J. Phys. Chem. 1994, 98, 4320-4326
4320
A Practical Approach to Solving the Double-Layer Problem That Includes Effects of Ion Size and Correlation Stanley J. Miklavic' and Phil Attardf Department of Food Technology, Chemical Center, University of Lund, P.O. Box 124, S-221 00, Lund, Sweden Received: July 20, 1993; In Final Form: February 7 , 1994"
We consider the hypernetted chain approximation to the interaction free energy between charged walls in a monovalent electrolyte, emphasizing the practical advantage of using the mean spherical approximation for the bulk ion correlation functions (the MSA theory). The decay rate of the interaction potential is largely determined by the bulk ion closure assumed, and in comparison with other closures, the asymptotic decay of the MSA theory is more rapid than given by the classical Debye length, agreeing well with the decay in the hypernetted chain (HNC) and the improved H N C (including bridge functions) models and with exact calculations. Common to both the MSA and H N C theories, the magnitude of the asymptotic interaction potential is underestimated, due to the inaccuracy in the hypernetted chain wall-ion closure approximation. Because of the semianalytical nature of the MSA closure, the MSA is numerically more appealing for practical implementation in an analysis of experimental data. For such purposes the Poisson-Boltzmann theory and the MSA theory provide lower and upper estimates, respectively, of the surface charge. By using both in comparison, one obtains a far better appreciation of the true value of the surface charge operating in real circumstances.
Introduction Classical DLVO theory' has long remained the theoretical cornerstone for our understanding of, for example, the interaction between colloidal particles, be they solid (such as silica or latex particles), "soft" (such as micelles, vesicles, or lamellae2), or flexible biological macromolecules such as DNA.3s4 In many cases though, DLVO theory, based as it is on the mean-field, Poisson-Boltzmann (PB) descriptionof the electrical double layer, has come up short in its predictivecapacity.5-8 This has stimulated enormous effort to establish the limits of validity of the PB model and to scrutinize the origin of some of the physical phenomena which appear in real experimental situations and are present only in more exact, but sophisticated, treatments of the double layer.9 The most notable of these effects are a substantial attraction between highly charged macroscopicparticles in strongly coupled electrolytes1° (multivalent ions and/or high concentrations) and the appearance of an inversion of the double layer with increasing concentration.9 At present, a most frustrating situation exists in which many of these systems of technical and fundamental interest clearly lie outside the regime of validity of the PB model, while access to analysis using potentially valid mathematical theories" or numerical simulation12is not immediately forthcoming. This situation has been made all the more obvious over the past 20 years with the ability of direct partic1e~urface'~J~ or surfacesurface interactions1Sz3being measured simply and accurately, with the possibility of theoretical analysis remaining considerably more remote. Ironically, even for the simplegeometriccase of planar charged walls (one or two), the PB equation lacks an analytical solution for asymmetric electrolytes where one or more of the ions have valency greater than two. Furthermore, for cases involving the two charged walls, these analytical solutions are expressed in terms of elliptic functions.2k26 For other cases, a numerical treatment is necessary, a pursuit which is not always straightforward, due to the inherent numerical instabilityof the nonlinear equation which worsens progressively as valency increases. ~ _ _ _ _
~~
~~~
* To whom correspondence should be addressed.
Department ofPbysics, FacultyofScience, AustralianNationalUniversity, Canberra, ACT, 0200, Australia. Abstract published in Advance ACS Absrrocrs. March IS, 1994.
