J. Phys. Chem. 1982, 86, 3251-3257
325 1
Electrical Double Layers. 4. Limitations of the Gouy-Chapman Theory G. M. Torrle Depertment of Metl"atlcs and Computer Sclence, Royal Mllltary College of Canada, Kingston, Ontarlo, Canada K7L 2W3
and J. P. Vallead lash Mllkr Chemical Laboratories, Unlverslty of Toronto, Toronto, Ontario, Canada M5S 1A 1 (Received: February 18, 1982)
This paper reports Monte Carlo (MC) calculations of the charge distribution and mean potential of the diffuse double layer for the restricted primitive model of 2:l and 2 2 aqueous electrolytes next to a uniformly charged plane surface. In the latter case the parameters that we have used in the primitive model correspond also to 1:l electrolytes in nonaqueous solvents. The MC results are compared with the modified Gouy-Chapman (MGC) and modified Poisson-Boltzmann (MPB) theories of these systems for surface charges up to 25 pC cm-2 and electrolyte concentrations up to 0.5 M. When the counterions in the 2:l system are singly charged, the success of the MPB theory and the relatively mild shortcomingsof the MGC theory are simiiar to those found previously for 1:l electrolytes. For doubly charged counterions in both the 2:2 and 2:l systems, however, the MC results show the MGC theory to be totally inadequate; the MGC greatly overestimates the surface potential even at quite low surface charges and electrolyte concentrations. In fact, the MC diffuse-layer potential as a function of surface charge has an extremum, and oscillations occur in the ion densities at concentrations as low as 0.05 M. The MPB theory is successful in predicting some of these qualitative features but differs quantitatively from the MC results in several respects.
I. Introduction A very old problem in physical chemistry, that of the surface behavior of a system of charged particles (or electrical double layer), has recently been the subject of a fresh burst of theoretical activity.'-1° This activity has been based on the modern methods of the statistical mechanics of the liquid state, particularly computer simulation?1° modified Poisson-Boltzmann theory,"' and integral equations.l4S6ts One objective of this work is to assess the accuracy of the classical double-layer theory of Gouy'l and Chapman12in a way free of the ambiguities involved in comparing a theory of so simple a model-point ions, planar charged surface, continuum solvent-with properties of real systems. The computer experiments must be carried out with finite-sized ions but apart from this complication provide numerically exact results for the model (Le., the Hamiltonian) implicit in the Gouy-Chapman theory. For parameters appropriate to 1:l aqueous electrolytes at moderate concentrations (up to 0.1 M) and surface charges of experimental interest (130pC cm-2),ion size turns out not to be an important considerationg and the discrepancies between the computer experiment results and the classical theory can be largely attributed to the mean field approximation of the latter. In fact, for the conditions mentioned above the errors in the theory are small for the ion densities and not very large, typically 10% or less, for the diffuse-layer potential? This success of the so-called modified Gouy-Chapman (MGC) theory was somewhat surprising, since the Pois(1)L. Blum, J. Phys. Chem., 81, 136 (1977). (2)D.Henderson and L. Blum, J. Chem. Phys., 69,5441 (1978). (3)D.Hendersonand L. Blum, J. Electroaml. Chem., 93,151 (1978). (4)T.L. Croxton and D. A. McQuarrie, Mol. Phys., 42, 141 (1981). (5)C. W.Outhwaite, Chem. Phys. Lett., 7,636 (1970). (6)S.Levine and C. W. Outhwaite, J. Chem. SOC.,Faraday Trans. 2, 74,1670 (1978). (7)C. W.Outhwaite, L. B. Bhuiyan, and S. Levine, J. Chem. Soc., Faraday Trans. 2,76,1388 (1980). (8)S . L. Carnie, D. Y. C. Chan, D. J. Mitchell, and B. W. Ninham, J. Chem. Phys., 74,1472 (1981). (9)G.M. Torrie and J. P. Valleau, J. Chem. Phys., 73,5807 (1980). (10)G.M. Torrie, J. P. Valleau. and G. N. Patev. J. Chem. Phvs.. 76, 4615 (1982). (11)G.Gouy, J. Phys. Radium (Paris),9,457 (1910). (12)D.L. Chapman,Philos. Mag., 26,475 (1913). 0022-3654/82/2086-3251$01.25/0
son-Boltzmann equation on which MGC theory is based is well known to be inconsistent with the postulates of equilibrium statistical mechanics.13 That the partial cancellation of errors behind this relatively good agreement of theory with the computer results may not extend to stronger ionic interactions such as those in charge-asymmetric systems is suggested by some recent calculations of Bhuiyan, Outhwaite, and Levine.14 The present paper is a report of a further investigation of this question by means of Monte Carlo calculations for the restricted primitive model of 2:l and 2:2 electrolytes at a planar charged surface with parameters which correspond to aqueous solutions at 25 "C. It is worth noting that the results for the 2:2 aqueous model apply equally to the 1:l electrolyte in a nonaqueous solvent of lower dielectric constant, since in the restricted primitive model the interactions are the same in each case. In the following section we describe the models that we have studied and discuss those aspects of the Monte Carlo calculation that differ from our previous simulations of 1:l electrolytes. The results of these calculations are presented in section 111, where they are compared with the classical GouyChapman theory and with the recent modified PoissonBoltzmann (MPB) calculation of Bhuiyan et al.14 The final section is a summary of our findings.
