The interpretation of electrokinetic potentials and the inaccuracy of the DLVO theory for anatase sols

The Interpretation of Electrokinetic Potentials and the Inaccuracy of the DLVO Theory for Anatase 50ls 1 JOSEPH T. WEBB,2 P. D. BHATNAGAR,3

AND

DALE G. WILLIAMS

The Institute of Paper Chemistry, Appleton, Wisconsin 54911

Received February 26, 1973; accepted May 20, 1974 The surface charge density, electrophoretic mobility, and colloidal stability were measured for hydroxide-dispersed anatase (TiO.) sols at 6.2% solids. The results of r calculations from electrophoretic mobility indicated that the Wiersema-Loeb-Overbeek method of correcting for retardation and relaxation effects is more accurate than Henry's method. At a given electrolyte concentration, the hydrodynamic shear plane was at a constant distance from the surface over a wide range of pH and '/I•. The distance decreased with increasing salt content and ranged between 12 A at 0.01 M Na+ to 25 A at 0.0004 M Na+. The resulting relationships between r and '/I. and the location of the shear plane were best explained by the LyklemaOverbeek case III, immobilized water at the surface, with small corrections for viscosity variation with the electric field outside the ordered region. Dispersion stability was not satisfactorily indicated by any of the double-layer potentials [r, '/I., or '/10] when they were substituted into the DLVO theory of colloidal stability. The DLVO theory gave inadequate quantitative results based on the large and widely varying values of the Hamaker constant required. A more exact mathematical description of double-layer interaction for spherical particles is apparently needed, and/or the effect of water structure on double-layer interaction and the Poisson-Boltzmann equation needs clarification. INTRODUCTION

The concepts of zeta potential (r) and the electric double layer have proven very useful in predicting stability or coagulation tendencies of many colloidal systems. Unfortunately, determinations of r cannot be relied upon to give fundamental information about the electric double layer. The relationship of r to other double-layer potentials and to colloidal stability needs to be clarified, especially since the development of instruments which permit 1 A portion of a thesis submitted by J. T. Webb in partial fulfillment of the requirements of The Institute of Paper Chemistry for the degree of Doctor of Philosophy from Lawrence University, Appleton, WI, January, 1971. 2 Present address, Westvaco Corp., North Charleston, SC. 3 Head of Science Department, Regional College, Bhopal (M.P.), India.

convenient measurements of electrophoretic mobility (1-3). Besides the limited data of Bhatnagar and Williams (4), Li and de Bruyn (5) are the only investigators who have determined the relationship between .\ and 1/;., the potential at the outer Helmholtz plane (O.H.P.), over a wide range of values. Based on double-layer models of Stern (6) and Grahame (7), the hydrodynamic slipping plane (r plane) should be essentially equivalent to the O.H.P. (1/;. plane). Li and de Bruyn found from studies on quartz that rand 1/;. were similar at low potentials, but r approached a constant or limiting value as 1/;0 increased beyond 100 mV. Their results supported qualitatively the theoretical predictions of Lyklema and Overbeek (8) and Hunter (9), who analyzed the effects of viscosity and permittivity variation in calculating zeta potential from electrophoretic mobility. 346

Journal of Colloid and Interface Science. Vol. 49. No.3. December 1974

Copyright © 1974 by Academic Press. Inc. All rights of reproduction in any form reserved.

347

ELECTROKINETIC POTENTIALS

This study was, therefore, proposed to further evaluate the validity of the LyklemaOverbeek-Hunter treatise with respect to double-layer structure and location of the slipping plane. In addition, the various electricdouble-layer potentials were used to test quantitatively the Derjaguin, Landau, Verwey, and Overbeek (DLVO) (10, 11) theory of the stability of lyophobic colloids for spherical particles. Aqueous, hydroxide-dispersed anatase (Ti0 2) sols were studied since TiO z particles are essentially spherical, narrow particle-sizedistribution pigments are available, and the charge development by hydroxyl ion adsorption is fairly well understood (12, 13). The research program required two groups of information: (1) values of r, 1f~, and 1fo under widely varied conditions of pH and electrolyte (NaCl) concentration and (2) measurements of relative colloidal stability at these conditions.

:>

...-::

f="1,

...•

3

1£.

"

w

2

w.L.o.!kT

FIG. 1. Unitless mobility vs reduced potential for the W.O.L. method.

DOUBLE-LAYER POTENTIALS

Zeta potential for spherical particles is calculated from electrophoretic mobility u with von Smoluchowski's (14) expression:

[lJ which assumes that viscosity 7J and permittivity ~ are constants throughout the double layer and double-layer thickness, 11K, is small compared to particle radius, a, i.e., Ka» = 1. The Debye-Huckel parameter, K, is defined for 1: 1 electrolytes by Eq. [2]. = 87reWcll000~kT,

Ka~50

AND >.~:; 70 ohm-I em 2.

y =e ~

THEORY

Z K

FOR 1:1 ELECTROLYTES 5

[2J

Here, e is the elementary charge, N is Avogadro's number, c is the electrolyte concentration (moles/liter), k is the Boltzmann constant, and T is the absolute temperature. Values of Ka ranged between 4 and 50 for the dispersions used in this study, and, when 0.1 < Ka < 100, Eq. [lJ must be corrected for retardation and relaxation effects. Henry (15), Overbeek (16), and Booth (17) proposed corrections for these effects, but the most com-

plete treatment available was derived by Wiersema, Loeb, and Overbeek (18). Henry's correction to Eq. [lJ involves a simple multiplication factor, !(Ka), which varies from 1.5 (for Ka < 0.1) to 1.0 (for Ka> 500).

