Precise Measurements of Particle Surface Area with the Microscope F R A N C E S D. PIDGEONI and C H A R L E S G. DODD2 Surface Chemistry Laboratory, Petroleum Experiment Station, Bureau o f Mines, Bartlesville, Okla.
The projected-area method has been modified for direct microscopic measurement of the specific surface areas of nonporous particulate matter, such as convex, irregularly shaped, crystalline particles. The new graticule projected-area method combines the accuracy and advantages of projected-area measurements with the speed of eye-piece graticule comparison procedures. The need for measuring or assuming “shape factors” and for determining suitable statistical methods for averaging size-distribution data, as in measurements of particle “diameters,” is eliminated. The method is accurate and direct and requires only 2 to 6 hours per sample compared with several days by earlier procedures. IIIicroscopic surface-area measurements based on the projected-area principle require samples mounted in randoni orientation. Jlounting techniques have been developed which permit extension of method to subseive-size particles. Coniparisons of results on crushed quartz pow-ders with surface areas measured by gas adsorption and liquid permeation indicate good agreement for samples that do not have an appreciable number of particles below 1 to 3 microns. An extension of the method to particles smaller th.an 1 micron is suggested.
M
ICROSCOPIC determinations of the specific surface area of nonporous particulate matter generally depend on measurements of a characteristic linear dimension, such as an average particle “diameter.” One of the most widely used diameters is that suggested by Martin, Blythe, and Tongue (16). Applications to specific surface-area calculations involve empirical “shape factors” which vary from a theoretical value of 6.0 for spheres to an experimental value of 55.6 for mica Aakes (11). In addition to selecting an appropriate shape factor, it is necessary to determine the proper statistical method of averaging the size-distribution data resulting from microscopic observations (6). These uncertainties are avoided by the more tedious procedures based on areal and volumetric measurements. The projected-area method suggested by Kenrick ( 1 3 ) and Tooley and Parmelee (20) has been shown experimentally ( 2 , 14, 80) and theoretically (4,20, 2 1 ) to be accurate for irregularly shaped particles having a negligible amount of re-entrant surfaces. This paper presents a modification of the projected-area method developed in this laboratory for direct measurement of specific surface areas of irregularly shaped particles, such as crushed quartz. I t is faster than previously described procedures and yet is capable of comparable accuracy. Particle surface area is related to projected area by the equation:
s = -4 2 A n
S approaches the absolute surface area of the particle of average area as n increases, provided there are no re-entrant surfaces and only a negligible amount of submicroscopic roughness. 2 A represents the summation of areas of the projected images of n convex particles mounted in random orientation, a type of mounting not normally obtained on microscope slides. 1 2
Present address, Box 776, Bartlesville, Okla. Present address, Continental Oil C o . , Ponca City, Okla.
Photomicrographs or tracings from projections of fields from microscope slides are required for making particle surface-area measurements by the projected-area method. Areal measurenient,s may be made planimetrically or by cutting and weighing images on enlarged photomicrographs, a more rapid procedure. Although the method is accurate, its use in this laboratory required several days for a given sample. The projected-area method presented another serious difficulty, which was not apparent to earlier workers who applied it to relatively coarse, sieve-size powders. It has not been possible to develop a random orientation method of mounting particles appreciably finer than 50 microns so that all particles lie in the same plane. Thus it has been extremely difficult to obtain photomicrographs or tracings of entire fields. Patterson and Cawood (17 ) introduced a comparison eyepiece graticule for measuring diameters of projected particle images. Later modificatioas of the grsticule were developed by Fairs ( 7 )and May (16),both of whom measured sample slides with the particles lying in stable positions. A11 of these graticules consist of a central rectangular field or fields, a-ith a series of opaque disks and/or open circles. The graticules have been used to make particle-size anslyscs of smokes (17 ) , aerosols ( 1 6 ) , and powders and dusts ( 7 ) . The Fairs design has proved most satisfactory for measurements of this kind. The three Fairs graticules in Figure 1 contain a series of graduated circles covering a range of diameters of 128 to 1 and, except for the two smallest progression. circles, are