Phase field method

The phase field method is a method for the numerical simulation of processes in which two or more phases and the interfaces between them, the phase boundaries, are to be described. The phase field method is used when it is to be calculated how structures and the course of the interfaces change over time. It was used in particular for solidification processes , but also for many other phenomena such as pattern formation at the interface between two liquids or for dynamic processes in fracture mechanics .

The upper picture shows a schematic structure with two phases, the lower graphic shows the value of the order parameter φ along the line in the upper picture. During the transition between the phases, the phase field function φ changes continuously and gradually.

To describe the structure or the distribution of the phases, the phase field method uses a function that is continuous in time and space, the phase field function φ, also called order parameter . For example, when describing two phases, it can assume values ​​between zero (first phase) and one (second phase).

The spatial area in which the phase field function changes describes the phase boundary, the position of which is thus implicitly given so that it does not have to be explicitly followed again. This makes the mathematical description easier.

The development of the phase field function over time is described with differential equations. In addition, the respective other processes, e.g. B. diffusion and heat conduction , can be described with appropriate equations, z. B. with the heat conduction equation .

Comparison with other methods

As with the level set and volume of fluid methods , the solid-liquid phase distribution is described by a characteristic function in the value range [0.1]. This function is carried out in addition to the physical description, for example the temperature, and indicates the percentage distribution of the two phases liquid-solid.

In contrast to the level set method, the phase field method is inherent , i.e. H. There is a direct domain differential equation for the description of the interface movement , which is based on thermodynamic principles. It can therefore not be used directly for general chemical diffusion processes with surface dynamics or other free-form tasks for general Stefan problems .

A modification of the phase field method is the phase field crystal method , which enables an anisotropic description of the atomic solidification structure of multi-component alloys , while the phase field represents a mesoscopic description of the solidification.

Applications

The phase field method is suitable for simulating the dendritic solidification of metals . Multiple diffusion equations can be coupled in order to take into account , for example, the dissolution behavior of different metals during the solidification of steel .

Multi-parametric further developments of the phase field method allow the simulation of the eutectic solidification of binary alloys such as titanium-iron.

Web links

• Hermann-Josef Diepers: Direct microscopic simulation of the solidification morphology. ACCESS eV affiliated to RWTH Aachen University, accessed on December 2, 2014 . 
• Modeling, analysis and numerics of local and non-local phase field models. Weierstrass Institute for Applied Analysis and Stochastics WIAS, accessed on December 2, 2014 . 

literature

• Nikolas Provatas and Ken Elder: Phase-Field Methods in Materials Science and Engineering. Wiley-VCH 2010, ISBN 978-3-527-40747-7

Individual evidence

1. ↑ Nele Moelans, Bart Blanpain, Patrick Wollants: An introduction to phase-field modeling of microstructure evolution . In: Calphad - Computer Coupling of Phase Diagrams and Thermochemistry . tape 32 , no. 2 , June 2008, p. 268-294 , doi : 10.1016 / j.calphad.2007.11.003 .