0022-3654/94/2098-4320%04.50/0
It is quite evident that theoretical research in this field will continueunabated in pace at least until simpleand accurate means of calculating properties of the electrostatic double layer are developed for use at the experimental laboratory level. In quite recent times, efforts in a new direction have been made by Henderson,Z7Lozada-Cassou and co-workers,2*J9 and Attard et a1.,'0 who have stressed a compromise between "exactnessn of modeling and expedience of calculation. In this theme, direct use is made of the electrolyte properties of a single double layer in the evaluation of the interaction of two opposing double layers. This is commonly referred to as the singlet approach, although there aredifferencesin implementationbetween thevarious works. Lozada-Cassou treats the two charged walls as a molecular dumbbell which is, in principle, a more accurate approach than that taken by Attard et al. and Henderson, who treat the walls as two distinct molecules. Generally, this treatment is obviously more approximate than either direct simulation12 or inhomogeneous integral equation theory," but with the twofold aim of improving upon the PB theory and remaining simple to evaluate, it seems an appropriate approach to take. In a previous paper,31we derived the interaction free energy between charged walls in an electrolyte within the singlet level of approximati~n~~ and explored its accuracy by comparison with more sophisticatedtheories. Henderson27 independently derived a similar expression from the same premises, valid for bodies of small but finite curvature. Characteristic of our work was the use of the hypernetted chain approximation to effect closure of the integral equation system for the bulk ion-ion correlation functions. Considering that other closure approximationsexist: we shall, in this paper, present results found using a simpler theory, namely, one based on the mean spherical approximation for the bulk ion correlation functions. This model possesses an analytical solution which leaves, as the main numerical task in the current theory, the iterative solution of the wall-ion equations for the isolated double layer. The calculations are therefore sufficiently tractable for the analysis of experimental data.13J6,2G23The benefit of this approach is that it includes realistic effects of ion size and of electrostatic correlations while remaining no more complicated than the traditional numerical calculations of the nonlinear Poisson-Boltzmann equati0n.~3 (Interested readers should also be aware of Henderson and Lozada-Cassou's simpler 0 1994 American Chemical Society
Solution of the Double-Layer Problem
The Journal of Physical Chemistry, Vol. 98, No. 16,1994 4321
but cruder calculationemployinga phenomenologicaltheory based on the linear PB model of the electrical double layer neglecting ionic sizes.49 There is also the density functional approach of Stevens and RobbinssOon point charge electrolytes.) In addition to comparing the mean spherical model to the hypernetted chain theory (with and without bridge diagrams34.35), we also give attention here to the decay length of the force and the effective surface charge, key quantities deduced from experiments on interacting double layers.
and
is the short-ranged contribution to the bulk ion-ion direct correlation function. The first term in (3) involves the mean electrostatic potential, $(z),
Analysis
As in our previous paper, we consider a restricted primitive model electrolyte with ionic species of charge valency, z,, bulk density,pa, and equal hard-sphereradius, R. Equation 1 expresses the interaction free energy per unit area, F ( z ) , acting between two hard walls, each uniformly charged at a density, u, and separated by a distance z + 2R (with z being the width of the region available to the centers of the ions). This expression was derived by HendersonZ7and Attard et al.,30 who considered the large radius, R, infinite dilution limit of two interacting solute species. The interaction free energy (or interaction potential) consists of the direct electrostatic interaction between the walls, the wall-wall bridge function, and the Omstein-Zernike wallwall series function."
$(O) - ~ T U Z / C -R < z < 0 = $(O) + ~ T U R / Cz < -R
(8) Expressing the wall-ion direct correlation function, CJz), as the sum of a short-range contribution, C,(z), and the wall-ion electrostatic potential
with
7
z 1 0 (1)
Here, k T = 1/j3 is the thermal energy and V(z) is given by
V(Z) = ( ~ T U ' / E ) ( % - IZ
+ 2RI) + d ( 0 )
(2)
we have shown31 that eq 1 can thus be rewritten @F(z)
+
~ u $ ( z R ) -D(z) C p Y K H Y ( z ' )c",(z
- z') dz'
z 1 0 (1 1)
Y
The choice of the arbitrary constants in eq 2 (i.e., W ,which is related to the lateral size of the plates in the limiting procedure, and $(O), the value of the mean electrostatic potential at the plane of closest approach of the ions to the wall) has no physical significance but is made so that the interaction potential decays to zero at large separations. In (1) H 7 ( z ) is the wall-(?) ion total correlation function and C,(z) is the wall-(?) ion direct correlation function. D ( z ) is the wall-wall bridge function which makes eq 1 formally exact. From liquid-state theory it is found that the ion density profile adjacent to a charged wall p,(z) = p,(l H,(z)) can be approximated by the solution of the integral equation,36
+
ln[l
+ H,(z)] = - B e z , W + C P Y L C Y ( l Z- Yl)HYWdY
z 2 0 (3)
Quite clearly, all dependent quantities on the right-hand side of (1 1) are properties of the electrolyte adjacent to an isolated wall. Hence, to arrive at the interaction potential between surfaces, for any arbitrary separation, one does not require any numerical considerationsbeyond evaluating one-dimensional integrals over the same functions, once these have been determined. It is interesting to note that the first term of (1 1) has the form of a purely electrostatic energy of interaction of one wall of charge with the potential of the other, used by Brenner and Parsegian in their analysis of the interaction of charged r0ds.3~ In view of the realistic properties of the double layer, this term can obviously be attractive as well as repulsive, depending on whether the mean potential profile is monotonic or oscillatory.9 Considering the exact result (4), the integral in eq 11 can be broken up into three terms, according to the argument of C,,
Y
together with the impenetrability condition, H , ( z ) = -1
z30
-
-
15
0
200
30
45
Separation (A)
60
75
m e 2. Comparison of interaction potentials: solid line, NLPB long dashed line, HNC; short dashed line, HNCD dotted line, MSA; dotdashed line, PB. The parameters here are e/# = 250 A2,e = 78.5,and R = 2.125 A. The salt concentration is 0.1 M.