arranged in a Heywood (10) has shown that, although there is a general tendency to overestimate projected areas with the Fairs graticules, the error is reduced to negligible proportions when experience is gained in their use. All prior applications of these graticules that have come to the writers’ attention involved determination of a characteristic diameter of particles mounted in stable positions. Fairs obtained a diameter that he found agreed Tvell with the effective diamet,er of an equivalent sphere as measured by sedimentation ( 7 ) . He was able to determine this diameter by comparing projected particle areas with the circles on his graticules without using a shape factor, but he did not attempt to calculate part’icle surface areas. Use of the Fairs comparison eyepiece graticules for specific surface-area determinations in this laboratory has reduced the time required for complete measurements and calculations based on the project’ed-area principle from several days to about 2 to 6 hours per sample. In place of measuring areas of part,icle images on photomicrographs, the Fairs graticules are employed to estimate directly the projected area of each particle. (Fairs graticules may be purchased from Elliott Scales, Ltd., Long Hill, Woldingham, Surrey, England, and P. C. Smethhurst, Smethhurst Highlight Co., Sidcot Heaton, Bolton, Lanes., England. Several American firms recently have initiated graticule production on custom orders.) The new graticule projected-area method also avoids the use of an empirical shape factor, thereby redacing the uncmt,ainty of surface-area calculations based on measurement of a diameter. The actual time required for analyzing a given sample depends on the over-all range of particle size and the extent of irregularity of particle shapes. The open circles, rather than the opaque disks, on the Fairs graticules are used at. all times in accordance n-ith Heywood’s suggestion ( I O ) , although this slightly increases the time required. Slides are mounted by methods that permit random orientation of particles. The number of particles per gram of powder is determined by the count’ing method employed by Amberg ( I ) , suitablv modified
1823
4;
1824
ANALYTICAL CHEMISTRY
when applied to come, sieve-size particles. The specific surface area of an unknown sample is calculated by the equation:
where So is sample specific surface area in square centimeters per cubic centimeter, is particles per gram, and p is true density of the solid particles in grams per cubic centimeter. Workers who prefer a projection rather than a visual method may use enlarged scale drawings of the Fairs comparison graticules with a camera lucida or a projection microscope.
in size ranges below 50 microns are difficult to mount by the method described above, partly on account of moisture adsorption, with the resultant formation of aggregates, and partly because the finer particles tend to drift with air currents and fail to contact the slide. An alternative procedure has been developed to meet this problem. Approximately 7 to 8 ml. of glycerol jelly is melted in a small beaker and a suitable amount of thoroughly dried sample is added. Adequate dispersion may be obtained by careful stirring with a microspatula. When the mixture has started to gel, a small amount is spread on a clean slide. After the mount has set, it ic protected with a cover slip thinly contcd with glycerol jelly.
PROJECTED-AREA MEASUREMENTS
Preparation of Slides. A suitable mounting procedure is essential to success. 911 particles must lie in random orientation. To satisfy this requirement, slides generally are coated with a thin film of a transparent, viscous and tacky plastic medium in which dispersed sample particles are retained in their initial orientations. The following procedures have been used successfully to mount quartz powder samples.
Most probable
:: I , , o
Apparent limits of probable precision
o* 195
, ,
I
NO. 1
2
3
4
I I
I
I
I
,
(
‘
I
1
6 7 S 9 I O I I 12 13 14 15 16 17 Hundreds of particles measured
5
Figure 2. Precision of Projected-.hea Measurements Effect of total number of particles measured on observed precision for S-2 crushed quartz powder fraction
0 0 0 NO. 2
Figure’ 1. Forms of Comparison Eyepiece Graticules Designed hy Fairs PROCEDURE FOR SIEVE-SIZE P.4RTICLES. A thin film Of Canada balsam or Kaphrax A [distributed by R. P. Cargille Co., New York, N. Y. Others have used Farrant’s medium ( 2 0 ) or grease (Z).] is spread on a slide and heated until the material is sufficiently viscid, which may be determined by scratching the film with a fine wire until there is no tendency for the troughs to fill in. Particles sprinkled on the film at this stage are caught and held in random orientation. After a suitable hardening period, cover glasses coated with glycerol or warm glycerol jelly (8) are placed carefully over the slides. Mineral powders PROCEDURE FOR SUBSIEVE-SIZE PARTICLES.
