Contributions to Mineralogy and Petrology

, Volume 156, Issue 4, pp 413–429 | Cite as

Quantitative textural analysis of packings of elongate crystals

  • John F. Rudge
  • Marian B. Holness
  • Graham C. Smith
Original Paper

Abstract

The spatial distribution of grains in a solidifying igneous rock controls the physical properties of the crystal mush, and in turn is controlled by the rate of crystal growth and accumulation. A predominant non-spherical habit for igneous minerals brings into question the use of spherical particles in reference packings used for quantification of spatial distribution. Furthermore, variations of crystal clustering/ordering with length scale require spatial statistics which take into account the distribution of particles beyond nearest neighbours. Using random close packings of spherocylinders, we demonstrate the importance of aspect ratio for the aggregation index (usually known as R) and show that packings of spherical particles have more structure than packings of rods. The spatial distribution functions demonstrate that the plagioclase grains in the colonnade from the Holyoke basalt are clustered on a length scale of 0.5 mm. Understanding the controls on grain spatial distribution in igneous rocks will depend on the application of these techniques to well-understood environments.

Keywords

Cumulates Textural analysis Spatial statistics Point patterns Random packing 

References

  1. Aste T, Saadatfar M, Senden TJ (2005) Geometrical structure of disordered sphere packings. Phys Rev E 71:061302. doi:10.1103/PhysRevE.71.061302 CrossRefGoogle Scholar
  2. Baddeley A, Jensen EBV (2004) Stereology for statisticians. Chapman & Hall/CRC, Boca Raton. ISBN-10-1584884053Google Scholar
  3. Baddeley A, Turner R (2005) Spatstat: an R package for analyzing spatial point patterns. J Stat Softw 12:1–42. URL http://www.jstatsoft.org, ISSN 1548-7660Google Scholar
  4. Berryman JG, Blair SC (1986) Use of digital image analysis to estimate fluid permeability of porous materials: application of two-point correlation functions. J Appl Phys 60:1930–1938. doi:10.1063/1.337245 CrossRefGoogle Scholar
  5. Bezrukov A, Stoyan D, Bargiel M (2001) Spatial statistics for simulated packings of spheres. Image Anal Stereol 20:203–206Google Scholar
  6. Blair SC, Berge PA, Berryman JG (1996) Using two-point correlation functions to characterize microgeometry and estimate permeabilities of sandstones and porous glass. J Geophys Res 101:20359–20375. doi:10.1029/96JB00879 CrossRefGoogle Scholar
  7. Blumenfeld R, Edwards SF, Ball RC (2005) Granular matter and the marginal rigidity state. J Phys Condens Matter 17:S2481–S2487. doi:10.1088/0953-8984/17/24/007 CrossRefGoogle Scholar
  8. Boorman S, Boudreau A, Kruger FJ (2004) The Lower Zone-Critical Zone transition of the Bushveld complex: a quantitative textural study. J Petrol 45:1209–1235. doi:10.1093/petrology/egh011 CrossRefGoogle Scholar
  9. Campbell IH (1978) Some problems with the cumulus theory. Lithos 11:311–323. doi:10.1016/0024-4937(78)90038-5 CrossRefGoogle Scholar
  10. Carlson WD (1989) The significance of intergranular diffusion to the mechanisms and kinetics of porphyroblast crystallization. Contrib Mineral Petrol 103:1–24. doi:10.1007/BF00371361 CrossRefGoogle Scholar
  11. Clark PJ, Evans FC (1954) Distance to nearest neighbor as a measure of spatial relationships in populations. Ecology 35:445–453CrossRefGoogle Scholar
  12. Daniel CG, Spear FS (1999) The clustered nucleation and growth processes of garnet in regional metamorphic rocks from north-west Connecticut, USA. J Metamorphic Geol 17:503–520CrossRefGoogle Scholar
  13. Denison C, Carlson WD, Ketcham RA (1997) Three-dimensional quantitative textural analysis of metamorphic rocks using high-resolution computed X-ray tomography: part I. Methods and techniques. J Metamorphic Geol 15:29–44. doi:10.1111/j.1525-1314.1997.00006.x CrossRefGoogle Scholar
  14. Dixon PM (2002a) Nearest neighbor methods. In: El-Shaarawi AH, Piegorsch WW (eds) Encyclopedia of environmetrics. Wiley, UK. doi:10.1002/9780470057339.van007
  15. Dixon PM (2002b) Ripley’s K function. In: El-Shaarawi AH, Piegorsch WW (eds) Encyclopedia of environmetrics. Wiley, UK. doi:10.1002/9780470057339.var046
  16. Donaldson CH (1976) An experimental investigation of olivine morphology. Contrib Mineral Petrol 57:187–213. doi:10.1007/BF00405225 CrossRefGoogle Scholar
