prediction of the viscosity of multimodal suspensions from

TRANSACTIONS OF THE SOCIETY OF RHEOLOGY 12:2,281-301 (1908)

Prediction of the Viscosity of MultimodalSuspensions from Unimodal Viscosity Data

R. .J. FARRIS,* Aerojet-Generol Corporation, Sacramento,Calijomia 95809

Synopsis

A theoretical treatment of particle-particle interaction is described from whichthe viscosity-concenl.ration behavior of multi modal suspensions of rigid particlescan be related to the viscosity-concent.ration behavior of the unimodal components. From this theory, the viscosity of multimodal suspensions can becalculated and shows excellent agreement with existing experimental data.Blend ratios that will produce minimum viscosities are simply derived from thetheory and agree well with experimental results. Another important feature ofthis theory is that it predicts and defines a lower limit for the viscosity at anyconcentration and indicates that this lowest viscosity can be obtained with avariety of solids combinations.

Introduction

Perhaps the most basic work in the rheology of suspensions wasdue to Einstein,' who derived a formula for the relative viscosity ofdilute suspensions of uniform-sized spherical particles. Since thepublication of his basic analysis, numerous equations have beendeveloped in efforts to extend Einstein's formula to suspensions ofhigher concentrations.>:" The various resulting formulas, boththeoretical and empirical, differ considerably from each other as doexperimental data at high concentrations; the experimental dataprobably differ because of varying latitudes of the particle size distributions of the monodispersed sizes used," thereby changing themaximum possible concentrations in each study.

Attempts to extend this work to understand and predict the behavior of a suspension when two or more monodisperscd sizes areblended together has been unsuccessful to date. Geometrically, it

* Present address: Department of Civil Engineering, The University of Utah,Salt Lake City, Utah S41l2.

281

282 It J. FARRIS

is easy to understand that through the use of multimodal size distributions the limiting concentrations could be increased, but theblend ratios and size ratios of such systems had to be determinedexperimentally, always with the question, "It there another blendthat will reduce the viscosity even more?"

This paper does not attempt to describe mathematically theviscosity-concentration behavior of suspensions of particles of unimodal size distribution. Instead it treats the way in which two ormore size distributions interact when they are combined in the samesuspension. The result of which is a simple means to understandand predict the viscosity of multimodal suspensions from the observed viscosity-concentration behavior of its unimodal components.Through the use of this model, optimum blend ratios for multimodalsuspensions can be derived as well as the minimum possible viscosityfor any concentration.

Theory

In the literature there are references2.6·7 indicating that the finerparticles in a bimodal suspension behave essentially as a fluid towardthe coarser particles. The most illustrative work was performed byFidleris and Whitmore who investigated the settling velocity of alarge sphere in a 20% suspension of uniform-sized small spheres.The results of their investigation showed that if the size ratio, R12

(small to large) was 1/10 or less, then the large sphere encountered thesame resistance to motion when passing through a suspension ofsmaller spheres as when it passed through a pure liquid of the sameviscosity and density as the suspension. When the size ratio became greater than 1/10, the falling sphere appeared to encounter thesame resistance but followed a zig-zag random path instead of alinear path. This was true even when the falling sphere was smallerthan the suspension spheres.

Using this concept, the viscosity of a multimodal suspension ofparticles can be calculated from the unimodal viscosity data of eachsize as long as the relative sizes in question are sufficient to have thiscondition of zero interaction between coarse and fine, since then thebehavior of each size is completely independent of the other. Forthe purpose of clarity and to minimize the mathematics, a systemcontaining two sizes (bimodal) will be described first followed bythe general case of N sizes.

VISCOSITY OF MULTIMODAL SUSPENSIONS 283

Consider the two parts of the volume making up a suspensionhaving only very fine particles: V I = volume of liquid; VI = volumeof fine particles. The volume concentration of the fine filler in theliquid is:

(1)

We know that the liquid is going to be stiffened and viscosity increased by the presence of the filler. This stiffening factor is definedas H(</» and is simply the relative viscosity of a unimodal system tothat of the liquid alone. Figure 1 illustrates H(</» versus the concentration <P for unimodal rigid spherical particles. From the datait appears that H</> is independent of size. Therefore from the definition of H (</» we may write

(2)

where '70 = viscosity of liquid; '71 = viscosity when filled with fines;<PI = VII (V I + Vf); '7r = relative viscosity, compared to the pureliquid.

