# Kinetics of Precipitation Processes at Non-Zero Input Fluxes of Segregating Particles

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## Abstract

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## 1. Introduction

## 2. Precipitation Kinetics at Non-Zero Input Fluxes of Segregating Particles: Numerical Computations

#### 2.1. Some General Considerations

#### 2.2. Basic Kinetic Equations

#### 2.3. Results of the Numerical Solution of the Kinetic Equations

## 3. Theoretical Analysis

#### 3.1. Number of Clusters in Dependence on the Rate of Input Fluxes of the Segregating Component

#### 3.2. Coarsening in Closed Systems: Lifshitz-Slezov-Wagner−Approach

#### 3.3. Coarsening at Constant Input Fluxes of the Segregating Component: Alternative Approach

#### 3.3.1. Basic Ideas

#### 3.3.2. Derivation of the Kinetic Equations Modeling Coarsening in a Closed System

#### 3.3.3. Account of Additional Factors like Elastic Stresses on Coarsening

#### 3.3.4. Application to Coarsening at Non-Zero Input Fluxes of Segregating Particles

#### 3.3.5. Possible Further Applications of the Method: General Theoretical Approach to the Description of Coarsening in Open Systems and at Time-Dependent Boundary Conditions

## 4. Summary of Results and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Evolution of characteristic properties of a system undergoing a precipitation process for the case of zero input fluxes of segregating particles. (

**Top**): Change of the supersaturation, $\mathrm{\Delta}\mu /{k}_{B}T$, as a function of time (cf. Equation (1)). (

**Center**): Change of the average (full curve) and the critical cluster sizes (dotted curve) in the course of the transformation. (

**Bottom**): Change of the number of clusters in the system. Herein the dotted curve counts all clusters in the system, while the full curve refers to clusters with a radius $R>0.6$ nm. In these computations, the aggregation processes were determined for small cluster sizes by kinetically−limited growth, going over continuously to diffusion−limited growth for larger cluster sizes (the figures were taken with permission from [20,21], where further details can also be found).

**Figure 2.**Cluster size distribution function $\phi (u,{t}^{\prime})$ in reduced variables $u=R/{R}_{c}$ for different moments of (dimensionless) time, ${t}^{\prime}$ (cf. Equation (19)). In the course of the evolution, a time−independent shape of the distribution develops as predicted first by Lifshitz and Slezov (see [14,20,21,24,25] for details). In the numerical computations shown in the figure, the aggregation processes are determined for small cluster sizes via kinetic−limited growth going over continuously to diffusion−limited growth for larger cluster sizes. The figures were taken with permission from [20,21], where further details are also given.

**Figure 3.**Time-dependence of the supersaturation for diffusion− (full curve) and kinetically−limited growth (dashed curve). (

**a**) shows the initial stage of the segregation process, while (

**b**) gives an impression of later stages. The values of the parameters were taken from [31], and were set equal to: ${c}_{eq}=8.84\xb7{10}^{25}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-3}$, ${v}_{\alpha}=4.89\xb7{10}^{-29}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}$, ${R}_{1}=2.269\xb7{10}^{-10}\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}$, $\sigma =0.08\phantom{\rule{3.33333pt}{0ex}}\mathrm{N}/\mathrm{m}$, $T=457$ °C $=730\phantom{\rule{3.33333pt}{0ex}}\mathrm{K}$, ${k}_{B}T={10}^{-20}\phantom{\rule{3.33333pt}{0ex}}\mathrm{J}$, $D=5.8\xb7{10}^{-19}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}/\mathrm{s}$. With these parameters, Equation (19) yields $t\cong 6.84{t}^{\prime}$, and, in a dimensionless time scale, ${t}^{\prime}$, the input flux of segregating particles is equal to $\mathrm{\Phi}={10}^{27}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-3}$.

