246
P-ART 1C 1,E -S I ZL: (.'.iLlLl\i
Bcl I Te I C pii < I ! i i
r,
'? b
,
Ir1ion-s
I ; L c . ,463 i l - c s t Street,
-\-rti:
1-orl; 14, S e w E'ork
R e c c i w d J l a r c l : 28, lo,$!+
5ediment at ion i-11e t li ior the clctcrniirintion of particle-size distributions are applicable t o sus1xnsion.s of ;)articles ranging in 4ze all the n-ay from molecular dimensions t o sieve size. of 50 111ici'ons or niorc. Tlit, theorct,ical hasis for the determination of tlie particle-size distribution in a s u s p ~ n ~ i oofnparticles settling under gravity according t o Stokes' law wa.. estahlishcd by Od6n and a number of efficient applications are ,.lilaljlc (Z), Fvr ..ii.pensions containing a large proportion of particle- as small as 1 micron, tlie rate of settling under gravity lseconies so slon- that use of the strongcr settling iorcr afforded by the centrifuge is desirable. The de\-elopmc-iit of tlic ult~racciitrii'uge(8) and its application t o s hai-e opened a n-hole new field in fundamental chemistry. is a highll- spcciulized tool, l i o w w r , and it is primarily designed for the study of particles iiiialler than 0.1 micron. -4 variety of ccntriiugal methods h n ~ r iI ~ e e n applied to the intermediate particle-:ize range frmi 0.1 t o 2 microns. Continuow sedimentation in t h e hollow? rotating-cylinde~~ y p cor' centrifuge (1'1 rillon-; part'iclc-size fractionation as well as ana1ysi.s i(2~1l o be carried o a t . \-?e of the beaker-type centrifuge is more convenient for control tests or in caw. n-here separation of particle-size fractions is not necessary. .inalysis of sedinientation data niay be simplified by t8heiiee of a long-nrnietl wntriiuge (4),so that tlic centrifugal force on all particles i s approximately the wnie ant1 thc aiialysis reduces to that for gravity consists in floating the 3ettling. .Inothei device for knplifying thc anal) cm a relatiT,-ely long cohinin of a denser liquid (3) so suspension to be centrifu that all particles settle t tially the sanic chtnnce in reaching the bottom of the centrifuge tub ti. Direct application of CklPn'7 line of reasoning on grtivitational sedimentation t o centrifugal seclimeiitatioII in a beaker-type or simple cup-type centrifuge has been carried out by Romrmlter and T-end1 (6). Theii. anal) apparently solves the problem of calculating the particle-size distribution in a suspension from experimental values of the n-eight fraction of the suqsended particles sediniented bj- a beaker-type centrifuge running at constant qieccl, free from vibration, for a series of time inter\-ah. ieir treatment, hnmver, appears t o involve an inheir solution iavalid, ns is shown below. The purcorrect step which rend pose of this pape- is t o pre.sent an alternative treatment n-hich offers a coniplete solution to the problem in t h o r ~ - . On the basis of t,he treatment presented, it is proposed that experimrnt:il \-ahice ? vtleterniined for the weight fraction of suspended particles d i n i e n t d 1)J- a 1x.ntker-type centrifugc running at conqtant speed for a suitable %t;'xcdtiiiic interval \\.it11 \-nr;\-ing quantities of suspension in the centrifuge t i i l w x . Pnrticle-;ize a n a l y s c ~1 - 1 1 3 ~ -a l v ~tic made by determination :IT--
P.1RTICLE-SIZE DISTRIBUTIOSS
217
of a series of sedimentatioii-time curves with various quantities of suspension in the centrifuge tubes. THEORETICAL
The weight fraction sedimerzted Particles settling under centiifugal force follon- paths radial to the centrifuge axis instead of parallel paths as in gravity settling. For strictly convection-free centrifuging, therefore, sector-shaped centrifuge tube< are essential (8). Figure 1 illustrates the condition? of sedimentation in a sector-shaped tube. The centrifuge rotates free from i-ibration about it. axis, A , at constant speed. The iuspension fills the qector-shaped tube from the bottom, R cm. distant from the axis, to the meniscus, S em. from thc asis. -1particle S cni. from the axis moves outward a t a speed determined by the balance 1,et.ir-wn its continually increasing centrifugal force and the Stokes' lan- force re.iqting its motion through the dispersion medium.