TABLE 1: Comparison of Different Integral Equation Closure Theories, with Nonlinear Pofseoa-Boltzmann Model e/#
NLPB PB MSA
HNC NLPB PB MSA
HNC HNCD NLPB PB MSA
HNC HNCD NLPB PB MSA
HNC HNCD
= IO00 A2,
p
= 0.01M, R = 2.125 A
0.00879 8.356 0.00549 8.417 0.00534 8.311 0.00544 R = 2.125 A 0.0380 2.630 -0.9279 12.87 0.0261 2.500 -0.9355 12.94 0.0233 2.458 -0.9267 12.79 0.0243 2.470 -0.9291 12.77 0.0305 e / u = 60 A=, p = 0.1 M, R = 2.3 A 0.123 5.344 -0.9952 208.3 0.207 4.338 -0,9990 196.8 0.143 3.828 -0.9963 208.19 0.142 5.457 -0.9997 205.14 0.336 e / u = 60 A2, p = 0.5 M, R = 2.3 A 0.141 3.771 -0.9770 42.44 0.150 2.986 -0.9944 43.92 0.098 2.782 -0.9923 42.58 0.110 3.251 -0.9952 42.60 0.184
2.236 -0.8931 2.228 -0.8747 2.2037 -0.8897 e/# = 250 A2, p = 0.1M,
A), again agreeing with the exact NLPB theory, which at this stage can no longer be considered reliable in representing the force behavior. At the surface charge e / u = 250 A2 (0.064C m-2) and electrolyte concentration of 0.1 M, the nonlinear PB is now at best qualitative?1tu being higher than predicted by the self-consistent anisotropicintegral equationtheory (AHNC). The MSA bulk ion closure introduces corrective terms, mainly associated with finite ion size.36 The HNC bulk ion closure provides a (strong) correction for ion-ion correlation as well as ion size. Of these, it has been found by comparison with Monte Carlo data for the isolated wall system that the HNC/MSA approximation agrees best with the exact numerical simulation, lying between the PB/S and HNC/HNC models.39 At this concentration this property of the MSA, of being intermediate, iscamed through totheinteractionpotential. At largeseparations it is therefore probably a better approximation to the exact result than either of the singlet PB or singlet HNC. Results for the still higher surface charge ( e / u = 60 Azor u = 0.267 C m-2) with 0.1 M salt present are compared in Figure 3 with the integrated osmotic pressure data obtained with the AHNC theory.* Here it can be seen by how much the singlet HNC underestimates the magnitude of the repulsion in the asymptotic region and how it predicts an attraction at shorter separations (< 30 A) which is still nonexistent in the AHNC
Miklavic and Attard
4324 The Journal of Physical Chemistry, Vol. 98, No. 14. 1994 9 -
-
9
7 -
.-2
5 -
B
3 -
I 6
P
3
3
1 -
5
-d
0
25
50
Separation (A)
15
Figure 3. Comparison of interaction potentials. Curve descriptions are as in Figure 2; the symbolsreprtsent integratedself-consistent anisotropic HNC osmotic pressure data.* The parameters here are e / a = 60 A2,c = 78.5, and R = 2.3 A. The bulk concentrationis 0.1 M. The vertical axis is plotted on an arcsinh scale.
