Oct., 1963
SURFACE POTENTIALS OF IONIZED MONOLAYERS
2095
THE DISCRETE-ION EFFECT AND SURFACE POTEXTIALS OF IOKIZED MONOLAYERS BY S. LEVINE, Department of Mathematics, University of Manchester, England
J. MINGINS, Basic Research Division, Unilever Research Laboratory, Port Sunlight, England AND
G. Af.
BELL
Department of Mathematics, Chelsea College of Science and Technology, London, England Received March 10, 1963 The theory of the discrete-ion effect for adsorbed counter-ions in the electric double layer is applied to fully ionized monolayers a t A-W and 0-W interfaces. This effect may be described in terms of a two-dimensional self-atmosphere or fluctuation potential acting a t an adsorbed ion and is responsible for a negative term in the adsorption energy of the counter-ions which increases with their surface density. If the Stern theory is assumed, this term appears as a variation in the specific (chemical) adsorption potential with surface charge and this has been calculated for monolayers of long-chain sulfates. A second phenomenon predicted by the discrete-ion theory is the occurrence of a maximum in the magnitude of the potential a t the plane of adsorbed ions when the surface charge is increased at constant ionic strength. A corresponding maximum is found a t the sIipping plane defining the potential and this behavior in the potential of 0-W emulsions stabilized by SDS and DTAB monolayers has been observed. The presence of a maximum in the interfacial potential with variation in surface charge, observed for monolayers of long-chain sulfates, also can be attributed to the discrete-ion effect. The calculation of the “adsorption-plane” self-atmosphere potential depends on the model used for the ‘‘inner region” at the interface. I n the case of an ionized monolayer, the adsorbed counter-ions and primary head-group ions are assumed to be situated in the same plane, embedded in an inner region having a dielectric constant smaller than that of the aqueous substrate. The various phenomena described above are reproduced by the theory if the specific binding between counter-ions and head-group ions is assumed small, so that the counterions are mobile. A detailed comparison is made between the theory and the experiments of Mingins and Pethica on surface potentials of sodium octadecyl sulfate at the A-W interface in the presence of 0.01 M NaC1.
1. Introduction Recently, two of the authors1 modified the SternGrahame-Devanathan theory of the electric double layer by introducing the so-called discreteness-of-charge effect (more briefly the discrete-ion effect) for an adsorbed counter-ion. This effect, which also may be expressed in terims of a “self-atmosphere” or “fluctuation” potential in the adsorption plane, seems t o account for a number of hitherto unexplained phenomena a t the mercury-aqueous electrolyte interface and also for certain features of colloid stability for which the classical Derjaguin-Landau-Verwey-Overbeek theory does not account.2 The discrete-ion effect should be a general characteristic of electric double layers and the present paper is concerned with its manifestations in ionized monolayers a t oil-water (0-W) and air-water (A-W) interfaces. In a recent paper,a calculations on a simple model of the interfacial region showed that this effect contributes a significant negative term to the surface pressure of an ionized monolayer. Here we shall consider three other properties of ionized monolayers which are inconsistent, with the Gouy-Stern theories of the double layer a t the monolayer and which find a plausible explanation when the discrete-ion effect is invoked. These phenomena are: (i) the apparent variation in specific adsorption potential @ of an adsorbed counter-ion with area, per film molecule A; (ii) the occurrence of a maximum in the electrophoretic mobility of oil-in-water emulsions with change in concentration of a fully ionized surfactant, when the aqueous phase is held a t constant ionic Eitrength; and (iii) the presence of (1) 8. Levine, G. M. Bell, and D. Calvert, Can. J . Chem., 40,518 (1962). (2) S. Levine, G. M. Bell, and D. Calvert, Nature, 191, 699 (1961); S. Levine and G. M. Bell, J. Coolloid Sci., 17,838 (1962); J. Phus. Chem., 67, 1408 (1963). (3) G. M. Bell, 8, Letins, and B, A. Psthice, Frann. F Q M ~ QSoc., U 88, 904 (lQ64),
