Probing the Structure of Colloidal Doublets by Electrophoretic Rotation

Langmuir 1996, 12, 675-685

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Probing the Structure of Colloidal Doublets by Electrophoretic Rotation Darrell Velegol and John L. Anderson* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213

Stephen Garoff Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received June 12, 1995. In Final Form: September 25, 1995X A doublet formed by the coagulation of two colloidal spheres rotates toward alignment with an applied electric field. The rotation rate is proportional to the difference in the zeta potentials of the two spheres and a geometric coefficient N. This coefficient can be calculated from the electrohydrodynamic equations and depends on the kinematic boundary condition imposed on the spheres. If the doublet rotates as a single rigid body, so that the surfaces of the two spheres do not move relative to one another, then the value of N is a factor of 2 or 3 smaller than if the two spheres are free to rotate relative to one another. Thus, one can evaluate the rigidity of the doublet by determining the value of N experimentally. We have tracked the motion of individual doublets formed from micrometer-size polystyrene latex spheres differing in zeta potential and determined the value of N for different solution conditions. The influence of Brownian motion was taken into account when interpreting the rotational trajectories of the doublets. Over a range of electrolyte concentration, pH, sphere size, and electric field, all the values of N agree with the theoretical values for rigid-body rotation. This result is surprising because in many of the experiments the conditions were such that the two spheres should have been in a secondary minimum with a gap between 5 and 20 nm according to classical DLVO theory. In these cases the freely-rotating condition on each sphere is expected. Possible explanations for the rigid-body behavior of the doublets are suggested.

Introduction The mechanical and material properties of composite media and coatings depend on pair interactions between colloidal particles, since these interactions control the morphology in the early stages of particle aggregation. These pair interactions are generally modeled by DLVO theory, which allows for van der Waals and electrostatic forces. For two spheres, DLVO-based theories assume that forces act along the line between the centers. Typical energy and force profiles, plotted against the gap separating the two spheres, can be found in many texts (for example, refs 1 and 2). The qualitative features and quantitative predictions of this theory, including the existence of a “secondary minimum”, have been demonstrated for macroscopic surfaces of mica using the surface forces apparatus.3 There have been some studies, however, that can only be explained by the existence of nonDLVO forces, such as hydration forces,4 which are very short range and important at higher electrolyte concentrations when electrostatic repulsions are small and the system is not in a secondary minimum. Much less progress has been made toward the direct determination of forces acting between two colloidal spheres. One recently developed technique, based on total internal reflection microscopy (TIRM), permits the measurement of the potential energy between a flat surface and a single sphere of order 10 µm diameter.5 Another * To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, January 1, 1996. (1) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989; Chapter 8. (2) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992; Chapter 12. (3) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1, 1978, 74, 975. (4) Ducker, W. A.; Xu, Z.; Clarke, D. R.; Israelachvili, J. N. J. Am. Ceram. Soc. 1994, 77, 437. (5) Prieve, D. C.; Frej, N. A. Langmuir 1990, 6, 396.

0743-7463/96/2412-0675$12.00/0

technique based on atomic force microscopy (AFM) has been used to measure the forces between a particle and a flat surface.6 Continual refinement of this technique might make feasible its application for measuring forces between two fixed particles with large interparticle forces.7 A new technique, colloidal particle scattering,8 probes the interaction between a stationary sphere fixed to a surface and a second sphere convected by shear flow. The experimental trajectories of the second sphere are consistent with the DLVO theory, including a secondary minimum in some cases. However, it is unclear from ref 8 how many trajectories, if any, were in the vicinity of the secondary minimum. Little has been done to measure forces that are not along the line of centers between two spheres. Tangential forces determine the rotational rigidity of two spheres. If the two spheres act as a single rigid body (i.e., no relative movement between the spheres), then the spheres make a rigid doublet. However, the spheres might remain in a doublet and yet rotate individually such that the torque on each is zero; this is a freely-rotating doublet. To identify rigid versus freely-rotating doublets, Mason and co-workers9-11 tracked the rotation of doublets in simple shear flows. The doublets follow Jeffery-like orbits with a period of rotation (T) that depends on the relative size of the spheres in the doublet, the separation between the spheres, and the shear rate (γ). For a rigid doublet of equal-size spheres, Tγ ) 15.62 if the gap (δ) between (6) Larson, I.; Drummond, C. J.; Chan, D. Y. C.; Grieser, F. J. Am. Chem. Soc. 1993, 115, 11885. (7) Li, Y. Q.; Tao, N. J.; Pan, J.; Garcia, A. A.; Lindsay, S. M. Langmuir 1993, 9, 637. (8) van de Ven, T. G. M.; Warszynski, P.; Wu, X.; Dabros, T. Langmuir 1994, 10, 3046. (9) van de Ven, T. G. M.; Mason, S. G. J. Colloid Interface Sci. 1976, 57, 505. (10) van de Ven, T. G. M.; Mason, S. G. J. Colloid Interface Sci. 1976, 57, 517. (11) Takamura, K.; Goldsmith, H. L.; Mason, S. G. J. Colloid Interface Sci. 1979, 72, 385.

© 1996 American Chemical Society

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the spheres is zero and Tγ varies by less than 2% when the gap is less than 10% of the sum of the radii of the two spheres. However, if the doublet is freely-rotating, then Tγ rises from 15.62 at contact (δ ) 0) to values between 18 and 20 for gaps predicted by DLVO theory for doublets of micrometer-size spheres in a secondary minimum (δ ≈ 10-50 nm for their experiments). On the basis of the rotation rate, van de Ven and Mason10 attempted to classify a doublet as either rigid or freely-rotating, with δ determined from the amount by which the experimental value of Tγ exceeded 15.62. Such an interpretation is difficult when Tγ ≈ 16-17, as it was for many doublets in their experiments. Takamura et al.11 attributed values of Tγ in this range to freely-rotating doublets with spheres that are only 2 nm apart, an unrealistic value according to classical DLVO theory. But perhaps other factors (e.g., surface roughness and charge heterogeneity on the surface) caused a freely-rotating doublet to be rigid for part of the rotations, making Tγ lower for the doublet. One difficulty with the interpretation is that every value of Tγ must be assigned as either a rigid doublet (if Tγ ) 15.62) or a freely-rotating doublet with a particular gap (for all other Tγ). Between the purely rigid and purely freely-rotating cases there are no “impossible” values that would enable one to discover if factors such as charge heterogeneity caused the lower Tγ. Nevertheless, van de Ven and Mason found some doublets with Tγ ) 15.6 (rigid) and Takamura et al. found some with Tγ between 18 and 20 (freely-rotating), providing evidence that a doublet can indeed be either rigid or freely-rotating. In this paper we describe a more sensitive test of doublet rigidity based on “electrophoretic rotation”. A doublet formed by the coagulation of two colloidal spheres with different zeta potentials acts as a dipole and rotates toward alignment with an applied electric field. The rotation rate is proportional to the difference in the zeta potentials of the two spheres and a geometric coefficient N. This coefficient can be calculated from the electrohydrodynamic equations assuming either of two kinematic boundary conditions imposed on the spheres. If the doublet rotates as a rigid body, so that the surfaces of the two particles do not move relative to one another, then the value of N is a factor of 2-3 smaller than if the spheres can rotate relative to each other. Thus, an experimental determination of N provides evidence of whether or not the doublet is rigid. In addition to its importance in the basic understanding of colloidal forces, the question of rotational rigidity of colloidal aggregates is central to the evolution of the structure and ultimately the mechanical strength and reliability of consolidated media. The phenomenon of electrophoretic rotation was first demonstrated experimentally by Fair and Anderson.12 They were testing a theory for the electrophoretic rotation of rigid, heterogeneous doublets with very thin double layers.13-15 Even when the difference in zeta potential (ζ) was only two times kT/e ≈ 25 mV (k ) the Boltzmann constant, T ) the absolute temperature, e ) the charge on a proton), Fair and Anderson found strong rotation in rather modest fields (∼1 V/cm) upon switching the polarity of the applied field. However, they did not find good quantitative agreement with the theory, the experiments showing apparent rotation rates 50% higher than predicted from the theory. A significant problem with the experiments was the ambiguity in determining the time (12) Fair, M. C.; Anderson, J. L. Langmuir 1992, 8, 2850. (13) Fair, M. C.; Anderson, J. L. Int. J. Multiphase Flow 1990, 16, 663. Corrigenda in Int. J. Multiphase Flow. 1990, 16, 1131. (14) Keh, H. J.; Yang, F. R. J. Colloid Interface Sci. 1990, 139, 105. (15) Keh, H. J.; Yang, F. R. J. Colloid Interface Sci. 1991, 145, 362.

