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Measurements of Charge Nonuniformity on Polystyrene Latex Particles Jason D. Feick and Darrell Velegol* Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802-4400 Received October 4, 2001. In Final Form: February 8, 2002 The experimental technique of rotational electrophoresis, similar in principle to standard translational electrophoresis, has been developed to quantify the charge nonuniformity on Brownian colloidal particles. Experiments were conducted on doublets of 4.5 µm polystyrene latex particles under various solution conditions, finding variations in zeta potential on individual particle surfaces of 50-75% of the average zeta potentials. Significant charge nonuniformity was found for all systems studied, indicating that charge nonuniformity might be critical in many colloidal applications.
Colloidal systems are pervasive in industrial and natural processes, finding applications in coatings, electronic inks,1 photonic crystals,2 and drug delivery.3 For most colloidal processes, controlling the stability is critical; however, the classical model for colloidal stability (with Derjaguin-Landau-Verwey-Overbeek (DLVO) theory4) often fails, even qualitatively.5-8 One possible reason for this is that the DLVO model assumes a uniform charge or zeta (ζ) potential distribution on the particle surface, which might not always be the case (Figure 1).9-12 Recent theory shows that a nonuniform ζ potential distribution can dominate electrostatic interactions and lead to large attractive forces that cause suspension instability.13-18 There are two primary reasons for this: (a) for colloidalsize particles, the local ζ potential determines stability, not the area average ζ potential of the particles;18 (b) colloidal-size particles sample all orientations by Brownian rotation, so that an extreme ζ potential anywhere on a particle’s surface can control stability. This paper has two purposes: (1) to describe our experimental method of “rotational electrophoresis” for measuring the standard * To whom correspondence should be addressed. E-mail: velegol@ psu.edu. (1) Comiskey, B.; Albert, J. D.; Yoshizawa, H.; Jacobson, J. Nature (London) 1998, 394, 253. (2) Joannopoulos, J. D.; Villeneuve, P. R.; Fan, S. H. Nature (London) 1997, 386, 143. (3) Langer, R. Science 1990, 249, 1527. (4) Hunter, R. J. Foundations of Colloid Science, Vol. I and II; Clarendon: New York, 1986, with corrections 1992. (5) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge: New York, 1989, with corrections 1991; examples on p 277. (6) Velegol, D.; Anderson, J. L.; Garoff, S. Langmuir 1996, 12, 675. Using experimental conditions like those in this paper, polystyrene spheres have been found to form rigid doublets. (7) Velegol, D.; Anderson, J. L.; Garoff, S. Langmuir 1996, 12, 4103. (8) Behrens, S. H.; Borkovec, M.; Schurtenberger, P. Langmuir 1998, 14, 1951. Behrens, S. H.; Christl, D. I.; Emmerzael, R.; Schurtenberger, P.; Borkovec, M. Langmuir 2000, 16, 2566. (9) Gibb, A. W. M.; Koopal, L. K. J. Colloid Interface Sci. 1990, 134, 122. (10) Koopal, L. K. Electrochim. Acta 1996, 41, 2293. (11) Hiemstra, T.; Yong, H.; Van Riemsdijk, W. H. Langmuir 1999, 15, 5942. (12) Hiemstra, T.; Van Riemsdijk, W. H. Langmuir 1999, 15, 8045. (13) Czarnecki, J. Adv. Colloid Interface Sci. 1986, 24, 283. (14) Miklavic, S. J.; Chan, D. Y. C.; White, L. R.; Healy, T. W. J. Phys. Chem. 1994, 98, 9022. (15) Grant, M. L.; Saville, D. A. J. Colloid Interface Sci. 1995, 171, 35. (16) Holt, W. J. C.; Chan, D. Y. C. Langmuir 1997, 13, 1577. (17) Stankovich, J. M.; Carnie, S. L. J. Colloid Interface Sci. 1999, 216, 329. (18) Velegol, D.; Thwar, P. Langmuir 2001, 17, 7687.
