J. Phys. Chem. 1984,88, 5735-5739
5735
A Formulation of the Short-Range Repulsion between Spherical Colloidal Particles Donald L. Feke,* Naresh D. Prabhu, J. Adin Mann, Jr., and J. Adin Mann, I11 Department of Chemical Engineering, Case Institute of Technology, Case Western Reserve University, Cleveland, Ohio 44106 (Received: March 19, 1984; In Final Form: June 8, 1984)
With the assumption of pairwise additivity and the integration procedure of Hamaker, mean potential expressions for the short-range repulsion between colloidal particles were constructed from the repulsive parts of interatomic potentials. These expressions have more realistic qualitative characteristics than other formulations used at the same approximation level. The combination of short-range repulsion with van der Waals and electrostatic forces results in a total interparticle potential diagram that may exhibit one of three general shapes regardless of the details of the individual potentials. Values of physical parameter sets that give rise to each of the three types of curve have been calculated for a specific but realistic DLVO potential. Under these conditions, noninfinitely steep Born repulsion can eliminate the primary minimum from the interparticle potential for colloidal systems.
Introduction In the DLV01g2representation of interparticle forces in colloidal systems, the interaction potential consists of an attractive contribution from van der Waals forces and a repulsive part due to electrostatic effects between the charged particles. Combining these two portions results in a net potential that is predicted to approach a value of negative infinity upon particle contact. Such an infinitely deep primary minimum is not consistent with the observations made in spontaneous repeptization experiment^"^ or studies of reversible coagulation.6-12 Often, for realistic modeling of the interaction, DLVO theory must be modified to account for short-range repulsive forces that develop as particles begin to contact. For colloidal particles having adsorbed polymers, the short-range forces are known as steric or osmotic forces. Structural or hydration forces are the terms associated with the forces that develop as particles with adsorbed fluid layers interact. Direct mea~ u r e m e n t ' ~and , ' ~ Monte Carlo or molecular dynamics simulation (ref 15-17, for example) of solid surfaces interacting in liquids show spatial oscillations in these short-range forces. However, when the precise details of the spatial variation of the short-range forces are not important, it is often convenient to resort to a mean potential formulation.18-20In the past, ad hoc models for the mean
(1) Derjaguin, B.; Landau, L. Acta Physiochem. 1941, 14, 633. (2) Verwey, E. J.; Overbeek, J. Th. G. "Theory of the Stability of Lyophobic Colloids"; Elsevier: Amsterdam, 1948. (3) Kruyt, H. R.; Klompt, M. A. M. Kolloid Beih. 1943, 54, 507. (4) Troelstra, S. A.; Kruyt, H. R. Kolloid Beih., 1943, 54, 225. (5) Frens, G.; Overbeek, J. Th. G.J . Colloid Interface Sci. 1971,36, 286. (6) Westgren, A. Ark. Kemi, Mineral. Geol. 1918, 6, 7 . (7) Goodeve, C. F. Trans. Faraday SOC.1939,35, 342. ( 8 ) Gillespie, Th. J . Colloid Sci. 1960, 15, 313. (9) Eniinstiin, B. V.; Turkevich, J. J. Am. Chem. SOC.1963, 85, 3317. (10) Wiese, G. R.; Healy, T. W. Trans. Faraday SOC.1970, 66, 480. (1 1) Long, J. A.;'Osmond, D. W. J.; Vincent, B. J. Colloid Interface Sci. 1973, 42, 545. (12) Ottewil, R. H.; Walker, T. J . Chem. SOC.,Faraday Trans. 1, 1974 70, 917. (13) Israelachvili, J. N. Adu. Colloid Interjace Sci. 1982, 16, 31. (14) Pashley, R. M. J. Colloid Interface Sci. 1981, 80, 153. (15) Snook, I. K.; van Megen, W. J . Chem. Phys. 1978, 70, 3099. 1980, 72, 2901. (16) Chan, D.Y. C.; Mitchell, D. J.; Ninham, B. W.; Pailthorpe, B. A. J . Chem. Soc., Faraday Trans. 2 1979, 75, 556. 1980, 76, 176. (17) Ninham, B. W. J . Phys. Chem. 1980,84, 1423. (18) Agterof, W. G. M.; van Zomeren, J. A. J.; Vrij, A. Chem. Phys. Lett. 1976, 43, 363. (19) Vrij, A.; Niewenhuis, A.; Fijnaut, H. J.; Agterof, W. G. M., Discuss. Faraday SOC.1978,65, 101.
