Particle form factor

This internal interference is described by the particle form factor, P. It allows the direct calculation of the radius of gyration, of the macromolecules... 

The above derivation can also be applied to colloidal or polymer-based liquids and is then used to calculate the so-called form factors of soft matter samples. The major difference between a monoatomic liquid and a polymer chain in the melt or in solution is that the total structure factor consists of two parts. The first is the inter-particle structure factor and the second the intra-particle structure factor. This second part is also often called the particle form factor P(q). Using Eq. (2.38) it is straightforward to calculate P(q) for a given soft matter sample. A good example is the form factor of a single polymer coil in a melt [88, 92]. The pair correlation function of such a coil is given by... 

Here we consider how to obtain the single-particle form factor even when labeled molecules are present at a high concentration. For this purpose let us first consider a single-component, bulk polymer, consisting of N molecules in volume Vy each molecule with Z = v/vu segments. The amplitude of scattering from such a polymer is... 

In summary, the diflBculties in determining aggregate form factors, particle form factors, phase shifts, and distribution functions combine to make the Rayleigh-Debye approach too complicated for practical application. On the other hand, the coalesced-sphere approach using the Jobst approximations to the Mie scattering eflBciencies allows rapid correlation of turbidity with particle size distributions. Consequently, a coalesced-sphere approach was adopted for experimentation in the E. coli-PEI system. 

F(q,d) is the particle form factor dependent on the particle size d and the scattering vector q... 

Fig. 3. Single particle form factor scattering for regular particles.

Fig, 16. Calculated single particle form factors for spherical styrene domains of radius 100 A. 

Figure 8.2 Single particle form factors for a Gaussian coil and a rigid rod of the same radius of gyration plotted as Kratky plots.

In Equation 11.17, A/ is the scatterer s molar mass, P(q) is the particle form factor, which sums the phase differences over all scattering centers of the particle. and A are the second and third virial coefficients, respectively, and the magnitude of the scattering vector, q, is given by... 

For non-interacting spheres D pp = D independent of q. The product of the two experimentally accessible quantities I(q) Dapp(q) exhibits the q-dependence of the particle form factor... 

A particularly recurrent shape for filler particles is the spherical one. The particle form factor P(q) for a spherical particle of radius tq and contrast Ap can be written as [112,120]... 

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