Phase boundary crystal-melt

The hybrid mesh is applied for computation of the Cz configuration. The unstructured mesh is used for thermal computation in the crystal while the structured one is applied in the melt. Both domains, crystal and melt, are given by different regions or group of regions. The melt and crystal domains consist of regions filled with different materials. Only single phase materials are allowed for them. The line separating the crystal and melt domains is the region boundary between the single region "melt" downstairs and the single region "crystal" upstairs. Both crystal and melt domains can consist of several regions but only one region from each domain may stay in the direct contact with another region from the opposite domain.

The same principles are valid for modeling of the configuration of the Vertical Gradient Freeze configuration (VGF) of crystal growth, where the crystal and melt domains are interchanged vertically. The instructions for computation of the phase transition in the VGF configuration coincide mainly with these of the Cz method.

The shape of the phase boundary crystal-melt is looked for by the phase tracking method. The position of the phase boundary is known always precisely and coincides with the current position of the region boundary shared by the crystal and melt.

This feature is different from the enthalpy method where the location of the phase boundary is looked for inside of the region filled with the two phase material. Its location corresponds always to the defined phase transition temperature. In the phase tracking method the looked position of the phase transition is located at the isotherm passsing through the triple point. The temperature of the triple point can be regulated by means of the power controlled heaters and the definition of the control point in the triple point position. If the thermal computation with the phase tracking and controlled triple point temperature converges, then the temperature of the triple point should achieve the prescribed temperature value of the phase change (it means the netered parameter of the control point) and all vertexes of the deformed shared region boundary between the melt and the crystal should have the same triple point temperature. The enthalpy method is implemented on the unstructured mesh in CrysMAS. It can be applied especially successful for the VGF configuration because it allows the simulation of the time-dependent solidification with the position of the triple point crystal-melt-crucible gliding freely along the crucible wall. The complete VGF solidification process can be simulated by means of the enthalpy method.

The application of the enthalpy method to the Cz method is more elaborated in comparison to the phase tracking method because of several reasons.

First, the advantage of the time-dependent simulation with changing aspect ratio between the crystal and the melt in the VGF method cannot be applied for Cz because of more complex change in the crystal and melt shape and movement of the crucible with respect to the furnace.

Second, the melt flow in the Cz configuration leads to much thinner convective and concentration boundary layer under the phase boundary crystal-melt, while the heat transfer in the Cz configuration is much more convection dominated than in the VGF case. Much higher spatial resolution is therefore required in the melt layer under the phase boundary. In the enthalpy method the phase boundary is crossing arbitrary the mesh triangle elements. During the iterative solution of the fluid flow and heat transfer problem the phase boundary changes its position in the shared crystal-melt domain. Therefore a fine mesh should be available everywhere in the melt. A high mesh density increases the computational time necessary for solution of Navier-Stokes equations. Third, the released latent heat at the phase boundary crystal - melt is accounted for as a heat source inside of the triangle mesh. The iterative movement of the phase boundary within the mesh changes the position of the heat source. The jumping behaviour of the heat source from one iteration to the next causes the unstable convergence behaviour for high values of the latent heat density at the phase boundary. The crystal pulling velocity in the Cz method is usually higher than by the VGF method, therefore the ratio of the latent heat to the conductive heat flux in the crystal is higher too. The stability of the enthalpy method can be improved a bit again by the mesh refinement in the area around the phase boundary. But for high enough pulling rates no converged solution could be obtained with the enthalpy method even without consideration of the melt flow.

The phase tracking method in combination with hybrid mesh overcomes problems of the enthalpy method by a high crystal pulling rate and by strong domination of the convective transport. The only drawback of this method is the anchoring of the triple point position and fixed shape of the outer crystal boundary.

The application of the phase tracking in CrysMAS is only possible after the phase transition crystal-melt was accounted for already by the structured mesh generation. The creation of the phase transition is described in section Structured mesh generation and adjustment. Once the structured mesh with phase transition is generated, the latent heat at the phase transition will be accounted automatically for in each thermal computation and the phase tracking procedure can be activated optionally too. The latent heat density in W/m≤ at the phase interface is given by the product of the crystal pulling velocity in m/s and the specific solidification heat in W/m≥ The crystal pulling velocity is entered in the dialog Computation -> Process Parameters -> Czochralski tab in the Growth rate [m/s]: dialog field. A positive value of the growth velocity means in the Cz and VGF configurations the crystallization. In opposite the enthalpy method requires an opposite sign for the pulling velocity in the Cz configuration.

The second value in product to be prescribed is the specific latent heat. Again in opposite to the enthalpy method the latent heat is not a material property of the two phase material but is entered as a process parameter in the same dialog window Computation -> Process Parameters -> Czochralski tab in the Latent Heat [J/m≥]: dialog field.

Latent heat dialog field in the Czochralski parameters set. 

Figure†108.† Latent heat dialog field in the Czochralski parameters set.

This dialog field is visible if only the phase transition on the hybrid mesh is defined and the structured mesh with the phase transition is already generated.

The latent heat is accounted for in the Stefan boundary condition. The latent heat is balanced with the conductive heat fluxes computed on the unstructured mesh in the crystal and on the structured mesh on the melt in the formulation of Stefan boundary condition on the hybrid mesh.

The phase tracking can be activated for the hybrid mesh with the phase transition. For activation open the dialog window Computation -> Numerical Parameters -> Forward tab and check the Track interface in the Other dialog box. This checkbox causes shift of the nodal position towards the current position of the phase boundary crystal-melt if the enthalpy method on the unstructured mesh is applied. Or it activates motion of the region boundary between the crystal and melt towards the isotherm crossing the triple point if the phase tracking is activated.

The phase tracking starts to work if the normed residual of the enthalpy equation is reduced under the thereshold defined in the same dialog window in the dialog box Start Residual. This measure can avoid too quick and too large mesh distortion due to the phase tracking at the beginning of the thermal computation while the temperature distribution is still far away from the equilibrium state. Typical advised values for Start Residual are 0.01 or less.

Another important parameter for the application of the phase tracking method is the Underrelaxation. The underrelaxation is entered in the same dialog window in the dialog Underrelaxation field. The underrelaxation means that after the current axial position of the target isotherm in the crystal and in the melt are estimated, the vertexes of the shared region boundary are not moved exactly to the new found positions. The axial shift vectors are multiplied with the underrelaxation value and the positions of the boundary vertexes are changed by the prescribed shift fraction. Accordingly the positions of the mesh elements, i. e. nodes of the structured mesh in the melt and nodes of the unstructured mesh in both melt and crystal, will be adjusted. The mesh will be not generated again. Instead of it the axial shift of the phase boundary is interpolated for each shifted node inside of the deformated crystal and melt regions. The nodal positions are then changed by the interpolated shift value.

An underrelaxation value of 0.01 and less are advised for the stable phase tracking execution.

Note

Multiple blocks of the structured mesh under the phase boundary crystal-melt should be avoided. A side of the structured mesh block at the phase boundary should either contain the whole phase boundary length or the block side should terminate at the triple point. In the VGF case only the second option is possible. Both options are possible by the Cz configuration. By the first option the melt region can consist of the single block. By the second option at least two blocks should be contained in the melt region.

As a summary for the Cz configuration, the shared region boundary crystal-melt cannot be intersected by the other block boundary in any location other than the triple point.

Related Procedures

Structured mesh generation and adjustment.