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Limitations of Differential Electrophoresis for Measuring Colloidal Forces: A Brownian Dynamics Study† Gretchen L. Holtzer and Darrell Velegol* Department of Chemical Engineering and the Materials Research Institute, The Pennsylvania State University, University Park, Pennsylvania 16802 Received March 1, 2005. In Final Form: May 9, 2005 Differential electrophoresis experiments are often used to measure subpiconewton forces between two spheres of a heterodoublet. The experiments have been interpreted by solving the electrokinetic equations to obtain a simple Stokes law-type equation. However, for nanocolloids, the effects of Brownian motion alter the interpretation: (1) Brownian translation changes the rate of axial separation. (2) Brownian rotation reduces the alignment of the doublet with the applied electric field. (3) Particles can reaggregate by Brownian motion after they break, forming either heterodoublets or homodoublets, and because homodoublets cannot be broken by differential electrophoresis, this effectively terminates the experiment. We tackle points 1 and 2 using Brownian dynamics simulations (BDS) with electrophoresis as an external force, accounting for convective translation and rotation as well as Brownian translation and rotation. Our simulations identify the lower particle size limit of differential electrophoresis to be about 1 µm for desired statistical accuracy. Furthermore, our simulations predict that particles around 10 nm in size and at ambient conditions will break primarily by Brownian motion, with a negligible effect due to the electric field.
Introduction Over the past 30 years, various techniques have been used to measure surface and interparticle forces.1 Direct methods include the surface force apparatus (SFA),2 atomic force microscopy (AFM),3 total internal reflection microscopy (TIRM),4 and optical laser trapping.5 In recent years, the technique of differential electrophoresis6-8 has been developed to measure subpiconewton forces between nearly touching Brownian colloids. The purpose of this article is to answer a key question: What is the accuracy of the force measurement as a function of particle size? In differential electrophoresis, we start with two or more interacting particles with different zeta (ζ) potentials. For this study, we will focus on doublets of particles (1 and 2), as shown in Figure 1. When the electric field (E0 ) E0ix) is applied, the two spheres will try to separate by electrophoresis. When the tendency to separate by electrophoresis exceeds the ability of the existing colloidal forces to hold the two particles together, the doublet breaks into two singlets. Experimentally, this is straightforward to see using video microscopy for large particles or indirectly using static light scattering for small particles. To interpret the force applied with the electric field (Fep, also called the electrophoretic separation force), the electrokinetic equations have been solved to give6
Fep ) 8.76πa|ζ2 - ζ1|E0
(1)
where is the permittivity of the fluid and a is the particle †
Part of the Bob Rowell Festschrift special issue. * Corresponding author. E-mail:
[email protected].
(1) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: San Diego, CA, 1992. (2) Israelachvili, J. N.; Adams, G. E. J. Chem. Faraday Trans. 1978, 74, 975. (3) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239. (4) Prieve, D. C.; Frej, N. A. Faraday Discuss. Chem. Soc. 1990, 90, 209. (5) Crocker, J. C.; Matteo, A.; Dinsmore, A. D.; Yodh, A. G. Phys. Rev. Lett. 1999, 82, 4352. (6) Velegol, D.; Anderson, J. L.; Garoff, S. Langmuir 1996, 12, 4103. (7) Anderson, J. L.; Velegol, D.; Garoff, S. Langmuir 2000, 16, 3372.
Figure 1. Colloidal doublet in an applied electric field. The e vector is along the line of centers, and θ is the angle from the applied field (E0) direction to e. The radii (a) of particles 1 and 2 are the same.
