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Electrical Double Layer Interaction between Dissimilar Spherical Colloidal Particles and between a Sphere and a Plate: Nonlinear Poisson-Boltzmann Theory Jim Stankovich and Steven L. Carnie* Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia Received May 4, 1995. In Final Form: November 6, 1995X Recently the double layer forces and interaction free energies between spherical colloidal particles, according to the Poisson-Boltzmann equation, have been accurately calculated by us and others. In this paper we conclude our investigations in this area by extending the calculation of double layer interactions to unequal spherical particles. We also consider the case of a sphere and a plate, a geometry relevant to atomic force microscope measurements when a colloidal particle is attached to the cantilever tip. We compare the results against those of the linearized Poisson-Boltzmann equation and the Deryaguin approximation over a wide range of conditions. The Deryaguin approximation is found to be highly accurate for the force in sphere/plate geometry and acceptably so for dissimilar spheres under typical colloidal conditions. We conclude with comments on the double layer forces acting on a bare atomic force microscope tip.
1. Introduction With the advent of atomic force microscope (AFM) experiments1 and detailed kinetic modeling,2 it is necessary to assess the accuracy of common approximations made in the classical DLVO theory of colloidal stability.3,4 In particular, the assumption of thin double layers is increasingly being violated in experimental systems. This process of checking the validity of classical theory has been going on since the early days of digital computing.5 With current computing resources, it is much easier to obtain accurate numerical results for the force and interaction free energy between two uniform spherical particles, i.e., for cases with nice geometry. Two recent contributions in this area are the calculation of double layer forces between identical spherical particles using a spline collocation scheme6 and the calculation of interaction free energies for spherical particles of equal size by orthogonal collocation.7 Both these papers solve the nonlinear boundary value problem arising from the Poisson-Boltzmann equation, together with boundary conditions on the surface of the particles. In this paper we employ similar methods to those of our previous work6 to study double layer interactions between dissimilar spherical particlessappropriate for heterocoagulationsand between a sphere and a platesa geometry relevant to atomic force microscope measurements when a colloidal particle is attached to the cantilever tip.1 Although the computational burden of these calculations is only moderate, it is useful to compare with two common approximations. If the potential throughout the double * To whom correspondence should be addressed. E-mail:
[email protected]. X Abstract published in Advance ACS Abstracts, February 1, 1996. (1) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239. (2) Kim, S.; Zukowski, C. F. J. Colloid Interface Sci. 1990, 139, 198. (3) Russell, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (4) Verwey, E. J. W.; Overbeek, J. Th. Theory of the Stability of Lyophobic Colloids; Elsevier: New York, 1948. (5) Hoskins, N. E.; Levine, S. Philos. Trans. R. Soc. London 1956, 248, 433, 449. (6) Carnie, S. L.; Chan, D. Y. C.; Stankovich, J. J. Colloid Interface Sci. 1994, 165, 116. (7) Palkar, S. A.; Lenhoff, A. M. J. Colloid Interface Sci. 1994, 165, 177.
layer is small compared to kT/e, about 25 mV at room temperature, the Poisson-Boltzmann equation can be linearized. The resultant linear boundary value problem is a much easier problem to solve, and several boundary methods have been used.2,8,9 Such methods can produce double layer forces8 or interaction free energies10 in a few seconds on personal computers, as opposed to minutes on a workstation for the nonlinear problem. Thus it is desirable to know how large the region of applicability of the linear theory is. Results in ref 6 show that the linear theory gives accurate results for the force between two identical spherical colloidal particles at potentials of up to 40 mV. Carnie et al.11 calculated the force and interaction free energy between dissimilar spheres and between a sphere and a plate using the linear theory. One aim of the current paper is to perform similar calculations based on the nonlinear theory to determine the range of potentials at which these linear calculations are accurate. The validity of the Deryaguin approximation,3 which was designed for particles with thin double layers and constant potential boundary conditions, will also be assessed. Since it relates interactions between spheres to interactions between parallel plates, it reduces the calculation to a one-dimensional boundary value problem, which again can be solved much faster than the full nonlinear problem.12 Its accuracy has already been tested for identical particles6 and for a special case of dissimilar particles.11 The linearized version, which produces an analytic formula13,14 has been extensively tested against the linearized Poisson-Boltzmann equation.11 We will only consider constant potential and constant charge boundary conditions, with the dielectric constant of the particles assumed to be zero in the constant charge case. These two cases represent the two extremes of a system’s possible behavior. Other recent papers have (8) Glendinning, A. B.; Russel, W. B. J. Colloid Interface Sci. 1983, 83, 95. (9) Yoon, B. J.; Lenhoff, A. M. J. Comput. Chem. 1990, 11, 1080. (10) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 155, 297. (11) Carnie, S. L.; Chan, D. Y. C.; Gunning, J. S. Langmuir 1994, 10, 2993. (12) McCormack, D.; Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1995, 169, 177. (13) Hogg, R.; Healy, T. W.; Fuerstenau, D. W. Trans. Faraday Soc. 1966, 62, 1638. (14) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 161, 260.
