Structure metrics for disordered materials

Disordered (or amorphous) structures are ubiquitous in physics, materials science, biology and elsewhere. There may be various reasons why one might be interested in disordered spatial geometries, be this because they form in a particular system of interest (such as granular materials) or because the disorder is a desired design element (such as in isotropic photonic band gap materials based on hyperuniform structures).

“Sloppy language, sloppy thinking!” (*)

Regardless of the motivation to study disordered materials, when we want to analyse their structure, we need a language that’s concise and useful to provide characterisation of the structural motifs that make up the structure. The search for structure metrics for disordered materials is really the quest for the “Language of Shape for disordered Structures”. Given that physics is a quantitative science, what’s needed are quantitative structure metrics that capture quantitatively the characteristics of a spatial structure, to create a filter to decide if two given structures are of the same nature; to correlate the structure to the resulting physical properties; to reduce a structure to a short compressed description of it that captures the relevant features, etc. That language of shape is developing (and has been for quite a few decades now, in a multitude of disciplines including computer science, physics, geography, stochastic geometry, spatial statistics and so forth…. Clearly, we are aiming for concise and precise structure metrics that are quantitatively accurate. No sloppy language there. Yet, it is an interesting question how a concise language develops over time … is it through continuous refinement of the initially sloppy ways of talking about something that is unknown?

We have worked on the development of structure metrics for a number of years now. One of our contributions, the development of the Minkowski tensor method into a demonstrated and versatile method for shape description, was greatly influenced, co-develMorphometryBusinessCardoped and inspired (and indeed suggested) by Klaus Mecke, based on his work on scalar Minkowski functionals. For a hands-on introduction to this method, check out https://morphometry.org/, an online tool developed by Fabian Schaller, Sebastian Kapfer and Michael Klatt.

My approach to the structure metrics for disordered media is heavily influenced by exposure to stochastic and integral geometry, two of the mathematical disciplines involved with the characterisation of geometries. This collaboration was formalised through the DFG-funded research group “Geometry and Physics of Spatial Random Systems” (GPSRS) that was active formally from 2011 to 2018, and continues to exist forth.

Articles listed below are (give or take) those focused on method development, rather than application to a particular physical problem. Please see section “Physics of disordered materials and packing problems” and “Non-equilibrium driven systems, transport processes and dynamics” for references to specific results.

Minkowski Tensor Structure Metrics

  • G.E. Schröder-Turk, W. Mickel, S.C. Kapfer, F.M. Schaller, B. Breidenbach, D. Hug, K. Mecke, “Minkowski tensors of anisotropic spatial structure”, New Journal of Physics 15, 083028 (2013)
  • G.E. Schröder-Turk, W. Mickel, S.C. Kapfer, M.A. Klatt, F.M. Schaller, M.J.F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger and K. Mecke, “Minkowski Tensor Shape Analysis of Cellular, Granular and Porous Structures”, Advanced Materials 23, 2535-2553 (2011)
  • G.E. Schröder-Turk, S. Kapfer, B. Breidenbach, D. Hug, C. Beisbart and K. Mecke, “Tensorial Minkowski functionals and anisotropy measures for planar patterns”, Journal of Microscopy, 238 (1), 57-74 (2010)
  • M.A. Klatt, G. Last, K. Mecke, C. Redenbach, F.M. Schaller and G.E. Schröder-Turk, “Cell shape analysis of random tessellations based on Minkowski tensors“, (arxiv version), Springer Lecture Notes in Mathematics “Tensor Valuations and their Applications in Stochastic Geometry and Imaging”, vol 2177 (2016)
  • W. Mickel, G.E. Schröder-Turk and K. Mecke, “Tensorial Minkowski Functionals of Triply-Periodic Minimal Surfaces”, J. Roy. Soc. Interf. Focus, doi: 10.1098/rsfs.2012.0007 (2012)

Bond-orientational order metrics and metrics for sphere assemblies

  • W. Mickel, S.C. Kapfer, G.E. Schröder-Turk, and K. Mecke, “Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter”, Journal of Chemical Physics 138, 044501 (2013)
  • S.C. Kapfer, W. Mickel, K. Mecke, and G.E. Schröder-Turk, Jammed Spheres: Minkowski Tensors Reveal Onset of Local Crystallinity , Phys. Rev. E 85 , 030301(Rapid Communication) (2012)
  • G.E. Schröder-Turk, W. Mickel, K. Mecke, G. Delanay, M. Saadatfar, T. Senden, T. Aste, “Disordered Spherical Bead Packs are Anisotropic”, Europhys. Lett. 90 (3), 34001 (2010)

Hyperuniformity-related questions

Interface Tensor metrics

Pore-size distributions

  • W. Mickel, S. Münster, L.M. Jawerth, D.A. Vader, D.A. Weitz, A.P. Sheppard, K. Mecke, B. Fabry and G.E. Schröder-Turk, Robust Pore Size Analysis of Filamentous Networks From 3D Confocal Microscopy, Biophysical Journal 95 (12), 6072-6080 (2008)

Mean-intercept length analyses for synthetic structure models

Local or density-resolved analysis

Set Voronoi methods for aspherical particle assemblies

(*) I couldn’t quite establish the origin of this quote. I first heard it from Mohan Srinivasarao, but realise that its roots go back alongway, possibly to George Orwell’s famous “But if thought corrupts language, language can also corrupt thought. A bad usage can spread by tradition and imitation even among people who should and do know better.”

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