T. I?. FORD
1168
vergence of P’(x,t) since each term of the series representing Pz(z,t)is in absolute value equal to or less than the corresponding terms of the series representing P’(z,t). It is a simple matter now to show that the series representations of dPf2/bt. bP‘/bt and b”P’/bx2 are all uniformly convergent. The series representation of bPz/bt is equal to the sum of the two uniformly convergent series representing HP and -HP2. The terms of the series representing bP‘/ bt are in absolute value less than the terms of the uniformly convergent series representing dPz,l&, hence the series representing bP’/bt is uniformly convergent by the M-test. Finally, the series representing b2P’/bz2is equal to the sum of the two
Yol. 64
uniformly convergent series representing (l/cr)(bP’/dt)and ( l / a ) (V2/Vl)(dPz/bt).We have shown then that the term-by-term differentiation both with reof the series representing P’ and PI2, spect to x and t (twice with respect to x) is justified; furthermore, we have shown that the solutions P’(x,t) and Pz(z,t) satisfy the partial differential equations-equations 1B and 2B. That the solutions satisfy the boundary conditions-equations 3B and 4B-may be verified by inspection. The authors have verified that t,he solutions also satisfy the initial conditions-equations 5B and GB-by substitution; consequently, the validity of the series of residues is established. Furthermore, the solution is unique.
VISCOSIT’T’-CONCE~JTRATIO~ AND FLUIDITY-CONCENTRATION RELATIONSHIPS FOR STJSPENSIOKS OF SPHERICAL PARTICLES I X NEWTOSIAN LIQUIDS BY T. I?. FORD’ Dair!~I’rotlucls Laboratory, Eastern UfilizafionResearch and Development Division, &p-icultural Eeseareh Service, United Slates Department of Agriculture, Washington, U . C. Received March 7, 1.960
It mas established by Bingham? that plots of fluidity against concentration tend to be linear over considerable concentration ranges, whereas viscosity plots are always curved. Many illustrations, for proteins, have been given by Treffers.3 DeBruijn4 and Vand6b have made use of such plots. In general, however, the fluidity function has received little attention, despite the first order reciprocal relationship between the linear equation t+,140 = 1 - 2 . 3 2 , (I) and the Einstein equation as usually xvritten 7 ’70 = 1 2.3Cv (11). It is the purpose of this paper to shorn that the consensus of present theoretical and empirical equations for the flow of suspensions of rigid spherical part iclcs alone rnakes I a closer approximatioil than IT, a r i d by re-csaminatioii of availahlc data to support thic; conclusion t o the estciit it may even tie suggestcd that equation I and its exacat reciproral, q170 = 1 2.5Cv G.%X‘,‘ . . . (111)may describe acvurately the flow of s u s p e n F i o i i s of rigid spheres in Kewtonian liquids ut low aiid moderate cwncentmtions. Collected Viscosity Equations and Their Reciprocal or Fluidity Forms Yarious equations relating viscosity t o concentration for suspeimons of spheres are listed in Table I together with the reciprocal or fluidity forms of each. For ready comparison all equations arc given in Table I as power series, Y = 1 aC, bCv2 cCv3 (I) Naval Research Laboratory, Washington, n C.
+
+
+
+
+ +
+
(2) E. C. Bingliam, “Fluidity and Plasticity,” McCraw-11111 Book Co., Inc., New York, N. Y., 1922. (3) H. P. Treffers, J . A m . Chern. Soc., 62, 1405 (1940). (4) H. DeRruijn, R P Ct. r a ~chim., . 61, 883 (1942); Proceedings of t h e International Congress of Rheology, Amsterdam, 1949, p. 11-95 ( 5 ) V. Vand, (a) TIEIS JOURNAL, 62, 277 (1948); (b) 62, 300 (1948).