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greatersthe double layers are thin compared to the particle size. This is the most common case experimentally, and the classical theory makes use of this fact. As techniques for producing submicron particles improve, systems with κa approaching 1 are becoming increasingly common. The Poisson-Boltzmann equation
∇2ψ ) sinh ψ
Figure 1. Geometric parameters for two dissimilar spheres. Table 1. Scales Used for Length, Potential, Surface Charge Density, Energy, and Force in the Text, Together with Typical Values for 1 mM 1:1 Electrolyte at 25 °C in Watera scale ) x(0 potential kT/e surface charge density 0κ(kT/e) energy 0κ-1(kT/e)2 length
force
κ-1
0(kT/e)2
kT)/(2ne2)
(1)
for the dimensionless electrostatic potential ψ (expressed as a function of dimensionless position, with the scale being the Debye length) in the region E external to the particles is to be solved numerically. Suppose that the surface ∂Sj of each particle has uniform potential ψj and uniform (dimensionless) surface charge density σj ) σj(ψj,κaj) when it is in isolation. Since we are assuming that the particles are spheres or infinite plates, a particle with a uniform surface potential will have a uniform surface charge density and vice versa. Boundary conditions to (1) are imposed by assuming that the potential on each surface remains constant
typical value 2.57 ×10-2 V 1.85 ×10-3 C/m2 4.41 ×10-21 J ) 1.07kT 4.58 ×10-13 N
a Here is the dielectric constant of the solvent, is the 0 permittivity of free space, k is the Boltzmann constant, T is the absolute temperature, n is the ion density in bulk solution, and e is the charge on an electron.
discussed the effect of a nonzero dielectric constant10 and more complicated charge regulation boundary conditions.6 Assuming that the particles have zero dielectric constant means that it is not necessary to solve for the potential within the particles. Although strictly not physical, it appears to be a good approximation for low dielectric particles in a polar solvent, at least for separations of relevance to colloidal stability.10 The methods we use have largely been described elsewhere,6 so we concentrate on those features that require special treatment in the present work. In comparing with linearized Poisson-Boltzmann results, we freely use results from an earlier paper.11 Although this makes the paper less self-contained, it means we can focus on the new results, which turn out to confirm the intuition developed from our previous work. 2. Method of Calculation Consider two spherical colloidal particles S1 and S2, with radii a1 and a2, respectively and smallest distance of separation h, surrounded by electrolyte E (Figure 1). If aj ) ∞, then particle j is an infinite plate. In what follows, potential is scaled by kT/e, distances by the Debye length κ-1, and force and free energy as in ref 11. The Debye length sets the scale of the double layer interaction, so it is a natural length scale for the problem. The other geometric quantities then become dimensionless radii κa1 and κa2 and a dimensionless separation κh. For convenience, we collect the scales used to produce dimensionless quantities in Table 1 and give some typical values. Double layer interactions are important when they are larger than other interactions, in particular thermal interactions (kT ); this typically occurs for κh in the range 0.3-2. At smaller separations they are dominated by other forces; at larger separations they are quite weak and well described by a superposition approximation.6 If the dimensionless radii, κa1 and κa2, are largessay, 10 or
(2)
ψ|∂Sj ) ψj
9.63 ×10-9 m
or that the surface charge density remains constant
∇ψ‚n|∂Sj ) σj
(3)
(where n is the unit normal directed into the particles and we have used a dimensionless surface charge density σj which has absorbed the dielectric constant of the solvent) as the particles approach one another. As in ref 6, bispherical coordinates, Newton-Raphson iteration, and collocation with bicubic Hermite basis functions were employed to solve this boundary value problem. Our domain E corresponds to the rectangular domain