11. Monte Carlo Method The Monte Carlo calculations are carried out in the grand canonical ensemble for a collection of ions confined to a rectangular prism of dimensions W X W X L. In our previous MC work on 1:l electrolytes, all of the surface charge resided on the impenetrable wall at z = 0; the cell is considered to repeat periodically in the x and y directions and is closed by a second impenetrable wall at z = L that carries no charge. This abrupt truncation of the system at z = L will perturb the ion densities, of course; but, when the electrolyte is symmetric in all respects, the cation and anion densities near z = L are affected equally and no charge separation occurs in that region. In our (13)J. G.Kirkwood, J. Chem. Phys., 2, 767 (1934). (14)L. B. Bhuiyan, C. W. Outhwaite, and S. Levine, Mol. Phys., 42, 1271 (1981).
0 1982 American Chemical Society
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earlier calculations for the 1:l restricted primitive model, we found it easy to use a cell with a large enough value for L that the properties of the double layer at z = 0 were unaffected by these small effeda at the opposite boundary. In the 2:l system, however, the charge asymmetry will necessarily result in a charge separation (i.e., a second, different double layer) at the uncharged wall. As long as one is forced in any case to study two double layers, they might as well both correspond to the parameters of interest. Therefore, we have used a cell in which the constant excess charge on the mobile ions is neutralized by an equal and opposite charge that is divided equally between the walls at the two sides of the cell. Two identical double layers will form, one at each side of the central cell; it must, therefore, be long enough to ensure complete separation of these two layers as demonstrated by the appearance of a substantial region of neutral bulk electrolyte in the center of the cell. Of course, symmetry about the midpoint of the cell, within the statistical noise, becomes an additional criterion for convergence to equilibrium, and the results are obtained by averaging the properties of the left and right halves of the cell. In all other respects the Monte Carlo algorithm for this two-plate geometry is analogous to that described previously for the 1:l system? In particular, the interaction of central cell ions with the potential due to the nonuniform charge distribution outside the central cell, is taken into account by representing the external charge distribution by a set of closely spaced planes each carrying a uniform charge density equal to the mean charge density in the same plane observed in the central cell over all previous configurations. Each central cell ion then interacts with the portion of each of these planes that lies outside the central cell. For the two-plate geometry, however, in the infinite slab consisting of the central cell plus all of its periodic replications, there is no net field exerted on an ion by the equal surface charges on the two walls; hence, the wall charge contributions to and the wall-ion interactions within the central cell are both omitted from the Hamiltonian. As a technical aside we mention that we found convergence to equilibrium to be was forced to be symmetric about considerablyfaster if z = '/& (by using the average of the charge distributions in the two halves of the cell to compute When this was not done, the mean dipole that occurs as a fluctuation over a block of configurations leads to a n e t reaction force in that, in turn, tends to induce a mean dipole of opposite sign over the subsequent block of configurations. The slow convergence caused by the resulting oscillatory behavior can largely be avoided by requiring +ext to be symmetric while the system is being aged. Although this restriction was enforced throughout the calculation, it becomes redundant once equilibrium is attained. For the 2 1 system, then, the ith configuration of the MC calculation consists of N+' doubly charged cations and N_' = 2(N+' + ah3 singly charged anions with equal neutralizing surface charges of uniform density u = H e / W at the two walls. H and u are therefore positive when the counterions are singly charged and negative when they are doubly charged. The model can equally well be interpreted as describing a 1:2 electrolyte by interchanging these relationships. For the symmetric 2:2 system the algorithm used previously for 1:l electrolytes may be used without modification to generate one doubIe layer at the z = 0 cell wall. In this more strongly interacting system, however, we have found comparable efficiency with either geometry. The type of algorithm used for each 2:2 calculation is indicated
+&
Torrie and Valleau
TABLE I: Results of GCMC Experiments To Obtain Activity Coefficients for Selected Concentrations of the Restricted Primitive Model for
e z l ( e k T d )= 1.6809. d = 4.25 a electrolyte 2:1
(concn),
In y ?
