A reassessment of the applicability of the DLVO theory as an explanation for the Schulze-Hardy rule for colloid aggregation

A Reassessment of the Applicability of the DLVO Theory as an Explanation for the Schulze-Hardy Rule for Colloid Aggregation SIMON BERNERS HALL, JOHN RALPH DUFFIELD, AND D A V I D R A Y M O N D W I L L I A M S 1 School of Chemistry and Applied Chemistry, University of Wales College of Cardiff, P.O. Box 912, Cardiff CF1 3TB, United Kingdom Received March 12, 1990; accepted October 16, 1990 Shortcomings in the application of the DLVO theory in explaining the observations of the SchulzeHardy rule occur because of inappropriate simplifications. Previous authors ( 14, 15) have commented that an approximation valid for high surface potentials is difficult to reconcile with the aggregation of low potential particles. In addition to this, we suggest that a further simplification, assuming only weak electrostatic interaction between particles, is also inappropriate in explaining the Schulze-Hardy rule, We derive a new expression that determines the critical ionic strength for the coagulation of low potential colloids in terms of the interracial charge density. The new equation is more general than previous expressions since the ionic strength may be used to describe nonideal unsymmetrical electrolytes, and the charge density is a more fundamental parameter than surface potential. The concept of restabilization in coalescing emulsions is described and discussed in terms of the new expression for critical aggregation ionic strength. © 1991 Academic Press, Inc. 1. INTRODUCTION Previous authors ( 1-3 ) have sought to provide a theoretical rationale for the empirical S c h u l z e - H a r d y rule for colloid aggregation (4). The S c h u l z e - H a r d y rule observes that the critical coagulation concentration (c.c.c.) o f an ideal electrolyte sufficient to coagulate a colloid depends approximately to the inverse sixth power on the charge n u m b e r o f those ions o f opposite sign (counterions) to the colloid particles. The c.c.c, is independent o f the specific character o f the counterions, independent o f the charge n u m b e r o f the coions, and only moderately dependent on the nature o f the sol. The rule has also been demonstrated for emulsion systems (5, 6). This paper, for the m o s t part, will be concerned with aggregation o f emulsions leading to coalescence, t h o u g h the initial aggregation processes will be similar for flocculation o f sols. To whom correspondence should be addressed.

Deryaguin and Landau (7) and Verwey and Overbeek (8) developed a theory ( D L V O ) o f colloid stability by c o m b i n i n g long-range attractive van der Waals forces with long-range repulsion caused by the overlap o f double layers. The net free energy o f interaction is calculated as a function of interparticle separation, H , and the stability o f the colloid is represented by the magnitude o f any energy barrier. Figure 1 shows the net energy o f interaction for several systems where (i) a positive energy barrier (several multiples o f the thermal energy k T ) prevents coalescence; (ii) an energy bartier prevents coalescence b u t a secondary energy m i n i m u m (attraction) at larger interparticle separation is sufficient for flocculation; (iii) the energy barrier is reduced to zero; and (iv) there is no energy barrier and an attractive energy at all particle separations results in coalescence. These potential energy curves do not consider any structural forces imposed by solvent molecules adjacent to the interface as

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HALL, DUFFIELD, AND WILLIAMS

