fuzzy_sphere

Scattering from spheres with a Gaussian interface.

Parameter

Description

Units

Default value

scale

Scale factor or Volume fraction

None

1

background

Source background

cm-1

0.001

sld

Scattering length density of sphere

10-6Å-2

1

sld_solvent

Scattering length density of solvent

10-6Å-2

3

radius

Pseudo-radius of sphere (read the definition in the model help)

Å

60

fuzziness

Std dev from the average thickness of the fuzzy interface (must be << radius)

Å

10

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

For information about polarised and magnetic scattering, see the Polarisation/Magnetic Scattering documentation.

Definition

Scattering from spheres with a Gaussian interface.

The scattering intensity \(I(q)\) is calculated as:

\[I(q) = \frac{\text{scale}}{V}(\Delta \rho)^2 A(q)^2 S(q) + \text{background}\]

where the amplitude \(A(q)\) describes the radial scattering length density profile of a homogeneous sphere convoluted with a Gaussian in order to give a function with a gradual drop-off in scattering length density (SLD) towards the interface (i.e. a sphere with a diffuse, or “fuzzy”, interface):

\[A(q) = \frac{3\left[\sin(qR) - qR \cos(qR)\right]}{(qR)^3} \exp\left(\frac{-(\sigma_\text{fuzzy}q)^2}{2}\right)\]

Here \(A(q)^2\) is the form factor, \(P(q)\). The \(scale\) is equivalent to the volume fraction of spheres, each of volume, \(V\). And the contrast \((\Delta \rho)\) is the difference in SLD between a sphere and the surrounding medium.

In this model, \(R\) represents the radius at which the SLD has decreased to half of its value at the core, not the overall radius of a sphere. This is a frequent source of confusion when applying this model.

\(\sigma_\text{fuzzy}\) is then the width of the fuzzy interface; strictly, the standard deviation from the average thickness of the interface.

From Reference [1]:

The inner regions … that display a higher … [SLD] are described by a radial box profile extending to a radius of approximately \(R_\text{box} \sim R - 2 \sigma_\text{fuzzy}\). The profile approaches zero at \(R_\text{SANS} \sim R + 2 \sigma_\text{fuzzy}\). Therefore, the overall size of the fuzzy sphere is approximated by … \(R_\text{SANS}\).

For this model to give meaningful results it is important \(\sigma_\text{fuzzy} \ll R\). It is for the User to ensure that this condition is maintained, especially if applying polydispersity to one or both length scales.

This model has been widely applied to the scattering from polymer microgel particles as illustrated below, where \(R_\text{h}\) is the hydrodynamic radius.

../_images/fuzzy_sphere_geometry.png

Figure shows the fuzzy_sphere model applied to a microgel particle (adapted from [2], Fig 5).

Although the fuzzy sphere model often provides a good description of scattering data from such systems, advances in measurement techniques have highlighted that the real-space density profile can be far more complex than this model assumes [3].

This model is not suitable for describing spherical particles decorated with so-called polymer ‘brushes’ (where the SLD profile follows a parabolic decay) or spherical particles with terminally-attached polymer chains (where the SLD profile is expected to exhibit a maximum before the Gaussian decay).

To model more complex SLD profiles, see the onion and spherical_sld models.

For 2D data: The 2D scattering intensity is calculated in the same way as 1D, where the \(q\) vector is defined as

\[q = \sqrt{{q_x}^2 + {q_y}^2}\]
../_images/fuzzy_sphere_autogenfig.png

Fig. 93 Figure 2: 1D plot corresponding to the default parameters of the model.

Source

fuzzy_sphere.py \(\ \star\ \) fuzzy_sphere.c \(\ \star\ \) sas_3j1x_x.c

References

  1. M Stieger, J. S Pedersen, P Lindner, W Richtering, Langmuir, 20 (2004) 7283-7292

  2. E Ponomareva, B Tadgell, M Hildebrandt, M Krüsmann, S Prévost, P Mulvaney, M Karg, Soft Matter, 18 (2022) 807-825

  3. F Scheffold, Soft Matter, 20 (2024) 8181-8184

Authorship and Verification