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Analytical Expressions for Calculating the Depletion Interaction Produced by Charged Spheres and Spheroids Martin Piech and John Y. Walz* Department of Chemical Engineering, Yale University, P.O. Box 208286, New Haven, Connecticut 06520 Received May 31, 2000. In Final Form: August 15, 2000 Analytical expressions for predicting the depletion interaction in charged systems where long-range electrostatic effects are important are presented. The approach used was to model the nonadsorbing materials as hard particles but with an effective size representing the hard-core size and the effective size of the charged double layer. This effective size was then substituted into analytical expressions for predicting the depletion interaction between two hard spherical particles in solutions of nonadsorbing hard spheres or spheroids. We also found the effective thickness of the charged double layer providing the most accurate approximation to be the surface-to-surface separation distance between the particle and nonadsorbing species at which the electrostatic repulsion equals 0.5 kT.
Introduction The depletion interaction between two colloidal particles, which arises whenever a smaller, nonadsorbing material is present, was first successfully explained by Asakura and Oosawa in 1954.1 Since that time, a substantial amount of theoretical and experimental research has been performed toward understanding this interaction in a variety of colloidal systems.2,3 In 1994, Walz and Sharma4 developed a model for calculating the depletion interaction between two charged, spherical particles immersed in a solution of like-charged, nonadsorbing spherical macromolecules. The model calculates the force exerted on one of the particles by integrating the individual particle/macromolecule interaction force over an equilibrium distribution of macromolecules. When the particles and macromolecules were treated as simple hard spheres, the Walz and Sharma model was shown to match the well-known Asakura and Oosawa results. As expected, the presence of charge significantly increased the magnitude and range of the depletion interaction. In addition, interaction energy profiles, measured with the technique of total internal reflection microscopy,5 and dispersion stability studies,6 all performed with charged particles and macromolecules, supported the results of this force balance model. Recently, Piech and Walz7 extended this model to calculate the depletion interaction produced by nonadsorbing charged spheroidal macromolecules. One of the drawbacks of the Walz and Sharma approach is that numerically integrating the electrostatic force expression over an equilibrium configuration of macromolecules is cumbersome and relatively time-consuming. This fact has motivated us to develop analytical expressions that can be used to approximate the depletion interaction in systems where long-range electrostatic effects are important. The approach is to determine an effective size of the nonadsorbing species, representing both the hard-core size plus the size of the charged double (1) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (2) Seebergh, J. E.; Berg, J. C. Langmuir 1994, 10, 454. (3) Jenkins, P.; Snowden, M. Adv. Colloid Interface Sci. 1996, 68, 57. (4) Walz, J. Y.; Sharma, A. J. Colloid Interface Sci. 1994, 168, 485. (5) Sharma, A.; Walz, J. Y. J. Chem. Soc., Faraday Trans. 1996, 92, 4997. (6) Sharma, A.; Tan, S.; Walz, J. Y. J. Colloid Interface Sci. 1997, 190, 392. (7) Piech and Walz J. Colloid Interface Sci. Submitted.
layer. We show here that such an approach can successfully match the rigorously calculated interactions for both spherical and spheroidal macromolecules. Theory Force Balance Model. Spherical Macromolecules. We first provide an overview of the Walz and Sharma model. A schematic of the system is depicted in Figure 1a, where two spherical particles of radius R, separated by gap width h, are immersed in a solution of nonadsorbing spherical macromolecules at bulk number density Fb. The approach assumes that both the surfaces of the particles and macromolecules are hard and continuous (i.e., nonporous) with charge groups distributed uniformly over the surfaces. Summing the individual particle-macromolecule interactions over the entire system leads to the following expression for the force exerted on particle 1, F1(h)
F1(h) )
∫xFb exp{-
}
E(x) ∇1E1(x) dx kT
(1)
where E(x) denotes the total interaction energy between a macromolecule at position x with particles 1 and 2, and ∇1E1(x) is the gradient of the interaction energy with respect to the surface of particle 1 (i.e., the force of interaction between the particle and macromolecule). The total interaction energy, E(x), is calculated as the sum of the interaction energies with particles 1 and 2. Once the depletion force is known, the depletion energy can be calculated using
EDep(h) ) -
∫∞h F1,C-C(h′) dh′
(2)
where F1,C-C is the component of the force F1 acting along the line of centers between the two particles and h′ is a dummy integration variable. In simple, uncharged systems, E(x) is frequently modeled as a hard sphere interaction of the form8,9,10 (8) Henderson, D. J. Colloid Interface Sci. 1988, 121, 486. (9) Mao, Y.; Cates, M. E.; Lekkerkerker, H. N. W. Physica A 1995, 222, 10. (10) Biben, T.; Bladon, P.; Frenkel, D. J. Phys.: Condens. Matter 1996, 8, 10799.
