Heterodyne Scattering: Multi-Component Models¶

When a sample contains multiple scattering populations with different dynamics — for example, a flowing region and a static (arrested) region — the intensity correlation function receives contributions from cross-correlations between all component pairs. This page develops the general \(N\)-component heterodyne scattering formula from He et al. PNAS 2025 and details the special cases most relevant to colloidal suspensions undergoing yielding.

Physical Motivation¶

In shear banding, the sample separates into a flowing band and a slow (or arrested) band coexisting in the scattering volume. Pure homodyne analysis (single component) cannot distinguish:

A single component with intermediate effective \(\dot{\gamma}\) and \(D\).
Two coexisting populations with different \(\dot{\gamma}_1\), \(\dot{\gamma}_2\), and scattering weights.

The heterodyne formula resolves this ambiguity by modelling the mixture scattering from \(N\) distinguishable populations explicitly.

Note

“Heterodyne” here refers to the presence of multiple scattering components with different mean velocities, not to optical heterodyne detection with a local oscillator. The scattered field from different components interferes, producing oscillatory patterns in \(c_2\).

N-Component General Formula¶

Consider \(N\) scattering populations. At time \(t\), component \(n\) carries a fraction \(x_n(t)\) of the total scattering weight, has mean velocity \(\langle v_n(t)\rangle\), and transport coefficient \(J_n(t)\).

The N-component second-order correlation function is:

(1)¶\[c_2(\mathbf{q}, t_1, t_2) \;=\; 1 + \frac{\beta}{f^2(t_1, t_2)} \sum_{n=1}^N \sum_{m=1}^N x_n(t_1)\,x_n(t_2)\,x_m(t_1)\,x_m(t_2) \cdot A_{nm}(t_1, t_2)\]

where the cross-correlation amplitude \(A_{nm}\) is:

(2)¶\[A_{nm}(t_1, t_2) \;=\; \exp\!\left(-\frac{q^2}{2}\int_{t_1}^{t_2}\left[J_n(t_1,t')+J_m(t_1,t')\right]dt'\right) \cos\!\left(q\cos\phi\int_{t_1}^{t_2}\left[\langle v_n(t')\rangle - \langle v_m(t')\rangle\right]dt'\right)\]

and the normalization factor is:

(3)¶\[f^2(t_1, t_2) \;=\; \left[\sum_{n=1}^N x_n(t_1)^2\right] \left[\sum_{n=1}^N x_n(t_2)^2\right]\]

Physical interpretation: Each term \(A_{nm}\) represents the interference between components \(n\) and \(m\). The cosine factor oscillates when the two components drift apart (relative velocity \(\Delta v_{nm}\) non-zero), producing characteristic fringes in the \(c_2\) matrix. The \(f^2\) factor normalizes for the fact that mixing reduces the maximum achievable contrast.

Two-Component Case: Static + Flowing¶

The simplest heterodyne case has two components:

Component 1 (index s): static (arrested), \(\langle v_s\rangle = 0\), transport \(J_s(t)\).
Component 2 (index f): flowing, \(\langle v_f(t)\rangle = \dot{\gamma}(t)\,r\cos\phi\), transport \(J_f(t)\).

With scattering weights \(x_s + x_f = 1\), the correlation function becomes:

(4)¶\[c_2 \;=\; 1 + \frac{\beta}{\left(x_s^2 + x_f^2\right)^2} \Bigl[ x_s^4\,A_{ss} + x_f^4\,A_{ff} + 2x_s^2 x_f^2\,A_{sf} \Bigr]\]

with:

\[\begin{split}A_{ss} &= \exp\!\left(-q^2 \mathcal{D}_s\right), \\ A_{ff} &= \exp\!\left(-q^2 \mathcal{D}_f\right), \\ A_{sf} &= \exp\!\left(-\frac{q^2(\mathcal{D}_s + \mathcal{D}_f)}{2}\right) \cos\!\left(q\cos\phi\int_{t_1}^{t_2}\langle v_f(t)\rangle\,dt\right)\end{split}\]

The cross term \(A_{sf}\) generates oscillations in \(c_2\) as a function of lag time \(\tau = t_2 - t_1\), with a period set by the inverse of the characteristic velocity \(\langle v_f\rangle\).

Three-Component Case: Two Flowing + One Static¶

The three-component case models a shear-banded state with two flowing bands and one static band:

Component s: static (\(v = 0\))
Component f1: slow-flow band (\(\dot{\gamma}_1\))
Component f2: fast-flow band (\(\dot{\gamma}_2 > \dot{\gamma}_1\))

The \(c_2\) matrix for three components is:

(5)¶\[c_2 = 1 + \frac{\beta}{f^2} \Bigl[ x_s^4\,A_{ss} + x_{f1}^4\,A_{f1f1} + x_{f2}^4\,A_{f2f2} + 2x_s^2 x_{f1}^2\,A_{sf1} + 2x_s^2 x_{f2}^2\,A_{sf2} + 2x_{f1}^2 x_{f2}^2\,A_{f1f2} \Bigr]\]

The three cross-terms \(A_{sf1}\), \(A_{sf2}\), \(A_{f1f2}\) each oscillate at a different frequency determined by the pairwise velocity differences \(\Delta v_{ij} = \langle v_i\rangle - \langle v_j\rangle\).

This multi-frequency beating pattern is a direct fingerprint of shear banding and can be distinguished from single-component laminar flow by the appearance of additional fringes in the two-time \(c_2\) matrix.

Normalization Factor f²¶

The normalization factor \(f^2\) defined in (3) ensures that \(c_2 \leq 1 + \beta\) always holds. Its geometric interpretation is:

\[f^2 = \left\langle x^2(t_1)\right\rangle \left\langle x^2(t_2)\right\rangle \;\equiv\; P_2(t_1)\,P_2(t_2)\]

where \(P_2(t) = \sum_n x_n(t)^2\) is the participation ratio (inverse Herfindahl index) of the weight distribution. For a single component \(x_1 = 1\), \(f^2 = 1\) and Equation (1) reduces to the homodyne formula. As the weight disperses over many components, \(f^2 \to 1/N^2\) and the effective contrast \(\beta/f^2\) increases accordingly.

Oscillatory Patterns as Diagnostic¶

The presence of oscillatory fringes in measured \(c_2(t_1, t_2)\) at fixed \(t_1 - t_2 = \tau\) is a direct diagnostic of multi-component scattering:

No oscillations: single component, homodyne model applies.
One oscillation frequency: two-component (static + flowing) system.
Multiple frequencies: three or more components, or shear banding.

The frequency of oscillations gives the velocity difference between components:

\[\nu_{nm} \;=\; \frac{q\cos\phi\,|\langle v_n\rangle - \langle v_m\rangle|}{2\pi}\]

from which the individual component velocities can be extracted.

Comparison to Homodyne¶

Feature	Homodyne (single component)	Heterodyne (multi-component)
Number of parameters	7 (laminar_flow mode)	7 + 2(N-1) per additional component
\(c_2\) shape	Monotone decay from diagonal	Oscillatory patterns
Applicable to	Homogeneous laminar flow	Shear banding, mixed phases
Analysis mode in package	`laminar_flow`	Not yet in current version
Paper reference	He et al. PNAS 2024	He et al. PNAS 2025

Note

The multi-component heterodyne model is described theoretically in He et al. PNAS 2025 for attractive suspensions exhibiting shear banding. The current homodyne package implements the single-component laminar flow model. Multi-component support is planned.