Powder Sampling and Particle Size Determination
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Powder Sampling and Particle Size Determination By
Terence Allen Formerly Senior Consultant E.I. DuPont de Nemours and Company Wilmington, Delaware, USA
2003
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1^^ edition 2003
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ISBN:
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Contents Acknowledgements
x vi i
Preface
xix
Editor's foreword
xxi
1 1.1 1.2 1.3
1.4
1.5
1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13
Powder sampling Introduction Sample selection Sampling stored material 1.3.1 Sampling stored non-flowing material 1.3.2 Sampling from heaps 1.3.3 Sampling stored bulk free-flowing powders 1.3.4 Sampling from sacks and drums 1.3.5 Sampling from trucks and railcars Sampling flowing streams 1.4.1 Sampling from a conveyor belt 1.4.2 Point samplers 1.4.3 Sampling from falling streams 1.4.4 Stream sampling ladles 1.4.5 Traversing cutters 1.4.6 Sampling dusty material 1.4.7 In-line sampling Sample reduction 1.5.1 Scoop sampling 1.5.2 Cone and quartering 1.5.3 Table sampling 1.5.4 Chute splitting 1.5.5 Spinning rifflers 1.5.6 Commercial rotary sample dividers 1.5.7 Miscellaneous sampling devices Slurry sampling Reduction of laboratory sample to measurement sample Number of samples required Theoretical statistical errors on a number basis Practical statistical errors on a number basis Theoretical statistical errors on a weight basis Practical statistical errors on a weight basis Experimental tests of sampling techniques
1 2 6 7 9 10 11 12 12 13 14 14 18 19 19 23 24 25 26 27 27 28 30 30 35 36 38 42 45 46 46 49
vi
Contents
1.14
Weight of sample required 1.14.1 Gross sample 1.14.2 Sampling by increments
2 2.1 2.2 2.3 2.4 2.5
Data presentation and interpretation Introduction Particle size Average diameters Particle dispersion Particle shape 2.5.1 Shape coefficients 2.5.2 Shape factors 2.5.3 Applications of shape factors and shape coefficients 2.5.4 Shape indices 2.5.5 Shape regeneration 2.5.6 Fractal dimensions characterization of textured surfaces 2.5.7 Other methods of shape analysis 2.5.8 Sorting by shape Determination of specific surface from size distribution data 2.6.1 Determination of specific surface from a number count 2.6.2 Determination of specific surface from a surface count 2.6.3 Determination of specific surface from a volume (mass) count Tabular presentation of particle size distribution Graphical presentation of size distribution data 2.8.1 Presentation on linear graph paper Standard forms of distribution fiinctions Arithmetic normal distribution 2.10.1 Manipulation of the normal equation The log-normal distribution 2.11.1 Relationship between number mean sizes for a log-normal distribution 2.11.2 Derived mean sizes 2.11.3 Transformation between log-normal distributions 2.11.4 Relationship between median and mode of a log-normal equation 2.11.5 An improved equation and graph paper for log-normal evaluations 2.11.6 Application Johnson's 5*^ distribution Rosin-Rammler-Bennet-Sperling formula Other distribution laws 2.14.1 Simplification of two parameter equations. 2.14.2 Comments The law of compensating errors
2.6
2.7 2.8 2.9 2.10 2.11
2.12 2.13 2.14 2.15
50 50 51
56 57 63 68 69 74 76 78 82 83 84 88 88 89 89 90 90 93 95 95 96 96 99 100 102 105 106 107 108 109 109 111 112 112 114 117
Contents
2.16
2.17
2.18 3 3.1 3.2 3.3
3.4 3.5 3.6 3.7
3.8 3.9 3.10 3.11
3.12 3.13
Evaluation of nonlinear distributions on log-normal paper 2.16.1 Bimodal intersecting distributions. 2.16.2 Bimodal non-intersecting distributions. 2.16.3 Other distributions 2.16.4 Applications of log-normal plots 2.16.5 Curve fitting 2.16.6 Data interpretation Alternative notations for frequency distribution 2.17.1 Notation 2.17.2 Moment of a distribution 2.17.3 Transformation from qjix) to q^{x) 2.17.4 Relation between moments 2.17.5 Means of distributions 2.17.6 Standard deviations 2.17.7 Coefficient of variation 2.17.8 Applications 2.17.9 Transformation of abscissae Phi-notation Particle size analysis by image analysis Introduction Standards Optical microscopy 3.3.1 Upper size limit for optical microscopy 3.3.2 Lower size limit for optical microscopy Sample preparation Measurement of plane sections through packed beds Particle size Calibration 3.7.1 Linear eyepiece graticules 3.7.2 Globe and circle graticules Training of operators Experimental techniques Determination of particle size distribution by number Conditions governing a weight size determination 3.11.1 Illustrative example of the calculation of a size distribution by weight Semi -automatic aids to microscopy Automatic aids to microscopy 3.13.1 Beckman Coulter RapidVUE 3.13.2 Micromeretics OptiSizer PSDA™ 5400 3.13.3 Oxford VisiSizer 3.13.4 Retsch Camsizer 3.13.5 Malvern Sysmex Flow Particle Image Analyzer 3.13.6 Sci-Tec PartAn - video Image Analyser
vii
117 117 122 122 123 123 125 125 125 126 126 127 128 129 130 130 132 136
142 144 145 145 146 147 151 151 153 154 154 156 157 15 8 160 162 164 167 167 167 168 168 168 169
via Contents
3.14
3.15 3.16
3.17 3.18 3.19 3.20
Quantitative image analysis 3.141 Calibration of image analyzers 3.14.2 Experimental procedures 3.14.3 Commercial quantitative image analysis systems 3.14.4 Confocal laser-scanning microscopy 3.14.5 On-line microscopy 3.14.6 Flatbed scanners 3.14.7 Dark field microscopy 3.14.8 Phase contrast microscopy 3.14.9 Polarized light microscopy (PLM) 3.14.10 Dipix 1440F power scope imaging microscope 3.14.11 Transmission wide field phase contrast microscopy Electron microscopy Transmission electron microscopy (TEM) 3.16.1 Specimen preparation for TEM 3.16.2 Replica and shadowing techniques 3.16.3 Chemical analysis Scanning electron microscopy Other scanning electron microscopy techniques Errors involved in converting a number to a volume count Evaluation of procedures
4 Particle size analysis by sieving 4.1 Introduction 4.2 Standard sieves 4.3 Tolerances for standard sieves 4.4 Woven-wire and punched plate sieves Electroformed micromesh sieves 4.5 4.6 Mathematical analysis of the sieving process 4.7 Calibration of sieves Sieving errors 4.8 4.9 Methods of sieving 4.10 Amount of sample required 4.11 Hand sieving 4.12 Machine sieving 4.13 Wet sieving 4.13.1 Manual 4.13.2 Wet sieving by machine 4.14 Air-jet sieving 4.15 The Sonic Sifter 4.16 The Seishin Robot Sifter 4.17 Automatic systems 4.17.1 The Rotex Gradex 2000 particle size analyzer 4.17.2 Labcon automatic sieve system 4.17.3 Gilson Compu-Sieve« analysis system
169 170 170 180 183 184 185 185 186 186 187 187 187 188 188 192 192 193 196 199 200
208 210 212 213 214 218 221 224 227 229 230 231 234 234 235 237 239 239 240 240 241 241
Contents
4.18 4.19 4.20 4.21 4.22 4.23 4.24
Ultrasonic sieving The sieve cascadograph Felvation Self organized sieves (SORSl) Shape separation Correlation with light scattering data Conclusions
5 5.1 5.2 5.3 5.4
Fluid classification Introduction Assessment of classifier efficiency Systems Counter-flow equilibrium classifiers in a gravitational field elutriators Theory for elutriators Water elutriators Air elutriators Counter-flow centrifugal classifiers; Zig-zag gravitational classifiers Zig-zag centrifugal classifiers The Warmain Cyclosizer Cross-flow gravitational classification 5.12.1 The Humboldt particle size analyzer TDS Cross-flow centrifugal classifiers 5.13.1 Analysette9 5.13.2 The Donaldson Acucut classifier Cross-flow elbow classifier Micromeretics classifier; Fractionation methods for particle size measurement Hydrodynamic chromatography Capillary hydrodynamic fractionation ; Capillary zone electrophoresis Size exclusion chromatography Field flow fractionation 5.21. r Sedimentation field flow fractionation (SFFF) 5.21.2 Centrifugal field flow fractionation 5.21.3 Time-delayed exponential SFFF 5.21.4 Thermal field flow fracfionation 5.21.5 Magnefic field flow fractionation 5.21.6 Flow field flow fractionation 5.21.7 Steric field flow fractionation 5.21.8 Multi-angle light scattering (MALS) The Matec electro-acoustic system EAS-8000 Continuous split fractionation Classification by decantafion;
5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13
5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21
5.22 5.23 5.24
ix
241 241 243 243 244 245 245
251 251 260 261 262 264 265 266 267 267 268 268 268 269 269 269 270 270 271 272 275 276 276 277 278 279 279 282 282 282 284 284 285 285 287
Contents
Interaction between particles and fluids 6.1 Introduction 6.2 Settling of a single homogeneous sphere under a gravitational force 6.2.1 Relationship between settling velocity and particle size 6.2.2. Calculation of particle size from settling velocity in the laminar flow region 6.3 Size limits for gravity sedimentation 6.3.1 Upper size limit 6.3.2 Lower size limit 6.4 Time for terminal velocity to be attained 6.5 Errors due to the finite extent of the fluid (wall effects) 6.6 Errors due todiscontinuity of the fluid 6.7 Viscosity of a suspension 6.8 Non-rigid spheres 6.9 Non-spherical particles 6.9.1 Stokes' region 6.9.2 Relationship between fiber diameter and Stokes diameter 6.9.3 Transition region 6.10 Relationship between drag coefficient and Reynolds number in the transition region 6.11 The turbulent flow region 6.12 Concentration effects 6.13 Hindered settling 6.13.1 Low concentration effects 6.13.2 High concentration effects 6.14 Electro-viscosity 6.15 Dispersion of powders 6.15.1 Dry powder dispersion 6.15.2 The use of glidants to improve flowability of dry powders 6.15.3 Wet powder dispersion 6.15.4 Role of dispersing agents 6.15.5 Wetting a powder 6.15.6 Determination of contact angle (0) 6.15.7 Deagglomerating wetted clumps 6.15.8 Suspension stability * 6.15.9 Tests of dispersion quality 6.16 Powder density 6.17 Liquid viscosity 6.18 Standard powders 6.19 National Standards 7 7.1 7.2 7.3
295 295 297 297 299 300 301 302 306 308 309 311 312 312 312 317 319 322 325 326 332 333 334 335 336 336 337 338 338 339 340 342 343 344 347 350 350 352
Gravitational sedimentation methods of particle size determination Introduction 359 Resolution of sedimenting suspensions 362 Concentration changes in a suspension settling under gravity 364
Contents
1.4
Homogeneous incremental gravitational sedimentation 7.4.1 The pipette method of Andreasen 7.5 Theory for the gravity photosedimentation technique 7.5.1 The Beer Lambert law 7.5.2 The extinction coefficient 7.5.3 Turbidity measurements (Turbidimetry) 7.5.4 The photosedimentation technique 7.5.5 Commercial photosedimentometers 7.5.6 Sedimentation image analysis 7.5.7 Transmission fluctuation spectrometry 7.6 Theory for concentration determination with the x-ray gravitational sedimentation technique 7.6.1 X-ray sedimentation 7.7 Relationship between density gradient and concentration 7.8 Hydrometers and divers 7.8.1 Introduction 7.8.2 Theory 7.8.3 Depth of immersion 7.8.4 Experimental procedure 7.8.5 Divers 7.9 Homogeneous cumulative gravitational sedimentation 7.9.1 Introduction 7.9.2 Theory 7.9.3 Sedimentation balances 7.9.4 Sedimentation columns 7.10 Line-start incremental gravitational sedimentation 7.10.1 Photosedimentation 7.11 Line-start cumulative gravitational sedimentation 7.11.1 Introduction 7.11.2 Methods 8 8.1 8.2 8.3 8.4
Centrifugal sedimentation methods of particle size determination Introduction Stokes' equation for centrifugal sedimentation 8.2.1 General theory Homogeneous, incremental, centrifugal sedimentation 8.3.1 General theory Variable time method (r and S constant, t variable) 8.4.1 General theory 8.4.2 The Simcar pipette disc centrifuge (r constant, S assumed constant, t variable) 8.4.3 Worked example 8.4.4 The Ladal x-ray disc centrifuge(r constant, S constant, t variable) 8.4.5 Discussion of the Kamack equation
xi
365 365 366 366 369 370 370 372 373 374 374 375 378 379 379 380 381 383 384 384 384 384 386 387 387 387 388 388 388
392 394 394 395 395 397 397 403 404 406 406
xii Contents
8.5
8.6.
8.7
8.8
8.9 8.10 8.11 8.12
8.13 8.14 8.15 8.16 8.17 8.18 8.19 9 9.1 9.2
Variable time and height method {S constant, both r and t vary) 8.5.1 Stokes diameter determination 8.5.2 Mass frequency undersize determination 8.5.3 DuPont/Brookhaven scanning x-ray disc centrifugal sedimentometer 8.5.4 Worked example Variable inner radius (Both S and / vary, r remains constant) 8.6.1 Stokes diameter determination 8.6.2 Ladal pipette disc centrifuge 8.6.3 Worked example 8.6.4 Mass frequency undersize determination Photocentrifuges 8.7.1 Introduction 8.7.2 Disc photocentrifuges. 8.7.3 Homogeneous mode Line-start incremental centrifugal sedimentation 8.8.1 Line-start, incremental centrifugal technique 8.8.2 Discussion of line-start theory 8.8.3 BI-DCP disc (photo)centrifuge particle size analyzer Cuvette photocentrifuges Homogeneous, cumulative, centrifugal sedimentation 8.10.1 General theory Variable time method (variation of P with t) Sedimentation distance small compared with distance from centrifuge axis 8.12.1 Hosokawa Mikropul Sedimentputer 8.12.2 Alpine long-arm centrifuge Variable inner radius (variation of P with S) 8.13.1 Alternative theory (variation of P with S) Variable outer radius (variation of P with R) Line-start cumulative centrifugal sedimentation 8.15.1 MSA analyzer Particle size analysis using non-invasive dielectric sensors Supercentriftige Ultracentriftige Conclusions Stream scanning methods of particle size measurement Introduction The electrical sensing zone method (the Coulter principle) 9.2.1 Introduction 9.2.2 Operating principle 9.2.3 Theory for the electrical sensing zone method 9.2.4 Effect of particle shape and orientation 9.2.5 Pulse shape
406 406 407 407 408 410 410 412 413 415 417 417 418 419 422 422 425 428 429 431 431 433 434 434 435 435 437 438 439 439 439 440 442 442
447 449 449 450 452 455 457
Contents xiii
9.2.6 9.2.7
9.3 9.4
9.5
9.6
9.7
9.8 9.9
Effect of coincidence Multiple aperture method for powders having a wide size range 9.2.8 Calibration 9.2.9 Carrying out a mass balance 9.2.10 Oversize counts on a mass basis using the Coulter Counter 9.2.11 Apparatus 9.2.12 Limitations of the method Fiber length analysis Optical particle counters 9.4.1 Light blockage 9.4.2 Optical disdrometer 9.4.3 Light scattering Commercial instruments 9.5.1 Aerometrics 9.5.2 Canty Vision Climet 9.5.3 Contamination Control Systems 9.5.4 9.5.5 Danfoss VisionSensor 9.5.6 Faley Status 9.5.7 Flowvision 9.5.8 Galai 9.5.9 Kane May Kowa 9.5.10 9.5.11 Kratel Malvern 9.5.12 Pacific Scientific Hiac/Royco, Met One 9.5.13 9.5.14 Particle Measuring Systems 9.5.15 Partikel Messetechnik Particle Sizing Systems 9.5.16 9.5.17 Polytec 9.5.18 Rion Spectrex 9.5.19 Dwell time Brinkmann 201 analyzer 9.6.1. Focused Beam Reflectance Measurement (FBRM) 9.6.2 Lasentec Messetechnik Optical Reflectance Method (ORM) 9.6.3 9.6.4 Procedyne Aerodynamic time-of-flight measurement 9.7.1 Thermo Systems Incorporated 9.7.2 Ancillary equipment Laser Doppler velocimetry (LDV) Laser phase Doppler principle
459 460 461 463 464 465 466 467 468 469 470 470 474 474 474 474 475 476 476 476 477 478 478 478 479 480 484 486 486 488 488 489 492 492 493 496 497 497 497 500 501 501
xiv Contents
9.10 9.11 9.12 9.13
9.14 9.15 9.16 9.17 9.18 9.19 9.20 10 10.1 10.2
9.9.1 TSI Aerometrics phase Doppler particle analyzer 9.9.2 Discusion 9.9.3 Differential phase-Doppler anemometry 9.9.4 Bristol Industrial Research Association 9.9.5 Dantec Particle Dynamic Analyzer Hosokawa Mikropul E-Spart Analyzer Shadow Doppler velocimetry Other light scattering methods Interferometers 9.13.1 Mach Zehnder type interferometer 9.13.2 The TSI LiquitrakTM interferometer Flow ultramicroscope. 9.14.1 ISPA image analysis system Measurement of the size distribution of drops in dispersions Dupont electrolytic grain size analyzer Light pressure drift velocity Impact size monitor Monitek acoustic particle monitors Erdco Acoustical Counter
Field scanning methods of particle size measurement Introduction Single point analyzers 10.2.1 Static noise measurement 10.2.2 Ultrasonic attenuation 10.2.3 (J-ray attenuation 10.2.4 X-ray attenuation and fluorescence 12.2.5 Counter-flow classifiers 10.2.6 Hydrocyclones 10.2.7 The Cyclosensor 10.2.8 Automatic sieving machines 10.2.9 Gas flow permeametry 10.2.10 Correlation techniques 10.3 Light scattering and attenuation 10.3.1 Introduction 10.3.2 Effect of extinction coefficient on turbidity 10.3.3 Transient turbidity 10.3.4 Holography 10.3.5 State of polarization of the scattered radiation 10.3.6 Forward^ackward intensity ratio (FBR) 10.3.7 Optical back-scattering 10.3.8 Transmission fluctuation spectroscopy 10.4 Light scattering theory 10.4.1 The Rayleigh region (dX) 10.4.2 The Rayleigh-Gans region (D < X)
502 502 503 504 504 505 506 507 508 508 509 510 510 510 512 512 513 513 514
524 525 525 526 527 527 527 528 528 529 530 531 5 31 531 532 535 536 537 538 539 539 539 539 540
Contents
10.5
10.6 10.7 10.8 10.9 10.10 10.11
10.12 10.13 10.14
10.15 10.16 10.17 10.18
xv
10.4.3 High order Tyndall spectra (HOTS) 10.4.4 Light diffraction 10.4.5 Early commercial light scattering equipment Multi angle laser light scattering; (MALLS) 10.5.1 Theoretical basis for MALLS instruments 10.5.2 Commercial instruments 10.5.3. Discussion Malvern (Insitec) Ensemble Particle Concentration Size (EPCS) Systems Optical incoherent space frequency analysis 10.7.1 Retsch Crystalsizer Pulse displacement technique (PDT) Small angle x-ray scattering (SAXS) Near infra-red spectroscopy (NIR) Ultrasonic attenuation 10.11.1 Introduction 10.11.2 Theoretical basis for ultrasonic instruments 10.11.3 Discussion Matec Acoustosizer (ACS) Ultrasonic attenuation and velocity spectrometry Photon correlation spectroscopy (PCS) 10.14.1 Introduction 10.14.2 Principles 10.14.3 Through dynamic light scattering 10.14.4 Particle size 10.14.5 Concentration effects 10.14.6 Particle interaction 10.14.7 Particle size effects 10.14.8 Polydispersity 10.14.9 The controlled reference method 10.14.10 Multi-angle measurements 10.14.11 Commercial equipment 10.14.12 Discussion 10.14.13 Spectral turbidity 10.14.14 Diffusion wave spectroscopy (DWS) 10.14.15 Photon migration
542 543 543 544 547 552 563
Turbo-Power Model TPO-400 in-line grain size analyzer Concentration monitors Shape discrimination Miscellaneous 10.18.1 Back-scatter intensity 10.18.2 Spectroscopy; photo-acoustic (PAS) and photo-thermal (PTS) 10.18.3 Transient electric birefringence
603 603 604 604 604
568 572 573 574 575 576 576 576 576 581 584 584 586 586 587 588 589 590 590 591 591 593 594 596 601 602 603 603
605 605
xvi Contents
10.18.4 10.18.5 10.18.6 10.18.7 10.18.8 Appendix
Crossed lasers Frequency domain photon migration Laser induced incandescence (LII) Spectral transmission and extinction Turbiscan multiple light scattering measurements
606 606 607 607 608
Manufacturers and suppliers
623
Author index
628
Subject index
653
Acknowledgements I would like to express my grateful thanks to Dr. Brian H. Kaye for introducing me to the fascinating study of particle characterization. After completing a Masters degree at Nottingham Technical College under his guidance I was fortunate enough to be offered a post at the then Bradford Institute of Technology (now Bradford University). At Bradford, Dr. John C. Williams always had time for helpful advice and guidance. John became a good friend and, eventually, my PhD supervisor. After 22 years at Bradford I retired from academic life and looked for other interests. It was then that I met Dr. Reg Davies who had been a postgraduate student with me at Nottingham. Reg was working for the DuPont company in America, who were in need of someone with my background, and I was fortunate to be offered the position. In my ten years with DuPont I have seen the development of the Particle Science and Technology (PARSAT) under Reg's direction. It has been my privilege to have been involved in this development since I consider this group to be pre-eminent in this field. Since my retirement Reg has also retired and the new Group leader is Dr. Arthur Boxman. My special thanks go to Dr. John Boughton who gave advice on the electron microscopy section. My thanks are also due to holders of copyright for permission to publish and to the many manufacturers who have given me details of their products. Terence Allen Cape St. Francis, 6312 Eastern Cape South Africa and Hockessin, DE, USA
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Preface Although man's environment, from the interstellar dust to the earth beneath his feet, is composed of finely divided material his knowledge of the properties of such material is surprisingly slight. For many years scientists have accepted that matter can exist as solids, liquids or gases: Although the dividing line between the states may often be rather blurred; this classification has been upset by powders which at rest are solids, when aerated may behave as liquids and when suspended in a gas take on some of the properties of the gas. Superficially one would consider DuPont to be a chemical company but this is a misconception. Two-thirds of DuPont's products are sold in the form of powders and eighty percent, contain powders. It is therefore essential that employees of this and similar companies should have an understanding of powder properties and behaviour. It is now widely recognised that powder technology is a field of study in its own right. The industrial applications of this new science are far reaching. The size of fine particles affects the properties of a powder in many ways. For example, it determines the setting time for cement, the hiding power of pigments, the activity of chemical catalysts, the taste of food, the potency of drugs and the sintering shrinkage of metallurgical powders. Particle size measurement is to powder technology as thermometry is to the study of heat. In making a decision on which particle sizing technique to us, the analyst must consider the purpose of the analysis. What is generally required is not the size of the particles, but the value of some property that is size dependent. In such circumstances it is important whenever possible to measure the size dependent property, rather than to measure the "size" by some other method and then deduce the required property. For example, in determining the "size" of boiler (fly) ash with a view to predicting atmospheric pollution the terminal velocities of the particles should be measured; in measuring the "size" of catalyst particles, the surface area should be measured, since this is the property that determines the reactivity. The cost of the equipment as well as the ease and the speed with which the analysis can be carried out have then to be considered. The final criteria are that the method shall measure the appropriate property of the particles, with accuracy sufficient for the particular application at an acceptable cost, in a time that will allow the result to be used. Terence Allen
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Editor's Foreword Particle science and technology is a key component of chemical product and process engineering and in order to achieve the economic goals of the next decade, a fundamental understanding of particle processes has to be developed. In 1993 the US Department of Commerce estimated the impact of particle science and technology to industrial output to be one trillion dollars annually in the United States. One third of this was in chemicals and allied products, another third was in textiles, paper and allied products and the final third was in food and beverages, metals, minerals and coal. It was Hans Rumpf in the 1950's who related changes in the functional behavior of most particle processes to be a consequence of changes in the particle size distribution. By measurement and control of the size distribution, one could control product and process behavior. This book is the most comprehensive text on particle size measurement published to date and expresses the experience of the author gained in over fifty years of research and consulting in particle technology. Previous editions have found wide use as teaching and reference texts. For those not conversant with particle size analysis terminology, techniques and instruments, the book contains basis information from which instrument selection can be made. For those familiar with the field, it provides an update of new instrumentation - particularly on-line or in - process instruments upon which the control of particle processes is based. Overall, the book continues to be the international reference text on particle size measurement and is a must for practitioners in the field. Dr. Reg Davies Particle Engineering Research Center, University of Florida
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Powder sampling 1.1 Introduction There are many instances where estimates of population characteristics have to be made from an examination of a small fraction of that population and these instances are by no means confined to the field of powder technology. Regrettably, there are many workers in this field who still believe that sample selection procedure is unimportant. This results in the analyst being presented with hastily taken biased samples on which a great deal of energy is devoted to get precise results which do not reflect the characteristics of the bulk powder. Non-representative sampling results in incorrect analyses, process failure, unacceptable products and customer dissatisfaction. It is essential that the samples selected for measurement should be representative of the bulk in particle size distribution and the relative fractions of their various constituents, irrespective of whether a physical or chemical assay is to be carried out, since these characteristics are frequently inter-dependent. The magnitude of the problem may be realized when one considers that the characteristics of many tons of material may be assumed on the basis of analyses carried out on gram, or even milligram, measurement samples so the chances of carrying out the measurement on a non-representative sample are considerable. Two case studies illustrate problems that may arise. In case 1 a binary mixture of fine and coarse granules was blended to a satisfactory end-point. The blend was next emptied into a bucket elevator and thence to a hopper through a central feed point. The material was then fed, in 60 kg lots, to an extruder to produce an unacceptable finished product. The process started with a mixing operation followed by a segregating operation in the hopper. The central region (core) of the hopper contents was rich in fines and the outer regions rich in coarse, hence the initial feeds to the extruder contained an excess of fines whereas later feeds contained an excess of coarse. Since a bimodal mixture such
Powder sampling and particle size determination
as this has a high tendency to segregate it is better to weigh-feed the two streams concurrently into the extruder: This also has the advantage of eliminating process steps resulting in lower operating costs and a less expensive plant. The second case was a tabletting operation where a wellmixed fine powder was emptied into a core-flow hopper in such a way that segregation occurred. The resulting tablets were well outside specification. The solution was to replace the hopper with a properly designed mass-flow hopper. The total sampling error is made up of errors due to the primary sampling, subsequent sample dividing and errors in the analysis itself Sampling is said to be accurate when it is free from bias, that is, the error of sampling is a random variable about the true mean. Sampling is precise when the error variation is small irrespective of whether the mean is the true mean or not. The ultimate that may be obtained by representative sampling may be called the perfect sample; the difference between this sample and the bulk may be ascribed wholly to the expected difference on a statistical basis. Errors in particle size analysis may be due to: • •
instrument limitations; improper procedure e.g. inadequate dispersion, particle fracture during handling; • operator errors e.g. improper instrument set-up or poor calibration; • incorrect sampling. Two types of sampling errors are possible [1] •
Errors due to segregation of the bulk; this depends upon the previous history of the powder and can be minimized by suitable mixing and building up the sample from a large number of increments.
•
Statistical errors that cannot be prevented. Even for an ideal random mixture the quantitative distribution in samples of a given magnitude is not constant but is subject to random fluctuations. It is the only sampling error, which cannot be suppressed and occurs in ideal sampling. It can be estimated beforehand and reduced by increasing the sample size.
1.2 Sample selection Samples are withdrawn from a population in order to estimate certain characteristic of that population and to establish confidence limits for that
Powder sampling
3
characteristic. The characteristic may be particle size, composition or quality; a measure of the spread of the distribution may also be required. The objective may be to set up limits between which the quality of a final product is acceptable, to decide whether the characteristics of a given lot meets preset criteria, or it may be to estimate the variability within a lot or between lots. If the material comes in containers, or can be viewed as discrete units, the objective may be to estimate the number of units outside of specification. The value of the estimate is largely dependent on the sampling technique adopted. It is of little value, and could impart false information, if it is biased or imprecise. It is usually impossible to measure the size distribution of a bulk powder and so it is necessary to carry out measurements on a sample extracted from the bulk. This sample is itself frequently too large and has to be further sub-divided. The probability of obtaining a sample that perfectly represents the parent distribution is remote. If several samples are taken their characteristics will vary and, if these samples are representative, the expected variation can be estimated from statistical analysis. However, the sampling equipment will introduce a further variation, which may be taken as a measure of sampler efficiency. Imposed on this there may also be operator bias. The stages, in reducing from bulk to measurement samples may be conveniently divided into the five stages illustrated below: bulk or process stream (10" kg)
gross sample (>kg)
laboratory sample (
test sample (g)
measurement sample (mg)
The gross sample is either one of a series of spot samples that have been extracted in order to determine the variability of the bulk or process stream with location or time or it is made up of sub-samples to be representative of the bulk as a whole. In some cases the gross sample is too large to send to the laboratory and has to be reduced in quantity. This reduction needs to be carried out in such a way that the laboratory sample is fully representative of the gross sample. When this reduction is unnecessary the gross sample is also the laboratory sample. The laboratory sample may be required for a number of tests, so it is sometimes necessary to further sub-divide it into test samples. Finally, the test sample may be used in its entirety or further subdivided to form the measurement sample.
4
Powder sampling and particle size determination
The object of sampling is to gain knowledge of the characteristics of the whole from measurements impracticable to apply to the whole; bias at any of the reduction stages adversely affects the final analysis. Problems arise due to inhomogeneities in the parent distribution. If the bulk powder is homogeneous, or can be mixed prior to sampling in order to generate a homogeneous powder, sampling problems do not arise. In order to establish homogeneity it is necessary to examine samples taken from the bulk, either at random or according to some pre-determined pattern. If homogeneity is established a single sample is representative of the bulk. The definition of homogeneity requires specification of the portion or sample size between which the variability is sufficiently small to be neglected. Temporal or spatial variability of inhomogeneous powders may be random, i.e. there is no discernible pattern and it is impossible to predict the value at any one point from knowledge of the value at any other point. In this case it is necessary to examine a number of samples in order to establish a mean and a standard deviation [2]. Non-random variation may be regular cyclic, which can cause problems if the sampling sequence follows the same cycle, or irregular cyclic in which case the larger the portion the greater the smoothing out of irregularities.
Fig. 1.1 Segregation of powders when poured into a heap. Powders may be defined as free-flowing powders or cohesive. Freeflowing powders tend to segregate during handling and storage hence spot samples are rarely representative. Cohesive powders tend to retain their characteristics during handling but if they are segregated during manufacture or packaging they will tend to remain segregated. Handling of free-flowing powders can result in size segregation; hence the distribution of particle size in a powder depends upon its previous history. If free-flowing powder is poured into a heap there is a tendency
Powder sampling 5 for the fine particles to percolate through the coarse and for the coarse particles to roll on the fines. This leads to an excess of fines in the center of the heap and an excess of coarse on the outside. (Figure 1.1) shows a cross-section through a heap of binary powder with the fine particles tending to remain in the center. Sizes; coarse (black) approximately 1 mm diameter; fine (white) approximately 0.2 mm diameter. For markedly greater disparities in size the segregation seems to be size independent.
'j(^MtboTDc dust 2 ^
Powder
Fig. 1.2 Segregation due to vibration For powders that are vibrated, the percolation process lifts larger particles to the powder surface (Figure 1.2). i.e. vibration causes small particles to percolate under large ones thus lifting them. Even when the larger particles are denser than the smaller particles in which they are immersed they can be made to rise to the surface. This can be demonstrated by placing a one-inch diameter steel ball in a beaker and then filling the beaker with sand to completely cover the ball; gently vibrating the beaker will cause the ball to rise to the surface. A recent computer modeling study [3,4] with 50,000 small spheres and 250 larger spheres whose diameters were four times as great, showed that 60 vibrations are enough to bring most of the larger spheres to the surface. Further, the process seems to function provided the ratio of diameters is greater than 2.8:1. The situation can be quite complex since material is often fed into containers or on to belt conveyors from the side, hence the feeding itself can generate both vertical and horizontal segregation. Generally little is known in advance concerning the homogeneity of a system and uniformity is rarely constant and often non-random. Conditions often
6
Powder sampling and particle size determination
necessitate the use of inferior sampling procedures; some principles can however be laid down and should be adhered to whenever possible. These principles are embodied in: The golden rules of sampling • A powder should always be sampled when in motion. •
The whole of the stream of powder should be taken for many short increments of time in preference to part of the stream being taken for the whole of the time.
Observance of these rules, coupled with an understanding of the manner in which segregation takes place, leads to the best sampling procedure. Any method that does not follow these rules should be regarded as a second-best method liable to lead to errors. Finally, the need for care and skill in abstracting samples cannot be over-emphasized. 1.3 Sampling stored material Sometimes a sample has to be obtained from a powder at rest; an indication that the sampling is taking place at the wrong time since it must have previously been in motion. In such cases a sample should be compounded from incremental samples and a lower sampling efficiency is to be expected. In one case a binary mixture of equal weights of particles of two sizes, having a diameter ratio of 2.8:1, was poured into a heap and the concentration of fines in different parts of the heap was found to vary between 1% and 70%. Any attempt to find the composition of the heap from measurements based on sampling from the heap had, therefore, little chance of giving an accurate answer. There are a very large number of possible systems from which a gross sample has to be extracted, so it is impossible to lay down instructions which will meet all situations. Essentially, the solution depends on whether the powder is stationary or moving and whether it is cohesive or free-flowing. It is usual to assume that the powder was mixed before storage. If this assumption is not true then the homogeneity of the powder will depend on its previous history. Thus, a non-flowing material, which has been segregate during manufacture, or prior to storage, will remain segregated. For a free-flowing material, segregation can occur during transfer from the mixer to the storage container.
Powder sampling 7
Inner tube Type2
Inner tube split into separate containers Shaft of inner tube
Outer tube case of thief sampling device
TVpe3
Fig. 1.3 Sampling spears 1.3.1 Sampling stored non-flowing material Non-flowing material is composed of very fine cohesive powders, sticky material, moist material or fibrous solids. These may be stored in small containers such as drums or bags or large containers such as trucks or
8
Powder sampling and particle size determination
railway wagons. Surface sampling is usually carried out with a scoop because of its simplicity; a pre-supposition is that the powder at the sampling point is representative of the bulk i.e. that the powder was mixed before storage. Accuracy is increased by taking more than one sample and these should be analyzed separately in a preliminary examination and combined in later analyzes if the variation between samples is at an acceptable level. Sampling accuracy is improved if samples from the body of the material are included and this may be carried out with the aid of a sampling spear (thief). The flow characteristics of the powder dictate the type of sampling thief required. Three types are available (Figure 1.3): Type 1 consists of an inner and outer tube with a closed end and longitudinal window so that the sampling chamber runs the full length of the spear; this generates a core sample and is suitable for free flowing powders. In type 2 the sampling chamber is at the end of the spear; this generates a spot sample and is suitable for compacted powders. In type 3 there are separate chambers along the length of the spear; this is suitable for compacted powders and generates several samples simultaneously. With a Type 2 sampling spear it is usual to extract several samples at different depths, typically top, middle and bottom, and blend these to obtain a composite sample. The spear is thrust into the powder with the inner chamber closed off and, when in position, the outer tube is rotated to allow powder to fall into the inner chamber. When the chamber is full, the inner tube is turned to the closed position and the spear withdrawn. Possible segregation throughout the bed may be investigated with type 3, an average value for the length of the spear with type 2 and a spot sample with type 3. A biased sample may be obtained if the bulk has a wide size range, due to percolation of fines through the coarse particles leading to an excess of fines in the withdrawn sample. Frequently the spears are vibrated to facilitate filling and this can lead to an unrepresentative quantity of fines entering the sample volume. The sampling chambers also have a tendency to jam if coarse particles are present since they can get lodged between the inner and outer chamber. Fragile material may suffer particle fracture and fine powder may become compacted so that it does not flow into the tube opening(s). Self-burrowing probes (Figure 1.4) are also available for sampling stored powder. The accuracy (or inaccuracy) of the sampling spear is comparable to that of scoop sampling and its use is to be deprecated.
Powder sampling
"SelfbuiTOwing serrated probe
9
{ Sample
Fig. 1.4 Self-burrowing sampler for stored powders.
(O Discard
(E)
(F)
^°>
Discard (D) Fig. 1.5 Coning and quartering process. 1.3.2 Sampling from heaps In industry it is common to sample small heaps by coning and quartering. The heap is first flattened at the top and then separated into four equal segments with a sharp-edged board or shovel. The segments are drawn apart and, frequently, two opposite quadrants are recombined and the
10
Powder sampling and particle size determination
operation repeated until a small enough sample has been generated. This practice is based on the assumption that the heap is symmetrical and, since this is rarely so, the withdrawn sample is usually non-representative. This method is no more accurate than scoop or thief sampling, that are simpler to carry out, but gross errors are to be expected (Figure 1.5). Coning and quartering should never be used with free-flowing powders. The solution to the problem of sampling from a heap lies in the fact that the powder must have been poured to form the heap, and this is the time the sample should be collected. 1.3.3 Sampling stored bulk free-flowing powders It is practically impossible to sample representatively, stationary freeflowing powder because of the severe segregation that has almost certainly occurred. There is only one sound piece of advice to give regarding sampling such material. Don't! Never! Well hardly ever! If there is no alternative several samples should be taken and analyzed separately so that an estimate can be made of the degree of segregation. The act of forming a heap will result in segregation and the distribution of particle sizes within the powder will depend upon how the heap was formed. Examination of the cross-section of a heap of free-flowing powder reveals marked segregation, fine particles being concentrated in a region near the axis and coarse particles in the outer part. The nature of segregation within containers will depend upon how the containers were filled and whether they have been transported i.e. upon the previous history of the powders. Table 1.1 Recommended number of containers to be sampled from a packaged lot (ISO 3954) Number of containers in lot 1 to5 6 to 11 12 to 20 21 to 35 36 to 60
Number of containers to be sampled All 5 6 7 g
Number of containers in lot 61 to 99 100 to 149 150 to 199 200 to 299 300 to 399
Number of containers to be sampled 9 10 11 12 13
For every additional 100 containers one additional container shall be sampled.
Powder sampling 11
1.3.4 Sampling from bags and drums Suppose an analysis is required from several tons of material that is available in bags or small containers. Should a bag be selected at random with the hope that it is representative of the bulk? Having selected the bag how should a sample be withdrawn for subsequent analysis? In an analysis of this problem Kaye and Naylor [5] suggest that several containers should be selected in order to obtain a representative sample. These may be selected systematically, i.e. the 100th, 200th, 300th and so on or, preferably, using a table of random numbers. These containers should be examined individually in order to determine whether the variation between containers is at an acceptable level and combined to determine an average. In either case it is necessary to obtain a representative sample from each container. For this purpose Kaye and Naylor recommend a thief sampler but warn that this may easily give a biased sample. The International Standards Organization (ISO) recommends a much higher number of samples depending on the number of containers (Table 1.1). It should be ascertained whether there is segregation in the containers by examining several samples from different depths, e.g. top, bottom, center. Samples should also be taken from the front and rear of the containers. For this purpose a sampling thief may be used subsequently if it is determined that the material does not segregate. For segregated material the whole of each bag should be sampled using a full stream or Vezin type splitter so that the golden rules of sampling are obeyed. This is the only way of obtaining a representative sample from each bag. It would probably have been easier and certainly preferable to obtain a sample as the containers were being filled. 2ft WPL
la Fig. 1.6 Sampling points for a railcar or a truck.
12
Powder sampling and particle size determination
1.3.5 Sampling from trucks andrailcars It is very difficult, in fact practically impossible, to obtain a satisfactory sample of free flowing powder from a truck or railcar, because of the severe segregation that has almost certainly occurred as the container was filled and in its subsequent motion. In sampling from a truck or a wagon it is recommended that eight samples be extracted [6]. No increment should be taken at less than twelve inches below the surface; this avoids the surface layer in which segregation can have occurred due to vibration (Figure 1.6). Care needs to be taken to prevent powder sliding down the slope created due to removal of surface material. This method of obtaining a sample is mentioned because it is possible that there may be circumstances in which there is no alternative but to use it, but this must not be taken to imply that such method will give satisfactory sampling. Every effort should be made to avoid this method and to use one, which satisfies the two 'golden rules'. If a powdered material is in a container the container has been filled and presumably is going to be emptied. At both these times, the powder will be in motion, and a more satisfactory sampling procedure can then be used. Several relevant standards are described in [7]. ASTM C322 [8] provides a detailed procedure for obtaining samples of 4.5 kg from bulk and bagged shipments of up to 30 tons. ASTM D1900 [9] gives examples of sampling from hopper cars with a sampling thief ASTM D75 [10] provides procedures for sampling from a flowing aggregate stream, conveyor belts, stockpiles, transportation units and roadways. This standard also includes a table giving the minimum number of samples required for an analysis based on the maximum nominal size of the aggregates. ASTM B215 [11] describes two different methods for selecting representative samples of metal powders, one using a chute splitter and the other a sample thief A table giving the number of containers to be sampled, based on the number of containers in the lot is included in the standard. 1.4 Sampling flowing streams Most powder systems are transported at some time during their manufacture as flowing streams: Hoppers are emptied by screw or belt conveyors, powders are transferred to bagging operations by screw or pneumatic conveyors and many solids are transported through pipes. A general rule in all sampling is that whenever possible the sample should be taken when the powder is in motion. This is usually easy with
Powder sampling 13 continuous processes; with consignment sampling it may be possible during filling and emptying of storage containers. 1.4.1 Sampling from a conveyor belt When a sample is to be collected from a conveyor belt, the best position for collecting the increments is where the material falls in a stream from the end of the belt. If access at such a point is not possible the sample must be collected from the belt. The whole of the powder on a short length of the belt must be collected. It must be borne in mind that the particles at the edge of the belt may be different to those at the center, and particles at the top of the bed may not be the same as those at the bottom. If the belt can be stopped, inserting into the stream a frame consisting of two parallel plates shaped to fit the belt may collect the sample, the whole of the material between the plates is then swept out. A scoop, such as the one shown in Figure 1.7, can be used to scoop out an increment, but this operation can be hazardous if the belt is moving.
Fig. 1.7 Scoop suitable for sampling coal with a 2 in maximum size, from a moving belt (BS 1017). Bristol Engineering Company manufactures a belt conveyor system, which has only one moving part in contact with the material. An arm sweeps across the belt to remove the sample (Figure 1.8). When sampling from a continuous stream the sampling may be continuous or intermittent. In continuous sampling a portion of the flowing stream is split off and frequently further divided subsequently. In intermittent sampling the whole stream is taken for many short increments of time at fixed time intervals. These increments are usually compounded and samples for analysis taken from this gross sample. Consignment sampling is carried out on a single consignment (e.g. a truck or wagon load).
14
Powder sampling and particle size determination
Fig. 1.8 Automatic sampler for belt conveyor 1.4.2 Point samplers Samples can be extracted from the product stream by the projection of a sample tube, containing a nozzle or orifice, into the flow. The particles impact on the tube and fill the open cavity. The sampling head is out of the stream when not sampling. The snorkel type sampler (Figure 1.9) is available for vertical or inclined applications and can be pre-programmed on sampling frequency. It is not possible to sample non-homogeneous streams representatively with this type of device. With the auger type sampler (Figure 1.10) a slot inside the process stream is rotated to capture a cross-section of the process stream that is then delivered by gravity into a sample container. This type of device does not collect a representative sample unless the stream is homogeneous and has the added disadvantage that it obstructs flow. 1.4.3 Sampling from falling streams In collecting from a falling stream of powder, care should be taken to offset the effects of segregation. Each increment should be obtained by collecting the whole of the stream for a short time. Care must be taken in putting the sampler in and out of the stream. Figure 1.11 shows correct and incorrect ways of doing this. Unless the time during which the
Powder sampling 15 receiver is stationary in its receiving position is long compared with the time taken to insert and withdraw the sampler, the method shown in Figure 1.1 la will lead to an excess of coarse particles as the surface region of the stream, usually rich in coarse particles, is sampled for a longer time than the rest of the stream
Direction of material flow Slot opening
Sample dischaige
Fig. 1.9 Snorkel type point sampler
Powder flow
Sample
•efe^sS* Fig. 1.10 Auger sampler
/6
Powder sampling and particle size determination
Fig. 1.11 Sampling from falling streams, (a) bad sampling technique, (b) good sampling technique, (c) sampling procedure to be adopted for high mass flow rate. The method shown in Figure 1.1 lb is not subject to this objection. If this method is not possible due to some obstruction, the ratio of stationary to moving time for the receiver should be made as large as possible. In many cases it is not possible to collect the whole of the stream, as this would give too large an amount to be handled. The best procedure is to pass a sample collector of the form shown in Figure 1.11c through the stream. Figure 1.12 shows a ladle designed for manual withdrawal of samples from a moving stream of coal (BS1017). The recommended dimensions for different sizes lumps are ^ = 4.5 in; / / = 6 in; Z = 6, 7.25 and 8.5 in
Powder sampling 17
Fig. 1.12 Ladle suitable for sampling from a falling stream of powder.
Fig. 1.13 Stream sampling cup respectively for lumps of size 2, 2.5 and 3 in. An alternative design is shown in Figure 1.13. The width of the receiver, b, is chosen to give an acceptable weight of sample but it must not be made so small that the biggest particles have any difficulty in entering the receiver. Particles that strike the edges of the receiver are likely to bounce out and not be collected, so that the effective width is (b'd) where d is the particle diameter. The effective width is therefore greater for small particles than for large ones. To reduce this error to an acceptable level the ratio of the box width to the diameter of the largest particle should be made as large as possible with a minimum value of 3:1. The depth (a) must be great enough to ensure that the receiver is never full of powder. If the receiver fills before it finishes its traverse through the powder, a wedge shaped heap will form and this will
/ 8 Powder sampling and particle size determination be size selective. As more powder falls on top of the heap the fine particles will percolate through the surface and be retained, whereas the coarse particles will roll down the sloping surface and be lost. The length of the receiver (c) should be sufficient to ensure that the full depth of the stream is collected. /. 4.4 Stream sampling ladles With the designs shown in Figure 1.14a and 1.14b uniform increments are withdrawn to give a representative sample. However the design shown in Figure 1.14c will give a biased sample if the inner and outer arcs of the container are significantly different in length and the powder is horizontally segregated across the belt. Several commercial on-line samplers, based on these principles, are available.
*::::'-*\
(a)
- - T r t*. 1 \ \ «•. •*• I * i l V*. >•••*. «• • * ! 1 • • i*.l
'^-.•»1
lijjj|^j\ /l^:i;ljiil=!ij:i;k i^;ij!j!j!j|il;!!!iV /giisiii:M:!i!i!K
/ililinim
Fig. 1.14 (a) Cross-sectional sampler straight path action, in line (b) crossline (c) oscillating or swinging arc path
Powder sampling 19
1.4.5 Traversing cutters With large tonnages, samples taken from conveyors can represent large quantities of material that need to be further reduced. Often, a traversing cutter is used as a primary sampler, and the extracted sample is further cut into a convenient quantity by a secondary sampling device. It must be borne in mind that the secondary sampler must also conform to the golden rules of sampling. This equipment is satisfactory for many applications but it has limitations, which restrict its use. These are: • Although comparatively readily designed into new plant it is frequently difficult and expensive to retrofit an existing plant. The main reason for this is due to space requirements. • The quantity of sample obtained is proportional to product flowrate and this can be inconvenient when the plant flow-rate is subject to wide variations. On the other hand, where a plant's daily average is required, this is a necessary condition. • It is difficult to enclose the sampler to the extent required to prevent the escape of dust and fume when handling dusty powders. Commercial samplers are available which combine a traversing type sampler with an unacceptable table sampler. An alternative design is the radial cutter or Vezin sampler shown in Figure 1.15. These samplers vary in size from a 15 cm laboratory unit to a 152 cm commercial unit. 1.4.6 Sampling dusty material Figure 1.16 shows a sampler designed to sample a dusty material, sampling taking place only on the return stroke. This is suitable provided the trough extends the whole length of the stream and does not overfill. The sampler shown in Figure 1.17 was designed to extract a constant volume of homogeneous granular material for chemical assay and cannot be recommended when a physical assay is required [12]. The slide valve sampler (Figure 1.18) is suitable for collecting size-representative samples [13].
20
Powder sampling and particle size determination
Primary cutter Traversing cutter drive Secondary sample hopper
Vezin sampler
Motor-driven rotary cutter
Discharge
Sample •:•••; ••.«'
Fig. 1.15 Schematic of a primary and secondary system based on Denver Equipment Company's type C and Vezin samplers. A variant of this problem is encountered in sampling from pre-weighed batches (Figure 1.19). The sampler is, essentially, a screw conveyor which extracts a sample continuously while the container is being filled. This system suffers from two drawbacks in that it limits the discharge opening thus reducing throughput and takes part of the stream for the whole of the time thus contravening the golden rules of sampling. Similar samplers are available to sample flowing streams continuously; these are essentially Archimedes screws (Figure 1.20). Many variations of this design are possible and the reader is referred to Cornish et. al [14] for a comprehensive treatment. Figure 1.21 shows an industrial sampler for free-flowing material including granules, powders and pellets. This can be mounted on screw conveyors, drag conveyors or angular gravity chutes with manual or automatic control.
Powder sampling 21
Normal position
Dischaiging sample
I Sample Fig. 1.16 Full-stream trough sampler Process stream
Sampling position Fig. 1.17 Constant volume sampler
Process stream
Dischaigc position
22
Powder sampling and particle size determination
Process stream
Nonnal pdsiticm
Sampling position
Fig. 1.18 Slide-valve sampler • . • . A / . 4^. • . 4t ^ ^ f ^ ^^ € f ^ .
Hopper feed conveyor
Hopper dischai^ge device Sampling device Sampler receiver
Fig. 1.19 Sampling from a hopper
Hopper
Powder sampling 23
i
Fig. 1.20 Archimedian screw sampler "•
Direction of powder flow
Sliding gate mil
Sample
I HI I
dischaige ii*'i
Piston
Fig. 1.21 Sampler for screw conveyor. 1.4.7 In-line sampling The diverter valve shown in Figure 1.22a was used for extracting a sample from a pilot plant granulator operation. The sample was fed to a hopper and thence, by a vibratory feeder, to a low-angle laser light scattering instrument. The granules were then returned to the process stream. This system allowed on-line analyses every minute so that the process could be optimized. A more substantial design was required for continuous plant operation (Figure 1.22b) with a moving piston operated rectangular flap weighing around 50 lb. Heuer and Schwechten [15] designed a sampler for installation with a fluidized bed jet mill where the entire classifier product passed through a Sympatec laser measuring device. This system is suitable for use with
24
Powder sampling and particle size determination
From granulator
To analyzer To dryer
; 1 To dryer
Fig. 1.22 Sampler for on-line particle size measurement: (a) diverter valve sampler (b) moving flap sampler low throughput pharmaceutical or food products but sample splitting is required for heavy throughputs. For this purpose a rotating pipe sampler was designed [16] which, in order to sample the whole process stream, rotated in a spiral path. Commercial samplers are available from Gustafson Intersystems and Forratechnic 1.5 Sample reduction The gross sample is frequently too large to be handled easily and may have to be reduced to a more convenient weight. ASTM C702 describes sample splitting and cone and quartering together with a miniature stockpile method designed for use with damp, fine aggregates only [17]. Obviously the method employed should comform to the two golden rules mentioned earlier. Usually the amount of material to be handled is small enough that getting it in motion poses no great difficulty. There is a natural tendency to remove an aliquot with a scoop or spatula and this must be avoided since it negates the effort involved in obtaining a representative sample from the bulk. To obtain the best results, the material should be made as homogeneous as possible by pre-mixing. It is common then to empty the material into a hopper and this should be done with care. A homogeneous segregating powder, when fed to a hopper from a central inlet, will segregate since, in essence, it is being poured into a heap. In a core flow hopper the central region, which is rich in fines.
Powder sampling 25
empties first, followed by the material nearer the walls that has an excess of coarse. The walls of the hopper should have steep sides (at least 70°) to ensure mass flow and should be filled in such a way that size segregation does not occur. This can best be done by moving the pour point about, so that the surface of the powder is more or less horizontal. Several sampledividing devices are available and these are discussed briefly below.
Fig. 1.23 Scoop sampling. 1.5.1 Scoop sampling The method consists of plunging a scoop into the powder and removing a sample (Figure 1.23). This is particularly prone to error since the whole of the sample does not pass through the sampling device and, since the sample is taken from the surface, where it is not representative of the bulk. It is sometimes stated that a satisfactory sample can be obtained by shaking the bottle containing the laboratory sample in order to mix it, then extracting the analysis sample with a scoop. Kaye used five modes of shaking and six operators in order that both operator and technique bias could be evaluated [5,18]. The samples consisted of coarse and fine particles mixed in known proportions. He found that moving the bottle rapidly around in a circle in a vertical plane, with a brisk horizontal shaking superimposed on this motion, introduced no bias. Similarly, if the bottle is placed in the palm of the hand and shaken in a circular motion in a plane at about 30° to the horizontal no bias occurs. The other three methods introduced a definite bias. Kaye concluded: (1) Shaking a bottle containing free-flowing particles of different sizes is not an effective method for mixing them. Anyone who doubts this should try the experiment of putting equal volumes of two powders of different sizes and colors into a bottle and examining the mixture after shaking. Pockets of segregated material form and cannot be
26
Powder sampling and particle size determination
broken up by further shaking. In particular the surface region will be rich in large particles. A sample removed with a scoop will include the surface region, where the composition is very likely to be different from that of the body of the powder. (2) A further objection to the use of a scoop it is liable to be sizeselective favoring the collection of fine particles. The reason for this is that, when the scoop is removed from the material, some particles will flow down the sloping surface of the powder retained in the scoop; the finer particles tend to be captured in the surface craters and retained, whereas coarse particles are more likely to travel to the bottom of the slope and be lost. The effect is particularly important if a flat blade (such as a spatula) is used for the removal of the sample. 1.5.2 Cone and quartering This method consists of pouring the powder into a heap and relying on its radial symmetry to give four identical samples when the heap is flattened and divided by a cross-shaped cutter (Figure 1.5). The method would give reliable results if the heap were symmetrical about a vertical axis and if the line common to the two cutting planes coincided with the axis. In practice, the heap is unlikely to be symmetrical and symmetry of cutting would be very difficult to achieve without precision equipment. Departures from symmetry will result in differences in the amount of material in the four samples. Since severe size segregation will certainly occur with freeflowing powders in forming the heap, this departure from symmetry will generate differences in the size distributions of the four samples. Further, each operation reduces the sample size to a quarter if one sample is retained and a half if opposite quadrants are combined: In order to reduce the sample size still further it is necessary to repeat the exercise thus compounding the error. The method is very dependent on the skill of the operator and should be avoided. If coning and quartering is possible, this implies that the amount of material to be divided is such that it can be easily moved by hand, so that it is just as easy to feed it into the hopper of a device such as a spinning riffler in which increments are collected from a stream in an acceptable manner.
Powder sampling 27
Sample Fig. 1.24 Table sampling 1.53 Table sampling In a sampling table the material is fed to the top of an inclined plane in which there is a series of holes. Prisms placed in the path of the stream break it into fractions. Some powder falls through the holes and is discarded, while the powder remaining on the plane passes on to the next row of holes and prisms, and more is removed, and so on. The powder reaching the bottom of the plane is the sample (Figure 1.24). The objection to this type of device is that it relies on the initial feed being uniformly distributed, and a complete mixing after each separation, a condition not in general achieved. As it relies on the removal of part of the stream sequentially, errors are compounded at each separation; hence its accuracy is low. Its advantages are its low price and its lack of moving parts. 1.5.4 Chute splitting The chute splitter consists of a V-shaped trough along the bottom of which is a series of chutes alternately feeding two trays placed on either side of the trough (Figure 1.25). The material is repeatedly halved until a sample of the desired size is obtained. When carried out with great care this method can give satisfactory sample division but it is particularly prone to operator error, which is detectable by unequal splitting of the sample.
28
Powder sampling and particle size determination
Fig. 1.25 Line diagram of chute splitter. In a review of factors affecting the efficiency of chute rifflers Batel [19] stated that there is little to be gained by increasing the number of chutes. Kaye and Naylor [5] point out that this is only true if the chute widths are kept constant; increasing the number of chutes by reducing the chute width increases the efficiency. They tested this hypothesis, using four rifflers of the same overall width having 4, 8, 16 and 32 chutes and found that the efficiency increased as the number of chutes increased. It follows that two narrow, oppositely directed chutes intercepting the flow of powder would provide a very efficient sampling device. An equivalent selection procedure is to have one narrow stream oscillating between two reception bins. This in fact forms the basis of the British Standards oscillating hopper sample divider and the I d oscillating paddle divider (section 1.5.6). 7.5.5 Spinning rifflers The rotary sample divider or spinning riffler was first described in 1934 [20] and conforms to the golden rules of sampling. The preferred method of using this device is to fill a mass flow hopper in such a way that segregation does not occur. The table is then set in motion and the hopper outlet opened so that the powder falls into the collecting boxes. The use of a vibratory feeder is recommended to provide a constant flowrate
Powder sampling 29 (Figure 1.26). In 1959 Pownall [21] described the construction and testing of a large laboratory spinning riffler and a year later Hawes and MuUer [22] described the construction of a small instrument. They also examined the riffler to find how various factors influence its efficiency using quartz and copper sulfate crystals of the same size.
Moss feed hopper,
Spinning riffler
Vibratory feeder
Fig. 1.26 Line diagram of the spinning riffler. Their conclusions were that the efficiency: (a)
is dependent on the relative proportions of the mixture increasing as the proportion of copper sulfate was raised from 1% to 5%.
(b)
increases with increasing particle size.
(c)
is reproducible under similar experimental conditions.
(d)
is not affected by a combination of variables.
(e)
is not affected by the number of presentations of the sample containers to the feed. (Since the minimum number of such
30 Powder sampling and particle size determination presentations was 100, this statement applies only to numbers greater than this). The mixtures in these experiments contained particles all the same size hence the main cause of segregation was not present and the conclusions may not apply in the more usual case where segregation occurs. Several commercial versions of this instrument are available, some of which were designed for free-flowing powders, some for dusty powders and some for cohesive powders. They handle quantities from 40 liters down to a few grams. 1.5.6 Commercial rotary sample dividers These are available from many manufacturers and can be divided into several groups; micro-dividers for small samples and larger versions for greater quantities; closed dividers for dusty and cohesive material (Figure 1.27) and open dividers for granular free-flowing material. Kaye and Naylor [5] suggest that samples of a given size may be obtained directly by controlling the amount of time, during each oscillation, that the feed is directed to each hopper or container. For efficient sampling the number of increments should be high (>35). In the Pascal turntable sampler the powder falls through a hopper outlet on to a cone whose position can be varied in order to alter the hopper outlet size. The powder slides down the cone into containers on a revolving table (Figure 1.28) 1.5.7 Miscellaneous sampling devices] In the oscillating hopper sample divider [23] ](Figure 1.29), the feed hopper is pivoted about a horizontal axis so that it can oscillate while emptying. Two collectors are placed under the hopper outlet so that the powder falls into them alternately so that at each step the sample is halved. The contents of one box are retained so that at each step the weight of the sample is halved. The oscillating paddle sample divider (Figure 1.30) works in a similar way. A similar unit may be installed in a feed pipe down which particulate material is flowing. In this unit the splitter cones are mounted within the feed pipe and the sample falling through the segmental slots passes out of a side pipe, while the remainder flows over the cone and continues down the feed pipe. Since the whole of the stream is not taken for many short intervals of time, non-representative sampling is possible.
Powder sampling 31
Fig 1.27 The Retsch sample divider
Lid provides handle 5 times diameter of laigest particle Rotation
Side plate bent over to form clip Fig. 1.28 Pascal turntable sample divider.
32
Powder sampling and particle size determination
Fig. 1.29 Oscillating hopper sample divider
Fig.1.30 Oscillating paddle sample divider
Glen Creston rotary sampler incorporates this design (Figure 1.31) that contains an internal inclined pipe, which is rotated by a geared motor. The powder is fed into this pipe and with each revolution a proportion falls into the sample column. The opening in the sample column can be adjusted from outside the divider by means of a sliding plate. By this method it is possible to vary the ratio of sample to total throughput. Appropriate dividers are available for different particle sizes and sampling volumes. Retsch rotary divider PT 100 consists of a feeder unit with a dividing head that rotates at 110 rpm; this means that with a dividing head with 10 outlets the feed is split into 1100 aliquot parts per minute. The feed can range from a few grams to 5000 ml depending on the size of the collecting vessels. Retsch rotary divider PK 1000 (Figure 1.32) is designed for representative sampling and dividing of large quantities of bulk material. Depending on the lower cone, from one to six samples can be taken. The sample collecting vessels have a maximum capacity of 0.5 liters and the reject container 30 liters. Gustafson manufacture an automatic sampler for free flow^ing materials from screw or drag conveyors (Figure 1.33).
Powder sampling 33
Fig. 1.31 The Glen Creston rotary sampler
Figure 1.32 Retsch PK 1000 laboratory rotary sample divider 1 feed hopper, 2 on/off switch, 3 time setting controls, 4 upper cone, 5 lower cone, 6 stand, 7 display, 8 start button, 9 sample slot adjustment, 10 sample vessel, 11 reject container
34
Powder sampling and particle size determination
2
I ••90%
^
^
V
^
• KM]« • CZ3*
/0\
Fig. 1.33 Sampling from screw or drag conveyors.
|Motor| Sample
Sample
Fig. 1.34 (a)Sloping trough cutter (b) Vertical pipe cutter A sample splitter described by Fooks [24] consists of a feeder funnel through which the sample is fed. It passes on to the apex of two resting cones, the lower fixed and the upper adjustable by means of a spindle. Segments are cut from both cones and by rotation of the upper cone the effective area of the slits can be varied to vary the sampling proportion. Material, which falls through the segmental slots, is passed to a sample pot. The residue passes over the cone and out of the base of the unit.
Powder sampling 35
Pumped sample flow Rotating slot
Main flow
Fig. 1.35 (a) Osborne's rotating slot slurry sampler, sampling tank, (c) Cross's slotted pipe slurry sampler
(b) Osborne's
1.6 Slurry sampling Slurry process streams vary in flowrate, solids concentration and particle size distribution. Any sampling technique must be able to cope with these variations without affecting the representativeness of the extracted sample. For batch sampling, automatic devices are available where a sampling slot traverses intermittently across a free-falling slurry. Unfortunately it is difficult to improvise with this technique for continuous sampling, since such samplers introduce pulsating flow conditions into the system. Clarke [13] discusses the problems of sampling from liquid streams in open channels and describes a parallel-sided scoop for extracting a sample of a size that is proportional to flow rate.
36
Powder sampling and particle size determination
Denver Equipment Company manufactures a sloping trough cutter (Figure 1.34a) and a vertical pipe cutter (Figure 1.34b) for sampling slurries. Osborne [25] described a sampler that consists of a narrow slot continuously rotated on an axis parallel to the sluny flow (Figure 1.35a). Cross [26] used a slotted pipe mounted vertically in the overflow compartment next to the vortex fmder of a hydrocyclone (Figure 1.35c). Since most continuous size analyzers require a constant volume flowrate, further subdivision is often necessary. Osborne's solution (Figure 1.35b) was to feed the sample stream to a well-agitated tank and withdraw a representative sample at a constant flow-rate. Hinde and Lloyd [27] are more interested in extracting samples for continuous on-line analysis. They state that the process streams from industrial wet classifiers can vary in volume flow rate, solids concentration and particle size distribution. Any sampling technique should be able of coping with these variations without affecting the representativeness of the extracted sample. Autometrics [28] analyze the whole of the extracted sample (Figure 1.36a). Gaps are left below the weirs to prevent sanding. Subdivision of the sample stream, for calibration purposes, can be accomplished very successfully by siphoning from a vertical fast- flowing stream (Figure 1.36b). Due to their simplicity side-wall samplers are superficially attractive (Figure 1.37) but serious errors in concentration and particle size distribution can arise unless the particles are fine, the concentration is high and a very high sampling velocity is used. A projection extending from the pipe wall only marginally improves sampling efficiency [29]. 1.7 Reduction of laboratory sample to measurement sample The methods described in section 1.5 are capable of reducing the gross sample of many kilograms down to a laboratory sample of up to about 1 kg and further reducing this to a few grams. In many cases this reduction is sufficient to generate a measurement sample whilst for some analytical techniques a further reduction is required. Coning and quartering a paste [30] has been used for preparing samples down to about 20 mg in weight but the efficiency of the method is operator dependent. Lines [31] compared three methods of sampling powder prior to Coulter analysis and found a spread of results of ±6% with cone and quartering, ±3.25% with random selection and ±1.25% with cone and quartering a paste.
Powder sampling 37
Pumped sample flow ^
Main flow Flow under weirs to prevent sanding (a)
(b) Overflow
Main flow
Fig. 1.36 (a) Autometrics'design for direct sampling (b) Autometrics' design for extracting a small sample
Stirrer Vent fill tube
Air supply Cover
PTFE seal ring QVF plastic flange Suspension
QVFpipe section
PTFE seal ring
Base plate Stainless steel capillary
10 ml centrifuge tubes
Plinth
4-
Fig. 1.37 Microscal suspension sampler, 100 ml to 10 ml.
38
Powder sampling and particle size determination
In microscopy the required sample consists of a few milligrams of material. This may be extracted from the laboratory sample by incorporating it in a viscous liquid in which it is known to disperse completely. The measurement sample may then be extracted with a measuring rod [2]. Sampling an agitated suspension with a syringe or pipette is a good method for fine powders but, with coarse powders, concentration gradients and particle segregation due to settling and the centrifugal motion of the liquid due to the action of the stirrer may lead to selective sampling. Alternatively the measurement sample may be withdrawn with a sample divider as illustrated in [32]. The Microscal suspension sampler [33] (Figure 1.37) is designed to eliminate these errors. It consists of a glass cylinder closed at either end by stainless steel plates. Around the periphery of the base plate are ten equidistant holes leading to ten centrifuge tubes via stainless steel capillary tubes. The cover plate has a central hole through which passes a stirrer, a scalable inlet for introduction of the suspension, and a gas orifice which enables the gas pressure to be varied. In operation 100 cm^ of suspension is placed in the cylinder and, whilst undergoing agitation, gas is introduced under pressure in order to blow the suspension into the centrifuge tubes. Tests indicate that this system gives significantly less variation than syringe withdrawal, i.e. 1.0% standard deviation as opposed to 3.0%. 1.8 Number of samples required The basic assumption, in analyzing data statistically, is that the samples are representative of the populations from which they are withdrawn. However, there is an uncertainty associated with any measured property and this can be estimated using the confidence interval (CI). This can be expressed in general terms as: CI = estimate ± M (standard deviation of the estimate) where A/ is a multiplier that is determined by the chosen confidence interval (usually 90, 95 or 99%) and the amount of information available to calculate the standard deviation of the estimate. Given a random set of n samples with a measured mean x^ withdrawn from a powder whose standard deviation cr is known, the true mean will lie in the interval [34]: /^ = ^ ^ ± - 7 =
(1-1)
Powder sampling 39 At the 95% confidence level M= 1.96, i.e. there is a five in one hundred chance that the true mean lies outside these limits. The 90% confidence interval is smaller (M= 1.645) and is less likely to have the true value of x^ within its limits. The 99%) confidence interval is larger (M= 2.576) but is more likely to contain the true value of x^ within its limits. For pharmaceutical applications, a value / = 2 is used to denote working quality and a value / =^ 3 (99.9% confidence level) is used for total quality [35] The statistical reliability of analytical data can be improved by increasing the homogeneity of the sample (reducing a), increasing the size of the sample, or increasing taken (increasing n). In most instances the population standard deviation is not known and must be estimated from the sample standard deviation (s). Substitution of s for a in equation (1.1) with M = 1.96 does not result in a 95% confidence interval unless the sample number is infinitely large (in practice n>30). When s is used, multipliers, whose values depend on sample number, are chosen from the /^-distribution and the denominator in equation (1.1) is replaced by ^{n -1) . Assuming a normal distribution of variance, the number of samples required, to assume at the 95%) confidence level that the median is known to +.4, is given by ^ts^' n-
(1.2)
where ^ = |/i-x^| is the maximum allowable difference between the estimate to be made from the sample and the actual value. Example 1 In many sampling procedures, sub-samples are taken at different levels and locations to form a composite sample. If historical evidence suggests that the standard deviation between samples is 0.5, and it is necessary to know the average quality of the lot to within 0.3, the number of subsamples required, at the 95% confidence level, is given by equation (1.2) [36].
/; =
0.3
40
Powder sampling and particle size determination
Example 2 16 samples, withdrawn at random from an unmixed powder, gave a median x^^ = 3.13 |iim by multi-angle laser light scattering (MALLS) with a standard deviation of ^ == 0.80 |Lim. Then, in microns, the median lies between the limits: // = 3.13±2.14- ^-^ V16-1 /i = 3.13±0.4 The multiplier ^ = 2.14 is obtained from a / table for n = 16-1 degrees of freedom at the 95% confidence level. Thus we are 95% confident that the median lies in the confidence interval (CI): CI = 2.69 < 3.13 < ± 3.57 Based on this data, the number of samples required to give an estimate to within ±0.10, (£' = 0.10) is: M=
^2.14x0.8^^ V
0.10
«=293 After mixing, 16 samples gave a median jc^ of 3.107 |a,m with a standard deviation s, of 0.052 fim. Then: /i = 3.107±2.14
^0.052^
//= 3.107 ±0.029 Thus we are 95% confident that the true median lies in the confidence interval: CI-3.078 < 3.107 < 3.136 i.e. the mixing step increases the measurement precision by a factor of fourteen.
Powder sampling 41 A single sample, run 16 times on a MALLS instrument gave a median of 3.11 jLim with a standard deviation of 5^ ^ 0.030 |Lim. The total variation Isf ] is the sum of the variation due to the measuring procedure Is^ ) and the variation due to the sampling procedure (^^). It is possible to isolate the sampling error from the measurement error. The standard deviation due to sampling (sj is 0.042 |Lim and the standard deviation of the measurement technique (s^) is 0.030 |Lim giving a total standard deviation (s^) of 0.052 |Lim. As can be seen from this example, there is little to be gained in using a measurement technique substantially more accurate than the sampling that preceded it. (1.3)
+ s: 0.042
Further, the number of samples required, after mixing, in order to assume at the 95% confidence level that the median is known to ±0.10 |Lim is: n-\
2.14x0.052 V 0.10
n = 2.2 i.e. 3 samples. Compounding increments from the unmixed powder whilst it is in motion, for example, by riffling in a spinning riffler can attain the same accuracy. Table 1.2 Means and standard deviations of active ingredients in simulated sampling trials Sample weight (g) 1 3 5 9
Mean (mg) 98.20 99.45 99.50 100.20
Standard deviation (mg) 5.78 4.78 3.74 3.10
Expected standard deviation (mg) 8.31 4.83 3.74 2.79
In the pharmaceutical industry there has been a tendency by federal agencies to request that blending validation be carried out using samples the size of a dosage unit. A theoretical experiment using random numbers was carried out to assess the effect of changing sample size [37]. A set of
42
Powder sampling and particle size determination
assays of a dosage weighing 1 g containing 100 mg of drug substance was extracted from a bulk of 100 g. Four 1 g samples were averaged for each of 10 measurements and the standard deviation for each group of four determined. The experiment was repeated with 3 g, 5 g and 9 g samples. The means and averages for each set of 10 groups are given in Table 1.2. The expected standard deviation is based on the premise that it is inversely proportional to the square root of sample size. Assuming the 5 g sample gives accurate data it can be seen that the 9 g sample is slightly worse than expected and the 1 g sample is much better than expected. 1.9 Theoretical statistical errors on a number basis The ultimate that can be obtained by representative sampling may be called the 'ideal' sample. A powder may be considered as made up of two components A and B. The probability that the number fraction {p) of the bulk in terms of ^ shall be represented by the corresponding composition ip) of an ideal sample can be computed from the number of particles of .4 and B in the sample {n) and the bulk (N).
PO-P) r
1- n
(1.4)
where a; is the theoretical standard deviation, [the variance Var(o;) is defined as the square of the standard deviation]. For a normal distribution of variance, the spread of data about the mean is described by the probability equation d^ dp
1 CTI^IITT
exp
jp-p) 2a'
2\
Using the transformation a, j = {p- p) dip
1
dy
yjlK
(
y-^
(1.5)
(1.6)
(1.7)
Powder sampling 43
0.5 0.4 H
0.3 H
0.2 H
0.1 H 0.0
Fig. 1.38 Normal probability function (relative).
Fig. 1.39 Normal probability function (cumulative).
44
Powder sampling and particle size determination
Table 1.3 Cumulative normal distribution
y -4.0 -3.0 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 3.0 4.0
(%) 0.003 0.13 2.28 6.68 15.87 30.85 50.00 69.15 84.13 93.32 97.72 99.87 99.997
y -3.50 -2.50 -1.75 -1.25 -0.75 -0.25 0.00 0.25 0.75 1.25 1.75 2.50 3.50
d(t>IAy 0.13 2.15 8.60 18.38 29.96 38.30 39.89 38.30 29.96 18.38 8.80 2.15 0.13
Table 1.4 Variation in the number of black balls in samples taken from a bulk containing 4000 black balls and 8000 similar white balls.
Upper limit 3857 3873 3889 3905 3921 3937 3953 3969 3985 4001 4017 4033 4049 4065 4081
N umber of black balls Lower limit Median value (x) 3864 3872 3880 3888 3904 3896 3912 3920 3928 3936 3944 3952 3960 3968 3984 3976 3992 4000 4008 4016 4032 4024 4040 4048 4064 4056 4072 4080 4096 4088
Frequency of occurrence (A«) 0 1 6 10 43 103 141 195 185 160 90 37 17 11 1
Powder sampling 45
This differential equation is presented graphically in Figure 1.38 and the integrated version in Figure 1.39. From Table 1.3, 68.26% of all occurrences lie within ±1 a; (between y = -\ and j = +1) from the mean, 95.44% within ± 2 q and 99.94% within ±30;. Example 3 Consider a bulk made up of 8000 white balls and 4000 black balls from which 750 are extracted. Substituting in equation (1.4): 750 12,000, a; =0.0167 Hence: « ± wo; = (2/3 ± 0.0167) of 750 for the white balls
n ±na^ =n± 12.5 where n = 500 or 250 for the white and black balls respectively. Thus 68 times out of 100 there will be between 487.5 and 512.5 white balls in the sample; 95 times out of 100 there will be between 475 and 525 and 26 times out of 10, 000 there will be either more than 537.5 or less than 462.5. 1.10 Practical statistical errors on a number basis The mean, on a number basis, can be calculated from experimental data using the following equation: 3c=^^
(1.8)
The standard deviation can also be calculated using the following equation:
46
Powder sampling and particle size determination
If the true mean is not known, the experimental mean, determined using equation (1.8), must be used. In this case the denominator in equation (1.9) is replaced by {n - 1), which has a negligible effect when n is large. Example 4 Assume that the sampling operation described in example 1 is carried out // ^^ 1000 times with the results presented in Table 1.4. Then, from equation 1.9, the standard deviation <7^ = 35.9, which is 2.87 times the theoretical value. If the true mean (4000) is not known the experimental mean, which equals 3985 by equation (1.8), must be used and the denominator in equation (1.9) is replaced by {n -1). The resulting standard deviation a;= 32.4 is very similar to the previous value due to the large number of samples taken. 1.11 Theoretical statistical errors on a weight basis Instead of the number fraction it is more convenient to assess sample and bulk composition in terms of weight fractions P and Pi giving [38]. Var(P,)^-^
1[P^^^^(\-P^)^A\-
\
(1.10)
w M'l and vi'^ are the average weights of individual grains of components A and B. The theoretical standard deviation (a;) of the sample is given by the square root of the variance, and the weight of the sample divided by the weight of the bulk, (w/W), in many practical cases tends to zero. This function may be used as a basis from which to assess the efficiency of a real, non-ideal sampling device. In this case the variance of the sample assay Var(P^) will be greater than Var(P^) due to experimental deficiencies. 1.12 Practical statistical errors on a weight basis The experimental percentage standard deviation can be calculated from experimental data using the following equation:
Powder sampling 47
\——^AJ
L
(1.11)
where P is the true (known) percentage by weight of component A or B, Arij is the number of samples in a narrow weight percentage range centered on P^ which is obtained by sampling and n the total number of samples taken, it being assumed that n is large otherwise it would be necessary to replace the denominator in equation (1.11) with n-\. If P is not known it can be determined using the experimental value. (1.12) The efficiency of a non-ideal sampler can be defined as: C^^^^r^ VariP^)
(1.13)
and should approximate to unity when experimental sampling errors are low. The maximum sample error can be expressed as [22]: E=± ^
(1.14)
Example 5 Consider a binary powder with a bulk of 800 g made up of a 40:60 mixture of particles of weight 0.05 g and 0.10 g. Determine the required gross sample weight, assuming perfect sampling, for a 2% standard deviation. From equation (1.10). 2 0.40x0.60/. . . . _ . _ . . , x ( . w (0.02) =—^—^^(0.40x0.10 + 0.60x0.05) 1 w V 800; w = 40 g.
48 Powder sampling and particle size determination Table 1.5 Sampling of a 60:40 binary sand mixture using a spinning riffler Coarse percentage {P^
Frequency (Aw^)
^rt,(P,-P)
59.8 59.9 60.0 60.1 60.2 60.3
3 3 5 2 2 1
0.12 0.03 0.00 0.02 0.08 0.09
Experimental data (g) W=%00 w^=2.84x 10-4 Wfi= 0.214 X 10-4
Calculated data using equations (1.10) and (1.11) Var(P;) = 0.0057 o;- 0.075% Var(i'„) = 0.0213 cr„= 0.146%
£"= 0.486% C = 2.7
Maximum sample error [equation (1.14)] Sampling effectiveness [equation (1.13)]
Table 1.6 Reliability of selected sampling methods using a 60:40 sand mixture Percentage Sampling technique standard deviation a% Cone and quartering 6.81 Scoop sampling 5.14 Table sampling 2.09 Chute slitting 1.01 Spinning riffling 0.146 0.075 Random variation
Estimated maximum sample error (E%) 22.7 17.1 7.0 3.4 0.42 0.25
Efficiency (C%) 0.013 0.022 0.130 0.56 36.3
Example 6 Any powder can be considered as being made up of two components, the fraction above and the fraction below a certain size. For a reduction of 16 to 1 during sampling with average oversize weight H^^ = 0.10 g, and
Powder sampling 49 average undersize weight Wg = 0.05 g and sample weight w = 50 g, equation (1.10) yields: Var(/^)-—5^
^[0.10P +0.05(1-P)]
800
The maximum value of the variance, obtained by differentiation, equating the differential to zero and solving the resulting equation, occurs for P=\N3 =57.7% giving Var(P,) = 0.000361 and a;= 1.90%. This means that if 50 g samples were withdrawn from this bulk powder of 800 g and replaced without loss, the maximum standard deviation at any percentage level would be equal to or less than 1.90% provided no sampling bias existed. 1.13 Experimental tests of sampling techniques Example 7 Binary mixtures of coarse and fine sand (60:40 ratio) were examined [39] with a spinning riffler to give the data in Table 54. The experiment was repeated using other sampling techniques to give the data in Table 1.6. In every case 16 samples were examined to give the standard deviations shown in column 2. It may be deduced that very little confidence can be placed in the first three techniques and that the spinning riffler is so superior to all other methods that it should be used whenever possible. Since the spinning riffler proved far superior to the other methods it was examined further in order to determine its optimum operating conditions. It was found that a minimum of 35 presentations is required to give optimum results. If the speed of rotation is made too great the efficiency will fall off due to air currents that result in powder loss. Hatton found [40] that Var(P^^)oc(l + L/V)^-^ where Var(P^) is the square of the standard deviation, L is the linear flow rate of the feed and V the peripheral velocity of the spinning disc. He found that this equation fitted Allen's data, which suggests that to reduce variations it is necessary to reduce L/V. Several other articles [13,41-44] prove conclusively the superiority of the spinning riffler over all other sampling methods investigated, and it is recommended in ASTM Standard F577B-78.
50
Powder sampling and particle size determination
1.14 Weight of sample required 1.14.1 Gross sample Particle size analysis is carried out on a sample extracted from the bulk that, irrespective of the precautions taken, never represents the bulk exactly. The limiting (minimum) weight of the gross sample may be calculated, using a simple formula to give an error within pre-designated limits, provided the weight of the gross sample is much smaller than that of the bulk. The limiting weight is given by:
i-2
M. +
(1.15)
df x]0^
CT,
where: M^ p 'J
is the limiting weight in (g) is the powder density (g cm~^)
Gj Wi
is the variance of the tolerated sampling error is the fractional mass of the coarsest size class being sampled
di
is the arithmetic mean of the cubes of the extreme diameter in the size class (cm^).
Table 1.7. Minimum sample mass required for sampling from a stream of powder Upper sieve size (cm) 0.0600 0.0420 0.0300 0.0212
Lower sieve size (cm) 0.0420 0.0300 0.0212 0.0150
Mass % in class (lOOw/) 0.06 2.5 19.2 35.6
Sample weight required 37.5 kg 0.474 kg 14.9 g 1.32 g
Example 8 Determine the minimum sample of quartz (p == 2.65 g cm~^) to give an expected sampling error a; = 0.05 (= 5%) if the coarsest size range of 105 to 75 |Lim contains 10% of the total weight.
Powder sampling 51
M, = 1.56g Since M^ is proportional to cfi it increases rapidly with increasing particle size. This equation is applicable when the coarsest class covers a size range of not more than 2:1 and w^ is less than 50% of the total sample. Example 9 Assume that a sieve analysis of the above powder is carried out, with the results shown in Table 1.7 and the tolerable sampling error for each fraction is ±5%. The amount required for each fraction, in order to keep within this limit, equation (1.15) is given in column 4. F^or a sieve analysis, in order to reduce the errors at the coarse end of the distribution, repeat analyses should be made using only the coarsest sieves. For the example above an error of 100% in the coarsest size range may be acceptable, i.e. (0.1 ±0.10) g; this reduces the required weight to 94 g. Table 1.8. Minimum incremental mass required for sampling from a stream of powder Maximum particle size (mm) 250-150 150-100 100-50 50-20 20-10 10- 0
Minimum mass of increment (kg) 40.0 20.0 12.0 4.0 0.8 0.3
1.14.2 Sampling by increments For sampling a moving stream of powder the gross sample is made up of increments. In this case the minimum incremental weight is given by: M^ =
l^o^fo
(1.16)
52
Powder sampling and particle size determination
where: M//Q WQ VQ
is the average mass of the increment, is the average rate of flow, is the cutter width for a traversing cutter, is the cutter velocity.
If M^o is too small, a biased sample deficient in coarse particles, results. For this reason w^ should be at least ?>d where d is the diameter of the largest particle present in the bulk. ISO 3081 suggests a minimum incremental mass based on the maximum particle size in mm. These values are given in Table 1.8. Secondary samplers then reduce this to analytical quantities. Example 10 Determine the minimum increment weight for a powder falling from a belt conveyer at a rate of 3 metric tons per hour if the size of the largest particle is 1.0 mm and the sampling cutter speed is 6 cm s~^
M^
r .3x10^ . . . n 3 .kgh" .u-iV3^10-^m^ 3600 s h~^
0.06 ms"*
M,-42g Since the flowrate is 833 g s~^ this is not a practical amount, hence a twostage sampler is required. Sampler 1, say, can sample for 2 s to generate 1.67 kg of powder, which is fed to a hopper to provide a feed to a second sampler that reduces it by a factor of 40 to generate the required 42 g. The minimum number of increments required to give an acceptable accuracy for the sampling period is 35, hence, the gross sample weight is given by: A/,.= 1.47 kg The gross sample can be reduced to a laboratory sample of about 10 g, using a Vezin type sampler for example, and finally to a measurement sample of about 1 g using a rotary riffler. If the particle size analysis is carried out on less than 1 g the final reduction is usually effected by dispersing the powder in a liquid and pipetting out the required aliquot.
Powder sampling 53 Gy [45] proposed an equation relating the standard deviation, which he calls the fundamental error a)r, to the sample size:
-I-
\w
W
Cd^
(1.17)
where Wis the mass of the bulk and w = nco is the mass of « increments, each of weight &> which make up the sample, C is the heterogeneity constant for the material being sampled and d is the size of the coarsest element. For the mining industry [46] he expressed the constant C in the form C = clfg where: C=
\-P
p
(1.18)
P is the investigated constant; p is the true density of the material; / is the relative degree of homogeneity, for a random mixture / = 1, for a perfect mixture / == 0; / is a shape factor assumed equal to 0.5 for irregular particles and 1 for regular particles; g is a measure of the width of the size distribution, g = 0.25 for a wide distribution and 0.75 for a narrow distribution (i.e. d^^^y^ < 2d^^^). For the pharmaceutical industry Deleuil [35] suggested C = 0.1/c with the coarsest size being replaced by the 95% size. For W»w equation (1.17) can be written. w(9^-0.1/
]-P
IP' ,
.3
pfd'
T3
(1.19)
where: 0 = t^-^ and t = 3, (99.9% confidence level) for total quality, • For t/95 = 100 )im, p = \.5, P = 10-^ (1000 ppm), 6 = 0.2, / = 0.03 (random), w = 1000 g. • For dg^ = 100 |Lim, /? - 1.5, P = 0.05, 0= 0.05, / = 1 (homogeneous), w = 4 g. • For ^95 = 20 |um, p^ 1.5, P - 10"^ (100 ppm), 0 = 0.05, / - 0.03 (random), w = 8000 g.
54
Powder sampling and particle size determination
Deleuil points out that a sample of this weight is never prepared because the lot is considered to be perfectly homogeneous (/ = 1). The product from industrial grinding circuits oscillates due to variation in hardness and particle size distribution of feed. Heiskanen and Niemelia [47] demonstrated that, using automatic sampling, on-line analysis and autocorrelation procedures, they could map out a frequency of oscillation. References 1 2 3 4 5 6 1
8 9 10 11 12 13 14 15
16
17 18 19
Sommer, K. (1981), Aufbereit Tech., 22(2), 96-105, 2 Cornish, D.C., Jepson, G. and Smurthwaite, M.J. (1981), Sampling for Process Analyzers, Butterworth, 4,38 Julian, R., Meakin, P. and Pavlovich, A. (1992), Phys. Rev. Let. 69, 5 Maddox, J. (1992), Nature, 358, 5 Kaye, B.H. and Naylor, A.G. (1972), Particle TechnoL, 47-66, 11, 25, 28,30 ASTM D451 -63 (1963), Sieve analysis of granular mineral surfacing for asphalt roofing and shingles, 12 Jillavenkatesa, A., Dapkunas, S.J. and Lum,L-S. H. (2001), Particle size characterization. National Institute of Standards and Technology, NIST Sp. Publ. 960-1,72 ASTM C322-82 (1982), Standard practice for sampling ceramic whiteware clays, 12 ASTM D1900-94 (1994^, Standard practice for carbon black-sampling bulk shipments, 12 ASTM D75-97 (1997), Standard practice for sampling aggregate, 12 ASTM 8215-96 (1996), Standard practice for sampling finished lots of metal powders, 12 Hulley, B.J. (1970), Chem Engr., CE 410-CE 413, 7P Clarke, J.R.P. (1970), Measurement and Control. 3, 241-244, 19,36,49 Cornish, D.C., Jepson, G. and Smurthwaite, M.J. (1981), Sampling for process analyzers, Butterworth, 20 Heuer, M. and Schwechten, D. (1995), Partec 95, 6th European Symp. Particle Characterization, 301-314, Numberg, Germany, publ. NUmbergMesse GmbH, 23 Witt, W. and Rothele, S. (1998), 7th European Symp. Particle Characterization, 611-624, Numberg, Germany, publ. NUmbergMesse GmbH, 24 ASTM DC702-98 (1998), Standard practice for reducing samples of aggregates to testing size, 24 Kaye, B.H. (1961), Ph.D. thesis. University of London, 25 Batel. W. (1960), Particle size measuring techniques, Springer Verlag, Germany, 28
Powder sampling 55
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
36 37 38 39 40 41 42 43 44 45 46 47
Wentworth, C.K., Wilgers, W.L. and Koch, H.L. (1934), A rotary type of sample divider, J. Sed Petrol., 4, 127, 28 Pownall, J.H. (1959), The design and construction of a large rotary sampling machine, AERE-R-2861, Harwell, Oxfordshire, UK, UKAEA, 28 Hawes, R. and Muller, L.D. (1960), A small rotary sampler and preliminary studies of its use, AERE, R3051, Harwell, UKAEA, 29, 46 BS3406 (1961), Methods for determination of particle size ofpowders. Part I Sub-division of gross sample down to 2 ml, 30, 36 Fooks, J.C. (1970), Sample splitting devices, Br. Chem. Engr., 15(6), 799, 34 Osborne, B.F. (1972), CM Bull, 65, 97-107, 36 Cross, H.E. (1967), Automatic mill control system, Parts I and II, Mining Congress J., 62-67, 36 Hinde, A.L. and Lloyd, PJ.D. (1975), Powder Technol,. 12, 37-50, 36 Hinde A.L (1973), J. S. Afr. Inst. Min. MetalL, 73, 26-28, 36 Nasr-el-Din, H., Shook, C.A. and Esmail. M.N. (1985), Can. J. Chem. Engr,, 63, 746-753, 36 Burt, M.W.G (1967), Powder Technol, 1, 103,, 36 Lines, RW. (1973), Powder Technol., 7(3), 129-136, 36 BS616 (1963), Methods for sampling coal tar and its products, 38 Burt, M.W.G., Fewtrel. C.A. and Wharton, R.A. (1973), Powder Technol, 7(6), 327-330, 38 Herden, G. (1960), Small particle statistics, Butterworths, 38 Deleuil, M. (1994), Handbook of powder technology, No 9, Powder Technology and Pharmaceutical Processes, Ch. 1, ed. D. Chulia, M. Deleuil and Y. Pourcelet, Elsevier, 38,53 Davies, R. (1982), Kirk Othmer, Encyclopedia of Chemical Technology, 3rd ed., p 528, John Wiley, 39 Carstensen, J.T. and Rhodes, C.T., (1993^, Drug Dev. Ind Pharm., 19(20), 2699-2708, 41 Stange, K. (1954), Chem. Engr. Tech., 26, 331, 46 Allen, T. and Khan, A.A. (1970), Chem. Engr., 108-112, 49 Hatton, T.A. (1978), Powder Technol, 19, 227-233., 49 Montgomery, J.R. (1968), Analyt. Chem., 40(8), 1399-1400, 49 Kaye, B.H. (1962), Powder Metall, 9, 213, 49 Scott, K.J. (1972), The CSIR rotary sample divider, CSIR Special Report, Chem., Pretoria, South Africa, 49 Charlier, R. and Goosens, W. (1970/71), Powder Technol, 4, 351-359, 49 Gy, P. (1953), R. Ind Min., (French), 36, 311-345, 53 Gy, P. (1982^, Sampling of particulate material, theory and practice, Elsevier, Amsterdam, 2nd ed, 53 Heiskanen, K. and Niemelia, O. (1993), Part. Part. Syst. Char act., 10, 7013,54
Data presentation and interpretation 2.1 Introduction I he behavior and properties of particulate material are, to a large extent, dependent on particle morphology (shape, texture etc.) size and size distribution. Therefore proper measurement, informative data presentation and correct data interpretation are fundamental to an understanding of powder handling and end-use properties. In this chapter the following questions will be addressed: What is meant by particle size? What is meant by particle diameter? For a single particle? For an assembly of particles? How is the average size of an assembly of particles defined? What is meant by particle shape? What is meant by particle size distribution? As well as answering these questions, methods of presenting data will be covered together with data analysis and interpretation. Physical characterization differs from chemical assay in that frequently a unique value does not exist. The determined amount of copper in an ore sample should not depend upon the analytical procedure employed whereas the measured size distribution is method dependent. Only homogeneous, spherical particles have an unambiguous size. The following story illustrates the problem. Some extra terrestrial beings (ETB) were sent to earth to study humans. Their homes were spherical and the more important the ETB the bigger the sphere. The ETB who landed in the Arctic had no problem in defining the shape of the igloos as hemispherical with a single (base) diameter. The ETB who landed in North America classified the wigwams as conical but required two dimensions, height and base diameter, to describe their size. The ETB
Data presentation and interpretation
57
who landed in New York classified the skyscrapers as cuboid with three dimensions mutually perpendicular. The one who landed in London gazed about him despairingly before committing suicide. One of the purposes of this chapter is to reduce the possibility of similar tragedies. 2.2 Particle size Ihe size of a spherical homogeneous particle is uniquely defined by its diameter. For regular, compact particles such as cubes or regular tetrahedra, a single dimension can be used to define size. With some regular particles it may be necessary to specify more than one dimension: For a cone the base diameter and height are required whilst for a cuboid three dimensions are needed. Derived diameters are determined by measuring size-dependent properties of particles and relating them to single linear dimensions. The most widely used of these are the equivalent spherical diameters. Thus, a unit cube has the same volume as a sphere of diameter 1.24 units; hence this is the derived volume diameter. The diameter therefore depends upon the measured property. Consider a cube of side 1 cm; its volume V = 1 cm^ and its superficial surface area S ^ 6 cm^, d^ is the diameter of a sphere having the same volume as the cube and d^ is the diameter of a sphere having the same surface area. V^^-dl 6 ' ]2
5' = 7C(i;
so that
d,V
^6^'"
O
so that
1.241
\TIJ
<^5 = - I
=1.382
The surface to volume ratio is of fundamental importance since it controls the rate at which a particle interacts with its surroundings. This is given by: S —=
V
nd] —^
{^/6)dl
„. S 6J2 Thus — = — ~
V
dl
.
6 I.e. 5^„ =
d,.
Hence, for a unit cube d^^= 1. Thus a sphere of diameter 1.241 cm has the same volume as the cube, a sphere of diameter 1.382 cm has the same superficial surface area and a sphere of diameter 1cm has the same surface to volume ratio. Definitions of the symbols used are given in Table 2.1.
58
Powder sampling and particle size determination
If one were dealing with crystals of known shape it would be more sensible to relate the dimension to that shape, but this is not common practice; for the unit cube this procedure would make all the above-derived diameters equal to unity. A spherical homogeneous particle settling in a fluid rapidly reaches a constant 'terminal' velocity that is uniquely related to the diameter of the sphere. If an irregularly shaped particle is allowed to settle in a fluid, its terminal velocity may be compared with that of a sphere of the same density settling under similar conditions. The size of the particle, defined as its free-falling diameter, is then equated to the diameter of that sphere. In the laminar flow region (i?^<0.25) irregularly shaped particles settle in random orientation and a single particle generates a range of equivalent diameters depending on its orientation. The Stokes diameter is some average of these. Outside the laminar flow region, such particles orientate themselves to give maximum resistance to motion and the free falling diameter that is generated will be the smallest of these (Figure 2.1). Thus the free-falling diameter for a non-spherical particle is smaller in the intermediate region than in the laminar flow region.
''s/^n^ 236 Mm
^ 5 . ^ = 277 Mm
Oo o ^a,max = 252Mm
d, = 204Mm
d^^^l'25m
Fig. 2.1 Stokes diameter for an irregular particle of volume diameter 204 |Lim. With maximum resistance to drag the particle will fall at the same speed as a sphere of diameter 236 jim. With minimum resistance to drag the particle will fall at the same speed as a sphere of diameter 277 \im.
Data presentation and interpretation
59
Table 2.1 Definitions of particle diameters Symbol d^
Diameter Volume
d^
Surface
d^^,
Surfacevolume (Sauter) Drag
d^
df
Freefalling
d^f
Stokes
d^
Projected area
d
Projected area
d^
Perimeter
d^
Sieve
• df^
Feret
*^M
Martin
y^
Unrolled
Definition Diameter of a sphere having the same volume as the particle Diameter of a sphere having the same external surface area as the particle Diameter of a sphere having the same ratio of external surface area to volume as the particle Diameter of a sphere having the same resistance to motion as the particle in a fluid of the same viscosity and at the same velocity {d^ approaches d^ when Re is small) Diameter of a sphere having the same freefalling speed as a particle of the same density in a fluid of the same density and viscosity Free-falling diameter in the laminar flow region Diameter of a circle having the same projected area as the particle in stable orientation Diameter of a circle having the same projected area as the particle in random orientation [for convex particles, mean value for all orientations d == d^], Diameter of a circle having the same perimeter as the projected oufline of the particle Width of the minimum square aperture through which the particle will pass The distance between pairs of parallel tangents to the projected outline of the particle in some fixed direction Chord length, parallel to some fixed direcfion, which divides the particle projected outline into two equal areas Chord length through the centroid of the particle oufline
Formula
V^^d^ s = 71^3
-[dll4) Fij =3;rd^rju
*st =
Wdd
P = ltd,.
statistical diameters, often defined in terms of the mean value for a particular particle.
60
Powder sampling and particle size determination
Fig. 2.2 The projected area of a particle is orientation dependent. Martin's diameter {dj^^) is 246 ^im, the Feret diameter {dp) is 312 |Lim and the projected area diameter in stable orientation (climax) *^ ^52 |Lim (the particle is the same as in Figure 2.1) For irregular particles, the assigned size depends upon the method of measurement, hence the particle sizing technique should, whenever possible, duplicate the process one wishes to control. Thus, for paint pigments the projected area is important since this controls hiding power, whereas for chemical reactants the total surface area should be determined. The projected area diameter may be determined by microscopy for each individual particle, but surface area is usually determined for the powder as a whole. The magnitude of this surface will depend upon the method of measurement, permeametry for example, giving a much lower area than gas adsorption. Further, both of these methods depend upon the size of the gas molecules used in the determinations, since less surface may be accessible for larger molecules. The sieve diameter, for square mesh sieves, is the length of the minimum square aperture through which the particles can pass, though this definition needs modification for sieves which do not have square apertures. Microscopy is the only widely used particle sizing technique in which individual particles are observed and measured. A single particle can have an infinite number of linear dimensions and it is only when these are averaged that a meaningful value results. For an assembly of particles, each linear measurement quantifies the particle size in only one direction. If the particles are in random orientation, and if sufficient particles are counted, the size distribution of these measurements reflects the size distribution of the particles perpendicular to the viewing direction. Because of the need to count a large number of particles in order to generate meaningful data these diameters are called statistical diameters.
Data presentation and interpretation
61
Fig. 2.3 (a) Definition of unrolled diameter df^[ d^^ 2R] (b) unrolled curve
Projected area diameter (stable orientation) 2S2^m
Martin diameter 246 ^m
Fig. 2.4 Particle size of a quartz particle by microscopy using Feret, Martin and projected area diameter If the projected areas of the particles are compared with the areas of series of circles the projected area diameters generated describe the particles in two dimensions for the orientation in which they are measured. In microscopy this is usually the projected area in stable orientation but in certain cases the particles may be in a less stable orientation which generates a lower value (Figure 2.2). For a single particle the expectation of a statistical diameter and its coefficient of variation may be calculated from the following equations [1] (Figure 2.3).
62
Powder sampling and particle size determination
Fig. 2.5 Electron microscope photomicrographs of two paint pigments showing how particles can be aggregates of finer particles [2]. E{dj^)^-
\d^i\e
(2.1)
TV i
(2.2)
(2.3)
E{d^)=^{dpddp
_2 _p,j2.
E {dp) E{dp)
E(d^) = ^fd^dd, ^M
(2.4)
(2.5)
IT J
E\d^)
(2.6)
Data presentation and interpretation 63
R6 is known as the shape descriptor (polar signature) [3]. Provided the ^increments are small enough a precise description of the boundary is obtained. If the image has a perimeter that is encountered more than once by R the descriptor can no longer regenerate the shape. For such a particle it is necessary to use equal length steps around the perimeter to define particle shape An illustration of two statistical diameters, Feret and Martin, and the projected area diameter is given in Figure 2.4. Anomalies can occur due to the state of aggregation of the particles. Figure 2.5a shows a single particle of Prussian blue about 1 |Lim in diameter. The nitrogen adsorption surface area is 61.3 m^g"^ from which the surface-volume mean diameter is calculated as 0.051 |im. This is the diameter of the primary particles of which the aggregate is made up. Similarly the micronized Prussian blue Figure 2.5b has approximately the same surface-volume mean diameter. With the red oxide Figure 2.5c the diameter is 0.21 fim that is approximately the same as the solid particle seen in the micrograph. 2.3 Average diameters The purpose of an average is to represent a group of individual values in a simple and concise manner in order to obtain an understanding of the group. It is important therefore that the average should be representative of the group. All averages are a measure of central tendency, which are only modestly affected by the relatively few extreme values in the tails of the distribution. The mode, the most commonly occurring (most popular) value in a distribution, passes through the peak of the relative distribution curve, i.e. it is the value at which the frequency is a maximum (Figure 2.6). More than one high density region may be present in which case the distribution is said to be multi-modal, i.e. bimodal, trimodal and so on. The median divides the distribution into two equal parts, i.e. it is the 50% size on the cumulative distribution curve. The mean {x) is the center of gravity of the distribution (Figure 2.7) i.e. For the mean, the moment of the sums of the elementary areas of the relative distribution, of width djc, about the ordinate, equals the sum of the moments of the elements about the ordinate:
^(x-x)—hx-Y^i^-x)—bx 0 dx Y dx
64 Powder sampling and particle size determination
%/^in
80
60
40
20
Mode - • Median Mean
Particle size (^m)
dx
Fig. 2.6 Definitions of some average diameters
30
40
Particle size (^m) Fig. 2.7 Finding the center of gravity of a distribution by taking moments
Data presentation and interpretation 00
65
00
\
(2.7) 0
For a number distribution d^ = dN and: IjcdTV X =
IdiV 0
For a mass (volume) distribution d^ = dK = x^dTVgiving: 00
°^
/
A
n
V
T
\
°°
/
/I
A
^ - V - = - V - — = i — - = %M ZdF
Z^dTV
Zx^dTV
0
0
0
(2.8)
[The symbol x denotes size, as opposed to diameter, and includes a shape coefficient. This artifact is found to be useful for general treatment of data]. The mode and the median may be determined graphically but the above summation has to be carried out for the determination of the mean. For a slightly skewed distribution the approximate relationship, meanmode ^ 3(mean-median) holds. For a symmetrical distribution, all three averages coincide. These means represent the distribution in only two of its properties. The characteristics of a particle size distribution are its total number, length, surface, volume (mass) and moment. Note that: A system of unequally sized particles may be represented by a system of uniform particles having two, and only two, characteristics of the original distribution. The size of the particles in the uniform system is then the mean size of the non-uniform system with respect to these two characteristics.
66
Powder sampling and particle size determination
As a simple illustration, consider a system of (spherical) particles containing one particle of each diameter from one to ten (Figure 2.8). This distribution can be represented in number and length by a mean diameter of 5.5 (i.e. the number-length mean diameter dj^^ = 5.50) but ten particles each of diameter 5.5 will not have the same surface or volume as the original distribution, i.e. Spheres each of diameter 8.37 will have the same volume and the same moment as the original distribution (i.e. the mean of the mass distribution), but not the same number or length. Mean sizes are defined in Table 2.2 and values for the example above are given in Table 2.3. The size increases systematically with the order of the distribution; i.e. the mean of the volume distribution is greater than the mean of the surface distribution; the mean of the surface distribution is greater than the mean of the length distribution and the mean of the length distribution is greater than the mean of the number distribution.
Table 2.2 Definitions of mean sizes
Number, length Number, surface Number, volume
IdZ ZdTV
^NL
•^A^^* -
XMV \jsJV
\YAS_ 'zdTV
'
—
Length, surface
^LS
-
Length, volume
^LV
-
Surface, volume
^SV
Volume, moment
^VM
^NL
"ZdK AN
Zd^ ZdZ IdF Idi ZdF
-'
ZdM ZdF
-'
IxdTV ZdA^
^NS - ^
IdA^ 1/3
ni/3 ^NV
-
Hs
ZdTV
IjcdA^
^LV - ^
\YyAN_ ' XxdTV
__ Y.x^dN l/dTV YJX AN
Data presentation and interpretation 67
Fig. 2.8 The homogeneous distribution that represents in number and length a heterogeneous distribution of 10 particles of size 1 to 10 with unit separation in size. Table 2.3 Calculated values of mean sizes for a selection of particles of diameter 1 to 10 with one particle in each size class Number (AO
Length {L)
iS)
10
10
L=Y^xAN
S=Y. x^dN x=\ 5 = 385 Xf^y =6.71 HM = 7-72
x=l
A^=10 %L =5.50 1x^^=7.00 x^y = 7.86 \^VM "8.37
Surface
i =55 ^Ns"^ 6.20 Xi^y=lA2
,
Volume {V) 10
,
F = Z x^dN x=\
F= 3,025 %M=7.09
Moment {M) •0
A
M = S jc^dTV x=l
M = 25,333
%M"8.11
The arithmetic mean size of a number distribution (x^) is the sum of the sizes of the separate particles divided by the number of particle; it is most significant when the distribution is normal.
TAN
-X NL
(2.9)
The geometric mean size of a number distribution {x^ is the wth root of the product of the sizes of the n particles examined; it is of particular value with log-normal distributions:
68
Powder sampling and particle size determination \\IN
^^
(ric"^)'
A^lnjc^=ZlnjcdA^
lnx^=
^
(2.10)
The harmonic mean size of a number distribution is the number of particles divided by the sum of the reciprocals of the sizes of the individual particles; this is related to specific surface and is of importance where surface area of the sample is concerned [4]. ZdiV
Y^ANIx
X^
^_^ -^^ dA^ ^"^ N
^
(2.11)
X
The method of sizing may also be incorporated into the symbol. Hence, for particle sizing by microscopy, the arithmetic mean diameter becomes ^a ~ ^NL a' ^^^ volume-moment mean diameter calculated from the results of a sedimentation analysis isd^^ = <^vM,St • ^^^ mean diameter of a cumulative mass percentage distribution obtained by sieving is d^ or J^^ ^ . International Standards Organization is preparing a Standard, ISO/FDIS 9276-2 Calculation of average particle sizes/diameters and moments from particle size distributions. For a discussion of this draft Standard see Alderliesten [5]. 2.4 Particle dispersion The spread of a distribution may be expressed in terms of a range, i.e. the difference between its minimum and maximum sizes; the inter-quartile range is the difference between the 25% and 75% size (25^75); the interpercentile range between two percentages, usually 10 and 90 doXgo). The least significant of these is the first, since a stray undersize or oversize
Data presentation and interpretation 69 particle can greatly affect its value. The most significant are the standard deviations. The standard deviation and geometric standard deviation are statistical measures of spread. The former is more commonly used with powders having a narrow size range and is the difference between the 50% and the (50±16)% sizes. The latter is more commonly used with powders having a wide size range and is the ratio of the 50% and the non-fractional (50±]6)%sizes. The standard deviation is defined as:
^ ^ \{l:{x-xf^<|>~^ T^(f>
(2.12)
Hence: ^2^Ix-A<^_-2
(2.13)
where o^ is called the variance. The geometric standard deviation (
l^kzlM
,2.4)
where: (X= Incr^ and z=\nx 2.5 Particle shape F^article shape is a fundamental powder property affecting powder packing and thus bulk density, porosity, permeability, cohesion, flowability, caking behavior [6] attrition, interaction with fluids and the covering power of pigments, although little quantitative work has been carried out on these relationships. Davies [7] gives other examples where information on shape is needed to describe powder behavior. Many papers have been written on shape determination but there are few articles that relate the measurements to powder behavior and end-use properties. Hawkins [8] critically reviews nearly 300 articles on particle shape measurement and Singh and Ramakrishnan [9]review seventy-six articles on powder characterization by particle shape assessment.
70
Powder sampling and particle size determination
Table 2.4 Qualitative terms for particle shape Acicular Angular Crystalline Dentritic Fibrous Flaky Granular Irregular Modular Spherical
needle-shaped sharp-edged or having a rough polyhedral shape freely developed in a fluid medium of geometric shape having a branched crystalline shape regularly or irregularly thread-like plate-like having approximately an equi-dimensional irregular shape lacking any symmetry having rounded, irregular shape global shape
Rod
Needle
Cube
Prism
Hake
Rake
Fig. 2.9 Form and proportions Qualitative terms [10] may be used to give some indication of particle shape but these are of limited use as a measure of particle properties (Table 2.4). Such general terms are inadequate for the determination of shape factors that can be incorporated as parameters into equations concerning particle properties where shape is involved as a factor. In order to do this, it is necessary to be able to measure and define shape quantitatively. Particle shape analysis can be carried out using pattern recognition techniques [11-14] in which input data are categorized into classes. The potential use of these techniques [15] and the use of the decision function in morphological analysis have been introduced. There are two points
Data presentation and interpretation 71
regarding the assessment of particle shape. One is that the actual shape is unimportant and all that is required is a number for comparison purposes. The other is that it should be possible to regenerate the original particle shape from the measurement data. The numerical relationships between the various 'sizes' of a particle depend on particle shape, and dimensionless ratios of these are called shape factors; the relations between measured sizes and particle volume or surface area are called shape coefficients. Heywood [16] recognized that the word 'shape' in common usage refers to two distinct characteristics of a particle, form and proportions. The former refers to the degree to which a particle approaches a definite form, such as a cube, tetrahedron or sphere, and the latter by the relative proportions of the particle which distinguish one cuboid, tetrahedron or spheroid from another of the same class (Figure 2.9). When three mutually perpendicular dimensions of a particle may be determined, Heywood's ratios [17] may be used: Elongation ratio n == L/B Flakiness ratio m = B/T (a)
(b)
(c)
(2.15) (2.16)
thickness T is the minimum distance between two parallel planes which are tangential to opposite surfaces of the particle, one plane being the plane of maximum stability. breadth B is the minimum distance between two parallel planes which are perpendicular to the planes defining the thickness and are tangential to opposite sides of the particle. length L is the distance between two parallel plane which are perpendicular to the planes defining thickness and breadth and are tangential to opposite sides of the particle.
Consider a particle circumscribed by a rectangular parallelepiped of dimensions Z by 5 by 7, then the projected area of the particle is given by: A = ~dl=a,BL
(2.17)
where a^ is the area ratio. The particle volume equals the projected area multiplied by the mean thickness: a^jl^a^BLp.T
(2.18)
72 Powder sampling and particle size determination
Fig. 2.10 Heywood's dimensions where/7,. is the prismoidal ratio (see Figure 2.10). Combining equations (2.17) and (2.18), and eliminating B, L and T from the resulting equation using equations (2.15) and (2,16), gives: 7CV7C
«v,« =
Pf.
T mV ^
If the particle is equi-dimensional i.e. B = L = T and n = m volume coefficient takes on a special value a^ where: _ 7CV7C
Pf.
(2.19)
1, then the
(2.20)
8 Thus, a^ may be used to defme particle form. When the particle is not equi-dimensional, the appropriate value of a^^ is a/m^n which substantiates the earlier statement that shape is a combination of form and proportions. Heywood classified particles into tetrahedral, prismoidal, sub-angular and rounded. Values of a^ and/?^ are given in Table 2.5 [18].
Data presentation and interpretation
73
Table 2.5 Values of a^ and/?^ for particles of various shapes Shape group
Pr 0.40-0.53 0.53-0.90 0.55-0.80 0.62-0.75
«« 0.50-0.80 0.50-0.90 0.65-0.85 0.72-0.82
. rtetrahedral ^"^"'^"•Iprismoidal sub-angular Rounded
Table 2.6 Values of a g^ and C for particles of various shapes Shape group Geometrical forms tetrahedral cubical spherical Approximate forms , ^tetrahedral ^"^"'^'Vrismoidal sub-angular Rounded
«^a
C
0.328 0.696 0.524
4.36 2.55 1.86
0.38 0.47 0.51 0.54
3.3 3.0 2.6 2.1
a^^ can be calculated using equation (2.19) combined with direct observation to determine the shape group into which the particle fits and the values of w and n. This is practical down to sizes as small as 5 |Lim by measurements of the number, mean size, weight and density of closely graded fractions. Indeed, a^^ may be determined directly by weighing a known number of particles of known mean size. a^^ is more difficult to determine, but Heywood developed the following relationship on the basis of a large number of experimental measurements: 4/3
^
«,,^=1.57 + C
(2.21)
V f" J
in which C is a constant depending on geometrical form. Table 2.6 shows the values of a^^ and C for various geometrical forms and also for irregular particles.
74 Powder sampling and particle size determination
Macroscopically, shape may be derived using shape coefficients or shape factors. Microscopically particle texture may be defined using fractals or Fourier transforms. The introduction of quantitative image microscopy has made such approaches to particle texture analysis practical. Laser diffraction not only provides a particle size distribution it can also provide a value of volume concentration. For spherical and granular particles, this volume is in accord with measured volume concentration. For plate-like particles, the measured volume is an over-estimation that is related to the thinness and aspect ratio. The thickness calculated, using this method, has been used to rank several graphite powders. Results agree closely with those obtained by the BET method [19]. By using a laser diffraction particle size analyzer as a light scattering spectrometer, the fractal dimensions of clusters can be determined. This follows on the work of Prod and Kratky [20] who used neutron and x-ray diffraction scattering. They demonstrated that the plot of log scattering against wavenumber exhibited fractal behavior. 2.5.1 Shape coefficients There are two especially important properties of particles, surface and volume, and these are proportional to the square and cube respectively of some characteristic dimension. The constants of proportionality depend upon the dimensions chosen to characterize the particles; the projected area diameter is used in the following discussion. The surface area and volume of a particle are given by the following equations: Surface of a particle, S = %d^^ - cCs,a^a ^ ^a Volume of a particle, V = —dl =a^ ^d^ = x\ (2.22) 6 where a^ and a^ are the surface and volume shape coefficients, the additional suffix denoting that the measured diameter is the projected area diameter, (for a sieve analysis for example, S - oc^^d^ and so on). It must be noted that, in general, particles do not have a unique surface area, the measured surface depending on the method of measurement that is the degree of discrimination. At a low level of discrimination the area of the convex envelope of the particle is measured; at higher levels the areas of concavities in the surface are included. Similarly, unless the particles
Data presentation and interpretation
75
are homogeneous, their measured volume also depends on the measurement technique. The surface area per unit volume (volume-specific surface) is the ratio of iS to V\ For example, for a single particle: S
6d^
6
•^v
(2.23)
dsy
<^s,ada
^sv,a
^v
^v,a4
(2.24)
da
where d^y, the surface-volume diameter is the diameter of a sphere having the same surface to volume ratio as the particle, asv_a is the surface-volume shape coefficient by microscopy. Similarly the volume-specific surface of a single particle by microscopy is: S.,a=6/d,
(2.25)
and the volume-specific surface of a single particle by sieving is defined as: ^vj^^ldA
(2.26)
Thus, the manner in which the surface area changes from sample to sample can be investigated on the assumption that shape is size independent. Surface-volume shape coefficients have been determined for quartz and silica from surface area measurements using nitrogen adsorption giving \A
76 Powder sampling and particle size determination
also be determined though this will differ from coefficients based on gas adsorption measurements. Hence, when any shape coefficient is quoted, the method of obtaining it should also be stated. 2.5.2 Shape factors The size of a particle may be expressed by a single dimension using one of the diameters defined in Table 2.1. The differences between these dimensions increase as the particle diverges in shape from a sphere. For a population of particles whose shape is not size dependent, distributions obtained using different methods of analysis may be homologous. Multiplying the sizes of one distribution by a constant (shape) factor will therefore generate the other distribution. One of the earliest defined shape factors is the sphericity xj/^ that was defined by Wadell as [22-24]. ^^ ~
surface area of a sphere having the same volume as the particle surface area of the particle (2.27)
Ww
This is less than unity for non-spherical particles. The reciprocal of the sphericity has been termed the coefficient of rugosity [25] or the coefficient of angularity [26]. At low Reynolds number and with convex particles, the drag diameter equals the surface diameter and Stokes diameter may be defined as:
dst^
dst-¥wd, Thus, for non-spherical particles d^
"5V ~
\^s J
(2.28)
Data presentation and interpretation 77
ds.=y^wd.
(2-29)
Hence, for non-spherical particles d^^< d^^
¥\ =
fB^^
(2.30)
\^J
where L is the longest dimension of the particle, the breadth B is measured perpendicular to this and C is the particle thickness. (Note: these definitions differ from Heywood's). For microscope analysis Laird [29] prefers the definition: V,-^
(2.31)
Where S^ is the surface area of a sphere with a diameter equal to the equivalent diameter {S^ = 4A) and S is the surface area of the particle computed from the measured surface area. For rounded images, whose principle dimensions in two directions at right angle to each other are a and b. Heywood [16] quotes the semiempirical formula for the equivalent diameter: 1/2
d = ^-xOJ7ab\ ^' \n
(2.32)
For rectangular images the following equation yields a result that is only 1% different to the one calculated above:
d,={abf^
(2.33)
Particles usually rest on microscope slides in the position of greatest stability. Cylindrical particles, of length kd where d is the cross-sectional diameter, would be expected to rest with the axes horizontal for k>\ and vertical for k<]. Laird [29] found that this was so and that a region existed, 0.85
78 Powder sampling and particle size determination
For discs or cylinders with A:<0.85: (2.34)
VL--^^^
^^
1 + 2)5:
For cylinders with k>\.S\ ¥ L - - ^
(2.35)
Laird also determined sphericity from sieving and sedimentation studies. For two-dimensional images the proximity of the image to the outline of a circle is defined by circularity where: , . An X cross - sectional or projected area of particle outline circularity = (perimeter of particle) _ AnA /?2
I'
I
d
\2
(2.36)
\dcj
This has a maximum value of unity for a circle and decreases with increasing departure from sphericity. (It can also be used inverted). It is also known as compactness or roundness. Fuchs [30] introduced a dynamic shape factor, y/p', to relate volume and Stokes diameters: y/f^
(d.
(2.37)
\^st J From equation (2.28), for non-spherical particles d^>d^f, so that i//>\. 2.5.3 Applications of shape factors and shape coefficients As an illustration of the application of shape coefficients and shape factors consider a cuboid of side x, x, kx, where A: is a variable. In this case, from equation 2.27:
¥w=^
[A.Snk^] — (l + 2yt)
(2-38)
Data presentation and interpretation
and, assuming
d^^ = ^dl/d^
sv,St
(4.571^2 yA25
79
(see Table 2.1 and equation 2.28). (2.39)
^sv,st is the surface-volume shape coefficient by Stokes diameter (d^^). Both l/i//^ and a^^ are plotted against k in Figure 2.11; the shape of the curve will vary according to the ordinate units employed, e.g. if Stokes diameter were chosen for the ordinate instead of k. From this graph it can be seen that both factors are at a minimum when the shape is most compact (i.e. a cube) and increase as the particles become either rod-shaped or flaky. 2.5
2 h
1.5 h
0.1
10
Fig. 2.11 Relationship between shape factors and particle dimensions. If an analysis is carried out by two different techniques, provided tht the particle shape is not size dependant, the two results can be brought into coincidence by multiplying by a shape factor provided that particle shape does not change with particle size. For example, if the medium Stokes diameter is 29.2 |im and the median Coulter diameter is 32.0 |im for particles of cuboid form, multiplying the Stokes diameter by [32.0/29.2]
80
Powder sampling and particle size determination
will yield the Coulter distribution. Thus the shape factor for this powder, relating Stokes and Coulter diameter is 0.913. From equation 2.28, d^ = 38.4 |Lim and, from equation 2.27 i//^= 0.694. Thus the particles are discs (k = 0.329) or cylinders (k = 3.55) (Figure 2.10). It can be deduced from this that unless the particles are grossly irregular, the difference between Stokes diameter and Coulter diameter is very small. Ellison [31] obtained a value of 0.9 for the ratio of the sizes of silica particles determined by settling experiments and mounted in agar in random orientation. Hodkinson [32] found, from measurements on quartz particles by light scattering, a diameter ratio of 0.8 between particles in a liquid suspension and settled particles. Cartwright [33] attempted to find the magnitude of the difference in mean projected diameters between quartz particles in random and stable orientation by microscopy. He used four different mounting techniques and found no significant differences. He attributed this to the difficulty of mounting particles in random orientation. These factors for the mean ratio of projected diameters for random and stable orientation are indicative of the properties of the powder and therefore of use to the analyst. Respirable coal mine dust samples from three different US mines were classified into four fractions using a Bahco classifier [34]. Shape factors were determined as ratios of the following mean diameters. From microscopy:
Y.ndl da
(2.40)
From photosedimentation
dp =
(2.41) K-^wj
From krypton gas adsorption
'es.^'" ^BET
yTlN^y
(2.42)
Data presentation and interpretation 81
dsy=-—2pAp
(2 A3)
p is from density measurements. N\^, is the number of particles per gram of powder, Ap is the projected area by photosedimentation. It was postulated that a relationship might exist between shape and the incidence of pneumoconiosis. For narrowly classified mica flakes and carbon fibers in the 1 to 100 ]xm size range it was found that particle shape could be estimated from the ratio of median sizes by laser diffraction and sedimentation [35]. Austin found that conversion between Sedigraph and sieve analyses depends not only on a mean shape factor, but also on size distribution. He generated an equation that applies for overall conversion when the sieve distribution followed a Schuman form (Equation 2.97) [36] Hausner [37] proposed a method of assessing particle shape by comparing the particle with an enveloping rectangle of minimum area. If the rectangle length is a and its width b, three characteristics are defined: Elongation ratio x = a/h Bulkiness factor j^ = A/ab Surface factor z = C^I\2.6A
(2.43)
where A is the projected area of the particle and C is the perimeter. Medalia [38] represents the particle in three dimensions as an ellipsoid with radii of gyration equal to those of the particle and defines an anisometry in terms of the ratio of these radii. Church [39] proposed the use of the ratio of the expected values of Martin's and Feret's diameters as a shape factor for a population of elliptical particles. Cole [40] introduced an image-analyzing computer (the Quantimet 720) to compare longest chord, perimeter and area for a large number of particles. Other parameters have been proposed by Pahl et. al [7,41], Beddow [42] and Laird [29]. Barreiros et. al. [43] tested three different particle shape materials; glass beads, crushed glass and mica using four particle sizing instruments; Coulter Multisizer, Sedigraph, Malvern 2600C and microscope. Not surprisingly they found wide variations between the derived distributions.
82
Powder sampling and particle size determination
In essence the Coulter and the Sedigraph median diameters were almost identical in all cases. For glass beads all the instruments generated similar median values. For crushed glass the median diameter with the Malvern and microscope were, respectively 28%, 45% greater, and for the mica the ratios were 1.96 and 2.43. They also found that the Malvern broadened the distribution. These differences are to be expected since, unless the particles are greatly irregular, the volume diameter is similar to the Stokes. The projected area diameter by microscopy neglects the smallest particle dimension and one would expect a much greater median size for flaky particles. The Malvern does not measure particle size as such but measures the forward light flux distribution that it converts to the distribution of (opaque) spheres that would generate the same light flux distribution. They also calculated shape factors based on surface area determination by krypton adsorption. Unfortunately the equations they applied are incorrect. 2.5.4 Shape indices Tsubaki et. al. [44,45] argue that many of the proposed shape factors have little practical relevance to the analysis of real powders until the advent of electronic equipment and the computer. They define six shape indices based on the following diameters (see Table 2.1): d^, d^, dp^, dj^. The shape indices are: i//^^, y/^f,, y/^j^, y/^^, y/^^^, y/^^ where, for example, y/^^ = djd^. They define statistical diameters and coefficients of variation according to equations (2.1) to (2.4). Further, elongation Z was also studied where: Z = —^^^ dp .
(2.44)
^ min.
dp
is the minimum value of the Feret diameter and dp
is the diameter
perpendicular to this. Three more indices were added later, y/p^^, y/^^^ and K.
The arithmetic average of breadth and length is defined as follows:
^F=iK.„+^F.,,)
(2.45)
Data presentation and interpretation
83
The dynamic shape factor K\S defined as the ratio of the resistance to motion of a given particle divided by the resistance of a spherical particle of the same volume. Under laminar flow conditions: 'd^
(2.46)
K^st J
Since this is a squared term that, for comparison purposes, they wished to reduce to unit power they introduced the shape factor y/^^y={d^/d^). [Note — 1/7
K ~y/w where if/^ is the Wadel sphericity factor]. For non-re-entrant particles, according to Cauchy's theorem, d^ = E{dp) and i//p^^ has a maximum value of 1.0 for circles, rectangles and other convex shapes; it is therefore very useful for indicating the extent of concavities, y/j^^, y/^p, ¥RF^ ¥R^ ¥F ^^^ ^ where found to show mainly the slimnness of the particles, the best indicators being y/p and Z in that order. y/^f^ and y/^p^ were found to correspond poorly with particle morphology. 2.5.5 Shape regeneration Briefly, this method consists of finding the center of gravity (G) of a particle and its perimeter (particle outline), from which a polar coordinate system is set up. A fixed angular coordinate interval, say InlN radians, is chosen and the distances of G to the boundary at these A^ points are determined so that the shape of the particle is represented by these distances {A^ and their angular coordinates {0^. Thus for 7V(= 8, 16, 32) measurements, determined on the image of the particle, the Fourier coefficients {A^ and {6^ can be estimated. For each particle a Fourier representation is obtained, giving pattern vectors of 8, 16 or 32 elements, which are subject to feature extraction, classification and recognition [46]. In 1969 and 1971 Meloy presented papers in which fast Fourier transforms were used to process particle silhouettes as signals [47,48] and this work was extended in 1977 [49,50]. One of their main conclusions was that particles have 'signatures' which depend on A^ and not on 6^ and they proposed the equation:
84
Powder sampling and particle size determination
"^n -A ' •
(2.47)
A plot of A^ against log n yields a straight line of slope s that depends on particle shape, rounder particles having lower s values [51]. Beddow [42] showed how a number of particle silhouette shapes could be analyzed and reproduced by Fourier transforms. Gotoh and Finney [52] proposed a mathematical method for expressing a single, three-dimensional body by sectioning as an equivalent ellipsoid having the same volume, surface area and projected area as the original body. The sliape of individual particles can be characterized using Fourier grain analysis or morphological analysis [53-55]. The method has been used to analyze beach sand silhouettes, to relate attrition rate in a milling operation to particle shape [56,57] and it has been extended to measuring the shape mix in powders [58]. Particle shape grouping by the co-ordinate detection function with Fourier analysis has been discussed by Shibata et. al [59]. Orford and Whalley [60] discuss various quantitative grain form analysis techniques. However most of these shape descriptors either do not have the pre-requisites of ideal shape parameters, namely, invariance to translation, rotation, scale and starting point, or are reported as a singlevalue factor that results in low discriminating power. To overcome these problems an image analysis technique of using Fourier descriptors was developed [61]. The technique, referred to as the Zahn-Roskies (ZR) Fourier descriptor representation, is based on the Fourier expansion of the angular bend of the periphery of a particle as a function of its arc length. The authors state that the technique of applying image analysis, ZR Fourier descriptors and neural networks, in conjunction with fiber optic imaging techniques seems promising and well suited to on-line monitoring and control of particulate processes and in particular crystallization since both size and shape parameters are made available. 2.5.6 Fractal dimensions characterization
of textured surfaces
Rough (textured) particles do not have a unique surface. The measured surface depends upon the method of measurement and will increase as the degree of scrutiny increases. For example, corrugating a one-acre field by plowing it will increase its superficial area to V2 acres. Texture is difficult to define and quantify. Davies [7] for example, defined texture as the number of asperities possessed by a particle outline.
Data presentation and interpretation
85
He generated a shape distribution histogram by plotting asperity frequency against particle perimeter and defined a mean texture for a powder in terms of asperities per mm. Using these definitions it is possible to monitor particle abrasion and quantify the potential of conveyed powders to generate dust [62]. Texture has also been defined in terms of roughness or rugosity where the rugosity coefficient is defined as the perimeter of the particle outline divided by the perimeter of the convex hull [63]. Mandelbrot introduced a new geometry in a book, which was first published in French in 1975, with a revised English edition [64] in 1977. In 1983 he published an extended and revised edition that he considered to be the definitive text [65]. Essentially he stated that there are regions between a straight line that has a dimension of 1, a surface that has a dimension of 2 and a volume that has a dimension of 3, and these regions have fractional dimensions between these integer limits, Kaye has presented an excellent review of the importance of fractal geometry in particle characterization [66]. If an irregular outline is enclosed by a polygon of constant side length X, the perimeter P^^ will increase as the side length decreases. For a polygon with n sides: Pp^ = nl
(2.48)
Mandelbrot showed that: P;^ = k2}-^
(2.49)
Hence plots of log P^ against log /I will have a slope of \-D, The parameter D is characteristic of the texture of the particle and was called by Mandelbrot the fractal dimension. The fractal dimension for the outline of a particle lies between 1 and 2, the more irregular the outline, the higher the value. As an illustration of the technique consider a map of the British coastline (Figure 2.12). The fractal dimension is found to depend upon the degree of scrutiny [67,68] (Figure 2.13), having a dual value of 1.41 for large step lengths decreasing to 1.28 for short step lengths. The method has found applications in determining the ruggedness of particle profiles and can be modified for use with image analyzers.
86
Powder sampling and particle size determination
Fig. 2.12 Fractal analysis of the British coastline.