+ . . .,6mhere Y designates either relative viscosity,
or relative fluidity, 4/40, and C, is the volume fraction of the dispersed phase. The several pairs of equations are exactly equivalent and are obtained, the one from the other, by long division. The over-all purpose of the table is to show that without regard to theoretical or experimental background all of these equations are morc ncarly linear in the fluidity form. It is seldom noted that Einstein’s derivation7 really led to the equation 1;/?0 = (1 0.5C,)/ (1 - C,)*, from which the series given in Table I are derived. This equation takes no account of interactions betmeen particles. It applies, therefore, over concentration rangcs in wliich interactions can be disregarded; but these ranges may he considerably wider thaii those to ivhirh the simplified form, 11, applies. Interactions betIvWJl particah halve been considered by various authors, ill particulnr Simha and co-worhers (cf. ref. S), D e B r ~ l j n ,and ~ lya11d.5a De13r~ijii,~ with Burgers,9 give a derivation which is unique in utilizing the fluidity fuuctioii and is semi-niclependent of Einstein’s. DeBruijn obtains +/+o = 1 - 2 . X ” 2.57iCv2. He c.valu:ztes the constant IC in an empirical may by making & ‘& = 0 at C, = 0.74, the volume fraction occupied by 7/70,
+
+
( C ) A catalog of \ iscosity equations converted t o poner series I S also given by Bredbe and de Rooys. Their list includes several earlier forms omitted here. H. L. Bredee and J. deBooys, KoZZozd Z., 79, 31 (1037). ( 7 ) A. Einstein, (a) Ann. P h y s t k , 19, 286 (1908) b) 3 4 , 591 (1911). (8) R. Simha, (a) THISJOURNAL. 4 4 , 25 (1940); (b) J. Applied Physics, 2 3 , 1020 (1952). (9) ,I. M . Burgers, “First and Second Report on T‘isrosity and
Plasticity,” N. V. Noord-Hollandsche Uetgeversmastschappij, Amsterdam, Holland, 1938.
TABLEI EXPRESSIONS 1Lma.rrsc: I ~ E L A TVISCOSITY ~ V E ASD RELATIVE TiLciDrrY TO VOLUMECOKCEWRATIOS ~01% SIX’ RIGIDSPHERICAL PARTICLES Author
Year
Ref.
Einstein DeBruijn Vand Simhn” Iiynch“
19U6-11 104‘2 1948 1952 19.56
(7)
Theoretical equations Viscosity form
q/qo =
(4)
(ja) (8b) (10)
1 1 1 1 1
+ 2.5Cv + 4Cv2 + 5 . 5 C V 3. . . + 2.5c,. + 4.7c,2 + 7.77cv3 . . . + 2 . K ” + 7.340C,2 + 2.5CVf 3.73Cv2+ 6.!+4C;l3.. + 2.5C“ + ( 7 . 5 t o 6.75)C”Z . .
Fluidity form
+ 2.25Cv2- 1 . 2 5 C Y 3.. - 2 . 5 4 , + 1.552c,,2 . . . 1 - 2 . X ” - 1.O!fOC,.’ . . . 1 - 2 , SC, + 2 , ~ l C , z- t i , 9~iC,.”~.. 1 - 2 . 5 C , - ( 1 . 2 5 to0.30)C,2 . . .
+/&
1 - 2.5Cv
1
,
,--
+ +
+ + +
+ + +
Empirical equations
>
.Irrheiiius lY8T i l l ) 1 2.5C, 3 . 125Cv2 2 . 6OCv3.. 1 -2.5Cv+3.123Cv2-22.606,3... Vancl 1948 (3,) 1 f 2 . 5 C V 7 . liC,,? 16.2CY3. . . 1 - 2 . X “ - 0.92c,,2+ I.OC”3 1 - 2.5C, 1 .65CY2 1. 0 c v 3. . . Robinson 1949 (16) 1 2 . .5C, 4 . 6CV2 8 .4CV3 a Both Simtm’s and Kynch’s second and higher order coefficients change with concentmtioii. The equations given are for dilute solutions. See text.
