Mono Gauss Coil

#mono_gauss_coil model #conversion of DebyeModel.py #converted by Steve King, Mar 2016 This Debye Gaussian coil model strictly describes the scattering from *monodisperse* polymer chains in theta solvents or polymer melts, conditions under which the distances between segments follow a Gaussian distribution. Provided the number of segments is large (ie, high molecular weight polymers) the single-chain form factor P(Q) is that described by Debye (1947).

To describe the scattering from *polydisperse* polymer chains see the `poly-gauss-coil` model.

Definition

$$ I(q) = \text{scale} \cdot I_0 \cdot P(q) + \text{background}
$$
where

$$ I_0 = \phi_\text{poly} \cdot V \cdot (\rho_\text{poly} - \rho_\text{solv})^2 \\ P(q) = 2 [\exp(-Z) + Z - 1] / Z^2 \\ Z = (q R_g)^2 \\ V = M / (N_A \delta)
$$
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

#. P Debye, *J. Phys. Colloid. Chem.*, 51 (1947) 18. #. R J Roe, *Methods of X-Ray and Neutron Scattering in Polymer Science*, Oxford University Press, New York (2000). #. http://www.ncnr.nist.gov/staff/hammouda/distance_learning/chapter_28.pdf

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