Method for Particle Size Distribution for the Entire Subsieve Range A. E. JACOBSEN AND W. F. SULLIVAN Titanium Division, National Lead Company, Research Laboratory, Sayreville, iV. J . than a pan has been employed for the determination of particle size distribution for the subsieve size range down to 1.0 micron. The use of a buoyancy factor increases the convenience of the method and decreases the time for a complete determination. Examples show the use of the cylindrical cup and centrifugal sedimentation methods for obtaining composite particle size distribution curves for the complete subsieve range.
Sedimentation procedures are based on fundamental principles and have proved suitable for the ,determination of particle size distribution for the subsieve size range. The theory underlying the use of a sedimentation pan, specific gravity cylinder, and cylindrical cup is discussed. The mathematical equivalence of the sedimentation and specific gravity methods is shown. A modified sedimentation balance which uses a tall cylindrical cup rather
F
do = density of suspension outside the cup a t time t o Vi = volume of ith element of cup wall d: = density of suspension inside the cup a t time t o V,! = volume of ith element of suspension inside the cup
REQUENTLY it is highiy desirable to determine the particle size distribution of finely divided material for the entire subsieve size range. The microscopic method appears to be the only direct means for such determinations. The count procedure, however, is laborious and of doubtful value for particulate materials of the size bordering on the resolving power of the optical system. Sedimentation procedures serve as alternate methods, but unfortunately no single method has been found which will accommodate the entire subsieve range. From a practical point of view gravitational methods are satisfactory only down to about 1.0 micron and for material with finer particle size the centrifugal sedimentation methods are necessary. ’ A preferred simple and rapid gravitational method has been developed which is fundamentally based on OdBn’s balance method, but differing essentially in that a tall cylindrical cup is employed in place of the balance pan and the weighings are done manually rather than by elaborate equipment. (Since this work was completed it has come to the authors’ attention that an apparatus which also employs a cylindrical cup has been patented by Obenshain and Welton, 12). Complete sedimentation is avoided by the use of a buoyancy factor in the calculation of the weight settled. Centrifugal sedimentation methods have been worked out and where the particle size range requires the use of the two methods it is possible to combine the data to obtain a complete cumulative distribution curve.
n
summation from 1t o n layers
=
i=l
GENERAL THEORY
-1mathematical theory of the various general methods available for determining the particle size distribution of materials was first given by O d h (IS, 16). The following treatment is concerned with the use of a cylindrical cup, but is also designed to emphasize the relationship betrveen the sedimentation and specific gravity methods for determining particle size.
General Case. Figure 1 shows a cylindrical cup which is suspended from the arm of an analytical balance by a thin wire and immersed in a fluid. In particle size work the fluidsare usually restricted t o suspension and suspension medium, and in general the latter may be taken as different inside and outside the cup. The cup may further be thought of as divided into small elements of volume in which the wall elements of volume are to be distinguished from the elements of volume inside the cup. Then, according to Archimedes’ principle, we have initially
c w,n
wt,
=
i=l
n
&Vi i=l
+
n
i=l
-
Figure 1. Cylindrical Cup Attached to Balance
For the sake of simplicity both the suspending wire and the bottom of the cup may be thought of as being extremely thin and hence have been neglected. “. At any later time, t , during which sedimentation has t a k z place the weight on the balance arm is given by n
*
n
doV:
(1)
i=l
n
n
n
where W t = weight on the balance arm a t time t di = density of suspension outside the cup a t the ith layer a t timet d l = density of suspension inside the cup a t the ith layer a t time t and in which it is assumed that there are no additional buoyant forces nor convection currents due to depletion of material underneath the pan.
where Wto = weight on the balance arm a t time t o Wi = weight of i t h element of volume of cup wall
855
V O L U M E 19, NO. 1 1
856 By subtracting Equation 1 from Equation 2 we obtain n
To,
-
n
Wr T
Dm
2b
25
io
(P~ICRON~)
&
qb
Cumulative Weight Per Cent Curve for Quartz 0. Sedimentation balance 0 . Microscopic c o u n t
DISPERSIONTECHNIQUE. In cases where a complete distribution waa required, the same suspension was used with both the cylindrical cup and the centrifuge. The manner of preparation of the suspension depends on the nature of the suspended material and the suspension medium and examples of suitable dispersion techniques are given below. T h e particle size distributiorl obtained is dependent on the degree of dispersion achieved, and different distributions may be obtained using different dispersion techniques (8). In all cases the temperature was held a t approximately 25' C. Variation of Rim Height. I n order to show that it is necessary to keep the distance between the top of the rim and the surface of the suspension at a minimum to prevent loss of sedimented material, a series of experiments was made varying this distance. The suspension used contained 30 grams per liter of commercial barytes in water and had 0.3% sodium hexametaphos hate (based on the weight of barytes) as a dispersing agent. Lata, obtained by allowing settling to proceed until the supernatant suspension was only slightly turbid, drying the pan plus sedimented material, and weighing the latter, are given in Table I. The suspension height was 12.0 cm. in all cases. When the procedure was carried out with other materials using both a flat pan and a cylindrical cup, losses resulted with the
Figure 3.