FIG.1 . C'eiitrifugnl sedimentation i n a iecto?-shsped tube
dX = outn-arc1 velocity of particle, dt dl = density of particle, do = density of suqienqion medium, 7 = coefficient of i-iscosity of mediuni, D = equivalent spherical diameter of particle. and w = speed of rotation of centrifuge in radian. per second. For convenience, let where
1- = -
so that
dx = bD2X dt
(la)
The diameter, D,, of a particle which starts a t the top of the suspension (S= S ) and just reaches the bottom of the tube (X = R ) at the end of a fixed time, t
seconds, may be determined by integration of equation l a between the limits X = S t o X = Xandf = Otot = t. In XIS
=
bDk
(2)
At the end of t seconds all particles of diameter D 2 D, have reached the bottom of the tube. In addition, all particles of diameter D < D, have been partially sedimented. For each particle of diameter D < D, there exists a starting point, X = XO, such that the particle just reaches X = R a t the fixed time, f. The distance SOis determined by similar integration of equation l a : In R / s o = bD2f
(3)
xo= Re-bD2t
(34
or Particles of diameter D will he sedimented if a t the beginning of the centrifuging they are at a point X 2 X o . The volume t o an)- point X within flat sectorshaped tubes is proportional t o R2 - X 3 . The xdimented fraction of particles of diameter D is therefore:
Define the particle-size distribution function F(D) 40 that F(D) dD is the dD. The weight fraction of particles with diameters between D and D weight fraction represented by the scdinwnted fraction of particles with diameter. between D and D dD < D,, is:
+
+
The weight fraction 01 all particle. uith diameter. D 2 D , (100 ppr cent Gedimented is jD;F(D dD. The total weight fraction -edimented is:
Incor,.cct i m of the slope io tlic sccli,,zeiztntio,i-fi,ne curve The treatment to thi- point is identical in principlr with that of Romn-alter and T-eiidl ( 6 ) . -It thii point they advocate the determination of p , the weight iraction seclimt~nted,aitei centrifuging for a seyies of time intervals and use of the slope, b p 8,oi the sedimentation-time curve for calculation of the distribution fimction. T h y Jifi'erentiate equation 6 with reqxct t o time t o obtain:
249
PARTICLE-SIZE DISTRIBUTIOKS
From equation 2, e-2bDft = S2/R2and therefore
so that the first two terms of equation 7 cancel, leaving:
From equations 3 and 3a it, follows that e- 2 b D 2 t
=
1
S2/R2 and 2bD2 = - I n R 2 / S 2 t
when D = D,,(Xo = S). Romwalter and Vend1 substitute these particular values for the two functions in equation 7a, obtaining: D,
s2 RV R2Z F Z
In R2/S2F(D) dD
Since only F ( D ) varies v i t h D in equation 7b,
apparently gives the distribution function in terms of R , S , t and the slope t o the sedimentation-time cun-e. Since e-2bD2t and 2bD2 are functions variable with D, there is no reason for substituting the particular values at D = D,,in equation 723, and equation 7c is therefore invalid. Equation 7a is the correct expression for the slope of the sedimentation-time cun-e, but an exact solution for the distribution function seems t o be extremely difficult if not impossible t o obtain.
Suspension lecel a s a variable f o r particle-size analysis The complications arising from differentiation of equation 6 with respect to time may be avoided if, instead of determination of the fraction sedimented after centrifuging for a series of time intervals, the fractions sedimented after a given time interval with the centrifuge tubes filled n-ith suspension t o a series of levels are determined; that is, if S instead of t is varied (figure 1). On increasing S , an increasingly large fraction of the suspended particle< will be sedimented in a given time. Differentiate equation 6 with respect to S to obtain:
+
lDm (RZ -
2R2SS2)Z (1 - e - " D 2 t ) F ( D )
dD
(8)
As for equation 7, the first tn-o term5 of equation 8 cancel, leaving:
Since R2 2s - S2 is independent of D,
From equation 6, the right-hand term of equation 8b is I-"
and by definition of F ( D ) ,
lDm
F ( D ) dD
+
F ( D ) dD
=
1
Dm
so t h a t
Combining equations Ga and 8b n-e obtain:
R2 - S2dp --
as + (1 - p )
28
=
jDm F ( D ) dD 0
Thus, if p is determined for a qeries of values of S , measurement of the slope Dlfl
of the p 2's.
