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.


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],


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


\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:

RHO_1.25.TEMP_0600 RHO_1.50.TEMP_0600
RHO_1.75.TEMP_0600 RHO_2.00.TEMP_0600

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:

RHO_1.50.TEMP_0550 RHO_1.50.TEMP_0600
RHO_1.50.TEMP_0650 RHO_1.50.TEMP_0700

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).


1 Comment »

  1. […] we present simulation results of sintering on a longer time scale than the previous simulations. Average grain size and number of grains are plotted as a function of […]

    Pingback by Sintering Simulations: Part 2 « Kris's Research Notes — December 18, 2012 @ 8:33 pm

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