Origins of dendritic growth

1. In a wide range of problems in pattern formation, there is a moving boundary that forms interesting spatio-temporal patterns. When diffusion of a quantity to the surface is involved, this often gives rise to an instability causing the surface to undulate. A well known example of this is dendritic growth. It appears is a variety of context and has been analyzed theoretically and experimentally in great detail. One of the best known examples of dendritic growth are snowflakes Links to an external site., but their study has relevance to many other things as we shall see. We shall use a phase field Links to an external site. model of this system.

Note that this model explains the morphology of a phase change that happens in simple physical systems such as ice and water, not dendrites in biological systems, such as neurons. The fact that there is such complicated behavior that comes out of such simple rules is interesting and the possible relationship to biological systems is discussed below.

hw4/dend has the scipy script dend.py, which illustrates the growth of a dendrite from an initial seed by updating a system similar to the reaction diffusion equation we looked at to study spots on the coats of animals. The physics is that of a substance, such as ice immersed in water below the freezing point. As the ice freezes, it releases latent heat the heats up the surrounding water, and must diffuse away before further growth can take place. The higher the curvature of the ice's surface, the lower the freezing temperature must be. That is, the freezing temperature isn't just 0 degrees C, but a bit lower owing to the curvature of the surface. This prevents the surface from bending too rapidly and becoming unstable at very short length scales.

This physics is similar to that of diffusion limited aggregation (DLA). One can think of the diffusion of heat in mathematical analogy to the diffusion of a particle. So "anti-heat" particles come in from infinity and hit the surface of the ice. When they hit, they've cooled down the material, allowing it to grow. This is also similar to the diffusion of nutrients to a colony of bacteria as we also saw.

What's the difference between DLA and dendritic growth? One of the biggest ones is that DLA is random but effects of randomness are much smaller with dendrites. The equation we simulate for dendrites is deterministic.

Run dend.py to see the beginning stages of dendritic growth. Note the four fold symmetry. To speed up the computation , also included is quart_den.py that uses the four fold symmetry to simulate only one quadrant of the growth. You can see the beginning of dendrites in this simulation. Increasing the lattice size, and the amount of computer time will lead to other interesting instabilities like side branching Links to an external site..

The model has two diffusion like terms. One for the diffusion of temperature, and the other one for the "diffusion" of the ice phase, phi. In order to see the four fold symmetry, the "diffusion" term for phi was made anisotropic, to favor movement in some directions rather than others.

In the code, for the quarter dendrite, are commented several cases where the anisotropy has been varied. Run the simulation. Is there any case where a dendritic instability is not easily seen? What does that suggest to you about the ingredients necessary for dendritic growth?

2. We shall explore the basic mechanism where by the surface in dendritic growth, and other systems where diffusion is important, will have an instability, causing it to undulate. This also related to the fractal nature of Diffusion Limited Aggregation. The code hw4/mullins/mul_sek.py, models a surface, say of ice, that has a small sinusoidal undulation in it. The right of it is a large body of water below freezing. Why doesn't it freeze? It does, but not instantaneously. That's because when it freezes, latent heat must diffuse away from it before more freezing can occur. It occurs at the surface of the ice.

If the ice were completely flat, it would remain flat, and heat would be uniformly shed from the interface between ice and liquid. If sinusoidal undulations are added, what will happen? The attached code models this situation.

You can change the initial frequency of undulation in the line where freq is defined. You can change the vertical height, and hence the lowest period of oscillation by changing nx. What happens as you change freq? You should stick to integral multiples of the lowest mode otherwise you won't really be modeling sin waves. The instability you find is known as the Mullins Sekerka instability.

3. There is research suggesting a relationship between the above physical process and some biological systems that exhibit a branching morphology.

• Aside from the morphogenesis of bacteria, give two examples where an instability similar to the Mullins-Sekerka instablity has been suggested.
• The original physical model of dendritic growth has a diffusion field that is crucial to the model. Do the examples that you found include a diffusion field, or do they get their unstable behavior by positing some other mechanisms?
• In at least one of the examples you gave, can you think of any experiments that could validate or invalidate the theoretical hypotheses used?