Interparticle Effects

6.4. Interparticle Effects#

6.4.1. Necessity of Removal#

It is said that monodispersity is essential in SAXS analysis. In this context, “mono” relates to decomposition, while “dispersity” pertains to the removal of interparticle effects.

We have already discussed decomposition using matrices. Matrices can also be used for the latter purpose, as described here. Note that, in terms of form and structure factors, our goal can be interpreted as obtaining the form factors \(P(q)\) for each component.

6.4.2. Matrix Representation#

As we have seen in the previous chapter, scattering intensity is known to be expressed as a function of scattering angle and concentration as follows.

\[ I(q,c) = cP(q)S(q,c) \qquad \qquad \qquad (1) \]

where

  • \(c\): oncentration

  • \(P(q)\): form factor (single particle scattering)

  • \(S(q,c)\): structure factor (accounts for interparticle interactions)

When the solution is dilute, interparticle interactions are ignorable and the structure factor reduces to \( S(q,c)=1 \).

Genarally, the structure factor can be expanded as follows.

\[ S(q,c) = 1 + cS_1(q) + c^2S_2(q) + ... \qquad (2) \]

where coefficients \(S_1(q), S_2(q), ... \) are related to pair, triplet, etc. interactions.

Using the simplest approximation \( S(q,c) \approx 1 + cS_1(q) \), i.e., considering only pairwise interactions and ignoring other multiple interactions, we get the quadratic approximation

\[ I(q,c) \approx cA(q) + c^2B(q) \qquad \qquad \qquad (3) \]

where

  • \(A(q)=P(q)\) : single particle term

  • \(B(q)=P(q)S_1(q)\) : pairwise interaction term.

This approximation (3) can be expressed in matrix notation as follows.

\[ M = P \cdot C \]

Or in element-wise notation,

\[\begin{split} \begin{pmatrix} d_{11} & d_{12} & \dots & d_{1n} \\ d_{21} & d_{22} & \dots & d_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ d_{m1} & d_{m2} & \dots & d_{mn} \end{pmatrix} = \begin{pmatrix} a_{1} & b_{1} \\ a_{2} & b_{2} \\ \vdots & \vdots \\ a_{m} & b_{m} \end{pmatrix} \begin{pmatrix} c_{1} & c_{2} & \dots & c_{n} \\ c_{1}^2 & c_{2}^2 & \dots & c_{n}^2 \end{pmatrix} \end{split}\]

where

  • \(d_{ij}\) : measured data

  • \( a_i \) : single particle factor

  • \( b_i \) : interparticle factor

  • \( c_j \) : concentration

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