Hydrodynamic interactions play a crucial role in diverse low Reynolds number fluids, including microbial suspensions, particle-laden suspensions, and protein diffusion in membranes. While far-field hydrodynamic interactions are well-understood in two and three dimensions, near-field interactions remain less explored, hindering our comprehension of dense fluid suspensions. This study addresses this gap by experimentally examining the near-field hydrodynamic correlations in quasi-two-dimensional colloidal fluids. The research is driven by the need to better understand the behavior of these complex systems, which have implications for numerous applications in material science, biology, and engineering. Previous research has established the presence of monopole-like hydrodynamic interactions in three dimensions, resulting in drag forces on particles in pairs, both longitudinally and transversely. In contrast, two-dimensional asymptotic far-field solutions reveal a dipolar flow profile with longitudinal drag and transverse antidrag coupling. However, the near-field behavior, especially the transverse antidrag coupling and potential phase differences between transverse and longitudinal correlations, is less understood. This study aims to elucidate these near-field interactions and their influence on transport properties, specifically the validity of the ubiquitous Stokes-Einstein relation (SER) in confined geometries. The SER links bulk viscosity to microscopic self-diffusivity and is fundamental to various processes; however, its breakdown in reduced dimensions remains unclear. This research directly explores the microscopic origins of SER breakdown in two dimensions by examining direction-dependent dynamics and identifying conditions that either violate or restore the SER.

The study builds upon a substantial body of work exploring hydrodynamics in low Reynolds number fluids. Purcell's seminal work on life at low Reynolds numbers established the importance of hydrodynamic interactions. Einstein's theory of Brownian motion provides the foundation for understanding particle diffusion in fluids. Previous research on quasi-two-dimensional suspensions has shown anomalous hydrodynamic interaction and correlated particle dynamics. Studies on interfacial flow around Brownian colloids and the behavior of two-dimensional microfluidic dipoles have also contributed to this understanding. The impact of hydrodynamic interactions on colloidal crystallization and the dynamics of active systems, such as sperm cells and bacterial suspensions, has been extensively studied. Furthermore, previous research on the Stokes-Einstein relation (SER) has highlighted its validity in dilute and dense three-dimensional suspensions, while the origins of its breakdown in reduced dimensions, particularly in two dimensions, have been a subject of ongoing investigation. Several theoretical and computational studies have explored the relationship between the SER and long-wavelength fluctuations in two-dimensional liquids. The current research directly addresses the unresolved issue of SER breakdown in reduced dimensions by focusing on the near-field hydrodynamic interactions in quasi-two-dimensional colloidal fluids.

The experimental setup involved optical video microscopy to observe hydrodynamic interactions in quasi-two-dimensional aqueous colloidal suspensions. Micron-sized polystyrene latex beads were suspended in water, and experiments were conducted at varying packing area fractions (φ). The body-frame (L,T) axes were defined along and perpendicular to the line connecting a particle pair at the initial time (t₀). The displacement correlation data were analyzed using several techniques. Single particle displacement distributions were examined to assess the spatial symmetry of particle motion. Conditional probability distributions were used to reveal hydrodynamic interactions, examining the probability of the displacement of one particle in a pair given the displacement of its partner, both along the longitudinal (L) and transverse (T) directions. The hydrodynamic displacement field was derived from ensemble-averaged displacement correlations. Longitudinal and transverse displacement correlation functions (HL and HT) were calculated and normalized by the packing fraction-dependent single-particle diffusivity (Dself). These functions revealed the presence of dipolar decay profiles in the far-field and oscillatory spatial modulation at higher packing fractions. The spatial modulation of HL and HT were compared to identify phase differences. The relative change in particle separation (Zrel) was also analyzed. To study the Stokes-Einstein relation (SER), single particle diffusion (D(r,θ)) and relaxation (τ(r,θ)) were measured as a function of particle separation (r) and angle (θ). The power-law relationship between D and τ was used to extract SER exponents (ξ). The SER was also examined along the direction perpendicular to the center-of-mass displacement (CM⊥) of a particle pair, where hydrodynamic correlations are minimal. Specific methods included standard tracking algorithms for particle trajectory analysis, and calculation of self-intermediate scattering functions to determine structural relaxation times. The experiments were performed at various packing fractions to assess the impact of particle density on hydrodynamic interactions and the SER.

The experiments revealed several key findings. Firstly, the displacement and relaxation of particle pairs exhibited direction-dependent dynamics in the body frame, attributable to the difference in strength and phase between longitudinal drag and transverse antidrag hydrodynamic interactions. This direction dependence was particularly prominent in the near-field. Secondly, the Stokes-Einstein relation (SER) was found to break down in the quasi-two-dimensional system, with the SER exponent (ξ) being less than -1. This breakdown was linked to the anisotropic hydrodynamic interactions. The magnitude and anisotropy of the relaxation time (τ) highlighted the differing contributions of longitudinal and transverse hydrodynamic modes. The observed phase lag (~0.25σ) between the longitudinal and transverse hydrodynamic correlations was also reflected in the SER exponents along those directions. Thirdly, while the SER was violated in the body frame and even in the lab frame where ξ was found to be -1.16 ± 0.03, it was surprisingly recovered (ξ ≈ -1) along the direction perpendicular to the center-of-mass displacement of a particle pair. This recovery occurred in the far-field and also at specific particle separations in the near-field where hydrodynamic correlations were weakest. These findings highlight that the near-field hydrodynamic correlations directly drive the breakdown of the SER, while the recovery of SER in specific directions emphasizes the critical role of these correlations in determining the overall dynamics.

The findings of this study offer significant insights into the near-field hydrodynamic interactions in quasi-two-dimensional colloidal fluids and their impact on particle transport and the validity of the Stokes-Einstein relation. The direction-dependent dynamics observed directly demonstrate the anisotropic nature of hydrodynamic interactions in this confined geometry. The breakdown of the SER, a commonly observed relationship in many fluid systems, is explained mechanistically through the interplay of longitudinal drag and transverse antidrag, along with their spatial phase lag. The fact that the SER can be recovered along specific directions where hydrodynamic correlations are minimal suggests that the SER validity is directly coupled to the strength and directionality of the hydrodynamic interactions. The results emphasize that simplified models of hydrodynamic interactions that assume isotropic behavior might be inadequate in characterizing systems with strong near-field interactions. The study's insights are crucial for improving our understanding of transport phenomena in dense, confined systems, which is crucial for a range of applications.

This research provides a comprehensive experimental investigation of near-field hydrodynamic interactions in quasi-two-dimensional colloidal fluids. The direction-dependent dynamics and the breakdown of the Stokes-Einstein relation are directly linked to the contrasting magnitudes and phase shift between longitudinal and transverse hydrodynamic modes. The recovery of the SER under specific conditions, where hydrodynamic correlations are minimal, provides a mechanistic understanding of the observed anomalous behavior. Future studies could explore these near-field hydrodynamics with anisotropic particles and investigate their impact on self-assembly, and other phenomena in dense, confined systems. The experimental insights will hopefully guide the development of improved simulation techniques for dense suspensions in confined geometries.

The study is limited to quasi-two-dimensional systems using spherical colloidal particles. The generalization of these findings to other geometries and particle shapes requires further investigation. The analysis focuses primarily on near-field interactions, with less emphasis on far-field behavior at very large particle separations. The use of optical microscopy imposes limits on the range of packing fractions that can be accurately studied, as well as the dynamic range and resolution of the measurements.