Summary of key points
+++++++++++++++++++++

.. image:: n.128/images/grain_growth.0.png
   :width: 25%

.. image:: n.128/images/grain_growth.3.png
   :width: 25%

.. image:: n.128/images/grain_growth.28.png
   :width: 25%

Modeling and Analysis
.....................

* Model *normal* isotropic grain growth using the *potts* model
* *spparks* can run *serial* or *parallel*, 2- or 3D
* Visualize grains using *paraview* or render images directly from *spparks*
* Post process *dump* files for analysis

.. _potts-rules-of-thumb:

*Rules of Thumb* for *Potts* Model
..................................

* Use large number of spin values :math:`Q` (thousands)
* Use temperatures :math:`T_f\leq kT\leq T_r`

 * :math:`T_r, T_f` depend on lattice and dimensionality
   
   * :math:`T_f`, *mesh pinning* temperature
   * :math:`T_r`, *grain boundary roughening* temperature


 * Trends as :math:`kT` increases:

         * Rate of grain growth increases 
         * Standard deviation :math:`\sigma` may decrease

* For :math:`kT\gt T_r`

         * Model exhibits *grain boundary roughening*
         * Excessive single site grains

* For :math:`kT\lt T_f`

         * Grain exhibit *mesh pinning*
         * Grains lock onto underlying lattice;

* Visualize grains
* Plot distributions