Langmuir 1996, 12, 4103-4110
4103
Determining the Forces between Polystyrene Latex Spheres Using Differential Electrophoresis Darrell Velegol and John L. Anderson* Department of Chemical Engineering, Colloids, Polymers, and Surfaces Program, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Stephen Garoff Department of Physics, Colloids, Polymers, and Surfaces Program, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received January 12, 1996. In Final Form: May 28, 1996X The attractive force holding two polystyrene latex spheres in a doublet was measured by the method of differential electrophoresis. The two spheres of each doublet had different surface chemistries (e.g., sulfate and carboxylate) and different ζ potentials ζ1 and ζ2. The doublet acted as a dipole, and an applied electric field (E∞) caused the doublet to rotate such that the less negative sphere pointed in the direction of the field. Once the doublet was aligned, the tendency of the spheres to translate at different velocities produced a tension, the “electrophoretic displacement force”. This force, proportional to ζ2 - ζ1 and the applied electric field E∞, is calculated from solutions to the electrostatic and hydrodynamic equations. For our systems (5 µm diameter spheres, ζ2 - ζ1 ≈ 40 mV, E∞ ≈ 200 V/cm) the electrophoretic displacement force was 20-50 pN, which is more than a factor of 10 greater than the maximum attractive force predicted by DLVO theory for doublets in a secondary minimum. In no case could we break the doublets with the electrophoretic displacement force. We conclude that DLVO theory is inadequate for our colloidal system, either because the doublets were in a primary minimum (even though DLVO theory predicted an insurmountable energy barrier) or because the depth of the secondary minimum was more than a factor of 10 greater than predicted.
Introduction The microstructure and stability of colloidal suspensions ultimately depend on the forces between colloidal particles.1 The theory generally used to describe these forces is DLVO theory, which accounts for van der Waals and electrostatic forces.2 The qualitative and quantitative predictions of this theory, such as the existence of a “secondary minimum”, have been demonstrated for macroscopic surfaces of mica using the surface forces apparatus.3 The veracity of the electrostatic part of DLVO theory has been established for a spherical particle and a flat surface using total internal reflection microscopy (TIRM).4 Direct measurements of the forces between two colloidal particles are scarce. Simple shear flow has been used to produce extensional forces to break colloidal doublets.5-7 Measurements of the period of Jeffrey orbitals for doublets in simple shear have also been used to deduce the gap between two particles.1,8,9 However, a rigorous quantitative comparison between the experimental data and the DLVO model was not made in these flow experiments. A * To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, July 15, 1996. (1) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989; Chapters 5 and 8. (2) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992; Chapters 10 and 12. (3) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (4) Walz, J. Y.; Prieve, D. C. Langmuir 1992, 8, 3073. (5) Tha, S. P.; Goldsmith, H. L. Biophys. J. 1986, 50, 1109. (6) Tha, S. P.; Shuster, J.; Goldsmith, H. L. Biophys. J. 1986, 50, 1117. (7) Tees, D. F. J.; Coenen, O.; Goldsmith, H. L. Biophys. J. 1993, 65, 1318. (8) van de Ven, T. G. M.; Mason, S. G. J. Colloid Interface Sci. 1976, 57, 517. (9) Takamura, K.; Goldsmith, H. L.; Mason, S. G. J. Colloid Interface Sci. 1979, 72, 385.
S0743-7463(96)00037-6 CCC: $12.00
recently developed technique called “colloidal particle scattering” is based on measuring the trajectories of particles suspended in a flowing fluid as they approach a single particle that is fixed to a flat surface.10 By examining the difference between the measured trajectory and that predicted by hydrodynamics alone, the parameters of the DLVO model can be inferred. The experimental results from this technique are consistent with the DLVO model as applied to the system studied.10 Another technique, based on inducing a magnetic dipole in ferrofluid droplets to produce attractive forces, has been used to determine the gap and nonmagnetic forces acting between ferrofluid oil droplets in water.11,12 This method allows measurements of the force with resolutions of 10-13 N and gaps as small as 20 nm; however, it is restricted to particles with a significant magnetic susceptibility. We have developed an experimental technique, called “differential electrophoresis”, which can be used to determine the normal and tangential forces between the surfaces of two colloidal particles. The concept as applied to a doublet of two spheres is shown in Figure 1. A doublet formed by the aggregation (e.g., by Brownian flocculation) of two particles with different ζ potentials acts as a dipole and rotates by electrophoresis toward alignment with an applied electric field. The electrophoretic rotation of a doublet provides information about the tangential forces acting between the surfaces of two particles. Previous work by the authors13 found that, for doublets composed of polystyrene latex spheres with carboxylate and sulfate surface groups, the doublets rotated as rigid bodies even though they should have been in a secondary minimum. The rigid body rotation indicates either that the spheres (10) van de Ven, T. G. M.; Warszynski, P.; Wu, X.; Dabros, T. Langmuir 1994, 10, 3046. (11) Leal-Calderon, F.; Stora, T.; Mondain-Monval, O.; Poulin, P.; Bibette, J. Phys. Rev. Lett. 1994, 72, 2959. (12) Mondain-Monval, O.; Leal-Calderon, F.; Phillip, J.; Bibette, J. Phys. Rev. Lett. 1995, 75, 3364. (13) Velegol, D.; Anderson, J. L.; Garoff, S. Langmuir 1996, 12, 675.
© 1996 American Chemical Society
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Figure 1. Schematic of a colloidal doublet in an applied electric field E∞. Sphere 1 is always taken to be the smaller sphere, so that β ) a1/a2 e 1. e is the unit vector from the center of sphere 1 to sphere 2, and θ is the angle from the applied field E∞ to e. δ is the gap between the spheres.
