Modified Poisson-Boltzmann Equation and Free Energy ofElectrical Double Layers in Hydrophobic ColloidsBY S. LEVINE* AND G. M. BELL?Received 21st June, 1966Corrections are obtained to the Poisson-Boltzrnann (P.B.) equation for the potential distributionin the diffuse part of the electric double layer for an aqueous medium. In addition to effects ofion-size, variation in dielectric constant and self-atmosphereimage terms, these corrections include theeffect of medium compressibility and so-called cavity potentials. Numerical integration of thecorrected P.B. equation for a single plate and for two parallel plates shows that the potential dropsmore rapidly with distance from a plate surface than is predicted by the P.B. equation. The freeenergy of the double layers with the modified P.B.equation is determined, in the absence of specificadsorption of counter-ions.1. INTRODUCTIONThe numerous publications (see ref. (1)) on corrections to the Poisson-Boltzmann(P.B.) equation for the potential in the diffuse part of the electric double layer fallinto two broad categories. The more fundamental approach, based on statisticalmechanics 2-6 has been limited in its scope because of the mathematical difficulties.Although it has been indicated that owing to ion-size effects the classical P.B. equationceases to be an acceptable approximation in 1-1 electrolytes for concentrationsexceeding roughly 0-1 M, the approximations made in these studies appear tomake this result inconclusive.16 Most of the investigations are in the second category,which is based largely on the method of local thermodynamic balance.In arecent paper (to be referred to as paper I) we have extended this work to includeimage-self-atmosphere potentials? the important so-called cavity potentials (both ofwhich, although treated approximately, are based on statistical mechanics) and alsothe effect of Compressibility of the dispersion medium. In this paper, we presentsome numerical solutions of the modified P.B. equation for a single charged plateand for two equal charged parallel plates, immersed in al a ~ g e volume of aqueous 1-1electrolyte. For simplicity, no account is taken of tire Stern inner region con-taining adsorbed counter-ions. A brief discussion of previous work is imluded.The determination of the double-layer Helmholtz free energy, with all thecorrections considered in this paper, is lengthy, particularly since it may be expressedin a number of equivalent and physically meaningful forms.17 Here we quote withoutproof one particular form for the free energy for a single plate and two parallelplates immersed in a large volume of dispersion medium and give some numericalresults.2.NOTATION AND PRELIMINARY RELATIONSe, k, N, T proton charge, Boltzmann’s constant, Avogadro’s number, absolutetemperature T = 298°K.* Department of Mathematics, University of Manchester, Manchester.f Department of Mathematics, Chelsea College of Science and Technology, London, S.W.3.670 FREE ENERGY OF ELECTRICAL DOUBLE LAYERSx distance measured from (i) plate wall for a single plate (ii) medianplane for two parallel plates.2h separation of two plates at x = kh.el charge on ion of speciesj(j = I , .. ., p ) ; el = e, e2 = - e for 1-1electrolyte type.nj,n(?),ng volume number densities of ion j or of solvent (water) ( j = 0) atposition x, at median plane and in electrolyte bulk (at x = KI forsingle plate) respectively; np = n; = n for 1-1 electrolyte; c =concentration in molesll. of 1-1 electrolyte.