# Do “random” “2D” polymers constrained to a box actually form “space filling” curves?

Background: The self avoiding walk model is (allegedly) a reasonable model for polymers, at least with respect to certain "universal" (i.e. lattice independent) constants -- see section 1.1. in this survey.

Consider an $$n \times n$$ square grid, with lower left corner marked $$a$$, and upper right corner marked $$b$$. Let $$S_n$$ denote the set of self avoiding walks in the grid from $$a$$ to $$b$$.

There is a family of probability distributions on $$S_n$$, parametrized by a single number called the "fugacity." (Allgedely, a measurement of pressure.) In particular, for $$x > 0$$, define $$\nu_x$$ as the probability distribution on $$S_n$$ that awards a walk $$\omega \in S_n$$ mass proportional to $$x^{|\omega|}$$, where $$|\omega|$$ is the length of the walk.

This family of probability distributions is known to have a phase transition around certain values of $$x$$. In particular, for $$x > 1 / \mu$$, where $$\mu$$ is the connective constant of the square lattice, a walk is "space filling" with high probability. (My usage of space filling here is heuristic, it is made precise in the linked article.)

Mathematically, this is all very interesting. I'm wondering if there is empirical evidence that complements these results. This leads to my imprecise question, but first some caveats.

Caveat one: These results I mentioned concern $$2D$$ self avoiding walks. The closest analogue in our 3D world is presumably polymer systems under tight constraints, such as in this. I'll call these 2D-ish in the question.

Caveat two: I ask this question as a mathematician extremely ignorant of chemistry. Please take that into account in your answers, if possible.

Question: Is there a physical process that produces a "random" (that is, $$\nu_x$$ distributed) polymer chain in a 2D-ish medium or 3D medium, and if so, is it "space filling" for sufficiently high pressure / fugacity $$x$$?

Totally naive thoughts:

I debated whether to include these -- I decided to err on the side of sounding ignorant but at least communicating a better picture of what I'm curious about, and where the boundaries of my knowledge are.

• As far as making the process, what if just chucked some monomers in a solvent along with a polymer strand and waited for it to grow by random rerouting attachments? Is there any incentive for a long polymer path to replace one of the monomers in our starting strand? I guess one hard part is insuring that you get one chain, and not one chain and many polymer loops. Maybe there are physical reasons why the loops would like to merge into the chain, at least under high pressure? Or maybe it's not a problem that there are extra polymer loops, as long as we can somehow point to our favorite polymer chain.

• Taking inspiration from the algorithm in that last link, one process I can imagine is sticking a long polymer into solvent in a 3D container, shaking it up (very gently), and then squeezing down one of the sides of the container. It seems hard to ensure that this can be squeezed thinly enough to make criss-crossing of the polymer impossible. Maybe instead of physically squeezing the polymer to one side, we can squeeze it using some other force (magnets???) ? Or maybe criss-crossing is inevitable if you try to do something like this?

• Very interesting question. This is largely beyond what I can answer, but I should mention that restricting a "normal" linear 1D polymer to a 2D plane may not be so difficult. For a century we've known a marvellous method for producing layers a single molecule thick: placing hydrophobic molecules on the surface of a hydrophillic liquid, allowing it to spread, then carefully squeezing the surface layer together. This is the principle of the Langmuir-Blodgett trough. I'm sure this has been applied to the study of polymer packing. – Nicolau Saker Neto Jun 19 at 22:57
• Hm. The self-avoiding walk is a model which is correct (=rather good) for polymers in a good solvent. Actually that it's self-avoiding doesn't make a big difference in that case. However random walk (and self-avoiding walk even less) does not fill space. The larger the molecule gets, the lower the space-filling. I'm rather convinced this is the same in any dimensionality. So, the question is: What is your question? – Karl Jun 20 at 0:47
• @Karl The mathematical model for self avoiding walks *constrained to lie in an $n \times n$ grid * is space filling at supercritical fugacity. My question is whether this is an empirical fact as well. – Lorenzo Jun 20 at 1:14
• Is that an artefact? Obviously a self-avoiding walk of $n^2$ steps must fill an $n \times n$ rectangle: But the last steps would be unnaturally (unphysically) constrained. – Karl Jun 20 at 1:19
• @Karl There is no constraint on the length of the walk, just that it is drawn from the distribution $\nu_x$. Not every such walk is space filling, it's just that they are "space filling" with very high probability. (And my question is whether $\nu_x$ is an empirically supported model.) – Lorenzo Jun 20 at 1:22