846
Langmuir 2006, 22, 846-847
Comments Comment on Electrodipping Force Acting on Solid Particles at a Fluid Interface
In a recent paper,1 the interface deformations caused by glass spherical particles of radius R ) 200...300 µm trapped at a planar interface between water and a nonpolar fluid (air, oil) have been studied experimentally and theoretically. See Figure 1 of ref 1 for the definition of the relevant quantities in this context. Experimental results for the interface deformation ζ(r) have been reported for lateral distances up to r ≈ 2R and have been analyzed using a linearized capillary model taking into account the electrostatic effects caused by the charges on the particle surface. With this model, the authors claim to have obtained excellent agreement with their experimental data, and they predict a longranged, logarithmic interface deformation ζ(r). Using a superposition approximation, it is further asserted that the capillaryinduced effective interaction potential U(d) between two spheres a distance d apart is attractive and decays logarithmically. In a recent publication,2 we have shown, under the same physical assumptions (small interface deformation and mechanical isolation of the experimental system), that the deformation ζ(r) is short-ranged and neither ζ(r) nor U(d) decays logarithmically but much faster. In this comment, we point out the corresponding flaws in the theoretical analysis peformed in ref 1. Under the assumption of small meniscus deformations, the radially symmetric ζ(r) is given by the solution of the following equations (eqs 3.10 and 3.12 in ref 1):
1 d2ζ 1 dζ + - q2ζ ) pel(r) 2 r dr γ dr
(1)
dζ F (r ) ) dr c 2πγR ζ(r f ∞) ) 0 Here, γ is the surface tension of the meniscus, q is the inverse capillary length induced by gravity, F is the (electrostatic and gravitational) force acting on the particle, pel(r) is the electrostatic pressure acting on the meniscus, and rc is the radius of the circular three-phase contact line. The general solution to eq 1 is given by5
ζ(r) ) -I0(qr)
pel(s) K (qs) + γ 0
∫r∞ ds s
[
K0(qr) A -
c
]
pel(s) I (qs) (2) γ 0
∫rr ds s
where the integration constant A is determined by the boundary condition at r ) rc. In the intermediate asymptotic regime (1) Danov, K. D.; Kralchevsky, P. A.; Boneva, M. P. Langmuir 2004, 20, 6139. (2) Oettel, M.; Domı´nguez, A.; Dietrich, S. Phys. ReV. E 2005, 71, 051401. (3) Nikolaides, M. G.; Bausch, A. R.; Hsu, M. F.; Dinsmore, A. D.; Brenner, M. P.; Gay, C.; Weitz, D. A. Nature 2002, 420, 299. (4) Oettel, M.; Domı´nguez, A.; Dietrich, S. J. Phys.: Condens. Matter 2005, 17, L337. (5) This differs from eq 3.11 in ref 1 by a term ∝ I0(qr) in order to satisfy the boundary condition ζ(r f ∞) ) 0; in the limit qr , 1, this term gives a constant shift in ζ(r) which does not affect the following discussion.
qr , 1 (q-1 ≈ 3 mm), eq 2 reduces to
ζ(r) =
1 γ
∫r∞ ds spel(s) ln rs +
(
)
F + F(men) qr ln 2πγ C
(3)
where C ) 2e-γe = 1.12 and
F(men) :) 2π
∫r∞ ds spel(s)
(4)
c
is the total electrostatic force acting on the meniscus. We note that eq 3 together with the definition (eq 4) is mathematically equivalent to eq 3.14 in ref 1 up to an additive constant. The intermediate asymptotics for ζ(r) as given in eq 3 hold for arbitrary pel(r) for which the integral in eq 4 exists. One notices that the amplitude of the logarithmic deformation is determined by the total force acting on the system “particle + interface”. The total force on the particle is given by the sum of a gravitational and an electrostatic part, F ) F(g) + F(el), where |F(g)| , |F(el)|. (See Table 1 in ref 1.) Using the second assumption of mechanical isolation of the experimental system (note that gravity is irrelevant), one derives the equality (eqs 6.3-6.6 in ref 1, and ref 2)
F(el) ) -F(men)
(5)
(i.e., the electrostatic force on the particle is counterbalanced by the electrostatic force on the interface). Thus, F + F(men) ) F - F(el) ) F(g), and F(g) is negligible compared to F(el). Consequently, the prefactor of the logarithm in eq 3 is also negligible, and the interface deformation is given by the first term in eq 3. For a power-law decay of the pressure, pel ∝ r-n (eq 3.9 of ref 1), one has ζ(r) ∝ r-(n - 2), describing a much shorter-ranged interface deformation. This result has to be compared with the conclusion in ref 1, where the logarithmic term in the expresssion for the interface deformation comes with a prefactor F - λF(el) (eq 3.15) and the constant λ < 1 is determined by the electrostatic solution for pel(r) and depends in general on the contact angle and on the dielectric permittivities of the nonpolar liquid and the colloid. (For the explicit example