Fickian yet non-Gaussian diffusion of a quasi-2D colloidal system in an optical speckle field: experiment and simulations

Abstract

We investigate a quasi-2D suspension of Brownian particles in an optical speckle field produced by holographic manipulation of a laser wavefront. This system was developed to study, in a systematic and controllable way, a distinctive instance of diffusion, called Fickian yet Non Gaussian diffusion (FnGD), observed, during the last decade, for colloidal particles in a variety of complex and biological fluids. Our setup generates an optical speckle field that behaves like a disordered set of optical traps. First, we describe the experimental setup and the dynamics of the particles, focusing on mean square displacements, displacement distributions and kurtosis. Then, we present Brownian Dynamics simulations of point-like particles in a complex energy landscape, mimicking that generated by the optical speckle field. We show that our simulations can capture the salient features of the experimental results, including the emergence of FnGD, also covering times longer than the ones so far achieved in experiments. Some deviations are observed at long time only, with the Gaussian restoring being slower in simulations than in experiments. Overall, the introduced numerical model might be exploited to guide the design of upcoming experiments targeted, for example, to fully monitor the recovery of Gaussianity.

Figure 1

Schematic representation of experimental setup. In figure M1, M2, M3 and M4 are dichroic mirrors, O1 and O2 objectives, L1, L2, L3 and L4 lenses, R1 a polarizer, PBS a polarized beam splitter, and F1 and F2 infrared filters.

Figure 2

(a) a typical image of the optical speckle field with a power of 0.67 W acquired in the sample plane, and (b) the corresponding SLM mask. (c) histogram of pixel intensity and (d) the radial average of the auto-correlation function of the optical speckle field in (a). The solid lines are the fitting curves obtained with Eqs. (1) and (2).

Figure 3

Optical speckle fields for different laser power on the sample (a) 0.15 W, (b) 0.43 W, (c) 0.61 W, and (d) 0.67 W. The white line in panel (a) has a length of 50 μm.

Figure 4

Sample image acquired without one of the two IR filters (a) and including the recorded particle trajectories (b). The scale bar is 50 μm.

Figure 5

(a–c) Optical potential landscapes obtained from the trajectories recorded at laser power P = 0.43 W, 0.61 W, and 0.67 W respectively. The colorbars are in unit of $k_{B}T$; (d) auto-correlation function radial mean of the potential field at power 0.67 W.

Figure 6

(a) TAMSD for laser power 0.43 W (red lines), 0.61 W (green lines) and 0.67 W (blue lines); (b) MSD versus $\mathrm{\Delta }{t}_n$, where the gray solid line with slope 1 is a guide for the eye; (c) TAMSD$/\mathrm{\Delta }{t}_n$ and (d) MSD$/\mathrm{\Delta }{t}_n$ as a function of $\mathrm{\Delta }{t}_n$; (e) $\alpha$ versus $\mathrm{\Delta }{t}_n$; (f) zoom of  the panel (e) to show the transition between region I and II. In all panels, the dotted lines indicate ${\tau }_{I/II}$, and the dashed lines ${\tau }_{II/III}$.  Data in panel (a) and (c) have been shifted for clarity.

Figure 7

DDs in semi-logarithmic scale for (a) $\mathrm{\Delta }{t}_n=0.1$s, (b) $\mathrm{\Delta }{t}_n=50$s, and (c) $\mathrm{\Delta }{t}_n=410$s at different optical powers. Black curves in panel (c) are guides to the eyes corresponding to exponential decay,  with a slope close to the one of the DD at P=0.67W (blue); Panel (d) represents the kurtosis k values of the DDs as a function of the lag time $\mathrm{\Delta }{t}_n$.

Figure 8

(a) simulated optical speckle field of $2000\mathrm{\mu }\text{m}\times 2000\mathrm{\mu }\text{m}$ and zoom of a region of $78\mathrm{\mu }\text{m}\times 78\mathrm{\mu }\text{m}$ (inset), (b) histogram of pixel intensity, and (c) auto-correlation function along x and y direction.

Figure 9

(a) force field (red arrows) of simulated optical speckle field around a few grains; (b) a typical simulated trajectory; (c) simulated and experimental MSD for the lowest optical power $P=0.43W$, in the inset ${\chi }_{\mathit{rid}}^2$ as a function of C; (d) simulated (black lines) and experimental (colored lines) for different optical powers, as indicated.

Figure 10

Comparison between the experimental (colored line) and the simulated (black line) DD at $\mathrm{\Delta }t=410$s at optical power $P=0.43\text{W}$ (a), $P=0.61\text{W}$ (b),and $P=0.67\text{W}$ (c); (d) comparison between the experimental (colored lines) and the simulated kurtosis (black lines).