This technique has proved to be the only satisfactory method for obtaining both random orientation and complete dispersion of fine quartz powders. Particles mounted in this manner lie in various planes. Continual readjustment of the fine focus is necessary when the slide is measured, and the possibility of using a 2-nim. oil-immersion objective is eliminated. Measurement of Projected Area. The microscope used for precise particle surface-area measurements should be fitted with a mechanical stage and centering adjustments for both condenser and objective. Best results are obtained by the use of apochromatic objectives, compensated eyepieces. and an aplanatic condenser of sufficient aperture to accommodate all objectives Chamot and AIason (6) have discussed the need for optical perfection in such work. A method of illumination similar to that of Kohler (18) should be used to obtain the utmost in resolution from the optics. Thr objectives, eyepieces, and comparison graticules utilized for each sample are chosen according to the particle-size range of the ponder. All particles in the field of the microscope must be measured by comparison of the area of the projected image of each particle with circles on the comparison graticules. Measurements are made of the number of particles having projected areas approximately equal to each circle and of those falling bet\\ecn two consecutive circles. The slide is moved by means of the mechanical stage, measuring all particles in a large enough area of the slide or slides to obtain the required precision. To determine the number of particles that must be measured, a graph such as that presented in Figure 2 may be made of mean projected diameter versus the corresponding number measured. It is seen that approximately 1800 particles had to be measured to obtain a precision of 1% for this particular sample. In general, for one series of measurements, the number that must be measured varies from approximately 300 for 40- to 70-mesh sand to 600 or more for finer sieve-size samples and may be appreciably higher than 1000 for subsieve-size powders. In addition, at least three separate series of mcasurernents should be made on each sample.
V O L U M E 2 6 , NO. 11, N O V E M B E R 1 9 5 4
1825 to the spatula). The use of a graph such as that shown in Figure 3 is helpful in determining the number of particles that must be counted. Procedure for Particles Having A11 Dimensions Smaller Than 100 Microns. A standard hemacytometer or bloodcounting cell having a depth of 0.1 mm. is used unlees vel\ fine particles are . For the latter a Petroff-E1aussc.r bacteria-counting c g z k f r with a depth of 0.02 mm. permits the use of a 2-mm. oil-immersion objective. For coarser saniples, the count may be made with either a special 4-mm. achromatic objective having a 0.65 K.A. and a long working distance, a 4.3-mm. fluorite oil-immersion objective of 1.00 N.A., or an 8-mm. objective of approximately 0.50 N.A. For counts of particles per gram, the best optics are not essential, as it is necessary only to obtain separate images of each particle.
I
2
Figure 3.
3 4 5 6 7 8 9 IO II 12 Thousands of particles counted
13
Precision of lMeasurement of Particles per Gram
Effect of total n u m b e r o f particles counted on observed precision for S-2 crushed quartz powder fraction
Calibration of Fairs Eyepiece Graticules. The accuracy of projected-area measurements made with eyepiece graticules depends on the graticule calibration. The most precise calibratioii may be obtained by a photomicrographic method utilizing a stage micrometer as a standard of length. For the work descmt)ed in this paper, a stage micrometer ruled in divisions of 0.01 mm. and calibrated by the Sational Bureau of Standards was eniploj ed. Three photomicrographs of the calibrated stage micrometer were made vith each applicable objective, one for each of the three eyepiece graticules. Except for the replacemmt of the binocular body of the microscope with a monocular tube of equal tube length, the microscope was set up exactly 2s for projected-area measurements. On the resulting photomicrographs the ratios of graticule circle diameters to the appropriate divisioiis of the stage micrometer were determined by measurement with a coordinate comparator, which could be read nith greater accuracy than necessary. For all objective-ocular combinations, the resulting precision of the calibration ranged from &0.05Yo for the largest circle ( S o . 128, Figure 1 ) to zk49;b foi the smallest circle (Yo. 1 ) . DETERMINATION OF PARTICLES PER G R A l I