X
curve allows calculation of
F ( D ) d D , the weight fraction of
particles x-ith diameters less than D,. To each value of S there corresponds a value of D , (equation 2 ) . so the distribution of particle sizes may be determined in the range of D,, covered by the p 1's. S curve. J u t as in gravity vdinicntation ( 5 ) . second derivatives of the fraction hedimented are required t o obtain the distribution function itself. T o obtain F(D,) in terms of vcond derivative? of p , differentiate equation 8c with respect to S:
-+
€2' - S' d ' p - RL 3s' d~p ~. _ - -dD, 2s as? 2s2 as as F(Dm)
From equation 2
dDm __ = dS
-_
D, 2 s In R / S
so that In R/S R' Drn
-[
-1
+ 3s' ap d2p - (R2- S ) s
2
as
as3
=
F(D,)
(9)
P.IRTICLE-SI%I,: DIbTRIBL-TIOSS
25 1
Similarly, the distribution function ma!- lie ohtained in terms of azp/aXat and difirrentintion of equation Sc TI ith respect to tinie:
@,’ai ljj-
From equation 2 -
dt
-
D?n -_ 2t
so that (loa)
Three distinct methods are therefore available for calculating the distribution particle sizes in a wy~ensionif the TT eight fraction .etliinented is determined with qector-shaped centrifuge tubes filled t o a serie. of levels. First, the weight fraction of particle. smaller than a known diameter ma)- he calculated from equation Sc and the distribution function determined from the .lope of the cumulative weight per cent cim’e. Second, the distribution function may be calculated directly in terms of the first and second derivatives of the fraction sedinieiited uith respect t o the length of the column of suqpension centrifuged by use of equation 9a. Or third, from sedimentation-time c i i r w - at a serie, of levels, the distribution function may be calculated by use of equation loa. In all cases the range of particle sizes corered is that of D,,a. calculated from equation 2. 01
Comparison of e p a t z o n s for centrifugal a t d grai Ity sedimentation The height of the column of suspension containing the particles to be sedimented may also be used a4 a yariable for gravitational sedimentation, although less convenient experimentally than ,sedimentation-time determinations. It is oi interest that analpi4 similar to the aboye gives the following equations for gray it at ional sedinien t at ion :
and
where h is the height of the uniform column of ;u>pension containing particles settling under gravity, p is the m i g h t fraction sediniented after time t , and D, is the size of the smallest particle. 100 per cent wlimented. Centrifugal sedimentation reducek to graT-itationnl sedimentation when the length of the cen-
trifuge arm is w r y great comparcd nith the length of the column of su,\pension. Referrnce t o equation 8c +hoi\; that if ( R 8 1 2 s approache. u n i t - , equation 8c reduces to eqiiation 11, 2iiicc X - 5‘ = 11 and (1s = -dh Similarly, equation 9a reducec: to equation 12, Qinwthe limit a- S approaches l? of
+
I: _ _In_ R / S = 1 R - 8 and equation 10a reduce- t o cqiiation 13 Cy12 tidricnl ccrzt ,+i jitqe t u be s Tlicb :i\e of a -imple I)cclic~-ty11~ ccntiiiuqe with cylindrical tubes is often de-irable for par: i r k - .izc tsontroi tc+t*. l l a n y ordinary laboratoi5- centrifuges such as the 1n:crnation:tl clinical centrifuge run remarkably free from vibration when properly balanced. h i t they are cle+ywd for cylindrical centrifuge tubes and require modification if wctor-shaped tithes are t o he used. In addition, sector-shaped tubes require careful machining for accuracy and are practically impossible to obtain as gla., w x e l s . If cylindrical tubes are used, a fraction of the suspended particles wttling under centrifugal force will strike the walls of the tube obliquely. oning t o the path.. of d i m e n t a t i o n radiating froin the axis of rotation. Tn-o eflects Tthich partially balance each other may result: ( 1 ) Some particles stiike the cylindrical walls of the tube before they would reach the bottom of the tub. through free sedimentation, agglomerate with Qimilar