Table of Contents
Journal of Nanoparticles
Volume 2013 (2013), Article ID 640436, 11 pages
http://dx.doi.org/10.1155/2013/640436
Review Article

Small Angle X-Ray Scattering Technique for the Particle Size Distribution of Nonporous Nanoparticles

1Department of Industrial and Manufacturing Engineering, Florida State University, 2525 Pottsdamer Street, Tallahassee, FL 32310, USA
2North Carolina State University, Department of Mechanical & Aerospace Engineering, 911 Oval Drive, Raleigh, NC 27695, USA

Received 22 July 2013; Revised 19 November 2013; Accepted 20 November 2013

Academic Editor: Frank Hubenthal

Copyright © 2013 A. Agbabiaka et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. O. V. Salata, “Applications of nanoparticles in biology and medicine,” Journal of Nanobiotechnology, vol. 2, article 3, pp. 1–6, 2004. View at Publisher · View at Google Scholar · View at Scopus
  2. D. Pantarotto, C. D. Partidos, J. Hoebeke et al., “Immunization with peptide-functionalized carbon nanotubes enhances virus-specific neutralizing antibody responses,” Chemistry & Biology, vol. 10, no. 10, pp. 961–966, 2003. View at Publisher · View at Google Scholar · View at Scopus
  3. C. Mah, T. J. Fraites Jr., I. Zolotukhin et al., “Improved method of recombinant AAV2 delivery for systemic targeted gene therapy,” Molecular Therapy, vol. 6, no. 1, pp. 106–112, 2002. View at Publisher · View at Google Scholar · View at Scopus
  4. T. J. Schmidt, M. Noeske, H. A. Gasteiger et al., “Electrocatalytic activity of PtRu alloy colloids for CO and CO/H2 electrooxidation: stripping voltammetry and rotating disk measurements,” Langmuir, vol. 13, no. 10, pp. 2591–2595, 1997. View at Google Scholar · View at Scopus
  5. M. T. Reetz, R. Breinbauer, P. Wedemann, and P. Binger, “Nanostructured nickel-clusters as catalysts in [3 + 2] cycloaddition reactions,” Tetrahedron, vol. 54, no. 7, pp. 1233–1240, 1998. View at Publisher · View at Google Scholar · View at Scopus
  6. M. T. Reetz, R. Breinbauer, and K. Wanninger, “Suzuki and Heck reactions catalyzed by preformed palladium clusters and palladium/nickel bimetallic clusters,” Tetrahedron Letters, vol. 37, no. 26, pp. 4499–4502, 1996. View at Publisher · View at Google Scholar · View at Scopus
  7. A. Balamurugan, K.-C. Ho, and S.-M. Chen, “One-pot synthesis of highly stable silver nanoparticles-conducting polymer nanocomposite and its catalytic application,” Synthetic Metals, vol. 159, no. 23-24, pp. 2544–2549, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. M. Bruchez Jr., M. Moronne, P. Gin, S. Weiss, and A. P. Alivisatos, “Semiconductor nanocrystals as fluorescent biological labels,” Science, vol. 281, no. 5385, pp. 2013–2016, 1998. View at Publisher · View at Google Scholar · View at Scopus
  9. R. L. Edelstein, C. R. Tamanaha, P. E. Sheehan et al., “The BARC biosensor applied to the detection of biological warfare agents,” Biosensors and Bioelectronics, vol. 14, no. 10-11, pp. 805–813, 2000. View at Publisher · View at Google Scholar · View at Scopus
  10. R. Elghanian, J. J. Storhoff, R. C. Mucic, R. L. Letsinger, and C. A. Mirkin, “Selective colorimetric detection of polynucleotides based on the distance-dependent optical properties of gold nanoparticles,” Science, vol. 277, no. 5329, pp. 1078–1081, 1997. View at Publisher · View at Google Scholar · View at Scopus
  11. G. Schön and U. Simon, “A fascinating new field in colloid science: small ligand-stabilized metal clusters and their possible application in microelectronics—part II: future directions,” Colloid & Polymer Science, vol. 273, no. 3, pp. 202–218, 1995. View at Publisher · View at Google Scholar · View at Scopus
  12. I. Brigger, C. Dubernet, and P. Couvreur, “Nanoparticles in cancer therapy and diagnosis,” Advanced Drug Delivery Reviews, vol. 54, no. 5, pp. 631–651, 2002. View at Publisher · View at Google Scholar · View at Scopus
