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Measuring Colloidal Forces Using Differential Electrophoresis John L. Anderson,*,†,‡ Darrell Velegol,§ and Stephen Garoff‡,| Department of Chemical Engineering, Department of Physics, and Colloids, Polymers, and Surfaces Program, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, and Department of Chemical Engineering, Pennsylvania State University, University Park, Pennsylvania 16802 Received May 11, 1999. In Final Form: September 28, 1999 Differential electrophoresis is a technique for studying the forces holding two Brownian particles in a doublet configuration. The technique is based on the concept that particles having different zeta potentials (ζ) try to move at different velocities in an electric field but the colloidal forces holding the particles together frustrate the electrophoretic motion. A doublet with a dipole moment of ζ will rotate in an electric field; by measuring the angular velocity or stationary-state angular distribution, one is able to make conclusions about the rigidity of the doublet (i.e., whether or not there are forces acting tangentially to the surfaces of the two particles). At a sufficiently high electric field, the doublet aligns with the field and an electrophoretic displacement force tries to pull apart the doublet; this force, which is typically in the range 0.1-50 pN for micron size particles, can be calculated precisely from the electric field and known properties of the system. The doublet breaks if the electrophoretic displacement force exceeds the maximum attractive colloidal force. The electrophoretic rotation velocity and the displacement force are both proportional to the applied electric field and the difference in the ζ potentials between the two particles. Experimental results are reported for mixed doublets of a silica/polystyrene latex, which demonstrate the existence of tangential forces between the particles’ surfaces and a time-dependent nature of the colloidal interactions.
1. Introduction A major focus of colloid science has been on understanding the forces between two colloidal particles suspended in a fluid. Models for the stability of suspensions and the morphology of flocs require this understanding in order to be qualitatively and quantitatively reliable. Until recently the major obstacle to testing theories of colloidal forces has been a scarcity of experimental methods capable of directly measuring forces between two solid surfaces with a gap of liquid that is nanometers thick between them. Previously, theories of colloidal forces were tested from macroscopic observations of a suspension, such as the initial coagulation rate, the collection efficiency of particles on filter media, or the extent to which a flocculated suspension could be resuspended by changing the composition of the suspension. Because such phenomena depend in a complex way on more than colloidal forces (e.g., hydrodynamic forces, distribution of particles in suspension), the measurements are only tenuously connected to the theory of colloidal forces, and any agreement between experiment and theory must be viewed strictly as a consistency check. The term “colloidal forces” generally refers to forces operating between two bodies separated by distances of order 1-100 nm. The text by Israelachvili1 provides an excellent review of the origin of these forces and methods to measure them. The classical DLVO theory has been * To whom correspondence should be addressed. E-mail: johna@ andrew.cmu.edu. † Department of Chemical Engineering, Carnegie Mellon University. ‡ Colloids, Polymers, and Surfaces Program, Carnegie Mellon University. § Department of Chemical Engineering, Pennsylvania State University. | Department of Physics, Carnegie Mellon University. (1) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic: New York, 1992.
the underpinning for models of colloidal stability but has been brought into question by recent experiments.2,3 It is becoming apparent that one general theory cannot explain the forces in all systems; rather, the behavior of a colloidal system is highly dependent on its detailed chemistry and microscopic environment. Some of the forces involve the structural properties of the liquid molecules themselves, and these cannot be accounted for by a model based only on the continuum dielectric properties of the fluid and particles.2 It is not clear whether many-body effects or long-range forces play an additional role to pair potentials in producing ordered arrays of particles.4 In addition, understanding the forces due to surface heterogeneity5-7 could be central to describing the behavior of many colloidal systems. The forces between heterogeneous surfaces would depend not only on the average properties of the surfaces but also on the precise geometric arrangement of these surfaces relative to each other, and thus the effective surface forces could be temporally altered by Brownian motion. The focus of this paper is on the direct determination of the colloidal forces operating between two Brownian particles suspended in a conducting liquid such as an aqueous solution. We have developed a new technique, differential electrophoresis,3,8,9 based on an old phenomenon, electrophoresis. Before discussing the principles behind differential electrophoresis, we review in section 2 the existing techniques for determining forces between surfaces separated by colloidal length scales. The theoretical and experimental bases of differential electro(2) Israelachvili, J. N.; Wennerstrom, H. Nature 1996, 379, 219. (3) Velegol, D. Determining the Forces between Colloidal Particles Using Differential Electrophoresis. Ph.D. Thesis, Carnegie Mellon University, 1997. (4) Larsen, A. E.; Grier, D. G. Nature 1997, 16, 385. (5) Czarnecki, J. Adv. Colloid Interface Sci. 1986, 24, 283. (6) Miklavcic, S. J. J. Chem. Phys. 1995, 103, 4794. (7) Holt, W. J. C.; Chan, D. Y. C. Langmuir 1997, 13, 1577. (8) Velegol, D.; Anderson, J. L.; Garoff, S. Langmuir 1996, 12, 675. (9) Velegol, D.; Anderson, J. L.; Garoff, S. Langmuir 1996, 12, 4103.
