Skip to main content
SpringerLink
Log in

Second-order analysis by variograms for curvature measures of two-phase structures

  • Statistical and Nonlinear Physics
  • Published:
The European Physical Journal B - Condensed Matter and Complex Systems Aims and scope Submit manuscript

Abstract.

Second-order characteristics are important in the description of various geometrical structures occurring in foams, porous media, complex fluids, and phase separation processes. The classical second order characteristics are pair correlation functions, which are well-known in the context of point fields and mass distributions. This paper studies systematically these and further characteristics from a unified standpoint, based on four so-called curvature measures, volume, surface area, integral of mean curvature and Euler characteristic. Their statistical estimation is straightforward only in the case of the volume measure, for which the pair correlation function is traditionally called the two-point correlation function. For the other three measures a statistical method is described which yields smoothed surrogates for pair correlation functions, namely variograms. Variograms lead to an enhanced understanding of the variability of the geometry of two-phase structures and can help in finding suitable models. The use of the statistical method is demonstrated for simulated samples related to Poisson-Voronoi tessellations, for experimental 3D images of Fontainebleau sandstone and for two samples of industrial foams.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and its Applications (John Wiley & Sons, Chichester, 1995)

  • S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer-Verlag, New York, 2002)

  • R. Hilfer, Statistical Physics and Spatial Statistics - The Art of Analyzing and Modeling Spatial Structures and Pattern Formation, edited by K.R. Mecke, D. Stoyan, 554 Lect. Notes Phys. (Springer, Berlin, 2000) pp. 203–241

  • K.R. Mecke, Integralgeometrie in der Statistischen Physik: Perkolation, komplexe Flüssigkeiten und die Struktur des Universums (Harry Deutsch, Frankfurt, 1994)

  • K. Mecke, Acta Physica Polonica B, 28, 1747 (1997)

    Google Scholar 

  • K.R. Mecke, Int. J. Mod. Phys. B 12, 861 (1998)

    Article  Google Scholar 

  • K.R. Mecke, Statistical Physics and Spatial Statistics - The Art of Analyzing and Modeling Spatial Structures and Pattern Formation, edited by K.R. Mecke, D. Stoyan, 554 Lect. Notes Phys. (Springer, Berlin, 2000) pp. 111–184

  • M. von Laue, Münchener Berichte (1912) p. 303

  • Lord Rayleigh, Proc. Roy. Soc. London Ser. A 84, 25 (1911)

    Google Scholar 

  • A. Guinier, Ann. Phys. 12, 161 (1939)

    Google Scholar 

  • P. Debye, H.R. Anderson, H. Brumberger, J. Appl. Phys. 28, 679 (1957)

    Article  Google Scholar 

  • G. Porod, Kolloid-Zeitschrift 125, 51; 108 (1952)

    Article  Google Scholar 

  • R. Kirste, G. Porod, Kolloid-Zeitschrift 184(1), 1 (1962)

    Google Scholar 

  • J.P. Hansen, I.R. McDonald, Theory of Simple Liquids (Academic Press, 1976)

  • U. Sonntag, D. Stoyan, H. Hermann, Phys. Stat. Sol. 68, 281 (1981)

    Google Scholar 

  • P. Smith, S. Torquato, J. Comp. Phys. 76(1), 176 (1988)

    Article  Google Scholar 

  • K.R. Mecke, J. Stat. Phys. 102, 1343, (2001)

    Article  Google Scholar 

  • C.H. Arns, M.A. Knackstedt, K.R. Mecke, Phys. Rev. Lett. 91, 215506 (2003)

    Article  PubMed  Google Scholar 

  • P.-M. König, R. Roth, K. Mecke, Phys. Rev. Lett. 93, 1–4 (2004)

    Google Scholar 

  • C.L.Y. Yeong, S. Torquato, Phys. Rev. E 57, 495 (1998)

    Article  Google Scholar 

  • C.L.Y. Yeong, S. Torquato, Phys. Rev. E 58, 224 (1998)

    Article  Google Scholar 

  • J.P. Chilès, P. Delfiner, Geostatistics. Modelling Spatial Uncertainity (J. Wiley and Sons, New York, 1999)

  • R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, Vol. 44 (Cambridge University Press, Cambridge, 1993)

  • D.A. Klain, G.-C. Rota, Introduction to Geometric Probability (Cambridge University Press, Cambridge, 1997)

  • A. Tscheschel, D. Stoyan, J. Microsc. 211, 80 (2003)

    Article  PubMed  Google Scholar 

  • A. Bezrukov, M. Bargiel, D. Stoyan, Part. Part. Charact. 19, 111 (2002)

    Article  Google Scholar 

  • S. Torquato, G. Stell, J. Chem. Phys. 79, 1505 (1983)

    Article  Google Scholar 

  • M. Kerscher, K. Mecke, J. Schmalzing, C. Beisbart, T. Buchert, H. Wagner, Astron. Astrophys. 373, 1 (2001)