η2 e η e η1 and 0 e θ e π
(4)
in bispherical coordinates (η, θ), where the parameters η1, η2, and b are related to κa1, κa2, and κh via the equations:
b ) κa1 sinh η1 -
b ) κa2 sinh η2
b b ) κa1 + κa2 + κh tanh η1 tanh η2
(5) (6) (7)
Notice that in the sphere-plate case, if κa2 ) ∞, then η2 ) 0 and η1 and b are given by
b ) κa1 sinh η1
(8)
b ) κa1 + κh tanh η1
(9)
Rather than solving for the potential directly, more accurate results are obtained by solving for the difference
u ) φ1 + φ2 - ψ
(10)
between the potential φ1+ φ2 given by the superposition approximation (where φj is the potential surrounding particle j in isolation) and the true potential ψ. This
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approach was also used by Ledbetter et al.15 The potential φj surrounding a single sphere was calculated numerically, while the potential φ2(z) at a scaled distance z from an isolated plate with surface potential ψ2 is
(
φ2 ) 4 arctanh e-z tanh
)
ψ2 4
∫0
F ) 2π
{[
The interaction free energy U is
1 ∂ψ 2 ∂ψ + 2 2 ∂θ ∂η (cosh η - cos θ)
2
}
∂ψ ∂ψ sin θ sinh η sin θ dθ (12) ∂η ∂θ cosh η -cos θ For the sphere-plate case, this integral was evaluated over the surface η ) η1/2; in all other cases it was evaluated over the surface η ) 0 (more accurate results were obtained integrating over a surface between the particles than over one of the particles). A 23 × 23 grid sufficed to calculate the force with four-figure accuracy for κh g 0.1. This grid was coarser than the 29 × 29 grid used in ref 6; gradients in u tend to be less steep than gradients in ψ so that u is easier to approximate using the basis functions of the collocation method. 2.2. Calculation of Interaction Free Energy. The free energy of interaction was calculated in two ways: (1) using an expression for free energy involving volume and surface integrals of potential and (2) integrating force with respect to separation. Note that only if we are content with the total interaction free energy, and not the various thermodynamic contributions as in ref 7, do we have the option of integrating the force. Otherwise we must either perform the volume integrals below or carry out a charging procedure coupled with surface integrals.16 In this sense, the free energy is actually harder to calculate than the force for the nonlinear problem, the reverse of the situation in the linear case.10 2.2.1. Integration of Potential. The (dimensionless) free energy H of our system is
[
]
∫E 2|∇ψ|2 + ψ sinh ψ - (cosh ψ - 1) dV ∑ψj∫∂S ∇ψ‚n dA 1
cp
j
(15)
U ) H - H∞
H∞ )
[| | [| | 1
∫E∪S
2
∫E∪S
1
2 1 2
] ]
∇φ1 2 + φ1 sinh φ1 - (cosh φ1 - 1) dV + ∇φ2 2 + φ2 sinh φ2 - (cosh φ2 - 1) dV ψj∫∂S ∇φj‚n dA ∑ cp j
[1 - cosh η cos θ] +
H)
(14)
where
(( ) ( ) )] ×
b2(cosh ψ - 1)
∫E[- 21|∇ψ|2 - (cosh ψ - 1)] dV
(11)
2.1. Calculation of Force. The dimensionless force F between the two particles is given by integrating the stress tensor over a surface separating the two particles. Evaluating this integral over a surface of constant η, we have π
H)
(16)
is the free energy of the particles in isolation. Substituting (13) and (16) into (15) gives
U)
∫E[21(|∇ψ|2 - |∇φ1|2 - |∇φ2|2) + (ψ sinh ψ - φ1 sinh φ1 - φ2 sinh φ2)
-(cosh ψ - cosh φ1 - cosh φ2 + 1)] dV 1 - ∫S ∇φ1 2 + φ1 sinh φ1 - (cosh φ1 - 1) dV 2 2 1 - ∫S ∇φ2 2 + φ2 sinh φ2 - (cosh φ2 - 1) dV 1 2
[| | [| |
- ∑ψj∫∂S (∇ψ - ∇φj)‚n dA cp
j
] ]
(17)
Evaluating the volume integral over the electrolyte external to the particles and the surface integrals in (17) directly can result in large cancellation errors. (Note that, in the sphere-plate case, direct evaluation is impossible because both H and H∞ are infinite.) To illustrate the problem, consider an interaction between two large spheres. The potential behind the first sphere is virtually unchanged by the presence of the second sphere, so ψ and φ1 are almost equal there and the pair of large terms |∇ψ|2 and |∇φ1|2 almost cancel one another, while |∇φ2|2 is small. This cancellation can be avoided by noting that
|∇ψ|2 - |∇φ1|2 - |∇φ2|2 ) |∇u|2 - 2∇u‚∇(φ1 + φ2) + 2∇φ1‚∇φ2 (18) (13)
