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Electrophoresis of Spheroidal Particles Having a Random Distribution of Zeta Potential Jason D. Feick and Darrell Velegol* Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802-4400 Received July 21, 2000. In Final Form: October 3, 2000 Charge nonuniformity on colloidal particles can be a dominant factor in determining the bulk stability of a dispersion. However, experimental measurements of such nonuniformity are lacking. One straightforward technique for measuring the charge nonuniformity on particles is “electrophoretic rotation”, and the calculations in this paper enable the interpretation of electrophoretic rotation experiments for spheroids. For systems with infinitesimal double layers, it is shown that if the charge (or ζ potential) is randomly distributed over the individual particles in a dispersion, where each particle is covered with N equal-area patches, then the spheroids will on average rotate with an angular velocity proportional to σζ/xN, where σζ is the standard deviation of zeta potential on the patches. This is true for any random distribution of ζ potential, which emphasizes that “random” implies “nonuniform”. Whereas standard translational electrophoretic mobility gives the average zeta potential (〈ζ〉) on particles, the rotational electrophoretic mobility gives the standard deviation of ζ potential (σζ).
Introduction The stability of colloidal suspensions has traditionally been described by the fifty-year-old DLVO theory. This theory predicts the stability of a suspension based on the charge or zeta (ζ) potential on the particle’s surface.1 But the classical DLVO theory, which assumes a uniform ζ potential on the particle surface, often fails to accurately predict the stability of many colloidal suspensions.2,3,4 Recently, theory has shown that the distribution or nonuniformity of ζ potential on colloidal particles is important in predicting colloidal stability.5-10 Charge nonuniformity might arise, for instance, from different amounts of selective adsorption onto different crystal planes of the same material.11,12 A schematic of a nonuniformly charged particle is shown in Figure 1. The figure shows a particle whose average ζ potential (roughly -50 mV) might be sufficient to keep the suspension stable if the charge were uniformly distributed; however, if on two different particles, the patches of -20 mV were to approach each other (e.g., by Brownian motion), the particles could aggregate into an irreversible primary energy minimum. Thus, due to the small area of interaction between colloidal * To whom correspondence should be adressed. E-mail: velegol@ psu.edu. (1) Hunter, R. J. Foundations of Colloid Science; Clarendon Press (Oxford University Press): New York, 1986 (with corrections in 1992); Vols. I and II. (2) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989 (with corrections 1991). They list a few examples on p 277. (3) Velegol, D.; Anderson, J. L.; Garoff, S. Langmuir 1996, 12, 675. (4) Velegol, D.; Anderson, J. L.; Garoff, S. Langmuir 1996, 12, 4103. (5) Czarnecki, J. Adv. Colloid Interface Sci. 1986, 24, 283. (6) Miklavic, S. J.; Chan, D. Y. C.; White, L. R.; Healy, T. W. J. Phys. Chem. 1994, 98, 9022. (7) Grant, M. L.; Saville, D. A. J. Colloid Interface Sci. 1995, 171, 35. (8) Holt, W. J. C.; Chan, D. Y. C. Langmuir 1997, 13, 1577. (9) Stankovich, J. M. Electrical Double Layer Interactions Computed Using the Poisson-Boltzmann Theory. Ph.D. Thesis, University of Melbourne, Australia, 1998. (10) Stankovich, J. M.; Carnie, S. L. J. Colloid Interface Sci. 1999, 216, 329. (11) Koopal, L. K. Electrochim. Acta. 1996, 41, 2293. (12) Hiemstra, T.; Yong, H.; Van Riemsdijk, W. H. Langmuir 1999, 15, 5942.
