West University of Timisoara

Blv. V Parvan 4, Timisoara 300223, Romania   http://www.uvt.ro 

International program

Hubert Curien (PHC) - Brancusi

 Bilateral cooperation between Romania and France

Research Project Code 19550 PK, 2009-2010

Mathematical and computational modelling of crystal growth processes based on capillarity

A. ROMANIAN PARTNER

                 A.1. Institution: West University of Timisoara (UVT)

                                          Faculty of Mathematics and Computer Science                                                            Results: 2009

                 A.2. Address: Blv V Parvan 4, Timisoara, 300223

                 A.3. Phone/Fax/e-mail: + 40 256 592 221/ Fax: +40 256 592 316

                                          lilianabraescu@balint1.math.uvt.ro or braesculiliana@yahoo.com

                 A.4. Coordinator: Assoc. Prof. Liliana BRAESCU

                 A.5. Members of the research team

                                         Senior researchers and internal collaborators

                                    Assoc. Prof. Liliana BRAESCU

                                    PhD Student: Simona EPURE

B. FOREIGN PARTNER

B.1. Institution: Institut National Polytechnique de Grenoble (INPG)

           Science et Ingénierie des Matériaux et Procédés (SIMAP) laboratory 

                 B.2. Address:  SIMAP-EPM, ENSEEG BP. 75, 38402 Saint Martin d’Hères, France

                 B.3. Phone/Fax/e-mail: Office Phone +33 4 76 82 52 13, E-mail: thierry.duffar@grenoble-inp.fr

                 B.4. Coordinator:  Prof. Thierry DUFFAR

B.5. Members of the research team: Prof. Thierry DUFFAR

                                                                        To be defined PhD student

C. DESCRIPTION OF THE PROJECT

                

Crystals can be obtained by different growth methods. Before its utilization in engineering, the crystals are constrained to some supplementary mechanical processes (cutting, polishing) for bringing them to the desired form. These processes are generating defects and material losses, so the final product has low quality and is more expensive. For this reason those growth methods are preferred which allow obtaining the crystal directly in the final desired form (without additional machining) and with minimal defects. Techniques of crystal lateral surface shaping without contact with the container walls are preferred: Dewetted Bridgman (DW), Edge-defined film-fed growth (EFG), Czochralski, Floating-zone; the absence of contact between the crystallizing substance and crucible walls allows improving crystal structures and decreasing the mechanical stress level.

In this project we focus on two growth methods in which the crystal shape is depending on a liquid meniscus: EFG and DW, because there are new techniques that need quick improvement.

The EFG method is known to be capable of pulling up a crystal according to the geometry of a die which is immersed into a crucible filled with a melt. In this case, the melt goes up to the upper surface of the die through the slit. Then, a seed crystal, which is deposited onto the melt on the upper surface of the die, is pulled up at any desired rate, thereby obtaining a crystal according to the geometry of the die (Figure 1).

Figure 1: Schematic EFG crystal growth system

 

The classical Bridgman method involves heating a polycrystalline material above its melting point in a crucible and slowly cooling it from one end where a seed crystal is located (Figure 2(a)). Single crystal material is progressively formed along the length of the crucible. The disadvantage of this technique is that the crystal contacts the crucible wall, which generally results in increasing the mechanical stresses, impurity level, and defect density in the grown crystals. The disadvantage can, however, be overcome by the dewetting solidification technique.

The phenomenon of dewetting is characterized by the Bridgman growth of a crystal without contact with the crucible walls due to the existence of a liquid meniscus at the level of the solid-liquid interface which creates a gap between the crystal and the crucible (Figure 2(b)). On the ground the dewetting has been obtained by introducing into the ampoule a pressure P_c with the aim to detach the meniscus away from the ampoule walls (Figure 2(b)).

           Figure 2: Schematic Bridgman (a), dewetted Bridgman (b) crystal growth systems, and

           photograph of an ingot showing attached and detached regions (c)

 

In parallel, mathematical and computational modelling has become very important in order to improve growth processes in general and is also needed for these two techniques. We will perform qualitative and numerical studies of the nonlinear systems of ordinary differential equations and of partial differential equations which allow the prediction of the shape and of the compositional uniformity of the crystals obtained by both methods.

For the calculation of the meniscus shape, its surface z=z(x,y) will be given by the Young-Laplace equation describing the equilibrium under pressure or electromagnetic field. This will be transformed in a nonlinear system of differential equations. From qualitative and numerical studies of the solution, the dependence of the form of the meniscus (convex, concave, convex-concave) on the pressure, surface tension, contact angle, wetting angle, and on the gravity, will be determined. The stability of the growth processes will be studied. From these, the automatic control of the processes will be possible and will constitute premises for the elaboration of new production technologies and control processes for the improvement of the crystal quality.

In parallel, crystal growth experiments by these techniques will be performed and results compared to numerical predictions.