a maximum in the magnitude of the surface potential AT’ of an ionized monolayer a t 0-W and A-W interfaces when A is varied a t constant ionic strength of the substrate. (Henceforward, the “maximum’?of any potential will refer to the maximum of its magnitude, irrespective of sign.) A variation in @ (-3kT) of adsorbed sodium ions has been calculated by A n d e r ~ o n ,mho ~ applied the Stern model to his data on the electrophoresis in various aqueous salt solutions of n-decane droplets stabilized by sodium dodecyl sulfate (SDS). The results of Stigter and Mysels5 on the electrophoretic mobility of SDS micelles and Davies and RideaP on cetyl trimethyl ammonium chloride were also considered by Anderson. Recently, Mingins and Pethica’ have calculated a variation in @ (-3kT) from their measurements on sodium octadecyl sulfate (SODS) monolayers spread a t the A-W interface. A change in ch (-7kT) was shown a t an earlier date by Grahame8,9for counter-ions adsorbed at the mercury-elec trolyte interface. A maximum in electrophoretic mobility and therefore in the ill potential was shown by Anderson’s results4 on n-decane dispersions in aqueous solutions of SDS at fixed KaCl concentrations. Haydonloobtained similar results with the same system and also found a maximum in the { potentials of petroleum ether droplets stabilized in various SaCl solutions by dodecyl trimethylammonium bromide (DTAB). A maximum in the /p!potential when the surface charge is varied a t constant ionic strength has been observed for AgBr sols by Ottewill (41 P. J. Anderson, ibid., 66, 1421 (1959). ( 5 ) D. Stigter and K. J. Mysels, J. Phys. Chem., 69, 45 (1955). (6) J. T.Davies and E. K. Rideal, J. Collozd Sei. Suppl., 1, 1 (1954). (7) J. Mingins and B. A. Pethioa, Trans. Faraday Soc., 59, 1892 (1963). (8) D.C. Gmhame, Chem. Rev., 41,441 (1947). (9) D. C. Grahame and B. A. Soderberg, J. Chen. Phys., 42,449 (1951). (10) D.A. Heyden, Proc. Rou. Soc. (London), 8388, 319 (1960).
2096
S. LEVINE,J. MIXGINS, AND G. M. BELL
and Woodbridge” and for the glass-electrolyte interface by Greenberg, Ghang, and Jarnulowski.12 Grahame calculated a corresponding maximum in the potential ,$,j of his “outer Helmholtz plane” (O.H.P.) a t the mercury-electrolyte interface in the presence of adsorbed halide ions. (The O.H.P. is the limit of the diffuse layer in the aqueous phase and defines the distance of nearest approach to the interface of the coions.) Parry and Parsons13also obtained a maximum a t the same interface with adsorbed aromatic sulfonate ions. Recently, Lyklema14 has calculated a minimum in the surface excess of co-ions for the negative AgIsimple 1-1 electrolyte system when the PI is changed a t constant ionic strength, particularly at 0.1 M . This indicates a maximum in the potential j$d’ of the O.H.P. lllaxima at A = 75 A.2 in l A V - A plots a t constant ionic strength have been reported by Phillips and Rideal16for SODS spread a t both 0-W and A-IT7 interfaces. Pethica and Few16 have found maxima in IAV, for the same molecyle a t the A-IV interface at somewhat smaller A (65 AZ). I n unpublished work on SODS at the A-IV interface, RIingins and Pethica also find maxima a t A = 65 At the 0-’VI;interface Phillips and Rideal15 found no maximum with sodium docosyl sulfate spread on HC1 substrate and Stenhagen17 obtained a similar behavior for the same film spread at the A-W interface. However, both sets of data were restricted to low A and it would appear from the plots of these authors that the sign of AV changes a t higher A (at low acid concentrations). Since necessarily AV -+ 0 as A 00, this change in the sign of AV indicates a maximum in AVl. Pethica and also obtained a maximum in lAV a t A = 60-70 A.2 for adsorbed films of SDS a t the A-ITr interface in the presence and absence of KaCl although in the latter case the ionic strength in the substrate mas not kept constant. Haydon and Phillips1*observed no maximum in AV for SDS adsorbed at the petroleum