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A numerical study was carried out to describe the flow field structure of an oxide melt under 1) the effect of internal radiation through the melt (and the crystal), and 2) the impact of surface tension-driven forces during Czochralski growth process. Throughout the present Finite Volume Method calculations, the melt is a Boussinnesq fluid of Prandtl number 4.69 and the flow is assumed to be in a steady, axisymmetric state. Particular attention is paid to an undulating structure of buoyancy-driven flow that appears in optically thick oxide melts and persists over against forced convection flow caused by the externally imposed rotation of the crystal. In a such wavy pattern of the flow, particularly for a relatively higher Rayleigh number , a small secondary vortex appears nearby the crucible bottom. The structure of the vortex which has been observed experimentally is studied in some details. The present model analysis discloses that, though both of the mechanisms 1) and 2) end up in smearing out the undulating structure of the flow, the effect of thermocapillary forces on the flow pattern is distinguishably different. It is shown that for a given dynamic Bond number, the behavior of the melt is largely modified. The transition corresponds to a jump discontinuity in the magnitude of the flow stream function.

Refractory oxide crystals such as gadolinium gallium garnet (GGG) and yttrium aluminum garnet (YAG) are widely used as solid-state laser hosts and materials for epitaxial films in magneto-optical devices [

Accounting for the internal radiation within the melt and crystal is of crucial importance in numerical modeling of Cz growth of oxides because they are often semitransparent to infrared radiation [

Tsukada et al. [_{3}Ga_{5}O_{12}(GGG) and Tb_{3}Ga_{5}O_{12}(TGG) crystals. In contrast to GGG, the authors in [^{−}^{1}∙K^{−}^{1}. For both of the cases, the melt was assumed to be opaque, and the thermocapillary effect as well as the meniscus influence on the crystal/melt interface was ignored.

In this paper, we report the results of a numerical simulation of the Czochralski growth of a large-diameter GGG crystal. The main subject of the present model calculations is, in a first step, to explain the effect of internal radiation on the flow and thermal fields. The results obtained for semitransparent material are compared with the case in which both the melt and the crystal are assumed to be opaque to the thermal radiation. The convective behavior of the melt is discussed, and it is shown that the thermal stratification of the fluid depends on the intensity of buoyant forces in optically thick melt. The undulating pattern of the thermal field disappears with contribution of the internal radiation in heat transfer in the melt. Throughout the first step, thermocapillary flow is neglected.

In the second step, the material is assumed to be opaque, and the radiative heat exchange occurs between exposed surfaces in the Cz enclosure. The structure and properties of a small vortex which appears in the melt of high Rayleigh number is studied. It is shown that there is a critical thermocapillary coefficient at which the undulating structure [

The Cz growth system can be characterized by coexisting vertical and horizontal temperature gradients and the differential rotation rates of the crystal and crucible. The general feature of the fluids motion in the Cz crucible is described as follows. The buoyancy-driven hot flow ascends along the crucible wall and then accompanied by the surface/tension-driven flow, travels along the melt free surface towards the crystal rim. The fluid is being cooled down along the path and more intensively adjacent to the crystal/melt interface (CMI) due to the larger heat conductivity of the oxide crystal. This creates a stream of cold fluid which descends along the centerline towards the crucible bottom. If the crucible is at rest and the crystal rotation rate is sufficiently low, the flow is essentially buoyancy driven with a unicellular meridional circulation, namely the Hadley cell circulation [

In the model, the crystal pulling rate is, as usual, much smaller than the characteristic velocity of the buoyancy-driven flow. Therefore, the system is assumed to be in a quasi-steady state. In the melt model, the fluid motion is substantially determined by the natural convection for which the Rayleigh number does not exceed the relevant critical value estimated for relatively high Prandtl number fluids [

In the present modeling of Cz-oxide growth process with bulk radiation incorporated in the governing equations, the influence of thermal convections on the flow pattern is studied. The externally imposed rotational effect is, however, assumed to be secondary in the present analysis. The crystal rotation, when accounted for in the melt, breeds a small zonal flow beneath the CMI. For experimentally reasonable rates of rotation, the flow pattern remains almost unchanged.

