Home Small-angle scattering from spherical particles on randomly oriented interfaces
Article
Licensed
Unlicensed Requires Authentication

Small-angle scattering from spherical particles on randomly oriented interfaces

  • Oskar Paris , Barbara Aichmayer and Peter Fratzl EMAIL logo
Published/Copyright: January 11, 2022
Become an author with De Gruyter Brill
Author Information
International Journal of Materials Research
From the journal International Journal of Materials Research Volume 97 Issue 3

Abstract

Small-angle scattering (SAS) has long been used to study nucleation and growth of particles in solid or liquid matrices. In some special cases, nucleation of particles occurs on interfaces, e. g., grain boundaries in crystalline solids or membranes in a solvent. Clearly, the position of the particles is constrained in this case to a narrow region close to the interface, which leads to correlations between the particles. In the present paper, we derive simple expressions for the analysis of SAS data from particles located on planar interfaces, and compare the analytical approximations with computer simulations.

Keywords: Small-angle scattering; Structure factor; Interfaces; Plates

Dedicated to Professor Dr. Gernot Kostorz on the occasion of his 65th birthday

Prof. Dr. Peter Fratzl Department of Biomaterials Max-Planck-Institute of Colloids and Interfaces Research Campus Golm, D-14424 Potsdam, Germany Tel.: +49 331 567 9400 Fax: +49 331 567 9402

References

[1] A. Guinier, G. Fournet: Small-angle scattering of X-rays, Wiley, New York, 1955.Search in Google Scholar

[2] O. Glatter, O. Kratky: Small-angle X-ray scattering, Academic Press, New York (1983).Search in Google Scholar

[3] D.I. Svergun, M.H.J. Koch: Rep. Prog. Phys. 66 (2003) 1735.10.1088/0034-4885/66/10/R05Search in Google Scholar

[4] J.S. Pedersen: Adv. Colloid Interface Sci. 70 (1997) 171.10.1016/S0001-8686(97)00312-6Search in Google Scholar

[5] G. Kostorz: J. Appl. Crystallogr. 24 (1991) 444.10.1107/S0021889891002467Search in Google Scholar

[6] G. Kostorz, in: R.W. Cahn, P. Haasen (Eds.), X-ray and neutron scattering, Physical Metallurgy, 4th Edition, Vol. North Holland, Amsterdam (1996) 1115.10.1016/B978-044489875-3/50017-XSearch in Google Scholar

[7] G. Kostorz: Z. Kristall. 218 (2003) 154.10.1524/zkri.218.2.154.20667Search in Google Scholar

[8] P. Fratzl: J. Appl. Crystallogr. 36 (2003) 397.10.1107/S0021889803000335Search in Google Scholar

[9] O. Paris, I. Zizak, H. Lichtenegger, P. Roschger, K. Klaushofer, P. Fratzl: Cell. Mol. Biol. 46 (2000) 993.Search in Google Scholar

[10] J.S. Pedersen, P. Vyskocil, B. Schönfeld, G. Kostorz: J. Appl. Crystallogr. 30 (1997) 975.10.1107/S0021889897004706Search in Google Scholar

[11] J. Brunner-Popela, O. Glatter: J. Appl. Crystallogr. 30 (1997) 431.10.1107/S0021889896015749Search in Google Scholar

[12] B. Weyerich, J. Brunner-Popela, O. Glatter: J. Appl. Crystallogr. 32 (1999) 197.10.1107/S0021889898011790Search in Google Scholar

[13] P. Fratzl, H.S. Gupta, O. Paris, A. Valenta, P. Roschger, K. Klaushofer: Progr. Colloid Polym. Sci. 130 (2005) 33.Search in Google Scholar

[14] B.H. Tian, G.A. Zickler, C. Lind, O. Paris: Acta Mater. 51 (2003) 4149.10.1016/S1359-6454(03)00233-7Search in Google Scholar

[15] M.J. Portmann, R. Erni, H. Heinrich, G. Kostorz: Micron 35 (2004) 695.10.1016/j.micron.2004.04.002Search in Google Scholar PubMed

[16] J.T. Russell, Y. Lin, A. Boker, L. Su, P. Carl, H. Zettl, J.B. He, K. Sill, R. Tangirala, T. Emrick, K. Littrell, P. Thiyagarajan, D. Cookson, A. Fery, Q. Wang, T.P. Russell: Angew. Chem. Int. Edit. 44 (2005) 2420.10.1002/anie.200462653Search in Google Scholar PubMed

[17] W. Shenton, D. Pum, U.B. Sleytr, S. Mann: Nature 389 (1997) 585.10.1038/39287Search in Google Scholar

[18] B. Aichmayer, M. Mertig, A. Kirchner, O. Paris, P. Fratzl: Adv. Mater. (2006) submitted.Search in Google Scholar

[19] B.E. Warren: Phys. Rev. 59 (1941) 693.10.1103/PhysRev.59.693Search in Google Scholar

[20] W. Ruland: Acta Crystallogr. 22 (1967) 615.10.1107/S0365110X67001252Search in Google Scholar