The detailed analyses of Overbeek and Booth resulted in expressions in which electrophoretic mobility is related to zeta potential through an infinite power series. The power functions are presented in the respective articles (16, 17). Wiersema-Loeb-Overbeek (W.L.O.) obtained precise solutions of the appropriate differential equations of Overbeek (16) and Booth (17), and representative data for applying the W.L.O. method are illustrated in Fig. 1. Unitless mobility, E, is calculated from u, and the corresponding reduced potential, y, is interpolated from the graph. The W.L.O.-corrected r is then calculated from y. Compared to the

,Journal of Colloid and Interface Science, VoL 49. No, 3, Deye'!!ber )9(4

348

WEBB, BHATNAGAR AND WILLIAMS

W.L.O. method, Henry's (15) expression underestimates the retardation and relaxation effects and Overbeek's (16) and Booth's (17) equations overestimate them (18). For Ka < 0.1 or Ka> 100, E essentially becomes a linear function of y, and corrections for these effects are negligible. Wiersema et al. (18) found limited support for their theoretical results from past electrophoresis experiments in the critical Ka region, since a maximum empirically determined E was observed in several studies, and in most cases, this maximum was in good agreement with the maximum predicted E. The electrical potential at the O.H.P., 1/;., is determined from surface charge density, (J", from the following equations (9) :

[4J (J"1

(J"2=

1+ (N/Mn) exp [-(vey;.+
,

-(€/41r)(dl/;jdr).plane,

[5J

[6J

where (J"1 is the Stern layer charge density, esu/cm2 ; (J"z is the Gouy-Chapman diffuse layer charge density, esu/cm2 ; v is the counterion valence; N 1 is the number of adsorption spots on 1 cm2 of surface; N is Avogadro's number; M is the molecular weight of the solvent; N / M is the number of available positions for counterions in the solution near the surface, no';cm3 ; n is the bulk-phase counterion concentration, no';cm3 ; is the specific chemical adsorption potential for counterions, ergs; I/; is the electrical potential at some point within the electric double layer (I/; = 1/;. at the o plane); 0 is the thickness of the Stern layer; and r is the distance from the center of a spherical particle. Stern (6) originally used N / M, the number of water molecules per unit volume of water, as a measure of the available positions for counterions near the surface. However, the number of hydrated counterions that can be contained per unit volume was used in this study as more appropriate to the defined quantity. The value used for the hydrated JmlYnal of Colloid and Interface Science, Vol. 49, No.3,

sodium radius was 4.1 X 10-8 em (19). The counterions that do not form compounds with some degree of insolubility with the coions or the surface, and are electrostatically adsorbed tend to become specifically adsorbed only when the surface is highly oppositely charged (7). Therefore, = 0 is probably a reasonable assumption for sodium counterions at a hydroxylated surface. The values for (dl/;/dr). plane were determined from the numerical solution of the Poisson-Boltzmann equation for spherical particles as published by Loeb et al. (20). The calculation of 1/;. employed a trial-anderror solution of (d1/;/dr).plane and the corresponding 1/;. that caused (J"1 plus (J"2 to equal the experimentally determined (J" (21). Surface potential, 1/;0, was calculated from the Nernst equation (10, p. 47) and knowledge of the concentration of the charge determining ion which is hydroxyl in this case and so may be expressed as pH giving the following equation: 1/;0 = (kT/e)(pH o - pH),

[7J

where e is the elementary charge and pH o is the value at the point of zero charge. Since the ionic strength is the same at the two pH's, the activity coefficients will cancel out.

THE DLVO

THEORY

The total potential energy of interaction, V T, between two colliding particles is equal to the sum of the repulsive potential energy, V R, and the attractive potential energy, VA:

[8J This theory predicts the functional form of these potential energies. For spherical particles and Ka > 3, the appropriate expression for determining V Risgiven byEq. [9J (11, p. 258): VR

= (aN), G(y, KH 0),

[9J

where y = vey;o/kT and H 0 is the distance between particle surface. The 1/;0 should probably be replaced by 1/;. or r because the potential drop across the diffuse layer is conceivably much more important than 1/;0 in double-layer

December 1974

349

ELECTROKINETIC POTENTIALS

interaction (10, p. 131; 11, p. 263). However, the correct potential to use in calculating V R has not been satisfactorily determined. The values of the function, G (y, KH o), are tabulated in Verwey and Overbeek (10, p. 141). For computer calculations, this table was put in the form of polynomial equations of y for specific values of KHo (21). Equation [9J is based on a mathematical model introduced by Derjaguin (22), who divided the interacting spheres into concentric rings making the interaction like that between infinitely large plates. Hamaker's expression for the attractive potential energy between approaching spherical particles of equal radius, a, at distance H 0 + 2a between their centers (10, p. 160) is

AG--+-+In(S2_ - )J ,

VA =-6

2

2-4

4

2

S2

S2

[lOJ

+

where S = (H o 2a)ja. The A iscaUed the London force constant or the Hamaker constant, which must be approximately equal to 10-12 erg to satisfy coagulation theory (10, p. 104). This constant is often estimated by selecting a value that forces the calculated function of V T vs H 0 to have no potential energy barrier at the experimental conditions for the onset of rapid coagulation. Some independent methods have been used to determine A, and they are summarized by Napper (23) and Ottewill and Shaw (24) and criticized by Padday (25). Recently, Fowkes (26) developed a method which requires surface tension and heats of immersion data, and for anatase he lists A = 3.5 X 10-13 erg. However, Bhatnagar and Williams (4) found that A must equal .5 X 10-12 erg for aqueous anatase dispersions in order for their data to fit the DVLO stability theory. The appropriate value of A is discussed in more detail later. EXPERIMENTAL

free of special surface treatments, was used. Preparation of the Ti0 2 dispersions consisted of the following steps: (1) Dispersing in distilled water at 73% solids with a Hamilton Beach Malt Mixer (Model No. 25) and storing at 52% solids. (2) Diluting to 7.5% solids, allowing settling for 24 hr, and siphoning off the usable fraction which had particles less than 1.0 JLm in diameter. (3) Dialyzing against 0.01 N NaOH sol.utions to replace the original polyphosphate dISpersing agent with hydroxyl ions. (4) Preparing the final sols at the desired pH and 1: 1 electrolyte concentration by the proper additions of Hel and NaCl. To minimize variation between sols, a large batch of the pigment was originally dispersed and diluted to 52% solids. This master sol was stored on a revolving platform and gently rotated at 15 rpm. The final dispersions tested had a solids content of 6.5 g of Ti0 2 j100 ml of sol. This solids concentration was sufficient for accurately measuring adsorbed hydroxyl ions and electrophoretic mass-transport, yet low enough for preventing double-layer overlap (27). The tested dispersions also had a pH adjusted between 6 and 12 and a 0.01, 0.002, or 0.0004 M sodium ion concentration. The sodium or counterion concentration (as NaCl NaOH) was held constant, but the chloride and hydroxyl content varied as a result of the added acid. The effect of varying the chloride ion concentration was insignificant, because chloride ions are not specifically adsorbed on Ti0 2 for the conditions used in this study (28, 29) and this was confirmed using the Volhard titration method for chloride concentration of the supernatants. The pH was changed to vary surface charge density and thus provide a range of double-layer potential values.

