# Directional Crystallization in the Presence of a Mushy Layer with Applications to the Earth’s Inner Core Boundary

^{1}

^{2}

^{3}

^{*}

*Crystals*2023)

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations and Analytical Solutions

## 3. Discussion

^{5}m due to uncertainties in $\eta $. It is shown in Figure 2 that an increase in the absolute value of the liquidus slope causes an increase in the thickness. At the same time, smaller velocities $|\overline{V}|$ give larger differences in the two curves in Figure 2.

## 4. Conclusions

^{2}m. It is significant that the primary interdendritic spacing reduces when the crystallization and melt velocities increase, as well as when the dynamic viscosity of the melt is estimated to be larger.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The mushy region width h as a function of average fluid velocity $\overline{V}$ plotted for (

**a**) different temperature gradients and solid fractions; (

**b**) different solidification velocities and solid fractions. Parameters used for calculation: (

**a**) ${g}_{l}=-5.4\xb7{10}^{-4}$ K/m, ${\varphi}_{*}=0.043$ (solid lines) and ${g}_{l}=-3.7\xb7{10}^{-4}$ K/m, ${\varphi}_{*}=0.432$ (dashed lines); ${m}_{\sigma}=-1.1\xb7{10}^{4}$ K, ${u}_{s}=3.2\xb7{10}^{-12}$ m/s, $\eta ={10}^{19}$ Pa-s (1), $\eta ={10}^{17}$ Pa-s (2), $\eta ={10}^{15}$ Pa-s (3), $\eta ={10}^{13}$ Pa-s (4), $\eta ={10}^{10}$ Pa-s (5); the critical fluid velocities are shown by vertical lines ${\overline{V}}_{c}$ (we evaluate here and later in the paper $\kappa $ as $0.05/(\sigma \left({\varphi}_{*}\right)-{\sigma}_{h})$); (

**b**) ${u}_{s}=3.2\xb7{10}^{-12}$ m/s, ${\varphi}_{*}=0.432$ (solid lines), ${u}_{s}=6\xb7{10}^{-12}$ m/s, ${\varphi}_{*}=0.664$ (dashed lines) and ${u}_{s}=2\xb7{10}^{-11}$ m/s, ${\varphi}_{*}=0.889$ (dotted lines); ${m}_{\sigma}=-1.1\xb7{10}^{4}$ K, ${g}_{l}=-3.7\xb7{10}^{-4}$ K/m, $\eta ={10}^{19}$ Pa-s (1), $\eta ={10}^{17}$ Pa-s (2), $\eta ={10}^{15}$ Pa-s (3), and $\eta ={10}^{13}$ Pa-s (4).

**Figure 2.**(

**a**) Solid fraction versus spatial coordinate in a mushy region for different viscosities. Parameters used for calculation: $\eta ={10}^{15}$ Pa-s (solid lines) and $\eta ={10}^{14}$ Pa-s (dashed lines), ${m}_{\sigma}=-1.1\xb7{10}^{4}$ K, ${g}_{l}=-3.7\xb7{10}^{-4}$ K/m, $\overline{V}=-{10}^{-6}$ m/s, ${u}_{s}=3.2\xb7{10}^{-12}$ m/s (1), ${u}_{s}=6\xb7{10}^{-12}$ m/s (2), and ${u}_{s}=2\xb7{10}^{-11}$ m/s (3); (

**b**) mushy region thickness versus the mean fluid velocity for different liquidus slopes. Parameters used for calculation: ${m}_{\sigma}=-1.1\xb7{10}^{4}$ K (solid line) and ${m}_{\sigma}=-{10}^{2}$ K (dashed line), ${g}_{l}=-3.7\xb7{10}^{-4}$ K/m, ${u}_{s}=2\xb7{10}^{-11}$ m/s, and $\eta ={10}^{15}$ Pa-s.

