Rpa

Definition

Calculates the macroscopic scattering intensity for a multi-component homogeneous mixture of polymers using the Random Phase Approximation. This general formalism contains 10 specific cases

Case 0: C/D binary mixture of homopolymers

Case 1: C-D diblock copolymer

Case 2: B/C/D ternary mixture of homopolymers

Case 3: C/C-D mixture of a homopolymer B and a diblock copolymer C-D

Case 4: B-C-D triblock copolymer

Case 5: A/B/C/D quaternary mixture of homopolymers

Case 6: A/B/C-D mixture of two homopolymers A/B and a diblock C-D

Case 7: A/B-C-D mixture of a homopolymer A and a triblock B-C-D

Case 8: A-B/C-D mixture of two diblock copolymers A-B and C-D

Case 9: A-B-C-D tetra-block copolymer

.. note:: These case numbers are different from those in the NIST SANS package!

The models are based on the papers by Akcasu *et al.* [1] and by Hammouda [2] assuming the polymer follows Gaussian statistics such that $R_g^2 = n b^2/6$ where $b$ is the statistical segment length and $n$ is the number of statistical segment lengths. A nice tutorial on how these are constructed and implemented can be found in chapters 28, 31 and 34, and Part H, of Hammouda's 'SANS Toolbox' [3].

In brief, the macroscopic cross sections are derived from the general forms for homopolymer scattering and the multiblock cross-terms while the inter, polymer cross terms are described in the usual way by the $\chi$ parameter.

USAGE NOTES:

**Component D is assumed to be the "background" component (ie, all contrasts
* Only one case can be used at any one time. * The RPA (mean field) formalism only applies only when the multicomponent polymer mixture is in the homogeneous mixed-phase region. are calculated with respect to component D).** So the scattering contrast for a C/D blend $\rho_{C/D} = [\rho_C - \rho_D]$`2`. * Depending on which case is being used, the number of fitting parameters can vary.

.. Note:: * In general the degrees of polymerization, the volume fractions, the molar volumes, and the neutron scattering lengths for each component are obtained from other methods and held fixed while The *scale* parameter should be held equal to unity. * The variables are normally the segment lengths ($b_a$, $b_b$, etc.) and $\chi$ parameters ($K_{ab}$, $K_{ac}$, etc).

References

A Z Akcasu, R Klein and B Hammouda, *Macromolecules*, 26 (1993) 4136
B. Hammouda, *Advances in Polymer Science* 106 (1993) 87
B. Hammouda, *SANS Toolbox* https://www.ncnr.nist.gov/staff/hammouda/the_sans_toolbox.pdf.

Authorship and Verification

**Author:** Boualem Hammouda - NIST IGOR/DANSE **Date:** pre 2010
**Converted to sasmodels by:** Paul Kienzle **Date:** July 18, 2016
**Last Modified by:** Paul Butler **Date:** March 12, 2017
**Last Reviewed by:** Steve King **Date:** March 27, 2019