  17. Donev A, Cisse I, Sachs D, Variano EA, Stillinger FH, Connelly R, Torquato S, Chaikin PM (2004) Improving the density of jammed disordered packings using ellipsoids. Science 303:990–993. doi:10.1126/science.1093010 CrossRefGoogle Scholar
  18. Dunbar NW, Jacobs GK, Naney MT (1995) Crystallization processes in an artificial magma: variations in crystal shape, growth rate and composition with melt cooling history. Contrib Mineral Petrol 120:412–425. doi:10.1007/s004100050085 CrossRefGoogle Scholar
  19. Finney JL (1970) Packings and the structure of simple liquids. I. The geometry of random close packing. Proc R Soc Lond A 319:479–493CrossRefGoogle Scholar
  20. Gaillot P, Darrozes J, de Saint Blanquat M, Ouillon G (1997) The normalised optimised anisotropic wavelet coefficient (NOAWC) method: an image processing tool for multi-scale analysis of rock fabric. Geophys Res Lett 24:1819–1822CrossRefGoogle Scholar
  21. Higgins MD (2006) Quantitative textural measurements in igneous and metamorphic petrology. Cambridge University Press, Cambridge. ISBN-13: 9780521847827. doi:10.2277/0521847826
  22. Hirsch DM (2008) Controls on porphyroblast size along a regional metamorphic field gradient. Contrib Mineral Petrol 155:401–415. doi:10.1007/s00410-007-0248-y CrossRefGoogle Scholar
  23. Hirsch DM, Ketcham RA, Carlson WD (2000) An evaluation of spatial correlation functions in textural analysis of metamorphic rocks. Geol Mat Res 2:1–42Google Scholar
  24. Irvine TN (1987) Layering and related structures in the Duke Island and Skaergaard intrusions: similarities, differences, and origins. In: Parsons I (ed) Origins of Igneous Layering. Reidel, Dordrecht, pp 185–245Google Scholar
  25. Jerram DA, Cheadle MJ (2000) On the cluster analysis of grains and crystals in rocks. Am Mineral 85:47–67Google Scholar
  26. Jerram DA, Higgins MD (2007) 3D analysis of rock textures: quantifying igneous microstructures. Elements 3:239–245. doi:10.2113/gselements.3.4.239 CrossRefGoogle Scholar
  27. Jerram DA, Cheadle MJ, Hunter RH, Elliott MT (1996) The spatial distribution of grains and crystals in rocks. Contrib Mineral Petrol 125:60–74. doi:10.1007/s004100050206 CrossRefGoogle Scholar
  28. Jerram DA, Cheadle MJ, Philpotts AR (2003) Quantifying the building blocks of igneous rocks: are clustered crystal frameworks the foundation? J Petrol 44:2033–2051. doi:10.1093/petrology/egg069 CrossRefGoogle Scholar
  29. Ketcham RA, Meth C, Hirsch DM, Carlson WD (2005) Improved methods for quantitative analysis of three-dimensional porphyroblastic textures. Geosphere 1:42–59. doi:10.1130/GES00002.1 CrossRefGoogle Scholar
  30. Kretz R (1966) Grain-size distribution for certain metamorphic minerals in relation to nucleation and growth. J Geol 74:147–173Google Scholar
  31. Kretz R (1969) On the spatial distribution of crystals in rocks. Lithos 2:39–65. doi:10.1016/S0024-4937(69)80005-8 CrossRefGoogle Scholar
  32. Kretz R (2006) Shape, size, spatial distribution and composition of garnet crystals in highly deformed gneiss of the Otter Lake area, Québec, and a model for garnet crystallization. J Metamorphic Geol 24:431–449. doi:10.1111/j.1525-1314.2006.00647.x CrossRefGoogle Scholar
  33. Lochmann K, Oger L, Stoyan D (2006) Statistical analysis of random sphere packings with variable radius distribution. Solid State Sci 8:1397–1413. doi:10.1016/j.solidstatesciences.2006.07.011 CrossRefGoogle Scholar
  34. Maaløe S (1987) The origin of rhythmic layering. Mineral Mag 42:337–345CrossRefGoogle Scholar
  35. Marsh BD (1996) Solidification fronts and magmatic evolution. Mineral Mag 60:5–40CrossRefGoogle Scholar
  36. Mattfeldt T (2005) Explorative statistical analysis of planar point processes in microscopy. J Microsc 220:131–139. doi:10.1111/j.1365-2818.2005.01521.x CrossRefGoogle Scholar
  37. McBirney AR, Nicolas A (1997) The Skaergaard layered series. Part II. Magmatic flow and dynamic layering. J Petrol 38:569–580CrossRefGoogle Scholar
  38. McKenzie D (1984) The generation and compaction of partially molten rock. J Petrol 25:713–765. doi:10.1093/petrology/25.3.713 Google Scholar
  39. Mock A, Jerram DA, Breitkreuz C (2003) Using quantitative textural analysis to understand the emplacement of shallow-level rhyolitic laccoliths—a case study from the Halle Volcanic Complex, Germany. J Petrol 44:833–849. doi:10.1093/petrology/44.5.833 CrossRefGoogle Scholar
  40. Morishita R (1998) Statistical properties of ideal rock textures: relationship between crystal size distribution and spatial correlation of minerals. Math Geol 30:409–434CrossRefGoogle Scholar
