Langmuir 2000, 16, 5749-5753
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Debye Length in Multivalent Electrolyte Solutions Mika M. Kohonen, Marilyn E. Karaman, and Richard M. Pashley* Department of Chemistry, The Faculties, The Australian National University, Canberra, A.C.T 0200, Australia Received December 10, 1999. In Final Form: March 20, 2000 We present the results of direct force measurements between charged surfaces in solutions of multivalent electrolytes (MgSO4, K4Fe(CN)6, and Th(NO3)4). For surface separations greater than approximately 3 nm the measured forces between the surfaces agree, within experimental uncertainty, with the predictions of the Derjaguin-Landau-Verwey-Overbeek theory of colloidal interactions. In particular, we find that various proposed corrections to the electrostatic screening lengths in solutions of multivalent electrolytes are either invalid or beyond the limits of measurement by present techniques.
Introduction The interaction between two charged surfaces in solution is an important component of the total force between colloidal particles and hence is central to the study of the structure, stability, and rheological properties of colloidal dispersions.1 The surface layer of charge together with a diffuse cloud of counterions and co-ions in the adjacent solution is referred to as the electrical double layer (EDL). When two similar electrical double layers overlap, a repulsive force is generated. In addition to this repulsive force there is always a van der Waals force (usually attractive) between the surfaces. According to the classic Derjaguin-Landau-Verwey-Overbeek (DLVO) theory2 of colloidal interactions, the net force between two surfaces is given by the sum of the electrostatic component and the van der Waals component. Theoretical treatments of the electrostatic component of the force between charged surfaces differ in sophistication, ranging from the mean-field Gouy-Chapman theory (see, for example, ref 1) to modern statistical mechanical theories in which the techniques of liquid-state physics are applied to the double layer (for a recent review, see Attard3). Despite their differences, these theories all predict that at sufficiently large distances the electrical double layer properties are described by an exponential function of distance. For example, if we consider a flat charged surface and denote the mean electrostatic potential next to the surface by ψ(z), then ψ(z) ∼ exp(-κz), where the potential in the bulk solution is taken as zero, z is the distance from the surface, and κ is a decay constant. Similarly, if we consider the interaction of two flat double layers, then Eint(L) ∼ exp(-κL), where Eint(L) denotes the interaction energy per unit area of surface and L denotes the separation between surfaces. The inverse of the decay length, κ-1, is referred to as the screening length and is an important parameter in the description of the double layer. The screening length characterizes the range over which the perturbation due to an electrical double layer extends. In the classical GouyChapman theory the screening length, referred to as the * To whom correspondence should be addressed. Fax 61-2-6279 9614; e-mail
[email protected]. (1) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991. (2) Derjaguin, B. V.; Landau, L. D. Acta Physicochim. URSS 1941, 14, 633-52. Verwey, E. J.; Overbeek, W. J. Th. The Theory of Stability of Lyophobic Colloids; Elsevier: Amsterdam, The Netherlands, 1948. (3) Attard, P. Adv. Chem. Phys. 1996, 92, 1-159.
Debye length (or Debye-Huckel parameter) and denoted by κD-1, is given by the formula
κD-1 )
(∑ ) e
2
kT 2 B iZi ni
1/2
(1)
where k is Boltzmann’s constant, T is the absolute temperature, e is the proton charge, is the permittivity of the solvent, Zi and niB are the charge and bulk concentration of ion species i, respectively, and the sum extends over all ion species in solution. The classical Gouy-Chapman theory of electrical double layer interactions, which forms the basis of the DLVO treatment of double layer forces, has been the subject of many experimental studies.1 These studies have largely confirmed the validity of classical theory for monovalent electrolyte systems at low to moderate concentrations. In contrast, a number of experimental and theoretical studies have been published, suggesting that the Debye length may not accurately describe the decay length in solutions of multivalent electrolytes. Long ago it was observed that the effectiveness of multivalent electrolytes in destabilizing colloidal dispersions was reduced in the presence of moderate amounts of a second electrolyte.4 It was suggested that this could be due to the large deviations from ideal behavior observed in solutions of multivalent electrolytes, implying that activities should be used in place of concentrations in the formula for the Debye length (eq 1). The use of activities in place of concentrations in the formula for the Debye length has also been suggested in a study of surface forces in solutions of multivalent surfactants5 and in the calculation of the crystalline swelling of montmorillonite.6 Soil scientists working on co-ion exclusion have also considered corrections to the screening length due to the nonideal behavior of electrolytes (Quirk, J. P., private communication). At concentrations normally encountered in colloidal systems, the activity of an ion is less than its concentration (i.e., the activity coefficient, γi, is less than 1). If activities were used in eq 1 in place of concentration, the predicted Debye lengths would be longer, and the electrical double (4) Overbeek, J. Th. G. The interaction between colloidal particles. In Colloid Science; Kruyt, H. R., Ed.; Elsevier: Amsterdam, 1952; Vol. 1. (5) Rutland, M. W. Ph.D. Thesis, Australian National University, April 1992. (6) Quirk, J. P.; Marcelja, S. Langmuir 1997, 13, 6241-6248.
