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Langmuir 2007, 23, 7456-7458
Ion-Specific Forces between a Colloidal Nanoprobe and a Charged Surface E. R. A. Lima,†,‡ E. C. Biscaia, Jr.,‡ M. Bostro¨m,*,§,⊥ and F. W. Tavares† Escola de Quı´mica, UniVersidade Federal do Rio de Janeiro, Cidade UniVersita´ ria, CEP 21949-900, Rio de Janeiro, RJ, Brazil, Programa de Engenharia Quı´mica, COPPE, UniVersidade Federal do Rio de Janeiro, 21945-970, Rio de Janeiro, RJ, Brazil, DiVision of Theory and Modeling, Department of Physics, Chemistry and Biology, Linko¨ping UniVersity, SE-581 83 Linko¨ping, Sweden, and Institute of Physical and Theoretical Chemistry, UniVersity of Regensburg, D-93040 Regensburg, Germany ReceiVed March 9, 2007. In Final Form: May 14, 2007 We investigate the effect of ion-specific potentials on the force between a nanoprobe attached to a cantilever tip, and a charged surface. The probe is treated as a spherical nanoparticle with constant charge. A modified PoissonBoltzmann equation in bispherical coordinates is used to address this problem in a more quantitative way. We predict that the ion-specific series of measured forces depend on the sign and magnitude of surface charge densities.
The force between charged objects in an aqueous electrolyte solution is a central preoccupation of colloid science and biotechnology. There is an enormous amount of literature devoted to force measurements, either via the surface force apparatus technique of Israelachvili and colleagues,1-3 by atomic force microscopy, or by the osmotic measurements pioneered by Parsegian.4-6 These force measurements are often compared with Derjaguin-Landau-Verwey-Overbeek (DLVO) theory using surface potential or surface charge as adjustable parameters. Forces between charged surfaces in electrolyte solutions have been found to be highly ion specific.7-11 The reason for this ion specificity, as pointed out by Ninham and co-workers,7 is to a large extent due to previously neglected ion-specific nonelectrostatic (NES) potentials acting between ions and between ions and charged interfaces.7,10,11 For air-water interfaces, there are also vital contributions from ion-specific forces due to the static polarizabilities of ions and water molecules.12 The theoretical results presented here are the first that quantitatively consider the experimentally accessible geometry of a sphere and a plate interacting in a salt solution. This geometry is relevant to atomic force microscope measurements when a colloidal particle is attached to the cantilever tip.14 The spherical
nanoprobe is assumed to be attached to a cantilever tip. Our results predict that NES forces acting between ions and surfaces will give rise to observable ion-specific forces between a nanoprobe and a planar surface. The force can even change sign when one salt is replaced with another. Notably, the ion-specific Hofmeister series of measured force curves is predicted to depend on the surface charge densities of the interacting objects. This effect remains to be confirmed experimentally. The original DLVO1 theory fails to predict any such ion specificity. The problem lies in7 the inconsistency built into the DLVO theory, which separates forces between particles into electrostatic doublelayer and van der Waals forces (Hamaker). The electrostatic forces are handled by a nonlinear Poisson-Boltzmann description or derivatives thereof. The van der Waals forces are treated in a linear (Lifshitz) theory. When ion-specific NES forces are treated at the same nonlinear level as the electrostatic forces, the origin of ion-specific effects finally comes into sight. The NES potential between an ion solution and both nanoprobe and plate can (at large ion-surface separations) be approximated with6
* Corresponding author. E-mail:
[email protected]. † Escola de Quı´mica, Universidade Federal do Rio de Janeiro. ‡ Programa de Engenharia Quı´mica, COPPE, Universidade Federal do Rio de Janeiro. § Linko ¨ ping University. ⊥ University of Regensburg.
