Electrostatic disjoining pressure in symmetrical films with adsorptive charge regulation

Electrostatic Disjoining Pressure in Symmetrical Films with Adsorptive Charge Regulation CHR. ST. VASSILIEFF, B. G. TENCHOV,* L. S. GRIGOROV, AND P. RICHMOND t Department of Physical Chemistry, 1126 Sofia, Bulgaria, *Central Laboratory of Biophysics, 1113 Sofia, Bulgaria, and tFood Research Institute, Norwich NR4 7UA, England Received February 22, 1982; accepted July 7, 1982 The case o f a film containing k strong 1:1 electrolytes sandwiched between nonpolar phases is considered at low electric surface potentials. An analytical equation for the electrostatic disjoining pressure under adsorptive charge regulation is obtained. It gives the well-known limiting cases o f constant surface charge or potential depending on the value o f a dimensionless parameter ~1,. T h e calculated balance between van der Waals' attraction and electrostatic repulsion is in agreement with experimental evidence for the dependence on electrolyte concentration o f the transition between c o m m o n black a n d Newton water foam films stabilized with nonionic surfactants.

INTRODUCTION

An important position in the theoretical description of colloidal disperse systems is occupied by the theory of Deryaguin and Landau (1) and of Verwey and Overbeek (2). This explains the influence of the concentration of electrolyte on the stability of lyophobic colloids from the point of view of a balance between the van der Waals' forces of attraction between the colloidal particles and the electrostatic forces of repulsion between the electrical double layers at the surface of the particles. On viewing the problem within the framework of a model of a thin plane-parallel film (1-3), the particles (in the case of a foam-the bubbles; in the case of an emulsion--the droplets of a disperse phase) are regarded as two semiinfinite phases separated by a layer (film) of dispersion medium with thickness h. This model is applicable to all cases in which the radii of curvature of the surfaces of the particles are much greater than the distance between them. In the thin-film model the force acting on unit area perpendicular to the surfaces of the film is used as 0021-9797/83 $3.00 Copyright© 1983 by AcademicPress, Inc. All fightsof reproduction in any form reserved.

a measure of the force of attraction or repulsion of the particles. This disjoining pressure we represent as the sum of two components: ILw, the van der Waals' disjoining pressure, and Ilel the electrostatic component. We neglect structural forces which can occur at short separations (4): I I = ILw + II~l + .

• ..

[1]

Here we shall focus primarily on the electrostatic disjoining pressure at a symmetrical film in equilibrium with a solution ofk strong univalent electrolytes, which is given by the expression (1-3) IIel= 2kT ( c o s h ~

k - l) /~c?~,

[2]

where ~o is the electric potential! in the plane parallel to the surfaces of the film and equidistant from them, c~ is the bulk concentration of the electrolyte i, e is the electronic charge, k is the Boltzmann constant, and T is the absolute temperature. Consequently, In all equations throughout the paper physical quantities are in cgs and cgse units; [c] = particles/cm ~.

Journalof Colloidand InterfaceScience.Vol. 93, No. 1, May 1983

DISJOINING PRESSURE the problem of determining IIe~ and its dep e n d e n c e on the thickness o f the film amounts to finding the relation ~0(h). The electrostatic problem of finding xI'0 has been solved for the case o f constant surface potential (~s = const) (1-3, 5) or constant surface charge Ors = const) (5); the selection of one of the two limiting cases is determined from kinetic considerations (2, 6). It is assumed that with a sufficiently slow approach of the particles thermodynamic equilibrium exists in which ~ = const, and with a sufficiently fast contact thermodynamic equilibrium cannot be established and rr~ remains constant. However, in general (7, 8) b o t h • ~ and ¢~ may vary with a change in the thickness of the film. Various models have been published to account for this. Thus Muller a n d Martynov (9, 10) use Stern's model, Dunning et al. (11) regard the change in the dissociation of an adsorbed ionogenic surfactant as a consequence of the change in the ionic concentration close to the surface, and Healy et al. (12) consider the effect of charge regulation in latex dispersions. Here we consider the influence on IIe~ of changes both in the surface potential and in the surface charge with a change in h for foams and emulsion films where the surface ions may be mobile. We shall limit ourselves to the case of low surface potentials in order to obtain an analytical expression for IIel which makes it possible for us to exhibit more directly the influence of various factors on the electrostatic disjoining pressure. Comparison will be made with some recent experimental data (13) for foam films. THEORY

