Centrifugal Sedimentation Method for Particle Size Distribution‘ A. E. JACOBSEN
AND
W. F. SULLIVAN, Titanium Division, National Lead Company, Research Laboratory, Sayreville,
The centrifugal sedimentation method for particle size distribution of materials in a dispersed system is reviewed. A preferred method has been used which is analogous to Od6n’s method of tangential intercepts for gravitational sedimentation, This procedure yields the same results as those obtained by the variable suspension height method but is more convenient from the practical point of view. Examples are given to show practical applications of the beakertype centrifuge to the study of relative dispersion of titanium dioxide pigment in paint systems. The method is of limited usefulness for the determination of specific surface area or the relative efficiency of Ilght-diffusing properties of materials which are either aggregated or lnegularly shaped, since Stokes’ law is based on the assumption of an equivalent spherical particle. Electron micrographs supplement the sedimentation studies.
N.J.
Since the usual method of obtaining data on a polydisperse suspension is to determine the sedimentation-time curve, it is apparent that in time 4 all particles havin a diameter equal to or greater than D, will settle through hei $t h. In addition, a certain fra,ction of the particles having a t i m e t e r less than Dm m11 settle through a height less than h. The deduction of the sedimented “fines” may be accomplished by OdBn’s method of tangential intercepts in which
F ( D ) d D = (1
- p ) + t d_p dt
where = total weight fraction sedimented F ( D ) d 8 = weight fraction of particles having a diameter between D and D d D
+
For materials with a particle size range of approximately 0.1 to 1 micron, the ordinary beaker-type centrifuge has been em-
T
ployed to determine the distribution. The essential point of divergence from gravitational sedimentat.ion is that the force acting on the particle increases with the settling distance. As first shown by Svedberg and Nichols (1.2) the gravitational equation must then be modified to
(46).
where z1 = distance from the center of rotation to a point in the suspenuion zr= distance from the center of rotation to a second point in the suspension w = angular velocity in radians per second
HE determination of the particle size distribution of finely
divided material is of practical and fundamental importance in physicochemical studies. In particular, centrifugal sedimentation has proved very valuable in determining the relative distribution of pigments in paint vehicles ( 7 ) . The theory underlying the application of sedimentation data for the calculation of the particle size distribution hsll been worked out (3, 8, 9) and verified by comparison with the results obtained by other methodi From the practical point of view the use of a suitable dispersion technique for the particulate material is of significance, since the sedimentation data are of little value unless a satisfactory dispersion has been attained. Hence, cognizance must be taken of the nature of the dispersing agent used, and the particular technique employed in preparing the dispersion. A further limitation is the effect of particle shape. The concept of ‘‘equivalent spherical diameter” has been used in all applications of Stokes’ law, but it is known from electron microscope studies that the particles are rarely spherical, often being irregular aggregates of primary particles. I n extreme conditions poor correlation may exist between specific surface as calculated from particle size distribution data and specific surface as determined by independent methods.
The calculation of the particle size distribution from the sedimentation-time curve was apparently first accomplished by RomWalter and Vend1 (IO) who arrived at the equation
THEORETICAL
The gravitational method of sedimentation has proved suitable for determining the particle size distribution of particles in suspensions where the size range is from about 1to 50 microns. The procedure used involves the application of Stokes’ law (4, 6, 1 1 ) and OdBn’s method of tangential intercepts (8,9). The equating of the frictional force, as ven by Stokes’ law, to the gravitational force on the particle yiefls the relation
where D = equivalent spherical diameter of the particle 9 = coefficient of viscosity of the suspension medium h E= height of suspension dl = density of particle 4 = density of suspension medium g = gravitational acceleration t = time of settling 1 Three papers presented in the Symposium on Meaaurement and Crestion of Partiole 8i.e at the Twelfth Annual Chemical Engineering Symposium of the Divinion of Industrial and Engineering Chemhtry are presented here, pages 860-78. Other papers in the symposium will be published in the July h u e Of the INDVBTRIAL EDITION.
Figure 1.