M
-0.239 -0.569 -0.881
0.00493 0.0491 0.496
electrolyte 2:2
(concn),
In y +
M
-0.685 -1.469 -2.402
0.00493 0.0515 0.501
in the table of results in the following section as one-plate (1P) or two-plate (2P). For both the 2:l and 2:2 systems the electrolyte is characterized by a bulk concentration with its associated chemical potential and a dimensionless inverse temperature P* = e 2 / ( & T d ) with c the dielectric constant of the continuum solvent, k Boltzmann's constant, T the absolute temperature, and d the ion hard-sphere diameter. For all of the calculations reported in this paper, /3* = 1.6809 corresponding to c = 78.5, T = 298 K, and d = 4.25 A. This value of d is implied wherever bulk densities are expressed in molarity. The applied surface charge and the resulting mean electrostatic potential are reported in the same dimensionless units used previously u* ad2/e $* = $e/kT For d = 4.25 A, u = 88.7a* pC cm-2; at 298 K IC, = 25.7$* mV.
111. Results and Discussion The particular parameters and concentrations used by Bhuiyan et al.I4 in the MPB theory do not correspond exactly to any of the 2:l electrolyte systems for which the bulk properties have been obtained in previous MC work. To obtain the activity coefficient-bulk concentration relationships needed in the grand canonical Monte Carlo (GCMC) procedure for the double-layer problem, then, we had first to carry out conventional GCMC experiment^'^ on the bulk systems of interest. These results are summarized briefly in Table I. The principal results of this paper, together with the controlling parameters of the MC experiments, are given in Table 11. As in previous MC calculations for double layers, we have measured the average local ion concentrations c++(z) and c-(z) (c--(z) for 2:2 electrolytes) and the local mean electrostatic potential $(z), where z is measured from the charged wall. In the two-plate geometry used for most of the calculations reported here, the results represent averages of those at the two charged walls in the MC cell. The zero of $ ( z ) is defined to be at the midpoint of the cell, Le., in the region of the bulk electrolyte. Hence, it can be determined as described in ref 9 (cf. eq 9). To allow for equilibration, 1 X lo5-3 X lo5 configurations were discarded from the initial stage of each run. The quantity shown in parentheses after the diffuse-layer potential $*('/2d) is one standard deviation of the mean, determined by treating segments of 4 X 104 steps as independent samples. The fraction of Monte Carlo moves in which a change of the number of particles was attempted was varied from 0.20 at low concentrations to at most 0.35 at high concentrations. We consider first the dependence on surface charge of the total diffuse-layer potential +*('12d) for 2:l systems. This is shown in Figure 1. For a 2:l electrolyte of point (15)J. P.Valleau and L.K. Cohen, J. Chem. Phys., 72,5935(1980).
