Back in 2009 I was asked to make ultra-narrow linear poly(ethylenimine) as part of my Ph.D.. Previously we’d been working with broader materials (with Ɖ = 1.25 at a degree of polymerisation of 500, already far better than our competitors), and assaying the effects of partial deacylation on toxicity. I will never publish that work, and so can simply say even moderate degrees of acylation made the polymer non-toxic, but transfection efficiency drops through the floor. Why is an interesting question as the polymers still complex DNA; a thorough understanding of the mechanism of polymer mediated transfection is necessary (and I may get round to submitting a manuscript on it).
Now we never got quite to the theoretical dispersity, but my efforts were submitted to Macromolecules and published in 2015. The theoretical dispersity is the Poisson distribution, and for DP = 500 the Ɖ value (Mm/Mn) would be 1.002 and the distribution of species would be according to this figure:
Figure 1: Theoretical molecular weight distribution of a DP = 500 polymer
For this DP my dispersities were higher, and my best value was ca. Ɖ = 1.05 due to a side-reaction. However, the above is absolutely the best that can be done by standard synthesis, and few polymerisation systems achieve anything like it. Narrower polymers are achievable via sequential deprotection-addition steps (like in a peptide synthesiser) or fractionation of already narrow polymers.