+

MATERIALS

A water-dispersible pigment grade of the anatase form of titanium dioxide, RG-grade manufactured by The Glidden Company and

CHARACTERIZATION OF THE

Ti0 2

The particle size distribution of the Ti0 2 was evaluated by two methods: (1) actual

Journal of Colloid and inlerface Science, Vol. 49, No.3. December 1974

350

WEBB, BHATNAGAR AI'lD WILLIAMS

measurement with a mierocomparator of single particle diameters from electron micrographs and (2) centrifugal sedimentation analysis of dispersions according to Martin (30). Electron micrographs showed that the particles were essentially spherical (21). The measurement of individual particles revealed that about 90% (by weight) were between 0.1 and 0.2 ,urn in diameter. The distribution for the suspensions was not quite as narrow; around 70% remained dispersed as single particles. The remaining particles in suspension were small aggregates having an effective diameter usually less than 0.4 ,urn. The mean-surface-area average diameter was 0.1226 ,urn for the individual particle measurements and 0.183 /lm for the dispersions. The average particle size for the master sol increased only about 0.5% during 18 rna of storage on the rotator. The specific surface area of the Ti0 2 was determined by the B.E.T. method (Ar and N 2 adsorption), and the values were 7.1 to 9.8 m 2jg, respectively. An area of 22.7 m 2jg was obtained with the Ross and Olivier gas (Ar) adsorption method (31). Similar differences have been documented by Barber (32) who attributes the lower B.E.T. areas to a heterogeneous energy site distribution on Ti0 2 • He has shown that the Ross and Olivier method gives a more reasonable value for the true area of such a surface. The geometric specific surface area, which was calculated from electron micrograph measurements, was 12.73 m 2jg. Pits and indentations in the particle surfaces, as observed in their electron micrographs, could account for the 80% increase of the Ross and Olivier area over the geometric area. However, according to a refinement of the electrie-double-layer model on metallic oxides as elaborated by Berube and de Bruyn (28,33), the surface disparities may actually be somewhat smoothed out by the double-layer development within the "surface region" of a highly ordered, hydrogen bonded, water atmosphere. For this reason, the geometric area was used to determine the surface charge per unit area (21). The effects of the surface area selection on the final results are discussed later.

GENERAL PROCEDURES

The water used in the work was purified by doubly distilling and then filtering through a Millipore filter (0.45 /lm). Using the Millipore filter not only removed minute particles, but the strong vacuum required also removed dissolved gases. For the suspensions with the lowest ionic concentration, triply distilled water was prepared and filtered as before. The specific conductance was usually less than 1 X 10-6 Q-l em-I. All NaOH solutions and basic Ti0 2 sols were prepared and handled in a COdree or nitrogen atmosphere. The sample rotator, on which the Ti0 2 dispersions were continually agitated during dialysis and storage, was enclosed in an essentially CO 2-free atmosphere to minimize CO 2 diffusion through the polyethylene and polypropylene bottles containing the samples. The molybdenum blue colorimetric test for phosphate (34) was used to follow the removal of the original polyphosphate dispersing agent. Dialysis was continued until phosphate could not be detected in the dialyzate (less than 10-5 M Na 2HP0 4). The pH measurements were taken in a nitrogen atmosphere on a Corning Research pH Meter, Model 12, with an accuracy of ±0.01 pH unit. The pH of a suspension was always determined on its supernatant, which was decanted after centrifugation in a manner that prevented contact with air. Accurate knowledge of sodium ion concentrations was necessary in order to determine hydroxyl ion adsorption and calculate surface charge density. The concentrations were determined with flame photometry on the Beckman Model DU Spectrophotometer. An oxygenhydrogen flame was employed, and maximum intensity was at a wavelength of 587 nm. The 95% confidence limits for five measurements were ±0.5% for Na+ concentrations near 0.002 and 0.0004 M and ± 1% for concentrations around 0.01 M. Centrifugation at 6500 rpm for 10 min was used to remove the dispersed pigment so that the sodium ion concentration could be determined on the supernatant. Trace

Joumal of Colloid anti Interface Science, Vol. 49. No.3, December 1974

351

ELECTROKINETIC POTENTIALS

amounts of Ti0 2 in the decanted solution had no effect on the measured sodium concentration. ELECTROPHORETIC MOBILITY

The Electrophoretic Mass-Transport Analyzer, Model MIC 1201, manufactured by Numinco was used to determine electrophoretic mobility. The principles, basic operations, and diagrams of mass-transport cells have been presented by Sennett and Olivier (1) and Ross and Long (2). SURFACE CHARGE DENSITY

Surface charge density was calculated from the amount of hydroxyl ion adsorption, determined by an indirect method like that used by Holtzman (35) and Bhatnagar (4). This method is based on the fact that the change in the sodium ion concentration as the pH is reduced equals the change in hydroxyl ion content of the surface. The difference between the change at a given pH and that for the sol at the isoelectric point (LP.) is the hydroxyl ion adsorption at that pH. Thus, the adsorption density p of hydroxyl ions at a given pH is p

= (f1[Na+JI.p. - f1[Na+JgivenpH)/SC, [l1J

where S is the specific surface area and C is the solids concentration. In this work S was equal to 1.273 X 105 cm 2/g, and C was equal to 66.1 g Ti0 2/1iter of suspending fluid. Before Eq. [11J could be used, each measured f1 [Na+J had to be adjusted to an equivalent f1 [Na+J at a constant bulk-phase sodium ion concentration [see Ref. (21) for details].

was used as the stability indicator. However, for the rapidly coagulating sols the rate was much greater, but decreased with time. For these cases, the initial sedimentation rate was used as a measure of relative stability. The average particle size for a given sol was determined from its sedimentation rate by a graphical interpolation from a plot of sedimentation rate vs mean-surface-area particle radius. The latter parameter was determined by centrifugation sedimentation analysis (30) of selected sols. The values of particle radius a and Ka for specific sols are tabulated in Ref. (21). RESULTS AND DISCUSSION ELECTROPHORETIC MOBILITY AND ZETA POTENTIAL