1000
Fig. 2.13 Relationship between step length {X) and perimeter {P^^for the British coastline The texture of surfaces can also be described using fractals, the more irregular the surface, the higher its fractal dimension; a fractal surface is defined as being a surface in which increasing and similar detail is
Data presentation and interpretation
87
revealed with increasing magnification, i.e. the dimension is scale invariant [69]. This parameter has been used to examine rice hull (husks) [70]. Because of its high energy content, rice hull can be a source of energy for the rice milling process. The major obstacle has been its non-uniform flow in the reactor, attributable in part to its irregular geometry. Kaupp [71] fitted a polynomial to describe the surface irregularities as observed with an electron microscope by tracing the circumference with a computer guided image analyzer. Chan and Page developed an algorithm to calculate the boundary fractal dimensions of boundary segments [72]. The algorithm was validated by accurate calculation of the fractal dimension of the Koch island fractal. The algorithm was applied to measurements on three atomized copper powders having major differences in the roughness of their profiles. The fractal dimension was found to vary significantly reaching a peak at a magnification of 800 to 2000 and falling appreciably as the magnification was increased to 14,000. This sensitivity to magnification indicates that these powders do not have the self-similar profile characteristics of true fractals. This does not exclude fractals as a useful shape descriptor at some magnifications since they have been shown to be more sensitive than conventional shape factors in correlating particle packing, flow and inter-particle friction [73]. However the results indicate that care needs to be taken in their evaluation. Fan et. al [74] determined surface area by nitrogen gas adsorption and stated that the fractal dimension could be determined by adsorbing different sized molecules on similar surfaces or, keeping the adsorbate the same, but changing the size of the adsorbent. They adopted the latter procedure by using between sieve fractions. In the first case the specific surface S is related to the fractal dimension D^f hy equation (2.50) and in the second the relationship given in equation (2.51) applies. ^^^_0.5(D-'/-2)
(2.50)
5 = r-<^-^'-3>
(2.51)
where cris the cross-sectional area of the adsorbate molecule and r is the particle radius. Avnir et. al [75] used molecules of different sizes and foundl
88
Powder sampling and particle size determination
The fractal surface of particles has been linked to their shedding propensity [79] and to erosion during pneumatic conveying [80]. For a recent review of the application of fractals to particle morphology readers are referred to [81]. Pore wall roughness has also been determined by fractal geometry using mercury intrusion/retraction [82]. Pore wall roughness has also been determined by fractal geometry using mercury intrusion/retraction. [83]. Fractal analysis has been found to be a useful tool for not only characterizing the irregularities in surfaces but also for correlating these with the physical description and flow properties of pharmaceutical solids [84]. Size and shape factors in combination with fractal dimension and rheological tests provide, prior to preformulation assays, a reliable technique for the appropriate selection of materials and detection of undesirable properties related to shape-surface characteristics. 2.5.7 Other methods of shape analysis Gotoh and Finney [85] proposed a mathematical method for expressing a single, three-dimensional body by sectioning as an equivalent ellipsoid with the same volume, surface area and average projected area as the original body. Micro-cuboids in three-dimensional turbulent flow have been examined using Fraunhofer diffraction [86]. Both dynamic and static three dimensional particle shape features could be obtained and served as a basis for particle shape analysis by pattern recognition. The use of wedge-shaped photodetectors to measure forward light scattering intensity has also been explored for determination of crystal shape [87]. 2.5.8 Sorting by shape Particle may be sorted by shape by taking advantage of their behavior on a sloping, vibrating surface. Rounded particle tend to roll, granular particles to hop and flaky particles to shuffle. Ridgeway and Rupp [88,89] used a table designed for sorting industrial diamonds (Jeffrey-Galion) for their work whereas others used a rotating disc [90,91]. Slotted sieves have also been used for shape sorting. These have the advantage that they sort quantitatively on the basis of two of the particles' principal dimensions and they sort more selectively than a sorting table [92]. Identical square mesh sieves have also been used to sort different shaped particles by residence
Data presentation and interpretation time [93]. The Jeffrey-Galion table has also been used to sort milled pasta and milled gelatin by shape [94]. 2.6 Determination of specific surface from size distribution data Surface to volume (or mass) ratio is a fundamental property of a powder since it governs the rate at which the powder interacts with its surroundings. As examples, small crystals in a mother-liquor, with high surface to volume ratios, dissolve or grow more rapidly than large ones. Medication, in powder form, can pass through the body relatively unadsorbed if the active ingredients are composed of large particles, whereas fine particles are rapidly adsorbed; in the former case there is little reaction to the medication whereas in the latter it could be toxic. Specific surface can be determined directly by permeametry, gas diffusion, gas adsorption and adsorption from solution and can also be calculated from size distribution data. 2.6.1 Determination of specific surface from a number count Consider a number count carried out by microscopy, where the measured diameter is the projected area diameter {d^. For an assembly of particles: max
S=-^ ^
'-^^
(2.52)
y max
where there are An^ particles of projected area diameter da,-. Assuming that the surface-volume shape coefficient by projected area, ocsv.a, is size independent over the size range under consideration:
^^-y-^s.,"^
H^rdlr
(2-53)
89
90
Powder sampling and particle size determination
Thus if .S*^, is determined by some independent procedure, the manner in which the surface-volume shape coefficient by microscopy, asv,a, changes with projected area diameter can be investigated. Alternatively, assuming ^sva ^ ^' ^^ volume-specific surface by microscopy, S^^ can be determined. 2.6.2 Determination of specific surface from a surface count Consider a size distribution obtained by a surface analysis method, e.g. photosedimentation, where the measured diameter is the Stokes diameter {d^^ and the surface fraction between two diameters centered on d^^^is Sf.. Let
Pr = s,/s
where
S=
r=max
I s, r=min
then
Sr ^PrS =
and
Vr = (^y,p,Mrdlt,r
therefore
Sr
s,p,r
a,p,An,dl
Vr
Pr ^^v,p,r^St,r ^
Assuming that a^^^, the surface-volume shape coefficient by photosedimentation, is size independent over the size range under consideration: ^^^ Z
(2.54)
Pr^St^r
2.6.3 Determination of specific surface from a volume (mass) count Consider a particle size distribution determination by mass where the fractional masses (volume =V^ of particles of mean diameters {d^ are determined. For a sieve analysis, for example, the measured diameters are
Data presentation and interpretation
91
20 dW/dx
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Particle size (x) in microns Fig. 2.14 Frequency distribution presented as a histogram and as a continuous curve. The ordinate is presented as percentage per micron so that the area under the curve is 100
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Particle size (x) in |im Fig. 2.15 Cumulative percentage undersize curve.
92 Powder sampling and particle size determination
dF(jc)/(k
10
15
20
25
30
35
Particle size(x) in microns
40
Fig. 2.16 Frequency distribution presented as continuous curve with the abscissa on a linear scale. The right ordinate is presented as percentage/micron so that the area under the curve is 100. 1000
100
dF(xydlog(x) 4-
-•
800
f -4 «0
•if
I
i
\
600
f -| 400
200
10
FBfticle Size (x) in Microns
30
40
Fig. 2.17 Frequency distribution presented as continuous curve with the abscissa on a logarithmic scale. The right ordinate is presented as percentage log-micron so that the area under the curve is 100.
Data presentation and interpretation 93 sieve diameters {d^ r^ ^Ar) ^^^ the fractional volume residing between two sieves of mean aperture d^ ^ is V^.
y
q^ =-^
r=max
where V = Y. K
y
r—im\r\
then
3 V^ =qf.V = a^^^An^d:
and
5*^ =a^^^An^d^^
therefore
5*^ = ^sv,x,r
q,V ^x.r
and:
5,= a , , ,
E - ^
(2.55)
Example: Determination of specific surface from sieve analysis data Table 2.7 Calculation of mass specific surface from sieve analysis data Sieve size (|Lim) ^^ (^V) 105 150 210
Mean sieve Mass fraction size residing on sieve [qrldA,) 89 125 178
Surface area factor (cm^ cm-3)
0.30 0.50 0.20
33.7 40.0
rL2
From the data in Table 2.7, the volume specific surface by sieving is 509 cm^ cm"^. For a powder of density 2500 kg m-^ this becomes 204cm2^i. 2.7 Tabular presentation of particle size distribution Having defined the relevant particle size the frequency of occurrence of each size can be found. Number distributions can be determined by microscopy, electrical and light sensing zone methods; surface
94
Powder sampling and particle size determination
distributions by photosedimentation and mass distributions by sieving and x-ray sedimentation. Table 2.8 shows an example of size analysis data (^ can be number, surface or weight). Column 1 gives the interval of the size classes; column 2 gives the mean size of each of these classes; column 3 gives the percentage frequency of occurrence within these classes; these sets of data are accumulated in column 4; percentage/|Lim is presented in column 5, percentage/log(|Lim) (base 10) in column 6 and percentage/ln(|Lim) (base e) in column 7. Table 2.8 Tabular presentation of size distribution data Size limits (urn) 1-^ 3-5 5-7 7-9 9-11 11-13 13-15 15-17 17-19 19-21 21-23 23-25 25-27 27-29 29-31 31-33 33-35
Mean size (nm) 2~ 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34
Relative frequency (%) d^
Cumulative frequency (%) <^
0^00 0.04 0.22 0.88 2.70 6.48 12.10 17.60 19.95 17.60 12.10 6.48 2.70 0.88 0.22 0.04 0.01
0^00 0.04 0.27 1.14 3.84 10.32 22.42 40.02 59.97 77.57 89.67 96.15 98.85 99.72 99.95 99.99 100.00
% per Hm
d(t) Ax 0.00 0.02 0.12 0.44 1.35 3.24 6.05 8.80 9.98 8.80 6.05 3.24 1.35 0.44 0.12 0.02 0.01
% per log(|Lim) d(l) dlog;c
% per ln(|im) d^ dlnx
0 1 3 13 39 85 141 179 176 133 78 35 12 3 1 0
0 2 8 31 89 195 324 413 405 306 179 81 28 8 1 0
For narrow size distributions it is usually preferable to use an arithmetic size progression (as in Table 2.8) and present the frequency data in the manner given in column 5. For wide size ranges a geometric size progression (e.g. 2, 2V2, 4) is usually preferred with frequency data as presented as in column 6. For data manipulation the presentation in column 7 is preferred. The relationships between these data are as follows:
Data presentation and interpretation 95 d^
d^ d^ = X— = log(^) dln(x) dx dlog(jc)
0.434-
d^
dlog(x)
(2.56)
2.8 Graphical presentation of size distribution data 2,8.1 Presentation on linear graph paper The simplest presentation of this data is in the form of a histogram on a relative and/or cumulative basis (Figures 2.14 and 2.15). Since the distribution is continuous, straight lines can be drawn through the histogram to produce one of two curves: frequency/micron and cumulative percentage distribution (either undersize or oversize). It is common practice to normalize particle size distributions (i.e. summate to 100%). In order to compare distributions it is equally essential to normalize the frequency distribution so that the area under this curve is 100%. The ordinate should always be plotted as percentage per micron if the size is plotted on a linear scale (Figure 2.16). For the linear scale, the relationship between size and frequency may be written:
f{x)
6F{x) dx
JdF(x)= j/(x>k
(2.57)
[(/) = ¥{x)] may be replaced by W for a mass distribution, 5* for a surface distribution and A^ for a number distribution. Alternatively, the frequency can be plotted with a logarithmic scale for the abscissa. In this case the ordinate is calculated as percentage per log(micron) in order to normalize the area under the frequency curve to 100 (Figure 2.17). This type of presentation is particularly useful for size analysis data since many powders are logarithmically distributed and many instruments present data with sizes in a geometric progression. A linear plot of geometrically tabulated data compresses the data points at the fme end of the distribution so that detail is lost« If further data manipulation is envisaged Naperian logarithms are preferred (i.e. to base e rather than base 10) since data in this form lend themselves to computer manipulation, particularly if the notation popular in Germany is used [95]. If a log to the base 10 abscissa is used the area under the frequency curve decreases from 100 to 43.4 units in this case [see equations (2.24)]. It is common practice to plot size distribution data in such a way that a straight line results, with all the advantages that follow from such a
96
Powder sampling and particle size determination
presentation. Some of the mathematical expressions for achieving this are given below. 2.9 Standard forms of distribution functions Particle size, like other variables in nature, tends to follow well-defmed mathematical laws in its distribution. This is not only of theoretical interest since data manipulation is made much easier if the distribution can be described by a mathematical law. Experimental data tends to follow the Normal law or Gaussian frequency distribution in many areas of statistics and statistical physics. However, the log-normal law is more frequently found with particulate systems. These laws suffer the disadvantage that they do not permit a maximum or minimum size and so, whilst fitting real distributions in the middle of the distribution, fail at each of the tails. 2.10 Arithmetic normal distribution The normal law or Gaussian frequency distribution occurs when the measured value of some property of a system is determined by a large number of small effects, each of which may or may not operate. If a large number of the measurements of the value are made and the results plotted as a frequency distribution the well known Gaussian bell-shaped curve results. The equation representing the normal distribution is: d^ y^
1
— = —T=^y^\i
dx
0-V271
(jc-jc)
' ~ 2 ^
and ^=j/(jc)djc
(2.58)
(2.59)
0
i.e. the distribution is normalized (sums to unity or 100%) (J is the standard deviation, I' is the mean size, (j) is the general term for the frequency; number, length, surface or volume (mass).
Data presentation and interpretation
97
The mean size, x, is defined as: X =
(2.60)
—
The standard deviation a; is defined as:
cr = .
\Y.{x-xf^(|) ZA^
(2.61)
• -, then adt = dx and equation (2.57) becomes: Let ^ = x-x d^ At
exp
^ /^^
(2.62)
Hence: exp
(2.63)
W
A plot of d^/d/ against ^ results in the well-known 'dumb-bell' shape of the normal probability curve [Figure 2.18]. Table 2.9 Tabular solution to the normal probability equation t 0 0.5 1.0 1.5 2.0 3.0
Integral {(/)) 0.5000 0.6915 0.8413 0.9332 0.9987 0.9997
98
Powder sampling and particle size determination
Fig. 2.18 The normal probability curve; relative frequency against standard deviation. 68.26% of the distribution lies within 1 standard deviation of the mean [-!
99.9 99 -
\^ CD
C
2.
95 90 80 ~ 70 50 30 ?0 10 5
\
^
^
^
.
^
1 .1 1—1
5
1
L—1
1
1
^16 . i . . .
10
•^50 1_ j
'
'
'
15
Particle size in microns Fig. 2.19 Graphical presentation of size analysis data on normal probability paper
1
20
Data presentation and interpretation
99
2.10.1 Manipulation of the normal equation The fraction undersize the mean size 3c is obtained by inserting the limits jc = -00, jc = 3c, i.e. / = - 00, / = 0, in equation (2.62) to give ^ = 0.5, i.e. the mean and median are coincident. Similarly it can be shown that the mode is the same as the median. Writing -X'^ =-t^ / 2 , exp(-X ) is defined as:
Qxp{-X^)= lim 1--
•X^
Expanding by the binomial theorem: (-X1 Qxp[- X^)=l + n\ \ n n\ .+(n-r)\r\
)
n(n -1) -XI 2!
+
n(n - \){n - 2) -XI \ n 3!
)
r-x^y
The fraction lying within one standard deviation of the mean is obtained by inserting the integral limits (jc = 3c, / = 0, X = 0 ) and (x = x + a, t =\, X = \H2). Therefore: (jc+a)
I/V2
fd^ = - L {exp{-X^yX
1/V2
x+a)
^
X^ 3
\ X^ 2! 5
\X^ 3! 7
(-ir^X^'^-i r-\ Ir-\
JO
100 Powder sampling and particle size determination
Thus: \7
11 + 2!5
4i
3!7
V^, + ....
^(x + c r ) - ^ ( x ) = 0.5642[0.707-0.118 + 0.018-0.002 +
]
^(x + cr)-^(3c) = 0.3413 The fractional area under the curve between the mean size and one standard deviation from the mean, 0.8413-0.5000 - 0.3413. 68% of the distribution falls within one standard deviation from the mean (see Table 2.9) and all but 0.26% lies within two standard deviations from the mean. Therefore the standard deviation equals the difference between the 84.13% size and the mean (median) size. Equation 2.63 is the basis for arithmetic probability graph paper and the solution is widely available in tabular form. A fundamental property of the normal distribution is that differences from the mean are equally likely, i.e. the probability of finding particles 10 units larger than the mean is the same as finding particles 10 units smaller. An example, plotted on probability axes, is shown in Figure 2.19 with a mean of 12 jiim and a standard deviation of 3 M^m so that the (84%-50%) size interval equals the (50%-16%) size interval = 3 |Lim. The distribution is completely described by these two parameters. Although it might be expected that this type of distribution would be common, it seems to occur only for narrow size ranges of classified material. Most distributions are skewed, usually to the right. 2.11 The log-normal distribution According to the normal law, differences of equal amounts in excess or deficit from a mean value are equally likely. With the log-normal law, it is ratios of equal amounts that are equally likely. In order to maintain a symmetrical bell-shaped curve, it is therefore necessary to plot the relative frequency against size in a geometric progression. The equation of the log-normal distribution is obtained by replacing x with z = In jc, in equation (2.58). Then:
A(t)
y^ dinjc
1 exp
Data presentation and interpretation 101
y=
or:
1 -exp dim: ^J2nln(J,
\lrvc - live J IXxi^c
(2.64)
Xg is the geometric mean of the distribution (i.e. the arithmetic mean of the logarithms) and a^ is the geometric standard deviation; a^ is the standard deviation of z. is the product of the group data in which the frequency of particles of size jc is d^ -^1
^ Q
-^O
"X
[(/) = TV (number), S (surface) or ^(weight)] Zzd^ z
=•
Zd^ _ Zzd^ z= lnx«
Thus:
Xlnxd^
x,=[U:
(2.65)
Since the particle size is plotted on a logarithmic scale, the presentation of data on a log-probability graph is particularly useful when the range of sizes is large. The geometric standard deviation can be read from the graph, as with the arithmetic distribution, and is given by: log cr =logX5o-logX]16 log cr =logXg4-logx50 logag =Jlog(x84/X5o)
(2.66)
/ 02 Powder sampling and particle size determination
The geometric mode, the geometric mean and the median coincide for a log-nomial distribution. In Figure 2.20 the median size x is 18 |Lim, the 84% size (7^^ is 25.2 |im and the 16% size xJa is 12.86 |Lim. The ratio of the 84% size to the 50% size is the same as the ratio of the 50%) size to the 16% size and equals the geometric standard deviation cr^ = 1.40. The distribution is completely described by the two numbers (18 and 1.40). 99.99 99.9 99 N C/5
c3 (D
tJ) C^
C (D (D
Ou
95 90 80 70 50 30 >0
10
s [
.1
"
h i 10
.01
^16
•^50 20
^84 40
Particle size in microns Fig. 2.20 Graphical presentation normalprobability paper. 2.11.1 Relationship distribution
between
number
of
size
analysis
data
mean
sizes for
a
on
log-
log-normal
Consider a log-normal distribution by number such that:
d^ = —T= 0
|exp
^27tlnc7^J
In jc - In X
g
12K
dlrLx: = l
(2.67)
\r\a„
i.e. the distribution is normalized. dN^ is the number of particles in a narrow size range centered on x^. XQ and x^ are the smallest and largest
Data presentation and interpretation 103
particles present in the distribution and Cg is the geometric standard deviation [which is the same for number, length, surface and volume (mass) distribution]. The number-length mean diameter is defined as:
^NL ~
r=0 00
f'=0
Normalizing i.e. equating the denominator to unity gives:
—J'
^NL
Inlna
exp
In jc - In jcg
Inlna
g 0
jcdlruc
(2.68)
x^dlnx
(2.69)
x^dlnx
(2.70)
g J
Also
X^'(W. ^13 = '-^^ =I^.'dA^. /•=0
Inx-lnx.
\ I^'^P
•^A'i-
lliXxiG ^
Y.^]^^r 3
_ r=0
= Ix'cW. r=0
r=0
.3
_.
In JC - In X
1 2n\n(j
exp g 0
2K
In cr^
104 Powder sampling and particle size determination
Making the substitution: X
In X - In X gN
^l2\n(
V2lncr^dZ = d(lnjc)
so that oo
Then: x.rr NL =
,2
JQxp(^[2X\nag-X^)iX V7C
(2.71)
0
2 '^ - " " g ^ Jexp(2V2 A' In
a^-X^^X
(2.72)
_ V Jexp(3V2Xlno-g-X2)lY
(2.73)
4^0
X)
,3
4^
Making the transformations: \\ = X
V2
hi a^ in equation (2.71)
^2 = X - V2 In (j^ in equation (2.72) }^=X
Then
X\x\<j in equation (2.73)
^gN f1 -^exp^lln2^^jjexp(-r,2)dl^
(2.74)
tls =^exp(2ln2.Tjfexp(-r22)ir2
(2.75)
^NL
^NV
y/n
exp
In^cT I}exp(-r32>ir3
(2.76)
Data presentation and interpretation 105
The integration yields a value / = VTT giving: \nx^^^^\nXgj^+05\n^cTg
(2.77)
ln.x-;v^-lnx^;v'+10'n^^g
(2-78)
\ux^y = \nxg^ + 1.5 In^ a^
(2.79)
Similarly ln.v,;^v/ - In V + 2.0In^ cj^.
(2.80)
2.11.2 Derived mean sizes If the number size distribution of a particulate system is found to be lognormal, equations (2.77) to (2.80) can be used to determine other mean sizes. For example, the volume-moment mean size is the mean size of a volume (mass) distribution: r=oo
I 4AN, r=0
3
Therefore: In xyj^ - 4In x^j^ - 3.0In x^vj/
(2.81)
Substituting from equations (2.79) and (2.80): \\\xyj^^ = \nxg^j + 3.5In^ a^
(2.82)
Similarly, the mean size of a surface distribution is given by: \nxsy = Inx^^ +2.51n^ a^
(2.83)
106 Powder sampling and particle size determination
Using this equation, the volume specific surface of the particulate system may be determined since:
2.11.3 Transformation between log-normaldistrihutions If the number distribution is log-normal, the mass distribution is also lognormal with the same geometric standard deviation. Using the same treatment as was used to derive equation (2.77) gives, for a mass analysis: \nxyf,^ - \nx^y + 3.5In^ a^
(2.84)
Comparing with equation (2.82) gives: \x\x^y - Injc^yv +3.01n^ a^ Since the geometric expressed geometric
(2.85)
relations between the number average sizes and the number mean are known [equations (2.77) to (2.80)], these can now be as relationships between number average sizes and the weight mean jc^^ to give a similar set of equations.
\nx^f ^\nx^y-2.5\n^cj^
(2.86)
Injcy^.s =lnjc^j -2.01n^cr^
(2.87)
\nx^y=\x\Xgy-\.5\n^c7g
(2.88)
\nx^,f^\nx^y-\mn'-cj^
(2.89)
Other average sizes can be derived from the above using a similar procedure to that used to derive equations (2.84) and (2.85) to give: Inx^y - In x^y - 0.5In^ a^
(2.90)
Similarly, for a surface distribution, the equivalent equation to equation (2.80) is: InA-^^^ -IHA^V +2.01n2cr
(2.91)
Data presentation and interpretation 107
Substituting this relationship into equations (2.86) to (2.89) yields the equivalent relationships relating surface average sizes with the surface geometric mean diameter. 2.11.4 Relationship between median and mode of a log-normal equation The log-normal equation may be written: d^_ 1 -exp(-X^) dx X v27r In cr, where
(2.92)
Incr =lnx-lnjc
so that: AX _ 1 dx X
V21n(
At the mode {x = x^) and: —y = 0 djc
I.e.
d^ -0 dx dx
— = 0 where Y = -j=^ expl-X^) Ax ^2%x\n(jg so that
dx
2(lnx:^
iexp(-X^) = 0
-\nxg)
V2Ir\Org
x^42\
r\Gr
J^--mv-«"-gy
In x^ = In Xg - In cj^
These relationships are summarized in Table 2.10.
(2.93)
108 Powder sampling and particle size determination
Table 2.10 Relationship between average sizes for lognormal equations In %^ = In x^+ 0.5 In^cr^ In Xj^^ = In x^+ 1.0 In^cr^ \nxj^y=\nx^^+ 1.5 In^cr^ \n Xj^j^= \n x^r+ 2.0 \n^ag In Xi^^^ In Xgyy-H 1.5 In^cr^ In x^,/= In Xgy^/+ 2.0 In^cr^ In x^^-= In x^-^ 2.51n2o^
In x^i^ In x^-^ 2.5 In^cr^ In Xsj^= In Xgyv+ 3.0 In^cr^ In Xyj^= In Xg^+ 3.5 In^cr^ In Xgs= In Xgyv+ 2.0 In^cr^ Inx^^- lnXg^+3.01n2(T^ lnx^= '"^g~^ 10 ^^^^g
2.1 J. 5 An improved equation and graph paper for log-normal evaluations Using equation (2.59) and the relationship [94] : x-^ -exp(lnjc)
(2.94)
the log-normal equation can be written: d(f> _
dx
1
yf2nx^\na.
-exp
2 ' " ^^
exp
In
jx/x^)
(2.95)
V " 21nH y
This form of the log-normal equation is more convenient for use since the variable (x) only appears once. Equation (2.95) may be written in the simplified form:
dl - ^exp -Z>ln^(x/x^)
(2.96)
dx The relationship between geometric mean and mode [equation (2.93)] takes the form: C
:exp
J_ lb
{1S)1)
Data presentation and interpretation 109 This modified form of the log-normal equation simplifies parameter determination from log-probability plots of experimental data. The graph paper may be furnished with additional scales of b and C both being determined by drawing a line parallel to the distribution through the pole (0.25 ^m, 50%). 2.11.6 Application Consider a log-normal equation with a geometric mean x^ = 6.75 jiim and a standard deviation a^ = 1.64. According to equation (2.63) the mode x^ will be 5.27 |Lim making: -^=0.1344exp -2.021n^(jc/5.27) djc
This form is particularly useful when further mathematical computation is envisaged, such as for grade efficiency GJx) since:
'
^ dFix)
G,(x) = Er^
(2.98) dx
where (d^^dx:) is the frequency of the coarse stream, (d^dx) is the frequency of the feed and E^ is the total efficiency (see Chapter 5). 2.12 Johnson's Sg distribution Johnson's S^ distribution is a bounded log-normal distribution, i.e. a truncated log-normal equation with a minimum and maximum size so that the whole size distribution is directly considered. This differs from conventional correlation analysis as summarized by Herdan [97]. It has been presented, in a series of papers by Yu and Standish [98-101] as a function that can represent all unimodal size distributions of particles. It may be written: dF(z)
110 Powder sampling and particle size determination
y
_
V-^max
Q"z
-^min /
27c(^-^min)(^max-^)
exp
^ 1^
vV2y
,2^
//^+o-Jn
(^-^min)
(2.99) (^min<^<^max)
The log-normal distribution is a special case of the SQ distribution which can be obtained by extending the size limits to ±<x). t^fj^ +(T^ In 2-z„
If we let
V ^max
(7 :(' \Z
—Z
^y
•
)
so that
d/ =
then:
( t^\ AF{z) AF{t) 1 y - —^^— = = .— expl dr dt V 2y
zV max
min
Az
If the theoretical fraction undersize the actual smallest size is negligible it is usually safe to assume that z^^^ =0 so that:
'(z)= I exp •00
where
(
,i\
V
y
dz
^-(J^ In z-z^ V ^max
In ^J
This reduces to the log-normal equation if <^is simplified t o: ^-G^{^xvz-\xvz^ Distributions that fit the Sg distribution will also fit the log-normal distribution but differences will occur in the end regions. These differences arise since the log-normal distribution does not admit a
Data presentation and interpretation 111
minimum or maximum size that must exist in any real system [102]. These small discrepancies can lead to large differences in their transformed size distributions. The modified log-normal equation has been applied to particle size determination in acoustic spectroscopy using the Pen Kem 8000 Acoustophor [103]. The authors state that the unmodified equation overestimates the fraction of large particles and under-estimates the median size. They also state that the median size generated using the truncated log-normal distribution for fitting the acoustic spectrum is not the 50% size. 2.13 The Rosin-Rammler-Bennett-Sperling formula This formula was derived originally for broken coal and has since been found to apply to many other materials [104]. The equation may be written: i?=100exp(-Z)x«)
(2.100)
where R is the weight percentage retained on a sieve of aperture x, \og{\miR)^bx^\oge Taking logs gives: log[log(l 00/i?)] ^ log b +wlog x + log(loge) i.e.
log[log(100/ /?)] ^n\ogx + constant
A plot of the log of the log of reciprocal percentage weight retained against the log of particle size generates a straight line. The slope /? is a measure of the particle size dispersion (size range) of the powder. The peak of the frequency curve for « = 1 is at (100/^) = 36.8% and, denoting the mode by x^ equation 2.69 gives b = \lx^. The sieve aperture for R = 36.8% is used to characterize the degree of comminution of the material and, since the slope of the line on the RosinRammler graph depends on the particle size range, the ratio of tan~' n and x^ is a form of variance. This equation is useful for monitoring grinding operations for highly skewed distributions, but should be used with caution since the device of taking logs always reduces scatter hence, taking logs twice is not to be recommended. An alternative form of the Rosin-Rammler equation is: 7?=100exp[-(jc/jCo)«]
(2.101)
/12 Powder sampling and particle size determination
2.14 Other distribution laws Yu and Standish [100] list the laws presented in Table 2.11 that have been used for particulate systems. The cumulative form of the above distributions may also be presented as cumulative two-parameter equations: Gates-Gaudin-Schumann [105-107]
F (JC) - {bxY
Gaudin-Meloy [108]
F (x) = [ 1 - (1 - to) «]
Roller [109-111]
F{x) - a^x exp
Svensson [112]
Fix) = xEpiy)
where:
b^
(2.102) (2.103) (2.104) (2.105)
Ep{y) = — j(l//7)exp(-j;)dj;
y = {x I x^Y
where x^ is the mode of the distribution.
Three and four parameter equations have also been proposed, e.g.: Harris [113]. F{x) = \-{\-bx'Y
(2.106)
For s = 1 this degenerates to the Gaudin-Meloy equation. 2.14.1 Simplification of two parameter equations. Tarjan [114] converted a two-parameter size distribution function from the form (f> ^f(x) to the form (/> ^fx/x^ 5), where JCQ 5 is the median. The result is an easy to handle function with a high degree of correspondence to the more complicated logarithmic function (t)j^ below:
Data presentation and interpretation 113
(2.107)
^^ =(/>(bxr where ^ «2 1 u (p = 271 f——-a\ exp
(2.108)
Table 2.11 The properties of some commonly used density distribution functions Name
-00,+ OO]
1
Normal
Log-nomial
Range of x
Equation -exp
Ax
llncj
dlnx
1 -exp V27ilncr^,
V2 o0,+ 00]
V2ln(T„
Rosin-Rammler [104]
0,+ 00]
(^x Gates-Gaudin-Schumann [105-107]
^ = nb{bxr' Ax
^' ^maxl
Gaudin-Meloy[108]
(t> =
^' -^maxJ
Roller [109-111] Harris [113] Martin [115]
=C
\-{\-bx)" ^0.5
b '
^' -^maxJ
' = Cx
0,+ 00]
(/>^Cx'e-'"
Gamma function [116,117] Weinig[118] Heywood[119] Griffith [120] Klimpel-Austin [121] Beta function [122]
^' ^maxJ
+
0,+ 00]
(/> = Cx''-^e-'
0,+ 00]
= Cx'e-''"
0,+ 00]
(t>=^Cx-''e-'"''^
0,+ 00]
(t> = Cx^ \~n{\~Cxf
^' ^maxJ
p-\
\-n{\-Cxy-'
^' ^maxJ
114 Powder sampling and particle size determination
Let the parameter b in equations (2.100), (2.102) and (2.103) be expressed in terms of JCQ 5 when ^ = 0.50. Rosin-Rammler (RR)
0.50=exp(-AjCo 5),
Gates-Gaudin-Schumann (GGS)
0.50=(^jCo 5)
Gaudin-Meloy (GM):
0.50=l-(l - bxQ^)"
Substituting back for b gives:
RR
X
^^^=l-exp V
GGS
CGS=^-^{XIXQS)"
GM
'l>GM = 1 -
In 2
(2.109)
^0.5 J
-(l-(x/Xo.5)(l-^)
(2.110)
(2.111)
2.14.2 Comments All the above equations are attempts to fit a straight line relationship to a frequency distribution. This procedure is worthwhile only if some benefit is derived. Narrow range size distributions are usually best presented as cumulative percentage undersize by mass or number on linear graph paper: The curve should then be differentiated (rather than using raw data) in order to smooth out experimental error. A simple visual presentation of this kind is preferred, for reasons of clarity, for presenting information to non-mathematicians. These distributions can also be plotted on arithmetic normal probability paper to examine deviations from normality. Wide size distributions are best plotted using a logarithmic scale for the size axis. Log-probability paper is extremely useful for examining data. Multimodality is easy to discern as is the removal of fines (by classification) from a grinding process. Grinding processes often generate parallel distributions on log-probability paper and the time to reach a desired endpoint can be predicted. Simple mathematical relationships exist for
Data presentation and interpretation 115
Table 2.12 Illustration of the law of compensating errors Mean " %in size interval (cl) (c2)
0.84 1.09 1.30 1.54 1.83 2.18 2.59 3.08 3.67 4.36 5.19 6.17 7.34 8.72 10.37 12.34 14.67 17.45 20.75 24.68 29.34 34.90 41.50 49.35 58.69
0.00 0.02 0.05 0.14 0.32 0.68 1.34 2.41 3.94 5.90 8.06 10.07 11.50 11.99 11.42 9.94 7.91 5.74 3.81 2.31 1.28 0.65 0.30 0.13 0.05 0.02
(c3)
0.25 X coarse (c4)
0.25 X fine (c5)
0.01 0.03 0.07 0.16 0.34 0.67 1.20 1.97 2.95 4.03 5.04 5.75 5.99 5.71 4.97 3.95 2.87 1.91 1.16 0.64 0.32 0.15 0.06 0.02 0.01
0.00 0.01 0.03 0.08 0.17 0.34 0.60 0.99 1.47 2.02 2.52 2.87 3.00 2.86 2.49 1.98 1.44 0.95 0.58 0.32 0.16 0.08 0.03 0.01 0.00 0.00
0.02 0.00 0.01 0.01 0.03 0.08 0.17 0.34 0.60 0.99 1.47 2.02 2.52 2.87 3.00 2.86 2.49 1.98 1.44 0.95 0.58 0.32 0.16 0.08 0.03 0.01
0.5c2
Unbiased error % (c6) = c2+c3+c4 0.02 0.07 0.16 0.36 0.76 1.44 2.52 4.05 5.95 8.02 9.93 11.26 11.72 11.19 9.80 7.87 5.80 3.92 2.43 1.38 0.72 0.34 0.15 0.06 0.02
Biased error % (c7) = 0.75c2+c4 0.00 0.03 0.07 0.18 0.41 0.85 1.61 2.79 4.43 6.44 8.57 10.43 11.62 11.85 11.05 9.43 7.37 5.26 3.44 2.06 1.12 0.56 0.26 0.11 0.04 0.01
116 Powder sampling and particle size determination
True distribution
25% wrongly placed in the size category below
Fig. 2.21 Illustrations of the law of compensating errors
Mean size {x) in microns Fig. 2.22 Effect of displacing half the distribution, without bias, in each size interval, together with misplacing 25% in each size interval into the next finest interval.
Data presentation and interpretation 11V
log-noiTnal distributions so that size distributions may be rapidly converted from number to surface to volume. As demonstrated by Yu and Standish [100] distribution transformations can lead to unacceptable errors, particularly with distribution laws with no limiting maximum or minimum sizes. Even direct transformation of a number count to a volume count for example is unacceptable, unless the distribution is narrow, the count is excessively high or special procedures are employed, as with carrying out volume counts in microscopy. 2.15 The law of compensating errors In any method of particle size analysis it is always possible to assign the wrong size to some of the particles. If this error is without bias, possibility of assigning too great a size is equally as probable as assigning too small a size. This will modify the distribution but will have little effect on the central region. An illustration is shown, in Table 2.12 and Figure 2.21, with 25% in each size category wrongly placed in the size categoiy below and 25% in the size category above, together with a biased distribution in which 25% in each size interval is displaced to the adjacent finer interval above. These are plotted as percentage per In (^im) against particle size in Figure (2.22). The original log-normal distribution has a median size of 7.29 |im with a geometric standard deviation of 1.78; the unbiased distribution has a median size of 7.29 |am with a geometric standard deviation of 1.80 and the biased distribution has a median size of 6.99 |im (an error of 4%) with a standard deviation of 1.79. For measurements in arithmetic progression of sizes the effect is small, provided sizing is carried out at 10 or more size intervals, and for a log-normal distribution the position of the mode is only slightly affected. 2.16 Evaluation of nonlinear distributions on log-normal paper A bimodal distribution is detectable when plotted on log-probability axes by a change in the slope of the line. It is also possible to deduce other features. 2.16.1 Bimodal intersecting distributions. Figure 2.23 shows relative plots of two intersecting (overlapping), lognormal distributions with medians of 16 and 10 jim and standard deviations of 1.4 and 2.0 respectively. Relative plots of blends of these parent distributions are shown in Figure 2.24 and these look to be mono-
118 Powder sampling and particle size determination
Particle size(jc) in microns Fig. 2.23 Parent distributions for bimodal log-normal mixtures. AW 120 dlog(x) 100 c o 80
ebD O^ Ui
60
a. cd C (D O
intersecting
Mix ratios ^ -10:90 -30:70 - • -x-- 70:30 -90:10
40 20
0^
Particle size(x) in microns(log scale) Fig. 2.24 Frequency plots of bimodal intersecting log-normal distributions in various mix ratios.
Data presentation and interpretation
119
99.99 99.9 -o— Parent
•^ 99 95 C 90 80 cd 70 +-» C D 50 O 30 OH 20 10 5 1
- I — Parent 2 X- - 30:70 Mix -^— 70:30 Mix
.1 .01
I I I !
I
I
10 Particle size(x) in microns
100
Fig. 2.25 Cumulative plots of bimodal intersecting log-normal distributions 200 1 1 1 r 1 1—1—T 1 • ' ' ' •'• » 1 '••r»
dln(jc) 150
1
/ \
X 1
*
1 aicnt 1
•
I aicni z
a
/v.jyj
H
100 h
H
50
—kA*k.A A Ikd
10 Particle size(jc) in microns
TlV
IOC
Fig. 2.26 Frequency plots of bimodal non-intersecting log-normal distributions in the ratios 30:70, 50:50 and 70:30.
120 Powder sampling and particle size
99.999 99.99 99.9
-I
1
1
1
1
determination
1—r
99 95 (u 90 .N 80 ^ 70
-8 50 § 30 so 20
^
•'^— Parent "•— Parent -»—30:70 • ^ ^ 50:50 -^—70:30
10
.01 .001
•
1
I
i
•
»
10
100
Particle size(x) in microns Fig. 2.27 Log-probability plots of mixtures of two non-intersecting lognormal distributions.
100
Particle size in microns (log scale) Fig. 2.28 Cumulative plots of trimodal non-intersecting log-normal distribution together with parent distributions (1:1:1 mixture).
Data presentation and interpretation 121
Particle size {%) in microns
Fig. 2.29 Relative plot of trimodal non-intersecting log-normal distribution together with parent distributions (1:1:1 mixture).
Particle size in microns Fig. 2.30 Andreasen analyses monitoring a grinding operation.
122 Powder sampling and particle size determination
Particle size in microns Fig. 2.31 Effect of classification on a log normal distribution. (Full line = parent distribution, dotted line = sub 6 |im particles removed). modal distributions. A log-probability plot (Figure 2.25) shows the parent distributions together with mixtures: It can be seen that the mixtures generate curves that are asymptotic to the parent with the wider distribution (spread). Thus a log-probability plot picks out bimodalities that would not otherwise be detected. 2.16.2 Bimodal non- intersecting distributions. Figure 2.26 presents relative plots of blends of two non-intersecting lognormal distributions with medians of 8 and 13 |Lim and standard deviations of 1.3 and 1.5 respectively. The areas under the two quite distinct curves give the proportions of the two components. On a log-probability plot (Figure 2. 27) the mixtures are asymptotic to both parents and have a point of inflection where the two distributions overlap. 2.16.3 Other distributions Figure 2.28 shows a trimodal distribution together with the parent distributions (1:1:1 mixture). This may be easily resolved into its component parts if the parent distributions do not intersect Figure 2.29).
Data presentation and interpretation 123
Table 2.13 Andreasen analyses monitoring a grinding operation Median size 5.30 4.10 3.75
Grinding time (T hours) 9 13 15
Tx^ 47.7 53.3 56.3
3.42
16
54.7
2.16.4 Applications of log-normal plots Figure 2.30 shows Andreasen analyses monitoring a grinding operation. Since, in this case, the product of the median and the grinding time approaches a constant value it is possible to predict the grinding time required to attain a desired end point median size (Table 2.13). Figure 2.31 shows a log-normal distribution of geometric mean size 10 |im. The distribution is deficient in sub 6 |im particles, probably due to classification during a comminution operation, and is asymptotic to this size. If (x-6) is taken as the particle size the parent distribution is obtained. A similar sort of plot occurs when there is a deficiency of coarse particles. 2.16.5 Curve fitting An application of curve fitting using Kalaidograph [123] is shown in Figure 2.32. The experimental data are plotted as a differential lo-gprobability graph and the program asked to fit a trimodal log-normal equation to the graph: Estimates are made of the nine variables (percentages under the 3 modes, 3 means and 3 geometric standard deviations) and the curve fitting procedure determines the true values. The time to carry out this iteration depends upon how near the estimates are to the true values. The 34% widths of the modes are obtained by estimating the ratios of the two sizes having 8% of the distribution above and 8% below so that these two sizes contain 84% of the distribution. The geometric standard deviation is the square root of the ratio of these sizes. Thus, the percentages under each mode can be determined. The program can be used for other distributions having nine or less variables.
124 Powder sampling and particle size determination
dW^ /dln(x)F
0.1
1
Particle size(jc) in microns Fig. 2.32 Curve fitting a trimodal log-normal equation to Microtrac SPA data
dWj W/
/^ln(jc)i 100 |~
(b)
Particle size(x) in microns
Fig. 2.33 (a) Size distribution of a slurry pumped at 25 gpm (b) size distribution of a slurry pumped at 100 gpm
Data presentation and interpretation 125 2.16.6 Data interpretation The curve fitting procedure is a useful aid in data interpretation. It has been used, for example, to demonstrate that a unimodal log-normal distribution with a substantial sub-micron fraction appears as a bimodal log-normal distribution when measured using a gravitational sedimentation technique; this bimodality being due to the effects of thermal diffusion (Brownian motion) [124]. Figures 2.33 illustrate the differences in the size distribution of a product pumped at different flow rates. Increasing the flow rate caused the slurry to plug on-line filters more rapidly. The differences are subtle: The main effect of increasing the flow rate is to decrease the size of the coarse mode (agglomerates) from 1.65 ^lm to 1.22 |Lim while the fine mode remains constant at 0.47 ± 0.01 fim and the spread (o-p decreases from 2.17 to 2.12. The percentage under the fine mode remains constant at 58% ± 1%. 2.17 Alternative notations for frequency distribution The notation given here is widely used but alternative notations have also been developed {German Standard DIN 66141, (1974)} [125-129] Although elegant, the German notation requires memorization and is most suitable for frequent usage and computer applications. 2.17.1 Notation Let the fractional number smaller than size x be: X
aW= l^oWd^
(2-112)
•^min
dx Hence ^oC^) is the fractional number in the size range x to x + dx. Further:
eoWmax=
j9oWd^ =l
(2-113)
126 Powder sampling and particle size determination The subscript may be varied to accommodate other distributions, namely qr and Qr where: r = 0 for a number distribution; r = 1 for a length distribution; r = 2 for a surface distribution; r = 3 for a volume distribution. 2.17.2 Moment of a distribution The moment of a distribution is written as: •^max
A/,,=
I x''q,{x)6x
(2.114)
•^min ^max
Note: MQ^ = j ^^(jc)dx: = l -^min
For an incomplete distribution: X
Mk.r{^e^^u)- J x V W d x
(2.115)
2.17.3 Transformation from q/^x) to qj(x) If ^X^) is known qf{x) may be determined using the following: •^max
-^max
-^max
J q^{^x)&K- J x^qQ[x)dx= xmin
Hence:
^ i n
J x^~^qf{^x)Ax -"-min
q^ ix)
^^-^ j ^'^~^9^(^)djc
x' 'q^ {x)
qAA- J
^r-Ut
(2-'16)
The denominator is necessary in order to normalize the distribution function.
Data presentation and interpretation 127 Examples. To convert from a number to a volume (mass) distribution put / = 0, r = 3. To convert from a surface to a mass distribution put / = 2, r = 3. (a) Effect of particle shape This transformation is derived with the assumption that particle shape does not change with size. More correctly, a shape coefficient a{x) needs to be introduced: ,.(.)=
^^y"^'^')
(2.117)
j a{x)x^~^q^{x)Ax
2.17.4 Relation between moments Putting ^ = 0 in equation (2.116) Substituting x^^^'q^^ix) for x^q^ix) in equation (2.114) gives: max
J x^q^[x^Ax ^"^k.r ^r.O _
^k+rfi
^k,r
(2.118)
More generally: _ ^k,r
J^k+r-t,l
(2.119)
Examples. To determine the surface-volume mean diameter from a number distribution, put / = 0, r = 2, A: = 1.
128 Powder sampling and particle size determination
To determine the surface-volume mean diameter from a surface distribution, put t='2,r = 2,k= 1. Ml 2 ' ^0,2
^ 1 2 -
2.17.5 Means of distributions (a) Distribution means: IIIOA
X, •^nax
X|^=
—=
^ = A/, ^
(2.120)
(b) Arithmetic means •^max
M
^
-^min IIIilA
*_
(^*,o) -
^kfi ^0,0
,0=^;^
(2.121)
Data presentation and interpretation 129 (c) More generally: I X q^[^x)dx ^
(^^,o) =
^min UIOA
(2.122) 2.17.6 Standard deviations For a number distribution, the variance is defined by: 0 - 0 = 1 ( x - x , o ) ?oWd^
iiifiA
iiiaA
0-0= I x'qQ{x)
^0 = ^ 2 , 0 - ( ^ 1 , 0 )
(2.123)
More generally: (T,=
I {x-x,)
q,{x)
(2.124) Alternatively:
M 2,r M,i,/-
-M,
130 Powder sampling and particle size determination
Replacing r by r + 1, and putting / = r and A: = 1 in equation (2.119), and substituting in the above equation gives: al^M^^,[My,^,-M^^,]
(2.125)
From equations (2.121) and (2.125): ^^r=\r\_\r^\ "^u]
(2.126)
2.17.7 Coefficient of variation 2
O^
C]^-^^
{2A21)
C,^=%^-1
(2.128)
2.17.8 Applications (a) Calculation of volume-specific surface surface area S = ^
volume %iax
I C
a,{x)x^qo{x)dx
— "^niin niax
J a^{x)x
qo{x)dx
where a^ and a^ are the surface and volume shape coefficients. Assuming that these are independent of particle size and defining the ratio of a^ to a^ as a^y, the surface-volume shape coefficient is:
Data presentation and interpretation 131
M20
a
^3,0
(2.129)
^1,2
Also, since M] 2 =^12
(2.130) ^1,2 (a) Calculation of the surface area of a size increment f
9
^v v^e^^u)''
^v\^e^^u)~'^s
^2,o('^e'^«)
% 0
^2,0
^3,o(^e'^J
Now: Sr(^,)=
1 ?.(^)d^
J jv'^^oC^)'^
a^)^ Qrixi)
M.r,0
_^r,o(^min»^/)
M-,0
132 Powder sampling and particle size determination
so:
QA\)-QA^e)-
7^3,0 (^e'^«)
M 3,0
Hence: (2.131)
• J i , — OC^
%0
ft(^J-ft(^e)
The application of this equation enables a surface area to be calculated from a summation of increments, i.e. M 2,0 M 3,0 "2 _
^5V
j A: 9Q(x)dx:+ ^x^q^{x)&K + •
M 3,0 ^.= a.
M 3,0
/=A7
(2.132) /=1
e^ = mm, e^ = max 2.17.9 Transformation of abscissae Suppose, in an analysis, <^, which is a function of x, is measured e.g. (^= ln(x). Since the amount of material between sizes x and x ^ Ax is constant, there must be a simple relationship between q{^ and q{x). Let jc - / ^ ) so that ^ - ^x); then: q,{^)A^^q^{x)(\x
Data presentation and interpretation 133 i.e. the quantity in the interval d^ is the same as the quantity in the interval dx. The new ordinate ^^(<^) is calculated from qf.(x) and the differential ratio djc/di^. q;(^ = q,(x)^
(2.133)
For example, suppose the following relationship holds: ^ - x *
dx and q;(^ = -L^q^(x)
(2.134)
Similarly for ^ = ln(x) d<^
-1
•
— =x dx
41 / ^
djc
I.e. dln(x) = — X
and 9*[ln(jc)] = x9^(A:) In general, suppose we have the q^^ix) distribution and wish to find the q^ix^) distribution. From equation (2.118):
^^ r-uM
Substituting in equation (2.136) gives: g;(^=
./"^ \ ; _ , M he''
(2.135)
Example. In a Coulter Counter analysis, the pulse height V is proportional to particle volume, i.e. ^= V = p^x^
134 Powder sampling and particle size determination
(a) Calculation of M^ Q
= (-^r O )'^
By definition:
^r,o=
\x'qQ(x)dx
Using the transformation from the above equation: •^max
M.r,0
^ / • o = — {p''x'-qQ{x)dx
Pr J. (2.136)
A/,. Pr
x'' a* (x^ Also, since qr{x) = — , substituting in equations (2.135) and M.rfi (2.136):
q,{x) = x'-ql{4)^^ ^
<Jr(x)
P' ^2/3,0 ( ^
(2.137)
Data presentation and interpretation 135 (b) Calculation ofMj^ ^ From equation (2.114): •^max
1
f^^^^^ ^min
^
^a+r)/3.0(^) ^-'--(...)/3,0V9;
(2.138)
(b) Calculation of volume specific surface c _ ^2,0
which, using equation (2.138), can be written:
^ v - ^ - / ^ % ^
(2.139)
Thus, specific surface can be determined from moments calculated directly from Coulter Counter data. (b) Calculation of mean size xj^ y Equation (2.121) may be written:
Substituting for M^^ from equation (2.140):
'"^'--tT'i^^
(2.140)
136 Powder sampling and particle size determination
2.18 Phi-notation In geological literature dealing with particle size distribution [130,131] a very advantageous transformation of particle size is commonly used. This transformation replaces scale numbers based on linear millimeter values by the logarithms of these values. Because it is a logarithmic transformation it simplifies computation. The transformation is: ^ - - l o g 2 X , or X , - 2 - ^
(2.141)
where X^ is a dimensionless ratio of a given particle size, in millimeters, to the standard particle size of 1mm. Phi - values can be found if the common decadic logarithms of X/ are multiplied by (loglO^)"^ = 3.322 For easy manipulation a conversion chart [132] or a conversion tale [133,134] can be used. The standard deviation cr^used in this notation is defined as: ^^ =0.5(^4-«>16)
(2-142)
The skewness is defined as: ^,^84+^16-2^50
(2.145)
^84-^16
Other statistical measurements used in geology for particle size distribution characterization (moment, quartile and others) have been defined [135,136]. References 1 2 3 4 5
Tsubaki,J. and Jimbo, G. (1979), Powder TechnoL, 22(2), 161-70, 61 Growl, V.T. (1963), Paint Research Station Report, No 235, Teddington, London, 62, 75 Anon. (1985), Image Analysis, Principles and Practice, Joyce Loebl, 63 Morony, M.J., Facts From Figures, Pelican, 68 Alderliesten, M. (1990), Part. Part. Systems Charact., 1, 233-241; (1991), 8,237-241,^5
Data presentation and interpretation 137
6 I 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Heyd, A. and Dhabbar, D. (1979), Drug, Cosmet. Ind, 125, 42-45, 146-147, 69' Davies, R. (1975), Powder Technol.,\2, 111-124, 69, 81, 84 Hawkins, A.E. (1993), The Shape of Powder-Particle Outlines, John Wiley, 69 Singh, P. and Ramakrishnan, P. (1996), Kona, 14, 16-30, publ. Hosokawa Powder Technology Foundation,, 69 British Standard 2955, Glossary of terms relating to powders, 70 Beddow, J.K., Sisson, K. and Vetter, Q.V. (1976), Powder MetalL Int., 2, 69-76, 70 Tracey, V.A. and Llewelyn, D.M. (1976), Powder MetalL Int., 8(3), 126128, 70 Beddow, J.K. (1978) Powtech 77, publ. Heyden, 70 Beddow, J.K., Philip, G.C. and Nasta, M.D. (1975), Plansbeer, PulvermetalL, 23, 3-14, 70 Beddow, J.K. et. al. (1976),. 8thA.GM. Fine Particle Soc., Chicago, 70 Heywood, H., (1947^, Symposium on Particle Size Analysis, Inst. Chem. £:^2gr.v., Suppl. 25, 14, 77.77 Heywood, H. (1963), J. Pharm. Pharmac. Suppl., 15, 56T, 71 Heywood, H. (1973), Harold Heywood Memorial Lectures, Loughborough University, U.K., 72 Ward-Smith, R.S. and Wedd, M. (1997), Part. Part. Syst. Charact., 14, 306, 74 Prod, G. and Kratky, (1949), Rec. Trav. Chim. Pays Bas, 68, 1106, 74 Fair, G.L. and Hatch, L.P. (1933), J. Am. Water Wks Ass., 25, 1551, 75 Wadell, H. (1932), J. GeoL, 40, 250-80, 43, 459, 76 Wadell, H. (1934), J. Franklin Inst., 217, 459, 76 Wadell, H. (1934), Physics, 5, 281-91, 76 Robertson, R.H.S. and Emodi, B. (1943), Nature, 152, 539, 76 Davies, C.N. and Rees, W.J. (1944), J. Iron and Steel Inst., 150, 19P, 76 Krumbein, W.C. (1934), J. Sediment. Petrol., 4, 65, 77 Krumbein, W.C. (1941), J. Sediment. Petrol, 11(2), 64-72, 77 Laird, W.E. (1971) Particle Technol, Proc. Seminar, Indian Inst. TechnoL, Madras, eds. D. Venkateswarlu and A. Prabhakdra Rao, 67-82, 77, 81 Fuchs, N.A. (1964) Mechanics of Aerosols, Pergamon Press, London, 78 Hesketh, H.E. (1977), Ann Arbor Science, Mich. U.S.A. p8, 78 Ellison, J. McK (1954), Nature, 173, 948, 80 Hodkinson, J.R. (1962), PhD thesis. London University, 80 Cartwright, J. (1962) Ann. Occup. Hyg., 5, 163, 80 Stein, F. and Com, M. (1976), Powder Technol., 13, 133-141, 80 Endoh, S., Kuga, Y., Ohyo, H., Ikeda, C. and Iwata, H. (1998), Part. Part. Syst. Characterisation, 15, 145-149, 81 Austin, L.G. (1998), Part. Part. Systems Charact., 15, 108-1 \\,81 Hausner, H.H. (1966), Plansbeer, PulvermetalL, 14(2), 74-84, 81 Medalia, A.I. (1970/1971), Powder TechnoL, 4, 117-138, 81 Church, T. (1968/1969), Powder TechnoL, 2, 27-31, 81
138 Powder sampling and particle size determination
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69
Cole, M. (June 1971), Am. Lab., 19-28, 81 Pahl, M.H., Schadel, G. and Rumpf, H. (1973), Aufbereit, Tech., 5, 257-264, 81 Beddow, J.K. (1974), Report A390-CLME'74-007, The University of Iowa, 81,84 Barreiros, F.M., Ferreira, PJ. and Figueiredo, M.M. (1996), Part. Part. Syst. Charact., 13(6), 368-373, 81 Tsubaki,J. and Jimbo, G. (1979), Powder TechnoL, 22(2), 161-70, 82 Tsubaki,J., Jimbo, G. and Wade, R. (1975), J. Soc. Mat. Sci., Japan, 24(262), 622-626, 82 Massacci, P. and Bonifazi, G. (1990^, Proc. Second World Congress Particle Technology, Sept., Kyota, Japan, Part 1, 265-271, 83 Meloy, T.P. (1969) Screening, AIME, Washington, DC, USA, 83 Meloy, T.P. (1971), Eng Found Conf, Deerfield, USA, 83 Meloy, T.P. (1977), Powder TechnoL 16(2), 233-254, 83 Meloy, T.P. (1977), Powder TechnoL 17(1), 27-36, 83 Gotoh, K. and Finney, J.L. (1975), Powder TechnoL, 12(2), 125-130, 84 Beddow, K. and Philip, G. (1975), Plansbeer, 23(1), 84 Meloy, T.P., Clark, N., Dumey, T.E. and Pitchumani, B. (1985), Chem. Eng. Sci., 1077-1084,54 Alderliesten, M., (1991), Part. Part. Charact., 231-241, 84 Shibata, T. and Yamaguchi, K. (1990), Second World Congress Particle Technology, Sept., Kyota, Japan, Part 1, 257-264, 84 Schwartcz, H.P. and Shane,K.C. (1969), Sedimentology, 13, 213-31, (54 Fairbridge, C , Ng, S.H. and Palmer, A.D. (1986), Fuel, 65, 1759-1762, 84 Shibata, T., Tsuji, T., Uemaki, O. and Yamaguchi, K. (1994), First International Particle Technology Forum, Am. Inst. Chem. Engrs., Part 1, 95-100,(54 Orford, J.D. and Whalley, W.B. (1991), Principles, Methods and Applications of Particle Size Analysis, pp. 88-108, ed. J.P.M. Syvitski, Cambridge University Press, New York, 84 Hundal, H.S., Rohani, S., Wood, H.C. and Pons, N.M. (1997), Powder TechnoL, 9\(3),2\l-227, 84 Johnston, J.E.and Rosen, L.J. (1976), Powder TechnoL, 14, 195-201, 85 Hawkins, A.L, (1993), The Shape of Powder-Particle Outlines, John Wiley, p58, 85 Mandelbrot, B.B. (1977), Fractals, Form, Chance and Dimension, W.H. Freeman & Company, San Francisco, 85 Mandelbrot, B. B.(1983), The Fractal Geometry of Nature, W. H. Freeman & Company, 85 Kaye, B.H. (1991), PSA '91, Proc. 25th Anniversary Conf Particle Characterization Group, An. Div. Royal Soc, Chem., ed N. Stanley-Wood and R. Lines, 85 Mandelbrot, B.B. (1967), Science, 155, 636-638, 85 Kaye, B. H. and Clark, G.G.(1989), Particle Charact., 6(1), 1-12, 85 Fairbridge, C , Ng, S.H. and Palmer, A.D.(1986), Fuel, 65, 1759-1762, 87
Data presentation and interpretation 139
70 71 72 13 74 75 76 77 78 79 80 81 82 83 84 85 86 87
88 89 90 91 92 93 94 95
Fan, L.T., Boateng, A.A. and Walawender, W.P.(1992), Can. J. Chem. Eng., 70, 388-390, April, 87 Kaupp, A.(1984), Gasification of Rice Hulls, Theory and Practice, Friedr. Vieweg & Son, Braunschweig/Weisbaden, Germany, 112-138,, 87 Chan, L.C. and Page, N.W.(1997), Part. Part. Charact., 14(2), 67-72,, 87 Chan, L.C. and Page, N.W., (1998), Particle Fractal and Load Effects on Internal Friction in Powders, Powder TechnoL, in the press, 87 Fan, L.T., Boateng, A.A. and Walawender, W.P.(1992), Can. J. Chem. Eng., 70, 388-390, April, 87 Avnir, D., Farin, D. and Pfeifer, P. (1983), J. Chem. Phys., 79, 3566, 87 Richter, R., Sander, L.M. and Cheng, Z. (1984), J. Colloid Interf Sci., 100(1), 203-209, (^7 Kurd, A.J. and Flower, W.L. (1988), J. Colloid Interf Sci., 122(1), 87 Schaeffer, D. W. (1989), Science, 243, 1023-1027, 87 Kasper, G., Chesters, S. Wen,, H.Y. and Lundin, M. (1989), Applied Surface Science, 40, 185-192,55 Zaltash, A., Myler, C.A., Dhodapkar, S. and Klinzing, G.E. (1989), Powder TechnoL, 59, 199-207, 88 Allen, M., Brown, G.J. and Miles, N.J. (1995), Powder TechnoL 84, 1-14, 88 Tsakiraglou, CD. and Pasyatakes, A.C. (1993), J. Colloid Interf ScL, 159, 287-301,55 Wettimuny, R. and Penumadu, D. (2003), Part. Part. Syst. Charact., 20, 1824, 88, 55 Fini, A., Femandez-Hervas, M.J., Holgado, M.A. and Rabasco, A.M. (1998J, Partec 98, 1st European Symp. Process Technology in Pharmaceutical and Nutritional Sciences, 17-26, Numberg, Germany, 55 Gotah,K. and Finney,J.L. (1975), Powder TechnoL, 12(2), 125-30, 55 Neese, Th., Diick, J. and Thaufelder, T. (1955), 6th European Symp. Particle Size Characterization, Partec 95, Numberg, Germany, publ. NUmbergMesse GmbH, 315-325, 55 Heffels, C , Heitzmann, D., Kramer, H. and Scarlett, B. (1995), 6th European Symp. Particle Size Characterization, Partec 95, Niimberg, Germany, publ. NumbergMesse GmbH.267-276, 88 Ridgeway, K. and Rupp, R. (1969), J. Pharm., Pharmac. SuppL, 21, 30-39, 55 Ridgeway, K. and Rupp, R. (1970/71), Powder TechnoL, 4, 195-202, 55 Riley, G.S. (1968/69), Powder TechnoL 2, 62, 55 Viswanathan, K., Aravamudhan, S., Mani, B.P. (1984), Powder TechnoL 39, 83-91,55 Whiteman, M. and Ridgeway, K.,. (1988), Powder TechnoL 56, 83-94, 55 Meloy, T.P. and Makino, K. (1983), Powder TechnoL, 36, 253-258, 89 Lenn C.P. and Holt, C.B. (1982), Proc. Fourth Particle Size Analysis Conf, 1981, publ. John Wiley, pp233-240, edM.G. Stanley-Wood and T. Allen, 89 Anon (1974), German Standard DIN 66141, Representation of Particle Size Distribution, 95
140 Powder sampling and particle size determination
96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 I 18 119 120 121 122 123 124 125 126 127 128 129 130 131
Svarovsky, L. (1973), Powder Technol, 7(6), 351-352, 108 Herdan, G. (1968), Small Particle Statistics, Butterworths, 109 Yu. A.B. and Standish, N. (1987), Powder Technol., 52, 233, 109 Yu. A.B. and Standish, N. (1990^, World Congress Powder Technology, Part 1, Soc. Powder Techn., Kyota, Japan, 109 Yu, A.B. and Standish, N. (1990), Powder Technol., 62, 101-118, 109,112,117 Yu, A.B. (1994), First Int.Powder Technology Forum, Am. Inst. Chem. Engrs, Denver, 120-125, 109 Harris, C.C. (1968), Am. Inst. Meek Engrs., 241, 343, / / / Dukhin, A.S., Goetz, P.J. and Hackley, V. (1998), Colloids and Surface Chem.,\U^, 1-9,7// Rosin, P. and Rammler, E., (1933), J. Inst. Fuel, 7,29, 109, 110,111,113 Gaudin, A.M. (1926), Trans. AIME, 73, 253, 112, 113 Gates, A.O. (1915), Trans. AIME, 52, 875-909., 112, 113 Schuman, R. (1940), Trans AIME, Tech. publ 1189,/12,113 Gaudin, A.M. and Meloy, T.P. (1962), Trans AIME, 43-50, 112, 113 Roller, P.S. (1937), Proc. ASTM., 37,675, U2, 113 Roller, P.S. (1937), J. Franklin Inst., 223, 609, 112, 113 Roller, P.S. (1941), J. Phys. Chem., 45, 241, 112, 113 Svensson, J. (1955), Acta. Polytec. Scand.,\63, 53, /72 Harris, C.C.(1969), Trans. AIME, 244, 187-190, 109, 112, 113 Tarjan.G. (1974), Powder Technol., 10, 73-6, 112 Martin, G. (1924), Trans. Ceram. Soc, 23, 61, 109, 113 Svensson, K. (1955;,. Tekn. Hpgsk. Handl., No 88, 109, 113 Evens, I. (1958), Proc. Sci. in the Use of Coal, Inst. Fuel, London, Paper 14, 109,//J Weinig, A.J. (1933), Col School of Mines Quarterly, 28, 57, 109, 113 Heywood, H. (1933), Proc. Inst. Meek Engrs., 125, 383, 109, 113 Griffith, L. (43), Can. J. Res., 21, 57, 109, 113 Klimpel, R.R. and Austin, E.G. (1965), Trans AIME, 232, 82, 113 Popplewell, E.M., Campanella, O.H. and Peleg, M. (1988), Powder Technol., 54, 119, / / i Kalaidograph, Abelbeck Software, Synergy Software, 2457 Perkiomen Ave., Reading, PA 19606-9976, 123 Allen, T. and Nelson, R.D.Jr. (1994), First Int. Techn. Forum, Am. Inst. Chem. Engrs., Denver, Part 1, 113-119, 725 Rumpf, H. and Ebert, K.F. .(1964), Chem. Ing Tech, 36,523-37, 125 Rumpf, H. and Debbas, S. (1966), Chem. Eng Sci., 21,583-607, 125 Rumpf, H., Debbas, S. and Schonert, K. (1967), Chem. Ingr. Tech., 39,3, 116-9,/25 Rumpf, H. (1961), Chem. Ingr. Tech., 33(7), 502-8, 125 Eeschonski, K., Alex, W.,and Koglin, B. {\91A),Chem. Ingr. Tech., 46(3), 23-26, 125 McManus, D.A. (1963), J. Sediment. Petrol, 33, 670,136 Krumbein, W.C. (1964), J. Sediment. Petrol, 34, 195, 136
Data presentation and interpretation 141
132 133 134
135 136
Krumbein, W.C. and Pettyjohn, F.J. (1938), Manual of Sedimentary Petrology, p244, Appleton-Century-Crofts, New York., 136 Page, H.G. (1955), J. Sediment, Petrol, 25, 285, 136 Griffiths, J.C. and Mclntyre, D.D. (1958), A Table for the Conversion of Millimeters to Phi Units, Mineral Ind. Exp. Sta., Penn. State University., 136 Folk, R.L. (1966), Sedimentology, 6, 73, 136 Griffiths, J.C. (1962) Sedimentary Petrolgraphy (ed. H.B. Milnes) Macmillan, New York, Ch. 16,136
Particle size analysis by image analysis 3.1 Introduction A microscope examination should always be carried out whenever a sample is prepared for particle size analysis. Such an examination allows an estimate of the particle size range of the powder under test and its degree of dispersion. If the dispersion is incomplete it can be determined whether this is due to the presence of agglomerates or aggregates and, if agglomeration is present, may indicate the need for an alternative dispersing procedure. Microscopy is often used as an absolute method of particle size analysis since it is the only method in which the individual particles are observed and measured [1-3]. It is particularly useful in aerosol science where particles are often collected in a form suitable for subsequent optical examination. It is useful not only for particle size measurement but also for particle shape and texture evaluation, collectively called morphology, with sensitivity far greater than other techniques. Reports have also been presented of the use of microscopy to relate particle size to processing characteristics of valuable mineral ores [4,5]. Particle shape may be defined either qualitatively or quantitatively. The former includes the use of such terms as acicularity, roundness and so on. The latter, more definitively, compares perpendicularly oriented diameters, for example, to obtain shape factors. The introduction of automatic image analyzers allows for the determination of complex shape factors which were previously unobtainable. These factors are of great value in defining crystal morphology and relating this to operating (attrition during conveying, compaction and so on) and end-use properties. A size analysis by number is simpler to perform than an analysis by mass since, in the former, the statistical reliability depends solely on the number of particles measured. For a mass analysis the omission of a single 10 |Lim particle leads to the same error as the omission of a thousand 1 |Lim
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particles since they both have the same volume. For a particle size analysis by mass, 25 particles in the largest size category have to be counted, in order to obtain an estimated standard error of less than 2%. If all the particles in this area were counted the final count would run into millions. It is obvious therefore that the area to be examined must decrease with decreasing particle size and the results obtained must be presented as particles per unit area. The problem may be likened to determining the size distribution of a number of differently sized homogeneous balls in a container. If the balls are of size 2, 2V2, 4, 4^2 64, 64^2, in line with the size ratios often adopted in optical microscopy, and the relative frequency of the top size category is found to be 8 in 1000 particles this can be readily converted to a mass frequency when the number of balls in the other size categories is known. If the estimated mass of the 25 particles is 10% of the sample then the forecasted percentage standard error is (10/V25) i.e. 2%. If, on completing the analysis, the mass percentage of the coarsest fraction is greater than 10% then it is necessary to count more coarse particles in order to maintain this level of accuracy. The errors in converting from a number to a volume (mass) distribution are greatest when the size range is wide. For a narrowly classified powder, ranging in size from say 10 to 30 |Lim, it is necessary to use an arithmetic grading of sizes, probably a 2 |Lim interval in this case, but the same rules still apply and the direct conversion of a number distribution to a weight distribution can still give rise to considerable error at the coarse end of the size distribution. Using closer size intervals adds little to analytical accuracy but can greatly increase the computation time. The images may be viewed directly or by projection. Binocular eyepieces are preferred for particle examination but monoculars for carrying out a particle size analysis since, by using a single eyepiece, the tube length can be varied to give stepwise magnification. Most experienced operators prefer direct viewing but projection viewing, less tiring to the eye, is often used for prolonged counting. Projection may be front or back. With the former the operation is carried out in a darkened room due to the poor contrast attainable. Back projection gives better illumination but image definition is poor; this can be rectified by using a system whereby two ground-glass screens are placed with their faces in contact and one is moved slowly relative to the other [6]. Some automatic counting and sizing devices work from photographic negatives or positives. The principle objection that can be leveled against photographic methods is that only particles in focus can be measured accurately and this can lead to serious bias. Although photographic
144 Powder sampling and particle size determination
methods are often convenient and provide a permanent record, the processing time may well offset any advantage obtained. This is particularly true when a weight count is required since, from statistical considerations, a large number of fields of view are required for accurate results. Light microscopy is best suited for the size range 0.8 to 150 |Lim, with a resolution of around 0.2 |Lim depending on the wavelength of the light source. Scanning electron microscopy (SEM) operates in the size range from 0.1 |im to 1000 |Lim with a resolution of 10 nm and transmission electron microscopy (TEM) from 0.01 |Lim to 10 jum with a resolution of 5 nm. Back scattered electrons and x-rays contain information on the chemistry and average atomic number of the material under the beam. Groen et. al [7] determined the optimum procedure for automatic focusing of a microscope. Kenny [8] examined the errors associated with detecting the edge of the particle image and outlined a technique, suitable for automatic image analysis for minimizing this error. The shape and texture of construction aggregates are important parameters that have a direct bearing on the strength and durability of their asphalt and concrete end products. Typically, a batch of material is rejected if more than a specific fraction of particles have elongation and flatness ratio that exceed some limit. In the ASTM procedure [9] the measurements are carried out on 100 particles using specially designed calipers. More recently this has been replaced by image analysis which reduces the measurement time to less than 10 minutes [10]. In addition, this procedure is capable of conducting other useful particle characterization measurements without the need for additional image processing time. One such measurement incorporated into the design is roughness defined as "surface irregularity" and "jaggedness". Examples of determining both a number and a mass distribution are given below. Although the examples relate to manual counting, the conditions also govern size analyses by automatic image analyzers. 3.2 Standards Relevant national standards are available covering particle size analysis by microscopy. BS 3406 Part 4 [11] is the British Standard guide to optical microscopy. The American standard ASTM E20 was discontinued in 1994 [12]. ASTM 175-82 [13] is a standard defining terminology for microscope related applications. ASTM E766-98 [14] is a standard practice for calibrating the magnification of an SEM. NF XI1-661 [15] is the French standard for optical microscopy. NF Xl 1-696 [16] covers
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general image analysis techniques. ISO/CD 13322 [17] is a draft international standard on image analysis methods. 3.3 Optical microscopy Optical microscopy is most often used for the examination of particles from about 3 |Lim to 150 |im in size, although a lower limit of 0.8 jum is often quoted. Above 150 |Lim a simple magnifying glass is suitable. The most severe limitation of optical transmission microscopy is its small depth of focus, which is about 10 |Lim at a magnification of lOOx and about 5 |Lim at lOOOx. This means that, for a sample having a wide range of sizes, only a few particles are in focus in any field of view. Further, in optical transmission microscopy, the edges of the particles are blurred due to diffraction effects. This is not a problem with particles larger than about 5 |im since they can be studied by reflected light, but only transmission microscopy, with which silhouettes are seen, can be used for smaller particles. A two dimensional array of latex spheres is often used for measuring more or less uniformly sized lattices. Hartman [18-20] investigated the errors in this method which comprise focusing, image distortion, misreading of photomicrographs, distortions in the photographic material, anisotropy, other array defects, non-uniformity of particle size, coating of solutes on the lattices and contact deformation. Hartman introduced a new method, the center finding technique in which the latex spheres acts as lenses enabling the center-to-center distance to be determined with high accuracy (10 ± 0.4) |Lim for 10 |Lim particles. The National Physical Laboratory [21] introduced an NPL certified stage graticule [22] to test linearity over the complete image field. 3.3.1 Upper size limit for optical microscopy The method is preferably limited to sub-200 mesh sieve size (75 \\m) but larger particles may be counted and sized provided their fractional weight is less than 10% of the total weight of the powder. When the fractional oversize weight exceeds 10%, these particles should be removed and a sieve and microscope analyses merged. Alternatively such large particles can be sized using a simple magnifying glass.
146 Powder sampling and particle size determination
3.3.2 Lower size limit for optical microscopy The theoretical limit of resolution of an optical microscope is expressed by the fundamental formula:
NA where di^ is the limit of resolution, i.e. particles in closer proximity than this appear as a single particle, A is the wavelength of the illuminant, the numerical aperture of the objective NA = jusind where ju is the refractive index of the immersion medium, ^is the angular aperture of the objective and/is a factor of about 0.6 to allow for the inefficiency of the system. For A = 0.6 |um the resolving power is a maximum with NA = 0.95 (dry) and NA = 1.40 (wet) giving lower size limits, d^^^^ = 0.38 |Lim and 0.26 |um respectively. The images of particles having a separation of less than these limits merge to form a single image. The resolution of the human eye is around 0.3 mm, therefore the maximum effective magnification with white light is:
28 /mi Particles smaller than the limit of resolution appear as diffuse circles; image broadening occurs, even for particles larger than d^^^, and this results in oversizing. Some operators routinely size down to this level but the British Standard BS 3406 Part 4 [11] is probably correct in stipulating a minimum size of 0.8 |am and limited accuracy from 0.8 to 2.3 |u.m. Powders containing material smaller than this are usually imaged by transmission or scanning electron microscopy and the resulting negatives or prints examined. Charmain [23] in an investigation into the accuracy of sizing by transmission optical microscopy, showed that for two-dimensional silhouettes greater than 1 |Lim in diameter, the estimated size under ideal conditions was about 0.13 |am too high; a 0.5 |Lim silhouettes gave a visual estimate of 0.68 fim and all silhouettes smaller than 0.2 |Lim appeared to have a diameter of 0.5 |j,m (Figure 3.1). The measurements were made with the circular discs immersed in oil. Due to less precise focusing with three-dimensional particles, real particles are subject to greater errors.
Image analysis 147 Rowe [24] showed that wide differences in particle sizing can occur between operators because of this effect. 3.4 Sample preparation Great care has to be taken in slide or grid preparation since the measurement sample is so small that it is difficult to make it representative of the bulk. Many particulate systems contain agglomerates and aggregates and, if it is necessary that they retain their integrity, the dispersing procedure needs to be very gentle. Further, since it is usually impossible to measure every particle in the measurement sample it is necessary that it be dispersed uniformly. Small regions selected at random or according to some predetermined plan must therefore be representative of the whole. The analysis is suspect if the regions in one area of the measurement sample give a very different size distribution to those in another area. The simplest procedure is to extract samples from an agitated suspension; for less robust materials a procedure detailed in reference [3] may be used in which an air jet circulates the suspension through a sampling tube that can be closed and withdrawn to provide samples for analysis. Slides may be of three main types: dry, temporary and permanent. For very easily dispersed material, the particles may be
1 1.5 2 True disc diameter ((un) Fig. 3.1 Oversizing of small discs by optical microscopy [23]
148 Powder sampling and particle size determination
shaken from a fine brush or the end of a spatula on to a slide. Humphries [25] describes a microsample splitter that assists the free flow of grains in order to provide the very small samples needed for microscopy: A diagram of the device is reproduced in a book by Hawkins [26]. Hawkins also describes a moving pavement version of the spinning riffler designed for preparation of representative samples of free flowing particles on microscope slides [27]. A novel method of mounting particles on regularly spaced adhesive circles has also been developed. This method of mounting results in an ordered array rather than the random chaos of usual methods and greatly facilitates particle analysis [28,29]. Some acceptable procedures for easily dispersed powders are described by Green [30] and Dunn [31]. For a temporary slide the powder can be incorporated into a viscous liquid, such as glycerin or oil, in which it is known to disperse completely. Some operators work the powder into the liquid with a flexible spatula; others roll it in with a glass rod. Either of these procedures can cause particle fracture and a preferable alternative is to use a small camelhair brush. A drop of this liquid can then be transferred to a microscope slide and a cover slip gently lowered over it. Rapid pressing of the cover slip must be avoided as it causes preferential transfer of the larger particles to the edge of the cover slip. It is undesirable for liquid to spread outside the limits of the cover slip; improved spreading is best effected with highly viscous liquids by pre-warming the microscope slide. Sealing the cover slip with amyl acetate (nail varnish is a good substitute) makes the slide semi-permanent. If low viscosity liquids are used it is necessary to have a well, or depression, on the slide to contain the dispersion. The method of Orr and Dallevalle [32] for the production of permanent slides is to place a small representative sample of the powder to be analyzed in a 10 ml beaker, add 2 to 3 ml of a solution containing about 2% coUoidon in butyl acetate, stir vigorously and place a drop of suspension on the still surface of distilled water in a large beaker. Prior to adding the suspension the surface is cleaned by allowing a drop of butyl acetate to fall on it. As the resulting film expands, it sweeps any particles on the surface to the walls of the beaker. As the drop of suspension spreads, the volatile liquid evaporates and the resulting film may be picked up on a clean microscope slide and completely dried. A dispersing agent may be added to prevent flocculation. Permanent slides may also be produced by using the alternative combinations of Canada balsam or polystyrene in xylol, dammar in turpentine, gum arabic in glycerin, styrex in xylene, rubber in xylene and gelatin in water [33]. With a 1% solution this may be formed by dropping
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it on to the cleaned surface of distilled water; with a 0.5% solution it may be cast directly on to a microscope slide; spreading is accelerated if the slide is first washed in a detergent. Dullien and Mehta [34] use Cargill's series H compound, having a refractive index of 2.0, as a mounting medium for salt particles. This gives a transparent yellow background for the particles and, since it has a higher refractive index than salt, the particles appear as dark spots. A range of systems is necessary in order to select one where the difference in refractive index gives an easily detectable image. Millipore recommend filtering a dilute suspension through a 0.2 jiim PTFE membrane filter that is then placed on a dry microscope slide. The slide is then inverted over a watch glass half filled with acetone, the vapors of which render the filter transparent after two to three minutes. Harwood [35] describes two methods for dispersing difficult powders. One involves the use of electrical charges to repel the particles then fixing the aqueous solution with a gelatin-coated slide to overcome Brownian motion. The other, for magnetic materials, involves heating the sample to a temperature above the Curie point then dispersing it and fixing it on a slide to cool. Allen [36] mounted the powder directly into clear cement, dispersing it by using sweeping strokes of a needle and spreading the film on a microscope slide to dry. Lenz [37] embedded particles in solid medium and examined slices of the medium. Particles may also be suspended in a filtered agar solution that is poured on to a microscope slide where it sets in seconds [38]. Variations in analyses between these procedures may occur due to particles settling on the slide with preferred orientations. Ellison showed that if particles were allowed to fall out of suspension on to a microscope slide they would do so with a preferred orientation. Also, if the dispersing is not complete, the presence of floes will give the appearance of coarseness [39]. Pidgeon and Dodd [40], who were interested in measuring particle surface area using a microscope, developed methods for preparing slides of particles in random orientation. For sieve size particles, a thin film of Canada balsam was spread on the slide and heated until the liquid was sufficiently viscid, determined by scratching with fine wire until there was no tendency for the troughs to fill in. Particles sprinkled on the slide at this stage were held in random orientation. After a suitable hardening time, a cover glass coated with glycerol or warm glycerol jelly, was placed carefully on the slide. Sub-sieve powders were dispersed in a small amount of melted glycerol jelly. When the mixture started to gel a small amount was spread on a dry slide. After the mount had set, it was protected by a cover slip coated with
150 Powder sampling and particle size determination
glycerol jelly. With this technique, it is necessary to refocus for each particle since they do not lie in the same plane. Several of these techniques were examined by Rosinski et. al. [41] in order to find out which gave the best reproducibility. The sizing of fibrous particles by microscopy presents serious problems including overlapping. In order to minimize this it is necessary to work with only a few particles in the field of view at any one time. Timbrell [42,43] showed that certain fibers showed preferred orientations in a magnetic field, e.g. carbon and amphibole asbestos. He dispersed the fibers in a 0.5% solution of colloidon in amyl acetate and applied a drop to a microscope slide, keeping the slide in a magnetic field until the film had dried. For SEM examination an aqueous film may be drained through a membrane filter held in a magnetic field. In order to reduce overlapping to an acceptable level it is necessary to use a far more dilute suspension than for more compact particles. Various means of particle identification are possible with optical microscopy. These include dispersion staining for identification of asbestos particles [44] and the use of various mounting media [45]. Proctor et. al. [46,47] dispersed particles in a solidifying medium of Perspex monomer and hardener. This was poured into a plastic mold that was slowly rotated to ensure good mixing. Microscope analyses were carried out on thick sections; a lower size limit of 5 |j,m was due to contamination. Zeiss [48] describes a method for measuring sections of milled ferrite powder. The powder was mixed in 40:60 volume ratios with epoxy resin using a homogenizing head rotating at 25,000 rpm. The mixture was then poured into a 0.5 in diameter mold and cured at 60°C and 1000 psi to eliminate air bubbles. The casting was then polished in a vibratory polisher using 0.3 and 0.5 |im alumina in water. A photomicrograph of the polished section was used for subsequent analysis. Automatic and quantitative microscopes tend to give erroneous results for transparent particles. To overcome this problem Amor and Block [49] a silver staining technique to make the particles opaque. The particles are dry-mounted on to a thin film of tacky colloidon on a microscope slide. Silver is then deposited from solution using the silver mirror reaction. Preliminary sensitizing the crystalline surface ensures that much more silver is deposited on the particles than on the colloidon. A method of staining particles in aqueous solution prior to deposition on a membrane filter for analysis is also given. Hamilton and Phelps [50] adapted the metal shadowing technique for the preparation of transparent profiles of dust particles. The process consisted of evaporating in vacuo a thin metal film in a direction normal to
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a slide containing particles. The particles are then removed by a jet of air or water, leaving sharp transparent profiles. 3.5 Measurement of plane sections through packed beds When the size distribution of particles embedded in a continuous solid phase is required, the general approach is to deduce the distribution from the size of particle cross-section in a plane cut through the particle bed. The problem has occupied the attention of workers in diverse fields of science, who have tended to work in isolation and this has led to much duplication of effort. The historical development of this technique has been reviewed by Eckhoff and Enstad [51] and the relevant theory of Scheil by Dullien et. al. [52]. A theoretical analysis [53] has been criticized on several grounds [54]. Dullien et. al. [55-57] examined salt particles embedded in a matrix of Wood's metal using the principles of quantitative stereology. They then leached out the salt particles and examined the matrix using mercury porisimetry. Poor agreement was obtained and this they attribute to the mercury porosimetry being controlled by neck diameter. Nicholson [58]considered the circular intersections of a Poisson distribution of spherical particles to estimate the particle size distribution. Saltzman et. al. [59] generated a computer based imaging system for slices through a packed bed and found good experimental agreement. 3.6 Particle size The images seen in a microscope are projected areas whose dimensions depend on the particles' orientation on the slide. Particles in stable orientation tend to present their maximum area to the microscopist, that is the smaller dimensions of the particles are neglected, hence the sizes measured by microscopy tend to be greater than those measured by other methods. Any one particle has an infinite number of linear dimensions hence, if a chord length is measured at random, the length will depend upon the particle orientation on the slide. These orientation dependent measurements are known as statistical diameters, acceptable only when determined in such numbers as to typify a distribution. They are measured parallel to some fixed direction and are acceptable only when orientation is random; i.e. the distribution of diameters measured parallel to some other direction must give the same size distribution. They are representative of the two largest particle dimensions, since the smallest is perpendicular to the viewing plane if the particles are in stable orientation.
152 Powder sampling and particle size determination
Acceptable statistical diameters are: Martin's diameter {dfj) is the length of the line which bisects the area of the particle's projected area. The line may be in any direction, which must be maintained constant throughout the analysis [60,61]. Feret's diameter {dp) is the distance between two tangents on opposite sides of the particle parallel to some fixed direction [62]. Longest dimension. A measured diameter equal to the maximum value of Feret's diameter. Perimeter diameter {d^ is the diameter of a circle having the same circumference as the particle. Projected area diameter (dj takes into account both dimensions of the particle in the measurement plane, being the diameter of a circle having the same projected area as the particle. It is necessary to differentiate between this diameter and the projected area diameter for a particle in random orientation (dp) since, in this case, the third and smallest dimension of the particle is also included. The easiest diameter to measure is the Feret diameter but this is significantly larger than the other two diameters for most powders. It is probably best to reserve this diameter for comparison purposes and for rounded particles. Of the other two diameters, the projected area diameter is preferred since two dimensions are included in one measurement and the projected area is easier to estimate using globe and circle graticules than the length of the chord that bisects the image. It has been shown [63,64] that the relationship between specific surface and Martin's diameter is: ^v-:^
(3.2)
Since the surface-volume diameter is inversely proportional to S^, the constant of proportionality being a minimum of six for spherical particles, Martin's diameter is systematically different to the surface-volume diameter. Experiments confirm that, on the whole, dj^
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sieve. He determined the projected area with a planimeter and calculated the mean projected diameter; he next estimated the diameter using both the opaque and transparent circles on a globe and circle graticule and also determined Feret and Martin diameters. His conclusion, based on an examination of 142 particles, was that the Feret diameter was greatly different to the other diameters for elongated particles, but that the Martin and projected area diameters are sufficiently in agreement for all practical purposes. This was disputed by Walton [66] who showed that the Feret diameter, averaged over all particle orientations, was equal to the other diameters. Herdan [67] examined Heywood's data more rigorously and found that: (a) (b)
The Feret diameter was significantly different from the other four diameters. The Martin diameter showed significant difference from that obtained using the globe and circle graticule if the planimeter data were accepted as standard.
He concluded that there was no definite advantage to be gained by laboriously measuring profiles. As one might expect, the projected area diameters gave the best estimate of the true cross-sectional areas of the particles. This does not rule out the use of the other diameters if they are conveniently measured, since the cross sectional-area diameter of a particle is not necessarily its optimum dimension. 3.7 Calibration It is necessary to use a calibrated eyepiece scale when carrying out a microscope analysis. The simplest form consists of a glass disc that is fitted on to the field stop of the ocular. Engraved upon the disc is a scale that is calibrated against a stage micrometer placed in the object plane; typically this is a microscope slide on which is engraved a linear scale. The image of the scale is brought into coincidence with the ocular scale by focusing. With a single tube microscope the magnification be varied somewhat by racking the tube in or out. The stage graticule is then replaced by the microscope slide containing the sample. The microscope slide is made to traverse the eyepiece scale and particles are sized as the cross the reference line. Linear eyepiece graticules labeled 0 to 100 may be used to scan the sample so that the linear dimensions yields a size distribution as a function of the Martin or Feret diameter. Special graticules are also available containing globes (opaque images) and circles
154 Powder sampling and particle size determination
(transparent images). The former are designed for the sizing of opaque images and the latter for transparent images. 3.7.1 Linear eyepiece graticules These are linear scales, typically 10 mm, divided into 100 divisions of 100 |Lim, or 2 mm divided into 100 divisions of 20 |im each. They are placed in the focus of the microscope eyepiece so that they are coincident with the image of the microscope slide on the microscope stage. Calibration is effected using a stage graticule, 10 mm (100 x 100 |um), 1 mm (100 X 10 |Lim) or 100 |Lim (50 x 2 |Lim), which is placed in the object plane. Kohler illumination should be used [11] to give uniform illumination of the viewing plane. Using an oil immersion objective, it is possible to resolve down to about 1 |Lim, although a 15% oversizing is to be expected at this level due to diffraction effects. Ocular graticules having a linear scale are satisfactory for the measurement of linear dimensions of particles. Particle sizes obtained with a linear eyepiece graticule are best classified arithmetically hence it is most suited to particles having a narrow size range. 3.7.2 Globe and circle graticules Linear eyepiece graticules have been criticized on the grounds that the dimensions measured are greater than those determined by other methods. To overcome this objection, grids inscribed with opaque and transparent circles have been developed. For best results, opaque images are measured using the (opaque) globes while transparent images are best measured using the (transparent) circles. This permits direct comparisons between the projected areas of the particles and the areas of the circles. According to Cauchy the projected area is a quarter of the surface area for a random dispersion of convex particles [68] hence this measurement is fundamental to the properties of the powder. The earliest of these graticules by Patterson and Cawood has 10 globes and circles ranging in diameter from 0.6 to 2.5 |Lim when used with a +2 mm lOOx objective-eyepiece combination and is suitable for thermal precipitator work [69]. Fairs [70] designed graticules covering a size range of 128:1 using reference circles with a root two progression in diameter except for the smaller sizes. He considered this system to be superior to the Patterson-
Image analysis 155 Cawood where the series is much closer. He also described a graticule, having nine circles in a V2 progression of sizes, for use with the projection microscope [71]. This was incorporated in a projection screen instead of being in the eyepiece and was adopted by the British Standards Organization [11]. Watson [72] developed a graticule designed specifically to measure particles in the 0.5 to 5 |Lim (respirable dust) size range. May's graticule, [73] covers 0.25 to 32 |Lim in a root two progression of sizes (the lower limit is highly suspect).
a
O O o o
Fig. 3.2 The British Standard Graticule (Graticules Ltd) [11]. Diagram from BS 3406 (1963), Confirmed April 1993: Methods for the Determination of the Particle Size of Powders: Part 4, Optical Microscope Method. (Reproduced by permission from British Standards Institution, 2 Park Street, London Wl, from whom copies of the complete standard may be obtained.)
156 Powder sampling and particle size determination
[74] designed a graticule in which the diameters of eleven circles are arranged in a constant ratio of 1.2589 (logjol-2589 = 0.1). He claimed that the choice of this ratio and the novel system of marking the circles facilitated the rapid calculation of size parameters from measured data. The British Standard graticule is an improvement on Fair's graticule with seven globes and circles in a root two progression of sizes (Figure 3.2). With a +2 mm objective and a 20x eyepiece combination the minimum measurable size is about 1 |j.m. Circle 7 has a diameter eight times greater than circle 1; the grid has dimensions of 64 by 45.3 units with fiduciary marks at 60.4 units (circle 1 has diameter 1 unit). The graticule is divided into halves, quarters, eighths and sixteenths. Detailed setting up procedures are given in BS 3406 but, as an example, if the distance between the fiduciary marks is 566 )am the diameter of circle 7 is equal to 75 |j.m.
Particle thickness may be measured by a stereophotogrammetric method developed by Aschenbrenner [75, see also 32 p 19]. Hamilton and Holdsworth [76] compared the Watson graticule with a line graticule and with the Patterson-Cawood graticule. As the sizing of particles by visual comparison with reference circles some distance away from the particles may be subject to appreciable operator errors the line graticule was included in order to determine whether more consistent resuhs were obtained by this method. With this type of graticule, particles are sized as they pass the reference lines and the diameter so measured is the Feret diameter. They found systematic differences in the mean counts of operators for the size range of 1 to 5 |im, but no evidence that this was affected by the type of graticule used. It was also found that the Feret diameter over-estimated the size of coal dust. Their conclusions were that all three graticules gave equally accurate and consistent results but the operators preferred the Watson and line graticules on the grounds that they were less trying to use. 3.8 Training of operators Although the use of linear diameters such as Martin's or Feret's give the most reproducible analyses the projected area diameter is more representative hence the globe and circle graticule is the most popular. When comparing an irregular profile with a circle, untrained operators have a tendency to oversize the profile. A method of correcting this is to compare the analysis of a trained operator with that of a trainee. When the trainee recognizes the bias he readily corrects it. Heywood [77] produced a set of hand-held test cards pre-calibrated by counting squares. The trainee
Image analysis
157
is required to compare each of the profiles with reference circles and assign it to a size group. Watson and Mulford [78] extended the technique by inscribing a number next to each profile and reducing them photographically so that they could be examined under a reversed telescope, giving more realistic conditions. In a series of tests with nine operators, four were over estimating and five were under estimating, seven of the nine being badly biased. The nine operators were trained microscopists who were aware of the natural tendency to oversize and mentally corrected. The nine observers were consistent with their bias but reduced it only slightly on a second test. Fairs [79] used a projection microscope for training purposes. This technique can also be used for size analysis [80] but is not recommended for particles smaller than 2 \im. Hamilton et. al [81 c/Y.82] demonstrated the need to train operators and showed that gross count differences on the same samples at different laboratories were much reduced after interlaboratory checks. Nathan et. al [83] compared three commonly used microscopic measurement techniques and confirmed oversizing by untrained operators. They suggested that unskilled operators produced the best results with a line graticule and that experienced operators perform best with an image splitting device. 3.9 Experimental techniques The microscope should be set up using Kohler illumination; monochromatic illumination produces a better image if small particles are to be measured. The microscope needs a vernier stage capable of moving the slide in two directions at right angles. The limits of the uniformly dispersed field are determined using a low magnification objective. Since it is impracticable to examine every particle on the slide, a sample is selected. Either isolated fields areas or strips are selected from the whole of the viewing area in order to smooth out local concentration variations (Figure 3.3). For a number count at least five strips 5 to 20 mm long need to be examined, the length of the strips depending on the density of particles on the slide, the whole area being such that at least 625 particles are counted. For a weight count, the requirement is that the area contains at least 25 particles in the largest size category. If these are few in number it is advisable to use strip scanning for counting the coarsest size categories. For particles that are present in quantity, the use of an eyepiece graticule containing a field area, as opposed to a linear scale, is recommended. In
158 Powder sampling and particle size determination
either case the total area scanned must be determined in order that particle density on the slide (counts per mm^) may be calculated. When counting isolated fields, particles overlapping two adjacent sides are not counted. With strip counting, particles overlapping one of the edges are ignored; i.e. in Figure 3.3, shaded particles are included in count, unshaded particles are excluded otherwise the measured distribution will be weighted to the coarse sizes.
Fig. 3.3 Treatment of edge particles; (a) counting isolated fields, (b) counting strips 3.10 Determination of particle size distribution by number Little difficulty is likely to be experienced in carrying out a number count since the same total area is used for all size classes and the minimum number of particles, namely 625, that it is necessary to count is specified in advance [equation (3.3)]. Let there bey classes of particles of interest. For
Image analysis 159 the r^^ class (r = 1, 2...^')The number size distribution is calculated from the recorded values of the number m^ of particles of mean size dj. in 7 size classes found in sample area A^ = w^ a^. For each class the number of particles per unit area is calculated m^l A^ Then/?^ is calculated where:
P,-~^Y^^
(3.3)
Y^mrlA^ 100/7^ is the number percentage in the r* class. The use of the British Standard graticule is illustrated in Table 3.1. The magnification is set so that the largest particle present in the field of view is smaller than circle 7. The magnification is then increased in order to accurately size the smaller particles. If there are particles present, which are smaller than the smallest circle on the graticule, a second increase in magnification may be necessary. Particles smaller than 2.3 |im are usually considered below the limit of resolution of the technique although this limit can be extended with loss of precision in the smaller size ranges. Sizes are often selected as an extension of sieve sizes i.e. 75, 75(V2/2), 37.5 and SO on. The class that contains the highest number concentration of particles (usually the smallest size class) is taken as the control size class. The dispersion technique should give about 3 particles per field of view for these particles and at least 96 fields of view should be examined. At lower magnifications it is usual both to increase the area of the field of view and decrease the number of fields examined. The expected standard error s{p^ of the percentage p^. by number in each size class, out of the total number in all size classes, is given by:
^,^^,
/M00-£^
(3.4)
where Sm^ is the total number of particles of all size classes. The standard error is a maximum when p^ =50, hence s{p^ will always be less than 2% if m^ > 625. A statistical analysis has been carried out to determine the number of particles that need to be counted in order to construct a 95% confidence band over the whole sample. The measured size was the equivalent area diameter [84].
160 Powder sampling and particle size determination Table 3.1 Illustrative example of the calculation of a size distribution by number using the British Standard globe and circle graticule Number Cumulative Number Expected particles number frequency standard error counted undersize (mm^) SJPr) (urn) i^r) >75 100.0 630 0 \ ^ 7-6 32 X 0.25 X 0.32 75-53 100.0 4 630 99.4 626 6-5 0.45 53-37 (75/8)2 X 0.064 X 8 5-4 37-27 0.0453 = 2.039 98.1 618 0.40 6 >7 19* 97.1 7-6 0.66 27-19 18 612 86.8 6-5 1.05 594 19-13 47 • 5-4 0.96 86.8 13-9.4 128 X 0.50 X 547 39 4-3 9.4-6.6 (26.52/8)2 X 0.06 154 1.71 80.6 508 X 0.0453 = 2.039 3-2 1.95 39.4 354 248 6.6-3.3 <2 1.49 106 <3.3 16.8 106 • At high magnification, circle 7 is mac e equal to 26.52 ^im = 75^2/4 making field area (26.52/8)2 x 0.064 x 0.0453 = 0.0319 mm2 • It is necessary to examine at least 100 fields and count over 625 particles and a quick check reveals that this can be done using half this Circle Size number limits
Total area examined
QfpQ
•
•
At low magnification, circle 7 is made equal to 75 |j.m making field area (75/8)^ x 0.064 x 0.0453 = 2.039 mm^ and, in order to examine at least 25 fields of total area 2.039 mm^, it is necessary to examine 32 fields using a quarter of the graticule area. This is an oversize check that should roughly agree with the count at low magnification.
3.11 Conditions governing a weight size determination The percentage by weight in each class is: [m^d^ /n^a^)
^^=7
(3.5)
Direct conversion of a number count to a mass count leads to unacceptable errors. The omission of a single particle, in the largest size class (of
Image analysis 161 average size 63 |Lim in the previous example) is equivalent to the omission of over 10,000 particles in the smallest class (of average size 2.9 |Lim) i.e. the mass of a 63 ^m particle is more than 10,000 times greater than the mass of a 2.9 |Lim particle. The expected standard error s{qj) of the percentage q^ in each size class, out of the total weight in all size classes, is given by:
s{q,) = -^]\-^
(3.6)
q^ is usually a maximum for the coarsest size range. Assuming this contains 10% by weight of the material and that a standard error of 2% is satisfactory, then m^ = 20 particles. This weight percentage should be calculated for each size class to confirm that in no case does it exceed 2%. For a weight distribution, the number of particles to be counted is governed by the number in the control size class. The number to be counted to achieve a given accuracy is: s(M^) = ^
(3.7)
s(M^) is the standard deviation expressed as a percentage of the total by weight; M^ is the percentage by weight in the given size range; m^ is the number of particles counted in the size range . If it is estimated that 10% by weight of particles lie in the top size range and an accuracy s{M) of 2% is required then it is necessary to count 20 particles in the top size range. In order to cater for errors in the estimate it is suggested that 25 particles be counted. It is The number density (N^) is then calculated (particles mm"^). The total count is maintained at about 700 by reducing the area examined for the smaller sizes. Briefly, the sample area required for particles of size d^ is: '^r^
4 = K^OJ
^
(3.8)
where suffix '0' denotes the control class (the class containing the most particles by weight), which is usually the top size class.
162 Powder sampling and particle size determination
A more comprehensive mathematical procedure to determine the number of particles to be counted in order to minimize errors has been proposed by Masuda and linoya [85]. This procedure mandates a much higher count than the British Standard procedure presented here [86]. 3.11.1 Illustrative example of the calculation of a size distribution by weight This example is based on using the British Standard procedure [11]. At minimum magnification (circle 1 = 15 \\m) the number of scans required to give 150 particles in the three top size categories is determined (Table 3.2). Number of scans n^= 5 Length of scan = 10 mm Width of scan = (64/8) x 72 |im - 0.60 mm Area scanned A^ = n^a^ = 5 x 10 x 0.60 = 30 mm^ Number of particles recorded = m^ Table 3.2 Preliminary examination of 5 scans, area 30 mm^ Class
Number of particles {m^ 9 47 103
Size limits (jim)
1 2 3
106-75 75-53 53-37.5
Number density 0.300 1.567 3.433
Number of scans required in order to count 25 particles in the top size class «j=(25/9)x5 « 14. For other classes: f ^ ^6 "1
class 2:
class 3:
"2 =
V^,
1.567 x l 4 « 1 0 scans 0.300
V 3.433 0.300
x l 4 » 3 scans
Image analysis 163 Nine more scans are completed for the top (control, «i= n^ class size, during five of which particles in the second size class are recorded. The modified A^ values are determined and the above calculations repeated to ensure that a sufficient area has been scanned. The process is then repeated at a higher magnification (Table 3.3). Calibration for class 4 is that 60.4 units = 283 |um making the diameter of circle 7 equal to: (8/60.4) x 283 = 37.5 |Lim. Area of graticule - ^ X ^ X 0.283^ = 0.0636 mm^ 60.4 60.4 In class 6, particle density is high hence half the graticule is examined. The requisite number of fields is examined and the calculations carried out to ensure a sufficient area has been covered. The process is repeated at a higher magnification (Table 3.4). Circle 7 is made equal to 13.25 |Lim making the distance between the calibration marks 100 |Lim and the field area 0.00795 mm^. 75 more fields need to be examined for class 7 only. Table 3.3 Preliminary examination of 25 fields of total area 1.59 mm^
Class
Size limits
4 5 6
37.5-26.5 26.5-18.8 18.8-13.3
Number counted
(|im)
21 44 79
13.2 27.7 99.4
Required area (mm)^ [equation (3.7)] {Ar) 6.6 =107 fields 1.77 = 28 fields 0.79 = 25 X (0.5) fields
Table 3.4 Preliminary examination of 25 x (1/4) fields of total area 0.0497 mm^ Class
Size limits
7 8 9 10 11
13.3-9.4 9.4 - 6.6 6.6-4.7 4.7-3.3 3.3-2.3
()Lim)
Number counted (w^) 9 8 7 6 4
N
= - ^ 181 161 141 121 81
Required area (mm)^ [equafion (3.7)] {A,) 0.193 0.020 < 0.020 < 0.020 < 0.020
164 Powder sampling and particle size determination
The data are tabulated and the standard error of the control size class determined to ensure that it is less than 2%. An accuracy factor is calculated for each size class and provided it is always less than the value for the control size class the standard error for the other size classes will be better than this. The completed table is shown as Table 3.5. Although the technique appears onerous, a skilled microscopist can carry out a weight analysis in about an hour. 3.12 Semi-automatic aids to microscopy Semi-automatic aids to counting and sizing have been developed to speed up analyses and reduce the tedium of wholly manual methods. The advantage of these aids over fully automatic systems was that human judgment was retained. The operator could select or reject particles, separate out agglomerates, and discriminate over the choice of fields of view. Many such aids were developed and these differed widely over the degree of sophistication, price, ease of use, mode and speed of operation; they have however been supplanted by quantitative image analyzers. The Zeiss-Endter analyzer [87,88] allowed a direct comparison between the projected area of the particle and the area of a reference circle that consisted of a spot of light adjustable in size by an iris diaphragm. The instrument was designed to work with a photomicrograph that could be obtained from an electron microscope to extend the lower size down to around 0.01 |Lim. Exnor et. al [89] applied the instrument to size and shape determination of lead powder. A modified instrument, that was rugged and simpler but not as versatile, was described by Becher [90]. Crowl [91] used a projector, a transparent screen and a large transparent graticule to facilitate sizing from electron micrographs. The size was recorded via an electrical contact as the appropriate circle was touched by an electrical contact. The basic module for the Digiplan [92] was an electronic planimeter with a built-in microprocessor. The image structure under analysis was traced out and stored in different count channels. Measured parameters were area and lengths that could be extended to other functions such as maximum diameter, form factor, centroid and so on. The Chatfield comparator [93] was devised for the size classification of sub-10 |Lim particles from 35 mm film records and was based on the projection of a photograph on to a translucent screen which was back illuminated by a variable size light spot. When a match was made the operator used a foot switch to record the size.
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166 Powder sampling and particle size determination
The Lark counter [94] was designed for measuring particles in a loose powder, recording the sizes by the use of a series of pinholes in a chart. By ruling lines on the chart corresponding to a scale of sizes, and counting the pinholes between them, the size distribution in terms of the Feret diameter could be determined at a rate of about a thousand particles per hour. The Watson eyepiece [95-97] used an image shearing principle whereby two images of the particle were produced and separated by the use of rotating mirrors. The distance moved was recorded so that the particle (shear) size could be found. An improved optical and mechanical system was claimed for the Vickers image splitting eyepiece in which the mirrors were replaced by prisms [98].Timbrell [99] modified a normal microscope optical system by the introduction of a small mirror to reflect the light from the objective into the eyepiece. The mirror was mounted on the diaphragm of a loudspeaker vibrating at 50 Hz and illuminated at the extremities of the vibration. Because of the persistence of vision two images were produced. The energizing current controlled the amplitude of the mirror so that this was therefore proportional to particle size. The system gave high precision down to 2 |Lim [100]. A particular feature of Timbrell's device was that the images could be made to rotate in any direction so that they could actually 'dance' around each other thus permitting measurement of maximum and minimum diameters. The images could also be made to vibrate at two different amplitudes so that the overlap varied by down to 2 |Lim so that particles could be classified into narrow size categories [82]. An application of this system to difficult particulates has been presented by VQYvyet.aL [101]. Powder blends used for the manufacture of pharmaceutical products often consist of many ingredients and homogeneity can only be determined by using a technique that can discriminate between the ingredients. Manual microscopy is usually the answer where individual particles are identified and placed into size classes by comparing their images with a British Standard eyepiece graticule [102]. Manual methods are slow and tedious and automated methods, such as image analysis, are not always preferable. The Zeiss AxioHOME (Highly Optimized Microscope Environment) was developed by pathologists to count, measure and analyze cell structure in biological thin sections. The AxioHOME is a light microscope coupled to a personal computer that allows the microscopist to make measurements on particles whilst still observing the real image. It is highly suited to particle size analysis because the measurements can be exported directly to a spreadsheet [103].
Image analysis 167 3.13 Automatic aids to microscopy The need to count and size large numbers of samples of airborne dust stimulated the development of automatic microscopes. These instruments may be classified as spot or slit scanners [104]. The spot scan methods [105-107] were based on the flying spot microscope of Roberts and Young [108] In this the scan was produced by a moving spot of light from a cathode ray tube which was projected through a microscope on to a specimen. When a particle interrupted the beam a photocell was activated and the particle counted. Special memory devices prevented the same particle from being counted twice [109]. However, certain designs of instrument were considered suspect, and there was always the possibility of re-entrant particles being counted twice [110]. In some instruments direct scanning was abandoned in favor of scanning a photographic image [111]. This extended the range of the instrument down to about 0.08 |im but added an extra operation [91]. The theory of slit scanning has been covered by Hawksley [112] and the procedure was implemented in a commercial instrument [113]. In operation an image of the sample slide is projected on to a slit using a conventional microscope. The sHde is mechanically scanned and the signals produced as the particle images pass over the slit are recorded. The reproducibility and accuracy were very good for spherical opaque particles down to 2 |Lim in size [114]. 3.13.1 Beckman Coulter RapidVUE The RapidVUE analyzer can characterize fibers, powders, crystals, polymers and other materials at rates of 1200 particles per second. In the RapidVUE, fibers tend to be aligned with the direction of flow, making possible good measurements of fiber length, width and aspect ratio. The analyzable size range is 20 |im to 2500 |Lim. 3.13.2 Micromeretics OptiSizer PSDA™ 5400 The OptiSizer^^ PSDA System delivers continuous, on-line particle size analysis for materials with particles 45 ]xm or greater. It consists of a CCD camera (with an internal image processor), a vibratory feeder, light source, monitor, keyboard and mouse. The camera captures high-speed images of the particles as they pass from the feeder through a controlled light source. The images are digitized and processed by the internal CPU. The software converts collected data into particle counts, diameters, areas and volumes. Form factoring is also available based on particle shape. Resuhs are
168 Powder sampling and particle size determination
displayed in both tabular and graphical formats. Digital outputs enable online statistical process control based on minimum and maximum particle size means. If the material varies from the acceptable range, an alarm can be triggered or it can be configured to control the process equipment. 3.13.3 Oxford VisiSizer VisiSizer is a direct imaging system that combines a pulsed laser to freeze subject motion, a digital camera to capture frames at high or low speed and a computer to display and analyze images. The VisiSizer range offers image exposures from 30 nanoseconds to 1 microsecond, image capture from 30 to 10,000 frames per second, with digital cameras of varying resolution. Software uses Particle Droplet (including bubbles) Image Analysis to rapidly analyze particles and droplets. Images can be analyzed directly or fed to a hard disc. Software analyzes 3 frames per second and the laser is pulsed twice per frame to give velocity as well as size. Slightly out of focus images can be accurately measured by examining edge gradient. 3.13.4 Retsch Camsizer The Camsizer measuring system is based on digital image processing. The dry powder flows between a light source and camera and the particles are detected as projected areas, digitized and processed. During the measurements two digital full-frame cameras perform the particle analysis: The basic camera detects the large particles and a zoom camera registers the small. The theoretical measurement limits are 15 |Lim to 90 mm with a practical range of 30 |Lim to 30 mm. Electronic processing of the images takes place internally in 10,000 size classes and saved in a 1000 size classes. Graphical representations of the results are available while the measurement process is still running. The computer calculates all standard particle distributions together with microscope shape factors such as Martin and Feret diameter, longest chord and so on. 3.13.5 Malvern Sysmex Flow Particle Image Analyzer FPIA-2100 automated particle shape and size analyzer Particles are sampled from a dilute suspension and held in an agitated chamber to ensure it is kept in suspension. The particles are introduced into a sheath flow through a jet nozzle. The sheath flow transforms the suspension into a flat, narrow stream through hydrodynamic focusing. The
Image analysis 169 sheath flow ensures that the largest projected area of the particle is oriented towards the video camera and all the particles are in focus. The cell is illuminated with a stroboscope and images of the particle are captured at 30 Hz. The particle images are processed in real time by digitizing, edge highlighting, binarization, edge extraction, edge tracing and image storage. If there are 7 particles per image a total of 25,000 particles are captured and analyzed. Typically, comprehensive particle data are generated in less than 5 minutes. A computer calculates the area and perimeter of each of the captured particle images and then calculates the particle diameter and circularity. A high power field (x20) covers the particle size range 0.7 |Lim to 40 |Lim in 3 size classes and a low power field (x5) covers the size range 4 |Lim to 160 |Lim in 4 size classes. Once the measurement is complete, particle size and distribution data are displayed in graphical and tabular form. A typical result report includes three plots; particle size distribution, circularity distribution, and a scattergram of particle size versus circularity. Individual particle images are captured and displayed and these can be classified by the operator into various categories. The instrument is covered by US patent No 5,721,433. 3.13.6 Sci-Tec Part An - video Image Analyser Particles are dropped between a video camera and a synchronized strobe light. When the strobe flashes, the camera takes an image of the particle which is digitized by a computer frame grabber. For samples with a wide size range two cameras can be used simultaneously, one with high magnification and one with low. Particles are measured for area, perimeter and minimum and maximum diameters. From these measurements shape factors, aspect ratios and volumes are calculated. Thousands of particles can be measured in each digital image and images can be acquired at 20 frames per second. Total analysis time is 3 to 5 minutes. Size and shape are displayed in particle size and sieve size bins. Shape and aspect ratios are displayed by size fraction. Totals and averages are also displayed. Size range covered is from 10 jum to 6 inches. 3.14 Quantitative image analysis Manual methods of obtaining data from images are slow and tedious and this can give rise to considerable error. The introduction of fully automated image analysis systems has virtually eliminated manual methods and also supplanted semi-automatic systems. All image analysis systems use scanning techniques for converting images into electrical signals that are
170 Powder sampling and particle size determination
processed to yield data on the images. If the microscope is fitted with an automatic stepping stage and autofocus, it is possible to measure large numbers of fields of view, thus greatly improving the statistical accuracy. However, image analysis systems cannot discriminate artifacts as readily as human operators and are unable to adjust focus during measurements in the field of view, which can cause problems with particulate systems having a wide size range. With automatic image analysis, not only is it necessary for particles to be singly dispersed, they should also be clearly separated from each other so that the analyzer is not confused by touching particles [115]. 3.14 1 Calibration of image analyzers The procedure for calibration of image analyzers varies from machine to machine. It usually involves indicating on a screen the dimensions of an imaged artifact of known dimensions. This may be a grid, grating, micrometer or ruler and the dimensions are usually expressed as piXEls per unit length. The calibration should be carried out both parallel to, and perpendicular to, the scan direction. Image analyzers may also suffer from localized distortion, which may be detected by comparing a square grid with an overlaid software generated pattern. The National Physical Laboratory [21] introduced a certified graticule to test the linearity of a scanner over the whole field, the resolution obtained and the effect of gray level detection on the measured size distribution. It has four separate fields: A major field divided into smaller proportional fields, an array of equidistant monosize circles, a root-two by diameter array and a log-normal number/diameter distribution in an equi-centered array. The value of this, commercially available graticule has been illustrated by its use on a Quantimet 900 [22]. Polystyrene spheres (5, 7, 10 and 15 |Lim) have also been used for calibration and the results compared with SEM and Coulter data [116]. 3.14.2 Experimental procedures Automatic image analysis is a six step process (Figure 3.4); (1) image formation, (2) image scanning, (3) feature detection, (4) feature analysis by count, shape, size or other selected parameter, (5) data processing and analyzing; (6) data presentation. Image formation is a crucial step in image analysis. Quantitative image analyzers consist of a high linearity television camera that can be interfaced with a microscope, macroviewer or videotape. An electron probe interface
Image analysis 171 can be used to accept image inputs from scanning electron microscopes, microprobes or acoustic microscopes [118-120] The macroviewer is used to analyze large objects such as 8 in by 10 in photographs. Large objects and photographs can be illuminated with incident light; transparencies and slides by transmitted light. The electron probe stores whole images generated by the SEM and converts them into a proper form for analysis. Signals from the image received by the camera are processed by a central processing unit that contains circuits for measuring areas of features, number counts and size distributions based on selected diameters. Parameters such as ratio of maximum to minimum diameters may also be determined. The area under examination is displayed on a screen and interaction is either via a teletype keyboard or menu driven. Using this unit, objects can be selected for examination, objects can be deleted and touching particles can be separated. Image editing and classification features such as picture enhancement using gray level detection to enhance contrast are often available (Figure 3.5). Surface texture analysis, fractal dimension determination, shape regeneration using Fourier analysis, angularity and roundness measurement may also be carried out. The resulting distributions are presented graphically in a wide variety of modes and the data may be stored for further interrogation. To outputting devices
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Without shading correction 1009fr50%10%:: 0%- ' h - - h - - 4 - ~ h - h - ' h H - - l - - h h With shading correction Fig. 3.5 Gray level shading correction the objective of the gray level detection function is to define in the images the boundaries of features that are to be counted and measured (Bausch & Lomb, Omnicon). Images require a great deal of computer memory because an image is actually an array of numbers; i.e. intensity at every location / = j{x,y). For a 512 by 512 digital image, over a quarter of a million numbers must be stored. Consequently, until recently, most image analysis was done on large computers or with dedicated image processing hardw^are. It is now possible to do a great deal of processing on a workstation computer. Most digital image operations fall into one of two categories [121] namely image processing and image analysis. Image processing operations turn one image into another image; for example, edge finding or crispening. Operations that quantify some aspect of the image, such as area fraction or average particle size, are called image analysis. In general one would prefer to do little or no processing in order to save time and not alter the original image. However it is often necessary to do some processing to prepare an image for analysis [122]. An image can be digitized from a video camera attached to a light microscope or on a copy stand to digitize either the particles themselves or a photomicrograph. Images can also be digitized from one of the contrast-bearing signals from a microscope. Once digitized, the images can be stored for later analysis and processing or for archival purposes. A digital image has a discrete number of pixels (picture elements), each o{ which contains an intensity value, usually scaled from 0-255. The
Image analysis 173 number of pixels determines the resolution with 512 by 512 being the most common. Theoretically, image resolution is limited only by computer memory; however it is usually not possible to display more than 1024 by 1024 and PC class computers may not be able to display 512 by 512. For most practical purposes 512 by 512 is adequate but higher resolution is useful for analyzing large and small features at the same time. Each pixel in the image is digitized to a limited number of gray levels depending on the hardware used. The human eye has a range of 1010 brightness levels that it can adapt to but it can only discriminate about 20 levels at a time. Older image systems provide 64 or fewer levels but newer ones commonly provide 256. Such a large number of gray levels is necessary for discriminating phases on the basis of their brightness but may not be necessary for particle characterization. The computer compares several focal planes and the sharpest image selected. The software then corrects the brightness until it is approximately constant over the whole image. The image is then binarized: All pixels below a certain gray level become white and the rest become black or vice versa; with this procedure particles can be differentiated from background. The height of the binarization threshold has an influence on the number of black pixels and its selection is a multiple step procedure [123]. Some of the binary particles may contain bright spots. This happens especially with transparent particles in transmission microscopy. These spots have to be closed. Although the human eye can only discriminate about 20 levels of intensity simultaneously, it can distinguish about 350,000 different shades of color. The significance of this is that pseudo color can be used in an image to convey details that would otherwise be lost in a gray scale image. In general, the goal of image processing is improving the image, but the only thing certain is that the image is changed. IMo information can be extracted that was not present in the original image and artifacts can be introduced. One example of a need for image processing would be separating overlapping particles (Figure 3.6). •
•
Resolution is determined by the number of pixels (picture elements): The measured particle area is the number of elements multiplied by the elemental area; the particle perimeter is the number of edge element multiplied by the length of the sides. Erosion In order to separate touching images, the edge pixels are removed together with any touching pixels within the image.
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• •
The residual pixels make up smaller non-touching images; the smaller image disappears completely. Dilation In order to create new non-touching images a peripheral layer of pixels is apphed.
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175
/ 76 Powder sampling and particle size determination
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Image analysis 177
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Erosion and dilation results in some loss of information. Simple operations such as erosion and dilation or complex operations such as artificial intelligence can accomplish this task. If these fail, the operator, using a light pen, can cut features apart. Gray level erosion and dilation has been studied to characterize the particle size of powders in bulk. The procedure was tested using mixtures of steel balls, from 0.3 to 1 mm, in diameter and ground pea kernels [125]. The results, using a Trydin image analysis system, agreed reasonable well with laser diffraction data. Other forms of image processing operate on the entire gray scale image and transform the contrast. Some uses for these operations include background smoothing, crispening, edge detection, noise removal and the enhancement or suppression of periodic information. The systems available today are both more sophisticated and simpler than before. The trend is towards more automation, greatly increased speed, more sensitivity and improved camera technology, particularly the useofCCDs. Specifically, many systems are based around or incorporate parallel processing hardware so that image pixels can be processed simultaneously, providing faster throughput and allowing highly complex algorithms to be supported. Fibers can be aligned, touching particles can be separated, particles can be selected for measurement and deleted holes can be filled. A light pen is often provided so that features can be selected for acceptance or rejection. Particles can be dilated or eroded for image enhancement, and fibers may be stretched or shrunk to create a better image. The application of quantitative image analysis to size (area) and perimeter determination is demonstrated in Figure 3.7. The image analyzer sees only two images since the larger images overlap. The larger image The application of quantitative image analysis to size (area) and perimeter determination is demonstrated in Figure 3.7. The image analyzer sees only two images since the larger images overlap. The larger image (Figure 3.7a) has an area of 552 square units made up of 325 LHS (left hand side) and 227 RHS (right hand side) and the smaller image has an area of 66 square units (assuming the area occupied by one pixel is one square unit). The pixels that overlap the edges of the image define the perimeters (Figure 3.7b). The larger particle(s) have a perimeter of 134 units LHS and 73 units RHS and the smaller particle has a perimeter of 36 units. Thus, as in fractal geometry, the smaller the pixel the greater the perimeter. The contour following algorithm depends upon the chosen connectivity of the pixels (i.e. the number of touching pixels within the image). Some other parameters that can be measured by quantitative image microscopy are illustrated in Figure 3.8; shape factors are illustrated in
Image analysis 179 Figure 3.9 and editing features in Figure 3.10. Serra [126-128] developed the method of mathematical morphology in image analysis and the theory was extended by Matheron [129,130]. The basic operation involves erosion and dilation procedures to produce a modified image picture from the original. The major disadvantage of this procedure is the requirement for large memory space. This problem was resolved by use of a compressed data technique for binary image pictures [131]. Davidson et. al [132] compare image analysis to other methods of particle size measurement. They used a Magiscan image analyzer with Genias^^ particle sizing software. The images were generated with a Zeiss universal microscope equipped with Optovar and bright field optics and acquired using a Dage-TI camera model 70 with a green filter. Calibration was with a Bausch & Lomb ruled stage graticule. The gray levels were converted to binary and touching particles were separated manually. Various parameters were measured and data compared with data generated by sieving, using a Hiac PA 720 (operated with dry powder) and the Coulter Counter TA II. It was deduced that the cut size for the sieves was controlled by the breadth of the particle (as expected) and the measured parameter by Hiac was also controlled by breadth rather than equivalent spherical diameter as one would expect. If multiple pictures of the particles, taken from different directions, are available then a three dimensional image can be reconstructed and the 3-D convex hull reconstructed [133]. Automated image analysis has also been used to characterize the degree of mixing in a drum mixer. Different colored glass beads were placed at the front and rear end of the mixer and were set in place after mixing using solidification techniques. The bed was then sliced and the slices scanned with a video camera to produce digital images. The mixture quality of elements within each slice was next determined using an image analyzer. This procedure enabled a spacial determination of mixture quality both along the axis and within bed depth [134]. For automatic image analysis of fibers it is necessary to generate an image with a limited number of overlapping fibers. A method of generating such a distribution with dry fibers by impacting the particles on an inclined plate and subsequent sedimentation on to a microscope slide has been described [135]. For wet sample preparation the suspension was first dispersed, then deposited on a membrane filter for examination by reflected light or transmitted light after first making the filter transparent by a chemical agent. An alternative method of preparation, by allowing the fibers to settle out of suspension on to a microscope slide, was also used.
180 Powder sampling and particle size determination
Schafer, [136] in a discussion on the accuracy of image analysis systems, states that the accuracy of area and equivalent diameter measurements is sufficient for most practical purposes. On the other hand he suggests improvements in the determination of perimeters and shape factors which he found to be unsatisfactory. 3.14.3 Commercial quantitative image analysis systems American Innovation Videometric 150 readily measures and counts specific features in an image. The system incorporates threshold monitoring so that images within selected brightness levels are selected. Editing is provided with the use of an adjustable paintbrush with the mouse to include or exclude any parts of the image. In this way objects that appear separated may be connected or objects that wrongly appear as one may be separated. The erosion routine reduces every detected feature by one pixel at a time over the entire boundary. This process breaks bridges and opens pores. During dilation pixels are added one at a time to smooth surfaces and fill in bays. Analytical Measuring Systems features Quickstep, a sophisticated control system for precise positioning of microscope stages and macrostages in X, 7 and Z axes with the added facility for automatic focus. As a 'standalone' system it offers versatile semi-automatic and fully automatic control for scanning selected areas of a specimen. Alternatively it can be used in conjunction with a microcomputer to run application programs or as an integral part of an AMS image analysis system. VIDS V is an interactive image analyzer that provides a means of quantifying images not suitable for gray level analysis. An image, which can be viewed in color or monochrome, may be analyzed by tracing around, or along, features; or by placing dots across them using a digitizing pad and cursor. The standard software provides all the measurements and statistical tests. A low cost alternative to VIDS V is found in Measuremouse, which features a high resolution CCD camera to view the sample and display the image on its graphic monitor. Objects selected for measurement are inscribed using a mouse-driven cursor and their size and shape determined via an Amstrad personal computer. Comparison of measured parameters and the generation of size or shape distributions are carried out using spreadsheets such as Super Calc and the results provided via a high speed dot matrix printer. Optomax V is a fully automated image analysis system with a high spatial precision of 704 by 560 pixels. These systems are available in the USA from Optomax Inc.
Image analysis
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Artek market a range of image analyzers. The Artek Omnicon 3600 is an advanced, easy to use system with turnkey application software. A precision scanner converts optical images into video signals that are presented on a high resolution monitor screen. The required measurement is selected from more than 20 parameters. The measure key is pushed and the answer appears ready to be printed out, saved or exported to any of a wide range of data bases and spreadsheet programs Automatix produces automated image analysis packages comprising data collection, spreadsheet analysis and charting to be used with the Macintosh computer. This has been used, in conjunction with ancillary equipment, to produce a computer digital analysis system [137] for around $4,000 [138]. Later, a more sophisticated system, costing around $20,000, was described for digital examination of film. Boeckeler manufacture simple systems which require more operator input but are less expensive than more automated systems. The VIA-20 Video Image Marker can be used to mark details on a video image; the VIA-50 includes a more varied array of positionable markers with an option for multiple overlay storage; the VIA-100 can perform horizontal, vertical and point-to-point measurements, and the VIA-150 combines the capabilities of the VIA-50 and VIA-100. The VIA-160A1 video area measurement system can be combined with a video microscopy system to give cell areas, chord lengths and number counts. Buehler Omnimet II is a high resolution automatic image analysis system. Compix C-Imaging 1280 System has an image input of 1024 x 1024 resolution and uses the latest hardware to achieve high speed processing and analysis. The 1280 x 1024 non-interlaced 19 in. color monitor provides viewing quality for user comfort. The system is complemented by Simple© imaging software to give user flexibility in an uncomplicated framework. The system can enhance, identify and count 1000 objects in less than 1 s, measure the data and perform summary statistics in less than 3 s, and provide histograms at the touch of a button. The C-Imaging 640 system has a color or monochrome input with a resolution of 640 by 640. The system benefits from high speed processing and analysis hardware. Data Translation market Global Lab Image that captures and digitizes images, then displays, processes and analyzes them. A complete set of algorithms is provided which work with any imaging device. The programs are used with Microsoft Windows^^ for scientific word processing, statistics, spreadsheet analysis or data plotting. Hamamatsu C-IOOO is a computer compatible video camera that can be mounted on a microscope, macroviewer, optical bench or tripod. The
182 Powder sampling and particle size determination
number of particles can be tabulated based on specified parameters. Information can be input and analyzed by the computer. Hitech Olympus Cue-3 is a color image analysis system with a wide range of image enhancement and processing capabilities. Joyce Loebl Magiscan is a total image analysis system which can be interfaced to a wide range of optical and electron microscopes. The general purpose menu and results programs provide flexible means of extracting particle size and shape information [139]. Leco offer several image analysis systems. The AMF-100 is dedicated to microhardness measurement and the 2005 is a top of the line, high performance system. The 2001 is a mid-range general purpose image analyzer. The 3001 operates using Microsoft Windows^"^. The system can interface with a broad range of peripherals and has the ability to archive images for later processing. Leco IA32 is a versatile image analyzer suitable for users with varying levels of expertise. Although designed primarily for the metallographer its other applications include the measurement of porosity, grain size, area fractions, number count, particle size, fiber length and dendrite arm spacing. Leica Quantimet 500 costs less than $15,000 complete and can be used with any microscope. The single display combines menus and images through PC based Windows^"^ graphical user interface. LeMont Oasy image analysis system acquires, enhances, measures and classifies images. The Omega system forms a high quality bridge between a PC and SEM. Maztech Microvisions Spy grain grader is an image analysis benchtop instrument that uses linear CCD arrays to determine objective grade and non-grade factors in grains and seeds [140]. Millipore nMC System offers speed and precision in counting particles, determining size distributions and characterizing shapes. Nachet 1500 is a processor specifically designed for image analysis. The image is acquired with a TV camera and directed to the Nachet 1500 for image processing and measurements. The results are sent to an Apple He type computer to drive the various peripheral devices associated with the topic being studied. The microcomputer is also used for setting up the instruction sequence for image processing and for running programs. Nikon analytical microscopy workstation Microphot SA is adaptable for complex multi-imaging requirements run from a host computer. Oncor Instrument Systems (Tormerly American Innovision) designs and manufactures its own hardware and software which incorporates multiple
Image analysis
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true color analysis models, multiple thresholds and low light true color CCD cameras. Optomax V is a simple to use automatic image analysis system employing dual microprocessors combined with high speed measurement circuitry. The system uses up to four high performance TV cameras to provide direct viewing of macroscopic or microscopic images. It counts and measures image features such as area, perimeter, Feret diameters, horizontal and vertical intercepts and so on. Outokumpu Imagist is a medium priced unit which bridges the gap between the limited capability PC systems and the high cost, hardware based systems. PharmaVision 830 (Malvern) measures dry powders in the size range 0.5 JLim to 2000 |im. A range of optical configurations is available together with a range of presentation plates (microscope slides). A comprehensive list of size and shape parameters can be measured together with custom parameters. Shapespeare Corporation's Juliet features single particle size and shape measurement together with fractal analysis as well as conventional classical measurements. It also measures size and shape distributions for sets of particles with image manipulation and data graphing capability. Tracor Northern manufacture complete x-ray microanalysis systems with video image collection and storage. The TN-8500 Image Analysis System features dedicated imaging hardware and software combined in an integrated system for advanced applications from the simplest particle size analysis to the most complex imaging applications such as fast Fourier transformation and 3-D reconstruction. Carl Zeiss offer a family of upgradeable products including parallel processing software. The Videoplan-Vidas-Ibas line of digital image processing systems offers three different levels of automation. The videoplan takes advantage of the user's experience in recognizing complex structures. Measurements are performed by tracing contours with a cursor on a digitizing tablet or on a TV on-line image. The evaluation is supported by automatic data processing which includes data management. The Vidas system digitizes the images, which are then processed by Ibas. The Ibas system supports advanced procedures such as texture analysis and pattern recognition. 3.14.4 Confocal laser-scanning microscopy(CLSM) In the CLSM a laser illuminates a pinhole that functions as a point light source. The divergent beam from this source is focused on the surface or
184 Powder sampling and particle size determination
inside the particle to be examined. Reflected light from this location travels back to the lens, is focused and brought to the analyzer pinhole by means of a semi-transparent mirror and thence to a photomultiplier. By moving the illumination spot through the analyzed volume the complete particle can be registered in all three dimensions. The method not only has the advantage of a three-dimensional registration of objects but also a considerable increased optical resolution. The Leica CLSMg\\QS a lateral resolution of 0.25 jum. (CLSM) has been used to generate three-dimensional information on particle size, shape and porosity [141]. The CLSM has been used to measure particle size distribution in situ and ex situ using computer based image analysis system [142]. A model was developed to assess processing conditions to produce a floe with desirable characteristics in an enhanced actinide removal. Ferreira et.al. [143] present some additional methods of measuring wood pulp fibers and compares these with data from CLSM 3.14.5 On-line microscopy On-line systems, covered more fully in Chapter 10, have been developed for measuring particle contamination in molten polymers [144] and for measuring particle size and shape. Microscopy is combined with a lightscanning device for particle size and shape determination in the Galai instrument. Castellini et. al. [145] use a pulsed semiconductor laser as a light source to illuminate a flowing stream of particles, in the 2 to 400 |im size range, and determine their shape using a shadowgraph technique. Herpfer and Jeng [146] used streak particle imaging velocimetry for planar measurement of droplet velocities and sizes. They encountered problems due to out-of-focus effects of the PIV imaging procedure. Kato et. al. [147] applied flow visualization and image processing to measure the velocity and size of glass beads of size, 50, 100 and 200 |Lim, falling through a pipe using a stereo-imaging technique. Malot and Blaisot [148] determined particle size and sphericity of low density sprays by image analysis. Particle size distribution, in the 125 to 2000 |im size range, was determined by incoherent back-light imaging and shape by morphological analysis. Blandin et. al. [149] measured agglomeration in a crystallizer using a video camera focused on a thick black screen inside the crystallizer and connected to an image analysis system. This system films and measures particles circulating between the screen and the transparent crystallizer wall.
