# Axial Vibration Control Technique for Crystal Growth from the Melt: Analysis of Vibrational Flows’ Behavior

^{1}

^{2}

^{*}

## Abstract

**:**

_{3}, CdTe.

## 1. Introduction

## 2. Numerical Model

**D**, (2) thickness—

**d**, and (3) radius of the edge curvature between a cylindrical generatrix and a faceplate—

**R**. It had been demonstrated that the

**R**-value sufficiently influenced on average flows’ generation [18].

#### 2.1. Model of Fluid Motion under the AVC Action

#### 2.2. Material Properties

_{3}, Ge and CdTe melts were used as model liquid phases. Their physicochemical properties are presented in Table 2.

## 3. Results

#### 3.1. AVC Flows Velocities

_{3}were obtained at various values of the amplitude (

**A**) and frequency (

**f**) of disk oscillation, as well as at various radii of the disk edge (

**R**) (Table 3).

**Aω**. It can be seen that near the edge of the disk the speed of movement of the melt increases by approximately 10 times the speed of movement of the disk, and in the area between the disk and the side wall the speed of movement of the melt is opposite to the movement of the disk. However, over time, due to the viscosity of the melt, a thin layer of liquid should appear on the side of the disk, which will move in the same direction as the movement of the disk, which will lead to a break in the existing streamlines and the formation of a controlled vortex flow of the melt.

**Aω**) at the edge of the disk showed that in the case of more viscous CdTe the thickness of the boundary layer on the side surface of the disk is significantly greater comparing to Ge melt (Figure 3). However, the scale of the edge vortex is almost the same in both cases.

_{i}) between the melt velocity and amplitude

**A**, angular frequency

**ω**, disk edge radius

**R**, melt density ρ and dynamic viscosity µ:

_{i}using regression analysis, we revealed (Figure 5) that the speed of the melt motion should be proportional to the expression, which for brevity we will denote as

**I**and will call the intensity of oscillations:

**I**intensity is divided into a linear region at low values of the amplitude and frequency of oscillations and a logarithmic region at high values amplitude and frequency of oscillations. The linear region corresponds to the laminar flow regime without the vortex shedding from the edge of the disk. The logarithmic region coincides with the beginning of the transition to the turbulent regime. At the edge of the disk, one can observe a loss of flow stability, which leads to the development of small vortices. These vortices, with further development, form a vortex street, which transport into a secondary vortex under the disk.

^{−0.5}m

^{1.5}s

^{−1}.

_{3}at

**f**= 25 Hz and

**A**= 500 μm when

**R**changed from 0.1 mm to 0.8 mm, it is clear that the melt flow regime begins to become turbulent. This is explained by the appearance of an uneven velocity field in the secondary vortex, as well as an increase in the dissipation energy in the area under the disk (Figure 7 and Figure 8). Separation vortices begin to form here, which are transferred from the edge of the disk to the region of the secondary vortex, leading to a change in the speed of the secondary vortex. This is most clearly visible in the velocity field of CdTe melt at

**R**= 0.1 mm, where several small vortices (blue dots) can be seen that leave the disk edge and propagate into the central flow. Also, in the central region under the disk, a reverse flow occurs more actively, which also leads to a redistribution of velocity on the axis. It can be seen that with a decrease in the radius of the disk edge, this regime stabilizes, gradually reducing to a laminar flow regime and more potential type of the flow.

#### 3.2. AVC Thermodynamic Control

**R**influence on generation and on parameters of steady vibrational flows as well as on viscous dissipation energy (

**P**

_{w}), determined by the expression:

**P**

_{w}determines structure characteristics of the melt of a complex chemical composition [17]. Numerical simulation showed that the maximum (averaged over the oscillation period) velocity of vibrational flows was inversely to

**R**and demonstrated a non-linear increase with AVC amplitude (

**A**). Integral viscous dissipation energy for the whole melt volume, which can be processed by integrating Equation (9) over the volume, demonstrated the cubic dependence vs. the oscillating velocity and inverse proportionality to edge radius:

**P**

_{w}~

**A**

^{3}

**ω**

^{3}

**R**

^{−1}. The maximum viscous dissipation (Figure 9) near the disc edge is also proportional to oscillating velocity and inverse proportionality to edge radius

**A**

^{3}

**ω**

^{3}

**R**

^{−1}, but it increases faster with 3.8 number exponents.