The Journal of Physical Chemistry. Vol. 86, No. 16, 1982 3253
Limitations of buy-Chapman Theory
TABLE XI: Parameters and Resulting Diffuse-Layer Potential for the Monte Carlo Calculations Described in This Papef cell dimensions, A electrolyte 2:l
molarity 0.05
0.50 0.005 0.05
0.50
2:2 (1P) (1P)
(-1 (2P)
(-1
(2P)
0.005 0.05
a*
0.04294 0.18 0.18 0.0989 0.20 -0.01 - 0.02 - 0.05 - 0.05 -0.0975 -0.126 -0.20 -0.284 -0.05 -0,099 -0.1704 - 0.1704 -0.24 -0.02 -0.05 - 0.20 -0.20 - 0.284 - 0.284 - 0.1704 - 0.1704 -0,1704
L 127.5 127.5 170.0 63.8 63.8 255.0 255.0 297.5 85.0 127.5 85.0 76.5 76.5 51.0 55.3 63.8 51 .O 55.3 170.0 170.8 48.0 106.3 48.0 106.3 65.8 46.0 63.8
1 0 - 3 steps ~ ( N , + N-) 560 185 560 273 560 311 2000 155 600 258 520 166 600 198 600 462 480 135 400 117 520 98.6 520 154 600 281 560 308 2000 197 1800 245 1680 324 560 328 560 156 600 198
W 119.6 100.2 100.2 63.4 63.00 255.0 255.0 301.7 120.2 86.1 84.7 95.0 112.8 85.0 63.4 63.5 81.1 75.6 340.0 300.6 134.4 134.4 134.4 112.8 92.1 92.1 92.1
520 600 520 1000 1200 640 600
rii * ( ‘ / 2 d ) 1.73 (0.02) 3.99 (0.05) 3.95 (0.05) 1.04 (0.02) 1.94 (0.03) -0.91 (0.02) -1.37 (0.01) -1.87 (0.03) -1.05 (0.01) -1.21 (0.06) -1.26 (0.03) -1.18 (0.03) - 1.02 (0.03) -0.40 (0.02) -0.50 (0.01) -0.45 (0.02) -0.46 (0.01) -0.35 (0.03) - 1.47 (0.02) - 1.96 (0.02) -1.33 (0.02) -1.32 (0.03) -1.14 (0.03) - 1.14 (0.03) -0.63 (0.04) -0.58 (0.03) - 0.60 (0.02)
145 312 190 282 333 234 366
0.50 (1P) (2P) For 2 : l electrolytes positive surface charge U * corresponds to singly charged counterions. For 2:2 electrolytes the symbols 2P and 1P denote experiments carried out in two-plate and one-plate geometries, respectively (cf. section 11). The statistical uncertainty in j, * ( l / l d ) is one standard deviation of the mean.
(-1
ions with either positive or negative surface charge, Grahamel6 has shown that the Poisson-Boltzmann equation (PBE), that is, the Gouy-Chapman theory, may be solved analytically; the results are dashed lines in Figure 1. The solid lines are the solutions of the MF’B theory of Bhuiyan et al.14 This theory takes the PBE as a starting point but attempts to include some of the fluctuation terms neglected in the mean field of the PBE and the effect of excluded volume (i.e., finite ion size) on the ion-ion correlation functions. (With the exception of their Figure 10, Bhuiyan et d.14use u > 0 throughout; hence, they use the terminology “1:2”for our u < 0. Their parameter $J’(O) is -4?r/3*@, roughly -21u*.) The solid (open) symbols are our Monte Carlo results from Table I1 for 2:l (2:2) electrolytes. The right half of Figure 1 corresponds to positive surface charge and hence singly charged countenons. The picture here is very reminiscent of that of 1:l electrolyte^.^ The MGC theory, through its neglect of ion-ion correlations, tends to overestimate the extent to which other counterions screen the surface charge from a given counterion. Thus, it predicts a double layer which is too thick and a potential drop which is too high. This defect of the theory is more pronounced at higher concentrations and increases with surface charge density. Both the nature and the magnitude of these shortcomings of the MGC theory are similar to those found in 1:l electrolytes. Apparently the neglect of ion-ion correlations in the classical theory is no worse an approximation when only the co-ions are doubly charged; for surface charges low enough that a significant fraction of the ions near the wall are co-ions, the overall particle concentration near the wall is also low enough for a mean field treatment to be appropriate. Of course, the MGC theory is by no means exact, and more sophisticated (16)D.C. Grahame, J . Chem. Phys., 21,1054 (1953).