close-packed uniform spheres. This makes k = 0.6209 and gives the equat,ions attributed to DeBruijn in tho table. It will be seen in the discussion of experimental results, liowever, that for rigid particles it is more probable that 4/40 = 0 at C, = 0.5236, the volume fraction in cubical packing. Imposing this condition makes k = 0.222 and the two equations become v/v0 = 1 2.5CV 5.123CV2 9.99Cv3 :t~id6/40 = 1 - 2.5Cv 1.127Cv2, respectively. DeBruijn’s and Vaiid’s equations both represent attempts t o fit trhe complete concentration range. Recently SimhaSband more recently Il;gnchlo have deduced that the constants of t,he higher power terms must change with concentration. The equations attributed t o Simha in the table are espansions of one of his theoretical equations (7a) applicable a t 1 1 conceiitrations, ~ ~ up to C, = 0.065. This equation contains a parameter defined by t’he relationship f 3 = 8Cma,.where C,,,. may be either that corresprinding to cubical packingLC, = 0.5236 (.f = 1.61), ortoclosc packing, C, = 0 . l - l (.f = 1.81). In evaluatiiig the coiistaiits for Table I cubical pecliing has been assunred. In his discussion Simhn notes that to fit csperiniental data over the whole concent~rationrange, .f niust actually change from a value around 1.2 t,o less than 2 . His final evaluation thus bec!mios scmi-einpirical. Kyiich’s derivative differs from Simha’s hiit’ thc net result is ahout, tlic sanic. Ia,titx(3r(1 iii thc f-liiiclity form. Alccw(liiig to Siinhh’s ~ i . t ~ ~ t n i tc ihi tscor~iclortler c-oiistniit 6.3.5 is possible, rc~liiiriiigoiily t,li:it f 1)c giveii ilic value 32/23 or :.~GT. IiyncIi’s o~)~crvatioit is of interest in t h light of statistical aiialyscs to he prcsented in thc iiext sed’ioiiof this paper. The empiricul cquat,ioii due to ~4\.rrht:nius1’ is usually written in 7,‘vu = kC,. For iiiclusioii in the table it. has heen rewritten as ~ , ’ 1 7 ~= ekCv.,exp?>aded as the series 7 , ’ = ~ ~1 kC, iCv2//‘>! kCV3/3! . . ,, aiid the constaiit IC arbitrarily assigned the value 2.5. The secorid arid third order constants iii the empirical eciuaticm assigned to Robinqon iii Table I are averages Irm-ed oii three extrapolated cxperi-
+
+
+
+
+
+ +
+
(11)) G. J. Kynch, f r o c . R o y . SOC.( L o n d o n ) , A237, DO (195b) (11) S.hirlienius, Z. p h y s t k . Chem , 1, 285 (1887).
+
-
niental values of sediment volumes for glass spheres, 1.V’ 1.81 aiid 1.88, given by him. The reciprocals of these numbers, 0.565, 0.553 aiid 0.532, compare well with the volume fraction of uniform spheres in cubical packing, (4/3 7 3))/(8ra)or 0.5236. Review of Experimental Results J-iscosity measurements which are sufficiently extensive and rigorous for mathematical analysis and on systems acceptable as models of suspensions of spherical particles are scanty. No data yet reported are adequate in 311 respects. Thir arises i ] ~ part from uncertainties as t o concentration and shape in molecular and colloidal solutions, arid in part from uncertainties in correcting results on visible particles for effects due t o sive alone. The experiments here considered were selected for \~arious reasons, some merely because prominence previously given them demands their inclusion. In w e r y case the original data were carefully scrutinized and recalculated. Insofar as justified. statistical methods iverc applied sinw the purpose was refinement of piecision in establishing thr constants in the type equation Y = 1 a(’, bCv3 ccy3 . . . . Bancelin’s Experiments on Gamboge Sols.-I3:mcelin’s cxperimc~nts1~ arc the first iii point of t imc and apparciitly ivcre performed \-c~-ycarefully. Ile prcp:trcci nionodispcrie stispc~iisioiiroi gnmbogr 11v wntrifuqntion (thv nicthotl of .JC:LII I’crrin), niitl for cwh f rwtion tlc~tcvniiletlviscositiw at vwiouq voiicwltrnt ioiis. 13aiic~li1i’s ol)jccat was to tclst the Ein\tciii cqn::tion which as tlic.11 published was 7 ’70 = 1 He o1)taiiied as hih a:-(~agcrcwlt 7/17” = 1 2.0(‘,. 1Te thcrcforc itrote to Einstcitt who infornmrd hiin that thc cqu;~tioiishould ha\-e been 11 ’vn = 1 2.5Cb. Banccliii’s papcr gives values for n sispciision containing particles 0.30 p iii diameter, at six eonceutrations k~ttveeii 0.0024 aiid 0.0211 nil. per mi. Least scluarcs analysis of these data, using the type equation Y = 1: aC,, gives
+
+
+
+ +
+
(IL..
