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Measurement of Aqueous Film Thickness between Charged Mercury and Mica Surfaces: A Direct Experimental Probe of the Poisson-Boltzmann Distribution Jason N. Connor and Roger G. Horn* Ian Wark Research Institute, University of South Australia, Mawson Lakes, South Australia 5095, Australia Received August 23, 2001. In Final Form: September 25, 2001 A stable aqueous electrolyte film is formed between a mercury drop and a flat mica surface due to electrical double-layer repulsion when a negative potential is applied to the mercury. Film thickness has been measured as a function of applied potential while keeping the film pressure constant. By making measurements in this way, it is possible to map the data directly according to the Poisson-Boltzmann equation. An excellent fit to the data is obtained, providing direct evidence for this classical equation and its use as the basis of the Gouy-Chapman model of the diffuse double layer in electrolyte solutions.
The Gouy-Chapman model1,2 of the diffuse double layer in an electrolyte solution adjacent to a charged surface is based on the Poisson-Boltzmann (PB) equation which describes the spatial distribution of electrostatic potential within a medium in which mobile charged species are present. This model (with extensions to allow for adsorption of ions in a Stern layer) is universally used in colloid science and electrochemistry, and it successfully describes many phenomena including double-layer capacitance,3 colloid stability,4,5 electrokinetics in moderate concentrations of simple electrolytes,5 and direct measurements of double-layer repulsion between surfaces.6,7 Notwithstanding these numerous successes, there is no direct evidence for the Gouy-Chapman model because it is difficult to probe the ionic distribution in the diffuse double layer without disturbing it. While the X-ray fluorescence measurements of Bedzyk et al.8 were able to achieve the latter, they could do no more than demonstrate qualitative consistency with the Gouy-Chapman-Stern model. The experiments we report here provide a method of mapping out the profile of electrical potential in an electrolyte adjacent to an electrode interface and hence testing the quantitative correctness of the nonlinear Poisson-Boltzmann equation. Measurements are made of the thickness of an aqueous electrolyte film between a mica surface and a mercury drop while the potential applied to the mercury is varied. The key feature of our experiments is that they are conducted at constant pressure in the film, which puts the data in an ideal form for comparison with a numerical solution of the PB * To whom correspondence should be addressed. E-mail: roger.
[email protected]. (1) Gouy, G. J. Phys. Radium 1910, 9, 457. (2) Chapman, D. L. Philos. Mag. 1913, 25, 475. (3) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry: An Introduction to an Interdisciplinary Area; Macdonald: London, 1970. (4) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (5) Hunter, R. J. Foundations of Colloid Science, 2nd ed.; Clarendon Press: Oxford, 2001. (6) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (7) Pashley, R. M. J. Colloid Interface Sci. 1981, 83, 531. (8) Bedzyk, M. J.; Bommarito, G. M.; Caffrey, M.; Penner, T. L. Science 1990, 248, 52.
equation obtained using the algorithm of Chan, Pashley, and White.9 The experimental arrangement is to bring a flat mica sheet down toward a drop of mercury held at the end of an insulating capillary, as shown schematically in Figure 1. The mercury drop protrudes from an insulating capillary of diameter ∼1 mm. The upper surface of the mica is coated with a semireflecting silver layer, and optical interference between the silver and mercury surfaces, observed as fringes of equal chromatic order (FECO) in the reflected beam,11 allows the thickness of the aqueous layer that separates the mercury and mica to be measured with a resolution of 0.3 nm12 and the shape of the mercury drop to be determined. Pressure (P0) in the mercury at the apex of the drop is controlled by an external syringe and is measured by finding the radius of curvature (R) of the drop when it is far from the mica and then using the Laplace equation P0 ) 2γ/R, where γ is the mercury/water interfacial tension.10 The potential of the mercury is controlled using a Wenking MP87 three-electrode potentiostat (Bank Electronik, Germany) with a saturated calomel reference electrode (SCE), a platinum gauze counter electrode in the aqueous phase, and a platinum wire in contact with the mercury. Electronic grade mercury (99.9999%, Aldrich, USA) is washed under nitric acid with an agitating stream of ultra-high-purity oxygen followed by extensive rinsing in water. Water for rinsing, and for preparation of the electrolyte solution, is prepared by passing through multistage reverse osmosis and deionizing and organic adsorption cartridges followed by subboiling distillation in an all-quartz apparatus (Quartz & Silice, France). If a repulsive force is present between the mica and mercury surfaces, there is a positive disjoining pressure in the aqueous film.13 When this pressure is equal to the known internal pressure in the protuberant mercury drop, the pressure difference across the interface is zero and (9) Chan, D. Y. C.; Pashley, R. M.; White, L. R. J. Colloid Interface Sci. 1980, 77, 283. (10) Miklavcic, S. J.; Horn, R. G.; Bachmann, D. J. J. Phys. Chem. 1995, 99, 9, 16357. (11) Tolansky, S. An Introduction to Interferometry, 2nd ed.; Wiley: New York, 1973. (12) Connor, J. N. Ph.D. Thesis, University of South Australia, 2001. (13) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Surface Forces; Consultants Bureau: New York, 1987; Chapter 2.