Velegol et al.

Figure 1. Schematic of a colloidal doublet in an applied electric field.

Figure 2. N versus gap (δ) for equal size spheres undergoing electrophoretic rotation as rigid-body doublets or freely-rotating doublets. The curves are from eq 5.

required for a doublet to align, because Brownian motion prevented complete alignment. In this paper we report measurements of the electrophoretic rotation rates of colloidal doublets that were formed by Brownian coagulation of two different populations of polystyrene latex spheres in water. The difference in zeta potential (∆ζ ) ζ2 - ζ1) of the two spheres was of order kT/e. We use microelectrophoresis and video microscopy to record the electrophoretic rotation of the doublets. The trajectories were recorded for individual doublets whose axis was in the plane of view. The electric field was turned on in a step change (with a transient much faster than the time scale of the rotation). After the doublet rotated into alignment with the field, the field was reversed in a step change. This was repeated so that many rotations of the same doublet were recorded. The videotapes were analyzed frame-by-frame to obtain data for θ versus time for many rotations of each doublet, where θ is the orientation angle of the doublet’s axis with respect to the direction of the electric field. Since the rotations included a deterministic contribution due to electrophoresis and a stochastic contribution due to Brownian motion, a moments analysis of the orientational probability equation was fit to the data to obtain the geometric coefficient N describing the electrophoretic rotation. N was thus determined for doublets of two size ratios, various values of ∆ζ between the particles, and different solution conditions to correlate N with the physicochemical properties of the solution and then to compare it to theory. These experiments address two basic questions: (1) Does the hydrodynamic theory give the correct value of N for doublets expected to be in a rigid-body configuration? and (2) Is the value of N measured for doublets under conditions where they are predicted to be in a secondary minimum consistent with free rotation of the spheres? Before explaining the experimental system, we first describe the dynamics of a colloidal doublet in an electric field. The analysis is for thin, unpolarized electrical double

Probing the Structure of Colloid Doublets

Langmuir, Vol. 12, No. 3, 1996 677

layers. We then develop a model for rotational trajectories that includes both electrophoretic rotation and Brownian motion. Theory of Electrophoretic Rotation Consider a doublet of two spheres with a gap δ between them, as shown in Figure 1. The unit vector e points from the center of sphere 1 to sphere 2, where sphere 1 is smaller than sphere 2. The double layer on each sphere is assumed to be thin and unpolarized, which means we require κai . cosh(zeζi/2kT) where κ-1 is the Debye screening length of the solution.12 The angular velocity of e is given by

Ω)-

(U2 - U1) × e L

(1)

where U1 and U2 are the velocities of each sphere center and L is the distance between the centers. Keh and Yang14,15 represent the spheres’ velocities as follows:

U1 )

ζ1 (ζ2 - ζ1) p n (I - ee)]‚E∞ E + [M12ee + M12 η ∞ η (2a)

U2 )

ζ2 (ζ2 - ζ1) p n (I - ee)]‚E∞ E [M21ee + M21 η ∞ η (2b)

where is the permittivity and η is the viscosity of the suspending fluid, ζ1 and ζ2 are the zeta potentials on the spheres, I is the identity tensor, and E∞ is the applied electric field. The Stokes-flow equations must be solved for the two spheres with the appropriate electroosmotic “slip velocity” on their surfaces16 to determine the Mcoefficients. These coefficients depend only on the relative size of the spheres and the gap between them. Substituting eq 2 into eq 1 gives

Ω)-

(ζ2 - ζ1) n n - M21 )E∞ × e (1 - M12 ηL

(3)

Defining the rotation parameter N in terms of the coefficients in eq 3, we have n n N ) 1 - M12 - M21

Ω)-

(ζ2 - ζ1) NE∞ × e ηL

(4a) (4b)

The interaction effects disappear at infinite gap, so N f 1 as δ f ∞. At finite gaps the values of the mobility coefficients Mnij depend on the kinematic condition of the doublet, that is, whether the spheres move as independent bodies (freely rotating) or as a single rigid body. Fair and Anderson12,13 derived the above expression for doublets that rotate as a rigid body and obtained values of N for doublets of different size ratios β ) a1/a2. An empirical expression that fits the values of N as a function of β when δ ) 0 for rigid-body rotation (“rb”) is

N ) Nrb )

{

0.5β + 1.44β2 0.67 - 1.19 exp(-3.72β)

for β < 0.30 (5a) for 0.30 e β < 1.0

Equation 5a is accurate to 1%. Under rigid-body conditions the value of N varies less than 4% when the gap is smaller than 10% of the mean sphere radius and less than 0.4% when the gap is smaller than 1% of the mean sphere radius. (16) Anderson, J. L. Annu. Rev. Fluid Mech. 1989, 21, 61.

Instead of being locked into a rigid-body condition, the particles of a doublet can be two torque-free hydrodynamic bodies, such that the hydrodynamic torque is zero on both spheres but their angular velocities can differ. We call this a “freely-rotating” doublet. Our calculations for the freely-rotating case are based on the values of Mnij determined by Keh and Yang15 for this kinematic condition, who assumed the double layers do not overlap (κa . 1). Because Mnij < 0 for this case, N exceeds 1 at all gaps for freely-rotating doublets. The following empirical expression is a good fit to the theoretical values of N for freely-rotating doublets (“fr”) when 0.2 e β e 1.0 and 0.001 e λ e 0.2, where λ ) 2δ/(a1 + a2 + δ):

N - 1 ) Nfr - 1 ) (0.174 + 0.341β 2

0.2184β2)λ-(0.2177-0.0254β+0.0125β ) + (0.057 - 0.8887β + 0.55β2) (5b) Figure 2 compares N for rigid-body versus freelyrotating doublets. N is sensitive to the gap for the freelyrotating case but varies little for rigid doublets. The curves in Figure 2 are for a 1:1 size ratio of the spheres. The differences in N between the rigid-body and freely-rotating cases are more pronounced for other size ratios. Figure 2 clearly indicates that by finding an experimental value for the coefficient N, one can determine the rigidity of a doublet. In addition, we now see that 0.64 < N < 1 should never occur if the doublet is purely freely-rotating or purely rigid. These “impossible” values provide the possibility of finding whether heterogeneities are causing the doublet to behave more rigidly. Analysis of Doublet Rotations with Brownian Motion. The scalar form of eq 4 gives the variation of the orientation angle versus time:

Ω)-

[

]

(ζ2 - ζ1)E∞ dθ N sin θ ) dt ηL

(6)

where θ is the angle between the applied electric field (E∞) and e. Values for the parameters in the square brackets of eq 6 can be independently determined, as long as it is assumed that the zeta potentials of the spheres do not change when the spheres form the doublet. L is approximately L ) a1 + a2 since δ is much smaller than the particles’ radii. Thus, if electrophoresis were the only force causing changes in orientation, then data for θ(t) could be used to determine the value of N from eq 6. The rotation of a doublet due to electrophoresis will always be affected to some extent by Brownian motion.12 The effect becomes stronger as θ f 0 or π. Brownian motion can be included in the analysis by using the orientational probability equation developed by Brenner and Condiff:17

∂f ) ∇s ‚ (Dr∇sf) - ∇s ‚ (Ω × ef) ∂t

(7)

f(θ,φ) is the probability density of the doublet being at any particular orientation e(θ,φ), where φ is the azimuthal angle in spherical coordinates. Dr is the rotational diffusion coefficient, and ∇s is the dimensionless gradient in (θ,φ) space. The first term on the right side of eq 7 represents the diffusion of orientational probability due to Brownian rotation, while the second term describes changes in orientation arising from electrophoretic rotation with Ω given by eq 4b. In the absence of Brownian motion, all rotations would follow a deterministic curve set by the electrophoretic rotation; however, Brownian (17) Brenner, H.; Condiff, D. W. J. Colloid Interface Sci. 1974, 47, 199.

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Velegol et al.

motion tends to change the θ(t) from rotation to rotation, giving a spread of the value of θ(t) for many rotations (see Figure 4 for sample data). After defining a dimensionless time τ ) tDr and expressing all the terms of eq 7 in scalar form, we have

1 ∂2f ∂f 1 ∂ ∂f ) sin θ + ∂τ sin θ ∂θ ∂θ sin2 θ ∂φ2

(

)

(

Pe 2f cos θ +

∂f sin θ (8) ∂θ

)

Figure 3. Schematic of microelectrophoresis cell. The 10 mm length is aligned with gravitational acceleration to within 0.1°.

where the Peclet number is defined as

Pe ) -

(ζ2 - ζ1)E∞N ηLDr

(9)

For an axisymmetric doublet f does not depend on φ. The boundary conditions on θ reflect the fact that there is no orientation flux at the boundaries:

θ ) 0, π: ∂f/∂θ ) 0

(10)

By its definition, f must be normalized over all angles:

∫02π∫0πf sin θ dθ dφ ) 1

(11)

The initial condition is obtained by defining t ) 0 for each rotation as the time when θ reaches a known value θ0:

t ) 0:

f)

δD(θ-θ0) 2π sin θ0

(12)

where δD(θ-θ0) is the Dirac delta function. The value of θ0 and the electric field are the same for all rotations of the same doublet. By solving eq 8 for f(τ,θ) one could directly calculate the average orientation θ(τ) and the standard deviation of the orientations from θ2(τ), where the overbar denotes an ensemble average over infinite rotations of the same doublet. However, it is easier to use the method of moments18-20 to find the average functions cos θ(τ) and

cos2 θ(τ). The first moment is defined by

cos θ(τ) )

∫02π∫0πcos θ′ f(θ′,φ′,τ) sin θ′ dθ′ dφ′

(13)

The time derivative of eq 13 is

∂ cos θ(τ) ) ∂τ

∫02π∫0πcos θ′

∂f(θ′,φ′,τ) sin θ′ dθ′ dφ′ (14) ∂τ

Using the above two expressions in eq 8 and doing several integrations by parts gives

∂ cos θ(τ) ) -Pe - 2 cos θ(τ) + Pe cos2 θ(τ) (15) ∂τ In a similar manner, ordinary differential equations (ODEs) can be found for all the moments cosn θ(τ). Thus, the partial differential equation eq 8 is reduced to an infinite set of coupled ODEs. The infinite set is closed by (18) Aris, R. Proc. R. Soc. London A. 1956, 235, 67. (19) Brenner, H. PhysicoChem. Hydrodyn. 1980, 1, 91. (20) Brenner, H.; Edwards, D. A. Macrotransport Processes; Butterworth-Heinemann: Boston, 1993; Chapter 2. (21) van de Ven, T. G. M. Colloidal Hydrodynamics; Academic Press: New York, 1989; Chapter 4. (22) Fair, M. C. Electrophoresis of Nonspherical and Nonuniformly Charged Colloidal Particles. PhD Thesis, Carnegie Mellon University, Department of Chemical Engineering, Pittsburgh, PA, 1990. (23) Jeffrey, D. J.; Onishi, Y. J. Fluid Mech. 1984, 139, 261. (24) Jeffrey, D. J.; Onishi, Y. J. Appl. Math. Phys. (ZAMP) 1984, 35, 634.

setting an odd central moment to zero and numerically solving a small number (e.g., 4-6) of coupled equations. The ODEs for the moments up to the fourth order, as well as the closure equation, are given in Appendix A. To calculate Dr, the Stokes-Einstein equation21 is used:

D ) R-1kT

(16)

where D is the diffusion coefficient matrix of which Dr is a part. R is the hydrodynamic resistance matrix, and kT is the thermal energy. We have calculated R from expressions given by Fair22 and Jeffrey and Onishi,23,24 assuming that the doublet rotates as a rigid body. The following equation is a good fit to the calculations for touching doublets (δ ) 0) with β ) a1/a2 e 1,

Dr )

kT K πηa23 r

Kr ) 8 - 1.72β + 23.64β2

(17a) (17b)

Equation 17b is accurate to better than 1% for all size ratios. A precise value of Dr is not essential to the determination of N from data for θ(t), but is merely convenient for nondimensionalizing time. This is because Pe . 1 in all our experiments, so that the electrophoretic rotation mainly determines the time dependence of the first moment of cos θ. In this case the second moment helps us to understand the deviations from the expected value of cos θ. We note that the electrophoretic rotation discussed in this paper is distinct from the weak rotation caused by dielectric polarization.25,26 Rotation by dielectric polarization is an E∞2 effect that causes little rotation for E∞ < 100 V/cm and diameters less than 10 µm. Electrophoretic rotation is proportional to E∞ and is significant even for fields of 1 V/cm. Even at the highest electric field in this work (13 V/cm), rotation due to electrophoretic rotation is at least 100 times more significant than that due to dielectric polarization. We also note that convection from Joule heating had an insignificant effect on the rotations because any induced shear rates would have been too small (less than 0.001 s-1). Experiments The microelectrophoresis apparatus consisted of three main parts: (1) the microscope and microelectrophoresis cell, (2) the current source, and (3) the video microscopy equipment. Figure 3 shows the microelectrophoresis cell. The microscope was an Olympus IMT-2 microscope positioned on its back with a metal support. The microelectrophoresis cell (Rank Brothers, England) was a rectangular glass capillary (1 × 10 mm cross section, about 5 cm long) with two ports leading to electrode compartments. (25) Baloch, M. K.; van de Ven, T. G. M. J. Colloid Interface Sci. 1989, 129, 91. (26) van de Ven, T. G. M.; Baloch, M. K. J. Colloid Interface Sci. 1990, 136, 494.