nation with counterion adsorption to explain the Schulze-Hardy rule. > Several factors suggest the need for a reapo= praisal of the application of the DLVO theory ¢a to the Schulze-Hardy rule. First, the previous _= derivations (14-16) assume that the relative electrostatic interaction is small, so that the .,=, repulsion energy can be determined by the simple addition of two G u o y - C h a p m a n double layers. This assumption is valid only for large interparticle separations, whereas such studies clearly involve consideration of close interparticle approaches. Second, the use of Interpart[cle Separation H the electrolyte ionic strength in calculations FIG. 1. Total interactionenergycurves for (i) stability, would be useful in describing emulsion sta(ii) flocculation,(iii) transition between stabilityand cobility in nonideal, unsymmetrical electrolytes. alescence, and (iv) coalescence. Third, the interfacial charge density is a more useful parameter than potential since it better suggested by Ninham and co-workers (9-11 ) reflects the chemistry at the interface. and van Megen and Snook ( 12, 13). 3. A LOW POTENTIAL DLVO MODEL FOR THE SCHULZE-HARDY RULE 2. THE DLVO RATIONALE FOR THE SCHULZE-HARDY RULE AT HIGH The net potential energy, V, for the interPOTENTIALS AND SMALL INTERACTIONS action of two identical spherical particles is Hunter (14) and Overbeek ( 15 ) have given the sum of an electrostatic term, VR, and a the derivation for the interaction between par- van der Waals term, VA. Hamaker (19) has allel thick plates and have shown that at high derived an expression for VA, potentials the c.c.c, depends inversely to the sixth power on the charge number of the counterions. This is in agreement with the observations of the Schulze-Hardy rule and Freundlich's (17) and Ostwald's (18) subsequent interpretations. This relationship has been shown to hold for spherical particles also (14, 16). Hunter (14) and Overbeek (15) comment that the high surface potential approximation is difficult to reconcile with the more usual aggregation of low potential particles. As a result, Hamaker attraction constants determined using these relationships are generally higher than those that would be expected from theory. In low potential derivations, the c.c.c, depends to the fourth power on the surface potential and inversely to the square of the counterion charge number ( 14, 15). Overbeek (15 ) has developed a model using the low potential derivation in combiJournal of ColloM and Interface Science, Vol. 143, No. 2, May 1991

-Aa VA- 12H'

[1]

where a is the radius of the spherical particles and H is assumed to be small compared to a. Healy and co-workers (20, 21 ) have shown that the repulsion energy under conditions of constant charge is given by VR = --27rEa~bZln[1 -- exp(--KH)],

[2]

where ~ is the permittivity of the continuous phase and the inverse Debye length K is given by

= (2e2LIrn] 1/2

K \ ~--Z-~/

[31

and Im is the ionic strength of the electrolyte with units mol m -3 . It is assumed in the derivation of Eq. [ 2 ] that the Debye-Hfickel approximation is valid and that Ka is large.

RESTABILIZATION

The conditions for the transition between stability and aggregation exist when V = 0 and d V~ d H = 0 for the same H , thus

-Aa -- 27remp21n[1 - exp(--KH)] = 0. [4] 12H Since

dVA dH

--VA H

[5]

and

413

IN E M U L S I O N S

Debye-Htickel approximation is valid and Ka >1: q = --.

Substituting [3], [7], and [10] into [4], solving for ionic strength, and using more comm o n units for this (mol dm-3), we obtain the critical aggregation ionic strength, /ca, /ca = 5.4741 X 10 -3- El/3kZq4/3 e2LA2/3 ,

dVR dH

[10]

EK

[11]

which, at 25°C, simplifies to the relationship K exp ( - KH)

z

[1 - e x p ( - K H ) ] . l n [ 1 -- exp(--KH)] × VR,

[6]

then equating [ 4 ] - [ 6 ] and solving for KH, we obtain KH = [1 -- e x p ( K H ) ] - l n [ 1 -- exp(--KH)]

KH = In 2.

[7]

A similar expression m a y be derived using the repulsion energy under conditions of constant potential (20, 21 ), VR = 27rea~21n[1 + e x p ( - - K H ) ] ,

[8]

~H = in 7r.

[9]

/ca = 1.2914 × 10 -12" q4/3 A2/3 .