    Article  Google Scholar 

  • U. Brodatzki, K. Mecke, Computer Physics Communications 147, 218 (2002)

    Article  Google Scholar 

  • Arthur Sakellariou, T.J. Sawkins, T.J. Senden, A. Limaye, Physica A 339(1-2), 152 (2004)

  • M. Saadatfar, M.A. Knackstedt, C H. Arns, A. Sakellariou, T. Senden, A.P. Sheppard, R.M. Sok, H. Steininger, W. Schrof, Physica A 339(1-2), 131, (2004)

  • T. Mattfeldt, D. Stoyan, J. Microsc. 200(2), 158 (2000)

    Article  Google Scholar 

  • D. Marcotte, Computers & Geosciences 22(10), 1175 (1996)

    Google Scholar 

  • K. Koch, J. Ohser, K. Schladitz, Adv. Appl. Prob. 35, 603 (2003)

    Article  Google Scholar 

  • W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in Fortran 90 - The Art of Parallel Scientific Computing, Vol. II (Cambridge University Press, Cambridge, 1996)

  • N.A. Seaton, E.D. Glandt, J. Chem. Phys. 85(9), 5262 (1986)

    Article  Google Scholar 

  • M. Knackstedt, C. Arns, M. Saadatfar, T. Senden, A. Sakellariou, A. Sheppard, R. Sok, W. Schrof, H. Steininger, Adv. Eng. Mat. 7(4), 238 (2005)

    Article  Google Scholar 

  • A.P. Sheppard, R.M. Sok, H. Averdunk, Physica A 339(1-2), 145 (2004)

    Google Scholar 

  • P. Spanne, J.F. Thovert, C.J. Jacquin, W.B. Lindquist, K.W. Jones, P.M. Adler, Phys. Rev. Lett. 73, 2001 (1994)

    Article  PubMed  Google Scholar 

  • F.M. Auzerais, J. Dunsmuir, B.B. Ferrèol, N. Martys, J. Olson, T.S. Ramakrishnan, D.H. Rothman, L.M. Schwartz, Geophys. Res. Lett. 23(7), 705 (1996)

    Article  Google Scholar 

  • J.T. Fredrich, Phys. Chem. Earth (A) 24(7), 551 (1999)

    Google Scholar 

  • J.-F. Thovert, F. Yousefian, P. Spanne, C.G. Jacquin, P.M. Adler, Phys. Rev. E 63, 61307 (2001)

    Article  Google Scholar 

  • W.B. Lindquist, A. Venkatarangan, J. Dunsmuir, T.F. Wong, J. Geophys. Res. B 105, 21509 (2000)

    Article  Google Scholar 

  • C.H. Arns, M.A. Knackstedt, W. Val Pinczewski, W.B. Lindquist, Geophys. Res. Lett. 28(17), 3361 (2001)

    Article  Google Scholar 

  • C.H. Arns, M.A. Knackstedt, K.R. Mecke, Morphology of Condensed Matter — Physics and Geometry of spatially complex Systems, edited by K.R. Mecke, D. Stoyan, Lect. Notes Phys. 600 (Springer, Berlin, 2002) pp. 37–74

  • C.H. Arns, M.A. Knackstedt, W.V. Pinczewski, E.G. Garboczi, Geophys. 67(5), 1396 (2002)

    Article  Google Scholar 

  • C.H. Arns, M.A. Knackstedt, W.V. Pinczewski, N. Martys, J. Petr. Sc. Eng. 45(1-2), 41 (2004)

    Google Scholar 

  • W. Oh, W. Brent Lindquist, IEEE T. Patt. Anal. Machine Intelligence 21, 590 (1999)

    Article  Google Scholar 

  • W.B. Lindquist, A. Venkatarangan, Phys. Chem. Earth (A) 25, 593 (1999)

    Google Scholar 

  • D.J. Bergman, K.-J. Dunn, L.M. Schwartz, P.P. Mitra, Phys. Rev. E 51, 3393 (1995)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT, 0200, Australia

    C. H. Arns

  2. Institut für Stochastik, Friedrich-Schiller-Universität Jena, 07740, Jena, Germany

    J. Mecke

  3. Institut für Theoretische Physik Universität Erlangen-Nürnberg, Staudtstrasse 7, 91058, Erlangen, Germany

    K. Mecke

  4. Institut für Stochastik, TU Bergakademie Freiberg, 09596, Freiberg, Germany

    D. Stoyan

Authors
  1. C. H. Arns
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Mecke
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. Mecke
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. Stoyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. Stoyan.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arns, C., Mecke, J., Mecke, K. et al. Second-order analysis by variograms for curvature measures of two-phase structures . Eur. Phys. J. B 47, 397–409 (2005). https://doi.org/10.1140/epjb/e2005-00338-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjb/e2005-00338-5

Keywords

Search

Navigation