At the surface of a solution of an electrolyte and in a film, where there are two phase boundaries, an equilibrium distribution in the concentration of each of the ions is established which is due primarily to electrostatic interactions. The equilibrium ion distribution is determined self-consistently in the equations

9 d2~ dx 2

4~r -

p,

[31

where o is charge density 2k

p = e ~ zjcj,

[4]

j=l

z s denoting dimensionless ion charge. These also determine the potential ~0(h) and, from Eq. [2], disjoining pressure. Now we use approximations contained in Stern's model (see (2, 3) and Appendix 1) and disregard the intrinsic dimensions of the ions not only in the diffuse part of the electrical double layer but also in the surface (Stern) layer. Thus for the chemical potential of an ion tz; (referred to one particle) close to the phase boundary and in the bulk solution we can write (14) ~zj = ~o~ + k T In c~(x) + chj(x) + zje~(x), u~ = ~Zoj + k T In c~ + ck~ + z j e ~ ~,

[5] [6]

where u0j is the standard chemical potential for the standard state of an ideal gas, xI, is the electric potential, and 4~j is the potential not due to purely electrostatic (Coulomb) interactions 2 between the ions. F r o m the condition for chemical (diffusional) equilibrium, ~j(x) = uY and Eqs. [5] and [6] we obtain cj(x) = c~ exp[-2x4~j(x)/kT] X exp[-zjeql(x)/kT].

[7]

We have i n t r o d u c e d the symbol A4~j(x) = ~j(x) - ~ and chosen our zero o f potential as ~ = 0. If in Eq. [7] we put A~bj = 0 we obtain the condition for diffusional equilibrium used by G o u y and Chapman. We assume within a surface layer of thickness 6 (see Fig. 1) that AGj = const and therefore cj(x) = c~ e x p [ - z j e ' ~ ( x ) / k T ] , - I < x < l,

[8a]

2 This may include such effectsas van der Waals' and image potentials (26). Journal of Colloid and Interface Science, VoL 93, No. I, May 1983

10

VASSILIEFF

ET AL.

~T

Cs~ = c~ exp[-A4~sJkT] e x p [ - z j e ~ J k T ] , l < [x] < l + 6.

%

[8b]

In Appendix 1, Eq. [8b] is compared with the condition for diffusional equilibrium used by Stern. In order to solve the problem analytically, we shall assume that [zex~/kT] ~ I, and linearize the exponentials in 9 , i.e.,

q(x) = c~[ 1 - zjeg(x)/kT], - l < x < l,

[9a]

:×

-t -~ •

0

-h

FIG. I. Schematica] p l o t o f ~ ( x ) in a s y m m e t r i c a l f i l m w i t h thickness h = 2L T h e n o t a t i o n s ~s = ~ ( - Z ) = ~(1) a n d ~o = ~(0) a r e u s e d .

csj = c~ exp(-A(gsj/kT) × (1 - z j e 9 J k T ) .

[9b]

The substitution o f Eq. [9a] in [2, 3] gives the well-known linearized Poisson-Boltzm a n n (PB) equation which yields (1-3, 16) the following expressions for the surface potential and the midplane potential: 9s(h) = 4_~ as(h) coth (Kh/2),

[10]

EK

9o(h) = 4_7ras(h) 1 . EK sinh (Kh/2)

EK

1

90 = ~

[ 121

As expected, at the same boundary transition Eq. [11] gives 9o ~ 0. Equation [10] is satisfied by any dependences of 9s and as on the thickness o f the film. Thus if it is assumed that the surface charge does not change when the surfaces of the film are brought closer together, the sur-

sinh (rh/2) '

~s = a~ = const.

[13]

Similarly if we assume constant surface potential we obtain 1

[ 11 ]

Here O-sis the electric charge on unit area o f the surface layer, the surface charge, and K denotes inverse Debye length. 3 At h ~ from Eq. [10] we obtain the well-known relationship between the surface potential and the charge for a phase boundary of a solution of an electrolyte: 47r 92 ° = - - cr~° .

face potential must increase and from Eqs. [ 11 ] and [ 12] we obtain

90 = 9 ~ cosh (Kh/2) ' 9s = 9 ~ = const.