Image analysis
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Danfoss QueCheck Vision System performs a continuous analysis directly from the production line, typically at 0.5 s intervals. The final results of the measurement are available after 100-300 frames so that an equivalent sieve analysis is completed every 3 min. The particle size distribution is documented via an interface with database and printer. The material is fed in a fine stream, by means of a vibratory feeder, past a vision camera that calculates the size distribution as it falls. The system has been applied to measuring the size distribution of sugar crystals. Online image processing has also been applied to monitoring granule size distribution and shape in fluidized bed granulation [150]. 3.14.6 Flatbed scanners The size range of most CCD-based imaging systems does not exceed two orders of magnitude. A sensor array of 500 by 500 pixels does not give a correct reproduction of objects smaller than 4 by 4 pixels and any object greater than 500 pixels will be larger than the imaging area. Due to this limitation it is necessary to examine a large number of images for samples having a wide size distribution. Optical systems have the further limitation in that large objects are not in focus at the same magnification as small ones. Flatbed scanners offer a large imaging area, usually 12 by 8 inches, an optical resolution of at least 300 pixels per inch and grayscale or color imaging. Scanner images therefore contain far more information than images from a CCD camera. A black and white full page image at 1200 ppi is equivalent to 500 images taken with a 500 by 500 pixel camera. However optical magnification is not possible and this limits the scanner to a size range of two orders in magnitude. This system is particularly suitable to determine the size and shape of fragile agglomerates of irregular shape with a wide range of sizes [151]. 3.14.7 Dark field microscopy DFM is considered to be the most sensitive light microscope method. The light is directed through an outer ring on to a mirror (Figure 3.11) and reflected on to the surface. An object on the surface will reflect light back into an inner tube and focused with a lens to the observer. If the surface is clean the rays are reflected back into the outer ring and the surface looks dark. Best results are obtained using sample carriers with a flat polished surface.
186 Powder sampling and particle size determination
Fig. 3.11 Paths of rays in dark field microscopy To get information about the lowest detection limit Schmidt et. al [152] carried out experiments using monosize, spherical, fluorescent particles of known diameter. They deposited the particles on a wafer surface and observed them with a fluorescent and DFM. With both methods the light off the particles was detected and the sub-|im particles appeared larger than using a usual light microscope with a lower detection limit of 0.08 |Lim. Schmidt et. al [153] investigated the use of dark field microscopy (DFM) to measure fibrous particles. Large areas were examined for statistical reliability. A high correlation was found between DFM and light microscopy for fiber lengths greater than 1 |Lim. 3.14.8 Phase contrast microscopy In phase contrast microscopy particles are sampled on to a membrane filter that is treated with acetone and glycerin triacetate to make the filter transparent. Problems associated with the preparation step, and low depth of focus limit thus the applicability of the method [153]. 3.14.9 Polarized light microscopy (PLM) The morphology and size of high explosive crystal grains are known to affect their processing characteristics and shock sensitivity. Particle size distributions are normally obtained by light diffraction assuming a spherical model. PLM and SEM reveal this assumption to be incorrect. By combining PLM, SEM and LD a quantitative measure of size and morphology can be obtained. Mean sizes and aspect ratios are determined and refined to give best fit to LD data and a comparison made between these three procedures [154].
Image analysis 187 3.14 10 Dipix 1440Fpower scope imaging microscope The 1440F is a fluorescent microscope, powered by a high intensity mercury arc lamp, coupled with a high speed image analyzer which was designed specifically for automatic image analysis of agriculture and food products. Certain molecules absorb energy when excited by high intensity radiation. The excited molecules emit a portion of the absorbed energy when returning to ground state. Two types of spectra are associated with molecules which fluoresce - excitation and emission. The spectra have characteristic shapes and wavelength maxima that are specific to the absorbing components. This unique "fingerprinf feature can be used for identification purposes. 3.14 11 Transmission wide field phase contrast microscopy A phase contrast wide field transmission microscope combining the advantages of interferomic and confocal techniques has been developed [155]. Confocal operation is achieved by superimposing speckle illumination of a reference beam in a Mach-Zehnder interferometer with a matched speckle pattern of the object beam. The technique was applied to both dry powders and suspensions and gave good agreement with modelled results. Data acquisition time is less than a millisecond. 3.15 Electron microscopy When a solid is bombarded with high energy electrons the interaction produces secondary electrons (elastic), back-scattered electrons (inelastic), low loss electrons, Auger electrons, photo electrons, electron diffraction, characteristic x-rays, x-ray continuum, light, hole electron pairs and specimen current. These interactions are used to identify the specimen and elements of the specimen and can also be used to physically characterize particulate systems. Transmission electron microscopy (TEM) is used for very fine particles or thin specimens of crystalline materials. The elastically scattered electrons give amplitude contrast which is proportional to mass thickness. The inelastically scattered electrons suffer small energy losses and give high phase contrast. Bright field images form when electrons, which are inelastically scattered through small angles, combine with unscattered electrons, whereas dark field images are formed only from the scattered electrons. Scanning transmission electron microscopy (STEM) allows
188 Powder sampling and particle size determination
simultaneous viewing of both images and this is one of its major advantages. The skill level required to operate and prepare specimens for a TEM is significantly higher than that associated with a SEM [156]. For scanning electron microscopy (SEM) the two most important interactions are: The generation of secondary electrons, which are the result of elastic collisions and typically less than 50 eV. The images formed by these are the most common and are marked by great depth of field. Backscattered electrons, which are less applicable to particle sizing but have niche uses; the contrast is closely related to the atomic number of the sample and typically the voltage is greater than 50 eV. 3.16 Transmission electron microscopy (TEM) TEM is used for the direct examination of particles in the 0.01 to 5 |Lim size range [157]. The TEM operates by flooding the sample with an electron beam, most commonly at 100-200 keV, and generating an image on a fluorescent screen or a photographic plate beyond the sample. Analyses can be performed directly on the screen images, but this ties down the instrument for long periods of time, hence it is more usual to analyze photographic images. TEMs operate in the magnification range from about 600x to l,000,000x. Many particle size studies can be carried out at magnifications of less than 4,000x and several relatively inexpensive instruments are available giving magnifications up to 10,000x. Resolution capability of the best instrument currently available is sub-0.2 nm for line-to-line and approximately 0.2 to 0.3 nm for point-to-point. Calibration is usually effected with narrowly classified polystyrene lattices available from Dow Chemicals or Duke Scientific, but diffraction gratings are necessary for high accuracy work. Single crystals of stable, well-characterized materials such as gold may be used as diffraction gratings and lattice fringe imaging can give direct calibration. Absolute calibration is uncertain to within 5% [158]. Some TEMs have a closed circuit television system fitted so that the images can be fed directly to an automatic image analysis system. 3.16.1 Specimen preparation for TEM Specimens for TEM are often deposited on or in a thin (10 to 20 nm) membrane that rests on a grid. These grids are usually made of copper and
Image analysis
189
form a support for the film, which is usually only self-supporting over a small area (Figure 3.12). Specialty grids are available from a number of suppliers to fit particular needs. Examples would be nylon or beryllium grids for special characterization needs, or locator grids with numbered and/or lettered field markers.
3 mm
Fig. 3.12 Electron microscope grid. Since most materials are opaque to the electron beam, even when only a few hundred nanometers thick, special problems arise in the production of suitable mounted specimens. Specimen support films are usually made of plastic or carbon, though other materials have also been used. Suitable film solutions may be made up of 2% w/v formvar (polyvinyl formal) in ethylene dichloride or chloroform. Films may be produced in the following manner [159,160]. A dish about 20 cm in diameter is filled with clean distilled water and a large circle of 200-mesh steel gauze is placed on the bottom of the dish. A number of grids are placed on the wire gauze, then two drops of the film solution are dropped on to the surface of the water and the film that results after the solvent has evaporated is removed with a needle; this ensures that the water surface is clean. A second film, formed the same way, is removed by raising the wire gauze containing the grids that are then allowed to dry. A pre-examination of the grids in the electron microscope is desirable as this enables dirty films to be rejected, and the film polymerizes in the electron beam, thus greatly increasing its strength. An alternative procedure is to clean a microscope slide with detergent and polish with a clean cloth without rinsing away the detergent, so as to form a hydrophilic layer at the surface, which facilitates stripping. The slide is dipped in a solution of formvar in ethylene dichloride (0.3% to 0.7% depending on the film thickness required) and allowed to drain dry. The film may be floated on to a water surface and mounted on grids as before. If individual grids are required, the film may be cut into small
190 Powder sampling and particle size determination
squares with a needle or razor blade. The mounting operation is easier if a special jig is used [161]. The jig is a brass cylinder about 1 in. long and 0.5 in. diameter with a hole of the same diameter as the specimen drilled and tapped through its axis. A set-screw in the threaded hole is adjusted so that its end is flush with the face of the plug and withdrawn slightly to leave a shallow recess. The specimen grid is held in this recess and the membrane is lifted from the water surface with a wire loop of slightly larger diameter than the jig, surplus water being carefully removed with blotting paper. The wire loop is then lowered over the jig and, when the membrane is dry, the grid is raised by means of the screw, surplus membrane being removed by scoring round the grid with a needle. Special apparatus has also been described for producing plastic films of uniform thickness suitable for the preparation of replicas [162]. Carbon films are prepared under vacuum (10-3 mmHg) by electrical discharge from two pointed hard graphite rods [163]. Films are best deposited on microscope slides cleaned with detergent and placed about 10 to 15 cm from the source. A thickness indicator, consisting of a drop of vacuum oil on a piece of white glazed porcelain, is placed beside the slide. During the discharge, the porcelain not covered with oil takes on a brownish color, changing to a light chocolate shade as the film thickness increases from 5 to 10 nm, the latter being a suitable thickness for general use. Evaporation is completed in about half a second with a current of 50 A. The film may then be floated on to the surface of distilled water and picked up as a whole or scored into small squares. A simple method of producing a suitable specimen is to place a few milligrams of the powder on a microscope slide, add a drop of 1% to 2% w/v suspension of formvar in a suitable solvent and rub out the slide with a glass rod. Further solvent is added if required and the dispersion is spread over the slide and allowed to dry. The film is removed from the slide and mounted on the grid as before. Another method of preparing the sample is to disperse it in linseed oil, which is then thinned with white spirit. The dispersion is spread over a microscope slide that is immersed in white spirit for a few minutes to remove the oil. After drying the slide a thin layer of carbon is deposited on the specimen to form a supporting film. Finally this is floated off on water as before and picked up on a grid for examination [91]. A dispersed sample may also be obtained by means of an aerosol sampling device. A suitable technique is to form a sandwich of plastic film particles and 20 nm thick carbon. The underlying plastic may then be washed away with solvent and the specimen examined after shadowing 159,160,162].
Image analysis 191 A suspension of the powder may be made up and a drop placed on a grid by means of a pipette or hydrometer syringe. However, this often produces an uneven deposit. Spraying the suspension on to the grid often produces a more uniform deposit; several suitable spray guns have been described [157,164]. Timbrell [82] modified the Hamilton and Phelps [165] method for the preparation of transparent profiles to facilitate electron microscopy. The metal film is floated on to a water surface and picked up on a grid. In cases of difficulty, the slide is first dipped into 1% hydrofluoric acid to release the edge of the film and the process is completed in water. Although the metal film is strong enough to be floated off whole, it is preferable to score it into small squares as described earlier and then remove the separate pieces. In order to obtain reliable results for particle size analysis, as many separate grids as possible should be prepared and a large number of electron micrographs taken from each. Many other methods have been used and the preparation of the particle dispersion is the most important aspect of sizing by electron microscopy. In an analysis of the errors involved in electron microscopy, Cartwright and Skidmore [166] examined the optical microscopy specimens produced by thermal precipitation. The specimens were then stripped from the microscope slides and the fines were counted, using an electron microscope. Good agreement was obtained by examining 1000 particles in the electron microscope using about 60 fields of view (60 micrographs) and almost 4000 particles in the optical microscope. To obtain accurate magnification calibration, four overlapping micrographs along the bar of a readily identifiable grid square were taken and the total length of the image of the grid bar was measured from the micrographs directly, using an optical microscope. The surface areas of dust samples as determined by optical and electron microscope have also been compared [167]. Pore size distributions of thin films of AI2O3, as measured by TEM, have also been compared with those determined by gas adsorption/desorption [168]. It has also been suggested that electron microscope gives a truer estimate of surface area than gas adsorption techniques [169]. Further information can be obtained in a recent review of specimen preparation for TEM [170]. Two other effects, which may influence apparent particle size, relate to the effect of the flooding electron beam on the particles. One of these is charging of non-conducting particles, which spreads the beam around the particles and makes sharp focusing impossible. The other is the gradual accumulation of vacuum pump oil (and other things) on the charged sample in the sample chamber. This often appears as a slowly developing skin
192 Powder sampling and particle size determination
around the particle thus increasing its apparent size. A view at reduced magnification will reveal a discolored region. This effect limits the time one can spend in a particular field of view, but the contamination is usually slow enough to not seriously jeopardize the study. The higher vacuum systems found in newer instruments reduces this effect. 3.16.2 Replica and shadowing techniques Replicas are thin films of electron-transparent material that are cast on opaque specimens in order that their surface structure may be studied. The basic procedure is to form a film on the substance to be examined, separate the two and examine the film. If a reverse of the original is unsatisfactory, a positive replica may be obtained by repeating the process. One method is to deposit the specimen on a formvar-covered grid, vacuum-deposit about lOnm of carbon, remove the formvar by rinsing with chloroform and finally remove the specimen with a suitable solvent [171]. Instead of a backing film, it is sometimes possible to prepare a carbon replica of a dried suspension deposited on a microscope slide, the replica being washed off the slide in a water bath or a bath of hydrofluoric acid. The carbon film may be strengthened immediately after being deposited by dipping the slide in a 2% w/v solution of Bedacryl 122X in benzene, which is removed by a suitable solvent after the film has been deposited on a grid range [157]. Numerous variations of these techniques have been used [32,67,160,172174]. In order to determine surface characteristics and particle thickness, it is usual to deposit obliquely a film of heavy metal on to a specimen or its replica. The metal is applied by deposition in a hard vacuum by a small source in order that a nearly parallel beam may reach the specimen. The technique was originated by Williams and Wyckoff [175] in 1946 and has been used extensively since. Surface topography can also be studied using 3-D imaging. Two successive photographs of the same field are made with the specimen tilted by 11"^ to 14"^ between photographs. This stereo-pair can then be viewed with stereo viewing lenses for a remarkable sense of the third dimension. 3.16.3 Chemical analysis When particles are bombarded with electrons, they emit radiation that depends upon their chemical composition. The Auger process [176,177] is a secondary electron process that follows the ejection of an electron from an inner shell level. The hole is filled by an electron falling to the vacant
Image analysis 193 level, which provides the energy for another electron to be emitted. The energy of this Auger electron is characteristic of the molecule involved. The process can be studied by using monochromatic or polychromatic radiation or electron beams. There have been many studies on metal surfaces using vacuum ultra violet techniques and the energy distribution curves (EDCs) obtained give information on the band structure of the metals. The use of soft x-rays is known as electron spectroscopy for chemical analysis (ESCA), or x-ray photo-electron spectroscopy (XPS). In addition to ejecting electrons from the valence shell orbits, the x-rays have sufficient energy to eject electrons from some of the inner shells. These are essentially atomic in nature and the spectra produced are characteristic of the atom concerned, rather than the molecule of which it forms a part [178]. These, and other techniques, may be applied in electron microscopy to permit chemical assay and particle size analysis to be run concurrently [179]. 3.17 Scanning electron microscopy In SEM a fine beam of electrons of medium energy (5-50 keV) is caused to scan across the sample in a series of parallel tracks [180]. These electrons interact with the sample, producing secondary electron emission (SEE), back-scattered electrons (BSE), light cathodoluminescence and x-rays. Each of these signals can be detected and displayed on the screen of a cathode ray tube like a television picture. Examinations are generally made on photographic records of the screen, although the images can be processed on-line [181]. The SEM is considerably faster and gives more three-dimensional detail than the TEM. Some instruments can take samples as large as 8 in. square and parts viewed at magnifications varying from 20x to 100,000x at resolutions of 5 to 7 nm. The latest instruments are capable of resolutions down to 0.7 nm. The depth of focus is nearly 300 times that of the optical microscope. Because of its great depth of focus the SEM can give considerable information about the surface texture of particles. In both the SEE and BSE modes, the particles appear as being viewed from above. In the SEE mode the particles appear to be illuminated diffusely, and particle size and aggregation behavior can be studied but there is little indication of height. In the BSE mode the particles appear to be illuminated by a point source of light and the resulting shadows give a good indication of height. Several of the current methods of particle size analysis can be adapted for the measurement of images in SEM
194 Powder sampling and particle size determination
photographic records. There is also active interest in the development of analysis techniques that will make use of the three-dimensional image presentation. Many of the image modification procedures associated with automatic image analysis are applicable also with images based on back-scattered electrons. This permits thresholding, classification, boundary enhancement, separation of touching particles, etc. to be carried out before the image is measured [182]. There are three ways of producing a stereographic image in an SEM: horizontally shifting the object, tilting the object and tilting the electron beam. The first is the method used for stereography of aerial photographs, though it is the viewpoint that shifts in this case. The second method suffers the defect that it is difficult to position the object accurately after tilting. Kolendik [183] examined both these methods and preferred the former whereas Kramarenko [184] stated that the first method was more effective and easier for high accuracy measurement. The tilting method has been described by Shoji et. al [185]; essentially it generates a binocular image by striking the specimen at oblique angles. ASTM provides standard reference materials for calibrating SEM's: SRM 484 g can be used to calibrate the magnification in the range lOOOx to 20,00x, SRM 2090 is a new standard made of a silicon chip with welldefined line separations and SRM 2069b is a standard of graphitized rayon fibers. ASTM E766-98 [186] discusses standard practice for calibrating the magnification of an SEM. Le Mont Scientific B-10 system features an energy-dispersive x-ray detector. Particles are loaded and interrogated to find size and shape; various software options are available. The Bausch and Lomb system has also been applied to electron beam microscopy [ 187,188].Tracor Northern describe an integrated system for the collection and processing of analytical and image data from SEM and STEM [189,190].Various sample preparation methods have been described. Krinsley and Margolis [191] mounted sand grains in Duco cement on the SEM target stub; this technique is not suitable for size analysis and cannot be used for fine particles [192]. Willard and Hjelmstad [193] tried to improve this technique, in order to mount fine coal, by using doublebacked adhesive tape. They had limited success with particles smaller than 30 |im and found that the method was unsuitable for larger particles. White et. al. [194] prepared specimens of alumina by aspirating droplets of alumina suspension in NH4OH solution on to aluminized glass slides and the fixing the slides to the SEM stub. This procedure is very slow and has been criticized as being unsuitable for particles which readily agglomerate
Image analysis 195 or which swell in aqueous solutions. Turner et. al [192] prepared an adhesive layer by immersing about 3 ft. of Scotch Tape in 150 cm^ of carbon tetrachloride or chloroform, agitating long enough for the adhesive layer to be dissolved and then removing the tape. A few drops of the solution were deposited on the SEM stub, the thickness of the adhesive layer on the stub being controlled by adjusting the concentration and the number of drops applied. A suitable amount of powder was dispersed in acetone (0.5% w/v) and a drop taken on a glass rod and dropped on to the stub. The mounted specimen was then coated with a 15 nm layer of aluminum. They also describe a method for studying agglomerates by freeze drying. Once the images are recorded, the quality of the analysis depends on the quality of the particle dispersion, the contrast and focus (sharpness of image). From this point on, the particle sizing proceeds as in the optical process described earlier. King and Schneider [195] state that the available commercial image processing and image analysis software systems do not usually include adequate algorithms for the effective analysis of multi-phase mineralogical textures. Filtering algorithms are usually inadequate for the accurate removal of image noise without compromising the integrity of phase edges. Although most systems offer good algorithms for the analysis of binary images, algorithms for higher order images are almost non-existent and these are essential for mineralogical analysis. They describe the development of an image analysis system that overcomes these limitations. It is based on SEM equipped with secondary electron and back-scattered electron detectors, an image memory for the storage of digital images captured at slow scan speeds and a SUN workstation for image processing and image analysis. A scanning electron microscope connected to an image processor allows one to obtain, automatically, a wide variety of parameters describing the shape and granulometric properties of powder particles. This method offers numerous advantages over other methods with automatic and numerous measurements with a saving in time and yielding more accurate data with a smaller number of observations [196]. Particle size and shape have been determined using a special computer program that was based on obtaining a pixel matrix of the particle boundary by the digitizing of the particle image [197] It was concluded that fractal analysis is a useful tool for correlating the fractal parameter with the physical description and flow properties of pharmaceutical solids.
196 Powder sampling and particle size determination
3.18 Other scanning electron microscopy techniques The scanning transmission electron microscope (STEM) uses a fine beam of electrons to scan the specimen as with the SEM, but detects transmitted electrons for display on a cathode ray tube (CRT). The performance is similar to the TEM, but with certain advantages. Since the sample is irradiated with a scanning beam, there is less beam damage to sensitive samples. In addition, for easily damaged samples, focusing can be done on a small region of the sample and instantly "transplanted" to an undamaged part for image recording. The CRT display of the image usually gives more flexibility in manipulating data, transforming it and so on. Use of the annular dark field detector also allows simultaneous bright field and dark field imaging of the same sample field. Additional advantages enjoyed by the dedicated STEM units are their clean, high vacuum systems in use and their very bright electron source in the field emission gun. The high vacuum (10~ii torr in the gun chamber and 10~^ to 10~^ torr in the sample chamber) allows prolonged observation of the sample without contamination, and the bright source allows viewing of data collection at TV scan rates. The scanning tunneling electron microscope (STM), invented in 1981, allows examination of non-conductive surfaces [198] down to atomic resolution and can operate in ambient and aqueous environments [199,200]. Early references to the instrument use the acronym STEM, which produced confusion with the scanning transmission electron microscope. More recently, the consensus has been to use STM, as the acronym for the tunneling instrument. Since the introduction of the STM a number of variations have been devised, such as ATM (atomic force microscope). The basic concept is that piezoelectric actuators move a miniature cantilever arm (with a nm-sized tip) across the sample while a non-contact optical system measures the deflection of the cantilever caused by atomic scale features. The deflection is proportional to the normal force exerted by the sample on the probe tip and images are generated by raster scanning the sample [201]. One application of this technique was to measure the thickness and size distribution of sub-micron clay particles with diameters in the 0.1 to 1 |Lim size range and thickness from 0,01 to 0.12 |im [202]. MFM (magnetic force microscope), LFM (lateral force, or friction force microscope), etc. None of the above finds wide use for particle size determination. The AFM has however been used to determine the shape, size and types of particle on a polished silicon wafer surface [203].
Image analysis 197
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198 Powder sampling and particle size determination
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Image analysis 199 and STM have been applied to nanoparticles and these studies have demonstrated that the techniques have potential for very small particles [204-206]. While atomic resolution is not possible with AFM; STM showed atomically resolved clusters on supports in some cases. An interesting feature of STM is scanning tunneling spectroscopy which can be used to distinguish between metals and semiconductors and to measure the band gap of the latter [207]. A detailed review of the capabilities of these technique has been published by Springer [208]. TEM and SEM are sometimes unable to clearly differentiate between agglomerates, particles and grains in soil samples and sample preparation is difficult and tedious. ATM has been used to overcome these limitations with a sample of submicron oxisol. The thickness and diameter of soil particles deposited in freshly cleaved mica were measured for each individual particle. Assuming cylindrical shape and a constant density the mass particle size distribution was obtainable [209]. 3.19 Errors involved in converting a number to a volume count This example consists of TEM number count data on 662 droplets in 40 size classes linearly spaced between 0.015 |Lim and 0.250 |um. The resulting number distribution is noisy (Figure 3.13a) due to the large number of size classes and the linear presentation of what is essentially a log-normal distribution. The conversion to a volume distribution and comparison with Microtrac UPA data are shown in Figure 3.13b. The poor agreement is due, in part, to the errors generated when converting a number distribution of a wideranging population into a volume distribution. A single 0.25 |Lim droplet has the same volume as 4630 droplets of size 0.015 |um hence overcounting by a single large particle is equivalent to over-counting 4630 small particles. Reducing the number of classes to 20, and plotting logarithmically, greatly reduces the noise (Figure 3.13c). Comparison between UPA and TEM data (Figure 3.13d) illustrates that the difference between the two sets of data is due to overcounting large particles in the TEM analysis. This error occurs when all droplets in the photomicrograph are counted. It is essential that droplets overlapping two adjacent boundaries be neglected since the probability of large droplets overlapping is greater than the probability for small droplets. This example illustrate four points:
200 Powder sampling and particle size determination
1. Number distributions, for powders having a wide size range, should never be converted into volume distributions unless a statistically acceptable number of particles in the largest size category have been counted. For a microscope count, the correct procedure for carrying out a size distribution by weight should be followed. 2. Dividing the distribution into too many size categories can result in a noisy distribution unless sufficient particles are measured. The size ranges should be arranged logarithmically for powders having a wide size range. 3. When sizing photomicrographs, a frame should be drawn in the photograph, displaced from the edge by a distance equal to the radius of the largest particle in the distribution. All particles overlapping two adjacent sides of the frame should be left uncounted, otherwise the large particles are over-counted. Data should be presented in such a way as to highlight the required information. Incorrect presentation can completely hide relevant information 3.20 Evaluation of procedures Several alumina powders have been analyzed using SEM, TEM, gas adsorption, x-ray Sedigraph and laser light scattering and the results compared [210]. References 1 2 3 4 5 6 1
Schmidt, F., Schmidt, K.G. and Fissan, H. (1990), J. Aerosol Sci., 21, Suppl. 1,S535-S538, 142 McCrone, W.C. (1991), Phys. Meth. Chem., 2nd ed., eds. B.W. Rossiter and J.F. Hamilton, Wiley, N.Y., 142 Mayette, D.C, McMahon, B. and Daghlian, C.P. (1992), Ultrapure Water, 9(4), 20, 22-24, 26-29, 32-34, 142, 147 Sutherland, D.N. (1993), Part. Part. Syst. Charact., 10(5), 271-274, 142 Lane, G.S. and Richmond, G.D. (1993), Proc. 18th Int. Min. Proc. Cong., Sydney, 897-904, 142 Welford, G,. A. (1960), Optics in Metrology, 85, 143 Groen, F.C.A., Young, l.T. and Ligthart, T. (1985), Cytometry, 6(81), 144
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9 10 11
12 13 14 15 16 17 18 19 20 21
22 23 24 25 26 27 28 29 30 31 32 33
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Kenney, L.C. (1985), Particle Size Analysis, 247-260, Proc. 5th International Conf., Analyt. Div. Chem. Soc, ed. PJ. Lloyd, publ. John Wiley & Sons, 144 ASTM D4791-898 (1989), Standard test method for flat and elongated particles, 144 Broyles, D.A., Rimmer, H.W. and Adel, G.T. (1996), Kona, 14, 130-138, 144 BS 3406 Part 4 (1993), Methods for determination of particle size distribution, Part 4, Guide to microscopy and image analysis methods, 144, 146, 154, 155, 162 ASTM E20 Practice for particle size analysis of particulate substances in the range 0.5 jum to 75 jum, 144 ASTM E175-82 (1995), Standard terminology of microscopy, 144 ASTM E766-98 (1998), Standard practice for calibrating the magnification of an scanning electron microscope, 144 NF 11-661 Test methods for particle size analysis-Determination of particle size ofpowders-Optical microscope, 144 NF XI1-696 Test methods for particle size analysis through image analysis, 144 ISO/CD 13322 (Draft Standard), Particle size analysis-Image analysis methods, 145 Hartman, A.W. (1984), Powder TechnoL, 39, 49, 145 Hartman, A.W. (1985), Powder TechnoL, 42, 259, 145 Hartman, A.W. (1986), Powder TechnoL, 46, 109, 145 Wilson, R. and Elkington, A. A. (1985), Int. Conf Royal Soc. Chem. Particle Size Analysis Group, 261-270, ed. P.J. Lloyd, publ. John Wiley & Sons, 145, 170 National Physical Laboratory. Reference Stage Graticule for Image Analysis calibration, Teddington, Middlesex, U.K., 145, 170 Charman, W.N. (1961), Ph.D. thesis, London Univ., 146, 147 Rowe, S.H. (1955), Microscope, 15, 216, 147 Humphries, D.W. (1961), J. Sediment. Petrol., 31, 471-473, 148 Hawkins, A.E. (1993), The Shape of Powder Particle Outlines, 22, John Wiley & Sons, 148 Hawkins, A.E. and Davies, K.W. (1984), Part. Part. Syst. Charact., 4, 2227, 148 Caimcross, A., Flaherty, D.M. and Klabunde, U. (1993), Proc. Microsc. Soc. of America, 51st A.G.M., August, Cincinnatti, Ohio, 148 Allen, T. (1994), Powder TechnoL, 79, 61-68, 148 Green, M. (1921), J. Franklin Inst., 192, 657, J48 Dunn, E. (1930), Ind. Engg Chem., Analyt, Ed, 2, 59, 148 Orr, C. and Dallevalle, J.M. (1959), Fine Particle Measurement, Macmillan, N.Y., 148,156, 192 ASTM (1976), Annual Book of Standards, Part 4. Particle Size Measurement, Microscopy, E20-69, Reapproved 1984, 148
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Particle size analysis by sieving 4.1 Introduction Sieving has been used since early Egyptian times for the preparation of foodstuffs. The simplest sieves were made of woven fabric, but punched plate sieves are recorded in early Egyptian drawings and, by 1556, Agricola is illustrating woven wire sieves [1]. The technique is particularly useful since particles are sorted into categories solely on the basis of size, independently of other properties (density, surface, etc.). It can be used to classify dry or wet powders and generates narrowly classified fractions. Such sieves were used solely for powder classification, and the inception of test sieving did not arise until sieve aperture sizes were standardized. Standard apertures were first proposed by Rittinger [2] who, in 1867, suggested a V2 progression of aperture sizes based on 75 jiim thus, similarly shaped particles passing consecutive apertures, are in a 2:1 surface ratio. Modem standards are based on a fourth root of two progression, apart from the French AFNOR series, adopted in 1938, which is based on a sieve aperture of 1 mm in a tenth root often progression; this is known as the RIO or Renaud series. Woven wire sieves are often referred to by their mesh size, which is the number of wires per linear inch. The ASTM range is from 400-mesh to 4.24 in. The aperture for the 400-mesh sieve is 37.5 jam hence the wire thickness is 26.0 |im making the percentage open area 35. Generally, woven wire sieves have pseudo-square apertures (i.e. the weaving process generates trapezoidal apertures in three dimensions) but punched plate and electroformed sieves are available with round and rectangular apertures. A variety of other shapes is also readily available. Woven wire sieves are available, ranging in size from around 125 mm down to 20 ]xm.
Sieving 209
Plane of maximum stability
Limiting particle dimensions
Spheres of equal volume and surface Fig. 4.1 Equivalent particle diameters (after Heywood). Sieve Diameter = 144 |im
Fig. 4.2 Sieve size of particle shown in Figure 2.1. Electroformed micromesh sieves extend the range down to 5 |Lim or less, and punched plate sieves extend the upper range. Sieving consists of placing a powder sample onto a sieve containing openings of a fixed size, and agitating the sieve in such a way that particles that can pass through the openings do so. To speed up the analysis, several sieves are stacked on top of each other, with the sieve containing the coarsest openings on top and a 'catch' pan on the bottom. This 'nest' of sieves is vibrated until the residue on each sieve contains particles that can pass through the upper sieve but cannot pass through the lower sieve. Fractionation by sieving is a function of two particle dimensions only, maximum breadth and maximum thickness for, unless the particles are excessively elongated, the length does not affect the passage of particles through the sieve apertures (Figure 4.1). Particles, having two dimensions
210 Powder sampling and particle size determination
smaller than the openings, will pass through when the sieve is vibrated, whereas larger particles will be retained. The sieve size d^ is defined, on the basis of woven wire sieves, as the minimum square aperture through which a particle can pass (Figure 4.2). The sieve surface consists of a go, no-go barrier; particles much smaller than the openings pass through rapidly and larger particles pass through more slowly. Since the apertures have a range of sizes, the final particle that can pass through, will pass through the largest opening when its two smaller dimensions are in a preferred direction. Since this will take a very long time sieving is usually deemed complete when not more than 0.2% of the original weight of sample passes through in a 2 min. sieving operation. Sieving is probably the most widely used and abused method of particle size analysis because the equipment, analytical procedures and basic concepts are so deceptively simple. Its popularity is due the relative simplicity of the technique, low capital investment and low level of expertise required to carry out the analyses. Sieve analysis presents three major difficulties [1]. With woven wire sieves, the weaving process produces three-dimensional apertures with considerable tolerances, particularly for fine-woven mesh [2]. The mesh is easily damaged in use [3]. The particles must be efficiently presented to the sieve apertures. Sieve analyses can be highly reproducible even when using different sets of sieves. Although most of the problems associated with sieving have been known for many years, and solutions proposed, reproducibility is rarely achieved in practice due to the failure to take cognizance of these problems [4]. With micromesh sieves, near-monodisperse powders can be generated in the 1 to 10 |Lim size range [3]. An overview of recent developments in fine sieving in Japan has been presented in two reviews [5,6]. 4.2 Standard sieves A variety of sieve apertures is currently available, and these are classified as coarse (4 mm to 100 mm), medium (0.2 mm to 4 mm) and fine (less than 0.2 mm). The fine range extends down to 37 |im with woven wire sieves but the tolerances are liberal and this has limited their acceptance. Complete instructions and procedures on the use and calibration of testing sieves are contained in ASTM STP447B [7]. Contents include analytical methods, information relating to wire mesh, perforated plate and micromesh sieves, dry and wet testing and other methods. This publication also contains a list of all published ASTM standards on sieve analysis
Sieving 211 procedures for specific materials or industries. Standard frames are available with nominal diameters of 3, 6, 8, 10 or 12 in. Non-standard frames are also available. There are also standards available that define standard test methods for characterizing particular classes of materials. Standards pertaining to specifications for sieves and standard techniques to be followed for sieve analysis are available from every major standards organization [8]. ASTM El919-97 [9] has a comprehensive list of the pertinent standards available from ASTM (America), BSI (Britain), DIN (Germany), AFNOR (France), JSA/JIS (Japan). International standards are also available from ISO. Three ASTM standards define specifications for various types of sieves, El 1-95 [10], E161-96 [11] and E323-80 [12]. El 1-95 covers the requirements for woven wire sieves, El61-96 covers the requirements for electroformed sieves and E323-80 defines the specifications for perforated plate sieves. These standards give information on aperture openings, wire diameters, frame diameters and methods of measuring aperture openings together with methods for maintaining and cleaning sieves. British Standard Specifications BS 410 [13] adopts a primary size of 75 |Lim (200-mesh) with a fourth root of two progression in size and suggest that alternate sieves should normally be used for an analysis (i.e. a root two progression of sizes). Thus, the specific surface area of particles on consecutive sieves is in a 2:1 progression. BS 1796 describes the methods to be used in sieving with woven wire and perforated plate sieves [14]. The relevant German Standards for perforated plate, woven wire and textile cloth are, respectively: DIN 4187 [15], DIN 4188 [16] and DIN 4195 [17]. The French AFNOR Standard NFX 11-501 [18] is based on a woven wire sieve opening of 1 mm in a tenth root often progression of size. Now: (lO^) if =1.2593=2
(4.1)
Hence, similarly shaped particles passing consecutive apertures are in a 2:1 volume ratio as opposed to the 2:1 surface ratio in the British Standard. Series R20 defines intermediate sieves to give a twentieth root of ten progression and may be included for narrow size distributions. International sieve standards available from the International Organization for Standardization, Geneva are: ISO 3310-1 for wire cloth [19], ISO 3310-2 for perforated plate [20], and ISO 3310-3 for electroformed [21].
212 Powder sampling and particle size determination
As well as these, several industries have their own specifications; aggregates [22], porcelain [23], ceramic powders [24], plastics [25], coating powders [26], concrete [27] and metals [28]. The standard for glass spheres [29] is designed specifically for sieve analysis of glass spheres used in reflective road and pavement markings and other industrial uses. ISO 2395 [30] and ASTM E1638-94 [31] describe the terminology used in sieve analysis. Most test sieves are certified and manufactured to ISO 9002 standards. Table 4.1 Nominal apertures and permissible variations for a selection of US woven wire sieves Standard (mm)
\
nsxi 63.0 31.5 16.0 8.0 4.0 2.0 1.0 (l-^m) 850 300 150 125 106 90 75 38 20
Alternative (in) 5 2.500 1.250 0.625 0.312 0.157 0.0787 0.0394 (in) 0.0331 0.0117 0.0059 0.0049 0.0041 0.0035 0.0029 0.0015 0.0008
Tolerance (±mm) 3J 1.9 1.0 0.5 0.25 0.13 0.070 0.040 (t^m) 35 14 8 7 6 5 5 3 3
Intermediate (mm)
Maximum (mm)
i3o!o 65.6 32.9 16.7 8.41 4.23 2.135 1.080 (^tm)
925 337 174 147 126 108 91 48 29
66.2 33.2 17.0 8.58 4.35 2.215 0.135
im)
1
970 363 192 163 141 122 103 57 35
4.3 Tolerances for standard sieves The apertures for the British Standard 400 mesh are 37.5 |iim with a nominal wire thickness of 26 )Lim. On the basis of Table 4.1, in which a selection of sieve dimensions are shown, the (nominal) 75 \xm sieve has a
Sieving 213 median aperture size in the range 70-80 |um and not more than 5% of the apertures shall fall in the (intermediate to maximum) size range of 91-103 |Lim. This implies that there is a probability of having a 103 |im aperture in a nominal 75 jum sieve. {The percentage of undersize apertures is not relevant since only particles smaller than the nominal aperture size can pass through them. Their sole (detrimental) contribution is to reduce the effective open area of the sieve and thus increase sieving time}. The relative tolerance increases with decreasing nominal size leading to poor reproducibility when analyses are carried out using different nests of (uncalibrated) sieves. Electroformed sieves with square or round apertures and tolerances of ±2 |j,m are also available. Woven wire sieves, having apertures ranging from 20 [im to 125 mm, are readily available in 100, 200, 300 and 450 mm diameters frames as well as 3, 8, 12 and 18 in diameters. Microplate sieves are available in 100 and 200 mm diameters with round or square apertures ranging from 1 to 125 mm. Endecottes' test sieves have at least five intermediate and one final inspection in which wire cloth dimensions are inspected by projector. 4.4 Woven-wire and punched plate sieves Sieve cloth is woven from wire and the cloth is soldered and clamped to the bottom of cylindrical containers [10,13]. Although the apertures are described as square, they deviate from this shape due to the threedimensional structure of the weave. In the weaving process, the weft wires are 'crimped' on to the warp wires for added strength; vigorous cleaning, for example with a wire brush, can separate the wires leading to oversize apertures. Fine sieves are usually woven with phosphor bronze wire, medium with brass, and coarse with mild steel. Heavy-duty sieves are often made of perforated plate giving rise to circular holes [12]. Various other shapes, such as slots for sieving fibers, are also available. Special purpose sieves are available in stainless steel and the flour industry uses nylon or silk. Bates discusses screen mesh fabric selection and how the material should be fixed to the frame, particularly with regard to stretching the material and adjusting the tension [32]. Wahl [33] describes the production of highly wear resistant sieves in soft annealed plates of chromium steel by punching, plasma cutting or mechanical working followed by heating and strain hardening. A method of preparing a metal sieve cloth holder has been described in which parallel grooves were cut on a metal mandrel [34]. These were
214 Powder sampling and particle size determination
filled with an insulating material to produce a network of conducting and non-conducting lines. After passivating the mandrel, copper wire was wound perpendicular to the linear net to give the pattern, and the whole immersed in a nickel plating bath. Two thousand five hundred holes per cm^ were produced with a nickel deposition of 25 to 30 |Lim. The cylindrical sieve cloth containers (sieves) are formed in such a way that they will stack, one on top of the other, to give a snug fit (Figure 4.3). Due to the method of manufacture, woven wire sieves have poor tolerances, particularly as the aperture size decreases. Tolerances are improved and the lower size limit extended with electroformed micromesh sieves.
qc:
IP
Fig. 4.3 Stacking of sieves. 4.5 Electroformed micromesh sieves Micromesh sieves [11] were first described by Daescher et. al [35] and are available in standard sizes ranging from 500 jum down to 3 jum. Etched sieves usually have round or square apertures, ranging in size from 500 to 1200 |Lim, but other aperture shapes are available. They are available in 3, 8 and 12 in diameter frames, as well as custom-made sizes. Other aperture sizes are also available and Zwicker reports using a 1 \xm sieve [36].
Sieving
215
,.'......'...4mmmmmmmuuu v
•
I •
•
•
•
"• • • •
•
•
• •
••
• •
o
Q •
• qj
••
••
•
Q •
• ®
• q •
•
• i
•
•
• o
• •
•
Q •
d
•
•
• •
• Q
•
•
•
qP •
Q
0 •
-
I
Fig. 4.4 Electroformed micromesh sieves (a) 30 ~tm square (b) 20 ~tm square (c) 75 lain round (d) 45 lain round
216 Powder sampling and particle size determination
The photo-etching process is as follows. A fully degreased metal sheet is covered on both sides with a photosensitive coating and the desired pattern is applied photographically to both sides of the sheet. Subsequently the sheet is passed through an etching machine and the unexposed metal is etched away. Finally, the photosensitive coating is removed. A supporting grid is made by printing a coarse line pattern on both sides of a sheet of copper foil coated with photosensitive enamel. The foil is developed and the material between the lines etched away. The mesh is drawn tautly over the grid and nickel-plated on to it. The precision of the method gives a tolerance of ±2 jum for apertures from 300 to 500 |Lim reducing to ±1 jum for apertures from 5 to 106 |Lim. Some typical sieve meshes are illustrated in figure 4.4. For square mesh electroformed sieves, the pattern is ruled on to a waxcoated glass plate with up to 8000 lines per inch, with each line 0.0001 in wide, and the grooves are etched and filled. The lower aperture size limit for off-the-shelf sieves is about 5 |Lim but apertures down to 1 |Lim have been produced. The percentage open area decreases with decreasing aperture size, ranging from 31.5% for 40 |Lim aperture sieves, to 2.4% for 5 |Lim aperture sieves; this leads to greatly extended sieving time when using the smaller aperture sieves. These were originally available only in 3 in diameter frames but this has been extended to 8 in and 12 in frames. Buckbee Mears also make 10 and 20 cm diameter frames with apertures down to 3 jam. Sieves are available with a greater open area; this reduces the sieving time but the sieves are more fragile. Burt [37] examined samples of the finer sieve cloth and found that the average width of metal between openings for each grade of cloth was one-third to one-half that recommended in ASTM El61-70. Veco manufacture round and square aperture sieves. The apertures for the former are in the shape of truncated cones with the small circle uppermost. This reduces blinding but also reduces the open area and therefore prolongs the sieving time. Where thicker sieves are required, the Veco sieves are subjected to further electrodeposition on both sides to produce biconical apertures. The tolerances with micromesh sieves are much better than those for woven-wire sieves; the apertures being guaranteed to ±2 jiim of nominal size apertures except for the smaller aperture sieves. Each type of sieve has advantages and disadvantages [38]; e.g. sieves having a large percentage open area are structurally weaker but measurement time is reduced.
Sieving
50
60
217
100
70
W,
W(iiominal)
110
W^
Fig 4.5 Effective aperture width distribution with increasing sieving time.
100
' • •"•—•
1
Region 2
h
j
h
^r
CO
a
Region 1
10 L
^
1
Transition region
]X 1
1
^
V J
1 i.....L Lnii.li
10
1
1
1
I 1 1 iil_
_ __.!
100
Sieving time in minutes Fig. 4.6 The rate at which particle pass through a sieve.
1
1—1 1 t i l l
looq
218 Powder sampling and particle size determination
Size and shape accuracy are improved by depositing successive layers of nickel, copper and nickel on a stainless steel plate followed by etching through a photolithographic mask additional layers of copper and nickel. The holes are filled with dielectric, after which the additional nickel is removed down to the copper layer [39]. A description of electroforming methods for making multilayer matrixes for precision screens has been patented [40]. A process, which is claimed to give better bonding of the mesh on to the support grid, has also been described [41]. Niklas [42] discusses edge weaknesses in nickel electroformed sieves due to acute angle comers. Additives used during photo-etching increases these defects [43] Stork [44] also describe sieve preparation by electrodepositing a thin metal skeleton on to a substrate, removal of the skeleton from the substrate, followed by deposition of two or more layers of metal on both sides of the substrate. Additives encourage growth perpendicular to the surface of the skeleton. 4.6 Mathematical analysis of the sieving process The tolerances on sieve cloth are extremely wide, particularly for small aperture cloth. For example, the British Standard Specification (BS 410) for a 200-mesh sieve requires a median diameter of 75 plus or minus 33 |Lim. It is clear that oversize apertures are more undesirable than undersize, since the latter are merely ineffective whilst the former permit the passage of oversize particles. In order to reduce differences between analyses using different sets of sieves (differences of up to 42% have been recorded) manufacturers such as ATM make specially selected sieves [45] available that can reduce the differences by a factor of 10. The nominal wire thickness for a 75 |Lim sieve is 52 |Lim hence, at the commencement of a sieving operation, the nominal open area comprises 35% of the total area [i.e. (75/127))^]; the apertures may range in size from 42 to 108 |Lim (Figure 4.5). The number of particle that can pass through the smaller apertures decreases as sieving progresses and this results in a decrease in the effective percentage of available open area. Thus, the effective sieve size increases, rising, in the example given above, to 84 )Lim and then to 94 \xm and, eventually to the largest aperture in the sieve cloth. Hence, the mechanism of sieving can be divided into two regions with a transition region in between [46], an initial region that relates to the passage of particles much finer than the mesh openings and a second region that relates to the passage of near-mesh particles (Figure 4.6). Nearmesh particles are defined as particles that will pass through the sieve
Sieving 219 openings in only a limited number of ways, and the ultimate particle is the one that will pass only through the largest aperture in only one orientation. The first region is governed by the law: P = afi
(4.2)
where P is the cumulative weight fraction through the sieve, t is the sieving time, a is the fraction passing through the sieve in unit time or per tap for hand sieving and Z? is a constant nearly equal to unity. Whitby assumed a to be a function of several variables; total load on sieve {W), Particle density (/T^), mesh opening (5), percentage open area (^Q), sieve area {A\ particle size {d) and bed depth on sieve (7). This function reduces to:
a=f
(4.3)
p^SA^ ' d' A' d' S^
an identity with seven variables and two dimensions; hence a is a function of five dimensionless groups. A^A is constant for any sieve, A/Sis so large that it is unlikely to have any appreciable effect, and the effect of variation in T/d is negligible so that the equation reduces to: /
' w s^
(4.4)
Ps^^o d
Whitby found aW A^S
f
h
— ^1
vMm
(4.5)
^^S^gpj
where k^d^ is a linear function of the geometric mass mean of the particle size distribution, Cj and h are constants and a is the geometric standard deviation at a particular size on the distribution curve. This expression was found to hold for wheat products, crushed quartz, St Peter's sand, glass beads and other similar materials. Whitby suggested that the end-point of sieving be selected at the beginning of region 2. This can be done, by plotting the time-weight curve on log-probability paper, and selecting the end-point at the beginning of
220 Powder sampling and particle size determination
region 2. It is difficult to do this in practice and an alternative procedure is to use a log-log plot and define the end-point as the intersection of the extrapolation for the two regions (Figure 4.6). Using the conventional rate test, the sieving operation is terminated some time during region 2. The true end-point, when every particle capable of passing through a sieve has done so, is not reached unless the sieving time is unduly protracted. The second region refers to the passage of 'near-mesh' particles. These are defined as particles that will pass through the sieve openings, in only a limited number of ways, relative to the many possible orientations with respect to the sieve surface. The passage of such particles is a statistical process, that is, there is always an element of chance as to whether a particular particle will or will not pass through the sieve. In the limit, the largest aperture through which the ultimate particle will pass in only one particular orientation controls the sieving process. In practice there is no 'end-point' to a sieving operation, so this is defined in an arbitrary manner. The rate method is fundamentally more accurate than the time method but it is more tedious to apply in practice and, for most routine purposes, a specified sieving time is adequate. Several authors have derived equations for the rate of sieving during region 2 where the residual particles are near mesh. The general relationship is of the form: '-^ = k{R,-Rj'
(4.6)
where 7?^ is the residue on the sieve at time t and R^ is the ultimate endpoint. Kaye [47] and Jansen and Glastonberry [48] assumed m = 1 and plotted \og(Rf - R^) against t, which yields a straight line if the (assumed) value for R^ is correct. In practice, this value of 7? is of limited practical value, since it cannot apply to the nominal aperture of the sieve. As sieving progresses, the smaller apertures become ineffective since all the particles finer than these apertures will have passed through the sieve. The largest aperture in the sieve therefore controls the sieving operation and the final particle to pass through the sieve will only do so when presented to this aperture in its most favorable orientation, i.e. for a 75 |im sieve, the true end-point could be 100 jam or more.
Sieving 221 4.7 Calibration of sieves It is not widely realized that analyses of the same sample of material, by different sieves of the same nominal aperture size, are subject to discrepancies that may be considerable. These discrepancies may be due to non-representative samples, differences in the time the material is sieved, operator errors, humidity, different sieving actions and differences in the sieves themselves which may be due to wear. Ideally, sieves should have apertures of identical shape and size. However, due to the methods of fabrication, woven wire sieves have a range of aperture sizes and weaving gives a three dimensional effect. Fairly wide tolerances are accepted and specified in standards but even these are sometimes exceeded in practice. Leschonski [4] examined eight 50 |Lim woven wire sieves with the results presented in Figure 4.7. The median varied between 47.3 and 63.2 |um (almost certainly an incorrectly labeled 63 }xm sieve) and the standard deviation between 1.8 and 8.6 |Lim. Ilantzis [49] gave a rigorous statistical analysis of microscopic measurements on a set of 13 different sieves (Table 4.2). Most of the sieves had rectangular openings with wider variations on the warp than the weft, and with a coefficient of variation rising from 3% to 5% with large aperture sieves and up to 10% with small apertures. Only four of the sieves met the standard specifications. All aperture widths shown in Figure 4.7 are for nominal 50 |Lim sieves but comparative tests on the same material using these sieves could yield enormous differences in the residual weight. In order to obtain agreement between different sets of sieves it is therefore necessary to calibrate them and thenceforth to monitor them to detect signs of wear. One way of standardizing a single set of sieves is to analyze the products of comminution. It is known that the products are usually lognormally distributed hence, if the distribution is plotted on a logprobability paper, a straight line should result. The experimental data are fitted to the best straight line by converting the nominal sieve aperture to an effective sieve aperture. The traditional way of determining the median and spread of aperture sizes for a woven wire sieve is to size a randomly selected set of apertures using a microscope. Due to the method of manufacture, the measurements for the warp and weft will tend to differ. The limiting size may also be determined by using spherical particles. These are fed on to the sieve which is then shaken and the excess removed. Many spheres will have
222 Powder sampling and particle size determination
40
50
60
Aperture size in microns Fig. 4.7 Cumulative distribution by number of aperture widths of eight 50 |j.m woven wire sieves. Table 4.2 Means and standard deviations of 13 different sieves as reported by Ilantzis [50] Nominal aperture width (urn) 40 50 63 80 100 125 160 200 250 315 400 500 630
Weft
Warp Median (urn) 44.1 53.2 62.9 82.6 103.2 121.5 167.7 200.8 258.9 299.3 414.5 512.9 639.0
Standard deviation 4.85 6.12 4.20 5.68 4.44 4.89 10.48 11.51 14.60 22.63 22.00 28.70 20.70
Median (|Lim)
45.2 50.8 64.0 84.3 106.1 123.8 155.2 214.8 260.9 295.9 467.6 500.5 634.0
Standard deviation 4.56 5.19 3.75 4.02 2.91 5.59 6.79 16.46 14.40
17.29 26.40 25.00 43.20
1
Sieving 223 jammed into the sieve cloth and may be removed for examination. In such an examination, Kaye et al [51] showed that both methods yielded similar data for a 65-mesh Tyler sieve, with 90% of the apertures falling between 0.91 and 1.07 of the nominal size. Kaye recommends monitoring sieves every few months using glass beads to find out whether the sieve apertures are deteriorating or enlarging. These beads are available from the National Institute of Standards and Technology. Table 4.3 NBS Standard reference materials designated for calibration of sieves SRM 1003 b 1004b 1017b 1018b 1019b
Particle diameter in microns 10 to 60 40 to 150 100 to 400 220 to 750 750 to 2450
Standard polystyrene spheres are also available for calibration purposes from Gilson and Duke Standards. Stork Screens Inc. measure apertures and screen wire dimensions using wax impressions and the impressions are examined with an automatic image analyzer. Sieves may also be calibrated by a counting and weighing technique applied to the fraction of particles passing the sieve immediately prior to the end of an analysis. These will have a very narrow size range and the average particle size may be taken as the cut size of the sieve. A minimum number {n) of particles need to be weighed to obtain accurate volume diameters {d^)\ let this weight be m and the particle density be /? then:
< -
(4.7)
\f^.
Particles larger than 250 |im can easily be counted by hand and, if weighed in batches of 100, d^ is found to be reproducible to three significant figures. For particles between 100 and 250 |Lim in size, it is necessary to count in batches of 1000 using a magnifier. For sizes smaller than this, the Coulter Principle can be used to obtain the number concentration in a known suspension. An alternative procedure using tacky dots has also
224 Powder sampling and particle size determination
been described [52]. Sieve analyses are then plotted against volume diameter in preference to the nominal sieve diameter. The method is tedious and time consuming and BCR have prepared quartz samples by this method that may be used as calibration material [53]. The quartz is fed to a stack of sieves and the analytical cut size is read off the cumulative distribution curve of the calibration material. A method for measuring accurately the aperture and wire diameter using wax impressions has also been described [54]. Rideal et. al. [55] describe the preparation, measurement and use of microspheres for calibrating individual test sieves. Broad distribution glass microspheres were first sieved at 75 |Lim and the undersize re-sieved on a 53 |am sieve using a production sonic sifter from Gilson. This was repeated three times to give about 90% by weight of powder between 53 and 75 |im. The resulting samples were then analyzed using a nest of electroformed sieves in the 45 to 75 [im aperture range and a calibration graph of percent, passing against mean aperture size was the generated. This graph was then used to calibrate a 63 |am sieve. Results were confirmed by microscope analysis of the sieve mesh. The calibration method gave a mean aperture of 59.6 |Lim with a standard deviation of 0.08 |Lim; by microscopy the sieve apertures were 60.5 |im; the near mesh method gave 63.5 |Lim. This last method is expected to generate a larger size since the particles measured are those trapped in the sieve mesh at the end of the analysis. Centerline sieve measurements service from ATM evaluate the wire and opening sizes in wire cloth and electroformed mesh used in test sieves. Service provides a data set of opening sizes for verifying compliance with applicable standards. 4.8 Sieving errors Hand sieving is the reference technique by which other sieving techniques should be judged. For instance, in the French standard NFX 11-57, it states: If sieving machines are used, they must be built and used in such a way that the sieve analysis must, within the agreed tolerances, agree with the analysis obtained by hand sieving. The apertures of a sieve may be regarded as a series of gauges that reject or pass particles as they are presented at the aperture. The probability that a
Sieving 225 particle will present itself to an aperture depends upon the following factors: •
•
•
•
• •
The particle size distribution of the powder. The presence of a large fraction of near-mesh particles reduces the sieving efficiency. An excess of fines also reduces sieving efficiency. It is recommended that fines should be removed before carrying out the sieve analysis. This is effected by pre-sieving, by hand, on the finest sieve to be used in the subsequent analysis. If this is not done, the fines will have to pass through the whole nest of sieves, thus prolonging the analysis and increasing the risk of high powder loss. Since small particles often adhere to large ones it may be necessary to carry out this operation by wet sieving. The number of particles on the sieve (load). The smaller the sieve loading the more rapid the analysis; too low a load however leads to errors in weighing and unacceptable percentage losses. The physical properties of the particles. These include adhesion; stickiness due to the presence of water, e.g. high humidity, cohesion, i.e. the tendency of particles to stick together to form granules and other surface phenomena. Brittle particles are best sieved using a gentle sieving action, as is found with the air jet sieve and sonic sifter. Coating the powder to reduce cohesivity can reduce granule formation. The method of shaking the sieve. Sieve motion should minimize the risk of aperture blockage and preferably include a jolting action to remove particles that have wedged in the sieve mesh. Particles shape. Elongated particles sieve more slowly than compact particles. The geometry of the sieving surface, e.g. fractional open area. Table 4.4 The effect of sieving time and load on the amount passing a 200-mesh sieve Sample weight {W) (g) 500 250 125 62.5
5 83.6 67.2 58.6 56.6
Sieving time (?) in minutes 40 20 10 Percentage retained on sieve {P) 73.8 76.5 80.7 61.6 57.8 64.3 55.2 53.2 58.0 53.2 52.3 55.0
226 Powder sampling and particle size determination
Whether or not the particle will pass the sieve when it is presented at the sieving surface will then depend upon its dimensions and the angle at which it is presented. The size distribution given by the sieving operation depends also on the following variables. •
Duration of sieving. Sieving time is closely related to sieve loading, a reduction in the latter resulting in a reduction in the former. Variation of sieve aperture. Wear. Errors of observation and experiment. Errors of sampling. Effect of different equipment and operation.
Shergold [56] investigated the effects of sieving time and sieve loading. The tests were carried out with 14, 52 and 200-mesh sieves, using samples of sand specially prepared so that 50% of each sample could pass through the appropriate sieve. Some of the results using the 200-mesh sieve are given in Table 4.4. These results are of the form P = k\nt. Shergold showed that the smaller the sieve aperture, the greater the effect of overloading and the greater the discrepancies between the results for different loadings. He also showed that, although in general there is no end-point for sieving, the approach to the true percentage is faster for small apertures. Since it is evident, that a reduction in sieve loading is more effective than prolonging the sieving, he recommended that the sample should be as small as is compatible with convenient handling 100-150 g for coarse sand and 40-60 g for fine sand with a sieving time of 9 min. Heywood [57] also investigated the effect of sample weight on sieving time using 20, 50 and 100 g samples of coal dust on 60, 100, 150 and 200-mesh sieves. He found that neither the time required to attain the endpoint of 0.1% per min. nor the residual percentage on the sieve was affected by the weight of the sample. Table 4.5 Aperture errors as a percentage of nominal apertures Different sieves, different methods Different sieves, same method Same sieve, same method Machine sieving Hand sieving
8.30 3.71 0.61 0.80
Sieving
227
Other variables that have been investigated include sieve motion [58,59], open area [60], calibration [61-63], static [64], humidity [65] and accuracy for a particular application [66-70]. Heywood, for example, describes experiments carried out at seven laboratories, in which 50 g samples of coal dust were sieved on BS sieves of 72, 100, 150 and 200-mesh. The average percentage retained on a specified sieve, in all the trials, was taken to correspond to the nominal sieve aperture, whilst the percentage retained on that sieve for a single trial was taken to correspond to an effective sieve aperture. From a graph of the cumulative percentage passing through the sieve against the nominal sieve aperture, an effective sieve aperture may be read. Heywood found that the average passing through the 52-mesh sieve (295 |Lim) was 76.9%. In a particular analysis, 74% passed through the sieve making the effective aperture 280 |Lim, an aperture error of 15 |Lim. Heywood, by averaging over all the sieves, arrived at the values given in Table 4.5 for the standard deviation of aperture errors expressed as a percentage of the nominal aperture (Table 4.5). Although differences between analyses are inevitable, standardization of procedure more than doubles the reproducibility of a sieving operation. Sieve calibration increases this even further. A useful statistical analysis of Heywood's data may be found in Herdan [71]. 4.9 Methods of sieving Sieving procedures are standardized in BS 1796 [14]. Dry sieving by machine is used for coarse separation but other procedures are necessary as the powder becomes finer and more cohesive. Conventional dry sieving is not recommended for brittle material since attrition takes place and an endpoint is difficult to define. If the rate of passage of particles does not decrease with sieving time, it may be due to particle attrition, deagglomeration or a damaged sieve [72]. Machine sieving is performed by stacking sieves in ascending order of aperture size and placing the powder on the top (coarsest aperture) sieve. Conventional dry sieving machines use a vibratory action; the most aggressive machine action is performed with Pascal Inclyno and Tyler Rotap sieves, which combine a gyratory and jolting movement. Electromagnetic drives on some of the later machines causes a 3dimensional action that spreads the powder over the entire sieving surface. The amplitude of vibration can be adjusted over a range from around 0.2 to 3 mm and the frequency of vibration is typically around 3000 Hz. Other sieves employ a circular horizontal movement.
228 Powder sampling and particle size determination
Automatic machines are also available which use an air jet to clear the sieves or ultrasonics to effect passage through the apertures. The sonic sifter combines two actions, a vertically oscillating column of air and a repetitive mechanical pulse. Wet sieving is frequently used with cohesive powders. Dry sieving is often possible with the coarser micromesh sieves, and this may be speeded up and the lower limit extended, with air jet and sonic sifting. Agglomeration of the particles can sometimes be reduced by drying or adding about 1% of a dispersant such as stearic acid or fumed silica. If this is unsuccessful, wet methods may have to be used [35]. The effect of using stearic acid and triethanolamine on sieving sub-100 |im limestone has also been reported [73]. Some materials tend to form granules when sieved. Coating the particles in order to reduce cohesiveness can often reduce this effect. Powders may, for example, be shaken in a container with 1% fatty acid (stearic acid is often used) or fumed silica. Alternatively, the powder may be sieved wet. The addition of 0.1% sub-sieve carbon black has been found useful (although rather messy) for eliminating electrostatic charge. Brittle powders are best sieved using a gentle sieving action as is found, for example, with the Air Jet Sieve or the Sonic Sifter. For reliable data, around 5% of the sample should be retained on the coarsest sieve and a similar amount should pass through the finest. Unless the size distribution is narrow, alternate sieves (a root two progression of aperture sizes) is recommended. With machine sieving, sieve motion should minimize the risk of aperture blockage and preferably include a jolting action to remove particles that are wedged in the sieve mesh. The time of sieving is closely related to sieve loading, a reduction in the latter resulting in a reduction of the former. It is usual in routine analyses to machine sieve for 20 min. If more accurate data are required it is preferable to sieve for 10 min. and weigh the residues. Repeat 2 min. sieving should then be carried out until the amount passing any given sieve is less than 0.2% of the initial load. With brittle material care should be taken that particle breakage does not occur. Granular particles pass through the sieve more rapidly than elongated particles although spherical particles are inclined to block the apertures. Several variants of microsieving(5-30 |Lim) using ultrasound have been examined [74]. Efficient sieving was only possible by using a liquid column and high suspension transport. The required conditions were determined for quartz and chalk to give an undersize free residue.
Sieving 229 Ultrasonics are frequently used as an aid to sieving or for cleaning blocked sieves; the danger of rupturing the delicate mesh is possible under these conditions, readily occurring at low frequencies circ.50 Hz and sometimes at frequencies as high as 20 kHz. A recommended safe frequency, according to Colon, is 40 kHz though Craw^ley [75] reports damage to an 11 |am sieve at a frequency of 80 kHz and a power level of 40 W and subsequently recommends a frequency of 800 kHz and a power level of 20 W. Daescher [76] confirmed Rosenberg's [77] findings that the rate of cavitation is less in hydrocarbons than in alcohol and about six times greater in water than in alcohol. Saturating the alcohol with carbon dioxide was found to reduce the rate. He tested three ultrasonic cleaners, 40 kHz, 60 W; 40 kHz, 100 W; 90 kHz, 40 W, and found that the higher frequency cleaner produced the least amount of erosion. Veco recommend 15 to 20 s at a time, with a low power 40 kHz ultrasonic bath containing an equi-volume mixture of isopropyl or ethyl alcohol and water, with the sieve in a vertical position, and this is in accord with ASTM El61-70. Alpine recommend a cleaning time varying from 10 to 20 min. for 10 |Lim sieves down to 2 to 4 min. for sieves in the size range 50 to 100 |im. In a comparison between round and square hole micromesh sieves [78] it was found that the effective apertures of the round mesh sieves, using sand, was 21% greater than the square mesh apertures. 4.10 Amount of sample required In determining the amount of sample to be used, it is necessary to consider the type of material, its sievability, and the range of sizes present [79]. Two opposing criteria must be met; it is necessary to use sufficient material for accurate weighing and a small enough sample that the sieving operation is completed in a reasonable time. The natural tendency is to use too large a sample though, in practice, the smaller the sample, within limits, the more reproducible the data. Table 4.6 Amount of sample required for a sieve analysis on an 8 in diameter sieve (a) Based on particle density 1
Density (g cm^^) Sample weight (g)
1.5 25
1.5-3.0 50
+3 100
230 Powder sampling and particle size determination
(b) Based on median diameter Median, (mm) Sample weight (g)
> 2 2-1 500 200
1-0.5 100
0.50-0.25 75
0.25-0.075 < 0.075 50 25
As a rough guide, the amounts recommended for 8 in diameter sieves are given in Table 4.6a. Alternatively, the sample weight may be based on the median particle size (Table 4.6b), but this neglects to take into account that the narrower the distribution, the smaller the sample required [80]. 4.11 Hand sieving Hand sieving is time consuming but necessary for dependable dry sieving data. A representative sample is obtained and the whole of the sample used in the analysis. The preferred method of sampling is with a spinning riffler or, failing that, a chute splitter. Coning and quartering induces segregation and should never be used with free-flowing powders. It is recommended that, for a dry sieving operation, the fines be removed prior to the sieve analysis. This is effected by pre-sieving, usually by hand, on the finest sieve to be used in the subsequent analysis. If this is not done, the fines have to pass through the whole stack of sieves, thus increasing the time of the sieving operation and increasing the risk of high powder loss. Since small particles often adhere to large ones, this pre-sieving may be carried out wet, as recommended in ASTM C925 [24], using water (with wetting agent if necessary) or some other liquid in which the powder is insoluble. The smallest aperture sieve to be used should be rested on a catch pan, the tared sample placed on the mesh and the whole sealed with a lid. The sieve should be slightly inclined to the horizontal and rapped with a cylindrical piece of wood about 20 cm long and 3 cm in diameter; this should be wrapped in duct tape to eliminate splintering. (The heel of the hand is recommended in some standards and is an acceptable alternative if the hand can take it!). The rate of tapping should be about 150 per min. and the sieve should be rotated 1/8 of a turn every 25 raps. After about 10 min. the residue is transferred to the coarsest sieve that is nested in a second catch pan for subsequent weighing. It is suggested that white card be placed on the bench (approximately 60 cm square) so that accidental spillage may be recovered. The process is repeated in 2 min. cycles until less than 0.2% of the original charge passes through the sieve. The powder is not normally removed from the sieve unless excessive blinding is
Sieving 231 occurring; both the sieve and the residue are weighed and the residual weight determined. At the end of the sieving operation the sieve is upturned on a white sheet of paper and the fine particles adhering to it removed with a soft brush and added to the sieve residue. The process is repeated with sieves of increasing fineness and the residue weights collated. Finally, the process is repeated with the finest sieve and the fines are added to the dust collected initially. Brushing is not recommended for sieves with aperture less than 200 |Lim due to the possibility of damage. Sieves should be washed and dried after use. Ultrasonics should be used to remove particles clogging the apertures or these may be leached out if this can be done without damage to the sieve. The results may be expressed in terms of the nominal size, although it is preferable to use calibrated sieves. A reference set of sieves should be used after every fiftieth analysis for comparison purposes in order to detect wear. In essence, the smaller the sieve loading, the more rapid is the sieving operation. The low weights however lead to errors in weighing and intolerable percentage losses. 4.12 Machine sieving Machine sieving is carried out by stacking the sieves in ascending order of aperture size and placing the powder on the top sieve. A closed pan, a receiver, is placed at the bottom of the stack to collect the fines and a lid is placed at the top to prevent loss of powder. A stack usually consists of five or six sieves in a root two progression of aperture size. The stack of sieves is clamped on to a test sieve shaker that is vibrated for a fixed time and the residual weight of powder on each sieve is determined. Results are usually expressed in the form of a cumulative percentage of the nominal sieve aperture. The three essentials required in a test sieve shaker are: An effective sieving action in order that an end-point is reached this end-point to be reached in a short time and reproducible results. For routine control purposes it is usual to machine sieve for 20 min., after which time the sieving operation is deemed to be complete. It is recommended in BS 1796 [14] that sieving be continued until less than 0.2% of the sample passes through in any 2-minute sieving period. In ASTM D452 [81] a 20 min. initial sieving period is recommended followed by 10 min. periods during which the amount passing should be less than 0.5% of the total feed (Figure 4.8). For coarse aggregates, sieving is deemed complete when the rate falls below 1% per min [22].
2i2 Powder sampling and particle size determination 100
End point Ou
10 End-point criteria BS 1796 0.2% in 5 min. ASTM D452 0.5% in 10 min. 1
•
1
I
I
I I I I i411
I • I
I I I I II I
I
I
10 100 Sieving time in minutes
I
I • • I ll
1000
Fig. 4.8 Amount passing through a sieve as a function of time. It is generally recommended that if losses during sieving exceed 0.5% of the total feed, the test should be discarded. Preliminary hand sieving on the finest sieve should be carried out for the removal of dust. This dust would otherwise pass through the whole nest of sieves and greatly prolong the sieving time; it would also percolate between sieves in the nest and increase powder loss. Extreme care needs to be taken when weighing the sieve residue. The powder retained on each sieve should be emptied on to a sieving paper and the underside of the sieve brushed lightly to remove any particles adhering to the sieve. In BS 1796 [14] which applies to the sieving of material from 3350 to 53 |im in size, it is suggested that the sieving operation be carried out in 5 min. stages at the end of which the sieves should be emptied and brushed in order to reduce aperture blockage. This procedure can however lead to excessive powder loss. While developing a protocol for a system that will serve as a standard, NIST recommend sieving a given quantity of the material for varying amounts of time, e.g. 10 min. to 30 min. at 5 min. intervals. Sieving time is selected based on the time period after which no significant difference in mass change is observed. Sieving for a fixed period of time is more convenient than sieving until less than a certain fraction of the total mass passes through the sieve in a given period of time. The time and effort required, if the latter procedure is adopted, can be considerable and would
Sieving 233 prove to be too onerous if numerous samples were to be analyzed. Test sieving procedures covering a broad band of materials are also described in ISO 2591 test sieving procedures [82]. The sieving action of many commercial machines is highly suspect and frequently subsequent hand sieving will produce a sieving rate far greater than is produced on the machine. For the vibratory test sieve shaker a rapid vertical movement is needed to help keep the apertures clear and prevent blinding. The best type of sieving action is found with the types of shakers exemplified by the Pascal Inclyno and the Tyler Ro-tap, which combine a gyratory and a jolting movement, although the simpler vibratory sieves may be suitable in specific cases. ASTM B214 suggests 270 to 300 rotations per min. for granular materials combined with 140 to 160 taps to reduce blinding of sieve apertures. Tyler also markets a sieve enclosure to reduce noise levels from approximately 85 dB to 60 dB. The Endecott Octagon digital sieve shaker has a controller, which is used to set the sieving time and amplitude of vibration. These laboratory shakers are fitted with a clamping device to ensure that the nest of sieves is held firmly without over tightening. The Endecott EFL 2000 series are rugged shakers ideal for heavy-duty applications. The Endecott Star 2000 is a dedicated sieve test analyzer and recorder that incorporates a precision balance linked to a microprocessor and printer. The Star memorizes the sieve weight before and after sieving and makes the necessary calculations to generate a size distribution. The test results can be stored for future reference as a master. Current test results can then be compared with stored data for up to five masters. Modifications to the methods may be necessary for materials that are not free-flowing, are highly hygroscopic, very fragile, have abnormal particle shapes or have other properties that cause difficulty in sieving. For example, in ASTM C92 the fines are first removed by washing through the finest sieve; the residue is then dried and analyzed in the dry state. Vorti-siv manufacture gyratory and ultrasonic lab sieves that include a de-blinding kit for sieving powders as small as 5 |Lim. Units operate at 1,750 or 3,450 rpm and have an 8, 10 or 12 in diameter screen. Options include stainless steel or explosive proof construction and continuous discharge chutes. Large-scale sieving machines, to take a charge of 50 to 100 kg, are needed for the coarse range [83]. A wide range of commercial sieve shakers is available for the medium range and these usually classify the powder into five or six fractions with a loading of 50 to 100 g. Special sieving techniques are used with the finer micromesh sieves.
234 Powder sampling and particle size determination
4.13 Wet sieving 4.13.1 Manual Several manual methods of wet sieving using micromesh sieves have been described. Mullion [84] uses an 80 kHz, 40 W ultrasonic bath in which the micromesh sieve rests on a support, which, in turn, rests on a beaker in the bath. Sieving intervals are 5 min. with an initial load of 1 g. and the operation is deemed complete when no further powder can be seen passing through the sieve. Colon [85] rinses the fines through the sieve aperture with a suitable liquid after 0.5 to 1 g of sample has been dispersed in a small volume of the liquid. Sieving continues by moving the sieve up and down in a glass beaker filled with the same liquid so that the direction of flow through the sieve is continuously reversed. If necessary, ultrasonics may be used. After a standardized time, the operation is repeated using a second beaker containing fresh sieving liquid. Sieving is deemed complete when the amount passing through the sieve is visibly negligible. Niedick [86] disperses about 1 g of powder in 1 liter of liquid and pours this through a sieve in a retort stand. The suspension passing through is channeled with a funnel into a second container. For woven wire and coarse micromesh sieves, the sieves may be mechanically rapped to facilitate sieving. The residual powder is then rinsed off the sieve and weighed or the pre-tared sieve is dried and weighed. With fine micromesh sieves and water as the dispersant, surface tension prevents the suspension from passing through the sieve, so that after filling the sieve with suspension an ultrasonic probe is used to initiate flow. Alternatively, a low surface tension liquid may be used. The procedure is then repeated with sieves of increasing fineness. Daescher [87] describes a method using a set of tared sieves mounted on a special funnel held in a filter flask. 1 to 3 g of powder are placed on the top sieve and washed through each sieve in turn with a suitable polar liquid or hydrocarbon containing a trace of dispersant. At the same time, alternate pulses of pressure and suction are applied to the filter flask. This pulsating action orientates the particles in such a way as to speed up the sieving action. A full analysis can be completed in less than an hour.
Sieving
235
Separating funnel
Non-rigid tube
Shower head or spray nozzle
Sieving machine
li
FUter unit Fi Low pressure connection
Fig. 4.9 The Retsch wet sieving machine. loos [88] describes a device that makes it possible to carry out particle size analysis from 60 |Lim down to 5 |j.m by means of sieves arranged vertically above one another and subjecting this nest of sieves to ultrasonics. A method of wet sieving clays is described in ASTM C325 [89] in which the fines are washed out first and the rest are washed through a nest of sieves. In ASTM D313 and CI 17 washing through a 200-mesh sieve is suggested for the removal of fines and in D185-72 the use of a camel-hair brush is recommended to facilitate passage through a 325-mesh sieve. Other investigators have described similar methods. [90,91]. Ultrasonic agitation is also required for cleaning fine micromesh sieves. Veco recommend 15 to 20 s at a time in a low power 40 kHz ultrasonic bath containing an equ7al volume mixture of isopropyl or ethyl alcohol and water with the sieve in a vertical position 4.13.2 Wet sieving by machine Several writers have proposed automated wet sieving procedures [92,93]. In most of these methods, a stack of sieves is filled with a liquid and the
236 Powder sampling and particle size determination
sample is fed into the top sieve. Sieving is accomplished by rinsing, using vibration, using a reciprocating action, applying vacuum, applying ultrasonics or a combination of these [94]. Commercial equipment is available in which the sample is placed in the top sieve of a stack of sieves and sprayed with water whilst the stack is vibrated (Figure 4.9). In the Retsch water jet sieve a spray ring is pushed over each analysis sieve and a spray arm with 34 nozzles rotates in each ring due to water pressure. This ensures that the whole sieve surface is evenly sprayed. Up to five analysis sieves of diameter 200 mm can be clamped in the spray tower. A wet sieving device for the size range 10 to 100 jiim that includes a sieve vibrator of variable amplitude is commercially available (Figure 4.10). The device consists of a variable-amplitude vibrator (1) with support (2), microsieve (3), regulator with voltmeter for amplitude (4) and timer (5).
^
£^
B
O ^C^ o DtD
Fig. 4.10 The Alpine wet sieving device. Hosokawa Mikropul Micron Washsieve is one version of a wet sieving process where water is sprayed on to the surface of a vibrating sieve (Figure 4.11). The machine consists of a sprinkler section, a sieving section and an electromagnetic section. The sprinkler rotates through the force of water to give an even spray whilst the sieve is vibrated to prevent blockage.
Sieving 237 Gallenkamp Gallie-Porritt apparatus (BS 4398) consists of a metal funnel terminating in a short cylindrical outlet in which a wire sieve cloth is soldered. Water, at a pressure greater than 2 bar, is supplied by a nozzle to discharge a spreading jet through the sieve. A similar arrangement is provided for another tube to give a gentle stream of water to keep the level of the water in the funnel constant throughout the test. About 25 g of powder is slurried and introduced into the funnel at the commencement of the test, which continues until the water issuing from the apparatus, is clear. The residual mass is determined in order to find the mass percentage undersize.
Fig. 4.11 The Hosakawa Mikropul Micron Washsieve. 4.14 Air-jet sieving The principle of operation of this instrument (Figure 4.12) is that air is drawn upwards, through a sieve, from a rotating slit so that material on the sieve is fluidized. At the same time, a negative pressure is applied to the bottom of the sieve that removes fine particles to a collecting device (a filter paper). With this technique, there is a reduced tendency to blind the apertures, and the action is very gentle, making it suitable for brittle and fragile powders. Sieving is possible with some powders down to 10 |Lim in size but with others, balling occurs [95,96].
238 Powder sampling and particle size determination
Vacuum tight lid
Rotating arm
Fig. 4.12 Mode of action of the Alpine Air Jet Sieve Top diaphragm
^
Fines collector ^ holder
Fines collector Sample prior to sieving
Sample in lift position
Sample in sift position
Sample sifted
Fig. 4.13 Mode of action of the Sonic Sifter [103]. The reproducibility is much better than by hand or machine sieving. Size analyses are performed by removal of particles from the fine end of the size distribution by using single sieves consecutively. The end-point of sieving can be determined by microscopic examination of the cleanness of the sample [97] or by adopting the same criteria as are used in conventional dry sieving. Since only one sieve is in operation at any one time, a full analysis may be unduly protracted although with sieves coarser than 30 )Lim the run time is less than 3 min.
Sieving 239 Sieving is more protracted with finer mesh sieves, and a sieving time of 20min. is usual with a 1 g load on a 3in diameter 20 \m\ sieve. Similar instruments are marketed by Alpine, Micron Powder Systems and by MicroPul. Jones [98] has presented a discussion of this technique and Lauer [83,99] has appraised it by microscopic examination of the powder fractions. 4.15 The Sonic Sifter ATM Sonic Sifter [100] is produced as a laboratory (LP3) and as an industrial (P60) model. It is claimed to be able to separate particles in the 2000 to 20 |im size range for most materials and 5660 to 5 |Lim in some cases. It combines two motions to provide particle separation, a vertical oscillating column of air and a repetitive mechanical pulse. The Sonic Sifter moves the air in the sieve stack (Figure 4.13). The oscillating air sets the sample in periodic vertical motion, which reduces sieve blinding and breaks down aggregates and yet produces very little abrasion, thus reducing sieve wear and particle breakage. Gilson GA-6 Autosiever uses a unique double tapping action together with sonic sifting. The GA-6 features a memory for storing up to 10 programs. Its performance in the BCR round-robin exercise with the Whitehouse 20-100 |Lim reference standard has been reported. The average sample loss after sieving five samples from the 100-stage riffle was 0.4% and the standard deviation was 1.46% [101]. A downward air flow has been found to improve sieving rate for 10, 25 and 45 |Lim electroformed sieves using a sonic sifter operating at 107 dB and 78 Hz [102]. The sieving rate also increased with decreasing feed rate, especially for sub-10 |am sieves [103]. 4.16 The Seishin Robot Sifter Seishin Robot Sifter is an automated version of the ATM in that it includes a robot arm that transports the empty sieves to a balance where they are weighed and the data stored. At the end of this operation, a nest of sieves has been assembled on the balance ready to receive a sample of powder on the top sieve. The robot arm then transfers the nest of sieves to the sieving position where sieving is automatically performed. Upon completion of sieving the sieves are transferred to the balance and reweighed. The data is stored in a CPU and can be printed out as required (Figure 4.14).
240 Powder sampling and particle size determination
^
-^—•
®
Sieving mechanism
n* (2)
Robot mechanism
Balance
Fig. 4.14 The Seishin Robot Sifter 4.17 Automatic systems 4.17.1 The Rotex Gradex 2000 particle size analyzer Rotex use an octagonal compartment drum that can be arranged with up to 14 testing sieves (Figure 4.15). The operator selects the preset cycle for the sample to be analyzed and then loads the sample into the drum, via a feed chute, on to the sieve having the smallest opening. The operator then initiates the automatic cycle, which imparts a reciprocating motion to the drum, separating the smallest size fraction from the sample. At the completion of the cycle for this separation, the reciprocating motion stops and the material, which passed through the sieve, is weighed. The sample then passes to the next coarsest sieve and the operation is repeated. Up to six samples can be loaded into an automatic dispenser on top of the analyzer. The cycle continues for each sieve in the cycle and the material passing through each sieve is accumulated on an electronic scale. A computer reads and stores the weight of each fraction, then tares the balance before the cycle proceeds to the next sieve. The computer uses this data to calculate then print out the size distribution. When the
Sieving
241
Autotime computer controlled cycle is selected, the analysis is stopped when the throughput rate decreases to a set value. Test times vary from 5 to 15 min depending on the analysis required [35]. 4.17.2 Labcon automatic sieve system Labcon PI 14 computer system consists of an Epsom Pine computer mounted in a purpose made portable case containing the necessary power supplies and interface components to allow control of a sieve shaker and electronic balance. Firmware in Epsom is included to provide a variety of methods of data handling and presentation procedures. 4.17.3 Gilson Compu-Sieve« analysis system This system includes a computer with interface for electronic scale or balance for direct data collection. Any electronic balance with RS-232 interface can be connected to the system. A record of test data is produced by color plotter. Fifteen sieves and pan can be handled. 4.18 Ultrasonic sieving Retsch ultrasonic sieving apparatus was developed for accurate sieving with micromesh sieves in the 5-100 |am range. The ultrasonic sieving apparatus has two power stages specifically geared to the gentle sieving (40 W) and intensive cleaning (80 W) of the sieves. The sieve tower with up to five sieves is filled with the sieving liquid, inserted in the sieve fixture of the device and lowered into an ultrasonic bath. Sieve loading is 0.2 to 0.5 g unless the sub-5 |Lim fraction is small, in which case it can be increased to around 2 g. The sieves are held in the stack and sealed from each other with radial packing rings. Gravitational action carries the suspension through the stack through progressively smaller meshes. The Vibrosonic sieve employs ultrasonic vibration of the separator mesh for industrial separation at high rates down to 400-mesh. 4.19 The sieve cascadograph This technique was designed to classify particles according to shape and to provide a shape distribution profile for powders having a narrow size distribution [104].
242 Powder sampling and particle size determination
Fig. 4.15 Rotex Gradex 2000 The cascadograph consists of a stack of sieves of the same nominal aperture size in a shaking device (a Tyler Ro-Tap sieve shaker). Each stack consists of twenty 8 in diameter sieves specially made by the Newark Wire Cloth Company; Newark sieve cloth is claimed to have very uniformly sized apertures and a low tendency to 'blind'. The narrow size distribution is prepared by pre-sieving for 1 h and then washing to remove fines. The feed for the 100-mesh cascadograph is the 100 to 120-mesh pre-sieved fraction with a maximum loading of 2.5 g. Samples are removed from the cascadograph after 20, 40 and 80 s with subsequent time doubling until completion. A plot of weight fraction removed against sieving time plotted on a log scale provides a shape signature of the powder. This was extended to continuous sieve cascadography to generate narrow distributions of spherical particles [105,106]. Theoretically, classification by sieving gives a size range of 1.2:1, whereas the new process can reduce this to a 1% variation. It is
Sieving 243 claimed to be economical with operating costs of less than a dollar per pound [107]. 4.20 Felvation The term felvation has been applied to a technique for grading powders using an elutriation process, with the sieves acting as stops for the coarse powder in suspension [108]. The powder to be classified is dispersed in a premixing unit which is connected to a felvation column. A needle valve is opened to allow liquid from the header tank to carry suspended particles into the conical base of the column, where they are fluidized. The flow rate in increased gradually until the finer particles are elutriated up the column to an Vi in square or 3/4 in round micromesh or woven wire sieve cloth. The particles continue into the next felvation column, where they meet, and if fine enough pass through, the sieving surface. The flow rate is increased in steps until larger and larger particles are elutriated and continued until the ascending particles are too large to pass through the sieves. The end point is when the liquid above the sieves becomes clear. The powder passing the finest sieve is collected on a suction filter and the various fractions are contained in the bodies of the appropriate columns. Burt used 0.5 to 1.0 g samples for the micromesh sieves and found that the efficiency increased with decreasing sample size and then only when three felvation units were used. Separation efficiency increased if the fluid flow was pulsed two or three times a second so a pulsator was added to the equipment. The technique was found to be unsuitable with micromesh sieves, which were too fragile, but was more successful with woven wire sieves. The technique was not proposed as an alternative to standard sieving methods, but may be useful if only small quantities are available, or with hazardous materials, where small samples are desirable for safety reasons. The technique has also been used to grade 5 kg samples [109] in the size range 45 to 64 |Lim. 4.21 Self organized sieves (SORSI) This system was developed in an attempt to eliminate some of the problems associated with nest sieving and to automate the process [110,111]. A hexagonal sieving chamber is supported on discs and rotated. A steady stream of powder is fed to the entry port and, as the sieve rotates, the powder gently moves across the surface. As the sieve rotates further, the powder is picked up by a vane and dumped out of the other side of the
244 Powder sampling and particle size determination
sieve. An air jet automatically cleans the mesh every cycle in order to eliminate blinding and, while in a vertical position, it is interrogated by a laser beam to see if there are any damaged, distorted or blinded apertures. A very low rate of feed and a series of mesh sizes can be used in sequence, and the fractions passing the sieves can be collected and weighed automatically to give a continuously generated sieve analysis of the powder. Kaye postulated that partitioning of sieves should improve performance [112]. For new sieves the system isolates the effects of extremes of aperture size to give a distribution more nearly related to nominal size than a conventional sieving operation. The better mechanical support should also reduce wear and tear. Kaye suggests the reduction of an 8 in diameter sieve to a honeycomb of 0.5 in square partitions. A variation of this system is the installation of 0.5 in diameter sieves mounted on a spinning riffler to obtain small, representative powder samples, which can be sieved in situ without removing the miniature sieves from the riffler system [113]. By mounting several different aperture sieves around the riffler, a confederation of miniature sieves could replace nest sieving. 4.22 Shape separation Wire mesh screens have been used as a length classifier of fibrous aerosols [114]. Glass fiber aerosols were sieved through a nest of sieves and the length distribution of the sieve residues decreased with decreasing aperture size. The rate of passage through the sieves depended only on the ratio of fiber length to aperture size. This rate was simulated using a Monte Carlo method and the experimental data were in good accord with theory. If a log-normal distribution is assumed for fiber length, and the rate of passage through the sieves determined for a few sieves, a fiber length distribution can be obtained using a calibration chart. Kaye and Yousufzai [115] discuss the use of trapped particles, i.e. the particles blinding the sieve mesh at the completion of sieving, for determination of elongation ratio. A general formula was developed [116] on the probability of particles of different shape passing through a sieve and the formula has been applied to spherical and needle shaped particles. A set of sieves with rectangular apertures was used in combination with a set of sieves with square apertures to separate according to shape. The separation was superior to that obtained using a vibrating deck [117].
Sieving 245 4.23 Correlation with light scattering data In many industries, particle size measurements have been carried out historically by sieve analysis and light scattering instruments are increasingly replacing this. In order to correlate with historic data banks some manufacturers have software to manipulate the data so as to present the size distribution in terms of sieve diameter. Table 4.7 Particle size distribution of BCR 68 by four laboratories Equivalent volume diameter (|xm) 160 250 320 400 500 630
Lab. 1 4.7 21.9 44.0 68.6 89.6 96.8
Mass percentage undersize Lab. 4 Lab. 3 Lab. 2 4.4 4.3 4.4 24.5 23.0 24.6 47.6 44.5 45.4 66.9 70.6 70.0 88.5 87.7 88.6 96.0 97.0 98.1
Tests carried out on pneumatically conveyed salt particles showed that Insitec data gave size distributions that were over 30% coarser than sieve data, the difference being attributable to particle shape effects. The results for both instruments showed that the particles attrited with number of passes through the system, with the Insitec being more sensitive than sieving [118]. 4.24 Conclusions According to Hey wood [1] sieving is the Cinderella of particle size analysis methods; it does most of the hard work and gets little consideration. This was reiterated by Leschonski [119] who also quotes the chairman of the Institution of Mining and Metallurgy as stating, in 1903, that screening is not a scientific means of measurement. However, measuring particle size distributions by sieving is simple and inexpensive and can give reproducible results, even when using different sets of sieves, if proper care is taken. Although most of the problems encountered in sieving have been known for years and solutions proposed, reproducibility is rarely achieved, owing to the failure to take advantage of this knowledge.
246 Powder sampling and particle size determination
For accuracy, it is necessary that the sieves be calibrated and, if the sieves are dedicated to a single powder, the calibration should be carried out with the powder under test. It is also necessary that the sieves be checked, on a regular basis using a calibration powder, so that worn sieves can be rejected. Normally, if a sieve analysis is plotted on log-probability paper, a smooth curve results; any points lying off the curve should be viewed with suspicion. For reproducibility, a standard operating procedure should be adopted. Table 4.7 illustrates the reproducibility that can be attained using calibrated sieves. The data is taken from the certification report on reference materials of defined particle size issued by the Commission of the European Communities and refers to data generated by four laboratories with BCR 68. References 1 2 3 4 5 6 7
8
9
10 11 12 13
Heywood, H. (1970), Proc. Particle Size Analysis Conf., ed. M.J. Groves and J.L. Wyatt-Sargent, See. Analyt. Chem., 1-18, 208, 210, 245 Rittinger, P.R. von (1887), Aufbereit, 222 & 243, 208, 210 Rudolph, A., Peters, C. and Schuster, M. (1992), Aufhereit Tech., 33, 384386,389-391,2/0 Leschonski, K. (1977), Proc. Particle Size Analysis Conf., Chem. Soc, Analyt. Div., ed. M.J. Groves, publ. Heyden, London, 205-217, 210, 221 Miwa, S. (1984), Shikizai Kyokaishi, 57(16), 334-341, 210 Miwa, S. (1984), Funtai to Kogyo, 16(1), 21-25, 210 STP 447B, Manual on testing sieving methods, (1998) Guidelines for establishing sieve analysis procedures (4th edition), Available from ASTM 1916 Race Street, Philadelphia, PA 19103, 2/0 Jillavenkatesa, A., Dapkunas, S.J. and Lum, L-S. H. (2001), Particle Size Characterization, NIST Sp. Publ. 960-1, NIST Recommended Practice Guide, National Institute of Standards and Technology, 211 ASTM E1919-97, Standard guide for worldwide published standards relating to particle and spray characterization, American Society for Testing Materials, West Conshohoken, 211 ASTM-11 -95, Standard specification for wire cloth sieves for testing purposes, 211,213 ASTM E161 -87, Reapproved (1992), Standard specifications for precision electroformed sieves (square aperture), 211,214 ASTM 323-80, Reapproved (1990), Standard specifications for perforated plate sieves for testing purposes, 211,213 BS 410 (1986), (Replaced 2000) Specifications for test sieves, 211, 213
Sieving
14
15 16 17 18 19 20 21 22 23
24
25
26 27 28 29 30 31 32 33 34 35
247
BS 1796 Test Sieving, Part I, (1989), Method using test sieves of woven wire and perforated plate, available from BSI Sales Dept., Linford Wood, Milton Keynes, MK14 61E, UK, 21 f 227, 231, 232 DIN 4187, Perforated plate sieves, 211 DIN 4188, Woven wire test sieves, 211 DIN 4195, Textile cloth sieves, 211 AFNOR-NFX 11-501,(1957) Woven wire test sieves, 211 ISO 3310-1,(1990) Test Sieves - Technical Requirements and Testing - Part 1, Test sieves of metal wire cloth, 211 ISO 3310-2 1990, Test Sieves - Technical Requirements and Testing- Part 2, Test sieves of perforated plate, 211 ISO 3310-3 1990, Test Sieves - Technical Requirements and Testing- Part 3, Test sieves of electroformed sheets, 211 ASTM C136-96a Standard test methodfor sieve analysis of fine and coarse ^gg^^g^tes, 212, 231 ASTM C285-88 (Re-approved 1994) Standard test methodfor sieve analysis of wet milled and dry milled porcelain enamel, for determination of the fineness of frit in wet-or dry-milled porcelain enamels and other ceramic coatings on metals by use of the number 200 or No 325 mesh, 212 ASTM C925-79 (Re-approved 1995) Standard test methodfor precision electroformed wet sieve analysis of non-plastic ceramic powders, for particle size distribution determination of pulverized alumina and quartz for particle sizes from 45 pm to 5 pm by wet sieving, 212, 230 ASTM C1921 -96 Standard test methodfor particle size (sieve analysis) of plastic materials, applicable to particle size determination of plastic materials in powdered, granular or pelleted forms by dry sieving with a lower limit of measurement of about 38 pm, 212 ISO 8130-1 (1992) Coating Powders-Part 1, Determination ofparticle size distribution by sieving, 212 ISO 6274 (1982), Concrete - Sieve analysis of aggregates, 212 ISO 4497 (1983), Metallic powders, Determination of particle size by dry sieving, 212 ASTM D1214-89 (Re-approved 1994), Standard test methodfor sieve analysis of glass spheres, 212 ISO 2395 (1990), Test sieves and test sieving-vocabulary, 212 ASTM El638-94, Standard terminology relating to sieve analysis, 212 Bates, R.H. (1981), Proc. Filtech. Conf, 299-308, Uplands Press., Croydon, U.K., 213 Wahl, W. (1976), Ger. Offen, 2 413 521, 213 Stork-Brabent, B.V. (1975), Neth. Appl, 74 07348, 213 Daescher, M.W., Sebert, E.E. and Peters, E.D. (1958), Symp. Particle Size Measurement, ASTM Sp. Publ. 234, 26-56, 214, 228, 241
248 Powder sampling and particle size determination
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53
54 55 56 57 58 59 60 61 62
Zwicker, J.D. (1966), Report No. AW-FR-2-66, Aluminum Company of Canada Ltd., Arvida, Canada, 214 Burt, M.W.G. (1970), Proc. Soc. Analyt. Chem., 7(9). 165-168, 216 Colon, F.J. (1965), Chemy Ind., 206, Feb., 216 Serkowski, S. and Paulowski, S. (1988), Polish Pat., 153 180, June, 218 Sundukov, V.K., Chechetkin, A.Y. and Zhukov, A.N. (1993), USSR Pat., SU 1,788,095 (CI C25D1/108), Jan., 218 Heidenreich, E. (1977), Proc. Particle Size Analysis Conf, 471 -472, Analyt. Div. Chem. Soc., ed. M.J. Groves, publ. Heyden, 218 Niklas, U. (1990), Sonderb. Prokt. Metallogr., 21, 55-60, 218 Thuis, H.H.W. (1990), (Stork Screens B.V.), Int. Appl. WO 90 06 381 (CI 23F1/04, June, 27(5 Stork Screens (1982), Neth. Appl. NL 82 04 381, 218 Anon., (1991), Particle Post, November, publ. ATM Test Sieves Inc., 218 Whitby, K.T. (1958), Symp. Particle Size Measurement, ASTM Sp. Publ. No 234, 3-25, 218 Kaye, B.H. (1962), Powder Metall, 10, 199-217, 220 Jansen. M.L. and Glastonberry, J.R. (1968), Powder TechnoL, 1, 334-343, 220 Ilantzis, M. A. (1961), Ann. Inst. Tech. Batim. Trav. Publics., 14(161), 484, 221 Ilantziz, M.A., (1961), Ann.Inst. Tech. Batim. Trav. Publics, 14(161), 484, 222 Kaye, B.H. and Yousufrai, M.A.K (1992), Powder and Bulk Engineering, 2, 29-35, 223 Allen, T. (1994), Powder Technology, 79(6), 61 -68, 224 Community Bureau of Reference (BCR), Commission of the European Communities, Directorate General for Science, Research and Development, 200 rue de la Loi, B-1049, Brussels, Belgium, 224 Calboreanu, G. (1991), Trans. Am. Foundrymen's Soc, 99, 111-116, 224 Rideal, G.R., Storey, J. and Morris, T.R. (2000), Part. Part. Syst. Charact., 17, 77-82, 224 Shergold, F.A. (1946), Trans. Soc. Chem. Eng,, 65, 245, 226 Heywood, H. (1945/46), Trans. Inst. Min. Metall., 55, 373, 226 Fahrenwald, A.W. and Stockdale, S.W. (1929), U.S Bureau of Mines Investigation 2933, 227 Kaye, B.H. (1962), Powder Metall., 10, 199-217, 227 Weber, M. and Moran, R.F. (1938), Ind Engng Chem., Analyt. Div., 19, 180,227 Carpenter, F.G. and Dietz, V.K. (1950), J. Res. Natnl. Bureau Stand., 47, 139,227 Moltinin, E. (1956), Ind. Miner aria, (Rome), 7,11 \, Appl. Meek Rev., 11,345,227
Sieving
63 64 65 66 67 68 69 70 71 72 73 74 75 76 11 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94
249
Carpenter, F.G. and Dietz, V.K. (1950), J. Res. Natnl Bureau Stand., 5, 328, 227 Allen, M. (1958), Chem. Engng., 65(19), 176, 227 Fritz, S.S. (1937), Ind. Engng Chem., 9, 180, 227 MacCalman, D. (1937), Ind Chem., 13, 464, 227 MacCalman, D. (1938), Ind Chem., 14, 64, 227 MacCalman, D. (1939), Ind Chem., 15, 161, 227 Heywood, H. (1956), Instn. Min. Metall. Bull. 55, 477, also (1956), Inst. Min. Metal, 55, 373, 227 Ackerman, L. (1948), Chem. Engng. Min. Review, 41, 211, 227 Herdan, G. (1960), Small Particle Statistics, p 121, Butterworths, 22 7 Gupta, V.S., Fuerstenan, D.W. and Mika, T.S. (1975), Powder Technol, 11(3), 257-272, 227 Husemann, K. and Hermann, R. (1996), Schuettgut, 2(4), 581-588 (Ger.), 228 Bernhardt, C. (1982), Chem. Tech., Leipzig, 34(10), 501-511, 228 Crawley, D.F.C. (1968), J. Scient. Instrum., Series 2, 576-578, 229 Daeschner, M.W. (1969), Powder Technol, 2(6), 349-355, 229 Rosenberg, L. D. (1960), Ultrasonic News, 4, 16, 229 Besancon, P., Chastang, J. and Lafaye, A. (1993), Part. Part. Syst. Char act., 10, 222-225, 229 Tyler Rotap Operating Instructions, 229 Endecotts Test Sieving Manual, London, 230 ASTM D452-91 Standard Test Methodfor Sieve Analysis ofSurfacingfor Asphalt Roofing Products, 231 ISO 2591-1 (1988), Test Sieving-Part 1 Methods using test sieves of woven wire cloth and perforated metal plate, 233 Lauer, O. (1966), Grain Size Measurements on Commercial Powders, Alpine A.G., Augsberg, Germany, 233, 239 Mullin, J.W. (1971), Chemy Ind., 50, 1435-1436, 234 Colon. F.J. (1970), Proc. Soc. Analyt. Chem., 7(9), 163-164, 234 Niedick, E.A. (1969), Z Zukerind., 19(9), 495-506, 234 Daescher, M.W. (1969), Powder Technol, 2(6), 349-355, 234 loos, E. (1965), Staub Reinhalt Luft, 25(12), 540-543, 235 ASTM C325-81 (Re-approved 1997) Standard test methodfor sieve analysis of ceramic whiteware clays, 235 Jones, T.M. (1970), Proc. Soc. Analyt. Chem., 7(9), 159-163, 235 Suhm, H.O. (1969), Powder Technol, 2(6), 356-362, 235 Irani, R.R. and Callis, C.F. (1963), Particle Size Measurement, Interpretation and Application, Wiley, N.Y., 235 Peterson, J.L. (1969), U.S. Patent 3 438 490, Method and apparatus for wet sizing finely divided materials, 235 Colon, F.J., (1970), Proc. Soc. Analyt. Chem., 7(9), 163-164, 236
250 Powder sampling and particle size determination
95 96 97 98 99 100 101 102 103 104 105 106 107
108 109 110 111 112 113 114 115 116 117 118 119
Brown, O.E., Bobrowski, G.S. and Kovall, G.E. (1970), ASTMSp. publ 473, 82-97, 237 Malhetra, V.M. and Zaldems, N.G. (1970), ASTMSp. Publ. 473, 98-105, 237 Lauer, O. (1958), Staub, 18, 306, 238 Jones, T.M. (1970), Proc. Soc. Analyt. Chem., 7(9), 159-163, 239 Lauer, O. (1960), Staub, 20, 69-71, 239 US Patent 3 045 817, 239 Rideal, G. (1996), Am. Lab., Shelton, Conn, 28(17, 46-50, 239 Yamomoto, H., Utsumi, R. and Kushida, A. (1986), Nagoya Kogyo Gijutsu Shikensho Hokoku, 35(5), 208-213, 239 Yamomoto, H., Utsumi, R. and Kushida, A. (1986), Nagoya Kogyo Gijutsu Shikensho Hokoku, 35(4), 159-164, 238,239 Kaye, B.H. (1978), Powder TechnoL, 19(3), 121-123, 241 Meloy, T.P., Clark, N.N., Dumey, T.E. and Pitchumani, B. (1985), Chem. Engg ScL, 40(7), 1077-1084, 242 Dumey, T.E. and Meloy, T.P. (1985), Int. J. Mineral Proc, 14, 313-317, 242 Meloy, T.P. and Williams, M.C. (1991), Particle Size Analysis, Proc. Conf. Particle Size Analysis Group, An. Div. Royal Soc. Chem., ed. N.G. Stanley Wood and R. Lines, Sp. Publ. 102, 514-521, 243 Kaye, B.H. (1966), Symp. Part. Size Analysis, Soc. Analyt. Chem., London, 243 Burt, M.W.G. (1970), Proc. Soc. Analyt. Chem., 7(9), 165-168, 242,243 Kaye, B.H. (1977), Dechema Monogram, 79, 1589-1615, Part B, 1-19, 243 Kaye, B.H. (1977), Proc. Powder Technol. Conf, Powder Advisory Center, Chicago, U.S., 243 Kaye, B.H. (1978), Powder Technol., 19(3), 121-123, 244 Kaye, B.H. (1985), Proc. Conf. Particle Size Analysis, 439-447, Bradford, UK, ed P.J. Lloyd, publ. John Wiley & Sons Ltd., 244 Myojo, T. (1991), J. Soc. Powder Technol, Japan, 28(8), 495-500, 244 Kaye, B.H. and Yousufzai, M.A.K. (1992), Powder and Bulk Engineering, 2, 29-35, 244 Kutepov, A.M. and Sokolov, N.V. (1983), Teor. Osn. Khim TekhnoL, 17(4), 510-518,2^^ Whiteman, M. and Ridgeway, K. (1986), Drug Dev. Ind Pharm., 12(11-13), 1995-2013,2^^ Holve, D.J. (1994), Powder and Bulk Engineering 8(82), 245 Leschonski, K. (1977), Particle Size Analysis, Proc. Conf., Analyt Div. Chem. Soc, Bradford, publ. Heyden, 245
Fluid classification 5.1 Introduction Fluid classification is a process for separating dispersed materials based on the movement of the suspended particles to different points under the effect of different forces. The fluid is usually water or air and the field force gravity, centrifugal or coriolis. The other forces of importance are the drag forces due to the relative flow between the particles and the flow media, and the inertia forces due to accelerated particle movement. The classification process is defined in terms of sorting and sizing. The former includes processes such as froth flotation where particles are separated on the basis of chemical differences and classification based on particle density. The latter, which is covered here, is based only on differences in particle size for uniform density material. A wide range of equipment is available, ranging in capacity from many tons per hour with the larger units to a few grams per hour with laboratory machines. The results of classification processes may be presented as size distributions, the accuracies of which depend on the sharpness of cut. In an ideal system the cut size is well defined and there are no coarse particles in the fme fraction and vice versa. In practice, however, there is always overlapping of sizes. The cut size may be predicted from theory but this usually differs from the actual cut size due to the difficulty of accurately predicting the flow patterns in the system. It is therefore necessary to predict the future performance of classifiers based on their past performance 5.2 Assessment of classifier efficiency [1,2] Consider a single stage of a classifier where W, W^, WJWCQ the weights of the feed, coarse stream and fine stream respectively and F(x), FJ^x), Fpc)
252 Powder sampling and particle size determination are the cumulative fraction undersize of feed, coarse and fine respectively; jc is particle size. Then: W=W^^Wf
(5.1)
And, for any element of the distribution, of width Ax W-^-^ dx
= W^-^^-L + Wf—^-'^ Ax ^ djc
(5.2)
The total fine efficiency is defined as: Ef^-^
(5.3)
The total coarse efficiency is defined as: E,=-^
(5.4)
AndE^ + Ef=\
(5.5)
The total efficiency has no value in determining the effectiveness of a classification process since it only defines how much of the feed ends up in one or other of the two outlet streams and not how much of the desired material ends up in the correct outlet stream. To discover this, it is necessary to determine the grade efficiency, which is independent of the feed provided the classifier is not overloaded. amouut of dcslrcd matcHal iu product of size x r. A nr- ' Grade etnciency = amount of desired material in feed of size x amount of coarse product of size X ^ , r-r- ' Coarse grade eiiiciency = amount of feed of size x
0M-w,S6i}.,^^(') ^
dx
Ax
Fluid classification 253
Gc{x)
GJX)
' ^ ^
W dF{x) =E , ^ ^ ^
(5.6)
' dF{x)
Similarly, the fine grade efficiency is defined as:
^
W
dF(x)
G / « =^/-Ti7Y
(5.7)
Hence, from equations (5.2), (5.6) and (5.7): G,{x) = \-Gy{x)
(5.8)
These equations are used to determine the grade efficiency of a classifier provided the total efficiency and the size distributions of two of the streams are known. Results are usually plotted as grade efficiency curves of G^(x) or Gj(x) against x [3]. Since the classifier separates on the basis of Stokes diameter it is preferable to carry out the size determinations, for grade efficiency evaluations, on the same basis. Table 5.1 presents the data for the grade efficiency plots of Figure 5.1. Column 1 gives the medians and column 2 the size limits of the ranges; columns 3 and 4 give the mass percentage undersize of the feed and the coarse stream, (data for the fine stream are not necessary since the size distribution of the fines can be deduced from these data); column 5 gives the mass frequency distribution for the feed [f(x) = dF(x)/dx]; column 6 gives the feed fraction that ends up in the coarse stream and column 7 gives the coarse grade efficiency. These data form the basis of Figures 5.1. It is preferable to know the total coarse efficiency as a check against the derived value from Figure 5.1b but this is not essential. The grade efficiency curve is best determined by plotting F^{x) against F{x) and differentiating by taking tangents (Figure 5.1b) since this allows
254 Powder sampling and particle size determination
Table 5.1 Example of grade efficiency calculation Mean size (3c)|Lim 065 0.77 0.92 1.09 1.30 1.54 1.83 2.18 2.59 3.08 3.67 4.36 5.19 6.17 7.34 8.72 10.37 12.34 14.67 17.45 20.75 24.68 29.34 34.90 41.50 49.35 58.69 69.79 83.00 98.70 117.38 139.58 1 166.00
Size limits
F{x) (%) OOO 0.01 0.03 0.07 0.14 0.28 0.51 0.91 1.55 2.54 4.02 6.12 8.99 12.77 17.52 23.26 29.93 37.34 45.26 53.38 61.36 68.89 75.71 81.64 86.58 90.53 93.57 95.80 97.38 98.45 99.15 99.59 99.85
G,{x) 1
Ax) (%)
(%/^m)
(%/|Lim)
(%)
0.00 0.00 0.00 0.00 0.00 0.01 0.03 0.07 0.16 0.35 0.71 1.38 2.53 4.38 7.19 11.24 16.68 23.58 31.83 41.09 50.88 60.63 69.76 77.80 84.38 89.45 93.17 95.81 97.60 98.77 99.51 99.95
0.074 0.125 0.210 0.311 0.528 0.718 1.052 1.422 1.833 2.312 2.763 3.171 3.516 3.725 3.788 3.685 3.446 3.099 2.671 2.207 1.753 1.334 0.975 0.683 0.459 0.297 0.183 0.109 0.062 0.034 0.018 0.009
0.000 0.000 0.000 0.000 0.000 0.031 0.053 0.089 0.167 0.297 0.474 0.740 1.070 1.451 1.855 2.238 2.5308 2.701 2.714 2.562 2.280 1.908 1.502 1.113 0.766 0.496 0.306 0.183 0.104 0.057 0.030 0.015
0.00 0.00 0.00 0.00 0.00 2.61 3.00 3.75 5.45 7.70 10.29 14.01 18.25 23.37 29.37 36.43 44.05 52.27 60.96 69.62 78.01 85.78 92.38 97.65 99.95 100 100 100 100 100 100 100
(jc) | i m
0?7i 0.85 1.01 1.20 1.42 1.69 2.01 2.39 2.84 3.38 4.02 4.78 5.68 6.76 8.03 9.55 11.36 13.51 16.06 19.10 22.72 27.01 32.12 38.20 45.43 54.02 64.24 76.40 90.85 108.04 128.48 152.79 181.70
Fluid classification 255
AW dx 3 I-
Fine fraction Feed Coarse fracticMi
(a)
20
40 60 Particle size (x) in microns
80
100
256 Powder sampling and particle size determination
100 80 60 40 h 20
20 (c)
30
40
60
Particle size (x) in microns
Fig. 5.1 Graphical determination of grade efficiency curve. experimental errors to be smoothed out. The tangent at FJ^x) = 100% in Figure 5.1b has a slope of AFJix)IF{x) = 100/60 hence, from equation (5.5), E^ = 60/100. Since this tangent merges with the curve at x = 58 |Lim all particle coarser than 58 |Lim are collected with the coarse fraction. Differentiating this curve at selected values of F(jc) and multiplying by 60 gives G^(x), the relevant diameters being determined from Table 5.1. The 50% size on the grade efficiency curve is called the equiprobable size since particles of this size have an equal chance of being in either the coarse or the fine stream; for this example e = 16.7 |Lim. Figure 5.1a shows how the feed is split between the coarse and fine fraction i.e. Wf{x)
= w/^{x)^w/pcy Two other cut sizes are also used [4]: The analytical cut size, x^ is that at which the feed is split in proportions given by the total efficiency. This definition implies that the amount of displaced material in the coarse stream is balanced by the amount of displaced coarse material in the fine stream. This is also the condition that the analytical cut size is equal to the equiprobable size. As this rarely happens in practice the two sizes are usually different. The analytical cut size is less useful than the
Fluid classification 257 equiprobable size since it is somewhat dependent on the size distribution of the feed, however it is favored in industry since it is easily obtained. Intersection cut size x is defined as the size at which the cumulative percentage oversize of the coarse stream is equal to the cumulative percentage undersize of the fine stream. This size requires just two size determinations on samples from the two streams; it is however even more sensitive to changes in the feed size distribution than the analytical cut size. The grade efficiency is often expressed as a single number. This number is known as the sharpness index, i//, and is a measure of the slope of the grade efficiency curve: 75(^25=^
(5.9)
X75 and ^25 are the particle sizes at which the grade efficiencies are 75% and 25% respectively. For perfect classification if/= 1, while values above 3 are considered poor. Alternatively 90^^10 has been used. These ratios are not always adequate to define the sharpness of cut [5]. In many cases it is important to keep the amount of fines in the coarse or the amount of coarse in the fines as small as possible. For these cases a measure of the effectiveness of a separation process is given by the following: For the coarse yield: _ weight of particles coarser than e in the coarse fraction weight of particles coarser than e in the feed max
•^jngx
e
Similarly for the fine yield:
if,f=Ef^
(5.11)
258 Powder sampling and particle size determination
The grade efficiency can be calculated directly from cumulative size distribution data using a simple geometric construction [6]. The method consists of plotting a square diagram ofFJipc) against F{x) directly above a square diagram of Fpc) [which will also be against F{x) (Figure 5.2)]. Consider the coarse versus feed diagram. Equation (5.7) can be written: G,{x) = E,{Xma)^
(5.12)
For large x, (coarse sizes), GJ^x) = 1 and: (tana)
=—
(5.13)
Hence, a tangent through FJ^x) = F{x) = 100% intersects the F(x) axis at the point whereF(x) =l-E^(= 49%). Equations (5.7) and (5.8) can be combined to give:
GJx)=\-Ef
^;;
G,(x) = l - ( l - £ ^ ) ( t a n / ? ) ^
(5.14)
Consider the fines versus feed diagram. At small sizes G^{x) = 0 and: (tanyff)
=—^—
(5.15)
Hence, a tangent through F^{x) = F(x) = 100% intersects the F{x) axis at the point where F{x) =" \ - E^{= 40%). Plotting the two square diagrams as shown in Figure 5.2, using the data presented in Table 5.1, the total efficiency is represented by a single point R^ and its straight line connections to F^(x) = 100%) for coarse and Fj(x) = F(x) = 0 for fines. Some classifiers give grade efficiency curves where the grade efficiency does not reach the jc-axis but runs parallel to it at a constant value x [7].
Fluid classification 259
100
20
40
60
80/7(^jj100
Fig. 5.2 An example of a square diagram The corrected curve is given by: (5.16) In the limit at small sizes, as jc approaches 0, G^ (x) approaches r.
260 Powder sampling and particle size determination
Equation (5.13), for coarse versus feed becomes: r = £,(tan«)^^^
(5.17)
This corresponds to a line passing through the 7?j point and parallel to the tangent to the curve at Fjipc) = 0, F{x) = 0. This intersects the F^{x) = 100, F{x) = 100 axis at a value r. (- 27%). Similarly for the fine versus feed plot for jc -^ 0, G^(x) —>r and equation (5.16) becomes:
The line passes through F{x) = 0, Fpc) = 0 and is parallel to a line through /?! which intersects the F(x) = 0,Fj(x) = 0 axis at a value T. Particle size distribution and classifier selectivity have been determined, using kernel density estimations, to data from (two) classifier flow streams. The procedure has been applied to hydrocyclones using platey particles whose sizes were determined with a Sedigraph 5100 and spheroidal particles whose size distributions were determine using the Malvern Mastersizer and the Coulter Counter [8]: Svarovsky's equation was used [9,10]. 5.3 Systems Classifiers may be divided into two categories, counter-flow equilibrium and cross-flow separation. Counter-flow can occur either in a gravitational or centrifugal field; the field force and the drag force act in opposite directions and particles leave the separation zone in one of two directions according to their size. At the 'cut' size, particles are acted upon by two equal and opposite forces, hence stay in equilibrium in the separation zone. In gravitational systems these particles remain in a state of suspension, while in a centrifugal field the equilibrium particles revolve at a fixed radius that is governed by the rate at which material is withdrawn from the system. They would therefore accumulate to a very high concentration in a continuously operated classifier, if they were not distributed to the coarse and fine fractions by a stochastic mixing process.
Fluid classification 261
In a cross-flow classifier, the feed material enters the flow medium at one point in the classification chamber, at an angle to the direction of fluid flow with a component of velocity transverse to the flow and is fanned out under the action of field, inertia and drag forces. Particles of different sizes describe different trajectories and so can be separated according to size. 5.4 Counter-flow equilibrium classifiers in a gravitational fieldelutriators Elutriation is a process of grading particles by means of an upward current of fluid, usually air or water. The process is therefore the reverse of gravity sedimentation and Stokes' law applies. The grading is usually carried out in one or a series of containers, the bodies of which are cylindrical and the bases inverted cones. In a single stage elutriator, the cut size is reduced in steps by increasing the volume flow rate. At each step the operation is considered complete when, for air elutriators, the rate of change in weight for the residue is negligible, say 0.2% of the initial weight in half an hour. For water elutriators, the endpoint is reached when there are no visible signs of further classification taking place. In a multiple stage elutriator, the cut size is reduced in steps by successively increasing the cross-sectional areas of the elutriation chambers. Water was used as the flow medium in some of the early elutriators. These instruments suffered from flocculation problems, and took several hours to generate a size distribution; they are no longer widely used. In most elutriators Stokes' law does not apply since the ratio of tube length to tube diameter is too small for laminar flow conditions i.e. the fluid disturbances at inlet and outlet overlap. Combined with this, the tube shape is not always conducive to laminar flow. Due to the viscosity of the fluid, a parabolic velocity front exists which is flattened in the case of large diameter tubes. The cut is not sharp therefore since the upward force on the particle depends upon its axial position in the tube. Roller [11] showed that the effect of the uneven cut is the removal of some coarse above the theoretical cut-point, while leaving behind some of the fines. Thus, while the separate fractions are not accurately sized, the final mass fraction is often reasonably close to the correct value. This was confirmed by Stairmand [12] who pointed out that the method was not applicable to bimodal distributions. A particular advantage of elutriation is the production of closely graded fractions that are often useful for further investigations.
262 Powder sampling and particle size determination
5.5 Theory for elutriators For streamline flow the velocity profile is parabolic, the velocity at a point at a distance r from the axis of the tube being given by Poisieulle's equation:
1"^-^)
^r|L where
(5.19)
/? is the pressure drop across the elutriator tube 77 is the fluid viscosity L is the length of the tube a is the radius of the tube
The volume flow rate through the tube is given by: a
r
(5.20) STJL
giving an average velocity: v„ =
Q
P ^2 a
na^
(5.21)
STJL
By putting r = 0 in equation (5.19), the maximum velocity may be found: P
2 ^y
(5.22)
From equations (5.19) and (5.20):
f v = v„
f
r']
1' a^
V
i-
= '^Vm
J
V
r'] «^ . J
(5.23)
Fluid classification 263
If it is assumed that there is no radial flow of particles, the possibility of a particle being elutriated depends on its position in the tube. Since v^ is the velocity usually taken as the elutriation velocity, from equation (5.23): — =] - — ^ 2v^
(5.24)
Thus, a particle of terminal velocity v can ascend if it is in a coaxial circle of radius r. Assuming a homogeneous distribution of particles, the fraction of this size contained in a cylinder of radius r is:
-4 F =\ - - 2v^
(5.25)
Since, from Stokes' equation, v is proportional to particle size squared: ^ D^ 2 yD^j
2
(5.26)
where D^ is the theoretical cut size. The collection efficiency may be calculated using equation (5.26). The cut, however, is better than one would expect from the theoretical grade efficiency (Table 5.2) due to a radial flow of particles from the outer to the inner areas. This is due to the pressure difference across the particle in a radial direction. At A (Figure 5.3) the particles rotate in the same direction as the gas flow; this increases the gas velocity at A and at B the reverse occurs. The energy to accelerate is drawn from the pressure energy of the fluid; hence the pressure is lower at A than at B therefore the particles move from the regions of low velocity to the regions of high velocity. The cut velocity is therefore v^^^ since if elutriation is carried out to completion; all particles smaller than ^2D^ will be removed. Experimental results [13] are in general agreement with the above, the top size elutriated being approximately ^2D^ and the fraction of undersize retained is of the order ofO.5.
264 Powder sampling and particle size determination
Fig. 5.3 The pressure gradient across a particle Table 5.2 Theoretical efficiency of elutriators D/D^ 0.1 % elutriated lOOF 99.5
0.2 98.0
0.3 95.5
0.4 92.0
0.5 87.7
0.6 82.0
0.7 75.5
D/D„ 0.8 % elutriated lOOF 68.0
0.9 59.5
1.0 50.0
1.1 39.5
1.2 28.0
1.3 15.5
<2 2.0
5.6 Water elutriators Early elutriators used water as the fluid medium but due to the problems of dispersion and leakage these have been largely replaced by air elutriators. The Andrews kinetic elutriator [14] consisted of four chambers vertically arranged in series; the suspension was contained in a feed tube at the top of the system that fed into a circulating tube followed by a coarse classifying tube and a collecting tube. In operation, the suspension was admitted to the circulating tube and then water was passed through the system in reverse order. Classification was considered complete when the liquid in the feed tube was clear. Other water elutriators have been described by Schone, [15 cit. 11] Roller [11] Andreasen [16] and Blythe et, al [17]. For a full description of these instruments readers are referred to earlier editions of this book.
Fluid classification 265
5.7 Air elutriators Air elutriators are especially useful for powders that are, in practice, subject to grading by airflow; e.g. fine dust, which contains particles of different densities with settling velocities, which are not uniquely related to physical dimensions. The major problems encountered in air elutriation are the difficulty in break up agglomerates and preventing particles from sticking to the walls of the elutriator tubes. The three main types of air elutriator are the up-blast, the down-blast and the circulating. The Gonell elutriator [18] consists of three cylindrical brass tubes in series, of decreasing diameter, together with ancillary equipment to provide and measure a dry air-flow. The sample tube has a down-blast arrangement to prevent choking. Particle adhesion to the walls is reduced by the use of mechanical rappers and of antistatic agents. Fines are collected in glass containers at the top of each tube. The analysis time can be reduced by altering the shape of the dust reservoir, and commencing the analysis with a high velocity blast to carry the dust to the top of the tube [19]. The circulating type of elutriator, as developed by Roller is similar to the up-blast but the sample of powder is caused to circulate in a U-tube at the base of the elutriator tube. The elutriator has four chambers, to each of which is attached a paper extraction thimble collector for the fines. Ancillary equipment, including mechanical rappers, is included and the instrument is contained in a sound proof cabinet. In the miniature elutriator [20] a high velocity jet blows downwards into a thimble at the base of the elutriator tube. The tube is much smaller than that of other elutriators being 14 in long and 1 in diameter. The Sepor Haultain Infrasizer [21] consists of six elutriating tubes in series, air-flow entry being into the smallest. Air enters through a conical seating supporting a golf ball, which, by rotation and impact, breaks down agglomerates. An elutriator has also been described which contains perforated baffles to de-agglomerate the powder and a flat velocity profile [22]. Weilbacher and Rumpf [23] investigated the velocity distribution in a Gonell elutriator and found that the flow at the lower end of the tube was characterized by strong turbulence and instability and that this region governed the separation. As a result of their investigation a new elutriator was designed, which had a classifier chamber only 1 cm high, sitting below a conical section. Powder, spread on a filter paper resting on a porous plate, became fluidized when pressure was applied below the plate. Fine
266 Powder sampling and particle size determination
particles were entrained in the air, whose velocity increased when it reached the conical section, so that the particles were carried away leaving the coarse fraction behind. Leschonski and Rumpf [5] showed that to achieve a residue of 60% for a cut size of 10 |Lim the analysis time was reduced from 1000 min to 200 min. The instrument was commercialized as the Analysette 8 [24]. The elutriation process is time consuming with an ill-defined cut size and has been largely supplanted by micromesh sieving.
Fig. 5.4 Simplified schematic diagram of the Bahco microparticle classifier; showing its major components; 1, electric motor; 2, threaded spindle; 3, symmetrical disc; 4, sifting chamber; 5, container; 6, housing; 7, top edge; 8, radial vanes; 9, feed point; 10, feed hole; 11, rotor; 12, rotary duct; 13, feed slot; 14, fanwheel outlet; 15, grading member; 16, throttle. 5.8 Counter-flow centrifugal classifiers; The Bahco Classifier; is a centrifugal elutriator (Figure 5.4). The sample is introduced into a spiral air current created by a hollow disc rotating at 3500 rpm. Air and dust are drawn through the cavity in a radially inward direction against centrifugal forces. Separation into different sizes fractions is made by altering the air velocity that is effected by changing the air inlet gap by the use of spacers. Since no two instruments perform identically, instrument calibration is necessary. 5 to 10 g of powder are required for the sample, which can be graded in the size range 5 to 100 \ux\ [25]. ASME Power Test Code 28 Committee undertook to recommend
Fluid classification 267
standard tests for measurement of the significant properties of fly ash; after investigating many devices for particle size determination, the committee selected the Bahco as the standard instrument [26,27]. 5.9 Zig-zag gravitational classifiers Zig-zag classifiers are, in essence, a series of similar air elutriators in series. Thus, instead of a single classification, the powder is subjected to multiple classifications thereby increasing the sharpness of cut as well as reducing the running time. Several versions of the Alpine Multi-plex zigzag classifiers; are available and these may be categorized as gravitational or centrifugal counter-flow classifiers [28]. In the gravitational laboratory classifier (Figure 5.4) the material is fed into a zig-zag shaped sifting tube in which there is an upward air flow. The fine material is carried upward and the coarse downward and at every change in direction the material is graded to give good grade efficiency. This instrument operates in the 0.1 to 6 mm size range at a feed rate of up to 50 kg h'^.