Figure 1. Schematic of a nonuniform distribution of ζ potential on the surface of a prolate spheroid. This spheroid has been placed in an electric field (E∞). Each patch has a different ζ potential (units mV) for the chosen length scale (L).
particles and Brownian motion, the local ζ potential on the particle is critical. Measurements do not exist for the charge nonuniformity on colloidal particles. One method for obtaining this measurement is the experimental technique of electrophoretic rotation.13-16 In systems with thin double layers, a uniformly charged particle does not rotate;17 however, a nonuniformly charged particle does in general rotate.3,13,14,18,19 Thus, whereas translational electrophoresis can be used to measure the average zeta potential (〈ζ〉) on colloidal particles, electrophoretic rotation can be used to measure the standard deviation of ζ potential (σζ) on the particles. The purpose of this paper is to provide calculations that will enable the measurement of charge nonuniformity from the electrophoretic rotation of colloidal spheroids. (13) Fair, M. C.; Anderson, J. L. J. Colloid Interface Sci. 1989, 127, 388. (14) Fair, M. C.; Anderson, J. L. Int. J. Multiphase Flow 1990, 16, 663. Corrigenda Int. J. Multiphase Flow 1990, 16, 1131. (15) Fair, M. C.; Anderson, J. L. Langmuir 1992, 8, 2850. (16) Velegol, D.; Feick, J. D.; Collins, L. J. Colloid Interface Sci. 2000, 230, 114. (17) Morrison, F. A., Jr. J. Colloid Interface Sci. 1970, 34, 210. (18) Teubner, M. J. Chem. Phys. 1982, 76, 5564. (19) Anderson, J. L. J. Colloid Interface Sci. 1985, 105, 45.
10.1021/la001031a CCC: $19.00 © 2000 American Chemical Society Published on Web 11/28/2000
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In standard electrophoresis experiments, the measured quantity is the translational velocity (U) of the particles. This is then related to the ζ potential by the Smoluchowski equation17,20
U)
ζE∞ Ω)0 η
(1)
or some other similar equation, such as the model by O’Brien and White.21 In eq 1, is the fluid permittivity; η is the fluid viscosity; and E∞ is the uniform applied electric field. The Smoluchowski equation contains several important assumptions:17 (1) While the particle can have any shape, it is rigid and nonconducting. (2) The electrical double layer (EDL) is infinitesimally thin [i.e., κR/cosh(Zeζ/2kT) . 1, where κ-1 is the Debye length; R is the local radius of curvature on the particle; Z is the valence of solution ions; and kT is the thermal energy].22 (3) The fluid around the particle is unbounded. (4) The ζ potential is uniformly distributed over the particle surface. In this paper, we relax the fourth assumption, assuming that the particles in fact have a random distribution of ζ potential on their surface. Modeling a Random Charge Distribution 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 EDL must be infinitesimally thin, and the correlation length (d) of charge variation must have (κd . 1). The equations for U and Ω are13
U) Ω)
∫ ∫ (n‚r)(I - nn)ζdS‚G‚E
3ηV
(2)
∞
S+ p
∫ ∫ (n‚r)r × (I - nn)ζdS‚G‚E
H‚ ηV
∞
S+ p
(3)
where for a spheroid with semi-axes (a,b,a), the surface vector (r) and the surface normal (n) at the edge of the double layer (Sp+) are defined by23
r ) xix + yiy + ziz
(4)
n ) ∇f/|∇f|
(5)
x2 y2 z2 f) 2+ 2+ 2-1)0 a b a
(6)
4 2 πa b 3
For uniformly charged particles, eq 3 shows that the electrophoretic angular velocity of a particle is zero. However, particles that are randomly charged will in general have a finite angular velocity due to electrophoresis,16 and it is this rotation that can be used to measure the charge nonuniformity on the colloidal particles. Electrophoretic rotation experiments are difficult to perform on colloidal spheres since the particle would have to be tagged (e.g., by a fluorescent marker). Colloidal doublets are much easier to observe rotating in an experiment. On the other hand, developing a model to interpret the standard deviation of the zeta potential (σζ) from the electrophoretic rotation experiments is more difficult for a doublet than a sphere, and thus we will approximate a doublet as a spheroid (see Figure 2).