ether-water interface with no excess electrolyte. However, more recently, Haydonlg does find iAV1 increasing with A for SDS a t n-heptane-KaC1 solution interfaces, a t salt concentrations of 0.25 and 0.50 M , indicating that ‘AVI has a maximum a t some A > 80 A.2. Crisp’s results20 on a-bromopalmitic acid on citrate buffer a t pH 7 containing 0.06 &?sodium ions also gave a maximum in 1 AV; a t A = 650A2. Naxima in IAVI a t extremely low A (-25 A.)2 for weakly ionized fatty acids have been obtained by various authors but these possibly are due to other effects such as changes in dipole orientation, and since the degree of dissociation is unknown, we shall only be concerned with fully ionized monolayers. The above data on maxima in IAVl have been restricted to anionic surfactants. In contrast, no maxima in AV have been found for cationic films of 0-W or A-IV (11) R. H. Ottewill and R. F. Woodbridge, to be published. (12) S. A. Greenberg, T. N. Chang, and R. J. Jarnulowski, J . PoZymeT Sci., 68, 157 (1962). ( 1 3 ) J. M. Parry and R. Parsons, Trans. Faraday Soc., 59, 241 (1963); see also M. A. V. Devanathan and M. J. Fernando, ibid., 58, 368 (1962). (14) J. Lyklema, ibid., 59, 418 (1963). (15) J. N. Phillips and E. K. Rideal, Proc. Roy. Soc. (London). A232, 159 (1955). (16) B. A. Pethica and A. V. Few, Discussions Faraday Soc., 18, 258 (1954). (17) E. Stenhagen, Trans. Faraday Soc., 36, 496 (1940). (18) D. A. Haydon and J. N. Phillips, %bid., 64, 698 (1958). (19) D. A. Haydon, Rolloid-Z., 186, 148 (1962). (20) D J. Crisp, “Surface Chemistry,” Rutterworthn, London, 1949, p. 65.
Vol. 67
interfaces as illustrated by the data of Davies21on cetyl and octadecyl TABS and of Haydon and Phillips,1s Haydon,Ig and Powell and A41exander22 on DTAB, although Haydonlo did find a maximum in the mobility of emulsion droplets stabilized by DTAB ions. This distinction in the behavior of cationic and anionic films can be attributed to differences in the size of the dipole contributions to A V and is discussed in section 2 . It is important to note here that for soluble surfactants such as SDS and DTAB the contribution of the surfactant to the ionic strength of the bulk solution is negligible except in the absence of salt. Thus in the presence of excess salt, the leveling-off of AV for DTAB and the occurrence of a maximum in jAV1 for SDS as A increases cannot be due to the increasing ionic strength of the bulk solution. Hitherto, no authors hare attempted to correlate or adequately explain the three phenomena described above. This paper shows that all three are related and arise from the discrete-ion effect. To achieve this we introduce a model of fully ionized monolayers that takes into account the penetrability of the plane of the headgroups by the counter-ions, the lowering of the dielectric constant in the interfacial region, and the mobility of the adsorbed counter-ions. K e shall shorn that with this model our theory of the discrete-ion effect predicts a maximum in I+OI (where $o is the mean potential a t the plane containing the centers of the head-group ions) with variation in A at constant ionic strength. A corresponding maximum will o c c ~ in ~ rthe potential at any parallel plane on the diffuse layer side, such as the O.H.P. or the slipping-plane which defines the .t potential. Because of the absence of precise information’ about the composition of the interfacial region for the systems quoted in this and the following sections where maxinia in IAV, lfl, and ;$o’ occur, we shall assess quantitatively the data in ref. 7 only. The apparent variation in with change in A is shown to be a natural consequence of neglecting the discrete-ion contribution to the free energy of adsorption of a counter-ion. I n the particular case of the SODS film on 0.01 M N a c l studied by Mingins and Pethica7 (who estimate $0 and obtain strong evidence for the mobility of the adsorbed counter-ionsZ3)the discrete-ion effect appears to account for the whole of the apparent variation in a. 2. Components of the Surface Potential AV.-It is usual to separate the surface potential of an ionized monolayer into the two terms given in the equation
