10.1.4: Polymer Chain Morphology

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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.

How does the random walk model compare to reality?

By the nature of a random walk, a model molecule may overlap itself. Real polymers have a finite volume, so a molecule cannot ‘crash into’ itself or other chains.
The random walk model does not take account of any complicating forces, such as the interaction of electrons in bulky side groups, which tend to inhibit bond rotation.
Atoms in the polymer backbone are not freely jointed.
As a result of these simplifications, performing a random walk where each segment is a single C-C bond gives an underestimate of the end-to-end distance, real polymer chains are stiffer than predicted by the model.
To take account of this, random walk segments are modelled as being several C-C bonds in length.We can then use a quantity called the Kuhn length, l, to represent the average length assigned to a model segment.
The Kuhn length varies for different polymers: it is longer for a stiffer molecule.
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

Calculating the root mean square end-to-end distance of a random walk ‘molecule’

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,

You can now test this model using the simulation below.