Fig. 5.5 The zig-zag centrifugal classifier 5.10 Zig-zag centrifugal classifiers In the centrifugal version (Figure 5.5) a feed rating worm (b) feeds the unclassified material (c) into a classifying chamber. Radially arranged
268 Powder sampling and particle size determination
blades, on the outer face of the classifier rotor (d), speed the inflow of material up to the peripheral velocity of the rotor suspending it, extra air being admitted through (e). The dust-air mixture is then sucked in to the zig-zag shaped rotor channels where classification takes place. Fine material is sucked into the classifier center (g), where it leaves via a cyclone. The coarse material (f) is expelled by centrifugal force. At the periphery it is flushed by the incoming air before being discharged. Cut size is in the range 1 to 100 jam at a feed rate of 0.5 to 3 kg h-^ Sample inlet
M C3
%
0 Uteri
Fig. 5.6 Principle of the Warmain Cyclosizer. 5.11 The Warmain Cyclosizer The Warmain Cyclosizer [29] hydraulic cyclone elutriator (Figure 5.6). Using inverted cyclones as separators with water as the flow medium, samples of between 25 to 200 g are reduced to five fractions having cut sizes (for quartz) of 44, 33, 15 and 10 |Lim. The cyclones are arranged in series and during a run the oversize for each cyclone is trapped and subjected to elutriating action for a fixed time period. At the end of a run the trapped materials are extracted by opening the valves at the apex of each cyclone in turn and, after decantation, the solids are recovered by filtration and evaporation. 5.12 Cross-flow gravitational classification 5.12.1 The Humboldt particle size analyzer TDS This classifier [30] combines classification with particle size analysis. The material is fed into the classifying chamber with the aid of a special feeder and injector at rates between 40 and 60 g min"^ At the outlet of the
Fluid classification 269
injector the particles enter a high-speed air stream that breaks up any agglomerates. An acceleration channel forms the suspended stream into a flat jet that enters the separation chamber at an average speed of 20 to 80 m s"^ Cross-flowing separation air rapidly forces the fine particles downwards whereas the coarser particles tend to fly straight on. Whilst separation proceeds a photometer scans the classification zone to generate a concentration profile in 5 s. 5.13 Cross-flow centrifugal classifiers 5.13.1 Analysette 9 Cross-flow is used preferably in spiral classifiers which were investigated by Rumpf [31 cit. 5] and subsequently improved by Kxxxxvpi et.al, [32,33]. As a result the Analysette 9 centrifugal transverse flow classifier was developed although, due to constructional difficulties, it was never available commercially. The feed material is dispersed and accelerated in a tube and enters a flat classification chamber almost tangential ly at a point on the circumference. The air leaves the chamber together with the fine fraction on spiral paths through an orifice at the center while the coarse fraction gathers at the circumference, constituting a ring of coarse material that spins along the peripheral wall of the classification chamber. The rotating ring of material enters the constant velocity jet of air transversely at a very acute angle with a definite velocity and is distributed fanwise under the influence of drag and inertia forces. The procedure repeats itself very frequently, i.e. of the order of 2000 times per minute. This high number of classifications coupled with the dispersion of agglomerates by friction forces in the rotating ring leads to a sharpness of cut at a very low size not previously obtainable on spiral classifiers. The coarse material is removed intermittently through an outlet and is collected on a filter. A good sharpness of cut is obtained in the size range 2 to 12 |am. In this range it is claimed to be superior to the Bahco but the Bahco permits separation at coarser sizes with equally good sharpness. Cut size may be varied by altering the amount of coarse material in the classifier and this can be estimated theoretically [34]. 5.13.2 The Donaldson Acucut classifier The Donaldson Acucut was patented in 1970 [35] and its mode of operation was described by Schaller and Lapple [36]. A vaned rotor
270 Powder sampling and particle size determination
produces a centrifugal field while, at the same time, air is drawn into the center of the rotor [37]. All but about 5% of the air intake, induced by a positive displacement pump downstream of the classifier, enters the classification zone through a very narrow gap formed between the rotor and the stator. This leads to a very high turbulence in the preclassification zone. The material enters the classifier through a venturi type nozzle with the remaining 5% of air. Between planes 1 and 2 the ratio of centrifugal force to drag force is kept very nearly constant by a diverging radial cross-section. This zone is the classification zone. The smaller particles are carried out through the middle and the larger ones move towards the stator, where they undergo de-aggregation until they reach the exit. The cut size of these machines ranges from 0.5 to 50 |Lim. Air
a2
Fig. 5.7 Principle of the cross-flow elbow classifier 5.14 Cross-flow elbow classifier In the cross-flow air classifier (Figure 5.7) the main air is introduced at (aj) and secondary air at {dij). Both streams are bent round a solid wall (b) and the resulting flow follows the bend without leaving the wall or forming vortices. The so-called Coanda effect helps to maintain the flow round the bend for approximately 90° and this is enhanced by the application of suction. 5.15 Micromeretics classifier; The Micromeretics model 1001 [38] works on the following principle. A de-agglomerated stream of particles is sucked from a dispersion device into the center of a rotor where it divides into two streams. Air of either
Fluid classification 271
Stream flows radially outwards to the rotor wall, while the particles follow curvilinear paths depending on their size and density. At the wall, they contact either of two paper or plastic films, which can be removed once a sufficient deposit is collected. The film can then be cut into segments, each containing discrete sizes, and the particles can be scraped off into a series of separate containers. 5.16 Fractionation methods for particle size measurement Sub-micron particle sizing can be divided into two categories; nonfractionation and fractionation methods. The most commonly employed non-fractionation method is radiation scattering and absorption, which includes light and ultrasonic attenuation, quasi-elastic light scattering and diffraction. Although suitable for rapid determination of an average particle size or an estimate of the size distribution of broad size distributions, these methods are not capable of high resolution. The shortcomings of the non-fractionation techniques have led to an interest in fractionation methods [39]. Commercially available fractionation methods include hydrodynamic chromatography (HDC), field flow fractionation (FFF) and disc centrifugation (DSC). One advantage of fractionation methods over nonfractionation methods is that the particles are separated physically according to size, prior to detection, which allows much higher resolution in determining the size distribution [40]. Eluant
Latex suspension Solid spheres Detector Recorder (Optical scattering) N.
1 Fig. 5.8 Experimental arrangement for HDC experiment.
272 Powder sampling and particle size determination
HDC and FFF are based on the separation by size of colloidal particles when they travel in Poiseuille flow through a narrow conduit. The larger particles travel faster than the smaller ones in HDC whilst the opposite occurs in FFF. Although FFF exhibits high efficiency of size separation, it is necessary to know the particle density in order to determine the particle size distribution, whereas HDC is density independent. 5.17 Hydrodynamic chromatography Hydrodynamic chromatography (HDC) offers a means of obtaining size information on colloidal particles in suspension with the same ease that is characteristic of chromatographic methods for size analysis of molecules in solution. The basic equipment consists of a column packed with stationary, solid beads in the size range 10 to 50 |Lim [41,42] (Figure 5.8). Means are provided for injecting about 0.2 cm^ of colloidal suspension, containing about 0.01% polymer by weight, into the flowing stream at the entrance of the column and monitoring the colloid in the column effluent. Particle separation occurs due to a hydrodynamic interaction between the particles and the velocity profile near solid surfaces. HDC is a fast technique but its resolution is low.
.---h^.-./" Particle exclusion layer R Large
|
.•-••ii__
i ".7 ? ^''^' Particle exclusion layer
"i—f Small Fig. 5.9 Capillary model of hydrodynamic chromatography separation
Fluid classification 273
Optical Density
Time Fig. 5.10 Typical HDC calibration chromatogram using latex spheres. The marker is used to define the emergence of the elute. Particle velocity increases with particle size. If a marker dye, e.g. potassium dichromate, is injected with the suspension, it is found that larger particles elute before the smaller and the dye is the last to elute. Thus, the velocity of a particle through the column increases with increasing particle size. The explanation for this is that the fluid velocity is highest at the center of the streamlines and large particles, because of their size, are confined to high velocity regions [43]. Small particles, on the other hand, can sample the lower velocities near the wall. Brownian motion ensures that the particles sample all the available radial positions (Figure 5.9). A distribution of elution times due to axial dispersion also occurs, leading to a Gaussian distribution of velocities for a monosize distribution. A typical calibration chromatogram is shown in Figure 5.10. The technique has been evaluated by Langhorst et. al [44,45] and reviewed by Miller and Lines [46]. HDC suffers from poor resolution and particle size discrimination. In principle, more accurate particle size determinations are possible but require special software with correction for the extensive band dispersion [47,48]. It has also been shown that the molecular weight and concentration of non-ionic surfactants adsorbed on latex particles have a significant effect on their separation factor [49].
274 Powder sampling and particle size determination
The application of HDF to mixtures of monodisperse particle samples, with diameters ranging from 20 nm to 1000 nm, has been described. The distributions were compared to distributions obtained by electron microscopy. HDC and SFFF and gave comparable results [50]. The rate at which particles are transported through the bed is also found to depend on the size of the bed particles, the ionic strength of the liquid and the flow velocity as well as the particle size of the eluting particles. By operating at low ionic strength and high zeta potential, the double layer repulsion force predominates over the van de Waals attraction force and particles are repelled from solid surfaces. In such cases the rate of transport of particles through the column is enhanced; again, large particles are eluted before small ones. This is known as enhanced electrostatic chromatography. With potential barrier chromatography, the double layer is suppressed by the use of high ionic strength and the van de Waal's attractive force predominates. Large particles, due to their greater attraction to the packing, are retarded more than small ones and elute later. In size exclusion chromatography, otherwise known as porous wall hydrodynamic chromatography [51], the packing is made up of particles having closely controlled pore size distributions. These selectively retard polymer molecules in solution with sizes small enough to enter the pores and this imposes an extra separation effect by steric exclusion to give better resolution [52,53]. As before, large particles flow through the voids between the particles and are eluted first [54,55]. The upper limit for this technique is 400 nm. Optimization of parameters for fine particulate packing material carried out on special test mixes was found to shorten analysis time without the usual pressure drop. The use of 1.5 |Lim particles gave faster and more efficient separation for a wide range of analyses, and using non-porous material was found to offer advantages over the usual porous material [56]. An extension of the technique to powders such as cement, flour and chalk; has also been described [57]. The packed column consisted of 50 and 250 |Lim diameter particles. An instrument, which operated in the size range 30 to 1577 nm, was marketed as the Micromeretics Flow Sizer 5600 but it is no longer available. It was automated with operating programs on floppy discs and a data terminal was included. Calibration was effected with monosize lattices. Polymer Laboratories particle size distribution analyzer (PL-PSDA) is an integrated, automated system operating on the principle of packed
Fluid classification 275
column hydrodynamic chromatography. Samples are prepared in 2 ml vials that are loaded into a carousel-based autosampler, providing continuous unattended operation for up to 84 samples. Analysis time is less than ten minutes and covers the size range 20 nm to 3 |Lim. 5.18 Capillary hydrodynamic fractionation Capillary hydrodynamic chromatography (CHDC), in which a long narrow capillary replaces the packed bed, is an extension of HOC [58]. When rigid spheres are transported in Poiseuille flow through a straight cylindrical tube, they undergo radial displacement and move in trajectories parallel to the tube axis at fixed distances between the axis and the wall. The term capillary hydrodynamic fractionation was coined to describe the inertial hydrodynamic focusing of particles into annular rings, the radial positions of which are a function of particle radius [59]. The capillaries can be up to 200 ft long, with an internal diameter of 0.015 in, to fractionate particles in the 0.5 to 30 \xm size range. The resolution is higher than that of HDC with comparable analysis time. CHDF has also been used to characterize mini-emulsion stability and droplet size [60]. A set of capillary tubes is used to perform separation of particles in the 0.0015 to 1.1 |Lim size range. In Poiseuille flow the velocity profile is parabolic and the particles close to the wall travel more slowly than those near the center. Since the larger particles are excluded from the regions near the wall, they attain greater velocities than those of the smaller particles. Consequently, their average velocity will exceed the average velocity of the eluant and also that of the smaller particles. Separation of particles based solely on hydrodynamic effects was first discussed in 1970, [61,62] and the analytical separation was later documented [58,6366]. A limitation is a commonly observed phenomenon of particle retention by the packing material, which leads to inaccuracies in the calculated size distribution [67]. A full analysis of the efficiency of particle separation in CHDF gives the appropriate criteria for the development of a steady state radial concentration profile [68]. Particle transit time is a logarithmic function of particle size. Pressures of up to 30,000 Pa are required and give a separating range from 0.2 to 200 nm [69]. Matec CHDF-1100 uses CHDF to obtain high-resolution particle size distributions in the size range 0.015 |Lim to 1.1 |im and is capable of resolving size differences as small as 10% in diameter within that range. The analysis is particle density independent and a run takes less than 10
276 Powder sampling and particle size determination
min. The system consists of modular type components, including a solvent delivery pump, pressure gauge, high pressure safety release valve, pre-filter, sample injection valve, separation micro-capillary, ultraviolet flow-through detector, microcomputer and printer. The injected samples of less than 0.1 ml are at a volume concentration of 1% to 3%. After injection, the output of the detector, which measures the UV light absorbed and scattered by the particles, is monitored. The output is used to determine the relative amount of different size modes present and is pre-calibrated. Approximately one minute after the sample injection a marker is injected. The use of a marker allows accurate measurement of particle flow rate from which particle size can be calculated. CHDF has also been used for the characterization of the size and stability of mini droplets of oil; (50-500 nm) [70]. 5.19 Capillary zone electrophoresis A wide range of ionizable species can be separated by capillary zone electrophoresis (CZE) with high resolution. In CZE small fused silica or glass capillaries are used as the separating medium [71,72]. Capillary forces between a buffer and the capillary wall stabilize a liquid buffer in which the separation occurs, eliminating the need for a semi-rigid gel that is used in conventional slab-gel electrophoresis. The very narrow capillaries permit the use of very high voltages (30 kV) with resultant small electric currents. Joule heating is rapidly dissipated from the narrow capillary while permitting rapid separations. Since the process takes place in a capillary that is transparent to ultra-violet light, a portion of the capillary can be used as an optical measurement cell thus permitting online detection [73]. Electrical conductivity; has also been used in a capillary system for drop-size concentration measurement [74]. 5.20 Size exclusion chromatography This is an effective and relatively simple method for characterizing silica sols and other colloids [75]. It has also been used to determine the particle size distributions of polymer lattices [76,77]. Separations are performed in a column packed with particles having pores substantially of the same size. A carrier liquid is passed through the column; as a mixture of colloidal particles passes through the bed, the larger ones exit first since they are too large to sample the pore volume. Intermediate sized colloids enter the pores and are retained according to the volume that can be
Fluid classification 277
accessed by the colloid; the smaller the colloid, the larger the volume that can be accessed and the greater the delay. The limitation of SEC is that calibration standards are required and silica sols larger than about 60 nm cannot be analyzed. The method has lower resolution and particle discrimination than FFF methods, which results in poorer analytical precision. 5.21 Field flow fractionation Field flow fractionation (FFF) comprises a group of classification methods in which particles with different properties are eluted from the classifier at different times. The elution time depends upon the applied force field, the most common being sedimentation. While gravity has had limited use for particles larger than a micron, the sedimentation force is usually generated in a centrifuge. In FFF a narrow plug of suspension is introduced into a flow stream of carrier liquid that sweeps the sample into a long flat, ribbon like channel. When the sample reaches the channel, flow is stopped momentarily, and a force field applied, under which large particles accumulate into thinner, more compact layers at the outer wall, whereas small particles form thicker, more diffuse layers. The particles do not however touch the wall because shear-induced hydrodynamic lift forces oppose the driving force of the field [78]. Because of their smaller sizes the centers of the smaller particles approach the wall more closely than the centers of the larger particles. The larger particles sample higher velocity streams than the smaller ones and are eluted first. The different velocities, as indicated by the length of the arrows in Figure 5.10, give rise to the separation. This mechanism is applicable in the size range 1-100 |Lim where Brownian effects are negligible. Giddings et. al [79] showed that, using a gravitational field, it was possible to resolve seven polystyrene lattices in 3 min. As a result of its versatility, FFF in one form or another has been applied to the characterization of particles or molecules whose sizes range over five orders of magnitude from particles as small as 0.005 |Lim to as large as 500 \xm [80]. The total mass range covered is over 15 decades. Most of the reported data are for aqueous suspensions; non-aqueous suspensions have also been used [81]. Materials analyzed by FFF range from high-density metals and lowdensity latex microspheres to deformable particles such as emulsions and biological cells. The particles need not be spherical since separation is based on effective particle mass.
278 Powder sampling and particle size determination
The degree of compression of the layer and its interaction with the field determines the differential migration and separation of the particles. At the outlet end of the channel, the particles pass through a detector that is used to determine the relative concentrations of the separated fractions. The determination of concentration is usually based on light scattering by the particles as they pass through the flow cell of a chromatographic ultraviolet detector. The measured concentration versus elution time is converted into a size distribution using instrument software. The force field can be gravitational, centrifugal, thermal, electrical or magnetic but not all of these have been commercialized. Giddings and his co-workers [82] have presented an overview of particle separation and size characterization by FFF. In this they describes how it works, the applicable size range, the properties that can be characterized and the underlying theory together with a number of applications. Gravitational sedimentation FFF interacts with particle mass; cross flow FFF with hydrodynamic diameter; temperature gradient with particle diameter and thermal diffusion coefficient and electrical FFF is proportional to particle charge. 5.21.1 Sedimentation field flow fi-actionation (SFFF) Sedimentation field flow fractionation (SFFF) is one of a number of field flow fractionation methods that were originally devised by Giddings [83,84]. In GFFF a gravitational field is applied. Large particles settle through a liquid that is moving at right angles to the settling direction. Standardless conversion of fractograms is performed by single run analysis. All parameters are obtained from sample specifications and previous instrument calibration by semi-empirical models. Fractograms can be converted directly to particle size distributions although retention exhibits substantial dependence on flow rates compared to other FFF methods [85]. SFFF can measure gold particles down to 15 nm. Particle size distribution was determined from peak broadening caused by polydispersity of the sample. Peak broadening due to instrument imperfections was also detected. The results were compared with data from SEM and PCS. SEM gave a mean diameter of 20 nm; sizing by PC was not possible due to aggregation [86].
Flu id class ification 2 79 5.21.2 Centrifugal field flow fractionation In (CFFF) a centrifugal field is applied. Large particles accumulate in thin compact layers at the wall and small particles form more diffuse layers (Figure 5.11). Interaction with the streamlines of fluid passing through the channel separates the thicker, more diffuse fine particle layer, eluting it first [87]. SFFF separations are carried out in a very thin open channels, shaped like ribbons, that are suspended in a centrifuge. Liquid mobile phase is delivered at a constant flow rate through a sampling loop containing the powder to be analyzed. The sample is swept from this loop into the channel within the centrifuge. Following separation in the channel, the sample flows back through the seal into a detector (typically a turbidimeter) that is used to determine the relative concentrations of the separated fractions. Since the retention time can be related with particle properties, including size, the detector signal at any time indicates the relative amount of particulate material having a specific value of the relevant property. A computer controls the pump output, sample valve actuation and rotor speed. It also acquires data from the detector and transforms this output to a true concentration profile. A plot of detector signal against time called the fractogram then provides the information necessary to generate the corresponding distribution curve. This method has an extraordinary ability to probe aggregation phenomena and to track particle size distribution changes caused by aggregation in colloidal samples [88] I Sedimentation W field l\r.^'\\\ -^^^'^'^^ v^Wity/ ^ ^ flow^rofile y ^ fl<>^ profile ^. ^. ;V. •-.Vvectors Medium velocity = d ^ I ^ ^ ^ ^ ^ ^ w velocity ^ Velocity Heavy Medium Light vector particles particles particles Fig. 5.11 Centrifugal sedimentation field flow fractionation. 5.21.3 Time-delayed exponential SFFF In the Dupont SFFF a Sorvall centrifuge is used to contain the SFFF rotor and develop the necessary centrifugal field. The speed of the motor is allowed to decay exponentially in order to speed up the analysis. Mobile
280 Powder sampling and particle size determination
phase is pumped through a rotating seal into the rotor. The separating channel is formed between two parallel surfaces 250 |Lim apart housed in a titanium rotor that can operate at up to 32,000 rpm. Particle sizes or molecular weights are determined from the output of a pre-calibrated UV spectrophotometer. Polystyrene lattices A - 0.091 ^m B = 0.176 Mm C = 0.220 Mm D = 0.312 Mm E = 0.481 Mm
- 10 000 rpm n
IDE ^ - ^ field
B
.J5
10
15
20
25
Retention time, tpmin
Fig. 5.12 A typical SFFF fractogram. With the liquid mobile phase off and the channel rotating at an appropriate speed, the sample mixture is injected into the channel. The channel is rotated in this mode for a relaxation or pre-equilibrium period that allows the particles to be forced towards the accumulation wall at approximately their sedimentation equilibrium position. Particles denser than the mobile phase are forced towards the outer wall. Diffusion opposite to that imposed by the centrifugal force causes the particles to establish a specific mean thickness near the accumulation wall as a function of particle mass. Liquid mobile phase is then restarted with a parabolic velocity front. Small particles are engaged by the faster moving central streamlines and are eluted first. Large particles near the wall are intercepted by the slower streamlines and are eluted later. Thus particles are eluted from the channel in order of increasing mass.
Fluid classification 281 Constant force field provides for highest resolution of particles in the sample with resulting highest precision. However, characterization of samples with wide size distributions is difficult and time consuming. Force field programming [89,90] removes these limitations to ensure that the entire distribution can be analyzed in a convenient time. In time delayed exponential decay the initial force field is held constant for a time equal to r and after this the force field is decayed exponentially with a time constant r. In this mode, a log-linear relationship is obtained of particle mass against retention time. This simple relationship permits a convenient calculation of the quantitative information needed for the sample. Retention time is given by: /^:=3rln-^
(5.27)
where /] = ^^^^p— tQWcoQrAp
(5.28)
r = time of initial force field and delay constant; W= channel thickness; Ap = density difference between the particle and the liquid; IQ = solvent marker retention time; o)Qr= centrifugal force field; dp= particle diameter; k:= Boltzmann's constant; T= absolute temperature. A typical fractogram is shown in Figure 5.12. The instrument operates in the size range 0.01 ]im to 1.0 |Lim. A recent paper describes the use of programmable SFFF to speed up analyses [91]. In summary, flow chromatographic methods offer the advantage of both separation and size analysis of fine particles and macromolecules. Separation is relatively gentle, and has application in the biological sciences. The driving force can be centrifugal as described here or electrical, thermal or magnetic. While many methods are still classed as academic, sedimentation field flow fractionation is commercially available. The former, because of a packed column system, has less flexibility in the size analysis of broad industrial and biological particle systems, whereas the latter, by virtue of open channel separation, has greater potential. At low ionic strengths in the eluant, the retention of SFFF deviates from the prediction of standard theory, resulting in an underestimation of particle size due to particle-particle interaction and a modified theory has
282 Powder sampling and particle size determination
been proposed [92]. The influence of zone broadening on the measured particle size distribution has also been investigated [93]. 5.21.4 Thermal field flow fractionation In thermal field flow fractionation; (TFFF), a temperature gradient is applied. The primary potential advantage of this technique is that it can be used to size particles in the range 0.01 \xm to 0.001 |am, an order of magnitude smaller than SFFF. Fffractionation market a TFFF polymer fractionator channel module with 286/16 MHz IBM compatible PC, super VGA color monitor workstation to include data acquisition software, hardware and data analysis software. A linear UV detector and single channel high performance pump are optional. 5.21.5 Magnetic field flow fractionation In magnetic field flow fractionation; the separation depends upon how strongly particles are attracted to one pole of a magnet [94].
Sample loading, relaxation and focusing Sample inlet
Elution
Carrier In
U IU U U U Cross-flow outlet
Sample inlet Glass platcy
lercwit
U U U U i U Cross-flow outlet
Fig. 5.13 Asymmetrical channel for flow field fractionation. 5.21.6 Flow field flow fractionation When a simple cross-flow of carrier liquid is used, the method is known as flow FFF and this has been represented as having the widest range of application of any single FFF technique [95,96]. This technique has been used to separate and characterize particles in the 0.01 to 50 |Lim size range,
Fluid classification 283 using the normal mode for the size range 0.01 to 2 |Lim and a sterichyperlayer mode for particles larger than 0.3 |Lim [97]. In this method, the sample flow is perpendicular to the flow of the mobile phase [98]. This is accomplished in a rectangular channel of which one or both walls is a semi-permeable membrane so that solvent can flow through the membranes but particles are retained within the channel. Alternatively, a narrow tubular porous channel can be used [99]. The remainder of the apparatus is similar to that described above. A more advantageous design [100] is shown in Figure 5.13, in which a single membrane is used as the accumulation wall and a flat glass plate is used at the top of the channel. The force field is created by flow through the bottom membrane. The mobile phase is introduced at one end and, due to loss of solvent through the porous frit, the liquid velocity decreases as it proceeds down the channel thus generating an asymmetrical velocity profile. A range of particle sizes from 10 to 1000 nm can be characterized in 0.5 to 1 hour [101]. This instrument is available from FFFractionation as the high resolution asymmetric flow SedFFF system. In an evaluation of these two techniques, to measure the particle size of paint components having a broad size range, Schauer [102] states that they each have their own merits. In the case of symmetrical FFF, the set-up for the channel flow and cross-flow is simpler, thinner channels can be used and the distinct fractionation in the steric hyperlayer can be achieved. The characteristics of asymmetrical FFF are the simpler construction of the fractionating channel, the superior focusing and the minimization of zone broadening in the channel. Of the two, he prefers the latter.
Velocity vectors
^^rtr^r/
i
w y B
Sedimentation field
Parabolic
AOW profilC
Accumulation wall
Fig. 5.14 Sedimentation/steric field flow fractionation
284 Powder sampling and particle size determination
5.21.7 Steric field flow fractionation When particle diameters reach the layer thickness, the zone as a whole will sample faster flow lines, and the particle velocities will depend on how far they extend into the faster flow lines. The elution order becomes reversed under these circumstances and the larger particles elute first. Because the particle extends through liquid layers of different velocities, a lift force operates to raise the particles into the higher velocity streamlines. The first SFFF separations took place in flat horizontal channels with gravity as the field [103,104] and dense particles tended to be retained (Figure 5.14). Fractionation time is reduced by centrifugation; by performing the fractionation in channels spun to a field of 15 times gravity, the lift forces generated by very high channel flows could be offset by the increased settling forces. Sedimentation SFFF is reviewed in an article by Moon and Lee [105] who discuss particle retention in FFF, the calibration process and recent findings on hydrodynamic lift. It is difficult to determine number distributions of sub-|Lim particles except by counting from transmission electron micrographs. Wyatt [106], using FFF followed by multi-angle light scattering, generated data that agreed with TEM's with a resolution that exceeded that of TEM data. Flow FFF and Thermal FFF have been used as complementary techniques in a study of core shell latex particles. Flow and sedimentation FFF have been used to determine the size and density of core shell particles and the shell thickness and particle density as a function of pH [107]. Flow, sedimentation and electrical, and other particle size measuring techniques have been used to characterize parenteral emulsions [108]. 5.21.8. Multi-angle light scattering (MALS) The MALS'FFF analytical technique is based on the following rationale: Light scattering is the method of choice when all the particles are the same size since an ensemble of identical particles produces a scattering pattern the same as a single particle but greatly enhanced. Using FFF as a fractionator to separate particles into "slices" each of which contains particles with a narrow distribution of sizes permits the subsequent measurement of size in each slice and hence the determination of the size distribution of the unfractionated ensemble [109]. The Wyatt Dawn instrument is an ideal detector for FFF and similar fractionation techniques. It does not require calibration and produces a
Fluid classification 285
linear detector response over a wide range of intensities, and spans the broad range of scattering angles required for accurate particle size analysis. The Dawn can be coupled to cross-flow, capillary hydrodynamic, thermal and sedimentation FFF to determine accurate particle size data. Conventionally, the calculation of quantitative sizes following fractionation requires a mass detector, usually a UV detector. The resulting particle size distributions have been found by subsequent MALS analyses to be far too broad. 5.22 The Matec electro-acoustic system EAS-8000 It has long been recognized that dispersion properties can be a strong function of particle concentration and there is an increasing interest in obtaining direct analysis of particle size and zeta potential in non-dilute systems. The EAS-8000 measures the electro-kinetic properties of concentrated dispersions and the Acoustosizer measures particle size and zeta potential over a wide concentration range. The ESA is used for routine characterization, as an aid in the preparation of high solids dispersions for coatings and castings and for characterizing non-aqueous liquid toners used in electrostatic imaging. 5.23 Continuous split fractionation A split-flow separation cell is a thin (generally sub-millimeter) rectangular channel in which various physical forces are utilized to drive components across the thin dimension of the channel, generally from one major wall towards the opposite wall [110,111]. At the same time, a film of fluid flowing lengthwise through the channel causes the rapid displacement of entrained components. By positioning a flow splitter at the downstream end of the channel, the suspended material can be separated into two substreams containing particles of different sizes. The separation process is characterized by a high resolving power, a high speed, a relatively simple theoretical description, and a great deal of versatility resulting from the large variety of driving forces and flow configurations that can be used. The methodology differs from field flow fractionation (FFF) in that in can be used for continuous split fractionation (CSF) whereas FFF is limited to batch operations. A comparison between the two techniques has been published [112].
286 Powder sampling and particle size determination
Wall A
WallB
Fig. 5.15 Edge view of split cell. Carrier flow is in the downward direction. A representation of the parabolic velocity profile across the cell thickness, and the position of the inlet and outlet splitting planes (ISP and OSP) are shown. A number of different driving forces have been used to implement CSF including gravitational sedimentation; [113] diffusive transport; [114,115] electrically driven transport; [116] and hydrodynamic lift forces [117]. The concept of utilizing hydrodynamic lift forces in the transport mode is illustrated in Figure 5.15. This figure represents the relevant features of separation viewed from one edge of the channel. The system is characterized by an inlet splitter that serves to compress the incoming feed particles into a narrow band near wall A. The thin particle-containing lamina formed just beyond the inlet splitter is subject to loss of material by various transport processes including sedimentation, diffusion and lift forces. The first can be eliminated by turning the channel on one end and the second is negligible for particles larger than about a micron. Large particles are driven more rapidly from wall A than small particles by lift forces. A full description of the procedure may be found in [118] where its use is described for classifying polystyrene particles in the 1 to 60 jiim size range together with separation of red blood cells from serum. Typically, cells have dimensions 330 jiim thickness, 2cm breadth and lengths from 2 to 10 cm. FFFractionation market a split flow thin cell fractionator SPLITT for fractionation of micron sized particles and preparation of narrow size cuts.
Fluid classification 287 5.24 Classification by decantation; In this method a homogeneous suspension is allowed to settle for a predetermined time. The supernatant liquid is then decanted and replaced by fresh dispersing liquid and the suspension re-agitated. The process is repeated until the supernatant liquid is clear. The decanted liquid will contain only particles smaller than where h is the depth at which the liquid is siphoned off at times t and A: is a constant. Herdan [119] suggested that six repeats would be sufficient to remove substantially all the particles smaller than a selected size whilst Allen [120] found that 15 repeats were necessary to produce a clear supernatant. The process can be repeated for shorter times so that the particles removed become progressively coarser Consider a vessel containing a depth /zj of a homogeneous suspension, the siphoning off being carried out at a depth //2. No particles coarser than Stokes diameter \.Q. D = hl{h2/i) will be removed. Particles of size xD, where x<\, will have fallen a distance h in the same time where: xD = kyjhit Hence from these two equations: X^ =h/h2 The fraction F | of particles of size xD removed is:
The fraction of particles of this size still in suspension will be: 1-Fi - l-X If the suspension is made up to its original volume, re-dispersed and a second fraction removed after a further time t, a further fraction of particles of size xD will be removed where:
'^-'i'-'tH'--')!:
X{\-X)
288 Powder sampling and particle size determination
The total fraction removed will be:
,...=(i-.^)^.0-.^)AJ(,_..)^ '^
h
^
F,+F2=X+X-^ The fraction of these particles remaining in suspension will be: 1 -(Fi+F2)=l
-2X+X^={\-X)^
After n decantings the fraction remaining will be:
i-iiv=(i-x)" n
(5.29)
'-('-^)^ Therefore the fraction removed will be:
l^.=i r=i
(5.30)
-('--')''/?,
V
Table 5.3 Increase in the number of decantings, required to give the same degree of separation, as the ratio of decant height to suspension height decreases from 0.9 to 0.8. Relative particle size (jc) Ratio increase of decants {m/n)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.4 1
1.3 7
1.3 1
1.2 7
1.2 3
1.2 0
1.1 7
1.1 5
1.1 4
0.9 5 1.1 3
Fluid classification 289
Table 5.4 Percentage of particles removed after a known number of decantings. Relative particle size
Number of decantings 1
(X)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.98 0.99
89.10 86.40 81.90 75.60 67.50 52.70 45.90 32.40 17.10 8.78 3.56 1.79
2 Percenta ge 98.81 98.15 96.72 94.05 89.44 82.02 70.73 54.30 31.28 16.78 7.00 3.55
4 8 of particles removed 100 99.99 100 99.97 100 99.89 99.65 100 98.88 99.99 99.90 96.77 91.43 99.27 95.64 79.12 77.69 52.77 52.04 30.74 25.20 13.51 13.46 6.97
16 100 100 100 100 100 100 99.99 99.81 95.02 77.00 44.05 25.11
The effect of the ratio {h2/h^) is small over a wide range compared with x hence there is little to be gained from making (//2 - h^) very small with consequent risk of disturbing the solids which have settled out. For example, if (//2/^i) is reduced from 0.9 to 0.8, the number of decantings to achieve the same separation will increase from nto m where (Table 5.3): Some values of the number of decantings required to achieve the same separation are presented in Table 5.3 and the percentages of various relative sizes removed are given in Table 5.4 for //2/^i^ 0.9. A large number of decantings are required to remove particles whose size is near the cut size. Hence the wider the size range of the original suspension, the fewer decantings required. Several descriptions of suitable apparatus have been described [121-123] including a fully including automatic system [124].
290 Powder sampling and particle size determination
References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
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Richards, J.C. (1966), The efficiency of classifiers, BCURA Monthly Bull. 30(4), 251 Wessel, J. (1967), Aufhereit Technik, 2(53), 251 Allen, T. and Baudet, MJ. (1977), Powder TechnoL, 18(2), 253 Svarovsky, L.(ed), (1978), Solid-Liquid Separation, Butterworths, 256 Leschonski, K. and Rumpf, H. (1968-1969), Powder TechnoL, 2, 175-185, 257,266 Gibson, K.R. (1977), Powder TechnoL, 18(2), 165-170, 258 Leschonski, K. (1976), Proc. Comminution and Air Classification, Univ. Bradford Short Course, 258 West, R.M., Cullivan, J.C. and Williams, R.A. (2000), Part. Part. Syst. Charact., 17, \39-\45, 260 Svarovsky, L. (1984), Hydrocyclones, Technomic, Penns., USA, 260 Svarovsky, L. (1992), Hydrocy clones. Holt. Reinhart and Winston, London, 260 Roller, P.S. (1931), Ind Engng. Chem., Analyt. Ed, 3, 212-216, 261,264 Stairmand, C.J. (1951), Engineering 171, May, 585-587, 261 Stairmand, C.J. (1947), Symp. Particle Size Analysis, Inst. Chem. Engrs, 25, 77, 263 Andrews, L. (1972), Proc. Inst. Engng. Inspection, 25, also Min. Mag, May, 301,2^^ Schone, E., (1867), Ober Schlammanalyse und einen neuen Schlammapparat, Berlin, 264 Andreasen, A,H,M. (1930), Ber. dt. Keram., Ges., 11, 675, 264 Blythe. H.N. Pryor, E.|J. and Eldridge, A. (1953), Symp. Recent Developments in Mineral Dressing, Inst. Min MetalL, London, 23-25 Sept., pi 1,2^^ Gonell, H.W. (1928), Z Ver. dt. Ing., 11, 945, 265 Hughes, T.H. (1957), 2ndConf. Pulverized Fuel, Inst. Fuel, London, 265 Stairmand, C. J. (1951), Engineering, 111, May, 585-587, 265 Haultain, H.E.T. (1937), Trans. Canad Min. MetalL, 40, 265 Neuzil, L., Bafrnec. M. and Bena, J. (1977), Czech Pat., 169 912 (cl 01N15100), June 15,2^5 Weilbacher, M. and Rumpf, H. (1968), Aufbereit. Technik, 9(7), 3230\-330, 265 BS 3406, (1963) reconfirmed (1983), British Standard Methods for Determining Particle Size Distribution, Part 3 Air Elutriation Methods, 266 Stein, F. and Com, M. (1976), Powder TechnoL, 13, 133-141, 2(5(5 Crandall, W.A. (1964), Development of standards for determining the properties of fine particulate material. Winter Annual Meeting ASME, NY, 232, 267
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69
Ramos, J.G. dos and Silebi, C.A. (1991), Particle Size Distribution II, ed. T. Provder, Am. Chem. Soc, ACS Symp., series 472, 292-307, 274 Nagy, D.J., Silebi, C.A. and McHugh, A.J. (1978), ACS Conf., Florida, 274 Nagy, D.J, Silebi, C.A. and McHugh, A.J. (1980), Polymer Colloids II, ed R.M. Fitch, Plenum Press, New York, 274 Penlids, A., Hamielic, A.C. and McGregor, J.F. (1983) J. Liqu. Chromatogr., 6, Suppl. 2, 274 Kirkland, J.J. (1979), J. Chromatog, 1, 273, 274 Singh, S. and Hamielic, A.E. (1978), J. Liqu. Chromatogr., 1,187, 274 Kohne, A.P., Mayr, G. and Welsch, T. (1997) LaborPraxis, 21(9), 28-30 (Ger), Vogel, 274 Kawahashi, M. et. al. (1975), Japan Kokai, 54, 391, 274 Silebi, C.A. and Ramos, J.G.dos. (1989), J. Colloid Inter/. Sci., 130, 14, 275 Noel, R.J., Goodwin, K.M., Regnier, F.E., Ball, D.M., Orr, C. and Mullins, M.E. (1978), J. Chromatogr, 166, 373-383, 275 Muller, CM., Venkatesan, J., Silebi, C.A. and Sudoi, E.D. (1994), J. ColloidInterf. Sci., 162, 11-18, 275 DiMarzio, E.A. and Guttman, CM. (1970), MacramoL, 3, 681, 275 Guttman, CM. and DiMarzio, E.A. (1970), MacramoL, 3, 131, 2 75 Silebi, C.A. and Ramos, J.G.dos. (1989), J. Colloid Interf. Sci., 130, 14, 275 Silebi, C.A. and Ramos, J.G.dos (1989), J. Colloid Interf. Sci., 133, 302, 275 Silebi, C.A. and Ramos, J.G.dos. (1989), J.Amer, Inst. ChemE., 35, 1351, 275 Ramos, J.G. dos and Silebi, C.A. (1990), J. Colloid Interf Sci., 136, 3, 275 Ramos, J.G.dos and Silebi, C.A. (1992), CHDF Application Note 501 Matec Applied Science, 2 75 Ramos, J.G. dos, Jenkins, R.D. and Silebi, C.A. (1991), Particle Size Distribution II, ed T. Provder, Am. Chem. Soc. Symp. Series, 472, 264-278, 275 Jaeger, N.C de. Trappers, J.L. and Larden, P. (1986), Part. Part. Syst.
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Miller, CM., Venkatesan, J., Silebi, E.D. Sudal. E.D. and Asser, M.S. el. (1984), J. Colloid Interf Sci., 162, 11-18, 276 Jorgenson, J.W. and Lukacs, K.D. (1981), Anal. Chem., 53, 1298, 276 McCormick, R.M. (1988), Anal. Chem., 60, 2322, 276 Jorgenson, J.W. and Lukacs, K.D. (1981), Analyt. Chem., 53, 3036, 276 Hocqs, S., Milot, J.F., Gourdon, C and Casamatta, G. (1994), Chem. Eng. ^-c/., 49(4), 481-490, 27(5 Yau, W.W., Kirkland, J.J. and Bly, D.D. (1979), Modern Size Exclusion Chromatography, John Wiley and Sons, Ch 12 & Chi3, 27(5 Singh. S. and Hamielec, J. (1978), J. Liqu. Chromatogr., 1, 187, 276
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77 78
79 80 81
82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97
98 99 100
Kirkland, J.J. (1979), J. Chromatogr., 48, 273, 276 Williams, P.S., Moon, M.H. and Giddings, J.C. (1992), Proc. Conf. PSA '91 ed. N.G. Stanley-Wood and R. Lines, publ. Royal Soc. Chem., 280-289, 277 Giddings, J.C, Ratanathanawongs, S.K. and Moon, M.H. (1991), Kona Powder and Particle, No 9, 200-277, 277 Caldwell, K,D. (1984), Modern Methods of Particle Size Analysis, 211250, ed. H.G/ Barth. John Wiley & Sons, 277 Ratanathanawongs, S.K., Lee, I. and Giddings, J.C. (1991), Particle Size Distributions II, ed. T. Provder, Am.. Chem. Soc, ACS Symp. series, 472, 229-246, 277 Giddings, J.C, Myers, M.N., Moon, M.H. and Barman, B.N. (1994), Chem. Eng. ScL, 9(4), 481-490, 278 Giddings, J.C, Yang, F.J. and Myers, M.N. (1974), Anal. Chem., 46, 19171977, 278 Caldwell, K,D., Brimhall, S.L., Gao, Y. and Giddings, .J.C (1988), J. Appl. Polymer Sci., 36, 703, 278 Reschiglian, P., Melucci, D., Tolsi, G. and Zattoni, A. (2000), Chromatographia, 51(1,2), 87-94, 4, 278 Anger, S., Caldwell, K., Niehus, H. and Muller, R.H. (1999), Pharm. Res., 16(11), 1743-1747,27^ Kirkland, J.J. (1982^, Science, 218, 8, 279 Barman, B.N. and Giddings, J.C. (1991), Particle Size Distribution II, ed. T. Provder, Am. Chem. Soc., ACS Symposium series 472, 217-228, 279 Kirkland, J.J., Yau, Doemer, W.A. and Grant, J.W. (1980;, J. Colloid Interf Sci., 60, 574, 281 Kirkland, J.J., Rementer, S.W. and Yau, W.W. (1981), Analyt. Chem., 53, 1730,2^7 Mori. Y. and Harada, M. (1998), World Congress Particle Technology, Inst, Chem. Engrs., Rugby, UK, 345-356., 281 Mori, Y., Kimura, K. and Tanigaki, M. (1991), Proc. Second World Congress Particle Technology, Kyota, Japan, Part I, 305-312, 282 Mori, Y., Kimura, K. and Tanigaki, M. (1991), Proc Conf PSA '91, ed. N.G. Stanley Wood and R. Lines, publ. Royal Soc. Chem., 280-289, 282 Giddings, J.C. (1981), Analyt. Chem., 53, 1170A, 282 Schunk, J. (1984), Separation Science, 19(10), 653, 282 Wahlund, K.G. and Litzen, A. (1989), J. Chromatog., 461, 73, 282 Ratanathanawongs, S.K., Lee, I. and Giddings, LC. (1991), Particle Size Distribution II, ed T. Provder, Am. Chem. Soc, ACS Symposium series 472, 229-246, 283 Giddings, J.C. (1981), Anal. Chem., 53, 1170A, 283 Granger, J. and Dodds, J. (1992), Separation Sci. and Technol, 27(13), 1691-1709,255 Wahlund, K.G. and Litzen, A. (1989), J. Chromatog., 461, 73, 283
294 Powder sampling and particle size determination
101 102
103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124
Ratanathawongs, S.K. and Giddings, J.C. (1993), ACS Symposium Ser., 1993-521 (Chromatography of Polymers), 13-29,2^5 Schauer, T. (1995), 6th European Symp. Particle Size Characterization, Partec '95, Numberg, Germany, publ. NumbergMesse GmbH, 157-166, 283 Giddings, J.C. and Myers, M.N. (1978), Sep. Sci. TechnoL, 13, 637, 284 Giddings, J.C, Myers, M.N., Caldwell, K.D. and Pav. J.W. (1979), Sep. Sci. TechnoL, 14,935, 284 Moon, M.H. and Lee, S. (1997), J. Microcolumn Sep., 9(7), 284 Wyatt, P.J. (1998), J. ColloidInterf. Sci., 197(1), 9-20, 284 Ratanathawongs, S.K. (1992), 3rd Int. Symp. FFF, FFF Research Center, Utah, 284 Ratanathanawongs, S.K., Shiundu, P.M. and Giddings, J.C. (1995), Colloids and Surfaces, A105, 243-250, 284 Caldwell, K.D., Li, J.M. and Li, J.T. (1993), Proc. Am. Chem. Soc, Div. Polymeric Material, Am. Chem. Soc, 69, p404, 284 Wyatt, P.J. (1998), J. Colloid and Interface Sci., 197, 9-20, 285 Giddings, J.C. (1979), Sep. Sci. TechnoL, 20, 749-768, 285 Giddings, J.C. (1980), Sep Sci. TechnoL, 23, 931-943, 285 Giddings, J.C. (1986), Chemical Separations. 1, 3-20, ed. J.D. Navratil and C.J. King, Litarven, Denver, 286 Springston, S.R., Myers, M.N. and Giddings, J.C. (1985), Sep. Sci. TechnoL, 20, 749-768, 286 Williams, P.S., Levin, S., Lenczycki, T. and Giddings, J.C. (1992), Ind Eng. Chem. Res., 31, 2172-2181, 286 Levin, S. and Giddings, J.C. (1991), J. Chem. Tech. Biotech., 50, 43-56, 286 Levin, S., Myers, M.N. and Giddings, J.C. (1989), Sep Sci. TechnoL, 24, 1245-1259, 286 Zhang, J., Williams, P.S., Myers, M.N. and Giddings, J.C. (1994), Sep Sci. TechnoL, 29(18), 2493-2522, 286 Herdan, G. (1960), Small Particle Statistics, Butterworths, 287 Allen, T. (1960), M.Sc. Thesis, London University, 287 Davies, R.J., Green, R.A. and Donnelly, H.F.E. (1937), Trans. Ceram. Soc, 36, \Sl,289 Birchfield, H.P., Gullaston, D.K. and McNew, G.L. (1948), Analyt. Chem., 20,1168,2(^9 Johnson, E. and King, J. (1951), Analyst, 76, 66, 289 Horsfall, F. and Jowett, A. (1960), J. Scient. Instrum., 37(4), 120, 289
Interaction between particles and fluids 6.1 Introduction The interaction between particles and fluids is important in many industrial processes. The most relevant size distribution in such cases is, therefore, based on the sedimentation behavior of particles in fluids and the applicable size is the settling or Stokes diameter. The simplest case to consider is the settling of a single homogeneous sphere, under gravity, in a fluid of infinite extent. Many experiments have been carried out to determine the relationship between settling velocity and particle size under these conditions and a unique relationship between drag factor (C^)) and Reynolds number (Re) has been found that reduces to a simple equation, known as the Stokes equation, at low Reynolds number. At high velocities, due to the onset of turbulence, the drag on the sphere increases above that predicted by Stokes* equation and particles settle more slowly than the law predicts. However, settling velocities can be related to particle diameters by applying Newton's equation which is available as an empirical relationship between C^ and Re. The upper size limit for Stokes' equation is limited, due to the onset of turbulence, to Reynolds number smaller than 0.25. Difficulties arise with non-spherical particles that fall in random orientation in the laminar flow region, but orientate themselves to give maximum resistance to drag in the turbulent region. Thus, in the laminar flow region a single particle will have a range of settling velocities (usually narrow) depending on its orientation whereas, once the boundary condition is exceeded, the measured size distribution becomes finer in a manner difficult to predict.
296 Powder sampling and particle size determination
After agitation, a finite time is required before the particles commence to settle uniformly, but this is much greater than the time the particles are accelerating to a constant velocity, known as the terminal velocity, under a gravitational force. If the concentration is monitored at a fixed depth below the surface, for an initially homogeneous suspension, it will remain constant until the largest particle in the suspension has fallen from the surface to the measurement zone. The concentration will then fall, being at all times proportional to the concentration of particles smaller than the diameter given by Stokes' equation for that particular time and depth of fall. Samples removed from this depth should not contain particles with diameters larger than the Stokes diameter. In practice this is not true due to particle-particle interaction; pairs of equally sized spheres, for example, falling in close proximity will interfere with each other and fall with a different velocity to that predicted by Stokes' equation. For unequally sized particles the situation is more complex; the larger particle may actually pick up the smaller one so that it revolves round the larger one like a satellite. Assemblies of particles tend to diverge due to the rotation effect caused by the greater velocities of the streamlines on the envelope of the assembly. A cluster of particles may act as a single large particle of appropriate density and reduced rigidity and have a much greater velocity than that of the particles of that it is composed. Interactions between particles are usually negligible provided the concentration is low. It is usually recommended that the volume concentration be no higher than 0.2%; if it is necessary to use a higher concentration it should be established that the measured size distribution is no finer than that obtained at the lower concentration. At volume concentrations as low as 1% the suspension may settle en masse. The rate of fall of the interface gives an average size for the particles using a modified Stokes equation that is very similar to the permeametry equation for a fluid passing through a fixed bed of powder. This is in fact an analogous system in that a moving bed of particles is settling through a stationary liquid. In some cases multiple interfaces occur due to smaller particles penetrating the initial interface to form a second or even a third interface. Settling velocities are also reduced due to the proximity of container walls, though this effect is usually quite small. As particle size decreases to the sub-micron level, the particles begin to move in a random manner due to molecular bombardment (Brownian or thermal diffusion) to give a variable settling rate for particles of the same size. Indeed, a proportion of the particles can actually rise during a time
Interaction between particles and fluids 297
interval, although the concentration at a fixed depth will fall. Impressed on this effect there are convection currents that may be set up by surface evaporation or temperature fluctuations. All sedimenting systems are basically unstable due to the preferred paths up the sedimentation tank by the fluid displaced by the settling particles. This fluid tends to rise up the walls of the containing vessel, carrying with it some of the finer particles, and dissipate itself as convection currents at the top of the sedimentation column. This leads to an overestimation of the fines percentage in an analysis. For these reasons, for accurate analyses below about a micron, centrifugal techniques should be used. Correction for the finite extent of the fluid is negligible in most cases and errors due to discontinuities in the fluid are only of importance for gas systems. Similarly the increased viscosity of a suspension over that of a pure liquid has negligible effects at low concentrations. Reproducible data are possible at high Reynolds numbers, high concentrations and with submicron particles. These data may be highly inaccurate and in general precise values of erroneous sizes and percentage undersize are of limited worth. 6.2 Settling of a single homogeneous sphere under a gravitational force 6.2.1 Relationship between settling velocity and particle size When a particle falls under gravity in a viscous fluid three forces act upon it; a gravitational force W acting downwards and a buoyant force U and a drag force F^ acting upward. The resulting equation of motion is: W-U-Fr.=m— ^ dt mg-m'g-F^=m-—
(6.1)
m is the mass of the particle, m' is the mass of the same volume offluid,u is the particle velocity and g is the acceleration due to gravity. Small particles accelerate rapidly to a constant or terminal velocity {du/dt = 0) and the motive force balances the drag force.
298 Powder sampling and particle size determination
For a sphere of diameter D and density p^ falling in a fluid of density ycy, the equation of motion becomes:
^D-fU-P/k^^
(6.2)
The drag coefficient C^, is given by: _
drag force
C^) —
cross-sectional area x dynamic pressure of the sphere on the particle F D = C ^ X ^ ^ ^
(6.3)
Combining equations (6.2) and (6.3) gives:
C„=iklAv)4 3
Pf
(6.4)
u'
Dimensionless analysis of the general problem of particle motion under equilibrium conditions gives a unique relationship between two dimensionless groups, drag coefficient (C^) and Reynolds number (Re) where: Re = ^—
(6.5)
rj is the viscosity of the fluid. The experimental relationship, for spheres, between the Reynolds number and the drag coefficient is shown in Figure 6.1 and Table 6.1. The graph is divided into three regions; a laminar flow or Stokes region, an intermediate region and a turbulent flow or Newton region. In the laminar flow region the drag on the particle is due entirely to viscous forces within the fluid. As the Reynolds number increases, eddies build up behind the
Interaction between particles andfluids
299
particle thus increasing the drag. Further increase in Reynolds number leads to fully turbulent motion. Table 6.1 The experimental relationship between drag coefficient; and Reynolds number for a sphere settling in a liquid [1] Re
CD
0.01 0.02 0.05 0.08 0.10 0.20 0.50 0.80 1.00
2400 1200
484 304 244 123 51.4 33.3 27.2
Re
2 5 10 20 50 100 200 500 1000
CD
15.00 7.12 4.35 2.74 1.56 1.10 0.808 0.568 0.460
Re
CD
2000 5000 10000 20000 50000 100000 200000 500000 800000
0.420 0.410 0.420 0.450 0.480 0.480 0.440 0.200 0.220
6,2.2. Calculation ofparticle size from settling velocity in the laminar flow region From Figure 6.1 and Table 6.1 it can be seen that, as the Reynolds number approachO.Szero, C^e approaches 24, i.e. in the limit: CD =
24 Re
(6.6)
This is equivalent to stating that the drag on a spherical particle falling in a fluid of infinite extent is due entirely to viscous forces within the fluid. Combining equation (6.3), (6.5) and (6.6) gives, for low Reynolds numbers: (6.7)
F^ = 3nDT]Us,
Inserting equation (6.7) in equation (6.2) gives the relationship between the particle diameter and its Stokes velocity: D
\ST]USt
\(Ps'Pf)
(6.8) g
300 Powder sampling and particle size determination
where u^^ is the terminal veIocity,in the laminar flow region, for a sphere of diameter D, Hence, for low Reynolds Numbers, if the diameter of a settling sphere is known then its velocity can be predicted and vice-versa. This equation is known as Stokes equation. Terminal velocities will be about 3.5% lower than predicted by Stokes' equation at Re = 0.25 and the derived diameter will be 1.7% smaller than the physical diameter. This difference will increase with increasing Reynolds number. The assumptions made in the derivation of Stokes' law are: • the particle must be spherical, smooth and rigid and there must be no slip between it and the fluid • the particle must move as it would in a fluid of infinite extent • the terminal velocity must have been reached • the settling velocity must be low so that inertia effects are negligible • the fluid must be homogeneous compared with the size of the particle 10000 (24IRe) 1000
3' 100 I 10
I
1 k
Intermediate region
Laminar flow| region
Turbulent region
(Stokes)
0.1 0.001 0.01
"IM11
0.1
I I mini
I 1 mill
i i mini
1 10 100 1000 Reynolds number (Re)
Fig. 6.1 Experimental relationship between drag coefficient and Reynolds number for a sphere settling in a liquid. 6.3 Size limits for gravity sedimentation The upper size limit by gravity sedimentation is fixed by the unreliability of measurements taken in the first 30 s or so of the analysis and by the
Interaction between particles andfluids
301
breakdown in Stokes' law due to the onset of turbulence; the extent of this breakdown is well documented. The lower size limit is due to diffusional broadening (or Brownian diffusion) and is a matter of some debate. This diffusion is due to bombardment by fluid molecules that causes the particles to move about in a random manner with displacements in all directions that ultimately exceed the displacements due to gravity. Brownian motion is not the only limiting criterion. Gravity sedimentation is inherently unstable, due to factors such as return flow and vibration, so should be used with caution for very fine powders. 6.3.1 Upper size limit Stokes equation is usually considered valid for Reynolds numbers less than 0.25 and the correction term is neglected. Combining equations (6.5) and (6.8) and applying the condition that Re ^ 0.25 gives the upper size limit (upper critical diameter D^^. xl/3
Due
(6.9)
Pfg[ps-Pf)j
For example, for silica (p^ = 2500 kgm~^) sedimenting in water {pj = 1000 kgm~^, r| = 0.001 Pas, g = 9.81 ms"^) the upper size limit is 67.4 |Lim. The upper size limit for silica settling in air is 33 |im. The upper critical diameter can be increased by increasing the liquid viscosity; alternatively a correction term may be applied. For routine applications the diameters obtained using Stokes equation at Re greater than 0.25 are accepted as correct. The following relationship is an empirical fit to experimental data for spherical particles, between drag coefficient and Reynolds number for Reynolds number between 0.01 and 1.0: 24 / \ C^ =—(0.998 + 0.157?^-0.0137?^2J
(6.10)
The resultant errors at Reynolds numbers of 0.25 and 1.0 are given in Table 6.2 where % error is defined as 100[(i)-i)^^)/Z)].
302 Powder sampling and particle size determination
Table 6.2 Differences between Stokes diameter {D^^ and true diameter for a sphere of diameter {D) at two values of Reynolds number Re
CD
0.25 99.33 1.00 27.24
CjyRe^
6.208 27.240
Re/Cj^
0.0025 0.0367
D 68.14 111.6
u ( m m s"^) 3.67 8.96
%1
(mm s"^) Error 1.69 66.109 3.80 104.7 10.17 6.16 i (jim)
Since C^, and Re are functions of the two variables u and D, it is necessary to derive relationships that are functions of only one of these variables. Cj-^e^ is a function of D only and Re/C^) is a function of w. For Reynolds Numbers less than 1.0, C^ may be determined using equation (6.10) so that CjyRe^ and Re/C^^ can be calculated. u and D can be calculated for the case of silica under the conditions quoted above, i.e. a spherical silica particle of diameter 68.14 |Lim will settle, under the conditions under consideration here and Re = 0.25, with a velocity of 3.67 mm s"^ instead of the 3.80 mm s~^ derived using Stokes' equation. Applying Stokes* equation, the Stokes diameter is 66.11 |um, an undersizing of 1.69%. For Re = I, sometimes quoted as the upper limit for the application of Stokes equation, a silica sphere of diameter 111.6 |Lim will settle with a velocity of 8.96 mm s'^ rather than the 10.17 mm s"^ obtained using Stokes equation; this gives a Stokes diameter of 104.7 /am and an undersizing of 6.16%. The effect is greater for non-spherical particles since they settle with random orientation in the laminar flow region but orientate to give maximum drag in the turbulent flow region. The time required for a particle to reach its terminal velocity is negligible, but excessively short sedimentation times should be avoided since concentration measurements fluctuate due to the initial agitation, up to a time of about 30 seconds. 6.3.2 Lower size limit Stokes' law breaks down for very small particles settling under gravity due to diffusional broadening (or Brownian diffusion). This diffusion is due to bombardment by fluid molecules that causes the particles to move about in a random manner with displacements, in all directions, that ultimately exceed the displacement due to gravity settling.
Interaction between particles andfluids
303
The equations to quantify this effect were developed by Mason and Weaver in 1924 [2]. For monosize distributions Brownian motion leads to broadened measured distributions [3] but the effect is reduced as the width of the distribution increases [4,5]. Chung and Hogg [6] carried out theoretical and experimental studies of clay particles using centrifugal and gravitational sedimentation. Agreement between theory and practice was not too good. The root mean square displacement due to Brownian motion in the time interval /is [7]:
^hlff
^^^^
(6.11)
371T/D
k is the Boltzmann constant (1.3806 x 10-^3 J K-i) Thus the root mean square displacement in 1 s for a 1 |Lim particle settling in water, viscosity 0.001 Pa s, at an absolute temperature 300 K is 0.938 \xm\ this is almost the same as the distance settled under gravity by a quartz particle (density 2650 kg m"^) in 1 s (0.90 |im). A comparison of Brownian movement displacement and gravitational settling displacement is given by Fuchs [8]. For a size determination to be meaningful the displacement of the particles due to Brownian diffusion must be much smaller than their displacement due to gravity, hence the condition:
a ^hlff « h'
(6.12)
where h equals w^/, so that, in combination with Stokes equation [Equation (6.8)], the lower size limit for gravitational sedimentation is given by: Kin »
-2 '^ (P.-Pf)s't
(6-13)
— {^Ps-Pf)^^
(6.14)
or: D\ »— ^min
"
304 Powder sampling and particle size determination
Thus, the lower size limit depends upon the nature of the sedimenting system, e.g. density, temperature etc. It also increases if the analysis time is reduced or the measurement height is decreased. The measured distributions for monodisperse powders will be broadened, the effect increasing as the measurement time is reduced [9]. A corollary to this is that analyses of fine powders, in which a scanning system is used, will give different results at different scan speeds. For an estimate of the lower size limit, the displacement of the smallest particle by Brownian diffusion should be at least ten times smaller than its settling distance [10]. Other criteria could be selected since the error is both a function of the size and the spread of the distribution. It is reasonable however, for the sake of simplicity, to stipulate that if more than 10% of the distribution is smaller than the lower size limit, gravitational sedimentation should not be used. Correspondingly, equations (6.13) and (6.14) take the equality sign and their right hand sides are multiplied by 100. Equation (6.13) becomes:
Example: For quartz particles settling in water: Pg = 2500 kg m~^ Pf= lOOOkgm-3 yt = 1.3806 X 10-23 J K-l
Hence, in meters: 2.59x10"^ ^mm = ~~^
r=300K g = 9.81ms-2 7] = 8.905 X 10-2 Pa s
^ ^mm
1.024x10-^ 37^
For quartz and a run time of 30 min, / = 1800 s and Z)^^„ = 0.58 |im: For a run time of 15 min, / = 900 s and D^j^= 0.66 |um. These values are much lower than those quoted in national standards (BS 3406 Part 2:1984, DIN 66111:1983, AFNOR, ASTM) that vary from 1 |Lim to 3 |im. The applied criterion is different since the requirement here is that if 10% of the distribution is smaller than the lower size limit, gravitational sedimentation should not be used whereas the Standards simply state a lower size limit.
Interaction between particles and
fluids
305
One common way of monitoring the changing concentration in a settling suspension is to scan upwards, from far below the top of the sedimentation column. Assuming the initial reading is taken at a depth of 2.5 cm at time / = 10 seconds, the maximum size that can be measured, from Stokes equation, is 52.2 |am. Assuming a beam width of 100 |Lim, Heywood's criterion [1] is that the final measurement depth must be at least 0.7 |Lim. The logarithmic relationship between measurement height and time, using the above boundary conditions is: For a scan time of 30 min:
h = 0.0330-0.0080 log(/)
(6.16)
For a scan time of 15 min:
h = 0.0342-0.0092 log(0
(6.17)
This hyperbolic scan compresses small diameters into shorter times, to give similar resolutions at all heights, hence is the preferred scanning procedure. For sub-micron powders and monomodal log-normal parent distributions, calculations indicate that the measured distributions would be bimodal log-normal and broadened, i.e. indicating an excess of coarse as well as an excess of fines [11]. The error increases with decreasing size and decreasing spread (geometric standard deviation). The theoretical effect for a powder with a density of 4250 kg m"^, a geometric mean of 0.228 |Lim and a geometric standard deviation of 1.30, is an excess of 13% at the 10% level and 10% at the 90% level. As expected, due to the inherent instability of sedimenting suspensions, experimental results give even larger differences. Brownian motion is normally negligible when compared with even the most feeble convection currents [12 pi6] and this further limits the smallest size at which gravitational sedimentation gives meaningful results. Muta and Watanabe [13] examined samples extracted during Andreasen analyses and detected particles of 1 \xm, for example, when Stokes' equation predicted a maximum size of 0.89 |um. They concluded that gravitational Andreasen sedimentation analysis was unsuitable for powders of wide size range containing sub-micron particles. They surmised that this might be due to re-distribution of fine particles caused by disturbance to the suspension in the course of long-term sedimentation. Allen [14] reported the presence of convection currents that appeared in sedimenting suspensions. He concluded that these were caused by a basic instability in sedimenting suspensions that defines the lower size for that gravitational sedimentation could be used. The liquid displaced by the
306 Powder sampling and particle size determination
sedimenting particles tended to flow towards the surface up the sides of the walls and stirrer. This he attributed to the lower velocity of particles near such vertical surfaces which generated a radial pressure gradient, causing the particles to rotate in such a way to cause such a flow. The effect is most apparent when there are large particles together with fines. The large particles initiated a return flow of about 15 |Lim s"^, and particles in the vicinity of the walls, having a velocity smaller than this, were entrained and carried towards the surface. Allen and Baudet [15] compared several centrifugal techniques with the Andreasen and the Sedigraph x-ray sedimentometer. As an example, for kaolinite, the centrifugal techniques gave the percentage smaller than 0.2 |Lim as 31 to 36.6; the Andreasen gave 48; the Sedigraph gave 80. The high percentage of fines as given by the Andreasen was attributed to the long settling times required. In the Sedigraph high concentrations were required, since the material was a poor x-ray absorber, and the high concentrations resulted in hindered settling. Other cases are known where Andreasen results are comparable with centrifugal results. For example, for quartz particles of size range 0.2 to 0.8 |Lim, the author has found that the Andreasen results, using a volume concentration of 0.2%, agree with pipette centrifuge results. 6.4 Time for terminal velocity to be attained For laminar settling, equation (6.1) may be written: '^[p^-p^)gD'-3nriDu^^p,D'^
-.. . . . • , • (ps'Pf) Simplifying by putting p = — = pg-Xu at
7 ^=l^f J{pg-Xu) 0 0
(6.18)
. ,. 18/7 and X = j PsD'
Interaction between particles and fluids 307
Pg u = X (l-exp(-X<))
(6.19)
As / approaches infinity, u approaches the Stokes velocity %^ = pg/X as given in equation (6.8). Theoretically a particle never reaches its terminal velocity but for practical purposes it can be assumed that the velocity is sufficiently near the terminal velocity for the error to be neglected. From equation (6.19) the velocity is 0.99 times the terminal velocity when: l-exp(-XO = 0.99 A.6pD^ I877
(6.20)
For spheres of density 2650 kg m"^, in water 77 = 0.001 Pa s and in air 77 = 18 X 10"^ Pa s, the times taken to reach 99% of the terminal velocity for particles of different diameters are shown in Table 6.3. From Table 6.3 it can be seen that the assumption that a particle from rest reaches its terminal velocity instantaneously does not introduce any appreciable errors. Table 6.3 Time for particles to reach 99% of their terminal velocities Diameter (urn) 5 10 50
Time (ms) Water 0.0017 0.068 1.693
Air 0.94 3.76 94.06
The distance covered, h, during the acceleration time is given by integration of equation (6.19):
^Ah^Ust\i}^-exp{-Xt)^t
308 Powder sampling and particle size determination
h - u^^t 1
(l-exp(-Xir)
(6.21)
Xt
At 99% of the terminal velocity [l - exp(-X/)] = 0.99 as before and Xt = A.6 making h = OJSusft Since / is very small, the distance fallen in achieving a velocity equal to 0.99 times the settling velocity at low Reynolds numbers is also very small. 6.5 Errors due to the finite extent of the fluid (wall effects) When the fluid is contained in a vessel, the fluid streamlines about the particle, impinge on the walls and reflect back on to the particle causing increased drag. Also, since the fluid is stationary at the vessel walls, there is a distortion of the flow pattern that reacts back on the particle. Both effects increase the drag on the particle, leading to a reduced estimate of particle size. The modified form of equation (6.7) is [16]: F ^ = 3nDT]u \ + k
(6.22)
where L is the distance from the center of the particle to the walls of the containing vessel. For a single wall k = 0.563, for two walls k = 1.004 and for a circular cylinder k = 2.104. Each extra wall increases the drag by an approximately equal amount, thus the effect on a sphere in a cylinder is much the same over a large part of the central area. Replacing equation (6.7) by equation (6.22) modifies Stokes' equation to give:
D = D.'St ] - ^ k
D
(6.23)
where D is the true diameter of the spherical particle and D^^ the diameter obtained using the unmodified Stokes equation.
Interaction between particles andfluids
309
For a 100 |am sphere settling centrally In a 0.5 cm diameter container (L = 0.25 cm) the error in particle size is given by:
r
100x10 -6
^
0.5
D=:D'St 1 + 2.104 0.25x10 -2 V D = 1.041D^^, an error of 4%. The modified form of equation (6.10) due to the bottom of the container is: 9 D Fp = 3nDT]u l + — 16 B
(6.24)
where B is the distance between the particle and the container bottom. The correction term is negligible at a distance greater than 1000 particle diameters from the bottom and small (0.6%) for a distance of 50 diameters. A sphere moving near a vertical wall will also rotate, as if it were rolling on the wall, at an angular velocity given by [10]. 0)=-
3u_ 2D
V
T]
1+ 4 L
(6.25)
Between two parallel walls w i t h D « L, where the sphere is located such that its distance from one wall is three times its distance from the other, rotation is in the opposite direction to rolling in the near wall with angular velocity: CD =
80
W
D 1 + 0.326
(6.26)
where L is the distance to the nearest wall. 6.6 Errors due todiscontinuity of the fluid The drag factor needs to be modified, for particles settling in a gas, to take account of the molecular nature of the gas. At low pressures, if the mean free path of the gas molecules (A) is much greater than the size of the
310 Powder sampling and particle size determination
particles, the resistance to particle motion is due to molecular bombardment by individual molecules acting independently. This is much smaller than the Stokes drag leading to increased velocity: 4.49/1 U = Ust
(6.27)
CD
where C depends on the nature of the molecular reflections and lies between 1 and 1.4 [17]. At high pressures, for particles much larger than A, the discontinuity effect gives rise to 'slip' between the particle and the gas leading to the following modification to the Stokes velocity:
u-u
St
2A_ 2-f D f
.(6.28)
,
where / the fraction of molecules undergoing diffuse reflection at the particle surface, is of the order of 0.9. Cunningham [18] introduced a correction for slip of the form:
u = uSt
2 AX D
(6.29)
where A is approximately equal to unity. Experimental data, analyzed by Davies [19] gave the more accurate empirical equation:
U = UsA\
2.514 +O.SOOexp -0.55
A
(6.30)
For air at 20°C and a pressure of 1 atmosphere A. = 6.62 x 10~* m so that, with D in )iim, equation (6.30) becomes: u = UsAl- —[0.1663 + 0.0529exp(-8.32Z))]
(6.31)
Interaction between particles andfluids
311
The mean free path is here defined as: A ^ ^ j ^ ^ PV8M
(6.32)
where R is the molar gas constant, T the absolute temperature and M the molecular weight of the gas. Lapple [20] gives the following values for the square bracketed term in equation (6.31) for spherical particles in air at a temperature of 70°F
(2rC): D{\im) [ ]
0.1 1.88
025 OL68
050 0,33
To 0.166
lOO 0.0166
The second term in the square bracketed term becomes negligible for D > 1 |Lim. The correction is only of importance for sedimentation in a gas. 6.7 Viscosity of a suspension When discrete particles are present in a fluid, they cannot take part in any deformation the fluid may undergo, and the result is an increased resistance to shear. Thus, a suspension exhibits a greater resistance to shear than a pure fluid. This effect is expressed as an equivalent viscosity of a suspension. As the concentration of solids increases, so the viscosity increases. Einstein [21] deduced the equation: Tj^ = 7](l + kc)
(6.33)
where 77 is the viscosity of the fluid, rij the viscosity of the suspension, c the volume concentration of solids and k a constant that equals 2.5 for rigid, inertialess, spherical particles. The equation is found to hold for very dilute suspensions but requires modification for c greater than 1%. In the very dilute suspensions used in sedimentation analysis, the effect is smaller than errors inherent in the determination of // by conventional methods.
312 Powder sampling and particle size determination
6.8 Non-rigid spheres Non-rigid particles, e.g. liquid droplets, will deform in such a way that the drag will be reduced. The diameter {D^^ calculated from the increased terminal velocity will therefore be greater than the true diameter D. It has been shown that the drag is [7]: F^=3nDrj2U^'^^^^'^^A 37,+3%
(6.34)
where TJI and 7/2 are the viscosities of the drop and fluid respectively. For the case of a gaseous bubble rising slowly through a liquid, in the laminar flow region, rj^ « 772 and thus: F^ = 2nDT]2U
(6.35)
This is identical to the drag force on a solid sphere at whose surface perfect slip occurs. Comparing with Stokes' equation gives, for a droplet of diameter D: Ds,'DfJ^
(6.36)
For a raindrop falling in air; TJ^ =10~^ Pa s; 772 = 0.180 x 10"^ Pa s making Dst=\.026D. Experimentally, small bubbles behave like solid spheres having terminal velocities closely approaching Stokes' that, according to Levich [22] may be attributed to impurities at the interface. 6.9 Non-spherical particles 6.9.1 Stokes' region Homogeneous symmetrical particles can take up any orientation as they settle slowly in a fluid of infinite extent. Spin-free terminal states are attainable in all orientations for ellipsoids of uniform density and bodies of revolution with fore and aft symmetry, but the terminal velocities will depend on their orientation. A set of identical particles will, therefore, have a range of settling velocities according to their orientation. This
Interaction between particles and
fluids
313
range is fairly limited [23] being less than 2:1, for discs and cylinders having length: diameter ratios up to 10:1. Particles that are symmetrical in the sense that the form of the body is similarly related to each of three mutually perpendicular co-ordinate planes, as, for example, a sphere or a cuboid, not only fall stably in any orientation but fall with the same velocity in any orientation. An orienting force exists for less symmetrical particles. Asymmetric particles, such as ellipsoids or discs, do not generally fall vertically, but tend to drift to the side. Thin, flat, triangular laminae fall edgeways unless equilateral. Few particles possess high symmetry and small local features exert an orienting influence. Not all bodies are capable of attaining steady motion; with unsymmetrical bodies spiraling and wobbling may occur. For an oblate spheroid [12 p 144] of eccentricity s and equatorial diameter a, the maximum drag force, for 6: tending to 0, is: F^-371^7X1 -0.2^)w
(6.37)
For a sphere of equal volume to the above oblate spheroid F^-37ra/Xl-0.33^)w
(6.38)
Hence a sphere of equal volume has a smaller resistance than the spheroid; the same is also true for a sphere of equal surface.
Fig. 6.2 Direction of motion of a disc settling in a fluid
314 Powder sampling and particle size determination
A prolate spheroid, settling with its axis of revolution parallel to the direction of motion, behaves as a long thin rod when the major diameter a greatly exceeds the equatorial diameter b [7]. For this limiting case: F^ =, '^^^" , ^ (lna/6 + 0.1935)
^
(6.39)
Because of the logarithmic term the drag changes but slowly with the ratio of major to minor axes. A thin circular disc of diameter a, thickness b, will have both a horizontal (u^) and a vertical velocity (wj component of velocity (Figure 6.2), where: nabg Ap . ^ ^ 128 77 z.,^ ^ ^ ^ ( 5 - C O S 2^) "^ 128 7 ^ ^
(6.40)
(/> is the angle between the normal and the vertical to the plane of the disc. If /] is the angle between the downward vertical and the direction of motion of the disc, then: sin2^
tan/?-5^ 5 - cos 2(p
^^ ^IX
(6.41)
This angle is a maximum when the disc orientation is ^ = 39.2° J3= 11.5° corresponding to a maximum ratio of horizontal to vertical velocities of 0.204. When ^ = 0°, the disc will fall with its face horizontal and the drag force will be: F^^=Sarju^
(6.42)
When (/> = 90°, the disc will fall with its face vertical and the drag force will be: ^D,=j7«"v
(6-43)
^L,
Interaction between particles and fluids 315
|J Averaging ^ e r all orientations: Fjy^ = 6arjm
(6.44)
A disc, falling with its face horizontal, will have a velocity: u=^e^M^ab
(6.45)
A disc, falling with its face vertical, will have a velocity: .
=^^^a6 // 64
(6.46)
These velocities may be compared with that derived using Stokes equation: "st=^^7^^s,
(6.47)
A disc falling with its face horizontal will have a Stokes diameter given by: 4, ~ab
(6.48)
A disc falling with its face vertical will have a Stokes diameter given by: dl=^ab
(6.49)
The Stokes diameters for a falling disc will range from: lib ^stu ^f^^0.94J— a ^ a
(6.50)
to
(6.51)
- ^ = IA5J— a \a
These expressions were first derived by Lerman et. a/.(138). For a disc of diameter 10 and thickness 1 the Stokes diameters for horizontal and vertical orientation are 4.20 and 5.14. These equations were confirmed experimentally by Konert and van der Berghe [139].
316 Powder sampling and particle size determination
The ratio of velocities is given by: (6.52) with an average velocity u^ = 1.125w;^. The ratio of Stokes diameters is the square root of this i.e. (6.53)
ds,Jds,^=\ 225
With an average diameter d^^ = lAlSd^^ . Thus, if the average Stokes diameter is taken as unity the measured diameter range will be from 0.94 to 1.15. These equations are similar to the Stokes equation with the drag diameter of the same order as the disc diameter. Pettyjohn and Christiansen [24] made an extensive experimental study of isometric particles and proposed the following relationship: dl=d;
xOM3log
W 0.065
(6.54)
where the sphericity \i/ is the ratio of the surface area of a sphere of the same volume as the particle to the surface area of the particle:
\l/.
'ay
(6.55)
\^sj
Hawksley [17] showed that this was equivalent to stating that the drag diameter closely approximates the surface diameter. For non-spherical particles, equation (6.3) becomes:
^D-CD
nd,2 ^ Pf"
2\
(6.56)
d^ the drag diameter, is the diameter of the cross-sectional area of the particle perpendicular to the direction of motion. The modified form of equation (6.2) is: ^D=^(ps-Pf)g^. d„ is the volume diameter.
(6.57)
Interaction between particles and
fluids
317
The Stokes* equation for non-spherical particles is therefore:
ds. = H ^
(6-58)
dgf, the Stokes diameter, is:
(6.59) For non-re-entrant particles the drag diameter (d^) is the same as the surface diameter d^ and the Stokes diameter is defined as: (6.60)
Since the drag diameter is otherwise indeterminable, it is usual in practice to assume that, in the Stokes region, it is equal to the surface diameter. This holds at low Reynolds numbers but as Reynolds number increases, d^ > d^. For irregular, compactly shaped particles the range of settling diameters is limited (Figure 6.3). 6.9.2 Relationship between fiber diameter and Stokes diameter The Stokes diameter of a solid cylinder, diameter d^, depends upon its orientation. When the aspect ratio (length/diameter) (3 is large the effective Stokes diameter of a long thin cylinder flowing with its major axis parallel to the lines of flow is [7]: d'st = .r. \:"\.s 3(ln2y5 + 0.5) For the case where the axis is perpendicular to the lines of flow
(6-61)
318 Powder sampling and particle size determination
The resistance of a cylinder falling with arbitrary orientation, neglecting periodic oscillation is [7]: (6.63)
F^ ~ 3nT]v^ {d'^f cos^ 4- d^t sin^)
where ^ is the angle relative to the vertical flow lines. The gravitational force on a cylinder of density p^ and diameter d^ settling in a fluid of density p^is given by: (6.64)
F,=^P4{p.-Pf)g
Equating equations (6.63) and (6.64) and inserting the values for Stokes diameter from equations (6.61) and (6.62) gives: (ps-P/)gd. cos^ STJ [In2y9 -0.5
(6.65)
2sin^ ln2y0-fO.5
^amar=27.7nm
^*.™,= 23.6nm
25.0 nm
20.4 urn
22.5 urn
Fig. 6.3 Range of Stokes diameters for a particle of volume diameter 20.4 ^m
Interaction between particles andfluids
319
From Stokes Law, the settling velocity is related to the Stokes diameter by the equation:
_(Ps-Pf)gdl
^c-
~
(606)
18/7 Equating equations (6.65) and (6.66) gives: ds, = — (
^ :T7T cos^ 3d, 2sin^ nr\c /h In 2/?-0.5 ln2>5 + 0.5
(6-6^>
From Figure (6.5) it can be seen that, for a cylinder settling in a liquid, the Stokes diameter is larger than the cylinder diameter for cylinders with an aspect factor greater than 1.4, the difference increasing with increasing aspect ratio. Further, the settling velocity is not greatly affected by particle orientation, but the cylinder does settle more slowly, i.e. smaller Stokes diameters, when not either vertically or horizontally orientated. Henn [25] assumed an average orientation of 63% and stated that if the aspect ratio was greater than 20 equation (6.47) could be approximated by: dst^iXnipf^d,
(6.68)
with a positive error of less than 7%. Thus, he states, the Stokes diameter of a fiber depends directly on its diameter but only weakly on the aspect ratio, the influence of which drops off as it increases. Calculated values were found to be in good agreement with experimental values for phosphate fibers. Henn underlines the importance of accurate knowledge of fiber diameters in determining whether they are likely to be retained in the lung, i.e. with an aerodynamic diameter between 3.5 |Lim and 7 |Lim. 6.9.3 Transition region In the transition region particles fall with their largest cross-sectional area horizontal to give maximum resistance to drag [19]. Hawksley [17] proposed that the Cj^Re relationship be used with the following definitions:
320 Powder sampling and particle size determination
_
4
(Ps-Pf)pfud,g
J
Re =
(6.69)
U
ud. v^/
(6.70)
Vr
where y/ is the sphericity. It is more convenient to extend Heywood's technique [1] to nonspherical particles than to use these equations. C^^Re^ is evaluated in terms of volume diameter d^ and, if this is known, the free-falling velocity zymay be determined. Heywood determined u for particles of different projected area diameters d^ as seen in a microscope. The volumes of such particles are given by: (6.71)
V = a^ d'^
0.0 4 10
20
30
40
50
60
70
80
90
Cylinder orientation (degrees) Fig. 6.4 Stokes diameters for cylinders of different aspect ratios settling in different orientations (• >5 =.1, - /? = 2, A >0 = 4, n >0 = 8, /? - 16, —>0 = 32, • /? = 32)
Interaction between particles and fluids 321
10
§ o
£
2
10
8 u ^
,0 10'
.-2 10 10 -3
10-1 10^ 103 Reynolds number based on d^
105
Fig. 6.5 Drag coefficient versus Reynolds number for particles of different volume shape coefficients by microscopy (a^a) compared with data for a sphere Table 6.4 Best values of parameters for some regular shapes Shape Sphere Octahedron Cube Tetrahedron
y^ 1.000 0.846 0.806 0.670
A 0.1806 0.2559 0.2734 0.4531
B 0.6459 0.5876 0.5510 0.4484
C 0.4251 1.2191 1.4060 1.9450
D 6880.5 1154.1 762.39 101.18
^is Wadell's sphericity factor. Hence modified values of P^? and u/Q were determined for various values of a^^. These values may then be used to determine the settling velocities of non-spherical particles in the transition zone and vice versa. Figure 6.5 displays the relationship between drag coefficient and Reynolds number for particles of different shapes. As the velocity increases the particles tend to rotate to give maximum resistance to drag.
322 Powder sampling and particle size determination Haider and Levenspeil [26] presented the following empirical equation relating drag coefficient and Reynolds number for spherical and nonspherical particles, where Reynolds number is based on volume diameter: ^ Cr. = — ( l + .ii?^M + ^ Re^ ' l + D/Re
(6.72)
The values of the constants are given in Table 6.4 and apply for Re< 25000. Squires and Squires [27] studied 18 discs, in both vertical and horizontal orientation, having a range of diameter to thickness ratios. A detailed report on these and other studies has been presented by Ganser [28]. 6.10 Relationship between drag coefficient and Reynolds number in the transition region Both equations (6.4) and (6.5) contain D and u and need to be expressed in terms of a single variable in order to determine Difuis known or uifD is known. Stokes neglected the terms due to inertia and obtained a very simple relationship between settling velocity and particle size for particles settling with low velocities. Several attempts at theoretical solutions for the relationship between CQ and Re at higher velocities have been made. Oseen [29] partially allowed for inertial effects to obtain:
c
3 ^ -^ ^ 1 + —i?e
(6.73)
16 The equation is more complicated than Stokes', and in practice is found to be equally inaccurate, since the second term overcorrects Stokes' equation and gives a value of C^as much in excess as Stokes' is too low. Proudman and Pearson [30] pointed out that Oseen's solution could only be used to justify Stokes' law and not as a first order correction. They obtained a first order correction: 241
3
9
->
Cn = — \ \ + — Re+ — Re^ \nR, " \ 16 16 % Re
(6.74)
Interaction between particles and
fluids
323
1000
Reynolds number^^^) Fig. 6.6 Comparison between equations relating drag coefficient to Reynolds number and experimental data Goldstein [31] solved the equation without approximation and obtained the solution for Reynolds Number less than 2 in the form of the series:
c
-^
19
\ + — Re+ Re' 16 1280
(6.75)
Schillar and Nauman [32] fitted an empirical equation to the experimental values and obtained for Re<700: (6.76)
C^=^(l
+ 0A5Re'''')
A comparison between these equations and experimental data is presented in Figure 6.6. Davies [33] analyzed the published data and expressed the result in the form of Re as a function of CpRe^ that is suitable for calculating the velocity when the diameter is known.
324 Powder sampling and particle size determination
For Re < 0.4 or C ^ e ^ < 130; Re ^ £D^— 2.34 X10"'* (c^Re^)% 24 V / -6.9\x\0-^
2.02 x 10"^ (c^^e^ f V /
(677)
iCpRe^)^
For 3
(6.78)
A method for simplifying the calculation was proposed by Heywood [I] who used the dimensionless groups:
CRe^A^A^lZfflgO^^PD' 3
^ =1
(6.79)
;;
"'
u^^-Qu^
(6.80)
Experimental data were embodied in tables presenting C^Re^ in terms of Re/Cj) and vice versa Since the former expression is independent of velocity and the latter is independent of particle diameter, the velocity may be determined for a particle of known diameter and the diameter determined for a known settling velocity. Heywood also presented data for non-spherical particles in the form of correction tables for four values of volume-shape coefficient from microscopic measurement of particleprojected areas. Pettyjohn and Christiansen [24] restricted their calculations to the Stokes regime (Re < 0.25) and defined a Stokes shape factor (K^) from modification to equations (6.4) and (6.5) so that: Co=-^
(6.81)
Interaction between particles andfluids
(a)
325
(b)
Direction of flow
(c) Fig. 6.7 Formation of vortices behind a spherical particle as the relative velocity between the fluid and the particle increases. This expression is the usual equation for Stokes' law for a sphere when KY=\ modified for non-spherical shapes. 6.11 The turbulent flow region For low Reynolds number, the drag is due mainly to viscous forces and the streamlines about a settling particle are all smooth curves. With increasing Reynolds number, the boundary layer begins to detach itself from the rear of the particle and form vortices (Figure 6.7a). Further increases in C^ causes the vortices to increase in size and move further downstream (Figure 6.7b). At very high Reynolds numbers, the wake becomes fully turbulent and the vortices break up and new vortices are formed (Figure 6.7c). For Re > 500, C^ = 0.44 and is roughly constant. This effect can be demonstrated with a table tennis ball. If this is immersed in water, and released from a position near the surface, it will break through the surface at a high velocity, whereas if it is released several inches below the surface it will barely penetrate it. In the first case the ball's momentum is low but its velocity is high; in the second case the momentum is high but the velocity is low. Since momentum is the product of mass times velocity, the mass in the second case must be greater than
326 Powder sampling and particle size determination
the mass in the first case. The increase in mass is due to the water contained in the turbulent eddies behind the ball adhering to it as it breaks through the surface. 6.12 Concentration effects Stokes equation applies to the settling of a single spherical particle. This requirement is never fulfilled in sedimentation analyses where particles are separated by finite distances and mutually affect each other. In order to reduce interaction effects as much as possible it is recommended that a volume concentration no greater than 0.25% should be used. If it is necessary to use a higher concentration analyses should be carried out at two different concentrations in order to determine if the concentration effects are negligible. Most of the theoretical work on particle-particle interaction has been limited to the study of pairs of spheres. If the particles are close together they may behave as a single particle and a correction applied, provided that their center-to-center distance is small. As the center-to-center distance increases, their separate effects must be considered, together with field reflections from the container walls. The net effect in the first case is a reduction in drag on the individual particles so that they fall with a greater terminal velocity than for a single sphere. Happel and Brenner [7] plot the ratio of the drag force exerted on either sphere to that exerted on a single sphere, against L/D, the ratio of inter-particle separation (spheres touching when L = D\ and against particle diameter for the cases where spheres are falling (a) parallel, and (b) perpendicular to their line of centers. The results are shown in Table 6.5. Particles settling side by side will also rotate as shown in Figure 6.8. For all practical purposes the interaction becomes negligible for separations greater than about 10 diameters (0.3% by volume assuming a perfect dispersion). For more than two spheres, interaction becomes more complex; assemblies of spheres will diverge more slowly, that is repel each other, due to the rotation effect. For three identical spheres in a vertical line, with the top two closer together so that they fall more rapidly, the center sphere will join forces with the lower sphere and, leaving its original companion behind, fall as a doublet [34]. A large sphere, falling in a vertical line close to a small sphere, can pick it up so that it revolves as a satellite. For two identical spheres, falling in the same vertical line, the retardation on the leading sphere is smaller than that on the leading sphere so that they will move towards each other [29].
Interaction between particles and
fluids
327
Table 6.5 Ratios of drag force for pairs of spheres and single spheres falling (a) parallel, and {b) perpendicular to their line of centers, where L is the center to center separation of spheres of diameter D. L/D (a) (b)
1 0.65 0.70
2 0.73 0.83
3 0.80 0.87
4 0.83 0.91
5 0.86 0.93
6 0.90 0.94
7 0.91 0.95
Fig. 6.8 Direction of rotation of two spheres falling close together. It is important to distinguish between two cases; an assembly of particles that completely fills the fluid, and a cluster of particles. The descent of a single particle creates a velocity field that tends to increase the velocity of nearby particles. To balance this, the downward motion of the particle must be compensated for an equal volume upflow. If the particles are not uniformly distributed, the effect is a net increase in settling velocity, since the return flow will predominate in particle sparse regions. On the other hand, a system of uniformly distributed particles will be retarded by much the same extent. A cluster of particles in an infinite fluid can be treated as a single large particle of appropriate density and reduced rigidity that is as a liquid drop. A very large increase in settling rate could arise that is important at low concentrations. For a dilute assembly of uniform spheres of diameter Z), the settling velocity is retarded by a factor: ^ =
u
\
r
(6.82)
\+\3{DIL)
where u^ is the settling velocity of a particle in the presence of other particles, u is the free-falling velocity and L is the inter-particle separation. In a simple cubic assembly the volume concentration is given by C-
71 —
6
D
328 Powder sampling and particle size determination
hence:
^ =
^—--
(6.83)
Famularo [35] investigated the problem further and found values of the constant for different assemblies as follows: cubic, 1.91; rhombohedral, 1.79; random, 1.30. Burgers [36] considered a random assembly of particles and replaced the second term in the denominator with 6.88c. The numerical constant has been questioned by Hawksley [17 pi31] who suggested that in practice the particles would accelerate to an equilibrium arrangement with a reduced constant of 4.5. The form of expression has also been criticized by Happel and Brenner [7]. Maude and Whitmore [37] suggest that: ^
=( l - c /
(6.84)
where 4.67 > p >4.2 for Re < 1 and >0 is a function of particle shape and size distribution. Richardson and Zaki [38] also propose a relationship similar to this with >5= 4.65. For 0.6<£<0.95, where the porosity £=\-c Brinkman [39] proposed:
U
4^
\
\\-£
(6.85)
Steinour [40] developed, on theoretical grounds, a modification of Stokes' law that resulted in the equation: 3
"^^-^eis) U
(6.86)
\-£
He also derived a general expression for 6{£)\ -s. e(s) = ^^\Q-^^'-'^
(6.87)
£
Substituting in equation (6.86) gives: ^-^^10'^^^-^^
(6.88)
Interaction between particles and fluids 329
Steinour obtained the result ^ = 1.82 from the slope of a plot of \og(u^/u) against e. Steinour, working with tapioca in oil and spherical glass beads in 0.1% aqueous sodium hexametaphosphate, reported that d(s) was sensibly constant over the range 0.3 <^< 0.7 giving: ^ =0.123-^
(6.89)
Powers [41] found that in order to represent his experimental data he had to introduce a factor w^ to compensate for the liquid dragged down by the settling particles. His final equation was:
II
_
For spherical particles, S^ = 6/D, hence:
^ =0.10^::^ u
(6.90)
l-e
Steinour [40] defined the immobile liquid per unit total volume as a(l-£), where a = w^(\-Wj). To correct equation (6.87) [£-a{l-s)] must be substituted for each value of ^giving:
^ =0 . 1 2 3 ^ : 4 ^
(6-91)
u (l + a ) ( l - ^ ) In terms of Wj this becomes:
ajLj'^^f-.f
(6.92)
330 Powder sampling and particle size determination
Similarly substitution into equation (6.88), and eliminating a, gives: -1.82
10
\-£
Vi->^/J
(6.93)
Bed expansion in particulate fluidization may be described by the Richardson-Zaki equation [38]. — = —^^-^
uu = U:e"
(6.94)
where u^ is the superficial gas velocity and n is an exponential parameter given by: -0.1 n = 4.45 + 1 8 ^ ^ Re-""'
(6.95)
for ]
(6.96)
where D = particle diameter and D^ = bed diameter [42] These relationships have also been applied to sedimentation where the problem of dealing with particles that are aggregated into sedimentation units with properties similar to floes has been recognized [43]. Such units effectively immobilize a relatively large volume of fluid, thus reducing the apparent porosity. In this situation Scott [44] replaced s in equation (6.94) by an 'effective' porosity. Uf,=us,{\-kcf^'
(6.97)
where c is solids concentration and k is the volume of aggregates per unit weight of solid, k= 1/(1 -w^). Equation (6.97) is also applicable to rough irregular particles [45] in that a layer of fluid may be considered to be immobilized in the surface irregularities. The linear rate of settling of the interface (u^) increases with increasing initial liquid volume fraction e. Additionally, since (l-s)
Interaction between particles andfluids
331
decreases to zero as s increases, a plot of u^(\-s) against £ will pass through a maximum at a porosity Sy. The quantity u^(\-s)p^ is known as 'solid flux' and indicates the mass transfer of solid per unit cross-section per unit time down the sedimentation column. Applying this concept to equation (6.97) a graph of 'solid flux' against porosity has a slope with D«Dg so that u^ = %^. u,{l-s)p,=us,{\-s)p,s''
(6.98)
For maximum mass transfer
de
[u,{\-e)p,]
=0
(6.99)
Defining this point as s^, gives s^ = [n/{n+\)] so that the Richardson-Zaki equation takes the form: u,=us/''^'-''^^
(6.100)
The physical significance of «, as a function of the porosity at which solids flux is a maximum was pointed out by Davies et. at [46] In a later paper [47] they propose a relationship of the form: u^^UstlQ'^^'^^''"^
(6.101)
where p^ is the density of the solids. For ^tending to unity this is equivalent to Steinour's equation (6.98). Multiplying both sides of equation (6.101) by {\-s)ps and differentiating the right hand side and equating to zero for maximum porosity as before gives [48]: ^1=1
^
(6.102)
This is equivalent to equation (6.100) and s^ may be determined from a plot of log u^ against c.
332 Powder sampling and particle size determination
Equation (6.101) is criticized by Dixon et. al [49] as being equivalent to equation (6.103) at low concentrations and incorrect at high concentrations; they also criticize the use of equation (6.102) to evaluate £-|. In a reply Davies and Dollimore [50] defend their approach. Instantaneous concentration profiles of the batch sedimentation of noncolloidal hard spheres, at concentrations from 4% to 60%, have been determined using non-invasive magnetic resonance imaging [51]. Measured profiles commonly had two distinct interfaces, the upper one between the clarified fluid and the settling suspension, and the lower one between the settling suspension and the sediment. The upper interface was found to keep spreading due to polydispersity, especially at low initial concentrations. It was determined that the falling velocity of the upper interface was given by equation (6.73) with p =5.0. 6.13 Hindered settling At very low concentrations particles settle singly and, under laminar flow conditions, Stokes' law is valid. At higher concentrations particle-particle interaction occurs that, on average, should lead to reduced settling rates. At very high concentrations particles tend to settle en masse, and the rate of fall of the interface {Uf^ is given by equations of the form presented in the preceding section. With closely graded powders, the interface between suspension and clear liquid is sharp, becoming more diffuse for powders with a wide size range that, in some cases form more than one interface. This phenomenon is due to fines being swept out of the bulk of the suspension by return flow of liquid displaced by the sedimenting solids; These form a suspension of fines over the main suspension; the fines supernatant being, in turn, subject to hindered settling. Multiple interfaces may form and persist until, at some critical condition, mechanical interlocking occurs [52]. The equations developed to represent such systems have much in common with permeametry equation developed to describe the flow of fluid through a fixed bed of powder. A review on hindered settling, with many references, states that particles in suspension experience settling velocities that differ from their terminal velocities due to multi-particle hydrodynamic interactions; these give rise to reduced sedimenting velocities [53].
Interaction between particles andfluids
333
6.13.1 Low concentration effects Several investigators have shown that settling rates for selected particles increases with concentration, when the concentration is low, and this is probably due to particle-particle interaction. Kaye and Boardman [54] visually determined the velocities of 900 |Lim red glass marker spheres, in the presence of glass spheres of diameter 850, 400 or 100 |im, settling in a 62 mm tube filled with liquid paraffin. A maximum settling rate of 1.6 times the value found in very dilute suspensions was reported at volume concentrations around 1%, after that the ratio decreased. Twenty measurements were carried out for each experimental point and considerable scatter was found, particularly around the maximum values. It was also found that this enhancement was greatest for powders having a narrow size range. Johne [55] used 200 |Lim glass spheres in a 35 mm diameter tube filled with 148 cP motor oil. The sedimentation velocity was determined by following a single 200 |Lim radioactive marker sphere. A maximum settling rate 2.4 times the values found in very dilute suspensions was reported at volume concentrations around 1%. Koglin [56] continued Johne's work with an emphasis on wall effects and attributed the difference between the two earlier investigations to these effects. Since Boardman visually determined the settling time between two marker lines on the sedimentation tube, he selected particles in the center of the tube, whereas Johne's method was non-selective. The increasing rate of sedimentation at increasing concentrations, in the low concentration region, was also demonstrate by Jovanovic [57] who showed that the weight-mean diameter of aluminum powder, as determined with a Sartorius balance, increased as the powder volume concentration was increased from 0.02% to 0.05%. This he attributed to particle agglomeration. Barford [58] carried out similar experiments with polishing powders in the 15 to 30 |uim size range. The maximum settling rate found was 1.12 to 1.40 times the Stokes rate at volume concentrations between 0.1% and 0.2%) and this he attributed to cluster formation. Jayaweera et a/[59] in experiments with collections of 2 to 7 mm spheres, also found settling rates higher than the Stokes velocity for a single sphere. Davies and Kaye [60] used a Cahn sedimentation balance to investigate cluster formation and found that, for a powder having a narrow size range, cluster formation was not eliminated at volume concentrations as low as 0.47%); for a powder having a wide size range it was not important at volume concentrations as high as 0.414%).
334 Powder sampling and particle size determination
The laws of probability predict that, even in dilute suspensions, the unequal distribution of particle separation will result in cluster formation giving rise to enhanced settling. It is therefore desirable to use as low a concentration as is feasible when carrying out sedimentation size analysis. 6.13.2 High concentration effects Several investigations have been carried out on the settling behavior of spherical particles in concentrated suspensions [7] with best agreement with equations of the form (6.84). Equations similar to (6.85) indicate too high a concentration dependency on settling rate. Apparatus for carrying out hindered settling experiments [61] was described in I960, modified apparatus was described in 1970 [62] and experimental procedure in 1977 [63]. There are various ways of dealing with experimental data [47,64] 1 Apply equation (6.89) combined with Stokes equation to find an 'uncorrected particle radius'. 2 Apply equation (6.92) and plot [u^{\-6)\ against s to fmd w^ and equation (6.87) to fmd d(^e) and a 'corrected particle radius' 3 Apply equation (6.88) and plot u^s^, against £. 4 Plot log u^ against c and extrapolate to zero concentration. Dollimore and Heal [65] applied method 2 to determine the thickness of the film on the outside of the particles. Ramakrishna and Rao [66] criticize this approach and prefer method 3. In a later discussion [67] they stated that their criticism was because equation (6.92) was being used outside its range of validity (i.e. ?ii£> 0.8) and in this region equation (6.89) must be used. Pierce [68], in the same discussion, suggested the use of equation (6.87). Dollimore and co-workers [47] in later investigations found wide variations in A an.d O^s). Dollimore and Horridge [69] found A = 20.2 for china clay. Thompson [cit. 45] found 24.3<4<36.6 for calcium carbonate in aqueous sodium chloride; Dollimore and Owens [70] found ^ = 81 and A = 206 for non-porous silica in water and w-heptane respectively. General conclusions are that, at low concentrations, hindered settling is more likely with high polarity systems, particularly high polarity solids [46]; it is more likely with dense solids in viscous liquids; the stability of some systems is due to the presence of electrical double layers:
Interaction between particles and
fluids
335
Davies and Dollimore [71] conclude that hindrance is directly related to surface density of charge on the particle, particle density and liquid viscosity. It is inversely related to the difference in viscosity betw^een the suspension and the bulk liquid and the cation-stability constant. Sarmiento and Uhlherr [72] also considered temperature effects on settling behavior and suggest that at low concentrations the change in settling velocity with temperature change can be predicted solely from the change in viscosity of the suspending liquid; at high concentrations this is no longer possible since the floe characteristics undergo change. Sediment volume changes have also been used to predict particle size [10,43]. The general conclusions that can be drawn are that settling behavior is extremely complex in the high concentration region and several equations may apply according to the range of porosity considered and the presence or absence of flocculation. The determined particle size decreases with the addition of dispersing agents and it is suggested that the size so determined is floe size [73]. The technique is therefore a useful and simple one for determining floe size. 6.14 Electro-viscosity Charged particles in weak electrolytes have associated with them an electrical double layer. When these particles settle under gravity the double layer is distorted with the result that an electrical field is set up that opposes motion. This effect was first noted by Dom [74] and was studied extensively by Elton et. al [75-78] and later by Booth [79,80]. For a spherical particle of diameter D, the electrical force F^ is equal to the product of the electrical field E and the charge q. For a surface charge per unit area crthe force on the sphere is D^GE. If the sum of this force and the viscous force is set equal to the gravitational force, equation (6.6) is modified to: ' (ps-Pf)sD^ ^st = -rr^ I877
DaE z— 3/7
,^.^.. (6.103)
That is, Stokes velocity is diminished by the term (DoE/3 77), The specific conductivity of a solution (k) is equal to the current density divided by the electrical field.
336 Powder sampling and particle size determination
For A^ spheres of diameter D per unit volume of suspension: k^^^-^u'st
(6.104)
But A^ is equal to the total weight of particles in suspension {m) divided by the volume of suspension {V) and the weight of a single particle.
A^-, "^^J ,
(6.105)
Substituting in equations (6.103) and (6.104) and rearranging gives:
Uc, = St^^St
' ,
Imcj
V
P,7lVk i-s'i-'-J
1+
^
(6.106)
Pavlik and Sansone [81] found that the size distribution of spherical particles, in the size range 5 to 40 )am, obtained by sedimentation in double-distilled deionized water plus a wetting agent was significantly different from that obtained in O.lN KCl plus a wetting agent. Coulter Counter data agreed with the latter data. These results were later confirmed [82]. The electro-viscous effect needs to be eliminated, for Stokes' law to apply, by the addition of non-ionic sedimentation liquids. The magnitude of this effect was determined [83] for a 0.1% suspension of 1.10 \xm polystyrene settling in water. For a zeta potential of 50 mV the electric field strength is 1 mV cm"^ making the correction term negligible at a volume concentration of 0.04%. 6.15 Dispersion of powders 6.15.1 Dry powder dispersion Dry powders are made up of primary particles that stick together to form clumps. If the forces holding the primary particles together are weak, the clumps are defined as agglomerates; strongly bonded clumps are called aggregates. If the particle size distribution of a powder is to be determined in a dry state, the amount of energy employed is critical. In order to determine the size distribution of products held together by weak bonds,
Interaction between particles andfluids
33 7
such as fertilizer granules, gentle treatment is required. The same is true for weak crystals that will either attrit, i.e. asperities removed due to collision, or comminute, i.e. break into pieces. For pigments, such as titanium dioxide, high shear forces are required to break down the floes, but they must not be so great that the aggregates are broken down. Quality granules have a defined size range, fines, that form dust, and coarse lumps are detrimental to the final product, and the determined size distribution should reflect this. Similarly for pigments, particles much smaller than the wavelength of light have low hiding power and particles coarser than about a micron generate surface irregularities that affect surface gloss. In order to predict end-use properties, the dispersing procedure should not destroy the coarse aggregates or generate fines, but should be vigorous enough to break down the agglomerates. The demarcation between individual particles and groups of particles is often very blurred. In some systems there are aggregates that are tightly bound together with some chemical bonding or sintering, and strong mechanical forces are required to break the bonds; these aggregates may be associated with agglomerates that are weakly bound together. In others there may be a continuous gradation in the strength of clumps. Several of the manufacturers of Multi-angle laser light scattering instruments (MALLS) have developed dry powder feeders. The Sympatec Rodos and the Aerosizer dry powder dispenser are particularly suitable for dispersing fine cohesive powders. 6.15.2 The use ofglidants to improve flowability of dry powders Glidants are often added to powders, that are to be analyzed in the dry state, in order to improve their flow properties [84,85]. Various mechanisms have been proposed to explain their mode of action: • Reduction of interparticle friction by coating the particles. • Reduction of surface roughness. • Reduction of interparticle attraction by creating a physical barrier between particles. • Reduction of static electrical charge on particles. An optimum concentration of glidant exists for most systems and amongst the techniques to estimate this concentration are angles of repose [86], flow through an orifice [87], vibrating funnel [88] and bulk density.
338 Powder sampling and particle size determination
These additives are particularly useful for micromesh sieving and classification. In the absence of other data a mass concentration of about 1% is used. Fatty acid (usually stearic), fine silica (Aerosil 2000 from Degussa, Frankfurt, Germany), purified talc and magnesium stearate are used. York [89] examined three typical glidants with lactose mixtures using the Jenike shear cell [90] to assess flowability and found that fine silica produced the greatest increase in flowability, while magnesium stearate and purified talc gave smaller increases. All three systems indicated an optimum mass concentration (1.75%, 2.25% and 0.85% respectively). He also concluded that the angle of internal friction is not a useful indicator of flowability. 6.15.3 Wet powder dispersion In many particle size analysis procedures it is necessary to incorporate the powder into a liquid or the sample may be provided in slurry form. In the latter case it is necessary to know whether it is desirable to determine the size distribution as delivered or whether loosely bound floes need to be broken down. In many cases, drying a suspension and redispersing the powder will alter the resulting size distribution. Starting with a dry powder and a liquid medium, three stages in the dispersing process may be distinguished. First there is the process of wetting defined as the replacement of the solid-air interface by a liquid-air interface. Second there is the process of deagglomeration of the particle clusters and finally there is the process of dispersion stabilization [91]. 6.15.4 Role of dispersing agents The molecules of a dispersing agent adsorb on to the particle surface to change particle-particle interaction. This may be accomplished by: (1)
(2) (3)
The adsorption of a strongly hydrated hydrophilic colloid thus increasing the affinity of the particle to water so that it exceeds any mutual attraction between particles The adsorption of an ionic electrolyte to create strong mutual repulsion on the particle surface The adsorption of a non-ionic polymer of sufficient chain length to create steric hindrance i.e. preventing the particles from entering each others attractive field.
Interaction between particles and fluids
339
A discussion on the selection of dispersing agents and the optimum concentrations to be used has been published [92]. For a fuller discussion of dispersion, readers are referred to Nelson [93] and Conley [94]. 5LA
AIR LIQUID »SA ^^
ZL^ SOLID
1
^
^SL
Fig. 6.9 The spreading of a liquid on the surface of a solid. 6.15.5 Wetting a powder If liquid is placed on a solid it will spread if: YsA^ YsL^ YLA C O S ^
(6.107)
where y^^, y^^, y^^ are the interfacial tensions between the solid and the liquid, solid and air and liquid and air, and d is the angle of contact between the solid and the liquid (Figure 6.9) [95]. {y^^ and y^^ will fall rapidly to y^y and y^^y as the solid surface becomes saturated with vapor but, for simplicity, the former suffixes will be retained.) The liquid will spread on the surface if the spreading coefficient is positive, i.e. the liquid will wet the solid. The spreading coefficient is defined by:
SLs>rsA-rsL-rLA
(6-i08)
The work of adhesion is defined as the work necessary to separate unit area of interface between two phases (Figure 6.10).
^A^rsA + rLA -YsL
(6-109)
The work of cohesion is defined as the work necessary to separate unit area of one phase Wc-2y,,
(6.110)
340 Powder sampling and particle size determination
Hence the spreading coefficient may be defined as: (6.111)
SLS=W,-WC
and, if positive, infers a greater affinity between the liquid and solid than between the liquid and itself. The energy of immersion is defined as the surface energy loss per unit area of surface on immersion.
w, = rsA-rsL
(6-112)
Combining equations (6.107) and (6.109) gives: W^ = ra^^co^e)
(6.113)
The ease of displacement of air from the surface of the powder is enhanced if W^ is increased. This is frequently accomplished by the use of surfactants to reduce the contact angle 0 to zero if possible. At equilibrium, from equation (6.107): cos(^)^^^-^~^^^
(6.114)
YLA
The addition of surfactant usually causes a reduction in /^^ and, if absorbed, a reduction in y^^^. Both effects lead to better wetting. The change in y^^ is negligible in most cases so the dominating factor is y^^, the surface tension of the liquid phase. 6.15.6 Determination of contact angle (9) The difficulty in the use of these equations in practice is the experimental one of determining .^for fine powders. Bartell and Walton [96] developed a method in which a pressure was applied to prevent liquid from penetrating a plug of powder. The required pressure is given by the Laplace equation:
r
Interaction between particles and
Liquid
Liquid
Liquid
fluids
341
Liquid
!• « • • • • • • !
Solid
Solid
Liquid
Liquid
Adhesion
Cohesion
Fig. 6.10 Work of adhesion and work of cohesion. For a liquid that wets the solid: AP:
2rL
(6.116)
where yl^ is the interfacial tension between the liquid and its saturated vapor so that: (6.117) YLA
The principle of this method is to obtain the effective capillary radius r using a non-wetting liquid and repeat the measurement with a wetting liquid. The method is difficult to use and the following simpler method has been described [97]. The distance of penetration L in time M n a horizontal capillary, or in general when gravity can be neglected, is given by the Washburn equation [98]. (6.118) t
27]
where // is the viscosity of the liquid.
342 Powder sampling and particle size determination
For a packed bed of powder this becomes: ^'-riA^'^
27
'• [K^]
(6.119)
where the bracketed term is an unknown factor dependent on the packing. If several liquids are used with the same powder, uniform high values of (r/K?-) are found and it is assumed that these correspond to cos{9) = 1. This factor is then used to obtain values of .^or other liquids. Heertjes and Kossen [99] present a full discussion of their techniques together with descriptions of the required apparatus and experimental procedure. They considered both the above methods unsuitable for the determination of cos(^ and proposed a new method, the h-s method. Briefly this consists of determining the height of a drop of liquid placed on top of a cake of the compressed powder previously saturated with the liquid. The theory was presented in an earlier paper [100]. An analysis has been presented [101] on the wetting of fine powders by aqueous solutions of anionic wetting agents in terms of adhesion tension, spreading coefficient and capillary forces involved in the displacement of air from externally wetted aggregates. This study revealed how the film preceding the bulk liquid can retard or markedly accelerate the submersion of powder in a liquid. 6.15.7 Deagglomerating wetted clumps The next stage in the dispersion process is the breakdown of agglomerates. For easily wetted material, penetration of liquid into the voids between particles may provide sufficient force to bring about disintegration. Often however, mechanical energy is required and this is usually introduced by spatulation or stirring, though the use of ultrasonics is now widely practiced. For powders that are difficult to disperse, the dispersing liquid (e.g. 0.1% sodium hexametaphosphate in distilled water) may be added to the powder to form a paste. More liquid is added to this paste while it is sheared using a flexible spatula so that it becomes a slurry and eventually a suspension. Care needs to be taken, with this procedure, to ensure that aggregates and particles are not fractured.
Interaction between particles andfluids
343
Clumps of particle may contain occluded air that is difficult to displace; this is best dealt with by adding the liquid whilst the powder is under vacuum. The effect, on dispersion and de-agglomeration in water, of electrostatic repulsion force ^^ arising from the surface potential and the double layer l/x around particles has been investigate. Several suspensions of polystyrene latex in an agglomerated state were prepared where ^^and l/x were controlled by the pH and electrolyte concentration respectively. These were accelerated in a convergent nozzle to give an external force and the resulting dispersions were examined by optical microscopy. It was found that the dispersion was enhanced with an increase in ^^and l/x. 6.15.8 Suspension stability The stability of the wetted, dispersed system depends on the forces between particles. The random motion of particles brings them into close contact and, under certain circumstances, causes them to flocculate. The frequency of collisions depends upon the concentration, viscosity and temperature. Whether two approaching particles will combine or not, depends on the potential barrier between them. The potential energy can be considered to consist of two terms, an attractive London-van der Waals force and a repulsive force due to the electrical double layer that surrounds particles. This double layer consists of an inner layer of ions at the surface of the particle and a cloud of counter-ions surrounding it. The sum of the two potentials generate an increasing repulsive force as the particles move closer together that reaches a maximum, or potential barrier, before decreasing and becoming an attractive force. If the potential barrier exceeds 15 kT, where k is the Boltzmann constant and T is absolute temperature, the system is stable [103, p. 192]. With large particles the net potential energy curve may show a minimum at appreciable distances of separation (Figure 6.11). This concept cannot account for the action shown by many non-ionic, surface-active agents, and in these cases it has been suggested that the size of the agent could lead to repulsion due to steric hindrance, i.e. the molecules extend so far out into the media that two approaching particles do not get close enough to flocculate. In non-aqueous media of low dielectric constant ionic charge stabilization is unlikely to be very important. In such cases stabilization depends on steric or entropic repulsion and polymeric agents are preferred
344 Powder sampling and particle size determination
against the long-chain paraffin types that are so successful in aqueous media. In practice it is necessary to decide whether an agent is required to wet out the solid so that it will disperse in the liquid concerned or whether the problem is one of stabilization. The minimum of wetting agent to ensure adequate dispersion is added. Any combination that causes foaming should be rejected or used in combination of an anti-foaming agent such as fumed silica. The present state of the art is such that dispersing agents are chosen almost at random, but with some knowledge of the surface chemistry involved [104]. Increasing Repulsion
Distance
Increasing Attraction
Fig. 6.11 Net interaction potential between particles: A, force of attraction, R, force of repulsion, B, resultant force. 6.15.9 Tests of dispersion quality A simple test for wetting efficiency is to make up suspensions using the same concentration of powder but different wetting agents and allow the suspension to settle out. About a gram of powder dispersed in 10 ml of liquid is suitable. Slow settling, a clear interface between the clear liquid and the turbid lower layers and a small depth of sediment indicate the best
Interaction between particles and
fluids
345
agent or, if several concentrations of the same agent are being tested, the best concentration [105,106]. It is important that no vibration be imparted to the containers during settlement that may take several hours. Another test for the degree of flocculation of a paste is to measure the difference between the smear and flow points. The test is made by adding known quantities of dispersing medium to a known weight of powder and working it in with a spatula. The difference is noted between the amount required to smear and to flow; the better the dispersion, the smaller the difference [107]. It has been found that for pigments in solvents, a high dielectric constant leads to a more dispersed system. In general, polar liquids disperse polar solids and non-polar liquids disperse non-polar solids. For polar solids suspended in non-polar liquids, it is possible to use the difference in polarity to anchor a stabilizing molecule to the powder surface. The effectiveness is characterized by the heat of wetting, which can be determined by calorimetry. For sub-micron metallic powders, precoating with gelatin aids dispersion in aqueous systems. Coating of commercial powders in order to stabilize them is widely used e.g. aluminum oxide coated titanium dioxide. In such cases the interfacial properties will be those of the coating rather than those of the bulk powder. Many weird and wonderful methods for obtaining stable, well-dispersed systems have been proposed. Some of the more valuable are incorporated into National Standards for particle systems. A more general one is to be found in BS 3406 [107]. Effectiveness of deflocculants in dilute suspensions may be studied by light absorption. If the optical density of an agitated suspension falls with time this is an indication of either flocculation or dissolution, whereas an increasing optical density indicates breakdown of floes. Dilute suspensions may be studied using the Coulter principle. It is sometimes found that the count level at the lower sizes decreases with time and this may be attributed to flocculation or dissolution. A problem arises in that the suspending liquid needs to be electrically conducting and one would expect this to reduce the potential energy barrier and decrease suspension stability [108]. A microscope examination should always be carried out. A thick paste should be made up as described earlier and a drop removed; this should be diluted to a reasonable concentration by the incorporation of more liquid. A drop of this should be placed on a microscope slide, without a cover slip, and examined for signs of agglomeration or break-up of aggregates.
346 Powder sampling and particle size determination
Particles smaller than a few microns will vibrate in a random manner due to thermal jostling (Brownian motion). Note what happens when particles collide; in a good dispersion they repel each other; in a poor dispersion they form floes. Although a visual examination of particles is always useful as an indication of particle size, a false impression may be obtained as to their state of dispersion due to the wide difference in environment between the testing situation and the actual analysis. Dispersion of magnetic suspensions may be effected by subjecting the suspension to a high frequency magnetic field [109]. For the Autometrics PSM 200, 400 Hz was selected at a peak field strength of 800 Oersted [110]. A range of non-ionic fluorochemical surfactents (FC-170C, FC-433, FC-430, FC-431) manufactured by 3M Corporation has been found to be particularly useful for dispersing sub-micron powders. The addition of 0.01% of any of these surfactants to water reduces the interfacial tension from 7.2 x 10-^ N m-i (72 dyn cm-i) to 2.5 x 10-^ N m-^ FC-430 is a 100% active, viscous liquid, FC-431 is a solution of 50% active solid in ethyl acetate. These surfactants exhibit good solubility in most liquids. The final criterion is the practical test; if size analyses are carried out with two systems, provided there is no dissolution, the one showing the finer distribution will have the best dispersion. Koglin [111] has reviewed the methods of assessing the degree of agglomeration in suspension. He states that the only direct method is microscopy. Estimates of degree of flocculation are possible by rheological properties or sediment volume. The Coulter principle is only suitable for aggregates since floes disintegrate due to the shear experienced as they pass through the aperture. He states that the degree of agglomeration can be determined by comparing the size distribution of the agglomerated suspension with the size distribution of a completely dispersed system. The methods he used are photo-sedimentation and the sedimentation balance; the review cites 57 references. Dispersion of pigments is often effected by prolonged milling [112-114]. It is argued that pigment is made up of agglomerates that are readily reduced to their constituent aggregates (1-20 |Lim in size). These aggregates may be further reduced to primary crystals (0.3-1 iiim) with the application of considerably more force. Further reduction to crystallites is effected by crystal fracture (0.001 - 0.6 |j.m). Milling times of around 20 h are required to attain stable conditions.
Interaction between particles and
fluids
347
6.16 Powder density The effective density of a particle is the particle mass divided by the volume of liquid it displaces (Archimedes density). Its true density is the particle mass divided by the volume it w^ould occupy if it were compressed so as to eliminate all the pores and surface fissures. Its apparent density is its mass divided by its volume, excluding open pores but including closed pores. Quoted density values in standard reference works are of the materials true density. If density is determined using a gas pyknometer, the volume measured would include closed pores but exclude open pores i.e. the measured density would be the apparent density. If the suspending liquid penetrates all the cracks and fissures on the particle surface, the measured volume would be the same as that determined by gas pyknometry but the total mass would be greater due to the included liquid that will remain with the particle as it falls in the liquid, hence its sedimentation density will be intermediate between the apparent density and the true density and greater than the effective density. These differences are usually not highly significant for coarse particles unless they are highly porous. In density determination the volume of fluid, displaced by a known weight of powder, is determined. Since weight can be measured accurately, the problem is that of accurate determination of volume. With pyknometers (or density bottles) the fluid is a liquid, usually water with surfactant, unless the powder is water miscible. With gas pyknometers the fluid is usually dry air or helium. The usual methods for determining density using a liquid pyknometer bottle are described in British Standards, BS 733 (1952) and BS 3483 (1978). Essentially, the pyknometer bottle is weighed empty (Mj), then full of liquid (M2), next about one third full of powder (M3) and finally the bottle is topped up with water to the fill mark (M4). Great care is required in this final step that the liquid is fully wetted and all the air removed. Variations in recorded weight also arise depending on how much liquid escapes when the ground glass stopper is inserted in the liquid filled container. The relevant equation for powder density ip^) determination is:
^^
^^
{M,-M,)-{M,~M,)
348 Powder sampling and particle size determination
where p^ is the density of air and p^ is the density of water with dispersant. Wilkes and Allen [115] found that using this procedure with BCR 66 quartz gave a density of 2630 kg m"^ reproducible to ±20 kg m"^. By taking extraordinary care, i.e. letting the bottle, after it had been filled, stand in a constant temperature bath 2°C above ambient temperature until the liquid no longer escaped through the capillary; topping up the bottle plus powder under vacuum and taking each reading six times they re-determined the density and standard deviation, in g cm"^, to be: (2.611 ±0.002). Repeat measurements gave 2.614 and 2.613 g cm"^. Their data were as follows (all values in cgs units; bracketed values are standard deviations): Ml =21.1287 (0.0002), M2= 45.8154 (0.0004), M3 = 29.9503 (0.0002) M4= 51.2690 (0.0005), M5-45.7945 (0.0003), p,,= 0.0072, p^= 0.0012 p = 0.9980, p^ = 00012, If liquids other than water are used they should have a low evaporation rate under vacuum and a high boiling point (>170°C); aromatic or aliphatic compounds are suitable. The use of oil presents a particular problem in that it is difficult to ensure that the outside of the bottle is completely oil free. Burt et. al [116] consider the liquid pyknometer method to be unsuitable for powders that are predominantly smaller than 5 |im due to absorbed gases present on their surfaces. In order to remove these gases, the powder may need treatment at high temperature or under high vacuum. For particles with rough surfaces it is also possible that air trapped within surface pits and cracks cannot easily be removed. They propose the use of a centrifugal pyknometer in that a suspension, prepared in the usual way for sedimentation analysis, is placed and centrifuged and the density determined. The density so obtained is slightly lower than that obtained by other methods. A special apparatus has been described that is claimed to be simpler to operate than most other methods for the determination of the apparent density of porous carbons [117]. The apparatus is designed so that the solid can be thoroughly outgassed in order that degassed dilatometric liquid can be brought into contact with it without exposing the latter to the atmosphere. Other procedures involve the use of mercury [118] to
Interaction between particles andfluids
349
determine envelope volume and volatile liquids to determine pore structure [119]. May and Marienko [120] used a 1 cm^ micropyknometer, with ethylene glycol as the fill liquid, for measuring the density of small amounts of material. Stein et. al [121] found this method time consuming and difficult and developed a method in that air was used as the fill 'liquid'. Their micropyknometer had a 2 mm bore stem, accurately calibrated with mercury, at 0.30 cm^ and 0.50 cm^. A known weight of powder was placed in the pyknometer and the neck sealed with a mercury plug. This was forced down to the 0.50 cm^ level in a pressure chamber at pressure P^ and then to the 0.30 cm^ level at pressure P2- The volume of powder (F) is given by: ^^P,(0.3)-P.(0-5) Pi-Pi
The volume of powder could be determined to ±0.5 mm^ corresponding to an accuracy of ±3% for a 25 mg sample of density 150 kg m~^. Gas pyknometers are commercially available and a description of these has been presented by Thudium [122]. He criticizes instruments in that the measurement consists of an absolute volume change [123,124] or a variable containing the absolute volume change [125] since the relationship between the measured parameter and particle volume is nonlinear. In instruments having only one chamber, or a variable containing the absolute volume change, or having one chamber as pressure reference only [125,126] every fluctuation in temperature gives an error of measurement. In systems having two chambers [127] only differences between the two chambers gives an error. Essentially, the two chamber instruments consist of two cylinders, containing linked pistons on a screw thread, separated by a diaphragm linked to a volume readout. When a handle is turned, the pistons are driven into the cylinders until the measurement piston hits a stop, at zero reading. With powder in the measurement cylinder, the reference cylinder will record a volume when the measurement cylinder hits the stop. Thudium criticizes this type of pyknometer [122] on the grounds that if the pressure was reduced, rather than increased, the large volume change could be reduced by 50%; moreover, sorption effects would also be reduced. An instrument embodying these recommendations has been described [128]. Thudium was particularly interested in micropyknometers for measuring
350 Powder sampling and particle size determination
volumes smaller than 5 mm^ and considered that the best to be one described by Hanel [126] that would measure a volume of 20 to 40 mm^ with an accuracy of ±10%. He then described a micropyknometer similar to the Beckmann with expansion using a micrometer syringe, thus reducing the pressure change to less than 10 mbar. Micromeretics GeoPyc^'^ 1360 measures envelope density; this includes the volume of pores and surface crevices. When absolute density is inputted, specific pore volume and percent porosity are determined. The instrument uses a flowing dry medium, DryFlo^^, to surround the powder and generate the displacement volume. 6.17 Liquid viscosity The viscosity of the suspending medium for a sedimentation analysis should have a value that fulfills the following conditions. 1. The largest particle in the suspension should settle under laminar flow conditions, i.e. the Reynolds number should be less than 0.25. 2 The free-falling velocity of the largest particle should be restricted so that it takes at least 1 min for it to reach the measurement zone. For the first condition, the relationship between Stokes diameter and viscosity is given by inserting a value Re = 0.25 in Stokes equation giving:
.^fllZfllfl^
(6.122)
For a particle of density p^ = 2700 kg m-^ and size 75 jum, settling in a suspending medium of density Pf= 1000 kg m"^, the required viscosity 7] = 0.00125 Pa s. Alternatively, for comparison purposes, a size frequency distribution may be plotted or tabulated against free-falling diameter. 6.18 Standard powders Many commercially available particle size distribution measuring instruments may be considered to be sub-standard in that they need to be calibrated at regular intervals using standards powders of known size
Interaction between particles and fluids 351 distribution. Standard powders are also used for control purposes to ensure that there is no 'drift' in instrument response. Such powders are used, for example, to monitor sieving operations to determine the effect of wear. Narrowly classified powders are required to calibrate sensing zone instruments. Some of the light-scattering instruments are pre-calibrated by the manufacturer, but with the electrical sensing zone instruments calibration is carried out on a regular basis by the user. In both cases standard powders are also useful for control charting to ensure that instrument response remains within tolerance. Early calibration materials were often pollen since these have a very narrow size range. Arguments persist as to whether these swell in liquids or change size over time. Polymer latices are also available as calibration material. When instruments are used regularly to test a limited range of products the product itself can be used for calibration purposes. Thus a manufacturer of titanium dioxide may set aside an easily dispersed, stable sample for sample splitting and subsequent use for instrument evaluation. A method of preparing narrowly classifies powder has been described by Muta et. al [129]. A fine copper powder of approximately the required size is mixed with twenty times as much calcium carbonate and heated for 10 minutes at 1100°C in a hydrogen atmosphere. The calcium carbonate is then dissolved out using a solution of 10 parts water to 1 part concentrated nitric acid by volume. The copper spheres so produced had a mean diameter of 6 |am and 80% by weight lay in the size range 1.4 to 10.5 |Lim. This method was also used by Colon et. al. [130] for the production of glass spheres. Several methods are available for the preparation of narrowly classified powders. The simplest is separation between consecutive sieves. Glass and metal spheres of diameter greater than 50 |Lim and standard deviations of 10% to 15% produced by this method are commercially available. A much smaller standard deviation is attainable by separation using two sieves of the same nominal diameter, since no two sieves are identical, however the rate of production is much lower. Even narrower size fractions are obtainable by collecting the particles wedged in the sieve mesh after a sieving operation but the quantities that are produced are very small. Colon et. al. used a combination of sedimentation and sieving to produce glass spheres in commercially viable quantities. Sieve size range metal spheres are readily available in commercial quantities. Glass spheres are also available and these may be preferred due to their lower densities [131]. Glass beads in the size range 1 to 30 |J,m were carefully treated and examined to provide NBS Standard 1003 [132, 133].
