Scattering from an adsorbed layer on particles

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

None

1

background

Source background

cm-1

0.001

second_moment

Second moment of polymer distribution

23

mg·m-2

1.9

density_shell

Bulk density of polymer in the shell

g·cm-3

0.7

500

volfraction

Core particle volume fraction

None

0.14

sld_shell

Polymer shell SLD

10-6-2

1.5

sld_solvent

Solvent SLD

10-6-2

6.3

The returned value is scaled to units of cm-1 sr-1, absolute scale.

Definition

This model describes the scattering from a layer of surfactant or polymer adsorbed on large, smooth, notionally spherical particles under the conditions that (i) the particles (cores) are contrast-matched to the dispersion medium, (ii) $$S(Q) \sim 1$$ (ie, the particle volume fraction is dilute), (iii) the particle radius is >> layer thickness (ie, the interface is locally flat), and (iv) scattering from excess unadsorbed adsorbate in the bulk medium is absent or has been corrected for.

Unlike many other core-shell models, this model does not assume any form for the density distribution of the adsorbed species normal to the interface (cf, a core-shell model normally assumes the density distribution to be a homogeneous step-function). For comparison, if the thickness of a (traditional core-shell like) step function distribution is $$t$$, the second moment about the mean of the density distribution (ie, the distance of the centre-of-mass of the distribution from the interface), $$\sigma = \sqrt{t^2/12}$$.

$I(q) = \text{scale} \cdot (\rho_\text{poly}-\rho_\text{solvent})^2 \left[ \frac{6\pi\phi_\text{core}}{Q^2} \frac{\Gamma^2}{\delta_\text{poly}^2R_\text{core}} \exp(-Q^2\sigma^2) \right] + \text{background}$

where scale is a scale factor, $$\rho_\text{poly}$$ is the sld of the polymer (or surfactant) layer, $$\rho_\text{solv}$$ is the sld of the solvent/medium and cores, $$\phi_\text{core}$$ is the volume fraction of the core particles, $$\delta_\text{poly}$$ is the bulk density of the polymer, $$\Gamma$$ is the adsorbed amount, and $$\sigma$$ is the second moment of the thickness distribution.

Note that all parameters except $$\sigma$$ are correlated so fitting more than one of these parameters will generally fail. Also note that unlike other shape models, no volume normalization is applied to this model (the calculation is exact).

The code for this model is based originally on a a fortran implementation by Steve King at ISIS in the SANDRA package c. 1990 [1].

Source

adsorbed_layer.py

References

Authorship and Verification

• Author: Jae-Hi Cho Date: pre 2010