Expressions for Electrophoretic Mobilities The experimentally observable quantities in electrophoresis are the translational and angular electrophoretic velocities. Thus, eqs 2 and 3 are the starting point. We model a randomly charged spheroid as a particle covered with N equal-area patches, each with a ζ potential taken from a Gaussian [〈ζ〉,σζ] distribution. We approximate eqs 2 and 3 as
U)
Ω)
N
∑(ni ‚ri)(I - nini)ζiAi‚G‚E∞
3ηVi)1
(8)
N
ηV
(ni ‚ri)ri × (I - nini)ζiAi‚G‚E∞ ∑ i)1
H‚
(9)
where for equal-area patches,
and G and H are constant geometric tensors defined in ref 13. Note that geometrically, Sp+ is the same as S. The volume (V) of a spheroidal particle with semiaxes (a,b,a) is24
V)
Figure 2. Colloidal doublet approximated as a spheroid. If the radius of the spheres in the doublet is R, the equivalent spheroid has a/R ) 0.99993 and b/R ) 2.39192.
(7)
Equations 2 and 3 were developed from the Stokes equations of hydrodynamics and the electrostatic equations. (20) Anderson, J. L. Annu. Rev. Fluid Mech. 1989, 21, 61. (21) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2. 1978, 74, 1607. (22) Dukhin, S. S.; Derjaguin, B. V. Electrokinetic Phenomena; Matijevic, E., Ed.; Surface and Colloid Science; Vol. 7; Wiley: New York, 1974. (23) Kreyszig, E. Advanced Engineering Mathematics, 6th ed; Wiley: New York, 1988. (24) Weisstein, E. W. CRC Concise Encyclopedia of Mathematics; CRC Press: New York 1999.
Ai ) A/N
(10)
∫∫ζ(x)dA ζi )
Ai
(11)
Ai
∫∫n(x)dA ni )
Ai
(12)
Ai
As N f ∞, eqs 8 and 9 are equivalent to eqs 2 and 3. The dyads 〈UU〉 and 〈ΩΩ〉 are measurable quantities, and these are what we seek. The ensemble average of x, indicated by 〈x〉, is defined as Q
〈x〉 ) lim
Qf∞
∑ xR R)1 Q
(13)
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Langmuir, Vol. 16, No. 26, 2000 10317
over Q particles that are independent and identically distributed. The dyads are given by
UU )
( )∑ 2 N
[
3ηV
N
ΩΩ )
( ) ηV
(nj ‚rj)(I - njnj)ζjAj‚G‚E∞] ∑ j)1
(14)
(ni ‚ri)ri × (I - nini)ζjAj‚G‚E∞] ∑ i)1 N
[H‚
(nj ‚rj)rj × (I - njnj)ζjAj‚G‚E∞] ∑ j)1
(15)
1
N
3V
∑ ζiAiI‚E∞ 3ηV i)1 A ΩSmol ) 0
(17) (18)
The dyads of eqs 17 and 18 are
( ) ( )
3ηV
2 N N
∑ ∑ i)1 j)1
3V
2
ζiζjAiAj(I‚E∞)(I‚E∞)
A
(19) ΩSmolΩSmol ) 00
(20)
Since ΩSmolΩSmol ) 00, it will be ignored throughout the remainder of the discussion. Combining eqs 14, 15, 19, and 20 gives
UU - USmolUSmol ) 2N N (ni‚ri)(nj‚rj)ζiζjAiAj[(I - nini)‚G‚E∞] 3ηV i)1 j)1 3V 2 [(I - njnj)‚G‚E∞] ζiζjAiAj(I‚E∞)(I‚E∞) (21) A
( ) ∑∑ {
( ) ( )
ΩΩ - ΩSmolΩSmol ) ΩΩ )
}
N
2
ηV
∑ i)1
[H‚
(ni ‚ri)ri ×
N
(I - nini)ζiAi‚G‚E∞][H‚
(nj ‚rj)rj × ∑ j)1
(I - njnj)ζjAj‚G‚E∞] (22) Substituting eqs 16 into eqs 21 and 22 and converting
2 2σζ EE T 〉) η N k m ikjm
()
(25)
2 2σζ EE W ηb N k m ikjm
(26)
Smol
Uj
