Electroviscous effects in dispersions of monodisperse polystyrene latices

Electroviscous effects in dispersions of monodisperse polystyrene latices

Electroviscous Effects in Dispersions of Monodisperse Polystyrene Latices J. S T O N E - M A S U I ~ AND A. W A T I L L O N Faculty of Sciences, Free...

2MB Sizes 0 Downloads 23 Views

Electroviscous Effects in Dispersions of Monodisperse Polystyrene Latices J. S T O N E - M A S U I ~ AND A. W A T I L L O N

Faculty of Sciences, Free University of Brussels, Belgium Received March 14, 1968 The contribution of electroviscous effects to the viscosity of monodisperse polystyrene latices has been studied. The sols were prepared by emulsion polymerization using as emulsifier a weak acid soap and a strong acid soap, respectively. Particle sizes were determined by electron microscopy and turbidimetry. Excellent agreemerit between the two methods was obtained. Zeta potentials and charges of the electrokinetie unit were deduced from eleetrophoretie mobility measurements at different ionic strengths. The validity of the Einstein equation was demonstrated for perfectly stable latices when the primary electroviseous effect can be considered as negligible. Influences of f potential, particle charge, ionic strength, particle size, and rate of shear on the first- and second-order electroviscous effects have been investigated. For the primary eleetroviseous effect, a good agreement with the Booth theory was obtained. For the second-order effect, the results indicated that whereas Street's equation yielded too low viscosities, values predicted by the recent treatment of Chan, Blachford, and Goring were too high. However, this last theory represents a more realistic approach than the Street one. INTRODUCTION The first theoretical approach to the viseosity of suspensions of rigid, uncharged spheres was derived b y Einstein (1). He proposed the relation (nrel 2

1)~.0 = KE~nsto~n,

in which the constant K E i n s t e i n = 2 . 5 ; = volume fraction; and nrd = relative viscosity of the suspension. Another value for K was derived b y Happel (2) (K~ppe~ = 5.5), who considered possible long-range interparticle interactions. F o r more concentrated suspensions various relations have been proposed (3-5). Furthermore, for suspensions of undeformable charged particles, the viscosities exceed those of suspensions of uncharged ones. This is due to eleetroviseous effects. A first eleetroviseous effect existing even in

extremely dilute suspensions results from the distortion, under a gradient of shear, of the ionic atmosphere surrounding the partiele. In more concentrated suspensions a second eleetroviseous effect, due to interactions between double layers, is superimposed on the first one. An extensive review of the subject has been given b y Conway and Dobry-Duelaux (6). Theoretical treatments of the first electroviscous effect have been proposed b y Smoluchowski (7), Finkelstein and Chursin (8), and Booth (9). T h e y extended the Einstein equation in order to allow for electrical effects. Smoluehowski's equation, valid for Ka > 10, can be written:

Chargd de reeherehes F.N.R.S. Journal of Colloid and Interface Science, Vol. 28; No. 2, October 1968



! ( °Y1 187




a = particle radius; e = dielectric constant of the dispersion medium; X = conductivity of the dispersion medium; 1/K = double layer thickness in the Debye-Hilckel theory; and ~" = eleetrokinetic potential at the surface of shear. The Finkelstein and Chursin equation can be expressed as: KF.c. = 2.5 1 ÷ 3.008 ~rekT(Ka)~a2~o with f:~ = mean friction coefficient of the ions and Ze = charge of the eleetrokinetic unit. Booth derived the expression:





where q* is a function which depends on the ionic mobility of the ions and [email protected]) = the relaxation function given by Booth (9). By introduction of the Debye-Hfickel relation Ze = ea~(1 + Ka)valid only if ~ < 25 millivolts Eq. [1] becomes: KBoot

= 2.S

1 + q -7


• (1 + ~a)~iz(,~a)}J. If all ionic mobilities are made equal, Eq. [2] then becomes: KBooth = 2.5

2 term. This implies the nonexistence of a first-order electroviscous effect. Another approach to the theory of the second electroviscous effect has been published by Chan, Blachford, and Goring (11a), which is valid for medium rate of shear and ionic strength. Many experimental studies relative to the electroviscous effects ( 12-19 ) have been performed. Nevertheless, in most of these works, the experimental conditions required by the various theories are not completely fulfilled. Frequently, the shape and size of the particles are not well defined or vary with the composition of the dispersion medium. It then becomes very difficult to evaluate the contributions to the viscosity by the geometrical and electrical factors. A quantitative verification of the theoretical relations is possible only with highly monodisperse sols containing stable, rigid, spherical, and lyophobic particles. Moreover an accurate experimental technique is required. For our study, we chose the synthetic latex dispersions stabilized by an emulsifier. On these systems the influence of shape, size, heterodispersity, and state of aggregation on the viscosity has been investigated (2023). Furthermore, the importance of the ionic strength on the viscosity of concentrated latices (~ > 10 %) has been demonstrated (24-26). Finally, an interesting attempt concerning the first electroviscous effect has recently been published (19). EXPERIMENTAL




•~r(Ka)2(1 ÷ Ka)2{[email protected])}J. Street 10) proposed for the viscosity of charged suspensions the equation: t 2

Vr~l = 1 ÷ 2.5~ + K ~ , K'




2a2~0X \ ~ /

(1 +

According to this treatment, the electrostatic contribution would influence only the




Latices were prepared by emulsion polymerization under a N2 atmosphere using Na persulfate as initiator. The emulsifiers were, respectively, Na stearate and Na dodecyl sulfate. Successive seedings allowed us to obtain suspensions with particle diameters ranging from 50 m~ to 1000 m~. Particle sizes were determined by electron microscopy and by a turbidimetric setup taking into account Heller and Tabibian's (27)

Journal of Colloid and Interface Science, Vol. 28, No. 2, October 1968




@ O~

@ '@ FIG. 1. E l e c t r o n micrography of a polystyrmm latex sol. Modal, number, weight, Z-average diameters are 88 mr*, 88 mu, 90 m~,, and 92 m~, respectively. S t a n d a r d deviation ~ = 3%. Particle diameter from t u r b i d i m e t r y is 83 m~.

recommendations in order to avoid errors occurring especially during the determination of large diameters. As internal standard for electron microscopy we used the very monodisperse Dow PS latices (28) 2. These results were in closer agreement with turbidimetrie data when we used for the Dow PSL standards, particle diameters obtained from optical methods by DeSeli4 and Kratohvil (29). For example, one sample had a diameter of 83 m~, as determined from turbidimetry, and modal, number-, weight-, and Z-average diameters of 88 m~, 88 m~, 90 m~,, and 92 m~, respectively, as obtained from electron microscopy (Fig. 1). For each sol we determined the frequency distribution curve and the corresponding standard deviation. For all the systems investigated the standard deviation ranged between 0.03 and 0.6. Owing to their high monodispersity, in this work we always determined the diameters by means of turbidity and the standard deviations by electron microscopy. The results obtained will be given further in the paper when necessary. Our thanks are due to Dr. J. W. Vanderhoff, who sent us the PSL samples.