Langmuir, Vol. 12, No. 3, 1996 679

Table 1. Summary of Experimental Conditions and Experimental Values of the Rotation Coefficient N Defined by Eq 4a latexb particles

d1 (µm)

d2 (µm)

ζ1 (mV)

ζ2 (mV)

KCl (M)

pH

E∞ no. of T (°C) (V/cm) rotationsc

1: A/S* 2: A/S*

2.51 ( 0.12 2.75 ( 0.09 2.51 ( 0.12 2.75 ( 0.09

+74 ( 10.9 -113 ( 4.9 -18 ( 4.5 -62 ( 4.4

0.010 0.100

5.18 4.66

20.0 21.0

1.98 1.04

3: c/A*

1.17 ( 0.03 2.51 ( 0.12

-46 ( 4.7

+8 ( 8.5

0.020

4.41

22.8

1.67

4: S/C

4.20 ( 0.24 4.34 ( 0.24 -114 ( 2.4

-56 ( 2.5

0.010

3.13

20.5

1.46

5: S/C

4.20 ( 0.24 4.34 ( 0.24 -108 ( 2.9

-40 ( 2.3

0.023

3.08

21.8

1.58

6: S/C

4.20 ( 0.24 4.34 ( 0.24

-68 ( 2.7

-21 ( 2.3

0.100

3.13

20.5

2.02

7: S/Cf

4.20 ( 0.24 4.34 ( 0.24

-83 ( 5.5

-39 ( 1.2

0.023

2.92

22.8

1.30 6.50 13.00

a: 8 b: 8 c: 8 d: 8 e: 8 f: 1 g: 2 h: 4 i: 4 j: 4 k: 4 l: 4 m: 4 n: 4 o: 4 p: 4 q: 4 r: 2 q: 2 q: 4 r: 2 s: 2

Ped

Ne

305 ( 7.9 77.5 ( 3.9 82.3 ( 1.4 76.4 ( 9.9 82.0 ( 2.1 74.7 ( 4.5 75.6 ( 3.8 510 ( 6.0 521 ( 6.3 427 ( 9.1 520 ( 7.2 520 ( 7.5 550 ( 7.2 560 ( 14.2 595 ( 7.9 587 ( 4.4 148 ( 3.0 134 ( 2.6 713 ( 8.9 1388 ( 7.9 1372 ( 5.4 1487 ( 7.5

0.569 ( 0.043 0.594 ( 0.092 0.631 ( 0.093 0.586 ( 0.115 0.629 ( 0.093 0.458 ( 0.076 0.464 ( 0.075 0.644 ( 0.047 0.657 ( 0.048 0.540 ( 0.040 0.652 ( 0.045 0.653 ( 0.045 0.690 ( 0.048 0.625 ( 0.056 0.664 ( 0.057 0.655 ( 0.056 0.707 ( 0.096 0.638 ( 0.086 0.677 ( 0.091 0.660 ( 0.088 0.652 ( 0.087 0.707 ( 0.095

a The particles are polystyrene latices; the electrolyte is KCl. During any particular experiment the temperature remained constant to within 1 °C. The particle radii (ai) are taken as the nominal sizes provided by the manufacturer; these values agree with the sizes we determined from sedimentation-rate measurements. All particle concentrations were in the range 106-108/mL. All particles except those designated by an asterisk were cleaned by dialysis as explained in the text. b The particle types are A (amidine surface), c (carboxyl surface), C (carboxylate surface), and S (sulfate surface). Experiment “1” is the first of the seven listings. c The letters a-s indicate different doublets, and the numbers following the letters indicate the number of rotations recorded for that doublet. Thus, doublets b-e are four different doublets in experiment 2. d The Peclet number was determined by fitting the moments analysis (see Appendix A) to the data for θ(t).e The rotation coefficient N was calculated from the Pe using eqs 9 and 17. For high Peclet numbers, the final value for N depends little on the actual value of Dr. f A total of 12 doublets was examined at a field of 13 V/cm. Three were analyzed in detail and appear in the table. The other nine were analyzed using fewer data and interpreted using the deterministic model of eq 6. All 12 doublets gave values of N to within experimental error of N ) 0.63.

The electric field was produced using a dc current source (Keithley Model 220). The images from the microscope were captured by a Sony 77 CCD camera and recorded with a Panasonic SVHS video recorder. The resulting video tape was analyzed frameby-frame using a thermal printer (Mitsubishi Video Copy Processor P71U). The final magnification of the doublets ranged from 705 to 2820, while the field of view ranged from 250 to 62 µm. All experiments were done at temperatures between 20 and 25 °C; the temperature was constant to within (1 °C during any one experiment. All glassware, including the electrophoresis cell, was soaked overnight in concentrated nitric acid. The glassware was rinsed with deionized water from a MilliQ Water Purification System (Millipore) which typically had a conductivity of 1 µS/cm. This water was also used in the experiments to make solutions, to perform dialysis, and for any other operations. The doublets were formed by the Brownian coagulation of polystyrene spheres. These spheres had amidine, carboxyl, or sulfate charge groups chemically bound to the surface. We purchased from Interfacial Dynamics Corporation (Portland, OR) the amidine (2.51 µm diameter, Catalog No. 2-122-45.233), carboxyl (1.17 µm, Catalog No. 10-249-93), and sulfate (2.75 µm, Catalog No. 10-295-22, and 4.20 µm, Catalog No. 2-356-97) spheres. We obtained carboxylate spheres (4.34 µm, Lot No. 435893 of Catalog No. 17140-5) from Polysciences (Warrington, PA). Some of the particles were used as received from the manufacturer, and some were cleaned with dialysis (Table 1 specifies experiments for which the particles were not cleaned). The cleaning involved diluting the particles in about 15 mL of deionized water (typically 106-107 particles/mL) and dialyzing them in 6000-8000 MW tubing (Spectrum Medical Industries, Los Angeles, CA) against 500 mL of deionized water. The particles were dialyzed for roughly 24 h with the 500 mL of water being changed 2-3 times. Before mixing the particles for coagulation, all particles were sonicated for roughly 15 s to break any homoaggregates. To form the doublets, solutions were mixed with the proper salt concentration and pH, and two different types of particles (e.g., amidine and sulfate or carboxylate and sulfate) were added. Typically the particles were allowed to

coagulate for 24 h. For the data analysis we used the sizes and standard deviations of size provided by the manufacturer; however, we did verify these values with sedimentation-rate experiments. The sizes of the spheres were too small (e.g., 1-4 µm) to be accurately determined by direct measurement in the microscope. After the coagulation process, the suspension was added to the electrophoresis cell. The zeta potentials of the particles were determined by examining the electrophoretic mobility across nine cross sections through the cell.27 In this way the effect of electroosmosis from the walls was easily and accurately included. The zeta potential was determined from the average mobility of approximately 20 particles. The mobility measurements were converted to zeta potentials using the Smoluchowski equation since for most experiments κa ) 400-2000 and the zeta potentials were generally less than 4kT/e in magnitude. Interfacial Dynamics reported their surface charge densities as 18.9 µC/cm2 for the 1.17 µm diameter carboxyl sphere, 24.2 µC/cm2 for the 2.51 µm amidine, 9.8 µC/cm2 for the 2.75 µm sulfate, and 5.1 µC/cm2 for the 4.20 µm sulfate. The pK for sulfate groups is less than 2. For carboxyl groups it is about 5, and for amidine groups it is about 12. The stability of the zeta potentials was checked by examining the mobilities before and after the set of rotations; these values were the same to within measurement uncertainty. One “rotation” yielded a set of data for θ versus time for one doublet at one electric field. These doublets were chosen such that they rotated in the plane of view (i.e., the doublet appeared as two connected circles in two dimensions). Except when θ was nearly 0 or π, we could visually determine when the doublet rotated out of the plane; for example, the doublet changed shape in two dimensions or the brightness of the Poisson spots changed. The error in measuring θ was (1°, determined by measuring the angle of a doublet in a single frame several times. For each doublet there were 2-8 rotations performed at the same applied field, which was reversed in direction to start each new rotation. If there were s rotations of the doublet, then the experimental (27) Hunter, R. J. Zeta Potential in Colloid Science: Principles and Applications; Academic Press: New York, 1981; Chapter 4.

