Random Walks, Polymer Chains and Membranes

The random walk is perhaps the simplest example of a physical fractal, as well as being one of the most fertile subjects of study in physics. Consider a walker 

When it contains an infinite number of steps, the random walk becomes the simplest model available for describing linear polymers. The step then repre- 

However, the Df — 2 random walk fails in one important respect to produce a correct model of the coil formed by a linear polymer soaking in a good solvent (see Chap. 7 on polymers) there is nothing to stop the walker from coming back through a previously visited site. In a non-crosslinked molecule whose constituent monomers repel each other by hard core interactions, this would clearly be impossible. The structure of such a molecule could only be correctly described by a self-avoiding random walk, rather than the simple 

Putting together the energy and entropy terms and minimising with respect to R, we find that the radius of an A-unit polymer varies as 

The fractal exponent is thus significantly smaller than that of a simple random walk, and the structure of the coil is considerably more open. This result is in good agreement with experiment, as shown by the example of polystyrene dissolved in benzene, at sufficiently low concentrations that the polymers cannot penetrate each other (which would totally change the type of solution). 

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