[14]

In the general case both 9s and gs m a y change with a decrease in the thickness of the film. In order to determine unambiguously the disjoining pressure we need the relation between 9s and ~s. We shall now obtain it by using the condition of the diffusional equilibrium of the ions in the surface layer [9b]. Since we are considering the case o f a film in equilibrium with a solution o f k strong univalent electrolytes ([z[ = 1) with a bulk concentration c~, the density o f the charges in the surface layer will be given by the expression (obtained from Eqs. [9b] and [4]) k

ps = e ~ cT[exp(--A~+/kT) i-1 k

3 F o r t h e c a s e o f 1:1 e l e c t r o l y t e s s t u d i e d h e r e

e29s ~ cF

- exp(--Aq~JkT)] -- k-T- i=]

k Kz = 87re 2 ~

c~/EkT.

Journal of Colloid and Interface Science, Vol. 93, No. l, May 1983

× [exp - AO[/kT) + exp(-A4~-[/kT)]. [15]

DISJOINING

It remains only to take into account the fact that in a surface layer with thickness ~ the connection between the charge in unit volume 0s and the charge in unit area is 0s~ --- as. Thus, from Eq. [15] we obtain the required second equation connecting ~s and

PRESSURE

11

We can now also derive equations giving the equilibrium change in the surface potential and the surface charge with a change in the thickness of the film. From Eqs. [ 10] and [21] we obtain as(h) = ~

ffs _

as

~x

kT~,_

41r e

eK

~

°~s,

[161

k

l+,I, • dh) = ~I'~° tanh (Kh/2) + ¢i,"

i=1

X [exp(-A4~+/kT) - exp(-A49[/kT)], [17] r6/k

~0 = ~ o

k

, r = 2 ~ c~o Z c~ i=1

i=1

X [exp(--A4aJ-/kT) + exp(-A4~7/kT)]. [18] Assuming that the nonelectrostatic adsorption potentials A~bjdo not depend on the film thickness the charge at a single interface at h ~ oo is given by an expression analogous to [16] ~K

a~ - G

k T ~, e

~K

- ~

q'xI'~ °.

[191

Combining the last equation with [12] yields a simple formula connecting the dimensionless surface potential at a single interface with the parameters of the model eq'~ ~* k~- - 1 + q,"

[201

It follows logically from [20] that at Aq5+ = A~bF (i.e., in the absence of a difference between the non-Coulomb adsorption potentials of the positive and negative ions), ~* = 0 and correspondingly (see also Eq. [12]), ¢;o = ,iqo = O. From Eqs. [16] and [19] if we exclude the first c o m m o n summand, which takes into account the adsorption due to nonelectrostatic interactions, we obtain EK

as(h) = ~ o + ~

[23]

Substituting [22] into [11 ] gives the required expression for the electric potential in the center of the film:

• * = 2 ~ c~° ~ c~ i=1

[22]

and from Eqs. [22], [10], and [12] similarly

where the following dimensionless parameters have been introduced r~lk

l+q, 1 + • coth (Kh/2)

4 [ ~ , ~ _ %(h)].

[21]

1

1+~

cosh (Kh/2) tanh (Kh/2) + q,

. [241

Combining [24] with [20] and [2] (after linearization) we finally get IIel = k

cosh (Kh/2)[tanh (Kh/2) + ~1 k

X ~ c~°.

[251

/=1

At • ~ tanh (Kh/2) (which corresponds to the case Cs = eonst, see below) Eq. [25] coincides with the result of Muller and Derjaguin (15) (written for a symmetrical film), which was derived from the Gibbs-Duhem equation.

Constant Charge or Constant Potential? Analysis of Eqs. [24] and [20] Noting that with a decrease in the thickness of the film, tanh (Kh/2) diminishes monotonically from 1 to 0, we see that regardless of the value of the parameter q, at large values o f h (Kh/2 > 2), tanh (Kh/2) tends to 1 and expression [24] tends to a limit given by [ 14], which was derived on the assumption G = const. At these large values of h, the difference between the cases G = const and as = const is insignificant (7) (sinh × (Kh/2) - cosh (Kh/2)--compare with Eqs. [13] and [14]). Depending on the value of the parameter from Eq. [24] we obtain the two limiting Journal of Colloid and InterfaceScience, Vol. 93, No. 1, May 1983

12

VASSILIEFF ET AL.