Here the first term is related to the nanoprobe, the second is related to the planar surface, rs is the nanoprobe radius (here taken to be 100 nanometers), r1 is the distance between the ion and the center of the nanoprobe, r2 is the distance between the ion and the planar surface, and Bs and Bp are the ion-nanoprobe and ion-plate dispersion constants (that depend on the ionic excess polarizability and the dielectric properties of water, the nanoprobe, and the surface). The dependence of the dispersion constant by the ionic excess polarizability and dielectric properties is discussed elsewhere.10 Here, the dielectric properties of the nanoprobe and plate are treated as being the same, therefore Bs ) Bp for each ion. The ionic dispersion constants B are nondimensionalized by the factor kTrs3, where T is the temperature and kB is Boltzmann’s constant. For sodium, chloride, bromide, and iodide, the following constants were taken from the literature:10 0.138, 1.086, 1.348, and 1.735, respectively. For the sake of comparison, we also consider the artificial case without any NES potentials (which is the electrostatic part of the DLVO result). In two important papers, Carnie and co-workers13,14 used the nonlinear Poisson-Boltzmann equation in bispherical coordinates
(1) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. See also Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic: London, 1992, and references therein. (2) Pashley, R. M. J. Colloid Interface Sci. 1981, 83, 153. (3) Pashley, R. M.; McGuiggan, P. M.; Ninham, B. W.; Brady, J.; Evans, D. F. J. Phys. Chem. 1986, 90, 1637. (4) Dubois, M.; Zemb, Th.; Fuller, N.; Rand, R. P.; Parsegian, V. A. J. Chem. Phys. 1998, 108, 7855. (5) Parsegian, V. A.; Rand, R. P.; Fuller, N. L. J. Phys. Chem. 1991, 95, 4777. Tsao, Y.-h.; Evans, D. F.; Rand, R. P.; Parsegian, V. A. Langmuir 1993, 9, 233. (6) Parsegian, V. A. Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists; Cambridge University Press: Cambridge, U.K., 2006. (7) Ninham, B. W.; Yaminsky, V. Langmuir 1997, 13, 2097. (8) Bostro¨m, M.; Williams, D. R. M.; Ninham, B. W. Phys. ReV. Lett. 2001, 87, 168103. Edwards, S. A.; Williams, D. Phys. ReV. Lett. 2004, 92, 248303 and references therein. (9) Zhang, Y.; Cremer, P. S. Curr. Opin. Chem. Biol. 2006, 10, 658. (10) Tavares, F. W.; Bratko, D.; Blanch, H. W.; Prausnitz, J. M. J. Phys. Chem. B 2004, 108, 9228. (11) Bostro¨m, M.; Williams, D. R. M.; Ninham, B. W. Langmuir 2001, 17, 4475.
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-Bs (r1 - rs) [1 + (r1 - rs) /(2rs )] 3
10.1021/la700690g CCC: $37.00 © 2007 American Chemical Society Published on Web 05/31/2007
3
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+
-Bp (r2)3
(1)
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Langmuir, Vol. 23, No. 14, 2007 7457
(without including ion-specific potentials) to consider the interaction between spherical particles and between a spherical particle and a surface. To obtain the self-consistent electrostatic potential, we solve the nonlinear Poisson-Boltzmann equation in bispherical coordinates now with NES potentials included15
1 ∇2ψ ) [exp(ψ - U-) - exp(-ψ - U+)] 2
(2)
for the dimensionless electrostatic potential ψ ) φez/(kT) and the dimensionless NES potential (expressed as a function of dimensionless position, with the scale being the Debye length κ). The boundary condition is that the surface charge density remains constant (∇ψ‚nj ) σ, where nj is the unit normal directed into the particles, and we use the dimensionless surface charge σ). The ion-specific Poisson-Boltzmann equation is solved using the finite volume method described in detail by Lima et al.15 Colloidal probes attached to atomic force microscope cantilevers are typically on the order of 100 nm. Most used equivalent tip radii are typically around 20 or 200 nm. We use a probe radius of 100 nm, but the results are qualitatively similar for both smaller and larger values. In Figure 1, we show a three-dimensional figure that illustrates the concentration profile of nonpolarizable anions that are counterions for the nanoprobe and co-ions for the plate. In this figure, κ‚h ) 0.5 (h ) 4.81 Å), and the planar surface is located at κ‚z ) 0. We consider a nanoprobe with σs ) 0.03 C/m2 and a planar surface with σp ) -0.01 C/m2 immersed in 0.1 M salt solutions. Given that both surfaces have constant charge density, both surface potentials vary due to the presence of the other surface. The presence of the sphere increases the concentration of undesirable co-ions close to the plate, giving rise to a repulsive contribution to the attractive force. Once the electrostatic potential is known, we use the following expression for the Poisson-Boltzmann contribution to the doublelayer force, based on that presented by refs 14 and 15:
f)π
{[
∫0π
b (exp(ψ - U-) - exp(-ψ - U+) - 2) 2
(cosh(η) - cos(θ))2
+
∂ψ - ( ) ) × [1 - cosh(η) cos(θ)] + ((∂ψ ∂θ ) ∂η ] 2
Figure 1. Concentration profile of nonpolarizable anions around a sphere with σs ) 0.03 C/m2 and close to a plate with σp ) -0.01 C/m2.
Figure 2. Force F between a sphere with radius ) 100 nm and σs ) 0.03 C/m2, and a plate with σp ) -0.01 C/m2 interacting in different 0.1 M salt solutions. We consider four cases: nonpolarizable ions (solid line), NaCl (dash line), NaBr (dash-dot line), and NaI (dash-dot-dot line).