I - 0.30
1
I -0.20
I
I
1
0
-0.10
1
I
0.10
,
I
020
U*
Flgure 1. Surface charge dependence of the diffusalayer potential. The counterions are singly charged for u* > 0, doubly charged for u* < 0. 0 ,A)MC results for 2 1 electrolytes at 0.005,0.05and 0.05 M, respectively. Open symbols are MC results for 2:2 electrolytes. (---) buy-Chapman theory for 2:l; (-) MPB theory for 2:l.
m,
theories are certainly able to do much better. The MPB theory (solid curves) is essentially exact here for 0.05 M as it is for the same parameter range in 1:l systems. At 0.50 M the MPB potential is lower than the MC values by a small but statistically significant amount. The left half of Figure 1 corresponding to doubly charged counterions is much more startling. A t both 0.05 and 0.50 h.I the MC results show a broad but unmistakable extremum in the u* dependence of +*(1/2 d). This behavior is quite unlike anything observed in the MC results for 1:l systems and, at these concentrations and surface charge densities, must arise from electrostatic rather than excluded volume effects. Such behavior does not contra-
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The Journal of Physical Chemistry, Vol. 86, No. 16, 1982
dict the rigorous result of Blum et al." that the total potential +*(O) is a monotone function of a* since +*(O) equals +*(1/2d) plus a trivial linear potential drop of 2u/3*u* over the charge-free region between z = 0 and z = 1/2d. It is immediately apparent from the figure that the classical theory has broken down completely; it not only fails to show this qualitative behavior but is seriously in error even for quite low concentrations and surface charges. At 0.005 M and u* = -0.01 ( E 0.9 NCcm-2) the MGC diffuse-layer potential is 5% too large; this error increases to 36% at the same concentration and the still modest surface charge of 4.4 pC cmW2(a* = -0.05). By contrast, the MPB calculations lie relatively close to the MC results and so show the same broad extremum in +*(1/2d)at the highest concentration; however, the MPB theory does not have the quantitative accuracy here that it does for singly charged counterion systems. The absolute error in the MPB potential is quite insensitive to concentration and to surface charges less negative than -0.20; presumably, then, the principal shortcoming of the theory in this regime lies not in the treatment of the excluded volume effect but rather in the estimation of the fluctuation terms. This defect is apparently being gradually cancelled at more negative a* by a compensating underestimation of the excluded volume effect so that at 0.05 M and a* = -0.284 the MC and MPB potentials happen to coincide. The magnitude of the error in the MPB +*, though fairly small and nearly constant, is largest in relative terms at the high concentration and moderate surface charges where the MPB potential is as small as 45% of the MC value. To gain further insight into the behavior of these systems, it is necessary to study the more detailed information contained in the local ion densities and mean electrostatic potential as functions of distance from the charged surface. For singly charged counterions at 0.05 M the behavior of these functions is so similar to that found in 1:l systems that we have not displayed the results. Specifically, the MGC double layer is somewhat too thick though this is easier to see as an integrated effect in + ( z ) than in the rapidly varying ion densities themselves; the MPB theory is essentially exact. At 0.50 M however, some small but interesting new features are found. At both 8 = 0.099and 8 = 0.20 (Figure 2) the counterion density appears to have a very shallow minimum around 2.5d and the co-ion density a corresponding maximum. As a result +*(%)appears to change sign around 2.0d and go through a similar shallow minimum. In the MC results this behavior is scarcely distinguishable from the statistical noise; however, the MPB results exhibit the same behavior at the same points, though with more pronounced oscillations. This type of behavior is not observed in 1:l systems except at much higher surface charges and for different reasons having to do with packing effects at high densities. Of course, the classical theory (the dashed lines in Figure 2) cannot predict oscillatory behavior. (In this and subsequent figures we have found the equations of M a r m u P convenient for computing the Gouy-Chapman +*(z) of the charge-asymmetric system.) The exaggerated oscillations and more compressed double layer in the MPB theory are typical of its behavior for charge-asymmetric systems: where excluded volume effects are not important, the theory tends to overestimate the deviations from MGC predictions. Certainly, the small discrepancy between the MPB and MC $* at 0.5 M, u* = 0.20 is nevertheless larger (17)L.Blum, J. Lebowitz, and D.Henderson, J.Chem. Phys., 72,4249 (1980). (18)A. Marmur, J. Colloid Interface Sci., 71,610 (1979).
Torrie and Valleau 7
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Figure 2. (A) Local concentration divided by the bulk concentration for counterions and co-ions In a 2:l system at 0.5 M and a surface charge a* = 0.20 (A,0 )Monte Carlo results; (---) Gouy-Chapman theory; (-) MPB theory. Some typical statistical uncertainties in the MC data are lndlcated. (B) Mean electrostatic potential $ ' ( z ) for a 2:l electrolyte at 0.5 M and a surface charge 8 = 0.20. All symbols as in part A.