Sedimentation of Titanium Dioxide in D u k h
Oil ZP
Vwirble ruspmdon height method
360
361
ANALYTICAL EDITION
June, 1946
~
where R = distance from center of rotation to bottom of cup S = distance from center of rotation to surface of suspension However, this equation was shown by Brown (3) to be invalid and this conclusion has been confirmed by the authors. Thus, since R and S are constants, the weight fraction of particles having diameters between 0 and D, is given by
F(D)dD = K X t @ dt
(4')
Therefore the value of the integral must run from 0 to 1 as the partide diameter is increased from 0 to 03. Direct application of the sedimentation-time curve, however, shows that the term
t@ ' and hence the integral dt
F ( D ) d D go through a maxi-
loDm
mum, since a t low values of t , t'%5
dt
equals a small number and, at
high valuesof t, it is again a small number since the slope dP - apdt
proaches 0. Brown was unable to find a mathematical solution for the distribution function F(Df when the time of centrifugingwaaused as a variable, but did arrive a t a satisfactory solution when the suspension height ( R - 8)was varied. His equation for cylindrical centrifuge tubes is:
XDW
F(D)ri?D = (1 - p )
+ ( R- S) a dP
L
I
4 Figure 2.
Cumulative Weight Per Cent Curve for Titanium Dioxide in Dutch Oil ZP
(5)
0 Variable suspension heisht method 650 r.p.m.
r
A Variable suspension height method: 350r.p.m. 0 Modified grrvitational method
This equation has been used by the authors, and results have been expressed as a cumulative curve,
F ( D ) d D vs. D ,
rather than as the distribution function itself, F ( D ) us. D,, since the latter would involve second derivatives, a procedure which is probably not justified by the experimental accuracy achieved. In actual practice, however, it has been found somewhat more convenient to use a modified method for the determination of the cumulative weight per cent curve. The principle of this method, which has already been pointed out in the work of both Brown (9) and Martin (Y), is that centrifugal sedimentation can be made t o approach gravitational sedimentation by making the height of suspension ( R 1S) small. This means that the centrifugal force on the particle is taken t o be approximately constant over the settling distance. Hence, the mathematical procedure is identical with the gravitational calculations, except that the gravitational acceleration, g, is replaced by the centrifugal acceleration, d R . The results obtained by this method have been found t o check closely those obtained by the variable suspension height method for many pigment dispersions. There are two reasons for preferring to use a small height of suspension with variable time rather than variable suspension height and constant time. First, the former method requires considerably less sample, which is an important factor where limited quantities are available and, secondly, a wider particle size range can be covered without having to readjust the centrifuge conditions. Thus, using the variable suspension height procedure, two or more analyses may be necessary at different centrifuge speeds in order to cover adequately the particle size range usually encountered. From the physical viewpoint the requirement of sector-shaped tubes is also more closely approached when the suspension height is made small. On the negative side it must be considered that the relative experimental error involved in measuring the height is larger when the height is small. However, by carefully calibrating the centrifuge cups on a volume basis, the error seems t o have been minimized as judged by the
reproducibility of the results, and their agreement with other methods. EXPERIMENTAL
APPARATUS.The apparatus used was an International centrifuge, siee 1, Type SB, equipped with 115-volt Transtat volt e regulator whch waa designed to improve the constancy of rotational speed. The centrifuge head holds eight chromiumplated cylindrical cups (9.20 X 3.58 cm.) each having a cover to prevent solvent evaporation. The distance, R, from the center of rotation to the bottom of the centrifuge cup was 21.3 cm. The centrifuge speed w&s measured with a General Radio Strobotso and the time of acceleration and deceleration allowed for by the method of Marshall (6). For the centrifuge speed used in the experiments described, the correction was one minute. The pigment dispersions were repared by mixing titanium dioxide with an organic vehicle in tge ratio of 100 to 54 grams in a mechanical mixer for 60 minutes. The mixture w a s then put through a three-roll laboratory paint mill with the rolls set at 0.075 mm. (0.003 inch) and finally diluted to approximately 37' solids with mineral spirits. The pigment suspension was a n a l y r d by pipetting a 10-ml. aliquot into a weighed crucible, burning off the organic matter, and weighing the*ignitedresidue as titanium dioxide. In the case of the centrifuged samples the supernatant suspension was carefully separated from the sedimented material and mixed before sampling. Since the suspension is dilute, the per cent sedimented may be calculated in the following manner:
%
yosedimented
=
lOOp =
weight of sediment weight of total
loo =
weight of total - weight in suspension weight of total ~
(CTi0i)initirl
- (CTiOdsuapcnaian x
(CTiOz)idwd
x loo
100
where (Cao,) = concentration of Ti02 in grams per liter.
