The unified scattering function was proposed in 1995 as a universal approach to describe small-angle X-ray, and neutron scattering (and in some cases light scattering) from disordered systems that display hierarchical structure.
Concept The concept of universal descriptions of scattering, that is scattering functions that do not depend on a specific structural model, but whose parameters can be related back to specific structures, have existed since about 1950. The prominent examples of universal scattering functions are Guinier's Law,
and Porod's Law,
where G, Rg, and B are constants related to the scattering contrast, structural volume, surface area, and radius of gyration. q is the magnitude of the scattering vector which is related to the Bragg spacing, d, q = 2π/d = 4π/λ sin(θ/2). λ is the wavelength and θ is the scattering angle (2θ in diffraction). Both Guinier's Law and Porod's Law refer to an aspect of a single structural level. A structural level is composed of a size that can be expressed in Rg, and a structure as reflected in a power-law decay, -4 in the case of Porod's Law for solid objects with smooth, sharp interfaces. For other structures the power-law decay yields the mass-fractal dimension, df, which relates the mass and size of the object, thereby partially defining the object. For instance, a rod has df = 1 and a disk has df = 2. The prefactor to the power-law yields other details of the structure such as the surface to volume ratio for solid objects, the branch content for chain structures, the convolution or crumpled-ness of various objects. The prefactor to Guinier's Law yields the mass and volume fraction under dilute conditions. Above the overlap concentration (generally 1 to 5 volume percent) structural screening must be considered. In addition to these universal functions that describe only a part of a structural level, a number of scattering functions that can describe a single structural level have been proposed for some disordered systems, most interestingly Debye's scattering function for a Gaussian polymer chain derived during World War II,
where x = q2Rg2. Eq. 3 reverts to Eq. 1 at low-q and to a power-law, I(q) = Bq−2 at high-q reflecting the two dimensional nature of a random walk or a diffusion path. Eq. 3 refers to a single structural level, corresponding to a Guinier regime and a power-law regime. The Guinier regime reflecting the overall size of the object without reference to the internal or surface structure of the object and the power-law reflecting the details of the structure, in this case a linear (unbranched), mass-fractal object with mass-fractal dimension, df = 2 (connectivity dimension of 1 reflecting a linear structure; and minimum dimension of 2 indicating a random conformation in 3d space). In the 1990s it became apparent that single structural level functions similar to Eq. 3 would be of great use in describing complex, disordered structures such as branched mass-fractal aggregates, linear polymers in good solvents (df ~ 5/3), branched polymers (df > 2), cyclic polymers, and macromolecules of complex topology such as star, dendrimer, and comb polymers, as well as polyelectrolytes, micellar and colloidal materials such as worm-like micelles. Further, no analytically derived scattering functions could describe multiple structural levels in hierarchical materials. The observation of multiple structural levels is extremely common even in the case of a simple linear Gaussian polymer chain describe by Eq. 3 which is statistically composed of rod-like Kuhn units (level 1) which follow I(q) = Bq−1 at the highest-q. Common examples of hierarchical materials are silica, titania, and carbon black nano-aggregates composed of solid primary particles (level 1) displaying Porod scattering at highest q, Eq. 2, which aggregate into fairly rigid mass-fractal structures at intermediate nanoscales (level 2), and which agglomerate into micron-scale solid or network structures (level 3). Since these structural levels overlap in a small-angle scattering pattern, it was not possible to accurately model these materials using Eq. 1 and various power-law functions such as Eq. 2. For these reasons, a global scattering function that could be expanded to multiple structural levels was of interest. In 1995 Beaucage derived the Unified Scattering Function,
where "i" refers to the structural level starting with the smallest size, highest q. qi* is defined by,
and k has a value of 1 for solid structural levels (: 3 < P i {\displaystyle 3<P_{i}} ) and approximately 1.06 for mass-fractal structural levels (: 3 > P i {\displaystyle 3>P_{i}} ). Eq. 4 recognizes that all structures display the behavior of Eq. 1 at largest sizes, that is all structures exhibit a size, and if the structure is randomly arranged that size manifests as a Gaussian function in small-angle scattering governed by the radius of gyration with larger objects displaying a smaller standard deviation, or larger Rg. At high-q Eq. 1 fails to describe the structure because it reflects an object with no surface or internal structure [8]. The second term in Eq. 4 gives the missing information concerning the surface or internal structure of the object by way of the power Pi and the prefactor Bi (as well as how Pi and Bi relate to Gi, and Rg,i). Beaucage realized that the problem of obtaining a generic multi-level scattering function lay in Eq. 2 since a power-law could not extend infinitely to low-q and yield a finite intensity at q => 0. Also, such a function would over power Eq. 1 in the range of q where Eq. 1 is appropriate. Reference provides one of several possible derivations of Eq. 4, using Eq. 2 as an example of a power-law regime. A vector, r, can be visualized as the vector connecting interference points between an incident beam and the scattered beam. r = 2π/q where q = 4π/(λ sin θ/2) is the scattering vector in inverse space. Scattering occurs when two fringe points separated by r contain scattering material. If material is located at |r|/2 destructive interference occurs. So within a solid object there is always material at a position |r|/2 that negates scattering form material separated at |r|. Only at the surface do conditions of contrast occur.
… excerpt ends here. Continue reading the full article.
