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General expressions for the scattering functions of polydisperse particles with an arbitrary number of concentric shells of lamellar, cylindrical, or spherical symmetry are derived. For shells consisting of dense polymer layers or brushes with algebraic density profiles, φ( r ) ∼ r α, it is possible to obtain closed analytical expressions by taking advantage of the mathematical properties of hypergeometric functions. The same formalism can be employed to derive a closed expression for the form factor of polydisperse excluded volume chains assuming a des Cloizeaux-type segment distribution function.
Förster et al. (Thu,) studied this question.