poly_mic_exv_disph

Parameter Descriptions
$scale$: polymer volume fraction (if I(q) on absolute scale)
$N_{agg}$: number of polymer chains in 1 micelle
$v_{core}$: volume of 1 core chain
$v_{corona}$: volume of 1 corona chain
$sld_{core}$: neutron scattering length density of micelle core polymer
$sld_{corona}$: neutron scattering length density of micelle corona polymer
$sld_{solvent}$: neutron scattering length density of solvent
$radius_{core}$: radius of the core
$s$: width of the corona profile/2, fitting parameter for the radial density profile
$\alpha$: shape parameter of the radial density profile; as $\alpha$ goes up, volume fraction at surface increases and decays more steeply
$t_{itf}$: Width of the core-corona interface
$\xi_{eff}$: effective blob size of polymer chains in the corona
$\nu$: excluded volume parameter (polymer-solvent interactions)
$v_{cc}$: effective chain-chain interaction parameter in the corona
$f_{core}$: volume fraction of solvent in the core

For block copolymer micelles, the small angle scattering tends to show features from the particles and the polymer chains on different length scales, via different regions in q. A fundamental framework for describing both the polymer and particle scattering was developed by Jan Skov Pedersen and colleagues. The model framework can be adapted for differences in the shape of the nanoparticle (e.g. spherical, ellipsoidal, cylindrical), the concentration profile of the corona (or shell), and the types of polymer physics describing the chain interactions. In this model, we describe a spherical micelle created from a diblock copolymer; the model specifically includes a defined, though malleable, density profile for the corona and can model both idealized chains (e.g. $\theta$ condition) and polymer chains with excluded volume interactions (inter-chain and intra-chain).

A schematic and other details of the theory can be found in the reference by Rehmann et al, which will be linked below after publication.

The scattering equation is:
\[I(q) = \frac{\phi_{poly}}{N_{agg} \cdot \ v_{poly,total}}\lbrack N_{agg}^{2} \cdot \beta_{core}^{2} \cdot \left( {\ A}_{core}\left( q,\ r_{core}\ \right) \cdot e^{\frac{{- q^{2} \cdot \left( \frac{t_{itf}}{4} \right)}^{2}}{2}} \right)^{2}\]
\[+ \ 2 \cdot N_{agg}^{2}{\cdot \beta}_{core}{{\cdot A}_{core}\left( q,\ r_{core} \right) \cdot \beta_{corona} \cdot A}_{corona}\left( q,\rho_{corona\ profile} \right)e^{- q^{2} \cdot \left( \frac{t_{itf}}{4} \right)^{2}}\]
\[+ \ N_{agg} \cdot \left( N_{agg} - P_{chain,eff}(q = 0) \right) \cdot \beta_{corona}^{2}\left( A_{corona}\left( q,\rho_{corona\ profile}\ \right) \cdot e^{\frac{{- q^{2} \cdot \left( \frac{t_{itf}}{4} \right)}^{2}}{2}} \right)^{2}\]
\[+ \ N_{agg} \cdot \beta_{corona}^{2} \cdot P_{chain,\ eff}\left( q,\nu,\ \xi_{eff},\ v_{cc} \right)\rbrack\]

Each of the terms is described below.

$\phi_{poly}$ is the volume fraction of the total polymer, usually described by scale in the SasView model if using absolute units, or incorporated into the scale if using arbitrary units.

$N_{agg}$ is the aggregation number or the number of polymer chains per one micelle. Note that this is not a fit parameter, but is calculated from the core radius ($r_{core}$), the volume fraction of solvent in the core ($f_{core}$), and volume of the core polymer ($v_{poly,core}$) using:
\[N_{agg} = \frac{4}{3}\pi\cdot\frac{r^3_{core}}{v_{poly,core}}\cdot (1 - f_{core})\]

$v_{poly,total}$ is the volume of a single polymer chain, generally calculated from the volume of each block (e.g. = $v_{poly,core} + v_{poly,\ corona}$). An estimate of this volume can be calculated using $v_{poly,i} = M_{i} \cdot 10^{24}/(\rho_{m,i} \cdot N_{Avo})$, in which $M_i$ is the molar mass and $\rho_{m,i}$ is the mass density of the polymer block. These volumes are set parameters in the SasView code and should not be fit, like the SLDs.

