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Evaluating Randomness of Charge Distribution on Colloidal Particles Using Stationary Electrophoresis Angles Darrell Velegol* and Jason D. Feick Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16803 Received December 4, 2002. In Final Form: March 7, 2003 The method of “rotational electrophoresis” has previously been used to measure charge nonuniformity on colloidal particles. In this technique, one measures the angular velocities of many particles in electrophoresis and interprets these angular velocities in terms of electrokinetic theory, assuming a random distribution of charge on the individual particles. In the present paper, we verify whether the charge distribution is in fact statistically random by examining the stationary angle to which spheroidal particles (or similar particles, such as doublets) rotate. Calculations are provided for the average stationary angle to which spheroids (axis ratio of 0.1-10) rotate. Estimates are given for how strongly the spheroids remain at a steady angle despite Brownian motion. In addition, experimental data are reported for polystyrene latex particles, showing that the charge distribution on the particles is random for about half of the particles.
Introduction A central problem in colloidal processes is maintaining the stability of the suspended particles. For 50 years, the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory of colloidal forces has been used to predict colloidal forces and stability,1 but this model is often found not to explain data for forces and stability.2-5 This model has traditionally assumed that individual particles have a uniform charge density on their surface;6-9 however, if particles are nonuniformly charged, the predicted stability of a colloidal suspension will in general be greatly reduced.10-15 There are two primary reasons for this: (a) for colloidalsize particles, the local surface potential (for this paper, we assume this to be equal to the zeta potential ζ) determines stability, not the area average surface potential of the particles;15 (b) colloidal-size particles sample all orientations by Brownian rotation, so that an extreme surface potential anywhere on a particle’s surface can control stability. (1) Hunter, R. J. Foundations of Colloid Science, Vol. I and II; Clarendon: New York, 1986, with corrections 1992. (2) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989, with corrections 1991; examples on p 277. (3) Velegol, D.; Anderson, J. L.; Garoff, S. Langmuir 1996, 12, 675. Using experimental conditions such as those in this paper, polystyrene spheres have been found to form rigid doublets. (4) Velegol, D.; Anderson, J. L.; Garoff, S. Langmuir 1996, 12, 4103. (5) 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. (6) Gibb, A. W. M.; Koopal, L. K. J. Colloid Interface Sci. 1990, 134, 122. (7) Koopal, L. K. Electrochim. Acta 1996, 41, 2293. (8) Hiemstra, T.; Yong, H.; Van Riemsdijk, W. H. Langmuir 1999, 15, 5942. (9) Hiemstra, T.; Van Riemsdijk, W. H. Langmuir 1999, 15, 8045. (10) Czarnecki, J. Adv. Colloid Interface Sci. 1986, 24, 283. (11) Miklavic, S. J.; Chan, D. Y. C.; White, L. R.; Healy, T. W. J. Phys. Chem. 1994, 98, 9022. (12) Grant, M. L.; Saville, D. A. J. Colloid Interface Sci. 1995, 171, 35. (13) Holt, W. J. C.; Chan, D. Y. C. Langmuir 1997, 13, 1577. (14) Stankovich, J. M.; Carnie, S. L. J. Colloid Interface Sci. 1999, 216, 329. (15) Velegol, D.; Thwar, P. Langmuir 2001, 17, 7687.
Figure 1. Schematics of random charge nonuniformity on colloidal doublets: (a) randomly distributed charges on two particles; (b) local area average of the charge density of (a) with a small length scale (L1); (c) local area average of (a) with a larger length scale (L2). Regions that are brighter have a higher local charge density. Note that the variations among regions are smaller when a larger length scale is used (i.e., with a larger L, variations tend to “gray out”).
