Boundary conditions for velocity components

The computation of the melt flow in the Cz configuration requires a set of boundary conditions for different velocity components at the interface of the melt volume. Some boundary conditions can be set automatically before the CFD computation starts, sometimes boundary conditions should be supplied by user. In any case one should be familiar with the required input data and control the settings responsible for the velocity boundary values.

Marangoni effect

The shear force parallel to the melt interface is acting provided a temperature gradient at the melt interface is imposed. The force magnitude is directly proportional to the temperature gradient value. In cylindric symmetric geometries this force has generally two independent components: radially and azimuthally directed. For the 2D axisymmetric model no azimuthal temperature variation (gradient) is assumed. Therefore no Marangoni force in azimuthal direction is considered. The direction of the considered Marangoni force coincides with the cross-section line of the free melt interface with the azimuthal plane.

The Marangoni coefficient is a thermocapillary coefficient or a temperature derivative of the specific surface tension. The Marangoni coefficient is the material parameter. For the contact interface melt - gas or melt - liquid encapsulant the Marangoni coefficients of both media in contact are considered and the maximum value of the materials pair will be applied as an effective Marangoni coefficient in the Marangoni boundary condition. Therefore it is advised to set the Marangoni coefficients of all materials other than melt equal to zero and define the melt Marangoni coefficient.

The Marangoni coefficient may be either positive or negative by different liquid-liquid surfaces (or more precisely by the interface between two inmiscible liquids). The positive sign of the Marangoni coefficient causes the melt drive in direction opposite to the projection of the local temperature gradient on the free melt interface. For no acting forces other than Marangoni the positive Marangoni coefficient causes the melt movement in direction from hotter to colder melt. A classical example of the melt interface with positive Marangoni coefficient is the silicon melt. In the Si Cz configuration the Marangoni effect is typically responsible for the surface melt acceleration radially inwards towards the triple point crystal - melt - gas.

Axial and radial velocities

The axial and radial velocity components at the interface between the melt and solid bodies like crystal and crucible are set automatically to zero value. In opposite to the fluid flow computation on the unstructured mesh, no care should be taken about the no slip boundary conditions everywhere. The manual setting of the zero velocities at any boundary is allowed but is not required.

The cylindrical symmetry axis doesn't require any manual settings for any variable too. The symmetry boundary condition is assumed for the axial velocity component there, the azimuthal and radial velocity components will be automatically set to zero.

The only type of the melt boundary where manual settings may be neccessary is the free melt interface in the Cz configuration or the liquid encapsulant - gas interface in the Liquid Encapsulated or Vapor Controlled Czochralski configuration (LEC or VCz).

Cz crystal growth configuration. Melt and gas regions are meshed structurally. The Marangoni and shear interaction boundary conditions will be accounted automatically for velocity components at the free melt interface without any manual settings. The fluid flow should be computed in both melt and gas. 

Figure 109.  Cz crystal growth configuration. Melt and gas regions are meshed structurally. The Marangoni and shear interaction boundary conditions will be accounted automatically for velocity components at the free melt interface without any manual settings. The fluid flow should be computed in both melt and gas.

If the area over the free melt interface is meshed structurally and the fluid flow is computed in this area (it means in gas or in the liquid encapsulant), then again no manual settings are required for any velocity component at the free melt interface. In this case the shear interaction of both moving fluids at the shared interface and the Marangoni effect are considered as default boundary condition for the velocity component parallel to the shared liquid interface. In the figure above the scheme of the Cz configuration without manual boundary conditions at the boundary between the melt and the gas regions is demonstrated.

Another case of the automatic setting of the boundary conditions is the LEC or VCz configuration with liquid encapsulant floating over the melt. Here the automatic setting of boundary conditions is supported at the melt - liquid encapsulant and at the liquid encapsulant - gas interfaces as demonstrated in the following figure.

LEC crystal growth configuration. Melt, liquid encapsulant and gas regions are meshed structurally. The Marangoni and shear interaction boundary conditions will be accounted automatically for velocity components at the free melt interface and at the upper liquid encapsulant interface without any manual settings. The fluid flow should be computed in each of melt, liquid encapsulant and gas medium. 

Figure 110.  LEC crystal growth configuration. Melt, liquid encapsulant and gas regions are meshed structurally. The Marangoni and shear interaction boundary conditions will be accounted automatically for velocity components at the free melt interface and at the upper liquid encapsulant interface without any manual settings. The fluid flow should be computed in each of melt, liquid encapsulant and gas medium.

The velocity component perpendicular to the shared interface is set equal to zero. The velocity component parallel to the shared interface is computed. Its value is the same for the melt and the adjacent medium interface. Both media as mentioned above should be meshed structurally. By such meshing the mesh lines are terminated at the melt - adjacent medium interface and match there precisely. Ventral nodes of the structured mesh of two adjacent regions coincide at the shared boundary. The nodal boundary velocities are computed therefore in the same positions for both media.

