DETERMINATION OF PARTICLE SIZEDISTRIBUTIONS IN POLYDISPERSED SYSTEMS
Sept., 1955
845
DETERMINATION OF PARTICLE SIZE DISTRIBUTIONS IN POLYDISPERSED SYSTEMS BY MEANS OF MEASUREMENTS OF ANGULAR VARIATION OF INTENSITY OF FORWARD-SCATTERED LIGHT AT VERY SMALL ANGLES1i2 BY
J. H. CHIN,^
c. M. SLIEPCEVICH*AND M. TRIBUS~
Departmant of Chemical and Metallurgical Engineering, University of Michigan, Ann Arbor, Michigan Received February 16, 2066
An experimental technique for the determination of the particle size distributions in polydispersed systems, based on the angular variation of the intensity of forward-scattered light a t very small angles, is described. The experimental apparatus consists of a monochromatic, parallel light source, a dis ersion cell, a lens-moving-pinhole receiving unit and a photomultiplier-potentiometer measuring system. An example o f a n analysis of a polydispersion of glass spheres in water is given. Close agreement was obtained between the distributions obtained by the experimental technique and the distributions obtained by microscopic counting.
In the previous paper, an integral formula for computing the relative pa.rticle size distribution N(D)D2 is derived. According to the integral formula, the distribution N ( D ) D 2is given by N ( D ) D ~=
clam 2 (ess)
[-Jl((~e)Yl(ote)(ote)lde (1)
where N(D) is the no. distribution function of the dispersion and is the number of particles of av. diameter D per unit volume of dispersion per unit range of particlediameter, or f N dD = where ni is the number of particles of
designed in such a way that the focal plane of the lens is traversed by the turning of a screw, the radial distance r may be expressed as this turn number v measured from the central position where the transmitted parallel light is focused. The quantities 5, e3, e35, and d/dO(B3S) may be expressed as a function of v. For a given a, -J1. (d) Yi(ae)(ao) is also a function of v . Equation 1 may then be rewritten as N(D)D2 =
En[, i
size i per unit volume of dispersion C is a proportionality constant e is the forward-scattering half angle 5 is the radiant flux passing through a small pinhole moving on the focal plane of a receiving lens in front of which is placed the dispersion cell CY is the ratio of the particle circumference to the wave length of the incident light in the surrounding medium, or CY = ?rD/X, where X is the wave length of the incident light in the surrounding medium J l ( d )is Bessel function of the first kind and of order 1 Y1(ole) is Bessel function of the second kind of order 1
For small angles, the half-angle 0 is proportional to the radial distance, r , between the moving pinhole and the optical axis of the receiving lens e = r/f (2) where f is the focal length of the receiving lens. The value of 0 in the medium is less than that in air because of the difference in index of refraction. However, the value of a in the medium is increased by the same amount because of the shortening of the wave length of the incident light. Therefore, the product of a and 0 is independent of the index of refraction of the medium and it is convenient to use the value of a and 6 in air for the computation, If a traversing unit for moving the pinhole is ( 1 ) Presented before the twenty-ninth National Colloid Symposium which was held under the auspices of the Division of Colloid Chemistry of the American Chemical Society in Houston, Texas, June 20-22, 1955. (2) This work was performed in partial fulfillment of the requirements for the Degree of Doctor of Philosopliy in Chemical Engineering at the University of Michigan. (3) Graduate Fellow, Department of Chemical and Metallurgical Engineering, University of Michigan, Ann Arbor, Michigan. (4) Department of Chemical Engineering, University of Oklahoma, Norman, Oklahoma. ( 5 ) Department of Engineering, University of California, Los Angeles, California.
Clam $
(09)h(D,u)dv
(3)
Yl((ye)(oIe)
(4)
where h(D,v) =
noting that D = Xajn and that v is proportional to
e.
