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Force Measurements between Colloidal Particles of Identical Zeta Potentials Using Differential Electrophoresis Gretchen L. Holtzer and Darrell Velegol* Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16803 Received October 23, 2002. In Final Form: February 19, 2003 The technique of differential electrophoresis can be used to measure subpiconewton forces between Brownian colloidal particles. Previously, this technique was limited by two factors: (1) forces could be measured only between particles with different average zeta potentials and (2) the particles had to be large enough for observation under a light microscope. This paper focuses on overcoming factor 1 by using a third particle as a “handle”, allowing forces to be measured between particles with identical zeta potentials. Semianalytical solutions to the electrokinetic equations for three spheres were developed, with results similar to those for two spheres, enabling the ready interpretation of experiments. Interparticle forces were then measured for charged polystyrene latex particles 1.5 and 4.5 µm in diameter over a range of aqueous solution conditions. One important finding was that while nearly-touching particles aggregate irreversibly in the absence of the polyelectrolyte sodium polystyrene sulfonate (NaPSS), the addition of trace quantities of NaPSS enables the particles to form triplets that break with forces of O(10 pN).
Introduction Bulk solution properties such as stability, ordering, and rheology are governed by interparticle forces. Such forces have usually been described by classical DerjaguinLandau-Verwey-Overbeek (DLVO) theory, which has been the standard model for describing interparticle forces for over half a century.1,2 This model assumes that two forces control colloidal interactions: electrostatic forces, which are typically repulsive for like-charged particles, and van der Waals (or dispersion) forces, which are typically attractive. Unfortunately, DLVO theory often fails to accurately describe particle interactions and forces.2-5 This may be due to the presence of other effects in the colloidal system such as depletion,6-17 solvation,2,18-24 and steric repul(1) Verwey, E. J. W.; Overbeek, H. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (2) Israelachvili, J. N. Intermolecular & Surface Forces, 2nd ed.; Academic Press: San Diego, CA, 1992. (3) Hunter, R. J. Foundations of Colloidal Science, Volumes I and II; Clarendon Press: New York, 1992. (4) Behrens, S. H.; Borkovec, M.; Schurtenberger, P. Langmuir 1998, 14, 1951. (5) Behrens, S. H.; Christl, D. I.; Emmarzael, R.; Schurtenberger, P.; Borkovec, M. Langmuir 2000, 16, 2566. (6) Thwar, P. K.; Velegol, D. Langmuir 2002, 18, 7328. (7) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (8) Peich, M.; Walz, J. Y. Langmuir 2000, 16, 7895. (9) Chattergee, A. P.; Schweizer, K. S. J. Chem. Phys. 1998, 109, 10464. (10) Sharma, A.; Walz, J. Y. J. Colloid Interface Sci. 1994, 168, 485. (11) Ye, X.; Narayan, T.; Tong, P.; Huang, J. S.; Lin, M. Y.; Carvalho, B. L.; Fetters, L. J. Phys. Rev. E 1996, 54, 6500. (12) Goetzelmann, B.; Evans, R.; Dietrich, S. Phys. Rev. E 1998, 57, 6785. (13) Bechinger, C.; Rudhardt, D.; Leiderer, P.; Roth, R.; Dietrich, S. Phys. Rev. Lett. 1999, 83, 3960. (14) Hanke, A.; Eisenriegler, E.; Dietrich, S. Phys. Rev. E 1999, 59, 6853. (15) Tuinier, R.; Vliegenthart, G. A.; Lekkekerker, H. N. W. J. Chem. Phys. 2000, 113, 10768. (16) Mendez-Alcaraz, J. M.; Klein, R. Phys. Rev. E 2000, 61, 4095. (17) Ramakrishnan, S.; Fuchs, M.; Schweizer, K. S.; Zukowski, C. F. Langmuir 2002, 18, 1082. (18) Henderson, D.; Lozada-Cassou, M. J. Colloid Interface Sci. 1986, 114, 180.
