Forces between dissimilar colloidal plates for various surface conditions. 11. Equilibrium adsorption effects1

Forces between dissimilar colloidal plates for various surface conditions. 11. Equilibrium adsorption effects1 G. M. BELL Mathetnatics Departtnent, Chelsea College (University of London), Manresa Road, London, SW3 6LX, England AND G. C. PETERSON Unileuer Research, Port Sunlight Laboratory, Quarry Road East, Bebington. Wirral, Merseyside, L63 3JW, England Received July 7, 1980 This paper is dedicated to Dr. Sam Leuine on the occasion of his 70th birthday G. M. BELL and G. C. PETERSON. Can. J. Chem. 59, 1888 (1981). A method previously developed by the authors is used to study the effects of adsorption of ions on the electric double layer interaction between dissimilar colloidal plates immersed in : electrolyte. For adsorption models which permit the total charge on a plate to change sign, the double layer force remains finite at all plate separations, including zero. For weak adsorption of the ions on the plates the force between two dissimilar plates tends to be repulsive at small separations, looking rather like a weakened constant surface charge density model. Conversely for strong ionic adsorption the force tends to be attractive at small separations, rather as in the constant surface potential model. n this paper we discuss threeadsorption models: () fixed primary charge density on the plates with secondary adsorption of both counter-ions and co-ions; (2) fixed primary charge density on the plates with secondary adsorption of the counter-ions only, but including the effects of a Stern layer and self-atmosphere potentials; (3) zero primary charge on both plates with equilibrium adsorption of both anions and cations from solution, the net charge density on the platesarising from differential adsorption of the ion types. G. M. BELL^^ G. C. PETERSON. Can. J. Chem. 59, 1888 (1981). On utilise une methode que 'on a mise au point anterieurement pour Ctudier les effets d'adsorption d'ions sur 'interaction electrique de la double couche entre des plaques colloidales dissemblables immergees dans un electrolyte de type 1: 1. Dans le cas des modeles d'adsorption qui permettent a la charge totale sur une plaque de changer de signe, la force de la double couche demeure finie pour toutes les separations de plaque, y compris zero. Pour 'adsorption faible d'ions sur les plaques, la force entre deux plaques dissemblables tend a &tre repulsive pour de petites separations et se comporte comme un modele de densite de charge superficielle constante faible. Dans le cas d'une adsorption ionique forte, la force devient au contraire attractive pourde petites separations et elle se comporte plut8t comme un modele de potentiel superficiel constant. Dans cet article nous discutons de trois modkles: () un modele de densite de charge primaire fixe sur les plaques avec une adsorption secondaire a la fois des ions opposes et des co-ions; (2) un modele de densite de charge primaire fixe sur les plaques avec seulement une adsorption secondaire des ions opposes mais incluant les effets d'une couche de Stern et des potentiels d'auto-atmosphere; (3) un modele de charge primaire egale a zero sur les deux plaques avec une adsorption equilibree des anions et des cations a partir de la solution; la densite de charge nette sur les plaques provient d'une adsorption differentielle des deux types d'ions. [Traduit par le journal] 1. ntroduction We consider the electrical double-layer forces between dissimilar parallel plates, the left and right hand plates being labelled 1 and 2 respectively. The term "dissimilar" means here that when the width H of the diffuse region between the plates is infinite the reduced potential a,, at the diffuse region boundary adjacent to plate 1 is unequal to the corresponding potential a,, at plate 2. t has often been assumed that the plate potentials 0, and a, remain equal to a,, and a,, for all values of H. However the "constant potential" assumption implies that the electrical force approaches an 'The authors much appreciate being able to participate in this issue in honour of Dr. Sam Levine's devoted work and many important contributions to theoretical chemistry. G.M.B. has had the privilege of collaboration with him over many years and would like to express his grateful thanks. infinite attractive value as H -+ 0, except when a,, = a,. This occurs because the electric field and hence the Maxwell tension term in the force must become indefinitely large if a, and a, remain unequal as H -+ 0. An alternative assumption that the