Consider now what would happen when coarse particles are addedto the suspension of fines. With the assumption that the fine par-

0.6

5L8·240.35 -45261.64-12

Particle Size, /f

0.2 0., 0.4Volume Fraction 01 Solllls, •

Investigatorso ChDngc RobinsDn• Sweeny• Williams100

COMPARISON OF THE RELATIVE VISCOSItYOF MONODISPERSED SYSIIMS

i~

§S;~ 10'=----+----t----+---j-;:;-l<~rs_r--_j

i

Figure 1.

284 R. J. FARRIS

ticles behave as a fluid toward the coarse, the coarse particles can beconsidered simply as stiffening this already stiffened fluid by anadditional factor of H(<f>c). H(<f>c) is the ratio of the new viscositywith coarse and fine particles, n., to that of the fluid containing onlythe fine particles, 7Jf.

H(¢c) = 'T/c/7Jf = ('T/c!'T/o) . ("10/"1/)

= (7Jcl7Jo) . [l/H(¢f)] = [7Jr/H(¢f)] (3)

where 7Jc/7Jo becomes the relative viscosity of this bimodal suspension.Hence

(4)

In eq. (4), <Pc is the concentration of coarse in the apparent liquidwhose volume is now given by (VI + Vf)' Thus,

(5)

where V c is volume of coarse particles.Since H(rp) = 1 when rp = 0, eq. (4) is valid for all values of Vf

and V c includin~ zero. Naturally this line of reasoning can beapplied again and again to obtain:

N

II H(<Pi)i=l

(6)

where II denotes product and rp, is concentration of each size III

apparent liquid.In this system of notation the sizes are always ordered from

smallest to largest as i goes from 1 to N. The liquid volume isalways designated as Yo. Thus rpm, the concentration of the mthvalue of i, is written as

V m,pm = -------------(V o + V1 + V2 • • • V m-l) + V m

(7)

All that is left now is to describe the total filler concentration, cPr,in terms of the concentration of each component, ,pi. This totalconcentration is clearly

Vi + V2 + ... + V N<pr = ---------Vo + V1 + V 2 ••• V N

N

LVii=l-N--

LVi.=0

(8)


Recalling that each value of tPi involves the same Yo, but not thesame value of apparent liquid, e.g., (Vo + VI + V2 + ... + V i-I)is the apparent liquid for V i, it is clear that the total concentrationis not simplySe; The general solution for the total concentrationmay be expressed as

N

(1 - <Pr) = II (1 - tPi)i=l

(9)

Equation (9) is not readily apparent but is simply verified by substitution as follows. From the definition of tPm and tPr we may writethe following two equations,

(10)

(12)

N

LVi1 i=1 Vo

- -N-- = -N-- =

LVi LVii=O i=O

substituting eqs. (10) and (11) into eq. (9) term for term we obtain

(Vo + VI + ~2 ••• + VJ(

Vo ) ( Vo + VI ) ( Vo+ VI + V2 )

= Vo+ VI Vo+ VI + V2 Vo+ VI + V2 + Va

.. (Vo + VI + ... VN - 1)Vo+ VI + V2 + V N

Careful examination of the right side of eq. (12) reveals that thedenominator of each term is cancelled by the numerator of the following term. Hence, all terms cancel except the numerator of the firstterm, Yo, and the denominator of the last term, Vo+ VI + V2 • • • VN,

and the proof is complete.Expanding eq. (9) for the bimodal and trimodal cases yields:

tPr(bimodal) = 1 - (1 - <PI) (1 - tP2) = <PI + tP2 - <PltP2 (13a)

I/>r(trimodal) = 1 - (l - 1/>1)(1 - 1/>2)(1 - cf>2)

= tPl + <P2 + <Pa - tPl<P2 - <PltPa - <P21/>a + 1/>1cf>2l/>a (13b)

286 R. J. FARRIS

In both eqs. (6) and (9) the product notation can be eliminatedby expressing 7/r and c/>T as logarithms as done in eqs. (14) and (15).