**Figure 3.**(

**a**) Primary interdendritic spacing versus the mean fluid velocity for different solidification rates. Parameters used for calculation: ${u}_{s}=2\xb7{10}^{-11}$ m/s (solid lines) and ${u}_{s}=3.2\xb7{10}^{-12}$ m/s (dashed lines), ${m}_{\sigma}=-1.1\xb7{10}^{4}$ K, ${g}_{l}=-3.7\xb7{10}^{-4}$ K/m, ${b}_{1}=0.86$, $\eta ={10}^{15}$ Pa-s (1), and $\eta ={10}^{14}$ Pa-s (2); (

**b**) mean horizontal dimension $\Xi $ and mean radius of a chimney’s $\alpha $ dependency on the solid fraction at the inner core boundary for different fluid velocities: ${u}_{\xi}={10}^{-9}$ m/s, ${\theta}_{\Xi}=6000$ K, ${\theta}_{0}=5000$ K, $\overline{V}=-0.5\xb7{10}^{-6}$ m/s (1), $\overline{V}=-{10}^{-7}$ m/s (2), and $\overline{V}=-0.5\xb7{10}^{-7}$ m/s (3).

**Figure 4.**Schematic illustration of the mushy region near the inner core boundary. Velocity flow lines are shown as blue solid lines with arrows. Growing dendrites are illustrated by dash-dotted lines. The dashed curve connecting the dendrites shows the curved thickness of the mushy region.

**Table 1.**Physical parameters characterizing the solidification conditions of the Earth’s core [12].

Parameter | Symbol | Value | Units |
---|---|---|---|

Chemical diffusivity in the liquid phase | ${D}_{l}$ | 10${}^{-9}$ | m^{2}/s |

Thermal diffusivity in the liquid phase | ${\lambda}_{l}/\left({\rho}_{h}{c}_{l}\right)$ | $6\xb7{10}^{-6}$ | m^{2}/s |

Thermal diffusivity in the solid phase | ${\lambda}_{s}/\left({\rho}_{s}{c}_{s}\right)$ | $7\xb7{10}^{-6}$ | m^{2}/s |

Thermal conductivity in the liquid phase | ${\lambda}_{l}$ | 63 | J/(m s K) |

Thermal conductivity in the solid phase | ${\lambda}_{s}$ | 79 | J/(m s K) |

Solute partition coefficient | ${k}_{0}$ | 0.25 | - |

Latent heat of solidification | ${Q}_{V}$ | $6.84\xb7{10}^{9}$ | J/m^{3} |

Thermal (heat) capacity in the liquid phase | ${c}_{l}$ | 860 | J/(kg K) |

Acceleration of gravity | g | 4.4 | m/s^{2} |

Clapeyron slope | ${m}_{p}$ | $9\xb7{10}^{-9}$ | K/Pa |

Permeability constant | ${\Pi}_{0}$ | 10 | m^{2} |

Capillary constant | ${d}_{0}$ | ${10}^{-9}$ | m |

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**MDPI and ACS Style**

Alexandrov, D.V.; Alexandrova, I.V.; Nikishina, M.A.; Malygin, A.P.; Toropova, L.V.
Directional Crystallization in the Presence of a Mushy Layer with Applications to the Earth’s Inner Core Boundary. *Crystals* **2023**, *13*, 1361.
https://doi.org/10.3390/cryst13091361

**AMA Style**

Alexandrov DV, Alexandrova IV, Nikishina MA, Malygin AP, Toropova LV.
Directional Crystallization in the Presence of a Mushy Layer with Applications to the Earth’s Inner Core Boundary. *Crystals*. 2023; 13(9):1361.
https://doi.org/10.3390/cryst13091361

**Chicago/Turabian Style**

Alexandrov, Dmitri V., Irina V. Alexandrova, Margarita A. Nikishina, Alexey P. Malygin, and Liubov V. Toropova.
2023. "Directional Crystallization in the Presence of a Mushy Layer with Applications to the Earth’s Inner Core Boundary" *Crystals* 13, no. 9: 1361.
https://doi.org/10.3390/cryst13091361