theory. The singlet HNCD, on the other hand, serves to overcompensatefor the attraction at short separationsbut provides a very good account of the pressure profile asymptotically. A point here is that the NLPB seems to have the correct order of magnitude, close to Kjellander and Marcelja's integral equation results, although in finer detail the AHNC is more repulsive at short separations as a result of finite size effects and less at larger separationsdue to the more effective screeningof surface charges by counterions implicit in that theory. There are too few AHNC points to implicate a decay different from that of the NLPB as is suggested by the singlet HNCD. Again, we can point out the merging of the singlet PB with the NLPB (beginningat a distance of about 40 A = 4 Debye lengths). Between about 50 and 80 A both of these agree with HNCD, but beyond 80A the latter, with its very slightly different decay rate, diverges. Except for the HNCD, all other models predict exponential decay with decay length equal to the Debye length of this concentration. Over the separation range examined, Le., of 25 A and beyond, the singlet MSA lies much closer to the HNC theory in magnitude than it does to the grouping of the other three. This can only be a reflection on the hypernetted chain approximation for the wallion profiles, present in both theories, which is unsuitable in some regimes of concentration depending on surface chargea9 At this concentration, the (erroneous) attractive interaction predicted by the uncorrected wall-wall interaction potential is largely insensitive to the nature of the bulk ion-ion closure relations. It is only by correcting (improving) the hypernetted chain wall-wall potential, say, by including the bridge function term, that this error, characteristically present at separations of between 0.5 and 3 Debye lengths, can be removed. We have already found, by comparisonof numerically derived HNC and HNCD pressures with those of the AHNC," that at theeven higher concentrationof 0.5 M (K-1= 4.3 A) the prediction of the singlet HNC (as well as the exact NLPB theory) begins to exhibit qualitative differences to the more accurate data for separations less than 10 Debye lengths. In particular, the true force decays slightly more rapidly than either of these a p proximations, which either still have the classical Debye length decay or are close to it. On the other hand, the singlet HNCD has both the correct slope and magnitude in this range. Naturally, the interaction potentials behave accordingly. In Figure 4 show these in comparison with integrated AHNC pressures. Once again, the singlet PB is all but identical to the NLPB solution (now not shown) beyond 15 A separation (4 K-1). Unlike the 0.1 M concentrationcase, the effect of wall-ion closure approximation is here much more significant to the transition point (repulsion -attraction) as well as the magnitude of the repulsive maximum prior to the transition: this is especially true of the PB closure
12.5
0
100
25
separation
(A)
31.5
50
Figure 4. Comparison of interaction potentials. The parameters here are e / a = 60 A2, t = 78.5, and R = 2.3 A; salt concentrationis 0.5 M: solid line, HNC; long dashed line, HNCD; short dashed line, MSA, dotted line, PB. The symbols represent integrated anisotropic HNC osmotic pressures (unpublished data, courtesy of Drs.R. Kjellander and S. Marcelja). 12
h
,
I
8
3 2
4
I
E
6
'i
1
0
-8 -4
.-
0
IO
20 Separation
30
(A)
40
50
Figure 5. Comparison of interaction potentials. Curve descriptions art as in Figure 2. The parameters here are e / a = 48 AZ,t = 78.5, and R = 2.3 A; salt concentrationis 0.5 M. The vertical axis is plotted on an arcsinh scale. (c$ = 0) compared to the other three approximate closures. As we said above, we also begin to see that it is the nature of the S;C ; that determines the decay rate of the force. The exponential decay of the singlet PB, which is identical to NLPB, shows the classical Debye length, but it is clear that both the HNCD and MSA forces decay noticeably more sharply, while the decay of the HNC profile appears less different. Once more, it is only by correcting the hypernetted chain wallwall approximation that the exaggerated attraction at short separations can be removed. However, it appears that in this regime a quantitativelyaccurate theory will require further bridge diagrams in addition to the first one that is used here. In Figure 5 we go to the extreme in surface charge density, e / a = 48 A2 (0.5 C m-2), which is the value appropriate to muscovite mica, used more often than not as substrate in surface force measurements.1c19 It is again the case that the singlet PB calculation is identical to the self-consistenf NLPB theory, diverging only at extremely short separations (less then 12 A), demonstrating the utility of the singlet PB approach. In fact, all curves show the same qualitative features as for the previous surface charge which is therefore not implicated in any of the qualitative effects mention above. In absence of any exact data for comparison, we conjecture, on the basis of the previous case, that the HNCD model provides a good account of the true force profile. The MSA model reproduces the correct decay length, in good agreement with the decay of the HNCD.