Sedimentation Curve for Quartz 0 . Small t i m e male 0. Large t i m e scale
N O V E M B E R 1947
*
selected because of its almost spherical particles and the micioscopic projection count method was maae according to usual procedure (f ), Aroclor 4465 (Monsanto Chemical Company) wa5 used for mounting the powder and over 2000 particles were counted. The results are shown in the form of a cumulative distribution curve in Figure 2. Before use with the sedimentation balance, the quart6 powder was dispersed in water at a concentration of 30 grams per liter by agitating for 10 minutes in a Hamilton Beach mixer, using 0.3y0of sodium hexametaphosphate as a dispersing agent. The sedimentation results are summarized in Table I11 and the cedimentation-time curve is given in Figure 3. Because of thc.
Table IV.
Cumulative Distribution of Quartz
Microns
Timea Min.
40.0 30.0 20.0 15.0 10.0 7.0 5.0 3.0 2.0
1.2 2.2 5.0 8.8 19.9 40.6 79 6 221.0 497.0
D "C
Tangential Intercept b
100
hD"
F(D)dDc
70
%
10.5 16.0 26.5 34.0 49.0 58.5 65.5 79.5 89.0
89.5 84.0 73.5 66.0 51.0 41.5 34.5 20.5 11.0
a From Equation 17 using 7 = 0.00894 poise, h = 12.0 om., d e = 2.65 grams per cc., d m = 0.997 gram per cc.. and Q = 980 cm. per aeo.1 we have:
t =
l80'00894 ' 'O = 1'1?4 (2.65 - 0.997) 980Drn* D m'
seconds, If t is expressed in
minutes and D , in microns, then
Table 111. Wr
Time Min.
Sedimentation of Quartz,
-
Wt,
Grams
0 0.5 1.0 1.5 3.0 5 10 16 26 36 57 108 162 249 352 432 1320
W,"
100 X p b
(?Tam8
7%
0 0.864 1.057 1.375 2.030 2,335 2.895 3.295 3.660 3.935 4.255 4.655 4.865 5.070 5.190 5.250 5.760
0 15.1 18.4 24.0 35.4 40.8 50.5 57.5 63.8 68.7 74.2 81.2 84.9 88.4 90.6 91.6 100.5
a From Equation 15, using dn warns per cc., we have: W a =
= d m = 0.997 gram per cc, and d, = 2.65 1 1 - 0.997/2.65 ("" - wto)
b
16'-
Wa = 1.60 ( T V t - W t o ) Froin Equation 12; W , is calculated from Equation 16 as follows: = o l-o30 o
x
(Lg)' x
x
12.0 = 5.73 grams.
h c
From Figure 3. From Equation 18'.
wide particle size range covered it is necessary t o use two time scales on the abscissa. Two small errors are introduced in the calculations of Table 111, neither of which is significant in practice. The first occurs in calculating the W , column from Equation 15 in which the suspension medium density rather than the variable d, is used. The second error is caused by the use of the inside cup diameter rather than the outside diameter in calculating Woa, The calculation of the particle size distribution from these data is illustrated in Table IV and the cumulative curve i. shown in Figure 2. Considering the wide differences in tcchnique used, the agreement with the microscopic count results is good. Reproducibility of Results. Since the determination of particle size with the sedimentation balance involves only the opera-
P
,i
O'J
'
Figure 4.
~
JO 2
I
/00
4
I
,
I
,
,
250 300 350 /50 200 6 8 /O 12 /4 r t M E ( M I N U T E S )
I
400
450
/6
Sedimentation Curve for Coarse Rutile Powder 0 . Run 1 0. R u n 2
4 9S
f0
15
Dm
20
2.5
30
35
40
(MKRONJ)
Figure 5 . Cumulative Weight Per Cent Curve for Coarse Rutile Powder
V O L U M E 19, NO. 1 1
860 tion of weighing with the analytical balance, the results are highly reproducible. This is illustrated by the case of a coarse rutile powder which was dispersed by a technique similar t o that used for the quartz powder. I n this case duplicate runs were carried out starting from the powder in each case. The sedimentationtime curves, as shown in Figure 4, show that all operations including the dispersions were reproducible in this case. The cumulative distribution curve is shown in Figure 5. Applications. I n addition t o the two previous examples, which also serve to illustrate the application of the sedimentation balance, the particle size distributions of other materials have been determined.
