1.7.22. 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. and by Hammouda 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 and 39 of Boualem Hammouda’s ‘SANS Toolbox’.

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

References

A Z Akcasu, R Klein and B Hammouda, Macromolecules, 26 (1993) 4136.

  1. 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