were not in a secondary minimum (i.e., DLVO theory failed) or, if the spheres were in a secondary minimum, that there existed tangential forces between the surfaces sufficient to produce an internal “restraining torque”.13 This paper focuses on determining the attractive force acting along the line of centers between two spheres. After a doublet aligns by electrophoretic rotation, the two spheres try to translate at different velocities; however, they cannot separate because they are constrained by colloidal forces to be in a doublet. This produces a tension (i.e., a normal force) within the doublet, which we call the “electrophoretic displacement force”. This force can be calculated exactly from the electrostatic and hydrodynamic equations, and it is proportional to the applied electric field (E∞) and the difference in ζ potentials (ζ2 - ζ1) on the spheres. At fields of about 200 V/cm, we were able to exert electrophoretic displacement forces as high as 50 pN on a doublet. This is more than an order of magnitude higher than the maximum attractive force predicted by DLVO theory for the systems studied. In this paper we report our attempts to break doublets of carboxylate/ sulfate latices which should have been in a secondary minimum according to the classical DLVO theory. In addition to the electrophoresis techniques, we also performed aggregation rate and resuspension experiments to connect the results to more traditional techniques. Dynamics of Colloidal Doublets in Electrophoresis Consider a doublet of two spheres with a gap δ between them, as shown in Figure 1. The unit vector e points from the center of sphere 1 to sphere 2, where sphere 1 is smaller than sphere 2. For this configuration the electrostatic and hydrodynamic equations have been solved numerically to give the electrophoretic velocities Ui of spheres 1 and 2.14-16 The results, valid for thin, unpolarized double layers, give U1 and U2 for arbitrary e and δ. The differential motion of the two spheres can be represented in terms of a rotation of e and a displacement of the gap δ between the surfaces of the particles:
motion between the particle surfaces.13 Measurements of the field necessary to break the doublet when it is aligned with the field (i.e., |dδ/dt| > 0) probe the interparticle forces acting along the line of centers. Electrophoretic Rotation. The angular velocity Ω is very sensitive to the kinematic condition of the spheres in the doublet (i.e., whether they behave as two hydrodynamically distinct rigid spheres or as a single rigid doublet).13 In scalar form the rotation rate of the doublet is expressed as17
Ω)-
dθ (ζ2 - ζ1)E∞ sin θ ) N dt ηL
(3)
where θ is the angle from E∞ to e in Figure 1, t is time, and η are the dielectric permittivity and viscosity of the liquid, and N is a dimensionless coefficient determined from solutions of the electrostatic and hydrodynamic (Stokes) equations. As L approaches infinity, N approaches 1. If the doublet rotates as a single rigid body, N is less than unity; but if the spheres act as separate hydrodynamic bodies (so they can rotate relative to each other), N is greater than unity. Approximate formulae for N are given in ref 13 for both the rigid-body and freelyrotating conditions. The rigidity of a doublet can thus be determined by experimentally finding N. For carboxylate/ sulfate doublets under various conditions (e.g., salt, pH, ζ potentials), such measurements have shown that the doublets rotate as rigid bodies, even though DLVO calculations predicted a secondary minimum for the doublets.13 If the spheres of a doublet are in a secondary minimum with a gap of fluid between them and yet rotate as a single rigid body (i.e., N < 1), then tangential forces must exist to balance the hydrodynamic force on the spheres.13 These tangential forces give rise to a “restraining torque” Ti that must be applied to sphere i in order to keep the doublet in a rigid-body conformation as it rotates. The restraining torque is determined from the grand hydrodynamic resistance matrix for two spheres.18 The results can be represented by the following empirical expressions:13
Ti ) 8πa22(ζ2 - ζ1)E∞gi(λ,β) ln λ
(4a)
g1 ) (0.555 - 0.582β + 0.396β2) + (0.104 - 0.231β + 0.191β2)(ln λ) + (0.0133 - 0.0132β + 0.0106β2)(ln λ)2 (4b) g2 ) (0.0462 - 0.329β + 0.652β2) + (0.0127 - 0.0824β + 0.133β2)(ln λ) + (0.00164 - 0.0107β + 0.0198β2)(ln λ)2 (4c)
where the angular velocity Ω ) e × (U2 - U1)/L and L is the center-to-center distance between the spheres. The difference in particle velocities is proportional to (ζ2 ζ1)E∞. Measurements of the rotation rate of e can be used to probe the tangential forces that resist the relative lateral
where λ ) 2δ/(a1 + a2) and β ) a1/a2. The functions gi in eqs 4b and 4c are within 5% of the precise numerical results for 0.0002 e λ e 0.2 and 0.2 e β e 1. In previous experiments,13 which demonstrated rigidbody rotation of doublets that were predicted to be in a secondary minimum, the restraining torques must have been in the range 100-2500 kT if indeed the spheres were in a secondary minimum. There is no well-established theory for the tangential forces between two colloidal spheres that would predict such a restraining torque. Electrophoretic Displacement. Once a doublet has nearly aligned, the two particles try to translate at
(14) Fair, M. C.; Anderson, J. L. Int. J. Multiphase Flow 1990, 16, 663. Corrigenda in Int. J. Multiphase Flow 1990, 16, 1131. (15) Keh, H. J.; Yang, F. R. J. Colloid Interface Sci. 1990, 139, 105. (16) Keh, H. J.; Yang, F. R. J. Colloid Interface Sci. 1991, 145, 362.
(17) Fair, M. C.; Anderson, J. L. Langmuir 1992, 8, 2850. (18) Kim, S.; Karrila, S. J. Microhydrodynamics: Principles and Selected Applications; Butterworth-Heinemann: Boston, 1991; Chapters 7 and 11.
de )Ω×e dt
(1)
dδ ) (U2 - U1)‚e dt
(2)
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Langmuir, Vol. 12, No. 17, 1996 4105
different velocities by electrophoresis because they have different ζ potentials. However, when the spheres are held in the doublet by attractive forces, they must move at the same velocity. This constraint produces an internal tension in the doublet which can be calculated a priori from the electrohydrodynamic equations. If there were no colloidal forces between the particles, they would translate according to the following equations, which were derived assuming zero hydrodynamic force and torque on the spheres (including the double layers)15
ζ1 (ζ2 - ζ1) p n (U1)ep ) (I - ee)]‚E∞ E∞ + [M12ee + M12 η η (5a) (U2)ep )
ζ2 (ζ2 - ζ1) p n (I - ee)]‚E∞ E∞ [M21ee + M21 η η (5b)
where the MRij’s are mobility coefficients and I is the identity tensor. The relative velocity along e is dδ/dt, and the electrophoretic part (U2 - U1)ep ) (U2 - U1)ep‚e is given by
(U2 - U1)ep )
(ζ2 - ζ1)E∞‚e kep η
(6)
p p - M21 ). Note that E∞‚e can be either where kep ≡ (1 - M12 positive or negative, depending on the direction of e. Keh and Yang15 provide numerical values for the Mpij as a function of λ and β. Loewenberg and Davis19 published the results of a lubrication analysis for the near-contact, axisymmetric electrophoresis of two spheres. Equation 6 describes the separation of the spheres due to electrophoresis only. Any additional forces between the spheres (e.g., colloidal forces) will produce an additional particle velocity determined by hydrodynamics. If an interparticle force Fint between the spheres acts along e such that F2 ) -F1 ) Fint (where Fi ) Fie), then we can define a hydrodynamic mobility coefficient khyd such that, in the absence of electrophoresis,
(U2 - U1)hyd )
Fint k 6πηa2 hyd
(8)
( ) kep khyd
(9)
where E∞ is the magnitude of the applied electric field. Equation 9 is valid for thin, unpolarized double layers; it was derived from the electrostatic and hydrodynamic equations (i.e., without assuming any particular model (19) Loewenberg, M. L.; Davis, R. H. J. Fluid Mech. 1995, 288, 103.
