Polymer chain morphology
A single polymer chain can exist in any one of its possible conformations, from a tight coil to a straight chain. The probability of it having a particular end-to-end distance increases with the number of possible conformations that would achieve that size. There is only one possible conformation that will produce a straight chain, but as the molecule becomes more coiled the number of possibilities increases. A polymer chain will therefore tend to coil up to some extent.
The expected end-to-end distance of a chain can be estimated using a model in which a molecule is considered as being made up of a large number n of segments. Each segment is rigid, but is freely jointed at both ends, so that it can make any angle with the next segment. A model ‘molecule’ can then be built by adding each of the successive segments at a random angle, a procedure called a random walk.
To illustrate this, here are some example Kuhn lengths (expressed as a multiple of the length of a C-C bond).
Polymer | Kuhn length / C-C bond lengths | Notes |
Poly(ethene) | 3.5 | PE is very flexible (due to low torsional barriers) |
Poly(styrene) | 5 | PS has large side-groups which inhibit flexibility |
DNA | 300 | DNA is very stiff due to its double helix structure |
In two dimensions, we can estimate the distance from end to end of a molecule modelled by a random walk, given the Kuhn length and the number of segments. Each segment is represented by a vector,
Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.
You can now test this model using the simulation below.
Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.
Although this is a two-dimensional model, extending it to three dimensions gives the same result.