1042
JAMES V . ROBISSON
(3) COSSOR,FOLKERS, A S U ADKISS: J . Am. (’hem. SOC.54, 1139 (19323. (4) ECKERT ASU WEETXIS : Intl. E n g . Cheni. 39, 1512 (10471, (5) EISESSCHITZ: Z. p h p i k . Cheni. A158, 75 (1031); A163, 133 (1933). et a l . : U.S.Bur. Mines Tech. Paper No. 525 (1032). (6) FIELDXER (7) FWHSASII SASDHOFF: Fuel 19, 69 (lcj40i. (8) HAYSOX A S D BOWIIAS: Ind. E n y . Cheni.. +11ial.Ed. 11, 4-10 (1939). (9) HOWARD: J . Phys. Chcm. 40, 1103 (1036). (10) KIEBLER:Ind. E n g . Cliem. 32, 1380 ( 1 0 4 0 ~ . (11) KUHS:Z . physik. Chem. A161, 1 (1932). ~GH ST.MX: J . Phys. Chem. 40, 1117 (1036). (12) L O U G H B O R OAXI) (13) MITCHELL: I n d . E n g . Chem. 38, 8-13 (19461. J. P h y s . Chem. 50, 12 (1046). (14) MOREPA S D TAIIBLYS: (15) SEUKORTH: J . .Im. S O C . 69, 1653 (194T). (16) SCHL-LZ: Z. physik. Cheni. A179, 321 (193Tj. Treatise on Physical Chemistr!], p . 1616. D. Van Sostrand Conipnriy. h e . , (17) TAYLOR: Ken. York (1931).
THE VISCOSITY O F SUSPESSIOXS OF SPHERES JXMES Y.ROBISSOS The N e a d Corporatiori, Chillicothe, Ohio Receiicd Dctember 15, 1948 INTRODUCTION
Several equations t o express the viscosity of suspensions have been proposed, most of which involve arbitrary constants “characteristic of the materials.” The -4rrhenius equation is 9 = 7oeiic
in which 1; is empirical and c is the n-eight of solute per cubic centinieter of solvent (1).Weltmann and Green (12), modifying the hrrhenius equation, proposed that C = (vo .4)eED
+
in which U is the plastic viscosity, p is the volume per cent of solid, and J. and B are constants of the materials. Einstein (4) derived theoretically the relationship for the viscosity of suspensions of spheres at low concentration, (17
- ?lo)
’170
=
vsp
=
and later found the constant t o be 2.5. Eirich, Bunzl, and Nargaretha ( 5 )esperimentally confirmed the Einstein equation for concentrations of spheres u p to 5 t o 10 per cent by volume, depending on the viscosimeter used. Hatschek (8) proposed a theoretical law in the same form as the Einstein equation, with a constant of 4.5 instead of 2.5, applicable t o a concentration of 40 per cent by volume of dispersed phase, and another equation applicable at
VISCOSITY O F SGSPESSIOSS O F SPHERES
1043
high concentrations: (7 - 1) 7 = P I 3 . Guth and Siniha ( 7 ) derived an equation for solutions and suspensions, in n-hich the relative viscosity, 7 ’v0, is espressed as a power series of the volume concentration of the suspended material. 1-and (11) more recently made a theoretical derivation n-hich he reduced to a similar poner series expressing the relative viscosity; this equation fitted his experimental results fairly well at lox concentrationb but deviated above approximately 2.5 per cent by volume concentration. S o n e of these equations have been applicable oi’er the entire range of suspension concentration. Sorton, ,Johnson, and Lan-rence (10) have set up the equation 7 = qo(1 -
e)
+
/:le
+ kzc”
and hare empirically determined the parameters, by which the f l o behavior ~ of clay suspensions may be characterized, at any concentration. -4s further indicated by Bredbe and de Booys (a), the relationships n-hich are valid over a wide range of concentration involve parameters n-ithout direct physical significance. An attempt is made herein t o demonstrate a relationship with two parameters, one of which has simple physical significance, valid over the entire range of suspension concentration. DERIVATIOS
The Einstein equation states that at lo^ suspension concentrations the specific viscosity, (7 - 70) qn, is directly proportional t o the volume concentration of the suspended solid, 7Eil= 7c.T’. This equation n-as derived for very dilute suspensions, in Ti-hich the volume occupied by the spheres was negligible. To extend the equation to higher concentrations, the concept is introduced that the specific viscosity is not only proportional t o the volume concentration, but that it is also inversely proportional to the volume of free liquid in the suspension. The volume of “free liquid” in the suspension is that liquid outside of the suspended particles; in case the particles are highly aggregated, there TI ill be liquid entrapped within the aggregates n-hich TT ill contribute nothing to the fluidity. The free liquid therefore is not the differeiice betveen the total volume and the actual volume of solid, but that between the total volume and an effective volume of the solid. K h a t this effective volume is may be deducecl from a consideration of the upper concentration limit of the suspension. K h e n the concentration of solid becomes sufficiently high, a point is reached at TI hich each particle is in firm contact n ith its neighbors, that is, there is just sufficient liquid to fill the voids between the packed particles. Under such conditions fluid flo~v is impossible; unless sufficient stress is applied to change the condition of the solid particles, the “suspension” has become a porous solid and its r-ibcosity is infinite. Under these conditions the “suspended” particles constitute a packed sediment, n-hose specific volume dl depend upon the particle-size distrihution and the degree of aggregation. The effective volume of the particles is then the packed-sediment volume ; the data presented beloTr show that the pncked-sediment volume appears to be the effective volume of the particles at any concentration.