10.1021/la991621c CCC: $19.00 © 2000 American Chemical Society Published on Web 05/26/2000
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layer force would be expected to decay less rapidly. The activity coefficients γi for solutions of monovalent ions are very close to unity at concentrations suitable for accurate surface force measurements whereas for multivalent electrolytes the γi can deviate markedly from unity, even in relatively dilute solutions. (For example, at a concentration of 10-3 M, the values of γi for monovalent and divalent ions are approximately 0.96 and 0.75, respectively.) The nonideal behavior of electrolytes is often partially attributed to ionic association.7 The concept of ion pairs was introduced to explain inadequacies of the classical theories of activity coefficients and conductivities of solutions of multivalent ions. The anomalous behavior of multivalent electrolyte solutions could be explained by postulating that free ions existed in equilibrium with ion pairs. The equilibrium constants for such ion-pairing reactions were calculated by assuming that the classical theories of electrolyte solutions described the behavior of free ions accurately and that any remaining deviations from ideal behavior were entirely due to the formation of ion pairs. This concept of ion binding has been invoked to partially explain the results of measurements of the forces between charged mica surfaces in micellar and polyelectrolyte solutions. Pashley et al.8 measured the forces between charged mica surfaces in solutions of CTAB micelles and found that the decay length of the interaction was much longer than predicted using the formula for the Debye length. Similar conclusions were drawn by Marra and Hair,9 who measured the forces between mica surfaces in micellar solutions of sodium dodecyl sulfate and in solutions of sodium polystyrenesulfonate. In both cases deviations in the screening lengths were partially attributed to ion binding by the highly charged micelles and polyelectrolytes. The description of the nonideal behavior of electrolytes in terms of an ion-pairing contribution and a separate electrostatic contribution is of course somewhat arbitrary, and in this investigation no distinction is made between the two. Activity coefficients of ions were calculated using the equations and ion-pairing constants given by Davies.7 For concentrations less than 0.1 m, the values calculated in this manner have been shown to agree with experimental values to within 1.5%.7 Another proposed correction to the Debye length arose from theoretical studies of bulk electrolytes. According to Mitchell and Ninham,10 the screening length in solutions of asymmetric electrolytes is given by an asymptotic expansion, the leading term of which is the Debye length, κD-1. For an electrolyte of the form Czv11Azv22, where zi and ni denote the valency and stoichiometric number of ion i, respectively, they proposed that the effective screening length, κeff-1, is given by -1
κD
)1+ -1
κeff
3 3 2 Λ ln (3) (v1z1 + v2z2 ) + O(Λ3 ln Λ), 8 (v z 2 + v z 2)2 1 1
2 2
Λ)
e2 -1
(2)
κD kT (7) Davies, C. W. Ion Association; Butterworth: London, 1962. (8) Pashley, R. M.; McGuiggan, P. M.; Horn, R. G.; Ninham, B. W. J. Colloid Interface Sci. 1988, 126, 569-578. (9) Marra, J.; Hair, M. L. J. Colloid Interface Sci. 1989, 128, 511522. (10) Mitchell, D. J.; Ninham, B. W. Chem. Phys. Lett. 1978, 53, 397399.