2.01\,
2.4
,
Figure 3. Mean electrostatic potential $ ' ( z ) for a 2:l electrolyte at 0.005 M and a surface charge a. = -0.05. All symbols as in Figure 2.
than the corresponding discrepancy would be for the analogous 1:l system. As suggested by the behavior in Figure 1,the situation is quite different for doubly charged counterions. In Figure 3 is shown the potential profile for a 0.005 M system with a* = -0.05. Both the MGC and MPB theories correctly predict the monotone behavior of J/* shown by the MC results, but the quantitative predictions of both theories are rather unsatisfactory. The MGC profile is charac-
The Journal of Physical Chemistry, Vol. 86, No. 76, 1982 3255
Limitations of Gouy-Chapman Theory
1
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1
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t
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3 1
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Figure 4. Local ion concentrations for a 2 1 electrolyte at 0.05M, u* = -0.20. All symbols as In Figure 2.
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teristically too extended while the MPB theory has overestimated the error in the classical theory, predicting too compressed a layer. Errors of this magnitude in the classical theory for 1:l electrolytes are not to be found except at much higher surface charges. For surface charges between -0.15 and zero, the behavior of 0.05 M systems is similar to that of the 0.005 M system shown in Figure 3, but a t u* = -0.20 there has developed a maximum in Figure 4). This maximum the co-ion density near 2.W (6. occurs inside the envelope of the counterion density so the charge density and mean potential are still monotone. The MPB theory predicts monotone ion densities here as, of course, does the classical theory. A t the still larger surface
Flgure 6. Local ion concentrations for a 2:l electrolyte at 0.5 M, u* = -0.1704: (A,0)MC results for a cell with L = 15d, W = 14.9d; (A,0 ) MC resub for a cell with L = 12d, W = 19.1~3;(---) Gouy-Chapman theory; (-) MPB theory. (6)$ * ( z ) for 2:l electrolyte at 0.5 M, u* = -0.1704. All symbols as in part A.
charge u* = -0.284 this tendency toward oscillatory behavior in the ion densities has produced a very small charge inversion in the region 3d-4d (Figure 5A) with the result that $*(z) now has a shallow minimum near 2d (Figure 5B). Prediction of oscillatory behavior at so low a concentration would be a very stringent test of theory; in fact, the MPB density and potential profiles remain monotone, confirming our earlier observation that the coincidence of the MPB and MC values of +*('lzd)results from a cancellation of errors. The rather surprising minimum in the MPB co-ion density near the wall shown in Figure 5A is not found in the MC results. The nature of the MC sampling procedure makes it difficult to get precise values for the low co-ion density very close to the wall; nevertheless, it seems likely that the MPB minimum is an artifact of the theory. Oscillatory behavior is much more pronounced at 0.50 M where we found $*(z) to be nonmontone at all surface charges investigated. The behavior becomes quite pronounced at higher surface charges as shown in Figure 6 for u* = -0.17. The MPB theory is qualitatively correct in predicting oscillations in the ion densities here, as'it does for all 0.5 M systems that we studied. Quantitatively, it is far less accurate than it would be for a 1:l system at comparable Q* where no such oscillations occur. The MPB double layer is much thinner than that found in the computer experiments with the consequence that the MPB diffuse-layer potential is less than half the correct value. Here the MC sampling of the co-ion density very close to the wall is certainly adequate to show the minimum in the
The Journal of Physical Chemistty, Vol. 86, No. 76, 1982
3256
rr-74 1
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Torrie and Vaileau
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7
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Flguro 7. Local ion concentrations for a 2:2 electrolyte at 0.05 M, u* = -0.284: (A,0 )MC results from a single-plate (1P) experiment (cf. section 11); (-) a smooth curve through the MC results for the 2: 1 system at the same concentratbn and surface charge (cf. Figure 5A); (- - -) Gouy-Chapman theory for the 2:2 system.
