# Mathematical Modeling of the Solid–Liquid Interface Propagation by the Boundary Integral Method with Nonlinear Liquidus Equation and Atomic Kinetics

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## Abstract

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## 1. Introduction

## 2. The Model and BIE for the Interface Function

#### 2.1. The Heat and Mass Transfer Model

#### 2.2. The Green’s Function Technique

#### 2.3. Stationary Growth

#### 2.4. The Parabolic Cylinder Reference Frame

## 3. Numerical Examples

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Dendrite tip velocity V and radius $\rho /2$ as functions of the melt undercooling $\Delta $ for the TiAl alloy. The colored lines represent the effect of quadratic liquidus ${m}_{2}$. The parameters used in the calculation are: solidification temperature ${T}_{d0}=1748$ K, hypercooling ${T}_{Q}=272$ K, impurity concentration ${C}_{d\infty}=55$ at.%, distribution coefficient ${k}_{0}=0.8$, liquidus slope ${m}_{1}=-8.8$ K (at.%)${}^{-1}$, diffusion coefficient in liquid ${D}_{C}=8.27\times {10}^{-9}$ m${}^{2}$ s${}^{-1}$, temperature diffusivity ${D}_{T}=7.5\times {10}^{-6}$ m${}^{2}$ s${}^{-1}$, kinematic viscosity of the liquid $\nu =0.5\times {10}^{-7}$ m${}^{2}$ s${}^{-1}$, fluid velocity ${U}_{\infty}=0.1$ m s${}^{-1}$, latent heat of solidification $Q=12268.8$ J mol${}^{-1}$, heat capacity ${c}_{p}=45$ J (mol K)${}^{-1}$, capillary length ${d}_{c}\approx {d}_{0}=7.8\times {10}^{-10}$ m, kinetic coefficient ${\beta}_{k}=0.2$ m (s K)${}^{-1}$, power exponent of the atomic kinetics $n=1$, surface energy anisotropy ${\epsilon}_{c}=0.01$, selection constant ${\sigma}_{0}=0.05$, selection parameter responsible for fluid flow $b=0.1$. The coefficient ${m}_{2}$ shown in the figure inserts is measured in K (at.%)${}^{-2}$.

**Figure 3.**Dendrite tip velocity V and radius $\rho /2$ as functions of the melt undercooling $\Delta $ for the pure Ni alloy. The colored lines represent the effect of different powers of the atomic kinetics exponent n. The parameters used in the calculation: solidification temperature ${T}_{d0}=1728$ K, hypercooling ${T}_{Q}=435$ K, temperature diffusivity ${D}_{T}=1\times {10}^{-5}$ m${}^{2}$ s${}^{-1}$, kinematic viscosity of the liquid $\nu =7.14\times {10}^{-7}$ m${}^{2}$ s${}^{-1}$, fluid velocity ${U}_{\infty}=0.001$ m s${}^{-1}$, capillary length ${d}_{c}\approx {d}_{0}=4.2\times {10}^{-10}$ m, kinetic coefficient ${\beta}_{k}=0.01$ m (s K)${}^{-1}$, surface energy anisotropy ${\epsilon}_{c}=0.02$, selection constant ${\sigma}_{0}=0.05$, selection parameter responsible for fluid flow $b=0.1$.

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**MDPI and ACS Style**

Titova, E.A.; Alexandrov, D.V.; Toropova, L.V. Mathematical Modeling of the Solid–Liquid Interface Propagation by the Boundary Integral Method with Nonlinear Liquidus Equation and Atomic Kinetics. *Crystals* **2022**, *12*, 1657.
https://doi.org/10.3390/cryst12111657

**AMA Style**

Titova EA, Alexandrov DV, Toropova LV. Mathematical Modeling of the Solid–Liquid Interface Propagation by the Boundary Integral Method with Nonlinear Liquidus Equation and Atomic Kinetics. *Crystals*. 2022; 12(11):1657.
https://doi.org/10.3390/cryst12111657

**Chicago/Turabian Style**

Titova, Ekaterina A., Dmitri V. Alexandrov, and Liubov V. Toropova. 2022. "Mathematical Modeling of the Solid–Liquid Interface Propagation by the Boundary Integral Method with Nonlinear Liquidus Equation and Atomic Kinetics" *Crystals* 12, no. 11: 1657.
https://doi.org/10.3390/cryst12111657