+ = $(x) mean electrostatic potential at x ; for single plate $0 = $(O) is platepotential and $(a) = 0 ; for two plates $0 = $( & h) and t,hm = $(O)is potential at median plane.v = e$/kT, V m = e$mlkT, ~0 = e$o/kT,s = sinh (q/2), s, = sinh (q,,,/2), (2.1 )k = exp (-urn), 0 = arc sin (exp (- (y - ym)/2)),EO dielectric constant of bulk electrolyte,KO Debye-Hiickel parameter for bulk electrolyte, given byPj = 1IC: = (4n/sokT) nge? = 8nne2/&,kT (1-1 type), (2.2)1-1 type) (2.3) I rc0x = In [tanh (go/4)/tanh (q/4)],KOX = k3[F(k,z/2)- F(k,@)],are solutions of P.B.equations for one plate and two plates respectively,where F(k, CD) is elliptic integral of the first kind ; when = yo, CD = CD0,x = h, half-plate separation.is the density of ion j at x and at median plane by P.B. equation.v j = ng exp (- ej$lkT), vy) = nj. exp (- ej@,/kT), (2.4)E = d$/dx, electric field at x.are the electric fields for one plate and two plates respectively in P.B.approximation.4n 8nne2K 2 =- C v.e? = - cosh qEokTj=l I EokT (1-1 type)for the local K at x in P.B.approximation.unit volume) no, nl, . . ., np of solvent andp ion types.bulk electrolyte; hj = uS/u& ratio of partial volumes.x(nj) logarithm of the number of configurations available to numbers (peruj, j = 0, 1, . . ., p , partial volume of solvent molecule or ion j ; u j = u? iS . LEVINE AND G . M. BELLis volume fraction statistics formula of Flory-Huggins.P l Px(nj) = xo(no)- C n j ( h n j - 1 ) - - C C ( ~ z ~ - n ~ ) ( n ~ - n I t g ) v ~ ~ (2-9)j = 1 2 i = l j = 1is a formula based on imperfect gas theory; for hard-sphere model ofhydrated ions, exclusion volume u ~ j = 4na$3, where aif = distanceof nearest approach between ions i and j . Comparison of (2.8) and(2.9) gives vij = hihpvo.A fraction of actual ion charge in Debye-Hiickel fictitious chargingprocess.~ o ( u 0 ) Helmholtz free energy per molecule of pure solvent at molecularvolume VO.{;(uo) Helmholtz free energy of interaction of ion j with the immediate solventneighbourhood ; a separate coupling parameter A', distinct from dand equal to 1, is assumed in the interaction energy between an ion andmolecules in its hydration shell.n(2.10)is the Helmholtz free energy per unit volume in completely dischargedstate d = 0.is the corresponding contribution to chemical potential of solvent orion i.Pj = Op*(nj) = -f*+ C njp7 (2.12)is the local pressure at x in the completely discharged state ;1 = 0.q - v ; = hi(Pi/Po)(vo-v~), i = 1 , .. ., p (2.13)gives the relation between partial volumes based on the assumptionthat ratio of compressibilities of a hydrated ion pi and a water mole-cule PO is constant ; PO = 0.46 x 10-10 c.g.s.units(2.14)is the self-atmosphere-image (Debye-Huckel) potential at ion i, chargeel( = h i ) , at position x and charging stage A. For a single plate (of infinitethickness)#IDH) = 41DH)(e; ,A) = - (ef /Eo)g(Arc,x)(2.15)for small or large values of E ~ / E O ; d, common distance of nearestapproach between two ions; E ~ , dielectric constant of plate medium(Ep = 10);f = (EO- E ~ ) / ( E o + gp) reflection coefficient in electrostatic imaging72 FREE ENERGY OF ELECTRICAL DOUBLE LAYERSg(rC0) = q / ( l + ~ o a ) value of g in bulk electrolyte.For two plates, if q(u) = exp ( - ~ K ' u ) / u ,( 2 - ( m - n ) 2 ) f "+"q(mh-x+nh+x) (2.16)- -1 03n = l r n = n , n f ldescribes multiple screened image reflections.is the mean charge density at x and stage A.4 c a v ( 4 = - (2.a21&o>P(4 (2.17)is the main term in the so-called cavity potential, due to removal ofdiffuse layer charge distribution from spherical exclusion volume of ion.+fav(A) = (47ca3/3&O)p(l)g(l~,~) (2.18)is an additional term in the cavity potential, due to the reduction ineffective charge of ion by an amount (4nd3/3)p(A).(2.19)is the standzrd Debye-Hiickel local free energy density associated withthe potential (2.14) ; integration is carried out at constant nj.