The deetroviseous effects are greatly influenced by the ionic strength. It is therefore important to purify the suspensions. They were successively passed through an Iena glass filter and a mixed-bed ion-exchange resin (Amberlite MB 1). Columns of these resins allowed us to obtain only small volume fractions of purified effluent. Moreover, the volume fraction threshold still decreased with the particle size. As an example, for 50 m~ particles, the available purified sol did not exceed a value of ~ of about 0.05. Beyond this value, owing to the second electroviscous effect, the viscosity of the flowing suspension inside the resin became sufficiently large to block the column. The purification was controlled by surface tension measurements using a drop volume method, the accuracy of which amounted to 4-0.03 dyne. Calibration curves in good agreement with previous data on sodium lauryl sulfate (30) showed that the limit of soap detection was about 10-5 mole/1. From measurements on all our purified systems, we concluded that the soap content was equal to or probably smaller than this value however large the volume fraction of the sol. The surface tension of the pure sols was measured for a period of two weeks without change. Therefore, no appreciable desorption of the emulsifier trapped at the particle surface was observed. The purified latices were stable even at ~o = 0.04 owing to the ionization of the surface emulsifier and to the widely expanded double layers. For a few samples prolonged dialysis against bidistilled water has been performed without change in any property of the systems. Furthermore, electrophoretic measurements have shown that the particle charge is not at all negligible. These facts seem to disagree with several remarks made by Dobry (13b), Sieglaff and ~.iazur (31), and Chan and Goring (32), who assumed that at low ionic strength soap desorption reduces the

Journal of Colloid and Interface Science, Vol. 28, No. 2, October I968



surface charge and renders the particles unstable. The preparation of electrolyte solutions (HC104, NaOH, NaC104) and all the manipulations involving purification of suspensions, electromotive forces, eleetrophoretie and viscosity measurements were performed in a CO2-free atmosphere. Volume fractions were determined from dry weights, using a density of 1.057 (33) and correcting for the electrolyte present, when necessary. I I I . DILUTION OF THE LATICES

The first-order electroviscous effect can be measured experimentally by extrapolation of the relation n~,/~ against ~ to the volume fraction ~ = 0. For each volume fraction investigated, the charge of the electrokinetic unit and the thickness of the double layer must be kept constant. For a given sample, these conditions will be fulfilled if dilution is performed at constant pH and ionic strength. For all our suspensions, the ionic strength is determined by the salt concentration in the interparticle medium outside the ionic atmospheres. The ionic strength defined in this manner is generally used for hydrophobic systems (14, 34b) and for micellar solutions of ionic surfactants (17, 35). However, when the electrolyte concentration is extremely small and the volume fraction relatively large, an overlapping of the ionic atmosphere can occur. The situation is then complicated by the fact that the ionic strength will increase with the volume fraction. Another definition of the ionic strength, widely used for polyelectrolyte solutions, considers that the macroion bearing Z charges is assimilated, in its contribution to the ionic strength, as Z small monovalent ions.

According to Tanford (36) this procedure is more reasonable when the charges are capable of independent motion; this is not the case for our dispersions. This "isoionic dilution" method (36, 37) is nevertheless

adopted for polystyrene latices by Chan and Goring (19). IV. TECHNIQUES OF MEASUREMENTS

1. Surface Tension Surface tension measurements were performed using a drop volume apparatus based on that of Gaddum (38) and an Agla micrometer syringe whose barrel was immersed in a water bath maintained at 25 ° 40.01°C. The drops were collected in a glass vessel under a saturated vapor atmosphere. The Harkins and Brown (39) instructions were used for the grinding of the glass tip (diameter d = 4.873 ram), the drop formation, and the computation of the results with the use of their correction factors.

2. Electromotive Force The electromotive forces were obtained by means of a potentiometrie technique iacluding a Vibron electrometer (Electronic Instruments). Platinized platinum wires were used as H + reversible electrodes. The "suspension effect" (40) was eliminated by using as reference a calomel electrode of the Van Laar type (41). Titrations of the PSL dispersions were performed under a N2 atmosphere with HCI04 and CO2-free NaOH, respectively. It was assumed that ion activity coefficients were unaffected in colloidal dispersions.

3. Viscosity Viscosities were measured, respectively, with a variable shear viscometer and a modified Ostwald viseometer (42). The variable shear viscometer consisted of an horizontal capillary and a pressure system. The device is described in more detail elsewhere (43). Shearing stress rl~ between 0.7 and 17.2 dynes/cm2 corresponding to rates of shear, as defined by Kroepelin (44), ranging between 49 and 1286 sec-1 could b e investigated. The rate of shear for o~r Ostwald type viscometer was 640 sec-1. Capillary end effects, drainage, and kinetic energy corrections were negligible. Absolute error

Journal of Colloid and Interface Science, Vol. 28, No. 2, October 1968

MONODISPERSE POLYSTYRENE LATICES on vsp was 4 X 10-3 for the variable shear viscometer and 10-3 for the other one. All measurements were performed at 25 ° ± 0.003°C. Straight lines in vs,/~ graphs were obtained by the least squares method. Owing to the weak density of PSL (33), sedimentation during viseometrie measurements was negligible. 4. Electrophoretic Mobility

Electrophoretic mobilities were determined using a technique, based on the transport method (analytical method) (45), especially adapted for the study of small particles at low ionic strengths as encountered here. The method permits the study of very dilute sols (¢ < 2 X 10-~) of small particle size (PSL 2a ~ 50 m~) whatever the ionic strength. Under these conditions, surface conductance does not contribute to the sol conductivity. Possible electroosmotic current does not perturb mobility measurements even at low ionic strength. For particles of larger size and higher ionic strength, the analytical method gave the same results as those obtained by a moving boundary technique specially adapted for measuring hydrophobic colloid mobilities (45). R E S U L T S A N D DISCUSSION


As the transport method allowed us to work at very low volume fraction, no interaction between particles could occur. Therefore the measured electrophoretic mobilities could be related to ~ potentials and charges at the slipping surface, by means of existing theories. Zeta potentials were calculated by the Booth (46b) and by the Wiersema (47) mobility:f potential relationships. The charges Ze were then calculated from the f potentials obtained in the two afore-mentioned ways using the numerical solutions of the Poisson-Boltzmann equation ob-


rained by Loeb, Overbeek, and Wiersema (48a). These must be used here because eleetroviseous effects are important only at sufficiently large ~ potentials (~- >> 25 my). For low electrolyte concentrations (c < 10.2 moles/l) and for potentiMs f -<_ 100 my, as encountered in this work, the corrections to the Poisson-Boltzmann equation due to ion size, dielectric saturation, self-atmosphere effect, etc., can be considered as negligible (49, 50). Booth proposed another treatment relating directly the mobility to the charge (46c). However, our measured mobilities were rather large and this method did not allow us to derive any charge value. For electrolyte concentrations below 10.2 moles/1 and f potentials smaller than 100 my, the viseoelectrie effect (51) may be considered as negligible (52-55). In Table I are reported the electrophoretic mobilities determined at various pH's and ionic strengths. The corresponding ~" potentials and charges obtained in the different ways mentioned above are also listed. From examination of Table I, it is possible to deduce a few general trends. a) Consideration of cases 1 and 11 shows us that at the same acid pit, particles bearing weak acid groups have a ~ potential and a specific charge much smaller than those bearing strong acid groups although their neutralization equivalent Ne is about 3 times larger (case 1: Ne = 0.095 milliequivalent/gin and ease 11: N~ = 0.035 millieqnivalent/gm). b) At a given ionic strength, for a weakly acid emulsifier, comparison of cases 1, 3, and 5, and of cases 2 and 7, respectively, shows that the ~" potential and the sPecific charge decrease with pH. c) Cases 4 to 9 indicate that at constant pH the specific charge increases with ionic strength owing to increasing ionization of the earboxyl groups. d) Electrophoretic results are obtained in conditions where relaxation effect plays a very important role. For many measured