Results of electrophoretic mobility determinations are shown in Fig. 2. Reproducible mobility values could not be obtained below pH 8 because the samples coagulated and disrupted mass transport. The 95% confidence limits for the average of 3 to 5 determinations were ±0.1 to ±0.2 X 10-4 cm seel/V cm-l. All the values were around 4 X 10-4 em seel/V cm-1 even though the electrolyte concentration and charge density varied considerably. This apparent contradiction of the effect of salt concentration on mobility resulted because of the pronounced retardation and relaxation effects for the Ka region (between 4 and 50) of the systems studied. Because of retardation and relaxation effects, Eq. [lJ was unsatisfactory for calculating zeta

RELATIVE COLLOIDAL STABILITY AND AVERAGE PARTICLE SIZE

Using an automatic analytical balance as a sedimentation balance, the relative colloidal stability of each Ti0 2 sol was evaluated by measuring the rate of weight change caused by the pigment sedimentation into a pan held within the sol. For stable suspensions, the amount of settling was linear with time for at least several hours, and the sedimentation rate

pH

FIG. 2. Electrophoretic mobility vs pH of sol supernatant at various counterion concentrations.

Journal of Colloid and Interface Science. Vol. 49. No.3, December 1974

352

WEBB, BHATNAGAR AND WILLIAMS

80

o [No +] - o.OIM a [N a +] =0.002M • [No'] =0.0004M 70

~

.j I

60

50

7

8

9

10

pH

FIG. 3. Henry's zeta potential vs pH[for Ti02 suspensions.

potential. The results of zeta potential calculations using Henry's correction (14) (Eq. [3J) are shown in Fig. 3. In comparison to general double-layer concepts and Li and de Bruyn's (5) findings on Si0 2, the results indicated that Henry's method was inadequate for two reasons: (1) SH did not increase with decreasing electrolyte concentration, and (2) SH did not decrease with decreasing pH (i.e., surface charge density). The results using the W.L.O. correction (17) are shown in Fig. 4. The variation of SW.L.O. with pH is quite similar to that predicted by Lyklema and Overbeek (8), thus providing the best indirect evidence to date on the validity of the W.L.O. theoretical treatment of relaxation and relaxation effects. The maximum electrophoretic mobility determined for the 0.0004 M Na+ concentration sols was very close to the maximum predicted in Fig. 1 by the W.L.O. theory, e.g., E was 3.20 and Ka

was 6.33. The measured mobilities were above the maximum predicted by the equations of Overbeek (16) and Booth (17). Therefore, the W.L.O. method was further supported by the results of this study. The range of some of the values are rather large because a small change in mobility makes a big difference in the interpolated y and the resulting SW.L.O (see Fig. 1). Only the point at pH 11.13 for the 0.01 M data does not fit reported zeta-potential relationships (5). This sharp increase in S was also found by Bhatnagar and Williams (4). It is felt that this increase may be the result of the extremely high surface charge densities that are approached in this reglOn. CHARGE DENSITY AND POTENTIAL

AT THE

O.H.P.

Figure 5 presents a sample of sodium ion concentration change data that were collected for determining surface charge density. As acid was added, the bulk-phase sodium ion concentration increased in direct proportion to the number of surface hydroxyl groups neutralized or desvrbed. There would be no further increase at the isoelectric point. An examination of all the data like that in Fig. 5 indicated that this point is between pH 5.5 and 6. Berube and de Bruyn (28) reported that the isoelectric point for anatase was at pH 5.9; therefore, pH 5.9 was used as the basis for calculating surface charge density by Eq.

[11].

1.0

INITIAL [No-t]

= 0.832

mM

a L--4~~--}--~~~8:---~--;;;IO--""b.L---;'2!;­ pH

pH

FIG. 4. W.L.O.--corrected zeta potential vs pH for Ti02 suspensions.

FIG. 5. Increase in bulk phase Na+ concentration as Hel is added (sol JI, solids = 6.50 g!100 cc).

Journal of Colloid and Interface Science. Vol. 49. No.3, December 1974

353

ELECTROKINETIC POTENTIALS RELATIONSHIP BETWEEN DOUBLE

20

N

E

~

a

O.OIM

'i.

>-f-

iii

zw

a

~

a:

LAYER POTENTIALS

[Na+]

10

~ u

w

~a:

:::>

'" I

pH

FIG. 6. Surface charge density on anatase as a function of pH at various counterion concentrations.

The results of surface charge density determinations are shown in Fig. 6, and, as expected, charge density increased with increasing pH. However, charge density also increased with increasing counterion concentration. These findings agree with those reported by Berube and de Bruyn (28, 33) on TiO z, by Tadros and Lyklema (36) on SiO z, and Blok and de Bruyn (37) on ZnO. To explain the influence of counterion concentration and different counterions on surface charge density, Berube and de Bruyn (28) proposed the concept of structured or "relatively ordered liquid water" at the surface of metallic oxides. Tadros and LykJema (36) used the term, "porous gel structure," to account for the same phenomena. A greater counterion concentration tends to disrupt ordered regions and allows greater surface adsorption. The results of '1h determinations are given in Fig. 7 as a function of surface charge density. These data show that at a given charge density,1/;o increases as ionic concentration decreases, which is consistent with doublelayer concepts. In calculating 1/;0' specific adsorption potential was assumed equal to zero, and a hydrated Na+ radius of 4.1 X 10-8 cm (19) was used. The changes in 1/;/j values were usually less than 5% when adsorption potential and hydrated radius were varied within reasonable limits up to 2 kT and 0.0556 molejcm3, respectively. The changes were negligible when 1/;0 was less than 100 mV.

The results of electric-double-layer potential determinations are summarized in Table 1. The values of r were interpolated from the smoothed data in Fig. 4. At high salt content and pH, IT and 1/;0 are extremely high compared to those on silver iodide or mercury (36). Tadros and Lyklema (36) also found high IT and 1/;0 values in conjunction with medium 1/;0 and!: values for Si0 2• The value of ITljlT indicates the relative magnitude of the Stern layer. The following equation has been given (10, p. 43) as describing the potential gradient in the Stern layer: IT

[12J

= (E'j47/"o)(1/;0 -1/;.),

where E' is the dielectric constant of the medium in the layer and 0 is the thickness of the layer. The results of applying the data in Table I to this equation are presented in Fig. 8. The linear functions indicated are statistically of high significance. The abrupt deviations from linearity may be due in part to the zero point of charge being at pH's a little higher than the value of 5.9 used in the calculations. The maximum change possible is still within the measured limits as identified by the data as given in Fig. 5. The finite values of the intercept, which are larger than the experimental uncertainties of IT, may be a result of the Stern 200

~

,:'

100

I

o L..-..J---'------l----l._L.....--'---L.--J....---'----'

o

~

~

-SURFACE CHARGE DENSITY, !,coul./cm 2

FIG. 7. if;; vs surface charge density for TiO z sols at various counterion concentrations.