[21] G. Vignaud, A. Gibaud, J. Wang, S.K. Sinha, J. Daillant, G. Grübel, Y. Gallot: J. Phys. Condens. Matter 9 (1997) L125.10.1088/0953-8984/9/9/003Search in Google Scholar

[22] G. Renaud, R. Lazzari, C. Revenant, A. Barbier,M. Noblet, O. Ulrich, F. Leroy, J. Jupille, Y. Borensztein, C.R. Henry, J.-P. Deville, F. Scheurer, J. Mane-Mane, O. Fruchart: Science 300 (2003) 1416.10.1126/science.1082146Search in Google Scholar PubMed

[23] M. Goulian, N. Lei, J. Miller, S.K. Sinha: Phys. Rev. A 46 (1992) R6170.10.1103/PhysRevA.46.R6170Search in Google Scholar

[24] R. Perret, W. Ruland: J. Appl. Crystallogr. 1 (1968) 257.10.1107/S0021889868005455Search in Google Scholar

[25] O. Paris, D. Loidl, H. Peterlik, M. Muller, H. Lichtenegger, P. Fratzl: J. Appl. Crystallogr. 33 (2000) 695.10.1107/S0021889899013436Search in Google Scholar

[26] E.W. Kaler, in: H. Brumberger (Ed.), Small-Angle Scattering From Complex Fluids, Modern Aspects of Small-Angle Scattering, NATO ASI Series C, Vol. 451, Kluwer Academic Publishers, Dordrecht (1995) 329.10.1007/978-94-015-8457-9_13Search in Google Scholar
Received: 2005-09-29
Accepted: 2005-12-11
Published Online: 2022-01-11

© 2006 Carl Hanser Verlag, München

Articles in the same Issue

  1. Frontmatter
  2. Professor Dr. Gernot Kostorz 65 years
  3. Dislocation micromechanisms under single slip conditions
  4. Characterisation of short-range order using dislocations
  5. Between microscopic and mesoscopic descriptions of twin–twin interaction
  6. Influence of the thermoelastic effect on the acoustic properties of pure metals at low temperatures
  7. Recent progress in the area of bulk metallic glasses
  8. Formation of the ABC6-type ordered structure in fcc alloys
  9. Short-range order in Fe-21.9 at.% Al
  10. Criteria for developing castable, creep-resistant aluminum-based alloys – A review
  11. Phase decomposition and precipitation of metastable A2 phase in B2 ordered Co–Al–Fe alloys
  12. Atomic migration and ordering phenomena in bulk and thin films of FePd and FePt
  13. Late-stage coarsening of oil droplets of excess oil in microemulsions following a temperature quench
  14. Small-angle scattering from spherical particles on randomly oriented interfaces
  15. Ripening of L12 Ni3Ti precipitates in the framework of the trans-interface diffusion-controlled theory of particle coarsening
  16. Texture evolution in equiaxed polycrystalline L10-ordered FePd during coarsening at 600 °C
  17. Modulated structures in amorphous films of Cr-silicide prepared by electron-beam-deposition
  18. Early stages of nucleation and growth of Guinier –Preston zones in Al–Zn–Mg and Al–Zn–Mg–Cu alloys
  19. Experimental and theoretical characterization of Al3Sc precipitates in Al–Mg–Si–Cu–Sc–Zr alloys
  20. Ag2Al plates in Al–Ag alloys
  21. A critical analysis of the composite model as applied to high-temperature creep of Al and an Al–Mg alloy
  22. Damage behaviour of an Al2O3 particle-reinforced 6061 alloy induced by monotonic and cyclic deformation
  23. Deformation behaviour of ultrafine-grained magnesium with 3 vol.% graphite
  24. Press/Presse
  25. Conferences/Konferenzen
  26. Frontmatter
  27. Editorial
  28. Professor Dr. Gernot Kostorz 65 years
  29. Articles BBasic
  30. Dislocation micromechanisms under single slip conditions
  31. Characterisation of short-range order using dislocations
  32. Between microscopic and mesoscopic descriptions of twin–twin interaction
  33. Influence of the thermoelastic effect on the acoustic properties of pure metals at low temperatures
  34. Recent progress in the area of bulk metallic glasses
  35. Formation of the ABC6-type ordered structure in fcc alloys
  36. Short-range order in Fe-21.9 at.% Al
  37. Criteria for developing castable, creep-resistant aluminum-based alloys – A review
  38. Phase decomposition and precipitation of metastable A2 phase in B2 ordered Co–Al–Fe alloys
  39. Atomic migration and ordering phenomena in bulk and thin films of FePd and FePt
  40. Late-stage coarsening of oil droplets of excess oil in microemulsions following a temperature quench
  41. Small-angle scattering from spherical particles on randomly oriented interfaces
  42. Ripening of L12 Ni3Ti precipitates in the framework of the trans-interface diffusion-controlled theory of particle coarsening
  43. Articles AApplied
  44. Texture evolution in equiaxed polycrystalline L10-ordered FePd during coarsening at 600 °C
  45. Modulated structures in amorphous films of Cr-silicide prepared by electron-beam-deposition
  46. Early stages of nucleation and growth of Guinier –Preston zones in Al–Zn–Mg and Al–Zn–Mg–Cu alloys
  47. Experimental and theoretical characterization of Al3Sc precipitates in Al–Mg–Si–Cu–Sc–Zr alloys
  48. Ag2Al plates in Al–Ag alloys
  49. A critical analysis of the composite model as applied to high-temperature creep of Al and an Al–Mg alloy
  50. Damage behaviour of an Al2O3 particle-reinforced 6061 alloy induced by monotonic and cyclic deformation
  51. Deformation behaviour of ultrafine-grained magnesium with 3 vol.% graphite
  52. Notifications/Mitteilungen
  53. Press/Presse
  54. Conferences/Konferenzen
https://doi.org/10.3139/ijmr-2006-0046
Keywords for this article
Small-angle scattering; Structure factor; Interfaces; Plates