radius. Because the force is applied only in one dimension, scalar terms can be used in place of vector quantities. A number of assumptions were used to obtain eq 1: (a) the electrical double layer around the particles is infinitesimally thin, (b) the particles are nearly touching, (c) the doublet axis is aligned with the applied electric field, and (d) Brownian motion is negligible. A cos θ factor could be easily added to account for the angle of the doublet with respect to E0, but still the assumption is that the doublet remains at a single orientation. In this article, only assumption (a) will be kept. The other assumptions are relaxed and the pertinent equations are solved so that we can determine the accuracy of our differential electrophoresis force measurements by modeling. The differential electrophoresis experiment has at least three time scales. The experimental time scale has typically been on the order of seconds, although one result from the present research is that we will have to decrease field application times for small-particle systems. A second time scale is obtained from the Smoluchowski rapid aggregation time9
tagg )
ηa3W 2kTφ
(2)
where η is the fluid viscosity, φ is the particle volume
10.1021/la0505566 CCC: $30.25 © 2005 American Chemical Society Published on Web 06/17/2005
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Differential Electrophoresis Limitations Due To Brownian Motion Analyzing the limitations of differential electrophoresis due to Brownian motion will require further modeling, which is described in this section. The electrophoretic separation force can be calculated in terms of the electric field using the electrokinetic equations. For doublets of equal-sized particles, the following equation was developed for the electrophoretic separation force as a function of the gap between the particles (δ, as shown in Figure 1):6,7
( )
Fep(δ) ) 6πa
Figure 2. Reduction of heterodoublet population due to reaggregation. (1) We start with a sample having many heterodoublets. (Only heterodoublets are shown, although a number of singlets and homodoublets might also exist.) (2) When a critical electric field is applied, many of the doublets will break. Some of these will stay as singlets, some will reform as heterodoublets, and some will form into homodoublets (either dark-dark or light-light in the plot). (3, 4) As the differential electrophoresis process is continued, heterodoublets continue to break apart, and an increasing number of resulting singlets will form homodoublets, effectively terminating the experiment. Higher-order aggregates have been neglected in this scenario.
fraction in suspension, and W is the stability ratio () 1 for rapid aggregation). Thus, if heterodoublets break and reaggregate much more quickly than our experimental time (especially if they are able to reaggregate into homodoublets), then the differential electrophoresis experiments will appear to yield no results. A third time scale is that required for breaking. Although eq 1 can give an estimate of this time, we show in this article that the time can be altered significantly by Brownian motion. For very small interparticle energies (∼kT), no field is required to break a doublet because Brownian motion can cause breaking. At this point, light-scattering measurements are helpful because one examines thousands of doublets at once and therefore obtains statistical measurements. The use of light scattering for our force measurements will be addressed in a future publication.
kep(δ)
khyd(δ)
|ζ2 - ζ1|E0 cos θ
(3)
θ is the angle from E0 to e (see Figure 1), and the interaction coefficients (kep and khyd) are for electrokinetic and hydrodynamic interactions. These coefficients depend on the gap distance (δ) between the particles. Previously, Fep was balanced against the maximum attractive colloidal force, which we call the critical force (Fc), to interpret the experiments. This force occurs at a separation of δ where d2Φ/dδ2 ) 0 and Φ is the interparticle potential. Because Fc denotes colloidal attraction, it will be negative in value. Thus, when (Fep + Fc) > 0, the doublet breaks. As a result, the force required to break the doublet is associated with some critical electric field (Ec), as given in eq 1. We will nondimensionalize all quantities. Thus, we define a Peclet number in terms of the critical colloidal force as
Pec )
|Fc|δ0 kT
(4)
where δ0 is the equilibrium gap between the colloids (i.e., when the colloidal force is zero, not the maximum value Fc) and kT is the thermal energy with the Boltzmann constant (k) and the absolute temperature (T). When Pec , 1, we expect that a doublet can break apart relatively easily by Brownian motion, even in the absence of an applied electric field. Thus, a small Pec makes the interpretation of differential electrophoresis experiments difficult. Example calculations based on our results will be given in the Discussion section. We also define an electrophoretic Peclet number (Peep)
Peep )
F0epδ0 kT
(5)
The Smoluchowski rapid aggregation time is given by eq 2. If we start with a suspension containing many heterodoublets and break them using differential electrophoresis, then the particles can reaggregate quickly. Although some of the particles will reaggregate into heterodoublets, many of the particles will aggregate into homodoublets if repulsive forces cannot keep them apart. Thus, the population of heterodoublets will be diminished (as shown in Figure 2). Because homodoublets have ζ1 ) ζ2, we see that eq 1 yields Fep ) 0. Therefore, the homodoublets cannot be broken by differential electrophoresis alone.
where Peep gives the relative strength of the electrophoretic separation force at θ ) 0 and δ ) δ0 versus the effect of Brownian translation. If Peep , 1, then diffusive translation (i.e., Brownian translation) dominates, and separation of the two particles in the doublet is slow. If Peep . 1, then electrophoretic translation dominates, and this is the driving force behind experiments involving differential electrophoresis. We account for the translational and rotational motion of the particles using a Brownian dynamics simulation (BDS).9-12 For translation, both Brownian and electrophoretic translation will be considered, as well as the interparticle colloidal forces (which can be described as a net force through the hydrodynamics equations). The orientation of the doublet changes with both electro-
(8) Holtzer, G. L.; Velegol, D. Langmuir 2003, 19, 4090. (9) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, with corrections 1991.
