Modelling Solidification of Binary Alloys

  • Researcher: Ferran Brosa Planella
  • Academic Supervisors: Colin Please and Robert Van Gorder
  • Industrial Supervisors: Kjetil Hildal and Aasgeir Valderhaug

Background

Silicon is common in many objects that we see in our day-to-day lives, from the silicone in cooking utensils to the microprocessors in our smartphones. Even though silicon is very common in the Earth’s crust, we do not find it as pure silicon and therefore some process is required to obtain silicon in a form that can be used for these applications.

In this process, quartz (silicone dioxide) is reduced in a furnace with coal, charcoal and woodchips at temperatures around 2000°C, and molten silicon is extracted at the bottom of the furnace. After some refining, the melt is poured into iron moulds and left to solidify. At this stage, the silicon is around 99% pure, and when it solidifies a structure of pure silicon grains appears and the impurities get trapped between these grains. The distribution of the silicon grains and the impurities has an impact on the quality of the final product and, therefore, understanding how silicon solidifies is the first step towards a better control of the final product.

Figure 1: Casting of silicon in an iron mould. Reproduced from The Si Process Drawings, by Thorsteinn Hannesson (Elkem Iceland, 2016).

Outcomes

The mathematical model we use to study the solidification process is an extended Stefan problem which includes the diffusion of the impurities in the silicon and its effects on the melting temperature of silicon. Even though the motivation of the project is solidification of silicon, the model can be used to model any binary alloy.

We studied the stability of the self-similar solutions of the one-dimensional solidification problem. By introducing small perturbations to the solutions and studying how they evolve in time, we determined that, regardless of the material and the cooling rates, the perturbation to the freezing front always grows unboundedly in time and therefore it is unstable. Physically, this means that the only way to prevent the appearance of silicon grains in the cast is by controlling the velocity of the freezing front. The details of this analysis are presented in [1].

Figure 2: Silicon sample, where each different tone of grey is a silicon grain. Image courtesy of Elkem.

In order to better understand the behaviour of the system near the end of the solidification process we consider finite geometries. While the mathematical analysis in semi-infinite geometries shows that self-similar solutions are allowed, if we consider a finite domain then the effects of the boundary come into play and this type of solution is no longer valid. Therefore, we use asymptotic techniques to find approximations of the solutions in the distinguished limit of heat transport being much faster than mass transport. In this analysis we need to distinguish and study separately different time regimes and layers in the cast, and in each of them we observe different physical phenomena driving the behaviour of the problem. We performed the analysis for a one-dimensional planar and a three-dimensional spherically symmetric geometry, finding qualitative differences between both geometries. The results for the planar problem have been submitted for publication [2].

To validate the asymptotic solutions, we implemented a finite volume scheme in order to solve the full problem, finding very good agreement between numeric and analytic solutions for both geometries. In addition, for the planar geometry we found good agreement between the analytical solutions and experimental data provided by Elkem.

Lay report

Publications

[1] F. Brosa Planella, C.P. Please and R.A. Van Gorder. Instability in the self-similar motion of a solidification front. IMA Journal of Applied Mathematics, 83(1):106-130 (2018).

[2] F. Brosa Planella, C.P. Please and R.A. Van Gorder. Extended Stefan problem for solidification of binary alloys in a finite planar domain. Submitted for publication

 

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