cases o f constant surface potential or charge. Thus, at ~, >> 1 Eq. [24] tends to [14], which was derived on the assumption ¢/s = const, and at • ~ 1 plus the more severe condition 4' ~ tanh (Kh/2) [24] tends to [13], which was derived on the assumption as = const. For 4, of the order of unity the dependence of ~0 on h lies between the limits of the constant surface potential and surface charge. These qualitative consequences o f Eq. [24] are shown in Fig. 2. These calculations were performed at different values o f ~ : (a) • = 0.001, (b) ¢, = 0.1, (c) ¢) = 1, (d) 4" = 10, (e) 4, --- 100. Curve a practically coincides with the predictions of Eq. [13], which was derived for constant charge and curve e with those o f Eq. [14], derived for constant potential. This inclusion o f possible changes of~P0 on h between the limiting cases of constant potential and constant charge is a consequence of the simultaneous solution of [ 10] and [211. Equation [21] requires an inverse dependence of (rs on ffs which Eq. [10] limits between these cases. It follows also from [21] that depending on the value of 4" a large potential change may correspond to negligibly small change in the charge and vice versa. Using different equilibrium conditions and solving the full PB equation numerically, many authors (9, 11, 12) obtained the same qualitative picture as shown in Fig. 2. The other interesting question is how the value o f • affects the dependence of ~ F on the electrolyte concentration. From Eq. [20] we get for ~, ¢ 1

e~/kT

~- ~*

[261

~_ ~*/4".

[27]

and at (I, >> 1

e~/kT

From the last two equations it follows that in the general case of more than one electrolyte in the system (cf. Eqs. [17] and [18]) ~I,~° varies with electrolyte concentrations even in the case • >> I which corresponds to surface potential constant with h. T h e "Nernstian" behavior ~ ( C e 0 = const is obtained from [27] only when i = 1. If • ~ 1, Journal o f Colloid and Interface Science, Vol. 93, No. 1, May 1983

II b

a) O~s= const b) cp :OA c) ~p=r d) @ = ~0

e) ~s: const i 3

2

I

c~ d

i

aeh/2 L

2

FIG. 2. Dependences of ¢/o(h)/¢/~ ° calculated from Eq. [24] for different • values.

which corresponds to the case as(h) = const with i = 1, one gets q ~ ~ c w2 and a~ ~ c. It should be noted here that the value of ,1~ itself depends on the electrolyte concentration.

Interplay between van der Waals' Attraction and Electrostatic Repulsion in Water Foam Films Let us examine first the main limitations imposed on the above analytical formulae. Since 4'* < • (compare [17] and [18]) Eq. [20] does not allow a value for q ~ higher than about 30 mV (at 300°K e q ~ / k T < 1 corresponding to qs~ < 26 mV). This is a formal consequence o f the use of the linearized Poisson-Boltzmann equation. Gregory [14] has made a comparison between the resuits from the linearized and the full Poisson-Boltzmann equation for the cases ~s = const and O-s = const. He found that the linearized equation gives satisfactory agreement with the full equation for ~I's = const providing q ~ < 60 mV, regardless o f ~h; and for as = const providing qs~ < 30 m V and Kh > 0.4.

DISJOINING PRESSURE I 5 L ~""

which express the limiting cases o f Eq. [25] (with i = 1, cf. Eqs. [26] and [27] at • >> 1 and • < 1, respectively. The result for ~os = const is shown by the full line and for as = const by the broken line. As already proved for the given pairs of q ~ and c values the true dependence lies between these two limiting cases. The van der Waals' disjoining pressure was calculated from a formula which incorporates retardation effects (27)

o)c=lO-gmollI b)c=lO-~molll c)c=I0-molll d)c~_I0ImoI/l

1

b °

5

<

h.106c ~.,,,

2 3 5

FIG. 3. Disjoining pressure isotherms 17 = II~ + l~el at lp~ = 30 mV and differentelectrolyteconcentrations ([17] = dyn/cm2);full lines 0Ps = const) from Eqs. [28] and [30]; broken lines (~s = const) from Eqs. [29] and [30].