10.1021/la0155505 CCC: $20.00 © 2001 American Chemical Society Published on Web 10/20/2001
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Figure 1. Schematic diagram of the apparatus used to measure the thickness of a film of electrolyte solution between mercury and mica surfaces. A mercury drop protrudes from an insulating capillary under a pressure P0 applied by an external syringe. As the mica is moved down toward the mercury using a twostage differential spring drive with sub-nanometer control,6 the double-layer force exerted on the drop flattens it, giving a uniform aqueous film thickness at which the disjoining pressure in the film equals P0.10 The film thickness is measured by optical interferometry11 between the mercury and a partially reflecting (20 nm) silver reference surface deposited on the upper surface of the flat mica sheet, which is of uniform thickness ∼5 µm. A platinum electrode contacts the mercury phase and a SCE is used as a reference electrode in the aqueous phase.
the top of the drop must have zero curvature: it is flattened by the repulsive force.10 The FECO technique allows this flattening to be observed and the aqueous film thickness to be measured. Because the mica surface is negatively charged in water, repulsion occurs when the mercury is charged negatively by applying an appropriate potential to it. Making the mercury surface increasingly negative increases the magnitude of the repulsive force; hence the aqueous film thickness increases if disjoining pressure is kept constant.14 Data showing this behavior are presented in Figure 2, which plots the variation of film thickness with applied potential measured in 1 mM KCl solution. The aqueous film thicknesses observed in our experiments can be explained entirely by repulsive forces between the electrical double layers of the mica and mercury surfaces.15 In contrast to two earlier reports,16,17 our data give no indication of additional forces such as hydration effects, nor do we see any indication of longrange hydrophobic attraction. There are several previous reports of indirect and direct disjoining pressure measurements between mercury drops and other surfaces14,16-23 as a function of potential applied to the mercury, as well as reports of double-layer forces measured between other conducting surfaces with control of potential,24-27 but none has presented comprehensive (14) Matsumoto, M.; Takenaka, T. Bull. Inst. Chem. Res., Kyoto Univ. 1982, 60, 269. (15) There is also expected to be a van der Waals contribution to the disjoining pressure, but a calculation using Lifshitz theory shows that this is no more than a few tens of pascals at the film thicknesses measured under the present conditions, compared to the experimental disjoining pressure which is ∼103 Pa. (16) Derjaguin, B. V.; Gorodetskaya, A. V.; Titievskaya, A. S.; Yashin, V. N. Kolloid. Zh. 1961, 5, 535. (17) Gupta, A.; Sharma, M. M. J. Colloid Interface Sci. 1992, 149, 392. (18) Gorodetskaya, A. V.; Frumkin, A. N.; Titievskaya, A. S. Zh.. Fiz. Khim. 1947, 21, 675. (19) Watanabe, A.; Gotoh, R. Kolloid-Z. Z. Polym. 1963, 191, 36. (20) Usui, S.; Yamasaki, T.; Shimoiizaka, J. J. Phys. Chem. 1967, 71, 3195. (21) Usui, S.; Yamasaki, T. J. Colloid Interface Sci. 1969, 29, 629. (22) Usui, S.; Sasaki, H.; Hasegawa, F. Colloids Surf. 1986, 18, 53. (23) Porter, J. D.; Zinn, A. S. J. Phys. Chem. 1993, 97, 1190. (24) Ishino, T.; Hieda, H.; Tanaka, K.; Gemma, N. Jpn. J. Appl. Phys. 2 1994, 33, L1552.