  13. J. Baltrusaitis, P. M. Jayaweera, and V. H. Grassian, “Sulfur dioxide adsorption on TiO2 nanoparticles: influence of particle size, coadsorbates, sample pretreatment, and light on surface speciation and surface coverage,” Journal of Physical Chemistry C, vol. 115, no. 2, pp. 492–500, 2011. View at Publisher · View at Google Scholar · View at Scopus
  14. A. Kasuya, G. Milczarek, I. Dmitruk et al., “Size- and shape-controls and electronic functions of nanometer-scale semiconductors and oxides,” Colloids and Surfaces A, vol. 202, no. 2-3, pp. 291–296, 2002. View at Publisher · View at Google Scholar · View at Scopus
  15. A. J. Maira, K. L. Yeung, C. Y. Lee, P. L. Yue, and C. K. Chan, “Size effects in gas-phase photo-oxidation of trichloroethylene using nanometer-sized TiO2 catalysts,” Journal of Catalysis, vol. 192, no. 1, pp. 185–196, 2000. View at Publisher · View at Google Scholar · View at Scopus
  16. H. Yano, J. Inukai, H. Uchida et al., “Particle-size effect of nanoscale platinum catalysts in oxygen reduction reaction: an electrochemical and195Pt EC-NMR study,” Physical Chemistry Chemical Physics, vol. 8, no. 42, pp. 4932–4939, 2006. View at Publisher · View at Google Scholar · View at Scopus
  17. J. D. Morse, “Research challenges for integrated systems nanomanufacturing: report from the National Science Foundation Workshop,” Internano 2008.
  18. C. Li, “Structure controlling and process scale-up in the fabrication of nanomaterials,” Frontiers of Chemical Engineering in China, vol. 4, no. 1, pp. 18–25, 2010. View at Publisher · View at Google Scholar · View at Scopus
  19. J. Grobelny, F. W. DelRio, N. Pradeep, D.-I. Kim, V. A. Hackley, and R. F. Cook, “Size measurement of nanoparticles using atomic force microscopy,” Methods in Molecular Biology, vol. 697, pp. 71–82, 2011. View at Google Scholar · View at Scopus
  20. W. D. Pyrz and D. J. Buttrey, “Particle size determination using TEM: a discussion of image acquisition and analysis for the novice microscopist,” Langmuir, vol. 24, no. 20, pp. 11350–11360, 2008. View at Publisher · View at Google Scholar · View at Scopus
  21. S. Sahoo, C. K. Chakraborti, S. C. Mishra, and U. N. Nanda, “Scanning electron microscopy as an analytical tool for particle size distribution and aspect ratio analysis of ciprofloxacin muco adhesive polymeric Suspension,” Ijrras, vol. 6, no. 1, pp. 94–100, 2011. View at Google Scholar
  22. T. Walther, Photon Correlation Spectroscopy in Particle Sizing, John Wiley & Sons, Chichester, UK, 2000.
  23. H. K. Kammler, G. Beaucage, D. J. Kohls, N. Agashe, and J. Ilavsky, “Monitoring simultaneously the growth of nanoparticles and aggregates by in situ ultra-small-angle X-ray scattering,” Journal of Applied Physics, vol. 97, no. 5, Article ID 054309, 11 pages, 2005. View at Publisher · View at Google Scholar · View at Scopus
  24. M. Harada, N. Tamura, and M. Takenaka, “Nucleation and growth of metal nanoparticles during photoreduction using in situ time-resolved SAXS analysis,” Journal of Physical Chemistry C, vol. 115, no. 29, pp. 14081–14092, 2011. View at Publisher · View at Google Scholar · View at Scopus
  25. L. C. McKenzie, P. M. Haben, S. D. Kevan, and J. E. Hutchison, “Determining nanoparticle size in real time by small-angle X-ray scattering in a microscale flow system,” Journal of Physical Chemistry C, vol. 114, no. 50, pp. 22055–22063, 2010. View at Publisher · View at Google Scholar · View at Scopus
  26. H. Brumberger, Modern Aspects of Small Angle Scattering, NATO Science Series C, Springer, New York, NY, USA, 1994.
  27. L. A. Feigin and D. I. Svergun, Structural Analysis by Small Angle X-Ray and Neutron Scattering, Platinum Press, New York, NY, USA, 1987.