$\beta_i$ is the excess scattering length where $i$ denotes either the core or corona and is defined as $\beta_i =v_i\cdot(\rho_{SLD,i} -\rho_{SLD,0})$. $\rho_{SLD,0}$ is the solvent scattering length density.

$A_{core}$ is the amplitude of the "shape function" $F(q)$ for the core. Since we are assuming this is a spherical particle, it is:
\[A_{core} = \ 3 \cdot \frac{j_{1}\left( q \cdot r_{core} \right)}{q \cdot r_{core}}\]

$t_{itf}$ is the thickness of the interfacial region between the core and corona; the exponential function allows for a smooth decay between the scattering length density profiles of the core and the corona. For most systems, this term is close to 0 and a reasonable starting case is to set the parameter to 0.

$A_{corona}$ is the amplitude of the "shape function" ($F(q)$) for the corona. It describes the scattering from the particle aspects of the corona. It is defined as:

\[A_{corona}\left( q,\rho_{corona\ profile}\ \right) = \frac{4 \cdot \pi\int_{}^{}{\rho_{corona,\ profile}(r) \cdot j_{0}(q \cdot r) \cdot r^{2}dr\ \ }}{4 \cdot \pi\int_{}^{}{\rho_{corona,\ profile}(r) \cdot r^{2}dr}}\]

The corona profile can be whatever density profile one wants, but in this specific code, we use the combination of 2 cubic b-splines below:
\[\rho_{corona,\ \ profile} = (1 - \alpha) \cdot \rho_{1}\left( r,\ r_{core},\ s \right) + \alpha \cdot \rho_{2}(r,\ r_{core},\ s)\]
for \(r_{core} \leq r < r_{core} + s\):
\[\rho_{1}(r) = \frac{4\left( r - \left( r_{core} + s \right) \right)^{3}\ - \left( r\ - \left( r_{core} + 2s \right) \right)^{3}}{4s^{3}}\]
\[\rho_{2}(r) = \frac{- \left( r - \left( r_{core} + s \right) \right)^{3}\ }{4s^{3}}\]
For \(r_{core} + s \leq r < r_{core} + 2s\):
\[\rho_{1}(r) = \frac{- \left( r\ - \left( r_{core} + 2s \right) \right)^{3}}{4s^{3}}\]
Elsewhere:
\[\rho_{1}(r) = 0\]
\[\rho_{2}(r) = 0\]
Note that the profile ends at $r_{core} + 2s$, the thickness of the corona is $2s$, and $\alpha$ is associated with steepness of the decay of the volume fraction profile.

$P_{chain,eff}$ is the form factor of the effective polymer chain. This describes the scattering of the polymer chains at high q. It is calculated with:

\[P_{chain,eff} = \dfrac{P_{exv}}{1 + v_{cc}\cdot P_{exv}} \]

$v_{cc}$ describes the effective concentration of chains in the corona. It allows the chain scattering to be modulated as if the chain is in a semi-dilute solution; as concentration increases, $v_{cc}$ increases, and the chain scattering contributes less to the polymer micelle scattering. To model dilute chain scattering, simply set $v_{cc}= 0$.

