.. _rpa: 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 |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. **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. [#Akcasu]_ and by Hammouda [#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'[#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: * 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 = [SLD(component C) - SLD(component 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\ :sub:`a`, b\ :sub:`b`, etc) and $\chi$ parameters (K\ :sub:`ab`, K\ :sub:`ac`, etc). .. figure:: img/rpa_autogenfig.png 1D plot corresponding to the default parameters of the model. **References** .. [#Akcasu] A Z Akcasu, R Klein and B Hammouda, *Macromolecules*, 26 (1993) 4136. .. [#Hammouda] B. Hammouda, *Advances in Polymer Science* 106 (1993) 87. .. [#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:** Paul Butler **Date:** March 12, 2017