sion.2,25 Even DLVO contributions themselves can be altered if particles have rough surfaces26,27 or charge nonuniformity,28-34 and the assumption of additivity of the two contributing forces has come into question.35 A key limitation in verifying correct models for colloidal forces between Brownian particles at close separation has been the absence of reliable techniques for measuring these forces. Over the past 30 years, various techniques have been used to measure surface forces and interparticle forces.6 Among the direct methods are the surface force apparatus,2,36 atomic force microscopy,20-22,37 total internal reflection microscopy,38 and laser trapping.39 In the present work, colloidal forces are measured using the technique (19) Horn, R. G.; Israelachvili, J. N. J. Chem. Phys. 1981, 75, 1400. (20) Cleveland, J. P.; Shaffer, T. E.; Hansma, P. K. Phys. Rev. B 1995, 52, R8692. (21) Franz, V.; Butt, H. J. Phys. Chem B 2002, 106, 1703. (22) Jarvis, S. P.; Uchihashi; Takayuki; Ishida; Takao; Tokumoto; Hiroshi; Nakayama; and Yoshikazu J. Phys. Chem. B 2000, 104, 6091. (23) Raghavan, S. R.; Walls, H. J.; Khan, S. A. Langmuir 2000, 16, 7920. (24) Qin, Y.; Fichthorn, K. A., in preparation. (25) Safinya, C. R.; Roux, D.; Smith, G. S.; Sinha, S. K.; Dimon, P.; Clark, N. A.; Bellocq, A. M. Phys Rev. Lett. 1986, 57, 2718. (26) Czarnecki, J. Adv. Colloid Interface Sci. 1986, 24, 283. (27) Suresh, L.; Walz, J. Y. J. Colloid Interface Sci. 1997, 196, 177. (28) Miklavic, S. J.; Chan, D. Y. C.; White, L. R.; Healy, T. W. J. Chem. Phys. 1994, 98, 9022. (29) Miklavic, S. J. J. Chem. Phys. 1995, 103, 4794. (30) Grant, M. L.; Saville, D. A. J. Colloid Interface Sci. 1995, 171, 35. (31) Holt, J. C. W.; Chan, D. Y. C. Langmuir 1997, 13, 1577. (32) Stankovich, J.; Carnie, S. L. J. Colloid Interface Sci. 1999, 216, 329. (33) Cordoso, A. H.; Leite, C. A. P.; Galembeck, F. Colloid Surf. A 2001, 181, 49. (34) Feick, J. D.; Velegol, D. Langmuir 2002, 18 (9), 3454. (35) Ninham, B. W. Adv. Colloid Interface Sci. 1999, 83, 1. (36) Israelachvili, J. N.; Adams, G. E. J. Chem Faraday Trans. 1978, 74, 975. (37) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239. (38) Prieve, D. C.; Frej, N. A. Faraday Discuss. Chem. Soc. 1990, 90, 209. (39) Crocker, J. C.; Matteo, A.; Dinsmore, A. D.; Yodh, A. G. Phys. Rev. Lett. 1999, 82, 4352.
10.1021/la026736m CCC: $25.00 © 2003 American Chemical Society Published on Web 04/03/2003
Force Measurements between Colloidal Particles
Figure 1. A colloidal triplet in a uniform electric field. Particles 2 and 3 are identical here (material B), while particle 1 is composed of material A.
