Small-angle scattering behavior of thread-like and film-like systems

Film-like and thread-like systems are, respectively, defined by the property that one of the constituting homogenous phases has a constant thickness (δ) or a constant normal cross section (of largest chord δ). The stick probability function of this phase, in the limit δ → 0, naturally leads to the definition of the correlation function (CF) of a surface or of a curve. This CF closely approximates the generating stick probability function in the range of distances larger than δ. The surface and the curve CFs, respectively, behave as 1/r and as 1/r2 as r approaches zero. This result implies that the relevant small-angle scattering intensities behave as Ps/q2 or as Pc/q in the intermediate range of the scattering vector magnitude (q) and as P/q4 in the outermost q range. Similarly to P, pre-factors Ps and Pc simply depend on some structural parameters. Depending on the scale resolution it may happen that a given sample looks thread like at large scale, film like at small scale and particulate at a finer scale. An explicit example is reported. As a practical illustration of the above results, the surface and the curve CFs of some simple geometrical shapes have been explicitly evaluated. In particular, the CF of the right circular cylinder is evaluated. Its limits, as the height or the diameter the cylinder approaches zero, are shown to coincide with the CFs of a circle and of a linear segment, respectively.

Film-like and thread-like systems are, respectively, defined by the property that one of the constituting homogenous phases has a constant thickness (δ) or a constant normal cross section (of largest chord δ). The stick probability function of this phase, in the limit δ → 0, naturally leads to the definition of the correlation function (CF) of a surface or of a curve. This CF closely approximates the generating stick probability function in the range of distances larger than δ. The surface and the curve CFs, respectively, behave as 1/r and as 1/r2 as r approaches zero. This result implies that the relevant small-angle scattering intensities behave as PS}/q^2 or as P C/ q in the intermediate range of the scattering vector magnitude (q) and as l Pq^4 in the outermost q range. Similarly to P, pre-factors PS and PC simply depend on some structural parameters. Depending on the scale resolution it may happen that a given sample looks thread like at large scale, film like at small scale and particulate at a finer scale. An explicit example is reported. As a practical illustration of the above results, the surface and the curve CFs of some simple geometrical shapes have been explicitly evaluated. In particular, the CF of the right circular cylinder is evaluated. Its limits, as the height or the diameter the cylinder approaches zero, are shown to coincide with the CFs of a circle and of a linear segment, respectively.