where cp denotes that the surface integral over ∂Sj is only included if particle j is subject to constant potential boundary conditions. Overbeek16 derived this result by summing various thermodynamic components of the free energy, and Reiner and Radke17 identified this expression for H as the functional which is minimized when (1) is satisfied, as did Sharp and Honig18 in the biophysical context. In the case when the particles interact under constant potential conditions, (13) can be simplified through the divergence theorem and (1) to give: (15) Ledbetter, J. E.; Croxton, T. L.; McQuarrie, D. A. Can. J. Chem. 1981, 59, 1860. (16) Overbeek, J. Th. G. Colloids Surf. 1990, 51, 61. (17) Reiner, E. S.; Radke, C. J. J. Chem. Soc., Faraday Trans. 1990, 86, 3901. (18) Sharp, K. A.; Honig, B. J. Phys. Chem. 1990, 94, 7684.
Because u and φ2 are small behind the first sphere, all the terms on the right-hand side are much smaller there than |∇ψ|2. Hence it is preferable to solve for u and substitute (18) into the expression (17) for U. The other components of (17) involving significant cancellation error can similarly be expressed in terms of u. Palkar and Lenhoff7 employed a 50 × 50 collocation grid to evaluate energies to four-figure accuracy. Perhaps because we used a spline collocation algorithm rather than spectral collocation, we found that a 50 × 50 grid (requiring around 30 min of CPU time on a workstation to solve (1) for one set of boundary conditions) was insufficient for four-figure accuracy in some cases, particularly for some sphere-plate interactions at low separations. It was found that energies could be calculated more efficiently at low separations by integrating force (calculated on a much coarser 23 × 23 grid) with respect to separation. 2.2.2. Integration of Force. To integrate the force F(κh) and calculate U(κh) for 0.1 e κh e 2 a polynomial
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Figure 2. The interaction free energy between two equal-sized spheres, scaled by ψ21, for κa ) 10 at various surface potential ratios. The surface potential of sphere 1, ψ1, ranges from 0swhere the linear theory holds exactlysto 2. Results are shown for (a) constant potential interaction with ψ2/ψ1 ) 3, (b) constant charge interaction with ψ2/ψ1 ) 3, (c) constant potential interaction with ψ2/ψ1 ) -3, and (d) constant charge interaction with ψ2/ψ1 ) -3. N
1
∑ ckTk-1(u)] - 2 c1 k)1
p(u) ) [
(19)
(where Tn(u) ) cos(n arccos u) is the Chebyshev polynomial of degree n)19 in u ) log(κh) was fitted to
(κh)F(κh) ) euf(eu)
(20)
for log 0.1 e u e log 2. This fit was motivated by the fact that the linear Deryaguin approximation of Hogg, Healy, and Fuerstenau13 for the two-sphere (or sphere-plate) force is of order 1/(κh) for small κh, suggesting that (κh) F(κh) should be well approximated by a polynomial. Some testing showed that an accurate fit is obtained with a polynomial p(u) of degree 11 (requiring calculation of the force at 12 separations). Then the interaction free energy at separation κh is given by
U(κh) ) U(2) +
∫κhF(κh) d(κh) 2
) U(2) +
∫
≈ U(2) +
∫log(κh)p(u) du
were made with calculations of the force and interaction free energy on the basis of the following: 1. The nonlinear Deryaguin approximation. Here the force FD(κh) between the two particles is expressed in terms of the nonlinear interaction free energy Up(κh) between two plates at the same separation subject to the same boundary conditions
FD(κh) )
2π(κa1)(κa2) Up(κh) κa1 + κa2
(23)
and the interaction free energy UD is expressed as an integral of Up with respect to separation
UD(κh) )
2π(κa1)(κa2) κa1 + κa2
∫κh∞Up(κh) d(κh)
(24)
(21) In the sphere-plate case we multiply by the coefficient
log 2
euf(eu) du log(κh) log 2
(22)
U(2), the energy at κh ) 2, was calculated by integrating the potential (17), or by evaluating the integral ∫2∞ F(κh) d(κh) using 10-point Gauss-Laguerre quadrature. 3. Results Comparisons of the force F and interaction free energy U calculated from the solution of the nonlinear problem (19) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes, 2nd ed.; Cambridge University Press: Cambridge,1992.
