# Mixed Oscillation Flow of Binary Fluid with Minus One Capillary Ratio in the Czochralski Crystal Growth Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Formulation and Numerical Methodology

#### 2.1. Physical Model and Basic Assumption

_{c}is filled with GeSi melt, and the height of the melt is noted as h. The radius of the growth crystal is marked as r

_{s}. The radius ratio (η = r

_{s}/r

_{c}) is 0.3. The temperature and concentration at the sidewall (T

_{c}, C

_{c}) are higher than that along the crystal/fluid interface (T

_{s}, C

_{s}). The bottom is thermally adiabatic, and no-slip and impermeable boundary conditions are applied. The flat surface is not deformable. The flow is assumed to be laminar and the test fluid is incompressible. The linear variations of surface tension σ and density ρ satisfy the following conditions:

_{T}= −(∂ρ/∂T)

_{C}/ρ

_{r}, β

_{C}= −(∂ρ/∂C)

_{T}/ρ

_{r}are the thermal and solutal expansion coefficients. γ

_{T}=− (∂σ/∂T)

_{C}and γ

_{C}= −(∂σ/∂C)

_{T}are the temperature and concentration coefficients of surface tension, respectively. The subscript r represents the reference value.

#### 2.2. Mathematical Formulation

_{c}, ν/r

_{c}, r

_{c}

^{2}/ν, and ρν

^{2}/r

_{c}

^{2}, the control equations are transformed to the non-dimensional form:

**V**, the temperature is in the form of Θ = (T−T

_{s})/(T

_{c}−T

_{s}), and the concentration is Φ = (C−C

_{s})/(C

_{c}−C

_{s}). In addition, Gr

_{T}= gβ

_{T}(T

_{c}–T

_{s})r

_{c}

^{3}/ν

^{2}is the thermal Grashof number, and R

_{ρ}is the buoyancy ratio, which is defined as R

_{ρ}= β

_{C}(C

_{c}–C

_{s})/[β

_{T}(T

_{c}–T

_{s})]. Pr = ν/a denotes the Prandtl number and Le = α/D is the Lewis number, where ν is kinematic viscosity, α is thermal diffusivity, and D is mass diffusivity of the working fluid.

#### 2.3. Numerical Procedure and Method Validation

^{−6}and 2 × 10

^{−5}. The criteria for convergence is defined as follows:

^{R}× 35

^{Z}× 160

^{θ}used in this work is appropriate for accurate simulations.

## 3. Results and Discussions

#### 3.1. Basic Two-Dimensional Flow and Stability

_{T}, both Re

_{CON}and Gr

_{T}can be determined. For brevity, only the values of Re

_{T}are given in Figure 2, which shows the typical flow structures for the basic two-dimensional steady flow. The marked positive or negative sign indicates the clockwise or counter-clockwise circulation directions. For the case of a mixture with minus one capillary ratio, the soluto and thermal-capillary forces are equal, but act in contrary directions, and the overall capillary effect is zero. However, even without rotation, the existence of buoyancy breaks the equilibrium between the capillary forces. The fluid always moves from the crucible sidewall to the crystal, producing a counterclockwise vortex, as shown in Figure 2a. Because of the large Lewis number of the melt, the heat diffusion speed is faster than that of mass, and the isotherms are almost kept as conductive state. For the small thermocapillary Reynolds number of Re

_{T}= 394, the iso-concentration lines bent slightly, and the flow rate is small on the free surface, as illustrated in Figure 3.

_{s}= −561, as shown in Figure 2b, a clockwise cell forms under the crystal and dominates the flow field. Compared with the natural convection displayed in Figure 2a, the mixed natural and forced flow is stronger. When rotation rate is increased to −1400, as shown in Figure 2c, the clockwise cell becomes stronger and the maximum stream function increases from 0.12 to 0.80. The enhanced crystal rotation induces a uniform concentration distribution underneath the crystal–fluid interface. When the crucible rotation is introduced, as illustrated Figure 2d, a primary counterclockwise rolling cell appears. The surface flow rate near the crystal is greatly increased. For the case of counter rotation of crucible and crystal, two counter-circulations are formed and rival each other. The competition of the vortex creates the zigzag profile of radial velocity. Away from the crystal/fluid interface, thermal and soluto-capillary forces are imposed oppositely and counterbalance each other, causing the surface fluid flows to be rather sluggish.

_{T}or rotation rate exceed threshold values, the basic flow rapidly loses the stability and transforms into 3D oscillatory state. For the crystal growth, the oscillations should be avoided as much as possible. The amplitudes of oscillations are calculated and the dichotomy method is utilized to obtain the critical values of thermocapillary Reynolds numbers Re

_{T,cri}. Figure 4 gives the variations of critical thermocapillary Reynolds numbers at different rotation rates. Without rotation, Re

_{T,cri}approximately equals 1010; this value is larger than that of fluid mixture with a capillary ratio of −0.2, but still smaller than that of pure silicon melt [35]. This indicates that the introduction of the solutocapillary effect makes it easier for the flow to lose stability.