352 Powder sampling and particle size determination
Narrowly classified latices are available from Dow Chemicals [134] but doubt has been expressed on the accuracy of their sizing. A 3.49 ^m polystyrene latex was independently sized by electron microscopy and found to have a mean diameter of 3.40 |im [135]. This standard was used by Coulter Electronics as a standard to size other Dow latices [136] that are available from them as standard suspensions. The Community Bureau of Reference has five quartz samples prepared as reference materials for the calibration of apparatus, these are described in chapter 11. These powders were analyzed by gravity sedimentation except for BCR 68 which was analyzed by sieving. Analyses were carried out at five laboratories and the results compared to give a measure of the quality of the standards. The National Physical Laboratory has surface area and pore size distribution standards. AC fine test dust is available from General Motors for calibrating classifiers and ASME also produce standards for this purpose. The particle size distributions of 15 International Atomic Energy Agency and 16 National Institute of Standards and Technology reference materials (RMs) were measured by laser diffraction to determine their potential as reference or quality control materials in that less than 100 mg are required. Most of the materials are commercially available as environmental and biological reference materials (RMs) [137]. NIST produce two standard powders for gravity sedimentation. SRM 659 is a silicon nitride powder that has equiaxed particles in the size range 0.2 to 10 jam. SRM 1978 is a zirconium oxide powder of granular irregulary shaped particles in the size range 0.33 |Lim to 2.19 |Lim. 6.19 National Standards BS 3406, British Standard Method for determinig Particle Size Distribution is a comprehensive standard that includes methods for both incremental and cumulative methods of particle size deterination. Part 1, 1984, covers Recommendations for Gravitational Liquid Sedimentation Mehods for Powders and Suspensions. Part 5, 1985 covers Recommendations for Centrifugal Liquid Sedimentation Methods for Powders and Suspensions. ASTM C958, 1997 is a Standard Test Method for Particle Size Distribution of Alumina or Quartz by X-ray Monitoring of Gravity Sedimentation covering the size range from 2.5 |Lim to 10 |im. ASTM B761, 1998 is a Standard Test Method for Particle Size Distribution of
Interaction between particles andfluids
353
Refractory Metals and their Compounds by X-ray Monitoring of Gravity Sedimentation. DESf 66111, 1973 is a German stadard that covers the principles of gravity sedimentation methods and DIN 66115 covers test methods. AFNOR produce the French standard NFX 11-681 for gravity sedimentation and NFX 11-683 for variable height sedimentation using xray adsorption. Japan has a standard JIS Z8820 covering general rules for determination of particle size by sedimentation. ISO is compiling a standard ISO/WD 13317-1, 1996 covering general principles and guidelines. Another draft ISO/WD 13317-3, 1996 covers gravitational x-ray analysis. References 1 2 3 4 5 6 I 8 9
10 II
12 13
Heywood, H. (1962), Proc. Symp. Interaction Particles and Fluids, Inst. Chem. Engrs. London, 299,305,320.324 Mason, M. and Weaver, W. (1924), Phys. Review, 23, 412-426, 303 Berg, S. (1958), Symp. Particle Size Measurement, ASTM Pub. 234, 143, 303 Okuyama, K., Kousaka, Y., Miyazaki, T. and Yoshida, T. (1977), J. Chem. Eng. Japan, 10, 46, 303 Moore, D. W. and Orr, C. (1973), Powder Technol, 8, 13, 303 Chung, H. S. and Hogg, R. (1985), Powder Technol. ,4\,2\\-2\6, 303 Happel, J. and Brenner, H. (1965), Low Reynolds Number Hydrodynamics, Prentice Hall, 303,312,314,317,318,326,328,334 Fuchs, N.A. (1964), Mechanics of Aerosols, Trans., ed. C.N. Davies, Pergamon, Oxford, 303 Bernhardt, C. (1992), Particle Size Analysis, ed N.G. Stanley-Wood and R.W. Lines, 477-487, Proc. Conf, Bradford, U.K., publ. RoyalSoc. Chem., 304 Brugger, K. (1976), Powder Technol., 14, 187-188, 304,309,335 Allen, T. and Nelson R.D. Jr. (1995), 6th European Symp. Particle Size Characterization, Partec 95, Numberg, Germany, publ. N^mbergMesse GmbH., 143-155,505 Boothroyd, R.G. (1971), Flowing gas solids suspensions. Chapman & Hall, London, 305,313 Muta, A. and Watanabe, S. (1970), Proc. Conf. Particle Size Analysis, Bradford, U.K., (eds. M.J. Groves and J.L. Wyatt-Sargent), Soc. Anal. Chem., pp 178-193, 196, 197, 305
354 Powder sampling and particle size determination
14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
Allen, T. (1970), Proc. Conf. Particle Size Analysis, Bradford, U.K., (eds. M.J. Groves and J.L. Wyatt-Sargent), Soc. Anal. Chem., discussion, pp.194, 195, 198, i05 Allen, T. and Baudet, M.G. (1977), Powder Technol, 18, 131-138, 306 Lorentz, H. (1906), Abk u, Th. Phys., 82, 541, 308 Hawksley, P.G.W. (1951), British Coal Utilization Assoc. Bull, (BCURA), \S{A\ 310,316,319,328 Cunningham, E. (1910), Proc. Royal Soc, A83, 357, 310 Davies, C.N. (1954), Proc. Phys. Soc, 57, 259, 310,319 Lapple, C. E. (1950), Perry's Chemical Engineering Handbook, 311 Einstein, A. (1911), Ann. Phys. Leipzig, 19, 289, 34, 591, 311 Levitch, V.G. (1962), Physiochemical Hydrodynamics, Prentice Hall, Englewood Cliffs, NJ., USA, 312 Heiss, F. and Coull, J. (1952), Chem. Eng Progr., 48(3), 133-140, 313 Pettyjohn, E.S. and Christiansen, E.B. (1948), Chem. Eng. Prog., 44(157) 203, 316,324 Henn, A.R. (1996), Part. Part. Syst. Charact., 13(4), 249-253, 319 Haider, A. and Levenspeil, O. (1989), Powder Technol, 58, 63, 322 Squires, L. and Squires, W. Jr. (1937), Trans. Am. Inst. Chem. Eng, 33(1), 322 Ganser, G.H. (1993), Powder Technol, 11, 143-152, 322 Oseen, C.W. (1927), Neuere Methoden und Ergebrisse in der Hydrodynamik, Leipzig Akademische Verlag, 322,326 Proudman, 1. and Pearson, J.R.A. (1957), J. FluidMech., 2, 237, 322 Goldstein, S. (1938), Modern Developments in Fluid Dynamics, Clarendon Press, 323 Schillar, L. andNauman, A.Z. (1933), Ver. dt. Ing 11, 318, 323 Davies, C.N. (1947), Trans. Inst. Chem. Engrs., Suppl. 25, 39, 323 Kynch, G.jj. (1959), J. Fluid Mech. 5, 193, 326 Famularo, J. (1962), Engng ScL Thesis, New York Univ., 328 Burgess, J.M. (1941), Proc K. Ned Akad Wet., 44, 1045, 328 Maude, A.D. and Whitmore, R.L. (1958), Br. J. Appl Phys., 4, 477, 328 Richardsoi^, J.F. and Zaki, W.N. (1954), Chem. Eng ScL, 3, 65, 328,330 Brinkman, H.C. (1947), Appl ScL Res. A\, 27, (1948); Al, 81, (1949), 328 Steinour, R H . (1944), Ind Engng Chem., 36, 618, 840, 901, 328,329 Powers, T.C. (1939), Proc. Am. Concr. Inst., 35, 465, 329 Capes, C.E (1974), Powder Technol, 18, 283-284, 330 Michaels, i . S . and Bolger, J.C. (1962), Ind Eng Fund., 1, 24, 330,335 Scott, K.J. (1968), Ind Eng Fund, 1, 484, 330 Whitmore, k L . (1957), J. Inst. Fuel, 30, 238, 330 Davies, L, Dollimore, D. and Sharpe, J.H. (1976), Powder Technol, 13, 123-132, j p Davies, L, Dollimore, D. and McBride, G.B. (1977), Powder Technol, 16, 45-49, 3311334
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61 62 63 64
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68 69 70 71 72 13
fluids
355
Davies, L. and Dollimore, D. (1977), Powder Technol, 19, 1-6, 331 Dixon, D.C., Buchanon, J.E. and Souter, P. (1977), Powder Technol,\% 283-284, 332 Davies, L. and Dollimore, D. (1977), Powder TechnoL,\H, 285-287, 332 Chang, D., Lee, T., Jang, Y., Kim, M. and Lee, S. (1997), Powder TechnoL, 92(1), 81-87, J i 2 Davies, R. and Kaye, B.H. (1971/2), Powder TechnoL, 207-222, 332 Davis, R.H. (1996), Adv. Fluid Mech., 7, 161-198, 332 Kaye, B.H. and Boardman, R.P. (1962), Proc. Symp. Interaction between Fluids and Particles, Inst. Chem. Engrs., London, 333 Johne, R. (1966), Diss. Karlsruhe (1965), 333 Koglin, B. (1970), Proc. Conf. Particle Size Analysis, eds. M.J. Groves and J.L. Wyatt-Sargent, Soc. Analyt. Chem., Bradford, U.K., 223-235, 333 Jovanovic, D.S. (1965), KolloidZ. Polymery 203(1), 42-56, 333 Barford, N. (1972), Powder Technol, 6(1), 39-44, 333 Jayaweera, K.O.L.F., Mason, B.J. and Slack, B.W. (1964), J. Fluid Meek, 20, 121-128, 5iJ Davies, R. and Kaye, B.H. (1970), Proc. Conf. Particle Size Analysis, eds. M.J. Groves and J.L. Wyatt-Sargent, Soc. Analyt. Chem., Bradford, U.K., 223-235, 333 Dollimore D. and Griffiths, D.L. (1960), Proc. 3rd Int. Cong. Surface Activity, Cologne, V2 Section B674, 334 Christian, J.R., Dollimore, D. and Horridge, T.A. (1970), J. Phys. E, 3, 744, 334 Sarmiento, G. and Uhlherr, P.H.T. (1977), Proc. 5th Australian Chem. Eng. Conf, Canberra, 334 Dollimore and McBride, G.B. (1970), Proc. Conf Particle Size Analysis, eds. M.J. Groves and J.L. Wyatt-Sargent, Soc. Analyt. Chem., Bradford, U.K., 223-235, 334 Dollimore, D. and Heal, G.R. (1962), J. Appl. Chem., 12, 455, 334 Ramakrishna, V. and Rao, S.R. (1970), J. Appl. Chem., 15, 413,334 Ramakrishna, V. and Rao, S.R. (1970), Proc. Conf. Particle Size Analysis, eds. M.J. Groves and J.L. Wyatt-Sargent, Soc. Analyt. Chem., Bradford, U.K., 223-235, 334 Pierce, T.J. (1970), Proc. Conf Particle Size Analysis, eds. M.J. Groves and J.L. Wyatt-Sargent, Soc. Analyt. Chem., Bradford, U.K., 223-235, 334 Dollimore, D. and Horridge, T.A. (1971), Trans. Br. Ceram. Soc, 70, 191, 334 Dollimore, D. and Owens, N.F. (1972), Proc. 6th Int. Cong Surface Activity, Zurich, 334 Davies, L. and Dollimore, D. (1978), Powder TechnoL, 19, 1-6, 335 Sarmiento, G. and Uhlherr, P.H.T. (1979), Powder Technol, 22, 139-142, 335 Dollimore, D. (1972), Powder Technol, 8, 207-220, 335
356 Powder sampling and particle size determination
74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92
93 94
95 96 97 98 99 100 101 102 103 104 105 106
Dom, E. (1880), Wied Ann. 10, 46, cit. Adamson, H.A. (1934), Electokinetic Phenomena, Chem. Catal. Co. N.Y, 335 Elton, G.A.H. (1948), Electroviscosity, 1, Proc. Royal Soc, A194, 259; Electroviscosity, 2, A194, 275, 335 Dulin, C.I. and Elton, G.A.H. (1952), J. Chem. Soc, 286, 341 Elton, G.A.H. and Hierschler, F.F. (1954), Br J. AppL Phys., Suppl. 3, S60, 335 Elton, G.A.H. and Hierschler, F.F. (1962), J. Chem. Soc., 2953, 335 Booth, F. (1950), Proc. Royal Soc, A203, 533, 335 Booth, F. (1954), J. Chem. Phys., 22, 1956, 335 Pavlik, R.E. and Sansone, E.B. (1973), Powder TechnoL, 8, 159-164, 336 Sansone, E.B. and Civic, T.M. (1975), Powder TechnoL, 12(1), 11-18, J i d Siano, D.B. (1979), J. Colloid Interf. ScL, 68(1), 111 -127, 336 Jones, T.M. (1969), Symp. Powders, Soc. Cosmetic Chem., Dublin, 337 Peleg, M. and Mannheim, C.H. (1973), Powder TechnoL 7, 45, 337 Pilpel., N. (1970), Mftg. Chem. Aerosol News, 4, 19, 337 Jenike, A.W. (1961), BulL 108, Eng. Exp. St.,. Utah State Univ., 337 Gold, G. et al (1966), J. Pharm. ScL, 55, \29\, 337 York, P. (1975), Powder TechnoL, 11, 197-198, 339 Jenike, A.W. (1961), BulL 108, Eng. Exp. St.,. Utah State Univ., 338 Parfitt, G. (1977), Powder TechnoL, 17(2), 157-162, 338 Gabrys-Deutscher, E. and Leschonski, K. (1995), 6th European Symp. Particle Size Characterization, Partec 95, Numberg, Germany, publ. NumbergMesse GmbH.225-234, 339 Nelson, R.D. (1988), Dispersing Powders in Liquids, Elsevier, 339 Conley, R.F. (1996), Practical Dispersion, A guide to Understanding and Formulating Slurries, VCH Publishers Inc., 220 East 23rd Street, New York, NY 10010, i i P Adamson, A.W. (1963), Physical Chemistry of Surfaces, Wiley, N.Y., 339 Bartell, F.E. and Walton, C.W. (1934), J. Phys. Chem., 38, 503, 340 Crowl, V.T. and Wooldridge, W.D.S. (1967), Wetting, SCI Monogram No. 25, 200, 341 Washburn, E.D. (1921), Phys. Rev., 17, 374, 341 Heertjes, P.M. and Kossen, N.W.F. (1966), Powder TechnoL, 1, 33-42, 342 Kossen, N.W.F. and Heertjes, P.M. (1965), Chem. Eng ScL, 20, 593, 342 Carino, L. and Mollet, H. (1975), Powder TechnoL, 11, 189-194, 342 Kousaka, Y., Endo, Y., Horiuchi, T. and Sasaki, Y. (1996), Kona, 14, 162167, publ. Hosokawa Powder Technology Fopundation, 343 Crowl, V.T. (1967), Pigments, An Introduction to their Physical Chemistry, ed. D. Patterson, Elsevier, 343 Bryant, D.P. (1968), Proc Soc Analyt. Chem., 5(8), 165-166, 344 Rossi, C. and Baldocci, R. (1951), J. AppL Chem., 1, 446, 345 Buzagh, A. von (193 7), Colloid Systems, 345
Interaction between particles and
107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
127 128 129 130 131 132 133 134 135
fluids
35 7
British Standard, BS3406; Recommendations for gravitational liquid sedimentation methods for powders and suspensions, 345 Groves, M.J. (1968), Proa Soc. Analyt. Chem., 5(8), 165-166, 345 Hartig, H.E., Oristad, N.I. and Foot, N.J. (1951), Univ. Minn. Mines Exp 5'r., Info. Circular No 7., i^d Hathaway, R.E. and Guttrals, D.L. (1976), Can. Min. Metall. Bull.,69(166), 64-71,346 Koglin, B. (1977), Powder Technol, 17(2), 219-227, 346 Karpenko, I., Dye, R.W. and Engel, W.H. (1962), TAPPI, 45(1), 65-69, 346 Carr, W. (1970), JOCCA, 53, 884, 346 Carr, W. (1977), Powder Technol, 17(2), 183-190, 346 Wilkes, R. and Allen, T. (1990), Particle Size Measurement, p252-254. Chapman and Hall, 348 Burt, M.W.G., Fewtrell, C.A. and Wharton, R.A. (1973), Powder Technol, 8, 223-230, 348 DoUimore, D. et. al (1970), J. Phys. E., 3, 465-466, 348 Bond, R.L. and Spencer, D.H.T. (1957), Proc. 1st Ind. Conf. Ind. Carbon and Graphite, Soc. Chem. Ind, London, pp 231-251, 348 Spencer, D.H.,T. (1967), Porous Carbon Solids, (ed. R.L. Bond), Academic Press, NY, pp 87-154, i^P May, I. and Marienko, J. (1966), Am. Mineralog., 51, 931-934, 349 Stein, F., Pankala, S. and Buchino, J. (1971/2), Powder Technol, 1, 45, 349 Thudium, J. (1976), J. Aerosol Scl, 7(2), 167-174, 349 Baranowski, J. (1973), Ochrona Powietrza, 2, 30, 349 Juda, J. (1966), Staub, 26, 197, 348 Krutzch, J. (1954), Chemiker-Zeitung, 78, 49, 348 H^nel, G. (1972), Bestimmung Physikalischer Eigenschaften Atmospherischer Schwebetulchen als Funktion der Relativen Lufteuchtigkeit, Diss. Universitat Mainz, 348,350 Mtiller, G. (1964), Methoden der Sedimentuntersuchung, E. Schweltzerbartsche Verlagbuchandlung,, Stuttgart, 348 Keng, E.Y.H. (1969/70), Powder Technol, 3, 179-180, 348 Muta, A., Saito, N. and Ueharqa, Y. (1967), Particle Size Analysis, Proc. Conf., Soc. Analyt Chem., London, p. 215, 351 Colon, F.J., Fijman, W.J., Tuinman, P.C. and Veldt. C. (1973), Powder Technol, S, 301-310, 351 Gaudin, A.M. and Bowdish, F.W. (1944), Mining Tech., 8(3), 1-6, 351 Hunt, CM. and Woolf, A.R. (1969), Powder r^chnol., 3(1), 9-23, 352 Carpenter, F.G. asnd Dietz, V.R. (1951), J. Res. Nat. Bur. Stand, 47(3), 139-147,352 Bradford, E.B. and Vanderhoff, J.W. (1966), Symp. Particle Size Distribution, Pittsburgh, PA, 352 Matthews, B.A. and Rhodes, C.T. (1970), J. ColloidInterf ScL, 32, 339, 352
358 Powder sampling and particle size determination
136
137 138 139
Hartfield, J.G. and Wood, W.M. (1970), Particle Size Analysis, Proc. Conf. Soc. Analyt. Chem., Bradford, 293-300, ed M.J. Groves and J.L. WyattSargent, 352 Fajgeli, A. and Zeisler, R. (1998), Fresenius, J. Analyt. Chem., 360(3-4), 442-445, 352 Lerman, A., Lai, D., and Dacy, M.F. (1974), Suspended Water in Solids, 1747, ed. R.D. Gibbs, Plenum Press New York, 315 Konert, M. and van der Berghe, J. (1997), Sedimentology, 44, 523-535, 315
Gravitational sedimentation methods of particle size determination 7.1 Introduction Gravitational sedimentation methods of particle size determination are based on the settling behavior of a single sphere, under gravity, in a fluid of infinite extent. Many experiments have been carried out to determine the relationship between settling velocity and particle size and a unique relationship has been found between drag factor and Reynolds number. This relationship reduces to a simple equation, the Stokes equation, which applies at low Reynolds numbers. Thus at low Reynolds numbers the settling velocity defines an equivalent Stokes diameter which, for a homogeneous spherical particle, is its physical diameter. At low Reynolds number, flow is said to be laminar i.e. the fluid flow lines around the particle are unbroken. As the Reynolds number increases, turbulence sets in leading to increased drag on the particle so that it settles at a lower velocity than predicted by Stokes' equation. It therefore follows that, if the settling velocity of a homogeneous, spherical particle is known, its particle size can be deduced and, conversely, if the size is known the settling velocity can be determined. The drag force on a particle is orientation dependent, hence nonspherical particles settling with their largest cross-sectional area perpendicular to the flow direction will settle more slowly than similar particles settling with minimum area perpendicular to flow. It follows that an assembly of identical non-spherical particles, settling under laminar flow conditions, will have a range of settling velocities according to their orientation. Sedimentation techniques can be classified according to the principles outlined in Table 7.1. Table 7.2 lists the various procedures that have been developed according to the principle applied. Techniques in current use are
360 Powder sampling and particle size determination
described here; descriptions of techniques, which are no longer used, can be found in earlier editions of this book [1]. Analytical procedures for some of these techniques are covered more fully in British Standards BS 3406 [2,3]; German Standards DIN 66111 [4] and DIN 66115 [5]; French Standards AFNOR NFX 11-681 [6] and NF 11-683 [7]; two International Draft Standards ISO/WD 13317-1 [8,9] and a Japanese Standard JIS Z8820 [10]. Two American Standards for gravitational X-ray analysis are also available, one for ceramic material in ASTM C958 [11] and one for Refactory Metals in ASTM B761 [12] (see [13]). Table 7.1 Principles of sedimentation techniques Suspension type Homogeneous Line-start
Measurement principle Incremental Cumulative
Force-field Gravitational Centrifugal
In the homogeneous, incremental, gravitational technique, the solids concentration (or suspension density) is monitored at a known depth below the surface for an initially homogeneous suspension settling under gravity. The concentration will remain constant until the largest particle present in the suspension has fallen from the surface to the measurement zone (Figure 7.1). At the measurement zone the system will be in a state of dynamic equilibrium since, as particles leave the zone, similar particles will enter it from above to replace them. When the largest particle present in the suspension settles through the measurement zone, the concentration will fall since there will be no particles of this size above the zone. Thus, the concentration will be of particles smaller than the Stokes diameter and a plot of concentration against Stokes diameter is, in essence, the mass undersize distribution. In the homogeneous, cumulative, gravitational technique, the rate at which solids settle out of suspension is determined for an initially homogeneous suspension settling under gravity (Figure 7.2). This technique is typified by the sedimentation balance, in which the balance pan can be in the suspension (Figure 7.3), or suspended in a clear liquid (Figure 7.4). With the former set-up, correction has to be made for the particles that do not fall on the pan; errors are also introduced since the particle free zone below the pan leads to convection currents. The latter technique also suffers from problems due to the motion of the pan as particles settle on it. In this system, the amount settled out consists of two parts, all particles larger than Stokes diameter and a fraction of particles
Gravitational sedimentation methods
361
t^O • :: -TTTI
• • •: ii I
• • •:f ©•:::!
©©•;
©©•;::!
^ o » >: I
«k •
•
•
: :
O 0 •
Measurement zone Fig. 7.1 Homogeneous, incremental, gravitational sedimentation
Fig. 7.2 Homogeneous, cumulative, gravitational sedimentation
"V
\/ • • h
••
• © 0
•©•I
© • ••.'
•'rtr !©• .© , 0
» •
o
^kovCo'r^Particle free zone Fig. 7.3 Balance pan in suspension
Clear liquid Fig 7.4 Balance pan in clear liquid
fiSDStfOPi ocxxx)oooci popcxxxl
pooo \ Measurement zone Fig. 7.5 Line-start, incremental gravitational sedimentation.
ooooooooootxoooo
OOOOOCXXX)
boooooq
I Innono Fig. 7.6 Line-start, cumulative gravitational sedimentation.
362 Powder sampling and particle size determination
smaller than this. The amount undersize is determined by carrying out an integration of the second fraction. With the incremental, gravitational, line start technique (Figure 7.5) the suspension is floated on top of a container of clear liquid and, provided the particles fall independently, the largest particles present in the suspension will reach the measurement zone first and the measured concentration will be the concentration of this size band in the measurement zone. This technique can also be used in the cumulative mode (Figure 7.6). In this presentation, some of the methods for sedimentation particle size analysis in current use are described. Although operating procedures are not covered here, it is stressed that two factors, more than anything else, lead to incorrect analyses. The first is incorrect sampling, since analyses are carried out on from a tenth of a gram up to a few grams and these samples must be representative of the bulk for the analyses to be meaningful. The second is dispersion: it has been said rightly that the most important factor in obtaining accurate sedimentation data is dispersion-the second most important factor is dispersion and the third is also dispersion!! 7.2 Resolution of sedimenting suspensions The size range of particles within a detector is controlled by the height of the detector beam (A/z), hence the measurement gives the concentration between an upper and lower size limit. Assuming Stokes' law to apply, equation (6.8) may be written as: '
di=k\
(7.1)
Differentiating with respect to h with t constant: 2dst
^dd,.\ 'St dh
k _dl t
The size resolution is given by: ^ ^ =^ dst 2/?
(7.2)
For a hydrometer, assuming A/? = h: When 50 ^m particles are at a measurement depth of 10 cm, the bottom of the hydrometer bulb is at a
Gravitational sedimentation methods
363
depth of 15 cm and the top is at a depth of 5 cm, so that particles of size 35.4 |Lim will be entering the measurement zone and particles of size 70.7 |im will be leaving it. If the weight frequency of particles in the 35.4 to 50 |im range is balanced by the weight frequency in the 50 to 70.7 |Lim range the effect is balanced out, otherwise a bias results. The effect of such a bias is to mask peaks in multi-modal distributions. If the thickness of the measurement zone is less than one sixth of the settling depth, the size resolution of around 8% leads to tolerable errors. Table 7.2 Commercial sedimentation particle size analyzers Homogeneous, incremental gravitational sedimentation Andreasen pipette Leschonski pipette Fixed depth pipette Side-arm pipette Wagner photosedimentometer EEL photosedimentometer Bound Brook photosedimentometer Seishin Photomicrosizer Ladal wide angle scanning photosedimentometer Retsch Paar Lumosed 1 Kemsis K200 photosedimentometer ICI x-ray sedimentometer Ladal x-ray sedimentometer Micromeretics Sedigraphs 5000 & 5100 Quantachrome Microscan X-ray sedimentometer Hydrometers Divers Suito specific gravity balance Line-start, incremental, gravitational sedimentation MSA analyzer
Homogeneous, cumulative, gravitational sedimentation Oden Balance Svedberg and Rinde automatic recording sedimentation beam balance Cahn balance Gallenkamp balance Mettler H20E balance Sartorious Recording Sedibel balance Palik torsion balance Kiffer continuous weighing chain link balance Rabatin and Gale spring balance Shimadzu balance ICI sedimentation column BCURA sedimentation column Fisher Dotts apparatus Decanting p-Back-scattering
Line-start, cumulative, gravitational sedimentation Werner and Travis method Granumeter Micromerograph MSA analyzer
'
'
364 Powder sampling and particle size determination
7.3 Concentration changes in a suspension settling under gravity Let a mass m^ = p^v^ of powder be dispersed in a mass my= p^Vy of fluid, p and V being density and volume respectively. Initially the mass concentration will be uniform and equal to:
di=k[^]
(7.3)
where C(kO) is the concentration at depth h, time / = 0. Consider a small horizontal element at depth h. At the commencement of sedimentation, the particles leaving the element are balanced by the particles entering it from above. When the largest particles present in the suspension leave the element, after settling from the surface, there are no similar particles entering to replace them. The concentration will then fall and become equal to a concentration smaller than that ofd^^ where d^^ is the size of the particle that settles at a velocity of M. The concentration of the suspension at depth h at time / may be written:
1
C(h,t) = -^-^-=
\f(d)dd
^.+v
(7.4)
where is the mass and v^ is the volume of solids in a volume Vj of fluid at a depth h from the surface of the suspension at time t from the commencement of sedimentation. C(//,0) = — ^ — = •*
•'
f f{d)dd
(7.5)
"mm
From equations (7.3 to 7.5):
C{h,t) ^m, ^ d^^ C(/?,0) m, V
^ ^
f f{d)Ad
Mtnin
.^g.
Gravitational sedimentation methods
365
It is assumed that the difference between v^ and v^ is negligible compared with Vj-, Thus a graph of C(^,0/C'(/2,0) against d^^ gives the percentage undersize Stokes diameter by weight. 7.4 Homogeneous incremental gravitational sedimentation 7.4.1 The pipette method ofAndreasen In the pipette method (Figure 7.7), concentration changes occurring within a settling suspension are followed by drawing off definite volumes, at predetermined times and known depths, by means of a pipette. The method was first described in 1922 by Robinson [14] who used a normal laboratory pipette. Various modifications were later suggested which complicated either the operating procedure or the apparatus [15]. Andreasen was the Scale graduated in cm and mm
h 1
Bulb funnel Stopcock Suction tube Safely bulb Three way tap Outlet tube
2o| cm
I
m
1
m
Pipet
— l em L 1
L
II—
cm
P
Sedimentation tube Constant temperature bath
CM
10 cm jO
Fig. 7.7 (a) The fixed position pipette (b) the variable height pipette.
366 Powder sampling and particle size determination
first to leave the pipette in the sedimentation vessel for the duration of the analysis. The apparatus described by Andreasen and Lundberg [16] is the one in general use today. Although, theoretically, errors can be reduced by the use of more complicated construction and operation, it is highly debatable as to whether this is worthwhile for routine analyses since conventional apparatus is reproducible to ±2% if operated with care [17]. This technique is a standard procedure since both the Stokes diameter and the mass undersize are determined from first principles. The method is versatile, since it can handle any powder that can be dispersed in a liquid, and the apparatus is inexpensive. The analysis is, however, time consuming and intensive. 7.5 Theory for the gravity photosedimentation technique 7.5.1 The Beer Lambert law Consider a sedimentation container of width L measured in the direction of the light beam, containing the suspension of powder under analysis. Let the incident light intensity falling on an element of thickness 8Z be / and the emergent intensity be 7-57. If the area of the light beam is A, the reduction in flux due to the presence of particles may be attributed to a fall in the overall intensity of the light beam, or a reduction in the area of the light beam (Figure 7.8). The emergent flux may be written: (7-67)^ = {A-^A)I ^ =^ 7 A
(7.7)
where 5^ is the effective cross-sectional area of particles in the beam perpendicular to the direction of propagation. The equation holds, provided that the beam of light becomes homogeneous again between adjacent particles. Let there be n^ particles of diameter d^ in unit mass (1 kg) of powder and let the powder concentration in suspension be c (kg m"^); then, at time /, the following expression holds.
8^ = -^^8^Z'^A«A'
(7-8)
Gravitational sedimentation methods
367
where k^ is a shape coefficient (k^ = n/4 for spheres) J^j^^ and d^^ are the diameters of the smallest and largest particles in the beam at time t and K^ is the extinction coefficient for a particle of diameter d^. The extinction coefficient is defined as: K =
light obscured by a particle of diameter d^ light which would be obscured if the laws of geometric optics held
(7.9)
From equations (7.7) and (7.8)
h (7.10) Integrating for time / gives: '' T \
In
x=St
= C^ Z \
t J
^x^x^xd
(7.11)
jc=min
where /Q is the emergent light intensity with clear liquid in the beam and /^ is the emergent light intensity at time t.
l-dl
^
Fig. 7.8 For a light beam of cross-sectional areav4 intersecting a suspension the reduction in light flux, 5/ is proportional to the cross sectional area of particles in the beam 5^.
368 Powder sampling and particle size determination
The optical density {E^ of the suspension at time / is defined as: ^ J ^
E, =cZlogio e X K,k,n,dl
(7.12)
Consider the small fall (A^"^) in the optical density as the sedimentation time changes from / to / +A/ so that the maximum Stokes diameter in the beam changes from with an average value d^. ^^-cL\og,,{e)k,n,dl
(7.13)
The cumulative distribution undersize by surface, assuming that K is constant for the restricted size range under consideration, is: x=^St x=0
x=St _
^
x=0 x=ma\ AEy
x=0
x=0
(7.14) It is therefore necessary to know how K varies with d in order to determine the size distribution. If this correction is not applied, the method is only valid for comparison purposes. Theoretical values of K may be used but this will also introduce errors, since the effective K values depend upon the optical geometry of the system. Calibration may also be against some external standard. The cumulative distribution undersize by weight is given by: x=St
Z ^x4 jc=0
\ ^ ^
^x^x ^x
jc=max
Z "x4
^
^X
(7.15)
Gravitational sedimentation methods
369
The surface area of the powder is derivable from the initial concentration of the suspension and the maximum optical density (E^^^):
Combining with equation (7.13), bearing in mind that n^ is the number of particles of size d^ in unit weight (W= \) of powder: x=max p
Sw= fcllog,o(e)
Y— ; ^ K^
(7-16)
where a^ is the surface shape coefficient, which may be assumed constant for a powder having a narrow size range. Summating equation (7.16): (7.17)
C>^r —
kK^LXog^^ie)
c
where K^ is the mean value for the extinction coefficient. For non-reentrant (convex) particles, the ratio of the surface and projected area shape coefficients {a/k) is equal to 4. For re-entrant particles, the surface obtained by making this assumption is the envelope surface area. Making this assumption equation (7.16) simplifies to: S^^
^'^ """^^ K^L c
(7.18)
7.5,2 The extinction coefficient The extinction coefficient varies with the optical properties of the solid and liquid that make up the suspension. Knowing these properties, it is possible to generate a relationship between the extinction coefficient and particle size using Mie or a boundary condition theory. Since unit area of 0.2 |Lim Ti02 cuts of 10 times as much light as unit area of 0.1 |im Ti02, (Figure 7.9), if no correction is applied the measured distribution for submicron Ti02 will be heavily weighted towards the coarser particles. The relationships hold for an infinitely small solid angle between the detector and the suspension so a correction for the geometry of the analyzer may be
370 Powder sampling and particle size determination
required. Alternatively, the instrument may be calibrated against some external standard. If no correction is made for variation in extinction coefficient {K =" \) the derived distribution is only a size dependent response and the method becomes a fingerprint method (i.e. useful for comparison purposes only).
0.2
0.4 0.6 Panicle size (JC) in microns
0.8
1
Fig. 7.9 Extinction curve for titanium dioxide in water for white light. 7.5.3 Turbidity measurements (Turbidimetry) Turbidimetry measurements, using monochromatic light, yields data that can be used to determine particle size distributions. It requires simple optical technology, but complex computational software to handle the Mie theory conversion. The sensitive diameter range for latex-water suspensions was found to be 0.1 to 10 |Lim. Different types of sensors have been conceived and applied to various experimental situations. The method is particularly useful in crystallization experiments. Other applications include agglomeration, attrition and nucleation studies. Applications of the equipment and software to studies of emulsions, fumes and aerosols are also envisaged [18]. 7.5.4 The photosedimentation technique The photosedimentometer combines gravitational settling with photoelectric measurement. The principle of the technique is that a narrow
Gravitational sedimentation methods
371
horizontal beam of parallel light is projected through the suspension at a known depth on to a photocell. Assuming an initially homogeneous suspension, the attenuation at any time will be related to the undersize concentration.
ac»6.0
Fig. 7.10 Polar light scattering diagrams [19]. The outer curve magnifies the inner by a factor of 10 in order to show fine detail, x = (nD/A) where D = particle diameter and A is the wavelength of light. Superficially, the attenuation is related to the random projected areas of the particles. The relationship is more complex than this however, due to the breakdown in the laws of geometric optics so that complex diffraction, scattering, interference and absorption effects have to be considered. For small particles, an amount of light flux, equal in magnitude to that incident upon the particle, is diffracted away from the forward direction (Figure 7.10), making their effective obscuration area twice their projected
3 72 Powder sampling and particle size determination
area. As the particle size increases, the diffracted light is contained in a decreasing solid angle in the forward direction. No matter how small the light detector, most of the diffracted light is accepted and the effective obscuration area becomes the same as the projected area. For partially transparent particles, some of the incident light is absorbed and some refracted to cause constructive and destructive interference in the transmitted beam. It cannot therefore be assumed that each particle obstructs the light with its geometric cross-sectional area. These effects are compensated for by inclusion of an extinction coefficient {K) in the equation, making the apparent area K times the geometric area. Early experimenters [20,21] were either unaware of, or neglected, this correction. Some research workers used monochromatic light and determined K theoretically [19,22] others used empirical calibration by comparison with some other particle sizing technique. Rose and Lloyd [23] attempted to define a universal calibration curve. Allen [24,25] designed a wide angle scanning photosedimentometer (WASP) which accepted forward scattered light so that K was constant down to a size of around 3 fim. Weichert [26] determined a relative extinction coefficient by the use of different wavelengths and speeded up the analysis by the use of different settling heights 7.5.5 Commercial photosedimentometers Kemsis K200 is a white light, narrow angle, scanning photosedimentometer. Caron et. al [27] described an application of this instrument for studies of turbidity and sedimentation measurements of solid-liquid dispersions. Sedimage 1000 uses a white light source and a linear sensor with 2048 detectors to determine the concentration gradient within a settling suspension. A linear image sensor, 28.6 mm in length, measures the transmitted light along this distance. The length of the image sensor limits the measurable height of particle sedimentation. The instrument continuously monitors the changing concentration of the settling suspension and the analysis is deemed complete when the smallest particle present passes the upper detector. After about 5 minutes the particle size distribution of a carborundum powder having a median size of 5 |Lim can be accurately determined [28] whereas conventional scanning photosedimentometers take around 20 minutes to carry out an analysis A method of determining the particle size distribution from a single measurement, with a digital image acquisition system, on a sedimenting
Gravitational sedimentation methods
3 73
suspension has been presented [87]. Individual particles illuminated by a laser light sheet are tracked by a continuously operating CCD camera. The projected areas, shape factors and the centers of gravity are detected during the sedimentation process from a series of images with a constant time spread. As the algorithm is based on single particle tracking, the heterogeneity of the sample can be taken into account. From these measured particle characteristics particle sizes and settling rates are determined.
-6^
Reflected ?• \ /' Light
\-
Cuvet
Fig. 7.11 The Retsch Paar Lumosed, depths of beams are marked in mm. Retsch PAAR Lumosed (Figure 7.11), operates in the gravitational size range, with three light sources at different depths to speed up the analysis [29,30]. A range of cuvette photocentrifuges which also operate in the gravitational mode are also available commercially. With these instruments a K factor, obtained either theoretically or experimentally, can be inserted in the software algorithm. A photosedimentometer has also been described for measuring freefalling diameters up to 20 cm in size with an application using coke. The system has been used to accurately measure a range of materials [31]. 7.5.6 Sedimentation image analysis The basic idea of this method is the analysis of particles settling under gravity using a digital image acquisition system. A continuously operating CCD camera, with a frame grabber, tracks individual particles illuminated
374 Powder sampling and particle size determination
by a laser light sheet. The laser light sheet is arranged vertically at about one-third from the bottom of the sedimentation vessel. Since the particles in the beam need to be representative of the whole sample, the analysis is over when the largest particles settle below the measurement volume. Consequently, short measurement times are not only desirable but necessary. The projected areas, shape factors, circularities and centers of gravity are determined, together with the settling velocities, during the sedimentation process from a series of images with constant time spread. The capacity of this method was demonstrated first for nearly spherical particles [32]. However it was found that problems arise when particles deviate strongly from spherical shape [33]. Later developments overcame these problems and allowed the determination of particle shape [34]. For every settling particle the projected area diameter is obtained from several images at different positions of the particle and a mean value calculated. Consequently the influence of particle shape on the sizing technique is greatly reduced. The number of detected particles is in the range of 3000 to give a reliable (number) distribution. The sizing technique was verified with BCR materials and applied to the particle size analysis of soils. 7.5.7 Transmission fluctuation spectrometry If an extinction measurement is made with high spatial and temporal resolution, the transmitted signal shows significant fluctuations. The strength of fluctuation is related to the physical properties of the suspension and the process of spatial and temporal averaging. This connection has been exploited to calculate the particle size distribution and particle concentration from transmission measurements. A theory of temporal transmission fluctuations has been developed for the case where the beam diameter was much smaller than the particle diameter [35]. This was expanded to provide an analytical solution to the use of a focused laser beam where the beam diameter is different for each monolayer according to the variation of the beam cross-section along the path length [36]. 7.6 Theory for concentration determination gravitational sedimentation technique
with
the
x-ray
A natural extension to the use of visible radiation is to use x-rays. In this case the x-ray density is proportional to the weight of powder in the beam. The Beer-Lambert law takes the form: I = IQQXP(-BC)
(7.19)
Gravitational sedimentation methods
3 75
where J5 is a constant related to the atomic number of the powder in suspension, c is the powder concentration, / is the emergent flux with suspension in the beam and /Q is the emergent flux with clear suspension liquid. E, the x-ray density, is defined as: E = -\og{III^)
(7.20)
For powders of low atomic number, c needs to be high in order to obtain a large enough signal. Thus for silica powders, atomic number 13, a volume concentration of around 3% may be necessary and this can lead to hindered settling. 7.6.1 X-ray sedimentation Brown and Skrebowski [37] first suggested the use of x-rays for particle size analysis and this resulted in the ICI x-ray sedimentometer [38,39]. In this instrument, a system is used in which the difference in intensity of an x-ray beam that has passed through the suspension in one half of a twin sedimentation tank, and the intensity of a reference beam which has passed through an equal thickness of clear liquid in the other half, produces an inbalance in the current produced in a differential ionization chamber. This eliminates errors due to the instability of the total output of the source, but assumes a good stability in the beam direction. Since this is not the case, the instrument suffers from zero drift that affects the results. The 18 keV radiation is produced by a water-cooled x-ray tube and monitored by the ionization chamber. This chamber measures the difference in x-ray intensity in the form of an electric current that is amplified and displayed on a pen recorder. The intensity is taken as directly proportional to the powder concentration in the beam. The sedimentation curve is converted to a cumulative percentage frequency using this proportionality and Stokes equation. The introduction of more stable x-ray sources and detectors resulted in the development of simpler, commercially viable systems. Kalshoven [40,41] described an x-ray instrument which used a special program for scanning the sedimentation tank. As the concentration measurements by means of x-ray attenuation are very rapid, the scanning greatly speeds up the analysis, reducing the measurement time down to a few minutes. In this instrument it was done in such a way that the concentration, and hence the cumulative mass percentage undersize, is recorded as a function of Stokes diameter rather than time. An x-ray tube was used as a source and
376 Powder sampling and particle size determination
a scintillation counter as a detector. The difference in intensity between a measurement beam and a reference beam, in which the emitted beam is alternately split, was measured by a rotating wedge that automatically set the difference to zero. Sub-micron particles can be measured if the sedimentation tank is spun in a centrifuge for some time. The time integral of the centrifugal force is measured and the tank is scanned after the centrifugation. The inventor claimed that volume concentrations in the range 0.01% to 1% could be used, depending on the atomic number of the analyzed material. Experimentally it was found that readings could be taken at short distances below the surface without seriously affecting the results. When the centrifuge was used the results were independent of the time of centrifugation but no comparison analyses were presented. Several commercial instruments utilizing these principles were developed. Oliver et. al. [42] patented a gravitational x-ray particle size analyzer that incorporated the absorption technique and improved the system described by Kalshoven. This instrument was described by Hendrix and Orr [43] and is available commercially as the Micromeretics' Sedigraph 5000 (Figure 7.12). The instrument automatically presents results as a cumulative mass percentage distribution , and the sedimentation tank is driven in such a way that the concentration is recorded directly as a function of Stokes diameter. An air cooled, low power x-ray tube is used for generation of x-rays. These are collimated into a narrow beam that passes through approximately 0.14 in, (3.6 mm) thickness of suspension. The sedimentation tank is only 1.375 in (35 mm) high, is closed at the top and, in use, completely filled with suspension. Filling and emptying of the tank is accomplished with a built-in circulating pump. The transmitted radiation is detected as pulses by a scintillation detector, these are amplified and discriminated to eliminate low energy extraneous noise. The pulses are next clipped to constant amplitude and fed to a diode pump circuit which, in conjunction with an operational amplifier with a diode feedback, gives a voltage proportional to the logarithm of the x-ray intensity and is therefore proportional to the powder concentration. The instrument can analyze powders with atomic numbers greater than 13, but rather high initial volume concentrations of powder have to be used for powders having low x-ray adsorption (0.5% to 3%). This is due for the need for the initial decrease in x-ray intensity to be greater than 20% of the intensity with clean liquid in order to obtain reasonable resolution of the resulting attenuation curve. An absolute system is used here, in which the initial intensity is first measured with clean liquid in the cell and the zero set; the suspension is then introduced. This assumes an excellent stability of the source that, in the case of x-ray tubes, may be unreliable. The authors
Gravitational sedimentation
methods
377
claim a good reproducibility and present several comparison analyses with microscopy etc. The range is claimed to be 300 to 0.1 |Lim for most powders. This lower limit is unreal since it is generally accepted that gravitational sedimentation is limited to particle coarser than around a micron due to the effects of Brownian motion. The acceptance by the manufacturers of this lower size has affected the instrument design in that only a 35 mm fall-height is possible and this restricts the upper size limit [44].
Outlet
SUt movement
X-iay tube
Detector Relative concentration signal
Inlet Digital position translator
Pump
Sample or
pure liquid
CeU positioning signal
Digital program computer
1 ^
50 5 0.5 Particle size in microns
Digital-to-position translator
Fig. 7.12 The Micromeretics Sedigraph. Sedigraph 5100 was later designed with three scanning speeds, slow, standard and fast. Micromeretics' Windows-compatible operating software permits automatic overlaying of plots, saves and recalls sets of run conditions, and can operate as many as four Sedigraphs from a single computer. The temperature of the suspension is controlled automatically with heaters located in the mixing chamber. The mixing chamber is situated outside the instrument, thus eliminating the need to switch off the x-rays when changing samples, which applies to the earlier version. The Sedigraph 5100 is also available with the MasterTech 051 Autosampler so that as many as eighteen samples can be analyzed unattended.
378 Powder sampling and particle size determination
Quantachrome Microscan reduced the time for an analysis from about 45 to about 25 minutes. In this instrument the source and detector scanned up the sedimentation tank rather than the other way round; this was claimed to reduce vibration. In 1970 Allen and Svarovsky [45-47] developed an instrument in which the traditional x-ray tube was replaced by an isotope source. Allen and Svarovsky's design was incorporated in the Ladal x-ray scanning gravitational sedimentometer and the x-ray centrifugal sedimentometer, which are no longer commercially available. A later design of Allen's is available as the Brookhaven BI-XDC. This can operate in both the gravitational and centrifugal mode which greatly increases the size range covered. In the centrifugal mode the size range of nominal 0.2 |im titanium dioxide was found to be much narrower than in the gravitational mode and a 'phantom' bimodality appeared in the gravitational analysis which was ascribed to thermal diffusion i.e. Brownian motion. Many industries have large data banks on product size distributions by sieve analysis and want to continue using this form of presentation. In order to accommodate this need Cho et. al [48] converted Sedigraph data to sieve data using wet screened powder in the 38 to 53 \xm size range and fitted the data to a logarithmic distribution to give the slope and median size. This procedure must be use with caution since the conversion factors are shape dependent and a new calibration is required for each product. A review of these and other methods of size analysis is contained in a thesis by Svarovsky [49], papers by Svarovsky and Allen [50], and by Allen and Davies [51]. 7.7 Relationship between density gradient and concentration Following from equation 7.6: Let ^/z,/) be the density of the suspension at depth h and at time /. Then: {h,t)--
Ps^'s+Pf^f (/>(h,t) = ">"/
Gravitational sedimentation methods
379
^^h,t)=^''-^^'^'^-'^>'^ V,+Vf (t>{h,t) = pf+Ps
^s+^f
Ps Also
Ps
Therefore
CXM.^.^Ml^ C(/z,o)
(7.21)
"^ (l>{KO)-Pf
where ^ /s the mass fraction undersize d^^. A plot of 100 xN^(/?,/)-py)/(^(/2,0)-/7y^M against d^^ gives a cumulative mass percentage undersize curve. 7.8 Hydrometers and divers 7.8.1 Introduction The changes in density of a settling suspension may be followed with a hydrometer, a method widely used in soil science and in the ceramic industry. A suspension of known concentration is made up and the hydrometer inserted. Some operators leave the hydrometer in the suspension throughout the analysis and some remove it after each reading and replace it slowly before the next. Objections can be raised to either procedure since, in the former, particles settle on the hydrometer causing it to sink to a lower level than it would otherwise sink whereas, in the latter,
380 Powder sampling and particle size determination
the suspension is disturbed after each reading. To minimize errors some operators re-shake the container after each reading. 7.8.2 Theory With the hydrometer immersed, its weight W equals the weight of suspension displaced. Let the length of stem immersed in clear suspending liquid be Z, i.e. the same as would be immersed at infinite time; the length immersed at the commencement of the analysis be L^^ and the length immersed at time t be L^. Then, at the commencement of the analysis: W = V(f>{h,Q) + L^ap^
(7.22)
At time / = oo (clear liquid in the container) W = Vpj+Lapf
(7.23)
During the analysis, at time t W = V(t){h,j) + L,ap2
(7.24)
where Fis the volume of the hydrometer bulb, a the cross-sectional area of the stem and hj the depth of the hydrometer bulk at time t. Since the density of the suspension around the stem (pj, pj^ p^ varies negligibly compared with the variation in L equation (7.21) can be written: L,-L
^ (/>{h,,t)-py
LQ-L
(/>{h^,0)-pf
= ^^^LZJ^
(7.25)
WQ-W
where w is the specific gravity marked on the hydrometer stem. If the suspension is made up of W gram of powder making up 1 L of suspension, equation (7.25) can be written:
Gravitational sedimentation methods
381
Fig. 7.13 Depth of immersion using a hydrometer.
^
1000 A ( A - / ^ / )
W
(7.26)
Ps-Pf
where p^ is the density of the suspension at time /. An equivalent formula fpr powders that are present as slurries in water removes the necessity for drying out the slurry. A specific gravity bottle is filled with water and weighed; the water is replaced with the slurry under test and the bottle is re-weighed; the difference in weights being /SW. The sample is then taken out of the bottle and used for the analysis. The equivalent formula is:
mQ(p,-pf)
^w
(7.27)
7.8,3 Depth of immersion With the hydrometer technique, both density and depth of immersion vary with each reading. If the temperature is maintained constant at the hydrometer calibration temperature the density may be read directly from the hydrometer stem otherwise a correction needs to be applied [52].
382 Powder sampling and particle size determination
150-170
did
Fig. 7.14 Hydrometer (Calibration in gml"^ at20°C. All dimensions in mm). It is clear that a hydrometer with a long bulb does not measure density at a point; it only measures the average density of the suspension displaced by the hydrometer. The difficulty lies in determining the point of reference below the surface to which this density refers, for when the hydrometer is placed in the suspension the liquid level rises in the container, thus giving a false reference point (Fig. 7.13).
Gravitational sedimentation methods
383
If the cross-sectional area of the container is A, the depth to be used in Stokes equation, from geometrical considerations, is [53]: L = L^+-\ L2--\ 2V Aj
(7.28)
Several workers, who claim that corrections have to be applied for the density gradient about the bulb, and the displacement of suspension by the stem, have challenged this simple formula. Johnson [54] for example, gives the sedimentation depth, in cm, as:
^ 2
-0.5
(7.29)
7.8.4 Experimental procedure The changes in density of a sedimenting suspension may be followed with a hydrometer (Figure 7.14), a method still used in the ceramic industry. In order to achieve sufficient accuracy in the specific gravity readings, it is necessary to use a concentration of at least 40 g L'^, which is well into the hindered settling region. The only justification for this that has been advanced is that the method gives reproducible results. Most hydrometers are calibrated to be read at the bottom of the meniscus and this is usually not possible when the hydrometer is immersed in a suspension. The readings are, therefore, taken at the top of the meniscus and an experimentally determined correction, which is usually of the order of 0.003 g ml"^ applied. It is usual to disregard calibration errors although these may be substantial. Good quality hydrometers are usually guaranteed to ±0.0005 g ml"^ which corresponds to an error of around ±1.5% under normal operating conditions. Johnson [54] recognized this error and suggested that it should be determined at several points by calibration in a series of dilute suspensions of common salt. A correction to meniscus reading error and density should also be applied if a wetting agent is used. The resolution, see Section 7.2, is particularly poor for the hydrometer method of size analysis where the height of the measurement zone is of the same magnitude as the depth of immersion in the suspension. Although the hydrometer cannot be recommended as an absolute instrument, it is useful for control work with wide size range continuous distributions. It is of little use with discontinuous size distributions since these give sharp
384 Powder sampling and particle size determination
boundaries in the settling suspension, which lead to peculiar results. The method for carrying out a hydrometer analysis is given in BS 1377 [55]. 7.8.5 Divers Divers overcome many of the objections associated with the hydrometer technique. These miniature hydrometers were developed by Berg [56] for use with both gravitational and centrifugal sedimentation, but have never been widely used. Basically, divers are small objects of known density that are immersed in the suspension so that they find their density level. Berg's divers for example, were hollow glass containers that contained mercury to give the desired density. The density was then adjusted to the desired value by etching with hydrofluoric acid. Various modified divers were later developed, the final ones, by Kaye and James [57], being metal coated polythene spheres which were located with search coils. 7.9 Homogeneous cumulative gravitational sedimentation 7.9.7 Introduction The principle of this method is the determination of the rate at which particles settle out of a homogeneous suspension. This may be done by extracting the sediment and weighing it; allowing the sediment to fall on to a balance pan or determining the weight of powder still in suspension by using a manometer or pressure transducer. One problem associated with this technique is that the sediment consists both of oversize (greater than Stokes diameter) and undersize particles so that the sedimentation curve of amount settled (P) against time (0 has to be differentiated to yield the weight {W) larger than Stokes diameter. Several balance systems, based on this equation, have been described. 7.9.2 Theory The theory given below was developed by Oden [58] and modified by Coutts and Crowthers [59] and Bostock [60]. Consider a distribution of the form:
W=
\ d=dst
f{d)dd
Gravitational sedimentation methods
385
where W is the mass percentage having a diameter greater than Stokes diameter. The weight percentage P, which has settled out in time t, is made up of two parts: One consists of all the particles with a falling speed equal or greater than u^^, the other consists of particles with a smaller falling speed which have settled out because they started off at some intermediate position in the fluid column (Figure 7.2). If the falling velocity of one of these particles is u, the fraction of particles of this size that will have fallen out at time t is ut/h, where h is the height of the suspension. Hence: "max
P= ^f{d)dd+
'^V
ljfid}dd
(7.30)
Differentiating with respect to time and multiplying by /:
'^'Ijm^
(7.31,
I.e. dP P = W-\-t— dt
(7.32)
Since P and / are known, it is possible to determine fusing this equation. It is preferable, however to use the equation in the following form [61]. dP P^W + -^^^ (7.33) din/ Several methods of applying this equation have been suggested. The most obvious is to tabulate / and P and hence derive dP, dt and finally W. Alternatively, P may be plotted against t and tangents drawn. A tangent drawn at point {P^^t^ will intercept the abscissa at W^, the weight percentage oversize dr. Another method is to tabulate P against t at times such that the ratio of {t/dt) remains constant, i.e. at time intervals in a geometric progression; a simple expression relating W and P then develops [62].
386 Powder sampling and particle size determination
Weighing mechanism Thermal jacket \::::1
Adjustable balance clamp \ mmttrntrnM RMiMniiRRlMii
L-J
(a)
Pressure equalizing tube Sedimentatioii column Balance pan
I
(b)
Fig. 7.15 (a) Sedimentation balance with pan in the suspension, (b) Sedimentation balance with pan in clear liquid (Leschonski modification of the Sartorius balance) Many powders have a wide size distribution and, in such cases, the time axis becomes cramped at the lower end or unduly extended at the upper end; in such cases equation (7.33) should be applied. Evaluation proceeds from a plot of P against In t; tangents are drawn every half-unit of In t\ the point where the tangent cuts the ordinate line one In t unit less than the value at which it is tangential gives the weight percentage oversize W at that value [63]. 7.9.3 Sedimentation balances In the Gallenkamp balance [64,65] the pan is placed below a sedimentation chamber with an open bottom and the whole assembly is placed in a second chamber filled with sedimentation liquid so that all the powder falls on to the pan. The weight settled is determined from the deflection of a torsion wire, and either, the run continues until all the powder has settled out of suspension, or a second experiment is carried out to determine the supernatant fraction. Problems arise during the charging operation with leakage into the clear water reservoir and particle adhesion to the premixing tube.
Gravitational sedimentation methods
387
In the Sartorius balance [66-68] the pan is suspended in the suspending liquid and a correction has to be applied for the particles which fall between the rim of the pan and the sedimentation vessel. In this instrument, when 2 mg of sediment has deposited, electronic circuitry activates a stepby-step motor which twists a torsion wire to bring the beam back to its original position. A pen records each step on a chart. The manufacturers suggest that about 8% of the powder does not settle on the pan. Leschonski [69] and Leschonski and Alex [70] reported losses of between 10% and 35%, depending on the fineness of the powder; the difference was attributed to the pumping action of the pan as it re-balances. Leschonski modified the instrument (Figure 7.15) by placing the pan at the bottom of a sedimenting column surrounded by a second column of clear liquid so that all the powder settled on to the pan. This eliminated powder losses and resulted in more accurate analyses [71]. The manufacturers of the Cahn micro-balance make available an accessory to convert it into a sedimentation balance [72]. The balance pan is immediately below the sedimentation cylinder in order to eliminate convection currents. Shimadzu also make a beam balance [73] that operates using a simple compensating system that is prone to considerable error. Yodshida et.al [74] describe an improved sedimentation balance. They compared the results using this balance with those from microscope counting, using three kinds of standard reference beads, and found good agreement. Fukui et al investigated data reduction and sedimentation distance for sedimentation balances [75}. 7.9.4 Sedimentation columns Sedimentation columns (ICI, BCURA) have also been described in which the sediment is extracted, dried and weighed. A full description of these and other sedimentation columns may be found in [1] 7.10 Line-start incremental gravitational sedimentation 7.70.7 Photosedimentation The Horiba cuvette photo(centri)fuge has been operated in this mode [76] but is not recommended since it is very difficult to make up a stable twolayer system in a cuvette.
388 Powder sampling and particle size determination
7.11 Line-start cumulative gravitational sedimentation 7.11.1 Introduction If the powder is initially concentrated in a thin layer floating on the top of a suspending fluid, the size distribution may be determined by plotting the fractional weight settled against the free falling diameter. 7.11.2 Methods Marshall [77] was the first to use this principle. Eadie and Payne [78] developed the Micromerograph, the only method in which the suspending fluid is air. Brezina [79,80] developed a similar water based system, the Granumeter, which operated in the sieve size range, and was intended as a replacement for sieve analyses. The Werner and Travis methods [81,82] also operate on the layer principle but their methods have found little favor due to the basic instability of the system; a dense liquid on top of a less dense liquid being responsible for a phenomenon known as streaming in which the suspension settles en masse in the form of pockets of particles which fall rapidly through the clear liquid leaving a tail of particles behind. Whitby [83] eliminated this fault by using a clear liquid with a density greater than that of the suspension. He also extended the size range covered by using centrifugal settling for the finer fraction. The apparatus enjoyed wide commercial success as the (Mines Safety Appliances) MSA Particle Size Analyzer although it is less widely used today [84]. The MSA analyzer can be operated in the gravitational mode, although it is more usually used in the centrifugal mode. Several papers have been published on applications of this equipment. The line-start technique has also been used to fractionate UO3 particles by measuring the radioactivity at the bottom of a tube, the settled powder being washed out at regular intervals without disturbing the sediment [85]. References 1 Allen, T. (1990), Particle Size Measurement, Chapman & Hall, 4th ed., 360, 388 2 British Standard 3406, Determination of particle size distribution, Part 2 Gravitational methods, 360 3 British Standard 3406, Determination of particle size distribution, Part 6, Centrifugal methods, 360 4 DIN 66111, Particle Size Analysis, 360
Gravitational sedimentation methods
389
5 DIN 66115 Particle Size Analysis, 360 6 NFX 11-681 Test Methods for Particle Size Analysis-Particle Size Analysis by Gravity Sedimentation in a Liquid Medium, 360 7 NF 11-683 Test Methods for Particle Size-Analysis-Particle Size by Variable Height Gravity Sedimentation in a Liquid-Method Using X-ray Adsorption Measurements, 360 8 ISO/WD 13317-1 Determination of Particle Size Distribution by Gravitational Liquid Sedimentation Methods- Part 1 (1996) General Principles and Guidelines, 360 9 ISO/WD 13317-3 Determination of Particle Size Distribution by Gravitational Liquid Sedimentation Methods- Part 3 (1996) The X-ray Gravitational Technique, 360 10 JIS Z8820 General Rules for the Determination of Particle Size distribution by Sedimentation in Liquid, 360 11 ASTM C958 (1997), Standard Test Method for Particle Size-Analysis of Alumina or Quartz by X-ray Monitoring of Gravity Sedimentation, 360 12 ASTM B761 (1998), Test Method for Particle Size-Analysis of Refractory Metals and their Compounds by X-ray Monitoring of Gravity Sedimentation, 360 13 Jillavenkatesa, A, Dapkunas, S.J. and Lum, L-S, H. (2001), Particle Size Characterization, NIST Sp. Publ. 960-1, 360 14 Robinson, G. W. (1922), J. Agr. ScL, (12)3, 305-321, 365 15 Andreasen, A.H.M. (1928), Kolloid Beith, 27, 405, 365 16 Andreasen, A. H.M. and Lundberg, J. J. V. (1953), Ber Dt. Keram. Ges., 11(5), 312-323,5^5 17 Allen, T. (1969), Powder Technology, 2(3), 132-140, 365 18 Crawley, G., Coumil, M. and Benedetto, D.D. (1997), Powder Technol 91(3), 197-208,577 19 Vouk, V. (1948), Ph D thesis, London University, 372, 373 20 Jarrett, B.A. and Heywood, H. (1954), Br. J. Appl Phys., Suppl. No 3, S21, 373 21 Morgan, V.T. (1954), Symp. Powder Metall, Iron and Steel Inst., Preprint Group 1,33-38, 575 22 Lewis P.C. and Lothian, G.F. (1954), Br. J. Appl. Phys., Suppl. No 3, S571, 373 23 Rose, H.E. and Lloyd, H.B. (1946), J. Soc. Chem. Ind, 65, 52, 373 24 Allen, T. (1968), Powder Technol., 2, 132-140,, 373 25 Allen, T. (1968), Powder Technol., 2, 141-153, 373 26 Weichert, R. (1981), Proc. Particle Size Analysis Conference, 303-310, ed. Allen, T. and Stanley-Wood, N.G. publ.John Wiley, 373 27 Caron, P., Fancompre, B., Membrey, F. and Foissy, A. (1996), Powder Technol., H9,9\-\00, 373 28 Ma, X. and Wan, M. (1997), Part. Part. Syst. Charact., 14, 267-271, 373 29 Staudinger, G., Hangl, M. and Peschtl, P. (1986), Proa Partec Nurnberg, publ. NiimbergMesse, 374
390 Powder sampling and particle size determination
30 Staudinger, G., Hangl, M. and Peschtl, P. (1986), Part. Part. Syst. Charact., 3, 158-162,57^ 31 Marquard, H. von (1998), Aufbereit Tech., 39(9), 461-466, Eng., 374 32 Hubner, T., Will, S. and Leipertz, A. (1998), Proc. 7'^ European Symp. Particle Charact., Partec '98, Numberg, 243-253, 375 33 Hubner, T., Will, S. and Leipertz, A. (1999), Part. Part. Syst. Charact., 16, 8591,575 34 Hubner, T., Will, S. and Leipertz, A. (2001), Part. Part. Syst. Charact., 18, 7078, 375 35 Breitenstein, M, Krauter, U. and Riebel, U. (1999), Part. Part. Syst. Charact., 16, 249-256,, 375 36 Breitenstein, M, Riebel, U. and Shen, J. (2001), Part. Part. Syst. Charact., 18, 134-141,575 37 Brown, J.F. and Skrebowski, J.N. (1954), Br. J. Appl. Phys., Suppl No 3, S27, 376 38 Conlin, S.G. et. al. (1967), J. Scient. Instrum., 44, 606-610, 376 39 Nonhebel, G. (1964), ed. Gas Purification Processes, Newnes, 376 40 Kalshoven, J. (1965), British Patent, BP 1 158 338, 376 41 Kalshoven, J.(1967), Conf. Proc. Particle Size Analysis, Soc. Analyt. Chem., London, 376 42 Oliver, J.P., Hicken, G.K. and Orr, C. (1969), US Patent 3,449,567, 377 43 Hendrix, W.P. and Orr, C. (1970), Proc. Conf. Particle Size Analysis, Soc. Analyt. Chem., London, pp 133-146, ed. M.J. Groves and J.L. Wyatt-Sargent, 377 44 Borothy, J. (1975), Chimia, 29(5), 240-242, 378 45 Allen, T. (1970), Br. Patent, 1764/70 3, 379 46 Allen, T. and Svarovsky, L. (1970), J. Phys E, 3, 458-460, 379 47 Allen, T. and Svarovsky, L. (1970), Proc. Particle Size Analysis Conf. Bradford, Publ. Soc. Anal. Chem., 379 48 Cho, H., Yildirim, K. and Austin, L.G. (1998), Powder TechnoL, 95(2), 109117, 57P 49 Svarovsky, L. (1972), PhD thesis, Univ. Bradford, Yorkshire, UK., 379 50 Svarovsky, L. and Allen, T. (1973), Paper presented at Heywood Memorial Symp., Univ. Loughborough, UK., 379 51 Allen, T. and Davies, R. (1989), 4th European Symp. Particle Charact., Numberg, Germany, Preprints 1, publ. NumbergMesse 17-46, 379 52 Stairmand, C. (1947), Symp. Particle Size Analysis, J. Inst. Chem. Engrs., 25, 110,5^2 53 Edwald, P. (1942), Ind Engg Chem., Analyt. edn., 14, 66, 383 54 Johnson, R. (1956), Trans. Ceram. Soc, 55, 237, 384 55 British Standard 1377, The Hydrometer Method of Particle Size Measurement, 384 56 Berg, S. (1940), Ingen Vidensk., Skr. B., No 2, Phys. Suppl. No 3, S27, 385 57 Kaye, B.H. and James, G.W. (1962), Br. J. Appl. Phys., 13, 415, 385 58 Oden, S. (1916), KolloidZ., 18, 33-47, 385
Gravitational sedimentation methods
391
59 Courts, J. and Crowthers, E. M. (1925), Trans. Faraday Soc, 21, 374, 385 60 Bostock, W. (1952), JScient. Instrum., 29, 209, 385 61 Gaudin, A. M., Schumann, R. and Schlechter, A. W. (1942), 1 Phys, Chem., 46, 903, 386 62 Kim, S. C , Schlotzer, G. and Palik, E. S. (1967), Powder Technol, 1, 54-55, 386 63 Bostock, W. (1952), JScient. Instrum., 29, 209, 386 64 Bostock, W. (1952), JScient. Instrum., 29, 209, 387 65 Cohen, L. (1952), Instrum. Pract., 13, 1036, 387 66 Gerstenberg, H. (1957), Chem. Eng. Tech., 8, 589, 387 67 Bachman, D. (1959), Dechema Monograph, 31, 23-51, 387 68 Gerstenberg, H. (1959), Dechema Monograph, 31, 52-60, 387 69 Leschonski, K. (1962), Staub, 22, 475-486, 388 70 Leschonski, K. and Alex, W. (1970), Proc. Int. Symp. Particle Size, Bradford, 236-254, publ. Soc Anal. Chem. ed Groves and Wyatt-Sargent, 388 71 Pretorius, S.T. and Mandersloot, W.G.B. (1967), Powder Technol., 1, 23-27, 388 72 Kaye, B.H. and Davies, R. (1970), Proc. Conf. Particle Size Analysis, Bradford, 207-222, publ. Soc Anal. Chem. ed Groves, M. W. G. and WyattSargent, J., 388 73 Suito, E. and Arakawa, M. (1950), Bull. J Chem. Res.,Kyota University, 23, 7, 388 74 Yoshida, H., Masuda, H., Futui, K, and Tokunaga, Y. (2001), Adv, Powder Technol., 12(1), 79-94, 388 75 Fukui, K., Yoshida, H., Shiba, M. and Tokunaga, Y. (2000), J Chem. Japan, 33(3), 393-399, 388 76 Hofftnan, R.L. (1991), J Colloid Interf. ScL, 143, 232, 388 11 Marshall, C.E. (1930), Proc. Royal Soc, A126, 427, 389 78 Eadie, F.A. and Payne, R.E. (1954), Iron Age, 174, 99, 389 79 Brezina, J.J. (1969), Sediment. Petrol, 16, 27-31, 389 80 Brezina, J. (1970), Proc. Conf. Particle Size Analysis Bradford, publ. Society Anal. Chem. ed. Groves, M.W.G. and Wyatt-Sargent, J.L, 389 81 Werner, D. (1925), Trans. Farad Soc, 21, 381, i^P 82 Travis, P.M. (\940),ASTMBull., 29, 102, 389 83 Whitby, K.T. (1955), Heat. Pip. Air Cond, Jan., Part 1, 231, June, Part 2, 139, 389 84 Whitby, K.T., Algren, A.B. and Annis, J.C. (1958), ASTM Spec Publ. 234, 111,389 85 Imris, P. and Landsperky, H. (1956), Silikaty, 1956, 9(4), 327, 389
8
Centrifugal sedimentation methods of particle size determination 8J Introduction Gravitational sedimentation particle size measurement techniques are of limited value for particles smaller than about a micron due to the long settling times required. In addition, gravitational sedimentation devices generate inaccurate data due to the effect of convection, thermal diffusion and Brownian motion. Centrifuging the suspension in order to speed up the settling process reduces these errors. As with gravitational sedimentation, various options are available (Table 8.1). These may be categorized as: • • • •
variable time (/ varies, all other parameters remain constant); variable measurement radius (r varies, all other parameters remain constant); variable surface radius (S varies, all other parameters remain constant) combinations of these variables e.g. variable time and height {/ and (r/S) var}', all other parameters remain constant}.
Centrifugal techniques may be classified as incremental or cumulative, homogeneous or line-start. Incremental, line-start techniques are restricted to photocentrifuges and the attenuation has to be corrected for the breakdown in the laws of geometric optics unless the data are being used solely for comparison purposes. This correction can be considerable for powders having a wide size range, for example a 0.10 |nm particle may cut off less than a tenth of the light that one would expect from its geometric size whereas a 1 |Lim particle may cut off more than one would expect.
Centrifugal sedimentation methods 393 Table 8.1 Commercial sedimentation particle size analyzers Homogeneous, incremental centrifugal sedimentation Simcar centrifuge Ladal pipette centrifuge I .adal x-ray centrifuge Brookhaven scanning x-ray centrifuge Brookhaven Bl-DCP, disc photocentrifuge Kaye disc photocentrifuge Coulter photofuge Technord photocentrifuge Horiba cuvette photocentrifuges LUM Lumifuge 114 Seishin cuvette photocentrifuge Shimadzu cuvette photocentrifuge
Homogeneous, cumulative, centrifugal sedimentation Alpine centrifuge Hosokaw a M i kropu 1 Sedimentputer Line-start, incremental, centrifugal sedimentation Joyce-Loebl disc photocentrifuge Brookhaven BI-DCP, disc photocentrifuge CPS disc photocentrifuge Line-start, cumulative, centrifugal sedimentation MSA analyzer
A further problem arises, due to the presence of a range of particle sizes in the light beam at any one time, so that, even excluding extinction coefficient problems, the response may not be proportional to the projected area of the particles, in homogeneous, incremental, centrifugal techniques, (Figure 8.1) matters are also more complex than for homogeneous, incremental, gravitational sedimentation. The particles move in radial paths, hence the number of particles smaller than Stokes diameter entering the measurement zone is less than the number leaving, so that the measured concentration of these particles is smaller than their original concentration. In this presentation, some of the methods for centrifugal sedimentation particle size analysis in current use are described. Although operating procedures are not covered here, it is stressed that two factors, more than anything else, lead to incorrect analyses. The first is incorrect sampling, since analyses are carried out on from a tenth of a gram up to a few grams and these samples must be representative of the bulk for the analyses to be meaningful. The second is dispersion; it has been said rightly that the most important factor in obtaining accurate sedimentation data is dispersion the second most important factor is dispersion and the third is also dispersion!
394 Powder sampling and particle size determination
Fig. 8.1 Homogeneous, incremental, centrifugal sedimentation (the radial dilution effect). 8.2 Stokes* equation for centrifugal sedimentation S. 2. / Genera! theory A particle settling in a centrifugal field is acted upon by a drag force and a centrifugal force. The force balance in the laminar flow region is given by:
-,(Ps-
Pf yi ^^^ = l(Ps-
PfVWr
- ^ncl,,^
(8.1)
where: r dr/dt Pg,pr // d^, rfj a)
= radial distance from the axis of the centrifuge to the particle; = outward velocity of the particle; = density of the particle and suspension medium respectively; ^ coefficient of viscosity of the medium; = volume diameter of the particle; =^ drag diameter of the particle; =•'• speed of rotation of the centrifuge in rad.s ^
At the terminal velocity, the outward radial acceleration, d^r/dt- ^ 0 so that this equation becomes:
Centrifugal sedimentation methods 395
(ps~Pf)dWr dr — ^u,.^ d/ ' I8/7
(8.2)
Thus, the settling velocity is not constant as in gravitational sedimentation but increases with increasing radius. Comparing with Stokes equation for gravitational settling (settling velocity under gravity ^ 11^^: CO r u^. -^ - — u <^f -Gii^f K
(8.3)
G, the separation factor, is a measure of the increased rate of settling in a centrifugal field. Rewriting equation (8.2) in integral form:
\Ar ^[ps-Pf)dW V At ] r~ I877 J' . r _(ps~Pf)dW^ S I8/7
<,,= ( ^ ' ^ ^
\(p,'Pf)
(8.4)
CO t
I is the time for a particle of Stokes diameterrf^yto settle from the surface of the fill liquid at radius Sio measurement radius r which, for cumulative techniques, is equal to R, the distance from the axis to the inside radius of the centrifuge. 8.3 Homogeneous, incremental, centrifugal sedimentation 8.3.1 General theory The largest particle, of Stokes diameter d^^, present in the measurement zone at time t and radius r will have originated from the surface at radius S.
396 Powder sampling and particle size determination From equation (8.4) the following relationship holds:
~ " ^XP
I8/7
(8.5)
xp(^4/) exp where
k -
I8/7
Particles in the measurement zone of size d^ will have originated from radius r^ where r > r^ >S and: — = exp[fa/;^/|
(8.6)
The particles originally at radius r^, in an annular element of thickness Ar^, move in diverging (radial) paths and at radius r occupy an annular element of thickness Ar. There will be a fall in the concentration of particles of this size in the measurement zone therefore, since the same number of particles will occupv a greater volume. The fractional increase in volume is given by: rAr r,Ar,
r
(8.7)
\ny
since, from equation (8.6), Ar/Ar^ = r/r^. For a polydisperse system, with a weight fraction ofJ{d)dd in the size range dto d + dd, the fractional concentration dQ of this weight fraction at r is given by: dO
f/V
f(d)dd
Centrifugal sedimentation methods 397 Hence, the mass undersize J^^ is:
Q{ds,)= J p l fid)dd
(8.8)
Combining witii equation (8.6) gives:
Q(ds,)=
j cxp(~2kdft)f(d)dd
(8.9)
cl,=0
Substituting for k from equation (8.5):
Q{d„)= j exp
-9
^ d.^ ^^ r.^\ In
\^St J
d,=0
S)
f(d)dd
(8.10)
Various solutions to these equations have been proposed: • • •
In the variable time method the concentration is measured as a function of/' ^ ai^t and all other variables are kept constant. In the variable height method the concentration is measured as a function of {r/S) and all other variables are kept constant. For scanning systems (variable time and height) both co^t and r vary and in pipette withdrawal systems co^t and S vary.
8.4 Variable time method (r and S constant, t variable) 8.4.1 General theory Differentiation of equation (8.9) gives:
Ad
exp(~2kdh)f{d)
where f(d) =
dF(d) dd
(8.11)
398 Powder sampling and particle size determination
The boundary conditions are that Q ^1 when / =^ 0 for all r^, i.e. initially the concentration is the same throughout the suspension; (> == 0 when r- ^ S for / > 0, i.e. the surface concentration falls to zero on start-up; with the additional condition that F{d) '-^ 0 when d^^ 0. Thus: Qsr ' • « s y ) - jexp(2fo/^/)dO
&
,
.2
dQ
(8.12)
0
d^f is the diameter of the particle that settles from the surface, radius .S', to the measurement radius r in time /. The expression was first developed by Berg [1] and later by Kamack [2]. Berg solved equation (8.12) graphically by plotting (r/S) against Q and determining the area under the curve, deriving the following formulae:
J F(d)dd - c, -f ^" " 5c, - 4F ^ --; 1/••(c/jdc/ - c\ f 1 +2x^
for x < -
for c^ < 0.15
(8,13)
(8.14)
where r =5'+jc and c^ is the concentration at radius x at the time required for a particle of diameter d^ to fall from the surface to measurement radius x. These equations are applicable for powders having a wide size range, losing accuracy for monodisperse powders where volume concentration changes rapidly with changing r. Equation (8.13) is used for the calculation of the smallest F(d), that is the smallest c,, and equation (8.14) for higher values of F(d). F{d) ^ 0 when J = 0, hence F(d/2) may be estimated by joining F{d) to 0 for a small value of d since most functions are linear towards the origin. The F(d) curve is then built up step-by-step. Berg determined concentration using pipette withdrawal or small divers.
Centrifugal sedimentation methods 399 Kamack offered the following solution to equation (8.12). If g is plotted as a iiinction of v^ - {rJSy with f^ o9^t as parameter, a family of curves is obtained whose shape depends on the particle size distribution function. The boundaiy conditions are that Q = 1 when / ' - 0 for all r^ (i.e. the suspension is initially homogeneous) and Q^^ 0 for r- = S when t'>0 (i.e. the surface region is particle free as soon as the centrifuge bowl spins). Hence all the curves, except for t'>0, pass through the point Q^O, r- - S, and they will all be asymptotic to the line /' =- 0, which has the equation ig ^ 1. Furthermore, from equation (8.12), the areas under the curves are each equal to F{d^f), Let Q\ be the smallest experimentally determined concentration so that h-^h^^r ^'^d let Q be determined at a fixed sampling distance r for various values of t. Then one point is known on each curve in addition to the common point j / - 1 , ^ = 0. Such a set of points is illustrated by the black circles in Figure 8.2. To each point corresponds a known value of d^^ obtained from equation (8.4). Further, the area included between each curve, the concentration axis and the ordinates g = 0 and g<^^^ is equal to F(^^Y). Thus F{d^) is approximated by the trapezoidal rule for, first of all.
Gi-2
QiiQz
Q2.3
Q3
Fig. 8.2 Theoretical diagram for homogeneous, incremental centritiige technique (variable time method).
400 Powder sampling and particle size determination y)Q^. Now considering the curve for /2 ? a point can be found on it corresponding to d^, i.e. a point such that the area under the curve up to this point is F^, which is now known. If the ordinate at this point is called y^ j ^"^ ^^^ abscissa F^ ^ then, by equation (8.4):
FY=^Q.5{\+
dx-
97 In J, 2
y.^pf)
and
9ri In V
d-, -
(O^tn
^Ps-Pf)
(8.15)
CO t-,
so that >'| 2 - y^'^^ '"^^^ ^^ where y - {r/S)^ Equating areas: (8.16) SO both v'l 2 ^Jid Q^ 2 ^^^ known. Applying the trapezoidal rule: (8.17) Substituting for g , 2 from equation (8.16):
^2=i(.>' + .Vl,2)(?2 +
y+y'u 1 + V| 2
(8.18)
Proceeding in a like manner gives the general formulae: Pn -Fr,-\ =\(y K-\ -K~2
+ yn-\,r,)(Qn
(8.19)
-Qn-u)
^\\yn~\.n^yn-2ji){Qn-\.n
~Qn2.n)
(8.20)
and so on. By considering this series of equations with successive elimination of the Q functions, there is obtained a general equation in recursive form:
Centrifugal sedimentation methods 401
/-I
y+yi-^.i
y+y,-\.i
y}^u^y,i,i
y}.i^y,hu
{d,ld,f
where y ^ ^ ~y
'"it
(8.21)
(8.22)
9ri[ny
(8.23)
i= 1,2,3,....,m; ; ; Q I = 1 Equations (8.21) are a set of linear equations which express the desired values of F^ explicitly in terms of the measured values of Q^. The coefficients of the equations depend upon the values of d^ (corresponding to the value of t\) at which the concentrations are measured; more exactly, the coefficients depend on the ratios of the values of d^y as shown in equation (8.22). Consequently, if the values of d^ are chosen in a geometric sequence when carrying out a particle size analyses, the coefficients are considerably easier to calculate and the equations themselves are also simplified. A ratio of v 2 is recommended. The coetHcients depend also on the values of y that is, the dimensions of the centrifuge bowl employed. The modified form of equation (8.21) for experimental points in a 2:1 progression in time giving a ^2:1 progression in diameter is: i-\
M
ys
yB
F,/=-7z?a+2
(8.24)
l(-./) r>2
1,2 +. y^
where A = 1 forj = \, A-y^ W h e n / - 1, F i = i ( l + j ; ) 0 |
y'
+A
for j V 1 and j / ^ = y + y^
402 Powder sampling and particle size determination
1 A
When / = 2,
^ 2 y-^y'
y + y'
Qi^
yA J^ yl
y^y yl + ]
And so on. In a later paper Kaniack [3] replaced his earlier approximate solution with an exact solution which is equally practical to use and which has more accuracy and generality. Starting with equation (8.11) he derived the exact solution: 00
/^((i^>,) = exp - 2 In
fr\
Q{ds,) - ^h{x)Q{ds, exp(-x))dx
(8.25)
v^>y
in which h{x) is a function which depends on the ratio r/S and=^ \n(d/d^^. For full details readers should refer to the original paper. It is preferable, wlien using the Kamack equation, to smooth out the experimental data. This can be done manually by plotting Q against d^^on log-probability paper or using a smoothing equation during computer data collection. Alternative solutions to equation (8.9) are available if the shape of the distribution curve is assumed. Svarovsky and Friedova [4] assumed a fit by the 3-parameter equation of Harris [5] the parameters being found by means of a curve-fitting technique applied to the measured concentrations. This method is applicable to variable height, variable time and variable height and time. The main disadvantage of this method is that it is unsuitable for multi-modal distributions. Svarovsky and Svarovska [6] derived an alternative version of equation (8.25) and later [7,8] developed a new data analyzer for data evaluation. Their equation was:
F(7):
C(T)-
f//(Z)c(7e^)dZ
(8.26)
where Z = ln(//7); / is time as a variable, T is the time at which a measurement is made. a = 2\n{r/S)
Centrifugal sedimentation methods 403
H{z) = K{Z)- \K{z-r)H{rw 0
ForZ>0 K(Z) - aexp(a)[- Z ~ aexp(-Z)] and K{Z) - 0 for Z< 0 Function H{Z) is a convolution integral which is readily computed digitally for different values of a and then used in equation (8.26) for evaluation. Alex [9] suggested iteration by Neumann's series, Langarm approximations or substitution of polynomial functions for the distribution curves in order to solve the problem. Lloyd et. al [10] suggested a complicated solution involving high order differentials; they also speculated on the 'on-line' evaluation of experimental data. Truly on-line evaluation is, however, impossible and only a delayed quasi-on-line evaluation can be performed [11]. 8A.2 The Simcar pipette disc centrifuge (r constant, S assumed constant, t variahle) The Simcar centrifuge was developed by Slater and Cohen [12] to conform with Kamack's theory and was, in essence, a centrifugal version of the gravitational pipette. One problem with this instrument was the amount of suspension required so that the liquid level did not alter appreciably during an analysis (about 2.5 liter of liquid containing about 5 g of powder). The amount removed at each withdrawal was about 40 ml but variable, hence the error due to assuming a constant liquid level, S, increased as more samples were withdrawn. Since no correction was available for the fall in surface with each extraction, it was inadvisable to withdraw more than four samples in each analysis. The analysis was carried out in duplicate to give eight points on the distribution curve so that a single evaluation from about 5 |Lim to 0.2 |am took up to a full day. Concentrations were determined by drying and weighing. Using the modified equation to take into account the changing liquid level and a more reproducible withdrawal volume would allow more data points to be taken in a single run.
404 Powder sampling and particle size determination 8.4.3 Worked example (a) Deterniination of Ffactors for a cenirifiige without scanning Centrifuge dimensions: /? ==^ 8 cm, h ^- 0.70 cm, r -1 cm. Hence, for a fill volume of 100 cm\ S == 4.304 cm. Applying equations (8.22) and (8.23) gives the following>^^ values for a 2:1 progression in time: Table 8.2 yij values for a centrifuge operating under fixed radii conditions y
2.645
y\2
>2,3
y\A
^4,5
X5,6
JV.,?
yi^
1.626
1.275
1.129
1.063
1.031
1.015
1.008
i.e. V, 2 = y^''^, y2 3 ^ y^^^ ^"d so on. From these yij values the following F^ values are determined: Table 8.3 F/values for a centrifuge operating under fixed radii conditions h\ ^1.823g>, / s =2.136{?2-0-626F, - 1.823(?2-1I4I^, Fy =2.136^3-1.008^2-0-200^1 ^4 = 2.136^4-1.008^3-0.174(;2-0.0.023(;, ^5 =2.136(^5-1.008^4-0.174^3+0.021 Q2^^S)A(>Q^ Ff, = 2A36Qg-\ .008^5-0.174^4+0.021 Qn^+O.Ol 1Q2+OM1 g, F, = 2.136^-1.008(^,_,-0.174(^,_2+0.021 g,„3+0.021 ^,,4+ Particle size is determined using equation (8.23) and measured concentration is converted to mass undersize using equation (8.21). The results are presented in Table (8.4). Material; quartz p^ = 2650 kg m~-^ pj= 1000 kg m^^ N= 1500 rpm ( « = 5Qn rad s ') 7= 0.001 Pa s , 9 X 0.001 X n(7/
f
^(2650-1000)x(507t)^/,
Centrifugal sedimentation methods 405
Table 8.4 Mass percentage iindersize determination for homogeneous, incremental centrifuge technique (variable time method)
I
1 2 3 4 5 6 7 8 9
Time (7) (min)
Size iixm)
Measured (%) cone.
dst 0.118 0.167 0.237 0.335 0.473 0.669 0.947 1.339 1.893
(0
256 128 64 32 16 8 4 2 1
Mass % Iindersize ' (Z^)
!
6.9 18.7 34.5 65.5 88.4 97.3 99.4 100.0 100.0
256 128 64 32 16 78.2 87.9 93.6 96.7
Table 8.5
Qus
e,.6 QiJ
Q\s
e,,9 QiAO
\Q' . ' • \
C>6,7 ^6,8 ^6,9
66.10 ^6
L^6
5.3 6.1 6.5 6.7 6.8 6.9 6.9 6.9 6.9 3.8 6.9 86,9 91.8 94.4 95.9 78.2 97.3
22,3 QiA Ql.S
Qis Qij
22,8 22,9 22,10 22 ^2
27,8 27,9 27,10 27 ^7
14.2 16.3 17.5 18.1 18.4 18.6 18.7 18.7 10.8 18.7 93.0 96.2 97.8 87.9 99.4
23,4 23.5 23,6 23.7 23,8 23,9 23,10 23 ^^
27.2 30.6 32.4 33.5 34.0 34.1 34.4
28,9 28,10 28
96.6 98.3 93.6 100
^8
21.6 34.5
24,5 24,6 24,7 24,8 24,9 24,10 24
n
29,10 29 ^9
51.9 58.2 61.8 63.6 64.5 65.0 41.7 65.5
98.6 100 100
25.6 25,7 25,8 25,9 25.10 25 ^^5
2,0 ^10
74.2
m3 , 84.7 86.6 i 87.6 62.7 88.4
100 100
406 Powder sampling and particle size determination
8.4.4 The Ladal x-ray disc centrifuge (r constant, S constant, t variable) This is an extension of the Allen and Svarovsky [13] x-ray gravitational sedimentometer. The x-rays are generated by an isotope source and, after passing through the suspension, they are detected by a scintillation counter. The signal from the counter passes to a pre-amplifier, and thence to a ratemeter, and a trace is recorded by a pen recorder. The attenuation of the x-ray beam is proportional to the mass concentration at the measurement radius that has to be converted to the size distribution using the Kamack equation. A size range of about 8:1 is covered in about an hour. 8.4.5 Discussion of the Kamack equation The Kamack treatment builds up the concentration gradients within the centrifuge for each of the measurement times. The Qjj values (Table 8.5) may be determined using equation 8.19 and, in combination with the y^ j values, generate the concentration gradients. In Figure 8.2 the black circles are the measured concentrations at a fixed measurement and surface radius and the white circles give the calculated concentration gradient for / ^ 7. In essence we are building up the concentration gradient between the measurement radius and the surface for each measurement time. For example, at / = 2 the concentration at the measurement radius is 3.8% and, from the first measurement we deduce that the concentration at a radius y, Q\2 "" '^ 5.3%. In figure (8.2) the concentration gradient for / "= 7 is displayed as an example 8.5 Variable time and height method {S constant, both r and / vary) 8.5.1 Stokes diameter determination. The approximation due to Kamack can be modified, for the scanning mode of operation, by replacing the constant {r/S) with the variable (r/S) where r^ is the position of the source and detector at time / i.e. equation (8.4) becomes:
Centrifugal sedimentation methods 407
97 In J,
'Isij
(8.27)
\{Ps'-pfW', C-..2
V. here y- =• (r/S)" and c/<,-, ^ is the largest part icle present at the measurement radius .'•, at time /. 8.5.2 Mass frequency undersize determination. Equation (8.12) becomes: Q,
F{d,,)=
\cxp(2kd^t)iQ
I.e.
(8,28)
dQ 0
'•
J
modifying equation (8.21) to: /--I
2
tS[v,-,i,, + r,,, (
where v,
\
and
yj.^n
>'.M+>Vi..|
/d, \
K^U
/ = 1,2,3,...., w; V Q / ^ 1
(8.29)
8.5.3 DuPont/Brookhaven scanning x-ray disc centrifugal sedimentometer (BI-XDC) (S constant, r variable, t variable) This instrument was developed as a centritiigai version of the Allen and Svarovsky's [13] x-ray gravitational sedimentometer in order to reduce the analysis time and measure down to smaller sizes [14,15], The x-rays are generated by an air cooled low power x-ray tube and, alter passing through the suspension, they are detected by a scintillation counter. The signal is
408 Powder sampling and particle size determination
then processed to generate the size distribution. The attenuation is proportional to the mass concentration at the measurement radius, which has to be converted to the size distribution using the Kamack equation. A size range of about 8:1 is covered in about an hour. This instrument was designed [16] to fill a need for fast, reproducible sedimentation analyses in the sub-micron size range. The heart of the instrument is a hollow, x-ray transparent, disc which, under nomial operating conditions, contains 20 ml of suspension at a concentration of around 0.2% by volume. The centrifuge speed is selectable in the range 750 to 6000 ipm. The default condition is for the source and detector to remain stationary for 1 minute at a radial position of 48.00 mm and then to scan towards the surface. Total run time is normally 8 min. A commercial version is available from Brookhaven as the Bl-XDC (Figure 8.17). The instrument can operate in the gravitational or centrifugal mode and the analyses can be blended to cover a total size range of 0.05 |im to over 100 jiim. A size range of 15:1 is covered in a standard 8 min analysis. Weiner et. al. [17] describe its application to accelerated size analyses down to 10 nm. A computer controlled particle size measurement device with very high resolution was presented by Foerdeneuther [18] based on the Brookhaven optical and x-ray disc centrifuges. 8.5.4 Worked example (a) Delermination of Ffactors for a centrifuge with scanning Hyperbolic scan - The analyzer is scanning upwards, at a vaiying rate, from an initial radius of 7 cm to just below the surface at S =• 4.304 cm so that the following relationship holds: 18.7 r,^ =
,0.24
/ in seconds, /-/ in |im. The hyperbolic scan gives similar resolution at all sizes hence is preferable for centrifugal particle size analysis. S - 4.304 cm 77 - 0.001 Pa s
p^ = 2650 kg m"^ pj-- 1000 kg m"^
N- 1500 rpm (^ - 5071 rad s"^)
Centrifugal sedimentation methods 409 From equation (8.15) y = (7/4.304)2 = 2.645 From equation (8.27):
/is X o.oobTR'irviysoi)"
dSlj
4130x(507r)'x/.
where y,- = (r/S)^ Applying equation (8.29): F, = F, = F3 = F4 = F5 = Ff, = Fj -
1.020^, 1.065^2-0.059^1 1.138^3-0.105F2-0.030F, 1.235e4-0.]55F3-0.061F2-0.017F, !.367e5-0.227F4-0.094F3-0.035F2-0.010F, 1.563^6-0.356F5-0.132F4-O.O51 F3-O.0109F2-0.005F, 1.940g7-0.683F6-0.169F5-O.O57F4--O.O21 F3-O.OO8F2-O.OO2F1
Ihus the F values as given in Table (8.6) and they, values given in Table (8.7) can be determined. Table 8.6 Centrifuge particle size analysis, homogeneous mode with scanning Time
Radius
(0
F(secs.)
r^iixm)
7 6 5 4 3 2 1
60 120 180 240 300 360 420
1 5.927 5.377 5.019 4.757 4.553 4.388
Stokes diameter dfiiym) 1.893 1.086 0.740 0.532 0.384 0.263 0.143
Measured concentration
C>,(%) 95.6 88.6 75.7 55.8 32.5 11.5 0.9
Percentage undersize F,(%) 99.9 97.2 87.0 62.6 35.7 12.2 0.92
410 Powder sampling and particle size determination Applying equation (8.28) to the V/values given in Table 8.5, the following Vj j values are determined: Table 8.7 j - for centrifuge particle size analysis, homogeneous mode with scanning V|
1.038 >'2
Vi.2 I-OH \y\3 '-005 >i4 1.003 j;,5
1.001
1.119
>'3
1.222
yA
1.360
>'2.3
1.054
>'2,4
1.028
y^A
1.110
>'2.5
1.014
J^3,5
1.055
3^4,5
1.175
1.007
>'3.6
1.025
>'4,6
1.077
1.002
>'3,7
1.008 y^.i
V,.6 "-001 yifi \y\j '-^00 yi.i
1.025
V,
1.561
3-6
1-896
.V,6 '-229 Vj 1.070
Vy
1.234
8.6. Variable inner radius (Both 5' and t varj% r remains constant) H.6.1 Stokes diameter determination Let the time of the first withdrawal be /j; the largest particle present in the withdrawn sample at this time will have fallen from the surface at radius ^S* to the measurement zone at radius r. Equation (8.6) will apply and may be written: JrA =^A:ln — ' ' S
(8.30)
The liquid level will then fall to S^ where: S
-S'^
nh
(8.31)
where v is the volume extracted (10 cm^) and h is the thickness of the centrifuge bowl (1.02 cm). The fall in the inner radius can therefore be determined:
AS^S^
-S
(8.32)
Centrifugal sedimentation methods 411
Let the time for the second withdrawal be ^2; the largest particle present in the withdrawn sample will have fallen from S to x^ 2 '^ ^'^^^ ^i' ^ distance Axi 2 due to the withdrawal of the first sample and from Xj 2+^^^i 2 to r in a time t2-ti. Hence: dh^ - y t l n ^
(8.33)
dUt2-h)-k\n
-—
(834)
Adding equations (8.33) and (8.34) gives:
dhi = ^ l n ~ 1 . ^ . 2 ^V ^12 J
(8.35)
For the third withdrawal; in time t^ particles of size ^3 will fall from the surface at radius S to x^ 3, hence: d^t^=k\n^
(8.36)
These partiqles will then fall a distance ZVT, 3 due to the withdrawal of the first sample where, from equation (8.31): (^1,3 -^\3f
-^13 =3.1207
(8.37)
In the next time increment, particles of size d^ will fall from radius Xj 3 H-Axj 3 to radius X2 3 hence:
^3 0 2 - ^ i ) = ^>n
"^ ^2,3 ^ ^^2,3
(8-38)
412 Powder sampling and particle size determination These particles will then fall a distance A.\:2 3 due to the withdrawal of the second sample where: (8.39)
(.To 3 - A x o j f - x f 3 =3.1207 and dj {t2,-t\) = kIn
(8.40) ^23 + ^^^2.3
Adding equations (8.36), (8.38) and (8.40) gives:
dh, =A:ln —
Axi1,3
1+-
-V
Ax.2,3 1+-
M,3 )
'•2,3
V" (8.41) ;
The bracketed terms are the correction terms for the fall in level due to each extraction. Repeating this gives, for the «th withdrawal: A\-|l.n
./,^/„^^ln S
I+V
X2ji
J
Zix;n--\,n
(8.42)
^n-lM J
This differs from the variable time equation in that the Stokes diameter reduces more rapidly thus, effectively, making this technique into a scanning technique. 8.6.2 Ladal pipette disc centrifuge The Ladal pipette centrifuge was developed as a centrifugal version of the Andreasen gravitational pipette [11]. This pipette centrifuge (Figure 8,3) was designed by Allen and Svarovsky [19] to operate with a reduced volume of suspension (150 ml), as compared to the Simcar. The modified Kamack equation, given above, was derived to correct for the changing surface radius as samples were extracted. One of the consequences of this M\ in surface radius with time, is a reduction in the measurement time down to about an hour. At 500 rpm the measured size range for quartz is
Centrifugal sedimentation methods 413
approximately 8 |Lim to 0.8 |am in 1 h and these sizes are halved if the speed is doubled. 8.6.3 Worked example (a) Determination of ratio of Stokes diameters for constant r, variable S and t. Using a feed volume of 150 cm^ gives AS = 4.146 cm; hence equation (8.33) becomes: dh
=)tln(7/4.146)
letting (k/^i) be equal to 1.91 makes d^ = unity. Second extraction, for withdrawals in a 2:1 progression in time,
Equation (8.33) gives:
d^ = 1.91/«(jCi2/4.146)
Fig. 8.3 Line diagram of the Ladal pipette centrifuge.
414 Powder sampling and particle size determination
Equation (8.34) gives:
^ | -1.91ln[7/(x|2 ^ ^^n)[
Hence
x^2 + -^1.2^^1,2 = 29.022
and from equation (8.31):
(xj 2 + ^X\ 2 ) --Af2-3.1207
Solving simultaneously gives v, 2 =^5.244 and zitj 2 "^ 0.2896; hence cA ^0.670. Assuming that this progression in sizes continues then ili^'-'"-0.670^^0.449. Substituting this value in equation (8.36) gives: x,3-4.146exp(0.449^/1.9]) ,v, 3 - 4.608 Substituting in equation (8.37): A.v, 3=0.327 From equation (8.38):
^23 -4.935exp(0.449^/1.9lj .V2 3 - 5.484
From equation (8.39): z\x:2 3^0.278; Substituting these values into equation (8.41) gives a more accurate value for J3, (/^ --^ 4/,). ,2 1.91 '' 7 (. 0.327 V ' r . 0.278'^'^ a- =" In 1 + _: 1+ 4.146 4.608; I 5.484 4 V ^ , ^ , J c/3 = 0.440
This iteration is repeated until the assumed value and the derived value are sensibly equal. The derived diameter ratios are shown in Table 8.8. Table 8.8 Ratio of particle sizes for extraction in a 2:1 progression in time Ratio of times
1 2
Ratioof sizes
1
0.670
4
8
16
32
64
0.438
0.282
0.179
0.112
0.070
Centrifugal sedimentation methods 415 8.6.4 Mass frequency undersize determination I ,et the final sample withdrawn be of concentration (?, and let the surface be at .S'l immediately prior to this withdrawal, then: ^1=^(1+>'1 )Q^
(8.43)
where: Vj =
(8.44)
(8.45)
and:
(8.46)
y.^^y^^''^
Hence, by the trapezoidal rule: /'V
/^i4(>'2+>i2)(a-a2) f
where y2 -
^2
r \^i J
Substituting for (9, 2 gives:
^2^7(v2+J^i,2)ft +
>'2 + y\i + > 1,2
Proceeding in a like manner gives the general formula:
(8.47)
416 Powder sampling and particle size determination By successively eliminating the Q functions, this gives a general equation in recursive form as before: /-I 7=1
f
yi
y+yi-\,i
yj.\..+yj.,
.v+3^M,,
IF,
(8.48)
yj^+yn.,
\2
r
and y,.x.,=y,
/"'
\^i J
A numerical solution is given below for a feed volume of 150 cm^ and a 2:1 progression in time, K , values are given in Table (8.5) and these are inserted into the general equation to give the F values presented in Table 8.9. The dimensions of the centrifuge bowl are such that)', ^1.364. Table 8.9 Tabulated V/, values for a pipette centrifuge J, 2 1.1654 y|3
1.0793
V2 3 1-2216
.F,.4 10391
y2.4 1-1057
y,,; 1.0195
^2,5 1 0 5 1 9 >'3.5 '••352 >'4.5 '-3616
>1.6 '0099 yiA "0263 >;|7 1.0053 >'2 7 1.0140
>'3 4 1.2863 V 3 , 1.0632 74.6 ''715 .)'.s.6 ' 4 6 0 y.^^ 1.0353 V4^ 1.0882 .V5,7 '-224 .V6,7 1-601
Calculation of F values; using a feed volume of 150 cm^ as before. Equation (8.43) gives
F, = l.l82e, Equation (8.47) gives F. =J-(1.494 + 1.1654)C?T + 1 - ^"^^^^ F, ^ 2^ ^^2 2.1654
Centrifugal sedimentation methods 417
F2 = 1.330g2-0.228F, Equation (8.48) gives: f;^:^ 1(1.651 + 1.222)^3 +
"2.873 2.873" /s + 2302 2.302 _
2.873 F, 2.079
^ 3 - 14366(^3- 0.249/^- 0.133Fi and so on, giving the general equations for the conditions / ^ 7, V "= 150 cm^ presented in Table 8.10. An alternative approach to the procedure outlined above has been presented by Dumm and Hogg [20]. Table 8.10 Table of F values for a pipette withdrawal centrifuge F,-1.18220^ F2-1.330g2-0-228Fi F3 - 1.4366e3-0.2486F2-0.133F, F4 - 1.5658eF4-0.3093/^V0-1509F2-0.0757F, F5=1.7255e5-0.3836F4-0.1950F3-0.0877F2-0.0427F, F^=^1.9373g5-0-4713F3-0.2595FF4-0.1200F3-0.0521Fv~0.0249F{ F7^2.2255|27-0.5755F6-0.3489F5~0.1717FF4-0.0760F3-0.0323F2 .0153F 8.7 Photocentrifuges 8.7.1 Introduction In the photocentrifuge method the concentration of a suspension is monitored using a light beam. The light can come from either a white light source (an incandescent bulb) or a monochromatic coherent source (a laser) and the detector may be either a photodiode or photomultiplier. The signal from the detector is usually digitized and converted to a size distribution via a computer. Photocentrifuges are available in both disc and cuvette configuration. The former are normally used in the line-start mode and the latter in the homogeneous mode. The line-start mode has a much higher resolution
418 Powder sampling and particle size determination
than the homogeneous mode so that muhimodal distributions are closely defined. The homogeneous mode can be run using a gradient procedure, with acceleration over time, which greatly speeds up the analysis. Both modes suffer the disadvantage that the laws oi^ geometric optics do not apply, and the correction required can introduce large errors, especially with size distributions having a wide size range. For the examination of paint pigments, end-use properties may be more closely related to the attenuation curve than the derived size distribution. It is therefore arguable that the measured relationship between attenuation and Stokes diameter should be used to define the powder rather than size distribution. 8.7,2 Disc
photocentrifuges.
The first disc photocentrifuge was developed by Kaye [21]. In this instrument, concentration changes within a suspension are followed using a white light beam. The instrument is usually used in the line-start mode and intuitively, one would expect that attenuation would be proportional to the projected area of the particles in the beam so that the curve of attenuation against Stokes diameter would be a differential surface distribution. However, Treasure [22] derived a relationship which showed that the attenuation, due to the finite width of the light beam, was proportional to the volume (mass) of particles in the beam [23,24]. In the line-start mode it is necessary to use a spin liquid that is denser than the suspension, otherwise the suspension can break through the interface and settle en-rnasse in a phenomenon known as streaming. In order to eliminate this effect a buffer layer technique is often used (Figure 8.4). Fractionation Buffer layer
vSusjKnsion Fig. 8.4 The line-start technique.
Spin liquid
Centrifugal sedimentation methods 419
The spin liquid may consist of 15 ml of 10% aqueous glycerol on which is floated 0.5 ml of water. For a more viscous suspension the concentration of glycerol can be increased. It is however necessary- that the buffer layer be less dense than the fill liquid so that inversion does not occur. An interface forms between the two liquids and this may be broken up by a momentary change in the speed of the centrifuge although this is not always necessary to eliminate streaming. Typically, 0,25 ml of dilute suspension is then introduced; this tends to break through the air-water interface and spread out on the diffuse interface between the buffer liquid and the spin liquid, which is the starting radius for the subsequent sedimentation process [25]. If streaming persists it may be eliminated by using much smaller volumes of buffer liquid and suspension, e.g. 0.1 ml. Coll and Seartes [26] used 20 ml of sucrose solution topped by 1 ml of/?dodecane to prevent evaporation. Several injections of 0.25 ml of colloid sample, ^ (001%) concentration, were then injected through the oil layer. In the external gradient method, a hypodermic syringe is used to form the gradient. For spin conditions requiring 15 ml of aqueous spin liquid exactly 15 ml are drawn into a 25 ml syringe. Air bubbles are expelled and, with the needle pointing down, an additional 1 ml of methanol is drawn in and the entire volume injected into the disc. Finally 1 ml of a suspension containing < 0.5% by volume solids in an 80:20 water/methanol solution are added. An application of this technique is described by Devon et. al. [27]. A density method with correction for light scattering has also been published [28]. It must be stressed that the raw curves are not size distributions and calibration is required to convert to absolute values [29]. The importance of the correction for the breakdown in the laws of geometric optics is stressed by Weiner et, al. [30] who show excellent agreement between theot7 and experiment when this is done correctly. They also use the Brookhaven disc photocentrifuge to characterize ASTM carbon blacks.[31] This method has been used to characterize void-containing latex particles [32]. Commercial instruments are available from Joyce-Loebl, CPS Instruments and Brookhaven. 8.7.3 Homogeneous mode This is the preferred mode with cuvettes; it is still however necessar>^ to correct for radial dilution effects.
420 Powder sampling and particle size determination (a) Stokes diameter determination For a constant (centrifuge) speed operation equation (8.4) is applied. In the gradient mode (centrifuge speed increasing with time in order to speed up the analysis) co\s replaced by an expression relating centritiige speed with time. (h) Mass frequency undersize determination The light cut off by particles in the light beam is related to the concentration by the equation: \n^-± = b''f ^
where
K,.n,d^
(8.49)
r=min
/Q
is the intensity of the emergent light beam when no particles are present / is the intensity of the light beam at time / after the start of sedimentation b is a constant depending on the dimensions of the light beam, the geometry of the system and the shape of the particles n^ is the number of particles in the beam of diameter d^, dc^f is the diameter of the largest particle in the light beam K^, is the extinction coefficient for a particle of diameter d^.
Kamack's equation is applied with the assumption that the attenuation is proportional to the product of the extinction coefficient and the crosssectional area of the particles in the beam i.e. ln(/(//) is replaced by the optical density D where: D = \og^ I The data are then converted to a mass distribution by summating the product of AD and d<^i.
Centrifugal sedimentation methods 421
8.7.4 Worked example assuming that the conditions in section 8.4.3 applies for a centrifuge without scanning Material; titanium dioxide Initial concentration 3.48 g in 100 cm-^ Density of powder p^ = 4125 kg m ^ Density of liquid pj- 1000 kg m"^ Liquid viscosity 77^ 0.000891 Pa s Centrifuge speed A'^^ 3000 rpm = 314 rad s"^ Table 8.11 Conversion of attenuation of homogeneous centrifuge into a cumulative surface undersize distribution assuming constant extinction and shape coefficients Time
Emergent log(!o/I) log(Io/I) intensity (7) min. (/) (%) 0 0.778 15.0 92.4 1 0.719 17.2 2 83.3 0.648 20.2 69.5 0.541 4 25.9 0.391 50.3 8 36.6 29.2 0.227 16 53.4 12.7 32 0.099 71.7 64 0.031 4.0 83.8 0 90.0
(0 8 7 6 5 4 3 2 1
FTom Stokes equation, at T = 1 min.: 18x 0.000891 xin{7/4.304) J^ L = 0.65 |.im
J, =r V
3125x314'x60
Stokes diameter D, (iiim)
0.65 0.46 0.325 0.23 0.163 0.115 0.081 0.058
Surface (%) undersize F,{S) 99.7 98.3 92.3 75.5 48.8 22.6 7.3 0
422 Powder sampling and particle size determination Table 8.12 Conversion of cumulative undersize distribution by surface into a cumulative undersize distribution by mass Relative surface ^S 03 i.4 6.0 16.8 26.7 26.2 !5.3 13
Mean diameter d 0.784 0.56 0.392 0.278 0.197 0.139 0.098 0.070
Stokes diameter SA5J 20.964 14.790 9.922 6.113 3.034 1.111 0.278 0.050 0.00
Mass % undersize.
di{\im)
Pji^
0.65 0.46 0.325 0.23 0.163 0.115 0.081 0.058
100.00 98.75 94.65 82.21 57.58 29.88 10.63 2.67 0.00
8.8 Line-start incremental centrifugal sedimentation 8.8.1 Line-start, incremental centrifugal technique In the line-start technique the centrifuge disc is filled with clear liquid and allowed to attain its running speed. A small volume of buffer liquid is then introduced. The suspension is then introduced at time / ^ 0. The start radius is assumed to be at the mid-point of the suspension radius. (a) Stokes diameter determination Since all the particles emanate from the same starting point, the Stokes diameter is determined using equation (8.4) with r as the measurement radius and S as the midpoint of the suspension layer {b) Mass frequency undersize determination There is some disagreement as to whether the optical attenuation using the line-start technique is proportional to surface or volume distribution or whether it varies in a complex manner with d<^f. The consensus is that a volume relationship applies. A wedge shaped detector window (with radial sides, circular inner and outer sides), centered on r and spanning r^ to ^2 where r2>ri, will view an
Centrifugal sedimentation methods 423
annular section of the disc. At time / == 0 there is only clear liquid in the window. As time progresses, the largest particles present in the suspension will enter the window and at time t the diameter of the particles at the center of the window will be: t/^ - p i n - ^
(8.50)
The suspension in the window will contain particles with diameters in the size range J,=(i^^(l -p) to d2='d^f(]-^0) where c/, is the diameter of the particle just entering the window and d^ the diameter of the particle just leaving it: .n d, ^ ds, (1 f ^) -- ^ y^ .In-1
.; „ . , /I .^2 d2ds, {\'P)r,\-.- ^ ^y . In j
(8.51)
If we substitute to eliminate t/^y we find that p and ^are dependent only on geometric factors and not on material properties. 6 ^\ - — V — r
>0- L _ l l
Z-1
(8,52)
If the detector window is fixed, both r, and r2 will remain constant during a run, so that ^ and 6 will be constants independent of time. Note that while the ratio ofd^/d2 remains constant with time the difference between d[ and (^2 decreases since the value of (i^^^ decreases with time: d2~d^-d^,{e+fj)
(8.53)
i.e. the difference is proportional to the Stokes diameter]. The optical density is proportional to /?^
(8.54)
424 Powder sampling and particle size determination
1
Attenuation D 60
• C\ \
40
-
/
20
.
•10
1
.
.
1
^^
.
.
20
1
.
r
30
40
for I
50
d^
Fig. 8.5 Elemental area under attenuation curve for photocentrifuge in linestart mode.
120
2.5
%/nm
^C'
100
r\l
Photocentrifuge trace 2
Mass percentage
m X
3'
80
o
Extinction curve
1.5 o' o
60
1 3 n W
40
3
0.5
20 01
0
JL 0.2
0.4 0.6 0.8 1.0 1.2 Particle size in microns
1.4
1.6
Fig. 8.6 Converting the line-start attenuation curve into a mass frequency curve
Centrifugal sedimentation methods 425 The elemental area under the experimental curve of attenuation (D) versus time {t) (Figure 8.5) is given by: M^Y^^dt
(8.55)
Since ta\ldl,\ from equation (8.54) the elemental area under the attenuation curve becomes:
the relative number of particles in the size range d\ to ^2
AA=
I
ri{dst)dds,
(8.56)
Multiplying each ordinate by c/J^ or d^^ gives the area or volume of distribution. A full solution is generated if the extinction factor is introduced into the equation. Alternatively the recorder trace of attenuation against time can be converted to attenuation against Stokes diameter and then normalized, i.e. the area under the curve is made equal to 100 (Figure 8.6). It is assumed (hat this curve is the mass frequency distribution (dJV/dd^^ versus dc^^) uncorrected for the breakdown in the law of geometric optics. The normalized curve of the product of d^f/dJ^yand extinction coefficient is the corrected distribution. This has the effect, in the above example, of reducing the measured median size of titanium dioxide from 0,49 |Lim to 0.45 fxm. 8.8.2 Discussion of line-start theoiy Although equation (8.56) is the recommended one for the line-start technique [33] there is considerable disagreement as to whether it is correct and it is often assumed that the attenuation is proportional to
426 Powder sampling and particle size determination
projected area rather than particle volume. An alternative treatment, in agreement with the volume proportionality, is presented below. The number fraction of particles in the field of view is given by:
^^i^s,)-- S fOMds,,
(8^7)
whereX^) ^ dN/dd^^ Treasure assumed that the relative number concentration in the light beam, centered on d^^ may be expressed by the linear function n{d^^y-= f-^ ^(^Sf)Thus the total cross-sectional area of particles in the beam is proportional to: {\+0)d Sf
f
{\-p)dst I
K^St J
>d^dd
(8.58)
The solution to this integral, if higher order terms are neglected, gives the proportionality: Dx(^;-f/i)(/ + g ) 4 D^{f^g)dl
(8.59)
since {0 ^p) is a constant. The optical density of the suspension is therefore proportional to the total particle volume in the beam as before i.e. DocKsfn{dst)di
(8.60)
Other published solutions state that the amount of light cut off is proportional to particle volume [34,35] for line-start and to particle surface for homogeneous mode.
Centrifugal sedimentation methods 421
T^elson e/. al [36] challenged these derivations. Using a general series expansion, they stated that the solution to equation (8.53) takes the form
/(^) = (;0+0)Xg/i;'
(8.61)
Thus, the fraction of particles in the field of view varies in a complex fashion with 6/^^. The conclusion drawn from equation (8.56) is at odds with published data on polystyrene lattices and silver bromide, in which a volume proportionality is found [37,38]. However these distributions were narrow, and with narrow distributions the difference between volume and surface distributions is small. The conclusion is also at variance with data published on BCR 66 quartz powder, ranging in size from 0.3 to 3 )im. In this case, the median for the attenuation curve was 1.52 ^m which reduced to 1.14 \xx\\ with extinction factor correction [39] and a correction of this magnitude could hide the effect. Centrifugal photosedimentation yields an attenuation curve; particles at the fme end of the distribution, at say 0.1 \xm, obscuring the light by perhaps one twentieth of their geometric area whereas at the coarse end, say 1 |am, the ratio can be greater than two. The correction for extinction coefficient modifies the shape of the curve considerably, making decisions as to correct theory to apply difficult. The unmodified attenuation curve may be more relevant to end-use properties, for hiding power of pigments for example, than the derived size distribution. Since the introduction of a correction for extinction coefficient has such an enormous effect on the shape of the distribution curve for wide distributions the correction should be applied with caution. Putman et. al. [40] used a CONTIN-based approach towards data analysis of photosedimentometry using the Shimadzu SA-CP2~10 to accurately determine size distributions. Contin is a mathematical FORTRAN IV program, which analyzes an experimental signal consisting of a sum of exponential curves, and determines its individual functions each with a weight.
428 Powder sampling and particle size
determination
SjnMplflTTFV
n BaBB 1=1
BI-DCP
Fig. 8.7 The Brookhaven disc photocentrifuge. 8.8 J BI-DCP disc (photo)centrifuge particle size analyzer The technology that Brookhaven developed for the x~ray centrifuge has been transferred to their photocentrifuge (Figure 8.7). The revised software is for analysis by the homogeneous start technique plus a scanning head detector. The high-resolution line-start technique can be used, but this is not amenable to scanning since the low concentrations necessarily employed generate a noisy baseline. An additional benefit of the new software is that it allows for particles whose density is lower than that of the surrounding liquid, thus making it suitable for emulsion sizing.
Centrifugal sedimentation methods
429
Ceotrifofalcett
OfMian
Option
Fig. 8.8 Block diagram of the Horiba cuvette photocentrifuge. 8.9 Cuvette photocentrifuges In these instruments (Figure 8.8) the disc is replaced with a rectangular cell containing a homogeneous suspension. Unless corrections are applied for radial dilution effects and the breakdown in the laws of geometric optics, the derived data are suitable only for comparison purposes. For example, a 50:50 mixture of 0.25 and 0.60 |im spherical silica particles was recorded as 54.4:45.6 with no correction for radial dilution, and this increased to 70:30 without extinction coefficient correction with the smaller particles grossly under-counted. A computer program to correct for the light scattering of small particles reduces these errors [41]. Bowen et. al [42] report on a method of programming the Horiba CAPA-700 to generate accurate sub-|im measurement of alumina and quartz powders. Using the manufacturer's correction for light scattering was found unsatisfactory. It was also found that the light scattering correction was strongly affected by the shape of the particles [43]. An alternative procedure is to use several wavelengths and deconvolute the resulting set of linear equations that develop in order to find the size distribution. This procedure was applied by Niemann and Weichert who used a modified Phillips-Twomey algorithm [44,45]. Their photo-
430 Powder sampling and particle size determination
centrifuge used white light from a short arc xenon high-pressure lamp. Two light beams are generated, one passing through the suspension at a depth of 2 mm and the other at a depth of 20 mm. Four cuvettes are used, two containing clear liquid and two containing suspension. The light beams are collected with fiber optic guides after passing through the cuvettes and then separated into four wavelengths. Additional photodetectors monitor the intensity of the lamp at the same four wavelengths. The speed of the wheel accelerates continuously over 20 minutes to a final speed of 3,000 rpm and maintained at this speed until all the particles have settled. This system enables a broad size distribution from below 0.05 |j.m to 10 |um (e.g. quartz in water) to be analyzed in 30 minutes. histruments are available from Horiba, Seishin, Shimadzu and LUM. They can be run in the gravitational, centrifugal, gravitational followed by centrifugal or gradient mode. In the gradient mode, the centrifuge accelerates over the analysis time to reduce the measurement time. The simpler instruments operate at constant speed and an analysis can take 45 min, which can be reduced to a few minutes in the more sophisticated versions. Horiha CAPA-700 covers the size range 0.01 to 300 |im, automatically selecting the best of five operating conditions, involving the three modes enumerated above, at centrifuge speeds from 300 to 10.000 rpm. Horiba CAPA-SOO is a more economical version covering the size range 0.04 to 300 |am. Seishin offer three versions of their micron photo-sizers covering the size range 0.1 to 500 |Lim, the SKC-2000, the SKC-3000 and the SKC5000. Shimadzu SA-CP3 operates at 120, 240 or 480 rpm and in any of four modes to cover the size range 0.02 to 150 |Lim. Shimadzu SA-CP4 operates in the range 500 to 11,000 rpm to cover the size range 0.01 to 500 jiim. L.U.M, LUMiFuge^^ 114 (Laboratoiy, Environmental, Medical Diagnostics & Technology) is a photocentrifuge covering the size range 0.1 to 300 jam with a sample volume of 0.1 to 2ml. No sample pre-dilution is required and eight simultaneous analyses can be carried out. The photocentrifuge operates in the gradient mode with at speeds accelerating from 300 to 3000rpm.
Centrifiigal sedimentation methods 431
Fig. 8.9 Berg's conoidal ceiitrifiige tubes 8.10 Homogeneous, cumulative, centrifugal sedimentation The shape of the centrifuge tube is immaterial for instruments operating in the incremental mode but the shape is important for instruments operating in the cumulative mode since particles travel in radial paths. The disadvantages of cylindrical tubes are that sedimenting particles strike the walls of the tube, agglomerate with other particles on the walls and reach the bottom more quickly than freely sedimenting particles. The oblique force of the suspension on the walls also set up convection currents within the suspension. The advantages of cylindrical tubes instead of sector or conoidal-shaped tubes (Figure 8.9) is that they are easier to construct and may be used in ordinary !aborator> centriliiges. 8.10.1 General theory Equation (8,6) may be written (for r. '-=--S and r ^= R): S -^ i?exp(- kdj^i)
(8.62)
At time /, all particles greater than d^^^ will have reached the bottom of the container (r "= R). In addition, partial sedimentation will have taken place for particles smaller than d^^^. For each of these smaller sizes, a starting point XQ exists, beyond which all the smaller particles will have reached R where, from equation (8.62): .^Q
^Rcxp^-kd^tj
(8.63)
432 Powder sampling and particle size determination The volume fraction of suspension lying between R and XQ for shallow bowl or flat sector shaped tubes is equal to: /? -An
R'
R'-S'
R'-S'
1 ~ exp(--2fo/^/)
(8.64)
If the particle size distribution is defined such that the weight fraction in the size range d to d'rM is f{d)dd then the weight of particles with diameters greater than d^^ that have completely settled is: 00
W^
\f{d)Ad
(8.65)
The weight fraction of particles smaller than d^^^ which have completely settled is:
W,^ - ~ 2 — T j h ~ exp(~2yW'/) \f{d)Ad
(8.66)
The total weight fraction deposited is:
P^W + - ~ — y j[1 -exp(-2)tJ^/)J/(J)dJ
(8.67)
R —S n The weight fraction oversize can be determined if the weight traction deposited is measured for different values of the variables S, R and t. Similarly, the weight fraction of particles still in suspension at time / will consist of particles smaller than J^^that have originated in the volume between the surface S and radius XQ. By comparison with equations (8.66) and (8.67) this fraction is:
1 - P - - ^ — y 11 exp(-2^^^/) - Qxp{-2kdll)Y{d)dd
Cenlrifiigal sedimentation methods 433
<^Sl
~P =
l-exp(-a) Q^
exp
ad 2\
(8.68)
exp(-a) \f{d)M
d St J
where a - 21og 8,11 Variable time method (variation of P with t) Romwalter and Vendl [46] derived a solution to equation (8.67) by differentiating with respect to time and substituting back in the original equation. Brown [47] drew attention to an error in their derivation and stated that an exact sohition for the distribution function appeared to be difficult, if not impossible, to obtain by the above method. The following solutions were derived by Robison and Martin [48,49] who used sector-shaped tubes. Their analyses agreed closely with those obtained using the variable height method.
} f{d)M = 1 -
M{6-M)p 8
^ Mds, d/> ^ M(M-2)(A/--4) 8 dJ 8
St
^
(8.69) where
l^ds,) = ~ ] Pd^'~'Ad d' 0
and
M
4[R^~S^) S^\n{RIS)
An exact solution to equation (8.67), as given by Kamack [50], is as follows:
Pids,)-
exp(a) -
(8.70)
ddP 2 M
where
q{d) = pid) +
and
h\ {x) = /2(x)exp( ~2x)
434 Powder sampling and particle size determination
X ~\x\
^St
and the 'resolvent kernel' h{x) is a function that depends on the apparatus constant. Muschelknautz [51,52] designed a centrifuge in which the displacement of two diametrically opposed bodies floating in a dispersion was measured. The bodies are fixed on a common rod and are immersed at different depths in two chambers. The differential force yields the size distribution directly. Sokalov et. al [53] described a centrifugal sedimentometer with a float measurement system. 8.12 Sedimentation distance small compared with distance from centrifuge axis The simplest procedure with a homogeneous, centrifugal system is to make r-S small compared with S and assume that the particles fall with constant velocity. Equation (8.2) becomes:
(ps~Pf]4t 18/7
,(r^S
(8.71)
This approach has been used by several investigators and applies to the Alpine sedimentation centrifuge and the Mikropul. 8.12.1 Hosokawa Mikropul Sedimentputer The Hosokawa Mikropul Sedimentputer [54-56] has a sealed suspension in a cell that is rotated. As the particles settle, out the center of gravity changes, which creates an inbalance that causes the cell to vibrate (Figure 8.10). By detecting the amplitude and angular velocity of the vibration, the size distribution is obtained. Muschelknautz designed a centrifuge in which the displacement of two diametrically opposed bodies floating in a dispersion was measured [5760]. The bodies are fixed on a common rod and are immersed at different depths in two chambers. The differential force yields the size distribution.
Centrifugal sedimentation methods 435
, Angular vdodtygigiialg J 'ndtioaieter
Pkkiqp
>y lliOixa \ fof^ilre 1 scn^Mr - ^ S i g n a l for revolution Rotor
^iiLipjjprJ
-»1measuAsoicavi
P
Signal for an^plitude
(a) Center of gravity
B<^<MC measuring (unifonn misprision)
^1 DfaiplacctTOnt of ^Pccntier of gravity dear
••JL^^M^T^Jtm.£StDttdi^ measuring <soaiepaiticles have settled out)
(b) Fig. 8.10 The Mikropul Sedimentputer. (a) Schematic of the system, (b) The settling of particles causes the center of gravity to shift. 8.12.2 Alpine long-arm
centrifuge
The Alpine sedimentation centrifuge is a long-ami centrifuge of diameter 400 mm with a 50 mm high measuring cell. The rate at which sediment settles out is determined by measuring pressure changes at the bottom of the cell using a diaphragm arrangement. 8.13 Variable inner radius (variation of P with S) Brown [61] avoided the complications arising from the differentiation of equation (8.66) with respect to time by considering the fraction sedimented in a given time interval with the centrifuge tubes filled with suspension to a series of levels. On increasing .S* an increasingly large fraction of suspended particles will be deposited in a given time. Equation (8.67) may be written:
•tJ6 Powder sampling and particle size determination
R^ P=W+ R^ - S^
(8.72)
Differentiating equation (8.72) with respect to S, keeping R and / constant, and substituting baci< forP-W, gives: :)A 8f_ 8S «•'
2R^S
.-/
(R^-S^f
Substituting for / from equation (8.72) gives:
S
(8.73)
8S
Thus, a rigorous solution is obtained by keeping R and / constant, and determining the weight fraction deposited for varying heights of suspension. Just as in gravitational sedimentation, second derivatives of the fraction sedimented are required in order to obtain the distribution. In order to obtain ¥(dgf) in terms of the second derivatives of P, it is necessary to differentiate equations (8.61) and (8.66) with respect to .S', eliminating dd^/dS from the two resulting equations. 5d St [ 5S , RJ
(8.74)
\st 2S\n(R/S)
giving, in combination witii the differential of equation (8.66): f(cl:„)bch, 8S
_\n{RIS) R^ ^3S^ 5/ S 8S 2S
dS^
(8.75)
Combining these two equations gives: Fids,)8ds,=-
\n(R/S) d.
R^ + 3SHP
5S
,_2
^2, d^F
3S'
(8.76)
Centrifugal sedimentation methods 43 7 Similarly, the distribution function may be derived in terms ot(8P/8t) and 2
(5 P/6S5t) by differentiating equations (8.63) and (8.67) with respect to time: , 2t dP ^ (dst) ==
R^ ~S
5 P
(8.77)
Three distinct methods are therefore available for calculating the mass distribution of particles in suspension for a series of sector-shaped tubes filled to a series of levels. First, the mass fraction of particles larger than a known diameter may be calculated from equation (8.77) and the distribution function determined from the slope of the cumulative mass deposited against S. Secondly, the distribution function may be calculated directly in terms of the first and second derivatives of the fraction sedimented with respect to the length of the column of suspension centrifuged by use of equation (8.77). Thirdly, from the sedimentation time curve at a series of levels, the distribution functions may be calculated by the use of equation (8.77). 8.13.1 Alternative theory (variation ofP with S) An alternative approach is given by Murley [62] as follows: If the inner radius of the centrifliging suspension is decreased by a small amount 8S, then the extra weight of sample introduced into the centrifuge is 2KSTC5S, T is the thickness of the suspension in an axial direction and c is the weight of solid per unit volume of suspension. For this extra amount of material added, all that with a particle size less than d^^ will reach the collecting plane at R and all that with a particle size greater than d^f\m\\ be at a smaller radius than R at the end of the running time t. The extra weight of sample deposited at radius R due to this change in radius ^S' of the top surface of the sample is therefore: L\P - -InSTchS ] f(d)dd
(8.78)
The negative sign occurs because the added layer causes a decrease in S. This equation applies to an apparatus where the liquid is run off and the deposited layer is retained for analysis. In the type of apparatus where the
438 Powder sampling and particle size determination overlaying liquid is removed for estimation of the weight of solids the following equation is applicable: 0
^P = -InSTcdS J fid)dd
(8,79)
By plotting P against S and finding the slope of the curve, yW)dc/ may be evaluated. Equations (8.78) and (8.79) can also be derived by differentiating equation (8.69) which may be written:
P-y-
j / ( ^ ) d j L - / l y ][\-exp(-2kd^f{d)dd
I 1 - P - -~^
(8,80)
^st ^ ] {R' Qxp(~-2kii^() - S^} f(d)dd
(8.81)
where IP is the mass fraction still in suspension. The mass fraction of powder that has been sedimented is P where P = n{R^~S^)Tc{\-PX i.e. P = nTc l^R'Qxpi-2kd^t)-S^^f{d)dd
(8.82)
0
Differentiating this with respect to S leads directly to equation (8.65). 8.14 Variable outer radius (variation of P with R) Donaghue and Bostock [63] differentiated equation (8.68) with respect to R, giving:
X/(d)dd = P-,J'^'~''^'^'^^ is'
dR
(8.83)
Slope measurements of the P-R curve permit calculation of the mass fraction oversize. The apparatus developed for this determination consisted of a stepped centrifuge so that the suspension is contained in a space consisting of a
Centrifugal sedimentation methods 439
number of rings, each of which has the same inner radius .S*, but progressively larger outer radii R from top to bottom. Particles are deposited on detachable surfaces that are removed and dried before weighing, the supernatant being removed before the centrifuge is stopped. The advantage of this instrument, over the Simcar which was developed later, was that six points on the distribution curve were determined simultaneously. The main disadvantage, which prevented the centrifuge from being accepted, was loss of sediment due to movement of supernatant liquid during its withdrawal.
Fig. 8.11 MSA special centrifuge tube. 8.15 Line-start cumulative centrifugal sedimentation 8.15.1 MSA analyzer The MSA analyzer (Figure 8.11) operates in this mode [64]. Objections that can be leveled at this technique are: • The amount of deposited sediment is determined by its height and, since the settled volume is not independent of size, errors are introduced. The lower part of the sedimentation cell has sloping wails, hence some particles adhere to this section and others slide down the sloping walls into the measurement zone, so that large particles are frequently fbund at a level where only small particles should be present. 8.16 Particle size analysis using non-invasive dielectric sensors The use of capacitance measurement is based upon the principle that the concentration changes as particles settle through a suspension and will alter the effective dielectric constant between the sensing electrodes. A complete capacitance transducer consists of capacitance sensing electrodes together with capacitance sensing electronics, which is essentially a capacitance to voltage converter.
440 Powder sampling and particle size determination The analyzer described by Simons and Williams [65] consisted of an array of eight pairs of capacitance electrodes mounted vertically down the side of a sedimentation tube of length 26 cm and diameter 2.5 cm. The electrodes were embedded in acrylic and flush mounted to the inside wall of the tube. Soda glass spheres at a volume concentration of 1.65%, in size ranges Uom 20 to 60 \m\ and 2 to 18 jixm, gave results comparable with Andreasen and Elzone. Liquid out
turfoct
\ \
\\ \\ t I
\1 \ 1 \ t
FIX
\ \
\ M B \ I-
Fttd nozzte Sutptntion in
Fig. 8.12 Diagrammatic section of the Hauser-Lynn centrifuge 8.17 Supercentrifuge The supercentrifuge rotates at speeds between 8000 and 50 000 rpm and may be used to determine the size distribution of particles too small to be analyzed with conventional centrifuges. Several ways of using Hauser's [66,67] supercentrifuge (Figure 8.12) for particle size analysis have been described. The usual procedure involves successive fractionation of the suspension and W'cighing of the fractions collected on a removable liner on the bowl [68™70]. The
Centrifugal sedimentation methods 441
suspension is fed into the bottom of the bowl and the particles move in a spiral path until they reach the wall. The liquid is then discharged in an annular layer over the overflow dam wall. If Q is the rate of flow of the suspension that causes a particle of diameter D to be deposited at a height h above the feed inlet, it can be shown that [67]. Q^khD'-
(8.84)
If the removable liner is divided into identical strips, dried and weighed, the weight deposited on each strip may be used to find the size distribution [71]. The constant k may be evaluated from curves given by Hauser and Lynn [72] or a nomograph developed by Saunders [73] may be used. This method of determining size distributions cannot be recommended for routine analyses and a method developed by Bradley is to be preferred [74]. This method is applicable to the Sharjiles supercentrifuge. If D^^^ is the diameter of the smallest particle retained in the centrifuge, it can be shown that: P - r + {-^-~\f{d)dd
(8.85)
This is identical to equation (8.65), but the relationship between XQ and D is more complicated than for a centrifuge without flow. Bradley derives the empirical solution for the Sharpies supercentrifuge as:
^-^
^-^
-4.1xlO^V
(8-86)
This gives, for the supercentrifuge as well as for batch centrifuges: ff ,, p_.._^^r_^_^!. 2S dS which is the same as equation 8.72.