Experimental Work The experimental apparatus consists of a collimatedsource unit, a dispersion unit, and a receiving-measuring unit. A General Electric type C-H3 mercury lamp and a Wratten 77-A filter are used to obtain the monochromatic green light (A 5460 A.). The Wratten 77-A filter transmits 68% of the green line and a negligible percentage of the yellow lines. With a receiver of S-9 response, the net spectral purity of the green line is over 95%. An effective point source is used to obtain a parallel peam of light. The parts are mounted on optical benches. The light sent out by the mercury lamp is condensed by a condensing lens onto a pinhole after passing through the Wratten 77-A filter. An iris diaphragm is placed behind the condensing lens to prevent the unwanted radiation from getting to the collimating lens. A shutter is placed just next to the pinhole, near which the smallest cross-section of the light cone is located. The pinhole is a t the focal point of a collimating lens. The collimating lens is an achromatic, coated, telescope objective, 54 mm. in diameter and 508 mm. in focal length. A diaphragm is placed in front of the collimating lens to limit the diameter of the parallel beam. The inside of the collimated-source-unit enclosure is blackened to minimize reflections and scattering. A small blower is used to supply a stream of air to provide cooling for the lamp. With a '/,,-inch pinhole and a collimating lens of 508 mm. focal length, the divergence of the parallel beam is 2.68 minutes, according to equation 2 . The dispersion unit consists of a brass cell and a variablespeed electric stirrer. The cell is provided with two plainglass windows held in place by screws and rubber gaskets. The camera is behind the dispersion cell. The camera body is a wood box painted flat-black inside. The front of the camera box is fitted with a lens holder with fine threads for focusing. The camera lens is an achromatic, coated, telescope objective, 83 mm. in diameter and 914 mm. in focal length. The quality of this lens is not excellent as several small bubbles are observed in the optical glass. There is no shutter in the camera, the shutter being placed
J. H. CHIN,C. M. SLIEPCEVICH AND M. TRIBUS
846
a t the source unit. The apparatus is placed in a dark room and the experiments are carried out in complete darkness. The radiation is collected by a Du Mont 6291 photomultiplier. The Du Mont 6291 is a ten-stage, head-on type multiplier phototube with S-9 spectral response. The multiplier phototube is operated by a Furst Electronics Model 710 PR, 300 to 1500-volt variable negative power supply. The circuit diagram for the photomultiplier is shown in Fig. 1. Two 45-volt “B” batteries are used in series to provide the potential difference between dynode No. 10 and the anode. The anode current is determined by measuring the potential drop across a 5100-ohm resistor in the anode circuit with a potentiometer. Precision resistors of low temperature coefficients are used in the multiplier circuit to minimize temperature effects during measurement. The voltage drop between the cathode and the first dynode is made twice as large as that between dynodes to better the collection efficiency and to obtain higher multiplication on the first dynode. A 20-megohm and a 500-ohm resistor in series are used across the cathode and dynode No. 10 so that the cathode supply voltage can be determined more accurately by measuring the potential across the 500-ohm resistor with a potentiometer. :ATHOOE IYNODE I
I
R P E; ;+M ;
303- 1500 VOLTS
I
I“’ I E:
1
2
3 4
----
5
--_-
6
----
POWER SUPPLY
7
VOl. 59
then idled for about one hour to allow the recovery of possible changes of sensitivity from prolonged operation. Then a weighed sample of the glass bead dispersion w a ~ added to the distilled water to obtain the desired concentration. The dispersion cell was maintained at the same position as before. The stirrer was turned on and the focal plane was traversed again as before. The gain was adjusted by increasing the cathode supply voltage a t suitable turn numbers so that the signals could be measured conveniently by means of a potentiometer. To obtain relative values for the readings, at suitable turn numbers, the readings were tied-in by measuring the signal both before and after the gain was changed. The method of correction for dust particles on optical surfaces and in the distilled water is to subtract the reading for distilled water from that for the dispersion when the incident beam to the front window of the dispersion cell is of the same illumination for both traversings. To compute the size distribution from the readings, a smooth curve was drawn throu h the data points for the dispersion and distilled water. %he values a t each turn of the knurled-head screw as read from the smooth curve were used for further computations. Because distribution curves are not in general smooth, the 5-curves will be expected to have slight irregularities. These slight irregularities were disregarded because they are indistinguishable from experimental errors. The 5-readings after multiplication by 0’ are to be differentiated. The differentiation of a smoothed curve will give more re resentative results. Large irregularities, of course, shouyd be handled differently. For instance, the 5-curve for narrow-cut dispersions exhibits maxima and minima which can be easily distinguished from experimental irregularities. Figure 2 shows the net 5curves for two runs for the polydispersion.
