/BfCuaE0.png)

it seems that they have the same distribution. on average the green cluster has: 0.300041 while the red cluster has: 0.301360


recently the configuration is shifting to a pure crystal state: 

![](https://i.imgur.com/c63vkGl.png)

![](https://i.imgur.com/sQdyGk4.png)




**Poly = 7% p.f. = 0.847602**




this case looks more extreme than the previous one: 

![](https://i.imgur.com/bHj0J32.png)

![](https://i.imgur.com/VuA0AtB.png)




we will apply the same method


![](https://i.imgur.com/sOgINXy.png)

![](https://i.imgur.com/0TnssYT.png)

![](https://i.imgur.com/oJxkuKY.png)


in this case the stripes only partially coincides with the boundary of the different clusters. 

![](https://i.imgur.com/FqWf0sr.png)

![](https://i.imgur.com/XzSAYrn.png)

![](https://i.imgur.com/MFMQQwE.png)


only longer simulation time will tell if the system will fall to a unique crystal phase. it may be possible that the free energy landscape at this value of pressure/packing fraction is extremely rough


also this configuration is slowly shifting toward a more crystalline state:

![](https://i.imgur.com/68gNeNs.png)

![](https://i.imgur.com/BaCIyWy.png)


**with and without sann algorithm**

without: 

![](https://i.imgur.com/iUn9NBz.png)

with: 

![](https://i.imgur.com/VHNUEuq.png)