of differential electrophoresis,6,40,41 where interparticle forces are determined by breaking apart aggregates in the presence of an electric field according to differences in the average zeta (ζ) potentials between particles. This method is especially well-suited for quickly measuring subpiconewton forces between nearly-touching Brownian particles, and it has been used to study colloidal doublets.6,40,41 However, differential electrophoresis between two particles has been subject to two important limitations: (1) the two particles must have different average ζ potentials; (2) the particles must be visible in a light microscope. This paper focuses on overcoming barrier 1. A future paper will discuss barrier 2. The principle behind differential electrophoresis is that, since the two particles (here denoted A and B) have different average ζ potentials, they tend to separate in an electric field. The “displacement force” (i.e., the force tending to separate the particles) has been calculated in terms of the electric field using the electrokinetic equations.40,41 For two particles with the same average ζ potential (A and A), there is no tendency to separate and therefore no displacement force. This has been the source of the difficulty until now. Recently, it was realized that if a third particle was incorporated, forming a colloidal triplet (A-B-B), one could measure the force between two “B-type” particles by using A as the “handle”. Experimentally we had seen such behavior,6 although no systematic experiments had been performed with triplets. But more importantly, the electrokinetic theory required to interpret the measurements had not been derived. This paper has three purposes: (1) to develop the equation required to interpret differential electrophoresis for a linear, axisymmetric triplet, (2) to validate the technique for triplets of particles that has been previously tested on doublets, and (3) to show that even a trace amount of sodium polystyrene sulfonate (NaPSS) greatly reduces the attractive forces between nearly-touching polystyrene particles. Differential Electrophoresis Theory for Three Spheres The raw data in a differential electrophoresis experiment consist of observations of breaking aggregates (e.g., doublets or triplets) and the critical electric field at which they break. To make these observations useful, one must know the force applied to pull the particles apart as a function of the applied electric field (E∞). The physical situation is shown in Figure 1. The three particles shown (1, 2, and 3) are made of two different materials (A and B). (40) Velegol, D.; Anderson, J. L.; Garoff, S. Langmuir 1996, 12, 4103. (41) Anderson, J. L.; Velegol, D.; Garoff, S. Langmuir 2000, 16, 3372.
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Several assumptions can be made to simplify the problem. (1) The three spheres have the same radius (a) and are each uniformly charged34 with zeta potentials (ζ1, ζ2 ) ζ3). (2) The electrical double layers (EDLs) of the particles are infinitesimal compared to the particle radius (i.e., κa f ∞), which is a good approximation as long as κa/cosh(Zeζ/2kT) . 1.42 Here κ-1 is the Debye length, Z is the solution ion valence (Z:Z electrolyte), and kT is the thermal energy. (3) The gaps (δ1 and δ2) between the particles are much smaller than the particle radius a (i.e., δi/a , 1).43,44 As long as an electric field is applied and the triplet does not break, we see that
U2 - U1 ) U3 - U2 ) 0
(1)
That is, the triplet is stable, even though an electrokinetic flow field exists around the particles. At this point, there are two forces acting on the particles: (a) electrophoretic forces, which tend to pull the particles apart; and (b) attractive colloidal forces, which tend to pull the particles together. The colloidal forces manifest themselves through hydrodynamic flows. Since both the electrokinetic equations and hydrodynamic equations are linear in velocity, the overall velocity of sphere R can be decomposed into an electrophoretic velocity and a hydrodynamic velocity. This is convenient, since solutions to both problems exist in the literature and can be applied here with little modification. hyd UR ) Uep R + UR
(2)
The electrophoretic motion of three spheres with an infinitesimal EDL along a line parallel to E∞ is given by45 3
Uep R )
MRiUi0 ∑ i)1
(3)
where the MRi’s are the electrophoretic mobility coefficients of sphere R relative to sphere i. These coefficients are constrained by the following relation:45 3
MRi ) 1 ∑ i)1
(4)
The colloidal forces on the spheres can be denoted F12 between spheres 1 and 2 and F23 between spheres 2 and 3. The key observation that makes the present analysis tractable is that the dominant hydrodynamic interactions between the particles result from the lubrication hydrodynamics (i.e., the hydrodynamics in the thin gaps between the particles).43,44 Furthermore, the lubrication force acting on particle 1 will appear to come from the combined force on particles 2 and 3, since particles 2 and 3 are not separating. Similarly, the lubrication force acting on particle 3 will appear to come from the combination of spheres 1 and 2. Thus, the forces acting on the individual spheres are
F1 ) F12
(5)
(42) Dukhin, S. S.; Derjaguin, B. V. Surface and Colloid Science, Volume VII; Wiley: New York, 1974. (43) Brady, J. F. J. Fluid Mech. 1994, 272, 109. (44) Kim, S.; Karrila, S. J. Microhydrodynamics; ButterworthHeinemann: Boston, MA, 1991. (45) Keh, H. J.; Yang, F. R. J. Colloid Interface Sci. 1990, 139 (no. III), 105.