10.1021/la000764s CCC: $19.00 © 2000 American Chemical Society Published on Web 09/22/2000
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molecule interaction depends on both the distance of separation, x, as well as the macromolecule orientation, Ω (see Figure 1b). Piech and Walz7 give the following general expression for predicting the depletion force in such systems
F1(h) )
∫Ω ∫xFb exp{-
}
E(x,Ω) ∇1E1(x,Ω) dx dΩ (6) kT
where E(x,Ω) denotes the total interaction energy between a macromolecule in a given orientation with particles 1 and 2. Again, the depletion energy can be calculated by integrating this force, as given in eq 2. In purely hard systems, E(x,Ω) is given by
E(x,Ω) ) EHS(x,Ω) ) +∞ for particle-macromolecule overlap (7) 0 otherwise
{
while in charged systems, the total interaction energy between a particle and macromolecule will be
E(x,Ω) ) EHS(x,Ω) + EElec(x,Ω) Figure 1. 1. Schematic defining the variables used in the depletion force equation. Two spherical particles of radius R are interacting across gap width h in a solution of macromolecules at bulk concentration Fb. The vector x defines the position of a macromolecule relative to the center of particle 1. (a) The macromolecules are charged spheres of hard radius a and effective radius aeff. (b) The macromolecules are charged spheroids of hard dimensions R (semimajor axis) and β (semiminor axis) and effective dimensions Reff and βeff.
E(x) ) EHS(x) )
{
+∞ for |x| < R + a 0 for |x| g R + a
(3)
where a represents the macromolecule radius. When the particles and macromolecules are charged, the expression for E(x) can be written as
E(x) ) EHS(x) + EElec(x)
(4)
where EElec(x) is the electrostatic free energy of interaction between a colloidal particle and a macromolecule at position x. An appropriate expression for the electrostatic interaction between two spheres is the linear superposition approximation (LSA) described by Bell et al.11
(kTe)
EElec(x) ) r0
2
(ax) exp[-κ(x - R - a)]
YpartYmacroR
(5) where x is the center-to-center distance between the two spheres, Ypart and Ymacro represent the dimensionless effective surface potentials of the particle and macromolecule, respectively, r is the relative dielectric constant of the medium, 0 is the permittivity of free space, e is the charge of a proton, and κ-1 is the Debye length. Explicit expressions for calculating the effective surface potentials can be found in Bell et al. The effective surface potentials are bounded such that 4 tanh(ψ0e/4kT) < Y < ψ0e/kT, where ψ0 denotes the actual surface potential. For particles with radius of curvature much greater than the Debye length in a 1:1 electrolyte solution, Y ≈ 4 tanh(ψ0e/4kT). Spheroidal Macromolecules. When the nonadsorbing macromolecules are nonspherical, the particle-macro(11) Bell, G. M.; Levine, S.; McCartney, L. N. J. Colloid Interface Sci. 1970, 33, 335.
