Yielding Dynamics of Colloidal Suspensions¶

This page summarizes the physics of yielding transitions in charged colloidal suspensions as studied by He et al. (PNAS 2025) using the homodyne XPCS framework. Two fundamentally distinct yielding mechanisms are identified — ductile Andrade creep in repulsive systems and brittle shear banding in attractive systems — and connected to measurable signatures in the two-time correlation function \(c_2\).

Overview: Repulsive vs Attractive Suspensions¶

Charged colloidal suspensions near the glass or gel transition exhibit qualitatively different rheological responses depending on the inter-particle potential:

Property	Repulsive (electrostatic)	Attractive (depletion/van der Waals)
Microstructure	Ordered, charge-stabilized glass	Fractal gel network
Yielding type	Ductile (gradual, homogeneous)	Brittle (abrupt, heterogeneous)
Flow field	Homogeneous laminar flow	Shear banding
\(c_2\) signature	Homodyne + Andrade creep	Heterodyne + oscillations
Non-affine motion	Gaussian (small)	Non-Gaussian (power-law tails)
PNAS 2025 section	Section II	Section III–IV

Repulsive Suspensions: Andrade Creep¶

Under a step-stress protocol, repulsive colloidal glasses exhibit Andrade creep, first identified in polycrystalline metals and now established in soft matter:

(1)¶\[\gamma(t) \;\propto\; t^{1/3} \quad \Rightarrow \quad \dot{\gamma}(t) \;\propto\; t^{-2/3}\]

The \(t^{1/3}\) scaling arises from the cooperative, thermally-activated rearrangement of particle cages under a sub-yield stress. Each rearrangement event relieves a small amount of stored elastic energy, producing a decreasing strain rate.

Mapping to homodyne parameters:

\[\dot{\gamma}_0 > 0, \quad \beta_\gamma = -2/3, \quad \dot{\gamma}_\mathrm{offset} = 0\]

The corresponding shear integral is:

\[\Gamma(t_1, t_2) \;=\; \frac{\dot{\gamma}_0}{1/3}\!\left(t_2^{1/3} - t_1^{1/3}\right) \;=\; 3\dot{\gamma}_0\!\left(t_2^{1/3} - t_1^{1/3}\right)\]

Diffusion during creep: Simultaneously, the mean-squared displacement of particles grows anomalously with an exponent \(\alpha < 1\):

\[\mathrm{MSD}(t) \;\propto\; t^\alpha, \quad \alpha \approx 0.3 \text{–} 0.5\]

This is sub-diffusion driven by caged motion — particles rattle within increasingly disrupted cages but do not undergo long-range diffusion until the yield stress is exceeded.

XPCS signature: The two-time \(c_2\) matrix shows a diagonal ridge whose width grows as \(t^{1/3}\) — the fingerprint of Andrade creep directly observable without bulk rheometry.

Attractive Suspensions: Shear Banding and Delayed Yielding¶

In attractive gels (formed by depletion or van der Waals forces), yielding is fundamentally different:

Delayed yielding: The gel remains apparently solid until a critical accumulated strain \(\gamma_c \approx 0.3\) is reached, at which point it abruptly fluidizes (“brittle failure”).

\[\gamma(t) \;\approx\; 0 \quad (t < t_\mathrm{yield}), \qquad \gamma(t) \;\gg\; 0 \quad (t > t_\mathrm{yield})\]

Shear banding: During and after yielding, the flow field becomes heterogeneous: a fast-flowing band (near the rotor) coexists with a slow or static band (near the stator). This breaks the laminar flow assumption.

Resolidification: After yielding, the sample may partially re-arrest in the static band due to network reformation. This manifests as a non-monotone evolution of the correlation function.

Bond Dynamics and Interfacial Layers¶

The microscopic origin of the two yielding mechanisms is revealed by tracking the bond lifetime distribution:

Repulsive glass: Cage lifetime \(\tau_\alpha\) decreases monotonically under stress as bonds are thermally activated. No preferential bond breaking.
Attractive gel: Network strands break irreversibly at the applied stress (“bond rupture” mechanism). The failed bonds concentrate at the interface between the two shear bands.

The interfacial layer (few-particle-diameter thickness) between bands is the locus of all plastic deformation. XPCS probes this layer directly because the beam coherence length (\(\sim`nm) is much smaller than the layer thickness (:math:\)sim mu`m).

Non-Affine Displacements¶

Affine displacements follow the macroscopic strain field exactly. Non-affine displacements are the residuals:

\[\delta\mathbf{u}_i(t) \;=\; \mathbf{u}_i(t) - \mathbf{E}(t)\cdot\mathbf{r}_i(0)\]

where \(\mathbf{E}(t)\) is the macroscopic strain tensor. The distribution \(P(\delta u)\) characterizes the heterogeneity of the flow:

Gaussian \(P(\delta u)\): homogeneous flow, homodyne XPCS model valid.
Power-law tails in \(P(\delta u)\): shear banding, heterodyne model required.

For repulsive suspensions, He et al. PNAS 2025 find:

\[P(\delta u) \;\sim\; \exp(-\delta u^2 / 2\sigma^2)\]

confirming the Gaussian ansatz underlying the homodyne single-component model.

For attractive suspensions:

\[P(\delta u) \;\sim\; |\delta u|^{-\mu}, \quad \mu < 3\]

indicating a population of rare but large displacements (“particle avalanches”) not described by a single effective \(D\).

Connection to XPCS Measurements¶

The mapping from microscopic dynamics to XPCS observables is:

Repulsive (Andrade creep):

\[c_2^\mathrm{exp}(q, t_1, t_2) \;=\; 1 + \beta\, e^{-q^2\mathcal{D}(t_1,t_2)} \mathrm{sinc}^2\!\left(\tfrac{qh\cos\phi}{2\pi}\Gamma(t_1,t_2)\right)\]

with \(\mathcal{D}\) and \(\Gamma\) reflecting the Andrade creep laws. This is exactly the homodyne laminar_flow model implemented in this package.

Attractive (shear banding): The measured \(c_2\) matrix shows oscillatory patterns described by the two-component heterodyne formula. The oscillation frequency increases abruptly at \(t = t_\mathrm{yield}\) as the fast band velocity jumps.

Practical XPCS diagnostic criteria:

If \(c_2(t_1, t_2)\) is smooth and decays monotonically away from the diagonal: use homodyne laminar_flow mode.
If \(c_2(t_1, t_2)\) shows oscillatory fringes: use heterodyne multi-component analysis (currently outside the scope of the homodyne package; see He et al. PNAS 2025).
If \(c_2\) is featureless (no angle dependence): use static mode (no flow).

Summary of Physical Signatures¶

Observable	Repulsive (Andrade)	Attractive (banding)
Strain \(\gamma(t)\)	\(\propto t^{1/3}\) (smooth)	Step-like at \(t_\mathrm{yield}\)
\(c_2\) matrix shape	Ridge, width \(\propto t^{1/3}\)	Oscillatory fringes
\(\alpha\) (diffusion exp.)	\(0.3 < \alpha < 1\) (sub-diffusive)	Multi-modal MSD
Non-Gaussian param. \(\alpha_2\)	\(\approx 0\)	\(\gg 0\)
Analysis mode	`laminar_flow`	Heterodyne (future work)