Complex Nanoparticle Diffusional Motion in Liquid-Cell Transmission Electron Microscopy

Liquid-cell transmission electron microscopy (LCTEM) is a powerful in situ videography technique that has the potential to allow us to observe solution-phase dynamic processes at the nanoscale, including imaging the diffusion and interaction of nanoparticles. Artefactual effects imposed by the irradiated and confined liquid-cell vessel alter the system from normal “bulk-like” behavior in multiple ways. These artefactual LCTEM effects will leave their fingerprints in the motion behavior of the diffusing objects, which can be revealed through careful analysis of the object-motion trajectories. Improper treatment of the motion data can lead to erroneous descriptions of the LCTEM system’s conditions. Here, we advance our anomalous diffusion object-motion analysis (ADOMA) method to extract a detailed description of the liquid-cell system conditions during any LCTEM experiment by applying a multistep analysis of the data and treating the x/y vectors of motion independently and in correlation with each other and with the object’s orientation/angle.


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2. The Motion Position Smear full width half maximum (FWHM) is set as half the average velocity due to motion in positive and negative x-direction or y-direction: (FWHM) = [(1/2)*V x ] or [(1/2)*V y ] 3. One standard deviation in position error due to motion "smear" is then taken as: Boothroyd et al. 5 • The Boothroyd 5 PSF was measured using the focused beam method: central spot of diffraction pattern through vacuum (using: 100 mm camera length, small selected arear diffraction SAD aperture, energy filter on) is recorder directly on an Ultrascan1000 CCD camera, where the TEM beam diameter is smaller than CCD pixel size (entire TEM beam is incident on one pixel, but residual signal is transferred into adjacent pixels due the camera's PSF). A line scan is then taken across the diffraction spot to measure the blurring of the signal into adjacent CCD pixels.
3. Using ImageJ à the Average Signal and Noise (σ) of each micelle and of its local background were measured (Tables S1-S4) and plotted ( Figure S2) 4. Using ImageJ à Created a series of 10 simulated LCTEM images for S3 ( Figure S3): 6. Using ImageJ à Created a series of 10 simulated LCTEM images for S10 ( Figure S5): 7. Using ImageJ à Created a series of 10 simulated LCTEM images for S11 ( Figure S6):     (Table S6) Video Frame from S11 S13 • The 4 largest 0.7-AR ellipses (70 x100 pixels) and the 4 largest 0.8-AR ellipses (80 x 100 pixels) were tracked in the reference and simulated images • For the LCTEM simulated images, the center of mass position error is σ = 2.54 nm • For the LCTEM simulated images, the major axis orientation angle error is σ = 9.06˚ 3. MOTA tracking algorithm was applied to all 10 of the simulated [and reference] LCTEM images for video S10 (Table S7) • The 4 largest 0.7-AR ellipses (70 x100 pixels) and the 4 largest 0.8-AR ellipses (80 x 100 pixels) were tracked in the reference and simulated images • For the LCTEM simulated images, the center of mass position error is σ = 3.32 nm • For the LCTEM simulated images, the major axis orientation angle error is σ = 7.86˚ 4. MOTA tracking algorithm was applied to all 10 of the simulated [and reference] LCTEM images for video S11 (Table S8) • The 4 largest 0.7-AR ellipses (70 x100 pixels) and the 4 largest 0.8-AR ellipses (80 x 100 pixels) were tracked in the reference and simulated images • For the LCTEM simulated images, the center of mass position error is σ = 2.54 nm • For the LCTEM simulated images, the major axis orientation angle error is σ = 9.06˚  Table S7. Final center of mass (x, y) position error (σ) and major axis orientation angle error (σ) for the MOTA-extracted trajectories from LCTEM video S10.

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Section III: Effects of static and dynamic errors on the estimate of the scaling exponent of a diffusing particle.
Static localization error, which stands for the random error in the measurement of an immobilized particle's position, has an impact on the mean squared displacement (MSD) of a diffusing particle where D is the diffusion coefficient in units of L 2 /T, = ∆ stands for the time, ∆ is the elapsed time between two measurements, and is the static localization error, practically expresses the standard deviation in measured positions of an immobile particle. 7,8 The additional effect of motion blur (dynamical error), wherein a particle's average position over the camera frame interval ∆ is measured, has been taken into account and the MSD reads where R is the "motion blur coefficient". micelles under different dose rates in x-and y-axes. We should notice that equation (S.1) has been used S17 twice, namely, x-trajectory of the S6_2_A micelle, and for the y-trajectory of the S11_NP_C micelle. Equation (S.1) for the rest of the recorded trajectories gives negative values of the square of the static error, which has not has a physical meaning. , the blur coefficient, R, the diffusion coefficient, D. When equation (S.2) is used for fitting then we notice γ=---, which means that the actual value of the scaling exponent is one, γ=1. Furthermore, if the blur coefficient is not used for fitting then we notice R=----. We keep the same description in Tables S10 and S11. For the lowest dose rate, the mean value of σ 2 cam / σ 2 fit in x-axis is 0.4, and in y-axis is 0.33. It means that the static error as well the motion blur contribute to the particle's motion roughly about 40% and 33% in each axis respectively . However, these values are accompanied by scaling exponents less than one in three cases while in the rest three the scaling exponent is one. By using equation (S.4) the variation of the S18 scaling exponent is illustrated in Figure S7. All motions except one start superdiffusive and then converge to a constant value describing sub or normal diffusion.
There is no a single trajectory where its anomalous character can be attribute to the static error and/or motion blur. Initially, micelles undergo a super diffusive motion which slows down over time and converges either to sub-diffusion or to normal diffusion in the long time limit. Notice that the normal diffusion here is described by scaling exponents a bit higher than one, close to 1.1. x-axis y-axis   By analyzing the effects of static localization error and of motion blur on micelles anomalous motion for these sets of experiments one can safely conclude that their contribution on particles motion is negligible.
Furthermore, there is no single experiment where its anomalous character can be solely associated either to static error and/or of motion blur. We see that the lower the dose rate the higher the contribution of these factor, probably connected to contrast effects, without of course to be considered as dominant sources of anomalous diffusion.

