Structure function and screening length

The structure function of a homogeneous solution is related to the density-density fluctuations of monomers in the solution. However, the definition of the concentration is model-dependent. On a lattice, the monomer concentration is the number cp of monomer per site. For the standard continuous model, it is expressed as an area per unit volume, which is denoted by . For a Kuhnian chain, the quantity = CA 1/V represents the monomer concentration. However, the definition of the structure function should not really depend on the model under consideration, and therefore we shall define this quantity in an intrinsic manner. 

In the following, the structure function is defined in the framework of the standard continuous model but generalizing to other models is trivial. For the standard continuous model, the area concentration (r) plays the role of 

If the point of position vector r belongs to a large volume, the average is a constant 

The structure function H(q) plays a very important role because it is directly measurable by radiation scattering, as was indicated in Chapter 7. We shall now examine its analytical properties, and this study will enable us to give a rigorous definition of the screening length e. 

let us consider the elementary situation where the grand ensemble is made of independent Brownian chains (b = 0 in the standard continuous model). Then, it is conspicuous that the only correlations that exist and contribute to H(q) are those corresponding to points on the same chain. In this case, for a monodisperse ensemble, we have 

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