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The influence of particle-size distribution on the apparent viscosity of non-newtonian dispersed systems
The Influence of Particle-Size Distribution on the Apparent Viscosity of Non-Newtonian Dispersed Systems C. P A R K I N S O N ,
S. M A T S U M O T O ,
AND P. S H E R M A N
Unilever Research Laboratory, Welwyn, Hertfordshire, England Received September 8, 1969; accepted November 19, 1969 Monodisperse suspensions of poly(methylmethacrylate) spheres in Nujol, with diameters (D) of 0.1 ~, 0.6/~, 1.0/4 and 4.0 ~, respectively, have been blended together in various proportions to give suspensions with different modal size distributions. At the maximum rate of shear employed (1467.6 sec-1) the monodisperse suspensions were not completely defioceulated, and the degree of permanent flocculation increased as the particle size decreased. The viscosity (~r~ICZi))at the maximum rate of shear of suspensions containing i size fractions was given by "~rel(E~) ~
~reI(1) X ~rel(2) X Yrel(3) ' ' "
~
~/rel(i)
where w~(~), ~rel(~), Vr~(3), "'" Vrel(1)are the relative viscosities at the same rate of shear for the appropriate concentrations of size fractions 1, 2, 3, ... i when they are dispersed independently in Nuj el. This equation proved valid also for W/O emulsions over the whole volume concentration range (up to e ~-~ 0.6) examined, and for O/W emulsions up to a volume concentration ¢ of N0.4. Deviations from the equation in more concentrated O/W emulsions can be attributed to deformation of particles when they are closely packed. The emulsions did not exhibit permanent flocculation, so that the preceding equation could now be modified to a Mooney type equation
where e~, ~2, ~ , • • • el are the volume concentrations of each size fraction, and the values of ]c~, k2, ]c3 , .. • k~ are derived from an empirical equation of the general form k = 1.079 + exp (0.01008/D) + exp (0.00290/D~). In the past the methodology adopted to calculate the viscosities of dilute binary and tertiary particle suspensions has been justified by observations made in particle settling studies. This cannot be justified since particles in shear flow interact in a different way. INTRODUCTION
persions, i t is difficult to c o n t r o l t h e p a r t i c l e size d i s t r i b u t i o n w i t h a n y e x a c t i t u d e d u r i n g m a n u f a c t u r i n g processes. A s u i t a b l e w a y of i n v e s t i g a t i n g p a r t i c l e size d i s t r i b u t i o n effects w o u l d be to " s y n t h e size" m o d e l s y s t e m s w i t h different size ranges by blending together varying proportions of p o l y m e r b e a d s w i t h different d i a m eters. T h i s a p p r o a c h has b e e n u t i l i z e d b y s e v e r a l w o r k e r s ( 4 - 9 ) , b u t as t h e y r e s t r i c t e d t h e i r a t t e n t i o n to b i m o d a l size d i s t r i b u t i o n s their observations represent only the pre-
I t is now well e s t a b l i s h e d t h a t t h e rheologi e a l p r o p e r t i e s of m o n o d i s p e r s e suspensions c o n t a i n i n g s p h e r i c a l p a r t i c l e s are influenced b y p a r t i c l e size ( 1 - 3 ) . H o w e v e r , m a n y disp e r s e d s y s t e m s of p r a c t i c a l i n t e r e s t , a n d e s p e c i a l l y emulsions, a r e n o t m o n o d i s p e r s e so t h a t i t is n e c e s s a r y to e s t a b l i s h w h a t a d d i t i o n a l effect is e x e r t e d b y t h e p a r t i c l e - s i z e distribution. The latter problem has not been s t u d i e d in a n y d e t a i l p r e s u m a b l y b e c a u s e w i t h t h e e x c e p t i o n of p o l y m e r b e a d disJournal of Colloid and Interface Science, Vol. 33, No. 1, M a y 1970
150
PARTICLE-SIZE DISTRIBUTION AND VISCOSITY OF DISPERSED SYSTEMS 151 liminary phase of the study of continuous size distributions. Furthermore, all these workers, with one exception (7), used spheres with relatively large diameters (several microns to > 100 ~) so that particle diameter effects were minimized (10). Because of the authors' interests in the rheologieal properties of emulsions which contain significant numbers of particles < 1 ~ in diameter model systems have been studied which contained up to four different sizes of polymer spheres within the range 0.1 ~-4.0 ~ diameter. It was envisaged that the results of these studies would lead to an expression for the influence of particle-size distribution on viscosity. This expression would then be used to calculate the viscosities of emulsions when their particle-size distributions were known in some detail. EXPERIMENTAL a. Polymer Suspensions. Monodisperse suspensions of poly(methylmethaerylate) of 0.1 ~, 0.6 ~, 1.0 t~, and 4.0 ~ diameter were obtained from I.C.I. (Paints Division) Slough, England (11). An amphipathic copolymer, in which the soluble stabilizing component was the hexamer of 12-hydroxystearic acid, was used to stabilize these suspensions. The dispersion medium was eyclohexane, which was removed under vacuum until a thick paste developed and then Nujol was added. The residual cyclohexane was then evaporated under vacuum. The volume concentration of particles in each preparation was derived from specific gravity determinations. Suspensions with bimodal, trimodal, and tetramodal size distributions were prepared by mixing monodisperse suspensions with different particle sizes in varying proportions. The volume concentration of particles in these polymodal suspensions was kept constant at 0.117 for all studies. Higher concentrations of particles were not used because the suspensions then exhibited dilatant flOW.
b. Emulsions. The O/W and W/O emul-
sions were prepared using distilled water and Nujol as the two fluid phases. Span 20 (sorbitan monolaurate) and Span 85 (sorbitan trioleate) were employed as the respective emulsifying agents at 1.5 % concentration (expressed as wt/wt % of continuous phase) in the commercial quality provided by HoneywilI-Atlas Ltd., London. The emulsions were homogenized either by a handoperated valve homogenizer or a laboratory scale "Homozenta" emulsifying machine (E. Zehnder, Ztirich, Switzerland). c. Particle-Size Analysis. Particle-size distributions were determined with a Coulter centrifugal disk photosedimentometer (12, 13) using a disk rotation speed of 1500 rpm. Analyses with this instrument confirmed the very satisfactory monodispcrsity of the polymer suspensions. Oil-in-water emulsions were well diluted with aqueous phase containing the appropriate concentration of Span 20, and then injected from a syringe into the rotating centrifuge disk ("homogeneous" technique). Because the water drops in the W/O emulsions had a higher density than the oil phase a "three-layer" technique had to be applied in this case when introducing the diluted emulsion sample into the centrifuge disk. A first layer of pure cyclohexane was introduced and 2 ml petroleum spirit (boiling range I00°-120°C) was then injected from a syringe. This formed a thin layer on the first liquid. Finally, the diluted emulsion was introduced, and the water drops moved radially outwards through the layer of cyclohexane. The particle sizes of the poly(methyl methacrylate) spheres were determined using the original suspensions in cyclohexane and applying the "homogeneous" technique. d. The Influence of Particle Size on the Scattering Coefficient. Interpretation of optical density data is not absolute for particles the diameters of which approximate to the wavelength of the incident light because the scattering coefficient of particles depends upon a number of factors (14, 15), viz., the ratio of particle size to wavelength, the Journal of Colloid and Interface Science, Vol. 33, No. 1, May 1970
152
PARKINSON, MATSUMOTO, AND SHERMAN
ratio of the particle refractive index to that of the suspending fluid, the effect of light absorption by the particles, and the cone angle of the light beam to the photocell. Several theoretical treatments have been proposed based upon a consideration of these factors but they all relate to the use of monochrom~tic light. The centrifugal photosedL mentometer has a white light source (tungsten lamp) so that it is difiqcult in this case to compensate theoretically for the influence of particle size on the scattering coefficient. An estimate of the particle-size correction was derived experimentally using the poly(methyl methacrylate) suspensions and also monodisperse polystyrene latexes (16), obtained from the Dow Chemical Co., Midland, Michigan. The suspensions were diluted with the appropriate continuous phase fluid to obtain a weight of particles of (6.2 ~ 0.2) X 10-4 gm per 100 ml suspending fluid.
