Sizing Particles with X-Ray Sedimentation
An exact method of determining particle size by sedimentation was developed in the 1850s by George Gabriel Stokes, who derived an equation (known as Stokes' law) describing the settling rates of spheres in a fluid as a function of size. Since early in the 20th century, it has been known that the attenuation of X-radiation is proportional to the mass of the absorber. In the mid-'60s, Olivier and Hickin,1 working in a Freeport Kaolin laboratory, combined these two principles and developed an instrument that uses sedimentation to determine particle size and X-ray absorption to measure the time-dependent change in a mass concentration of solids settling from suspension. The technology was acquired by Micromeritics Instrument Corp. and introduced as a commercial instrument* in 1967. Implementation of the X-ray sedimentation technique for particle sizing has evolved considerably over the years.
Since the introduction of the X-ray sedimentation instrument, a number of other types of analytical instruments have been developed based on different techniques of sizing and quantity determination. Scientists and technologists newly entering the field of particle technology often wonder why X-ray sedimentation continues to be the method of choice in a number of applications. The reasons for remaining with this technique vary from user to user, but two reasons seem to prevail: first, the direct way in which size and quantity are determined, and second, the ability of the user to understand how variations from the theoretical model affect reported results. Because of these advantages, X-ray sedimentation continues to be widely used throughout the ceramic and related industries for accurate particle size analysis.
Objective Assessments of the X-Ray Sedimentation MethodThe X-ray sedimentation method has been objectively compared to various other particle sizing techniques and has been found to be a reliable and repeatable method, producing results that compare favorably with "reference" techniques.2-19 One of the more recent comparisons was published by the National Institute of Standards and Technology (NIST) as a guide to particle size characterization.20 The report covers the most commonly used techniques for particle sizing in ceramics manufacturing-including sieving, sedimentation, microscopy and laser light diffraction. It points out that each method has a unique set of strengths, limitations and sources of error. These should be weighed against the application and environment in which the instrument is to be used in order to best employ its strengths and minimize the influence of the weaknesses and sources of error.
When evaluating a particle sizing method, it is also very important to know what characteristics of the particle the instrument is measuring, and to understand the specific set of circumstances that must prevail for the measurements to reliably relate to the linear dimensions of the particle.21 When this evaluation is applied to the X-ray sedimentation method, it quickly becomes apparent why the technique remains so popular.
Determining Size and Relative MassThe X-ray sedimentation method is based on two well-established and well-understood physical phenomena-gravitational sedimentation and low energy X-ray absorption.
Stokes' law describes the gravitational sedimentation of spherical particles as a function of particle diameter. The law states simply that the terminal settling velocity of a spherical particle in a fluid medium is directly proportional to the square of the diameter of the particle. The Stokes settling model applies rigorously, provided that laminar flow around the particle is maintained as the particle settles and displaces the liquid. This condition can be confirmed by calculating the Reynolds number of the largest particle in the suspension-a Reynolds number less than 0.3 indicates sufficient laminar flow around the particle to satisfy the theoretical model within acceptable tolerance. Smaller particles will have a smaller Reynolds number value, thus also satisfying the conditions.
Classifying particles by settling velocity is equivalent to classifying them by particle size. The settling velocity is determined by measuring the time required for particles to fall a known distance. The simplest case, at least from the point of appreciating the technique, is to imagine introducing all particles simultaneously at the same level at the top of the settling cell. The particles will separate by settling velocity as they fall. At any instant, a specific range of velocities exists within a defined vertical column of liquid below the level of introduction. If all particles have the same density, then they will also be separated by size.
This removal of larger particles from the view continues until all particles have had time to settle below the window, leaving only clear liquid visible. If the depth of the window from the top of the sedimentation cell is known, then at any time during the sedimentation process, the minimum settling velocity of the particles no longer visible can be determined. Any particles settling at a lesser velocity will still be visible in the window, and all particles having a velocity greater than the minimum settling velocity will also still be represented in the window. Using Stokes' law, this partitioning can be converted to one of size.