(10) Ermak, D. L.; McCammon, J. A. J. Chem. Phys. 1978, 69, 1352. (11) Bevan, M. A.; Prieve, D. C. J. Chem. Phys. 2000, 113, 1228. (12) Sholl, D. S.; Fenwick, M. K.; Atman, E.; Prieve, D. C. J. Chem. Phys. 2000, 113, 9268.
Rapid Aggregation Limitation
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Holtzer and Velegol
phoretic rotation and Brownian rotation (as discussed above). To simplify our analyses, we let a1 ) a2 ) a and the ζ potential on each sphere be uniform (although ζ1 * ζ2). In addition, we examine one doublet at a time, as we would do with a video microscopy differential electrophoresis measurement. The details of our simulation, starting with the theory behind each of the sub-issues above, are presented below. Interparticle Force Dependence. In this portion of the calculation, we determine the extent of separation (∆δhyd) between particles in a doublet due to hydrodynamics in a given time (∆t), as given by the following relationship:
∆δhyd ) (U2 - U1)hyd∆t
(6)
The particle velocities (Ui) depend on the forces between the particles, and because our particles are in the creeping flow regime (low Reynolds numbers), we have13,14
(U2 - U1)hyd )
F(δ) k (δ) 6πηa hyd
(7)
In this equation, η is the fluid viscosity, a is the particle radius, and F(δ) is the colloidal force between particles in the doublet. The interaction coefficient (khyd) accounts for the proximity of the particles, either by lubrication hydrodynamics or reflection calculations.13 An equation for khyd(δ) can be obtained by performing an empirical fit of known, tabulated values. This equation, as well as a plot of khyd(λ) (where λ ) δ/2a), can be found in the Appendix. To model F(δ), we chose a model that has similarities to DLVO theory with van der Waals attraction and electrostatic repulsion. We used
F(δ) )
( )( ) ( 5 5 3 2
2/3
|Fc|
δ05 δ5
-
)
δ02 δ2
δ0 c Uhyd(δ0)
)
6πηaδ0
∆δhyd ) ∆xhyd )
(10)
Fckhyd(δ0)
5 5 3 2
2/3
)
1 1 khyd(x)∆τ - 2 5 x x khyd(x0)
(11)
Thus, we have the dimensionless attraction distance (∆xhyd) due to colloidal forces for a dimensionless time step (∆τ). Electrophoretic Translation and Separation. In this portion of the calculation, we want to determine the separation due to electrophoresis. For small time steps, we have
∆δep ) (U2 - U1)ep∆t
(12)
Here, we assumed that the thickness of the double layer (κ-1) is much smaller than the particle radius (a), and we therefore assumed infinitesimal double layers.16 Thus, (U2 - U1)ep is given by
(U2 - U1)ep )
(ζ2 - ζ1)E0 cos θ kep(δ) η
(13)
The coefficient kep(δ) can be calculated from known mobility coefficients (Mpij) for the near-field case,17 and O(L-7) reflection results for the far-field case.18 Matching of the near-field and far-field cases occurs at λ ) 4.5. The resulting empirical fit to the exact calculations for kep is given in the Appendix (accurate to within 5% of known values), along with a plot as a function of λ ) δ/2a. As before, we nondimensionalize the gap change due to electrophoresis using the equations above:
∆xep )
(9)
F(δ) khyd(δ) δ0∆τ
( )( ) (
(8)
Using fully expanded and detailed forms of the force models is not critical, but it is critical that the equation have (a) zero force at infinite separations, (b) a secondary minimum well, and (c) a large barrier closer to contact. The 2-5 force profile used is analogous to the 6-12 energy profile commonly used in molecular dynamics simulations.15 The use of different exponents changes the numerical results slightly, but qualitatively we obtain the same conclusions discussed at the end of the article. To simplify the calculations, we have chosen to nondimensionalize the parameters and variables in the simulations. We define two spatial dimensionless scaling variables, x ) δ/δ0 (dimensionless gap distance) and R ) a/δ0 (dimensionless particle radius). Therefore, δ/a ) x/ . We also define a dimensionless time, τ ) t/t , where R 0 the time scale (t0) is given by the following equation:9