+
AV = 4 ~ p / ’ A $0 (2.1) where p is the effective dipole moment of the film molecule. By means of this equation and an estimated p of 400 mD. (assumed independent of A ) Davies21b coalculated a maximum in $o ( = 160 mv.) a t A = 100 for cetyl TAB spread at the 0-IV interface in 0.01 M KaCl. We have re-examined some of Davies’ AV-A plots21afor octadecyl TAB spread a t the 0-JV interface in the presence of 0.033 ill NaC1. Choosing Davies’ value of 400 mD. for p , a plot of AV - 4 ~ , u / Avsb A shows a maximum in $o of about 170 mv. at A = 75 A2. From their AV-A data on the anionic film of SODS a t the A-IT interface on 0.01 I l l NaCl substrate, where (21) (a) J. T. Davies, Proc. Roy. SOC.(London), A208, 22 (1951): (b) Trans. Faraday Soc , 48, 1052 (1952). (22) B. D.Powell and A. E. Alexander, J. Collord Sci., 7, 493 (1952). (23) J. hIingins and B. A. Pethica, t o be published.
Oct., 1963
-
SURFACE POTENTIALS OF IONIZED MONOLAYERS
/AVi has a maximum a t A 65 filingins and Pethica7 obt$ned a maximum in lt+bol ($0 = -63 mv. 180 A2)using a fixed value of - 120 mD. for p. at A Over the range of A studied, it is reascnable to assume that both the quaternary ammonium ion and the sulfate ion have dipole moments independent of A. Also €or each type of film p and $0 have the same sign. With the SODS film, the maximum in lATI1 occurs a t smaller A t h m that in go , when A varies a t constant ionic strength. This readily follows from (2.1) if p is independent of A and has the same sign as $0. The argument cart be presented more simply for a cationic film where go> 0. We differentiate (2.1) with respect to 1/A to obtain d(AV)/d(l/A) =
+ d$o/d(l/A)
(2.2) and suppose that the second term on the right is zero a t some A = A,, where $0 is a maximum. Since $0 -t. 0 as 1/A + 0, d$o/d(l/A) 2 0 according as 1/A 5 l/Am. If F :> 0, thus having the same sign as $0, then AV can only have a maximum for l/A > l/Am. However, if p is too large (which apparently is the case with quaternary ammonium ions), so that the term d$a/d(l/A) cannever cancel withthe 47rp term, then AV mill not have a maximum. If p and $0 are opposite in sign and a maximum does occur in AV, then 1/A < 1/ A,. If the AV-l/A curves level off a t low A so that the left-hand side of (2.2) tends to zero and if $0 has the same sign as p, then necessarily $0 will show a maximum. Such leveling off is obtained with the TAB films studied by the: authors quoted above. Equation 2.2 can be appIied in a simiIar manner to anionic films and for the long-chain sulfate ion p is small enough to allow ‘AV/to reach a maximum. Recently Haydon and Taylorz4 have developed a theory of ionized monolayers at 0-W interfaces, a detailed study of which is presented elsewhere.26 However, since H a ~ i d o n ’ sconclusions ~~ about p are highly relevant to the present work, we shall briefly discuss them here. Haydon and Taylor24 replaced (2.1) by A V = 4n/.~/A $i (2.3) 47rp
+
where 9, is the mean potential in the plane of the interface and difYers from $0 because of counter-ion penetration behind the plane of the head-groups. Haydonlg estimated \L, from their theory and then calcula1,ed values of p from his AV data on SDS and DTAB by means of (2.3). He concluded that the p for SDS and DTAB depend on the substrate salt concentration and furthermore that U , for SDS changes sign, having t>he same sign as in 0.05 and 0.10 M NaCl, but the opposite sign in 0.25 and 0.50 M NaC1. Equation 2.3 is quite valid provided il.i is calculated properly but it will be shown in Appendix A that the formulation of their theory leading to a value for $; is incorrect. Consequently his results for p are questionable and indeed his conclusion that p for SDS actually changes sign with varying electrolyte concentration seems improbable. 3. Model of the Monolayer Region.-In the Stern theory of the double layer, the plane of the primary (24) D. A. Haydon and F H. Taylor, P A L Trans. Roy. Sac. (London), A263, 225, 255 (1960); Proc. Intern. Conor. Surface Actzrzty, drd, Cologne, 11, 341, 1960; Trans. Paruduv Sac., 68, 1233 (1961). (25) S. Leuine, G. M. Bell and B. A. Pethica. to be published.