The X-ray sedimentation instrument employs a window similar to that described above. Through this vertically narrow window is projected a beam of low energy (soft) X-ray. The particles in the volume of liquid along the path of the X-ray beam absorb some portion of the X-ray energy, with the degree of energy attenuation predictably related to the quantity of particles in the measurement zone. The relationship between mass concentration and photon attenuation is described rigorously by the Beer-Lambert-Bouguer law.22 During an analysis, as the suspension begins to separate by size, the X-ray sedimentation instrument shifts the location of this window upward in a precisely defined manner to reduce the distance, and therefore the time required, for smaller particles to settle past the window, as illustrated in Figure 1. The result is that the actual analysis time is measured in minutes, even though the sedimentation of small particles in a liquid can be slow.
The straightforward relationships between settling velocity and size, and between X-ray attenuation and mass, are easily expressed as elementary algebraic equations. Being able to understand details of the theory of operation provides the analyst considerable insight into how best to prepare and analyze a sample, how to interpret the analysis data, and how to relate the reported size deviations to manufacturing processes.
Particles That Deviate from the Theoretical ModelA general assertion that can be made about essentially all particle-sizing instruments is that the most accurate determination of particles size is accomplished when analyzing spherical particles. This is because the vast majority of sizing techniques use an idealized spherical particle model. Whatever is being measured-volume displacement, settling velocity, projected image dimensions or the pattern of scattered light-is related to the size sphere that most closely exhibits the same result as the particle being measured.
Non-spherical particles introduce various degrees of deviation from the theoretical model. The deviation produced in some techniques might be as simple as a shift in reported equivalent spherical size, while the deviation produced in other techniques can be far more complicated. As an example for comparing techniques, consider a collection of identical particles having a 10:1 length-to-width ratio. The settling velocity of any individual particle depends on its orientation at the moment. Sizing such particles by the sedimentation technique results in a broadening of the distribution of equivalent spherical sizes, even though the particles are identical. Since orientation has no effect on X-ray absorption (assuming the particle is smaller than the measuring zone), mass is accurately registered. The light scattering technique of sizing, on the other hand, is far more complex. The angle and intensity of scattered light from each particle will vary, depending on the orientation of the particle relative to the direction and polarization of the incident light. The deconvolution algorithm in the instrument will have to determine what distribution of spherical particles best reproduces the measured scattering pattern. The net result on the reported particle size caused by a tumbling, oblong particle is far more difficult to predict or comprehend, and affects both reported size and quantity.
The ability to understand what effects are to be expected when analyzing non-spherical particles and, in some cases, to be able to apply mathematical corrections to the reported data, are additional reasons why the X-ray sedimentation technique remains so widely used.
Practical AnalysisParticle size is an important determination in ceramic manufacturing and other industrial applications. As a result, particle sizing equipment is available in a variety of configurations to accommodate sizes from nanometers to millimeters and to handle essentially any material, liquid or solid, suspended in liquid or gas. The X-ray sedimentation instrument is but one of many choices of automated sizing instruments.21 However, compared to more "modern" sizing techniques, the direct and easily comprehended technique of size and mass measurement used by the X-ray sedimentation instrument provides the analyst with a higher degree of understanding about how the reported data relates to the size, shape and material under study. Combined with its time-proven reliability and ruggedness, these benefits explain why the X-ray sedimentation instrument is the most relied upon method of particle sizing in a variety of applications.
For more information about X-ray sedimentation, contact Micromeritics Instrument Corp., One Micromeritics Dr., Norcross, GA 30093-1877; (770) 662-3633; fax (770) 662-3696; or visit www.micromeritics.com.
References1. Olivier, J.P. and Hickin, G.K., "An Instrument for Rapid, Automatic Determination of Particle Size Distribution," Freeport Kaolin Co., Gordon, Ga.
2. le Doussal, H.; Martin, D.; and Astier, J., "Analyse Granulometrique: Etude Comparative des Resultats Obtenus par l'Hydrometre de Bouyoucos et le SediGraph 5000," l'Industrie Cheramique, June 1974, No. 674, pp. 449-453.
3. Steger, E.H., "Evaluation of the SediGraph for Particle Size Analysis of Superfine Ammonium Perchlorate," presented at the JANNAF Propellant Characterization Subcommittee Meeting, Gainesville, Va., Nov. 1980, and printed in CPIA Publication 333, 1981, pp. 19-35.