2
}
sin(θ)dθ ∂ψ ∂ψ sinh(h) sin(q) 2 (3) ∂θ ∂η cosh(η) - cos(θ) Here, b ) κrs sinh(η0), where η0 is the value of η at the surface of the sphere, and, in the evaluation of the force, we do the integration on the surface η ) η0/2. This plane was chosen to give good accuracy.14 The bispherical coordinates (η,θ) are related to the cylindrical coordinates (F,z) in a way described by Carnie et al.13 In eq 3, the dispersion van der Waals interactions between ions and surfaces were taken into account indirectly because the electrostatic potential obtained from the Poisson-Boltzmann equation depends on their magnitudes. Because our calculations are for low salt concentrations and for large distances, here we neglected any contribution from the direct dispersion interaction between ions and surfaces. An expression for this additional term, which can be thought of as being due to the ions “pulling” on the surfaces via dispersion interaction, is used by Edwards (12) Jungwirth, P.; Tobias, D. J. J. Phys. Chem. A 2002, 106, 379. (13) Carnie, S. L.; Chan, D. Y. C.; Stankovich, J. J. Colloid Interface Sci. 1994, 165, 116. (14) Stankovich, J.; Carnie, S. L. Langmuir 1996, 12, 1453. Butt, H.-J.; Cappella, B.; Kappl, M. Surf. Sci. Rep. 2005, 59, 1. Sharp, J. M.; Duran, R. S.; Dickinson, R. B. J. Colloid Interface Sci. 2006, 299, 182.
and Williams8 for two plates. We also have not taken into account the direct dispersion interaction between the sphere and the plate (the Hamaker contribution to the double-layer force). In the examples considered here, the charge of the nanoprobe is positive, and the charge of the planar surface is negative. As a very interesting example, we consider in Figure 2 the force between a nanoprobe and the planar surface with the same charges as those in Figure 1, immersed in different 0.1 M salt solutions. In this figure, κh is the dimensionless closest approach between the sphere and the planar surface. We consider four different solutions: nonpolarizable (NP) ions, NaCl, NaBr, and NaI. For nonpolarizable salt, the force is repulsive at short separations, whereas it is attractive at larger separations. For polarizable ions, the effect may be attraction at all separations. This is interesting since it demonstrates that both the sign and the magnitude of the force measured between a nanoprobe and a surface can depend on the choice of salt and on the separation between the nanoprobe and the planar surface. The predicted force curves follow an ion-specific Hofmeister series that becomes more attractive as we go from NP < NaCl < NaBr < NaI. This series can be manipulated by changing the magnitude or the sign of the surface charges. We show in Figure 3 the case when σs ) 0.01 C/m2 and the planar surface with σp ) -0.03 C/m2. Here, the predicted force becomes more attractive as we go from NaI < NaBr < NaCl < NP. Note that, in this case, we changed only the
7458 Langmuir, Vol. 23, No. 14, 2007
Figure 3. Same as Figure 2, but σs ) 0.01 C/m2 and σp ) -0.03 C/m2.
magnitude, not the sign of the surface charges. In the case of two surfaces with same charge, the force becomes more attractive as we increase the polarizability of the counterions. For co-ions, the exact opposite series is seen. The inversion of the Hofmeister series has been experimentally and theoretically observed in protein suspensions when the pH of the salt solution crosses the pI of the protein.16 With asymmetric surface charges, the same effect can be expected, but now the ions are counterions at one surface and co-ions at the other. The relative magnitude of the surface charge densities are now the key parameter that decides what Hofmeister series will be observed. In Figure 2, the nanoprobe has a larger positive charge, and the counterion effects dominate. The interaction becomes more and more attractive with increasing NES potentials. In Figure 3, we reduced the magnitude of the positive nanoprobe surface charge and enhanced the negative surface charge of the planar surface. Here, the negative surface plays a more determining role and the predicted force curve follows the opposite series. Calculations with constant surface potential instead of constant surface charge, similar dependence of the nanoprobe-planar surface force on salt type, and salt concentration are observed but without inversion of the Hofmeister series. (15) Lima, E. R. A.; Biscaia, E. C.; Tavares, F. W. Phys. Chem. Chem. Phys., in press. (16) Bostro¨m, M.; Tavares, F. W.; Finet, S.; Skouri-Panet, F.; Tardieu, A.; Ninham, B. W. Biophys. Chem. 2005, 117, 217 and references therein.
Letters
Figure 4. Force F between a sphere with radius ) 100 nm and σs ) 0.03 C/m2, at a fixed probe surface distance h ) 2 nm. Here, the plate surface charge density varies.
In Figure 4 we consider the case where the sphere-surface distance and the (positive) probe surface charge density are both constant, and we vary the plate surface charge density. One can have direct or reversed Hofmeister sequences depending on the charge of the plate surface. We note that one can tune the order for the Hofmeister series by varying the surface charges of the interacting objects. This prediction should be possible to observe experimentally. The main conclusion of this letter is that DLVO theory often fails to give correct sign and magnitude for the force between a nanoprobe attached to a cantilever and a charged surface. Earlier results could only predict qualitative behavior. The use of a typical experimental geometry gives novel results that can be more quantitatively compared with force measurements. We observe finally that, in general, the NES potentials depend on the surface properties and will be different for the nanoprobe and the surface. This can be taken care of within our methodology. Acknowledgment. E.R.A.L., F.W.T. and E.C.B. thank Capes, CNPq, and FAPERJ, the Brazilian agencies for scholarship and for supporting part of this project. M.B. thanks the Swedish Research Council for financial support. LA700690G