+
+
7/10 = 0.9994
+ 2.715Cv
(S.E.E., 0.00069)
+/do = 1.0000
- 2.549C,
(S.E.E., 0.00052)
and Here, the numbers in parentheses are the standard errors of estimate of Y , a convenient measure of the (12) R f . Bancelin, Compt r e n d , 152, 1383 (1911)
T. F. Fonn
1170
C,
Vol. 64
(mI./mI.).
Fig. I.-Fluidity-concentration plots for suspensions of 125-205p diameter glnss beads, data by Vand.6b Points for stirred suspensions (Couette) indioated by opon triangles, for nonatirred suspensions by open squares. The method of extrapolation is described in the text.
fit.Is These results show that t.o a close approximation, for these data k = 1, a necessary check, and that the experimental first order constaut is much closer to 2.5 than t o 2.9 as reported by Bancelin and repeatedly quoted since. The data cover a low concentration range. Nevertheless, the measures of deviation given ahove indicate that over this concentration range the fluidity function is the more linear. It is possible that for Bancelin's gamboge the experimental frst order constant was actually about 2.55, considering that errors in effect,ivevolume concentration, due to slight departure from the spherical shape, for example, would be positive. If the concentrations be corrected accordingly by the factor 2.55/2.5, the data are now fit by '/*o = + 2.5cv + 6.75cv' o.m53) and
and G o l d s ~ h m i d t , ~ ~ V aRobinson,1E nd,~~ and Eveson and Whitmore (ref. 10). The experiments by Eirich, et al., which were on yeast cells and mushroom spores as well as glass beads, are generally credited with establishing the first order constant within limits of 10.3 by use of capillary viscosimeters, and within limits of +0.2 by use of the Couette viscosimeter. The glass spheres used by Eirich, et al., were from 0.0125 to 0.0205 cm. in diameter. Subsequent to their experiments it has been pointed out by Vand (tide infra) that use of particles of this size requires appreciable corrections of both C , and +/& - 1, by factors which are related to the dimensions of the viscosimeters. Such corrections cannot be applied with assurance to Eirich's capillary data, but they can be applied t o the Couette data, and the Couette data are for a low concentration range, C , = 0.01 to 0.08,not adequately covered by other workers with glass beads. A large scale plot of these data discloses 4/90 = 1 - 2.5'2, - O.lCvl (S.E.E., O.OM)52) that they do not extrapolate to +/& = 1 at C = 0; On Suspensions of spheres.- and least squares analysis indicates that the C, Suspensions of glass sphereshavebeenused asmodel values are toolow by the amount AC" = systems by Eirich, Bund and Margaretha,14 Eirich (13) The standard emor of edtimete is the aquare r o d of the mean of the squared deviations. S, = Other meajlures of the fit, mean deviations. ataodard deviationd. and coefficients of correlation, agreed with the atsndard emom of eatirnate.
d/Zd'/N.