1044
JAMES V . ROBIA-SON
The ~euimentvolume is expressed as “relative sediment volume” ( S ’ ) , that is, the yolume 11 liich x sediment I\ ill occupy hose particles thernselves occupy unit 1 ojiiiiie. I t is to be undeistood that in packing into the sediment the particle. in no ]vise changc their agglomeration or shape, the sediment constituting the ~ma1le.t volume 11 hich such agglonici Lite< can occupy without changing. Tlic relati1 e sediment volume ( 3 )is c\prcssed as 1-olume per unit volume of solidi, TI hilc in unit volume of suspenyion the volume of d i d pai t icles is 1’. The d i m c J n t volume in unit volume of suspension is therefore LS’T’, and the volumc 01 ~ i e eliquid is ( I - X I * ) . Thcl cqu:ition applicahle to any convcntration oi susperi-ion then hecwmes VD1, = /, I- t 1 - #I7). To proi P the eyuntion by inca‘ii,if 1 and S ’ are constants, v i t h intercepts equal t o 1 1, and 1 S’, respectivcbly. *ill of the data reported are hantllecl in this manner. F(ii mot: suspcnhions, it i b Common cxpciience that tlic agglomcrJtion 71 hich dctclriniiie+ $’changes with the rate of >hear. If 5“ nere not a constant for a given suspcn,-mn. expeiimciital proof of the relationship TI-ouldbe excessively difficult. For th15 rwson glass sphere. I\ ere chosen, since the assumption can reasonably be made that they will remain completely disaggregated. The value of 8’ also depend.; irpon the particle-size distiibution, but it will remain constant for any partic ul,u vollection of spheres. I: XPCI?I3I LST \ L \IL1 HOD
wity nicaiurcments \\-ere made on a rotational viwosinictcr designed and ninnur’actured by the Sun Chemical Company, S e w I’ork, and described by Burlitlahl t t nl. (3). Thc viicosiineter I\ as modified by the substitution of helices of n i \ i ~ i c\\ire in place of straight torbion wires, with the attendant advantages of great r m g e of deflectioii without danger of inducing a permanent set in the torsioii del ice. The ends oi the helix \\ere tied together with a thread, coaxial n-ith tlw lwlix, t o hold the veight of the loner suspension off the hearings. The cup ha+ :t radius of 0.5994 em. and a depth of about 8 em., 11hile the bob has a radius (>f O..i499 cni. arid an effective length of 4.3GG em. Becauie of the small clearance llrtn-een the side alls of cup and bob, the large ratio of length t o radius, niicl the large clearaiicc of the m d of the hob from the bottom of the cup, it ~ i concluded a ~ that the correction for end effect was negligible (cf. Lindsley and Fi5cher (9)). The speed of rotation is varied through a Graham transmission, n ho.e ili:ii -ettings are calibrated in R.P.\I. The reproducibility of the speed for a given dial ietting was within 1 per cent, except at the lowest speed for which data are giren, ivhere the reproducibility is less satisfactory. A11 viscosity values are averages of those computed from data taken a t seven rates of shear, equally spaced ovei the entiie range of the instrument. As shoir-n in the tables, the maximuni rate of shear is G39 em. /see./ cm. The rate of shear is computed at the center of the annular clearance by multiplying the revolutions per minute of the cup by the instrumeiit constant 1.212. The viscosity is computed 1,- dividing
VISCOSITY O F SUSPEKSIOXS O F SPHERES
1045