Journal of Colloid and Interface Science. Vol. 49. No.3. December 1974

354

WEBB, BHATNAGAR AND WILLIAMS TABLE I ELECTRIC-DOUBLE-LAYER POTENTIALS.

[Na+]. M

0.01

0.002

0.0004

pH

5.9 6.5 7.0 7.5 8.0 8.5 9.0 9.2 9.5 10.0 10.5 11.0 11.5 5.9 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 5.9 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

fT.I'Cj

em'

10.2

12.4 15.0 17.7 20.6

o

0.19 0.37 0.95

1.74

3.13 5.19 6.91 8.74 10.81

o

0.004

0.014 0.034 0.068 0.116 0.175 0.202 0.234 0.283 0.330 0.376 0.422 0.0007 0.0009 0.0044

0.0139 0.0413 0.0962 0.147 0.195 0.249

0.09 0.17

0.0001 0.0002

0.29

0.0004

0.57

0.0014 0.0074 0.0154 0.0308

1.31

1.89 2.58

-fl, mY

mY

o

0 23 53 77 95 110 123 127 132 138

0

mY

o

0.6 1.5 2.7 4.1 5.8 7.9 8.8

-to

-f.,

35 65 94 124 153 183 195 212 242 271 301 329

o

35 65 94 124 153 183 212 242 271

o

35 65 94 124. 153 183 212

144

149 154 0 17 35 65 94 123 146 158 167 175 181

0 15 29 46 76 119 138 156 171

64 66 67 67 67 67 71 0

59 73 82 86 88 89 90

0

61

84 100 104 105

model equations employed. Verwey and Overbeek (9, p. 44) indicate a value for E' /21ro of 107 em. The values found here are 10.0, 8.5, and 5.2 X 107 em for the salt concentrations 0.01, 0.002, and 0.0004 M, respectively. Assuming that 0 = 5 X 10-8 em, the e' values are then 65, 55, and 34. These values appear to be reasonable for partially oriented water and they decrease with the extent of increase of oriented water layer as discussed below. The potential distribution in the diffuse layer was determined from the complete solu-

tion of the Poisson-Boltzmann equation according to Loeb et al. (20). From the potential distributions, the location of the 5 plane was determined and its distance from the Y;o plane was calculated. The results are shown in Fig. 9. The distance between Y;o and 5 planes is essentially independent of pH and Y;o at a given counterion concentration. The wave pattern of the 0.0004 M data is believed to be the result of systematic error introduced with the use of smoothed 5W.L.O. values (Fig. 4). A change of one Angstrom unit (A) in the distance between the 5 and Y;o planes roughly corresponds to an error of 1 mV for the 0.0004 M data. The ±3-A change is therefore well within the ± 7-mV range for some of the 5W.L.O. values, thus the wave pattern is not significantly different from a straight line. From double-layer concepts and the results of Li and de Bruyn (5) on the relationship between 5 and Y;o, one would expect the distance between the hypothetical and actual shear plane to decrease as Y;o decreased. However, for Y;o > 100 mV the distance is uniform for a given salt concentration. This uniformity supports the concept of water structuring near the surface, which could physically control the location of the hydrodynamic plane of shear even though electrostatic conditions change drastically. Surface roughness could also have an effect; however, the distance decreases with increasing ionic strength, which IS consistent with the structuring concept since increased electrolyte concentration (except for certain ionic species) generally tends to disrupt the structuring of water (38). The distances shown in Fig. 8-12, 20, and 25 A-are, therefore, estimates of water structure thickness at the Ti0 2 interface, and the values are the first determined from doublelayer theory. The thicknesses correspond to 5 to 10 molecular water layers. This thickness range is consistent with water structure configuration hypothesized by Berube and de Bruyn (28), although others have predicted thicknesses corresponding to hundreds of molecular diameters (21).

Journal of Colloid and Interface Science, Vol. 49. No.3, December 1974

355

ELECTROKINETIC POTENTIALS 22

Cone.

18

0

0.01

0

0.002

11

0.0004

Slope

Intercept

0./11

1.02

M. M. M.

0.094

1.73

0.058

-0.67

14 N

~

-; 0

~

10

b 6

0

80

0 (+0

100

120

140

160

180

millivolts

-tal,

FIG. 8. The surface charge density vs. the difference between >/;0 and >/;,.

Fig. 10. The agreement between this research work and Li and de Bruyn's results is remarkably good and believed to be very meaningful, especially since their r values were obtained on another material (Si0 2) by a different method (streaming potential measurements). A comparison of the two curves in Fig. 11 suggests that certain assumptions in the theoretical treatment are invalid. In their treatment, Lyklema and Overbeek (8) analyzed the effect of viscosity variation on the relationship between 1/;0 and observed zeta potential (calculated from Eq. [1J) for three cases:

The relationships between rand 1/;0 at various ionic concentrations are presented in Fig. 10, and the theoretical predictions of Lyklema and Overbeek (8) are also shown. These authors accounted for viscosity variation in calculating r from electrophoretic mobility. Li and de Bruyn's experimental results were similar to those found in this work. Both sets of empirical data followed the theoretical results only in the sense that r became independent of 1/;0 at high 1/;0 values. The maximum or limiting values of zeta potential, rlimit, for the e},,-perimental results are compared to the theoretical predictions in

"

" " "

0

n

0.oo04M

0

"

0 0

" " 0

0.0021;1

0

[No+J

~

0

"

" 0

0

0

a a

O.OIM

FIG. 9. Distance between the O.H.P. and the plane of shear vs. ,po for various Na+ concentrations. Journal of Colloid and Interface Science, Vol. 49. No.3. December 1974

356

WEBB, BHATNAGAR AND WILLIAMS

the following expression:

200

'fj

~

100

/ o "-...J--'---'----'-_'----.J..--'--'---'----'-_~ o 100; 200 -"a,mv

FIG. 10. Relationships between rand ,p. at various ionic strengths (- - - theoretical predictions of Lyklema and Overbeek (7); - - experimental results).