COMPARJSONS OF METHODS. I n applying the variable suspension height method it was found convenient t o modify Equation 6 to
362
INDUSTRIAL A N D ENGINEERING CHEMISTRY Table
-
R 8 Cm.
l.w
3 . 0 5 . 0
7.W
carried out with the previous suspension, so that a comparison between the two methods might be made. I n order to apply the intercept method, Equation 2 was changed to
I. Sedimentation oi.Titanim Dioxide
1 0x P)
(1
-
2X
IW(1
- P)
%
%
0.94 21.1 46.6 56.0
1.9 42.2 93.2 112.0-
Tangential lnteroepta
100 F ( D ) d D
%
%
L
5.4 56.7 83.0 83.5
0.17 0.30 0.40
--14.5 3.5 10.0 29.5
D"2b
Vslues taken from Figure 1. 6 From Equation 3. wing XI = 8. Xs = R , d = 4.20 grama per 0.80 wsim8 per 0 0 . and 7 = 0.0086 Doiae at 25' C. me have:
0.46
0
18
x
(4.20
"r
F(D)dD
0.WSB
x
00..
do
-
21.3 2 . 3 0 3 1 0 ~2 1 , 3 - ( R - - S )
- 0.80)03')*(
= 2(1 - p )
"r
F(D)dD = 1 - tangential intercept
(2')
and Equation 1 to
The results are summariaed in Table 11. The cumulative weight per cent curve is also shown in Figure 2. The close agreement with the variable suspension height method is evident. It should he mentioned in connection with the precision of the modified gravitational method that a cheek analysis on a portion of the original suspension showed an ahsolute enor not greater than 1%.
X 60
- tangential intercept
Vol. 18, No. 6
(5')
-
where the tangential intercept is that obtrtined f r o n t h e lOo(1 p ) Venus ( R - S ) curve. The r a u l t s summarized in Table I were obtained using a centrifuge speed of 650 r.p.m. with a time of 30 minutes. The dispedon medium was a linseed oil vehicle (National Lead Company, Dutch Oil 22). These results, however, cover only the particle size interval from 0.11 to 0.46 micron and leave an appreciablefraction of the distribution unaccounted for. Hence, with the variable suspension height method, a second analysis was necessary using different conditioneviz.. 350 r.p.m. and a time of 30 minutes. The end results are expregsed together with those of Table I in the form of a cumulative,weight per cent curve (Figure 2). The modified gravitational method, using (R - S) = 1cm. was
I Figure 4. Elochon Micrograph Titanium Dioxide, x 15,000
of
' 7
I
Figure 3. Sedimentation Cune ofTitanium Dioxide in Dutch
Oil ZP
Modified g n v i b l i o d nelhod
Figure 5.
Cumulative Weight Per Cent Cune for Titanium Dioxide in Dutch Oil ZP
0 Modinid praritdionil method A Eleclmn
micrograph oovnt mlhod
ANALYTICAL EDITION
June, 1946
Table
II. Sedimentation of Titanium Dioxide
Tangential 100 F(D)dD Dmb Time 100 X p Intercept" Min. % % % Ir 0 2 10.3 100 1.24 2.5 5 17.8 0.78 97.5 29.2 7.5 10 0.55 92.5 54.3 0.39 80.5 19.5 20 52.0 80.8 0.28 48.0 40 29.5 70.5 60 89.0 0.23 87.0 94.5 0.18 13.0 90 94.5 97.0 0.15 120 5.5 a Values taken from Figure 3. b From Equation 1' using 7 = 0.0096poise, ( R S ) = 1 om., dl = 4.20 g r a w per cc., do = 0.80 gram per cc., R = 20.8 cm.,o = radians per second.
-
Table
111.