the shearing stress, in dynes per square centimeter, by the shearing rate in centimeters per second per centimeter, the quotient then being in poises. The viscosimeter is jacketed with a large thermostated bath, filled I\ i t h transformer oil and maintained at 35.0"C. for all of these experiments. K i t h each 5uspension, after the material had been placed in the cup and the torsion spring adjusted, measurements nere taken at seven points from the lowest to the highest speed of nhich the viscosimeter is capable, and 11ei-e again taken at the same points with descending speed. -it least t h e e "round trips" from lo\vc-t to highest speed and back were made with each suspension, each iound tiip requiring about 2 niin. =It least one round trip n as required before temperature equilibrium u-as reachcd, as shown by declining viscosity during the initial runs. Kitli suspensions of high viscosity, heating of the contents of the cup TI as noticeable, sufficient to elevate the temperature of the oil bath in the immediate vicinity of the cup a fell tenths of a degree. Careful opeiation ]\as iequired to ebtimate, by The fluctuations in viscosity, which values corresponded to 33.0"C. By nialiiiig the run5 a t high speeds a> brief as possible, and by alloning a period of a fen minute., between runs. 11-ith the viscosimeter turned off or iunning slon-ly, fairly Ftahle 7-alues could be obtained. The glass spheres n ere prepared by entraining finely powdered :m(l screened Pyrex glass in an oxygen stream supporting the combustion of a g:i- f l ~ m ethe , flame being contained in a small-diameter pipe connected to a collect L g chamber and a nater pump in such a n-ay that complete fusion resulted, but tlie collccted pon-der of spheres n-as cooled rapidly t o prevent sintering. The zphpie., I\ crc cleaned by boiling in a mixtuie of concentrated nitric and sulfuric acids. and washed by successively stirring with distilled m t e r , settling, an(! i1cc:inting until the supernatant water had the same pH as the distilled water !icing uwd. The spheres 11ere separated into tlyo fractions by a sedimentatioii procedure. The fraction above 10 microns diameter was used in these experiments, iii 11 hich there vere very fen- individuals above 30 microns, n i t h the most betnern 1 0 and 20 microns. The suspensions n ere prepared by 11-eighing the desired quant it y of glass spheres in a 150-ml. beaker, weighing in the desired quantity of l ~ ~ u i c and l. stirring for a fen- minutes x i t h a stirring rod. Weighing n-as done on a clirrct reading balance accurate to the nearest 0.1 g., and 20 to 40 g. of each suspcn5ion TT as prepared (7 cc. being required to fill the viscosinicter cup). The gre:~tcb-t concern m s for absorption of moisture from the air by those liquids of a liygi*o-copic nature, so manipulations were conducted n-ith the greatest reasonable dispatch. From the first mixing of the suspension components to the completion of t I1c viscosimeter runs required about 15 min. I n all of the liquids used dispersion of the spheres was easy, so nothing more elaborate than the few minutes' stirring n-ith a qlass rod x i s required. Experiments wbsequent t o those described proved that ivith smaller spheres de-aeration hy evacuation i q essential; the erratic rwult? n these data may be due to air bubbles. After use, the spheres were recovered by repeated washings and decantations sith benzene or water, for the types of suspension media used. They were dried i n a n-arm shelf, as it Tyas learned by experience that heating caused ihc. ~ptic~rcs
to stick together. After many uses the >pheres darkened, presumably owing to t h e ahra+ion of steel from the visco4meter. The dark color \vas removed by treatment Jvith a mixture of concentrated nitric and sulfuric acids. I n order t o calculate the volume concentration from the 71 eight concentration, accurate values of the densities of the suspension media were necessary. These were obtained by measurement of the density using a pycnometer method at 33.0"C. Assuming that there is no sir-elling of the spheres, which is certainly
YISCOSITI
li!CIC
1007'
)e,. cctlt b y ;eight
60.0 65 .0 72.o
0 .o
1.25
9.1 16.5 20 . s 28.4 37.3 42.4 50.4
1 .62
~
'
__-
-
Concentration by weight in S.A.E. 30 motor oil (per cent!
I
.