t0 )
δ0. Combining eqs 6-9 and the dimensionless parameters gives
Peepkep(x) cos θ ∆τ
(14)
Peckep(x0)
Equation 14 is sufficient for calculating ∆xep for any given ∆τ. The ratio of the Peclet numbers gives Fep0/|Fc|, which is the relative strength of the applied field to the critical colloidal force. Brownian (Diffusional) Translation and Separation. The next component of the translational calculations for this simulation is the separation due to Brownian (or diffusional) translation of the doublet. Because Brownian motion is a random thermal motion, the increase or decrease in gap distance due to Brownian motion will also be random. For small time steps (∆t), a solution to the diffusion equation is19,20
p(∆δbr) )
1
x4πD(δ)∆t
[
exp -
∆δbr2
]
4D(δ)∆t
(15)
Here, we have chosen the characteristic time as that required for the colloidal forces to alter the separation by
In the above equation, p(∆δbr) is the probability distribution function of ∆δbr based on a Gaussian distribution with an average of 0 and a standard deviation of 2D(δ)∆t. Equation 15 does not include the effects of drift due to convective motion11 (accounted for separately in eqs 11
(13) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media; Martinus Nijhoff Publishers: Hingham, MA, 1983. (14) Kim, S.; Karrila, S. J. Microhydrodynamics: Principles and Selected Applications; Butterworth-Heinemann: Boston, 1991. (15) Frenkel, D.; Smit, B. Understanding Molecular Simulation; Academic Press: New York, 1996.
(16) Anderson, J. L. Annu. Rev. Fluid Mech. 1989, 21, 61. (17) Keh, H. J.; Yang, F. R. JCIS 1990, 139, 105. (18) Chen, S. B.; Keh, H. J. AIChE J. 1988, 34, 1075. (19) Van de Ven, T. G. M.; Mason, S. G. J. Colloid Interface Sci. 1976, 57, 517. (20) van de Ven, T. G. M. Colloidal Hydrodynamics; Academic Press: San Diego, CA, 1989.
|Fc|khyd(δ0)
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and 14). Also, in eq 15, D(δ) is the translational diffusion coefficient, which takes the following form for a sphere:14
D(δ) )
kTkhyd(δ) 6πηa
(16)
When we nondimensionalize this expression for the diffusion coefficient and multiply by the time increment (needed to calculate p), we obtain
kTkhyd(δ)δ0 khyd(δ)δ02 D(δ)∆t ) ∆τ ) ∆τ Fckhyd(δ0) Peckhyd(δ0)
(17)
Because
p(∆xbr) ) p(δ0∆xbr)δ0 ) p(∆δbr)δ0 we find that
p(∆xbr) )
(
Peckhyd(x0)
4πkhyd(x)∆τ
) ( 1/2
exp -
(18)
)
∆xbr2Peckhyd(x0) 4khyd(x)∆τ
(19)
By taking a Gaussian pseudorandom number,21 we thus obtain the Brownian translational step for any given set of parameters. By choosing a small ∆τ (e.g., ∆τ e 0.01), we ensure that the dimensionless movements ∆xbr , 1. We also account for diffusional drift due to a gradient in the translational diffusion coefficient.9-11 For sufficiently small time steps, this drift is given by
∂D ∂δ
∆δ∇D ) ∆t
(20)
We can nondimensionalize this equation to give
dkhyd(x) ∆τ ∆x∇D ) Peckhyd(x0) dx
(22)
where the rotational electrophoretic velocity for a doublet (Ωep) has been shown to be22,23
Ωep )
(ζ2 - ζ1) N(δ) E0 sin θ η(2a + δ)
∆θep ) -
(23)
for either rigid or nonrigid (i.e., freely rotating) doublets. Here, N(δ) is a dimensionless coefficient that characterizes hydrodynamic interactions between the spheres in a doublet, and its value approaches unity as the gap (δ) between the particles approaches infinity. For rigid and touching doublets consisting of equal-sized spheres, N ) 0.64. For freely rotating spheres in which the two spheres (21) From algorithm 712, Trans. Math. Software 1992, 18, 434. (22) Velegol, D.; Anderson, J. L.; Garoff, S. Langmuir 1996, 12, 675. (23) Fair, M. C.; Anderson, J. L. Int. J. Multiphase Flow 1990, 16, 663. Corrigenda Int. J. Multiphase Flow 1990, 16, 1131.
