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Investigations into methods of photometric measurement of surface area and average diameters of fine.. Charles, Richard Joseph Thor 1949

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UNIVERSITY OF BRITISH COLUMBIA DEPARTMENT O F  MINING AND METALLURGY  -INVESTIGATIONS INTO METHODS OF PHOTOMETRIC MEASUIIEMENT OF SURFACE AREA AND AVERAGE DIAMETERS OF FINE PARTICLES  Richard Joseph Thor Charles  A thesis submitted i n p a r t i a l f u l f i l l m e n t of the requirements f o r the degree of MASTER OF APPLIED SCIENCE in the department of MINING AND METALLURGY  The University of B r i t i s h Columbia  October, 1949  ABSTRACT The problem of surface area and p a r t i c l e size determination has been examined.  A survey of the methods for  measuring the surface areas and average sizes of aggregates of fine p a r t i c l e s was made and a method selected for Improvement. An apparatus employing the principles of fine p a r t i c l e measurement established i n turbldimetry was constructed. Experimental work with the apparatus was conducted i n an attempt to establish a single method of p a r t i c l e size measurement. The results warrant the conclusion that the apparatus w i l l measure average surface diameters of single component materials, constant assaying materials, and mixtures containing one semi-transparent material.  A value for the  r e l a t i v e surface area of these materials can also be determined.  INDEX  -I  II  INTRODUCTION  1  PRACTICAL AND THEORETICAL CONSIDERATIONS  OF THE PHOTOMETRIC METHOD  6  III  THE APPARATUS  13  IV  PREPARATION OF SAMPLES FOR CALIBRATION  20  V  EXPERIMENTAL RESULTS  31  VI  CONCLUSIONS  42  VII  ACKNOWLEDGEMENTS  .45  VIII BIBLIOGRAPHY  46  LIST OF ILLUSTRATIONS  Figures 1  B e l l ' s Apparatus  2  Reflectance C o e f f i c i e n t versus Refraction Index  Page 9  10  3-5  Author's Apparatus  15-16  6  Schematic Diagram Of Author's Apparatus  16  7  Electronic Balancing C i r c u i t  18  8  Spectral S e n s i t i v i t y Graph of 929 Phototube  18  9  Anode C h a r a c t e r i s t i c Curves  19  10  Frequency D i s t r i b u t i o n Graphs Of Size Groups  25  11  Shape Factor D i s t r i b u t i o n Graph  25  12-17 Size Groups Of Quartz  27-28  18-23 Size Groups Of P y r i t e  29-30  24.  Quartz C a l i b r a t i o n Graph  33  25  Chalcopyrite C a l i b r a t i o n Graph  33  26  P y r i t e C a l i b r a t i o n Graph  34  27  C a l i b r a t i o n Graphs Corrected For S p e c i f i c Gravity  34  28  Quartz-Chalcopyrite Mixtures C a l i b r a t i o n Graph  36  29  S p e c i f i c Surface Area Graph  42  INVESTIGATIONS INTO METHODS OF. PHOTOMETRIC MEASUREMENT OF SURFACE AREA AND AVERAGE DIAMETERS OF FINE PARTICLES  I - INTRODUCTION  The measurement of f i n e l y divided material i s a problem f o r which there has been no single general s o l u t i o n .  P a r t i c u l a t e material  can have such varied physical c h a r a c t e r i s t i c s that i t i s doubtful i f any single method of measuring aggregates of p a r t i c l e s w i l l ever be developed. In the selection o f a method f o r p a r t i c l e size measurement, the type of material,  the size range of the material,  and the intended  use of the measurement must be considered. Insofar as the selection i s affected by the type of material, p a r t i c l e shape i s the most important physical c h a r a c t e r i s t i c to be considered.  P a r t i c l e size i s not an absolute term but must be defined  i n r e l a t i o n t o the r e g u l a r i t y of p a r t i c l e shape. of p a r t i c l e s can be treated  In order that aggregates  s t a t i s t i c a l l y i t i s necessary that  irregular  shapes be considered as equivalent geometric shapes.  The equivalent  p a r t i c l e diameter, which i s generally referred to as  the p a r t i c l e  diameter, i s considered as the diameter of a hypothetical sphere or c i r c l e which has at l e a s t one physical c h a r a c t e r i s t i c equal to the corresponding c h a r a c t e r i s t i c of the related p a r t i c l e .  I t i s readily  seen that as p a r t i c l e diameters based on the length, surface, or volume of a p a r t i c l e need not have the same numerical value the inherent p a r t i c l e shape of a material must be considered before a method of p a r t i c l e s i z e  2  measurement can be selected. Two methods o f p a r t i c l e size measurement that are directlydependent on p a r t i c l e shape i l l u s t r a t e the importance of t h i s characteristic.  I f , f o r instance, a mixture of two types of materials,  one having a compact, granular form and the other a needle-like structure, were screened i n an attempt to determine the size d i s t r i bution, the r e s u l t s would have l i t t l e meaning as the needle-like structure of the one material would prevent i t from passing through i n t e r s t i c e s apparently larger than those necessary.  On the other hand  i f sedimentation rates were the sole basis f o r measurement, s i z e determinations of materials i n which the p a r t i c l e form varied greatly and e r r a t i c a l l y from the spherical form could not be r e l i a b l e .  Methods  involving sedimentation are dependent on the use of Stoke's law which states that the diameter of a p a r t i c l e i s proportional to the square root of the rate of s e t t l i n g .  Stoke's law, however, holds true only  when the p a r t i c l e s dealt with are spheres and can be used only as an approximation even when the degree of deviation from s p h e r i c i t y i s constant. The e l e c t r i c a l , magnetic, and o p t i c a l properties, and the s p e c i f i c gravity of a material must also be considered before a method of p a r t i c l e size measurement i s selected. An equally important f a c t o r i n the s e l e c t i o n of a suitable method of p a r t i c l e size determination i s the size range of the material to be examined.  The f i e l d of f i n e p a r t i c l e technology i s generally  considered to deal with p a r t i c l e s of which the upper s i z e l i m i t i s 10^" microns and the lower size l i m i t i s 10"  1  microns.  Screening  procedures are f a i r l y s a t i s f a c t o r y f o r size measurements of p a r t i c l e s between the l i m i t s of 10^" and 50 microns.  