The boundary condition is formulated and solved for the parallel velocity component. It includes a balance of the viscose shear forces from both sides from the interface plus the Marangoni effect. The driving effect of the gas stream over the free melt interface analogiously to the slide drived cavity effect and action of the Marangoni effect are accounted for automatically.

If the modeling of the Marangoni effect is not desired, then the Marangoni coefficients of the melt material and of the material in the adjacent medium above the melt should be set equal to zero.

The liquid-liquid shear interaction can be accounted for in the model only if conditions described above are satisfied. In the other cases, if no structured mesh is generated in the area over the free melt interface or if it is generated but no fluid flow is considered there, the shear interaction of coarse cannot be accounted for. Then a zero shear boundary condition or at most the Marangoni effect may be imposed manually at the free melt interface. It is then advised to set the Marangoni boundary condition at such boundary.

Cz crystal growth configuration. Only melt is meshed structurally. The Marangoni boundary condition should be applied for velocity components at the free melt interface. 

Figure 111.  Cz crystal growth configuration. Only melt is meshed structurally. The Marangoni boundary condition should be applied for velocity components at the free melt interface.

The Marangoni boundary condition is set for all 3 velocity components or the variables group UVW in the dialog window Settings-> Boundaries in the drop-down variables list Variables. If no Marangoni interaction should be considered then the Marangoni coeffficient of the melt material should be set equal to zero. Then a pure zero shear stress boundary condition will be prescribed and executed. Otherwise the Marangoni effect will be computed. The total shear stress at the free melt interface will be set proportionally to the temperature gradient at the melt interface in the pure Marangoni boundary condition.

LEC crystal growth configuration. Melt and liquid encapsulant regions are meshed structurally. The Marangoni boundary condition should be applied for velocity components at the free liquid encapsulant interface. Correct boundary conditions will be applied automatically at the interface between the melt and the liquid encapsulant. 

Figure 112.  LEC crystal growth configuration. Melt and liquid encapsulant regions are meshed structurally. The Marangoni boundary condition should be applied for velocity components at the free liquid encapsulant interface. Correct boundary conditions will be applied automatically at the interface between the melt and the liquid encapsulant.

Another case where the explicit setting of the boundary conditions is required is the situation in the scheme above for the LEC configuration. The fluid flow is computed in the melt and liquid encapsulant media but not in the gas region over the liquid encapsulant. Then the automatic boundary condition will be applied for the interface between the melt and the liquid encapsulant. The Marangoni boundary condition at the upper boundary of the liquid encapsulant should be set explicitely.

Azimuthal velocity

The azimuthal velocity boundary conditions are of Dirichlet type everywhere at boundaries liquid-solid. They are required at the crystal - melt and melt - crucible interfaces in the Cz and LEC crystal growth configurations.

The boundary conditions for the azimuthal velocity at the interfaces between two inmiscible fluids like between melt and gas or between melt and liquid encapsulant of liquid encapsulant and and gas satisfy to the same rules as radial and axial velocity components. Such interfaces between two structurally meshed regions are processed automatically also for azimuthal velocity component. The diffusive azimuthal momentum transport through the interface separating inmiscible fluids is considered in the boundary condition. The additional necessary requirement is, the azimuthal momentum transport equation should be solved in each of adjacent regions. Otherwise the shear free boundary condition will be automatically applied in one of the regions where the azimuthal velocity is considered.

If the Marangoni boundary condition is prescribed explicitely at the external boundary of the computational domain of azimuthal velocity, then no shear stress boundary condition will be applied for azimuthal velocity.

A tool for the automatic setup of the boundary conditionss for the azimuthal velocity at the interfaces melt-crystal and melt-crucible is available in CrysMAS. The work of manual setting of Dirichlet boundary conditions for W can be saved by means of this tool. The automatic procedure is released by checking the checkbox "automatic boundary setup for structured mesh" in the Czochralski tab of the Process Parameters dialog window.

Initialization of the automatic boundary conditions setup for azimuthal velocity component. The usage of the hybrid mesh in the Czochralski crystal growth configuration is assumed for application of the automation tool. 

Figure 113.  Initialization of the automatic boundary conditions setup for azimuthal velocity component. The usage of the hybrid mesh in the Czochralski crystal growth configuration is assumed for application of the automation tool.

If the structured mesh with phase transition is available and the checkbox automatic boundary setup for structured mesh is checked, then the values of the rotation rate in Hz from the same dialog will be used for setting of the Dirichlet values of the linear azimuthal velocity in m/s. The geometry will be analysed and all boundaries crystal-melt and crucible - melt will be identified. According the nodal velocity values will be supplied after the button Apply or OK of the dialog window will be pressed.

Note

The melt flow is computed in 2D always in the unmovable coordinate system. The linear azimuthal velocity at the interface crystal - melt will be defined from difference of the crucible and crystal rotational rates in Hz.