The double layer potential Øδ as a rate determining factor in the coagulation of electrocratic colloids

Colloids and Surfaces, 30 (1988) 295-305 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

295

The Double L a y e r P o t e n t i a l as a Rate D e t e r m i n i n g Factor in the Coagulation of Electrocratic Colloids* G. FRENS** and J.J.F.G. HEUTS

Philips Research Laboratories, 5600 JA Eindhoven (The Netherlands) (Received 11 March 1986; accepted in final form 20 September 1987)

ABSTRACT Measurements of double-layer potentials Ca in AgI hydrosols at equal coagulation rates reveal that O~ is low and inversely proportional to the counterion charge. It is also found to be proportional to the square root of the double-layer thickness so that AgI is seen to follow the Eilers-Korff criterion. AgI is a rather typical model system for electrocratic colloids. This implies that phenomena regarding the stability and the coagulation of electrocratic colloids such as the Schulze-Hardy rule and the linear log W-log c relationship could result from the effect of added electrolyte on ~ rather than on the range of the double-layer repulsion.

INTRODUCTION

Hydrosols like colloidal metals, oxides, AgI, As2S3 and many polymer latices as well are electrocratic colloids. Such colloids owe their stability to the repulsion of electrical double layers. This paper deals with the various modes in which the double-layer repulsion can affect the coagulation rate. It could create an energy barrier and prevent (a fraction of) the collisions [ 1-7]. Or, it may diminish the depth of the potential energy minimum at which aggregated particles are kept together. From a shallow minimum there is a chance for particles to escape, and this breaking away (repeptization) from an aggregate reduces the fraction of eventually effective Brownian collisions [ 8,9 ]. The repulsion between particles in an electrocratic colloid arises from the overlap of the diffuse charge layers which extend from the particles into the surrounding medium. These layers have two characteristic parameters: the potential difference 0~ across the diffuse layer and the Debye length l/K, which is a characteristic of the electrolyte medium in which the particles are embedded. The potential difference ~a across the diffuse layer plays a different role *Dedicated to the memory of Professor G.D. Parfitt. **Present address: Laboratory of Physical Chemistry, Technical University Delft, Julianalaan 136, 2628 BL Delft, The Netherlands.

0166-6622/88/$03.50

© 1988 Elsevier Science Publishers B.V.

296 in the different versions of the theory for coagulation rates. In the traditional DLVO theory [6,7] there is nothing very specific about ¢}5. It is taught that the addition of salt to a colloid causes coagulation because it reduces 1/K. An appealing feature of this classical model is that a theoretical derivation of the Schulze-Hardy (SH) rule can be based on it. This rule describes the experimental observation that electrocratic colloids are extremely sensitive to coagulants with polyvalent counterions, whereas the valency of the co-ions is hardly important for the coagulation power of a salt. The quantitative DLVO expression for the SH rule is that z6Cc is constant; z is the counterion valency and C~ the critical coagulation concentration. The shrinking range of the repulsion makes the energy barrier just disappear at Co. Above this concentration the coagulation is rapid and its rate is independent of the salt concentration. Actually, the DLVO derivation of the SH rule leads to the expression:

~4 C~ = L-A-~-z6

(1)

where L is some constant, 7 = tanh (ze~)5/4kT) and A is the Hamaker constant of van der Waals attraction. Assuming that ~5 is high (05 >> 150 mV), 7 approaches unity and Eqn (1) reduces to the SH rule. Therefore, the only condition for ~5 in the classical DLVO theory is, that it must be large in colloids which follow the SH rule. Since the rule is obeyed by practically all electrocratic sols (with the exception of some polymer latices) it must then be assumed that in all these colloids ¢5 > 150 mV at Co. However, such ~5 values are improbably high. Other experiments, such as those examining the effects of electrolytes on electrode kinetics [ 10 ] and the determination of electrophoretic mobilities [ 11,12 ], suggest that 25 < ~5 < 50 mV at salt concentrations of the order of Ca. At such low potentials, ~ zeOs/4kT. Now C~ becomes inversely proportional to z 2 instead of z 6. It is possible though to obtain the SH rule from considerations of a vanishing energy maximum when 05 is low, by postulating [13,14] (as a new independent condition) that z and 05 are inversely proportional at equal rates of coagulation. Typical for colloids with low ¢~5and weak van der Waals attraction ( such as latices) [15] would be the third version of the theory. It describes the rate of slow coagulation as a net balance of aggregation and (spontaneous) repeptization [ 8 ]. Aggregated particles are kept together in a potential well. However, if this is only a small energy minimum the particles have a chance to escape from the aggregate through their Brownian motion. Equal depths of the energy minimum would then correspond to equal probability of escape, i.e. to equal rates of slow coagulation. Equal depths for electrolytes with different z values correspond to equal values of ~5 at equal coagulation rates. In summary: three versions of the theory for coagulation rates in electrocratic colloids predict different behaviour of 05 in the same experimental set-