N(x)Peep sin θ
(24)
∆τ
(2R + x)Peckep(x0)
Brownian (Diffusional) Rotation. The final component of the rotational calculations is for Brownian rotation. Similar to the results for translational motion, a solution of the diffusion equation for the Brownian rotation is19
p(∆θbr) )
1
x4πDr(δ)∆t
(
exp -
∆θbr2
)
4Dr(δ)∆t
(25)
p(∆θbr) is the probability distribution function of ∆θbr based on a Gaussian distribution with an average of 0 and a standard deviation of 2Dr(δ)∆t. The term Dr(δ) represents the rotational diffusion coefficient, which varies with the gap between the particles as20
Dr(δ) )
(21)
Electrophoretic Rotation. We have examined the various types of translational motion. Because the electrophoretic displacement depends also on the orientation of the doublet with respect to the applied field, we must also examine the rotation of the doublet. We can neglect the φ-directional effects because they do not influence the breaking of doublets, as seen in eq 3. We have the following θ dependence for electrophoretic rotation
∆θep ) -Ωep∆t
can rotate relative to each other torque-free even as the doublet as a whole (i.e., the e vector) rotates differently, N actually increases for smaller gaps. The theory has been well established,22 and we approximate the known values for N(δ) to within 1% by using an empirical fit to the exact values (Appendix), just as we did for khyd(δ) and kep(δ) previously.22 Typical values of N for freely rotating doublets (which we have in our analysis) are 1.10-1.60 for the gaps of interest. Figure 11 showing N and simple expressions that enable a quick evaluation of N are given in the Appendix for both the near-field case (λ e 0.20) and the far-field case (λ > 0.20). The near-field case was obtained using numerical calculations, and the far-field case was obtained from O(L-7) reflection results.17,18 Substituting eq 23, the Peclet numbers, and the dimensionless time into eq 22 gives
kT πηa3Kr(δ)
(26)
where Kr(δ) accounts for the system geometry including the dimensionless gap. Expressions are known for both the near-field and far-field cases, and these appear in the Appendix.20 Upon nondimensionalizing our equations, we have to rewrite Dr(δ)∆t in terms of known parameters. We obtain
Dr(δ)∆t )
6 ∆τ Peckhyd(δ0)R2Kr(δ)
(27)
which gives us
p(∆θbr) )
(
) (
Peckhyd(x0) Kr(x)R2 24π∆τ
1/2
)
∆θbr2Peckhyd(x0) Kr(x)R2 exp 24∆τ (28)
Simulation Procedure We now have all five motions characterized: (1) colloidal forces acting through hydrodynamic translation along e, (2) electrophoretic separation acting along e, (3) Brownian translation acting along e (both random motion and drift due to a gradient in the diffusion coefficient), (4) electrophoretic rotation, and (5) Brownian rotation. In this section, we briefly describe the simulation procedure. The equations from the previous section, particularly eqs 11, 14, 19, 24, and 28, were incremented for a single doublet for a small time step (∆τ ) 0.0001 for Pec < 1 and ∆τ )
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Holtzer and Velegol Table 1. Parameters Used in This Study parameter
definition
x0 xmax
dimensionless equilibrium gap distance dimensionless maximum gap distance (for breaking) critical colloidal Peclet number applied electrophoretic Peclet number dimensionless particle radius number of independent doublets dimensionless time step
Pec Peep R N ∆τ
Figure 3. Average breaking time for 10 000 doublets with a critical colloidal Peclet number of 0.1. For R ) 0.1, we varied the time step from 10-5 to 10-2. Convergence occurs at ∆τ ) 10-4, as seen by the fact that the curves for ∆τ ) 10-4 and 10-5 appear to be the same. For R ) 100, we varied the time step from 10-4 to 10-2, with convergence occurring at ∆τ ) 10-3. (Again, the curves appear as one.)