680


Velegol et al. Table 2. Comparison between Experimental and Theoretical Values of Na expt 1 2 3 4 5 6 7a 7b 7c

Figure 4. Typical rotation data for one doublet (eight rotations), taken from experiment 1 in Table 1. For this doublet N ) 0.57 and Pe ) 305. The latex system is amidine/sulfate. The thick line is the average angle cos-1[cos θ(τ)] from the best fit to the moment analysis of eq 8. The two thin lines represent approximately the region of one standard deviation, and in particular cos-1[cos θ(τ) ( σcosθ(τ)], where σcosθ(τ) ) [cos2 θ(τ) 2

cos θ(τ) ]1/2. Roughly two-thirds of the data should fall within the two thin lines. value for the moment of cosn θ(τ) at each time t is s

∑ cos

cosn θ(τ) ) (

n

θi(τ))/s

(18)

i)1

By comparing the experimental data for the first moment versus time with the moment analysis described in Appendix A, a bestfit value of Pe was obtained using a least-squares criterion. Values of θ starting at 20-40° and going to 140-160° were used in the fitting (or starting at 140-160° and going to 20-40°, depending on the direction of the field and the sign of ∆ζ). Since out of plane motion below our detection limit causes large errors in the measured θ near 0 and π, we neglected data at these extreme angles.10 The experimental value of N was then determined from eq 17 for Dr and eq 9 for Pe using the zeta potentials obtained by tracking the motion of single particles of each type.

Results and Discussion Sample data for eight rotations of one amidine/sulfate doublet are plotted in Figure 4. The dark line represents the best-fit average line of the data (i.e., cos-1[cos θ(τ)]. The light lines in Figure 4 represent the region of one standard deviation (i.e., cos-1[cos θ(τ) ( σcosθ(τ)], where 2

σcosθ(τ) ) [cos2 θ(τ) - cos θ(τ) ]1/2) in the data predicted from the moment analysis of eq 8, with Dr given by eq 17. Approximately 68% of the data should fall within these curves. This percentage is approximately correct for all doublets, so that the spread of the data is generally accounted for by rotational diffusion. A summary of all the results is presented in Table 1. At high Peclet numbers (Pe > 100) the primary sources of uncertainty in determining N are the sizes and zeta potentials of the particles. The uncertainty in N from the statistical analysis of the least-squares fit for Pe is negligible compared to these effects. The mean sizes are given by the manufacturer (verified by our sedimentationrate experiments) along with a standard deviation. The standard deviation of zeta potentials is determined from the mobility experiments. We then take the percentage variance of N to be the sum of the percentage variance of the sphere size, the percentage variance of the mobilities, and the percentage variance of Pe. The calculated percentage standard deviation is listed as a ( quantity in Table 1.

latex (1)b

A/S A/S (4) c/A (2) S/C (3) S/C (3) S/C (3) S/C (2) S/C (1) S/C (3)

a1/a2

KCl (M)

E (V/cm)

Nexpc

0.91 0.91 0.47 0.97 0.97 0.97 0.97 0.97 0.97

0.010 0.100 0.020 0.010 0.023 0.100 0.023 0.023 0.023

2.0 1.0 1.7 1.5 1.6 2.0 1.3 6.5 13.0

0.57 ( 0.04 0.61 ( 0.10 0.46 ( 0.08 0.61 ( 0.05 0.67 ( 0.05 0.65 ( 0.06 0.67 ( 0.09 0.68 ( 0.09 0.67 ( 0.09

Nrbd (κδ)DLVO 0.63 0.63 0.46 0.64 0.64 0.64 0.64 0.64 0.64

0.0 0.0 0.0 8.8 7.8 5.3 7.4 7.4 7.4

Nfr xxx xxx xxx 1.45 1.53 1.73 1.54 1.54 1.54

aN exp is the experimental determination, which is the arithmetic mean of the values in Table 1 for all doublets of the same chemistry at the same KCl concentration and electric field. Nrb is the prediction for a rigid-body rotation from eq 5a. The predicted gap based on DLVO theory is discussed in the text. Nfr is the predicted value of N for a doublet with this value of δ, calculated from eq 5b. b The number in parentheses is the number of doublets observed, and Nexp is the arithmetic average of the determinations of N from the experiments for these doublets. c These error bars on the experimental value of N are the averages of the error bars on the doublets in that experiment. d The theoretical value of N for both the rigid and freely rotating cases also has variance, typically 4%, due to the uncertainty in the sizes of the spheres.

The experimental values of N are compared with the theory in Table 2. This table shows the expected values of N for a rigid-body doublet (Nrb) and a freely rotating doublet (Nfr). The most striking feature is that the experimental values of N for all the doublets agree with the theory for rigid-body rotation. This agreement is expected for those doublets with the designation “xxx” in the column for Nfr because they are predicted to sit in a primary minimum according to DLVO theory as discussed below. Note that the values of the size ratio β are quite different in experiments 1 and 2 versus 3. The conclusions we draw from Table 2 are (1) the electrohydrodynamic model for the rigid-body rotation of doublets is quantitatively accurate and (2) all the doublets in our experiments behaved as rigid bodies while rotating toward alignment with the electric field. The surprising result is that the S/C (sulfate/carboxylate) doublets, which are predicted to be in a “secondary minimum” with a separation of 5-7 times the Debye screening length, also rotated as rigid bodies. The gap for the secondary minimum is predicted to be 5-21 nm depending on the KCl concentration. Thus, the surfaces of the spheres should have been able to rotate relative to each other so as to give a value of N close to Nfr if there were no tangential force hindering the relative motion of the two surfaces. Clearly, N for every doublet agrees well with the rigid-body value and cannot be in agreement with the freely-rotating value. Nfr/Nrb > 2, making the difference between rigid and freely-rotating more than 10 times the error bar. Furthermore, there was no detectable effect of ionic strength or electric field on the value of N (see experiment 7). To place these results in perspective, we first review the model for the configuration of colloidal doublets at equilibrium according to classical DLVO theory and then consider the hydrodynamic torque that must be overcome to keep a doublet in a rigid-body state when δ > 0. Finally, we suggest possible origins of the restraining torque that would allow a doublet to be in a secondary minimum yet rotate as a rigid body. DLVO Theory for Equilibrium Particle Separation. Hogg et al.28 developed an equation for the potential energy of mean force, Ψ(δ), between two spheres having different radii and zeta potentials. The model is based on (28) Hogg, R.; Healy, T. W.; Fuerstenau, D. W. Faraday Soc. Trans. 1966, 62, 1638.