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Figure 2. Variation of thickness of a 1 mM aqueous KCl film between a flat mica sheet and a flattened drop of mercury, as a function of the potential applied to mercury with respect to SCE. Symbols show experimental data measured at a constant disjoining pressure of 926 Pa in the film. The solid line is obtained by solving the Poisson-Boltzmann equation at this pressure (see text and Figure 3), for a mica surface potential of -85 mV.7 The dashed line between B and D, which is a continuation of the PB solution, is experimentally unstable. Points A to D are explained in the text.
constant-pressure data as we do. As we will now explain, it is this feature which provides the unique opportunity to test the PB equation directly. Double-layer repulsion between flat surfaces is readily calculated by solving the one-dimensional PB equation
zeψ d2ψ 2n0ze sinh ) 2 kT dx
(1)
Here ψ is the electrical potential measured with respect to the background (Galvani) potential of the aqueous phase which is a z:z electrolyte solution of number concentration n0 and dielectric permittivity , x is the spatial coordinate perpendicular to the surfaces, e is the electronic charge, k is Boltzmann’s constant, and T is temperature. Several methods are available to solve this equation, some of which make approximations that are valid for small potentials.5 One method that avoids any such approximation is the numerical algorithm presented by Chan, Pashley, and White,9 which uses the clever device of setting a value of pressure and solving for the corresponding film thickness, rather than the other way around. After integrating the PB equation once, it can be shown4,5 that the disjoining pressure Π is equal to the excess osmotic pressure at a particular position xm in the film where the magnitude of potential has a minimum value, ψm. The osmotic pressure, which is proportional to the concentration of ions at that position, is simply related to ψm through
[ ( ) ]
Π ) 2n0kT cosh
zeψm -1 kT
(2)
Hence by setting a value for the pressure, the value of ψm is established. Together with the condition of zero gradient at the position of minimum-magnitude potential (PMMP), (25) Raiteri, R.; Grattarola, M.; Butt, H.-J. J. Phys. Chem. 1996, 100, 16700. (26) Hillier, A. C.; Kim, S.; Bard, A. J. J. Phys. Chem. 1996, 100, 18808. (27) Frechette, J., Vanderlick, T. K. Langmuir, in press.
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Figure 3. The Poisson-Boltzmann (PB) equation describes the spatial distribution of potential between two plates, shown here as the magnitude although potentials are actually negative in the present experiments. The form of ψ(x) can be computed using the Chan-Pashley-White algorithm9 which solves from the PMMP where ψ ) ψm, dψ/dx ) 0 and we have set xm ) 0, by integrating numerically until ψ reaches the value ψd1, which is the surface potential at one surface, e.g., the left-hand one in this diagram. This establishes the distance |x1|. The process is then repeated to find x2 corresponding to a potential ψd2 at the second surface, and the surface-surface separation is equal to |x1| + |x2|. To analyze the present data, we take the first surface to be mica, whose potential (ψd1) does not change during the measurement, and the second surface to be mercury, whose potential is varied using the electrode system. The curve then shows how the mica-mercury separation, given by the distance from the left-hand axis to the right-hand branch of the curve, increases as the magnitude of the mercury potential (ψd2) increases. Rotating this figure through 90° puts it in the same form as the plot in Figure 2, where it is seen that the PB curve matches the data.
this fixes the initial conditions for a first-order differential equation which is integrated numerically from the PMMP until the potential reaches a known value ψd corresponding to a surface potential, thereby determining the distance between the PMMP and the surface. In this context, “surface potential” means the potential at the boundary between the diffuse layer and the Stern layer of adsorbed counterions, and the “surface” is the outer Helmholtz plane (OHP).5 The process is repeated for a second surface bearing a different potential (of the same sign); the separation, or film thickness, between the two surfaces is just the sum of the distances from the PMMP to each surface. The procedure is illustrated in Figure 3. For the present purposes we wish to fix one potential corresponding to the mica surface and to calculate how the film thickness varies when potential on the second (mercury) surface is varied. This is simply given by the Poisson-Boltzmann curve plotted in Figure 3: as potential on the right-hand surface increases, the total separation is given by the distance from the left-hand axis to the right-hand branch of the curve. Rotating the figure by 90° would show separation as a function of varying potential on one surface. This is exactly the form in which our data are plotted in Figure 2. Indeed, if it is assumed that the mercury diffuse layer potential ψd differs by a constant amount of -405 mV from the potential applied to the interior of the mercury (with respect to SCE) so that the PB curve is shifted by this amount, then the calculated PB curve overlays the data exactly (solid curve in Figure 2). Making this shift is equivalent to assuming that