101 190 274 363 453 540 639
S.40 23.0 __~~_
0 ,306 0.206 0.215 0.136 0 ,090 0 ,074 0.02s
VISCOSITY IS POISES
SHE.4R
D
0.37 1.00 1.21 2.60 5.21 7.15 22.3
2.25 2.40 3.R5 0.46
I
RAIL O€
'I - 71
puis.?,
per ceiii b y ~ ~ u l z r m e
0 .o 21) .0 33.3 40 .(I 50.0
_ _'I71.
- 'I1
'I
'I
0.0
1.22 1.25 1.26 1.25 L25 1.26 1.26
33.3
20.0
___
~
1 1
1 ~
I
~,~
1.595 1.65s 1 1.625 1.G0S , 1.028 , 1.603 , 1.585
2.97 2.26 2.27 2.24 2.25 2.24 2.21
1 617
2 35
'
50.0
40.0
,
' '
2.43 2.47 2.44 2.47 2.48 2A6 2.48
___
, ,
3.80 3.02 3.90 3.83 3.3 3.68 3.72
1
60.0
1
6.92 6.51 6.2s 6.18 6.27 6.49
~
~
,
~
9.58 8.30 S.24 8.16 8.40 6.40
6.5G
8.50
6 46
8 40
I
Avei age
1 25
i2.0
65.0
-1
~
i
I
-
j '
,
25.4 24.5 23.9 23.5 22.9 22.6 22.2
I
2 16
3 b5
23 6
the case in the systems studied, the volume concentration is calculated from the JTeight concentration by the formula: 1 I- = ((1 - c) c)(d,/di) 1. The density of the Pyrex glass (d,) is 2.23, and the density of the suspension medium is di.
+
RESULTS
The complete results of measurements of the viscosity of suspensions of glass spheres in tn-o motor oils, castor oil, polyethylene glycol, corn syrup, and sucrose solution are presented in tables 1 t o 12. The liquids were selected because of
1047
VISCOSITY O F SUSPESSIOSS O F SPHERES
TdBLF: 3
Glass s p h e r e s i n S.,.4.E. -Yo. 50 moior oii Density of oil
0.S897 g . / c c . a t 35.0‘c‘.
=
VISCOSITY
1007’
‘I-’I
~
n
‘I
-
’I1
~
p e r rent b y tolume
poises
0.0 9.1 16.6 21 . 0 28.5 37.4 -12.5 45.9
2.31 2.94 4.11 4.76 6.55 12. i 1 17.28 23.4
0.63 1.SO 2.15 4.24 10.40 14.97 21.1
0.332 0.213 0.195 0,155 0.053 0.065 0.045
T.1BLE 4 Glass .spheres in S . A . E . S o . 50 motor. oil VISCOSITT I S POISES R 4 T I TIE SHEAR
Concentration by weight in S..-I.E. 50 motor oil (per c e n t )
D 0.0 .~
101 1CO l
271 363
:*53 54 6 639
20.0 ~
2.2s 2.30 2.30 2.26 2.27 2.26 2 26
40.0
33.3 ~
~
50.0
,
60.0
65.0
3.04 2.07 2.97 2.94 2.92 2.8i 2.S4
4.’16 4.12 4.09 4.05 4.12 4.09 4.11
4.79 4.81 4.79 4.76 4.73 4.66 4.i9
6.61 6.46 6.56 6.45 6.52 6.60 6.66
2.04
4.13
4.i6
6.55
~
’ ~
~
~
~
,
1
1
68.0
_________
~~
13.08 ’ 12.53 12.51 , 12.55 12.69 12.80 12.S6
17.7s 17.10 17.16 17.10 17.06 17.18 17.58
12.71
17.28
~
i
~
1
~
,
25.2 23.8 23.15 22.9 22.9 22.6 23.15 23.4
ThBLE 5 Glass s p h e r e s i n cnsfor o i l Density of oil = 0.9521 g. c c . n t 35 0°C 1007-
:01c
i
VIscosxrs ‘I
I ~~
1
p e r cent b y t,olume
0 .0 17.6 22.2 30.0 40.9 44.3
poises I
3.55 6.45 7.57 11.06 20.15 34.2
2.90 4.02 7.51 16.60 30.7
0.216 0.196 0.142 0.087 0.051
their demonstrated Seu-tonian flon- characteristics and because their viscosities n-ere high enough t o make the spheres settle at a rate negligible for the measurements being made. Furthermore, it was desired to hare liquids representing a
1048
JAMES V. ROBIFSOS
TABLE 6 Gluss spheres in castor oil \ I S C O S I T Y I I POISES
Concentration by xeiglit in castor oil i7er cent)
I
D
;. ~
101
33.3
0.0
_ _ _ ~ ~ _ ~
'
'
,iverngr. , , ,
j
, ~
~
~
...