(10)
The expression for the displacement force (eq 9) could have been written for an arbitrary doublet direction e by omitting the absolute value signs and keeping the notation E∞‚e from eq 6. Equation 9 is the main result in this paper. It provides a way to determine the displacement force Fdisp from E∞ and (ζ2 - ζ1). If we define the critical attractive force Fcrit (defined here as positive) as the maximum attractive force able to hold two spheres together (e.g., from colloidal forces), then eq 9 can be used to experimentally determine Fcrit or a lower bound for Fcrit. In particular, if an electric field is applied such that Fdisp exceeds Fcrit, then the doublet should break into two spheres. If the electric field does not break the doublet, Fdisp is a lower bound for Fcrit. We define Ecrit as the electric field for which Fdisp ) Fcrit. The ability to determine Fcrit experimentally provides a way to test predictions from DLVO theory (see the section Colloidal Forces and DLVO Theory). To obtain the value of Fcrit from eq 9, a visible separation of the spheres in the doublet is required. Thus, the time necessary for two spheres to separate must be estimated. For a1 ) a2 ) a and a gap δ < 0.1a, the coefficient kep (see Appendix 1) can be approximated by kep ) 11.68/(2/λ + 11). Equation 6 can be used to estimate the separation time:
(ζ2 - ζ1)Eexcess dδ ) U2 - U1 ) kep dt η
(11)
where Eexcess ) E∞ - Ecrit. Integrating eq 11 from the equilibrium gap δ1 to a gap δ2 that is visible during the experiment and using the approximation for kep, we have
2a ln τsep )
As long as the doublet is stable (i.e., does not separate into two singlets), then U2 - U1 ) 0. By combining eqs 6-8, we calculate the interparticle force Fint necessary to stabilize the doublet against separation by electrophoresis. We identify the “electrophoretic displacement force” (Fdisp) as the force tending to separate (i.e., destabilize) the particles by electrophoresis, which means that Fdisp ) -Fint as long as the doublet does not break. When the doublet is aligned with the field,
Fdisp ) 6πa2|ζ2 - ζ1|E∞
kep ≈ -0.17 + 2.3β - 0.67β2 khyd
(7)
The linearity of the Stokes equations means we can add the relative velocities in eqs 6 and 7:
U2 - U1 ) (U2 - U1)ep + (U2 - U1)hyd
for the colloidal forces). Note that the displacement force is always positive at alignment. The ratio kep/khyd is O(1) and is relatively constant for gaps λ e 0.1. The mobility coefficients kep and khyd are given in Appendix 1. For λ e 0.1 and 0.2 e β e 1.0,
δ2 + 11(δ2 - δ1) δ1 η 11.68 (ζ2 - ζ1)Eexcess
(12)
Typical parameter values for our experiments are a ) 2.5 µm, δ1 ) 10 nm, δ2 ) 1 µm, r ) 80, ζ2 - ζ1 ) 2kT/e ) 50 mV, Eexcess ) 5 V/cm, and η ) 1 cP. Equation 12 gives τsep ≈ 0.2 s for these values. Thus, once the critical field is exceeded, the displacement between the spheres grows quickly and should be easily visible. It should be emphasized that eqs 3 and 9 of this paper, as well as the coefficients N and kep, are derived from the classical equations of electrokinetics (again, no model for the colloidal forces is necessary in our measurements). These equations are supported experimentally by diffusiophoresis experiments,20 dielectric response measurements,21,22 and mobility measurements.23 In ref 21 certain polystyrene latex particles purchased from Interfacial Dynamics Corporation (Portland, OR), the supplier of the particles used in the present study, were shown to exhibit classical electrokinetic behavior. In addition, Russell et (20) Ebel, J. P.; Anderson, J. L.; Prieve, D. C. Langmuir 1988, 4, 396. (21) Gittings, M. R.; Saville, D. A. Langmuir 1995, 11, 798. (22) Russell, A. S.; Scales, P. J.; Mangelsdorf, C. S.; White, L. R. Langmuir 1995, 11, 1553. (23) Russell, A. S.; Scales, P. J.; Mangelsdorf, C. S.; Underwood, S. M. Langmuir 1995, 11, 1112.