particle\ at the wall, and the resulting aggregate is sedimented faster than under free iediiiientatioii, and ( 2 ) convection currents result from the oblique torce of the suqienqion 011 the wall- of the tube. Conyection current5 from this and other sources (tmipcraturc gradient., x-ibration, etc.) may cause c:oiiie particles to be sedimenred faster than under conyection-free zedimentation, but it seems prol~ablethat the iict efYect i y to decrease the fraction of the wspended particlei which are sedimentd. If the length of the column of su3pension is kept reaionably small compared t o the di\tancc of thc bottom of the tuhe from the avis of rotation,-for example, lee. than f(S > 3 17, hgiire l),--cylindiicai tube- will probably gii-e particleiize analyse.. of scihcient accuincy for control tt-ts. ninst he modihctl .lightly for cylindrical centrifuge The f oi cgoing ail alj tube.. -4t equation 4, the ~rdimentedi1 action oi particlec of size D beconies
R - So ___ R -S
R (1 - p * t ) I2 - S Carrying this change through t o equation 8c u-e obtain : -
It i. cIoul,tfiil that cdcniation of the pnrticle-size distribution function itself is juqtified if cylindrical tube- arc u-e(!. IIon-c.i-cr, f o r cj-linclricol tube.. equations Sa and 10a become 2s
x’s
D,,
[2
3
- ( R - AS)
--
d2 d 5‘2
=
F(D,,,)
(9%’)
PAIRTICLE-SIZE DISTRIBUTIOSS
and
[” - ( E - s) --] a‘ p D, asat 2t
=
F(Dd
at
253
( 1Oa‘)
Another way of using the length of the centrifuge arm as a variable iq of possible interest in connection with cylindrical tubes. Instead of centrifuging for a fixed time with the quantity of suspension in the tubes variable, fis the quantity of suqpension as ~ c l as l the time and vary the length of the centrifuge arm (change R with R - S constant, figure 1). K i t h cylindrical tulles this may be readily accomplished by placing in the bottom of the centrifuge cups small blocks of known thickness. The tubes containing suspension may thus be held at a variable distance from the asis of rotation. The analogue of equation 8c’ is
With sector-shaped centrifuge tuhej, use of R as a variable would require a different set of tubes for er-ery change in R , and no advantage over fixing R and varying S is apparent. It \Till be noted that equations 8c’ and loa’ for centrifugal sedimentation in cylindrical tubes are identical in form with equations 11 and 13, respectively, obtained for gravitational sedimentation, since R - X = h and dX = -dh. Equation Oa‘ reduces t o equation 12 as S approaches R. However D,, the diameter of the smallest particles 100 per cent sedimented, is calculated from equation 2 for the centrifugal case and from the equation
for the gravitational cabe
Asstiniption of coiwectiorz-free centrifuging Schlesinger (7) has developed a method of particle-size analysis in n.hich, instead of the assumption of vibration-free centrifuging, vibrations sufficiently violent t o keep the concentration of the wspension uniform during centrifuging are aqsumed. Contiiiuous sedimentation occurs on a filter paper mat placed a t the bottom of the centrifuge tubes. His method is primarily designed for meaiurement of the particle size in monodisperse sols, but he suggests variation of the length 3f the column of suqpension t o determine particle-size distributions. Precision sedimentation, free froin coni-ection currents, is undoubtedly obtained only viith the most carefully balanced centrifuge, with sector-shaped tubes, and with precautions to maintain a uniform temperature. Hon-ever, even without such precaution5 suspcniion- in cylindrical tubes in our centrifuge showed very mailied concentration gradients after centrifuging. At high speeds and n i t h the tubes filled t o a high level, the effect of convection c nia~kecl. But it seemq likely thal the assumption of convection-free sedimentation is adequate for many purposes and much more nearly accurate unless the centrifuging is carried out at high qpeeds in a deliberately unbalanced centrifuge.