[12]

Thus, an emulsion with potential 20 m V in an electrolyte of ionic strength 0.05 mol dm -3 and a critical aggregation ionic strength of 0.150 mol d m -3 at 25°C can be shown to have a H a m a k e r constant o f A = 2.67 × 10 -21 J. This new expression does not predict the empirical Schulze-Hardy rule in a simple way, unlike previous derivations (14-16), since both the ionic strength and the interfacial charge density need to be measured, or calculated by chemical speciation.

where

The values for KH given in [7] and [9] differ significantly from KH = 1 found in previous derivations ( 14-16). This suggests that the weak interaction approximation is inappropriate since the position of the potential energy m a x i m u m is influenced. Overbeek (22) has considered the choice of using either constant charge or constant potential relationships and has concluded that the real situation will lie somewhere between the two. In this paper, only the constant charge case will be considered, although the general features will be similar for constant potential conditions. The Stern potential, ~b, m a y be expressed in terms of surface charge density, q, when the

4. R E S T A B I L I Z A T I O N D U R I N G T H E AGGREGATION AND COALESCENCE OF MONODISPERSE EMULSIONS

There appears to have been no previous consideration given to the consequences of coalescence in emulsions, namely, the stability characteristics of new particles arising from the coalescence of the original dispersion. If these characteristics are to be assessed, conservation of charge and volume of the discontinuous phase must be assumed. Thus, if two particles in a monodisperse system, each with surface charge density, ql, and radius, al, aggregate and then coalesce, a new particle with charge density, q2, and radius, a2, will form such that a2 = 21/3- al

[13]

Journal of Colloid and Interface Science, Vol. 143, No. 2, May 1991

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HALL, D U F F I E L D , A N D W I L L I A M S

and q2 = 21/3" ql.

[14]

The increase in charge density is due to a decrease in surface area to volume ratio as one goes to larger spherical particles. This increase is significant since the magnitude of/ca according to [12] depends markedly on charge density. If successive generations of particles are considered (each particle formed only by coalescence of two particles from the previous generation), it can be shown for the nth generation that an

= 2 (n-l)/3 °

al

[15]

and qn = 2(n-1~/3" ql.

[16]

Substitution of [16] into [12] yields an expression for/ca for the nth generation, Ina = /ca" 24(n-1)/9,

to detect with confidence for small particles, even with the most sophisticated particle analysis devices. The inadvertent measurement of coalescence of many-generation and, hence, more stable particles will adversely affect any meaningful interpretation. It is because of the second feature that we feel it is inappropriate to discuss the empirical Schulze-Hardy rule with respect to the present or previous theoretical considerations for emulsion systems, or to use these methods for determining the Hamaker constant. Equation [ 12 ], however, may be applicable to some sols. An inadequacy common to both this treatment of the DLVO theory and previous works. is that interactions are constrained to those between identical particles. Any successful model must include interactions between dissimilar particles since even monodisperse emulsions become polydisperse during coalescence.

[17] 5. C O N C L U S I O N S

where Ic~aand/ca are the critical aggregation ionic strengths for the nth and first generations, respectively. The model is simple since interactions are only considered between particles of the same generation and since the possibility of charge regulation due to changes in adsorption of potential determining ions is ignored. Two main features, however, are demonstrated. First, [ 17 ] shows that most emulsions (except those with very low interfacial charge density) are expected to restabilize, preventing separation into two continuous phases. This is due to a marked increase in/ca with each successive generation, which will, within a few generations, usually exceed the electrolyte ionic strength and, thus, terminate further coalescence. Second, [ 15 ] and [ 17 ] show that the visual technique for measuring/ca or c.c.c, is impractical for emulsions since it requires an observation of the coalescence of solely firstgeneration particles. The difficulty arises from detecting the increase in particle size since the 26% increase in radius between generations is impossible to detect by eye alone and difficult Journal of Colloid and Interface Science, Vol. 143, No. 2, May 1991