1

Ilvw - 67rh3 {0.34 × 10-13 + 6.1 × 10-13

× We expect therefore that our results have a similar quantitative validity (cf. Appendix 1). Other specific effects connected with discreteness of charges (17, 18) or with "steric" interactions (19) are not taken into account here: they appear at very small thicknesses (h < 10 nm). Consequently, it is interesting to see the result o f the balance between electrostatic and van der Waals' contributions to the disjoining pressure at h > 10 n m and small ~ with various concentrations of electrolyte. The next figures show graphically the dependence of log [111 on h for an aqueous foam film at room temperature for two fixed values of q*~ (30 m V in Fig. 3 and 10 m V in Fig. 4). In order to discover transitions from a negative disjoining pressure to a positive one (II~ > [Ilvwl) at 11 < 0, the values of lglIII have been plotted along the negative branch of the axis of ordinates. The value of II~ was calculated from the formulas (16) . [ e ~°' ~ 2 F Khl-2 11~' = c k T t k T - ) Lc°sh 2 .] ' ~I'~ = const, Xa~

e s

2

as = const,

[28] Kh

13

-2

[29]

[( 1-

3.4 × 10 -17 + 7.8 × 10-nh

)']}

"

[3o] Generally speaking, these results illustrate changes in 11 with electrolyte concentration only for the case ~s = const, which for one electrolyte present in the system is consistent with the assumption ~2(ce~)= const. According to Eq. [20] in the general case ~ varies with electrolyte concentration, a fact that complicates somewhat the interpretation of experimental data. Data for ~ in foam films were obtained indirectly (13) from measured changes in the equilibrium film thickness with electrolyte concentration and assuming ~s = const with both c and h. A comparison is possible for films containing nonionic surfactants and one strong 1:1 electrolyte, where values q/s---30 m V were found. Here a detailed analysis o f the dependence of ~ on c is impossible due to the many still unresolved questions concerning the nonelectrostatic adsorption potential in that case (19). Fortunately Kretzschmar and Fruhner (20) proved (measuring directly potential differences) that in films stabilized with nonionic surfactants ~s does not change with h. We have already shown that in this case of one electrolyte present in the system if ~s does not change with h it does not change with c either. It follows from the results shown in Fig. Journal of Colloid and Interface Science, VoL 93, No. 1, May 1983

14

VASSILIEFF ET AL.

3 for ~ -- 30 m V that at c > 10 - l M i n the whole investigated thickness with range 10100 n m IIel ~ III,~l. Thus c o m m o n equilibrium films with h > 10 n m should be obtained at c < 10 -2 M and the transition to Newton ones (if at h < 10 n m other "steric" factors stabilize the film) m u s t take place in the range 10 -2 < c < 10 -1M. This conclusion is confirmed fully by the "critical" electrolyte concentration for transition from c o m m o n to Newton foam. films stabilized with nonionic surfactants found experimentally by Exerova et al. (13). The same experimental finding has been interpreted (13) as a discrepancy between D L V O theory and experiment. There the "critical" electrolyte concentration has been estimated theoretically by means of the widely used "classical D L V O " assumption that the transition occurs due to the lowering in the m a x i m u m in the II(h) curve with increasing c. But from Figs. 3 and 4 it can be seen that such lowering o f the m a x i m u m does not take place. In fact the height of the m a x i m u m even increases with c (compare curves a, b, and c in Fig. 3 and a and b in Fig. 4). The changes in the equilibrium thickness with c are mainly due to changes of the position o f the m a x i m u m . Thus experimental evidence concerning the influence of the concentration of a strong 1:1 electrolyte on the properties o f water foam films stabilized with nonionic surfactants is in qualitative agreement with the theoretical calculation of the balance between IIvw and l - l e l at ~ ( c ) = const = 30 m V which corresponds to the case ¢ >> 1. If one assumes that ¢ ~ 1 (see the end of the previous section) 92 ° m u s t vary with c according to ~go ~ cl/2. If further it is assumed that for the greatest electrolyte concentration (c = 10 -1 M ) studied xI,2 (10 -1 M ) = 30 m V then due to the decrease o f ,I~g° with decreasing c the balance of IIvw and IIe~ obtained in the same way as above predicts H~I ~ {Ilvw[ for 10 -1 < C < 10-4 M a n d l0 < h < 100 nm. The latter picture contradicts the experimental facts. Journal of Colloid and Interface Science, Vol. 93, No. I, M a y 1983

6 5

__

a) c= I0"/" moil[ b) c= 10.3 rno[/I c) c-~lO-2 mot/I

3 2 1

h,cm 2 3 /. 5 5

FIG. 4. Disjoining pressure isotherms II = ILw + IIe~ at ff~ = 10 m V a n d different electrolyte concentrations ([II] = dyn/cm2): full lines (¢s = const) from Eqs. [28] and [30]; broken lines (as = const) from Eqs. [29] a n d