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MPB co-ion profile to be spurious. The two sets of symbols used in Figure 6 correspond to the results of separate MC experiments using different cell dimensions. Recently, Jan~ovici'~ has shown that, in systems of charged particles with a surface, correlation functions will have a component parallel to the surface that decays slowly (algebraically). If the amplitude of the slowly decaying component were at all large, the implications for computer simulation using small periodic systems would have to be carefully considered. The results shown in Figure 6 are typical of a number of similar checks that we carried out; viz., the measured properties normal to the surface are apparently quite insensitive to cell size and geometry within reasonable limits. The similarity of the u* > 0 portion of Figure 1 to the corresponding picture for 1:l electrolyte makes it clear that it is the strength of the interaction between counterions rather than the charge asymmetry between counterions and co-ions that causes the breakdown of the classical theory. To investigate this aspect of the problem further, we have carried out MC experiments on 2:2 electrolytes at concentrations and surface charges corresponding to a limited number of the 2:l systems with u < 0. The total diffuse-layer potentials for these 2:2 systems are shown as open symbols in Figure 1. For the surface charges of these 2:2 experiments the classical theory predictions of diffuse-layer potential are not distinguishable on the figure from the 2:l curves. By and large the 2:2 MC results conform to the physically reasonable expectation that, at all but the smallest surface charges, the double-layer properties are nearly independent of co-ion charge except, of course, for the co-ion density profile itself. Thus, the breakdown of classical theory and the broad shallow extremum in #* as a function of a* for 2:2 systems are much the same as for 2:l systems. The MC 2:2 electrostatic potentials, however, are consistently more negative than their 2:l counterparts. This seems to arise from the greater ease of accumulating co-ion charge near the first layer of counterions when each co-ion carries a double charge. In Figure 7, for example, the ion densities for the 2:2 system at 0.05 M, u* = -0.284 are compared with smooth curves representing the MC results for the analogous 2:l system. The 2:2 ion densities show oscillations similar to those in the 2:l system, but the 2:2 co-ion distribution is more compact with a higher peak and lies entirely beneath the counterion profile with the result that the oscillation in (19)B.Jancovici, preprint.
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1
3
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2
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4
/d
Figure 8. (A) Local ion concentration for a 2:2 electrolyte at 0.5 M, u* = -0.1704. All symbols as In figure 7 but the 2 2 MC results were for a 2:2 electrolyte at obtained in the two-plate geometry. (B) $*(z) 0.5 M, u* = -0.1704. All symbols as in Figure 7 but the 2 2 MC results were obtained in the two-plate geometry.
#*(z) found in the 2:l system does not occur here. Figure
8 is a similar comparison of 2:2 and 2:l systems at 0.5 M and u* = -0.17. Here the 2:2 and 2:l co-ion profiles are more similar than at the lower concentrations. A charge inversion does occur in the 2:2 system (Figure 8A) and hence there is also an oscillation in #* (Figure 8B) albeit reduced from that in the 2:l system.
IV. Conclusion We have used a Monte Carlo method to generate numerically exact results for the restricted primitive model of 2:l and 2:2 electrolytes next to a uniformly charged plane wall over the concentration range 0.005-0.50 M and for positive and negative surface charges up to 25 pC cm-2. When the counterions in the 2:l system are singly charged, a Gouy-Chapman theory is found to work about as well as it does for the comparable 1:l system. The much more accurate MPB theory of Outhwaite and collaborators is nevertheless not quite as accurate here as in the 1:l case. When the counterions in the 2:l system are doubly charged, however, the classical theory fails altogether, even for quite low concentrations and surface charges. At both 0.05 and 0.50 M the Monte Carlo results show the diffuse-layer potential to be nearly independent of the surface charge though with a shallow extremum near u = 10 NC cm-2. Oscillations in the ion densities occur at high but physically reasonable surface charges at concentrations as low as 0.05 M and are found at virtually all negative u at 0.50 M. The MPB theory is successful in predicting most
J. Phys. Chem. 1982, 86, 3257-3263
of these qualitative features, particularly at 0.50 M, but tends to overestimate the constriction of the double layer relative to the classical theory. The behavior of 2:2 systems-hence the failure of Gouy-Chapman theory-is quite similar to that of 2:l systems with doubly charged counterions. However, there are some small systematic differences between 2:2 and 2:l systems that are much larger than the corresponding differences in the classical theory. For the symmetrically charged case, the strength of the ionic interactions depends only on the ratio q 2 / c , q being the ionic charge. The model can, therefore, be interpreted equally well as a 1:l electrolyte in a solvent of dielectric constant of about 20. Thus, the failure of Gouy-Chapman theory for systems characterized by this interaction strength may well have important consequences for the analysis of experimental data not only in aqueous 2:2 systems but also in nonaqueous systems. Similarly, the
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marked overestimation of the double-layer thickness in these systems by the Gouy-Chapman theory implies a similar overestimation of the range of the repulsive force of two overlapping layers responsible for the stabilization of charged colloid particles. Of course, a better understanding of these phenomena in real systems will come from more sophisticated models, particularly those that take account of the discrete nature of the solvent, but it appears that significant errors can arise in the GouyChapman theory even in regimes where a primitive model might well be considered appropriate. Acknowledgment. This research has been supported in part by the Natural Sciences and Engineering Research Council and in part by the Department of National Defence under DRB grant no. 3610-645. We are grateful to Chris Outhwaite for providing us with detailed numerical results of the MPB theory.
