Direction-dependent dynamics of colloidal particle pairs and the Stokes-Einstein relation in quasi-two-dimensional fluids

The study examines how near-field hydrodynamic interactions govern particle transport in low Reynolds number fluids confined to two dimensions. While far-field interactions are well understood in both 2D and 3D, near-field behavior—especially the transverse (antidrag) component and its relation to structural organization—remains unclear. In 3D, monopole-like interactions produce drag in both longitudinal (L) and transverse (T) directions, whereas in 2D far-field flows are dipolar with longitudinal drag and transverse antidrag. Prior experiments in dense 2D suspensions revealed that near-field longitudinal drag correlations oscillate with separation in phase with structural pair correlations, but the nature of transverse antidrag in the near-field and any phase differences with longitudinal modes have been unexplored. The authors aim to experimentally resolve these near-field hydrodynamic modes in quasi-2D colloidal suspensions, quantify their directional (body-frame) effects on particle displacement and relaxation, and assess how these anisotropic interactions influence and can break the Stokes-Einstein relation (SER), including conditions under which SER may be recovered.

Background highlights include: (i) Hydrodynamic interactions in confined geometries depend on dimensionality and boundaries; 2D far-field solutions are dipolar with longitudinal drag and transverse antidrag, whereas in 3D both directions show drag. (ii) In dense 2D suspensions, near-field longitudinal drag correlations exhibit oscillatory modulations in phase with the pair correlation function (g(r)), but transverse near-field behavior has been less characterized. (iii) The SER connects microscopic diffusivity and macroscopic viscosity; although derived for dilute systems, it often holds in dense 3D systems. However, in 2D, simulations and experiments report anomalous exponents (ξ < −1), with proposed origins including long-wavelength Mermin-Wagner fluctuations. Approaches that remove such long-wavelength correlated motions can recover ξ ≈ −1, though these generally assume isotropy of D and τ, which may not hold in the near-field of 2D systems. The present work directly probes anisotropic near-field hydrodynamics to clarify the microscopic basis for SER breakdown in 2D.

Quasi-2D colloidal suspensions of polystyrene microspheres (diameter a = 1.04 μm, ~3% polydispersity) in water were prepared in a wedge-shaped cell. Particles sedimented into a thin quasi-2D region; the region was equilibrated for at least six hours. Experiments were conducted at seven packing area fractions 0.15 ≤ φ ≤ 0.61 at T = 22 °C. Imaging was performed at 10 frames per second for 20 minutes; typical fields of view at φ = 0.15 contained ~1100 particles. Standard particle-tracking algorithms were used to obtain trajectories with dynamic spatial resolution of ~20 nm. Control experiments verified that the wedge geometry did not affect measured pair dynamics. For comparison, an open (3D) cell was also used to assess the sign of correlation amplitudes. Analyses focused on particle pairs with separation vector r at initial time t0. Conditional displacement distributions P(Δrj|Δri) were computed in the body-frame, whose longitudinal (L) axis lies along r and transverse (T) axis is orthogonal to r. Longitudinal and transverse hydrodynamic displacement correlation functions were defined as HL,T(r,t) = ⟨Δri(t) Δrj(t)⟩ / Dself, averaging over initial times and unique pairs; normalization by the φ-dependent single-particle diffusivity Dself enabled comparison across densities. Polar displacement fields around a moving particle were reconstructed by aligning individual particle displacements and ensemble-averaging to map the induced displacement field r(r,θ). Single-particle diffusivity in the lab frame, Dself, was obtained from mean squared displacements. Body-frame, pair-referenced diffusivities D(r,θ) were extracted from the linear regime of mean squared displacements along a probing direction at angle θ to L. Structural relaxation times in the lab frame, τ, were measured from self-intermediate scattering functions F_s(q,t) at q = 2π/α, where α is the position of the first peak in g(r) at φ = 0.61; the direction of q was along the x-axis. Body-frame relaxation times τ(r,θ) were measured from F_s(q,t; r, r + r′) using pair-aligned coordinates and reading off the decay time to 1/e. At high φ (≥ 0.58), short-time subdiffusivity limited extraction of D(r,θ) on very short timescales. SER exponents ξ were obtained from power-law fits of D versus τ across φ at fixed r and direction (L or T), as well as from lab-frame Dself versus τ and from directions orthogonal to pair center-of-mass motion for ξCM.

