Abstract
A stochastic network model is developed which describes the 3D morphology of the pore space in fibre-based materials. It has the form of a random geometric graph, where the vertex set is modelled by random point processes and the edges are put using tools from graph theory and Markov chain Monte Carlo simulation. The model parameters are fitted to real image data gained by X-ray synchrotron tomography. In particular, they are specified in such a way that the distributions of vertex degrees and edge lengths, respectively, coincide to a large extent for real and simulated data. Furthermore, the network model is used to introduce a morphology-based notion of pores and their sizes. The model is validated by considering physical characteristics which are relevant for transport processes in the pore space, like geometric tortuosity, i.e., the distribution of shortest path lengths through the material relative to its thickness.
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Abdel-Ghani MS, Davies GA (1985) Chem Eng Sci 40:117
Armatas GS (2006) Chem Eng Sci 61:4662
Berhan L, Yi YB, Sastry AM (2004) J Appl Phys 95:5027
Benes V, Rataj J (2004) Stochastic geometry: selected topics. Kluwer Academic Publishers, Boston
Blunt MJ, Jackson MD, Piri M, Valvatne PH (2002) Adv Water Resour 25:1069
Casella G, Berger RL (2002) Statistical inference, 2nd ed. Duxbury, Pacific Grove
Daley DJ, Vere-Jones D (2003) An introduction to the theory of point processes volume I: elementary theory and methods. Springer, New York
Diestel R (2005) Graph theory. Springer, Heidelberg
Eichhorn SJ, Sampson WW (2005) J Royal Soci Interface 2:309
Eichhorn SJ, Sampson WW (2010) J Royal Soc Interface 7:641
Fourard C, Malandain G, Prohaska S, Westerhoff M (2006) IEEE Trans Med Imaging 156:B1339
Gelfand AE, Diggle PJ, Fuentes M, Guttorp P (2010) Handbook of spatial statistics. Chapman & Hall/CRC, Boca Raton
Gostick JT, Ioannidis MA, Fowler MW, Pritzker MD (2007) J Power Sources, 173:277
Hartnig C, Kuhn R, Krüger P, Manke I, Kardjilov N, Goebbels J, Müller BR, Riesemeier H (2008) MP Mater Test 50:609
Illian J, Penttinen A, Stoyan H, Stoyan D (2008) Statistical analysis and modelling of spatial point patterns. Wiley, Chichester
Inoue G, Matsukuma Y, Minemoto M (2007) Proceedings of the 2nd European Fuel Cell Technology and Applications Conference, EFC2007-39024
Inoue G, Yoshimoto T, Matsukuma Y, Minemoto M (2008) J Power Sources 175:145
Jähne B (2005) Digital image processing. 6th revised and extended ed. Springer, Berlin
Jungnickel D (1999) Graphs, networks and algorithms. Springer, Berlin
Keil F (1999) Diffusion und Chemische Reaktionen in der Gas/Feststoff–Katalyse. Springer, Berlin
Kendall WS, Molchanov I (eds) (2010) New perspectives in stochastic geometry. Springer, Berlin
Kroese DP, Taimre T, Botev ZI (2011) Handbook of Monte Carlo methods. Wiley, Hoboken
Lahiri SN (2003) Resampling methods for dependent data. Springer, New York
Leóny León CA (1998) Adv Colloid Interface Sci 76(77):341
Maheshwaria PH, Mathur RB, Dhamia TL (2008) Electrochim Acta 54:655
Manke I, Hartnig C, Grünerbel M, Lehnert W, Kardjilov N, Haibel A, Hilger A, Banhart J (2007) Appl Phys Lett 90:174105-1
Mathias MF, Roth J, Fleming J, Lehnert W (2003) In: Vielstich W, Lamm A, Gasteiger H (eds) Handbook of fuel cells, Vol III, Chap 42. Wiley, Chichester
Møller J, Waagepetersen RP (2004) Statistical inference and simulation for spatial point processes. Chapman & Hall/CRC, Boca Raton
Münch B, Holzer L (2008) J Am Ceram Soc 91:4059
Ohser J, Schladitz K (2009) 3D images of materials structures—processing and analysis. Wiley-VCH, Weinheim
Piekaar HW, Clarenburg LA (1967) Eng Sci 22:1399
Saito T, Toriwaki J-I (1994) Pattern Recogn 27:1551
Sampson WW (2003) J Mater Sci 38:1617. doi:https://doi.org/10.1023/A:1023298820390
Sampson WW, Urquhart SJ (2008) J Porous Mater 15:411
Schlather M (2001) Bernoulli 7:99
Schulz VP, Becker J, Wiegmann A, Mukherjee PP, Wang C-Y (2007) J Electrochem Soci 154:B419
Shen L, Chen Z (2007) Chem Eng Sci 62:3748
Sinha PK, Mukherjee PP, Wang C-Y (2007) J Mater Chem 17:3089
Sinha PK, Wang C-Y (2007) Electrochim Acta 52:7936
Soille P (1999) Morphological image analysis. Springer, Berlin
Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications, 2nd ed. Wiley, Chichester
Stoyan D (1984) Math Nachr 116:197
Thiedmann R, Fleischer F, Hartnig C, Lehnert W, Schmidt V (2008) J Electrochem Soc 155:B391
Thiedmann R, Hartnig C, Manke I, Schmidt V, Lehnert W (2009) J Electrochem Soc 156:B1339
Thulasiraman K, Swamy MNS (1992) Graphs: theory and algorithms. Wiley, Chichester
Tripathi A, Vijay S (2009) Discret Math 265:417
Weil H (1954) Ann Math Stat 25:168
Yoneda M, Takimoto M, Koshizuka S (2007) ECS Trans 11:629
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This research has been supported by the German Federal Ministry for Education and Science (BMBF) under Grant No. 03SF0324C/E/F.
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Thiedmann, R., Manke, I., Lehnert, W. et al. Random geometric graphs for modelling the pore space of fibre-based materials. J Mater Sci 46, 7745–7759 (2011). https://doi.org/10.1007/s10853-011-5754-7
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DOI: https://doi.org/10.1007/s10853-011-5754-7