The scattering coet~.cient (Q~) of particles with diameter Di is given by Q~ = In
1 log10 eN~D~
,
[1]
where I and I0 are the intensities of the incident and emergent beams, respectively, 1 is the optical path length of the optical disk which contains Ni particles per unit volume of suspension fluid. Thus, in order to evaluate the absolute optical density ln(/0/I) from the recorder response vs. centrifugation time curves (Fig. 1) it is necessary to know the value of Q~ for some samples. When the ratio of particle diameter to the incident wavelength exceeds 100 then Qi ~ 1, irrespective of whether a monochromatic or white light source is used. We have assumed Q~ ~ 1 for the largest size particles used in this study, viz., 1.099 u diameter monadisperse polystyrene latex particles, and on this basis the particle-size dependence of
I0
G .1
f ®
_c
II ' -
I
In I l
II fl
o
I
fE
I I
4OO
I
I
I
1
I
350
300
250
200
150
Centrifugation
o.~o
o.~s o!4o
Diameter (.,u) of polystyrene latex particle
0:20
o!ls
50
0
time (mirt)
o:2s
o.'2o
e
I00
Dlarecter Gu') of
o!2s
o'-3
o!s o!6 all I
0-4
I
II
o~o6,
poly(methyl methacrylate)spheres
FIG. 1. Recorder response vs. centrifugation time for monodisperse suspensions. (~)--1.099 # polystyrene latex monodisperse suspension (~)--0.714 # polystyrene latex monodisperse suspension (~)-0.500 t~polystyrene latex monodisperse suspension (~)--0.357 ~ polystyrene latex monodisperse suspension @--0.234 ~ polystyrene latex monodisperse suspension @--1.00 tt poly(methyl methacrylate) monadisperse suspension (~)--0.60 ~ poly(methyl methacrylate) monodisperse suspension. Journal of Colloid and Interface Science,
VoL 33, No. 1, May 1970
PARTICLE-SIZE DISTRIBUTION AND ¥ISCOSITY OF DISPERSED SYSTEMS
153
0-3
~
0"~ z ~0
o_ ×
°o
o'.2
o'-~
o!~
o!~
,.o
;;2
Particle d~Gme~er (fl~
FIG. 2. Optical density of suspensions of small spheres (calculated by assuming that Q~ = 1).
optical density was derived (Figs. 2 and 3). Figure 4 shows a plot of Qi against D i . This curve also includes published data for glass spheres, quartzite powders, sand, etc. (1719), with diameters generally in excess of the polymer particle diameters. The Q~ data for the polystyrene and poly(methyl methacrylate) spheres fall reasonably well on a single smooth curve, and these data merge sarisfactorily with the data of other investigators for larger size particles. It is evident from Fig. 4 that Q~is greatly influenced by particle diameter in the submieron range and that its value falls to < 10-2 when particle diameter is ~0.2 ~. e. Viscosity Measurements. A Weissenberg rheogoniometer model R.16 (Sangamo Weston, Bognor Regis, England) was used for all viscosity determinations. The rates of shear employed ranged from 0.1462 to 1467.6 sec-~. Cone diameters of 3.75 em and 7.5 cm were used for the poly(methyl methacrylate) dispersions and the emulsions, respectively. For both series of tests the cone angle was 1°32 t. A constant temperature of 25.0 ° ± 0.1°C was maintained by means of a water jacket, and all samples were left between the cone and plate for 1 hour to at-
o
o'.2
o'.4
o'.6
o'.B
~'.o
~.2
Particle didmctcr ~ )
FIG. 3. Optical density of suspensions of small spheres (calculated by assuming that Qi = 1 for D = 1.099 u). O--polystyrene latex; A--poly(methyl methaerylate).
rain this temperature before they were tested. At the highest rates of shear employed there was a possibility that frictional heating effects might invalidate the result, but calculation showed this effect to be negligible under the conditions employed. A more serious problem was the danger of sample being thrown out from between the cone. and plate at very high rates of shear, which would have lowered the shear stress readtrigs. This danger was minimized by ensuring complete concentricity and alignment of the platens, and by checking that the shear stress readings were independent of time. All viscosity data quoted relate to the maximum rate of shear. RESULTS
a. P oly (Methyl M ethacrylate ) Suspensions. Figure 5 shows the influence of particle volume concentration (~) on the relative viscosity (~roI) at high rates of shear of monodisperse suspensions in Nujol which were prepared from each of the four available Journal of Colloid and Interface Science, Vol, 33, No. 1, ~[ay 1970
15~
PARKINSON, MATSUMOTO, AND SHERMAN
x
&
0"2
0"4
t 4
O" 0"8 I
~
i 8 I0
i 20
30
Partlcl¢ dla~t¢i" (fl)
FIG. 4. Particle-size dependence of the scattering coefficient when a white light source is used. © polystyrene latex A - - p o l y ( m e t h y l methacrylate) X--glass spheres (18) [:]--silica sand (18) ~ - - d e s e r t sand (17) @--quartzite (19). 2"4
l'S
1.6
1"41
I:I
z.*"
/
0-02
0"04
0'06
O' 8
0"I0
FIG:. 5. Influence of particle size on the Vro~- - ~ relationship. [] acrylate) spheres X X 0.6 ~ poly(methyl methacrylate) spheres A acrylate) spheres © © 4.0 ~ poly(methyl methaerylate) spheres. Journal o/Colloid and Interface Science, ¥ol. 33, No. 1, ~ a y 1970
o~.12 [] 0.1 ~ poly(methyl methA 1.0 ~ poly(methyl meth-
PARTICLE-SIZE DISTRIBUTION AND VISCOSITY OF DISPERSED SYSTEMS 155 sizes. The marked influence of particle size on the viscosity data is very noticeable, and the effect becomes progressively greater as the particle size decreases. Dm-ing viscosity measurements with these suspensions it was observed that their viscosities did not reach absolutely steady values even at the highest rates of shear eraployed, although the decrease in viscosity with further rise in rate of shear (-~) was quite small. Attempts were made to extrapolate the viscosity data to infinitely high shear rate b y plotting different functions of 1/~ against various functions of viscosity (7), but the only instance when a linear plot was obtained was for 7~/2 vs. #-~i2. A typical series of plots is shown in Fig. 6. The plots, which are for three different concentrations of 0.1 diameter particles, accord with Casson's (20) theory for the influence of # on 7. Suspensions with bimodal size distributions were prepared by mixing monodisperse suspensions with different particle sizes in the ratios 1/1, 1/3, and 3/1. The 7~ol curves of these suspensions (Fig. 7) showed pronounced minima when the smaller size particles constituted ~-~25% of the total particle volume concentration. For the monodisperse suspensions of 0.1 t~ diameter particles, 7~e~ =
/ o J2
o!4
~ j
~-~o'6
ols
~'2°`o*'~*'
,'-o
FIG. 6. Plot of ~/~ vs. q-~f2 for monodisperse suspensions of 0.1 t~ diameter poly(methyI methacrylate) spheres.
