Research and development of most technological processes is usually carried out on a significantly smaller scale than the following industrial application. In the crystal growth in particular, the difference between crystal dimensions (or furnace dimensions) in the research and in the industry may reach an order of magnitude. Directional solidification is one of the main methods for the production of crystalline silicon ingots for photovoltaic (PV) applications. While there are many research studies with ingot diameters (widths) in the range from 10 to 22 cm [1], the industry is currently working with ingots between 83 and 100 cm [2]. Furthermore, the PV industry is continuously trying to use larger ingots to decrease the production costs. Consequently, the transfer of crystal growth processes between different scales is of great importance.

All crystal growth processes include a variety of interacting physical phenomena as can be seen on an example of directional solidification of silicon. The solidification is a thermal process in the first place. It occurs in a furnace in an inert gas atmosphere, where all heat transfer mechanisms are present: heat conduction in solid parts (e.g., insulation); heat radiation and convection in fluid parts (e.g., gas and melt). Convection in the melt is of particular importance because it influences the shape of the crystallization interface as well as the transport of various impurities in the melt. The convection is driven by buoyancy forces, but additional Lorentz forces may be induced by the magnetic fields of heaters or dedicated inductors.

We investigated the scaling laws of the coupled phenomena of heat transfer, phase change, fluid flow, magnetic fields, and species transport [3]. In the first step, the relevant dimensionless numbers are derived from the fundamental physical equations and from typical boundary conditions, following a well-known technique, e.g., [4]. Inevitably, several dimensionless numbers share the same parameters, such as the characteristic size or velocity, but with different ratios and powers. Therefore, a perfect up-scaling that simultaneously considers all coupled phenomena is not realistic. However, it might be possible to neglect some phenomena or some interactions or to adjust the geometric proportions to obtain a better scaling of the key processes. These conclusions are demonstrated by several numerical calculations for directional solidification of silicon (see Fig. 1 for an example).

Fig. 1. Example for scaling the temperature field due to heat conduction and radiation between a research scale (ingot size L₁=22 cm) and industrial scale (L₂=88 cm) silicon crystallization furnace. A perfect scaling can be achieved if thermal conductivities of all furnace parts are adjusted as L₂/L₁ and all heat fluxes (white numbers, in Watts) are adjusted as (L₂/L₁)², which obviously cannot be realized in a direct way.

[1] Kakimoto, K., Liu, L., and Nakano, S. Analysis of temperature and impurity distributions in a unidirectional-solidification process for multi-crystalline silicon of solar cells by a global model. Materials Science and Engineering B, 134 (2006) 269–272.
[2] Chunduri, S. K. Preparing for bigger loads. Photon International, 06 (2011) 236–259.
[3] Dadzis, K. Modeling of directional solidification of multicrystalline silicon in a traveling magnetic field. PhD thesis, TU Bergakademie Freiberg, 2012.
[4] Ruzicka, M. C. On dimensionless numbers. Chemical Engineering Research and Design, 86 (2008) 835–868.

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1. The Formation of Subgrain Boundaries During Block-Casting of Multicrystalline Silicon studied by Synchrotron X-Ray Topography