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Modifications of the Potts Model |
The standard MC Potts model simulates normal grain growth as it is characterized by uniform grain boundary energies and mobilities of the grain boundary faces. The other structural elements of a three dimensional grain boundary network - namely triple lines and quadruple points - are assumed to have no direct influence on the migration kinetics of the boundaries. The same holds for the triple points of a two dimensional grain boundary network. However, recently it has been demonstrated by experimental, theoretical and simulation studies that triple lines and quadruple points have finite mobilities different from the adjoining grain boundaries (see here). Therefore, the kinetics of grain growth can also depend on the mobility of these boundary junctions. In particular, it can be assumed that a limited junction mobility yielding a junction drag can be represented by a principle expression giving an effective mobility of such a grain boundary in terms of the intrinsic mobilities mgb, mtj and mqp of grain boundary, triple lines and quadruple points, .Here a is the average grain boundary junction spacing and the mobility of the grain boundary can be calculated with the Huang-Humphreys-relation yielding for high angle grain boundaries mgb = 1. The intrinsic mobilities of triple and quadruple junctions, mtj and mqp, control the kinetics in case mtj << mgb and/or mqp << mgb, respectively. Using this general idea of reduced mobilities of the boundary junctions in nanocrystalline grain structures two Potts model modifications can be implemented: 1. Use of an effective mobility. 2. Direct modification of mobilities and energies of triple and quadruple junctions. The implementation of an effective mobility works such that the mobility of a boundary depends on its (above defined) length a. It holds for triple junction controlled growth that if a is small against mgb:mtj then the junctions control the kinetics and the average grain diameter and radius, respectively, increases linearly with time. However, for long-time annealing and therewith for large grains the classical parabolic growth law follows. On the other hand, using the direct modification approach, each grain feature is assigned an own specific mobility such that all lattice points associated with grain boundaries are characterized by the mobility mgb, associated with triple lines by mtl and associated with quadruple points by mqp, where each has a value between zero and one (one being the maximum value in the Potts model). Hence, the transition probability changes and depends in the modification on the grain feature of a lattice point. In particular, the choice of mobilities resp. the ratio of the values determines the growth kinetics; for further information see [Zöllner (2011, 2012), Schäfer and Zöllner (2012)]. Influence on growth kinetics: In case of normal grain growth - as it may occur for grain structures on the micrometer scale - only the grain boundaries control the coarsening kinetics. Regarding the Potts model simulation this means that the mobility of the boundaries is set equal to one and triple lines as well as quadruple points are considered to be part of the boundaries: mgb = mlt = mqp = 1. Such kinetics is supposed to follow a square-root law in time. This is shown indeed in Figure 1a. In case of triple line controlled grain growth - as it may occur in nanocrystalline grain microstructures - the mobility of the triple lines (together with the energy of the grain boundaries) controls the kinetics, hence, mgb = 1 and mtl = mqp < 1. Here it follows from the above considerations of an effective mobility that the average (linear) grain size should increase linearly with time, which is true according to Figure 1b, where also the deviating behavior for long annealing times can be observed. In case of quadruple point controlled grain growth - as it may also occur in nanocrystalline grain microstructures - the mobility of the quadruple points (also together with the energy of the grain boundaries) controls the growth kinetics, hence, mgb = mtl = 1 and mqp < 1. In this case we expect and observe an exponential increase with time - see Fig. 1c, where it is apparent that for larger grain sizes the kinetics changes over linear back to parabolic! 
Figure 1: Temporal development of the average grain size for: a - normal grain growth; b - triple line controlled grain growth; c - quadruple point controlled grain growth. |
| [Zöllner (2011)] | D. Zöllner: A Potts model for junction limited grain growth, Computational Materials Science, Vol. 50, (2011), p. 2712-2719. |
| [Zöllner (2012)] | D. Zöllner: Grain microstructure evolution in two-dimensional polycrystals under limited junction mobility, Scripta Materialia, Vol. 67, (2012), p. 41-44. |
| [Schäfer and Zöllner (2012)] | S. Schäfer and D. Zöllner: Parallel potts model simulation of nanocrystalline grain growth, CD-ROM Proceedings of the 6th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012), September 10-14, 2012, Vienna, Austria, Eds.: Eberhardsteiner, J.; Böhm, H.J.; Rammerstorfer, F.G., Publisher: Vienna University of Technology, Austria, ISBN: 978-3-9502481-9-7. |
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