Hydrogen Isotope Effects in the Dehydration of Polystyrene-Divinylbenzene Type Ion-Exchange Resins and the Structure of Electrolyte Solutions. 1. Li+ Form of Variously Cross-Linked Dowex 50W Resins A. R. Gupta,’ Deokl Nandan, and S. K. Sarpal Chemlstry Dlvlslon, Bhabha Atomlc Research Centre, Trombay, Bombay 400 085, I d l a (Recelvd: October 29, 7987; In Flnal Form: March 75, 7982)
Hydrogen isotope effects in the dehydration of the lithium form of strong acid polystyrene-divinylbenzene type ion exchangers (Dowex 50 W) of varying cross-linking (2-12% DVB content) have been determined by a Rayleigh distillation type technique. Resins containing varying amounts of water (4-35 mol of water per equivalent of resin, n,), prepared either as fully swollen resins or by equilibrating with different water activities isopiestically, were used in these experiments. The single-stage separation factors, a,were dependent only on n, and were independent of resin cross-linking. At n, = 4, a was lower than aw,the separation factor for pure water distillation at the same temperature. In the region of n,= 10-12,a was greater than a, and reached a maximum value. Beyond n, 12,a decreased gradually with n,but remained greater than a,. These results have been interpreted in terms of the hydration of lithium ions in the resin phase. When only the primary (first)hydration shell of lithium ions is complete (n, N 4.0), the separation factors are low because of the absence of any hydrogen bonding between the water molecules. Values of a greater than a, at n, N 12 are consistent with the predictions of quantum mechanical calculations on lithium ion-water clusters, namely, (i) the formation of a second hydration shell whose water molecules are hydrogen bonded to those in primary hydration shell, (ii) these hydrogen bonds are stronger than in bulk water, and (iii) the water molecules in the third hydration layer are not much affected by lithium ions and form hydrogen bonds similar to those in bulk water. a values have been resolved into separate contributions from water molecules in the first and second hydration shells and the remaining water. a for the last category is close to but larger than a,. These data indicate that the counterions in resin phase water behave like single ion solutions, provided counterion-water interactions are much greater than counterion-ionogenic group interactions. The resin network seems to influence the outermost layers (beyond the second hydration shell) of water molecules via hydrophobic interactions.
Introduction Ion-exchange resins based on the polystyrene-divinylbenzene (PS-DVB) network, containing strong acid (sulfonic acid groups in cation exchangers) and strong base (quaternary ammonium groups in anion exchangers) ionogenic groups, absorb substantial quantities of water depending upon the nature of the counterions and degree of cross-1inking.l In moderately and low cross-linked resins
(18% DVB content) the counterions have been considered to be fully hydrated and the excess water in the resip phase (i.e., over and above the hydration water) has been called yfree water” or “osmotically imbibed water”. The state (degree) of hydration of the counterions and ionogenic groups in the exchanger phase has been investigated by many workers using different techniques. Glueckauf and Kitt2 studied water sorption isotherms and the enthalpy
(1) Helfferich, F. ‘Ion Exchange”;McGraw-Hill; New York,1962.
(2) Glueckauf,E.; Kitt, G. P. h o c . R. SOC.(London),Ser. A 1955,228, 322.
0022-365418212086-3257~01.2510
0 1982 American Chemical Society
