# Visual Analysis of Stochastic Trajectory Ensembles in Organic Solar Cell Design

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## Abstract

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## 1. Introduction

- We propose a set of linked visualization techniques that enable the investigation of dense charge-transport trajectory ensembles by exploiting trajectory abstraction and relating trajectories to a solar cell’s morphology.
- We propose novel geometric measures to analyze the efficiency of individual trajectories and trajectory ensembles based on the concept of charge flow lines.
- We discuss how these components are integrated into a single visualization framework, which supports domain experts when visually analyzing organic solar cell simulations.

#### 1.1. Application Task Characterization

#### 1.2. Organic Solar Cell Design

#### 1.3. Some Details about the Data

## 2. Related Work

## 3. Trajectory Exploration Framework

#### 3.1. Spatial Views

#### 3.1.1. Quadrant I: Ensemble Visualization, Microscopic View

#### 3.1.2. Quadrant II: Ensemble Visualization, Macroscopic View

#### 3.1.3. Quadrant III and IV: Inspection of Individual Trajectories

#### 3.2. Statistical Plots and Efficiency Measures

- (i)
- Trajectory-based measuresThese measures have the purpose to equip the macro-level charge flow lines with measures that are commonly related to flow. A central measure is the effective velocity, which describes the macro-level charge velocity. The measures can be adapted to the chosen level of detail via a scale parameter r. The unit for r is intermolecular distance.
- Escape time ${t}_{e}({M}_{i},r)$. The escape time ${t}_{e}({M}_{i},r)$ of a charge from molecule ${M}_{i}$ with respect to scale r is defined as the time the charge needs to leave the r-neighborhood of the molecule, see Figure 8c. It is high in regions where the charge is trapped for a longer time. The dwell time at a molecule corresponds to the escape time for $r=1$.
- Effective velocity ${v}_{e}({M}_{i},r)$. The effective velocity ${v}_{e}({M}_{i},r)=r/{t}_{e}({M}_{i},r)$ is directly related to the escape time. Low velocity hints at low efficiency in the charge transport, this can be due to traps in the morphology or a strong inter charge interaction.
- Effective distance traveled ${d}_{eff}\left(t\right)$—The effective distance is the Euclidean distance of the current charge position to the start position as function of time. This measure is related to the escape time but allows a stronger focus on the geometry of the trajectory.
- Tortuosity $T\left(t\right)$—Tortuosity sets the actual path length $l\left(t\right)$ of the trajectory in relation to the effective distance traveled $T\left(t\right)=l\left(t\right)/{d}_{eff}\left(t\right)$.

- (ii)
- Charge pair related measuresThe morphology of the material is not the only critical aspect for the efficiency of the charge transport. There is also a strong interaction between individual charge pairs influencing their transport. If charge pairs come very close to each other, this comprises the risk of recombination, which means that the charge is lost for the entire process.
- Pair distance ${d}_{p}\left(t\right)$—This distance measure keeps track of the Euclidean distance of a hole and an electron created in one CT state. In the optimal case this would be a monotonously increasing function of time.
- Minimal distance to charge of other kind ${d}_{pmin}\left(t\right)$—In the case of multiple CT states a recombination is not only possible with the own ‘partner’ (geminate recombination) but with all charges of the complementary type (nongeminate recombination). In this way it is a generalization of ${d}_{p}$.
- Minimal distance to charge of same kind ${d}_{min}$—Charges of the same type interact with each other and can thus reduce the effective transport. This measure gives an overview over the distribution of the charges within the material. Charges of the same type interact with each other and can thus reduce the effective transport. This measure gives an overview over the distribution of the charges within the material.