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al. have suggested reasons why some past experiments showing nonclassical behavior could be in error.22 Induced Electrical Polarization Force. When an electric field is applied to a doublet, an attractive force results from an induced dipole on each sphere. In our experiments the polystyrene latex spheres are dielectrics and the electrolyte solution is a conductor. These definitions follow from the characteristic time of the experiment (seconds) and the characteristic free charge relaxation times24 of the aqueous solution (microseconds) and latex particles (years). When an electric field is applied, positive charges accumulate on one side of the sphere and negative charges on the other. For two spheres the negative charges of one sphere are adjacent to the positive charges of the second sphere, inducing an attractive force. This force can be calculated by integrating the Maxwell stress tensor around one sphere. The electric potential (φ) outside the spheres (plus their double layers) is determined from Laplace’s equation
∇2φ ) 0
far field (r f ∞):
φ f -E∞‚r
∫∫(fEE - 21fE‚EI)‚n dS
(13c)
(14)
F1 ) 4
2
|
θ - f⊥ sin2 θ)e + fΓ sin 2θeθ] (15)
In eq 15 L is the center-to-center distance between the spheres. The unit vector eθ is for the θ coordinate (see Figure 1). The f coefficients are given in ref 27. For λ < 3 these coefficients can be approximated as
f| ) 0.86 + 0.124(ln λ) + 0.012(ln λ)2
For λ > 3 all f coefficients approach the limiting value of 1. By using the electric potential calculated from a method of reflections,28 we have obtained a force along e that agrees with eq 15 for large λ (i.e., where f| ) 1). We emphasize that p does not affect the final result for F1 for an insulating sphere in a conducting fluid. For the case of polystyrene spheres in water, the attractive dipole force predicted by eq 15 when the doublet is parallel to the applied field is the following for λ < 0.01:
Fdipole ) -0.66fa2E∞2
(17)
(13b)
where E is the local electric field at the surface S1, which we place at the outer surface of the double layer. The solution to this problem is mathematically the same as that for a dielectric particle in a dielectric fluid with R ) p/f ) 0; this problem has been solved.26,27 (We note that while p cannot physically be zero, the mathematics for the solved problem are the same as those for the one we are trying to solve when p ) 0.) For equal size spheres, the dipole force on sphere 1 due to sphere 2 is26
(La) [(2f cos
(16c)
Fdipole aE∞ ) Fdisp 41.0|ζ2 - ζ1|
S1
3πfa2E∞2
fΓ ) 0.973 + 0.0183(ln λ) + 0.00244(ln λ)2
Compared to the electrophoretic displacement force, the induced dipole force is small:
The unit normal n points out of the spheres; r is the position vector from the center of sphere 1. The permittivity of the fluid f is assumed uniform, even in the double layer, since the fields in the double layer do not exceed 106 V/cm and should not orient the water molecules to a great extent.25 If the doublet in Figure 1 is aligned with E∞, and sphere 1 is at the origin, then the polarization force F1 on sphere 1 is24
F1 )
0.0078(ln λ)3 (16b)
(13a)
The boundary conditions are
on each sphere: ∇φ‚n ) 0
f⊥ ) 1.065 - 0.126(ln λ) + 0.070(ln λ)2 +
(16a)
(24) Woodson, H. H.; Melcher, J. R. Electromechanical Dynamics. Part II: Fields, Forces, and Motion; Krieger Publishing: Malabar, FL, 1968; Chapter 8. (25) Hunter, R. J. Zeta Potential in Colloid Science: Principles and Applications; Academic Press: New York, 1981; Chapter 5 (p 208) and Chapter 8 (p 330). (26) Klingenberg, D. J.; Zukoski, C. F., IV. Langmuir 1990, 6, 15. (27) Klingenberg, D. J. M.S. Thesis, University of Illinois, 1988.
| |
(18)
For ζ2 - ζ1 ) 40 mV, a ) 2.5 µm, and E∞ ) 200 V/cm, the ratio is 0.03. It should be noted that this is not the case for conducting spheres. Indeed, the electric field intensification in the gap makes the induced dipole force large (sometimes dominating) for conducting spheres.29 Colloidal Forces and DLVO Theory An equation giving the potential of mean force Ψ as a function of gap δ between the spheres was derived by Hogg et al.30 The input parameters for their model are the electrostatic surface potentials ψ1 and ψ2 on the spheres, the Debye parameter κ, and an effective Hamaker constant A. Their result assumes that the Hamaker constant is independent of the separation δ between the spheres, uses the Derjaguin approximation for slightly curved surfaces,2 and is valid for low surface potentials |ψi| < kT/e. In this paper we calculate the potential of mean force Ψ(δ) in a more rigorous way. First, the analysis of Ohshima et al.31 is used to consider larger ζ potentials. Second, the retarded Hamaker constant A(δ) is calculated using the full Lifshitz theory (including the correction for solution conductivity).32,33 The results for A(δ) calculated from Lifshitz theory are summarized in Appendix 2. These results are listed for both plane-plane interactions and sphere-sphere interactions at various salt concentrations. Ψ(δ) is calculated for two spheres using the Derjaguin approximation, and the colloidal force between the spheres is then determined using Fcol ) -∇Ψ. In all calculations we assume ζi ) ψi. Sample calculations for Ψ(δ) and F(δ) are plotted in Figure 2, and the DLVO calculations for our experimental conditions are summarized in Table 2. The ζ potentials (ζS and ζC) were determined experimentally from the electrophoretic mobilities of the single particles. (28) Chen, S. B.; Keh, H. J. AIChE J. 1988, 34, 1075. (29) Davis, M. H. Q. J. Mech. Appl. Math. 1964, XVII, 499. (30) Hogg, R.; Healy, T. W.; Fuerstenau, D. W. Faraday Soc. Trans. 1966, 62, 1638. (31) Ohshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 89, 484. (32) Parsegian, V. A. Long Range van der Waals Forces. In Physical Chemistry: Enriching Topics from Colloid and Surface Science; van Olphen, H., Mysels, K. J., Eds.; Theorex: La Jolla, CA, 1975. See especially Section 4. (33) Prieve, D. C.; Russel, W. B. J. Colloid Interface Sci. 1988, 125, 1.
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Langmuir, Vol. 12, No. 17, 1996 4107
Table 1. Mobility Coefficients for Displacement β ) 1.0
β ) 0.5
β ) 0.2
λ ) 2δ/ (a1 + a2)
kep
khyd
kep/khyd
kep
khyd
kep/khyd
kep
khyd
kep/khyd
0.001 0.005 0.01 0.05 0.1 0.5 1 5 10 50 100 ∞
0.005 71 0.027 94 0.054 46 0.226 17 0.373 27 0.778 17 0.908 09 0.994 06 0.998 84 0.999 99 1.000 00 1.000 00
0.003 96 0.019 28 0.037 44 0.157 46 0.270 49 0.747 35 0.984 21 1.574 19 1.750 80 1.942 32 1.970 59 2.000 00
1.439 79 1.449 33 1.454 71 1.436 36 1.379 99 1.041 24 0.922 66 0.631 47 0.570 50 0.514 84 0.507 46 0.500 00
0.005 05 0.025 74 0.050 48 0.212 51 0.353 90 0.755 99 0.888 71 0.992 15 0.998 45 0.999 98 1.000 00 1.000 00
0.006 69 0.032 45 0.062 89 0.261 33 0.444 37 1.186 45 1.607 10 2.431 65 2.667 74 2.923 10 2.960 79 3.000 00
0.754 93 0.793 00 0.802 65 0.813 18 0.796 42 0.637 19 0.552 99 0.408 01 0.374 27 0.342 10 0.337 75 0.333 33
0.004 79 0.023 23 0.045 36 0.192 09 0.323 26 0.716 16 0.824 10 0.986 38 0.997 30 0.999 97 1.000 00 1.000 00
0.021 33 0.102 52 0.196 60 0.771 06 1.251 90 2.913 26 4.088 57 5.287 27 5.584 70 5.903 88 5.950 98 6.000 00
0.224 35 0.226 63 0.230 70 0.249 12 0.258 21 0.245 83 0.201 56 0.186 56 0.178 58 0.169 37 0.168 04 0.166 67
Table 2. Predictions from DLVO Theory for the Experimental Conditions in This Paper (see Figure 2 for Definitions of Some Quantities, Such as Predicted Fcrit)a doublet typeb
KCl (M)
ζC (mV)
ζS (mV)
predicted DLVO Fcrit (pN)
predicted DLVO δcritκ
predicted DLVO Ψmin/kT
predicted DLVO δminκ
type 1 (a-d) type 1 (e-f) type 1 (g-j) type 2 (k-n)
0.001 0.005 0.020 0.009
-61 ( 7 -55 ( 3 -33 ( 6 -19 ( 6
-84 ( 5 -94 ( 4 -78 ( 5 -121 ( 6
≈0 0.07 1.46 0.12
150 12.3 9.8 10.4
-0.40 -0.65 -5.94 -0.67
78 11 8.0 9.0
a The pH for these experiments is 3.0. The predicted electrostatic repulsion barrier is from 200 kT to 7000 kT. b Type 1 indicates a doublet composed of a 5.0 µm sulfate sphere and a 5.1 µm carboxylate sphere. Type 2 indicates a doublet composed of a 2.75 µm sulfate sphere and a 1.17 µm carboxyl sphere. The letters in parentheses are labels for all doublets examined. For example, (a-d) means four doublets a, b, c, and d were examined under the conditions listed.