0022-3654/84/2088-5735$01.50/0
Born potential between macroscopic particles have been directly adapted from interatomic potentials with little or no theoretical justification. Often, these formalisms fail or render the mathematics intractable when they are used to predict properties of colloidal systems (see ref 21, for example). Here we present expressions for the short-range Born repulsion between colloidal spheres as calculated through the Hamaker method of pairwise integration. Our formulas, which have better mathematical characteristics than the ad hoc models, have been used in the prediction of coagulation rates.22 Also, they should be particularly appropriate for the examination of reversible coagulation or adsorption p h e n ~ m e n a ~and ~ -the ~ ~ simulation of structure and ordering within colloidal systems.1x-20The rationale and validity of the Hamaker approach as applied to the calculation of Born repulsion are also described. Since Born repulsion governs particle interactions at small separations, it should be included in all realistic descriptions of colloid behavior. The combination of van der Waals and electrostatic forces with Born repulsion determines the nature of the overall interparticle potential. In our formulation with a mean potential Born repulsion, the net interparticle potential falls into one of three general categories: potentials that exhibit only a primary minimum; potentials that show only a secondary minimum; or potentials that feature both a primary and a secondary minimum. For use in predicting the stability states of dispersions, we have calculated the ranges of the governing parameters that result in each type of interparticle potential.
Theory Since the short-range Born repulsion between colloidal particles originates from the repulsion that develops between atoms, we first review interatomic potentials. To extend the analysis to macroscopic particles, the assumption of pairwise additivity and the integration procedure of Hamaker were used. Models for Interatomic Potentials. The study of the atomatom pair interaction potential, E , has received a great deal of interest from quantum and statistical mechanicians. Precise descriptions of the interaction potentials must be based on quantum mechanical considerations. However, for simplicity, a plethora of approximate analytic forms have been proposed (see F i t t ~ * ~ ) ) . The most widely used forms are the hard-sphere potential (20) Niewenhuis, E. A.; Pathomamanoharan, C.; Vrij, A. J . Colloid Interface Sci. 1981, 81, 196. (21) Souidi, F., Ph.D. Thesis, Case Western Reserve University, 1982. Diss. Abstr. 43 (4), 1222-B. (22) Prabhu, N., M.S. Thesis, Case Western Reserve University, 1984. (23) Ruckenstein, E.; Prieve, D. C. AIChE J . 1976, 22, 276. (24) Prieve, D. C.; Ruckenstein, E. J. Colloid Interface Sci. 1976.57, 547. (25) Determan, H. "Gel Chromatography";Springer-Verlag: New York, 1968. (26) Ruckenstein, E.; Prieve, D. C. AIChE J . 1976, 22, 1145. (27) Small, H. J . Colloid Interface Sei. 1974, 48, 147. (28) DiMarzio, E. A.; Guttman, C. M. J. Polym. Sci. 1968, 137, 276. (29) Fitts, D. D. Annu. Rev. Phys. Chem. 1966, 17, 59.
0 1984 American Chemical Society
5736 The Journal of Physical Chemistry, Vol. 88, No. 23, 1984
Feke et al.
In this case (see ref 3 1 , for example) the square-step potential
and Lennard-Jones m-n potential
-%( !!)"'"-"[(4)" - (J)"]
E = n-m
m
(3)
In each of the eq 1-3, r is the interatomic separation, u the separation at which the potential becomes 0, and e the depth of the primary well. The Lennard- Jones potential (eq 3) contains an attractive part due to van der Waals forces and a repulsive contribution known as Born forces. Since the asymptotic form for the pair potential is known to decay as r4, the parameter m in (3) is typically chosen to be 6. There is much less agreement on a value for the exponent of the repulsive part of the Lennard-Jones potentials since the f " dependence may be viewed simply as an appropriate fitting function. Values of n ranging from 8 to 20 have been advanced, and typically, for lack of better information, a value of n = 12 is used. With this value, (3) reduces to the familiar Lennard-Jones 6-12 potential
E = 4e[
(J)" - (;)6]
(4)
The two adjustable parameters in this formulation are usually determined on the basis of experiment. Typically, u = 0(5 A) and e = O(lW2'J). Although the Lennard-Jones potential neglects multiple-body interactions which may be important in fully describing phenomena in condensed media, it continues to be a useful formulation. Models for Interparticle Potentials. The similarity between simple fluids and dispersions of colloidal spheres has often been exploited. In the past, Born repulsion has been incorporated to the interparticle potential by directly taking the repulsive part of an appropriate interatomic potential and rewriting it for macro~ *an~ r-12 ~~ scopic spheres. A hard-sphere repulsive p ~ t e n t i a l or repulsive potential (based on center-to-center separation)20between the particles have been extensively used. Although they have found some utility in various applications, there is no real theoretical justification for ascribing these forms to the repulsive parts of the interparticle potentials. In fact, in some instances it appears that a hard-sphere repulsive potential cannot be used, for example, in the simulation of Brownian dynamics in colloidal dispersions.21 It is our purpose to present at the approximation level of DLVO a more realistic and usable formulation of the repulsive part of the interaction potential for colloidal spheres having interfaces with constant properties. As a first approximation to the interactions of macroscopic particles, the assumptions of pairwise additivity (or, more precisely, pairwise integrability) of the interaction potentials between all molecules in the system can be invoked.30 In general, the energy of interaction, V, between two macroscopic particles of the same composition can be written as
where qp is the number density of molecules in each particle, and v1 and v2 are the volumes of the particles. In (3,E , represents the energy of interaction between a molecule in one particle and a molecule in the other particle. When the two particles are immersed in fluid, the interaction energy must be modified to account for fluid-particle and fluid-fluid molecular interactions. (30) Langbein, D. "Theory of van der Waals Attraction"; Springer-Verlag: Berlin, 1974.
