Angles between Successive Steps

Figure 4.8: Histogram of angles between successive steps. Solid lines: Ideal 1D (red) and 2D (blue) random walk. Broken lines: Example distributions of real tracking data. Red: Typical distribution for a molecule following a one-dimensional structure. Blue: Random diffusion in 2D with a slight excess of steps forward.

for 1D diffusion values accumulate at0° and180° (red line). As said above, most of the tracked trajectories in a structured material lie in between 1D and 2D behaviour, like the dotted lines show in the figure. For any kind of random diffusion the number of angles below and above 90° should be equal (solid lines, black dotted line). Deviations from randomness are immediately visible from the graphs, when the curve at an angle of 90° is below or above 0.5. It is not known what the exact distribution - somewhere between 1D and 2D - should be. Nonetheless, independent of its precise shape, the absence of memory implicit in the model of random diffusion requires that P(180°− α) = 1−P(α). That can be checked by plotting both (α,P(α)) and (180°−α,1−P(α)) The curves should be identical, but in practice deviations of the two curves are found.

It has to be tested, if the deviations are significant compared to just statistical noise.

This is done using a Kolmogorov-Smirnov test¹⁴⁵ is applied to test if the deviation of the two graphs is significant. This test searches the maximum distance between two data sets, here the cumulative distribution of angles and its inverse, and calculates the probability that this distance might originate from pure statistical spread of the data points for a given level of significance.

An overview of the ratio of steps forward, backward and to the side for all molecules in one movie can be plotted in a stacked column chart, like shown in Figure 4.9b.

Here, angles between 0° and 60° were considered as steps forward, between 60° and 120° as sideward and 120° to 180° as steps backward. The percentage of each of the categories are plotted in one column per molecule in one movie. In order to obtain a clearer picture and to better compare the data from different movies, the columns were sorted by decreasing ratio of forward/backward steps. Note that molecules which

Figure 4.9: Cumulative distribution of angles and stacked histogram of forward and back-ward steps.(a) Examples for cumulative distributions of angles between successive steps for one individual molecule. Solid lines: Ideal 1D (red) and 2D (dark blue) random walk.

Dotted lines: Example distributions of real tracking data. Black: Equal amount of forward and backward steps. Pale blue: Excess of forward steps. Green: Inverse of the distribution.

For a random distribution of angles the original distribution and its inverse should overlap perfectly. (b) Stacked column chart of forward, intermediate and backward steps for the indi-vidual molecules tracked in one movie, sorted by decreasing forward/backward ratio (black line).

are immobile within the positioning accuracy (green columns) exhibit a large excess of ’steps’ backward, due to a tracking artefact which is described above and in Ref.¹¹⁸ The colour coding in this graph was done according to the classification of molecules into different populations described above. Interestingly, the order of molecules in the stacked histogram follows roughly the grouping into different populations.

Angular distribution for immobile molecules

The angular histogram for the tracked data of an immobile molecule shall be discussed here in a bit more detail, as the outcome of such a plot is fairly puzzling at first sight.

Figure 4.10a depicts such a graph. The U-shaped distribution is striking, implying a very large excess of ’steps’ backward (±180°) and hardly any ’steps’ forward. Tracked data from simulated immobile trajectories, i.e. molecules withD = 0 and a gaussion noise added to the position according to the positioning accuracy of experimental data, show similar distributions. Indeed, the U-shaped distribution of angles is an arte-fact due to the gaussian-distributed positioning error. In the following calculation the forward-backward asymmetry is derived for the 1D case.P(x)is the positioning-error gaussian distribution centred at the true molecular position. If a value ofx_iis measured

Figure 4.10: Histogram of angles for an immobile molecule. (a) Typical U-shaped histogram for an immmobile molecule, with high excess of backward steps. (b) For the case of a gaussian distribution of points around the true centre, the probability is much higher for the subse-quent ’step’ to point in the direction of the centre (backward) than another step away from the centre (forward).

at stepi, then

P_forward =P(xi−1 < xi)P(xi+1> xi) +P(xi−1 > xi)P(xi+1 < xi) (4.2.6) P_backward =P(xi−1 < x_i)P(x_i+1< x_i) +P(xi−1 > x_i)P(x_i+1 > x_i) (4.2.7)

Since the molecule is immobile the probability distribution is the same for alli:

P_forward = 2P(x < x_i)P(x > x_i) (4.2.8)

P_backward = 2P(x < x_i)P(x > x_i) (4.2.9)

Settingp:=P(x < x_i)leads toP(x > x_i) = 1−p, and

P_forward = 2p(1−p) (4.2.10)

P_backward =p²+ (1−p)², (4.2.11)

and the forward-backward asymmetryAis:

A= P_forward

P_backward = 2p(1−p)

p²+ (1−p)² (4.2.12)

A ≤ 1 ⇔ P_backward ≥ P_forward holds regardless of the particular value of p, which can be seen as follows:

P_backward ≥ P_forward (4.2.13)

⇔P_backward −P_forward ≥ 0 (4.2.14)

⇔p²+ 1−2p+p²−2p+ 2p² ≥ 0 (4.2.15)

⇔4p²−4p−1 ≥ 0 (4.2.16)

⇔4(p−¹/2)² ≥ 0 (4.2.17)

An analogous consideration holds for the 2D case (Figure 4.10b).