N

In 7fr = L In H(epi)[=1

N

In (1 - epT) = LIn (1 - cP,)i=l

(14)

(15)

Figure 2 illustrates the excellent agreement between the calculatedand measured relative viscosity data for a bimodal system of nearzero size ratio (i.e., one filler fraction has very coarse particles compared to the other, or R 12 = 0) where this assumption of no interaction should be well justified. Also illustrated in this figure arethe measured viscosity data for four other size ratios, the 1: 1 corresponding to the monomodal data (also illustrated in Fig. 1) fromwhich the values of H(c/>l) and H(cP2) were obtained. The data inFigure 2 indicate that if the size ratio is 1: 10 or less, this assumptionof no interaction found for a simple falling sphere by Fidleris andWhitmore, still holds for the blending of two sizes.

Before proceeding further it should be pointed out that the datain Figure 2 correspond to a particular blend situation, namely for a

CALCULATED AND MEASURED VISCOSITIES OF MONOMODAl AND BIMODAlSUSPENSIONS WITH VOlUME fRACTION OF SMALl SPHERES • 251>

1D,ooo r-t--t--

ro~ ~ ~ • ~ ~ M ~ ~ n ~ a ~Volume Fraction ofSolids••

Figure 2.


fixed fraction of fine spheres (based on total volume, not the sameas cPl)' Now using a bimodal blend of RI2 = 0, lines correspondingto R12 = 0.477 or the other nonzero size ratios in Figure 2 can becalculated, but the corresponding blend ratios will be different. Infact the line for R12 = 0,477 can be calculated two ways still assumingno interaction. First by having a system of almost all fines orsecond by having a system of almost all course. Similarly, thecurves for R12 = 0.313 and R12 = 0.138 can be calculated assumingno interaction by selecting more optimized blends, but again thecorresponding blend ratios will be different. In doing so bimodalblends with slightly lower viscosities than illustrated for R I2 = 0can be found and this will be discussed later.

From the above discussion it appears that all possible lines between monomodal and best bimodal can be calculated assuming nointeraction just by changing the blend ratios.

The curve, H(¢) versus ¢, in Figure 1 becomes very steep simplybecause of the mutual crowding of particles. Since in this methodof analysis H(¢) is used, we are already accounting for mutual crowding in its most severe case, when R I2 = 1. By defining a crowdingfactor, j, the behavior of all size ratios can be accounted for. Thiscrowding factor permits one to find that blend ratio for zero interaction that behaves equivalently to any particular blend ratio withinteraction, (R 12 > 0.1). The crowding factor is defined as thatfraction of one size that behaves as though it were the other size.Since poorer packing can be obtained by using an excess of the coarseor an excess of the fine fraction if we desire to shift toward poorerpacking the crowding factor, j, must operate on the size of the fillerhaving the lowest concentration. Logically this factor j varies fromo to 1 as the size ratio varies from 0 to 1.

The similarities between blend ratios and size ratios are shown inFigure 3 were the calculated viscosity data for different blend ratiosassuming R I2 = 0 are illustrated. In a bimodal system one obtainsthe same viscosity regardless of whether ¢I and ¢2 are interchangedat any concentration of the two given sizes. There are two blendratios that will result in the same viscosity, the second being thereciprocal of the first. The equivalence of size ratios and the effectof the crowding factor, j, are illustrated in Figure 4 showing themeasured viscosity data for different size ratios and the calculateddata for different values of f. No attempt was made to make the

288 H. .1, FAIUU::;

curves coincide with each other and no data exists to show that f isindependent of the blend ratio. This approach provides a simple,but effective, means of calculating what will result when similarsizes are blended together, however more experimental data areneeded so additional correlations can be made.