(14) F. Eirioh, M. Bund and H. Maresretha. Kolloid. Z., 74, 270 (1036), 115) F. Eirieh and 0. Goldwhmidt. ibid.. 81,T (1037). (16) J. V. Rob;naon, (a) T ~ i Jouarm~. s Sa, 1042 (1940); (b) SS, 455 (1051).
Sept., 1960
VISCOSITY-CONCENTRATION-FLUIDITY RELATIOXSHIPS FOR SPHERES
0.0010, approximately. The C, values adjusted by this amount and the observed 4/60- 1 values were corrected by the use of formulas later developed by Tand.5b Thus corrected the data are fit by the equation 4/40 = 1 - 2.50CV 0.95Cv2, S.E.E., 0.0015. The data as given are fit by the equation 6/40= 1 - 2.39C" 3.29Cv2 S.E.E., 0.0018. These experiments by Eirich, et al., provide a second close check of the first order constant, 2.5. \*andsb u\ed the same glass beads as Eirich, et al., loaned t o him by Prof. Eirich, but he suspended them in a more viscoui liquid, made other refinemeiits in tee hiiique, and extended the concentration range almost to saturation. Yand gives an average diameter of 0.013 cm for these beads. Like Eirich, ct al., he used two Ostwald viscosimeters, of different tiore diameters, and a Couette viscosimeter, the Couette machine only at concentrations of 35% and highcr. TTaiid multiplied all of this C, and - 1 values by specific correction factors for each instrument, and hiially combined his corrected data as a composite plot. His final equation is designed to fit a smooth curl-e through these points. Inclusioii of the Couette results has been criticized by I. The data themselves, howver, provide sufficient reason for their reexaminatim. With both the Couette and Ostwald instruments, at concentrations begiiining immediately abol-e C, = 0.35, different I were obtaiiled depending on whether the suspeiisions were stirred or iiot. These kiiown deviations are greater than the total deviations at lower coiicentrations. This fact is itself sufficieiit to exclude most of the Couette data, and some of the 0stn.ald data, or at Ie:i5t t o diminish the weight to be given these data. It 15 uotcd also that Taiid'S corrected values for hir two Ostwald viscosimeters fall more ncarly on sinooth rurves if plotted separately than if combiiied. This was confirmed by comparison of the fit of 3eroiid order equations obtained by least squares analysis. The deviation of the two curves indicates L: more empirical analysis than used by \-and. Aimethod of blind extrapolatioii in the direction of aero rorrectioiit 12 illustrated hy Fig. 1. Here smooth ciii~ebthrough the original uncvrrected Ostwald point- are drawn in the two receding verti( ~ tp1:ziiey l aiid thc corrertcd CouPtte values and the cxtrapo1:~tcd O*twald curve are hho~viiin the plane of the p t t p c ~ ~ The positions of the receding verticd plaiiei : ~ r vdetermined as iollows: at C, = 0 by requirinjq thiht the first order constants, 2.145 sild 2 405, fouiiJ 111 the uiual way by plottiiig (+,'+" 1) C, agaiitit (', . .hall extrapolate to 2 5 for ail illfinite capil1,iry; :tiid at ( ' y = 0.35 by placiiig the experinieiitd O*tvtlld $ $0 - 1 value... 0 760 aiid 0.680, and the corrected Couette value, (3.830, on a straight line. The two Ostmald curves thus placed iii ..pace ailti extrapolated into the plane of the paper give a third c ~ r \ - efor Jvhich the corrections are certainly redtized. T h e assiimptions involved are: (1) That the fir5t order (wiist:iiit i q ill facbt 2.5. Thi5 is supported euperinientally hy Himceliii's aitd Eirich's results. ( 2 ) That the ('ouettc fluidity used at C, = 0.35 is
