Annals of Applied Statistics

Spatial modeling of the 3D morphology of hybrid polymer-ZnO solar cells, based on electron tomography data

O. Stenzel, H. Hassfeld, R. Thiedmann, L. J. A. Koster, S. D. Oosterhout, S. S. van Bavel, M. M. Wienk, J. Loos, R. A. J. Janssen, and V. Schmidt

Full-text: Open access


A spatial stochastic model is developed which describes the 3D nanomorphology of composite materials, being blends of two different (organic and inorganic) solid phases. Such materials are used, for example, in photoactive layers of hybrid polymer zinc oxide solar cells. The model is based on ideas from stochastic geometry and spatial statistics. Its parameters are fitted to image data gained by electron tomography (ET), where adaptive thresholding and stochastic segmentation have been used to represent morphological features of the considered ET data by unions of overlapping spheres. Their midpoints are modeled by a stack of 2D point processes with a suitably chosen correlation structure, whereas a moving-average procedure is used to add the radii of spheres. The model is validated by comparing physically relevant characteristics of real and simulated data, like the efficiency of exciton quenching, which is important for the generation of charges and their transport toward the electrodes.

Article information

Ann. Appl. Stat., Volume 5, Number 3 (2011), 1920-1947.

First available in Project Euclid: 13 October 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Marked point process parameter estimation spatial statistics stochastic geometry adaptive thresholding segmentation model fitting simulation model validation exciton quenching polymer solar cells


Stenzel, O.; Hassfeld, H.; Thiedmann, R.; Koster, L. J. A.; Oosterhout, S. D.; van Bavel, S. S.; Wienk, M. M.; Loos, J.; Janssen, R. A. J.; Schmidt, V. Spatial modeling of the 3D morphology of hybrid polymer-ZnO solar cells, based on electron tomography data. Ann. Appl. Stat. 5 (2011), no. 3, 1920--1947. doi:10.1214/11-AOAS468.

Export citation


  • Baddeley, A. J., Howard, C. V., Boyde, A. and Reid, S. (1987). Three-dimensional analysis of the spatial distribution of particles using the tandem-scanning reflected light microscope. Acta Stereologica 6 87–100.
  • Baddeley, A. J., Gregori, P., Mateu, J., Stoica, R. and Stoyan, D., eds. (2006). Case Studies in Spatial Point Process Modeling. Lecture Notes in Statist. 185. Springer, New York.
  • Ballani, F., Daley, D. J. and Stoyan, D. (2005). Modelling the microstructure of concrete with spherical grains. Computational Materials Science 35 399–407.
  • Beil, M., Fleischer, F., Paschke, S. and Schmidt, V. (2005). Statistical analysis of the three-dimensional structure of centromeric heterochromatin in interphase nuclei. J. Microsc. 217 60–68.
  • Blayvas, I., Bruckstein, A. and Kimmel, R. (2006). Efficient computation of adaptive threshold surfaces for image binarization. Pattern Recognition 39 89–101.
  • Brabec, C., Scherf, U. and Dyakonov, V. (2008). Organic Photovoltaics: Materials, Device Physics, and Manufacturing Technologies. Wiley-VCH, Weinheim.
  • Daley, D. J. and Vere-Jones, D. (2008). An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure, 2nd ed. Springer, New York.
  • Diggle, P. J. (2003). Statistical Analysis of Spatial Point Patterns, 2nd ed. Arnold, London.
  • Gelfand, A. E., Diggle, P. J., Fuentes, M. and Guttorp, P. (2010). Handbook of Spatial Statistics. CRC Press, Boca Raton, FL.
  • Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008). Statistical Analysis and Modelling of Spatial Point Patterns. Wiley, Chichester.
  • Kendall, W. S. and Molchanov, I., eds. (2010). New Perspectives in Stochastic Geometry. Oxford Univ. Press, Oxford.
  • Koster, L. J. A. (2010). Charge carrier mobility in disordered organic blends for photovoltaics. Phys. Rev. B 81 205318.
  • Møller, J. and Waagepetersen, R. P. (2004). Statistical Inference and Simulation for Spatial Point Processes. Monogr. Statist. Appl. Probab. 100. Chapman & Hall/CRC, Boca Raton, FL.
  • Ohser, J. and Mücklich, F. (2000). Statistical Analysis of Microstructures in Materials Science. Wiley, New York.
  • Oosterhout, S. D., Wienk, M. M., van Bavel, S. S., Thiedmann, R., Koster, L. J. A., Gilot, J., Loos, J., Schmidt, V. and Janssen, R. A. J. (2009). The effect of three-dimensional morphology on the efficiency of hybrid polymer solar cells. Nature Materials 8 818–824.
  • Shaw, P. E., Ruseckas, A. and Samuel, I. D. W. (2008). Exciton diffusion measurements in poly(3-hexylthiophene). Advanced Materials 20 3516–3520.
  • Stoica, R. S., Gregori, P. and Mateu, J. (2005). Simulated annealing and object point processes: Tools for analysis of spatial patterns. Stochastic Process. Appl. 115 1860–1882.
  • Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd ed. Wiley, Chichester.
  • Thiedmann, R., Haßfeld, H., Stenzel, O., Koster, L. J. A., Oosterhout, S. D., van Bavel, S. S., Wienk, M. M., Loos, J., Janssen, R. A. J. and Schmidt, V. (2011). A multiscale approach to the representation of 3D images, with application to polymer solar cells. Image Anal. Stereol. 30 19–30.
  • van Bavel, S. S., Sourty, E., de With, G. and Loos, J. (2009). Three-dimensional nanoscale organization of bulk heterojunction polymer solar cells. Nano Lett. 9 507–513.
  • Yang, X. and Loos, J. (2007). Toward high-performance polymer solar cells: The importance of morphology control. Macromolecules 40 1353–1362.
  • Yanowitz, S. D. and Bruckstein, A. M. (1989). A new method for image segmentation. Computer Vision, Graphics, and Image Processing 46 82–95.