CALCULATED VISCOSITIES OF BIMODAL SUSPENSIONS OF ZEROSIZE RP.T10 FOR DIFfERENT BlEND RP.T10S

I I Iecle, • 0 I

~or ellD • 0 0cIO,·.1

0clOf • 0.3 00'0, • 0.8I c/ jOc -0.1 Of10c • 0.3 0' • 0.8j

;ff V VV

0c'Of • 1~ /~<:!

t.> V?/ /' ~

~ / /' l:::.-./

f.--:.--

~~---~ ---~

10.54 .50 .58 .00 ,~2 .64 .~ .68 .70 .72 .74 .16 .78Volume Fraction 01 Solids, V

100

10,000

~

Fignre ;l.

CALCULATED AND MEASURED VISCOSITIES OF MONOMODAL AND BIMODALSUSP£NSIONS WITH VOLUME FRACTION OF SMAll SPI£RES • m

.78.74 .76.72

.........-+_-+--<>'9'_-1-_+_'::; 'r-l----I

,00 ,62 .64 .66 .68 .70Volume fraction ~I Solids, ~

Figure 4.

.58

VISCOSITY OF MULTI MODAL SUSPENSIONS 289

Optimization of Multimodal System

In order to optimize the blend ratios in a multimodal system(assuming no or equal interaction) all that is necessary is to determinethe minimum value of the product of the values of H(c/>i). This canbe achieved by differentiating "1T or In "1r with respect to c/>i and settingthe differential equal to zero. The solution of the minimum minimacan be obtained as follows:

N

In 11. = Lin H(c/>i)i=l

definition from theory (14)

N

In (1 - c/>T) = LIn (1 - <Pi) definition from theory (15)i=l

From the definition of a total derivative we may write

o In 7]T 0 In "1r 0 In "1rd In 7]r = -",- dc/>I +~ d<P2 ... + -",-- d<PN

U<PI ~2 Uc/>N

~ 0 In "1r d=L....-- <P

i~1 Oc/>i t

(16)

the partial derivatives in eq. (16) can be obtained from eq. (14).Doing this and setting eq. (16) equal to zero we obtain

(17)

The values of d¢i can be obtained by differentiating eq. (15) forconstant ¢T

din (1 - ¢T)N

L d In (1 - <Pi),~I

i: -dc/>i = 0 (18)i~1 (1 - ¢i)

(19)

since In (l - <PT) is constant. Solving eq. (18) for d<PN we obtain

N-l dcPidc/>N = - (1 - cPN) L (1 _ )

FI cP,

Substituting this into the last term of eq. (17) yields

290 R. J. FARRIS

There are many solutions to eq. (20), but all of these solutions butone are for secondary minima. The primary solution is for theblends that will produce the lowest possible relative viscosity for anyconcentration and that occurs when each term in the series is zero.This occurs when

Figure fi illustrates the relative viscosity of the best multimodalsuspensions using the above solution. It is interesting to note thatthere is a lower limit to the viscosity of a suspension and that thislower limit can be achieved with only one component at low concentrations. The equation being independent of the concentrationof each size, <Pi.

The functional form of this lower limit may therefore be found bytreating each term in eq. (20) as the differential equation

oIn H(<p,) _ (1 - <pn) 0 In H(<Pn) = 0O<p, 1 - <P' O<Pn

and since <p, is independent of <Pn we have

(21)

(1 - <Pi) 0 In H (<Pi)

O<Pi

(1 - <Pn) olnH(¢n)

O¢nconstant = K (22)

and one arrives at the solution

H(<Pi) = (1 - <p,)-K when <Pi ::; 0.25 (23)

and for all values of <Pi the following inequality can be demonstrated

(24)

applying eq. (24) to eq. (6) yields

1)r ~ (1 - <pl)-K(l - <pz)-K ..... = (1 - ¢T)-K (25)

Equation (25) may also be arrived at by recognizing that theviscosity decreases as the number of components increases foroptimized systems. For a fixed value of <PT the lowest viscosity willtherefore occur in the limit as the number of components becomesinfinitely large.