2-6
2.4
2.2
2-0 × 1.8
1"6'
b4
1"5
o~ 0-1 ~
diameter spheres in blmodal suspension
FIG. 7. R e l a t i o n s h i p b e t w e e n ~r~l a n d c o m p o s i t i o n of suspensions with bimodal size distribu-
tions. X spheres • spheres • spheres.
X mixture of 0.1 ,~ and 0.6 ~ diameter • mixture of 0.1 tt and 1.0 t~diameter • mixture of 0.1 ~ and 4.0 ~ diameter
17.76 (Table I). This value has not been inserted in Fig. 7 as a drastic reduction in the 7~ei scale would have been necessary, and then the minima in the three curves would not have been so prominent. When suspensions with trimodal size distributions were prepared b y mixing monodisperse suspensions with particle sizes 0.1 ~, 0.6 ~, and 1.0 ~, respectively, with the total volume concentration of particles being held constant at 0.117, no minimum was observed in 7rel as the concentration of the 0.1 g diameter particles was increased (Table 1). Instead, the value of 7rel increased curvilinearly as the proportion of the 0.i g particles increased. Similarly when tetramodal size distributions were prepared from suspensions with particle sizes 0.i g, 0.6 #, 1.0 ~, and 4.0 g, respectively, with the total volume concentration of particles still maintained constant at 0.117, 7tel again increased: curvilinearly as the proportion of 0.i Journal of Colloid and Interface Science, Vol. 33, N'o. I, May 1970
156
PARKINSON, MATSUMOT0, AND SHERMAN TABLE I
VISCOSITY D A T A FOR SUSPENSIONS C O N T A I N I N G 0 . 1 #, 0 . 6 /~, AND 1.0 /z POLY(METI~YL METHACRYLA'rE)
SPI~EI~ES Ratio of O.1 #/0.6 ~/1,0 I~ spheres
0:50:50 10:45:45 20:40:40 30:35:35 40:30:30 50:25:25 60:20:20 70:15:15 80:10:10 90:5:5 100:0:0
Relative viscosity contribution of constituent size fractions (derived from Fig. 5)
Relative viscosity of suspension
0.1 ~
0.6 ~
l.O/~
Calculateda
Experimental
1.00 1.22 1.31 1.52 2.18 4.67 6.38 7.24 9.36 12.82 17.76
1.18 1.16 1.14 1.12 1.09 1.06 1.03 1.01 1.01 1.00 1.00
1.11 1.09 1.07 1.06 1.04 1.03 1.02 1.01 1.00 1.00 1.00
1.31 1.54 1.60 1.80 2.47 5.10 6.70 7.39 9.45 12.82 17.76
1.32 1.38 1.66 1.74 3.91 6.18 7.56 8.90 10.12 13.46 17.76
Product of contributions by 0.1 ~, 0.6 ~, and 1.0 ~ fractions in accordance with Eq. [4].
I D
(:0
FIG. 8. Size-frequency analysis of 50% (w/w) O / W emulsiom particles increased ( T a b l e I I ) a n d there was no evidence of a m i n i m u m in Vre~ • b. Emulsions. T y p i c a l particle-size distrib u t i o n d a t a for the O / W and W / O emulsions are given in Figs. 8 and 10, respectively, and the corresponding experimental ~ r e l , ~ d a t a are given in Figs. 9 and 11, respectively. T h e O/W emulsions contained no particles < 0 . 5 ~ diameter, whereas some of t h e W / O emulsions contained relatively high proportions of particles < 0 . 5 ~. Journal of Colloid and Interface Science, Vol. 33, No. 1, May 1970
DISCUSSION Viscosities derived f r o m the e x t r a p o l a t e d d a t a for the monodisperse suspensions (Fig. 6) were higher t h a n was to be expected f r o m NIooney's e q u a t i o n (21) ~rel = exp
(%) 1 2
'
[2]
where a = 2.5 w h e n the particles are completely deflocculated, and k is a h y d r o d y namic interaction coefficient which increases
PARTICLE-SIZE DISTRIBUTION AND VISCOSITY OF DISPERSED SYSTEMS 157
5[
u
7
o'-i
o!2
o!~
o!s
d4
,j
o'.6
x ×
~p
Fro. 9. Comparison of experimental and theoretical VrelVS. ~ relationships for O/W emulsions. - - experimental data; . . . . . theoretical data.
x !
2c
.2
o:a
o-'4
o'.~
FIG, 11. Comparison of experimental and theoretical ~e~ vs, ~ relationships for W/O emulsions. O--experimental data; X--calculated values.
I
'
D(2)
FIG. 10. Size-frequency analysis of 50% (w/w) W/O emulsion. in value as the particle size decreases according to the empirical equation (22) /~ = 1.079 -t- exp (0.01008/D) + exp (0.00290/D2).