Journal of Colloid and Interface Science, Vol. 28, No. 2, October 1968



SYS-tem pH

loni . . . . position (millimoles/t)



Mobility ) fB Ozcm/vsec ~B (my) (~C/cm2)


t w (my)

304 334 380 341 274 262 --

62.2 61.4 108.5 91.5 90.5 86.4 90.5


87.9 86.9






0.611 ] 380

~W ~-) (~C/cm


Emulsifier: Na Stearate 1 2 3 4 5 6 7 8 9 10 11

4 4 10 6 6 6 6 6 6 6 4

0.1 HC104 0.1 HC104 + 0.1 NaC104 0.1 NaOH 0.05 NaC10.~ 0.1 N aC10.,. 0.1 KC104 0.2 NaC104 1 NaC104 10 NaC104 0.1 NaC10~ 0.1 HC104

70 70 56 63 50 50 63 57 57 237

1.13 1.61 0.914 0.73 0.81 0.81 1.45 2.94 9.28 3.84

3.11 2.97 4.27 3.95 3.85 3.84 3.72 4.15 4.20 3.70

64] 62 100 94 90 89 -----

Emulsi ier:Na DodecylSulfate I 56 0.91 430 f - I

0.371 0.348 0.611 0.437 0.558 0.534 ---

0.289 0.345 0.687 0.424 0.559 0.517 0.620 0.349

278 332 428 331 274 254 483 2080 3850

fB: electrokinetic potential (Booth equation (46b)). ~B: specific charge calculated from fB and the numerical solutions of the Poisson-Boltzmann equation (48a). ZB: number of ionic charges of the electrokinetic unit. ~-w: f from Wiersema's U - f relationship (47). ~w and Zw: specific charge and number of ionic charges as obtained from fw. mobilRies the use of Booth's approach does not permit us to obtain a finite value for the f potentiak As indicated b y Wiersema (47) this is due to an overestimation of the relaxation effect. On the other hand, the Wiersema treatment allows one to obtain, in the majority of cases, finite values for the potentials. However, for case 8, where the experimental mobility was 4.15 tL cm/vsec, Wiersema's theory predicts a maximum value of 4.02. I t thus seems that Wiersema's treatment still overestimates slightly the relaxation effect. Experimental data mentioned b y Wiersema (47) and calculations from o t h e r results on polystyrene latices (19, 26) present the same trend. Other U - ~ relationships obtained from rearrangements of Booth's expressions Mso give values lower than Wiersema's. For example, in case 3, we obtained from the Wiersema, Booth, Hunter (56), and Stigter and Mysels (57) treatments: 108.5 my, 100 my, 97 mv, and 91 my, respectively.


1. Validity of the Einstein Equation for Polystyrene Latices The theories of the electroviscons effects add to the Einstein equation a term taking into account the electrical contribution to the viscosity. We, therefore, checked at first the validity of the Einstein relation on the smM1 size systems especially prepared for the study of the electroviscous effects (2a < 100m~). For these systems, large ionic strengths are needed in order to cancel the electroviscous contribution. The values of the coefficient K -- ( ~ , / ~ ) ~ 0 are reported in Table I I (systems 12, 13, 14, 15). As shown in this table, a significant deviation exists between the experimental and the Einstein theoretical K value. Table I I indicates Mso that this deviation cannot be assigned to aa electroviscous effect. The discrepancy could be explained b y different reasons: a particle asymmetry, a volume

Journal of Colloid and Interface Science, VoL 28, No. 2, October t968




Sys- Dintern meter(m~)



Total electrolyte concentration (millimoles/l),


Experimental K = (~sp)~_ ~ 0





Smolu Finkel- Booth stein chowskiK ChursinK Eq.K[ll

Booth Booth Eq.K[2] Eq.K[3]

a) I n Presence of Indifferent Electrolyte

12 13 14 15

58 58 100 100

10 NaOH 10 NaOH 10 NaOH 10 NaOH

+ 40 NaC104 ÷ 30 NaC10~ -~ 110 NaC104 ÷ 60 NaC104

3.5 3.6 3.6 3.8

+ 0.2 4- 0.1 4- 0.3 4- 0.2

21 19 57 43

2.5 2.5 2.5 2.5

2.5 2.5 2.5 2.5

2.5 2.5 2.5 2.5

2.5 2.5 2.5 2.5

2.5 2.5 2.5 2.5

2.9 2.5 2.5

3.0 2.7 2.5

2.7 2.6 2.5

2.6 2.6 2.5

b) In Absence of Indifferent Electrolyte 16 17 18

100 237 430

10 NaOH 10 NaOH 10 NaOH

2.8 4- 0.2 2.7 4- 0.1 2.6 4- 0.2

fraction increase, or the presence of small aggregates. Owing to the high sphericity of the latices a particle asymmetry cannot explain this discrepancy. Indeed, in order to explain the K values obtained, an axial ratio of about 3 would be needed (58, 59). A volume fraction increase due to the presence of a superficial layer either of emulsifier (22) or of bound water (60) is also unable to explain the difference: a layer of 3-6 m~ thick would be necessary to explain the experimentally too large viscosity. It is difficult to admit such a thickness for an emulsifier layer. Saunders (22) indeed considered that in presence of large amounts of soap, the layer of emulsifier surrounding the latex particles is not larger than 1 m~. Furthermore, such an immobilization of bound solvent is not realistic for two reasons. Even for large electrolyte concentrations, in the ease of soap mieelles, Stigter (52) and Mukerjee (53) consider that the water thickness amounts to less than a uniform monolayer. Moreover, interaction studies (55, 61) have shown that polystyrene latices present a purely hydrophobie character. The introduction into the model of a very thin water layer would cause a complete discrepancy between calculated and measured data.