- (iii)
- Morphology related measuresThe morphology is a critical parameter for the design of the solar cells. While a large interfacing surface is advantageous for the creation of CT states, a complex morphology can crate traps for the charge transport.
- Distance to interface ${d}_{interface}\left({M}_{i}\right)$—This distance is the shortest distance of molecule ${M}_{i}$ to the material interface. It is computed once for each morphology. As distance metric we use a Manhattan metric following the molecular grid structure. Thus the distance roughly corresponds to the minimal number of charge transitions necessary to reach the interface. Since recombination of charges only happens at the material interface it is favorable that the charges keep a certain distance to the interface.
- Morphology Composition plots—Through these plots the morphology composition (acceptor-donor ratio) can be investigated. It displays the acceptor-donor material ratio in the neighborhood surrounding a charge. For a single trajectory, the ratio is plotted for the entire transportation (Figure 7b). For ensembles, the morphology ratio at a specific time (Figure 7c) is plotted. The acceptor-donor ratio is computed for a spherical region. The spherical region is divided into eight octants illustrating the distribution in each octant. (Figure 7d).

## 4. Technical Details

#### 4.1. Charge Flow Lines

#### 4.2. Ribbon Computation

## 5. Use Cases

#### 5.1. Scenario 1—Simple Planar Interface One CT

#### 5.2. Scenario 2—Complex Interface Exploration

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The visualization system has become an essential part of the scientific process in the applications and the specific tasks toward the system have been developing continuously. New hypotheses that are developed during the visual exploration trigger new tasks and new visualization methods as specific efficiency measures used for statistical plots and line abstraction.

**Figure 2.**Our visualization parameter space can be roughly divided into four quadrants micro-level vs. macro-level of detail, ensemble vs. single trajectory). The parameter space can be investigated using several visualization techniques, which are associated with the four identified task groups. To move between the single and the ensemble level, brushing-and-linking is realized using plots.

**Figure 3.**Illustration of the different organic solar cell setups. A simple organic solar cell consists of two layers while single (

**a**) or multiple excitons can be considered (

**b**). More complex morphologies reduce interface distances and can also be considered with single (

**c**) or multiple excitons (

**d**). We use a color scheme assigning red to donor material and trajectories and blue to acceptor material and trajectories for all visualizations.

**Figure 4.**Screenshot of the system with annotations: It supports the exploration of the trajectory ensemble on different levels of detail. Single trajectories can be selected in qualitative plots (here effective distance a charge has traveled), which then can be inspected as isolated flow lines or with respect to the charge coverage volume within the morphology. An overview of the ensemble distribution provides contextual information.

**Figure 5.**The simulation uses periodic boundary conditions, meaning that charges leaving the volume on one side will enter it again on the opposite side. Thus the trajectories are disrupted and the resulting density distribution misleading. The system supports a periodic continuation of the volume to get an untangled representation. The image on the left (bottom) represents raw data in the original simulation volume in contrast to the same data in the extended volume shown on the right. The image on the top left shows a slice through the morphology. The image in the center results from a periodic continuation of the morphology with one selected flow line. The original volume is highlighted by a red square.

**Figure 6.**Trajectory visualization on different levels of abstraction. (

**a**,

**c**) visualization of multiple charge trajectories together with ribbon arrows giving a hint of the general trend of the charge movement. (

**b**,

**d**) show one selected trajectory with micro and macro abstraction level and rendering options. (

**e**–

**h**) show single trajectory abstractions. (

**e**) The stripe pattern is a measure for the effective velocity of the charge. The time the charge needs for one stripe is constant. (

**f**) Arrow representation added to simplified trajectory representation. (

**g**) Direct rendering of raw trajectory represented using tube rendering. (

**h**) Charge coverage volume visualization of single trajectory.

**Figure 7.**Interactive plots of morphology composition of single trajectory and ensemble. (

**a**) selected distance measure at a certain time step, with one selected trajectory pair highlighted (

**b**) acceptor-donor ratio along selected trajectory. (

**c**) morphology composition of all trajectories at a specific time step t = 0.4. (

**d**) stack and radial plot for octant acceptor-donor ratio of selected trajectory at specific time.

**Figure 8.**Different simplifications can be applied to a trajectory. The red lines are the acceptor trajectories and the grey-blue lines the donor trajectories. (