lim κa2f∞
2π(κa1)(κa2) κa1 + κa2
) 2πκa1
(25)
The numerical code described in ref 12 was used to calculate Up(κh). 2. The linear theory. Force Fl and interaction free energy Ul were calculated as described in ref 11. 3. The linear Deryaguin approximation (Hogg-HealyFuerstenau approximation). Closed form expressions for the force Fl,D and interaction free energy Ul,D are given in refs 11, 13, and 14. These results are derived by applying (23) and (24) to the linear interaction free energy between two plates.
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Figure 3. As in Figure 2 but for κa ) 1. Results are shown for (a) constant potential interaction with ψ2/ψ1 ) 3 and (b) constant charge interaction with ψ2/ψ1 ) 3.
Figure 4. As in Figure 2b but for (a) unequal-sized spheres, κa1 ) 10, κa2 ) 1, and (b) a sphere and a plate κa1 ) 10, κa2 ) ∞.
In our previous work, we have used several methods to demonstrate the accuracy of the various approximations. For identical particles, the interactions are always repulsive, so it is easy to construct relative error “maps”.6 For dissimilar particles, the interactions can change sign, so we resorted to simply plotting force and energy curves versus separation.11 Since the qualitative behavior of the force and energy curves is similar to those determined from the linearized Poisson-Boltzmann equation, we first concentrate on the effect of the nonlinearity on the interaction free energy curves, which are shown in Figure 2. Results for the force curves are similar. The results for the linear theory are as in Figures 17 and 18 of ref 11. They show U/ψ21 as a function of separation between two particles of radius κa ) 10. In each figure the ratio of potentials at infinite separation ψ2/ψ1 is held constant at either 3 or -3 as ψ1 varies between 0 (the linear theory) and 2; i.e., we are ranging from very low potentialsswhere the linear theory is accuratesup to dimensionless potentials of 2 and (6 keeping the ratio constant. If the linear theory held for all these potentials, there would be a single curve on each graph, so the spread between the curves is a measure of the effect of nonlinearity. We have shown four casessrepulsive double layers with ψ2/ψ1 ) 3 under constant potential (Figure 2a) and constant charge (Figure 2b) conditions and attractive double layers with ψ2/ψ1 ) -3 with the same boundary conditions (Figure 2c,d). Comparing the spread of curves in Figure 2a,c with Figure 2b,d, we see that for given values of ψ1 and ψ2 the linear theory is more accurate for the constant potential case than for the constant charge case, as is the case for identical particles.6 For example, in Figure 2a (constant potential) the linear energy at κh)1 is around 60% of its true value, while at the same point on Figure 2b (constant charge) the linear energy is roughly
40% of its true value. Also, the linear theory is more accurate when ψ1 and ψ2 are of opposite sign than when they have the same sign. In the worst case, when constant charge conditions apply and ψ1 and ψ2 are of the same sign, the absolute value of the potential on each particle increases as they approach, and the problem becomes highly nonlinear at small separations. However, when constant charge conditions apply and ψ1 and ψ2 are of opposite sign, the potential on one particle changes sign as the particles approach and remains of small enough magnitude to make the linear theory reasonably accurate. As seen in refs 6 and 11, the performance of the linearized theory improves for lower values of κa. This is seen in Figure 3, which shows results for repulsive double layerssψ2/ψ1 ) 3sbut with κa ) 1 for both spheres. The effect of curvature is seen again in Figure 4 where we show the energy curves in the worst casesconstant charge with ψ2/ψ1 ) 3sfor κa2 ) 1 (Figure 4a) and κa2 ) ∞ (Figure 4b) with κa1 held constant at 10. The linear theory is seen to improve markedly as κa is reduced for one of the particles. If that particle becomes a plate, on the other hand, relatively little change is seen, implying that at κa ) 10 the effect of nonlinearity is already at its maximum. We now proceed to assess the performance of the Deryaguin approximation, a task we could only do indirectly in ref 11. We start with sphere/plate geometry, where the existence of a plate naturally improves the validity of the Deryaguin approximation. We show force curves because of the relevance to AFM force measurements. In ref 6 we found that the Deryaguin approximation for identical spheres was more accurate for the constant potential case than for constant chargesas is the case for the linear Deryaguin approximation compared to the linear theory.11 In Figure 5 we show force curves
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Figure 5. The force between a sphere and a plate at constant charge with κas ) 10: (a) ψp/ψs ) 3 and (b) ψp/ψs ) -3. Results are shown for the Poisson-Boltzmann equation (solid line), linearized Poisson-Boltzmann equation (dotted line), Deryaguin approximation (solid symbols), and linear Deryaguin approximation (open symbols).