16 39 70

2.6 2.5 2.5

In our opinion, the difference between the Einstein and experimental K values can be related to the existence of small agglomerates due to the slow coagulation of the sol ia presence of significant concentrations of electrolyte. As the number of particles amounted to about 1014/ml, the coagulation time Tr~pia was very short, in the range of 10-a see (62). For slow coagulation, in order to explain the presence of multiplets during viscosity measurements, the coagulation time Tslo,,must have a value of about an hour ( T~low 103-104 sec). The relation between Tr~pid, Tslow, and the stability factor W allows us to estimate a value for W of about 106-107. B y means of the Reerink equation (63), and taking for the parameters values in the range of the experimental ones (f = 50 my, a = 4.10 .6 em, c = 100 mM/1) we eMeulated the value W ~-~ 107, which is of the right order of magnitude. The influence of aggregates on the value of the Einstein constant has also been proposed by Gillespie (23). Systems containing aggregates can present a non-Newtonian behavior. For our systems, we always observed a Newtonian behavior shown by curves A and B of Fig. 2. This fact is not in contradiction with the presence of small agglomerates. As a matter of fact, we could observe non-Newtonian

Journal of Colloid and Interface Science, VoI. 28, No. 2, October 1968




° U







]0£I ocHIA~)/NO STRESS, T ( dyne/co~~)

FIO. 2. Flow curves for polystyrene latices after addition of various amounts of electrolytes to the purified latices. Emulsifier: Na stearate. Curve A: 10-3.moles/1 NaOH -t- 5 X 10-3 moles/1 NaC104; f = 0.01206; 2a = 100 m#. Curve B: 10-3 moles/l NaOH + 10-1 moles/1 NaC104; q~= 0.01377; 2a = 100 m#. Curve C: 10-3 moles/1 NaOH; ~ = 0.009762; 2a = 100 mt~. Curve D: Without addition of electrolyte to the purified latex. ~ = 0.00929; 2a = 50 mt~. Curve E: Without addition of electrolyte to the purified latex, e = 0.03506; 2a = 50 mtt.

But under these electrolyte conditions, the decrease of the eleetroviscous contribution could be achieved only by choosing larger particle sizes. The results are given in Table II and the Newtonian behavior of system 16 is shown by curve C of Fig. 2. System 18, where particles are large, presents a K value in close agreement with the Einstein one. For systems 16 and 17, the slight deviation from 2.5 can be explained by a very small first-order eleetroviscous effect, as calculated in Table II. F r o m these attempts, we can conclude that the Einstein relation is valid when latices are perfectly stable. This is in fair agreement with other works on latices but which were performed in the presence of large amounts of soaP (22, 23). 2. Phenomenology of the Electroviscous Effects

For all the following systems, the ionic strength is very far from t h e coagulation range and the sols are always p e r f e c t l y stable. The viscosity of latices increases rapidly with decreasing ionic strength at least in the effects only when the rate of shear induced range of low electrolyte concentrations oweither orientation or rupture of the muling to the electroviscous effects. Furthertiplets. F r o m the Scheraga tables (64), the non- more, the variations of viscosity in any Newtonian effects of orientation can be direetiou due to an increase or a decrease o f measured, for aggregates having an axial ionic strength are perfectly reversible. A series of viscosity measurements has ratio not larger than 3, only if the rate of been performed in the presence of increasing shear exceeds 12,000 see-1 (65) corresponding to 10 times the value accessible to our concentrations of alkaline perchlorates. The results, plotted on Fig. 3, correspond to apparatus. On the other hand, rupture of systems 4 to 9 of Table I, for which particle the multiplets did not seem to occur during sizes and electrokinetic potentials were flow as the viscosity remained unchanged nearly identical, although the corresponding after exposure of the latices to an ultrasonic beam (Branson Sonifier S-75) for one hour. charge increases with ionic strength. The In the previous measurements we did not first- and second-order electroviscous effects, succeed in checking the Einstein equation determined, respectively, b y the ordinate to owing t o / t h e presence of aggregates. In the origin and by t h e slope of these lines, another set of experiments, in order to decrease progressively with the double layer improve the sol stability, we worked in the thickness. Finally, curve 0 corresponding to presence of peptiZing electrolyte only (10: a system of particle diameter = 50 m~ h a s n ~ I / 1 N a O H ) . The calculated stability fac- been obtained on the purified sol, without tors then increased to extremely large values. addition of electrolyte. Non-Newtonian beJournal of Colloid and Interface Science, VoI. 28, No. 2, October 1968





5 ~


8 I

FIG. 3. V~,/~ as a f u n c t i o n of the volume fraction ~ for p o l y s t y r e n e latices at v a r i o u s concent r a t i o n s of alkaline perehlorate. Emulsifier: N a s t e a r a t e . Curve 0 : 0 mole/1 NaC104; 2 a = 50 mtt. Curve 4 : 5 X 10-5 moles/1 NaCtO4; 2 a = 63 mt~. Curve 5 : 1 0 -4 moles/1 NaC104; 2a = 50 m~. Curve 6:10 -4 moles/1 KCtO4; 2a = 50 m~. Curve 7: 2 X 10-4 moles/1 NaC104; 2a = 63 m,. Curve 8: 10-~ moles/1 NaC10~; 2a = .57 m#. Curve 9:10 -x

moles/1 NaC10~; 2a = 57 m~. havior can be expected for sols where double layers are widely expanded. This effect would be the largest for system 0. However, no shear dependence of the viscosity was observed as shown by curves D (~ = 0.00929) and E (~ = 0.0356) inFig. 2. This is true at least for rates of shear between 50 and 1,300 see -~. Curve 0 on Fig. 3 presents an abnormal ~ p / ~ - ~ relation. In the liquid phase only H + and O H - ions coming from the dissociation of superficial groups and of water are present. I n this ease, where the double layer overlapping is very important, the ionic strength cannot be defined simply from the electrolyte concentration far from the particle surface. There is also a contribution of the ionic atmospheres in the formulation of the effective ionic strength. An increase in volume fraction then results in an increase oLionic strength, and consequently in a decrease of double layer thickness. Therefore, for system 0, the abnormal behavior of ,~,/.~ with increasing


can be attributed to a decrease of the secondorder electroviseous effect according to the decrease with ~ of the double layer thickness. An extrapolation of vsp/e values to e = 0 is impossible. Moreover, the 1/~ value needed in order to apply theories of electroviscous effect is not accessible. We would like to point out that for curves 4 to 9, without streaming of the sol, no overlapping of the atmospheres occurred and the solvent dilution method then gave a perfectly normal viscosity behavior. We also considered the influence of the other experimental parameters on the electroviscous effects. Viscosity curves were obtained on systems 5 and 10 of Table I having the same ionic strength and p H and about the same f potentials but different particle sizes (50 m,~ and 237 m~ in diameter). For system 10 the viscosity curve can be assimilated to curve 9 on Fig. 3. I t can be concluded that the electroviscous effects decrease markedly ~4th increasing particle size. Furthermore, systems 1 and 3 on Fig. 4 show that for ~a values remaining m the same range, the electroviscous effects decrease with the f potential or with the charge of the electrokinetic unit. The same trend is observed









r 0,015

Fro. 4. ~,p/e as a function of the volume fraction # for polystyrene latices at different pH. Emulsifier: N a s t e a r a t e . Curve 1: p H = 4; 10-4 moles/1 HC104; 2 a = 70 m~. Curve 2: p}I = 4; 10-4 moles/1 ttC104 + 10-4 moles/1 NaC104; 2 a = 70 m~. Curve 3: p H = 10; 10-~ m o l e s / l N a O H ; 2 a = 56 m~.