Recent experiments from our lab have quantified a nonuniform charge distribution on the surfaces of individual polystyrene latex spheres (Figure 1).16 These experiments used rotational electrophoresis, in which we measure the angular velocities of many colloidal doublets and interpret the angular velocities using electrokinetic theory. Since the electrokinetic equations are more straightforward to solve for spheroids, we approximated the doublets by spheroids with an axis ratio of 0.418.17 In solving the electrokinetic equations, we assumed a random charge distribution on the particles, and therefore the interpreted results give the standard deviation of ζ potential (σζ) over the particle surfaces. (16) Feick, J. D.; Velegol, D. Langmuir 2002, 18, 3454. This paper also contains references to other measurements of charge nonuniformity, including some taken with atomic force microscopy. (17) Feick, J. D.; Velegol, D. Langmuir 2000, 16, 10315.
10.1021/la026954f CCC: $25.00 © 2003 American Chemical Society Published on Web 05/02/2003
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In this paper, we describe a modification of this technique, which can be used to verify a key assumption in the interpretation: that particles have a statistically random distribution of charge. The experimental technique involves observing the stationary angle of the doublets in electrophoresis. While uniformly charged particles will not rotate in electrophoresis,16-18 nonuniformly charged particles will on average rotate. The rotation of a doublet will stop when its dipole moment of ζ potential (approximately) aligns with the direction of the applied electric field (E∞). This paper has three purposes: (1) for an ensemble of randomly charged spheroids, to calculate the expected average stationary angle of the particles in an applied electric field, (2) for the same particles, to give estimates for the average Peclet number (strength of rotational electrophoresis compared to Brownian motion), and (3) to report data for polystyrene latex particles with sulfate charge groups.
with the spheroid axis of symmetry along iy. The value of H is then given by
H)
exex 2
a +b
2
+
eyey
ezez
+
2
2
a +b
2a
∫∫
h ) b2H )
exex + ezez 1+β
f)
n ) ∇f/|∇f|
(3)
x2 y2 z2 + + -1)0 a2 b2 a2
(4)
and G and H are constant geometric tensors defined in ref 19. Equation 1 uses the spheroid volume (V) and the fluid viscosity (η) and permittivity (). Equation 1 was developed from the Stokes equations of hydrodynamics and the electrostatic equations (i.e., the electrokinetic equations), assuming (a) the particle is rigid3 and nonconducting, (b) the EDL is infinitesimally thin [i.e., κR/ cosh(Zeζ/2kT) . 1,where κ-1 is the Debye length, R is the radius of the particle, Z is the valence of solution ions, and kT is the thermal energy],21 and (c) the fluid around the particle is unbounded. In modeling the electrophoresis, we fixed the spheroid axes along the x-y-z axes of the coordinate system (18) Morrison, F. A., Jr. J. Colloid Interface Sci. 1970, 34, 210. (19) Fair, M. C.; Anderson, J. L. J. Colloid Interface Sci. 1989, 127, 388. (20) Kreyszig, E. Advanced Engineering Mathematics, 6th ed.; Wiley: New York, 1988. (21) Dukhin, S. S.; Derjaguin, B. V. Electrokinetic Phenomena; Matijevic, E., Ed.; Surface and Colloid Science, Vol. 7; Wiley: New York, 1974.