Langmuir, Vol. 12, No. 3, 1996 681

Figure 5. Potential of mean force (Ψ) versus separation (δ) for the latices of experiment 1 (A/S, 0.01 M KCl) and experiment 6 (S/C, 0.10 M KCl). See Appendix B for the calculation of Ψ. Table 3. Estimated Times for Two Spheres To Move by Brownian Motion into a Primary Minimum from a Secondary Minimum, As Calculated from Eq 19a expt

Ψmax/kT

Ψmin/kT

hmaxκ

hminκ

τD (years)

4 5 6 7

5530 2680 330 2300

-13.8 -24.1 -72.2 -25.5

0.24 0.44 0.95 0.39

8.8 7.8 5.3 7.4

10320 10150 1020 10130

a See Table 1 for experimental conditions and Appendix B for the calculation of the energy minima and maxima.

the classical DLVO theory and makes use of the lowpotential approximation in the double layer and the Derjaguin method for summing the interactions across the gap between the two particles. Ohshima et al.29 extended this model to higher zeta potentials. The input parameters to the model are the Debye length (κ-1), the Hamaker constant (A) of the particle-solvent-particle system, and the surface potentials (assumed to be the zeta potential here) of the particles. This theory is summarized in Appendix B. The profile Ψ versus δ for the latex particles in experiments 1 and 6 (see Tables 1 and 2) is plotted in Figure 5. The profile for experiment 1 has only a primary minimum, which means these particles should essentially make contact with each other when they form a doublet. The particles of experiment 6, however, have a deep secondary minimum that should hold the particles at a gap of δ ) 5.3κ-1 ) 5.1 nm. Table 3 gives the predicted value of the energy barrier (Ψmax) and energy minimum (Ψmin) for the doublets in experiments 4-7. Prieve and Ruckenstein30 developed a model for the kinetics of coagulation for two equal spheres situated in a secondary minimum. The characteristic time τD for a doublet to move from the secondary to the primary minimum is obtained from eqs 2, 3a, and 4d of their paper:

τD )

1 ) e+(Ψmax-Ψmin)/kT K′′

πkT 2hmaxD∞ (γmaxγmin)1/2 a

(

)

(19)

where D∞ ) kT/6πηa and γ(h) ) |d2Ψ(h)/dh2|. The estimates of τd are listed in Table 3. Thus, according to the conventional DLVO theory for colloidal forces, there should have been essentially no doublets residing in a primary minimum in experiments 4-7 and all should have been in a secondary minimum. (29) Ohshima, H.; Healy, T. W.; White, L. R. J. Colloid Interface Sci. 1982, 89, 484. (30) Prieve, D. C.; Ruckenstein, E. J. Colloid Interface Sci. 1980, 73, 539.

One indication of whether or not a doublet is irreversibly bound in a primary minimum is the ability to resuspend the doublets into singlets when the ionic strength of the solution is lowered. We performed such resuspension experiments with S/C and A/S latices. Doublets were formed in 0.023 M KCl solutions. These suspensions were then dialyzed against deionized water for roughly 24 h. The dialysis water was replaced several times until its conductivity became almost constant, indicating that the suspension had a salt concentration near to that of the dialysis water. Examination of the dialyzed suspensions involved viewing a single frame of particles at each of nine cross sections throughout the cell. This gave a total aggregate count before and after dialysis. This same measurement was repeated three times for each sample so that we could estimate the reliability of the measurement. The results of the resuspension experiments are shown in Figure 6a-c. For the A/S doublets there was no increase in the number of single particles, thus indicating these doublets were in a primary energy minimum at 0.023 M electrolyte. With the S/C mixtures there was a significant increase in the number of single particles and a decrease in the number of doublets, a result consistent with the notion that these doublets were in a secondary minimum. However, the resuspension results for S/C suspensions at 1.5 M KCl, shown in Figure 6c, are not consistent with what we expect from DLVO theory. For this high electrolyte concentration, the model of Hogg et al.28 predicts no secondary minimum so one would assume that the doublets were in a primary minimum. Yet dialysis against deionized water resulted in significant resuspension of the aggregates into single particles. The number of total spheres before and after the S/C resuspensions remained the same to within 1%. While resuspension experiments might not correlate well with predictions from DLVO theory, the experiments do show that the S/C particles were not irreversibly bound together. Hydrodynamic Torque on Doublets Rotating as Rigid Bodies. If the doublets in experiments 4-7 were actually in a secondary minimum, with the values of δ listed in Table 3, then other forces must be present to oppose the hydrodynamic torque tending to rotate each sphere. In Appendix C we derive expressions for the torque on the spheres of a doublet as it rotates rigidly by electrophoresis. The following empirical expression gives the hydrodynamic torque (Th) on each sphere necessary to keep a doublet rigid when it is oriented perpendicular to the electric field:

Th ) 8πa2(ζ2 - ζ1)E∞g(λ) ln λ

(20a)

For β ) 1 an empirical expression for g is

g(β)1) ) 0.38083 + 0.063614 ln λ + 0.01049(ln λ)2 (20b) where λ ) δ/a for 0.0002 e λ e 0.2. Calculations of Th for the particles of the S/C doublets in experiments 3-7, with δ ) δmin, are listed in Table 4. Since the spheres did not rotate relative to each other, if the gaps existed as shown in this table, then lateral stresses acting on the surfaces of the spheres must have been present to oppose Th. The purpose of varying the electric field in experiment 7 was to see if a freely rotating situation could be attained at higher fields where Th is greater. At E∞ ) 13 V/cm the hydrodynamic torque on each sphere was -2300kT, assuming the spheres were positioned at the predicted secondary minimum. Yet even at this high field the experimental value of N agrees with the theory for rigidbody doublets.

682


Velegol et al. Table 4. Hydrodynamic Torque (Th) on Each Sphere of the System in Experiments 4-7a expt κ-1 (nm) expected δ/a E∞ (V/cm) ∆ζ (mV) torque (kT) 4 5 6 7

3.0 2.0 1.0 2.0

0.004 12 0.003 65 0.002 48 0.003 47

1.46 1.58 2.02 13.00b

58 68 47 44

-325 -429 -430 -2320

a The sphere sizes are 4.20 µm (sulfate) and 4.34 µm (carboxylate); thus, the size ratio is almost 1:1 and the average sphere diameter is 4.27 µm. The value for δ/a is the expected value from Table 3. The calculation is done for when the doublets rotated as rigid bodies and the orientation was 90° with respect to the applied electric field (as in the derivation in Appendix C). Th was calculated from eq 20. b Other fields scale proportionately. For example, when the field for experiment 7 is 1.30, the torque is -232kT. See eq 20.

resuspension experiments indicate that the S/C doublets were not irreversibly connected. One final explanation in this category involves any steric layer on the carboxylate spheres. If the steric layer extends out to where the secondary minimum might be, the steric layer could rub against the sulfate sphere and cause rigidity. The second type of restraining force concerns heterogeneity in the interaction potentials. If the carboxylate and sulfate spheres have a heterogeneous charge distribution on them, the spheres might fall into a local potential well. Even small wells of 1kT could cause large rigidity torques. If the length scale of heterogeneity were 10 nm while the sphere radius were 2 µm, then the torque caused by a 1kT well would be approximately 1kT/(10 nm/2 µm) ) 200kT. This heterogeneity effect has been examined for large length scales by Grant and Saville.32 Surface roughness could add to this effect, though a quantitative estimate is not possible without a detailed model. Kostoglou and Karabelas33 have performed calculations that show a surface roughness of only half a Debye length can cause changes in the potential comparable to the potential itself. It is possible that a local potential well could even allow the spheres to fall into a primary minimum. The ideas expressed above are speculative in the absence of models for lateral colloidal forces. Such forces must exist in our system if the S/C doublets were in a secondary minimum. Until these forces are recognized, however, the most obvious explanation of our results, namely that N ) Nrb for all the S/C doublets, is that these doublets were not in a secondary minimum. If this conclusion is correct, the implication is that the standard DLVO-based model for calculating the potential energy between two charged colloidal spheres is in error, at least for our colloidal system. Figure 6. Frequency of aggregates before and after dialysis against deionized water. See Table 1 for properties of the latices. (a) A/S, 0.023 M KCl; this system produced large aggregates for which it is difficult to count the spheres in them; hence, the aggregates are subjectively counted as small, medium, or large. The most reliable numbers are for the singlets. (b) S/C, 0.023 M KCl; mass balance of total spheres goes from 327 ( 8.2 before to 333 ( 22.5 after. (c) S/C 1.50 M KCl; mass balance of total spheres goes from 283 ( 64.6 before to 270 ( 26.2 after.

