Multiscale Modeling

Prediction of the Grain Nucleation and Growth Using Cellular Automaton

Mahdi Torabi Rad

Spring 2014

Abstract

In the present project we have used cellular automaton method along with the well-known finite

volume method to model the grain growth and nucleation in a solidifying system. The cellular

automaton is used to simulate the phenomenon happening at the microscopic scale and the finite

volume is used to simulate the heat flow at the macroscopic scale. The results for nucleation of

the grains and the growth of a horizontal grain and an inclined grain, and the solid

fraction/temperature profiles are presented and discussed.

Introduction

In the 1990’s, the cellular automaton (CA) models were developed to predict phenomenon such

as equiaxed grain nucleation, columnar grain growth and, in general, the prediction of the grain

structure in the solidification processes. This technique was initially developed for 2D problems;

but later, it was extended to 3D and was coupled with finite element (CAFE) method to calculate

the flow of heat in the macroscopic scale as well. Detail of the CA and CAFE can be found in

[1].

Finite volume method has been used excessively to predict temperature and solid fraction fields

in the solidification literature. In this study we will used a combined cellular automaton finite

volume (CAFV) to predict the grain nucleation/growth and temperature/solid fraction fields in a

solidifying system. The CA will be used for prediction of the grain nucleation/growth, happening

at the microscopic scale; while FV will be used to solve for the energy transport happening at the

macroscopic scale.

The approach is shown schematically in Fig. 1, illustrating a portion of a FV mesh along with the

underlying microstructure in sub-Fig. 1a; and the cellular automaton sites in sub-Fig. 1b. The

temperature field is obtained by solving the energy transport equation using the FV mesh. Then,

the known temperature field is interpolated to the cellular sites using a classical interpolation

scheme. Knowing the temperature at each cellular site, the nucleation and growth models are

applied at each cellular site.

Nucleation and growth modeling at the level of CA cells is carried out as follows. At the start of

solidification, all the cells are inactive with solidification index Is = 0. This indicated that they

are all liquid. We randomly select a certain number of nucleation sites in the domain, based on

the nucleation law. Each nucleation site is assigned a critical nucleation undercooling. If the

temperature, from the macroscopic FV calculation, at the site location is less than this critical

undercooling, then the site will be activated i.e. a new grain will be generated. At the time of

activation, we will assign a random number to the grain, representing its crystallographic

orientation. Then the solidification index Is will be set to this random number. For all the active

site we will calculate the grain length at each time step and once an active grain touches its

inactive neighbor, the neighbor will be also activated and its crystallographic orientation will be

set the crystallographic orientation of the activating parent site. In this process of activating an

inactive neighboring site through the growth, we should be very careful to make sure that the

grain orientation does not depend on the cellular grid. This will be discussed in more detail in the

next section. Finally, the variation of the solid fraction within each cell is calculated using

truncated Scheil macrosegregation model.

Figure 1. A schematic showing the coupling between the finite volume mesh (top figure) and the

cellular automaton sites (bottom figure).

Model Description

The model consists of six equations discussed below.

Energy equation

2HT

t

(1)

where H is the enthalpy, t is the time, is the thermal diffusivity, and T is the temperature. The

enthalpy is related to the temperature and solid fraction through

p sH = c T g L (2)

where pc is the specific heat, sg is the solid fraction and L is the latent heat.

The solid fraction at the microscopic scale is calculated from

2

1 1s k

p l f s

Hg

c T T k g L

(3)

where is lT the liquidus temperature of the alloy, fT is the melting point of the pure metal, k is

the partition coefficient.

The grain nucleation is modeled by a Gaussian distribution as

max 1

exp22

mn T Tdn

d T TT

(4)

vv v CA

CA

Np n V

N

(5)

where n is the total density of the grains, mT is the mean undercooling, and T is the standard

deviation, and pv is the probability that a cell nucleates during a time step.