_{3}melt, the most stable values of viscous dissipation energy were obtained, since at

**f**= 25 Hz,

**A**= 500 μm its flow, despite the formation of separation vortices, is still stable. Under these conditions, CdTe melt is in a stronger turbulent regime, which is expressed in variable values of the maximum dissipation energy. In the case of Ge-melt at

**f**= 25 Hz,

**A**= 500 μm, an extremely unstable turbulent regime of oscillatory flow is achieved, so we calculated the dissipation energy for the parameters

**f**= 25 Hz,

**A**= 100 μm.

_{3}melt; and ~330 Joule/mole per second for CdTe melt in one cubic nanometer near the disk edge. These values match the approximation for energy of cluster formation in a liquid phase [21]. For Ge at

**R**= 0.1 mm we are able to introduce only ~2–3 Joule/mole per second in one cubic nanometer near the disc edge. This energy is enough to organize the laminar flow motion in the melt for desired heat-mass transfer, but it is insufficient to change the Ge melt thermodynamic state with respect to components’ composition.

## 4. Discussion

**I**(Equation (9)) is primarily a mathematical correlation, but it is also possible to speculate on its physical interpretation. Using the data approximation in Figure 6, we can conduct the linear regression equation in the form:

**R**, since the flow velocity near the edge of the disk is 10 times greater than the oscillatory flow velocity, as stated earlier. In the case of direct proportionality, the missing parameter is included in b.

^{−5}kg

^{0.5}m

^{−0.5}. This value, together with the oscillation intensity

**I**, can be used when estimating the oscillatory contribution of melt motion in real crystal growth processes, where the main driving force of the process is thermal convection or diffusion.

## 5. Conclusions

**I**, at which a stable mode of oscillatory flow is maintained, which ranges from 10 to 50 kg

^{−0.5}m

^{1.5}s

^{−1}. An analysis of the obtained dependence was also carried out and a possible relationship between the parameters was identified, however, to confirm the obtained result it is necessary to collect a large sample of data.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Grid of calculation area (

**left**) with grid extension near vibrating disk (

**right**): 1—stationary zones, 2—stress-strained zone, 3—rigid zones.

**Figure 2.**Normalized velocity field with streamlines (

**left**) and velocity vectors (

**right**) for the central position of a disk moving downward in a germanium melt (

**f**= 25 Hz,

**A**= 80 μm,

**R**= 0.1 mm).

**Figure 3.**Normalized instantaneous velocity with velocity vectors (arrows) at the edge of the disk (

**f**= 25 Hz,

**A**= 100 μm,

**R**= 0.1 mm) for Ge and CdTe melts with residual fluid movement when the oscillating disk stops at bottom dead center.

**Figure 4.**Velocity field averaged over the oscillation period, normalized to the oscillatory component, with streamlines (

**left**) and velocity vectors (

**right**) for the central position of a disk moving downward in a germanium melt (

**f**= 25 Hz,

**A**= 80 μm,

**R**= 0.1 mm) after 90 s of calculation.

**Figure 5.**Dependence of the modulus of the maximum speed of melts along the central axis on the of oscillation intensity (

**I**).

**Figure 6.**Dependence of the modulus of the maximum velocity of melts along the central axis on the intensity of oscillations

**I**on the scale of laminar stationary oscillatory flow of the melt.

**Figure 7.**Numerical simulation of momentum viscous dissipation rate (left half of the simulated volume, see Section 3.2) and velocity contours colored by velocity magnitude and supported by velocity vectors for NaNO

_{3}melt activated by oscillation (

**f**= 25 Hz,

**A**= 500 μm) of cylindrical disc with an edge radius 0.1 mm (

**left**) and 0.8 mm (

**right**).

**Figure 8.**Numerical simulation of momentum viscous dissipation rate (left half of the simulated volume, see Section 3.2) and velocity contours colored by velocity magnitude and supported by velocity vectors for CdTe melt activated by oscillation (