Measured p r o j e c t e d
diomeler, d p , microns
Figure 4. Curnulatire Particle Size-Distribution Curves of Five Crushed Quartz Powder Fractions
TITOequidimensional weighing bottles, one of which contains the powder sample, are dried at 110' C. Using the second bottle as a tare and weighing by difference, approximately 0.1 gram of the dried sample is transferred to a clean, graduated test tube containing 20 ml. of a liquid sufficiently viscous to retard settling of particles after agitation and nonvolatile to eliminate evaporation difficulties. Satisfact.ory liquids are corn sirup] glycerol alone or mixed with glycerol jelly, or a white mineral oil alone or mixed with a soluble terpene resin. Rubber gloves or tongs are used for all maiiipulat,ions of t.he weighing bottles. Filtered air is bubbled t.hrough t'he liquid to distribute the powder sample uniformly, as the use of a stirring rod may lead to mechanical disintegration of t'he coarser particles. The suspension is transferred to the (,ells of the count'ing chambers by means of a 2-mm. glass capillary tube. The cells must be scrupulously clean, Blank counts should be made, especially in the case of particles smaller than 10 microns, as the chambers may become contaminated by particles of dust from the air during the short time elapsing before a cover glass can be put in place. After a small amount is ejected from the capillary, the cells are filled by means of a blood-counting technique, such as that described by Fairs ( 7 ) . After all particles have settled in the cell, enough of each half is counted to give a total of at least 500 to 600 particles. The effective volume of suspension is determined by the number of ruled areas counted. A check count of the suspension should lie made on a second cell and then upon a second suspension prepared in the same manner. The number of sample suspensions that must be prepared and the total number of particles that
Procedure for Particles Larger Than 100 Microns. Slides may be prepared conveniently by ruling the concavities of niicroculture slides into small squares (about 1 mm. on edge) with a diamond-point pencil. hft'er the powder is dried at 110" C. in a weighing bottle, a sniall sample is placed on each of approximately six ruled concavities (depending on the sample size.) Weighing is done by difference. Rubber gloves or tongs are used for all manipulations of the weighing bottle. To disperse the sample, a- small amount of turpentine, Table I. Results of Graticule Projected-.hea JIeasurements glycerol, or ot,hrr liquid capable of wetSurface Area of ting the powder is mixed carefully with Of Of 'u-0. of Calcd. Specific it in each c0ncavit.y. Care must be Desig- Particles Average Areaa, Particles per Gramatb, Particles Surface Area". b I c , .vm Counted iSo. Sq.Cm./Cc. t,alren to avoid breaking the particles nation Measured 8.Sq. Microns by allowing the microspatula to grind s-1 2312 20.9 =k 0.3 (7.71 i 0.04) X 10'0 4547 (4.27 f 0 07) X 104 them against the glass (9, 19). All pars-2 1778 215.8 i 2.6 ('3.57 i 0.06) X 109 12275 (2.05 i 0.05) X 104 E-1 2388 1919 i 22 11.16 i 0.03) X 108 6184 13.90 i 0.21) x 103 ticles on the slides are counted as they M-1 2320 831'3 + 43 (2.10 = 0.00) X 10; 5387 (3.51 i 0.12) x 103 are moved across the field of the microRI-2 2228 9814 =t54 (1.16 0.02) X 10' 4053 (2.98 i 0.06) X 103 scope. The ruled lines are used as a The probable precision of each mean measurenient was calculated according to the equation: guidps, counting all particles that touch 0.6745 one line and neglecting t.hose that touch E = ___ d m X d(rn--tl)z + (m--tr)? 7 . . . , , + ( r n - - t n ) 2 the nest. The spatula is esamined as given by Burington (51, where E is the probable error of the mean of n measures, t l , t z , .,,.,., t,,, the arithmetic mean of which is m. niici,oscopically foi, any adhering parb lleasurrd density of 2.65 grams per cc. for quartz powder was used for calculations. ticles. The particles per gram, Xu, are C These results are somewhat higher than those presented in Table 111 of ( I S ) , as a result of imcalculated from the weight Of the sample prored graticule calibrations (described in text) and measurement of additional samples after ( 1 8 ) was submitted for publication. taken and the total number of particles in the sample (including any adhering
ANALYTICAL CHEMISTRY
1826