(8)
Piech and Walz assumed that the macromolecules could be modeled as general spheroids (prolate or oblate) and calculated this electrostatic interaction by integrating the total stress tensor over a midpoint plane between the spherical particle and macromolecule. The potential distribution around the particle was calculated using a solution to the Poisson-Boltzmann equation in spherical coordinates given by Bell et al.,11 while the potential around the spheroid was calculated using a two-term perturbation expansion described by Hsu and Liu.12 Hard System Approximation. As mentioned earlier, our goal here is to develop analytical expressions that can accurately approximate the depletion interaction in charged systems and thus eliminate the need for the cumbersome integrals required by eqs 1 and 6. The approach used was to determine if the interactions in charged systems predicted with the rigorous force-balance model (eqs 1 or 6) could be matched by treating the charged macromolecules as hard spheres or spheroids with an effective size in place of the actual size. This effective size would account for the actual hard sphere or spheroid size plus the thickness of the surrounding ion cloud next to each surface. Thus, for spherical macromolecules
aeff ) a + δ
(9)
where a is the physical macromolecule size and δ is an adjustable parameter reflecting the double layer thickness (on the order of the solution Debye length, κ-1). Similarly, for spheroidal macromolecules
Reff ) R + δ βeff ) β + δ
(10)
where R and β represent the spheroid semimajor and semiminor axes, respectively. (Note that for both the prolate and oblate spheroids, the semimajor axis refers here to the longest axis.) In the case of purely hard sphere systems, the depletion interaction energy between two spheres of radius R is given by the well-known Asakura and Oosawa potential1 (12) Hsu, J. P.; Liu, B. T. J. Colloid Interface Sci. 1997, 192, 481.
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ESS,Sphere(h) FbπRkTa2
)
{
h2 h3 4a h 2h - 2+ - 2 for h < 2a (11) a 3R R 12a2R 2a 0 for h g 2a
where Fb is the bulk number density of macromolecules. For the case of either prolate or oblate spheroidal macromolecules, Piech and Walz7 developed the following approximate expressions for the depletion interaction between two large spheres of radius R
{
ESS,Prolate(h)
)
FbπRkTR2
h 2 4 β2 h2 + - R 3 2R2 3 R2
for h < 2β
hβ2 ln[((R2 - β2)1/2 + R)/β]
R2(R2 - β2)1/2 h 2 h2 4 β2 + - - 2R 3 2R 3 R2 2 2 2 2 1/2 (h + 8β )(h - 4β )
for 2β e h < 2R
+ 12R2(R2 - β2)1/2 hβ2 ln[2((R2 - β2)1/2 + R)/(h + (h2 - 4β2)1/2)] R2(R2 - β2)1/2
0
{
for h g 2R (12a)
ESS,Oblate(h) FbπRkTR2
) 2
h 2β 4 hβ - 2- + for h < 2β 2 2 3 3 R 2R R h π β - tan-1 2 (R2 - β2)1/2 2 (R - β2)1/2 2 2 2 2 1/2 h π (h + 8R )(4R - h ) - for 2β e h < 2R (R2 - β2)1/2 2 12R2 h h tan-1 (4R2 - h2)1/2 0 for h g 2R (12b)
[
[
(
])
])
Equations 12a and 12b were derived using the Derjaguin approximation to account for particle curvature and are thus valid only in the limit of R . R. The procedure used here was to determine if the interactions predicted using the rigorous force-balance model could be matched using eqs 11 (spherical macromolecules) or 12 (spheroidal macromolecules) by substituting the effective macromolecule sizes (aeff, Reff, βeff) for the actual sizes. The thickness of the ion cloud, δ, providing the best fit between the two methods was determined by minimizing a normalized root-mean-square deviation, srms, defined as
srms ≡
1
EElec(0), to facilitate comparisons of srms for different systems (this normalization does not affect the regressed value of δ). Results and Discussion
2
(
Figure 2. Comparison of the rigorous and analytical predictions of the depletion interaction between two 5 µm radius, charged, spherical particles in solution with charged, spherical macromolecules. The hard sizes of the macromolecules are either 10 or 30 nm, all surfaces (particles and macromolecules) are assumed to have constant surface potentials of -50 mV, and the solution Debye length is 10 nm.