2097
(potential-determining) ions is assumed to be distinct from the plane of the adsorbed counter-ions. However, this model is not particularly suited to ionized monolayers a t 0-W and A-W interfaces. Pethica and Few16 have pointed out that the high surface charge densities of the ionic head groups attainable with fully ionized monolayers would lead to improbably high potential drops in the Stern region situated between the above two planes. Several authors6r26-z8have suggested that the plane of the head-group ions of an adsorbed monolayer is penetrable by substrate ions (in particular counter-ions) and such penetration would reduce the potential drop in the Stern region. An important feature of the Haydon-Taylorz4 double layer model is the penetration of diffuse layer ions behind the head-group ions and a variant of their model is used by van Voorst Vader,29who assumes that the head-group charge occupies a shallow zone, into which the counter-ions may penetrate. For calculating the discrete-ion effect, a model more amenable to mathematical analysis is one in which the primary and adsorbed charges are in the same plane and this simplification will be made here. We shall also adopt some features of the general model of the “inner region” at the mercury-aqueous electrolyte interface proposed by Grahame8 and Devanathan.aO This inner region is assumed to be a homogeneous medium of dielectric constant €1 and of thickness d = /3 y where p is the distance from the interface of the common plane containing the centers of the head-group and counter-ions (for brevity the adsorption plane). The O.H.P. (limit of the diffuse layer) is situated a t distance d from the interface, forming one boundary of the inner region (see Fig. 1). A value for EI appreciably less than the dielectric constant e of the bulk aqueous phase seeoms necessaryz6for values of ,8 lying in the range 2-5 A. (see Appendix A). The value for el < e is due to the immersion of the film molecule, to the thin layer of partly oriented water a t the interface, and to the bound water surrounding the head-group ions. Each head-group carries a negative charge -e, where e is the magnitude of the electronic charge and uo (= - e / A ) is the primary surface charge density. The positive charge density of the adsorbed counter-ions is denoted by uo and the net charge in the diffuse layer per unit area of interface by Ud. The electrolyte in the substrate is a 1-1 valency type, the volume density of each ion type is n, and the GouyChapman theory governs the ion distribution in the diffuse layer. Thus Ud is related to the potential $d a t the O.H.P. by
+
2ne1cT
1 e$d sinh - 2iiT
‘Iz
(3.1)
where k is Boltzmann’s constant and T is the absolute temperature. Since we assume that the counter-charge either resides in the diffuse layer or on the adsorption plane, the potential $0 a t this plane is given by the electrostatic relation (3.2), (26) R. G . Aickin and R. Palmer, Trans. Faraday SOC.,40, 116 (1944). (27) J. H. Phillipa and E. K. Rideal, Proc. Roy. SOC.[London), 8332, 140
(1956). (28) A. J. Payens, Philips Res. Rspt., 10, 425 (1955). (29) F. van Voorat Vader, Proc. Intern. Conor. Surface Actiuity, 9rd, Cologne, I f , 277 (1960). (30) M. A. V. Devanathan, Trans. Faraday Soc., 60, 373 (1964).