[3]
This implies that the actual values of a are higher t h a n the theoretical value, and that there was some residual, and permanent,
flocculation of particles. Suspensions of poly(methyl methacrylate) particles in ntetradecane exhibit this phenomenon (11), and Gillespie (23, 24) has observed similar behavior with polystyrene latexes. The ratio of particle sizes in suspensions with bimodel, trimodal, and tetramodal size distributions influences their viscosities even when the volume concentration of particles is identical in all eases (Tables I and I I ) . Eveson et al. (4) and Eveson (5) suggested that a suspension with a bimodal size distribution can be regarded as a system irt which the larger particles are suspended in a continuous phase formed by suspension of the smaller particles in the fluid medium. B y extension of this line of reasoning a suspension with a trimodM, tetramodal, o r / - m o d a l size distribution can be regarded as a suspension of the large size fractions in a conJournal of Colloid and Interface Science, ¥ol. 33, :No. 1, M a y 1970
158
PARKINSON, MATSUMOTO, AND SHERMAN
VISCOSITY
TABLE II DATA FOR SUSPENSIONS CONTAINING0.1 ~, 0.6 t~, 1.0 ~, AND 4.0 ( M E T H Y L METItACRYLATE) SPHERES
Ratio of 0.1 #/0.6 t~/1.0 #/4.0 t* spheres
0:33:33:33 10:30:30:30 25:25:25:25
40:20:20:20 55:15:15:15 70:10:10:10 85:5:5:5 100:0:0:0
Relative viscosity contributions of constituent size fractions(derived from Fig. 5)
Relative viscosity of suspension
0.1 tL
0.6 tt
1.0 t~
4.0/z
Calculateda
Experimental
1.00 1.22 1.41 2.18 5.52
1.11 1.09 1.06 1.03 1.01 1.01 1.00 1.00
1.05 1.04 1.03 1.02 1.01 1.00 1.00 1.00
1.02 1.01 1.00 1.00 1.00 1.00 1.00 1.00
1.19 1.40 1.54 2.29 5.63 7.31 11.27 17.76
1.26 1.48 1.56 2.46 5.80 7.47 11.30
7.24 11.27
17.76
17.76
a Product of contributions by 0.1 ~, 0.6 ~, 1.0 ~, and 4.0 ~ size fractions in accordance with Eq. [4]. tinuous phase formed by a suspension of the smallest size fraction in the fluid medium. On this basis ~rel(~i) =
~rel(ll ~
~rel(2) X
~rel(3) " ' "
[4] X
~rel(i) ~
~]rel(1), ~]rel(2), ~rel(3) " ' " ~rel(i) represent the experimentally derived viscosities for the appropriate concentrations of the different size fractions in the fluid medium. Chong (6) suggested that particles in bimodal suspensions interact in this way only when the ratio of the particle sizes is less t h a n 1/10; otherwise the smaller particles behave like ball bearings between the larger particles. If this view is correct then both mechanisms should be possible in dispersions with a continuous polymodal size distribu,tion. The general form of Eq. [4], as applied to bimodal and trimodal size distributions, has been justified (4-6, 25) on the basis of particle settling studies (25) with spherical particles having size ratios of 1/10 to 1/100. Actually, particle interactions in settling studies differ from those found between particles in shear flow. In the latter, for example, all particles are involved in temporary doublet formation and rotation, and in floeculation-deflocculation reactions under the influence of both shearing forces and Browninn motion when submieron particles are present. Particle settling studies involve where
Journal of Colloid and Interface Science, Vol. 33, No. 1, May 1970
movement under the influence of gravity of single size particles while other size particles remain suspended in the fluid medium. The main differences between the present study and those of previous workers (4-6) are that wider ranges of size fractions have been examined now, and that these sizes included submicron fractiorLs. This latter difference is important since small particles, and especially those smaller than 1~, have a profound influence on viscosity at a constant particle volume concentration (7, 22). Furthermore, the ultimate aim was to calculate the viscosities of dispersions, and emulsions, from the experimentally determined partieie-size distributions, each term on the right-hand side of Eq. [4] being calculated by Eq. [2] rather than being measured experimentally as was done by Eveson and his coworkers (4, 5). If the dispersions of poly(methyl methaerylate) particles had not shown some degree of irreversible floceulation such calculations would have been possible with the value of k for each size fraction being derived from Eq. (3). However, since residual flocculation was observed values of a larger than 2.5 have to be used in Eq. (2). In general, a was found to decrease with increasing particle size from ~ 5 . 1 for the 0.1 ~ diameter particles to ~ 2 . 5 for the 4.0 ~ diameter particles. Insufficient experimental data were available to derive a reliable value for the latter size
PARTICLE-SIZE DISTRIBUTION AND VISCOSITY OF DISPERSED SYSTEMS 159 particles so the validity of Eq. [4] was examined by deriving vro~(1), ~ro1(2), vrd(3), - " ~¢1(~) directly from Fig. (5). This approach assumes that at the highest rate of shear employed the irreversible flocculates contain particles of only one size. Tables I and II compare experimental and calculated viscosities for dispersions (~ = 0.117) with trimodal and tetramodal size distributions. The agreement between the data is reasonably satisfactory. The particles in the emulsions did not exhibit residual floeculation at high rates of shear so that ~]rel(1) , ~rel(2) , 'i~rel(3) " ' " ~rel(1) could be calculated directly from Eq. [2]. Equation [4] now becomes
exp (1 _
h exp ( :
/ 2.5~ • " exp ~1 = IIexp
[5]
1-- /~i/
and from the particle-size distribution in each emulsion the relevant values of k could be calculated. The O / W emulsions, which did not contain any particles <0.5 diameter (Fig. 8), showed good agreement between experimental and calculated viscosity data for values of ~ up to --~0.4 (Fig. 9). When ~ exceeded 0.4 the discrepancy between the two sets of data grew progressively larger as ~ increased. This is due to particle deformation as the particles pack more closely together (27) with increasing concentration of particles. For the W / O emulsions, some of which contained relatively high proportions of particles smaller than 0.5 ~ in diameter (Fig. 10), satisfactory agreement between experimental and calculated viscosity data (Fig. 11) extended to higher values of ~ t h a n for the O / W emulsions. Some data are given in Fig. 11 for the higher particle concentrations (~ = 0.468 and 0.568) to indicate the reproducible vis-
cosity values obtained for duplicate samples (A and B) having particle-size distributions which showed no obvious differences. iV[any factors influence the viscosity of an emulsion (28). It is possible that some of these factors act, at least partly, by altering the particle-size distribution. Consequently, if other effects are to be identified quantitatively it is necessary to know from the outset how particle size and/or particle size distribution affect viscosity. In the absence of such knowledge incorrect conclusions may be drawn. ACKNOWLED GMENT The authors are indebted to Drs. Walbridge and Waters of I.C.I. (Paint Division) Ltd., Slough, England, for providing the range of (poly) methacrylate suspensions used in this study and for a most helpful discussion about their properties. REFERENCES 1. SWEENY, K. I-I., AND GECKLER, R. D., J .
Appl.
Phys. 25, 1135 (1954). 2. SAUNDERS, F. L., or. Colloid Sei. 16, 13 (1961). 3. S~ERMAN, P., Proe. $th Intern. Congr. Rheol.
3,605 (1965). 4. EVESON, G. F., WARD, S. G., AND WttlTMORE, R. L., Discussions Faraday Soc. 11, 11 (1951). 5. EVESON, G. F., in C. C. Mill, ed, "Rheology of Disperse Systems," p. 61. Pergamon Press, London, (1959). 6. CHONG,J. S., Doctoral thesis, University of Utah, (1964). 7. SAVNDV.RS,F. L., J. Colloid Interfac. Sci. 23, 230 (1967). 8. FARRIS, R. J., Trans. Soc. Rheol. 12, 281 (1968). 9. LEWIS, T. B., AND NIELSEN, L. E., Trans. Soc. Rheol. 12, 421 (1968). 10. SHERMAN, P., Proc. Intern. Congr. Surface Activity 8rd Cologne 11, 596 (1960). 11. WALEIm)GE,D. J., AND WATtlRS, J. A., Discussions Faraday Soc. 42, 294 (1966). 12. GRovEs, M. J., KAyE, B. It., AND SCARLETT, B., Brit. Chem. Eng. 9,742 (1964). 13. FRESHWATER, D. C., SCARLETT, B., AND GROVES, M. J., Amer. Perfumer 81, 43 (1966). 14. VAN DE HULST, I-I. C., "Light Scattering by Small Particles." Wiley, New York, (1957). 15. WALSTRA,P., Brit. J. Appl. Phys. 15, 1545 (1964). Journal of Colloid and Interface Science, Vo]. 33~ No. 1, May 1970
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PARKINSON, MATSUMOTO, AND SHERMAN
16. BRADFORD, E. B., AND VANDERHOFF, J. W., J. Appl. Phys. 26,864 (1955). 17. RosE, H. E., AND LLOYD,I-I. B., J. Soc. Chem. Ind. 65, 52 (1946). 18. ROSE, It. E., Or. Appl. Chem. (London) 2,
80 (1952). 19. SUGANUM&, A., "Particle Size Analysis of Powders." Yokando, Tokyo, (1965). 20. CASSON, N., in C. C. Mill, ed, "Rheology of Disperse Systems." p. 84. Pergamon Press, London, (1959). 21. MOONEY, M., J. Colloid Sci. 6, 162 (1951). 22. MATSUMOTO, S., AND SHERMAN, P. dr. C o l l o i d Interfac. Sci. 30, 525 (1969). 23. GILLESPIE, T. ft., in P. Sherman, ed, "Rheol-
Journal of Colloid and Interface Science,
Vol. 33, No. 1, May 1970
24. 25.