[301. Small changes of ~ in the range 10-30 m V cannot, however, be excluded because according to the data in Fig. 4 at ~F2~ = 10 m V the transition to Newton films should also occur at c > 10 2 m . Further speculations are uncertain due to the lack of information for &0 in that example. Unfortunately, the analytical equations obtained here cannot be used for a quantitative interpretation of the arguably m o r e interesting experimental results ( 13, 21-23) for films stabilized with ionic surfactants where (13) ~ s - 100 mV. Nevertheless, in order to establish to some extent the influence on the electrostatic disjoining pressure of the possible nonfulfillment of the condition xI,s = const we shall estimate the values of the p a r a m e t e r • for films stabilized with an ionic surfactant. We consider then a film containing two univalent strong electrolytes, an ionic surfactant (assumed to be completely dissociated) with a bulk concentration cl = 10 -5 M, and a second, non-surface active, electrolyte with a concentration in the range of 1 0 - 5 < c2 < 10 1 M. For this case it is reasonable to assume that 2x0~ < 0 (assuming surfactant is cationic), and A0~ = 2x0~ = Aq~y = 0 (hydrated ions which are not connected with a hydrocarbon tail are not adsorbed specifically). Then from [18] we obtain

DISJOINING TABLE I E s t i m a t e o f • f r o m Eq. [31] w i t h V a l u e s o f A c k t / k T C o r r e s p o n d i n g to 8, 10, a n d 12 M e t h y l e n e G r o u p s in the H y d r o p h o b i c Test C2 (M)

C2 (cm-s)

Kt5

q}(8)

q}(10)

{1}(12)

10 5 10 -3 10 -I

6 × 1015 6 × 1017 6 × 1019

10 -3 10 -2 10 -1

1 10 -1 10 -2

10 1 10 -1

102 10 1

K6 2(c?° + c~)

× [cF(e -~+/kT + 1) + 2c~].

[31]

The specific adsorption of the surface-active cation is due mainly to the fact that it is more favorable for the nonpolar tail to be present in the nonpolar gas phase and not in the polar solvent (water in this case). The energy of adsorption is about k T per methylene group (24). The results of the estimates are given in Table 1, where the last three columns correspond to different values of ACf/kT. The calculation was performed on the basis of Eq. [31 ]. For the thickness of the surface layer we used a value of 6 = 10-s cm. This estimate gives values of • from 10-2 to 102 covering all the cases of charge regulation shown in Fig. 2. CONCLUDING

REMARKS

Interpretation of experimental data on foam and emulsion films from the point of view of the interplay between van der Waals' attraction and electrostatic repulsion needs a careful analysis of the dependence of the electric surface potential on both electrolyte concentration and film thickness. For a given electrolyte concentration the mechanism of charge regulation limits the value of the electrostatic disjoining pressure between the cases of constant surface potential and constant surface charge. In the framework of the model used and at small surface potentials (xI,~ < 30 mV) the position of the true de-

15

PRESSURE

pendence between the above limiting cases is determined by the value of the parameter q?. It is clear that depending on the concrete characteristics of the system studied, which in the general case must include effects connected with the size of ions and uncharged molecules, possible changes in the dissociation of the electrolytes, etc., the criterion for passing from one limiting case to the other will also change. Within the framework of Stern's model the equations obtained are limited above all by the assumption of the low surface potentials. This disadvantage can be circumvented by computer calculation according to the full PB equation combined with the condition of diffusional equilibrium of the ions in the surface layer. Such a detailed quantitative theoretical investigation is meaningful only if carried out in combination with appropriate experimental work, the model taking into account all the special features of the particular system, which could not pass into the framework of Stem's model. APPENDIX

1: A N A L Y S I S O F S T E R N ' S

MODEL

The basic assumptions in Stem's model are that the non-Coulomb interactions appear only in the surface layer and that the surface concentration ns (number of ions per cm 2) of the potential-determining ions in this layer is given by an equation analogous to Langmuir's adsorption isotherm

Ns(c~/Nj) exp[-(zjext% + A(~j)/kT] n~j = 1 + (c~/Nj) exp[-(zjeq/~ + AOj)/kT] '

[Al.1] where Ns is the number of sites accessible for adsorption (per cm 2) in the surface layer, and Nj is the number of accessible sites (per cm 3) in the bulk phase. This equation is obtained from the following considerations (2). If only one ion is adsorbed, the number of sites free for adsorption in the surface will be Ns - nsj, and in the bulk Nj - c~. Since the difference in the potential energies of the interaction between an ion in the surface and in an inJournal of Colloid and Interface Science, Vol. 93, No. 1, May 1983

16

VASSILIEFF

finite bulk phase is z~e'~s + A~j, if we apply Boltzmann's theorem we obtain //sj _

Ns -

c?