Solution of the Double-Layer Problem
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4325
* 01 0
I 50
100
I50 Separation (A)
200
250
300
Figure 6. Comparison of experimental surface force measurements for the interaction of two mica surfaces across a 7 mM NaCl electrolyte4' using MSA theory. The parameters (surface charge and ion diameter) used in the MSA calculations are e/o = 85 AZ and R = 2.7 A. The nonlinearPoisson-Boltzmanncurve (NLPB)has been calculated assuming a surfacecharge (potential)of e / u = 150 A* (160 mV). The exponential asymptotic decay of the MSA curve has been extrapolated to smaller separations (passed the region where the MSA theory predicts an attractiion) in order to ensure a correct fit to the data.
estimate of the actual surface charge, complementing the lower estimate given by the Poisson-Boltzmann theory. The singlet model, eq 11 without bridge function, gives the exact asymptotic description of electrostatic forces corresponding to the properties of a single electrical double layer and bulk electrolyte, however these be calculated! An example of this is given by the singlet PB system in which the assumptions (of both the bulk electrolyte and those of the isolated electrical double layer) are consistent with those of the full, two-wall NLPB problem. That the two are equivalent asymptotically is therefore not surprising; the decay of the force is characterized by the Debyelengthwhich isdetermined by the bulksolution properties, while the magnitude of the force is governed by the double layer which has an expanded, monotonic form as a result of the neglect of the electrostatic and hard-core correlations. Turning to the corrected theories, in particular to the HNC and MSA models, it is evident that the hypernetted chain approximation to the wall-ion distribution function is largely responsible for the prediction, generally, of weaker forces at high electrolyte concentrations. At the same time, it is by comparing these two theories that one first appreciates the dependence of the force decay on the nature of the bulk direct pair correlation function. It is necessary for the singlet theory to be successful that accurate models for the wall-ion and bulk ion distribution functions be supplied (such as with the HNCD profiles). It is our feeling that it should be in this direction that steps be taken in future work.
We should point out that the bridge diagram correction to the hypernetted chain wall-wall potential of mean force is a relatively short-ranged function which contributes nothing to the longrange nature of the force (Le., for separations beyond about 5 Summary and Conclusions Debye lengths). Calculations we have performed with and without the bridge function in eq 11 using the same wall-ion profiles In summary, we have given consideration to an alternative, confirm this (data not shown). Since in the majority of cases approximate approach to calculating the interaction free energy experimental concern is with the behavior of the force over much between charged planar surfaces with an intervening monovalent larger separations, one can overlook the lack of a better correction electrolyte. The advantage of this approach is that one need only to the wall-wall potential on first consideration. Keep in mind provide information about the nature of the electrical double the manner of analysis of surface force experiments where the layer adjacent to an isolated charged wall, in order to obtain the exponential decay and the magnitude of the asymptotic force are interaction potential between two walls. Within the approximaused to characterize bulk (concentration) and surface ( a ) tion there is scope to implement different orders of closure for properties, respectively. As an example of the application of the bulk electrolyte ion correlations, and we have studied the effects present theory to an experimental situation, we show in Figure these have on the interaction potential. 6 results of surface force measurements of the electrical doubleThere are two main conclusions to be drawn from our analysis. layer interaction between two crossed cylindrical mica surfaces For those wishing to pursue with the PB model of the double immersed in a 7 mM NaCl s0lution.~5 We compare this layer, considerable numerical saving can be achieved in calculating experimental data with the interaction free energy results obtained the double-layer interaction free energy by using the singlet with the MSA model, using the Derjaguin appro~imation~6.