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, 9 (1), 13222

Nanomotor Tracking Experiments at the Edge of Reproducibility

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Nanomotor Tracking Experiments at the Edge of Reproducibility

Filip Novotný et al. Sci Rep.

Abstract

The emerging field of self-propelling micro/nanorobots is teeming with a wide variety of novel micro/nanostructures, which are tested here for self-propulsion in a liquid environment. As the size of these microscopic movers diminishes into the fully nanosized region, the ballistic paths of an active micromotor become a random walk of colloidal particles. To test such colloidal samples for self-propulsion, the commonly adopted "golden rule" is to refer to the mean squared displacement (MSD) function of the measured particle tracks. The practical significance of the result strongly depends on the amount of collected particle data and the sampling rate of the particle track. Because micro/nanomotor preparation methods are mostly low-yield, the amount of used experimental data in published results is often on the edge of reproducibility. To address the situation, we perform MSD analysis on an experimental as well as simulated dataset. These data are used to explore the effects of MSD analysis on limited data and several situations where the lack of data can lead to insignificant results.

Conflict of interest statement

The authors declare no competing interests.

Figures

Figure 1
Figure 1
Basic characterization of the used 100 nm latex size-standard sample. (Left) FEG-SEM sample of the drop-casted 100 nm latex sphere colloid. (Middle) Single frame from the .avi video file taken by Nanosight NS300 instrument with the 100 nm latex spheres. Overlaid are the measured particle tracks picked for the visualizations in this work. (Right) 100 nm latex size-standard sample size distribution as analyzed by the NTA 3.2 software (Malvern). The median particle size derived by the NTA 3.0 software (FTLA method) is 104.3 nm and the corresponding experimentally derived diffusion coefficient is 470.8 × 104 nm2/s.
Figure 2
Figure 2
Demonstration of the spread of diffusion coefficients derived from the MSD analysis on experimental 100 nm Latex sphere sample. (A) Plot of MSD functions of all 244 measured 100 nm latex particle tracks (in colors) with the averaged MSD function derived from the whole particle assembly (black). The red line represents the fit of the first one-fourth of Δt values by the Eq. (1). (B) The subplots a–j (blue) correspond to MSD analyses of a subset of 24 tracks of the 100 nm latex particle tracks. The subplots k, l (purple) correspond to biased selection of 24 tracks from the measured particle set based on the lowest respective highest diffusion. (C) Resulting diffusion coefficients from the MSD analysis. The indicated ∑all value (431.6 × 104 nm2/s) is a diffusion coefficient derived by MSD analysis of the whole track ensemble. Blue bars a–j are values of diffusion coefficients corresponding to the respective subplots a–j in B. The purple bars k, l are diffusion coefficient values corresponding to the MSD analysis of the biased selection of tracks from subplots k, l. The low number of particle tracks analyzed provides ±30% deviation in the derived diffusion coefficient. As an extreme situation, the two biased subsets that were provided change the results, respectively, to −55% and +65% with respect to the value derived from the analysis of the whole experimental dataset of 244 tracks.
Figure 3
Figure 3
Effect of projecting the three-dimensional Brownian paths to a set of three orthogonal two-dimensional planes. (A,B,D) Projections of a set of 10 three-dimensional Brownian paths to three orthogonal planes XY, XZ, and YZ. Insets in the two-dimensional projections are the calculated MSDs (color) together with averaged MSDs (black) for each track and projection. (C) Original 3D Brownian paths. (E) Result of MSD analysis yielding the diffusion coefficient for the separate projections and increasing the number of original 3D tracks. The values of diffusion coefficient D differ concerning the selected 2D projections of a small number of analyzed tracks (5, 10, 20). For an increasing number of tracks used for MSD analysis, the value of the diffusion coefficient converges to a common value.
Figure 4
Figure 4
Plot of XY projection MSD analysis results for non-overlapping subsets of the simulated 1,000 three-dimensional track ensemble. The depicted value Dth corresponds to the theoretical value of the diffusion coefficient of the 104.3 nm nanoparticle. Overlaid over the point plot is the 95% confidence band derived from published analytical solution based on the total amount of recorded positions and the fraction of the track length used as the maximum Δt.
Figure 5
Figure 5
Effect of subsampling the simulated self-propelled particle trajectory, demonstrating the amount of subsampling needed to unravel the proposed parabolic trend at the very beginning of the averaged MSD curve used to determine propulsion speed. (A) Small segment of one simulated self-propelled particle track and visualization of the subsampling of the initial simulated particle location. (B) MSD analysis done on the self-propelled particle track in sub-image A. Depending on the amount of subsampling, the parabolic trend of the beginning of the MSD function appears. Subsampling by order of magnitude (10x) gives confidence in the parabolic trend. (C) Schematic depiction of the simulated self-propelled particle. The original particle displacements are augmented by a constant-length step in the direction of the respective displacement. (D) Effect of an increasing amount of particle self-propulsion demonstrated on the MSD analysis of sets of 10 simulated particle paths with x0, x0.5, x,1 and x2 added speedup, and x10 subsampling. Here, we demonstrate that considering most of the averaged MSD function, the self-propulsion exhibits itself as an enhanced diffusion, effectively increasing the MSD slope. Only when looking at the very beginning of the MSD function (displacements at very short time differences), can one see the parabolic behavior.
Figure 6
Figure 6
Demonstration of the effect of self-propulsion and convection to an MSD analysis on the same set of 10 two-dimensional tracks. (A) Visualization of the original 10 Brownian particle tracks. (B) Plot of the individual MSD functions together with the averaged MSD from the whole ensemble of Brownian tracks. (C) Visualization of the 10 original Brownian tracks, where a constant speedup was added to the particle displacements, simulating self-propulsion. (D) Result of the added propulsion on the MSD analysis. In this data representation, an increased slope of the averaged MSD is apparent. The increased slope can be interpreted as enhanced diffusion. (E) Visualization of the 10 original Brownian tracks where a constant directional shift was added to the particle displacements, simulating collective motion such as convection of the medium. (F) Result of the added collective motion on the MSD analysis. In this data representation, an upward tilting slope is apparent. The classic method of deriving the diffusion coefficient by fitting with the linear function is not possible in this scenario.
Figure 7
Figure 7
Selected 10 tracks from the simulated ensemble (left) and their corresponding calculated MSD functions (colors match) (right). The thick black line is the combined average of the MSDs (multiple time origins) of the individual particles together with the standard deviation symbolized by the vertical lines. The averaged MSD function is fitted by the formula y = 4DΔt,which yields the diffusion coefficient D.

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