Test of the diffusing-diffusivity mechanism using near-wall colloidal dynamics

The mechanism of diffusing diffusivity predicts that, in environments where the diffusivity changes gradually, the displacement distribution becomes non-Gaussian, even though the mean-square displacement grows linearly with time. Here, we report single-particle tracking measurements of the diffusion of colloidal spheres near a planar substrate. Because the local effective diffusivity is known, we have been able to carry out a direct test of this mechanism for diffusion in inhomogeneous media.

Figure 1

Schematic diagram of experimental setup. (a) Details of sample. (b) Microscope and feedback loop used to track the beads.

Figure 2

Intensity profile of the central region of a bead stuck to the microscope slide. The stuck bead was kept at height $z_{\mathrm{ref}}$ by a feedback loop, while the Brownian bead diffused above the stuck bead. Inset shows camera image of both stuck and Brownian (freely diffusing) beads. The thin green line indicates the extent of the linear region of the intensity curve.

Figure 3

Brownian diffusion near a horizontal substrate. Sphere of diameter $2a$ diffuses a height $z$ above the surface, in a vertical gravity field $g$.

Figure 4

(a)–(c) Time series of vertical displacements and (d)–(f) histograms. Data from run 10.

Figure 5

(a) Probability density function $P_B\left(z\right)$ of bead height above substrate, $z$. Solid line is a fit to the Boltzmann distribution, Eq. (4). (b) Vertical diffusivity of the bead vs $z$. Solid line is a plot—not fit—of Eq. (3), using material parameters inferred from (a). Data from run 10, with parameters from Table 1.

Figure 6

Diffusing diffusivity and Boltzmann regimes for short and long time intervals, with $\overline{\mathrm{\Delta }t}=\mathrm{\Delta }t(D_0/{\ell }_g^2)$. (a) Excess kurtosis is nonzero for all intervals. Light gray curves show results from 13 individual runs (Table 1). Solid symbols represent the unbiased estimate based on the 13 runs, with the standard error of this estimate shown, too. Uncertainties in the scaled time interval $\overline{\mathrm{\Delta }t}$ are $\approx 2\%$, which is smaller than the symbol size. Pink bars represent the diffusing-diffusivity limit at short time intervals and the Boltzmann limit at long time intervals. (b) Mean-square displacement (MSD). Data from run 10. Dotted line shows linear MSD behavior for short time intervals, with the slope obtained theoretically using parameters from Table 1. Dashed line shows MSD for the difference between two position measurements drawn from the Boltzmann distribution for $U\left(z\right)$, as described by Eq. (8).

Figure 7

Displacements at starting-point intervals of $\approx 0.01$ $\mu \mathrm{m}$ close to the substrate. Right: The nearly Gaussian height conditional distributions. Left: The non-Gaussian distribution from all heights explored by the bead. $\mathrm{\Delta }t=0.033$ s. Data from run 10.

Figure 8

Excess kurtosis of the vertical displacement distribution as a function of the time interval obtained in simulations, with the vertical diffusivity either (a) given by Eq. (3) or (b) constant and equal to the theoretical average. All other parameters roughly match the experimental conditions. The gray curves are the binned results for 13 individual realizations, and the solid circles with error bars are unbiased estimates based on these realizations. The horizontal red lines are the theoretical short- and long-time-interval limits. For details, see the text.

Figure 9

Scaled horizontal and vertical diffusion coefficients, as a function of height $z$. Solid lines are plots—not fits—of Eqs. (1) and (A1), based on parameters from Table 1. The shaded vertical strip depicts the typical range of heights explored, as estimated in Appendix pp3. Horizontal strips show corresponding diffusion constant variations. Data from run 10.

Figure 10

(a)–(c) Time series of horizontal displacements and (d)–(f) histograms. Data from run 10.

Figure 11

Results for horizontal diffusion. (a) Excess kurtosis: individual runs (gray curves) and unbiased estimates (solid symbols with error bars). (b) MSD as a function of scaled time intervals $\overline{\mathrm{\Delta }t}=\mathrm{\Delta }t(D_0/{\ell }_g^2)$ for horizontal and vertical motions. Data from run 10.

Figure 12

Particle potential (thick solid curve), generated using parameters from run 11. The potential minimum, at $z_0$, divides the electrostatic and gravitational regimes. The harmonic approximation is indicated by the dashed curve. The dotted line is $k_{B}T$ above the minimum. The typical fluctuation scale is of order $\delta z$.