April, 1959
PARTICLE SIZEDISTRIBUTION MEASUREMENTS OF POWDERED SUBSTANCES
53 1
THE AGREEMENT BETWEEN INDEPENDENT METHODS FOR PARTICLE SIZE DISTRIBUTION MEASUREMENTS ON FINELY DIVIDED POWDERS INCLUDING PHOSPHATES BY D. P. AMES,R. R. I R A N IAND ~ C. F. CALLIS Research Department, Inorganic Chemicals Division, Monsanlo Chemical Company, St. Louis 26, Missouri Received AuOust 8, 1068
The particle size distributions of various powdered substances as measured by a microscopic electronic sizing and counting technique and by an automatized liquid sedimentation balance are shown to exhibit log normal distribution behavior. The results of these two independent methods are shown t o agree with the predicted values based on the transformations involving log normal distribution behavior. Screen analysis is critically evaluated. The geometric mean diameters on a weight basis, M,, for the samples investigated are in the size range 4-70 1.1. The geometric standard deviations, ug,of these samples are found to lie between 1.3 and 3.5. The average deviation between the observed and calculated M gand u, values are 6 and 5%, respectively.
Introduction The interest in the measurement of particle size distributions has increased over the past few years2+ due to the necessity for better characterization of powdered materials. This study was initiated in order to correlate independent methods for these measurements. In addition to making it possible to evaluate the soundness of these methods, such a correlation would further test the mathematical transforms involving log normal d i s t r i b u t i ~ n . ~ - lThe ~ methods chosen for this correlation are direct sizing and counting, liquid sedimentation and screen analysis. The search for a direct sizing and counting method revealed that particle size distributions could be obtained automatically using flying spot scanning.6~’~Flying spot scanning is a method of converting a two dimensional density distribution into a varying voltage-time relationship. An intense light spot slowly scanning a cathode ray tube is optically imaged down through a conventional microscope. This imaged down light spot scans a fixed microscopic field. The transmitted spot intensity alteration as received by a photomultiplier is amplified, counted and monitored. Particle sizing is performed by pulse width selection; all pulses below the preset size are not recorded during the count. Since the determination of particle size distributions by liquid sedimentation and screen analysis has been thoroughly reviewed by various no additional description is necessary. (1) Presented at the September, 1958, meeting of the American Chemical Society in Chicago. (2) “Symposium on Particle Sire Analysis,” Suppl. to Trans. Inst. Chem. Engrs. (London), 1947. (3) J. M. Dalla Valle, “Micromeritics,” 2nd Ed., Pitman Publishing Co., New York, N. Y.,1948. (4) G. Herdan, “Small Particle Statistics,” Elsevier Publishing Co., New York, N . Y., 1953. (5) J. J. Hermans, “Flow Properties of Disperse Systems,” Interacience Publishers, Inc., New York, N. Y., 1953. (6) “Conference on the Physics of Particle Size Analysis,” Suppl. No. 3, Brit. J . A p p l . Phys., 1954. (7) H. E . Rose, “The Measurement of Particle Size in Very Pine Powders,” Chemical Publishing Co., New York, N. Y., 1954. (8) R. D. Cadle, “Particle Size Determination,” Intersoienco Publishers, Inc., New York, N. Y., 1955. (9) H.Green, J . Franklin Inst., 204, 713 (1927). (10) T. Hatch and S. P. Choate, ibid., 207, 369 (1929). (11) T.Hatch, i b i d . , 216, 27 (1933). (12) F. Kottler, ibid., 260, 339 (1950). (13) F. Kottler, ibid., 261, 617 (1951). (14) J. Z. Young and F. Roberts, Natu-e. 167, 231 (1951).
Experimental Sources of Materials.-All powdered samples investigated in this work were Monsanto pre arations except for the glass beads, which were rocured From the Minnesota Mining and Manufacturing 8ompany. The solvents used as dispersion fluids were standard C.P. grade and no further purification was erformed. Procedures.-%he particle number size distribution measurements were made with a Flying-Spot Particle Resolver manufactured by Cinema Television (Cintel) Co. Ltd. (London).’* The microscope utilized as an integral component of the resolver is a Baker Model 4, and the apochromatic optics were purchased from Bausch and Lomb Optical Company. Prior to making the measurements, the powder must be dispersed on a microscope slide. These dispersions were obtained by one of the following methods: (1) touching a powder coated brush to a clean slide, (2) evaporating the solvent from a liquid dispersion, (3) counting the particles as a dispersion on a covered slide. The instrument calibrations14 were checked (Fig. 1) by visually sizing and counting several powders as given by ASTM Designation E 20-51T, 1951. The resolving power of the “Cintel” is 0.6 1.1 and is a stable characteristic. The reproducibility of the count and size measurements as obtained by the Flying-Spot Resolver is shown in Table I. Triplicate counts of different representative microscopic fields were made in all of the reported cases in this article.
