Background

Most of this information can be found in more detail in the work of P. Salgi and R. Rajagopalan, "Polydispersity in colloids: implications to static structure and scattering", Adv. Colloid Interface Sci. 43, 169-288 (1993). A basic understanding of the principles of small-angle scattering and light scattering by particles is recommended.

Definitions and Naming Conventions

The scattered intensity \(I(Q)\) in dependence of the scattering wavevector \(Q\) of a multicomponent system of spherical particles with \(n\) distinct species can be written as

\[ I(Q) \propto \sum_{\alpha, \beta=1}^{n} (x_\alpha x_\beta)^{1/2} \, F_\alpha(Q) \, F_\beta(Q) \, S_{\alpha \beta}(Q), \]

where \(x_\alpha\) is the number fraction of the species \(\alpha\).

The scattering amplitude \(F_\alpha(Q)\) of the species \(\alpha\) is the Fourier transform of the scattering contrast \(\Delta\rho_\alpha(r)\) of the particle. Because the particles are spherically symmetric, the 3D Fourier transform can be formulated as a 1D Fourier-Bessel integral:

\[ F_\alpha(Q) = 4\pi \int\limits_{0}^{\infty} \Delta\rho_\alpha(r) \, r^2 \, \dfrac{\sin(Q \, r)}{Q r} \,\mathrm{d} r. \]

An important quantity is the size average of the squared amplitude

\[ \langle F^2(Q) \rangle = \sum_{\alpha=1}^{n} x_\alpha \, F^2_\alpha(Q). \]

The size-averaged form factor can then be expressed as

\[ P(Q) = \dfrac{\langle F^2(Q) \rangle}{\langle F^2(0) \rangle}. \]

The partial structure factors \(S_{\alpha \beta}(Q)\) describe the interparticle correlations between the particles of the species \(\alpha\) and the species \(\beta\). They form the matrix \(\mathbf{S}\), which is related to the matrix of weighted direct correlation functions \(\mathbf{\tilde{c}}\) with elements \(\tilde{c}_{\alpha\beta} = \sqrt{x_i x_j} \, c_{\alpha\beta}\) by the Ornstein-Zernike equation:

\[ \mathbf{S} = [\mathbf{1} - \rho \mathbf{\tilde{c}}]^{-1}, \]

where \(\rho\) is the total number density. The partial structure factors defined here follow the property \(S_{\alpha\beta}(Q\to\infty) = \delta_{\alpha\beta}\).

Radius of Gyration

The radius of gyration \(R_\mathrm{G}\) describes the root mean square distance of the scattering centers from the center of mass. For a single particle, it can be calculated from the radial contrast profile \(\Delta\rho_\alpha(r)\) according to

\[ R_{\mathrm{G}, \alpha}^2 = \dfrac{4\pi \int\limits_{0}^{\infty} \Delta\rho_\alpha(r) \, r^4 \,\mathrm{d} r} {4\pi \int\limits_{0}^{\infty} \Delta\rho_\alpha(r) \, r^2 \,\mathrm{d} r} \]

In a single component system, \(R_\mathrm{G}\) is related to the initial slope of the form factor

\[ P(Q) = 1 - \dfrac{R_{\mathrm{G}}^2}{3} Q^2 + \mathcal O (Q^4), \]

which leads to the well-known Guinier Law for small wavevectors

\[ P(Q) \approx \exp{\left(- \frac{R_{\mathrm{G}}^2}{3} Q^2 \right)}. \]

For particle mixtures, the Guinier approximation is still valid, however, with an averaged, apparent radius of gyration

\[ \langle R^2_\mathrm G \rangle = \langle F^2(0) \rangle^{-1} \sum_{\alpha=1}^{n} x_\alpha \, F^2_\alpha(0) R_{\mathrm{G}, \alpha}^2, \]

such that

\[ P(Q) \approx \exp{\left(- \frac{\langle R_{\mathrm{G}}^2 \rangle }{3} Q^2 \right)}. \]

Effective Structure Factors

There are multiple ways to define single effective, structure factors:

• The measurable structure factor

  \[ S_{\mathrm{M}}(Q) = \langle F^2(Q) \rangle^{-1} \sum_{\alpha, \beta=1}^{n} (x_\alpha x_\beta)^{1/2} \, F_\alpha(Q) \, F_\beta(Q) \, S_{\alpha \beta}(Q), \]

  which is simply the intensity of a suspension divided by the intensity of a non-interacting suspension with the same scattering properties.

• The average number-number structure factor

  \[ S_{\mathrm{NN}}(Q) = \sum_{\alpha, \beta=1}^{n} (x_\alpha x_\beta)^{1/2} \, S_{\alpha \beta}(Q), \]

  which is just the sum of all partial structure factors weighted with the size distribution. This effective structure factor describes the overall correlation between all existing particles.

• The "compressibility" structure factor

  \[ S_{\mathrm{\kappa}}(Q) = \dfrac{1}{\sum_{\alpha, \beta=1}^{n} x_\alpha x_\beta \, S_{\alpha\beta}^{-1}(Q)}, \]

  where \(S_{\alpha\beta}^{-1}(Q)\) denotes the \(\alpha\beta\) element of the inverse of the partial structure factor matrix. According to the Kirkwood-Buff theory of solutions, the isothermal compressibility \(\kappa_{\mathrm{T}}\) is related to the zero-\(Q\) limit of the partial structure factors by

  \[ \rho k_\mathrm B T \kappa_{\mathrm{T}} = \dfrac{1}{\sum_{\alpha, \beta=1}^{n} x_\alpha x_\beta \, S_{\alpha\beta}^{-1}(0)}. \]

  The compressibility structure factor extends this definition to finite \(Q\) and the relation

  \[ S_{\mathrm{\kappa}}(Q\to0) = \rho k_\mathrm B T \kappa_{\mathrm{T}} \]

  is satisfied.

Apparent Diffusion Coefficient

Of interest in the context of dynamic light scattering is the apparent Stokes-Einstein diffusion coefficient, which for mixtures of particles depends on the scattering amplitudes and also changes with the wavevector:

\[ \langle D_0(Q) \rangle = \langle F^2(Q) \rangle^{-1} \sum_{\alpha=1}^{n} x_\alpha \, F^2_\alpha(Q) D_{0, \alpha}, \]

with

\[ D_{0, \alpha} = \dfrac{k_{\mathrm{B}} T}{6 \pi \eta_0 R_{\mathrm{h}, \alpha}}, \]

where \(k_{\mathrm{B}} T\) indicates the thermal energy, \(\eta_0\) the viscosity of the suspension medium and \(R_{\mathrm{h}, \alpha}\) the hydrodynamic radius of the particles of species \(\alpha\).

An apparent hydrodynamic radius can then be defined as

\[ R_{\mathrm{h, app}}(Q) = \dfrac{k_{\mathrm{B}} T}{6 \pi \eta_0 \langle D_0(Q) \rangle}. \]