based theory. It is also possible that “hairy layers” are important; however, experimental evidence indicates that the average thickness of a polymer layer would be less than 1 nm for the salt concentrations in this study (1-20 mM).36 The particles used in ref 36 were polystyrene latex particles purchased from Interfacial Dynamics Corporation. Rates of Aggregation. During aggregation single particles are depleted as they form doublets and higher aggregates. The aggregation rate can be found experimentally for the colloidal system and compared to the aggregation rate predicted by the Smoluchowski equation for diffusion-limited flocculation.37 At short times, when mostly doublets are forming and not higher aggregates, one can experimentally determine the stability ratio W using the equation37 8 /3kT 1 1 - ) t c c0 ηW
Figure 2. Curves for the potential of mean force and the force between two spheres, according to classical DLVO theory. These curves are calculated for a doublet of type 1 in Table 2 at 0.020 M KCl. In the figure, kT/κ-1 ≈ 2 pN. The critical force Fcrit occurs at a gap δcrit, for which d2Ψ/dδ2 ) 0.
Using the more rigorous form of DLVO theory, we can predict what Fcrit should be for our experimental conditions and test the DLVO model. Figure 2 shows Fcrit, δcrit (the gap at which Fcrit occurs), Ψmin (the potential of the secondary minimum), and δmin (the gap at which Ψmin occurs) for a typical doublet in our experiments (see Table 2; the doublet in Figure 2 is for 0.020 M KCl). Although the effects of surface roughness34 and charge heterogeneity35 could be important to the experiments described in this paper, the current models describing these effects are insufficient to make quantitative predictions, and such effects are generally not included in DLVO-
(19)
where c is the singlet concentration (number of singlets per volume suspension) and c0 is the concentration at t ) 0. W depends on Ψ(δ) and is predicted to grow exponentially with the electrostatic energy barrier between the spheres.38 Sample calculations given in ref 13 show that characteristic flocculation times over the electrostatic energy barrier into the primary minimum were predicted to be more than 10100 years for certain conditions. In all our experiments, the energy barrier opposing flocculation into a primary minimum was essentially infinite. Experiments The differential electrophoresis experiments were performed in a microelectrophoresis apparatus which included (1) a (34) Kostoglou, M.; Karabelas, A. J. J. Colloid Interface Sci. 1995, 171, 187. (35) Grant, M. L.; Saville, D. A. J. Colloid Interface Sci. 1995, 171, 35. (36) Seebergh, J. E.; Berg, J. C. Colloids Surf. A 1995, 100, 139. (37) Hiemenz, P. C. Principles of Colloid and Surface Chemistry, 2nd ed.; Marcel Dekker: New York, 1986; Chapter 11. (38) Prieve, D. C.; Ruckenstein, E. J. Colloid Interface Sci. 1980, 73, 539.
4108 Langmuir, Vol. 12, No. 17, 1996 microscope and microelectrophoresis cell, (2) a current source, and (3) video microscopy equipment. An Olympus IMT-2 microscope was positioned on its back with a metal support. The microelectrophoresis cell (Rank Brothers, England) was a rectangular glass capillary (1 × 10 mm2 cross section, about 5 cm long) with two ports for platinum electrodes. A Keithley 220 current source produced a dc constant current, and images from the microscope were captured by a Sony 77 CCD camera connected to a Panasonic SVHS video recorder. The resulting videotape was analyzed frame-by-frame using the images obtained from a thermal printer (Mitsubishi Video Copy Processor P71U). The final magnification of the doublets ranged from 470 to 705, and the field of view ranged from 375-705 µm. All experiments were done at temperatures between 20 and 25 °C, and during any single experiment the temperature remained constant to within (1 °C. All glassware was soaked overnight in concentrated nitric acid and rinsed with deionized water from a MilliQ water purification system (Millipore). The conductivity of the water was typically less than 1 µS/cm. This water was also used to make the suspensions and solutions and to perform cleaning of the particles. The polystyrene latex particles were cleaned using a combination of sedimentation/decantation and dialysis.9,39 The particles were centrifuged, and the supernatant was decanted; this process was repeated eight times. After this the particles were dialyzed for seven to eight days in 6000-8000 MW dialysis tubing, with the water being replaced daily. The tubing had been boiled three times prior to use. The latex particles (Interfacial Dynamics Corportation (IDC), Portland, OR) were of various sizes and had various surface groups: 1.17 µm diameter carboxyl (Catalog # 10-249-93, surface charge density 18.9 µC/cm2), 2.75 µm sulfate (Catalog # 10-295-22, surface charge density 9.8 µC/cm2), 5.00 µm sulfate (Catalog # 553,1, surface charge density 5.98 µC/ cm2), and 5.10 µm carboxylate40 (Catalog # 433, surface charge density 118 µC/cm2). The sizes of other IDC spheres had previously been verified using sedimentation techniques.13 The doublets were formed by the Brownian coagulation of the latices. Before mixing the particles for coagulation, they were sonicated for 15-30 s to break any homoaggregates. To make the suspension, aqueous solutions were mixed with the proper salt concentration and the proper pH, and two different types of particles (e.g., carboxylate and sulfate, which have different ζ potentials) were added. Since carboxyl groups have a pKa of 4 and sulfate groups have a pKa of 2, the difference in ζ potentials of the spheres was enhanced by adding enough hydrochloric acid to obtain a pH of 3.0. The concentration of the particles was typically 106 to 107 per mL. The suspension was poured into the electrophoresis cell and sonicated again to assure that very few aggregates existed at the beginning of the experiment. The cell with the electrodes was closed to minimize natural convection during the aggregation process. After the cell was sonicated, it was mounted onto the microscope, and using the video equipment, 4-10 microscope frames at 9 cross sections (i.e., 36-90 frames total) through the cell were recorded. By sampling the particle concentrations in this way, the number of singlets, doublets, and higher aggregates in the cell could be counted at any time. The suspension was allowed to aggregate for 1-5 h, and the aggregates were counted again in a similar manner. For one experiment an additional count was made at an intermediate time to check that the number of doublets was proportional to time. These aggregate counts were later used to examine the aggregation kinetics and to obtain an absolute stability ratio W. The aggregate counts were also used to check that the selection process of examining doublets was not biased by using preexisting, irreversibly bound doublets. The ratio of the number of doublets at the time of the experiment to the number of doublets at time zero was typically 3-4. In most cases three or four doublets were sampled, so the probability that all sampled doublets were irreversibly bound from a time before the experi(39) Zukoski, C. F., IV; Saville, D. A. J. Colloid Interface Sci. 1985, 107, 322. (40) According to IDC, carboxyl particles have COOH groups attached directly to the polymer colloid. Carboxylate particles have COOH groups on polymer strands, which are then grafted onto the polymer colloid. In addition, the density of COOH groups on the carboxylate particles is many times greater than that on the carboxyl.