The previous applications of the DLVO theory to the Schulze-Hardy rule have been shown to depend on assumptions inappropriate for most colloids. We have derived a new expression (Eq. [ 12]) that predicts the critical aggregation ionic strength for monodisperse colloids. Although this new expression does not predict the Schulze-Hardy rule in a simple manner, it should be applicable to mixed nonideal electrolytes and, because it embodies interfacial charge density, may be used to account for ions which are specifically bound at the interface. Both the new and the previous relationships have been shown to be inadequate in the description of emulsion coalescence for two reasons. First, coalescence does not result in a well-defined experimental characteristic, as does, say, the flocculation of sols. Second, coalescence leads to polydisperse emulsions which need to be considered as interactions between dissimilar particles. The concept of restabilization in coalescing emulsions is introduced and shown to be sig-

RESTABILIZATION IN EMULSIONS nificant. I n a s u b s e q u e n t p a p e r a n a l t e r n a t i v e a p p r o a c h to s u c h e m u l s i o n c o a l e s c e n c e is described. ACKNOWLEDGMENT We gratefully acknowledge the support provided by the Wolfson Research Award Scheme (1989). REFERENCES 1. Overbeek, J. Th. G., in "Colloid Science" (H. R. Kruyt, Ed.), Vol. I, p. 306. Elsevier, Amsterdam, 1952. 2. Deryaguin, B. V., Trans. FaradaySoc. 36, 203 (1940). 3. Deryaguin, B. V., Trans. FaradaySoc. 36, 730 (1940). 4. Overbeek, J. T. G., in "Colloid Science" (H. R. Kruyt, Ed.), Vol. I, p. 302. Elsevier, Amsterdam, 1952. 5. Burnham, W. R., Hansrani, P. K., Knott, C. E., Cook, J. A., and Davis, S. S., Int. J. Pharm. 13, 9 (1983). 6. Davis, S. S., in "Advances in Clinical Nutrition" (I. D. A. Johnson, Ed.), Chap. 5. Lancaster MTP Press, Lancaster, 1984. 7. Deryaguin, B. V., and Landau, L., Acta Physicochim. URSS 14, 633 ( 1941 ). 8. Verwey, E. J. W., and Overbeek, J. Th. G., "Theory of the Stability of Lyophobic Colloids." Elsevier, Amsterdam, 1948.

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9. Mitchell, D. J., Ninham, B. W., and Pailthorpe, B. A., J. Chem. Soc. Faraday Trans. 2 74, 1098 (1978). 10. Mitchell, D. J., Ninham, B. W., and Pailthorpe, B. A., J. Chem. Soc. Faraday Trans. 2 74, 1116 (1978). 11. Chan, D. Y. C., Mitchell, D. J., Ninham, B. W., and Pailthorpe, B. A., Mol. Phys. 35, 1669 (1978). 12. Van Megan, W., and Snook, I. K., Phys. Lett. A 74, 332 (1979). 13. Van Megan, W., and Snook, I. K., J. Chem. Phys. 70, 3099 (1979). 14. Hunter, R. J., "Foundations of Colloid Science," Vol. I. Chap. 7. Clarendon Press, Oxford, 1987. 15. Overbeek, J. Th. G., Pure Appl. Chem. 52, 1151 (1980). 16. Shaw, D. J., "Introduction to Colloid and Surface Chemistry," 3rd ed., Chap. 8. Butterworths, London, 1983. 17. Freundlich, H., Z. Phys. Chem. 73, 385 (1910). 18. Ostwald, Wo., Kolloid Z. 73, 320 (1935). 19. Hamaker, H. C., Physica 4, 1058 (1937). 20. Hogg, R., Healy, T. W., and Furstenau, D. W., Trans. Faraday Soc. 62, 1638 (1966). 21. Wiese, G., and Healy, T. W., Trans. Faraday Soc. 66, 490 (1970). 22. Overbeek, J. Th. G., 9". Colloid Interface Sci. 58, 408 (1977). 23. Healy, T. W., Chan, D., and White, L. R., PureAppl. Chem. 52, 1207 (1980).

Journal of Colloid and Interface Science, Vol. 143, No. 2, May 1991