Journal of Colloid and Interface Science,

Vol. 28, No. 2, October 1968



when, for the same pH, the ionic strength increases (curves 1 and 2). The extrapolation to ~ - 0 of curves 1 and 2 gives values of (~sp/~)~0 = 2.5. No first-order electroviscous effect has been observed. The sols are still stable and the values obtained confirm the validity of the Einstein relation. Nevertheless, the second-order effect is still important because the atmospheres are expanded. We also studied the influence of the nature of the emulsifier. The comparison of system 2 on Fig. 4 (Na stearate) and system 11, curve C on Fig. 5 (Na dodeeyl sulfate) which have been studied at the same acid pH and the same Ka, shows that the electroviscous effects are more important when a strong acid soap is used. Moreover, as also shown by curve C on Fig. 5, in presence either of I-ICI04 or of NaCI04, the viscosity is not influenced by the pH for a strong acid soap because in this range of pH the dissociation of superficial groups can be considered as constant. The behavior is very different for a weak acid soap, as shown by comparing system 5 on Fig. 3 and system 1 on Fig. 4. Summarizing these results, we can conelude that the observed electroviseous effects decrease if, for a constant f potential, Ka increases or if, for a constant Ka, charge and potential decrease. Furthermore, beyond a given ionic strength, the eleetroviseous effects vanish however large is the potential or the charge. Another explanation for the large viscosities of dialyzed latices has been proposed by Fryling (24). He assumes that the effective volume fraction increases according to the existence around the particle of an icelike hydrated shell. Under this assumption, for our system 0, the Mooney equation (4) gives for the effective volume fraction a value about 10 times larger than the volume fraction of the dry latex. This corresponds to an ieelike hydrated shell 30 m~ thick. Such a value seems to us rather improbable.

3. Comparison with Theories First-Order Electroviscous Effect. For each system investigated we compared the experimental values of (W,/~)~o with those calculated by means of existing theories. The two values of the charge ( ~ and w ) and of the f potential (fB and fw), deduced, respectively, from the eleetrophoretie treatments of Booth and Wiersema as explained before, were introduced ia the theoretical equations. The results are listed in Table III. First, it can be seen that the Smoluehowski and Finkelstein-Chursin theories predict a far too large electroviseous effect for all systems. However, the Smoluchowski treatment is valid only for Ka > 10. In this range of Ka values, the calculated viscosity increments are of the order of the experimental uncertainties as can be seen from Table II. The Finkelstein-Chursin theory, which contains no restriction concerning the magnitude of Ka, is in complete disagreement with experimental results. Finally, Eq. [1], which is the more elaborated treatment of Booth relating the viscosity to the charge of the eleetrokinetic unit, gives K values in good agreement with the experimental ones. Equations [2] and [3], which are approximations of Eq. [1], relate the viscosity to the f potential. They give slightly too low K values when the experimental viscosity increments are important. We would now like to compare our results with literature data. Various studies on proteins (12, 16) indi= cate that a more reasonable order of magnitude for the primary electroviscous effect is given by the Booth theory than by the Smoluehowsld equation. However, definite conclusions could not be drawn from these works because of important uncertainties concerning the experimental parameters. An interesting attempt (59) to interpret the viscosity measurements on AgI sols (14) and emulsions (60) shows that these are in support of the Booth theory. The re-

Journal of Colloid and Interface Science, Vol. 28, No. 2, October 1968




Ionic composition



Experimental /~sp\

. ISmoluchowski] I

Finkelstein Chursin

Booth Eq. [1] I Boo~ Eq.

Booth Eq. [3]


K(~B) K(~ W) K(¢ B) K(a W) ]K(¢B~ K (~ W) K (~-B)I(KtW) K (~-B) K (~"Wi Emulsifier: Na Stearate

3 4 5 6 7 8 9 10

0.1 HCIO4 1.13 0.5zh 0.1 0.1 HCIO4 + 0.1 1.61 2.5=5 0.1 ! NaC104 0.1 NaOH 0.914 3.6~: 0.3 0.05 NaCIO4 0.73 5.1 :t: 0.8 0.1 NaC104 0.81 3.7~: 0.4 0.1 KCIO4 0.81 3.3~: 0.1 0.2 NaC104 1.45 2.8~: 0.2 1 NaC104 2.94 2 . 8 ~ 0.2 10 NaC104 9.28 2.8:L 0.2 0.1 NaC10~ 3.84 3.0:E 0.2

7.4 5.8

7.4 5.8

35.5 19.5

35.5 19.5

2.7 2.7

2.7 2.7

2.7 2.7 2.7 2.7

2.6 2.6

31.3 80.3 65.5 51.6

~6.4 '6.2 15.6 ~8.7 :2.6

321.0 707.3 471.3 352.4 --

406.5 664.5 471.3 331.3 91.8

3.5 3.7 3.6 3.3

3.8 3.7 3.6 3.3 3.4

3.1] 3.2 3.31 3.3 3.2 3.2 3.1 3.0 - - 3.1

2.9 3.2 3.2 3.1 --

3.0 5.1


3.7 10.7

3.0 3.2

--!2.7 - - I 2.8


2.6 2.6 0 3.2 3.2 3.0 3.1 3. 2.8 2.8

Emulsifier: Na Dodecyl Sulfate

11 I 0.1 HCIO4



02 I -

207 1 -

sults of Rutgers and Nagels (18) present the same trend. Nevertheless measurements at very low ionic strength and extrapolations to ~ = 0 have not been performed. Viscometric studies published b y D o b r y ( 13 ) and Donner ( 15 ) unfortunately did not confirm either theory. The Parker and Wasik (17a) investigations on micelles are in very good agreement with Eq. [2] of Booth. This agreement is maintained if, in place of ~B, the electrokinetic potential ~,w calculated according to Wiersema is introduced in Eq. [2]. However, if the more elaborated equation of Booth [1] is used, the calculated values are larger than the experimental ones. This divergence is inconclusive o~4ng to the uncertainties in the extrapolations at ~ = 0 of the viscosity results obtained on sodium dodecyl sulfate micelles ( 17b ). Finally, sulfonated latices, composed of a 80/20 styrene-divinylbenzene copolymer, were recently investigated b y Chan and Goring (19). Their system is to some extent comparable with ours. Their experimental values are considerably larger t h a n the