4. Butters, G. and Wheatley, A.L., "A Critical Review of Laser Diffraction Particle Sizing," Particle Size Analysis, 1981.
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8. Weber, O.; Gonthier, E.; and Faugures, J-C, "Grain-Size Data on Marine Sediments: A Comparison Between SediGraph and Malvern Method," Bulletin de l'Institut de Geologie du Bassin d'Aquitaine, No. 50, 1991, pp. 107-114.
9. Syvitski, James P., Principles, Methods, and Application of Particle Size Analysis, Cambridge University Press, Cambridge, 1991.
10. Cooper, John J., "Particle-Size Measurements," Ceram. Eng. Sci. Proc., 12 [1-2], 1991, pp.133-143; and Allen, T., "The DuPont/Brookhaven Scanning X-Ray Disc Centrifuge," Particle Size Analysis, The Royal Society of Chemistry, 1992.
11. Jimbo, G.; Tsubaki, J.; and Yamamoto, H., "Comparisons of the Measured Results of Particle Size by Several Kinds of Measuring Instruments (Report of Japanese Working Parties)," Particle Size Analysis, The Royal Society of Chemistry, 1992.
12. Boning, Charles W., "Water Resources Division (WRD) Policy on Publication of Sediment Size Data Determined by Use of the SediGraph," Chief Office of Surface Water, Office of Surface Water Technical Memorandum No. 93.11.
13. Buchan, G.D.; Grewal, K.S.; Claydon, J.J.; and McPherson, Robin J., "A Comparison of SediGraph and Pipette Methods for Soil Particle-Size Analysis," Aust. J. Soil Res., Vol. 31, 1993, pp. 407-417.
14. Davies, R. and Allen, T., "Evaluation of Instruments for Particle Size Analysis," E.I. DuPont de Nemours & Co., Engineering Services Division, 1989.
15. Cramp, A.; Lee, S.V.; Herniman, J.; Hiscott, R.N.; Manley, P.L.; Piper, D.J.W.; Deptuck, M.; Johnston, S.K.; and Black, K.S., "Data Report: Interlaboratory Comparison of Sediment Grain-Sizing Techniques: Data from Amazon Fan Upper Levee Complex Sediments," Flood, R.D., Piper, D.J.W., Klaus, A., and Peterson, L.C. (Eds.), Proceedings of the Ocean Drilling Program, Scientific Results, Vol. 155, 1997.
16. Rodney Stevens, et al., "Data and Method Priorities Regarding Sediment Investigations of Harbour Sites," Dept. Earth Sciences, Göteborg University, Sweden (http://hjs.geol.uib.no/Hsense/publications/public/98-D2.html), Oct. 15, 1998.
17. Molinarolia, Emanuela; De Falcob, Giovanni; Rabittic, Sandro; and Asunta, Rosana, "Stream-Scanning Laser System, Electric Sensing Counter and Settling Grain Size Analysis: A Comparison Using Reference Materials and Marine Sediments," Sedimentary Geology, Vol. 130, No. 3-4, Feb. 2000, pp. 269-281.
18. Dalkey, Ann and Leecaster, Molly K., "Comparison of Sediment Grain Size Analysis Among Two Methods and Three Instruments Using Environmental Samples," Southern California Coastal Water Research Project Authority (SCCWRP) Annual Report: 1999-2000.
19. Skinner, John, "Pipet and X-ray Grain Size Analyzers: Comparison of Methods and Basic Data," Federal Interagency Sedimentation Project, May 2000.
20. Jillavenkatesa, Ajit; Dapkunas, Stanley J.; and H. Lum, Lin-Sien, "Particle Size Characterization," NIST Recommended Practice Guide, Special Publication 960-1, Jan. 2001.
21. Webb, P.A., "Interpretation of Particle Size Reported by Different Analytical Techniques," 2002, online at www.micromeritics.com.
22. Also known as Beer's law or the Beer-Lambert law-see www.chem.vt.edu/chem-ed/spec/beerslaw.html.