  28. I. Pilz, O. Glatter, and O. Kratky, “Small-angle X-ray scattering,” Methods in Enzymology, vol. 61, pp. 148–249, 1979. View at Publisher · View at Google Scholar · View at Scopus
  29. B. Chu and B. S. Hsiao, “Small-angle X-ray scattering of polymers,” Chemical Reviews, vol. 101, no. 6, pp. 1727–1761, 2001. View at Publisher · View at Google Scholar · View at Scopus
  30. R. Bienert, F. Emmerling, and A. F. Thünemann, “The size distribution of 'gold standard' nanoparticles,” Analytical and Bioanalytical Chemistry, vol. 395, no. 6, pp. 1651–1660, 2009. View at Publisher · View at Google Scholar · View at Scopus
  31. O. Glatter, “The interpretation of real-space information from small-angle scattering experiments,” Journal of Applied Crystallography, vol. 12, no. 2, pp. 166–175, 1979. View at Google Scholar
  32. J. J. Müller, G. Damaschun, and G. Hübner, “Small angle X-ray scattering studies on the structure and symmetry of yeast pyruvate decarboxylase in solution,” Acta Biologica et Medica Germanica, vol. 38, no. 1, pp. 1–10, 1979. View at Google Scholar · View at Scopus
  33. S. Hansen, “Calculation of small-angle scattering profiles using Monte Carlo simulation,” Journal of Applied Crystallography, vol. 23, no. 4, pp. 344–346, 1990. View at Google Scholar
  34. S. J. Henderson, “Monte Carlo modeling of small-angle scattering data from non-interacting homogeneous and heterogeneous particles in solution,” Biophysical Journal, vol. 70, no. 4, pp. 1618–1627, 1996. View at Google Scholar · View at Scopus
  35. B. C. McAlister and B. P. Grady, “Simulation of small-angle X-ray scattering from single-particle systems,” Journal of Applied Crystallography, vol. 31, no. 4, pp. 594–599, 1998. View at Google Scholar · View at Scopus
  36. H. B. Stuhrmann, “Ein neues Verfahren zur Bestimmung der Oberflächenform und der inneren Struktur von gelösten globulären Proteinen aus Röntgenkleinwinkelmessungen,” Zeitschrift für Physikalische Chemie, vol. 72, pp. 177–198, 1970. View at Google Scholar
  37. A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princton University Press, 1996.
  38. D. I. Svergun and H. B. Stuhrmann, “New developments in direct shape determination from small-angle scattering. 1. Theory and model calculations,” Acta Crystallographica Section A, vol. 47, no. 6, pp. 736–744, 1991. View at Publisher · View at Google Scholar
  39. D. I. Svergun, V. V. Volkov, M. B. Kozin, and H. B. Stuhrmann, “New developments in direct shape determination from small-angle scattering. 2. Uniqueness,” Acta Crystallographica Section A, vol. 52, no. 3, pp. 419–426, 1996. View at Publisher · View at Google Scholar · View at Scopus
  40. D. I. Svergun, V. V. Volkov, M. B. Kozin, H. B. Stuhrmann, C. Barberato, and M. H. J. Koch, “Shape determination from solution scattering of biopolymers,” Journal of Applied Crystallography, vol. 30, no. 5, pp. 798–802, 1997. View at Google Scholar · View at Scopus
  41. F. Spinozzi, F. Carsughi, and P. Mariani, “Particle shape reconstruction by small-angle scattering: integration of group theory and maximum entropy to multipole expansion method,” The Journal of Chemical Physics, vol. 109, no. 23, pp. 10148–10158, 1998. View at Publisher · View at Google Scholar · View at Scopus
  42. V. L. Shneerson and D. K. Saldin, “Molecular shapes from small-angle X-ray scattering: extension of the theory to higher scattering angles,” Acta Crystallographica Section A, vol. 65, no. 2, pp. 128–134, 2009. View at Publisher · View at Google Scholar · View at Scopus
  43. P. Chacón, F. Morán, J. F. Díaz, E. Pantos, and J. M. Andreu, “Low-resolution structures of proteins in solution retrieved from X-ray scattering with a genetic algorithm,” Biophysical Journal, vol. 74, no. 6, pp. 2760–2775, 1998. View at Google Scholar · View at Scopus