N~ - c7

ET

AL.

when (cj/Nfl exp(-- . ) ~ 1. In this case [AI.1] and [A1.4] become

Ns

nsj

n~j = ~ c? exp[-(zje~s + Ag~fl/kT].

× exp[-(zje~s + Ag)fl/kT].

[A1.5]

[A1.2]

X exp[-(z~e~ + AO~)/kT]

A comparison of [A1.5] with the condition which we used for the diffusional equilibrium in the surface layer [8b] shows that csj(N~/Nj) = nsj, i.e., Ns/Nj = 6 (6 is the thickness of the surface layer). The latter can easily be shown on the basis of geometrical considerations. The limits of the validity of [A 1.5], or [8b], are determined by the values of c~/Nj and the full energy of adsorption Ej = zje'~s + A~j (including Coulomb and non-Coulomb contributions). If we assume that on adsorption there is no change in the hydrate shell of the ion and that the ionic radius together with the hydrate shell is R = 5 × 10-8 cm, for N~ we obtain 2 × t021 cm -3 from N~-1 = VI = (4/3)7rR 3. Comparison with the values of c° (in cm 3) given in Table 1 in the interesting range of concentrations show that cT/N j varies in the interval 3 × 10-6-3 × 10-2. In estimating the adsorption energy, it must be borne in mind that for the potential-determining ion ezgs and Aq~ have opposite signs (Aq~j < 0 in the case of positive adsorption). Consequently for values of exp(Ej/kT) that are not too large (how large will be determined by c~ in the particular case) it is safe to use [A1.5] or [8b] instead of [AI.1] or [A1.4]. A similar estimate with Nj = Mj/NA is given by Verwey and Overbeek (2). The main qualitative consequence lost by using Eq. [8b] instead of Langmuir's adsorption isotherm in Stem's model is the limitation of the value of a~ due to the saturation of the surface layer with charges of the same sign. An estimate by means of the equation (25) (e~

These problems concerning the difference in the intrinsic dimensions of the ions disappear

(which is the nonlinearized analog of [7])

The solution to Eq. [A1.2] for nsj at c~ ~ N~ gives [A 1.1 ]. Contradictions in relation to the value of Nj exist in the literature: Stern himself (2) proposes Nj = Mj/NA (M is the molecular weight, and NA is Avogadro's number), Scheludko (3) has used Nj = l, and Muller more correctly assumes Nj = V~ 1 (V is the intrinsic volume of the ion). Problems also appear in relation to the magnitudes Ns, particularly if ions with different volumes are adsorbed--one must take into account the different changes in the filling of the surface with the adsorption of one ion or another. Hence to use the substitution of [A 1.1 ] in [A1.3] (as is done in (3)) in order to obtain the surface charge 2k

as = e ~ Zjnsj

[A1.3]

1=i

cannot be correct--the filling of the surface will be determined by all the adsorbed ions. To a certain extent, the effect of the difference in the ionic volumes was taken into account by Muller (10) within the framework of Stem's model. He took into account the differences in N~ and in the filling due to the adsorption of all the ions in the solution but neglected the differences in areas occupied by different ions in the surface layer. Thus, instead of [A 1.1 ] he obtained

N~(c?/Nfl n~j =

× e x p [ - ( z j e ~ + AOfl/kT] 2k

.