Now that the materials, experimental method, theoretical method and data analysis have been introduced, we are ready to discuss - in the following three chapters - the experimental results obtained by tracking the diffusion of single dye molecules in sol-gel glasses and mesoporous thin films.

Dye 9A1 in Porous Sol-Gel Materials

In this chapter diffusion within different types of sol-gel glasses is characterised us-ing sus-ingle-molecule microscopy and pulsed-field gradient NMR. First, the macroscopic analysis of the host matrices by adsorption measurements and ensemble fluorescence spectra of the dye used for single-molecule tracking are presented. The main section focusses on single-molecule data collected with wide-field microscopy from two differ-ent sample seriesM3andM22. Single-molecule techniques are especially well-suited for such heterogeneous pore systems. Tracking of single molecules yields the complete distribution of the diffusion coefficients, and, as will be shown here, single-molecule tracking can even distinguish non-Brownian behaviour of individual dye molecules, when the overall diffusion appears to be normal. In this way single molecules act as probes for spatial heterogeneities. In order to obtain a significant statistic to derive the ensemble behaviour, a high number of molecules has to be tracked. A complemen-tary method, which provides information about the ensemble diffusion is pulsed-field gradient NMR. Therefore, the last part of this chapter reports on PFG NMR of the self-diffusion of ethylene glycol in different sol-gel systems, including those studied by SMT.

This project was done in collaboration with different groups: The synthesis of the sol-gel glasses was done by Virginie Latour, Christophe Cantau and Thierry Pigot at the University of Pau (France), sorption measurements were conducted by Pierre Mocho, University of Pau and Tina Reuther from the group of Prof. Bein at the LMU Munich.

The pulsed-field gradient NMR measurements were done in cooperation with Bärbel Krause from the group of Prof. Kärger at the University of Leipzig.

5.1 Synthesis of the Sol-Gel Glasses

A major goal of this work was to elucidate the influence of pore size on molecular diffusion in cast silica xerogels and on spatial heterogeneities in these systems. Two types of xerogel were doped with a new streptocyanine dye: in one of the gels (M3) the pores are about the same size as the dye molecule, in the other (M22) they are much bigger. M3and M22were studied by SMT; these, and a series of similar samples with porosities in the same range were investigated by PFG NMR spectroscopy.

The porous sol-gel materials were synthesized as monoliths by the conventional method of hydrolysis and condensation of tetramethyl orthosilicate (TMOS) in the presence of methanol under neutral conditions (cf. Chapter 2.1¹⁴⁶). Samples suitably clear and crack-free for single-molecule detection were obtained. Porosity depends on the TMOS/methanol/water molar ratio, which was 1/5/4 or 1/5/20 for the two gels in-vestigated using single-molecule spectroscopy.⁹³ The resulting gels are called hereM22 and M3, respectively, after their mean mesopore sizes in nanometres, determined by the adsorption measurements described below. The initial ratio of water to TMOS (wa-ter is also produced during condensation of the silicate) is stoichiometric forM22, but corresponds to a large excess of water forM3. The solutions in methanol were added to unbuffered water (reverse osmosis,5−10S cm⁻¹), and TMOS (Aldrich, 99%) was added in one step. This solution was stirred magnetically for two minutes. Samples of1mL of the sol were closed in polypropylene Eppendorf^rmicrotubes. Gelation was achieved in 10 days in an oven at50°C. The microtubes were then opened for drying the gels at atmospheric pressure for6h at60°C and60h at80°C, after which the sam-ples were kept in the closed tubes until use. Residual water and methanol are present in the dried monoliths. The loss of weight on drying under primary vacuum for four hours at 200°C was about 15% of the dry weight, corresponding to the common mi-cropore volume of the gels, see below in Section 5.2.

Table 5.1: Synthesis recipes for the different sol-gel glasses (molar ratios).

Gel M3 M4 M22 SG8 SG26 SG21

TMOS 1 1 1 1 1 1

APTES - - - - 0.03 0.06

Methanol 5 5 5 5 5 5

Water 20 8 4 4 4 4

In order to gain a broader overview of the ensemble diffusion phenomena in such sol-gel glasses, a larger variety of samples was used for the NMR measurements. In addition toM3 andM22described above, another monolithic sol-gel glass, M4, with

a mean mesopore diameter of about 3− 5nm was used. It was synthesized using the same procedure as for M3 and M22 with a TMOS/methanol/water molar ratio of 1/5/8. In order to obtain larger mesopore diameters and higher degrees of meso-porsoity, samples were synthesized by co-condensation of TMOS and aminopropyl triethoxy silane (APTES ) in two different ratios (3% APTES: SG26 and 6% APTES:

SG21). A summary of the different samples and their syntheses is provided in Ta-ble 5.1. The synthesis of these sol-gel glasses has been published in reference.¹⁴⁷ The purity of all materials was checked by fluorescence spectroscopy on an Edinburgh In-struments FS920 spectrofluorimeter.