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## poly_gauss_coil

Scattering from polydisperse polymer coils

Parameter | Description | Units | Default value
--- | --- | --- | ---
scale | Scale factor or Volume fraction | 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

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

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. 117 Figure 1: 1D plot corresponding to the default parameters of the model.

Source

`poly_gauss_coil.py`

References

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

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

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

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

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