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Holtzer and Velegol
F2 ) -F12 + F23
(6)
F3 ) -F23
(7)
Thus, the sum of the forces is zero, as required. These forces manifest themselves through hydrodynamics and can include the effects of van der Waals, electrostatic, solvation, depletion, and steric forces. Also, because of the local lubrication approximation, the effects produced by the “outer” solution of the fluid are negligible.43,44 Using two-sphere hydrodynamics, the following can be written:
E∞ [(M21 - M11)ζ1 + (M22 - M12 + M23 η 1 M13)ζ2] - [2(xa12 - xa11)F12 + (x′11a - x′12a)F23] η
U2 - U1 )
E∞ [(M31 - M21)ζ1 + (M32 - M22 + M33 η 1 M23)ζ2] - [(xa11 - xa12)F12 + 2(x′12a - x′11a)F23] (12) η
U 3 - U2 )
The above equations are solved then for F12 and F23, giving
F12 ) (2 - 2M12 - 2M13 - M21 -
1 ) - [a11‚F1 + a12‚(F2 + F3)] Uhyd 1 η 1 ) - [a21‚F1 + a22‚(F2 + F3) + Uhyd 2 η a′11‚(F1 + F2) + a′12‚F3] (8) 1 ) - [a′21‚(F1 + F2) + a′22‚F3] Uhyd 3 η The primes on the a tensors indicate that the gap used is δ2; otherwise the gap used is δ1. The other bolded terms in the equation are the vector quantities FR ) FRiz and UR ) URiz. The mobility tensors (a) can be further decomposed into44
a11 ) a22 ) xa11ee + ya11(I - ee)
(9)
a12 ) a21 ) xa12ee + ya12(I - ee)
6πaxa12
a E∞aπ(ζ1 - ζ2) δ1
F23 ) (1 - M12 - M13 + M21 2M31)
()
a E∞aπ(ζ1 - ζ2) (13) δ2
The particle mobility coefficients in the above expressions were determined numerically using a FORTRAN computer code graciously provided by Professor Huan Jang Keh.45 This code is used to compute these coefficients as a function of interparticle gap distance, and the necessary modifications were made to find the electrokinetic interaction coefficients for smaller gaps. Keh and Yang calculated and validated values for triplets in axisymmetric motion. By performing a linear fit of the particle mobility coefficients in the above expression, one can develop a simplified form of the force expression:
() ()
Q1 ) 2 - 2M12 - 2M13 - M21 - M31 ≈ 12.42
where I is the identity tensor and e is a unit vector parallel to their line of centers. As above, for the primed tensors (a), one uses the gap (δ2) with the same functional form. Since there exists only axisymmetric motion in this system, (I - ee)‚F ) 0. Finally, values for the x aij coefficients are known:44
() ()
6πaxa11 ≈ 0.775 + 0.93
()
M31)
δ1 a
Q2 ) 1 - M12 - M13 + M21 - 2M31 ≈ 7.94
(10)
The same functional form appears for the x ′ij variables, except that the gap δ2 is used. Since only motion in the z-direction is being considered, all vector and tensor quantities can be converted to scalars. Thus, the expressions for hydrodynamic velocity become a
1 ) - [xa11F12 - xa12F12] Uhyd 1 η 1 Uhyd ) - [xa12F12 - xa11F12 + x′11aF23 - x′12aF23] 2 η
(11)
1 ) - [x′12aF23 - x′11aF23] Uhyd 3 η Now eqs 3 and 11 can be solved for F12 and F23. Making substitutions for the electrophoretic and hydrodynamic velocity terms, one finds
δ2 a
(14)
The numerics for the fits are shown in Figure 2. We are interested in the displacement force, which is the tension we are applying using electrophoresis. These are opposite of F12 and F23, and so for triplets they are
) -F12 ) 12.42πa(ζ2 - ζ1)E∞ Fdisp 1
δ1 ≈ 0.775 - 1.07 a
δ1 a
Fdisp ) -F23 ) 7.94πa(ζ2 - ζ1)E∞ 2
(15)
These equations make sense physically. For instance, consider the case when ζ1 ) 0 mV and ζ2 ) ζ3 ) 100 mV. If particles 2 and 3 were tightly bound (e.g., by colloidal forces), then the separation between spheres 1 and 2 would be like separating 100 mV (both spheres 2 and 3) from 0 mV (sphere 1), a difference of 100 mV. On the other hand, if particles 1 and 2 were tightly bound, then the separation between spheres 2 and 3 would be more like separating 100 mV (sphere 3) from 50 mV () average of 0 and 100 mV for spheres 1 and 2). That is, more force is exerted by the technique in the first case. This is what we see in (15). The second result compares quite well with the result for doublets:40,41