range/value 1 10 0.1-1000 (0-1.2)Pec 0.1-100 10 000 0.0001-0.01
The parameters were chosen for each simulation as given in Table 1. The number (N) of doublets studied for each simulation was 10 000; higher N changed the resulting average breaking times very little. Before running the simulations to obtain results, we first performed a number of calculations to ensure that our code gave results that match known physics. Known analytical results exist for all five types of motion studied. For instance, in checking rotational Brownian motion, we compared 〈∆θ2〉 ) 2Dr(δ)t to our simulation results for various values of δ. All simulated changes in x and θ matched known theory, and this was checked for various values of ∆τ from 0.0001 to 1. Values of ∆τ e 0.01 gave accurate comparisons to known theory. For simulations that ranged from near field (e.g., x ) 1 for R ) 100) to far field (e.g., x ) 100 for R ) 1), we numerically integrated known expressions from the Appendix, and we again obtained the same x(τ) and θ(τ) behavior as from our simulations. The match for hydrodynamics was always within 0.5% for 10 000 doublets. Analytical integration of the electrophoretic separation for small gaps gives
()
2 ln τbreak )
x1 + 10.75(x2 - x1) x2 ηa 11.478 (ζ2 - ζ1)E0 cos θ (29)
which can be nondimensionalized to give
τbreak )
Figure 4. Average breaking time for 10 000 doublets with a critical colloidal Peclet number of 100. The solid lines represent ∆τ ) 0.01, and the lines with large dashes represent ∆τ ) 0.001. For both cases, ∆τ ) 0.01 was found to be a sufficiently small time step for solution convergence.
0.01 for Pec g 1). Checks were made to ensure that the time step did not affect the ∆x or ∆θ result by more than 1%.9 We also checked this by running the code for 0.00001 e ∆τ e 0.01 until the resulting average breaking times converge to within 5%. Example plots for this check are shown in Figures 3 and 4. The simulation was run sequentially for N colloidal doublets. The goal of each simulation was to find the time τ required to “break” the doublet. The simulation was started at x0 ) 1 (i.e., at the equilibrium gap distance), and the doublet was considered to be “broken” when x ) xmax ) 10. Although we ran our simulations in series (typical computer execution time for 10 000 doublets was less than 5 min, and the longest runs took about 72 h), the code can be easily parallelized.
()
x + 10.75(x - x0) x0 Peep 2R + 10.75 Pec x0
2R ln
( )(
)
(30)
Comparisons of analytical and simulation results matched to within 3.7%. For larger gaps (x), numerical integration was used for comparison instead of the equations above, giving results to within 2% (better than for analytical theory because exact values for k coefficients are used for all gaps). For another check, we isolated the effects of electrophoretic rotation, verifying both the near-field and farfield cases. The known result for rotation is22
τrotation ) -
(
)
(1 - cos θ)sin θ0 2aη (31) ln (ζ2 - ζ1)E0 cos θ N(δ) sin θ(1 - cos θ0)
which can be nondimensionalized to
τrotation ) -
2PecRkep(x0) PeepN(x)
(
ln
)
(1 - cos θ)sin θ0
sin θ(1 - cos θ0)
(32)
We varied ∆x from 0.1 to 10 000 and ∆τ from 0.0001 to 1. For all simulations, the error was small ( 0.1, the error (difference in the simulated result and numerically integrated result) increases significantly, sometimes by more than 100%. For the far-field case, the results matched to within 0.005% for all values of ∆x and ∆τ ) 0.01. Results Figures 5-9 show the simulated results for the breaking time (i.e., the time to go from x ) 1 to x ) 10) for various values of critical colloidal Peclet numbers (Pec), electrophoretic separation Peclet numbers (Peep), and the radius (R). Discussion Figures 5-9 become more meaningful when actual numbers are put into the equations. For example, let us say that we have a doublet consisting of spheres with radii a ) 1 µm. We will assume that the doublet breaks at a force of Fep ) -Fc ) 4 pN, which we can estimate using known colloidal force models. We will also assume that the overall colloidal force is dominated by depletion
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Figure 7. Average breaking time for 10 000 doublets with a critical colloidal Peclet number of 10.