r.u Partirlc-iizc analykcz of w+ptn4onYc,f 1 !ai ium a n d s:rontiiim carbonates in alcohol illustrate the application oi cquatioii 8c'. Thc wqwnsions aneiyzed were prepared by atlding a L 4 c ; c alcohol t o P e 0 1 Lntrated cai%onate $uspension which had been ball niillctl i u r 48 hi. in amyl acctate containing 2 per cent of pyroxylin. The resulting wspcnsion> contaiiictl 15 g. of carbonates per litrr in alcohol niixed n-ith 2.8 per ccmt amy1 acetate and 0.05 per cent pyroxylin. Cylindrical tubes n ith flat bottom< 11 crc ri;ecl in an International clinical centrifuge. The centrifuge speed IT ab>nir-a+uretln ith a General Radio strobatac, and the time of centrifuging n a s correctcvl to con-taiix qpeed by the method of Marshall (3). -1fter centrifuging, the suspension ant1 iediment ere carefully separated, and p , the weight fraction vdimented. tlereiiiiined from the volume of 0.1 S hydrochloric acid found equivalent .!;I titiation t o each of the separated fractions. The suspensions n-crt centrifuged at 500 E.P.U. f o i an effectiT.e time of 400 see. with the tubes filled to a series of levels. Figure 2 shows a plot of 1 - p , the weight fraction remaining in suspenqion, against R - S , the length of the cylindrical column of suspension centrifuged. The slope of the curves was measured with a tangent meter at each experimental point, giying ap/aS ai. each point. D, is calculated at each point from equation 2. LVLI
1111
x,
which on substitution for the constants' 7 = 0.0110 poise, R = 15.25 cm., dl - dl = 3.64 for barium carbonate, di - do = 2.91 for Ctrontium carbonate, w = 27r ( ~ . ~ . 3 1 . ) / 6=0 52.4 radians per second, anti i = 400 qec., gives
For SrC03, D,, (in microns) = 3
.
7
8 15.25 4 / 7 15.25 - ( R - S )
A convenient' method for application of the equation
(R - S)
$ + (1 - p ) = 1
D,
0
F(D) dD
is illustrated in figure 2 at the point R - S = 3 on the strontium carbonate curve. The tangent APB is d r a m so that the intercept, B , at R - S = 2 X 3 DTn
may be read.
The intercept B gix-es the value of
F ( D ) dD, the Tveight frac-
1 The viscosity and deiisity of dcohol a t 25-C meie used. The viscosity and density data were taken from HandbooX of Chemzstri/ a/zd Phgsics, 21st edition, Chemical Rubber Publishing Company, Clei eland, Ohio
255
P-IRTICLE-SIZE DISTRIBUTIOSS
tion of particles with diameters smaller than D,, since BC/PC = ap/aS and PC = R - X, SO that BC = ( R - S)a p
as
R - 5 IN C M FIG.2 . Sedimentation of barium carbonate and strontium carbonate suspensions in alcohol. l O O ( 1 - p ) is the weight per cent of suspended particles remaining in suspension after centrifuging a t 500 R.P.V. for 400 see.; R - S is the length of the column of suspension in the cylindrical centrifuge tubes.
which is the term added to 1 - p to give
6””
F(D)d D (equation
Sc’)
If it is not convenient t o extend tangents as far as 2(R - S), (1 - p ) - tangential intercept A = ( R
- S)ap/&S
and therefore,
2(1
- p)
D7Il
- tangential intercept A
F(D) dD
=
Results are summarized in table 1. Column 5 gives the weight per cent of particles with diameters less than D,, listed in column 6. The resulting cumulative weight per cent curves are shown in figure 3. A suspension of uniform particles 1 micron in diameter would give a vertical line a t D, = 1 in figure 3. Thus the particles of strontium carbonate are distributed over a wider size range than those of barium carbonate; 52 per cent by weight of the barium carbonate particles lie in the size range 0.6-1.6 microns as TABLE 1 Particle-six distribution R-S
lOO(1
-p)
100
3 as
Barium carbonate suspension Cnt.
per cent
per cenl cm.-1
1.oo 2.00 3.00 4.00 5.00 7.00
5.4 17.2 26.2 33 .O 38.8 48.2
,
13.4 10.4 7.6 6.2 5.3 4.04
4.00 5.00 7.00
27.5 30.0 34.0
, I I
2.9 2.3 2.0
~
~
'I ~
I
per cent
per cenl
13.4 20.8 22.8 24.8 26.5 28.3
18.8 38.0 49.0 57.8 65.3 i6.5
11.5 14.0
39.1 41.5 48.0
1
cm.