F23 ) 8.76πa(ζ1 - ζ2)E∞
(16)
The equations have the same functional form, and the only difference is a 9.4% smaller prefactor for the triplet equation (7.94 instead of 8.76). Thus, the presence of the third particle has relatively little effect on the interpreta-
Force Measurements between Colloidal Particles
Figure 2. Solutions for the electrokinetic parameters in eq 3. The results were computed from the code of Keh and Yang for small gap distances. These were used to obtain the QR prefactors in (14) and (15). The slope was determined using a linear regression analysis with a 95% confidence level, yielding Q1 ) 12.42 ( 0.28 and Q2 ) 7.94 ( 0.20.
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Figure 3. Time evolution of a triplet aggregate breaking under the influence of an electric field. The solution conditions for this case were 4.5 µm sulfated and carboxylated PSL particles, 10 mM KCl, 10 µM NaPSS, and pH ) 2.5.
tion of the interparticle force between the two particles. The next section discusses the use of (15) to measure interparticle forces experimentally. Experiments Description of Apparatus. The apparatus used for the differential electrophoresis experiments consisted of three main parts: (1) an inverted light microscope with a mounted capillary electrophoresis cell, (2) a current source, and (3) imaging and video equipment. These have been described previously.6 The capillary electrophoresis cell, developed by the Velegol laboratory group at Penn State, consists of a rectangular borosilicate glass capillary (0.2 mm × 2 mm × 50 mm, Vitrocom) mounted to a glass slide and two reservoirs (fit for insertion of the electrodes) with Omnifit 2-way control valves. The optical light microscope (Nikon TE300) was mounted on a hinge setup so that the objective could face horizontally; this enabled longer experiments since the wider direction of the capillary could be placed parallel to gravity. An automated microscope stage (Prior ProScan) provided motion of the electrophoresis cell on the microscope. A current was applied to the system using a pair of platinized electrodes connected to a direct current (dc) source meter (Keithley model 2410). Imaging of the particles under the microscope was performed using a Cohu 4010 CCD camera and recorded using a JVC VHS HR-S3600U recorder. The resulting videotapes were analyzed using Scion Image PC-based software, and final images and movies were produced using MovieStar 4.24 software. A 40× objective lens was used for all of the experiments, and the experimental temperatures ranged from 23.3 to 25.4 °C (and were constant to within (0.5 °C during each experiment). Sample Preparation. The solutions used in the experiments were prepared in the following manner. Glass beakers (100 mL) were sonicated using an Aquasonic (VWR) model 50T sonicator containing soap (Aquasonic Cleaning Solution) and deionized (DI) water (Millipore Milli-Q). They were then rinsed 10 times with fresh DI water. Ethanol (Pharmco, 200 proof, CAS no. 6417-5) was then added to each beaker for 24 h to prevent bacterial contamination. After the ethanol was removed and each beaker was rinsed 10 times, 80 mL of DI water was measured into each beaker. Potassium chloride (KCl) salt (J.T. Baker, catalog no. 3046-01) and/or a minute amount of hydrochloric acid (HCl) (36.5-38.0%, J.T. Baker, catalog no. 9535-02) was added to half of the solutions. Next, NaPSS polyelectrolyte (J.T. Baker, catalog no. 24,305-1, MW ) 70 000, valency ) 340) was added in concentrations of either 0, 0.1, 1, 5, 10, or 15 µM. The polystyrene latex (PSL) particles used in the experiments (Interfacial Dynamics Corp., batch nos. 695,1; 740,1; 736,3; and 1228,1) were either 1.5 or 4.5 µm in diameter and contained either sulfate or carboxyl surface groups. The particles were monodisperse in each solution, and this was verified by taking scanning electron microscope images of select solution samples. Two drops each of