+
+
1171
approximately correct. This seems justified within limits by the apparent absence of stirring errors at this concentration, and by the fact the Couette corrections are in any case small (roughly onefifth to one-tenth of the Ostwald corrections). (3) That the effective or total corrections vary iiot only with bore diameter, but also with concentration. This is supported by the observations that Vand's ~ $ i = +~ 1 correctioiis, for wall effects, are in fact averages of yalues which apparently vary, approximately linearly, by as much as 12y0over the range C, = 0.05 to 0.35. The extrapolations actually were performed analytically. Interpolated fluidities for the two Ostwald viscosimeters at concentrations increasing by 5% were taken from large scale plots, and also calculated from the second order equations fit to the original data by least squares. Using the analytically interpolated Ostwald values and referring the extrapolation at C, = 0.35 to the corrected Couette values (Fig. 11, the extrapolated curve is fit over the range C, = 0.05 to 0.35 by the equation +/+o
= 1- 2
5c,
- 0.11C"Z + 16C,3
Iieferriiig the extrapolatioiis t o the uucorrected Couette va.lues the equation is
+
+/40 = 1 - 2 . x , - O.OSC,~
i.w,3
The graphically interpolated values give essentially the same con3taiitq. The uncertainty in the second order constant is 0.10, approximately. Although the type equation I.' = 1 f 2.5C,. bC,' cCv3is used heie, the extrapolated curve3 are fit equally well. over most or a11 of the range C, = 0.05 to 0.35, by equations of the forin 4 c $ ~ = 1 - %.XI dC, While the above described extrapolations, based on the uncorrected data, wein to avoid many assumptions, they also include a certain error. This is because in the polynomial equations used the coiicentration and all multipliers of it are involved as polvers higher than the first. It is noted, however, that the C, corrections (multipliers) as calculated by T'and are less than half his 4, 4po- 1 corrections for wall effects; and it is noted also that it is only in the latter and not in the concentration corrertiom that any reason for variation with concentration is apparent. Accordingly, I-and'h corrected C, values were combined with his uncorrected 0, 00- 1 values, and the various maiiipulations repeated. Referring the extrnpolatioii\ to the rorrected C, and corrected e 0" - 1 Couette values at C, = 0.35, fur the range C, - 0.05 t o 0.35 the equation obtaiiied ib
*
+
+
+
4/+J = 1 - ?Si,+ O.OO(',~
+ I 4(',3
ljeferriiig the extrapolatiou.: to the corrected C,. aiid uucorrected 4,'4" - 1 Couette 1-alues the equation is +/+o
=
1 - 2.5c,
+ O.llC,* + 1 5 ( ' > 3
The uncertainty in the second order constant is 0.10, approximately. The foregoing calculations all indicate that within the apparent accuracy of the data, in ai1 equation of the form = 1 - 2.37, bC,r2 cC,3 . . ., the second order constant "b" is numerically small, and may be zero.
+
+
+
1172
T. F. FORD
1101. 64
other three by addition of
Sept., 1960
T~ISCOSITY-CONCENTRATION-FLUIDITY RELATIONSHIPS FOR SPHERES
0.2
0.4
0.6
1173
0.8
Fig. 4.-Fluidit,y-concentration plots for rubber latex (main plot) and asphalt emulsions (inserted box). Latex data are by Rhodes and Smith23; asphalt data by Eilew24 Independent sets of data are indicated by different types of points.
