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16.4: 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.

    This page titled 16.4: Polymer Chain Morphology is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Dissemination of IT for the Promotion of Materials Science (DoITPoMS).

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