  44. D. I. Svergun, M. V. Petoukhov, and M. H. J. Koch, “Determination of domain structure of proteins from X-ray solution scattering,” Biophysical Journal, vol. 80, no. 6, pp. 2946–2953, 2001. View at Google Scholar · View at Scopus
  45. D. I. Svergun, “Mathematical methods in small-angle scattering data analysis,” Journal of Applied Crystallography, vol. 24, no. 5, pp. 485–492, 1991. View at Publisher · View at Google Scholar · View at Scopus
  46. C. G. Shull and L. C. Roess, “X-ray scattering at small angles by finely-divided solids. I. General approximate theory and applications,” Journal of Applied Physics, vol. 18, no. 3, pp. 295–307, 1947. View at Publisher · View at Google Scholar · View at Scopus
  47. P. B. Elkin, C. G. Shull, and L. C. Roess, “Silica-Alumina, gels,” Industrial & Engineering Chemistry, vol. 37, no. 4, pp. 327–331, 1945. View at Google Scholar
  48. M. H. Jellinek and I. Fankuchen, “X-ray diffraction examination of Gamma alumina,” Industrial & Engineering Chemistry, vol. 37, no. 2, pp. 158–163, 1945. View at Google Scholar
  49. A. L. Patterson, “The diffraction of X-rays by small crystalline particles,” Physical Review, vol. 56, no. 10, pp. 972–977, 1939. View at Publisher · View at Google Scholar · View at Scopus
  50. L. C. Roess and C. G. Shull, “X-ray scattering at small angles by finely-divided solids. II. Exact theory for random distributions of spheroidal particles,” Journal of Applied Physics, vol. 18, no. 3, pp. 308–313, 1947. View at Publisher · View at Google Scholar · View at Scopus
  51. P. Mittelbach and G. Porod, “Zur Röntgenkleinwinkelstreuung kolloider Systeme: Die mittleren Durchschußlängen und die Kohärenzlänge eines kolloiden Systems; Kennzahlen zur Ermittlung von Teilchenform und Polydispersitätsgrad,” Kolloid-Zeitschrift & Zeitschrift für Polymere, vol. 202, no. 1, pp. 40–49, 1965. View at Publisher · View at Google Scholar · View at Scopus
  52. B. Sjöberg, “Small-angle X-ray investigation of the equilibria between copper(II) and glycyl-l-histidylglycine in water solution. A method for analysing polydispersed systems,” Journal of Applied Crystallography, vol. 7, no. 2, pp. 192–199, 1974. View at Google Scholar
  53. P. Schmidt, “The small angle X-ray scattering from polydisperse solutions of ellipsoidal particles,” Acta Crystallographica, vol. 11, no. 10, pp. 674–676, 1958. View at Google Scholar
  54. Y. Mori, M. Furukawa, T. Hayashi, and K. Nakamura, “Size distribution of gold nanoparticles used by small angle X-ray scattering,” Particulate Science and Technology, vol. 24, no. 1, pp. 97–103, 2006. View at Publisher · View at Google Scholar · View at Scopus
  55. M. L. Lavčević and A. Turković, “The measurements of particle/crystallite size in nanostructured TiO2 films by SAXS/WAXD method,” Scripta Materialia, vol. 46, no. 7, pp. 501–505, 2002. View at Publisher · View at Google Scholar · View at Scopus
  56. L. C. Roess, “A simple method of obtaining a particle mass distribution by inverting the X-ray intensity scattered at small angles,” The Journal of Chemical Physics, vol. 14, no. 11, pp. 695–697, 1946. View at Google Scholar · View at Scopus
  57. J. Riseman, “Particle-size distribution from small-angle X-ray scattering,” Acta Crystallographica, vol. 5, no. 2, pp. 193–196, 1952. View at Google Scholar
  58. V. Luzzati, “Sur deux problemes relatifs a la diffusion des rayons X aux petits angles: determination de la distribution des masses et correction du polychromatisme,” Acta Crystallographica, vol. 10, no. 1, pp. 33–34, 1957. View at Google Scholar
