Patterns evolving from interfaces

Does this apply to bacteria?

To understand this instability we can perform a kind of "linear stability" analysis

This says a tiny ripple with

wave-number k is put on the interface. Does this grow or shrink with time?

The coefficient A depends on

microscopic length scales and on a much longer diffusion length.

For small enough k, this expression is positive meaning the ripple will grow.

Is a flat interface stable?

Heat can escape more quickly

Patterns evolving from interfaces

The tip will grow more quickly

making it elongate. This is

called the "Mullins-Sekerka instability"

D is called the "Fractal Dimension"

The ice crystal grows but

releases "latent heat". This

heat must diffuse away

before more ice can form.

We saw how spots on cows, stripes on fish, etc, could be understood by the diffusion and interaction of chemicals. But there are lots of other kinds of patterns. Here we'll talk about mechanisms that involve physics at "interfaces". We'll start with the snowflake.

The interface velocity

is proportional to the local

temperature gradient.

https://deutsch.physics.ucsc.edu/phys180/dla_explained.html

https://deutsch.physics.ucsc.edu/phys180/dla_c.html

Simple physical laws determine

the way the ice grows. Diffusion

of heat (temperature) is important.

The flow of heat is proportional to the

negative gradient of temperature

The local temperature

depends on the radius of

curvature of the interface

Think about a poker with one

end a fire and the other in water.

And it's not hard

to prove that the temperature field

is diffusional:

Sorry Dwight.

You can "supercool" water!

Yes it will eventually freeze

but that could also take the lifetime of the Universe!

In the case of DLA,

it's the diffusion of particles to the cluster that determines the morphology.

So an initial nucleation

site of ice forms and

the snowflake grows from there.

In the case of snowflakes, it's the diffusion of heat (anti-heat particles) diffusing from infinity to the surface.

The instability (Mullins-Sekerka) is the same for the two cases!

What causes the

regular patterns

we all marvel at?

Temperature < freezing

A snowflake starts with water below freezing.