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Theory and numerics of phase field models for solidification of binary alloys

Croné, Philip LU (2015) In TFHF-5205 FHL820 20151
Solid Mechanics
Abstract
The phase field method emerged as a tool to solve free boundary problems with a sharp interface. In previous methods, this involved the computationally inefficient and complicated tracking of a moving interface. The phase field method overcomes this limitation by introducing an auxiliary variable called the order parameter which varies continuously over the whole domain and takes on finite values in the bulk phases and then varies rapidly but continuously across the interface. This method gives the location of the interface implicitly and therefore allows one to avoid explicit tracking of the interface. Although phase field equations lack a real physical interpretation, they can nowadays be tuned- in comparison to some corresponding... (More)
The phase field method emerged as a tool to solve free boundary problems with a sharp interface. In previous methods, this involved the computationally inefficient and complicated tracking of a moving interface. The phase field method overcomes this limitation by introducing an auxiliary variable called the order parameter which varies continuously over the whole domain and takes on finite values in the bulk phases and then varies rapidly but continuously across the interface. This method gives the location of the interface implicitly and therefore allows one to avoid explicit tracking of the interface. Although phase field equations lack a real physical interpretation, they can nowadays be tuned- in comparison to some corresponding thermodynamically consistent equations such as the Stefan problem- so that they can be used for quantitative simulations. This report explores different phase field models for phase transitions and provides results from their implementation. The first four models contain some details about the underlying physics, these models are mainly in this report to build up some conceptual understanding of the phase field approach and how it is applied to model solidification. Moreover, these models involve growth of magnetic zones, spinodal decomposition, non-isothermal solidification in a pure material and isothermal solidification in an alloy. The last model in this report, on which the main emphasis was put, deals with non-isothermal solidification in a Ni-Cu alloy. In this model, the influence of a temperature field along with prescribed boundary temperatures and the effects of a thermal fluctuation term were studied and compared to the isothermal case. Using supersaturation and under cooling as a driving force for solidification, it was found that a single crystal grows into a dendrite-like structure with the smaller side branches only present when a thermal fluctuation source was included in the model. It was also found that prescribed boundary temperatures have a big effect on the direction and rate of solidification. In the case of the temperature-concentration coupled model for solidification, some numerical issues, such as time step restriction due to instability, were present due to the fact that the added temperature field evolves much faster than the phase and concentration field. These numerical issues were overcome by using an Alternating Direction Implicit (ADI) method, which is a semi-implicit finite difference method. The code for the numerical simulations were written in C++ and most of the programs were parallelized using the OpenMP framework. (Less)
Popular Abstract
Have you ever stared in awe at the beautiful pattern found in a snowflake? These intricate yet structured patterns evolve during a process commonly known as solidification.

In today's modern society, solidification plays a large part in tailoring the properties of nearly all of our engineering materials. It is therefore of great importance for scientists and engineers to understand and be able to simulate these remarkable patterns. In contrast to the snowflake, these beautiful solidification patterns usually form on a micro scale, which is why they are not visible to the naked eye. Although solidification is a very complex process, sophisticated mathematical methods along with today's powerful computers enable relatively realistic... (More)
Have you ever stared in awe at the beautiful pattern found in a snowflake? These intricate yet structured patterns evolve during a process commonly known as solidification.

In today's modern society, solidification plays a large part in tailoring the properties of nearly all of our engineering materials. It is therefore of great importance for scientists and engineers to understand and be able to simulate these remarkable patterns. In contrast to the snowflake, these beautiful solidification patterns usually form on a micro scale, which is why they are not visible to the naked eye. Although solidification is a very complex process, sophisticated mathematical methods along with today's powerful computers enable relatively realistic simulations to be carried out.

Despite the complexities of these patterns, the underlying driving force is simply to minimize the overall excess energy so that the structure eventually reaches a state of balance, also know as equilibrium. Modeling solidification involves a set of equations which are traditionally very hard to solve, mainly because of the presence of a moving boundary between the liquid and solid phases. This boundary is in reality extremely thin (in the order of nanometers) and it is constantly changing its shape. Luckily a relatively new method called "The Phase Field Method" exists which addresses these problems and drastically reduces the difficulty (compared to older methods) with which simulations can be performed. (Less)
Please use this url to cite or link to this publication:
author
Croné, Philip LU
supervisor
organization
course
FHL820 20151
year
type
H2 - Master's Degree (Two Years)
subject
keywords
ADI, Solidification, Binary alloy, Non-isothermal, Phase field
publication/series
TFHF-5205
language
English
id
7862017
date added to LUP
2016-08-19 14:16:25
date last changed
2016-08-19 14:17:09
@misc{7862017,
  abstract     = {The phase field method emerged as a tool to solve free boundary problems with a sharp interface. In previous methods, this involved the computationally inefficient and complicated tracking of a moving interface. The phase field method overcomes this limitation by introducing an auxiliary variable called the order parameter which varies continuously over the whole domain and takes on finite values in the bulk phases and then varies rapidly but continuously across the interface. This method gives the location of the interface implicitly and therefore allows one to avoid explicit tracking of the interface. Although phase field equations lack a real physical interpretation, they can nowadays be tuned- in comparison to some corresponding thermodynamically consistent equations such as the Stefan problem- so that they can be used for quantitative simulations. This report explores different phase field models for phase transitions and provides results from their implementation. The first four models contain some details about the underlying physics, these models are mainly in this report to build up some conceptual understanding of the phase field approach and how it is applied to model solidification. Moreover, these models involve growth of magnetic zones, spinodal decomposition, non-isothermal solidification in a pure material and isothermal solidification in an alloy. The last model in this report, on which the main emphasis was put, deals with non-isothermal solidification in a Ni-Cu alloy. In this model, the influence of a temperature field along with prescribed boundary temperatures and the effects of a thermal fluctuation term were studied and compared to the isothermal case. Using supersaturation and under cooling as a driving force for solidification, it was found that a single crystal grows into a dendrite-like structure with the smaller side branches only present when a thermal fluctuation source was included in the model. It was also found that prescribed boundary temperatures have a big effect on the direction and rate of solidification. In the case of the temperature-concentration coupled model for solidification, some numerical issues, such as time step restriction due to instability, were present due to the fact that the added temperature field evolves much faster than the phase and concentration field. These numerical issues were overcome by using an Alternating Direction Implicit (ADI) method, which is a semi-implicit finite difference method. The code for the numerical simulations were written in C++ and most of the programs were parallelized using the OpenMP framework.},
  author       = {Croné, Philip},
  keyword      = {ADI,Solidification,Binary alloy,Non-isothermal,Phase field},
  language     = {eng},
  note         = {Student Paper},
  series       = {TFHF-5205},
  title        = {Theory and numerics of phase field models for solidification of binary alloys},
  year         = {2015},
}