Persistent correlation of constrained colloidal motion

We have investigated the motion of a single optically trapped colloidal particle close to a limiting wall at time scales where the inertia of the surrounding fluid plays a significant role. The velocity autocorrelation function exhibits a complex interplay due to the momentum relaxation of the particle, the vortex diffusion in the fluid, the obstruction of flow close to the interface, and the harmonic restoring forces due to the optical trap. We show that already a weak trapping force has a significant impact on the velocity autocorrelation function $C\left(t\right)=⟨v\left(t\right)v\left(0\right)⟩$ at times where the hydrodynamic memory leads to an algebraic decay. The long-time behavior for the motion parallel and perpendicular to the wall is derived analytically and compared to numerical results. Then, we discuss the power spectral densities of the displacement and provide simple interpolation formulas. The theoretical predictions are finally compared to recent experimental observations.

Figure 1

(Color online) Double-logarithmic representation of the normalized VACF for the motion (a) parallel and (b) perpendicular to the wall. The parameter ${\tau }_p/{\tau }_f=0.5$ corresponds approximately to a silica bead in water. Full lines correspond to weakly trapped particles ${\tau }_f/{\tau }_k={10}^{-5}$, whereas the dotted lines are without trapping $\left({\tau }_f/{\tau }_k=0\right)$. The wall is gradually approached, ${\tau }_w/{\tau }_f=16,8,4,2$, corresponding to $h/a=4,2.83,2,1.41$ and the initial decay shifts to the left. The straight lines are guides to the eye for the $t^{-5/2}$ power law.

Figure 2

(Color online) Semilogarithmic representation of the power spectral density for a weakly trapped Brownian particle ${\tau }_f/{\tau }_k={10}^{-5}$ close to a limiting wall for the ratio ${\tau }_p/{\tau }_f=0.5$ [(a) lateral direction and (b) vertical direction]. The distances to the wall are the same as in Fig. 1. The normalization is chosen as the zero-frequency limit of the power spectrum in bulk $S_0\left(\omega =0\right)=2k_{B}T/{\mu }_0{K}^2$. The increase in the initial value is due to the wall according to Eq. (22).

Figure 3

(Color online) Power spectral density with respect to a Lorentzian [Eq. (23)] in the frequency regime where fluid inertia plays a role for the (a) parallel and (b) perpendicular motions. The parameters correspond to the ones of Fig. 2. Also included is the harmonic-oscillator prediction according to Uhlenbeck and Ornstein.

Figure 4

(Color online) (a) Three-dimensional (3D) lateral view of the experiment; spheres are drawn to scale. A silica particle of radius $a=1.5\mu \text{m}$ trapped by the laser focus is placed next to the surface of a significantly larger silica sphere. This $100\mu \text{m}$ sphere is immobilized between the two cover glass surfaces of the sample chamber. (b) Optical image of the probing particle’s position relative to the wall created by the big sphere. The $3\mu \text{m}$ probing particle was placed at a distance $h=11.5\mu \text{m}$ away from the $100\mu \text{m}$ 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 5

(Color online) Log-log plot of both normalized VACF, $C_{\parallel }\left(t\right)/C_{\parallel }\left(0\right)$ and $C_{\perp }\left(t\right)/C_{\perp }\left(0\right)$, for a sphere $\left(a=1.5\mu \text{m},{\tau }_p=1\mu \text{s},{\tau }_f=2.25\mu \text{s}\right)$ trapped in a weak optical potential $\left(k\approx 2\mu \text{N}/\text{m},{\tau }_k=14\text{ms}\right)$. The increasingly anisotropic VACF is measured at three distances from the wall ($h=9.8$, 6.8, and $4.8\mu \text{m}$, corresponding to ${\tau }_w=96$, 46, and $23\mu \text{s}$, respectively). Positive correlations are represented by full symbols and negative ones by open symbols. The characteristic power laws are represented by thick lines as guide for the eyes.

Figure 6

(Color online) Normalized time-dependent diffusion coefficients, $D_{\parallel }\left(t\right)/D$ and $D_{\perp }\left(t\right)/D$, with the bulk diffusion constant $D=k_{B}T/6\pi \eta a$ for the direction parallel and perpendicular to the wall at the same distances $h$ as in Fig. 5. The experimental data are represented by symbols and the full lines correspond to the theoretical fits.