considered in ref 1, λ ≈ 0.48.) The error in the analysis of ref 1 resides in the evaluation of F(men) (eq 4), with the interfacial stress pel(r) replaced by its asymptotic behavior that is valid for r . rc (eq 3.9). This is certainly an invalid approximation because the main contribution to the integral in eq 4 stems precisely from the region r ≈ rc (eq 3.7 and eq 6 below). Nevertheless, we have shown that the vanishing of the logarithmic prefactor follows solely from the exact relationship in eq 5, independently of the explicit expression for pel(r), which is not needed to reach this conclusion. In ref 1, the solution to eq 1 has been computed in toroidal coordinates using a Runge-Kutta second-order scheme with a minimal step size of ∆r/rc ≈ 0.01 close to the contact line. We calculated the solution to eq 1 numerically as well as using a Runge-Kutta fourth-order scheme combined with a shooting algorithm to incorporate the boundary condition at infinity. For
10.1021/la0514260 CCC: $33.50 © 2006 American Chemical Society Published on Web 12/15/2005
Comments
Langmuir, Vol. 22, No. 2, 2006 847
Figure 1. Plot of the meniscus height ζ vs r, computed using a fourth-order Runge-Kutta routine with fixed step size of ∆r/rc. The curves show the sensitivity of the numerical solution to the step size chosen. Parameters have been chosen as in Figure 2 of ref 1.
pel(r), we used the fitting formula provided in ref 1 (eq 3.7),
pel(r) ) -
[ ] []
2λF(el) r -1 πrc2 rc
µ-1
r rc
-(µ+5)
(6)
λ ) (3 + µ)(2 + µ)(1 + µ)µ/24 which satisfies eq 5 exactly. Our numerical solution is in complete accordance with our eq 3. We find, however, that rather small step sizes for r f rc are necessary to treat the integrable singularity present in eq 6 properly. For the parameters pertaining to Figure 2 in ref 1, we find the necessary step size to be smaller than ∆r/rc ) 10-6 for r f rc. Larger step sizes introduce an artificial violation of eq 5, which leads to a spurious logarithmic term in the numerical solution; see Figure 1. These findings are independent of the exact behavior of pel(r) near the contact line which we have checked explicitly for different parametrizations of pel(r). This has profound consequences for the potential U(d) between two colloidal particles. To compute U(d), the superposition approximation is employed. In ref 1, the superposition approximation is defined as U(d) ) Fζ(d), which can be derived in this form only for the case pel ) 0. For nonzero pel, the correct implementation of the superposition approximation yields U(d) ) (F + F(men))ζ(d) as derived in ref 2. The absence of logarithmic
contributions in ζ(d) also leads to an absence of logarithmic contributions in U(d). Consequently, the discussion of the relevance of this capillary potential for micro- and nanocolloids in section 3.7 is also misleading, and contrary to the conclusions presented in ref 1, the electrodipping force as analyzed there does not explain the experimental results obtained in ref 3. Furthermore, we note that in the case of vanishing logarithmic contributions in ζ(d) the superposition approximation for U(d) is invalid. As discussed in ref 4, a dominant attractive contribution in U(d) is found by going beyond the superposition approximation. Besides the differences in analyzing the long-range meniscus deformation between ref 1 and the present comment, we noticed an inconsistency in the data presentation in ref 1. The data points in the inset of Figure 2 do not match the data in Table 1 (the line marked with the superscript a). From the latter, one extracts tan Ψ ≈ 0.23, whereas upon digitizing Figure 2 we have found tan Ψ ≈ 0.31. This is confusing because the numerical solution of the authors fits the data points in Figure 2 quite well, which apparently are not the correct ones. Finally, we mention that the linearized Young-Laplace equation in toroidal coordinates given by eq 6.2 is wrong. The correct equation reads
(
x14
)
rc2 1 dζ d2ζ + ) p + q2rc2(ζ - ζ∞) γ el dx12 x1 dx1
(7)
In conclusion, the meniscus deformation measured in ref 1 cannot be explained in terms of a logarithmic decay if one sticks to the assumption of mechanical isolation. The possible appearance of logarithmic deformations is linked to mechanical nonisolation of the experimental system that could be a result of, for example, electric stray fields or finite size effects. This must be taken into consideration in designing and analyzing experiments with colloids at fluid interfaces. M. Oettel,* A. Domı´nguez, and S. Dietrich
Max-Planck-Institut fu¨r Metallforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany ReceiVed May 31, 2005 In Final Form: September 16, 2005 LA0514260
