Non-isothermal grain-growth in polycrystalline material using Monte-Carlo

 Rituraj Nandan

            Graduate student, Department of Materials Science and Engineering, Penn State, University Park

            Most crystalline solids are composed of millions of small crystals or grains. These crystals have the same crystal structure, however they differ in their crystallographic orientation. Therefore grains are identified by a number denoting their orientation. When a polycrystalline material is uniformly heated, grains grow in size uniformly. However, when subjected to non-uniform heating, grains grow at different rates at different locations in the sample.

            Grain-growth during materials processing affects properties of the material. Numerical modeling provides insight into the kinetics and can help us predict useful information like spatial variation of hardness in the plate.

            Here, we try to model grain-growth in a thin metallic plate, which is heated by a torch (approximated as a point heat source) moving with a constant velocity in the positive x-direction.. In the plate, regions behind the torch have been subjected to much high temperature, compared to regions ahead of the torch. Hence, we should observe large grains behind the torch and smaller grains in front of the torch.

Temperature distribution in the plate

            The temperature field is used obtained using Rosenthal’s equation for moving point heat source, on a thin plate.

T₀ = initial temperature (K)

q₀ = net power (W)

d = plate thickness (m)

k = thermal conductivity of the plate (W/m-K)

v = torch velocity (m/s)

x = distance along the x-axis (m)

a = k/(ρC_p) = thermal diffusivity (m²/s)

r = radial distance from the torch

K₀ = modified Bessel function of the second kind and zero order

Model for Grain-growth

            At the beginning, each grid point is assigned a random number between 0 and n-1. Grains may have a large number of orientations and a larger value n gives realistic results. A value of n equal to 32 has been used here.

The probability for selection of a site is given as:

In the MC simulation the dimensionless grain size changes with the number of iterations or the MC simulation time steps (t_MCS). The variation with t_MCS is largely independent of material properties and real-time grain growth dynamics and is only dependent on the grid system in the MC model. Thus, in order to quantitatively predict the grain growth for a specific material under given thermal cycle, a relation between the simulation steps and real time is required.  

Real time is converted to Monte Carlo time using the following equation. The data used is for pure copper.

K₁ = model constant = 1.01

n₁ = model constant = 0.42

A = Accommodation probability =1

N = Average number of atoms per unit area at the grain boundary = 1.53 x 10¹⁹ /m²

N_a = Avogadro’s number = 6.022 x 10²³

h = Boltzmann’s constant = 6.62 x 10^-34

R = Gas constant = 8.314 J/mol-K

T = Absolute temperature

ΔS_f  = Activation entropy of grain boundary migration = 9.48 J/mol-K

Q = Activation enthalpy for grain boundary migration = 99.4 kJ/mol

γ = Grain boundary energy = 1.77J/m²

λ = L₀ = grid-size = initial grain size

Potts model describes the local interaction energy.

Grain boundary (GB) migration is shown. GB lies between adjacent sites having different orientations.

S_i = orientation at a random site i

S_j = orientation of nearest neighbor of i

n = number of nearest neighbors (n is usually taken as 8 for 2D lattice)

δ = 1    when S_i = S_j

         0    when S_i ≠ S_j                      

Results and discussion

        Elliptic temperature contours are obtained  for the quasi-steady state. The eccentricity increases with increase in weld velocity. Corresponding the the temperature contours, t_MCS is larger near the heat source and behind it, where torch has already passed. Hence these grains are larger compared to grains in front of the torch and far away in the y-direction.

        These results are for 200 x 200 grid in a 10cm x 10cm plate.

Temperature contours	Monte-Carlo time step

Distribution of grains

Java code

GMC.java

            The code has three key subroutines. These subroutines are called by a task handler called callroutine. The temperature is calculated in subroutine temperature. Monte-carlo time steps are calculated based on the thermal profile in subroutine tmc. The new orientation is calculated in subroutine potts.

public static void temperature(double t[][])    //quasi-steady thermal profile

public static double[][] tmc(double t[][]) // monte-carlo time at each location (i,j)

public static void potts(int o[][],double t[][],double tmcs[][]) //update orientation using potts model for energy reduction

 Other subroutines and their  are as follows:

public static int energy(int ii,int jj,int o[][])    //energy of location (ii,jj)

public static double grainsize(int o[][]) //measurement of grain size distribution and average grain size

public static void displayD(double tmcs[][],String filename) //write temperature/tMCS to file for viewing using software Tecplot

public static void display(int o[][]) //write orientation to file for viewing using software Tecplot

public static double bessi0(double x) // Returns the modified Bessel function I0(x) for any real x.

public static double bessk0(double x) // Returns the modified Bessel function K0(x) for positive real x.

The bessel function bessk0 is called from subroutine temperature while bessel function bessi0 is called by bessk0. They were obtained from "Numerical Recipes in C" and

 were validated by comparing results with Mathematica output.

Important variables in the code are:

                t[nx][ny]    :     temperature

	tmcs[nx][ny] :     monte-carlo time

	o[nx][ny]    :     orientation

References:

1.         Mishra, S. and T. DebRoy, Non-isothermal grain growth in metals and alloys. Materials Science and Technology, 2006. 22(3): p. 253-278.

2.         Shi, Y.W., et al., HAZ microstructure simulation in welding of a ultra fine grain steel. Computational Materials Science, 2004. 31(3-4): p. 379-388.

3.         Li, M.Y. and E. Kannatey-Asibu, Monte Carlo simulation of heat-affected zone microstructure in laser-beam-welded nickel sheet. Welding Journal, 2002. 81(3): p. 37s-44s.

4.         William H. Press, Saul A. Teukolsky, Numerical Recipes in C (available online at http://www.nr.com/)

Links to:

        My PSU web-page

        My google-page

        Revised

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