plate charge densities, o, and o, respectively, remain constant implies that the electrical force approaches an infinite repulsive value as H -, 0, except when o, = -0,. This happens because the total charge in the diffuse region per unit plate area is - (o, + o,) and hence if the latter remains nonzero as H -, 0 the ionic concentrations in the diffuse region and the osmotic term in the force become infinitely large. Double layer models are hardly realistic for very small H since, for instance, real ions have a finite size. Nevertheless an indefinite increase in force is unphysical and it seems likely that some mechanism operates to moderate the force by reduc- 0008-404218 11131888-10$01.W/O 01981 National Research Council of CanadaConseil national de recherches du Canada

BELL AND PETERSON 1889 region X plate 1 plate 2 FG. 1. Surface charges and diffuse layer. ing the magnitude of the double-layer free energy. Here ) is the potential in V, e, is the proton charge, One possible mechanism is equilibrium specific k Boltzmann's constant, T the absolute temperature, adsorption of ions from the diffuse layer onto a K the Debye-Hiickel constant for the bulk electrolyte plane at or near the diffuse layer boundary. n this while the x-coordinate is perpendicular to the plates paper we shall use several models to study the effect (see Fig. 1). We assume throughout that the electro- of such adsorption. For definiteness and compar- lyte is of the 1-1 type and the Poisson-Boltzmann ability we fix the values of a,, and a,, in most of equation for the diffuse region can then be expressed ) our illustrations of the theory at 2 and f 0.8 respec- as tively. The methods used here were developed in Bell and Peterson (), which will henceforth be referred [2] d2+/dc2 = sinh + to as, where they were applied mainly to plates A first integral of [2] can be immediately obtained with unequal constant charge densities. and expressed in the physically perspicuous form 1 nteractions between similar plates with a fixed primary charge and equilibrium adsorption of 131 F' = cash 4 - - f(d+/dc)2 counter-ions were first considered by Bierman (2) H~~~ F+ is the reduced force per unit plate area and then by Levine and Bell (3, 4) and Melville and defined for a 1-1 electrolyte by Smith (5). Models of this type are considered in 9 3 and 9 4 below. Stern layer and self-atmosphere effects [4] 9' = F(2nkT) are discussed in 4. and Smith (5) F being the electrical double-layer force per unit considered the assumption that the potential stays 'Onstant at the plane of primary charge rather than the diffuse region boundary. Chan et al. discussed the interaction between similar (6) and dissimilar (7) "amphoteric" surfaces in which all surface ions are in equilibrium with the medium by way of association-dissociation reactions and gave a comprehensive survey of possible cases. The model we introduce in 9 5 yields a similar adsorption isotherm to theirs and [5] d$/dc = kg(+) we discuss its relation to the results deriving from the other models. area and,2 the number of ions of each type per unit volume in the bulk electrolyte medium. n [3] the term cash + - represents the repulsive osmotic pressure term in the force while -f(d+kdc)z represents the attractive Maxwell tension term. Equation [31 can also be expressed as where 0 0' llo,la : p : 8 ---:-!! : [6] g(4) = (2 cosh + - 2-2 ~')"~ > 0 2. Definitions and Method To obtain the diffuse layer width H a second in-, We define a reduced (dimensionless) potential + tegration is necessary. This yields the relations and distance 5 by ' i E -H- diffuse region i i E i i ' Q, +x plate 1 plate 2 [] = e,)/kt 5 = KX

1890 CAN. J. CHEM. Vc The form [8] is to be used when the curve of against 5 has a turning point between the plates, the value of + at the turning point being denoted by +,. This can only occur when F' > 0. For F' < 0 and when F' > 0 but there is no turning point the alternative form [7] must be used. The reduced width KH can be obtained in terms of elliptic integrals, different formulas being necessary for different conditions. These are given in forms suitable for the dissimilar plate problem in the Appendix to. However the results given in the present paper were all obtained by numerical integration. We now consider conditions on the plates. t is assumed that the plate surface charge densities o, and o, are functions of the diffuse layer boundary potentials @, and @, respectively. Reduced densities f, and f, are defined by the relations where E denotes the dielectric permittivity of the diffuse layer aqueous medium and is assumed