26.
27. 28.
ogy of Emulsions," p. 115. Pergamon Press, London, (1963). GILLESPIE, T. J., J. Colloid Sci. 18, 32 (1963). FIDLERIS, V., 2~ND WHITMORE, R. L., Rheol. Acta 1, 573 (1961). GOLDSMITH, H. L., AND MASON, S. G., in F. Eirieh, ed, "Rheology. Theory and Applications," Vol. IV, p. 163. Academic Press, New York, (1967). SHERMAN, P., J. Colloid Interfac. Sci. 27, 282 (1968). SHERMAN, P., in P. Sherman, ed, "Emulsion Science," Chapt. 4. Academic Press, London, (1967).
S. M A T S U M O T O ,
AND P. S H E R M A N
Unilever Research Laboratory, Welwyn, Hertfordshire, England Received September 8, 1969; accepted November 19, 1969 Monodisperse suspensions of poly(methylmethacrylate) spheres in Nujol, with diameters (D) of 0.1 ~, 0.6/~, 1.0/4 and 4.0 ~, respectively, have been blended together in various proportions to give suspensions with different modal size distributions. At the maximum rate of shear employed (1467.6 sec-1) the monodisperse suspensions were not completely defioceulated, and the degree of permanent flocculation increased as the particle size decreased. The viscosity (~r~ICZi))at the maximum rate of shear of suspensions containing i size fractions was given by "~rel(E~) ~
~reI(1) X ~rel(2) X Yrel(3) ' ' "
~
~/rel(i)
where w~(~), ~rel(~), Vr~(3), "'" Vrel(1)are the relative viscosities at the same rate of shear for the appropriate concentrations of size fractions 1, 2, 3, ... i when they are dispersed independently in Nuj el. This equation proved valid also for W/O emulsions over the whole volume concentration range (up to e ~-~ 0.6) examined, and for O/W emulsions up to a volume concentration ¢ of N0.4. Deviations from the equation in more concentrated O/W emulsions can be attributed to deformation of particles when they are closely packed. The emulsions did not exhibit permanent flocculation, so that the preceding equation could now be modified to a Mooney type equation
where e~, ~2, ~ , • • • el are the volume concentrations of each size fraction, and the values of ]c~, k2, ]c3 , .. • k~ are derived from an empirical equation of the general form k = 1.079 + exp (0.01008/D) + exp (0.00290/D~). In the past the methodology adopted to calculate the viscosities of dilute binary and tertiary particle suspensions has been justified by observations made in particle settling studies. This cannot be justified since particles in shear flow interact in a different way. INTRODUCTION
persions, i t is difficult to c o n t r o l t h e p a r t i c l e size d i s t r i b u t i o n w i t h a n y e x a c t i t u d e d u r i n g m a n u f a c t u r i n g processes. A s u i t a b l e w a y of i n v e s t i g a t i n g p a r t i c l e size d i s t r i b u t i o n effects w o u l d be to " s y n t h e size" m o d e l s y s t e m s w i t h different size ranges by blending together varying proportions of p o l y m e r b e a d s w i t h different d i a m eters. T h i s a p p r o a c h has b e e n u t i l i z e d b y s e v e r a l w o r k e r s ( 4 - 9 ) , b u t as t h e y r e s t r i c t e d t h e i r a t t e n t i o n to b i m o d a l size d i s t r i b u t i o n s their observations represent only the pre-
I t is now well e s t a b l i s h e d t h a t t h e rheologi e a l p r o p e r t i e s of m o n o d i s p e r s e suspensions c o n t a i n i n g s p h e r i c a l p a r t i c l e s are influenced b y p a r t i c l e size ( 1 - 3 ) . H o w e v e r , m a n y disp e r s e d s y s t e m s of p r a c t i c a l i n t e r e s t , a n d e s p e c i a l l y emulsions, a r e n o t m o n o d i s p e r s e so t h a t i t is n e c e s s a r y to e s t a b l i s h w h a t a d d i t i o n a l effect is e x e r t e d b y t h e p a r t i c l e - s i z e distribution. The latter problem has not been s t u d i e d in a n y d e t a i l p r e s u m a b l y b e c a u s e w i t h t h e e x c e p t i o n of p o l y m e r b e a d disJournal of Colloid and Interface Science, Vol. 33, No. 1, M a y 1970
150
PARTICLE-SIZE DISTRIBUTION AND VISCOSITY OF DISPERSED SYSTEMS 151 liminary phase of the study of continuous size distributions. Furthermore, all these workers, with one exception (7), used spheres with relatively large diameters (several microns to > 100 ~) so that particle diameter effects were minimized (10). Because of the authors' interests in the rheologieal properties of emulsions which contain significant numbers of particles < 1 ~ in diameter model systems have been studied which contained up to four different sizes of polymer spheres within the range 0.1 ~-4.0 ~ diameter. It was envisaged that the results of these studies would lead to an expression for the influence of particle-size distribution on viscosity. This expression would then be used to calculate the viscosities of emulsions when their particle-size distributions were known in some detail. EXPERIMENTAL a. Polymer Suspensions. Monodisperse suspensions of poly(methylmethaerylate) of 0.1 ~, 0.6 ~, 1.0 t~, and 4.0 ~ diameter were obtained from I.C.I. (Paints Division) Slough, England (11). An amphipathic copolymer, in which the soluble stabilizing component was the hexamer of 12-hydroxystearic acid, was used to stabilize these suspensions. The dispersion medium was eyclohexane, which was removed under vacuum until a thick paste developed and then Nujol was added. The residual cyclohexane was then evaporated under vacuum. The volume concentration of particles in each preparation was derived from specific gravity determinations. Suspensions with bimodal, trimodal, and tetramodal size distributions were prepared by mixing monodisperse suspensions with different particle sizes in varying proportions. The volume concentration of particles in these polymodal suspensions was kept constant at 0.117 for all studies. Higher concentrations of particles were not used because the suspensions then exhibited dilatant flOW.
