Core-Chain-Chain (CCC) Model


This form factor describes scattering from spherical cores (nanoparticle, micellar, etc.) that have chains coming off normal from their surface. In the case of
the Core-Chain-Chain (CCC) Model, these chains have two different regions of conformation, size, and scattering length density. The transition from region 1 (near
the nanoparticle core, CPB) to region 2 (SDPB) is given by the parameter $r_{c}$, which is the distance from the center of the nanoparticle to the junction point. See
fig. 1 of reference 1 for a schematic of the geometry.

The scattering intensity is the sum of 8 terms, and an incoherent background term $B$. This model returns the following scattering intensity:
I(q) = \frac{scale}{(V_{core} + N_{c}V_{c})} \times \left[ P_{core}(q) + N_{c}P_{CPB}(q) + N_{c}P_{SDPB}(q) + 2N_{c}F_{core}(q)j_{0}(qr_{core})F_{CPB}(q) + 2N_{c}F_{core}(q)j_{0}(qr_{c})F_{SDPB}(q) + N_{c}(N_{c} - 1)F_{CPB}(q)^{2}j_{0}(qr_{core})^2 + N_{c}(N_{c} - 1)F_{SDPB}(q)^{2}j_{0}(qr_{c})^2 + N_{c}^{2}F_{CPB}(q)j_{0}(qr_{core})j_{0}(qr_{c})F_{SDPB}(q) \right] + B

where $N_{c}$ is the number of chains grafted to the nanoparticle, $r_{core}$ is the nanoparticle radius, and $r_{c}$ is the position of the junction between the CPB and
SDPB regions. The terms $`P_{core}(q)$, $P_{CPB}(q)$, and $P_{SDPB}(q)$ are the form factors of the spherical core, polymer in the CPB region, and polymer in the SDPB
region, respectively. $F_{core}(q)$, $F_{CPB}(q)$, and $F_{SDPB}(q)$ are the form factor amplitudes of the respective regions. For the nanoparticle core, these terms are related as:

P_{core}(q) = \left| F_{core}(q) \right|^{2} = \left| V_{core}(\rho_{core} - \rho_{solv})\frac{3j_{1}(qr_{core})}{qr_{core}} \right|^{2}
where $\rho_{core}$ is the scattering length density of the nanoparticle core, $\rho_{solv}$ is the scattering length density of the solvent/matrix, $V_{core}$ is the
volume of the nanoparticle core, and $j_{1}(.)$ is a spherical Bessel function.

Scattering from the CPB and SDPB regions is described by the form factor amplitudes and form factors of these regions. For a given region $i$, these terms read:

P_{i}(q,N_{i}) = V_{i}^{2}(\rho_{i} - \rho_{solv})^{2} \left[ \frac{1}{\nu_{i} U_{i}^{1/2\nu_{i}}}\gamma \left( \frac{1}{2\nu_{i}}, U_{i}\right) - \frac{1}{\nu_{i} U_{i}^{1/\nu_{i}}}\gamma \left(\frac{1}{\nu_{i}}, U_{i} \right) \right]
F_{i}(q,N_{i}) = V_{i}(\rho_{i} - \rho_{solv})\frac{1}{2\nu_{i} U_{i}^{1/2\nu_{i}}}\gamma \left( \frac{1}{2\nu_{i}}, U_{i}\right)

where $N_{i}$ is the degree of polymerization of the portion of polymer in region $i$, $V_{i}$ is the volume of polymer in region $i$, $\nu_{i}$ is the excluded volume parameter of the polymer in region $i$, $\rho_{i}$ is the scattering length density of polymer in region $i$, $U_{i} = q^{2} b^{2} N_{i}^{2\nu_{i}}/6$, $b$ is the polymer's Kuhn length, and $\gamma$ is the lower incomplete gamma function.

Example Data:


Created By mjahore
Uploaded Aug. 23, 2018, 12:09 a.m.
Category Sphere
Score 0
Verified This model has not been verified by a member of the SasView team
In Library This model is not currently included in the SasView library. You must download the files and install it yourself.


No comments yet.

Please log in to add a comment.