Figure 1. Colloidal interactions between nonuniformly charged spheres: (a) schematic of randomly distributed charges on two particles; (b) local area average of charge density of (a) with small length scale (L1); (c) local area average of (a) with larger length scale (L2). Regions that are brighter have a higher local charge density. Note that the variations among regions are larger when a smaller length scale is used (i.e., with a larger L, variations tend to “gray out”). A smaller L gives more regions (N), and for a sphere 8πR2 ) NπL2 if we assume circular regions.
deviation of ζ potential (σζ) on individual particles given a length scale (L) of observation and (2) to demonstrate that charge nonuniformity was significant in every system we studied. The literature contains few experimental measurements of charge nonuniformity. Atomic force microscopy (AFM)19 has been used to map local electrostatic forces20 between an AFM tip and various surfaces, providing a measure of surface charge distributions.21-25 But challenges exist for mapping charge distributions on particles with AFM. For example, the particle is fixed, not Brownian, and so the electrostatic forces would be difficult to map over all orientations of the particle surface. Also, even if the (19) Binnig, G.; Quate, C. F.; Gerber, C. Phys. Rev. Lett. 1986, 56, 930. (20) Butt, H.-J. Biophys. J. 1991, 60, 777. Butt, H.-J. Biophys. J. 1991, 60, 1438.
10.1021/la011515m CCC: $22.00 © 2002 American Chemical Society Published on Web 03/30/2002
Measurements of Charge Nonuniformity
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we divide each spheroid into N equal-area (A ˆ ) regions and take the local area average zeta potential (ζi,j) as
ζi,j )
∫S ∫ζ(x) dS i,j
(1)
A ˆ
in the ith region on the jth spheroid (Si,j). Then the “suspension average” zeta potential (ζ0) and standard deviation of zeta potential (σζ) are defined by Figure 2. Rotational electrophoresis of a colloidal doublet. In this schematic, the ζ potential dipole (D) of the doublet is not parallel with the center-to-center vector (e).
electrostatic forces could be distinguished from other potentially nonuniform colloidal forces (a difficult task), the effective “electrostatic size” of the tip limits the spatial resolution of the method. We have measured charge nonuniformity on Brownian particles using rotational electrophoresis.26-30 The method is in principle similar to standard translational electrophoretic measurements, involving two steps: (1) the electrophoretic angular velocities (Ω) of many particles are measured using a standard microelectrophoresis cell and video microscopy and (2) the angular velocities are interpreted using classical electrokinetic theory. The essential physics that enables our measurement is that randomly charged particles have on average a small but finite net electric dipole moment (D) and higher order moments, causing them to rotate in an electric field.6,26,27,31,32 On the other hand, uniformly charged particles with thin electrical double layers (EDLs) have zero dipole moment and will not rotate by electrophoresis (regardless of shape).33 Thus, rotational electrophoresis of the particles enables us to quantify their charge nonuniformity. We note that using standard translational electrophoresis to measure charge nonuniformity is extremely challenging since random charge nonuniformity contributes a negligible effect to the translational velocity (u).30 Theory The raw data that we obtain are the angular velocities (Ω ) Ωix) of M doublets in an electric field (E∞ ) E∞iz). Most doublets that we observe have their e vectors (along line of centers; see Figure 2) in the image plane (y-z plane). The Ω’s are interpreted by a “patchy spheroid” model29,30 derived from the standard electrokinetic equations.31,34 In this model (≈doublet model, but easier to calculate35), (21) Butt, H.-J. Biophys. J. 1992, 63, 578. (22) Radmacher, M.; Cleveland, J. P.; Fritz, M.; Hansma, H. G.; Hansma, P. K. Biophys. J. 1994, 66, 2159. (23) Cappella, B. P.; Frediani, B. C.; Miccoli, P.; Ascoli, C. Nanotechnology 1997, 8, 82. (24) Rotsch, C.; Radmacher, M. Langmuir 1997, 13, 2825. (25) Heinz, W. F.; Hoh, J. H. Biophys. J. 1999, 76, 528. (26) Fair, M. C.; Anderson, J. L. J. Colloid Interface Sci. 1989, 127, 388. (27) Fair, M. C.; Anderson, J. L. Int. J. Multiphase Flow 1990, 16, 663. Corrigenda. Int. J. Multiphase Flow 1990, 16, 1131. (28) Fair, M. C.; Anderson, J. L. Langmuir 1992, 8, 2850. (29) Velegol, D.; Feick, J. D.; Collins, L. J. Colloid Interface Sci. 2000, 230, 114. (30) Feick, J. D.; Velegol, D. Langmuir 2000, 16, 10315. (31) Teubner, M. J. Chem. Phys. 1982, 76, 5564. (32) Anderson, J. L. J. Colloid Interface Sci. 1985, 105, 45. (33) Morrison, F. A., Jr. J. Colloid Interface Sci. 1970, 34, 210. (34) Russell, A. S.; Scales, P. J.; Mangelsdorf, C. S.; Underwood, S. M. Langmuir 1995, 11, 1112. Russell, A. S.; Scales, P. J.; Mangelsdorf, C. S.; White, L. R. Langmuir 1995, 11, 1553. These papers address the validity of the standard electrokinetic model experimentally.