for a sphere and a plate with κa ) 10 and ψp/ψs ) (3 under constant charge conditions. The Poisson-Boltzmann result is compared to the three approximations mentioned above. Two features stand outsthe good performance of the Deryaguin approximation and the poor performance of the linear theory for ψp/ψs ) 3, i.e., where strong repulsions occur. We ascribe the accuracy of the Deryaguin approximation to three factors. 1. One surface is a plate, which naturally helps what is a low-curvature approximation. This effect was also seen in ref 11. 2. The large value of κa also improves the accuracy, as is well-known. 3. The performance is better at high surface potentials due to the larger gradients near the surface. For identical spheres, this effect is quantified and discussed in ref 6. From our previous work, we had surmised that the Deryaguin approximation would prove fairly accurate under these conditions; from Figure 5 it seems we were actually overly pessimistic. The Deryaguin approximation performs better than we might reasonably have expected. A more stringent test of the Deryaguin approximation comes from a lower value of κa; in Figure 6 we show corresponding curves for κa ) 1. The linear theory becomes more accurate, as expected, but the Deryaguin approximation is still remarkably good. The fact that one surface is flat has compensated for the relatively thick double layer around the sphere. From Figure 6a one can see that the comparison based on the linear theory is overly pessimistic; the same is true in Figure 6b, although here the Deryaguin approximation is qualitatively wrong at
Figure 6. As for Figure 5 but with κas ) 1.
small separations. It should be noted that the absolute errors in Figure 6a,b are similar; the relative error is much greater in the latter case because the force is much smaller. From Figures 5 and 6 we also see that the linear Deryaguin approximation is almost never very good for constant charge interactions; one of the other two approximations is always better. Sometimes it fails because the linear theory is inaccurate (Figure 5a), but sometimes it fails even when the linear theory is fairly good (Figure 6a). For constant potential interactions, the situation is much simpler and the tentative conclusions in ref 11 are borne out; the Deryaguin approximation (and, for these potentials, the linear Deryaguin approximation) is accurate for values of κa down to 1. Figure 7 shows force curves for κa ) 1; the results for κa ) 10 are almost described by a single curve. Again, the absolute errors of the various approximate curves in Figure 7a,b are similar. The accuracy of the Deryaguin approximation improves if the plate rather than the sphere has the lower surface potential, since the surface with the lower potential controls an interaction (see Figure 13 of ref 11). The improved performance of the Deryaguin approximation compared to that of the linear Deryaguin approximation is also observed for “mixed” boundary conditionssone surface having constant potential, the other constant chargesbut is not shown here. The other way to fully test the validity of the Deryaguin approximation is to have both values of κa small. In Figures 8 and 9 we show free energy curves between two equal-sized spheres with κa ) 10 and 1, respectively. The results where both spheres have thin double layers are striking; the Deryaguin approximation is again more accurate than we would have expected from ref 11 for both boundary conditions. With κa ) 1 in Figure 9 we would certainly not expect the Deryaguin approximation
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Figure 7. As for Figure 6 but at constant potential.
