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 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

$q = \sqrt{q_x^2 + q_y^2}$

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