〈ΩiΩj〉 )
( )
where
Tikjm )
Wikjm )
A
USmolUSmol )
〈UiUj - Ui
(16)
∫∫AζdS‚E∞ ) 3ηV∫∫ A ζIdS‚E∞ ) η A
∫∫
Smol
3V
2
Putting these into indicial notation and separating out the electric field, we obtain
where δij is the Kronecker delta. Equation 16 is a mathematical statement that the ζ potentials on different patches are uncorrelated, and therefore that their products average to zero. We want to compare the averages of these dyads to the averages that result from a uniform distribution of ζ potential, which are obtained from eq 1:
USmol )
2
b2A σζ [H‚(n‚r)r × V2 N A (I - nn)‚G‚E∞][H‚(n‚r)r × (I - nn)‚G‚E∞]dA (24)
We now assume that the particles are randomly charged, where the correlation length of charge variation (d) is much smaller than the patch size (L).25 For a Gaussian distribution, this gives
〈ζiζj〉 ) ζ20 + σζ2δij
}
( )
〈ΩΩ〉 ) ηb
[H‚
{
∫∫
( )( )
N
2
〈UU - USmolUSmol〉 ) 2 2 A3 σζ 1 (n‚r)2[(I - nn)‚G‚E∞] η 9V2 N A A2 3V 2 (I‚E∞)(I‚E∞) dA (23) [(I - nn)‚G‚E∞] A
( )( )
(ni ‚ri)(I - nini)ζiAi‚G‚E∞]
i)1
[
back to the integral form yields
( ) ∫∫ { A3 9V2
A
1 (n‚r)2{[(δil - ninl)Glk][(δjn 2 A 3V 2 δikδjm dA (27) njnn)Gnm]} A
( ) ∫∫ b2A V2
( )
}
(n‚r)2[Hil(lpqrp(δqs - nqns)Gsk] ×
A
[Hjt(tuvru(δvw - nvnw)Gwm] dA (28) Note the order of the subscripts in eqs 25 and 26. We point out that the σζ2/N portion of these equations is reminiscent of the stochastic problem of the fluctuations in the number of ideal gas molecules in a finite volume.25 This type of behavior occurs for all random processes that have no correlations. Similar results have been obtained for a spherical particle, including results that give 〈xU‚U〉 and 〈xΩ‚Ω〉 in addition to 〈U‚U〉 and 〈Ω‚Ω〉.14 Some of these results are
U)
1 〈ζ〉I - P2 ‚E∞ η 2
[
]
9 4 Ω ) P1 × E∞ ηa
(29)
(30)
σζ 〈xP1‚P1〉 ) 0.92 xN
(31)
σζ 〈x(P‚iz)‚(P‚iz)〉 ) 1.302 xN
(32)
where eq 32 is written assuming that the electric field points in the z direction. For a sphere, it has been shown that G ) (3/2)I and H ) I/2a2.13 It is interesting that for a sphere, the complete electrophoretic motion can be (25) Landau, L. D.; Lifshitz, E. M. Statistical Physics; Pergamon Press: London, 1958; Chapter XII.
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In order that the points are distributed evenly over the surface, we assign a number (Nk) points to each slice:
( )
Nk ) int
2πRk L
(36)
The total number of points on the spheroid is thus given by M
N)
∑ Nk
(37)
k)0
where at the poles of the spheroid we set N0 ) 1 and NM ) 1 (a single point). Then, the coordinates of each patch are found from eqs 4 through 6 to be (the kth point on the mth slice) Figure 3. Discretization of a spheroid. The origin of the x-y-z axes is at the center of the spheroid, and the axes rotate with the spheroid.
