poly_gauss_coil

Scattering from polydisperse polymer coils

Parameter	Description	Units	Default value
scale	Source intensity	None	1
background	Source background	cm-1	0.001
i_zero	Intensity at q=0	cm-1	70
rg	Radius of gyration	Å	75
polydispersity	Polymer Mw/Mn	None	2

The returned value is scaled to units of cm^-1 sr^-1, absolute scale.

This empirical model describes the scattering from polydisperse polymer chains in theta solvents or polymer melts, assuming a Schulz-Zimm type molecular weight distribution.

To describe the scattering from monodisperse polymer chains, see the mono_gauss_coil model.

Definition

\[I(q) = \text{scale} \cdot I_0 \cdot P(q) + \text{background}\]

where

\[ \begin{align}\begin{aligned}I_0 &= \phi_\text{poly} \cdot V \cdot (\rho_\text{poly}-\rho_\text{solv})^2\\P(q) &= 2 [(1 + UZ)^{-1/U} + Z - 1] / [(1 + U) Z^2]\\Z &= [(q R_g)^2] / (1 + 2U)\\U &= (Mw / Mn) - 1 = \text{polydispersity ratio} - 1\\V &= M / (N_A \delta)\end{aligned}\end{align} \]

Here, \(\phi_\text{poly}\), is the volume fraction of polymer, \(V\) is the volume of a polymer coil, \(M\) is the molecular weight of the polymer, \(N_A\) is Avogadro’s Number, \(\delta\) is the bulk density of the polymer, \(\rho_\text{poly}\) is the sld of the polymer, \(\rho_\text{solv}\) is the sld of the solvent, and \(R_g\) is the radius of gyration of the polymer coil.

The 2D scattering intensity is calculated in the same way as the 1D, but where the \(q\) vector is redefined as

\[q = \sqrt{q_x^2 + q_y^2}\]

Fig. 98 1D plot corresponding to the default parameters of the model.

References

O Glatter and O Kratky (editors), Small Angle X-ray Scattering, Academic Press, (1982) Page 404.

J S Higgins, H C Benoit, Polymers and Neutron Scattering, Oxford Science Publications, (1996).

S M King, Small Angle Neutron Scattering in Modern Techniques for Polymer Characterisation, Wiley, (1999).

http://www.ncnr.nist.gov/staff/hammouda/distance_learning/chapter_28.pdf