M
ζ0 ) M
σζ2 )
N
∑ ∑ζi,j j)1 i)1
(2)
MN N
(ζi,j - ζ0)2 ∑ ∑ j)1 i)1
(3)
MN
Using these definitions, our spheroid model shows that with e ) iy, we have30
[ ] ( )[ ] ( ) [ [ ]()[ 〈Ux2〉 ζ0 〈Uy2〉 ) η 〈Uz2〉
2
Ex2 2 2σζ Ey2 + η N Ez2
][ ] ][ ]
2 0.4743 0.0699 0.2853 Ex 0.1578 0.0118 0.1578 Ey2 0.2853 0.0699 0.4743 E 2 z
2 〈Ωx2〉 2 0.0600 0.1805 0.8847 Ex 2σζ 2 〈Ωy 〉 ) 2.1567 0 2.1567 Ey2 ηR N 0.8847 0.1805 0.0600 E 2 〈Ωz2〉 z
(4)
(5)
where is the fluid permittivity; η is the fluid viscosity; R is the radius of each sphere in the doublet; N is the number of equal-area regions on each spheroid (Figure 1); and 〈x2〉 is the ensemble average of x2 for all M particles measured. Equations 4 and 5 were developed semianalytically from the hydrodynamic and electrostatic equations,26,27,28,32 and they are the analogue of the Smoluchowski equation for translational electrophoresis (indeed, these equations give the Smoluchowski result for uniformly charged doublets). Note that eq 4 has two terms, one due to the average zeta potential (ζ0) and the other due to the standard deviation of zeta potential (σζ). For randomly charged particles, the first term completely dominates, which is why the translational velocity cannot be used to detect charge nonuniformity. It is important to note that σζ2/N is constant for randomly charged particles.30,36 For uniformly charged (σζ ) 0) spheroids, (5) predicts no rotation.33 However, for σζ * 0, eq 5 enables us to quantify charge nonuniformity. The assumptions used to derive (5) are as follows: (a) The particle is rigid6 and nonconducting. (b) The electrical double layer (EDL) is infinitesimally thin [i.e., κR/cosh(Zeζ/2kT) . 1,where κ-1 is the Debye (35) Nir, A.; Acrivos, A. J. Fluid Mech. 1972, 59, 209. (36) Velegol, D. Electrophoresis of Randomly-Charged Particles. Electrophoresis, accepted for publication.