b. Emulsions. The O/W and W/O emul-
sions were prepared using distilled water and Nujol as the two fluid phases. Span 20 (sorbitan monolaurate) and Span 85 (sorbitan trioleate) were employed as the respective emulsifying agents at 1.5 % concentration (expressed as wt/wt % of continuous phase) in the commercial quality provided by HoneywilI-Atlas Ltd., London. The emulsions were homogenized either by a handoperated valve homogenizer or a laboratory scale "Homozenta" emulsifying machine (E. Zehnder, Ztirich, Switzerland). c. Particle-Size Analysis. Particle-size distributions were determined with a Coulter centrifugal disk photosedimentometer (12, 13) using a disk rotation speed of 1500 rpm. Analyses with this instrument confirmed the very satisfactory monodispcrsity of the polymer suspensions. Oil-in-water emulsions were well diluted with aqueous phase containing the appropriate concentration of Span 20, and then injected from a syringe into the rotating centrifuge disk ("homogeneous" technique). Because the water drops in the W/O emulsions had a higher density than the oil phase a "three-layer" technique had to be applied in this case when introducing the diluted emulsion sample into the centrifuge disk. A first layer of pure cyclohexane was introduced and 2 ml petroleum spirit (boiling range I00°-120°C) was then injected from a syringe. This formed a thin layer on the first liquid. Finally, the diluted emulsion was introduced, and the water drops moved radially outwards through the layer of cyclohexane. The particle sizes of the poly(methyl methacrylate) spheres were determined using the original suspensions in cyclohexane and applying the "homogeneous" technique. d. The Influence of Particle Size on the Scattering Coefficient. Interpretation of optical density data is not absolute for particles the diameters of which approximate to the wavelength of the incident light because the scattering coefficient of particles depends upon a number of factors (14, 15), viz., the ratio of particle size to wavelength, the Journal of Colloid and Interface Science, Vol. 33, No. 1, May 1970
152
PARKINSON, MATSUMOTO, AND SHERMAN
ratio of the particle refractive index to that of the suspending fluid, the effect of light absorption by the particles, and the cone angle of the light beam to the photocell. Several theoretical treatments have been proposed based upon a consideration of these factors but they all relate to the use of monochrom~tic light. The centrifugal photosedL mentometer has a white light source (tungsten lamp) so that it is difiqcult in this case to compensate theoretically for the influence of particle size on the scattering coefficient. An estimate of the particle-size correction was derived experimentally using the poly(methyl methacrylate) suspensions and also monodisperse polystyrene latexes (16), obtained from the Dow Chemical Co., Midland, Michigan. The suspensions were diluted with the appropriate continuous phase fluid to obtain a weight of particles of (6.2 ~ 0.2) X 10-4 gm per 100 ml suspending fluid.
The scattering coet~.cient (Q~) of particles with diameter Di is given by Q~ = In
1 log10 eN~D~
,
[1]
where I and I0 are the intensities of the incident and emergent beams, respectively, 1 is the optical path length of the optical disk which contains Ni particles per unit volume of suspension fluid. Thus, in order to evaluate the absolute optical density ln(/0/I) from the recorder response vs. centrifugation time curves (Fig. 1) it is necessary to know the value of Q~ for some samples. When the ratio of particle diameter to the incident wavelength exceeds 100 then Qi ~ 1, irrespective of whether a monochromatic or white light source is used. We have assumed Q~ ~ 1 for the largest size particles used in this study, viz., 1.099 u diameter monadisperse polystyrene latex particles, and on this basis the particle-size dependence of
I0
G .1
f ®
_c
II ' -
I
In I l
II fl
o
I
fE
I I
4OO
I
I
I
1
I
350
300
250
200
150
Centrifugation
o.~o
o.~s o!4o
Diameter (.,u) of polystyrene latex particle
0:20
o!ls
50
0
time (mirt)
o:2s
o.'2o
e
I00
Dlarecter Gu') of
o!2s
o'-3
o!s o!6 all I
0-4
I
II
o~o6,
poly(methyl methacrylate)spheres
FIG. 1. Recorder response vs. centrifugation time for monodisperse suspensions. (~)--1.099 # polystyrene latex monodisperse suspension (~)--0.714 # polystyrene latex monodisperse suspension (~)-0.500 t~polystyrene latex monodisperse suspension (~)--0.357 ~ polystyrene latex monodisperse suspension @--0.234 ~ polystyrene latex monodisperse suspension @--1.00 tt poly(methyl methacrylate) monadisperse suspension (~)--0.60 ~ poly(methyl methacrylate) monodisperse suspension. Journal of Colloid and Interface Science,
VoL 33, No. 1, May 1970
PARTICLE-SIZE DISTRIBUTION AND ¥ISCOSITY OF DISPERSED SYSTEMS
153
0-3
~
0"~ z ~0
o_ ×
°o
o'.2
o'-~
o!~
o!~
,.o
;;2
Particle d~Gme~er (fl~
FIG. 2. Optical density of suspensions of small spheres (calculated by assuming that Q~ = 1).
optical density was derived (Figs. 2 and 3). Figure 4 shows a plot of Qi against D i . This curve also includes published data for glass spheres, quartzite powders, sand, etc. (1719), with diameters generally in excess of the polymer particle diameters. The Q~ data for the polystyrene and poly(methyl methacrylate) spheres fall reasonably well on a single smooth curve, and these data merge sarisfactorily with the data of other investigators for larger size particles. It is evident from Fig. 4 that Q~is greatly influenced by particle diameter in the submieron range and that its value falls to < 10-2 when particle diameter is ~0.2 ~. e. Viscosity Measurements. A Weissenberg rheogoniometer model R.16 (Sangamo Weston, Bognor Regis, England) was used for all viscosity determinations. The rates of shear employed ranged from 0.1462 to 1467.6 sec-~. Cone diameters of 3.75 em and 7.5 cm were used for the poly(methyl methacrylate) dispersions and the emulsions, respectively. For both series of tests the cone angle was 1°32 t. A constant temperature of 25.0 ° ± 0.1°C was maintained by means of a water jacket, and all samples were left between the cone and plate for 1 hour to at-
o
o'.2
o'.4
o'.6
o'.B
~'.o
~.2
Particle didmctcr ~ )
FIG. 3. Optical density of suspensions of small spheres (calculated by assuming that Qi = 1 for D = 1.099 u). O--polystyrene latex; A--poly(methyl methaerylate).