(i) The electric field causes the viscosity to vary from zero fluidity at the ift. plane to bulk fluidity at the r. plane, therefore, the plane of shear is actually a slipping layer of finite thickness. (ii) The surface influences viscosity by immobilizing liquid at the surface to form a structured layer with the ift. plane within the structured region and r. plane at the outer edge. (iii) Viscosity variation affects the description and location of the plane of shear by a combination of Cases (i) and (ii), i.e., there is a structured region and a finite slipping layer and the r. plane is at the outer edge of the slipping layer. For Case (i), Lyklema and Overbeek assumed that the strong electric field within the double layer influenced viscosity according to 200

'-

.................THEORETICAL

'-

o o

'-

RESULTS OF THIS STUDY

10-3

tcr- 2

CONCENTRATION, n'lol.s/lit..

FIG. 11. Limiting values of zeta potential.

[13J

where 'fjo is the bulk-phase viscosity, f is the viscoelectric constant, diftj dx is the potential gradient, and x is the distance from the surface. Using this correction, they solved Eq. [1] for electrophoretic mobility in terms of the true zeta potential [ift.for Case (i)]. Their use of the flat-plate approximation for the potential gradient introduces less than 2% error when Ka > 4 (20). Lyklema and Overbeek estimated from values on organic liquids for f for water equals 1.02 X 10-11 cm2jV2 and found that Eq. [1J had to be corrected only for viscosity variation and not for changes in permittivity. The results of the relationship between rand ifi. for Case (i) were shown in Fig. 10. The r limit in Fig. 11 was also obtained by applying Case (i) with the above f value and utilizing the following expression, which is valid for c < 0.035 M (9): lim

!{to"''''

t.= (kTjeB) In [(1+B)/(1-B)J,

[14J

where B = [1 - f(32'lrcNkT/1000f)Jt. To test Case (i), the viscoelectric constant was determined for the experimentally observed tlimit values with Eq. [14]. The results are tabulated in Table II. It is obvious from a comparison of the f values that, either Case (i) is invalid, or the viscoelectric constant is not really a constant as the salt concentration changes. The effect of salt concentration on f is unknown (7); however, assuming Case (ii) or (iii) instead of Case (i) is much more realistic for the anatase-water interface based on the evidence for the water-structure concept indicated in Fig. 9. The limiting zeta potential value for Case (ii) is dependent on the thickness of the immobilized layer, D, and is described by the expression (8) exp (etlimit/2kT) =

RESULTS OF Ll AND DE BRUYN

10-4

= 710[1 + f(dift/dx)2],

1 + exp (-ill) 1 - exp (-KD)

[15J

10-'

Values of D were obtained from Fig. 9, and each theoretical tlimit was calculated with Eq.

Journal of Colloid and Interface Science, Vol. 49, No.3, December 1974

357

ELECTROKINETIC POTENTL<\LS TABLE

II

COMPARISON BETWEEN THEORETICAL PREDICTIONS AND EXPERIMENTAL RESULTS FOR LIMITING ZETA POTENTIAL VALUES

Case (j) 0 structuring and slipping layer of finite thickness Conen, M

-!'"Iimit. mV (Extrapolated, Fig. 11)

Theore- Expori· mental tical

0.035 0.01 0.001 0.0001

52 75 128 186

50 73 102 120

f X 10", cm'/V' Original Estimatea.

Calcnlated b

1.2 1.2

1.02 1.02 1.02 1.02

3.2 14.3

Case (ii) Structuring with discrete slipping plane Concn, M

0.01 0.002 0.0004

-flimit.

mV

Theore- Experitical c mental

84 99 127

73 94 110

D, A Experimental d

12 20 25

Calculated'

14 22 36

• Value of f used by Lyklema and Overbeek (8) to calculate the theoretical Ilimit for Case (i). b Determined f value which causes theoretical Ilimit to equal experimentalllimil. c Calculated from Eq. [14J with the e},:perimentally observed D. d Structure thickness from Fig. 10. e Calculated D value which causes theoretical llimit to equal experimental Ilim;t.

[15]. The results are compared with the experimentally observed limiting zeta potentials in Table II, Case (ii). The theoretical .IUndt values are higher, but in much better agreement with the experimental determinations than in Case CO, although only the 0.002 M data are statistically identical. When the experimental .\limit was used to calculate D for Case (ii), the difference between the empirical and calculated D were significantly different only for the 0.0004 M data. The difference between the .IUmit values listed for Case (ii) may have occurred because viscosity variation at the edge of the structured layer should also have been considered. This means that Case (iii) should be assumed to describe hydrodynamic slipping plane phe-

nomena. The electric field is still sufficiently strong [t#/d~ 1.5 to 3.0 X 103 V/cm (21)J outside the surface immobilized layer to affect viscosity, provided the viscoelectric constant, j, is greater than about 0.4 X 10-11 cm2jV'2. In the above considerations of the viscoelectric effect, the value of j was assumed to be around 10-11 cm2jV2. However, this estimate of the viscoelectric constant does not agree with that found by Hunter (9) and indicated by Stigter (39). Hunter found that j = 1 to 2 X 10-13 cm2jV2 when he applied the Lyklema-Overbeek treatment to Haydon's (40) data. If the viscoelectric constant is really that low, then the effect of permittivity variation becomes significant (9). Hunter analyzed the effect of changing permittivity on the calculation of zeta potential, and he found that correcting for both varying permittivity and viscosity (for j:::::10-13) gives essentially the same results as correcting only for viscosity variation when j:::::10-11 cm2jV2. TEST

OF

THE DLVO THEORY

The results of sedimentation experiments are shown in Fig. 12, and the pH values at which 1000

~

,

\

I

Na+ CONCN.! M o

o

0.1 0.01

0.002

t.

0.0004

•

12

FIG. 12. Sedimentation rate vs pH of sol supernatant.

Journal of Colloid and Interface Scielt'" Vol. 49, No.3, December 1974

358

WEBB, BHATNAGAR AND WILLIAMS TABLE

III

CONDITIONS AT THE ONSET OF RAPID COAGULATION AND ESTIMATED VALUES OF THE HAMAKER CONSTANT

pH

-"'0

0.01 0.01 0.01

9.2 9.2 9.2

195

0.002 0.002 0.002

8.5 8.5 8.5

153

0.0004 0.0004 0.0004

8.0 8.0 8.0

124

0.01 0.002 0.0004

9.2 8.5 8.0

Na+ Conen, M

Potential, mV

-"'.