  59. J. H. Letcher and P. W. Schmidt, “Small-angle X-ray scattering determination of particle-diameter distributions in polydisperse suspensions of spherical particles,” Journal of Applied Physics, vol. 37, no. 2, pp. 649–655, 1966. View at Publisher · View at Google Scholar · View at Scopus
  60. O. L. Brill and P. W. Schmidt, “Small-angle X-ray-scattering determination of diameter distributions,” Journal of Applied Physics, vol. 39, no. 5, pp. 2274–2281, 1968. View at Publisher · View at Google Scholar · View at Scopus
  61. O. L. Brill, C. G. Weil, and P. W. Schmidt, “Determination of particle-diameter distributions in silica and gold suspensions,” Journal of Colloid And Interface Science, vol. 27, no. 3, pp. 479–492, 1968. View at Google Scholar · View at Scopus
  62. G. Walter, R. Kranold, T. Gerber, J. Baldrian, and M. Steinhart, “Particle size distribution from small-angle X-ray scattering data,” Journal of Applied Crystallography, vol. 18, no. 4, pp. 205–213, 1985. View at Publisher · View at Google Scholar · View at Scopus
  63. I. S. Fedorova and V. B. Emelyanov, “Solution of inverse problems of scattering in the Rayleigh-Debye approximation. II. Determination of diameter distribution functions of thin spherical shells,” Journal of Colloid and Interface Science, vol. 59, no. 1, pp. 106–112, 1977. View at Google Scholar · View at Scopus
  64. I. S. Fedorova, “Solution of inverse scattering problems in the Rayleigh-Debye approximation. I. Determination of the diameter distribution of assemblies of long cylinders,” Journal of Colloid and Interface Science, vol. 59, no. 1, pp. 98–105, 1977. View at Google Scholar · View at Scopus
  65. P. W. Schmidt, V. B. Emel'yanov, and I. S. Fedorova, “Solution of inverse scattering problems in the Rayleigh-Debye approximation. III. Calculation of the length distribution for assemblies of thin cylinders,” Journal of Colloid and Interface Science, vol. 67, no. 2, pp. 226–233, 1978. View at Google Scholar · View at Scopus
  66. P. W. Schmidt, I. S. Fedorova, and V. B. Emel'yanov, “Solution of inverse scattering problems in the Rayleigh-Debye approximation. IV. Calculation of the diameter distribution function for assemblies of thin circular discs,” Journal of Colloid and Interface Science, vol. 67, no. 2, pp. 234–239, 1978. View at Google Scholar · View at Scopus
  67. I. S. Fedorova, “Solution of inverse scattering problems in the Rayleigh-Debye approximation. V. Determination of the diameter distribution functions of helical structures,” Journal of Colloid and Interface Science, vol. 73, no. 1, pp. 208–211, 1980. View at Google Scholar · View at Scopus
  68. I. S. Fedorova and P. W. Schmidt, “A general analytical method for calculating particle-dimension distributions from scattering data,” Journal of Applied Crystallography, vol. 11, no. 5, pp. 405–411, 1978. View at Google Scholar
  69. M. Mulato and I. Chambouleyron, “Small-angle X-ray and neutron scattering of polydisperse systems: determination of the scattering-particle-size distribution,” Journal of Applied Crystallography, vol. 29, no. 1, pp. 29–36, 1996. View at Google Scholar · View at Scopus
  70. M. Mulato, D. Tygel, and I. Chambouleyron, “On the retrieval of particle size distributions from small-angle scattering data: the influence of statistical data dispersion,” Journal of Applied Crystallography, vol. 31, no. 2, pp. 149–153, 1998. View at Google Scholar · View at Scopus
  71. R. W. Hendricks, J. Schelten, and W. Schmatz, “Studies of voids in neutron-irradiated aluminium single crystals: II. small-angle neutron scattering,” Philosophical Magazine, vol. 30, no. 4, pp. 819–837, 1974. View at Google Scholar
  72. C. Vonk, “On two methods of determination of particle size distribution functions by means of small-angle X-ray scattering,” Journal of Applied Crystallography, vol. 9, no. 6, pp. 433–440, 1976. View at Google Scholar