to be accurate. Even here, the Deryaguin approximation is a great improvement on the linear Deryaguin approximation. Of course, at these small values of κa, the linear theory itself is very good (see Figure 3 with ψ1 equal to 1). The major discrepancies seen in Figure 9 occur at large separations, because the Deryaguin approximation produces the wrong asymptotic behavior at large separations. This can be remedied in a heuristic manner by the replacement20
a1a2 a1a2 f a1 + a2 a1 + a2 + h
(26)
which has no effect for large κa or small separations but improves the curves considerably. To see this, in Figure 10 we show the Poisson-Boltzmann result, the Deryaguin approximation, and the Deryaguin approximation modified by the above replacement. The Deryaguin results for the force are substantially more accurate than the energy curves shown in Figures 8 and 9. 4. Discussion The numerical results presented above represent good news for experimental colloid scientists. They show that, for the case of a spherical colloidal particle attached to an AFM tip approaching a uniform surface, the Deryaguin approximation can be used reliably for interpretation of double layer force curves, as is done in ref 21, for example. Interaction free energy curves for two spherical particles can also be calculated reliably with the Deryaguin approximation (with some corrections at large separations if required) over a wide range of boundary conditions. (20) Sader, J. E.; Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1995, 171, 46. (21) Drummond, C. J.; Senden, T. J. Colloids Surf. A 1994, 87, 217.
Figure 8. The interaction free energy between two equalsized spheres for κa ) 10. Results are shown for both constant potential and constant charge boundary conditions. PoissonBoltzmann results are shown as lines, the Deryaguin approximation as filled symbols, and the linear Deryaguin approximation as open symbols: (a) ψ2/ψ1 ) 3 and (b) ψ2/ψ1 ) -3.
Since the accuracy improves as the surface potentials increase in magnitude or the particles increase in size, we hope that readers can assess the reliability of the Deryaguin approximation for their own systems from the results presented here and in ref 6. By contrast, the linearized Poisson-Boltzmann theory is rather limited in accuracy for constant charge interactions. For constant potential interactions at moderate surface potentials or for low values of κa, it can be useful. It plays a complementary role to the Deryaguin approximation, in that those cases where the Deryaguin approximation is worst (low κa, low surface potentials) are where linearization works bestssee, for example, Figure 6b or compare Figure 9b with Figure 2d. The linearized Deryaguin approximation (or variations of it22 ) is widely used in applications such as heterocoagulation and deposition23 because of its analytic formula. It appears to be useful for constant potential interactions at moderate potentials but cannot be recommended except for qualitative purposes for constant charge interactions. Even then it can sometimes give qualitatively wrong answers; see Figure 6b. Finally, it is necessary to mention other work on the double layer interaction between the silicon nitride tip of the atomic force microscope and a surface. Butt24,25 was the first to examine this question, initially with the aim of avoiding double layer effects which can interfere with (22) Parsegian, V. A.; Gingell, D. Biophys. J. 1972, 12, 1192. (23) Elimelech, E. J. Colloid Interface Sci. 1994, 164, 190. (24) Butt, H.-J. Biophys. J. 1991, 60, 777. (25) Butt, H.-J. Nanotechnology 1992, 3, 60.
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a
b
b
Figure 9. As for Figure 8 but with κa ) 1.
the imaging of surfaces. The force on a sharp conical tip was calculated from the potential calculated by a finitedifference scheme in cylindrical coordinates. The results were then compared with a formula applying the Deryaguin construction to an approximation to the linearized Poisson-Boltzmann equation,22 i.e., something similar to a linear Deryaguin approximation. Under the conditions of interest (salt concentration g0.01 M), this approximation was adequate and was used for interpretation in subsequent work.26,27 All these calculations were based on the conical tip with half-angle 45°. Recently, the effect of the tip shape and the half-angle of a conical tip was studied.28 It was found that the shank of the tip could not be neglected (or equivalently the tip approximated by a sphere) when the tip radius was comparable to the Debye length. Both these papers24,28 claim to solve for the force without using any approximation beyond the Poisson-Boltzmann equation. Unfortunately, both claims are false. Neither paper correctly accounts for the stress acting on the tip. The Deryaguin construction for spheres consists of two major steps: (1) neglect of transverse components of the stress tensor and (2) approximation of the subsequent integral over the sphere. In ref 28, the neglect of transverse components is explicit (their equation 2), but the subsequent integration is performed over the cone with a spherical tip. In refs 24 and 25 an attempt is made to include transverse components but not through the component of the stress tensor normal to the tip surface. For a sharp tip of half-angle 45°, this has the effect of the contribution from the transverse components having the wrong sign. These comments are made precise in the Appendix. (26) Butt, H.-J. Biophys. J. 1991, 60, 1438. (27) Butt, H.-J. Biophys. J. 1992, 63, 578. (28) Arai. T.; Fujihira, M. J. Electroanal. Chem. 1994, 374, 269.