**f**= 25 Hz,

**A**= 500 μm) of cylindrical disc with an edge radius 0.1 mm (

**left**) and 0.8 mm (

**right**).

**Figure 9.**Numerical simulation of momentum viscous dissipation in NaNO

_{3}, CdTe (

**f**= 25 Hz,

**A**= 500 μm) and Ge (

**f**= 25 Hz,

**A**= 100 μm) melts activated by oscillation of cylindrical disc with different curvature of a sharp edge (

**R**).

Parameter | Value, mm |
---|---|

Simulating area length | 100 |

Simulating area height | 100 |

Disk diameter, D | 80 |

Disk thickness, d | 10 |

Edge radius, R | 0.1; 0.2; 0.4; 0.8 |

Distance from the disk to the bottom and top walls | 45 |

Disk oscillation amplitude, A | 0.080–0.500 |

Frequency oscillation, f | 10–30 Hz |

Property | Value | ||
---|---|---|---|

NaNO_{3} | Ge | CdTe | |

Melting temperature, T_{m} (K) | 579.95 | 1211.4 | 1369 |

Density, ρ (kg/m^{3}) | 1903 | 5490 | 5661.8 |

Dynamic viscosity, μ (Pa s) | 2.2 × 10^{−3} | 7.14 × 10^{−4} | 2.33 × 10^{−3} |

**Table 3.**Modules of the maximum melt velocity along the central axis at various values of amplitude, frequency, disk edge radius and intensity.

f, Hz | A, μm | R, mm | I, kg^{0.5} m^{1.5} s^{−1} | v, mm·s^{−1} |
---|---|---|---|---|

Ge | ||||

10 | 125 | 0.1 | 8.00 | 0.126 |

10 | 200 | 0.1 | 32.77 | 2.240 |

20 | 100 | 0.1 | 16.39 | 0.744 |

20 | 110 | 0.1 | 21.81 | 1.199 |

25 | 80 | 0.1 | 13.11 | 0.535 |

25 | 100 | 0.1 | 25.61 | 1.370 |

25 | 110 | 0.1 | 34.08 | 2.217 |

25 | 125 | 0.1 | 50.01 | 3.635 |

30 | 110 | 0.1 | 49.08 | 3.376 |

25 | 100 | 0.2 | 12.80 | 0.587 |

25 | 100 | 0.4 | 6.40 | 0.096 |

25 | 100 | 0.8 | 3.20 | 0.060 |

25 | 50 | 0.05 | 6.40 | 0.096 |

CdTe | ||||

10 | 125 | 0.1 | 2.49 | 0.064 |

10 | 200 | 0.1 | 10.20 | 0.657 |

10 | 300 | 0.1 | 34.42 | 2.683 |

20 | 100 | 0.1 | 5.10 | 0.217 |

25 | 80 | 0.1 | 4.08 | 0.091 |

25 | 100 | 0.1 | 7.97 | 0.474 |

25 | 125 | 0.1 | 15.56 | 1.114 |

25 | 500 | 0.1 | 996.03 | 24.86 |

25 | 500 | 0.2 | 498.01 | 20.18 |

25 | 500 | 0.4 | 249.01 | 14.55 |

25 | 500 | 0.8 | 124.50 | 8.757 |

NaNO_{3} | ||||

25 | 500 | 0.1 | 611.57 | 21.67 |

25 | 500 | 0.2 | 305.79 | 16.27 |

25 | 500 | 0.4 | 152.89 | 9.655 |

25 | 500 | 0.8 | 76.45 | 4.857 |

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**MDPI and ACS Style**

Nefedov, O.; Dovnarovich, A.; Kostikov, V.; Levonovich, B.; Avetissov, I.
Axial Vibration Control Technique for Crystal Growth from the Melt: Analysis of Vibrational Flows’ Behavior. *Crystals* **2024**, *14*, 126.
https://doi.org/10.3390/cryst14020126

**AMA Style**

Nefedov O, Dovnarovich A, Kostikov V, Levonovich B, Avetissov I.
Axial Vibration Control Technique for Crystal Growth from the Melt: Analysis of Vibrational Flows’ Behavior. *Crystals*. 2024; 14(2):126.
https://doi.org/10.3390/cryst14020126

**Chicago/Turabian Style**

Nefedov, Oleg, Alexey Dovnarovich, Vladimir Kostikov, Boris Levonovich, and Igor Avetissov.
2024. "Axial Vibration Control Technique for Crystal Growth from the Melt: Analysis of Vibrational Flows’ Behavior" *Crystals* 14, no. 2: 126.
https://doi.org/10.3390/cryst14020126