described using only three moments (the monopole 〈ζ〉, the dipole P1, and the quadrupole P2).19 This is not the case for a general spheroid; higher order moments are important, and thus the angular velocity cannot be described purely in terms of a single moment like the dipole. It is important to realize that whereas the results shown for both spheroids and spheres have been derived assuming a Gaussian distribution of charge, the result applies for any other random distribution, including the Poisson distribution. This is according to the central limit theorem of statistics. We also emphasize that the randomness need not occur only by the simple Poisson placement of charges on the surface. Indeed, longer-range correlations are allowable, as long as L/d . 1 (i.e., the patch size is greater than the correlation length of charge variation). To evaluate the integrals in eqs 27 and 28, the spheroid was divided into N patches, each represented by a point on the spheroid surface. In the limit of large N, these patches become square, and the point becomes the center of the patch. The spheroid was divided into M slices (the top and bottom poles are each considered to be a slice here), each with an equal arc length (L) between them (see Figure 3). Figure 3 also shows the ellipse that results from projecting the spheroid onto the y-z plane. The length (P) along the ellipse from -b up to any point c along the y axis is
P(a,b,c) )
x
∫-bc
4
2
2
2
b + h (a - b ) dh b2 - h2 b
(33)
Thus, the perimeter of the total ellipse is 2P(a,b,b). The arc length (L) separating each slice is
L)
P(a,b,b) M-1
(34)
The y coordinate (Yk) of each slice was determined by finding the roots of eq 33 such that P(a,b,Yk) ) kL. The radius (R) of each cross-sectional slice is
Rk )
x(
a2 1 -
)
Yk2 b2
(35)
xk,m ) Rksinφk,m, yk,m ) Yk, zk,m ) Rkcosφk,m
(38)
where m goes from 0 to M - 1, and φk,m is the angle of each point on the spheroid (see Figure 3) defined by the equation
φk,m )
2mπ , m ) 0,...,Nk - 1 Nk
(39)
The result of this discretization is a spheroid approximately covered with N equal area patches. Near the top and bottom of the spheroid, the distribution of points is less precise than near Yk ) 0, where the patches are very nearly square. The normals for each point on the spheroid can now be found from eq 5 to be
xk,m
nxk,m ) a2
x
x2k,m + z2k,m a4
(40) +
y2k,m
+
y2k,m
+
y2k,m
b4
yk,m
nyk,m ) b2
x
x2k,m + z2k,m a4
(41) b4
zk,m
nzk,m ) a
2
x
x2k,m
+ z2k,m a4
(42) b4
Finally, we set e ) iy in order to evaluate the constant geometric tensors G and H. Thus, our coordinate system is fixed to the particle, which means the electric field rotates with respect to the particle. Since we assume that 1/|Ω| . b2F/η, the time-independent Stokes equations and electrostatic equations are still sufficient to describe our system. The r and n vectors are listed above with subscripts (k,m), indicating the kth point on the mth slice of the spheroid. Once we had calculated these values, we renumbered the r and n values consecutively from 1 to N so that we could numerically evaluate the integrals in eqs 27 and 28 by summing over the N patches. Convergence of eqs 27 and 28 was determined by increasing the number (N) of points used to cover the spheroid. As a higher N was used, the T and W mobility tensors converged to constant values. We estimated the accuracy of the values by the percent change between the two highest values of N for each a/b ratio. In all cases,
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Table 1. Numerical Results for T and W in eqs 27 and 28. An “Equivalent Doublet” is Approximated by a Prolate Spheroid with a Size Ratio (a/b) of 0.418 a/b
0.1
0.2
0.418
0.5
0.667
1
1.5
2
T1111 T1122 T1133 T1212 T1221 T1313 T2121 T2222
0.7860 0.1412 -0.2288 0.0220 0.0422 0.5074 0.0810 0.1006
0.6790 0.0694 -0.1529 0.0389 0.0697 0.4159 0.1246 0.0400
