Broadband boundary effects on Brownian motion

Brownian motion of particles in confined fluids is important for many applications, yet the effects of the boundary over a wide range of time scales are still not well understood. We report high-bandwidth, comprehensive measurements of Brownian motion of an optically trapped micrometer-sized silica sphere in water near an approximately flat wall. At short distances we observe anisotropic Brownian motion with respect to the wall. We find that surface confinement not only occurs in the long time scale diffusive regime but also in the short time scale ballistic regime, and the velocity autocorrelation function of the Brownian particle decays faster than that of a particle in bulk fluid. Furthermore, at low frequencies the thermal force loses its color due to the reflected flow from the no-slip boundary. The power spectrum of the thermal force on the particle near a no-slip boundary becomes flat at low frequencies. This detailed understanding of boundary effects on Brownian motion opens a door to developing a 3D microscope using particles as remote sensors.

Figure 1

Schematic of setup to study the Brownian motion of an optically trapped glass microsphere in water near a boundary. (a) Simplified schematic showing a glass microsphere ($3\mu \mathrm{m}$ diameter) trapped at the focus of a 1064 nm optical tweezer near a boundary. The trapping beam is used to detect the horizontal motion of the particle using a high bandwidth balanced detection system [21]. (b) Two $80\pm 1\mu \mathrm{m}$ diameter cylindrical fibers are sealed in the chamber with their axes in the vertical and horizontal directions, respectively, providing well-defined no-slip cylindrical boundaries. (c) The geometry of the sphere and cylindrical boundary to scale. The sphere-wall separation $h$ refers to the distance between the center of the sphere and the surface of the boundary. The perpendicular ($\perp$) and parallel ($\parallel$) directions referred to in the text are as marked.

Figure 2

Measurements of the absolute sphere-wall separations in the perpendicular direction with a $3\mu \mathrm{m}$ diameter silica microsphere. (a) Obtained from fitting the data recorded by the AC-coupled detector to the no-slip flat-wall theoretical VACF. The green squares with error bars represent the experimental data and statistical errors obtained by averaging from 10 measurements at each position. The horizontal axis denotes the distance measured by the strain gauge after subtraction of the fitted offset (discussed in the text). The red line is the $y=x$ line; the VACF fitting gives reliable results as long as the sphere-wall separation is smaller than $7\mu \mathrm{m}$. (b) Obtained from measuring the hindered diffusion coefficient using the DC-coupled detector with sphere-wall separations from $2.5\mu \mathrm{m}$ to $30\mu \mathrm{m}$. The red line is the flat-wall theory, and the blue dashed line is the modified cylindrical-wall theory (see text). The green squares with error bars represent the experimental data and statistical errors obtained by averaging from 10 measurements at each position.

Figure 3

Experimental and theoretical mean square displacements (MSDs) at four positions in the perpendicular direction with respect to the wall. (a) With a sphere-wall separation ($h$) of $30\mu \mathrm{m}$ (${\tau }_w=0.9$ ms). (b) $h=6.1\mu \mathrm{m}$ (${\tau }_w=37\mu \mathrm{s}$). (c) $h=4.6\mu \mathrm{m}$ (${\tau }_w=21\mu \mathrm{s}$). (d) $h=3.1\mu \mathrm{m}$ (${\tau }_w=9.6\mu \mathrm{s}$). The blue circles are the experimental data (${\tau }_p=1\mu \mathrm{s},{\tau }_f=2.3\mu \mathrm{s},{\tau }_k=190\mu \mathrm{s}$); the black lines are the unbounded theoretical predictions [33] and the red dashed lines correspond to the bounded theoretical predictions at various sphere-wall separations. The MSD becomes suppressed as the sphere approaches the wall. The insets show higher resolution of the suppression of the MSD.

Figure 4

Experimental and theoretical velocity autocorrelation functions (VACFs) at the same positions as in Fig. 3 in the perpendicular direction with respect to the wall. The plots show the absolute value (normalized to $〈{\left(v_{\perp }^{*}\right)}^2〉=k_{B}T/m_{\perp }^{*}$) on a log-log scale. The sharp cusplike features correspond to zero crossings. (a) With a sphere-wall separation ($h$) of $30\mu \mathrm{m}$ (${\tau }_w=0.9$ ms). (b) $h=6.1\mu \mathrm{m}$ (${\tau }_w=37\mu \mathrm{s}$). (c) $h=4.6\mu \mathrm{m}$ (${\tau }_w=21\mu \mathrm{s}$). (d) $h=3.1\mu \mathrm{m}$ (${\tau }_w=9.6\mu \mathrm{s}$). The blue circles are the experimental data (${\tau }_p=1\mu \mathrm{s},{\tau }_f=2.3\mu \mathrm{s},{\tau }_k=190\mu \mathrm{s}$); the black lines are the unbounded theoretical predictions [33] and the red dashed lines correspond to the bounded theoretical predictions at various sphere-wall separations. The VACF decays faster as the sphere approaches the wall.