Figure 10. As for Figure 9 but with the Deryaguin approximation modified through (26).
In summary, neither of these papers has actually tested the Deryaguin approximation for a conical tip. In light of the very good performance of the Deryaguin approximation reported here for sphere-plate geometry, however, we do not think any of the conclusions in refs 24 and 28 need to be altered. The effect of the tip shank in ref 28 is physically reasonable, and since Butt found the linear Deryaguin approximation to be adequate for his purposes, presumably the error is not great under these conditions; i.e., the term with the wrong sign is small anyway. How well any of these results apply to an actual pyramidal tip is an open question. 5. Conclusions A numerical method for the calculation of force and interaction free energy between dissimilar spheres and between a sphere and a plate based on the nonlinear Poisson-Boltzmann theory has been presented. The method requires considerably more computational effort than that required for the calculation of force and interaction free energy using the Deryaguin construction or the linear theory. It appears that at least one of these two theories provides an accurate method of calculating interactions in most cases. In particular, the Deryaguin approximation is accurate for both constant charge and constant potential interactions under most colloidal conditions and should be the method of choice; in the rare conditions where it fails, the linear theory generally performs well. In general terms, the accuracy of the linear theory improves as particle radius and/or surface potential decrease, while the Deryaguin theory improves as particle radius and/or surface potential increase. Both theories are particularly good in the case of constant potential interactions for moderate surface potentials, when the
+
+
Nonlinear Poisson-Boltzmann Theory
analytic expression of the linear Deryaguin approximation is also accurate. The linear Deryaguin approximation cannot be recommended for constant charge interactions. The Deryaguin approximation is particularly accurate for double layer forces; a simple modification for the interaction free energy extends its validity to values of κa around 1, i.e., fairly thick double layers. It is most accurate for the force in sphere/plate geometry so that experiments with colloidal particles attached to an AFM tip can be reliably analyzed using the Deryaguin approximation. Acknowledgment. J.S. is supported by an Australian Postgraduate Research Award. S.L.C. acknowledges the support of the Australian Research Council through the Advanced Mineral Products Centre. Appendix Both the papers referred to above24,28 attempt to calculate the double layer force on a conical tip, possibly with a spherical end section. Consider an axisymmetric surface S described by rotating the curve r ) r(z) about the z-axis, where r is a cylindrical coordinate. Then the force acting on the surface is given by
F)
∫ST‚nˆ dS
where T is the total stress tensor and n ˆ is the unit normal pointing into the surface. Note that the equations in the Appendix are not scaled, so as to allow easier comparison with other work. For the axisymmetric surface S parametrized by z and the azimuthal angle φ, we have
dS ) rx1 + r′2 dφ dz n ˆ )
r′(z)zˆ - rˆ
x1 + r′2
Langmuir, Vol. 12, No. 6, 1996 1461
so that
∫zz dz r(r′T‚zˆ - T‚rˆ )
F ) 2π
2
1
which already differs from ref 24, equation 1. Here z1 and z2 are whatever limits of integration are appropriate for the surface in question. We now express the total stress tensor as the sum of an osmotic pressure term and a Maxwell stress
1 T ) Π + 0E2 1 - 0EE 2
(
)
where 1 is the unit dyadic. Then the force becomes
F ) 2π
∫zz dz r(r′[(Π + 210E2)zˆ - 0EEz] 2
1
[(Π + 21 E )rˆ - EE ]) 2
0
0
r
Since by symmetry all components parallel to rˆ must integrate to zero, we get
∫zz dz (rr′[Π + 210E2 - 0Ez2] + r0EzEr)
Fz ) 2π
2
1
If at this stage we neglect transverse electric fields, we recover the method of ref 28. Specializing for the sake of simplicity to a conical tip of half-angle 45° for which r′(z) ) 1, we get
∫h∞dz r(Π + 0[21E2 - Ez2 + EzEr])
Fz ) 2π
which differs from the corresponding expression in ref 25 by a sign in the last term. For other surfaces, it is clear the differences will be more complicated. LA950384K