(2.20)(2.21)gives the condition of self-consistency implied in the method of localthermodynamic balance and satisfied by (2.14).is a particular form of condition of self-consistency with +cav(A), sincethis potential is independent of ei ; the self-consistency condition is notsatisfied by &&).(2.23)are parts of the self-atmosphere potential and corresponding free energydensity which satisfy the self-consistency condition.a f c a v ( 2 ) l a n i = ei#cav(n) (2.22)#LA> = $f""(~eiJ> + 4cav(R), feI(1) = fDH(2) + fcav(A>(2.24)is a pressure term associated with self-atmosphere-cavity potential atR = 1 .(2.25)(2.26)n(nj> = Pel(nj) + ~ " ( n j).PE = cO+ C (nj-ng)(aeidng)ni,.=o-aE2j = S.LEVINE AND G . M. BELL 73gives the dependence of dielectric constant of medium at x on ion con-centrations and electric field ; dielectric saturation parameter a - 3 x 10-7e.s.u.7(2.27)is the energy of polarization of a hydrated ion or solvent moleculefor moderate field E.(2.28)is the form of E equivalent to (2.26) for 1-1 electrolyte,8 where as-suming a variation of E with concentrations at constant pressure,4nkT 4nkTJ I - h l J o = --6,(s2-si), J2-h2J0 = -- 6,(s2 - s:).(2.29)EON EONHere we assume 61 = 8 2 = lO36,6 = - 7.5.t = d In &;Id In PO; PO, density and dielectric constant of pure solvent ;t = 1.34.9noJo m -4nkTt(s2-s2) (1-1 type).is the (electro-)chemical potential of ion i at stage A, which is uniformthroughout diffuse layer ; pLp(A) chemical potential in bulk electrolyte.Po = A+ Jo (2.31)is the chemical potential of solvent.I = -hE4"(2.32)r0 = (1/8n)(&E2+I)- n j J j - I ? ( n j ) + I I ( n ~ ) ) -j = Ois the first integral of the modified P.B.equation for system of twoparallel plates, obtained from (2.30)-(2-31) in paper I for a single plate,where n(n(7)) is replaced by rI(n;) and limits of integration (0,x) bySo = e2/&kTa (2.34)z(x) = 3Eln (1 + x) - x + +x2]/x3. (2.35)S = Jcosh q, S m = Jcosh q,, V = 1 + K o a S , V , = 1 + KOaSm. (2.36)G(V, v,) = $( v 4- v:) -$( v3 - v:) i- 3( v2 - V:) - 4( V - V,) + In (VlV,). (2.37)3 ( S 2 - s:,2 (I -I- uoa)'H(S,S,) = S 3 ~ ( S x o a ) - S:z(S,lcoa) - - (2.38)l(q) = Ag(xoS,x) sinh q/rco f, A g ( d , x ) = g(rc',x) - d / ( l + d a ) (2.39)where x is regarded as a function of g, given by the solution of the modified P.B.equation74(2.40)(2.41)(2.42)where x is held constant in the integration.3.SOLUTION OF MODIFIED P.B. EQUATIONEqn. (2.33) can be expressed in a form suitable for numerical integration, viz.In arriving at (3.1) from (2.33), it is assumed that the electrolyte is sufficiently dilute(&q<no) and that the difference between the ion densities y1i given by the refinedtheory and the densities vi given by the P.B. equation are small compared withvi. The equation for a single plate corresponding to (3.1) was derived inpaper I. A first integral of the P.B. equation is recovered by retaining the firstterm only on the right-hand side of (3.1). The second, third, fourth and fifth termsrepresent respectively the corrections due to (ii) volume (ion-size) effect (iii) depend-ence of dielectric constant on ion concentration and contribution of polarizationenergy of ions to electro-chemical potentials, (iv) dielectric saturation and (v) com-pressibility effect of medium, PO being the compressibility of the pure solvent.Theseventh term is due to the self-atmosphere-image (Debye-Hiickel) potential and thesixth and last terms account for the two cavity potentials $cav and $La", respectively.Considering a 1-1 electrolyte, we choose the Flory-Huggins formula (2.8) andwrite hl = hz such that hTvo equals the exclusion