  73. A. N. Tichonov and Y. Arsenin, Solution of Ill Posed Problems, WH Winston, Washington, DC, USA, 1977.
  74. O. Glatter, “Determination of particle-size distribution functions from small-angle scattering data by means of the indirect transformation method,” Journal of Applied Crystallography, vol. 13, no. 1, pp. 7–11, 1980. View at Google Scholar
  75. D. I. Svergun, A. V. Semenyuk, and L. A. Feigin, “Small-angle-scattering-data treatment by the regularization method,” Acta Crystallographica Section A, vol. 44, no. 3, pp. 244–250, 1988. View at Google Scholar
  76. H. G. Krauthaeuser, W. Heitmann, A. Kops, and G. Nimtz, “Small-angle X-ray scattering analysis of particle-size distributions of mesoscopic metallic systems with consideration of the particle form factor,” Journal of Applied Crystallography, vol. 27, no. 4, pp. 558–562, 1994. View at Publisher · View at Google Scholar · View at Scopus
  77. P. Moore, “Small-angle scattering. Information content and error analysis,” Journal of Applied Crystallography, vol. 13, no. 2, pp. 168–175, 1980. View at Google Scholar
  78. D. Tatchev and R. Kranold, “Maximum-entropy method as a routine tool for determination of particle size distributions by small-angle scattering,” Journal of Applied Crystallography, vol. 37, no. 1, pp. 32–39, 2004. View at Publisher · View at Google Scholar · View at Scopus
  79. V. Goertz, N. Dingenouts, and H. Nirschl, “Comparison of nanometric particle size distributions as determined by SAXS, TEM and analytical ultracentrifuge,” Particle and Particle Systems Characterization, vol. 26, no. 1-2, pp. 17–24, 2009. View at Publisher · View at Google Scholar · View at Scopus
  80. J. Skilling, Maximum Entropy and Bayesian Methods, Kluwer Academic Publisher, Dordrecht, The Netherlands, 1989.
  81. J. A. Potton, G. J. Daniell, A. D. Eastop et al., “Ferrofluid particle size distributions from magnetisation and small angle neutron scattering data,” Journal of Magnetism and Magnetic Materials, vol. 39, no. 1-2, pp. 95–98, 1983. View at Google Scholar · View at Scopus
  82. S. Hansen and J. S. Pedersen, “Comparison of three different methods for analysing small-angle scattering data,” Journal of Applied Crystallography, vol. 24, no. 5, pp. 541–548, 1991. View at Publisher · View at Google Scholar · View at Scopus
  83. S. Hansen, “Bayesian estimation of hyperparameters for indirect Fourier transformation in small-angle scattering,” Journal of Applied Crystallography, vol. 33, no. 6, pp. 1415–1421, 2000. View at Publisher · View at Google Scholar · View at Scopus
  84. C. S. Tsao and T. L. Lin, “Analysis of small-angle scattering data from spherical particles by both the indirect transform method and the maximum-entropy method,” Journal of Applied Crystallography, vol. 30, no. 3, pp. 353–361, 1997. View at Google Scholar · View at Scopus
  85. T.-L. Lin and C.-S. Tsao, “The analysis of small-angle scattering data from polydisperse rodlike particles by indirect transform and maximum-entropy methods,” Journal of Applied Crystallography, vol. 29, no. 2, pp. 170–177, 1996. View at Google Scholar · View at Scopus
  86. P. R. Jemian and A. J. Allen, “The effect of the shape function on small-angle scattering analysis by the maximum-entropy method,” Journal of Applied Crystallography, vol. 27, no. 5, pp. 693–702, 1994. View at Google Scholar