(8.87)
442 Powder sampling and particle size determination
Hence, the weight fraction oversize is calculable by measurement of/^ for different values of S at constant W and Q. The quickest analytical procedure is to calculate P from gravimetric or chemical analysis of feed and overflow suspensions. Choice of flowrate and speed can be made in accordance with prior knov/ledge of approximate size and use of derived theoretical expressions, or by trial and error to establish the rate at which P approaches unity with maximum S. The main disadvantages of the technique are the need for large samples and the uncertainty of end effects in the bowl. A big advantage is the ability to use an item of standard equipment without modification for a size below the range of most of the specially designed centrifuges. 8 J 8 IJllracentrifuge The rotor of the ultracentrifuge is spun at speeds of up to 60 000 rpm in a vacuum to minimize air drag [75-77]. It may be used, therefore, to measure the size distribution of ver>' tine particles. McCormick [78], for example, describes its use for determining the size distribution of polystyrene (0.088 < (^ < 0.511) |Lim and Brodnyan [79] uses it for determining emulsion particle size. It has also been used in combination with light scattering for polymer size distribution determination [80]. In a round-robin test series Mueller [81] found the ultracentrifuge to be the most satisfactory size analysis method for the sub-|Lim range; a sample containing nine monodisperse components with 10% diameter difference being resolved. An analytical ultra-centrifugation technique has been used in combination with a scanning optical absorption system for particle size distribution determination. The system was demonstrated for colloidal platinum (0,4 to 2) nm and unstabilized zinc (4-9) nrn during particle growth [82]. A review of examples of colloid analysis of nanosize particles by ultracentrifugation with a focus on multicomonent mixtures has been published.[83] This is claimed to be the first fractionating analytical technique with almost atomic resolution. 8.19 Conclusions Sedimentation techniques are widely used for particle size analysis since the determined size distribution relates to unit operations such as classification. The distribution also relates to many end-use properties
Centrifugal sedimentation methods 443
such as the hiding power and gloss of pigments. Thermal diffusion limits the use of gravity sedimentation to powders containing a limited amount of sub-micron material, but the technique is extended into the sub-micron size range with the use of centrifuges. Gravitational and centrifugal sedimentation using a pipette is attractive due to its versatility and low capital cost, but the analysis requires a skilled operator and is time consuming. Mass distributions are also determinable using x-ray systems that are available as gravity and centrifugal sedimentometers. These are essential where speed and running costs are more important than capital costs. Photosedimentometers are also available for gravitational and centrifugal sedimentation. Centrifugal photosedimentometers are available with disc and cuvette cells. The former are usually used in line-start mode which gives high resolution, whereas the latter operate in the homogeneous mode. These instruments need to be calibrated, or corrected, for the breakdown in the laws of geometric optics. The modified distribution greatly alters the raw curve for sub-micron powders with a wide size range, thus limiting the accuracy of this technique. Accuracy is not always important; detecting changes which may affect powder handling or final product may be all that is necessary. References 1 2 3 4 5 6 7 8 9 10
11
12
Berg, S. (1940), Ingen. Vidensk. Skr. B, number 2,398 Kamack, H.J. (1951), Anal Chem., 23(6), 844-850,i9r^ Kamack, H.J. (1972), Br. J. AppL Phys., 5, 1962-1968,^02 Svarovsky, L. and Friedova, J. (1972), Powder TechnoL, 5(5), 213-211,402 Hanis, C.C. (1969), AMIE Trans., 244, 187,402 Svarovsky, L. and Svarovska, J. (1975), J. Phys D, 5, 1962-1968,402 Svarovsky, L. and Svarovska, J. (1975), J. Phys £, 9, 959-962,402 Svarovsky, L. and Svarovska, J. (1976), Dechema Monogram, Nurnberg, Nos. 1589-1615,279-292,402 Alex, W. (1972), Dissertation, Univ. Karlsruhe, Germany,40J Lloyd, P.J., Scarlett, B. and Sinclair, I. (1970), Proa Symp, Particle Size Analysis, Bradford, 267-275, publ. Soc. Analyt. Chem., (1972), ed. M.!.Groves and J.L. Wyatt-Sargent,40J Allen, T. and Svarovsky, L. (1976), Dechema Monogratth Numbers 79(1589-1615), 279-292, Verlag Chemie, GmbH, Weinheim/Bergstrasse, Nurnberg, 403, 412 Slater, C. and Cohen, L. (1962), 1 Sclent, fnstrum., 39, 614,40J
444 Powder sampling and particle size determination
13 14 15 16 17
18 19 20 o
22 23 24
25 26 27 28 29 30 31 32 33
34 35
36
Allen, T. and Svarovsky, L. (1974), Powder TechnoL, 10(1/2), 23-28, 406, 407 Allen, T. (1992), Centrifugal particle analyzer. US Patent 5,095,451J07 Allen, T. (1992), Proc Conf Particle Size Analysis, PSA 91 Loughborough University, Anal. Div. Chem Soc, publ. Heyden,^(?7 Allen. T. (1991), Proc. Int. Symp. Particle Size Analysis, publ. Royal Soc. Chem., ed. Stanley-Wood, N.G. and Lines, R., pp 498-513,'^0<^ Weiner, B.B., Tschamuter, W., Fairhurst, D. (1994), Proc. 2nd Int. Conf. Tungsten Refract,. Met., (publ 1995), 727-735, ed. A. Bose and R.J, Dowding, R.J., Princeton, NJ, \]S\,408 Foerderreuther, H. (1998), GT. Lahor-Faschz, 42(3), 234, GQV.,408 Allen, T. and Svarovsky, L. (1972), Proc. Soc. Anal. Chem,, 9(2), 38-40,^/2 Dumm, T.F. and Hogg, R. (1986), Part. Part. Char act., 3, 122,^^/7 Kaye, B. H. (1962), British Patent 895 222,4/.^ Treasure, C.R.G. (1964), Tech. Paper No 50, Whiting and Industrial Powders Res. Council, Welwyn, \J.K.,418 Coll, H. and Haseler, S.C. (1984), J. ColloidInterf Sci., 99, 59\A18 Devon, M.J., Meyer, E., Provder, T., Rudin, A, and Weiner, B.B. (1991), Particle Size Distribution II, ed. T. Provder, Amer. Chem. Soc. ACS Symposium Series 472, 154-168,4/(^ Jones, M.H. (1969), US Patent 3, 475,968,^79 Coll, H. and Searles, C.G. (1987), ./. Colloid Interf. Sci., 115(1), 121-129, 419 Devon, M.J., Provder, T. and Rudin, A. (1991), Particle Size Distribution II, ed T. Provder, Am. Chem Soc, 134-153,^/9 Hansen, F.K. (1991), Particle Size Distribution II, ed T. Provder, Am. Chem Soc, 169-183,4/9 Allen. T. (1988), Powder TechnoL, 50(3), 193-200,-^/9 Weiner, B.B., Fairhurst, D. and Tscharnuter, W.W. (1991), Particle Size Distribution II, ed T. Provder, Am. Chem Soc, 134-153,4/9 Weiner, B.B., Tscharnuter, W.W. and Bemt, W. (2002), Dispersion Science and Technology^ 23(5), 671-678,4/9 Cooper, A.A., Devon, M.J. and Rudin, A. (1989), J. Coating Technology, 61,169,419 BS 3406, Part 6, (1985), Determination of particle size distribution. Recommendations for centrifugal liquid sedimentation methods for particle size distribution J25 Coll, H. and Haseler, S.C. (1984), J. Colloid Interf Sci., 99(2), 591-592,426 Devon, M. J., Meyer, E., Provder, T., Rudin, A. and Weiner, B. B. (1990), Particle Size Distribution II, Assessment and Characterization, I, Am. Chem. Soc. Symposium Series 472, Ch. 10,426 Nelson, R.N.Jr., Khalili, M. and Allen, T. (1995), Poranai Symp. Particle Size Analysis and Powder Technology, Hungary, May,427
Centrifugal sedimentation methods 445
37 38 39 40 41 42 43
44
45 46 47 48 49 50 51 52 53 54 55 56
57 58 59 60 61 62 63
Oppenheimer, L.E. (1983), J. Colloid Inter/. ScL 92(2), 350-357,270 Coll, H. and Seaiies, C.G. (1987), J. Colloid Interf Sci., 115, 121-129,^27 Weiner, B.B., Fairhurst, D. and Tscharnuter, W.W. (1991), Particle Size Distribution II, Am. Chem. Soc, Ch. 12, ed. T Provder,-^27 Putman, B., Meeren Pahl van der and Vanderdeelen, J. (1999). Part. Part. Syst, Characterization, 14, 73-78,427 Bowen, P., Dirksen, J.A., Humphrey-Baker, R. and Jelinek, R. (1993), Powder TechnoL, 74, 61-11,429 Bowen, P., Heraud, C , Humphry-Baker, R. and Sato, E., (1994), Powder TechnoL, 81,234-240,429 Bowen, P., Humphry-Baker, R. and Eriksson, P.A. (1996), The 5th World Congr. of Chem. Eng., San Diego CA, 6, pp. 518-523, publ. Amer. Chem. SoQ,429 Miemann, J. and Weichert, R. (1995), 4th International Congress Optical Particle Sizing, Partec 95, Numberg, Germany, publ. NurnbergMesse GmbH.375-385,429 Niemann, J. and Weichert, R. (1995), Part. Part. Syst. Charact,, 289294,429 Romwalter, A. and Vendl, M. (1935), KolloidZ., 72, 1,433 Brown, C. (1944), J. Phys. Chem., 48, 246,433 Martin, S.W. (1939), fnd Engng Chem., Analyt. ed., 11, 41\-415,433 Martin, S.W. and Robinson, H.E. (1948), J. Phys. Colloid Chem., 52, 854881, (1949), J. Phys. Colloid Chem, 53, 860-886,43 J Kamack, H.J. (1972), Br. J. Appl Phys., 5, 1962-1968,4Ji Muschelknautz, E. (1974), Ger. OJfen., 2 324 421,434 Muschelknautz, (1975), Dechema Monogram, Nurnberg, 79(1589-1615), Part B, 267-278, Verlag Chemie GmbH, (1976), Weinheim'Bergstrasse,4i4 Sokalov, V et al (1975), Zh. Prike. Khim., (Leningrad), 48(7), \65\,434 Kaya, N., Yokoyama, T. and Arakawa, M. (1986), Kona, No 4, %2,434 Arakawa, M., Shimomura, G.Imamura, A. Yawaza, N., Kaya, N. and Kitai, H. (1984), J. Soc. Powder TechnoL, Japan, 21,16S,434 Kaya, N., Yokoyama, T. and Arakawa, M. (1990), Proc. Second World Congress Particle Technology, Sept., Kyota, Japan, Part 1, 518-525, Soc. Powder Technol. Japan,-/i4 Muschelknauts, E. (1974), Ger. Offen., 2 324 421,434 Muschelknauts, E. (1967), Verh. dt. Ing Z , 109(17), 151-161,434 Muschelknauts, E. (1976), Dechema Monogram, 79, Nr. 1589-1615, Nurnberg, Part B, 261-21S,434 Muschelknauts, E. (1993), Int. Chem. Eng., 33(3), 426-438,454 Brown, C. (1944), J. Phys. Chem,, 48, 246,4J5 Murley, R.D. (1965), Nature, 207, 1089,4i7 Donaghue, J.K. and Bostock, W. (1955), Trans. Inst. Chem. Engrs,, 33, 12J38
446 Powder sampling and particle size determination
64 65 66 67 68 69 70 11 72 73 74 75 76 77 78 79 80 81 82 83
ASTM C678-75 (Reapproved 1991), Standard test method for determination of particle size of alumina or quartz using centrifugal sedimentation,439 Simons, S.J.R. and Williams, R.A. (1992), Powder TechnoL, 73, 85-90,^^0 Hauser, E.A.and Read, C.E. (1936),./ Phys. Chem., 40, 1 \69J40 Hauser E.A. and Schachman, H.K. (1940), J. Phys. Chem., 44, 584,^^5, 440 Fancher, G., Oliphant, S.C. and Houssiere, C.R. (1942), Ind. Engng. Chem., Analyt. ed.. 14,552.440 Mcintosh, J. and Seibie, F.E. (1940), Br. J. Exp. Path., 21, 143,440 Schachman, H.K. (1948), J. Phys. Colloid Chem., 52, 1034-1045,440 Irani, R.R.and Callis, C.E. (1963), Particle Size Measurement, p.86 MacMillan,NY.,^^/ Hauser, E.A. and Lynn, J.F. (1940), Ind Engng. Chem., 32, 660,441 Saunders, E. (1948), Analyt. Chem., 20, 379,4^/ Bradley, D. (1962), Chem. Proc. Engng, 43, 591, ^/ seq, 634,441 Svedberg, T. (1938), Ind Engng Chem.. Analyt Edn., 10, 113,442 Svedberg, T. and Peterson, K.O. (1940), The Ultracentrifuge. Oxford Univ. Press,'/-/2 Alexander, .1. ed. (1926), Colloid Chem.. Chemical Catalogue Co.. NY Ch. 6 by T. Svedberg,^'^2 McCormick, H.W. (1964), J. ColloidSci., 19, \13,442 Brodnyan, J.G. (1960), J. Colloid Sci., 15, 563,^^2 Lombard, G.A. and Carr, W. (1975), J. Oil Colour Chem. Assoc, 58(7), 246251,^^2 Mueller, H.G. (1997), Progr. Colloid Polym. Sci., 107, 180-1 S6,442 Colfen, H. and Pauck, T. (1997), Colloid Polym. Sci.. 275(2), 175-180,442 Coffen, H. (2003), Polymeric Materials Science and Engg. 87, 345-347,442
Stream scanning methods of particle size measurement 9.1 Introduction It is convenient to divide particle size measurement techniques involving the interaction between particles and an external field into two categories, stream scanning and field scanning. In the former, particles are examined one at a time and their interaction is taken as a measure of their size. In the latter, the interaction of an assembly of particles is interpreted in terms of the size distribution of the assembly. The most widely used stream scanning technique employs the Coulter principle (Figure 9.1a) where the interrogating field is electrical and particle size (volume) is proportional to the change in electrical impedance as the particles pass through the field. Particle projected area can be measured by the amount of light cut off as a particle passes through a light beam (Figure 9.1b). With a small diameter rotating or scanning beam the pulse length is a measure of a random chord length (Figure 9.1c). Scanning beams, in concert with backscattering detectors, are also used for chord size determination (Figure 9.1g). These can be used with high concentration slurries since the beam does not have to traverse the suspension. If the incident beam is absorbed in a light trap the light scattered in the forward direction (Figure 9.Id) or at right angles (Figure 9.1e) is size dependent. The signal is greatly enhanced with the aid of an elliptical mirror (Figure 9. If). Interferometers (Figure 9.1h) determine particle size through the phase shift between a split laser beam, one passing through the particle and the other through the surrounding liquid. If particles are accelerated through a
448 Powder sampling and particle size determination
. Electrodes I
CyliiidriairpMicle apettme
^. . Signal
(b) Light blockage
(a) Electrical lesistaiice Lt^beam
Pmicle
"
Signal
Dwell jChofd time length
(c) Dwell time
((0 Low-angk forwvd acatter
IUghtto«|E»**tic^^ Photo-detector
PhottHlelietgQ
y\t^
fLight Signal (0 Right angle acatter
(e) Solid angle acatter SigSI
Dwell .Chord time leogdi Particle
AO Particle
Measmement 4 Doppler signal vtdume I Signal
ttt
-JL-
Si (i)rmie of
(VUme) - JSST"' flight
RefeveiKC bmn
(h) Imetferometty
(g) Back-scaoerdweUliiiie Accetemiiig P"««* Fbotodeiectofs
i^oianzediaaer ndS" AOo^izd / v ^ Beamsplitter Ibstbeam '^ TV > Test
.
Frequency Deam
fovsize
Q) P h w I>oi)plcr method
Fig. 9.1 Principles of streaming systems.
Stream scanning methods
449
nozzle, the time takes them to pass through two laser beams is a measure of aerodynamic size, (Figure 9.1i). In the phase Doppler method (Figure 9.1j) particle size is determined from the interference pattern as a particle passes through the intersection of two laser beams. Stream scanning is generally limited to low-concentration suspensions, and is best suited to the determination of particle size distribution by number and contamination monitoring. The Hiac/Royco light blockage method is allowed in USP XXI (1985) for the evaluation of particle burden in large volume dextrose solutions. The United States Pharmaceutical method, USP XXII (1988), for determining particulates in injectable solutions is based on microscopy and suffers several disadvantages [1]. Conversion of number distribution to mass (volume) distribution can result in gross errors unless the width of the distribution is narrow. For example, if the range is 10:1 the omission of a single 10-unit particle (volume=1000) is equivalent to the omission of 1000 one-unit particles. In order to obtain accurate volume distribution data it may therefore be necessary to size millions of small particles in order to get a statistically acceptable count at the coarse end of the distribution. 9.2 The electrical sensing zone method (the Coulter principle) 9.2.1 Introduction The Coulter technique is a method of determining the number and size distribution of particles suspended in an electrolyte by causing them to pass through a small orifice on either size of which is immersed an electrode. The changes in electrical impedance as particles pass through the orifice generate pulses whose amplitudes are proportional to the volumes of the particles. The pulses are fed to a pulse height analyzer where they are scaled and counted and, from the derived data, the size distribution of the suspended phase is determined. The Coulter principle was patented in 1949 [2] and described in 1956 [3] as a method for counting and sizing blood cells. Kubitschek [4,5] introduced modifications which permitted the counting of bacterial cells, and pointed out that the method could be applied to the measurement of cell volumes as well as number counting. Modified instruments were soon developed with which particles could be sized as well as counted. In 1998 the company was acquired by Beckman and renamed Beckman Coulter. Since analyses may be carried out rapidly and with good reproducibility using semi-skilled operators, the method has become very popular in a
450 Powder sampling and particle size determination
wide range of industries. In a recent Coulter bibliography 1432 industrial references and 436 pharmaceutical references are cited [6]. The manufacturers claim more than 6000 documented references using the Coulter principle. This type of counting device is designated in ASTM 3365-74T as a 'Tentative method of test for concentration and size distribution of airborne particulates collected in liquid media'. It is specified that the method is suitable for particulate matter greater than 0.6 |Lim in diameter collected by a Greenburg-Smith or midget impinger. The original development of the method was carried out by Anderson et. al. [7] and was extended by others. The Coulter principle is also standard for dry toners [8,9] and an accepted method for aluminum oxide powder [10], chromatography media [11], polymeric powders [12], plutonium [13], filter evaluation [14], catalytic material [15] and comparing particle size distribution using alternative types of particle counters [16]. In ASTM method C-21 it states that the experience of several laboratories indicates that the method is capable of a repeatability of 1% and a reproducibility of 3% at the 95% confidence level. Operating procedures for this technique are also covered in BS3405 [17]. The method is also the subject of an international standard [18]. 9.2.2 Operating principle The operating principle of the instrument may be followed by referring to Figure 9.2. The sample to be analyzed is dispersed in an electrolyte that is placed in a beaker. The glass sample tube, on either side of which there is a platinum electrode, is immersed in the electrolyte. A controlled vacuum initiates flow of suspension through a sapphire orifice let into the glass tube and unbalances a mercury siphon. The instrument can then operate in one of two modes: 1 The system may be isolated from the vacuum source by closing tap A and flow continues due to the balancing action of the mercury siphon. The advancing column of mercury activates a counter by means of start and stop probes placed so that a count is carried out whilst a known volume of electrolyte (0.05 ml, 0.50 ml, 2.0 ml) passes through the aperture.
Stream scanning methods
451
I\itc^h0l4 iiitiPi
Fig. 9.2 Line diagram of the Coulter Counter. 2 Data may be acquired in a preset count mode in which counting is initiated manually and ends when the preset count number (up to 1 million) has been reached. In the latest versions of the instrument the mercury column is replaced by a small vacuum pump. A current is passed through the aperture between the two electrodes. Particles drawn through the aperture cause momentary changes in the electrical impedance due to each particle displacing its own volume of electrolyte within the aperture itself. These changes in impedance are detected and presented as voltage pulses, the heights of which are proportional to the volumes of electrolyte displaced by the particles, and hence to the volumes of the particles themselves, provided they are neither unduly plate-like nor porous. The particle-generated pulses are amplified, sized and counted and, from the derived data, the particle size distribution is determined. Particle size analysis may be performed in the overall size range of 0.4 to 1200 |Lim. To achieve this, a number of different sensors (aperture tubes) are required. The operating range of each sensor is from around 2% to 60% of the aperture diameter, e.g. from 2 to 60 |Lim for a 100 ^m aperture tube. Smaller particles generate pulses that are lost in electronic noise generated within the aperture and in the electronic circuitry. Larger particles give increasingly non-linear response and tend to block the aperture if they are greater than about half the aperture diameter. For
452 Powder sampling and particle size determination
area of orifice iAj
Crc>ss-s€€lional area of element
m Fig. 9.3 The passage of a particle through the orifice of a Coulter Counter. powders with a wider size range, an extrapolation or a two-tube technique may be necessary. Alternatively the Coulter analysis can be combined with some other technique, e.g. the coarse end of the distribution can be analyzed by sieving and the two analyses combined after correction for shape factor. Care should be taken with powders having a wide size range since the uncounted fraction may form a substantial part of the distribution. 9.2.3 Theory for the electrical sensing zone method The basic assumption underlying the Coulter principle is that the voltage pulse generated when a particle passes through the aperture is directly proportional to particle volume. The relationship between particle size and instrument response may be determined using a simplified theory. Figure 9.3a shows a particle passing through the aperture and Figure 9.3b shows an element of the particle and aperture. The resistance of an element without a particle, 87?Q, is: PfU 5^ = The resistance of an element with a particle included, 67?, is that of two resistors 5i?j, 8/?2 in parallel 1 M
1
1
6i?, • + 8i?2
Stream scanning methods
453
so that:
bR =
pfhL
(9.1)
Ps^L
Pji p^, are the resistivities of the fluid and particle respectively, A is the cross-sectional area of the orifice, a is the cross-sectional area of the particle and Fis the volume of the particle. Thus the change in resistance of the element, 5(A/?), due to the presence of the particle is given by: 5(A/?) =8i?o - 87? abL S(A/?) = - ^
PL
1 - Pf
(9.2)
"5 J
The external resistance of the circuit is sufficiently high to ensure that a small change. A/?, in the resistance of the aperture due to the presence of a particle will not affect the current /; the voltage pulse generated is therefore /A/?. In practice, it is found that the response is independent of the resistivity of the particle. If this were not so, the whole technique would break down since a different calibration factor would be required for each electrolytesolid system. This independency is attributed to oxide surface films and ionic inertia of the Helmholtz electrical double layer and associated solvent molecules at the surface of the particles, their electrical resistance becoming infinite [19]. The terms involving {p/p^ may therefore be neglected and the preceding equation becomes: 5(A.) =
-
^
^
(9.3)
AH\-^ The response therefore, is not proportional to the volume of the particle, but is modified by the a/A term. For rod-shaped particles whose length is smaller than the aperture length, this leads to an oversizing of about 6% in
454 Powder sampling and particle size determination
terms of the diameter at the upper limit of each aperture with distortion of the measured size distribution [20,21]. This error decreases as a/A decreases. For spherical particles of diameter d = 2b the change in resistance due to an element of thickness 5/ at a distance / from the center of the sphere may be determined and this can be integrated to give the resistance change due to the particle [22,23].
(^'-n
A/?-
AR
_JPfd' 3nD'
4fd^^
1+ 5 KDJ
24 fd^' 35 v ^ y
\69 fd
+ ••
(9.4)
This gives a limiting value of two-thirds the Maxwellian value. Recognizing this, Gregg and Steidley multiplied their solution by three halves. This procedure has been questioned [22]. The complete solution is:
AR = -
Apf
sm~\d/D)
d_
nD
4\-(d/Df
D
(9.5)
This equation may be written: PfV
(9.6)
Hence, the instrument response is proportional to the volume of the sphere, F, modified by the function F^. This equation results from a simple integration of the area available for conduction. Several approximations have been derived for F^. [22,24-26] and these have been compared to find the one that agrees best with experiment [27]. The initial experimental data by de Blois and Bean were with PVC spheres, down to 0.09 |Lim in diameter, using a pore in a plastic sheet for an aperture. They concluded that the technique was applicable down to
Stream scanning methods
455
0.015 |Lim. In a later paper [28] they reported measurements down to a diameter of 0.06 fim, using a pore in a Nuclepore filter, and they also measured the osmotic velocity of the liquid in the pore. The error in assuming a linear relationship between resistivity change and particle volume for spherical particles is about 5.5% for {d/D) = 0.40. The principle may therefore be applied to higher ratios of (d/D) than 0.40 provided corrections are applied and aperture blockage does not become too troublesome. For non-spherical particles, F is modified by the inclusion of a shape factor [29]. In general, as the ratio of (d/D) increases, the resistance pulse generated is greater than predicted by assuming proportionality and oversizing of the larger particles occurs. 9.2.4 Effect of particle shape and orientation It has been claimed that particle shape, roughness and the nature of the material has little effect on the analysis [30] but there is considerable evidence that the size measured is the envelope of the particle. Comparison with other techniques gives good agreement for homogeneous spherical particles; for non-spherical particles results may differ [31,32]. For porous particles the measured volume may be several times the skeletal volume, and the apparent volume for floes is greater than the volume of the particles that make up the floes [20]. Since flaky particles rotate as they pass through the sensor, the measured volume may be the volume swept out by the particle and this can lead to oversizing. With extreme shapes such as rods, this may cause a change in size distribution, as apertures of different sizes are used, if the whole of the rod cannot be accommodated in the sensing zone. It has been reported that silica containing large pores can be undersized by as much as 100% due to pore filling by the electrolyte [33]. This particular effect has been used to measure the amount of particulate material within a floe [34] and the porosity of porous samples [35]. Anomalous results have also been reported for fly ash [21]. Ratios of 1.31:1 have been reported for non-extreme shapes with higher values for flaky particles [36]. For these reasons, it may be worthwhile to calibrate the analyzers with the test material or carry out a mass balance routinely as recommended in BS3406 Part 5. Model experiments have been carried out by Marshall [37], Lloyd et. al. [38] and Eckhoff [39] but no firm conclusions can be drawn from them since the models used differ widely from the commercial instruments. This work was extended by Harfield et. al. [40,41] using a large two-
456 Powder sampling and particle size determination
dimensional model and they obtained a linear response for particles of diameter up to 77% of orifice diameter. Kubitscheck [42] found that the output pulse due to the passage of a particle through the aperture was round topped not rectangular as expected theoretically (the pulse height should remain constant as long as the particle is in the aperture). This implies that the amplifier does not reach full response to the presence of each particle before it leaves the aperture and this can result in an undersizing of coarse particles. Kubitscheck found it necessary to construct apertures five times as long as they were wide, as opposed to the conventional 0.75 ratio, in order to produce flattopped pulses. Eckhoff [39] stated that the pulses were round-topped due to the shape of the electric field and rejected the assertion that roundtopped pulses were produced due to the instrument not reaching full response. Pulses deviate from the ideal, single modal shape when they are generated from coincident particles. These pulses take on a variety of shapes always of longer duration and often having multiple peaks.
35 yun \\\\\\\\\\'
Fig. 9.4 Shape of pulses generated by particles not passing centrally through the aperture
Stream scanning methods
457
9.2.5 Pulse shape Grover et. ah [26] conducted experimental studies of the pulses generated in the aperture and determined that the potential field was dense at the inlet and outlet edges. Particles traveling parallel to and near the walls pass regions of high potential gradient and generate M-shaped pulses. This does not affect the analysis of powders having a wide range of particle size but grossly skews narrow distributions to generate too coarse an analysis. Thom et. al. [43] used a magnified version of the Coulter Counter and, by drawing spheres through the aperture on nylon threads, mapped out the generated pulses on different streamlines (Figure 9.4). Pulse distortion was eliminated when the edges of the orifice were rounded to form a conical
Fig. 9.5 Hydrodynamic focusing. Suspension is focused on the axis of the measurement opening M with a probe D. The suspension streams through the axis after dilution with particle free electrolyte E. entrance and exit. They also made a central filament capillary for the injection of particles into the center of the orifice to eliminate pulse distortion. They called this set-up 'hydrodynamic focusing' (Figure 9.5). A secondary benefit of this arrangement is that the transit time is approximately constant for all particles. This work resulted in the development of an instrument available commercially as the Telefunken particle detector MS PDl 1105/1, which was reported on by Polke [44]. In
458 Powder sampling and particle size determination
1973, Coulter purchased the rights to manufacture this instrument that they made available as the model TF Coulter counter. In 1969 Coulter's patented a conical entrance and exit but in 1970 they filed another patent which indicated a preference for cylindrical apertures over the contoured variety, possibly because of plugging difficulties. They also considered channeling the particles down the center of the aperture but rejected this approach as impractical. In 1973 they were issued a patent for a rounded orifice. In 1973 IITRI patented a trumpet-shaped orifice that included a flow straightener. Orifice plugging was eliminated by the use of a screen in the flow straightener [36,45,46]. The use of a flow directional collar protected by a micromesh sieve whose openings are equal to 40% of the orifice diameter has two advantages: By protecting the orifice it permits the use of a multi-tube system to operate in the same suspension thus eliminating the need for the conventional two-tube analysis method for wide-ranging powders. It also permits the detection and measurement of particles having different aspect ratios; this is particularly useful for the detection of fibers in the presence of other shapes. Two types of pulse discrimination have been described; one type rejects all pulses having a rise time greater than a preset minimum; the other type accepts only those pulses generated by particles passing through the aperture on nearly central paths. A partial correction is to use an 'edit' switch so that grossly distorted pulses are not counted, but this can lead to the rejection of considerable information and is not necessarily free of bias. A better solution is to reshape the pulses electrically so as to provide information from them all. A full correction is to force the particles into a central streamline using 'hydrodynamic focusing' [49] but this complicates both the experimental set up and the analytical procedure. One effect of eliminating distorted pulses is the production of a narrower size distribution. Another is that a mass balance calibration is not possible due to count loss. The rejected pulses are oversize thus their inclusion tends to skew resulting size distribution to the right. Elkington and Wilson [48] examined narrow size distributions of particles and resolved an additional 'artifact' peak, on the coarse side of the main or normal distribution, which was generated by particles moving non-axially through the aperture. They used a Coulter ZQ, a Coulter Channelyzer ClOO with 'edit' on and 'edit' off and a Coulter TF. A comparison between the TF system, the 'edit' system and the standard system has been given by Lines [49], who also discusses the effect of using a long aperture tube.
Stream scanning methods
459
A later development was the use of four electrodes in a tube rather than two; the outer two electrodes are used for current injection and the inner two yield voltage measurements [50]. It is claimed that this arrangement reduces errors of false counts and oversizing. The signal shape, together with signal peak, gives particle size and shape characteristics. Theoretical modeling and experiment showed that aspect ratio along with particle diameter can be measured for example for cylindrical particles. 9.2.6 Effect of coincidence Two types of error arise due to more than one particle being in the measurement zone at any one time. Primary Coincidence. Two or more particles in the measurement zone give rise to two or more overlapping pulses. Depending on their proximity and electrical resolution, these pulses may not be resolved, leading to loss of count. Secondary Coincidence Two or more very close particles, which individually give rise to pulses below the threshold level, may collectively generate a pulse above the level. Thus if the concentration is too high, oversize particles begin to appear. A coincidence correction can be applied for primary coincidence but it is preferable to use a concentration so low that this effect is negligible. Under this condition, secondary coincidence errors are also reduced. The correction for primary coincidence is derived through Poisson probabilities of finding multiple particles in different parts of the sensing zone simultaneously. This yields the following relationship between true count A^ and observed count n:
^V
V
n = — 1 - exp s
(9.7)
If higher-order terms are neglected this gives: N =n+
N^
(9.8)
• 2v
V is the monitored volume for each count and s is the sensing zone volume which is slightly larger than the volume of the aperture.
460 Powder sampling and particle size determination
The equation developed on the basis of transit time distribution, takes the form: n = NQxA~-N\
(9.9)
V
givmg: ^r2
'
N = n-¥-N'
(9.10)
V
Note, these equations are of the form N = n-hpN^
(9.11)
The manufacturers propose the more convenient empirical equation: N = n + pn^
(9.12)
/7, the coincidence factor, can be calculated from:
D Vrsoo^ xlO"^
p = 2.5\ ' 100
V V
(9.13)
;
D is the orifice diameter in micrometers and v the volume of the suspension in microliters monitored for each count. The factor 2.5 was determined experimentally using a 100 |Lim diameter aperture tube 75 ]xm long with V = 500 \xL If it is assumed that the sensing zone comprises the volume of the aperture plus a hemisphere at the entrance and exit, s = lAlD^ so that s/v = 2.22xlO~^ which is not greatly different to the value of 2.5x10"^ found experimentally. 9.2.7 Multiple aperture methodfor powders having a wide size range (a) General If the size range of the powder is too wide to be covered by a single orifice, two or more aperture tubes can be used. As a general rule, if there is more than 2% of the distribution in the smallest size interval it is advisable to use a smaller orifice to determine the fine end of the distribution.
Stream scanning methods
461
An aperture tube should first be selected to give zero count in the top size channel and a smaller aperture tube should then be selected to size the fine end of the distribution. The ratio of the orifice diameters should be less than 5:1 to ensure that the overlapping counts will match. An analysis is first carried out w^ith the large aperture tube using a know^n volume of electrolyte: In order to facilitate calculation of dilution factors it is recommended that the weight of the beaker and the suspension be determined prior to the analysis. For an instrument that can provide a number count, the dilution factor can be calculated by counting the diluted suspension using the large aperture tube at two size settings at around one third of the orifice size. The ratio of this count, corrected for coincidence, to the corrected count on the original suspension, is the dilution factor. {b) Sieving technique The weight of the beaker and suspension is determined and the suspension is then poured through a micromesh sieve with openings approximately half the diameter of the orifice of the smaller aperture tube. The filtrate is collected in a clean, weighed beaker. The suspension beaker is next rinsed with clean electrolyte and the rinsings poured through the sieve and collected. The dilution factor is calculated from the weights of the original and final suspensions. (c) Sedimentation technique An analysis is carried out using the larger aperture tube as before. Using Stokes' equation, the time is calculated for particles coarser than the upper size limit for the smaller aperture tube to settle below the orifice. After this time has elapsed, an analysis is carried out with the smaller aperture tube, after diluting if necessary. With multichannel models it is necessary to calibrate the two aperture tubes so that the channel size levels coincide. The two sets of data should coincide in the overlap region to generate the combined analysis. An application of multiple aperture technique, using wet sieving to remove oversize, has been described for fly ash in the size range 1 to 200 |.im [51]. 9.2.8 Calibration Calibration is usually effected with the use of narrowly classified (monosize) spherical latex particles. Since the calibration particles are nearly monosize the pulses on the oscilloscope will be nearly uniform in
462 Powder sampling and particle size determination height. Rapid calibration may be made by observing the threshold level, t^, required to screen out the single height pulses displayed on the oscilloscope and to give a count n^. More accurately, a full count, Uj-, is taken at a visually determined threshold level of/^ ==0.5/^ and an oversize count, n^, at threshold setting t^ =1.5/^. The threshold setting, to give a count equal to half the difference between these two counts n^= 0.5(wy+Wo) is used for calibration purposes. An alternative procedure is to plot the number count against the instrument response and differentiate this to find the mode which is assumed to occur at t^. Calibration may also be made using powders under test provided the size range remains within the range of a single aperture tube. This procedure is a primary calibration procedure and is recommended in a recent standard as being superior to latex calibration [17,20,21,52-54]. This procedure cannot be used with the Coulter Multisizer due to count loss. One paper reported a count loss of over 30% which generated a mass loss of over 15%, which implies that the loss was preferentially of fines [55]. In a comparison between the TAII and Multisizer II weight percentage errors of ±10% were reported [56]. The volume of particles in a metered volume of suspension is: v^=
(9.14) VsPs
where: v = metered suspension volume, V^ = total suspension volume, w = total weight of the powder and p^= particle density. If t is the average threshold setting as the pulse count changes by An then! yp=^k'^nT Hence: 3_6
Vp
n'Znt p^6Vp^_l^ 7t Vs Ps ^nt
(9.15)
Stream scanning methods
463
If the calibration constant as determined by equation (9.15) is significantly smaller from that using monosize particles it is likely that the whole range of powder has not been examined. In that case, equation (9.15) may be used to determine the fraction undersize by comparing the experimental value of X A^^ with the expected value. If it is significantly larger, it is possible that the particles are porous and the envelope sizes are being measured, or that the particles are flocculating. A third possibility is that the assumed powder density is incorrect. Calibration materials in general use consist of pollens, latex spheres and glass spheres. Several investigators have discouraged the use of pollens due to non-sphericity, surface irregularities and changing size in dispersing media [57-59]. Various pollens, lattices and glass beads are available from Coulter, Dow, Duke Standards and CTI-TNO [60]. Alliet [61] describes a red bead latex available from Coulter UK, with a mean size of 18.99 jiim and a standard deviation of 0.18 |uim, as being a particularly promising latex. It is generally agreed that the standard deviations of the size distributions of the Dow latex particles measured by the Coulter counter are greater than those measured by microscopy and those quoted by Dow [62,63]. This may be due partly to the quality control methods used by Dow [36] and partly to the effects discussed in the section on pulse shape. Spherical hollow carbon particles (1-300) |Lim have also been used for calibration purposes [64]. 9.2.9 Carrying out a mass balance Although it is common practice to calibrate the Coulter Counter using a standard powder, it is possible to calibrate the instrument with the powder being examined. This is the preferred British Standard method [17]. It is reiterated that this procedure cannot be carried out with some instruments due to count loss. Essentially one balances the volume of particles passing through the measuring aperture with the known volume in the measurement sample. This serves a multiple purpose in that: • It indicates if part of the distribution has been missed; (This occurs if the size range of the powder is greater than the detection range of the aperture. This is not always obvious from the appearance of the determined distribution. In particular, one mode of a bimodal distribution could easily be lost).
464 Powder sampling and particle size determination • Particle dissolution or growth is detected; • It checks on the accuracy of the calibration and exposes measurement errors. This procedure should be used as routine since it indicates whether all the powder is accounted for and allows for correction for powder outside the measuring range of the aperture used. In an extreme case, it has been found that less than 5% of the total distribution was being measured and decisions were made based on these incorrect distributions. Alternatively, if the whole size range cannot be covered using a single aperture tube, a two-tube technique is required. This is not possible if the fraction unaccounted for is below the limit of the technique and the alternative mass balance procedure, as used with BCR 66 standard quartz powder, is as follows. Disperse w gram of powder in Vj cm^ of liquid. Pipette out V2 cm^ of suspension and add it to electrolyte to make up V^ cvn?. Determine the corrected Coulter count on a metered volume of v cm^. 9.2.10 Oversize counts on a mass basis using the Coulter Counter In many powder additives, the presence of oversize particles results in faults in the finished product. A technique has been developed [65] for determining the number concentration of these oversize particles using the Coulter Counter Multisizer II. The normal procedure for determining the number oversize is to carry out a mass balance and present the derived Coulter data graphically as counts/gram against particle size. The problem with using this procedure is that, in some cases, there are only one or two oversize particles in the presence of millions of smaller ones. It is therefore necessary to filter out many of the smaller particles and carry out a count on the oversize residue. This poses problems in that some oversize particles may be lost in the process and, even more likely, large contaminant particles may be introduced in the filtration process. A precision transparent sieve with ±2% tolerance, introduced by CoUimated Holes Inc., was tested as a simple alternative to straightforward Coulter counting and gave good, reproducible results when used in combination with a specially designed filter holder to reduce contamination. This sieve has an additional advantage in that it can be examined under a microscope to determine the nature of the oversize particles.
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9.2.11 Apparatus The original Coulter Counters have been replaced by Beckmann instruments although they are still widely used. These comprise the Coulter Counter model TAll, Coulter Multisizer TAIIE, the Multisizer TAIIE + edit analyzer, the Coulter Counter Model D Industrial and the Coulter Counter Z series Beckman Coulter Zl operates in the 1 to 120 \\m size range and 1 to 60 |Lim using ampoule insertable aprture tubes. Metered volumes include 0.10 ml, 0.50 ml and 1.00 ml. The mercury manometer is replaced with an oil displacement pump. Beckman Coulter Z2 uses the same technology as the Zl. It performs channelyzation of particle data into 256 channels while displaying size distribution data. The advanced user interface allows the operator to view the data in a variety of ways. Sample volumes as small as 10 ml can be handled using Accuvette II vials or less than 2 ml using ampule insertable aperture tubes. Beckman Coulter Multisizer i is a new generation instrument introduced by the new owners of the company. The counter provides number, surface and volume distributions in the size range 0.4 to 1200 |im in a single run. The instrument is mercury free thus eliminating potentially hazardous spills. The calibration system is fully automated with this instrument and the parameters are stored for future reference. Advanced digital processing circuitry allows the user to increase the resolution by a factor of up to one hundred. Malvern Sysmex SD-2000 particle counter and sizer delivers highperformance particle size analysis from 1 to 120 |Lim by combining electrozone sensing with hydrodynamic sheath flow focusing. Malvern Sysmex CDA 500 employs a mercury-free vacuum pump and is designed for counting and sizing cells and particles in the size range 1 |u,m to 60 |im. Micromeretics market a similar range of instruments covering a size range from 0.4 to 1200 jiim, under the trade mark Elzone (previously marketed by Particle Data Inc.). Their product line is listed below with brief notes on their principal differences. Micromeretics Elzone 5370 has one analysis station and is used primarily for particle and biocell counting and concentration analysis of particle-liquid systems. It reports particle count as a function of total liquid volume or total time. It can also provide fast low-resolution particle
466 Powder sampling and particle size determination
sizing. It is fully self-contained featuring an internal microprocessor, control keypad and 18 cm diagonal video monitor. Micromeretics Elzone 5380 has one analysis station features complete sizing and counting capability and reports particle size distribution as a function of number, area or volume. Micromeretics Elzone 5382 has two analysis stations has complete sizing and counting capability. In addition it has automated valves that nearly eliminate the need for operator intervention after the start of an analysis. Aperture tubes are available in 22 sizes ranging from 12 to 1900 jiim in diameter. The orifice is drilled in a synthetic jewel that is permanently sealed into the wall of the tube. For diameters greater than 480 |Lim the holes are drilled through a ceramic insert. The effective range of particle sizes a typical orifice can cover is 3% to 70% of the diameter. All models are available with a choice of sample dispersing options including propellor and magnetic stirrers and hydropulsers. Mercury manometers are available in eight sizes ranging from 10 to 5000 |LIL. Micromeretics Elzone 5380 and 5382 are operated via a separate control module running in a Windows^^ environment. Macro commands allow repetitive analyses to be scripted asnd executed with a few simple strokes. Collected data are presented in graphical and tabular form. 9.2.12 Limitations of the method The primary limitation of this technique is the need to suspend the powder in an electrolyte. For powders insoluble in water a 0.9% saline solution is often used and dispersion effected using ultrasonics. The manufacturers also provide a list of electrolytes for use with water-soluble materials but these can cause cleaning difficulties. The narrow size range covered using one aperture tube may necessitate the use of a two-tube procedure that can be onerous. For a powder having a wide size range, the large particles may cause troublesome aperture blockage with the smaller aperture. One procedure to cope with this is to sieve out the course particles prior to the analysis using the smaller aperture and the second procedure is to allow the coarse particles to sediment out below the aperture level. Particles, which do not pass along the axis of the cylindrical aperture, generate mis-shapen pulses that can greatly deform narrow size distributions. This effect can be reduced by editing out the mis-shapen pulses, or using a technique known as
Stream scanning methods
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hydrodynamic focusing, in which the particles are fed through the central streamline. 9.3 Fiber length analysis This is of fundamental importance in the pulp and paper industry which uses a light obscuration method - the Kajaani FS-200. This instrument is also used in the chemical industry for measuring the length of man-made fibers. The Advanced Fiber Information System (APIS) [66] contains a mechanism for opening a hand fed ribbon of fibers so that individual fibers can be presented aerodynamically to an electro-optical system for measuring fiber length. Length measurements for 10,000 individual fibers can be obtained in 5 min. The assumption made when the Coulter counter was first developed was that the variability in speed of the fibers, as they passed through the sensor, was small and could be assumed constant [67]. Analysis showed that this assumption was incorrect and a later instrument included a sensor to measure fiber speed [68]. Since the fibers are aligned with flow, the pulse length is a measure of fiber length and the pulse height is a measure of fiber width. The results showed excellent agreement with Suter-Webb measurements. The APIS is widely used for cotton fiber measurement in the dry state. Workplace exposure to asbestos fibers is usually assessed by personal sampling on a membrane filter that is subsequently examined using phase contrast or electron microscopy. Lilienfeld et. al [69] described an instrument that monitors asbestos fiber concentration in sampled air by laser light scattering from fibers oscillating in phase with an electric field and this instrument has also been used to monitor silicon carbide dust [70]. Light scattering has also been used to detect fibers in air down to 1 |j.m [71] and magnetically aligned fibers on a membrane filter [72]. Rood [73] aligned fibers using a simplified Prodi instrument, then passed them through a corona discharge so that a downstream precipitator deposits the fibers on to a removable glass slide where they retain their alignment. He then used the difference in light scattered parallel to and perpendicular to fiow in order to determine fiber length distribution. The Coulter principle has also been applied to fiber length determination [74,75] in which the aperture length was made greater than the fiber length. This approach is interesting in that the pulse duration is a measure of fiber length and pulse height is a measure of fiber volume.
468 Powder sampling and particle size determination
hence an estimate can be made of fiber thickness. An alternative approach to fiber measurement involves a flow collar upstream of the sensing zone to provide selective screening and fiber alignment with the aperture axis [36,45,48]. Elzone used a long flow tube upstream of the sensing zone to provide laminar flow and fiber alignment [76]; this flow is caused to join a clear liquid sheath that centralizes the fibers in the aperture. 9.4 Optical particle counters Optical particle counters, in the stream-scanning mode, have been used for many years to determine particulate contamination levels in liquids and in aerosols. Particle size may be determined in one of the following ways: • by the amount of light cut off by a particle as it passes through a sensitive zone in a light beam; • by collecting and measuring the light it scatters over a specific solid angle in the forward direction; • by collecting and measuring the light it scatters over a specific solid angle at an angle to the incident beam (usually a right angle); • by measuring the phase shift as a particle passes through a crossed laser beam; • by measuring the "time of flight" between two laser beams. Optical particle counters are available which range from simple to highly sophisticated, together with considerable design differences to cater for the wide range of applications. Instrument response depends on particle size, particle shape, particle orientation, wavelength of light, liquid flow rate and relative refractive index between the particle and its surroundings. In addition, the amount of light collected is determined by the geometry of the collecting system. The efficiency of the photo-detector will further determine the degree to which the light pulses can be converted to electronic signals that can be detected [77]. The sensitivity to particle shape is minimized with forward light scattering detectors and particles with aspect ratios of up to two-to-one can be measured [78]. Small off-axis collection angles limit the size of the sample volume thus reducing the problem of coincidence in particle flows of high number density [79]. Differentiation between particles of similar sizes is hampered by pick-up of
Stream scanning methods
469
stray light that causes noise. Stray light is light, reflected from internal surfaces of the sensor, which falls on the detector. This phenomenon is less of a problem with properly designed off-axis detectors than with coaxial systems. KnoUenberg and Veal [80] discuss operation, design and performance of optical counters in general and a review of extinction optical particle counters has been presented by Sommer [81] {cit. 82). A comprehensive review of laser-based techniques for particle size measurement, covering both stream scanning and field scanning methods, contains 167 references [83]. 2S00^m
1/4 in typically
HiacCMH-150 sensor Particles
1 in typically
Photodiode ISO ^ s q u a r e window
CoUimated Ught
Direction of flow Fig. 9.6 Light blocking principle (Hiac/Royco). 9.4.1 Light blockage In the light blockage technique (Figure 9.6) a narrow area of uniform illumination is established across the flow channel of the sensor so that the passage of a small particle causes an amount of light, proportional to the cross-sectional area of the particle, perpendicular to the beam, to be cut off. For a larger particle, having a diameter comparable with the width of the illuminated zone, the dependence of pulse height on particle diameter is linear. In both cases, the pulse height increases monotonically with particle diameter. Light blockage sensors are available for a variety of size ranges from 1 jum to 3 mm with a dynamic range for each sensor of 100:1 or less. Light blockage is the method of choice for sizing above a micron or so since the instrument response is less affected by variations in relative refractive index and particle morphology than light scattering whereas light scattering provides the higher sensitivity required for smaller particles.
470 Powder sampling and particle size determination
Light obscuration particle counters determine particle size from the projected areas of particles, giving information on only two dimensions. Umhauer [84,85] determines three projected areas of each particle by measuring in three mutually orthoganol directions. An application of this system to agglomerates of two to four spheres has been reported [86]. This work was later extended to measurements on several regular figures [87]. Umhauer et.al [88] designed a 90° single particle light scattering counter to measure the size and concentration of particles in gas flows at high temperature. With this counter, in-situ measurements could be carried out in pipes with cross-sections of around 60 cm^ with protection against heat and dust precipitation on the optical windows. 9.4.2 Optical disdrometer The prototype of a novel disdrometer has been described which features low cost, easy handling, robustness and flexibility [89]. The instrument was developed from one used for classifying raindrops into 20 size classes in the field of meteorology [90]. The size and velocity of particles in the diameter range of 0.3 to 10 mm are measurable with this prototype and the lower size can be reduced to 10 \xm with a second sensor. The disdrometer consists of an optical sensor, electronics and a PC. The commercially available optical sensor produces a horizontal sheet of light 160 cm long, 30 mm wide and 1 mm high. Within the receiver, the sheet is focused on a single photodiode. The transmitter and receiver are mounted in the same housing and the light sheet is folded to keep the instrument small. Particles passing through the sheet cause a decrease in received light due to light blockage and this results in a decrease in the primary voltage of 5 V. The voltage decrease is a linear function of particle cross-sectional area and the pulse duration the particle velocity. Particle size distribution was determined using a vibratory feeder at a distance of 115 cm from the disdrometer. A first analysis was made with semolina of around 0.5 mm diameter and a second was made with limestone of diameter around 1.7 mm. The data were then compared with that from a flatbed scanner (see Ch. 3). The measured distributions with the disdrometer were slightly broader with very similar medians. 9.4.3 Light scattering The relationship between particle size and scattered light intensity at any angle may be obtained for spheres and simple shapes using Mie theory.
Stream scanning methods
471
Particles larger than about 20 |Lim scatter light in proportion to the square of their diameter. For smaller particles, the amount of scattered light increases significantly to the sixth power of particle diameter for 0.2 |Lim particles. The scattered intensity does not necessarily increase monotonically with particle size at all angles. In order to maintain a monotonic increase, small angles or small differences in refractive index need to be used. The main disadvantage of low angle scattering is that the background scatter from the cell walls is higher than with 90° scatter and the main advantage is improved size resolution. A further advantage of low angle configurations is that the intensity is less dependent on relative refractive index than 90° scatter. The fluctuations that occur with a single well-defined angle are damped out by the use of large collection angles. These instruments cover the size range 0.05 |Lim to 30 |Lim with a dynamic range varying from 20:1 at the small end to 40:1 for particles bigger than about half a micron. Usually, basic commercial light scattering counters pick up the light scattered in the forward direction, and great care is taken to damp out the direct beam. Greater sensitivity is obtained by incorporating a lightcollecting device such as an elliptical mirror. These instruments find a ready market in measuring low-level contamination in pharmaceutical suspensions such as intravenous liquids, and for contamination measurements in industrial liquids, and are being increasingly used for on-line process monitoring. Often, two versions of an instrument are manufactured, one for measuring particles in suspension and the other for measuring particles in air. Simple collecting systems generate a non-monotonic response in the sub-micron size range. Figure 9.7a shows a calibration curve for a co-axial collection system that illustrates this feature. Between the sizes 0.8 |a,m to 1.8 |Lim the calibration curve changes direction so that the same output signal occurs for as many as three different particle sizes. For this reason, this type of system is limited to instruments with common size thresholds at 0.3 |Lim, 0.5 |j.m and 5 |Lim. Any size threshold between 0.8 |im and 1.8 |im is ambiguous. 9.7b shows the calibration curve for an off-axis collection system where the output signal for a 0.49 |j.m particle is the same as that for a 0.61 |im particle. Figure 9.7c shows the calibration curve for the Climet elliptical mirror system.
472 Powder sampling and particle size determination
10000.
'-'
,^.^,,„,^,,.,,
,
,^...,um
Particle size in microns
(a)
' ""'^ (b)
Paruclc size io microns
lOOOOr
0,2
1
1
PariicJc size in microns (c)
Fig. 9.7 Calibration curves for light scattering instruments after Chandler [91]. (a) Simple coaxial system, (b) simple off axis system, (c) Climet elliptical mirror system. Less sophisticated systems tend to use white light illumination, others use gas lasers or solid-state laser diodes. Laser diodes are smaller and more robust than gas lasers, resulting in smaller instruments. Their spatial properties have increased the sensitivity of particle sensors by a factor of two, enabling dynamic size ranges of 700:1 to be measured. Fast analog to
Stream scanning methods
473
digital conversion now counts every particle in semi-concentrated suspensions and provides 64,000 channels of particle size information [92]. The wavelength of white light ranges from 400 nm to 800 nm, heliumneon is 633 nm and solid state 780 nm. This variation results in differences in measured size when particles with a difference in refractive index from the calibration material are analyzed. Although the instruments are usually factory calibrated it is hardly surprising that instrument-toinstrument variability is high. Volumetric sensors examine the entire stream of liquid passing through the flow cell at relatively low flowrates. In situ sensors focus the light on only a small portion. Focusing intensifies the illumination resulting in lower detection limits. Volumetric sensors can detect particles as small as 0.2 |um while in-situ sensors can get down to 0.05 |Lim. The sensitivity of volumetric sensors is limited since the amount of scattered light varies as the sixth power of particle size, hence a 0.1 |Lim particle scatters 1/64 of the light scattered by a 0.2 |a.m particle. Light scattered from the interface between the sensor wall and the fluid also reaches the detector to give a noisy base line. Hydrodynamic focusing is found to reduce errors when measuring the size of particles in light scattering and phase Doppler instruments as well as electrical sensing zone instruments [93]. In order to increase the maximum number concentration per unit volume the size of the measuring zone has to be reduced. This is realized by using a highly focused beam. This results in a border zone error i.e. particles passing through the measurement beam at the borders generates a smaller pulse than it would be if it passed through the center. Umhauer [94] corrected for this by using a white light source and two detectors at right angles to the beam. Lindenthal and Molter [95] designed a sensing zone shaped like a T. The photomultiplier registers the pulse height for the determination of particle size and the pulse duration in combination with this to determine whether the particle is in the border zone. The Palas PCS 2000 was run with and without border correction, using filter test dusts AC fine and AC coarse. As expected the measurement without border correction was shifted towards the fines. The new instrument was evaluated using a monodisperse aerosol generated by a Sinclair-La Mer generator. The advantage of this set-up is that the correction for border zone error is no longer needed thus eliminating the need for a second PM. An instrument employing a laser source and two photodiode arrays at 49.3^ and 126.4° has been described [96]. The measurements are based on
474 Powder sampling and particle size determination Mie theory for horizontally and vertically polarized light. spheres were used for calibration.
Polystyrene
9.5 Commercial instruments 9.5.1 Aerometrics Aerometrics Eclipse is a white light blockage system designed for both liquid and gas systems. The probe head includes a light source, flow cell and photodiode. The processor can accommodate two heads covering the size ranges: 2-100 |Lim, 10-500 |Lim, 20-1000 |Lim and 50-2500 |Lim. The flow cells are compatible with most liquids and can be specially adapted for high-pressure environments. 9.5.2 Canty Vision Canty Vision is an in-line system that uses lighted video images and a microprocessor-based image analysis system to visually verify particle size, length, width and distribution. The microprocessor can monitor up to eight applications, under process conditions, with a lower limit of 1 |Lim.
Lionp
Lkhttnpand buu» hri^itness detector Fig. 9.8 The Climet elliptical mirror optical system 9.5.3 Climet Climet manufacture the following range of instruments for liquid-borne and air-borne particle counting. CI 200 series collect forward scattered light at angles, relative to the incident radiation, from 15° to 105°. The sample volume is located at the
Stream scanning methods
475
primary focal point of an elliptical mirror and the scattered light is picked up by a photomultiplier located at the secondary focal point (Figure 9.8). Its most distinct advantage over other systems is its high degree of monotonicity. Its disadvantage is that a large detector is required to collect light properly from particles at the extreme edge of the view volume and this drives up the price. CI 220 Liquid Particle Analyzer counts particles drawn from a syringe. Four size ranges are available via panel switches: 2 to 20, 5 to 50, 8 to 80 and 20 to 200 jiim. CI 221 On-line Monitor is designed for high purity liquids for counting particles in the size range 2 to 200 |Lim for flow rates of 120 to 750 ml min* 1. Particle concentrations up to 100 particles ml"l can be handled directly. Higher concentrations are sampled at proportionately lower flow rates down to 120mlmin-i. CI 7000 operates using light obscuration with white light and a fiber optics bundle for light collimation. Sensors are available to cover a variety of sizes from 1 to 1000 |Lim, each one encompassing a 50:1 dynamic range. The sampler accommodates sample sizes from 0.1 to 500 ml. The electronic niodule contains the electronic components for particle counting, data display and printing. Data are stored in 3000 size ranges simultaneously and can be recovered in six size ranges for display and printing. ()ther size ranges can be recovered by resetting the size thresholds. 9.5.4 Contamination Control Systems Contamination Control Systems AWK analyzers measure vibratory fed dry powders in the size range of 20 |um to 10 mm and drops in the 30 |um to 3 mm range at a rate of up to 10,000 particles per second. The analysis system consists of a sensor comprising measurement optics, control electronics and a personal computer that is connected to the sensor via a parallel or serial interface and undertakes evaluation and data display. The measuring principle is based on a combination of light scatter and obscuration. Particles crossing the measurement cell produce an impulse of height proportional to size. The impulses are examined to eliminate "bad" signals and this automatically eliminates coincident pulses. The data can also be processed to give an equivalent sieve distribution. The measurement time is selectable in 0.01 s steps up to 10 min and the
476 Powder sampling and particle size determination
generated pulses are classified into 32 size categories for final presentation in 8, 16 or 32 channels. 9.5.5 Danfoss VisionSensor Danfoss VisionSensorQueCheck system was developed for manufacturers whose products have to be checked, inspected and sorted at high speed. Using this system the lengths of components on a conveyor belt can be determined and out-of-spec components rejected. The instrument has also been used for size analysis of sugar crystals from 40 |im to several mm in size. By means of a vibratory feeder, the sugar is fed in free fall, past a vision camera and sized every 5s. The final result of the measurement is available after 100-300 frames and documented via an interface with a database and printer. Measured data includes particle count and size or projected area. 9.5.6 Faley Status Foley Status 8000 measures particles in non-corrosive liquids from 1 to 150 )Lim, using either of two available sensors based on the light extinction principle. Counts are collected in up to six preset particle sizes. Faley Status also manufacture several counters for airborne particulates. 9.5.7 Flowvision Flowvision Analyzer uses fiber optics to channel white light through a flowing liquid. As particles pass through the light beam they generate images that are converted to video and then analyzed using a high-speed digital computer. The computer first enhances the image and then classifies according to size. Particles are detected in the size range 2 to 1000 |im; the optimum range for sizing is 25 to 600 |Lim.
V4
Pholixiicxie
Fig. 9.9 The laser optic measuring system of the Galai Particle Analyzer.
Stream scanning methods
477
9.5.8 Galai Galai CIS-1 Model 2010 (Figure 9.9) scans the sample with a shaped and focused laser beam using a rotating wedge prism [97]. The time spent by the scanning beam on a particle is interpreted as particle size. A CCD TV microscope is incorporated into the basic unit, permitting the operator to observe the sample whilst it is being measured. The images can be enhanced, processed and analyzed to obtain an independent measure of size distribution, particle shape and state of dispersion. The instrument operates in the size ranges 0.7 to 150, 2 to 300, 5 to 600 and 12 to 1200 ]im. Interchangeable sample cells allow a wide variety of sample presentations; these consist of a spectrophotometer cell, a microscope slide, a liquid flow-through cell, an aerosol flow through cell and a thermoelectric cooled flow-through cell. Galai CIS-100 combines particle size analysis with dynamic shape characterization, covering the size range from 0.5 to 3600 |Lim in 300 discrete size intervals by changing a single lens. The Galai Video Microscope uses a synchronized strobe light and a black and white CCD camera to capture images at 1 or 30 times per second. The included software automatically analyses up to 30,000 particles with up to 1800 non-overlapping particles per frame. The measurement cell modules comprise systems for dry powders, aerosols, fibers in laminar flow and cells for particles in regular and opaque liquids. Galai Dynamic Shape Analyzer DSA-10 is a complete shape characterization system for particles in motion. All particles are classified by: maximum and minimum diameters, area and perimeter, aspect ratio, shape factor and more. A video microscope camera synchronized with a strobe light takes still pictures continuously of particles in dynamic flow, generating shape information on tens of thousands of particles, in the 1 to 6,000 |im range in minutes. Galai CIS-1000 is an on-line particle size analyzer. A bypass from the process line feeds the sample into the sensor unit where it is sized and either drained off or fed back to the line. Full compatibility with the laboratory instrument is maintained since it uses the identical combination of laser-based time-of-transit particle sizing using the 1001 sensor and dynamic shape analysis using the 1002 sensor. The size range covered is from 2 |Lim to 3600 |Lim with measurement of size, area, volume, shape, concentration and estimated surface area with a cycling time of 300 s.
478 Powder sampling and particle size determination 9.5.9 Kane May Kane May manufacture a series of instruments based on the light blocking principle. These consist of an on-line sampler, a small volume sampler and a large volume sampler. Each sensor operates over a size range of 75:1 using a variety of sensors and data may be processed into up to 10 size channels. 9.5.10 Kowa Kowa Nanolyzer^^ PC-500 uses a He-Ne source and 90^ light scattering to size from 0.1 |im in the size ranges: 0.1 to 0.2 |Lim, 0.2 to 0.5 lim and +0.5 |Lim at a flow rate of 20 ml min"^ and a maximum pressure of 70 psi. The axis of the measuring cell is parallel to the flow path and a mask placed in front of the photomultiplier precisely defines the measuring area. The image of the mask area is centered on the laser focus, where beam intensity is constant, in order to ensure even spatial sensitivity across the measuring area. Moreover, with detection occurring along the flow path, particles flow through the mask image, so that the measuring area is not only precisely defined but is also of equal size for all particle diameters. This configuration guarantees accurate measurement of particle size and concentration in each size range. The analyzer is intended for use with clear liquids. Kowa Nanolyzer^^ PC-30 is of similar design and intended for monitoring ultra clean water. It classifies particles in the size ranges, 0.08 to 0.1 ]um, 0.1 to 0.15 |im, 0.15 to 0.2 |um and +0.2 |Lim, in concentrations of less than 30 particles ml~^. Particles are sized according to Mie theory. 9.5.11 Kratel Kratel Partascope operates using a light blocking detection system. Sensors are available, each one covering a 50:1 size range, to give an overall size range of 1 to 8000 |Lim. Data are presented in 4, 8, 16 or 32 channels using a multichannel analyzer module and this can be expanded to 64 channels using a computer. As with Hiac/Royco, a range of samplers is available. A sub-micron sensor is also available for the size range of 0.4 to 20 |im using near-forward laser light scattering. Kratel Partograph measures size, extinction, light scattering and fluorescence of particles. Hydrodynamic focusing is used to allow single
Stream scanning methods
479
particle centering in the beam. The optical design allows the simultaneous measurement of the particle size by forward and right angle light scattering as well as an extinction measurement. Particle size is also determined by time of flight from pulse width. Size range covered is 0.5 |Lim to 200 |Lim at concentrations up to 10"^particles ml"^, at flow rates of 10-50 JLIL min"^. beam block
laser beam
n
fiber optic cable to signal processor
measurement volume
fiber optic coupler Fig. 9.10 Light scattering geometry used in the Malvern Insitec PCSV instrument 9.5.12 Malvern Malvern Autocounters [98] use light extinction sensors for particle measurement. Although designed primarily for counting particles in hydraulic and fluid power systems the ALPS 15OH can be used for any liquid samples. Eight size thresholds are available in the 2 to 100 |Lim, 3 to 150 |im and 5 to 250 \xm range with automatic verification to principal contamination standards. The flowrate can be adjusted between 1 and 30 ml min-i with particle counts up to 10,000 ml~^. Simple, in-field calibration is an important feature of all models of this instrument. Malvern ALPS 100 system liquid particle counter is a modular system that can be used with an autosampler for multiple samples or with an online sampler for direct measurement of flowing liquids. It can be used with sample volumes down to 0.5 ml and uses a built-in multi-channel analyzer to perform size distribution analyses of low concentration dispersions. Suitable for both aqueous liquids and solvents, it measures up to 50 size bands in the 2 to 100 |Lim or 3 to 150 |Lim size range with output from a built-in thermal printer or external dot matrix printer. Malvern Autocounter 300A Air Particle Counter is designed for general cleanroom and environmental monitoring. It is a 0.3 jum to +5 |Lim, 1 cfm
480 Powder sampling and particle size determination
(cubic feet per minute) laser particle counter with eight preset or user defined size thresholds. Malvern Insitec PCSV-P (particle counter sizer velocimeter-probe) is a non-intrusive U-shaped instrument, with the transmitter and receiver mounted on opposite sides of the tube, designed primarily for basic particle research and filter testing. The instrument is intended for in-situ monitoring of particle physical characteristics at medium (< 100 ppm) and very low (< 1 ppb) particle concentrations in a size range from 0.2 to 200 |im. In this instrument (Figure 9.10) an He/Ne laser beam is used to illuminate the particles and particle size is calculated from the amplitude of the scattered light measured in the near forward direction [99-101]. Employing near-forward scattering creates some degree of insensitivity for particles with aspect ratios less than 2. A separate response function is used for light absorbing and non-absorbing particles. Holve and Self [102,103] developed an intensity deconvolution algorithm that corrected for the intensity profile of the sample volume based on the assumption that trajectories of the particles through the sample volume are random. Two beam widths are used to measure the wide size range covered. Measurements on the two beam widths are made sequentially and combined to give the full distribution. Information on the velocity of the particle is found by measuring the transit time of the particle through the sample volume [104]. The instrument has been used in large scale pulverized coal boilers [105-108] char fragmentation and fly ash formation during pulverized coal combustion [109] and for coal slurries [110,111]. 9.5.13 Pacific Scientific Hiac/Royco, Met One Hiac/Royco is a division of Pacific Scientific together with Met One. Hiac manufacture a wide range of counters for liquid borne systems whereas Royco and Met One manufacture counters to monitor airborne contamination. A brief description of a selection from the Hiac range is presented below. Hiac PC4000 portable liquid particle counter is a contamination measurement tool, designed to run on-line analyses of hydraulic systems and fluids. The fully self-contained counter operates in the light-blocking mode using a laser diode and reports contamination levels at 4, 6, 10, 14, 21,38 and 70 ^m at a flow rate of 60 ml min"^ Hiac 8011 liquid particle counting system provides fast results for particle contamination analysis. Composed of a light blocking or dual
Stream scanning methods
481
mode sensor, the system pressurizes fluid in an ABS-2 (automatic bottle sampler) for precise volumetric sampling of industrial fluids. Hiac 8012 liquid particle counting system analyzes viscous, dark, dirty fluids without dilution. The system includes the Hiac SDS (syringe driven sampler), 8000A counter and an HRLD (Hiac/Royco laser diode) sensor. Hiac BR'8 liquid particle counting system is designed for testing filter efficiency simultaneously sampling upstream and downstream. High resolution, large dynamic range laser diode sensors provide an accurate profile of the particle size distributions in the size range 0.5 to 400 |Lim at a concentration up to 18,000 particles per ml and a flow rate of up to 200mlmin-i. Hiac ChemQuaP^ on line sampler is designed to provide a solution for the connection of a particle sensor to bulk chemical distribution systems and distribution valve boxes for pre-hookup qualification. A built-in air pump eliminates the need for an external gas supply for enclosure and line purging. For continuous monitoring applications, the sampler is also compatible with Particle Vision® Online facility monitoring software. The light source is a laser diode with count display at 4 channels 0.1, 0.2, 0.3, 0.5 ^im at a flow rate of 100 ml min~i. Hiac SDS (syringe driven sampler) is designed for particle contamination determination of high viscosity fluids and applications where only small amounts of sample are available. A liquid crystal display allows the user to view the distribution in tabular or histogram format Hiac 3000A liquid syringe particle sampler incorporates precision stepper motors for sample testing, eliminating the need for pressure airlines. The microprocessor-controlled sampler is designed to interface with the 8000A counter or with the liquid particle counting system and software. Sample testing parameters, such as volume and number of runs, are entered by the operator either at the 8000A counter or within the software. The counter and software then automatically give testing instructions to the 3 000A sampler. Hiac ABS-2 automatic bottle sampler is a pressure sampling device used for batch analysis of volatile or viscous fluids. The ABS-2 delivers liquid to the sensor at flow rates of 10 to 200 ml min"^. A check valve in the sample introduction line eliminates backflow and minimizes sample cross contamination, and an automatic drain mechanism allows unattended sample analysis of multiple runs. The sampler has a built-in pressure/vacuum chamber that subjects samples to pressures up to 60 psi, accommodating viscosities up to 80 centistokes. In addition, vacuum levels of 23 psi are used to degas samples which contain entrained air.
482 Powder sampling and particle size determination
Hiac SDS syringe driven sampler is designed for high viscosity fluids and applications where only small amounts of sample are available. The sample syringe is close to the sensor, permitting particle analysis with as little as 3 ml of available sample. Hiac 8000A eight-channel particle counter incorporates state-of-the-art electronics to process and display real-time sample data. An image of the particle size distribution data is presented using. The LCD displays the data as a histogram or in tabular format. Parameters of the analysis, alarm conditions and standards protocol are presented as text against a contrastadjustable backlit screen. With a single keyboard entry small volume parenterals, product cleanliness levels and the cleanliness of components used in hydraulic systems can be tested. Hardcopy printout is provided through an internal 40 column thermal printer. Hiac 2000 liquid particle counter provides an inexpensive means to transfer up to four channels of particle size information data from sensor to host computer system. Data can be viewed in real-time via a liquid crystal display. The 2000 interfaces to all Hiac liquid sensors including the MicroCount, submicron and HRLD laser sensors. Applications include point-of-use monitoring for corrosive chemical delivery systems, DI water lines, wet process tools, hydraulic oil systems and parts cleaning Hiac 215W on line liquid particle counter is designed for automated particle monitoring in industrial environments. The counter is rugged with a vibration isolated laser diode and clog-free' sample cell on the inside. Particle size data from 2 to 400 |Lim (1-300 |im optional) is presented in six channels using light blockage. PCX interfacing Pacific Scientific liquid counter/controller interfaces Pacific Scientifics complete family of liquid sensors to their Particle Vision® Online software. The iPCX is designed for online continual monitoring applications. Hiac/Royco manufacture two counters that are compatible with a wide range of samplers. The 4100 series microprocessor based counter produces particle count data simultaneously in six channels; this is expanded to thirty-two channels with the 4300 series counter. Detection limits range from 0.25 to 25 |Lim for air samplers and 0.5 to 9000 |Lim for liquid sensors. A range of sensors is available for liquid systems, each one covering a 50:1 size range. The particles to be counted are suspended in a liquid having a refractive index different from the particles. The particles are then forced through a sensor containing a small rectangular cell with windows on opposite sides. A collimated beam of light from a high intensity, quartz halogen lamp is directed through the stream of liquid on to
Stream scanning methods
483
a sensor. The particles pass through the sensor in random orientation and produce pulses proportional to their average projected area. These pulses are scaled and counted. The sub-micron size range is covered using nearforward laser light scattering. The shear forces in the sensing zone are small enough for the instrument to be used for sizing floes [98]. Hiac/Royco 4100 series are for liquid-borne systems. Systems 4101 and 4102 are for airborne monitoring using models 1100 and 1200 airborne sensors respectively. Hiac/Royco Optisizer [112] is a single particle counting system which operates with composite scattering/extinction sensors, a 64,000 channel high speed digital counter, an automatic sample dilution system and a software package. The software controls the counter/sampler and acquires, archives and reduces the data. Near forward light scattering is used to measure particles smaller than 3 |Lim and light extinction for particles above 3 |im. Combining the two optical sizing techniques into a single sensor allows sizing over a dynamic range exceeding 700:1. The model HPS-200 uses a He-Ne laser and covers the size range 0.2 |um to 200 |Lim; the model HPS-350 uses a laser diode and covers the size range 0.7 |Lim to 350 |Lim. Hiac/Royco also offer the Dynacount laser diode sensors for sizing particles in liquids with a dynamic range of greater than 150:1 at number concentration levels up to 12000 ml~i. HRLD-150 covers the size range of 1 to 150 |am at particle concentrations up to 12,000 mh^ and flow rates from 10 to 75 ml xmxr^. HRLD-400 covers the size range of 2 to 40 |Lim at particle concentrations up to 8,000 mh^ and flow rates between 10 and 200mlmin-i. Hiac/Royco Model 5100 series is a He-Ne laser-based aerosol particle counter which uses a solid-state photodiode detector to collect light scattered, at 90° to the incident beam, over an angular range from 60° to 120°. Its standard dynamic range is 40:1, from 0.25 to 10 |Lim at 1 cfm, with automatic collection and averaging of up to 10 sample runs. Applications include qualifying and monitoring class 100,000 to class 1 clean-rooms, downstream filter testing, pharmaceutical and semiconductor manufacture, aerospace research and inhalable particle monitoring. The sample stream is focused so that all particles in the stream are counted. Pacific Scientific Met One Model 211 series of sensors are laser-based light-scattering sensors for liquidbome particles. Their rugged design, flexibility, and low cost make them ideal for industrial day-to-day use. Pacific Scentific Met One Model 100 series of sensors are laser-based light-blocking sensors for liquid by Met One. These have a simpler optics
484 Powder sampling and particle size determination
design than light-scattering sensors and therefore cost less. They are effective when counting dark-colored particles but not as effective as lightscattering sensors when counting light-colored or small particles. They remain calibrated over a large range of flow rates, handle higher concentrations of particles and have a longer service life than light scattering sensors. Pacific Scientific Met One 210 Liquid Particle Counter is used to measure particles in clean fluids used in electronic, pharmaceutical and other manufacturing processes. It classifies particles in six size ranges in the 0.4 to 25 |Lim size range using laser diode based forward light scattering. Maximum count rate is 8000 particles per minute at a fluid flow rate of 100 ml min"^. Pacific Scientific Met One 2500 batch sampling system is computer controlled for greater flexibility. Flowrate and sample volume are controlled automatically to ensure precise particle counting and sizing of liquid-borne particles. The sampler comes with either a 500 ml or 1000 ml sample capacity and uses either light scattering or light extinction sensors. The system's software displays and prints average counts per ml for up to sixteen size ranges in both graphic and numeric format. Sensors are available for various size ranges from 0.5 to 400 |Lim. 9.5.14 Particle Measuring Systems Particle Measuring Systems (PMS) manufacture a range of laser-based particle sizing instruments. These are based on volumetric or in-situ sampling. The former perform particle sizing on the whole of the fluid passing through them and have relatively low flow rates. Sizing may be accomplished using either light obscuration or light scattering. Volumetric sampling may be continuous or batch; batch sampling is accomplished using a liquid sampler to deliver a sample from a vial, bottle or tank to the sensor; continuous sampling is accomplished by measuring a side-stream flow from the primary line. In situ sensors accommodate a wide range of flow rates and perform the measurements directly in-line with the main flow of fluid. Particle sizing is accomplished using light scattered from particles within an optically defined area. The ratio between this area and the flow cross-sectional area yields the fraction sampled. Particles are classified into either 8 or 16 size intervals. Their Ultra DI monitors are desgned specifically for deionized water systems for contaminents as small as 0.03 |Lim.
Stream scanning methods
485
S;implc outlet P;srjhalic mirrors
Capillary Kcfcrejwx phoicdiodc
Rcfca-iKC preumplifjcr
Signal
ConNJeiising
\
' Laser %mxnn%
wimdow
Signal preamplifier
/ Sumpic m\t\
Fig. 9.11 Particle Measuring Systems' Liquid Volumetric Probe, designed for continuous monitoring of process waters, product liquids and semiconductor chemicals. PMS CLS'700 Corrosive liquid sampler was developed for batch sampling process chemicals at temperatures up to 150°C. Up to 15 sizing thresholds are available from 0.2 to 5 |im. Simultaneous multipoint measurements at 0.3 |Lim, 0.4 |im, or 0.5 |Lim can be made. Sensitivity is greatly improved with the use of parabolic mirrors (Figure 9.11). PMS CLS'900 Corrosive liquid sampler is a compression sampler designed for integration into wet benches and process tools. The compressive sampler eliminates bubbles from process chemicals PMS CLS'IOOO Corrosive liquid sampler is the latest offering from PMS for monitoring the chemicalswithin ultra-pure processing environments. PMS HSLIS liquid optical counter provides continuous real-time monitoring of contamination levels in DI watwer and process chemicals providing size sensitivity down to 0.5 \xm in DI water and 0.65 |Lim in process chemicals. PMS APSS'200 Automated parenteral sampling system for pharmaceutical fluids is designed to size and count suspended particlate matter in a wide range of liquids. PMS APSS Automated particle sampling systems are used to size and count suspended particles in liquids. These systems are ideal for applications where precise small volume sampling is needed. PMS LiQuilaz® counting spectrometer for liquids is designed to measure particles in liquids for a wide range of applications.
486 Powder sampling and particle size determination
Nano in-situ sensor and LS-50 liquid sampler are designed for particle measurements in process chemicals and DI water. They are ideal for sampling main process supply lines or point-of-use delivery lines where continuous monitoring of contamination levels is necessary. They can be operated either in-line or batch mode. PMS Pressurized Automated Liquid Sampler (PALS) is designed for the fields of hydraulic oil contamination monitoring together with pharmaceutical particle size measurement of viscous and volatile liquids. This batch sampler has a self-sealing sample chamber, programmable volumetric settings from 0.1 to 100 ml per cycle, automatic flush cycle, computer controlled and monitored sample chamber pressure and corresponding liquid flow rate. The heart of the system is the Particle Sizing System AccuSizer 580 pulse height analyzer. Data are collected in 32 channels that can be expanded to 128 logarithmetically spaced channels for high-resolution particle size distribution. Light extinction sensors cover the size range from 0.5 to 2500 |im. 9.5.15 Partike I Messetechnik Partikel Messtechnik (PMT) manufacture a range of microprocessor controlled 16 and 32 channel analyzers. PMT RBG-1000 is a dry powder dispersing system covering the size ranges 1.5 |im to 500 |im and 20 |Lim to 2000 |im. A dry system, with online capability, is also available for granules and free flowing powders covering the micron size ranges, 20-2000, 80-8000, 200-15000. The SAS sampling system covers the same size ranges as the RBG-1000 but is intended for suspensions and emulsions; this can also be used on-line. PMT-2120 counter uses light blockage and can count up to 3,000 particles per second. It covers the size range 1-12,000 |Lim with selectable 16 or 32 channels. 9.5.16 Particle Sizing Systems Particle Sizing Systems (PSS) is a designer and manufacturer of particle sizing instruments that are used for research and development, US? quality assurance, contamination and in-line monitoring. PSS Accusizer^^ 780/SPOS Single particle optical sizer consists of a patented autodilution stage, single particle optical sensor, pulse height analyzer/counter (PHA), systems computer/processor and software control. The autodiluter performs a continuous dilution of the concentrated sample
Stream scanning methods
487
suspension prior to its passage through the optical sensor. The PHA module continuously monitors the pulse rate during autodilution and when it falls below the coincidence level (typically 10,000 particles per second for particles smaller than 100 |Lim) the PHA unit starts to collect data. The resulting size distribution is displayed in real time as absolute counts versus diameter for each diameter channel (8 to 512) logarithmetically displayed over the total size ranges covered by the sensors (e.g. 0.5 to 400 |Lim, 1.0 to 400 |Lim, 2.0 to 1000 |Lim, 3.0 to 2500 |Lim). Additional derived distributions are calculated from the measured number distribution. The instrument can be combined with the Nicomp 380 to extend the lower size limit to 0.003 |Lim. Ancillary equipment comprises: A syringe injection sampler, the 780/SIS, which includes 32 userdefmed channels that can be converted to 512 channels to give contamination monitoring and particle size analysis in a single system. A dry powder sampler, the 780/DPS, that has an air-flow design that minimizes cell wall contact. The flow channel has a 1 cm diameter allowing for a size range from 5 to 5000|Lim. Air-flow can be controlled to change the shear forces applied to the particle stream in order to optimize particle deagglomeration. In cases where no shear forces are required the particles are fed through the sensor under gravity. Fillers
rxc Wni 10 air
Punip^ Row direciiofi Conccntr;ilcdl ^iampk
=^
Fresh dilueril
Sensor outpyi
V
GD.V . Ligli
]/ Dm\n m colleclur
Fig. 9.12 Schematic diagram of the Particle Sizing Systems Accusizer autodilution apparatus and optical particle counter.
488 Powder sampling and particle size determination
The autodilution system is shown in Figure 9.12. A few drops of concentrated liquid suspension containing several grams of powder are manually mixed into the mixing chamber. Filtered diluent is caused to flow into the mixing chamber, the resulting positive pressure causing some of the suspension to exit through the sensor at a rate of 25-100 ml min"^. The PHA starts counting when the count rate falls to 20,000 s"! [113,114]. PSS Accusizer 780/OL is specifically designed for on-line applications. Simple customizable computer controlled liquid sampling devices are used to grab an aliquot of concentrated in-line suspension and inject it into the autodilution unit. Multi-port systems are available to monitor multiple points in an on-line process. One great virtue of the instrument is its ability to measure minute amounts of oversize particles in the tail of a distribution. The oversize fraction from a homogenizer, for example, is more pertinant for control purposes than the mean size. 9.5.17 Polytec Polytec HC (high concentration) optical counter series is for counting and size analysis of particles, droplets and bubbles in gases or liquids in the size range 0.4 to 300 |im. The HC system uses a white light source and a relatively large aperture which smoothes out irregularities in the intensity of the transmitted beam. The near linear relationship between intensity and particle size provides sufficient resolution to allow classification of the particles into 128 size channels in a size range of 30:1 at number concentrations up to 10^ particles ml~^ and at a velocity of 0.1 m s~i to 10 m s~i. A correction can be applied for coincidence errors. The optical design of the HC system (Figure 9.13) defines a small rectangular measurement volume, at right angles to the flow direction, by imaging rectangular apertures with the transmitting and receiving optics. A particle passing through the illuminated measurement volume scatters light that is focused on to a photodetector. Each optical signal is converted into an electrical pulse that is electronically processed. 9.5.18 Rion Rion Laser Based Liquid-borne Particle Counter uses an optional sampler and 'sideways' scatter to allow off-line, on-line and automatic measurement down to 0.2 |im using a range of configurations. A range of sensors is available for use with either corrosive or aqueous liquids.
Stream scanning methods
489
L^i^ps J Phmoniulliplier
^^
\\ / \
live I Lighi Objective - J 4 k 4-^ source
ih)
Fig. 9.13 (a) Block diagram of the Polytec HC Particle Counter (b) optics of Polytec HC white light counter 9.5.19 Spectrex Spectrex PC 2000 Laser particle counter counts and sizes particles, from 0.5 to 100 |xm in diameter, in both flowing and in-situ liquids. The PC 2000 uses near-forward scattering from a revolving laser beam (900 rpm.) for particle sizing and counting of in-situ and flowing liquids [115]. A He-Ne laser beam is focused to a small, well-defined volume (10 ml) within the liquid. Total particle count within the range 0.5 to 100 \\xx\ can be determined in less than a minute. Readout is in 1 |Lim steps from 0.5 to 17 )Lim and in five channel steps from 17 to 100 |im. Distribution may be presented on a number or a volume basis. A small vial attachment permits
490 Powder sampling and particle size determination
inspection of vials and ampoules down to 5 ml size at concentrations up to 1000 particles ml-i. Spectrex provides three sealed calibration standards containing a precise number of NIST traceable polystyrene spheres of know^n size in suspension. Supercount software is a custom interfacing electronics software to provide an easy means of analyzing and saving data generated by the PC 2000. Supercount provides instant sizing information and histogram in addition to indicating: absolute count, mean size, mass ditribution, standard deviation anmd total suspended solids.
spring loaded moiiniing its^Hcmbly
Beaker
Sensor pr(*e assembly
Fm it lolling lable
nDrjp (a)
Moyntintg bracket
^czr~
\zr
phoTOlciccU^r
source
(b)
Beaker
Adju.siable phitfcirni
Fig. 9.14 The Spectrex SPC-510, (a) schematic of particle counter, (b) laser beam optics. 1 on-off switch, 2 digital readout, 3 count button, 4 threshold setting dial, 5 Illuminate button, 6 prism, 7 secondary lens, 8 target, 9 bottle, 10 sensitive zone, 11 scanning laser beam, 12 prism, 13 beam splitter and beam strength monitor, 14 scanner, 15 mirror, 16 lamp.
Stream scanning methods
491
The dilution factor can be automatically computed to give absolute counts for liquids as dense as sludge. Supercount is also loaded with a special program for hydraulic fluids and phi classification. Spectrex SPC-510 (Figure 9.14) uses both diffuse vertical illumination for visual identification of large particles and a scanning laser for detection of small particles. The instrument is widely used for quantitative particle counts in bottles [116] including in situ examination of bottled beer [117]. Spectrex ILI-IOOO Particle Counter combines the Prototron with a Particle Profile Attachment (multichannel analyzer). The instrument has been used [118] for examining volcanic ash. AC Fine Dust was used for calibration in eight 5 jum steps, which indicated that accurate data was obtained for sizes above 2 |Lim. It has also been shown to correlate well with the more tedious filtration and counting method for large volume parenteral liquids [119]. Although semi-transparent containers or liquids reduce the amount of transmitted light flux, the instrument gives valid data for particulates in oil [120]. Spectrex PC 50 Partascope viewer consists of a small, battery-operated "Black Box" which internally generates two parallel, scanning laser beams. Samples of water, oil or other liquid are introduced by beaker or bottle. Through Fraunhofer diffraction, particles in suspension, as small as 1 |im diameter, are made visible directly to the human eye. The two laser beams permit comparison of a sample with standards, or of pre- and post-filter samples. Sample flow
Beam Wock
Mow cell
Rcflcciar llKfluiiiMltiplit:r
Ismt bmm
He-Ne Laser
Fig. 9.15 Spectrex PCT-1, principle of measurement.
492 Powder sampling and particle size determination
^ ^ ^ T I ^ ^ / lOCUd
h I
. DJensuring < "^— 1 d>ne
^ ^ J ! /accepled
1^--
1
——————— Chord interactions. rpicrOSCOpe
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scanning circle " (measuring zor>e) accept)
Fig. 9.16 The laser optic measuring system of the Brinkmann particle size analyzer Spectrex PCT-1 laser particle counter is a compact liquid particle counter designed specifically to monitor real-time particle counts in ultrapure water. The water to be monitored is fed to a rectangular cross-section Pyrex glass cell (Figure 9.15) and illuminated by a He-Ne laser at right angles to the direction of liquid flow. When a particle passes through the sensing zone, a pulse of scattered light is emitted and detected by a photomultiplier positioned at right angles to both the laser beam and liquid flow direction. Two channels of information are generated; total count coarser than 0.11 |Lim and counts above 0.2, 0.3, 0.5 or 0.7 jiim as set in a four-position switch. Measuring time is selectable from 1 second to 1 h. 9.6 Dwell time 9.6.1. Brinkmann 201 analyzer The Brinkman Model 2010 analyzer (figure 9.16) scans the sample with a shaped and focused laser beam using a rotating wedge prism. The time
Stream scanning methods
493
spent by the scanning beam on a particle is interpreted as particle size. A CCd TV microscope is incorporated into the basic unit, permiting the operator to observe the sample while it is being measured. The images can be enhanced, processed and analyzed to obtain an independent measure of size distribution, particle shape and state of dispersion. The instrument operates in the size ranges 0.7 to 150, 2 to 300, 5 to 600 and 12 to 1200 |Lim. The measuring cells are standard, a spectrophotometer cell, a microscope slide, a liquid flow-through cell, an aerosol flow-through cell and a thermo-electric cooled flow-through cell. This instrument has been compared with the Coulter and automated image analysis for the size determination of grain based products [121]. The electrical sensing zone and the image analysis techniques gave similar mean sizes whereas the time of transition gave significantly coarser 9.6.2 Lasentec Focused Beam Reflectance Measurement (FBRM) Lasentec 's Lab-Tec 100 uses a scanning infrared laser beam to measure the particle size distribution of particles in suspension. The beam is highly focused and illuminates individual particles in its path (Figure 9.17a). The back-scattered light pulses are picked up by a non-scanning stereoscopic detection system (Figure 9.17b). The size of each particle is determined by measuring the time that the particle is in the beam hence the size is recorded as a random chord length. The laser diode and detectors are stationary while the lens, which focuses the light beam, is vibrated normal to the laser detector plane. The vibrating action causes the beam spot (focal point) to be scanned up and down normal to the direction of fluid flow. The beam amplitude is 3 mm and measurements are carried out in the central 1.5 mm where the velocity is maintained constant; since the frequency is 400 Hz the scan rate is 1.2 m s~i. Since the focal point is only about a millimeter in front of the window it can operate with very high (40% by volume) concentration slurries [122]. Lasentec Labtec 1000 is a laboratory instrument that covers the size range from 0.7 to 250 \\m in 28 size channels. The data are generated as 'scanned counts' an empirical frequency distribution created from classification of chords from randomly oriented particles. Software can convert these chords to a spherical equivalent distribution on the assumption that the chords were generated from an assembly of spherical particles: this software contains a filter system to reject improbable data that would tend to skew the distribution to a coarser size. A discrimination
494 Powder sampling and particle size determination loop sorts impulse for short rise times; only pulses from particles that pass directly through the focal spot have short rise times and will be accepted. Spnni loaded „ l^<:^^t
m Stcrco,scopic phoiodcicctor Laser source
m
Beaker Adjustable pJaiform
Fig. 9.17 (a) Sampling configuration using standard glass beakers on a magnetic stirrer plate (b) The Lasentec Lab-Tec 100 measuring geometry Lasentec Partec 100 was found to give reasonably accurate data for the chord length distribution of sherical particles but the accuracy deteriorated progressively as the shape became more ellipsoidal [123]. Lasentec's Focussed Beam Reflectance Measurement Model (FBRM M400L) operates at a wavelength of 791 nm and the beam rotates @ 75 s'^ at a peripheral speed of 1.9 ms-^to describe a circle of diameter 8 mm. It has been used with a 3D-model of chord length distribution for particles of a general shape as well as particles with the same shape but different sizes [124]. The focal point can be changed in the axial position but it is
Stream scanning methods
recommended that the distance from the window should lie between and d^^Jl for best results [130].
Scanning focusing lens
495
d^Jl
Slurry How
Scanning focal poiiil
Fig. 9.18 Optical arrangement of the Lasentec Par-Tec 200/250 Other conversions are also available, including calibration against known standards. Monnier et. al. [125] investigated the influence of variables such as temperature, stirrer speed, focal point, concentration and the position of the focal point, on the measured size distribution and concluded that the most important, especially for small sizes, was the focal point position. They concluded that, due to this, it was not suitable for precise analyses [126]. Lasentec Partec 200 (Figure 9.18) is an on-line version that gives a reasonably constant response over a concentration range of 5-30% by volume [127]. The Partec 200 gives much coarser data than the Lab-Tec 100, together with a lower count at similar concentrations. This can be attributed to the reflected light being collected by the same optics that are used to focus the light i.e. both the source and detector move in unison. Thus, a much smaller volume around the focal point is investigated on reflection. The beam moves in a circular path compared to the vertical vibrations of the laboratory system and different hardware is used to determine the pulses [128]. The size range of 1.9 to 1000 |Lim is covered in a V(1.5) progression of sizes.
496 Powder sampling and particle size determination
The latest on-line version covers the size range 0.8 to 1000 |Lim in 38 channels. In an investigation of this instrument it was found that the measured chord lengths data were sensitive to the location and orientation of the probe. Normalizes chord length data were in good agreement with predictions from the measured size distributions [129]. Langston and Jones [130], starting from an assumed uniform distribution, used Bayes' theorem to determine particle size distributions from chord distributions of non-spherical 2-dimensional images. Usung numerical simulations the derived distributions are found to be quite accurate and robust for a number of different types of particle shapes [131]. With others,[132,133] they also determined particle and droplet sizes from chord size distributions. They concluded that the probability apportioning method worked well as an iterative procedure to predict particle size distributions from chord distributions. Heath et. al [134] discuss methods of estimating average particle size by FBRM. They stated that the mode average of the square weighted chord length gave results that were comparable with other techniques for the size range 50 to 400 jiim. The results were not affected by instrument focal position, suspension flow velocity and solids fraction from O.lv to 20% w/v. 9.6.3 Messetechnik Optical Reflectance Method (ORM) ORM system consists of a sensor probe, a signal processing unit and a personal computer running DISPAS™ software for analysis [135]. The instrument is intended for on-line analysis but can also be used off-line. Light from a 1 mW semiconductor laser is transmitted to the sensor through optical fibers. Within the sensor the beam passes an optical system, a focusing lens, a window and finally a sapphire lens before entering the particulate system. The beam is focused inside the sample and is made to rotate eccentrically. A detector picks up the light reflected back from particles and transforms it into electrical pulses. Bad pulses, selected on the criteria of symmetry and slope of the flanks, are rejected and the accepted pulses are converted into rectangular shaped signals. The dwelltimes of these pulses are stored in 128 logarithmic channels that, for special requirements, can be increased to 2,000. Since the scanning speed of the laser beam is known these, effectively, give a chord length distribution. The size distribution is extracted from this chord distribution in the DISPAS^'^ software. The instrument measures particles in the 0.1 jLim to mm size range and has been used to measure droplets from 0.1 to 125 |Lim [136].
Stream scanning methods
497
Sample flow Scanning spot
Particle field Sample chamber
Fig. 9.19 Sample chamber of the Procedyne Analyzer. 9.6.4 Procedyne Procedyne PA-110 [137] uses a high intensity focused laser beam to scan a flowing suspension via an oscillating mirror, thus scanning the suspension horizontally and vertically (Figure 9.19). As particles interrupt the beam a pulse is generated at a photodiode, the length of which is related to particle size. The instrument is designed for the on-line analysis of process slurries at a sampling rate of 250 cm^ s~i with a maximum mass concentration of 2% and a particle size range of 3 to 2000 |Lim. There are five memory registers, four calibrated for particle size and the fifth for total particle count. Hinde [138] reported on the use of the instrument for controlling mill circuits with disappointing results. 9.7 Aerodynamic time-of-flight measurement 9.7.1 Thermo Systems Incorporated Time-of-flight aerosol beam spectrometry was first described by Dahneke in 1973 [139]. A commercial instrument, the Aerosizer, [140] was developed by Amherst Process Instruments Inc. which is now a division of Thermo Systems Inc.The TSI Model 3603 replaces the Aerosizer.
498 Powder sampling and particle size determination
Sheath
air
x
Photomultipliers
Fig. 9.20 TSI Amherst Aerosizer. ( TSI) Model 3603 aerosol time-of-flight mass spectrometerATOFM is based on the precise laser based measurement of a particle's aerodynamic time of flight (Figure 9.20). Particles to be measured are first suspended in air using a disperser. The particle laden transport air carries particles that are then surrounded by sheath air. The suspension expands through a converging nozzle into a partial vacuum and accelerates through a measurement region to near supersonic speeds with a barrel shock element; the smaller the particle, the faster the acceleration. The particle's velocity is determined by measuring the time of flight as they cross tw^o laser beams in the measurement region. As particles pass through the beams, they scatter light that is picked up and converted to electrical signals by two photomultipliers. One photomuhiplier detects light as the particle passes through the first beam and the other detects light as it passes through the second beam. Particles are measured, with 500-channel resolution, at rates up to 100,000 per second in the size range 0.2 to 700 |Lim. The Aerosizer is widely used in the pharmaceutical industry since it measures the particles aerodynamic diameter. This parameter is of great importance in the design of nasal inhalers.
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Aerosol in 4 Outer nozzle 5 L min-^l Inner nozzle ( 1 L min~l Filter Flowmeter
HH 3
Focusing optics Laser
Sheath air valve 4 L min""l
Photo multiplierj L tube Scattered light Pressure transducer
Flowmeter [
Internal vacuum pump
Fig. 9.21 Schematic of the Thermo-Systeins APS showing the accelerating nozzle and the two spot laser velocimeter This technique has been extended by inclusion of a pulsed ionization laser to vaporize the particles after their size has been determined by aerodynamic time of flight. This causes the particle to vaporize and the resulting fragments are partly ionized. Positive ions are accelerated into the flight tube of a mass spectrometer where their chemical composition is determined [141] 757 Model 3800 Time-of-Flight Mass Spectrometerwsis developed in cooperation with the University of California as the first single airborne particle mass spectrometer to be offered commercially. It uses an aerodynamic sizing technique to size individual particles in real time. It then desorbs and ionizes the particle for chemical analysis in a bi-polar, time-of-flight mass spectrometer. It operates in the 0.3 to 3 |Lim with an optional disperser to extend the range to 10 |Lim. The instrument can save positive and negative mass spectra at a rate up to 10 particles per second [142]. TSI Aerodynamic Particle Sizer APS 335counts and sizes airborne solids and non-volatile liquids at number concentrations up to 600 particles
500 Powder sampling and particle size determination
cm~^ at 0.5 |am and 45 particles cm~^ at 30 |Lim, and sorts them into 58 size channels (Figure 9.21). The aerosol is sampled into an inner tube, surrounded by a sheath of filtered air and accelerated through a nozzle. Two timers are used, a 2 ns resolution timer for sizes from 0.5 to 15 |Lim (small particle processor, SPP) and a 66.67 ns timer for sizes from 5 to 30 )Lim (large particle processor LPP). Calibration is effected using polystyrene latex spheres of known density. The major factors influencing accuracy are particle density, particle shape, particle deformation and the particle counting process. The SPP is known to generate phantom counts and the LPP is known to reduce them. Sreenath et. al [143] provide insights on how to operate the APS to avoid counting errors and the advantages and limitations of different correction methods are discussed. A full description of the equipment is given in a paper by Blackford et. al. [144]. Although designed for the size range 0.5 to 30 |Lim its overall efficiency falls from 95% for 3 |um particles down to less than 45% for particles smaller than 10 |im [145]. The APS Model TSI 3310 has been used to calibrate an optical particle counter and good agreement was found [146]. TSI Aerodynamic particle size spectrometer model 3i27provides two measurements, aerodynamic diameter and light-scattering intensity, in the size range 0.5 to 20 jim. The generation of paired data makes the instrument of particular interest to aerosol scientists. 9.7.2 Ancillary equipment TSI Aerodisperser is designed to ensure that particles are properly separated from each other, even in the sub.jum region, by controlling the applied shear forces. TSI Aero-Dilution model enables high concentrations of powders and aerosols to be analyzed. Since only a small sample is required (typically 0.1 g) it is particularly useful for expensive or research material. TSI Aero-breather has been developed for sampling dry powder inhalers. Ban* and Cheng [147] have carried out an evaluation of this instrument. 757 Model 3433 small-scale powder disperser breaks up agglomerated powder, by lifting particles from a turntable, and feeds them to the APS. TSI also manufacture a fluidized bed aerosol generator to generate high concentration aerosols. The instrument has been used to monitor mixing structure of cohesive dry powders [148]. Both the resolution of the Aerosizer and de-agglomeration with the Aero-disperser has been
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demonstrated using albuterol [149]. In comparison tests the Aerosizer was found to agree well with image analysis, however the Malvern Mastersizer and the Coulter LS were found to give dissimilar data [150,151]. At variance with these data Kaye et. al, [152] found very good agreement between the Coulter Counter and the Aerosizer with Ti02 coated PMMA spheres. Naqui et. al. [153] describe a phase Doppler technique, for measuring particle velocity and statistcal information about particle size of irregular particles, based on a phase shift signal. The technique works on near backscatter which leads to a robust set-up under conditions of limited optical access. Preliminary measurements in a crystallizer were presented and good agreement with TSI Aerodynamic particle sizer was found. 9.8 Laser Doppler velocimetry (LDV) Measuring particle size from the peak amplitude of a LDV signal was first done on combustion flows [154,155]. A later paper discussed the errors associated with this method and proposed procedures to correct for them [156]. Types of error include: 1
The possibility of more than one particle being in the sample volume at the same time: 2 Particles only partly in the control volume being counted as smaller particles; 3 The dependence of the light scattering signal magnitude on the particle's location in the sample volume; 4 The dependence of the sample volume size on the particle's apparent diameter since larger particles have a larger effective sample volume due to the Gaussian intensity distribution of the laser beam. The top-hat technique eliminates this problem by creating a beam with a constant intensity distribution over most of the beam width [157]. This volume is defined by the boundary where the illumination intensity falls below 1/e^ = 0.135 of the peak intensity. Various methods of creating this profile have been presented [158,159]. 9.9 Laser phase Doppler principle The laser phase Doppler particle analyzer (PDPA) simultaneously measures particle velocity, size and flux and may be considered an extension of laser Doppler velocimetry (LDV). It is particularly useful for
502 Powder sampling and particle size determination determining the local volume and mass flux within a spray cone. The basic principle is as follows. The phase Doppler method is based on the principles of light scattering interferometry. Measurements are made at a small non-intrusive optical probe defined by the intersection of two laser beams. As a particle passes through the probe volume it scatters light from the beams and creates an interference fringe pattern. A receiving lens located at an off-axis collection angle projects a portion of this fringe pattern on to several detectors. Each detector produces a Doppler burst signal with a frequency proportional to the particle velocity. The phase shift between the Doppler bursts from the different detectors is proportional to particle size. The system measures sample volume simultaneously with particle size; this enables accurate determination of particle number concentration and volume flux. No calibration is required since the particle velocity and size are dependent only on the laser wavelength and optical configuration. The instrument calibration is verified periodically using microspheres. The technique was introduced by Durst and Zare [160] and improved by others [161-163]. 9.9.1 TSI Aerometrics phase Doppler particle analyzer This was developed by Aerometrics in 1983, in collaboration with Lewis Research center, for research into pollution reduction from gas turbines. It is particularly relevent to measurements of small, spherical particles such as are found in fuel injection systems, medical nebulizers and bubbles in water. Aerometrics was later acquired by TSI who currently produce the TSI/Aerometrics PDPA 2D System. This instrument measures sizes in the 0.5 to 10,000 \im range using various optical configurations. The optical transmitter and receiver can be traversed together to move the location of the optical probe for spatial mapping of the flow field and the particle size distributions. 9.9.2 Discusion It has been assessed that the response of PDPA is very sensitive to the roughness of conducting particles [164] (for surface irregularities equal to the wavelength of light the errors are about 23%) to inhomogeneities in droplets i.e. spray dried dairy milk powders [165] and to the periodic oscillations of droplets [166]. If these conditions do not apply, the scattered light bears irregular and ambiguous information about the phase difference A^ that leads to incorrect particle size determination. The
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various errors associated with mass flux measurements in sprays are highlighted in a paper by Dullenkopf e^. al [167]. Dual mode PDA combines a planar PDA with a standard PDA so that two-phase differences are measured per scattering event. These are validated against each other to see if the signal is acceptable or not. This permits the rejection of distorted events caused by multiple scattering due to poor surface quality [168]. Dual beam frequency biased LDV permits simultneous measurement of particle size and electric charge in a twocomponent developer [169]. An approximate mathematical correction of a measured polydisperse size distribution has been carried out using a deconvolution technique [170]. This was verified with suspension droplets; the advantage being that it can measure such distributions; the disadvantages being that the correlation between particle size and velocity is lost and at least 5,000 data points are required for deconvolution [171] A modified PDA has been described for on-line fiber diameter measurement [172]. Petrak [173] measured particle size and velocity using an optical probe with a fiber optical spatial technique. In order to determine particle size in a free particle-laden air stream spatial filtering was modified by fiber optical spot scanning. He found good agreement with LDV in the size range 10-1000|Lim at velocities from 0.01 to 20 ms~^ An in-line device is available from PARSUM. Digital Particle Imaging Velocimetry (DPIV) is being studied for in-situ measurements of two-phase flows for mass transit [174]. A knowledge of droplet/particle size and spatial distribution is required. DPIV was found to be useful for large particles but failed for micron sized particles. 9.9.3 Differential phase-Doppler anemometry The two detectors of a standard PDA system have been replaced by a charge coupled device (CCD) line scanner sensor with 256 pixels. Hence, scattered light can be measured with high spatial resolution. Instead of two signals, 256 are detected which leads to 128 phase difference values rather than one. By statistically evaluating the scattered signals even severely distorted signals can be analyzed and need not be discarded. This is called differential phase Doppler anemometry. (DPDA). It has the potential to measure the size of glass spheres with surface defects or inhomogeneous composition. The smallest particles measured were 25 |Lim water droplets but measurements down to 10 |Lim were deemed possible [175].
504 Powder sampling and particle size determination
9.9.4 Bristol Industrial Research Association In the phase-Doppler (PD)-Lisatec a measuring volume with a Gaussian distribution (that is without real fringes) is generated and the velocity and phase information is derived using sophisticated patented polarization techniques. The optical processing employed directly filters the signal and removes the Gaussian envelope. This allows the dynamic range for velocity measurements to be greatly increased from a usual 5:1 or 10:1 to at least 30:1. The instrument covers the size range from 1 |Lim to several mm at flow rates from 1 cm s"^ to 500 m s"^ An optional fiber link between the laser and the optics allows the laser to be remote from the experiment. This instrument was developed by the Industrial Optics Group at AEA Technology who have published over 50 papers on this subject. Laser two-focus velocimetry is applied in the BIRAL L2F. The measurement volume is formed by focusing two laser beams to two small waists of about 10 |Lim diameter. The result is a light gate operating with concentrations orders of magnitude greater than possible with the LDV system. Particle velocity is determined by the time it takes the particle to pass from one beam waist to the other. 9.9.5 Dantec Particle Dynamic Analyzer The Dantec Particle Dynamic Analyzer measures particle size, one to three velocity components and concentration using the phase-Doppler principle. The Doppler effect is a phase shift in light scattered from moving particles and incident illumination with the frequency shift being proportional to particle velocity. If a standard laser Doppler anemometer is combined with a second photodetector, the photodetector signals, under certain conditions, are a direct measure of particle size. A third photodetector is included to extend the dynamic range and, in two-dimensional flows, a color separator and a fourth photodetector are added to allow two velocity directions to be measured. A fifth detector is needed for measurement of threedimensional flows. Its applications include droplet sizing, spray characterization, fuel injectors and agricultural sprays. The size range covered is 0.5 to 10,000 ]xm with a dynamic range of 40:1 with the possibility of extending this by a factor of 2.5 by varying the aperture spacing. Velocity maximum is >500 m s"^; up to three components being measured using a combination of near forward scatter, near backward scatter and side scatter.
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9.10 Hosokawa Mikropul E-Spart Analyzer Hosokawa Micron E-Spart Analyzer carries out simultaneous measurements of the size of a particle and its electrostatic charge. Characterization of electrostatic charge and aerodynamic size of particles is of critical importance in many electrokinetic processes [176]. A number of instruments are available that can characterize aerodynamic size distribution of particles. Likewise, instruments are available to estimate the net average electrostatic charge of particles. However, the choice of instruments for real-time simultaneous measurement of aerodynamic size and electrostatic charge distribution of particles on a single particle basis is limited. Electrical Single Particle Aerodynamic Relaxation Time (E-SPART) Analyzer simultaneously measures size in the range from submicron to 100 |Lim and particle charge distribution from zero to saturation levels [177,178]. The operating principle depends upon the phenomenon that when an airborne particle is subjected to an oscillatory external force, such as Particle inlet
Photomultiplier hb signal processor
Transducer Electrode
LDV signal
Acoustic or electric excitation Laser beam from laser Doppler velocimeter (LDV)
Particle outlet
Fig. 9.22 (a) Schematic view of E-Spart relaxation cell (b) principle of particle measurement. Individual particles are subjected to acoustic and/or electric excitation and the resultant response is measured by LDV to determine aerodynamic size and electrostatic charge.
506 Powder sampling and particle size determination
acoustic excitation, the resultant oscillatory motion of the particle lags behind the external driving field. The particle vibrates at the same frequency as the acoustic field but with a phase lag due to particle inertia. The phase lag increases with increasing particle size and so can be related to particle size. To determine this phase lag, the analyzer uses a differential laser Doppler velocimeter (LDV) to measure the velocities of individual particles subjected to a combination of an acoustic and a DC electric field. Simultaneously a charged particle will have its vertical position shifted by the electric field by an amount related to the charge. The maximum count rate varies from 10 to 2,000 particles per second depending on particle size that, typically, can range from 0.3 |Lim to 75 |Lim. The particles are sampled in a laminar flow field through the LDV sensing volume. As each particle passes through the sensing volume it experiences the acoustic excitation and the superimposed DC electric field in a direction perpendicular to the direction of the laminar air flow. A typical sampling configuration is shown in Figure 9.22. A full description of the theoretical basis for the instrument has been published [179] together with some typical applications. In a further paper, the instrument was applied to the size and charge analyses of toners [180]. The size range for toner was limited to 2 \xm to 25|Lim; a lower acoustic frequency is used in the improved E-Spart that increases the upper size limit, for glass beads, to 50 |im [181]. 9.11 Shadow Doppler velocimetry The SDV is based on the imaging of a conventional LDV probe volume on to a linear photodiode array and has the advantage over other sizing methods in that it is tolerant of the optical misalignment and fouling which are inevitable when passing laser beams through windows in furnaces. The technique has been used for simultaneous size and velocity measurements of irregular particles in confined reacting fiows. [182]. The instrument was developed by Hardalupas et. al. [183]. The transmitting optics are identical to a conventional LDV but the receiving optics, in contrast, allow collection of the transmitting light beam so that an image of the LDV measuring volume is formed. This image is magnified by a microscope objective and projected on to a 32-element linear photodiode array. Particle size is derived by measuring the size of the shadows of the particle on the array and, because the presence or absence of the shadow is a binary phenomenon, the method is independent
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of signal intensity. The analog outputs of the diode array are subsequently digitized and stored for processing and display. The accuracy of SDV was assessed by Morikita et. al. [184] who showed that the maximum difference between the arithmetic means of irregular particles by SDV and microscopic measurements was about 10%. Hishida et al. [185] recorded a maximum difference of 4% owing to beam wandering due to temperature gradients and concluded that the maximum error with increasing flame size cannot exceed 15%. 9.12 Other light scattering methods A simple light scattering photometer was designed, to measure the angular distribution of intensity of scattered, polarized, He/Ne light, by micron and sub-micron particles [186]. The photometer used an ellipsoidal reflector and simple optical components to collect the scattered light and focus it on a 512 element photodiode array. The intensity ratio method is based on measurements of light scattered at two angles and applies to the size range 0.1 to 10 |Lim [187]. Due to the possibility of large errors [188] the method has found little application. Using Scanning Flow Cytometry the size distribution of submicron spherical particles is determined from the scattered light intensity ratios at two angles. In one example the ratio at 6T and 15"^ was used to determine sizes between 1 and 15|Lim at a flow rate of 500 particles per second [189]. This was extended to 0.5 to 14 |im using a parametric solution based on analytical approximating equations [190]. The ratio of the polarized light intensity scattered from two different coaxial beams illuminating a particle can be used to determine particle size. Azzizy and Hess [191] used two coaxial beams of different wavelengths at 30*^ from the forward axis polarized in different directions. The ratio of these two parameters gives a unique curve that is a function only of particle size. They found errors of a similar magnitude to those found with intensity ratio methods. Hess [192] described an arrangement in which an LDV velocity system is positioned concentrically inside a larger sizing beam. The signal from the smaller beam is used to trigger the larger beam so that only particles passing centrally through the larger beam are counted. Wang and Henken [193] applied such a system for measuring particles in the 10 to 200 |Lim range.
508 Powder sampling and particle size determination
9.13 Interferometers 9.13.1 Mach Zehnder type interferometer Unlike light scattering instruments, interferometers do not measure the scattered light, but the phase shift in light waves. They can distinguish between gas bubbles and particles because bubbles have a lower refractive index than the surrounding liquid and therefore produces phase signals of opposite polarity to those of particles. This makes these instruments particularly useful for examining the reagents used for semiconductor cleaning since these reagents often have high vapor pressures and tend to form bubbles that can be counted as particles using light scattering or obscuration instruments. Interferometers separate a laser beam into two beams and then recombine them to create a signal whose intensity depends on the phase difference between them. When a particle with a refractive index greater than that of the surrounding liquid passes through the beam the wave front is retarded and when a gas microbubble passes through it the wave front is advanced. The magnitude of the phase signal depends on particle size and the pulse can be calibrated with particles of known size.
Beani spHlier 1
M i d ss^iaic
\1
^
J l%.Ho«Jcicil£>r
B
Fig. 9.23 Schematic diagram of a Mach Zehnder type interferometer. Figure 9.23 is a schematic of a Mach Zehnder type interferometer the design of which allows for a polarized light beam to be split in two. The incoming beam with intensity /Q is divided into two equally intense beam
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with a beam splitter. The beam following path 1 undergoes a 90"" phase change and a second beam splitter combines beams 1 and 2 to form new beams A and B. The two beams are now 180° out of phase; when one is bright the other is dark. The intensities of the beams I^ and /^ are measured using separate detectors. Due to their inherent complexity, large size and susceptibility to vibration, interferometers have remained laboratory and research tools. The variation of the polarization ratio with time has been used to determine the droplet size distribution in fuel sprays. Although polarization ratio is generally applied to an assembly of droplets it can be used for single droplets provided the incident beam is circularly polarized [194] 9.13.2 The TSI Liquitrak^^ interferometer The TSI Liquitrak^^ interferometer [195-197] uses a dual beam interferometer to detect slight differences in the refractive indices of particles relative to the surrounding media. It is less sensitive to vibration than the Mach Zehnder type of interferometer because it does not separate the two light beams. Instead the beams are overlapped and polarized at 90° to each other, hence the beams do not interfere and are effectively separated from each other until combined by the second beam splitter. The narrow separation of the beams reduces vibration sensitivity dramatically because any vibration interference affects both beams equally. The dual beam interferometer also has the advantage of allowing flowrate measurement. The instrument's flow-rate ranges from 4 to 40 ml min"^of which 1/200 is examined. Flow-rate is measured each time a particle signal is processed hence particle concentration can be measured in a fluctuating flow. The interferometer has several advantages over dark field scattering instruments. Because it is a bright field instrument it is less sensitive to the stray light scattered by interfaces between the instrument's capillary cell wall and the liquid medium. The instrument can also identify signals created by bubbles thus avoiding false counts. It can also measure flowrate and it is more sensitive to particles with a refractive index near that of the liquid than dark field instruments because it can look at forward light without noise interference from the incident laser beam. A drawback is that its inspected volume is small (0.5%) compared to the full flow stream since a highly intensive beam is required in the
510 Powder sampling and particle size determination
viewing volume, rendering the instrument less useful for low concentration contamination measurements. An evaluation of this procedure is available in articles by Blackford and Grant [198] and Grant [199] and the instrument is available from Thermal Systems Inc. as Model 7750. 9.14 Flow ultramicroscope. In the flow ultramicroscope [200] dispersed particles are injected into a stream of liquid and hydrodynamically focused so as to pass through a laser beam. The scattering is detected by a photomultiplier and processed electronically as a series of pulse heights. The detector can be at right angles to the incident beam, with either a narrow or wide receiver angle, or forward angle scattering may be used. Since the scattered light intensity is highly dependent on particle size, the dynamic range of the photomultiplier can be exceeded by samples of relatively low polydispersity. The range is greatly increased by using a feedback system from the photomultiplier to the laser. The instrument measures number concentration and size distribution for spheres and particles of simple shapes in the size range 0.1 to 5 )Lim. Such an instrument has been used to measure particle flocculation [201]. 9.14.1 ISP A image analysis system ISPA 800 series characterizes the size and shape of 5 ju.m to 10 mm particles by measuring the projected areas of the particles in a strobed image. The unit, manufactured by Greenfield, can process 15 images per second and can analyze particles moving at 30 m sec~i. 9.15 Measurement of the size distribution of drops in dispersions The most common method is direct photography [202-205]. The method is simple, easy and accurate and covers a wide size range, controlled by microscope magnification, from a lower limit of 1 |Lim. The technique is only suitable for low concentration systems, particularly in the case of high opacity continuous phases. The procedure requires many pictures and lengthy analysis times. Direct image analysis of the data has met with limited success.
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Light attenuation is a simple, widely used method for determining interfacial area, i.e. surface-volume mean diameter if droplet concentration is known but cannot be used for size distribution determination [206-208]. Light scattering has been used for measurement of small drop sizes below 10 |Lim in diameter [209] and also for drop sizes below 800 |Lim. Although on-line measurement is possible the technique is only suitable for volume concentrations smaller than 0.05 [210] Laser Doppler velocimetry (see section 9.6) has also been used for the measurement of a broad size range of drop sizes in solid-liquid and liquid spraying systems [211,212]. Drop size distribution in dilute suspensions of electrical conducting liquids may be determined using the Coulter principle but the need to add what may be undesirable conductive materials limits its applicability [213215]. The use of chemical means to measure interfacial area has been used extensively for gas-liquid dispersions. Chemical reaction methods for determining the interfacial area of liquid-liquid systems involve a reaction of a relatively unchanging dispersed-phase concentration diffusing to the continuous phase. The disadvantage of this approach is that the mass transfer can affect the interfacial tension, and hence the interfacial area [216-218]. Drop stabilization methods rely on the immediate stabilization of drops by encapsulation with thin polymer films or surfactants [219-221] a photomicrographic method has been employed usually after encapsulation of drops. However this method cannot always be used due to incompatibility of the encapsulating materials with some systems. The method also has the disadvantage of the influence of the chemical treatment on drop size. A special sampling apparatus has been developed to withdraw a sample of dispersed phase from the mixing vessel to stabilize drops with a surfactant and to force the dispersed sample through a capillary with a photometer assembly to measure both droplet size and concentration [222]. The capillary method employs a fine bore capillary of the order of the drop size for sampling from the liquid dispersion. As drops pass through the capillary, they are transformed into cylindrical slugs of equivalent volume. A laser beam is split into two rays using a beam splitter and a plane mirror and the rays pass directly through the capillary precisely 0.1 mm apart. The emergent beam is collected by a x 10 microscope objective lens and focused on to a photodiode. From the measurement of the passage time of a slug at one detector and its travel time between two detectors, the velocity and diameter of the drop can be calculated. The
512 Powder sampling and particle size determination
method can be used to obtain broad drop size distributions in the range above 50 |am in real time and automatically [223-227]. The scintillation method uses short-range radioactive particles for measuring interfacial area. This technique is limited by the necessity of high immiscibility between the phases as well as the availability of suitable isotopes and target materials [228]. The Lasentec particle/droplet size analyzer can be used for laboratory and in-line analysis in the +1 \xm size range over a wide range of operating conditions. 9.16 Dupont electrolytic grain size analyzer The EGSA provides a rapid (4 to 10 min) absolute measurement of the charge required to electrolytically reduce/oxidize AgX crystals. This electrolytic decomposition is singularly effective as a basis for measuring the particle size distribution of photographic emulsion grains since the charge is directly related to grain volume. Grains are electrolytically reduced as they are rotated under a measuring electrode and the generated pulses are sorted according to their integral size and stored in 256 logarithmically distributed channels. Three size ranges cover an overall range of 0.05 |im to 2 |Lim [229]. 9.17 Light pressure drift velocity The motion of individual Brownian particles is observed using a confocal tracking microscope. Particles are trapped in a strongly focused laser beam. By evaluating light-pressure-drift-velocity and the back-scattered light intensity the particle size is determined to ±2%. The method was demonstrated on a mixture of seven polystyrene latices between 300 and 450 nm that were divided into six size classes. A discussion of the method is presented together with a suggestion as to potential applications [230]. Tuch et. al. [231] ran a Mobile Aerosol Spectrometer (0.1 to 2.5 |Lim) and an Electrical Aerosol Spectrometer (0.5 to 10 |im) side by side for 6 weeks and found both to be reliable with almost identical results. Total number counts agreed with results from a Condensation Particle Counter.
Stream scanning methods
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9.18 Impact size monitor Size distributions in pneumatic conveying systems are usually monitored by taking grab samples or by using the Malvem/Insitec optical diffraction particle size analyzer [232]. CSIRO has developed and patented a technique to measure particle size from measurements of the peak compression of an ultrasonic transducer subject to impact by the particles [233]. Each impact produces an independent pulse, the duration and amplitude of which conveys information about the particle size and velocity. The impact times for sub-mm particles is typically less than a |is; this short duration allows the measurement of tens of thousands of impacts per second. The instrument has been tested using several grades of ballotini from 50 to 165 ^m in size and was able to differentiate between particles of size 157 and 165 |Lim [234]. Fluid dynamic modelling of this instrument was carried out to determine the size range of particles which could be monitored and it was determined to be applicable to the size range 50 |Lim to 200 |Lim at load rates of up to 1 kg m"^ [235]. Impact sensors have been used previously to detect particles in streams such as sand in oil pipelines [236]. Quantitative measurements have also been carried out at low impact rates by dropping particles, in a vacuum, on to a sensor consisting of a hit plate in point contact with an ultrasonic transducer [237]. High-velocity air-laden dust has also been measured using transducers [238]. 9.19 Monitek acoustic particle monitors Monitek Micro Pure Systems acoustic particle monitorsusQs a focused acoustical beam to sense discontinuities in a flowing liquid and can detect the size and amount of suspended solids, entrained gases, fibrous material in any liquid, or oil droplets in water. The sensor mounts in-line without restricting the process flow and the acoustical beam is focused to a point approximately 0.8 to 1.5 in from the sensor tip. A piezoelectric crystal that acts both as a transmitter and receiver generates the high frequency. The transmitter emits hundreds of pulses per second and monitors the echoes; this high sampling rate makes the instrument insensitive to liquid flowrate. The amplitude of the echo is size sensitive so that a lower limit size threshold can be set; this limit can range from 0.2 |Lim to a few millimeters
514 Powder sampling and particle size determination
9.20 Erdco Acoustical Counter Audible sounds may be produced by particles exiting from a high-velocity laminar flow tube into a low velocity tube. The phenomenon was first reported by Langer [239] and has since been investigated by Langer [240]. and others. The sensing zone of the Erdco counter is a capillary at the exit of a glass tube. As particles enter this section they interact with the boundary layer, resulting in a toroidal vortex that moves as a shock wave that is reflected back on the capillary. The pressure wave is detected by a transducer at the outlet of the capillary, whose displacement is measured by an optical probe. The displacement is proportional to particle size, which is measurable down to 4 \xm. References 1 2 3 4 5 6
I 8 9 10 II 12 13 14
15 16
Groves, M.J. (1991), Particle Size Distribution II, ed. T. Provder, Am. Chem. Symp. No 472,123-131,^^9 Coulter, W.H. (1953), US Patent No. 2,656,508. Appl., 1949, 449 Coulter, W.H. (1956), Proc. Soc. National Electronics Conf., 12, 1034, 449 Kubitschek, H.E. (1958), Nature, 182, 234-235, 449 Kubitschek, H.E. (1960), Research, 13, 128, 449 Coulter Counter and other scientific instruments, Industrial Bibliography January 1992, Coulter Electronics Ltd., Northwell Drive, Luton, Beds LU3 3RH, England, 450 Anderson, F.G., Tomb, T.F. and Jacobson, M. (1968), US Bureau Mines, R.L, 1\05,450 ASTM F751-83 (1997), Measuring particle size of wide-size range dry toners, 450 ASTM 577-83 (1997), Particle size measurement of dry toners, 450 ASTM C-690-86 (1997), Particle size distribution of alumina or quartz by electronic counting, 450 ASTM E 1772-95( 1995) (1997), Particle size distribution of chromatography media by electrical sensing zone technique, 450 ASTM 3451-92 (1992), Testing polymeric powders and powder coating, 450 ASTM C757-90 (1996), Nuclear grade plutonium dioxide powder, sinterable, 450 ASTM F-662-86 (1992), Measurement of particle count and size distribution in batch samples for filter evaluation using an electrical resistance particle counter, 450 ASTM D-4438-85 (1997), Particle size distribution of catalytic material by electronic counting, 450 ASTM F-660-83 (1993), Comparing particle size in the use of alternative particle counters, 450
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18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
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Cummings, P. G., Smith, A. L., Staples, E. J. and Thompson, L. G. (1984), SolidLiquid Separation, ed J Gregory, publ. John Wiley for Soc. Chem. Ind.,London, Ch. 12, 510 Chen, T.M. and Middleman, S. (1967), AIChE Journal, 13, 989, 510 Coulaloglou, C. A. and Tavlarides, L. L. (1976), AlChE J., 22N2, 289, 510 Hazett R. L., Sechter, R. S. and Agarwel, J. K. (1985), Ind Eng. Chem. Fund., 24, 101,510 Kirou, V., Tavlarides, L. L., Bonnet, J. C. and Tsouris, C. (1988), AIChE J., 34, 2^3,510 Calderbank, P. M. (1958), Trans. Inst. Chem. Engrs., 36, 443, 511 McLaughlin, C. M. and Rushton, J. H. (1973), AIChE, J., 19, 817, 511 Hancil, V., Rodd, V.and Reznickova, R. (1986), ISEC, 81, 1986, 511 Sullivan, D. M. and Lindsey, E. E. (1962), Ind Eng Chem.l, 87-93, 511 Rebelein, F. and Blass, E. (1986), ISEC '86,11, 387, Munchen, Germany, 511 Semiat, R. and Duckler, R.E. (1981), ISEC'81, 27, 214, Munchen, Germany, 511 Plawsky, J.L. and Hatton, T.A. (1986), ISEC '86, IV-S9, Munchen, Germany, 511 Sprow, F. B. (1961), AIChE J., (13)5, 995, 511 Hoffer, M. S. and Reznick, W. (1975), Chem. Eng Sc, 30, 473, 1975, 511 Tamano, K. (1986), Inst. Chem. Engrs., 26, 698, 511 Nanda, A. K. and Sharma, M. M. (1966), Chem. Eng Sci., 21, 707, 511 Femandes, J. B. and Sharma, M. M. (1967), Chem. Eng Sci., 22, 1267, 511 Puranik, S. A. and Sharma, M. M. (1970), Chem. Eng Sci., 25, 257, 511 Madden, A. J. and McCoy, B. J. (1964), Chem. Eng Sci., 19, 506, 511 Mylnek, Y. and Reznick, W. (1972), AIChE. J., 6, 505, 511 Tanaka, M. (1985), Can J. Chem. Eng., 63, 723, 511 Verhof, F.H., Ross, S.L. and Curl, R.L. (1977), Ind Eng Chem. Fund., 16, 371, 511 Janjua, K.M. (1982), PhD thesis. University of London, 512 Goldmann, G. and Blass, E. (1984), CHISA '84, 14, 512 Pietzsch, W. and Blass, E. (1987), Chem. Eng Technol, 10, 512 Bae, J.H. and Tavlarides, L.L. (1989), AIChE Journal, 35(7), 1073-1084, 512 Smith, T.N. (1974), Chem. Eng Sci., 29, 583, 512 Mitsis, T.J., Plebuch, R.R. and Gordon, K.F. (1960), AIChE Journal, 6, 505, 512 Holland, A.B. and Sawers, J.R. (1973), Photographic Science and Engineering, 17(3) 295-8, 512 Garbow, N., Mueller, J., Schaetzel, K. and Palberg, T. (1997), Physica A., Amsterdam, 235(1/2), 291-305, 512 Tuch, Th., Mirme, A., Tamm, E., Heinrich, J., Heyder, J., Brand, P., Roth, C.H., Wichman, H.E., Pekkannen, J. and Kreyling.W.G. (1999), Atmos. Environ. 2000, 34(1), 139-149,572 Godek, A.R. (2000), Cement Industry Technical Conf., Sat Lake City, May, 57 J Coghill, P.J. (2000), Australian Patent Provisional Application No PQ 1176, CSIRO Minerals, 57 J Coghill, P.J. (2001), Part. Part. Syst Charact.., 18(3), 114-119, 57i
Stream scanning methods
235 236 237 238 239 240
523
Coghill, P.J. (2002), World Congress Particle Technology, Sydney, Australia, July, 513 Beattie, A.G., Corrales, N. and Bohon, W.M. (1993), U.S Pat. 5.257.530, Atlantic Richfield Company, 513 Buttle, D.J.and Scruby, C.B. (1990), Wear, 37, 63-90, 513 Wills, B.G., Taggard Jr, R.B., Knudsen, K.L. and Gee-Clough, D. (1974), U.S. Pat. 3.805.59, Hewlett-Packard, 513 Langer, G. (1965), J. Colloid Sci., 20(60, 602-609, 514 Langer, G (1972), Powder TechnoL, 6, 5-8., 514
10 Field scanning methods of particle size measurement 10.1 Introduction Field scanning methods are those in which the size distribution of an assembly of particles is inferred from the interaction between the assembly and a measurement probe. In the simplest systems, the powder (or slurry) is probed or classified in order to generate a single point on the distribution curve. For example, one might monitor the 100-mesh percentage oversize from a mill in order to control a continuous milling operation. If the percentage increases, the residence time in the mill is increased in order that the product size remains unchanged. It is commonly found that comminution shifts the whole distribution to a finer size distribution, to form a homologous family of curves, and plotting particle size against milling time on a log-log scale generates a straight line. Knowledge of two points on the distribution curve allows one to generate the whole distribution. An alternative method for plotting such distributions is the Gaudin-Schuman plot where the cumulative weight finer than a given size is plotted against that size, with each scale on a logarithmic basis. For the majority of milled material the relationship between the two variables is linear except at the coarse ends of the distributions. The distribution is characterized by two parameters; a distribution modulus, n (slope), and a size modulus, k. Again, n remains constant for consecutive grinding of the same material. Djamarani and Clark [1] state that many industrial processes are defined by a coarse ( Q and a fine fraction (F), for example, oversize and undersize. In their example they use sieve sizes of 1400 |um and 180 |um that they fit to a Rosin-Rammler distribution. They define a curve of C+F against C/F from which the Rosin-Rammler constants can be read. Field scanning instruments are ideally suited to on-line analysis. Rapid control of crystal size from a crystallizer; granule size from a granulator;
Field scanning methods
525
product size from a milling operation; particle size from a reactor etc. can yield enormous dividends in terms of less wastage (i.e. more material in specification) and superior product quality. One problem associated with implementing this technology is the need to build an interface between the process and the measuring instrument. This often requires a dilution step that may alter the size of the particles. In the case of crystallizer control, for example, it may be necessary to remove two streams from the crystallizer and filter one so that the mother liquor can be used as the diluent. Process streams often operate at high flowrates and these have to be split in order to obtain an acceptable flowrate in the measuring device. This reduction has to be carried out with care in order to minimize sampling errors. Ideally, the whole of the process stream should be examined. Inserting a probe directly into a process line is usually not feasible due to servicing and downtime problems. It is therefore preferable to use a side-stream that can be isolated from the process stream. A sophisticated on-line analyzer can cost around $100,000 and the interface can easily double this cost. However, a 1% increase in yield can pay back this investment inside a year, making on-line size analysis very attractive. At the present time retroactive fitting of size analyzers is often necessary and one is often faced with space limitations. Designing these units into new process lines greatly reduces cost and makes their introduction more attractive. 10.2 Single point analyzers 10.2.1 Static noise measurement This technique has been applied to the measurement of the average size of milled silica powder (size range 2 to 5 |Lim) suspended in air [2]. A continuous sample is drawn from the product stream into a sampling probe and diluted with an air injector that also provides the driving force. The sample stream is then passed through a 'uni-flow' cyclone that splits the sample into two streams; a low concentration 'fine' stream and a high concentration 'coarse' stream. As the relative mass flow rates of the two streams depend strongly on the size distribution of the feed (at a given flow rate), an average size may be found from a measure of the two concentrations. Most particles suspended in air carry an electric charge, particularly if they have passed through a highly turbulent process. A probe inserted into the stream will detect this charge as an AC voltage that is strongly
526 Powder sampling and particle size determination
dependent on concentration. The system was calibrated by feeding in samples of known mean sizes and recording the signals these generated for comparison with signals from unknown samples. 10.2.2 Ultrasonic attenuation The attenuation of ultrasound by a slurry depends upon the particle size distribution and concentration of the solid phase. In order to separate these two variables it is necessary to carry out analyses at two different wavelengths, one of which is strongly dependent on concentration and the other on particle size distribution. The attenuation is also dependent on the spacing of the transmitter and receiver and other physical parameters in a predictable manner [3,4]. The first commercial ultrasonic on-line particle size analyzer was developed in the 1970's and was based on the measurement of ultrasonic attenuation at two frequencies with an empirical model to predict particle size and concentration [5]. Instruments based on this patent are available as the Denver Autometrics PSM-100, 200, 300 and more recently 400. These are pre-calibrated for the selected mesh size (100, 200, 300 and 400) and the mesh read-out is proportional to the mass percentage less than this. These instruments can operate at extremely high concentrations, up to 60% by weight, and have found their widest application in mineral processing plants for improved grinding circuit control. A major problem in the early development of the Autometrics' system was that traces of air could lead to substantial attenuation losses. The air must be stabilized or removed to allow accurate measurement of particle size. Removing the air with a device that utilizes a combination of centrifugal force and reduced pressure solved the problem. The need to remove air increases the cost of the overall system significantly and makes it an expensive instrument when compared with other instrumentation often installed in grinding circuits. Nevertheless, it appears to be perfectly compatible with other approaches when its inherent reliability and longterm stability as an accurate size analyzer is taken into account. Several articles have been written describing applications of the PSM systems [6-9]. The limitations of the system 100 are: (a) the percentage solids should be less than 60% by weight (b) the particle size distribution should be within the range 20% to 80% less than 270 mesh and (c) the slurry particles should not be magnetized. The PSM systems 200 and 400 are later instruments designed to overcome these limitations.