0.4743 -0.0301 -0.0964 0.0699 0.1050 0.2853 0.1578 0.0118
0.4145 -0.0529 -0.0921 0.0809 0.1139 0.2533 0.1604 0.0214
0.3189 -0.0831 -0.0921 0.1031 0.1282 0.2055 0.1595 0.0618
0.2000 -0.1000 -0.1000 0.1500 0.1500 0.1500 0.1500 0.2000
0.1204 -0.0693 -0.1003 0.2315 0.1774 0.1104 0.1360 0.5029
0.0939 -0.0067 -0.0876 0.3288 0.2036 0.0908 0.1261 0.8936
0.1713 0.5437 0.0544 1.2694 0.3619 0.0585 0.1032 4.7537
5
0.3548 1.5764 0.2561 4.2022 0.6287 0.0493 0.0941 16.6357
10
W1111 W1212 W1221 W1313 W1331 W2121 W2222
1.8637 0.2424 -4.8967 7.9184 -4.1911 266.7457 0.0000
1.1636 0.4972 -4.3262 7.0802 -4.7529 62.3964 0.0000
0.3431 1.0326 -3.3130 5.0619 -4.3757 12.3392 0.0000
0.2007 1.2013 -2.9998 4.3732 -3.9719 8.2254 0.0000
0.0551 1.4624 -2.4574 3.1958 -3.0857 4.2611 0.0000
0.0000 1.6875 -1.6875 1.6875 -1.6875 1.6875 0.0000
0.0136 1.6380 -1.0432 0.7136 -0.6863 0.6834 0.0000
0.0191 1.4819 -0.7109 0.3530 -0.3149 0.3686 0.0000
0.0071 0.9745 -0.2046 0.0347 -0.0204 0.0600 0.0000
0.0020 0.7817 -0.0868 0.0069 -0.0030 0.0169 0.0000
these accuracies are better than 1%, and in most cases better than 0.1%. Numerical Results Table 1 gives results for T and W at aspect ratios from a/b ) 0.1 to 10. Although the fourth-order tensors T and W have 81 components, the numerics show that in fact T has only 8 independent and nonzero components, whereas W has only 6. In addition, T has 21 nonzero components, whereas W has only 16. Useful symmetry relations are
Tikjm ) T(4-i)(4-k)(4-j)(4-m)
(43)
Wikjm ) W(4-i)(4-k)(4-j)(4-m)
(44)
T1331 ) T1313, T2112 ) T1221
(45)
W1133 ) -W1111, W2112 ) W1221
(46)
In addition,
It is convenient that T and W are nondimensional, because this enables these tensors to be used for any size spheroids. If we are interested only in diagonal terms (i.e., not terms such as 〈UxUy〉, but rather terms such as 〈UxUx〉), the above equations can be simplified to
[] [ [] [ 〈U2x 〉
2
〈U2y 〉 ) 〈U2z 〉
2σζ
T1111 T1212 T1313
(η) N TT
〈Ω2x 〉
2121 3131
][ ] ][ ] E2x
T2222 T2323 ‚ E2y T3232 T3333 E2 z
2 W1111 W1212 W1313 Ex 2 σ 2 ζ W 2 〈Ω2y 〉 ) 2121 W2222 W2323 ‚ Ey ηb N W3131 W3232 W3333 E2 〈Ω2z 〉 z
( )
(47)
(48)
Several methods were used to verify the results in Table 1. First, the results for a randomly charged sphere have been calculated analytically.16 These match the coefficients in Table 1 for a size ratio of 1. Second, the T and W mobility tensors have the appropriate symmetries. For example, for spheroids aligned with e ) iy (as in this paper) and for E∞ ) E∞iy, 〈Ω1Ω1〉 ) 〈Ω3Ω3〉 . This is seen in Table 1 since W1212 ) W3232. Third, we checked that the area we obtained from covering a spheroid with “patches” (i.e., NL2) well-
Figure 4. Particles charged “by halves”: (a) spheroid and (b) doublet.
{
approximated the exact result for a spheroid, which is
x
sin-1 e a2 , e) 2πa2 + 2πab 1 - 2 if a < b e b 2 e ) 0 if a ) b 4πa A) 1+e ln b2 2πa2 + πb2 1 - e , e ) 1 - 2 if a > b a e (49)
(
)
x
The approximate area always matched the exact area to within better than 0.02%. A fourth test examined how well our numerical scheme matched analytical results for a sphere and a spheroid. Analytical results exist, giving the translational mobility of a sphere with ζ1 around the equator and ζ2 at the top and bottom; analytical results also exist for the rotation of a sphere in which one-half has ζ1 and the other half has ζ2.19 Our numerical scheme reproduced the translational results to within 0.6% and the rotational results to within 0.001%. Analytical results also exist for the rotation of a spheroid with an axisymmetric charge distribution.13 These are
Ωspheroid )
9 β × E∞ ηb 4 1
()
(50)
where if the top of the spheroid has zeta potential (ζ2) and the bottom has (ζ1) (see Figure 4) then
1 β1 ) G⊥(ζ2 - ζ1) 3
[1 + ((R2 - 1)/(R2 + 1))z2] dze [1 + (R2 - 1)z2] (51)
∫01z
(R2 - 1 + 2R2 ln R) 1 ) G⊥(ζ2 - ζ1) 6 (R2 + 1)(R2 - 1) where G⊥ is defined in ref 13 and R is the axis ratio (a/b).