I- 13.2 I-I 2.9 I-


values calculated from the Booth equations [2] and [3]. The values of the electroviscous contribution to the intrinsic viscosity and to the K value are listed in Table IV, for the system having an ionic strength = 10-s. The use of Eq. [1] improves the situation but a deviation of about a factor 2.4 still remains. This difference in behavior between their systems and ours cannot be explained in a simple manner. Their values for (V~p/~)~0 are identical whatever method is used to change the volume fraction, isoionic dilution (11c) or solvent dilution (66b). T h e y are also systematically higher than ours. One essential fact to be pointed out is the great difference between their particle sizes determined b y various experimental techniques. Number-, weight-, and Z-average diameters of 51 m;~, 59 mt~, and 61 m~, respectively, were measured by electron microscopy. The diameter calculated from the radius of gyration obtained b y light scattering is 121 m~. Finally, on the assumption of the validity of the Einstein equation, an hydrodvnamic diameter of 76 m~ was derived. Such a value implies a particle swelling b y at least a factor

Journal of Colloid and Interface Science, Vol. 28, :No. 2, October 1968



of 2. In this situation, according to Gregor, Cutoff, and Bregman (67) it could be supposed that the ionic strength influenced the degree of swelling. Nevertheless, Chan and Goring showed t h a t f o r electrolyte concentrations ranging from 10-5 moles/1 to 1.3 moles/l, swelling remained constant because internal sites are blocked by divalent cations and are unable to make any exchange with the solution. Therefore, the viscosity increment with decreasing ionic strength cannot be related to an increase of swelling but to an electroviscous effect. However, for these kinds of

copolymers, according to Calmon (68), swelling should be negligible. The true diameter would then be the one obtained by electron microscopy and consequently the Einstein constant would be 9.7. Such a value cannot be explained. In our case, we have measured 2.5 for the Einstein constant, although not being indispensable for the study of the electroviscous effects. Therefore no ambiguity exists in the interpretation of our results. Second-Order Electroviscous Effect. For each system, we deduced from the measured slope of the n~,/¢ - ~ relation, the experimental value of the coefficient K' defined by the equation:

TABLE IV EXPERIMENTAL AND THEORETICAL ELECTROVIS~s, _ K + K'~; [4] ~o COUS EFFECT IN SULFONATEBLATICES (19) COMPOSED O F A 80/20 STYRENE-]~}IVINYLK' values are then compared with those calBENZENE COPOLYMER culated by means of the theoretical treatA[,fl AK ments of Street (10) and of Chan, Blach(dl/gm X 102) ford, and Goring ( 1 la). Deduced from Eq. [2] in0.49 0.69 As shown in Table V and Fig. 5, the Street troducing ~-B equation gives far too low K ' values. MoreDeduced from Eq. [2] in0.72 1.05 troducing ~-w over, as the Street relation does not take Deduced from Eq. [1] in1.17 1.67 into account the existence of a primary troducing ~ w electroviscous effect, the treatment is not Experimental 2.8 3.98 valid. In Fig. 5, we also give for comparison, TABLE V


Ionic composition (millimoles/l-


Experimental K'

ChaR, Blachford, and Goring

K' (~-B)


11.06 11.2 52.8 116 103 80

11.06 11.2 62.1 110 103 76 60

K" from ~i~IIand eq." 29 in reference llb

Emulsifier: Na Stearate 1 2 3 4 5 6 7 8 9 10

0.1 HC104 0.1 HC10~ 7- 0.1 N a C 1 0 4 0.1 N a O H 0.05 N a C 1 0 ~ 0.1 N a C 1 0 4 0.1 I(C104 0.2 N a C 1 0 ~ 1 NaC10~ 10 N a C 1 0 4 0.1 N a C 1 0 ~


0.1 HC104

1.13 1.61 0.914 0.73 0.81 0.81 1.45 2.94 9.28 3.84

319 172 592 1069 422 438 197 36 ---

732 361 4150 2450 3370 3535 579

28 31

Emulsifier: Na Dodecyl Sulfate



Journal of Colloid and Interface Science, Col. 28, No. 2, October 1968



MONODISPERSE POLYSTYRENE LATICES values of v~p/~ versus ¢ for dispersions of uncharged particles. We used Eq. [4] with K = 2.5 and K' = 10.05. This K' value, taken from the work of Manley and Mason (69), is obtained from considerations relative to the formation of transient doublets which rotate and disrupt during flow. A recent theoretical treatment concerning the seeond-order electroviseous effect has been published by Chart, Blaehford, and Goring. It is also based on the concept of the collision doublet but it takes into account



o lO~rr~/g t~aClO, 10

5 -A ooc5




FIG. 5. n~,/~ as a function of the volume fraction ~ for polystyrene latices. Emulsifier: N a dodecyl sulfate. Curve A: calculated by the Manley and Mason equation (69) for dispersions of uncharged particles. Curve B: calculated by the Street equation (10). Curve C: experimental results. Curve D: calculated by the Chan, Blachford, and Goring t r e a t m e n t (11).


the ability of the liquid to flow between the surfaces of the two charged particles during the rotation of the doablet under a gradient of shear. A collision distance a~ is defined as half the closest distance of approach between the two particle surfaces; ae can be estimated in two different manners the resuits of which are called aeI and aii, respectively. A quantitative expression for a i in terms of the Huggins interaction coefficient k' has been derived (llb). 1J is related to K' of Eq. [4] by the expression k' = K ' / K 2. It is thus possible to calculate a i from experimental K' values. On the other hand, aIe*can be calculated by equilibrating the electrostatic repulsive force and the hydrodynamic compressive force during flow. In the estimation of the electrostatic repulsive force the approximation of small potentials is used because no theory is available to calculate repulsions for small Ka and large potentials. Values of a i and ~ii corresponding to our systems are reported in Table VI. Van der Waals forces carl be completely neglected at the ae distances obtained owing to the very low Hamaker constant of the polystyrene embedded in water (55). In this treatment it has been assumed that particles are spheres without Brownian motion. This assumption is not at first sight justified. Therefore, we estimated the ira-


Ionic composition (millimoles/l)

I 2

0.1 HC104 0.1 HC10~_ ~-0.1 NaClO4 0.1 NaOH 0.05 NaClO4 0.1 NaC104 0.1 KC104 0.2 NaCtO4

Diameter (m~) avI (10-6 cm)

~cII (10_8 cm)