Field scanning methods
52 7
10.2.3 P-ray attenuation Instruments have been described that employ p-ray attenuation [10-15]. The accuracy of these devices is limited by their sensitivity to changes in feed density [16]. In order to calibrate it is necessary to re-circulate slurry samples in a closed loop at a number of dilutions for each slurry system, sieve analyses being carried on representative sub-samples. The signals from a scintillation counter can be used to control mill feed rate in order to compensate for changes in feed ore, grindability and feed particle size. Accuracies of 2% to 3% have been reported on Cornish granite, nepheline systems and copper and iron pulps in the size range 20 to 105 ^im [17,18]. 10.2.4 X-ray attenuation and fluorescence This sensor is based on the comparison of the absorption of two x-ray beams, one of which is sensitive and the other insensitive to variations in particle size [19-21]. Each sensing head is specific to a particular system since the relationship between the two beams is dependent on the composition of the solids in the slurry stream. The technique is limited to x-ray opaque material. Von Alfthan [22] describes an on-stream x-ray fluorescence system that consists of two flow cells through which the slurry passes. In the classifying flow cell, the slurry flows in a straight path behind a window; it then strikes an obstacle that causes slurry mixing as it enters a turbulent flow cell. X-rays excite the slurry in both cells and the resulting fluorescent radiation is a measure of particle size. The system, sold as the Courier 300, measures both x-ray scattering and x-ray fluorescence and is intended primarily as a composition monitor. The measured data can be analyzed to give chemical composition, solids content and maximum particle size. 10.2.5 Counter-flow classifiers Two instruments have been developed for on-line measurement of flowing powders coarser than 100 |Lim in size [23-26]. In the first instrument a side stream of solid particles from a process line is fed into an air elutriator that separates it into an oversize and undersize stream. The particle flow rate into the elutriator is measured and the cut size for the elutriator adjusted so that the flow of oversize particles out equals 50% of the inlet flow. The elutriator cut size is then equal to the average size of the powder. In the second instrument the flow rate is varied and the signal ratio of the two
528 Powder sampling and particle size determination
flowmeters is inputted as the j;-axis of an x-y recorder. The x-axis is reduced to the cut size for the elutriator. A sweep time of 40-60 s at flow rates of 2.4-3 g s"^ gives a cumulative distribution in the size range 100700 ^m. 10.2.6 Hydrocyclones An ideal on-stream sizing device would sample the whole of the stream and not include any special instrumentation. The nearest approach is to use a classifying hydrocyclone as these are easily installed and often form part of an industrial plant. By measuring the flow rates and pulp densities, and assuming a size distribution law for the feed, a computer program can be written to give the modulus and index of the feed. Under normal operating conditions the present state of the theory of cyclone operation renders this impracticable [27] although it can be used under favorable conditions [28]. Lynch et. al. [29] proposed that the percentage less than some chosen mesh size in the cyclone overflow could be related directly to the d^^^Q parameter of the cyclone, provided that the size distribution of the feed to the cyclone does not change appreciably. In closed production circuits there may be marked changes in the size distribution of the cyclone feed and an empirical relationship has been developed [30]. The application of this technique requires very thorough analysis of the circuit and repeated checking of empirical equations. An alternative approach has been to accept the inherent difficulties of sampling and install smaller, more precise, classifiers alongside the production classifiers [31]. Tanaka has also investigated the use of hydrocyclones for on-line analysis [32]. 10.2.7 The Cyclosensor This is a batch size analyzer [33] (Figure 10.1). An extremely dilute sample of milled ore is introduced, at a constant flow rate, to a coarse separator in the form of a tangentially fed cylindrical screen. The coarse fraction is allowed to settle and the fine fraction is further separated with an efficient hydrocyclone into a fine and a very fine fraction. The very fine fraction is discarded and the fine fraction is allowed to settle. The ratio of the times taken to fill the coarse and fine fraction collection vessels to indicated levels can be related directly to the particle size distribution. The cyclosensor has a sensitivity whereby a change of ±1.8% passing 100
Field scanning methods
Very fine
Metal core
Cylindrical screen
Coarse solids
529
Efficient cyclone
Fine solids
Fig. 10.1 The Cyclosensor. mesh can yield a 7% change in the ratio of the settling times. The reproducibility is such that for the same feed rate of the same solids the ratio of times remains constant to better than 1% and an increase in feed rate of 30% has no effect on the ratio. A patent has been issued on an instrument operating in a similar way [34]. It consists of a particle suspension sampler, a settler and a weight or volume sensor. Particle size distribution is determined from sensor output and the time for the settling particles to pass the sensor. 10.2.8 Automatic sieving machines This automatic wet sieving machine determines a single point on the size distribution curve in a few minutes without the need to dry samples [1,16]. The sieving vessel is first filled with slurry and topped up with water to a precise level to allow accurate determination of the mass of solids added (wi) by application of Archimedes' principle. The fine fraction is next removed from the vessel through a discharge valve. Screening is hastened by propeller agitation and with ultrasonics to maintain the sieve mesh free
530 Powder sampling and particle size determination
of pegged material. The weight of the residue (^2) is determined by further application of Archimedes' principle, and the fraction coarser than the screen size is given directly by (W2AV1). It is interesting to note that the capacity of the rapid wet sieving device, expressed as screen charge mass per unit screen area, is more than an order of magnitude higher than that normally recommended for conventional dry test sieving [16]. A description has been given of a technique using a two-cell compartment divided by a screen [35]. The slurry density in the two compartments is determined using nuclear gauges to provide a single point on the distribution curve. A fully automatic sieving machine that can determine seven points on the size distribution curve has also been described [36]. In this technique a pulsating water column is used with the application of ultrasonics and the charges are dried and weighed automatically. 10.2.9 Gas flow permeametry Air, whose pressure varies sinusoidally with a specific amplitude and frequency, is forced through a moving bed of powder [37-40]. At a known height in the bed the attenuated and retarded air pressure is tapped by a pressure transducer, so that the amplitude and pressure drop are measured after being separated into the pulsating and steady flow components. The amplitude attenuation of the pulsating pressure is related to bed porosity and specific resistance. Using the Carman-Kozeny permeametry relationship the average size of particles can be evaluated. The bed is packed into a test cylinder and discharged by a vibratory feeder at the bottom after measurements have been taken. This enables a new bed to be packed and measured within minutes. Air permeability has also been used to determine the surface area of cement [41]. A porous piston compresses the sample of cement into a cell. Air is passed through a bottom porous plate, through the sample and porous piston, into the atmosphere. The inlet pressure is automatically adjusted and recorded to give a known air flowrate and the surface area is evaluated from the inlet pressure. The cell is emptied automatically becoming ready for the next test. Weiland [42] used a similar idea but based on the Blaine permeability method. An automatic weigher produced a packed bed of powder, of known voidage, in a standard cell. Air was drawn through the bed by the passage of water from one reservoir to another. After a certain volume of air had passed through the bed, measured by a certain volume of water flowing, the time required was converted to an electrical proportionality
Field scanning methods
531
signal. The measurements were repeated every 4 min and the signals used to control the feeder to a grinding mill. 10.2.10 Correlation techniques Correlation techniques can be used with signals from the attenuation of radiation, such as light, but these are mainly used for low concentration systems. The signals from two sensors in close proximity, situated in a flowing stream of slurry, are cross-correlated to give an autocorrelation function. Stanley-Wood et. al [43,44] found that this function obtained from alternating current transducers, initially designed to measure mass flow rate, gave a measure of the particle size of a sand/water mixture. A measure of mean size could be achieved by allowing the normalized signal from a correlator to be divided in two and passed through either high or low pass filters. This results in an inequality, due to variations in frequencies from large and small particles; the ratio of this inequality can be used to determine mean size after calibration. The particle size was between 70 and 2000 \\xxv with a concentration between 10% and 30% by weight. k Reflected
Vay Incident i^y\
Fig. 10.2 Interaction of a ray of light with a spherical particle. 10.3 Light scattering and attenuation 10.3.1 Introduction When light strikes a particle, some of it is absorbed, some is refracted, some diffracted and some transmitted (Figure 10.2). The amount absorbed depends upon the optical properties of the particles and surroundings and
532 Powder sampling and particle size determination
is also a function of the cross-sectional area of the particles. The absorption, or turbidity, can be used to determine a mean particle size and, in conjunction with sedimentation, a size distribution. For very small particles however, the laws of geometric optics no longer hold and a correction has to be applied in the form of an extinction coefficient, K, which is defined as the effective particle cross-section divided by its geometric cross-section. This may be determined theoretically over the whole particle size refractive index domain using Mie theory or, over limited ranges, with modified theories. Other interactions between the particles and the incident radiation, such as state of polarization, light flux at a fixed angle to the direction of the incident beam and angular spectra can be used for particle size determination. Interaction between the incident and diffracted radiation gives rise to interference phenomena with characteristic maxima and minima in intensity [45]. In order to describe fully the scattering pattern it is necessary to assume that the particles are optically homogeneous and spherical and, in order to give independent, incoherent scattering, in a dilute random arrangement. 10.3.2 Turbidity measurements Turbidity has been widely used for determining the particle size distribution (PSD) of particles in suspension, since it is experimentally simple, can be used over a wide size range and does not disturb the system under investigation. It is also fast, reproducible and inexpensive. If a light beam falls on an assembly of macroscopic particles the attenuation is given by: /-/oexp(-aA2Z)
(10.1)
where / is the transmitted intensity when a light beam falls on a suspension of particles of projected area a and number concentration n and traverses it by a path of length L\ /Q is the transmitted intensity when no particles are present. Turbidity gives a measure of the attenuation of a beam of light passing through a suspension. More generally, equation (10.1) may be written: I = lQQxp{-KanL)
(10.2)
Field scanning methods 533 where the extinction coefficient, K, may be evaluated theoretically thus permitting the determination of particle size. If c^ is the volume concentration: (10.3)
^v=~^d^
where d^ is the mean volume diameter. The projected area (a= an) of an assembly in random orientation is: (10.4)
a = —nd} 4 ' where d^ is the mean surface diameter; Hence:
^
I = IQ exp
3KrO 2d,sv
(10.5)
J
where d,,, is the surface-volume mean diameter. Hence: / 1 \ / = /Q exp —Kc^S^L V
/ = /Q exp(-
(10.6)
4 TI)
(10.7)
where x is the turbidity. For a suspension of non-spherical, non-monosize non-adsorbing, isotropic particles, in the absence of multiple scattering: X --
2d„,
(10.8)
The surface volume mean diameter for a suspension of spherical particles is given by:
Y,d^f(d)M _ 0
Y^d^f(d)M 0
(10.9)
534 Powder sampling and particle size determination
Equation (10.8) can be written in the form: 00
T = ~\Kd^f{d)dd 4o
(10.10)
This is a Fredholm integral equation of the first kind. The regularized solution to this equation has been applied to the measurement both for the moments and the size distribution of a wide range of latices [46]. K has been given by van de Hulst [45] in terms of particle size/refractive index domain. Mie theory applies to the whole domain but in the boundary regions simpler equations have been derived. For dilute suspensions of particles smaller than 0.04 |Lim in diameter, the turbidity can be calculated from the equation:
r=
3271^ ( w - i f ^ - ^ /
(10.11)
c is the mass concentration, A^ is the Avogadro Number and/is very nearly equal to unity [45 p396]. Turbidity measurements have been carried out on non-uniform latices and it is suggested that this is one of the most useful of the light scattering techniques for average size determination [47]. PSD can be estimated from the turbidity at different wavelengths provided the other variables are known. Kourti et. ah [48] assumed a log-normal PSD and observed that the parameters of the estimated distribution were so highly correlated that an infinite number of distributions could explain the data. However, all the alternative solutions were found to have the same weight average (surface-volume mean) diameter. With turbidity ratio, the ratio of the turbidities at two wavelengths, one of which is chosen as basis, is used. This has been successfully applied with large particles [49], (0.65
Field scanning methods
535
mono-modal distributions. Turbidity has also been used on-line, in conjunction with an Anton Paar vibrating tube densitometer for measuring concentration, in order to determine the particle size distribution of polyvinyl acetate. Samples were withdrawn from the process line using a Bristol Engineering Isolock sampler and a dilution system. Turbidity measurements were carried out using a Bausch and Lomb Single Beam Spectronic20 [54]. Raphael and Rohani [55] developed a method for on-line estimation of solids concentration or mean size of crystals in a crystallizer. The method was only applicable to slurries in the absence of background particles. Later, an on-line double-sensor turbidimeter was proposed [56]. The technique is based on measuring the transmittance of an infrared light beam through the suspension, once in the presence of soluble particles and a second time when the particles have been dissolved. Crawley et. al [57] applied the above equations to determine particle size distributions from turbidity measurements. The problems arise in finding a particle size distribution from the measured extinction coefficient due to the ill-defined inversion problem. Scholtz et.al. [58] focused on the problem of analyzing spectra of colloidal solutions, for which the size distribution was known from other methods like electron microscopy and light scattering; they termed this 'transmission spectroscopy'. Turbidimeters can be used to determine the product of particle concentration and particle size. Small measuring zones additionally allow measurement of the standard deviation of the fluctuating extinction signal. For monodisperse particles the particle size and particle concentration can be calculated from the measured mean and standard deviation of the extinction signal [59,60]. Variation in the size of the measuring light beam influences the standard deviation of the extinction signal and allows the determination of the particle size distribution. This can be effected by the use of a small angle photometer with the variation in the size of the light beam being realized by an axially shiftable flow cell in combination with a convergent laser beam [61]. An alternative approach is to use two light beams and a flowing suspension [62]. 10.3.3 Transient turbidity Transient turbidity is an optical technique for measuring the size of magnetic particles [63,64]. It does this by aligning particles in an electric field, removing the field, and following their return to random orientation induced by Brownian motion. Their relaxation is measured by turbidity and this can be related to particle size distribution if assumptions are made
536 Powder sampling and particle size determination
about the particle geometry and the shape of the size distribution. The technique is rapid (less than a second) and reproducible. Its most serious limitation is that the specific conductance of the sample must be less than 100 |Limho cm"^ Transient electrical birefringence operates in a similar manner 10.3.4 Holography Conventional methods of sampling aerosols are frequently unsatisfactory because they are too slow to monitor dynamic aerosols which results in the collection of non-representative and modified samples. Hologram systems, which overcome these objections, record and reconstruct large volumes of aerosols containing particles in the size range 3-100 |Lim. These holograms are called far-field or Fraunhofer holograms [65-67] because they are recorded at a distance from the object, effectively in its far field. The effective sampling depth is A9{D'^IX) that, for 50 |im particles and a ruby laser, gives a depth of 18 cm, which is over 3 orders of magnitude greater than a microscope. Prototype instruments based on the use of Fraunhofer holograms have been described [68-70]. Visual comparison of the holographically recorded radiation pattern of a particle with Mie theory has also been used for particle sizing [71]. Holography has also been used to locate sub-micron particles in a 3-dimensional volume [72] and, in conjunction with an image analyzing computer, to size the droplets in sprays [73]. In-line holography has been used to characterize the spray produced by a commercial rotary device; a description of the optical system used to record and reconstruct the images has been given [74]. A simple method of laser diffractometry has been described for sizing droplets with radii greater than 1.5 |Lim [75]. Under partially polarized laser illumination, at a 90° angle to a camera receiver, well-focused droplets appear as small circular dots. Out of focus droplets give large diffraction haloes crossed by a row of dark, parallel stripes, the number of which is indicative of particle size. Yule et. al. [76] suggest these holographic techniques offer no significant advantage over the relatively simple two-dimensional spark photography technique for measuring particles in sprays. Tyler [77] has reassessed the application of Fraunhofer holography to particle size determination.
Field scanning methods 53 7 10.3.5 State of.polarization of the scattered radiation When a system, containing isotropic and monosize spherical particles of diameter D, is irradiated with unpolarized incident radiation of wavelength 1^ the horizontal and vertical components of the scattered radiation are, in general, functions of the three parameters; refractive index m, angle of observation 6 and x = D/A^. The scattered intensity increases with the square of the particle size for vertically polarized light whereas it increases linearly for horizontally polarized light. If the intensities of the horizontal and vertical components are, respectively, H and V, then the polarization ratio R = H/V is a function of m, x and 6. Forfixedm and x, R depends only on 0, hence particle size can be determined from the positions of maxima and minima. For Rayleigh scattering 7? == 0 at ^= 90°. As R increases, theory shows that X is a periodic function of diameter for monosize particles, and this has been used to measure particle size [78] specifically the size of aerosols in the size range 0.1 to 0.4 |im [79]. It has also been used to determine the sizes of sulfur solutions [80]: In this work, transmission and polarization methods yielded results in accord with high order Tyndall spectra (HOTS) for sizes in the range 0.365 to 0.62 |Lim. In the limited region where (0.45<J<2.8) |Lim and (1.06<m<1.12) the fluctuations in R at 90° are smoothed out and the following identity results [81]. R^\.^9{m-2f7tm^
(10.12)
where
(10.13)
D =^
where there are n{D) particles in the beam in the size range D to D+dD. Maron, Elder and Pierce [82] review and extend earlier work on R measurements at 90° on monodisperse polystyrene latices and found appreciable differences between theoretical and experimental values. They showed that the discrepancy is due to inherent anistropy in the latex particles believed to be due to non-random orientation of the polymer chains in the colloidal latex particles. The size range of applicability they give to this technique is 0.135 to 1.117 |am.
538 Powder sampling and particle size determination
This procedure has been used to determine droplet size in sprays. Oscillations in the curve relating x and D can be smoothed out by the use of an incident laser beam having a broad spectral bandwidth [83]. An accumulation of independent scattering intensities from multiple scatterers can be used to measure the mean droplet size of a group [84]. This procedure has been applied to water sprays and the experimental data confirmed by phase Doppler anemometry [85]. The applicability of the polarization ratio technique is strongly influenced by the complex refractive index of the dispersed media and is limited to media having a relative refractive index below about 1.44 [86].
Fig. 10.3 Design of the fiber optic FBR-sensor 10.3.6 Forward/backward intensity ratio (FBR) The physical principle of FBR is the increasing lack of symmetry in the spatial intensity pattern of light scattered in the forward and backward directions which becomes significant outside the Rayleigh region. Depending on the sensor configuration and the light source a size range from 0.05 to 10 |Lim can be detected and sized in a matter of milliseconds. This technique is highly pertinent in processes where changing particle size needs to be monitored with a high time resolution. A prototype instrument (Figure 10.3) has been described using the ratio of light scattered at 60° and 120° from the direction of the incident light using a collecting angle of 10° [87]. For a realistic upper volume concentration level of 0.1% numerous particles are present in the measuring volume and the derived size is the mean size for these particles.
Field scanning methods
539
A simple calibration has been carried out with latex spheres and a practical application on time dependent growth of calcium carbonate [88]. A further experiment was carried out to monitor the wet grinding of submicron color pigment using diluted, extracted samples. Further work was proposed to investigate the effect of particle shape. A redesign of the FBR-sensor was performed to reduce the measurement volume and increase the upper volume concentration level to 1%. The improvements also permitted a low number concentration to be monitored (<10^ particles cm"^) so that the instrument could be used in a stream-scanning mode and as an in-line counter to monitor particles in gases. 10.3.7 Optical back-scattering The mixing process of pesticide dispersion in a spray tank mounted on a tractor was monitored using optical back-scattering probes mounted in the tank [89]. A granulated powder was fully dispersed in 20 s whereas a powder product took 4 min. 10.3.8 Transmission fluctuation spectroscopy A transmission signal measured on a flowing suspension of particles shows significant fluctuations which contain complete information on particle size and concentration. Details have been published in three parts [90-92]. In parts one and two the basic properties were studied for a beam of uniform intensity. The theory was extended in part three to a Gaussian beam with experimental tests to follow. 10. 4 Light scattering theory 10.4.1 The Rayleigh region (d« X) In the Rayleigh region the intensity of the scattered radiation in a direction making 6 with an incident beam of unit intensity is given by [45]: f
Io^\a
i2
2^Y(l + cos^^) ^^ I ^
\^m J
Ir"
^
(10.14)
where a is the particle polarizability, r is the distance from the particle to the point of observation and X^ is the wavelength of the incident radiation in the medium surrounding the particle.
540 Powder sampling and particle size determination
The intensity is the sum of two terms: L, = \cc\
2n
\4
2n
h=M \^mj
and
2r'
cos^<9
1?
which refer respectively to the intensities of the vertically and horizontally polarized components. For spheres: 2
3(m-\)V
(10.15)
a-\a
Equation (10.15) has been applied to very small particles but is more relevant to the size determination of molecules. For m-\->Q (10.16)
a = (»,^-l)^.2(™-,)^
where m is the refractive index of the particle relative to that of the surrounding medium and Fis the volume of the particle. 10.4.2 The Rayleigh-Gans region (D < A) The Rayleigh-Gans equation for the angular dependence of the intensity of the scattered light is given for spherical particles of low refractive index by the equation [45]:
—{smu - ucosu)
j0=h
(l + cos^^)
(10.17)
Again, the two terms in the fmal brackets refer respectively to the intensities of the vertically and horizontally polarized components, /Q is the intensity of the incident beam, D is particle diameter and: u=
2TID .
e
sm — A^ 12
and
k=
2n_
(10.18)
Equation (10.17) reduces to equation (10.14) when the middle term is equal to one. The scattering pattern is however modified by this term, thus
Field scann ing m ethods
541
enabling a size determination to be carried out in the Rayleigh-Gans region. Differentiating equation (10.17) with respect to u and putting d/^dw = 0, for minimum intensity gives: sin u-ucos u = 0
(10.19)
and, for maximum intensity: 3wcos u - u^sin u -3sin u =0
(10.20)
The first minimum is at w = 4.4934 radians corresponding to: - ^ s i n 4 = i : ^ = 0.715 X^ 2 2n
(10.21)
Similarly the first maximum occurs at: — s i n ^ = 0.916
(10.22)
A graphical solution for all maxima and minima has been determined by Pierce and Maron [93] who, together with Elder [94,95] extended equations (10.21) and (10.22) beyond the Rayleigh-Gans region, {m-l^' 0) to 1.00 < w < 1.55, deriving the following formulae: — s i n ^ = 1.062 - 0.347m
(10.23)
—sin ^ = 1.379 - 0.463m
(10.24)
These equations give the positions of the first intensity minimum and maximum respectively. The range of validity of these equations has been investigated by Kerker et.al. [96]. The angular positions determined experimentally were found to depend on concentration hence it is necessary to take reading at several concentrations and extrapolate to zero concentration. This concentration dependency has also been noticed with depolarization and dissymetry methods becoming negligible only when particle separation exceeds 200 radii [97].
542 Powder sampling and particle size determination
10.4.3 High order Tyndall spectra (HOTS) When a dilute suspension of sufficiently large, mono-disperse, spherical particles is irradiated with white light, vivid colors appear at various angles to the incident beam. The angular positions of the spectra depend on m and A hence they may be used to determine particle size in colloidal suspensions. Since the red and green bands predominate it is usual to observe the ratios of the intensities of the vertical component of the red and green light in the scattered radiation as a function of 9. When these ratios, R = ^red^^CTeen ^^^ plotted against d, curves showing maxima and minima appear, the maxima being the red order, the minima the green. Smaller particles yield only one order but the number of orders increases with increasing particle size. HOTS have been studied extensively in monodisperse sulfur solutions by La Mer et.al. [98-101], by Kenyon [102], in aerosols by Sinclair and La Mer [103], in polystyrene latices by La Mer and Plessner [105] and in butadiene latices by Maron and Elder [106]. The following equations derived by Maron and Elder for the angular positions of the first red and green order, 0^^ and 0^^ are particular examples of equation (10.22). Dsin ^ 1 = 0.2300
Dsin
^.1
-0.3120
(10.25)
(10.26)
Pierce and Maron [107] show that the angular positions of the red orders are identical with the angles at which minima occur in the intensity of the scattered light when the incident light has a wavelength of y^. Similarly, the angular positions of the green orders coincide with the angles at which minima occur with incident light of wavelength A/. Consequently, for the same effective incident wavelengths, minimum intensity and HOTS represent equivalent measurements. For equations (10.25) and (10.26), solving with equation (10.23) with m= 1.17 gives: 1.062 - 0.347(1.17) = 0.2300/^ .*. /^= 0.3506 jLim (0.4673 jum in vaccuo)
Field scanning methods
543
1.062-0.347(1.17) = 0.3120/A; .'. A'^=0. 4756 |am (0.6340 jum in vaccuo) This method is qualitative unless A^ and /^ are known. The above equations yield weight average diameters. With increasing m the evaluation of D becomes more difficult due to reduction in the intensity of the maximum and broadening of the peak. The method has been used for the size range 0.26 to 1.01 |Lim [108]. 10.4.4 Light diffraction The far-field diffraction pattern of an assembly of particles yields information concerning their size distribution. For particles dispersed on a transparent slide the geometrically scattered part of the incident beam may be eliminated by coating the slide with aluminum and then removing the particles. The particle size distribution of the particles can then be determined from the resulting far-field diffraction pattern [109-112]. This effect was utilized in the Talbot diffraction size frequency analyzer (DISA) [113] that used a series of interchangeable filters to determine the number size distribution of the resulting apertures. Talbot later described a filter that transmitted amounts of light proportional to particle volume [114]. He also incorporated the principle in a slurry sizer, the Talbot Spacial Period Spectrometer [115,116]. Gucker et. al. [117] developed equations from the Lorentz-Mie theory relating the size of the Airy points to particle size. Davidson and Haller [118] applied these equations to 0.07 to 0.50 \\m latices deposited on microscope slides and obtained poor agreement that they attributed to strong particle-slide interaction. Robillard et.al. [119,120] used Mie scattering at two wavelengths and a similar set up to determine the mean diameter, the polydispersity and the refractive index of a Dow latex. The experimental intensity curves were in good agreement with theory. 10.4.5 Early commercial light scattering equipment Most of the equipment described in the preceding section has been individually designed but commercial equipment was also developed. The Sinclair-Phoenix aerosol photometer brought the light to a focus at the sample cell, with a diaphragm stop placed in the optical path so that a diverging cone of darkness encompassed the light-collecting lens of the
544 Powder sampling and particle size determination
photomultiplier housing. Hence the only light reaching the detector was that scattered by the aerosol under test. Solids concentrations as low as 10"^ \xg 1"^ were detected, the mass concentration being displayed on a meter or recorder. The Differential I light scattering photometer was designed for the study of monosize particles in suspension [121]. A cuvette containing the particles in suspension is illuminated by monochromatic light and a scanning detector system records the intensity of the scattered light as a function of scattering angle. The light scattering patterns are matched against computer generated curves available as an Atlas of Light Scattering Curves [122]. The analytical approach is described in [123,124]. The Brice-Phoenix light scattering photometer and the Absolute light scattering photometer were designed for the measurement of attenuation, dissymetry and depolarization. The former has also been used to determine the particle size distribution of an aerosol at various levels in a flowstream through a column [125]. The Brice-Phoenix model 7 photo-nephelometer received light at two photocells at right angles to the incident beam whilst the model 9 had a third photocell to measure the transmitted light. 10.5 Multi angle laser light scattering; (MALLS) Early instruments employed low (forward) angle laser light scattering (LALLS) but these have been replaced by multi-angle instruments. MALLS instruments use Lorenz-Mie (often referred to as Mie) theory or Fraunhofer diffraction theory. Mie theory is the classical theory relating light scattering to particle size. Major assumptions in the derivation include plane light waves and spherical particles [126,127] but the theories of light scattering are continuously being upgraded [128]. Mie theory requires knowledge of both the real and imaginary (absorption) parts of the refractive index. Fraunhofer theory is limited to particles that are opaque or large compared with the wavelength of light and, since only diffraction is considered, no knowledge of the particle refractive index is needed. The diffracted interference pattern is large compared with the geometric image with smaller particles diffracting the light through larger angles than larger particles. The optical system is arranged on an optical bench whose length, therefore, needs to be greater for large particles than for small ones. Assemblies of monosize, spherical particles give enhanced diffraction patterns. Azzopardi [129] states that this phenomenon was used for particle sizing more than 180 years ago, in what was called Young's
Field scanning methods
545
eriometer, and in 1918 a commercial instrument was developed for sizing blood cells [130,131]. Refractive index consists of two parts, a real part that describes the refracted light and an imaginary (complex) part that describes absorption. In the case of very small particles, or particles where the complex part of the refractive index is near zero, light is transmitted through the particles and interferes with the diffracted radiation. The interaction between the transmitted and diffracted radiation results in 'anomolous scattering' and can be catered for in the Fraunhofer theory if both parts of the refractive index are known. Non-spherical particles are measured over all orientations and this causes a broadening in the measured size distribution. Textured particles tend to give enhanced weighting to the fine end of the distribution. The measurement volume is controlled by the width of the laser beam (10-25 mm) and the path length in the sample cell. The number of large particles in a broad distribution is low, which requires an extended measurement time in order to obtain an accurate estimate of their frequency. Since large particles are present in a small number of "sweeps" they are lost in the averaging process. They can, however, be distinguished by comparison of the sweeps using cross-correlation and principle component studies [132]. Synthetic fibers have been found to give a bimodal log-normal distribution using MALLS and rod shaped particles a trimodal log-normal distribution. Miihlenweg and Hirleman [133,134] found that surface roughness had little effect on measurements; in contrast to this a particle's deviation from sphericity, i.e. an elongated or a flattened shape, greatly affects the measurement result. Laser diffraction data has been inverted to provide size and shape distributions. The method was investigated using simulated diffraction patterns of elliptical particles; the limitation is that all particle sizes need to have the same shape distribution [135]. Other investigations of the response of MALLS to anisotropic particles have been carried out [136]. MALLS instruments collect light, scattered from particles in a collimated laser beam, by an array of detectors in the focal plane of the collecting lens. Early instruments used forward scattering angles up to 14° which limited the lower size to about a micron. Later instruments allowed angles up to about 40° through the use of converging incident beams and larger lenses, which extended the lower size to 0.1 |um. Some instruments incorporate additional information, such as polarization ratios and intensities at higher angles using extra detectors, in order to improve the characterization of particle size in the sub-micron size range, with
546 Powder sampling and particle size determination
extension down as low as 0.04 |im. The optical system of the Helos acts as a telescope, which extends its upper size to 3,500 jim while retaining a lower size limit at the sub-micron level. Samples may be introduced directly into the laser beam, as is the case with aerosols and metered dose inhalers, passed through a sample cell whose windows are transparent to the laser beam, or suspended in a cuvette under agitation. Dry powders are either blown through the beam or allowed to fall through it under gravity whereas particles in suspension are recirculated through the beam via a pump. Stirrer T^^mTt
Particles in suspension
[^Qi:^r=>j
;.jSample cell Pump
= : ^ ^
\ Receiver
h
Printer Central processor Fig. 10.4 The principle of low angle laser light scattering instruments. The basic instrument consists of an analyzer that includes an optical bench that has a low-power visible wavelength laser, a lens train, a photodetector, a receiver and an amplifier/analog-to-digital converter linked to a microprocessor and monitor (Figure 10.4). Misalignment of the photodetector and incident laser beam is found to have severe effects on results [137]. If the inclination is limited to < 2.5° and the beam eccentricity to less than 0.10 |Lim the resuhing error is less than 0.5%. The possible size range of a MALLS instrument is linearly dependent on the wavelength X and the focal length of the Fourier lens/. The lower limit for a He-Ne 632.8 nm source is 0.1 |Lim and this can be extended to 0.05 |Lim with a 325 nm He-Cd source; however this is 30 times more expensive. At the upper limit, a He-Ne laser at a wavelength of 1152 nm
Field scanning methods
54 7
extends the size limit to 6 mm. A discussion of the various alternatives has been presented by Witt and Rothele [138]. 10.5.1 Theoretical basis for MALLS instruments The angular intensity distribution of scattered light \{0,/J) at angle 6, for particles of diameter d and refractive index // is given by Mie scattering theory. For a polydisperse distribution: l{e,n) = ^^h{a,9,^}x'^nda
(10.27)
0
which is also known as a non-homogeneous Fredholm equation of the first kind. 7(0) is the intensity of the incident beam, n{a) is the number of particles in the laser beam with sizes between a and a+da, where a = nD/A, X is the wavelength of light in the surrounding medium. The solution depends on the exact form of the kemal function h{a,6,ij). For a ring detector the scattered light arriving at the /th ring of the detector is given by: /(^^,;/) = Zv^ \h{a,e^^)x^nda
(10.28)
Similar equations apply to each ring of the detector. This results in a linear set of / equations with k (discrete size classes) unknowns. The vector of scattered energy can be expressed as a matrix equation: S=CW
(10.29)
where S is the real vector of scattered energy, W is the real vector of size distribution of particles with diameters between the limits for the discrete size classes and C is the coefficient matrix of / rows and k columns. The element cik indicates that unit mass of particles in the Ai:h size interval produces a scattering signal on the /th ring of the detector. For this reason the particle sizes are divided into k discrete size classes that are determined by the following equation [139]: afi' =1.357
(10.30)
548 Powder sampling and particle size determination
where a = nDj/Ji, 6^ = rjf, is the maximum radius of each ring of the detector and/is the focal length. Equation (10.30) indicates that the maximum of scattered energy is reached only in one position of the detector, for representative diameters of each size class. The scattering matrix depends upon two parameters, the focal length of the receiving lens and the relative refractive index. Equation (10.30) is ill-posed so that the scattering matrix C is highly illconditioned. This implies high-frequency oscillations in the solution W. Thus more sophisticated methods are required in order to find a solution. Integral transformations are used successfully [140] giving close solutions to equation (10.30). Alternatively a function constrained inversion in which the size distribution is assumed to follow a given form with two degrees of freedom or a model independent function can be employed. Alvares et. al. [141] successfully applied a method known as the Tikhonov Regularization method and L-curve criterion to generate data in close accord with the Malvern software. Fraunhofer theory applies to the scattering of light in the near forward direction by large particles. The scattering pattern for a single spherical particle consists of a series of light and dark concentric rings that decrease in intensity with increasing angular position. These rings are produced through constructive and destructive interference of light diffracted from the edge of the particle with changing light path length. The angular distribution of light flux I{d) for a single opaque spherical particle, as given by the Fraunhofer equation, is shown in equation (10.31) in terms of the first order spherical Bessel function J^iO).
I{9) = I{Q)
2Ji(ad)'
(10.31)
For a distribution of particle sizes this becomes:
/(,)=/(o)|MM)
2
f{D)AD
(10.32)
Figure 10.5 presents a two-dimensional graph of the pattern for latex beads in air (/I = 1.55) of diameter 50 and 25 |Lim. The diffraction pattern is independent of the optical properties of the particles and is a unique function of particle size being defined by the size parameter a=^nD/X, where D is the particle diameter. The scale of the curve on the vertical
Field scanning methods 549
(0) degrees Fig. 10.5 The forward intensity distributions for single particles: note that 84% of the energy lies within the first minimum. 2.0
200^01
2
4
6
10
Scattering angle (degrees) Fig. 10.6 Intensity distribution from three particles of size 10, 60 and 200 |Lim. The intensity scale is times 1 for the 200 |Lim particle, times 100 for the 60 |Lim particle and times 105 for the 10 |Lim particle [142].
550 Powder sampling and particle size determination
axis decreases with increasing particle size, compressing to smaller angles on the horizontal axis, so that the 50 |Lim particle results in a curve compressed by a factor of two compared to the 25 \im particles. Over a particle size range of 1000:1 the scale changes a thousand fold. Figure 10.6 shows the relative intensities from three opaque particles, in air, of sizes 10, 60 and 200 iiim; this illustrates how the scattered flux in the forward direction falls off rapidly with decreasing particle size. The effect is also illustrated in Figure 10.7a together with the resulting diffraction pattern for a monosize distribution (Figure 10.7b). Information on the particle size distribution can be found by measuring the scattered light flux at several radial locations to characterize the series of annular rings from the particle's diffraction pattern. It is then necessary to invert the relationship between the scattering pattern and the particle size distribution. This can be done using iterative methods or analytical inversion techniques. In early instruments, the detectors consisted of a series of half rings [143,144] (Figure 10.8) so that a matrix equation developed. Sliepcevich and co-workers [145,146] inverted this equation to obtain the particle size distribution. The equation was solved by assuming the distribution fitted a standard equation and carrying out an iteration to obtain the best fit. A matrix inversion was not possible due to the large dynamic range of the coefficients and experimental noise that could give rise to non-physical results. An inversion procedure that overcame these problems was developed by Philips [147] and Twomey [148] that eliminated the need to assume a shape for the distribution curve. rx-rr . A^u. Diffracted light
Multi-angle detector
Laser beam
(a)
(b)
Fig. 10.7 (a) Particle size determines diffraction angle (b) the diffraction pattern from a monosize distribution.
Field scanning methods 551
Fig. 10.8 Representation of a typical photosensitive silicon detector. The thin lines represent insulating gaps. The mathematics of the single-event Fraunhofer diffraction of an on-axis laser beam by a sphere was used in the development of this technique [149]. The assumption of single scattering is adequate for accurate measurement so long as the light obscuration by the particle field lies within the range 5-50%. Although measurement at low concentration is desirable it is not always possible; with industrial sprays, for example, size measurements may have to be made for light obscuration values in the range 90-99%. The more that multiple scattering occurs, the more the particle size distribution is biased to smaller sizes if single event theory is wrongly used. Boxman et. al suggest that more information can be obtained if the fluctuations in the signals from each detector are examined together with the mean values [150]. They note that this approach can identify whether the inaccuracy is due to insufficient sampling of the detector array or imperfections in the optical model. More recently Knight et. al [151] developed an analytical inversion method that gave improved resolution and accuracy in size distribution measurement. Considerable differences exist between instruments, both in hardware and software, so that there is a lack of agreement in data generated by different instruments. A commercially used means of measuring the scatter pattern is with a logarithmic line array detector, which has detector elements in a geometric size progression with each element larger than the preceding one by a constant multiple. The important feature of the logarithmically measured scatter pattern is that for two different particle sizes the shape of the elements remains the same but the position on the ln(^ axis is shifted. This shift invariant function X-^) permits the use of iterative deconvolution
552 Powder sampling and particle size determination
to determine the particle size distribution from the scatter pattern measurement. Typically, the pattern is measured 1000 times in 20 s and the results averaged. Mathematically, a coordinate transformation converts the linear scatter function I{0) to the logarithmic scatter functionX^). This function is a convolution of the volume distribution of particles and the single sized shift invariant response functions for each size shifted according to the size. A method of solving the deconvolution is to divide the particles into size intervals and assume that each one will generate a diffraction pattern according to its average size, the intensity of which depends upon the number of particles in that size range. The diffraction pattern can then be manipulated by matrix methods to yield the size distribution. The measured data always contain random and systematic errors which have to be accounted for in the deconvolution step. Several mathematical procedures have been developed which can generate different solutions using the same initial data [148,152]. 10.5.2 Commercial instruments Several instrument manufacturers have applied this principle (Table 10.1). A low power laser illuminates a suspension of particles to be examined. The scattered light is focused by a convergent optical system on to a multielement, solid-state detector that usually consists of 32 semi-circular rings or a linear array. An additional detector at the center is used to align the laser and measure the intensity of the unscattered light. The electrical output of the rings is proportional to the amount of light flux falling on them and the signals can be interrogated to obtain the size distribution generating them. A total size range of 0.1 |Lim to 3600 |um can be measured, though only a part of this range can be measured at one time. The measureable size range can be extended by the use of off-axes detectors and white light sources with 90° detection of polarized light. The powder to be measured is dispersed in a bath, where it is stirred or/and ultrasonically agitated whilst being circulated through a glass measurement cell. The cell is illuminated with a laser beam and the forward scattered light is focused on the multi-element detector with a Fourier transform lens. The signal that is derived from each detector element according to the intensity of the light falling on it is collected and digitized for high speed computer processing. Each instrument uses its specialized software to generate the size distribution from the signals and, since the algorithms differ, the size distributions generated also differ.
Field scanning methods
553
Table 10.1 Commercial multi-angle laser light scattering instruments Instrument
Number of 1 Size Ranges Intervals | (jam) 84 0.40 - 948 92 0.40 -2000 116 0.04 -2000 100 0.70 - 400 100 0.50 -2000 100 0.10 - 500 0.04 -2000 62 0.16 -1250 42 0.1 - 600 80 0.02 - 1000 80 0.02 - 1000 87 0.02 -2000 0.04 - 700 0.7 - 700 3.2 -2000 0.02 -3000 0.02 -3000 25 -1500 0.7 -1000 31 1.00 - 500 100 0.10 - 600 100 0.05 -3500 100 0.05 - 900 100 0.10 -2000 100 0.10 - 600 32 0.30 - 300 0.05 -550 0.02 -2000 0.5-200, 1-400, 2.2 5-850 0.60-170. 0.9-25f , 1.8-510, 3.61020, 1-500
Beckman Coulter LSIOOQ Beckman Coulter LS200 Beckman Coulter LS230 Cilas 920, (U.S. agent,Quantachrome) Cilas 940 Cilas 1064 Cilas "2 in 1" dry/wet Fritsch Analysette 22 (U.S. agent, Gilson) Horiba LA-300 Horiba LA-900 Horiba LA-910 Horiba LA-920 Leco Lecotrac LT-100 Leco Lecotrac LT-150 Leco Lecotrac LTL-200 Microtrac S3000 Premium Enhanced Standard Malvern Insitec ESPA systems Malvern Mastersizer E Malvern Mastersizer S ; long bed Malvern Mastersizer S ; short bed Malvern Mastersizer X; long bed Malvern Mastersizer X Malvern Mastersizer Micro Malvern Mastersizer Microplus Malvern Mastersizer 2000 Malvern Spraytec droplet sizer NittoLDSA-1300A Nitto LDSA-300 Seishin SK Laser Micron Sizer PRO-7000 1.00 -192 1 0.40-50 Shimadzu SALD-201V; Model 1 0.4-200 Model 2 Shimadzu SALD-2001 0.03-70 1 Sympatec Helos 0.10-35, 0.25-87.5 , 0.50 -175, 31 Latest version has a stated upper size limit of 0.50-350, 0.50-75, 0.50-1750; 0.50-2625; 16-350 0 8750|^m.
554 Powder sampling and particle size determination
Since one lens is only suitable for a limited range of particle sizes, several lenses are usually available to encompass a wider range. The technique can be used to analyze powders in suspension, liquids, droplets and particles dispersed in air. The radial symmetrical diffraction pattern can be measured in other ways, for instance by a rotating optical filter with windows at different radial distances, such as was used in the Microtrac Small Particle Analyzer [153] Some instruments combine MALLS with 90° scattering at three wavelengths and orthogonal polarities to extend the size range to a lower size than is usually assumed possible for forward light scattering alone. Others use off-axis lasers and detectors to extend the size range Beckman Coulter LS series uses a dark-field reticule having an array of 500 i^m holes for automatic alignment purposes. The LS uses reverse Fourier lens optics incorporated in a binocular lens system to optimize light scattering over the widest dynamic size range. Results are calculated from either Fraunhofer or Mie theories. An application of this instrument to the measurement of micro-silica mixtures has been presented [154]. Beckman Coulter LS 200 uses 118 detectors to measure particles from 0.4 to 2000 |im in a single scan, using 92 channels, without changing optics or settings with a resolution into 116 size channels. Beckman Coulter LS 230 has eight more detectors in its diffraction arrays with an additional 42 data points provided by the polarization intensity differential scattering (PIDS) system which extends the lower size limit down to 0.04 |Lim. The PIDS system uses three wavelengths at two polarizations, obtained by filtering light from an incandescent bulb at 0.450, 0.600 and 0.900 ]xm. Measurements are made at several scattering angles at both polarizations. The difference in scattered intensity between the two polarizations is highly sensitive to particle size, wavelength of light and angle of measurement. Beckman Coulter LS 13 320 is new to the series designed to meet the requirements of ISO Standard 13320-1 [155]. It comes with an "Autodock" system that enables switching between modules in seconds together with a number of functions that simplify the measurement process. Aqueous liquid module is a fully automated module, for large sample volumes, with high sample throughput and an optional auto-prep, station Universal liquid module is designed for operating with water or organic solvents. It incorporates full auto-rinse and fill capability, minimizing fluid usage. Tornado dry powder system uses a "hypersonic" high shear system to ensure maximum dispersion without sample attrition. This system is stated
Field scanning methods
555
to be preferable to positive pressure systems that tend to offer poor control and a tendency to mill the sample. Micro liquid module has a 15 ml capacity and is ideal when sample is at a premium or when hazardous material is being analyzed. In a series of lectures, the improvements effected using the Coulter double lens system and PIDS are discussed [156]. Scattering intensity falls off rapidly as 6 increases; the effect of this is reduced using PIDS which also enables the lower limit of measurement to be reduced to 0.05 |am. PIDS system has been applied to size determination of several pigments [157]. The importance of the imaginary part of the refractive index, in calculation of particle size distribution, was stressed. Ancillaries include a micro-volume module, a hazardous fluid module and a dry powder module. Their small volume module uses less than 125 ml of diluent. The micro volume module has a total volume of 12 ml; this has an internal stirring mechanism to keep particles suspended during analyses. The automatic hazardous fluids module can be supplied with a microtip eight power ultrasonic probe to keep the particles dispersed. The dry powder module analyzes both free flowing and cohesive powders covering the size range from 0.375 ^m to 2000 jiim. Cilas 920 measures suspensions or dry powders in the size range 0.3 to 400 Jim without changing the optics. It includes automatic feeding and analyses of up to 28 samples and has the capability for displaying particles in the 10 to 400 |Lim range using a CCD camera. Cilas 1064 has high resolution, giving information over the size range 0.04 to 500 |um for suspensions and 0.3 to 500 \xm for dry powders in one shot, due to a double sequenced laser system (two lasers at different incident angles) and 64 separate silicon detectors to give 100 size channels of data. The benefits of such a system include coverage of a wide size range with a small optical bench structure. The diffraction pattern is compared to a known powder and the size distribution calculated. Cilas 1180 covers the size range of 0.04 to 2,500 |im, for both wet and dry systems, using three lasers with a custom silicon detector and a CCD camera, with fast response and high resolution. A wide range of accessories includes free fall, small liquid volume unit, alcohol recirculator, integrated Windows base and particle visualization. Coulter LSI00 uses (PIDS) to extends the lower limit to 0.1 |um and 126 detectors to measure the scattered light. Multiple optical trains and a wideangle detector array are used so that the instrument can simultaneously capture light diffracted at high angles from small particles and at small angles by large particles. Measurements from 0.1 to 1000 |Lim can be made in a single run [158].
556 Powder sampling and particle size determination
Fritsch Analysette 22. The outstanding feature of this instrument is its convergent laser beam, with the broad measurement range of 0.16 to 1250 |im that can be covered in a single pass with no need to change optical elements. Moreover, by moving the measurement cell in the beam the complete range can be divided into over 400 individual size ranges starting at 0.16 to 23.7 |Lim. The instrument is available in four versions; A(dvanced), C(onvenient fully automated), E(conomical) and P(rofessional). Horiba LA-300 is a compact instrument with a built-in auto-alignment function. This feature not only maintains ideal operating conditions but self-calibrates with the touch of a button. Measurements are performed using a 650 nm laser diode, six wide-angle detectors and a 36 channel ringshaped silicon diode array detector Horiba LA-700 is a more sophisticated instrument that employs side and back scattering detectors as well as forward scattering detectors. By using a tungsten lamp in conjunction with a He-Ne laser the lower limit is reduced to 0.04 |Lim to cover the range 0.04 |Ltm to 262 jum with no change in lens. Horiba LA-900 extends the upper range to 1000 |am by the use of a long focal length lens to cover the range 0.04 to 1000 |Lim with no need to change lenses. Horiba LA-QIO extends the lower size limit down to 0.02 |j,m with no optical changes. Small angle light intensity is measured using a condenser lens to condense the light on to a ring-type detector. Detectors are located at the side and rear to measure light scattered at larger angles. The measurable range is extended with the use of a short wavelength tungsten lamp in addition to the He-Ne laser beam. Up to three different dispersing media can be stored in the analyzer's reservoir unit that automatically injects the right amount of dispersant into the sample. It employs an autocycle agitation, ultrasonic dispersion, circulation and measurement. The analyzer uses an iterative deconvolution method to calculate the size distribution. The results are displayed in an 80-channel histogram. Horiba LA-920 has the functionality of the LA-910 with improved software. It has a ring shaped detector in 75 sections together with 12 wide-angle detectors to give 87 sensing elements that extend the upper size limit to 2000 |im. The optical system includes a He-Ne laser for large particles together with a shorter wavelength tungsten lamp for small particles. The unit comes with a computer, printer, monitor and Windowsbased operator interface. Horiba Powder Jet Dry Feeder can be teamed up with the LA-910 or LA 920 to disperse and de-agglomerate powders by subjecting them to a
Field scanning methods
55 7
high shear rate where high and low velocity air meets at the nozzle exit followed by expansion in a low pressure region. An optional vacuum sample cell is available for friable powders. A high performance industrial vacuum is also included. Horiba Autoreservoir, available for the LA-910 and LA-920, is a threechamber tank with a built-in pump. The unit supplies liquid to the analyzer in response to commands. Concentrated surfactant solutions can be added to two of the tanks and then automatically diluted during the fill sequence. This automates the supply of dispersant solution for fill, sample dilution and rinses. Horiba Autofill or Autoreservoir supplies liquid to fill the pumping system. The system then circulates the dispersant solution and records the baseline. Horiba Autosampler or Slurry Sampler then supplies the sample to the analyzer. Built-in ultrasonics aid in dispersing the sample. The concentration is checked and diluted until it is in the optimum range before measurement begins. After the results have been calculated, the data is saved or printed. The sample dispersion is then drained and rinse cycles prepare the system for the next sample. The Autosampler can make continuous measurement of up to 24 different samples. Minute quantities of material can be analyzed in the optional Fraction Cell. The unit includes a PowderJet dry feeder capable of feeding millimeter size powders and dispersing cohesive powders to sizes below Ijiim. Horiba claim a higher resolution than similar devices by using repetitive calculations instead of the more conventional matrix method. A description of the Horiba instruments and a discussion of the requirements and physical basis for carrying out particle size measurements with these instruments have been presented by Boeck [159]. Insitec EPCS is covered in detail in section 10.7. They are laser-based instruments for in-line particle measurements that provide information on particle volume concentration and size distribution. Unlike other instruments operating on this principle, the EPCS can perform direct measurements of particle-laden gas flow stream provided the concentration is within operating limits. Lecotrac-LT-100, manufactured by Leco Corp, uses three lasers and two detector arrays to measure light intensities from 0 to 160 degrees. Lasers are self-aligning and the instrument uses Windows based software. Lecotrac-LT-150 is a basic cost-effective analyzer with a more limited size range. Lecotrac-LT-200 is designed to analyze coarser particles than the LT150,
558 Powder sampling and particle size determination
Low angle detector array Scattering particles
High angle collector lens
High angle detector array
Fig. 10.9 Low angle on-axis and high angle off-axis array as used with the Microtrac Full Range Analyzer. The Leco units handle dry or wet samples and an automated small volume recirculator with autosampler permits unattended particle size analysis. Microtrac SRA 150 (Standard Range Analyzer) is a low cost instrument which covers the size range of 2 to 700 jiim using either a small volume (250 ml) recirculator or a dry powder feeder (10-20 g min"^). Microtrac FRA (Full Range Analyzer) has extended the lower size by using an off-axis low angle array with a separate collector lens in conjunction with the logarithmic line array detector (Figure 10.9). Microtrac XI00 system uses three lasers to measure scattered light at angles up to 160° with resolution from 0.04 to 704 |um. The use of three lasers provides superior resolution in the sub-micron region so that modes smaller than 0.5 jam can be more readily detected and resolved. An automated small-volume re-circulator accessory provides efficient sampling. Microtrac Ultrafine Particle Analyzer operates using photon correlation spectroscopy in the controlled reference mode, extends Microtrac's lower size to 0.003 with an upper limit of 6.5 |a,m. Microtrac S3000 is available in a number of upgradeable configurations to satisfy a wide range of application requirements. A photograph of this instrument is presented in Figure 10.10. The modular design provides theability to update and expand a system inexpensively. The S3 000 uses a Tri-laser source and logarithmic array detectors for high accuracy and
Field scanning methods 559
Fig. 10.10 The Leeds and Northrup Microtrac X3000 consistency. Conversion from a wet system to a dry powder system is accomplished in seconds. The instrument is designed to occupy the minimum of bench space. There are two types of Mastersizer instruments; the Mastersizer Micro and the Mastersizer E, which are low cost instruments for repetitive analyses; and the modular series of Mastersizer S and Mastersizer X, the ultimate in resolution and dynamic size range, which are required when samples in the form of aerosols, suspensions and dry powders need to be measured. Mastersizer X provides a selection of small size ranges using a variety of interchangeable lenses whereas the Mastersizer S provides a wider dynamic size range covered in a single measurement. For powders which are to be suspended in a solvent, emulsions, suspensions and particles in liquids there are small volume cells which require as little as 15 ml of dispersant. Where a material is either valuable or toxic the Malvern Small Volume Flow Cell, with a sample volume of 50-80 ml and full sample recovery, can be used. The X-Y sampler is a 40-sample accessory
560 Powder sampling and particle size determination
for either dry or wet samples. Malvern also offer a free fall dry powder feeder, a dry powder feeder, an automated dry powder feeder and an aerosol mounting unit. The Malvern lower size limit is extended to the range 0.01 to 3 |Lim with the addition of the Autosizer, which operates using fixed angle photon correlation spectroscopy, and this is extended to 0.001 to 5 |Lim with the more sophisticated System 4700. Malvern Mastersizer £" is a fixed configuration model to lower cost. A range of small volume accessories are available along with the standard Mastersizer E tank that is suitable for volumes of 1 liter. The wide size range of 0.1 |Lim to 600 |im makes it ideal for routine analysis and quality control. Earlier versions comprise the Series 2600 and 3600 and Mastersizer R/IP on-line particle size analyzer. The Malvern 2200 was a droplet and spray particle analyzer and the Fibersizer 600 was designed for fast automatic measurement of fibers in the 4 to 64 |am size range with a resolution of 0.25 \xm. Malvern Ma^/^r^/z^r X operates in the size range 0.1 to 2000 jLim. A wide choice of modules can handle dry powders, suspensions, emulsions, aerosols and sprays in manual or fully automatic mode. Malvern Mastersizer S uses Mie theory to cover the size range 0.05 to 900 |Lim using a single lens, and up to 3500 |Lim with an extended range system. Scattering angles from 0.01° to 150° are detected in order to cover this wide size range. The system handles measurements of particles, droplets or gas bubbles. Malvern Mastersizer Micro, which uses a Tolded optics' configuration, is the most compact MALLS instrument presently available. In its basic form it has a size range of 0.3 to 300 |Lim. Malvern Mastersizer Microplus extends this range to 0.05 to 550 ^m. This instrument is intended where low sample throughput, a limited size range or well-defined user needs makes the flexibility of the more sophisticated versions unnecessary. Roberts has presented a discussion of low cost sizing using these two instruments [160]. Malvern Mastersizer 2000 uses a standard operating procedure to eliminate user variability. A redesigned optical system has removed the need for adjustments between samples. An integrated optical system covers the full size range from 0.02 \im to 2000 |Lim. Reverse Fourier optics, using wide angle dual wavelength detection system provides improved resolution and extended range, have been enhanced by additional
Field scanning methods
561
forward and back scattering detectors, combined with a high-stability monochromatic blue light source. Malvern Spraytec droplet size analyzer measures size and concentrations in continuous sprays, aerosols and dry powder inhalers. The multi-scattering algorithm allows the unit to measure higher concentrations than with laser diffraction. Sampling is carried out every 0.4 ms. Measurements can consist of a single detector scan or averages of multiple scans, Micromeritics Saturn DigiSizer 5200 measures in the size range 0.1 to 1000 |Lim. A light beam from a solid-state laser is coupled to a beam splitter that directs a portion of the light on to a reference photodiode. The remainder of the light is directed by fiber optics on to a collimater and thence to the sample cell. A Fourier lens intercepts the scattered light, projecting a fraction of the scattering pattern on to a CCD array. The CCD array is composed of over 1.3 million detector elements that accumulate electrical charge in proportion to the intensity of the light on its surface and its exposure time. The operating software scans the detector array and determines which element contains the center of the beam. The software then maps all the other pixels according to their angular position. This eliminates the need for accurate optical alignment. The range of light intensity in a scattering pattern can scan more than 10 orders of magnitude which is greater than the dynamic range of any light detector. To compensate for this, multiple exposures are taken at each CCD position using different quantities of light and different exposure times. After the first set of measurements, the scattering pattern is shifted by increasing the angle of incidence relative to the sample cell, thereby exposing the CCD to a new range of angles. The software re-maps the CCD array accordingly. Another series of exposures is taken, the series normalized, and the angle of incidence again increased. The process continues until the scattering pattern is determined. The result is a digital representation of the scattering pattern expressed as intensity versus scattering angle. The resolution gained by this system is such that modes spaced as closely as 1.1 and 1.3 |um can be resolved. The MasterTech 052 Autosampler can line up 18 samples to run consecutively without operator involvement. Nitto LDSA-1300A covers the size range 1 to 2000 |im and the LDSA2300A covers the size range 1 to 500 |am. Seishin PRO-7000 laser micron sizer operates in the measurement range 1 to 192 jLim with ancillaries for dry powder feeding and automated
562 Powder sampling and particle size determination
analyses on 20 samples. The PRO-8000 is an on-line system for either wet or dry processes Shimadzu SALD 201V Model 7 is a compact unit, consisting of the analyzer and a batch cell, covering the size range 0.4 to 50 |Lim and featuring reliable analyses with small (2 ml) samples. Shimadzu SALD 201V Model 2 also has a flow-through cell and provides a wider measuring range of 0.4 to 200 (im. Shimadzu Said 2001 covers the size range from 0.03 to 700 |um using a 76 element sensor to capture the forward scattered light while side and back-scattered light is captured by three other sensors. Shimadzu Said 3001 covers the size range from 0.1 to 2000 |Lim Shimadzu Wingsald measurements and data processing software allows flexible test parameter set-up and real-time monitoring of sample dispersion. Windows 95 software features 3-D graphing and a superimposed comparison of up to 12 graphs. Sympatec Helos is available in 4 configurations. BF: 0.1-875 |im, KF: 0.1- 8750 jLim, Vario: 0.1-8750 |Lim, Mytos: 0.5-1750 |Lim. Ancillary modules permit the sizing of aero-dispersions, suspensions and sprays. The measurements are carried out using a multi-element detector consisting of 31-semi-circular elements with the beam auto-aligned on the central element. Sample couplers are available for on-line use. The ancillary Rodos is a unique dispersing system for both cohesive and free-flowing dry powders [161]. The powder is fed from a hopper into a rotating channel where the excess is skimmed off and the remainder compacted. It is then transported into an eductor using suction and a rotating brush. Impacting the powder against inclined targets in the eductor can carry out further dispersion. The brush disperser has been compared with a pin mill disperser with sub-micron powder feed [162]. Ancillary Equipment: Most instruments are based on the analysis of circulating wet suspensions. This set-up is unsuitable for brittle material and many manufacturers provide agitator cells to reduce breakage. Waste disposal is a major problem with some materials and this is reduced by the use of small volume cells (< 125 ml). Cells are also provided for corrosive liquid systems. Instruments can be set up to analyze falling streams of dry powders; this is particularly suitable for brittle or soft (wet granules) materials. Deagglomerators are often provided for sticky dry powders. Activity Software imports data from Micromeretic's Sedigraph 5100, Leeds and Northrup's Microtrac and Cilas Model 920. In operation data such as particle size distribution, surface area, filler costs or other properties are entered into the software. Target values are next assigned to the data components with the values scaled to reflect each component's
Field scanning methods
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importance. Software automatically analyzes the data and displays the result as a table or a graph. 10.5.3. Discussion MALLS are being used increasingly for rapid off-line and also on-line size analysis. The latter requires an automatic sampling device that takes part of the process slurry (or powder) and dilutes it to the required concentration for the analyzer. The dilution unit can be quite complex; when the instrument is hooked up to a crystallization unit for example it is necessary to filter some of the mother liquor and use that as the diluent. The Mastersizer on-line analyzers, the MS200/1P series have the same 0.1-600 |im overall and sub-ranges as the laboratory analyzers. For a wider overall range the MS200/IP systems extend from 0.1 to 2000 jam. Sympatec manufacture an on-line system extending from 5 to 3500 |Lim or more. A cell has also been described for measuring the fines outlet in an air classifier [163]. The instruments tend to be easy to operate and yield highly reproducible data. Data presentation varies from instrument to instrument and generation to generation, each new model being an improvement on its predecessor [164,165]. The general tendency with non-spherical particles is to oversize the coarse end of the distribution and assign an excess of particles to the fine end and, in so doing, broaden the distribution. The instruments are particularly useful with powders that are difficult to disperse since such powders can be incorporated into a liquid containing a dispersant and close loop circulated whilst undergoing ultrasonic probing. Analyses are taken every few minutes until the measured size distribution stabilizes. In an examination of glass beads sieved into five fractions in the size range 70 to 400 mm it was found that the accuracy of the determined medians was good. The distribution widths and the resolution of binary mixtures were compared with image analysis. These showed some discrepancies for both mono-and bi-modal distributions with undersizing in both cases [166]. Comparison tests between the PSS Accusizer 770 (light blockage), API Aerosizer (time of flight) and MALLS instruments (Malvern Mastersizer and Coulter LS) indicated that the MALLS instruments gave results that differ from each other and from microscopy whereas the other two instruments gave results in agreement with each other and with microscopy [167]. The choice of the Fraunhofer or Mie theory model affected the calculated size distribution in an unpredictable manner and the use of Mie
564 Powder sampling and particle size determination
theory did not assure more accurate results. For glass microspheres 1095 \\m in diameter the Mie model with the Mastersizer yielded results closely matching the Aerosizer but not the Coulter instrument, three other irregularly shaped powder systems were also examined and gave disparate results. [168]. Kaye et. al. [169] compared sieving, image analysis and MALLS using four ferrite powders and discussed the causes of observed differences. They found that the particle size distribution by MALLS was larger than by image analysis and much larger than by sieving. In particular they found large particles that were not present using image analysis; they ascribed this to artifacts from the data processing deconvolution. In a comparison with an x-ray centrifuge (XRC) it was found, predictably, that the XRC determined mixing ratios of bimodal mixtures with small errors: The errors with the Sympatec were larger but the reproducibility was higher [170]. The shape of the detector disc in the Malvern Mastersizer X allowed it to be used in an innovative way. Particles are aligned in the flow cell hence the scattering pattern differs in the vertical and horizontal plane. The detector is an annular disc divided into 32 light sensitive segments radiating outwards. This can be rotated and the two sets of signals compared to generate data on fiber length distribution. [171]. The effect of multiple scattering has been studied by several researchers [172,173]. Solutions, in the form of empirical equations, have been provided for Rosin-Rammler and log-normal distributions [174]. This has since been extended to a 'model-independent' case [175]. Fraunhofer theory does not apply if the refractive indices of the particles and the fluid are similar. Fraunhofer theory applies when a plane wave front of monochromatic light is diffracted by an opaque particle, much larger than the wavelength of light, and focused by a lens on to a detector. For low relative refractive indices it is possible for light to pass through the particle and the transmitted light then interferes with the diffracted light to produce the so-called 'anomalous diffraction patterns'. If no correction is applied an erroneously high percentage of small particles will be predicted. Experimental and theoretical verification of this effect has been reported [176]. An experimental study has also been carried out on the on-line monitoring of clear and opaque particles. [177]. Jones discussed ISO 13320 [178] and stated that application of this guideline ensured high reproducibility and highlighted the weakness of Fraunhofer theory as opposed to the Mie theory [179]. The need for maintaining constant temperature was emphasized since this affects the length of the light path in the measurement cell and the
Field scanning methods
565
particle. Considerations such as this are particularly important when studying crystal growth [180]. A standard test method for calibration verification of laser diffraction particle sizing instruments using photomask reticules has been introduced by the American Society for Testing and Materials [181-183]. The test sample consists of a two dimensional array of thin, opaque, circular discs deposited on a transparent disc. The test method is designed to verify instrument performance on an ongoing basis, to compare one instrument performance with that of another and to provide error limits for the instrument tested. These photomask reticules have been tested to determine the effect of having overlapping images as opposed to nonoverlapping images [184]. Seselj [185] reported on the use of wet and dry laser light scattering to determine the size distribution of magnetic material. Muehlbacher [186] used a tri-laser system to determine accuracy and reproducibility with +0.022 |Lim polystyrene samples. Classically, the particle size distribution of soils and sediments are determined by sieve for the coarse fraction and Andreasen pipette for the fine fraction. From the point of view of laboratory efficiency MALLS is deemed superior. Although MALLS data for standard material were in good agreement with certified measurements; tests on the mean diameters of sediment, except for the sub-|Lim mean diameter, showed poor agreement. Platey clay particles, for example, were 2 |im in median size by pipette and 8 |Lim by laser light scattering. The authors discuss the possibility of calibrating the laser instrument to give similar data to traditional data [187]. Veran stated that in industry, measurements are usually carried out by x-ray sedimentation and laser diffraction and they hardly ever agree. He compared the CILAS 1064, the x-ray Sedigraph 5000ET and 5100 and found that the application of an extinction coefficient brought the laser diffraction into agreement with photosedimentation [188]. Bordes et.al. [189] reported experimental results on the use of the Turbiscan On-Line Formal action particle analyzer for monitoring a wet grinding process. Many manufacturers have large data banks using, for example, sieve analysis and they need to continue generating data in the same form. To cater for this need, some manufacturers of MALLS instruments install a conversion program, so that the instruments provide data that can be compared with historical data. However, Austin [190] compared sieve analyses with Sedigraph and Microtrac data and found that the conversion
566 Powder sampling and particle size determination
from Microtrac to sieve size depends not only on a mean shape factor but also on the size distribution being measured. Laser diffraction not only provides a particle size distribution by volume but it can also be used to provide a value for volume concentration. Since laser diffraction assumes that all particles are spherical, for plate-like particles, this leads to an overestimation of volume concentration. Particle thickness and aspect ratio can be derived using this information [191]. The Sympatec has been applied on-line to the analysis of a polymer in a styrene/water suspension [192]. A recent review contains 35 references and covers fundamentals, sample preparation, operations and evaluation of results [193]. Figure 10.11a illustrates the installation of a low-angle laser light scattering instrument in a pan granulation process. A major problem in retro-fitting equipment such as this is the dearth of suitable installation positions. In cases where there is insufficient room for a commercial sampler, a special unit has to be designed (Figure 10.11b). Since segregation is almost certainly present it is necessary to design a system that takes the whole of the stream for very short time intervals. This 'bite' is fed to a hopper and thence to the analyzer via a vibratory feeder such that the complete sample is fed through the system in about 20 s. The sampler senses when the concentration is adequate and commences analyzing; when the concentration fell below the required level the analysis is terminated. During the measurement time several hundred analyses are carried out; the instrument generates a weighted average and stores the data. In this illustration, six parameters were forwarded to the control room for display; the 10%, median and 90% sizes, the mass percentage within specification, the mass percentage oversize and the mass percentage undersize. Figure 10.9c and 10.9d show a process going out of specification due to generation of excess fines. An increase in yield of 1% can be worth hundreds of thousands of dollars per year and pay-back is rapid. Using laboratory type equipment to measure the dried product gives a time lag of hours. Minute by minute control of such variables as the slope of the pan, the amount of water and the positions of the sprays maximizes the yield. The use of the Sympatec for on-line particle size distribution of fines in an air classifier outlet has also been reported [194]. It is well known that free-flowing powders segregate, but not so widely recognized that cohesive powders may segregate during manufacture and need to be mixed prior to assay. A ceramic paste, used as a dielectric in electrical capacitances, needed a median size by MALLS of between 2.8 and 3.2 |xm. Too fine a distribution caused blistering during sintering and
Field scanning methods
567
too coarse a distribution led to electrical leakage. The paste was ballmilled then dried and 16 samples taken from the dried powder had a mean size and standard deviation of (3.1±0.17) |im. The selection criterion was that if any three consecutive samples fell outside the range of 2.8 |im to 3.2 |Lim the batch was rejected and on this criterion there was a 50% chance of this happening. After mixing the standard deviation fell to 0.012 |Lim. It was conventional to mill for 145 hours to reach the desired end-point. The implementation of these rapid response instruments permitted the extraction of samples after 8 hours and 12 hours and plotting the measured median on log-probability paper allowed one to determine the milling time required to attain the desired end-point thus eliminating wastage.
Pan granulator
Loss-in-wcighl feed control and % moisture Particle size analyzer Drier
Vibratoiy feeder
P]C=]| (Granule size analysis every 30 seconds)
Under
100
.s
Product
Over
(a)
1 — iI — 1• • f— i — r 9| w r sI i 5 g M"Il*|.a . -
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ibl
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.-4-,
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. •
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30
40
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Fig. 10.11 On-line installation of particle size analyzer
568 Powder sampling and particle size determination
Determination of refractive index is difficult for light scattering determination. A new method of refractive index determination has been proposed and applied to several powders and its usefulness confirmed [195]. 10.5.4. Neural networking Empirical methods such as neural networking can be used in place of optical methods to estimate the size distribution of concentrated suspensions. The method determines particle size distribution and suspension concentration based on measured spectra on known size distributions. The feasibility of the inversion of laser diffraction data for size and shape distribution has been investigated by computer simulation. Size and shape density distributions can be represented by only four parameters making this technique an order of magnitude less accurate than conventional inversion techniques [196]. Three particle systems were examined by image analysis and their aspect ratio and particle size distributions were measured [197,198]. The data were then used as a reference method for neural networking using a Malvern Mastersizer X and concentrations from 2 g 1'^ to 60 g 1"^ Particle shapes ranged from an ellipsoidal cracking catalyst, needle shaped asbestos and monoclinic sucrose crystals. Neural networking has also been applied to particle sizing of slurries by reflectance spectroscopy [199]. The method is based on measured optical reflectance spectra measurement on known size and concentration slurries. 10.6 Malvern (Insitec) Ensemble Particle Concentration Size (EPCS) Systems Insitec now forms part of Malvern Instruments but is still based in California for process and laboratory R & D . The EPCS are laser-based instruments for in-line particle measurements that provide information on particle volume concentration and size distribution. EPCS instruments are part of the larger group of electro-optical instruments (MALLS) whose principle of operation is based on light scattering from a group (or ensemble) of particles. Unlike other instruments operating on this principle, the EPCS can perform direct measurements of particle laden flow stream provided the concentration is within operating limits.
Field scanning methods
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Malvern (Insitec) EPCS-P instrument is designed for in-line measurements in powder or spray processing systems under hostile conditions [200,201]. Applications include process powder sizing, mass emission monitoring and fossil energy combustors. It has a general capability from 1 to 500 |Lim at concentrations up to 1000 g m"^. EPCS-P has a gas-purged window with a 9 cm by 4 cm aerosol access region. The 5 mW He-Ne laser beam is approximately 1 cm in diameter. All particles in the beam, over the 9 cm length, scatter light into a logarithmetically scaled solid-state ring detector. Particle measurements are based on the analysis of light scattered into each of the 32 detector rings from all particles in the laser beam. Malvern (Insitec) EPCS-F is designed for powders in the size range of 0.2 |Lim to 1000 |Lim [202]. Particle measurements are made at rates up to 500 per second with immediate display of particle size distribution and characteristic diameters. Specific values or points on the particle size distribution are continuously fed back to the user or to a process control system. Particles with different refractive indices and aspect ratios up to 2:1 can be measured.
Flexible coupling
Puige inlet
EPCS-F optical head
Fig. 10.12 EPCS-F optical head installation. The instrument consists of an optical head with a purge gas over the lenses to reduce coating by the powder stream, an interface box, computer and
570 Powder sampling and particle size determination
software. As particles pass through the laser beam, light scattered in the forward direction is collected by the receiver lens and focused on to an annular ring detector. The detector is scanned at high speed and the signal level on each of 32 rings is measured and stored. Once a sufficient number of detector scans are acquired the software uses a non-linear inversion technique to solve for the relative particle concentration. The size distribution is determined from theory defined by the relative refractive index between the particles and carrier with no assumptions on the shape of the distribution. Figure 10.12 shows the optical head inserted in the process line. The head is installed directly into the line, preferably via a flexible coupling for vibration isolation. The interface box is a NEMA 4.9 rated explosion proof enclosure, weighing 50 lb, which can be bolted to a wall or floor within 20 ft of the head. 1 TRANSMISSION! 1
100% Val: 87.40% Sig: 17.9mA Avg: 84.39% Max: 92.07% Mini 62.64% Rms: 3.80%
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100% r Val: Sig: Avg: Max: Min: Rms:
o%| 1 1 Data
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Dv(50)
27.77% 5.48mA 46.26% 95.54% 25.13% 19.90% LJ<
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00:15:00 00:30:00 00:4iX) Test Tunc (00:26:46): Record #81
Fig. 10.13 EPCS process control display format. The process control display is divided into three sections. Six process control variable displays are shown at the top of the screen. Variable displays reflect the most recent measurements of the particulate. A time
Field scanning methods
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history of the point values for the process control variables is displayed on the lower portion of the screen. The third portion is the system status flags, shown in the middle of the screen. The particle size distribution may also be viewed whilst in the process control mode but this can slow down the processing time considerably. Figure 10.13 depicts the screen. The six variable display boxes are positioned at the top of the screen and reflect the current values only. The process control plot shows a time history of all six variables. The window bar is on the bottom of the time axis and shows the relationship of the current window to the total planned duration of the test. System status flags on the left light up when the control and error flags are on. The error status shows the error number if it is active. The other flags are for display only and indicate the status of the program. Discrete data point, extracted from the log file, can be viewed. The data can also be viewed in tabular form and as a size distribution curve. Data can also be integrated over any selected range. A Statistical Process Control (SPC) option enables the file data to be viewed in standard control chart format either as an X or R chart. Malvern {Insitec) ECPS2 is designed to monitor and control particle size distributions from 0.5 to 1,500 |im, at concentrations up to 10,000 ppm, directly in pneumatic powder flow streams. Up to one thousand size distribution measurements per second are carried out at flow velocities from static to ultrasonic. Discrete data point, extracted from the log file, can be viewed. The data can also be viewed in tabular form and as a size distribution curve. Data can also be integrated over any selected range. A Statistical Process Control (SPC) option enables the file data to be viewed in standard control chart format either as an X or R chart. Various interface arrangements have been described, [203]: E (external) the transmitter and receiver units are external to the flow (sprays, pipe flows with windows, free jets etc.) for systems less than a meter in size and for high pressure/temperature environments. P (probe) for large-scale systems (e.g. combustion furnaces etc.). The probe is gas purged and can be used at high temperatures. F-(flange) is intermediate to the probe and external systems being used for production streams in the 4 to 16 inch diameter range. E-(eductor) is a slipstream or bypass system which uses a gas driven aspirator to withdraw a portion of the flow stream isokinetically, pass it through the optical system and re-inject it back into the flow stream. The system is intended for high temperature/concentration applications and multiple flow streams.
572 Powder sampling and particle size determination
A comparison between the Insitec ensemble sizer (field scanning system) and the Insitec single particle counter showed good agreement with gas atomized zinc powder [204]. The ensemble sizer has been used successfully to characterize the atomization and dispersion of droplets and solids dispersed in a pneumatic transport device [205]. The instrument has also been used for on-line control of grinding circuits [206] and in side lines in cement plants to measure particle size [207]. The Malvern instrument has been used for on-line monitoring for process optimization with special regard to pneumatic and mechanical conveying. Control of particle size in the range 0.5 to 1000 |Lim was found feasible. The application of on-line monitoring led to fast system adaptation with associated risk and cost reduction [208]. 10.7 Optical incoherent space frequency analysis The method for obtaining particle size distributions using optical incoherent space frequency analysis is detailed in [209,210] and resulted in the development of a low price, robust, on-line dry powder measuring system, the Jenoptik PSLZ, (Particle Sizing using Incoherent light) which covered 32 size intervals in the range 1.5 |Lim to 2 mm. This was later expanded to 0.7 to 2500 |Lim [211]. For particles smaller than 50 |um a dispersion nozzle had to be fitted. The basis of the method is to replace the coherent light source used in MALLS with an infinite number of point light sources that emit light in an incoherent way with respect to each other. The intensity of radiation resulting from this two dimensional radiator is measured using a point detector located on the optical axis. This is equal to the surface integral of the Fourier power spectrum and can be measured with the aid of wedgering detectors in a coherent optical set-up. In this way, the incoherent arrangement consists of several switchable two-dimensional light emitters together with a single detector on the optical axis. This allows the measurement of the same intensity characteristics as in a coherent arrangement. Figure 10.14 shows the principle set-up. The incoherent light source consists of a 25 W halogen lamp L with voltage stabilization, including condenser system Ol, diffusing screen S and filter F. The variable geometry, binary two-dimensional emitter consists of an addressable liquid crystal modulator (LCD) with back lighting. The structural organization of the transparent electrodes within the LC modulator can create ring segments with transmission controlled by the help of the corresponding voltage. The response time of the individual LC segments is less than 3 ms
Field scanning methods
5 73
when the electrically controlled birefringence of liquid crystals is utilized [212]. A photomultiplier, with a micro pinhole in front of it, serves as a point light detector with diameter variable up to 300 \xm with a resolution of 1.2 jam. The LCD modulator is projected on to the detector by transformer objective 02. A piezoelectic x-y stage adjusts the central segment of the LC modulator precisely on the micro-pinhole. The material is fed through the optical path with sample feeding device P.
L
01 S
„ P
LCD — 02
V
Ph
A
o-
PMT
x-y piezo stage
T
PC
Fig. 10.14 Experimental set-up of PSI-Z. A personal computer (PC) controls the measuring arrangement. The evaluation algorithm in the PC calculates the ring intensity values of the percentage of certain particle size classes by comparison between measured values and predetermined theoretical values. The intensity values are converted into a size distribution by an iteration process. The apparatus and software are described in a paper on the application of this technique to the particle size measurement of bulk material [213] and control of a grinding process [214]. 10.7.1 Retsch Crystalsizer This is a later development of the Jenoptic system described above. A high resolution liquid crystal display system makes it possible to measure freeflowing dispersible materials in the size range 0.7 to 2500 |Lim. Its
574 Powder sampling and particle size determination
measuring principle is based on the diffraction of incoherent white light by a stream of particles. The light source used is a halogen lamp whose light is modulated by a liquid crystal display (LCD). The LCD produces ringshaped light beams with various diameters. Each diameter only results in a signal when particles of the correct matching size are found within the sample. These particles scatter the light at the right angle on to a detector. During measurement the individual rings of the LCD are switched through in rapid succession. Each ring is only active for a few milliseconds at a time. The cycle is repeated until the complete sample has been measured. The Crystalsizer is based on the principle of incoherent light diffraction thus turning the traditional optical structure back to front. Thus the same physical effect is measured as with traditional devices, but without using coherent laser light. The size distribution is determined using Fraunhofer diffraction theory. The robust construction of the instrument makes it suitable for use 'on-site'. Typically, samples are removed from the process and fed to the Crystallizer sequentially. 10.8 Pulse displacement technique (PDT) PDT is based on the detection of scattered refraction and reflection pulses that sweep past a detector at different times as a particle traverses a narrow laser sheet. In conjunction with Mie scattering and time-of-flight velocity measuring technique, detailed distributions of particle size from 2 \im to 6000 (am are provided together with particle velocities from 0.5 m s'^ to 150 m s"^ A miniaturized particle size velocimeter developed by Metrolaser is the first of its kind to utilize this technique. Particles larger than the focused laser beam are measured with PDT whiles smaller particles are measured by Imax. In principle the entire size range could be measured with a single optical configuration but in most applications small particles are present in large numbers while large particles are present in small numbers. This creates incompatible probe volume requirements and number density constraints. For this reason the instrument incorporates two optical configurations for large and small particles respectively. PDT is implemented using a single off-axis receiver that measures the intensity of the scattered light from a particle sequentially passing two laser sheets with wavelengths of 0.5145 |Lim and 0.4880 )Lim. The laser sheets have waists of 20 |Lim that is smaller than the diameter of the particles being measured by PDT. As a particle traverses either sheet, a set of double pulses results when the laser light is first refracted and then reflected by the particle. The two pairs of pulses are sorted according to wavelength and sent to one of two detectors. The temporal separation of
Field scanning methods
5 75
either of the two refraction or reflection pulses is proportional to the velocity of the particle. The temporal separation of the refraction and reflection pulses, each of the same wavelength, is proportional to particle size. The Imax technique measures diameters smaller than PDT. This technique uses a laser beam elongated in one dimension to form a sheet where the intensity along the central portion is constant. As the particles traverse the sheet a single pulse is produced the size of which is related to particle diameter by Mie theory. Hess and Wood presented detailed analytical results [215,216]. The instrument is small enough to fit in the palm of a hand and operates with negligible flow disturbance. It is rugged enough to stand harsh environments, high moisture levels, high noise and high vibration levels [217]. 10.9 Small angle x-ray scattering (SAXS) This effect depends upon the difference of electron density between particles and their surroundings, and the measured sizes are of primary particles rather than the external aggregate size. Thus samples are relatively easy to prepare and do not require pre-dispersion. The operating size range is from 1 to 300 nm and it breaks down at concentrations over about 3% due to inter-particle interference. A necessary requirement is near sphericity of the particles. The electron density also needs to be high. SAXS cannot distinguish between pores and particles and is therefore not suitable for porous particles. The theory is similar to that for MALLS, the forward scattered flux being related to the size and size distribution of the particles in an x-ray beam [218]. Three approaches have been described for calculating size distributions from SAXS data, the successive logarithmic graphical (SLG) [219,220], the dividing distribution function (DDF) [219] and Monte Carlo fitting. Monte Carlo fitting of SAXS of nanosphereswas tested with several mono and bimodal distributions. [221] The method has been used extensively on metallic and ceramic powders, colloidal suspensions and precipitates. Colloidal gold is measured conventionally by TEM which is a complex operation requiring considerable time and a skilled operator and cannot be used in-situ. Using SAXS the mean size can be estimated from the wavelength of peak absorbance using Mie theory; particle size distribution can be estimated by fitting experimental and theoretical spectrum for particles larger than 60 nm; and, for narrow size ranges, the mean size can be determined. [222].
576 Powder sampling and particle size determination
10.10 Near infra-red spectroscopy (NIR) NIR has been used to determine the mass median diameter of a micronized active compound contained in a lactose monohydrate at a concentration of 4% by weight and a size between 8 and 20 |im [223]. Multivariate statistical analysis was applied to zero order NIR spectra using particle size distributions by low angle laser light scattering as a reference technique. Due to its speed, simplicity and low operating costs it was demonstrated that this is a viable alternative to other methods used to carry out this type of analysis. Merck & Co. [224] have a patent on a visible and IR spectroscopy method to monitor particle size distribution of on-line monitoring of optically dense samples. 10.11 Ultrasonic attenuation 10.1 LI Introduction In-line measurements of particle size distributions are essential in order to maximize production capacity and product quality. Ultrasonic attenuation is emerging as a technique, with capabilities beyond those of light scattering. In addition to the needs of industry for compact, robust instrumentation, this method is capable of operating at high concentrations, thus eliminating the need for an expensive dilution step, which may alter the very properties one wishes to measure [225,226]. Originally developed as the Denver Autometric's PSM single point device; the scope of the technique was greatly enhanced by the use of a range of frequencies to generate a series of relationships between particle size, mass concentration and wavelength. These can be solved by nonlinear mathematical programming techniques to generate the full particle size distribution. The program is linked to the PSM instrument via a computer based signal processor for on-line data analysis and graphics display and is marketed by Proassist as the SD-21. 10.11.2 Theoretical basis for ultrasonic instruments A mathematical model, Allegra-Hawley [227,228], predicts the attenuation of ultrasonic waves as a function of frequency for each particle size distribution and concentration. Some mechanical, thermodynamic and transport properties of both phases are needed. The relationship between the size, concentration and frequency is obtained from the solution of the
Field scanning methods
577
ultrasonic wave propagation equations synthesized in matrix form [229]. The theory assumes spherical particles and single scattering conditions. At slightly higher concentrations (5% to 10% by volume) a multiple scattering correction has to be made [230]. In the case of even higher concentrations steric effects dominate and the scattering model fails. Coupled phase theories agree with experimental results at concentrations up to 25% by volume. An alternative approach has been to combine existing models with empirical correction factors [231]. Heat losses occur at the surface of each particle when a suspension is radiated with ultrasonics due to velocity differences between the viscous liquid and suspended particles and this results [232] in absorption of energy. A viscous term predominates when the ratio of wavelength to particle size is so high that particles do not follow the liquid movement. The viscous losses term can be derived from Stokes' equation for the effect of viscosity on a spherical pendulum swinging in a viscous liquid. A scattering losses mechanism is an apparent absorption of energy due to a redistribution of energy. Energy losses occur because of interference between radiated and scattered waves or simply because a scattered beam goes outside the wave path of the main beam and is not picked up by the detector. Essentially, in the diffraction region the wavelength is so small compared to particle diameter that sound behaves in the same way as light. Each particle casts a shadow so that the attenuation is proportional to the square of the particle diameter. 10.11.3 Development of commercial instruments Proassist: Herbst and Alba [233], in developing the Proassist, measured the attenuation of the sound at ten different frequencies from 0.5 MHz to 6 MHz to retrofit five points on the size distribution instead of only one as with the original Autometrics instrument. They discretized the resulting equation for narrow size fractions and used a mathematical programming technique to find the set of fractions and concentrations which gave the best fit. Ultraspec: Alba reported on an extension of this technology and termed it the Ultraspec particle size analyzer [234]. The prototype laboratory instrument covered the size range 0.01 |im to 1,000 |Lim at a volume concentration range from 0.1 to 70 employing a continuous frequency band of 1 MHz to 2000 MHz. In conjunction with the DuPont company, the technique was extended to the measure of undiluted emulsions and an online system was developed.
578 Powder sampling and particle size determination
Further development was carried out by Scott et.al. [235] who state that on-line measurement of PSD on concentrated slurries of sub-|Lim particles is a difficult challenge. First in the design of the flow cell, since the slurries heavily attenuate the ultrasonic beam, a typical design will use a small-bore flow cell that frequently plugs. Secondly is that surprisingly little information is contained in the attenuation spectrum so that it is difficult to estimate a meaningful PSD from the spectrum. The authors addressed these issues with the development of a new ultrasonic instrument that differed significantly from their earlier instrument based on Alba's design. This instrument uses a flow cell designed for reliable operation with concentrated slurries. A short wide-band pulse, that covers a wide range of frequencies, is sent to the transmitting transducer thereby launching an ultrasonic wave through the slurry. The attenuated wave is detected by a receiving transducer, amplified, digitized and Fourier transformed into its frequency components. Once the attenuation spectra is measured the PSD is estimated by a curve fitting procedure based on the Allegra-Hawley [236] model of ultrasonic attenuation. The instrument has been applied to on-line comminution studies on pigments and to droplet size studies in emulsion of chlorobenzene in water. Malvern Ultrasizer SV: After the original research work Alba worked with Malvern Instruments and this led to the development of the Malvern Ultrasizer [237]. The analyzer covers the size range 0.01 |Lim to 1,000 |im, at a volume concentration range of 0.5% to 50% for powders and up to 80% for emulsions, employing a frequency band of 1 MHz to 150 MHz. The Ultrasizer SV features a range of interchangeable measurement cells that can be exchanged in seconds. Fixed stirred batch cells are available with capacities from 450 ml. Flow cells are available that allow the instrument to be interfaced into particle streams and reactors. Pen Kem System 7000 Acoustopheretic Titrator: Pendse and co-workers used discrete frequencies in the development of the Acoustophoretic titrator. This instrument is based on the measurement of colloidal vibration potential arising from the motion of suspended solids relative to the suspending medium when subjected to a sound field, was the first commercial instrument capable of monitoring zeta potential of concentrated solids. Pen Kem Acoustophor^^ 8000 system (now Dispersion Technology): The system was extended for the measurement of sub-micron particle size distributions in concentrated slurries [238] which resulted in a prototype instrument, the Acoustophor^'^ Penn Kem 8000 system. This instrument consists of two flow-through cells for acoustic and electro-acoustic measurements respectively. With this system the acoustic attenuation is
Field scanning methods
5 79
measured at several discrete frequencies between 1 and 100 MHz with a variable transducer separation. This set-up makes a dynamic range up to 3,000 dB cm"^ attainable. At these frequencies viscous energy dissipation of the sound wave is the dominant phenomenon for sub-micron, rigid particles. Dispersion Technology DT-100 Acoustic Spectrometer measures sound attenuation in suspensions with volume concentrations from 0.1 to 50% to generate size distributions from 0.005 to 100 |Lim. No information about the sample is required. A variable path length between the source and detector eliminates any need for calibration. A pulse technique, which automatically adjusts the accumulated number of pulses to reach a target signal-to-noise ratio, provides the extremely large dynamic range needed to characterize samples over a wide size range of particle size and concentration. Dispersion Technology DT-HOO combines the DT-100 with the DT200 to give both size and zeta potential. Ultrasound induces a motion of particles relative to the liquid. This motion disturbs the double layer shifting a screening cloud of counter-ions. This displacement creates a dipole moment the sum of which creates an electric field that is measured by two electrode sensors. This field depends upon the value of the zeta potential which can be calculated using the appropriate theory. Sympatec Opus: The OPUS^^ (Figure 10.15) is based on ultrasonic attenuation in the regime where attenuation is proportional to the total projected area concentration of the particles and the attenuation is governed by the Lambert-Beer law. For this to be valid, the particles must be considerably larger than the wavelength of the incident radiation. The original Sympatec Opus; (Figure 10.16) system used 20 discrete frequencies in the 1.7 to 8 MHz range with typical measurement times of 2 to 5 min to cover the size range 5 |am to 3000 |Lim at volume concentrations from 1% to 40%. The measurement range has now been extended to 0.01 |Lim to 3000 |Lim at volume concentrations from 1 to 70%. The early flowthrough cuvette for on-line analysis had to be arranged in a bypass but a new sensor, called the OPUS Probe, is now available which can be installed in a pipe or vessel [239]. Riebel and Loffler [240] obtained an acoustic attenuation spectrum with one transducer pair to infer the particle size distribution. Solids concentrations and particle size distribution were obtained using both the Phillips-Twomey algorithm (PTA) and the relaxation method. The PTA gives a least squares solution by simple linear matrix operations to yield a numerical inversion from the attenuation spectra to the size distribution
580 Powder sampling and particle size determination
Computer control of measurements evaluation DC voltmeter
I
I IS. RF
I
Reference
generator
Signal processor
signal
I
t
Measurement cell
Power amplifier
Emitter
RF filter
Receiver
Fig. 10.15 Schematic representation of the experimental apparatus for ultrasonic spectroscopy.
Transmitter
Receiver
Fig. 10.16 Simplified presentation of the Sympatec Opus.
Field scanning methods
58 I
and concentration. This works well in the presence of systematic errors such as concentration fluctuations whereas errors arising from inaccurate extinction data often give negative values. Iterative solution algorithms use a priori knowledge to correct for this. However,Riebel and Loffler showed that the relaxation method, though slow, gave the most reliable results but that its use on-line requires a larger computing capacity. Narrow and broad size ranges of glass beads were analyzed. They found that concentration effects were less important than with laser diffraction with little deviation from linearity until the volume concentration exceeded 10%. If results are plotted in terms of the surface-volume diameter, even fibers give good agreement with microscopy. In later papers they extended the theory to cover multiple scattering effects [24 1,2421. They also investigated neural network recognition of particle size distribution by ultrasonic spectroscopy [243] for measuring high concentration suspensions. This work formed the basis for an on-line ultrasonic size analyzer (OPUS) which is available from Sympatec (Figure 10.15). The problem reduces to finding the size distribution and the solids concentration given measurements of the total attenuation. The problem is difficult because of inherent instabilities in the inversion of the transform. 10. I I . 3 Discussion
Data, using the Malvern Ultrasizer has been presented from two applications, chemical mechanical polishing slurries used in the semiconductor industry and monitoring a crystallization growth system [244]. The technique has been extended to measurement of undiluted emulsions [254] and the on-line system has also been tested [246]. The flow cell for the on-line system consists of two transducers, one stationary and one that can be moved to give different acoustic path lengths. Three distinct concentration regimes were found. For concentrations below 5% by volume the attenuation at each frequency, from 2 to 50 MHz, was found to be proportional to slurry concentration. In the intermediate regime, 5 to 10% by volume, the observed attenuation was higher than expected. A third regime, at greater concentrations, was found to have attenuation significantly lower. Although the attenuation spectra could be predicted using the A-H model, it was found necessary to assume a log- normal
582 Powder sampling and particle size determination
particle size distribution to determine the particle size from the attenuation data [247]. The originators of the Penn Kem system claim that it is capable of measuring particle size distributions in the size range 0.01 to 100 |Lim for slurry concentrations at volume concentrations as high as 50%. They report experimental work with an on-line system using titanium dioxide at volume concentrations from 3.5% to 42.3%. Quantitative comparison of data was carried out at eighteen frequencies and eleven concentrations by volume [248,249]. Theoretical work resulted in the development of a unified coupled phase model which successfully predicted the experimental data for suspensions, emulsions and aerosols [250]. To illustrate the capability of this instrument, two high-purity alpha aluminas having log-normal distributions were analyzed separately and as a 50:50 mixture. Results clearly demonstrated the ability of this technique to resolve the original component distributions from a mixture of the two powders [251]. A high-density system (silica spheres) was examined by image analysis using an SEM, by sedimentation, laser diffractometry, dynamic light scattering and ultrasonic attenuation using the AcoustoPhor. Image analysis and acoustic attenuation both gave narrow distributions whereas the other techniques gave wider distributions, the widest being with laser diffractometry (MALLS) [252]. A comparison between the Matec Acoustosizer and the AcoustoPhor with DuPont R900 titanium dioxide gave median sizes of 0.300 and 0.311 |Lim respectively. These compare well with typical DuPont values, obtained with the Brookhaven x-ray centrifuge, of 0.308 |Lim. Dukhin and Goetz [253] found a non-linear attenuation spectra for rutile titanium dioxide dispersions in the volume concentration range \% to 42% and developed a theory to account for this. Dukhin et al. [254] state that viscous losses are dominant in high density contrast dispersions e.g. rutile whereas thermal losses are dominant in low-density contrast dispersions e.g. latices. A report of two years experience of operating this instrument was reported by Hinze et.al [255]. A personal computer (PC) controls the measuring arrangement. The evaluation algorithm in the PC calculates the ring intensity values of the percentage of certain particle size classes by comparison between measured values and predetermined theoretical values. The intensity values are converted into a size distribution by an iteration process. Ultrasonic measurement has also been used to determine the particle size distribution in emulsions down to 20 nm in size. The attenuation was measured in the frequency range 100 kHz to 185 MHz with computer
Field scanning metho&
583
controlled small volume cylindrical resonators and computer assisted V H F and UHF pulse send-receive apparatus. Concentrated (25% w/v) aqueous emulsions of F-dimethyladamantane-trimethyl-bicyclononane, among others, were studied as well as perfluorochemical emulsions [256]. For a review of ultrasonic particle sizing readers are referred to a paper by Riebel and co-workers [257]. A comparison with x-ray sedimentation showed good agreement for nominal 1 pm silica [ 2 5 8 ] . In a comparison between the Acoustosizer and the AcoustoPhor it was found that both instruments were capable of zeta potential measurement. Good agreement was found between the techniques for particle size measurement and also between measurements made with diluted and concentrated suspensions. This was expected for monodisperse spherical particles but excellent agreement was also found with non-spherical particles and broader distributions [259]. The potential of ultrasonic spectroscopy for in .situ measurement of a crystallization process has also been examined 12601. Babick et.aZ. [26 1 ] compare sound spectroscopic determination of particle size distributions of sub-micron emulsions with dynamic light scattering and laser diffraction using olive oil in water droplet suspensions at volume concentrations from 1 YOto 60%. An experimental method has been developed to measure the intrinsic attenuation spectra for polystyrene latex using the Malvern Ultrasizer. [262]. Sipnrl From C a k d Amplifier
FSA Measurement Acoustic Delay Rod
584 Powder sampling and particle size determination
10.12 Matec Acoustosizer (ACS) Matec [263,264] in collaboration with the University of Sydney introduced the AcoustosizerTMESA-8000, in which the sound waves are generated by the particles themselves as they are exposed to a high voltage alternating electric field. When an alternating electric field is applied across a sample, the double boundary layer surrounding each particle is distorted, so that the particles are displaced relative to the liquid. This motion represents a pressure wave, which is detected by ultrasonic transducers. The amplitude of the wave depends on surface potential, the electrical double layer around the particles and particle size. The phenomenon is called the ESA effect, an acronym for electro-acoustic size analyzer, and yields the average particle size (0.1 to 10 pm), breadth of distribution and zeta potential at volume concentrations from 1% to 40%. ACS measurements provide a means of investigating moderately concentrated suspensions directly without changing the surface equilibria and hence the state of agglomeration. The measurement cell contains about 400 ml of suspension and two electrodes that are used to apply the electric field to the sample. Behind the electrodes there are two glass rods that serve to guide the sound waves to the transducers (Figure 10.17). When comparing results with other methods it is often necessary to dilute the suspensions without changing their state of agglomeration. Despite this, under electrokinetically stable conditions, excellent agreement was found in a comparison with the Mastersizer S, the Ultra Fine Particle Analyzer, the Sedigraph 5000D and scanning electron microscopy [265]. It was stressed however that sample preparation procedure was important. The original Acoustosizer used a single frequency whereas a later development has a range of 13 frequencies between 0.3 and 13 MHz. This allows the measurement of the dynamic mobility spectrum and the determination of the zeta potential and particle size. In order to invert the mobility spectrum into a size distribution a log-normal distribution of particle size is assumed. A comparison with photon correlation spectroscopy for determining particle size and laser Doppler anemometry for particle charge confirmed the results using ACS [266]. These and additional sedimentation measurements confirmed that changes in particle size and zeta potential due to dilution effects are likely to occur in aqueous and non-stabilized systems.
Field scanning methods
585
10.13 Ultrasonic attenuation and velocity spectrometry For liquids, the velocity of ultrasound depends on the compressibility and density of the liquid. For suspensions, the velocity depends also on the drag of particles in the liquid under the influence of the ultrasonic wave. At low frequencies, small particles tend to move in phase with the liquid and the ultrasonic velocity may differ widely from that in the pure liquid. As particle size and ultrasonic frequency increases, the particles tend to lag more and more behind the movement of the liquid and the ultrasonic velocity approaches that of the suspension acting as a uniform fluid. There is a transition frequency range between complete entrainment and no entrainment of the particles that can be used to obtain particle size information. The hydrodynamic model of Marker and Temple [267 ] can be used to calculate ultrasonic velocity. This model takes into account the effects of fluid viscosity, of concentration, density and elastic modulus of both particles and fluid and can predict ultrasonic velocities accurately for volume fractions between 5% and 20%. Ultrasonic velocity measurements in the 50 kHz to 50 MHz can be used to determine particle size distributions in the range of about 0.1 to 30 |Lim. In a comparison between ultrasonic velocity and attenuation measurements Harkins et. al [268] reported excellent agreement between predicted and measured values for the former and discrepancies for the latter. CSIRO Minerals has developed a particle size analyzer (UltraPS) based on ultrasonic attenuation and velocity spectrometry for particle size determination [269]. A gamma-ray transmission gauge corrects for variations in the density of the slurry. UltraPS is applicable to the measurement of particles in the size range 0.1 to 1000 |iim in highly concentrated slurries without dilution. The method involves making measurements of the transit time (and hence velocity) and amplitude (attenuation) of pulsed multiple frequency ultrasonic waves that have passed through a concentrated slurry. From the measured ultrasonic velocity and attenuation particle size can be inferred either by using mathematical inversion techniques to provide a full size distribution or by correlation of the data with particle size cut points determined by laboratory analyses to provide a calibration equation. A resonant technique has been applied to the measurement of phase velocity and attenuation of acoustic waves in water suspensions. Results were compared with theoretical predictions from three different approaches and with data from the Microtrac XI00. A tri-modal titanium dioxide (due to aggregation of 0.3 |Lim primary particles) and a mono-
586 Powder sampling and particle size determination
modal alumina were examined at a frequency of about 75 kHz and at a volume concentration of 10%.[270]. Aeration in a slurry cause large ultrasonic attenuations making accurate particle size measurement impossible. Aerated slurry can be fed to a holding tank and ultrasonic measurements made once the aeration level has dropped low enough to allow meaningful measurements. Clearance times were found to be short enough for a baffled tank passive de-aeration system to be feasible. The method was used, in combination with a gamma ray transmission technique, for particle size applications in alumina plants [271] and has been tested on a variety of mineral and paint slurries giving good agreement with laser diffraction measurements. For composite samples the method discriminated separate Ti02 and CaC03 components and accurately determined their proportions. In addition, in combination with ultrasonic attenuation measurements, the size fractions of iron ore slurries below 10 and 30 i^im were determined to within 1.3% and 1.0% respectively when compared with laser diffraction measurements [272]. According to Coghill et. al velocity measurements are complementary to attenuation methods but better suited to the finer size fractions. A description of the analyzer and the results of plant feasibility tests and on-line installation has been presented [273]. 10.14 Photon correlation spectroscopy (PCS) 10.14.1 Introduction Particle sizing of sub-micron powders can be performed on a routine basis using photon correlation spectroscopy. The success of the technique is based mainly on the facts that it provides estimates of average size in a few minutes, no severe sample preparation procedure is required and userfriendly commercial equipment is available. The main drawback is its low resolution. In order to partially overcome this problem, advanced data inversion procedures are required. Other limitations of the technique are; the need to use low concentrations in order to avoid multiple scattering, which results in too low an estimate of particle size; and the conflicting need for high concentrations in order that the number of particles in the measurement zone is sufficiently high for statistical significance. There are also reservations about its ability to separate accurately multimodal distributions and determine wide size distributions. Its strong point is the accuracy with which narrow size distributions may be determined on an
Field scanning methods
Lens
Light J stop
Low power laser «
Scattering angle Lens
Detector electronics
587
Collecting optics \ Detectorl
TJ Correlator
o
\ Printer |
[^Computation unit
Control keyboard
Display
Fig. 10.18 Block diagram of a fixed angle photon correlation spectrometer [284] absolute basis, i.e. without calibration, in only a few minutes. The technique is also referred to as quasi-elastic light scattering and dynamic light scattering [274]. 10.14.2 Principles The technique involves passing a collimated laser beam into a dilute suspension and measuring the radiation scattered at an angle 6 (usually 90°) with respect to the incident beam (Figure 10.18). Particles in a fluid are in constant motion as a result of collisions with molecules of the suspending medium. As the particles become smaller the movement becomes more rapid and gives rise to the phenomenon known as Brownian motion. The incident light is of wavelength X whilst the scattered light is of wavelength X + hX, where the frequency shift is an (optical) Doppler shift the magnitude of which depends upon the velocities of the particles and the angle of observation. The Doppler shifts are too small to be measured directly and are sensed from the interference of light scattered from pairs of particles and summed over the whole distribution. The velocity differences between the paired particles, ranging from a few microns to thousands of microns per second, generate 'beat' frequencies ranging from 1 to 10,000 Hz. The signal generated by the detector resembles a noise signal due to the constantly changing diffraction pattern caused by destructive and
588 Powder sampling and particle size determination
constructive interference as the particles change their position. Analysis of the intensity fluctuations yields a diffusion coefficient that is related to particle size. The basic technique is only applicable to dilute suspensions where multiple scattering does not occur and this technique is sometimes referred to as through dynamic light scattering. The introduction of the controlled reference method has extended it to more concentrated systems [275]. In the through sample technique the low frequency signal is deconvulated using the autocorrelation function, whereas in the controlled reference method the signal is transformed into a frequency spectrum and the particle size determined from iterative deconvolution of the spectrum. This greatly simplifies photon correlation for process control since the remote sampling, dilution and wash cycles are eliminated. The signal is fed to a correlator and the autocorrelation function of the scattered intensity is interpreted in terms of average particle size and polydispersity index. 10.14.3 Through dynamic light scattering The autocorrelation function of the scattered intensity G{t) is defined as the product of the light intensity at the detector at time t and at a short time later t + r. G(r)=
(10.33)
where t is effectively zero at the commencement of an analysis. The symbol <•••> refers to an average value of the product /(/) x / ( / + r) for a large number of times r. The normalized first order autocorrelation function G{T) can be calculated from the measured function: G{T) =A+BQxp{-2rT)
(10.34)
where A and B can be considered as instrument factors with B < A. The ratio B/A is often designated as the intercept, as a percentage merit or as a signal to noise ratio. The decay rate Fis linked to the translational diffusion coefficient D by: r=K^D
(10.35)
Field scanning methods
589
The modulus of the scattering vector, K, is defined as: i^^^sin^
A)
(10.36)
2
where n = refractive index of liquid medium, A,Q = wavelength of light in vacuum. Note, that with PCS the diffusion coefficient D is determined and not the particle size. The latter quantity can only be determined by relating the diffusion coefficient to the particle size. There is no general relationship that applies to all situations and the frequently used StokesEinstein expression only applies to non-interacting, spherical particles.
D^ = '^ = J ^
(10.37)
where DQ is the diffusion coefficient for a single particle in an infinite medium; T is the absolute temperature; k is Boltzmann constant; 77 is liquid viscosity; jc is particle size (x is used in this section to avoid confusion with diffusion coefficient) and/o is the friction coefficient for a single particle. 10.14.4 Particle size For homogeneous spherical particles, which are small compared to the wavelength of light, the average diffusion coefficient is the z average D^. However the diameter calculated from this (JC^) is not a 2 average but a harmonic z average i.e. an average intermediate between the volumemoment and the z average [276].
^P \^z J
(10.39)
590 Powder sampling and particle size determination
SO that:
^m-^r^
(10.40)
i
where n^ represents the number of particles of size x^. 10.14.5 Concentration effects The particles must scatter independently; otherwise the diffusion coefficient, and particle size, cannot be determined unambiguously from the decay rate of the autocorrelation function. The net effects of multiple scattering are that the instrument factor B/A decreases, and the autocorrelation factor decays faster, leading to too low an estimate for particle size. Thus, multiple scattering limits the application of the technique to low concentration dispersions (< 0.01% by volume), although techniques have been developed to overcome this condition. 10.14.6 Particle interaction Since most colloidal dispersions are stabilized by particle interactions, the use of equation (10.51) may lead to biased estimates of particle size that are often concentration dependent. The effect may be taken into account by expanding the diffusion coefficient to a concentration power series that, at low concentrations, gives: D,^D^{\-^kj,c)
(10.41)
The equation reduces to the Stokes-Einstein equation for spherical particles. Since the friction coefficient for a non-spherical particle always exceeds the friction coefficient for a spherical particle, over estimation of particle size will occur if equation (10.41) is applied. The virial coefficient kD is positive for repulsive particle interaction and negative for attractive interaction. Thus if particle interaction is neglected the apparent size will be concentration dependent, increasing with increasing concentration for attractive interactions and decreasing with repulsive interactions. In such cases, the diffusion coefficient should be determined at a range of concentrations and D^ determined by extrapolating to zero concentration.
Field scanning methods
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The effect of particle interaction is proportional to the average interparticle distance that, for a fixed volume concentration, decreases w^ith particle size. Hence, the effect of interaction reduces as particle size increases. However, small particles scatter much less light than large particles and it is necessary to use a higher concentration for reliable PCS measurements. In these cases the concentration needs to be increased to volume fractions up to 0.1% and, again, particle sizes can only be determined from extrapolations to zero concentration. 10.14.7 Particle size effects PCS relies on uneven bombardment of particles by liquid molecules that causes the particle to move about in a random manner and this limits the technique to particles smaller than 2 or 3 |im. In order to avoid bias due to number fluctuations, it is necessary that there is at least 1000 particles present in the measuring volume and, for a typical value of the scattering volume of 10"^ cm^, effects of number fluctuations are to be expected for particle diameters greater than around 0.5 |Lim. Number fluctuations lead to an additional time decaying term in the autocorrelation function. Since the characteristic decay time of this additional term is usually much slovs^er than the decay attributed to Brownian motion, the average particle size, which is proportional to the average decay time, will be overestimated if the effect of number fluctuations is neglected [277]. Loss of large particles due to sedimentation effects can usually be considered negligible. Stokes' law predicts that a 1 ^m particle of density 3000 kg m'^ sediments in water at a rate of about 1 |Lim s"^ so that, in 3 min, there will be no particles larger than 1 |am at a depth of 0.2 mm below the surface. Since the measuring volume is usually situated several mm below the surface, this effect is only important for unduly protracted measurement times. 10.14.8 Polydispersity For monodisperse samples, a plot of G(r) against r gives a straight line with a constant slope which is inversely proportional to particle size. For polydisperse samples, the relationship is multi-exponential and a plot of G{T) against racquires curvature, the degree of which increases with increasing polydispersity [278].
592 Powder sampling and particle size determination
The autocorrelation function for a polydisperse system represents the weighted sum of decaying exponential functions, each of which corresponds to a different particle diameter. For such a system:
G(r)-
JF(r)exp(-rr)dr
(10.42)
F{r) is the normalized distribution of decay constants of the scatterers in suspension. Given G{T) it is necessary to invert equation (10.42) in order to determine F{r), Unfortunately, the inversion is ill-posed in that there are an infinite number of distributions which satisfy this equation within the experimental error to be found in G{T). A large number of algorithms have been suggested for the inversion and an evaluation of their performance can be found in Stock and Ray [279]. The autocorrelation function can also be analyzed by the method of cumulants. In this approach G(r) is fitted to a low order polynomial. For a third order cumulants fit: r 1 A
G(r) = - r r + - 1 W^^\ 2!
^ \^ - U^T^ v3/y
(10.43)
An average particle size is obtained from the average decay rate 7" using equations (10.41-10.43) and an indication of spread (or polydispersity) is given by o^. An advantage of the cumulants approach is that it is computationally very fast. A chi-squared fitting error parameter serves to test whether the assumed Gaussian shape in diffusivities is reasonable. The calculated values of mean size and polydispersity are reasonable (chi-squared approaching unity) for approximately symmetrical distributions having a coefficient of variation within 25% of mean size. Commercially available instruments usually employ both approaches. For highly skewed distributions or distributions having more than one mode, an inversion algorithm must be used [280] whereas for narrowly classified mono-modal distributions the cumulants approach is satisfactory. The relative second moment, ^^2//^^ , a dimensionless quantity, is a measure of polydispersity. It is the intensity-weighted variance divided by the square of the intensity-weighted average of the diffusion coefficient distribution. The relative second moment is also called the polydispersity
Field scanning methods
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index that characterizes the spread of the decay rates and hence the spread of particle size about the average value. Most inversion methods (e.g. Contin [281] and maximum entropy method) [282], require prior knowledge of the distribution. The singular value analysis and reconstruction method (SVR) reduces the inversion problem to a well-conditioned problem, thus eliminating the need for prior knowledge [283]. Other methods of translating the polydispersity index into size distribution information have been proposed [284]; but the reliability of the transformations is in question. Finsey details these procedures in a review containing 67 references [285]. A later review contains 292 references [286]. 10.14.9 The controlled reference method In the controlled reference method laser light is guided into the sample cell by an optical waveguide. Particles within 50 \xxn of the tip of the wave guide (a fiber optic probe) scatter light, some of which is reflected back into the fiber and transmitted back through the guide. The reflected light from the interface between the guide tip and the suspension is also transmitted back. If these two components are coherent they will interfere with each other and result in a component of signal that has the difference or beat frequency between the reflected and scattered components. The difference frequencies are the same as the desired Doppler shifts. The received signal resembles random noise at the output of the silicon photodiode as a result of the mixing of the Doppler shifts from all the particles scattering the laser light. The photodiode output is digitized and the power spectrum of the signal is determined using fast Fourier transform techniques. The spectrum is then analyzed to determine the particle size distribution [287,288]. The iterative deconvolution of the frequency spectrum used with this technique supplies more detail about complex distributions than is possible with autocorrelation analysis. The controlled reference method has been shown to give a more constant measured size, over the concentration range 1 to 1,000 ppm, than the through dynamic method and has also been operated successfully with polystyrene at a concentration of 25% [289]. For a single particle size the power function takes the form of a Lorenzian function. The COQ term depends inversely on size so the power spectrum plots for different sizes show a shift to higher frequencies as the particle size decreases. In terms of the Brownian motion, smaller particles move more rapidly than large ones. An assembly of particles will have a
594 Powder sampling and particle size determination
power spectrum P{co) which is the sum of Lorenzian functions weighted by the volume concentration of each size (equation 10.44). An addition weighting occurs since the scattering efficiency S{a) is size dependent. The analysis routine must deconvolute the combined power spectrum to determine the volume distribution. The optical properties of the particles and the suspending medium together with the viscosity and its temperature coefficient must be known. Pco = S(a)
f^\
(10.44)
SnkT (Or,
3Z^Tja
rj = viscosity; A = wavelength in fluid; T = absolute temperature and a = particle radius. One advantage of this system over conventional PCS is that since the light is reflected back rather than transmitted through the suspension, higher concentrations can be monitored. Measurement of bimodality for mixtures of sizes ranging from less than 0.1 |Lim to sizes greater than 0.1 |im is difficult because of the rapid decrease in scattering efficiency as the size decreases. Broad distributions can be measured accurately. A laboratory made fiber optic dynamic light scattering instrument has been described together with a description of its use to study the kinetics of aging processes in emulsions [290]. The method is particularly useful for this purpose since it permits measurements in concentrated emulsions. Weiner et. al. [291] determined the particle size distribution as a function of concentration for a number of colloid suspensions. The results showed the advantage of using single mode fiber optics as a practical tool. A comparison made between this and a previous design was presented and several limiting features summarized. A recent review with 54 references covers basic physics, and experimental methods [292]. Applications of the technique to the determination of mean particle diameter, polydispersity and higher order moments are discussed. 10.14.10 Multi-angle measurements The resolution of PCS can be improved by the simultaneous analysis of data collected at more than one scattering angle, combined with the
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additional constraint of the angular dependence of particle scattering power [293]. The basic idea behind multi-angle measurements is that, although the particle scattering power for a monodisperse sample varies over several orders of magnitude as a function of scattering angle, the decay time of the intensity autocorrelation constant remains constant and is proportional to particle size. Consider a mixture of particles, some with a particle size comparable to the wavelength of light and others of a smaller size, e.g. 500 nm and 100 nm respectively. In this case the decay time at small angles will be characteristic of the larger particles, whereas the decay time at larger angles will be characteristic of the small particles. Both features of the size distribution can be recovered from the simultaneous analysis of the autocorrelation functions collected at several angles. Experimental multi-angle PCS results [294-297] indicate that this technique is an excellent tool to detect minor amounts of relatively large particles in the presence of small ones. On the other hand equivalent or better results can be obtained from multi-angle static light scattering data [298,299]. Multi-angle instruments are also available to generate the angular variation of scattered light intensity for derivation of molecular weight, radius of gyration, translational and rotational diffusion coefficients and other molecular properties [300]. The utility of combining Static Light Scattering (SLS), i.e. single angle, with Dynamic Light Scattering (DLS), i.e. multi-angle for obtaining more robust and reproducible particle size distributions, has been demonstrated. The results were excellent but the measurements were tedious and the method required high quality photometer equipment. Bryant [301] combined SLS measurements with only a few angles of DLS data, and also recording DLS data at many angles and iteratively reconstructing the SLS data during the data analysis. The results, by doing DLS at only ten scattering angles and perhaps as few as five, were as good as those obtained with simultaneous multi-angle DLS and SLS. The advantages of this approach are shorter measurement times and a lower quality photometer system. An instrument has been described which performs zeta potential measurements via electrophoretic light scattering (ELS) and particle size distribution using multi-angle dynamic light scattering. The device uses a single optical fiber/coUimetor and a high resolution stepping motor [302].
596 Powder sampling and particle size determination
Table 10.2 Commercial photon correlation spectroscopy equipment Amtec Brookhaven BI 90 Brookhaven BI-200SM Brookhaven BI-Foqels Brookhaven 90 plus Horiba LB-500 Beckman/Coulter N4 Plus LecotracLTU-150 LecotracLTU-251 Leco Trilaser Leco DLS Leeds & Northrup UPA Malvern System 4700 Malvern Autosizer Hi-C Malvern Autosizer Malvern Autosizer 4800 Malvern Zetasizer 3 Malvern Zetasizer nano zs Malvern HPPS Malvern CGS-3 Microtrac UPA Nicomp380 Nicomp Model 3SOMA Otsuka Photol DLS-700 (Munhall) Wyatt Technology Dawn
Multi-angle Fixed angle Multi-angle Fiber optics Dual angle Forward scatter Fiber optics: multi-angle Fixed angle Fixed angle Fixed angles Controlled reference method Fiber optics Multi-angle Fiber optics Fixed angle Multi-angle Multi-angle Multi-angle Back Scatter Multi-angle Fiber optics Fixed angle Multi-angle Fiber optics; multi-angle Multi-angle
10.14.11 Commercial equipment Commercial particle sizing equipment usually operates at a fixed angle of 90"^ (Table 10.2). Multi-angle instruments generate two moments of the size distribution that renders direct evaluation of size distribution possible, provided a suitable model (e.g. log-normal) can be selected [303]. Multiangle goniometers using different wavelengths increase flexibility. Amtec spectrophotometers are designed to measure angular dependent intensity and correlation function either separately or concurrently. The photon correlation option enables sizing to be carried out from 5 nm to 3 |Lim. Rotation is continuously variable between 10° and 160° with angular resolution of 1/60° in the manual model and 1/100° in the step motor version.
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Beckman-Coulter Model N4 Plus system operates in the 3 nm to 3 |Lim size range, determining average particle size, standard deviation and diffusion coefficient, typically in 1 min. For complete size distributions, the Size Distribution Processor analysis resolves the components of polydispersed sample and reports results in intensity % or weight % as a function of particle diameter or molecular weight. The intensity of scattered light depends on both particle size and angle of observation therefore multiple angles lead to greater sensitivity and increased size range. To take advantage of this the N4 Plus uses six fixed angular positions Brookhaven BI-90 is designed for routine sub-micron particle sizing in quality control. Operation of the instrument is fully automatic and a series of repeat measurements including data processing can be set up easily. Brookhaven ZetaPlus is an electrophoresis instrument with the capability of particle sizing by photon correlation spectroscopy. Brookhaven BI-200SM is a multi-angle instrument that yields more information on particles and molecules. Brookhaven 90 Plus sub-micron particle size instrument is a dual-angle instrument with measurements at 15° and 90° to generate size distributions in the 10 nm to 1 |Lim size range together with zeta potential determinations. Brookhaven BI-Foqels is designed for on-line process control using a fiber optic probe for remote sensing in colloidal dispersions at concentrations from 0.001% up to 40% by volume. Fiber optic sensing greatly simplifies the application of photon correlation spectroscopy to process control by eliminating the need for sample dilution and wash cycles. Since glass fibers are inherently rugged they may be used in hostile environments. In this instrument, visible light from a laser diode is transmitted into the sample via a monomode fiber and the scattered light is collected by a second monomode fiber at an angle of 153°. It is claimed that the troublesome homodyne signals that arise in single fiber optics design are eliminated by the use of two fibers [304,305]. The measurement range is 0.002 to 2 |Lim. Brookhaven BI-200SM goniometer system is a precision instrument designed for macromolecular studies and sub-micron particle sizing. It is based on a special turntable to measure angular intensity and photon correlation measurements. Horiba LB-500 operates in the size range from 0.003 to 6 mm, at concentrations up to 20%. The focal point of the lens is positioned on the inner wall of the cell concentrating maximum light in order to measure high concentration suspension. In addition a pinhole filter removes errant
598 Powder sampling and particle size determination
beams assuring that only light scattered from the focal point will be sent to the detectors. The LB-500 can be combined with the LA-910 and 920 to extend their size range. Lecotrac LTU-150 operates in the 0.003 to 6.5 jiim size range using the controlled reference method. Lecotrac LTU-251 Coupled to an external probe, this analyzer allows fast in-line measurement. The external probe is constructed of rugged, flexible steel and can be attached to a process line via a 3/8-inch NPT pipethread mounting hardware. The probe tip consists of a sapphire window embedded in a stainless steel sleeve using Teflon® seals. Leco Tri-laser uses 3 angular detectors to cover the size range 0.02 to 2000 |Lim. Leeds and Northrup Microtrac Ultrafine Particle Analyzer (UPA) uses the controlled reference method, using a sapphire tipped waveguide that collects back-scattered light within 100 \xm of probe tip, to cover the size range 0.003 to 6.5 )Lim. Leeds and Northrup Microtrac Series 9200 Ultrafine Particle Analyzer Model 9230 operates in the 0.003 jiim to 6 |Lim size range (Figure 10.19) and gives reproducible results in the concentration range 2 to 2000 ppm (2%). Malvern Autosizer Hi-C. Particles in suspension scatter light emitted from the tip of the launching fiber (Figure 10.20). Scattered light backtracks to the fiber and is collected again and passes to the detector via the directional coupler. This prevents light from the source reaching the detector directly. Some light is reflected back from the tip of the fiber without being emitted. This acts as a reference signal that is mixed with the scattered portion. Since the particles in the sample are in random motion due to Brownian motion the light they scatter is frequency shifted due to the Doppler effect. This shift is revealed by comparison with the reference signal resulting in a low frequency spectrum that is related to particle size. The instrument covers the size range 0.015 to 1 |Lim at solids concentrations from 0.01% to 50%. One reported application [306] was the measurement of casein micelles in cheese making as they grew in size from 200 to 1200 nm. Malvern System 4700 comprises a variable angle spectrometer with computer controlled automatic operation, combining photon correlation spectroscopy and angular intensity measurements with full Mie theory calculations to give accurate size distributions in the 1 nm to 5 |Lim size range.
Field scanning methods 599
Surface guided wave K-splitter Fiber optic connection Laser diode silicon pliotoetector
Fig. 10.19 Line diagram of the Microtrac Ultrafine Particle Analyzer (UPA). Solid state laser
Directional coupling
• Light • backscattered by particle Light reflected from end of fiber Index matched termination
Fig. 10.20 The optical unit for the Malvern Autosizer Hi-C. Malvern Auto-sizer 4800 is a sensitive, high performance multi-angle light scattering spectrometer designed for both static and dynamic light scattering measurements operating in the size range 1 nm to 5 ^im. Malvern Zetasizer II consists of a light scattering spectrometer and digital autocorrelator with integral microcomputer. In addition to measuring electrophoretic mobility the movement of charged colloidal particles under the influence of an applied electric field the Zetasizer II also determines particle size by Brownian motion. The Zetasizer III combines both photon correlation spectroscopy and angular intensity measurements with full Mie theory calculations to give accurate size distributions in the 2 nm to 3 |Lim size range. Malvern Zetasizer nano zs uses non-invasive back-scatter technology to measure particles in the 0.6 nm to 6 |Lim size range together with molecular weight and zeta potential measurements.
600 Powder sampling and particle size determination
Malvern High Performance Particle Sizer (HPPS) combines high sensitivity with a high concentration capability using dynamic light scattering with back-scatter optics. The HPPS covers the size range 0.6 nm to 6 ^im at concentrations from 0.00001 vol% to 20 vol% at speeds up to 20°/s. Malvern Compact Goniometer System (CGS-3) is a multi-angle light scattering spectrometer. The detector scans a sample cell from an angle of 12° to 152° with an angular resolution of 0.025°. Nicomp380/DLS Submicron particle sizer uses dynamic light scattering with a size range of 0.003 to 5 |Lim with unique modular options. An Autodilution unit that eliminates the need for manual dilution of concentrated samples [307]; an Auto-sampler that permits automatic batch analysis of as many as 76 samples; high-power laser diodes, multi-angle options and zeta-potential accessory. Nicomp Model 780/SPOS provides two analysis modes, Gaussian and Nicomp's proprietary algorithm. This algorithm enables the instrument to analyze complex multi-modal distributions with high resolution and reproducibility. Its capabilities can be enhanced by adding one or more accessory modules including a zeta potential analyzer with multi-angle particle sizing capability Nicomp Model 380/MA is a multi-angle option that uses a precision stepper motor to vary the scattering angle (0.9 degrees per step). This capability permits an improved analysis of polydisperse distributions of larger particles (>0.1 |Lim). Nicomp Model 380/L is designed for on-line monitoring either by itself or in combination with the Particle Sizing System AccuSizer. These instruments are available from Particle Sizing Systems. Otsuka Photal is a dynamic light scattering spectrophotometer that provides sub-micron sizing in the 3 nm to 3 |Lim size range and also provides information on the shape of polymers. Wyatt QELS is a compact add-on instrument that can be connected to a Dawn® Eos or a miniDawn Tristar in order to determine particle sizes and their distributions. The combined system gives the ability to size very small particles (under 10 nm), as well as very large particles on-line after they have been separated by some kind of fractionation. The Wyatt QELS comes with the ability to measure the correlation function at 18 angular positions. In the microliter batch mode, the system collects data from small samples that have been injected into special blackened cuvettes. Unfractionated samples produce complex correlation functions whose
Field scanning methods
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departure from exponential decay arises from the presence of heterodisperse components in the suspension. In the flow mode, the instruments may be used to collect, simultaneously, the dynamic and classical light scattering data from which the molar mass and root mean square radii are calculated from each slice (See section on fractionation). For the mini-Dawn Tristar, dynamic light scattering signals are only collected from the 900 location, whereas for the Dawn Eros dynamic light scattering signals can be collected at any one of its 18 detector locations. 10.14.12 Discussion The basic theory and discussion of results are covered in papers by Thomas [308]. who uses a Brookhaven Instrument Fiber Optics QuasiElastic Light Scattering System (BI-FOQELS) with dynamic light scattering obtained using the BI-DLS and diluted samples. An autodilution unit has been described to analyze on-line particle growth during a polymerization process [309]. The results compared favorably with off-line dynamic light scattering and on-line turbidimetric data [310]. Several data analysis software packages are available and average sizes generated by these are not comparable [311-313] De Jaeger et.al. [314] carried out inter-laboratory tests using polystyrene lattices with particle sizes ranging from 30 nm to 2 \xm. They concluded that reliable particle sizes could be obtained for diameters less than 0.5 |Lim. In the range 0.5 to 1 |um this was only possible within a very narrow range of concentration. For the largest size investigated (about 2 jLim) the measurements were less reliable. In comparison tests, using standard latex particles, it was found that DLS gave data that is approximately 0.005 |im larger (0.064 |Lim) than electron microscopy and light turbidity methods, i.e. 0.059 ± 0.002 |Lim. In a comparison between the Coulter N4 and the Malvern Autosizer it was found that both instruments gave good results with monodisperse latices [315. For bimodal distributions, the two modes were not always detected and, if they were, the locations of the modes were incorrect [316]. Koehler and Provder [317] sized monodisperse PMMA latexes with a range of instruments: Disc centrifugal sedimentation (DCP), sedimentation field flow fractionation (SFFF), hydrodynamic chromatography (HDC), photon correlation spectroscopy (PCS), turbidimetry and transmission electron microscopy (TEM). TEM gave the smallest sizes, DCP and SFFF were in fair agreement in the center and PCS the highest sizes.