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mercury is a perfectly polarizable electrode, i.e., no current crosses the mercury/aqueous interface, and that the interfacial dipole potential3 is independent of the potential applied to the mercury. These are good approximations for the mercury/aqueous interface at moderate surface charge and low electrolyte concentrations. The potentials labeled A, B, and C in Figure 2 provide information about the double layer at the mercury/aqueous interface. C is the point of zero electronic charge, σ0, on the mercury, which is determined in situ from capacitance measurements, or from observing a change in drop shape due to the maximum in interfacial tension that occurs when σ0 ) 0.3,5,28 At this point, which we measure at -430 mV (compared to a value of -438 mV calculated by Conway29 from the data of Grahame30), the charge in the diffuse double layer must be balanced by the charge in a layer of adsorbed Cl- ions. The diffuse layer charge can be calculated from the potential gradient at the mercury surface OHP using the Poisson equation, giving 1.9 mC m-2. Point B (-420 mV) is the PMMP, and here the diffuselayer charge on the mercury surface is zero. The position of this point depends on the disjoining pressure; it is shifted from point A by an amount ψm which is calculated from eq 2. Point A (-405 mV) is the amount by which the PB curve was shifted to match the data. In the limit of zero disjoining pressure, points A and B should coincide, and there would be no charge in the diffuse layer of an isolated mercury surface at this potential. In other words, at -405 mV with respect to the SCE reference electrode, the surface potential at the OHP of the mercury would be equal to the Galvani potential in the aqueous phase far from the surfaces. If there were no specific adsorption, the (electronic) surface charge of mercury would also be zero at this potential. However, since there is specific adsorption of Cl-, the electronic charge on the mercury, which is related to the electrocapillary curve or capacitance measurements through the Lippmann equation,3,5 must be balanced by the adsorbed charge at point A. In principle this provides a second method of determining the adsorbed charge. Alternatively, we can reverse the procedure and use the above data to estimate a value of 7.6 µF/m2 for the capacitance, on the assumption that the amount of adsorbed charge does not vary significantly between potentials A and C. Point D in Figure 2 is the mica OHP surface potential, and the difference D - A (-85 mV) can be obtained independently from force measurements between two mica surfaces.7 In summary, we have shown that the data of film thickness vs potential at constant disjoining pressure provide a direct measurement of the nonlinear PB equation. Since the constant pressure measurements fix the potential at the PMMP, this effectively decouples the two sides of the aqueous film so that the diffuse ionic layer on the mica side is not influenced by that on the mercury side. Regardless of whether the surface charge or surface potential of the mica surface remain constant or regulate along a disjoining pressure isotherm, they both remain fixed if the pressure is held constant, as it is here. As the mercury potential is increased in magnitude, the total number of counterions in the diffuse layer on the mercury side of the PMMP increases. The thickness of this layer (and hence the total separation between the two surfaces) increases, tracing out the PB equation between the boundary conditions set by the minimum-magnitude (28) Grahame, D. C. Chem. Rev. 1947, 41, 441. (29) Conway, B. E. Electrochemical Data; Elsevier: Amsterdam, 1952; Chapter 6. (30) Grahame, D. C.; Coffin, E. M.; Cummings, J. I.; Poth, M. A. J. Am. Chem. Soc. 1952, 74, 1207.
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potential (fixed by the pressure) and the mercury surface potential. Other data sets measured at different pressures and different electrolyte concentrations also fit the corresponding PB solutions, using the same potential offset.12,31 The present experiments produce a satisfying endorsement of the PB equation, but it is recognized that under more extreme conditions such as high electrolyte concentration, highly asymmetric electrolytes, or large-sized ions, the distribution of ions in a diffuse double layer may not follow the simple PB form. Extensions of the present experimental technique to those conditions may be able to detect departures from the PB equation. However, for (31) Connor, J. N.; Horn, R. G. Article in preparation.
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the present conditions of low concentrations of simple electrolyte, it can be concluded that the Gouy-ChapmanStern model based on the nonlinear PB equation gives a very accurate description of the distribution of ions in an electrolyte adjacent to a charged surface. Acknowledgment. This work was supported by the Australian Research Council. We are grateful to D. Y. C. Chan, S. J. Miklavcic, B. A. Pethica, C. J. Radke, and W. Schmickler for helpful discussions, to O. I. Vinogradova for drawing our attention to refs 16 and 18, and to T. N. Khmeleva for translating them. LA0155505