297

ting. In the classical DLVO concept ~ is high, and its precise value irrelevant, for different z and equal coagulation rates. Low double-layer potentials indicate the condition that z ~5 is constant at equal rates of coagulation in different salts. In systems where slow coagulation is a balance of aggregation and repeptization 05 is low, but constant for different z at equal coagulation rates. Experimental determination of ~ in different electrolytes should be a method to decide which model applies in a given system.

Experimental strategy We performed experiments to measure ~ in electrolytes with different z values at salt concentrations which produce equal coagulation rates in a given colloid. For this we measured electrokinetic zeta potentials, which are practically equal to ¢a [ 16 ], in AgI hydrosols with different salt concentrations, and also the coagulation rates at these concentrations. AgI is, of course, a well-known, well-documented [ 17 ] electrocratic colloid*. It does obey the SH rule [18]. It can be repeptized after coagulation with univalent counterions [ 19 ]. AgI is also very insoluble in water, so that an aged sol can be kept clean and does not change for relatively long periods of time. The rates of coagulation were determined in negative AgI sols, as a function of electrolyte concentrations in KNO3, Ba (NO3) 2 and La (NO3) 3 solutions. From these data log W-log C diagrams were constructed [ 17 ]. W is the retardation factor, measuring the fraction of Brownian collisions which remains ineffective in a coagulating sol. Subsequently, at salt concentrations which correspond to equal values of W in solutions of K +, Ba 2+ and La 3+, the electrophoretic mobilities of the AgI particles were determined; from the latter ¢~ can be computed [ 25 ]. For this the radius a of the particles must be known, so that corrections for electrophoretic retardation and time of relaxation effects can be applied in the interpretation of electrokinetic mobilities of particles in different electrolytes. EXPERIMENTAL PROCEDURE

Sol preparation and characterization Colloidal AgI was prepared by slowly adding 240 ml 0.04 M AgNO3, with stirring, at room temperature, to a solution of 250 ml 0.04 M KI. The fresh sol *In essence we have repeated the experiments which are described in the Ph.D. theses by Troelstra [ 12 ] and by Reerink (log W-log C curves ) [20 ] using modern techniques and in one colloid.The meticulous observations of these authors have remained valid through the years. W e found the same results in our experiments and have shared some dilemmas in speculations or interpretations of experimental data with these predecessors. It was not possible to use the experiments of these authors as the basis for our investigation because their reported data result from experiments with different time scales (see below, Fig. 3 ) and with different colloids of unknown particle size.

298 a(nm) 100

/.0 20 10

1

10

50

90

99

99g % particles

Fig. 1. Particle size distribution in sols A, B and C. was dialyzed until the conductivity was 10 -s Y2-1 m-1, which took between 10 and 15 days, and then aged for 5 days at 80°C. The aged sol was filtered to remove coarse material and t h e n diluted to the experimentally expedient optical extinction of E = 0.7 at )~= 800 n m in a 1-cm cuvette. This diluted sol was stored in quartz bottles. It remained stable for m a n y months. To prevent sedimentation the storage bottles were regularly agitated. In all preparations of sols and electrolytes we used permanganate-treated, triply-distilled water and P.A. reagents. Three AgI sols (A, B and C ) which had been made in this way were used in experiments. The experiments with each sol were carried out independently, with freshly prepared solutions, etc. The sols A and B were used to establish experimental procedures and their reproducibility. Sol C was made afterwards. With this sol all relevant experiments were reproduced in a relatively short period of time. This was done to make sure that no slow changes in the sols during the period of experimentation would affect the consistency of the set of experimental data. To exclude the possibility of dilution effects on the doublelayer charge in all the experiments with sol C, the sol and the added electrolytes were at pI= 6.4. For electron micrography the sols were nebulized ultrasonically, the mist droplets precipitating on formvar sample carriers. Particle size distributions were measured on micrographs (magnification: 50,000 X and 100,000 × ) using an electronic digitizer. The average radii in sols A, B and C were 19, 16 and 33 nm, respectively (Fig. 1).