Sedimentation, microscopy,  3  turbidimetry and elutriation are accepted methods for size determination in the narrow range from 1 to 50 microns. For particles that are in the size range below 1 micron the centrifuge and the electron microscope are widely used to determine size-distributions. The above methods of size determination are not completely restricted to the size ranges given as i t is often possible to obtain experimental results in one size range and extrapolate these results, by means of smooth curves or other acceptable mathematical procedures, into the ranges bounding the initial range. However i t is apparent that the complete size range with which particle size technology is concerned, is too large to be encompassed by one single general method of particle size measurement. Finally, i t is essential that any method for determining particle size be chosen with some regard for the intended use of the measurement.  It has already been indicated that a number of values  could be obtained for the equivalent particle diameter depending on whether i t was determined on a length, area, or volume basis. If i t is desired to obtain a particle diameter that is a measure of the surface then a method should be used which determines particle size by measuring particle surface. Similarly, i f i t is desired to obtain a particle diameter that is a measure of the volume then a method should be used that determines particle size by measuring particle volume. There are two other methods of arriving at equivalent particle diameters, each method giving an equivalent particle diameter expressed in terms that will serve a specific purpose. One of these methods is dependent on a volume-surface basis and the other on a weight basis.  4  Most methods of p a r t i c l e s i z e measurement, including sedimentation,  e l u t r i a t i o n , and centrifuging, give equivalent p a r t i c l e  diameters as a function of volume.  Microscopy may give equivalent  diameter as a function of length or area.  Turbidimetry gives the  equivalent diameter as a function of area. I f i t i s desired to obtain a value f o r s p e c i f i c surface of a material i t i s not necessary t o be concerned with p a r t i c l e diameter. Methods involving s o l u b i l i t y and gas adsorption are the best means f o r determining s p e c i f i c area but they give no i n d i c a t i o n of the s i z e distribution.  However, i n studying rates of oxidation and s o l u t i o n ,  c a t a l y s i s , adsorption, crushing and grinding, covering power and other s i m i l a r operations, s p e c i f i c surface i s the prime c r i t e r i o n , whereas, 5/-ze p a r t i c l e &haps i s r e l a t i v e l y unimportant.  The reverse i s true i n the  study of ceramics, pigments, abrasives, cements, and other s i m i l a r f i e l d s . I t i s thus apparent that a great many factors must be considered before a method of determining average p a r t i c l e s i z e , s i z e - d i s t r i b u t i o n , or s p e c i f i c surface area can be selected.  The problem of p a r t i c l e size  measurement, i n general, can become very complex i f the materials dealt with have i n d e f i n i t e and abnormal physical properties and i f the range of sizes encountered i s large. A method of p a r t i c l e size measurement that i s rapid, accurate and suitable f o r a wide range of materials and s i z e s , i s desired i n the research laboratory and i n industry.  The method should be simple and,  therefore, not require the services of s p e c i a l l y trained technicians. I t should be capable of determining a l l types of average p a r t i c l e sizes, s i z e - d i s t r i b u t i o n s , and i f possible, s p e c i f i c surface area.  As suggested  previously, i t i s u n l i k e l y that a method of p a r t i c l e size measurement  5  could f u l f i l l a l l of these r e q u i s i t e s .  However, the methods i n use at  present can be improved and extended so that they may be applicable to an ever widening range of conditions. A survey of some of the disadvantages of the common methods of p a r t i c l e size determination indicates where improvement i s necessary. Sedimentation,  e l u t r i a t i o n and the ordinary turbidimetry  methods of p a r t i c l e size determination are dependent on the laws of s e t t l i n g hypothesized  for truly spherical particles.  P a r t i c l e shape  i s often involved i n an indeterminate manner and thus a f f e c t s the results.  Further, these methods tend to be tedious and require con-  siderable knowledge and  skill.  P a r t i c l e size determination by microscopy i s tedious and exacting, and i t i s often impractical to examine more than a very minute part of the whole sample. Centrifuging has the great disadvantage that i t i s applicable only to the sub-micron p a r t i c l e size range. Solution and adsorption methods of surface area are time consuming and require considerable apparatus and  determination skill.  Permeability methods of surface area determination are greatly affected by p a r t i c l e shape and are consequently  subject to serious e r r o r .  As i t would appear that the best approach to the problem of improving the methods of f i n e p a r t i c l e measurement would be the developement of a method which inherently measures surface area but which gives r e s u l t s capable of being interpreted as average p a r t i c l e size^and as s o l u b i l i t y , adsorption, permeability and turbidimetry methods meet the above s p e c i f i c a t i o n s , turbidimetry was selected f o r i n i t i a l  study  since i t appeared the simplest of the four. An apparatus that would measure, with a high degree of accuracy,  6  the surface area of the powders of a few of the simple minerals has been described by John W. B e l l  . His apparatus consisted, e s s e n t i a l l y , of  a photometric device that would measure the t u r b i d i t y of f i n e p a r t i c l e suspensions.  Turbidity, i n t h i s case, was considered as a function of  surface area. B e l l ' s method of surface area determination appeared to be suitable f o r extension and improvement so that i t might approach a single general method of p a r t i c l e size measurement.  II - PRACTICAL AND THEORETICAL CONSIDERATIONS OF THE PHOTOMETRIC METHOD B e l l ' s method of surface area measurement by t u r b i d i t y  2) determination i s a modification of the method evolved by Wagner 1933.  1  in  In Wagner's device a suspension of p a r t i c l e s i s placed i n a glass  sedimentation tube, 8 inches i n depth and !§• inches square, and s t i r r e d . At a fixed distance from the tube i s placed a standard -lamp which passes a beam of l i g h t through the suspension.  