to be uniform. Bearing in mind that the left and right hand plates are labelled 1 and 2 respectively the electrostatic boundary conditions can now be written in dimensionless form as For any given 9' the possible values of the boundary potentials can be obtained by combining [5] with [lo] to yield At KH = co the signs are unambiguous since (d+/dk), and @, have the same sign while (d+/dc), and @, have opposite signs. Hence putting F' = 0 in [ll] we obtain t can be assumed without loss of generality that For definiteness we shall in most of our calculations adjust the parameters in f, and f, to make [12] yield Qlm = 2, @,, = k0.8. n order to obtain force-separation curves we thus have to solve the non-linear differential equation [2] and, for adsorption models, the boundary conditions [lo] are also nonlinear. For each case we have to pick the correct signs in [].] and, if 9' > 0, decide between [7] and [8]. This can be done during the whole process of taking the plates from infinite to zero separation with the aid of a graph of the type introduced in, in which the ordinate is - d+/dk and the abscissa is 4. t should be stressed that the aim of this graphical method is to give a general picture of the behaviour of the plate potentials and the electrical force F' during this process and is not intended as a substitute for accurate calculation. ndeed either numerical techniques or tables are always essential to evaluate the integrals [7] or [8]. n this method it is necessary to plot the curves of f (4) and - f,(+), which by [lo] give the values of -d+/dc at the diffuse region boundaries, against +. These are termed "plate curves" and if they intersect this corresponds to a solution of the equation and gives the potential @, = 0, = cdo in the mathematical limiting case of zero separation KH = 0. Since it corresponds to finite values of the boundary potential and the electrical field at zero separation the existence of a solution to [13] is necessary if the force is to have a finite limit. f the charge densities on both plates are positive at KH = ca then a solution of [13] implies that either (i) f, or f, changes sign during the approach of the plates or (ii) f1(ao) = f2(qo) = 0. Condition (ii) must occur for symmetrical plates since f,(@,) is then equal to f,(@,). However if fi(+) is a continuous monotonic function of + then A(@,) = 0 implies that fi(+) changes sign at + = a,. The only other possibility for a continuous function is thata(+) has both a zero and a turning point at + = Qo but this is unlikely for any physically reasonable isotherm. Hence for a solution of [13] to exist for all initial values @,, and @,, it must be possible to change the sign of the charge on at least one plate by an appropriate change of the potential a t the diffuse layer boundary adjacent to the plate. This places some restriction on the type of adsorption model which will give a finite force at KH = 0. f it is assumed that the primary ions are all of the same sign and that the number of adsorption sites is equal to the number of primary ions then the plate charge density cannot change sign under any circumstances. This implies that the limiting force is infinite even in the case of similar plates. The other element of the graphical method is to draw isodynamic slope curves which are plots of f g(+) against + for fixed 5'. By [5], these show the variation of reduced electric field -d+/dc with reduced potential in the medium. By [ll], the intersections of the plate and isodynamic slope curves give the possible values of the boundary potential for

BELL AND PETERSON 1891 given 9'. The segment of an isodynamic slope curve between its intersections with two plate curves represents the behaviour of the reduced field - d+/dc and the reduced potential + between the two plates at the given 9' value. The most important isodynamic slope curves are those for 9' = 0 which, by [6], are in fact plots of f 2 sinh (+/2) against +. These are the only isodynamic slope curves which pass through the origin and they divide the plane into regions of positive and negative 9' (see Figs. 2, 6, 8, and 10). The two regions containing the +-axis correspond to positive 9' while the two containing the -d+/dc axis correspond to negative F+. By [12] the intersection of an f, plate curve with the 2 sinh $12 plot gives the potential @, at KH = a~ and that of a - f2 plate curve with the - 2 sinh $12 plot gives @,,. The general behaviour of 9' can be deduced from the regions through which the plate curves pass from the KH = a~ points to the intersection corresponding to KH = 0. Generally