VISCOSITY OF MULTI:\1ODAL SUSPENSIONS 291

LO0.80.4 0.6Volume Fraction 01 Solids, •

Figure 5.

o.z

to MonomodallMeasuredlD Bimodal (Calculatedl -I----H---kRf----t

o Trimodalo Telramodal:It Octamodal• Infinite llllIdat~·Il·'fJ

100

COMPARISON OF CALCULAlED RElATIVEVISCOS IlY FOR BEST MULTIMODAL SYSlEMS

~~:>~ 1D1-----1-----1--J~~+---+---_1'li~

Taking the optimized case cPl = cP2 = cP3 . . . = ¢, and using eqs.(6) and (15) we have

¢ = -(liN) In (1 - cPr) as N -. 00 (26)

and

7Jr (Minimum) Lim [H(<t»]VN~oo

Lim IH[-(lIN) In (1 - <t>r)]}N1'1_ 00

(27)

since cP -. 0 as N -. 00 only first-order terms need be considered andlinear approximation for H(¢) is appropriate [i.e., H(<t» = 1 + K cPas <t> -. 0], substituting this into eq. (27) yields

7Jr (Minimum) = Lim [11'1_ 00

(KIN) In (l - cPr)]N

= e- K In (I-<pr) = (1 - <t>r)-K

and for the general case one may write the inequality

(29)

292 It J. FAHRIS

Choosing K 3, this equation fits exactly the behavior of all thedata in Figure 5 at low filler concentrations, and the data at highfiller concentrations approach this function as N increases. Thesedata point out that, for all practical purposes, this lowest viscositycan be obtained with very few components even for quite highloadings. Also since the only variably in eq. (29) is the constant K,and the viscosity at low concentrations must be included in thesolution, K must be a function of particle shape since the viscosityfor any concentration is a function of particle shape.

Table I gives the blend ratios for the best bimodal, trimodal, andtetramodal systems. These best blend ratios are dependent uponfiller concentration. The values in Table I are mathematical solutions of the equations. At low filler concentrations, however, itwould be difficult if not impossible to observe any difference in theviscosity experimentally while at very high filler concentrations,suspensions can only be made if highly optimized systems are used.Figure 6 illustrates the relative viscosities for bimodal systems versusblend ratio for a number of concentrations. As discussed above,

COMPARISON OF CALCULATED VISCOSITY FOR BIMODALSUSPENSIONS FOR VARIOUS BLEND RATIOS AND CONCENTRATIONS

_,301--+--+---1-----1_,201- --1- - -+-- -+- ---1

.0 .20 .40 .60 .80Coarse Fraction 01 Total Filler

Figure 6.

TABLE IOptimum Multimodal Blend Ratios for Conditions of Zero Interaction <....

tn

Bimodal by volume Trimodal by volume Tetramodal by volume 00in....

Total volume Fine, Coarse, Fine, Medium, Coarse, Very fine Fine, Medium, Coarse, >-3-<% solids % % % % % % % % % 0":j

64 37.0 63.0 22.5 32.0 45.5 16.5 21.5 27.0 35.0 s=66 36.5 63.5 22.0 32.0 46.0 16.0 21.0 27.5 35.5 C

t'"68 36.0 64.0 21.5 32.0 46.5 15.5 20.5 27.5 37.0 >-3....70 35.5 64.5 21.5 31.5 47.0 15.0 20.0 27.5 37.5 s=72 34.5 65.5 21.0 31.0 48.0 14..5 20.0 27.5 38.0 0

t:I74 33.5 66.5 20.0 31.0 49.0 14.0 20.0 27.5 38.5 ;.-76 33.0 67.0 19.0 31.0 50.0 13.5 19.5 27.5 39.5 t"'