**a**) One raw trajectory pair, (

**b**) charge flow line with abstraction level $r=4$ in comparison with Gaussian smoothing $n=4$. The charge flow line is colored with respect to time (same color for donor and acceptor). (

**c**) The escape time measures the time a charge needs to leave the r-neighborhood of a molecule for the first time. (

**d**) charge flow line computation.

**Figure 9.**

**Top row**illustrates various single trajectory representations on a synthetic data. Velocity is color mapped.

**Bottom row**represents ribbon representation, which uses desired curvature range for placement of arrows and co-ordinate frame correction applied.

**Figure 10.**Scenario 1: Simulation evaluation (SE) of the flat morphology with one charge pair. The images show different rendering options for the entire ensemble (

**a**) trajectories embedded in a charge coverage volume visualization, trajectories reaching the electrode are displayed as spheres colored by time they need to reach the electrode, (

**b**) density projection of trajectories; for one selected trajectory embedded in the charge coverage volume (

**c**) original trajectory colored by progression time (

**d**) flow line displaying the effective velocity as stripe texture.

**Figure 11.**The plots of the derived measures for the ensemble highlighting one selected trajectory, the x-axis of all plots is time, which is also encoded in the color of the trajectories. The y-axis are (

**a**) effective distance travelled from start point; (

**b**) distance between the charge pairs; (

**c**) shortest distance to the interface; (

**d**) escape time for a radius of 10 units; (

**e**) parallel coordinates of all the measures (

**a**) through (

**e**).

**Figure 12.**These images show examples of simulations ((

**a**) Flat Interface, (

**b**) Complex Interface) where the charge transport shows an unexpected behavior. The interaction between the two charges (donor in blue/acceptor in red) is so strong that they stick together for the entire simulation time. They never reach the electrode. The strength of this effect only became visible to the physicist through these visualizations. This observation led to a reconsideration of several simulation parameters and the introduction of the ‘distance between charge pairs’ as additional efficiency measure.

**Figure 13.**Overview visualizations for different time steps (columns). The selected time step is highlighted in the plots in the last row as vertical lines. The upper rows show different volumetric visualizations of the ensemble focusing on the different tasks. (

**a**) charge coverage volume within context morphology (Task OE); (

**b**) progression of the trajectory ensemble with time (Task OE, ME); (

**c**) density projection of the trajectory ensemble (Task ME); (

**d**) summary plot distance to start position (Task OE, CI).

**Figure 14.**Three selected charge trajectories are analyzed as above using one of the summary plots. Plot uses time of charge transport along x-axis and y-axis as distance from the start point of charge transport (negative distance is used to differential acceptor and donor pairs in red and blue respectively). This measure allows the user to inspect if charge flow digresses. In this case Trajectory 1 wobbles in a small coverage region for some time before they split apart to reach the electrode. Trajectory 2, on the other hand has shorter transport time for the acceptor and longer transport time for the donor. Trajectory 3 jumps rapidly from start to end. Context rendering.

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**MDPI and ACS Style**

Kottravel, S.; Volpi, R.; Linares, M.; Ropinski, T.; Hotz, I.
Visual Analysis of Stochastic Trajectory Ensembles in Organic Solar Cell Design. *Informatics* **2017**, *4*, 25.
https://doi.org/10.3390/informatics4030025

**AMA Style**

Kottravel S, Volpi R, Linares M, Ropinski T, Hotz I.
Visual Analysis of Stochastic Trajectory Ensembles in Organic Solar Cell Design. *Informatics*. 2017; 4(3):25.
https://doi.org/10.3390/informatics4030025

**Chicago/Turabian Style**

Kottravel, Sathish, Riccardo Volpi, Mathieu Linares, Timo Ropinski, and Ingrid Hotz.
2017. "Visual Analysis of Stochastic Trajectory Ensembles in Organic Solar Cell Design" *Informatics* 4, no. 3: 25.
https://doi.org/10.3390/informatics4030025