TABLE I CINTEL FLYING-SPOT RESOLVERREPRODUCIBILITY WITH DICALCIUM PHOSPHATE DIHYDRATE~ Ull
Total no. of particles counted
2.28 2.07 040 2.68 2.00 680 2.58 1.98 720 2.78 1.96 1040 2.54 1.94 800 2.53 2.10 750 2.45 2.05 820 2.67 2.06 690 2.52 2.11 700 2.77 1.93 780 Av. 2.58 2.02 790 yostand. dev. f5.5 h3.0 a M , and (I have , the same meaning as defined by equations 5 and 6 . The particle-weight size distribution measurements were made by a Rostock’6 type liquid sedimentation balance. The balance, manufactured by A. Gallenkamp and Company (London), was modified in order to omit the required continuous o erator attention. The modifications are: (1) the core 0?033S-L Schaevitz’G linear variable differential (14) W.K. Taylor, page 5173 of ref. 6. (15) W. Bostock, J . Sci. Instr., 29, 209 (1952). (16) Schaevits Engineering, Camden, N. J., Bulletin AA-1 A.
D. P.AMES,R.R. IRAN^ AND C. E’. CALLIB
532 60 40
agglomeration of the powdered particles. The volume fraction of the powder in the dispersion was not allowed to exceed 0.2’#,. The inter-agreement in the particle size data when di erent solvents are employed further roves that proper deagglomeration is achieved (see Fig. 37. The reproducibility of a typical weight size measurement is shown in Table 11. The particle densities were measured pyonometrically,” and the fluid densities and viscosities were obtained from the International Critical Tables. The screen analysis waa performed with standard U. 8. Screens and a “Ro-Tap Sieve Shaker.”
20
d
c
Vol. 63
lo
3i 5
TABLE I1 THEREPRODUCIBILITY OF THE SEDIMENTATION APPAIULTUS WITH ANHYDROUS MONOCALCIUM PHOSPHATE IN ISOBUTYL
2
ALCOHOL”
0.05 0.2 1
6
20 40 60 80
95
99
% by no. greater than. Fig. 1.-Cintel calibration check on two apatite sand samples: 0,Cintel data, sample 1; 0, microscopic counting based upon ASTM E20-51T, 1951, sample 1; 0,Cintel data, sample 2; H,microscopic counting based upon ASTM E20-51T 1951,sample 2.
Mdd
a
43.3 41.8 43.1 40.9 41.8 43.5 41.8 39.2 43.1 39.3 41.8 f3,9
1.48 1.58 1.45 1.55 1.58 1.60 1.62 1.58 1.67 1.70 1.58 *4.8
Av. yo stand. dev. a M aand u, have the same meaning as defined by equations 5 and 6.
Treatment of the Data The liquid sedimentation data were interpreted to give particle size distributions by the methods outlined by Odenlg and modified by Bostock16
DISPLACEMENT
u6
w = p - - ddInP t where P is the settled weight at time t, and W is the weight of that fraction of settled particles whose diameter d would allow sedimentation through the entire height of the column h as given by Stokes’ equation
q and
CORE’ RECORDER
Fig. 2.-Sedimentation apparatus. transformer is fastened to the balance beam, while the transformer body is attached to the balance assembly housing; (2) the output of the transformer secondary coils is amplified with a Daytronicl? Model 300 Displacement Indicator and recorded on a 0-100 MV Brown Electronik Minneapolis Honeywell recorder with the proper chart speed; (3) two solenoids and a se uence switching arrangement are installed to facilitate %e operation of the balance and (4)the original aluminum balance pan was replaced with a stainless steel pan to avoid aluminum dissolution in alkaline media. The block diagram of the modified balance is shown in Fig. 2. For the sedimentation experiments, the dispersions were obtained by the gradual dilution of the weighed sample powder, and examined microscopically to verify thc do__
(17) Daytronic Corporation, Dayton, Oliio.
p2 are the viscosity and density of the sedimentation fluid, respectively, g the acceleration due to gravity, and pi the particle density. For some of the powders investigated in this work, it was shown that the particle density is independent of particle size. It was found that the recorder reading in the sedimentation set-up is linearly proportional to the settled weight; thus, it can be shown easily that the recorder reading may be used directly to calculate the size data without converting to actual weight. Since the Flying-Spot Particle Resolver gives the number of particles greater than a chosen diameter, the various percentages of these particles readily are calculated by normalizing the total number of counted particles. Results and Discussion Electronic Sizing and Sedimentation Compari-
(18) F’. Daniels, J. H. Mathews, J. W. Williams and Staff, “Experimental Physical Chemistry,” 4th Ed., McGraw-Hill nook C o . . Inc., New York, N. Y.. 1949, p. 430. (ID) 8. Oden, Proc. Roy. SOC.Edinburgh, 86. 219 (1015).