eyey
2b2β2 ezez
+ (5)
(6)
2
+
eyey 2β2
(7)
r b
(8)
ds )
dS b2
(9)
E)
E∞ E∞
(10)
ζ σζ/xN
(11)
χ)
p
(2)
b (1 + β )
+
To nondimensionalize (1) further, we define
(n‚r)r × (I - nn)ζ dS‚G‚E∞ (1) S +
r ) xix + yiy + ziz
2
4 4 V ) πa2b ) πβ2b3 3 3
x)
where for a spheroid with semiaxes (a,b,a), the surface vector (r) and the surface normal (n) at the edge of the double layer (Sp+) are defined by20
2
where β ) a/b e1. Note that geometrically, Sp+ is the same as S. The volume (V) of a spheroidal particle with axes (a,b,a) is22
For spheroidal particles, expressions have been developed that relate the electrophoretic translational (U) and angular (Ω) velocities to an arbitrary ζ potential distribution on the surface. The electrical double layer (EDL) must be infinitesimally thin, and the correlation length (d) of charge variation must have (κd . 1). The equation for Ω of a spheroid in an applied electric field (E∞) is19
H‚ ηV
exex
)
b2(1 + β2)
Stationary State Angles for Random Charge Nonuniformity
Ω)
2
Now (1) can be rewritten as
Ω)
σζE∞
∫∫S
h‚
4 πηβ2bxN 3
(n‚x)x × (I - nn)χ ds‚G‚E
+
p
(12)
If we define
A)
∫∫S
3 h‚ 4β2
(n‚x)x × (I - nn)χ ds‚G
+
p
cep )
σζE∞ πηbxN
(13)
(14)
then we have
Ω ) cepA‚E
(15)
In this case, the value of A is dimensionless, and its elements are expected to be O(1). To find the stationary state angles, we need Ω ) 0, and therefore we need
A‚E ) 0
(16)
For rotational electrophoresis, this can occur if A ) 0 (highly improbable), if E ) 0 (trivial solution, not of interest here), or if |A| ) 0 (the case we are interested in). In experiments, the raw data that we obtain are the stationary angles (i.e., where the rotation stops) and (22) Weisstein, E. W. CRC Concise Encyclopedia of Mathematics; CRC Press: New York, 1999.
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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).
angular velocities (Ω ) Ωix) of M doublets undergoing rotational electrophoresis (E∞ ) E∞iz). The 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” model17,23 derived from the standard electrokinetic equations.24,25 In this model (which can approximate shapes such as doublets or disks accurately, with simpler calculations17,23,26), we divide each spheroid into N regions of equal area (A1) and take the area average of the local zeta potential (ζi,j) as
∫∫ζ(x) dS ζi,j )
Si,j
A1
therefore that 0 is an eigenvalue of A. Indeed, since (16) represents an eigenvalue problem, we can use the eigenvalue of 0 to obtain the eigenvector E that gives Ω ) 0. Since we fixed the axes on the doublet, the eigenvector gives the angle of the electric field at which the doublet remains at a stationary angle. Our task, then, is set out as follows. We will (1) randomly charge a spheroid of axis ratio β ) a/b, (2) place it with the unique (b) axis pointing along iy, (3) find the eigenvector E that gives Ω ) 0, and (4) repeat the process for a statistical number of trials (e.g., 10 000 trials). We can thus find the average angles between the spheroid axis of symmetry and the stationary state angle. We will let 0.1 e β e 10. To find the angles (θ,φ) that the eigenvector E makes with iy (i.e., the spheroid axis of symmetry), we use spherical coordinates:
x ) sin θ cos φ y ) sin θ sin φ z ) cos θ
E ) (x,y,z) is a unit vector (i.e., x2 + y2 + z2 ) 1). For M trials, the ensemble average of θ is M
(17)
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
(20)
〈θ〉 )
θi ∑ i)1
(21)
M
For a sphere, we know that the direction should be random (i.e., equal probability in every direction), and therefore
M N
ζ0 )
∑ ∑ζi,j j)1 i)1 MN
(18)
∫02π ∫0π/2 θ sin θ dθ dφ ) 1 ) 57.30° 〈θ〉 ) ∫02π ∫0π/2 sin θ dθ dφ
(22)
M N
2
σζ )
∑ ∑(ζi,j - ζ0)2 j)1 i)1 MN
(19)
It is important to note that σζ2/N is constant for randomly charged particles, regardless of the choice of N.17,27 Of course, for extremely large N (small regions) it is unlikely that the charge distribution remains random. The randomness of the charge distribution is an important assumption that we test experimentally in this paper. For uniformly charged (σζ ) 0) spheroids, the particle will not rotate.17,18 In computing A, we assigned ζ potentials randomly to the regions on a spheroid and looked at M trials. We used a Gaussian distribution, although we could have used any random distribution (e.g., uniformly distributed random, Poisson, Weibull) due to the central limit theorem. One always finds that the determinant of A is zero and (23) Velegol, D.; Feick, J. D.; Collins, L. J. Colloid Interface Sci. 2000, 230, 114. (24) 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. (25) Teubner, M. J. Chem. Phys. 1982, 76, 5564. (26) Nir, A.; Acrivos, A. J. Fluid Mech. 1972, 59, 209. (27) Velegol, D. Electrophoresis 2002, 23, 2023.
