## Random Walks

## Kuhn Lengths

The coil size of a Gaussian chain (e.g. *R*_{N} and *R*_{g}) scales quadratically with the Kuhn length. The stiffer the chain (a larger Kuhn length), the larger the coil size. To demonstrate this in class, we constructed oligomers with *N* = 10, i.e. 10 monomer bond lengths, so 11 monomers. Holding *N* constant, we varied the Kuhn length by selecting different materials to represent the bond lengths.

The bead and string oligomer is somewhat misleading as a single Kuhn length comprises the bead plus the string, which clearly has internal flexibility. A better rendition would be an unknotted string of beads in which each bead represented one Kuhn length. Nevertheless, this model shows the flexibility of a chain with a small Kuhn length. The paperclip model has a slightly longer Kuhn length, followed by the toothpick model. It is debatable whether the freely-jointed chains exhibit true random walk behaviour, as both chains appear to be somewhat extended. It is generally known that the statistics of small numbers favours outliers, and extended chains are definitely an outlier, but clearly more probably when *N* is small. The shish kebab coil is quite large and covers the desk because the Kuhn length is large compared to the previous models. The pipe cleaner model sported by Qianshuo features the largerst Kuhn length of the models constructed and the coil is clearly quite space-filling. Unfortunately, the polymer seems to have undergone branching during synthesis, likely due to impurities in the reactor or poorly controlled free-radical synthesis (e.g. backbiting.)