I
.
i
I
PARTICLE SIZEDISTRIBUTION MEASUREMENTS OF POWDERED SUBSTANCES
April, 1959
533
200
100 i
#a 50
3 R
20
0.05 0.2 1 5 20 40 60 80 95 yo by wt. greater than. Fig. 3.LInsoluble sodium metaphosphate: 0,sediment? tion m ethyl alcohol; . , sedimentation in hexane; 0,sedmentation in isobutyl alcohol.
0.05 0.2 1 5 20 40 60 80 95 % by wt. greater than. Fig. 4.-The particle size distribution of a sodium isethionate: D, m e e n analysis; 0, sedimentation data in Zethylhexanol. 200
son.-The size frequency relationship of powdered substances ordinarily can be represented by a lognormal distribution 1 exp[-1/2 (d) = d Z l 0 g u
(logd log -logM)’] Q
(3)
where f(d) represents the probability of occurrence of a diameter d, and M is the geometric mean diameter
140
200
j
270
50
400
n
8
3
z
20
OI
f(d) = 1
0.05 0.2 1
(4)
5
20 40 60 80 90
98
% by wt. greater than.
d-0
Fig. 5.-The comparison between screen analysis and the more precise methods on anhydrous monocalcium phosphate. Screening: 0,sample no. 18; A, sample no. 3 sedimentasample 110. 18; 0 sample no. The geometric standard deviation u can be repre- tion in isobutyl alcohol; ., 3 C i t e 1 data converted to weight. sented by log M
log u =
Zni log d Zni
---
-
C
(5)
Zni (log d log M)*]% Zni
100
(6)
Furthermore, it can be shown readily from the a 50 following definite integrals that M may be eval- 3 uated by locating on a log probability plot, e.g., Fig. 1, the diameter above which 50% of the par20 ticles lie; whereas, u represents either of the ratios in equation 9
270 325
$
$
z
0.050.2 1 5
JoMf(d) d In d =
l/2
l z f ( d ) d In d = S,M/’f(d) d in d = 0.3413 u = E
$ ;
20 40 60 80
95 99 99.9
(71
% by wt. greater than.
(8)
Fig. 6 . S c r e e n classification: 0,sedimentation data in %butanol for -270 mesh, +325 mesh anhydrous mono, sedimentation data in 2rethylhexanol calcium phosphate; . for -270 mesh, +325 mesh sodium isethionate.
diameter a t 15.870/, cumulative oversize
M M diameter a t 84.130/, cumulative oversize
The geometric mean diameters on a number and weight basis, Mnand M,, respectively, have been shown to be interrelatedii In M , = In M, + 3 Ina un (lo) It should be noted from equation 10 that the errors in M , and un are magnified when computing M,. Table I11 shows that the calculated values of M , according to equation 10 agree very well with the measured ones for various powders. In addition, u, and an are identical. It is interesting to note that the majority of the particles for the calcium phosphates lie in the sub-
sieve range with the exception of monocalcium phosphate monohydrate which has an M, and u, of 71.5 p and 3.54, respectively. For tricalcium phosphate, with an M , of 4.4 p, it was not feasible to employ sedimentation techniques due to the failure of Stokes’ law for extremely fine particles.’ All the powders that were investigated in this work did not differ markedly from sphericity. Unless this was true an agreement between the sedimentation and the electronic sizing technique would not be expected; the sedimentation technique yields particle size by computing the diameter of a sphere that falls with the same velocity as the particle, while the direct counting technique yields a random average of the projections of diameters of a particle resting under equilibrium.
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Vol. 63
GRAHAM WILLIAMS TABLE I11 AQREEMENTBETWEEN PARTICLE SIZEDISTRIBUTIONS FROM SEDIMENTATION AND ELECTRONIC SIZING Powder
Sand Glass beads Anhydrous monocalcium phosphate Monocalcium phosphate monohydrate Anhydrous dicalcium phosphate Dicalcium phosphate dihydrate Tricalcium phosphate Calcium pyrophosphate Insoluble sodium metaphosphate I n microna. Calculated from the
Dispersion fluid
Water Butyl alcohol Isobutyl alcohol 2-Ethylhexanol Ethanol 0 . 1 % Na hexametaphosphate in water Ethanol Ethanol Ethanol measured M , and u,, and equation 10.