602 Powder sampling and particle size determination
In an evaluation of a range of instruments Lange [318] stated that turbidimetry appeared the most reliable approach to average size determination, whereas ultracentrifugation, DCP and TEM with image analysis were superior for determining size distributions and polydispersity. Lee et. al. [319] compared FFF, PCS and TEM for sizing acrylic latexes. They stated that FFF is a useful tool for accurately determining mean size and size distribution due to its simplicity and ease of operation. PCS provided reasonable data for small diameters and narrow distributions but suggested that multi-angle analysis was needed for larger particles and broader distributions. Multi-angle PCS was also preferred by Bryant et.al [320,321] for measuring multi-modal PSD's, and they showed that accuracy increased with increasing number of angles. Four instruments were evaluated using eight polystyrene latexes, ranging in size from 0.039 to 0.804 |Lim, from Duke Standards [322]. The instruments were: Matec CHDC-2000, Brookhaven BI-DCP, Coulter N4Plus (PCS) and Hitachi H-7000 FA (TEM). TEM gave average sizes close to nominal values, CHDF gave values slightly larger than nominal except for the lowest size where it gave a value of 0.046 |Lim, PCS and DCP gave reasonable values. The measurement time was unduly protracted with the DCP for the finest fraction, the generated size being 0.70 jiim for a run time of 2 hours reducing to 0.056 |am when this was extended to 6 hours. It was found that both PCS and CHDC gave much broader distributions than DCP and TEM. For polydisperse (bi- and tri-modal) CHDF and DCP provided the most accurate distributions whereas PCS failed to capture the whole distribution. Photon correlation spectroscopy measurements for growth rate, together with a quartz crystal microbalance for mass deposition, have been integrated into a single platform to permit simultaneous in-situ real time measurement at times and temperatures representative to those found in aviation fuel systems [323]. Kroner et. a/.[324] compared DLS with static light scattering for determining soot particle size in a premixed flame. They concluded that static is preferable to dynamic since the latter procedure requires detailed information about the flame. 10.14.13 Spectral turbidity Beckman DU 7500 spectrophotometer has been used to determine the size distribution of "monodisperse" latex in the size range 200 to 800 nm, to
Field scanning methods
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Study agglomeration during crystallization and attrition of potassium sulphate. A polychromatic light beam from a uv source passes through a silica fiber to a sensor immersed in the suspension. After crossing the measurement zone the transmitted light passes through a fiber to a holographic grating that splits the light into a spectrum that falls on to a 256 photodiode array. Using Mie theory the spectral turbidity yields the particle size distribution of the powder in suspension [325]. 10.14.14 Diffusion wave spectroscopy (DWS) DWS addresses dynamic light scattering in the multiple scattering concentration range. Pine et. al. [326] describe the theory for the technique and it has been applied to the determination of mean size and polydispersity [327,328]. The method has also been used for on-line measurement of concentrated suspensions [329]. 10.14.15 Photon migration In photon migration, an intensity-modulated light beam is launched into the sample and the photons diffuse through the sample and are detected [330]. The transmitted signal is attenuated and phase shifted relative to the incident beam. Measurement of the phase shift and attenuation as a function of modulation frequency yields scattering coefficient and particle size. Mie scattering theory is applied to generate particle size. The theory can be extended to higher concentration regimes. The particle size distribution is determined by determining the scattering coefficient at one wavelength over a range of known particle concentrations. 10.15 Turbo-Power Model TPO-400 in-line grain size analyzer Nisshin developed this instrument for the cement industry. At preset times it automatically samples a few kilograms of material and feeds it into a turbo classifier. The fines are fed into a micron line that determines the Blaine number for the powder. 10.16 Concentration monitors Monitek instruments measure the concentration of suspended solids in a liquid by shining a light through the stream and detecting the amount of light that is scattered by the suspended solids. The scattered light is seen as turbidity hence the name turbidimeters. Suspended solids scatter light
604 Powder sampling and particle size determination
in all directions so that there are many potential viewing angles; 90° sidescatter instruments are called nephelometers. Forward scatter instruments sample a more representative cross-section of a process stream and can monitor as wide range of particle sizes from 0.01 to 100 \\m. For accurate measurement Monitek uses a ratiod forward scatter intensity where the scattered intensity is divided by the direct beam signal. 10.17 Shape discrimination The morphology of particles is an important characteristic that can seriously affect powder handling and end-use properties. Off-line techniques are often not suitable for monitoring industrial processes since extracting the particles from the process can alter their shape, e.g. particles taken from a crystallizer may fracture or aggregate. To resolve the problem an in-line camera, capable of imaging crystals produced in a commercial crystallizer, was developed [331]. The instrument is based on a borescope and video camera that fits inside the housing of a laser backscatter probe, which installs in a standard ball valve. A strobe light is used to "freeze" crystal motion. Crystal features seen directly include shape, surface roughness, inclusions and transparency. The information content of the uv/vis spectrum of sub-micron and micron size particles yields information on the size, chemical composition and shape of the particles [332]. The angular dependence of the scattered intensity is given by the Rayliegh-Gans-Debye (RGD) theory. The form factors for various particle shapes were calculated as a function of the angle of observation ^and wavelength X of the incident light. Comparison of the scattering intensities for particles with different shapes showed that each differently shaped particle had a unique surface pattern thus suggesting the possibility of selecting combinations of X and ^to enable shape discrimination. 10.18 Miscellaneous 10.18.1 Back-scatter intensity A relationship has been derived between average particle size and concentration and the back-scatter intensity of a light beam entering a slurry of non-transparent particles [333]. Concentrations used were 0.3 to 2.5% and the average sizes were 40 to 1200 |im. The light source and detector were built into one tube that was mounted on the wall of a 1 liter
Field scanning methods 605
vessel that contained an agitated suspension of the particles under test. The derived relationship took the form: ib, -h.={hm-h.)exp(kcPd'^)
(10.45)
The suffixes refer to the back-scatter intensity from; b^ = total, b^ = container walls, b^ = maximum possible. For aluminum silicate, experiment yielded/? = 0.77, q = 1.67, k = 0.20. In a second paper [334] the technique was extended to on-line. A similar technique to the above, but using ultrasonics, has also been described [332,336]. 10.18.2 Spectroscopy photo-acoustic (PAS) and photo-thermal (PTS) The surface of a specimen may become heated through irradiation, the degree of heating depending on the material's absorption coefficient at the particular wavelength of radiation. A wavelength scan across a suitable part of the electromagnetic spectrum thus causes temperature changes that reflect the adsorption spectra at the point of illumination. In PAS the heating serves to increase the pressure inside a small chamber in which the sample is situated, being irradiated through a window. A recording of the pressure variations versus wavelength of illumination reflects the absorption spectrum of the material. In PTS one records the increase in thermal emission from the sample induced by irradiation. PAS techniques require samples to be placed in a spectrophone for analysis. PTS methods allow constant free on-stream inspection at a distance. Particle size may be inferred from the signal level. Since specific surface increases with decreasing size the signals also decrease. A discussion of these techniques has been presented by Kanstad and Nordal [337]. 10.18.3 Transient electric birefringence A dilute suspension of electrically and optically anisotropic colloidal particles becomes birefringent when subjected to an electric field. In random orientation the suspension is optically isotropic but when the grains align with a uniform electric field the suspension becomes anisotropic; in particular the effective refractive index of the ordered suspension parallel to the field direction differs from its refractive index
606 Powder sampling and particle size determination
perpendicular to the field. This double refraction, or birefringence, has been used for evaluating the size distribution of sols in solution. Experimentally, a colloidal system, in random orientation, is illuminated with polarized light. The system is subjected to an electric field that aligns the particles due to the interaction between the field and any permanent dipole or electrical polarizability of the particles. The birefringence grows as the particles align; when the field is removed the birefringence decays as the particles revert to random orientation. For a monodisperse suspension the decay rate can be described by a first order rate equation. For a polydisperse suspension the decay rate is a sum of exponentials. Measurement of the decay rate permits computation of particle size [338]. Haseler and Hinds used this procedure to determine the size distribution of anisometric silver halide crystals using an instrument that they developed called an electric field birefringece. They found that the technique was capable of good accuracy and high precision [339]. 10.18.4 Crossed lasers A description of a crossed laser beam technique for particle sizing and its application to shock tube experiments was presented by Waterson and Chou [340]. 10.18.5 Frequency domain photon migration A method for on-line monitoring of particle size distribution and volume fraction in real time using frequency domain photon migration measurements (FDPM) has been described. In FDPM the time dependence of the propagation of multiply scattered light provides measurement of particle size distribution and volume fraction. The technique has been applied to a polystyrene latex and a titanium dioxide slurry at volume concentrations in the range 0.3 to 1% [341]. FDPM consists of launching sinusoidally modulated light into a scattering medium at a single point source and detecting the modulated light at another point some distance away from the source. The photon density wave is attenuated and phase-shifted relative to the incident signal as it propagates through the scattering medium due to the scattering and adsorption properties of the sample. These properties can be measured by fitting the phase shift and modulation to the appropriate solution of the optical diffusion approximation, which describes the transport of light in random media. Particle size distribution and volume concentration can be
Field scanning methods
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determined from multi-angle measurements of FDOM and determination of the isotropic scattering coefficient. The technique is fast and the equipment relatively inexpensive. Moreover, since photon migration measures the time it takes for light to travel through the sample rather than the intensity of the detected signal, it is self-calibrating. 10.18.6 Laser induced incandescence (Lll) Aerosol particles are heated close to their evaporation temperature by a high energy laser pulse and start to emit thermal radiation. As the particles cool, mainly by heat transfer to the carrier gas, their thermal emission intensity decreases with time. Due to their higher heat capacity large particle cool more slowly than small ones and the cooling time can be used as a measure of particle size. Early applications mainly concentrated on average parameters for soot particles in flames [342] and this was extended to determination of a detailed characterization of the size distribution and internal structure of aerosols at room temperature [343]. The results showed good agreement with TEM and DMA data. Particle sizes were in the nanometer range at number concentrations down to 103 cm'^. Other accessible parameters are the diameters of agglomerates, the volume fraction and the mean number of primary particles [344]. LII has been applied ti in-situ measurement of primary particle size on the manufacture of carbon blacks [345]. The system has also been applied to the measurement of nanosize particles on soot and vehicle exhausts. Measurements on carbon black particles gave data which correlated well with product properties. Tests with titanium dioxide and metal powders were encouraging. [346] 10.18.7 Spectral transmission and extinction Cemi used spectral transmission and extinction using UV, visible and near IR to measure slurry particle size distributions with undiluted continuous flow [347]. The method uses multiple linear detector array spectrometers. It also uses multiple sample cells of different optical depths optimized for a specific spectral range, multiple optical paths and multiple linear detector arrays.
608 Powder sampling and particle size determination
10.18.8 Turbiscan multiple light scattering measurements Turbiscan On-Line monitors and quantifies the effects of process variables in dispersed systems. It operates in the volume concentration range from 0 to 60% and the size range of 0.1 to 5,000 |am. The vertical scan macroscopic analyzer consists of a reading head that moves along a flat bottomed cylindrical cell to scan the entire sample length. The optical sensor consists of a pulsed near-infra-red light source and two synchronous detectors: The transmission detector monitors light transmitted through the suspension and the back-scattering detector receives the back-scattered light at 135°. The optical sensors acquire transmission and backscattering signals in from 0.1 to 10 seconds, every 40 |Lim along the sample tube for a maximum of 80 mm, and these signals are digitized and displayed by the software indicating real-time changes in transmission and backscattering intensities. These parameters are directly related to particle size and volume concentration. Turbiscan Lab measures concentrated suspensions at up to 95% by volume. The mean particle diameter can be calculated over the size range 0.05 to 1000 |im. Turbiscan Ma 2000 measures the destabilization of concentrated dispersions and determines the mechanisms driving it. The ma 2000 is used to improve formulations, document stability tests and shorten stability test time. Anisotropic particles have been measured using a variety of techniques and correlation between them was found to depend on particle morphology [348]. For cylindrical glass fibers photocentrifuge data gave good correlation with image analysis using the Turbiscan, whereas for platelets laser diffraction gave the best correlation. The correlation between Turbiscan for flake-like mica was found to be very good whereas the photocentrifuge gave better agreement with rod-like copper oxalate. In both cases some information from the image analysis data was required in order to make assumptions to simplify the deconvolution data of size and shape from the collected data. References 1 2 3 4
Hinde, A.L. (1973), IFAC Symp. Automatic Control in Mining, Mineral and Metal Processing, Sydney, Inst. J. Engrs., Australia, 45-47, 524, 529 Svarovsky, L. and Hadi, R.S. (1977), Proc. Particle Size Analysis Conference, Analyt. Div. Chem. Soc, 525 Urick, R.J. (1948), J. Acoust. Soc. Am., 20, 283-289, 526 Babick, F., Hinze, F., Stinz, M. and Ripperge, S. (1998), Part. Part. Syst. Charact., 15(5), 230-236, 526
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5 6 I 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
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Appendix
Names and addresses of manufacturers and suppliers Aerometrics Inc., (Now part of TSI) 550 Del Rey Avenue, Sunnyvale, CA 94086, USA, (408) 738 6688 aerometrics.com Alpine 89, Augsburg 2, Postfach 629, Germany (Air jJet sieve see Gilson) American Innovation (Now Oncor) Amtec, Alpes Maritimes Technologic, ler C.A., Avenue du Docteur, Julien-Lefebvre, 06270 Villeneuve-Loubert, France.(93 73 40 20) Analytical Measuring Systems, London Road, Pampisford, Cambridge, Cambs. CB2 4EF, England (01223) 836001 Analysette (see Fritsch) Armco Autometrics, 7077 Winchester Circle, Boulder, Colorado, 80301, USA, (303) 530 1600 Artec Systems Corp. 170 Finn Court Farmingdale, NY 11735, USA, (516) 293 4420, [France: Nachet, 106 Rue Chaptal, 92304 Levallois Peret-Cedex, 757 31 05] ATM Corporation, 645 S. 94th PI., Milwaukee, WI 53214, USA, (414) 453 1100 Autometrics, 4946N 63"^^ Street, Boulder. CO 80301, USA Bahco (see Gilson) Bausch and Lomb Inc., 820 Linden Avenue, 30320 Rochester, NY 14624, USA Beckman Coulter, Particle Characterization Div., P.O. Box 169015, ra/c 195/10, Miami, FL 33116-9015, 800-523-3713, www beckmancoulter.com Boekeler Instruments Inc., 3280 East Hemisphere Loop, #114 Tucson AR 85706-5024, 602 573 7100 Brinkmann Instrument. Company, 1 Cantiague Road, PO Box 1019, Westbury, NY 11590207, USA, (516) 334 7506 Bristol Equipment Company, Yorkville, IL, 708 553 7161 Bristol Industrial and Research Associates Ltd (BIRAL), PO Box 2, Portishead, Bristol, BS20 9JB, UK Brookhaven Instrument Corporation, 750 Blue Point Road, Holtsville, NY 11742, USA, (516)758 3200 Buckbee Mears Co., 245 East 6th Street, St Paul,, MN 55101, USA, (612) 228 6400 Buehler, Optical Instruments Division, 41 Waukegan Road, Lake Bluff, IL, USA, 60044 (312)295 6500 Canty Vision, Dublin, Ireland, 43 0732 7701 77 Carl Zeiss, 7082 Oberkochen, Germany Cilas 8 Avenue Buffon, BP 6319, 45163 Orleans, France (+33) 2 38 64 59 55164, http://www.cilasus.com/analyzer Climet Instruments Co., 1320 W. Colton Ave., PO Box 1760, Redlands, CA 92373,USA, (714)793 2788 Collimated Holes Inc.,Campbell, CA 95008
624 Manufacturers
and suppliers
Compix Inc. Imaging Systems, 230 Executive Drive, #102-E, Mars, PA 16046, 412 772 5277 Contamination Control Systems GmbH, D-8999 Munchen 2, Germany, (089) 18 80 06/07 Coulter Inc., (now Beckman Coulter)Box 2145, 590 W.20th Street, Hialieah, FL 33010, USA, (305) 885 0131 CSIR, South African Council for Scientific and Industrial Research, PO Box 395, Pretoria, South Africa
Dantec Measurement Technology A/S 16-18 Tonsbakken, 2740 Skovlunde, Denmark. (045) 44 57 80 00, e-mail [email protected], Web site www.dantecmt.com Denver Equipment Company, 1855 Blake Street, Suite 201, Denver CO 80202, 1-800-3211135 Dipix Technologies Inc. 1051 Baxter Road, Ottawa, Ontario K2C 3P1, Canada Tel: 613596-4942 Fax: 613-596-4914 Toll Free: 800-724-5929 E-mail: Vision Products Division: [email protected] Dispersion Technology Inc., 3 Hillside Avenue, Mount Kisco, NY 10549, (914)241-4791, email [email protected]. Donaldson Company Inc., 1400 W. 94'*^ Street, Minneapolis, MI 55431, USA Dow Chemicals, Midland, MI, USA Duke Scientific Corp., 1135D San Antonio Road, PO Box 50005, Palo Alto, CA 94303, USA, (415) 424 1100 DuPont Company (E.I. DuPont de Nemours), Materials Characterization Systems, Concord Plaza, Quillen Building, Wilmington, DE 19898, USA, 302 772 5488 Endecottes Ltd, 9 Lombard Road, London, SW19 3BR, UK, (081) 542 8121, [US Distributor, CSC Scientific Company] Erdco Engineering Corp., 136 Official Road, Addison, IL 60101, USA, (708)328 0550 Faley International Corporation, PO Box 669, El Toro, CA 92630-0669, Flowvision, Kelvin Microwave Corp., Charlotte, NC, USA, (704) 357 9849 FFFractionation 4797 South West Ridge Blvd., Salt Lake City, UT 84118-8429, www.fffract.com Fritsch, Albert & Co., Industriestrasse 8, D6580 Idar-Oberstein 1, Germany, 06784/70 0 Galai Production, PO Box 221, Industrial Zone, Migdal, Haemek 10500, Israel, 972-654 3369 Gallenkamp Ltd, Portrack Lane, Stockton on Tees, Co. Durham, UK Gelman Hawksley, 12 Peter Road, Lancing, Sussex, UK Gilson Co. Inc., PO Box 677, Worthington, OH 43085-0677, USA, (614) 548 7298 Glen Creston, 16 Dalston Gardens, Stanmore, Middlesex HA7 IDA ,UK, 0181 206 0123 Greenfield 01 866 296 3049 Gustafson Inc., 1400 Preston Rd., Piano, Texas, USA, (214) 985-8877 Hamamatsu Photonics France, 49/51 Rue de la Vanne, 92120 Montrouge, France, 46 55 47 58 Hiac/Royco Div., Pacific Scientific, 11801 Tech Road, Silver Springs MD 20904 (301) 680 7000
Manufacturers
and suppliers
625
Hitech Instruments, Inc., PO Box 886, 4799 West Chester Pike, Edgemont PA 19028, 215 353 3317 Horiba Instruments Inc. 17671 Armstrong Avenue, Irvine CA 92614-9782, 800-446-7422, 949 250 0924, e-mail; [email protected]/horiba Hosokawa Micron International Inc., 10 Chatham Road, Summit, NJ 07901, USA, (908) 598 6360 Insitec Inc., (now Malvern) 2110 Omega Rd., Suite D, San Ramon, CA 94583, USA, (510) 837 1330 InterSystems Industrial Products, 17330 Preston Road, Suite 105D, LB 342, Dallas, TX 75252, USA Joyce-Loebl, Marquisway, Team Valley, Gateshead NEl 1 OQW, UK, (0191) 482 2111 Kane May Ltd, Northey International Division, Nortec House, Chaul End Lane, Luton, Beds. LU4 8E2, UK, 01582 584343 Kemsis, 34830 Jacou, France Kowa Instruments, 355 Shoreway Road, South Vermont Avenue, Torrence CA 90502, (213) 327 Kratel SA, CH-1222 Geneve-Vesenaz, 64 Ch de St Maurice, Switzerland, 022 52 33 74 Labcon Ltd., 24 Northfield Way, Aycliffe Industrial Estate, Newton Aycliffe, CoDurham, DL5 6EJ, UK, 01325 313379. Lasentec, Laser Sensing Technology Inc., 15224 NE 95th Street, Redmond, WA, 98052, USA, (206) 881 7117) Lecotrac (see Horiba) Leeds & Northrup Instruments, Sunneytown Pike, PO Box 2000, North Wales, PA 19454, USA., (215) 699 2000 Leica Inc., 111 Deer Lake Road, Deerfield, IL 60015, USA LeMont Scientific, Inc. 2011 Pine Hall Drive, Science Park, State College, PA 16801, USA, (814)238 8403) L.U.M. Gmbh, (Gesellschaft fur Labor-Umweltdiagnosik & Medizentechnik mbH), Rudower Chaussee (OWZ), 12489 Berlin, Germany, 49-(0)30-67 19 81 88, e-mail [email protected], web www.lum-gmbh.de Malvern Instruments, Inc., 10 Southville Rd., Southborough, MA 01772, (508)480-0200, www.malvem.co.uk Matec Applied Sciences, 75 South Street, Hopkinton, MA 01748, USA, (508) 435 9039 Maztech Microvisions Ltd., 1306 Wellington Street, Suite 202, Ottawa, Ontario, Canada KlY 3B2, 613 596 5775, www.maztech.com Met One, (Pacific Scientific) 481 California Avenue, Grants Pass, OR 97526, USA. (503) 479 1248) Mettler (Switzerland) Instruments A.G., CH-8606, Greifensee-Zurich, Switzerland Micromeretics Instrument Corporation, One Micromeretics Drive, Norcross, GA 300931877. (770)662-3633, www.micromeretics.com Micron Powder Systems, 10 Chatham Road, Summit, NJ 07901, USA, (908) 273 6360 Micropul Corporation, 10 Chatham Road, Summit, NJ 07901, USA, (201) 273 6360 Microscal Ltd, 79 Southern Row, London WIO 5AL, UK, (0181) 969 3935
626 Manufacturers
and suppliers
Microtrac Inc., 148 Keystone Drive, MontgomeryviHe, PA 18936, (215)619 9920, www. microtrac.com Millipore (U.K.) Limited, Millipore House, Abbey Road, London NWIO 7SP, UK, (01965) 9611/4 Monitek Technologies, Inc., 1495 Zephyr Avenue, Hayward, CA 94544, USA, (415) 471 8300 MSA, (Mines Safety Appliances), 201 Braddock Ave., Pittsburgh 8, PA, USA Munhall Company, 5655 High Street, Worthington, OH, 43085, USA, (614) 888 7700 Nachet Vision SA, 125 Boulevard Davout B.P. 128, 75963 Paris Cedex 20, France, 33(1) 43.48.77.10 National Physical Laboratory, Middlesex, UK Newark Wire Cloth Company, Newark, New Jersey Nicomp Particle Sizing Systems, 75 Aero Camino, Suite B, santa Bardera, CA 93117, USA, (805) 968 1497 Nikon Corporation, Tokyo, Japan, www.nikon.co.jp Nisshin Engineering Co. Ltd., 1000 Corporate Grove drive, Buffalo Drive, IL 60089, 708 215 6865,708 520 9520 Nitto Denko, Tokyo, 1-12 Shimohozimi, Ibaraki, Osaka, 567-868, 81 726 22 2981 Oncor Instrument Systems, 9581 Ridgehaven Court, San Diego, CA 92123, 619 560 9584 Optomax, 9 Ash Street, Hollis, NH 03049, USA, (603) 465 3385 Otsuka(Munhall) Outokumpu Mintec Oy, PO Box 84, SF-02201, Espoo, Finland, 358 0 4211 Oxford Lasers Inc., 29 King Street, Littleton, MA 01460-1528, (978)742 9000, e-mail [email protected] Paar USA Inc., 340 Constance Drive, Warminster, PA 18974, USA, (215) 443 7570 Pacific Scientific Co., Hiac/Royco Div., 11801 Tech. Rd., Silver Springs, MD 20904, USA, (301)680 7000 Particle Measuring Systems Inc., 1855 South 57th Court, Boulder CO 80301, USA, (303) 443 7100 http://www.pmeasuring.com Particle Sizing Systems, 75 Aero Camino, Santa Barbara, CA 93117, (805)968-1497Pascall Ltd, Gatwick Road, Crawley, Sussex, RHIO 2RS, UK Pascal Ltd.,Gatwick Road, Crawley, Sussex, RHIO 2RS, UK PenKem(now Dispersion Technology) Pharma Vision Systems AB, Lund, Sweden PMT Partikel-Messtechnik GmbH, Carl Zeiss Str 11, Postfach 15 16, D-7250 LeonbergGebersheim, Germany, 7152 51008 Polymer Laboratories, 413 253 9554, www.polymerlabs.com/partsize Polytec GmbH & Co., D-7517 Waldbronn, Karlsruhe, Germany, (07243 6 99 44) Procedyne Corporation, 11 Industrial Dr., New Brunswick, NJ 08901, USA, (908) 249 8347 Proassist Company, 1614 East 5600 South, Salt Lake City, UT 84121, USA Procedyne Corporation, 221 Somerset Street, New Brunswick, NJ. USA Quality Control Equipment, PO Box 6010, Des Moines, Iowa 50309, 515 266 2289 Quantachrome Corp., 1900 Corporate Drive, Boynton Beach, FL 33426, USA, (407) 731 4999
Manufacturers and suppliers
62 7
Retsch, Inc., 17651 Armstrong Ave., Irvine, CA 92614, (949) 752-2004, e-mail [email protected] Rotex Inc., 1230 Knowlton Street, Cincinnati, Ohio 45223-1845, (513) 541-1236, www.rotex.com Rotex Inc., 1230Knowlten Street, CincinnatiOH 45223, USA 513 541 1236 Sartorius-Werke GmbH, Postfach 19, D-3400 Gottingen, Germany, (0551) 308-1 Sci-Tec-Inc, 6660 North High Street, Suite 2A, Worthington, OH 43085, 614 888 0285 http://sci-tec-inc.com Sedimage Photosedimentometer, www. cs.technion,ac.il/labs Seishin Enterprise Co. Ltd, Nippon Brunswick Bldg., 5-27-7 Sendagaya, Shibuya-ku, Tokyo 151, Japan, (03 350 5771) Shapespeare Corporation, 901 Park Place, Iowa City, lA 52240,USA, 319 351 3736 Shimadzu Scientific Scientific Instruments Inc., 7102 Riverwood Dr., Columbia, MD 21046, USA, (410) 381 1222, 1 800 477 1227, e-mail [email protected] Spectrex Corp., 3594 Haven Avenue, Redwood City, CA.94063, USA, (415) 365 6567 Stork Screens, 3201 North Interstate 85, Charlotte, NC 28213 Sympatec GmbH, 3490 U.S. Route 1, Princeton, NJ 08540-5706, (609)734-0404, e-mail [email protected] Technord photocentrifuge Thermo-Systems Inc (TSI), 500 Cardigan Road, Shoreview, MN 55126, USA, 651 490 2811, 1 800 874 2811, e-mail [email protected] Tracor Northern, 255 West Beltline Hishway, Middleton, WI 53562, USA 609 831 6511 Turbiscan, Fullbrook Systems Ltd., Unit 4, Boume End Mills Industrial Estate, Hemel Hempstead, Herts., HPl 2UJ. 01442 876 777, e-mail sales@fullbrook,com Turboscan, Woods Center, ANSTO, Technology Park, New lUawara Road, Lucas Heights, NSW 2234, Australia, Tel 02 9543 0477, e-mail [email protected] Tyler, W.S., Inc., E Hwy 12, Benson, MN 56215, USA, (216) 974 1047 Veco N.V. Zeefplatenfabrik, Eerbeck (Veluive), The Netherlands Vibrosonic sieves, Russel Finex Inc., Mount Vernon, NY Vorti-siv, 36135 Salem Grange Road, Salem, OH 44460, USA, (330) 332 4958, Fax: 1543 Warmain International Pty, Melbourne, Victoria, Australia Wyatt Technology Corp, 30 South La Patera Lane, 8-7, Santa Barbara, CA 93117, USA, (805) 681 9009, e-mail:[email protected] Zeiss, Carl Inc., One Zeiss Drive, Thomwood, NY 10594, USA, (914) 747 1800 Zeiss; Carl, 7082 Oberkochen, Postfach 35/36, Germany, (07364 20 3288)
Author index The unbracketed numbers give the reference locations in the text. The numbesr in brackets give the chapter number followed by the number of the reference in that chapter. Abbot, D., 550(10.143), 55/(10.149) Abeynayake, C , 502(10.321) Ackerman, L., 226(4.70) Adamson, A.W., 339(6.95) Adel, G.T., 144(3.10) Adeyayo, A., 572(10.205) Adjadj, L.P., 55J(10.262) Agar, A.W., 190(3.\62) Agarwel, J. K., 5/0(9.204) Aigal, A. K., 5d<10.174) Aires, A.M., (50/(10.315) Aizu,Y., 502(9.163) Akers, R.J., 479,483(9.9S) Alba, F., 577(10.233-234, 10,237, 10.245) Al-Chalabi, S.A.M., 467(9.7\) Alderliesten, M., 68(2.51 84(2.55) Alex, W., /25(2.129), 388(7.70), 403(^.9) Alexander, J., 442(S.77) Alexander, K., 55^(10.176) Alfthan, C. von., 527(10.22) Algren, A.B., J^P(7.84) Allano,D., 50/(9.158) Allegra,J.R., 575(10.227), 575(10.236) Allen R.P., 149(3.36) Allen, M., 88(2.SIX 227(4.64) Allen,T., ^9(1.39),/25(2.124), 148(3.29% /57(3.114), 22^(4.52), 25J(5.3), 257(5.120), J05(6.11, 6.14), i05(6.15), 348(6.\\5l 360, 388(7.1), 365(7.17), 373(7.24-25), 379(7.45-47,7.50-5\),403,412(S.\), 406, 407(S.\3-\5), 408(^.16), 4/2(8.19), 4I9(S.29), 427(8.36), 454, 455, 452(9.21), 462(9.55), 464(9.65), 493(9.\22), 495(9.127-
128), 553(10.164-165), 577(10.229) Alliett, D.F., 453(9.58, 5.61), 50/(9.152), 554(10.169) Alvares,M.L., 545(10.141) Alvarez-Arenas, T.E.G., 555(10.270) Amor,A.F., /50(3.49) Anand, S., /P5(3.204) Andersen, J.L., 454(9.27) Anderson, F.G., 450(9.7) Anderson, R., /P/(3.170) Anderson, v., 525(10.5) Andreasen, A,H,M., 254(5.16), 355(7.15-16) Andrews, L., 254(5.14) Anger, S., 275(5.86) Annen,K.D., 450(9.101) Annis, J.C, i5P(7.84) Ansari,R., 597(10.304) Arakawa, M., 355(7.73) , 434(8.5456) Aravamudhan, S., 55(2.91) Amd, J., 507(10.345) Arrivo,S.M., 575(10.224) Aschenbrenner, B.C., /55(3.75) Asser, M.S.el., 275(5.70) Astrakharchik-Farrimond, E., /57(3.155) Atakan,S., 50/(9.154) Atkins, J.M.,4P/(9.120) Austin, L.G., 5/(2.36), 113(2.121), 3 7P(7.48), 555(10.190) Avnir, D., 57(2.75) Azzazy,M., 507(9.191) Azzopardi,B.J., 544(10.129) Babick, F., 525(10.4), 575(10.225), 575(10.225), 553(10.261)
Author index 629
Bachalo,W.D., 5(^2(9.161) Bachman, D., 387(1.61) Backus, R.C., 191(3.164) Bacon, C , 604(10.332) Bae,J.H., 512(9.226) Baert,L., ^^7(10.311-313) Bafrnec. M., 265(5.22) Bahri,P.A., ^Pd(9.134) Bailey, G.W., 192(3.\12) Bailey, M., 572(10.211) Baldocci,R., 545(6.105) Balgi,G.V., (50(5(10.341) Ball, D., 55/(10.151) Ball, D.M., 275(5.59) Bandyopadhyay, R., 501(9.153) Bannerjee, A., 5P<10.291) Baranowski, J., 349,349(6.123) Barberry, G., 75/(3.53) Barford, N., 333(6.5S) Barker, D., /50(3.48) Barman, B.N., 27i(5.48), 275(5.82), 279(5.88) Barnes, M.D., 542(10.100) Bamett, M.I, /57(3.83), 166(3.100) Baron, P., 467(9.69) Barr,E.B., 500(9.147) Barreiros, P.M., 81(2.43) Barrett Gultepe, M.A., 5<^5( 10.256) Barrett, P., 49(5(9.129) Bartell, F.E., 340(6.96) Bartlett, R.W., 5J0(10.35) Bassarear, J.H., 526(10.1) Batch, B.A., 454,455,462(9.20) Batchelder, J. S., 509(9.195-197) Batel.W., 25(1.19) Bates, R.H., 2/5(4.32) Bauckhage, K., 502(9.164-166), 505(9.171,9.175) Baudet,M.G., 50(5(6.15) Baudet, M.J., 255(5.3) Bauer, U., 5(52(10.162) Baxter, L.L., 450(9.109-110) Bayvel,J.P., 545(10.140) Bean, C.P., 455(9.22. 9.28) Beattie, A.G., 5/5(9.236)
Becher, P., 164(3.9) Beck, M., 179(3.135) Beck, M.S., 557(10.43-44) Beddow,J.K., 70(2.11,13-15), 57,54(2.42), 54(2.53) Beer, J., 550(10.143) Beer, J.M., 55/(10.149) Behringer, A.J., 4(55(9.58) Bell, H.A.,/(57(3.109) Bemer,G.G., (504(10.333), (505(10.334) Bena, J., 265(5.22) Benedetto, D. de., 603(10.325) Benedetto, D., 57/(7.18), 555(10.57) Benker, B., 5(52(10.162), 564 (10.170) Benson, J.P.,/94(3.189) Berbner, S., 470(9.88) Beretta, P., 509(9.194) Berg, R.H., 455(9.19), 455, 468(9.46), 468(9.16) Berg, S., 30 3(6.3), 385(1.56), 595(8.1) Berghe, J. van der, 5/5(6.139), 5(55(10.187) Berkner,L.S., 505(9.172) Berne, B., 59/(10.277) Bernhardt, C, 225(4.74), 304(6.9) Bemt,W., 4/9(8.31) Bertero, M., 595(10.284) Berthold,C., 5(54(10.171) Besancon, P., 229(4.78) Bexon, R., 55(5(10.37) Binnig,G.,/9(5(3.199, 3.200) Birch, M.W., 527(10.14) Birchfield,H.P., 259(5.122) Bishop, G.D., 55(5(10.73) Black, D.L., 468(9.19), 4(59(9.83), 50/(9.157) Blackford, D.B., /70(3.116), 500(9.144), 5/0(9.198) Blaisot, J-B., /54(3.148) Blanchard, J., 49/(9.119) Blandin,A-F.,/54(3.149) Blass, E., 5//(9.210), 5/2(9.224-225)
630
Author index
Block, M., J50(3.49) Blois, R.W. de, 455(9.22, 9.28) Ely, D.D., 276(5.15) Blythe.H.N., 2(5^5.17) Boardman, R.P., 333(6.54) Boateng, A.A., 57(2.70,2.74) Bobrowski, G.S., 238(4.95) Boccacci, P., 593(10.2S4) Boeck,M., 557(10.159) Bohan,J.F., 5(?/(9.149) Bohon, W.M., 5/J(9.26) Bohren,C.F., 5^^ (10.127) Bolger, J.C, 330J35(6.43) Bond,R.L., J^5(6.118) Bonferoni, M.C., 462(9.56) Bonifazi, G., 83(2.46) Bonin, M.P., 468(9.791 469(9.^3), 480(9.\05-\0Sl 501(9.157) Bonnet, J.C, 5/0(9.205) Booth, F., 5J5(6.79-80) Boothroyd, R.G., 305,313(6.12) Bordera,L., 5^(^(10.141) Bordes,C., 565(10.189) Boron, S., 507(9.188) Borothy, J., 378(7.44) Bossoutrot, J-M., 7<^4(3.149) Bostock, W., 385(7.60% 386(7.63), 387(7.641 438(S.63) Bott, S.E., 555(10.158), 595(10.294) Bottlinger, M., /W(3.141) Boughton, J.H., ^95(9.128), 577(10.229) Bowdish,F.W., 557(6.131) Bowen, P., ^29(8.41-43), 563(10.\66l 608(\ 0.34S) Bowen, W.E., 57(5(10.224) Boxman, A., 5^9(10.142), 55/(10.150), 577(10.231), 57(^(10.235), 55/(10.246), 552(10.247), dO^lO.331) Boyco, CM., 572(10.204) Bradford, E.B.,i52(6.134) Bradley, D., 441(^.74) Bradley, D.E., /90(3.163), /92(3.171) Bragg, C.K., 467(9.67-6^)
Brand, P., 5/2(9.231) Brandolin,A.,5J4(10.53) Bravman,J.,/9/(3.170) Breitenstein, M., J75(7.35-36), 555(10.62) Bremer, L., 59^(10.290) Brenner, H., 303,312,314,317,318, 326, 328,334(6.7) Brezina, J.J., 559(7.79-80) Bricaud, J., 5^5(10.108) BrimhalI,S.L., 275(5.84) Brinkman, H.C, 525(6.39) Brodnyan, J.G., 442(S.79) Brown, C , ^55(8.47), ^55(8.61) Brown, D. J., 5(5^10.175-176) 5(55(10.180) Brown, G.J., 55(2.81) Brown, J.F., (7.37) Brown, O.E., 257(4.96) Broyles, D.A., /^^(3.10) Brugger, K., 304,309,335(6.10) Brundle,CR.,/95(3.178) Bryant, D.P., 5^^(6.104) Bryant, G., 595(10.297,10.301), 602(10.320-321 Buchanon, J.E., 552(6.49) Buchino, J., 549,5^9(6.121) Buether,H., 500(9.146) Bumiller,M., 55/(10.244) Burbridge, A.S., 49(5(9.132-133) Burgess, J.M., 525(6.36) Burson, J.H., 270(5.38) Burt, M.W.G., 5(5(1.30), 55(1.33), 2/6(4.37), 242,245(4.108), 245(4.111), 545(6.116) Burtscher, H., /96(3.2057) Butler, L.E., 495(9.127) Butler, R.S.,/79(3.132) Buttle, D.J., 5/5(9.237) Buzagh, A. von., 545(6.106) Caimcross, A., /45(3.28) Calboreanu, G., 224(4.54) Calderbank, P.M., 511(9.206)
Author index 631
Caldwell, K,D., 277(5.80), 275(5.84,5.86), 2(^4(5.104,5.109) Callis,C.E., 447(8.71) Callis,C.F., 2J5(4.92) Campanella, O.H., 113(2.122) Canals, A., 54(^(10.141) Cannon, D.W., 554(10.263) Cantoni, P., 759(3.84) Cao, J., 5^4(10.175-176 Capes, C.E., 330(6.42) Caramella, C , 462(9.56) Carhart, R.R., 576(\0.22S Carino,L., 542(6.101) Caron, P., i73(7.27) Carpenter, F.G., 227(4.61,63), 352(6.133) Carr, W., 34(5(6.113-114), 442(8.80) Carr-Brion, K.G., 527(10.19-21) Carstensen,J.T., 47(1.37) Cartwright, J., 50(2.334), 790(3.161), 797(3.166) Casamatta, G., 276(5.74) Castellini, C , 754(3.145) Cauchy, A., 754(3.68) Causley,D., 767(3.110) Cavaliere, A., 509(9.194), Caveney,R.J., 792(3.174) Cemi,T., 607(10.347) Chan, L.C., 57(2.72-73) Chandler, D., 465(9.77), 472(9.91) Chang, C.C, 797(3.176) Chang, D., 332(6.51) Chang, J. S., 455(9.113-114), 497(9.139) Chang, Y.J., 595(10.294,10.301) Charlier,R., 49(1.44) Charman, W.N., 146, 747(3.23) Chastang, J., 229(4.78) Chatfield,E.J., 764(3.93) Chechetkin, A.Y., 275(4.40) Cheeseman, G.C.N., 534(10.47) Chen, T.M., 570(9.202) Cheng, Y.S., 500(9.147) Cheng, Z., 57(2.76) Chesters, S., 55(2.79)
Chigier, N.A., 507(9.154-155), 536(10.76), 547(10.139) Chin, J.H., 550(10.146) Chin, T.H., 530(10.35) Cho, H., 379(7.48) Cho,S.K., 544(10.128) Chou, H.P., 606(10.340) Christian, J.R., 334(6.62) Christiansen, E.B., 376,324(6.24) Chu,Y.F., 797(3.168) Chung, H. S., 303(6.6) Church, T., 57(2.39) Ciocca, C , 462(9.56) Civic, T.M., 336(6.82) Clanton, U.S., 497(9.118) Clark, G.G., 55(2.68) Clark, N., 54(2.54), 242(4.105) Clarke, J.R.P., 79,36,49(1.13) Cloake, P., 555(10.275) Coffen, H., 442(8.83) Coghill, P.J., 573(9.233-235), 555(10.269), 556(10.272-273) Cohen, E.D., 757(3.138) Cohen, L., 357(7.65), 403(8.12) Cole, M., 57(2.40) Colfen, H., 442(8.82) Coll, H., 475(8.23), 479(8.26), 426(8.34), 427(8.38) Collin, A., 572(10.206) Colon, F.J., 276(4.38), 234(4.85), 236(4.94), 357,349(6.130), 463(9.60) Concoran, J.F., 792(3.173) Conley, R.F., 339(6.94) Conlin, S.G., 376(7.38) Cooke, D.D., 544(10.125) Cooper, A.A., 479(8.32) Cooper, H.R., 497(9.137) Coppens, P., 545(10.135) Com, M., 50(2.34), 266(5.25), 455(9.31) Comillault, J., 550(10.144) Cornish, D.C., 5(1.2), 20(1.14) Corrales, N., 573(9.236) Coulaloglou, C. A., 570(9.203)
632 Author index
Coull, J., 313(623) Coulter, W.H., 449(92,3) Coumil, M., i77(7.18), 555(10.57), 603(10.325) Courts, J., 385(7.59) Cowan, M., ^55(9.40) Cox, ^.W., 536(10.76) Crandall, W.A., 267(5.26) Crawley, D.F.C., 229(4.75) Crawley, G., J7/(7.18), 555(10.57), 603(10.325) Crawley, G.M., 57^(10.237) Crestana, S., 196(3.202), /PP(3.208) Cross, H.E., 55(1.26) Cross, J.A., 5^5(9.176) Crowl, V.T., 52,75(2.2), 164,167,191(3.91), 341(6.97), 343(6.\03) Crowthers, E.M., 555(7.59) Cullivan, J.C, 25^(5.8) Cummings, P.O., 57^(9.200-201), 595(10.300) Cunningham, E., 5/6/(6.18) Cunningham, K.D., 555(10.273) Curl, R.L., 5/7(9.222) Cushman,C.R., 525(10.5) Cutie, S.G., 275(5.44) Dacy, M.F., 5/5(6.138) Daescher, M.W., 214(4.35), 229(4.76), 234(4.S7) Daghlian, C.P., 142,147(3.3) Dahneke, B.E., ^97(9.139-140) Dahwan, G.K., 151(3.57) Dallevalle, J.M., 148,156,192(3.32) Dankers, S., 5^7(10.345-346) Dapkunas, S.J., /2(1.7), /52(3.86), 188(3.\56), 2//(4.8), 550(7.13) Dart, P.W., 525(10.8) Davidson, CM., 595(10.306) Davidson, J.A., /75(3.125), /79(3.132), 5^5(10.118) Davies, C.N., 75(2.26), 310,319(6.\9), 323(6.33) Davies, K.W., 148(3.27)
Davies, L., 55/,(6.46-48), 331, 334(6.47), 332(6.50), 335(6.7\) Davies, R., 59(1.36), 69,81,84(2.7), 332(6.52), 333(6.60), 579(7.51), 555(7.72), ^55,455(9.45-46), 495(9.122), ¥95(9.127-128), 555(10.164-165), 577(10.229) Davies, R.J., 259(5.121) Davis, G.W., 480(9. 99) Davis, R.H., 552(6.53) Davison, J.A., /52(3.139) Deanne,R., 554(10.168) Debbas, S.,/25(2.126,127) DeCain, D.M., 5(?9(9.196) Delaye,M., 595(10.282) Delessio, A., 509(9.194) Deleuil,M., 55,55(1.35) Delly,J.G.,/50(3.45) Demayere,H., 507(10.314) Deppert, K., 799(3.204) Deriemaeker, L., 545(10.135), 455(9.60), 555(10.196), 595(10.283), 594(10.290), 595(10.298) Devaux, M.F., 759(3.84), /74(3.124) Devon, M.J., 4/5(8.24), 4/9(8.27, 8.32), 425(8.35) Dhabbar, D., 59(2.6) Dhadwal, H., 597(10.304-305 Dhodapkar, S., 55(2.80) Diaz, M.L.S., 525(10.6) Dickens,;.,/55(3.153,154) Dietz,V.R., 552(6.133) Dietz,V.K., 227(4.61,4.63) DiMarzio,E.A., 275(5.61-62) Dirksen, J.A., 429(8.41) Dixon, D.C., 552(6.49) Dobbs,C.L., 554(10.177), 55/(10.245) Dodd, C.G., /49(3.40) Dodds, J., 255(5.99) Doemer,W.A., 25/(5.89) Doicu, A., 502(9.164) Dollimore, D., 55/(6.46-48), 55/,554(6.47), 552(6.50),
Author index 633
ii^(6.61-62,6.64,6.69-71), 335(6.13X348(6.111) Dominick,J., 5(?2(9.163) Donaghue, J.K., 438(S.63) Donnelly, H.F.E., 259(5.121) Dom, E., J55(6.74) Draper, N., 525(10.30) Drummond, D.G., 189,190(3.159) Duck, J., 55(2.86) Duckler,R.E., 5/7(9.211) Dukhin,A.S., 7/7(2.103), 577(10.230), 576 (10.226)., 552(10.253), 552(10.254) Dulin, C.I., 341(6.16) Dullenkopf, K., 503(9.161) Dullien, F.A.L., /45(3.33), 151(3.52, 3.55-57) Dulog, L., 554(10.266) Dumm,T.F., 4/7(8.20) Duncan, A.A., 164(3.SS) Dunn, E.J48(3.3\) Dunn, P., 5J(5(10.70, 10.74) Dunn-Rankin, D., 480(9.111) Dumey, T.E., 54(2.54), 242(4.105106) Durst, F., 5(?2(9.160,9.162-163) Dye,R.W., J4(5(6.112) Dyson, J., 166(3.95-91) Eadie, F.A., 559(7.78) Ebert, F., 269(5.34), 500(9.146) Ebert, J., 45/(9.51) Ebert, K.F., 725(2.125) Eckert, J.J.D., 792(3.174) Eckhof, R.K., /5/(3.51), 455(9.30), 455, 456(9.39), 462(9.52) Edwald, P., J5i(7.53) Einstein, A., 3//(6.21) Elder, M.E., 5J7(10.82), 54/(10.9495), 542(10.106) Eldridge, A., 264(5.17) Eling,V.B., 6^/(10.309) Elizalde, O., 602(10.322) Elkington, A.A.,/7(?(3.115) Elkington, D.A., 455,465(9.48)
Ellis, J.R.,/92(3.172) Ellison, J.McK., 50(2.31), /49(3.38) Elterman, P.B., 467(9.69) Elton, G.A.H., 335(6.75-78) Em5di, B., 76(2.25) Endo,Y., 543(6.102) Endoh, S., 5/(2.35) Endter, F., /64(3.87) Engel, A., 536(10.71) Engel,W.H., 346(6.112) Enneking,U.,/7(?(3.115) Enstad,G.,/57(3.51) Epstein, P.S., 576(10.228) Eriksson, P.A., 429(8.43) Esmail.M.N., 36(1.29) Etter, A.A., 779(3.132) Ettmuller, J., 539(10.89) Etzler,F.M., 50/(9.150-151), 536(10.167), 564(10.168) Evens,!.,/09,/73(2.117) Exnor, H.E., 764(3.89) Fahrenwald, A.W., 227(4.58) Fair, G.L., 75(2.21) Fairbridge, C, 54(2.58), 57(2.69) Fairhurst, D., 405(8.17), 479(8.30), 427(8.39) Fairs, G.L., 754(3.70), 755(3.71), 757(3.79) Fajgeli, A., 352(6.137) Fakouhl,N., 755(3.151) Famularo, J., 325(6.35) Fan, L.T., 57(2.70,2.74) Fancher, G., 440(8.68) Fancompre, B., 373(1.21) Fandrey,C.W., 50/(9.153), 503(9.172) Farin, D., 57(2.75) Farone,W.A., 544(10.121) FathR.,/70(3.115) Fawell,P.D., 496(9.134) Fayed, E.,/94,/95(3.192), /95(3.196) Feller, U., 535(10.61)
634
Author index
Felton, P.G., 564{\0.\1A\ 5^5(10.183) Feret, R.L., 752(3.62) Fernandes, J. B., 57/(9.217) Fernandez, M.J., 7P5(3.197) Femandez-Hervas, M.J., <^<^(2.84) Ferrara, P.R., 7PJ(3.112) Ferreira, P.J., ^7(2.43), 75^(3.143) Fewtrel C.A., i^(1.33), 348{6A\6) Figueiredo, M.M., 601{\Q3\5) Figueiredo, M.M., ^7(2.43), 754(3.143) Fijman, W.J., i57,J4P(6.130) Filippov, A., (5^7(10.343) Fils, F., 463(9.59) Fini,A., 55(2.84), 7P5(3.197) Finney, J.L., 54(2.52), 55(2.85) Finsey, R., 545(10.135), 565(10.196), 559(10.276), 597(10.278), 592(10.280), 593(10.283,10.286), 594(10.290), 595(10.293, 10.295, 10.298-299), 507(10.314) Fisher, R.M., 794(3.190) Fissan, H., 742(3.1), 75(5(3.152-153) Flaherty, D.M., 745(3.28) Flower, W.L., 57(2.77) Foerderreuther, H., 405(8.18) Foissy, A., i7J(7.27) Folk, R.L., 75(5(2.135) Fooks,J.C., 54(1.24) Foot, N., 54(5(6.109) Foweraker, A.R., (505(10.338) Frake, P., 575(10.223) Frances, C, 565(10.189) Francini, F., 754(3.145) Frank, R., 539(10.89) Freud, P.J., 595(10.287-288) Friedova, J., 402(8.4) Friedrich,F., 555(10.259) Friedrich,H., 555(10.265) Fritz, S.S., 227(4.65) Fuchs,H., (502(10.324) Fuchs, N.A., 75(2.30), 505,(6.8), 556(10.75) Fucile,E., 507(9.186) Fuerstenan, D.W., 227(4.72)
Fukui, K, 555(7.74-75) Gabas,N., 545(10.136) Gabrys-Deutscher, E., 559(6.92) Gale, R.W., 4(57(9.72) Ganser, G.H., 522(6.28) Gao, Y., 275(5.84) Garbow, N., 572(9.230) Garcia, F., 5(55(10.189) Garcia-Rubio, H.S., 604(10.332) Garcia-Rubio, L.H., 554(10.53) Garcis, I., 545(10.141) Gates, A.O., 709,772,775(2.106) Gaudin, A.M., 709, 772,775(2.105, 2.108), 557(6.131),, 556(7.61) Gebauer, H., 764(3.87) Gee-Clough, D., 575(9.238) Geers,H., 575(10.239) Gelade, E., 59/(10.278), 595(10.283), 594(10.290), 595(10.295,10.298) Gerber, C.H., 796(3.200) Gerber,M., 496(9.136) Gerstenberg, H., 557(7.66), 555(7.68) Ghosal,S., 457(9.51) Gibbs, J., 556(10.73) Gibson, K.R., 258,266(5.6) Giddings, J.C, 275(5.48), 277(5.7879,5.81), 275(5.82-84), 279(5.88), 252(5.94), 255(5.97-98,5.101), 254(5.103-104,5.108), 255(111112), 256(5.113-118), 602(10.319) Gilder, R.L.van, 275(5.45) Gilmore,J.R., 500(9.144) Glaess,H.E., 750(3.41) Glastonberry, J.R., 220(4.48) Glatter,0., 602(10.324) Glennon,B., 496(9.129) G6bel,G., 502(9.164) Godek, A.R., 5/5(9.232), 572(10.207) Goetz, P.J., 709,//7(2.103), 576(10.226), 577(10.230), 552(10.254), 552(10.253) Gold, G., 557(6.88)
Author index 635
Goldmann, G., 572(9.224) Goldstein, S.,i2i(6.31) Gonell,H.W., 255(5.18) Gonzales, R.C., 772(3.122) Gooding, J.L.,^P7(9.118) Goodwin, K.M., 275(5.59) Goosens, W., ^P(1.44) Gordon, K.F., 572(9.228) Goren, S.I., 463(9.63) GossQn, P., 601(10310) Gossen, P.D., 555(10.54) Gotoh, K., 84(2.52\ 88(2.S5\ 7P^(3.185), 527(10.23-25), 550(10.37-39) Gouesbet, G., 5(^7(9.158-159), 502(9.162) Gourdon, C , 276(5.74) Graessley, W.W., 557(10.81) Graf, J., 458,468(9.45) Grammenoudis, P., 5<^7( 10.243) Granger, J., 2^5(5.99) Grant, D.C., 570(9.198-199) Grant, D.J.W., 507(9.153) Grant, J.W., 257(5.89) Gras,L., 545(10.141) Grasso, v., 507(9.186) Green, M., 745(3.30), ]49(3.39) Green,R.A.,25P(5.121) Greenleaves, 1., 468(9.1S) Gregg, E.L., 454(9.23) Gregory, J., 555(10.59) Grehan,G., 507(9.158-159), 502(9.162-163), 505(9.168) Griffith, L., 109,113(2.120) Griffiths, D.L., 554(6.61) Griffiths, J.C, 136(2.134,136) Groen, F.C.A., 744(3.7) Groen, P. de, 595(10.283), 595(10.295) Groener, H., 539(10.89) Grotovsky,R., 757(3.138) Grover, N.B., 454, 457(9.26) Groves, M.J., 545(6.108), 449(9.1) Gruelich,F., 450(9.111) Guardani, R., 555(10.197-198)
Gucker,F.T., 545(10.117) Guerra, v., 545(10.141) Guilietti,M., 555(10.197) Gullaston, D.K., 25P(5.122) Gultepe,M.E., 55 (10.256) Gumprecht,R.O., (10.145) Guntheradt, H.J., 7PP(3.208) Gupta,V.S., 227(4.72) Guruswamy, S., 755(3.74) Guthals, D.L., 525(10.9) Gutsch, A., 470(9.86-87) Guttman, CM., 275(5.61,62) Guttrals,D.L., 545(6.100) Gy, P., 55(1.45-46) Habib,Z.G., 450(9.110) Hackley v., 109,111(2.103) Hackley,V.A., 552(10.251), 555(10.258) Hadi, R.S., 525(10.2) Hagen, G.E., 257(5.27) Haider, A., 522(6.26) Hale, J., 525(10.5) Haller,H.S., 545(10.118) Hamida, A.A., 554(10.172,10.174) Hamielic, A.C., 274(5.53,5.55), 275(5.76), 554(10.48), 507(10.309) Hamilet, C.W.Jr., 552(10.254) Hamilton, J.F.,7P5(3.199) Hamilton, R.J., 750(3.50),755(3.76), 757,755,7P0(3.81), 7P7(3.165) Hampe, A., 457(9.43), 455(9.47) Han, W., 552(10.250) Hancil, v., 577(9.208) Hanel, G., 348,350, 349(6.126) Hangl,M., 574(7.29-30) Hansen, F.K., 47P(8.28) Happel, J., 303,312,314,317,318,326, 328,334(6.1) Harada,M., 257(5.91) Hardalupas,Y., 505(9.183) Harfield, J.G., 455(9.40-41), 455(9.57) Harker, A.H., 554^10.267-278) Harrigan, K.,4P5(9.121)
636
Author index
Harrigan,K.A., 7(52(3.140) Harris, C.C.,7(?P,/77(2.102), 109,112J13{2AU\402{^.5) Harris, G.W., 750(3.46) Harris, J.E.C., 7(56(3.101) Harrison, C.F., ^(55(9.112) Hart, W.H., 555(10.158) Hartfield, J.G., J52(6.136) Hartig,H.E., 5^(5(6.109) Hartman, A.W., 745(3.18-20) Harvill,T.L., 5(59(10.201), 57/(10.203) Harwood, M.G.,74P(3.35) Hasapidis, K., 455(9.113-114) Hasegawa, M., 529(10.34) Haseler, S.C, (50(5(10.339) Haseler, S.C, 475(8.23) Haseler, S.C, 42(5(8.3) Hatch ,L.P., 75(2.21) Hathaway, R.E., 54(5(6.100), 52(5(10.9) Hatton, T.A., 4P(1.40), 577(9.212) Haugen, P., 505(9.168) Haultain,H.E.T., 2(55(5.21) Hauser, E.A., 445,44(5(8.67) Hauser, E.A., 440(8.67,8.70) Hausner, H.H., 57(2.37) Hawes,R.,2P,4(5(1.22) Hawkins, A.E., (5P(2.8), 55(2.63), 745(3.26-27) Hawksley, P.G.W., 7(57(3.112), 570,57(5,579,525(6.17) Hawley,S.A., 57(5(10.227), 575(10.236) Hawlitschek, N., 575(10.212) Hay, W., 795(3.180) Hazett R. L., 570(9.204) Heal, G.R., 554(6.65) Heath, A.R., 49(5(9.134) Heertjes, P.M., 542(6.99) Heffels, C, 55(2.87), 555(10.88) Heidenreich,E., 275(4.41) Heidenreich,S., 500(9.146) Heinrich, J., 5/2(9.231) Heiskanen,K., 54(1.47) Heiss, F., 575(6.23)
Heitzmann, D., 55(2.87) Heller, W., 554(10.49) Hemm, A., 607(10.345) Hemmer, G., 470(9.88) Hendrix, W.P., (7.43) Henein,H., 572(10.204) Henken,K.R., 507(9.193) Henn, A.R., 579(6.25) Heraud, C, 429(8.42) Herboltzheimer, E., (505(10.326) Herbst,J.A., 577(10.233) Herdan, G., 55(1.34), 109(2.97% 153 J 92(3.611 227(4J]l 257(5.119) Hermann, R., 225(4.73) Hermann, S.P., 79(5(3.202), 799(3.208) Hernandez, M.P., 545(10.141) Herpfer, D.C, /54(3.146) Herziger,G., 55(5(10.71) Hesketh,H.E., 54(2.51) Hess, CF., 507(9.191-192), 575(10.215-217) Hessemann, R., 572(10.208) Heuer, M., 25(1.15), 5(55(10.163), 552(10.152), 56(5(10.194) Heyd, A., 69(2.6) Heyder, J., 5/2(9.231)72(2.18), 709,7/5(2.119), 752(3.61), 156(3.771 208,210,245(4.11 226(4.57), 227(2,69), 299J 20.324(6.1) Heywood, H., 7/(2.17), 77,77(2.16), 208, 210, 245(4.1), 226(4.57), 226(4.57), 575(7.20) Hicken, G.K., 577(7.42) Hierschler, F.F., 555(6.77-78) Higgs,D.M.J., 575(10.237) Hildred, K.L., 754(3.142) Hinde,A.L., 56(1.27-28), 497(9.138), 524, 529(10.1), 527,529,550 (10.16), 527(10.18) Hinds, I.e., 606(10.339) Hinze, F., 526(10.4), 576(10.225), 552 (10.252), 555(10.259)
Author index 637
Hipp, A.K., 577(10.232) Hiquily,N., 5^5(10.136) Hirleman, E.D., 5^5(10.133-134), 5^7(10.139), 5(5<10.173), 5^5(10.182-184) Hishida,K., 5(^7(9.184-185) Hjelmstad,K.E., 7P<3.193) Hoagland, D.A., 275(5.44) Hobbel, E.F, ^95(9.127-128), 577(10.229) Hocqs, S., 276(5.1 A) Hodkinson ,J.R., ^^(2.32), 468{9n^\ 507(9.187) Hofeldt,D.L., 555(10.83) Hoff,C., ^95(9.125) Hoffer, M.S., 577(9.214) Hoffman, R.L., 388(7.16) Hofmann, H., 555(10.59), 5^5(10.166) HogeKampS., 755(5(3.151) Hogg, R., 303(6.6\ 417(S.20) Holdsworth, J.F., 156(3.161 157,] 66J 90(3.S\) Holgado, M.A., 55(2.84), 795(3.197) Holland-Batt, A.B., 572(9.229), 527(10.10-11,10.14-15) Holt, C.B., 59(2.94) Holve, D.J., 245(4.118), 480(9.99104(9.111), 5(59(10.200-203) Hoomstra, J., 450(9.111) Horiuchi,T., 545(6.102) Horn, D.S., 595(10.296), 595(10.306), (505(10.327-329) Horridge, T.A., 554(6.62,6.69) Horsfall,F., 259(5.124) Houser,M.J., 502(9.161) Houssiere, C.R., 440(8.68) Huang,!., 779(3.131) Hubner, T., 575(7.32-34) Huffman, D.R., 544(10.17) Hughes, T.H., 265(5.19) Huller,D., 779(3.133) Hulley,B.J, 79(1.12) Hulst, H.C. van de., 532,534,539, 540(10.45) Humphries, D.W., 745(3.25)
Humphry-Baker, R., 429(8.41-43) Hundal,H.S., 54(2.61) Hunt, CM., 552(6.132) Hurd, A.J., 57(2.77) Husemann, K., 225(4.73) Hutchinson, H.L., 497(9.116) linoya, K., 7(52(3.85) Ikeda, C , 57(2.35) Ikladious, N.E., 554(10.266) Ilantzis, M.A., 227(4.49), 222(4.50) lies, P., 4(57(9.70) Imamura, A., 454(8.55) Imris, P., 559(7.85) Inn, E.C.Y., 557(10.80) loos, E., 255(4.88) Irani, R.R., 255(4.92), 447(8.71 Issoukis,M., 777(3.119,120) Iwai,S., 557(10.78) Iwata, H., 57(2.35) Jacobs, D.J., 545(10.116) Jacobson, M., 450(9.7) Jaeger, N.C.de, 275(5.69), 559(10.276), 597(10.278), 592(10.280), 607(10.314) James, G.W., 555(7.57) Jang, Y., 552(6.51) Janjua, K.M., 572(9.223) Jansen. M.L., 220(4.48) Jarrett, B.A., 575(7.20) Jayasoorija, U.A., 575(10.223) Jayaweera, K.O.L.F., 555(6.59) Jelinek,R., 429(8.41) Jeng, S., 754(3.146) Jenike, A.W., 557(6.87), 555(6.90) Jenkins, R.D., 275(5.68) Jenkinson, A., 525(10.27) Jennings, B.R., 555(10.64), (50(5(10.338) Jepson,G., 4(1.2), 20,(1.14) Jillavenkatesa, A., 72(1.7), 7(52(3.86), 755,(3.156), 2/7(4.8), 5(50(7.13) Jimbo, G., 57(2.1), 52(2.44-45)
638
Author index
Jochen,C.E., 577(10.231), 575(10.235), 557(10.246), 552(10.247), d^^(10.331) Joffe,A.D., 7P7(3.167) John, R., 754(3.141), 333(6.55) Johnson, E.,25P(5.123) Johnson, I., 542(10.99) Johnson, R., i5<7.54) Johnston, J.E., 55(2.62) Jones, A.R.,4<57(9.71) Jones, M.H., 479(8.25) Jones, R.M., 5(^4(10.179) Jones, T.F., 49(5(9.130-132) Jones, T.M., 2J5(4.90), 255(4.98), J J 7(6.84) Jongen,N., (5^5(10.348) Joosten, J., 595(10.283), 594(10.290),
595(10.295,10.298) Jorgenson, J.W., 27(5(5.71,5.73) Joss, J., 470(9.90) Jovanovic, D.S., 333(6.51) Jowett, A., 259(5.125) Joy, A.S., 525(10.27) Juda, J., 545,549(6.124) Julien,R., 5(1.3) Jung, T., 79(5(3.205) Junno, T., 79(5(3.204) KadambU.R., 505(9.174) Kaiser, F., 2(59(5.32) Kalshoven, J., 57(5(7.40-41) Kamack, H.J., 595(8.2), 402(8.3), 455(8.50) Kamp,D.,5(?(5(9.178) Kansted,S.O., (505(10.337) Karasikov, N., 477(9.97) Karpenko, I., 54(5(6.112) Karuhn, R., 455,455,4(55,4(55(9.36), 455(9.45-46), 468(9.16) Karworth,R., 496(9.135) Kasper, G., 55(2.79) Kato, H., 754(3.147) Kaupp, A., 57(2.71) Kawabati, T., 575 (10.222) Kawafune, I., 5(5(5(10.193)
Kawahashi, M., 274(5.57) Kay, D.H., 188,191,192(3.\51) Kaya, N., 454(8.54-56), 50(5(9.180) Kaye, B.H., 72 ,25(1.5), 25(1.18), 49(1.42), 55(2.66,2.68), 7(57(3.114), 220(4.47), 225(4.51), 227(4.59), 247(4.104), 245(4.108, 4.110-111), 244(4.112-113, 4,115), 552(6.52), 555(6.54,6.60), 555(7.57), 555(7.72), 475(8.21), 499(9.141), 500(9.148),507(9.152), 545(10.120), 5(54(10.169) Keeler,R., 757(3.137) Kelsall, D,F., 2(55(5.29), 525(10.33) Keng, E.Y.H., 270(5.38),545, 549(6.128) Kennedy, J., 5(55(10.183) Kenney,L.C., 744(3.8) Kenyon,A.S., 542(10.103) Kenyon,E.S., 542(10.102) Kerker, M., 547(10.96), 542(10.98), 544(10.121,10.125-126) Kern, R., 554(10.266) Khalili, M., 427(8.36), 4(52(9.55) Khan, A.A., 49(1.39) Khatchikian, P., 555(10.257) Kilham, L.B., 184,(3.144) Kim, M., 552(6.51) Kim, S.C, 55(5(7.62) Kimura, K., 252(5.92-93) King, J., 259(5.123) King.R.P., 795(3.196) Kinney, P.D., 500(9.145) Kinoshita,T., 568(10.195) Kippax,P.C., 575(10.237) Kirkland, J.J., 277(5.39), 274(5.54), 27(5(5.75,5.77), 275(5.87), 250(5.89-90) Kirou, v., 570(9.205) Kitai, H., 454(8.55) Klabunde, U., 745(3.28) Klein, A., 554 (10.46,10.50) Klein, J-P., 754(3.149), 495(9.125) Klein, R., 5(54(10.171)
Author index 639
Klimpel,R.R., 775(2.121) Klinzing, G.E., <^^(2.80) Knight, J.C., 5^(^(10.140), 557(10.151) Knollenberg, R.G., 469{9M) Knosche, C , 555(10.259,10.265) Knudsen, K.L., 575(9.238) Koch, H.L., 25(1.20) Kocova,S., 795(3.196) Kodama,H., 5(^7(9.156) Koehler,M.E., 607(10.317) Koglin, B., 725(2.129), 555(6.56), 346{6A\\) Kohne, A.P., 274(5.56) Kolendik,0., 794(3.183) Komonz, D.R., 467(9.15) Konert, M., 575(6.139), 5(55(10.187) Koser,0., 505(9.170) Kossen, N.W.F., 542(6.99) Kourti,T., 554(10.48, 10.51), (507(10.309-310) Kousaka, Y., 505(6.4), 545(6.102) Kovall, G.E., 257(4.95) Kozhenkov, V.I., 555(10.75) Kramarenko, I.B., 794(3.184) Kramer, H., 55(2.87) Kramer, J.F., 756(3.154) Kratky,N., 74(2.20) Kratohvil, J.P., 544(10.123-124) Krauss, M., 477(9.97) Krauter, U., 575(7.35), 555(10.62), 539(10.91), 557(10.241-243), 555(10.257) Kreyling.W.G., 572(9.231) Krinsley,D., 794(3.191) Kroner, G., 602(10.324) Krumbein, W.C, 77(2.27,28), 756(2.131,132) Krutzch, J., 545,549(6.125) Kubitschek, H.E., 449(9.4-5), 456(9.42) Kuga, Y., 57(2.35) Kuist, C.H., 554(10.46,10.50) Kushida, A., 255,259(4.103), 247(4.104) Kutepov, A.M., 244(4.116)
Kutsuwanda, N., 506(9.181) Kutz. S., 799(3.206) Kynch, G.J., 526(6.34) La Mer, V.K., 557(10.79-80), 542(10.98-99,10.101,10.104-105) Laethem, M. van, 607(10.312) Laethen, M. van, 607(10.314) Lafaye, A., 229(4.78) LaguericC, 545(10.136) Laird, W.E., 77,57(2.29) Lai, D., 575(6.138) Landsperky, H., 559(7.86) Lane,G.S., 742,747(3.5) Lang, J.S., 497(9.116) Lange, H., 756(3.158), 602(10.318) Langer, G., 574(9.239-240) Langer, R., 757,757(3.5) Langhorst, M.A., 275(5.44-45,47) Langobardi,G., 754(3.145) Langston, P.A., 496(9.130-133) Lapple, C.E., 269(5.35,36), 577(6.20 Larden, P., 275(5.69) Lark, P.D., 766(3.94) Lauer, O., 255,259(4.83), 255(4.97) 259(4.99), 267(5.28) Leal. G.P., 602(10.322) Lee, L, 277(5.81), 255(5.97) Lee, R. J , 796(3.190) Lee, S., 255(5.105), 552(6.51), 602(10.319) Lee, T., 552(6.51), 557(10.43-44) Lee, W., 555(10.84) Lee, W.P., 796(3.203) Lehman, P., 502(9.166) Leipertz, A., 575(7.33-34), 555(10.85-86), 607(10.344-346) Leiza,J.R., 602(10.322) LenczyckT., 256(5.115) Lenn C.P., 59(2.94) Lenz, F., 749(3.37) Leonardo, E.M., 595(10.288) Lerman, A., 575(6.138) Leschonski,K., 725(2.19), 270,22/(4.4), 245(4.119),
640
Author index
256{5A\ 258(5.71 268(5.30% 269(5.33), 339(6.92% 388(7.69-70% 552(10.152), 5^2(10.161-162), 5^<10.170) Levenspeil, O., J22(6.26) Levin, S., 25(5(5.115-117) Levitch, V.G., 372(6.22) LewisP.C, J7i(7.22) Ley, I., 555(10.157) Li,J.M.,25<5.109) Li, J.T., 25^(5.109) Licinio, P., 5Pi (10.282) Liebermann, A., 469(9.S2) Ligthart, T., 144(3.7) Lilge,D., 595(10.296) Lilienfeld, P., 467(9.69) Lim. M., 5d<9(10.199) Linck, E., I64(3.S9) Lindenthal, G., ^7i(9.95), 57i(10.214) Lindsey,E.E., 577(9.209) Lines, R.W., 36(\.3\X 273(5.46% 455(9.41% 458(9.49) Lisieki,D., 5^7(9.158) Litzen, A., 252(5.96), 25i(5.100) Livesey, A., 5Pi( 10.282) Llewelyn, D.M., 70(2.\2) Lloyd, H.B., i7J(7.23) Lloyd, P.J.D., i^(1.27), 403(S.\0% ^55(9.38), 527,529,530(10.16) Loffler, F., 470(9.^6% 579(10.240) Loffler, W., 572(10.210), 575(10.213214) Loffler-Mang, M., 470(9.89) Lohner,H., 502(9.166) Loisel, C, 759(3.84) Lombard, G.A., 442(8.80) Lorentz,H., 505(6.16) Lothian, G.F., 575(7.22) Luddecke,E., 595(10.296) Luhmann, J., 5(54(10.171) Lui,C.F., 575(10.218-219) Lukacs, K.D., 27(5(5.71,5.73) Lum, L-S,H., 72(1.7), 7(52(3.86), 755(3.156), 277(4.8), 5d0(7.13) Lundberg,J.J.V.,5(55(7.16
Lundin, M., 55(2.79) Luscombe, C.N., 576(10.223) Lynch, A.J., 528(10.28-30) Lynn, J.F., 447(8.72) Ma,X., 779(3.131), 575(7.28) Ma, Z., 545(10.132) MacCalman, D., 227(4.66-68) MacGregor, J.F., 534 (10.48, 10.51), 555(10.54), (507(10.309-310) Madden,A. J., 577(9.219) Maddox,J.,5(1.4) Maeda, M., 50(5(9.182), 507(9.184185) Makino, K., 59(2.93) Malhetra,V.M., 257(4.96) Malot, H., 754(3.148) Malotsev,V.P., 507(9.189) Maltsev,V.P., 507(9.190) Mandelbrot, B.B., 55(2.64,65,67) Manderick,C., 5(55(10.196) Mandersloot, W.G.B., 555(7.71) Mang,.T., 757(3.154) Mangin, D., 754(3.149) Mani,B.P., 55(2.91) Mannheim, C.H., 557(6.85) Margolis,S., 794(3.191) Marienko, J., 549(6.120) Marijnissen, J.C.M., 499(9.142) Markus, M.W., 607(10.343) Maron,S.H., 557(10.82) Maron,S.H., 547(10.93-95), 542(10.106-107) Marquard, H. von, 574(7.31) Marshall, C.E., 559(7.77) Marshall, K., 455(9.37) Marson,B., 799(3.207) Martelli,S., 575(10.221) Martens, A.E., 794(3.187) Martin, G., 775(2.115), 752(3.60) Martin, S.W., 455(8.48-49) Masaoka, K., 529(10.34) Mason, B.J., 555(6.59) Mason, M., 505(6.2) Massacci, P., 55(2.46)
Author index 641
Masuda, H., 7(52(3.85), J^<^(7.74) Matheron. G., 779(3.129-130) Mathes,K.N.,^P7(9.120) Matthews, B.A., 552(6.135), 463{9.62) Maude, A.D., 325(6.37) Maxim, L.D., 534 (10.46, 10.50) May, l.,i^P(6.120) May, K.R., (1965), 755(3.73) Mayette, D.C., 142,147(332) Mayr,G., 27<5.56) Mazumder, M.K., 50(5(9.177-181) Mazzotti, M., 494(9.124), 495(9A26) McAdam, J.C.H., 268(5.29) McBride, B., 331,334(6.Al\ 334(6.6A) McCarthy, J.J., 793(3.181), 79^3.189190) McCarthy, J.L., 553(10.256) McCormick, H.W., 442(S.1S) McCormick, R.M., 276(5.12) McCoy, B. J., 577(9.219) McCreath, CO., 550(10.143), 557(10.149) McCrone, W.C, 7^2(3.2), 750(3.44) McCulley,C.R., 750(3.41) McGowan, G.R., 273(5.47) McGregor, J.F., 27<5.53) McHugh,A.J.,27<5.51-52) Mcintosh, J., 440(S.69) McIntyre,D.D., 73(5(2.134) McKenzie, D. C , 488(9.U3-\ 14) McLaughlin, C. M., 577(9.207) McMahon, B., 142,147(3.3) McManus, D.A., 73(5(2.130) McNew,G.L., 259(5.122) McQuay, M.Q., 468(9.19), ¥(59(9.83), 507(9.157) McQuie, G.R., 52(5(10.7) Meakin, P., 5(1.3) Medalia, A.I., 57(2.38) Meeren P, van der., 427(S.40), 607(10.311-314) Mehta, P.N., 749(3.34), 757(3.55) Melcion,J.P., 174(3.124)
Meloy, T.P., 53(2.47-50,54), 709, 772,773(2.108), 59(2.93), 242(4.105-106), 243(4.107) Melton, L.A., (507(10.342) Melucci, D., 275(5.84) Melzer,D., 554(10.154) Membrey,F., 373(7.27) Menzel,U., 5(52(10.161) Merkus, H.G., 265(5.30). 462(9.56), 545(10.132), 549(10.142), 557(10.150) Metzger, K.L., 2(55(5.30) Meyer, E., 475(8.24), 42(5(8.35) Meyer, M.E., 534(10.46,10.50) Meyer, W., 597(10.304) Michael, A., 455(9.33) Michaels, A.S., 330,335(6.43) Middleman, S., 570(9.202) Mielke,A.F., 503(9.174) Mignon,H., 503(9.172) Mika,T.S., 227(4.72) Miles, N.J., 55(2.81) Millen, M.J., 55(5(10.272-273) Miller, CM., 276(5.70) Miller. B.V., 273(5.46) Milot, J.F., 276(5.74) Mirme, A., 572(9.231) Mitchel,P.G., 527(10.20) Mitschke, M., 502(9.165), 503(9.171) Mitsis, T.J., 572(9.228) Miwa, S., 270(4.5-6) Miyasaka,K., 507(9.156) Miyazaki, T., 303(6.4) Mizutani,Y., 507(9.156) Mollet,H., 342(6.101) Molter, L., 473(9.95) Moltinin, E., 227(4.62) Monnier,0., 495(9.125) Monredon,F.le., 774(3.124) Montague, C.E., 453(9.112) Montelius, L., 799(3.204) Montgomery, J.R., 49(1.41) Moon, M.H., 277(5.78,79), 275(5.82), 254(5.105), 602(10.319) Moore, D.W., 303(6.5)
642
Author index
Moran, P.A.P., 752(3.64) Moran, R.F., 227(4.60) Morbidelli,M., 577(10.232), 583(10.262) Morgan, B.B., 7(57(3.105) Morgan, J.J., ^55(9.34) Morgan, S.P., 757(3.155) Morgan, V.T., J7i(7.21) Mori,C. de, 5Pi(l0.284) Mori.Y., 257(5.91), 252(5.92,5.93), 575(10.222) Morikita, H., 50(5(9.182), 507(9.IM185) Morony,M.J., (55(2.4) Morris, T.R., 224(4.55) Morris, V.J., (505(10.338) Morse, T.H., 467(9.74) Morton R.R.A., 7P<3.187-188) Mougin, P., 55J( 10.260) Muehlbacher, R., 5(55(10.186) Mueller, H.G., 442(8.81) Mueller, J., 572(9.230) Muhlenweg, H., 545(10.132,10.134), 565(10.184) Mulford, D.F., 757(3.78) Muller, CM., 275(5.60) Muller,G., 545, J4P(6.127) Muller, L.D, 29,47(1.22) Muller, R.H., 275(5.86) Mullin, J.W., 2i4(4.84) Mullins,M.E., 275(5.59) Munch, K-U., 555(10.86), 555(10.85) Mundo,C., 505(9.167) Murley, R.D., 457(8.62) Muschelknauts, E., 454(8,51-52,8.5760) Muta, A., 505(6.13), 557,54P(6.129) Muzzio, F.J., 77P(3.134) Myers, M.N., 275(5.82-83), 254(5.103-104), 25(5(5.114,5.117118) Myler, C.A., 55(2.80) Mylnek,Y., 577(9.220) Myojo,T., 244(4.114)
Nagy,D.J., 274(5.51-52) Nakajima, Y., 527(10.23 -26), 550(10.37-40) Nakamura, K., 575(10.222) Nakamura.Y., 505(9.169), 50(5(9.181) Nanda, A. K., 577(9.216) Napper,D.H., 547(10.97) Naqui, A., 502(9.162) Naqui, A.A., 507(9.153), 502(9.162) 505(9.172) Nascrimento, C.A.O., 5(55(10.197198) Nasr-el-Din,H., 5(5(1.29) Nasta,M.D., 70(2.14) Nathan, I.E., 757(3.83) Nauman, A.Z., 525(6.32) Naylor,A.G., 72,25(1.5) Neese, Th., 55(2.86) Nelson, R.D.Jr., 725(2.124), 505(6.11), 55P(6.93), 427(8.36) Neri,F., 507(9.186) Neuzil, L., 2(55(5.22) Ng, S.H., 54(2.58), 57(2.69) Nicholson, W.L., 757(3.58) Nickel, K.G., 5(54(10.171) Nicoli,D.F., 455(9.113-114), 5(54(10.171), 5P5(10.302), (500(10.307), (507(10.309-310) Nicolson,G., 544(10.125) Niedick, E.A., 254(4.86) Niehus, H., 275(5.86) Niehuser,R., (507(10.316) Niemann, J., 42P(8.44-45) Niemelia,0., 54(1.47) Nijs, B.de, 5P4( 10.290) Niklas, U., 27P(4.42) Nishino, A., 754,(1.147) Noel, R.J., 275(5.59) Nonhebel, G., 57(5(7.39) Nordel,P.E., (505(10.337) Nunzio,D., 575 (10.221) Oakley, D.M., 555(10.64) O'Brien, R.W., 554(10.263)
Author index 643
Oden, S., i^5(7.58) Oechsle,S., 547(10.139) Ohyo, H., <^7(2.35) Oiu,H., 502(9.163) Oja,T., 5(^7(10.244) Okaya,K., 794(3.185) Okuyama, K., 303(6.4) Oliphant, S.C, 440(^M) Oliver, J.P., J77(7.42) Onifri, F., 502(9.163), 505(9.168) Onimaru,R.S., 568(10.198) Oppenheimer, L.E., 427(8.37) Orford, J.D., 84(2.60) Oristad,N.I., i4(5(6.109) Orr, C , 148,156,192(3.32), 270(5.38), 275(5.59), 503(6.5), 377(7.42-43) Osborne, B.F., 3^(1.25), 527(10.1213) Oseen, C.W., 322,326(6.29) Otsukia.S., 794(3.185) Ottewell,R.H., 547(10.97) Ovad, V.I., 473(9.93) Overbeck,E., 607(10.316) Ovid, v., 595(10.302) Owens, N.F., 334(6.70) Page, H.G., 736(2.133) Page, N.W., 57(2.72-73) Pahl,M.H., 57(2.41) Palberg, T., 572(9.230), 607(10.316) Palik, E. S., 386(1.62), 462(9.53) Palmer, A.D., 54(2.58), 57(2.69) Pankala,S., 349,349(6.121) Paolo, E., 575(10.221) Papadakis, M., 530(10.41) Papoloni,E., 754(3.145) Parfitt, G., 335(6.91) Parrent, G.B. Jr., 536 (10.67,10.72) Pasternak, S.H., 757(3.59) Pasyatakes, A.C., 55(2.82) Patel,S., 535 (10.84) Patitsas, A.J., 543(10.119-120) Pauck, T., 442(8.82) Paulowsski, S., 275(4.39) Paulson, CM., 535(10.63)
Pav. J.W., 254(5.104) Pavlik,R.E., 336(6.81) Pavlovich, A., 5(1.3) Payne, B.O., 766(3.98) Payne, R.E., 359(7.78) Pearson, J.R.A., 322(6.30) Pecora, A., 597(10.277) PeinertJ.C, 566(10.192) Pekkannen, J., 572(9.231) Peleg, M., 773(2.122), 337(6.85) Pendse,H.P., 575(10.238), 552(10.248-250) Penlids, A., 274(5.53) Penumadu D., 55(2.83) Perry, R.W., (3.101) Peschtl, P., 374(7.29-30) Peters, C , 270(4.3) Peters, E.D., 274,225,247(4.35) Peterson, J.L., 235(4.93) Peterson, K.O., 442(8.76) Petrak,D., 503(9.173) Petruk,W., 794(3.182) Pettyjohn, E.S., 373(6.24) Pettyjohn, F.J., 736(2.132) Pfane,B., 605(10.335-336) Pfeifer, P., 57(2.75) Phelps, B.A., 750(3.50), 79/(3.165) Philip, G.C., 70(2.14), 54(2.53) Phillips, B.L., 550(10.147) Phillips, D.S., 756(3.154) Pidgeon, F.D., 749(3.40) Pierce, P.E., 537(10.82), 547(10.9395), 542(10.107) Pierce, T.J., 334(6.8) Pietzsch, W., 572(9.225) Pijper, A., 545(10.130-131) Pike, E.R., 593(10.284), 594(10.292) Pilpel.,N., 337(6.86) Pine, D.J., 603(10.326) Pitchumani, B., 54(2.54), 242(4.105) Plantz,P.E., 593(10.289) Plawsky,J.L., 577(9.212) Plebuch, R.R., 572(9.228) Plessner, I., 542(10.105) Pochlein, G.W., 273(5.43)
644
Author index
Polke, R., 455(9.35), 457(9.44), 538(\0M) Pons, N.M., ^4(2.61), 555(10.56) Pope, L.J., 510(9200) Popplewell, L.M., 7/J(2.122) Porter,M.C., 4<^P(9.115) Porter,M.C., 4P/(9.117) Powers, T.C., 329(6.41) Powitz,H.,5(5<10.170) Pownall,J.H., 29(1.21) Prassas,H., 50(^(9.182) Pretorius, S.T., i^^(7.71) Proctor, T.D., 150(3.46-47) Prod, G., 74(2.20) Proudman, I., 322(6.30) Provder, T., 418(^.241 47P(8.27), 426(S.35X60J(\0.3\7) Provencher, S., 595(10.281) Prud'homme, R,K., 273(5.44) Pryor,E.\].,264(5.11) Pukhaber,M., 55(^(10.192) Puranik,S. A., 5/7(9.218) Putman, B., 427(8.40) Putman,R.E.J., 52(^(10.31) Quant, F.R.,50(?(9.144) Quieroz,M., 4^(^(9.106-108) Quinn, J.A., 454(9.27) Rabasco, A.M., 88(2.^4), J95(3.197) Raddle, M., 539(10.89) Radle, M., 559(10.88) Rafter, R.T., 5(^(5(10.271) Ragucci,R., 509(9.194) Rainey,S., 5^(5(10.273) Ramachandran, G., 500(9.143) Ramakrishna, V., 554(6.67), 554(6.6667) Ramakrishnan, P., 69(2.9) Rammler,E., 709,770,777,775(2.104) Ramos,dos, J.G., 277(5.40), 275(5.49), 274(5.50), 275(5.58,5.63-67) Rand, B., 200(3.211) Rao, S.P., (502(10.319) Rao, S.R., 554(6.66-67)
Rao, T.C., 525 (10.29) Raphael, M., 555(10.55) Ratanathanawongs, S.K., 277(5.7981), 2(^5(5.97), 255(5.101), 254(5.107,108) Ratsimba,B., 495(9.125) Read, C.E., 440(S.66) Rebelein,F., 577(9.210) Reed, R,A., 576(10.224) Rees, W.J., 76(2.26) Regnier, F.E., 275(5.59) Reichel, A., 49(5(9.136) Reichel, F., 572(10.210), 575(10.212) Reinhold,B., 572(10.209) Rementer, S.W., 257(5.90) Rennie, A. G., 5(54(10.175) Rennie,F.W., 495(9.127) Reschiglian, P., 275(5.85) Restarick, C.J., 525(10.33) Revell,R.S.M., 790(3.162) Reznick, W., 577(9.214,9.220) Reznickova,R., 577(9.208) Rheims, J., 505(9.175) Rhodes, C.T., 47(1.37), 552(6.135), 463(9.62) Rhodes, E., 757(3.52) Ricci,R.J., 497(9.137) Richards, J.C, 257(5.1) Richardson, J.F., 525,550(6.38) Richmond, G.D., 142,147(3.5) Richter, R., 57(2.76) Richter,S.M., (50(5(10.341) Ridder,F.D., 545(10.135) Rideal, G.R., 224(4.55), 259(4.101) Ridgeway, K., 55(2.88-92), 244(4.117) Riebel, U., 575(7.35-36), 555(10.62), 559(10.90-92), 557(10.240-243), 555(10.257) Rigby, O., 462(9.54) Rimmer,H.W., 744(3.10) Ripperge, S., 526(10.4), 555(10.61), 57(5(10.225), 552(10.252,10.255), 555(10.261) Rippinger,S., 555(10.60)
Author index 645
Rittinger, P.R.von., 208,210(42) Rivoire, A., 7<^5(3.149) Roberts, D., 550(10.160) Roberts, F., (3.107) Roberts, K.J., 5^5(10.260) Roberts, P., 77^3.124) Robertson, G., 545(10.140) Robertson, G.N., 55/(10.151) Robertson, R.H.S., 75(2.25) Robillard, F., 543(10.119-120) Robinson, G.W.,i(55(7.14) Robinson, H.E., 433(SA9) Rodd, v., 577(9.208) Rohani, S., 84(2.611 494(9.123), 535(10.55-56) Roher, H., 7Pd(3.200) Rohricht, W., 462(9.55) Roller, P.S.,70P,772,77J(2.109-111), 257,254(5.11) Romwalter, A., 429(S.46) Rood, A.P., 457(9.73) Rose, H.E., 373(1.23) Rosen, L.J., 55(2.62) Rosenberg, L.D., 229(4.77) Rosin, P., 109,110J11,113(2.104) Rosinski,J., 750(3.41) Ross, S.L., 577(9.222) Rossi, C , 545(6.105) RossiterB.W., 795(3.199) Roth, C.H., 572(9.231) Roth, P., 507(10.343) Rothele, S., 24(1.16), 547(10.138), 552(10.161) Rowe, S.H., 747(3.24) Rubin, B., 505(9.178) Ruckenstein, E., 797(3.168) Rudd,D.R., 575(10.223) Rudin, A., 475(8.24), 479(8.27,8.32), 425(8.35) Rudolph, A., 270(4.3) Ruf. A., 494(9.124) Rumpf, H.,57(2.41), 725(2.126-128), 257,255(5.5), 255(5.23), 259(5.3133), Rupp, R., 55(2.88,89)
Rushton, A.G., 479,455(9.98) Rushton, J. H., 577(9.207) Ruzek, J., 797(3.169) Ryan, H.M., 555(10.84) Sachweh,B., 555(10.88) Sahu, B.K., 757(3.54) Saito,N., 557,549(6.129) Saltzman, W.M., 757(3.59) Samuelson, H., 795(3.204) Sand, J.A., 525(10.8) Sandberg, P., 795(3.180) Sander, L.M., 57(2.76) Sanderson, M.S., 507(9.150-151), 555(10.167) Sansone,E.B., 555(6.81-82) Santoro, R.J., 555(10.84) Sarabia, E.R.F. de, 555(10.270) Sarid,D., 795(3.198) Sarka,B., 502(10.323) Sarmiento, G., 554(6.63), 555(6.72) Sarto, L., 475(9.92) Sasabe,S., 505(9.180) Sasaki, Y., 545(6.102) Sato, E., 429(8.42) Sattler, K., 799(3.207) Sauerbrey, G., 457(9.43), 455(9.47) Saunders, E., 447(8.73) Savaloni,H., 457(9.71) Sawers, J.R., 572(9.229) Sawyer, N.B.E., 757(3.155) Scarlett, B., 55(2.87), 405(8.10), 455(9.38), 499(9.142), 545(10.132), 549(10.142), 557(10.150) Schachman, H.K., 440(8.67,8.70) Schadel,G., 57(2.412) Schaeffer, D.W., 57(2.78) Schaetzel, K., 572(9.230) Schafer,M., 750(3.136) Schaller, R.E., 259(5.36) Schaub,S.A., 505(9.172) Schauer, T., 255(5.102), 554(10.266) Schillar, L., 525(6.32) Schindler, U., 255(5.30)
646 Author index
Schlechter, A. W., 386{1.6\) Schleicher,B., /P(5(3.205) Schlotzer, G., 35(5(7.62) Schmidt, F., 7^2(3.1),7(^(5(3.152-153) Schmidt, K.G., 7^2(3.l),75d(3.152) Schmidt-Ott, A., 742(3.1), 75(5(3.152154), 799(3.206-207) Schmitz, O.J., 554(10.266) Schneider, C.I., 795(3.195), Schofield, P., 554(10.268) Scholtz,S.M., 555(10.58) Schombacher, E.H., 502(9.166) Schone,E., 264(5.15) Schone,F., 503(9.167) Schonert, K., 725(2.127), 530(10.36) Schrag,K.R., 455(9.31) Schraml, S., 607(10.344-345) Schrodl,M., 565(10.199) Schroeter, S.R., 757(3.52) Schubert, H., 755(3.151) Schuman, R., 709,772,773(2.107), 356(7.61) Schunk, J., 252(5.95) Schure, M.N., 273(5.48) Schuster, M., 270(4.3) Schwartcz, H.P., 54(2.57) Schwartz, F.H., 496(9.136) Schwartz, J.A., 497(9.119) Schwechten,D., 23(1.15) Schwedes,D., 566(10.194) Schwenk,W., 530(10.36) Scott D.M., 796(3.201), 577(10.231), 575(10.235), 557(10.246), 552(10.247), 604(10.331) Scott, K.J., 49(1.43), 330(6.44) Scruby, C.B., 573(9.237) Scullion, H.J., 766(3.101) Searles, CO., 479(8.26), 427(8.38) Sebert, E.E., 274,225,247(4.35) Sechter, R. S., 570(9.204) See,C.W., 757(3.155) Segura,L.E., 556(10.270) Seibie, F.E., 440(8.69) Self, S.A., 457(9.51), 450(9.102-103) Semiat,R., 577(9.211)
Sem'yanov,K., 507(9.190) Serkowski, S., 275(4.39) Serra, J., 779(3.126-128) Seselj, A., 565(10.185) Sevick-Muraca, E.M., 603(10.330), 606(10.341) Shane, K.C., 54(2.57) Sharma,A., 552(10.248-249) Sharma, M. M., 577(9.216-218) Sharpe, J.H., 337(6.46) Shekunov, B.Y., 757(3.1556) Shen, J., 375(7.36), 539(10.90,10.92), 546(10.137), 605(10.348) Shenton-Taylor, T., 467(9.70) Shergold, F.A., 226(4.56) Shiba, M., 355(7.75) Shibata, T., 54(2.56,2.59) Shimizu, Y., 529(10.34) Shimomura, G., 434(8.55) Shinde,R.R., 606(10.341) Shiundu, P.M., 254(5.108) Shnellhammer, M., 563(10.166) Shofner, F.M., 467(9.68) Shoji,T., 794(3.185) Shook, C.A., 36(1.29) Siano, D.B., 336(6.83) Sieglmeier,M., 503(9.167) Silebi, C.A., 277(5.40), 273(5.49), 274(5.50-52), 275(5.58,5.60,5.6368), 276(5.70) Silva,S.de, 270(5.37) Silverman, B.A., 536(10.68, 10.72) Simecek, J., 455(9.32) Simmons, M.J.H., 496(9.132-133) Simons, S.J.R., 440(8.65) Sinclair, D., 537(10.79), 542(10.104) Sinclair, I., 403(8.10), 455(9.38), 479,453(9.98) Singh, P., 69(2.9) Singh, S., 274(5.55), 276(5.76) Sinn, C , 607(10.316) Sisson, K., 70(2.11) Skidmore,C.B., 756(3.154) Skidmore, J.W., 790(3.161), 797(3.166)
Author index 647
Skrebowski, J.N., 57^(7.37) Slack, B.W., 333(6.59) Slater, C , 40J(8.12) Sliepcevich, CM., 55(^(10.145-146) Small, H., 272(5.41-42) Smet, J.G.E. de, 545(10.132) Smith, A.L., 570(9.200-201) Smith, G.C., 77/(3.117-118) Smith, J.T., 495(9.127) Smith, T.N., 572(9.227) Smurthwaite, M.J., 4,38(12\ 20(1.14) Smythe, W.R., 454(9.24) Snabre, P., 565(10.189) Sneyer,S., (^07(10.314) Sneyers, R., i97(10.278), 592(10.280) Snow, W.S.,796(3.203) Sokalov, v., 454(8.53) Sokolov,N.V., 244(4.116) Somekh, M.G., 757(3.155) Sommer, H.T., 469(9.81), 483(9.112) Sommer, K., 2(1.1), 770(3.115), 779(3.135) Sommerfeld, M., 502(9.163) Souter, P., JJ2(6.49) Sowerby, B.D., 585(10.269), 556(10.271-273) Spaite, R.A., 267(5.27) Sparks, R., 755(3.150), 550(10.36) Sparks, R.G., 557(10.245) Spencer, D.H.T., 345(6.118), 549(6.119) Spielman, L., 465(9.63) Spink, M., 459(9.50) Springston, S.R., 256(5.114) Sprow,F.B., 577(9.213) Squires, L., 522(6.27) Squires, W. Jr., 522(6.27) Sreenath, A., 500(9.143) Stach, T., 555(10.85-86) Stairmand C.J., 267(5.12), 265(5.13), 265(5.20), 552(7.52) Standish, N., 709,772,777(2.98,2.100) Stange,K., 46,55(1.38) Stanley, F.W., 275(5.44)
Stanley-Wood, N.G., 527(10.17), 531(10.43-44) Staples, E.J., 570(9.200-201), 595(10.300) Staudinger, G., 374 (7.29-30) Steidley, K.D., 454(9.23) Steier, K., 550(10.36) Stein, F., 50(2.34), 266(5.25), 550(6.44) Steinheitz, A.R., 752(3.65) Steinour, H.H., 525,529(6.40) Stenhouse, J.I.T., 479,455(9.98) Stevens, R.J., 556(10.273) Stevenson, A.F., 554(10.49) Stieglmeier, M., 505(9.168) Stimson, B.P., 554(10.268) Stinz, M., 526(10.4), 555(10.60), 552(10.252), 552(10.255), 576(10.225), 555(10.259,10.265) Stock, R.S., 592(10.279) Stockdale, S.W., 227(4.58) Stoists, R.F., 275(5.43) Storey, J., 224(4.55) Stork-Brabent, B.V., 275(4.34) Storti, G., 577(10.232), 555(10.262) Strout,T.A., 575(10.238) Sudal. E.D., 276(5.70) Sudoi, E.D., 275(5.60) Sugarman, G.H., 275(5.44) Sugamuma, G., 465(9.64) Suhm,H.O., 255(4.91) Suito, E., 555(7.73) Sullivan, D.M., 577(9.209) Sun, Z., 605(10.330) Sunago,S., 506(9.181) Sundukov, V.K., 275(4.40) Sunshine, G.A., 572(10.205) Sutherland, D.N., 742(3.4) Suzuki. Y., 505(9.169) Svarovska, J., 402(8.6-8) Svarovsky, L., 705(2.96), 256(5.4), 260(5.9-10), 579(7.46-50), 402(8.4, 8.6-8), 406, 407(8.13), 405, 472(8.11), 472 (8.19), 525(10.2) Svedberg, T., 442(8.75-76)
648
Author index
Svensson, J., 109J12{2A2\) Svensson,K., 7(?P,77i(2.116) Swechten,D., 5dJ(10.163) Swift, J.D., ^9(5(9.134) Swithenbank, J., 55(?(10.143), 55/(10.149), 55^(10.172) Switzer,L., 5^/(9.152) Switzer, L., 5(5<10.169) Sylvester, R.W., ^P5(9.127) Szalkowski, F.J., /P5(3.179) Tadayyon, A., 5i5(10.56), 494{9.\2?>) laggard Jr, R.B., 5/i(9.28) Takahashi,K.,5i 7(10.78) Talbot, J.H., 5^^(10.109-116), 454(9.251 455(9.29) Tamano,K., 511(9.215) Tamm,E., 5/2(9.231) Tanaka, M., 5//(9.221) Tanaka, T., 527(10.23-26,10.32, 10.37-40) Tanigaki, M., 252(5.92-93) Tarjan. G.,/OP, //2(2.114) Tasaka, A., 463(9.64) Tate,C.H.,4P/(9.116) Tatschl,R., 502(10.324) Taubenblatt, M. A., 50P(9.195-197) Tavlarides, L. L., 5/0(9.203,9.205), 5/2(9.226) Tawashi,R.,/P5(3.196) Taylor, A.M.K.P., 505(9.182-183), 507(9.185) Taylor, D.S., 50/(9.154-155), 550(10.143), 55/(10.149) Taylor, N.J.,/P2(3.177) Taylor, R.G., 55^(10.268) Taylor, W.K., /57(3.106) Temple, J.A.G., 555(10.267-268) Terao,Y., 503(9.169) Thaufelder, T.,55(2.86) Thibert,R.,/P5(3.196) Thieme,C., 572(10.211) Thorn, R von, 457(9.431 458(9.41) Thomas, J., 5P5(10.297), 5P5 (10.303), 50/(10.308),502 (10.321)
Thomas, T.R.,/7P(3.132) Thompson, B.J., 535(10.65-69,10.72) Thompson, C.N., ^P/(9.119) Thompson, L.G., 5/0(9.200-201) Thornton, M.J., 452(9.54) Thorwirth,G., 572(10.209) Thudium, J., 349,349(6.122) Thuis, H.H.W., 2/5(4.43) Timbrell, V., /50(3.42-43), /57(3.82), /55(3.99-100), 457(9.72) Todd, W.F., 257(5.27) TodolU.L., 545(10.141) Tokunaga, Y., 355(7.74-75) Tolsi,G., 275(5.85) Tomb, T.F., 450(9.7) Tomkieff, S.L., /52(3.63) Tomlin, CD., 503(10.330) Torii, K.,/54(3.147) Tou, T.Y., /P5(3.203) Townson,P.S.,/54(3.142) Tracey,V.A., 70(2.12) Tracy, B.,/P/(3.170) Trainer, M.N., 5P3( 10.287-288) Trappers, J.L., 275(5.69) Travis, P.M., 35P(7.82) Treasure, C.R.G., 4/5(8.22) Treweek, G.P., 455(9.34) Tribus,M., 550(10.146) Tropea,C., 503(9.167-168) Tsakiraglou, CD., 55(2.82) Tschamuter, W.W., 405(8.17), 419 (8.30-31), 4/P(8.31), 427(8.39), 5P4( 10.291) Tschudi,T., 535(10.71) Tsouris, C , 5/0(9.205) Tsubaki, J., 5/(2.1), 52(2.44-45) Tsuji, T., 54(2.59) Tsujimoto,H., 505(9.180) Tuch,Th., 5/2(9.231) Tuinman, P.C, 35/,34P(6.130 Tuma, J., 543(10.117) Turbit-Daoust, C, 50/(9.152), 554(10.169) Turner, G.A., /P4,/P5(3.192-193)
Author index 649
Turner, T.D., 757(3.83) Tuyet,H.Le., 572(10.204) Tweedie, R., 5(^i( 10.260) Twomey, S., 550, 552(10.148) Tyler, G.A.,5J(5(10.77) Ueharqa,Y.,J^P,i5/(6.129) Uemaki, O., 84(2.59) Uhlherr, P.H.T., 334(6.63% 335(6.72) Umhauer, H., 470(9.84-88), 473(9.94) Ungut, A., 507(9.155) Unterforsthuber, K., 5P5( 10.296) Urick, R.J., 525^0.3) Utsumi,R., 25(^(4.102), 238,239(4.103) Vacassy,R., 555(10.58) Valley, R.B., 467(9.14) Vanderdeelen, J., 427(8.40), (^07(10.311-314) Vanderhallen, F., 568(10.196) Vanderhoff, J.W., 27i(5.43), J52(6.134) Vaussurd,C.,5(5i(10.166) Vaz, C.M.P., 7P(3.202) Vaz,M.H., 7(^4(3.143) VazPaulo, CM.?., 196(3.20^) Veal, D.L., 459(9.80) Veldt. C , i57,i49(6.130) Vendl, M., 433(S.46) Venkatedan, J., 273(5.49), 275(5.60), 276(5.10) Veram,J-M., 565(10.188) Verheijen, P.J.T., 4PP(9.142), 545(10.132), 54P(10.142), 557(10.150) Verhof,F.H., 577(9.222) Vetter,Q.V., 70(2.11) Vick,F.A., 7(57(3.104) Vigneau, E., 75P(3.84) Vilimpoc, v., 602.(10.323) Vincent, J.H., 500(9.143) ViswanathanK., ^5(2.91) Vogel, K., 607(10.345) Vouk,V., J 72, J7J(7.19)
Wade, R., ^2(2.45) Wadell H., 76(2.22-24) Waggeling,R.,4P6(9.136) Wahl, W., 27J(4.33) Wahlund, K.G., 252(5.96), 283(5.100) Walawender, W.P., 57(2.70,2.74) Waldie,B., 507(9.188) Waldvogel, A., 470(9.90) Wallace, T.P., 544(10.123-124) Wallach, M.L., 5J4(10.49) Walls, J.M., 5J6(10.70) Walls, J.M., 556(10.74) Walton, C.W., 340(6.96) Walton, W.H., 75i(3.66), 157,166,190(3.SI), 767(3.103), 189J90(3.\60) Wan, M., i7J(7.28) Wang, J.C.F., 507(9.193) Wang, N., 546(10.137) Ward, J., 556(10.68) Ward-Smith, R.S., 74(2.19), 566(10.191) Ware, R.E., 506(9.177-178) Washburn, E.D., 547(6.98) Watanabe, S., 505(6.13) Watano,S., 755(3.150) Water, E.R.,4P5(9.127) Waterhouse, J., 576 (10.223) Waterson, R.M., 606(10.340) Watson, H.H., 754(3.69), 755(3.72), 757(3.78) Weatherby,E. J., 565(10.180) Weaver, W., 50(6.2) Weber, M., 227(4.60) Webster, C.B., 526(10.6) Wedd, M., 74(2.19), 566(10.191) Weibel, E., 7P6(3.200) Weichert, R., 775(3.123), 575(7.26), 42P(8.44-45) Weiland,W., 530(10.42) Weilbacher, M., 265(5.23)
650
Author index
Weiner, B.B., 408(SA7% 4I8(S.24% ^/P(8.30-31), 42^(8.35), ^27(8.39), 587(102141594(10.291) Weinig, A.J., /^P,l 13(2.118) Weiss, E.L., 5PJ( 10.287) Weiss, M.,^PP(9.142) Weitz, D.A., d(?J( 10.326) Welford, G,.A., 143(3.62) Welsch, T., 274(5.56) Wen, H.Y., 88(2.19) Wensing, M., 5J5(10.85) Wensing,M., 55(^(10.86) Wentworth,C.K., 25(1.20) Werner, D.,i5P(7.81) Wemet, M.P.,50i(9.174) Wertheimer, A.W., 554(10.153) Wesley, R.K.A., ^55(9.28) Wessel, J., 25/(5.2) Wessely,B., 555(10.60-61) West, R., 45P(9.50) West, R.M., 25(^(5.8) Wettimuny, R., 88(2.S3) Whalley, W.B., 84(2.60) Wharton, R.A., 38(\.33% 348(6.\\6) Wharton, R.T., 455(9.41) Whitby, K.T., 218(4.46), i5P(7.83-84) White, E.W.,7P4(3.194) Whitelaw, J.H., 50(5(9.182-183), 507(9.IS5) Whiteman, M., 88(2.92), 244(4.1 \1) Whiten, N.J., 52(^(10.29) Whitmore, R.L., 328(6.31), 330(6.45) Wichman,H.E., 5/2(9.231) Wiesendanger, R., /PP(3.208) Wightman,C.,/7P(3.134) Wiittig, S.,50J(9.167) Wilcock,W.L., 554(10.153) Wilder,J.,/7P(3.134) Wilgers,W.L., 25(1.20) Wilkes, R., 545(6.115) Wilkinson, D., 5(55(10.199), 555(10.260) Will, S., 575(7.32-34), 607(10.344345) Willard,R.J.,/P4(3.193)
Williams, D.J., /P2(3.171) Williams, M.C., 245(4.107) Williams, P.S., 277(5.78), 256(5.115,118) Williams, R.A., /54(3.142), 260(5.SX 42(5(8.35), 45P(9.50) Williams, R.C., /P/(3.164), /P2(3.175) Willman, M.,5(?5(9.167) Wills, B.G., 5/5(9.238) Wilson, J.B., 557(10.80) Wilson, J.D., 50(5(9.17) Wilson, L.R., 275(5.44) Wilson, R.,/45, /70(3.21), 455,4(55(9.48) Wintz,R,/72(3.122) Witt, W., 24(1.16), 547(10.138), 5 7P( 10.239) Wollny, M., 470(9.89) Wood, C.P., 575(10.215-216) Wood, H.C., 54(2.61) Wood, R., 467(9.71) Wood, W.M., 552(6.136), 463(9.51) Wooldridge, W.D.S., 54/(6.97) Woolf,A.R., 552(6.132) Worlitschek,J.,4P4(9.124), 4P5(9.126) Woznicki,B., 572(10.205) Wreidt,K., 502(9.164) Wreidt, T., 502(9.165), 505(9.170, 9.175) Wu,C.,5P5( 10.296) Wu,J.S., (50/(10.310) Wu,K., 455(9.113-114) Wu,Y-S.,5P5(10.302) Wyatt, P.J., 254(5.106), 255(5.110) Wyckoff, R.W.G.,/P2(3.175) Xu, T.H., 502(9.163), 505(9.168) Yamaguchi, K., 54(2.56,2.59) Yamamoto, H., 25P(4.102), 255,25P(4.103) Yang, F.J., 275(5.83) Yau, W.W., 27(5(5.75), 25/(5.89-90)
Author index 651
Yawaza, N., 4J4(8.55) Yeager,E., 5^5(10.256) Yildirim, K. J7P(7.48) Yokoyama, T., ^5^(8.54,8.56), 506{9An) York, P., JiP(6.89) Yoshida, T., 303{6A\ 388(1.14-15) Yoshikawa,S., 506(9.180) Yoshiyama, H., 473(9.96) You, H.K., 196(3.203) Youichi,T.Jun., 505(9.169) Young, I.T., 144(3.1) Young,;., 7(57(3.108,3.110) Yousufrai M.A.K., 225(4.51), 2^<4.115) Yu, A.B., 109,112J 17(2.9S,2-l00l 709(2.101) Yu,L., 779(3.131) Yule, A.J., 507(9.154-155), 555(10.76)
Zackariah, K., 79^795(3.192), Zaki, W.N., 52<^,550(6.38) Zalderns,N.G., 257(4.96) Zaltash, A., 88(2.S0) Zare,M., 502(9.160) Zattoni, A., 27^(5.85) Zbuzek,B., 797(3.169) Zeisler,R., 552(6.137) Zeng,R., 200(3.210) Zhang, J., 2^(5(5.118) Zhang, J.Y., 575(10.218-220) Zhang, Y., ^59(9.50) Zhu,J.H., (505(10.326) Zhukov, A.N., 218(4.40) Ziema,M., 502(9.163) Zimmerman, I., 555(10.155) Zollars, R.L., 55<10.52) Zufall,J.H., 557(10.81) Zwicker, J.D., 27^4.36)
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Subject index The numbers shown in bold indicate that a section about the subject commences on that page. Absolute light scattering photometer, 544 Accusizer, 563 acoustic spectrometer, 579 acoustopheretic titrator, 578phase Doppler anemometry, 538 Acoustophor 578, Acoustosizer 584 Advanced Fiber Information System, 466 aerodynamic time-of-flight measurement, 496 Aerometrics; Eclipse, 473, 501 agglomerates, 336 aggregates 336 air jet; sieve 227, sieving 237 Alpine; 228, wet sieving device 235, long-arm centrifuge 435, multi-plex zig-zag classifiers 267 American Innovation Videometric 150 178 AMF 180 Amtec spectrophotometers, 596 Analysette; 8 266, 9 269 analytical cut size, 256 Analytical Measuring Systems, 178 Andreasen; 306, pipette 365 anomolous scattering', 545 Anton Paar vibrating tube densitometer, 535 aperture tubes, 465 API Aerosizer, 563 arithmetic; mean 67, normal distribution 96 Artek 179 asbestos; 150, 466 ATM Sonic Sifter 238
automatic sieving machines, 529 Automatix 179 Autometrics 36, 526, 576,Differential I light scattering photometer, 544 average diameters 63 back-scatter intensity, 604 Bahco classifier 80, 266 balances; Sartorius 387, sedimentation 387, Cahn 388 Bausch & Lomb, Omnicon 172 Beckman Coulter; RapidVUE 167, 464 LS series, 554, Model N4, 597 Beckman spectrophotometer, 602 Beer Lambert law 367, 375 Berg's conoidal centriftige tubes 431 Beta-ray attenuation, 527 BI-DCP disc (photo)centrifuge particle size analyzer, 428 Blaine permeability, 530 Boeckeler 179 Boltzmann constant, 343 borescope, 604 Brice-Phoenix light scattering photometer, 544 Brinkmann, 491 Bristol Engineering; 13, Isolock sampler, 535 Bristol Industrial Research Association, 503 British Standard Graticule 155 Brookhaven; BI-XDC 379, 90 Plus, BI-200SM, BI-200SM goniometer, BI-90, BI-Foqels, ZetaPlus, 597 Brownian motion 273, 535 Buehler Omnimet 179
654 Subject index
Cahn micro-balance 388 calibration; 153, of image analyzers, 170, of sieves, 220, 460, materials 462, curve 470 Canty Vision, 473 capillary hydrodynamic fractionation 275 capillary zone electrophoresis 276 Carl Zeiss 182 Carman-Kozeny, 530 cascadograph 241 cathodoluminescence 191 centrifuges; Alpine long-arm 435, DuPont/Brookhaven scanning x-ray disc 407, Ladal x-ray disc 406, pipette disc 412 Chatfield comparator 164 chromatography; hydrodynamic 272, potential barrier 274, size exclusion 274, 276 chute splitting 27 Cilas, 555 classification by decantation 287 classifiers; water 261, 264, efficiency 251, gravity counter-flow 261, Bahco microparticle 266, centrifugal counter-flow 266, Alpine zig-zag 267, zig-zag gravitational 267, zig-zag centrifugal 267, cross-flow gravitational 268, Warmain Cyclosizer 268, Humboldt 268,cross-flow centrifugal 269, Donaldson Acucut 269, cross-flow elbow 270, Micromeretics 270 Climet, 473 Coanda effect 270 coefficient of variation, 130 coincidence, 458 Collimated Holes Inc, 463 comminution, 524 Compix C-Imaging 1280 System 179 concentration effects; low 333, high 334
concentration monitors, 603 coning and quartering; 9,26, a paste 36 contact angle 340 Contamination Control Systems, 474 CONTIN 427 controlled reference method, 593 correlation techniques, 531 Coulter; Counter 82,177,260, principle, 345, 448, pulse shape 456,Beckman 464, LSI GO, 555 counter-flow classifiers, 527 Courier 300, 527 crossed lasers, 606 cross-flow centrifugal 269 crystallizer, 524 curve fitting, 123 cut velocity 263 cuvette photocentrifuges 429 cyclone, 525 Cyclosensor, 528 Dage-TI camera 177 Danfoss; QueCheck Vision System 183, VisionSensor475 Dantec Particle Dynamic Analyzer, 503 dark field microscopy 184 data interpretation 125 Data Translation 180 Dawn, Wyatt 284, Eos, 600 deagglomerating 342 decantation 287 density; effective 347, true 347, apparent 347 Denver Autometrics PSM, 526, 576 Denver Equipment Company 36 diameters; definitions of 59, mean sizes 66 diameters; Martin 58,61,152, Stokes 58, definitions 59, projected area 60,Feret, 60,61,unrolled 61, harmonic mean 61,68,152, average 63, mean 66,67, arithmetic mean 67, perimeter, 152, projected area 152, equivalent 209
Subject index 655
differential phase-Doppler anemometry, 502 Diffusion wave spectroscopy (DWS), 603 Digiplan, 164 Dilation 174 Dipix 1440F power scope imaging microscope 185 disc centrifugation 271 disc photocentrifuges 418 dispersing; agents 338, quality tests 344 Dispersion Technology DT-100 acoustic spectrometer, 579 dispersion; dry powders 336, wet, 338, quality; tests 344 distribution modulus, 524 distributions; arithmetic normal 96, log-normal 100, bimodal intersecting 117, non-intersecting 122 divers, 385 Donaldson Acucut classifier 269 drag; factor 295, coefficient 298 drop size distribution, 510 droplet size in sprays, 538 dry powder dispersion 336 Duke Standards 223 Dupont electrolytic grain size analyzer, 511 DuPont/Brookhaven scanning x-ray disc centrifiigal sedimentometer 407 dwell time, 491 dynamic light scattering 588, 595 electro-acoustic; Matec 285 electroformed micromesh sieves 214 electron microscope photomicrographs 62 electrophoretic light scattering, 595 electro-viscosity, 335 elongation ratio, 71 elutriators; 261, multiple stage 261, theory 262, water 264, Andrews
264, Roller 265,Gonell 265 miniature 265, air 265, Endecott; EFL 2000 232, Octagon 233, Star 2000 233 energy of immersion 340 equiprobable size 256 Erdco Acoustical Counter, 513 erosion 173 external gradient method 419 extinction coefficient 367, 370 Fairs graticule 154 Faley Status, 475 felvation, 243 Feret diameter 60, 61, 152 FFFractionation 283 fiber; 176,178, length, 466, diameter 317 Field flow fractionation; 277, sedimentation 278, centrifugal 279, time delayed exponential 279, thermal 282, magnetic 282, flow 282, steric 284 flakiness ratio 71 flatbed scanners 183,469 flow ultramicroscope, 509 Flowvision Analyzer, 475 form and proportions 70 forward/backward intensity ratio, 538 Fourier descriptors 84 fractal dimensions; 84, surface 88, analysis 194 fractionation; 271, field flow 271, 277, capillary hydrodynamic 275, sedimentation field flow 278, centrifugal field flow 278, timedelayed exponential 279, thermal field flow 281, flow field flow 282, steric field flow 283, split flow thin cell 285, centrifugal field flow 278, continuous split, 285 Fraunhofer theory, 544 frequency domain photon migration, 606 Fritsch Analysette, 556
656 Subject index
Galai; 182,476 Gallenkamp; Gallie-Porritt apparatus 237, balance 387 gas flow permeametry, 530 Gaudin-Schuman, 524 Genias 177 Gilson; 223, GA-6 Autosiever 238, Compu-Sieve analysis system 241 Glen Creston rotary sampler 32 Microscal suspension sampler 37 glidants, 337 Global Lab Image 180 globe and circle graticules 154 Gonell elutriator 265 grade efficiency 252 Gradex 240 granulator, 524 Granumeter 389 graticules. Fairs 154, globe and circle 154, linear 154, Patterson and Cawood 154, Watson 155, May's 155, Gustafson, 32 HamamatsuC-1000 180 hand sieving 229 harmonic mean size 68 Hauser-Lynn supercentrifuge 440 Helos, 546, 562 Hey wood's ratios 71 hydrocyclones, 528 HiacPA720 177 Hiac/Royco, 468 high order Tyndall spectra, 542 hindered settling 332 Hitech Olympus Cue-3 180 Holography, 536 Horiba; cuvette photocentrifuge, 429, CAPA-700 429, 430, CAPA-300 430, LA-300, 556, LB-500, 597 Hosokawa; Mikropul Micron Washsieve 237, Mikropul Sedimentputer 434 Mikropul ESpart Analyzer, 504
Humboldt particle size analyzer 268 hydrodynamic chromatography; 272, porous wall 274 hydrodynamic focusing, 456, curve, 470, 472 hydrometers and divers 380 ICI x-ray sedimentometer 376 impact size monitor, 512 in-line camera, 604 Insitec; 245, ensemble particle concentration sSize, 568 in-situ sensors, 472 interferometers, 507 intersection cut size 257 ISPA image analysis system, 509 Jeffrey-Galion 88 Jenoptik PSI-Z, 572 Johnson's S^ distribution 109 Joyce Loebl Magiscan 180 Kamack equation 406, 420 Kane May, 477 Kemsis K200 373 Kohler illumination 157 Kratel, 477 L.U.M. LUMiFuge™ 430 Labcon automatic sieve system 241 Ladal centrifuge; x-ray disc 406, pipette disc 412 laminar flow region 58 Lark counter 166 Lasentec, 492 laser Doppler velocimetry, 500 laser induced incandescence, 607 laser phase Doppler principle, 500 law of compensating errors 115, 117 Le Mont Scientific B-10 system 192 Leco Tri-laser, 598 Leco, 180 Lecotrac ; 557, LTU-150, LTU-251, 598
Subject index 657
Leeds and Northrup Microtrac; 559, Ultrafine Particle Analyzer, 598 Leica; Quantimet 500 180, Confocal laser-scanning microscopy 182 LeMontOasy 180 light blockage, 468 light diffraction, 543 light pressure drift velocity, 511 limit of resolution 146 linear eyepiece graticules 154 line-start; incremental centrifugal sedimentation 422, mode 418, theory 425 liquid viscosity 350 logarithmic line array detector,, 551 log-normal distribution 100 longest dimension 152 low concentration effects 333 lower size limit 302 Mach Zehnder, 507 machine sieving 227 Magiscan 177 magnetic field flow fractionation 282 magnetic suspensions 346 Malvern ; Sysmex 168, 464, 478, Mastersizer 260, Ultrasizer SV: 578, (Insitec) Ensemble Particle Concentration Size 568, Autosizer Hi-C, System 4700 598, Zetasizer II, 599, Auto-sizer 4800 599, Mastersizer, 560, compact goniometer system, Malvern High Performance Particle Sizer (HPPS) 600 mass balance, 462 Mastersizer, 559 Matec Matec; CHDF-1100 275, electroacoustic system 285, Acoustosizer 584 May's graticule 155 Maztech Microvisions Spy grain grade 181
mean size; 63, arithmetic 67, harmonic 68 mean, 68 means of distributions 128 Measuremouse 179 median 63 mercury intrusion/retraction 88 mercury porisimetry 151 Messetechnik, 485, 495 Micromeretics; OptiSizer 167, classifier 271, Flow Sizer 5600 274, Sedigraph 5000, 377, Elzone 5370, 464, Saturn DigiSizer, 561 Micromerograph 389 microsample splitter 148 Microscal suspension sampler 37,38 microscope examination 345 microscopy, optical 145 microspheres 223 Microtrac ; UPA 198, Ultrafine Particle Analyzer 558 Mie theory, 544 milling, 525 Millipore TIMC System 181 miniature elutriator 265 miniDawn Tristar, 600 mode 63 moment of a distribution 126 Monitek acoustic particle monitors, 470 Monitek, 603 MSA Particle Size Analyzer 389,439 multi angle; light scattering 284, laser light scattering 544, measurements, 594 Nachet 1500 181 National Standards 352 near infra-red spectroscopy, 576 nephelometers, 604 neural networking, 568 Newark sieve cloth 242 Nicomp, 600 Nikon 181 Nitto, 561
6^55 Subject index
normal probability function 43 Nuclepore filter, 454 Oncor Instrument Systems 181 on-line microscopy 182 optical back-scattering, 539 optical density 368 optical disdrometer, 469 optical incoherent space frequency analysis, 572 OptomaxV 179, 181 Optovar 177 Opus, 579 Otsuka Photal, 600 Outokumpu Imagist 181 Oxford VisiSizer 168 Paar Lumosed 374 Pacific Scientific Hiac/Royco, Met One, 479 Palas PCS, 472 particle measurement, 504 Particle Measuring Systems, 483 Particle Sizing Systems; 485, Accusizer, 563 particle, dispersion 68, shape 69,127 Partikel Messtechnik, 485 Pascal; turntable sample divider 31, Inclyno 227, 232 pattern recognition 70 Patterson and Cawood 154 Pen Kem Acoustophor 111, 578, acoustopheretic titrator, 578 perimeter diameter 152 permeametry; 530, gas flow Pharma Vision 830 181 phase contrast microscopy 184 phi-notation 136 photocentrifliges; 417, disc 418, Brookhaven BI-DCP disc 428, cuvette 429, Shimadzu SA-CP2-10 427, SA-CP3, SA-CP4 430, Horiba cuvette 429, CAPA-300 430, CAPA-700 429, 430 photo-etching process 214
photomask reticules, 565 photon correlation spectroscopy, 586 photon migration, 603 photosedimentometers; 371, commercial 373, Kemsis K200 373, Retsch Paar Lumosed,, 374 photosensitive silicon detector, 551 pigment formation 87 pipette method of Andreasen 365 polar signature 63 polarization intensity differential scattering (PIDS) system, 554 polarized light microscopy 185 Polymer Laboratories 274 Polytec, 487 pore wall roughness 88 powder density, 347 prismoidal ratio 72 Proassist, 576, 577 Procedyne, 496 projected area diameter 60, 152 pulse displacement technique, 574 pyknometer; liquid 347, centrifugal 348, gas 349 Quantachrome Microscan 379 Quantimet 81,170 quantitative image analysis 169 QueCheck, 475 Rayleigh region, 539 Rayleigh-Gans region, 540 reference material 223 replica and shadowing techniques 190 resolution; 173, of sedimenting suspensions 362 resolving power 146 Retsch; sample divider 31,32, Camsizer, 168, wet sieving machine 235, water jet sieve 236, ultrasonic sieving apparatus 241, Accusizer, 563 Reynolds number 295, 298, Rion, 487 Rosin-Rammler distribution, 524
Subject index 659
Rotex Gradex 240,242 sample dividers; 30, commercial rotary sample, selection 2, gross 3, 50, laboratory 3, preparation 147 sampler, point 14, auger 15, snorkel type 15, constant volume 21, fullstream trough 21, slide-valve 22, for screw conveyor, 23, diverter valve, 24, moving flap 24, oscillating hopper 30, oscillating paddle 30, miscellaneous 30 sampling, errors, 2, golden rules of 6, spears 7,8, stored non-flowing material 7, from heaps 9, from bags and drums 11, flowing streams 12, from a hopper, 22, from screw or drag conveyors 34, from a conveyor belt, 13, from falling streams 14, 16, dusty material 19, In-line 23, scoop 25, Table sampling 27, from screw or drag conveyors, 34, slurry 35, by increments 51 Sartorius balance, 387 Saturn DigiSizer, 561 scanning electron microscopy 191 scanning flow cytometry, 506 Sci-Tec PartAn video Image Analyser 169 Sedigraph, 81, 82, 260, 306, 378 Sedimage 1000 373 sedimentation; image analysis 374, balances 387, columns 388 Sedimentputer 434 Seishin; Robot Sifter 239, 430, 561 self organized sieves 243 self-burrowing probes, 8 separation factor 395 Sepor Haultain Infrasizer,265 shadow Doppler velocimetry, 505 shape discrimination, 604 shape; descriptors 63, 84, coefficients 74, factors 76, 78, coefficient, surface-volume 79 indices 82,
regeneration 83, analysis 88, separation 244 Shapespeare Corporation's Juliet 181 sharpness index,257 Shimadzu SA-CP2-10 427, SA-CP3, SA-CP4 430 Shimadzu, 562 sieve; size 209, standard 210, tolerances 212, woven-wire and punched plate 213, electroformed 214, calibration 21,cascadograph, 242 sieving; amount of sample required 229, errors 224, hand 230, wet 235, by machine 231 Simcar pipette disc centrifuge 403 Sinclair-Phoenix aerosol photometer, 543 size exclusion chromatography 274, 276 size limits for gravity sedimentation 300, upper 301, low 302 size modulus, 524 size; geometric mean 67, mean 102, mode 102 sizing 251 SKC-2000, the SKC-3000 and the SKC-5000, 430 sloping trough cutter 34 slurry sampling 35 small angle x-ray scattering (SAXS, 575 sonic sifter 227, 238,239 sorting; by shape, 88, 25 spectral transmission and extinction, 607 spectral turbidity, 602 Spectrex, 488 Spectroscopy photo-acoustic (PAS) and photo-thermal (PTS), 605 sphericity 76, 316, 320 spin liquid 419 spinning rifflers 28 spreading coefficient 340 stacking of sieves 214
660 Subject index
standard deviation; 69,129, geometric 69, 101 standard powders 350 standard reference material 223 standard sieves 210 standards 144 state of.polarization, 537 static noise measurement, 525 stepped centrifuge 438 stereology 151 stereophotogrammetric 156 Stokes diameter 58 Stokes' law 300 Stork Screens Inc 223 stream sampling cup 17, ladles 18 streaming 418 surface shape coefficient 369 suspension stability 343 Sympatec; laser measuring device 23, Helos 562, Opus, 579 synthetic fibers, 545 Talbot; diffraction size frequency analyzer, spacial period spectrometer, 543 temporary slide 148 terminal velocity 300, 306 test sample 3 test sieving procedures 232 texture 84 Thermo Systems Incorporated, 496 through dynamic light scattering, 588 tolerances for standard sieves 212 Tracor Northern 181, 193 transient electric birefringence, 605 transient turbidity, 535 transition region 319 transmission electron microscopy 186 transmission fluctuation spectrometry 375 transmission fluctuation spectroscopy, 539 transmission wide field phase contrast microscopy 185 traversing cutters 19
Trydin image analysis system 176 TSI Liquitrak, 508 turbidimeters, 603 turbidity measurements 371, 532 Turbiscan, 608 Turbo-Power Model TPO-400 in-line grain size analyzer, 603 Tyler Ro-tap 227, 232 ultracentrifuge 442 UltraPS, 585 ultrasonics; 228, sieving 241, attenuation, 526, 576 Ultraspec, 577 unrolled diameter 61 Veco216 velocity spectrometry, 585 vertical pipe cutter 34 Vezin type splitter 11,52 Vibrosonic241 Videoplan-Vidas-Ibas 182 viscosity,350 volumetric sensors, 472 Vorti-siv 233 Wadel sphericity factor 83 Warmain Cyclosizer 268 Watson eyepiece 166 weight of sample required 50 wet powder dispersion 338 work; adhesion 339, cohesion 339, adhesion and cohesion 341 Wyatt; Dawn 284, QELS, 600 x-ray; gravitational sedimentation 375, 376, attenuation and fluorescence, 527, centrifuge, 564 Zeiss AxioHOME 166 Zeiss universal microscope 177 Zeiss-Endter analyzer 164 zeta potential, 595