this can be seen most easily from the curve for the plate with the lowest potential at infinity. Examples will be found in subsequent sections while may be consulted for further details. An alternative graphical approach of Chan et al. (7) depends on the square of the charge density function. We believe our method to be better than that of Chan et al. since by using o rather than o2 as ordinate the plate curves assume a simpler form and the signs of the plate charges are clearly displayed. The simultaneous states of the two plate surfaces can be seen for any 9' value by using the appropriate isodynamic slope curve and it is easy to deduce the succession of 9' values and plate states between infinite and zero separation. 3. Constant Primary Charge with Adsorption We assume that a plate surface carries a fixed charge density o, of primary ions and a charge o, of ions adsorbed from the 1-1 electrolyte onto an adsorption plane coincident with the diffuse layer boundary or outer Helmholtz plane (see Fig. la). Each unit area of surface carries v+ and v- positive and negative adsorbed ions respectively randomly distributed over N, adsorption sites (Langmuir isotherm). The adsorbed ions remain in equilibrium with the electrolyte medium for all plate separations and the equilibrium equations can be written electrolyte chemical potential, divided by kt. t can be shown from [14] and [15] that the adsorbed charge density is given by n this section we now specialise to the case C+ = C- = C which means that co-ions are not hindered chemically from adsorption although they are less favoured than counter-ions by the electrical potential, at least at large plate separations where @ has the same sign as the charge on the primary ions. Equation [16] now becomes [17] o, = 2eoNsC sinh @/(l + 2C cosh @) t is convenient to introduce a primary charge parameter q defined by [81 o, = e,n,q The sign of q is that of the primary charge while the magnitude JqJ is the number of primary ions per adsorption site. The value ql = 1 means that there is one adsorption site per primary charge. Since lo,l < N,e, for all finite positive or negative values of @ it follows that the resultant charge density o, + o, cannot change sign. Hence, as remarked in 5 2 above, it is necessary that the number of adsorption sites exceed the number of primary ions if the limiting force 9' at H = 0 is to remain finite, which implies ql < 1. We now reintroduce the labels 1 and 2 to distinguish the two plates. t will be assumed that the adsorption constant C and the density of adsorption sites N, are the same for the two plates but that the primary charges are unequal. Combining [9] with [17] and [18] the following relations for the reduced plate charge densities f, and f2 are obtained: where h = e02ns/(~'kt) is a dimensionless constant. We "OW consider a specific example that the bulk medium contains 0.1 mol L- of electrolyte and that there are 10" adsorption sites per square [14] lnv+/(n, - v+ - v-) + @ = n C+ metre (i.e. the area per adsorption site is 10 A2) which gives h = 86.2 at 25 C. (The same value of h would correspond to a bulk electrolyte concentration [15] lnv-/(n, - v+ - v-) - @ = n C- of 0.01 mol L- if the area per adsorption site were Here @ denotes the diffuse-layer boundary potential changed to 31.6 A'.) Taking the adsorption constant while n C+ and n C- are constants which include C as 0.01 we use [12] to adjust the primary charge pathe chemical free energy of adsorption and the bulk rameters q, and q2 to give @, = 2, @,, = k0.8. This

CAN. J. CHEM. VOL. 59. 1981 --- FG. 2. sodynamic slope diagram for primary charge with adsorbed counter-ions and co-ions. The plate curves are shown as full lines and the constant force curves as broken lines labelled with the force values. The points representing the plates at W = co are circled while the points representing them at W = 0 are indicated by squares. Parameter values are h = 86.2, C = 0.01 while for cp1, = 2 the primary charge parameter ql = 0.09473 and for @zm = f 0.8,qz = + 0.02683. corresponds to one primary ion per 105.6 A2 on plate 1 and one per 372.7 A2 on plate 2. Figure 2 is the isodynamic slope diagram for the plate interactions with these parameter values. For the (+ve,+ve) interaction the general force behaviour 2 - can be seen by following the segment of the @,, = 0.8 plate curve which starts at the KH = as point (i.e. its intersection with the - 2 sinh (4/2), 9' = 0, curve) and ends at the KH = 0 point (i.e. its inter- section with the a, = 2 plate curve). nitially the KH+ plate point is in the positive 