10.1021/la990577y CCC: $19.00 © 2000 American Chemical Society Published on Web 03/07/2000
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phoresis are then presented. There follows a summary of our experimental findings to date on two different colloidal systems. The objective of this paper is to demonstrate that differential electrophoresis has promise as a technique to study the forces operating within colloidal aggregates and offers unique advantages over other techniques in certain situations, especially with respect to sensing tangential (shear) forces between two coagulated particles for which there is very little theory. Our experimental results for doublets of silica/polystyrene latex demonstrate the existence of such tangential forces between nontouching particles (i.e., particles not in a primary minimum). 2. Review of Existing Techniques Two major experimental techniques have allowed the direct measurement of forces between fixed solid surfaces with separations in the range 1-100 nm. The technical advance of both methods was the precise determination of the spacing between the surfaces, so that force (or energy) versus distance profiles are attainable. The surface force apparatus (SFA)1,10,11 is able to directly measure both normal and tangential forces as small as 100 nN between two macroscopic surfaces (crossed cylinders of radius about 1 cm), and gaps can be measured with Angstrom precision. The SFA has revolutionized the way we look at interfacial forces, and experimental results from this method have spurred the development of new theories for colloidal forces, in particular the role of molecular structure and packing.2 Modified atomic force microscopy (AFM),1,12 in which a colloidal particle is fixed to the AFM probe, is in concept very similar to SFA except that particles as small as several microns can be used. AFM allows the direct measurement of normal forces between a fixed particle and a flat surface13 or between two fixed particles,14 and a “lateral force mode” is also available.15 The precision is roughly 10 pN. Larson et al.16 and Hartley et al.17 have verified the Gouy-Chapman model for double-layer repulsion between a particle and a surface using this technique. Another technique, total internal reflection microscopy (TIRM),18-20 has been developed to study the behavior of a Brownian colloidal particle interacting with a fixed flat surface. The vertical position of a single particle near a flat boundary is measured roughly every 100 ms using light scattered from an evanescent wave. Brownian motion causes the particle to sample various vertical positions according to a Boltzmann distribution, and thus a statistical number of measurements taken at equilibrium are converted to a potential of mean force between the particle and the flat plate. A pair of two-dimensional optical traps21 have enhanced the capability of this technique, in particular by allowing experiments at nonequilibrium (10) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (11) Van Alsten, J.; Granick, S. Phys. Rev. Lett. 1988, 61, 2570. (12) AFM can be used in “imaging mode” or “force mode”. Here we refer to only the force mode, as described in ref 1. (13) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239. (14) Li, Y. Q.; Tao, N. J.; Pan, J.; Garcia, A. A.; Lindsay, S. M. Langmuir 1993, 9, 637. (15) Marti, A.; Hahner, G.; Spencer, N. D. Langmuir 1995, 11, 4632. (16) Larson, I.; Drummond, C. J.; Chan, D. Y. C. J. Am. Chem. Soc. 1993, 115, 11855. (17) Hartley, P. G.; Larson, I.; Scales, P. J. Langmuir 1997, 13, 2207. (18) Prieve, D. C.; Luo, F.; Lanni, F. Faraday Discuss. Chem. Soc. 1987, 83, 297. (19) Prieve, D. C.; Frej, N. A. Langmuir 1990, 6, 396. (20) Walz, J. Y.; Prieve, D. C. Langmuir 1992, 8, 3073. (21) Liebert, R. B.; Prieve, D. C. Biophys. J. 1995, 69, 66.