1 ~
' I
1
~
x
10-4
0.58 0.83 1.04 1.23 1.40 1.75
1.37 1.57 1.95
compared with 21 per cent for the strontium carbonate suspension. The difference between these two curves illustrates the superiority of determinations of particle-size distribution over measurements of the a\-erage particle size. Any determination of "average" particle size would evidently indicate a larger particle size for the strontium carbonate suspension, yet from the course of the curves in figure 3 it is probable that the strontium carbonate suspension contains a higher proportion of particles of diameter lew than 0.5 micron than does the barium carbonate suspension. Calculation of the distribution function itself from these data, either from the slope of the cumulative weight per cent curves of figure 3 or by application of equation ga', does not appear t o be justified. The approxiniationq introduced by the use of cylindrical centrifuge tubes probably affect the second derivatives
257
PARTICLE-SIZE UISTHIRCTIOSB
of the fraction sedimented too much t o give reliable values of the distribution function. The results on barium carbonate and strontium carbonatc bu.ipensioiis ?how, however, that the use of cylindrical centrifuge tubes permits effectiw compariyon of suspensions rkith respect t o the range of particle 4zes present, and t h r cnmulative w i g h t per cent curves are readily determined.
-z I-
1
Y
Dm I N MM X
3. Cuinulative weight per cent curves f o r hariuiii rarlioirate ant1 strontium suspensions.
(
bUM\I.iRY
1. -In analysis is made of the sedimentation oi particles from a .iiispension 1)y I: beaker-type centrifugc. Calculation of particle-size distributions is g ~ e a t l y simplified if sedimentation is nieasiired a? a function of distance of the suypension from the axis of rotation rather than a\ a function of the time of rmtrifuging. D I?
2. The weight fraction of paiticlcs nith diameter.: les3 than D,,z, nux\- 1w calculated from t h c equation:
a s + (1 - p ) = 1
R2 - S2dp ~2s
D ,n
0
F(D)dD
F ( D ) tlD.
258
CALLAWAY B R O W S
where R and S are the distances of the bottom and top, respectively, of the suspension from axis of rotation, and p is the weight fraction sedimented in flat sector-shaped tubes. D,,Lis the diameter of the smallest particles 100 per cent sedimented and is readily calculated from the distance S and various constants. 3. Formulae are derived for the particle-size distribution function in term? of ap/aS and a2p/aS2or in terms of ap/at and a2p/aSat. The latter formula allows calculation of F(0,) from sedimentation-time curves for centrifuge tubes filled t o a series of levels. 4. Approximations introduced by the use of cylindrical centrifuge tubes and the assumption of vibration-free centrifuging are discussed. If cylindrical tubes are used, the sedimentation equations become identical in form with those for gravitational sedimentation. 5. Particle-size analysis of alcohol suspensions of barium carbonate and of strontium carbonate from sedimentation data in cylindrical tubes filled t o a series of levels illustrates the method suggested. This analysis was undertaken a t the suggestion of Dr. L. A. Footen. advice and criticism are grarefully acknowledged.
His
REFERESCES
(1) HAUSER, E . A., A N D SCHACHTMAN, H. IC.: J. Phys. Chem. 44,584 (1940). (2) HIUSER, E. il.,A N D LYNS,J. E.: Ind. Eng. Chem. 32,659 (1940). (3) MARSHALL, C. E. : Proc. Roy. Soc. (London) A126,427 (1930). (4) NORTON, F. H., A N D SPCIL,S.: J. Am. Ceram. SOC.21, 89 (1938). (5) ODEN,S.: I n J. Alexander’s Colloid Chemistry, Volume 1, p. 861. The Chemical Catalog Co., S e w York (1926). (6) ROXWAI.TER, A , , A N D VESDI,,AT.: Kolloid-Z. 72, 1 (1935). (7) SCHLESISGER, M.: Kolloid-Z. 67, 135 (1934). (8) SVEDBERG, T., AND PEDERSEN, I