volume 4na3/3, which is the samefor all three types of ion-pairs. Then (3.1) can be expressed asPj = S . LEVINE AND G . M.BELL 75In the two integrals yl(q,qm), ~2(q,qm) defined by (2.40) and (2.41), x can be treatedas a function of y and it is sufficient to assume the solution of the P.B. equation,as given by the second relation in (2.3). For a single plate we need to put qm = 0and therefore Sm = 0, Sm = 1, Vm = 1 +Kod and we substitute the first relationin (2.3) for x as a function of q in yl(y1,0) and yz(q,O). If we use (2.9), then differentdistances of nearest approach between ion pairs can be assumed. For example,writing u11 = 2122 = 2012 = 8na3/3, the third term on the right-hand side of (3.2)is replaced by72-Because the expressions (2.8) and (2.9) for the configuration function x(nj) startfrom different premises concerning the structure of the solvent, it is possible that3 -l -4 \ 0.5\ 40.41 I 1 1 I0 5 10 15 20 25x(&FIG.1.-Potential-distance (-q-x) plots for single plate for equal exclusion volumes for the threeion-pair types in 1-1 electrolyte ; a = 4 A, 30 = 4. Modified P.B. equation-curves A,C ; P.B.equation-curves B,D ; concentration c = 0.1 M, curves A,B ; c = 0.01 34, curves C,D. Ratio ofslopes (plate charge densities) at intersection point of A,B (-q = 3, xo = 3-89A) equals 1.25;that of C,D (3 = 3.5, xo = 3-87 A) equaIs 1-08. Plots of C(X), curves A1,Cl. Fs/Fip-B.l = 1.112,1.044 for curves (A,B), (C,D) respectively. On scale used curves C and D are almost coincident andD is drawn broken to indicate that it lies above C.the compressibility term in (3.2) will also be altered when (2.8) is replaced by (2.9).Since we have not investigated this problem, we shall retain the compressibilityterm based on (2.8).Fortunately this term is small.Typical numerical solutions of the eqn. (3.2) for the potential in the diffuse layeras a function of distance have been obtained for a single plate and for two parallelplates. For a single plate (fig. 1, Z), numerical integration of (3.2) is initiated byintroducing the solution of the P.B. equation at large x at a specified wall potential( q g ) and then step-by-step integration by a Runge-Kutta technique proceeds towardssmaller x until a distance XO, say, slightly less than 4A, is reached from the platewall; xo is treated as the distance of nearest approach of the hydrated ions to th76 FREE ENERGY OF ELECTRICAL DOUBLE LAYERSwall.For comparison, the solution of the P.B. equation is included with the samepotential at x = XO. If we write (3.2) in the formthen, because (3.4) is a first integral of the P.B. equation when C(x) = 0, C(x) isa convenient function to describe the difference between the modified P.B. equationand the P.B. equation; and this is also plotted. It is observed that the potentialderived from (3.2) and (3.3) drops more quickly with distance x than the P.B.potential. This is consistent with the conclusions reached from cluster theoryconsiderations.2-5 In contrast, Sparnaay 10 and Steinchen-Sanfeld, Sanfeld and(dtf/dx)2 = 4ti;(s2 -~:)(1+ C(X)) (3 *4)i t I5 I0 15 2 0 Lx(&FIG. 2.-Similar plots to fig.1, with volume effect given by (3.3). Curves A,B, c = 0.1 M, ratioof slopes = 1-16 at intersection 3 = 2.8, xo = 3.82A; TO = 4. Curves C,D, c = 0.01 M; ratioof slopes = 1-20 at intersection 7 = 5.0, xo = 3-95A; 30 = 6. Plots of C(x), curves At,C1.FJF$P-B.) = 1-069, 1.078 for curves (A,B), (C,D) respectively.Hurwitz,ll calculate corrections to the P.B. equations which have the oppositesign. The reason for this disagreement is that they have omitted the self-atmosphere-image and cavity terms,* which dominate the sign of C(x) ; e.g., these