Anisotropic Memory Effects in Confined Colloidal Diffusion

The motion of an optically trapped sphere constrained by the vicinity of a wall is investigated at times where hydrodynamic memory is significant. First, we quantify, in bulk, the influence of confinement arising from the trapping potential on the sphere’s velocity autocorrelation function $C(t)$. Next, we study the splitting of $C(t)$ into $C_{\parallel }(t)$ and $C_{\perp }(t)$, when the sphere is approached towards a surface. Thereby, we monitor the crossover from a slow $t^{-3/2}$ long-time tail, away from the wall, to a faster $t^{-5/2}$ decay, due to the subtle interplay between hydrodynamic backflow and wall effects. Finally, we discuss the resulting asymmetric time-dependent diffusion coefficients.

Figure 1

(color) (a) 3D lateral view of the experiment; spheres are drawn to scale. A silica particle of radius $a=1.5\mu \mathrm{m}$ trapped by the laser focus is placed next to the surface of a significantly larger silica sphere. This $100\mu \mathrm{m}$ sphere is immobilized between the two coverglass surfaces of the sample chamber. (b) Optical image of the probing particle’s position relative to the wall created by the big sphere. The $3\mu \mathrm{m}$ probing particle was placed at a distance $h=11.5\mu \mathrm{m}$ away from the $100\mu \mathrm{m}$ sphere’s surface and gradually approached. The velocity correlation functions, as well as the diffusion coefficients for the motion parallel and perpendicular to the wall, are measured.

Figure 2

(color) Log-log plot for the measured normalized velocity autocorrelation of a sphere ($a=2.25\mu \mathrm{m}$) held by an optical trapping potential with $k=19\mu \mathrm{N}/\mathrm{m}$ (green circles) and with $k=5\mu \mathrm{N}/\mathrm{m}$ (red squares). The continuous red and green lines are fits to the model of Clercx and Schramm [23]. The hydrodynamic long-time tail $C(t)\propto t^{-3/2}$ (black line) emerges as the spring constant is decreased. Inset: Blowup of the region of anticorrelations on a log-linear scale.

Figure 3

Log-log plot of both normalized VACF, $C_{\parallel }(t)/C_{\parallel }(0)$ and $C_{\perp }(t)/C_{\perp }(0)$, for a sphere ($a=1.5\mu \mathrm{m}$, ${\tau }_p=1\mu \mathrm{s}$, ${\tau }_f=2.25\mu \mathrm{s}$) trapped in a weak optical potential ($k\approx 2\mu \mathrm{N}/\mathrm{m}$, ${\tau }_k=14\mathrm{ms}$). The increasingly anisotropic VACF is measured at four distances from the wall ($h=37.8$, 9.8, 6.8, and $4.8\mu \mathrm{m}$, corresponding to ${\tau }_w=1400$, 96, 46, $23\mu \mathrm{s}$, respectively). The characteristic power-laws from Eq. (2, 3) are represented by black lines as guides to the eye. The experimental data (squares and circles) are compared to the theory that includes hydrodynamic memory, wall effects, and harmonic restoring forces (continuous lines).

Figure 4

Normalized time-dependent diffusion coefficients, $D_{\parallel }(t)/D$ and $D_{\perp }(t)/D$, for the direction parallel and perpendicular to the wall at the same distances $h$ as in Fig. 3. The arrows at the right correspond to the asymptotic values given by Lorentz’s prediction in Eq. (4). The experimental data are represented by symbols, whereas the full lines correspond to the theoretical fits. The arrows at the top and the bottom indicate the respective time points of the shallow maxima.