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conditions. This technique has been used to verify the classical theory for electrostatic repulsion between a polystyrene latex sphere and a quartz surface at separations of order 100 nm,22 to observe DLVO secondary potential minima due to attraction by van der Waals forces,23 to measure polymer/colloid depletion effects,24,25 and to uncover still-unexplained long-range forces between surfaces coated with antibodies and antigens.21 Colloidal particle scattering is a hydrodynamic-based technique for measuring colloidal interactions.26,27 One particle is fixed to a flat surface, and a dilute suspension of particles flows past the fixed particle in a shear flow. The particle trajectories are measured, and differences between the measured trajectory and that predicted by hydrodynamics alone are used to infer parameters of a model for colloidal forces between the suspended and fixed particles. This technique has produced data for polystyrene latex spheres that compare well with DLVO theory. However, the technique requires considerable hydrodynamic calculations to compute the correct particle trajectory, and data analysis is complicated greatly by the effects of Brownian motion. Furthermore, the “inverse problem” must be solved, that is, a model force profile must be assumed to compare the predicted trajectory with the measured one. One key limitation of the experimental techniques referenced above is that none gives information about two Brownian particles in suspension. Brownian particles undergo random rotation. When surface heterogeneities (e.g., roughness or charge nonuniformity) are present on the particles, interaction forces can vary with time as the relative orientation of one sphere changes with respect to the other. Furthermore, none of the techniques except AFM is designed to determine forces that act tangentially to the surfaces of particles. Such forces should play an important role in determining how colloidal flocs evolve in morphology and how dense colloidal suspensions flow. Finally, while these techniques are ideal for measuring repulsive forces, they are not well suited for measuring attractive forces at Ω-1), the experimental measurement of θa is used with (21) and (30) (or eq 32, which gives accurate approximations) to compute Pee and thus the value of N. The standard deviation of the data can be compared with the above expression to determine if Brownian motion is indeed the cause of the fluctuations in the angle. Analysis for Negligible Gravitational Effects. When |Peg| is small there is only Brownian motion to counterbalance the alignment caused by the electric field: 42
f(θ,φ) )
Pee 4π sinh(Pee)
exp[-Pee cos θ]
(33)
Even for modest fields, say 100 V/m, the alignment is
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Figure 6. Schematic of electrophoresis cell.
Figure 5. Linear plot (see eq 34) of data to obtain N in the absence of gravity and neglecting Brownian motion. The doublet is composed of PS latex particles (2.51 µm amidine-coated particle (+) and 2.75 µm sulfate-coated particle (-)), and the salt concentration is 10 mM KCl (see ref 8 for details). Time (t) was reset to zero on each rotation when θ ) 90°, and the electric field was alternated in direction but kept at constant magnitude. Brownian motion is the cause of the scatter when the ordinate is outside the range -1 f +1 (Pee ) 305). From the slope of the plot we determine the experimental value of N to be 0.57 ( 0.04; the theoretical value for this size ratio is 0.63.