The excluded volume polymer chain scattering has two parameters which describe it: \(\nu\) is the excluded volume coefficient and \(\xi_{eff}\) is the effective size of the polymer. This scattering function is similar to the $excluded\_volume\_polymer$ model in SasView, although the effective polymer size (a diameter) is used here instead of the radius of gyration of the polymer chain (more details in the reference).

\[P_{exv}(q) = \frac{1}{\nu \cdot U^{\ \frac{1}{2} \cdot \nu}\ \ }\left\lbrack \gamma\left( \frac{1}{2\nu},\ U \right) - \frac{1}{U^{\ \frac{1}{2} \cdot \nu}} \cdot \gamma\left( \frac{1}{\nu},\ U \right) \right\rbrack\]
\[\gamma(x,\ U) = \int_{0}^{U}{e^{- t} \cdot t^{\ x - 1}\ dt\ }\ \]
\[U = q^{2} \cdot \left( \frac{\xi_{eff}}{2} \right)^{2} \cdot \frac{(2\nu + 1)(2\nu + 2)}{6} = q^{2} \cdot \frac{b_{k}^{2}n_{k}^{2 \cdot \nu}}{6}\]

in which \(\gamma(x,\ U)\) is the incomplete $\gamma$ function (defined in the interval above); U is a scattering variable defined as either a function of the effective polymer size (\(\xi_{eff}\)) and \(\nu\) or as a function of \(\nu\), the Kuhn length $b_k$, and the number of Kuhn segments $n_k$.
$P_{exv}(q)$ in this form can describe different solvent conditions through $\nu$: ideal solvents (e.g. $\theta$ condition) are $\nu = \frac{1}{2}$; poor solvents are $\nu = \frac{1}{3}$, and good solvents are described when $\nu = \frac{3}{5}$. This describes the scattering at high q as $I(q) \propto q^{-1/\nu}$. To model an ideal chain, simply set \(\nu = \frac{1}{2}\) to recapitulate the Deybe scattering equation for polymers with Gaussian chain statistics. This model allows one to fit the value for \(\nu\), but there tends to be issues with correlations between \(\nu,\ \xi_{eff}\), and the background. We recommend choosing $\nu$ and not fitting it.

Additional Practical Considerations

Due to the number of parameters and other considerations, we strongly suggest using this model with the advanced DREAM algorithm. In doing so, we suggest the following initialization parameters:

samples >= 1.5e6
this can be a little less if you use the burn feature, which we generally set to 1e4. The aim is to have > 1e6 samples after the convergence criteria is met.

initializer = lhs
i.e. latin hyper square, which uses the parameter range you defined for the fit to choose randomized starting parameters. Be sure to set the ranges for fit parameters to reasonable values.

One should also take a careful look at the parameter traces after the fit: there may be outlier chains in the result that require trimming after the fact or more than one set of parameters with similar likelihoods (see the reference for more information about bifurcation with this model). These features are easier to access with the sasmodels program than in the SasView application GUI, where one can re-set the fit, use burn after the fact, or select a subset of results.

References
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1. K. Rehmann and K. Weigandt *Generalized Form Factor Model for Block Copolymer Micelles* (2026) submitted.
2. J.S. Pedersen; C. Svaneborg; K. Almdal; I.W. Hamley; R.N. Young, *A Small-Angle Neutron and X-ray Contrast Variation Scattering Study of the Structure of Block Copolymer Micelles: Corona Shape and Excluded Interactions* Macromolecules 2003 36 (2), 416-433 DOI: 10.1021/ma0204913
3. B. Hammouda *SANS from homogeneous polymer mixtures: a unified overview.* chapter in: "Polymer Characteristics. Advances in Polymer Science," (1993) https://doi.org/10.1007/BFb0025862
4. C. Svaneborg and J.S. Pedersen *Block copolymer micelle coronas as quasi-two-dimensional dilute of semi-dilute polymer solutions* Physical Review E Vol 54 (2001)
5. J.S. Pedersen and M.C. Gerstenberg *The structure of P85 Pluronic block copolymer micelles determined by small-angle neutron scattering* Colloids and Surfaces A (2003)

Authorship and Verification
----------------------------

* **Author:** Kelsi M.S. Rehmann
* **Last Modified by: Kelsi M.S. Rehmann
* **Last Reviewed by:** Reviewer Name Here **Date:** Date Here