# rpa

Random Phase Approximation

Parameter |
Description |
Units |
Default value |
---|---|---|---|

scale |
Scale factor or Volume fraction |
None |
1 |

background |
Source background |
cm |
0.001 |

case_num |
Component organization |
None |
1 |

N[4] |
Degree of polymerization |
None |
1000 |

Phi[4] |
volume fraction |
None |
0.25 |

v[4] |
molar volume |
mL/mol |
100 |

L[4] |
scattering length |
fm |
10 |

b[4] |
segment length |
Å |
5 |

K12 |
A:B interaction parameter |
None |
-0.0004 |

K13 |
A:C interaction parameter |
None |
-0.0004 |

K14 |
A:D interaction parameter |
None |
-0.0004 |

K23 |
B:C interaction parameter |
None |
-0.0004 |

K24 |
B:D interaction parameter |
None |
-0.0004 |

K34 |
C:D interaction parameter |
None |
-0.0004 |

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

Warning

This model is not functioning correctly in SasView and it
appears it has not done so for some time. Whilst the
problem is investigated, a workaround for Case 0 below
(the most common use case) is to use the binary_blend
model available on the Model Maketplace . For further
information, please email help@sasview.org . *The
SasView Developers. February 2022.*

**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:

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.

**Component D is assumed to be the “background” component (ie, all contrasts 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).

**Source**

**References**

A Z Akcasu, R Klein and B Hammouda,

*Macromolecules*, 26 (1993) 4136Hammouda,

*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