The integral on θ is only to π/2 since the eigenvector does not distinguish between θ and π - θ with a different φ ) φ + π. Since the standard deviation of θ is σθ2 ) 〈θ2〉 - 〈θ〉2, we can do a similar calculation for 〈θ2〉 to find σθ ) (π 3)1/2 ≈ 21.56°. We also find that 〈φ〉 ) π and σφ ) π/x3 ≈ 103.92°. For the φ direction, these values hold for all axis ratios. Peclet Numbers for Rotational Electrophoresis To assess not only the stationary state angle but also the effect of Brownian motion (or shear) on the stationary angle, we determined the Peclet number for Brownian motion. Thus, once we know the stationary angle of the doublet, the question we then need to answer is, How strongly is it held at that angle relative to Brownian rotation? We define two Pe numbers for each spheroid, since the spheroid can rotate about its axis (the z axis) or perpendicular to it (about the x or y axes):
Pex )
x〈Ωx2〉 ) x〈Ωy2〉
(23)
x〈Ωz2〉
(24)
Dr,x
Pez )
Dr,y
Dr,z
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Table 1. Numerical Results for Stationary State Angles, Rotational Diffusion Coefficients, and Rotational Electrophoresis Coefficientsa β ) a/b 0.1 0.2 0.418 04 0.5 1 2 5 10
〈θ〉 σθ (deg) (deg) 20.2 28.3 43.0 45.8 57.4 70.3 80.7 87.3
19.9 23.3 23.6 23.6 21.5 16.7 9.8 4.6
D cD x ) cz
cD y
0.760 72 18.369 64 0.523 32 4.425 84 0.305 15 0.920 34 0.260 35 0.619 83 0.125 00 0.125 00 0.049 43 0.022 16 0.011 26 0.001 87 0.003 23 0.000 26
ep cep x ) cz
cep y
2.814 2.661 2.250 2.091 1.299 0.594 0.186 0.083
16.33 7.899 3.513 2.868 1.299 0.607 0.245 0.130
a Note in particular the results for a/b ) 1, for which the analytical results are 〈θ〉 ) 57.3° and σθ ) 21.6. This gives an indication of the accuracy of the numerical results, since the analytical results are different by 0.1°. The precision (taken from several ensemble trials) is found also to be roughly 0.1°.
where the x〈ΩR2〉 come from a previous paper17 and are evaluated with the electric field perpendicular to the axis of rotation:
σ E
x〈Ωx2〉 ) x〈Ωz2〉 ) ηbζxN∞ cepx σ E
x〈Ωy2〉 ) ηbζxN∞ cepy
cep x ) xW1313
(25)
cep y ) xW2323 ) xW2121
(26)
where Wijkl are the components of the electrokinetic mobility tensor as calculated in ref 17. The rotational diffusion coefficients are calculated as28
(
)
2 3kT Rxβ + Ry kT D ) cx Drx ) Drz ) 3 2 16πηb 1 + β πηb3
(27)
3kT kT D R ) cy 3 x 16πηb πηb3
(28)
Rx )
∫0∞ [β2 + γ][(β2 +dγγ)2(1 + γ)]1/2
(29)
Ry )
∫0∞ [1 + γ][(β2 +dγγ)2(1 + γ)]1/2
(30)
Dry )
Substituting (27) through (30) into the expressions for Pe, we find
Pex )
Pey )
σζb2E∞ cep x D kTxN cx
σζb2E∞ cep y kTxN
cD y
(31)
(32)
Theoretical Results Table 1 gives the results for axis ratios from 0.1 to 10. The values calculated previously for φ hold for all axis ratios (i.e., 〈φ〉 ) 180°, σφ ) 103°), and therefore they are not listed in Table 1. The results shown are for 10 000 trials, but the results are insensitive (often less than 1% variation) to the number of trials > 1000 or the number of points > 75. We note that in all cases, the spheroid direction for the maximum angular velocity is perpendicular to the spheroid direction for a zero angular velocity. (28) van de Ven, Theo, G. M. Colloidal Hydrodynamics; Academic Press: New York, 1989.