Turbidimetric determination of W Coagulation rates can be compared by examining the rate at which the turbidity increases from the m o m e n t (t---O) of addition of the coagulant. The retardation factor W is the ratio of the rate of rapid coagulation to the rate at a given electrolyte concentration. It can be obtained [ 21,22 ] as the ratio of the slopes at t = 0 of curves of turbidity versus time. Extrapolating to t = 0 means

299 E ~ox

O~

I ~

100 minor. I ' I K N 0 3

II - - - 0.2 m mo[.t -~La(NO z )3

..........

Et= o

t(min) I

I

I

I

I

I

0

1

2

3

~

5

Fig. 2. Optical extinction (E), recorded as a function of time (t) in coagulating sol C. At t = 0 the

coagulant is added and the sol is diluted 1.25× at that moment. comparing the rate of doublet formation in a sol in rapid coagulation with the rate at a lower concentration of salt b u t using the same concentration of particles. In each experiment 2.0 ml sol were placed in a clean 1.0-cm quartz cuvette. At t = 0, 0.5 ml of an electrolyte solution were added and mixed with the sol by bubbling air. The optical extinction E was recorded through this action as a function of time (Fig. 2). At the wavelength ~ = 8 0 0 nm, Beer's Law applies for AgI. It is only the extinction coefficient (scattering power) which differs between particles and aggregates. The coordinates of the E - t plot were digitized and the plot was fitted by an exponential function, extrapolating it back to the theoretical E-value at t = 0. The slope of this function at t = 0 was computed and used as a measure of the coagulation rate in calculating W. T h e average of the slopes at all the concentrations above Cc was used as a measure of the rate of rapid coagulation. The determination of E - t diagrams was carried out in triplicate at least for every electrolyte concentration examined, to ensure that reproducibility of mixing had been obtained in the experiment. It was observed that in the later stages of coagulation there was a considerably larger variation in the E - t diagrams with polyvalent than with monovalent counterions. However, this seemed not to affect the extrapolated slopes of the curves at t--O.

300

Electrophoresis The electrophoretic mobility of AgI particles was measured in sols B and C, at 25°C and for different electrolyte concentrations. We used a Rank Bros MkII microelectrophoresis apparatus with a He/Ne-laser focussed at 1/7th of the diameter from the bottom of the cylindrical cuvette. The field at the point of observation was calculated from the conductivity of the sol, the current and the calibrated bore of the cuvette. The motion of the particles was observed with a red-sensitive TV-camera and displayed on a monitor screen with a calibrated mesh. The measurements were made with AgI sols which had been diluted 10 X but were at the same pI and electrolyte concentrations as those in the corresponding determination of W. The optical setup did not allow detection of the smaller particles in the polydisperse AgI on the monitor, even in dark-field illumination. This was deduced by comparing the number of particles in the illuminated volume that were seen on the screen with that calculated from the sol concentration and the particle size distribution. In fact, only the 10% largest particles, representing 50% of the AgI volume, were seen and measured in the electrophoresis experiment. The average radius of these particles must be used in the interpretation of the electrophoresis data. It is the same ( 65 nm) i n sols B and C. Electrophoretic mobilities were measured by following at least 10 particles, each moving at least 10 X up and down the monitor screen upon current reversal, at each electrolyte concentration. In some cases the measurement was repeated after some time, taking a new sample from the original mixture of sol and electrolyte. OBSERVATIONS,DATA,INTERPRETATION

The determination of log W-log C diagrams It was found in the experiments that the initial slopes of the turbidity-time curves for rapid coagulation with K +, Ba 2+ and La ~+ counterions varied in a ratio of 1.50 : 1.00 : 1.20. We have calculated W in each electrolyte in relation to the slope for rapid coagulation in that same electrolyte, assuming that the Smoluchowski rate of rapid coagulation is equal in all three electrolytes. Figure 3 shows the resulting log W-log C diagram for sols A, B and C. Sol C, which is somewhat coarser than the other two, appears a little more stable, but generally the results of the three experiments are very reproducible. The slopes for C < Cc are not very dependent on z. Also, these parts of the diagrams, representing slow coagulation, are not really linear. They curve upwards, indicating a lower coagulation rate at low salt concentrations than would correspond to the usual linear extrapolation of data for moderately rapid coagulation rates.

301

W 10.000

x

x

5000 La(N03)3

x

ax BcI(N03) 2

~ KN03

2000 1000

•

~,

x x

A

x

eB

500

A " B x C

•

200 •

10£

!