  87. H. G. Krauthäuser, “Deducing material properties from indirect measurements,” Physica A, vol. 211, no. 2-3, pp. 317–326, 1994. View at Google Scholar · View at Scopus
  88. H. G. Krauthäuser, W. Lennartz, and G. Nimtz, “Real-space distributions from small-angle scattering data: structure interference method versus indirect transformation method,” Journal of Applied Crystallography, vol. 29, no. 1, pp. 7–15, 1996. View at Google Scholar · View at Scopus
  89. S. Martelli and P. E. Di Nunzio, “Particle size distribution of nanospheres by Monte Carlo fitting of small angle X-ray scattering curves,” Particle & Particle Systems Characterization, vol. 19, no. 4, pp. 247–255, 2002. View at Google Scholar
  90. P. E. Di Nunzio, S. Martelli, and R. R. Bitti, “Use of Monte Carlo methods in characterizing nanostructured materials by wide- and small-angle x-ray scattering,” Journal of Dispersion Science and Technology, vol. 25, no. 4, pp. 491–501, 2004. View at Publisher · View at Google Scholar · View at Scopus
  91. P. E. Di Nunzio, S. Martelli, and R. Ricci Bitti, “A Monte Carlo estimate of crystallite-size and microstrain distribution functions from X-ray line broadening,” Journal of Applied Crystallography, vol. 28, no. 2, pp. 146–159, 1995. View at Google Scholar
  92. B. R. Pauw, J. S. Pedersen, S. Tardif, M. Takatab, and B. B. Iversen, “Improvements and considerations for size distribution retrieval from small-angle scattering data by monte-carlo methods,” Journal of Applied Crystallography Short Communications, vol. 46, no. 2, pp. 365–371, 2012. View at Google Scholar
  93. G. Tóth, “Simultaneous Monte Carlo determination of particle size distribution and pair-correlation function of spherical colloids from a diffraction experiment,” Langmuir, vol. 15, no. 20, pp. 6718–6723, 1999. View at Publisher · View at Google Scholar · View at Scopus
  94. G. Tóth, “Monte Carlo determination of the radii and the pair-correlation function of spherical colloids,” Physica B, vol. 276-278, pp. 404–405, 2000. View at Publisher · View at Google Scholar · View at Scopus
  95. J. Teixeira, “Small-angle scattering by fractal systems,” Journal of Applied Crystallography, vol. 21, no. 6, pp. 781–785, 1998. View at Google Scholar
  96. N. K. Ailawadi, “Equilibrium theories of simple liquids,” Physics Reports, vol. 57, no. 4, pp. 241–306, 1980. View at Google Scholar · View at Scopus
  97. E. Kaler, “Small-angle scattering from colloidal dispersions,” Journal of Applied Crystallography, vol. 21, no. 6, pp. 729–736, 1988. View at Google Scholar
  98. J. T. Schelten and W. Schmatz, “Multiple-scattering treatment for small-angle scattering problems,” Journal of Applied Crystallography, vol. 13, no. 4, pp. 385–390, 1980. View at Google Scholar
  99. N. F. Berk and K. A. Hardman-Rhyne, “Analysis of SAS data dominated by multiple scattering,” Journal of Applied Crystallography, vol. 21, no. 6, pp. 645–651, 1998. View at Google Scholar
  100. J. P. Hansen and I. R. McDonald, Theory of Simple Liquids, Elsevier Academic Press, London, UK, 2006.
  101. L. Belloni, Interacting Monodispersed and Polydispersed Spheres, Elsvier, Amsterdam, The Netherlands, 1991.
  102. R. J. Baxter, “Percus-Yevick equation for hard spheres with surface adhesion,” The Journal of Chemical Physics, vol. 49, no. 6, pp. 2770–2774, 1968. View at Google Scholar · View at Scopus
  103. E. J. W. Verwey and J. T. G. Overbeek, Theory of the Stability of Lyophobic Colloids, Courier Dover Publications, Mineola, NY, USA, 1999.