-,

Assumed Radius. A-

A, erg

67

613" 613 613

7.0 X 10-12 5.5 X 10-12 2.5 X 10-12

73

613 613 613

2.0 X 10-11 1.7 X 10-11 0.9 X 10-11

61

613 613 613

7.8 X 10- 11 4.5 X 10-11 3.0 X 10-11

1100b 1100 1100

4.9 X 10-12 1.3 X 10-11 2.5 X 10-11

127

123

76 127 123 76

" Mean-surface-area radius determined from electron micrograph measurements. b Apparent average radius near coagulation conditions.

slowly coagulating (or slowly settling) Ti0 2 sols became rapidly coagulating suspensions are readily apparent. The pH values at the onset of rapid coagulation are listed in Table III, and the corresponding double-layer potentials at those conditions are also given. Each of the potentials was used to calculate an A value for the DLVO theory potential energy functions that would make V T = O.

The calculated values of A given in Table III are very large and are not constant, which suggests that the DLVO theory is inadequate for describing the colloidal stability of the Ti0 2 dispersions studied. Hamaker constants for specific solids are usually around 10-12 erg [(10), p. 104, (41)J, and the effective A is reduced when the material is immersed in a liquid. Napper (23) and VoId (42) point out

+60

+40

+20

61

'0

g

i

0

0.002M

-20

-40

-60 0

200

400 Ho, ANGSTROMS

600

FIG. 13. Total potential energy of interaction vs. distance between particle surfaces at pH 9.5. Journal of Colloid and Interface Science, Vol. 49, No.3, December 1974

ELECTROKINETIC POTENTIALS

that the presence of an adsorbed layer reduces the effective Hamaker constant in a manner similar to immersing in a liquid. However, the calculated Hamaker constants increase rather than decrease with increasing adsorbed-layer thickness. One of the possible consequences when the value of A is too large is illustrated in Fig. 13, which shows sample potential energy of interaction curves [see Ref. (21) for the full range of curves]. For the 0.01 M curve, there is a calculated, strong attractive potential energy between the particles even at fairly extensive distances. The existence of this secondary minimum suggests that the particles would be held in a very loose agglomerant and would be unstable because of the reduced kinetic energy of the effectively larger particles. However, the suspensions above pH 9.5 were found to be very stable, so the use of large A values in the DLVO theory was inconsistent with experimental observations. Table III also shows the effect of the potential used and the particle size on the calculated Hamaker constant. However, employing the possible range of values still did not eliminate the problem of the large and varying A values. In general, changes in the assumed or selected values of specific surface area, hydrated counterion radius, and specific adsorption potential affected only slightly f/j determinations and ultimately the calculated Hamaker constants. The surface area and hydrated radius that were considered the best to use also resulted in the lowest f/j values and thus the lowest A values. If the specific adsorption potential c/J had been assumed greater than zero, then the calculated A values would have been more reasonable. However, values of c/J much greater than 5 kT would have been necessary to cause A to equal 10-12 erg, and the calculated f/j values would have been lower than the S values. When V R is corrected for the retardation effect on the London attraction, as described by Overbeek (11), its value is reduced. This correction was applied to the data for 0.002 and 0.0004 M and the resulting Hamaker

359

constants were approximately double those obtained without this retardation correction. Errors in the equations for V R and V A are automatically compensated for in the selection of an A value that causes the DLVO theory to agree with conditions at the onset of rapid coagulation. The Van der Waals' attractive forces have been generally substantiated by direct measurement (43), but the repulsive forces are more difficult to quantify. A recent publication by Sanfeld et al. (44) indicates that the DLVO repulsive energy expression for spherical particles gives results that are too high, especially at salt contents greater than 0.05 M. However, the difference is less than 3% when c = 0.01 M of 1: 1 electrolyte and f ~ 100 mV, and the difference is negligible when c S 0.002 M. Changes in V R of 3% are not sufficient to compensate for the large discrepancy in the calculated A values. Possibly, one of the major sources of error is the mathematical model from which V R was derived. Both Eq. [9J and the equation of Sanfeld et al. (44) were determined using Derjaguin's (23) model of concentric rings. The integration assumes that the range of repulsion is small compared to particle dimensions, and Overbeek (11) points out that the method is applicable for calculating V R if Ka> 3. In this study to test the DLVO theory, Ka values ranged from 4 to 20. It is difficult to estimate just how accurate Eq. [9J is, although accuracy decreases with decreasing Ka. In this work, larger discrepancies in A did occur at the lower Ka conditions. The poor agreement of the DLVO theory with the experimental results suggests that a more complete analysis is needed for the interaction of double layers about spherical particles when double-layer thickness is significant compared to particle size. Another major possible source of error in the DL va theory that arises from this and other recent studies is the role of interfacial water structure on the interaction of repulsive and attractive energies. According to Napper (23), adsorbed layers should tend to reduce

Journal of Colloid and Interface Science, Vol. 49. No.3, December 1974

360

WEBB, BHATNAGAR AND WILLIAMS

the effective Hamaker constant and thus the attraction. Whether structuring contributes to a mechanical repulsive energy barrier or affects the Poisson-Boltzmann distribution to a large extent remains to be solved. CONCLUSIONS

For colloidal systems in the critical Ka region between 0.1 and 100, the Wiersema-LoebOverbeek (18) correction method of calculating zeta potential from electrophoretic mobility gives more accurate results than Henry's (15) method. The distance between the hydrodynamic slipping plane (r plane) and O.H.P. (1/;a plane) was determined from potential distributions and found to be constant for a given ionic strength over the range of pH and surface charge densities examined. The relationships between rand 1/;5 were in best agreement with the Lyklema-Overbeek (8) predictions when the electric double layer around each particle was described as follows: 1/;5 is located within an immobile liquid region at the surface; and is measured at the outer edge of the structured liquid, where the slipping layer may be of finite thickneSs rather than a discrete plane because of the effect of the electric field on viscosity. Dispersion stability was not satisfactorily explained by any of the double-layer potentials, 1/;0, 1/;5, or r. Application of the theory was inadequate based on the large and varying Hamaker constants required. A more exact mathematical description of the interaction of double layers about spherical particles is apparently needed, and/or the effect of water structure on double-layer interaction and the Poisson-Boltzmann equation needs clarification.

