RESULTADOS DE FUNCIONALIDAD FAMILIAR SEGÚN MODELO

Introduction: The particle tracking method in combination with widefield microscopy proved to be the method of choice for the characterization of molecular diffusion in mesoporous materials. In a second study, this appraoch was used to readdress the

question of diffusion in a structured mesoporous material with unidimensional pores, like the MCM-41 material. In this study TDI (fig. 76a) was incorporated into the channels of a spin-coated SBA-15 film (see fig.76c and 76d), which has hexagonally arranged

unidimensional cylindrical pores - very simialr to MCM-41 (cf. section 4.4.).

Figure 76: TDI in SBA-15 films. a) Structural formula of TDI. b) Structural formula of the template Pluronic P123. c) General appearance of a spin coated SBA-15 film d) TEM images of a thinned SBA-15 sample showing the pores arranged in (fingerprint) domains The SBA-15 material is synthesized (in the group of Prof. Dr. Thomas Bein, LMU) using P123 (a block co-polymer, fig. 76) as a supramolecular template for the formation of the unidimensional channels. TDI is added during synthesis at a concentration of ca. 10-9 M,

and the material is left uncalcined. The diameter of the hexagonally arranged cylindrical pores can be estimated from XRD data and are between 6 and 7 nm. TEM images of materials synthesized under identical conditions show that the pores are organized in domains, in which the pores lie parallel to the surface of the substrate (the microscopy cover-slide). The size of these domains are in the order of a few hundred nanometers (fig. 76d).

The main advantage of the spin-coated material over the monolithic material discussed in the previous study, is that the out-of-plane diffusion of the molecules is suppressed, because the thickness of the spin-coated films is considerably smaller than the focal depth of the microscopy apparatus (100-200 nm film thickness versus 3 µm focal depth).

Trajectories can thus be observed over a longer period of time, allowing for a significant improvement in the statistical quality of the data.

Fig 77: Trajectory map obtained from different sequences of images on different regions of the the same sample. The two trajectories of molecule A and molecule B are discussed in more detail.

Microscopy image sequences show the individual diffraction limited patterns, from the individual TDI molecules, which are mobile. The trajectories are obtained by tracking the emission patterns in a sequence of images, like shown before for 9A1 diffusing in the sol- gel-glass. The average positioning accuracy in this case is 30 nm, typical acquisition

times lie between 40 ms and 100 ms depending on the quality of the sample. Two sets of trajectories are shown as an overview map in figure 77. A quick characterization of the diffusion on a global level, that is, estimating an effective diffusion coefficient from all mobile trajectories assuming a 2D random walk, gives an effective diffusion coefficient of Deff = 4.4 · 10-10 cm2s-1. This value, however, only describes a fictitious average behavior. Most of the trajectories in these maps show particularly pronounced structural features. The analysis of individual trajectories, carried out here on two exemplary trajectories (figure 78), reveals (a) details that are not sensed by other techniques used to characterize diffusion and (b) that the simple model of a 2D random walk is not

describing the data well.

Fig 78: Two trajectories of TDI diffusing in SBA-15 films. The trajectory of molecule A shows regions in which the molecule appears to remain for a longer time, and regions in which the molecule appears to be moving faster. The trajectory of molecule B is visibly structured.

Step-size distribution: The trajectories of the two molecules are first analysed in order to obtain the step-size distributions between successive frames. As described in section 2.3.2. the individual step lengths are sorted and assigned a relative rank j/N. The plot of j/N versus step length corresponds to the complementary cumulative probability C(R,t) (equation 18 - given below, cf. also section 2.3.2)

C(R,t) = 1 - P(R,t) =

∫

∑

C(R,t) which is a Gaussian or a sum of Gaussians centered at origin, depending if

different distinguishable sub-populations of step sizes are present in the data. The curves for the two trajectories are shown in figure 79. The results of the fitting procedure are resumed in the table given below.

Figure 79: Step length distribution for the trajectories of molecules A and B (N = 482 and N=97, respectively). Below the histogrammes and the corresponding envelope curves are shown for the two trajectories.

molecule A molecule B δ t 42 ms 78 ms m 2 1 N 482 97 〈r12〉 27600 ± 1000 nm2 67600 ± 5000 nm2 〈r22〉 121100 ± 2000 nm2 - rmax, 1 83 ± 10 nm 143 ± 20 nm rmax, 2 174 ± 10 nm -

In the case of molecule A the distribution points to the presence of two distinguishable sub-populations of step lengths. These two populations of step lengths in the case of molecule A are not spatially separated, the longer steps occur on the entire trajectory (fig.80). This molecule alternates between fast and slow steps throughout the trajectory.

Figure 80: Distribution of fast steps in the entire trajectory. The steps contributing to the fast population in the histogram are highlighted in the trajectory to the right.

The trajectory of molecule B shows one population of step lengths. The smooth distribution curves (plotted in the lower graphs of fig. 73) are obtained using the 〈ri2〉

found via fitting. The curves envelope the histogrammes obtained in a conventional way (via data binning).

Mean squared displacements / MSD versus time: The mean squared displacements MSD for different times are obtained from a population analysis. The MSDs are plotted versus time for molecule A and molecule B in figure 81. A linear regression through the data, assuming 〈r2(t)〉 = 4 D·t, gives an effective diffusion coefficients for molecule A and B respectively: DA,E = 5.6 · 10-10 cm2s-1 and DB,E = 2.1 · 10-9 cm2s-1. Note, however, that this diffusion model is not justified, despite the fair agreement with the data: (a) molecule A shows sub-populations of slow and fast steps; and (b) the trajectory of molecule B does not appear to be isotropic.

Figure 81: MSD versus time plots for molecule A and B.

Angles between steps: The angles between successive steps can be used to check whether the trajectories show directional anisotropies. (Note: Step angles are defined as 0° for two consecutive steps in the same direction, ±180° for a forward step followed by a backward step, and negative angles in counter-clockwise direction.) The parameters inferred so far, the trajectory themselves, the step-length distributions and the MSD versus time behaviour, pointing to a deviation from the isotropic random walk model, but are not clearly convincing. The histogram of step angles for the two trajectories, shown in figure 82, is very clearly showing that the trajectories cannot be described by isotropic motion models.

the molecule is predominatly moving forward and backward. Significantly fewer steps have an angle of ±90° in this trajectory. In this case the trajectory bears more similarity to an unidimensional random walk along a tortuous channel than with an isotropic two- dimensional one. The trajectory reflects a real structural feature within the host material (a bundle of curved channels, for example).

Figure 82: Histograms (bin = 10°) for the angles between successive steps in the trajectories of molecules A and B.

Outlook: The analysis of the trajectories from individual molecules allows for a very thorough characterization of the diffusional behaviour, as details of the diffusion processes become revealed. While a wealth of information can be gained from the trajectories, on the other hand the interpretation becomes more complex, as many details remain unknown. (a) The underlying structure of the host matrix, for example, is an important unknown factor: What structural feature of the host is being mapped by the tortuous trajectory? Domain boundaries? Tortuous channels? One possibility to clarify this would be to superimpose TEM images with the trajectories, recorded in the same area of the sample. (b) Is there a more direct proof showing that the molecules are

diffusing in comparatively narrow cylindrical pores? This could for example be answered by looking simultaneously at the diffusional and orientational behavior. (c) What causes a molecule to predominatly move backward? TDI is an asymmetric molecule with a

flexible n-heptyl tail, which could be giving rise to a preponderance of backward steps. Simulations and measurements on symmetric TDI derivatives could provide the