Figure 8. Average breaking time for 10 000 doublets with a critical colloidal Peclet number of 100. Note that the curves are getting steep when Pec ) Peep.
forces that give an equilibrium gap of δ0 ) 10 nm. We then vary the volume fraction of particles from φ ) 10-5 to 10-3. If we let T ) 290 K, then this yields a viscosity η ) 1 cP for water. These values give Pec ≈ 10, R ) 100, and t0 ) 1.3 ms. We can estimate the breaking time by looking at values for Pec ) 10 (Figure 7). Figure 7 indicates that for Pec ) Peep ) 10, τbreak is ∼20, corresponding to an average breaking time of 25 ms. For this example, the rapid aggregation times (from eq 2) are plotted as a function of φ in Figure 10 below. The breaking time is also shown in Figure 10 for comparison. Here, we learn several important pieces of information: (1) If we run our experiment for O(1 s), then there is ample time to break doublets. (2) The singlets in suspension cannot reaggregate significantly after they break (i.e., tagg . 1 s). (3) The accuracy of our measurement can be estimated by considering the process that takes place during a differential electrophoresis experiment. Because we do not know the breaking time a priori, we might start at Peep ) 0 for 1 s (roughly 785t0, estimated by theory and then checked by trial and error), then Peep ) 1, then Peep ) 2, and so forth. Because 1 s corresponds to τ ) 785, we see
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Figure 9. Average breaking time for 10 000 doublets with a critical colloidal Peclet number of 1000.
Holtzer and Velegol
by Brownian motion (Peep ) 0)! This represents a 100% error! If Pec is small, then particles smaller than a ) 100 nm (i.e., 200 nm diameter) suffer the same problem, as we will see in the next example. Finally, let us look at an example with a ) 10 nm with -Fc ) 0.04 pN (keeping the other solution conditions the same). Here, Pec ) 0.1, R ) 1, and the size of the equilibrium gap (δ0) is now comparable to the size of the particle. Therefore, we will treat this example only as a rough estimate because our assumption of infinitesimal double layers will begin to break down. Now τbreak is about 1/ 10 of the previous value (roughly 2), and t0 ) 0.048 ms (very fast). This means that the time to break is also 0.096 ms. Furthermore, the rapid aggregation time is now as low as 0.12 ms (Figure 10), which is even closer to the breaking time. Not only can the particles reaggregate as fast as they can break apart, but they are doing so more quickly than we would ever be able to see. And detection with a light-scattering system would become much more challenging because the minimum data acquisition time for our system is >10 ms. Finally, because Pec ) 0.1 in this example, much of the breaking is due to Brownian motion. Therefore, the application of an electric field would alter this scenario little. Conclusions
Figure 10. Plot of breaking and reaggregation times for a ) 1 µm (-), a ) 100 nm (- - -), and a ) 10 nm (- - -) as a function of the colloidal particle volume fraction, φ. The data represent times calculated using the solution conditions given in the above discussion.
that the particles might break on average at roughly Peep ) 4. Thus, we would have a 60% error in the measurement for a ) 1 µm. Thus, this starts to represent the lower limit of differential electrophoresis with some desired accuracy for the video microscopy form of the experiments. The purpose of this article becomes even clearer when we look at smaller-sized spheres. Under similar conditions as in the previous paragraph, if we use a ) 100 nm with -Fc ) 0.4 pN (forces are usually predicted to scale linearly), then Pec ) 1 and R ) 10. Although τbreak is only about half of the previous value (roughly 10), we now have t0 ) 0.18 ms. This means that the time to break is 1.7 ms (more quickly than above). However, two factors are even more important for this example as compared to the previous example. First, the rapid aggregation time is now as low as 125 ms (Figure 10) and much closer to the breaking time. Therefore, the particles can often reaggregate nearly as fast as they break. Second, the accuracy is reduced even more from that of the previous case. If we let τ ) 285 (i.e., for Peep ) Pec, set tbreak < t < tagg), then we see that on average the particles would have already broken simply
Our calculations have guided us in two important ways. First, we know that experimentally we need to apply our electric fields for shorter times. For instance, for the 100 nm particles discussed above, we would need to apply E0 for roughly 1 s, not tens of seconds as we have commonly done in video microscopy. We had not appreciated this before in experiments. Second, for particles smaller than 1000 nm, we see that video microscopy yields an unacceptable accuracy. We have recently developed a light-scattering method for observing doublet breaking that will provide a way around this limitation. Essentially, when a doublet breaks into two singlets, the light-scattering signal drops. Because in light scattering we might observe 105 doublets simultaneously, we would see (even for secondary energy minima wells as low as kT) that many of the doublets will be close to their equilibrium gap. That is, we might expect that even for low Pec, if we were to average over N doublets simultaneously, fewer would be broken at any given time because by a Boltzmann distribution many doublets would be in a secondary minimum. For nanocolloids of size O(10 nm), even the lightscattering method might be limited in differential electrophoresis. This is because as the doublets break, whether by electrophoresis or by Brownian motion, they can reaggregate into either homodoublets or heterodoublets. Because the homodoublets are no longer subject to breaking by differential electrophoresis, a diminishing number of heterodoublets would be available to break, down to the limit where Brownian motion is breaking the doublets apart. Acknowledgment. We thank the Environmental Protection Agency (EPA STAR grant no. R-82960501), the Ben Franklin Center for Excellence in Nanoparticle Research, and the NSF NIRT (grant CCR-0303976) for financial support of this research. Also, G.L.H. thanks Mr. and Mrs. William and Josephine Weiss for Dissertation Year funding and the Pennsylvania Space Grant Consortium for a fellowship. D.V. thanks Bob Rowell for his continual encouragement and support over a number of years. Professor Rowell’s curiosity and excitement for new ideas has always been an inspiration.