Vola 67
S.LEVINE,J. MIXGINS,AND G. M. BELL
2098
I I
I
I
Phase
I I I
I I
I I
I
1
I
I
I
I
I
I
I
I
I I I
a: I 0
I I I
I
I I I
I
I
I
I I
I
I I
I
I
I
I
Aqueous
I
Phase
I I
I
I I I
I
I
I I l
I I
I
I I I I
I I I
I
I
I I
I I
I
I I I
I
I
I
1
I
I
I I
u-
I
I
I
I I
*
Fig. 1.-Model
I
I
II I
I
I
I I
I - 1 Non Aqueous
I
I I
I
I I
I
I
I
I
I
I
I
I I I
I
I
I I
I
1
I
I
I
I
d
of the monolayer region depicting the infinite set of image reflections of charge density u on the adsorption plane.
kT In [ v p / ( N l e-
ua)] =
- a - e#o
-
+ kT In ( n / 4
(3.4) From the relations (3.1)-(3.4) it is shown in Appendix B that the condition that the potential #dl a t the 0.H.P be a maximum with variation in uoa t fixed ionic strength in the substrate requires that cp(uo,u~)
Also the condition of electrical neutrality reads u0
+ + up
Ud
=
0
(3.3)
Various a ~ t h o r shave ~ ~ suggested -~~ that the counterions have only a small specific binding with the primary head groups. Recent experimental work by Kotin and N a g a s a ~ v a ~on ~ proton magnetic resonance, on Raman spectra, and Mukerjee3* Lapanje and on partial molal volumes support this view that the binding of counter-ions and the primary charge in various polyelectrolyte and colloid systems is mainly electrostatic in origin. These experimental results imply that the counter-ions can be considered as a very highly mobile m0nolayer.~5 This mobility of the counter-ion charge is also shown by unpublished work by Mingins and PethicaZ3on the entropies of compression of SODS spread at the A-W interface. We shall therefore assume that the specific binding is small and that the counterions and ionized monolayer molecules move largely as separate entities in the common adsorption plane. For simplicity, we shall choose a Langmuir type model of adsorption sites for the counter-ions, such that the density N1 of sites available to counter-ions may be considerably larger than the density of head-groups 1/A. If no is the volume density of water molecules in the electrolyte, (o(u0,ap) the self-atmosphere, discrete-ion, or fluctuation potential at the center of a n adsorbed counter-ion (a function of both uo and up), @ the specific adsorption potential of the counter-ion, and the activity coefficient of the electrolyte is equated to unity, then the adsorption isotherm for the counter-ions may be written as (31) J. T. Davies, Proc. Intern. Congr. Surface Activity, Srd, Cologne, 11, 309 (1960). (32) L. Kotin and S L Nagassws, J . Am. Chem. Soc., 83, IO26 (1961). (33) S. Lapanje and S. A. Rice, ibtd., 83,496 (1961). (31) P. Mukerjee, J . Phgs. Chem., 66, 843 (1962). (35) S, Lapanje. J. Haebig, H. T. Davies, and 5. A. Rice, J . Am. Cham. Soc.. 83, 1590 (1861).
