Grain Growth: Monte Carlo Simulation & Theory

Volumetric Rate of Change

Aside from the initial period of time in the quasi-stationary state the growth of each grain can be described by the average self-similar growth law [Hillert (1965); Hunderi and Ryum (1980); Mullins (1998); Streitenberger and Zöllner (2007); Zöllner and Streitenberger (2007)]

,
according to which

is directly linked to the volume change rate

with

.
The constant is calculated from the ratio

,
where R is the radius of a volume-equivalent sphere with

yielding

.
Here G(x) and H(x) are time-invariant dimensionless functions depending only on the relative grain size x and are interrelated by

.
The constant k is the kinetic constant of curvature driven grain boundary motion (see Chapter Curvature Driven Kinetics) and b is given by

.

Therefore, the volume change rate depends only on the scaled grain size x and is time-independent and self-similar [Mullins (1986); Glicksman (2005)]. In order to test this assumption

vs. x has been plotted for the 400th, 800th and 1200th time step with the numerical values given by the simulation. The simulation data, divided into classes, of all three time steps fall together. There are only fluctuations for very small and very large grains (cf. Figure 1).

Figure 1: Volume change rate vs. relative grain size after 400, 800 and 1200 time steps.

One can see that there is a non-linear relation between

and x. For further investigations the volume change rate can be plotted versus the relative grain size for all grains of the ensemble after 500 time steps.
Figure 2a shows that the simulation data have a very broad scattering band. To the eye both a quadratic and a linear least-squares fits are good approximations. Although the quadratic one seems to be slightly better. To test this assumption that a quadratic function

gives a better approximation to the simulation data than a linear fit:

The residual sums of squares (sum of all squared residuals; abbreviated as rss) have been calculated. The value for the quadratic fit is approximately 3.1% smaller than for the linear fit (cf. Figure 2a), which is a first promising indicator.
Furthermore, the residuals between the simulation data of the volume change rate and the least-squares fitted functions have been calculated for all grains and are plotted vs. the relative grain size (Figure 2b,c). The residuals of the linear fit show a strong parabolic correlation. A cubic least-squares fit to these residuals gives a polynomial with an obvious quadratic and linear term, which indicates that the volume change rate is an at least quadratic function of the grain size. The appropriate residuals between the simulation data of the volume change rate and the quadratic least-squares fitted function only fluctuate around zero. All terms of a cubic least-squares fit to these residuals are negligible, which also indicates a quadratic function .

Figure 2: a - Volume change rate vs. relative grain size after 500 steps for all grains of the ensemble together with linear and quadratic fit; b - Residuals of the linear fit vs. grain size; c - Residuals of the quadratic fit vs. grain size.

A detailed qualitative explanation for the non-linear behaviour of

is given in [Zöllner (2006)].

[Hillert (1965)]	M. Hillert. On the theory of normal and abnormal grain growth. Acta Metallurgica, 13:227, 1965.
[Hunderi and Ryum (1980)]	O. Hunderi and N. Ryum. The kinetics of normal grain growth. Journal of Materials Science, 1104, 1980.
[Mullins (1998)]	W.W. Mullins. Grain growth of uniform boundaries with scaling. Acta Materialia, 46:6219, 1998.
[Mullins (1986)]	W.W. Mullins. The statistical self-similarity hypothesis in grain growth and particle coarsening. Journal of Applied Physics, 59:1341, 1986.
[Glicksman (2005)]	M.E. Glicksman. Analysis of 3-d network structures. Philosophical Magazine, 85:3, 2005.

[Streitenberger and Zöllner (2007)]	P. Streitenberger and D. Zöllner. Topology based growth law and new analytical grain size distribution function of 3D grain growth. Materials Science Forum, Vols. 558-559, 1183-1188, 2007.
[Zöllner and Streitenberger (2007)]	D. Zöllner and P. Streitenberger. Normal Grain Growth in Three Dimensions: Monte Carlo Potts Model Simulation and Mean-Field Theory. Materials Science Forum, Vol. 550, 589-594, 2007.

[Zöllner (2006)]	D. Zöllner. Monte Carlo Potts Model Simulation and Statistical Mean-Field Theory of Normal Grain Growth. Shaker-Verlag, Aachen, 2006.

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