Sieve Analysis Evaluation.-For those powders where ca. 80% or more of the particles have diameters greater than 44 p (325 mesh U. S. Screen), the particle size distributions as determined by sieving and sedimentation are identical, provided there is no significant amount of large particles to cause turbulent flow. Figure 4 demonstrates the agreement for a sodium isethionate sample. However, Fig. 5 shows that when ca. 35% or more of the particles of a powder have diameters smaller than 44 p , sieve analysis is inaccurate and fails to show significant differences for two powders having different particle size distributions. The results plotted in Fig. 6 offer a plausible explanation for the above described behaviors. Thus, for the case of sodium isethionate the geo-
Measured m
Mn"
5.3 23.6 4.6 0.61 2.5 2.05 1.16 2.08 3.0
1.96 1.38 2.38 3.54 2.22 1.96 1.95 1.95 2.07
Measured Mg'
ug
Ca1cd.b Mg" ug
1.96 1.38 2.38 3.54 2.22 1.96 .. .. 1.95 8.4 2.15 7.9 1.95 11.2 2.24 10.1 2.07
21.2 31.0 45.0 70.0 18.8 9 .O
2.00 1.34 2.18 3.50 2.22 2.09
20.7 32.5 43.5 71.5 16.8 8.0 4.4
metric mean diameter of a sieve cut as determined by sedimentation approximately equalled the average diameter of the two screen openings. However, for the anhydrous monocalcium phosphate sample investigated, the actual geometric mean diameter of a sieve cut is significantly different from the average diameter of the two screen openings. Figure 6 further shows that a significant portion of the powder lies outside the screen opening 1imits.l' I n addition, this figure demonstrates that exactly the same screens give different size classifications for different powders. Consequently, accurate sieve analysis is obtained only when the sieves are pre-calibrated against the more precise methods for the individual powder under investigation.
THE MEASUREMENT OF DIELECTRIC CONSTANT AND LOSS FACTOR OF LIQUIDS AND SOLUTIONS BETWEEN 850 AND 920 MC./SEC. BY MEANS OF A COAXIAL TRANSMISSION LINE BY GRAHAM WILLIAMS The Edward Davies Chemical Laboratories, University College of Wales, Aberystwyth, Wales Received July 3,I068
A coaxial line method of measuring the dielectric properties of liquid systems in the frequency range 250-1000 Mc./sec. (or higher) is described. The method is essentially that of Roberts, Westphal and von Hippel. Data are presented for typical systems of high and low E' and e" values showing results which compare favorably with other methods in this region. Evaporation losses, access of moisture and temperature are readily controlled for the liquids.
(1) C. N. Works, T. W. Dakin and F. W. Boggs. Proc. Inst. Radio Eng., 88, No. 4, 245 (1946). (2) C. N. Works, J . A p p l . Phya., 18, 605 (1947). (8) 8. J. Reynolds, Uenerol Elect&c Rev., 80, No. 9, 34 (1947). (4) J. V. L. Parry,Proc. Inst. Blectr. Eno., 98, pt. 111, 303 (1951). (5) D. L. Hollowry and G. J. A. Cassidy, ibid., 99, pt. 111, 364
fication of an i,mpedancemethod of Cole12by Marcy and Wyman. lS However, elaborate equipment is required, and in some cases only moderate reproducibility of results could be achieved. The present apparatus offers an excellent alternative, since the effects of stray inductance and capacitance are eliminated by confining the electric field between the coaxial conductors and treating the system in terms of distributed circuits. The apparatus could be used for lower frequency work, the lower limit being fixed 'by the length of coaxial transmission line available, and the measurements could be extended
(1952). (6) J. H. Beardsley, Reu. Sci. Inatr., 84, No. 2, 180 (1953). (7) R. A. Chipman, J . A p p l . Phys., 10, 27 (1939). (8) W. L. G. Gent, Trans. Faraday Soc., 60, 383 (1954). (9) H. Linhart, 2. physik. Cham.. BSS, 2 1 (1937). (IO) J. B. Bateman and G. Potapenko, Phgs. Rev., 67, 1185 (1940).
(11) W. P. Conner and C. P. Smyth. J . Am. Chem. SOC.,64, 1870 (1942). (12) R. H. Cole, Rev. Sei. Instr., l a ( 6 ) . 298 (1941). (13) H. 0. Marcy and J. Wyman, J . Am. Chem. Soc., 63, 3388 (1941).
The frequency range 100 to 1000 Mc./sec. has proved in the past to be a difficult region for the measurement of dielectric constant and loss factor. The difficultiesarise from residual lead and electrode inductance, resistance and capacitance, and the correction of these factors. Measurements have been achieved using re-entrant cavities1-6 and parallel transmission lines7-" and also by a modi-