  41. Morishita R, Obata M (1995) A new statistical description of the spatial distribution of minerals in rocks. J Geol 103:232–240Google Scholar
  42. O’Driscoll B, Donaldson CH, Troll VR, Jerram DA, Emeleus CH (2007) An origin for harrisitic and granular olivine in the Rum layered suite, NW Scotland: a crystal size distribution study. J Petrol 48:253–270. doi:10.1093/petrology/egl059 CrossRefGoogle Scholar
  43. Philipse AP (1996a) The random contact equation and its implications for (colloidal) rods in packings, suspensions, and anisotropic powders. Langmuir 12:1127–1133. doi:10.1021/la950671o CrossRefGoogle Scholar
  44. Philipse AP (1996b) The random contact equation and its implications for (colloidal) rods in packings, suspensions, and anisotropic powders (correction). Langmuir 12:5971. doi:10.1021/la960869o CrossRefGoogle Scholar
  45. Philpotts AR, Dickson LD (2000) The formation of plagioclase chains during convective transfer in basaltic magma. Nature 406:59–61CrossRefGoogle Scholar
  46. Philpotts AR, Shi J, Brustman C (1998) Role of plagioclase crystal chains in the differentiation of partly crystallized basaltic magma. Nature 395:343–346. doi:10.1038/26404 CrossRefGoogle Scholar
  47. Pommerening A, Stoyan D (2006) Edge-correction needs in estimating indices of spatial forest structure. Can J For Res 36:1723–1739. doi:10.1139/X06-060 CrossRefGoogle Scholar
  48. Raeburn SP (1996) New methods in quantitative metamorphic petrology: 1. In situ determinations of iron valence in minerals; 2. The application of 3-D textural analysis to the study of crystallization kinetics. Ph.D. Thesis, Pennsylvania State UniversityGoogle Scholar
  49. Rasband W (1997-2007) ImageJ. U. S. National Institutes of Health, Bethesda. URL http://rsb.info.nih.gov/ij/
  50. R Development Core Team (2007) R: a language and environment for statistical computing. R foundation for statistical computing, Vienna, Austria. URL http://www.R-project.org, ISBN 3-900051-07-0
  51. Ripley BD (1976) The second-order analysis of stationary point processes. J Appl Probab 13:255–266CrossRefGoogle Scholar
  52. Ripley BD (1977) Modelling spatial patterns. J R Stat Soc B 39:172–192Google Scholar
  53. Shirley DN (1986) Compaction of igneous cumulates. J Geol 94:795–809CrossRefGoogle Scholar
  54. Stoyan D (2002) Simulation and characterization of random systems of hard particles. Image Anal Stereol 21:S41–S48Google Scholar
  55. Stoyan D, Penttinen A (2000) Recent applications of point process methods in forestry statistics. Stat Sci 15:61–78CrossRefGoogle Scholar
  56. Stoyan D, Stoyan H (1994) Fractals, random shapes and point fields: methods of geometrical statistics. Wiley, UKGoogle Scholar
  57. Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications. Wiley, UKGoogle Scholar
  58. Swanson SE, Fenn PM (1986) Quartz crystallisation in igneous rocks. Am Mineral 71:331–342Google Scholar
  59. Tepley FJI, Davidson JP (2003) Mineral-scale Sr-isotope constraints on magma evolution and chamber dynamics in the Rum layered intrusion, Scotland. Contrib Mineral Petrol 145:628–641. doi:10.1007/s00410-003-0481-y CrossRefGoogle Scholar
  60. Torquato S, Truskett TM, Debenedetti PG (2000) Is random close packing of spheres well defined? Phys Rev Lett 84:2064–2067. doi:10.1103/PhysRevLett.84.2064 CrossRefGoogle Scholar
  61. Wager LR, Brown GM, Wadsworth WJ (1960) Types of igneous cumulates. J Petrol 1:73–85. doi:10.1093/petrology/1.1.73 Google Scholar
  62. Weitz DA (2004) Packing in the spheres. Science 303:968–969. doi:10.1126/science.1094581 CrossRefGoogle Scholar
  63. Williams SR, Philipse AP (2003) Random packings of spheres and spherocylinders simulated by mechanical contraction. Phys Rev E 67:051301. doi:10.1103/PhysRevE.67.051301 CrossRefGoogle Scholar
  64. Wouterse A, Philipse AP (2006) Geometrical cluster ensemble analysis of random sphere packings. J Chem Phys 125:194709. doi:10.1063/1.2390700 CrossRefGoogle Scholar
  65. Wouterse A, Williams SR, Philipse AP (2007) Effect of particle shape on the density and microstructure of random packings. J Phys Condens Matter 19:406215. doi:10.1088/0953-8984/19/40/406215 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • John F. Rudge
    • 1
    • 2
  • Marian B. Holness
    • 3
  • Graham C. Smith
    • 3
  1. 1.Bullard LaboratoriesDepartment of Earth SciencesCambridgeUK
  2. 2.Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical PhysicsCentre for Mathematical SciencesCambridgeUK
  3. 3.Department of Earth SciencesCambridgeUK

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