diameter of each powder as measured microscopicdly. (The mean projected Specific Surihce Areas. 69. Cm./Cc. diameter is the diameter of a sphere Method of with a surface mea equal to that of the Measurement 5-1 6-2 E-1 M-l Microscopic' (gmtiwle particleofmersnsurfacearea.) Theliquid method) projected-area 4.27 X 1 0 4 2.04 X 101 5.QO X 10' 3.51 X 108 permeahility and gas adsorption m a s G&8adsowtion?, 8.22 X 10' 2.24 X 10' 7.3 X 108 4.1 X 103 urements have been described and disLiquid perrneabdityb water 6.30 X 10' 1.58 X 101 5.08 X 10' 2.90 X 10' cussed by Johansen et al. (12). The m Iso~oothne 5.21 x 10' 1.35 x io' 4.54 x io1 2.72 x 1 0 8 sults obtained with the finest quartz See Table I, notec. powder, S-1, are not shown in Figure 5 Results of Johanaen et G I . (la). because the graticule projected-area method, as employed in this work. is . . not suitable for application to the finest particles in the S1 fraction. Photomicrographs of the M-1, E-1, S-2, and S-1 quart8 powder fractious are shown in Figures 6, 7, 8, and 9. Samples El, S-2, and S-1 were mounted dry in turpentine under cover glasses coated with cellulose acetate solutions. The M-1 mmple was mounted in cellulose acetate only. The marked similarity of particle shape in the different sized fractions is apparent from an examination of Figures 6 through 9. (3) Apparently d l four of these samples would he characterized hy the mme "shape factor." A photomicrograph of the M-2 fraction is not available, but its particles possessed similar shapes. Many of the quartz particles possessed a high degree of irregularity and angularity. Table 11. Comparison of Specific Surface Areas of Q u a r t z Powders
a 6
RESULTS OBTAINED WITH QUARTZ POWDER SAMPLES
The graticule projected-area method of surface-area measurement was developed particularly for use with sized fractions of a commercially availahle crushed quartB powder. Five fractions were separated by meshing, elutriation, and sedimentation. Four were identical with the powders used for liquid permeability studies by Johansen et al. (12). Particle size-frequency ourves determined by the improved microscopic procedure are presented in Figure 4. Reference to the curve for S 1 emphasizes the p r e ponderance of line particles, more than 80% of which were smaller than 3 microns, that limited the application of the graticule projected-area method.
I 1
Figure 6. Photomicrograph of M-1 Crushed Q u a r t z Powder Fraction 100 Miuona is 7 mm. on the scale of this enlargement
0.050
0.100
0.150
R e s i p r o e o l 01 m e o n P r O I e C r e d diomeler, I/dpm,mlsronr-l
Figure 5.
Comparison of Specific Surfaoe Areas o f Crushed Q u a r t z Powder Fraotions
Tahulated results of the microscopic measurements are found in Table I, including the probable precisian of each mean result. Tahle I1 presents a comparison of calculated mean specific surface area8 of the quartz powders 8 8 measured by the graticule projectedarea microscopic method, by gns adsorption, and by water and isa-octane liquid permeability techniques, The same data are plotted in Figure 5 against the reciprocal of the mean projected
After the conclusion of this work, electron microgrnphs of sample S1 were made avnilahle through the courtesy of the Continental Oil Co. A revealing field is presented in the micrograph reproduced as Figure 10. It is apparent that many of the qua& particles in the S I fraction were as small as 0.1 mioron and thus would be invisible in the light microscope. This, undoubtedly, is the reason that the graticule protected-area method was found not to be applicable to the S-1 fraction. DISCUSSION
The specific surface-area comparisons presented in Table I1 and Figure 5 emphasize the inherent difficulties of determining
1827
V O L U M E 26, NO. 11, N O V E M B E R 1 9 5 4 accurately the absolute surface area of particulate matter in the size range between microscopic particles visible to the unaided eye and particles of colloidal dimensions. Mioroscopic methods are commonly employed for such tasks, but the self-consistency of gas adsorption and liquid permeability results appears superior for powder samples containing an appreciable number of particles with projected diameters smaller than 3 micram. Before development of the methods presented in this paper for mounting subsievesize particles in random orientation, the projected-area method was applicable only to sievesize particles. The graticule projected-area. method has extended the range of applicability down to projected diameters of 1 to 3 microns. Graticule projected-ares measurements may be made on particles smaller than 3 microns with decreasing accuracy.