Section IV: Preliminary analysis of raw data
The raw data monitor the coordinates (x-and y-axes expressed with respect to the lab frame) of each micelle in the course of time with nanometer resolution.

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For each micelle we construct the following time series: ! , { ! } with n=1,2,…,N, where N is the total number of data points, which correspond to the recorded coordinates {x(t), y(t)}. If τ is the minimum time lag between two consecutive measurements and T the total length of the recorded trajectory then T=N τ. Figure S8 shows the lateral displacement of a micelle (red colour), as well the evolution in time of each one of the recorded coordinates (x and y). One can easily notice that the motion depicting either the lateral displacement or the jumps taken in x-or y-axes contains some big jumps interrupted by periods of constrained motion in a small area or by periods of complete immobilization. A very first analysis of such a behaviour can be made by examining the probability of length steps taken by the micelle, P(l). The length step is given by the absolute value of the difference between two consecutive points either in x or in y axis, while, for lateral displacements the length is ! = ( We define the random variable ξ(i)=l(i)/<l>, where <l> is the mean, and we estimate the probability distribution P(ξ). Figure S9 shows the probability distribution P(ξ) versus ξ for some micelles under different radiation dose conditions. Figure S9. Probability distribution P(ξ) versus ξ for movements either in x or in y-axis or lateral displacements. Colour code and i.d. of micelles are exactly the same as those illustrated in Figure S8.
For 1.6 e-/Å 2 s radiation dose rate, the probability P(ξ) poses its maximum at value ξ=0, which underlines immobilization of the motion and it is a characteristic feature of Continuous Time Random Walk (CTRW) 9 .
Interestingly, the same probability for lateral displacements attains its maximum again at ξ=0 but the x-axis x-axis x-axis y-axis y-axis y-axis x-y x-y x-y S10_NP_D S6_NP_E S3_NP_E S22 value of the probability is significant reduced with respect to the corresponding probabilities for motion in x-and y-axes. This finding indicates that, there exist time periods where the motion is immobilized on the surface but there also exist time periods where the motion is immobilized in one axis and is free to move in the second one. For 2.6 e-/Å 2 s radiation dose rate, (i.d. S6_NP_E), the maximum of the probability distribution moved from zero to a value very close to it. A first sign of the role of the radiation dose is observed, the higher the dose rate the lower the number of immobilization events. 8 For lateral displacements the probability for immobilization events is zero. The same behaviour is also observed for 5.6 e-/Å 2 s radiation dose rate. In general, as the radiation dose increases, the probability of finding immobilization events decreases.
The shape of the probability of length steps suggests that CTRW 9 is likely the stochastic mechanism.
CTRW is an ageing random process, which means that the choice of the initial starting point of the where, = Δ is the running length, and = Δ is the total length of the trajectory, ≤ . Using equation (S.6) we take as minimum length = /2. The behavior of ! for motions in either x-or yaxes or for lateral motion is illustrated in Figure S10a. The distribution of trapping events of a given sequence of raw data can be further connected to the effects of the radiation dose rate since the latter continuously provides energy to the system. We further explore this point by checking ageing effects of the time-reversed time-series. We analyse the same trajectories S24 presented in Figure S3a, where now the coordinates read, x(j)=x(N-i+1), y(j)=y(N-i+1), for i=1,2,3,....,N, and the results are illustrated in Figure S3b.  Figure S9.
CTRW is a random process characterized by long-tailed distributions of waiting times, which has the form ψ~! !!! , with 0< <1, if we consider waiting times with infinity mean, or 1 < < 2 considering waiting times with infinity variance. 13 This probability distribution corresponds to time periods between successive jumps and can be easily obtained by using the raw data. The strict condition for the estimation of ψ considers as steps taken by the particle only those with length different than zero. This condition can relax considering also as steps contributing to waiting times those with displacements much smaller than the S25 size of the particle. We have used as cut-off a value of 0.1 { } for each micelle and the estimated waiting times probabilities are depicted in Figure 3 of the main text.
For low rate dose (1.6 e-/Å 2 s), all analysed raw data (lateral displacement, in red, or independent movements in either x-or y-axis, purple and green respectively) for the micelle with id. S10_NP_D give a scaling exponent in the range 0.64 ≤ ≤ 0.78, and 0.

Section V: Analysing Rotational movements
The available rotational time series have been analysed by using equation (5)   The examined micelles are of elliptical shape (aspect ratio ~0.7). It is also available the variation of the aspect ratio over time for each one of the analysed micelles. We obtain the moments for each one of these time series by applying equation (5) of the main text, and in the following we demonstrate the first, and the second moment, as well the variance for each micelle and for all dose rates. All depicted measures have not have time dependency, indicating thus that aspect ratio does not change in time. Figure S12. The variance and the first two moments (first and second) of the aspect ratio as a function of the lag time for different micelles under different irradiation conditions. In all cases, variance and moments are constant in time indicating that micelles retain their initial elliptical shape during the experiment.