- Conditional displacement distributions directly visualized dipolar hydrodynamic modes: co-diffusion (drag) along L and anti-symmetric diffusion (antidrag) along T in quasi-2D. A near-field linear superposition of drag and antidrag produced circumferential “mass void filling” motion of one particle around its partner. - Hydrodynamic correlations HL(r) and HT(r) decayed as 1/r^2 in the far-field at low φ = 0.15, with quasi-2D confinement yielding positive amplitude for HL and negative amplitude for HT. In an open (3D) cell both HL and HT were positive. - At higher density (φ = 0.61), HL and HT showed oscillatory spatial modulations in the near-field that deviated from dipolar decay; HL modulations were in-phase with g(r), while HT exhibited a spatial phase lag of ~0.25σ relative to HL. The fractional pair separation change Zrel(r,t) captured antidrag-induced pair rotation and separation and was in-phase with HT modulations. - A most-probable spatiotemporal evolution emerged for pair configurations: near r ≈ 1.25σ longitudinal drag dominates and pairs are most stable (Zrel minimum); around r ≈ 1.5–1.75σ transverse rotations strengthen, destabilizing the pair (Zrel maximum), after which drag regains dominance near r ≈ 2.0σ, with cycles attenuating at larger r. - Transport quantifiers showed anisotropy tied to hydrodynamics: D(r,θ) was isotropic at low φ, but τ(r,θ) was anisotropic for r < 2σ, with τ∥ > τ⊥ near r ≈ 1.25σ due to stronger longitudinal drag relative to transverse antidrag. At high φ, D(r) remained angularly isotropic yet oscillated with r in-phase with HL, while τ(r,θ) was strongly anisotropic and oscillatory due to the contrast and phase lag between HL and HT. - Stokes-Einstein analysis in the body frame revealed direction-dependent exponents: ξ∥ and ξ⊥ differed from −1 and from each other, reflecting the anisotropic τ(r,θ). The ~0.25σ phase lag between HL and HT appeared in ξ∥ and ξ⊥ versus r. In contrast, exponents from two random orthogonal lab-frame directions were in-phase and indistinguishable within error. - Traditional lab-frame measures yielded SER violation: φ-dependent Dself and τ produced ξself = −1.16 ± 0.03; body-frame ξ(r) converged to ξself in the far-field r > 8σ (ξ ≈ −1.18 ± 0.03). The inset to Fig. 3 also showed Dself τ ≈ 1.16 ± 0.03 (dimensionless). - A recovery of SER along special directions was demonstrated: along the direction perpendicular to the pair center-of-mass displacement (CM⊥), hydrodynamic correlations are minimized. The corresponding exponent ξCM approached −1.01 ± 0.02 for r > 8σ and oscillated around −1 in the near field, with values near −1 at r corresponding to extrema of Zrel where net correlations are weakest.

The experiments establish that near-field hydrodynamic interactions in quasi-2D colloidal fluids are intrinsically anisotropic and feature a phase-shift between longitudinal drag and transverse antidrag modes. These features directly cause direction-dependent particle displacement and relaxation in the body frame of pairs, producing spatially heterogeneous and anisotropic dynamics. The SER breakdown in 2D is thus linked to these anisotropic near-field hydrodynamic correlations: ξ∥ and ξ⊥ deviate from −1 in a direction-dependent manner, and even lab-frame measures yield ξself < −1. The far-field convergence of body-frame exponents to ξself underscores the persistent influence of hydrodynamics across scales. Importantly, by identifying directions where hydrodynamic correlations are minimized (perpendicular to the pair center-of-mass displacement), the study demonstrates routes to recover the SER (ξ ≈ −1). These insights clarify microscopic mechanisms behind SER anomalies observed in 2D and highlight that assumptions of isotropy underlying some fluctuation-filtering approaches may not hold in the near-field. The findings are relevant for understanding transport, restructuring, and relaxation in dense, confined fluids and can inform interpretations of anomalous diffusion-relaxation relations in reduced dimensions.

This work experimentally resolves near- and far-field longitudinal and transverse hydrodynamic modes in quasi-2D colloidal suspensions, uncovering contrasting magnitudes and a ~0.25σ phase lag between them. These anisotropic, phase-shifted interactions generate direction-dependent dynamics in the body frame of particle pairs and lead to SER violations both in body- and lab-frames (ξself ≈ −1.16 to −1.18). A key contribution is the identification of a direction where hydrodynamic correlations are minimized—perpendicular to the pair center-of-mass displacement—along which the SER is effectively recovered (ξCM → −1.01 ± 0.02 for r > 8σ). The study provides a mechanistic framework for understanding SER breakdowns in 2D and suggests strategies to restore SER behavior. Future research directions include extending these insights to anisotropic and active particles, exploiting near-field hydrodynamics for controlled self-assembly and relaxation, and developing computational methods that accurately incorporate hydrodynamics in dense, confined suspensions.

- Measurements are performed in quasi-2D confined geometries; generalization to other confinement or true 2D/3D systems may require caution. - At high packing fractions (φ ≥ 0.58), short-time dynamics become mildly subdiffusive, limiting reliable extraction of D(r,θ) on short timescales. - SER validity is strictly asymptotic (dilute, far-field); near-field interpretations rely on finite-r analyses with oscillatory hydrodynamic modulations. - Extraction of D and SER exponents involved time-window choices; systematic errors from different time-windows exceeded standard fitting errors and were used to quote uncertainties. - Wedge-shaped cell geometry was tested and found not to affect pair dynamics within experimental uncertainty, but residual geometric effects cannot be entirely excluded. - Large raw datasets (>300 GB) are not publicly hosted; access is available upon request, which may limit immediate reproducibility by all readers.