rpa

Random Phase Approximation

Parameter Description Units Default value
scale Source intensity 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 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.

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:

../_images/rpa_autogenfig.png

Fig. 115 Figure 1: 1D plot corresponding to the default parameters of the model.

Source

rpa.py \(\ \star\ \) rpa.c

References

[1]A Z Akcasu, R Klein and B Hammouda, Macromolecules, 26 (1993) 4136
[2]
  1. Hammouda, Advances in Polymer Science 106 (1993) 87
[3]
  1. Hammouda, SANS Toolbox https://www.ncnr.nist.gov/staff/hammouda/the_sans_toolbox.pdf.

Authorship and Verification