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Our numerical scheme reproduced the results from the analytical equations for various R to better than 0.05%. An “Equivalent Doublet” Spheroids can be used to approximate shapes ranging from needles (long prolate spheroids) to disks (flat oblate spheroids), and thus, eq 25 through 28 represent broadly applicable solutions. An important solution is the “equivalent doublet”. The concept was introduced by Nir and Acrivos when they showed that for particles undergoing Jeffery orbits in a uniform shear field, a prolate spheroid with an axis ratio of 0.5045 () 1/1.982) is hydrodynamically equivalent to a doublet.26 Only the ratio a/b was necessary since the rotation was caused by a shear flow. For electrophoresis, the approximation is more subtle. We seek two coefficients (the major and minor axes of the spheroid), and there are many items we can equate to a doublet to calculate these axes: surface area, total volume, total surface charge, and hydrodynamic mobility parallel or perpendicular to the major axis. We chose to keep the surface area and surface charge density equal on both the doublet and spheroid. We also chose to make the electrophoretic angular velocity the same when both the doublet and spheroid are charged “by halves”, as shown in Figure 4. The electrophoresis of rigid colloidal doublets has previously been studied theoretically13,14 and experimentally.15 For a doublet composed of two spheres with radius (R)
Adoublet ) 8πR2
(52)
The angular velocity of a doublet charged “by halves” is given by14
Ωdoublet )
(ζ2 - ζ1) N ˜ e × E∞ 2ηR
[] [ [] [ 〈U2y 〉 ) 〈U2z 〉
2σζ
0.4743 0.0699 0.2853
][ ] ][ ]
〈Ω2y 〉 )
2 y
0.2853 0.0699 0.4743
2σζ
2
0.3431 5.0619
(54)
E2z
1.0326 5.0619
(ηb ) N 12.3392 0
Nonuniform Distribution of Surface Charge Up to this point, we have considered how a random ζ potential affects electrophoretic motion. However, it is also useful to know how a nonuniform surface charge affects the electrophoresis of colloidal particles. Since surface forces are between the constant potential and constant charge limits, the constant charge assumption is important. For a flat plate in an unbounded fluid, the ζ potential is related to the surface charge density (Fs) by
Fs )
E2x
12.3392 ‚ E2y
1.0326 0.3431
E2z
(55)
These equations can be used directly to evaluate the standard deviation of the zeta potential on colloidal doublets from electrophoretic rotation data. As an example, consider a doublet with R ) 1 µm and e ) iy in an electric field of 10 V/cm pointing in the z direction. The fluid permittivity ) 7.1 × 10-10 C2/m2N; (26) Nir, A.; Acrivos, A. J. Fluid Mech. 1972, 59, 209.
4n∞Ze Zeζ sinh κ kT
(56)
which is from the Gouy-Chapman model for the electrical double layer.1 In this equation, n∞ is the bulk electrolyte concentration; Z is the valence of the Z:Z electrolyte in solution; e is the proton charge; k is the Boltzmann constant; and T is the temperature. This equation is also valid if the particle radius of curvature is much greater than the Debye length. Dimensionless coefficients can be defined:
Fˆ s )
E2x
(η) N 0.1578 0.0118 0.1578 ‚ E
〈Ω2x 〉
〈Ω2z 〉
2
scale of obervation”) is xκ-1R ) 100 nm (Debye length is 10 nm), and so N ) 8πR2/πL2 ) 800. Under these circumstances, the measured angular velocities tell us that σζ ) 58 mV; this large variation could easily cause two particles to fall into an irreversible potential energy well. We could have chosen a different length scale; however, a larger length scale would not be pertinent. A smaller length scale would be very useful in a full PoissonBoltzmann analysis of the interparticle forces. An advantage of having the parameter σζ2/N is that the standard ˆ) deviation (σζ) can be found for a patch of any area (A since N ) Aspheroid/A ˆ . Thus, our results are quite general.
