Onion model: form factor of core-shell spheroid

Introduction

Spheroid is often used to model particle shapes in small-angle X-ray or neutron scattering modeling. If the particle consists of a core wrapped within a thin shell, one needs to adopt the core-shell spheroid model. Here we will generalize the simple core-shell model to a multi-layered model (onion model). This model can then be used to calculate SAXS or SANS intensities from spheroidal objects with radially continuous density profiles. The majority of this post can be found in the following paper:

Z. Jiang and W. Chen, Generalized skew-symmetric interfacial probability distribution in reflectivity and small-angle scattering analysis J. App. Cryst. 50, 1653-1663 (2017)

Types of spheroid onions

The radial profile can be discretized along the radial directions into a concentric onion-like structure consisting of many thin independent slices with sharp interfaces. For a bi-axial spheroidal onion, principal radii (abbreviated for the lengths of the semi-principal axes) are defined as \(\mathbf{R}=[R_r, R_z]\) (with \(R_r\) the equatorial radius, and \(R_z\) the polar radius). There are three common types of concentric onions: confocal, iso-thickness, and similar. We will generalize them into arbitrary type later. Assume that we anchor a radial density profile \(\rho(z)\) along the polar semi-axis \(R_z\) of a spheroid core whose principle radii are  \(R_{r0}\) and  \(R_{z0}\). For convenience, we define \(z=0\), i.e., the origin of the profile axis, to be at \(R_{r0}\) and \(R_{z0}\) (henceforth referred to as the reference radii \(\mathbf{R}_0\)). By this method of choosing the reference to construct a spheroidal onion, we have \(R_z(z)=R_{z0}+z\) for all onion types, but the dependence of \(R_r(z)\) on \(z\) varies. We now define a shape parameter to distinguish these three onion types via the so-called 'eccentricity ratio' \(\epsilon_{zr}(z)=\frac{R_z(z)}{R_r(z)}\)

• For confocal: 

\(\epsilon_{zr}(z)=\frac{R_{z0}+z}{\sqrt{R_{r0}^2-R_{z0}^2+(R_{z0}+z)^2}} \)   (1)

• For iso-thickness: 

\(\epsilon_{zr}(z)=\frac{R_{z0}+z}{R_{r0}+z}\)   (2)

• For similar:  

\(\epsilon_{zr}(z)=\frac{R_{z0}}{R_{r0}}\) (invariant)   (3)

We then can calculate \(\mathbf{R}(z)\) for every onion slice with the help of \(\epsilon_{zr}(z)\). The concentric confocal and iso-thickness onion types are more realistic in many systems, such as giant unilamellar vesicles and thin layers grown, deposited or absorbed on the surfaces of large particles. As the onion size increases, the shape approaches a sphere with an eccentricity ratio of one. Hence, the curvature effect becomes identical everywhere on the surface. Before we present the scattering cross sections and form factors for all three onion shapes, a useful quantities \(R_\theta(z)\) is defined as the distance from the center to a point on an onion slice surface with a polar angle \(\theta\),

\(R_\theta(z)=R_z(z)\sqrt{\frac{\sin^2\theta}{\epsilon_{zr}^2(z)}+\cos^2\theta}\)   (4)

In addition, instead of a real-valued electron-density profile \(\rho(z)\), a complex form \(\tilde{\rho}(z)\) can be used to account for the absorptions.

Differential scattering cross-section

Let's first consider a mono-disperse onion. The scattering cross section of a randomly oriented onion of reference (core) radii \(\mathbf{R}_0\) is

\(\frac{d\sigma}{d\Omega}(q,\mathbf{R}_0)=[\widetilde{V}(\mathbf{R}_0)]^2P(q,\mathbf{R}_0)\)   (5)

where \(P(q,\mathbf{R}_0)\) is the normalized form factor and 

\(\widetilde{V}(\mathbf{R}_0)=\sqrt{\frac{d\sigma}{d\Omega}(0,\mathbf{R}_0)}\)   (6)

is the density-gradient weighted volume to ensure \(P(0,\mathbf{R}_0)=1\).