rain this temperature before they were tested. At the highest rates of shear employed there was a possibility that frictional heating effects might invalidate the result, but calculation showed this effect to be negligible under the conditions employed. A more serious problem was the danger of sample being thrown out from between the cone. and plate at very high rates of shear, which would have lowered the shear stress readtrigs. This danger was minimized by ensuring complete concentricity and alignment of the platens, and by checking that the shear stress readings were independent of time. All viscosity data quoted relate to the maximum rate of shear. RESULTS
a. P oly (Methyl M ethacrylate ) Suspensions. Figure 5 shows the influence of particle volume concentration (~) on the relative viscosity (~roI) at high rates of shear of monodisperse suspensions in Nujol which were prepared from each of the four available Journal of Colloid and Interface Science, Vol, 33, No. 1, ~[ay 1970
15~
PARKINSON, MATSUMOTO, AND SHERMAN
x
&
0"2
0"4
t 4
O" 0"8 I
~
i 8 I0
i 20
30
Partlcl¢ dla~t¢i" (fl)
FIG. 4. Particle-size dependence of the scattering coefficient when a white light source is used. © polystyrene latex A - - p o l y ( m e t h y l methacrylate) X--glass spheres (18) [:]--silica sand (18) ~ - - d e s e r t sand (17) @--quartzite (19). 2"4
l'S
1.6
1"41
I:I
z.*"
/
0-02
0"04
0'06
O' 8
0"I0
FIG:. 5. Influence of particle size on the Vro~- - ~ relationship. [] acrylate) spheres X X 0.6 ~ poly(methyl methacrylate) spheres A acrylate) spheres © © 4.0 ~ poly(methyl methaerylate) spheres. Journal o/Colloid and Interface Science, ¥ol. 33, No. 1, ~ a y 1970
o~.12 [] 0.1 ~ poly(methyl methA 1.0 ~ poly(methyl meth-
PARTICLE-SIZE DISTRIBUTION AND VISCOSITY OF DISPERSED SYSTEMS 155 sizes. The marked influence of particle size on the viscosity data is very noticeable, and the effect becomes progressively greater as the particle size decreases. Dm-ing viscosity measurements with these suspensions it was observed that their viscosities did not reach absolutely steady values even at the highest rates of shear eraployed, although the decrease in viscosity with further rise in rate of shear (-~) was quite small. Attempts were made to extrapolate the viscosity data to infinitely high shear rate b y plotting different functions of 1/~ against various functions of viscosity (7), but the only instance when a linear plot was obtained was for 7~/2 vs. #-~i2. A typical series of plots is shown in Fig. 6. The plots, which are for three different concentrations of 0.1 diameter particles, accord with Casson's (20) theory for the influence of # on 7. Suspensions with bimodal size distributions were prepared by mixing monodisperse suspensions with different particle sizes in the ratios 1/1, 1/3, and 3/1. The 7~ol curves of these suspensions (Fig. 7) showed pronounced minima when the smaller size particles constituted ~-~25% of the total particle volume concentration. For the monodisperse suspensions of 0.1 t~ diameter particles, 7~e~ =
/ o J2
o!4
~ j
~-~o'6
ols
~'2°`o*'~*'
,'-o
FIG. 6. Plot of ~/~ vs. q-~f2 for monodisperse suspensions of 0.1 t~ diameter poly(methyI methacrylate) spheres.
2-6
2.4
2.2
2-0 × 1.8
1"6'
b4
1"5
o~ 0-1 ~
diameter spheres in blmodal suspension
FIG. 7. R e l a t i o n s h i p b e t w e e n ~r~l a n d c o m p o s i t i o n of suspensions with bimodal size distribu-
tions. X spheres • spheres • spheres.
X mixture of 0.1 ,~ and 0.6 ~ diameter • mixture of 0.1 tt and 1.0 t~diameter • mixture of 0.1 ~ and 4.0 ~ diameter
17.76 (Table I). This value has not been inserted in Fig. 7 as a drastic reduction in the 7~ei scale would have been necessary, and then the minima in the three curves would not have been so prominent. When suspensions with trimodal size distributions were prepared b y mixing monodisperse suspensions with particle sizes 0.1 ~, 0.6 ~, and 1.0 ~, respectively, with the total volume concentration of particles being held constant at 0.117, no minimum was observed in 7rel as the concentration of the 0.1 g diameter particles was increased (Table 1). Instead, the value of 7rel increased curvilinearly as the proportion of the 0.i g particles increased. Similarly when tetramodal size distributions were prepared from suspensions with particle sizes 0.i g, 0.6 #, 1.0 ~, and 4.0 g, respectively, with the total volume concentration of particles still maintained constant at 0.117, 7tel again increased: curvilinearly as the proportion of 0.i Journal of Colloid and Interface Science, Vol. 33, N'o. I, May 1970
156
PARKINSON, MATSUMOT0, AND SHERMAN TABLE I
VISCOSITY D A T A FOR SUSPENSIONS C O N T A I N I N G 0 . 1 #, 0 . 6 /~, AND 1.0 /z POLY(METI~YL METHACRYLA'rE)
SPI~EI~ES Ratio of O.1 #/0.6 ~/1,0 I~ spheres
0:50:50 10:45:45 20:40:40 30:35:35 40:30:30 50:25:25 60:20:20 70:15:15 80:10:10 90:5:5 100:0:0
Relative viscosity contribution of constituent size fractions (derived from Fig. 5)
Relative viscosity of suspension
0.1 ~
0.6 ~
l.O/~
Calculateda
Experimental
1.00 1.22 1.31 1.52 2.18 4.67 6.38 7.24 9.36 12.82 17.76
1.18 1.16 1.14 1.12 1.09 1.06 1.03 1.01 1.01 1.00 1.00
1.11 1.09 1.07 1.06 1.04 1.03 1.02 1.01 1.00 1.00 1.00
1.31 1.54 1.60 1.80 2.47 5.10 6.70 7.39 9.45 12.82 17.76
1.32 1.38 1.66 1.74 3.91 6.18 7.56 8.90 10.12 13.46 17.76
Product of contributions by 0.1 ~, 0.6 ~, and 1.0 ~ fractions in accordance with Eq. [4].
I D
(:0
FIG. 8. Size-frequency analysis of 50% (w/w) O / W emulsiom particles increased ( T a b l e I I ) a n d there was no evidence of a m i n i m u m in Vre~ • b. Emulsions. T y p i c a l particle-size distrib u t i o n d a t a for the O / W and W / O emulsions are given in Figs. 8 and 10, respectively, and the corresponding experimental ~ r e l , ~ d a t a are given in Figs. 9 and 11, respectively. T h e O/W emulsions contained no particles < 0 . 5 ~ diameter, whereas some of t h e W / O emulsions contained relatively high proportions of particles < 0 . 5 ~. Journal of Colloid and Interface Science, Vol. 33, No. 1, May 1970
DISCUSSION Viscosities derived f r o m the e x t r a p o l a t e d d a t a for the monodisperse suspensions (Fig. 6) were higher t h a n was to be expected f r o m NIooney's e q u a t i o n (21) ~rel = exp
(%) 1 2
'
[2]
where a = 2.5 w h e n the particles are completely deflocculated, and k is a h y d r o d y namic interaction coefficient which increases
PARTICLE-SIZE DISTRIBUTION AND VISCOSITY OF DISPERSED SYSTEMS 157
5[
u
7
o'-i
o!2
o!~
o!s
d4
,j
o'.6
x ×
~p
Fro. 9. Comparison of experimental and theoretical VrelVS. ~ relationships for O/W emulsions. - - experimental data; . . . . . theoretical data.
x !
2c
.2
o:a
o-'4
o'.~
FIG, 11. Comparison of experimental and theoretical ~e~ vs, ~ relationships for W/O emulsions. O--experimental data; X--calculated values.
I
'
D(2)
FIG. 10. Size-frequency analysis of 50% (w/w) W/O emulsion. in value as the particle size decreases according to the empirical equation (22) /~ = 1.079 -t- exp (0.01008/D) + exp (0.00290/D2).
