[wereworm73]

If you add a polyene to BF₃ and no nucleophiles are present, would any cycloadditions occur along with polymerization reactions?

When the carbocation forms, it could attack a double bond from a neighboring polyene molecule & polymerize.  Or...could the carbocationic polyene also react with one of its own double bonds and cause a cycloaddition reaction (especially if the concentration of polyenes is very low)?   And if the polyene was highly conjugated, like beta-carotene, could cycloadditions still occur to form three-membered rings?

[Dude]

In a nutshell, yes to all of the above.

In George Odian's book "Principles of Polymerizaiton", there was a section that dealt with cationic initiators.  The jist of some work done by Joe Kennedy was that you can never actually dry solvents well enough to provide a nucleophile-free system.  Even in a dry-box, there will always be sufficient water to complex with BF3 and initiate polymerization.

There is a book entitled "Hydrocarbon Resins".  The most common monomer used to make hydrocarbon tackifier resins is 1,3-pentadiene (piperylene is a common name).  In the "Hydrocarbon Resins" book, the authors speculate that cyclization does occur.  Instead of the perhaps expected 1,4-addition reaction, the authors stated that terpene type structures form (ie 6-membered rings).  A few years back I recall a Macromolecules journal article by Macedo dealing with quantifying the amount of cyclization by IR.  A significant gel content is obtained with many polymerizations of 1,3-pentadiene unless very specific conditions are adhered to (indicating a multitude of side reactions including crosslinking).  Some of the side reactions can be reduced by polymerizing at lower temperatures.   Not familiar with any work done with beta-carotene.  Much of the work on 1,3-pentadiene polymerization was also done with AlCl3 due to cost and toxicity issues. 

[wereworm73]

Wow, now that's got to be one crazy polymer. With all those different side reactions, you'd probably get something different every time you make it.

I wonder if palladium or platinum absorbing the 1,3-butadiene molecules would orient them in a way that favors one type of reaction over the other.  Maybe Pd or Pt plus low temperatures would discourage the side-reactions even further?  Or if something very porous was used instead, could it work the other way around, perhaps favoring intra-molecular cycloadditions (because the butadienes can't get to each other as easily since they'd be inside little holes)?