VnR X kT

Emulsifier: Na Stearate

3 4 5 6 7

70 70

7.3 6.1

9.3 7.6

0.43 0.195

0.11 0.05

56 63 50 50 63

7.1 9.27 5.68 7.05 5.5

10.4 11.5 9.9 10.0 7.8

0.97 0.95 1.80 0.45 0.68

0. 088 0.27 0.065 0. 038 0.06



E m u l s i f i e r : Ara D o d e c y l S u l f a t e


0.1 HC104




Journal of Colloid and Interface Science, Vol. 28, No. 2, October 1968



portanee of the electrostatic repulsive energies VR! and V~ corresponding to the distances 3c± and ~i1, respectively. Calculations were also carried out using equation 84 of reference 70, strictly valid for small Ka and small potentials. The values are reported in Table VI. All the V ~R values, calculated at the distance ~, range between ]~T/4 and I¢T/25. Thus, at the distance for which the electrostatic repulsive force counterbalances the hydrodynamic compressive one, the repulsive energy is lower than kT. This energy would thus be unable to influence the particle path. It must be concluded that the distance ~ for which doublets are formed must be lower than ~i~. The calculation of repulsive energies at distances ~i generally leads also to values lower than kT. Furthermore, the ~ are always smaller than the ~11 distances. Chan, Blachford, and Goring also observed a large discrepancy between ~i and ~ioI for ionic strengths and rate of shear comparable with those investigated here. They suggest that this is due to the approximation made in the theory of a circular path of rotation for the interacting particles, Initial displacement across the streamlines would occur before the interacting spheres attain the distance of closest approach. However, if displacement occurs for a distance larger than ~, the approximation made in neglecting the Brownian motion seems to us still more questionable. In order to show on a v~,/w - w graph the difference in viscosity values due to the divergence between ~i and ~i , we expressed ~i] in terms of K t by means of equatiOn 29 of reference lla. K p values are listed in Table V, and it can be observed that they are larger than the experimental ones by a factor ranging between 2 and 8. As an example, we plotted on Fig. 5 the ~,p/w theoretical relation corresponding to the system 11. We can conclude that the Chan, Blachford, and Goring treatment predicts a second-order electr0viscous effect which is too large at least for the electrolyte concentrations studied but which represents a more realistic approach than Street's.

CONCLUSIONS We studied the electroviscous effects on very monodisperse polystyrene latices. The particles are perfectly rigid and compact spheres. The ionized sites responsible for t h e particle charge, carboxylic or sulfate groups, were firmly bound to the surface and do not desorb during purification. Different latices prepared in similar conditions led to coherent results. The Einstein constant K = 2.5 has been found to be the limiting value when the systems are very stable and when the: first-order electroviscous effect vanishes. In the range of the investigated rates of shear and volume fractions, all the systems presented a Newtonian behavior. The observed e]ectroviseous effects decrease if, for a constant ~- potential, Ka increases or if, for a constant Ka, charge and potential decrease. Beyond a given ionic strength, the electroviscous effects vanish however large the potential or the charge, At low volume fractions, for Na + and K + salts, only small differences in viscosities have been observed. For the primary electroviscous effect, agreement between the calculations from the Booth theory (Eq. [1]) and the experimental results is very good. A similar trend was deduced in previous literature data, particularly from viscosity results on aqueous solutions of ionic detergents. Sometimes, the discrepancy between theory and experiment is ascribed to the fact that Booth gives only the i"~ term of a series expansion in f and that for a f potential of about 100 my more terms of the series should be considered. The agreement we obtained suggests that in these conditions further terms ir~ the series are not essential. On the other hand, Loeb, Overbeek, and Wiersema (48a) have shown that with increasing potential and charge the double layer becomes more compact than the Debye-Htickel prediction, which is used in the Booth treatment (48b). However, an eventual correction to the theory of the primary electroviscous effect in this direction would not probably change the order of

Journal of Colloid and Interface Science, Vol. 28, No. 2, October 1968


19. C~AN, F. S., AND GORING, D. A. I., J. Colloid and Interface Sci. 22,371 (1966). 20. CHENG, P. Y., AND SCHACH~AN, g . K., or. Polymer Sei. 16, 19 (1955).

magnitude of the calculated effect. Moreover the use of the Poisson-Boltzmann equation to describe the ionic atmosphere, even for surface potentials of about 100 my, does not seem,to introduce too large an error (71), at least within the present limits of experimental accuracy. If the theory of the first-order eleetroviscous effect is to be improved, the use of a new statistical treatment of the double layer would be necessary. For the second-order eleetroviseous effect no existing treatment agrees well with the experimental facts, although the Chan, Blachford, and Goring approach is based on a more elaborated model than the previous attempts.

21. MA]~ON, S. H., AND KRIEGER, I. M., I n F.

Eirieh, ed., "Rheology," Vol. 3, Chapt. 4. Academic Press, New York, 1960; JOHNSON, P. H., AND KELSEY, R. I~., Rubber World 138,877 (1958). 22. SAUNDERS,P. L., J. Colloid Sci. 16, 13 (1961). 23. (a) GILLESt'IE, T., J. Colloid Sei. 18, 32 (1963). (b) GILLESPIE, T., In P. Sherman, ed., "Rheology of Emulsions." Pergamon Press, New York, 1963. 24. FRYLING, C. F., J. Colloid Sci. 18,713 (1963). 25. BI~ODNYAN, J. C., AND LLOYD KELLEY, E., J. Colloid Sci. 19, 488 (1964); ibid. 20, 7

(1965). 26. SCHALLER, E. J., AND HUMPHREY, A. E., 40th

Natl. Congr. Colloid Syrup., University of Wisconsin, Madison June 14-16, 1966; J. Colloid Sci. 22, 573 (1966).

REFERENCES 1. EINSTmN, A., Ann. Physik 19, 289 (1906); ibid. 34,591 (1911). 2. HAPPEL, J., J. Appl. Phys. 28, 1288 (1957). 3. EILERS, H., Kolloid-Z. 97,313 (1941). 4. MOONnY, M., J. Colloid Sci. 6, 162 (1951). 5. THOMAS, D. G., J. Colloid Sci. 20,267 (1965). 6. CONWAYi B. E., AND

D0~n¥-DvcLAvx, A., I n

F. Eirich, ed., "Rheology," Vol. 3, Chapt. 3. Academic Press, New York, 1960. 7. yON SMOLUCnOWSKL M., Kolloid-Z. 18, 190

(1916). B. N., AND CHURSIN, M. PI, Acta Physicochim. U.R.S.S. 17, 1 (1942). BOOTH, F., Proc. Roy. Soc. (London) A203, 533 (1950). STgEL% N., J. Colloid Sci. 13,288 (1958). (a) CHAN, F. S., BLANCttFORD, J., AND GORING, D. A. I., J. Colloid and Interface Sci. 22,378 (1966); (b) ibid., Eq. 29; (c)ibid, Fig. 1. BULL, H. 13, Trans. Faraday Soc. 36, 80 (1940). (a) ])OBRY, A., Kolloid-Z. 145, 108 (1956); (b) ibid., p. 112. HARMSEN, G. J., VAN SCI-IOOTEN, J., AND OVEnBE~K, J. TH.G., J. Colloid Sci. 8, 72 (1953); ibid. 10, 120 (1955); HARMSEN, G. J., VAN SCItOOTEN, J., AND \TAN DEn WAARDEN, M., J. Colloid Sci. 10, 315 (1955). DONNET,J. B., AND RmTZER, C., Compt. Rend. 248, 1997 (1959). TA~FOnD, C., ,XN9 BVZZEL~, J. G., J. Phys. Chem. 60, 225, 1204 (1956). (a) PARKE~, R. A., AND WASII(, S. P., J. Phys. Chem. 62, 967 (1958); (b) KUS~NER, L. M., DUNCAN, B. C., AND ~IOFFMAN, J. I., J. J. Res. Natl. Bur. Std. 49, 85 (1952) RP 2346. RUTGERS, A. J., AND NAGELS, P., J. Colloid. Sci. 13, 148 (1958).