r

ACKNOWLEDGMENTS The authors express their appreciation to other members of the Ph.D. Thesis Advisory Committee, Dr. C. L. Garey and Dr. T. M. Grace, for their many helpful discussions during the course of this research, and to Mr. N. Chang and Mr. G. Brown for some of the computer programs used in the calculations. One of us (JTW)

acknowledges a graduate fellowship and another (PDB) a postdoctoral fellowship from The Institute of Paper Chemistry during the course of this work. REFERENCES 1. SENNETT, P. AND OLIVIER, l P., U. S. pat. 3,208,919(1965); Tappi 52, 153 (1969). 2. Ross, S. AND LONG, R. P., Ind. Eng. Chem. 61, 58 (1969). 3. RIDDICK, T. M., Tappi 47, 1nA (1964). 4. BHATNAGAR, P. D., AND WILLIAMS, D. G., to be pUblished. 5. LI, H. C., AND DE BRUYN, P. L., Surface Sci. 5, 203 (1966). 6. STERN, 0., Z. Elektrochem. 30, 508 (1924). 7. GRAHAME, D. C., Chem. Re:o. 41, 441 (1947). 8. LYKLEMA, l, AND OVERBEEK, J. TH. G., J. Colloid Sci. 12, 501 (1961). 9. HUNTER, R. J., J. Colloid Interf(]£e Sci. 22, 231 (1966). 10. VERWE):, E. l W., AND OVERBEEK, J. TH. G., "Theory of the Stability of Lyophobic Colloids," p. 205. Elsevier, Amsterdam, 1948. 11. OVERBEEK, l TH. G., in "Colloid Science" (Kruyt, Ed.), pp. 245-77. Elsevier, Amsterdam, 1952. 12. AHMED, S. M. AND MAKSIMOV, D., J. Colloid Interface Sci. 29, 97 (1969). 13. AHMED, S. M., J. Phys. Chem. 73, 3546 (1969). 14. VON SMOLUCHOWSKI, M., Bull. Acad. Sci. Cracovie 182 (1903). 15. HENRY, D. C., Proc. Roy. Soc. London Ser. A 133, 106 (1931). 16. OVERBEEK, l TH. G., Advan. Colloid Sci. 3, 97 (1950). 17. BOOTH, F., Proc. Roy. Soc., London Ser. A 203, 514 (1950). 18. WIERSEMA, P. H., LoEB, A. L., AND OVERBEEK, l Ta. G., J. Cottmd Interfare Sci. 22, 78 (1966). 19. HARNED, H. S., AND OWEN, B. B., "The Physical Chemistry of Electrolytic Solutions," 3rd ed., pp. 396, 490, 510. Reinhold, New York, 1958. 20. LOEB, A. L., OVERBEEK, J. TH. G., AND WIERSEMA, P. H., "The Electrical Double-Layer Around a Spherical Colloidal Particle," p. 375. M.LT. Press, Cambridge, MA 1961. 21. WEBB, J. T., "An Investigation of Electric-DoubleLayer Concepts and Colloidal Stability of TiO. Dispersions," Doctor's Dissertation, p. 231. The Institute of Paper Chemistry, Appleton, WI, 1971. 22. DERJAGUIN, B., Kottoid Z. 69, 155 (1934). 23. NAPPER, D. H., Sci. Progr. (Oxford) 55, 91 (1967). 24. OTTEWILL, R. H., AND SHAW, IN., Discuss. Faraday Soc. 42, 154 (1966). 25. PADDAY, l F., Discuss. Faraday Soc. 42,164 (1966). 26. FOWKES, F. M., Ind. Eng. Chem. 56, 40 (1964).

Journal of Col/oid and Inter/a.. Science. Vol. 49. No.3. December 1974

ELECTROKINETIC POTENTIALS 27. LONG, R. P., "Electrophoresis in Concentrated Suspensions," Doctor's Dissertation, p. 120. Rensselaer Polytechnic Institute, Troy, N. Y., 1966. 28. BERUBE, Y. G. AND DE BRUYN, P. L., J. Colloid Interface Sci. 28, 92 (1968). 29. FLAIG-BAUMANN, R., HERMANN, M., AND BOEHM, H. P., Z. Anorg. Allg. Chem. 372, 296 (1970); C.A. 72, A83272. 30. MARTIN, S. W. "Symposium of New Methods for Particle Size Determination in the Subsieve Range," pp. 66-89. A.S.T.M., 1941. 31. Ross, S. AND OLIVIER, J. P., "On Physical Adsorption," p. 401. Interscience, New York, 1964. 32. BARBER, H. T., The determination of the energy site distribution of the surface of cellulose fibers by gas adsorption methods, Doctor's Dissertation, p. 105. The Institute of Paper Chemistry, Appleton, WI, 1969. 33. BERUBE, Y. G. AND DE BRUYN, P. L., J. Colloia Interface Sci. 27, 305 (1968). 34. VAN DEN HUL, H. ]., "The Specific Surface Area of

35.

36. 37. 38. 39. 40. 41.

42. 43. 44.

361

Silver Iodide Suspensions," Doctor's Dissertation, p. 75. University of Utrecht, 1966. HOLTZMAN, W., "The Application of the Verwey and Overbeek Theory to the Stability of Kaolinite-Water Systems," Doctor's Dissertation, p.160. The Institute of Paper Chemistry, ApIJleton, WI, 1959; J. Colloid Sci., 17, 363-82 (1962). TADROS, TH. F. AND LYKLEMA, J., Eleetroanal. Chem. 17, 267 (1968). BLOK, L. AND DE BRUYN, P. L., J. Colloid Interface Sci. 32, 533 (1970). FRANKS, F., Chem. Ind. (London) 1968,560 (1968). STIGTER, D., J. Phys. Chem. 68,3603 (1964). HAYDON, D. A., Proc. Roy. Soc., London Ser. A 258,319-28 (1960). "Colloid Stability in Aqueous and Nonaqueous Suspension," Discuss. Faraday Soc. 42, 7-322 (1966). VOLD, M. J., J. Colloid Sci. 16, 1-12 (1961). TABOR, D., J. Colloid Interface Sci. 31, 364 (1969). SANFELD, A., DEVILLEZ, E., AND TERLINCK, P., J. Colloid Inteljace Sci. 32, 33 (1970).

Journal of Colloid and Interface Science, Vol. 49, :-.ro. 3, December 1974