(53)
where N ˜ is the dimensionless rotation coefficient, found to be 0.6400 for a rigid doublet with R ) 1. Setting the areas in eqs 49 and 52 equal, as well as the angular velocities in eqs 50 and 53, allows us to solve simultaneously for a and b. An “equivalent doublet” in electrophoretic rotation has a/R ) 0.99993 and b/R ) 2.39192. The values in Table 1 are for 0.99993/2.39192 ) 0.4180. The simplified matrixes for an equivalent doublet can be written in the forms shown in eqs 47 and 48. Taking the values from the T and W tensors gives
〈U2x 〉
the viscosity η ) 1 cP; and the salt concentration is 1 mM KCl. We can measure U2, U3, and Ω1 in a straightforward manner.3,4 In this example, we find experimentally that 〈U2‚U2〉 ) 0; 〈U3‚U3〉 ) 1260 µm2/s2; and 〈Ω1‚Ω1〉 ) 1.86 s-2. All the doublets appear to have the same translational mobility to within experimental uncertainty, which in this example is roughly 5%. In this case, the translational velocities tell us that 〈ζ〉 ) -50 mV. For the angular velocities, we need to choose a number of patches (N). From the theory of colloidal forces, we can estimate that a length scale we wish to know about (i.e., the “length
Fsκ Zeζ , ζˆ ) 4n∞Ze kT
(57)
ζˆ Fˆ s ) sinh 2
(58)
These give
By choosing the Fˆ s from a random distribution, the corresponding ζ potentials can be found and the 〈ζ〉 and σζ computed; these can then be used in eqs 25 through 28. For small values of Fˆ s, 〈UU - USmolUSmol〉 and 〈ΩΩ〉 increase linearly with σFs2 since for small surface charge densities, the relationship between ζ and Fs is linear. However, whereas 〈UU - USmolUSmol〉 and 〈ΩΩ〉 are independent of 〈ζ〉 for a given σζ at high 〈Fˆ s〉, the dyads do depend on both 〈Fˆ s〉 and σFs due to the nonlinear relationship between 〈Fˆ s〉 and 〈ζˆ 〉 expressed in eq 58. At high 〈Fˆ s〉, we see that σζ will normally be small, causing 〈UU - USmolUSmol〉 and 〈ΩΩ〉 to level off as σFs in-
Electrophoresis of Spheroidal Particles
creases. The leveling off is shown clearly for a sphere in Figure 6 of ref 16. Conclusions This paper emphasizes that a random charge distribution implies a nonuniform charge distribution. Although previous work has extended the analysis to thick EDLs,16 the present paper deals with only infinitesimal EDLs. The key results in this paper, which are eqs 25 through 28 and Table 1, provide the means to interpret the standard deviation of the ζ potential (σζ) from readily measurable quantities in electrophoretic rotation experiments. This can be done for spheroids, as well as for shapes that can be approximated as spheroids (e.g., colloidal
Langmuir, Vol. 16, No. 26, 2000 10321
doublets, disklike particles such as clay). σζ depends on the length scale of observation, and as one focuses on small regions of a spheroid, a larger standard deviation is seen. Whereas translational electrophoresis can be used to find the average zeta potential on the surface of colloidal particles, rotational electrophoresis can be used to find the standard deviation. Acknowledgment. The authors thank the National Science Foundation for CAREER Grant CTS 9984443 and the Petroleum Research Fund for Type G Grant 35400G9. D.V. thanks John L. Anderson for his continued support and several interesting discussions. LA001031A