where the symbols are as defined above and the third term on the right-hand side of the equation denotes a quantity whose value is “of the order of” Λ3 ln Λ. The authors state that the above result is valid when RCA/κ-1 D , 1, where RCA ) RC + RA is the sum of the radii of the cation C and anion A; in this regime the effective screening length is calculated by neglecting the last term in eq 2. Similar results were later presented by Kjellander and Mitchell11 and by Knackstedt and Ninham.12 Equation 2 predicts that the screening length in asymmetric electrolyte solutions will be shorter than the Debye length. To date there have only been three experimental studies directed toward testing the implications of eq 2. Kekicheff and Ninham13 measured the forces between mica surfaces in solutions of the protein cytochrome C, which they modeled as a highly asymmetric electrolyte. The measured screening lengths in these solutions were significantly shorter than predicted by eq 1 and were in reasonable agreement with values calculated using eq 2. Nylander et al.14 also applied eq 2 in their interpretation of the forces measured in solutions of insulin. Finally, Miklavcic et al.15 measured forces in solutions of calcium binding proteins which they modeled as 1:8, 1:5, 1:4, or 1:1 electrolytes, depending on the solution conditions. In contrast to the study of Kekicheff and Ninham, the screening lengths measured by Miklavcic agreed with the Debye length, calculated using eq 1. The implications of the various proposed corrections to the Debye length are clearly of great interest, given the common occurrence of multivalent electrolytes in natural and industrial systems. The aim of this investigation was to test the validity of the proposed corrections by directly measuring the forces between charged surfaces in solutions of multivalent electrolytes. In particular, it was of interest to determine whether some of the effects observed in experiments with proteins, polyelectrolytes, and micelles could be observed in solutions of relatively simple ions. With these aims in mind, we describe below the results of measurements of the forces between charged surfaces in solutions of MgSO4, K4Fe(CN)6, and Th(NO3)4. K4Fe(CN)6 and Th(NO3)4 are examples of highly asymmetric (4:1) simple electrolytes which are sufficiently soluble and stable to be used in measurements of surface forces. K4Fe(CN)6, in particular, has been used in many classic studies of electrolyte theory16 and is useful in experiments with negatively charged surfaces because the multivalent ion ([Fe(CN)6]4-) is negatively charged. Materials and Methods All chemicals used were of analytical grade and were used without further purification. MgSO4‚7H2O and NaCl were obtained from BDH, K4Fe(CN)6‚3H2O was obtained from UNIVAR, and Th(NO3)4 was obtained from M&B Laboratory Chemicals. The water used in this investigation was purified in the following manner. Tap water was passed through a Memtec “Krystal Kleen” water purification unit containing a 5 µm lambs wool prefilter to remove particulate matter, a reverse osmosis (11) Kjellander, R.; Mitchell, D. J. J. Chem. Phys. 1994, 101, 603617. (12) Knackstedt, M. A.; Ninham, B. W. J. Phys. Chem. 1996, 101, 1, 1330-1335. (13) Kekicheff, P.; Ninham, B. W. Europhys. Lett. 1990, 12, 471477. (14) Nylander, T.; Kekicheff, P.; Ninham, B. W. J. Colloid Interface Sci. 1994, 164, 136-150. (15) Miklavcic, S. J.; Thulin, E.; Jonsson, B. J. Phys. Chem. 1996, 100, 5554-5561. (16) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworth: London, 1959.