_{s}, the value of Re

_{T,cri}increases. However, when the value of Re

_{s}reaches 1123, the forced convection driven by crystal rotation is predominant and the enhanced disturbance stimulates flow instability. As a result, Re

_{T,cri}undergoes a sharp drop. Particularly, when Re

_{s}is increased to 1680, no stable state is observed. This indicates the pure rotating flow is unstable. On the other hand, when the crucible starts to rotate, the large contact area allows the melt to rotate with the crucible synchronically, inducing a stabilization effect on the mixed convection. Therefore, Re

_{T,cri}experiences a monotonic increase with the increase of Re

_{c}.

#### 3.2. Three-Dimensional Steady Flow

_{T}= 3943 without rotation. It is observed that the surface fluctuations are presented as straight spokes, and the STD is composed of four vertical strips. These surface patterns indicate the fluctuations are stationary in time, but oscillate in space. For the further understanding of the fluctuations, the streamlines are plotted in cross sections of θ = 0 and θ = π/4 (which correspond to the dark and bright strips, respectively, shown in Figure 5). As illustrated in Figure 6a, influenced by the thermocapillary effect, the fluid near the sidewall is pushed inward, forming a counterclockwise rolling cell. This convective vortex stirs the melt, inducing a local uniform concentration distribution near the crystal. Meanwhile, it can be seen that, away from the crystal, the deformation of iso-concentration lines reflects the existence of a reversed vertical concentration gradient. Therefore, the solutal buoyancy force is generated and the lighter fluid in the lower part is brought to the surface, forming a clockwise cell. At the cross section of θ = π/4, the rolling cell generated by the thermocapillary effect is squeezed by the small vortex near the crystal.

_{T}= 3943 and Re

_{s}= −561, the surface pattern is still shown as a spoke pattern, as seen in Figure 7a. However, an annular band is observed around the crystal. The concentration fluctuations in the annular band are almost zero. Compared with Figure 5, the amplitude of the surface fluctuations is decreased from 0.24 to 0.038, but the wave number increases greatly. The natural convection still dominates the flow field. The corresponding local capillary ratio ${R}_{\sigma}^{\mathrm{i}}$ on the free surface and the flow structures in the cross section are plotted in Figure 8 and Figure 9a, respectively. Near the crystal, the combination of rotation and solutocapillary effects drives the mixture with a low concentration to move outside, leading to a uniform concentration gradient in the range of 0.3 < R < 0.6. When 0.6 < R < 0.82, the sparse iso-concentration lines are observed, indicating the small concentration gradient. Meanwhile, the absolute value of ${R}_{\sigma}^{\mathrm{i}}$ is smaller than unity, which means the thermocapillary effect is stronger than the solutocapillary influence, thus the counterclockwise cell is dominant in this area. Near the sidewall, the combination of solutocapillary and buoyancy forces generates a weak clockwise circulation in the top right corner of the crucible. With the increase of Res, as shown in Figure 7b, the occupied area of the annular band is expanded and the surface fluctuations near the sidewall are presented as a rosebud pattern. The centrifugal force further pushes the fluid forward and squeezes the rolling cell driven by the thermocapillary effect, as described in Figure 9b. For the case of Re

_{T}= 3943, Re

_{s}= −1123, the surface fluctuations are shown as a pulsating ring, the oscillation amplitude is greatly deduced, and the wave number is lowered to 4, as illustrated in Figure 7c. From the variation of the local capillary ratio (Figure 8), it is shown that the region where $\left|{R}_{\sigma}^{\mathrm{i}}\right|$ is smaller than unity coincides with the position of the surface circular ring. This confirms that the thermocapillary force is dominant in the pulsating ring area.

#### 3.3. Three-Dimensional Oscillatory Flow

_{s}= −1123, once Re

_{T}is greater than 7098, the rotating wave appears on the free surface. For a typical case of Re