78 32.0 68.0 18.5 30.5 51.0 13.0 19.0 27.5 40.5 inC

80 31.0 69.0 17.5 30.5 52.0 12.5 18.5 27.5 41.5 tn"tI

82 30.0 70.0 17.0 30.0 53.0 12.0 18.0 27.5 42.5 trj

84 28.5 71.5 16.0 29.5 54.5 11.5 17.5 27.5 43.5 Zm

86 27.5 72.5 15.0 29.0 56.0 11.0 17.0 27.5 44.5....0

88 14.0 28.5 57.5 9.5 16.5 27.5 46.5 Zin

90 12.5 28.0 59.5 9.0 15.5 27.5 48.0

'"'"..,

2g4 R. J. FARRIS

the behavior at low concentrations is practially independent of theblend ratio while at high concentrations marked decreases in viscosity can be made by using proper blends. The unimodal datafrom which Figure 6 was obtained are in Figure 1. If these unimodal systems contained a greater latitude of particle size distribution, the values calculated at the higher concentrations would belower."

Figures 7 and 8 illustrate iso-relative-viscosity lines for trimodalsystems at 65% and 75% total solids by volume. In Figure 7 thegradient is quite weak and it is apparent that any of the optimumbimodal systems (the minimum occurring along any of the axes) isabout as good as the best trimodal blend. This is not the case,however, in Figure 8 where the total solids is higher and the gradientvery strong. In each case the location of the minimum is predictedwell by the solution of eq, (20).

RELATIVE viscos lTV \IS. TRI MODALBLEND RATIOS. TOTAL SOLIDS65VOL. ~

• Predicted Minimumfrom theory, D1- Dz-...

MedIum

Figure 7


RELATIVE VISCOSITY vs.TRIMODAL BlIND RATIOS.TOTAL SOLI DS 75 VOL "

.. Predicted M1nlmuDlfrom theory, D1• P2• 0,

Medium

Figure 8.

Another point of interest in these figures is that in a trimodalsystem, if the per cent of the total solids that is the finest or coarsestis held fixed while the other two are adjusted, the odds are no changein viscosity will occur since, in Figures 7 and 8, the axes and the isoviscosity lines are apt to be parallel. This is not true when theextreme sizes are adjusted while holding the midsize fixed. Hence,changing the blend ratio and observing no change in viscosity is notan indication that better combinations of those same sizes do notexist.

Points of Special Interest

The theory of optimized systems described above indicates thatbesides having optimized blend ratios at each concentration, theviscosity of some highly concentrated suspensions can be reduced

296 R. J. FARRIS

markedly by adding more filler to the existing suspension therebyactually increasing the total filler concentration, CPT. At first thismay seem impossible, but it is predicted analytically from this theoryand has been verified experimentally. The physical reason for thismay be explained as follows. Consider a concentrated monodispersed suspension of coarse particles. For the purpose of simplicitythis monodispersed suspension of coarse particles shall be a bimodalsuspension with the concentration of fines, CPI, equal to zero (i.e.,CPI = 0).

From the definition of concentration of each size we have

CPI = VI/(Vn + VI)

4>2 = V2/cYo + VI + V 2)

(30)

(31)

(32)

and the total concentration 4>T, may be expressed as

CPT = CPI + CP2 - I/JiCP2 = 4>2 if CPI = 0

and the relative viscosity of this suspension is

17T = H(I/Ji)H(4>2) = H(Q>2) since H(Q>I) = 1 if CPI = 0 (33)

From eqs. (30) and (31) it is evident that adding fine particles tothis concentrated suspension of coarse particles increases Q>I andhence H(CPI) while it decreases CP2 [because VI appears in the denominator of eq. (24)] and therefore decreases H(Q>2). If 17T was originallyquite high it is possible to decrease H(Q>2) much more than H(CPI)was increased, the net result being a decrease of viscosity.

