Kris's Research Notes

December 14, 2012

Sintering simulations: Part 1

Filed under: Grains, KMC, Sintering — Kris Reyes @ 4:13 am

In this note, we present preliminary simulations of sintering, by which we mean the evolution of granular, porous material. Grains represent contiguous regions of atoms of a given orientation. A system evolves by surface diffusion of atoms and changes of atom orientation.

Model

Atoms are placed on a 2D hexagonal lattice and each are one of $S$ species, $\sigma_1, \hdots, \sigma_S$, representing different orientations. Pair-wise bonding energies are specified as

$\displaystyle \gamma(\sigma_i, \sigma_j) = \gamma(\sigma_j, \sigma_i) = \begin{cases} \gamma_1 & i = j, \\ \gamma_0 & i \neq j,\end{cases}$

where $\gamma_1, \gamma_0$ are parameters. That is, $\gamma(\sigma_i, \sigma_j) = \gamma_{\delta_{ij}}.$ Throughout this note, $\gamma_1 = 0.5$ eV and $\gamma_0 = 0.25$ eV.

All events $X\rightarrow Y$ between atom configurations $X, Y$ are parameterized by an event on an individual atom $x$, either moving to an unoccupied neighboring lattice site (surface diffusion) or changing orientation (spin flips). Rates for either event are of the form

$\displaystyle R(X, Y) = R_0 \exp\left[-\frac{E(X) - E(X\wedge Y) + \epsilon(X, Y)}{k_B T}\right],$

where $E(X)$ is the total bonding energy in a configuration $X$, $X\wedge Y$ represents an intermediate state and $\epsilon(X,Y)$ an additional energy barrier that depends on the type of event. The constants $R_0 = 10^{13} \text{sec}^{-1}$ and $k_B$ is the Boltzmann constant. For the surface diffusion of an atom $x$, $X\wedge Y$ represents the state with that atom removed and $\epsilon(X,Y) = 0$ so that the total activation energy is given by

$E(X) - E(X\wedge Y) + \epsilon(X,Y) = E(x),$

where $E(x)$ is the local bonding energy of atom $x$.

For the spin flip of an atom $x$ changing from species $\sigma_i$ to $\sigma_j$, we use the intermediate state $X\wedge Y$ defined by replacing $x$ with an atom of intermediate species $\sigma_i \wedge \sigma_j$ representing a superposition of the two species. The bonding energy between this intermediate species is given by averaging:

$\displaystyle \gamma(\sigma_i\wedge \sigma_j, \tau) = \frac{1}{2}\left(\gamma(\sigma_i, \tau) + \gamma(\sigma_j, \tau)\right).$

The additional energy barrier for spin-flips $\epsilon(X,Y)$ is a parameter $\rho$, which we vary in this preliminary study. More information about this type of event can be found here.

As an example, we consider the spin-flip of an atom of species $\sigma$ inside a neighborhood in which $n$ neighbors are of species $\sigma$ and $6-n$ atoms are of species $\tau$. Then the change in energy for spin flip $\sigma \rightarrow \tau$ of that atom is given by

$\displaystyle E(X) - E(X\wedge Y)$

$\displaystyle = n\gamma(\sigma,\sigma) + (6-n)\gamma(\sigma, \tau) - \frac{n}{2}\left(\gamma(\sigma,\sigma) + \gamma(\tau, \sigma)\right) - \frac{6-n}{2}\left( \gamma(\sigma, \tau) + \gamma(\tau, \tau) \right),$

$\displaystyle = \frac{n}{2}\gamma(\sigma, \sigma) + (3-n)\gamma(\sigma, \tau) - \left(3 - \frac{n}{2}\right)\gamma(\tau, \tau),$

$\displaystyle = (n-3)(\gamma_1 - \gamma_0).$

Then the rate prescribed to the spin-flip is given by

$\displaystyle R(X, Y)$

$\displaystyle = R_0 \exp\left[-\frac{(n-3)\left(\gamma_1 - \gamma_0\right) + \rho}{k_B T}\right],$

$\displaystyle = \omega \exp\left[-\frac{n\widetilde{\gamma}}{k_B T}\right],$

where

$\displaystyle \widetilde \gamma = \gamma_1 - \gamma_0$,

and

$\displaystyle \omega = R_0 \exp\left[\frac{3 \widetilde \gamma - \rho}{k_B T}\right].$

Compare this with the rates given by the standard Monte Carlo Potts model.

Simulation results

We simulated the sintering of a 2D system on a 256 x 256 hexagonal lattice with periodic boundaries. Clusters were constructed with a random sphere packing algorithm that generated a system with 12,000 initial grains occupying 80% of the lattice, each assigned one of $S = 7$ species at random. The same initial configuration was used for all simulations described here, and is depicted below (different colors represent different spins):

The simulation time, temperature and parameter $\rho$ were varied in independent trials.

Varying $\rho$

The parameter $\rho$ is the additional energy barrier for spin flips. Larger values of $\rho$ correspond to the regime where there are little to no spin flip, so that surface diffusion is dominant, while spin-flipping is dominant when $\rho$ is small.

We fixed the temperature at $T = 600K$ and simulated 100 seconds of sintering. We varied $\rho$ between 1.25 eV and 2.0 eV:

We note here that for the case $\rho = 1.25$ eV, surface diffusion consisted of 70.8% of the events in the first 10 seconds of simulation, while for $\rho = 2.00$ eV, surface diffusion dominated and consisted of more than 99.9% of the events. Hence surface diffusion alone is not sufficient for coarsening of grains.

Varying temperature

Now we fix $\rho = 1.50$ and vary temperature between 550 and 700 K:

We see that increasing the temperatures increases the rate at which the grains coarsen. We also notice that vacancies tend to accumulate near points of potential triple or higher-order junctions:

These vacancies tend to be fairly stable at such points. Such a phenomenon may suggest why sintering can be difficult with granular material.

Varying simulation time

Lastly, we fixed $\rho = 1.25$ eV and temperature at 600K. We varied simulation time between 10, 100 and 1000 seconds. The results are as follows:

As expected, increasing the simulation time allows the clusters to coarsen to larger grains. Movies of the 1000 second simulation (right-most above) are here and here (slower version).