The l i g h t absorbed  i s indicated  on the other side of the tube by means of a photoelectric c e l l .  The  standard lamp i s provided with a condenser to remove radiant heat. Wagner used Stoke's law to determine the s i z e - d i s t r i b u t i o n of p a r t i c l e s at any l e v e l i n the s e t t l i n g tube.  The general equation f i t t i n g the  observed data was s where S  d  d  •  c(log I  0  - log I  d  )  z  c(2 - l o g I ) d  i s the surface area of a l l p a r t i c l e s i n the sample smaller than  d, c a constant f o r the material observed, I l i g h t , and I  A  c  the strength of the incident  the i n t e n s i t y of the l i g h t i n microamperes transmitted  7  through the suspension at the end of the time i n t e r v a l required f o r a p a r t i c l e d i n diameter to s e t t l e from the surface of the suspension to the center l i n e of the l i g h t beam.  A f t e r a standard curve has been  derived, only a single reading i s required f o r determination of the surface of p a r t i c l e s less than a stated s i z e . sizes may be accomplished e a s i l y .  Integration over a l l  Normally the Wagner method can be  considered only as a means of obtaining a r e l a t i v e surface and the surface measured depends upon the standard used and the accuracy with which the surface was o r i g i n a l l y determined.  According to D a l l a V a l l a - ^ ,  the method i s i n no sense absolute, nor can i t be applied s a f e l y to d i f f e r e n t materials unless suitable standards are available f o r each. Moreover, the accuracy of the method i s affected by color, adsorption, birefringence, r e f r a c t i o n , and r e f l e c t i v i t y which are involved i n an indeterminate manner. B e l l ' s apparatus, i n i t s barest elements, consisted of two photo-electric c e l l s , a standard l i g h t source, and a c i r c u l a t i n g chamber f o r f i n e p a r t i c l e suspensions.  The r e c i p r o c a l of the weight of material  i n suspension, necessary t o cut o f f a c e r t a i n amount of l i g h t i n front of one of the photo-electric c e l l s was considered proportional to the s p e c i f i c surface of that material.  The amount of l i g h t to be cut o f f  was kept standard by upsetting the balance of the two photo-electric c e l l s a certain amount and then bringing them into balance again by the addition of material to the c i r c u l a t i n g chamber.  A schematic diagram  of B e l l ' s apparatus i s given i n Figure I . B e l l ' s method d i f f e r s from the Wagner method i n that t u r b i d i t y determinations are made on a well mixed suspension and consequently s e t t l i n g equations involving Stoke's law are not involved.  8  However, the method i s s t i l l dependent on the o p t i c a l properties of the materials under consideration.  I t i s necessary, therefore, to determine  whether or not these properties a f f e c t the r e s u l t s appreciably and, i f they do, t o determine whether they remain constant f o r any one material down through the size ranges concerned. I t i s a w e l l known f a c t that decrease i n p a r t i c l e size of most materials w i l l change t h e i r color sensed by the human eye.  However, i f  these materials are examined under a microscope i t w i l l be noted that even the smallest p a r t i c l e s r e t a i n the color and l u s t r e which i s evident to the eye at larger grain s i z e .  Although t h i s color change i s r e a d i l y  apparent to the human eye i t i s d i f f i c u l t t o state i n what manner i t would a f f e c t the operation of a photoelectric Reference to the l i t e r a t u r e , ^'  device. ^' ) on the subject of  r e f l e c t i v i t y with respect to f i n e p a r t i c l e s , indicates that even with quartz the degree of maximum l i g h t r e f l e c t i o n should be small enough t o make the change of l i g h t r e f l e c t i o n with decrease i n p a r t i c l e size a negligible  factor. According t o the Fresnel equation  (n  x  - n )  2  Q  R s  EQUATION 1 (ni + n ) Q  2  where R i s the r e f l e c t i o n c o e f f i c i e n t or f r a c t i o n a l part o f the incident  light  that i s r e f l e c t e d , n-^ i s the r e f r a c t i v e index of the s o l i d , and  n  Q  i s the r e f r a c t i v e index of the medium.  SCHEMATIC DIAGRAM OF BELL'S APPARATUS Figure 1  f o r quartz water  n-^ = 1.547 n  0  r  (mean f o r D l i n e )  1.335  R z .0054 i.e.  0.54$  of the incident l i g h t i s r e f l e c t e d from a quartz  water  interface.  A graph showing the r e l a t i o n between r e f r a c t i v e indices and the r e f l e c t i o n c o e f f i c i e n t s of some materials used as white pigments  4) i n the paint industry i s given i n Figure 2.  I t may be noted that the  order of magnitude of the r e f l e c t i o n c o e f f i c i e n t of the materials, selected  f o r t h e i r comparatively high degree of r e f l e c t i v i t y f o r use  i n paints, i s small.  Thus i t may be assumed that most materials used  i n t u r b i d i t y measurements w i l l exhibit a low r e f l e c t i o n c o e f f i c i e n t .  Figure 2  11  In the event that varying lustres and degrees of r e f l e c t i v i t y of mixture of minerals might a f f e c t t u r b i d i t y measurements the o p t i c a l system of an apparatus such as B e l l ' s should be designed so that suitable f i l t e r s might be introduced.  The use of l i g h t of a wave length f a r removed  from that enjoying maximum r e f l e c t i o n from mineral surfaces should make consistent r e s u l t s possible.  This l a t t e r p r i n c i p l e has been put to good  use i n the Fisher Electrophotometer  which analyzes solutions containing  dissolved substances that produce varying color i n t e n s i t i e s according to the amount dissolved. There are a number of reasons why no great error should be expected due to p o l a r i z a t i o n of the l i g h t passing through the f i n e p a r t i c l e suspensions.  Insofar as transmitted l i g h t i s concerned, a l l the l i g h t  passing a p o l a r i z i n g medium should be absorbed except that part v i b r a t i n g i n a c e r t a i n planej consequently,  as the l i g h t passes successively through  p a r t i c l e s with d i f f e r e n t orientations, i t seems probable that only a small f r a c t i o n of the i n i t i a l l i g h t w i l l remain unabsorbed. In order to i l l u s t r a t e that the type of r e s u l t s expected from an apparatus such as B e l l ' s could be predicted i n advance the following theory may be hypothecated. I f a number of p a r t i c l e s , considered as spheres of uniform diameter, are placed at random i n the path of a beam of l i g h t and i f i t i s assumed that the surface of the material i s such that i t w i l l absorb 100%  of the l i g h t  which s t r i k e s i t , an expression can be formulated which w i l l give the change i n l i g h t i n t e n s i t y as a function of the diameter of the p a r t i c l e s , the s p e c i f i c gravity of the material and the weight involved.  