4 ,
I
0.2 0.3
0.5
1.0
Dm
I
,,,,,
2.0 3.0 SO (MICRONS)
~
10.0
I
,
2ao
, , s
Figure 7. Cumulative Weight Per Cent Curves for Incompletely XI illed Titanium Dioxide Pigments 0 . Centrifuge data 0. Sedimentation balance data 325-mesh sieve d a t a A. Centrifuge data 0. Sedimentation balance data
X
I
5
I
/O
,
I
/5
Dm
Figure 6.
20
25
I
30
35
,
40
(MIC~ONS)
Cumulative Weight Per Cent Curves 0 . Ground ilmenite
0 . Commercial zinc dust 325-mesh sieve
@
. -
Figure 6 shows two examples, the first being a ground ilmenite ore dispersed in water using 0.3c, of sodium hexametaphosphate as a dispersing agent, and the second a commercial zinc dust dispersed in a mixture of 25% mineral oil and 7 5 7 , kerosene with 3y0 of stearic acid added as a dispersing agent. In the former example the weight per cent passing through a 325-mesh sieve (44microns) is also given. An illustration of the use of both the sedimentation balance and the centrifuge t o secure a complete distribution curve for material having an appreciable w i g h t fraction of particles below as well as above 1.0 micron is shown in Figure 7. Since the method of obtaining the centrifuge data has been described (B), only the end results are given here. In the first example an incompletely milled titanium dioxide pigment (Xo. 1) was dispersed in water using 0.3y0 of sodium hexametaphosphate as a dispersing agent, and in the second example another incompletely milled titanium dioxide pigment (KO. 2) vivas ground in a linseed oiltung oil vehicle (known t o the paint trade as VM-1215) and diluted with mineral spirits. The matching together of the curves in the 1.0 micron region is very good, and serves to substantiate the fundamental “correctness” of both methods. The weight per cent passing through a 325-mesh sieve is also repeated in Figure 7. The data presented represent only a small fraction of those collected with the centrifuge and sedimentation balance. I n some cases it has been found convenient with tlTe balance method
.
1
For incompletely milled titanium dioxide pigment 1
}
For incompletely milled t i‘ t a n‘i u m dioxide pigment 2
to select a high-viscosity suspension medium in order to secure .an accurate particle size distribution of very coarse material, whereas in other cases, where a rapid determination of the distribution be lo^ 10 microns was desired, cylindrical cups having approximately half the height (5.90 cm.) of those described previously n’ere used. LITERATURE CITED
(1) Am. SOC.Testing Materials, Standards, Part 11, p. 575 (1944). (2) Bishop, D. L . , Bur. Standards J . Research, 12, 173 (1934). (3) Calbeck. J. H., and Harner, H. R., I n d . Eng. Chem., 19, 58 (1927). (4) Coutts, R. H., and Cron-ther, E. M.,T r a m . Faraday Soc., 21, 374 (1935). ( 5 ) ‘Coutts, R. H., Crowther, E. M.,Keen, B. A . , and O d h , S., Proc. Royal Soc., (A) 1 0 6 , 3 3 (1924). (6) Gessner, H., ”Die Schlammanalyse,” Leipzig, Akademische l’erlagsgesellschaf~,1931. (7) H a h n , F. V. von, “Dispersoidanalyse,” Dresden, Theodor Steinkonff. 1928. (SI Jacobsen, A. E., and Sullivan, W.F., IND. EKG.CHEM.,AR-AL. ED.,1 8 , 3 6 0 (1946). (9) Johnson, H . W., Soil Sci., 16, 363 (1923). (10) Kanning, E. W., H a r t m a n , R. J., and Childs, F., J . Phus. Chem., 36, 2369 (1932). (11) McCarron, R. D., and Rowland, B., Paper Trade J . , 96, No. 22, 36 (June 1, 1933). (12) Obenshain. S . . and Welton. W. M., U. S. P a t e n t 2,397,038 (March 19, 1946). (13) O d h , S., in J. Alexander’s “Colloid Chemistry,” Vol. 1, p. 861, New York, Chemical Catalog Co., 1926. (14) O d h , S., KolZoidZ., 18, 3 3 (1916). (15) Ibid., 26, 100 (1920). (16) O d h , S.,SoilSci., 19, l ( 1 9 2 5 ) . (17) Svedberg, T . , and Rinde, H., J . Am. Chem. SOC.,45, 943 (1923). RECEIVEDSeptember 26, 1946. Presented before the Division of Analytical and Micro Chemistry at the 111th Meeting of the AMERXCAX CHEMICAL SOCIETY, Chicago, Ill.
