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Small‐Angle Scattering from Polymeric Mass Fractals of Arbitrary Mass‐Fractal Dimension
Author(s) -
Beaucage G.
Publication year - 1996
Publication title -
journal of applied crystallography
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.429
H-Index - 162
ISSN - 1600-5767
DOI - 10.1107/s0021889895011605
Subject(s) - fractal dimension , fractal , fractal derivative , scattering , physics , statistical physics , power law , debye , small angle scattering , integral equation , exponent , mathematical analysis , materials science , mathematics , condensed matter physics , optics , fractal analysis , statistics , linguistics , philosophy
The Debye equation for polymer coils describes scattering from a polymer chain that displays Gaussian statistics. Such a chain is a mass fractal of dimension 2 as evidenced by a power‐law decay of −2 in the scattering at intermediate q . At low q , near q ≃ 2 π / R g , the Debye equation describes an exponential decay. For polymer chains that are swollen or slightly collapsed, such as is due to good and poor solvent conditions, deviations from a mass‐fractal dimension of 2 are expected. A simple description of scattering from such systems is not possible using the approach of Debye. Integral descriptions have been derived. In this paper, asymptotic expansions of these integral forms are used to describe scattering in the power‐law regime. These approximations are used to constrain a unified equation for small‐angle scattering. A function suitable for data fitting is obtained that describes polymeric mass fractals of arbitrary mass‐fractal dimension. Moreover, this approach is extended to describe structural limits to mass‐fractal scaling at the persistence length. The unified equation can be substituted for the Debye equation in the RPA (random phase approximation) description of polymer blends when the mass‐fractal dimension of a polymer coil deviates from 2. It is also used to gain new insight into materials not conventionally thought of as polymers, such as nanoporous silica aerogels.