~~approach, adopting the single wall Poisson-Boltzmann doubleF/R = 2rF, where R is the average radius of curvature of the layer profile. However, as very little more effort is required to cylinders and F is the measured force between them. While a implement the HNC/MSA set of equations, with the substantial fit to the data using the NLPB theory can be made if one invokes reward of including some measure of ion correlations and finite a surface charge (potential) of e / a = 150 Az (160 mV) and no size, it seems that this model is to be recommended. Since their ion size, an asymptotic fit using the MSA theory can also be magnitude is similar but the MSA appears to give a better achieved using the more reasonable parameters of e / a = 85 Az representation of the force decay than does the HNC, the former (163 mV) and assuming a finite ion diameter of 5.4 A.48 Even is more desirable for the practical implementation of experimental so, MSA lines assuming a fully charged mica surface ( e / . = 48 analysis. In fact, with the MSA theory, in comparison with the A2) are not too far off this experimental line and lie within Poisson-Boltzmann model, one already has a more realistic view experimental uncertainty. Despite the artifactual attraction of the values of the effective surface charge as well as decay predicted by the hypernetted chain approximation to the walllength. wall interation potential, the singlet MSA model is without doubt On a more technical level, it is our understanding that the useful in the analysis of experimental data, it being a simple decay rate of the interaction potential is largely determined by matter to extrapolate (as we have done) the exponential tail to the type of closure assumed for the bulk electrolyte correlation smaller separation in order to ensure the fit. Although the function but is only significant, at least for monovalent electrolytes, quantitative accuracy of the MSAis not fully known in this case, a t high concentrations. It appears that the asymptotic decays of we can take comfort in the fact that a t such low concentrations the HNCD and MSA theories are more sharp than the customary the HNC/MSA theory remains in very good agreement with Debye length, while the HNC model, probably for fortuitous Monte Carlo simulation as surface charge increases (even while reasons, is less so. At high concentrations, the magnitude of the the HNC/HNC theory deteriorates; see, e.g., Figures 6 and 7 asymptotic interaction potential is underestimated in both the of ref 9) and is therefore likely to provide a good asymptotic HNC and MSA theories due to the inaccuracy involved in the account of the interaction potential here. Now, in regard to what hypernetted chain wall-ion closure approximation of the isolated wall problem. Finally, the hypernetted chain wall-wall apcan be said quantitatively of the experimental situation, it is clear from our results that the MSA theory provides a ready upper proximation predicts an attraction independent of the closure
4326 The Journal of Physical Chemistry, Vol. 98, No. 16, 1994
models in the narrow separation range of 0.5-3 Debye lengths. More accurate predictions require a better closure model at this level of the theory.
Acknowledgment. We thank Drs. Roland Kjellander and Stjepan Marcelja for providing raw AHNC data, both published and unpublished, with which we made our comparison. References and Notes (1) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (2) Jtinsson, B.; Wennerstrtim, H. J. Phys. Chem. 1987,91, 338. (3) Rau, D. C.; Lee, B.; Parsegian, V. P. Proc. Natl. Acad. Sci. U.S.A. 1984, 81, 2621. (4) Rau, D. C.; Parsegian, V. P. Biophys. J. 1992, 61, 246. (5) Torrie, G. M.; Valleau, J. P. Chem. Phys. Lett. 1979, 65, 343. (6) Guldbrand, L.; JBnsson, B.; Wennerstrtim, H.; Linse, P. J. Chem. Phys. 1984,80, 2221. (7) Kjellander, R.; Marcelja, S . Chem. Phys. Lerr. 1984, 112, 49. (8) Kjellander, R.; Marcelja, S . J. Phys. (Paris) 1988, 49, 1009. (9) Carnie, S. L.; Torrie, G. M. Adu. Chem. Phys. 1984, 56, 141. (10) Kjellander, R.; Akesson, T.; JBnsson, B.; Marcelja, S.J . Chem. Phys. 1992, 97, 1424. (11) Kjellander, R.; Marcelja, S . J. Chem. Phys. 1985,82, 2122. (12) Akesson, T.; Jtinsson, B. Electrochim. Acta 1991, 36, 1723. (13) Ducker, W. A,; Senden, T. J.; Pashley, R. M. Nature 1991,353,239. (14) Butt, H.-J. Biophys. J . 1991, 60, 1438. (15) Israelachvili, J. N.; Adams, G. E. J. Chem. SOC.,Faraday Tram. 1 1978, 74, 975. (16) Parker, J. L.; Christenson, H. K.; Ninham, B. W. Rev. Sci. Instrum. 1989, 60, 3135. (17) Israelachvili, J. N. Chemrracrs: Anal. Phys. Chem. 1989, 1, 1. (18) Israelachvili, J. N.; McGuiggan, P. M. J . Mater. Res. 1990,5,2223. (19) Pashley, R. J . Colloid Interface Sci. 1981, 83, 531. (20) Horn, R. G.; Clarke, D. R.; Clarkson, M. T. J . Mater. Res. 1988, 3, 413. (21) Grabbe, A.; Horn, R. G. J . Colloid Interface Sci., in press. (22) Parsegian, V. A.; Rand, R. P.; Fuller, N. L.; Rau, D. C. Methods Enzymol. 1986,127, 400. (23) Parsegian, V. A,; Rand, R. P.; Fuller, N. L. J. Phys. Chem. 1991, 95, 4777.