Let  I s i n i t i a l light intensity Q  I  s final light intensity  therefore,  1 - fr. light cut off  1  where the % of light cut off -  sectional area of particles sectional area of light beam  Let  0^ r sectional area of particles X z sectional area of light beam W z weight of material P - specific gravity of material d z average particle diameter g  N z number of particles S z surface area of particles if d  0.A ~ N  P  N  r 1/4 s  -  Vf s d  (4)  o _  )  -  W/d  3  s  P Equation 2  Equation 3 l  -  3AWP  (W^B)  x  13  In the apparatus, however, the sectional area of the p a r t i c l e s i n the suspension as they pass i n front of the photoelectric cell:; w i l l always be a c e r t a i n predetermined amount.  Therefore 0^ i s constant and i s  equivalent to the amount of l i g h t cut o f f . I t follows that,  d s  =  3  W  - K W/P  Equation 4.  / 3&P0A  /-  &  /  /an c  which i s the equation of a straight l i n e passing through the o r i g i n . I t must be remembered that the d  g  referred to above i s a surface mean  diameter since i t i s a c t u a l l y determined from the expression,  -V  ds  nd  r  Equation 5  2  £n  I t should be noted that i f by a suitable arrangement of f i l t e r s i n the o p t i c a l system of the apparatus i t was found possible t o obtain the same value f o r K (the slope of the c a l i b r a t i o n l i n e ) independently f o r d i f f e r e n t minerals i t should be possible t o measure, d i r e c t l y , the average surface diameters of mixtures of the d i f f e r e n t minerals.  I l l - THE APPARATUS  An apparatus, i n many respects s i m i l a r to B e l l ' s device, was constructed.  The apparatus consisted, e s s e n t i a l l y , of two phototubes, an  electronic c i r c u i t f o r balancing the phototubes, a c i r c u l a t i n g chamber and a l i g h t source.  The l i g h t source was mounted i n a tee-shaped tube and a  condenser lens mounted i n each arm of the tee.  The f o c a l lengths of the  lenses were such that the rays of light after passing through them were substantially parallel. A plastic block in which glass windows had been set was attached to the end of one arm of the tee.  On the other side of  the plastic block was mounted one of the phototubes. The circulating tubes passed vertically through the plastic block. On the end of the opposite arm of the tee was attached another plastic block. A slot was cut in this block so that a piece of ground glass screen could be inserted thus cutting off a portion of the light reaching a phototube mounted on the other side of the block. An electric eye vacuum tube (6E5), arranged in a bridge circuit, served to balance the outputs of the phototubes. The intensity of the standard lamp was controlled by an A. C. ammeter. Pictures and schematic diagrams of the apparatus are given in the following pages. The first important pre-requisite in design of the circulating chamber was that the outlines of the chamber be extremely smooth such that there would be no sharp breaks in the direction of motion of the water, otherwise, local eddy currents would be set up and settling would occur. Secondly, the condition of turbulence must be such that the film of media on internal surfaces having zero velocity be of a minimum thickness, otherwise, the very fine portion of a mineral powder would have a tendency to adhere to the surfaces of the circulating chamber and be removed from circulation. In accordance with these requirements the circulating chamber was built of clear plastic so that the motion of the fine particles could be observed. An impeller was installed which had sufficient power to prevent even the heaviest of minerals from settling out. The impeller shaft bearing  15  Figure U  Figure 5  Figure 6  17  was mounted outside the c i r c u l a t i n g chamber i n order to keep i t from being damaged by abrasive materials and to keep the chamber free from any l u b r i cating materials. In the f i r s t few t r i a l runs i t was found necessary to i n s t a l l b a f f l e s of varied shapes i n order to counteract a pulsating e f f e c t set up by the impeller.  The position of the b a f f l e s i s given i n Figure  6.  The c i r c u l a t i n g chamber could possibly be further improved f o r some trouble has been experienced with extremely f i n e material adhering to the  walls of the p l a s t i c tubes, e s p e c i a l l y at the bends.  probably due to the surface of the l u c i t e p l a s t i c .  This f a u l t i s  Although i t would have  been more d i f f i c u l t to construct the tubes of glass they would more s a t i s f a c t o r y .  have been  However, i f the p l a s t i c tubes are frequently cleaned  with soap solution, consistent r e s u l t s can be obtained. The electronic balancing c i r c u i t used i n the apparatus has proven highly s a t i s f a c t o r y .  Once balanced, even moderately rough handling of the  machine w i l l not cause any noticeable d r i f t or unbalance.  The c i r c u i t  employs two 929 phototubes and a 38 vacuum tube connected i n the high s e n s i t i v i t y arrangement shown i n Figure 7^ but requires neither a battery nor a microammeter. was taken,  According to the R.C.A. Manual from which t h i s c i r c u i t  of 1% unbalance i n the l i g h t on the phototubes, at i l l u m i n a t i o n  levels as low as .0001 lumen, w i l l cause the shadow angle on the 6E5 to open to 90° or close to 0°.  At higher illumination l e v e l s , the  tube 6E5  shadow angle gives f u l l response to an even smaller percentage unbalance because the r a t i o of photoelectric current to leakage current i s l a r g e r . Figures 8 and 9 give the spectral s e n s i t i v i t y curve and the anode characteri s t i c curves of the type 929 phototube; the spectral s e n s i t i v i t y curve shows that t h i s phototube i s more sensitive to b l u e - r i c h l i g h t than to red-rich  18  light.  The anode c h a r a c t e r i s t i c curves show that the response from the 929  phototube i s a l i n e a r function of l i g h t  intensity.  