Fundamental properties of a Gaussian chain

It is proportional to the number n of repeat units, and hence the molecular weight of the polymer. The tension-elongation relation (1.29) of the Gaussian chain gives the free energy 

By expanding the Laplace transformed partition function (1.14) in powers of the dimensionless tension, we find 

Because the energy of orientation measured from the reference direction parallel to the end vector is /I R// = /a cos 0, the orientational distribution function of the bond vector is proportional to exp[/acos0, / B7 ]. Because the tension is related to the end-to-end distance by (1.28b), the orientational distribution under a fixed R is given by the probability 

The orientational order parameter of the chain is then defined by 

We have seen that a chain has a Gaussian property irrespective of the details of the model employed when the number n of the repeat units is large. This is a typical example of the central limit theorem in probability theory. 

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