Velegol et al. ment was 1/27 to 1/256. Thus, most of the doublets sampled during the experiment were formed during the aggregation period in the cell. After the aggregation process, the electrophoresis experiments were performed. Once a doublet was located optically, electrophoretic rotations were done on that doublet. In the case of doublets with different size spheres, it was apparent before the rotation whether the doublet was a heterodoublet; however, when the spheres were of the same size, the only way to see if the doublet was a heterodoublet was to apply a field and check for electrophoretic rotation. If the doublet did not rotate, then that doublet was a homodoublet (see eq 3). When an electric field had been applied to a doublet and a homodoublet was found, we started the experiment over, emptying the cell and reloading it with fresh suspension. This helped to avoid the chance that the field had broken all the “weak” doublets and left the others, biasing the results for the normal force. After finding a heterodoublet, several electrophoretic rotations were recorded so that the doublet could be checked for rigidity. Whenever possible, the rotations were done at applied fields at least a factor of two less than that expected from DLVO theory to break the doublet. Next the electrophoretic displacement experiment was performed on the same doublet. Starting at an electric field of zero, the field was gradually increased. At high fields the doublets translated rapidly, so the stage was mechanically moved to keep the doublets in the field of view. Usually the field was increased to 100 V/cm in 5-10 s, and typically electric fields of over 100 V/cm were reached. The displacement experiments sometimes heated the cell significantly (especially at high salt concentrations), causing a large amount of convection. Although this convection was not high enough to apply significant hydrodynamic extension forces to any doublet (shear rates less than 0.01 s-1), the convection did make it more difficult to keep the doublet in the field of view for more than 10-15 s. After completing the displacement experiments, the cell was allowed to sit for a few minutes until the convection subsided. Then mobility experiments were performed using roughly 10 singlets of each type of sphere. The particle mobilities were converted to ζ potentials using the Smoluchowski equation, since for most experiments κa ) 250-1000 and the ζ potentials were generally less than 4kT/e in magnitude. The stability of the ζ potentials with time was checked for several cases by examining the mobilities before and after the experiments; these values were the same to within measurement uncertainty. Resuspension experiments were performed for several of our systems (see ref 13 for more details). After allowing the particles to aggregate for the duration of the experiment, a sample was pipetted slowly into dialysis tubing and placed in DI water. Dialysis was performed for roughly 24 h, with the water being changed two or three times to lower the KCl concentration to 10-5 to 10-4 M. The suspension in the tubing was then analyzed for aggregate counts as described before. The aggregate counts before and after dialysis were compared to see if the latex particles had resuspended.
Results and Discussion For each colloidal system we examined, we performed five experiments: electrophoretic rotation (to find N), electrophoretic displacement (to attempt to break doublets), electrophoretic mobility (to determine the ζ potentials of the particles), aggregation rate (to obtain stability ratios), and resuspension (to determine whether the doublets were irreversibly bound). As discussed below, the combined results from these experiments indicate that the doublets were not in a secondary minimum, contrary to the predictions from DLVO theory. Tables 2 and 3 summarize the results for all electrophoretic rotation and displacement measurements. In no case did a doublet break in the electrophoretic displacement experiments. Thus, in every experiment performed, the maximum colloidal attractive force (Fcrit) holding the doublet together was greater than the highest displacement force (Fdisp) achieved by electrophoresis. For example, for the type 1 doublets at 0.020 M KCl (see Table 3), the DLVO model underestimates Fcrit by at least a
Forces between Polystyrene Latex Spheres
Langmuir, Vol. 12, No. 17, 1996 4109
Table 3. Results of Electrophoretic Rotation and Displacement Experimentsa
doublet
Nb
Emax (V/cm)
a: type 1 b: type 1 c: type 1 d: type 1 e: type 1 f: type 1 g: type 1 h: type 1 i: type 1 j: type 1 k: type 2 l: type 2 m: type 2 n: type 2
0.59 ( 0.22 0.72 ( 0.27 0.68 ( 0.25 0.55 ( 0.21 0.73 ( 0.09 0.68 ( 0.09 0.55 ( 0.10 0.78 ( 0.14 0.81 ( 0.14 0.77 ( 0.13 0.31 ( 0.04 0.41 ( 0.04 0.42 ( 0.05 c
125 139 111 139 108 108 222 174 144 123 108 167 167 167
Fdisp (pN) 14 ( 5 16 ( 6 13 ( 5 16 ( 6 21 ( 3 21 ( 3 50 ( 8 39 ( 6 32 ( 5 28 ( 5 14 ( 1 22 ( 2 22 ( 2 22 ( 2
predicted DLVO Fcrit (pN)d 0
0.07 1.46
0.12
a The fourteen doublets analyzed are labeled a-n and are of type 1 or 2 as given in Table 2. Emax is the maximum field that was applied to the doublet during an electrophoretic displacement experiment. In all cases the applied electrophoretic displacement force Fdisp was much greater than the critical force Fcrit predicted from DLVO theory. b The theoretical values of N for rigid doublets are N ) 0.64 for a size ratio β ) 1 (type 1 doublets) and N ) 0.42 for β ) 0.43 (type 2 doublets). c No N value was measured for doublet n. d The critical force would change little if both ζ potentials were increased or decreased by one standard deviation (see Table 2). For example, if the ζ potentials for doublets g-j were -73 and -27 mV (instead of -78 and -33 mV), Fcrit would increase by only 10%. Ψmin would also increase by only 10%. For the doublets at the lower concentrations, the change is even less.