H,  =  6E5 H E A T E R  R  H  2  =  6H6 HEATER  Ro, R-, Ru  H  3  =  38 H E A T E R  R  =  50 O H M S ,  R  =  3 2 6 ' O H M S , 30 W A T T S »  C,. C C , 4  4  s  C , = A uf, 250 V .  2 l  3  C ,.6. C 5  7  =  Ri =  15.000 O H M S ,  R, =  100,000 O H M S . / j  R  3  Ro =  0  0.1 uf. 200 V .  S =  WATT i/  2  50,000 O H M S ,  Rio =  Vi W A T T  I MEGOHM,  =  =  '/  WATT  2  2 M E G O H M S , '/  2  WATT  i/ WATT ' 2  0.5 M E G O H M , DOUBLE-POLE  Vi  WATT  ON-OFF  SWITfcH  WATT  Form- Pf. ZOAJ ^  9 — A - C operated  sensitive  circuit  for matching  measurements.  Figure 7 SPECTRAL SENSITIVITY CHARACTERISTIC OF S4 PHOTOSURFACE IN L I M E - G L A S S B U L B  1  TYPE s° 929 .0400  °SENSITI U 1 T V  IM  1 1 A / 1 1 W Q ADIAN T FLUX AT 3750 A"  2  UJ  o a  100  UJ  a. 1 >-  t> 80  5 u > <  °  6 0  / / *  AO  -J UJ  OC  --  20  ./£.<  4000  8000  W A V E L E N G T H - A N G S T R O M  Figure 8  12000 UNITS ( A ° )  i TY PE 92!9  LIGHT F L U X - L U M E N S = 0.1  0.08  u c uc i  .  If  V v«—  i  o cc . o  )  o  k  Ul  o o z <  :  r  c  C.06  \  0  0  0.04  X r  Xo.o; /  ANODE  VOLTS  orJtt.  c  92C-6I5I  Figure 9 The procedure for determining surface area or particle size is as follows. The circulation chamber is first filled to a certain mark with distilled water which has been previously boiled to remove as much of the dissolved gasses as possible. Air removal is necessary since gas bubbles have a tendency to form under the action of the impeller. The impeller is started and the light source brought up to a certain intensity, determined by the A. C. ammeter. The machine is then brought to balance by adjusting the two phototube potentiometers until the electric eye is just closed. The potentiometer shown on the right in Figure 3 is used for coarse adjustment  and the potentiometer on the  l e f t f o r f i n e adjustment.  Once the i n i t i a l  balance i s obtained i t may be o f f s e t a fixed amount by i n s e r t i n g the ground glass screen i n the s l o t . The machine i s now brought to balance once more by  adding to the  c i r c u l a t i o n chamber the material on which the determination i s to be made. As given i n Equation 3 the weight necessary f o r t h i s f i n a l balance w i l l be a d i r e c t function of the average surface area diameter of the material.  IV - PREPARATION OF SAMPLES FOR CALIBRATION  I t was evident at the beginning of the work on f i n e p a r t i c l e measurement that success or f a i l u r e would depend primarily upon the standard powders used i n the c a l i b r a t i o n of the apparatus.  Therefore  i t was of great importance that the average p a r t i c l e size of these powders be determined accurately. The most desirable method of p a r t i c l e size determination would be a p p l i c a n t s to a number of minerals without regard to chemical composition and would also give some measure of p a r t i c l e shape.  Further, the use of  mineral powders, that had been sized and graded, would obviate the necessity of determining complicated size d i s t r i b u t i o n curves. The Haultain I n f r a s i z e r was used f o r the preparation of c a l i b r a t i o n samples since i t w i l l separate a -325 graded groups.  The s i x groups w i l l have average p a r t i c l e sizes ranging  from 10 microns to 50 About 200 single run. every 100  Mesh mineral powder into s i x uniformly  microns.  grams of material was charged to the I n f r a s i z e r f o r a  The time required f o r treatment was approximately one hour f o r  grams of charge.  The runs, f o r the samples to be used i n  21  c a l i b r a t i o n , were made a t a i r pressures varying from 24" I^O to 28" Ii^O. Since the I n f r a s i z e r i s , i n e f f e c t , an a i r e l u t r i a t i o n device for the grading of f i n e material, the shape of the i n d i v i d u a l p a r t i c l e s w i l l be an important factor i n e f f i c i e n c y of grading. spheres  Photographs of glass  , that had been graded by the I n f r a s i z e r , showed that the degrees  of sorting, according to s i z e , i s p a r t i c u l a r l y good.  In the accompanying  pictures of mineral powders, (quartz, chalcopyrite, p y r i t e ) , where the shape of the p a r t i c l e s i s not uniform, the closeness of s i z i n g i s s t i l l good.  remarkably  The standard deviations and standard errors, given i n Table I indicate  the e f f i c i e n c y of the I n f r a s i z e r . One or two photomicrographs of samples from each sized group of mineral powders were taken on 5" x 7" plates at magnifications varying from 80 t o 400  times.  Under these conditions from 150 to 250 p a r t i c l e s were included i n the area photographed.  A few milligrams of powder, taken from a group, were placed  on a clean glass s l i d e .  The addition of a drop of Xylene accompanied by  s l i g h t side tapping of the s l i d e served to spread the powder into a t h i n layer.  The Xylene, when dry, also provided the adhesion necessary to hold  the p a r t i c l e s on the s l i d e .  The photomicrographs were taken on a L e i t z  Metallograph using dark f i e l d illumination i n the case of quartz and d i r e c t illumination i n the case of pyrite and chalcopyrite. An eyepiece with a graduated scale was used i n the microscope i n order that a reference scale would appear on the photomicrographs.  This  scale was calibrated, with respect to the various objectives used, by comparing i t with a standard 1 mm scale mounted on the microscope specimen table. The photomicrographs were projected on a 4 foot by 5 foot screen and the value of each projected scale d i v i s i o n determined  i n inches.  A  22  standard inch r u l e r , graduated i n tenths, was used to measure the diameters of the projected images. Two hundred p a r t i c l e s were considered a representative sample of each size group.  Two length dimensions of each p a r t i c l e were taken.  These  measurements, which have been termed the c r i t i c a l p a r t i c l e dimensions, correspond to the longest and shortest axes of the p a r t i c l e .  The square  root of the product of these i n d i v i d u a l measurements was taken as the surface diameter of the p a r t i c l e under consideration.  