_{s}= −1123 and Re

_{T}= 7887, the surface fluctuations and the corresponding STD at different radial positions are displayed in Figure 10. It is noted that the surface pattern can be regarded as a combination of spokes and rotating curved arms. Near the crystal (R = 0.32), the STD is shown as oblique lines leaning to the left. This means that the curved arms move in a clockwise direction and the rotation rate is less than that of the crystal. At R = 0.75, the corresponding the STD is displayed as vertical lines. This verifies that the spokes can be considered as standing waves. Between the rotating and standing waves, the annular band is still observed; the concentration distribution at a monitoring point (R = 0.52) inside the annular space is exhibited in Figure 11a. Along the circumference, the concentration is almost uniform and no fluctuation is detected in the annular band. The characteristics of the time-dependent of concentration and azimuthal velocity are also analyzed in Figure 11b. It is noted that the concentration oscillation lags behind velocity. The hysteresis between the oscillations was also observed in the report by Smith and Davis [46], which is one of the typical characteristics of hydrothermal waves generated by temperature difference. Moreover, the propagating angle of 17° is another feature of the hydrothermal waves. As shown in Figure 10, the propagating angle of the concentration oscillations is about 16°. These two values of wave angles are very close. Therefore, the spiral concentration oscillation wave is considered as hydrosoultal waves. Meanwhile, the oscillation frequencies are obtained by fast Fourier transform (FFT), as shown in Figure 12, and the main dimensionless frequency

**F**

_{0}is 1427 with a secondary frequency F

_{1}of 2853, which satisfies F

_{0}= 1/2F

_{1}. This frequency relationship was also experimentally reported by Shen et al. [34].

_{T}, the natural convection is enhanced, and the oscillation amplitude is enlarged. The wave number also has a tendency to increase, as presented in Figure 13b, and the waves with weaker oscillations near the annular band appear. When Re

_{T}is increased to 15,773, as shown in Figure 13a, the grown flow instabilities lead to the increase of the wave number in the azimuthal direction to dissipate energy. The FFT analysis corresponding to the three typical cases is displayed. It is noted that a multiple relationship is observed among the frequencies, which is f

_{1}= 1/2f

_{2}= 1/3f

_{3}.

_{T}= 5442, Re

_{c}= 1871, the forced flow driven by crucible rotation plays a decisive role in the occurrence of instability. From the STD shown in Figure 14a, it can be observed that synchronous rotation is achieved between the crucible and oscillation waves. The wave number is 4. Near the crystal, the absolute value of $\left|{R}_{\sigma}^{\mathrm{i}}\right|$ is less then unity, as displayed in Figure 15, the thermocapillary and centrifugal forces are superimposed to enhance the rotating forced flow. Thus, a strong counterclockwise circulation is formed and occupies the left-half part of the crucible, as illustrated in Figure 16a. When R is greater than 0.6, the local capillary ratio oscillates around its equilibrium value of −1. Near the crucible sidewall, the solutal-buoyancy effect drives a clockwise circulation at the corner. When the Re

_{T}is increased to 19,717, the solutocapillary effect is enhanced, and $\left|{R}_{\sigma}^{\mathrm{i}}\right|$ is greater than one in the region of R < 0.5. As the solutocapillary force counteracts the centrifugal force, the strength of the rolling cell is weakened near the crystal, and a series of surface spiral waves appear and extrude the rotating wave outward. Through the comparison between Figure 14a,b, it is found that, with the increase of Re

_{T}, the rotating wave number increases twice, and the oscillation amplitude is reduced by half. This reveals that the enhanced natural convection has a depression effect on the mixed 3D oscillation flow.

## 4. Conclusions

- (a)
- For the mixture with capillary ratio of minus one, the total thermal and solutocapillary forces are counterbalanced. The introduction of buoyancy force leads to the disturbance of the balance, generating the mixed capillary-buoyancy convection. For a small concentration gradient, the flow is in two-dimensional steady state. Owing to the good fluidity and thermal uniformity, this state is important for the growth of high quality of crystals. The coupled capillary and buoyancy flow in the crucible is presented as a large counterclockwise circulation. When the rotations of the crystal and crucible are considered, the mixed natural and forced flow structures are more complex, and the directions of the rolling cells are associated with the competitions among the driving forces.
- (b)
- When the capillary force is greater than a certain value, the basic flow transits to three-dimensional state. The critical conditions for the mixed flow transitions at different rotation rates are obtained. Crucible rotation can obviously strengthen the flow stability. Influenced by the crystal rotation, the critical thermocapillary Reynolds number increases until it reaches a turning point. With the enhancement of crystal rotation driven flow, a dramatic decrease of the critical value is observed.
- (c)
- Once the instability is incubated, the basic mixed flow firstly bifurcates to the three-dimensional steady state, which oscillates spatially along the circumferential direction. Driven by the competition among the capillary-buoyancy forces, centrifugal, and Coriolis forces, the surface fluctuations are presented as spokes, rosebud, and pulsating ring. With the enhanced flow instabilities, three-dimensional unsteady oscillation occurs. Prosperous oscillation patterns are discussed, including the spiral hydrosoultal waves, superimposition of spirals and spokes, as well as rotating waves.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Flow structures (upper) and iso-concentration lines (down) when Re

_{T}= 394. (

**a**) Re

_{s}= 0, Re

_{c}= 0, Ψ

_{max(−)}= 0.012; (

**b**) Re

_{s}= −561, Re

_{c}= 0, Ψ

_{max(+)}= 0.120; (

**c**) Re

_{s}= −1400, Re

_{c}= 0, Ψ

_{max(+)}= 0.80; (

**d**) Re

_{s}= 0, Re

_{c}= 1871, Ψ

_{max(−)}= 0.015; (

**e**) Re

_{s}= −561, Re

_{c}= 936, Ψ

_{max(+)}= 0.083.

