Direct Measurement of Hydrodynamic Cross Correlations between Two Particles in an External Potential
Original entry: Tony Orth, APPHY 225, Fall 2009
Jens-Christian Meiners and Stephen R. Quake, PRL vol. 82 p.2211 
Soft Matter Keywords
hydrodynamic coupling, optical tweezer, colloid
Hydrodynamic interaction between two microspheres held in separate optical potentials is observed through time-delayed, negatively cross-correlated motion of the two spheres.
This paper investigates the perturbation to the usual Brownian motion experienced by an isolated colloid due to the presence of another microsphere. This interaction is studied by means of optically trapping two microspheres in close proximity and computing the cross correlation of their movement. A schematic of the experiment is shown in Figure 1. Two orthogonally polarized optical traps are created in the sample volume by focussing through a high numerical aperture lens. The orthogonal polarization of the traps ensures the ability to distinguish signals coming from the two traps - the position of the beam in the quadrant photodiode gives the displacement of the microsphere from the centre of the trap.
An isolated, optically trapped Brownian particle has a position autocorrelation function that decays exponentially. This result can be obtained by considering the Langevin equation for a Brownian particle in a conservative parabolic potential (solved by Laplace transform, hence the exponential decay). The time constant associated with the autocorrelation decay is <math>\tau_i = \gamma / \Kappa_i</math>, where <math>\Kappa_i</math> is the spring constant in the ith direction. The authors find that, even when the spheres are in close proximity, that this relation holds and so no deviation from single particle dynamics is observed from the autocorrelation curve itself (Figure 2).
The cross correlation is the particle positions tell a different story. As can be clearly seen in Figure 2, there is a time-delayed dip in the cross correlation in the longitudinal displacement of the spheres. The magnitude of the time delay is at first a bit puzzling because hydrodynamic interactions at low Reynolds number are instantaneous. The time-delay is in fact due to the time scale of the relaxation of the trap (ie. <math>\tau_i</math>). Near the beginning of the article, the authors state that the time-delay is due to the relaxation of the trap, however, the model that they end up deriving has a weak dependence on the separation of the beads. The data obtained does not show a pronounced shift in the time delay over a roughly 3-fold increase in separation of the beads. In fact, the experimental data suggest that the time-delay at 4.8 um is smaller than at 3.1 and 9.8 um, which does not follow the expected trend. In any case, the time-delay is accounted for relatively accurately by the trap relaxation.
The cross correlation minimum of this hydrodynamic coupling scales linearly with inverse distance within the length range studied (Figure 3). The effect of this interaction is still apparent at 9.8 um, and in special quasi 2D systems, it can extend more than 10 times as far . One limiting factor in to the range of this interaction in more confined geometries is the dissipation of momentum by the rigid boundaries. In , this is minimized by bounding the fluid with a flexible boundary (interface of water and air) and so hydrodynamic coupling decays slower than in 3D. Given that this interaction is relevant to brownian particle on a length scale around that of cellular entities, the authors suggest that perhaps cells or even proteins may use hydrodynamic coupling as a means to create cooperative motion.