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length; R is the radius of the particle; Z is the valence of solution ions; and kT is the thermal energy].37 (c) The fluid around the particle is unbounded. (d) The particles are randomly charged at all length scales. The distribution can be of any type (e.g., Gaussian, uniform, Poisson, Weibull). Many length scales are pertinent in the charge nonuniformity problem: (1) the particle size (e.g., for a sphere, R), (2) the Debye length (κ-1), (3) the correlation length (Lc) of charge density, (4) the length scale of electrostatic
interaction (Lei ) xRκ-1, from colloidal force theory), and (5) the length scale of observation (L), which is arbitrary. The final assumption above will not hold unless L . κ-1 and L . Lc. Using the techniques described in this paper, we cannot measure the correlation length (Lc) of the charge density, although we recognize that this could be an important parameter for some applications. Since eq 5 applies to an average over many doublets, Ω was measured for a statistical number of doublets (i.e., M ∼ 20), and then σζ2/N was calculated. We were able to measure doublet rotation only in the y-z plane (i.e., Ωx), and our electric field was applied only in the z direction. Therefore, eq 5 reduces to 2 2σζ (0.181 cos2 θ + 0.884 sin 2 θ)E∞2 (6) 〈Ωx 〉 ) ηR N 2
( )
where θ is the angle from E∞ to the e vector of the doublet. Since we use Ωx as the doublet passes through 90°, (6) reduces to
〈Ωx2〉 ) 0.884
2 2σζ E 2 ηR N ∞
( )
(7)
This is the form of eq 5 that we used to interpret our rotational electrophoresis experiments. We measure 〈Ωx2〉, and knowing the other parameters in the equation (i.e., , η, R, E∞), we can calculate σζ2/N. Although we do not need to specify an N or L (Figure 1), often for convenience we choose L ) xRκ-1 ) Lei, where R is the particle radii in the doublet. This value of L is the distance away from the point of closest approach where the gap between the particles has increased by κ-1, the decay length for the electrostatic forces. If each region on the spheroid has an area πL2, then N ≈ 2 × 4πR2/πL2 ) 8κR. We of course do not believe that the charge nonuniformity occurs in well-defined “patches”; rather, by defining regions and using the average surface potential in each region (as in eq 3), we characterize the distribution that actually exists and obtain (7). For a given measurement of 〈Ω2〉, one sees that observing smaller patches (larger N) results in a higher σζ. This is common to many physical phenomena, such as fluctuations of gas molecules in a control volume,38 in which the smaller the region observed, the larger the fluctuation. Experiments We measured the charge nonuniformity of polystyrene (PS) latex particles (4.3 µm sulfated, from Interfacial Dynamics Corp., Portland, OR; 4.5 µm carboxylated, from Polysciences Inc., Warrington, PA) under different conditions of salt concentration, water cleanliness, and particle cleanliness. We cleaned some of (37) Dukhin, S. S.; Derjaguin, B. V. Electrokinetic Phenomena; Matijevic, E., Ed.; Surface and Colloid Science Vol. 7; Wiley: New York, 1974. (38) Landau, L. D.; Lifshitz, E. M. Statistical Physics; Pergamon: London, 1958; Chapter XII.
the particles using sedimentation-decantation,39 while we purposely used other particles “as-is” from the manufacturer.40 We did not intentionally alter the charge distribution of the particles in any way. MilliQ deionized (DI) water (1 µS/cm) was used for most solutions, while tap water (270 µS/cm) was used for some. Glassware (including the electrophoresis cell) was cleaned using Alconox cleaner and 15.8 N nitric acid. Dilute colloidal suspensions (∼0.05%V) were made by mixing the particles into a KCl solution (1 and 10 mM) in an unbuffered pH of 4-6 for clean suspensions and 8-9 for “dirty” suspensions. The suspensions were allowed to sit for 8-72 h so doublets could form (Smoluchowski rapid flocculation time ≈ 3.5 h). While it is possible to measure the angular velocity of a sphere using video microscopy (e.g., ref 41 where the sphere was tagged or ref 42 which describes the classic experiments of Perrin), we chose the much simpler method of measuring the angular velocities of doublets formed from a homogeneous population of particles. This had the advantage that we did not need to introduce additives (e.g., fluorophores) that might alter the charge distribution of the particles. Experimentally, using doublets introduced the potential artifact that the two spheres in the doublet were each uniformly charged but different from each other, such that the doublet (i.e., its e vector) would align with E∞. While this was the case for many of the doublets, many other doublets that we measured did not align with E∞. Another potential artifact is that only those particles most likely to form doublets, perhaps connected by patches with low ζ potentials, did in fact form doublets. However, since electrokinetic calculations show that the gap region contributes little to the angular velocity of the doublet,43 anomalies in the gap region do not affect our measurement technique. The rotational electrophoresis experiments were done in a custom microelectrophoresis apparatus consisting of three main parts: (1) the microelectrophoresis cell and blackened electrodes, (2) a Keithley 2410 current source, and (3) a Nikon TE 300 video microscope. The cell consisted of two Pyrex glass cylinders connected by a borosilicate capillary tube (0.2 mm thick i.d. × 2 mm wide i.d. × 5 cm long, Vitro Com, Mountain Lakes, NJ), all fastened to a metal plate for mechanical stability. To minimize settling effects, the video microscope was placed on its back using an aluminum support system. The average ζ potentials of the particles were measured using the translational mobility and the Smoluchowski equation. All experiments were conducted at 20-25 °C, with variations of