The reasons underlying t h i s  procedure were, f i r s t l y , that the surface diameter must n e c e s s a r i l y be determined from some dimension to the second power denoting area and, secondly, that with one mineral there i s assumed to be a constant shape factor which, i f multiplied by the product of two p a r t i c l e dimensions i . e . the surface diameter squared, w i l l give the sectional area of the p a r t i c l e . Previously (Page 11.), i t has been shown that the sectional area of p a r t i c l e s i n a suspension i s a function of the t o t a l area of the p a r t i c l e s and that i t i s the sectional area that w i l l be measured by the photometric apparatus. The p a r t i c l e shape f a c t o r w i l l have some value between 1,  f o r a rectangle,  and fyu, f o r a c i r c l e . The products of the short and long dimensions of a l l the p a r t i c l e s measured were added and t h i s divided by the number of p a r t i c l e s measured. The square root of the resultant figure was taken as the average p a r t i c l e diameter f o r the sized group under consideration. would be mathematically expressed as follows.  The above procedure  23  a,b  z dimensions measured through the centroid of the p a r t i c l e  - average surface diameter •s (av)  d„(average) =1/ -1 (ab 3  r — N  N  2 number of p a r t i c l e s  The root mean square method of average p a r t i c l e s i z e c a l c u l a t i o n thus weights the surface area presented by the larger p a r t i c l e s and consequently the average diameter arrived at i n t h i s manner i s a better measure of the surface area of a group of p a r t i c l e s than the arithfmetic average diameter. The average surface diameters determined f o r the sized groups of the minerals, (quartz, chalcopyrite, and p y r i t e ) , are given i n Table 1. Standard deviations and standard errors of these average surface diameters are also given.  The formula, used f o r the c a l c u l a t i o n of the standard  deviation i s as follows;  £>- z standard deviation N  N  z number of p a r t i c l e s  and that used f o r the c a l c u l a t i o n of the standard error i s as follows;  2U  Figure 10 i s a frequency d i s t r i b u t i o n graph of the s i x groups of powders obtained from the Haultain I n f r a s i z e r .  For the sake of s i m p l i c i t y ,  the maximum ordinates of a l l the groups have been made equal.  I t w i l l be  noted that the grading of f i n e r p a r t i c l e s i s closer than that of the larger particles. Figures 12 to 16 and 18 to 22 are photomicrographs of samples from the s i x size groups obtained from the I n f r a s i z e r .  Figures 17 to 23  are photomicrographs of samples from the I n f r a s i z e r dust bag overflow. I t was of i n t e r e s t t o estimate the value of the shape factors f o r the minerals that were to be used i n the surface area and p a r t i c l e diameter determinations.  A frequency graph, showing the percentage  deviation of the largest measurement taken f o r the diameter of a p a r t i c l e from the root mean square diameter of that p a r t i c l e , i s given i n Figure 11 for the mineral quartz.  The peak of the frequency curve indicates that  the longest dimension of the average quartz p a r t i c l e would be 35% longer than the surface area diameter of the p a r t i c l e .  The shortest dimension  of the p a r t i c l e would be 35% l e s s than the surface area diameter of that same p a r t i c l e .  I f the average quartz p a r t i c l e cross-section was considered  an e l l i p s e , with the c r i t i c a l dimensions as outlined above, the value f o r the calculated shape f a c t o r would then be .895  25  Figure 11  MATERIAL  Quartz  GROUP NO.  AVERAGE SURFACE DIAMETER  0^, STANDARD DEVIATION  ^ -  STANDARD ERROR  6  18.7 f  5.7  5  28.6 < •  7.7  0.80 "  4  39.5 "  6.7  1.05 "  3  48.5 "  8.2  "  1.03 < '  2  74.3 v  9.7  v  1.19 -  2.5  •«  f*  0.61 f<  7 Chaloopyrite  Pyrite  «i  6  13.6  5  19.4 „  4.1  „  0.40 "  4  27.3 ,  5.0  u  0.48 -<  3  40.5 «  6.8  *<  0.87 «.  2  56.0 "  8.8  6-1  13.1 "  3.0  "  0.32 "  5-1  18.2  3.1  "  0.29  4-2  27.3 ''  4.3 < '  0.48 *  3-2  39.0  "  6.4  <.  0.58  2-2  51.9  *  8.1  «  1.26 ''  "  Table No. I  0.32  1.38 < '  Figure 12 Quartz Sample #2 156  X  Figure 13 Quartz Sample #3 156  X  Figure 14 Quartz Sample #4 174  X  Figure  15  Quartz Sample #5  126 X  Figure  16  Quartz Sample #6  408 X  Figure 17 Quartz Sample #7  247 X Lnira sizcr Dvst fiag Grcr f /<tu>  Figure 18 Pyrite Sample #  82  2-2  X  Figure 19 Pyrite Sample #  90 X  Figure 20 Pyrite Sample  170 X  4--2  3-2  * 1  30  Figure 21 Pyrite Sample 5-1 170 X  Figure 22 Pyrite Sample 6-1  205 X  Figure 23 Pyrite Sample 7-0  283 X  la i>  31  V - EXPERIMENTAL RESULTS The  sized groups of minerals which had been prepared i n the  I n f r a s i z e r machine and the weights-to-balance f o r each size group u>^fa. determined.  The results are given i n Table I I .  C a l i b r a t i o n graphs were drawn up showing the weight-to-balance versus the average surface p a r t i c l e diameter as given i n Table I I . c a l i b r a t i o n graphs f o r quartz, chalcopyrite, Figures TM^ 25± and  The  and p y r i t e are given i n  26.  In order that some c o r r e l a t i o n of the c a l i b r a t i o n graphs f o r the three minerals used might be obtained the s p e c i f i c gravity of the individual minerals must be taken into account as given i n formula 2. Figure 27 i s a graph i n which W/P  i s plotted against the average surface  diameters f o r the s i z e groups of the minerals. I t w i l l be noted that a l l the c a l i b r a t i o n graphs show, as expected, a straight l i n e r e l a t i o n s h i p between p a r t i c l e size and weight-to-balance.  I t i s of p a r t i c u l a r i n t e r e s t to note that  c a l i b r a t i o n l i n e s , f o r the three minerals investigated, the o r i g i n .  I t may  the  the  pass through  be concluded from t h i s f a c t that the effects of  p o l a r i z a t i o n , r e f l e c t i v i t y , r e f r a c t i o n , birefringence  etc., of the f i n e  p a r t i c l e s i n the apparatus, are n e g l i g i b l e . Since quartz exhibits a f a i r degree of transparency and minerals chalcopyrite  and p y r i t e are opaque, i t was  expected that  the the  slope of the c a l i b r a t i o n l i n e for quartz would be less than those f o r chalcopyrite  and pyrite and such has proven to be the case.  In the f i n a l c a l i b r a t i o n graph, Figure 27. i t i s seen that the c a l i b r a t i o n l i n e s for pyrite and chalcopyrite  coincide.  