Figure 3. Stationary state angular distribution of randomly charged spheroids in electrophoresis at four different axis ratios (a/b). Ten thousand spheroids were examined and then binned into 5° bins (i.e., 0-5, 5-10, 10-15, etc.). The “frequency” was found by dividing the number in each bin by 10 000. Notice that for a/b ) 10, most of the angles are between 85 and 90°, and therefore that bin is above 0.25 (the top of the figure).
Table 1 shows that prolate (needlelike) spheroids tend to align with the electric field (〈θ〉 near to zero), while oblate (disklike) spheroids tend to have their thin dimension perpendicular to the applied field (〈θ〉 near to 90°). Thus, in both cases the long axis aligns with the electric field. The values of 〈θ〉 show a great variation with a/b (ratio of axes). Since it is difficult experimentally to measure accurately small angles, the best possibility for verifying a random distribution is for a/b from 0.4 to 2. Also, the σθ tend to be especially small for oblate spheroids, indicating that there is little variation from the expected angle. It is also instructive to look at the expected distribution of angles. Thus, even though the average angle for a colloidal spheroid with a/b ) 0.418 is 43°, in fact many of the spheroids would be expected near 0°, and elsewhere. Figure 3 shows this distribution for several axis ratios. Table 1 combines with eqs 31 and 32 to give the Peclet numbers against Brownian motion in the system. In general, for Pe > 10, the doublet will have an easily measurable stationary average angle. On the other hand, for small Pe (e.g., from small particles), a greater amount of data will be required to obtain the statistics for the average stationary angle. The electrophoretic stationary angle can also be compared with shear-induced rotations (Jeffrey orbits).28 This gives a different type of Peclet number, where PeJeffrey ) 〈|Ωep|〉/〈|Ωshear|〉. Experimental Methods The stationary state angle experiments were conducted 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 mounted on a standard 25 mm × 75 mm microscope slide to increase mechanical stability. To minimize settling effects, the video microscope was placed on its back using an aluminum support system. We measured stationary angles of bare polystyrene latex (PSL) particles (4.3 µm sulfated, from Interfacial Dynamics Corp., Portland, OR). We cleaned the particles using a sedimentationdecantation technique.29 MilliQ deionized (DI) water (1 µS/cm) was used as the suspending medium. All glassware for the experiments was sonicated in a VWR brand aquasonic cleaning
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are composed of two spherical particles with different average charges, thus creating a dipole with a large magnitude. Such doublets do not have a random charge distribution. Stationary angles can range from 0 to 180°; however, since the angles 30 and 150 are equivalent, all angles were adjusted from 0 to 90°. Discussion and Conclusions
Figure 4. Experimental distribution of colloidal doublets in an applied electric field. Shown are the final steady angles of 36 doublets, which electrophoretically can be approximated by a spheroid with axis ratio a/b ) 0.418. Observations 1-14 were allowed to coagulate overnight. Observations 15-22 were allowed to coagulate for 15.5 h; doublets 23-32 were allowed to coagulate for 13 h; doublets 33-36 were allowed to coagulate for 6 h. solution for 10 min and cleaned using Alconox precision cleaner and DI water. The electrophoresis cell was also sonicated for 10 min, and then it was soaked overnight in 15.8 N nitric acid. Dilute colloidal suspensions (∼0.05%V) were made by mixing the particles into a KCl solution (10 mM) in an unbuffered pH of 5-6. The suspensions were allowed to sit for 6-24 h to allow doublets to form (Smoluchowski rapid flocculation time ≈ 3.5 h). When a doublet was found, an electric field (usually