Possible Origins of Restraining Torque. Possible mechanisms that could provide the restraining torque are of two main types: (1) mechanical restraint and (2) heterogeneity of charge. Mechanical restraints include tethering of polymer strands between the spheres, surface roughness, and irreversible binding of the particles. In light of the work by Seebergh and Berg,31 the tethering strands are unlikely at our high salt concentrations. The (31) Seebergh, J. E.; Berg, J. C. Colloids Surf. A 1995, 100, 139.

Conclusions Electrophoretic rotation is a sensitive probe of the rigidity of colloidal doublets. Almost complete alignment of micrometer-size doublets with ∆ζ ≈ kT/e occurs within seconds at fields of order 1 V/cm. Theory predicts that the rotation coefficient N should be less than unity for rigid-body doublets and greater than unity for freelyrotating doublets, and the value of N for freely-rotating doublets is a function of the gap between the particles. We have found experimentally that N is 0.64 for a size ratio of ≈1 and 0.46 for a size ratio of ≈0.5, in agreement with the theory for rigid doublets. The effects of Brownian rotation were included in our analysis, and they explain well the deviations from the expected mean rotation curves. (32) Grant, M. L.; Saville, D. A. J. Colloid Interface Sci. 1995, 171, 35. (33) Kostoglou, M.; Karabelas, A. J. J. Colloid Interface Sci. 1995, 171, 187.


Langmuir, Vol. 12, No. 3, 1996 683

All our experiments with latex particles indicate that the doublets were rigid under a variety of electrolyte concentrations, pH, and particle types at fields between 1 and 13 V/cm, even though many of the doublets should have been in a secondary minimum with a gap of 5-21 nm. At the highest fields, the torque required on each sphere to keep the doublet rigid was about 2300kT. One explanation for the rigid-body rotation of all the doublets is that the spheres were actually touching and there was no secondary minimum even though it is predicted for many of the doublets from DLVO-based models. To reconcile our observation of rigid-body rotation with a secondary minimum for these doublets, one must posit lateral colloidal forces between the surfaces of the two spheres. Possible origins of lateral forces are mechanical restraints (e.g., tethers and surface roughness) and surface heterogeneities (e.g., patches of different potentials and surface roughness); however, models based on these ideas are not currently available. Acknowledgment. This work was supported under NASA Microgravity Grant NAG8-964 and NSF Grant CTS-9420780. We are also grateful to Hercules, Inc., for funding some of this work. D.V. acknowledges support from a National Science Foundation Fellowship. J.L.A. also acknowledges the financial support of the University of Melbourne and very helpful discussions with the staff.

y4 ) z4 - 4z1z3 + 6z12z2 - 3z14 y5 ) z5 - 5z1z4 + 10z12z3 - 10z13z2 + 4z15 n

yn )

n!

(-1)izn-iz1i ∑ i)0i!(n - i)!

n

n!

(-1)izn-iz1i, ∑ i)1i!(n - i)!

zn ) -

(A5)

(A6)

From this all the moments of cosine can be calculated. The results are

cos θ(τf∞) )

Equation A1 is derived just as eq 15. Thus, the first four moments are

cos2 θ(τf∞) ) 1 +

(A2a)

n odd

Pe exp(-Pe cos θ) 4π sinh Pe

∂zn ) n(n - 1)zn-2 - nPe‚zn-1 ∂τ n(n + 1)zn + nPe‚zn+1 (A1)

∂z1 ) -Pe - 2z1 + Pe‚z2 ∂τ

(A4f)

The numerical solution to eqs A2 has been checked in two ways. First, the solution for t f ∞ is checked. By solving the convective-diffusive equation (eq 8) analytically for t f ∞, one obtains for the distribution function (with the definition of Pe from eq 9)

f)

Let us define zn t cos (τ). The moment analysis for the nth moment gives

(A4e)

For a Gaussian distribution of cos θ(τ), y3 ) 0. For a distribution that is close to Gaussian, the odd central moments will be close to zero. Note that the even central moments will always be positive. By setting an odd central moment yn to zero, we can estimate zn. For example, we can estimate z5 ) 5z1z4 -10z12z3 + 10z13z2 - 4z15 from eq A4e. This closes eqs A2, which we can then solve with the Runge-Kutta method. By estimating the nth central moment to be zero, we obtain in general

Appendix A: Moment Analysis of Eq 8 n

(A4d)

1 cosh Pe Pe sinh Pe

(A7a)

2 2 cosh2 Pe Pe2 Pe sinh2 Pe

(A7b)

For Pe > 5 eq A7a is approximately cos θ(τf∞) ) (1/Pe) - 1, eq A7b is cos2 θ(τf∞) ) 1 - (2/Pe) + (2/Pe2), and so 2

∂z2 ) 2 - 2Pe‚z1 - 6z2 + 2Pe‚z3 ∂τ

(A2b)

∂z3 ) 6z1 - 3Pe‚z2 - 12z3 + 3Pe‚z4 ∂τ

(A2c)

∂z4 ) 12z2 - 4Pe‚z3 - 20z4 + 4Pe‚z5 ∂τ

(A2d)

σcosθ(t) ) [cos2 θ(τ) - cos θ(τ) ]1/2 ) 1/Pe. The numerical results reproduce the results of eqs A7 to about one part in a million. The second check involves closing the set of ODEs at successively higher odd central moments. We tried closing the set at n ) 3, n ) 5, n ) 7, up to n ) 21. The calculated moments change little from n ) 3 to n ) 5; after n ) 5 the results have a negligible change.

Since τ ) 0 is the experimental time when each doublet is at some angle θ0, we are forcing the initial condition to

Appendix B: Summary of DLVO Theory

be that at τ ) 0, zn ) cosn θ(τ) ) cosn θ0. To solve eqs A2, we must close the equations by estimating the fifth moment z5. Just as eq 13 defines the moments zn, we can define central moments as

This summary of DLVO theory for two spheres with low zeta potentials comes primarily from the work by Hogg et al.28 Ohshima et al.29 extended the results to higher zeta potentials. The electrostatic potential of mean force V between two flat plates with constant surface potentials is

yn )

∫02π∫0π(cos θ′ - cos θ(τ))nf(θ′,φ′,τ) sin θ′ dθ′ dφ′

(A3)

For example, the second central moment is the familiar “mean of the squares minus the square of the mean”. Thus,

y1 ) 0

(A4a)

y2 ) z2 - z12

(A4b)

y3 ) z3 - 3z1z2 + 2z13

(A4c)

V)

[

)]

κ 2ψ1ψ2 cosh κδ + (ψ12 + ψ22) 1 2 sinh κh sinh κδ

(

(B1)

where h is the separation between the plates and ψ1 and ψ2 are the surface potentials of the plates. If two spheres have κa . 1, then the sphere geometry may be approximated as a series of concentric rings originating from the point of nearest approach. If we assume these rings are like parallel plates and multiply eq B1 by the area of the ring (i.e., the Derjaguin