Example. Take a suspension made up of 45 cc of liquid and 55 ccof a coarse spherical particles. Hence 4>2 = 55/(45 + 55) = 0.55and from Figure 1 we determine that H(0.55) which is the same as17T is 50. Now if 12 cc of fine particles are added to this suspensionthe various parameters would change as follows:

4>1' 12/(45 + 12) = 0.21

4>/ 55/(45 + 12 + 55) = 0.493

CPT' = (12 + 55)/(45 + 12 + .55) = 0.600 = CPI' + Q>2' - Q>I'Q>2'

17/ = H(q,/)H(4>2') = H(0.21)H(0.493) = 2 X 14 = 28

and we see that the relative viscosity of this original suspension isdecreased by nearly 50% while the total concentration of solids wasincreased about 10%.


COMPARISON a: IlEl.ATlVE V1SCOSITf WITH m WISE ADDITIONS

llDll:----,..-----,----""JT'-.--,..-I

100---~b§~

110

.1Il.flOlOl""""'-----.l.------.fi:------.l;:-------a!,

Figure 9 illustrates the relative viscosity of a suspension of 55vol-% coarse particles when fine particles are added to the originalsuspension, and when liquid in the original suspension is replacedwith fine particles. The one curve on the far right is for a suspensionof 65 vol-% coarse particles which remains full of voids because thereis not enough liquid to fill the interstices until a sufficient volume offine particles has been added to decrease the concentration of coarseuntil there was enough material (liquid + fines) to fill the intersticesand free up the close-packed monodispersed system.

The data in Figure 10 is presented as additional proof of thissoftening by increasing the concentration of fines in some systems.Instead of viscosity we see the stress-strain data on two filled elastomers, the second differing from the first only by the addition offine particles. The original material was 60 vol-% filler which wasthen raised to 65 vol-% by the addition of fines. Here we see thatthe modulus of elasticity was decreased by adding fines. Thedilation data presented is a measure of the microstructural failurewhich is caused by high internal stresses. These data also indicatethere was an increase in the internal freedom after the fines were

298 n. J. FARRIS

CHANGE OF TENSILE PROI'ERTIES AND DILATATIONBY THE ADDITION OF SMALL PARTICLE SIZE FILLERTO A COMPOSllE CONTAINING ONLY COARSE FILLER

2

,,,,•I

II

I

../Z,,~"

I'I

11809 binder + 234 9 coarse filler: [00 vol. Sl,21809 binder +234g coarse ,-

filler +559 fine filler l165 vol. '" ",

/1,

",,••,.

Slraln -Figure 10.

added. Figure 11 illustrates the viscosity-concentration behaviorof a suspension of fines when coarser particles are added. Accordingto the theory, best processing and least particle attrition will beobtained for multimodal systems when the fillers are pre blended, oradded in stages, fines first, then the next size, etc. The coarsestparticles should always be added last. Poor processing can result(until the fines are added) if the coarser particles are added first,even though the total solids combination may process quite well.

Remarks

It should be pointed out that the first apparent systematic studydealing with the viscosity of solutions and suspensions was done byArrhenius" in 1887. His experimental data indicated there waslittle difference between dilute suspensions of rigid particles anddilute solutions.

In 1906 Einstein published a theoretical analysis dealing with theviscosity of dilute solutions as part of a paper titled "A New Determination of Molecular Dimensions." In this paper he showed howthe size of solute molecules in an undissociated dilute solution couldbe found from the viscosity of the solution and of the pure solvent,and from the rate of diffusion of the solute into the solvent when thevolume of a molecule of the solute is large compared with the volume

VISCOSITY OF MULTI MODAL SUSPEKSIOKS 299

1000

COMPARISON OF RElATIVE VISCOSllY wlm 5lEP WI SE ADDITIONS, : : IfI ,'1: I,',,'1 ," II

I.. "~, JI

" "", : l

/;~, 1, ",

,.1 I

,1/,'"II

,;,:/ ""

~.:""....', ...."

/ .... -- Fines only••••• Adding coarse

»> I•20 .40 •Volume Fraction ofSolids,

Figure 11.

of a molecule of the solvent. He stated such solute molecules wouldbehave approximately, with respect to their mobility in the solvent,and in respect to their influence on the viscosity of the latter, as solidbodies suspended in the solvent which for he chose the spherical form.He then derived the appropriate hydrodynamic equations whichsatisfied such a problem and calculated the energy, W*, (per unitvolume of solution per unit of time) which was transformed into heatduring a flow process.