8 d .
I
I
I
9
,
i
10 ANODE
I Fig. 1.-Photomultiplier
-[I~,I +
]
DU MONT 6291
circuit.
The Du Mont 6291 tube, facing against a S/la-inch pinhole to limit the receiving area, is mounted in a traversing unit which traverses vertically on the focal plane of the camera lens. The traversing unit is designed in such a way that it can move the multiplier either continuously or in small steps of l/ge of an inch by the turning of a large knurledhead screw. Each turn of the knurled-head screw corresponds to ‘/24 of an inch or a value of 0 about 4 minutes for a camera lens of 914 mm. focal length. A feeler is provided so that each step ( l / d turn) may be felt in the dark and the position of the multiplier tube may be determined. I n order to test the validity of the theoretical method, a polydispersion of glass spheres (diameter from 2 to 40 p ) in water, having a known distribution as obtained from microscopic counting, was used. The polydispersion was prepared by mixing several narrow-cut dispersions which were obtained from Pyrex glass bead samples prepared by . . Gumprecht .E The light source wa8 regulated bv means of a manualadjust r6eostat and a costant-voltage Sola transformer. The multiple reflection effect between the dispersion cell and the camera lens may be eliminated by tilting the cell slightly with respect to the camera lens. I n order to correct for the effect of dust particles on the optical parts and in the distilled water, it was necessary to traverse the focal plane of the camera lens and record the readings for both distilled water and the dispersion. A definite procedure of experiment was followed. First, the dispersion cell was cleaned and the cell windows were aligned parallel to each other. After the optical parts were cleaned with a camel hair brush, the cell was aligned in front of the camera lens. Then the focal plane was traversed with distilled .water in the cell, the electric stirrer being turned on. Readings were obtained intermittently as a function of the number of turns on the knurled-head screw by remotely opening the shutter a t the source unit through a solenoid. The multiplier was (6) R . 0. Gumprecht and C. M . Sliepcevich, THISJOURNAL, ST, 90 (1953).
IO”
..
.’
Fig. 2.-5-Curve
for polydispersion.
The values of d/du (Os$) were obtained by differentiating the smoothed PF-curve. The slopes of the 039 versus Y curve were calculated by finding the average slope between successive points on the curve and calling it the slope at the mid-point values of Y between the corresponding pair of points on the curve. The slopes thus obtained were plotted against P and were used for graphical integrations after
Sept., 1955
DETERMINATION OF PARTICLE SIZEDISTRIBUTIONS IN POLYDISPERSED SYSTEMS 847 woo NO2 VS. D
-
4000
N.1 I
I
I
2000
1000
0 OP.
Fig. 5.-ND2
us. D for polydispersion.
IO
Fig. 3.-833-Curve
for polydispewion.
ND'VS 0
OP.
Fig. 6.-ND3 us. D for polydispersion.
IS0
12 5 bRUN4401
10 0
i 7s
-I--+
50
25
0
Fig. 7.-N
I
10
I-
20
I x)
I:3
TURN NO V.