Multivalent Electrolyte Solutions membrane to remove dissolved electrolytes, and an activated charcoal filter to remove organic contaminants and dissolved gases such as chlorine. The permeate was then distilled into a Pyrex receptacle housed in a laminar flow cabinet. The water produced by this method was found to have a pH of 5.6 due to dissolved carbon dioxide and a very low bubble persistence. All glassware was cleaned by soaking in 10% NaOH for 15 min and then rinsing thoroughly with purified water followed by rinsing with AR grade ethanol and finally with purified water again. Force measurements in solutions of MgSO4 and K4Fe(CN)6 were obtained using a Digital Instruments Nanoscope II atomic force microscope. The details of the force-measuring technique have been described elsewhere17 and will not be repeated in full here. Briefly, the forces are measured between a particle that has been attached to the AFM cantilever and a flat substrate attached to a piezoelectric crystal. The total force between the surfaces is detected by the deflection of the cantilever using a laser reflectance system. Knowledge of the cantilever spring constant and the deflection of the spring for a given piezoelectric displacement allowed us to determine the separation distance between surfaces relative to the constant compliance position, where both surfaces are in contact. Silica probes were prepared by attaching 5 µm silica spheres (obtained from Polysciences) using an epoxy adhesive (Epicote 1009) to commercial cantilevers (produced by Digital Instruments). The silica substrate consisted of a 1 cm × 1 cm piece of polished silicon wafer which had been previously oxidized to a depth of 30 nm (by heating to 910 °C in purified oxygen). The cantilever spring constant was determined using a gravimetric method18 and found to be 0.18 ( 0.04 N m-1. Both the flat substrate and colloid probe were cleaned prior to each experiment by exposing the surfaces to a water plasma (10 W, 18 MHz for 45 s, PH2O ) 0.04 Torr, PAr ) 0.02 Torr). The AFM fluid cell was cleaned by rinsing with ethanol, purified water, and ethanol and then blowing dry with high-purity nitrogen. Once forces were measured in a given solution the cantilever was withdrawn from contact, and the fluid cell was flushed with the next solution at least three times with solution volumes 5 times that of the cell. The solution was then left to equilibrate at least 15 min before measurements recommenced. After AFM investigations the colloid probe was coated with a thin coating of gold and viewed in the Cambridge S360 scanning electron microscope to obtain an accurate estimate of the probe radius. Forces were then scaled with the estimated probe radius to enable a comparison to be made with theoretical force profiles (see below). Forces between mica surfaces in solutions of Th(NO3)4 were measured using the surface force apparatus (SFA). The principles behind the SFA technique are adequately described elsewhere1,19 and will not be elaborated upon here. DLVO Analysis. The experimentally measured force curves were fitted using DLVO theory, in which the electrical double layer force between the surfaces is described in terms of the Gouy-Chapman theory. The calculation of the double layer force requires the solution of the nonlinear Poisson-Boltzmann equation.1 Solutions to this equation for symmetric and asymmetric electrolytes, under both constant charge and constant potential boundary conditions, were obtained numerically using an extended version of the algorithm of Chan et al.20 The retarded van der Waals interaction between mica surfaces and between silica surfaces was obtained from calculations by Christenson21 and Meagher,22 respectively. Finally, the measured forces were scaled by a radius (characterizing the surfaces used in the (17) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature. 1991, 353, 239-241. (18) Senden, T. J.; Ducker, W. A. Langmuir 1994, 10, 1003. (19) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975-1001. (20) Chan, D. Y. C.; Pashley, R. M.; White, L. R. J. Colloid Interface Sci. 1980, 77, 283-285. (21) Christenson, H. K. Ph.D. Thesis, Australian National University, June 1983. (22) Meagher, L. Ph.D. Thesis, Australian National University, Dec 1994.
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Figure 1. Forces measured between silica surfaces in a 1.05 × 10-3 M solution of NaCl. The open circles represent the experimental values, and the upper and lower solid lines are theoretical (DLVO) curves calculated using respectively constant surface charge and constant surface potential boundary conditions. The parameters used to generate the theoretical curves were Ψ0 ) -36 mV (where Ψ0 is the surface potential at infinite separation) and [1:1 electrolyte] ) 1.05 × 10-3 M. experiments) in order to allow a direct comparison with theoretical curves, which are calculated for the interaction of two flat surfaces (see, for example, Israelachvili1).
Results and Discussion In each experiment, forces were first measured in purified water and in solutions of NaCl in order to provide a check on the reliability of the results obtained within an experiment. In terms of the corrections being considered in this work, solutions of NaCl would not be expected to display any deviations from the classical theory of double layer interactions. Indeed, forces in solutions of simple monovalent electrolytes such as sodium chloride have been measured by a variety of techniques and have been found to agree with the predictions of the DLVO theory. Forces measured between silica surfaces in the presence of 1.05 × 10-3 M NaCl (Figure 1) were found to be totally repulsive. The measured forces agreed well with the DLVO curves calculated using the stated concentration of NaCl. Furthermore, the surface potential required to fit the curves (Ψ0 ) -36 mV) agrees closely with the value obtained by Ducker et al.17 in similar experiments. Figure 1 also illustrates the fact that, for an interaction described by an exponential, the plot of force versus separation gives a straight line with slope equal to the decay constant. In particular, for forces described by the Gouy-Chapman theory the slope is equal to the Debye length (at least for sufficiently large separations: see below). The theoretical curves predict an attraction at small separations; however, as already mentioned, the measured forces were totally repulsive. This behavior is commonly observed, and various hypotheses have been proposed to explain this in terms of short-range steric forces, caused either by solvation, roughness, or a polymeric gel layer.23 This deviation from the DLVO theory at small separations does not affect the conclusions which can be drawn from the rest of the curve regarding long-range double layer forces. (23) Vigil, G.; Xu, Z.; Steinberg, S.; Israelachvili, J. J. Colloid Interface Sci. 1994, 165, 367-385.