Effect of Milling
Milled
100
hD"
F(D)dD
%
' -
r"
Not Milled
D, P
100
F(D)dD
70 5.6 29.2 55.7 75.0 95.0
Dm P
0.20 0.30 0.40 0.50 1.00
Although the agreement between the two methods lends credence to the fundamental "correctness" of the distributions arrived a t by neans of the centrifuge, it w&s thought advisable to check the results by an independent method. A series of electron micrographs was taken of the pigment from which a particle size distribution w&s made by the count method ( 1 ) . Over 500 particles were counted. The specimens for the electron microscope were made by incorporating a small amount of the original paint with collodion which was used as a supporting film. A typical micrograph is shown in Figure 4. The particle size distribution is represented graphically in Figure 5 for comparison with the centrifuge data. The comparative results are in reasonable agreement, considering the divergence in technique.
3e%
INFLUENCE OF PARTICLE SHAPE. Electron micrograph studiea of various titanium dioxide pigments have shown that the pigment particles are invariably aggregates of primary particles. As an extreme caae it is possible t o have approximately the eame particle size distribution for two different pigments, one of which is composed of aggregates having very small primary psrticl& and the other composed of aggregates having large primary particles. The cumulative weight per cent distributions, using a linseed oil-tung oil vehicle (VM-1215), are summarized in Table V. Electron micrographs (Figure 6) indicate that the particulate material in the pigment composed of smallprimary particles haa a more open structure and hence considerably more specific surface. The particle shape may also be a factor where the number of particles in an aggregate is approximately the same, but the shapes of the primary particles differ widely. The following example shows the cumulative weight per cent distribution for a pigment composed of acicular primary particles and one having approximately spherical particles. The vehicle is again VM1215. The corresponding electron micrographs, as shown in Figure 7, illustrate the difference in particle shape. Surprisingly enough, those pigments have approximately the same hiding power ( 8 ) . It is therefore apparent that complete reliance on sedimentation data would again lead t o error in the evaluation of these products. CORRELATION WITH OTHER PROPERTIES
SPECIFIC SURFACE.An estimate of the specific surface from a study of electron micrographs in comparison with the specific
Table
/o""
IV.
100
%
F(D)dD
r"
Effect of Dispersing Vehicle
Dutch Oil 22
Linseed Oil-Tung Oil (VM-1216)
Dm P
100
F(D)dD
%
D, AI
PRACTICAL APPLICATIONS
EFFECT OF MILLING.To illustrate that the method of preparTable V. Influence of Particle Shape ing the dispersion is of fundamental importance and has t o be Small Primary Particles /o%rge Primary Particles considered with any data given on particle size distribution, two mF(D)dD Dm Dm dispersion procedures were used for the same pigment. I n both % Ir % Ir cases a 60-minute mix was carried out in a small mechanical mixer 11.2 0.20 13.1 0.20 using a linseed oil-tung oil vehicle (known to the paint trade as 42.1 0.30 38.2 0.30 59.9 0.40 59.1 0.40 VM-1215, which was developed by the Technical Service 75.9 0.50 69.7 0.50 Laboratories of the Titanium Pigment Corp., New York, K,Y.). 92.8 1.00 90.8 1.00 One half of the pigment paste was subsequently put through the laboratorv Daint mill and the other half was not. The cumulative weight per cent distributions, as determined by the modified gravita1 tional method, are given in Table 111. These data show that milling has the effect of producing a finer particle size d stribution. NATURE OF DISPERSING VEHICLE. Since in the practical preparation of paint dispersions, a wide variety of vehicles are available, the distribution obtained will depend also on the vehicle used. As an example another titanium dioxide pigment was dispersed according to the previously described procedure in both a Dutch Oil 2 2 and a linseed oil-tung oil vehicle (VM-1215). The cumulative weight per cent distributions, as determined by the modified gravitational method, are given in Table IV These data again emphasize the relatio&ship Figure 6. Electron Micrographs of Titanium Dioxide, X i 5,000 between the particle size distribution and the dispersion conditions. L e f t . L w e p t i m m p.cncl*r. R i g h t . Small primuy p.dlcl*s I
_
364
I N D U S T R I A L A N D ENGINEERING CHEMISTRY
Vol. 18, No. 6 SUMMARY
Figure 1. Electron Micrographs of Titanium Dioxide, X15.000 kit. Acicular wrtislet.
100
r
Table VI.