accuracy of the previous graticule calibration against secondary length standards. No other microscopic method of surface-area messurement is absolute (with the exception of other projectedarea procedures). I n view of its many advantages, i t is uufortunate that, iu its present state of development, the accuracy of the graticule projected-area. method decreases rather sharply as particle projected diameter deoreases below 3 microns. The method becomes inapplicable a t l micron. These factors are responsible for this limitation:
the light mioroscope. &en t6;s precision is most difficult to attain. [Fairs (7) found that use of a monochromatic sodium
ments of 3- and 0.5-mkron particle$; resiectbely. 2. The only successful method of obtaining random orienta tion on subsievesize sample mounts necessitates the use of objeotives having suitable depths of field. This eliminates the possibility of using ail-immersion objectives having numerical apertures much greater than 1.00 or objective magnifications much above 45 X and increases the minimum uncertainty discussed above. 3. With the available objectives only the Nos. 1, 2, and 4 circles of the Fairs graticules (see Figure 1 ) are comparable with the projected areas of particles smaller than about 3 microns. This may result in a distorted frequency distribution of measured projected areas. To minimize this difficulty a cumulative siaedistribution curve of projected area. v e r ~ uper ~ cent below given area. may be used to calculate the surface area of the particle of average surface and decrease the resultant error.
micrograph of E-1 Crushed Quartz Powder Fraction ~ mon . the scale of this enlargement
Microscopic methods of surfacearea measurement yield sizefrequency distribution curves (as in Figure 4 ) in addition to areas, the only result of adsorption or permeability procedures. Furthermore, the giectness of the microscopic method lends confidence in the belief that it comes closer to measuring the "true" geometric surface than the other techniques. Gas adsorption surfaoe-ares mcaaurements would be expected to be higher than those made by microscopic methods, beoause minute blind cracks and crevices ou particle surfaces would be accessible to adsorbed gas molecules but invisible under the microscope. Neither is it surprising that liquid permeability measurements of specifio surface are lower than the microscopic. s of flocculation and packing in liquid ised fully by Johausen et al. (le), it is ,liquid flow paths do not encounter the adjacent to points of contact of neighher hand, differences in calculated sur,m an unfortunate choice of assumed lculation of gas adsorption and liquid 3 such constants are assumed in the nicroscopic method. results depends solely on the maguix introduced by the observer and the L
~-.."~.,
The specific surface mea of the S I quart% .-Iculated from graticule projected-area measurements, is not consistent with the data shown in Figure 5. [Quantitative reference to Figure 2 of Johansen et al. (fB) indicates that the mioroscopic area is much too low.] The reason for this discrepancy has been discovend to be the vew " auwreoiable number of uarticles in the 9-1 fraction below about 0.5 micran. Neither the graticule _I
..
1828
ANALYTICAL CHEMISTRY
projected-area method nor any light microscope measurement method should be applied to particle surface-area determination if an appreciable number of samule particles is below the limit of resolution. A possible solution to this problem is suggested when samples consist solely of particles that may be characterized by the same shape factor. The graticule projected-area method may be applied to the measurement of it fraction hsving all particles larger than 3 microns. Additional sample slides of the coarsest fractions may be prepared with the particles in stable orientations snd one of the diameters such 8 s that of Martin et al. (15) measured. From these data the requisite shape factor necessary to reconcile results of the two methods may be calculated, and this factor may he assumed to apply to the finer fractions. In this manner measurements may be extended to finer powders by the use of Zmm. oil-immersion objectives. Probably mom important are the potentialities of utilizing this procedure with the electron microscope, for which random orientation mounts would be difficult to prepare.
