# Polydisperse HS

###### tags: `Polydisperse`

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

Antoine, Frank, Laura

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## Preliminary work

> Update 10/04/24

**Goal:** Find the functional form of $\beta \mu(R)$ for a range of densities.

*Canonical simulations:*

- Deterministic particle size distribution
- $p = 0.08 ; 0.04$
- $N = 4000$
- Equilibration: $50k \tau_\textrm{MD}$
- Acquisition: $100k \tau_\textrm{MD}$, 500 samples
- 10 iterations for each statepoint


### Pressure


### Imposed size distribution and mixing chem. pot.


### Surface foces


Time dependence of the surface forces for a choice of three particles of size $\sigma_i$ at the indicated statepoint.


These are fitted with a cubic polynomial to obtain the configurational chem. pot.

### Chem. pot. functionals


> Update 26/04/24

### Semigrand simulations


Use the functional forms of $\beta \mu^\textrm{trial}[R]$ obtained for the choice of $\rho_f^\textrm{trial}$ to recover the expected particle size distribution

> Will update soon with the correct particle size distribution for initial configurations: fluid; FCC

**Notes on tailored interaction field scheme:**

- 3rd order polynomial field
- Has real coefficients: 3 real roots
- Find roots using the trigonometric solution
- Relatively slow, will try optimization in the near future (fitting the inverse function on each branch of the well independently + Newton's method)

#### Low density simulations

Expect to recover ideal gas behavior in the form of $exp(-\beta \mu^\textrm{trial}[R])$ for the particle size distribution. Simtime is $10^5 \tau_\textrm{MD}$, binsize is $0.002333\sigma$.



> Update 06/05/24

### Direct coexistence (SG) simulations

**Simulation details:**

- HS, $8\%$ polydisperse
- $FCC_A$ crystal orientation: square crystal plane facing the fluid
- $N = 4000$
- simulation box has roughly $zsize = 3 xzise = 3 ysize$
- simtime: $10^5 \tau_\textrm{MD}$


Statepoint at $\{\rho = 0.9744,~P^\textrm{trial} = 14.37426\}$ ($P_{zz} = 14.591819$) is fully melted by the end of the simulation.

Statepoint at $\{\rho = 0.9772,~P^\textrm{trial} = 14.56878\}$ ($P_{zz} = 14.293955$) has a small amount of leftover coexisting fluid by the end of the simulation.

> Update 07/05/24

**Simulation details:**

- HS, $8\%$ polydisperse
- $FCC_B$ crystal orientation: hexagonal crystal plane aligned with the $x-z$ plane of the simulation box
- $N = 4000$
- simulation box has roughly $zsize = 3 xzise = 3 ysize$
- simtime: $7-9\times10^4 \tau_\textrm{MD}$


**coexistence!**


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## Phase diagram \( $p,~P_{coex}$ \)

### Coexistence determination pipeline

- Use systems of $N = 16000$ particles as reference (smaller comes fith significant finite size effects)
- Robust fits can be obtained for surface forces $f_i(R, \rho)$:
- quadratic in particle radius $R$
- each coeff. quadratic in density $\rho$
- propagates to chemical potential functions $\mu_\textrm{conf}(R, \rho)$ and $\mu_\textrm{tot}(R, \rho)$
- Robust quadratic fits can be obtained for fluid system pressure $P_f(\rho)$ (get $P^\textrm{trial} = P_f(\rho_f^\textrm{trial})$)
- Robust quadratic fits can be obtained for crystalline system pressure $P_\chi(\rho_\chi)$ (get $\rho_\chi^\textrm{trial}$ such that $P_\chi(\rho_\chi^\textrm{trial}) = P^\textrm{trial}$)
- Direct coexistence simulations to run for at least $10^5 \tau_\textrm{MD}$


| $N$ | $\beta P_\textrm{coex} \sigma^3$ $(\textrm{FCC}_a)$ | $\beta P_\textrm{coex} \sigma^3$ $(\textrm{FCC}_b)$ |
| -------- | -------- | -------- |
| 8000 | 14.7686 | |
| 16000 | 14.7781 | |
| 32000 | | |
| 64000 | | |

> WIP, will be updated