terms contributeto C(x0) the amounts 0-46 and 0.40 in curves A, of fig. 1 and 2 respectively. Further-more, our calculations suggest that even at 1-1 electrolyte concentrations c = 0.1 Mand potentials -50-75 mV, the corrections to the P.B.equation in the diffuse layerare not excessive, because of partial cancellation among the various effects. Ap-parently trustworthy information on the range of validity of the P.B. equation inthe diffuse layer requires simultaneous treatment of all the corrections to this equa-tion. Also, the suggestions2-5 that the P.B. equation is only adequate at verylow electrolyte concentrations are not borne out by the present work. However,since our treatment of certain corrections, e.g., the volume effect, probably shouldbe modified very close to the plate wall 12 we cannot claim conclusive evidence thatthe P.B. equation is a reasonable approximation at concentrations as high as 0.1 Min 1-1 electrolytes. Potential-distance plots for 2 parallel plates are shown in* see note added in proofS.LEVINE AND G . M. BELL 77fig. 3. A value is chosen for the potential at the median plane and the distance 2 hbetween the plates is fixed from the solution (2.3) of the P.B. equation by choosingthe wall potential yo at a given concentration c. Step-by-step integration of (3.2)from the median plane to a wall proceeds until a value xo just less than h, is reached.The corresponding solutions of the P.B. equation (curves B and D) are shown for4 i I I I I'3 -I -I 1 I0 2 4 6 8 10--I ----I__ L--- 112FIG. 3.-PotentiaLdistance plots for 2 parallel plates with volume effect given by (3.3); ym = 1.5,yo = 4, K o h = 1.23. Modified P.B. equation, curves A,C; P.B. equation, curves B,D.CurvesA,B, c = 0.1 M ; ends of curves nearest wall xo = 10 8, (~0x0 = 1.04, h-xo = 1-78 A) and 11-07 A(~0x0 = 1.12) respectively withcommony = 3.54, wheresloperatio = 1.12. Curves C,D, c = 001 M ;ends of curves nearest wall xo = 34A ( ~ 0 x 0 = 1.12, h-xo = 3.27A) and 353A (KOXO = 1-16)respectively with common q = 3-59, where slope ratio = 1.03.0.2n-0.1 3 ucomparison in fig. 3. If the latter curves are shifted in such a way that they inter-sect at xo with the corrected curves (A and C respectively), then again we find thatthe P.B. solution drops less rapidly with distance from the wall than does themodified solution.FREE ENERGY OF THE DOUBLE LAYERSIn an earlier paper 13 (paper II), we described a general method of obtaining thefree energy of the electric double layers of two or more colloidal particles of arbitraryshape situated in the dispersion medium.The electrical part of the free energyFe was determined by means of the Debye-Hiickel charging process, carried out atconstant temperature, volume and surface density of ions on the surfaces of theparticles. The volume (ion-size) effects and dielectric saturation were consideredin paper 11. In a later treatment of two parallel plates, Sparnaay 14 included theeffect of the dependence of dielectric constant on electrolyte concentration. Herewe include the additional corrections due to self-atmosphere-image and cavitypotentials and to compressibility of the dispersion medium. For simplicity, weignore the presence of an inner Stern region and assume that the charge on th78 FREE ENERGY OF ELECTRICAL DOUBLE LAYERSplates is due to adsorbed ions which are identical in type with one of the latticeions constituting the plate medium.The total free energy which concerns us isobtained by adding to Fe the so-called chemical free energy Fc, i.e., the free energyof the colloidal system in the completely discharged state ;1 = 0. As the volumeof the dispersion medium becomes