Fig. 4.-d/d~(835)-curve for polydispersion. being multiplied by the factor h(D,v). Figures 3 and 4 show the plot of 035, d/dv ( 8 9 ) for the polydispersion. For the polydispersion, only the first two loops, one positive and one negative, of d/dv (e3%)were used for computation. For runs #lo1 and #102, the 835 versus Y curves exhibit a rather obvious maximum and a less obvious minimum and then an apparently constant or slightly decreasing posit8iveslope up to Y = 50, the last turn of the screw. This latter slope gives a positive loop of d/dv (6%) after the first negative loop. I t is expected that d/du ( 8 3 5 ) has more oscillations before it approaches zero. I n other words, d/dv. ( 8 9 ) oscillates about the v-axis with decreasing amplitudes as v increases. Theoretically, better results will be obtained if more loops of d/dv (e3%)are used. However, for polydispersions in which the size ranges are not too narrow, the higher order loops of d/dr(OaF) have very small
us. D for polydispersion.
amplitudes compared to that of the first two and may be neglected in the computation without introducing too much error. Furthermore, as will be explained later, the readings above Y in the order of 30 were not as accurate as for smaller Y because of limitations of the experimental apparatus. Consequently, it was necessary to use only the first two loops of d/dv (035) for computation. According to equation 3, the areas from graphical integrations are proportional to N( D)D2. The relative N ( D ) D z versus D curve8 obtained were then compared with that computed from counting data, after being normalized to give the same area under both curves. This method of comparison was selected because it is the surface distribution which is of most importance in many applications. From the N( D ) D aversus D curve, the N ( D ) versus D and the N ( D)D8 versus D curves may easily be computed and compared with those obtained from counting data. Figures 5 to 7 show the comparisons. The specific surface (total particle surface per unit volume of particles, 8,) and the surface mean diameter ( D = 6/S,) of the polydispersion were also computed. Table I shows the comparisons. The agreement for the distribution curves, the specific surface, and the surface mean diameter between the integral method and the counting method is rather good, con-
848
J. H. CHIN,C. M. SLIEPCEVICH AND M. TRIBUS TABLE I
COMPARISON OF RESULTSOF THE INTEGRAL METHODA N D COUNTINQ METHOD Basis: (100) (1.885) counted particles; D in N . NDP 0 3 fdD f N D a d D ( s )
Ds
% deviation Sp DS
Counting 51660 965000 3215 18.68 0 Run #lo1 51660 1023000 3033 19.70 -5.66 Run #lo2 51660 1014000 3058 19.65 -4.89
0 +5.45 +5.20
sidering the possible errors in the counting method, especially near the toes of the distribution curve where only very few particles are counted. The close agreement of the results of runs #lo1 and #102, obtained for the same polydispersion but different concentrations, shows that the integral method is probably more accurate than the counting method. The dark current and noise introduced an error of not greater than 5% over most of the distance traversed and only in large distances traversed the error might have been as high as 10 to 20%. The use of a finite pinhole introduced but a very small error because the photomultiplier readings were approximately linear over a distance of the diameter of the pinhole for most of the distance traversed. Since a0 appears as a product in the function h ( D , v ) and e is inversely proportional to the focal length of the lens, f, a positive error in f corresponds to a shift of the distribution curves toward larger diameters. Because the error in f was not more than 3%, the shift of the distribution curve was very small. As to the counting method, the most serious possible
VOl. 59
error was due to the inadequate number of particles counted for the particles with diameters near the toes of the distribution curves. Unless a great number of particles are counted, an accurate statistical distribution cannot be obtained. Inaccurate focusing introduces errors, especially to the particles whose diameter occupies full divisions of the eyepiece micrometer of the microscope. A slight variation of focus may attribute the particle to either n or n 1 division of the eyepiece micrometer, corresponding to a possible error of one size group or 1.885 p in diameter. Again, this error may be compensated only if a great number of particles are counted. From the above considerations, it is reasonable to conclude that the integral method is more accurate than the counting method.
+
summary The experimental apparatus is satisfactory for this preliminary investigation although it is rather primitive in design. However, this investigation suggests the possibility that a commercial highspeed measuring and computing unit for the determination of particle size distributions in polydispersed systems may be designed according to this technique. Acknowledgment.-The Eastman Kodak Company provided a Fellowship Grant for the year 1951-1952 and the Horace H. Rackham School of Graduate Studies of the University of Michigan provided the Horace H. Rackham Predoctoral Fellowship Grants for the years 1952-1954.
c