Miklavic and Attard (24) Attard, P.; Mitchell, D. J.; Ninham, B. W .J. Chem. Phys. 1988,88, 4987. (25) Attard, P.; Mitchell, D. J.; Ninham, B. W. J . Chem. Phys. 1988,89, 4358. (26) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971,31,405. (27) Henderson, D.; Abraham, F. F.; Barker, J. A. Mol. Phys. 1976,31, 1291. Henderson, D. J . Chem. Phys. 1992,97, 1266. (28) Lozada-Cassou, M. J. Chem. Phys. 1984,80,3344. Lozada-Cassou, M.; Henderson, D. Chem. Phys. Lett. 1986, 127, 392. (29) Lozada-Cassou,M.;Diaz-Herrera, E. J . Chem. Phys. 1990,92,1194; 1990, 93, 1386. (30) Attard, P.; Berard, D. R.; Ursenbach, C. P.; Patey, G. R. Phys. RW. A 1991,44, 8224. (31) Attard, P.; Miklavic, S. J. J. Chem. Phys. 1993,99,6078. (32) Henderson, D.; Plischke, M.J. Phys. Chem. 1988,92, 7177. (33) Chan, D. Y.C.; Pashley, R. M.; White, L. R. J. Colloid Interface Sci. 1980, 77, 283. (34) Ballone, P.; Pastore, G.; Tosi, M. P. J. Chem. Phys. 1986,85,2943. (35) Baquet, R.; Rossky, P. J. J. Chem. Phys. 1982, 79, 1419. (36) Carnie, S.L.; Chan, D. Y. C.; Mitchell, D. J.; Ninham, B. W. J . Chem. Phys. 1981, 74, 1472. (37) Brenner, S. L.; Parsegian, V. A. Biophys. J. 1974, 14, 327. (38) Waisman, E.; Lebowitz, J. L. J . Chem. Phys. 1972, 56, 3086. (39) Carnie, S . L. Mol. Phys. 1985, 54, 509. (40) Abramowitz, M.;Stegun,I. A. HandbookofMathemrrticalFunctionr; Dover: New York, 1965; Chapter 25. (41) Jeffreys, H.; Jeffreys, B. Methods of MothemaffcalPhysics,3rd ed.; C U P Cambridge, 1956; Chapter 9. (42) Kjellander, R.; Mitchell, D. J. Chem. Phys. Phys. Left. 1992, 200, 76. (43) Attard, P. Phys. Rev. E 1993, 48, 3604. (44) Marcelja, S.Biophys. J . 1992, 61, 1117. (45) Petrov, P.; Miklavic, S. J.; Nylander, T. J. Phys. Chem. 1994, 98, 2602. (46) Derjaguin, B. V. Kolloid 2.1934, 69, 155. (47) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1991; Chapter 10. (48) This value has been argued to be appropriate for hydrated cations. Kekicheff, P.; Marcelja, S.; Senden, T.; Shubin, V. J. Chem. Phys. 1993,99, 6098. (49) Henderson, D.; Lozada-Cmu, M. J. Colloid Interface Sci. 1986, 114, 180. (50) Stevens, M. J.; Robbins, M. 0. Europhys. Lett. 1986, 12, 81.