infinitely large, Fe + Fc -+a. We therefore sub-tract the free energy that the system would have if no ions were adsorbed on thecolloidal surfaces and the systems were homogeneous up to these surfaces. Ignoringthe outer faces of the parallel plates, the resulting free energy per plate per unitarea of its inner face is given byFs =If we put E = EO, I I ( r z j ) = k T 2 q and retain only the first and third terms in (4.1),then Fs reduces to the Verwey-Qverbeek free energy.15The last term in (4.1) differs from the others in that integration with respect to;I has not been carried out.This is basically owing to the fact that the potentialq5LaV(jl) defined by (2.18) does not satisfy the self-consistency condition discussedin $2. To carry out computations with a 1-1 electrolyte it is convenient to express(4.1) in a form which corresponds to (3.2). This readsj = 1nPo% UokT --4n ( ) [ (s2 + ( s2 - si)t> (s2 - ( s2 - s i ) t I - $1 +It is possible to express a number of the integral terms in (4.2) in terms of ellipticintegrals by a method of successive approximations and use of the P.B.equation.14, 15Again, (3.3) can be substituted for the third term in the integrand when weassume different distances of nearest approach between ion-pairs. If the P.B.equation is assumed, then the first term only in the integrand is retained.Integration with respect to x then gives the Verwey-Overbeek expression for thefree energy, which we denote by Fip.B.), in terms of elliptic integrals. For a singleplate we put qm = 0, 12 = and (4.2) reduces toBecause of electrostatic imaging and ion-size, the potential at the limit of the diffuselayer ( ~ ( x o ) ) is less than the P.B. potential ( ~ 0 ) at the wall. Values of the ratioFs/FLp.B.) for the various examples of a single plate illustrated in fig.1 and 2 areincluded in the legends. The ratio is somewhat greater than 1, whereas Sparnaay 14obtained a value less than 1 because he omitted the self-atmosphere-image-cavitypotentials. Since his potential-distance curves diminished with distance from theplate walls more slowly than predicted by the P.B. equation, Sparnaay also calculateS. LEVINE AND G . M. BELL 79an increase in the double-layer repulsion between two parallel plates at a givenseparation, resulting from his corrections to the P.B. equation. From our presenttreatment of these corrections, we would expect, however, a decrease in repulsion;numerical computations of the complicated integrals in (4.2) for this case are stillin progress. The small departure from the P.B. solution obtained here suggeststhat in 1-1 electrolytes, modifications in diffuse layer theory will not greatly alterpresent-day stability theory of hydrophobic colloids.We are indebted to Manchester University Computing Service for facilities on theirAtlas electronic computer.1 Bell and Levine, Chemical Physics of Ionic Solutions (ed.Conway and Barradas (Wiley and2 Stillinger and Kirkwood, J. Chem. Physics, 1960, 33, 1282.3 Stillinger and Buff, J. Chem. Physics, 1962, 37, 1 ; 1963, 39, 191 1.4 Krylov and Levich, Zhur. Fiz. Khim., 1963, 37, 106.5 Krylov, Electrochim. Acta, 1964,9, 1247.6 Martynov and Derjaguin, Dokl. Akad. Nauk S.S.S.R., 1963, 152, 767; Martynov, Research7 Malsch, Physik. Z., 1928, 29, 770 ; 1929, 30, 837. * Hasted, Ritson and Collie, J.Chem. Physics, 1948, 16, 1.9 Jacobs and Lawson, J. Chem. Physics, 1952,20, 1161.10 Sparnaay, Rec. trav. chim., 1958, 77, 872.11 Steinchen-Sanfeld, Sanfeld and Hunvitz, (a) Kon. Acad. Wetens, Belg. A, 1966, 66, 41 ; (b)12 Levine and Bell, J. Physic. Chem., 1960, 64, 1188.13 Bell and Levine, Trans. Faraday SOC., 1957, 53, 143.14 Sparnaay, Rec. trav. chim., 