Articles in the same Issue

  1. Frontmatter
  2. Professor Dr. Gernot Kostorz 65 years
  3. Dislocation micromechanisms under single slip conditions
  4. Characterisation of short-range order using dislocations
  5. Between microscopic and mesoscopic descriptions of twin–twin interaction
  6. Influence of the thermoelastic effect on the acoustic properties of pure metals at low temperatures
  7. Recent progress in the area of bulk metallic glasses
  8. Formation of the ABC6-type ordered structure in fcc alloys
  9. Short-range order in Fe-21.9 at.% Al
  10. Criteria for developing castable, creep-resistant aluminum-based alloys – A review
  11. Phase decomposition and precipitation of metastable A2 phase in B2 ordered Co–Al–Fe alloys
  12. Atomic migration and ordering phenomena in bulk and thin films of FePd and FePt
  13. Late-stage coarsening of oil droplets of excess oil in microemulsions following a temperature quench
  14. Small-angle scattering from spherical particles on randomly oriented interfaces
  15. Ripening of L12 Ni3Ti precipitates in the framework of the trans-interface diffusion-controlled theory of particle coarsening
  16. Texture evolution in equiaxed polycrystalline L10-ordered FePd during coarsening at 600 °C
  17. Modulated structures in amorphous films of Cr-silicide prepared by electron-beam-deposition
  18. Early stages of nucleation and growth of Guinier –Preston zones in Al–Zn–Mg and Al–Zn–Mg–Cu alloys
  19. Experimental and theoretical characterization of Al3Sc precipitates in Al–Mg–Si–Cu–Sc–Zr alloys
  20. Ag2Al plates in Al–Ag alloys
  21. A critical analysis of the composite model as applied to high-temperature creep of Al and an Al–Mg alloy
  22. Damage behaviour of an Al2O3 particle-reinforced 6061 alloy induced by monotonic and cyclic deformation
  23. Deformation behaviour of ultrafine-grained magnesium with 3 vol.% graphite
  24. Press/Presse
  25. Conferences/Konferenzen
  26. Frontmatter
  27. Editorial
  28. Professor Dr. Gernot Kostorz 65 years
  29. Articles BBasic
  30. Dislocation micromechanisms under single slip conditions
  31. Characterisation of short-range order using dislocations
  32. Between microscopic and mesoscopic descriptions of twin–twin interaction
  33. Influence of the thermoelastic effect on the acoustic properties of pure metals at low temperatures
  34. Recent progress in the area of bulk metallic glasses
  35. Formation of the ABC6-type ordered structure in fcc alloys
  36. Short-range order in Fe-21.9 at.% Al
  37. Criteria for developing castable, creep-resistant aluminum-based alloys – A review
  38. Phase decomposition and precipitation of metastable A2 phase in B2 ordered Co–Al–Fe alloys
  39. Atomic migration and ordering phenomena in bulk and thin films of FePd and FePt
  40. Late-stage coarsening of oil droplets of excess oil in microemulsions following a temperature quench
  41. Small-angle scattering from spherical particles on randomly oriented interfaces
  42. Ripening of L12 Ni3Ti precipitates in the framework of the trans-interface diffusion-controlled theory of particle coarsening
  43. Articles AApplied
  44. Texture evolution in equiaxed polycrystalline L10-ordered FePd during coarsening at 600 °C
  45. Modulated structures in amorphous films of Cr-silicide prepared by electron-beam-deposition
  46. Early stages of nucleation and growth of Guinier –Preston zones in Al–Zn–Mg and Al–Zn–Mg–Cu alloys
  47. Experimental and theoretical characterization of Al3Sc precipitates in Al–Mg–Si–Cu–Sc–Zr alloys
  48. Ag2Al plates in Al–Ag alloys
  49. A critical analysis of the composite model as applied to high-temperature creep of Al and an Al–Mg alloy
  50. Damage behaviour of an Al2O3 particle-reinforced 6061 alloy induced by monotonic and cyclic deformation
  51. Deformation behaviour of ultrafine-grained magnesium with 3 vol.% graphite
  52. Notifications/Mitteilungen
  53. Press/Presse
  54. Conferences/Konferenzen
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 2.12.2025 from https://www.degruyterbrill.com/document/doi/10.3139/ijmr-2006-0046/html
Scroll to top button