{
EElec(0)
}
∫0L [EElec(h) - EHSHW(h)]2 dh L
1/2
(13)
where EElec(h) and EHSHW(h) are the energy profiles of the rigorous and analytical predictions, respectively, and L is the distance at which the depletion interaction (calculated rigorously) decays to 0.01% of its value at h ) 0. We normalize this deviation by the depletion energy at contact,
Spherical Macromolecules. The depletion interactions predicted using both the rigorous model and the analytical expression of eq 11 between two 5 µm radius spherical particles in a solution of spherical macromolecules are shown in Figure 2. The hard-sphere sizes of the macromolecules were either 10 or 30 nm (solid and broken curves, respectively), all surfaces (i.e., particles and macromolecules) had constant surface potentials of -50 mV, and the solution Debye length was kept at 10 nm (approximately equal to 1 mM of monovalent electrolyte). It is clear from this figure that using the hard sphere approximation with effective macromolecule radii of either 55.4 nm (for the 10 nm actual radius) or 85.8 nm (for the 30 nm actual radius) provides an accurate match to the rigorous results. The values of srms are 1.01% and 0.48% for the 10 and 30 nm macromolecule systems, respectively. The results shown in Figure 2 indicate that the depletion interaction in charged systems can be calculated using eq 11 provided the correct value of aeff is used. To make this approach more useful, we sought to develop some guidelines for determining the appropriate value of the effective thickness of the double-layer cloud, δ, needed to calculate aeff. We found that a good choice for δ is the particlemacromolecule surface-to-surface separation distance at which the interaction energy drops to 0.5 kT. For example, the values of δ given in Figure 2 correspond to particlemacromolecule electrostatic energies (calculated using eq 5) equal to 0.50 kT and 0.52 kT for the 10 and 30 nm macromolecule cases, respectively. This value of 0.5 kT is not completely surprising, since the characteristic energy for one component of the three-dimensional translation Brownian motion is 1/2 kT.13 (13) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989; p 67.
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Table 1. Values of δ and srms for Various System Parameters (Spherical Macromolecules) R (nm)
a (nm)
ψ0 (mV)
κ-1 (nm)
δ at 0.5 kT (nm)
srms (%)
100 5000 5000 5000 5000 5000
30 30 100 10 30 30
50 50 50 50 25 50
10 10 10 10 10 3
50.6 56.2 68.0 45.5 43.3 16.8
0.77 0.71 0.55 1.01 0.78 0.57
To further illustrate the accuracy of this 0.5 kT criterion, we present in Table 1 comparisons between the rigorous and analytical approaches for various system parameters for the spherical macromolecule system. In each case, the value of δ was determined as the distance, x, at which the electrostatic energy in eq 5 was equal to 0.5 kT. The approach is clearly quite good, as the maximum value of srms here is only 1%. In addition, there are several trends in these results that should be mentioned. 1. The accuracy of the approach improves as κa increases. This is as expected, as electrostatic effects become less important at larger κa values (i.e., the ratio a/aeff approaches 1 as κa f ∞). 2. For a given κ-1 value, δ decreases with decreasing macromolecule radius, a. This is consistent with the faster decay of electric potential away from surfaces of increasing curvature. For example, in the Debye-Hu¨ckel regime (i.e., low surface potentials), the potential around a spherical particle of radius a decays as (a/r) exp(-κr), where r is the distance from the sphere’s center.14 3. The value of δ drops when the surface potential is reduced, which is as expected. The exact dependence of δ on the various systems parameters can be obtained directly from eq 5 by substituting δ + R + a for x and setting the energy equal to 0.5 kT. This yields
e2 a (14) exp(-κδ) ) R+a+δ 2r0kTYpartYmacroR
(
)
which can be solved numerically for a unique value of δ. Finally, as can be seen from the data in Table 1, a very crude estimate for the value of δ would be approximately 5κ-1. Recently, Odiachi and Prieve used the experimental technique of total internal reflection microscopy to measure the depletion interaction between a colloidal sphere and a hard plate in solutions of charged nanometer-sized Laponite clay particles.15 The measured energy profiles were compared to a relatively simple model in which the depletion interaction was calculated as the osmotic pressure of the bulk solution and the area of overlap of the depletion layers. The depletion layer thickness, or “effective radius” of the platelike clay particles, was treated as an adjustable parameter, equal to the known radius of the Laponite particles plus a multiple, m, of the solution Debye length. For Laponite concentrations of 100, 200, and 300 ppm, the values of m were found to be 4.62, 4.73, and 4.73, respectively, which is consistent with the value of δ equal to approximately 5κ-1 calculated from the results in Table 1. Spheroidal Macromolecules. The depletion interactions between two 5 µm radius spherical colloidal particles in solution with prolate spheroidal macromolecules, computed using the rigorous and analytical models, are shown in Figure 3. For the numerical calculations, the (14) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1987; Vol. I, p 389. (15) Odiachi, P. C.; Prieve, D. C. Colloids Surf., A 1999, 146, 315.