This f a c t  MATERIAL  Quartz P  :  2.7  GROUP NO.  6 5  AVERAGE SURFACE DIAMETER  W, WEIGHT TO BALANCE 1.358  18.7 u  28.6  2.090  4  39.5  I  3.077  3  50.2  < '  3.930  2  73.7  W/P  .502 "  .774  "  1.139  1.455  5.685  *'  2.102  .748  ' <  .182  7 Chalcopyrite P  =  4.1  Pyrite P  z 5.0  6  5  13.6  •«  19.4 -  1.105  .270  4  27.3 -  1.345  3  40.5  2.090  2  56.0 "  6-1  13.1  5-1  18.2 '<  1.090  4-2  27.3 "  1.721  3-2  39.0  < '  2.540  >,  2-2  51.9  "  3.419  "  •«  "  Table No. I I  .729  ''  .328 .510  "  .146  ''  .218  .344 .508 .685  33  0  10  20  30  40  50  60  AVERAGE SURFACE DIAMETER (microns)  70  35  suggests that any opaque material, when corrected f o r s p e c i f i c gravity, would have the same c a l i b r a t i o n l i n e and that surface areas and average diameters of mixtures of opaque materials may be d i r e c t l y determined. A few t e s t s were made on mixtures of d i f f e r e n t s i z e groups of the same mineral to determine whether the apparatus would s a t i s f a c t o r i l y measure the average surface diameters of a material where the „ range of p a r t i c l e size was large.  Following are the r e s u l t s of a t e s t .  Mat'l & Individual Composite Fraction Av.Part. Av. Calc. Sample No. Wt.-to-Bal. Wt.-to-Bal. of B a l . Size Part. Size quartz  #6  1.358.800 ^  .590  18.7  quartz  #3  3.930  J221  50.2  1.465 " 2.265  M  31.5 p  .963 % Error of B a l .  = 3.7$  Table No. I l l The average surface p a r t i c l e size determined from the graph i s  30.0^/..  In order that the photometric method of f i n e p a r t i c l e measurement could be applied to mixtures of opaque and p a r t i a l l y transparent minerals a graph as shown i n Figure 28  was drawn up.  This graph i s the same as that of Figure 27 except that a s e r i e s of l i n e s representing the volume assay of quartz, chalcopyrite mixtures, has been drawn between the c a l i b r a t i o n l i n e s f o r pure quartz and pure chalcopyrite (or p y r i t e ) . The slopes of the l i n e s i n Figure 28 were determined i n the following manner. With reference to formula 2^ i t may be remembered that  36  Figure 28 0^ is the specific surface necessary to bring the apparatus to balance. This surface, with mixtures, is made up of a l l the surfaces presented by the components of the mixture. 0 (for 100* quartz) A  W/P  .5236f d :  (7Td/L) 2  3  A (W/Pd)  Similarly, 0^ (for 100$ chalcopyrite) - B (W/Pd) The variance in the constants A and B will depend on the relative transparency of the materials under consideration, i.e. quartz and chalcopyrite.  37  Therefore with mixtures,  0  A  =A  _ +  i P  l l d  B_»2_ 22 P  d  where, A - a constant B - a constant (not necessarily the same value as A) W^: amount of quartz i n the mixture used to balance the apparatus P^; s p e c i f i c gravity of quartz W r amount of chalcopyrite i n the mixture used t o 2  balance the apparatus ?2Z s p e c i f i c gravity of chalcopyrite d^= average surface diameter of the quartz p a r t i c l e s average surface diameter of the chalcopyrite p a r t i c l e s and with s p e c i f i c reference to quartz and chalcopyrite,  B = 2.24 A where Bs equals slope of the chalcopyrite l i n e As equals slope of the quartz l i n e .  and  K = _ l l _ , 2.24 2 l l h*2 W  P  d  This i s the equation of a straight l i n e passing through the points (W^/P-^, d]) and (W /P > d ) on the graph. 2  2  2  For  s i m p l i c i t y a h o r i z o n t a l l i n e was s e l e c t e d .  i.e.  • d  4j s 50/*  x :  and K  l  = V 1  + 2.24W /P  P  If  2  :  %  .1.41  2  p  r 1.41  =  The r a t i o  +  w  2.24  lAl  W /P 2  2  was given the values 9/1, 8/2, 7 / 3 ,  6/4,  V 2 P  5/5, 4/6, 3/7, 1/9,  and the values f o r  WJ/PT^  +  W /P 2  2  determined as  follows.  W  l/ l/ 2/ 2 P  W  P  W  l/ l p  w  2/ 2 p  W!/P W /P2 L F  2  9/1  1.130  .125  1.225  8/2  .904  .226  1.130  7/3  .718  .308  1.026  6/4  .565  .377  .942  5/5  .435  .435  .870  4/6  .323  .485  .808  3/7  .226  .528  .754  2/8  .141  .566  .707  1/9  .066  .600  .666  Table No. IV  The values f o r W-/P-_ ^  ^2/^2  w  e  r  Pl°' ' - along  e  fc  te(  the 50 j  l i n e and straight l i n e s drawn from each point through tffe o r i g i n . As mentioned before these l i n e s represent the volume assay of the mixture. A few runs of chalcopyrite-quartz mixtures were made to determine the r e l i a b i l i t y of the calculated  Mat'l  Size  Quartz  50.2  #3  Chalco.  #4  W  3.9307.  Comp.Wt.to-Bal.  %o  W/P  2.000 S-  27.3 1.345 -< .638  graph.  % Vol.  A  .740  50.9  83.3  .155  47.5  16.7  98.4  100.0  ~7895  % Error of B a l .  = 2%  Table No.V Average surface diameter  (.740/50.2 + .155/27.3)  .740 T50T2p  +  ,  - 39.4 f*-  .155 (27.3)3 (point X on the graph)  The value of the average diameter from the graph i s approximately 39 f-. W/P  ;900  .333  46.5  72.0  .130  53.4  28.0  .463  99T9  100.0  Size  W  quartz  28.6  1.9305  Chalco.  19.4 1.105 « .580 «  #5 #5  %o  Comp.Wt.to-Bal.  Mat'l  <J  Table No. VI  A  % Vol.  40  Average surface diameter  (.333/28.6  +  .130/19.4)  (point T on the graph)  The value of the average diameter from the graph i s approximately 23.6fJ . In summary the procedure i n using the graph of Figure 28 would be as follows. F i r s t l y , determine the weight assay of the material to be used and convert to a volume assay.  Secondly, determine the weight-  to-balance f o r the mixture and convert to the volume-to-balance i . e .  V  P  1  +  V 2 P  Lastly, read o f f the average surface diameter corresponding to the value f o r the toolume-to-balance from the correct volume assay l i n e on the graph. In Figure 28^ the l i n e s YXZ and RTU are of no p a r t i c u l a r i n t e r e s t except that they provide a secondary check on the v a l i d i t y of the graph.  The following might be mentioned, however, as a prac-  t i c a l use of such l i n e s .  I f i t was of interest to determine the  d i f f e r e n t i a l rate of size reduction of the sulphide f r a c t i o n of an ore as compared to that of the gangue f r a c t i o n then a l i n e such as YXZ or RTU would f u r n i s h t h i s information.  A t e s t on the ground  41  product of the ore would determine one point on the graph.  Then i f i t  was possible to disturb the volume assay of t h i s ground product, say by panning, another t e s t could be made and another point plotted on the graph. the 100$  The intersections of the l i n e j o i n i n g these two points with  sulphide l i n e and the 100$  gangue l i n e would then give the  average diameters of the sulphide and the gangue f r a c t i o n s of the ground product. Figure 22 i s a plot of s p e c i f i c surface area versus the r e c i p r o c a l of the weight to balance.  