strong for doublets of micron size. Setting N ) 0.64 (the value for rigid-body rotation), doublets of particles with a ) 1 µm and ζ2 - ζ1 ) 25.7 mV have Pee ) -13. Using eqs 21 and 33, we have 〈θ〉 ) 20°. Angles this small are difficult to measure accurately because the angle φ can take on any value uniformly by Brownian motion, and so θ and θa can be greatly different. When gravitational alignment is negligible, a more direct and accurate measure of Pee (and hence N) is obtained by changing the direction of the field abruptly and following the trajectory of the doublet with time. When |Pee| . 1, the doublet will rotate toward alignment much faster than the Brownian diffusion time, so that a doublet that begins with e in the viewing plane will remain in the viewing plane for that time interval. Over a range of θ such that |Pee sin θ| > 10, Brownian motion is negligible and eq 12 can be used directly with little error; integrating this equation gives the following relation, which is applied to the data in order to determine N:
[
ln
] [
] (
)
1 + cos θ0 (ζ2 - ζ1) 1 + cos θ E∞ t ) ln +N sin θ sin θ0 ηL
(34)
where θ0 is the angle at t ) 0 and L = a1 + a2 for doublets. Sample data for polystyrene latex doublets are shown in Figure 5. When |Pee| < 10, account should be taken of Brownian motion when interpreting the data. A method of moments has been developed for this purpose.8 Effect of Brownian Motion on the Measurement of Normal Forces. In the absence of the electrophoretic displacement force, the two spheres of a doublet undergo random motions in a potential well. This potential well is caused by the colloidal forces, and it might be shallow (e.g., as in a DLVO secondary energy minimum) or steep (e.g., particles in a primary minimum and experiencing Born repulsion as they come into contact). The effect of the electrophoretic displacement force is to make the potential well appear more shallow (i.e., it transforms the potential energy landscape that a particle experiences),
such that Brownian motion can more easily cause particles to jump out of the well. If the potential well is several kT or more deep, this does not affect the measurement of the forces significantly. However, if the potential well is small (e.g., 50 for all the experiments. To study colloidal forces along the line of centers, the doublet was rotated to near alignment and then the field was increased to produce the electrophoretic displacement force given by (7) with θ ≈ 0°. The system of PS particles (Interfacial Dynamics, Oregon) provided a good test of the theory for rigid body rotation when the doublet was composed of a positive (amidine surface-functionalized) and a negative (sulfate surface-functionalized) particle. These doublets should have been locked into a “primary minimum”, which implies that the doublet should behave as a rigid body. The experimental value of N (0.57 ( 0.04, see Figure 5) agrees well with the theoretical calculation for rigid-body rotation (N ) 0.63, see eq 47). Figures 7 and 8 show experimental results for N for doublets composed of particles that were both negatively charged. The DLVO theory predicts that these doublets should have been in a secondary minimum;3,8,9 however, the results show clearly that the
rotation was of the rigid-body type. Note that the experimental value of N is independent of the electrolyte concentration and the applied electric field, as it should be from its definition. Furthermore, the experimental values of N are in good agreement with the theoretical value for rigid-body rotation (N ) 0.64 for β ≈ 1). In all cases (more than 50 doublets), the doublets rotated as rigid bodies even though the DLVO theory predicted a secondary minimum for most of them. Electrophoretic displacement forces were applied to the doublets in an effort to break them. The displacement forces were at least 10 times, and generally as high as 30 times, the predicted breakage forces for a secondary minimum, and several different particle surface chemistries were studied (carboxylate, carboxyl, sulfate). These results, along with the rigid-body rotation we observed for all particles, suggest that the basic concept of “secondary energy minimum” is in doubt for this system. We performed a second set of experiments with doublets formed from one polystyrene particle and one silica particle.3,70 For these doublets the gravitational forces were significant because silica has a much greater density than polystyrene. Both |Peg| and |Pee| were typically greater than 25, so that the stationary-state analysis represented by eq 32 allowed determination of N from the measured value of 〈θa〉. The particles were 2.10 µm diameter silica (Bang’s Laboratories) and 4.20 µm diameter PS latex with sulfate groups (Interfacial Dynamics, Oregon). The ζ potential of the particles and the solution conditions are listed in Table 1. The particle concentrations ranged from 106 to 108 per milliliter. During any particular experiment, the temperature remained constant at ambient conditions to within 1 °C. Figure 9 shows one type of behavior we observed. This doublet had time-varying kinematic states, such that the angles jumped by large amounts on non-Brownian time scales. Note that N is proportional to Pee, so that a change from “rigid” to “free” rotation leads to a change in cot 〈θa〉 of about a factor of 2 to 3; however, the change in 〈θa〉 itself is less, as seen in the figure. Also note that, because |Peg| . 1, the doublet (i.e., the director e) remained essentially in the image plane, and hence 〈θa〉 was nearly equal to 〈θ〉. The data in Figure 9 exhibit two important temporal variations. The first is due to Brownian motion which appears as random fluctuations on the time scale of about 1 s. The other variation is a transition from one kinematic state to another (rigid f free f rigid, marked A f B and C f D in the figure); each state is sustained for 5 s or longer. The standard deviations within each state, as computed from the data using eq 37, are in reasonable agreement with the values predicted from eq 32. The brackets on the right side of Figure 9 show the predicted range of fluctuations due to Brownian motion ((σθ) about the mean angle. (70) Velegol, D.; Catana, S.; Anderson, J. L.; Garoff, S. Phys. Rev. Lett. 1999, 83, 1243.