[A1.4]

1 + E (c#Nj) i=l

Journal of Colloid and Interface Science, Vol. 93, No. i, May 1983

~r= = (2c~k~Tf/Z sinh \ 2 k T ]

DISJOINING PRESSURE

gives ~b~ of 200-500 mV at saturation of the surface layer (area per ion 80 A 2) and electrolyte concentrations in the range 10 -5-10-1 M. Those values are many times greater than the experimentally found ~ values in water foam films (13) in the range 10-100 mV. Consequently it is unlikely the effect of saturation of the surface with charges of the same sign to appear in foams. ACKNOWLEDGMENTS We are indebted to Drs. B. Radeov, D. Exerova, and V. Muller for useful discussions and advice. REFERENCES 1. Derjaguin, B., and Landau, L., Acta Physicochim. URSS 14, 1 (1941). 2. Verwey, E. J. W., and Overbeek, J. Th. G., "Theory of the Stability of Lyophobic Colloids." Elsevier, Amsterdam, 1948. 3. Scheludko, A., "Colloid Chemistry," Chaps. V, VI. Elsevier, Amsterdam, 1966. 4. Grimson, M., Richmond, P., and Rickayzen, G., Mol. Phys. 39, 61. 5. Honig, E. P., and Mul, P. M., J. Colloid Interface Sci. 36, 258 (1971). 6. Lyklema, J., PureAppl. Chem. 52, 258 (1980). 7. Clunie, J. S., Goodman, J. F., and Ingrain, B. T., "Surface and Colloid Science" (E. Matijevic, Ed.), Vol. 3, p. 167. Wiley-Interscience, New York, 1971. 8. Carrol, B. J., "Surface and Colloid Science" (E. Matijevic, Ed.), Vol. 9, p. 1. Wiley, New York, 1976. 9. Martynov, G. A., and Muller, V. M., "Surface Forces in Thin Films and Disperse Systems" (B. V. Deryagin, Ed.), p. 7. Nauka, Moscow, 1972. 10. Muller, V. M., "'Surface Forces in Thin Films and Disperse Systems" (B. V. Deryagin, Ed.), p. 245. Nauka, Moscow, 1974. [in Russian]

17

11. Dunning, A. J., Mingins, F., Pethica, B. A., and Richmond, P, J. Chem. Soc. Faraday Trans. 1 74, 2617 (1978). 12. Healey, T. W., Chan, D., and White, L. R., Pure Appl. Chem. 52, 1207 (1980). 13. Exerova, D. R., Izv. Khim. 11, 739 (1978). 14. Hill, T. G., "Introduction to Statistical Thermodynamics." Addison-Wesley, Reading, Mass., 1967. 15. Muller, V. M., and Derjaguin, B. V., Kolloidn. Zh. 37, 1116 (1975). 16. Gregory, J., J. Chem. Soc. Faraday Trans. H 69, 1723 (1973). 17. Richmond, P., J. Chem. Soc. Faraday Trans. 2 70, 1066 (1974); 71, 1154 (1975). 18. Muller, V. M., and Derjaguin, B. V., Kolloidn. Zh. 38, 1117 (1976). 19. Grigorov, L. S., Dissertation. Sofia University, Chemical Faculty, Sofia, 1978. [in Bulgarian] 20. Kretzchmar, G., and Fruhner, H., Z. Phys. Chem. (Leipzig) 256, 217 (1975). 21. Exerowa, D., Zacharieva, M., Cohen, R., and Platikanov, D., Colloid Polym. Sci. 257, 1089 (1979). 22. Manev, E. D., Kabakchieva, M. St., and Ivanova, M. A., God. S o f Univ. Khim. Fak. 67, 273 (1972/3). 23. Vrij, A., Joosten, J. G. H., and Fijnaut, H. M., "Advances in Chemical Physics," Vol. 48, p. 329. Wiley, New York, 1981. 24. Skvirskii, L. Ya., Maiofis, A. D., and Abramzon, A. A., "Physicochemical Principles of the Use of Surface-Active Agents," p. 187. FAN, Tashkent, 1977. [in Russian] 25. Adamson, A., "Physical Chemistry of Surfaces." Interscience, New York, 1976. 26. Richmond, P., J. Chem. Soc. Faraday Trans. 2 70, 1650 (1974). 27. Vassilieff, Chr. St., and Ivanov, I. B., Z. Naturforsch. A 31, 1584 (1976); Vassilieff, Chr. St., and Ivanov, I. B., Annu. Univ. Sofia Fac. Chim. 70(2), 111 (1975/76); Vassilieff, Chr. St., Ann. Univ. Sofia Fac. Chim. 72(2), 5 (1977/78).

Journal of Colloid and Interface Science, Vol. 93, No. 1, May 1983