Figure 3. Comparison of the rigorous and analytical predictions of the depletion interaction between two 5 µm radius, charged, spherical particles in solution with charged, prolate spheroids. The hard semimajor axis length of the macromolecules is fixed at 100 nm, while three different values of the semiminor axis length are considered. Again, all surfaces are assumed to have constant surface potentials of -50 mV and the solution Debye length is 10 nm.
macromolecule semimajor axis length was fixed at 100 nm, while the semiminor axes lengths were 50 nm (R/β ) 2), 20 nm (R/β ) 5), and 10 nm (R/β ) 10) for the three cases considered. As before, all surfaces were assumed to have constant surface potentials of -50 mV, while the solution Debye length was 10 nm. For the analytical calculations, the effective macromolecule dimensions were calculated according to eq 10. The parameter δ was determined as the particle-macromolecule surfacesurface separation distance at which the magnitude of the electrostatic interaction between the spheroid and the sphere equaled 0.5 kT for the orientation where the spheroid semimajor axis is perpendicular to a line connecting the centers of the sphere and spheroid. Piech and Walz7 showed that this will be the most probable orientation for the spheroid interacting with the sphere. The values of srms for macromolecules with aspect ratios of 2, 5, and 10 are 0.31%, 0.84%, and 0.22%, respectively, again showing that the simple hard sphere model can be used to predict the depletion interaction in these systems. The results in Figure 3 also indicate that as the value of β decreases (i.e., the prolate spheroid becomes thinner), the value of δ decreases. This trend is consistent with the behavior observed with spherical macromolecules (i.e., δ decreases with decreasing a) and arises from the faster decay of the electric potential away from more highly curved surfaces. Finally, it should be mentioned that we also investigated using two different values of δ for the case of spheroidal macromoleculessone for the semimajor axis and one for the semiminor axis. The value of δ added to the semiminor axis length was the same as that described above, while the value added to the semimajor axis length was calculated as the separation distance at which the particle-macromolecule interaction energy decreased to 0.5 kT when the semiminor axis is perpendicular to a line connecting the centers of the sphere and spheroid (i.e., rotated 90° from the orientation described above). For R/β values ranging from 2 to 10, the matches to the more
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rigorous calculations, as characterized by the srms values, were not significantly improved compared to the model using only a single value of δ. This finding is consistent with the prediction of Piech and Walz7 that the preferred orientation of the spheroid around the sphere is one in which the semimajor axis is perpendicular to the line of centers. Conclusions The use of effective dimensions to account for the effect of charge was found to provide an accurate and straightforward method for predicting the depletion interaction in charged systems. We should caution, however, that extending this approach to more detailed calculations of the impact of a nonadsorbing species on the interparticle interaction must be done carefully. For example, Walz and Sharma4 also considered the effect of electrostatic interactions between the macromolecules in solution on the depletion interaction through the use of a second virial coefficient. It was shown that these second-order effects could actually give rise to a longer-range repulsion between
the particles. Because this second virial coefficient depends on the simultaneous interaction of two macromolecules with the particles, the integrations required in charged systems are particularly cumbersome. Although the use of effective macromolecule sizes would greatly simplify the calculations, the procedure is a bit trickier, since the effective size for the macromolecule-macromolecule interaction will most likely be different from that of the particle-macromolecule interaction. Specifically, because of the different sizes of the particles and macromolecules, the distance at which the electrostatic interaction decays to 0.5 kT will be different for the different interaction pairs. Nonetheless, such a model would provide a powerful and relatively easy-to-use tool for understanding colloidal interactions in these relatively complex binary systems. Acknowledgment. Support for this work was provided by the National Science Foundation, through Grant CTS-9702773, and by the Petroleum Research Fund, administered by the American Chemical Society. LA000764S