Figure 4. The average critical force as a function of NaPSS concentration for all solutions studied in this paper. Each point represents the average of about 10 measurements of triplets. (A) 4.5 µm PSL particles, 10 mM KCl, pH ) 5; (B) 4.5 µm PSL particles, 10 mM KCl, pH ) 2.5; (C) 1.5 µm PSL particles, 10 mM KCl, pH ) 5; (D) 1.5 µm PSL particles, 0 mM KCl, pH ) 2.5. both sulfated and carboxylated PSLs were added to 80 mL of DI water, giving a volume fraction of each type of particle of roughly 1 × 10-4. The pH of the solutions was 5 for the solutions without HCl and 2.5 for the solutions containing HCl. The solution conductivities, measured with a Fisher Scientific AR50 conductivity meter, ranged from 1.067 to 3.338 mS/cm. The zeta potentials were measured using a ZetaPALS apparatus (Brookhaven Instruments).
Results Throughout the course of the experiments, videos were taken of the triplet aggregates breaking apart in the electric field. A number of these videos were converted into still frames for analysis using the Scion Image software program (see one example in Figure 3). At time ) 0 s in Figure 3, the triplet aligned as the electric field was turned on. At time ) 3 s, the triplet broke at a critical electric field strength and therefore critical force (here, 26 pN). Next, from time ) 5 s to time ) 8 s, the direction of the electric field was manually switched to see if the doublet rotated. Doing this enabled us to determine the type of particles broken apart (i.e., sulfated or carboxylated) by how the portions translated in the field after breaking. For every triplet we observed, a sulfated particle was separated from another sulfated particle, while the third carboxylated particle served as a handle for pulling them apart.
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Figure 5. Plot of the critical force of each triplet as a function of NaPSS concentration for 1.5 µm PSL particles. The solution had no KCl, and pH ) 2.5. Filled circles represent triplets that broke into a doublet and a singlet, while open circles represent triplets that did not break.
Figure 6. Plot of the critical force of each triplet as a function of NaPSS concentration for 4.5 µm PSL particles. The solution had 10 mM KCl, and pH ) 2.5. Filled circles represent triplets that broke into a doublet and a singlet, while open circles represent triplets that did not break.
Figure 4 summarizes the results for the average critical force, as a function of NaPSS concentration for various solutions. Each data point shown in this figure is an average over all triplets broken in a given experiment. More detailed statistics for each data point are shown in the remaining figures. Figures 5-8 show the detailed data for the points in Figure 5. Filled circles on the plots indicate triplets that broke at the indicated critical force. The breaks tended to occur in one of two ways: in some cases, the singlet broke linearly and quickly away from the doublet (e.g., Figure 3); in other cases, the singlet first rotated slightly around the doublet before breaking away. This might be due to the chain being noncoaxial at first and then straightening before breaking. The open circles in Figures 5-8 show triplets that did not break. This may have occurred if the particles were irreversibly bound, as observed in the doublet experiments for solutions of bare particles.6 If the particles were in a potential energy secondary minimum (i.e., weak attractive forces), it is likely that our technique would have broken the aggregates. The open circles also include cases where
Holtzer and Velegol
Figure 7. Plot of the critical force of each triplet as a function of NaPSS concentration for 1.5 µm PSL particles. The solution had 10 mM KCl, and pH ) 5. Filled circles represent triplets that broke into a doublet and a singlet, while open circles represent triplets that did not break.