Ignoring higher-order terms, he concluded that

W* = 252k(1 + </J/2 + .... (34)

where k = viscosity of pure solvent; </J = volume fraction of solutein the solvent.

The velocity gradients in eq. (34) are the velocity gradients thatwould exist if the spherical particles were not present.

300 H. J. FARRIS

He then treated the solution as a homogeneous liquid with viscosity, k*, and expressed W* as being

W* = 2li*2k* (35)

Again ignoring higher-order effects, he proved that for dilutesolutions

smce

(oV x/ox)* = (c)Vx/ox)(l - 1/»

(oVy/oy)* = (oV y/oy)(l - 1/»

(oVx/oz)* = (oV z/oz)(l - 1/»

(36)

is followed that

(38)

The above-mentioned point is of interest since even today there isstill considerable controversy as to the increase of the microscopicstrain or shearing rate due to the presence of filler. Einstein'smathematics on this point provides verification that the line fraction,area fraction, and volume fraction of solids in dilute suspensions areon the average equal.

Substituting eq. (38) into eq. (35), Einstein obtained

W* = 2li2k*(1 - 1/»2.

Equation (39) together with eq. (34) yields

k*/k = (1 + ¢/2 + ... )/(1 - ¢)2

(39)

(40)

Since Einstein was only interested in dilute solutions, he simplifiedthis to

k*/k = 1 + 2.5¢. (41)

His main assumption in solving the hydrodynamic equations wasthat there be no interaction between the suspension particles whichis similar to the assumption made in this text. Einstein's relationmust therefore satisfy the lowest possible viscosity for any concentration as well as being the analytical solution for dilute suspensionssince one analytical solution must be contained in the other. If all


the higher-order effects were accounted for, one would expectEinstein's equation to become

k*/k = (1 - </»-2,5 (42)

This equation known as the Brinkman-Roscoe"" equation, deviateslittle from eq. (40) which Einstein derived in 1911 and satisfies thefunctional form required by this text for the equation of the lowerbound for the viscosity at any concentration.

The experimental data used in Figures 1 and 2 of this paper were taken byJ. S. Chong when he was at Aerojet. His fine report "The Rheology of Suspensions" (Aerojet Technical Memorandum 251 SRP 1964) and his Ph.D. Thesis"Rheology of Concentrated Suspensions" (University of Utah, 1962) providea fine accumulation of facts pertinent to this subject.

References

1. A. Einstein, Ann. Phys., 19, 286 (1906); ibid., 34, 591 (1911); available inEnglish in the book, Investigation of the Brownian Movement (by A. Einstein),Dover, New York, 1956.

2. J. S. Chong, "Rheology of Concentrated Suspensions," Ph.D. Thesis,Univ. of Utah, 1962.

3. H. Eilers, Kolloid-Z., 97, 313 (1941).4. V. Vand, J. Phys. Colloid Chem., 52, 277 (1948).5. M. Monney, J. Colloid s«, 6, 162 (1951).6. V. Fidleris and R. N. Whitmore, Rheol. Acta, I, No. 4--6 (1961).7. H. W. Bree, F. R. Schwarzl, and L. C. Struick, 5th Meeting Mechanical

Behavior Group ICRPG, CPIA Pub. No. 119, 1, 133 (1966).8. J. V. Robinson, Proc. Phys. s«, 4, 338 (1949).9. K. H. Sweeny and R. D. Geckler, J. Appl. Phys., 25, 1135 (1954).

10. P. S. Williams, J. Appl. Chem., 3, 120 (1953).11. S. Arrhemius, Z. Phys. Chern., 1,286 (1877).12. H. C. Brinkmann, J. Chern. Phys., 20, 571 (1952).13. R. Roscoe, Brit. J. Appl. Phys., 3, 267 (1952).

Received May 29, 1967Revised February 12, 1968

prediction of the viscosity of multimodal suspensions from

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