**Figure 5.**Concentration oscillations (

**left**) and spatial-temporal diagram (STD) R = 0.65 (

**right**) when Re

_{T}= 3943 without rotation.

**Figure 6.**Streamlines and concentration distribution at θ = 0 (

**a**) and θ = π/4 (

**b**) when Re

_{T}= 3943.

**Figure 7.**Evolution of surface oscillation patterns (

**top**) and the corresponding STD (

**bottom**). (

**a**) Re

_{T}= 3943, Re

_{s}= −561; (

**b**) Re

_{T}= 3154, Re

_{s}= −1123; (

**c**) Re

_{T}= 3943, Re

_{s}= −1123.

**Figure 8.**Local capillary ratio distribution along the free surface for the cases shown in Figure 7.

**Figure 9.**Flow structures and concentration profiles in the meridian plane. (

**a**) Re

_{T}=3943, Re

_{s}= –561; (

**b**) Re

_{T}= 3154, Re

_{s}= –1123; (

**c**) Re

_{T}= 3943, Re

_{s}= –1123.

**Figure 10.**Snapshot of concentration fluctuations (

**left**) and STD (

**right**) at Re

_{s}= –1123, Re

_{T}= 7887.

**Figure 11.**Distributions of surface oscillations when at Re

_{s}= −1123, Re

_{T}= 7887. (

**a**) Azimuthal distribution of surface concentration fluctuations. (

**b**) Phase lag between the oscillations of concentration and velocity.

**Figure 12.**Fourier spectra analysis of the concentration oscillation when Re

_{s}= –1123, Re

_{T}= 7887.

**Figure 13.**Transitions of surface concentration patterns (

**top**) and the corresponding Fourier spectra analysis (

**bottom**) at Re

_{c}= 936, (

**a**) Re

_{T}= 9938, (

**b**) Re

_{T}= 12,726, and (

**c**) Re

_{T}= 15,773.

**Figure 14.**Snapshots of rotating waves (

**top**) and STD at R = 0.65 (

**bottom**) when Re

_{c}= 1871. (

**a**) Re

_{T}= 5442; (

**b**) Re

_{T}= 19,717.

**Figure 15.**Distributions of local capillary ratios along the free surface for the cases shown in Figure 14.

**Figure 16.**Streamlines on the R–Z plane when Re

_{c}is kept at 1871. (

**a**) Re

_{T}= 5442; (

**b**) Re

_{T}= 19,717.

Property | Symbol | Unit | Value |
---|---|---|---|

Kinematic viscosity | ν | m^{2}/s | 1.4 × 10^{−7} |

Thermal diffusivity | α | m^{2}/s | 2.2 × 10^{−5} |

Mass diffusivity | D | m^{2}/s | 1.0 × 10^{−8} |

Temperature coefficient of surface tension | γ_{T} | N/(m·k) | 8.1 × 10^{−5} |

Concentration coefficient of surface tension | γ_{C} | N/m | −0.54 |

Prandtl number | Pr | - | 6.4 × 10^{−3} |

Lewis number | Le | - | 2197.8 |

Mesh | m | F_{1} | F_{2} |
---|---|---|---|

60^{R} × 20^{Z} × 80^{θ} | 8 | 650 | 1165 |

80^{R} × 35^{Z} × 120^{θ} | 8 | 665 | 1177 |

100^{R} × 35^{Z} × 160^{θ} | 8 | 657 | 1171 |

120^{R} × 55^{Z} × 200^{θ} | 8 | 659 | 1167 |

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**MDPI and ACS Style**

Wu, C.; Chen, J.; Li, Y.
Mixed Oscillation Flow of Binary Fluid with Minus One Capillary Ratio in the Czochralski Crystal Growth Model. *Crystals* **2020**, *10*, 213.
https://doi.org/10.3390/cryst10030213

**AMA Style**

Wu C, Chen J, Li Y.
Mixed Oscillation Flow of Binary Fluid with Minus One Capillary Ratio in the Czochralski Crystal Growth Model. *Crystals*. 2020; 10(3):213.
https://doi.org/10.3390/cryst10030213

**Chicago/Turabian Style**

Wu, Chunmei, Jinhui Chen, and Yourong Li.
2020. "Mixed Oscillation Flow of Binary Fluid with Minus One Capillary Ratio in the Czochralski Crystal Growth Model" *Crystals* 10, no. 3: 213.
https://doi.org/10.3390/cryst10030213