expected. Cream can be concentrated to 72 to 74% by centrifuging at room temperature, and an actual intercept in this neighborhood seems possible. Fluidity plots for two natural rubber latexes and for two asphalt emulsions, derived from data by Rhodes and Smith,2zand Eilerslz3respectively, are shown in Fig. 4. Both of these sets of data have been used in connection with derivations of viscosity equations. In Fig. 4 the latex data are plotted as given by the authors, since a Hoeppler viscosimeter was used and Vand’s corrections therefore do not apply. The first order constant cal. culated for the latex curve is about 2.8, and the final intercept is in the neighborhood C, = 0.74. Both results are reasonable. Latex particles are often not spherical, may be pear shaped, and may even have tails : therefore, the first order constant could well include a shape factor. Latex particles also are plastic and deformable. These experiments are, therefore, of little value as a check of the Einstein equation. The asphalt data, inserted box of Fig. 4, were first corrected by use of Trand’s formulas. The corrections are necessarily approximate. Eilers’ particle sizes range from less than 1.G p to 9.7 p in diameter: a diameter ( I f 5.6 p haqed on the average particle volume was used. Eilers gives three capillary diameters apd apparently reports averagc viscosities obtained with the three instruments : therefore, averagc C, and qj,’qjo - 1 corrections mere appplied. Thus correct,ed a first order constant of 2.3, approximately, is indicated. This is impossible since thc i k o s i t y of the arphalt used is so high that Taylor’s correction for sympathetic flow, as applied by Levitwi and Lelghton, is insignificant. Remember(22) C Rhodecr and I€. r Smith. J iZubher Research Ins1 M a l a y a , 9, 171 (1939). (23) IT. Eilers, KoZZozd Z , 97, 313 (1941).
ing the errors involved, for the plot shown the 4/40 -1 values have been again multiplied by such a factor as to make a = 2.5; this makes C,,,, = 0.56, approximately. KO really accurate interpretation of these data can be attempted because of the scarcity of poi!& at low concentrations, which makes it impossible definitely to establish a first order constant. Discussion Limiting Equations, Viscosity and Fluidity.It appears that for dilute suspensions meeting the requirements of Einstein’s original derivation, the equation +/+o = 1 - 2.5Cv (1) is a closer approximation than 7/q0 = 1 2.5CV (11). Within the limits of accuracy of available measurements, at low and even moderate concentrations, low order terms in C, higher than the first tend to vanish when the fluidity function is used. Thus, the limiting fluidity equation describing the flow of suspensions of rigid spheres appears to be I, as given above. The limiting viscosity equation appears to be not I1 but the reciprocal of I, 7/70 = 1 f 2.5Cv 6.25Cv2 . . . (111). This last compares with Einstein’s original complete equation, 7/70 = 1 2.5CV 4.0Cv2
+
+
. . . (IV).
+
+
+
+
Complete Equations.-The distinction between limiting equations expanded to include hydrodynamic effects not considered by Einstein, and equations expanded merely to cover wide concentration ranges is not always clear. In some derivations the added effects have been imposed on 11, not on IV. Such derivations nullify the basic hydrodynamics considered by Einstein, and are inadequate to the extent I1 is inadequate. Empirical equations of the form 4/& = 1 2.5C, $- mCVp - nCvP+q (VI), where p is at least 5, can be made to fit experimental fluidity plots over
T. F. FORD
1174
the entire range of fluidities from 1 to zero. For Vand's extrapolated and Couette curve (Fig. 1) the equation 4 4,, = 1 - 2.5C, 111?,~- 11.5C,' is quite saticfactory. A4pparentlythe first two terms suffirc for flow subject to classical hydrodynamic analysis. The third and fourth terms can be interpreted as due t o appearance of effects resulting from the near approach of particles. The contribution of the third term becomes significant a t C, = 0.25, approximately, at which concentration the least distance between spheres in cubical array is 0.218 times their diameter. The contribution of the fourth term becomes significant at C, = 0.45, approximately, at which the separation distance is only 0.05 times the particle diameter. These transition coilcentrations may mark in turn the onset of inhibition of rotation and of interlocking. Equations such as J-I, in which the last term is negative and increases rapidly in absolute value in a certain micentration range, can be adjusted to = 0 at and above any specific concentramake tion. t g., C, = 0 .5236; palynomials in which the last term or terms are positive give fictitious positive fluidities at high concentrations; logarithmic sories give fluidities which approach zero asymptotically. In neither of the last two cases is the flow behavior indicated in accord with the facts. Correction Factors.