So, we always talk about diffusion as being very slow and measured, whether we use Fick’s first or second law to model it, it still contains the fundamental assumption that diffusion is laminar.

What if this was not true? What kind of material properties would be required to make diffusion non-laminar?

Is this even a sensical question?

Basically, particles travelling through the material would have to be moving in such a way such that their trajectories would not for streamlines. That you would get turbulent mixing. You might be able to get this with something akin to strung-together ion channels, but this might only get you longer laminar flows in difference directions.

At the transition from laminar to turbulent flow, what exactly happens? Some of the particles can basically no longer roll over each other when they want to go faster than their neighbours*.

So they spin off in a chaotic** direction.

So, what would you need to do to make this happen for diffusing molecules?

What is diffusion, exactly?

In its purest form, it is a small (relative to the bulk) concentration of particles ‘diffusing’ into a large bulk of solvent.

At the macro level, it is a (relatively) simple application of entropy.

At the molecular level, it is a (slightly less) simple application of entropy***. Basically, molecules move around the solution effectively at random (really chaotically, but random is good enough for this calculation). Since the particles doing the diffusing are all in one place to start, and move randomly, they are more likely to move into the bulk than out of it (more out of the bulk, if they move in each direction equally likely, more of them will move into the bulk.)

For this motion to be turbulent, the molecules would need to ‘want to’ move quickly enough that they wouldn’t roll off each other when moving past each other. (What really causes the turbulence in large flow is that the molecules of the wall of the pipe are unmoving, and so there is a limit to how fast everything can flow.)

So, all we need to do is ensmallen the pipe, or make molecules stationary when others are diffusing? Some kind of matrix? Is it really diffusion then?

*Statistical molecular dynamics is really powerful.
**Not random.
***You are probably most familiar with Entropy as the law that systems tend to disorder. At the molecular level, this manifests as system states that have more ways they can happen (effectively higher probability) happen more often. These states tend to be more ‘disordered’. For example, for the system xxxooo, only two of the many (20) states involve the x’s and o’s separated like that.
xxxooo,
xxoxoo, xxooxo, xxooox,
xoxxoo, xoxoxo, xoxoox,
xooxxo, xooxox, xoooxx,
oxxxoo, oxxoxo, oxxoox,
oxoxxo, oxoxox, oxooxx,
ooxxxo, ooxxox, ooxoxx,
oooxxx.
Indeed, for each rule you give about how they have to be structured, you restrict the number of options, and reduce the probability of what you are proposing existing in real life without out some kind of outside influence.