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Figure 2. Forces measured between silica surfaces in a 4.8 × 10-3 M solution of MgSO4. The open circles represent the experimental values, and the upper and lower solid lines are theoretical (DLVO) curves. The parameters used to generate the theoretical curves were Ψ0 ) -17.5 mV, [2:2 electrolyte] ) 4.8 × 10-3 M, and [1:1 electrolyte] ) 1 × 10-5 M.
Kohonen et al.
Figure 3. Forces measured between silica surfaces in a 9.42 × 10-4 M solution of K4Fe(CN)6. The two types of symbol represent the experimental results from two separate experiments, and the upper and lower solid lines are theoretical (DLVO) curves. The parameters used to generate the theoretical curves were Ψ0 ) -18 mV, [1:4 electrolyte] ) 9.42 × 10-4 M, and [1:1 electrolyte] ) 1 × 10-5 M.
Forces in Solutions of MgSO4. Forces measured between silica surfaces in the presence of 4.8 × 10-3 M MgSO4 were found to be totally repulsive (Figure 2). As shown in Figure 2, the measured forces were described well by theoretical curves calculated using DLVO theory. The mean decay length obtained by fitting theoretical curves to 13 measured force curves was 2.4 ( 0.2 nm. This value agrees with the calculated Debye length of 2.2 nm. The activity coefficients of the Mg2+ and SO42- ions in a 4.8 × 10-3 M solution of MgSO4 were calculated to be 0.55. Using these values, the decay length calculated using activities in the expression for the Debye length was found to be 3.0 nm. This value falls outside the measured range of screening lengths, thus suggesting that using the bulk activities of ions in the expression for the Debye length (eq 1) is incorrect. In light of these results, it is natural to enquire into the reason why the use of activities in the expression for the Debye length is not correct. In fact, it can be shown that it is the ratio of the activity coefficients of an ion in the bulk and in the double layer which describes the effect of nonideal behavior (see, for example, Ruckenstein and Schiby24 and Leodidis and Hatton25). The use of activities in the expression for the Debye length is equivalent to assuming that the activity coefficients of ions within the double layer are equal to one. The fact that eq 1 appears to provide a reasonable prediction of the decay length suggests that the activity coefficients of ions in the bulk and in the double layer are approximately equal. Forces in Solutions of K4Fe(CN)6. Forces measured between silica surfaces in solutions of potassium ferrocyanide were found to be totally repulsive. Typical force curves obtained in a 9.5 × 10-4 M K4Fe(CN)6 solution are shown in Figure 3. Analysis was carried out on approximately 40 force curves at this concentration, and good agreement between curves was observed. The general conclusion drawn from this analysis was that the forces fell within the predicted limits of the interaction energy, based on DLVO theory.
Using the simple-minded approach of measuring the slopes of the semilogarithmic plots of the experimental force curves, we obtained a value of 3.5 ( 0.6 nm for the screening length. (The slopes were obtained by fitting a straight line to the force curves for separations between 3 and 9 nm using least-squares analysis.) Screening lengths were also calculated by varying the concentration of 1:4 electrolyte in the DLVO program until a good fit between experimental and theoretical curves was obtained. The “observed Debye length” was then calculated from the best fit concentration. The values of the observed Debye length obtained in this manner were found to range between 2.7 and 3.8 nm, with the majority of values lying close to the expected Debye length of 3.1 nm. At 1 × 10-3 M the activity coefficients of K+ and [Fe(CN)6]4- ions were calculated to be 0.90 and 0.19, respectively. Using the calculated activities in eq 1 gives a calculated screening length of 5.2 nm. This value falls outside of the range of screening lengths observed experimentally, confirming the fact that the use of activities in place of concentrations in eq 1 is incorrect. For a 1 × 10-3 M K4Fe(CN)6 solution eq 2 predicts an effective screening length of 2.4 nm. This value also falls outside of the range of screening lengths observed experimentally, suggesting that the result of Mitchell and Ninham is not accurate under these conditions. As mentioned earlier, the condition for the validity of eq 2 is that RCA/κ-1 D ,1, where RCA ) RC + RA is the sum of the radii of the cation C and anion A. It is important to note that the value of RCA/κ-1 D characterizing the experiments in this investigation (ca. 0.3) falls within the range characterizing the experiments of Nylander et al.14 and of Kekicheff and Ninham.13 Thus, our results, like those of Miklavcic et al.,15 suggest that the use of eq 2 may not always give accurate estimates of the screening length. A theoretical study to test the accuracy of eq 2 at finite concentrations was carried out by McBride et al.26 Their results suggest that use of eq 2 overestimates the deviation in the screening length to be expected for solutions of
(24) Ruckenstein, E.; Schiby, D. Langmuir 1985, 1, 612-615. (25) Leodidis, E. B.; Hatton, T. A. Langmuir 1989, 5, 741-753.