and the potential I$‘o’ a t the adsorption plane likewise shows a maximum. The condition (3.5) is still valid if the “outer zone” of thickness y in the inner region is eliminated and the diffuse layer is assumed to extend right up to the common plane of head-groups and adsorbed counter-ions. In an earlier paper3 we were concerned with the chemical potential of an ionized monolayer molecule rather than with that of the adsorbed counter-ion and we assumed that the fluctuation potential for the molecule was a function of uo up. However, in the comparison with experiment a@ was taken as zero. If the potential is similarly regarded as depending on uo up only, then the left-hand side of (3.5) vanishes Such a or I#o. and there will be no maximum in form for cp implies a strong specific binding between each counter-ion and a head-group, resulting in an ion pair which behaves as one kinetic unit in any twodimensional thermal motion. A maximum in l+dl implies one in the IS1 potential as well as in J,I%J, and hence the experimental observations of maxima in both and $01 require small specific binding, which is consistent with the other evidence on counter-ion binding given earlier. I n determining cp we apply the Guntelberg-Riluller charging process in the theory of strong electrolytes to an adsorbed counter-ion and therefore imagine that such a n ion carries a charge e’ which will vary from 0 to its full value e . I n the adsorption plane surrounding a typical counter-ion of charge e’ there will be ttvo-
+
+
Oct., 1963
2099
SURFACE POTENTIALS O F IONIZED MONOLAYERS
dimensional mean charge distributions of ionic head groups and other counter-ions carrying their full charge and these are described by the functions ~ ( r ) and up(r), respectively, of the distance r from the chosen who originated the dision e‘. The Soviet creteness-of-charge effect, were only concerned with a distribution of counter-ions and replaced up(r) by a hexagonal array of ions. They ignored, therefore, the thermal niotion of the ions and their model refers to a two-dimensional solid a t absolute zero. The general fornis of the functions uo(r) and ap(r) will depend on whether we are dealing with a two-dimensional dilute gas or with a dense gas or liquid, and the two possibili) up(^) ties are illustrated in Fig. 2 . Since ~ ( r and become the mean densities uo and up, respectively, when the counter-ion e’ is not fixed at the specified position in the adsorption plane but is free to move in it, conditions of electrical neutrality must be satisfied. For a monolayer of infinite area these read 2a JOm
r[uo - uo(r)]dr = 0
(3.6)
2n
r [ u p - up(r)]dr = e’
(3.7)
JOm
I n this connection it is observed that up will change very slightly with e‘, the variation being of order l/Np, where Np is the total number of adsorbed counterions in the monolayer film. Also uo(r) and ap(r) will in general both involve the mean distributions (TO and up as parameters. 4. Simple “Cut-off” Approximation to the SelfAtmosphere Potential cp(oO,upp) .-Since an accurate determination of cp(O0,up) is difficult we shall examine here a simplified model. It is assumed that the electrolyte concentration is so small that the screening effect of the ions in the diffuse layer on the interaction between the ions in the adsorption plane can be disregarded. Also the potential will be expressed as a sum of separate functions of UO and up, namely cp(uo,fl#d = cpl(ud
+
(PII(u0)
(4.1)
A simple approximation to up(^) is the cut-off model employed in previous papers and “corrected” here to maintain the validity of (3.11) a t small e’. Let a be the distance of nearest approach (exclusion diameter) of two counter-ions. Then for e‘ > aa2up,we put up(r) =
upl r 0 r
> rB’ < I’~‘
where rp‘ is defined by the condition (3.11), which yields e’ (4.3) For e’ < au2up, we ernploy the adjusted cut-off approxiniation which is defined by arp’2up =
Fig. 2.-Radial charge distributions U O ( T ) and u p ( r ) : ( a ) dilute gas-like distributions; (b) dense gas-like or liquid-like distributions.
rp, say as e’ varies from 0 to e, but it is more realistic to regard the exclusion diameter u as the minimum value of Q‘. When e’ is so small that the charge e’ in the area aa2 “occupied” by the ion on the adsorption plane is actually less than the average aa2up, then there will be a compensating accumulation of charge in the region immediately surrounding the ion and this is accounted for in the first approximation by the ring of charge. The cut-off model described here, in common with the hexagonal array used by the Soviet scientists, suffers from the defect that temperature dependence has been excluded. I n a later paper it is hoped to discuss rather better models in which the effect of temperature is taken into account. It is convenient to introduce the quantities rp‘ would increase from zero to its final value
fi
=. (€1
where phase
€2
Krylov, Dok2. Alcad. Naulc SBSR, 186, 1425 (1960); V. G.Leviah and V. 8. Krylov, ihid., 142, 123 (1962).
€2)
> 0, -
+