The schematic in

melt aspect ratio

The mathematical model developed here, incorporates transport process in the melt and the crystal. In the ambient gas phase, assumed to be totally transparent to the thermal radiation, only the energy equation is solved. The semitransparent phases are bounded to diffuse-gray surfaces and, according to the spectroscopic measurements of the refractory oxides optical properties [

Description (units) | Symbol | Value |
---|---|---|

Melting point (K) | 2023 | |

Heat of fusion (J/g) | 455.1 | |

Dynamic viscosity | 0.40 | |

Kinematic viscosity | ||

Thermocapillary coefficient | ||

Growth angle (deg) | 17.0 | |

Thermal expansion coefficient | ||

Density | 5.65 | |

7.09 | ||

Thermal conductivity | 5.0 | |

20.0 | ||

0.0242 | ||

Heat capacity | 0.586 | |

Thermal diffusivity | ||

Emissivity | 0.5 | |

0.8 | ||

Refractive index | 1.80 | |

Absorption coefficient ^{e} | 4.0 | |

Absorption coefficient ^{e} | 0.4 | |

Gravitational acceleration | 9.81 |

e: Estimated values.

Description, symbol | Value (units) |
---|---|

Crucible radius, | 100 (mm) |

Crucible height, | 150 (mm) |

Melt height, | 100 (mm) |

Crystal radius, | 50 (mm) |

Crystal length, | 150 (mm) |

Crystal rotation rate, | 0.5 (rad/s) |

Pulling rate, | 10 (mm/h) |

Enclosure wall height, | 160 (mm) |

Crucible exposed wall, | 50 (mm) |

Crucible wall temperature, | 2143 (K) |

Ambient temperature, | 1350.0 (K) |

Driving temperature difference, | 120 (K) |

Name | Expression | YAG | GGG |
---|---|---|---|

Grashof number | |||

Rayleigh number | |||

Marangoni number | |||

Thermocapillary Reynolds number | |||

Dynamic Bond number | 181.60 | 54.20 | |

Rotational Reynolds number | |||

Planck number |

The fluid flow for the melt region is described by coupled Navier-stokes and heat equations. The present model involves the following assumption: 1) the melt is an incompressible Newtonian fluid which satisfies the Boussinesq approximation; 2) the fluid flow is laminar; 3) viscous dissipation is negligible; 4) the melt/gas interface is not calculated from the Young-Laplace equation, but instead, following Galazka et al. [

To estimate the contribution of the internal radiation to heat transfer in the melt, the crystal and melt are assumed to be absorbing/emitting mediums bounded by vanishingly thin semitransparent diffuse gray surfaces. As a disposal parameter [

A finite volume method (FVM) is applied to compute quasi-steady and axisymmetric solutions to the fully coupled equations governing heat transfer and melt hydrodynamics for Czochralski growth of garnet oxide GGG crystal. The radiative heat transfer strongly couples with fluid dynamics [_{N}-approximation which expands the radiation intensity by an orthogonal series of spherical harmonics [_{1}-approximation. Numerically, it has been shown, however, that the computational methods based on the P_{1}-approximation is valid only for optically thick materials [_{1}-approximation is not of course a suitable approach. Furthermore, the approximation may be substantially in error for multi-dimensional systems with large aspect ratios and/or when surface emission dominates over medium emission [

As given in _{1}-approximation are removed by the use of discrete ordinates (DO) method based on a discrete representation of the angular dependence of the radiation intensity. The DO method has been widely recognized to be one of the most appropriate methods in high-temperature applications such as Cz/oxide growth system. This is particularly because the DO method shares the same philosophy and computational grid as the fluid dynamics approach [

The equations describing the conservation of mass, momentum and energy for the two-dimensional (2D) model represented and restricted in the preceding sections, are expressed as follows.

where

To estimate the radiative heat flux

We consider the radiative heat transfer for the axisymmetric system (the melt and crystal) depicted in

where

where

In the present work, the optical properties on both sides of the semitransparent mediums (the crystal and the melt) surfaces are estimated with their refractive indices [

where

In the Cz configuration

The velocity boundary conditions are

o the melt free surface where boundary conditions can be expressed by components as

if thermocapillary convection is taken into account

o at the crystal walls and the crystal/melt interface

o at the melt centerline

For any variable

o At the melt free surface:

o at the sidewall of the crystal:

o at the crystal/melt interface:

o at the insulating enclosure wall:

o at the top enclosing surface:

o at the crucible submerged wall:

o at the crucible bottom:

o at the exposed portion of crucible wall:

o at the centerline:

where subscripts

The governing equations with boundary conditions for the fluid flow and heat transport in the system were numerically solved by employing control volume (CV) based finite differential technique. The SIMPLEC algorithm [

crystal sums up 13,806 (each with area

The present section is divided into two parts: 1) the results which explain the effect of internal radiation transfer on the convective flow and thermal fields in the melt dominated by the buoyancy forces the intensity of which is respected by the Rayleigh number Ra, and 2) the results which reveal the influence of thermocapillary forces on the convective pattern in the melt when the material does not participate in the radiative heat transport. For the cases in which the internal radiation is ignored the boundary conditions for temperature and radiative transfer between the exposed opaque surfaces, are expressed as given in reference [

The convective behavior of GGG melt in the model can be characterized by the dimensionless similarity parameters given and compared to YAG melt in

tions were carried out for opaque system with

The numerical simulations of the opaque _{1}) and without (case C_{2}) the internal radiation, respectively. This comparative representation of the variables for optically different mediums (C_{1} and C_{2}) in each figure, disclose rapidly the effect of internal radiation on the flow and thermal fields. Quantitatively, the simulation results are elucidated in

It is well known [

form of the non dimensional parameter is given by

C_{2} | C_{1} | Rayleigh number |
---|---|---|

As shown in

_{1}) melt. In the other words, compared to the opaque system (case C_{2}), the velocity field is more largely distributed in the semitransparent

melt (C_{1}). For each value of the Rayleigh number

significantly different for two optically distinguished cases C_{1} and C_{2}. More precisely, it is anticipated that, for opaque melt

To describe the details relevant to the discussion,

The vortex (named RFV hereafter) appears only in the opaque melt and its center, located at the point

Radial velocity profile along the vertical

This section is assigned to description of 1) the properties of the small secondary vortex (RFV) which appears in the optically thick melt model and 2) the role of thermocapillary forces on the flow field structure which lead to similar results on the fluid motion as the radiative transfer.

In this section, the melt is assumed to be opaque. For the present steady and axisymmetric model, the streamlines are defined in terms of the velocity components as

It has been shown [

The area of the normal cross-section of the vortex tube is approximated by

shown [

For the case

the boundary,

Contrarily, at the lower edge, around the point A, viscous dissipation has the lowest effect. Note that, vorticity intensification due to stretching of vortex lines, does not occur in the present 2D model.

Within the surveyed range of

Increasing the Rayleigh number, the non-uniformity of the vorticity distribution on the closed

flow (not shown here) and, as shown in

Surface tension gradient,

stratification in the vicinity the melt free, is removed in the presence of thermocapillary flow. The effect, as expected, depends on the intensity of Ma-flow or more precisely, on the ratio of the boundary to surface tension forces represented by

where

To investigate the influence of thermocapillary forces on the flow pattern in the present optically thick melt of

ven forces on the melt behavior is significantly different for two cases of

In the case with

For a negligibly small increment of the thermocapillary forces, that is, for

For two cases of

value

This can be inferred that more energy is stored in the wavy structure of the flow by increasing the Rayleigh number of the melt. The small secondary vortex (RFV) volume

Two-dimensional axisymmetric simulations of the Navier-stokes equations were used to investigate the behavior of the melt under a) the effect of internal radiative transfer and b) the influence of thermocapillary forces, during Cz growth of GGG crystals.

The results indicated that the two different mechanisms end up, however, in a similar pattern of the flow in the interior of the melt: the undulating structure of the flow, caused by vertical stratification of the melt, was smeared out when the melt assumed to be semitransparent (case C_{1}) and/or when the surface tension coefficient

_{2};

The wavy pattern of the flow found to be enhanced with increasing in the intensity of convective flow _{2}) in which more thermal energy is stored in the internal waves. The condition was deemed to be removed in the case C_{1}.

The properties of an elliptical-shape secondary vortex (RFV) which appears in the interior of the opaque melt of

center

In the presence of thermocapillary forces

RezaFaiez,YazdanRezaei, (2015) Radiative Heat Transfer and Thermocapillary Effects on the Structure of the Flow during Czochralski Growth of Oxide Crystals. Advances in Chemical Engineering and Science,05,389-407. doi: 10.4236/aces.2015.53040