factor of 50/1.46 ) 34. For the DLVO model to give an Fcrit this high, the Hamaker constant A(δ) would have to be more than a factor of 10 greater than predicted by Lifshitz theory (i.e., Aeff(δ)0) ≈ 35kT), corresponding to a secondary minimum of -170kT. This Hamaker constant would be greater than that of colloidal gold.1 For the electrophoretic rotation experiments, all values of N were less than unity and close to the theoretical predictions for rigid bodies. From DLVO theory we expected the doublets to be in a secondary minimum (i.e., with a gap of fluid between the spheres) and thus to be freely-rotating. Since DLVO theory was unable to predict correctly the normal force between the spheres, we did not calculate the restraining torque (from eq 4) necessary to make a doublet rigid if it were in a secondary minimum configuration. Table 4 lists the stability ratios W estimated from the measured flocculation rates. Note that W was O(1), indicating rapid flocculation kinetics. According to DLVO theory, none of the doublets examined should have been in a primary energy minimum38 (since the electrostatic repulsive barrier should have been 200kT up to 6900kT); thus, to obtain rapid flocculation, the doublets should have fallen into a secondary minimum. However, none of the doublets in these systems could be broken by electrophoretic displacement. In some cases (e.g., 0.001 M KCl) no secondary minimum was predicted from DLVO theory. For these cases rapid flocculation would not be expected, but it occurred anyway. Thus, the aggregation results are inconsistent with the DLVO model. The resuspension experiments do not have a clear interpretation. A significant number of the carboxylate/ sulfate doublets (see Table 4) resuspended, showing that they were not all irreversibly bound. Although this finding is consistent with a secondary minimum, it is possible that the resuspension occurred due to a steric layer on the surface of the carboxylate particles. On the other hand, all the carboxyl/sulfate doublets were irreversibly bound, even though DLVO theory predicted them to be in a
secondary minimum. Instead of resuspending, these systems formed large aggregates in which the number of spheres could not be counted. Thus, the resuspension numbers for the carboxyl/sulfate doublets do not appear in Table 4. We note that in several instances we visually saw aggregates forming; two spheres would collide to form a doublet, or a doublet and a sphere would collide to form a triplet. Whenever this situation occurred, we tried to break the aggregate with a high electric field but could not. In addition, all triplets (and higher aggregates) in these experiments remained rigid during observation (one study in the literature does report flexible triplets41). Collectively these results cannot be explained in terms of classical DLVO theory. One possible explanation is that the charge on the surface of the particles is sufficiently heterogeneous to allow the spheres to fall into a primary minimum due to local regions of low ζ potential. A brief discussion of this heterogeneity of charge has appeared before.13 Conclusions The technique of differential electrophoresis can be used to determine colloidal forces between two particles in a doublet. Previous measurements of the electrophoretic rotation of latex doublets indicated a rigid structure, which is inconsistent with the DLVO prediction of a secondary minimum. The same results were obtained in the present work. The focus of this paper is on using the electrophoretic displacement force to break doublets when they are aligned with an applied electric field. Displacement forces as large as 50 pN were applied without breaking the doublets, even though predictions of the critical force based on DLVO theory were about 1.5 pN or less. The rigid-body rotation of the doublets (N < 1) means that the surfaces of the particles were unable to move laterally relative to each other. If the doublets were in a secondary minimum, as predicted from DLVO theory, then there must have been tangential forces acting on the adjacent surfaces to hold their relative positions. The rotation and displacement experiments together imply that these latex doublets were not in a secondary minimum. Classical DLVO theory, even with the use of the Lifshitz theory for the Hamaker constant, is significantly in error for our particles. Differential electrophoresis provides important information about both the tangential and normal forces acting between two colloidal particles confined to a doublet which is suspended in a fluid. We are not aware of any other technique that has this capability. As such, differential electrophoresis is a promising tool for studying the forces which hold particles together. Acknowledgment. This work was supported under NASA Microgravity Grant NAG8-964 and NSF Grant CTS-9420780. We are also grateful to Hercules, Inc., for funding some of this work. D.V. acknowledges support from a National Science Foundation Ph.D. Fellowship. J.L.A. also acknowledges the financial support of the University of Melbourne and very helpful discussions with the staff there. Appendix 1: Mobility Coefficients The mobility coefficients kep for electrophoresis are calculated from ref 15. Empirical fits of the form Mpij ) (1 + aλ)/(b + cλ) were used to interpolate values of kep for (41) van de Ven, T. G. M.; Mason, S. G. J. Colloid Interface Sci. 1976, 57, 535.
4110 Langmuir, Vol. 12, No. 17, 1996
Velegol et al.
Table 4. Results of Aggregation and Resuspension Experiments for “Type 1” Doublets in Table 2a expt type
spheresb before/after
singlets before/after
doublets before/after
345 ( 20/383 ( 17 488 ( 18/402 ( 20 309 ( 18/288 ( 5 94.3 ( 5.9/72.3 ( 5.0
324 ( 21/239 ( 20 458 ( 16/220 ( 2 282 ( 21/179 ( 8 47.0 ( 1.7/53.0 ( 5.0
8.3 ( 1.2/44.0 ( 2.9 14.0 ( 2.1/45 ( 4.1 12.3 ( 2.3/34.0 ( 2.0 13.7 ( 1.7/6.7 ( 1.3
time of KCl expt (h) conc (M)
aggregation aggregation aggregation resuspension
4.0 3.0 3.5 24
0.001 0.005 0.020 0.020
number average sizec before/after
W
1.034 ( 0.010/1.287 ( 0.038 3.0 1.034 ( 0.010/1.391 ( 0.029 0.7 1.048 ( 0.009/1.284 ( 0.017 1.6 1.421 ( 0.020/1.181 ( 0.022
a The number of singlets and doublets are listed for “before” the experiment and “after” its completion. b The “spheres” column indicates the summation of all spheres seen in singlets, doublets, triplets, etc.; it serves as a mass balance for the experiments. c The number average size of the aggregates is the ratio of total spheres (from the “spheres” column) to the number of aggregates (e.g., a triplet is one aggregate).