9' region. Then, after passing through a maximum 9' value between 0.5-2cr and 1, it crosses into the negative region to reach the intersection with the @,, = 2 plate curve. Thus the - 3 - force is positive (repulsive) at large KH but, after reaching a maximum, it becomes negative (attractive) -1 - at small KH. The forceldistance curve itself, labelled "adsorbed chargew, is shown in Fig. 3. For the FG. 3. F~rce/~eparationplotsfor@~, = 2,cPz, = 0.8. The curve labelled "adsorbed charge" corresponds to the model ( + ve, - ve) plate interaction the relevant segments of Fig. 2 and is compared with curves calculated on the the @ZCU = -0.8 and the @, = plate curves constant potential and constant plate charge assumptions. can be seen in Fig. 2 to lie entirely in the negative 9' region. The forceldistance curve is shown in assumed in the model. This can be seen from Fig. 5 Fig. 4. where the limiting value 9,' of the force as KH -+ 0 n Figs. 3 and 4 the forceldistance curves for the is plotted against the logarithm of the adsorption present model are compared with those calculated on constant C. The primary charge densities are kept the assumptions of constant potential and constant constant on each curve at the values used in the total plate charge respectively for the same values of previous calculations, which give @,, = 2, a,, = @,, and a,. The "adsorbed charge" curves go to A0.8 at C = 0.01. The labels (+ve,+ve) and finite 9' values as KH decreases to zero while the (+ve,-ve) refer to the signs of the potentials and "constant potential" and "constant charge" curves plate charges at large KH. For small values of C, the go to infinity. At non-zero KH values, the "constant limiting force 9,' is positive whatever the sign of potential" behaviour is qualitatively closer to that the force at large KH. AS C -+ 0 (log,, C -+ - as) of the present model than is the "constant charge" 9,' rapidly approaches infinity. Physically this behaviour. However, the behaviour of the force at means that for very weak adsorption the behaviour small KH is highly sensitive to the parameter values is similar to that found with constant total plate 9

BELL AND PETERSON 1893 3 KH- FG. 4. AS Fig. 3 for Qlm = 2, Q,, = -0.8. FG. 5. Primary charge with adsorbed counter-ions and coions. Primary charge parameters q1 = 0.09473, q, = + 0.02683. The limiting force So+ at KH = 0 is plotted against the logarithm of the adsorption parameter. charges. As C increases 9,'becomes negative on both curves, which is the situation illustrated in Figs. 2, 3, and 4. For C > 0.1, go+ is nearly constant on both curves. t is noteworthy that even for very large C the "constant potential" situation of infinite attractive limiting force is not approached. There is a small intermediate range of C (-2.992 < - - log,, C < -2.204) where the sign of Po+ is the same as that of 9' at large separations for both the ( + ve, + ve) and the ( + ve, - ve) cases. 4. Counter-ion Adsorption: Stern Layer and Selfatmosphere Effects Our main aim in this section is to investigate how Stern layer and self-atmosphere contributions to the potential at an adsorbed ion affect the inter-plate electrical force. However, we first consider a modification to the model of the last section which consists in assuming that only ions of opposite sign to the primary ions can be adsorbed. We must thus put C+ = 0 in eq. [16] for the adsorbed charge density if the primary charge is positive and C- = 0 if it is negative. ntroducing indices 1 and 2 to distinguish the plates and taking q, as always positive, the reduced charge densities f, and f, are given by 1 fl(el) = [ql - 1 + c1 Cl- - exp exp (6,) C2+ exp (-62) f2(e2) = h[q2 + 1 + C,, exp (-6,) t has been assumed that the adsorption site density is the same for both plates and that h is defined as in 3. We now consider a specific example. We take h = 43.1, corresponding to an area 208L2 per adsorption site if the bulk electrolyte is 0.1 normal and for simplicity put C, - = C, - = C, + = C, giving these adsorption constants the common value of low3. As before, the primary charge parameters are adjusted to make a,, = 2, a,, = f 0.8. n the isodynamic slope diagram (Fig. 6) the plate curve intersection points both lie in the 9' > 0 region and the relevant segment of the a,, = 0.8 plate curve lies entirely in this region. On the other hand the corresponding segment of the a,, = - 0.8 curve passes from the 9+ < 0 region to the 9' > 0 region. Thus in the forceldistance plots shown in Fig. 7, 9' is monotonic for the (+ve,+ve) case but passes through a minimum, changes sign and becomes positive at small KH for the (+ve,-ve) case. We now suppose that the plane of adsorbed charge is separated by a distance y from the diffuse layer boundary and that the region between these two planes has a dielectric permittivity E, which is less than E (see Fig. lb). This region is known as the