If we define

\(\mathcal{F}[x]=\frac{3(\sin x-x\cos x)}{x^3}\)   (7)

we can write for a spheroidal onion

\(\frac{d\sigma}{d\Omega}(q,\mathbf{R}_0)=\int_0^{\pi/2}|F_\theta(q,\mathbf{R}_0)|^2\sin\theta d\theta\)   (8)

where

\(F_\theta(q)(q,\mathbf{R}_0)=\int_{-\text{min}(R_{r0},R_{z0})}^\infty \frac{\partial \tilde{\rho}(z)}{\partial z} V[\mathbf{R}(z)] \mathcal{F}[qR_\theta(z)] dz\)   (9)

with  \(V[\mathbf{R}(z)]= (4\pi/3) R_r^2(z)R_z(z) \) the volume of a spheroid of principle radii \(\mathbf{R}(z)\). For non-randomly oriented spheroidal onions, one should include the orientation distribution function in the integrand of Equation (8). 

Let's explore the meaning of \(\widetilde{V}(\mathbf{R}_0)\). By lettering \(q\rightarrow0\) and applying \(\lim_{x\rightarrow0}\mathcal{F}(x)=1\), we arrive at

\(\widetilde{V}(\mathbf{R}_0)=\Big| \int_{-\text{min}(R_{r0},R_{z0})}^\infty \frac{\partial \tilde{\rho}(z)}{\partial z} V[\mathbf{R}(z)] dz \Big|\)   (10)

which is obviously a volume weighted by the density gradient. Its meaning can be easily understood by calculating \(\widetilde{V}(\mathbf{R}_0)\) for a homogenous hard-core object with sharp interface whose radial density profile is a step function. The density gradient is then a Dirac delta function with a density contrast \(\Delta\tilde{\rho}\) across the interface. Hence, the scattering cross section in Equation (5) becomes the familiar form for an object of homogenous density and sharp interface  

\(\frac{d\sigma}{d\Omega}(q,\mathbf{R}_0)=|\Delta\tilde{\rho}|^2[V(\mathbf{R}_0)]^2P(q,\mathbf{R}_0)\)   (11)

Polydispersity

Both size and shape polydispersities can be accounted for by a joint-distribution function for \(\mathbf{R}_0=[R_{r0},R_{z0}]\), which is denoted by \(p(\mathbf{R}_0)=p(R_{r0},R_{z0})\) such that \(\int_0^\infty\int_0^\infty p(R_{r0},R_{z0})dR_{r0}dR_{z0} \). A bivariate Gaussian distribution is often used. The effective form factor is then given by

\(P(q)=\frac{\Big< \frac{d\sigma}{d\Omega}(q) \Big>_{p(\mathbf{R}_0)}}{\Big<\Big[ \widetilde{V}(\mathbf{R}_0)\Big]^2\Big>_{p(\mathbf{R}_0)}} \)   (12)

where the numerator is the scattering cross section, the denominator is the normalization factor so that \(P(0)=1\), and the bracket denotes the polydispersity weighted averaging. 

Resolution

The instrumentation smearing effect, particularly in many small-angle neutron scattering experiments, can be included by convoluting to a resolution function.

Spheroidal onion of continuous type dimension

Before we wrap up this post, let's generalize the above three spheroidal onion types into a continuous type. We define the eccentricity ratio in a general form such that

\(\epsilon_{zr}(z)=\frac{R_{z0}+z}{\Big[R_{r0}^d-R_{z0}^d+(R_{z0}+z)^d\Big]^{1/d}} =\Big[1+\frac{R_{r0}^d-R_{z0}^d}{(R_{z0}+z)^d}\Big]^{-\frac{1}{d}}\)   (13)

where \(d\in(0,\infty)\) is the type dimension. Specifically, \(d=2\) leads to a confocal spheroidal onion shown in Equation (1), \(d=1\) leads to an iso-thickness spheroidal onion shown in Equation (2), and \(d\rightarrow0\) leads to a similar spheroidal onion shown in Equation (3). The larger the type dimension, the faster \(R_r(z)\) approaches \(R_z(z)\), i.e., the faster the shape of a spheroidal onion gets close to a spherical shape, as seen in the figure below. The advantage of adopting a continuous type dimension is that \(d\) can be an adjustable parameter so that the spheroidal shape can be fitted as well with other parameters.