The grain growth is modeled as

m

V A T (6)

where A and m are constants, and V is the tip velocity of the grain. The length of the grain at each

time step is calculated using

N

t

tL t v T t dt

(7)

Correcting the location of dendrite tip

Figure 2 shows a schematic of the growth of an inclined grain. Here we want to make sure that

the initial angle of the grain will be preserved during the growth. To achieve this, at each time

step, a tip triangle will be determined by finding the position of points U and V for a given tip

position T. Then, the cells that lie inside the dendrite tip triangle will be also activated.

Figure 2: a) A schematic of the growth of an inclined grain; b) dendrite tip triangle and the cell

activated inside it.

Problem Statement

The problem studied here is 1D solidification of a binary Pb-18wt%Sn alloy as shown in Fig. 2.

The heat is extracted from the bottom and the top wall is adiabatic. The material properties are

listed in tables 1 and 2.

Figure 3: a schematic of the solidifying system

Table 1. Thermophysical properties of the Pb-18wt%Sn (adopted from [1])

Property Symbol Units Value

Specific heat pc J (kg K)

-1 176

Thermal conductivity k W (m K)-1

17.9

Reference density 0 Kg m

-3 9250

Latent heat of fusion L J kg-1

3.76 × 104

Melting point at C = 0 fT C 327.5

Eutectic composition eutT wt% 61.911

Equilibrium partition coefficient 0k - 0.310

Liquidus slope m C (wt%)-1

-2.334

Liquid mass diffusivity lD m

2s

-1 0

Table 2. Parameters used in the present simulation (adopted from [1])

Parameter Symbol Units Value

Reference composition (taken as initial melt

composition) refC wt. % 18.0

Reference temperature (taken as initial melt

temperature) refT C 285.488

External fluid temperature T C 25

Heat transfer coefficient Th Wm-2

K-1

400

Domain height H m 0.15

Numerical Implementation

We have developed a parallel-computing code in OpenFOAM (which is an open source C++

library for solution of continuum mechanics problems [2,3] to solve these equations. The finite

volume method introduced by Patankar is used for discretization of the energy equation [4]. At

each time step the algorithm is as follows

1. Solve the energy equation for a new temperature T.

2. For the new temperature calculate pv.

3. Scan all the cellular automaton cells in the domain, which are still inactive, and assign a

random number r for each site. If r < pv then the cell will be activated in the current time step.

4. For active cells, calculate the grain tip velocity using Eq. (6) and update the grain diagonal

using Eq. (7).

5. When an active growing grain reaches its inactive neighbor activate the neighbor and assign

the same crystallographic direction as the parent activating cell to it.

6. Correct the dendrite tip location.

7. Go to the next time step.

Results and discussion

Figure 4 shows the total density of nucleated grains (left panel) and change in the density of the

nucleated grains (right panel). Figure 5 shows the growth of horizontal and inclined grains. The

behavior in these figures is as what we expect.

Figure 4: the total density of nucleated grains (left panel) and change in the density of the

nucleated grains (right panel)

Figure 5: the growth of a horizontal grain (left panel) and an inclined grain (right panel)

Figure 6 shows the solid fraction and temperature profiles at t = 60s in the domain. As we expect,

closer to the cooling wall we have higher solid fractions and lower temperatures.

ActivationIndex

1.9

1.7

1.5

1.3

1.1

0.9

0.7

0.5

0.3

0.1

ActivationIndex

1.9

1.7

1.5

1.3

1.1

0.9

0.7

0.5

0.3

0.1

Figure 6) the solid fraction and temperature profiles at t = 60s in the domain

References

1. M. Rappaz, Ch.-A. Gandin, Probilistic modeling of microstructure formation in solidification

process, Acta Metall.Matter, Vol. 41, No. 2, pp. 345-360

2. OpenFOAM The Open Source CFD Toolbox User's Guide, Version 1.7.1, 2010.

3. OpenFOAM The Open Source CFD Toolbox Programmer's Guide, Version 1.7.1, 2010. 4. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, NEW YORK: McGraw-Hill, 1980.