Figure 10. Electron Micrograph of S-1 Crushed Q u a r t z Powder Fraction College of Science and Technology, London, England; C. E. Barnett, New Jersey Zinc Ca., Palmerton, Pa.; T. G. Rochow, American Cyanamid Co., Stamford, Conn.; R. P. Lovelsnd, Enstmen Kodek Co., Rochester, X. Y.; and P. A. Loren%and R. T. Johitnsen of this station. LITERATURE CITED
Figure 9. Photomicrograph of S-1 Crushed Q u a r t z Powder Fraction 50 M i s m ~ sis 24 mm. on the male of this enlargement
The authors are indebted to a reviewer who has called their attention to sample preparation techniques used in electron microscopy wherein finely divided solids may be dispersed in a plastic supporting membrane rather than on it. Random orientation of the sample particles should be attainable by these teehniques. The graticule projected-area method then could be applied to enlarged electron micrographs. Measurements of projected area could be made by using enlarged prints of the graticules on film transparencies. The high resolution and great depth of focus of the electron microscope thus would overcome deficiencies encountered when the light microscope is applied to very h e particles. This suggestion should he applicable to particles in the submicron particle sim range, but not to particles having one or more dimensions approaching the thickness of the supporting membrane. ACKNOWLEDGMENT
The writers are indebted to the following far valuable discussions and criticisms of this work: Harold Heywood, Imperial
(1) Amberg. C. R.. I . Am. Cerom. Soc.. 19, 207 (1936). (2) Barrett. H. M., Birnie. A. W,, and Cohen. M., J . Am. C h m . Soc., 62,2839 (1940). (3) Burington. R. S., "Handbook of hhthemittiod Tables and Formulas," 2nd ed., p. 261, Sandusky, Ohia'Handbook Publishers, Ino., 1940. (4) Cauchy. A,, Compt. rend., 13, 1060 (1841). ( 5 ) Charnot, E. M., and Mason, C. w., "Handbook of Chemical Microsoouy," Vol. I, 2nd ed.. pp. 14-21, 38f-7, New York, John Wiley & Sons, 1938. (6) Dallavalle, J. M., "Micromeritics," 2nd ed., pp. 41-64, New York, Pitmen Publishing Co.. 1948. (7) Fairs, G.L., Chemistry & Industnd, 62,374 (1943). (8) Gage, S. H., "The Microscope," 17th ed.. pp. 452-3, Ithaca,
N. Y., Cornstock Publishing Co.. 1E147. (9) Green, H., J . Franklin Inst., 192,637 (1921) (10) Hemood, A,,Bull. Inst. Mining Met. , NO.477 (1946). , ? . . . I . * % c,.. 909 ,.nl)l)\ (11) Hemood, H.. Proe. Imt. Mech. Engrs. lYvlrvvrrl, 1111,voy \ I a y y I . (12) Johansen, R. T.,Lorenu, P. B... Dodd. C. 0.:. Pideeon. I . F. D., and Davis. J. W., J . Pi'lys. Chem., 57, 40 (1953). (13) Kenrick, F. B.,J . Am. C,em. see., 62,2838 (1940). (14) Langille, A. C., Braid, P. E., and Kenrick, F. B., Con. J . Re search, 23B, 31 (1945). (15) Martin. G.,Blythe. C. E., and Tongue, H., Tmns. Ceram. Soc., 23, 61 (19234). (16) May, K. R., J . Sei. Instr., 22, 194 (1945). (17) Patterson, H. S., and Cawood, W., Tmns. Faraday Soc., 32,1084 (1936). (18) Shillaber, I2. P., "Photomiorography in Theory and Practice.' pp. 93-6, New York, John Wiley & Sons, 1944. (19) Silverman, L., and Franklin, W., J . Ind. Hug. & Tozicol., 24,5 1 (1942). (20) Tooley, F. V., and Parmelee, C. W.. J . Am. Cwam. Soc.. 23.304 (1940). (21) Vouk, V., Notwe, 162,330 (1946). '-
~~
~
-
RECEZVED for review September 19, 1953. A o o e p i e a ~ u g u a r ~1.~ 2 4 . rresented at the XIIth International Congress of Pure and Applied Chemistry, Section 14, Physical and Inorganio Chemistry. New York,N. Y..Septcmber 13, 1951.