9. 10. 11.

12. 13. 14.

15. 16. 17.



27. HELLER, W., AND TABIBIAN, R . M . , J. Colloid Sci. 19., 25 (1957). 28. BRADFORD, E. B., AND VANDERHOFF, J. W., J. Appl. Phys. 26,864 (1955). 29. DEgELI6, G., AND KnATO~VIb, J. P., J. Colloid Sci. 16, 561 (1961). 30. ~{ATIJEVIC, E., AND PETHICA, B. A., Trans. Faraday Soc. 54, 1382 (1958). 31. SIEGLAFF, C. L., AND MAZUR, J., J. Colloid Sci. 15, 437 (1960). 32. CHAN, F. S:, AND GOmNG, D. A. I., Can. J. Chem. 44,726 (1966). 33. PUGtt, T. L., AND HELLER, W., dr. Colloid Sci. 12, 173 (1957). 34. (a) K~UYT, It. R., ed., "Colloid Science," Vol. 1. Elsevier, Amsterdam, (1952); (b) ibid., p. 128. 35. STIGTER, D., J. Phys. Chem. 68, 3603 (1964). 36. TANFORD, C., "Physical Chemistry of Macromolecules," p. 468. Wiley, New York, 1961. 37. PALS, D. T. F., AND HEnMANS, J. J., Rec. Tray. Chim. 71, 433 (1952); FUJITA, H.~ AND HOMMA, T., J. Polymer Sci. 15, 277 (1955); DOTY, P., AND STEINER, R. W., J. Chem. Phys. 20, 85 (1952); REZANOWICII, A., AND GOnING, D. A. I., J. Colloid Sci. 15, 452,

472 (1960). 38. G.~DDI~M, J. H., Proe. Roy. Soc. (London). B109, 114 (1931). 39. HaRI(INS, W. D., AND :BRo~,VN,F. E., J. Am. Chem. Soc. 41,499 (1919). 40. PALLMANN, H., Kolloid Chem. Beih. 30, 334 (1930); D~ BlgUYN, I{., Thesis, Utrecht, Netherlands, 1938. 41. VAN LA.~n, J. A. W., Thesis, Utrecht, Netherlands, 1952; VAN LAAt¢, J. A. W., to N. V.

Journal of Colloidandfnterface Science, VoI. 28, No. 2, October 1968



Philips Gloeilampen Fabrieken, Eindhoven, Netherlands, Dutch Patent 79,472, November 15, 1955. 42. MARTIN, F., Bull. Soc. Chim. Belges 34, 82 (1925). 43. STONE-MASUI, J., Thesis, Brussels, Belgium, 1966. 44. KROEPELIN, I-I., KolIoid-Z. 47, 294 (1929). 45. WATILLON, A., JOSEP~-PETI% A. M., AND STONE-MASUI, J., To be published. 46. (a) BOOTIL F., Proc. Roy. Soc. (London) A203, 514 (1950); (b) ibid., Eq. 9.4; (e) ibid., Eq. 9.1. 47. WIERS~,MA, P. H., Thesis, Utrecht, Netherlands, 1964; WIERSE~A, P. I-I., LOEB, A. L., AND OVERB:EEK, J. TH. G., J. Colloid and Interface Sci. 9.2, 78 (1966). 48. (a) LOEB, A. L., 0VERBEEK, J. TH. G., AND WIERSEMA, P. It., "The Electrical Double Layer around a Spherical Colloid Particle." M. I. T. Press, Cambridge, Massachusetts, 1961; (b) ibid., Table 16. 49. LEVINE, S., AND BELL, G. M., J. Phys. Chem. 64, 1188 (1960); Discussions Faraday Soc. 42, 69 (1966). 50. HAYDON, D. A., Recent Progr. Surface Sci. 1, 94 (1964). 51. LYKLEMA, J., AND OVERBEEK, J. Tit. G., J. Colloid Sci. 16, 501 (1961). 52. STIG~0ER,D., J. Phys. Chem. 68, 3600 (1964). 53. MURKERJEE, P., J. Colloid Sei. 19,722 (1964). 54. HUNTER, R. J., J. Colloid and Interface Sci. 29., 231 (1966). 55. WATILLON, A., AND JOSEPH-PETIT, A. M., Discussions Faraday Soc. 42,143 (1966). 56. HI:NTER, R. J., J. Phys. Chem. 66, 1367 (1962).

57. STIGTER, D., AND MYSELS, K. J., J. Phys. Chem. 59, 45 (1955). 58. SIMI~A, R., J. Phys. Chem. 44, 25 (1940); MEttL, J. W., 0NCLEY, J. L., AND SIMHA,R., Science 92, 132 (1940). 59. MUKERJEE, P., J. Colloid Sci. 19., 267 (1957). 60. VAN DER WAARDEN, M., J. Colloid Sei. 9, 215 (1954). 61. WATILLON, A., AND JOSEPH-PETIT, A. M., Meeting of the American Chemical Society, Symposium on Coagulatior~ and Coagulant Aids, Los Angeles, California, 1963. 62. KRUVT, H. R., ed., "Colloid Sci.," Vol. 1, Chapter 7. Elsevier, Amsterdam, 1952. 63. REERINK, H., Thesis, Utrecht, Netherlar~ds, 1952; I n H. R. Kruyt, ed., "Colloid Science," Vol. 1, p. 285. Elsevier, Amsterdam, 1952. 64. SC~IERAGA, H. A., J. Chem. Phys. 23, 1526 (1955). 65. YANG, J. T., J. Am. Chem. Soc. 81, 3902 (1959). 66. (a) C~IAN,F. S., AND GORING, D. A. I., KolloidZ. 215, 42 (1967); (b) ibid., Fig. 3. 67. GREGOR, H. P., GUTOFF, F., AND BREGMAN, J. I., J. Colloid. Sci. 6, 245 (1951); HELFFERIC~I, F., "Ion Exchange," Fig. 5.5, p. p. 104. McGraw-Hill, New York, 1966. 68. CALMON, C., Anal. Chem. 9.4, 1456 (1952); ibid. 25, 490 (1953); HELr~ERICI~, F., "Ion Exchange," Fig. 5.4, p. 102. McGraw-Hill, New York, 1966. 69. MANLEY, R. ST. J., AND MASON, S. G., Can. J. Chem. 33,763 (1955). 70. VERWEY, E. J. W., AND OVERBEEK, J. TH. G., "Theory of the Stability of Lyophobiz Colloids." Elsevier, Amsterdam, 1948. 71. SAWyeR, W. M., AND REHFELD, S. J., J. Phys. Chem. 67, 1973 (1963); LA MEn, V. K., ibid., Discussion, p. 1976.

Journal of Colloidand InterfaceScience, Vol.28, No. 2, October1968