(26) McBride, A.; Kohonen, M. M.; Attard, P. J. Chem. Phys. 1998, 109, 2423-2428.
Multivalent Electrolyte Solutions
Figure 4. Electrical double-layer force between surfaces with Ψ0 ) +50 mV in a solution of 10:1 electrolyte with a Debye length of 5 nm. The upper and lower solid lines are theoretical curves calculated assuming constant surface charge and constant surface potential boundary conditions, respectively. The shaded line is an exponential with decay length equal to the Debye length.
asymmetric electrolytes. For a 1:4 electrolyte solution at 1 mM concentration they estimate a reduction in Debye length of about 17%. A numerical calculation of the double layer forces in solutions of asymmetric electrolytes reveals another possible complication involved in experimental studies directed toward testing eq 2. Figure 4 demonstrates the fact that for asymmetric electrolytes there is a large difference between the apparent decay length (obtained from fitting a straight line to a semilogarithmic plot of the force curve) and the Debye length. This figure shows clearly that the slopes of the interaction curves calculated on the basis of Gouy-Chapman theory only approach the value given by the Debye length in the limit of large separations and weak forces. This suggests that for asymmetric electrolytes the asymptotic regime of exponential forces may be at the limits of present force measuring techniques. Forces in Solutions of Th(NO3)4. Figure 5 shows the experimental and theoretical force curves for a 1 × 10-3 M solution of Th(NO3)4. The fact that a positive surface potential (Ψ0 ) +55 mV) was required to fit the forces suggests that the multivalent Th4+ ion has been adsorbed on the surfaces. The adsorption of multivalent cations on mica surfaces has previously been observed by Pashley27 in measurements of forces in solutions of Cr(NO3)3 and LaCl3. (27) Pashley, R. M. J. Colloid Interface Sci. 1984, 102, 23-35.
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Figure 5. Forces measured between mica surfaces in a 1.0 × 10-3 M solution of Th(NO3)4. The three types of symbol represent the experimental results from three separate positions of contact between the surfaces and the upper and lower solid lines are theoretical (DLVO) curves. The parameters used to generate the theoretical curves were Ψ0 ) +55 mV, [4:1 electrolyte] ) 1.0 × 10-3 M, and [1:1 electrolyte] ) 5 × 10-6 M.
The conclusions drawn from an analysis of the forces measured in solutions of K4Fe(CN)6 were supported by the measurements in Th(NO3)4 solutions. Because of the limited amount of experimental data, a full analysis of the Th(NO3)4 results was not attempted. The close agreement between the experimental results and the theoretical curves calculated using the nominal concentration of Th(NO3)4 suggests that the long-range doublelayer forces in such solutions are adequately described using the classical Gouy-Chapman theory. Conclusions Analysis of direct force measurements in solutions of MgSO4, K4Fe(CN)6, and Th(NO3)4 suggests that the use of activities in place of concentrations in the equation for the Debye length leads to incorrect conclusions regarding the double-layer forces in such solutions. The failure of corrections based on the use of activities in this case is probably due to the fact that, although it allows for deviations from ideality of the ions in bulk solution, it neglects these deviations for ions in the double layer. Our results represent the first test of the proposed use of activity corrections for screening lengths in solutions of simple asymmetric electrolytes and provide a contrast to previous studies which focused on protein solutions. LA991621C