Since no absolute values of surface  area were available f o r c a l i b r a t i o n of the apparatus, i t has been necessary to introduce a constant, B, termed the i r r e g u l a r i t y factor, which when m u l t i p l i e d by the calculated s p e c i f i c surface area of a material, w i l l give the absolute s p e c i f i c surface area of that material. B w i l l u s u a l l y have a value between 1.0  and 1.30.  The introduction of  such constants i n p a r t i c l e size work i s often necessary.  However, they  do not usually detract from the value of the data to which they p e r t a i n .  SPECIFIC 1  9  1.0  SURFACE  QUA!.TZ.  AREA  CIiALCGFYRITE  GRAPH FOR &  PYRITE  B  s OF V/EIGHT  0.8  0.6  %  a EE Hi O  0.4  0.2  0  8000  4000 SURFACE  AREA PER GRAM / B  igooo (cms.  16000  /gm.)  Figure 29  VI - CONCLUSIONS The r e s u l t s obtained show that i t i s possible t o determine, r a p i d l y and a c c u r a t e l y , the r e l a t i v e surface areas and average surface diameters of powders of the three minerals p y r i t e , quartz, and chalcop y r i t e , and that i t i s not d i f f i c u l t t o determine the surface areas and average surface diameters o f mixtures o f the powders o f these three minerals. Since the p h y s i c a l c h a r a c t e r i s t i c s of quartz, p y r i t e and chalcop y r i t e d i f f e r widely the r e s u l t s of these i n v e s t i g a t i o n s apparently  A3  warrant the conclusion that the photometric method of p a r t i c l e size measurement i s applicable to mixtures of many other minerals.  Certainly  i t seems possible that average surface diameters and r e l a t i v e surface areas of constant assaying materials can be d i r e c t l y determined from a c a l i b r a t i o n graph.  Moreover, i t should be possible to determine  surface areas and average surface diameters of opaque materials which vary i n assay.  The problem becomes more d i f f i c u l t when two or more  non-opaque materials, i n varying amounts, are included i n the mixture. I t i s suggested that a common ternary diagram might solve the problem of determining  surface areas and diameters of mixtures containing two  non-opaque materials, i n varying amounts, besides any number of opaque materials.  The apices of the t r i a n g l e would represent r e s p e c t i v e l y  100% opaque materials by volume, 100% non-opaque material of the f i r s t type by volume, and 100% non-opaque material of the second type by volume.  Contour l i n e s on the diagram would represent a function of the  o v e r a l l weight-to-balance and the average surface diameter of the mixture. The apparatus, i n i t s present condition, coupled with the Haultain I n f r a s i z e r w i l l give complete s i z e - d i s t r i b u t i o n s , of constant assaying materials, from approximately 75 microns down to the one micron range.  With respect to surface area i t seems probable that the apparatus  can measure surface areas of p a r t i c l e s down to the l i m i t s imposed by the wave length of l i g h t . The accuracy of the apparatus and indeed the complete method of size determination  appear s a t i s f a c t o r y .  Duplicate runs on the same  material usually vary within l e s s than one percent.  With the f i n e r  materials one milligram i s s u f f i c i e n t to close the e l e c t r i c eye from  44  45° to 0°. The standard errors given i n Table 1 f o r the c a l i b r a t i o n samples are s u f f i c i e n t to account f o r the experimental points l y i n g s l i g h t l y o f f the c a l i b r a t i o n l i n e s . Runs, such as the one given i n Table IV, i n which the components of a mixture are added separately to the machine to give a composite weight-to-balance reading compare within three percent or l e a s with the calculated weight-to-balance. In order to simplify the method of c a l i b r a t i n g the apparatus f o r constant assaying mixtures the following procedure has been developed. I f i t i s possible to assume that the c a l i b r a t i o n l i n e f o r the material w i l l be a straight l i n e , i t i s necessary to determine only one point, on the p a r t i c l e size versus weight-to-balance graph, other than the o r i g i n , to f i x the c a l i b r a t i o n l i n e . This point may be located by determining the weight-to-balance f o r a portion of the mixture which had been sized by screening. product caught between 325 mesh and a 400  The  mesh screens or between 270  and 325 mesh screens would be s u i t a b l e . I t would be assumed that the average p a r t i c l e size of the screened portion would be the mean of the openings of the screens used i n the preparation-. The s i z e of the openings i n the screens must be determined experimentally since prepared screens are r a r e l y standardized. I f t h i s method i s to be used i t i s necessary that there be no great difference between the assay of the screened product and the assay of the i n i t i a l material.  45  VII - ACKHOYfLEDGEMENTS  The author i s grateful to Mr. F. A. For*ward, Head of the Department of Mining and Metallurgy, f o r h i s consideration and i n t e r e s t , and to Associate Professor W. M. Armstrong under whose d i r e c t i o n the investigations were carried out, and to Professor H. M. Howard f o r h i s c r i t i c i s m and i n s t r u c t i o n during the past year. The author i s indebted to the Consolidated Mining and Smelting Company f o r a research grant obtained during the summer of 1949.  46  VIII - BIBLIOGRAPHY I.  B e l l , J . W;, Bneritus Professor of Mineral Dressing, McGill University  "A Method for the Measurement of Surface of Finely Divided Material", C.T.M. Trans., V o l . XLVII (1944)= 424-436.  2.  Wagner, L . A . ,  "A rapid method for the determination of the specific surface of Portland Cement." Proc. A . S . I . M . , 3 3 (Part I I ) : 553-570  3.  Dalla ' V a l l a , J . M.  4.  Barnett, C. E.  "Micromeritics" Pitman Publishing Corp. (1948). "Physics and Chemistry of .Pigments." Ind. and B i g . Chem., V o l . 41, No. 2 (1949)i 272-278.  5.  Memin, fi. 1.  "Proc. A . S . T . M . , 17, 1946. (1917)  6.  Mie, G.  Ann. Physik, 25, 377.  7.  Haul t a i n , H. H T.  "The Infrasizer - A Report on Some Research Hbrk i n Size Analysis i n the Sub-Sieve Sizes, carried on at the University of Toronto.* November-1946= 23-29.  8.  Burden, H. .and Barker, A.  (1908).  The Measurement of Grain--"-Size -of Tungsten and Tungsten Carbide Powders Used for the Manufacture of Hard-Metal", Journal of the Inst, of Metals, Oct. 48r 51-68.  

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