Figure 8. Plot of the critical force of each triplet as a function of NaPSS concentration for 4.5 µm PSL particles. The solution had 10 mM KCl, and pH ) 5. Filled circles represent triplets that broke into a doublet and a singlet, while open circles represent triplets that did not break.
the triplets were lost at the edge of the cell or at the cell wall. In each of these cases, the critical forces shown are minimum forces. Discussion There are two significant results from this work. First, the technique of differential electrophoresis has been extended so that forces can be measured between two colloidal particles with identical zeta potentials. The primary theoretical result is in eq 15. A key point is that the result is almost identical to the result for doublets, except that the coefficient is 9.4% smaller. The fact that this difference is small is extremely useful, because if we were to alter the experiments and “visualize” the breakup of aggregates using light scattering, the small difference indicates that it is not critical to know whether triplets are breaking into doublets and singlets, or whether doublets are breaking into singlets. A second important result is that the forces measured between particles in triplets are comparable to the results for particles in doublets under similar solution conditions.6
Force Measurements between Colloidal Particles
That is, the forces were O(10 pN). These forces were measured in the presence of NaPSS; for the concentrations of NaPSS used, no measurable difference in viscosity was expected nor seen using a capillary viscometer. However, no obvious trends occurred as we added KCl or altered the pH. This could be due to the complex interaction between the NaPSS structure and the traditional DLVO forces between the particles. The one change that gave a consistent result was that larger particles gave larger attractive forces, which we would expect. In the absence of NaPSS, we were not able to break a single triplet that was observed. This is similar to results of previous experiments we have done.6,40,41 However, in the presence of even a trace of amount of NaPSS (and the lowest quantity we used was 100 nM) we could still break aggregates. Such tiny amounts should produce only extremely small depletion forces (even considering Donnan equilibrium effects) and so suggest that the NaPSS is modifying the polystyrene particle surface, perhaps through adsorption. This could occur as parts of the polyelectrolyte in solution become hydrophobic through a “pearling” effect.46-48 Therefore, it is possible that the polyelectrolyte could adsorb to regions of the PSL particles with low surface charge density.34 (46) Chodanowski, P.; Stoll, S. J. Chem. Phys. 1999, 111 (13), 6069. (47) Dobrynin, A. V., Rubenstein, M., and Obukov, S. P. Macromolecules 1996, 29, 2974. (48) Pickett, G. T.; Balazs, A. C. Langmuir 2001, 17, 5111.
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Conclusions In this paper, colloidal forces were measured between particles with identical zeta potentials using the technique of differential electrophoresis. The electrokinetic equation for interpreting force results was solved for colloidal triplets, and the forms of the equations for both doublets and triplets were found to be similar, with only a 9.4% difference in the prefactor. The new technique was tested in suspensions of polystyrene latex particles, and the average attractive forces ranged between 0 and 60 pN, comparable to results of previous experiments with colloidal doublets.6 Furthermore, only trace amounts of NaPSS were required to significantly reduce the attractive forces between particles. A key advantage of the work in this paper is that it enables us to extend the technique of differential electrophoresis to nanoparticle systems using a modified technique for visualizing particle breakup. Acknowledgment. The authors thank Professor Huan Jang Keh for providing a copy of the code required to determine the electrokinetic interaction coefficients for triplets. They also thank Prasanna Thwar for experimental insights and Olatilewa Awe for zeta potential measurements. Finally, the authors thank the Environmental Protection Agency (EPA STAR Grant No. R-82960501) and the National Science Foundation (CAREER Grant No. CTS-9984443) for funding this research. LA026736M