1894 CAN. J. CHEM. VOL. 59, 1981 n a comprehensive theory allowance must be made for the "self-atmosphere" which each adsorbed ion creates by repelling other adsorbed ions and modifying the diffuse layer charge in its vicinity. Hence to obtain the potential at an adsorbed ion charge a self-atmosphere term a,.,. must be added to the adsorption plane mean potential. We shall use a simple expression obtained by Levine et al. (8) and used in refs. 3 and 4, which is FG. 6. Primary charge with adsorbed counter-ions. As Fig. 2. Parameter values are h = 43.1, C = while for cd,, = 2,q1 = 0.06189andforcD2, = +0.8,q2 = k0.02188. FG. 7. Force/separation plots for model and parameter values of Fig. 6. Note change of scale at F+ = 2. Curves for > 2 are shown as broken lines. F+ Stern layer and its introduction is intended to allow for finite ion size and decreased polarisability near the plate surface. The potential at the adsorption plane is denoted by @, and is related to the potential cd at the diffuse layer boundary by where p is the depth of the layer of permittivity E, behind the adsorption plane (see Fig. lb), d = P + y and g is a positive parameter depending on P, y, the electrolyte concentration and the permittivities. Combining [21] with [22], t can be seen that the o, terms in the Stern layer effect and the self-atmosphere effect tend to cancel out. There is some simplification if E/E, and the electrolyte concentration are large enough for the diffuse layer boundary to act as a perfect conductor in relation to the inner layer and if the effective adsorption plane self-atmosphere radius is considerably larger than y and p. Then g = 1 if the plate is metallic while if, on the other hand, it is of low dielectric permittivity and adsorbed ion image effects in the plate are neglected we can put pg/d = 1 (8). We shall make the latter assumption and then, using eq. [18] for the primary charge density parameter q, eq. [23] becomes E [241 a, + a,.,. = @ + A-Kyq El More refined theories. of the self-atmosphere effect have been developed (9, 10) but their incorporation in the present formalism would be difficult. We now introduce a specific example. Putting KY = 0.2 and E/E, = 5 we replace @, and @, in eq. [20] by @, + hq, and a, + hq, respectively. We take h = 43.1 as before but modify C to and adjust ql and 9, to make @,, = 2 and @,, = k0.8. Figure 8 is the isodynamic slope diagram for this case and it can be seen that the plate curve intersections are in the 9' > 0 region, just as they were before Stern and self-atmosphere effects were introduced into the adsorbed counter-ion model. One interesting feature is that for the (+ve, +ve) interaction it is the charge on the plate originally at the higher potential (a,, = 2) which changes sign. However, this makes little difference to the force/ distance curves shown in Fig. 9 which are very

BELL AND PETERSON 1895 FG. 8. Primary charge with adsorbed counter-ions: Stern layer and self-atmosphere effects incorporated. As Fig. 2. Parameter values are h = 43.1, C = while for a,, = 2, ql = 0.06847 and for a,, = k0.8, q, = k0.01958. similar to those of Fig. 7. However, it should be noted that the adsorption constant C has been reduced by a factor of 10. This indicates that the main effect of introducing the Stern and self-atmosphere potentials is to produce similar behaviour with a smaller adsorption constant. n other words, they increase the effective adsorption strength for counter-ions. 5. Complete Adsorption Equilibrium The theory of 9 3 is now modified in another way. We assume that there is no fixed primary charge and that all the plate charge is adsorbed from the electrolyte medium and in equilibrium with it. Equation [61 now gives the total plate charge density and it follows that ~ 5 1 C1+ exp(-ql) - C1- exp(q1) /(@') = hl + C1+ exp(-@1) + Cl- exp(q1) C2 + exp ( - 02) - C2 - exp (02) f2(e2) = 1 + C2 + exp ( - 0,) + C2 - exp (0,) For positive potentials at infinite separation FG. 9. Force/separation plots for model and parameter values of Fig. 8. Change of scale as on Fig. 7. Cl+ > Cl- exp (201,) C2+ > C2- exp (202,) while if the potential 0,, is negative C2- > C2+ exp 21@2,1 n a specific example we put the larger adsorption constants equal to 0.3 and adjust the others till, again, a,, = 2, a,, = f 0.8. Figure 