Technique · Simulations
Grain growth
Soap-foam or crystal grains swallowing one another in the Potts model, melting and regrowing in cycles.
Grain growth is a simulation that evolves step by step. Every piece on Synapse Drip is made from a recipe that layers techniques like this one, and you can change any of it in the Studio.
Good to know
- Cyril Stanley Smith, a metallurgist, used soap froths in the 1950s as a model of the grains in metals, because both coarsen in the same way.
- The Metropolis rule of accepting a worse move with probability exp(-energy / temperature) was published in 1953 by Nicholas Metropolis and four colleagues for simulating a liquid, and is still one of the most used algorithms in science.
- Annealing, heating a metal and cooling it slowly, makes its grains larger and its structure more even; the cycle here is a small annealing run over and over.
- Renfrey Potts proposed his model in 1952 as a generalisation of the Ising model of magnets, and physicists later found it describes the growth of crystal grains in metals as well as the coarsening of soap foam.
- In a flat foam or a sheet of metal, a grain with more than six neighbours grows and one with fewer than six shrinks, a rule worked out by John von Neumann and William Mullins in the 1950s.
The mathematics
- At temperature 0.60 a move that costs one unit of energy is accepted with probability 0.19, and one that costs two with probability 0.036; in the melting phase the temperature is 3.0, where costly moves are common enough to scramble the picture.
- In two dimensions a grain with n sides changes its area at a rate proportional to n minus six, the von Neumann-Mullins law, so the average grain has six sides and the typical grain area grows in proportion to time.
- Half the cells are tried at each step, in a checkerboard, so that no two neighbours change together and every cell sees its neighbours as they really are.
In art
- Stained-glass windows are patchworks of coloured regions with dark lead lines between them, just like this; here the pieces swallow one another instead of staying fixed.
- Foams, cracked mud, giraffe coats and dry riverbeds all share this look of irregular polygons with an average of six sides, which is why a grain pattern seems at once random and natural.
More ways to use it
Related techniques
Reaction-diffusion · Cellular automata · Sandpile · Slime mould · Flocking boids · Attractor cloud · Elementary automaton · Synchrony