Table 5. Hamaker Constants for Plane-Plane PS A(h)/kT at given KCl conc (mM)/κ-1 (nm) h (nm) 0.10 0.18 0.32 0.56 1.00 1.78 3.16 5.62 10.00 17.78 31.62 56.23 100.00 177.83 316.23 562.34 1000.00
0.0001/ 0.001/ 964 305 3.520 3.515 3.504 3.478 3.421 3.310 3.113 2.807 2.403 1.953 1.540 1.224 1.021 0.907 0.832 0.718 0.488
3.520 3.515 3.504 3.478 3.422 3.310 3.114 2.810 2.407 1.962 1.556 1.245 1.022 0.813 0.528 0.201 0.026
0.01/ 96.4
0.1/ 30.5
1/ 9.64
10/ 3.05
100/ 1000/ 0.964 0.305
3.520 3.515 3.504 3.478 3.423 3.313 3.118 2.819 2.424 1.983 1.557 1.151 0.718 0.296 0.067 0.016 0.004
3.520 3.516 3.505 3.481 3.427 3.322 3.135 2.840 2.424 1.890 1.254 0.635 0.257 0.110 0.045 0.015 0.004
3.521 3.518 3.510 3.490 3.444 3.343 3.135 2.746 2.121 1.373 0.793 0.449 0.235 0.110 0.045 0.015 0.004
3.526 3.527 3.526 3.511 3.444 3.249 2.832 2.230 1.660 1.187 0.770 0.448 0.235 0.110 0.045 0.015 0.004
3.542 3.548 3.527 3.417 3.141 2.733 2.372 2.044 1.638 1.187 0.770 0.448 0.235 0.110 0.045 0.015 0.004
3.543 3.455 3.224 2.901 2.681 2.547 2.349 2.043 1.638 1.187 0.770 0.448 0.235 0.110 0.045 0.015 0.004
the near field, and the O(L-7) reflection results were used to interpolate kep for the far field.28 The khyd’s defined by eq 7 are calculated from the results in Chapters 7 and 12 of the book by Kim and Karrila.18 The fluid is quiescent in the far field, with Ui ) Uie and Fi ) Fie (i.e., the hydrodynamic force on sphere i and the velocity of sphere i are along the line of centers). Since no external forces are applied, and only the forces internal to the doublet are accounted for (e.g., colloidal forces), we set F ) F1 ) -F2. There are no external torques, so Ωi ) 0. Using the grand resistance matrix formalism, the forces on the spheres are
F1 ) -η[A11‚U1 + A12‚U2]
(20)
F2 ) -η[A21‚U1 + A22‚U2]
(21)
where ARβ ) XARβee + YARβ(I - ee). These equations give
(U2 - U1)hyd )
A F(X11
A A A + X12 + X21 + X22 A A A A η(X11 X22 - X12 X21 )
(22)
If we define XAij ′ ) XAij /6πa2, then eq 7 is the same as eq 22 with
khyd )
A A A A ′ + X12 ′ + X21 ′+ X22 ′ X11 A A A A X12 ′X21 ′ - X11 ′X22 ′
Table 6. Effective Hamaker Constants between Two Spheres, As Used in Eq 24 Aeff(δ)/kT at given KCl conc (mM)/κ-1 (nm) δ (nm) 0.10 0.18 0.32 0.56 1.00 1.78 3.16 5.62 10.00 17.78 31.62 56.23 100.00 177.83
0.0001/ 0.001/ 964 305 3.463 3.420 3.349 3.236 3.066 2.824 2.509 2.138 1.755 1.409 1.135 0.938 0.794 0.662
0.1/ 30.5
1/ 9.64
10/ 3.05
100/ 1000/ 0.964 0.305
3.463 3.420 3.349 3.236 3.065 2.821 2.500 2.118 1.706 1.302 0.924 0.575 0.275 0.086
3.463 3.420 3.348 3.233 3.057 2.802 2.458 2.027 1.533 1.020 0.562 0.249 0.102 0.042
3.462 3.417 3.341 3.215 3.015 2.712 2.284 1.745 1.170 0.695 0.389 0.206 0.099 0.042
3.454 3.398 3.298 3.124 2.841 2.429 1.922 1.419 0.997 0.651 0.385 0.206 0.099 0.042
3.412 3.308 3.125 2.842 2.479 2.104 1.749 1.376 0.994 0.651 0.385 0.206 0.099 0.042
3.239 3.026 2.763 2.517 2.306 2.061 1.745 1.376 0.994 0.651 0.385 0.206 0.099 0.042
the gap δ where khyd was the same for both. No smoothing procedure was used at the matching point. Table 1 lists the values for kep and khyd. Appendix 2: Calculation of Hamaker Constants This appendix includes results for the retarded Hamaker constant between polystyrene surfaces immersed in univalent salt solutions ranging from 10-7 to 1.0 M for gaps of 0.1-1000 nm. Calculations for the Hamaker constant between two planes were done using the scheme in ref 32, especially section 4 of that paper (see also ref 33). Reference 32 also shows how to account for the solution conductivity. In order to calculate the effective Hamaker constant between two spheres, the Derjaguin approximation was used. The potential of mean force due to van der Waals forces is defined in two ways:2
Ψ)-
∫0a
1 6
(
)
Aeff a1a2 A(h) r dr ) 9 2 6δ a1 + a2 h
2
(24)
Here δ is the gap between the spheres, h is a local gap between Derjaguin rings, A(h) is the retarded Hamaker constant between two planes, and Aeff is the effective Hamaker constant between two spheres. Using the parabolic approximation between spheres, h ) δ + r2/2a1 + r2/2a2, one obtains
(23)
For different size ratios β ) a1/a2, the limiting value of khyd in the far field is 1 + 1/β. In determing the value of khyd, lubrication results were used for the near field and reflection results were used for the far field. All results are from ref 18. The locations of the near field and far field regions were determined by
3.463 3.420 3.349 3.237 3.066 2.824 2.508 2.136 1.749 1.393 1.098 0.857 0.638 0.410
0.01/ 96.4
∫δ∞
Aeff ) δ
A(h) h2
dh
(25)
Thus, Aeff is a function of the gap δ between the spheres. Tables 5 and 6 give the results for the plane-plane and sphere-sphere Hamaker constants at a temperature of 298 K. LA960037Q