Figure 5

Experimental and theoretical velocity power spectrum (VPSD) at the same positions as in Fig. 3 in perpendicular direction to the wall. (a) $h=30\mu \mathrm{m}$ ($F_w=177\mathrm{Hz}$). (b) $h=6.1\mu \mathrm{m}$ ($F_w=4.3\mathrm{kHz}$). (c) $h=4.6\mu \mathrm{m}$ ($F_w=7.5\mathrm{kHz}$). (d) $h=3.1\mu \mathrm{m}$ ($F_w=16.6\mathrm{kHz}$). The blue circles are the experimental data ($F_p=153\mathrm{kHz}, F_f=68\mathrm{kHz}, F_k=833\mathrm{Hz}$); the black lines are the unbounded theoretical predictions and the red dashed lines correspond to the bounded theoretical predictions at various sphere-wall separations.

Figure 6

Experimental and theoretical thermal force power spectral density (FPSD) at the same positions as in Fig. 3 in perpendicular direction to the wall. (a) $h=30\mu \mathrm{m}$ ($F_w=177\mathrm{Hz}$). (b) $h=6.1\mu \mathrm{m}$ ($F_w=4.3\mathrm{kHz}$). (c) $h=4.6\mu \mathrm{m}$ ($F_w=7.5\mathrm{kHz}$). (d) $h=3.1\mu \mathrm{m}$ ($F_w=16.6\mathrm{kHz}$). The blue circles are the experimental data ($F_p=153\mathrm{kHz}, F_f=68\mathrm{kHz}, F_k=833\mathrm{Hz}$); the black lines are the unbounded theoretical predictions [18, 21] and the red dashed lines correspond to the bounded theoretical predictions at various sphere-wall separations.

Figure 7

Experimental and theoretical position power spectrum (PSD) at the same positions as in Fig. 3 in perpendicular direction to the wall. (a) $h=30\mu \mathrm{m}$ ($F_w=177\mathrm{Hz}$). (b) $h=6.1\mu \mathrm{m}$ ($F_w=4.3\mathrm{kHz}$). (c) $h=4.6\mu \mathrm{m}$ ($F_w=7.5\mathrm{kHz}$). (d) $h=3.1\mu \mathrm{m}$ ($F_w=16.6\mathrm{kHz}$). The blue circles are the experimental data ($F_p=153\mathrm{kHz}, F_f=68\mathrm{kHz}, F_k=833\mathrm{Hz}$); the black lines are the sum of the unbounded theoretical predictions and a constant shot noise and the red dashed lines correspond to the sum of the bounded theoretical predictions at various sphere-wall separations and a constant shot noise. At high frequencies, the position PSDs flatten at $3\times {10}^{-15}\text{m}/\sqrt{\mathrm{Hz}}$, limited by the shot noise of the detection beam [21, 22].

Figure 8

Experimental and theoretical Brownian dynamics in the parallel direction with a sphere-wall separation of $2.9\mu \mathrm{m}$. (a) MSD. (b) VPSD. (c) Absolute VACF (normalized to $\langle {\left(v_{\parallel }^{*}\right)}^2\rangle =k_{B}T/m_{\parallel }^{*}$). (d) FPSD. The blue circles are the experimental data; the black lines are the unbounded theoretical predictions and the red dashed lines correspond to the bounded theoretical predictions.

Figure 9

Fluid fields of steady Stokes flow near a sphere both in free space and bounded fluid (a) pressure field. (b) Shear stress magnitude field. (c) Velocity magnitude field. (d) Vorticity field (azimuthal component). The top four figures are for the unbounded case and the bottom four figures are for the bounded case. The white half circle represents the sphere ($3\mu \mathrm{m}$ diameter) and the left boundary corresponds to the axis of cylindrical symmetry. The purple solid lines represent the no-slip wall and all other boundaries are open boundaries. The sphere is moving downwards at $1\mu \mathrm{m}/\text{s}$. In the bounded case, the sphere is at position with a sphere-wall separation of $3\mu \mathrm{m}$.

Figure 10

Study of the onset of boundary effects by observing the vorticity field (azimuthal component). The sphere (purple half circle, $3\mu \mathrm{m}$ diameter), initially at rest, receives a 1-$\mu \mathrm{s}$ impulse in the downward direction, with an acceleration of $2\text{m}/{\text{s}}^2$ in the first $0.5\mu \mathrm{s}$ and a deceleration of $2\text{m}/{\text{s}}^2$ in the second $0.5\mu \mathrm{s}$. Unbounded case (a) vorticity field at $\text{t}=1\mu \mathrm{s}$. (b) Vorticity field at $\text{t}=5\mu \mathrm{s}$. (c) Vorticity field at $\text{t}=10\mu \mathrm{s}$. (d) Vorticity field at $\text{t}=20\mu \mathrm{s}$. Bounded case with the sphere initially at a sphere-wall separation of $6\mu \mathrm{m}$. (e) Vorticity field at $\text{t}=1\mu \mathrm{s}$. (f) Vorticity field at $\text{t}=5\mu \mathrm{s}$. (g) Vorticity field at $\text{t}=10\mu \mathrm{s}$. (h) Vorticity field at $\text{t}=35\mu \mathrm{s}$. The left edge is the axis of cylindrical symmetry, and top and right boundaries are open boundaries. The white areas are out-of-range clippings: the white areas should be redder than their surrounding color. The bottom edge is an open boundary in the unbounded case and a no-slip boundary in the bounded case.