  104. J. K. Percus and G. J. Yevick, “Analysis of classical statistical mechanics by means of collective coordinates,” Physical Review, vol. 110, no. 1, pp. 1–13, 1958. View at Publisher · View at Google Scholar · View at Scopus
  105. J. B. Hayter and J. Penfold, “An analytic structure factor for macroion solutions,” Molecular Physics, vol. 42, no. 1, pp. 109–118, 1981. View at Google Scholar
  106. B. Weyerich, J. Brunner-Popela, and O. Glatter, “Small-angle scattering of interacting particles. II. Generalized indirect Fourier transformation under consideration of the effective structure factor for polydisperse systems,” Journal of Applied Crystallography, vol. 32, no. 2, pp. 197–209, 1999. View at Google Scholar · View at Scopus
  107. L. Blum and G. Stell, “Polydisperse systems. I. Scattering function for polydisperse fluids of hard or permeable spheres,” The Journal of Chemical Physics, vol. 71, no. 1, pp. 42–46, 1979. View at Google Scholar · View at Scopus
  108. C. Robertus, W. H. Philipse, J. G. H. Joosten, and Y. K. Levine, “Solution of the Percus-Yevick approximation of the multicomponent adhesive spheres system applied to the small angle X-ray scattering from microemulsions,” The Journal of Chemical Physics, vol. 90, no. 8, pp. 4482–4490, 1989. View at Google Scholar · View at Scopus
  109. L. Blum, “Solution of the Ornstein-Zernike equation for a mixture of hard ions and Yukawa closure,” Journal of Statistical Physics, vol. 22, no. 6, pp. 661–672, 1980. View at Publisher · View at Google Scholar · View at Scopus
  110. M. Kotlarchyk and S.-H. Chen, “Analysis of small angle neutron scattering spectra from polydisperse interacting colloids,” The Journal of Chemical Physics, vol. 79, no. 5, pp. 2461–2469, 1983. View at Google Scholar · View at Scopus
  111. J. S. Pedersen, “Determination of size distributions from small-angle scattering data for systems with effective hard-sphere interactions,” Journal of Applied Crystallography, vol. 27, no. 4, pp. 595–608, 1994. View at Publisher · View at Google Scholar · View at Scopus
  112. S. Hansen, “Monte Carlo estimation of the structure factor for hard bodies in small-angle scattering,” Journal of Applied Crystallography, vol. 45, no. 3, pp. 381–388, 2012. View at Google Scholar
  113. S. Hansen, “The structure factor in small-angle scattering and the effect of deviation from spherical symmetry,” Journal of Applied Crystallography, vol. 44, no. 2, pp. 265–271, 2011. View at Publisher · View at Google Scholar · View at Scopus
  114. J. Brunner-Popela and O. Glatter, “Small-angle scattering of interacting particles. I. Basic principles of a global evaluation technique,” Journal of Applied Crystallography, vol. 30, no. 4, pp. 431–442, 1997. View at Google Scholar · View at Scopus
  115. H. M. A. Ehmann, S. Spirk, A. Doliska et al., “Generalized indirect Fourier transformation as a valuable tool for the structural characterization of aqueous nanocrystalline cellulose suspensions by small angle X-ray scattering,” Langmuir, vol. 29, no. 11, pp. 3740–3748, 2013. View at Google Scholar
  116. J. Brunner-Popela, R. Mittelbach, R. Strey, K.-V. Schubert, E. W. Kaler, and O. Glatter, “Small-angle scattering of interacting particles. III. D2O-C12E5 mixtures and microemulsions with n-octane,” The Journal of Chemical Physics, vol. 110, no. 21, pp. 10623–10632, 1999. View at Publisher · View at Google Scholar · View at Scopus
  117. N. Lutterbach, H. Versmold, V. Reus, L. Belloni, and T. Zemb, “Charge-stabilized liquidlike ordered binary colloidal suspensions. 1. Ultra-small-angle X-ray scattering characterization,” Langmuir, vol. 15, no. 2, pp. 337–344, 1999. View at Google Scholar · View at Scopus
  118. G.-W. Lee, K. S. Jin, J. Kim et al., “Small angle X-ray scattering studies on structures of alkylthiol stabilized-silver nanoparticles in solution,” Applied Physics A, vol. 91, no. 4, pp. 657–661, 2008. View at Publisher · View at Google Scholar · View at Scopus