3.4 Finite exposure time problem
3.4.3 Application to a binary system with glassy dynamics
The question is if and how the recomputation described above improves the positioning. In order to answer this and to get more information on the finite exposure time error, we again have a look at data from a glassy binary sample of 1µm and 2µm particles, which is already described in section3.3.3.
In Figure3.27 we see the trajectories of a big and a small particle, once determined with the normal
3.4 Finite exposure time problem
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x [µm]
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x [µm]
2D recomputation
small (1µm) big (2µm)
Figure 3.27: Comparison of the resulting particle trajectories in a binary sample, once for the standard 3D SIFT and once with the 2D-recomputed𝑥, 𝑦- positions. The recompution is done by determining the coordinates in the appropriate 2D slices.
Figure 3.28: Normalized self part of the van Hove correlation function in 𝑥direction 𝐺𝑠(𝑥, 𝑡) for a fixed lag time 𝑡 = 4.5s. A comparison is shown of standard 3D SIFT (blue) to 3D SIFT with 2D recomputation (green) and 2D SIFT on an actual 2D measurement of the sample (red). The recomputed curve goes together with the real 2D curve, showing the importance and confirming the correctness of the 2D recomputation.
3D SIFT method and once with after the additional 2D recomputation. Trajectories consists of relatively small movements around an average position, an expected behaviour for glassy dynamics. The difference between the two panels is small but important: Trajectories cover a considerably larger region in the 2D-recomputed version, the particles seem to be a lot more mobile.
Especially for the dynamics of a glass this mobility around the average position, namely the size of the cage formed by the neighbouring particles, is an important quantity to be measured and also a quantity that can be predicted from theories like the mode coupling theory (MCT). Obviously, standard 3D-SIFT suffers from the long exposure time. The position of a big particle is an average over a time of more than half a second (see 2x2 binning in Table3.2), which means positions that are fairly distant from the equilibrium positions can never be seen, since a particle will not stay there for such a long time (cf. AppendixB). By using the 2D recomputation the effective exposure time is reduced to the time needed to capture a single 2D slice (61ms or36ms, see Table3.2). This means a reduction by a factor of 11 for big particles and a factor of 5 for small particles.
A good measure for the mobility of particles is the self-part of the van Hove correlation function𝐺𝑠(𝑟, 𝑡): It is proportional to the probability of finding a particle at the distance𝑟after the time𝑡if it was at the origin at𝑡 = 0. For a comparison of corrected and uncorrected results the self part of the van Hove function𝐺𝑠(𝑥, 𝑡)is used, neglecting the𝑦and𝑧coordinate. It is proportional to the probability of finding
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lag time
τ[sec]
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MS D
1.5 x(τ)2 +y(τ)2®[µ m
2] small big
2D ∆t=61ms 3D corr ∆t=4.1s 3D corr ∆t=50s
3D old ∆t=4.1s 3D old ∆t=50s
Figure 3.29:MSDs computed from𝑥and𝑦coordinates (for𝑧coordinates there is no correction for the finite exposure time problem). Green lines show the uncorrected 3D SIFT data, red lines corrected data.
Without the correction the mismatch between 2D (blue) and 3D measurements is about a factor of1.5.
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∆x
[µm]
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normalized frequqency [a.u.]
small:σx=0.044
big: σx=0.043
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Figure 3.30: Displacement between𝑥coordinates obtained from normal 3D SIFT and𝑥 coordinates obtained with the 2D recomputation. The distribution shows Gaussian character and is the same for small and big particles, meaning that they move similar distances during their effective 3D exposure time, which is approximately proportional to the particles size.
a particle at coordinate𝑥at time𝑡, if this particle was at𝑥= 0at time𝑡= 0. From Figure3.28one can conclude that 2D-recomputed𝑥coordinates lead to the same probabilities as the coordinates calculated from the measurement in a single 2D slice. Uncorrected 3D data totally underestimates the probability of larger moves, so that the particles appear much less mobile. Very big 𝑥moves (𝑥 > 0.5) also seem to be underestimated in the recomputed data. However, this is actually an error in the 2D measurement:
For long distances𝑥the statistic is not very good and a few wrong links will already have a quite big influence. They occur for example if a particle moving out of the focus of the 2D slice is erroneously connected with another particle just moving into the slice shortly later. Wrong links of that kind lead to an overestimation of the frequency of long distance moves.
Even more striking is the importance of the correction for the determination of the MSD. 2D measure-ments were done in order to get information about the MSD at the shortest possible lag times, but the comparison to MSDs from 3D measurements reveals a large mismatch of a factor of∼ 1.5(see Fig-ure3.29). After the 2D recomputation this mismatch disappears and by combining 2D and 3D measure-ments one obtains a smooth MSD curve spanning 6 orders of magnitude in lag times. This reduction for an MSD that is measured using longer exposure times (3D measurements), agrees with the theoretical discussion of Brownian motion in a harmonic potential (see AppendixB).
3.4 Finite exposure time problem For a quantitative measurement of the error due to the long effective exposure time in normal 3D SIFT it is a good idea to look at the distribution of the displacements between coordinates with and without the 2D recomputation as shown in Figure3.30. The Gaussian characteristic fits well together with the assumption of Brownian motion during the capture of the 2D slices. As shown in Table3.2, on average one expects small and big particles to move almost equal distances, which is the reason for the almost equal width of the difference distribution for both particle sizes presented in Figure3.30.