5C 2C

C w=lo

C w:lo

Cw:lo

1C

| '

o bs

l

lb

lOO

-C{mmol [-1}

Fig. 3. The log W-log C diagrams for sols A, B and C in the three electrolytes KN03, Ba(NOa)2 and La (NO3)3. The Cl0 values for sol C are indicated with an arrow.

The critical coagulation concentrations in these AgI sols are 200 m M for K +, 4.2 m M for Ba 2+ and 0.20 m M for La 3+, respectively. In the electrophoresis experiments we used the concentrations C10 for which W= 10. These are 55 m M for K +, 2.3 m M for Ba 2+ and 0.07 m M for La 3+ in sol B and in sol C 65 m M for K + , 2.6 m M for Ba 2+ and 0.10 m M for La 3+. Experimentally, these values for Clo have a precision of the order of 10%.

The determination of ~ in coagulating sols Electrophoresis in a coagulating colloid is a complicated operation and at first sight it would seem that at Co, or under coagulation conditions in general, electro-osmosis is a more reliable method to measure the zeta potential of AgI. However, as shown in Fig. 4, the electrophoretic mobility of particles is not constant for some time following the addition of an electrolyte. This is especially so when KNO3 is added to an AgI sol. Comparing time scales between Fig. 2 and Fig. 4 it is clearly seen that in coagulation rates at t = 0 the relevant electrokinetic data are those immediately after the addition of electrolyte. These cannot be obtained from an electro-osmosis experiment (in which a porous plug of solid AgI must be saturated with electrolyte before meaningful data can be obtained), or generally, from any type of experiment in which one waits for the electrophoretic velocity to become constant.

302 -0" (rn2V4sec -1 } I (a) •

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~'o

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o

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o

~

o

" °

i

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(b) -~ (m2V-lsec

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8

o

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-,.<

KNO3;-,60mmol-1 Sol C 2 Ba(NO3}2.2.51mmol14So[ C • 2.75mmol I-~SoiC o2 32retool L~Sot B 'El: La {NO3) 3 i 0.1 rhino( I Sol4C v O.0?mmol lqSol B

:\.

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÷

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÷

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o

o

I

%

v

4t(h) 0

1

2

v

3

4

~

6

24 t{h}

Fig. 4. Electrophoretic mobility (U) as a function of the time t after the addition of different electrolytes. Sols B and C; (a) different concentrations of KN03; (b) different counterions. Therefore we chose electrophoresis for measuring zeta potentials and settled for the concentration Clo at which W = 10 (moderately rapid coagulation) and diluted the sol, thus minimizing the complications with rapidly increasing "particle sizes" and sedimentation of flocs during the experiment. The electrophoretic mobilities which were used to calulate zeta potentials were averaged for each particle (moving 20 × over the screen) and for all the particles at a given salt concentration. The spread in mobilities of the individual particles was much less than had been expected from the assumption that all the particles had the same zeta potential, and then taking the electron micrographic particle size distribution as the basis for calculating electrophoretic retardation and relaxation effects. This observation corroborates the deduction (see above) that mobilities were measured on the larger particles ( a - - 6 5 nm) only. Methods for the calculation of ~5 from electrophoretic data have been described by Overbeek [23] and Wiersema et al. [24], and in a more recent version of the theory, which remains tractable up to higher potentials, by O'Brien [ 25 ]. These theories give mobilities at known ~5 values as a function of •a. Interpolating experimental mobilities at the pertinent Ka values, we have obtained 0~ at Clo for the AgI sols of the experiment. In certain areas the theoretical parameter field is highly non-linear, so that some interpolated val-

303 TABLE 1 Electrophoresis results

Sol B KN03 B a (NO3) 2 La (NO3) 3

Clo (mM)

~,~ (mY)

f~/x

55 2.3 0.07

60 _ 5 22 _ 7 15 +_2

1.05 0.45 0.98

65 2.6 0.10

60 ___5 37_ 7 17_ 2

0.98 1.05 0.90

Sol c KN03 Ba(N03) La (NO3)

ues are more precise than others. The resulting margins of error in ¢~ are also given in Table 1. In the first column of this table the sensitivity of coagulation rates for the counterion valency is seen, which may be formally expressed by the SH rule. The second column gives the measured double-layer potentials ¢~. They are low in the sense attached to the interpretation of Eqn (1). For W = 10 and z = 1, 2 and 3 we find ¢~=60, 30 and 16 mV, respectively. Since this result is obtained in two independent experiments with sols B and C it seems a good generalization to state that at equal coagulation rates z¢~ is equal: the potential of the diffuse charge layer turns out to be inversely proportional to the counterion valency when the coagulation rate has a given value in a given colloid. The last column has been constructed from the measured data. It gives the "Eilers and Korfff' criterion ¢~/K. According to these authors [ 26 ] ¢~/x should have a constant value at a given coagulation rate. The experimental confirmation of this notion for AgI is of a surprising quality in view of the fact that the Eilers and Korff criterion is so sensitive for comparatively small experimental errors in the diffuse layer potential. DISCUSSION