,
6.51
60.0
6 . 4*5
- . a--i
11 , O(i
~
!
I
21.1
11.33 10 .!)2
7.51 7.58 7 .oluiiiiiS divided 1)y the product of column 2 timcs column 3. ~
+ +
+
+
1.96 cc. per cubic centimeter. The values from the graphical interceptc. are 1 . 7 , 1.81, and 1.88, respectively. The above experimental check indicates that the effective volume of the >uspended particles, which determines the suspension viscosity, is clohely related t o the relative sediment volume; by postulate, the tn-o are identical. By implication in the derivation, S' varies only with the particle aggregation. and for the systems used such seems to be the case. HOTT ever, this restriction is not neceisary, and S' will be increased if the particles collect shells of material around them in their motion through a suspenbion medium containing large molecules. The absence of such an effect n-ith the spheres uhed is probably due t o their large +e relatilye to molecular dimen:'-ions. The extension of the equation to suspmqions of practical significance is com-
VISCOSITY O F SUSPESSIOSS OF SPHERES
105.5
plicated by the fact that S’ generally n-ill not be a constant for a given suspension, hut n-ill depend on the past history of the suspension and upon the shear t o n-hich it is being subjected. The hypothesis is advanced that the relationship is valid for any suspension, and affords a means for calculating the relative sedimentation volume of the suspended solid for the specific conditions under vhich viscosity measurements are made. The extension of the equation to suspensions of macromolecules should be similar and complicated by the same type of phenomenon. The parameter X’ should then be effective molecular size, that is, relative molecular packing in the same sense as relative sediment volume, n ith the complication that the molecular association responsible for the thixotropy of solutions of macromolecules changes analogously t o the aggregation of the particles of a suspension, with a correbponding change in S’. The physical significance of k is conjectural. The independence of k and S‘ shon-n by the particular systems chosen is unexpected and not t o be anticipated generally. The parameter k may be analogous to a frictional coefficient, and therefore is influenced by factors such as particle roughness and shape and the presence or absence of shells around the particles, derived from the suspension medium. These factors in turn are influenced by the particle size and particlesize distribution which determine s’.The fact that a single dispersed material, as herein, demonstrates a nearly constant value of S’ and a changing value of k aq the suspension medium changes indicates that the “frictional coefficient” can be changed without the formation of a film around the spheres of sufficient thickness t o greatly affect 8’. The intercept of the graph of 1’ qsi, against T’ on the T‘-axis represents the point at n-hich S’1’ equals unity. It is easily possible t o make up a suspension in n-hich the proportions are such that 1’ is greater than the intercept value. There is then insufficient liquid t o fill the voids betn-een particles. The Tvork of shearing may be sufficient to change X‘ and therefore move the T’-axis intercept, or some of the voids will be filled u-ith air and another phase introduced into the system. The appearance of scattered points at values of I’ greater than the 1’axis intercept is therefore to be expected. The author is indebted to The Mead Corporation for permission to publish this ~rorli. REFERESCES (1’1 . ~ R R H E S I TSVASTE: S, Z. physik Cheni. 1, 285 (1587). (2) B R E D ~ H. E , L.! ASD DE B o o ~ sJ.: , Iiolloid -Z. 79, 31-43 (1937). R . , C L - R ~ D JO., G., ASD BRADDICKS, R . , J R . : Rev. Sci. Instruments 18, (3) BL-CHDAHL, 168-72 (194T). (4) EISSTEIS, -1.: A n n . Physik 19, 289-306 (1906). (5) EIRICH, F . , BISZL,A I . , ASD ~ I A R G A R E T H H.: A ,Iiolloid-Z. 74, 276-85 (1936’1. (6) FISCHER, E A R LIC.,.IXD LISDSLEY, CHARLES H. : J. Colloid Sci. 3,114-17 (1946). ( 7 ) GL-TH,E., ASD SIXHA,R . : Iiolloid-Z. 74, 266-75 (1936). ( 8 ) IIATSCHEK, EXIIL: Trans. Faraday SOC. 9, 80-92 (1913). (9, LISDSLEY, CHARLES H., ASD FISCHER,E A R LI
