# Zeroth-Order Nucleation Transition under Nanoscale Phase Separation

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Spatial Phase Separation

## 3. System Statistical Operator

## 4. Averaging over Phase Configurations

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 5. Hamiltonian in Spin Representation

## 6. Free Energy of Mixture

## 7. Stability Conditions

## 8. Results and Discussion

- (i)
- At low temperatures, the system is a pure ferromagnet described by the free energy ${F}_{fer}$ and the order parameter ${s}_{fer}\equiv {s}_{1}$, with $w\equiv 1$. When increasing temperature, ${F}_{fer}$ gradually approaches ${F}_{par}$ corresponding to a paramagnet. The order parameter ${s}_{fer}\equiv {s}_{1}$ has the form typical of the ferromagnetic magnetization. This behavior, for instance, happens for $u<0.25$ and all $h>0$.
- (ii)
- For $u>0.25$, at low temperatures, below the nucleation temperature ${T}_{n}$, the system is a pure ferromagnet, with the free energy ${F}_{fer}$, the order parameter ${s}_{fer}\equiv {s}_{1}$, and $w\equiv 1$. At the nucleation temperature ${T}_{n}$, there appears a solution for the mixed state with the free energy F and the order parameters ${s}_{1}$ and ${s}_{2}$. The free energy F is lower than ${F}_{fer}$, but does not intersect it so that the nucleation is to be classified as a zeroth-order transition.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Free energies of the mixed state, F (solid line), ferromagnetic state, ${F}_{fer}$ (dash–dotted line), and of the paramagnetic state, ${F}_{par}$ (dashed line), for $u=0.3$ and different magnetic fields: (

**a**) $h=0.1$; (

**b**) $h=0.5$.

**Figure 2.**Order parameters ${s}_{1}$ (

**a**), ${s}_{2}$ (

**b**), and w as functions of dimensionless temperature T (

**c**), for $u=0.3$ and different fields: (1) $h=0.01$; (2) $h=0.1$; (3) $h=0.2$; (4) $h=0.3$; (5) $h=0.5$; (6) $h=1$. The corresponding nucleation temperatures are: (1) ${T}_{n}=0.15$; (2) ${T}_{n}=0.24$; (3) ${T}_{n}=0.34$; (4) ${T}_{n}=0.44$; (5) ${T}_{n}=0.66$; (6) ${T}_{n}=1.42$.

**Figure 3.**Free energies of the mixed state, F (solid line), ferromagnetic state, ${F}_{fer}$ (dash-dotted line), and of the paramagnetic state, ${F}_{par}$ (dashed line), for $u=0.6$ and different magnetic fields: (

**a**) $h=0.1$; (

**b**) $h=0.5$.

**Figure 4.**Order parameters ${s}_{1}$ (

**a**), ${s}_{2}$ (

**b**), and w as functions of dimensionless temperature T (

**c**), for $u=0.6$ and different fields: (1) $h=0.01$; (2) $h=0.1$; (3) $h=0.2$; (4) $h=0.3$; (5) $h=0.5$; (6) $h=1$. The corresponding nucleation temperatures are: (1) ${T}_{n}=0.14$; (2) ${T}_{n}=0.20$; (3) ${T}_{n}=0.26$; (4) ${T}_{n}=0.33$; (5) ${T}_{n}=0.46$; (6) ${T}_{n}=0.89$.

**Figure 5.**Free energies of the mixed state, F (solid line), ferromagnetic state, ${F}_{fer}$ (dash–dotted line), and of the paramagnetic state, ${F}_{par}$ (dashed line), for $u=0.8$ and different magnetic fields: (

**a**) $h=0.3$; (

**b**) $h=1$. For $h=0.3$, the mixed state is not stable. For $h=1$, the zeroth-order nucleation transition occurs at the nucleation temperature ${T}_{n}=0.72$.

**Figure 6.**Order parameters ${s}_{1}$, ${s}_{2}$, and w as functions of dimensionless temperature T, for $u=0.8$ and different fields: (

**a**) $h=0.1$ (solid line), $h=0.2$ (dashed line); $h=0.3$ (dash–dotted line); $h=0.5$ (dotted line); (

**b**) $h=0.01$; $h=1$; $h=2$. The corresponding nucleation temperatures are ${T}_{n}=0.14$, ${T}_{n}=0.72$, and ${T}_{n}=1.77$; (

**c**) $h=0.01$; $h=1$; $h=2$; (

**d**) $h=0.01$; $h=1$; $h=2$.

**Figure 7.**Free energies of the mixed state, F (solid line), ferromagnetic state, ${F}_{fer}$ (dash–dotted line), and of the paramagnetic state, ${F}_{par}$ (dashed line), for $u=1.5$ and different magnetic fields: (

**a**) $h=1$; (

**b**) $h=5$. For $h=1$, the mixed state is not stable. For $h=5$, the zeroth-order nucleation transition occurs at the nucleation temperature ${T}_{n}=4.55$.

**Figure 8.**Order parameters ${s}_{1}$ (

**a**), ${s}_{2}$ (

**b**), and w as functions of dimensionless temperature T (

**c**), for $u=1.5$ and different fields: (1) $h=0.01$; (2) $h=0.5$; (3) $h=1$; (4) $h=2$; (5) $h=5$; (6) $h=6$. The nucleation temperatures are ${T}_{n}=4.55$ for $h=5$ and ${T}_{n}=6.4$ for $h=6$.

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Yukalov, V.I.; Yukalova, E.P.
Zeroth-Order Nucleation Transition under Nanoscale Phase Separation. *Symmetry* **2021**, *13*, 2379.
https://doi.org/10.3390/sym13122379

**AMA Style**

Yukalov VI, Yukalova EP.
Zeroth-Order Nucleation Transition under Nanoscale Phase Separation. *Symmetry*. 2021; 13(12):2379.
https://doi.org/10.3390/sym13122379

**Chicago/Turabian Style**

Yukalov, Vyacheslav I., and Elizaveta P. Yukalova.
2021. "Zeroth-Order Nucleation Transition under Nanoscale Phase Separation" *Symmetry* 13, no. 12: 2379.
https://doi.org/10.3390/sym13122379