Interpretation of small-angle scattering curves proportional to a negative power of the scattering vector

577

376

The concepts of dilation symmetry and the fractal dimension are introduced, and from these basic concepts scattering functions are computed for surface and mass fractals. It is then shown how the fractal structure of various random media has been elucidated from scattering measurements, and how these observations relate to specific models of fractal geometry. Examples include colloidal aggregates, gels, soot and the fractal porosity of rocks of various sorts.

Section: Surface and Pore Fractalsmentioning

confidence: 99%

“…As a final, historical note, it should be pointed out that an early nonfractal interpretation of nonintegral scattering exponents was put forth by Schmidt (1982), who realized that a power-law distribution of spheres, platelets or rods could give rise to scattering functions with fractional exponents.…”

mentioning

confidence: 99%

Scattering from fractals

Martin¹,

Hurd²

1987

577

376

“…About four years earlier (Schmidt, 1982), I calculated the intensity of the small-angle scattering from independently scattering randomly oriented polydisperse systems of non-fractal scatterers in which the distribution of the diameters was proportional to a power law (in 1982 I had not yet heard about fractals). In this calculation, the diameter distribution p(s c) was defined so that p(~)d~: was equal to the probability that the diameter s c of a scatterer had a value between ~ and s c + d~: [thus p(~) is proportional to the number distribution of the diameters of the scatterers].…”

Section: Polydisperse Systems Of Mass and Surface Fractalsmentioning

confidence: 99%

Small-angle scattering studies of disordered, porous and fractal systems

Schmidt¹

1991

Self Cite

578

390

Small-angle X-ray and neutron scattering are important techniques for studying the structure of fractals and other disordered systems on a scale of lengths from about 10 to 2000 A. This review begins with a brief outline of some properties of fractals. The small-angle scattering from fractal systems is then discussed and the effect of polydispersity is considered. The intensity of small-angle scattering from fractals and other disordered systems is often proportional to a negative power of the quantity q = 4~,t-Isin(0/2), where 0 is the scattering angle and ,~ is the X-ray or neutron wavelength. From the magnitude of the exponent that describes this type of scattering, which is often called power-law scattering, much important information can be obtained. Some situations in which power-law scattering can be expected are described. To illustrate the scattering from fractals and disordered systems, several experimental investigations of mass-fractal silicas and porous solids are reviewed and some calculations of the small-angle scattering from model fractal systems are outlined.

“…At high q, the substructure to the mass-fractal morphology limits the mass-fractal power law. Often, the substructure is a threedimensional level with a power law described by Porod's law or surface-fractal scaling laws (Schmidt, 1992), in contrast to the one-dimensional rod substructure conventionally used for organic polymers.…”

Section: Introductionmentioning

confidence: 99%

“…The application of mass scaling laws to scattering from disordered materials has led to an understanding of weak power-law decays in terms of mass-fractal morphologies (Schmidt, 1992;Korberstein, Morra & Stein, 1980;Debye, Henderson & Brumberger, 1957;Fisher & Burford, 1967). These power-law decays display power-law slopes shallower than -3.…”

Section: Introductionmentioning

confidence: 99%

Small-Angle Scattering from Polymeric Mass Fractals of Arbitrary Mass-Fractal Dimension

Beaucage¹

1996

931

891

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 ~_ 2rc/Rg, 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 massfractal 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.