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particles. The fact that the entire system undergoes Brownian motion is another unique attribute of the technique. The temporal variations in the “rigidity” of the doublet cannot be studied by other methods, yet this phenomenon is important in the rearrangement of particles in flocs, especially in the presence of shear flow. The changes in the kinematic state of a single doublet with time might result from Brownian impulses acting on a spatially varying potential energy landscape,5-7 causing the particles to fall into local rotational energy minima. 7. Summary
Figure 9. θa versus time for a doublet composed of a silica particle and a PS latex particle (see Tables 1 and 2 and ref 70). Table 2. Electrophoretic Displacement Experiments for Si-PS Doubletsa experiment
doublet
Emax (V/cm)
Fdisp (pN)
broke doublet?
Fcrit (pN) from DLVO
3 mM
a b c d e f g h i j k l m n o p q r s t u v w x
8.7 290 350 14 430 14 350 290 54 0.9 210 120 140 250 120 185 280 310 310 14 310 110 140 230
1.00 ( 0.09 34 ( 3.2 40 ( 3.6 1.6 ( 0.15 50 ( 4.6 1.7 ( 0.15 40 ( 3.6 34 ( 3.2 7.8 ( 1.2 0.13 ( 0.02 32 ( 5.0 18 ( 2.8 25 (2.0 46 ( 3.7 23 ( 1.8 34 ( 2.7 53 ( 4.2 57 ( 4.5 57 ( 4.5 2.5 ( 0.2 57 ( 4.5 22 ( 2.0 27 ( 2.5 45 ( 4.1
YES no no no no no no no YES YES no YES YES no YES YES no no no YES no no no no
0.01
5 mM
7 mM
10 mM
0.04
0.08
0.15
a The doublets analyzed are labeled a-x. E max is the maximum field that was applied to the doublet during an electrophoretic displacement experiment, and Fdisp was calculated from eq 7 with θ ) 0. The doublets which broke are indicated by a YES, while those which did not break are indicated by a no. Fcrit was predicted from DLVO theory.
We examined 24 doublets, labeled a-x. Table 2 shows the salt concentrations of these doublets, whether the doublet broke, the field at which it broke, the maximum applied displacement force calculated from (7), and the DLVO prediction for when the doublet should have broken.70 Note that one-third of the doublets broke, in contrast to previous experiments with doublets of two PS latex particles where none broke.9 When the Si/PS doublets behaved in a rigid manner, the tangential forces holding them in that configuration were about 0.05 pN, leading to restraining torques of about 0.1 pN µm. Because some of the doublets were broken by the electrophoretic displacement force, the particles were probably not touching. Figure 9 illustrates an important attribute of differential electrophoresis, namely that we can focus on a single doublet to obtain both stationary-state and temporal data, rather than relying on ensemble averages over many
The two main features separating differential electrophoresis from other techniques to probe colloidal forces are that it deals with two particles undergoing Brownian motion and it addresses attractive forces rather than repulsive forces. For particles of micron size, the displacement force tending to break the doublets is in the range 0.1-50 pN, depending on the applied electric field and the difference in ζ potential. The limitations of this technique are that the two particles of a doublet must have a known difference in ζ potential (though both can be of the same sign). The displacement force scales with the particle size, as do colloidal forces, so the ability of the classical theory to model the adhesion between two “weakly coagulated” particles can be tested for submicron particles as well, assuming the optical resolution of the microscope is satisfactory. The main problem with studying smaller particles is their Brownian motion; Pee ∼ a2, so higher fields and larger values of ∆ζ are needed to make electrophoretic rotation dominant as the particles become smaller. It is important to remember that the values of N and Q, which appear in eqs 11 and 6, are essentially independent of the dynamics within the gap region between the particles;61,62 thus, the determinations of the kinematic state of rotation (rigid versus free) and the displacement (breakage) force are independent of any model for the colloidal forces. The major findings from our experiments to date are the presence of tangential forces between nontouching particles (Si/PS doublets), temporal shifts between rigidbody and freely-rotating kinematic states for a single doublet, and failure to break doublets with electrophoretic displacement forces at least 10 times greater than the predicted maximum attraction of two particles in a secondary energy minimum. Whether the latter