684


Velegol et al.

approximation), we find the total potential

Ψ)

aaA

∫0a2πrV(r) dr - 6(a 1+2 a )δ 1

(B2)

2

in Keh and Yang.15 They can either be NFij for freely rotating spheres or NRij for spheres in a rigid doublet. Now we set e ) iz, Ui ) Uiix, and Ω ) Ω‚(I - ee) ) Ωiy (i.e., no rotation about e). We also define

In eq B2 the second term is for the van der Waals forces; A is the Hamaker constant and δ is the separation between the spheres. Evaluating the integral of eq B2 gives the final result for the DLVO potential between two spheres:

(

Ψ ) π

)[

(

) ]

a1a2A 6(a1 + a2)δ

Two spheres can have a finite gap between them but behave like a rigid doublet if some restraining torque is applied to the spheres. This section details the calculation for how much restraining torque is required to make a doublet appear rigid. This torque depends on the gap between the spheres. We use the linearity of the Stokes equations to break the motion of the spheres into a “free” (F) part, a “rigid” (R) part, and an “extra” (E) part such that13 R

F

(U2 - U1) ) (U2 - U1) - (U2 - U1) R

(C3b)

Ω1E ) Ω1R - Ω1F )

(ζ2 - ζ1)E∞ R F )iy (C4a) (N - N12 ηL

Ω2E ) Ω2R - Ω2F )

(ζ2 - ζ1)E∞ R F )iy (C4b) (N + N21 ηL

F

(C1b)

Ω2E ) Ω2R - Ω2F

(C1c)

The goal is to determine the “extra” motions and then to find the torque necessary to produce them. The “extra” motions are thus those required to make a freely-rotating doublet appear rigid. Equations 2 apply for free spheres. It is important to realize that for freely-rotating spheres, each sphere will in general have an angular velocity different from the doublet axis e. These rotation rates are given by

(ζ2 - ζ1) N21 e × E∞ Ω2 ) η L

(C4c)

Now that we have these “extra” quantities, we can calculate the torque necessary to produce them. To do this we follow Kim and Karrila36 in using the grand resistance matrix:

[] [

A11 A21 B11 B21

F1 F2 T1 ) -η T2

Thus,

B ˜ 11 B ˜ 21 C11 C21

A12 A22 B12 B22

B ˜ 12 B ˜ 22 C12 C22

][ ] U1 U2 Ω1 Ω2

(C5)

T1 ) -η[B11‚U1 + B12‚U2 + C11‚Ω1 + C12‚Ω2] (C6a) T2 ) -η[B21‚U1 + B22‚U2 + C21‚Ω1 + C22‚Ω2] (C6b) Using the definitions in Kim and Karrila,

BRβ ) YBRβE‚e

(C7a)

CRβ ) XCRβee + YCRβ(I - ee)

(C7b)

(C1a)

Ω1 ) Ω1 - Ω1

(ζ2 - ζ1) N12 Ω1 ) e × E∞ η L

(ζ2 - ζ1)E∞ R [N - NF]ix η

(U2 - U1)E )

Appendix C: Electrophoretic Torque

E

n n F - M21 ) NF ) (1 - M12

(B3)

The Hamaker constant A depends on the materials of the spheres and the fluid between. If one considers “retardation effects”, then A also depends on the gap (δ) between the spheres. For a polystyrene-water-polystyrene system at zero separation (δ ) 0), A/kT ) 3.33.34,35 This drops to A/kT ) 2.65 at 10 nm. In 0.1 M univalent electrolyte, A/kT ) 2.94 at zero separation and drops to A/kT ) 1.83 at 10 nm. In our calculations we used a “nonretarded” A/kT ) 2.50.

E

(C3a)

The “extra” quantities are

a1a2 1 + e-κδ 2ψ1ψ2ln + a1 + a2 1 - e-κδ (ψ12 + ψ22) ln(1 - e-2κδ) -

n n R R R NR ) (1 - M12 - M21 ) ) N12 ) -N21

Here E is the permutation operator. With β ) a1/a2 and λ ) 2δ/(a1 + a2), to a first approximation

[

B Y11 ) 4πa22

B Y12 ) 4πa22

[

B Y21 ) -4πa22

B ) -4πa22 Y22

(C2b)

(34) Prieve, D. C.; Russel, W. B. J. Colloid Interface Sci. 1988, 125, 1. (35) Parsegian, V. A. Long Range van der Waals Forces. In Physical Chemistry: Enriching Topics from Colloid and Surface Science; van Olphen, H., Mysels, K. J., Eds.; Theorex: La Jolla, CA, 1975. (36) Kim, S.; Karrila, S. J. Microhydrodynamics: Principles and Selected Applications; Butterworth-Heinemann: Boston, 1991; Chapters 7 and 11.

[

ln

1 1 Y (β) + (1 + β)2B12 λ 4

(C8b)

β2(4β + 1)

[

5(1 + β)2

ln

-β2(4β + 1) 2

5(1 + β)

C Y11 ) 8πa23

[ [

(C8a)

5(1 + β)

β(4 + β)

[

] ]

1 Y (β) + B11 λ

2

5(1 + β)2

(C2a)

The coefficients Nij for the individual spheres are given

-β(4 + β)

ln

]

1 1 + (1 + β)2BY21(β) λ 4 (C8c)

ln

]

1 Y (β) + β2B22 λ

(C8d)

]

(C8e)

2β 1 Y (β) ln + C11 λ 5(1 + β)

] ]

C ) 8πa23 Y12

β2 1 1 Y (β) ln + (1 + β)3C12 λ 8 10(1 + β)

C ) 8πa23 Y21

β2 1 1 Y (β) (C8g) ln + (1 + β)3C21 λ 8 10(1 + β)

(C8f)


[

C Y22 ) 8πa23

2β3 1 Y (β) ln + β3C22 λ 5(1 + β)

Langmuir, Vol. 12, No. 3, 1996 685

]

(C8h)

B B B B For a 1:1 size ratio, Y11 ) - Y12 and Y22 ) - Y21 . Now we define primed variables such that

YijB

4πa2 YijB′ 2

)

torque on spheres 1 and 2:

[

B ′(NR - NF) (ζ2 - ζ1)E∞ Y11 ) + ηa2 2 -8πηa23

T1

F

(C9a) )

YijC

8πa2 YijC′ 3

)

(C9b) T2

Substituting the expressions from before (e.g., Ui ) Uiix), we obtain

) 3 -8πηa2

T2 ) -η[-(U2 -

+

C Y21 Ω1

+

C Y22 Ω2]iy

) (C10b)

Now substituting the velocities from eqs C4 into eqs C10 we obtain the final expressions for the electrophoretic

(ζ2 - ζ1)E∞ 1 f1(λ) ln ηa2 λ

[

]

[

B ′(NR - NF) (ζ2 - ζ1)E∞ -Y22 + ηa2 2

(C11a)

]

C F C F ′(NR - N12 ) Y22 ′(NR + N21 ) Y21 + 1+β 1+β

B C C T1 ) -η[(U2 - U1)Y11 + Y11 Ω1 + Y12 Ω2]iy (C10a)

B U1)Y22

]

C F C ′(NR - N12 ) Y12 ′(NR + N21) Y11 + 1+β 1+β

(ζ2 - ζ1)E∞ 1 f2(λ) ln ηa2 λ

[

]

(C11b)

Equations 20 in the text are an empirical fit to eqs C11 for a1 ) a2. LA9504568

Probing the Structure of Colloidal Doublets by Electrophoretic Rotation

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