# Condensation and Crystal Nucleation in a Lattice Gas with a Realistic Phase Diagram

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## Abstract

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**1968**, 49, 1778–1783), a two-dimensional lattice-gas system featuring a crystalline phase and two distinct fluid phases (liquid and vapor). In this system, a particle prevents other particles from occupying sites up to third neighbors on the square lattice, while attracting (with decreasing strength) particles sitting at fourth- or fifth-neighbor sites. To make the model more realistic, we assume a finite repulsion at third-neighbor distance, with the result that a second crystalline phase appears at higher pressures. However, the similarity with real-world substances is only partial: Upon closer inspection, the alleged liquid–vapor transition turns out to be a continuous (albeit sharp) crossover, even near the putative triple point. Closer to the standard picture is instead the freezing transition, as we show by computing the free-energy barrier relative to crystal nucleation from the “liquid”.

## 1. Introduction

## 2. Model and Method

## 3. Results

#### 3.1. Transfer-Matrix Phase Diagram of the MOVB Model

#### 3.2. The Liquid–Vapor Transition Is in Fact a Crossover

#### 3.3. Features of Crystal Nucleation from the “Liquid”

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. On the Dominant Eigenvalue of the Transfer Matrix

## Appendix B. Cluster Free Energy and Its Relation to the Cluster-Size Distribution

**c**). Hence, ${\sum}_{i=1}^{{N}_{s}}{\delta}_{{s}_{i}\left(\mathbf{c}\right),n}=S\left(\mathbf{c}\right){\delta}_{S\left(\mathbf{c}\right),n}=n{\delta}_{S\left(\mathbf{c}\right),n}$ and ${N}_{n}\left(\mathbf{c}\right)={\delta}_{S\left(\mathbf{c}\right),n}$. As a result, we have the following:

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**Figure 1.**The two stable crystals of the MOVB model (particular). (

**Left**) square crystal. The distance between two neighboring particles is ${r}_{4}=\sqrt{5}a$. A primitive unit cell is shown in red. (

**Right**) centered-rectangular crystal. Each particle in this crystal has two neighboring particles at distance ${r}_{3}=2a$ and other four particles at distance ${r}_{4}$. A primitive unit cell (red) and a non-primitive cell (blue) are shown.

**Figure 2.**Transfer-matrix data for the OVB model at two different temperatures (

**left**panel: $\eta =6.5$;

**right**panel: $\eta =9$) and for three sizes ($L=10$, black; $L=15$, blue; $L=20$, red). For each temperature, the reduced compressibility (main figure) and the density (inset) are plotted as a function of $\beta \mu $. The ideal-gas limit $\rho {k}_{\mathrm{B}}T{K}_{T}=1$ is recovered for $\mu \to -\infty $.

**Figure 3.**Transfer-matrix data for the MOVB model at various temperatures ($\eta $ values are between 1 and 15, see legend). Only results for $L=10$ are available. In the main figure, the $\beta \mu $ derivative of the density is plotted as a function of $\beta \mu $. In the inset, a few density plots are shown. Any peak of the density derivative signals a more or less steep rise in the density, which is, in turn, indicative of the possibility of a phase transition in the thermodynamic limit.

**Figure 4.**MOVB phase diagram according to the transfer-matrix analysis. (

**Left**) $\beta \mu $ vs. $\beta \u03f5$. Different types of “transition points” are marked with different symbols and colors. The purple dots were computed through scans made at fixed $\mu $ (see text). The two dashed lines represent extrapolations to infinite temperature of the low-T transition loci $\mu =-2.4\u03f5$ (between square crystal and vapor) and $\mu =4.1\u03f5$ (between c-ret crystal and square crystal), see Section 2. (

**Right**) T-P phase diagram. The brown dots are the $T=0$ transition pressures computed in Section 2.

**Figure 5.**Transfer-matrix results for the MOVB model ($L=10$) along a number of constant-$\mu $ lines (in the legend). (

**Left**) entropy density; (

**Right**) constant-$\mu $ specific heat per unit volume.

**Figure 6.**Compressibility data for the OVB and MOVB models near the supposed liquid–vapor transition point for $\eta =9$. We have considered square lattices of four different lateral sizes L: 50 (purple), 80 (blue), 100 (green), and 120 (magenta). The smooth lines through the data points are spline interpolants. The statistical uncertainties are negligible, i.e., smaller than the size of the symbols. In the inset, the maximum of ${k}_{\mathrm{B}}T{K}_{T}$ is reported vs. volume on a log-log scale for both models to show that in the thermodynamic limit no phase transition is likely to occur in either of the models.

**Figure 7.**MOVB model for $\eta =9$ and $L=120$. Probability distribution of the density across the apparent liquid–vapor transition ($\beta \mu $ values in the legend).

**Figure 8.**MOVB model on a $100\times 100$ lattice, for $\eta =6.5$ and $\beta \mu =-3.70$: typical configuration (particular) of the overcompressed liquid, with liquid-like (white circles) and solid-like particles (colored circles) well distinguished. For the present values of $\eta $ and $\beta \mu $ the reduced density is about $0.161$ and the energy per particle is $-0.201\u03f5$. The size of the maximum cluster in the configuration shown is 28. Different colors are used to represent particles belonging to solid clusters with different sizes. The white circles with the red contour are isolated solid-like particles. The dense grid in the background is the underlying square lattice. Notice the presence of clusters where the occurrence of two particles at distance ${r}_{3}$ apart induces a change in crystalline orientation.

**Figure 9.**MOVB model for $\eta =6.5$ and $\beta \mu =-3.75,-3.70,-3.65$ (from top to bottom): the (reduced) cluster free energy (blue, dark green, and red) is plotted together with the (reduced) cost of formation of the largest cluster shifted upwards by $\mathrm{ln}{N}_{s}$ (cyan, light green, and orange). In the inset, a magnification of the low-size region.

**Figure 10.**MOVB model for $\eta =6.5$ and $\beta \mu =-3.70$: typical configuration of the system near the top of the nucleation barrier. Cluster sizes and colors in the legend.

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**MDPI and ACS Style**

Prestipino, S.; Costa, G.
Condensation and Crystal Nucleation in a Lattice Gas with a Realistic Phase Diagram. *Entropy* **2022**, *24*, 419.
https://doi.org/10.3390/e24030419

**AMA Style**

Prestipino S, Costa G.
Condensation and Crystal Nucleation in a Lattice Gas with a Realistic Phase Diagram. *Entropy*. 2022; 24(3):419.
https://doi.org/10.3390/e24030419

**Chicago/Turabian Style**

Prestipino, Santi, and Gabriele Costa.
2022. "Condensation and Crystal Nucleation in a Lattice Gas with a Realistic Phase Diagram" *Entropy* 24, no. 3: 419.
https://doi.org/10.3390/e24030419