observation is due to additional forces that are attractive (e.g., hydrophobic), surface roughness, or heterogeneous surface chemistry is speculative at present. The observed switching between rigid and freely-rotating states with time for a single doublet suggests that rotational Brownian motion of both particles in the doublet is important. The existence of tangential forces has important implications for the structure of flocs; if the contacts are rigid, then the structure is determined by collision kinetics alone. While all our experiments have involved coagulated doublets, the principles of differential electrophoresis could be applied to particles that are tethered to each other or to aggregates of three or more particles. For higher order aggregates, the electrokinetic equations must be solved for the given configuration of particles to compute the displacement forces between each particle-particle contact. 8. Appendix Free versus Rigid Rotation of Doublets. A distinct advantage of electrophoretic rotation is that it directly probes the tangential forces between two particles near
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Now consider the rigid-body case. Because the two particles act as one, due to forces between them directed both along the line centers and tangent to the surfaces, mechanical equilibrium is achieved when the total force and torque on the doublet are zero:54
Figure 10. Rigid-body rotation versus free rotation of a doublet. The rotation arrows are drawn assuming particle 1 is more positive than particle 2 (ζ1 > ζ2).
contact. To understand why rotation, through the coefficient N, yields the rigid state of a doublet, we consider here an approximate analysis of a doublet’s rotation. Referring to Figure 10, the forces are in the z direction (the direction of the applied electric field for this orientation of the doublet), and the torques and angular velocities are in the y direction perpendicular to the x-z plane. If the particles are free to rotate such that each achieves mechanical equilibrium with the surrounding fluid, then the total force and torque on each particle must be zero:
Fi ) 0, Ti ) 0
Ωe )
U20 - U10 U2 - U1 )N L L
(39)
N ) 1 in the absence of interactions. Effects of particle interactions have been determined by a “method of reflections” technique.45 Using only the first nonzero reflections of the disturbances to the fluid velocity and the electric field caused by each sphere, corrections to the velocity of each sphere are determined and then (39) gives the following:
() ()
a2 1 N ) Nfr ) 1 + (1 + β3) 2 L
3
a2 +O L
(41)
T1 + T2 + LF2 ) 0
(42)
Ω1 ) Ω2 ) Ωe
(43)
Equation 43 is most important, as it forces each particle to rotate with the doublet. Neglecting hydrodynamic interactions between the spheres, the force and torque on each sphere are
Fi ) -6πηai(Ui - Uif)
(44)
Ti ) -8πηai3(Ωi - Ωif)
(45)
where Uif and Ωif are the velocities for the freely-rotating case. Combining (39) through (45), we have the following for rigid-body rotation:
[ ()
(38)
Because the double layers are very thin relative to the radius of the particles (κai . 1), the force and torque on the particles can be calculated at the outer edge of the double layer and hence the forces are all hydrodynamic in nature (i.e., electrical forces are zero at the edge of the EDL to first order in the electric field, although the electric field is not).39 If electrostatic and hydrodynamic interactions between the particles are neglected, then each particle would move (without rotation) at the velocity Ui0 given by (3). The angular velocity of the doublet is defined by (9), so N is defined by
F1 + F2 ) 0
N ) Nrb ) 1 +
]
3 4 a2 2(1 + β)(1 + β ) 3 L β
-1
Nfr (46)
The above equation indicates that N < 1 for doublets that rotate as rigid bodies. The essence of the rigid-body constraint is (43). The coupling between the two spheres prevents their surfaces from moving tangentially relative to each other, thus imparting an angular velocity to each of O(a2/L)2. This is why N < 1. Numerical Evaluation of N. Using (11) to define N, Velegol3 used numerical calculations and semianalytical models for the two-sphere electrophoretic problem48,49,50 to determine N as a function of the dimensionless separation (λ ) δ/a2) and the size ratio (β ) a1/a2 < 1). For λ < 0.1 the results for rigid doublets are given by the following empirical expressions:
N ) NR ) for β < 0.30 0.5β + 1.44β2 (47) 0.67 - 1.19 exp(-3.72β) for 0.30 e β e 1
{
with 1 for the case of free rotation.
with