where qf refers to the number density of molecules in the fluid, Effis the interaction energy between two fluid molecules, and Epf is the interaction energy between a fluid molecule and a molecule in a particle. Since the attractive and repulsive parts of E,,, E,f, and Eff should show identical dependences on separation, the attraction and repulsion energies calculated by (5a) or (5b) for macroscopic particles will have the same dependencies on interparticle separation. The absolute magnitude of the interaction is typically characterized by a single coefficient which represents an average of the individual molecular interactions, weighted by the molecular densities in each phase. Hamaker32first made use of the assumption of pairwise additivity for the attractive part of the interparticle potential. For simplicity, the angular dependence of the intermolecular potential and retardation effects were ignored; the interaction potential was assumed to depend only on the separation between the molecules in the pair. Evaluating ( 5 ) over spheres of radii al and u2 for the r-6 attractive potential, Hamaker found VA =
Here X = a2/al,R is the center-to-center separation made dimensionless on a l , and A = ?rzq2eu6is the Hamaker coefficient. The assumption of pairwise additivity is believed to be a good first approximation for interaction energies in condensed bodies. Based on experiment, the Hamaker expression (6)apparently exhibits the proper dependence of attraction energy on interparticle separation. However, no similar arguments can be made to justify the validity of a pairwise additive approximation for repulsive forces. It is probably more realistic to assume that Born repulsion between macroscopic bodies should contain only near-neighbor effects. To derive formulas for sphere interactions would require a detailed knowledge of the arrangement of atoms within the interfaces or to complete the integration of (5a) only over the volumes of the interfacial regions (including the Stern layers). Since the interatomic repulsion forces are short ranged, a useful first approximation to this calculation is to extend the integration over the whole volumes of the spheres and to ignore the detailed structure and extent of the interface. The degree of hardness in the repulsion between macroscopic particles is reflected through n in the interatomic potential. Integrating the repulsive part of the potential for arbitrary n over the same sgherical particles yields
-R2 - (n - 5)(X - l)R - (n - 6)[X2- (n - 5)X -R2
+
+
(R - 1 (n - 5)(X - l ) R - (n - 6)[X2 - (n - 5)X
R2 + ( n - 5)(X R2 - (n - 5)(X
+ 11 + 11
+
+
(R 1 l ) R + ( n - 6)[X2 + (n - 5)X + 11
+
( R + 1 + ,),-5 l ) R + (n - 6)[Xz+ (n - 5)X (R- 1 -
+ 11
+
+ +
]
(7)
This equation is valid for all real n except n = 7, 6, 5, 4, 3, or 2. Expressions for both the near-field, R - 1 - X
0
\
'.
01
-
!I
L
I .o
15
R
Figure 2. Comparison between models for short-range repulsion. Shown are representative potentials for the pairwise-summed potential for n = 12, the RI2potential," and the hard-sphere p o t e ~ ~ t i a l . ' ~Note * ' ~ that the summed potentials are softer than the hard-sphere but harder than the RI2potential.
far-field, R are
.
>> 1, limits of (7) have also been computed.
These
6 1 PRIMARY AND
PSECONDARY
MINIMA
I
R-I-X