[3]
This implies that the actual values of a are higher t h a n the theoretical value, and that there was some residual, and permanent,
flocculation of particles. Suspensions of poly(methyl methacrylate) particles in ntetradecane exhibit this phenomenon (11), and Gillespie (23, 24) has observed similar behavior with polystyrene latexes. The ratio of particle sizes in suspensions with bimodel, trimodal, and tetramodal size distributions influences their viscosities even when the volume concentration of particles is identical in all eases (Tables I and I I ) . Eveson et al. (4) and Eveson (5) suggested that a suspension with a bimodal size distribution can be regarded as a system irt which the larger particles are suspended in a continuous phase formed by suspension of the smaller particles in the fluid medium. B y extension of this line of reasoning a suspension with a trimodM, tetramodal, o r / - m o d a l size distribution can be regarded as a suspension of the large size fractions in a conJournal of Colloid and Interface Science, ¥ol. 33, :No. 1, M a y 1970
158
PARKINSON, MATSUMOTO, AND SHERMAN
VISCOSITY
TABLE II DATA FOR SUSPENSIONS CONTAINING0.1 ~, 0.6 t~, 1.0 ~, AND 4.0 ( M E T H Y L METItACRYLATE) SPHERES
Ratio of 0.1 #/0.6 t~/1.0 #/4.0 t* spheres
0:33:33:33 10:30:30:30 25:25:25:25
40:20:20:20 55:15:15:15 70:10:10:10 85:5:5:5 100:0:0:0
Relative viscosity contributions of constituent size fractions(derived from Fig. 5)
Relative viscosity of suspension
0.1 tL
0.6 tt
1.0 t~
4.0/z
Calculateda
Experimental
1.00 1.22 1.41 2.18 5.52
1.11 1.09 1.06 1.03 1.01 1.01 1.00 1.00
1.05 1.04 1.03 1.02 1.01 1.00 1.00 1.00
1.02 1.01 1.00 1.00 1.00 1.00 1.00 1.00
1.19 1.40 1.54 2.29 5.63 7.31 11.27 17.76
1.26 1.48 1.56 2.46 5.80 7.47 11.30
7.24 11.27
17.76
17.76
a Product of contributions by 0.1 ~, 0.6 ~, 1.0 ~, and 4.0 ~ size fractions in accordance with Eq. [4]. tinuous phase formed by a suspension of the smallest size fraction in the fluid medium. On this basis ~rel(~i) =
~rel(ll ~
~rel(2) X
~rel(3) " ' "
[4] X
~rel(i) ~
~]rel(1), ~]rel(2), ~rel(3) " ' " ~rel(i) represent the experimentally derived viscosities for the appropriate concentrations of the different size fractions in the fluid medium. Chong (6) suggested that particles in bimodal suspensions interact in this way only when the ratio of the particle sizes is less t h a n 1/10; otherwise the smaller particles behave like ball bearings between the larger particles. If this view is correct then both mechanisms should be possible in dispersions with a continuous polymodal size distribu,tion. The general form of Eq. [4], as applied to bimodal and trimodal size distributions, has been justified (4-6, 25) on the basis of particle settling studies (25) with spherical particles having size ratios of 1/10 to 1/100. Actually, particle interactions in settling studies differ from those found between particles in shear flow. In the latter, for example, all particles are involved in temporary doublet formation and rotation, and in floeculation-deflocculation reactions under the influence of both shearing forces and Browninn motion when submieron particles are present. Particle settling studies involve where
Journal of Colloid and Interface Science, Vol. 33, No. 1, May 1970
movement under the influence of gravity of single size particles while other size particles remain suspended in the fluid medium. The main differences between the present study and those of previous workers (4-6) are that wider ranges of size fractions have been examined now, and that these sizes included submicron fractiorLs. This latter difference is important since small particles, and especially those smaller than 1~, have a profound influence on viscosity at a constant particle volume concentration (7, 22). Furthermore, the ultimate aim was to calculate the viscosities of dispersions, and emulsions, from the experimentally determined partieie-size distributions, each term on the right-hand side of Eq. [4] being calculated by Eq. [2] rather than being measured experimentally as was done by Eveson and his coworkers (4, 5). If the dispersions of poly(methyl methaerylate) particles had not shown some degree of irreversible floceulation such calculations would have been possible with the value of k for each size fraction being derived from Eq. (3). However, since residual flocculation was observed values of a larger than 2.5 have to be used in Eq. (2). In general, a was found to decrease with increasing particle size from ~ 5 . 1 for the 0.1 ~ diameter particles to ~ 2 . 5 for the 4.0 ~ diameter particles. Insufficient experimental data were available to derive a reliable value for the latter size
PARTICLE-SIZE DISTRIBUTION AND VISCOSITY OF DISPERSED SYSTEMS 159 particles so the validity of Eq. [4] was examined by deriving vro~(1), ~ro1(2), vrd(3), - " ~¢1(~) directly from Fig. (5). This approach assumes that at the highest rate of shear employed the irreversible flocculates contain particles of only one size. Tables I and II compare experimental and calculated viscosities for dispersions (~ = 0.117) with trimodal and tetramodal size distributions. The agreement between the data is reasonably satisfactory. The particles in the emulsions did not exhibit residual floeculation at high rates of shear so that ~]rel(1) , ~rel(2) , 'i~rel(3) " ' " ~rel(1) could be calculated directly from Eq. [2]. Equation [4] now becomes
exp (1 _
h exp ( :
/ 2.5~ • " exp ~1 = IIexp
[5]
1-- /~i/
and from the particle-size distribution in each emulsion the relevant values of k could be calculated. The O / W emulsions, which did not contain any particles <0.5 diameter (Fig. 8), showed good agreement between experimental and calculated viscosity data for values of ~ up to --~0.4 (Fig. 9). When ~ exceeded 0.4 the discrepancy between the two sets of data grew progressively larger as ~ increased. This is due to particle deformation as the particles pack more closely together (27) with increasing concentration of particles. For the W / O emulsions, some of which contained relatively high proportions of particles smaller than 0.5 ~ in diameter (Fig. 10), satisfactory agreement between experimental and calculated viscosity data (Fig. 11) extended to higher values of ~ t h a n for the O / W emulsions. Some data are given in Fig. 11 for the higher particle concentrations (~ = 0.468 and 0.568) to indicate the reproducible vis-
cosity values obtained for duplicate samples (A and B) having particle-size distributions which showed no obvious differences. iV[any factors influence the viscosity of an emulsion (28). It is possible that some of these factors act, at least partly, by altering the particle-size distribution. Consequently, if other effects are to be identified quantitatively it is necessary to know from the outset how particle size and/or particle size distribution affect viscosity. In the absence of such knowledge incorrect conclusions may be drawn. ACKNOWLED GMENT The authors are indebted to Drs. Walbridge and Waters of I.C.I. (Paint Division) Ltd., Slough, England, for providing the range of (poly) methacrylate suspensions used in this study and for a most helpful discussion about their properties. REFERENCES 1. SWEENY, K. I-I., AND GECKLER, R. D., J .
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Phys. 25, 1135 (1954). 2. SAUNDERS, F. L., or. Colloid Sei. 16, 13 (1961). 3. S~ERMAN, P., Proe. $th Intern. Congr. Rheol.
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