Historically the Eilers and Korff criterion precedes the DLVO theory. It was based on a rather unconvincing dimensional argumentation and tried to do away with the idea of a "critical zeta potential". Overbeek [ 14 ] argued that the criterion is a direct consequence of the DLVO Eqn (1) when sols with somewhat small potentials are discussed. In a sol where z ~ and ¢~/K are both found to be constant at a given coagulation rate it is immediately seen that z6C must then also be a constant. The SH rule follows directly from these two potential dependent conditions. That SH behaviour exists for so many electrocratic sols, independent of their particular potentials and coagulation con-

304 centrations, could thus be a consequence of the mathematical elimination of the potential between two other, potential related properties of double-layer repulsion at equal coagulation rates. The scheme of low ¢~a,with z¢~ and ¢~/~: determining the rate of coagulation should then be applicable for all colloids with SH behaviour. Another consequence of ¢~ being low and z¢~ constant is that Eqn (1) is still an adequate description of the vanishing of the potential energy maximum at Co, but that tanh(zeO~/4kT)~ 1. At low potentials, 7 is of the order of zeO~/4 kT, and for z¢~ = 50 mV the value of 7 becomes 0.46. It has often been attempted to deduce a value of the Hamaker constant A from coagulationexperiments by applying Eqn (1) or analogous expressions. This, invariably, gave values for A which were higher than those expected from theoretical considerations or other types of experiment by an order of magnitude. However, using the correct values of zO~ the Hamaker constant for AgI is found to be A = 5 X 10 -20 J from the experiments described in this paper. This value compares well with the theoretical value A--4 X 10 -20 J, given by Visser [ 27]. It seems, therefore, that we have obtained a quantitative, experimental verification of the DLVO equation [Eqn (1)] by using a consistent set of measurements: all in one colloid and all with the same time scale of the experiment. CONCLUSIONS The experiments with AgI sols show that z¢~ and ¢~/K are both constant at a given rate of coagulation and that ¢~ is low. This result gives an unambiguous answer to the questions posed in the introduction of this paper. Coagulation proceeds at this rate, not just because salt addition has reduced the range of the repulsion, nor because a deepening minimum has tipped the balance of coagulation and repeptization. Addition of the different electrolytes lowered the potentials in such a way that equal energy barriers were obtained. Polyvalent counterions are found to be more potent coagulants since they are more effective in lowering ¢~. This strong effect of z on 0~ may in some colloids be related to specific adsorption of polyvalent ions, and in others to the change in the distribution of ions between the compact inner and the diffuse outer part of the double layer because of the higher counterion valency. But the repulsion between the particles in an electrocratic colloid is caused by the overlap of the diffuse charge layers and therefore the exact reasons for the lowering of 0~ are inconsequential for coagulation behaviour. At equal rates of coagulation, z¢~ and ~ / ~ have constant values for different values of z with one colloid, and as a consequence experiments on coagulation reveal the SH behaviour, which is so characteristic for most electrocratic sols and appears itself seemingly independent of their particular materials properties. On the basis of these observations, we conclude that the textbook notions of

305

deriving the SH rule and obtaining Hamaker constants by assuming high potentials and tanh (ze¢d4kT) = 1 must be abandoned. They obscure the strong dependence of coagulation phenomena on the actual value of ¢~ and misrepresent the reason why polyvalent counterions are strong coagulants. The potential dependent conditions which determine the stability of a given electrocratic colloid are that z ~ and ~ / ~ are both constants for a given rate of coagulation. From these, one may eliminate ¢~ to obtain the derivation of the SH rule, but that does not discount the lowering of the diffuse-layer potential as the dominant factor in the coagulation of a sol by adding salt to it. ACKNOWLEDGEMENT

Electron microscopy was done by Mrs G. van Leeuwen, whose expert contributions to the investigation are gratefully acknowledged.

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