**Figure 4.**Average cluster radius $\langle R\rangle $, critical cluster radius ${R}_{c}$, and their ratio $\langle R\rangle /{R}_{c}$ as functions of (dimensionless) time ${t}^{\prime}$ for (

**a**) diffusion−limited growth and (

**b**) for kinetically−limited growth. The input flux of segregating particles is equal to $\mathrm{\Phi}={10}^{25}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-3}$. The average size and number of clusters is determined, again, in line with Equation (7), taking ${i}_{min}$ equal to ${i}_{min}=20$.

**Figure 5.**Number of clusters per cubic meter as a function of (dimensionless) time, ${t}^{\prime}$, for diffusion− (full curve) and kinetically−limited growth (dashed curve). (

**a**) shows the initial stage while (

**b**) also illustrates the behavior in advanced stages of the process. The input flux of segregating particles is equal to $\mathrm{\Phi}={10}^{27}$ m ${}^{-3}$. The number of clusters is determined in line with Equation (7) taking ${i}_{min}$ equal to ${i}_{min}=20$.

**Figure 6.**Number of clusters per cubic meter as a function of (dimensionless) time, ${t}^{\prime}$, for diffusion− (full curves) and kinetically−limited growth (dashed curves) in dependence on the value of the input fluxes, $\mathrm{\Phi}$ (

**a**). The approach to the asymptotic stage can hereby be quite different in dependence on the value of the input flux, as illustrated for diffusion−limited growth in (

**b**) plotting the ratio of the average cluster size $\langle R\rangle $ and the critical cluster size, ${R}_{c}$. The average size and number of clusters is determined, again, in line with Equation (7) taking ${i}_{min}$ equal to ${i}_{min}=20$.

**Figure 7.**Asymptotic value of the number of clusters per cubic meter for diffusion− and kinetic−limited growth as a function of the input flux of monomers $\mathrm{\Phi}$.

**Figure 8.**Change of the Gibbs free energy, $\mathrm{\Delta}G(N,\langle R\rangle )$, in dependence on (average) cluster size, $\langle R\rangle $, for different values of the number of clusters, N (${N}_{1}<{N}_{2}<\dots <{N}_{c}$). In (

**a**), $\mathrm{\Delta}G(N,\langle R\rangle )$ is shown for one given value of the number of clusters. The arrows indicate the change of the position of the extremum values of $\mathrm{\Delta}G(N,\langle R\rangle )$ with increase of N. In (

**b**), the respective curves are shown for a set of different values of the number of clusters. The dotted curve describes nucleation, the dashed−dotted curve describes the stage of dominating independent growth, while the dashed curve refers to coarsening. The minimum value of the average cluster size and the maximum number of clusters, which may start to evolve via Ostwald ripening, is denoted by ${R}_{c}^{c}$, respectively, ${N}_{c}$.

**Figure 9.**Predictions of Equations (99) and (100) modeling the evolution in time of the average size and the number of clusters in coarsening in an open system in a volume of the solution equal to $1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{3}$ for both kinetic−limited (

**left column**) and diffusion−limited (

**right column**) growth modes and different input fluxes of segregating particles, as indicated in the figures.

**Figure 10.**The present paper is devoted to our teacher, colleague, and friend, the Corresponding Member of the Ukrainian Academy of Sciences, Vitali V. Slezov (9 March 1930–30 October 2013).

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**MDPI and ACS Style**

Schmelzer, J.W.P.; Tropin, T.V.; Abyzov, A.S.
Kinetics of Precipitation Processes at Non-Zero Input Fluxes of Segregating Particles. *Entropy* **2023**, *25*, 329.
https://doi.org/10.3390/e25020329

**AMA Style**

Schmelzer JWP, Tropin TV, Abyzov AS.
Kinetics of Precipitation Processes at Non-Zero Input Fluxes of Segregating Particles. *Entropy*. 2023; 25(2):329.
https://doi.org/10.3390/e25020329

**Chicago/Turabian Style**

Schmelzer, Jürn W. P., Timur V. Tropin, and Alexander S. Abyzov.
2023. "Kinetics of Precipitation Processes at Non-Zero Input Fluxes of Segregating Particles" *Entropy* 25, no. 2: 329.
https://doi.org/10.3390/e25020329