The centrifugal sedimentation method is based on proved fundamental principles, thus making i t suitable for determining cumulative particle size distribution of particulate material in dilute suspensions. A modified Centrifugal sedimentation prccedure has been employed in which the malytical results agree with the mults obtained hy the variable suspension height method and with electron micrographic count. Examples have been presented to show the practical use of the method for physicochemical studies dealing with the dispersion of titanium dioxide pigments in organic vehicles.
A ~ ~ o r l m a spheiioal hl~ parIidei
Right.
InRvence 01 Particle Shape
Acicular
r
Appmximately Spherical
F(dD)D
100
Dm
%
F(D)dD
D,
z
%
P
30.0
0.20
49.3
0.30 0.40 0.50 1.00
9.7 62.7
0.20 0.30
81.0 86.8 98.0
0.50
67.1 73.0 88.7
0.40
1.00
Table VII. Specific Surface Large Primary Particles
Small Primary Partid-
sq. m. P6l mom
CsnWugal sedimentation method Electron micrograph method
47 5
4.8 10
S =
,."C'"y"pn
Or
Dioxide, X I 5,000
surface oalouleted from centrifugal sedimentation data is given in Table VII. I n the case of the former a particle siec distribution by the count method was made considering as far as possible only the primary particles. Figure 8, which is a second electron micrograph of the pigment shown in Figure 6 (right), was used for the count. This differs from the latter figure in t h a t the dry pigment was vigorously ground in an agate mortar in order to desggrcgate i t as much as possible. The equation employed for the determination of specific surface from sise distribution data is
where W%
LI.=CLI"II
==mzF 6
per cent by weight of the total pigment per given fraction
D = mean diameter for the size intervals in microns 2 = summation of fractions in the size distribution d = density in grams per cc. = specific surface in square meters per gram
S
A limitation of the centrifugal method as far as disclosing the inherent nature of the material is clearly indicated here. HIDING POWER.From a practical point of view, aggregate shapeand size arcimportant factors involved in the lightdiffusing properties of paints. Ameasure of this.property can he made by determining the hiding power of the paint. Referring t o the pair ofpaints on which-data have been presented in Table V (which for all practical purposes show close agreement), the pigment with large primary particles has 10% greater hiding power tban the pigment in which the particles are composed of small primary particles. On this basis i t would he a fallacy to rely completely on sedimentation studies for depicting light-diffusing qualities of particulate materials.
Limitations of the sedimentation method for t'he determination of specific surface area. and the hiding power of particulate material which are aggregated or nouspherid in shape have heen emphasized. ACKNOWLEDGMENT
Appreciation is expressed to R. Dahlstrom, director of research, National Lead Company, Titanium Division, for his interest and suggestions. LITERATURE CITED
Am. Soc. Testing Materials, Standards, Part 11, p. 1575 (1944). Zbid., Part 11. p. 882 (1944). (3) Brown. C.. J. Phya. Chem.. 48, 246 (1944). (4) Gessner, H.. "Die Schlimmanalyse". Leipaig. Akademische (1) (2)
Verlagsgeselischaft. 1931. (5) Hahn. F. V. "on, "Dispersoidanalyse", pp. 271-2. Dresden, Theodor Steinkooff.1928. (6) Manhaii. C. E., P;oc. Roy. Soo. (London). A126. 427 (1930). (7) Martin, S. W., Am. SOC. Testing Materials. Symposium dn New Methods for Partiole Size Determinations in the Subaieve Range,Philadelphia, Pa.,1941. (8) Odh. S., in J. Alexander's "Colloid Chemistry", Vol. I. p. 861. New York. Chemioal Catalog Co.. 1926. (9) OdBn, S.. Kolloid. Z., 18, 33 (1916). (10) Romwalter. A,. and Vendl. M., Kolloid Z.. 72.1 (1935). (11) . . Svedberc. T.. "Colloid Chemistrv". . . New York. Chemical Catslog c o : ; 1928. (12) Svedberg. T.. and Nichols, J. B., J . Am. Chem. Soo., 45, 2910 (1923). P ~ a s a r r ~before ~ o the Diviaion of lnduatrial and Engineering Chemistry, Auamcan Camuicar. S O C I ~ T Symposium . on Msksurement and Creation of Particle Siae. Brooklyn. N. Y..December. 1945.