10 is the isodynamic slope diagram and it can be seen that as in the "strong adsorption" example given in 5 3 above, the plate curve intersections are both in the 9' < 0 region. However, as would be expected, the range of variation of the plate charge densities is much larger, especially in the ( + ve, - ve) interaction. The corresponding forceldistance curves are given in Fig. 11 and resemble those given for the adsorbed charge model in Figs. 3 and 4 except that the limiting force magnitudes at KH = 0 are much larger, due to the greater variation of plate charge. A similar form of charge density function to [25] was derived by Chan et al. (6, 7) from association/ dissociation surface equilibria. However, it may be noted that mathematically [25] is not as different as it first appears to be from the charge density function [19] based on constant primary charge plus equilibrium adsorbed charge. Both are monotonically decreasing functions of the diffuse layer boundary potential. Also, after some manipulation, fl(o,) as given by [25] can be transformed to x {c, + - C, - - 2 sinh - 0, C1 -(1 + 2C1 +) exp (@,) + C, +(1 + 2C1 -) exp (-40,) 2 1 + C1+ exp (-0,) + C,- exp (0,.

1896 CAN. J. CHEM. VOL. 59, 1981 FG. 10. Equilibrium adsorption of all plate charge. As Fig. 2. Parameter values are h = 86.2 and for a,, = 2, C, = 0.3, C- = 0.001611; for a, = 0.8, C+ = 0.3, C- = 0.05518; for a, = -0.8, C, = 0.05518, C- = 0.3. FG. 11. Force/separation plots for model and parameter values of Fig. 10. Change of scale as in Fig. 7. The a,, = 2, a, = -0.8 plot goes to a limiting value 9 + = - 161.5 as KH-t 0. ; and similarly, of course for f2(q2). Thus, as in the model of 3 above, the charge density can be expressed as a constant term minus an increasing function of the diffuse region boundary potential Q, which becomes zero at Q, = 0. 6. Discussion The electrical forces between two parallel plates in a 1-1 electrolyte have been investigated for several adsorption models in cases where the plate potentials at large separation differ considerably in magnitude. Some attention has been given to the limiting value of the electrical force as the width of the diffuse layer between the plates approaches zero. Although the models treated can hardly be regarded as realistic at "zero separation" this attention is justified for two reasons. One is that the existence of a finite value for the force limit affords a test of the physical reasonableness of the model and the other is that the sign of the force limit provides valuable evidence for the behaviour of the force at non-zero separations. t has been found that at small KH all the adsorption mechanisms treated reduce the interaction free energy magnitude sufficiently to give a finite force limit. An essential condition is that it should be possible for the plate charge to change sign for a sufficiently large change of potential. The behaviour of the electrical force at medium and small separations depends quite sensitively on the details of the model and the parameter values. There is some interaction between the two and it is found, for instance, that in the adsorbed counter-ion model the incorporation of Stern and self-atmosphere potentials has the same effect as decreasing the free energy of adsorption. Two main patterns of force behaviour have emerged for the dissimilar plates studied here. n what may be termed the "weak adsorption" situation there is no change in the sign of the force when it is repulsive at large separations and its magnitude increases steadily as the separation decreases. On the other hand, when the force.is attractive at large separations it passes through a turning point and then changes sign to become repulsive at small separations. n the "strong adsorption" situation the opposite occurs. The force which is repulsive at large separations passes through a turning point and then becomes attractive at small separations while the originally attractive force remains attractive at all separations. There can be an intermediate adsorption parameter range where neither the originally repulsive nor the originally attractive force changes sign but in the case considered here this range is small. t might be larger if the ratio between the potentials at infinity were nearer to 1. We have not attempted here to incorporate the dispersive forces which are necessary for a consideration of colloidal stability. However in the "strong adsorption" case it seems unlikely that there is an effective energy barrier to the approach of two particles.

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