1962, 81, 395.15 Verwey and Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier Publishing Co.,16 Bell and Levine, Conference on Statistical Mechanics, International Union of Pure and Applied17 Bell and Levine, 2. physik. Chem (Lpg.), 1966, 231, 289.18 Huckel and Krafft, 2. physik. Chem., 1955,3, 135.19 Sparnaay, 2. physik. Chem., 1957,10, 156.20 Kirkwood and Poirier, J.Physic. Chem., 1954, 58, 591.21 Loeb, J. Colloid Sci., 1951, 6, 75.22 Williams, Proc. Physic. SOC. A , 1953, 66, 372.Sons, 1966), pp. 409-461.in Surface Forces (ed. Derjaguin) (1966) (Consultants Bureau, New York), 2, 75.Electrochim. Acta, 1964,9, 929.Amsterdam, 1948).Physics, Copenhagen, July, 1966 (Benjamin).Note added in proofSparnaay has brought to our attention that in their extension of the Debye-Huckel theoryof electrolytes, Huckel and Krafft 18 obtained corrections which correspond to the cavitypotentials and 4Lv defined by (2.17) and (2.18), and which were interpreted as such byhim.19 In this case the cavity effect is due to the displacement of an ion’s self-atmospherecharge, rather than the diffuse layer mean charge, by the finite volume of a second ion.Theseauthors therefore preceded Buff and Stillinger 3 to whom we had attributed the concept ofcavity potentials in ref. (1) although these were later neglected by Sparnaay 10 in his theory ofthe diffuse layer. Also, the cavity effect described by is implicit in the work of Kirkwoodand Poirier 20 on strong electrolytes, as well as in the statistical mechanical treatment of ion-size effect in diffuse layer theory in ref. (2)-(6). Comparison of our results with those ofHuckel and Krafft can be illustrated by means of eqn. (3.4) for a single plate at large x.This reads(dq/dx)2 = 4k-;s2( 1 + C( GO)),where, for the particular unequal exclusion volumes treated above80 FREE ENERGY OF ELECTRICAL DOUBLE LAYERSto which Huckel and Krafft’s more general form reduces. The first term, which dominatesfor KO a< 1 , is due to the cavity potential &cav, the second is due to &iaV and the third to thevolume effect ; the last-named contribution is absent if the exclusion volumes for the threetypes of ion-pairs are equal. (We have made the reasonable assumption that the radius aof the exclusion volume 2112 for an oppositely charged ion pair equals the Debye-Hiickel‘‘ a ”.) The cavity potential which determines the (positive) sign of C( a), lowers theenergy of a counter-ion in the diffuse layer, but raises that of a co-ion. Thus the populationof counter-ions is increased, whereas that of co-ions is decreased. Because (in a 1-1 electro-lyte) the number of counter-ions exceeds that of co-ions, the net effect is an increase in the totalion concentration in the diffuse layer. Assuming equal exclusion volumes and neglectingthe self-atmosphere-image terms (our potential (2.14)) and consequently $cav, Stillinger andKirkwood 2 obtained an expression for the surface potential of a single plate, which corres-ponds at small IcOa to our C(m) = H#oca)2 when expanded in powers of q a . In paper 1 weshowed how an extension of the work of Loeb 21 and Williams 22 on the self-atmosphere-image effects of point ions in the diffuse layer leads naturally to cavity potentials when theion-size effect is included. The Loeb-Williams theory was mentioned by Sparnaay 10 andapplied in more detail by Hurwitz, Sanfeld and Steinchen-Sanfeld.llb In their more recentresults on the modified P.B. equation at a single plate, these authors lla introduced a meanactivity coefficient for ions in the diffuse layer to account very roughly for the Loeb-Williamsterm. The cavity potentials, however, are absent in their work