Limitations of Differential Electrophoresis
Langmuir, Vol. 21, No. 22, 2005 10081
Appendix A number of dimensionless coefficients were mentioned in the introductory sections of this paper. Their numerical derivations will be discussed here. See Figure 11 for a plot of these coefficients as a function of λ ) δ/2a. Expressions for khyd can be found in the book by Kim and Karrila14 as
{
khyd(λ) )
[
[1.2904 + 0.2326λ]
λ2(0.01248 ln(1/2λ) + 0.00676) + λ(0.12146 ln(1/2λ) +
[
0.1565) + 0.29034 ln(1/2λ) + 0.1613/λ + 0.89726
]
]
2 - 6γ + 18γ2 - 38γ3 + 186γ4 - 774γ5 + 2394γ6 10662γ7 + 39642γ8 - 152 230γ9 + 640 346γ10 2 356 902γ11 2 4 2 + 9γ + 153γ + 1385γ6 + 23145γ8 + 410 729γ10 3 206 641γ12 - 255 110 057γ14 - 564 293 001γ16 5 665 839 721γ18 - 76 884 845 801γ20 1 388 746 759 401γ18
[
]
λ < 0.26
λ g 0.26
(33)
where λ ) δ/2a and γ ) 1/[4(1+λ)]. This two-region fit uses lubrication hydrodynamics for the near-field result and reflection calculations for the far-field result. Matching of the near- and far-field results occurs at λ ) 0.26. The equation is accurate to within 0.02% of known, tabulated values for khyd(δ). Results for electrophoresis are given by
{
(1 - 0.21887λ) λ < 4.5 (2 + 22.589λ) kep(λ) ) 1 - 2(2 + 2λ)-3 - 13(2 + 2λ)-6 λ g 4.5 (34) 1-2
The near-field case was obtained using numerical calculations, and the far-field case was obtained from O(L-7) reflection results that we used for kep(δ).17,18 We then used Excel SOLVER to develop an empirical expression for N in terms of λ ) δ/2a ) x/2R, with β ) a2/a1 ) 1:
N(λ) )
{
-0.204704
0.258λ
1 + (2 + 2λ)
-3
+ 0.71737 1 + (2 + 2λ)-6 λ > 0.20 2
λ e 0.20
Figure 11. Dependence of various hydrodynamic and electrophoresis parameters on the dimensionless gap (λ ) δ/2a).
This empirical expression yields 1% error as compared to known, tabulated values for N. We see that N approaches 1 as λ (and therefore δ) becomes large. Both the near-field and far-field equations for Kr exist and can be found in the book by van de Ven.20 Therefore, we have the following results for Kr as a function of λ ) δ/2a:
{
Kr(λ) ) 8(1 + λ)2(1 - 1.35/ln(2λ))
λ e 0.22
(0.2673 - 0.724/ln(2λ)) 2
3
1
24(1 + λ) (2 - /4(1 + λ) - /8(1 + λ)3)
-1
(36)
λ > 0.22
Matching of the near- and far-field results occurs at λ ) 0.22.
(35) LA0505566
