Basic
Concepts |
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Here I explain some issues that arise
when the multiple particle tracking is performed on heterogeneous systems.
I give conceptual examples to illustrate limitations and show how the
latter can be overcome using some realistic assumptions. |
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Calculating
the msd |
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To calculate the mean-squared displacement
from a trajectory, this trajectory must be divided into displacements.
If we go back to the trajectory list I was showing in the previous section ![]()
At a lag time of 2 we can extract these displacements:
We can repeat this calculation for all the trajectories, and we can perform this at various lag time. Note that these displacements are overlapping on the same trajectory, meaning that they are not statistically independent. This can become an issue when using quadratic formula of the displacements in estimators and it can be solved by performing a block-average of the displacements. I don't discuss this particular issue in this tutorial, but you can refer to the original article on block-average by Flyvbjerg & Petersen (1989), or find a detailed explanation for the particle tracking case in the article by Savin & Doyle (2007). Both articles are referenced here ![]()
which is valid as long as the trajectory contains the times j and (j+τ). From a trajectory i of duration Ti, you can compute ni displacements, with ni = Ti+1-τ. The longer the trajectory and the smaller τ are, the more displacements can be extracted (and the more precise will be the estimate of the msd). For a given lag time, we obtain a list (or sample) of displacements coming from several trajectories. Ignoring the generating trajectory for each displacement, thus considering all displacements in the same sample, we can plot the distribution (or equivalently the histogram) of the displacements. This is called the van Hove correlation function and it is specified for a given lag time. I show an example of this distribution at 3 different lag times in the figure above (obtained for 0.5 μm-diameter spheres mixed in water). I was able to gather a lot of displacements in this experiment, so the distributions are nicely fitted by Gaussian distributions. The variance of this distribution is precisely the mean-squared displacement. It is a time average because the displacements were originated from various time along each trajectory, and an ensemble average because these displacements come from different trajectories. Repeating this calculation at several lag times, by building new displacements sample, and calculating the variance of the van Hove correlation function generated for each lag time, we can calculate msd(τ) as I did in the first section. As said before, for a given lag time, the more trajectories, and the longer they are, the more displacements the sample will contain. Consequently the ensemble mean-square displacement will be more precisely calculated. |
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Particles
dynamics in heterogeneous systems - distribution of msd |
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Microrheology is often performed in
complex and heterogeneous systems. It is actually one of the main strength
of microrheology to be able to probe a material locally, at a length
scale that is below some material structural size. In that case, a particle
labeled k and fluctuating around
a given location in the material will have a certain dynamical signature,
quantified by its individual mean-squared displacement msdk(τ).
This individual dynamic results from the local mechanical property of
the material at the given location of the particle. Another particle
labeled l located at another
position in the material will exhibit another mean-squared displacement
msdl(τ),
and if the local mechanical property of the material at this location
is different from the one of the particle k,
then msdk ≠ msdl.
This is illustrated in the figure on the right, where I draw the schematic
of a heterogeneous material. Its mechanical property (called ν in
the figure) change with the location in the sample. I show the corresponding
map of msd for the particle dispersed in the sample. In order to calculate
msdk, we would divide
the trajectory k into a sample
of displacements, and then caculate the variance of this sample to have
a time average individual msd. We would repeat the same procedure with
all trajectories to then build a list of msd values, all calculated at
a given lag time. Each of these msd values represents the individual
dynamic of the beads in the material at various location, thus mapping
the corresponding spatial variations of material property. |
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Quantifying
heterogeneity |
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It is now clear that the distribution
(or histogram) of individual msd from multiple particle tracking can
be efficiently connected to the spatial distribution of the material
property of a sample. For example, if the material is strictly homogeneous,
all particles, no matter where they are in the material, will have the
same dynamics. Thus from each particle, the same van Hove correlation
function can be calculated and all mean-squared displacement will have
the same value such that plotting their histogram will show a unique
peak. Another example is a material that exhibits only two different
values of mechanical property. Then particles embedded in a region of
property 1 will all have an msd of value msd1, the ones in
region 2 will exhibit msd2. The distribution of msd will show
two peaks, thus efficiently mapping the distribution of material property.
We give an experimental realization of such a bimodal fluid in the next section ![]() |
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Finite depth
of tracking |
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As we have seen in the first section ![]() ![]() |
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The assumption
of constant density |
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The main consequence of the finite
depth of tracking is that it is not possible to build a meaningful distribution
of msd. From the example above we see that many more msd will be extracted
from a "fast" fluid, whereas few msd will be reported from
the "slow" fluid. This is the case even if the heterogeneous
fluid has as much "fast" region as "slow" regions.
The msd distribution will be biased towards large msd. This bias gets
more intricate when we recall that msd calculated from short trajectories
(which are, again, numerous in the "fast" fluid) are poorly
resolved, producing in the measured distribution a large statistical
spread for large msd. We will see that in the experimental example presented
in the next section ![]()
are (almost) unbiased estimators of the ensemble mean and ensemble variance of the individual msd distribution that correctly map the spatial distribution of the material. This means that:
are the mean and variance we are looking for. These quantities M1 and M2 are the weighted sample mean and variance of the sample of individual msd. The weight wi is proportional to Ti, the duration of the trajectory i, and the set of weight is normalized. Thus the shorter trajectories (which give the less precise msd's) weight less than the longer ones. But since they are more numerous, we will actually get an unbiased estimation. More precisely, the main argument is that the sum Σi Ti for trajectories located in a sub-region of material remains the same from one location to another, even if the material is heterogeneous. We illustrate in the next section ![]() |