.. _rpa:
rpa
=======================================================
Random Phase Approximation
========== =============================== ======= =============
Parameter Description Units Default value
========== =============================== ======= =============
scale Scale factor or Volume fraction None 1
background Source background |cm^-1| 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 |Ang| 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 $\rho_{C/D} = [\rho_C - \rho_D]$\ :sup:`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).
.. figure:: img/rpa_autogenfig.png
1D plot corresponding to the default parameters of the model.
**Source**
:download:`rpa.py `
$\ \star\ $ :download:`rpa.c `
**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