-The various corrections evaluated and applied directly by T'and for his glass beads, and implicitly involved in the extrapolations of his data deqcribed here, are in general numerically unimportant with true colloidal systems, e.g.. gamboge, latex, sulfur sols, mushroom spores, fat suspensions, and protein solutions. If for visible particles, such as glass beads, the corrections are to a certain degree inadequate, this certainly may he due in part to the several assumptioiib and approximations necessary in their evaluation. It also may be due to omission of still other types of corrections. Correction of C, for real or presumed shells of bound water, for example, would be comparable with the aggregate of the other C, corrections used by T-and. Applications.-The tangent to +,'+o us. C, plots C, = 0 usually can be determined graphically with considerable accuracy. The slope of this line, the reciprocal of its intercept at 9/40 = 0, is the first order roiictant in the equation rp.l$~ = 1 - aC, . . . This constant is identical with the intrinsic ciscositii. Its evaluation by using fluidity rather than viscosity data is the more accurate to the extent the fluidity-coiiceiitratioii relationship is the more linear. Since it seems established that for rigid spherical
+
C#J
+(,
+
T-01. 64
particles the init,ial slope intercept, should fall at, C, = 0.4, as required t o make a = 2.5, and since it appears that the act'ual final intercept should be in the near neighborhood of C, = 0.5263, t,he volume fraction of spheres in cubical packing, it follows that if for a given set of data a suit.able multiplier of C,, the dry volume fraction, mill give a plot, with these intercept>s,then the particles must be spheres and the multiplier equal to 17, the voluminosity. Voluminosity, a term proposed by BredeelZ4is commonly defined as the rat,io of the effective or hydrodynamic volume of a dispersed part,icle to the dry volume of contained colloid. As such it may include shape factors as well as correct,ions of the volume for hydration, solvation, and for electroviscous effects (cf. ref.25). For spheres the shape factor is 2.5,/2.5or I, making T' in this c ~ ~ as rvolume correction only. DISCUSSION R. H . OTTERTILL (Cambridge University).-I think it i2 important to stress that one has to be careful about applylng such viscosity-concentration relationships to systems which contain charged particles in solutions of lov electrolyte concentration. For example, in some recent work of Professor Overbeek and myself on very concentrated sols of silyer iodide, containing near spherical particles of radius 25OA., the coefficients of C,2 and C,3 were found to vary considerably with rate of shear and the electrolyte concentration of the suspending medium. T. F. FoRD.-Reference to the electroviscous effect is madr in the paper. It is agreed that in many systems large apparent changes in V can be brought about' by changes in the ionic environment. Such changes are insignificant for the systems here discussed. J. G. BRODNYAN (Rohm and Hans Co.)-Is there any reason why you ignored Mooney's eqtiation which was derived using a functional method? T . F. FoRD.--hIooney's equation ( J . COLE. sei., 6 , 162 (1951)) is only one of many which \\-ere omitted. It could be included in Table I between \'and's and Simha's equations. The expanded forms are n/no = 1 2.5& (3.125 2.5k)CV ... and 4/c#q = 1 - 2.5Cy (3.125 - 2.5k)Cv' .. Mooney sets the value of k , for monodisperse systems, only roughly between the limits 1.35 and 1.91, the reciprocals of the volume fractions of uniform spheres in close packing and cubical packing, respectively. The actual value to be used is left for experimental determination. To fit Vand's data he makes k = 1.43, and to fit Eiler's data on (polydisperse) asphalt he makes k = 0.75. In any case it will be seen that Mooney's equation also is more nearly linear in the fluidity form. It was omitted in the interest of brevity and because it,would contribute little to the basic conclusion.
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(24) H. L. Bredee a n d L. -1.Van Bergen, Chrm. IPrekhlad, SO, 229 (1933). ( 2 5 ) J. T. G. Overbeek a n d H. G. Bungenberg de Jong, in H. G. Kruyt, "Colloid Srience," Elserier Publishing Co., Inc.. N e w Pork, N. T., 1949, Vol. 11, p. 209.
